LMFDB - Embedded newform 3627.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3627,2,Mod(1,3627)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3627, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3627.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3627 = 3^{2} \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3627.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:	$28.9617408131$
Analytic rank:	$0$
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} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 $ x^8 - 3x^7 - 7x^6 + 19x^5 + 21x^4 - 31x^3 - 29x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.75938$ of defining polynomial
Character	$\chi$	$=$	3627.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75938 q^{2} +1.09540 q^{4} +3.67909 q^{5} +4.14977 q^{7} +1.59152 q^{8} +O(q^{10})$	\(q-1.75938 q^{2} +1.09540 q^{4} +3.67909 q^{5} +4.14977 q^{7} +1.59152 q^{8} -6.47291 q^{10} +5.07739 q^{11} -1.00000 q^{13} -7.30102 q^{14} -4.99090 q^{16} +0.643026 q^{17} -7.24868 q^{19} +4.03010 q^{20} -8.93304 q^{22} -0.685475 q^{23} +8.53574 q^{25} +1.75938 q^{26} +4.54568 q^{28} -5.82822 q^{29} -1.00000 q^{31} +5.59782 q^{32} -1.13132 q^{34} +15.2674 q^{35} +8.72827 q^{37} +12.7532 q^{38} +5.85537 q^{40} +11.1619 q^{41} -1.81828 q^{43} +5.56180 q^{44} +1.20601 q^{46} -6.05989 q^{47} +10.2206 q^{49} -15.0176 q^{50} -1.09540 q^{52} +1.56129 q^{53} +18.6802 q^{55} +6.60446 q^{56} +10.2540 q^{58} +7.14043 q^{59} +1.54027 q^{61} +1.75938 q^{62} +0.133123 q^{64} -3.67909 q^{65} +2.10412 q^{67} +0.704374 q^{68} -26.8611 q^{70} -0.546141 q^{71} -9.58155 q^{73} -15.3563 q^{74} -7.94024 q^{76} +21.0700 q^{77} -4.15021 q^{79} -18.3620 q^{80} -19.6379 q^{82} -6.77183 q^{83} +2.36575 q^{85} +3.19904 q^{86} +8.08078 q^{88} +14.7212 q^{89} -4.14977 q^{91} -0.750873 q^{92} +10.6616 q^{94} -26.6686 q^{95} -12.2489 q^{97} -17.9819 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 9 q^{4} + 15 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 5 * q^2 + 9 * q^4 + 15 * q^5 - 4 * q^7 + 15 * q^8	$ 8 q + 5 q^{2} + 9 q^{4} + 15 q^{5} - 4 q^{7} + 15 q^{8} + 9 q^{10} + 5 q^{11} - 8 q^{13} - 2 q^{14} + 3 q^{16} + 11 q^{17} - 9 q^{19} + 31 q^{20} - 2 q^{22} + 19 q^{25} - 5 q^{26} + 16 q^{28} + 12 q^{29} - 8 q^{31} + 25 q^{32} + 22 q^{34} - 7 q^{35} - 9 q^{37} + 9 q^{38} + 55 q^{40} + 25 q^{41} + 7 q^{43} + 26 q^{44} + 5 q^{46} + 17 q^{47} + 11 q^{50} - 9 q^{52} + 15 q^{53} + 7 q^{55} + 14 q^{56} - 5 q^{58} + 15 q^{59} + 11 q^{61} - 5 q^{62} + 47 q^{64} - 15 q^{65} + 18 q^{67} + 16 q^{68} - 24 q^{70} + 7 q^{71} + 24 q^{73} - 48 q^{74} - 3 q^{76} + 49 q^{77} + 33 q^{79} + 16 q^{80} - q^{82} + 13 q^{83} + q^{85} - 19 q^{86} + 37 q^{88} + 23 q^{89} + 4 q^{91} - 22 q^{92} + 10 q^{94} - 43 q^{95} - 17 q^{97} - 52 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 9 * q^4 + 15 * q^5 - 4 * q^7 + 15 * q^8 + 9 * q^10 + 5 * q^11 - 8 * q^13 - 2 * q^14 + 3 * q^16 + 11 * q^17 - 9 * q^19 + 31 * q^20 - 2 * q^22 + 19 * q^25 - 5 * q^26 + 16 * q^28 + 12 * q^29 - 8 * q^31 + 25 * q^32 + 22 * q^34 - 7 * q^35 - 9 * q^37 + 9 * q^38 + 55 * q^40 + 25 * q^41 + 7 * q^43 + 26 * q^44 + 5 * q^46 + 17 * q^47 + 11 * q^50 - 9 * q^52 + 15 * q^53 + 7 * q^55 + 14 * q^56 - 5 * q^58 + 15 * q^59 + 11 * q^61 - 5 * q^62 + 47 * q^64 - 15 * q^65 + 18 * q^67 + 16 * q^68 - 24 * q^70 + 7 * q^71 + 24 * q^73 - 48 * q^74 - 3 * q^76 + 49 * q^77 + 33 * q^79 + 16 * q^80 - q^82 + 13 * q^83 + q^85 - 19 * q^86 + 37 * q^88 + 23 * q^89 + 4 * q^91 - 22 * q^92 + 10 * q^94 - 43 * q^95 - 17 * q^97 - 52 * q^98

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 $

$2$	−1.75938	−1.24407	−0.622033	−	0.782991i	$-0.713693\pi$
$2$	−1.75938	−1.24407	−0.622033	+	0.782991i	$0.713693\pi$
$3$	0	0
$3$	0	0
$4$	1.09540	0.547702
$5$	3.67909	1.64534	0.822671	−	0.568518i	$-0.192483\pi$
$5$	3.67909	1.64534	0.822671	+	0.568518i	$0.192483\pi$
$6$	0	0
$7$	4.14977	1.56847	0.784234	−	0.620466i	$-0.213056\pi$
$7$	4.14977	1.56847	0.784234	+	0.620466i	$0.213056\pi$
$8$	1.59152	0.562688
$9$	0	0
$10$	−6.47291	−2.04691
$11$	5.07739	1.53089	0.765445	−	0.643501i	$-0.222519\pi$
$11$	5.07739	1.53089	0.765445	+	0.643501i	$0.222519\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−7.30102	−1.95128
$15$	0	0
$16$	−4.99090	−1.24772
$17$	0.643026	0.155957	0.0779784	−	0.996955i	$-0.475153\pi$
$17$	0.643026	0.155957	0.0779784	+	0.996955i	$0.475153\pi$
$18$	0	0
$19$	−7.24868	−1.66296	−0.831481	−	0.555554i	$-0.812506\pi$
$19$	−7.24868	−1.66296	−0.831481	+	0.555554i	$0.812506\pi$
$20$	4.03010	0.901157
$21$	0	0
$22$	−8.93304	−1.90453
$23$	−0.685475	−0.142931	−0.0714657	−	0.997443i	$-0.522768\pi$
$23$	−0.685475	−0.142931	−0.0714657	+	0.997443i	$0.522768\pi$
$24$	0	0
$25$	8.53574	1.70715
$26$	1.75938	0.345042
$27$	0	0
$28$	4.54568	0.859053
$29$	−5.82822	−1.08227	−0.541137	−	0.840935i	$-0.682006\pi$
$29$	−5.82822	−1.08227	−0.541137	+	0.840935i	$0.682006\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.59782	0.989564
$33$	0	0
$34$	−1.13132	−0.194021
$35$	15.2674	2.58066
$36$	0	0
$37$	8.72827	1.43492	0.717460	−	0.696600i	$-0.245305\pi$
$37$	8.72827	1.43492	0.717460	+	0.696600i	$0.245305\pi$
$38$	12.7532	2.06884
$39$	0	0
$40$	5.85537	0.925815
$41$	11.1619	1.74319	0.871596	−	0.490225i	$-0.163086\pi$
$41$	11.1619	1.74319	0.871596	+	0.490225i	$0.163086\pi$
