LMFDB - Embedded newform 3640.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.13856$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.13856 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} +O(q^{10})$	$q+1.13856 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} -3.23607 q^{11} -1.00000 q^{13} -1.13856 q^{15} +4.38814 q^{17} +0.532397 q^{19} -1.13856 q^{21} -4.64341 q^{23} +1.00000 q^{25} -5.35543 q^{27} +3.58697 q^{29} +7.10016 q^{31} -3.68447 q^{33} +1.00000 q^{35} +6.37463 q^{37} -1.13856 q^{39} +0.361122 q^{41} +6.06745 q^{43} +1.70367 q^{45} +2.59266 q^{47} +1.00000 q^{49} +4.99618 q^{51} +9.53693 q^{53} +3.23607 q^{55} +0.606168 q^{57} +6.35090 q^{59} -10.6287 q^{61} +1.70367 q^{63} +1.00000 q^{65} -0.0839932 q^{67} -5.28682 q^{69} +13.9469 q^{71} -5.22140 q^{73} +1.13856 q^{75} +3.23607 q^{77} -7.42591 q^{79} -0.986489 q^{81} -0.195007 q^{83} -4.38814 q^{85} +4.08399 q^{87} +14.2541 q^{89} +1.00000 q^{91} +8.08399 q^{93} -0.532397 q^{95} -5.34192 q^{97} +5.51320 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9	$ 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 7 q^{17} - q^{19} - q^{21} + 2 q^{23} + 4 q^{25} + 4 q^{27} + q^{29} - q^{31} + 4 q^{33} + 4 q^{35} + 13 q^{37} - q^{39} + q^{41} - 6 q^{43} + q^{45} + 22 q^{47} + 4 q^{49} - 2 q^{51} + 14 q^{53} + 4 q^{55} + 2 q^{57} + 21 q^{59} + 12 q^{61} + q^{63} + 4 q^{65} - 7 q^{67} + 20 q^{69} - 4 q^{71} + 22 q^{73} + q^{75} + 4 q^{77} - 23 q^{79} - 16 q^{81} + 10 q^{83} + 7 q^{85} + 23 q^{87} + 15 q^{89} + 4 q^{91} + 39 q^{93} + q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) $ 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 - 4 * q^13 - q^15 - 7 * q^17 - q^19 - q^21 + 2 * q^23 + 4 * q^25 + 4 * q^27 + q^29 - q^31 + 4 * q^33 + 4 * q^35 + 13 * q^37 - q^39 + q^41 - 6 * q^43 + q^45 + 22 * q^47 + 4 * q^49 - 2 * q^51 + 14 * q^53 + 4 * q^55 + 2 * q^57 + 21 * q^59 + 12 * q^61 + q^63 + 4 * q^65 - 7 * q^67 + 20 * q^69 - 4 * q^71 + 22 * q^73 + q^75 + 4 * q^77 - 23 * q^79 - 16 * q^81 + 10 * q^83 + 7 * q^85 + 23 * q^87 + 15 * q^89 + 4 * q^91 + 39 * q^93 + q^95 - 8 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.13856	0.657350	0.328675	−	0.944443i	$-0.393398\pi$
$3$	1.13856	0.657350	0.328675	+	0.944443i	$0.393398\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.70367	−0.567890
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.13856	−0.293976
$16$	0	0
$17$	4.38814	1.06428	0.532140	−	0.846656i	$-0.321388\pi$
$17$	4.38814	1.06428	0.532140	+	0.846656i	$0.321388\pi$
$18$	0	0
$19$	0.532397	0.122140	0.0610701	−	0.998133i	$-0.480549\pi$
$19$	0.532397	0.122140	0.0610701	+	0.998133i	$0.480549\pi$
$20$	0	0
$21$	−1.13856	−0.248455
$22$	0	0
$23$	−4.64341	−0.968218	−0.484109	−	0.875008i	$-0.660856\pi$
$23$	−4.64341	−0.968218	−0.484109	+	0.875008i	$0.660856\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.35543	−1.03065
$28$	0	0
$29$	3.58697	0.666083	0.333042	−	0.942912i	$-0.391925\pi$
$29$	3.58697	0.666083	0.333042	+	0.942912i	$0.391925\pi$
$30$	0	0
$31$	7.10016	1.27523	0.637614	−	0.770356i	$-0.279922\pi$
$31$	7.10016	1.27523	0.637614	+	0.770356i	$0.279922\pi$
$32$	0	0
$33$	−3.68447	−0.641384
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	6.37463	1.04798	0.523992	−	0.851723i	$-0.324442\pi$
$37$	6.37463	1.04798	0.523992	+	0.851723i	$0.324442\pi$
