LMFDB - Embedded newform 3520.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,2,Mod(1,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.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:	$28.1073415115$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	3520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} -1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} -2.00000 q^{13} -1.56155 q^{15} +3.56155 q^{17} -1.56155 q^{19} -2.43845 q^{21} +3.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} +2.68466 q^{29} -2.43845 q^{31} +1.56155 q^{33} +1.56155 q^{35} -6.68466 q^{37} -3.12311 q^{39} +2.00000 q^{41} -6.24621 q^{43} +0.561553 q^{45} +4.87689 q^{47} -4.56155 q^{49} +5.56155 q^{51} -0.438447 q^{53} -1.00000 q^{55} -2.43845 q^{57} -7.12311 q^{59} -14.6847 q^{61} +0.876894 q^{63} +2.00000 q^{65} -10.2462 q^{67} +4.87689 q^{69} +8.68466 q^{71} -2.00000 q^{73} +1.56155 q^{75} -1.56155 q^{77} -9.36932 q^{79} -7.00000 q^{81} -3.12311 q^{83} -3.56155 q^{85} +4.19224 q^{87} -8.43845 q^{89} +3.12311 q^{91} -3.80776 q^{93} +1.56155 q^{95} -1.12311 q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9	$ 2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} - 4 q^{13} + q^{15} + 3 q^{17} + q^{19} - 9 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 7 q^{29} - 9 q^{31} - q^{33} - q^{35} - q^{37} + 2 q^{39} + 4 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} - 5 q^{49} + 7 q^{51} - 5 q^{53} - 2 q^{55} - 9 q^{57} - 6 q^{59} - 17 q^{61} + 10 q^{63} + 4 q^{65} - 4 q^{67} + 18 q^{69} + 5 q^{71} - 4 q^{73} - q^{75} + q^{77} + 6 q^{79} - 14 q^{81} + 2 q^{83} - 3 q^{85} + 29 q^{87} - 21 q^{89} - 2 q^{91} + 13 q^{93} - q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9 + 2 * q^11 - 4 * q^13 + q^15 + 3 * q^17 + q^19 - 9 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 7 * q^29 - 9 * q^31 - q^33 - q^35 - q^37 + 2 * q^39 + 4 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 - 5 * q^49 + 7 * q^51 - 5 * q^53 - 2 * q^55 - 9 * q^57 - 6 * q^59 - 17 * q^61 + 10 * q^63 + 4 * q^65 - 4 * q^67 + 18 * q^69 + 5 * q^71 - 4 * q^73 - q^75 + q^77 + 6 * q^79 - 14 * q^81 + 2 * q^83 - 3 * q^85 + 29 * q^87 - 21 * q^89 - 2 * q^91 + 13 * q^93 - q^95 + 6 * q^97 + 3 * 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 $

$2$	0	0
$2$	0	0
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	−1.56155	−0.358245	−0.179122	−	0.983827i	$-0.557326\pi$
$19$	−1.56155	−0.358245	−0.179122	+	0.983827i	$0.557326\pi$
$20$	0	0
$21$	−2.43845	−0.532113
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	2.68466	0.498529	0.249264	−	0.968436i	$-0.419811\pi$
$29$	2.68466	0.498529	0.249264	+	0.968436i	$0.419811\pi$
$30$	0	0
$31$	−2.43845	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$31$	−2.43845	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$32$	0	0
$33$	1.56155	0.271831
$34$	0	0
$35$	1.56155	0.263951
$36$	0	0
$37$	−6.68466	−1.09895	−0.549476	−	0.835510i	$-0.685172\pi$
$37$	−6.68466	−1.09895	−0.549476	+	0.835510i	$0.685172\pi$
$38$	0	0
$39$	−3.12311	−0.500097
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−6.24621	−0.952538	−0.476269	−	0.879300i	$-0.658011\pi$
$43$	−6.24621	−0.952538	−0.476269	+	0.879300i	$0.658011\pi$
$44$	0	0
$45$	0.561553	0.0837114
$46$	0	0
$47$	4.87689	0.711368	0.355684	−	0.934606i	$-0.384248\pi$
$47$	4.87689	0.711368	0.355684	+	0.934606i	$0.384248\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	5.56155	0.778773
$52$	0	0
$53$	−0.438447	−0.0602254	−0.0301127	−	0.999547i	$-0.509587\pi$
$53$	−0.438447	−0.0602254	−0.0301127	+	0.999547i	$0.509587\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−2.43845	−0.322980
$58$	0	0
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	0	0
$61$	−14.6847	−1.88018	−0.940089	−	0.340929i	$-0.889258\pi$
$61$	−14.6847	−1.88018	−0.940089	+	0.340929i	$0.889258\pi$
$62$	0	0
$63$	0.876894	0.110478
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	0	0
$69$	4.87689	0.587109
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	−1.56155	−0.177955
$78$	0	0
$79$	−9.36932	−1.05413	−0.527065	−	0.849825i	$-0.676708\pi$
$79$	−9.36932	−1.05413	−0.527065	+	0.849825i	$0.676708\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−3.12311	−0.342805	−0.171403	−	0.985201i	$-0.554830\pi$
$83$	−3.12311	−0.342805	−0.171403	+	0.985201i	$0.554830\pi$
$84$	0	0
$85$	−3.56155	−0.386305
$86$	0	0
$87$	4.19224	0.449455
$88$	0	0
$89$	−8.43845	−0.894474	−0.447237	−	0.894416i	$-0.647592\pi$
