LMFDB - Embedded newform 966.4.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,4,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, 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("966.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	966.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:	$56.9958450655$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 411x^{4} + 1741x^{3} + 37570x^{2} - 116091x - 993528 $ x^6 - 2x^5 - 411x^4 + 1741x^3 + 37570x^2 - 116091*x - 993528
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.62131$ of defining polynomial
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -1.62131 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -1.62131 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -3.24263 q^{10} +64.4429 q^{11} +12.0000 q^{12} -71.6835 q^{13} +14.0000 q^{14} -4.86394 q^{15} +16.0000 q^{16} +105.514 q^{17} +18.0000 q^{18} -32.6522 q^{19} -6.48525 q^{20} +21.0000 q^{21} +128.886 q^{22} +23.0000 q^{23} +24.0000 q^{24} -122.371 q^{25} -143.367 q^{26} +27.0000 q^{27} +28.0000 q^{28} +268.557 q^{29} -9.72788 q^{30} +66.4455 q^{31} +32.0000 q^{32} +193.329 q^{33} +211.028 q^{34} -11.3492 q^{35} +36.0000 q^{36} -294.578 q^{37} -65.3045 q^{38} -215.050 q^{39} -12.9705 q^{40} +298.620 q^{41} +42.0000 q^{42} -168.408 q^{43} +257.771 q^{44} -14.5918 q^{45} +46.0000 q^{46} +455.769 q^{47} +48.0000 q^{48} +49.0000 q^{49} -244.743 q^{50} +316.543 q^{51} -286.734 q^{52} -540.330 q^{53} +54.0000 q^{54} -104.482 q^{55} +56.0000 q^{56} -97.9567 q^{57} +537.113 q^{58} +406.589 q^{59} -19.4558 q^{60} +658.798 q^{61} +132.891 q^{62} +63.0000 q^{63} +64.0000 q^{64} +116.221 q^{65} +386.657 q^{66} -685.271 q^{67} +422.057 q^{68} +69.0000 q^{69} -22.6984 q^{70} +2.75224 q^{71} +72.0000 q^{72} +400.869 q^{73} -589.155 q^{74} -367.114 q^{75} -130.609 q^{76} +451.100 q^{77} -430.101 q^{78} +767.099 q^{79} -25.9410 q^{80} +81.0000 q^{81} +597.239 q^{82} -987.166 q^{83} +84.0000 q^{84} -171.072 q^{85} -336.816 q^{86} +805.670 q^{87} +515.543 q^{88} +689.859 q^{89} -29.1836 q^{90} -501.784 q^{91} +92.0000 q^{92} +199.337 q^{93} +911.537 q^{94} +52.9395 q^{95} +96.0000 q^{96} +1158.17 q^{97} +98.0000 q^{98} +579.986 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 42 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 42 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 42 q^{7} + 48 q^{8} + 54 q^{9} + 40 q^{10} + 79 q^{11} + 72 q^{12} + 52 q^{13} + 84 q^{14} + 60 q^{15} + 96 q^{16} + 140 q^{17} + 108 q^{18} + 93 q^{19} + 80 q^{20} + 126 q^{21} + 158 q^{22} + 138 q^{23} + 144 q^{24} + 142 q^{25} + 104 q^{26} + 162 q^{27} + 168 q^{28} + 143 q^{29} + 120 q^{30} + 130 q^{31} + 192 q^{32} + 237 q^{33} + 280 q^{34} + 140 q^{35} + 216 q^{36} + 151 q^{37} + 186 q^{38} + 156 q^{39} + 160 q^{40} + 412 q^{41} + 252 q^{42} + 250 q^{43} + 316 q^{44} + 180 q^{45} + 276 q^{46} + 666 q^{47} + 288 q^{48} + 294 q^{49} + 284 q^{50} + 420 q^{51} + 208 q^{52} - 96 q^{53} + 324 q^{54} + 51 q^{55} + 336 q^{56} + 279 q^{57} + 286 q^{58} + 514 q^{59} + 240 q^{60} + 422 q^{61} + 260 q^{62} + 378 q^{63} + 384 q^{64} + 277 q^{65} + 474 q^{66} + 669 q^{67} + 560 q^{68} + 414 q^{69} + 280 q^{70} - 357 q^{71} + 432 q^{72} + 430 q^{73} + 302 q^{74} + 426 q^{75} + 372 q^{76} + 553 q^{77} + 312 q^{78} + 750 q^{79} + 320 q^{80} + 486 q^{81} + 824 q^{82} + 222 q^{83} + 504 q^{84} + 601 q^{85} + 500 q^{86} + 429 q^{87} + 632 q^{88} + 763 q^{89} + 360 q^{90} + 364 q^{91} + 552 q^{92} + 390 q^{93} + 1332 q^{94} - 541 q^{95} + 576 q^{96} + 575 q^{97} + 588 q^{98} + 711 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 42 * q^7 + 48 * q^8 + 54 * q^9 + 40 * q^10 + 79 * q^11 + 72 * q^12 + 52 * q^13 + 84 * q^14 + 60 * q^15 + 96 * q^16 + 140 * q^17 + 108 * q^18 + 93 * q^19 + 80 * q^20 + 126 * q^21 + 158 * q^22 + 138 * q^23 + 144 * q^24 + 142 * q^25 + 104 * q^26 + 162 * q^27 + 168 * q^28 + 143 * q^29 + 120 * q^30 + 130 * q^31 + 192 * q^32 + 237 * q^33 + 280 * q^34 + 140 * q^35 + 216 * q^36 + 151 * q^37 + 186 * q^38 + 156 * q^39 + 160 * q^40 + 412 * q^41 + 252 * q^42 + 250 * q^43 + 316 * q^44 + 180 * q^45 + 276 * q^46 + 666 * q^47 + 288 * q^48 + 294 * q^49 + 284 * q^50 + 420 * q^51 + 208 * q^52 - 96 * q^53 + 324 * q^54 + 51 * q^55 + 336 * q^56 + 279 * q^57 + 286 * q^58 + 514 * q^59 + 240 * q^60 + 422 * q^61 + 260 * q^62 + 378 * q^63 + 384 * q^64 + 277 * q^65 + 474 * q^66 + 669 * q^67 + 560 * q^68 + 414 * q^69 + 280 * q^70 - 357 * q^71 + 432 * q^72 + 430 * q^73 + 302 * q^74 + 426 * q^75 + 372 * q^76 + 553 * q^77 + 312 * q^78 + 750 * q^79 + 320 * q^80 + 486 * q^81 + 824 * q^82 + 222 * q^83 + 504 * q^84 + 601 * q^85 + 500 * q^86 + 429 * q^87 + 632 * q^88 + 763 * q^89 + 360 * q^90 + 364 * q^91 + 552 * q^92 + 390 * q^93 + 1332 * q^94 - 541 * q^95 + 576 * q^96 + 575 * q^97 + 588 * q^98 + 711 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−1.62131	−0.145015	−0.0725073	−	0.997368i	$-0.523100\pi$
$5$	−1.62131	−0.145015	−0.0725073	+	0.997368i	$0.523100\pi$
$6$	6.00000	0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−3.24263	−0.102541
$11$	64.4429	1.76639	0.883193	−	0.469009i	$-0.155389\pi$
$11$	64.4429	1.76639	0.883193	+	0.469009i	$0.155389\pi$
$12$	12.0000	0.288675
$13$	−71.6835	−1.52934	−0.764670	−	0.644422i	$-0.777098\pi$
$13$	−71.6835	−1.52934	−0.764670	+	0.644422i	$0.777098\pi$
$14$	14.0000	0.267261
$15$	−4.86394	−0.0837242
$16$	16.0000	0.250000
$17$	105.514	1.50535	0.752675	−	0.658392i	$-0.228763\pi$
$17$	105.514	1.50535	0.752675	+	0.658392i	$0.228763\pi$
$18$	18.0000	0.235702
$19$	−32.6522	−0.394260	−0.197130	−	0.980377i	$-0.563162\pi$
$19$	−32.6522	−0.394260	−0.197130	+	0.980377i	$0.563162\pi$
$20$	−6.48525	−0.0725073
$21$	21.0000	0.218218
$22$	128.886	1.24902
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	24.0000	0.204124
$25$	−122.371	−0.978971
$26$	−143.367	−1.08141
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	268.557	1.71965	0.859823	−	0.510593i	$-0.170574\pi$
$29$	268.557	1.71965	0.859823	+	0.510593i	$0.170574\pi$
$30$	−9.72788	−0.0592020
$31$	66.4455	0.384967	0.192483	−	0.981300i	$-0.438346\pi$
$31$	66.4455	0.384967	0.192483	+	0.981300i	$0.438346\pi$
