LMFDB - Embedded newform 2166.4.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 350x^{6} + 948x^{5} + 37019x^{4} - 115308x^{3} - 1098530x^{2} + 2724222x + 7883581 $ x^8 - 2x^7 - 350x^6 + 948x^5 + 37019x^4 - 115308x^3 - 1098530x^2 + 2724222*x + 7883581
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 19 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$12.0376$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.38845 q^{5} -6.00000 q^{6} +16.0395 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.38845 q^{5} -6.00000 q^{6} +16.0395 q^{7} +8.00000 q^{8} +9.00000 q^{9} -16.7769 q^{10} -27.9688 q^{11} -12.0000 q^{12} +32.6032 q^{13} +32.0791 q^{14} +25.1654 q^{15} +16.0000 q^{16} -19.2640 q^{17} +18.0000 q^{18} -33.5538 q^{20} -48.1186 q^{21} -55.9375 q^{22} +79.7464 q^{23} -24.0000 q^{24} -54.6339 q^{25} +65.2064 q^{26} -27.0000 q^{27} +64.1582 q^{28} -141.470 q^{29} +50.3307 q^{30} -167.373 q^{31} +32.0000 q^{32} +83.9063 q^{33} -38.5279 q^{34} -134.547 q^{35} +36.0000 q^{36} -55.2319 q^{37} -97.8096 q^{39} -67.1076 q^{40} -42.0654 q^{41} -96.2373 q^{42} +110.445 q^{43} -111.875 q^{44} -75.4961 q^{45} +159.493 q^{46} +268.524 q^{47} -48.0000 q^{48} -85.7330 q^{49} -109.268 q^{50} +57.7919 q^{51} +130.413 q^{52} +111.653 q^{53} -54.0000 q^{54} +234.615 q^{55} +128.316 q^{56} -282.941 q^{58} +644.590 q^{59} +100.661 q^{60} +360.293 q^{61} -334.747 q^{62} +144.356 q^{63} +64.0000 q^{64} -273.490 q^{65} +167.813 q^{66} +94.7662 q^{67} -77.0559 q^{68} -239.239 q^{69} -269.094 q^{70} +1185.73 q^{71} +72.0000 q^{72} -817.058 q^{73} -110.464 q^{74} +163.902 q^{75} -448.606 q^{77} -195.619 q^{78} -970.784 q^{79} -134.215 q^{80} +81.0000 q^{81} -84.1309 q^{82} -1136.70 q^{83} -192.475 q^{84} +161.595 q^{85} +220.889 q^{86} +424.411 q^{87} -223.750 q^{88} -93.7280 q^{89} -150.992 q^{90} +522.940 q^{91} +318.986 q^{92} +502.120 q^{93} +537.048 q^{94} -96.0000 q^{96} +554.985 q^{97} -171.466 q^{98} -251.719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9	\( 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9} - 60 q^{10} - 6 q^{11} - 96 q^{12} + 48 q^{13} - 68 q^{14} + 90 q^{15} + 128 q^{16} - 36 q^{17} + 144 q^{18} - 120 q^{20} + 102 q^{21} - 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} - 216 q^{27} - 136 q^{28} - 380 q^{29} + 180 q^{30} - 48 q^{31} + 256 q^{32} + 18 q^{33} - 72 q^{34} + 762 q^{35} + 288 q^{36} - 168 q^{37} - 144 q^{39} - 240 q^{40} + 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} - 564 q^{46} - 468 q^{47} - 384 q^{48} + 222 q^{49} + 792 q^{50} + 108 q^{51} + 192 q^{52} + 1682 q^{53} - 432 q^{54} - 46 q^{55} - 272 q^{56} - 760 q^{58} + 292 q^{59} + 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} - 1120 q^{65} + 36 q^{66} - 2484 q^{67} - 144 q^{68} + 846 q^{69} + 1524 q^{70} + 1182 q^{71} + 576 q^{72} + 182 q^{73} - 336 q^{74} - 1188 q^{75} - 504 q^{77} - 288 q^{78} - 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} + 408 q^{84} - 2238 q^{85} - 1576 q^{86} + 1140 q^{87} - 48 q^{88} + 1304 q^{89} - 540 q^{90} - 624 q^{91} - 1128 q^{92} + 144 q^{93} - 936 q^{94} - 768 q^{96} - 1248 q^{97} + 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9 - 60 * q^10 - 6 * q^11 - 96 * q^12 + 48 * q^13 - 68 * q^14 + 90 * q^15 + 128 * q^16 - 36 * q^17 + 144 * q^18 - 120 * q^20 + 102 * q^21 - 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 - 216 * q^27 - 136 * q^28 - 380 * q^29 + 180 * q^30 - 48 * q^31 + 256 * q^32 + 18 * q^33 - 72 * q^34 + 762 * q^35 + 288 * q^36 - 168 * q^37 - 144 * q^39 - 240 * q^40 + 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 - 564 * q^46 - 468 * q^47 - 384 * q^48 + 222 * q^49 + 792 * q^50 + 108 * q^51 + 192 * q^52 + 1682 * q^53 - 432 * q^54 - 46 * q^55 - 272 * q^56 - 760 * q^58 + 292 * q^59 + 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 - 1120 * q^65 + 36 * q^66 - 2484 * q^67 - 144 * q^68 + 846 * q^69 + 1524 * q^70 + 1182 * q^71 + 576 * q^72 + 182 * q^73 - 336 * q^74 - 1188 * q^75 - 504 * q^77 - 288 * q^78 - 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 + 408 * q^84 - 2238 * q^85 - 1576 * q^86 + 1140 * q^87 - 48 * q^88 + 1304 * q^89 - 540 * q^90 - 624 * q^91 - 1128 * q^92 + 144 * q^93 - 936 * q^94 - 768 * q^96 - 1248 * q^97 + 444 * q^98 - 54 * 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
$5$	−8.38845	−0.750286	−0.375143	−	0.926967i	$-0.622406\pi$
$5$	−8.38845	−0.750286	−0.375143	+	0.926967i	$0.622406\pi$
$6$	−6.00000	−0.408248
$7$	16.0395	0.866054	0.433027	−	0.901381i	$-0.357445\pi$
$7$	16.0395	0.866054	0.433027	+	0.901381i	$0.357445\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−16.7769	−0.530532
$11$	−27.9688	−0.766627	−0.383314	−	0.923618i	$-0.625217\pi$
$11$	−27.9688	−0.766627	−0.383314	+	0.923618i	$0.625217\pi$
$12$	−12.0000	−0.288675
$13$	32.6032	0.695577	0.347788	−	0.937573i	$-0.386933\pi$
$13$	32.6032	0.695577	0.347788	+	0.937573i	$0.386933\pi$
$14$	32.0791	0.612393
$15$	25.1654	0.433178
$16$	16.0000	0.250000
$17$	−19.2640	−0.274835	−0.137418	−	0.990513i	$-0.543880\pi$
$17$	−19.2640	−0.274835	−0.137418	+	0.990513i	$0.543880\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−33.5538	−0.375143
$21$	−48.1186	−0.500017
$22$	−55.9375	−0.542087
$23$	79.7464	0.722969	0.361484	−	0.932378i	$-0.382270\pi$
$23$	79.7464	0.722969	0.361484	+	0.932378i	$0.382270\pi$
$24$	−24.0000	−0.204124
$25$	−54.6339	−0.437071
$26$	65.2064	0.491847
$27$	−27.0000	−0.192450
$28$	64.1582	0.433027
$29$	−141.470	−0.905876	−0.452938	−	0.891542i	$-0.649624\pi$
$29$	−141.470	−0.905876	−0.452938	+	0.891542i	$0.649624\pi$
$30$	50.3307	0.306303
$31$	−167.373	−0.969715	−0.484858	−	0.874593i	$-0.661129\pi$
$31$	−167.373	−0.969715	−0.484858	+	0.874593i	$0.661129\pi$
$32$	32.0000	0.176777
$33$	83.9063	0.442612
$34$	−38.5279	−0.194338
$35$	−134.547	−0.649788
$36$	36.0000	0.166667
$37$	−55.2319	−0.245407	−0.122704	−	0.992443i	$-0.539156\pi$
$37$	−55.2319	−0.245407	−0.122704	+	0.992443i	$0.539156\pi$
$38$	0	0
$39$	−97.8096	−0.401591