$42$	0	0
$43$	−1.81828	−0.277285	−0.138642	−	0.990343i	$-0.544274\pi$
$43$	−1.81828	−0.277285	−0.138642	+	0.990343i	$0.544274\pi$
$44$	5.56180	0.838472
$45$	0	0
$46$	1.20601	0.177816
$47$	−6.05989	−0.883926	−0.441963	−	0.897033i	$-0.645718\pi$
$47$	−6.05989	−0.883926	−0.441963	+	0.897033i	$0.645718\pi$
$48$	0	0
$49$	10.2206	1.46009
$50$	−15.0176	−2.12381
$51$	0	0
$52$	−1.09540	−0.151905
$53$	1.56129	0.214460	0.107230	−	0.994234i	$-0.465802\pi$
$53$	1.56129	0.214460	0.107230	+	0.994234i	$0.465802\pi$
$54$	0	0
$55$	18.6802	2.51884
$56$	6.60446	0.882559
$57$	0	0
$58$	10.2540	1.34642
$59$	7.14043	0.929604	0.464802	−	0.885415i	$-0.346125\pi$
$59$	7.14043	0.929604	0.464802	+	0.885415i	$0.346125\pi$
$60$	0	0
$61$	1.54027	0.197211	0.0986054	−	0.995127i	$-0.468562\pi$
$61$	1.54027	0.197211	0.0986054	+	0.995127i	$0.468562\pi$
$62$	1.75938	0.223441
$63$	0	0
$64$	0.133123	0.0166404
$65$	−3.67909	−0.456336
$66$	0	0
$67$	2.10412	0.257059	0.128530	−	0.991706i	$-0.458974\pi$
$67$	2.10412	0.257059	0.128530	+	0.991706i	$0.458974\pi$
$68$	0.704374	0.0854179
$69$	0	0
$70$	−26.8611	−3.21052
$71$	−0.546141	−0.0648150	−0.0324075	−	0.999475i	$-0.510317\pi$
$71$	−0.546141	−0.0648150	−0.0324075	+	0.999475i	$0.510317\pi$
$72$	0	0
$73$	−9.58155	−1.12144	−0.560718	−	0.828007i	$-0.689475\pi$
$73$	−9.58155	−1.12144	−0.560718	+	0.828007i	$0.689475\pi$
$74$	−15.3563	−1.78514
$75$	0	0
$76$	−7.94024	−0.910808
$77$	21.0700	2.40115
$78$	0	0
$79$	−4.15021	−0.466935	−0.233468	−	0.972365i	$-0.575007\pi$
$79$	−4.15021	−0.466935	−0.233468	+	0.972365i	$0.575007\pi$
$80$	−18.3620	−2.05293
$81$	0	0
$82$	−19.6379	−2.16865
$83$	−6.77183	−0.743305	−0.371652	−	0.928372i	$-0.621209\pi$
$83$	−6.77183	−0.743305	−0.371652	+	0.928372i	$0.621209\pi$
$84$	0	0
$85$	2.36575	0.256602
$86$	3.19904	0.344961
$87$	0	0
$88$	8.08078	0.861414
$89$	14.7212	1.56045	0.780224	−	0.625500i	$-0.215105\pi$
$89$	14.7212	1.56045	0.780224	+	0.625500i	$0.215105\pi$
$90$	0	0
$91$	−4.14977	−0.435015
$92$	−0.750873	−0.0782839
$93$	0	0
$94$	10.6616	1.09966
$95$	−26.6686	−2.73614
$96$	0	0
$97$	−12.2489	−1.24369	−0.621843	−	0.783142i	$-0.713616\pi$
$97$	−12.2489	−1.24369	−0.621843	+	0.783142i	$0.713616\pi$
$98$	−17.9819	−1.81645
$99$	0	0
$100$	9.35009	0.935009
$101$	18.9673	1.88732	0.943661	−	0.330914i	$-0.107357\pi$
$101$	18.9673	1.88732	0.943661	+	0.330914i	$0.107357\pi$
$102$	0	0
$103$	11.2464	1.10814	0.554068	−	0.832471i	$-0.313075\pi$
$103$	11.2464	1.10814	0.554068	+	0.832471i	$0.313075\pi$
$104$	−1.59152	−0.156062
$105$	0	0
$106$	−2.74690	−0.266802
$107$	13.0590	1.26246	0.631232	−	0.775594i	$-0.282550\pi$
$107$	13.0590	1.26246	0.631232	+	0.775594i	$0.282550\pi$
$108$	0	0
$109$	−13.8788	−1.32935	−0.664673	−	0.747134i	$-0.731429\pi$
$109$	−13.8788	−1.32935	−0.664673	+	0.747134i	$0.731429\pi$
$110$	−32.8655	−3.13360
$111$	0	0
$112$	−20.7111	−1.95702
$113$	−0.433714	−0.0408004	−0.0204002	−	0.999792i	$-0.506494\pi$
$113$	−0.433714	−0.0408004	−0.0204002	+	0.999792i	$0.506494\pi$
$114$	0	0
$115$	−2.52193	−0.235171
$116$	−6.38426	−0.592764
$117$	0	0
$118$	−12.5627	−1.15649
$119$	2.66841	0.244613
$120$	0	0
$121$	14.7799	1.34363
$122$	−2.70991	−0.245343
$123$	0	0
$124$	−1.09540	−0.0983703
$125$	13.0083	1.16350
$126$	0	0
$127$	3.09560	0.274690	0.137345	−	0.990523i	$-0.456143\pi$
$127$	3.09560	0.274690	0.137345	+	0.990523i	$0.456143\pi$
$128$	−11.4299	−1.01027
$129$	0	0
$130$	6.47291	0.567712
$131$	0.655715	0.0572901	0.0286450	−	0.999590i	$-0.490881\pi$
$131$	0.655715	0.0572901	0.0286450	+	0.999590i	$0.490881\pi$
$132$	0	0
$133$	−30.0804	−2.60830
$134$	−3.70194	−0.319799
$135$	0	0
$136$	1.02339	0.0877551
$137$	0.444583	0.0379833	0.0189917	−	0.999820i	$-0.493954\pi$
$137$	0.444583	0.0379833	0.0189917	+	0.999820i	$0.493954\pi$
$138$	0	0
$139$	1.98710	0.168544	0.0842718	−	0.996443i	$-0.473144\pi$
$139$	1.98710	0.168544	0.0842718	+	0.996443i	$0.473144\pi$
$140$	16.7240	1.41344
$141$	0	0
$142$	0.960867	0.0806342
$143$	−5.07739	−0.424593
$144$	0	0
$145$	−21.4426	−1.78071
$146$	16.8576	1.39514
$147$	0	0
$148$	9.56099	0.785909
$149$	−15.8870	−1.30152	−0.650759	−	0.759285i	$-0.725549\pi$
$149$	−15.8870	−1.30152	−0.650759	+	0.759285i	$0.725549\pi$
$150$	0	0
$151$	20.0014	1.62769	0.813847	−	0.581079i	$-0.197369\pi$
$151$	20.0014	1.62769	0.813847	+	0.581079i	$0.197369\pi$
$152$	−11.5364	−0.935729
$153$	0	0
$154$	−37.0701	−2.98719
$155$	−3.67909	−0.295512
$156$	0	0
$157$	0.109578	0.00874525	0.00437263	−	0.999990i	$-0.498608\pi$
$157$	0.109578	0.00874525	0.00437263	+	0.999990i	$0.498608\pi$
$158$	7.30178	0.580899
$159$	0	0
$160$	20.5949	1.62817
$161$	−2.84457	−0.224183
$162$	0	0
$163$	−5.63385	−0.441277	−0.220639	−	0.975356i	$-0.570814\pi$
$163$	−5.63385	−0.441277	−0.220639	+	0.975356i	$0.570814\pi$
$164$	12.2268	0.954750
$165$	0	0
$166$	11.9142	0.924721
$167$	−9.04293	−0.699763	−0.349881	−	0.936794i	$-0.613778\pi$
$167$	−9.04293	−0.699763	−0.349881	+	0.936794i	$0.613778\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.16225	−0.319230
$171$	0	0
$172$	−1.99175	−0.151870
$173$	−9.53376	−0.724838	−0.362419	−	0.932015i	$-0.618049\pi$
$173$	−9.53376	−0.724838	−0.362419	+	0.932015i	$0.618049\pi$
$174$	0	0
$175$	35.4214	2.67761
$176$	−25.3407	−1.91013
$177$	0	0
$178$	−25.9002	−1.94130
$179$	10.3075	0.770419	0.385209	−	0.922829i	$-0.374129\pi$