$32$	32.0000	0.176777
$33$	193.329	1.01982
$34$	211.028	1.06444
$35$	−11.3492	−0.0548104
$36$	36.0000	0.166667
$37$	−294.578	−1.30887	−0.654436	−	0.756117i	$-0.727094\pi$
$37$	−294.578	−1.30887	−0.654436	+	0.756117i	$0.727094\pi$
$38$	−65.3045	−0.278784
$39$	−215.050	−0.882965
$40$	−12.9705	−0.0512704
$41$	298.620	1.13748	0.568738	−	0.822518i	$-0.307432\pi$
$41$	298.620	1.13748	0.568738	+	0.822518i	$0.307432\pi$
$42$	42.0000	0.154303
$43$	−168.408	−0.597256	−0.298628	−	0.954370i	$-0.596529\pi$
$43$	−168.408	−0.597256	−0.298628	+	0.954370i	$0.596529\pi$
$44$	257.771	0.883193
$45$	−14.5918	−0.0483382
$46$	46.0000	0.147442
$47$	455.769	1.41448	0.707241	−	0.706972i	$-0.249939\pi$
$47$	455.769	1.41448	0.707241	+	0.706972i	$0.249939\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−244.743	−0.692237
$51$	316.543	0.869114
$52$	−286.734	−0.764670
$53$	−540.330	−1.40038	−0.700189	−	0.713958i	$-0.746901\pi$
$53$	−540.330	−1.40038	−0.700189	+	0.713958i	$0.746901\pi$
$54$	54.0000	0.136083
$55$	−104.482	−0.256152
$56$	56.0000	0.133631
$57$	−97.9567	−0.227626
$58$	537.113	1.21597
$59$	406.589	0.897177	0.448588	−	0.893738i	$-0.351927\pi$
$59$	406.589	0.897177	0.448588	+	0.893738i	$0.351927\pi$
$60$	−19.4558	−0.0418621
$61$	658.798	1.38279	0.691397	−	0.722475i	$-0.256996\pi$
$61$	658.798	1.38279	0.691397	+	0.722475i	$0.256996\pi$
$62$	132.891	0.272213
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	116.221	0.221777
$66$	386.657	0.721124
$67$	−685.271	−1.24954	−0.624770	−	0.780809i	$-0.714807\pi$
$67$	−685.271	−1.24954	−0.624770	+	0.780809i	$0.714807\pi$
$68$	422.057	0.752675
$69$	69.0000	0.120386
$70$	−22.6984	−0.0387568
$71$	2.75224	0.00460044	0.00230022	−	0.999997i	$-0.499268\pi$
$71$	2.75224	0.00460044	0.00230022	+	0.999997i	$0.499268\pi$
$72$	72.0000	0.117851
$73$	400.869	0.642714	0.321357	−	0.946958i	$-0.395861\pi$
$73$	400.869	0.642714	0.321357	+	0.946958i	$0.395861\pi$
$74$	−589.155	−0.925512
$75$	−367.114	−0.565209
$76$	−130.609	−0.197130
$77$	451.100	0.667631
$78$	−430.101	−0.624351
$79$	767.099	1.09247	0.546237	−	0.837631i	$-0.316060\pi$
$79$	767.099	1.09247	0.546237	+	0.837631i	$0.316060\pi$
$80$	−25.9410	−0.0362537
$81$	81.0000	0.111111
$82$	597.239	0.804318
$83$	−987.166	−1.30549	−0.652744	−	0.757579i	$-0.726382\pi$
$83$	−987.166	−1.30549	−0.652744	+	0.757579i	$0.726382\pi$
$84$	84.0000	0.109109
$85$	−171.072	−0.218298
$86$	−336.816	−0.422324
$87$	805.670	0.992838
$88$	515.543	0.624512
$89$	689.859	0.821629	0.410814	−	0.911719i	$-0.365244\pi$
$89$	689.859	0.821629	0.410814	+	0.911719i	$0.365244\pi$
$90$	−29.1836	−0.0341803
$91$	−501.784	−0.578036
$92$	92.0000	0.104257
$93$	199.337	0.222261
$94$	911.537	1.00019
$95$	52.9395	0.0571735
$96$	96.0000	0.102062
$97$	1158.17	1.21231	0.606157	−	0.795345i	$-0.292710\pi$
$97$	1158.17	1.21231	0.606157	+	0.795345i	$0.292710\pi$
$98$	98.0000	0.101015
$99$	579.986	0.588795
$100$	−489.485	−0.489485
$101$	−891.776	−0.878565	−0.439283	−	0.898349i	$-0.644767\pi$
$101$	−891.776	−0.878565	−0.439283	+	0.898349i	$0.644767\pi$
$102$	633.085	0.614557
$103$	1630.48	1.55976	0.779882	−	0.625926i	$-0.215279\pi$
$103$	1630.48	1.55976	0.779882	+	0.625926i	$0.215279\pi$
$104$	−573.468	−0.540703
$105$	−34.0476	−0.0316448
$106$	−1080.66	−0.990216
$107$	−970.859	−0.877163	−0.438582	−	0.898691i	$-0.644519\pi$
$107$	−970.859	−0.877163	−0.438582	+	0.898691i	$0.644519\pi$
$108$	108.000	0.0962250
$109$	1942.06	1.70656	0.853282	−	0.521450i	$-0.174609\pi$
$109$	1942.06	1.70656	0.853282	+	0.521450i	$0.174609\pi$
$110$	−208.964	−0.181127
$111$	−883.733	−0.755678
$112$	112.000	0.0944911
$113$	−812.216	−0.676167	−0.338084	−	0.941116i	$-0.609779\pi$
$113$	−812.216	−0.676167	−0.338084	+	0.941116i	$0.609779\pi$
$114$	−195.913	−0.160956
$115$	−37.2902	−0.0302376
$116$	1074.23	0.859823
$117$	−645.151	−0.509780
$118$	813.179	0.634400
$119$	738.599	0.568969
$120$	−38.9115	−0.0296010
$121$	2821.88	2.12012
$122$	1317.60	0.977783
$123$	895.859	0.656723
$124$	265.782	0.192483
$125$	401.066	0.286980
$126$	126.000	0.0890871
$127$	−2694.16	−1.88243	−0.941213	−	0.337813i	$-0.890313\pi$
$127$	−2694.16	−1.88243	−0.941213	+	0.337813i	$0.890313\pi$
$128$	128.000	0.0883883
$129$	−505.225	−0.344826
$130$	232.443	0.156820
$131$	1181.22	0.787811	0.393906	−	0.919151i	$-0.371124\pi$
$131$	1181.22	0.787811	0.393906	+	0.919151i	$0.371124\pi$
$132$	773.314	0.509912
$133$	−228.566	−0.149016
$134$	−1370.54	−0.883558
$135$	−43.7754	−0.0279081
$136$	844.114	0.532222
$137$	−418.361	−0.260898	−0.130449	−	0.991455i	$-0.541642\pi$
$137$	−418.361	−0.260898	−0.130449	+	0.991455i	$0.541642\pi$
$138$	138.000	0.0851257
$139$	−305.595	−0.186477	−0.0932383	−	0.995644i	$-0.529722\pi$
$139$	−305.595	−0.186477	−0.0932383	+	0.995644i	$0.529722\pi$
$140$	−45.3968	−0.0274052
$141$	1367.31	0.816652
$142$	5.50449	0.00325300
$143$	−4619.49	−2.70141
$144$	144.000	0.0833333
$145$	−435.414	−0.249374
$146$	801.738	0.454468
$147$	147.000	0.0824786
$148$	−1178.31	−0.654436
$149$	509.272	0.280008	0.140004	−	0.990151i	$-0.455288\pi$
$149$	509.272	0.280008	0.140004	+	0.990151i	$0.455288\pi$
$150$	−734.228	−0.399663
$151$	−3496.28	−1.88426	−0.942129	−	0.335250i	$-0.891179\pi$
$151$	−3496.28	−1.88426	−0.942129	+	0.335250i	$0.891179\pi$
$152$	−261.218	−0.139392
$153$	949.628	0.501783
$154$	902.200	0.472087
$155$	−107.729	−0.0558258
$156$	−860.202	−0.441482
$157$	−1927.50	−0.979816	−0.489908	−	0.871774i	$-0.662970\pi$
$157$	−1927.50	−0.979816	−0.489908	+	0.871774i	$0.662970\pi$
$158$	1534.20	0.772496
$159$	−1620.99	−0.808508
$160$	−51.8820	−0.0256352
$161$	161.000	0.0788110
$162$	162.000	0.0785674
$163$	−2047.43	−0.983848	−0.491924	−	0.870638i	$-0.663706\pi$
$163$	−2047.43	−0.983848	−0.491924	+	0.870638i	$0.663706\pi$
$164$	1194.48	0.568738
$165$	−313.446	−0.147889
$166$	−1974.33	−0.923119
$167$	2567.73	1.18980	0.594901	−	0.803799i	$-0.297191\pi$
$167$	2567.73	1.18980	0.594901	+	0.803799i	$0.297191\pi$
$168$	168.000	0.0771517
$169$	2941.52	1.33888
$170$	−342.143	−0.154360
$171$	−293.870	−0.131420
$172$	−673.633	−0.298628
$173$	−3481.98	−1.53023	−0.765117	−	0.643892i	$-0.777319\pi$
$173$	−3481.98	−1.53023	−0.765117	+	0.643892i	$0.777319\pi$
$174$	1611.34	0.702042