$40$	−67.1076	−0.265266
$41$	−42.0654	−0.160232	−0.0801161	−	0.996786i	$-0.525529\pi$
$41$	−42.0654	−0.160232	−0.0801161	+	0.996786i	$0.525529\pi$
$42$	−96.2373	−0.353565
$43$	110.445	0.391690	0.195845	−	0.980635i	$-0.437255\pi$
$43$	110.445	0.391690	0.195845	+	0.980635i	$0.437255\pi$
$44$	−111.875	−0.383314
$45$	−75.4961	−0.250095
$46$	159.493	0.511216
$47$	268.524	0.833367	0.416684	−	0.909052i	$-0.363192\pi$
$47$	268.524	0.833367	0.416684	+	0.909052i	$0.363192\pi$
$48$	−48.0000	−0.144338
$49$	−85.7330	−0.249950
$50$	−109.268	−0.309056
$51$	57.7919	0.158676
$52$	130.413	0.347788
$53$	111.653	0.289371	0.144685	−	0.989478i	$-0.453783\pi$
$53$	111.653	0.289371	0.144685	+	0.989478i	$0.453783\pi$
$54$	−54.0000	−0.136083
$55$	234.615	0.575189
$56$	128.316	0.306196
$57$	0	0
$58$	−282.941	−0.640551
$59$	644.590	1.42235	0.711174	−	0.703016i	$-0.248164\pi$
$59$	644.590	1.42235	0.711174	+	0.703016i	$0.248164\pi$
$60$	100.661	0.216589
$61$	360.293	0.756242	0.378121	−	0.925756i	$-0.376570\pi$
$61$	360.293	0.756242	0.378121	+	0.925756i	$0.376570\pi$
$62$	−334.747	−0.685692
$63$	144.356	0.288685
$64$	64.0000	0.125000
$65$	−273.490	−0.521881
$66$	167.813	0.312974
$67$	94.7662	0.172799	0.0863995	−	0.996261i	$-0.472464\pi$
$67$	94.7662	0.172799	0.0863995	+	0.996261i	$0.472464\pi$
$68$	−77.0559	−0.137418
$69$	−239.239	−0.417406
$70$	−269.094	−0.459470
$71$	1185.73	1.98198	0.990991	−	0.133925i	$-0.0427582\pi$
$71$	1185.73	1.98198	0.990991	+	0.133925i	$0.0427582\pi$
$72$	72.0000	0.117851
$73$	−817.058	−1.30999	−0.654996	−	0.755632i	$-0.727330\pi$
$73$	−817.058	−1.30999	−0.654996	+	0.755632i	$0.727330\pi$
$74$	−110.464	−0.173529
$75$	163.902	0.252343
$76$	0	0
$77$	−448.606	−0.663941
$78$	−195.619	−0.283968
$79$	−970.784	−1.38255	−0.691277	−	0.722590i	$-0.742952\pi$
$79$	−970.784	−1.38255	−0.691277	+	0.722590i	$0.742952\pi$
$80$	−134.215	−0.187571
$81$	81.0000	0.111111
$82$	−84.1309	−0.113301
$83$	−1136.70	−1.50324	−0.751618	−	0.659598i	$-0.770726\pi$
$83$	−1136.70	−1.50324	−0.751618	+	0.659598i	$0.770726\pi$
$84$	−192.475	−0.250008
$85$	161.595	0.206205
$86$	220.889	0.276966
$87$	424.411	0.523008
$88$	−223.750	−0.271044
$89$	−93.7280	−0.111631	−0.0558154	−	0.998441i	$-0.517776\pi$
$89$	−93.7280	−0.111631	−0.0558154	+	0.998441i	$0.517776\pi$
$90$	−150.992	−0.176844
$91$	522.940	0.602407
$92$	318.986	0.361484
$93$	502.120	0.559865
$94$	537.048	0.589280
$95$	0	0
$96$	−96.0000	−0.102062
$97$	554.985	0.580930	0.290465	−	0.956886i	$-0.406190\pi$
$97$	554.985	0.580930	0.290465	+	0.956886i	$0.406190\pi$
$98$	−171.466	−0.176742
$99$	−251.719	−0.255542
$100$	−218.536	−0.218536
$101$	−101.877	−0.100367	−0.0501837	−	0.998740i	$-0.515981\pi$
$101$	−101.877	−0.100367	−0.0501837	+	0.998740i	$0.515981\pi$
$102$	115.584	0.112201
$103$	−1885.45	−1.80368	−0.901841	−	0.432067i	$-0.857784\pi$
$103$	−1885.45	−1.80368	−0.901841	+	0.432067i	$0.857784\pi$
$104$	260.825	0.245923
$105$	403.641	0.375155
$106$	223.305	0.204616
$107$	644.911	0.582672	0.291336	−	0.956621i	$-0.405900\pi$
$107$	644.911	0.582672	0.291336	+	0.956621i	$0.405900\pi$
$108$	−108.000	−0.0962250
$109$	−1232.86	−1.08336	−0.541680	−	0.840584i	$-0.682212\pi$
$109$	−1232.86	−1.08336	−0.541680	+	0.840584i	$0.682212\pi$
$110$	469.229	0.406720
$111$	165.696	0.141686
$112$	256.633	0.216514
$113$	261.847	0.217987	0.108993	−	0.994042i	$-0.465237\pi$
$113$	261.847	0.217987	0.108993	+	0.994042i	$0.465237\pi$
$114$	0	0
$115$	−668.949	−0.542433
$116$	−565.882	−0.452938
$117$	293.429	0.231859
$118$	1289.18	1.00575
$119$	−308.985	−0.238022
$120$	201.323	0.153151
$121$	−548.749	−0.412283
$122$	720.586	0.534744
$123$	126.196	0.0925101
$124$	−669.494	−0.484858
$125$	1506.85	1.07821
$126$	288.712	0.204131
$127$	−2696.16	−1.88383	−0.941913	−	0.335858i	$-0.890974\pi$
$127$	−2696.16	−1.88383	−0.941913	+	0.335858i	$0.890974\pi$
$128$	128.000	0.0883883
$129$	−331.334	−0.226142
$130$	−546.980	−0.369026
$131$	−1389.43	−0.926682	−0.463341	−	0.886180i	$-0.653349\pi$
$131$	−1389.43	−0.926682	−0.463341	+	0.886180i	$0.653349\pi$
$132$	335.625	0.221306
$133$	0	0
$134$	189.532	0.122187
$135$	226.488	0.144393
$136$	−154.112	−0.0971689
$137$	−784.529	−0.489247	−0.244624	−	0.969618i	$-0.578664\pi$
$137$	−784.529	−0.489247	−0.244624	+	0.969618i	$0.578664\pi$
$138$	−478.479	−0.295151
$139$	−2714.27	−1.65627	−0.828134	−	0.560530i	$-0.810598\pi$
$139$	−2714.27	−1.65627	−0.828134	+	0.560530i	$0.810598\pi$
$140$	−538.188	−0.324894
$141$	−805.572	−0.481145
$142$	2371.47	1.40147
$143$	−911.871	−0.533248
$144$	144.000	0.0833333
$145$	1186.72	0.679666
$146$	−1634.12	−0.926305
$147$	257.199	0.144309
$148$	−220.927	−0.122704
$149$	−503.598	−0.276888	−0.138444	−	0.990370i	$-0.544210\pi$
$149$	−503.598	−0.276888	−0.138444	+	0.990370i	$0.544210\pi$
$150$	327.803	0.178434
$151$	308.359	0.166185	0.0830924	−	0.996542i	$-0.473520\pi$
$151$	308.359	0.166185	0.0830924	+	0.996542i	$0.473520\pi$
$152$	0	0
$153$	−173.376	−0.0916117
$154$	−897.212	−0.469477
$155$	1404.00	0.727563
$156$	−391.238	−0.200796
$157$	2.02349	0.00102861	0.000514307	−	1.00000i	$-0.499836\pi$
$157$	2.02349	0.00102861	0.000514307		1.00000i	$0.499836\pi$
$158$	−1941.57	−0.977613
$159$	−334.958	−0.167068
$160$	−268.430	−0.132633
$161$	1279.10	0.626130
$162$	162.000	0.0785674
$163$	−530.326	−0.254836	−0.127418	−	0.991849i	$-0.540669\pi$
$163$	−530.326	−0.254836	−0.127418	+	0.991849i	$0.540669\pi$
$164$	−168.262	−0.0801161
$165$	−703.844	−0.332086
$166$	−2273.39	−1.06295
$167$	3066.43	1.42088	0.710441	−	0.703756i	$-0.248495\pi$
$167$	3066.43	1.42088	0.710441	+	0.703756i	$0.248495\pi$
$168$	−384.949	−0.176783
$169$	−1134.03	−0.516173
$170$	323.190	0.145809
$171$	0	0
$172$	441.779	0.195845
$173$	1561.41	0.686194	0.343097	−	0.939300i	$-0.388524\pi$
$173$	1561.41	0.686194	0.343097	+	0.939300i	$0.388524\pi$
$174$	848.823	0.369822
$175$	−876.303	−0.378527
$176$	−447.500	−0.191657
$177$	−1933.77	−0.821193
$178$	−187.456	−0.0789350
$179$	1476.69	0.616610	0.308305	−	0.951287i	$-0.400238\pi$