$179$	10.3075	0.770419	0.385209	+	0.922829i	$0.374129\pi$
$180$	0	0
$181$	3.40674	0.253221	0.126611	−	0.991953i	$-0.459590\pi$
$181$	3.40674	0.253221	0.126611	+	0.991953i	$0.459590\pi$
$182$	7.30102	0.541187
$183$	0	0
$184$	−1.09095	−0.0804259
$185$	32.1121	2.36093
$186$	0	0
$187$	3.26489	0.238753
$188$	−6.63803	−0.484128
$189$	0	0
$190$	46.9201	3.40394
$191$	17.0195	1.23149	0.615745	−	0.787946i	$-0.288855\pi$
$191$	17.0195	1.23149	0.615745	+	0.787946i	$0.288855\pi$
$192$	0	0
$193$	−0.103108	−0.00742191	−0.00371095	−	0.999993i	$-0.501181\pi$
$193$	−0.103108	−0.00742191	−0.00371095	+	0.999993i	$0.501181\pi$
$194$	21.5504	1.54723
$195$	0	0
$196$	11.1957	0.799695
$197$	2.51791	0.179393	0.0896967	−	0.995969i	$-0.471410\pi$
$197$	2.51791	0.179393	0.0896967	+	0.995969i	$0.471410\pi$
$198$	0	0
$199$	−0.00157334	−0.000111531 0	−5.57657e−5	−	1.00000i	$-0.500018\pi$
$199$	−0.00157334	−0.000111531 0	−5.57657e−5		1.00000i	$0.500018\pi$
$200$	13.5848	0.960592
$201$	0	0
$202$	−33.3707	−2.34795
$203$	−24.1858	−1.69751
$204$	0	0
$205$	41.0656	2.86814
$206$	−19.7866	−1.37860
$207$	0	0
$208$	4.99090	0.346057
$209$	−36.8044	−2.54581
$210$	0	0
$211$	−16.9852	−1.16931	−0.584653	−	0.811283i	$-0.698769\pi$
$211$	−16.9852	−1.16931	−0.584653	+	0.811283i	$0.698769\pi$
$212$	1.71025	0.117460
$213$	0	0
$214$	−22.9757	−1.57059
$215$	−6.68962	−0.456228
$216$	0	0
$217$	−4.14977	−0.281705
$218$	24.4180	1.65380
$219$	0	0
$220$	20.4624	1.37957
$221$	−0.643026	−0.0432546
$222$	0	0
$223$	10.0015	0.669748	0.334874	−	0.942263i	$-0.391306\pi$
$223$	10.0015	0.669748	0.334874	+	0.942263i	$0.391306\pi$
$224$	23.2297	1.55210
$225$	0	0
$226$	0.763066	0.0507584
$227$	15.2887	1.01475	0.507375	−	0.861726i	$-0.330616\pi$
$227$	15.2887	1.01475	0.507375	+	0.861726i	$0.330616\pi$
$228$	0	0
$229$	−16.6053	−1.09731	−0.548654	−	0.836049i	$-0.684860\pi$
$229$	−16.6053	−1.09731	−0.548654	+	0.836049i	$0.684860\pi$
$230$	4.43702	0.292568
$231$	0	0
$232$	−9.27575	−0.608983
$233$	10.3751	0.679696	0.339848	−	0.940480i	$-0.389624\pi$
$233$	10.3751	0.679696	0.339848	+	0.940480i	$0.389624\pi$
$234$	0	0
$235$	−22.2949	−1.45436
$236$	7.82166	0.509146
$237$	0	0
$238$	−4.69474	−0.304315
$239$	−26.7111	−1.72780	−0.863900	−	0.503664i	$-0.831985\pi$
$239$	−26.7111	−1.72780	−0.863900	+	0.503664i	$0.831985\pi$
$240$	0	0
$241$	−8.83202	−0.568920	−0.284460	−	0.958688i	$-0.591814\pi$
$241$	−8.83202	−0.568920	−0.284460	+	0.958688i	$0.591814\pi$
$242$	−26.0034	−1.67156
$243$	0	0
$244$	1.68721	0.108013
$245$	37.6027	2.40235
$246$	0	0
$247$	7.24868	0.461223
$248$	−1.59152	−0.101062
$249$	0	0
$250$	−22.8865	−1.44747
$251$	−28.0408	−1.76992	−0.884959	−	0.465668i	$-0.845814\pi$
$251$	−28.0408	−1.76992	−0.884959	+	0.465668i	$0.845814\pi$
$252$	0	0
$253$	−3.48042	−0.218812
$254$	−5.44633	−0.341733
$255$	0	0
$256$	19.8432	1.24020
$257$	−19.0954	−1.19114	−0.595568	−	0.803305i	$-0.703073\pi$
$257$	−19.0954	−1.19114	−0.595568	+	0.803305i	$0.703073\pi$
$258$	0	0
$259$	36.2204	2.25062
$260$	−4.03010	−0.249936
$261$	0	0
$262$	−1.15365	−0.0712727
$263$	2.07863	0.128174	0.0640870	−	0.997944i	$-0.479586\pi$
$263$	2.07863	0.128174	0.0640870	+	0.997944i	$0.479586\pi$
$264$	0	0
$265$	5.74414	0.352860
$266$	52.9227	3.24490
$267$	0	0
$268$	2.30486	0.140792
$269$	19.6631	1.19888	0.599440	−	0.800420i	$-0.295390\pi$
$269$	19.6631	1.19888	0.599440	+	0.800420i	$0.295390\pi$
$270$	0	0
$271$	−7.58175	−0.460559	−0.230279	−	0.973125i	$-0.573964\pi$
$271$	−7.58175	−0.460559	−0.230279	+	0.973125i	$0.573964\pi$
$272$	−3.20928	−0.194591
$273$	0	0
$274$	−0.782189	−0.0472538
$275$	43.3393	2.61346
$276$	0	0
$277$	8.83217	0.530674	0.265337	−	0.964156i	$-0.414517\pi$
$277$	8.83217	0.530674	0.265337	+	0.964156i	$0.414517\pi$
$278$	−3.49606	−0.209680
$279$	0	0
$280$	24.2984	1.45211
$281$	−3.73446	−0.222779	−0.111390	−	0.993777i	$-0.535530\pi$
$281$	−3.73446	−0.222779	−0.111390	+	0.993777i	$0.535530\pi$
$282$	0	0
$283$	−3.46167	−0.205775	−0.102887	−	0.994693i	$-0.532808\pi$
$283$	−3.46167	−0.205775	−0.102887	+	0.994693i	$0.532808\pi$
$284$	−0.598245	−0.0354993
$285$	0	0
$286$	8.93304	0.528222
$287$	46.3192	2.73414
$288$	0	0
$289$	−16.5865	−0.975677
$290$	37.7256	2.21532
$291$	0	0
$292$	−10.4957	−0.614213
$293$	22.9317	1.33969	0.669843	−	0.742502i	$-0.266361\pi$
$293$	22.9317	1.33969	0.669843	+	0.742502i	$0.266361\pi$
$294$	0	0
$295$	26.2703	1.52952
$296$	13.8913	0.807413
$297$	0	0
$298$	27.9513	1.61917
$299$	0.685475	0.0396421
$300$	0	0
$301$	−7.54545	−0.434912
$302$	−35.1901	−2.02496
$303$	0	0
$304$	36.1774	2.07492
$305$	5.66678	0.324479
$306$	0	0
$307$	−5.99871	−0.342364	−0.171182	−	0.985239i	$-0.554759\pi$
$307$	−5.99871	−0.342364	−0.171182	+	0.985239i	$0.554759\pi$
$308$	23.0802	1.31512
$309$	0	0
$310$	6.47291	0.367637
$311$	0.121230	0.00687433	0.00343716	−	0.999994i	$-0.498906\pi$
$311$	0.121230	0.00687433	0.00343716	+	0.999994i	$0.498906\pi$
$312$	0	0
$313$	−29.3281	−1.65772	−0.828862	−	0.559454i	$-0.811011\pi$
$313$	−29.3281	−1.65772	−0.828862	+	0.559454i	$0.811011\pi$
$314$	−0.192788	−0.0108797
$315$	0	0
$316$	−4.54616	−0.255742
$317$	14.6861	0.824856	0.412428	−	0.910990i	$-0.364681\pi$
$317$	14.6861	0.824856	0.412428	+	0.910990i	$0.364681\pi$
$318$	0	0
$319$	−29.5922	−1.65684
$320$	0.489773	0.0273792
$321$	0	0
$322$	5.00466	0.278899