$175$	−856.599	−0.370016
$176$	1031.09	0.441597
$177$	1219.77	0.517985
$178$	1379.72	0.580979
$179$	1603.55	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$179$	1603.55	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$180$	−58.3673	−0.0241691
$181$	−649.439	−0.266699	−0.133349	−	0.991069i	$-0.542573\pi$
$181$	−649.439	−0.266699	−0.133349	+	0.991069i	$0.542573\pi$
$182$	−1003.57	−0.408733
$183$	1976.39	0.798356
$184$	184.000	0.0737210
$185$	477.603	0.189806
$186$	398.673	0.157162
$187$	6799.64	2.65903
$188$	1823.07	0.707241
$189$	189.000	0.0727393
$190$	105.879	0.0404277
$191$	−1771.58	−0.671136	−0.335568	−	0.942016i	$-0.608928\pi$
$191$	−1771.58	−0.671136	−0.335568	+	0.942016i	$0.608928\pi$
$192$	192.000	0.0721688
$193$	2249.13	0.838838	0.419419	−	0.907793i	$-0.362234\pi$
$193$	2249.13	0.838838	0.419419	+	0.907793i	$0.362234\pi$
$194$	2316.34	0.857235
$195$	348.664	0.128043
$196$	196.000	0.0714286
$197$	3244.34	1.17335	0.586674	−	0.809823i	$-0.300437\pi$
$197$	3244.34	1.17335	0.586674	+	0.809823i	$0.300437\pi$
$198$	1159.97	0.416341
$199$	−722.225	−0.257272	−0.128636	−	0.991692i	$-0.541060\pi$
$199$	−722.225	−0.257272	−0.128636	+	0.991692i	$0.541060\pi$
$200$	−978.971	−0.346118
$201$	−2055.81	−0.721422
$202$	−1783.55	−0.621239
$203$	1879.90	0.649965
$204$	1266.17	0.434557
$205$	−484.156	−0.164951
$206$	3260.96	1.10292
$207$	207.000	0.0695048
$208$	−1146.94	−0.382335
$209$	−2104.20	−0.696415
$210$	−68.0951	−0.0223762
$211$	−2768.24	−0.903192	−0.451596	−	0.892223i	$-0.649145\pi$
$211$	−2768.24	−0.903192	−0.451596	+	0.892223i	$0.649145\pi$
$212$	−2161.32	−0.700189
$213$	8.25673	0.00265606
$214$	−1941.72	−0.620248
$215$	273.042	0.0866109
$216$	216.000	0.0680414
$217$	465.119	0.145504
$218$	3884.12	1.20672
$219$	1202.61	0.371071
$220$	−417.928	−0.128076
$221$	−7563.63	−2.30219
$222$	−1767.47	−0.534345
$223$	−2891.69	−0.868349	−0.434175	−	0.900829i	$-0.642960\pi$
$223$	−2891.69	−0.868349	−0.434175	+	0.900829i	$0.642960\pi$
$224$	224.000	0.0668153
$225$	−1101.34	−0.326324
$226$	−1624.43	−0.478122
$227$	−1880.59	−0.549863	−0.274932	−	0.961464i	$-0.588655\pi$
$227$	−1880.59	−0.549863	−0.274932	+	0.961464i	$0.588655\pi$
$228$	−391.827	−0.113813
$229$	−1792.65	−0.517300	−0.258650	−	0.965971i	$-0.583278\pi$
$229$	−1792.65	−0.517300	−0.258650	+	0.965971i	$0.583278\pi$
$230$	−74.5804	−0.0213812
$231$	1353.30	0.385457
$232$	2148.45	0.607986
$233$	−3818.62	−1.07368	−0.536838	−	0.843686i	$-0.680381\pi$
$233$	−3818.62	−1.07368	−0.536838	+	0.843686i	$0.680381\pi$
$234$	−1290.30	−0.360469
$235$	−738.943	−0.205121
$236$	1626.36	0.448588
$237$	2301.30	0.630740
$238$	1477.20	0.402322
$239$	1838.99	0.497717	0.248858	−	0.968540i	$-0.419945\pi$
$239$	1838.99	0.497717	0.248858	+	0.968540i	$0.419945\pi$
$240$	−77.8230	−0.0209311
$241$	−1954.77	−0.522479	−0.261240	−	0.965274i	$-0.584131\pi$
$241$	−1954.77	−0.522479	−0.261240	+	0.965274i	$0.584131\pi$
$242$	5643.76	1.49915
$243$	243.000	0.0641500
$244$	2635.19	0.691397
$245$	−79.4443	−0.0207164
$246$	1791.72	0.464373
$247$	2340.63	0.602958
$248$	531.564	0.136106
$249$	−2961.50	−0.753724
$250$	802.133	0.202925
$251$	1619.50	0.407259	0.203630	−	0.979048i	$-0.434726\pi$
$251$	1619.50	0.407259	0.203630	+	0.979048i	$0.434726\pi$
$252$	252.000	0.0629941
$253$	1482.19	0.368317
$254$	−5388.32	−1.33108
$255$	−513.215	−0.126034
$256$	256.000	0.0625000
$257$	3499.52	0.849393	0.424696	−	0.905336i	$-0.360381\pi$
$257$	3499.52	0.849393	0.424696	+	0.905336i	$0.360381\pi$
$258$	−1010.45	−0.243829
$259$	−2062.04	−0.494707
$260$	464.885	0.110888
$261$	2417.01	0.573215
$262$	2362.43	0.557067
$263$	835.986	0.196004	0.0980021	−	0.995186i	$-0.468755\pi$
$263$	835.986	0.196004	0.0980021	+	0.995186i	$0.468755\pi$
$264$	1546.63	0.360562
$265$	876.044	0.203075
$266$	−457.131	−0.105370
$267$	2069.58	0.474368
$268$	−2741.08	−0.624770
$269$	4477.80	1.01493	0.507465	−	0.861672i	$-0.330583\pi$
$269$	4477.80	1.01493	0.507465	+	0.861672i	$0.330583\pi$
$270$	−87.5509	−0.0197340
$271$	−4870.79	−1.09181	−0.545904	−	0.837848i	$-0.683813\pi$
$271$	−4870.79	−1.09181	−0.545904	+	0.837848i	$0.683813\pi$
$272$	1688.23	0.376338
$273$	−1505.35	−0.333729
$274$	−836.721	−0.184482
$275$	−7885.96	−1.72924
$276$	276.000	0.0601929
$277$	−7987.01	−1.73247	−0.866233	−	0.499640i	$-0.833466\pi$
$277$	−7987.01	−1.73247	−0.866233	+	0.499640i	$0.833466\pi$
$278$	−611.191	−0.131859
$279$	598.010	0.128322
$280$	−90.7935	−0.0193784
$281$	2497.50	0.530208	0.265104	−	0.964220i	$-0.414594\pi$
$281$	2497.50	0.530208	0.265104	+	0.964220i	$0.414594\pi$
$282$	2734.61	0.577460
$283$	3869.04	0.812687	0.406344	−	0.913720i	$-0.366804\pi$
$283$	3869.04	0.812687	0.406344	+	0.913720i	$0.366804\pi$
$284$	11.0090	0.00230022
$285$	158.819	0.0330091
$286$	−9238.98	−1.91018
$287$	2090.34	0.429926
$288$	288.000	0.0589256
$289$	6220.25	1.26608
$290$	−870.829	−0.176334
$291$	3474.51	0.699929
$292$	1603.48	0.321357
$293$	−4748.61	−0.946814	−0.473407	−	0.880844i	$-0.656976\pi$
$293$	−4748.61	−0.946814	−0.473407	+	0.880844i	$0.656976\pi$
$294$	294.000	0.0583212
$295$	−659.209	−0.130104
$296$	−2356.62	−0.462756
$297$	1739.96	0.339941
$298$	1018.54	0.197996
$299$	−1648.72	−0.318889
$300$	−1468.46	−0.282605
$301$	−1178.86	−0.225742
$302$	−6992.55	−1.33237
$303$	−2675.33	−0.507240
$304$	−522.436	−0.0985650
$305$	−1068.12	−0.200525
$306$	1899.26	0.354814
$307$	−3812.94	−0.708847	−0.354424	−	0.935085i	$-0.615323\pi$
$307$	−3812.94	−0.708847	−0.354424	+	0.935085i	$0.615323\pi$
$308$	1804.40	0.333816
$309$	4891.43	0.900531
$310$	−215.458	−0.0394748
$311$	−5743.90	−1.04729	−0.523644	−	0.851937i	$-0.675428\pi$
$311$	−5743.90	−1.04729	−0.523644	+	0.851937i	$0.675428\pi$
$312$	−1720.40	−0.312175
$313$	−1739.71	−0.314167	−0.157084	−	0.987585i	$-0.550209\pi$
$313$	−1739.71	−0.314167	−0.157084	+	0.987585i	$0.550209\pi$
$314$	−3854.99	−0.692834
$315$	−102.143	−0.0182701
$316$	3068.40	0.546237
$317$	−8120.62	−1.43880	−0.719400	−	0.694596i	$-0.755583\pi$
$317$	−8120.62	−1.43880	−0.719400	+	0.694596i	$0.755583\pi$
$318$	−3241.98	−0.571702
$319$	17306.6	3.03756
$320$	−103.764	−0.0181268
$321$	−2912.58	−0.506431
$322$	322.000	0.0557278
$323$	−3445.28	−0.593499
