LMFDB - Embedded newform 33.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,4,Mod(1,33)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	33.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.94706303019$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{97}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 24 $ x^2 - x - 24
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.42443$ of defining polynomial
Character	$\chi$	$=$	33.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} -16.2733 q^{6} -7.69772 q^{7} +72.8199 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} -16.2733 q^{6} -7.69772 q^{7} +72.8199 q^{8} +9.00000 q^{9} -91.3954 q^{10} -11.0000 q^{11} -64.2733 q^{12} +24.8489 q^{13} -41.7557 q^{14} +50.5466 q^{15} +223.611 q^{16} -15.9420 q^{17} +48.8199 q^{18} +15.1511 q^{19} -360.977 q^{20} +23.0931 q^{21} -59.6687 q^{22} +17.7557 q^{23} -218.460 q^{24} +158.884 q^{25} +134.791 q^{26} -27.0000 q^{27} -164.919 q^{28} -128.547 q^{29} +274.186 q^{30} +219.395 q^{31} +630.402 q^{32} +33.0000 q^{33} -86.4763 q^{34} +129.698 q^{35} +192.820 q^{36} +92.0703 q^{37} +82.1863 q^{38} -74.5466 q^{39} -1226.93 q^{40} -459.942 q^{41} +125.267 q^{42} +64.9648 q^{43} -235.669 q^{44} -151.640 q^{45} +96.3146 q^{46} +497.408 q^{47} -670.832 q^{48} -283.745 q^{49} +861.855 q^{50} +47.8260 q^{51} +532.373 q^{52} -526.919 q^{53} -146.460 q^{54} +185.337 q^{55} -560.547 q^{56} -45.4534 q^{57} -697.292 q^{58} -578.443 q^{59} +1082.93 q^{60} -221.569 q^{61} +1190.09 q^{62} -69.2794 q^{63} +1630.68 q^{64} -418.675 q^{65} +179.006 q^{66} -860.745 q^{67} -341.548 q^{68} -53.2671 q^{69} +703.536 q^{70} +580.919 q^{71} +655.379 q^{72} +510.116 q^{73} +499.429 q^{74} -476.652 q^{75} +324.605 q^{76} +84.6749 q^{77} -404.373 q^{78} +1035.12 q^{79} -3767.59 q^{80} +81.0000 q^{81} -2494.92 q^{82} +606.211 q^{83} +494.757 q^{84} +268.605 q^{85} +352.397 q^{86} +385.640 q^{87} -801.018 q^{88} -23.4411 q^{89} -822.559 q^{90} -191.279 q^{91} +380.406 q^{92} -658.186 q^{93} +2698.15 q^{94} -255.279 q^{95} -1891.20 q^{96} +719.490 q^{97} -1539.16 q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 - 3 * q^6 + 24 * q^7 + 57 * q^8 + 18 * q^9	\( 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9} - 104 q^{10} - 22 q^{11} - 99 q^{12} + 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} + 106 q^{17} + 9 q^{18} + 50 q^{19} - 328 q^{20} - 72 q^{21} - 11 q^{22} + 134 q^{23} - 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} + 202 q^{28} - 198 q^{29} + 312 q^{30} + 360 q^{31} + 857 q^{32} + 66 q^{33} - 626 q^{34} + 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} - 90 q^{39} - 1272 q^{40} - 782 q^{41} + 546 q^{42} + 386 q^{43} - 363 q^{44} - 126 q^{45} - 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} + 1379 q^{50} - 318 q^{51} + 592 q^{52} - 522 q^{53} - 27 q^{54} + 154 q^{55} - 1062 q^{56} - 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} - 778 q^{61} + 568 q^{62} + 216 q^{63} + 809 q^{64} - 404 q^{65} + 33 q^{66} - 776 q^{67} + 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} + 513 q^{72} + 1296 q^{73} + 2358 q^{74} - 126 q^{75} + 728 q^{76} - 264 q^{77} - 336 q^{78} + 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} - 324 q^{83} - 606 q^{84} + 616 q^{85} - 1068 q^{86} + 594 q^{87} - 627 q^{88} - 756 q^{89} - 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} + 3722 q^{94} - 156 q^{95} - 2571 q^{96} - 452 q^{97} - 4467 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q + q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 - 3 * q^6 + 24 * q^7 + 57 * q^8 + 18 * q^9 - 104 * q^10 - 22 * q^11 - 99 * q^12 + 30 * q^13 - 182 * q^14 + 42 * q^15 + 201 * q^16 + 106 * q^17 + 9 * q^18 + 50 * q^19 - 328 * q^20 - 72 * q^21 - 11 * q^22 + 134 * q^23 - 171 * q^24 + 42 * q^25 + 112 * q^26 - 54 * q^27 + 202 * q^28 - 198 * q^29 + 312 * q^30 + 360 * q^31 + 857 * q^32 + 66 * q^33 - 626 * q^34 + 220 * q^35 + 297 * q^36 - 328 * q^37 - 72 * q^38 - 90 * q^39 - 1272 * q^40 - 782 * q^41 + 546 * q^42 + 386 * q^43 - 363 * q^44 - 126 * q^45 - 418 * q^46 + 266 * q^47 - 603 * q^48 + 378 * q^49 + 1379 * q^50 - 318 * q^51 + 592 * q^52 - 522 * q^53 - 27 * q^54 + 154 * q^55 - 1062 * q^56 - 150 * q^57 - 390 * q^58 - 172 * q^59 + 984 * q^60 - 778 * q^61 + 568 * q^62 + 216 * q^63 + 809 * q^64 - 404 * q^65 + 33 * q^66 - 776 * q^67 + 1070 * q^68 - 402 * q^69 + 304 * q^70 + 630 * q^71 + 513 * q^72 + 1296 * q^73 + 2358 * q^74 - 126 * q^75 + 728 * q^76 - 264 * q^77 - 336 * q^78 + 652 * q^79 - 3832 * q^80 + 162 * q^81 - 1070 * q^82 - 324 * q^83 - 606 * q^84 + 616 * q^85 - 1068 * q^86 + 594 * q^87 - 627 * q^88 - 756 * q^89 - 936 * q^90 - 28 * q^91 + 1726 * q^92 - 1080 * q^93 + 3722 * q^94 - 156 * q^95 - 2571 * q^96 - 452 * q^97 - 4467 * q^98 - 198 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	5.42443	1.91783	0.958913	−	0.283702i	$-0.0915625\pi$
$2$	5.42443	1.91783	0.958913	+	0.283702i	$0.0915625\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	21.4244	2.67805
$5$	−16.8489	−1.50701	−0.753504	−	0.657444i	$-0.771638\pi$
$5$	−16.8489	−1.50701	−0.753504	+	0.657444i	$0.771638\pi$
$6$	−16.2733	−1.10726
$7$	−7.69772	−0.415638	−0.207819	−	0.978167i	$-0.566636\pi$
$7$	−7.69772	−0.415638	−0.207819	+	0.978167i	$0.566636\pi$
$8$	72.8199	3.21821
$9$	9.00000	0.333333
$10$	−91.3954	−2.89018
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−64.2733	−1.54617
$13$	24.8489	0.530141	0.265071	−	0.964229i	$-0.414605\pi$
$13$	24.8489	0.530141	0.265071	+	0.964229i	$0.414605\pi$
$14$	−41.7557	−0.797120
$15$	50.5466	0.870071
$16$	223.611	3.49392
$17$	−15.9420	−0.227441	−0.113721	−	0.993513i	$-0.536277\pi$
$17$	−15.9420	−0.227441	−0.113721	+	0.993513i	$0.536277\pi$
$18$	48.8199	0.639275
$19$	15.1511	0.182943	0.0914713	−	0.995808i	$-0.470843\pi$
$19$	15.1511	0.182943	0.0914713	+	0.995808i	$0.470843\pi$
$20$	−360.977	−4.03585
$21$	23.0931	0.239968
$22$	−59.6687	−0.578246
$23$	17.7557	0.160971	0.0804853	−	0.996756i	$-0.474353\pi$
$23$	17.7557	0.160971	0.0804853	+	0.996756i	$0.474353\pi$
$24$	−218.460	−1.85804
$25$	158.884	1.27107
$26$	134.791	1.01672
$27$	−27.0000	−0.192450
$28$	−164.919	−1.11310
$29$	−128.547	−0.823121	−0.411560	−	0.911383i	$-0.635016\pi$
$29$	−128.547	−0.823121	−0.411560	+	0.911383i	$0.635016\pi$