$179$	1476.69	0.616610	0.308305	+	0.951287i	$0.400238\pi$
$180$	−301.984	−0.125048
$181$	−3585.47	−1.47241	−0.736205	−	0.676759i	$-0.763384\pi$
$181$	−3585.47	−1.47241	−0.736205	+	0.676759i	$0.763384\pi$
$182$	1045.88	0.425966
$183$	−1080.88	−0.436617
$184$	637.971	0.255608
$185$	463.310	0.184125
$186$	1004.24	0.395884
$187$	538.789	0.210696
$188$	1074.10	0.416684
$189$	−433.068	−0.166672
$190$	0	0
$191$	−2164.90	−0.820139	−0.410069	−	0.912054i	$-0.634496\pi$
$191$	−2164.90	−0.820139	−0.410069	+	0.912054i	$0.634496\pi$
$192$	−192.000	−0.0721688
$193$	2977.28	1.11041	0.555206	−	0.831713i	$-0.312639\pi$
$193$	2977.28	1.11041	0.555206	+	0.831713i	$0.312639\pi$
$194$	1109.97	0.410780
$195$	820.471	0.301308
$196$	−342.932	−0.124975
$197$	−4423.90	−1.59995	−0.799974	−	0.600035i	$-0.795153\pi$
$197$	−4423.90	−1.59995	−0.799974	+	0.600035i	$0.795153\pi$
$198$	−503.438	−0.180696
$199$	1471.38	0.524138	0.262069	−	0.965049i	$-0.415595\pi$
$199$	1471.38	0.524138	0.262069	+	0.965049i	$0.415595\pi$
$200$	−437.071	−0.154528
$201$	−284.299	−0.0997656
$202$	−203.753	−0.0709704
$203$	−2269.12	−0.784537
$204$	231.168	0.0793381
$205$	352.864	0.120220
$206$	−3770.91	−1.27540
$207$	717.718	0.240990
$208$	521.651	0.173894
$209$	0	0
$210$	807.282	0.265275
$211$	−59.8544	−0.0195287	−0.00976433	−	0.999952i	$-0.503108\pi$
$211$	−59.8544	−0.0195287	−0.00976433	+	0.999952i	$0.503108\pi$
$212$	446.610	0.144685
$213$	−3557.20	−1.14430
$214$	1289.82	0.412011
$215$	−926.459	−0.293879
$216$	−216.000	−0.0680414
$217$	−2684.59	−0.839826
$218$	−2465.72	−0.766052
$219$	2451.18	0.756325
$220$	938.458	0.287595
$221$	−628.067	−0.191169
$222$	331.391	0.100187
$223$	−483.339	−0.145143	−0.0725713	−	0.997363i	$-0.523120\pi$
$223$	−483.339	−0.145143	−0.0725713	+	0.997363i	$0.523120\pi$
$224$	513.265	0.153098
$225$	−491.705	−0.145690
$226$	523.694	0.154140
$227$	−1711.16	−0.500324	−0.250162	−	0.968204i	$-0.580484\pi$
$227$	−1711.16	−0.500324	−0.250162	+	0.968204i	$0.580484\pi$
$228$	0	0
$229$	−2137.54	−0.616823	−0.308412	−	0.951253i	$-0.599797\pi$
$229$	−2137.54	−0.616823	−0.308412	+	0.951253i	$0.599797\pi$
$230$	−1337.90	−0.383558
$231$	1345.82	0.383326
$232$	−1131.76	−0.320275
$233$	937.852	0.263694	0.131847	−	0.991270i	$-0.457909\pi$
$233$	937.852	0.263694	0.131847	+	0.991270i	$0.457909\pi$
$234$	586.857	0.163949
$235$	−2252.50	−0.625263
$236$	2578.36	0.711174
$237$	2912.35	0.798218
$238$	−617.971	−0.168307
$239$	−1841.70	−0.498449	−0.249225	−	0.968446i	$-0.580176\pi$
$239$	−1841.70	−0.498449	−0.249225	+	0.968446i	$0.580176\pi$
$240$	402.646	0.108294
$241$	−2972.53	−0.794513	−0.397257	−	0.917708i	$-0.630038\pi$
$241$	−2972.53	−0.794513	−0.397257	+	0.917708i	$0.630038\pi$
$242$	−1097.50	−0.291528
$243$	−243.000	−0.0641500
$244$	1441.17	0.378121
$245$	719.167	0.187534
$246$	252.393	0.0654145
$247$	0	0
$248$	−1338.99	−0.342846
$249$	3410.09	0.867894
$250$	3013.70	0.762413
$251$	4249.17	1.06855	0.534273	−	0.845312i	$-0.320585\pi$
$251$	4249.17	1.06855	0.534273	+	0.845312i	$0.320585\pi$
$252$	577.424	0.144342
$253$	−2230.41	−0.554247
$254$	−5392.33	−1.33207
$255$	−484.784	−0.119052
$256$	256.000	0.0625000
$257$	−5987.38	−1.45324	−0.726620	−	0.687039i	$-0.758910\pi$
$257$	−5987.38	−1.45324	−0.726620	+	0.687039i	$0.758910\pi$
$258$	−662.668	−0.159907
$259$	−885.894	−0.212536
$260$	−1093.96	−0.260941
$261$	−1273.23	−0.301959
$262$	−2778.87	−0.655263
$263$	−320.368	−0.0751132	−0.0375566	−	0.999295i	$-0.511957\pi$
$263$	−320.368	−0.0751132	−0.0375566	+	0.999295i	$0.511957\pi$
$264$	671.250	0.156487
$265$	−936.592	−0.217111
$266$	0	0
$267$	281.184	0.0644501
$268$	379.065	0.0863995
$269$	7577.76	1.71756	0.858781	−	0.512342i	$-0.171222\pi$
$269$	7577.76	1.71756	0.858781	+	0.512342i	$0.171222\pi$
$270$	452.976	0.102101
$271$	−1701.46	−0.381389	−0.190694	−	0.981649i	$-0.561074\pi$
$271$	−1701.46	−0.381389	−0.190694	+	0.981649i	$0.561074\pi$
$272$	−308.223	−0.0687088
$273$	−1568.82	−0.347800
$274$	−1569.06	−0.345950
$275$	1528.04	0.335071
$276$	−956.957	−0.208703
$277$	−7967.33	−1.72820	−0.864098	−	0.503324i	$-0.832110\pi$
$277$	−7967.33	−1.72820	−0.864098	+	0.503324i	$0.832110\pi$
$278$	−5428.54	−1.17116
$279$	−1506.36	−0.323238
$280$	−1076.38	−0.229735
$281$	−6285.85	−1.33446	−0.667228	−	0.744854i	$-0.732519\pi$
$281$	−6285.85	−1.33446	−0.667228	+	0.744854i	$0.732519\pi$
$282$	−1611.14	−0.340221
$283$	−6592.08	−1.38466	−0.692329	−	0.721582i	$-0.743415\pi$
$283$	−6592.08	−1.38466	−0.692329	+	0.721582i	$0.743415\pi$
$284$	4742.94	0.990991
$285$	0	0
$286$	−1823.74	−0.377063
$287$	−674.711	−0.138770
$288$	288.000	0.0589256
$289$	−4541.90	−0.924466
$290$	2373.44	0.480596
$291$	−1664.96	−0.335400
$292$	−3268.23	−0.654996
$293$	1051.46	0.209649	0.104824	−	0.994491i	$-0.466572\pi$
$293$	1051.46	0.209649	0.104824	+	0.994491i	$0.466572\pi$
$294$	514.398	0.102042
$295$	−5407.11	−1.06717
$296$	−441.855	−0.0867645
$297$	755.156	0.147537
$298$	−1007.20	−0.195790
$299$	2599.99	0.502880
$300$	655.607	0.126172
$301$	1771.48	0.339224
$302$	616.718	0.117510
$303$	305.630	0.0579471
$304$	0	0
$305$	−3022.30	−0.567398
$306$	−346.751	−0.0647793
$307$	−2096.61	−0.389771	−0.194886	−	0.980826i	$-0.562434\pi$
$307$	−2096.61	−0.389771	−0.194886	+	0.980826i	$0.562434\pi$
$308$	−1794.42	−0.331970
$309$	5656.36	1.04136
$310$	2808.01	0.514465
$311$	−9812.04	−1.78903	−0.894517	−	0.447034i	$-0.852480\pi$
$311$	−9812.04	−1.78903	−0.894517	+	0.447034i	$0.852480\pi$
$312$	−782.476	−0.141984
$313$	7508.78	1.35598	0.677990	−	0.735071i	$-0.262851\pi$
$313$	7508.78	1.35598	0.677990	+	0.735071i	$0.262851\pi$
$314$	4.04699	0.000727340 0
$315$	−1210.92	−0.216596
$316$	−3883.14	−0.691277
$317$	−3638.34	−0.644636	−0.322318	−	0.946632i	$-0.604462\pi$
$317$	−3638.34	−0.644636	−0.322318	+	0.946632i	$0.604462\pi$
$318$	−669.915	−0.118135
$319$	3956.75	0.694469
$320$	−536.861	−0.0937857
$321$	−1934.73	−0.336406
$322$	2558.19	0.442741
$323$	0	0
$324$	324.000	0.0555556
$325$	−1781.24	−0.304017
$326$	−1060.65	−0.180197