$89$	−8.43845	−0.894474	−0.447237	+	0.894416i	$0.647592\pi$
$90$	0	0
$91$	3.12311	0.327390
$92$	0	0
$93$	−3.80776	−0.394847
$94$	0	0
$95$	1.56155	0.160212
$96$	0	0
$97$	−1.12311	−0.114034	−0.0570170	−	0.998373i	$-0.518159\pi$
$97$	−1.12311	−0.114034	−0.0570170	+	0.998373i	$0.518159\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	−12.2462	−1.21854	−0.609272	−	0.792961i	$-0.708538\pi$
$101$	−12.2462	−1.21854	−0.609272	+	0.792961i	$0.708538\pi$
$102$	0	0
$103$	6.24621	0.615457	0.307729	−	0.951474i	$-0.400431\pi$
$103$	6.24621	0.615457	0.307729	+	0.951474i	$0.400431\pi$
$104$	0	0
$105$	2.43845	0.237968
$106$	0	0
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	14.4924	1.38812	0.694061	−	0.719916i	$-0.255820\pi$
$109$	14.4924	1.38812	0.694061	+	0.719916i	$0.255820\pi$
$110$	0	0
$111$	−10.4384	−0.990774
$112$	0	0
$113$	−10.8769	−1.02321	−0.511606	−	0.859220i	$-0.670949\pi$
$113$	−10.8769	−1.02321	−0.511606	+	0.859220i	$0.670949\pi$
$114$	0	0
$115$	−3.12311	−0.291231
$116$	0	0
$117$	1.12311	0.103831
$118$	0	0
$119$	−5.56155	−0.509827
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.12311	0.281601
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.24621	0.199319	0.0996595	−	0.995022i	$-0.468225\pi$
$127$	2.24621	0.199319	0.0996595	+	0.995022i	$0.468225\pi$
$128$	0	0
$129$	−9.75379	−0.858773
$130$	0	0
$131$	4.68466	0.409301	0.204650	−	0.978835i	$-0.434394\pi$
$131$	4.68466	0.409301	0.204650	+	0.978835i	$0.434394\pi$
$132$	0	0
$133$	2.43845	0.211440
$134$	0	0
$135$	5.56155	0.478662
$136$	0	0
$137$	−21.6155	−1.84674	−0.923370	−	0.383912i	$-0.874577\pi$
$137$	−21.6155	−1.84674	−0.923370	+	0.383912i	$0.874577\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	7.61553	0.641343
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−2.68466	−0.222949
$146$	0	0
$147$	−7.12311	−0.587504
$148$	0	0
$149$	−14.6847	−1.20301	−0.601507	−	0.798867i	$-0.705433\pi$
$149$	−14.6847	−1.20301	−0.601507	+	0.798867i	$0.705433\pi$
$150$	0	0
$151$	−4.87689	−0.396876	−0.198438	−	0.980113i	$-0.563587\pi$
$151$	−4.87689	−0.396876	−0.198438	+	0.980113i	$0.563587\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	2.43845	0.195861
$156$	0	0
$157$	5.80776	0.463510	0.231755	−	0.972774i	$-0.425553\pi$
$157$	5.80776	0.463510	0.231755	+	0.972774i	$0.425553\pi$
$158$	0	0
$159$	−0.684658	−0.0542969
$160$	0	0
$161$	−4.87689	−0.384353
$162$	0	0
$163$	23.8078	1.86477	0.932384	−	0.361469i	$-0.117725\pi$
$163$	23.8078	1.86477	0.932384	+	0.361469i	$0.117725\pi$
$164$	0	0
$165$	−1.56155	−0.121567
$166$	0	0
$167$	6.43845	0.498222	0.249111	−	0.968475i	$-0.419862\pi$
$167$	6.43845	0.498222	0.249111	+	0.968475i	$0.419862\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0.876894	0.0670578
$172$	0	0
$173$	12.2462	0.931062	0.465531	−	0.885032i	$-0.345863\pi$
$173$	12.2462	0.931062	0.465531	+	0.885032i	$0.345863\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	−11.1231	−0.836064
$178$	0	0
$179$	10.2462	0.765838	0.382919	−	0.923782i	$-0.374919\pi$
$179$	10.2462	0.765838	0.382919	+	0.923782i	$0.374919\pi$
$180$	0	0
$181$	−12.2462	−0.910254	−0.455127	−	0.890427i	$-0.650406\pi$
$181$	−12.2462	−0.910254	−0.455127	+	0.890427i	$0.650406\pi$
$182$	0	0
$183$	−22.9309	−1.69510
$184$	0	0
$185$	6.68466	0.491466
$186$	0	0
$187$	3.56155	0.260447
$188$	0	0
$189$	8.68466	0.631716
$190$	0	0
$191$	6.24621	0.451960	0.225980	−	0.974132i	$-0.427442\pi$
$191$	6.24621	0.451960	0.225980	+	0.974132i	$0.427442\pi$
$192$	0	0
$193$	25.8078	1.85768	0.928842	−	0.370477i	$-0.120806\pi$
$193$	25.8078	1.85768	0.928842	+	0.370477i	$0.120806\pi$
$194$	0	0
$195$	3.12311	0.223650
$196$	0	0
$197$	−19.3693	−1.38001	−0.690003	−	0.723806i	$-0.742391\pi$
$197$	−19.3693	−1.38001	−0.690003	+	0.723806i	$0.742391\pi$
$198$	0	0
$199$	−19.8078	−1.40414	−0.702068	−	0.712110i	$-0.747740\pi$
$199$	−19.8078	−1.40414	−0.702068	+	0.712110i	$0.747740\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	−4.19224	−0.294237
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−1.75379	−0.121897
$208$	0	0
$209$	−1.56155	−0.108015
$210$	0	0
$211$	20.6847	1.42399	0.711995	−	0.702184i	$-0.247792\pi$
$211$	20.6847	1.42399	0.711995	+	0.702184i	$0.247792\pi$
$212$	0	0
$213$	13.5616	0.929222
$214$	0	0
$215$	6.24621	0.425988
$216$	0	0
$217$	3.80776	0.258488
$218$	0	0
$219$	−3.12311	−0.211040
$220$	0	0
$221$	−7.12311	−0.479152
$222$	0	0
$223$	−4.87689	−0.326581	−0.163291	−	0.986578i	$-0.552211\pi$