$30$	274.186	1.66864
$31$	219.395	1.27112	0.635558	−	0.772053i	$-0.280770\pi$
$31$	219.395	1.27112	0.635558	+	0.772053i	$0.280770\pi$
$32$	630.402	3.48251
$33$	33.0000	0.174078
$34$	−86.4763	−0.436193
$35$	129.698	0.626369
$36$	192.820	0.892685
$37$	92.0703	0.409088	0.204544	−	0.978857i	$-0.434429\pi$
$37$	92.0703	0.409088	0.204544	+	0.978857i	$0.434429\pi$
$38$	82.1863	0.350852
$39$	−74.5466	−0.306077
$40$	−1226.93	−4.84987
$41$	−459.942	−1.75197	−0.875986	−	0.482336i	$-0.839788\pi$
$41$	−459.942	−1.75197	−0.875986	+	0.482336i	$0.839788\pi$
$42$	125.267	0.460218
$43$	64.9648	0.230396	0.115198	−	0.993343i	$-0.463250\pi$
$43$	64.9648	0.230396	0.115198	+	0.993343i	$0.463250\pi$
$44$	−235.669	−0.807464
$45$	−151.640	−0.502336
$46$	96.3146	0.308713
$47$	497.408	1.54371	0.771855	−	0.635799i	$-0.219329\pi$
$47$	497.408	1.54371	0.771855	+	0.635799i	$0.219329\pi$
$48$	−670.832	−2.01721
$49$	−283.745	−0.827245
$50$	861.855	2.43769
$51$	47.8260	0.131313
$52$	532.373	1.41975
$53$	−526.919	−1.36562	−0.682811	−	0.730596i	$-0.739243\pi$
$53$	−526.919	−1.36562	−0.682811	+	0.730596i	$0.739243\pi$
$54$	−146.460	−0.369086
$55$	185.337	0.454380
$56$	−560.547	−1.33761
$57$	−45.4534	−0.105622
$58$	−697.292	−1.57860
$59$	−578.443	−1.27639	−0.638194	−	0.769876i	$-0.720318\pi$
$59$	−578.443	−1.27639	−0.638194	+	0.769876i	$0.720318\pi$
$60$	1082.93	2.33010
$61$	−221.569	−0.465067	−0.232533	−	0.972588i	$-0.574701\pi$
$61$	−221.569	−0.465067	−0.232533	+	0.972588i	$0.574701\pi$
$62$	1190.09	2.43778
$63$	−69.2794	−0.138546
$64$	1630.68	3.18493
$65$	−418.675	−0.798927
$66$	179.006	0.333851
$67$	−860.745	−1.56950	−0.784752	−	0.619810i	$-0.787210\pi$
$67$	−860.745	−1.56950	−0.784752	+	0.619810i	$0.787210\pi$
$68$	−341.548	−0.609100
$69$	−53.2671	−0.0929364
$70$	703.536	1.20127
$71$	580.919	0.971020	0.485510	−	0.874231i	$-0.338634\pi$
$71$	580.919	0.971020	0.485510	+	0.874231i	$0.338634\pi$
$72$	655.379	1.07274
$73$	510.116	0.817871	0.408935	−	0.912563i	$-0.365900\pi$
$73$	510.116	0.817871	0.408935	+	0.912563i	$0.365900\pi$
$74$	499.429	0.784560
$75$	−476.652	−0.733854
$76$	324.605	0.489930
$77$	84.6749	0.125319
$78$	−404.373	−0.587002
$79$	1035.12	1.47418	0.737088	−	0.675797i	$-0.236200\pi$
$79$	1035.12	1.47418	0.737088	+	0.675797i	$0.236200\pi$
$80$	−3767.59	−5.26536
$81$	81.0000	0.111111
$82$	−2494.92	−3.35998
$83$	606.211	0.801690	0.400845	−	0.916146i	$-0.368717\pi$
$83$	606.211	0.801690	0.400845	+	0.916146i	$0.368717\pi$
$84$	494.757	0.642648
$85$	268.605	0.342756
$86$	352.397	0.441860
$87$	385.640	0.475229
$88$	−801.018	−0.970328
$89$	−23.4411	−0.0279186	−0.0139593	−	0.999903i	$-0.504444\pi$
$89$	−23.4411	−0.0279186	−0.0139593	+	0.999903i	$0.504444\pi$
$90$	−822.559	−0.963392
$91$	−191.279	−0.220347
$92$	380.406	0.431088
$93$	−658.186	−0.733879
$94$	2698.15	2.96057
$95$	−255.279	−0.275696
$96$	−1891.20	−2.01063
$97$	719.490	0.753126	0.376563	−	0.926391i	$-0.377106\pi$
$97$	719.490	0.753126	0.376563	+	0.926391i	$0.377106\pi$
$98$	−1539.16	−1.58651
$99$	−99.0000	−0.100504
$100$	3404.00	3.40400
$101$	1871.27	1.84355	0.921774	−	0.387727i	$-0.126740\pi$
$101$	1871.27	1.84355	0.921774	+	0.387727i	$0.126740\pi$
$102$	259.429	0.251836
$103$	−428.745	−0.410151	−0.205075	−	0.978746i	$-0.565744\pi$
$103$	−428.745	−0.410151	−0.205075	+	0.978746i	$0.565744\pi$
$104$	1809.49	1.70611
$105$	−389.093	−0.361634
$106$	−2858.24	−2.61902
$107$	1148.02	1.03723	0.518616	−	0.855008i	$-0.326448\pi$
$107$	1148.02	1.03723	0.518616	+	0.855008i	$0.326448\pi$
$108$	−578.460	−0.515392
$109$	−1828.32	−1.60662	−0.803308	−	0.595564i	$-0.796929\pi$
$109$	−1828.32	−1.60662	−0.803308	+	0.595564i	$0.796929\pi$
$110$	1005.35	0.871421
$111$	−276.211	−0.236187
$112$	−1721.29	−1.45220
$113$	1126.40	0.937722	0.468861	−	0.883272i	$-0.344665\pi$
$113$	1126.40	0.937722	0.468861	+	0.883272i	$0.344665\pi$
$114$	−246.559	−0.202565
$115$	−299.163	−0.242584
$116$	−2754.04	−2.20436
$117$	223.640	0.176714
$118$	−3137.72	−2.44789
$119$	122.717	0.0945332
$120$	3680.79	2.80008
$121$	121.000	0.0909091
$122$	−1201.89	−0.891916
$123$	1379.83	1.01150
$124$	4700.42	3.40412
$125$	−570.907	−0.408508
$126$	−375.801	−0.265707
$127$	661.304	0.462057	0.231029	−	0.972947i	$-0.425791\pi$
$127$	661.304	0.462057	0.231029	+	0.972947i	$0.425791\pi$
$128$	3802.31	2.62562
$129$	−194.895	−0.133019
$130$	−2271.07	−1.53220
$131$	622.186	0.414967	0.207483	−	0.978239i	$-0.433473\pi$
$131$	622.186	0.414967	0.207483	+	0.978239i	$0.433473\pi$
$132$	707.006	0.466189
$133$	−116.629	−0.0760378
$134$	−4669.05	−3.01003
$135$	454.919	0.290024
$136$	−1160.89	−0.731955
$137$	−1872.84	−1.16794	−0.583969	−	0.811776i	$-0.698501\pi$
$137$	−1872.84	−1.16794	−0.583969	+	0.811776i	$0.698501\pi$
$138$	−288.944	−0.178236
$139$	−954.058	−0.582174	−0.291087	−	0.956697i	$-0.594017\pi$
$139$	−954.058	−0.582174	−0.291087	+	0.956697i	$0.594017\pi$
$140$	2778.70	1.67745
$141$	−1492.22	−0.891261
$142$	3151.15	1.86225
$143$	−273.337	−0.159844
$144$	2012.50	1.16464
$145$	2165.86	1.24045
$146$	2767.09	1.56853
$147$	851.236	0.477610
$148$	1972.55	1.09556
$149$	−2047.01	−1.12549	−0.562745	−	0.826631i	$-0.690255\pi$
$149$	−2047.01	−1.12549	−0.562745	+	0.826631i	$0.690255\pi$
$150$	−2585.57	−1.40740
$151$	475.863	0.256458	0.128229	−	0.991745i	$-0.459071\pi$
$151$	475.863	0.256458	0.128229	+	0.991745i	$0.459071\pi$
$152$	1103.30	0.588749
$153$	−143.478	−0.0758138
$154$	459.313	0.240341
$155$	−3696.56	−1.91558
$156$	−1597.12	−0.819691
$157$	−647.466	−0.329130	−0.164565	−	0.986366i	$-0.552622\pi$
$157$	−647.466	−0.329130	−0.164565	+	0.986366i	$0.552622\pi$
$158$	5614.92	2.82721
$159$	1580.76	0.788442
$160$	−10621.5	−5.24817
$161$	−136.678	−0.0669054
$162$	439.379	0.213092
$163$	1093.23	0.525329	0.262665	−	0.964887i	$-0.415399\pi$
$163$	1093.23	0.525329	0.262665	+	0.964887i	$0.415399\pi$
$164$	−9853.99	−4.69188
$165$	−556.012	−0.262336
$166$	3288.35	1.53750
$167$	1123.25	0.520479	0.260240	−	0.965544i	$-0.416198\pi$
$167$	1123.25	0.520479	0.260240	+	0.965544i	$0.416198\pi$
$168$	1681.64	0.772270
$169$	−1579.53	−0.718951
$170$	1457.03	0.657346
$171$	136.360	0.0609809
$172$	1391.83	0.617014
$173$	46.0123	0.0202211	0.0101106	−	0.999949i	$-0.496782\pi$
$173$	46.0123	0.0202211	0.0101106	+	0.999949i	$0.496782\pi$