$323$	−4.66109	−0.259350
$324$	0	0
$325$	−8.53574	−0.473478
$326$	9.91206	0.548978
$327$	0	0
$328$	17.7644	0.980874
$329$	−25.1472	−1.38641
$330$	0	0
$331$	20.0348	1.10121	0.550605	−	0.834766i	$-0.314397\pi$
$331$	20.0348	1.10121	0.550605	+	0.834766i	$0.314397\pi$
$332$	−7.41789	−0.407110
$333$	0	0
$334$	15.9099	0.870552
$335$	7.74126	0.422950
$336$	0	0
$337$	1.25458	0.0683411	0.0341706	−	0.999416i	$-0.489121\pi$
$337$	1.25458	0.0683411	0.0341706	+	0.999416i	$0.489121\pi$
$338$	−1.75938	−0.0956975
$339$	0	0
$340$	2.59146	0.140542
$341$	−5.07739	−0.274956
$342$	0	0
$343$	13.3649	0.721636
$344$	−2.89383	−0.156025
$345$	0	0
$346$	16.7735	0.901747
$347$	−33.4450	−1.79542	−0.897711	−	0.440586i	$-0.854771\pi$
$347$	−33.4450	−1.79542	−0.897711	+	0.440586i	$0.854771\pi$
$348$	0	0
$349$	16.0739	0.860417	0.430208	−	0.902730i	$-0.358440\pi$
$349$	16.0739	0.860417	0.430208	+	0.902730i	$0.358440\pi$
$350$	−62.3196	−3.33112
$351$	0	0
$352$	28.4223	1.51491
$353$	33.7999	1.79899	0.899493	−	0.436935i	$-0.143936\pi$
$353$	33.7999	1.79899	0.899493	+	0.436935i	$0.143936\pi$
$354$	0	0
$355$	−2.00930	−0.106643
$356$	16.1257	0.854662
$357$	0	0
$358$	−18.1348	−0.958452
$359$	−8.58273	−0.452979	−0.226490	−	0.974014i	$-0.572725\pi$
$359$	−8.58273	−0.452979	−0.226490	+	0.974014i	$0.572725\pi$
$360$	0	0
$361$	33.5434	1.76544
$362$	−5.99374	−0.315024
$363$	0	0
$364$	−4.54568	−0.238259
$365$	−35.2514	−1.84514
$366$	0	0
$367$	−22.3679	−1.16759	−0.583796	−	0.811900i	$-0.698433\pi$
$367$	−22.3679	−1.16759	−0.583796	+	0.811900i	$0.698433\pi$
$368$	3.42114	0.178339
$369$	0	0
$370$	−56.4973	−2.93716
$371$	6.47901	0.336373
$372$	0	0
$373$	14.2950	0.740166	0.370083	−	0.928999i	$-0.379329\pi$
$373$	14.2950	0.740166	0.370083	+	0.928999i	$0.379329\pi$
$374$	−5.74418	−0.297024
$375$	0	0
$376$	−9.64445	−0.497375
$377$	5.82822	0.300169
$378$	0	0
$379$	3.86964	0.198770	0.0993852	−	0.995049i	$-0.468312\pi$
$379$	3.86964	0.198770	0.0993852	+	0.995049i	$0.468312\pi$
$380$	−29.2129	−1.49859
$381$	0	0
$382$	−29.9437	−1.53206
$383$	−17.3267	−0.885352	−0.442676	−	0.896682i	$-0.645971\pi$
$383$	−17.3267	−0.885352	−0.442676	+	0.896682i	$0.645971\pi$
$384$	0	0
$385$	77.5186	3.95071
$386$	0.181407	0.00923335
$387$	0	0
$388$	−13.4175	−0.681170
$389$	−31.8175	−1.61321	−0.806605	−	0.591091i	$-0.798697\pi$
$389$	−31.8175	−1.61321	−0.806605	+	0.591091i	$0.798697\pi$
$390$	0	0
$391$	−0.440778	−0.0222911
$392$	16.2664	0.821576
$393$	0	0
$394$	−4.42995	−0.223177
$395$	−15.2690	−0.768268
$396$	0	0
$397$	9.19501	0.461484	0.230742	−	0.973015i	$-0.425885\pi$
$397$	9.19501	0.461484	0.230742	+	0.973015i	$0.425885\pi$
$398$	0.00276811	0.000138753 0
$399$	0	0
$400$	−42.6010	−2.13005
$401$	8.67433	0.433175	0.216588	−	0.976263i	$-0.430507\pi$
$401$	8.67433	0.433175	0.216588	+	0.976263i	$0.430507\pi$
$402$	0	0
$403$	1.00000	0.0498135
$404$	20.7769	1.03369
$405$	0	0
$406$	42.5519	2.11182
$407$	44.3168	2.19670
$408$	0	0
$409$	−11.7332	−0.580171	−0.290085	−	0.957001i	$-0.593684\pi$
$409$	−11.7332	−0.580171	−0.290085	+	0.957001i	$0.593684\pi$
$410$	−72.2498	−3.56816
$411$	0	0
$412$	12.3193	0.606929
$413$	29.6312	1.45805
$414$	0	0
$415$	−24.9142	−1.22299
$416$	−5.59782	−0.274456
$417$	0	0
$418$	64.7527	3.16716
$419$	−20.6599	−1.00930	−0.504651	−	0.863324i	$-0.668379\pi$
$419$	−20.6599	−1.00930	−0.504651	+	0.863324i	$0.668379\pi$
$420$	0	0
$421$	4.02970	0.196396	0.0981978	−	0.995167i	$-0.468692\pi$
$421$	4.02970	0.196396	0.0981978	+	0.995167i	$0.468692\pi$
$422$	29.8833	1.45469
$423$	0	0
$424$	2.48483	0.120674
$425$	5.48870	0.266241
$426$	0	0
$427$	6.39176	0.309319
$428$	14.3049	0.691455
$429$	0	0
$430$	11.7696	0.567578
$431$	−6.35712	−0.306212	−0.153106	−	0.988210i	$-0.548928\pi$
$431$	−6.35712	−0.306212	−0.153106	+	0.988210i	$0.548928\pi$
$432$	0	0
$433$	23.5980	1.13405	0.567024	−	0.823701i	$-0.308095\pi$
$433$	23.5980	1.13405	0.567024	+	0.823701i	$0.308095\pi$
$434$	7.30102	0.350460
$435$	0	0
$436$	−15.2029	−0.728086
$437$	4.96879	0.237689
$438$	0	0
$439$	0.942932	0.0450037	0.0225018	−	0.999747i	$-0.492837\pi$
$439$	0.942932	0.0450037	0.0225018	+	0.999747i	$0.492837\pi$
$440$	29.7300	1.41732
$441$	0	0
$442$	1.13132	0.0538116
$443$	21.8875	1.03991	0.519954	−	0.854194i	$-0.325949\pi$
$443$	21.8875	1.03991	0.519954	+	0.854194i	$0.325949\pi$
$444$	0	0
$445$	54.1609	2.56747
$446$	−17.5964	−0.833212
$447$	0	0
$448$	0.552432	0.0260999
$449$	−13.8867	−0.655352	−0.327676	−	0.944790i	$-0.606265\pi$
$449$	−13.8867	−0.655352	−0.327676	+	0.944790i	$0.606265\pi$
$450$	0	0
$451$	56.6732	2.66864
$452$	−0.475092	−0.0223465
$453$	0	0
$454$	−26.8987	−1.26242
$455$	−15.2674	−0.715747
$456$	0	0
$457$	−22.4197	−1.04875	−0.524374	−	0.851488i	$-0.675701\pi$
$457$	−22.4197	−1.04875	−0.524374	+	0.851488i	$0.675701\pi$
$458$	29.2150	1.36513
$459$	0	0
$460$	−2.76253	−0.128804
$461$	−32.1892	−1.49920	−0.749599	−	0.661892i	$-0.769754\pi$
$461$	−32.1892	−1.49920	−0.749599	+	0.661892i	$0.769754\pi$
$462$	0	0
$463$	−20.1648	−0.937138	−0.468569	−	0.883427i	$-0.655230\pi$
$463$	−20.1648	−0.937138	−0.468569	+	0.883427i	$0.655230\pi$
$464$	29.0881	1.35038
$465$	0	0
$466$	−18.2537	−0.845587
$467$	−2.15941	−0.0999256	−0.0499628	−	0.998751i	$-0.515910\pi$
$467$	−2.15941	−0.0999256	−0.0499628	+	0.998751i	$0.515910\pi$
$468$	0	0
$469$	8.73163	0.403189
