LMFDB - Embedded newform 3720.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3720,2,Mod(1,3720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.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:	$29.7043495519$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.78292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 8x + 18 $ x^4 - x^3 - 10x^2 + 8x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.68461$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +2.68461 q^{7} +1.00000 q^{9} +3.24072 q^{11} -4.97040 q^{13} +1.00000 q^{15} +0.921359 q^{17} +1.24072 q^{19} -2.68461 q^{21} +6.57637 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.89176 q^{29} +1.00000 q^{31} -3.24072 q^{33} -2.68461 q^{35} +2.97040 q^{37} +4.97040 q^{39} +4.00000 q^{41} +5.65501 q^{43} -1.00000 q^{45} -8.86216 q^{47} +0.207146 q^{49} -0.921359 q^{51} +0.792854 q^{53} -3.24072 q^{55} -1.24072 q^{57} -7.00397 q^{59} +10.8957 q^{61} +2.68461 q^{63} +4.97040 q^{65} -12.3396 q^{67} -6.57637 q^{69} +15.5468 q^{71} +7.09891 q^{73} -1.00000 q^{75} +8.70008 q^{77} -1.04986 q^{79} +1.00000 q^{81} -16.3887 q^{83} -0.921359 q^{85} +3.89176 q^{87} -3.13248 q^{89} -13.3436 q^{91} -1.00000 q^{93} -1.24072 q^{95} -4.57157 q^{97} +3.24072 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.68461	1.01469	0.507344	−	0.861744i	$-0.330627\pi$
$7$	2.68461	1.01469	0.507344	+	0.861744i	$0.330627\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.24072	0.977114	0.488557	−	0.872532i	$-0.337523\pi$
$11$	3.24072	0.977114	0.488557	+	0.872532i	$0.337523\pi$
$12$	0	0
$13$	−4.97040	−1.37854	−0.689270	−	0.724504i	$-0.742069\pi$
$13$	−4.97040	−1.37854	−0.689270	+	0.724504i	$0.742069\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.921359	0.223462	0.111731	−	0.993738i	$-0.464360\pi$
$17$	0.921359	0.223462	0.111731	+	0.993738i	$0.464360\pi$
$18$	0	0
$19$	1.24072	0.284641	0.142320	−	0.989821i	$-0.454544\pi$
$19$	1.24072	0.284641	0.142320	+	0.989821i	$0.454544\pi$
$20$	0	0
$21$	−2.68461	−0.585831
$22$	0	0
$23$	6.57637	1.37127	0.685634	−	0.727946i	$-0.259525\pi$
$23$	6.57637	1.37127	0.685634	+	0.727946i	$0.259525\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.89176	−0.722682	−0.361341	−	0.932434i	$-0.617681\pi$
$29$	−3.89176	−0.722682	−0.361341	+	0.932434i	$0.617681\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−3.24072	−0.564137
$34$	0	0
$35$	−2.68461	−0.453782
$36$	0	0
$37$	2.97040	0.488331	0.244165	−	0.969734i	$-0.421486\pi$
$37$	2.97040	0.488331	0.244165	+	0.969734i	$0.421486\pi$
$38$	0	0
$39$	4.97040	0.795901
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	5.65501	0.862381	0.431191	−	0.902261i	$-0.358094\pi$
$43$	5.65501	0.862381	0.431191	+	0.902261i	$0.358094\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.86216	−1.29268	−0.646339	−	0.763050i	$-0.723701\pi$
$47$	−8.86216	−1.29268	−0.646339	+	0.763050i	$0.723701\pi$
$48$	0	0
$49$	0.207146	0.0295923
$50$	0	0
$51$	−0.921359	−0.129016
$52$	0	0
$53$	0.792854	0.108907	0.0544534	−	0.998516i	$-0.482658\pi$
$53$	0.792854	0.108907	0.0544534	+	0.998516i	$0.482658\pi$
$54$	0	0
$55$	−3.24072	−0.436979
$56$	0	0
$57$	−1.24072	−0.164337
$58$	0	0
$59$	−7.00397	−0.911840	−0.455920	−	0.890021i	$-0.650690\pi$
$59$	−7.00397	−0.911840	−0.455920	+	0.890021i	$0.650690\pi$
$60$	0	0
$61$	10.8957	1.39506	0.697528	−	0.716558i	$-0.254283\pi$
$61$	10.8957	1.39506	0.697528	+	0.716558i	$0.254283\pi$
$62$	0	0
$63$	2.68461	0.338229
$64$	0	0
$65$	4.97040	0.616502
$66$	0	0
$67$	−12.3396	−1.50753	−0.753763	−	0.657147i	$-0.771763\pi$
$67$	−12.3396	−1.50753	−0.753763	+	0.657147i	$0.771763\pi$
$68$	0	0
$69$	−6.57637	−0.791702
$70$	0	0
$71$	15.5468	1.84506	0.922531	−	0.385923i	$-0.126117\pi$
$71$	15.5468	1.84506	0.922531	+	0.385923i	$0.126117\pi$
$72$	0	0
$73$	7.09891	0.830864	0.415432	−	0.909624i	$-0.363630\pi$
$73$	7.09891	0.830864	0.415432	+	0.909624i	$0.363630\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	8.70008	0.991466
$78$	0	0
$79$	−1.04986	−0.118119	−0.0590595	−	0.998254i	$-0.518810\pi$
$79$	−1.04986	−0.118119	−0.0590595	+	0.998254i	$0.518810\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.3887	−1.79889	−0.899445	−	0.437034i	$-0.856029\pi$
$83$	−16.3887	−1.79889	−0.899445	+	0.437034i	$0.856029\pi$
$84$	0	0
$85$	−0.921359	−0.0999354
$86$	0	0
$87$	3.89176	0.417240
$88$	0	0
$89$	−3.13248	−0.332042	−0.166021	−	0.986122i	$-0.553092\pi$