$327$	3698.57	0.625479
$328$	−336.524	−0.0566506
$329$	4307.00	0.721741
$330$	−1407.69	−0.234820
$331$	1992.93	0.330940	0.165470	−	0.986215i	$-0.447086\pi$
$331$	1992.93	0.330940	0.165470	+	0.986215i	$0.447086\pi$
$332$	−4546.79	−0.751618
$333$	−497.087	−0.0818024
$334$	6132.86	1.00472
$335$	−794.942	−0.129649
$336$	−769.898	−0.125004
$337$	−598.670	−0.0967705	−0.0483852	−	0.998829i	$-0.515408\pi$
$337$	−598.670	−0.0967705	−0.0483852	+	0.998829i	$0.515408\pi$
$338$	−2268.06	−0.364989
$339$	−785.542	−0.125855
$340$	646.379	0.103102
$341$	4681.23	0.743410
$342$	0	0
$343$	−6876.68	−1.08252
$344$	883.557	0.138483
$345$	2006.85	0.313174
$346$	3122.81	0.485213
$347$	5326.79	0.824084	0.412042	−	0.911165i	$-0.364816\pi$
$347$	5326.79	0.824084	0.412042	+	0.911165i	$0.364816\pi$
$348$	1697.65	0.261504
$349$	8069.88	1.23774	0.618869	−	0.785494i	$-0.287591\pi$
$349$	8069.88	1.23774	0.618869	+	0.785494i	$0.287591\pi$
$350$	−1752.61	−0.267659
$351$	−880.286	−0.133864
$352$	−895.000	−0.135522
$353$	−11595.5	−1.74835	−0.874177	−	0.485608i	$-0.838598\pi$
$353$	−11595.5	−1.74835	−0.874177	+	0.485608i	$0.838598\pi$
$354$	−3867.54	−0.580671
$355$	−9946.47	−1.48705
$356$	−374.912	−0.0558154
$357$	926.956	0.137422
$358$	2953.39	0.436009
$359$	−5015.77	−0.737388	−0.368694	−	0.929551i	$-0.620195\pi$
$359$	−5015.77	−0.737388	−0.368694	+	0.929551i	$0.620195\pi$
$360$	−603.968	−0.0884220
$361$	0	0
$362$	−7170.95	−1.04115
$363$	1646.25	0.238032
$364$	2091.76	0.301204
$365$	6853.85	0.982869
$366$	−2161.76	−0.308735
$367$	−3854.86	−0.548289	−0.274145	−	0.961688i	$-0.588395\pi$
$367$	−3854.86	−0.548289	−0.274145	+	0.961688i	$0.588395\pi$
$368$	1275.94	0.180742
$369$	−378.589	−0.0534107
$370$	926.620	0.130196
$371$	1790.86	0.250611
$372$	2008.48	0.279933
$373$	−11951.8	−1.65909	−0.829543	−	0.558443i	$-0.811399\pi$
$373$	−11951.8	−1.65909	−0.829543	+	0.558443i	$0.811399\pi$
$374$	1077.58	0.148985
$375$	−4520.55	−0.622507
$376$	2148.19	0.294640
$377$	−4612.39	−0.630106
$378$	−866.135	−0.117855
$379$	14300.6	1.93818	0.969090	−	0.246706i	$-0.0793481\pi$
$379$	14300.6	1.93818	0.969090	+	0.246706i	$0.0793481\pi$
$380$	0	0
$381$	8088.49	1.08763
$382$	−4329.80	−0.579926
$383$	5827.22	0.777434	0.388717	−	0.921357i	$-0.372918\pi$
$383$	5827.22	0.777434	0.388717	+	0.921357i	$0.372918\pi$
$384$	−384.000	−0.0510310
$385$	3763.11	0.498145
$386$	5954.57	0.785180
$387$	994.002	0.130563
$388$	2219.94	0.290465
$389$	7130.95	0.929443	0.464722	−	0.885457i	$-0.346154\pi$
$389$	7130.95	0.929443	0.464722	+	0.885457i	$0.346154\pi$
$390$	1640.94	0.213057
$391$	−1536.23	−0.198697
$392$	−685.864	−0.0883708
$393$	4168.30	0.535020
$394$	−8847.79	−1.13133
$395$	8143.37	1.03731
$396$	−1006.88	−0.127771
$397$	−13937.1	−1.76192	−0.880962	−	0.473187i	$-0.843103\pi$
$397$	−13937.1	−1.76192	−0.880962	+	0.473187i	$0.843103\pi$
$398$	2942.77	0.370622
$399$	0	0
$400$	−874.142	−0.109268
$401$	−251.101	−0.0312703	−0.0156352	−	0.999878i	$-0.504977\pi$
$401$	−251.101	−0.0312703	−0.0156352	+	0.999878i	$0.504977\pi$
$402$	−568.597	−0.0705449
$403$	−5456.91	−0.674511
$404$	−407.506	−0.0501837
$405$	−679.464	−0.0833651
$406$	−4538.24	−0.554752
$407$	1544.77	0.188136
$408$	462.335	0.0561005
$409$	9956.74	1.20374	0.601869	−	0.798595i	$-0.294423\pi$
$409$	9956.74	1.20374	0.601869	+	0.798595i	$0.294423\pi$
$410$	705.728	0.0850083
$411$	2353.59	0.282467
$412$	−7541.82	−0.901841
$413$	10338.9	1.23183
$414$	1435.44	0.170405
$415$	9535.12	1.12786
$416$	1043.30	0.122962
$417$	8142.81	0.956247
$418$	0	0
$419$	1055.61	0.123079	0.0615393	−	0.998105i	$-0.480399\pi$
$419$	1055.61	0.123079	0.0615393	+	0.998105i	$0.480399\pi$
$420$	1614.56	0.187578
$421$	10960.6	1.26885	0.634427	−	0.772983i	$-0.281236\pi$
$421$	10960.6	1.26885	0.634427	+	0.772983i	$0.281236\pi$
$422$	−119.709	−0.0138088
$423$	2416.72	0.277789
$424$	893.220	0.102308
$425$	1052.47	0.120123
$426$	−7114.41	−0.809141
$427$	5778.93	0.654947
$428$	2579.64	0.291336
$429$	2735.61	0.307871
$430$	−1852.92	−0.207804
$431$	8433.76	0.942552	0.471276	−	0.881986i	$-0.343794\pi$
$431$	8433.76	0.942552	0.471276	+	0.881986i	$0.343794\pi$
$432$	−432.000	−0.0481125
$433$	6816.97	0.756588	0.378294	−	0.925686i	$-0.376511\pi$
$433$	6816.97	0.756588	0.378294	+	0.925686i	$0.376511\pi$
$434$	−5369.19	−0.593846
$435$	−3560.15	−0.392405
$436$	−4931.43	−0.541680
$437$	0	0
$438$	4902.35	0.534802
$439$	6648.82	0.722849	0.361424	−	0.932401i	$-0.382291\pi$
$439$	6648.82	0.722849	0.361424	+	0.932401i	$0.382291\pi$
$440$	1876.92	0.203360
$441$	−771.597	−0.0833168
$442$	−1256.13	−0.135177
$443$	−12248.2	−1.31361	−0.656804	−	0.754062i	$-0.728092\pi$
$443$	−12248.2	−1.31361	−0.656804	+	0.754062i	$0.728092\pi$
$444$	662.782	0.0708429
$445$	786.233	0.0837551
$446$	−966.678	−0.102631
$447$	1510.79	0.159862
$448$	1026.53	0.108257
$449$	−4396.26	−0.462076	−0.231038	−	0.972945i	$-0.574212\pi$
$449$	−4396.26	−0.462076	−0.231038	+	0.972945i	$0.574212\pi$
$450$	−983.410	−0.103019
$451$	1176.52	0.122838
$452$	1047.39	0.108993
$453$	−925.078	−0.0959469
$454$	−3422.32	−0.353782
$455$	−4386.66	−0.451977
$456$	0	0
$457$	4307.95	0.440957	0.220478	−	0.975392i	$-0.429238\pi$
$457$	4307.95	0.440957	0.220478	+	0.975392i	$0.429238\pi$
$458$	−4275.08	−0.436160
$459$	520.127	0.0528921
$460$	−2675.80	−0.271217
$461$	8592.47	0.868094	0.434047	−	0.900890i	$-0.357085\pi$
$461$	8592.47	0.868094	0.434047	+	0.900890i	$0.357085\pi$
$462$	2691.64	0.271053
$463$	3188.27	0.320025	0.160012	−	0.987115i	$-0.448847\pi$
$463$	3188.27	0.320025	0.160012	+	0.987115i	$0.448847\pi$
$464$	−2263.53	−0.226469
$465$	−4212.01	−0.420059
$466$	1875.70	0.186460
$467$	−11923.6	−1.18149	−0.590745	−	0.806858i	$-0.701166\pi$
$467$	−11923.6	−1.18149	−0.590745	+	0.806858i	$0.701166\pi$
$468$	1173.71	0.115929
$469$	1520.01	0.149653
$470$	−4505.00	−0.442128
$471$	−6.07048	−0.000593871 0
$472$	5156.72	0.502876
$473$	−3089.00	−0.300280
$474$	5824.70	0.564425
$475$	0	0
$476$	−1235.94	−0.119011
$477$	1004.87	0.0964570
