LMFDB - Embedded newform 2013.4.a.f.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.15474 q^{2} +3.00000 q^{3} -6.66657 q^{4} -7.99519 q^{5} -3.46423 q^{6} +12.3786 q^{7} +16.9361 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.15474 q^{2} +3.00000 q^{3} -6.66657 q^{4} -7.99519 q^{5} -3.46423 q^{6} +12.3786 q^{7} +16.9361 q^{8} +9.00000 q^{9} +9.23239 q^{10} +11.0000 q^{11} -19.9997 q^{12} -60.0821 q^{13} -14.2941 q^{14} -23.9856 q^{15} +33.7757 q^{16} +44.4638 q^{17} -10.3927 q^{18} +56.9732 q^{19} +53.3005 q^{20} +37.1359 q^{21} -12.7022 q^{22} +138.072 q^{23} +50.8083 q^{24} -61.0769 q^{25} +69.3794 q^{26} +27.0000 q^{27} -82.5230 q^{28} -41.6308 q^{29} +27.6972 q^{30} +222.766 q^{31} -174.491 q^{32} +33.0000 q^{33} -51.3442 q^{34} -98.9695 q^{35} -59.9991 q^{36} -119.878 q^{37} -65.7893 q^{38} -180.246 q^{39} -135.407 q^{40} -355.939 q^{41} -42.8824 q^{42} +419.845 q^{43} -73.3323 q^{44} -71.9567 q^{45} -159.437 q^{46} -606.447 q^{47} +101.327 q^{48} -189.770 q^{49} +70.5281 q^{50} +133.391 q^{51} +400.542 q^{52} -53.3436 q^{53} -31.1780 q^{54} -87.9471 q^{55} +209.646 q^{56} +170.920 q^{57} +48.0728 q^{58} +103.700 q^{59} +159.902 q^{60} -61.0000 q^{61} -257.237 q^{62} +111.408 q^{63} -68.7134 q^{64} +480.368 q^{65} -38.1065 q^{66} -313.628 q^{67} -296.421 q^{68} +414.216 q^{69} +114.284 q^{70} +716.138 q^{71} +152.425 q^{72} +660.702 q^{73} +138.429 q^{74} -183.231 q^{75} -379.816 q^{76} +136.165 q^{77} +208.138 q^{78} +768.543 q^{79} -270.043 q^{80} +81.0000 q^{81} +411.017 q^{82} -556.718 q^{83} -247.569 q^{84} -355.496 q^{85} -484.813 q^{86} -124.892 q^{87} +186.297 q^{88} +265.502 q^{89} +83.0915 q^{90} -743.734 q^{91} -920.466 q^{92} +668.298 q^{93} +700.290 q^{94} -455.512 q^{95} -523.473 q^{96} +1345.35 q^{97} +219.135 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) $ 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * 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 $

$2$	−1.15474	−0.408263	−0.204132	−	0.978943i	$-0.565437\pi$
$2$	−1.15474	−0.408263	−0.204132	+	0.978943i	$0.565437\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−6.66657	−0.833321
$5$	−7.99519	−0.715112	−0.357556	−	0.933892i	$-0.616390\pi$
$5$	−7.99519	−0.715112	−0.357556	+	0.933892i	$0.616390\pi$
$6$	−3.46423	−0.235711
$7$	12.3786	0.668383	0.334192	−	0.942505i	$-0.391537\pi$
$7$	12.3786	0.668383	0.334192	+	0.942505i	$0.391537\pi$
$8$	16.9361	0.748477
$9$	9.00000	0.333333
$10$	9.23239	0.291954
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−19.9997	−0.481118
$13$	−60.0821	−1.28183	−0.640915	−	0.767612i	$-0.721445\pi$
$13$	−60.0821	−1.28183	−0.640915	+	0.767612i	$0.721445\pi$
$14$	−14.2941	−0.272876
$15$	−23.9856	−0.412870
$16$	33.7757	0.527746
$17$	44.4638	0.634356	0.317178	−	0.948366i	$-0.397265\pi$
$17$	44.4638	0.634356	0.317178	+	0.948366i	$0.397265\pi$
$18$	−10.3927	−0.136088
$19$	56.9732	0.687923	0.343962	−	0.938984i	$-0.388231\pi$
$19$	56.9732	0.687923	0.343962	+	0.938984i	$0.388231\pi$
$20$	53.3005	0.595918
$21$	37.1359	0.385891
$22$	−12.7022	−0.123096
$23$	138.072	1.25174	0.625869	−	0.779928i	$-0.284744\pi$
$23$	138.072	1.25174	0.625869	+	0.779928i	$0.284744\pi$
$24$	50.8083	0.432134
$25$	−61.0769	−0.488615
$26$	69.3794	0.523324
$27$	27.0000	0.192450
$28$	−82.5230	−0.556978
$29$	−41.6308	−0.266574	−0.133287	−	0.991078i	$-0.542553\pi$
$29$	−41.6308	−0.266574	−0.133287	+	0.991078i	$0.542553\pi$
$30$	27.6972	0.168560
$31$	222.766	1.29064	0.645322	−	0.763911i	$-0.276723\pi$
$31$	222.766	1.29064	0.645322	+	0.763911i	$0.276723\pi$
$32$	−174.491	−0.963936
$33$	33.0000	0.174078
$34$	−51.3442	−0.258984
$35$	−98.9695	−0.477969
$36$	−59.9991	−0.277774
$37$	−119.878	−0.532646	−0.266323	−	0.963884i	$-0.585809\pi$
$37$	−119.878	−0.532646	−0.266323	+	0.963884i	$0.585809\pi$
$38$	−65.7893	−0.280854
$39$	−180.246	−0.740064
$40$	−135.407	−0.535245
$41$	−355.939	−1.35581	−0.677906	−	0.735149i	$-0.737112\pi$
$41$	−355.939	−1.35581	−0.677906	+	0.735149i	$0.737112\pi$
$42$	−42.8824	−0.157545
$43$	419.845	1.48897	0.744486	−	0.667638i	$-0.232695\pi$
$43$	419.845	1.48897	0.744486	+	0.667638i	$0.232695\pi$
$44$	−73.3323	−0.251256
$45$	−71.9567	−0.238371
$46$	−159.437	−0.511038
$47$	−606.447	−1.88211	−0.941057	−	0.338247i	$-0.890166\pi$
$47$	−606.447	−1.88211	−0.941057	+	0.338247i	$0.890166\pi$
$48$	101.327	0.304694
$49$	−189.770	−0.553264
$50$	70.5281	0.199484
$51$	133.391	0.366245
$52$	400.542	1.06818
$53$	−53.3436	−0.138251	−0.0691256	−	0.997608i	$-0.522021\pi$
$53$	−53.3436	−0.138251	−0.0691256	+	0.997608i	$0.522021\pi$
$54$	−31.1780	−0.0785703
$55$	−87.9471	−0.215614
$56$	209.646	0.500270
$57$	170.920	0.397173
$58$	48.0728	0.108832
$59$	103.700	0.228823	0.114411	−	0.993433i	$-0.463502\pi$
$59$	103.700	0.228823	0.114411	+	0.993433i	$0.463502\pi$
$60$	159.902	0.344053
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−257.237	−0.526922
$63$	111.408	0.222794
$64$	−68.7134	−0.134206
$65$	480.368	0.916651
$66$	−38.1065	−0.0710695
$67$	−313.628	−0.571878	−0.285939	−	0.958248i	$-0.592305\pi$
$67$	−313.628	−0.571878	−0.285939	+	0.958248i	$0.592305\pi$
$68$	−296.421	−0.528622
$69$	414.216	0.722691
$70$	114.284	0.195137
$71$	716.138	1.19704	0.598521	−	0.801107i	$-0.295755\pi$
$71$	716.138	1.19704	0.598521	+	0.801107i	$0.295755\pi$
$72$	152.425	0.249492
$73$	660.702	1.05931	0.529653	−	0.848214i	$-0.322322\pi$
$73$	660.702	1.05931	0.529653	+	0.848214i	$0.322322\pi$
$74$	138.429	0.217460
$75$	−183.231	−0.282102
$76$	−379.816	−0.573261
$77$	136.165	0.201525