$89$	−3.13248	−0.332042	−0.166021	+	0.986122i	$0.553092\pi$
$90$	0	0
$91$	−13.3436	−1.39879
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−1.24072	−0.127295
$96$	0	0
$97$	−4.57157	−0.464173	−0.232087	−	0.972695i	$-0.574555\pi$
$97$	−4.57157	−0.464173	−0.232087	+	0.972695i	$0.574555\pi$
$98$	0	0
$99$	3.24072	0.325705
$100$	0	0
$101$	17.5649	1.74777	0.873885	−	0.486132i	$-0.161593\pi$
$101$	17.5649	1.74777	0.873885	+	0.486132i	$0.161593\pi$
$102$	0	0
$103$	10.7539	1.05962	0.529808	−	0.848118i	$-0.322264\pi$
$103$	10.7539	1.05962	0.529808	+	0.848118i	$0.322264\pi$
$104$	0	0
$105$	2.68461	0.261991
$106$	0	0
$107$	11.4721	1.10905	0.554525	−	0.832167i	$-0.312900\pi$
$107$	11.4721	1.10905	0.554525	+	0.832167i	$0.312900\pi$
$108$	0	0
$109$	−5.33565	−0.511063	−0.255531	−	0.966801i	$-0.582250\pi$
$109$	−5.33565	−0.511063	−0.255531	+	0.966801i	$0.582250\pi$
$110$	0	0
$111$	−2.97040	−0.281938
$112$	0	0
$113$	7.71421	0.725692	0.362846	−	0.931849i	$-0.381805\pi$
$113$	7.71421	0.725692	0.362846	+	0.931849i	$0.381805\pi$
$114$	0	0
$115$	−6.57637	−0.613250
$116$	0	0
$117$	−4.97040	−0.459514
$118$	0	0
$119$	2.47349	0.226745
$120$	0	0
$121$	−0.497731	−0.0452483
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.7694	−1.31057	−0.655286	−	0.755381i	$-0.727452\pi$
$127$	−14.7694	−1.31057	−0.655286	+	0.755381i	$0.727452\pi$
$128$	0	0
$129$	−5.65501	−0.497896
$130$	0	0
$131$	14.7203	1.28612	0.643061	−	0.765815i	$-0.277664\pi$
$131$	14.7203	1.28612	0.643061	+	0.765815i	$0.277664\pi$
$132$	0	0
$133$	3.33085	0.288822
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	11.6502	0.995345	0.497673	−	0.867365i	$-0.334188\pi$
$137$	11.6502	0.995345	0.497673	+	0.867365i	$0.334188\pi$
$138$	0	0
$139$	−9.36923	−0.794687	−0.397344	−	0.917670i	$-0.630068\pi$
$139$	−9.36923	−0.794687	−0.397344	+	0.917670i	$0.630068\pi$
$140$	0	0
$141$	8.86216	0.746328
$142$	0	0
$143$	−16.1077	−1.34699
$144$	0	0
$145$	3.89176	0.323193
$146$	0	0
$147$	−0.207146	−0.0170851
$148$	0	0
$149$	13.2407	1.08472	0.542361	−	0.840146i	$-0.317530\pi$
$149$	13.2407	1.08472	0.542361	+	0.840146i	$0.317530\pi$
$150$	0	0
$151$	17.8171	1.44993	0.724967	−	0.688783i	$-0.241855\pi$
$151$	17.8171	1.44993	0.724967	+	0.688783i	$0.241855\pi$
$152$	0	0
$153$	0.921359	0.0744875
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−7.91202	−0.631448	−0.315724	−	0.948851i	$-0.602247\pi$
$157$	−7.91202	−0.631448	−0.315724	+	0.948851i	$0.602247\pi$
$158$	0	0
$159$	−0.792854	−0.0628774
$160$	0	0
$161$	17.6550	1.39141
$162$	0	0
$163$	−15.3255	−1.20039	−0.600193	−	0.799855i	$-0.704909\pi$
$163$	−15.3255	−1.20039	−0.600193	+	0.799855i	$0.704909\pi$
$164$	0	0
$165$	3.24072	0.252290
$166$	0	0
$167$	12.1957	0.943728	0.471864	−	0.881671i	$-0.343581\pi$
$167$	12.1957	0.943728	0.471864	+	0.881671i	$0.343581\pi$
$168$	0	0
$169$	11.7049	0.900375
$170$	0	0
$171$	1.24072	0.0948803
$172$	0	0
$173$	8.95493	0.680831	0.340415	−	0.940275i	$-0.389432\pi$
$173$	8.95493	0.680831	0.340415	+	0.940275i	$0.389432\pi$
$174$	0	0
$175$	2.68461	0.202938
$176$	0	0
$177$	7.00397	0.526451
$178$	0	0
$179$	6.67130	0.498637	0.249318	−	0.968422i	$-0.419793\pi$
$179$	6.67130	0.498637	0.249318	+	0.968422i	$0.419793\pi$
$180$	0	0
$181$	−12.9650	−0.963684	−0.481842	−	0.876258i	$-0.660032\pi$
$181$	−12.9650	−0.963684	−0.481842	+	0.876258i	$0.660032\pi$
$182$	0	0
$183$	−10.8957	−0.805436
$184$	0	0
$185$	−2.97040	−0.218388
$186$	0	0
$187$	2.98587	0.218348
$188$	0	0
$189$	−2.68461	−0.195277
$190$	0	0
$191$	15.5755	1.12701	0.563504	−	0.826114i	$-0.309453\pi$
$191$	15.5755	1.12701	0.563504	+	0.826114i	$0.309453\pi$
$192$	0	0
$193$	6.41429	0.461711	0.230855	−	0.972988i	$-0.425848\pi$
$193$	6.41429	0.461711	0.230855	+	0.972988i	$0.425848\pi$
$194$	0	0
$195$	−4.97040	−0.355938
$196$	0	0
$197$	8.66435	0.617309	0.308655	−	0.951174i	$-0.400121\pi$
$197$	8.66435	0.617309	0.308655	+	0.951174i	$0.400121\pi$
$198$	0	0
$199$	−6.16208	−0.436818	−0.218409	−	0.975857i	$-0.570087\pi$
$199$	−6.16208	−0.436818	−0.218409	+	0.975857i	$0.570087\pi$
$200$	0	0
$201$	12.3396	0.870370
$202$	0	0
$203$	−10.4479	−0.733296
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	6.57637	0.457089
$208$	0	0
$209$	4.02083	0.278126
$210$	0	0
$211$	−0.767228	−0.0528182	−0.0264091	−	0.999651i	$-0.508407\pi$
$211$	−0.767228	−0.0528182	−0.0264091	+	0.999651i	$0.508407\pi$
$212$	0	0
$213$	−15.5468	−1.06525
$214$	0	0
$215$	−5.65501	−0.385669
$216$	0	0
$217$	2.68461	0.182243