$174$	2091.88	0.911406
$175$	−1223.04	−0.528305
$176$	−2459.72	−1.05346
$177$	1735.33	0.736923
$178$	−127.155	−0.0535430
$179$	−831.975	−0.347401	−0.173700	−	0.984799i	$-0.555572\pi$
$179$	−831.975	−0.347401	−0.173700	+	0.984799i	$0.555572\pi$
$180$	−3248.79	−1.34528
$181$	−1810.63	−0.743553	−0.371776	−	0.928322i	$-0.621251\pi$
$181$	−1810.63	−0.743553	−0.371776	+	0.928322i	$0.621251\pi$
$182$	−1037.58	−0.422586
$183$	664.708	0.268506
$184$	1292.97	0.518037
$185$	−1551.28	−0.616499
$186$	−3570.28	−1.40745
$187$	175.362	0.0685762
$188$	10656.7	4.13414
$189$	207.838	0.0799895
$190$	−1384.75	−0.528737
$191$	458.898	0.173847	0.0869233	−	0.996215i	$-0.472296\pi$
$191$	458.898	0.173847	0.0869233	+	0.996215i	$0.472296\pi$
$192$	−4892.05	−1.83882
$193$	1778.91	0.663465	0.331733	−	0.943373i	$-0.392367\pi$
$193$	1778.91	0.663465	0.331733	+	0.943373i	$0.392367\pi$
$194$	3902.82	1.44436
$195$	1256.02	0.461260
$196$	−6079.08	−2.21541
$197$	5304.53	1.91844	0.959218	−	0.282666i	$-0.0912188\pi$
$197$	5304.53	1.91844	0.959218	+	0.282666i	$0.0912188\pi$
$198$	−537.018	−0.192749
$199$	−5138.40	−1.83041	−0.915205	−	0.402989i	$-0.867971\pi$
$199$	−5138.40	−1.83041	−0.915205	+	0.402989i	$0.867971\pi$
$200$	11569.9	4.09058
$201$	2582.24	0.906153
$202$	10150.6	3.53560
$203$	989.515	0.342120
$204$	1024.65	0.351664
$205$	7749.50	2.64024
$206$	−2325.70	−0.786597
$207$	159.801	0.0536568
$208$	5556.47	1.85227
$209$	−166.663	−0.0551593
$210$	−2110.61	−0.693551
$211$	−4262.36	−1.39068	−0.695339	−	0.718682i	$-0.744746\pi$
$211$	−4262.36	−1.39068	−0.695339	+	0.718682i	$0.744746\pi$
$212$	−11288.9	−3.65721
$213$	−1742.76	−0.560619
$214$	6227.38	1.98923
$215$	−1094.58	−0.347209
$216$	−1966.14	−0.619345
$217$	−1688.84	−0.528323
$218$	−9917.58	−3.08121
$219$	−1530.35	−0.472198
$220$	3970.75	1.21685
$221$	−396.141	−0.120576
$222$	−1498.29	−0.452966
$223$	−1377.80	−0.413740	−0.206870	−	0.978368i	$-0.566328\pi$
$223$	−1377.80	−0.413740	−0.206870	+	0.978368i	$0.566328\pi$
$224$	−4852.65	−1.44746
$225$	1429.96	0.423691
$226$	6110.06	1.79839
$227$	−1227.28	−0.358843	−0.179422	−	0.983772i	$-0.557423\pi$
$227$	−1227.28	−0.358843	−0.179422	+	0.983772i	$0.557423\pi$
$228$	−973.814	−0.282861
$229$	3890.28	1.12261	0.561304	−	0.827610i	$-0.310300\pi$
$229$	3890.28	1.12261	0.561304	+	0.827610i	$0.310300\pi$
$230$	−1622.79	−0.465233
$231$	−254.025	−0.0723532
$232$	−9360.74	−2.64898
$233$	−3218.14	−0.904837	−0.452419	−	0.891806i	$-0.649439\pi$
$233$	−3218.14	−0.904837	−0.452419	+	0.891806i	$0.649439\pi$
$234$	1213.12	0.338906
$235$	−8380.75	−2.32638
$236$	−12392.8	−3.41823
$237$	−3105.35	−0.851115
$238$	665.670	0.181298
$239$	−428.098	−0.115864	−0.0579318	−	0.998321i	$-0.518451\pi$
$239$	−428.098	−0.115864	−0.0579318	+	0.998321i	$0.518451\pi$
$240$	11302.8	3.03996
$241$	1231.16	0.329070	0.164535	−	0.986371i	$-0.447388\pi$
$241$	1231.16	0.329070	0.164535	+	0.986371i	$0.447388\pi$
$242$	656.356	0.174348
$243$	−243.000	−0.0641500
$244$	−4747.00	−1.24547
$245$	4780.78	1.24667
$246$	7484.77	1.93988
$247$	376.489	0.0969854
$248$	15976.3	4.09072
$249$	−1818.63	−0.462856
$250$	−3096.84	−0.783446
$251$	2838.22	0.713732	0.356866	−	0.934156i	$-0.383845\pi$
$251$	2838.22	0.713732	0.356866	+	0.934156i	$0.383845\pi$
$252$	−1484.27	−0.371033
$253$	−195.313	−0.0485344
$254$	3587.20	0.886145
$255$	−805.814	−0.197890
$256$	7579.90	1.85056
$257$	342.007	0.0830110	0.0415055	−	0.999138i	$-0.486785\pi$
$257$	342.007	0.0830110	0.0415055	+	0.999138i	$0.486785\pi$
$258$	−1057.19	−0.255108
$259$	−708.731	−0.170032
$260$	−8969.87	−2.13957
$261$	−1156.92	−0.274374
$262$	3375.01	0.795834
$263$	5895.00	1.38213	0.691067	−	0.722791i	$-0.257141\pi$
$263$	5895.00	1.38213	0.691067	+	0.722791i	$0.257141\pi$
$264$	2403.06	0.560219
$265$	8877.99	2.05800
$266$	−632.647	−0.145827
$267$	70.3234	0.0161188
$268$	−18441.0	−4.20322
$269$	−2496.18	−0.565779	−0.282890	−	0.959152i	$-0.591293\pi$
$269$	−2496.18	−0.565779	−0.282890	+	0.959152i	$0.591293\pi$
$270$	2467.68	0.556215
$271$	−2249.68	−0.504274	−0.252137	−	0.967692i	$-0.581133\pi$
$271$	−2249.68	−0.504274	−0.252137	+	0.967692i	$0.581133\pi$
$272$	−3564.80	−0.794662
$273$	573.838	0.127217
$274$	−10159.1	−2.23990
$275$	−1747.72	−0.383243
$276$	−1141.22	−0.248889
$277$	4082.59	0.885556	0.442778	−	0.896631i	$-0.353993\pi$
$277$	4082.59	0.885556	0.442778	+	0.896631i	$0.353993\pi$
$278$	−5175.22	−1.11651
$279$	1974.56	0.423705
$280$	9444.57	2.01579
$281$	−1033.79	−0.219468	−0.109734	−	0.993961i	$-0.535000\pi$
$281$	−1033.79	−0.219468	−0.109734	+	0.993961i	$0.535000\pi$
$282$	−8094.46	−1.70928
$283$	−7809.14	−1.64030	−0.820150	−	0.572148i	$-0.806110\pi$
$283$	−7809.14	−1.64030	−0.820150	+	0.572148i	$0.806110\pi$
$284$	12445.9	2.60044
$285$	765.838	0.159173
$286$	−1482.70	−0.306552
$287$	3540.50	0.728186
$288$	5673.61	1.16084
$289$	−4658.85	−0.948270
$290$	11748.6	2.37896
$291$	−2158.47	−0.434817
$292$	10928.9	2.19030
$293$	−1949.19	−0.388645	−0.194323	−	0.980938i	$-0.562251\pi$
$293$	−1949.19	−0.388645	−0.194323	+	0.980938i	$0.562251\pi$
$294$	4617.47	0.915973
$295$	9746.10	1.92353
$296$	6704.55	1.31653
$297$	297.000	0.0580259
$298$	−11103.9	−2.15849
$299$	441.209	0.0853371
$300$	−10212.0	−1.96530
$301$	−500.081	−0.0957614
$302$	2581.28	0.491842
$303$	−5613.81	−1.06437
$304$	3387.96	0.639187
$305$	3733.19	0.700859
$306$	−778.286	−0.145398
$307$	2364.09	0.439497	0.219748	−	0.975557i	$-0.429476\pi$
$307$	2364.09	0.439497	0.219748	+	0.975557i	$0.429476\pi$
$308$	1814.11	0.335612
$309$	1286.24	0.236801
$310$	−20051.7	−3.67375
$311$	−1989.17	−0.362686	−0.181343	−	0.983420i	$-0.558044\pi$
$311$	−1989.17	−0.362686	−0.181343	+	0.983420i	$0.558044\pi$
$312$	−5428.47	−0.985021
$313$	−3878.67	−0.700433	−0.350216	−	0.936669i	$-0.613892\pi$
$313$	−3878.67	−0.700433	−0.350216	+	0.936669i	$0.613892\pi$
$314$	−3512.13	−0.631214
$315$	1167.28	0.208790
$316$	22176.8	3.94792
$317$	2913.73	0.516251	0.258126	−	0.966111i	$-0.416895\pi$
$317$	2913.73	0.516251	0.258126	+	0.966111i	$0.416895\pi$
$318$	8574.71	1.51209
$319$	1414.01	0.248180
$320$	−27475.1	−4.79971
$321$	−3444.07	−0.598846
$322$	−741.402	−0.128313
$323$	−241.540	−0.0416087
$324$	1735.38	0.297562
$325$	3948.09	0.673847
$326$	5930.17	1.00749
$327$	5484.95	0.927580
$328$	−33492.9	−5.63822