$478$	−3683.39	−0.352457
$479$	2874.54	0.274199	0.137099	−	0.990557i	$-0.456222\pi$
$479$	2874.54	0.274199	0.137099	+	0.990557i	$0.456222\pi$
$480$	805.291	0.0765757
$481$	−1800.74	−0.170699
$482$	−5945.06	−0.561806
$483$	−3837.29	−0.361496
$484$	−2194.99	−0.206141
$485$	−4655.47	−0.435864
$486$	−486.000	−0.0453609
$487$	261.803	0.0243602	0.0121801	−	0.999926i	$-0.496123\pi$
$487$	261.803	0.0243602	0.0121801	+	0.999926i	$0.496123\pi$
$488$	2882.34	0.267372
$489$	1590.98	0.147130
$490$	1438.33	0.132607
$491$	−3367.59	−0.309526	−0.154763	−	0.987952i	$-0.549461\pi$
$491$	−3367.59	−0.309526	−0.154763	+	0.987952i	$0.549461\pi$
$492$	504.785	0.0462550
$493$	2725.28	0.248967
$494$	0	0
$495$	2111.53	0.191730
$496$	−2677.98	−0.242429
$497$	19018.6	1.71650
$498$	6820.18	0.613694
$499$	2959.52	0.265504	0.132752	−	0.991149i	$-0.457619\pi$
$499$	2959.52	0.265504	0.132752	+	0.991149i	$0.457619\pi$
$500$	6027.40	0.539107
$501$	−9199.29	−0.820347
$502$	8498.34	0.755577
$503$	−8431.54	−0.747404	−0.373702	−	0.927549i	$-0.621912\pi$
$503$	−8431.54	−0.747404	−0.373702	+	0.927549i	$0.621912\pi$
$504$	1154.85	0.102065
$505$	854.587	0.0753042
$506$	−4460.82	−0.391912
$507$	3402.10	0.298013
$508$	−10784.7	−0.941913
$509$	5152.39	0.448675	0.224338	−	0.974512i	$-0.427978\pi$
$509$	5152.39	0.448675	0.224338	+	0.974512i	$0.427978\pi$
$510$	−969.569	−0.0841828
$511$	−13105.2	−1.13452
$512$	512.000	0.0441942
$513$	0	0
$514$	−11974.8	−1.02760
$515$	15816.0	1.35328
$516$	−1325.34	−0.113071
$517$	−7510.28	−0.638882
$518$	−1771.79	−0.150286
$519$	−4684.22	−0.396174
$520$	−2187.92	−0.184513
$521$	−21068.9	−1.77168	−0.885841	−	0.463989i	$-0.846418\pi$
$521$	−21068.9	−1.77168	−0.885841	+	0.463989i	$0.846418\pi$
$522$	−2546.47	−0.213517
$523$	7615.04	0.636678	0.318339	−	0.947977i	$-0.396875\pi$
$523$	7615.04	0.636678	0.318339	+	0.947977i	$0.396875\pi$
$524$	−5557.73	−0.463341
$525$	2628.91	0.218543
$526$	−640.737	−0.0531130
$527$	3224.28	0.266512
$528$	1342.50	0.110653
$529$	−5807.51	−0.477316
$530$	−1873.18	−0.153521
$531$	5801.31	0.474116
$532$	0	0
$533$	−1371.47	−0.111454
$534$	562.368	0.0455731
$535$	−5409.80	−0.437170
$536$	758.130	0.0610937
$537$	−4430.08	−0.356000
$538$	15155.5	1.21450
$539$	2397.84	0.191619
$540$	905.953	0.0721963
$541$	6088.48	0.483853	0.241926	−	0.970295i	$-0.422221\pi$
$541$	6088.48	0.483853	0.241926	+	0.970295i	$0.422221\pi$
$542$	−3402.92	−0.269683
$543$	10756.4	0.850096
$544$	−616.447	−0.0485845
$545$	10341.8	0.812830
$546$	−3137.64	−0.245932
$547$	23451.4	1.83311	0.916554	−	0.399910i	$-0.130959\pi$
$547$	23451.4	1.83311	0.916554	+	0.399910i	$0.130959\pi$
$548$	−3138.12	−0.244624
$549$	3242.64	0.252081
$550$	3056.08	0.236931
$551$	0	0
$552$	−1913.91	−0.147575
$553$	−15570.9	−1.19737
$554$	−15934.7	−1.22202
$555$	−1389.93	−0.106305
$556$	−10857.1	−0.828134
$557$	−14161.3	−1.07726	−0.538631	−	0.842542i	$-0.681058\pi$
$557$	−14161.3	−1.07726	−0.538631	+	0.842542i	$0.681058\pi$
$558$	−3012.72	−0.228564
$559$	3600.85	0.272450
$560$	−2152.75	−0.162447
$561$	−1616.37	−0.121645
$562$	−12571.7	−0.943603
$563$	−4782.00	−0.357970	−0.178985	−	0.983852i	$-0.557281\pi$
$563$	−4782.00	−0.357970	−0.178985	+	0.983852i	$0.557281\pi$
$564$	−3222.29	−0.240572
$565$	−2196.49	−0.163552
$566$	−13184.2	−0.979102
$567$	1299.20	0.0962282
$568$	9485.88	0.700737
$569$	−9926.67	−0.731367	−0.365683	−	0.930739i	$-0.619165\pi$
$569$	−9926.67	−0.731367	−0.365683	+	0.930739i	$0.619165\pi$
$570$	0	0
$571$	−13656.1	−1.00086	−0.500430	−	0.865777i	$-0.666825\pi$
$571$	−13656.1	−1.00086	−0.500430	+	0.865777i	$0.666825\pi$
$572$	−3647.48	−0.266624
$573$	6494.69	0.473507
$574$	−1349.42	−0.0981250
$575$	−4356.86	−0.315989
$576$	576.000	0.0416667
$577$	14811.1	1.06862	0.534311	−	0.845288i	$-0.320571\pi$
$577$	14811.1	1.06862	0.534311	+	0.845288i	$0.320571\pi$
$578$	−9083.80	−0.653696
$579$	−8931.85	−0.641097
$580$	4746.87	0.339833
$581$	−18232.1	−1.30188
$582$	−3329.91	−0.237164
$583$	−3122.78	−0.221840
$584$	−6536.47	−0.463152
$585$	−2461.41	−0.173960
$586$	2102.92	0.148244
$587$	14292.4	1.00496	0.502478	−	0.864590i	$-0.332422\pi$
$587$	14292.4	1.00496	0.502478	+	0.864590i	$0.332422\pi$
$588$	1028.80	0.0721544
$589$	0	0
$590$	−10814.2	−0.754601
$591$	13271.7	0.923730
$592$	−883.710	−0.0613518
$593$	9657.63	0.668788	0.334394	−	0.942433i	$-0.391468\pi$
$593$	9657.63	0.668788	0.334394	+	0.942433i	$0.391468\pi$
$594$	1510.31	0.104325
$595$	2591.91	0.178585
$596$	−2014.39	−0.138444
$597$	−4414.15	−0.302611
$598$	5199.98	0.355590
$599$	−9217.95	−0.628773	−0.314387	−	0.949295i	$-0.601799\pi$
$599$	−9217.95	−0.628773	−0.314387	+	0.949295i	$0.601799\pi$
$600$	1311.21	0.0892168
$601$	18966.6	1.28730	0.643649	−	0.765321i	$-0.277420\pi$
$601$	18966.6	1.28730	0.643649	+	0.765321i	$0.277420\pi$
$602$	3542.96	0.239868
$603$	852.896	0.0575997
$604$	1233.44	0.0830924
$605$	4603.15	0.309330
$606$	611.260	0.0409748
$607$	−9556.83	−0.639044	−0.319522	−	0.947579i	$-0.603522\pi$
$607$	−9556.83	−0.639044	−0.319522	+	0.947579i	$0.603522\pi$
$608$	0	0
$609$	6807.36	0.452953
$610$	−6044.60	−0.401211
$611$	8754.74	0.579671
$612$	−693.503	−0.0458059
$613$	−14958.7	−0.985606	−0.492803	−	0.870141i	$-0.664028\pi$
$613$	−14958.7	−0.985606	−0.492803	+	0.870141i	$0.664028\pi$
$614$	−4193.22	−0.275610
$615$	−1058.59	−0.0694090
$616$	−3588.85	−0.234738
$617$	−25766.3	−1.68122	−0.840610	−	0.541641i	$-0.817803\pi$
$617$	−25766.3	−1.68122	−0.840610	+	0.541641i	$0.817803\pi$
$618$	11312.7	0.736350
$619$	−9724.16	−0.631417	−0.315708	−	0.948856i	$-0.602242\pi$
$619$	−9724.16	−0.631417	−0.315708	+	0.948856i	$0.602242\pi$
$620$	5616.02	0.363782
$621$	−2153.15	−0.139135
$622$	−19624.1	−1.26504
$623$	−1503.35	−0.0966784
$624$	−1564.95	−0.100398
$625$	−5810.90	−0.371898
$626$	15017.6	0.958822
$627$	0	0
$628$	8.09398	0.000514307 0
$629$	1063.98	0.0674465
$630$	−2421.84	−0.153157
$631$	4792.37	0.302348	0.151174	−	0.988507i	$-0.451695\pi$
$631$	4792.37	0.302348	0.151174	+	0.988507i	$0.451695\pi$
$632$	−7766.27	−0.488807