$218$	0	0
$219$	−7.09891	−0.479700
$220$	0	0
$221$	−4.57952	−0.308052
$222$	0	0
$223$	6.09808	0.408358	0.204179	−	0.978934i	$-0.434548\pi$
$223$	6.09808	0.408358	0.204179	+	0.978934i	$0.434548\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−1.93765	−0.128606	−0.0643031	−	0.997930i	$-0.520482\pi$
$227$	−1.93765	−0.128606	−0.0643031	+	0.997930i	$0.520482\pi$
$228$	0	0
$229$	21.1911	1.40035	0.700174	−	0.713972i	$-0.253106\pi$
$229$	21.1911	1.40035	0.700174	+	0.713972i	$0.253106\pi$
$230$	0	0
$231$	−8.70008	−0.572423
$232$	0	0
$233$	22.6515	1.48395	0.741974	−	0.670429i	$-0.233890\pi$
$233$	22.6515	1.48395	0.741974	+	0.670429i	$0.233890\pi$
$234$	0	0
$235$	8.86216	0.578104
$236$	0	0
$237$	1.04986	0.0681960
$238$	0	0
$239$	22.3551	1.44603	0.723015	−	0.690832i	$-0.242756\pi$
$239$	22.3551	1.44603	0.723015	+	0.690832i	$0.242756\pi$
$240$	0	0
$241$	6.79765	0.437875	0.218938	−	0.975739i	$-0.429741\pi$
$241$	6.79765	0.437875	0.218938	+	0.975739i	$0.429741\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.207146	−0.0132341
$246$	0	0
$247$	−6.16688	−0.392389
$248$	0	0
$249$	16.3887	1.03859
$250$	0	0
$251$	8.06715	0.509194	0.254597	−	0.967047i	$-0.418057\pi$
$251$	8.06715	0.509194	0.254597	+	0.967047i	$0.418057\pi$
$252$	0	0
$253$	21.3122	1.33989
$254$	0	0
$255$	0.921359	0.0576977
$256$	0	0
$257$	9.04986	0.564515	0.282258	−	0.959339i	$-0.408917\pi$
$257$	9.04986	0.564515	0.282258	+	0.959339i	$0.408917\pi$
$258$	0	0
$259$	7.97437	0.495504
$260$	0	0
$261$	−3.89176	−0.240894
$262$	0	0
$263$	14.9037	0.919000	0.459500	−	0.888178i	$-0.348029\pi$
$263$	14.9037	0.919000	0.459500	+	0.888178i	$0.348029\pi$
$264$	0	0
$265$	−0.792854	−0.0487046
$266$	0	0
$267$	3.13248	0.191705
$268$	0	0
$269$	13.5084	0.823622	0.411811	−	0.911269i	$-0.364896\pi$
$269$	13.5084	0.823622	0.411811	+	0.911269i	$0.364896\pi$
$270$	0	0
$271$	5.90408	0.358647	0.179324	−	0.983790i	$-0.442609\pi$
$271$	5.90408	0.358647	0.179324	+	0.983790i	$0.442609\pi$
$272$	0	0
$273$	13.3436	0.807591
$274$	0	0
$275$	3.24072	0.195423
$276$	0	0
$277$	10.6868	0.642106	0.321053	−	0.947061i	$-0.395963\pi$
$277$	10.6868	0.642106	0.321053	+	0.947061i	$0.395963\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	15.6838	0.935616	0.467808	−	0.883830i	$-0.345044\pi$
$281$	15.6838	0.935616	0.467808	+	0.883830i	$0.345044\pi$
$282$	0	0
$283$	4.24949	0.252606	0.126303	−	0.991992i	$-0.459689\pi$
$283$	4.24949	0.252606	0.126303	+	0.991992i	$0.459689\pi$
$284$	0	0
$285$	1.24072	0.0734939
$286$	0	0
$287$	10.7385	0.633871
$288$	0	0
$289$	−16.1511	−0.950065
$290$	0	0
$291$	4.57157	0.267990
$292$	0	0
$293$	7.11917	0.415906	0.207953	−	0.978139i	$-0.433320\pi$
$293$	7.11917	0.415906	0.207953	+	0.978139i	$0.433320\pi$
$294$	0	0
$295$	7.00397	0.407787
$296$	0	0
$297$	−3.24072	−0.188046
$298$	0	0
$299$	−32.6872	−1.89035
$300$	0	0
$301$	15.1815	0.875048
$302$	0	0
$303$	−17.5649	−1.00908
$304$	0	0
$305$	−10.8957	−0.623888
$306$	0	0
$307$	28.8616	1.64722	0.823609	−	0.567158i	$-0.191957\pi$
$307$	28.8616	1.64722	0.823609	+	0.567158i	$0.191957\pi$
$308$	0	0
$309$	−10.7539	−0.611769
$310$	0	0
$311$	−18.5118	−1.04971	−0.524854	−	0.851192i	$-0.675880\pi$
$311$	−18.5118	−1.04971	−0.524854	+	0.851192i	$0.675880\pi$
$312$	0	0
$313$	−4.17275	−0.235858	−0.117929	−	0.993022i	$-0.537625\pi$
$313$	−4.17275	−0.235858	−0.117929	+	0.993022i	$0.537625\pi$
$314$	0	0
$315$	−2.68461	−0.151261
$316$	0	0
$317$	−11.8250	−0.664160	−0.332080	−	0.943251i	$-0.607750\pi$
$317$	−11.8250	−0.664160	−0.332080	+	0.943251i	$0.607750\pi$
$318$	0	0
$319$	−12.6121	−0.706142
$320$	0	0
$321$	−11.4721	−0.640310
$322$	0	0
$323$	1.14315	0.0636065
$324$	0	0
$325$	−4.97040	−0.275708
$326$	0	0
$327$	5.33565	0.295062
$328$	0	0
$329$	−23.7915	−1.31167
$330$	0	0
$331$	3.80914	0.209369	0.104685	−	0.994505i	$-0.466617\pi$
$331$	3.80914	0.209369	0.104685	+	0.994505i	$0.466617\pi$
$332$	0	0
$333$	2.97040	0.162777
$334$	0	0
$335$	12.3396	0.674186
$336$	0	0
$337$	−17.1682	−0.935212	−0.467606	−	0.883937i	$-0.654883\pi$
$337$	−17.1682	−0.935212	−0.467606	+	0.883937i	$0.654883\pi$
$338$	0	0
$339$	−7.71421	−0.418979
$340$	0	0
$341$	3.24072	0.175495
$342$	0	0
$343$	−18.2362	−0.984661
$344$	0	0
$345$	6.57637	0.354060
$346$	0	0
$347$	−18.0079	−0.966717	−0.483359	−	0.875422i	$-0.660583\pi$
$347$	−18.0079	−0.966717	−0.483359	+	0.875422i	$0.660583\pi$
$348$	0	0
$349$	27.1271	1.45208	0.726041	−	0.687652i	$-0.241358\pi$