$324$	324.000	0.0555556
$325$	8772.00	1.49718
$326$	−4094.86	−0.695686
$327$	5826.17	0.985285
$328$	2388.96	0.402159
$329$	3190.38	0.534624
$330$	−626.892	−0.104574
$331$	4734.84	0.786255	0.393128	−	0.919484i	$-0.371393\pi$
$331$	4734.84	0.786255	0.393128	+	0.919484i	$0.371393\pi$
$332$	−3948.66	−0.652744
$333$	−2651.20	−0.436291
$334$	5135.46	0.841317
$335$	1111.04	0.181202
$336$	336.000	0.0545545
$337$	−17.2297	−0.00278505	−0.00139253	−	0.999999i	$-0.500443\pi$
$337$	−17.2297	−0.00278505	−0.00139253	+	0.999999i	$0.500443\pi$
$338$	5883.04	0.946732
$339$	−2436.65	−0.390385
$340$	−684.286	−0.109149
$341$	4281.94	0.680000
$342$	−587.740	−0.0929280
$343$	343.000	0.0539949
$344$	−1347.27	−0.211162
$345$	−111.871	−0.0174577
$346$	−6963.97	−1.08204
$347$	−8526.77	−1.31914	−0.659569	−	0.751644i	$-0.729261\pi$
$347$	−8526.77	−1.31914	−0.659569	+	0.751644i	$0.729261\pi$
$348$	3222.68	0.496419
$349$	−5115.47	−0.784599	−0.392299	−	0.919838i	$-0.628320\pi$
$349$	−5115.47	−0.784599	−0.392299	+	0.919838i	$0.628320\pi$
$350$	−1713.20	−0.261641
$351$	−1935.45	−0.294322
$352$	2062.17	0.312256
$353$	9813.85	1.47971	0.739857	−	0.672765i	$-0.234893\pi$
$353$	9813.85	1.47971	0.739857	+	0.672765i	$0.234893\pi$
$354$	2439.54	0.366271
$355$	−4.46225	−0.000667131 0
$356$	2759.44	0.410814
$357$	2215.80	0.328494
$358$	3207.09	0.473464
$359$	3879.64	0.570361	0.285181	−	0.958474i	$-0.407946\pi$
$359$	3879.64	0.570361	0.285181	+	0.958474i	$0.407946\pi$
$360$	−116.735	−0.0170901
$361$	−5792.83	−0.844559
$362$	−1298.88	−0.188584
$363$	8465.64	1.22405
$364$	−2007.14	−0.289018
$365$	−649.934	−0.0932030
$366$	3952.79	0.564523
$367$	−9960.15	−1.41666	−0.708332	−	0.705880i	$-0.750552\pi$
$367$	−9960.15	−1.41666	−0.708332	+	0.705880i	$0.750552\pi$
$368$	368.000	0.0521286
$369$	2687.58	0.379159
$370$	955.205	0.134213
$371$	−3782.31	−0.529293
$372$	797.346	0.111130
$373$	−10931.5	−1.51746	−0.758732	−	0.651403i	$-0.774181\pi$
$373$	−10931.5	−1.51746	−0.758732	+	0.651403i	$0.774181\pi$
$374$	13599.3	1.88022
$375$	1203.20	0.165688
$376$	3646.15	0.500095
$377$	−19251.1	−2.62992
$378$	378.000	0.0514344
$379$	10597.9	1.43635	0.718177	−	0.695860i	$-0.244977\pi$
$379$	10597.9	1.43635	0.718177	+	0.695860i	$0.244977\pi$
$380$	211.758	0.0285867
$381$	−8082.48	−1.08682
$382$	−3543.16	−0.474565
$383$	−1824.94	−0.243473	−0.121736	−	0.992562i	$-0.538846\pi$
$383$	−1824.94	−0.243473	−0.121736	+	0.992562i	$0.538846\pi$
$384$	384.000	0.0510310
$385$	−731.374	−0.0968163
$386$	4498.25	0.593148
$387$	−1515.67	−0.199085
$388$	4632.68	0.606157
$389$	3901.74	0.508550	0.254275	−	0.967132i	$-0.418163\pi$
$389$	3901.74	0.508550	0.254275	+	0.967132i	$0.418163\pi$
$390$	697.328	0.0905400
$391$	2426.83	0.313887
$392$	392.000	0.0505076
$393$	3543.65	0.454843
$394$	6488.68	0.829682
$395$	−1243.71	−0.158425
$396$	2319.94	0.294398
$397$	−13520.9	−1.70930	−0.854651	−	0.519203i	$-0.826229\pi$
$397$	−13520.9	−1.70930	−0.854651	+	0.519203i	$0.826229\pi$
$398$	−1444.45	−0.181919
$399$	−685.697	−0.0860346
$400$	−1957.94	−0.244743
$401$	13629.6	1.69734	0.848668	−	0.528925i	$-0.177405\pi$
$401$	13629.6	1.69734	0.848668	+	0.528925i	$0.177405\pi$
$402$	−4111.63	−0.510123
$403$	−4763.05	−0.588745
$404$	−3567.11	−0.439283
$405$	−131.326	−0.0161127
$406$	3759.79	0.459595
$407$	−18983.4	−2.31197
$408$	2532.34	0.307278
$409$	−4171.37	−0.504305	−0.252153	−	0.967687i	$-0.581138\pi$
$409$	−4171.37	−0.504305	−0.252153	+	0.967687i	$0.581138\pi$
$410$	−968.312	−0.116638
$411$	−1255.08	−0.150629
$412$	6521.91	0.779882
$413$	2846.13	0.339101
$414$	414.000	0.0491473
$415$	1600.50	0.189315
$416$	−2293.87	−0.270352
$417$	−916.786	−0.107662
$418$	−4208.41	−0.492440
$419$	−5180.77	−0.604050	−0.302025	−	0.953300i	$-0.597663\pi$
$419$	−5180.77	−0.604050	−0.302025	+	0.953300i	$0.597663\pi$
$420$	−136.190	−0.0158224
$421$	−11289.2	−1.30689	−0.653446	−	0.756973i	$-0.726677\pi$
$421$	−11289.2	−1.30689	−0.653446	+	0.756973i	$0.726677\pi$
$422$	−5536.48	−0.638653
$423$	4101.92	0.471494
$424$	−4322.64	−0.495108
$425$	−12911.9	−1.47369
$426$	16.5135	0.00187812
$427$	4611.58	0.522647
$428$	−3883.44	−0.438582
$429$	−13858.5	−1.55966
$430$	546.085	0.0612431
$431$	−14628.3	−1.63485	−0.817424	−	0.576037i	$-0.804599\pi$
$431$	−14628.3	−1.63485	−0.817424	+	0.576037i	$0.804599\pi$
$432$	432.000	0.0481125
$433$	−1585.57	−0.175976	−0.0879879	−	0.996122i	$-0.528044\pi$
$433$	−1585.57	−0.175976	−0.0879879	+	0.996122i	$0.528044\pi$
$434$	930.237	0.102887
$435$	−1306.24	−0.143976
$436$	7768.23	0.853282
$437$	−751.002	−0.0822089
$438$	2405.21	0.262387
$439$	1855.82	0.201762	0.100881	−	0.994899i	$-0.467834\pi$
$439$	1855.82	0.201762	0.100881	+	0.994899i	$0.467834\pi$
$440$	−835.856	−0.0905634
$441$	441.000	0.0476190
$442$	−15127.3	−1.62790
$443$	4914.00	0.527023	0.263511	−	0.964656i	$-0.415119\pi$
$443$	4914.00	0.527023	0.263511	+	0.964656i	$0.415119\pi$
$444$	−3534.93	−0.377839
$445$	−1118.48	−0.119148
$446$	−5783.38	−0.614016
$447$	1527.82	0.161663
$448$	448.000	0.0472456
$449$	2076.89	0.218296	0.109148	−	0.994026i	$-0.465188\pi$
$449$	2076.89	0.218296	0.109148	+	0.994026i	$0.465188\pi$
$450$	−2202.68	−0.230746
$451$	19243.9	2.00922
$452$	−3248.87	−0.338084
$453$	−10488.8	−1.08788
$454$	−3761.18	−0.388812
$455$	813.550	0.0838237
$456$	−783.654	−0.0804780
$457$	−9745.93	−0.997583	−0.498791	−	0.866722i	$-0.666223\pi$
$457$	−9745.93	−0.997583	−0.498791	+	0.866722i	$0.666223\pi$
$458$	−3585.30	−0.365787
$459$	2848.88	0.289705
$460$	−149.161	−0.0151188
$461$	−14911.7	−1.50652	−0.753261	−	0.657721i	$-0.771520\pi$
$461$	−14911.7	−1.50652	−0.753261	+	0.657721i	$0.771520\pi$
$462$	2706.60	0.272559
$463$	−4644.25	−0.466170	−0.233085	−	0.972456i	$-0.574882\pi$
$463$	−4644.25	−0.466170	−0.233085	+	0.972456i	$0.574882\pi$
$464$	4296.91	0.429911
$465$	−323.187	−0.0322311
$466$	−7637.25	−0.759203
$467$	11496.1	1.13914	0.569568	−	0.821944i	$-0.307111\pi$
$467$	11496.1	1.13914	0.569568	+	0.821944i	$0.307111\pi$
$468$	−2580.61	−0.254890
$469$	−4796.90	−0.472282
$470$	−1477.89	−0.145042
$471$	−5782.49	−0.565697
$472$	3252.71	0.317200
$473$	−10852.7	−1.05498
$474$	4602.60	0.446001
$475$	3995.70	0.385969
$476$	2954.40	0.284484
$477$	−4862.97	−0.466792