$633$	179.563	0.0112749
$634$	−7276.68	−0.455826
$635$	22616.6	1.41341
$636$	−1339.83	−0.0835342
$637$	−2795.17	−0.173860
$638$	7913.50	0.491064
$639$	10671.6	0.660661
$640$	−1073.72	−0.0663165
$641$	7757.10	0.477983	0.238992	−	0.971022i	$-0.423183\pi$
$641$	7757.10	0.477983	0.238992	+	0.971022i	$0.423183\pi$
$642$	−3869.46	−0.237875
$643$	25559.8	1.56762	0.783810	−	0.621001i	$-0.213274\pi$
$643$	25559.8	1.56762	0.783810	+	0.621001i	$0.213274\pi$
$644$	5116.39	0.313065
$645$	2779.38	0.169671
$646$	0	0
$647$	−16384.6	−0.995585	−0.497793	−	0.867296i	$-0.665856\pi$
$647$	−16384.6	−0.995585	−0.497793	+	0.867296i	$0.665856\pi$
$648$	648.000	0.0392837
$649$	−18028.4	−1.09041
$650$	−3562.48	−0.214972
$651$	8053.78	0.484874
$652$	−2121.30	−0.127418
$653$	15643.0	0.937455	0.468727	−	0.883343i	$-0.344713\pi$
$653$	15643.0	0.937455	0.468727	+	0.883343i	$0.344713\pi$
$654$	7397.15	0.442280
$655$	11655.2	0.695276
$656$	−673.047	−0.0400580
$657$	−7353.53	−0.436664
$658$	8614.01	0.510348
$659$	32774.1	1.93733	0.968663	−	0.248378i	$-0.0798976\pi$
$659$	32774.1	1.93733	0.968663	+	0.248378i	$0.0798976\pi$
$660$	−2815.37	−0.166043
$661$	13644.5	0.802887	0.401443	−	0.915884i	$-0.368509\pi$
$661$	13644.5	0.802887	0.401443	+	0.915884i	$0.368509\pi$
$662$	3985.85	0.234010
$663$	1884.20	0.110371
$664$	−9093.57	−0.531474
$665$	0	0
$666$	−994.174	−0.0578430
$667$	−11281.8	−0.654920
$668$	12265.7	0.710441
$669$	1450.02	0.0837981
$670$	−1589.88	−0.0916755
$671$	−10076.9	−0.579756
$672$	−1539.80	−0.0883913
$673$	14135.4	0.809631	0.404815	−	0.914398i	$-0.367336\pi$
$673$	14135.4	0.809631	0.404815	+	0.914398i	$0.367336\pi$
$674$	−1197.34	−0.0684271
$675$	1475.12	0.0841144
$676$	−4536.13	−0.258087
$677$	−8604.23	−0.488460	−0.244230	−	0.969717i	$-0.578535\pi$
$677$	−8604.23	−0.488460	−0.244230	+	0.969717i	$0.578535\pi$
$678$	−1571.08	−0.0889928
$679$	8901.71	0.503117
$680$	1292.76	0.0729045
$681$	5133.47	0.288862
$682$	9362.46	0.525670
$683$	−17472.7	−0.978881	−0.489440	−	0.872037i	$-0.662799\pi$
$683$	−17472.7	−0.978881	−0.489440	+	0.872037i	$0.662799\pi$
$684$	0	0
$685$	6580.99	0.367075
$686$	−13753.4	−0.765460
$687$	6412.62	0.356123
$688$	1767.11	0.0979224
$689$	3640.23	0.201280
$690$	4013.69	0.221447
$691$	9660.63	0.531849	0.265925	−	0.963994i	$-0.414323\pi$
$691$	9660.63	0.531849	0.265925	+	0.963994i	$0.414323\pi$
$692$	6245.63	0.343097
$693$	−4037.46	−0.221314
$694$	10653.6	0.582716
$695$	22768.5	1.24267
$696$	3395.29	0.184911
$697$	810.347	0.0440374
$698$	16139.8	0.875214
$699$	−2813.56	−0.152244
$700$	−3505.21	−0.189264
$701$	22673.6	1.22164	0.610819	−	0.791770i	$-0.290840\pi$
$701$	22673.6	1.22164	0.610819	+	0.791770i	$0.290840\pi$
$702$	−1760.57	−0.0946560
$703$	0	0
$704$	−1790.00	−0.0958284
$705$	6757.50	0.360996
$706$	−23191.1	−1.23627
$707$	−1634.05	−0.0869235
$708$	−7735.08	−0.410596
$709$	13285.1	0.703711	0.351855	−	0.936054i	$-0.385551\pi$
$709$	13285.1	0.703711	0.351855	+	0.936054i	$0.385551\pi$
$710$	−19892.9	−1.05151
$711$	−8737.06	−0.460851
$712$	−749.824	−0.0394675
$713$	−13347.4	−0.701074
$714$	1853.91	0.0971721
$715$	7649.18	0.400088
$716$	5906.77	0.308305
$717$	5525.09	0.287780
$718$	−10031.5	−0.521412
$719$	−28752.4	−1.49136	−0.745678	−	0.666306i	$-0.767874\pi$
$719$	−28752.4	−1.49136	−0.745678	+	0.666306i	$0.767874\pi$
$720$	−1207.94	−0.0625238
$721$	−30241.8	−1.56209
$722$	0	0
$723$	8917.60	0.458712
$724$	−14341.9	−0.736205
$725$	7729.08	0.395932
$726$	3292.49	0.168314
$727$	28931.3	1.47593	0.737967	−	0.674837i	$-0.235786\pi$
$727$	28931.3	1.47593	0.737967	+	0.674837i	$0.235786\pi$
$728$	4183.52	0.212983
$729$	729.000	0.0370370
$730$	13707.7	0.694993
$731$	−2127.60	−0.107650
$732$	−4323.51	−0.218308
$733$	10740.6	0.541219	0.270609	−	0.962689i	$-0.412775\pi$
$733$	10740.6	0.541219	0.270609	+	0.962689i	$0.412775\pi$
$734$	−7709.73	−0.387699
$735$	−2157.50	−0.108273
$736$	2551.89	0.127804
$737$	−2650.49	−0.132472
$738$	−757.178	−0.0377671
$739$	−936.700	−0.0466266	−0.0233133	−	0.999728i	$-0.507422\pi$
$739$	−936.700	−0.0466266	−0.0233133	+	0.999728i	$0.507422\pi$
$740$	1853.24	0.0920627
$741$	0	0
$742$	3581.71	0.177209
$743$	31780.2	1.56918	0.784591	−	0.620014i	$-0.212873\pi$
$743$	31780.2	1.56918	0.784591	+	0.620014i	$0.212873\pi$
$744$	4016.96	0.197942
$745$	4224.41	0.207745
$746$	−23903.5	−1.17315
$747$	−10230.3	−0.501079
$748$	2155.16	0.105348
$749$	10344.1	0.504625
$750$	−9041.10	−0.440179
$751$	−25394.4	−1.23389	−0.616947	−	0.787005i	$-0.711631\pi$
$751$	−25394.4	−1.23389	−0.616947	+	0.787005i	$0.711631\pi$
$752$	4296.38	0.208342
$753$	−12747.5	−0.616926
$754$	−9224.77	−0.445552
$755$	−2586.66	−0.124686
$756$	−1732.27	−0.0833361
$757$	26773.4	1.28546	0.642732	−	0.766091i	$-0.277801\pi$
$757$	26773.4	1.28546	0.642732	+	0.766091i	$0.277801\pi$
$758$	28601.1	1.37050
$759$	6691.23	0.319995
$760$	0	0
$761$	26354.6	1.25539	0.627696	−	0.778459i	$-0.283998\pi$
$761$	26354.6	1.25539	0.627696	+	0.778459i	$0.283998\pi$
$762$	16177.0	0.769068
$763$	−19774.5	−0.938249
$764$	−8659.59	−0.410069
$765$	1454.35	0.0687350
$766$	11654.4	0.549729
$767$	21015.7	0.989352
$768$	−768.000	−0.0360844
$769$	37499.2	1.75846	0.879230	−	0.476397i	$-0.158058\pi$
$769$	37499.2	1.75846	0.879230	+	0.476397i	$0.158058\pi$
$770$	7526.22	0.352242
$771$	17962.2	0.839029
$772$	11909.1	0.555206
$773$	−17461.2	−0.812464	−0.406232	−	0.913770i	$-0.633157\pi$
$773$	−17461.2	−0.812464	−0.406232	+	0.913770i	$0.633157\pi$
$774$	1988.00	0.0923221
$775$	9144.27	0.423834
$776$	4439.88	0.205390
$777$	2657.68	0.122708
$778$	14261.9	0.657216
$779$	0	0
$780$	3281.88	0.150654
$781$	−33163.5	−1.51944
$782$	−3072.46	−0.140500
$783$	3819.70	0.174336
$784$	−1371.73	−0.0624876
$785$	−16.9740	−0.000771755 0
$786$	8336.60	0.378316
$787$	3251.22	0.147260	0.0736299	−	0.997286i	$-0.476542\pi$
$787$	3251.22	0.147260	0.0736299	+	0.997286i	$0.476542\pi$
$788$	−17695.6	−0.799974
$789$	961.105	0.0433666
$790$	16286.7	0.733489
$791$	4199.91	0.188788
$792$	−2013.75	−0.0903479
