LMFDB - Embedded newform 7942.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
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} - 14x^{6} - 4x^{5} + 38x^{4} + 3x^{3} - 29x^{2} + 2x + 4 $ x^8 - 14x^6 - 4x^5 + 38x^4 + 3x^3 - 29x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.78054$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.39858 q^{3} +1.00000 q^{4} -2.89336 q^{5} +3.39858 q^{6} +3.18396 q^{7} -1.00000 q^{8} +8.55033 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.39858 q^{3} +1.00000 q^{4} -2.89336 q^{5} +3.39858 q^{6} +3.18396 q^{7} -1.00000 q^{8} +8.55033 q^{9} +2.89336 q^{10} +1.00000 q^{11} -3.39858 q^{12} -3.56418 q^{13} -3.18396 q^{14} +9.83330 q^{15} +1.00000 q^{16} +3.26312 q^{17} -8.55033 q^{18} -2.89336 q^{20} -10.8209 q^{21} -1.00000 q^{22} +3.73596 q^{23} +3.39858 q^{24} +3.37152 q^{25} +3.56418 q^{26} -18.8632 q^{27} +3.18396 q^{28} +9.04792 q^{29} -9.83330 q^{30} +7.31289 q^{31} -1.00000 q^{32} -3.39858 q^{33} -3.26312 q^{34} -9.21233 q^{35} +8.55033 q^{36} -2.00473 q^{37} +12.1132 q^{39} +2.89336 q^{40} -10.6209 q^{41} +10.8209 q^{42} -7.35835 q^{43} +1.00000 q^{44} -24.7392 q^{45} -3.73596 q^{46} -9.94250 q^{47} -3.39858 q^{48} +3.13758 q^{49} -3.37152 q^{50} -11.0900 q^{51} -3.56418 q^{52} +3.53887 q^{53} +18.8632 q^{54} -2.89336 q^{55} -3.18396 q^{56} -9.04792 q^{58} -6.11668 q^{59} +9.83330 q^{60} -4.92246 q^{61} -7.31289 q^{62} +27.2239 q^{63} +1.00000 q^{64} +10.3125 q^{65} +3.39858 q^{66} -1.62768 q^{67} +3.26312 q^{68} -12.6969 q^{69} +9.21233 q^{70} +5.84857 q^{71} -8.55033 q^{72} +0.284732 q^{73} +2.00473 q^{74} -11.4584 q^{75} +3.18396 q^{77} -12.1132 q^{78} +6.65900 q^{79} -2.89336 q^{80} +38.4571 q^{81} +10.6209 q^{82} -10.4959 q^{83} -10.8209 q^{84} -9.44138 q^{85} +7.35835 q^{86} -30.7501 q^{87} -1.00000 q^{88} -12.6869 q^{89} +24.7392 q^{90} -11.3482 q^{91} +3.73596 q^{92} -24.8534 q^{93} +9.94250 q^{94} +3.39858 q^{96} +15.5153 q^{97} -3.13758 q^{98} +8.55033 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + q^{5} + 4 q^{6} + 4 q^{7} - 8 q^{8} + 6 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + q^5 + 4 * q^6 + 4 * q^7 - 8 * q^8 + 6 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + q^{5} + 4 q^{6} + 4 q^{7} - 8 q^{8} + 6 q^{9} - q^{10} + 8 q^{11} - 4 q^{12} - 12 q^{13} - 4 q^{14} + 9 q^{15} + 8 q^{16} + 7 q^{17} - 6 q^{18} + q^{20} - 13 q^{21} - 8 q^{22} - 6 q^{23} + 4 q^{24} + 5 q^{25} + 12 q^{26} - 22 q^{27} + 4 q^{28} + q^{29} - 9 q^{30} - 8 q^{31} - 8 q^{32} - 4 q^{33} - 7 q^{34} + 5 q^{35} + 6 q^{36} - 12 q^{37} + 10 q^{39} - q^{40} - 18 q^{41} + 13 q^{42} - 4 q^{43} + 8 q^{44} - 19 q^{45} + 6 q^{46} + 13 q^{47} - 4 q^{48} + 4 q^{49} - 5 q^{50} - 14 q^{51} - 12 q^{52} + 17 q^{53} + 22 q^{54} + q^{55} - 4 q^{56} - q^{58} - 26 q^{59} + 9 q^{60} - 7 q^{61} + 8 q^{62} + 13 q^{63} + 8 q^{64} - 36 q^{65} + 4 q^{66} - 6 q^{67} + 7 q^{68} - 19 q^{69} - 5 q^{70} + 3 q^{71} - 6 q^{72} + 5 q^{73} + 12 q^{74} - 30 q^{75} + 4 q^{77} - 10 q^{78} - 16 q^{79} + q^{80} + 8 q^{81} + 18 q^{82} - 21 q^{83} - 13 q^{84} - 10 q^{85} + 4 q^{86} - 29 q^{87} - 8 q^{88} - 16 q^{89} + 19 q^{90} - 31 q^{91} - 6 q^{92} + 20 q^{93} - 13 q^{94} + 4 q^{96} + 15 q^{97} - 4 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + q^5 + 4 * q^6 + 4 * q^7 - 8 * q^8 + 6 * q^9 - q^10 + 8 * q^11 - 4 * q^12 - 12 * q^13 - 4 * q^14 + 9 * q^15 + 8 * q^16 + 7 * q^17 - 6 * q^18 + q^20 - 13 * q^21 - 8 * q^22 - 6 * q^23 + 4 * q^24 + 5 * q^25 + 12 * q^26 - 22 * q^27 + 4 * q^28 + q^29 - 9 * q^30 - 8 * q^31 - 8 * q^32 - 4 * q^33 - 7 * q^34 + 5 * q^35 + 6 * q^36 - 12 * q^37 + 10 * q^39 - q^40 - 18 * q^41 + 13 * q^42 - 4 * q^43 + 8 * q^44 - 19 * q^45 + 6 * q^46 + 13 * q^47 - 4 * q^48 + 4 * q^49 - 5 * q^50 - 14 * q^51 - 12 * q^52 + 17 * q^53 + 22 * q^54 + q^55 - 4 * q^56 - q^58 - 26 * q^59 + 9 * q^60 - 7 * q^61 + 8 * q^62 + 13 * q^63 + 8 * q^64 - 36 * q^65 + 4 * q^66 - 6 * q^67 + 7 * q^68 - 19 * q^69 - 5 * q^70 + 3 * q^71 - 6 * q^72 + 5 * q^73 + 12 * q^74 - 30 * q^75 + 4 * q^77 - 10 * q^78 - 16 * q^79 + q^80 + 8 * q^81 + 18 * q^82 - 21 * q^83 - 13 * q^84 - 10 * q^85 + 4 * q^86 - 29 * q^87 - 8 * q^88 - 16 * q^89 + 19 * q^90 - 31 * q^91 - 6 * q^92 + 20 * q^93 - 13 * q^94 + 4 * q^96 + 15 * q^97 - 4 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.39858	−1.96217	−0.981085	−	0.193579i	$-0.937991\pi$
$3$	−3.39858	−1.96217	−0.981085	+	0.193579i	$0.937991\pi$
$4$	1.00000	0.500000
$5$	−2.89336	−1.29395	−0.646975	−	0.762511i	$-0.723966\pi$
$5$	−2.89336	−1.29395	−0.646975	+	0.762511i	$0.723966\pi$
$6$	3.39858	1.38746
$7$	3.18396	1.20342	0.601711	−	0.798714i	$-0.294486\pi$
$7$	3.18396	1.20342	0.601711	+	0.798714i	$0.294486\pi$
$8$	−1.00000	−0.353553
$9$	8.55033	2.85011
$10$	2.89336	0.914960
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−3.39858	−0.981085
$13$	−3.56418	−0.988527	−0.494263	−	0.869312i	$-0.664562\pi$
$13$	−3.56418	−0.988527	−0.494263	+	0.869312i	$0.664562\pi$
$14$	−3.18396	−0.850948
$15$	9.83330	2.53895
$16$	1.00000	0.250000
$17$	3.26312	0.791423	0.395711	−	0.918375i	$-0.370498\pi$
$17$	3.26312	0.791423	0.395711	+	0.918375i	$0.370498\pi$
$18$	−8.55033	−2.01533
$19$	0	0
$19$	0	0
$20$	−2.89336	−0.646975
$21$	−10.8209	−2.36132
$22$	−1.00000	−0.213201
$23$	3.73596	0.779001	0.389501	−	0.921026i	$-0.372648\pi$
$23$	3.73596	0.779001	0.389501	+	0.921026i	$0.372648\pi$
$24$	3.39858	0.693732
$25$	3.37152	0.674305
$26$	3.56418	0.698994
$27$	−18.8632	−3.63023
$28$	3.18396	0.601711
$29$	9.04792	1.68016	0.840079	−	0.542465i	$-0.182509\pi$
$29$	9.04792	1.68016	0.840079	+	0.542465i	$0.182509\pi$
$30$	−9.83330	−1.79531
$31$	7.31289	1.31343	0.656717	−	0.754137i	$-0.271945\pi$
$31$	7.31289	1.31343	0.656717	+	0.754137i	$0.271945\pi$
$32$	−1.00000	−0.176777
$33$	−3.39858	−0.591616
$34$	−3.26312	−0.559621
$35$	−9.21233	−1.55717
$36$	8.55033	1.42505
$37$	−2.00473	−0.329575	−0.164788	−	0.986329i	$-0.552694\pi$
$37$	−2.00473	−0.329575	−0.164788	+	0.986329i	$0.552694\pi$
$38$	0	0
