LMFDB - Embedded newform 2166.4.a.ba.1.1

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:	$4$
Coefficient field:	4.4.184225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 45x^{2} + 46x + 449 $ x^4 - 2x^3 - 45x^2 + 46*x + 449
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.28229$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.2398 q^{5} +6.00000 q^{6} -18.0653 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.2398 q^{5} +6.00000 q^{6} -18.0653 q^{7} +8.00000 q^{8} +9.00000 q^{9} -38.4795 q^{10} +16.3023 q^{11} +12.0000 q^{12} +28.5122 q^{13} -36.1306 q^{14} -57.7193 q^{15} +16.0000 q^{16} +21.7450 q^{17} +18.0000 q^{18} -76.9590 q^{20} -54.1959 q^{21} +32.6046 q^{22} -115.106 q^{23} +24.0000 q^{24} +245.168 q^{25} +57.0245 q^{26} +27.0000 q^{27} -72.2611 q^{28} +295.362 q^{29} -115.439 q^{30} +250.965 q^{31} +32.0000 q^{32} +48.9068 q^{33} +43.4900 q^{34} +347.572 q^{35} +36.0000 q^{36} +126.485 q^{37} +85.5367 q^{39} -153.918 q^{40} -484.330 q^{41} -108.392 q^{42} +24.1636 q^{43} +65.2091 q^{44} -173.158 q^{45} -230.212 q^{46} +76.4066 q^{47} +48.0000 q^{48} -16.6454 q^{49} +490.336 q^{50} +65.2350 q^{51} +114.049 q^{52} -600.523 q^{53} +54.0000 q^{54} -313.652 q^{55} -144.522 q^{56} +590.724 q^{58} -605.562 q^{59} -230.877 q^{60} -440.230 q^{61} +501.930 q^{62} -162.588 q^{63} +64.0000 q^{64} -548.568 q^{65} +97.8137 q^{66} -898.734 q^{67} +86.9800 q^{68} -345.319 q^{69} +695.143 q^{70} -812.354 q^{71} +72.0000 q^{72} +794.385 q^{73} +252.969 q^{74} +735.504 q^{75} -294.505 q^{77} +171.073 q^{78} -637.672 q^{79} -307.836 q^{80} +81.0000 q^{81} -968.660 q^{82} -19.5995 q^{83} -216.783 q^{84} -418.369 q^{85} +48.3271 q^{86} +886.086 q^{87} +130.418 q^{88} +847.436 q^{89} -346.316 q^{90} -515.082 q^{91} -460.425 q^{92} +752.895 q^{93} +152.813 q^{94} +96.0000 q^{96} -1145.18 q^{97} -33.2909 q^{98} +146.720 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9} - 56 q^{10} - 72 q^{11} + 48 q^{12} - 8 q^{13} - 34 q^{14} - 84 q^{15} + 64 q^{16} + 33 q^{17} + 72 q^{18} - 112 q^{20} - 51 q^{21} - 144 q^{22} - 88 q^{23} + 96 q^{24} + 94 q^{25} - 16 q^{26} + 108 q^{27} - 68 q^{28} - 75 q^{29} - 168 q^{30} + 419 q^{31} + 128 q^{32} - 216 q^{33} + 66 q^{34} + 331 q^{35} + 144 q^{36} + 131 q^{37} - 24 q^{39} - 224 q^{40} - 1134 q^{41} - 102 q^{42} + 76 q^{43} - 288 q^{44} - 252 q^{45} - 176 q^{46} + 395 q^{47} + 192 q^{48} - 1033 q^{49} + 188 q^{50} + 99 q^{51} - 32 q^{52} - 625 q^{53} + 216 q^{54} + 413 q^{55} - 136 q^{56} - 150 q^{58} - 1328 q^{59} - 336 q^{60} - 1887 q^{61} + 838 q^{62} - 153 q^{63} + 256 q^{64} - 1817 q^{65} - 432 q^{66} - 99 q^{67} + 132 q^{68} - 264 q^{69} + 662 q^{70} - 1307 q^{71} + 288 q^{72} + 183 q^{73} + 262 q^{74} + 282 q^{75} - 513 q^{77} - 48 q^{78} - 2064 q^{79} - 448 q^{80} + 324 q^{81} - 2268 q^{82} + 816 q^{83} - 204 q^{84} - 867 q^{85} + 152 q^{86} - 225 q^{87} - 576 q^{88} + 207 q^{89} - 504 q^{90} - 183 q^{91} - 352 q^{92} + 1257 q^{93} + 790 q^{94} + 384 q^{96} - 1331 q^{97} - 2066 q^{98} - 648 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9 - 56 * q^10 - 72 * q^11 + 48 * q^12 - 8 * q^13 - 34 * q^14 - 84 * q^15 + 64 * q^16 + 33 * q^17 + 72 * q^18 - 112 * q^20 - 51 * q^21 - 144 * q^22 - 88 * q^23 + 96 * q^24 + 94 * q^25 - 16 * q^26 + 108 * q^27 - 68 * q^28 - 75 * q^29 - 168 * q^30 + 419 * q^31 + 128 * q^32 - 216 * q^33 + 66 * q^34 + 331 * q^35 + 144 * q^36 + 131 * q^37 - 24 * q^39 - 224 * q^40 - 1134 * q^41 - 102 * q^42 + 76 * q^43 - 288 * q^44 - 252 * q^45 - 176 * q^46 + 395 * q^47 + 192 * q^48 - 1033 * q^49 + 188 * q^50 + 99 * q^51 - 32 * q^52 - 625 * q^53 + 216 * q^54 + 413 * q^55 - 136 * q^56 - 150 * q^58 - 1328 * q^59 - 336 * q^60 - 1887 * q^61 + 838 * q^62 - 153 * q^63 + 256 * q^64 - 1817 * q^65 - 432 * q^66 - 99 * q^67 + 132 * q^68 - 264 * q^69 + 662 * q^70 - 1307 * q^71 + 288 * q^72 + 183 * q^73 + 262 * q^74 + 282 * q^75 - 513 * q^77 - 48 * q^78 - 2064 * q^79 - 448 * q^80 + 324 * q^81 - 2268 * q^82 + 816 * q^83 - 204 * q^84 - 867 * q^85 + 152 * q^86 - 225 * q^87 - 576 * q^88 + 207 * q^89 - 504 * q^90 - 183 * q^91 - 352 * q^92 + 1257 * q^93 + 790 * q^94 + 384 * q^96 - 1331 * q^97 - 2066 * q^98 - 648 * 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
$5$	−19.2398	−1.72086	−0.860428	−	0.509572i	$-0.829804\pi$
$5$	−19.2398	−1.72086	−0.860428	+	0.509572i	$0.829804\pi$
$6$	6.00000	0.408248
$7$	−18.0653	−0.975434	−0.487717	−	0.873002i	$-0.662170\pi$
$7$	−18.0653	−0.975434	−0.487717	+	0.873002i	$0.662170\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−38.4795	−1.21683
$11$	16.3023	0.446847	0.223424	−	0.974721i	$-0.428277\pi$
$11$	16.3023	0.446847	0.223424	+	0.974721i	$0.428277\pi$
$12$	12.0000	0.288675
$13$	28.5122	0.608298	0.304149	−	0.952624i	$-0.401628\pi$
$13$	28.5122	0.608298	0.304149	+	0.952624i	$0.401628\pi$
$14$	−36.1306	−0.689736
$15$	−57.7193	−0.993537
$16$	16.0000	0.250000
$17$	21.7450	0.310232	0.155116	−	0.987896i	$-0.450425\pi$
$17$	21.7450	0.310232	0.155116	+	0.987896i	$0.450425\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−76.9590	−0.860428
$21$	−54.1959	−0.563167
$22$	32.6046	0.315969
$23$	−115.106	−1.04353	−0.521767	−	0.853088i	$-0.674727\pi$
$23$	−115.106	−1.04353	−0.521767	+	0.853088i	$0.674727\pi$
$24$	24.0000	0.204124
$25$	245.168	1.96134
$26$	57.0245	0.430132
$27$	27.0000	0.192450
$28$	−72.2611	−0.487717
$29$	295.362	1.89129	0.945644	−	0.325203i	$-0.105433\pi$
$29$	295.362	1.89129	0.945644	+	0.325203i	$0.105433\pi$
$30$	−115.439	−0.702536
$31$	250.965	1.45402	0.727010	−	0.686627i	$-0.240909\pi$
$31$	250.965	1.45402	0.727010	+	0.686627i	$0.240909\pi$
$32$	32.0000	0.176777
$33$	48.9068	0.257987
$34$	43.4900	0.219367
$35$	347.572	1.67858
$36$	36.0000	0.166667
$37$	126.485	0.561998	0.280999	−	0.959708i	$-0.409334\pi$
$37$	126.485	0.561998	0.280999	+	0.959708i	$0.409334\pi$
$38$	0	0
$39$	85.5367	0.351201
$40$	−153.918	−0.608414
$41$	−484.330	−1.84487	−0.922434	−	0.386154i	$-0.873803\pi$