$223$	−4.87689	−0.326581	−0.163291	+	0.986578i	$0.552211\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	11.1231	0.738266	0.369133	−	0.929376i	$-0.379655\pi$
$227$	11.1231	0.738266	0.369133	+	0.929376i	$0.379655\pi$
$228$	0	0
$229$	−24.7386	−1.63477	−0.817387	−	0.576088i	$-0.804578\pi$
$229$	−24.7386	−1.63477	−0.817387	+	0.576088i	$0.804578\pi$
$230$	0	0
$231$	−2.43845	−0.160438
$232$	0	0
$233$	−8.93087	−0.585081	−0.292540	−	0.956253i	$-0.594501\pi$
$233$	−8.93087	−0.585081	−0.292540	+	0.956253i	$0.594501\pi$
$234$	0	0
$235$	−4.87689	−0.318134
$236$	0	0
$237$	−14.6307	−0.950365
$238$	0	0
$239$	15.6155	1.01008	0.505042	−	0.863095i	$-0.331477\pi$
$239$	15.6155	1.01008	0.505042	+	0.863095i	$0.331477\pi$
$240$	0	0
$241$	0.246211	0.0158599	0.00792993	−	0.999969i	$-0.497476\pi$
$241$	0.246211	0.0158599	0.00792993	+	0.999969i	$0.497476\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	4.56155	0.291427
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	−4.87689	−0.309061
$250$	0	0
$251$	11.6155	0.733166	0.366583	−	0.930385i	$-0.380528\pi$
$251$	11.6155	0.733166	0.366583	+	0.930385i	$0.380528\pi$
$252$	0	0
$253$	3.12311	0.196348
$254$	0	0
$255$	−5.56155	−0.348278
$256$	0	0
$257$	20.7386	1.29364	0.646820	−	0.762643i	$-0.276098\pi$
$257$	20.7386	1.29364	0.646820	+	0.762643i	$0.276098\pi$
$258$	0	0
$259$	10.4384	0.648614
$260$	0	0
$261$	−1.50758	−0.0933167
$262$	0	0
$263$	15.8078	0.974748	0.487374	−	0.873193i	$-0.337955\pi$
$263$	15.8078	0.974748	0.487374	+	0.873193i	$0.337955\pi$
$264$	0	0
$265$	0.438447	0.0269336
$266$	0	0
$267$	−13.1771	−0.806424
$268$	0	0
$269$	−21.6155	−1.31792	−0.658961	−	0.752177i	$-0.729004\pi$
$269$	−21.6155	−1.31792	−0.658961	+	0.752177i	$0.729004\pi$
$270$	0	0
$271$	4.49242	0.272895	0.136448	−	0.990647i	$-0.456431\pi$
$271$	4.49242	0.272895	0.136448	+	0.990647i	$0.456431\pi$
$272$	0	0
$273$	4.87689	0.295163
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	1.12311	0.0674809	0.0337404	−	0.999431i	$-0.489258\pi$
$277$	1.12311	0.0674809	0.0337404	+	0.999431i	$0.489258\pi$
$278$	0	0
$279$	1.36932	0.0819789
$280$	0	0
$281$	8.24621	0.491928	0.245964	−	0.969279i	$-0.420896\pi$
$281$	8.24621	0.491928	0.245964	+	0.969279i	$0.420896\pi$
$282$	0	0
$283$	1.36932	0.0813974	0.0406987	−	0.999171i	$-0.487042\pi$
$283$	1.36932	0.0813974	0.0406987	+	0.999171i	$0.487042\pi$
$284$	0	0
$285$	2.43845	0.144441
$286$	0	0
$287$	−3.12311	−0.184351
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	−1.75379	−0.102809
$292$	0	0
$293$	18.4924	1.08034	0.540169	−	0.841556i	$-0.318360\pi$
$293$	18.4924	1.08034	0.540169	+	0.841556i	$0.318360\pi$
$294$	0	0
$295$	7.12311	0.414723
$296$	0	0
$297$	−5.56155	−0.322714
$298$	0	0
$299$	−6.24621	−0.361228
$300$	0	0
$301$	9.75379	0.562199
$302$	0	0
$303$	−19.1231	−1.09859
$304$	0	0
$305$	14.6847	0.840841
$306$	0	0
$307$	15.6155	0.891225	0.445613	−	0.895226i	$-0.352986\pi$
$307$	15.6155	0.891225	0.445613	+	0.895226i	$0.352986\pi$
$308$	0	0
$309$	9.75379	0.554874
$310$	0	0
$311$	29.5616	1.67628	0.838141	−	0.545454i	$-0.183643\pi$
$311$	29.5616	1.67628	0.838141	+	0.545454i	$0.183643\pi$
$312$	0	0
$313$	−2.49242	−0.140880	−0.0704400	−	0.997516i	$-0.522440\pi$
$313$	−2.49242	−0.140880	−0.0704400	+	0.997516i	$0.522440\pi$
$314$	0	0
$315$	−0.876894	−0.0494074
$316$	0	0
$317$	−33.8078	−1.89883	−0.949417	−	0.314019i	$-0.898324\pi$
$317$	−33.8078	−1.89883	−0.949417	+	0.314019i	$0.898324\pi$
$318$	0	0
$319$	2.68466	0.150312
$320$	0	0
$321$	−24.9848	−1.39452
$322$	0	0
$323$	−5.56155	−0.309453
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	22.6307	1.25148
$328$	0	0
$329$	−7.61553	−0.419858
$330$	0	0
$331$	24.4924	1.34623	0.673113	−	0.739540i	$-0.264957\pi$
$331$	24.4924	1.34623	0.673113	+	0.739540i	$0.264957\pi$
$332$	0	0
$333$	3.75379	0.205706
$334$	0	0
$335$	10.2462	0.559810
$336$	0	0
$337$	14.6847	0.799924	0.399962	−	0.916532i	$-0.369023\pi$
$337$	14.6847	0.799924	0.399962	+	0.916532i	$0.369023\pi$
$338$	0	0
$339$	−16.9848	−0.922490
$340$	0	0
$341$	−2.43845	−0.132049
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	−4.87689	−0.262563
$346$	0	0
$347$	−27.1231	−1.45604	−0.728022	−	0.685553i	$-0.759560\pi$
$347$	−27.1231	−1.45604	−0.728022	+	0.685553i	$0.759560\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	11.1231	0.593707
$352$	0	0
$353$	−1.50758	−0.0802403	−0.0401201	−	0.999195i	$-0.512774\pi$
$353$	−1.50758	−0.0802403	−0.0401201	+	0.999195i	$0.512774\pi$