$39$	12.1132	1.93966
$40$	2.89336	0.457480
$41$	−10.6209	−1.65871	−0.829353	−	0.558725i	$-0.811291\pi$
$41$	−10.6209	−1.65871	−0.829353	+	0.558725i	$0.811291\pi$
$42$	10.8209	1.66970
$43$	−7.35835	−1.12214	−0.561069	−	0.827769i	$-0.689610\pi$
$43$	−7.35835	−1.12214	−0.561069	+	0.827769i	$0.689610\pi$
$44$	1.00000	0.150756
$45$	−24.7392	−3.68790
$46$	−3.73596	−0.550837
$47$	−9.94250	−1.45026	−0.725131	−	0.688610i	$-0.758221\pi$
$47$	−9.94250	−1.45026	−0.725131	+	0.688610i	$0.758221\pi$
$48$	−3.39858	−0.490542
$49$	3.13758	0.448226
$50$	−3.37152	−0.476806
$51$	−11.0900	−1.55291
$52$	−3.56418	−0.494263
$53$	3.53887	0.486101	0.243051	−	0.970014i	$-0.421852\pi$
$53$	3.53887	0.486101	0.243051	+	0.970014i	$0.421852\pi$
$54$	18.8632	2.56696
$55$	−2.89336	−0.390140
$56$	−3.18396	−0.425474
$57$	0	0
$58$	−9.04792	−1.18805
$59$	−6.11668	−0.796324	−0.398162	−	0.917315i	$-0.630352\pi$
$59$	−6.11668	−0.796324	−0.398162	+	0.917315i	$0.630352\pi$
$60$	9.83330	1.26947
$61$	−4.92246	−0.630256	−0.315128	−	0.949049i	$-0.602047\pi$
$61$	−4.92246	−0.630256	−0.315128	+	0.949049i	$0.602047\pi$
$62$	−7.31289	−0.928738
$63$	27.2239	3.42989
$64$	1.00000	0.125000
$65$	10.3125	1.27910
$66$	3.39858	0.418336
$67$	−1.62768	−0.198853	−0.0994263	−	0.995045i	$-0.531701\pi$
$67$	−1.62768	−0.198853	−0.0994263	+	0.995045i	$0.531701\pi$
$68$	3.26312	0.395711
$69$	−12.6969	−1.52853
$70$	9.21233	1.10108
$71$	5.84857	0.694098	0.347049	−	0.937847i	$-0.387184\pi$
$71$	5.84857	0.694098	0.347049	+	0.937847i	$0.387184\pi$
$72$	−8.55033	−1.00767
$73$	0.284732	0.0333254	0.0166627	−	0.999861i	$-0.494696\pi$
$73$	0.284732	0.0333254	0.0166627	+	0.999861i	$0.494696\pi$
$74$	2.00473	0.233045
$75$	−11.4584	−1.32310
$76$	0	0
$77$	3.18396	0.362846
$78$	−12.1132	−1.37154
$79$	6.65900	0.749197	0.374598	−	0.927187i	$-0.377781\pi$
$79$	6.65900	0.749197	0.374598	+	0.927187i	$0.377781\pi$
$80$	−2.89336	−0.323487
$81$	38.4571	4.27301
$82$	10.6209	1.17288
$83$	−10.4959	−1.15208	−0.576038	−	0.817423i	$-0.695402\pi$
$83$	−10.4959	−1.15208	−0.576038	+	0.817423i	$0.695402\pi$
$84$	−10.8209	−1.18066
$85$	−9.44138	−1.02406
$86$	7.35835	0.793471
$87$	−30.7501	−3.29675
$88$	−1.00000	−0.106600
$89$	−12.6869	−1.34481	−0.672406	−	0.740183i	$-0.734739\pi$
$89$	−12.6869	−1.34481	−0.672406	+	0.740183i	$0.734739\pi$
$90$	24.7392	2.60774
$91$	−11.3482	−1.18962
$92$	3.73596	0.389501
$93$	−24.8534	−2.57718
$94$	9.94250	1.02549
$95$	0	0
$96$	3.39858	0.346866
$97$	15.5153	1.57534	0.787671	−	0.616097i	$-0.211287\pi$
$97$	15.5153	1.57534	0.787671	+	0.616097i	$0.211287\pi$
$98$	−3.13758	−0.316944
$99$	8.55033	0.859340
$100$	3.37152	0.337152
$101$	1.55252	0.154482	0.0772408	−	0.997012i	$-0.475389\pi$
$101$	1.55252	0.154482	0.0772408	+	0.997012i	$0.475389\pi$
$102$	11.0900	1.09807
$103$	4.44169	0.437652	0.218826	−	0.975764i	$-0.429777\pi$
$103$	4.44169	0.437652	0.218826	+	0.975764i	$0.429777\pi$
$104$	3.56418	0.349497
$105$	31.3088	3.05543
$106$	−3.53887	−0.343726
$107$	−5.53992	−0.535564	−0.267782	−	0.963480i	$-0.586291\pi$
$107$	−5.53992	−0.535564	−0.267782	+	0.963480i	$0.586291\pi$
$108$	−18.8632	−1.81511
$109$	7.42933	0.711601	0.355800	−	0.934562i	$-0.384208\pi$
$109$	7.42933	0.711601	0.355800	+	0.934562i	$0.384208\pi$
$110$	2.89336	0.275871
$111$	6.81322	0.646683
$112$	3.18396	0.300856
$113$	−5.77179	−0.542964	−0.271482	−	0.962443i	$-0.587514\pi$
$113$	−5.77179	−0.542964	−0.271482	+	0.962443i	$0.587514\pi$
$114$	0	0
$115$	−10.8095	−1.00799
$116$	9.04792	0.840079
$117$	−30.4749	−2.81741
$118$	6.11668	0.563086
$119$	10.3896	0.952416
$120$	−9.83330	−0.897654
$121$	1.00000	0.0909091
$122$	4.92246	0.445658
$123$	36.0959	3.25466
$124$	7.31289	0.656717
$125$	4.71176	0.421433
$126$	−27.2239	−2.42530
$127$	−5.75300	−0.510496	−0.255248	−	0.966876i	$-0.582157\pi$
$127$	−5.75300	−0.510496	−0.255248	+	0.966876i	$0.582157\pi$
$128$	−1.00000	−0.0883883
$129$	25.0079	2.20182
$130$	−10.3125	−0.904463
$131$	7.17237	0.626653	0.313327	−	0.949645i	$-0.398557\pi$
$131$	7.17237	0.626653	0.313327	+	0.949645i	$0.398557\pi$
$132$	−3.39858	−0.295808
$133$	0	0
$134$	1.62768	0.140610
$135$	54.5781	4.69733
$136$	−3.26312	−0.279810
$137$	−17.1097	−1.46178	−0.730889	−	0.682497i	$-0.760894\pi$
$137$	−17.1097	−1.46178	−0.730889	+	0.682497i	$0.760894\pi$
$138$	12.6969	1.08084
$139$	−15.1403	−1.28418	−0.642090	−	0.766630i	$-0.721932\pi$
$139$	−15.1403	−1.28418	−0.642090	+	0.766630i	$0.721932\pi$
$140$	−9.21233	−0.778584
$141$	33.7904	2.84566
$142$	−5.84857	−0.490801
$143$	−3.56418	−0.298052
$144$	8.55033	0.712527
$145$	−26.1789	−2.17404
$146$	−0.284732	−0.0235646
$147$	−10.6633	−0.879496
$148$	−2.00473	−0.164788
$149$	−13.6230	−1.11604	−0.558019	−	0.829828i	$-0.688438\pi$
$149$	−13.6230	−1.11604	−0.558019	+	0.829828i	$0.688438\pi$
$150$	11.4584	0.935573
$151$	−2.18559	−0.177861	−0.0889304	−	0.996038i	$-0.528345\pi$
$151$	−2.18559	−0.177861	−0.0889304	+	0.996038i	$0.528345\pi$
$152$	0	0
$153$	27.9008	2.25564
$154$	−3.18396	−0.256571
$155$	−21.1588	−1.69952
$156$	12.1132	0.969829
$157$	14.0654	1.12254	0.561271	−	0.827632i	$-0.310313\pi$
$157$	14.0654	1.12254	0.561271	+	0.827632i	$0.310313\pi$
$158$	−6.65900	−0.529762
$159$	−12.0271	−0.953813
$160$	2.89336	0.228740
$161$	11.8951	0.937468
$162$	−38.4571	−3.02148
$163$	23.9298	1.87432	0.937162	−	0.348896i	$-0.113443\pi$
$163$	23.9298	1.87432	0.937162	+	0.348896i	$0.113443\pi$
$164$	−10.6209	−0.829353
$165$	9.83330	0.765522
$166$	10.4959	0.814640
$167$	−6.65271	−0.514802	−0.257401	−	0.966305i	$-0.582866\pi$
$167$	−6.65271	−0.514802	−0.257401	+	0.966305i	$0.582866\pi$
$168$	10.8209	0.834852
$169$	−0.296594	−0.0228149
$170$	9.44138	0.724121
$171$	0	0
$172$	−7.35835	−0.561069
$173$	−15.9832	−1.21518	−0.607591	−	0.794250i	$-0.707864\pi$
$173$	−15.9832	−1.21518	−0.607591	+	0.794250i	$0.707864\pi$
$174$	30.7501	2.33116
$175$	10.7348	0.811474
$176$	1.00000	0.0753778
$177$	20.7880	1.56252
$178$	12.6869	0.950925
$179$	−23.0475	−1.72265	−0.861325	−	0.508054i	$-0.830365\pi$