$78$	208.138	0.302141
$79$	768.543	1.09453	0.547265	−	0.836959i	$-0.315669\pi$
$79$	768.543	1.09453	0.547265	+	0.836959i	$0.315669\pi$
$80$	−270.043	−0.377397
$81$	81.0000	0.111111
$82$	411.017	0.553528
$83$	−556.718	−0.736238	−0.368119	−	0.929779i	$-0.619998\pi$
$83$	−556.718	−0.736238	−0.368119	+	0.929779i	$0.619998\pi$
$84$	−247.569	−0.321571
$85$	−355.496	−0.453635
$86$	−484.813	−0.607892
$87$	−124.892	−0.153906
$88$	186.297	0.225674
$89$	265.502	0.316216	0.158108	−	0.987422i	$-0.449461\pi$
$89$	265.502	0.316216	0.158108	+	0.987422i	$0.449461\pi$
$90$	83.0915	0.0973179
$91$	−743.734	−0.856753
$92$	−920.466	−1.04310
$93$	668.298	0.745153
$94$	700.290	0.768398
$95$	−455.512	−0.491942
$96$	−523.473	−0.556529
$97$	1345.35	1.40824	0.704120	−	0.710081i	$-0.251342\pi$
$97$	1345.35	1.40824	0.704120	+	0.710081i	$0.251342\pi$
$98$	219.135	0.225877
$99$	99.0000	0.100504
$100$	407.173	0.407173
$101$	8.21503	0.00809333	0.00404666	−	0.999992i	$-0.498712\pi$
$101$	8.21503	0.00809333	0.00404666	+	0.999992i	$0.498712\pi$
$102$	−154.033	−0.149524
$103$	−1410.35	−1.34919	−0.674593	−	0.738190i	$-0.735681\pi$
$103$	−1410.35	−1.34919	−0.674593	+	0.738190i	$0.735681\pi$
$104$	−1017.56	−0.959420
$105$	−296.909	−0.275955
$106$	61.5982	0.0564428
$107$	1426.67	1.28898	0.644491	−	0.764612i	$-0.277069\pi$
$107$	1426.67	1.28898	0.644491	+	0.764612i	$0.277069\pi$
$108$	−179.997	−0.160373
$109$	−1568.96	−1.37871	−0.689354	−	0.724425i	$-0.742105\pi$
$109$	−1568.96	−1.37871	−0.689354	+	0.724425i	$0.742105\pi$
$110$	101.556	0.0880274
$111$	−359.635	−0.307523
$112$	418.097	0.352736
$113$	−1062.33	−0.884389	−0.442195	−	0.896919i	$-0.645800\pi$
$113$	−1062.33	−0.884389	−0.442195	+	0.896919i	$0.645800\pi$
$114$	−197.368	−0.162151
$115$	−1103.91	−0.895133
$116$	277.534	0.222142
$117$	−540.739	−0.427276
$118$	−119.746	−0.0934198
$119$	550.400	0.423993
$120$	−406.222	−0.309024
$121$	121.000	0.0909091
$122$	70.4393	0.0522727
$123$	−1067.82	−0.782778
$124$	−1485.08	−1.07552
$125$	1487.72	1.06453
$126$	−128.647	−0.0909587
$127$	−2024.23	−1.41434	−0.707172	−	0.707042i	$-0.750029\pi$
$127$	−2024.23	−1.41434	−0.707172	+	0.707042i	$0.750029\pi$
$128$	1475.28	1.01873
$129$	1259.54	0.859658
$130$	−554.701	−0.374235
$131$	724.318	0.483084	0.241542	−	0.970390i	$-0.422347\pi$
$131$	724.318	0.483084	0.241542	+	0.970390i	$0.422347\pi$
$132$	−219.997	−0.145063
$133$	705.250	0.459796
$134$	362.160	0.233477
$135$	−215.870	−0.137623
$136$	753.043	0.474801
$137$	1445.33	0.901332	0.450666	−	0.892693i	$-0.351187\pi$
$137$	1445.33	0.901332	0.450666	+	0.892693i	$0.351187\pi$
$138$	−478.312	−0.295048
$139$	−345.834	−0.211031	−0.105515	−	0.994418i	$-0.533649\pi$
$139$	−345.834	−0.211031	−0.105515	+	0.994418i	$0.533649\pi$
$140$	659.787	0.398301
$141$	−1819.34	−1.08664
$142$	−826.955	−0.488708
$143$	−660.903	−0.386486
$144$	303.981	0.175915
$145$	332.846	0.190630
$146$	−762.941	−0.432476
$147$	−569.309	−0.319427
$148$	799.178	0.443865
$149$	650.444	0.357627	0.178814	−	0.983883i	$-0.442774\pi$
$149$	650.444	0.357627	0.178814	+	0.983883i	$0.442774\pi$
$150$	211.584	0.115172
$151$	−1192.03	−0.642423	−0.321211	−	0.947008i	$-0.604090\pi$
$151$	−1192.03	−0.642423	−0.321211	+	0.947008i	$0.604090\pi$
$152$	964.904	0.514895
$153$	400.174	0.211452
$154$	−157.235	−0.0822753
$155$	−1781.06	−0.922954
$156$	1201.62	0.616711
$157$	−667.806	−0.339470	−0.169735	−	0.985490i	$-0.554291\pi$
$157$	−667.806	−0.339470	−0.169735	+	0.985490i	$0.554291\pi$
$158$	−887.470	−0.446856
$159$	−160.031	−0.0798193
$160$	1395.09	0.689322
$161$	1709.14	0.836641
$162$	−93.5341	−0.0453626
$163$	1149.25	0.552249	0.276124	−	0.961122i	$-0.410950\pi$
$163$	1149.25	0.552249	0.276124	+	0.961122i	$0.410950\pi$
$164$	2372.89	1.12983
$165$	−263.841	−0.124485
$166$	642.866	0.300579
$167$	−1685.73	−0.781113	−0.390557	−	0.920579i	$-0.627717\pi$
$167$	−1685.73	−0.781113	−0.390557	+	0.920579i	$0.627717\pi$
$168$	628.938	0.288831
$169$	1412.86	0.643086
$170$	410.507	0.185202
$171$	512.759	0.229308
$172$	−2798.93	−1.24079
$173$	1886.58	0.829099	0.414549	−	0.910027i	$-0.363939\pi$
$173$	1886.58	0.829099	0.414549	+	0.910027i	$0.363939\pi$
$174$	144.218	0.0628343
$175$	−756.048	−0.326582
$176$	371.533	0.159121
$177$	311.099	0.132111
$178$	−306.587	−0.129099
$179$	2560.21	1.06905	0.534523	−	0.845154i	$-0.320491\pi$
$179$	2560.21	1.06905	0.534523	+	0.845154i	$0.320491\pi$
$180$	479.705	0.198639
$181$	1830.08	0.751540	0.375770	−	0.926713i	$-0.377378\pi$
$181$	1830.08	0.751540	0.375770	+	0.926713i	$0.377378\pi$
$182$	858.821	0.349781
$183$	−183.000	−0.0739221
$184$	2338.40	0.936898
$185$	958.451	0.380901
$186$	−771.712	−0.304219
$187$	489.101	0.191265
$188$	4042.92	1.56841
$189$	334.223	0.128630
$190$	525.999	0.200842
$191$	−3602.38	−1.36471	−0.682354	−	0.731022i	$-0.739044\pi$
$191$	−3602.38	−1.36471	−0.682354	+	0.731022i	$0.739044\pi$
$192$	−206.140	−0.0774838
$193$	2649.50	0.988161	0.494080	−	0.869416i	$-0.335505\pi$
$193$	2649.50	0.988161	0.494080	+	0.869416i	$0.335505\pi$
$194$	−1553.53	−0.574932
$195$	1441.10	0.529229
$196$	1265.11	0.461047
$197$	−2242.00	−0.810844	−0.405422	−	0.914130i	$-0.632875\pi$
$197$	−2242.00	−0.810844	−0.405422	+	0.914130i	$0.632875\pi$
$198$	−114.320	−0.0410320
$199$	4371.96	1.55739	0.778694	−	0.627404i	$-0.215883\pi$
$199$	4371.96	1.55739	0.778694	+	0.627404i	$0.215883\pi$
$200$	−1034.40	−0.365717
$201$	−940.885	−0.330174
$202$	−9.48624	−0.00330421
$203$	−515.332	−0.178173
$204$	−889.262	−0.305200
$205$	2845.80	0.969557