$354$	0	0
$355$	−8.68466	−0.460934
$356$	0	0
$357$	−8.68466	−0.459641
$358$	0	0
$359$	−9.75379	−0.514785	−0.257393	−	0.966307i	$-0.582863\pi$
$359$	−9.75379	−0.514785	−0.257393	+	0.966307i	$0.582863\pi$
$360$	0	0
$361$	−16.5616	−0.871661
$362$	0	0
$363$	1.56155	0.0819603
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	−34.7386	−1.81334	−0.906671	−	0.421839i	$-0.861385\pi$
$367$	−34.7386	−1.81334	−0.906671	+	0.421839i	$0.861385\pi$
$368$	0	0
$369$	−1.12311	−0.0584665
$370$	0	0
$371$	0.684658	0.0355457
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	−1.56155	−0.0806382
$376$	0	0
$377$	−5.36932	−0.276534
$378$	0	0
$379$	−16.8769	−0.866908	−0.433454	−	0.901176i	$-0.642705\pi$
$379$	−16.8769	−0.866908	−0.433454	+	0.901176i	$0.642705\pi$
$380$	0	0
$381$	3.50758	0.179699
$382$	0	0
$383$	6.63068	0.338812	0.169406	−	0.985546i	$-0.445815\pi$
$383$	6.63068	0.338812	0.169406	+	0.985546i	$0.445815\pi$
$384$	0	0
$385$	1.56155	0.0795841
$386$	0	0
$387$	3.50758	0.178300
$388$	0	0
$389$	−15.7538	−0.798749	−0.399374	−	0.916788i	$-0.630773\pi$
$389$	−15.7538	−0.798749	−0.399374	+	0.916788i	$0.630773\pi$
$390$	0	0
$391$	11.1231	0.562520
$392$	0	0
$393$	7.31534	0.369010
$394$	0	0
$395$	9.36932	0.471421
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	3.80776	0.190627
$400$	0	0
$401$	−11.5616	−0.577356	−0.288678	−	0.957426i	$-0.593216\pi$
$401$	−11.5616	−0.577356	−0.288678	+	0.957426i	$0.593216\pi$
$402$	0	0
$403$	4.87689	0.242935
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	−6.68466	−0.331346
$408$	0	0
$409$	−1.12311	−0.0555340	−0.0277670	−	0.999614i	$-0.508840\pi$
$409$	−1.12311	−0.0555340	−0.0277670	+	0.999614i	$0.508840\pi$
$410$	0	0
$411$	−33.7538	−1.66495
$412$	0	0
$413$	11.1231	0.547332
$414$	0	0
$415$	3.12311	0.153307
$416$	0	0
$417$	18.7386	0.917635
$418$	0	0
$419$	22.7386	1.11085	0.555427	−	0.831565i	$-0.312555\pi$
$419$	22.7386	1.11085	0.555427	+	0.831565i	$0.312555\pi$
$420$	0	0
$421$	33.6155	1.63832	0.819160	−	0.573565i	$-0.194440\pi$
$421$	33.6155	1.63832	0.819160	+	0.573565i	$0.194440\pi$
$422$	0	0
$423$	−2.73863	−0.133157
$424$	0	0
$425$	3.56155	0.172761
$426$	0	0
$427$	22.9309	1.10970
$428$	0	0
$429$	−3.12311	−0.150785
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	8.63068	0.414764	0.207382	−	0.978260i	$-0.433506\pi$
$433$	8.63068	0.414764	0.207382	+	0.978260i	$0.433506\pi$
$434$	0	0
$435$	−4.19224	−0.201002
$436$	0	0
$437$	−4.87689	−0.233293
$438$	0	0
$439$	17.7538	0.847342	0.423671	−	0.905816i	$-0.360741\pi$
$439$	17.7538	0.847342	0.423671	+	0.905816i	$0.360741\pi$
$440$	0	0
$441$	2.56155	0.121979
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	8.43845	0.400021
$446$	0	0
$447$	−22.9309	−1.08459
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	0	0
$453$	−7.61553	−0.357809
$454$	0	0
$455$	−3.12311	−0.146413
$456$	0	0
$457$	24.4384	1.14318	0.571591	−	0.820539i	$-0.306326\pi$
$457$	24.4384	1.14318	0.571591	+	0.820539i	$0.306326\pi$
$458$	0	0
$459$	−19.8078	−0.924547
$460$	0	0
$461$	−16.4384	−0.765615	−0.382807	−	0.923828i	$-0.625043\pi$
$461$	−16.4384	−0.765615	−0.382807	+	0.923828i	$0.625043\pi$
$462$	0	0
$463$	−17.3693	−0.807221	−0.403610	−	0.914931i	$-0.632245\pi$
$463$	−17.3693	−0.807221	−0.403610	+	0.914931i	$0.632245\pi$
$464$	0	0
$465$	3.80776	0.176581
$466$	0	0
$467$	−18.9309	−0.876016	−0.438008	−	0.898971i	$-0.644316\pi$
$467$	−18.9309	−0.876016	−0.438008	+	0.898971i	$0.644316\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	9.06913	0.417883
$472$	0	0
$473$	−6.24621	−0.287201
$474$	0	0
$475$	−1.56155	−0.0716490
$476$	0	0
$477$	0.246211	0.0112732
$478$	0	0
$479$	−32.9848	−1.50712	−0.753558	−	0.657381i	$-0.771664\pi$
$479$	−32.9848	−1.50712	−0.753558	+	0.657381i	$0.771664\pi$
$480$	0	0
$481$	13.3693	0.609588
$482$	0	0
$483$	−7.61553	−0.346519
$484$	0	0
$485$	1.12311	0.0509976
$486$	0	0
$487$	−14.2462	−0.645557	−0.322779	−	0.946474i	$-0.604617\pi$
$487$	−14.2462	−0.645557	−0.322779	+	0.946474i	$0.604617\pi$
$488$	0	0
$489$	37.1771	1.68121
$490$	0	0
$491$	−30.0540	−1.35632	−0.678158	−	0.734916i	$-0.737222\pi$
$491$	−30.0540	−1.35632	−0.678158	+	0.734916i	$0.737222\pi$
$492$	0	0
$493$	9.56155	0.430631
$494$	0	0
$495$	0.561553	0.0252399
$496$	0	0
$497$	−13.5616	−0.608319
$498$	0	0
$499$	−16.4924	−0.738302	−0.369151	−	0.929369i	$-0.620352\pi$
$499$	−16.4924	−0.738302	−0.369151	+	0.929369i	$0.620352\pi$
$500$	0	0
$501$	10.0540	0.449178
$502$	0	0