$329$	−3828.90	−0.641624
$330$	−3016.05	−0.503115
$331$	8104.46	1.34580	0.672902	−	0.739731i	$-0.265047\pi$
$331$	8104.46	1.34580	0.672902	+	0.739731i	$0.265047\pi$
$332$	12987.7	2.14697
$333$	828.633	0.136363
$334$	6093.02	0.998189
$335$	14502.6	2.36525
$336$	5163.88	0.838430
$337$	5919.19	0.956792	0.478396	−	0.878144i	$-0.341218\pi$
$337$	5919.19	0.956792	0.478396	+	0.878144i	$0.341218\pi$
$338$	−8568.07	−1.37882
$339$	−3379.19	−0.541394
$340$	5754.70	0.917919
$341$	−2413.35	−0.383256
$342$	739.677	0.116951
$343$	4824.51	0.759472
$344$	4730.73	0.741465
$345$	897.490	0.140056
$346$	249.590	0.0387805
$347$	−8540.59	−1.32128	−0.660638	−	0.750705i	$-0.729714\pi$
$347$	−8540.59	−1.32128	−0.660638	+	0.750705i	$0.729714\pi$
$348$	8262.11	1.27269
$349$	937.337	0.143767	0.0718833	−	0.997413i	$-0.477099\pi$
$349$	937.337	0.143767	0.0718833	+	0.997413i	$0.477099\pi$
$350$	−6634.31	−1.01320
$351$	−670.919	−0.102026
$352$	−6934.42	−1.05002
$353$	−211.118	−0.0318319	−0.0159160	−	0.999873i	$-0.505066\pi$
$353$	−211.118	−0.0318319	−0.0159160	+	0.999873i	$0.505066\pi$
$354$	9413.17	1.41329
$355$	−9787.82	−1.46333
$356$	−502.213	−0.0747675
$357$	−368.151	−0.0545788
$358$	−4512.99	−0.666254
$359$	−1376.31	−0.202337	−0.101169	−	0.994869i	$-0.532258\pi$
$359$	−1376.31	−0.202337	−0.101169	+	0.994869i	$0.532258\pi$
$360$	−11042.4	−1.61662
$361$	−6629.44	−0.966532
$362$	−9821.63	−1.42600
$363$	−363.000	−0.0524864
$364$	−4098.05	−0.590100
$365$	−8594.87	−1.23254
$366$	3605.66	0.514948
$367$	1030.45	0.146564	0.0732821	−	0.997311i	$-0.476653\pi$
$367$	1030.45	0.146564	0.0732821	+	0.997311i	$0.476653\pi$
$368$	3970.37	0.562418
$369$	−4139.48	−0.583991
$370$	−8414.81	−1.18234
$371$	4056.07	0.567603
$372$	−14101.3	−1.96537
$373$	9365.39	1.30006	0.650029	−	0.759909i	$-0.274757\pi$
$373$	9365.39	1.30006	0.650029	+	0.759909i	$0.274757\pi$
$374$	951.239	0.131517
$375$	1712.72	0.235852
$376$	36221.2	4.96799
$377$	−3194.24	−0.436370
$378$	1127.40	0.153406
$379$	−7120.23	−0.965017	−0.482509	−	0.875891i	$-0.660274\pi$
$379$	−7120.23	−0.965017	−0.482509	+	0.875891i	$0.660274\pi$
$380$	−5469.22	−0.738329
$381$	−1983.91	−0.266769
$382$	2489.26	0.333407
$383$	−1163.56	−0.155235	−0.0776176	−	0.996983i	$-0.524731\pi$
$383$	−1163.56	−0.155235	−0.0776176	+	0.996983i	$0.524731\pi$
$384$	−11406.9	−1.51590
$385$	−1426.67	−0.188857
$386$	9649.57	1.27241
$387$	584.684	0.0767988
$388$	15414.7	2.01691
$389$	−10958.9	−1.42838	−0.714188	−	0.699954i	$-0.753204\pi$
$389$	−10958.9	−1.42838	−0.714188	+	0.699954i	$0.753204\pi$
$390$	6813.22	0.884617
$391$	−283.062	−0.0366114
$392$	−20662.3	−2.66225
$393$	−1866.56	−0.239581
$394$	28774.0	3.67923
$395$	−17440.6	−2.22159
$396$	−2121.02	−0.269155
$397$	−2172.09	−0.274595	−0.137298	−	0.990530i	$-0.543842\pi$
$397$	−2172.09	−0.274595	−0.137298	+	0.990530i	$0.543842\pi$
$398$	−27872.9	−3.51041
$399$	349.888	0.0439005
$400$	35528.2	4.44102
$401$	7830.71	0.975180	0.487590	−	0.873073i	$-0.337876\pi$
$401$	7830.71	0.975180	0.487590	+	0.873073i	$0.337876\pi$
$402$	14007.2	1.73784
$403$	5451.73	0.673870
$404$	40090.9	4.93712
$405$	−1364.76	−0.167445
$406$	5367.55	0.656126
$407$	−1012.77	−0.123345
$408$	3482.68	0.422594
$409$	−10731.2	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$409$	−10731.2	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$410$	42036.6	5.06351
$411$	5618.52	0.674309
$412$	−9185.62	−1.09841
$413$	4452.69	0.530515
$414$	866.831	0.102904
$415$	−10214.0	−1.20815
$416$	15664.8	1.84622
$417$	2862.17	0.336118
$418$	−904.049	−0.105786
$419$	−7315.88	−0.852994	−0.426497	−	0.904489i	$-0.640252\pi$
$419$	−7315.88	−0.852994	−0.426497	+	0.904489i	$0.640252\pi$
$420$	−8336.10	−0.968476
$421$	−12495.7	−1.44657	−0.723284	−	0.690551i	$-0.757368\pi$
$421$	−12495.7	−1.44657	−0.723284	+	0.690551i	$0.757368\pi$
$422$	−23120.9	−2.66708
$423$	4476.67	0.514570
$424$	−38370.2	−4.39486
$425$	−2532.93	−0.289094
$426$	−9453.46	−1.07517
$427$	1705.58	0.193299
$428$	24595.8	2.77776
$429$	820.012	0.0922857
$430$	−5937.49	−0.665887
$431$	6075.01	0.678939	0.339470	−	0.940617i	$-0.389752\pi$
$431$	6075.01	0.678939	0.339470	+	0.940617i	$0.389752\pi$
$432$	−6037.49	−0.672405
$433$	5641.79	0.626160	0.313080	−	0.949727i	$-0.398639\pi$
$433$	5641.79	0.626160	0.313080	+	0.949727i	$0.398639\pi$
$434$	−9161.01	−1.01323
$435$	−6497.59	−0.716174
$436$	−39170.7	−4.30260
$437$	269.019	0.0294484
$438$	−8301.26	−0.905593
$439$	10897.0	1.18470	0.592351	−	0.805680i	$-0.298200\pi$
$439$	10897.0	1.18470	0.592351	+	0.805680i	$0.298200\pi$
$440$	13496.2	1.46229
$441$	−2553.71	−0.275748
$442$	−2148.84	−0.231244
$443$	−7720.83	−0.828054	−0.414027	−	0.910265i	$-0.635878\pi$
$443$	−7720.83	−0.828054	−0.414027	+	0.910265i	$0.635878\pi$
$444$	−5917.66	−0.632522
$445$	394.956	0.0420735
$446$	−7473.76	−0.793481
$447$	6141.04	0.649802
$448$	−12552.5	−1.32378
$449$	−7473.86	−0.785553	−0.392776	−	0.919634i	$-0.628485\pi$
$449$	−7473.86	−0.785553	−0.392776	+	0.919634i	$0.628485\pi$
$450$	7756.70	0.812565
$451$	5059.36	0.528240
$452$	24132.4	2.51127
$453$	−1427.59	−0.148066
$454$	−6657.29	−0.688198
$455$	3222.84	0.332064
$456$	−3309.91	−0.339914
$457$	11140.5	1.14033	0.570167	−	0.821529i	$-0.306879\pi$
$457$	11140.5	1.14033	0.570167	+	0.821529i	$0.306879\pi$
$458$	21102.6	2.15296
$459$	430.434	0.0437711
$460$	−6409.41	−0.649652
$461$	14328.8	1.44763	0.723817	−	0.689992i	$-0.242386\pi$
$461$	14328.8	1.44763	0.723817	+	0.689992i	$0.242386\pi$
$462$	−1377.94	−0.138761
$463$	11760.7	1.18049	0.590246	−	0.807223i	$-0.299031\pi$
$463$	11760.7	1.18049	0.590246	+	0.807223i	$0.299031\pi$
$464$	−28744.4	−2.87592
$465$	11089.7	1.10596
$466$	−17456.5	−1.73532
$467$	11854.9	1.17469	0.587343	−	0.809338i	$-0.300174\pi$
$467$	11854.9	1.17469	0.587343	+	0.809338i	$0.300174\pi$
$468$	4791.35	0.473249
$469$	6625.77	0.652345
$470$	−45460.8	−4.46160
$471$	1942.40	0.190023
$472$	−42122.1	−4.10769
$473$	−714.613	−0.0694671
$474$	−16844.8	−1.63229
$475$	2407.27	0.232533
$476$	2629.14	0.253165
$477$	−4742.27	−0.455207
$478$	−2322.19	−0.222206
$479$	−1324.68	−0.126359	−0.0631796	−	0.998002i	$-0.520124\pi$
$479$	−1324.68	−0.126359	−0.0631796	+	0.998002i	$0.520124\pi$
$480$	31864.6	3.03003
$481$	2287.84	0.216874
$482$	6678.34	0.631100
$483$	410.035	0.0386278