$206$	1628.59	0.550823
$207$	1242.65	0.417246
$208$	−2029.32	−0.676480
$209$	626.705	0.207417
$210$	342.853	0.112662
$211$	3179.60	1.03740	0.518702	−	0.854955i	$-0.326415\pi$
$211$	3179.60	1.03740	0.518702	+	0.854955i	$0.326415\pi$
$212$	355.619	0.115208
$213$	2148.41	0.691112
$214$	−1647.43	−0.526244
$215$	−3356.74	−1.06478
$216$	457.275	0.144045
$217$	2757.54	0.862644
$218$	1811.74	0.562875
$219$	1982.11	0.611591
$220$	586.306	0.179676
$221$	−2671.48	−0.813135
$222$	415.286	0.125550
$223$	598.240	0.179646	0.0898232	−	0.995958i	$-0.471370\pi$
$223$	598.240	0.179646	0.0898232	+	0.995958i	$0.471370\pi$
$224$	−2159.96	−0.644279
$225$	−549.692	−0.162872
$226$	1226.72	0.361063
$227$	5506.16	1.60994	0.804971	−	0.593315i	$-0.202181\pi$
$227$	5506.16	1.60994	0.804971	+	0.593315i	$0.202181\pi$
$228$	−1139.45	−0.330972
$229$	4972.52	1.43490	0.717452	−	0.696608i	$-0.245308\pi$
$229$	4972.52	1.43490	0.717452	+	0.696608i	$0.245308\pi$
$230$	1274.73	0.365450
$231$	408.495	0.116351
$232$	−705.063	−0.199524
$233$	5640.81	1.58602	0.793008	−	0.609211i	$-0.208514\pi$
$233$	5640.81	1.58602	0.793008	+	0.609211i	$0.208514\pi$
$234$	624.414	0.174441
$235$	4848.66	1.34592
$236$	−691.320	−0.190683
$237$	2305.63	0.631927
$238$	−635.571	−0.173101
$239$	5860.02	1.58600	0.792998	−	0.609224i	$-0.208519\pi$
$239$	5860.02	1.58600	0.792998	+	0.609224i	$0.208519\pi$
$240$	−810.130	−0.217890
$241$	−4373.20	−1.16889	−0.584445	−	0.811433i	$-0.698688\pi$
$241$	−4373.20	−1.16889	−0.584445	+	0.811433i	$0.698688\pi$
$242$	−139.724	−0.0371148
$243$	243.000	0.0641500
$244$	406.661	0.106696
$245$	1517.24	0.395646
$246$	1233.05	0.319579
$247$	−3423.07	−0.881800
$248$	3772.79	0.966017
$249$	−1670.15	−0.425067
$250$	−1717.93	−0.434607
$251$	1325.44	0.333311	0.166655	−	0.986015i	$-0.446703\pi$
$251$	1325.44	0.333311	0.166655	+	0.986015i	$0.446703\pi$
$252$	−742.707	−0.185659
$253$	1518.79	0.377413
$254$	2337.47	0.577424
$255$	−1066.49	−0.261906
$256$	−1153.86	−0.281703
$257$	7945.55	1.92852	0.964261	−	0.264956i	$-0.0853573\pi$
$257$	7945.55	1.92852	0.964261	+	0.264956i	$0.0853573\pi$
$258$	−1454.44	−0.350967
$259$	−1483.93	−0.356011
$260$	−3202.41	−0.763865
$261$	−374.677	−0.0888579
$262$	−836.401	−0.197225
$263$	3670.25	0.860523	0.430262	−	0.902704i	$-0.358421\pi$
$263$	3670.25	0.860523	0.430262	+	0.902704i	$0.358421\pi$
$264$	558.892	0.130293
$265$	426.493	0.0988650
$266$	−814.382	−0.187718
$267$	796.507	0.182567
$268$	2090.83	0.476558
$269$	−2061.79	−0.467323	−0.233661	−	0.972318i	$-0.575071\pi$
$269$	−2061.79	−0.467323	−0.233661	+	0.972318i	$0.575071\pi$
$270$	249.275	0.0561865
$271$	−4740.73	−1.06265	−0.531326	−	0.847167i	$-0.678306\pi$
$271$	−4740.73	−1.06265	−0.531326	+	0.847167i	$0.678306\pi$
$272$	1501.79	0.334778
$273$	−2231.20	−0.494647
$274$	−1668.98	−0.367981
$275$	−671.846	−0.147323
$276$	−2761.40	−0.602234
$277$	5347.32	1.15989	0.579944	−	0.814656i	$-0.303074\pi$
$277$	5347.32	1.15989	0.579944	+	0.814656i	$0.303074\pi$
$278$	399.350	0.0861561
$279$	2004.89	0.430214
$280$	−1676.16	−0.357749
$281$	−8696.62	−1.84625	−0.923126	−	0.384497i	$-0.874375\pi$
$281$	−8696.62	−1.84625	−0.923126	+	0.384497i	$0.874375\pi$
$282$	2100.87	0.443635
$283$	−3179.44	−0.667838	−0.333919	−	0.942602i	$-0.608371\pi$
$283$	−3179.44	−0.667838	−0.333919	+	0.942602i	$0.608371\pi$
$284$	−4774.18	−0.997520
$285$	−1366.53	−0.284023
$286$	763.173	0.157788
$287$	−4406.03	−0.906202
$288$	−1570.42	−0.321312
$289$	−2935.97	−0.597593
$290$	−384.351	−0.0778272
$291$	4036.04	0.813047
$292$	−4404.62	−0.882742
$293$	6338.44	1.26381	0.631904	−	0.775047i	$-0.282274\pi$
$293$	6338.44	1.26381	0.631904	+	0.775047i	$0.282274\pi$
$294$	657.405	0.130410
$295$	−829.098	−0.163634
$296$	−2030.27	−0.398673
$297$	297.000	0.0580259
$298$	−751.095	−0.146006
$299$	−8295.65	−1.60451
$300$	1221.52	0.235082
$301$	5197.11	0.995203
$302$	1376.48	0.262277
$303$	24.6451	0.00467268
$304$	1924.31	0.363048
$305$	487.707	0.0915607
$306$	−462.098	−0.0863280
$307$	−3590.37	−0.667471	−0.333735	−	0.942667i	$-0.608309\pi$
$307$	−3590.37	−0.667471	−0.333735	+	0.942667i	$0.608309\pi$
$308$	−907.753	−0.167935
$309$	−4231.06	−0.778953
$310$	2056.66	0.376808
$311$	1827.03	0.333123	0.166561	−	0.986031i	$-0.446734\pi$
$311$	1827.03	0.333123	0.166561	+	0.986031i	$0.446734\pi$
$312$	−3052.67	−0.553921
$313$	9581.75	1.73033	0.865164	−	0.501489i	$-0.167214\pi$
$313$	9581.75	1.73033	0.865164	+	0.501489i	$0.167214\pi$
$314$	771.144	0.138593
$315$	−890.726	−0.159323
$316$	−5123.55	−0.912095
$317$	8603.85	1.52442	0.762209	−	0.647331i	$-0.224115\pi$
$317$	8603.85	1.52442	0.762209	+	0.647331i	$0.224115\pi$
$318$	184.794	0.0325873
$319$	−457.938	−0.0803750
$320$	549.377	0.0959722
$321$	4280.00	0.744194
$322$	−1973.62	−0.341570
$323$	2533.24	0.436388
$324$	−539.992	−0.0925912
$325$	3669.63	0.626321
$326$	−1327.09	−0.225463
$327$	−4706.88	−0.795997
$328$	−6028.22	−1.01479
$329$	−7506.98	−1.25797
$330$	304.669	0.0508226
$331$	−3631.65	−0.603061	−0.301531	−	0.953456i	$-0.597498\pi$
$331$	−3631.65	−0.603061	−0.301531	+	0.953456i	$0.597498\pi$
$332$	3711.40	0.613522
$333$	−1078.91	−0.177549
$334$	1946.59	0.318900
$335$	2507.52	0.408957
$336$	1254.29	0.203652
$337$	−4063.33	−0.656806	−0.328403	−	0.944538i	$-0.606510\pi$
$337$	−4063.33	−0.656806	−0.328403	+	0.944538i	$0.606510\pi$
$338$	−1631.49	−0.262548
$339$	−3187.00	−0.510602
$340$	2369.94	0.378024
$341$	2450.43	0.389144
$342$	−592.104	−0.0936179
$343$	−6594.96	−1.03818
$344$	7110.54	1.11446
$345$	−3311.73	−0.516805
$346$	−2178.51	−0.338490