$41$	−484.330	−1.84487	−0.922434	+	0.386154i	$0.873803\pi$
$42$	−108.392	−0.398219
$43$	24.1636	0.0856956	0.0428478	−	0.999082i	$-0.486357\pi$
$43$	24.1636	0.0856956	0.0428478	+	0.999082i	$0.486357\pi$
$44$	65.2091	0.223424
$45$	−173.158	−0.573619
$46$	−230.212	−0.737890
$47$	76.4066	0.237129	0.118564	−	0.992946i	$-0.462171\pi$
$47$	76.4066	0.237129	0.118564	+	0.992946i	$0.462171\pi$
$48$	48.0000	0.144338
$49$	−16.6454	−0.0485289
$50$	490.336	1.38688
$51$	65.2350	0.179112
$52$	114.049	0.304149
$53$	−600.523	−1.55638	−0.778190	−	0.628028i	$-0.783862\pi$
$53$	−600.523	−1.55638	−0.778190	+	0.628028i	$0.783862\pi$
$54$	54.0000	0.136083
$55$	−313.652	−0.768960
$56$	−144.522	−0.344868
$57$	0	0
$58$	590.724	1.33734
$59$	−605.562	−1.33623	−0.668114	−	0.744059i	$-0.732898\pi$
$59$	−605.562	−1.33623	−0.668114	+	0.744059i	$0.732898\pi$
$60$	−230.877	−0.496768
$61$	−440.230	−0.924028	−0.462014	−	0.886873i	$-0.652873\pi$
$61$	−440.230	−0.924028	−0.462014	+	0.886873i	$0.652873\pi$
$62$	501.930	1.02815
$63$	−162.588	−0.325145
$64$	64.0000	0.125000
$65$	−548.568	−1.04679
$66$	97.8137	0.182425
$67$	−898.734	−1.63877	−0.819387	−	0.573240i	$-0.805686\pi$
$67$	−898.734	−1.63877	−0.819387	+	0.573240i	$0.805686\pi$
$68$	86.9800	0.155116
$69$	−345.319	−0.602485
$70$	695.143	1.18694
$71$	−812.354	−1.35787	−0.678935	−	0.734199i	$-0.737558\pi$
$71$	−812.354	−1.35787	−0.678935	+	0.734199i	$0.737558\pi$
$72$	72.0000	0.117851
$73$	794.385	1.27364	0.636820	−	0.771012i	$-0.280249\pi$
$73$	794.385	1.27364	0.636820	+	0.771012i	$0.280249\pi$
$74$	252.969	0.397393
$75$	735.504	1.13238
$76$	0	0
$77$	−294.505	−0.435870
$78$	171.073	0.248337
$79$	−637.672	−0.908148	−0.454074	−	0.890964i	$-0.650030\pi$
$79$	−637.672	−0.908148	−0.454074	+	0.890964i	$0.650030\pi$
$80$	−307.836	−0.430214
$81$	81.0000	0.111111
$82$	−968.660	−1.30452
$83$	−19.5995	−0.0259195	−0.0129598	−	0.999916i	$-0.504125\pi$
$83$	−19.5995	−0.0259195	−0.0129598	+	0.999916i	$0.504125\pi$
$84$	−216.783	−0.281583
$85$	−418.369	−0.533864
$86$	48.3271	0.0605959
$87$	886.086	1.09194
$88$	130.418	0.157984
$89$	847.436	1.00930	0.504652	−	0.863323i	$-0.331621\pi$
$89$	847.436	1.00930	0.504652	+	0.863323i	$0.331621\pi$
$90$	−346.316	−0.405610
$91$	−515.082	−0.593354
$92$	−460.425	−0.521767
$93$	752.895	0.839479
$94$	152.813	0.167675
$95$	0	0
$96$	96.0000	0.102062
$97$	−1145.18	−1.19872	−0.599358	−	0.800481i	$-0.704578\pi$
$97$	−1145.18	−1.19872	−0.599358	+	0.800481i	$0.704578\pi$
$98$	−33.2909	−0.0343151
$99$	146.720	0.148949
$100$	980.672	0.980672
$101$	846.280	0.833743	0.416871	−	0.908966i	$-0.363127\pi$
$101$	846.280	0.833743	0.416871	+	0.908966i	$0.363127\pi$
$102$	130.470	0.126652
$103$	−153.479	−0.146823	−0.0734114	−	0.997302i	$-0.523389\pi$
$103$	−153.479	−0.146823	−0.0734114	+	0.997302i	$0.523389\pi$
$104$	228.098	0.215066
$105$	1042.71	0.969129
$106$	−1201.05	−1.10053
$107$	−905.591	−0.818194	−0.409097	−	0.912491i	$-0.634156\pi$
$107$	−905.591	−0.818194	−0.409097	+	0.912491i	$0.634156\pi$
$108$	108.000	0.0962250
$109$	908.945	0.798726	0.399363	−	0.916793i	$-0.369231\pi$
$109$	908.945	0.798726	0.399363	+	0.916793i	$0.369231\pi$
$110$	−627.304	−0.543737
$111$	379.454	0.324470
$112$	−289.045	−0.243858
$113$	−380.502	−0.316766	−0.158383	−	0.987378i	$-0.550628\pi$
$113$	−380.502	−0.316766	−0.158383	+	0.987378i	$0.550628\pi$
$114$	0	0
$115$	2214.61	1.79577
$116$	1181.45	0.945644
$117$	256.610	0.202766
$118$	−1211.12	−0.944856
$119$	−392.830	−0.302611
$120$	−461.754	−0.351268
$121$	−1065.24	−0.800327
$122$	−880.460	−0.653386
$123$	−1452.99	−1.06514
$124$	1003.86	0.727010
$125$	−2312.00	−1.65434
$126$	−325.175	−0.229912
$127$	−364.339	−0.254566	−0.127283	−	0.991866i	$-0.540626\pi$
$127$	−364.339	−0.254566	−0.127283	+	0.991866i	$0.540626\pi$
$128$	128.000	0.0883883
$129$	72.4907	0.0494764
$130$	−1097.14	−0.740194
$131$	−2503.63	−1.66980	−0.834899	−	0.550403i	$-0.814474\pi$
$131$	−2503.63	−1.66980	−0.834899	+	0.550403i	$0.814474\pi$
$132$	195.627	0.128994
$133$	0	0
$134$	−1797.47	−1.15879
$135$	−519.473	−0.331179
$136$	173.960	0.109683
$137$	1553.08	0.968528	0.484264	−	0.874922i	$-0.339087\pi$
$137$	1553.08	0.968528	0.484264	+	0.874922i	$0.339087\pi$
$138$	−690.637	−0.426021
$139$	−652.442	−0.398125	−0.199062	−	0.979987i	$-0.563790\pi$
$139$	−652.442	−0.398125	−0.199062	+	0.979987i	$0.563790\pi$
$140$	1390.29	0.839290
$141$	229.220	0.136906
$142$	−1624.71	−0.960158
$143$	464.814	0.271816
$144$	144.000	0.0833333
$145$	−5682.69	−3.25463
$146$	1588.77	0.900600
$147$	−49.9363	−0.0280182
$148$	505.938	0.280999
$149$	2840.42	1.56172	0.780860	−	0.624706i	$-0.214781\pi$
$149$	2840.42	1.56172	0.780860	+	0.624706i	$0.214781\pi$
$150$	1471.01	0.800716
$151$	−786.715	−0.423987	−0.211993	−	0.977271i	$-0.567995\pi$
$151$	−786.715	−0.423987	−0.211993	+	0.977271i	$0.567995\pi$
$152$	0	0
$153$	195.705	0.103411
$154$	−589.011	−0.308207
$155$	−4828.50	−2.50216
$156$	342.147	0.175600
$157$	−2202.50	−1.11961	−0.559804	−	0.828625i	$-0.689124\pi$
$157$	−2202.50	−1.11961	−0.559804	+	0.828625i	$0.689124\pi$
$158$	−1275.34	−0.642158
$159$	−1801.57	−0.898577
$160$	−615.672	−0.304207
$161$	2079.43	1.01790
$162$	162.000	0.0785674
$163$	−2675.07	−1.28544	−0.642722	−	0.766099i	$-0.722195\pi$
$163$	−2675.07	−1.28544	−0.642722	+	0.766099i	$0.722195\pi$
$164$	−1937.32	−0.922434
$165$	−940.955	−0.443959
$166$	−39.1989	−0.0183279
$167$	250.094	0.115885	0.0579426	−	0.998320i	$-0.481546\pi$
$167$	250.094	0.115885	0.0579426	+	0.998320i	$0.481546\pi$
$168$	−433.567	−0.199110
$169$	−1384.05	−0.629974
$170$	−836.737	−0.377499
$171$	0	0
$172$	96.6543	0.0428478
$173$	1071.62	0.470946	0.235473	−	0.971881i	$-0.424336\pi$
$173$	1071.62	0.470946	0.235473	+	0.971881i	$0.424336\pi$
$174$	1772.17	0.772115
$175$	−4429.03	−1.91316
$176$	260.836	0.111712
$177$	−1816.69	−0.771471
$178$	1694.87	0.713685
$179$	1751.48	0.731351	0.365676	−	0.930742i	$-0.380838\pi$
$179$	1751.48	0.731351	0.365676	+	0.930742i	$0.380838\pi$