$179$	−23.0475	−1.72265	−0.861325	+	0.508054i	$0.830365\pi$
$180$	−24.7392	−1.84395
$181$	25.6166	1.90407	0.952034	−	0.305991i	$-0.0989879\pi$
$181$	25.6166	1.90407	0.952034	+	0.305991i	$0.0989879\pi$
$182$	11.3482	0.841185
$183$	16.7294	1.23667
$184$	−3.73596	−0.275418
$185$	5.80040	0.426454
$186$	24.8534	1.82234
$187$	3.26312	0.238623
$188$	−9.94250	−0.725131
$189$	−60.0597	−4.36870
$190$	0	0
$191$	21.5968	1.56269	0.781344	−	0.624100i	$-0.214534\pi$
$191$	21.5968	1.56269	0.781344	+	0.624100i	$0.214534\pi$
$192$	−3.39858	−0.245271
$193$	−2.48323	−0.178747	−0.0893733	−	0.995998i	$-0.528486\pi$
$193$	−2.48323	−0.178747	−0.0893733	+	0.995998i	$0.528486\pi$
$194$	−15.5153	−1.11393
$195$	−35.0477	−2.50982
$196$	3.13758	0.224113
$197$	15.4026	1.09739	0.548695	−	0.836023i	$-0.315125\pi$
$197$	15.4026	1.09739	0.548695	+	0.836023i	$0.315125\pi$
$198$	−8.55033	−0.607645
$199$	−4.35270	−0.308555	−0.154277	−	0.988028i	$-0.549305\pi$
$199$	−4.35270	−0.308555	−0.154277	+	0.988028i	$0.549305\pi$
$200$	−3.37152	−0.238403
$201$	5.53179	0.390183
$202$	−1.55252	−0.109235
$203$	28.8082	2.02194
$204$	−11.0900	−0.776453
$205$	30.7301	2.14628
$206$	−4.44169	−0.309467
$207$	31.9437	2.22024
$208$	−3.56418	−0.247132
$209$	0	0
$210$	−31.3088	−2.16051
$211$	13.5690	0.934128	0.467064	−	0.884223i	$-0.345312\pi$
$211$	13.5690	0.934128	0.467064	+	0.884223i	$0.345312\pi$
$212$	3.53887	0.243051
$213$	−19.8768	−1.36194
$214$	5.53992	0.378701
$215$	21.2903	1.45199
$216$	18.8632	1.28348
$217$	23.2839	1.58062
$218$	−7.42933	−0.503178
$219$	−0.967684	−0.0653900
$220$	−2.89336	−0.195070
$221$	−11.6304	−0.782343
$222$	−6.81322	−0.457274
$223$	2.84491	0.190509	0.0952545	−	0.995453i	$-0.469634\pi$
$223$	2.84491	0.190509	0.0952545	+	0.995453i	$0.469634\pi$
$224$	−3.18396	−0.212737
$225$	28.8276	1.92184
$226$	5.77179	0.383934
$227$	15.7467	1.04515	0.522573	−	0.852594i	$-0.324972\pi$
$227$	15.7467	1.04515	0.522573	+	0.852594i	$0.324972\pi$
$228$	0	0
$229$	7.94718	0.525165	0.262582	−	0.964910i	$-0.415426\pi$
$229$	7.94718	0.525165	0.262582	+	0.964910i	$0.415426\pi$
$230$	10.8095	0.712755
$231$	−10.8209	−0.711965
$232$	−9.04792	−0.594025
$233$	−16.3309	−1.06987	−0.534936	−	0.844893i	$-0.679664\pi$
$233$	−16.3309	−1.06987	−0.534936	+	0.844893i	$0.679664\pi$
$234$	30.4749	1.99221
$235$	28.7672	1.87657
$236$	−6.11668	−0.398162
$237$	−22.6311	−1.47005
$238$	−10.3896	−0.673460
$239$	11.3810	0.736175	0.368087	−	0.929791i	$-0.380013\pi$
$239$	11.3810	0.736175	0.368087	+	0.929791i	$0.380013\pi$
$240$	9.83330	0.634737
$241$	15.5308	1.00042	0.500212	−	0.865903i	$-0.333255\pi$
$241$	15.5308	1.00042	0.500212	+	0.865903i	$0.333255\pi$
$242$	−1.00000	−0.0642824
$243$	−74.1099	−4.75415
$244$	−4.92246	−0.315128
$245$	−9.07815	−0.579982
$246$	−36.0959	−2.30139
$247$	0	0
$248$	−7.31289	−0.464369
$249$	35.6711	2.26057
$250$	−4.71176	−0.297998
$251$	−3.34649	−0.211229	−0.105614	−	0.994407i	$-0.533681\pi$
$251$	−3.34649	−0.211229	−0.105614	+	0.994407i	$0.533681\pi$
$252$	27.2239	1.71494
$253$	3.73596	0.234878
$254$	5.75300	0.360975
$255$	32.0873	2.00938
$256$	1.00000	0.0625000
$257$	2.85508	0.178095	0.0890476	−	0.996027i	$-0.471618\pi$
$257$	2.85508	0.178095	0.0890476	+	0.996027i	$0.471618\pi$
$258$	−25.0079	−1.55693
$259$	−6.38297	−0.396618
$260$	10.3125	0.639552
$261$	77.3627	4.78863
$262$	−7.17237	−0.443111
$263$	−6.62361	−0.408429	−0.204215	−	0.978926i	$-0.565464\pi$
$263$	−6.62361	−0.408429	−0.204215	+	0.978926i	$0.565464\pi$
$264$	3.39858	0.209168
$265$	−10.2392	−0.628990
$266$	0	0
$267$	43.1175	2.63875
$268$	−1.62768	−0.0994263
$269$	21.0191	1.28156	0.640779	−	0.767725i	$-0.278611\pi$
$269$	21.0191	1.28156	0.640779	+	0.767725i	$0.278611\pi$
$270$	−54.5781	−3.32152
$271$	11.9407	0.725347	0.362673	−	0.931916i	$-0.381864\pi$
$271$	11.9407	0.725347	0.362673	+	0.931916i	$0.381864\pi$
$272$	3.26312	0.197856
$273$	38.5678	2.33423
$274$	17.1097	1.03363
$275$	3.37152	0.203311
$276$	−12.6969	−0.764266
$277$	−4.78708	−0.287628	−0.143814	−	0.989605i	$-0.545937\pi$
$277$	−4.78708	−0.287628	−0.143814	+	0.989605i	$0.545937\pi$
$278$	15.1403	0.908052
$279$	62.5276	3.74343
$280$	9.21233	0.550542
$281$	9.17404	0.547278	0.273639	−	0.961833i	$-0.411773\pi$
$281$	9.17404	0.547278	0.273639	+	0.961833i	$0.411773\pi$
$282$	−33.7904	−2.01219
$283$	4.72461	0.280849	0.140424	−	0.990091i	$-0.455153\pi$
$283$	4.72461	0.280849	0.140424	+	0.990091i	$0.455153\pi$
$284$	5.84857	0.347049
$285$	0	0
$286$	3.56418	0.210755
$287$	−33.8165	−1.99612
$288$	−8.55033	−0.503833
$289$	−6.35205	−0.373650
$290$	26.1789	1.53728
$291$	−52.7300	−3.09109
$292$	0.284732	0.0166627
$293$	−25.8664	−1.51113	−0.755567	−	0.655072i	$-0.772638\pi$
$293$	−25.8664	−1.51113	−0.755567	+	0.655072i	$0.772638\pi$
$294$	10.6633	0.621897
$295$	17.6978	1.03040
$296$	2.00473	0.116522
$297$	−18.8632	−1.09456
$298$	13.6230	0.789158
$299$	−13.3156	−0.770063
$300$	−11.4584	−0.661550
$301$	−23.4287	−1.35041
$302$	2.18559	0.125767
$303$	−5.27636	−0.303119
$304$	0	0
$305$	14.2424	0.815519
$306$	−27.9008	−1.59498
$307$	−5.10095	−0.291127	−0.145563	−	0.989349i	$-0.546499\pi$
$307$	−5.10095	−0.291127	−0.145563	+	0.989349i	$0.546499\pi$
$308$	3.18396	0.181423
$309$	−15.0954	−0.858748
$310$	21.1588	1.20174
$311$	5.89166	0.334085	0.167043	−	0.985950i	$-0.446578\pi$
$311$	5.89166	0.334085	0.167043	+	0.985950i	$0.446578\pi$
$312$	−12.1132	−0.685772
$313$	−9.49494	−0.536686	−0.268343	−	0.963323i	$-0.586476\pi$
$313$	−9.49494	−0.536686	−0.268343	+	0.963323i	$0.586476\pi$
$314$	−14.0654	−0.793757
$315$	−78.7684	−4.43810
$316$	6.65900	0.374598
$317$	−2.53267	−0.142249	−0.0711245	−	0.997467i	$-0.522659\pi$
$317$	−2.53267	−0.142249	−0.0711245	+	0.997467i	$0.522659\pi$
$318$	12.0271	0.674448
$319$	9.04792	0.506586
$320$	−2.89336	−0.161744
$321$	18.8278	1.05087
$322$	−11.8951	−0.662890
$323$	0	0
$324$	38.4571	2.13651
$325$	−12.0167	−0.666568
$326$	−23.9298	−1.32535
$327$	−25.2491	−1.39628
$328$	10.6209	0.586441
$329$	−31.6565	−1.74528
$330$	−9.83330	−0.541306