$347$	−3267.44	−0.505492	−0.252746	−	0.967533i	$-0.581334\pi$
$347$	−3267.44	−0.505492	−0.252746	+	0.967533i	$0.581334\pi$
$348$	832.603	0.128253
$349$	5724.08	0.877945	0.438973	−	0.898500i	$-0.355343\pi$
$349$	5724.08	0.877945	0.438973	+	0.898500i	$0.355343\pi$
$350$	873.041	0.133331
$351$	−1622.22	−0.246688
$352$	−1919.40	−0.290638
$353$	−5444.77	−0.820952	−0.410476	−	0.911871i	$-0.634637\pi$
$353$	−5444.77	−0.820952	−0.410476	+	0.911871i	$0.634637\pi$
$354$	−359.239	−0.0539360
$355$	−5725.66	−0.856018
$356$	−1769.99	−0.263509
$357$	1651.20	0.244792
$358$	−2956.39	−0.436452
$359$	9031.66	1.32778	0.663890	−	0.747831i	$-0.268904\pi$
$359$	9031.66	1.32778	0.663890	+	0.747831i	$0.268904\pi$
$360$	−1218.67	−0.178415
$361$	−3613.06	−0.526761
$362$	−2113.27	−0.306826
$363$	363.000	0.0524864
$364$	4958.16	0.713950
$365$	−5282.44	−0.757522
$366$	211.318	0.0301797
$367$	10331.2	1.46944	0.734719	−	0.678372i	$-0.237314\pi$
$367$	10331.2	1.46944	0.734719	+	0.678372i	$0.237314\pi$
$368$	4663.48	0.660599
$369$	−3203.45	−0.451937
$370$	−1106.76	−0.155508
$371$	−660.321	−0.0924048
$372$	−4455.25	−0.620952
$373$	−11821.2	−1.64097	−0.820483	−	0.571671i	$-0.806295\pi$
$373$	−11821.2	−1.64097	−0.820483	+	0.571671i	$0.806295\pi$
$374$	−564.786	−0.0780866
$375$	4463.16	0.614605
$376$	−10270.9	−1.40872
$377$	2501.26	0.341702
$378$	−385.942	−0.0525150
$379$	11809.1	1.60050	0.800251	−	0.599665i	$-0.204700\pi$
$379$	11809.1	1.60050	0.800251	+	0.599665i	$0.204700\pi$
$380$	3036.70	0.409946
$381$	−6072.70	−0.816572
$382$	4159.82	0.557160
$383$	8602.90	1.14775	0.573874	−	0.818944i	$-0.305440\pi$
$383$	8602.90	1.14775	0.573874	+	0.818944i	$0.305440\pi$
$384$	4425.83	0.588163
$385$	−1088.66	−0.144113
$386$	−3059.49	−0.403430
$387$	3778.61	0.496324
$388$	−8968.85	−1.17352
$389$	−2696.02	−0.351397	−0.175699	−	0.984444i	$-0.556218\pi$
$389$	−2696.02	−0.351397	−0.175699	+	0.984444i	$0.556218\pi$
$390$	−1664.10	−0.216065
$391$	6139.19	0.794047
$392$	−3213.96	−0.414106
$393$	2172.95	0.278908
$394$	2588.94	0.331038
$395$	−6144.65	−0.782712
$396$	−659.990	−0.0837519
$397$	4956.17	0.626557	0.313279	−	0.949661i	$-0.398573\pi$
$397$	4956.17	0.626557	0.313279	+	0.949661i	$0.398573\pi$
$398$	−5048.49	−0.635824
$399$	2115.75	0.265464
$400$	−2062.92	−0.257864
$401$	4508.68	0.561478	0.280739	−	0.959784i	$-0.409420\pi$
$401$	4508.68	0.561478	0.280739	+	0.959784i	$0.409420\pi$
$402$	1086.48	0.134798
$403$	−13384.2	−1.65438
$404$	−54.7661	−0.00674434
$405$	−647.611	−0.0794569
$406$	595.075	0.0727416
$407$	−1318.66	−0.160599
$408$	2259.13	0.274126
$409$	8871.38	1.07252	0.536261	−	0.844052i	$-0.319836\pi$
$409$	8871.38	1.07252	0.536261	+	0.844052i	$0.319836\pi$
$410$	−3286.16	−0.395834
$411$	4335.98	0.520384
$412$	9402.21	1.12431
$413$	1283.66	0.152941
$414$	−1434.94	−0.170346
$415$	4451.07	0.526492
$416$	10483.8	1.23560
$417$	−1037.50	−0.121839
$418$	−723.683	−0.0846806
$419$	11897.9	1.38723	0.693614	−	0.720347i	$-0.256017\pi$
$419$	11897.9	1.38723	0.693614	+	0.720347i	$0.256017\pi$
$420$	1979.36	0.229959
$421$	4631.05	0.536113	0.268056	−	0.963403i	$-0.413619\pi$
$421$	4631.05	0.536113	0.268056	+	0.963403i	$0.413619\pi$
$422$	−3671.61	−0.423534
$423$	−5458.02	−0.627372
$424$	−903.434	−0.103478
$425$	−2715.71	−0.309956
$426$	−2480.86	−0.282156
$427$	−755.096	−0.0855777
$428$	−9510.97	−1.07414
$429$	−1982.71	−0.223138
$430$	3876.17	0.434711
$431$	17473.5	1.95283	0.976416	−	0.215896i	$-0.0692671\pi$
$431$	17473.5	1.95283	0.976416	+	0.215896i	$0.0692671\pi$
$432$	911.944	0.101565
$433$	4165.44	0.462305	0.231152	−	0.972918i	$-0.425750\pi$
$433$	4165.44	0.462305	0.231152	+	0.972918i	$0.425750\pi$
$434$	−3184.25	−0.352186
$435$	998.538	0.110060
$436$	10459.6	1.14891
$437$	7866.39	0.861100
$438$	−2288.82	−0.249690
$439$	−17748.5	−1.92959	−0.964796	−	0.262998i	$-0.915289\pi$
$439$	−17748.5	−1.92959	−0.964796	+	0.262998i	$0.915289\pi$
$440$	−1489.48	−0.161382
$441$	−1707.93	−0.184421
$442$	3084.87	0.331973
$443$	12522.0	1.34297	0.671486	−	0.741017i	$-0.265656\pi$
$443$	12522.0	1.34297	0.671486	+	0.741017i	$0.265656\pi$
$444$	2397.53	0.256266
$445$	−2122.74	−0.226130
$446$	−690.814	−0.0733430
$447$	1951.33	0.206476
$448$	−850.578	−0.0897009
$449$	−5217.26	−0.548369	−0.274184	−	0.961677i	$-0.588408\pi$
$449$	−5217.26	−0.548369	−0.274184	+	0.961677i	$0.588408\pi$
$450$	634.753	0.0664945
$451$	−3915.33	−0.408793
$452$	7082.12	0.736980
$453$	−3576.08	−0.370903
$454$	−6358.20	−0.657280
$455$	5946.30	0.612674
$456$	2894.71	0.297275
$457$	13625.5	1.39469	0.697344	−	0.716737i	$-0.254365\pi$
$457$	13625.5	1.39469	0.697344	+	0.716737i	$0.254365\pi$
$458$	−5741.98	−0.585819
$459$	1200.52	0.122082
$460$	7359.30	0.745933
$461$	8774.29	0.886463	0.443231	−	0.896407i	$-0.353832\pi$
$461$	8774.29	0.886463	0.443231	+	0.896407i	$0.353832\pi$
$462$	−471.706	−0.0475016
$463$	3334.78	0.334731	0.167366	−	0.985895i	$-0.446474\pi$
$463$	3334.78	0.334731	0.167366	+	0.985895i	$0.446474\pi$
$464$	−1406.11	−0.140683
$465$	−5343.17	−0.532868
$466$	−6513.68	−0.647512
$467$	−3408.02	−0.337697	−0.168848	−	0.985642i	$-0.554005\pi$
$467$	−3408.02	−0.337697	−0.168848	+	0.985642i	$0.554005\pi$
$468$	3604.87	0.356058
$469$	−3882.29	−0.382234
$470$	−5598.95	−0.549490
$471$	−2003.42	−0.195993
$472$	1756.27	0.171269
$473$	4618.30	0.448942
$474$	−2662.41	−0.257993
$475$	−3479.74	−0.336130
$476$	−3669.28	−0.353322
$477$	−480.093	−0.0460837
$478$	−6766.81	−0.647504
$479$	3497.01	0.333575	0.166788	−	0.985993i	$-0.446661\pi$