$180$	−692.631	−0.286809
$181$	−45.6569	−0.0187495	−0.00937473	−	0.999956i	$-0.502984\pi$
$181$	−45.6569	−0.0187495	−0.00937473	+	0.999956i	$0.502984\pi$
$182$	−1030.16	−0.419565
$183$	−1320.69	−0.533488
$184$	−920.849	−0.368945
$185$	−2433.53	−0.967117
$186$	1505.79	0.593601
$187$	354.493	0.138626
$188$	305.626	0.118564
$189$	−487.763	−0.187722
$190$	0	0
$191$	−4572.58	−1.73225	−0.866127	−	0.499823i	$-0.833398\pi$
$191$	−4572.58	−1.73225	−0.866127	+	0.499823i	$0.833398\pi$
$192$	192.000	0.0721688
$193$	3703.26	1.38117	0.690586	−	0.723250i	$-0.257353\pi$
$193$	3703.26	1.38117	0.690586	+	0.723250i	$0.257353\pi$
$194$	−2290.36	−0.847621
$195$	−1645.71	−0.604366
$196$	−66.5817	−0.0242645
$197$	1839.32	0.665210	0.332605	−	0.943066i	$-0.392072\pi$
$197$	1839.32	0.665210	0.332605	+	0.943066i	$0.392072\pi$
$198$	293.441	0.105323
$199$	4362.59	1.55405	0.777025	−	0.629470i	$-0.216728\pi$
$199$	4362.59	1.55405	0.777025	+	0.629470i	$0.216728\pi$
$200$	1961.34	0.693440
$201$	−2696.20	−0.946147
$202$	1692.56	0.589545
$203$	−5335.80	−1.84483
$204$	260.940	0.0895562
$205$	9318.39	3.17475
$206$	−306.958	−0.103819
$207$	−1035.96	−0.347845
$208$	456.196	0.152074
$209$	0	0
$210$	2085.43	0.685278
$211$	3887.74	1.26845	0.634225	−	0.773149i	$-0.281319\pi$
$211$	3887.74	1.26845	0.634225	+	0.773149i	$0.281319\pi$
$212$	−2402.09	−0.778190
$213$	−2437.06	−0.783966
$214$	−1811.18	−0.578550
$215$	−464.901	−0.147470
$216$	216.000	0.0680414
$217$	−4533.75	−1.41830
$218$	1817.89	0.564784
$219$	2383.16	0.735336
$220$	−1254.61	−0.384480
$221$	619.999	0.188713
$222$	758.907	0.229435
$223$	3410.33	1.02409	0.512047	−	0.858958i	$-0.328888\pi$
$223$	3410.33	1.02409	0.512047	+	0.858958i	$0.328888\pi$
$224$	−578.089	−0.172434
$225$	2206.51	0.653781
$226$	−761.004	−0.223988
$227$	−4292.40	−1.25505	−0.627525	−	0.778597i	$-0.715932\pi$
$227$	−4292.40	−1.25505	−0.627525	+	0.778597i	$0.715932\pi$
$228$	0	0
$229$	−1852.15	−0.534468	−0.267234	−	0.963632i	$-0.586110\pi$
$229$	−1852.15	−0.534468	−0.267234	+	0.963632i	$0.586110\pi$
$230$	4429.23	1.26980
$231$	−883.516	−0.251650
$232$	2362.90	0.668671
$233$	604.211	0.169885	0.0849425	−	0.996386i	$-0.472929\pi$
$233$	604.211	0.169885	0.0849425	+	0.996386i	$0.472929\pi$
$234$	513.220	0.143377
$235$	−1470.04	−0.408064
$236$	−2422.25	−0.668114
$237$	−1913.02	−0.524320
$238$	−785.660	−0.213978
$239$	−7110.26	−1.92437	−0.962185	−	0.272398i	$-0.912183\pi$
$239$	−7110.26	−1.92437	−0.962185	+	0.272398i	$0.912183\pi$
$240$	−923.508	−0.248384
$241$	−28.1168	−0.00751519	−0.00375759	−	0.999993i	$-0.501196\pi$
$241$	−28.1168	−0.00751519	−0.00375759	+	0.999993i	$0.501196\pi$
$242$	−2130.47	−0.565917
$243$	243.000	0.0641500
$244$	−1760.92	−0.462014
$245$	320.254	0.0835113
$246$	−2905.98	−0.753165
$247$	0	0
$248$	2007.72	0.514074
$249$	−58.7984	−0.0149646
$250$	−4624.01	−1.16979
$251$	4155.34	1.04495	0.522476	−	0.852654i	$-0.325008\pi$
$251$	4155.34	1.04495	0.522476	+	0.852654i	$0.325008\pi$
$252$	−650.350	−0.162572
$253$	−1876.49	−0.466301
$254$	−728.679	−0.180005
$255$	−1255.11	−0.308227
$256$	256.000	0.0625000
$257$	−4305.97	−1.04513	−0.522566	−	0.852599i	$-0.675025\pi$
$257$	−4305.97	−1.04513	−0.522566	+	0.852599i	$0.675025\pi$
$258$	144.981	0.0349851
$259$	−2284.98	−0.548192
$260$	−2194.27	−0.523396
$261$	2658.26	0.630430
$262$	−5007.27	−1.18073
$263$	4735.18	1.11020	0.555102	−	0.831782i	$-0.312679\pi$
$263$	4735.18	1.11020	0.555102	+	0.831782i	$0.312679\pi$
$264$	391.255	0.0912123
$265$	11553.9	2.67831
$266$	0	0
$267$	2542.31	0.582722
$268$	−3594.94	−0.819387
$269$	−5555.76	−1.25926	−0.629630	−	0.776895i	$-0.716793\pi$
$269$	−5555.76	−1.25926	−0.629630	+	0.776895i	$0.716793\pi$
$270$	−1038.95	−0.234179
$271$	−889.827	−0.199458	−0.0997290	−	0.995015i	$-0.531798\pi$
$271$	−889.827	−0.199458	−0.0997290	+	0.995015i	$0.531798\pi$
$272$	347.920	0.0775579
$273$	−1545.25	−0.342573
$274$	3106.16	0.684853
$275$	3996.80	0.876422
$276$	−1381.27	−0.301243
$277$	718.326	0.155812	0.0779062	−	0.996961i	$-0.475177\pi$
$277$	718.326	0.155812	0.0779062	+	0.996961i	$0.475177\pi$
$278$	−1304.88	−0.281517
$279$	2258.68	0.484673
$280$	2780.57	0.593468
$281$	−6371.16	−1.35257	−0.676284	−	0.736641i	$-0.736411\pi$
$281$	−6371.16	−1.35257	−0.676284	+	0.736641i	$0.736411\pi$
$282$	458.440	0.0968074
$283$	322.116	0.0676601	0.0338300	−	0.999428i	$-0.489230\pi$
$283$	322.116	0.0676601	0.0338300	+	0.999428i	$0.489230\pi$
$284$	−3249.42	−0.678935
$285$	0	0
$286$	929.629	0.192203
$287$	8749.56	1.79955
$288$	288.000	0.0589256
$289$	−4440.15	−0.903756
$290$	−11365.4	−2.30137
$291$	−3435.54	−0.692079
$292$	3177.54	0.636820
$293$	−8788.72	−1.75236	−0.876182	−	0.481980i	$-0.839918\pi$
$293$	−8788.72	−1.75236	−0.876182	+	0.481980i	$0.839918\pi$
$294$	−99.8726	−0.0198119
$295$	11650.9	2.29946
$296$	1011.88	0.198696
$297$	440.161	0.0859958
$298$	5680.84	1.10430
$299$	−3281.94	−0.634780
$300$	2942.02	0.566191
$301$	−436.522	−0.0835903
$302$	−1573.43	−0.299804
$303$	2538.84	0.481362
$304$	0	0
$305$	8469.92	1.59012
$306$	391.410	0.0731223
$307$	−5862.73	−1.08991	−0.544957	−	0.838464i	$-0.683454\pi$
$307$	−5862.73	−1.08991	−0.544957	+	0.838464i	$0.683454\pi$
$308$	−1178.02	−0.217935
$309$	−460.438	−0.0847682
$310$	−9657.01	−1.76929
$311$	3932.00	0.716924	0.358462	−	0.933544i	$-0.383301\pi$
$311$	3932.00	0.716924	0.358462	+	0.933544i	$0.383301\pi$
$312$	684.294	0.124168
$313$	8991.00	1.62365	0.811823	−	0.583903i	$-0.198475\pi$
$313$	8991.00	1.62365	0.811823	+	0.583903i	$0.198475\pi$
$314$	−4405.00	−0.791682
$315$	3128.14	0.559527
$316$	−2550.69	−0.454074
$317$	7803.21	1.38256	0.691280	−	0.722587i	$-0.257047\pi$
$317$	7803.21	1.38256	0.691280	+	0.722587i	$0.257047\pi$
$318$	−3603.14	−0.635390
$319$	4815.08	0.845117
$320$	−1231.34	−0.215107
$321$	−2716.77	−0.472384
$322$	4158.85	0.719763
$323$	0	0
$324$	324.000	0.0555556
$325$	6990.29	1.19308
$326$	−5350.13	−0.908946
$327$	2726.83	0.461145
$328$	−3874.64	−0.652260
$329$	−1380.31	−0.231303
$330$	−1881.91	−0.313927