$478$	3677.98	0.351939
$479$	−13317.7	−1.27036	−0.635178	−	0.772366i	$-0.719073\pi$
$479$	−13317.7	−1.27036	−0.635178	+	0.772366i	$0.719073\pi$
$480$	−155.646	−0.0148005
$481$	21116.4	2.00171
$482$	−3909.53	−0.369449
$483$	483.000	0.0455016
$484$	11287.5	1.06006
$485$	−1877.76	−0.175803
$486$	486.000	0.0453609
$487$	−684.691	−0.0637091	−0.0318545	−	0.999493i	$-0.510141\pi$
$487$	−684.691	−0.0637091	−0.0318545	+	0.999493i	$0.510141\pi$
$488$	5270.38	0.488891
$489$	−6142.30	−0.568025
$490$	−158.889	−0.0146487
$491$	6471.39	0.594806	0.297403	−	0.954752i	$-0.403880\pi$
$491$	6471.39	0.594806	0.297403	+	0.954752i	$0.403880\pi$
$492$	3583.44	0.328361
$493$	28336.5	2.58867
$494$	4681.25	0.426355
$495$	−940.338	−0.0853840
$496$	1063.13	0.0962417
$497$	19.2657	0.00173880
$498$	−5922.99	−0.532963
$499$	−19304.3	−1.73182	−0.865912	−	0.500196i	$-0.833261\pi$
$499$	−19304.3	−1.73182	−0.865912	+	0.500196i	$0.833261\pi$
$500$	1604.27	0.143490
$501$	7703.19	0.686933
$502$	3239.00	0.287976
$503$	−5648.02	−0.500662	−0.250331	−	0.968160i	$-0.580539\pi$
$503$	−5648.02	−0.500662	−0.250331	+	0.968160i	$0.580539\pi$
$504$	504.000	0.0445435
$505$	1445.85	0.127405
$506$	2964.37	0.260439
$507$	8824.57	0.773004
$508$	−10776.6	−0.941213
$509$	16528.0	1.43927	0.719635	−	0.694352i	$-0.244309\pi$
$509$	16528.0	1.43927	0.719635	+	0.694352i	$0.244309\pi$
$510$	−1026.43	−0.0891197
$511$	2806.08	0.242923
$512$	512.000	0.0441942
$513$	−881.611	−0.0758754
$514$	6999.04	0.600611
$515$	−2643.51	−0.226189
$516$	−2020.90	−0.172413
$517$	29371.0	2.49852
$518$	−4124.09	−0.349811
$519$	−10446.0	−0.883481
$520$	929.771	0.0784099
$521$	18493.3	1.55510	0.777548	−	0.628824i	$-0.216463\pi$
$521$	18493.3	1.55510	0.777548	+	0.628824i	$0.216463\pi$
$522$	4834.02	0.405324
$523$	−2738.25	−0.228940	−0.114470	−	0.993427i	$-0.536517\pi$
$523$	−2738.25	−0.228940	−0.114470	+	0.993427i	$0.536517\pi$
$524$	4724.86	0.393906
$525$	−2569.80	−0.213629
$526$	1671.97	0.138596
$527$	7010.95	0.579510
$528$	3093.26	0.254956
$529$	529.000	0.0434783
$530$	1752.09	0.143596
$531$	3659.30	0.299059
$532$	−914.263	−0.0745081
$533$	−21406.1	−1.73959
$534$	4139.16	0.335429
$535$	1574.07	0.127202
$536$	−5482.17	−0.441779
$537$	4810.64	0.386582
$538$	8955.60	0.717664
$539$	3157.70	0.252341
$540$	−175.102	−0.0139540
$541$	−13782.3	−1.09528	−0.547641	−	0.836714i	$-0.684474\pi$
$541$	−13782.3	−1.09528	−0.547641	+	0.836714i	$0.684474\pi$
$542$	−9741.59	−0.772024
$543$	−1948.32	−0.153979
$544$	3376.45	0.266111
$545$	−3148.68	−0.247477
$546$	−3010.71	−0.235982
$547$	−2774.89	−0.216903	−0.108451	−	0.994102i	$-0.534589\pi$
$547$	−2774.89	−0.216903	−0.108451	+	0.994102i	$0.534589\pi$
$548$	−1673.44	−0.130449
$549$	5929.18	0.460931
$550$	−15771.9	−1.22276
$551$	−8768.98	−0.677987
$552$	552.000	0.0425628
$553$	5369.70	0.412916
$554$	−15974.0	−1.22504
$555$	1432.81	0.109584
$556$	−1222.38	−0.0932383
$557$	10132.7	0.770803	0.385401	−	0.922749i	$-0.374063\pi$
$557$	10132.7	0.770803	0.385401	+	0.922749i	$0.374063\pi$
$558$	1196.02	0.0907375
$559$	12072.1	0.913408
$560$	−181.587	−0.0137026
$561$	20398.9	1.53519
$562$	4995.00	0.374913
$563$	8465.88	0.633738	0.316869	−	0.948469i	$-0.397369\pi$
$563$	8465.88	0.633738	0.316869	+	0.948469i	$0.397369\pi$
$564$	5469.22	0.408326
$565$	1316.86	0.0980541
$566$	7738.08	0.574657
$567$	567.000	0.0419961
$568$	22.0179	0.00162650
$569$	1064.22	0.0784086	0.0392043	−	0.999231i	$-0.487518\pi$
$569$	1064.22	0.0784086	0.0392043	+	0.999231i	$0.487518\pi$
$570$	317.637	0.0233410
$571$	2593.41	0.190072	0.0950358	−	0.995474i	$-0.469703\pi$
$571$	2593.41	0.190072	0.0950358	+	0.995474i	$0.469703\pi$
$572$	−18478.0	−1.35070
$573$	−5314.74	−0.387481
$574$	4180.67	0.304003
$575$	−2814.54	−0.204130
$576$	576.000	0.0416667
$577$	10743.6	0.775149	0.387574	−	0.921838i	$-0.373313\pi$
$577$	10743.6	0.775149	0.387574	+	0.921838i	$0.373313\pi$
$578$	12440.5	0.895253
$579$	6747.38	0.484303
$580$	−1741.66	−0.124687
$581$	−6910.16	−0.493428
$582$	6949.02	0.494925
$583$	−34820.4	−2.47361
$584$	3206.95	0.227234
$585$	1045.99	0.0739256
$586$	−9497.22	−0.669499
$587$	8673.98	0.609903	0.304952	−	0.952368i	$-0.401360\pi$
$587$	8673.98	0.609903	0.304952	+	0.952368i	$0.401360\pi$
$588$	588.000	0.0412393
$589$	−2169.60	−0.151777
$590$	−1318.42	−0.0919972
$591$	9733.02	0.677433
$592$	−4713.24	−0.327218
$593$	−15930.9	−1.10321	−0.551604	−	0.834106i	$-0.685984\pi$
$593$	−15930.9	−1.10321	−0.551604	+	0.834106i	$0.685984\pi$
$594$	3479.91	0.240375
$595$	−1197.50	−0.0825088
$596$	2037.09	0.140004
$597$	−2166.67	−0.148536
$598$	−3297.44	−0.225489
$599$	−15654.2	−1.06780	−0.533900	−	0.845547i	$-0.679274\pi$
$599$	−15654.2	−1.06780	−0.533900	+	0.845547i	$0.679274\pi$
$600$	−2936.91	−0.199832
$601$	5663.20	0.384371	0.192185	−	0.981359i	$-0.438443\pi$
$601$	5663.20	0.384371	0.192185	+	0.981359i	$0.438443\pi$
$602$	−2357.71	−0.159623
$603$	−6167.44	−0.416513
$604$	−13985.1	−0.942129
$605$	−4575.15	−0.307449
$606$	−5350.66	−0.358673
$607$	−2359.03	−0.157743	−0.0788716	−	0.996885i	$-0.525132\pi$
$607$	−2359.03	−0.157743	−0.0788716	+	0.996885i	$0.525132\pi$
$608$	−1044.87	−0.0696960
$609$	5639.69	0.375257
$610$	−2136.23	−0.141793
$611$	−32671.1	−2.16322
$612$	3798.51	0.250892
$613$	−1509.04	−0.0994285	−0.0497142	−	0.998763i	$-0.515831\pi$
$613$	−1509.04	−0.0994285	−0.0497142	+	0.998763i	$0.515831\pi$
$614$	−7625.89	−0.501231
$615$	−1452.47	−0.0952344
$616$	3608.80	0.236043
$617$	8086.70	0.527647	0.263824	−	0.964571i	$-0.415016\pi$
$617$	8086.70	0.527647	0.263824	+	0.964571i	$0.415016\pi$
$618$	9782.87	0.636771
$619$	−20060.4	−1.30258	−0.651288	−	0.758831i	$-0.725771\pi$
$619$	−20060.4	−1.30258	−0.651288	+	0.758831i	$0.725771\pi$
$620$	−430.916	−0.0279129
$621$	621.000	0.0401286
$622$	−11487.8	−0.740544
$623$	4829.02	0.310546
$624$	−3440.81	−0.220741
$625$	14646.2	0.937354
$626$	−3479.43	−0.222150
$627$	−6312.61	−0.402076
$628$	−7709.99	−0.489908
$629$	−31082.1	−1.97031
$630$	−204.285	−0.0129189
$631$	15996.6	1.00921	0.504607	−	0.863349i	$-0.331637\pi$
$631$	15996.6	1.00921	0.504607	+	0.863349i	$0.331637\pi$
$632$	6136.79	0.386248
$633$	−8304.72	−0.521458
$634$	−16241.2	−1.01738
$635$	4368.08	0.272979