$479$	3497.01	0.333575	0.166788	+	0.985993i	$0.446661\pi$
$480$	4185.27	0.397980
$481$	7202.55	0.682761
$482$	5049.92	0.477215
$483$	5127.42	0.483035
$484$	−806.655	−0.0757565
$485$	−10756.3	−1.00705
$486$	−280.602	−0.0261901
$487$	2761.46	0.256948	0.128474	−	0.991713i	$-0.458992\pi$
$487$	2761.46	0.256948	0.128474	+	0.991713i	$0.458992\pi$
$488$	−1033.10	−0.0958327
$489$	3447.76	0.318841
$490$	−1752.03	−0.161527
$491$	624.134	0.0573662	0.0286831	−	0.999589i	$-0.490869\pi$
$491$	624.134	0.0573662	0.0286831	+	0.999589i	$0.490869\pi$
$492$	7118.67	0.652306
$493$	−1851.06	−0.169103
$494$	3952.76	0.360006
$495$	−791.524	−0.0718714
$496$	7524.08	0.681131
$497$	8864.80	0.800082
$498$	1928.60	0.173539
$499$	−18400.3	−1.65072	−0.825362	−	0.564604i	$-0.809029\pi$
$499$	−18400.3	−1.65072	−0.825362	+	0.564604i	$0.809029\pi$
$500$	−9917.99	−0.887092
$501$	−5057.20	−0.450976
$502$	−1530.54	−0.136078
$503$	−6889.28	−0.610691	−0.305346	−	0.952242i	$-0.598772\pi$
$503$	−6889.28	−0.610691	−0.305346	+	0.952242i	$0.598772\pi$
$504$	1886.81	0.166757
$505$	−65.6807	−0.00578763
$506$	−1753.81	−0.154084
$507$	4238.58	0.371286
$508$	13494.7	1.17860
$509$	−11128.4	−0.969070	−0.484535	−	0.874772i	$-0.661011\pi$
$509$	−11128.4	−0.969070	−0.484535	+	0.874772i	$0.661011\pi$
$510$	1231.52	0.106927
$511$	8178.59	0.708022
$512$	−10469.8	−0.903719
$513$	1538.28	0.132391
$514$	−9175.07	−0.787344
$515$	11276.0	0.964819
$516$	−8396.78	−0.716371
$517$	−6670.92	−0.567479
$518$	1713.56	0.145346
$519$	5659.74	0.478680
$520$	8135.57	0.686093
$521$	21845.5	1.83699	0.918493	−	0.395436i	$-0.129407\pi$
$521$	21845.5	1.83699	0.918493	+	0.395436i	$0.129407\pi$
$522$	432.655	0.0362774
$523$	2347.21	0.196246	0.0981228	−	0.995174i	$-0.468716\pi$
$523$	2347.21	0.196246	0.0981228	+	0.995174i	$0.468716\pi$
$524$	−4828.72	−0.402564
$525$	−2268.14	−0.188552
$526$	−4238.20	−0.351320
$527$	9905.01	0.818727
$528$	1114.60	0.0918687
$529$	6896.84	0.566848
$530$	−492.489	−0.0403629
$531$	933.296	0.0762742
$532$	−4701.60	−0.383158
$533$	21385.5	1.73792
$534$	−919.761	−0.0745355
$535$	−11406.5	−0.921766
$536$	−5311.65	−0.428038
$537$	7680.64	0.617214
$538$	2380.84	0.190791
$539$	−2087.46	−0.166815
$540$	1439.11	0.114684
$541$	17656.9	1.40320	0.701600	−	0.712571i	$-0.252469\pi$
$541$	17656.9	1.40320	0.701600	+	0.712571i	$0.252469\pi$
$542$	5474.32	0.433842
$543$	5490.24	0.433902
$544$	−7758.53	−0.611478
$545$	12544.1	0.985930
$546$	2576.46	0.201946
$547$	18979.2	1.48353	0.741764	−	0.670661i	$-0.233989\pi$
$547$	18979.2	1.48353	0.741764	+	0.670661i	$0.233989\pi$
$548$	−9635.36	−0.751099
$549$	−549.000	−0.0426790
$550$	775.809	0.0601465
$551$	−2371.84	−0.183382
$552$	7015.20	0.540918
$553$	9513.51	0.731566
$554$	−6174.77	−0.473540
$555$	2875.35	0.219913
$556$	2305.53	0.175856
$557$	−20680.3	−1.57316	−0.786581	−	0.617487i	$-0.788151\pi$
$557$	−20680.3	−1.57316	−0.786581	+	0.617487i	$0.788151\pi$
$558$	−2315.14	−0.175641
$559$	−25225.2	−1.90861
$560$	−3342.77	−0.252246
$561$	1467.30	0.110427
$562$	10042.4	0.753757
$563$	−17174.5	−1.28564	−0.642822	−	0.766016i	$-0.722236\pi$
$563$	−17174.5	−1.28564	−0.642822	+	0.766016i	$0.722236\pi$
$564$	12128.8	0.905520
$565$	8493.56	0.632437
$566$	3671.43	0.272654
$567$	1002.67	0.0742648
$568$	12128.6	0.895958
$569$	5833.02	0.429759	0.214880	−	0.976641i	$-0.431064\pi$
$569$	5833.02	0.429759	0.214880	+	0.976641i	$0.431064\pi$
$570$	1578.00	0.115956
$571$	−9756.68	−0.715069	−0.357534	−	0.933900i	$-0.616382\pi$
$571$	−9756.68	−0.715069	−0.357534	+	0.933900i	$0.616382\pi$
$572$	4405.96	0.322067
$573$	−10807.1	−0.787915
$574$	5087.83	0.369969
$575$	−8433.00	−0.611618
$576$	−618.421	−0.0447353
$577$	−18153.4	−1.30977	−0.654885	−	0.755729i	$-0.727283\pi$
$577$	−18153.4	−1.30977	−0.654885	+	0.755729i	$0.727283\pi$
$578$	3390.29	0.243975
$579$	7948.49	0.570515
$580$	−2218.94	−0.158856
$581$	−6891.40	−0.492089
$582$	−4660.59	−0.331937
$583$	−586.780	−0.0416843
$584$	11189.7	0.792867
$585$	4323.31	0.305550
$586$	−7319.26	−0.515966
$587$	3860.74	0.271465	0.135732	−	0.990746i	$-0.456661\pi$
$587$	3860.74	0.271465	0.135732	+	0.990746i	$0.456661\pi$
$588$	3795.34	0.266185
$589$	12691.7	0.887864
$590$	957.395	0.0668056
$591$	−6726.01	−0.468141
$592$	−4048.98	−0.281101
$593$	21803.0	1.50985	0.754925	−	0.655811i	$-0.227673\pi$
$593$	21803.0	1.50985	0.754925	+	0.655811i	$0.227673\pi$
$594$	−342.959	−0.0236898
$595$	−4400.56	−0.303202
$596$	−4336.23	−0.298018
$597$	13115.9	0.899158
$598$	9579.34	0.655064
$599$	−10510.6	−0.716944	−0.358472	−	0.933540i	$-0.616702\pi$
$599$	−10510.6	−0.716944	−0.358472	+	0.933540i	$0.616702\pi$
$600$	−3103.21	−0.211147
$601$	−23121.4	−1.56929	−0.784645	−	0.619945i	$-0.787155\pi$
$601$	−23121.4	−1.56929	−0.784645	+	0.619945i	$0.787155\pi$
$602$	−6001.32	−0.406305
$603$	−2822.66	−0.190626
$604$	7946.73	0.535344
$605$	−967.418	−0.0650102
$606$	−28.4587	−0.00190768
$607$	−28563.8	−1.91000	−0.954998	−	0.296611i	$-0.904144\pi$
$607$	−28563.8	−1.91000	−0.954998	+	0.296611i	$0.904144\pi$
$608$	−9941.31	−0.663114
$609$	−1546.00	−0.102868
$610$	−563.176	−0.0373809
$611$	36436.6	2.41255
$612$	−2667.79	−0.176207
$613$	−350.309	−0.0230813	−0.0115406	−	0.999933i	$-0.503674\pi$
$613$	−350.309	−0.0230813	−0.0115406	+	0.999933i	$0.503674\pi$
$614$	4145.96	0.272504
$615$	8537.40	0.559774
$616$	2306.10	0.150837
$617$	−22350.2	−1.45832	−0.729161	−	0.684342i	$-0.760090\pi$
$617$	−22350.2	−1.45832	−0.729161	+	0.684342i	$0.760090\pi$
$618$	4885.78	0.318018
$619$	23994.7	1.55804	0.779021	−	0.626998i	$-0.215717\pi$