$38$	0	0
$39$	−1.13856	−0.182316
$40$	0	0
$41$	0.361122	0.0563977	0.0281989	−	0.999602i	$-0.491023\pi$
$41$	0.361122	0.0563977	0.0281989	+	0.999602i	$0.491023\pi$
$42$	0	0
$43$	6.06745	0.925278	0.462639	−	0.886547i	$-0.346903\pi$
$43$	6.06745	0.925278	0.462639	+	0.886547i	$0.346903\pi$
$44$	0	0
$45$	1.70367	0.253968
$46$	0	0
$47$	2.59266	0.378178	0.189089	−	0.981960i	$-0.439447\pi$
$47$	2.59266	0.378178	0.189089	+	0.981960i	$0.439447\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.99618	0.699605
$52$	0	0
$53$	9.53693	1.31000	0.654999	−	0.755630i	$-0.272669\pi$
$53$	9.53693	1.31000	0.654999	+	0.755630i	$0.272669\pi$
$54$	0	0
$55$	3.23607	0.436351
$56$	0	0
$57$	0.606168	0.0802889
$58$	0	0
$59$	6.35090	0.826817	0.413408	−	0.910546i	$-0.364338\pi$
$59$	6.35090	0.826817	0.413408	+	0.910546i	$0.364338\pi$
$60$	0	0
$61$	−10.6287	−1.36087	−0.680436	−	0.732808i	$-0.738209\pi$
$61$	−10.6287	−1.36087	−0.680436	+	0.732808i	$0.738209\pi$
$62$	0	0
$63$	1.70367	0.214642
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−0.0839932	−0.0102614	−0.00513070	−	0.999987i	$-0.501633\pi$
$67$	−0.0839932	−0.0102614	−0.00513070	+	0.999987i	$0.501633\pi$
$68$	0	0
$69$	−5.28682	−0.636459
$70$	0	0
$71$	13.9469	1.65520	0.827598	−	0.561321i	$-0.189707\pi$
$71$	13.9469	1.65520	0.827598	+	0.561321i	$0.189707\pi$
$72$	0	0
$73$	−5.22140	−0.611119	−0.305559	−	0.952173i	$-0.598843\pi$
$73$	−5.22140	−0.611119	−0.305559	+	0.952173i	$0.598843\pi$
$74$	0	0
$75$	1.13856	0.131470
$76$	0	0
$77$	3.23607	0.368784
$78$	0	0
$79$	−7.42591	−0.835481	−0.417740	−	0.908566i	$-0.637178\pi$
$79$	−7.42591	−0.835481	−0.417740	+	0.908566i	$0.637178\pi$
$80$	0	0
$81$	−0.986489	−0.109610
$82$	0	0
$83$	−0.195007	−0.0214048	−0.0107024	−	0.999943i	$-0.503407\pi$
$83$	−0.195007	−0.0214048	−0.0107024	+	0.999943i	$0.503407\pi$
$84$	0	0
$85$	−4.38814	−0.475961
$86$	0	0
$87$	4.08399	0.437850
$88$	0	0
$89$	14.2541	1.51093	0.755466	−	0.655187i	$-0.227410\pi$
$89$	14.2541	1.51093	0.755466	+	0.655187i	$0.227410\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	8.08399	0.838271
$94$	0	0
$95$	−0.532397	−0.0546227
$96$	0	0
$97$	−5.34192	−0.542390	−0.271195	−	0.962524i	$-0.587419\pi$
$97$	−5.34192	−0.542390	−0.271195	+	0.962524i	$0.587419\pi$
$98$	0	0
$99$	5.51320	0.554097
$100$	0	0
$101$	6.39001	0.635830	0.317915	−	0.948119i	$-0.397017\pi$
$101$	6.39001	0.635830	0.317915	+	0.948119i	$0.397017\pi$
$102$	0	0
$103$	−1.62421	−0.160038	−0.0800191	−	0.996793i	$-0.525498\pi$
$103$	−1.62421	−0.160038	−0.0800191	+	0.996793i	$0.525498\pi$
$104$	0	0
$105$	1.13856	0.111112
$106$	0	0
$107$	9.90321	0.957380	0.478690	−	0.877984i	$-0.341112\pi$
$107$	9.90321	0.957380	0.478690	+	0.877984i	$0.341112\pi$
$108$	0	0
$109$	18.9713	1.81712	0.908560	−	0.417754i	$-0.137183\pi$
$109$	18.9713	1.81712	0.908560	+	0.417754i	$0.137183\pi$
$110$	0	0
$111$	7.25793	0.688892
$112$	0	0
$113$	12.3516	1.16194	0.580971	−	0.813924i	$-0.302673\pi$
$113$	12.3516	1.16194	0.580971	+	0.813924i	$0.302673\pi$
$114$	0	0
$115$	4.64341	0.433000
$116$	0	0
$117$	1.70367	0.157504
$118$	0	0
$119$	−4.38814	−0.402260
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	0.411160	0.0370731
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.763932	−0.0677880	−0.0338940	−	0.999425i	$-0.510791\pi$