$636$	−6483.96	−0.404254
$637$	−3512.49	−0.218477
$638$	34613.1	2.14788
$639$	24.7702	0.00153348
$640$	−207.528	−0.0128176
$641$	−27531.3	−1.69644	−0.848221	−	0.529643i	$-0.822326\pi$
$641$	−27531.3	−1.69644	−0.848221	+	0.529643i	$0.822326\pi$
$642$	−5825.15	−0.358100
$643$	8310.78	0.509713	0.254856	−	0.966979i	$-0.417972\pi$
$643$	8310.78	0.509713	0.254856	+	0.966979i	$0.417972\pi$
$644$	644.000	0.0394055
$645$	819.127	0.0500048
$646$	−6890.55	−0.419667
$647$	−30221.6	−1.83637	−0.918186	−	0.396150i	$-0.870346\pi$
$647$	−30221.6	−1.83637	−0.918186	+	0.396150i	$0.870346\pi$
$648$	648.000	0.0392837
$649$	26201.8	1.58476
$650$	17544.0	1.05867
$651$	1395.36	0.0840066
$652$	−8189.73	−0.491924
$653$	−9024.65	−0.540830	−0.270415	−	0.962744i	$-0.587161\pi$
$653$	−9024.65	−0.540830	−0.270415	+	0.962744i	$0.587161\pi$
$654$	11652.3	0.696702
$655$	−1915.12	−0.114244
$656$	4777.91	0.284369
$657$	3607.82	0.214238
$658$	6380.76	0.378036
$659$	−886.903	−0.0524262	−0.0262131	−	0.999656i	$-0.508345\pi$
$659$	−886.903	−0.0524262	−0.0262131	+	0.999656i	$0.508345\pi$
$660$	−1253.78	−0.0739447
$661$	−11543.7	−0.679269	−0.339634	−	0.940558i	$-0.610303\pi$
$661$	−11543.7	−0.679269	−0.339634	+	0.940558i	$0.610303\pi$
$662$	9469.69	0.555966
$663$	−22690.9	−1.32917
$664$	−7897.32	−0.461560
$665$	370.577	0.0216095
$666$	−5302.40	−0.308504
$667$	6176.80	0.358571
$668$	10270.9	0.594901
$669$	−8675.07	−0.501342
$670$	2222.08	0.128129
$671$	42454.8	2.44255
$672$	672.000	0.0385758
$673$	9913.98	0.567839	0.283920	−	0.958848i	$-0.408365\pi$
$673$	9913.98	0.567839	0.283920	+	0.958848i	$0.408365\pi$
$674$	−34.4594	−0.00196933
$675$	−3304.03	−0.188403
$676$	11766.1	0.669441
$677$	−13230.4	−0.751085	−0.375542	−	0.926805i	$-0.622544\pi$
$677$	−13230.4	−0.751085	−0.375542	+	0.926805i	$0.622544\pi$
$678$	−4873.30	−0.276044
$679$	8107.19	0.458211
$680$	−1368.57	−0.0771799
$681$	−5641.76	−0.317464
$682$	8563.88	0.480833
$683$	4436.52	0.248549	0.124274	−	0.992248i	$-0.460340\pi$
$683$	4436.52	0.248549	0.124274	+	0.992248i	$0.460340\pi$
$684$	−1175.48	−0.0657100
$685$	678.294	0.0378340
$686$	686.000	0.0381802
$687$	−5377.96	−0.298663
$688$	−2694.53	−0.149314
$689$	38732.7	2.14165
$690$	−223.741	−0.0123445
$691$	30773.4	1.69418	0.847088	−	0.531453i	$-0.178354\pi$
$691$	30773.4	1.69418	0.847088	+	0.531453i	$0.178354\pi$
$692$	−13927.9	−0.765117
$693$	4059.90	0.222544
$694$	−17053.5	−0.932772
$695$	495.466	0.0270418
$696$	6445.36	0.351021
$697$	31508.6	1.71230
$698$	−10230.9	−0.554795
$699$	−11455.9	−0.619887
$700$	−3426.40	−0.185008
$701$	25125.6	1.35375	0.676876	−	0.736097i	$-0.263333\pi$
$701$	25125.6	1.35375	0.676876	+	0.736097i	$0.263333\pi$
$702$	−3870.91	−0.208117
$703$	9618.62	0.516036
$704$	4124.34	0.220798
$705$	−2216.83	−0.118426
$706$	19627.7	1.04632
$707$	−6242.44	−0.332066
$708$	4879.07	0.258993
$709$	10421.3	0.552018	0.276009	−	0.961155i	$-0.410988\pi$
$709$	10421.3	0.552018	0.276009	+	0.961155i	$0.410988\pi$
$710$	−8.92449	−0.000471733 0
$711$	6903.89	0.364158
$712$	5518.88	0.290490
$713$	1528.25	0.0802711
$714$	4431.60	0.232281
$715$	7489.64	0.391743
$716$	6414.19	0.334790
$717$	5516.96	0.287357
$718$	7759.29	0.403306
$719$	230.442	0.0119528	0.00597639	−	0.999982i	$-0.498098\pi$
$719$	230.442	0.0119528	0.00597639	+	0.999982i	$0.498098\pi$
$720$	−233.469	−0.0120846
$721$	11413.3	0.589536
$722$	−11585.7	−0.597193
$723$	−5864.30	−0.301654
$724$	−2597.76	−0.133349
$725$	−32863.6	−1.68348
$726$	16931.3	0.865536
$727$	31888.3	1.62679	0.813393	−	0.581715i	$-0.197618\pi$
$727$	31888.3	1.62679	0.813393	+	0.581715i	$0.197618\pi$
$728$	−4014.28	−0.204367
$729$	729.000	0.0370370
$730$	−1299.87	−0.0659045
$731$	−17769.5	−0.899079
$732$	7905.57	0.399178
$733$	−19424.9	−0.978818	−0.489409	−	0.872054i	$-0.662787\pi$
$733$	−19424.9	−0.978818	−0.489409	+	0.872054i	$0.662787\pi$
$734$	−19920.3	−1.00173
$735$	−238.333	−0.0119606
$736$	736.000	0.0368605
$737$	−44160.8	−2.20717
$738$	5375.15	0.268106
$739$	−8537.90	−0.424995	−0.212498	−	0.977162i	$-0.568160\pi$
$739$	−8537.90	−0.424995	−0.212498	+	0.977162i	$0.568160\pi$
$740$	1910.41	0.0949028
$741$	7021.88	0.348118
$742$	−7564.62	−0.374267
$743$	18270.2	0.902110	0.451055	−	0.892496i	$-0.351048\pi$
$743$	18270.2	0.902110	0.451055	+	0.892496i	$0.351048\pi$
$744$	1594.69	0.0785810
$745$	−825.690	−0.0406053
$746$	−21863.1	−1.07301
$747$	−8884.49	−0.435163
$748$	27198.5	1.32952
$749$	−6796.01	−0.331537
$750$	2406.40	0.117159
$751$	18329.5	0.890615	0.445308	−	0.895378i	$-0.353094\pi$
$751$	18329.5	0.890615	0.445308	+	0.895378i	$0.353094\pi$
$752$	7292.30	0.353621
$753$	4858.51	0.235131
$754$	−38502.2	−1.85964
$755$	5668.56	0.273245
$756$	756.000	0.0363696
$757$	−14063.2	−0.675211	−0.337606	−	0.941288i	$-0.609617\pi$
$757$	−14063.2	−0.675211	−0.337606	+	0.941288i	$0.609617\pi$
$758$	21195.8	1.01566
$759$	4446.56	0.212648
$760$	423.516	0.0202139
$761$	−16590.0	−0.790258	−0.395129	−	0.918626i	$-0.629300\pi$
$761$	−16590.0	−0.790258	−0.395129	+	0.918626i	$0.629300\pi$
$762$	−16165.0	−0.768497
$763$	13594.4	0.645020
$764$	−7086.32	−0.335568
$765$	−1539.64	−0.0727659
$766$	−3649.88	−0.172161
$767$	−29145.7	−1.37209
$768$	768.000	0.0360844
$769$	−26273.8	−1.23207	−0.616033	−	0.787720i	$-0.711261\pi$
$769$	−26273.8	−1.23207	−0.616033	+	0.787720i	$0.711261\pi$
$770$	−1462.75	−0.0684595
$771$	10498.6	0.490397
$772$	8996.51	0.419419
$773$	17000.3	0.791021	0.395511	−	0.918461i	$-0.370568\pi$
$773$	17000.3	0.791021	0.395511	+	0.918461i	$0.370568\pi$
$774$	−3031.35	−0.140775
$775$	−8131.03	−0.376871
$776$	9265.36	0.428617
$777$	−6186.13	−0.285619
$778$	7803.47	0.359599
$779$	−9750.60	−0.448462
$780$	1394.66	0.0640214
$781$	177.362	0.00812615
$782$	4853.65	0.221952
$783$	7251.03	0.330946
$784$	784.000	0.0357143
$785$	3125.08	0.142088
$786$	7087.29	0.321623
$787$	17376.7	0.787054	0.393527	−	0.919313i	$-0.371255\pi$
$787$	17376.7	0.787054	0.393527	+	0.919313i	$0.371255\pi$
$788$	12977.4	0.586674
$789$	2507.96	0.113163
$790$	−2487.42	−0.112023
$791$	−5685.52	−0.255567
$792$	4639.89	0.208171
$793$	−47224.9	−2.11476
$794$	−27041.7	−1.20866
$795$	2628.13	0.117246
$796$	−2888.90	−0.128636
$797$	25283.3	1.12369	0.561845	−	0.827243i	$-0.310092\pi$