$619$	23994.7	1.55804	0.779021	+	0.626998i	$0.215717\pi$
$620$	11873.5	0.769118
$621$	3727.94	0.240897
$622$	−2109.75	−0.136002
$623$	3286.56	0.211353
$624$	−6087.95	−0.390566
$625$	−4260.00	−0.272640
$626$	−11064.5	−0.706429
$627$	1880.11	0.119752
$628$	4451.98	0.282887
$629$	−5330.24	−0.337887
$630$	1028.56	0.0650457
$631$	−11032.0	−0.696002	−0.348001	−	0.937494i	$-0.613139\pi$
$631$	−11032.0	−0.696002	−0.348001	+	0.937494i	$0.613139\pi$
$632$	13016.1	0.819231
$633$	9538.79	0.598946
$634$	−9935.24	−0.622364
$635$	16184.1	1.01141
$636$	1066.86	0.0665152
$637$	11401.8	0.709190
$638$	528.801	0.0328141
$639$	6445.24	0.399014
$640$	−11795.1	−0.728504
$641$	6632.85	0.408708	0.204354	−	0.978897i	$-0.434491\pi$
$641$	6632.85	0.408708	0.204354	+	0.978897i	$0.434491\pi$
$642$	−4942.30	−0.303827
$643$	−28488.9	−1.74727	−0.873635	−	0.486582i	$-0.838243\pi$
$643$	−28488.9	−1.74727	−0.873635	+	0.486582i	$0.838243\pi$
$644$	−11394.1	−0.697190
$645$	−10070.2	−0.614752
$646$	−2925.24	−0.178161
$647$	14980.0	0.910242	0.455121	−	0.890430i	$-0.349596\pi$
$647$	14980.0	0.910242	0.455121	+	0.890430i	$0.349596\pi$
$648$	1371.82	0.0831642
$649$	1140.70	0.0689926
$650$	−4237.47	−0.255704
$651$	8272.61	0.498048
$652$	−7661.58	−0.460201
$653$	−15151.3	−0.907987	−0.453994	−	0.891005i	$-0.650001\pi$
$653$	−15151.3	−0.907987	−0.453994	+	0.891005i	$0.650001\pi$
$654$	5435.23	0.324976
$655$	−5791.06	−0.345459
$656$	−12022.1	−0.715524
$657$	5946.32	0.353102
$658$	8668.63	0.513584
$659$	−25752.8	−1.52229	−0.761143	−	0.648584i	$-0.775362\pi$
$659$	−25752.8	−1.52229	−0.761143	+	0.648584i	$0.775362\pi$
$660$	1758.92	0.103736
$661$	−26068.0	−1.53393	−0.766966	−	0.641688i	$-0.778235\pi$
$661$	−26068.0	−1.53393	−0.766966	+	0.641688i	$0.778235\pi$
$662$	4193.61	0.246208
$663$	−8014.43	−0.469464
$664$	−9428.63	−0.551057
$665$	−5638.61	−0.328806
$666$	1245.86	0.0724865
$667$	−5748.04	−0.333680
$668$	11238.1	0.650918
$669$	1794.72	0.103719
$670$	−2895.54	−0.166962
$671$	−671.000	−0.0386046
$672$	−6479.88	−0.371975
$673$	−10290.4	−0.589397	−0.294698	−	0.955590i	$-0.595219\pi$
$673$	−10290.4	−0.589397	−0.294698	+	0.955590i	$0.595219\pi$
$674$	4692.10	0.268150
$675$	−1649.08	−0.0940340
$676$	−9418.93	−0.535897
$677$	33493.5	1.90142	0.950709	−	0.310085i	$-0.100357\pi$
$677$	33493.5	1.90142	0.950709	+	0.310085i	$0.100357\pi$
$678$	3680.17	0.208460
$679$	16653.5	0.941244
$680$	−6020.72	−0.339536
$681$	16518.5	0.929500
$682$	−2829.61	−0.158873
$683$	−25419.7	−1.42410	−0.712049	−	0.702129i	$-0.752233\pi$
$683$	−25419.7	−1.42410	−0.712049	+	0.702129i	$0.752233\pi$
$684$	−3418.34	−0.191087
$685$	−11555.7	−0.644553
$686$	7615.48	0.423849
$687$	14917.5	0.828443
$688$	14180.6	0.785798
$689$	3205.00	0.177214
$690$	3824.20	0.210992
$691$	−12198.0	−0.671541	−0.335770	−	0.941944i	$-0.608997\pi$
$691$	−12198.0	−0.671541	−0.335770	+	0.941944i	$0.608997\pi$
$692$	−12577.0	−0.690906
$693$	1225.48	0.0671750
$694$	3773.06	0.206374
$695$	2765.01	0.150911
$696$	−2115.19	−0.115195
$697$	−15826.4	−0.860067
$698$	−6609.84	−0.358433
$699$	16922.4	0.915687
$700$	5040.25	0.272148
$701$	11887.1	0.640472	0.320236	−	0.947338i	$-0.396238\pi$
$701$	11887.1	0.640472	0.320236	+	0.947338i	$0.396238\pi$
$702$	1873.24	0.100714
$703$	−6829.86	−0.366419
$704$	−755.847	−0.0404646
$705$	14546.0	0.777069
$706$	6287.31	0.335164
$707$	101.691	0.00540944
$708$	−2073.96	−0.110091
$709$	−15690.1	−0.831104	−0.415552	−	0.909569i	$-0.636412\pi$
$709$	−15690.1	−0.831104	−0.415552	+	0.909569i	$0.636412\pi$
$710$	6611.66	0.349481
$711$	6916.89	0.364843
$712$	4496.58	0.236680
$713$	30757.7	1.61555
$714$	−1906.71	−0.0999396
$715$	5284.05	0.276381
$716$	−17067.8	−0.890859
$717$	17580.1	0.915675
$718$	−10429.2	−0.542083
$719$	8641.76	0.448238	0.224119	−	0.974562i	$-0.428050\pi$
$719$	8641.76	0.448238	0.224119	+	0.974562i	$0.428050\pi$
$720$	−2430.39	−0.125799
$721$	−17458.2	−0.901773
$722$	4172.15	0.215057
$723$	−13119.6	−0.674859
$724$	−12200.4	−0.626275
$725$	2542.68	0.130252
$726$	−419.172	−0.0214283
$727$	−16834.1	−0.858791	−0.429396	−	0.903117i	$-0.641273\pi$
$727$	−16834.1	−0.858791	−0.429396	+	0.903117i	$0.641273\pi$
$728$	−12596.0	−0.641260
$729$	729.000	0.0370370
$730$	6099.86	0.309268
$731$	18667.9	0.944537
$732$	1219.98	0.0616009
$733$	17272.6	0.870364	0.435182	−	0.900342i	$-0.356684\pi$
$733$	17272.6	0.870364	0.435182	+	0.900342i	$0.356684\pi$
$734$	−11929.9	−0.599917
$735$	4551.73	0.228426
$736$	−24092.3	−1.20660
$737$	−3449.91	−0.172428
$738$	3699.16	0.184509
$739$	13863.7	0.690100	0.345050	−	0.938584i	$-0.387862\pi$
$739$	13863.7	0.690100	0.345050	+	0.938584i	$0.387862\pi$
$740$	−6389.58	−0.317413
$741$	−10269.2	−0.509108
$742$	762.501	0.0377255
$743$	11525.6	0.569089	0.284545	−	0.958663i	$-0.408158\pi$
$743$	11525.6	0.569089	0.284545	+	0.958663i	$0.408158\pi$
$744$	11318.4	0.557730
$745$	−5200.42	−0.255743
$746$	13650.5	0.669946
$747$	−5010.46	−0.245413
$748$	−3260.63	−0.159386
$749$	17660.2	0.861534
$750$	−5153.80	−0.250920
$751$	14259.1	0.692840	0.346420	−	0.938080i	$-0.387397\pi$
$751$	14259.1	0.692840	0.346420	+	0.938080i	$0.387397\pi$
$752$	−20483.2	−0.993278
$753$	3976.32	0.192437
$754$	−2888.32	−0.139504
$755$	9530.49	0.459404
$756$	−2228.12	−0.107190
$757$	22920.2	1.10046	0.550231	−	0.835012i	$-0.314540\pi$
$757$	22920.2	1.10046	0.550231	+	0.835012i	$0.314540\pi$
$758$	−13636.4	−0.653426
$759$	4556.37	0.217900
$760$	−7714.59	−0.368208
$761$	31866.6	1.51795	0.758976	−	0.651118i	$-0.225700\pi$