$331$	4567.99	0.758548	0.379274	−	0.925284i	$-0.376174\pi$
$331$	4567.99	0.758548	0.379274	+	0.925284i	$0.376174\pi$
$332$	−78.3978	−0.0129598
$333$	1138.36	0.187333
$334$	500.187	0.0819432
$335$	17291.4	2.82009
$336$	−867.134	−0.140792
$337$	−10244.1	−1.65589	−0.827944	−	0.560811i	$-0.810489\pi$
$337$	−10244.1	−1.65589	−0.827944	+	0.560811i	$0.810489\pi$
$338$	−2768.10	−0.445459
$339$	−1141.51	−0.182885
$340$	−1673.47	−0.266932
$341$	4091.30	0.649725
$342$	0	0
$343$	6497.10	1.02277
$344$	193.309	0.0302980
$345$	6643.84	1.03679
$346$	2143.24	0.333009
$347$	−10463.8	−1.61880	−0.809402	−	0.587254i	$-0.800209\pi$
$347$	−10463.8	−1.61880	−0.809402	+	0.587254i	$0.800209\pi$
$348$	3544.35	0.545968
$349$	−4023.52	−0.617118	−0.308559	−	0.951205i	$-0.599847\pi$
$349$	−4023.52	−0.617118	−0.308559	+	0.951205i	$0.599847\pi$
$350$	−8858.06	−1.35281
$351$	769.830	0.117067
$352$	521.673	0.0789922
$353$	−3273.39	−0.493556	−0.246778	−	0.969072i	$-0.579372\pi$
$353$	−3273.39	−0.493556	−0.246778	+	0.969072i	$0.579372\pi$
$354$	−3633.37	−0.545513
$355$	15629.5	2.33670
$356$	3389.74	0.504652
$357$	−1178.49	−0.174712
$358$	3502.96	0.517143
$359$	−11181.2	−1.64379	−0.821893	−	0.569642i	$-0.807082\pi$
$359$	−11181.2	−1.64379	−0.821893	+	0.569642i	$0.807082\pi$
$360$	−1385.26	−0.202805
$361$	0	0
$362$	−91.3138	−0.0132579
$363$	−3195.71	−0.462069
$364$	−2060.33	−0.296677
$365$	−15283.8	−2.19175
$366$	−2641.38	−0.377233
$367$	7343.64	1.04451	0.522255	−	0.852790i	$-0.325091\pi$
$367$	7343.64	1.04451	0.522255	+	0.852790i	$0.325091\pi$
$368$	−1841.70	−0.260884
$369$	−4358.97	−0.614956
$370$	−4867.06	−0.683855
$371$	10848.6	1.51815
$372$	3011.58	0.419739
$373$	1947.63	0.270361	0.135180	−	0.990821i	$-0.456839\pi$
$373$	1947.63	0.270361	0.135180	+	0.990821i	$0.456839\pi$
$374$	708.986	0.0980236
$375$	−6936.01	−0.955131
$376$	611.253	0.0838376
$377$	8421.44	1.15047
$378$	−975.525	−0.132740
$379$	−4998.26	−0.677424	−0.338712	−	0.940890i	$-0.609991\pi$
$379$	−4998.26	−0.677424	−0.338712	+	0.940890i	$0.609991\pi$
$380$	0	0
$381$	−1093.02	−0.146974
$382$	−9145.17	−1.22489
$383$	−60.4794	−0.00806881	−0.00403441	−	0.999992i	$-0.501284\pi$
$383$	−60.4794	−0.00806881	−0.00403441	+	0.999992i	$0.501284\pi$
$384$	384.000	0.0510310
$385$	5666.21	0.750070
$386$	7406.51	0.976636
$387$	217.472	0.0285652
$388$	−4580.72	−0.599358
$389$	−13025.1	−1.69768	−0.848841	−	0.528648i	$-0.822699\pi$
$389$	−13025.1	−1.69768	−0.848841	+	0.528648i	$0.822699\pi$
$390$	−3291.41	−0.427351
$391$	−2502.99	−0.323738
$392$	−133.163	−0.0171576
$393$	−7510.90	−0.964058
$394$	3678.65	0.470374
$395$	12268.6	1.56279
$396$	586.882	0.0744746
$397$	−1406.48	−0.177807	−0.0889036	−	0.996040i	$-0.528336\pi$
$397$	−1406.48	−0.177807	−0.0889036	+	0.996040i	$0.528336\pi$
$398$	8725.18	1.09888
$399$	0	0
$400$	3922.69	0.490336
$401$	−12553.6	−1.56333	−0.781665	−	0.623699i	$-0.785629\pi$
$401$	−12553.6	−1.56333	−0.781665	+	0.623699i	$0.785629\pi$
$402$	−5392.41	−0.669027
$403$	7155.57	0.884477
$404$	3385.12	0.416871
$405$	−1558.42	−0.191206
$406$	−10671.6	−1.30449
$407$	2061.99	0.251127
$408$	521.880	0.0633258
$409$	6624.84	0.800922	0.400461	−	0.916314i	$-0.368850\pi$
$409$	6624.84	0.800922	0.400461	+	0.916314i	$0.368850\pi$
$410$	18636.8	2.24489
$411$	4659.23	0.559180
$412$	−613.917	−0.0734114
$413$	10939.6	1.30340
$414$	−2071.91	−0.245963
$415$	377.089	0.0446037
$416$	912.392	0.107533
$417$	−1957.32	−0.229858
$418$	0	0
$419$	8837.39	1.03039	0.515196	−	0.857072i	$-0.327719\pi$
$419$	8837.39	1.03039	0.515196	+	0.857072i	$0.327719\pi$
$420$	4170.86	0.484565
$421$	−2167.46	−0.250916	−0.125458	−	0.992099i	$-0.540040\pi$
$421$	−2167.46	−0.250916	−0.125458	+	0.992099i	$0.540040\pi$
$422$	7775.47	0.896929
$423$	687.659	0.0790429
$424$	−4804.18	−0.550264
$425$	5331.18	0.608471
$426$	−4874.13	−0.554348
$427$	7952.89	0.901328
$428$	−3622.36	−0.409097
$429$	1394.44	0.156933
$430$	−929.802	−0.104277
$431$	−8512.46	−0.951348	−0.475674	−	0.879622i	$-0.657796\pi$
$431$	−8512.46	−0.951348	−0.475674	+	0.879622i	$0.657796\pi$
$432$	432.000	0.0481125
$433$	15189.2	1.68579	0.842894	−	0.538079i	$-0.180850\pi$
$433$	15189.2	1.68579	0.842894	+	0.538079i	$0.180850\pi$
$434$	−9067.51	−1.00289
$435$	−17048.1	−1.87906
$436$	3635.78	0.399363
$437$	0	0
$438$	4766.31	0.519961
$439$	7930.92	0.862237	0.431118	−	0.902295i	$-0.358119\pi$
$439$	7930.92	0.862237	0.431118	+	0.902295i	$0.358119\pi$
$440$	−2509.21	−0.271868
$441$	−149.809	−0.0161763
$442$	1240.00	0.133440
$443$	−6766.59	−0.725712	−0.362856	−	0.931845i	$-0.618198\pi$
$443$	−6766.59	−0.725712	−0.362856	+	0.931845i	$0.618198\pi$
$444$	1517.81	0.162235
$445$	−16304.5	−1.73687
$446$	6820.67	0.724143
$447$	8521.26	0.901660
$448$	−1156.18	−0.121929
$449$	5064.96	0.532361	0.266181	−	0.963923i	$-0.414238\pi$
$449$	5064.96	0.532361	0.266181	+	0.963923i	$0.414238\pi$
$450$	4413.03	0.462293
$451$	−7895.68	−0.824375
$452$	−1522.01	−0.158383
$453$	−2360.15	−0.244789
$454$	−8584.79	−0.887454
$455$	9910.05	1.02108
$456$	0	0
$457$	−8306.44	−0.850238	−0.425119	−	0.905137i	$-0.639768\pi$
$457$	−8306.44	−0.850238	−0.425119	+	0.905137i	$0.639768\pi$
$458$	−3704.29	−0.377926
$459$	587.115	0.0597041
$460$	8858.46	0.897886
$461$	−2196.28	−0.221890	−0.110945	−	0.993827i	$-0.535388\pi$
$461$	−2196.28	−0.221890	−0.110945	+	0.993827i	$0.535388\pi$
$462$	−1767.03	−0.177943
$463$	−4257.93	−0.427393	−0.213696	−	0.976900i	$-0.568550\pi$
$463$	−4257.93	−0.427393	−0.213696	+	0.976900i	$0.568550\pi$
$464$	4725.79	0.472822
$465$	−14485.5	−1.44462
$466$	1208.42	0.120127
$467$	−4618.14	−0.457606	−0.228803	−	0.973473i	$-0.573481\pi$
$467$	−4618.14	−0.457606	−0.228803	+	0.973473i	$0.573481\pi$
$468$	1026.44	0.101383
$469$	16235.9	1.59852
$470$	−2940.09	−0.288545
$471$	−6607.49	−0.646406
$472$	−4844.49	−0.472428
$473$	393.921	0.0382928
$474$	−3826.03	−0.370750
$475$	0	0
$476$	−1571.32	−0.151305
$477$	−5404.71	−0.518794
$478$	−14220.5	−1.36073
$479$	−1727.99	−0.164831	−0.0824155	−	0.996598i	$-0.526263\pi$