$470$	39.2251	1.80932
$471$	0	0
$472$	11.3642	0.523078
$473$	−9.23211	−0.424493
$474$	0	0
$475$	−61.8728	−2.83892
$476$	2.92299	0.133975
$477$	0	0
$478$	46.9949	2.14950
$479$	43.6098	1.99258	0.996291	−	0.0860496i	$-0.0274244\pi$
$479$	43.6098	1.99258	0.996291	+	0.0860496i	$0.0274244\pi$
$480$	0	0
$481$	−8.72827	−0.397975
$482$	15.5388	0.707775
$483$	0	0
$484$	16.1900	0.735907
$485$	−45.0648	−2.04629
$486$	0	0
$487$	1.18940	0.0538970	0.0269485	−	0.999637i	$-0.491421\pi$
$487$	1.18940	0.0538970	0.0269485	+	0.999637i	$0.491421\pi$
$488$	2.45137	0.110968
$489$	0	0
$490$	−66.1572	−2.98868
$491$	0.630068	0.0284346	0.0142173	−	0.999899i	$-0.495474\pi$
$491$	0.630068	0.0284346	0.0142173	+	0.999899i	$0.495474\pi$
$492$	0	0
$493$	−3.74770	−0.168788
$494$	−12.7532	−0.573792
$495$	0	0
$496$	4.99090	0.224098
$497$	−2.26636	−0.101660
$498$	0	0
$499$	−11.4173	−0.511106	−0.255553	−	0.966795i	$-0.582258\pi$
$499$	−11.4173	−0.511106	−0.255553	+	0.966795i	$0.582258\pi$
$500$	14.2494	0.637251
$501$	0	0
$502$	49.3343	2.20190
$503$	−25.6104	−1.14191	−0.570955	−	0.820982i	$-0.693427\pi$
$503$	−25.6104	−1.14191	−0.570955	+	0.820982i	$0.693427\pi$
$504$	0	0
$505$	69.7827	3.10529
$506$	6.12338	0.272217
$507$	0	0
$508$	3.39094	0.150448
$509$	37.4325	1.65917	0.829583	−	0.558384i	$-0.188578\pi$
$509$	37.4325	1.65917	0.829583	+	0.558384i	$0.188578\pi$
$510$	0	0
$511$	−39.7613	−1.75894
$512$	−12.0519	−0.532623
$513$	0	0
$514$	33.5959	1.48185
$515$	41.3764	1.82326
$516$	0	0
$517$	−30.7684	−1.35319
$518$	−63.7253	−2.79993
$519$	0	0
$520$	−5.85537	−0.256775
$521$	−37.2529	−1.63208	−0.816040	−	0.577996i	$-0.803835\pi$
$521$	−37.2529	−1.63208	−0.816040	+	0.577996i	$0.803835\pi$
$522$	0	0
$523$	14.8086	0.647537	0.323768	−	0.946136i	$-0.395050\pi$
$523$	14.8086	0.647537	0.323768	+	0.946136i	$0.395050\pi$
$524$	0.718274	0.0313779
$525$	0	0
$526$	−3.65710	−0.159457
$527$	−0.643026	−0.0280107
$528$	0	0
$529$	−22.5301	−0.979571
$530$	−10.1061	−0.438981
$531$	0	0
$532$	−32.9502	−1.42857
$533$	−11.1619	−0.483474
$534$	0	0
$535$	48.0454	2.07718
$536$	3.34876	0.144644
$537$	0	0
$538$	−34.5948	−1.49149
$539$	51.8941	2.23524
$540$	0	0
$541$	−27.9305	−1.20083	−0.600413	−	0.799690i	$-0.704997\pi$
$541$	−27.9305	−1.20083	−0.600413	+	0.799690i	$0.704997\pi$
$542$	13.3392	0.572966
$543$	0	0
$544$	3.59954	0.154329
$545$	−51.0614	−2.18723
$546$	0	0
$547$	34.4275	1.47202	0.736008	−	0.676973i	$-0.236709\pi$
$547$	34.4275	1.47202	0.736008	+	0.676973i	$0.236709\pi$
$548$	0.486999	0.0208036
$549$	0	0
$550$	−76.2501	−3.25131
$551$	42.2469	1.79978
$552$	0	0
$553$	−17.2224	−0.732373
$554$	−15.5391	−0.660193
$555$	0	0
$556$	2.17668	0.0923117
$557$	−4.37431	−0.185346	−0.0926728	−	0.995697i	$-0.529541\pi$
$557$	−4.37431	−0.185346	−0.0926728	+	0.995697i	$0.529541\pi$
$558$	0	0
$559$	1.81828	0.0769050
$560$	−76.1981	−3.21996
$561$	0	0
$562$	6.57033	0.277152
$563$	−43.6108	−1.83798	−0.918988	−	0.394285i	$-0.870992\pi$
$563$	−43.6108	−1.83798	−0.918988	+	0.394285i	$0.870992\pi$
$564$	0	0
$565$	−1.59567	−0.0671305
$566$	6.09038	0.255998
$567$	0	0
$568$	−0.869196	−0.0364706
$569$	−9.54889	−0.400310	−0.200155	−	0.979764i	$-0.564145\pi$
$569$	−9.54889	−0.400310	−0.200155	+	0.979764i	$0.564145\pi$
$570$	0	0
$571$	−15.7147	−0.657639	−0.328819	−	0.944393i	$-0.606651\pi$
$571$	−15.7147	−0.657639	−0.328819	+	0.944393i	$0.606651\pi$
$572$	−5.56180	−0.232550
$573$	0	0
$574$	−81.4930	−3.40145
$575$	−5.85104	−0.244005
$576$	0	0
$577$	−1.09693	−0.0456660	−0.0228330	−	0.999739i	$-0.507269\pi$
$577$	−1.09693	−0.0456660	−0.0228330	+	0.999739i	$0.507269\pi$
$578$	29.1819	1.21381
$579$	0	0
$580$	−23.4883	−0.975299
$581$	−28.1016	−1.16585
$582$	0	0
$583$	7.92729	0.328315
$584$	−15.2493	−0.631019
$585$	0	0
$586$	−40.3456	−1.66666
$587$	2.59363	0.107051	0.0535253	−	0.998566i	$-0.482954\pi$
$587$	2.59363	0.107051	0.0535253	+	0.998566i	$0.482954\pi$
$588$	0	0
$589$	7.24868	0.298677
$590$	−46.2193	−1.90282
$591$	0	0
$592$	−43.5619	−1.79038
$593$	−13.7627	−0.565165	−0.282582	−	0.959243i	$-0.591191\pi$
$593$	−13.7627	−0.565165	−0.282582	+	0.959243i	$0.591191\pi$
$594$	0	0
$595$	9.81735	0.402472
$596$	−17.4027	−0.712844
$597$	0	0
$598$	−1.20601	−0.0493174
$599$	−10.5758	−0.432114	−0.216057	−	0.976381i	$-0.569320\pi$
$599$	−10.5758	−0.432114	−0.216057	+	0.976381i	$0.569320\pi$
$600$	0	0
$601$	34.3691	1.40194	0.700972	−	0.713189i	$-0.252750\pi$
$601$	34.3691	1.40194	0.700972	+	0.713189i	$0.252750\pi$
$602$	13.2753	0.541060
$603$	0	0
$604$	21.9097	0.891492
$605$	54.3766	2.21072
$606$	0	0
$607$	14.5224	0.589444	0.294722	−	0.955583i	$-0.404773\pi$
$607$	14.5224	0.589444	0.294722	+	0.955583i	$0.404773\pi$
$608$	−40.5768	−1.64561
$609$	0	0
$610$	−9.97000	−0.403674
$611$	6.05989	0.245157
$612$	0	0
$613$	−36.8553	−1.48857	−0.744287	−	0.667860i	$-0.767210\pi$
$613$	−36.8553	−1.48857	−0.744287	+	0.667860i	$0.767210\pi$
$614$	10.5540	0.425924
$615$	0	0
$616$	33.5334	1.35110
$617$	27.1376	1.09252	0.546259	−	0.837616i	$-0.316051\pi$
$617$	27.1376	1.09252	0.546259	+	0.837616i	$0.316051\pi$
$618$	0	0
$619$	21.8315	0.877481	0.438740	−	0.898614i	$-0.355425\pi$
$619$	21.8315	0.877481	0.438740	+	0.898614i	$0.355425\pi$
$620$	−4.03010	−0.161853
$621$	0	0
$622$	−0.213289	−0.00855212
$623$	61.0899	2.44751
$624$	0	0
$625$	5.18015	0.207206
$626$	51.5992	2.06232
$627$	0	0