$349$	27.1271	1.45208	0.726041	+	0.687652i	$0.241358\pi$
$350$	0	0
$351$	4.97040	0.265300
$352$	0	0
$353$	15.4028	0.819808	0.409904	−	0.912129i	$-0.365562\pi$
$353$	15.4028	0.819808	0.409904	+	0.912129i	$0.365562\pi$
$354$	0	0
$355$	−15.5468	−0.825137
$356$	0	0
$357$	−2.47349	−0.130911
$358$	0	0
$359$	27.8789	1.47139	0.735695	−	0.677312i	$-0.236855\pi$
$359$	27.8789	1.47139	0.735695	+	0.677312i	$0.236855\pi$
$360$	0	0
$361$	−17.4606	−0.918980
$362$	0	0
$363$	0.497731	0.0261241
$364$	0	0
$365$	−7.09891	−0.371574
$366$	0	0
$367$	−24.2729	−1.26704	−0.633518	−	0.773728i	$-0.718390\pi$
$367$	−24.2729	−1.26704	−0.633518	+	0.773728i	$0.718390\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	2.12851	0.110506
$372$	0	0
$373$	−17.3714	−0.899456	−0.449728	−	0.893166i	$-0.648479\pi$
$373$	−17.3714	−0.899456	−0.449728	+	0.893166i	$0.648479\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	19.3436	0.996246
$378$	0	0
$379$	0.236183	0.0121319	0.00606594	−	0.999982i	$-0.498069\pi$
$379$	0.236183	0.0121319	0.00606594	+	0.999982i	$0.498069\pi$
$380$	0	0
$381$	14.7694	0.756659
$382$	0	0
$383$	−34.8126	−1.77884	−0.889419	−	0.457092i	$-0.848891\pi$
$383$	−34.8126	−1.77884	−0.889419	+	0.457092i	$0.848891\pi$
$384$	0	0
$385$	−8.70008	−0.443397
$386$	0	0
$387$	5.65501	0.287460
$388$	0	0
$389$	16.7795	0.850757	0.425378	−	0.905016i	$-0.360141\pi$
$389$	16.7795	0.850757	0.425378	+	0.905016i	$0.360141\pi$
$390$	0	0
$391$	6.05920	0.306427
$392$	0	0
$393$	−14.7203	−0.742543
$394$	0	0
$395$	1.04986	0.0528244
$396$	0	0
$397$	−7.56488	−0.379670	−0.189835	−	0.981816i	$-0.560795\pi$
$397$	−7.56488	−0.379670	−0.189835	+	0.981816i	$0.560795\pi$
$398$	0	0
$399$	−3.33085	−0.166751
$400$	0	0
$401$	−12.1775	−0.608118	−0.304059	−	0.952653i	$-0.598342\pi$
$401$	−12.1775	−0.608118	−0.304059	+	0.952653i	$0.598342\pi$
$402$	0	0
$403$	−4.97040	−0.247593
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	9.62624	0.477155
$408$	0	0
$409$	−8.09973	−0.400506	−0.200253	−	0.979744i	$-0.564176\pi$
$409$	−8.09973	−0.400506	−0.200253	+	0.979744i	$0.564176\pi$
$410$	0	0
$411$	−11.6502	−0.574663
$412$	0	0
$413$	−18.8030	−0.925233
$414$	0	0
$415$	16.3887	0.804488
$416$	0	0
$417$	9.36923	0.458813
$418$	0	0
$419$	−35.4588	−1.73228	−0.866138	−	0.499805i	$-0.833405\pi$
$419$	−35.4588	−1.73228	−0.866138	+	0.499805i	$0.833405\pi$
$420$	0	0
$421$	−8.86216	−0.431915	−0.215958	−	0.976403i	$-0.569287\pi$
$421$	−8.86216	−0.431915	−0.215958	+	0.976403i	$0.569287\pi$
$422$	0	0
$423$	−8.86216	−0.430893
$424$	0	0
$425$	0.921359	0.0446925
$426$	0	0
$427$	29.2508	1.41555
$428$	0	0
$429$	16.1077	0.777686
$430$	0	0
$431$	−25.2503	−1.21626	−0.608131	−	0.793837i	$-0.708081\pi$
$431$	−25.2503	−1.21626	−0.608131	+	0.793837i	$0.708081\pi$
$432$	0	0
$433$	−13.9867	−0.672157	−0.336079	−	0.941834i	$-0.609101\pi$
$433$	−13.9867	−0.672157	−0.336079	+	0.941834i	$0.609101\pi$
$434$	0	0
$435$	−3.89176	−0.186596
$436$	0	0
$437$	8.15944	0.390319
$438$	0	0
$439$	−7.74299	−0.369553	−0.184776	−	0.982781i	$-0.559156\pi$
$439$	−7.74299	−0.369553	−0.184776	+	0.982781i	$0.559156\pi$
$440$	0	0
$441$	0.207146	0.00986411
$442$	0	0
$443$	−1.09947	−0.0522374	−0.0261187	−	0.999659i	$-0.508315\pi$
$443$	−1.09947	−0.0522374	−0.0261187	+	0.999659i	$0.508315\pi$
$444$	0	0
$445$	3.13248	0.148494
$446$	0	0
$447$	−13.2407	−0.626265
$448$	0	0
$449$	−3.58514	−0.169193	−0.0845967	−	0.996415i	$-0.526960\pi$
$449$	−3.58514	−0.169193	−0.0845967	+	0.996415i	$0.526960\pi$
$450$	0	0
$451$	12.9629	0.610398
$452$	0	0
$453$	−17.8171	−0.837120
$454$	0	0
$455$	13.3436	0.625558
$456$	0	0
$457$	32.2884	1.51039	0.755193	−	0.655502i	$-0.227543\pi$
$457$	32.2884	1.51039	0.755193	+	0.655502i	$0.227543\pi$
$458$	0	0
$459$	−0.921359	−0.0430054
$460$	0	0
$461$	−20.3236	−0.946564	−0.473282	−	0.880911i	$-0.656931\pi$
$461$	−20.3236	−0.946564	−0.473282	+	0.880911i	$0.656931\pi$
$462$	0	0
$463$	0.0325804	0.00151414	0.000757069	−	1.00000i	$-0.499759\pi$
$463$	0.0325804	0.00151414	0.000757069		1.00000i	$0.499759\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−30.0149	−1.38892	−0.694462	−	0.719529i	$-0.744358\pi$
$467$	−30.0149	−1.38892	−0.694462	+	0.719529i	$0.744358\pi$
$468$	0	0
$469$	−33.1271	−1.52967
$470$	0	0
$471$	7.91202	0.364567
$472$	0	0
$473$	18.3263	0.842645
$474$	0	0
$475$	1.24072	0.0569282
$476$	0	0
$477$	0.792854	0.0363023
$478$	0	0
$479$	−1.21466	−0.0554994	−0.0277497	−	0.999615i	$-0.508834\pi$