$503$	−2.24621	−0.100154	−0.0500768	−	0.998745i	$-0.515947\pi$
$503$	−2.24621	−0.100154	−0.0500768	+	0.998745i	$0.515947\pi$
$504$	0	0
$505$	12.2462	0.544949
$506$	0	0
$507$	−14.0540	−0.624159
$508$	0	0
$509$	3.36932	0.149342	0.0746712	−	0.997208i	$-0.476209\pi$
$509$	3.36932	0.149342	0.0746712	+	0.997208i	$0.476209\pi$
$510$	0	0
$511$	3.12311	0.138158
$512$	0	0
$513$	8.68466	0.383437
$514$	0	0
$515$	−6.24621	−0.275241
$516$	0	0
$517$	4.87689	0.214486
$518$	0	0
$519$	19.1231	0.839411
$520$	0	0
$521$	−40.7386	−1.78479	−0.892396	−	0.451253i	$-0.850977\pi$
$521$	−40.7386	−1.78479	−0.892396	+	0.451253i	$0.850977\pi$
$522$	0	0
$523$	−36.4924	−1.59570	−0.797851	−	0.602855i	$-0.794030\pi$
$523$	−36.4924	−1.59570	−0.797851	+	0.602855i	$0.794030\pi$
$524$	0	0
$525$	−2.43845	−0.106423
$526$	0	0
$527$	−8.68466	−0.378310
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	−4.56155	−0.196480
$540$	0	0
$541$	−36.5464	−1.57125	−0.785626	−	0.618701i	$-0.787659\pi$
$541$	−36.5464	−1.57125	−0.785626	+	0.618701i	$0.787659\pi$
$542$	0	0
$543$	−19.1231	−0.820651
$544$	0	0
$545$	−14.4924	−0.620787
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	8.24621	0.351940
$550$	0	0
$551$	−4.19224	−0.178595
$552$	0	0
$553$	14.6307	0.622160
$554$	0	0
$555$	10.4384	0.443087
$556$	0	0
$557$	−13.5076	−0.572334	−0.286167	−	0.958180i	$-0.592381\pi$
$557$	−13.5076	−0.572334	−0.286167	+	0.958180i	$0.592381\pi$
$558$	0	0
$559$	12.4924	0.528373
$560$	0	0
$561$	5.56155	0.234809
$562$	0	0
$563$	−11.1231	−0.468783	−0.234392	−	0.972142i	$-0.575310\pi$
$563$	−11.1231	−0.468783	−0.234392	+	0.972142i	$0.575310\pi$
$564$	0	0
$565$	10.8769	0.457594
$566$	0	0
$567$	10.9309	0.459053
$568$	0	0
$569$	46.1080	1.93295	0.966473	−	0.256769i	$-0.0826578\pi$
$569$	46.1080	1.93295	0.966473	+	0.256769i	$0.0826578\pi$
$570$	0	0
$571$	−31.4233	−1.31502	−0.657512	−	0.753444i	$-0.728391\pi$
$571$	−31.4233	−1.31502	−0.657512	+	0.753444i	$0.728391\pi$
$572$	0	0
$573$	9.75379	0.407470
$574$	0	0
$575$	3.12311	0.130243
$576$	0	0
$577$	−24.7386	−1.02988	−0.514941	−	0.857225i	$-0.672186\pi$
$577$	−24.7386	−1.02988	−0.514941	+	0.857225i	$0.672186\pi$
$578$	0	0
$579$	40.3002	1.67482
$580$	0	0
$581$	4.87689	0.202328
$582$	0	0
$583$	−0.438447	−0.0181586
$584$	0	0
$585$	−1.12311	−0.0464347
$586$	0	0
$587$	44.6847	1.84433	0.922167	−	0.386793i	$-0.126417\pi$
$587$	44.6847	1.84433	0.922167	+	0.386793i	$0.126417\pi$
$588$	0	0
$589$	3.80776	0.156896
$590$	0	0
$591$	−30.2462	−1.24416
$592$	0	0
$593$	34.4924	1.41643	0.708217	−	0.705995i	$-0.249500\pi$
$593$	34.4924	1.41643	0.708217	+	0.705995i	$0.249500\pi$
$594$	0	0
$595$	5.56155	0.228001
$596$	0	0
$597$	−30.9309	−1.26592
$598$	0	0
$599$	21.5616	0.880981	0.440491	−	0.897757i	$-0.354805\pi$
$599$	21.5616	0.880981	0.440491	+	0.897757i	$0.354805\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	5.75379	0.234312
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−0.192236	−0.00780262	−0.00390131	−	0.999992i	$-0.501242\pi$
$607$	−0.192236	−0.00780262	−0.00390131	+	0.999992i	$0.501242\pi$
$608$	0	0
$609$	−6.54640	−0.265273
$610$	0	0
$611$	−9.75379	−0.394596
$612$	0	0
$613$	32.7386	1.32230	0.661150	−	0.750253i	$-0.270068\pi$
$613$	32.7386	1.32230	0.661150	+	0.750253i	$0.270068\pi$
$614$	0	0
$615$	−3.12311	−0.125936
$616$	0	0
$617$	−31.3693	−1.26288	−0.631441	−	0.775424i	$-0.717536\pi$
$617$	−31.3693	−1.26288	−0.631441	+	0.775424i	$0.717536\pi$
$618$	0	0
$619$	24.4924	0.984434	0.492217	−	0.870473i	$-0.336187\pi$
$619$	24.4924	0.984434	0.492217	+	0.870473i	$0.336187\pi$
$620$	0	0
$621$	−17.3693	−0.697007
$622$	0	0
$623$	13.1771	0.527929
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.43845	−0.0973822
$628$	0	0
$629$	−23.8078	−0.949278
$630$	0	0
$631$	−3.80776	−0.151585	−0.0757923	−	0.997124i	$-0.524149\pi$
$631$	−3.80776	−0.151585	−0.0757923	+	0.997124i	$0.524149\pi$
$632$	0	0
$633$	32.3002	1.28382
$634$	0	0
$635$	−2.24621	−0.0891382
$636$	0	0
$637$	9.12311	0.361471
$638$	0	0
$639$	−4.87689	−0.192927
$640$	0	0
$641$	7.17708	0.283478	0.141739	−	0.989904i	$-0.454731\pi$
$641$	7.17708	0.283478	0.141739	+	0.989904i	$0.454731\pi$
$642$	0	0
$643$	11.3153	0.446234	0.223117	−	0.974792i	$-0.428377\pi$
$643$	11.3153	0.446234	0.223117	+	0.974792i	$0.428377\pi$
$644$	0	0
$645$	9.75379	0.384055
$646$	0	0
$647$	4.87689	0.191731	0.0958653	−	0.995394i	$-0.469438\pi$
$647$	4.87689	0.191731	0.0958653	+	0.995394i	$0.469438\pi$
$648$	0	0
$649$	−7.12311	−0.279606