$628$	0.120032	0.00478980
$629$	5.61251	0.223785
$630$	0	0
$631$	−32.3195	−1.28662	−0.643309	−	0.765607i	$-0.722439\pi$
$631$	−32.3195	−1.28662	−0.643309	+	0.765607i	$0.722439\pi$
$632$	−6.60516	−0.262739
$633$	0	0
$634$	−25.8384	−1.02618
$635$	11.3890	0.451959
$636$	0	0
$637$	−10.2206	−0.404956
$638$	52.0637	2.06122
$639$	0	0
$640$	−42.0515	−1.66223
$641$	15.3890	0.607831	0.303915	−	0.952699i	$-0.401706\pi$
$641$	15.3890	0.607831	0.303915	+	0.952699i	$0.401706\pi$
$642$	0	0
$643$	10.6498	0.419986	0.209993	−	0.977703i	$-0.432656\pi$
$643$	10.6498	0.419986	0.209993	+	0.977703i	$0.432656\pi$
$644$	−3.11595	−0.122786
$645$	0	0
$646$	8.20061	0.322649
$647$	−38.8098	−1.52577	−0.762886	−	0.646533i	$-0.776218\pi$
$647$	−38.8098	−1.52577	−0.762886	+	0.646533i	$0.776218\pi$
$648$	0	0
$649$	36.2547	1.42312
$650$	15.0176	0.589038
$651$	0	0
$652$	−6.17135	−0.241689
$653$	18.6676	0.730520	0.365260	−	0.930906i	$-0.380980\pi$
$653$	18.6676	0.730520	0.365260	+	0.930906i	$0.380980\pi$
$654$	0	0
$655$	2.41244	0.0942618
$656$	−55.7078	−2.17502
$657$	0	0
$658$	44.2433	1.72478
$659$	38.2464	1.48987	0.744934	−	0.667138i	$-0.232481\pi$
$659$	38.2464	1.48987	0.744934	+	0.667138i	$0.232481\pi$
$660$	0	0
$661$	13.4345	0.522541	0.261270	−	0.965266i	$-0.415859\pi$
$661$	13.4345	0.522541	0.261270	+	0.965266i	$0.415859\pi$
$662$	−35.2487	−1.36998
$663$	0	0
$664$	−10.7775	−0.418249
$665$	−110.669	−4.29154
$666$	0	0
$667$	3.99510	0.154691
$668$	−9.90567	−0.383262
$669$	0	0
$670$	−13.6198	−0.526178
$671$	7.82053	0.301908
$672$	0	0
$673$	−16.8262	−0.648603	−0.324301	−	0.945954i	$-0.605129\pi$
$673$	−16.8262	−0.648603	−0.324301	+	0.945954i	$0.605129\pi$
$674$	−2.20727	−0.0850209
$675$	0	0
$676$	1.09540	0.0421310
$677$	27.1489	1.04342	0.521708	−	0.853124i	$-0.325295\pi$
$677$	27.1489	1.04342	0.521708	+	0.853124i	$0.325295\pi$
$678$	0	0
$679$	−50.8301	−1.95068
$680$	3.76515	0.144387
$681$	0	0
$682$	8.93304	0.342064
$683$	−14.3975	−0.550905	−0.275452	−	0.961315i	$-0.588828\pi$
$683$	−14.3975	−0.550905	−0.275452	+	0.961315i	$0.588828\pi$
$684$	0	0
$685$	1.63566	0.0624955
$686$	−23.5139	−0.897764
$687$	0	0
$688$	9.07484	0.345975
$689$	−1.56129	−0.0594805
$690$	0	0
$691$	−30.0423	−1.14286	−0.571431	−	0.820650i	$-0.693611\pi$
$691$	−30.0423	−1.14286	−0.571431	+	0.820650i	$0.693611\pi$
$692$	−10.4433	−0.396996
$693$	0	0
$694$	58.8423	2.23362
$695$	7.31073	0.277312
$696$	0	0
$697$	7.17737	0.271862
$698$	−28.2801	−1.07042
$699$	0	0
$700$	38.8008	1.46653
$701$	−1.37156	−0.0518029	−0.0259015	−	0.999665i	$-0.508246\pi$
$701$	−1.37156	−0.0518029	−0.0259015	+	0.999665i	$0.508246\pi$
$702$	0	0
$703$	−63.2685	−2.38622
$704$	0.675919	0.0254747
$705$	0	0
$706$	−59.4667	−2.23806
$707$	78.7102	2.96020
$708$	0	0
$709$	2.70050	0.101420	0.0507098	−	0.998713i	$-0.483852\pi$
$709$	2.70050	0.101420	0.0507098	+	0.998713i	$0.483852\pi$
$710$	3.53512	0.132671
$711$	0	0
$712$	23.4292	0.878047
$713$	0.685475	0.0256712
$714$	0	0
$715$	−18.6802	−0.698600
$716$	11.2909	0.421960
$717$	0	0
$718$	15.1003	0.563537
$719$	−11.6253	−0.433552	−0.216776	−	0.976221i	$-0.569554\pi$
$719$	−11.6253	−0.433552	−0.216776	+	0.976221i	$0.569554\pi$
$720$	0	0
$721$	46.6698	1.73808
$722$	−59.0154	−2.19633
$723$	0	0
$724$	3.73176	0.138690
$725$	−49.7482	−1.84760
$726$	0	0
$727$	−3.07478	−0.114037	−0.0570186	−	0.998373i	$-0.518159\pi$
$727$	−3.07478	−0.114037	−0.0570186	+	0.998373i	$0.518159\pi$
$728$	−6.60446	−0.244778
$729$	0	0
$730$	62.0205	2.29548
$731$	−1.16920	−0.0432444
$732$	0	0
$733$	28.7230	1.06091	0.530455	−	0.847713i	$-0.322021\pi$
$733$	28.7230	1.06091	0.530455	+	0.847713i	$0.322021\pi$
$734$	39.3535	1.45256
$735$	0	0
$736$	−3.83717	−0.141440
$737$	10.6834	0.393530
$738$	0	0
$739$	−41.9629	−1.54363	−0.771816	−	0.635846i	$-0.780651\pi$
$739$	−41.9629	−1.54363	−0.771816	+	0.635846i	$0.780651\pi$
$740$	35.1758	1.29309
$741$	0	0
$742$	−11.3990	−0.418471
$743$	−27.0052	−0.990725	−0.495363	−	0.868686i	$-0.664965\pi$
$743$	−27.0052	−0.990725	−0.495363	+	0.868686i	$0.664965\pi$
$744$	0	0
$745$	−58.4500	−2.14144
$746$	−25.1502	−0.920815
$747$	0	0
$748$	3.57638	0.130765
$749$	54.1920	1.98013
$750$	0	0
$751$	−20.4276	−0.745414	−0.372707	−	0.927949i	$-0.621570\pi$
$751$	−20.4276	−0.745414	−0.372707	+	0.927949i	$0.621570\pi$
$752$	30.2443	1.10290
$753$	0	0
$754$	−10.2540	−0.373430
$755$	73.5872	2.67811
$756$	0	0
$757$	12.6110	0.458355	0.229177	−	0.973385i	$-0.426396\pi$
$757$	12.6110	0.458355	0.229177	+	0.973385i	$0.426396\pi$
$758$	−6.80816	−0.247284
$759$	0	0
$760$	−42.4437	−1.53959
$761$	12.6269	0.457726	0.228863	−	0.973459i	$-0.426499\pi$
$761$	12.6269	0.457726	0.228863	+	0.973459i	$0.426499\pi$
$762$	0	0
$763$	−57.5938	−2.08504
$764$	18.6433	0.674490
$765$	0	0
$766$	30.4841	1.10144
$767$	−7.14043	−0.257826
$768$	0	0
$769$	−51.9432	−1.87312	−0.936560	−	0.350507i	$-0.886009\pi$
$769$	−51.9432	−1.87312	−0.936560	+	0.350507i	$0.886009\pi$
$770$	−136.384	−4.91495
$771$	0	0
$772$	−0.112945	−0.00406500
$773$	19.4706	0.700310	0.350155	−	0.936692i	$-0.386129\pi$
$773$	19.4706	0.700310	0.350155	+	0.936692i	$0.386129\pi$
$774$	0	0
$775$	−8.53574	−0.306613
$776$	−19.4944	−0.699808
$777$	0	0
$778$	55.9789	2.00694
$779$	−80.9088	−2.89886
$780$	0	0
$781$	−2.77297	−0.0992246
$782$	0.775495	0.0277316
$783$	0	0
$784$	−51.0101	−1.82179