$331$	17.8526	0.981266	0.490633	−	0.871366i	$-0.336766\pi$
$331$	17.8526	0.981266	0.490633	+	0.871366i	$0.336766\pi$
$332$	−10.4959	−0.576038
$333$	−17.1411	−0.939326
$334$	6.65271	0.364020
$335$	4.70946	0.257305
$336$	−10.8209	−0.590330
$337$	−31.8594	−1.73549	−0.867746	−	0.497008i	$-0.834432\pi$
$337$	−31.8594	−1.73549	−0.867746	+	0.497008i	$0.834432\pi$
$338$	0.296594	0.0161326
$339$	19.6159	1.06539
$340$	−9.44138	−0.512031
$341$	7.31289	0.396015
$342$	0	0
$343$	−12.2978	−0.664017
$344$	7.35835	0.396736
$345$	36.7368	1.97784
$346$	15.9832	0.859263
$347$	9.09301	0.488139	0.244069	−	0.969758i	$-0.421518\pi$
$347$	9.09301	0.488139	0.244069	+	0.969758i	$0.421518\pi$
$348$	−30.7501	−1.64838
$349$	−32.7116	−1.75101	−0.875507	−	0.483205i	$-0.839472\pi$
$349$	−32.7116	−1.75101	−0.875507	+	0.483205i	$0.839472\pi$
$350$	−10.7348	−0.573799
$351$	67.2320	3.58858
$352$	−1.00000	−0.0533002
$353$	13.6327	0.725598	0.362799	−	0.931867i	$-0.381821\pi$
$353$	13.6327	0.725598	0.362799	+	0.931867i	$0.381821\pi$
$354$	−20.7880	−1.10487
$355$	−16.9220	−0.898127
$356$	−12.6869	−0.672406
$357$	−35.3100	−1.86880
$358$	23.0475	1.21810
$359$	−18.2584	−0.963643	−0.481821	−	0.876270i	$-0.660025\pi$
$359$	−18.2584	−0.963643	−0.481821	+	0.876270i	$0.660025\pi$
$360$	24.7392	1.30387
$361$	0	0
$362$	−25.6166	−1.34638
$363$	−3.39858	−0.178379
$364$	−11.3482	−0.594808
$365$	−0.823832	−0.0431213
$366$	−16.7294	−0.874457
$367$	14.7998	0.772542	0.386271	−	0.922385i	$-0.373763\pi$
$367$	14.7998	0.772542	0.386271	+	0.922385i	$0.373763\pi$
$368$	3.73596	0.194750
$369$	−90.8122	−4.72749
$370$	−5.80040	−0.301548
$371$	11.2676	0.584985
$372$	−24.8534	−1.28859
$373$	−9.20525	−0.476630	−0.238315	−	0.971188i	$-0.576595\pi$
$373$	−9.20525	−0.476630	−0.238315	+	0.971188i	$0.576595\pi$
$374$	−3.26312	−0.168732
$375$	−16.0133	−0.826923
$376$	9.94250	0.512745
$377$	−32.2485	−1.66088
$378$	60.0597	3.08914
$379$	−4.62269	−0.237451	−0.118726	−	0.992927i	$-0.537881\pi$
$379$	−4.62269	−0.237451	−0.118726	+	0.992927i	$0.537881\pi$
$380$	0	0
$381$	19.5520	1.00168
$382$	−21.5968	−1.10499
$383$	13.3037	0.679788	0.339894	−	0.940464i	$-0.389609\pi$
$383$	13.3037	0.679788	0.339894	+	0.940464i	$0.389609\pi$
$384$	3.39858	0.173433
$385$	−9.21233	−0.469504
$386$	2.48323	0.126393
$387$	−62.9163	−3.19822
$388$	15.5153	0.787671
$389$	7.87125	0.399088	0.199544	−	0.979889i	$-0.436054\pi$
$389$	7.87125	0.399088	0.199544	+	0.979889i	$0.436054\pi$
$390$	35.0477	1.77471
$391$	12.1909	0.616519
$392$	−3.13758	−0.158472
$393$	−24.3759	−1.22960
$394$	−15.4026	−0.775972
$395$	−19.2669	−0.969422
$396$	8.55033	0.429670
$397$	−36.9541	−1.85467	−0.927335	−	0.374231i	$-0.877907\pi$
$397$	−36.9541	−1.85467	−0.927335	+	0.374231i	$0.877907\pi$
$398$	4.35270	0.218181
$399$	0	0
$400$	3.37152	0.168576
$401$	−23.8512	−1.19107	−0.595535	−	0.803329i	$-0.703060\pi$
$401$	−23.8512	−1.19107	−0.595535	+	0.803329i	$0.703060\pi$
$402$	−5.53179	−0.275901
$403$	−26.0645	−1.29836
$404$	1.55252	0.0772408
$405$	−111.270	−5.52906
$406$	−28.8082	−1.42973
$407$	−2.00473	−0.0993707
$408$	11.0900	0.549035
$409$	−9.08726	−0.449336	−0.224668	−	0.974435i	$-0.572130\pi$
$409$	−9.08726	−0.449336	−0.224668	+	0.974435i	$0.572130\pi$
$410$	−30.7301	−1.51765
$411$	58.1485	2.86825
$412$	4.44169	0.218826
$413$	−19.4752	−0.958314
$414$	−31.9437	−1.56995
$415$	30.3684	1.49073
$416$	3.56418	0.174748
$417$	51.4553	2.51978
$418$	0	0
$419$	−15.3823	−0.751472	−0.375736	−	0.926727i	$-0.622610\pi$
$419$	−15.3823	−0.751472	−0.375736	+	0.926727i	$0.622610\pi$
$420$	31.3088	1.52771
$421$	15.1283	0.737309	0.368655	−	0.929566i	$-0.379819\pi$
$421$	15.1283	0.737309	0.368655	+	0.929566i	$0.379819\pi$
$422$	−13.5690	−0.660528
$423$	−85.0117	−4.13341
$424$	−3.53887	−0.171863
$425$	11.0017	0.533660
$426$	19.8768	0.963035
$427$	−15.6729	−0.758464
$428$	−5.53992	−0.267782
$429$	12.1132	0.584829
$430$	−21.2903	−1.02671
$431$	0.864959	0.0416636	0.0208318	−	0.999783i	$-0.493369\pi$
$431$	0.864959	0.0416636	0.0208318	+	0.999783i	$0.493369\pi$
$432$	−18.8632	−0.907557
$433$	−2.11322	−0.101555	−0.0507775	−	0.998710i	$-0.516170\pi$
$433$	−2.11322	−0.101555	−0.0507775	+	0.998710i	$0.516170\pi$
$434$	−23.2839	−1.11766
$435$	88.9710	4.26583
$436$	7.42933	0.355800
$437$	0	0
$438$	0.967684	0.0462377
$439$	−0.593710	−0.0283362	−0.0141681	−	0.999900i	$-0.504510\pi$
$439$	−0.593710	−0.0283362	−0.0141681	+	0.999900i	$0.504510\pi$
$440$	2.89336	0.137935
$441$	26.8274	1.27749
$442$	11.6304	0.553200
$443$	−25.8773	−1.22947	−0.614734	−	0.788735i	$-0.710737\pi$
$443$	−25.8773	−1.22947	−0.614734	+	0.788735i	$0.710737\pi$
$444$	6.81322	0.323341
$445$	36.7078	1.74012
$446$	−2.84491	−0.134710
$447$	46.2987	2.18986
$448$	3.18396	0.150428
$449$	−4.94336	−0.233292	−0.116646	−	0.993174i	$-0.537214\pi$
$449$	−4.94336	−0.233292	−0.116646	+	0.993174i	$0.537214\pi$
$450$	−28.8276	−1.35895
$451$	−10.6209	−0.500119
$452$	−5.77179	−0.271482
$453$	7.42790	0.348993
$454$	−15.7467	−0.739031
$455$	32.8344	1.53930
$456$	0	0
$457$	4.31076	0.201649	0.100824	−	0.994904i	$-0.467852\pi$
$457$	4.31076	0.201649	0.100824	+	0.994904i	$0.467852\pi$
$458$	−7.94718	−0.371347
$459$	−61.5530	−2.87305
$460$	−10.8095	−0.503994
$461$	−25.0127	−1.16496	−0.582480	−	0.812845i	$-0.697917\pi$
$461$	−25.0127	−1.16496	−0.582480	+	0.812845i	$0.697917\pi$
$462$	10.8209	0.503435
$463$	22.8919	1.06388	0.531939	−	0.846783i	$-0.321463\pi$
$463$	22.8919	1.06388	0.531939	+	0.846783i	$0.321463\pi$
$464$	9.04792	0.420039
$465$	71.9099	3.33474
$466$	16.3309	0.756514
$467$	1.39654	0.0646242	0.0323121	−	0.999478i	$-0.489713\pi$
$467$	1.39654	0.0646242	0.0323121	+	0.999478i	$0.489713\pi$
$468$	−30.4749	−1.40870
$469$	−5.18246	−0.239304
$470$	−28.7672	−1.32693
$471$	−47.8024	−2.20262
$472$	6.11668	0.281543
$473$	−7.35835	−0.338337
$474$	22.6311	1.03948
$475$	0	0
$476$	10.3896	0.476208
$477$	30.2585	1.38544
$478$	−11.3810	−0.520554
$479$	23.8441	1.08947	0.544733	−	0.838609i	$-0.316631\pi$