$484$	2592.36	0.243459
$485$	−12122.6	−1.13497
$486$	−1318.14	−0.123029
$487$	18636.4	1.73408	0.867040	−	0.498239i	$-0.166020\pi$
$487$	18636.4	1.73408	0.867040	+	0.498239i	$0.166020\pi$
$488$	−16134.7	−1.49668
$489$	−3279.70	−0.303299
$490$	25933.0	2.39089
$491$	124.552	0.0114480	0.00572398	−	0.999984i	$-0.498178\pi$
$491$	124.552	0.0114480	0.00572398	+	0.999984i	$0.498178\pi$
$492$	29562.0	2.70886
$493$	2049.29	0.187212
$494$	2042.24	0.186001
$495$	1668.04	0.151460
$496$	49059.2	4.44117
$497$	−4471.75	−0.403592
$498$	−9865.04	−0.887677
$499$	10230.2	0.917768	0.458884	−	0.888496i	$-0.348249\pi$
$499$	10230.2	0.917768	0.458884	+	0.888496i	$0.348249\pi$
$500$	−12231.4	−1.09401
$501$	−3369.76	−0.300499
$502$	15395.7	1.36881
$503$	5150.81	0.456587	0.228294	−	0.973592i	$-0.426685\pi$
$503$	5150.81	0.456587	0.228294	+	0.973592i	$0.426685\pi$
$504$	−5044.92	−0.445870
$505$	−31528.8	−2.77824
$506$	−1059.46	−0.0930806
$507$	4738.60	0.415086
$508$	14168.1	1.23741
$509$	−22.7715	−0.00198296	−0.000991481	−	1.00000i	$-0.500316\pi$
$509$	−22.7715	−0.00198296	−0.000991481		1.00000i	$0.500316\pi$
$510$	−4371.08	−0.379519
$511$	−3926.73	−0.339938
$512$	10698.1	0.923429
$513$	−409.081	−0.0352073
$514$	1855.19	0.159201
$515$	7223.87	0.618100
$516$	−4175.50	−0.356233
$517$	−5471.49	−0.465446
$518$	−3844.46	−0.326093
$519$	−138.037	−0.0116747
$520$	−30487.8	−2.57112
$521$	21521.7	1.80976	0.904879	−	0.425669i	$-0.139961\pi$
$521$	21521.7	1.80976	0.904879	+	0.425669i	$0.139961\pi$
$522$	−6275.63	−0.526201
$523$	2923.36	0.244416	0.122208	−	0.992504i	$-0.461002\pi$
$523$	2923.36	0.244416	0.122208	+	0.992504i	$0.461002\pi$
$524$	13330.0	1.11130
$525$	3669.13	0.305017
$526$	31977.0	2.65069
$527$	−3497.60	−0.289104
$528$	7379.15	0.608213
$529$	−11851.7	−0.974088
$530$	48158.0	3.94689
$531$	−5205.99	−0.425462
$532$	−2498.71	−0.203633
$533$	−11429.0	−0.928792
$534$	381.464	0.0309130
$535$	−19342.9	−1.56312
$536$	−62679.3	−5.05100
$537$	2495.93	0.200572
$538$	−13540.3	−1.08507
$539$	3121.20	0.249424
$540$	9746.38	0.776699
$541$	21272.8	1.69056	0.845278	−	0.534327i	$-0.179435\pi$
$541$	21272.8	1.69056	0.845278	+	0.534327i	$0.179435\pi$
$542$	−12203.2	−0.967108
$543$	5431.89	0.429290
$544$	−10049.9	−0.792067
$545$	30805.1	2.42118
$546$	3112.75	0.243980
$547$	−18730.5	−1.46409	−0.732046	−	0.681256i	$-0.761434\pi$
$547$	−18730.5	−1.46409	−0.732046	+	0.681256i	$0.761434\pi$
$548$	−40124.5	−3.12780
$549$	−1994.12	−0.155022
$550$	−9480.41	−0.734992
$551$	−1947.63	−0.150584
$552$	−3878.91	−0.299089
$553$	−7968.04	−0.612723
$554$	22145.7	1.69834
$555$	4653.84	0.355936
$556$	−20440.1	−1.55909
$557$	−18885.0	−1.43659	−0.718297	−	0.695736i	$-0.755078\pi$
$557$	−18885.0	−1.43659	−0.718297	+	0.695736i	$0.755078\pi$
$558$	10710.9	0.812593
$559$	1614.30	0.122143
$560$	29001.8	2.18848
$561$	−526.086	−0.0395925
$562$	−5607.71	−0.420902
$563$	10285.1	0.769922	0.384961	−	0.922933i	$-0.374215\pi$
$563$	10285.1	0.769922	0.384961	+	0.922933i	$0.374215\pi$
$564$	−31970.0	−2.38685
$565$	−18978.5	−1.41315
$566$	−42360.1	−3.14581
$567$	−623.515	−0.0461820
$568$	42302.5	3.12495
$569$	18008.8	1.32683	0.663415	−	0.748251i	$-0.269106\pi$
$569$	18008.8	1.32683	0.663415	+	0.748251i	$0.269106\pi$
$570$	4154.24	0.305266
$571$	−7010.79	−0.513822	−0.256911	−	0.966435i	$-0.582705\pi$
$571$	−7010.79	−0.513822	−0.256911	+	0.966435i	$0.582705\pi$
$572$	−5856.10	−0.428070
$573$	−1376.69	−0.100370
$574$	19205.2	1.39653
$575$	2821.10	0.204605
$576$	14676.1	1.06164
$577$	−16398.9	−1.18318	−0.591589	−	0.806240i	$-0.701499\pi$
$577$	−16398.9	−1.18318	−0.591589	+	0.806240i	$0.701499\pi$
$578$	−25271.6	−1.81862
$579$	−5336.73	−0.383052
$580$	46402.4	3.32199
$581$	−4666.44	−0.333213
$582$	−11708.5	−0.833903
$583$	5796.11	0.411750
$584$	37146.6	2.63208
$585$	−3768.07	−0.266309
$586$	−10573.3	−0.745354
$587$	12823.5	0.901671	0.450836	−	0.892607i	$-0.351126\pi$
$587$	12823.5	0.901671	0.450836	+	0.892607i	$0.351126\pi$
$588$	18237.2	1.27907
$589$	3324.09	0.232541
$590$	52867.0	3.68899
$591$	−15913.6	−1.10761
$592$	20587.9	1.42932
$593$	16899.5	1.17029	0.585144	−	0.810929i	$-0.301038\pi$
$593$	16899.5	1.17029	0.585144	+	0.810929i	$0.301038\pi$
$594$	1611.06	0.111284
$595$	−2067.64	−0.142462
$596$	−43856.1	−3.01412
$597$	15415.2	1.05679
$598$	2393.31	0.163662
$599$	−15074.9	−1.02829	−0.514143	−	0.857704i	$-0.671890\pi$
$599$	−15074.9	−1.02829	−0.514143	+	0.857704i	$0.671890\pi$
$600$	−34709.7	−2.36170
$601$	−11418.8	−0.775014	−0.387507	−	0.921867i	$-0.626664\pi$
$601$	−11418.8	−0.775014	−0.387507	+	0.921867i	$0.626664\pi$
$602$	−2712.65	−0.183654
$603$	−7746.71	−0.523168
$604$	10195.1	0.686809
$605$	−2038.71	−0.137001
$606$	−30451.7	−2.04128
$607$	−17952.8	−1.20046	−0.600232	−	0.799826i	$-0.704925\pi$
$607$	−17952.8	−1.20046	−0.600232	+	0.799826i	$0.704925\pi$
$608$	9551.30	0.637100
$609$	−2968.54	−0.197523
$610$	20250.4	1.34412
$611$	12360.0	0.818384
$612$	−3073.94	−0.203033
$613$	12528.9	0.825507	0.412753	−	0.910843i	$-0.364567\pi$
$613$	12528.9	0.825507	0.412753	+	0.910843i	$0.364567\pi$
$614$	12823.8	0.842878
$615$	−23248.5	−1.52434
$616$	6166.01	0.403305
$617$	−8586.10	−0.560232	−0.280116	−	0.959966i	$-0.590373\pi$
$617$	−8586.10	−0.560232	−0.280116	+	0.959966i	$0.590373\pi$
$618$	6977.09	0.454142
$619$	18415.4	1.19576	0.597882	−	0.801584i	$-0.296009\pi$
$619$	18415.4	1.19576	0.597882	+	0.801584i	$0.296009\pi$
$620$	−79196.7	−5.13003
$621$	−479.404	−0.0309788
$622$	−10790.1	−0.695569
$623$	180.443	0.0116040
$624$	−16669.4	−1.06941
$625$	−10241.4	−0.655448
$626$	−21039.6	−1.34331
$627$	499.988	0.0318462
$628$	−13871.6	−0.881427
$629$	−1467.79	−0.0930436
$630$	6331.82	0.400422
$631$	2374.38	0.149798	0.0748989	−	0.997191i	$-0.476137\pi$
$631$	2374.38	0.149798	0.0748989	+	0.997191i	$0.476137\pi$
$632$	75377.1	4.74421
$633$	12787.1	0.802909
$634$	15805.3	0.990080
$635$	−11142.2	−0.696324
$636$	33866.8	2.11149
$637$	−7050.74	−0.438557
$638$	7670.21	0.475966
$639$	5228.27	0.323673
$640$	−64064.6	−3.95684
$641$	11086.0	0.683104	0.341552	−	0.939863i	$-0.389047\pi$
$641$	11086.0	0.683104	0.341552	+	0.939863i	$0.389047\pi$
$642$	−18682.1	−1.14848
$643$	19934.1	1.22259	0.611294	−	0.791403i	$-0.290649\pi$
$643$	19934.1	1.22259	0.611294	+	0.791403i	$0.290649\pi$