$479$	−1727.99	−0.164831	−0.0824155	+	0.996598i	$0.526263\pi$
$480$	−1847.02	−0.175634
$481$	3606.36	0.341862
$482$	−56.2335	−0.00531404
$483$	6238.28	0.587684
$484$	−4260.94	−0.400164
$485$	22033.0	2.06282
$486$	486.000	0.0453609
$487$	12555.5	1.16827	0.584134	−	0.811657i	$-0.301434\pi$
$487$	12555.5	1.16827	0.584134	+	0.811657i	$0.301434\pi$
$488$	−3521.84	−0.326693
$489$	−8025.20	−0.742152
$490$	640.508	0.0590514
$491$	9491.11	0.872358	0.436179	−	0.899860i	$-0.356332\pi$
$491$	9491.11	0.872358	0.436179	+	0.899860i	$0.356332\pi$
$492$	−5811.96	−0.532568
$493$	6422.65	0.586738
$494$	0	0
$495$	−2822.87	−0.256320
$496$	4015.44	0.363505
$497$	14675.4	1.32451
$498$	−117.597	−0.0105816
$499$	2930.59	0.262909	0.131454	−	0.991322i	$-0.458035\pi$
$499$	2930.59	0.262909	0.131454	+	0.991322i	$0.458035\pi$
$500$	−9248.01	−0.827168
$501$	750.281	0.0669064
$502$	8310.68	0.738892
$503$	−2343.87	−0.207769	−0.103885	−	0.994589i	$-0.533127\pi$
$503$	−2343.87	−0.207769	−0.103885	+	0.994589i	$0.533127\pi$
$504$	−1300.70	−0.114956
$505$	−16282.2	−1.43475
$506$	−3752.99	−0.329724
$507$	−4152.16	−0.363715
$508$	−1457.36	−0.127283
$509$	8976.91	0.781718	0.390859	−	0.920451i	$-0.372178\pi$
$509$	8976.91	0.781718	0.390859	+	0.920451i	$0.372178\pi$
$510$	−2510.21	−0.217949
$511$	−14350.8	−1.24235
$512$	512.000	0.0441942
$513$	0	0
$514$	−8611.93	−0.739019
$515$	2952.90	0.252661
$516$	289.963	0.0247382
$517$	1245.60	0.105960
$518$	−4569.96	−0.387630
$519$	3214.86	0.271901
$520$	−4388.55	−0.370097
$521$	−16201.3	−1.36236	−0.681182	−	0.732114i	$-0.738534\pi$
$521$	−16201.3	−1.36236	−0.681182	+	0.732114i	$0.738534\pi$
$522$	5316.52	0.445781
$523$	−16705.1	−1.39668	−0.698340	−	0.715766i	$-0.746078\pi$
$523$	−16705.1	−1.39668	−0.698340	+	0.715766i	$0.746078\pi$
$524$	−10014.5	−0.834899
$525$	−13287.1	−1.10456
$526$	9470.36	0.785033
$527$	5457.23	0.451083
$528$	782.509	0.0644969
$529$	1082.43	0.0889647
$530$	23107.8	1.89385
$531$	−5450.06	−0.445409
$532$	0	0
$533$	−13809.3	−1.12223
$534$	5084.61	0.412046
$535$	17423.3	1.40799
$536$	−7189.88	−0.579394
$537$	5254.44	0.422246
$538$	−11111.5	−0.890431
$539$	−271.358	−0.0216850
$540$	−2077.89	−0.165589
$541$	−21446.0	−1.70431	−0.852157	−	0.523286i	$-0.824706\pi$
$541$	−21446.0	−1.70431	−0.852157	+	0.523286i	$0.824706\pi$
$542$	−1779.65	−0.141038
$543$	−136.971	−0.0108250
$544$	695.840	0.0548417
$545$	−17487.9	−1.37449
$546$	−3090.49	−0.242236
$547$	14560.4	1.13813	0.569067	−	0.822291i	$-0.307305\pi$
$547$	14560.4	1.13813	0.569067	+	0.822291i	$0.307305\pi$
$548$	6212.31	0.484264
$549$	−3962.07	−0.308009
$550$	7993.60	0.619724
$551$	0	0
$552$	−2762.55	−0.213011
$553$	11519.7	0.885838
$554$	1436.65	0.110176
$555$	−7300.59	−0.558365
$556$	−2609.77	−0.199062
$557$	15148.7	1.15238	0.576188	−	0.817317i	$-0.304540\pi$
$557$	15148.7	1.15238	0.576188	+	0.817317i	$0.304540\pi$
$558$	4517.37	0.342716
$559$	688.957	0.0521284
$560$	5561.15	0.419645
$561$	1063.48	0.0800359
$562$	−12742.3	−0.956410
$563$	581.773	0.0435503	0.0217751	−	0.999763i	$-0.493068\pi$
$563$	581.773	0.0435503	0.0217751	+	0.999763i	$0.493068\pi$
$564$	916.879	0.0684532
$565$	7320.76	0.545109
$566$	644.232	0.0478429
$567$	−1463.29	−0.108382
$568$	−6498.83	−0.480079
$569$	13602.6	1.00220	0.501098	−	0.865390i	$-0.332930\pi$
$569$	13602.6	1.00220	0.501098	+	0.865390i	$0.332930\pi$
$570$	0	0
$571$	−21086.1	−1.54540	−0.772700	−	0.634771i	$-0.781094\pi$
$571$	−21086.1	−1.54540	−0.772700	+	0.634771i	$0.781094\pi$
$572$	1859.26	0.135908
$573$	−13717.8	−1.00012
$574$	17499.1	1.27247
$575$	−28220.4	−2.04673
$576$	576.000	0.0416667
$577$	5631.33	0.406300	0.203150	−	0.979148i	$-0.434882\pi$
$577$	5631.33	0.406300	0.203150	+	0.979148i	$0.434882\pi$
$578$	−8880.31	−0.639052
$579$	11109.8	0.797420
$580$	−22730.8	−1.62732
$581$	354.070	0.0252828
$582$	−6871.09	−0.489374
$583$	−9789.89	−0.695465
$584$	6355.08	0.450300
$585$	−4937.12	−0.348931
$586$	−17577.4	−1.23911
$587$	4246.10	0.298561	0.149281	−	0.988795i	$-0.452304\pi$
$587$	4246.10	0.298561	0.149281	+	0.988795i	$0.452304\pi$
$588$	−199.745	−0.0140091
$589$	0	0
$590$	23301.7	1.62596
$591$	5517.97	0.384059
$592$	2023.75	0.140499
$593$	5040.63	0.349062	0.174531	−	0.984652i	$-0.444159\pi$
$593$	5040.63	0.349062	0.174531	+	0.984652i	$0.444159\pi$
$594$	880.323	0.0608082
$595$	7557.95	0.520749
$596$	11361.7	0.780860
$597$	13087.8	0.897231
$598$	−6563.87	−0.448857
$599$	−8562.91	−0.584092	−0.292046	−	0.956404i	$-0.594336\pi$
$599$	−8562.91	−0.584092	−0.292046	+	0.956404i	$0.594336\pi$
$600$	5884.03	0.400358
$601$	2352.29	0.159654	0.0798269	−	0.996809i	$-0.474563\pi$
$601$	2352.29	0.159654	0.0798269	+	0.996809i	$0.474563\pi$
$602$	−873.043	−0.0591073
$603$	−8088.61	−0.546258
$604$	−3146.86	−0.211993
$605$	20494.9	1.37725
$606$	5077.68	0.340374
$607$	−15735.1	−1.05217	−0.526084	−	0.850433i	$-0.676340\pi$
$607$	−15735.1	−1.05217	−0.526084	+	0.850433i	$0.676340\pi$
$608$	0	0
$609$	−16007.4	−1.06511
$610$	16939.8	1.12438
$611$	2178.52	0.144245
$612$	782.820	0.0517053
$613$	6399.92	0.421681	0.210841	−	0.977520i	$-0.432380\pi$
$613$	6399.92	0.421681	0.210841	+	0.977520i	$0.432380\pi$
$614$	−11725.5	−0.770685
$615$	27955.2	1.83294
$616$	−2356.04	−0.154103
$617$	−21863.0	−1.42653	−0.713267	−	0.700892i	$-0.752785\pi$
$617$	−21863.0	−1.42653	−0.713267	+	0.700892i	$0.752785\pi$
$618$	−920.875	−0.0599402
$619$	8081.41	0.524749	0.262374	−	0.964966i	$-0.415494\pi$
$619$	8081.41	0.524749	0.262374	+	0.964966i	$0.415494\pi$
$620$	−19314.0	−1.25108
$621$	−3107.87	−0.200828
$622$	7864.01	0.506942
$623$	−15309.2	−0.984509
$624$	1368.59	0.0878002
$625$	13836.4	0.885528
$626$	17982.0	1.14809
$627$	0	0
$628$	−8809.99	−0.559804
$629$	2750.41	0.174350
$630$	6256.29	0.395645
$631$	21717.0	1.37011	0.685055	−	0.728491i	$-0.259778\pi$
$631$	21717.0	1.37011	0.685055	+	0.728491i	$0.259778\pi$
$632$	−5101.38	−0.321079
$633$	11663.2	0.732339
$634$	15606.4	0.977618
$635$	7009.80	0.438072