$479$	23.8441	1.08947	0.544733	+	0.838609i	$0.316631\pi$
$480$	−9.83330	−0.448827
$481$	7.14522	0.325794
$482$	−15.5308	−0.707407
$483$	−40.4265	−1.83947
$484$	1.00000	0.0454545
$485$	−44.8914	−2.03841
$486$	74.1099	3.36169
$487$	−16.7643	−0.759664	−0.379832	−	0.925055i	$-0.624018\pi$
$487$	−16.7643	−0.759664	−0.379832	+	0.925055i	$0.624018\pi$
$488$	4.92246	0.222829
$489$	−81.3271	−3.67774
$490$	9.07815	0.410109
$491$	−28.0202	−1.26454	−0.632268	−	0.774750i	$-0.717876\pi$
$491$	−28.0202	−1.26454	−0.632268	+	0.774750i	$0.717876\pi$
$492$	36.0959	1.62733
$493$	29.5245	1.32972
$494$	0	0
$495$	−24.7392	−1.11194
$496$	7.31289	0.328359
$497$	18.6216	0.835293
$498$	−35.6711	−1.59846
$499$	16.5289	0.739935	0.369967	−	0.929045i	$-0.379369\pi$
$499$	16.5289	0.739935	0.369967	+	0.929045i	$0.379369\pi$
$500$	4.71176	0.210716
$501$	22.6097	1.01013
$502$	3.34649	0.149361
$503$	37.4795	1.67113	0.835565	−	0.549392i	$-0.185140\pi$
$503$	37.4795	1.67113	0.835565	+	0.549392i	$0.185140\pi$
$504$	−27.2239	−1.21265
$505$	−4.49200	−0.199891
$506$	−3.73596	−0.166084
$507$	1.00800	0.0447668
$508$	−5.75300	−0.255248
$509$	−6.38991	−0.283228	−0.141614	−	0.989922i	$-0.545229\pi$
$509$	−6.38991	−0.283228	−0.141614	+	0.989922i	$0.545229\pi$
$510$	−32.0873	−1.42085
$511$	0.906574	0.0401045
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.85508	−0.125932
$515$	−12.8514	−0.566300
$516$	25.0079	1.10091
$517$	−9.94250	−0.437271
$518$	6.38297	0.280452
$519$	54.3202	2.38439
$520$	−10.3125	−0.452231
$521$	−30.9060	−1.35402	−0.677009	−	0.735975i	$-0.736724\pi$
$521$	−30.9060	−1.35402	−0.677009	+	0.735975i	$0.736724\pi$
$522$	−77.3627	−3.38607
$523$	−20.9318	−0.915283	−0.457642	−	0.889137i	$-0.651306\pi$
$523$	−20.9318	−0.915283	−0.457642	+	0.889137i	$0.651306\pi$
$524$	7.17237	0.313327
$525$	−36.4830	−1.59225
$526$	6.62361	0.288803
$527$	23.8628	1.03948
$528$	−3.39858	−0.147904
$529$	−9.04262	−0.393157
$530$	10.2392	0.444763
$531$	−52.2996	−2.26961
$532$	0	0
$533$	37.8548	1.63967
$534$	−43.1175	−1.86588
$535$	16.0290	0.692993
$536$	1.62768	0.0703050
$537$	78.3287	3.38013
$538$	−21.0191	−0.906198
$539$	3.13758	0.135145
$540$	54.5781	2.34867
$541$	35.9132	1.54403	0.772015	−	0.635604i	$-0.219249\pi$
$541$	35.9132	1.54403	0.772015	+	0.635604i	$0.219249\pi$
$542$	−11.9407	−0.512898
$543$	−87.0601	−3.73611
$544$	−3.26312	−0.139905
$545$	−21.4957	−0.920775
$546$	−38.5678	−1.65055
$547$	16.8206	0.719195	0.359598	−	0.933107i	$-0.382914\pi$
$547$	16.8206	0.719195	0.359598	+	0.933107i	$0.382914\pi$
$548$	−17.1097	−0.730889
$549$	−42.0886	−1.79630
$550$	−3.37152	−0.143762
$551$	0	0
$552$	12.6969	0.540418
$553$	21.2020	0.901600
$554$	4.78708	0.203384
$555$	−19.7131	−0.836775
$556$	−15.1403	−0.642090
$557$	19.8589	0.841447	0.420724	−	0.907189i	$-0.361776\pi$
$557$	19.8589	0.841447	0.420724	+	0.907189i	$0.361776\pi$
$558$	−62.5276	−2.64701
$559$	26.2265	1.10926
$560$	−9.21233	−0.389292
$561$	−11.0900	−0.468219
$562$	−9.17404	−0.386984
$563$	−28.8312	−1.21509	−0.607545	−	0.794285i	$-0.707846\pi$
$563$	−28.8312	−1.21509	−0.607545	+	0.794285i	$0.707846\pi$
$564$	33.7904	1.42283
$565$	16.6999	0.702568
$566$	−4.72461	−0.198590
$567$	122.446	5.14224
$568$	−5.84857	−0.245401
$569$	−19.5784	−0.820768	−0.410384	−	0.911913i	$-0.634605\pi$
$569$	−19.5784	−0.820768	−0.410384	+	0.911913i	$0.634605\pi$
$570$	0	0
$571$	11.7467	0.491583	0.245791	−	0.969323i	$-0.420952\pi$
$571$	11.7467	0.491583	0.245791	+	0.969323i	$0.420952\pi$
$572$	−3.56418	−0.149026
$573$	−73.3984	−3.06626
$574$	33.8165	1.41147
$575$	12.5959	0.525284
$576$	8.55033	0.356264
$577$	5.56978	0.231873	0.115936	−	0.993257i	$-0.463013\pi$
$577$	5.56978	0.231873	0.115936	+	0.993257i	$0.463013\pi$
$578$	6.35205	0.264210
$579$	8.43944	0.350731
$580$	−26.1789	−1.08702
$581$	−33.4185	−1.38643
$582$	52.7300	2.18573
$583$	3.53887	0.146565
$584$	−0.284732	−0.0117823
$585$	88.1749	3.64558
$586$	25.8664	1.06853
$587$	−28.4078	−1.17251	−0.586257	−	0.810125i	$-0.699399\pi$
$587$	−28.4078	−1.17251	−0.586257	+	0.810125i	$0.699399\pi$
$588$	−10.6633	−0.439748
$589$	0	0
$590$	−17.6978	−0.728605
$591$	−52.3469	−2.15326
$592$	−2.00473	−0.0823938
$593$	−37.7918	−1.55192	−0.775962	−	0.630780i	$-0.782735\pi$
$593$	−37.7918	−1.55192	−0.775962	+	0.630780i	$0.782735\pi$
$594$	18.8632	0.773967
$595$	−30.0609	−1.23238
$596$	−13.6230	−0.558019
$597$	14.7930	0.605437
$598$	13.3156	0.544517
$599$	−30.6873	−1.25385	−0.626924	−	0.779080i	$-0.715686\pi$
$599$	−30.6873	−1.25385	−0.626924	+	0.779080i	$0.715686\pi$
$600$	11.4584	0.467787
$601$	−4.57962	−0.186806	−0.0934032	−	0.995628i	$-0.529775\pi$
$601$	−4.57962	−0.186806	−0.0934032	+	0.995628i	$0.529775\pi$
$602$	23.4287	0.954881
$603$	−13.9172	−0.566752
$604$	−2.18559	−0.0889304
$605$	−2.89336	−0.117632
$606$	5.27636	0.214338
$607$	9.81836	0.398515	0.199257	−	0.979947i	$-0.436147\pi$
$607$	9.81836	0.398515	0.199257	+	0.979947i	$0.436147\pi$
$608$	0	0
$609$	−97.9069	−3.96739
$610$	−14.2424	−0.576659
$611$	35.4369	1.43362
$612$	27.9008	1.12782
$613$	7.20426	0.290977	0.145489	−	0.989360i	$-0.453525\pi$
$613$	7.20426	0.290977	0.145489	+	0.989360i	$0.453525\pi$
$614$	5.10095	0.205858
$615$	−104.439	−4.21137
$616$	−3.18396	−0.128285
$617$	−42.2256	−1.69994	−0.849970	−	0.526831i	$-0.823380\pi$
$617$	−42.2256	−1.69994	−0.849970	+	0.526831i	$0.823380\pi$
$618$	15.0954	0.607227
$619$	−9.21254	−0.370283	−0.185142	−	0.982712i	$-0.559274\pi$
$619$	−9.21254	−0.370283	−0.185142	+	0.982712i	$0.559274\pi$
$620$	−21.1588	−0.849759
$621$	−70.4722	−2.82795
$622$	−5.89166	−0.236234
$623$	−40.3946	−1.61838
$624$	12.1132	0.484914
$625$	−30.4904	−1.21962
$626$	9.49494	0.379494
$627$	0	0
$628$	14.0654	0.561271
$629$	−6.54167	−0.260833
$630$	78.7684	3.13821
$631$	−37.8291	−1.50595	−0.752976	−	0.658048i	$-0.771382\pi$
$631$	−37.8291	−1.50595	−0.752976	+	0.658048i	$0.771382\pi$
$632$	−6.65900	−0.264881
$633$	−46.1153	−1.83292
$634$	2.53267	0.100585
$635$	16.6455	0.660556
$636$	−12.0271	−0.476907
$637$	−11.1829	−0.443083