$785$	0.403147	0.0143889
$786$	0	0
$787$	−36.0333	−1.28445	−0.642225	−	0.766516i	$-0.721988\pi$
$787$	−36.0333	−1.28445	−0.642225	+	0.766516i	$0.721988\pi$
$788$	2.75813	0.0982542
$789$	0	0
$790$	26.8640	0.955777
$791$	−1.79981	−0.0639940
$792$	0	0
$793$	−1.54027	−0.0546964
$794$	−16.1775	−0.574117
$795$	0	0
$796$	−0.00172345	−6.10860e−5 0
$797$	17.0989	0.605673	0.302837	−	0.953042i	$-0.402066\pi$
$797$	17.0989	0.605673	0.302837	+	0.953042i	$0.402066\pi$
$798$	0	0
$799$	−3.89667	−0.137854
$800$	47.7815	1.68933
$801$	0	0
$802$	−15.2614	−0.538899
$803$	−48.6493	−1.71680
$804$	0	0
$805$	−10.4654	−0.368858
$806$	−1.75938	−0.0619714
$807$	0	0
$808$	30.1870	1.06197
$809$	33.4560	1.17625	0.588125	−	0.808770i	$-0.299866\pi$
$809$	33.4560	1.17625	0.588125	+	0.808770i	$0.299866\pi$
$810$	0	0
$811$	6.67383	0.234350	0.117175	−	0.993111i	$-0.462616\pi$
$811$	6.67383	0.234350	0.117175	+	0.993111i	$0.462616\pi$
$812$	−26.4933	−0.929731
$813$	0	0
$814$	−77.9700	−2.73285
$815$	−20.7275	−0.726052
$816$	0	0
$817$	13.1801	0.461114
$818$	20.6432	0.721772
$819$	0	0
$820$	44.9834	1.57089
$821$	17.0481	0.594983	0.297491	−	0.954725i	$-0.403850\pi$
$821$	17.0481	0.594983	0.297491	+	0.954725i	$0.403850\pi$
$822$	0	0
$823$	−36.2818	−1.26470	−0.632352	−	0.774682i	$-0.717910\pi$
$823$	−36.2818	−1.26470	−0.632352	+	0.774682i	$0.717910\pi$
$824$	17.8988	0.623535
$825$	0	0
$826$	−52.1324	−1.81392
$827$	−14.2203	−0.494488	−0.247244	−	0.968953i	$-0.579525\pi$
$827$	−14.2203	−0.494488	−0.247244	+	0.968953i	$0.579525\pi$
$828$	0	0
$829$	23.0164	0.799393	0.399696	−	0.916648i	$-0.369116\pi$
$829$	23.0164	0.799393	0.399696	+	0.916648i	$0.369116\pi$
$830$	43.8335	1.52148
$831$	0	0
$832$	−0.133123	−0.00461522
$833$	6.57213	0.227711
$834$	0	0
$835$	−33.2698	−1.15135
$836$	−40.3157	−1.39435
$837$	0	0
$838$	36.3485	1.25564
$839$	−40.5007	−1.39824	−0.699120	−	0.715004i	$-0.746425\pi$
$839$	−40.5007	−1.39824	−0.699120	+	0.715004i	$0.746425\pi$
$840$	0	0
$841$	4.96817	0.171316
$842$	−7.08976	−0.244329
$843$	0	0
$844$	−18.6056	−0.640432
$845$	3.67909	0.126565
$846$	0	0
$847$	61.3332	2.10743
$848$	−7.79225	−0.267587
$849$	0	0
$850$	−9.65669	−0.331222
$851$	−5.98301	−0.205095
$852$	0	0
$853$	49.0041	1.67787	0.838934	−	0.544234i	$-0.183180\pi$
$853$	49.0041	1.67787	0.838934	+	0.544234i	$0.183180\pi$
$854$	−11.2455	−0.384813
$855$	0	0
$856$	20.7838	0.710374
$857$	21.6945	0.741069	0.370534	−	0.928819i	$-0.379175\pi$
$857$	21.6945	0.741069	0.370534	+	0.928819i	$0.379175\pi$
$858$	0	0
$859$	36.1931	1.23489	0.617446	−	0.786613i	$-0.288167\pi$
$859$	36.1931	1.23489	0.617446	+	0.786613i	$0.288167\pi$
$860$	−7.32784	−0.249877
$861$	0	0
$862$	11.1846	0.380948
$863$	6.19578	0.210907	0.105453	−	0.994424i	$-0.466371\pi$
$863$	6.19578	0.210907	0.105453	+	0.994424i	$0.466371\pi$
$864$	0	0
$865$	−35.0756	−1.19261
$866$	−41.5178	−1.41083
$867$	0	0
$868$	−4.54568	−0.154291
$869$	−21.0722	−0.714827
$870$	0	0
$871$	−2.10412	−0.0712954
$872$	−22.0884	−0.748008
$873$	0	0
$874$	−8.74197	−0.295702
$875$	53.9816	1.82491
$876$	0	0
$877$	−15.6975	−0.530067	−0.265033	−	0.964239i	$-0.585383\pi$
$877$	−15.6975	−0.530067	−0.265033	+	0.964239i	$0.585383\pi$
$878$	−1.65897	−0.0559876
$879$	0	0
$880$	−93.2310	−3.14282
$881$	10.5437	0.355225	0.177613	−	0.984100i	$-0.443163\pi$
$881$	10.5437	0.355225	0.177613	+	0.984100i	$0.443163\pi$
$882$	0	0
$883$	11.6997	0.393727	0.196864	−	0.980431i	$-0.436924\pi$
$883$	11.6997	0.393727	0.196864	+	0.980431i	$0.436924\pi$
$884$	−0.704374	−0.0236907
$885$	0	0
$886$	−38.5084	−1.29372
$887$	−23.8193	−0.799774	−0.399887	−	0.916564i	$-0.630951\pi$
$887$	−23.8193	−0.799774	−0.399887	+	0.916564i	$0.630951\pi$
$888$	0	0
$889$	12.8460	0.430843
$890$	−95.2893	−3.19411
$891$	0	0
$892$	10.9557	0.366823
$893$	43.9262	1.46993
$894$	0	0
$895$	37.9223	1.26760
$896$	−47.4313	−1.58457
$897$	0	0
$898$	24.4318	0.815301
$899$	5.82822	0.194382
$900$	0	0
$901$	1.00395	0.0334465
$902$	−99.7094	−3.31996
$903$	0	0
$904$	−0.690266	−0.0229579
$905$	12.5337	0.416635
$906$	0	0
$907$	−2.24010	−0.0743814	−0.0371907	−	0.999308i	$-0.511841\pi$
$907$	−2.24010	−0.0743814	−0.0371907	+	0.999308i	$0.511841\pi$
$908$	16.7474	0.555781
$909$	0	0
$910$	26.8611	0.890438
$911$	−12.9228	−0.428152	−0.214076	−	0.976817i	$-0.568674\pi$
$911$	−12.9228	−0.428152	−0.214076	+	0.976817i	$0.568674\pi$
$912$	0	0
$913$	−34.3832	−1.13792
$914$	39.4447	1.30471
$915$	0	0
$916$	−18.1895	−0.600999
$917$	2.72107	0.0898577
$918$	0	0
$919$	−21.0966	−0.695913	−0.347956	−	0.937511i	$-0.613124\pi$
$919$	−21.0966	−0.695913	−0.347956	+	0.937511i	$0.613124\pi$
$920$	−4.01371	−0.132328
$921$	0	0
$922$	56.6328	1.86510
$923$	0.546141	0.0179764
$924$	0	0
$925$	74.5023	2.44962
$926$	35.4775	1.16586
$927$	0	0
$928$	−32.6253	−1.07098
$929$	−19.7434	−0.647760	−0.323880	−	0.946098i	$-0.604987\pi$
$929$	−19.7434	−0.647760	−0.323880	+	0.946098i	$0.604987\pi$
$930$	0	0
$931$	−74.0861	−2.42807
$932$	11.3649	0.372271
$933$	0	0
$934$	3.79921	0.124314
$935$	12.0119	0.392830
$936$	0	0
$937$	−6.87850	−0.224711	−0.112355	−	0.993668i	$-0.535839\pi$
$937$	−6.87850	−0.224711	−0.112355	+	0.993668i	$0.535839\pi$
$938$	−15.3622	−0.501594
$939$	0	0
$940$	−24.4219	−0.796556
$941$	−16.7174	−0.544973	−0.272486	−	0.962160i	$-0.587846\pi$
$941$	−16.7174	−0.544973	−0.272486	+	0.962160i	$0.587846\pi$