$797$	25283.3	1.12369	0.561845	+	0.827243i	$0.310092\pi$
$798$	−1371.39	−0.0608356
$799$	48090.1	2.12929
$800$	−3915.88	−0.173059
$801$	6208.74	0.273876
$802$	27259.3	1.20020
$803$	25833.1	1.13528
$804$	−8223.25	−0.360711
$805$	−261.031	−0.0114288
$806$	−9526.10	−0.416306
$807$	13433.4	0.585970
$808$	−7134.21	−0.310620
$809$	−17843.5	−0.775456	−0.387728	−	0.921774i	$-0.626740\pi$
$809$	−17843.5	−0.775456	−0.387728	+	0.921774i	$0.626740\pi$
$810$	−262.653	−0.0113934
$811$	15639.7	0.677171	0.338586	−	0.940936i	$-0.390052\pi$
$811$	15639.7	0.677171	0.338586	+	0.940936i	$0.390052\pi$
$812$	7519.59	0.324982
$813$	−14612.4	−0.630355
$814$	−37966.9	−1.63481
$815$	3319.53	0.142672
$816$	5064.68	0.217279
$817$	5498.91	0.235474
$818$	−8342.73	−0.356598
$819$	−4516.06	−0.192679
$820$	−1936.62	−0.0824754
$821$	30640.3	1.30250	0.651251	−	0.758862i	$-0.274244\pi$
$821$	30640.3	1.30250	0.651251	+	0.758862i	$0.274244\pi$
$822$	−2510.16	−0.106511
$823$	28366.2	1.20144	0.600719	−	0.799461i	$-0.294881\pi$
$823$	28366.2	1.20144	0.600719	+	0.799461i	$0.294881\pi$
$824$	13043.8	0.551460
$825$	−23657.9	−0.998378
$826$	5692.25	0.239781
$827$	36775.5	1.54632	0.773161	−	0.634210i	$-0.218674\pi$
$827$	36775.5	1.54632	0.773161	+	0.634210i	$0.218674\pi$
$828$	828.000	0.0347524
$829$	10331.0	0.432822	0.216411	−	0.976302i	$-0.430565\pi$
$829$	10331.0	0.432822	0.216411	+	0.976302i	$0.430565\pi$
$830$	3201.01	0.133866
$831$	−23961.0	−1.00024
$832$	−4587.74	−0.191168
$833$	5170.20	0.215050
$834$	−1833.57	−0.0761288
$835$	−4163.09	−0.172539
$836$	−8416.82	−0.348208
$837$	1794.03	0.0740869
$838$	−10361.5	−0.427128
$839$	4181.83	0.172077	0.0860386	−	0.996292i	$-0.472579\pi$
$839$	4181.83	0.172077	0.0860386	+	0.996292i	$0.472579\pi$
$840$	−272.381	−0.0111881
$841$	47733.7	1.95718
$842$	−22578.4	−0.924112
$843$	7492.50	0.306116
$844$	−11073.0	−0.451596
$845$	−4769.13	−0.194157
$846$	8203.83	0.333397
$847$	19753.2	0.801330
$848$	−8645.28	−0.350094
$849$	11607.1	0.469205
$850$	−25823.8	−1.04206
$851$	−6775.29	−0.272919
$852$	33.0269	0.00132803
$853$	21152.1	0.849044	0.424522	−	0.905418i	$-0.360442\pi$
$853$	21152.1	0.849044	0.424522	+	0.905418i	$0.360442\pi$
$854$	9223.17	0.369567
$855$	476.456	0.0190578
$856$	−7766.87	−0.310124
$857$	41901.4	1.67016	0.835079	−	0.550129i	$-0.185422\pi$
$857$	41901.4	1.67016	0.835079	+	0.550129i	$0.185422\pi$
$858$	−27716.9	−1.10284
$859$	−9532.34	−0.378626	−0.189313	−	0.981917i	$-0.560626\pi$
$859$	−9532.34	−0.378626	−0.189313	+	0.981917i	$0.560626\pi$
$860$	1092.17	0.0433054
$861$	6271.01	0.248218
$862$	−29256.6	−1.15601
$863$	34341.8	1.35459	0.677294	−	0.735712i	$-0.263152\pi$
$863$	34341.8	1.35459	0.677294	+	0.735712i	$0.263152\pi$
$864$	864.000	0.0340207
$865$	5645.39	0.221906
$866$	−3171.14	−0.124434
$867$	18660.7	0.730971
$868$	1860.47	0.0727519
$869$	49434.1	1.92973
$870$	−2612.49	−0.101806
$871$	49122.6	1.91097
$872$	15536.5	0.603361
$873$	10423.5	0.404104
$874$	−1502.00	−0.0581305
$875$	2807.46	0.108468
$876$	4810.43	0.185536
$877$	19570.0	0.753513	0.376756	−	0.926312i	$-0.377039\pi$
$877$	19570.0	0.753513	0.376756	+	0.926312i	$0.377039\pi$
$878$	3711.64	0.142667
$879$	−14245.8	−0.546644
$880$	−1671.71	−0.0640380
$881$	−36906.6	−1.41137	−0.705683	−	0.708527i	$-0.749360\pi$
$881$	−36906.6	−1.41137	−0.705683	+	0.708527i	$0.749360\pi$
$882$	882.000	0.0336718
$883$	29928.2	1.14062	0.570309	−	0.821431i	$-0.306824\pi$
$883$	29928.2	1.14062	0.570309	+	0.821431i	$0.306824\pi$
$884$	−30254.5	−1.15110
$885$	−1977.63	−0.0751154
$886$	9827.99	0.372661
$887$	22080.1	0.835826	0.417913	−	0.908487i	$-0.362762\pi$
$887$	22080.1	0.835826	0.417913	+	0.908487i	$0.362762\pi$
$888$	−7069.86	−0.267172
$889$	−18859.1	−0.711490
$890$	−2236.96	−0.0842505
$891$	5219.87	0.196265
$892$	−11566.8	−0.434175
$893$	−14881.9	−0.557674
$894$	3055.63	0.114313
$895$	−2599.85	−0.0970988
$896$	896.000	0.0334077
$897$	−4946.16	−0.184111
$898$	4153.79	0.154358
$899$	17844.4	0.662006
$900$	−4405.37	−0.163162
$901$	−57012.5	−2.10806
$902$	38487.8	1.42074
$903$	−3536.57	−0.130332
$904$	−6497.73	−0.239061
$905$	1052.94	0.0386752
$906$	−20977.7	−0.769245
$907$	32770.2	1.19969	0.599843	−	0.800117i	$-0.295230\pi$
$907$	32770.2	1.19969	0.599843	+	0.800117i	$0.295230\pi$
$908$	−7522.35	−0.274932
$909$	−8025.99	−0.292855
$910$	1627.10	0.0592723
$911$	788.100	0.0286618	0.0143309	−	0.999897i	$-0.495438\pi$
$911$	788.100	0.0286618	0.0143309	+	0.999897i	$0.495438\pi$
$912$	−1567.31	−0.0569065
$913$	−63615.8	−2.30600
$914$	−19491.9	−0.705398
$915$	−3204.35	−0.115773
$916$	−7170.61	−0.258650
$917$	8268.51	0.297765
$918$	5697.77	0.204852
$919$	−35723.4	−1.28227	−0.641135	−	0.767428i	$-0.721536\pi$
$919$	−35723.4	−1.28227	−0.641135	+	0.767428i	$0.721536\pi$
$920$	−298.322	−0.0106906
$921$	−11438.8	−0.409253
$922$	−29823.4	−1.06527
$923$	−197.290	−0.00703564
$924$	5413.20	0.192729
$925$	36047.9	1.28135
$926$	−9288.51	−0.329632
$927$	14674.3	0.519922
$928$	8593.81	0.303993
$929$	3988.73	0.140867	0.0704337	−	0.997516i	$-0.477562\pi$
$929$	3988.73	0.140867	0.0704337	+	0.997516i	$0.477562\pi$
$930$	−646.374	−0.0227908
$931$	−1599.96	−0.0563229
$932$	−15274.5	−0.536838
$933$	−17231.7	−0.604652
$934$	22992.2	0.805491
$935$	−11024.3	−0.385598
$936$	−5161.21	−0.180234
$937$	−14247.7	−0.496748	−0.248374	−	0.968664i	$-0.579896\pi$
$937$	−14247.7	−0.496748	−0.248374	+	0.968664i	$0.579896\pi$
$938$	−9593.80	−0.333954
$939$	−5219.14	−0.181385
$940$	−2955.77	−0.102560
$941$	38547.3	1.33539	0.667697	−	0.744434i	$-0.267280\pi$
$941$	38547.3	1.33539	0.667697	+	0.744434i	$0.267280\pi$
$942$	−11565.0	−0.400008
$943$	6868.25	0.237180
$944$	6505.43	0.224294
$945$	−306.428	−0.0105483
$946$	−21705.4	−0.745987
$947$	12852.0	0.441007	0.220503	−	0.975386i	$-0.429230\pi$
$947$	12852.0	0.441007	0.220503	+	0.975386i	$0.429230\pi$
$948$	9205.19	0.315370
$949$	−28735.7	−0.982929
$950$	7991.40	0.272921
$951$	−24361.9	−0.830691
$952$	5908.79	0.201161
$953$	−12220.5	−0.415382	−0.207691	−	0.978194i	$-0.566595\pi$
$953$	−12220.5	−0.415382	−0.207691	+	0.978194i	$0.566595\pi$
$954$	−9725.94	−0.330072
$955$	2872.28	0.0973245
$956$	7355.95	0.248858
$957$	51919.7	1.75374