$650$	0	0
$651$	5.94602	0.233043
$652$	0	0
$653$	−9.80776	−0.383807	−0.191904	−	0.981414i	$-0.561466\pi$
$653$	−9.80776	−0.383807	−0.191904	+	0.981414i	$0.561466\pi$
$654$	0	0
$655$	−4.68466	−0.183045
$656$	0	0
$657$	1.12311	0.0438165
$658$	0	0
$659$	39.8078	1.55069	0.775345	−	0.631538i	$-0.217576\pi$
$659$	39.8078	1.55069	0.775345	+	0.631538i	$0.217576\pi$
$660$	0	0
$661$	0.246211	0.00957651	0.00478825	−	0.999989i	$-0.498476\pi$
$661$	0.246211	0.00957651	0.00478825	+	0.999989i	$0.498476\pi$
$662$	0	0
$663$	−11.1231	−0.431986
$664$	0	0
$665$	−2.43845	−0.0945589
$666$	0	0
$667$	8.38447	0.324648
$668$	0	0
$669$	−7.61553	−0.294433
$670$	0	0
$671$	−14.6847	−0.566895
$672$	0	0
$673$	50.7926	1.95791	0.978956	−	0.204073i	$-0.0654181\pi$
$673$	50.7926	1.95791	0.978956	+	0.204073i	$0.0654181\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	13.6155	0.523287	0.261644	−	0.965165i	$-0.415735\pi$
$677$	13.6155	0.523287	0.261644	+	0.965165i	$0.415735\pi$
$678$	0	0
$679$	1.75379	0.0673042
$680$	0	0
$681$	17.3693	0.665594
$682$	0	0
$683$	2.93087	0.112147	0.0560733	−	0.998427i	$-0.482142\pi$
$683$	2.93087	0.112147	0.0560733	+	0.998427i	$0.482142\pi$
$684$	0	0
$685$	21.6155	0.825887
$686$	0	0
$687$	−38.6307	−1.47385
$688$	0	0
$689$	0.876894	0.0334070
$690$	0	0
$691$	−21.3693	−0.812927	−0.406464	−	0.913667i	$-0.633238\pi$
$691$	−21.3693	−0.812927	−0.406464	+	0.913667i	$0.633238\pi$
$692$	0	0
$693$	0.876894	0.0333105
$694$	0	0
$695$	−12.0000	−0.455186
$696$	0	0
$697$	7.12311	0.269807
$698$	0	0
$699$	−13.9460	−0.527487
$700$	0	0
$701$	−46.6847	−1.76326	−0.881628	−	0.471946i	$-0.843552\pi$
$701$	−46.6847	−1.76326	−0.881628	+	0.471946i	$0.843552\pi$
$702$	0	0
$703$	10.4384	0.393693
$704$	0	0
$705$	−7.61553	−0.286817
$706$	0	0
$707$	19.1231	0.719198
$708$	0	0
$709$	−34.4924	−1.29539	−0.647695	−	0.761900i	$-0.724267\pi$
$709$	−34.4924	−1.29539	−0.647695	+	0.761900i	$0.724267\pi$
$710$	0	0
$711$	5.26137	0.197317
$712$	0	0
$713$	−7.61553	−0.285204
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	24.3845	0.910655
$718$	0	0
$719$	−53.1771	−1.98317	−0.991585	−	0.129455i	$-0.958677\pi$
$719$	−53.1771	−1.98317	−0.991585	+	0.129455i	$0.958677\pi$
$720$	0	0
$721$	−9.75379	−0.363250
$722$	0	0
$723$	0.384472	0.0142987
$724$	0	0
$725$	2.68466	0.0997057
$726$	0	0
$727$	23.6155	0.875851	0.437926	−	0.899011i	$-0.355713\pi$
$727$	23.6155	0.875851	0.437926	+	0.899011i	$0.355713\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−22.2462	−0.822806
$732$	0	0
$733$	10.8769	0.401747	0.200874	−	0.979617i	$-0.435622\pi$
$733$	10.8769	0.401747	0.200874	+	0.979617i	$0.435622\pi$
$734$	0	0
$735$	7.12311	0.262740
$736$	0	0
$737$	−10.2462	−0.377424
$738$	0	0
$739$	6.73863	0.247885	0.123942	−	0.992289i	$-0.460446\pi$
$739$	6.73863	0.247885	0.123942	+	0.992289i	$0.460446\pi$
$740$	0	0
$741$	4.87689	0.179157
$742$	0	0
$743$	−1.56155	−0.0572878	−0.0286439	−	0.999590i	$-0.509119\pi$
$743$	−1.56155	−0.0572878	−0.0286439	+	0.999590i	$0.509119\pi$
$744$	0	0
$745$	14.6847	0.538004
$746$	0	0
$747$	1.75379	0.0641678
$748$	0	0
$749$	24.9848	0.912926
$750$	0	0
$751$	−29.1771	−1.06469	−0.532343	−	0.846529i	$-0.678688\pi$
$751$	−29.1771	−1.06469	−0.532343	+	0.846529i	$0.678688\pi$
$752$	0	0
$753$	18.1383	0.660995
$754$	0	0
$755$	4.87689	0.177488
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	4.87689	0.177020
$760$	0	0
$761$	12.7386	0.461775	0.230888	−	0.972980i	$-0.425837\pi$
$761$	12.7386	0.461775	0.230888	+	0.972980i	$0.425837\pi$
$762$	0	0
$763$	−22.6307	−0.819286
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	14.2462	0.514401
$768$	0	0
$769$	−24.7386	−0.892098	−0.446049	−	0.895009i	$-0.647169\pi$
$769$	−24.7386	−0.892098	−0.446049	+	0.895009i	$0.647169\pi$
$770$	0	0
$771$	32.3845	1.16630
$772$	0	0
$773$	25.3153	0.910530	0.455265	−	0.890356i	$-0.349545\pi$
$773$	25.3153	0.910530	0.455265	+	0.890356i	$0.349545\pi$
$774$	0	0
$775$	−2.43845	−0.0875916
$776$	0	0
$777$	16.3002	0.584766
$778$	0	0
$779$	−3.12311	−0.111897
$780$	0	0
$781$	8.68466	0.310762
$782$	0	0
$783$	−14.9309	−0.533586
$784$	0	0
$785$	−5.80776	−0.207288
$786$	0	0
$787$	−5.26137	−0.187547	−0.0937737	−	0.995594i	$-0.529893\pi$
$787$	−5.26137	−0.187547	−0.0937737	+	0.995594i	$0.529893\pi$
$788$	0	0
$789$	24.6847	0.878797
$790$	0	0
$791$	16.9848	0.603912
$792$	0	0
$793$	29.3693	1.04294
$794$	0	0
$795$	0.684658	0.0242823
$796$	0	0
$797$	40.2462	1.42559	0.712797	−	0.701370i	$-0.247428\pi$