$636$	−7206.28	−0.449288
$637$	−474.598	−0.0295201
$638$	9630.15	0.597588
$639$	−7311.19	−0.452623
$640$	−2462.69	−0.152104
$641$	18967.7	1.16876	0.584382	−	0.811479i	$-0.301337\pi$
$641$	18967.7	1.16876	0.584382	+	0.811479i	$0.301337\pi$
$642$	−5433.54	−0.334026
$643$	−19322.9	−1.18510	−0.592552	−	0.805532i	$-0.701880\pi$
$643$	−19322.9	−1.18510	−0.592552	+	0.805532i	$0.701880\pi$
$644$	8317.70	0.508949
$645$	−1394.70	−0.0851417
$646$	0	0
$647$	11022.5	0.669764	0.334882	−	0.942260i	$-0.391303\pi$
$647$	11022.5	0.669764	0.334882	+	0.942260i	$0.391303\pi$
$648$	648.000	0.0392837
$649$	−9872.04	−0.597090
$650$	13980.6	0.843636
$651$	−13601.3	−0.818856
$652$	−10700.3	−0.642722
$653$	7031.74	0.421399	0.210699	−	0.977551i	$-0.432426\pi$
$653$	7031.74	0.421399	0.210699	+	0.977551i	$0.432426\pi$
$654$	5453.67	0.326078
$655$	48169.3	2.87348
$656$	−7749.28	−0.461217
$657$	7149.47	0.424547
$658$	−2760.61	−0.163556
$659$	−26220.9	−1.54996	−0.774978	−	0.631989i	$-0.782239\pi$
$659$	−26220.9	−1.54996	−0.774978	+	0.631989i	$0.782239\pi$
$660$	−3763.82	−0.221980
$661$	9214.18	0.542194	0.271097	−	0.962552i	$-0.412614\pi$
$661$	9214.18	0.542194	0.271097	+	0.962552i	$0.412614\pi$
$662$	9135.98	0.536374
$663$	1860.00	0.108954
$664$	−156.796	−0.00916393
$665$	0	0
$666$	2276.72	0.132464
$667$	−33998.0	−1.97363
$668$	1000.37	0.0579426
$669$	10231.0	0.591260
$670$	34582.9	1.99411
$671$	−7176.76	−0.412900
$672$	−1734.27	−0.0995548
$673$	−4760.01	−0.272637	−0.136319	−	0.990665i	$-0.543527\pi$
$673$	−4760.01	−0.272637	−0.136319	+	0.990665i	$0.543527\pi$
$674$	−20488.3	−1.17089
$675$	6619.54	0.377461
$676$	−5536.21	−0.314987
$677$	−9064.23	−0.514574	−0.257287	−	0.966335i	$-0.582829\pi$
$677$	−9064.23	−0.514574	−0.257287	+	0.966335i	$0.582829\pi$
$678$	−2283.01	−0.129319
$679$	20688.0	1.16927
$680$	−3346.95	−0.188749
$681$	−12877.2	−0.724603
$682$	8182.60	0.459425
$683$	−21037.2	−1.17858	−0.589288	−	0.807923i	$-0.700592\pi$
$683$	−21037.2	−1.17858	−0.589288	+	0.807923i	$0.700592\pi$
$684$	0	0
$685$	−29880.8	−1.66670
$686$	12994.2	0.723208
$687$	−5556.44	−0.308575
$688$	386.617	0.0214239
$689$	−17122.3	−0.946743
$690$	13287.7	0.733121
$691$	−33207.3	−1.82817	−0.914086	−	0.405520i	$-0.867090\pi$
$691$	−33207.3	−1.82817	−0.914086	+	0.405520i	$0.867090\pi$
$692$	4286.48	0.235473
$693$	−2650.55	−0.145290
$694$	−20927.6	−1.14467
$695$	12552.8	0.685116
$696$	7088.69	0.386058
$697$	−10531.8	−0.572337
$698$	−8047.04	−0.436368
$699$	1812.63	0.0980831
$700$	−17716.1	−0.956581
$701$	12181.8	0.656347	0.328173	−	0.944618i	$-0.393567\pi$
$701$	12181.8	0.656347	0.328173	+	0.944618i	$0.393567\pi$
$702$	1539.66	0.0827789
$703$	0	0
$704$	1043.35	0.0558559
$705$	−4410.13	−0.235596
$706$	−6546.79	−0.348997
$707$	−15288.3	−0.813261
$708$	−7266.74	−0.385736
$709$	−19592.1	−1.03779	−0.518897	−	0.854837i	$-0.673657\pi$
$709$	−19592.1	−1.03779	−0.518897	+	0.854837i	$0.673657\pi$
$710$	31259.0	1.65229
$711$	−5739.05	−0.302716
$712$	6779.49	0.356843
$713$	−28887.6	−1.51732
$714$	−2356.98	−0.123540
$715$	−8942.91	−0.467757
$716$	7005.93	0.365676
$717$	−21330.8	−1.11104
$718$	−22362.3	−1.16233
$719$	12211.3	0.633387	0.316693	−	0.948528i	$-0.397427\pi$
$719$	12211.3	0.633387	0.316693	+	0.948528i	$0.397427\pi$
$720$	−2770.52	−0.143405
$721$	2772.65	0.143216
$722$	0	0
$723$	−84.3503	−0.00433890
$724$	−182.628	−0.00937473
$725$	72413.4	3.70947
$726$	−6391.41	−0.326732
$727$	22063.4	1.12556	0.562782	−	0.826606i	$-0.309731\pi$
$727$	22063.4	1.12556	0.562782	+	0.826606i	$0.309731\pi$
$728$	−4120.65	−0.209782
$729$	729.000	0.0370370
$730$	−30567.5	−1.54980
$731$	525.437	0.0265855
$732$	−5282.76	−0.266744
$733$	−25069.2	−1.26323	−0.631617	−	0.775281i	$-0.717608\pi$
$733$	−25069.2	−1.26323	−0.631617	+	0.775281i	$0.717608\pi$
$734$	14687.3	0.738580
$735$	960.762	0.0482153
$736$	−3683.40	−0.184473
$737$	−14651.4	−0.732282
$738$	−8717.94	−0.434840
$739$	9709.63	0.483322	0.241661	−	0.970361i	$-0.422308\pi$
$739$	9709.63	0.483322	0.241661	+	0.970361i	$0.422308\pi$
$740$	−9734.12	−0.483559
$741$	0	0
$742$	21697.2	1.07349
$743$	−23381.1	−1.15447	−0.577233	−	0.816580i	$-0.695867\pi$
$743$	−23381.1	−1.15447	−0.577233	+	0.816580i	$0.695867\pi$
$744$	6023.16	0.296801
$745$	−54649.0	−2.68750
$746$	3895.27	0.191174
$747$	−176.395	−0.00863984
$748$	1417.97	0.0693131
$749$	16359.8	0.798094
$750$	−13872.0	−0.675379
$751$	−1585.42	−0.0770344	−0.0385172	−	0.999258i	$-0.512263\pi$
$751$	−1585.42	−0.0770344	−0.0385172	+	0.999258i	$0.512263\pi$
$752$	1222.51	0.0592822
$753$	12466.0	0.603303
$754$	16842.9	0.813503
$755$	15136.2	0.729620
$756$	−1951.05	−0.0938612
$757$	30847.9	1.48109	0.740546	−	0.672005i	$-0.234567\pi$
$757$	30847.9	1.48109	0.740546	+	0.672005i	$0.234567\pi$
$758$	−9996.53	−0.479011
$759$	−5629.48	−0.269219
$760$	0	0
$761$	−28714.9	−1.36783	−0.683913	−	0.729564i	$-0.739723\pi$
$761$	−28714.9	−1.36783	−0.683913	+	0.729564i	$0.739723\pi$
$762$	−2186.04	−0.103926
$763$	−16420.3	−0.779104
$764$	−18290.3	−0.866127
$765$	−3765.32	−0.177955
$766$	−120.959	−0.00570551
$767$	−17265.9	−0.812825
$768$	768.000	0.0360844
$769$	21245.5	0.996270	0.498135	−	0.867100i	$-0.334018\pi$
$769$	21245.5	0.996270	0.498135	+	0.867100i	$0.334018\pi$
$770$	11332.4	0.530379
$771$	−12917.9	−0.603407
$772$	14813.0	0.690586
$773$	−135.106	−0.00628647	−0.00314324	−	0.999995i	$-0.501001\pi$
$773$	−135.106	−0.00628647	−0.00314324	+	0.999995i	$0.501001\pi$
$774$	434.944	0.0201986
$775$	61528.6	2.85183
$776$	−9161.45	−0.423810
$777$	−6854.94	−0.316499
$778$	−26050.2	−1.20044
$779$	0	0
$780$	−6582.82	−0.302183
$781$	−13243.2	−0.606760
$782$	−5005.97	−0.228917
$783$	7974.78	0.363979
$784$	−266.327	−0.0121322
$785$	42375.5	1.92668
$786$	−15021.8	−0.681692
$787$	32506.1	1.47232	0.736161	−	0.676807i	$-0.236637\pi$
$787$	32506.1	1.47232	0.736161	+	0.676807i	$0.236637\pi$
$788$	7357.29	0.332605
$789$	14205.5	0.640977
$790$	24537.3	1.10506
$791$	6873.87	0.308985
$792$	1173.76	0.0526615
$793$	−12552.0	−0.562084