$644$	−2928.26	−0.179176
$645$	3283.75	0.200461
$646$	−1310.21	−0.0797983
$647$	−30634.8	−1.86148	−0.930739	−	0.365684i	$-0.880835\pi$
$647$	−30634.8	−1.86148	−0.930739	+	0.365684i	$0.880835\pi$
$648$	5898.41	0.357579
$649$	6362.87	0.384845
$650$	21416.1	1.29232
$651$	5066.53	0.305028
$652$	23421.9	1.40686
$653$	−9818.07	−0.588378	−0.294189	−	0.955747i	$-0.595049\pi$
$653$	−9818.07	−0.588378	−0.294189	+	0.955747i	$0.595049\pi$
$654$	29752.7	1.77894
$655$	−10483.1	−0.625358
$656$	−102848.	−6.12125
$657$	4591.04	0.272624
$658$	−20769.6	−1.23052
$659$	16478.5	0.974070	0.487035	−	0.873383i	$-0.338078\pi$
$659$	16478.5	0.974070	0.487035	+	0.873383i	$0.338078\pi$
$660$	−11912.2	−0.702551
$661$	2958.12	0.174066	0.0870328	−	0.996205i	$-0.472262\pi$
$661$	2958.12	0.174066	0.0870328	+	0.996205i	$0.472262\pi$
$662$	43962.1	2.58102
$663$	1188.42	0.0696146
$664$	44144.2	2.58001
$665$	1965.07	0.114590
$666$	4494.86	0.261520
$667$	−2282.44	−0.132498
$668$	24065.1	1.39387
$669$	4133.39	0.238873
$670$	78668.2	4.53614
$671$	2437.26	0.140223
$672$	14558.0	0.835692
$673$	−29960.3	−1.71602	−0.858012	−	0.513630i	$-0.828300\pi$
$673$	−29960.3	−1.71602	−0.858012	+	0.513630i	$0.828300\pi$
$674$	32108.2	1.83496
$675$	−4289.87	−0.244618
$676$	−33840.6	−1.92539
$677$	−4514.73	−0.256300	−0.128150	−	0.991755i	$-0.540904\pi$
$677$	−4514.73	−0.256300	−0.128150	+	0.991755i	$0.540904\pi$
$678$	−18330.2	−1.03830
$679$	−5538.43	−0.313027
$680$	19559.7	1.10306
$681$	3681.84	0.207178
$682$	−13091.0	−0.735018
$683$	−13555.7	−0.759438	−0.379719	−	0.925102i	$-0.623979\pi$
$683$	−13555.7	−0.759438	−0.379719	+	0.925102i	$0.623979\pi$
$684$	2921.44	0.163310
$685$	31555.2	1.76009
$686$	26170.2	1.45653
$687$	−11670.8	−0.648137
$688$	14526.8	0.804986
$689$	−13093.3	−0.723972
$690$	4868.37	0.268603
$691$	−11471.3	−0.631535	−0.315768	−	0.948837i	$-0.602262\pi$
$691$	−11471.3	−0.631535	−0.315768	+	0.948837i	$0.602262\pi$
$692$	985.787	0.0541532
$693$	762.074	0.0417731
$694$	−46327.8	−2.53398
$695$	16074.8	0.877340
$696$	28082.2	1.52939
$697$	7332.40	0.398471
$698$	5084.52	0.275719
$699$	9654.41	0.522408
$700$	−26203.0	−1.41483
$701$	22229.0	1.19769	0.598843	−	0.800866i	$-0.295627\pi$
$701$	22229.0	1.19769	0.598843	+	0.800866i	$0.295627\pi$
$702$	−3639.35	−0.195667
$703$	1394.97	0.0748397
$704$	−17937.5	−0.960292
$705$	25142.3	1.34314
$706$	−1145.19	−0.0610480
$707$	−14404.5	−0.766248
$708$	37178.4	1.97352
$709$	−15081.2	−0.798851	−0.399426	−	0.916766i	$-0.630790\pi$
$709$	−15081.2	−0.798851	−0.399426	+	0.916766i	$0.630790\pi$
$710$	−53093.4	−2.80642
$711$	9316.06	0.491392
$712$	−1706.98	−0.0898480
$713$	3895.52	0.204612
$714$	−1997.01	−0.104673
$715$	4605.42	0.240885
$716$	−17824.6	−0.930358
$717$	1284.30	0.0668938
$718$	−7465.71	−0.388047
$719$	−7399.80	−0.383819	−0.191910	−	0.981413i	$-0.561468\pi$
$719$	−7399.80	−0.383819	−0.191910	+	0.981413i	$0.561468\pi$
$720$	−33908.3	−1.75512
$721$	3300.36	0.170474
$722$	−35960.9	−1.85364
$723$	−3693.48	−0.189989
$724$	−38791.7	−1.99127
$725$	−20424.0	−1.04625
$726$	−1969.07	−0.100660
$727$	1705.77	0.0870202	0.0435101	−	0.999053i	$-0.486146\pi$
$727$	1705.77	0.0870202	0.0435101	+	0.999053i	$0.486146\pi$
$728$	−13928.9	−0.709122
$729$	729.000	0.0370370
$730$	−46622.3	−2.36379
$731$	−1035.67	−0.0524017
$732$	14241.0	0.719074
$733$	37122.6	1.87061	0.935303	−	0.353847i	$-0.115127\pi$
$733$	37122.6	1.87061	0.935303	+	0.353847i	$0.115127\pi$
$734$	5589.60	0.281084
$735$	−14342.3	−0.719762
$736$	11193.2	0.560581
$737$	9468.20	0.473223
$738$	−22454.3	−1.11999
$739$	34256.3	1.70520	0.852598	−	0.522568i	$-0.175026\pi$
$739$	34256.3	1.70520	0.852598	+	0.522568i	$0.175026\pi$
$740$	−33235.3	−1.65102
$741$	−1129.47	−0.0559945
$742$	22001.9	1.08856
$743$	1567.88	0.0774160	0.0387080	−	0.999251i	$-0.487676\pi$
$743$	1567.88	0.0774160	0.0387080	+	0.999251i	$0.487676\pi$
$744$	−47929.0	−2.36178
$745$	34489.8	1.69612
$746$	50801.9	2.49328
$747$	5455.90	0.267230
$748$	3757.03	0.183651
$749$	−8837.17	−0.431112
$750$	9290.53	0.452323
$751$	−955.613	−0.0464325	−0.0232163	−	0.999730i	$-0.507391\pi$
$751$	−955.613	−0.0464325	−0.0232163	+	0.999730i	$0.507391\pi$
$752$	111226.	5.39360
$753$	−8514.65	−0.412073
$754$	−17326.9	−0.836881
$755$	−8017.75	−0.386484
$756$	4452.82	0.214216
$757$	−14015.4	−0.672918	−0.336459	−	0.941698i	$-0.609229\pi$
$757$	−14015.4	−0.672918	−0.336459	+	0.941698i	$0.609229\pi$
$758$	−38623.2	−1.85073
$759$	585.938	0.0280214
$760$	−18589.4	−0.887249
$761$	−36271.0	−1.72776	−0.863879	−	0.503699i	$-0.831972\pi$
$761$	−36271.0	−1.72776	−0.863879	+	0.503699i	$0.831972\pi$
$762$	−10761.6	−0.511616
$763$	14073.9	0.667770
$764$	9831.63	0.465571
$765$	2417.44	0.114252
$766$	−6311.64	−0.297714
$767$	−14373.6	−0.676665
$768$	−22739.7	−1.06842
$769$	−18163.6	−0.851749	−0.425874	−	0.904782i	$-0.640033\pi$
$769$	−18163.6	−0.851749	−0.425874	+	0.904782i	$0.640033\pi$
$770$	−7738.90	−0.362195
$771$	−1026.02	−0.0479264
$772$	38112.1	1.77680
$773$	−8345.65	−0.388321	−0.194160	−	0.980970i	$-0.562198\pi$
$773$	−8345.65	−0.388321	−0.194160	+	0.980970i	$0.562198\pi$
$774$	3171.57	0.147287
$775$	34858.4	1.61568
$776$	52393.2	2.42372
$777$	2126.19	0.0981683
$778$	−59445.7	−2.73937
$779$	−6968.65	−0.320510
$780$	26909.6	1.23528
$781$	−6390.11	−0.292774
$782$	−1535.45	−0.0702142
$783$	3470.76	0.158410
$784$	−63448.5	−2.89033
$785$	10909.1	0.496001
$786$	−10125.0	−0.459475
$787$	−22996.2	−1.04158	−0.520791	−	0.853684i	$-0.674363\pi$
$787$	−22996.2	−1.04158	−0.520791	+	0.853684i	$0.674363\pi$
$788$	113647.	5.13768
$789$	−17685.0	−0.797975
$790$	−94605.0	−4.26063
$791$	−8670.69	−0.389752
$792$	−7209.17	−0.323443
$793$	−5505.75	−0.246551
$794$	−11782.4	−0.526626
$795$	−26634.0	−1.18819
$796$	−110087.	−4.90194
$797$	−2743.82	−0.121946	−0.0609730	−	0.998139i	$-0.519420\pi$
$797$	−2743.82	−0.121946	−0.0609730	+	0.998139i	$0.519420\pi$
$798$	1897.94	0.0841934
$799$	−7929.68	−0.351104
$800$	100161.	4.42652
$801$	−210.970	−0.00930619
$802$	42477.1	1.87022
$803$	−5611.28	−0.246597
$804$	55322.9	2.42673
$805$	2302.88	0.100827
$806$	29572.5	1.29237
$807$	7488.53	0.326653
$808$	136266.	5.93293
$809$	41241.7	1.79231	0.896156	−	0.443738i	$-0.146348\pi$
$809$	41241.7	1.79231	0.896156	+	0.443738i	$0.146348\pi$
$810$	−7403.03	−0.321131