$797$	40.2462	1.42559	0.712797	+	0.701370i	$0.247428\pi$
$798$	0	0
$799$	17.3693	0.614482
$800$	0	0
$801$	4.73863	0.167431
$802$	0	0
$803$	−2.00000	−0.0705785
$804$	0	0
$805$	4.87689	0.171888
$806$	0	0
$807$	−33.7538	−1.18819
$808$	0	0
$809$	−26.4924	−0.931424	−0.465712	−	0.884936i	$-0.654202\pi$
$809$	−26.4924	−0.931424	−0.465712	+	0.884936i	$0.654202\pi$
$810$	0	0
$811$	47.8078	1.67876	0.839379	−	0.543547i	$-0.182919\pi$
$811$	47.8078	1.67876	0.839379	+	0.543547i	$0.182919\pi$
$812$	0	0
$813$	7.01515	0.246032
$814$	0	0
$815$	−23.8078	−0.833950
$816$	0	0
$817$	9.75379	0.341242
$818$	0	0
$819$	−1.75379	−0.0612823
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	19.5076	0.679991	0.339996	−	0.940427i	$-0.389574\pi$
$823$	19.5076	0.679991	0.339996	+	0.940427i	$0.389574\pi$
$824$	0	0
$825$	1.56155	0.0543663
$826$	0	0
$827$	28.4924	0.990779	0.495389	−	0.868671i	$-0.335025\pi$
$827$	28.4924	0.990779	0.495389	+	0.868671i	$0.335025\pi$
$828$	0	0
$829$	19.3693	0.672724	0.336362	−	0.941733i	$-0.390803\pi$
$829$	19.3693	0.672724	0.336362	+	0.941733i	$0.390803\pi$
$830$	0	0
$831$	1.75379	0.0608383
$832$	0	0
$833$	−16.2462	−0.562898
$834$	0	0
$835$	−6.43845	−0.222812
$836$	0	0
$837$	13.5616	0.468756
$838$	0	0
$839$	−44.4924	−1.53605	−0.768025	−	0.640420i	$-0.778760\pi$
$839$	−44.4924	−1.53605	−0.768025	+	0.640420i	$0.778760\pi$
$840$	0	0
$841$	−21.7926	−0.751469
$842$	0	0
$843$	12.8769	0.443504
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−1.56155	−0.0536556
$848$	0	0
$849$	2.13826	0.0733849
$850$	0	0
$851$	−20.8769	−0.715651
$852$	0	0
$853$	20.2462	0.693217	0.346609	−	0.938010i	$-0.387333\pi$
$853$	20.2462	0.693217	0.346609	+	0.938010i	$0.387333\pi$
$854$	0	0
$855$	−0.876894	−0.0299892
$856$	0	0
$857$	−29.4233	−1.00508	−0.502540	−	0.864554i	$-0.667601\pi$
$857$	−29.4233	−1.00508	−0.502540	+	0.864554i	$0.667601\pi$
$858$	0	0
$859$	−47.1231	−1.60782	−0.803910	−	0.594751i	$-0.797251\pi$
$859$	−47.1231	−1.60782	−0.803910	+	0.594751i	$0.797251\pi$
$860$	0	0
$861$	−4.87689	−0.166204
$862$	0	0
$863$	31.6155	1.07621	0.538103	−	0.842879i	$-0.319141\pi$
$863$	31.6155	1.07621	0.538103	+	0.842879i	$0.319141\pi$
$864$	0	0
$865$	−12.2462	−0.416384
$866$	0	0
$867$	−6.73863	−0.228856
$868$	0	0
$869$	−9.36932	−0.317832
$870$	0	0
$871$	20.4924	0.694359
$872$	0	0
$873$	0.630683	0.0213454
$874$	0	0
$875$	1.56155	0.0527901
$876$	0	0
$877$	15.7538	0.531968	0.265984	−	0.963977i	$-0.414303\pi$
$877$	15.7538	0.531968	0.265984	+	0.963977i	$0.414303\pi$
$878$	0	0
$879$	28.8769	0.973993
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	34.9309	1.17552	0.587759	−	0.809036i	$-0.300010\pi$
$883$	34.9309	1.17552	0.587759	+	0.809036i	$0.300010\pi$
$884$	0	0
$885$	11.1231	0.373899
$886$	0	0
$887$	13.7538	0.461807	0.230904	−	0.972977i	$-0.425832\pi$
$887$	13.7538	0.461807	0.230904	+	0.972977i	$0.425832\pi$
$888$	0	0
$889$	−3.50758	−0.117640
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	−7.61553	−0.254844
$894$	0	0
$895$	−10.2462	−0.342493
$896$	0	0
$897$	−9.75379	−0.325670
$898$	0	0
$899$	−6.54640	−0.218335
$900$	0	0
$901$	−1.56155	−0.0520229
$902$	0	0
$903$	15.2311	0.506858
$904$	0	0
$905$	12.2462	0.407078
$906$	0	0
$907$	46.0540	1.52920	0.764599	−	0.644507i	$-0.222937\pi$
$907$	46.0540	1.52920	0.764599	+	0.644507i	$0.222937\pi$
$908$	0	0
$909$	6.87689	0.228092
$910$	0	0
$911$	30.9309	1.02479	0.512393	−	0.858751i	$-0.328759\pi$
$911$	30.9309	1.02479	0.512393	+	0.858751i	$0.328759\pi$
$912$	0	0
$913$	−3.12311	−0.103360
$914$	0	0
$915$	22.9309	0.758071
$916$	0	0
$917$	−7.31534	−0.241574
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	24.3845	0.803496
$922$	0	0
$923$	−17.3693	−0.571718
$924$	0	0
$925$	−6.68466	−0.219790
$926$	0	0
$927$	−3.50758	−0.115204
$928$	0	0
$929$	7.94602	0.260701	0.130350	−	0.991468i	$-0.458390\pi$
$929$	7.94602	0.260701	0.130350	+	0.991468i	$0.458390\pi$
$930$	0	0
$931$	7.12311	0.233450
$932$	0	0
$933$	46.1619	1.51127
$934$	0	0
$935$	−3.56155	−0.116475
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	−3.89205	−0.127012
$940$	0	0
$941$	32.9309	1.07352	0.536758	−	0.843736i	$-0.319649\pi$
$941$	32.9309	1.07352	0.536758	+	0.843736i	$0.319649\pi$
$942$	0	0
$943$	6.24621	0.203405
$944$	0	0
$945$	−8.68466	−0.282512
$946$	0	0
$947$	−27.3153	−0.887629	−0.443815	−	0.896119i	$-0.646375\pi$
$947$	−27.3153	−0.887629	−0.443815	+	0.896119i	$0.646375\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	−52.7926	−1.71192