$794$	−2812.97	−0.125729
$795$	34661.7	1.54632
$796$	17450.4	0.777025
$797$	−8254.37	−0.366857	−0.183428	−	0.983033i	$-0.558720\pi$
$797$	−8254.37	−0.366857	−0.183428	+	0.983033i	$0.558720\pi$
$798$	0	0
$799$	1661.46	0.0735648
$800$	7845.38	0.346720
$801$	7626.92	0.336434
$802$	−25107.1	−1.10544
$803$	12950.3	0.569123
$804$	−10784.8	−0.473073
$805$	−40007.6	−1.75166
$806$	14311.1	0.625420
$807$	−16667.3	−0.727034
$808$	6770.24	0.294773
$809$	9189.24	0.399353	0.199676	−	0.979862i	$-0.436011\pi$
$809$	9189.24	0.399353	0.199676	+	0.979862i	$0.436011\pi$
$810$	−3116.84	−0.135203
$811$	12785.8	0.553601	0.276801	−	0.960927i	$-0.410726\pi$
$811$	12785.8	0.553601	0.276801	+	0.960927i	$0.410726\pi$
$812$	−21343.2	−0.922413
$813$	−2669.48	−0.115157
$814$	4123.97	0.177574
$815$	51467.6	2.21206
$816$	1043.76	0.0447781
$817$	0	0
$818$	13249.7	0.566337
$819$	−4635.74	−0.197785
$820$	37273.5	1.58738
$821$	6581.97	0.279796	0.139898	−	0.990166i	$-0.455323\pi$
$821$	6581.97	0.279796	0.139898	+	0.990166i	$0.455323\pi$
$822$	9318.47	0.395400
$823$	21262.6	0.900570	0.450285	−	0.892885i	$-0.351322\pi$
$823$	21262.6	0.900570	0.450285	+	0.892885i	$0.351322\pi$
$824$	−1227.83	−0.0519097
$825$	11990.4	0.506002
$826$	21879.3	0.921644
$827$	27797.1	1.16880	0.584401	−	0.811465i	$-0.301329\pi$
$827$	27797.1	1.16880	0.584401	+	0.811465i	$0.301329\pi$
$828$	−4143.82	−0.173922
$829$	460.755	0.0193036	0.00965179	−	0.999953i	$-0.496928\pi$
$829$	460.755	0.0193036	0.00965179	+	0.999953i	$0.496928\pi$
$830$	754.177	0.0315396
$831$	2154.98	0.0899583
$832$	1824.78	0.0760372
$833$	−361.955	−0.0150552
$834$	−3914.65	−0.162534
$835$	−4811.74	−0.199422
$836$	0	0
$837$	6776.05	0.279826
$838$	17674.8	0.728598
$839$	−31987.0	−1.31623	−0.658113	−	0.752919i	$-0.728645\pi$
$839$	−31987.0	−1.31623	−0.658113	+	0.752919i	$0.728645\pi$
$840$	8341.72	0.342639
$841$	62849.8	2.57697
$842$	−4334.92	−0.177424
$843$	−19113.5	−0.780906
$844$	15550.9	0.634225
$845$	26628.8	1.08409
$846$	1375.32	0.0558918
$847$	19243.8	0.780666
$848$	−9608.37	−0.389095
$849$	966.348	0.0390636
$850$	10662.4	0.430254
$851$	−14559.1	−0.586464
$852$	−9748.25	−0.391983
$853$	−5713.39	−0.229335	−0.114668	−	0.993404i	$-0.536580\pi$
$853$	−5713.39	−0.229335	−0.114668	+	0.993404i	$0.536580\pi$
$854$	15905.8	0.637335
$855$	0	0
$856$	−7244.72	−0.289275
$857$	−14860.3	−0.592322	−0.296161	−	0.955138i	$-0.595706\pi$
$857$	−14860.3	−0.592322	−0.296161	+	0.955138i	$0.595706\pi$
$858$	2788.89	0.110969
$859$	29495.6	1.17157	0.585784	−	0.810467i	$-0.300787\pi$
$859$	29495.6	1.17157	0.585784	+	0.810467i	$0.300787\pi$
$860$	−1859.60	−0.0737348
$861$	26248.7	1.03897
$862$	−17024.9	−0.672705
$863$	11431.9	0.450923	0.225461	−	0.974252i	$-0.427611\pi$
$863$	11431.9	0.450923	0.225461	+	0.974252i	$0.427611\pi$
$864$	864.000	0.0340207
$865$	−20617.7	−0.810431
$866$	30378.4	1.19203
$867$	−13320.5	−0.521784
$868$	−18135.0	−0.709150
$869$	−10395.5	−0.405804
$870$	−34096.2	−1.32870
$871$	−25624.9	−0.996863
$872$	7271.56	0.282392
$873$	−10306.6	−0.399572
$874$	0	0
$875$	41767.0	1.61369
$876$	9532.62	0.367668
$877$	−23593.6	−0.908437	−0.454218	−	0.890890i	$-0.650081\pi$
$877$	−23593.6	−0.908437	−0.454218	+	0.890890i	$0.650081\pi$
$878$	15861.8	0.609694
$879$	−26366.2	−1.01173
$880$	−5018.43	−0.192240
$881$	−13012.7	−0.497626	−0.248813	−	0.968552i	$-0.580040\pi$
$881$	−13012.7	−0.497626	−0.248813	+	0.968552i	$0.580040\pi$
$882$	−299.618	−0.0114384
$883$	−11832.9	−0.450971	−0.225485	−	0.974247i	$-0.572397\pi$
$883$	−11832.9	−0.450971	−0.225485	+	0.974247i	$0.572397\pi$
$884$	2480.00	0.0943567
$885$	34952.6	1.32759
$886$	−13533.2	−0.513156
$887$	50764.2	1.92164	0.960821	−	0.277169i	$-0.0893963\pi$
$887$	50764.2	1.92164	0.960821	+	0.277169i	$0.0893963\pi$
$888$	3035.63	0.114717
$889$	6581.90	0.248312
$890$	−32608.9	−1.22815
$891$	1320.48	0.0496497
$892$	13641.3	0.512047
$893$	0	0
$894$	17042.5	0.637570
$895$	−33698.1	−1.25855
$896$	−2312.36	−0.0862170
$897$	−9845.81	−0.366490
$898$	10129.9	0.376436
$899$	74125.5	2.74997
$900$	8826.05	0.326891
$901$	−13058.4	−0.482839
$902$	−15791.4	−0.582921
$903$	−1309.57	−0.0482609
$904$	−3044.01	−0.111994
$905$	878.428	0.0322651
$906$	−4720.29	−0.173092
$907$	14457.8	0.529286	0.264643	−	0.964346i	$-0.414746\pi$
$907$	14457.8	0.529286	0.264643	+	0.964346i	$0.414746\pi$
$908$	−17169.6	−0.627525
$909$	7616.52	0.277914
$910$	19820.1	0.722011
$911$	−9841.83	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$911$	−9841.83	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$912$	0	0
$913$	−319.516	−0.0115821
$914$	−16612.9	−0.601209
$915$	25409.8	0.918056
$916$	−7408.58	−0.267234
$917$	45228.9	1.62878
$918$	1174.23	0.0422172
$919$	16698.0	0.599363	0.299682	−	0.954039i	$-0.403120\pi$
$919$	16698.0	0.599363	0.299682	+	0.954039i	$0.403120\pi$
$920$	17716.9	0.634902
$921$	−17588.2	−0.629262
$922$	−4392.57	−0.156900
$923$	−23162.0	−0.825989
$924$	−3534.06	−0.125825
$925$	31010.0	1.10227
$926$	−8515.86	−0.302212
$927$	−1381.31	−0.0489410
$928$	9451.59	0.334336
$929$	−45149.9	−1.59453	−0.797266	−	0.603629i	$-0.793721\pi$
$929$	−45149.9	−1.59453	−0.797266	+	0.603629i	$0.793721\pi$
$930$	−28971.0	−1.02150
$931$	0	0
$932$	2416.84	0.0849425
$933$	11796.0	0.413917
$934$	−9236.27	−0.323576
$935$	−6820.36	−0.238556
$936$	2052.88	0.0716886
$937$	−21364.8	−0.744885	−0.372442	−	0.928055i	$-0.621480\pi$
$937$	−21364.8	−0.744885	−0.372442	+	0.928055i	$0.621480\pi$
$938$	32471.8	1.13032
$939$	26973.0	0.937413
$940$	−5880.18	−0.204032
$941$	21552.4	0.746639	0.373320	−	0.927703i	$-0.378219\pi$
$941$	21552.4	0.746639	0.373320	+	0.927703i	$0.378219\pi$
$942$	−13215.0	−0.457078
$943$	55749.4	1.92518
$944$	−9688.99	−0.334057
$945$	9384.43	0.323043
$946$	787.842	0.0270771
$947$	−17111.1	−0.587155	−0.293578	−	0.955935i	$-0.594846\pi$
$947$	−17111.1	−0.587155	−0.293578	+	0.955935i	$0.594846\pi$
$948$	−7652.06	−0.262160
$949$	22649.7	0.774753
$950$	0	0
$951$	23409.6	0.798222
$952$	−3142.64	−0.106989