$638$	−9.04792	−0.358211
$639$	50.0072	1.97825
$640$	2.89336	0.114370
$641$	−11.9082	−0.470347	−0.235173	−	0.971953i	$-0.575566\pi$
$641$	−11.9082	−0.470347	−0.235173	+	0.971953i	$0.575566\pi$
$642$	−18.8278	−0.743076
$643$	10.1857	0.401683	0.200841	−	0.979624i	$-0.435632\pi$
$643$	10.1857	0.401683	0.200841	+	0.979624i	$0.435632\pi$
$644$	11.8951	0.468734
$645$	−72.3569	−2.84905
$646$	0	0
$647$	−21.7034	−0.853249	−0.426625	−	0.904429i	$-0.640297\pi$
$647$	−21.7034	−0.853249	−0.426625	+	0.904429i	$0.640297\pi$
$648$	−38.4571	−1.51074
$649$	−6.11668	−0.240101
$650$	12.0167	0.471335
$651$	−79.1322	−3.10144
$652$	23.9298	0.937162
$653$	−36.9748	−1.44693	−0.723467	−	0.690358i	$-0.757453\pi$
$653$	−36.9748	−1.44693	−0.723467	+	0.690358i	$0.757453\pi$
$654$	25.2491	0.987320
$655$	−20.7523	−0.810858
$656$	−10.6209	−0.414676
$657$	2.43455	0.0949809
$658$	31.6565	1.23410
$659$	22.0889	0.860461	0.430231	−	0.902719i	$-0.358432\pi$
$659$	22.0889	0.860461	0.430231	+	0.902719i	$0.358432\pi$
$660$	9.83330	0.382761
$661$	17.7335	0.689752	0.344876	−	0.938648i	$-0.387921\pi$
$661$	17.7335	0.689752	0.344876	+	0.938648i	$0.387921\pi$
$662$	−17.8526	−0.693860
$663$	39.5267	1.53509
$664$	10.4959	0.407320
$665$	0	0
$666$	17.1411	0.664204
$667$	33.8027	1.30884
$668$	−6.65271	−0.257401
$669$	−9.66863	−0.373811
$670$	−4.70946	−0.181942
$671$	−4.92246	−0.190029
$672$	10.8209	0.417426
$673$	25.6661	0.989357	0.494679	−	0.869076i	$-0.335286\pi$
$673$	25.6661	0.989357	0.494679	+	0.869076i	$0.335286\pi$
$674$	31.8594	1.22718
$675$	−63.5978	−2.44788
$676$	−0.296594	−0.0114075
$677$	−9.56717	−0.367696	−0.183848	−	0.982955i	$-0.558855\pi$
$677$	−9.56717	−0.367696	−0.183848	+	0.982955i	$0.558855\pi$
$678$	−19.6159	−0.753343
$679$	49.4001	1.89580
$680$	9.44138	0.362060
$681$	−53.5165	−2.05076
$682$	−7.31289	−0.280025
$683$	−6.75907	−0.258629	−0.129314	−	0.991604i	$-0.541278\pi$
$683$	−6.75907	−0.258629	−0.129314	+	0.991604i	$0.541278\pi$
$684$	0	0
$685$	49.5044	1.89147
$686$	12.2978	0.469531
$687$	−27.0091	−1.03046
$688$	−7.35835	−0.280534
$689$	−12.6132	−0.480524
$690$	−36.7368	−1.39855
$691$	−27.2148	−1.03530	−0.517650	−	0.855593i	$-0.673193\pi$
$691$	−27.2148	−1.03530	−0.517650	+	0.855593i	$0.673193\pi$
$692$	−15.9832	−0.607591
$693$	27.2239	1.03415
$694$	−9.09301	−0.345166
$695$	43.8062	1.66166
$696$	30.7501	1.16558
$697$	−34.6573	−1.31274
$698$	32.7116	1.23815
$699$	55.5018	2.09927
$700$	10.7348	0.405737
$701$	3.27544	0.123712	0.0618559	−	0.998085i	$-0.480298\pi$
$701$	3.27544	0.123712	0.0618559	+	0.998085i	$0.480298\pi$
$702$	−67.2320	−2.53751
$703$	0	0
$704$	1.00000	0.0376889
$705$	−97.7676	−3.68214
$706$	−13.6327	−0.513075
$707$	4.94316	0.185907
$708$	20.7880	0.781261
$709$	−7.14942	−0.268502	−0.134251	−	0.990947i	$-0.542863\pi$
$709$	−7.14942	−0.268502	−0.134251	+	0.990947i	$0.542863\pi$
$710$	16.9220	0.635072
$711$	56.9367	2.13529
$712$	12.6869	0.475463
$713$	27.3207	1.02317
$714$	35.3100	1.32144
$715$	10.3125	0.385664
$716$	−23.0475	−0.861325
$717$	−38.6792	−1.44450
$718$	18.2584	0.681398
$719$	10.2157	0.380982	0.190491	−	0.981689i	$-0.438992\pi$
$719$	10.2157	0.380982	0.190491	+	0.981689i	$0.438992\pi$
$720$	−24.7392	−0.921974
$721$	14.1421	0.526681
$722$	0	0
$723$	−52.7825	−1.96300
$724$	25.6166	0.952034
$725$	30.5053	1.13294
$726$	3.39858	0.126133
$727$	−21.4081	−0.793984	−0.396992	−	0.917822i	$-0.629946\pi$
$727$	−21.4081	−0.793984	−0.396992	+	0.917822i	$0.629946\pi$
$728$	11.3482	0.420593
$729$	136.497	5.05543
$730$	0.823832	0.0304914
$731$	−24.0112	−0.888086
$732$	16.7294	0.618335
$733$	−0.266039	−0.00982636	−0.00491318	−	0.999988i	$-0.501564\pi$
$733$	−0.266039	−0.00982636	−0.00491318	+	0.999988i	$0.501564\pi$
$734$	−14.7998	−0.546269
$735$	30.8528	1.13802
$736$	−3.73596	−0.137709
$737$	−1.62768	−0.0599563
$738$	90.8122	3.34284
$739$	−0.150780	−0.00554655	−0.00277327	−	0.999996i	$-0.500883\pi$
$739$	−0.150780	−0.00554655	−0.00277327	+	0.999996i	$0.500883\pi$
$740$	5.80040	0.213227
$741$	0	0
$742$	−11.2676	−0.413647
$743$	49.9527	1.83259	0.916293	−	0.400509i	$-0.131167\pi$
$743$	49.9527	1.83259	0.916293	+	0.400509i	$0.131167\pi$
$744$	24.8534	0.911171
$745$	39.4162	1.44410
$746$	9.20525	0.337028
$747$	−89.7434	−3.28354
$748$	3.26312	0.119311
$749$	−17.6389	−0.644510
$750$	16.0133	0.584723
$751$	5.54418	0.202310	0.101155	−	0.994871i	$-0.467746\pi$
$751$	5.54418	0.202310	0.101155	+	0.994871i	$0.467746\pi$
$752$	−9.94250	−0.362566
$753$	11.3733	0.414467
$754$	32.2485	1.17442
$755$	6.32370	0.230143
$756$	−60.0597	−2.18435
$757$	32.4600	1.17978	0.589889	−	0.807484i	$-0.299172\pi$
$757$	32.4600	1.17978	0.589889	+	0.807484i	$0.299172\pi$
$758$	4.62269	0.167904
$759$	−12.6969	−0.460870
$760$	0	0
$761$	27.4535	0.995190	0.497595	−	0.867410i	$-0.334217\pi$
$761$	27.4535	0.995190	0.497595	+	0.867410i	$0.334217\pi$
$762$	−19.5520	−0.708295
$763$	23.6547	0.856356
$764$	21.5968	0.781344
$765$	−80.7269	−2.91869
$766$	−13.3037	−0.480682
$767$	21.8010	0.787188
$768$	−3.39858	−0.122636
$769$	−37.1204	−1.33860	−0.669298	−	0.742994i	$-0.733405\pi$
$769$	−37.1204	−1.33860	−0.669298	+	0.742994i	$0.733405\pi$
$770$	9.21233	0.331989
$771$	−9.70322	−0.349453
$772$	−2.48323	−0.0893733
$773$	37.3371	1.34292	0.671461	−	0.741040i	$-0.265667\pi$
$773$	37.3371	1.34292	0.671461	+	0.741040i	$0.265667\pi$
$774$	62.9163	2.26148
$775$	24.6556	0.885655
$776$	−15.5153	−0.556967
$777$	21.6930	0.778233
$778$	−7.87125	−0.282198
$779$	0	0
$780$	−35.0477	−1.25491
$781$	5.84857	0.209278
$782$	−12.1909	−0.435945
$783$	−170.673	−6.09935
$784$	3.13758	0.112057
$785$	−40.6963	−1.45251
$786$	24.3759	0.869458
$787$	28.9358	1.03145	0.515725	−	0.856754i	$-0.327523\pi$
$787$	28.9358	1.03145	0.515725	+	0.856754i	$0.327523\pi$
$788$	15.4026	0.548695
$789$	22.5109	0.801408
$790$	19.2669	0.685485
$791$	−18.3771	−0.653415
$792$	−8.55033	−0.303823
$793$	17.5445	0.623025
$794$	36.9541	1.31145
$795$	34.7988	1.23419
$796$	−4.35270	−0.154277
$797$	10.0927	0.357501	0.178750	−	0.983894i	$-0.442795\pi$