$811$	12832.9	0.555641	0.277820	−	0.960633i	$-0.410388\pi$
$811$	12832.9	0.555641	0.277820	+	0.960633i	$0.410388\pi$
$812$	21199.8	0.916215
$813$	6749.03	0.291142
$814$	−5493.72	−0.236554
$815$	−18419.7	−0.791675
$816$	10694.4	0.458798
$817$	984.292	0.0421493
$818$	−58210.4	−2.48812
$819$	−1721.51	−0.0734488
$820$	166029.	7.07069
$821$	16368.5	0.695817	0.347908	−	0.937529i	$-0.386892\pi$
$821$	16368.5	0.695817	0.347908	+	0.937529i	$0.386892\pi$
$822$	30477.3	1.29321
$823$	3869.53	0.163892	0.0819461	−	0.996637i	$-0.473886\pi$
$823$	3869.53	0.163892	0.0819461	+	0.996637i	$0.473886\pi$
$824$	−31221.2	−1.31995
$825$	5243.17	0.221265
$826$	24153.3	1.01743
$827$	7388.69	0.310677	0.155339	−	0.987861i	$-0.450353\pi$
$827$	7388.69	0.310677	0.155339	+	0.987861i	$0.450353\pi$
$828$	3423.65	0.143696
$829$	23990.1	1.00508	0.502539	−	0.864554i	$-0.332399\pi$
$829$	23990.1	1.00508	0.502539	+	0.864554i	$0.332399\pi$
$830$	−55404.9	−2.31703
$831$	−12247.8	−0.511276
$832$	40520.6	1.68846
$833$	4523.47	0.188150
$834$	15525.7	0.644616
$835$	−18925.6	−0.784367
$836$	−3570.65	−0.147720
$837$	−5923.68	−0.244626
$838$	−39684.5	−1.63589
$839$	−18228.3	−0.750074	−0.375037	−	0.927010i	$-0.622370\pi$
$839$	−18228.3	−0.750074	−0.375037	+	0.927010i	$0.622370\pi$
$840$	−28333.7	−1.16382
$841$	−7864.78	−0.322472
$842$	−67782.3	−2.77427
$843$	3101.36	0.126710
$844$	−91318.7	−3.72431
$845$	26613.3	1.08346
$846$	24283.4	0.986855
$847$	−931.424	−0.0377852
$848$	−117825.	−4.77137
$849$	23427.4	0.947028
$850$	−13739.7	−0.554433
$851$	1634.77	0.0658511
$852$	−37337.6	−1.50137
$853$	−21737.3	−0.872534	−0.436267	−	0.899817i	$-0.643700\pi$
$853$	−21737.3	−0.872534	−0.436267	+	0.899817i	$0.643700\pi$
$854$	9251.79	0.370714
$855$	−2297.51	−0.0918987
$856$	83599.0	3.33803
$857$	−18712.2	−0.745852	−0.372926	−	0.927861i	$-0.621645\pi$
$857$	−18712.2	−0.745852	−0.372926	+	0.927861i	$0.621645\pi$
$858$	4448.10	0.176988
$859$	30527.6	1.21256	0.606279	−	0.795252i	$-0.292661\pi$
$859$	30527.6	1.21256	0.606279	+	0.795252i	$0.292661\pi$
$860$	−23450.8	−0.929845
$861$	−10621.5	−0.420418
$862$	32953.4	1.30209
$863$	10906.4	0.430196	0.215098	−	0.976592i	$-0.430993\pi$
$863$	10906.4	0.430196	0.215098	+	0.976592i	$0.430993\pi$
$864$	−17020.8	−0.670209
$865$	−775.255	−0.0304734
$866$	30603.5	1.20087
$867$	13976.6	0.547484
$868$	−36182.5	−1.41488
$869$	−11386.3	−0.444481
$870$	−35245.7	−1.37350
$871$	−21388.5	−0.832058
$872$	−133138.	−5.17043
$873$	6475.41	0.251042
$874$	1459.28	0.0564768
$875$	4394.68	0.169791
$876$	−32786.8	−1.26457
$877$	21770.9	0.838256	0.419128	−	0.907927i	$-0.362336\pi$
$877$	21770.9	0.838256	0.419128	+	0.907927i	$0.362336\pi$
$878$	59109.8	2.27205
$879$	5847.58	0.224384
$880$	41443.4	1.58757
$881$	47206.9	1.80527	0.902634	−	0.430409i	$-0.141631\pi$
$881$	47206.9	1.80527	0.902634	+	0.430409i	$0.141631\pi$
$882$	−13852.4	−0.528837
$883$	−6059.68	−0.230945	−0.115473	−	0.993311i	$-0.536838\pi$
$883$	−6059.68	−0.230945	−0.115473	+	0.993311i	$0.536838\pi$
$884$	−8487.09	−0.322909
$885$	−29238.3	−1.11055
$886$	−41881.1	−1.58806
$887$	−37130.2	−1.40553	−0.702767	−	0.711420i	$-0.748052\pi$
$887$	−37130.2	−1.40553	−0.702767	+	0.711420i	$0.748052\pi$
$888$	−20113.6	−0.760101
$889$	−5090.53	−0.192048
$890$	2142.41	0.0806896
$891$	−891.000	−0.0335013
$892$	−29518.5	−1.10802
$893$	7536.30	0.282410
$894$	33311.6	1.24621
$895$	14017.8	0.523536
$896$	−29269.1	−1.09131
$897$	−1323.63	−0.0492694
$898$	−40541.4	−1.50655
$899$	−28202.5	−1.04628
$900$	30636.0	1.13467
$901$	8400.15	0.310599
$902$	27444.1	1.01307
$903$	1500.24	0.0552879
$904$	82024.1	3.01779
$905$	30507.0	1.12054
$906$	−7743.85	−0.283965
$907$	1182.94	0.0433064	0.0216532	−	0.999766i	$-0.493107\pi$
$907$	1182.94	0.0433064	0.0216532	+	0.999766i	$0.493107\pi$
$908$	−26293.8	−0.961001
$909$	16841.4	0.614516
$910$	17482.1	0.636841
$911$	37676.4	1.37022	0.685112	−	0.728438i	$-0.259753\pi$
$911$	37676.4	1.37022	0.685112	+	0.728438i	$0.259753\pi$
$912$	−10163.9	−0.369035
$913$	−6668.32	−0.241719
$914$	60431.1	2.18696
$915$	−11199.6	−0.404641
$916$	83347.1	3.00640
$917$	−4789.41	−0.172476
$918$	2334.86	0.0839454
$919$	8697.82	0.312203	0.156101	−	0.987741i	$-0.450107\pi$
$919$	8697.82	0.312203	0.156101	+	0.987741i	$0.450107\pi$
$920$	−21785.0	−0.780686
$921$	−7092.26	−0.253744
$922$	77725.7	2.77631
$923$	14435.2	0.514778
$924$	−5442.33	−0.193766
$925$	14628.5	0.519981
$926$	63795.3	2.26398
$927$	−3858.71	−0.136717
$928$	−81036.0	−2.86653
$929$	−17247.5	−0.609119	−0.304559	−	0.952493i	$-0.598509\pi$
$929$	−17247.5	−0.609119	−0.304559	+	0.952493i	$0.598509\pi$
$930$	60155.2	2.12104
$931$	−4299.06	−0.151338
$932$	−68946.7	−2.42320
$933$	5967.51	0.209397
$934$	64306.0	2.25284
$935$	−2954.65	−0.103345
$936$	16285.4	0.568702
$937$	−41812.4	−1.45779	−0.728896	−	0.684624i	$-0.759966\pi$
$937$	−41812.4	−1.45779	−0.728896	+	0.684624i	$0.759966\pi$
$938$	35941.0	1.25108
$939$	11636.0	0.404395
$940$	−179553.	−6.23018
$941$	37655.9	1.30451	0.652257	−	0.757998i	$-0.273822\pi$
$941$	37655.9	1.30451	0.652257	+	0.757998i	$0.273822\pi$
$942$	10536.4	0.364431
$943$	−8166.60	−0.282016
$944$	−129346.	−4.45959
$945$	−3501.84	−0.120545
$946$	−3876.37	−0.133226
$947$	−21244.4	−0.728986	−0.364493	−	0.931206i	$-0.618758\pi$
$947$	−21244.4	−0.728986	−0.364493	+	0.931206i	$0.618758\pi$
$948$	−66530.4	−2.27933
$949$	12675.8	0.433587
$950$	13058.1	0.445958
$951$	−8741.20	−0.298058
$952$	8936.24	0.304228
$953$	1324.27	0.0450130	0.0225065	−	0.999747i	$-0.492835\pi$
$953$	1324.27	0.0450130	0.0225065	+	0.999747i	$0.492835\pi$
$954$	−25724.1	−0.873007
$955$	−7731.91	−0.261988
$956$	−9171.77	−0.310289
$957$	−4242.04	−0.143287
$958$	−7185.62	−0.242335
$959$	14416.6	0.485439
$960$	82425.4	2.77111
$961$	18343.4	0.615735
$962$	12410.2	0.415927
$963$	10332.2	0.345744
$964$	26376.9	0.881268
$965$	−29972.6	−0.999847
$966$	2224.21	0.0740815
$967$	52267.1	1.73815	0.869077	−	0.494676i	$-0.164713\pi$
$967$	52267.1	1.73815	0.869077	+	0.494676i	$0.164713\pi$
$968$	8811.20	0.292565
$969$	724.619	0.0240228
$970$	−65758.1	−2.17667
$971$	−52489.8	−1.73479	−0.867394	−	0.497622i	$-0.834207\pi$
$971$	−52489.8	−1.73479	−0.867394	+	0.497622i	$0.834207\pi$
$972$	−5206.14	−0.171797
$973$	7344.07	0.241973
$974$	101092.	3.32566
$975$	−11844.3	−0.389046