$479$	−1.21466	−0.0554994	−0.0277497	+	0.999615i	$0.508834\pi$
$480$	0	0
$481$	−14.7641	−0.673184
$482$	0	0
$483$	−17.6550	−0.803331
$484$	0	0
$485$	4.57157	0.207585
$486$	0	0
$487$	32.7464	1.48388	0.741941	−	0.670466i	$-0.233906\pi$
$487$	32.7464	1.48388	0.741941	+	0.670466i	$0.233906\pi$
$488$	0	0
$489$	15.3255	0.693043
$490$	0	0
$491$	23.6859	1.06893	0.534466	−	0.845190i	$-0.320513\pi$
$491$	23.6859	1.06893	0.534466	+	0.845190i	$0.320513\pi$
$492$	0	0
$493$	−3.58571	−0.161492
$494$	0	0
$495$	−3.24072	−0.145660
$496$	0	0
$497$	41.7371	1.87216
$498$	0	0
$499$	41.2401	1.84616	0.923080	−	0.384607i	$-0.125663\pi$
$499$	41.2401	1.84616	0.923080	+	0.384607i	$0.125663\pi$
$500$	0	0
$501$	−12.1957	−0.544862
$502$	0	0
$503$	−18.6953	−0.833581	−0.416791	−	0.909002i	$-0.636845\pi$
$503$	−18.6953	−0.833581	−0.416791	+	0.909002i	$0.636845\pi$
$504$	0	0
$505$	−17.5649	−0.781627
$506$	0	0
$507$	−11.7049	−0.519832
$508$	0	0
$509$	24.9544	1.10608	0.553041	−	0.833154i	$-0.313467\pi$
$509$	24.9544	1.10608	0.553041	+	0.833154i	$0.313467\pi$
$510$	0	0
$511$	19.0578	0.843068
$512$	0	0
$513$	−1.24072	−0.0547791
$514$	0	0
$515$	−10.7539	−0.473874
$516$	0	0
$517$	−28.7198	−1.26309
$518$	0	0
$519$	−8.95493	−0.393078
$520$	0	0
$521$	−17.2199	−0.754417	−0.377209	−	0.926128i	$-0.623116\pi$
$521$	−17.2199	−0.754417	−0.377209	+	0.926128i	$0.623116\pi$
$522$	0	0
$523$	25.0401	1.09493	0.547464	−	0.836829i	$-0.315593\pi$
$523$	25.0401	1.09493	0.547464	+	0.836829i	$0.315593\pi$
$524$	0	0
$525$	−2.68461	−0.117166
$526$	0	0
$527$	0.921359	0.0401350
$528$	0	0
$529$	20.2487	0.880377
$530$	0	0
$531$	−7.00397	−0.303947
$532$	0	0
$533$	−19.8816	−0.861168
$534$	0	0
$535$	−11.4721	−0.495982
$536$	0	0
$537$	−6.67130	−0.287888
$538$	0	0
$539$	0.671303	0.0289151
$540$	0	0
$541$	−18.6377	−0.801299	−0.400649	−	0.916231i	$-0.631215\pi$
$541$	−18.6377	−0.801299	−0.400649	+	0.916231i	$0.631215\pi$
$542$	0	0
$543$	12.9650	0.556383
$544$	0	0
$545$	5.33565	0.228554
$546$	0	0
$547$	−45.7653	−1.95678	−0.978391	−	0.206763i	$-0.933707\pi$
$547$	−45.7653	−1.95678	−0.978391	+	0.206763i	$0.933707\pi$
$548$	0	0
$549$	10.8957	0.465019
$550$	0	0
$551$	−4.82859	−0.205705
$552$	0	0
$553$	−2.81848	−0.119854
$554$	0	0
$555$	2.97040	0.126086
$556$	0	0
$557$	−25.0967	−1.06338	−0.531691	−	0.846939i	$-0.678443\pi$
$557$	−25.0967	−1.06338	−0.531691	+	0.846939i	$0.678443\pi$
$558$	0	0
$559$	−28.1077	−1.18883
$560$	0	0
$561$	−2.98587	−0.126063
$562$	0	0
$563$	−41.1041	−1.73233	−0.866166	−	0.499756i	$-0.833423\pi$
$563$	−41.1041	−1.73233	−0.866166	+	0.499756i	$0.833423\pi$
$564$	0	0
$565$	−7.71421	−0.324539
$566$	0	0
$567$	2.68461	0.112743
$568$	0	0
$569$	31.1714	1.30677	0.653386	−	0.757025i	$-0.273348\pi$
$569$	31.1714	1.30677	0.653386	+	0.757025i	$0.273348\pi$
$570$	0	0
$571$	2.63077	0.110094	0.0550472	−	0.998484i	$-0.482469\pi$
$571$	2.63077	0.110094	0.0550472	+	0.998484i	$0.482469\pi$
$572$	0	0
$573$	−15.5755	−0.650678
$574$	0	0
$575$	6.57637	0.274254
$576$	0	0
$577$	−34.7694	−1.44747	−0.723734	−	0.690079i	$-0.757576\pi$
$577$	−34.7694	−1.44747	−0.723734	+	0.690079i	$0.757576\pi$
$578$	0	0
$579$	−6.41429	−0.266569
$580$	0	0
$581$	−43.9972	−1.82531
$582$	0	0
$583$	2.56942	0.106414
$584$	0	0
$585$	4.97040	0.205501
$586$	0	0
$587$	1.11222	0.0459060	0.0229530	−	0.999737i	$-0.492693\pi$
$587$	1.11222	0.0459060	0.0229530	+	0.999737i	$0.492693\pi$
$588$	0	0
$589$	1.24072	0.0511230
$590$	0	0
$591$	−8.66435	−0.356404
$592$	0	0
$593$	32.6510	1.34082	0.670408	−	0.741993i	$-0.266119\pi$
$593$	32.6510	1.34082	0.670408	+	0.741993i	$0.266119\pi$
$594$	0	0
$595$	−2.47349	−0.101403
$596$	0	0
$597$	6.16208	0.252197
$598$	0	0
$599$	−16.6899	−0.681932	−0.340966	−	0.940076i	$-0.610754\pi$
$599$	−16.6899	−0.681932	−0.340966	+	0.940076i	$0.610754\pi$
$600$	0	0
$601$	16.9037	0.689516	0.344758	−	0.938692i	$-0.387961\pi$
$601$	16.9037	0.689516	0.344758	+	0.938692i	$0.387961\pi$
$602$	0	0
$603$	−12.3396	−0.502508
$604$	0	0
$605$	0.497731	0.0202356
$606$	0	0
$607$	−12.5353	−0.508791	−0.254396	−	0.967100i	$-0.581877\pi$
$607$	−12.5353	−0.508791	−0.254396	+	0.967100i	$0.581877\pi$
$608$	0	0
$609$	10.4479	0.423369
$610$	0	0
$611$	44.0485	1.78201
$612$	0	0
$613$	−25.7681	−1.04076	−0.520381	−	0.853934i	$-0.674210\pi$
$613$	−25.7681	−1.04076	−0.520381	+	0.853934i	$0.674210\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	−1.84488	−0.0742719	−0.0371359	−	0.999310i	$-0.511823\pi$