$761$	31866.6	1.51795	0.758976	+	0.651118i	$0.225700\pi$
$762$	7012.41	0.333376
$763$	−19421.6	−0.921505
$764$	24015.5	1.13724
$765$	−3199.47	−0.151212
$766$	−9934.13	−0.468583
$767$	−6230.49	−0.293311
$768$	−3461.57	−0.162641
$769$	−22184.8	−1.04032	−0.520158	−	0.854070i	$-0.674127\pi$
$769$	−22184.8	−1.04032	−0.520158	+	0.854070i	$0.674127\pi$
$770$	1257.13	0.0588360
$771$	23836.7	1.11343
$772$	−17663.1	−0.823455
$773$	28803.7	1.34023	0.670114	−	0.742258i	$-0.266245\pi$
$773$	28803.7	1.34023	0.670114	+	0.742258i	$0.266245\pi$
$774$	−4363.32	−0.202631
$775$	−13605.9	−0.630628
$776$	22784.9	1.05404
$777$	−4451.79	−0.205543
$778$	3113.21	0.143463
$779$	−20279.0	−0.932695
$780$	−9607.22	−0.441018
$781$	7877.52	0.360921
$782$	−7089.19	−0.324180
$783$	−1124.03	−0.0513021
$784$	−6409.60	−0.291983
$785$	5339.24	0.242759
$786$	−2509.20	−0.113868
$787$	−18845.3	−0.853575	−0.426787	−	0.904352i	$-0.640355\pi$
$787$	−18845.3	−0.853575	−0.426787	+	0.904352i	$0.640355\pi$
$788$	14946.5	0.675693
$789$	11010.8	0.496823
$790$	7095.49	0.319552
$791$	−13150.2	−0.591111
$792$	1676.67	0.0752248
$793$	3665.01	0.164121
$794$	−5723.10	−0.255800
$795$	1279.48	0.0570798
$796$	−29146.0	−1.29780
$797$	−25201.7	−1.12006	−0.560032	−	0.828471i	$-0.689211\pi$
$797$	−25201.7	−1.12006	−0.560032	+	0.828471i	$0.689211\pi$
$798$	−2443.15	−0.108379
$799$	−26964.9	−1.19393
$800$	10657.4	0.470994
$801$	2389.52	0.105405
$802$	−5206.36	−0.229231
$803$	7267.72	0.319393
$804$	6272.48	0.275141
$805$	−13664.9	−0.598292
$806$	15455.4	0.675424
$807$	−6185.38	−0.269809
$808$	139.131	0.00605767
$809$	−1888.08	−0.0820538	−0.0410269	−	0.999158i	$-0.513063\pi$
$809$	−1888.08	−0.0820538	−0.0410269	+	0.999158i	$0.513063\pi$
$810$	747.824	0.0324393
$811$	−31190.4	−1.35049	−0.675243	−	0.737595i	$-0.735961\pi$
$811$	−31190.4	−1.35049	−0.675243	+	0.737595i	$0.735961\pi$
$812$	3435.50	0.148476
$813$	−14222.2	−0.613522
$814$	1522.72	0.0655665
$815$	−9188.51	−0.394920
$816$	4505.38	0.193284
$817$	23919.9	1.02430
$818$	−10244.2	−0.437871
$819$	−6693.61	−0.285584
$820$	−18971.7	−0.807952
$821$	−16920.9	−0.719299	−0.359650	−	0.933087i	$-0.617104\pi$
$821$	−16920.9	−0.719299	−0.359650	+	0.933087i	$0.617104\pi$
$822$	−5006.94	−0.212454
$823$	−2180.94	−0.0923729	−0.0461865	−	0.998933i	$-0.514707\pi$
$823$	−2180.94	−0.0923729	−0.0461865	+	0.998933i	$0.514707\pi$
$824$	−23885.9	−1.00984
$825$	−2015.54	−0.0850570
$826$	−1482.30	−0.0624402
$827$	20739.5	0.872046	0.436023	−	0.899935i	$-0.356387\pi$
$827$	20739.5	0.872046	0.436023	+	0.899935i	$0.356387\pi$
$828$	−8284.19	−0.347700
$829$	−12347.2	−0.517291	−0.258646	−	0.965972i	$-0.583276\pi$
$829$	−12347.2	−0.517291	−0.258646	+	0.965972i	$0.583276\pi$
$830$	−5139.84	−0.214947
$831$	16042.0	0.669662
$832$	4128.45	0.172029
$833$	−8437.86	−0.350966
$834$	1198.05	0.0497422
$835$	13477.8	0.558583
$836$	−4177.97	−0.172845
$837$	6014.68	0.248384
$838$	−13739.0	−0.566354
$839$	−8091.40	−0.332951	−0.166476	−	0.986046i	$-0.553239\pi$
$839$	−8091.40	−0.332951	−0.166476	+	0.986046i	$0.553239\pi$
$840$	−5028.48	−0.206546
$841$	−22655.9	−0.928938
$842$	−5347.67	−0.218875
$843$	−26089.9	−1.06593
$844$	−21197.0	−0.864491
$845$	−11296.1	−0.459878
$846$	6302.61	0.256133
$847$	1497.81	0.0607621
$848$	−1801.72	−0.0729614
$849$	−9538.32	−0.385576
$850$	3135.94	0.126543
$851$	−16551.8	−0.666733
$852$	−14322.5	−0.575918
$853$	−15327.7	−0.615254	−0.307627	−	0.951507i	$-0.599535\pi$
$853$	−15327.7	−0.615254	−0.307627	+	0.951507i	$0.599535\pi$
$854$	871.942	0.0349382
$855$	−4099.60	−0.163981
$856$	24162.2	0.964773
$857$	29782.8	1.18712	0.593559	−	0.804791i	$-0.297722\pi$
$857$	29782.8	1.18712	0.593559	+	0.804791i	$0.297722\pi$
$858$	2289.52	0.0910989
$859$	35769.0	1.42075	0.710374	−	0.703824i	$-0.248526\pi$
$859$	35769.0	1.42075	0.710374	+	0.703824i	$0.248526\pi$
$860$	22378.0	0.887305
$861$	−13218.1	−0.523196
$862$	−20177.4	−0.797270
$863$	−29925.6	−1.18039	−0.590197	−	0.807259i	$-0.700950\pi$
$863$	−29925.6	−1.18039	−0.590197	+	0.807259i	$0.700950\pi$
$864$	−4711.26	−0.185510
$865$	−15083.6	−0.592898
$866$	−4810.01	−0.188742
$867$	−8807.92	−0.345021
$868$	−18383.3	−0.718860
$869$	8453.98	0.330013
$870$	−1153.05	−0.0449336
$871$	18843.5	0.733050
$872$	−26572.1	−1.03193
$873$	12108.1	0.469413
$874$	−9083.66	−0.351555
$875$	18415.9	0.711511
$876$	−13213.9	−0.509652
$877$	30123.7	1.15987	0.579934	−	0.814663i	$-0.303078\pi$
$877$	30123.7	1.15987	0.579934	+	0.814663i	$0.303078\pi$
$878$	20495.0	0.787782
$879$	19015.3	0.729659
$880$	−2970.48	−0.113789
$881$	−7181.63	−0.274637	−0.137319	−	0.990527i	$-0.543848\pi$
$881$	−7181.63	−0.274637	−0.137319	+	0.990527i	$0.543848\pi$
$882$	1972.21	0.0752924
$883$	−9152.44	−0.348816	−0.174408	−	0.984674i	$-0.555801\pi$
$883$	−9152.44	−0.348816	−0.174408	+	0.984674i	$0.555801\pi$
$884$	17809.6	0.677603
$885$	−2487.29	−0.0944740
$886$	−14459.6	−0.548286
$887$	−44084.3	−1.66878	−0.834388	−	0.551177i	$-0.814179\pi$
$887$	−44084.3	−1.66878	−0.834388	+	0.551177i	$0.814179\pi$
$888$	−6090.82	−0.230174
$889$	−25057.2	−0.945324
$890$	2451.22	0.0923204
$891$	891.000	0.0335013
$892$	−3988.21	−0.149703
$893$	−34551.2	−1.29475
$894$	−2253.29	−0.0842966
$895$	−20469.4	−0.764488
$896$	18261.9	0.680901
$897$	−24886.9	−0.926367
$898$	6024.59	0.223879
$899$	−9273.92	−0.344052
$900$	3664.56	0.135724
$901$	−2371.86	−0.0877004
$902$	4521.19	0.166895
$903$	15591.3	0.574581
$904$	−17991.8	−0.661945
$905$	−14631.8	−0.537435
$906$	4129.45	0.151426
$907$	416.678	0.0152542	0.00762711	−	0.999971i	$-0.497572\pi$