$953$	15347.7	0.521680	0.260840	−	0.965382i	$-0.416000\pi$
$953$	15347.7	0.521680	0.260840	+	0.965382i	$0.416000\pi$
$954$	−10809.4	−0.366842
$955$	87975.4	2.98096
$956$	−28441.0	−0.962185
$957$	14445.2	0.487929
$958$	−3455.99	−0.116553
$959$	−28056.8	−0.944735
$960$	−3694.03	−0.124192
$961$	33192.4	1.11417
$962$	7212.71	0.241733
$963$	−8150.31	−0.272731
$964$	−112.467	−0.00375759
$965$	−71249.7	−2.37680
$966$	12476.6	0.415556
$967$	28429.3	0.945424	0.472712	−	0.881217i	$-0.343275\pi$
$967$	28429.3	0.945424	0.472712	+	0.881217i	$0.343275\pi$
$968$	−8521.89	−0.282958
$969$	0	0
$970$	44066.0	1.45863
$971$	469.588	0.0155199	0.00775994	−	0.999970i	$-0.497530\pi$
$971$	469.588	0.0155199	0.00775994	+	0.999970i	$0.497530\pi$
$972$	972.000	0.0320750
$973$	11786.5	0.388345
$974$	25111.1	0.826090
$975$	20970.9	0.688826
$976$	−7043.68	−0.231007
$977$	26778.2	0.876879	0.438440	−	0.898761i	$-0.355531\pi$
$977$	26778.2	0.876879	0.438440	+	0.898761i	$0.355531\pi$
$978$	−16050.4	−0.524780
$979$	13815.1	0.451005
$980$	1281.02	0.0417557
$981$	8180.50	0.266242
$982$	18982.2	0.616850
$983$	−34172.3	−1.10878	−0.554389	−	0.832258i	$-0.687048\pi$
$983$	−34172.3	−1.10878	−0.554389	+	0.832258i	$0.687048\pi$
$984$	−11623.9	−0.376582
$985$	−35388.1	−1.14473
$986$	12845.3	0.414886
$987$	−4140.92	−0.133543
$988$	0	0
$989$	−2781.38	−0.0894263
$990$	−5645.73	−0.181246
$991$	34846.5	1.11699	0.558494	−	0.829508i	$-0.311379\pi$
$991$	34846.5	1.11699	0.558494	+	0.829508i	$0.311379\pi$
$992$	8030.88	0.257037
$993$	13704.0	0.437948
$994$	29350.8	0.936571
$995$	−83935.2	−2.67429
$996$	−235.193	−0.00748232
$997$	49622.2	1.57628	0.788140	−	0.615497i	$-0.211044\pi$
$997$	49622.2	1.57628	0.788140	+	0.615497i	$0.211044\pi$
$998$	5861.19	0.185904
$999$	3415.08	0.108157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.ba.1.1	yes	4
19.18	odd	2		2166.4.a.x.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.x.1.1	✓	4	19.18	odd	2
2166.4.a.ba.1.1	yes	4	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.2398 q^{5} +6.00000 q^{6} -18.0653 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -19.2398 q^{5} +6.00000 q^{6} -18.0653 q^{7} +8.00000 q^{8} +9.00000 q^{9} -38.4795 q^{10} +16.3023 q^{11} +12.0000 q^{12} +28.5122 q^{13} -36.1306 q^{14} -57.7193 q^{15} +16.0000 q^{16} +21.7450 q^{17} +18.0000 q^{18} -76.9590 q^{20} -54.1959 q^{21} +32.6046 q^{22} -115.106 q^{23} +24.0000 q^{24} +245.168 q^{25} +57.0245 q^{26} +27.0000 q^{27} -72.2611 q^{28} +295.362 q^{29} -115.439 q^{30} +250.965 q^{31} +32.0000 q^{32} +48.9068 q^{33} +43.4900 q^{34} +347.572 q^{35} +36.0000 q^{36} +126.485 q^{37} +85.5367 q^{39} -153.918 q^{40} -484.330 q^{41} -108.392 q^{42} +24.1636 q^{43} +65.2091 q^{44} -173.158 q^{45} -230.212 q^{46} +76.4066 q^{47} +48.0000 q^{48} -16.6454 q^{49} +490.336 q^{50} +65.2350 q^{51} +114.049 q^{52} -600.523 q^{53} +54.0000 q^{54} -313.652 q^{55} -144.522 q^{56} +590.724 q^{58} -605.562 q^{59} -230.877 q^{60} -440.230 q^{61} +501.930 q^{62} -162.588 q^{63} +64.0000 q^{64} -548.568 q^{65} +97.8137 q^{66} -898.734 q^{67} +86.9800 q^{68} -345.319 q^{69} +695.143 q^{70} -812.354 q^{71} +72.0000 q^{72} +794.385 q^{73} +252.969 q^{74} +735.504 q^{75} -294.505 q^{77} +171.073 q^{78} -637.672 q^{79} -307.836 q^{80} +81.0000 q^{81} -968.660 q^{82} -19.5995 q^{83} -216.783 q^{84} -418.369 q^{85} +48.3271 q^{86} +886.086 q^{87} +130.418 q^{88} +847.436 q^{89} -346.316 q^{90} -515.082 q^{91} -460.425 q^{92} +752.895 q^{93} +152.813 q^{94} +96.0000 q^{96} -1145.18 q^{97} -33.2909 q^{98} +146.720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9} - 56 q^{10} - 72 q^{11} + 48 q^{12} - 8 q^{13} - 34 q^{14} - 84 q^{15} + 64 q^{16} + 33 q^{17} + 72 q^{18} - 112 q^{20} - 51 q^{21} - 144 q^{22} - 88 q^{23} + 96 q^{24} + 94 q^{25} - 16 q^{26} + 108 q^{27} - 68 q^{28} - 75 q^{29} - 168 q^{30} + 419 q^{31} + 128 q^{32} - 216 q^{33} + 66 q^{34} + 331 q^{35} + 144 q^{36} + 131 q^{37} - 24 q^{39} - 224 q^{40} - 1134 q^{41} - 102 q^{42} + 76 q^{43} - 288 q^{44} - 252 q^{45} - 176 q^{46} + 395 q^{47} + 192 q^{48} - 1033 q^{49} + 188 q^{50} + 99 q^{51} - 32 q^{52} - 625 q^{53} + 216 q^{54} + 413 q^{55} - 136 q^{56} - 150 q^{58} - 1328 q^{59} - 336 q^{60} - 1887 q^{61} + 838 q^{62} - 153 q^{63} + 256 q^{64} - 1817 q^{65} - 432 q^{66} - 99 q^{67} + 132 q^{68} - 264 q^{69} + 662 q^{70} - 1307 q^{71} + 288 q^{72} + 183 q^{73} + 262 q^{74} + 282 q^{75} - 513 q^{77} - 48 q^{78} - 2064 q^{79} - 448 q^{80} + 324 q^{81} - 2268 q^{82} + 816 q^{83} - 204 q^{84} - 867 q^{85} + 152 q^{86} - 225 q^{87} - 576 q^{88} + 207 q^{89} - 504 q^{90} - 183 q^{91} - 352 q^{92} + 1257 q^{93} + 790 q^{94} + 384 q^{96} - 1331 q^{97} - 2066 q^{98} - 648 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9 - 56 * q^10 - 72 * q^11 + 48 * q^12 - 8 * q^13 - 34 * q^14 - 84 * q^15 + 64 * q^16 + 33 * q^17 + 72 * q^18 - 112 * q^20 - 51 * q^21 - 144 * q^22 - 88 * q^23 + 96 * q^24 + 94 * q^25 - 16 * q^26 + 108 * q^27 - 68 * q^28 - 75 * q^29 - 168 * q^30 + 419 * q^31 + 128 * q^32 - 216 * q^33 + 66 * q^34 + 331 * q^35 + 144 * q^36 + 131 * q^37 - 24 * q^39 - 224 * q^40 - 1134 * q^41 - 102 * q^42 + 76 * q^43 - 288 * q^44 - 252 * q^45 - 176 * q^46 + 395 * q^47 + 192 * q^48 - 1033 * q^49 + 188 * q^50 + 99 * q^51 - 32 * q^52 - 625 * q^53 + 216 * q^54 + 413 * q^55 - 136 * q^56 - 150 * q^58 - 1328 * q^59 - 336 * q^60 - 1887 * q^61 + 838 * q^62 - 153 * q^63 + 256 * q^64 - 1817 * q^65 - 432 * q^66 - 99 * q^67 + 132 * q^68 - 264 * q^69 + 662 * q^70 - 1307 * q^71 + 288 * q^72 + 183 * q^73 + 262 * q^74 + 282 * q^75 - 513 * q^77 - 48 * q^78 - 2064 * q^79 - 448 * q^80 + 324 * q^81 - 2268 * q^82 + 816 * q^83 - 204 * q^84 - 867 * q^85 + 152 * q^86 - 225 * q^87 - 576 * q^88 + 207 * q^89 - 504 * q^90 - 183 * q^91 - 352 * q^92 + 1257 * q^93 + 790 * q^94 + 384 * q^96 - 1331 * q^97 - 2066 * q^98 - 648 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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