$617$	−1.84488	−0.0742719	−0.0371359	+	0.999310i	$0.511823\pi$
$618$	0	0
$619$	−35.4337	−1.42420	−0.712101	−	0.702077i	$-0.752256\pi$
$619$	−35.4337	−1.42420	−0.712101	+	0.702077i	$0.752256\pi$
$620$	0	0
$621$	−6.57637	−0.263901
$622$	0	0
$623$	−8.40950	−0.336919
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.02083	−0.160576
$628$	0	0
$629$	2.73680	0.109124
$630$	0	0
$631$	38.7176	1.54132	0.770662	−	0.637244i	$-0.219926\pi$
$631$	38.7176	1.54132	0.770662	+	0.637244i	$0.219926\pi$
$632$	0	0
$633$	0.767228	0.0304946
$634$	0	0
$635$	14.7694	0.586105
$636$	0	0
$637$	−1.02960	−0.0407942
$638$	0	0
$639$	15.5468	0.615021
$640$	0	0
$641$	26.8201	1.05933	0.529665	−	0.848207i	$-0.322318\pi$
$641$	26.8201	1.05933	0.529665	+	0.848207i	$0.322318\pi$
$642$	0	0
$643$	−7.11437	−0.280563	−0.140282	−	0.990112i	$-0.544801\pi$
$643$	−7.11437	−0.280563	−0.140282	+	0.990112i	$0.544801\pi$
$644$	0	0
$645$	5.65501	0.222666
$646$	0	0
$647$	−1.30787	−0.0514176	−0.0257088	−	0.999669i	$-0.508184\pi$
$647$	−1.30787	−0.0514176	−0.0257088	+	0.999669i	$0.508184\pi$
$648$	0	0
$649$	−22.6979	−0.890971
$650$	0	0
$651$	−2.68461	−0.105218
$652$	0	0
$653$	4.44787	0.174058	0.0870292	−	0.996206i	$-0.472263\pi$
$653$	4.44787	0.174058	0.0870292	+	0.996206i	$0.472263\pi$
$654$	0	0
$655$	−14.7203	−0.575171
$656$	0	0
$657$	7.09891	0.276955
$658$	0	0
$659$	−0.653196	−0.0254449	−0.0127225	−	0.999919i	$-0.504050\pi$
$659$	−0.653196	−0.0254449	−0.0127225	+	0.999919i	$0.504050\pi$
$660$	0	0
$661$	47.3223	1.84062	0.920312	−	0.391185i	$-0.127935\pi$
$661$	47.3223	1.84062	0.920312	+	0.391185i	$0.127935\pi$
$662$	0	0
$663$	4.57952	0.177854
$664$	0	0
$665$	−3.33085	−0.129165
$666$	0	0
$667$	−25.5937	−0.990990
$668$	0	0
$669$	−6.09808	−0.235765
$670$	0	0
$671$	35.3100	1.36313
$672$	0	0
$673$	−7.95627	−0.306692	−0.153346	−	0.988173i	$-0.549005\pi$
$673$	−7.95627	−0.306692	−0.153346	+	0.988173i	$0.549005\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	4.47829	0.172115	0.0860573	−	0.996290i	$-0.472573\pi$
$677$	4.47829	0.172115	0.0860573	+	0.996290i	$0.472573\pi$
$678$	0	0
$679$	−12.2729	−0.470991
$680$	0	0
$681$	1.93765	0.0742509
$682$	0	0
$683$	−50.8493	−1.94569	−0.972847	−	0.231450i	$-0.925653\pi$
$683$	−50.8493	−1.94569	−0.972847	+	0.231450i	$0.925653\pi$
$684$	0	0
$685$	−11.6502	−0.445132
$686$	0	0
$687$	−21.1911	−0.808491
$688$	0	0
$689$	−3.94080	−0.150133
$690$	0	0
$691$	25.3388	0.963933	0.481967	−	0.876190i	$-0.339923\pi$
$691$	25.3388	0.963933	0.481967	+	0.876190i	$0.339923\pi$
$692$	0	0
$693$	8.70008	0.330489
$694$	0	0
$695$	9.36923	0.355395
$696$	0	0
$697$	3.68544	0.139596
$698$	0	0
$699$	−22.6515	−0.856757
$700$	0	0
$701$	1.66296	0.0628092	0.0314046	−	0.999507i	$-0.490002\pi$
$701$	1.66296	0.0628092	0.0314046	+	0.999507i	$0.490002\pi$
$702$	0	0
$703$	3.68544	0.138999
$704$	0	0
$705$	−8.86216	−0.333768
$706$	0	0
$707$	47.1549	1.77344
$708$	0	0
$709$	19.6363	0.737458	0.368729	−	0.929537i	$-0.379793\pi$
$709$	19.6363	0.737458	0.368729	+	0.929537i	$0.379793\pi$
$710$	0	0
$711$	−1.04986	−0.0393730
$712$	0	0
$713$	6.57637	0.246287
$714$	0	0
$715$	16.1077	0.602393
$716$	0	0
$717$	−22.3551	−0.834866
$718$	0	0
$719$	30.7102	1.14530	0.572648	−	0.819801i	$-0.305916\pi$
$719$	30.7102	1.14530	0.572648	+	0.819801i	$0.305916\pi$
$720$	0	0
$721$	28.8701	1.07518
$722$	0	0
$723$	−6.79765	−0.252807
$724$	0	0
$725$	−3.89176	−0.144536
$726$	0	0
$727$	7.26414	0.269412	0.134706	−	0.990886i	$-0.456991\pi$
$727$	7.26414	0.269412	0.134706	+	0.990886i	$0.456991\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.21030	0.192710
$732$	0	0
$733$	−47.8074	−1.76581	−0.882903	−	0.469556i	$-0.844414\pi$
$733$	−47.8074	−1.76581	−0.882903	+	0.469556i	$0.844414\pi$
$734$	0	0
$735$	0.207146	0.00764071
$736$	0	0
$737$	−39.9893	−1.47302
$738$	0	0
$739$	0.673943	0.0247914	0.0123957	−	0.999923i	$-0.496054\pi$
$739$	0.673943	0.0247914	0.0123957	+	0.999923i	$0.496054\pi$
$740$	0	0
$741$	6.16688	0.226546
$742$	0	0
$743$	12.5817	0.461577	0.230789	−	0.973004i	$-0.425869\pi$
$743$	12.5817	0.461577	0.230789	+	0.973004i	$0.425869\pi$
$744$	0	0
$745$	−13.2407	−0.485102
$746$	0	0
$747$	−16.3887	−0.599630
$748$	0	0
$749$	30.7982	1.12534
$750$	0	0
$751$	−43.8090	−1.59861	−0.799307	−	0.600923i	$-0.794800\pi$
$751$	−43.8090	−1.59861	−0.799307	+	0.600923i	$0.794800\pi$
$752$	0	0
$753$	−8.06715	−0.293983
$754$	0	0
$755$	−17.8171	−0.648430
$756$	0	0
$757$	3.92533	0.142669	0.0713343	−	0.997452i	$-0.477274\pi$