$793$	11746.7	0.526024
$794$	−27874.2	−1.24587
$795$	2809.78	0.125349
$796$	5885.53	0.262069
$797$	3246.61	0.144292	0.0721460	−	0.997394i	$-0.477015\pi$
$797$	3246.61	0.144292	0.0721460	+	0.997394i	$0.477015\pi$
$798$	0	0
$799$	−5172.84	−0.229039
$800$	−1748.28	−0.0772640
$801$	−843.552	−0.0372103
$802$	−502.202	−0.0221114
$803$	22852.1	1.00428
$804$	−1137.19	−0.0498828
$805$	−10729.6	−0.469776
$806$	−10913.8	−0.476951
$807$	−22733.3	−0.991635
$808$	−815.013	−0.0354852
$809$	7901.78	0.343401	0.171701	−	0.985149i	$-0.445074\pi$
$809$	7901.78	0.343401	0.171701	+	0.985149i	$0.445074\pi$
$810$	−1358.93	−0.0589480
$811$	22332.4	0.966952	0.483476	−	0.875358i	$-0.339374\pi$
$811$	22332.4	0.966952	0.483476	+	0.875358i	$0.339374\pi$
$812$	−9076.49	−0.392269
$813$	5104.38	0.220195
$814$	3089.53	0.133032
$815$	4448.61	0.191200
$816$	924.670	0.0396690
$817$	0	0
$818$	19913.5	0.851172
$819$	4706.46	0.200802
$820$	1411.46	0.0601100
$821$	10068.8	0.428021	0.214010	−	0.976831i	$-0.431347\pi$
$821$	10068.8	0.428021	0.214010	+	0.976831i	$0.431347\pi$
$822$	4707.18	0.199734
$823$	−35561.9	−1.50621	−0.753104	−	0.657902i	$-0.771444\pi$
$823$	−35561.9	−1.50621	−0.753104	+	0.657902i	$0.771444\pi$
$824$	−15083.6	−0.637698
$825$	−4584.13	−0.193453
$826$	20677.9	0.871035
$827$	35425.4	1.48955	0.744777	−	0.667313i	$-0.232556\pi$
$827$	35425.4	1.48955	0.744777	+	0.667313i	$0.232556\pi$
$828$	2870.87	0.120495
$829$	31632.8	1.32528	0.662638	−	0.748940i	$-0.269437\pi$
$829$	31632.8	1.32528	0.662638	+	0.748940i	$0.269437\pi$
$830$	19070.2	0.797515
$831$	23902.0	0.997774
$832$	2086.60	0.0869471
$833$	1651.56	0.0686951
$834$	16285.6	0.676169
$835$	−25722.6	−1.06607
$836$	0	0
$837$	4519.08	0.186622
$838$	2111.22	0.0870297
$839$	−35877.6	−1.47632	−0.738160	−	0.674625i	$-0.764305\pi$
$839$	−35877.6	−1.47632	−0.738160	+	0.674625i	$0.764305\pi$
$840$	3229.13	0.132637
$841$	−4375.12	−0.179389
$842$	21921.2	0.897215
$843$	18857.5	0.770448
$844$	−239.418	−0.00976433
$845$	9512.77	0.387277
$846$	4833.43	0.196427
$847$	−8801.68	−0.357059
$848$	1786.44	0.0723427
$849$	19776.2	0.799433
$850$	2104.93	0.0849395
$851$	−4404.54	−0.177422
$852$	−14228.8	−0.572149
$853$	−41382.1	−1.66107	−0.830537	−	0.556964i	$-0.811966\pi$
$853$	−41382.1	−1.66107	−0.830537	+	0.556964i	$0.811966\pi$
$854$	11557.9	0.463117
$855$	0	0
$856$	5159.29	0.206006
$857$	22106.8	0.881159	0.440580	−	0.897714i	$-0.354773\pi$
$857$	22106.8	0.881159	0.440580	+	0.897714i	$0.354773\pi$
$858$	5471.22	0.217698
$859$	−20872.2	−0.829047	−0.414524	−	0.910038i	$-0.636052\pi$
$859$	−20872.2	−0.829047	−0.414524	+	0.910038i	$0.636052\pi$
$860$	−3705.84	−0.146940
$861$	2024.13	0.0801187
$862$	16867.5	0.666485
$863$	−2076.93	−0.0819231	−0.0409616	−	0.999161i	$-0.513042\pi$
$863$	−2076.93	−0.0819231	−0.0409616	+	0.999161i	$0.513042\pi$
$864$	−864.000	−0.0340207
$865$	−13097.8	−0.514842
$866$	13633.9	0.534988
$867$	13625.7	0.533740
$868$	−10738.4	−0.419913
$869$	27151.6	1.05990
$870$	−7120.31	−0.277472
$871$	3089.68	0.120195
$872$	−9862.86	−0.383026
$873$	4994.87	0.193643
$874$	0	0
$875$	24169.2	0.933792
$876$	9804.70	0.378162
$877$	−43101.3	−1.65955	−0.829776	−	0.558097i	$-0.811532\pi$
$877$	−43101.3	−1.65955	−0.829776	+	0.558097i	$0.811532\pi$
$878$	13297.6	0.511131
$879$	−3154.39	−0.121041
$880$	3753.83	0.143797
$881$	27654.6	1.05756	0.528778	−	0.848760i	$-0.322650\pi$
$881$	27654.6	1.05756	0.528778	+	0.848760i	$0.322650\pi$
$882$	−1543.19	−0.0589139
$883$	14726.2	0.561243	0.280622	−	0.959818i	$-0.409459\pi$
$883$	14726.2	0.561243	0.280622	+	0.959818i	$0.409459\pi$
$884$	−2512.27	−0.0955845
$885$	16221.3	0.616129
$886$	−24496.3	−0.928861
$887$	−19781.1	−0.748800	−0.374400	−	0.927267i	$-0.622151\pi$
$887$	−19781.1	−0.748800	−0.374400	+	0.927267i	$0.622151\pi$
$888$	1325.56	0.0500935
$889$	−43245.2	−1.63149
$890$	1572.47	0.0592238
$891$	−2265.47	−0.0851808
$892$	−1933.36	−0.0725713
$893$	0	0
$894$	3021.59	0.113039
$895$	−12387.2	−0.462634
$896$	2053.06	0.0765491
$897$	−7799.96	−0.290338
$898$	−8792.52	−0.326737
$899$	23678.4	0.878441
$900$	−1966.82	−0.0728452
$901$	−2150.87	−0.0795293
$902$	2353.04	0.0868598
$903$	−5314.45	−0.195851
$904$	2094.78	0.0770700
$905$	30076.6	1.10473
$906$	−1850.16	−0.0678447
$907$	−25912.0	−0.948614	−0.474307	−	0.880360i	$-0.657301\pi$
$907$	−25912.0	−0.948614	−0.474307	+	0.880360i	$0.657301\pi$
$908$	−6844.63	−0.250162
$909$	−916.889	−0.0334558
$910$	−8773.32	−0.319596
$911$	−46653.3	−1.69670	−0.848350	−	0.529437i	$-0.822403\pi$
$911$	−46653.3	−1.69670	−0.848350	+	0.529437i	$0.822403\pi$
$912$	0	0
$913$	31792.0	1.15242
$914$	8615.90	0.311804
$915$	9066.89	0.327587
$916$	−8550.15	−0.308412
$917$	−22285.9	−0.802556
$918$	1040.25	0.0374003
$919$	44547.1	1.59899	0.799495	−	0.600673i	$-0.205101\pi$
$919$	44547.1	1.59899	0.799495	+	0.600673i	$0.205101\pi$
$920$	−5351.59	−0.191779
$921$	6289.83	0.225035
$922$	17184.9	0.613835
$923$	38658.7	1.37862
$924$	5383.27	0.191663
$925$	3017.53	0.107260
$926$	6376.54	0.226292
$927$	−16969.1	−0.601228
$928$	−4527.05	−0.160138
$929$	3524.10	0.124458	0.0622292	−	0.998062i	$-0.480179\pi$
$929$	3524.10	0.124458	0.0622292	+	0.998062i	$0.480179\pi$
$930$	−8424.03	−0.297027
$931$	0	0
$932$	3751.41	0.131847
$933$	29436.1	1.03290
$934$	−23847.1	−0.835440
$935$	−4519.61	−0.158082
$936$	2347.43	0.0819745
$937$	−19144.8	−0.667486	−0.333743	−	0.942664i	$-0.608312\pi$
$937$	−19144.8	−0.667486	−0.333743	+	0.942664i	$0.608312\pi$
$938$	3040.02	0.105821
$939$	−22526.4	−0.782875
$940$	−9010.00	−0.312632
$941$	−2110.74	−0.0731224	−0.0365612	−	0.999331i	$-0.511640\pi$
$941$	−2110.74	−0.0731224	−0.0365612	+	0.999331i	$0.511640\pi$
$942$	−12.1410	−0.000419930 0
$943$	−3354.57	−0.115843
$944$	10313.4	0.355587
$945$	3632.77	0.125052
$946$	−6178.00	−0.212330
$947$	−18860.7	−0.647192	−0.323596	−	0.946195i	$-0.604892\pi$
$947$	−18860.7	−0.647192	−0.323596	+	0.946195i	$0.604892\pi$
$948$	11649.4	0.399109
$949$	−26638.7	−0.911200
$950$	0	0
$951$	10915.0	0.372181