$958$	−26635.4	−0.898277
$959$	−2928.52	−0.0986100
$960$	−311.292	−0.0104655
$961$	−25376.0	−0.851801
$962$	42232.7	1.41542
$963$	−8737.73	−0.292388
$964$	−7819.06	−0.261240
$965$	−3646.54	−0.121644
$966$	966.000	0.0321745
$967$	18248.8	0.606867	0.303434	−	0.952853i	$-0.401867\pi$
$967$	18248.8	0.606867	0.303434	+	0.952853i	$0.401867\pi$
$968$	22575.1	0.749576
$969$	−10335.8	−0.342657
$970$	−3755.51	−0.124312
$971$	58244.6	1.92498	0.962492	−	0.271311i	$-0.0874572\pi$
$971$	58244.6	1.92498	0.962492	+	0.271311i	$0.0874572\pi$
$972$	972.000	0.0320750
$973$	−2139.17	−0.0704816
$974$	−1369.38	−0.0450491
$975$	26316.0	0.864397
$976$	10540.8	0.345698
$977$	−29401.3	−0.962774	−0.481387	−	0.876508i	$-0.659867\pi$
$977$	−29401.3	−0.962774	−0.481387	+	0.876508i	$0.659867\pi$
$978$	−12284.6	−0.401654
$979$	44456.5	1.45131
$980$	−317.777	−0.0103582
$981$	17478.5	0.568855
$982$	12942.8	0.420591
$983$	18062.6	0.586070	0.293035	−	0.956102i	$-0.405335\pi$
$983$	18062.6	0.586070	0.293035	+	0.956102i	$0.405335\pi$
$984$	7166.87	0.232186
$985$	−5260.09	−0.170153
$986$	56673.1	1.83046
$987$	9571.14	0.308665
$988$	9362.51	0.301479
$989$	−3873.39	−0.124536
$990$	−1880.68	−0.0603756
$991$	7920.72	0.253895	0.126947	−	0.991909i	$-0.459482\pi$
$991$	7920.72	0.253895	0.126947	+	0.991909i	$0.459482\pi$
$992$	2126.26	0.0680532
$993$	14204.5	0.453945
$994$	38.5314	0.00122952
$995$	1170.95	0.0373082
$996$	−11846.0	−0.376862
$997$	15559.2	0.494248	0.247124	−	0.968984i	$-0.420515\pi$
$997$	15559.2	0.494248	0.247124	+	0.968984i	$0.420515\pi$
$998$	−38608.7	−1.22459
$999$	−7953.60	−0.251893

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.4.a.r.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.4.a.r.1.3	✓	6	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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -1.62131 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -1.62131 q^{5} +6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -3.24263 q^{10} +64.4429 q^{11} +12.0000 q^{12} -71.6835 q^{13} +14.0000 q^{14} -4.86394 q^{15} +16.0000 q^{16} +105.514 q^{17} +18.0000 q^{18} -32.6522 q^{19} -6.48525 q^{20} +21.0000 q^{21} +128.886 q^{22} +23.0000 q^{23} +24.0000 q^{24} -122.371 q^{25} -143.367 q^{26} +27.0000 q^{27} +28.0000 q^{28} +268.557 q^{29} -9.72788 q^{30} +66.4455 q^{31} +32.0000 q^{32} +193.329 q^{33} +211.028 q^{34} -11.3492 q^{35} +36.0000 q^{36} -294.578 q^{37} -65.3045 q^{38} -215.050 q^{39} -12.9705 q^{40} +298.620 q^{41} +42.0000 q^{42} -168.408 q^{43} +257.771 q^{44} -14.5918 q^{45} +46.0000 q^{46} +455.769 q^{47} +48.0000 q^{48} +49.0000 q^{49} -244.743 q^{50} +316.543 q^{51} -286.734 q^{52} -540.330 q^{53} +54.0000 q^{54} -104.482 q^{55} +56.0000 q^{56} -97.9567 q^{57} +537.113 q^{58} +406.589 q^{59} -19.4558 q^{60} +658.798 q^{61} +132.891 q^{62} +63.0000 q^{63} +64.0000 q^{64} +116.221 q^{65} +386.657 q^{66} -685.271 q^{67} +422.057 q^{68} +69.0000 q^{69} -22.6984 q^{70} +2.75224 q^{71} +72.0000 q^{72} +400.869 q^{73} -589.155 q^{74} -367.114 q^{75} -130.609 q^{76} +451.100 q^{77} -430.101 q^{78} +767.099 q^{79} -25.9410 q^{80} +81.0000 q^{81} +597.239 q^{82} -987.166 q^{83} +84.0000 q^{84} -171.072 q^{85} -336.816 q^{86} +805.670 q^{87} +515.543 q^{88} +689.859 q^{89} -29.1836 q^{90} -501.784 q^{91} +92.0000 q^{92} +199.337 q^{93} +911.537 q^{94} +52.9395 q^{95} +96.0000 q^{96} +1158.17 q^{97} +98.0000 q^{98} +579.986 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 42 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 42 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} + 18 q^{3} + 24 q^{4} + 20 q^{5} + 36 q^{6} + 42 q^{7} + 48 q^{8} + 54 q^{9} + 40 q^{10} + 79 q^{11} + 72 q^{12} + 52 q^{13} + 84 q^{14} + 60 q^{15} + 96 q^{16} + 140 q^{17} + 108 q^{18} + 93 q^{19} + 80 q^{20} + 126 q^{21} + 158 q^{22} + 138 q^{23} + 144 q^{24} + 142 q^{25} + 104 q^{26} + 162 q^{27} + 168 q^{28} + 143 q^{29} + 120 q^{30} + 130 q^{31} + 192 q^{32} + 237 q^{33} + 280 q^{34} + 140 q^{35} + 216 q^{36} + 151 q^{37} + 186 q^{38} + 156 q^{39} + 160 q^{40} + 412 q^{41} + 252 q^{42} + 250 q^{43} + 316 q^{44} + 180 q^{45} + 276 q^{46} + 666 q^{47} + 288 q^{48} + 294 q^{49} + 284 q^{50} + 420 q^{51} + 208 q^{52} - 96 q^{53} + 324 q^{54} + 51 q^{55} + 336 q^{56} + 279 q^{57} + 286 q^{58} + 514 q^{59} + 240 q^{60} + 422 q^{61} + 260 q^{62} + 378 q^{63} + 384 q^{64} + 277 q^{65} + 474 q^{66} + 669 q^{67} + 560 q^{68} + 414 q^{69} + 280 q^{70} - 357 q^{71} + 432 q^{72} + 430 q^{73} + 302 q^{74} + 426 q^{75} + 372 q^{76} + 553 q^{77} + 312 q^{78} + 750 q^{79} + 320 q^{80} + 486 q^{81} + 824 q^{82} + 222 q^{83} + 504 q^{84} + 601 q^{85} + 500 q^{86} + 429 q^{87} + 632 q^{88} + 763 q^{89} + 360 q^{90} + 364 q^{91} + 552 q^{92} + 390 q^{93} + 1332 q^{94} - 541 q^{95} + 576 q^{96} + 575 q^{97} + 588 q^{98} + 711 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 18 * q^3 + 24 * q^4 + 20 * q^5 + 36 * q^6 + 42 * q^7 + 48 * q^8 + 54 * q^9 + 40 * q^10 + 79 * q^11 + 72 * q^12 + 52 * q^13 + 84 * q^14 + 60 * q^15 + 96 * q^16 + 140 * q^17 + 108 * q^18 + 93 * q^19 + 80 * q^20 + 126 * q^21 + 158 * q^22 + 138 * q^23 + 144 * q^24 + 142 * q^25 + 104 * q^26 + 162 * q^27 + 168 * q^28 + 143 * q^29 + 120 * q^30 + 130 * q^31 + 192 * q^32 + 237 * q^33 + 280 * q^34 + 140 * q^35 + 216 * q^36 + 151 * q^37 + 186 * q^38 + 156 * q^39 + 160 * q^40 + 412 * q^41 + 252 * q^42 + 250 * q^43 + 316 * q^44 + 180 * q^45 + 276 * q^46 + 666 * q^47 + 288 * q^48 + 294 * q^49 + 284 * q^50 + 420 * q^51 + 208 * q^52 - 96 * q^53 + 324 * q^54 + 51 * q^55 + 336 * q^56 + 279 * q^57 + 286 * q^58 + 514 * q^59 + 240 * q^60 + 422 * q^61 + 260 * q^62 + 378 * q^63 + 384 * q^64 + 277 * q^65 + 474 * q^66 + 669 * q^67 + 560 * q^68 + 414 * q^69 + 280 * q^70 - 357 * q^71 + 432 * q^72 + 430 * q^73 + 302 * q^74 + 426 * q^75 + 372 * q^76 + 553 * q^77 + 312 * q^78 + 750 * q^79 + 320 * q^80 + 486 * q^81 + 824 * q^82 + 222 * q^83 + 504 * q^84 + 601 * q^85 + 500 * q^86 + 429 * q^87 + 632 * q^88 + 763 * q^89 + 360 * q^90 + 364 * q^91 + 552 * q^92 + 390 * q^93 + 1332 * q^94 - 541 * q^95 + 576 * q^96 + 575 * q^97 + 588 * q^98 + 711 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more