$757$	3.92533	0.142669	0.0713343	+	0.997452i	$0.477274\pi$
$758$	0	0
$759$	−21.3122	−0.773583
$760$	0	0
$761$	−36.5406	−1.32460	−0.662298	−	0.749241i	$-0.730419\pi$
$761$	−36.5406	−1.32460	−0.662298	+	0.749241i	$0.730419\pi$
$762$	0	0
$763$	−14.3242	−0.518569
$764$	0	0
$765$	−0.921359	−0.0333118
$766$	0	0
$767$	34.8126	1.25701
$768$	0	0
$769$	7.47211	0.269451	0.134726	−	0.990883i	$-0.456985\pi$
$769$	7.47211	0.269451	0.134726	+	0.990883i	$0.456985\pi$
$770$	0	0
$771$	−9.04986	−0.325923
$772$	0	0
$773$	−50.4862	−1.81586	−0.907932	−	0.419118i	$-0.862339\pi$
$773$	−50.4862	−1.81586	−0.907932	+	0.419118i	$0.862339\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−7.97437	−0.286079
$778$	0	0
$779$	4.96288	0.177814
$780$	0	0
$781$	50.3827	1.80284
$782$	0	0
$783$	3.89176	0.139080
$784$	0	0
$785$	7.91202	0.282392
$786$	0	0
$787$	11.9616	0.426386	0.213193	−	0.977010i	$-0.431614\pi$
$787$	11.9616	0.426386	0.213193	+	0.977010i	$0.431614\pi$
$788$	0	0
$789$	−14.9037	−0.530585
$790$	0	0
$791$	20.7097	0.736351
$792$	0	0
$793$	−54.1562	−1.92314
$794$	0	0
$795$	0.792854	0.0281196
$796$	0	0
$797$	−4.53005	−0.160463	−0.0802313	−	0.996776i	$-0.525566\pi$
$797$	−4.53005	−0.160463	−0.0802313	+	0.996776i	$0.525566\pi$
$798$	0	0
$799$	−8.16523	−0.288865
$800$	0	0
$801$	−3.13248	−0.110681
$802$	0	0
$803$	23.0056	0.811849
$804$	0	0
$805$	−17.6550	−0.622257
$806$	0	0
$807$	−13.5084	−0.475518
$808$	0	0
$809$	−39.0450	−1.37275	−0.686375	−	0.727248i	$-0.740799\pi$
$809$	−39.0450	−1.37275	−0.686375	+	0.727248i	$0.740799\pi$
$810$	0	0
$811$	−21.5287	−0.755974	−0.377987	−	0.925811i	$-0.623384\pi$
$811$	−21.5287	−0.755974	−0.377987	+	0.925811i	$0.623384\pi$
$812$	0	0
$813$	−5.90408	−0.207065
$814$	0	0
$815$	15.3255	0.536829
$816$	0	0
$817$	7.01629	0.245469
$818$	0	0
$819$	−13.3436	−0.466263
$820$	0	0
$821$	53.5824	1.87004	0.935019	−	0.354598i	$-0.115382\pi$
$821$	53.5824	1.87004	0.935019	+	0.354598i	$0.115382\pi$
$822$	0	0
$823$	11.7756	0.410471	0.205235	−	0.978713i	$-0.434204\pi$
$823$	11.7756	0.410471	0.205235	+	0.978713i	$0.434204\pi$
$824$	0	0
$825$	−3.24072	−0.112827
$826$	0	0
$827$	−22.9928	−0.799539	−0.399769	−	0.916616i	$-0.630910\pi$
$827$	−22.9928	−0.799539	−0.399769	+	0.916616i	$0.630910\pi$
$828$	0	0
$829$	18.4686	0.641440	0.320720	−	0.947174i	$-0.396075\pi$
$829$	18.4686	0.641440	0.320720	+	0.947174i	$0.396075\pi$
$830$	0	0
$831$	−10.6868	−0.370720
$832$	0	0
$833$	0.190856	0.00661277
$834$	0	0
$835$	−12.1957	−0.422048
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	3.72869	0.128729	0.0643643	−	0.997926i	$-0.479498\pi$
$839$	3.72869	0.128729	0.0643643	+	0.997926i	$0.479498\pi$
$840$	0	0
$841$	−13.8542	−0.477731
$842$	0	0
$843$	−15.6838	−0.540178
$844$	0	0
$845$	−11.7049	−0.402660
$846$	0	0
$847$	−1.33622	−0.0459129
$848$	0	0
$849$	−4.24949	−0.145842
$850$	0	0
$851$	19.5345	0.669633
$852$	0	0
$853$	−21.7718	−0.745451	−0.372726	−	0.927942i	$-0.621577\pi$
$853$	−21.7718	−0.745451	−0.372726	+	0.927942i	$0.621577\pi$
$854$	0	0
$855$	−1.24072	−0.0424317
$856$	0	0
$857$	42.3134	1.44540	0.722700	−	0.691162i	$-0.242901\pi$
$857$	42.3134	1.44540	0.722700	+	0.691162i	$0.242901\pi$
$858$	0	0
$859$	12.5140	0.426973	0.213486	−	0.976946i	$-0.431518\pi$
$859$	12.5140	0.426973	0.213486	+	0.976946i	$0.431518\pi$
$860$	0	0
$861$	−10.7385	−0.365965
$862$	0	0
$863$	−28.8078	−0.980627	−0.490314	−	0.871546i	$-0.663118\pi$
$863$	−28.8078	−0.980627	−0.490314	+	0.871546i	$0.663118\pi$
$864$	0	0
$865$	−8.95493	−0.304477
$866$	0	0
$867$	16.1511	0.548520
$868$	0	0
$869$	−3.40232	−0.115416
$870$	0	0
$871$	61.3329	2.07819
$872$	0	0
$873$	−4.57157	−0.154724
$874$	0	0
$875$	−2.68461	−0.0907565
$876$	0	0
$877$	−43.2418	−1.46017	−0.730085	−	0.683356i	$-0.760520\pi$
$877$	−43.2418	−1.46017	−0.730085	+	0.683356i	$0.760520\pi$
$878$	0	0
$879$	−7.11917	−0.240124
$880$	0	0
$881$	−11.2205	−0.378027	−0.189013	−	0.981975i	$-0.560529\pi$
$881$	−11.2205	−0.378027	−0.189013	+	0.981975i	$0.560529\pi$
$882$	0	0
$883$	−25.5728	−0.860594	−0.430297	−	0.902687i	$-0.641591\pi$
$883$	−25.5728	−0.860594	−0.430297	+	0.902687i	$0.641591\pi$
$884$	0	0
$885$	−7.00397	−0.235436
$886$	0	0
$887$	−12.1007	−0.406302	−0.203151	−	0.979147i	$-0.565118\pi$
$887$	−12.1007	−0.406302	−0.203151	+	0.979147i	$0.565118\pi$
$888$	0	0
$889$	−39.6501	−1.32982
$890$	0	0
$891$	3.24072	0.108568
$892$	0	0
$893$	−10.9955	−0.367949
$894$	0	0
$895$	−6.67130	−0.222997
$896$	0	0