$952$	−2471.88	−0.0841535
$953$	−20352.1	−0.691784	−0.345892	−	0.938274i	$-0.612424\pi$
$953$	−20352.1	−0.691784	−0.345892	+	0.938274i	$0.612424\pi$
$954$	2009.75	0.0682054
$955$	18160.1	0.615338
$956$	−7366.78	−0.249225
$957$	−11870.3	−0.400952
$958$	5749.08	0.193888
$959$	−12583.5	−0.423714
$960$	1610.58	0.0541472
$961$	−1777.12	−0.0596528
$962$	−3601.47	−0.120703
$963$	5804.20	0.194224
$964$	−11890.1	−0.397257
$965$	−24974.8	−0.833126
$966$	−7674.58	−0.255616
$967$	4107.74	0.136604	0.0683020	−	0.997665i	$-0.478242\pi$
$967$	4107.74	0.136604	0.0683020	+	0.997665i	$0.478242\pi$
$968$	−4389.99	−0.145764
$969$	0	0
$970$	−9310.93	−0.308202
$971$	15720.4	0.519560	0.259780	−	0.965668i	$-0.416350\pi$
$971$	15720.4	0.519560	0.259780	+	0.965668i	$0.416350\pi$
$972$	−972.000	−0.0320750
$973$	−43535.6	−1.43442
$974$	523.607	0.0172253
$975$	5343.72	0.175524
$976$	5764.68	0.189061
$977$	21779.1	0.713179	0.356590	−	0.934261i	$-0.383939\pi$
$977$	21779.1	0.713179	0.356590	+	0.934261i	$0.383939\pi$
$978$	3181.96	0.104037
$979$	2621.46	0.0855793
$980$	2876.67	0.0937671
$981$	−11095.7	−0.361120
$982$	−6735.19	−0.218868
$983$	2370.66	0.0769201	0.0384600	−	0.999260i	$-0.487755\pi$
$983$	2370.66	0.0769201	0.0384600	+	0.999260i	$0.487755\pi$
$984$	1009.57	0.0327073
$985$	37109.6	1.20042
$986$	5450.56	0.176046
$987$	−12921.0	−0.416697
$988$	0	0
$989$	8807.57	0.283179
$990$	4223.06	0.135573
$991$	−46028.0	−1.47541	−0.737703	−	0.675125i	$-0.764090\pi$
$991$	−46028.0	−1.47541	−0.737703	+	0.675125i	$0.764090\pi$
$992$	−5355.95	−0.171423
$993$	−5978.78	−0.191068
$994$	38037.3	1.21375
$995$	−12342.6	−0.393254
$996$	13640.4	0.433947
$997$	14463.3	0.459436	0.229718	−	0.973257i	$-0.426220\pi$
$997$	14463.3	0.459436	0.229718	+	0.973257i	$0.426220\pi$
$998$	5919.04	0.187739
$999$	1491.26	0.0472286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bi.1.4	yes	8
19.18	odd	2		2166.4.a.bh.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.bh.1.4	✓	8	19.18	odd	2
2166.4.a.bi.1.4	yes	8	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.38845 q^{5} -6.00000 q^{6} +16.0395 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.38845 q^{5} -6.00000 q^{6} +16.0395 q^{7} +8.00000 q^{8} +9.00000 q^{9} -16.7769 q^{10} -27.9688 q^{11} -12.0000 q^{12} +32.6032 q^{13} +32.0791 q^{14} +25.1654 q^{15} +16.0000 q^{16} -19.2640 q^{17} +18.0000 q^{18} -33.5538 q^{20} -48.1186 q^{21} -55.9375 q^{22} +79.7464 q^{23} -24.0000 q^{24} -54.6339 q^{25} +65.2064 q^{26} -27.0000 q^{27} +64.1582 q^{28} -141.470 q^{29} +50.3307 q^{30} -167.373 q^{31} +32.0000 q^{32} +83.9063 q^{33} -38.5279 q^{34} -134.547 q^{35} +36.0000 q^{36} -55.2319 q^{37} -97.8096 q^{39} -67.1076 q^{40} -42.0654 q^{41} -96.2373 q^{42} +110.445 q^{43} -111.875 q^{44} -75.4961 q^{45} +159.493 q^{46} +268.524 q^{47} -48.0000 q^{48} -85.7330 q^{49} -109.268 q^{50} +57.7919 q^{51} +130.413 q^{52} +111.653 q^{53} -54.0000 q^{54} +234.615 q^{55} +128.316 q^{56} -282.941 q^{58} +644.590 q^{59} +100.661 q^{60} +360.293 q^{61} -334.747 q^{62} +144.356 q^{63} +64.0000 q^{64} -273.490 q^{65} +167.813 q^{66} +94.7662 q^{67} -77.0559 q^{68} -239.239 q^{69} -269.094 q^{70} +1185.73 q^{71} +72.0000 q^{72} -817.058 q^{73} -110.464 q^{74} +163.902 q^{75} -448.606 q^{77} -195.619 q^{78} -970.784 q^{79} -134.215 q^{80} +81.0000 q^{81} -84.1309 q^{82} -1136.70 q^{83} -192.475 q^{84} +161.595 q^{85} +220.889 q^{86} +424.411 q^{87} -223.750 q^{88} -93.7280 q^{89} -150.992 q^{90} +522.940 q^{91} +318.986 q^{92} +502.120 q^{93} +537.048 q^{94} -96.0000 q^{96} +554.985 q^{97} -171.466 q^{98} -251.719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9	\( 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9} - 60 q^{10} - 6 q^{11} - 96 q^{12} + 48 q^{13} - 68 q^{14} + 90 q^{15} + 128 q^{16} - 36 q^{17} + 144 q^{18} - 120 q^{20} + 102 q^{21} - 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} - 216 q^{27} - 136 q^{28} - 380 q^{29} + 180 q^{30} - 48 q^{31} + 256 q^{32} + 18 q^{33} - 72 q^{34} + 762 q^{35} + 288 q^{36} - 168 q^{37} - 144 q^{39} - 240 q^{40} + 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} - 564 q^{46} - 468 q^{47} - 384 q^{48} + 222 q^{49} + 792 q^{50} + 108 q^{51} + 192 q^{52} + 1682 q^{53} - 432 q^{54} - 46 q^{55} - 272 q^{56} - 760 q^{58} + 292 q^{59} + 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} - 1120 q^{65} + 36 q^{66} - 2484 q^{67} - 144 q^{68} + 846 q^{69} + 1524 q^{70} + 1182 q^{71} + 576 q^{72} + 182 q^{73} - 336 q^{74} - 1188 q^{75} - 504 q^{77} - 288 q^{78} - 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} + 408 q^{84} - 2238 q^{85} - 1576 q^{86} + 1140 q^{87} - 48 q^{88} + 1304 q^{89} - 540 q^{90} - 624 q^{91} - 1128 q^{92} + 144 q^{93} - 936 q^{94} - 768 q^{96} - 1248 q^{97} + 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9 - 60 * q^10 - 6 * q^11 - 96 * q^12 + 48 * q^13 - 68 * q^14 + 90 * q^15 + 128 * q^16 - 36 * q^17 + 144 * q^18 - 120 * q^20 + 102 * q^21 - 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 - 216 * q^27 - 136 * q^28 - 380 * q^29 + 180 * q^30 - 48 * q^31 + 256 * q^32 + 18 * q^33 - 72 * q^34 + 762 * q^35 + 288 * q^36 - 168 * q^37 - 144 * q^39 - 240 * q^40 + 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 - 564 * q^46 - 468 * q^47 - 384 * q^48 + 222 * q^49 + 792 * q^50 + 108 * q^51 + 192 * q^52 + 1682 * q^53 - 432 * q^54 - 46 * q^55 - 272 * q^56 - 760 * q^58 + 292 * q^59 + 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 - 1120 * q^65 + 36 * q^66 - 2484 * q^67 - 144 * q^68 + 846 * q^69 + 1524 * q^70 + 1182 * q^71 + 576 * q^72 + 182 * q^73 - 336 * q^74 - 1188 * q^75 - 504 * q^77 - 288 * q^78 - 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 + 408 * q^84 - 2238 * q^85 - 1576 * q^86 + 1140 * q^87 - 48 * q^88 + 1304 * q^89 - 540 * q^90 - 624 * q^91 - 1128 * q^92 + 144 * q^93 - 936 * q^94 - 768 * q^96 - 1248 * q^97 + 444 * q^98 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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