$907$	416.678	0.0152542	0.00762711	+	0.999971i	$0.497572\pi$
$908$	−36707.2	−1.34160
$909$	73.9353	0.00269778
$910$	−6866.44	−0.250132
$911$	17463.6	0.635121	0.317561	−	0.948238i	$-0.397136\pi$
$911$	17463.6	0.635121	0.317561	+	0.948238i	$0.397136\pi$
$912$	5772.93	0.209606
$913$	−6123.90	−0.221984
$914$	−15733.9	−0.569400
$915$	1463.12	0.0528626
$916$	−33149.6	−1.19574
$917$	8966.06	0.322885
$918$	−1386.29	−0.0498415
$919$	50109.6	1.79866	0.899328	−	0.437275i	$-0.144056\pi$
$919$	50109.6	1.79866	0.899328	+	0.437275i	$0.144056\pi$
$920$	−18696.0	−0.669987
$921$	−10771.1	−0.385364
$922$	−10132.0	−0.361910
$923$	−43027.1	−1.53440
$924$	−2723.26	−0.0969574
$925$	7321.80	0.260259
$926$	−3850.81	−0.136658
$927$	−12693.2	−0.449729
$928$	7264.20	0.256960
$929$	43305.7	1.52940	0.764700	−	0.644386i	$-0.222887\pi$
$929$	43305.7	1.52940	0.764700	+	0.644386i	$0.222887\pi$
$930$	6169.99	0.217550
$931$	−10811.8	−0.380603
$932$	−37604.8	−1.32166
$933$	5481.08	0.192328
$934$	3935.39	0.137869
$935$	−3910.46	−0.136776
$936$	−9158.01	−0.319807
$937$	−1251.85	−0.0436459	−0.0218229	−	0.999762i	$-0.506947\pi$
$937$	−1251.85	−0.0436459	−0.0218229	+	0.999762i	$0.506947\pi$
$938$	4483.05	0.156052
$939$	28745.3	0.999005
$940$	−32323.9	−1.12159
$941$	10412.9	0.360734	0.180367	−	0.983599i	$-0.442271\pi$
$941$	10412.9	0.360734	0.180367	+	0.983599i	$0.442271\pi$
$942$	2313.43	0.0800167
$943$	−49145.1	−1.69712
$944$	3502.53	0.120760
$945$	−2672.18	−0.0919851
$946$	−5332.94	−0.183286
$947$	14898.9	0.511246	0.255623	−	0.966777i	$-0.417719\pi$
$947$	14898.9	0.511246	0.255623	+	0.966777i	$0.417719\pi$
$948$	−15370.6	−0.526598
$949$	−39696.4	−1.35785
$950$	4018.21	0.137229
$951$	25811.6	0.880123
$952$	9321.64	0.317349
$953$	−8895.87	−0.302377	−0.151189	−	0.988505i	$-0.548310\pi$
$953$	−8895.87	−0.302377	−0.151189	+	0.988505i	$0.548310\pi$
$954$	554.383	0.0188143
$955$	28801.7	0.975919
$956$	−39066.2	−1.32164
$957$	−1373.81	−0.0464045
$958$	−4038.15	−0.136186
$959$	17891.1	0.602435
$960$	1648.13	0.0554096
$961$	19833.7	0.665761
$962$	−8317.09	−0.278746
$963$	12840.0	0.429660
$964$	29154.2	0.974061
$965$	−21183.2	−0.706645
$966$	−5920.85	−0.197205
$967$	−30585.2	−1.01712	−0.508559	−	0.861027i	$-0.669822\pi$
$967$	−30585.2	−1.01712	−0.508559	+	0.861027i	$0.669822\pi$
$968$	2049.27	0.0680434
$969$	7599.72	0.251949
$970$	12420.8	0.411141
$971$	32491.2	1.07383	0.536916	−	0.843636i	$-0.319589\pi$
$971$	32491.2	1.07383	0.536916	+	0.843636i	$0.319589\pi$
$972$	−1619.98	−0.0534576
$973$	−4280.96	−0.141049
$974$	−3188.77	−0.104902
$975$	11008.9	0.361607
$976$	−2060.32	−0.0675709
$977$	9378.92	0.307122	0.153561	−	0.988139i	$-0.450926\pi$
$977$	9378.92	0.307122	0.153561	+	0.988139i	$0.450926\pi$
$978$	−3981.28	−0.130171
$979$	2920.53	0.0953426
$980$	−10114.8	−0.329700
$981$	−14120.6	−0.459569
$982$	−720.714	−0.0234205
$983$	−4924.45	−0.159782	−0.0798909	−	0.996804i	$-0.525457\pi$
$983$	−4924.45	−0.159782	−0.0798909	+	0.996804i	$0.525457\pi$
$984$	−18084.7	−0.585892
$985$	17925.3	0.579844
$986$	2137.50	0.0690383
$987$	−22521.0	−0.726292
$988$	22820.1	0.734823
$989$	57968.8	1.86380
$990$	914.007	0.0293425
$991$	−59980.1	−1.92263	−0.961317	−	0.275445i	$-0.911175\pi$
$991$	−59980.1	−1.92263	−0.961317	+	0.275445i	$0.911175\pi$
$992$	−38870.7	−1.24410
$993$	−10894.9	−0.348178
$994$	−10236.6	−0.326644
$995$	−34954.7	−1.11371
$996$	11134.2	0.354217
$997$	50780.4	1.61307	0.806535	−	0.591186i	$-0.201340\pi$
$997$	50780.4	1.61307	0.806535	+	0.591186i	$0.201340\pi$
$998$	21247.6	0.673930
$999$	−3236.72	−0.102508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.f.1.14	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.f.1.14	✓	38	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-1.15474 q^{2} +3.00000 q^{3} -6.66657 q^{4} -7.99519 q^{5} -3.46423 q^{6} +12.3786 q^{7} +16.9361 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.15474 q^{2} +3.00000 q^{3} -6.66657 q^{4} -7.99519 q^{5} -3.46423 q^{6} +12.3786 q^{7} +16.9361 q^{8} +9.00000 q^{9} +9.23239 q^{10} +11.0000 q^{11} -19.9997 q^{12} -60.0821 q^{13} -14.2941 q^{14} -23.9856 q^{15} +33.7757 q^{16} +44.4638 q^{17} -10.3927 q^{18} +56.9732 q^{19} +53.3005 q^{20} +37.1359 q^{21} -12.7022 q^{22} +138.072 q^{23} +50.8083 q^{24} -61.0769 q^{25} +69.3794 q^{26} +27.0000 q^{27} -82.5230 q^{28} -41.6308 q^{29} +27.6972 q^{30} +222.766 q^{31} -174.491 q^{32} +33.0000 q^{33} -51.3442 q^{34} -98.9695 q^{35} -59.9991 q^{36} -119.878 q^{37} -65.7893 q^{38} -180.246 q^{39} -135.407 q^{40} -355.939 q^{41} -42.8824 q^{42} +419.845 q^{43} -73.3323 q^{44} -71.9567 q^{45} -159.437 q^{46} -606.447 q^{47} +101.327 q^{48} -189.770 q^{49} +70.5281 q^{50} +133.391 q^{51} +400.542 q^{52} -53.3436 q^{53} -31.1780 q^{54} -87.9471 q^{55} +209.646 q^{56} +170.920 q^{57} +48.0728 q^{58} +103.700 q^{59} +159.902 q^{60} -61.0000 q^{61} -257.237 q^{62} +111.408 q^{63} -68.7134 q^{64} +480.368 q^{65} -38.1065 q^{66} -313.628 q^{67} -296.421 q^{68} +414.216 q^{69} +114.284 q^{70} +716.138 q^{71} +152.425 q^{72} +660.702 q^{73} +138.429 q^{74} -183.231 q^{75} -379.816 q^{76} +136.165 q^{77} +208.138 q^{78} +768.543 q^{79} -270.043 q^{80} +81.0000 q^{81} +411.017 q^{82} -556.718 q^{83} -247.569 q^{84} -355.496 q^{85} -484.813 q^{86} -124.892 q^{87} +186.297 q^{88} +265.502 q^{89} +83.0915 q^{90} -743.734 q^{91} -920.466 q^{92} +668.298 q^{93} +700.290 q^{94} -455.512 q^{95} -523.473 q^{96} +1345.35 q^{97} +219.135 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more