$897$	32.6872	1.09139
$898$	0	0
$899$	−3.89176	−0.129797
$900$	0	0
$901$	0.730503	0.0243366
$902$	0	0
$903$	−15.1815	−0.505209
$904$	0	0
$905$	12.9650	0.430972
$906$	0	0
$907$	10.1744	0.337835	0.168918	−	0.985630i	$-0.445973\pi$
$907$	10.1744	0.337835	0.168918	+	0.985630i	$0.445973\pi$
$908$	0	0
$909$	17.5649	0.582590
$910$	0	0
$911$	11.5186	0.381627	0.190813	−	0.981626i	$-0.438887\pi$
$911$	11.5186	0.381627	0.190813	+	0.981626i	$0.438887\pi$
$912$	0	0
$913$	−53.1111	−1.75772
$914$	0	0
$915$	10.8957	0.360202
$916$	0	0
$917$	39.5184	1.30501
$918$	0	0
$919$	−3.18101	−0.104932	−0.0524659	−	0.998623i	$-0.516708\pi$
$919$	−3.18101	−0.104932	−0.0524659	+	0.998623i	$0.516708\pi$
$920$	0	0
$921$	−28.8616	−0.951022
$922$	0	0
$923$	−77.2737	−2.54349
$924$	0	0
$925$	2.97040	0.0976662
$926$	0	0
$927$	10.7539	0.353205
$928$	0	0
$929$	−12.6995	−0.416658	−0.208329	−	0.978059i	$-0.566802\pi$
$929$	−12.6995	−0.416658	−0.208329	+	0.978059i	$0.566802\pi$
$930$	0	0
$931$	0.257011	0.00842318
$932$	0	0
$933$	18.5118	0.606049
$934$	0	0
$935$	−2.98587	−0.0976483
$936$	0	0
$937$	−12.4330	−0.406167	−0.203084	−	0.979161i	$-0.565096\pi$
$937$	−12.4330	−0.406167	−0.203084	+	0.979161i	$0.565096\pi$
$938$	0	0
$939$	4.17275	0.136173
$940$	0	0
$941$	46.8952	1.52874	0.764369	−	0.644779i	$-0.223050\pi$
$941$	46.8952	1.52874	0.764369	+	0.644779i	$0.223050\pi$
$942$	0	0
$943$	26.3055	0.856625
$944$	0	0
$945$	2.68461	0.0873305
$946$	0	0
$947$	−21.5255	−0.699485	−0.349743	−	0.936846i	$-0.613731\pi$
$947$	−21.5255	−0.699485	−0.349743	+	0.936846i	$0.613731\pi$
$948$	0	0
$949$	−35.2844	−1.14538
$950$	0	0
$951$	11.8250	0.383453
$952$	0	0
$953$	15.4604	0.500810	0.250405	−	0.968141i	$-0.419436\pi$
$953$	15.4604	0.500810	0.250405	+	0.968141i	$0.419436\pi$
$954$	0	0
$955$	−15.5755	−0.504013
$956$	0	0
$957$	12.6121	0.407691
$958$	0	0
$959$	31.2763	1.00996
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	11.4721	0.369683
$964$	0	0
$965$	−6.41429	−0.206483
$966$	0	0
$967$	−37.8037	−1.21569	−0.607843	−	0.794057i	$-0.707965\pi$
$967$	−37.8037	−1.21569	−0.607843	+	0.794057i	$0.707965\pi$
$968$	0	0
$969$	−1.14315	−0.0367232
$970$	0	0
$971$	32.0153	1.02742	0.513710	−	0.857964i	$-0.328271\pi$
$971$	32.0153	1.02742	0.513710	+	0.857964i	$0.328271\pi$
$972$	0	0
$973$	−25.1527	−0.806360
$974$	0	0
$975$	4.97040	0.159180
$976$	0	0
$977$	2.31456	0.0740495	0.0370247	−	0.999314i	$-0.488212\pi$
$977$	2.31456	0.0740495	0.0370247	+	0.999314i	$0.488212\pi$
$978$	0	0
$979$	−10.1515	−0.324443
$980$	0	0
$981$	−5.33565	−0.170354
$982$	0	0
$983$	23.9487	0.763846	0.381923	−	0.924194i	$-0.375262\pi$
$983$	23.9487	0.763846	0.381923	+	0.924194i	$0.375262\pi$
$984$	0	0
$985$	−8.66435	−0.276069
$986$	0	0
$987$	23.7915	0.757291
$988$	0	0
$989$	37.1895	1.18256
$990$	0	0
$991$	−42.0010	−1.33421	−0.667103	−	0.744965i	$-0.732466\pi$
$991$	−42.0010	−1.33421	−0.667103	+	0.744965i	$0.732466\pi$
$992$	0	0
$993$	−3.80914	−0.120880
$994$	0	0
$995$	6.16208	0.195351
$996$	0	0
$997$	26.8648	0.850817	0.425408	−	0.905002i	$-0.360130\pi$
$997$	26.8648	0.850817	0.425408	+	0.905002i	$0.360130\pi$
$998$	0	0
$999$	−2.97040	−0.0939793

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.r.1.4	✓	4
4.3	odd	2		7440.2.a.bx.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.r.1.4	✓	4	1.1	even	1	trivial
7440.2.a.bx.1.1		4	4.3	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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.68461 q^{7} +1.00000 q^{9} +3.24072 q^{11} -4.97040 q^{13} +1.00000 q^{15} +0.921359 q^{17} +1.24072 q^{19} -2.68461 q^{21} +6.57637 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.89176 q^{29} +1.00000 q^{31} -3.24072 q^{33} -2.68461 q^{35} +2.97040 q^{37} +4.97040 q^{39} +4.00000 q^{41} +5.65501 q^{43} -1.00000 q^{45} -8.86216 q^{47} +0.207146 q^{49} -0.921359 q^{51} +0.792854 q^{53} -3.24072 q^{55} -1.24072 q^{57} -7.00397 q^{59} +10.8957 q^{61} +2.68461 q^{63} +4.97040 q^{65} -12.3396 q^{67} -6.57637 q^{69} +15.5468 q^{71} +7.09891 q^{73} -1.00000 q^{75} +8.70008 q^{77} -1.04986 q^{79} +1.00000 q^{81} -16.3887 q^{83} -0.921359 q^{85} +3.89176 q^{87} -3.13248 q^{89} -13.3436 q^{91} -1.00000 q^{93} -1.24072 q^{95} -4.57157 q^{97} +3.24072 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + q^99

Introduction

L-functions

Modular forms

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Database

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