$797$	10.0927	0.357501	0.178750	+	0.983894i	$0.442795\pi$
$798$	0	0
$799$	−32.4436	−1.14777
$800$	−3.37152	−0.119201
$801$	−108.477	−3.83286
$802$	23.8512	0.842214
$803$	0.284732	0.0100480
$804$	5.53179	0.195091
$805$	−34.4169	−1.21304
$806$	26.0645	0.918082
$807$	−71.4351	−2.51463
$808$	−1.55252	−0.0546175
$809$	18.8934	0.664255	0.332128	−	0.943234i	$-0.392233\pi$
$809$	18.8934	0.664255	0.332128	+	0.943234i	$0.392233\pi$
$810$	111.270	3.90964
$811$	18.4074	0.646372	0.323186	−	0.946335i	$-0.395246\pi$
$811$	18.4074	0.646372	0.323186	+	0.946335i	$0.395246\pi$
$812$	28.8082	1.01097
$813$	−40.5814	−1.42325
$814$	2.00473	0.0702657
$815$	−69.2374	−2.42528
$816$	−11.0900	−0.388227
$817$	0	0
$818$	9.08726	0.317729
$819$	−97.0309	−3.39053
$820$	30.7301	1.07314
$821$	−1.51254	−0.0527879	−0.0263939	−	0.999652i	$-0.508402\pi$
$821$	−1.51254	−0.0527879	−0.0263939	+	0.999652i	$0.508402\pi$
$822$	−58.1485	−2.02816
$823$	−21.6329	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$823$	−21.6329	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$824$	−4.44169	−0.154733
$825$	−11.4584	−0.398930
$826$	19.4752	0.677631
$827$	6.54050	0.227435	0.113718	−	0.993513i	$-0.463724\pi$
$827$	6.54050	0.227435	0.113718	+	0.993513i	$0.463724\pi$
$828$	31.9437	1.11012
$829$	−7.88206	−0.273755	−0.136878	−	0.990588i	$-0.543707\pi$
$829$	−7.88206	−0.273755	−0.136878	+	0.990588i	$0.543707\pi$
$830$	−30.3684	−1.05410
$831$	16.2693	0.564374
$832$	−3.56418	−0.123566
$833$	10.2383	0.354736
$834$	−51.4553	−1.78175
$835$	19.2487	0.666128
$836$	0	0
$837$	−137.945	−4.76807
$838$	15.3823	0.531371
$839$	13.6826	0.472377	0.236188	−	0.971707i	$-0.424102\pi$
$839$	13.6826	0.472377	0.236188	+	0.971707i	$0.424102\pi$
$840$	−31.3088	−1.08026
$841$	52.8649	1.82293
$842$	−15.1283	−0.521356
$843$	−31.1787	−1.07385
$844$	13.5690	0.467064
$845$	0.858153	0.0295214
$846$	85.0117	2.92276
$847$	3.18396	0.109402
$848$	3.53887	0.121525
$849$	−16.0570	−0.551073
$850$	−11.0017	−0.377355
$851$	−7.48958	−0.256740
$852$	−19.8768	−0.680969
$853$	−10.0471	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$853$	−10.0471	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$854$	15.6729	0.536315
$855$	0	0
$856$	5.53992	0.189350
$857$	−55.7601	−1.90473	−0.952365	−	0.304960i	$-0.901357\pi$
$857$	−55.7601	−1.90473	−0.952365	+	0.304960i	$0.901357\pi$
$858$	−12.1132	−0.413536
$859$	43.4054	1.48097	0.740487	−	0.672071i	$-0.234595\pi$
$859$	43.4054	1.48097	0.740487	+	0.672071i	$0.234595\pi$
$860$	21.2903	0.725995
$861$	114.928	3.91673
$862$	−0.864959	−0.0294606
$863$	50.8470	1.73085	0.865425	−	0.501038i	$-0.167048\pi$
$863$	50.8470	1.73085	0.865425	+	0.501038i	$0.167048\pi$
$864$	18.8632	0.641740
$865$	46.2452	1.57238
$866$	2.11322	0.0718103
$867$	21.5879	0.733164
$868$	23.2839	0.790308
$869$	6.65900	0.225891
$870$	−88.9710	−3.01640
$871$	5.80135	0.196571
$872$	−7.42933	−0.251589
$873$	132.661	4.48989
$874$	0	0
$875$	15.0021	0.507162
$876$	−0.967684	−0.0326950
$877$	20.5267	0.693138	0.346569	−	0.938024i	$-0.387347\pi$
$877$	20.5267	0.693138	0.346569	+	0.938024i	$0.387347\pi$
$878$	0.593710	0.0200367
$879$	87.9091	2.96510
$880$	−2.89336	−0.0975351
$881$	16.4976	0.555818	0.277909	−	0.960607i	$-0.410359\pi$
$881$	16.4976	0.555818	0.277909	+	0.960607i	$0.410359\pi$
$882$	−26.8274	−0.903324
$883$	−26.2708	−0.884083	−0.442042	−	0.896995i	$-0.645746\pi$
$883$	−26.2708	−0.884083	−0.442042	+	0.896995i	$0.645746\pi$
$884$	−11.6304	−0.391171
$885$	−60.1472	−2.02183
$886$	25.8773	0.869365
$887$	52.1441	1.75083	0.875414	−	0.483373i	$-0.160589\pi$
$887$	52.1441	1.75083	0.875414	+	0.483373i	$0.160589\pi$
$888$	−6.81322	−0.228637
$889$	−18.3173	−0.614343
$890$	−36.7078	−1.23045
$891$	38.4571	1.28836
$892$	2.84491	0.0952545
$893$	0	0
$894$	−46.2987	−1.54846
$895$	66.6847	2.22902
$896$	−3.18396	−0.106369
$897$	45.2542	1.51099
$898$	4.94336	0.164962
$899$	66.1665	2.20678
$900$	28.8276	0.960921
$901$	11.5478	0.384712
$902$	10.6209	0.353637
$903$	79.6241	2.64973
$904$	5.77179	0.191967
$905$	−74.1181	−2.46377
$906$	−7.42790	−0.246775
$907$	−15.6077	−0.518245	−0.259123	−	0.965844i	$-0.583433\pi$
$907$	−15.6077	−0.518245	−0.259123	+	0.965844i	$0.583433\pi$
$908$	15.7467	0.522573
$909$	13.2746	0.440289
$910$	−32.8344	−1.08845
$911$	−42.3797	−1.40410	−0.702050	−	0.712127i	$-0.747732\pi$
$911$	−42.3797	−1.40410	−0.702050	+	0.712127i	$0.747732\pi$
$912$	0	0
$913$	−10.4959	−0.347364
$914$	−4.31076	−0.142587
$915$	−48.4040	−1.60019
$916$	7.94718	0.262582
$917$	22.8365	0.754129
$918$	61.5530	2.03155
$919$	−28.3127	−0.933950	−0.466975	−	0.884270i	$-0.654656\pi$
$919$	−28.3127	−0.933950	−0.466975	+	0.884270i	$0.654656\pi$
$920$	10.8095	0.356378
$921$	17.3360	0.571240
$922$	25.0127	0.823751
$923$	−20.8454	−0.686134
$924$	−10.8209	−0.355982
$925$	−6.75899	−0.222234
$926$	−22.8919	−0.752276
$927$	37.9779	1.24736
$928$	−9.04792	−0.297013
$929$	12.4525	0.408554	0.204277	−	0.978913i	$-0.434516\pi$
$929$	12.4525	0.408554	0.204277	+	0.978913i	$0.434516\pi$
$930$	−71.9099	−2.35802
$931$	0	0
$932$	−16.3309	−0.534936
$933$	−20.0233	−0.655532
$934$	−1.39654	−0.0456962
$935$	−9.44138	−0.308766
$936$	30.4749	0.996105
$937$	−18.6685	−0.609874	−0.304937	−	0.952373i	$-0.598635\pi$
$937$	−18.6685	−0.609874	−0.304937	+	0.952373i	$0.598635\pi$
$938$	5.18246	0.169213
$939$	32.2693	1.05307
$940$	28.7672	0.938283
$941$	−20.1823	−0.657924	−0.328962	−	0.944343i	$-0.606699\pi$
$941$	−20.1823	−0.657924	−0.328962	+	0.944343i	$0.606699\pi$
$942$	47.8024	1.55749
$943$	−39.6792	−1.29213
$944$	−6.11668	−0.199081
$945$	173.774	5.65288
$946$	7.35835	0.239241
$947$	46.2240	1.50208	0.751040	−	0.660257i	$-0.229553\pi$
$947$	46.2240	1.50208	0.751040	+	0.660257i	$0.229553\pi$
$948$	−22.6311	−0.735025
$949$	−1.01484	−0.0329430
$950$	0	0
$951$	8.60748	0.279117
$952$	−10.3896	−0.336730
$953$	−46.7013	−1.51280	−0.756402	−	0.654107i	$-0.773045\pi$
$953$	−46.7013	−1.51280	−0.756402	+	0.654107i	$0.773045\pi$
$954$	−30.2585	−0.979655