$942$	0	0
$943$	−7.65118	−0.249157
$944$	−35.6371	−1.15989
$945$	0	0
$946$	16.2428	0.528097
$947$	−42.1651	−1.37018	−0.685091	−	0.728457i	$-0.740238\pi$
$947$	−42.1651	−1.37018	−0.685091	+	0.728457i	$0.740238\pi$
$948$	0	0
$949$	9.58155	0.311030
$950$	108.858	3.53181
$951$	0	0
$952$	4.24684	0.137641
$953$	47.4062	1.53564	0.767819	−	0.640667i	$-0.221342\pi$
$953$	47.4062	1.53564	0.767819	+	0.640667i	$0.221342\pi$
$954$	0	0
$955$	62.6164	2.02622
$956$	−29.2595	−0.946320
$957$	0	0
$958$	−76.7260	−2.47890
$959$	1.84492	0.0595756
$960$	0	0
$961$	1.00000	0.0322581
$962$	15.3563	0.495108
$963$	0	0
$964$	−9.67464	−0.311599
$965$	−0.379346	−0.0122116
$966$	0	0
$967$	22.3941	0.720147	0.360074	−	0.932924i	$-0.382752\pi$
$967$	22.3941	0.720147	0.360074	+	0.932924i	$0.382752\pi$
$968$	23.5225	0.756043
$969$	0	0
$970$	79.2860	2.54572
$971$	−5.11176	−0.164044	−0.0820221	−	0.996631i	$-0.526138\pi$
$971$	−5.11176	−0.164044	−0.0820221	+	0.996631i	$0.526138\pi$
$972$	0	0
$973$	8.24602	0.264355
$974$	−2.09261	−0.0670515
$975$	0	0
$976$	−7.68731	−0.246065
$977$	19.3950	0.620500	0.310250	−	0.950655i	$-0.399587\pi$
$977$	19.3950	0.620500	0.310250	+	0.950655i	$0.399587\pi$
$978$	0	0
$979$	74.7455	2.38888
$980$	41.1901	1.31577
$981$	0	0
$982$	−1.10853	−0.0353745
$983$	40.4106	1.28890	0.644449	−	0.764647i	$-0.277087\pi$
$983$	40.4106	1.28890	0.644449	+	0.764647i	$0.277087\pi$
$984$	0	0
$985$	9.26362	0.295163
$986$	6.59361	0.209983
$987$	0	0
$988$	7.94024	0.252613
$989$	1.24638	0.0396327
$990$	0	0
$991$	−56.3062	−1.78863	−0.894313	−	0.447443i	$-0.852335\pi$
$991$	−56.3062	−1.78863	−0.894313	+	0.447443i	$0.852335\pi$
$992$	−5.59782	−0.177731
$993$	0	0
$994$	3.98738	0.126472
$995$	−0.00578849	−0.000183507 0
$996$	0	0
$997$	27.2985	0.864552	0.432276	−	0.901741i	$-0.357711\pi$
$997$	27.2985	0.864552	0.432276	+	0.901741i	$0.357711\pi$
$998$	20.0872	0.635851
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3627.2.a.q.1.1		8
3.2	odd	2		403.2.a.d.1.8	✓	8
12.11	even	2		6448.2.a.bf.1.6		8
39.38	odd	2		5239.2.a.j.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.d.1.8	✓	8	3.2	odd	2
3627.2.a.q.1.1		8	1.1	even	1	trivial
5239.2.a.j.1.1		8	39.38	odd	2
6448.2.a.bf.1.6		8	12.11	even	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:	\( N \)	\(=\)	\( 3627 = 3^{2} \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3627.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75938 q^{2} +1.09540 q^{4} +3.67909 q^{5} +4.14977 q^{7} +1.59152 q^{8} +O(q^{10})\)	\(q-1.75938 q^{2} +1.09540 q^{4} +3.67909 q^{5} +4.14977 q^{7} +1.59152 q^{8} -6.47291 q^{10} +5.07739 q^{11} -1.00000 q^{13} -7.30102 q^{14} -4.99090 q^{16} +0.643026 q^{17} -7.24868 q^{19} +4.03010 q^{20} -8.93304 q^{22} -0.685475 q^{23} +8.53574 q^{25} +1.75938 q^{26} +4.54568 q^{28} -5.82822 q^{29} -1.00000 q^{31} +5.59782 q^{32} -1.13132 q^{34} +15.2674 q^{35} +8.72827 q^{37} +12.7532 q^{38} +5.85537 q^{40} +11.1619 q^{41} -1.81828 q^{43} +5.56180 q^{44} +1.20601 q^{46} -6.05989 q^{47} +10.2206 q^{49} -15.0176 q^{50} -1.09540 q^{52} +1.56129 q^{53} +18.6802 q^{55} +6.60446 q^{56} +10.2540 q^{58} +7.14043 q^{59} +1.54027 q^{61} +1.75938 q^{62} +0.133123 q^{64} -3.67909 q^{65} +2.10412 q^{67} +0.704374 q^{68} -26.8611 q^{70} -0.546141 q^{71} -9.58155 q^{73} -15.3563 q^{74} -7.94024 q^{76} +21.0700 q^{77} -4.15021 q^{79} -18.3620 q^{80} -19.6379 q^{82} -6.77183 q^{83} +2.36575 q^{85} +3.19904 q^{86} +8.08078 q^{88} +14.7212 q^{89} -4.14977 q^{91} -0.750873 q^{92} +10.6616 q^{94} -26.6686 q^{95} -12.2489 q^{97} -17.9819 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 9 q^{4} + 15 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 5 * q^2 + 9 * q^4 + 15 * q^5 - 4 * q^7 + 15 * q^8	\( 8 q + 5 q^{2} + 9 q^{4} + 15 q^{5} - 4 q^{7} + 15 q^{8} + 9 q^{10} + 5 q^{11} - 8 q^{13} - 2 q^{14} + 3 q^{16} + 11 q^{17} - 9 q^{19} + 31 q^{20} - 2 q^{22} + 19 q^{25} - 5 q^{26} + 16 q^{28} + 12 q^{29} - 8 q^{31} + 25 q^{32} + 22 q^{34} - 7 q^{35} - 9 q^{37} + 9 q^{38} + 55 q^{40} + 25 q^{41} + 7 q^{43} + 26 q^{44} + 5 q^{46} + 17 q^{47} + 11 q^{50} - 9 q^{52} + 15 q^{53} + 7 q^{55} + 14 q^{56} - 5 q^{58} + 15 q^{59} + 11 q^{61} - 5 q^{62} + 47 q^{64} - 15 q^{65} + 18 q^{67} + 16 q^{68} - 24 q^{70} + 7 q^{71} + 24 q^{73} - 48 q^{74} - 3 q^{76} + 49 q^{77} + 33 q^{79} + 16 q^{80} - q^{82} + 13 q^{83} + q^{85} - 19 q^{86} + 37 q^{88} + 23 q^{89} + 4 q^{91} - 22 q^{92} + 10 q^{94} - 43 q^{95} - 17 q^{97} - 52 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 9 * q^4 + 15 * q^5 - 4 * q^7 + 15 * q^8 + 9 * q^10 + 5 * q^11 - 8 * q^13 - 2 * q^14 + 3 * q^16 + 11 * q^17 - 9 * q^19 + 31 * q^20 - 2 * q^22 + 19 * q^25 - 5 * q^26 + 16 * q^28 + 12 * q^29 - 8 * q^31 + 25 * q^32 + 22 * q^34 - 7 * q^35 - 9 * q^37 + 9 * q^38 + 55 * q^40 + 25 * q^41 + 7 * q^43 + 26 * q^44 + 5 * q^46 + 17 * q^47 + 11 * q^50 - 9 * q^52 + 15 * q^53 + 7 * q^55 + 14 * q^56 - 5 * q^58 + 15 * q^59 + 11 * q^61 - 5 * q^62 + 47 * q^64 - 15 * q^65 + 18 * q^67 + 16 * q^68 - 24 * q^70 + 7 * q^71 + 24 * q^73 - 48 * q^74 - 3 * q^76 + 49 * q^77 + 33 * q^79 + 16 * q^80 - q^82 + 13 * q^83 + q^85 - 19 * q^86 + 37 * q^88 + 23 * q^89 + 4 * q^91 - 22 * q^92 + 10 * q^94 - 43 * q^95 - 17 * q^97 - 52 * q^98

Introduction

L-functions

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