$976$	−49545.3	−1.62490
$977$	8324.11	0.272581	0.136291	−	0.990669i	$-0.456482\pi$
$977$	8324.11	0.272581	0.136291	+	0.990669i	$0.456482\pi$
$978$	−17790.5	−0.581675
$979$	257.852	0.00841777
$980$	102426.	3.33864
$981$	−16454.9	−0.535539
$982$	675.623	0.0219552
$983$	−44407.1	−1.44086	−0.720431	−	0.693527i	$-0.756056\pi$
$983$	−44407.1	−1.44086	−0.720431	+	0.693527i	$0.756056\pi$
$984$	100479.	3.25523
$985$	−89375.3	−2.89110
$986$	11116.2	0.359039
$987$	11486.7	0.370442
$988$	8066.05	0.259732
$989$	1153.50	0.0370870
$990$	9048.15	0.290474
$991$	−45124.7	−1.44645	−0.723226	−	0.690612i	$-0.757341\pi$
$991$	−45124.7	−1.44645	−0.723226	+	0.690612i	$0.757341\pi$
$992$	138307.	4.42667
$993$	−24313.4	−0.777001
$994$	−24256.7	−0.774020
$995$	86576.2	2.75844
$996$	−38963.2	−1.23955
$997$	5480.61	0.174095	0.0870474	−	0.996204i	$-0.472257\pi$
$997$	5480.61	0.174095	0.0870474	+	0.996204i	$0.472257\pi$
$998$	55492.9	1.76012
$999$	−2485.90	−0.0787291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.a.c.1.2	✓	2
3.2	odd	2		99.4.a.f.1.1		2
4.3	odd	2		528.4.a.p.1.1		2
5.2	odd	4		825.4.c.h.199.4		4
5.3	odd	4		825.4.c.h.199.1		4
5.4	even	2		825.4.a.l.1.1		2
7.6	odd	2		1617.4.a.k.1.2		2
8.3	odd	2		2112.4.a.bg.1.2		2
8.5	even	2		2112.4.a.bn.1.2		2
11.10	odd	2		363.4.a.i.1.1		2
12.11	even	2		1584.4.a.bj.1.2		2
15.14	odd	2		2475.4.a.p.1.2		2
33.32	even	2		1089.4.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.c.1.2	✓	2	1.1	even	1	trivial
99.4.a.f.1.1		2	3.2	odd	2
363.4.a.i.1.1		2	11.10	odd	2
528.4.a.p.1.1		2	4.3	odd	2
825.4.a.l.1.1		2	5.4	even	2
825.4.c.h.199.1		4	5.3	odd	4
825.4.c.h.199.4		4	5.2	odd	4
1089.4.a.u.1.2		2	33.32	even	2
1584.4.a.bj.1.2		2	12.11	even	2
1617.4.a.k.1.2		2	7.6	odd	2
2112.4.a.bg.1.2		2	8.3	odd	2
2112.4.a.bn.1.2		2	8.5	even	2
2475.4.a.p.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)

\(f(q)\)	\(=\)	\(q+5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} -16.2733 q^{6} -7.69772 q^{7} +72.8199 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} -16.2733 q^{6} -7.69772 q^{7} +72.8199 q^{8} +9.00000 q^{9} -91.3954 q^{10} -11.0000 q^{11} -64.2733 q^{12} +24.8489 q^{13} -41.7557 q^{14} +50.5466 q^{15} +223.611 q^{16} -15.9420 q^{17} +48.8199 q^{18} +15.1511 q^{19} -360.977 q^{20} +23.0931 q^{21} -59.6687 q^{22} +17.7557 q^{23} -218.460 q^{24} +158.884 q^{25} +134.791 q^{26} -27.0000 q^{27} -164.919 q^{28} -128.547 q^{29} +274.186 q^{30} +219.395 q^{31} +630.402 q^{32} +33.0000 q^{33} -86.4763 q^{34} +129.698 q^{35} +192.820 q^{36} +92.0703 q^{37} +82.1863 q^{38} -74.5466 q^{39} -1226.93 q^{40} -459.942 q^{41} +125.267 q^{42} +64.9648 q^{43} -235.669 q^{44} -151.640 q^{45} +96.3146 q^{46} +497.408 q^{47} -670.832 q^{48} -283.745 q^{49} +861.855 q^{50} +47.8260 q^{51} +532.373 q^{52} -526.919 q^{53} -146.460 q^{54} +185.337 q^{55} -560.547 q^{56} -45.4534 q^{57} -697.292 q^{58} -578.443 q^{59} +1082.93 q^{60} -221.569 q^{61} +1190.09 q^{62} -69.2794 q^{63} +1630.68 q^{64} -418.675 q^{65} +179.006 q^{66} -860.745 q^{67} -341.548 q^{68} -53.2671 q^{69} +703.536 q^{70} +580.919 q^{71} +655.379 q^{72} +510.116 q^{73} +499.429 q^{74} -476.652 q^{75} +324.605 q^{76} +84.6749 q^{77} -404.373 q^{78} +1035.12 q^{79} -3767.59 q^{80} +81.0000 q^{81} -2494.92 q^{82} +606.211 q^{83} +494.757 q^{84} +268.605 q^{85} +352.397 q^{86} +385.640 q^{87} -801.018 q^{88} -23.4411 q^{89} -822.559 q^{90} -191.279 q^{91} +380.406 q^{92} -658.186 q^{93} +2698.15 q^{94} -255.279 q^{95} -1891.20 q^{96} +719.490 q^{97} -1539.16 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 - 3 * q^6 + 24 * q^7 + 57 * q^8 + 18 * q^9	\( 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9} - 104 q^{10} - 22 q^{11} - 99 q^{12} + 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} + 106 q^{17} + 9 q^{18} + 50 q^{19} - 328 q^{20} - 72 q^{21} - 11 q^{22} + 134 q^{23} - 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} + 202 q^{28} - 198 q^{29} + 312 q^{30} + 360 q^{31} + 857 q^{32} + 66 q^{33} - 626 q^{34} + 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} - 90 q^{39} - 1272 q^{40} - 782 q^{41} + 546 q^{42} + 386 q^{43} - 363 q^{44} - 126 q^{45} - 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} + 1379 q^{50} - 318 q^{51} + 592 q^{52} - 522 q^{53} - 27 q^{54} + 154 q^{55} - 1062 q^{56} - 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} - 778 q^{61} + 568 q^{62} + 216 q^{63} + 809 q^{64} - 404 q^{65} + 33 q^{66} - 776 q^{67} + 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} + 513 q^{72} + 1296 q^{73} + 2358 q^{74} - 126 q^{75} + 728 q^{76} - 264 q^{77} - 336 q^{78} + 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} - 324 q^{83} - 606 q^{84} + 616 q^{85} - 1068 q^{86} + 594 q^{87} - 627 q^{88} - 756 q^{89} - 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} + 3722 q^{94} - 156 q^{95} - 2571 q^{96} - 452 q^{97} - 4467 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q + q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 - 3 * q^6 + 24 * q^7 + 57 * q^8 + 18 * q^9 - 104 * q^10 - 22 * q^11 - 99 * q^12 + 30 * q^13 - 182 * q^14 + 42 * q^15 + 201 * q^16 + 106 * q^17 + 9 * q^18 + 50 * q^19 - 328 * q^20 - 72 * q^21 - 11 * q^22 + 134 * q^23 - 171 * q^24 + 42 * q^25 + 112 * q^26 - 54 * q^27 + 202 * q^28 - 198 * q^29 + 312 * q^30 + 360 * q^31 + 857 * q^32 + 66 * q^33 - 626 * q^34 + 220 * q^35 + 297 * q^36 - 328 * q^37 - 72 * q^38 - 90 * q^39 - 1272 * q^40 - 782 * q^41 + 546 * q^42 + 386 * q^43 - 363 * q^44 - 126 * q^45 - 418 * q^46 + 266 * q^47 - 603 * q^48 + 378 * q^49 + 1379 * q^50 - 318 * q^51 + 592 * q^52 - 522 * q^53 - 27 * q^54 + 154 * q^55 - 1062 * q^56 - 150 * q^57 - 390 * q^58 - 172 * q^59 + 984 * q^60 - 778 * q^61 + 568 * q^62 + 216 * q^63 + 809 * q^64 - 404 * q^65 + 33 * q^66 - 776 * q^67 + 1070 * q^68 - 402 * q^69 + 304 * q^70 + 630 * q^71 + 513 * q^72 + 1296 * q^73 + 2358 * q^74 - 126 * q^75 + 728 * q^76 - 264 * q^77 - 336 * q^78 + 652 * q^79 - 3832 * q^80 + 162 * q^81 - 1070 * q^82 - 324 * q^83 - 606 * q^84 + 616 * q^85 - 1068 * q^86 + 594 * q^87 - 627 * q^88 - 756 * q^89 - 936 * q^90 - 28 * q^91 + 1726 * q^92 - 1080 * q^93 + 3722 * q^94 - 156 * q^95 - 2571 * q^96 - 452 * q^97 - 4467 * q^98 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more