$955$	−62.4873	−2.02204
$956$	11.3810	0.368087
$957$	−30.7501	−0.994009
$958$	−23.8441	−0.770369
$959$	−54.4764	−1.75914
$960$	9.83330	0.317369
$961$	22.4784	0.725109
$962$	−7.14522	−0.230371
$963$	−47.3681	−1.52642
$964$	15.5308	0.500212
$965$	7.18487	0.231289
$966$	40.4265	1.30070
$967$	8.83799	0.284210	0.142105	−	0.989852i	$-0.454613\pi$
$967$	8.83799	0.284210	0.142105	+	0.989852i	$0.454613\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	44.8914	1.44137
$971$	39.5477	1.26915	0.634573	−	0.772863i	$-0.281176\pi$
$971$	39.5477	1.26915	0.634573	+	0.772863i	$0.281176\pi$
$972$	−74.1099	−2.37708
$973$	−48.2059	−1.54541
$974$	16.7643	0.537164
$975$	40.8398	1.30792
$976$	−4.92246	−0.157564
$977$	−1.98723	−0.0635772	−0.0317886	−	0.999495i	$-0.510120\pi$
$977$	−1.98723	−0.0635772	−0.0317886	+	0.999495i	$0.510120\pi$
$978$	81.3271	2.60055
$979$	−12.6869	−0.405476
$980$	−9.07815	−0.289991
$981$	63.5232	2.02814
$982$	28.0202	0.894162
$983$	−34.2862	−1.09356	−0.546780	−	0.837276i	$-0.684147\pi$
$983$	−34.2862	−1.09356	−0.546780	+	0.837276i	$0.684147\pi$
$984$	−36.0959	−1.15070
$985$	−44.5652	−1.41997
$986$	−29.5245	−0.940251
$987$	107.587	3.42453
$988$	0	0
$989$	−27.4905	−0.874147
$990$	24.7392	0.786262
$991$	16.3156	0.518282	0.259141	−	0.965839i	$-0.416561\pi$
$991$	16.3156	0.518282	0.259141	+	0.965839i	$0.416561\pi$
$992$	−7.31289	−0.232185
$993$	−60.6733	−1.92541
$994$	−18.6216	−0.590641
$995$	12.5939	0.399255
$996$	35.6711	1.13028
$997$	−9.37948	−0.297051	−0.148525	−	0.988909i	$-0.547453\pi$
$997$	−9.37948	−0.297051	−0.148525	+	0.988909i	$0.547453\pi$
$998$	−16.5289	−0.523213
$999$	37.8156	1.19643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.bm.1.1	✓	8
19.18	odd	2		7942.2.a.br.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7942.2.a.bm.1.1	✓	8	1.1	even	1	trivial
7942.2.a.br.1.8	yes	8	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.39858 q^{3} +1.00000 q^{4} -2.89336 q^{5} +3.39858 q^{6} +3.18396 q^{7} -1.00000 q^{8} +8.55033 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.39858 q^{3} +1.00000 q^{4} -2.89336 q^{5} +3.39858 q^{6} +3.18396 q^{7} -1.00000 q^{8} +8.55033 q^{9} +2.89336 q^{10} +1.00000 q^{11} -3.39858 q^{12} -3.56418 q^{13} -3.18396 q^{14} +9.83330 q^{15} +1.00000 q^{16} +3.26312 q^{17} -8.55033 q^{18} -2.89336 q^{20} -10.8209 q^{21} -1.00000 q^{22} +3.73596 q^{23} +3.39858 q^{24} +3.37152 q^{25} +3.56418 q^{26} -18.8632 q^{27} +3.18396 q^{28} +9.04792 q^{29} -9.83330 q^{30} +7.31289 q^{31} -1.00000 q^{32} -3.39858 q^{33} -3.26312 q^{34} -9.21233 q^{35} +8.55033 q^{36} -2.00473 q^{37} +12.1132 q^{39} +2.89336 q^{40} -10.6209 q^{41} +10.8209 q^{42} -7.35835 q^{43} +1.00000 q^{44} -24.7392 q^{45} -3.73596 q^{46} -9.94250 q^{47} -3.39858 q^{48} +3.13758 q^{49} -3.37152 q^{50} -11.0900 q^{51} -3.56418 q^{52} +3.53887 q^{53} +18.8632 q^{54} -2.89336 q^{55} -3.18396 q^{56} -9.04792 q^{58} -6.11668 q^{59} +9.83330 q^{60} -4.92246 q^{61} -7.31289 q^{62} +27.2239 q^{63} +1.00000 q^{64} +10.3125 q^{65} +3.39858 q^{66} -1.62768 q^{67} +3.26312 q^{68} -12.6969 q^{69} +9.21233 q^{70} +5.84857 q^{71} -8.55033 q^{72} +0.284732 q^{73} +2.00473 q^{74} -11.4584 q^{75} +3.18396 q^{77} -12.1132 q^{78} +6.65900 q^{79} -2.89336 q^{80} +38.4571 q^{81} +10.6209 q^{82} -10.4959 q^{83} -10.8209 q^{84} -9.44138 q^{85} +7.35835 q^{86} -30.7501 q^{87} -1.00000 q^{88} -12.6869 q^{89} +24.7392 q^{90} -11.3482 q^{91} +3.73596 q^{92} -24.8534 q^{93} +9.94250 q^{94} +3.39858 q^{96} +15.5153 q^{97} -3.13758 q^{98} +8.55033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + q^{5} + 4 q^{6} + 4 q^{7} - 8 q^{8} + 6 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + q^5 + 4 * q^6 + 4 * q^7 - 8 * q^8 + 6 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + q^{5} + 4 q^{6} + 4 q^{7} - 8 q^{8} + 6 q^{9} - q^{10} + 8 q^{11} - 4 q^{12} - 12 q^{13} - 4 q^{14} + 9 q^{15} + 8 q^{16} + 7 q^{17} - 6 q^{18} + q^{20} - 13 q^{21} - 8 q^{22} - 6 q^{23} + 4 q^{24} + 5 q^{25} + 12 q^{26} - 22 q^{27} + 4 q^{28} + q^{29} - 9 q^{30} - 8 q^{31} - 8 q^{32} - 4 q^{33} - 7 q^{34} + 5 q^{35} + 6 q^{36} - 12 q^{37} + 10 q^{39} - q^{40} - 18 q^{41} + 13 q^{42} - 4 q^{43} + 8 q^{44} - 19 q^{45} + 6 q^{46} + 13 q^{47} - 4 q^{48} + 4 q^{49} - 5 q^{50} - 14 q^{51} - 12 q^{52} + 17 q^{53} + 22 q^{54} + q^{55} - 4 q^{56} - q^{58} - 26 q^{59} + 9 q^{60} - 7 q^{61} + 8 q^{62} + 13 q^{63} + 8 q^{64} - 36 q^{65} + 4 q^{66} - 6 q^{67} + 7 q^{68} - 19 q^{69} - 5 q^{70} + 3 q^{71} - 6 q^{72} + 5 q^{73} + 12 q^{74} - 30 q^{75} + 4 q^{77} - 10 q^{78} - 16 q^{79} + q^{80} + 8 q^{81} + 18 q^{82} - 21 q^{83} - 13 q^{84} - 10 q^{85} + 4 q^{86} - 29 q^{87} - 8 q^{88} - 16 q^{89} + 19 q^{90} - 31 q^{91} - 6 q^{92} + 20 q^{93} - 13 q^{94} + 4 q^{96} + 15 q^{97} - 4 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + q^5 + 4 * q^6 + 4 * q^7 - 8 * q^8 + 6 * q^9 - q^10 + 8 * q^11 - 4 * q^12 - 12 * q^13 - 4 * q^14 + 9 * q^15 + 8 * q^16 + 7 * q^17 - 6 * q^18 + q^20 - 13 * q^21 - 8 * q^22 - 6 * q^23 + 4 * q^24 + 5 * q^25 + 12 * q^26 - 22 * q^27 + 4 * q^28 + q^29 - 9 * q^30 - 8 * q^31 - 8 * q^32 - 4 * q^33 - 7 * q^34 + 5 * q^35 + 6 * q^36 - 12 * q^37 + 10 * q^39 - q^40 - 18 * q^41 + 13 * q^42 - 4 * q^43 + 8 * q^44 - 19 * q^45 + 6 * q^46 + 13 * q^47 - 4 * q^48 + 4 * q^49 - 5 * q^50 - 14 * q^51 - 12 * q^52 + 17 * q^53 + 22 * q^54 + q^55 - 4 * q^56 - q^58 - 26 * q^59 + 9 * q^60 - 7 * q^61 + 8 * q^62 + 13 * q^63 + 8 * q^64 - 36 * q^65 + 4 * q^66 - 6 * q^67 + 7 * q^68 - 19 * q^69 - 5 * q^70 + 3 * q^71 - 6 * q^72 + 5 * q^73 + 12 * q^74 - 30 * q^75 + 4 * q^77 - 10 * q^78 - 16 * q^79 + q^80 + 8 * q^81 + 18 * q^82 - 21 * q^83 - 13 * q^84 - 10 * q^85 + 4 * q^86 - 29 * q^87 - 8 * q^88 - 16 * q^89 + 19 * q^90 - 31 * q^91 - 6 * q^92 + 20 * q^93 - 13 * q^94 + 4 * q^96 + 15 * q^97 - 4 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

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Database

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