$952$	0	0
$953$	11.1771	0.362061	0.181031	−	0.983477i	$-0.442057\pi$
$953$	11.1771	0.362061	0.181031	+	0.983477i	$0.442057\pi$
$954$	0	0
$955$	−6.24621	−0.202123
$956$	0	0
$957$	4.19224	0.135516
$958$	0	0
$959$	33.7538	1.08997
$960$	0	0
$961$	−25.0540	−0.808193
$962$	0	0
$963$	8.98485	0.289533
$964$	0	0
$965$	−25.8078	−0.830781
$966$	0	0
$967$	58.5464	1.88273	0.941363	−	0.337397i	$-0.109546\pi$
$967$	58.5464	1.88273	0.941363	+	0.337397i	$0.109546\pi$
$968$	0	0
$969$	−8.68466	−0.278991
$970$	0	0
$971$	−10.2462	−0.328817	−0.164408	−	0.986392i	$-0.552571\pi$
$971$	−10.2462	−0.328817	−0.164408	+	0.986392i	$0.552571\pi$
$972$	0	0
$973$	−18.7386	−0.600733
$974$	0	0
$975$	−3.12311	−0.100019
$976$	0	0
$977$	−33.1231	−1.05970	−0.529851	−	0.848091i	$-0.677752\pi$
$977$	−33.1231	−1.05970	−0.529851	+	0.848091i	$0.677752\pi$
$978$	0	0
$979$	−8.43845	−0.269694
$980$	0	0
$981$	−8.13826	−0.259835
$982$	0	0
$983$	−27.1231	−0.865093	−0.432546	−	0.901612i	$-0.642385\pi$
$983$	−27.1231	−0.865093	−0.432546	+	0.901612i	$0.642385\pi$
$984$	0	0
$985$	19.3693	0.617158
$986$	0	0
$987$	−11.8920	−0.378528
$988$	0	0
$989$	−19.5076	−0.620305
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	38.2462	1.21371
$994$	0	0
$995$	19.8078	0.627948
$996$	0	0
$997$	43.8617	1.38912	0.694558	−	0.719437i	$-0.255600\pi$
$997$	43.8617	1.38912	0.694558	+	0.719437i	$0.255600\pi$
$998$	0	0
$999$	37.1771	1.17623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.2.a.bk.1.2		2
4.3	odd	2		3520.2.a.bp.1.1		2
8.3	odd	2		440.2.a.e.1.2	✓	2
8.5	even	2		880.2.a.o.1.1		2
24.5	odd	2		7920.2.a.bu.1.1		2
24.11	even	2		3960.2.a.w.1.2		2
40.3	even	4		2200.2.b.i.1849.3		4
40.13	odd	4		4400.2.b.t.4049.2		4
40.19	odd	2		2200.2.a.s.1.1		2
40.27	even	4		2200.2.b.i.1849.2		4
40.29	even	2		4400.2.a.bj.1.2		2
40.37	odd	4		4400.2.b.t.4049.3		4
88.21	odd	2		9680.2.a.bs.1.1		2
88.43	even	2		4840.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.e.1.2	✓	2	8.3	odd	2
880.2.a.o.1.1		2	8.5	even	2
2200.2.a.s.1.1		2	40.19	odd	2
2200.2.b.i.1849.2		4	40.27	even	4
2200.2.b.i.1849.3		4	40.3	even	4
3520.2.a.bk.1.2		2	1.1	even	1	trivial
3520.2.a.bp.1.1		2	4.3	odd	2
3960.2.a.w.1.2		2	24.11	even	2
4400.2.a.bj.1.2		2	40.29	even	2
4400.2.b.t.4049.2		4	40.13	odd	4
4400.2.b.t.4049.3		4	40.37	odd	4
4840.2.a.j.1.2		2	88.43	even	2
7920.2.a.bu.1.1		2	24.5	odd	2
9680.2.a.bs.1.1		2	88.21	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:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} -1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} -2.00000 q^{13} -1.56155 q^{15} +3.56155 q^{17} -1.56155 q^{19} -2.43845 q^{21} +3.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} +2.68466 q^{29} -2.43845 q^{31} +1.56155 q^{33} +1.56155 q^{35} -6.68466 q^{37} -3.12311 q^{39} +2.00000 q^{41} -6.24621 q^{43} +0.561553 q^{45} +4.87689 q^{47} -4.56155 q^{49} +5.56155 q^{51} -0.438447 q^{53} -1.00000 q^{55} -2.43845 q^{57} -7.12311 q^{59} -14.6847 q^{61} +0.876894 q^{63} +2.00000 q^{65} -10.2462 q^{67} +4.87689 q^{69} +8.68466 q^{71} -2.00000 q^{73} +1.56155 q^{75} -1.56155 q^{77} -9.36932 q^{79} -7.00000 q^{81} -3.12311 q^{83} -3.56155 q^{85} +4.19224 q^{87} -8.43845 q^{89} +3.12311 q^{91} -3.80776 q^{93} +1.56155 q^{95} -1.12311 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9	\( 2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} - 4 q^{13} + q^{15} + 3 q^{17} + q^{19} - 9 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 7 q^{29} - 9 q^{31} - q^{33} - q^{35} - q^{37} + 2 q^{39} + 4 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} - 5 q^{49} + 7 q^{51} - 5 q^{53} - 2 q^{55} - 9 q^{57} - 6 q^{59} - 17 q^{61} + 10 q^{63} + 4 q^{65} - 4 q^{67} + 18 q^{69} + 5 q^{71} - 4 q^{73} - q^{75} + q^{77} + 6 q^{79} - 14 q^{81} + 2 q^{83} - 3 q^{85} + 29 q^{87} - 21 q^{89} - 2 q^{91} + 13 q^{93} - q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9 + 2 * q^11 - 4 * q^13 + q^15 + 3 * q^17 + q^19 - 9 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 7 * q^29 - 9 * q^31 - q^33 - q^35 - q^37 + 2 * q^39 + 4 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 - 5 * q^49 + 7 * q^51 - 5 * q^53 - 2 * q^55 - 9 * q^57 - 6 * q^59 - 17 * q^61 + 10 * q^63 + 4 * q^65 - 4 * q^67 + 18 * q^69 + 5 * q^71 - 4 * q^73 - q^75 + q^77 + 6 * q^79 - 14 * q^81 + 2 * q^83 - 3 * q^85 + 29 * q^87 - 21 * q^89 - 2 * q^91 + 13 * q^93 - q^95 + 6 * q^97 + 3 * q^99

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