$127$	−0.763932	−0.0677880	−0.0338940	+	0.999425i	$0.510791\pi$
$128$	0	0
$129$	6.90819	0.608232
$130$	0	0
$131$	−11.3689	−0.993309	−0.496655	−	0.867948i	$-0.665438\pi$
$131$	−11.3689	−0.993309	−0.496655	+	0.867948i	$0.665438\pi$
$132$	0	0
$133$	−0.532397	−0.0461646
$134$	0	0
$135$	5.35543	0.460922
$136$	0	0
$137$	−21.1600	−1.80782	−0.903910	−	0.427723i	$-0.859316\pi$
$137$	−21.1600	−1.80782	−0.903910	+	0.427723i	$0.859316\pi$
$138$	0	0
$139$	−4.31553	−0.366038	−0.183019	−	0.983109i	$-0.558587\pi$
$139$	−4.31553	−0.366038	−0.183019	+	0.983109i	$0.558587\pi$
$140$	0	0
$141$	2.95191	0.248595
$142$	0	0
$143$	3.23607	0.270614
$144$	0	0
$145$	−3.58697	−0.297881
$146$	0	0
$147$	1.13856	0.0939072
$148$	0	0
$149$	14.0270	1.14914	0.574569	−	0.818456i	$-0.305170\pi$
$149$	14.0270	1.14914	0.574569	+	0.818456i	$0.305170\pi$
$150$	0	0
$151$	6.62171	0.538867	0.269434	−	0.963019i	$-0.413164\pi$
$151$	6.62171	0.538867	0.269434	+	0.963019i	$0.413164\pi$
$152$	0	0
$153$	−7.47595	−0.604395
$154$	0	0
$155$	−7.10016	−0.570299
$156$	0	0
$157$	−1.40103	−0.111814	−0.0559070	−	0.998436i	$-0.517805\pi$
$157$	−1.40103	−0.111814	−0.0559070	+	0.998436i	$0.517805\pi$
$158$	0	0
$159$	10.8584	0.861128
$160$	0	0
$161$	4.64341	0.365952
$162$	0	0
$163$	5.22772	0.409466	0.204733	−	0.978818i	$-0.434367\pi$
$163$	5.22772	0.409466	0.204733	+	0.978818i	$0.434367\pi$
$164$	0	0
$165$	3.68447	0.286836
$166$	0	0
$167$	4.65808	0.360453	0.180226	−	0.983625i	$-0.442317\pi$
$167$	4.65808	0.360453	0.180226	+	0.983625i	$0.442317\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.907029	−0.0693622
$172$	0	0
$173$	6.32717	0.481046	0.240523	−	0.970643i	$-0.422681\pi$
$173$	6.32717	0.481046	0.240523	+	0.970643i	$0.422681\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	7.23091	0.543508
$178$	0	0
$179$	5.34379	0.399414	0.199707	−	0.979856i	$-0.436001\pi$
$179$	5.34379	0.399414	0.199707	+	0.979856i	$0.436001\pi$
$180$	0	0
$181$	−2.40671	−0.178890	−0.0894449	−	0.995992i	$-0.528509\pi$
$181$	−2.40671	−0.178890	−0.0894449	+	0.995992i	$0.528509\pi$
$182$	0	0
$183$	−12.1015	−0.894569
$184$	0	0
$185$	−6.37463	−0.468672
$186$	0	0
$187$	−14.2003	−1.03843
$188$	0	0
$189$	5.35543	0.389550
$190$	0	0
$191$	−22.9650	−1.66169	−0.830844	−	0.556506i	$-0.812142\pi$
$191$	−22.9650	−1.66169	−0.830844	+	0.556506i	$0.812142\pi$
$192$	0	0
$193$	3.83389	0.275969	0.137985	−	0.990434i	$-0.455938\pi$
$193$	3.83389	0.275969	0.137985	+	0.990434i	$0.455938\pi$
$194$	0	0
$195$	1.13856	0.0815343
$196$	0	0
$197$	−4.84677	−0.345318	−0.172659	−	0.984982i	$-0.555236\pi$
$197$	−4.84677	−0.345318	−0.172659	+	0.984982i	$0.555236\pi$
$198$	0	0
$199$	−0.685099	−0.0485654	−0.0242827	−	0.999705i	$-0.507730\pi$
$199$	−0.685099	−0.0485654	−0.0242827	+	0.999705i	$0.507730\pi$
$200$	0	0
$201$	−0.0956316	−0.00674534
$202$	0	0
$203$	−3.58697	−0.251756
$204$	0	0
$205$	−0.361122	−0.0252218
$206$	0	0
$207$	7.91085	0.549842
$208$	0	0
$209$	−1.72287	−0.119174
$210$	0	0
$211$	−14.4523	−0.994938	−0.497469	−	0.867482i	$-0.665737\pi$
$211$	−14.4523	−0.994938	−0.497469	+	0.867482i	$0.665737\pi$
$212$	0	0
$213$	15.8795	1.08804
$214$	0	0
$215$	−6.06745	−0.413797
$216$	0	0
$217$	−7.10016	−0.481991
$218$	0	0
$219$	−5.94490	−0.401719
$220$	0	0
$221$	−4.38814	−0.295178
$222$	0	0
$223$	9.49853	0.636069	0.318034	−	0.948079i	$-0.396977\pi$
$223$	9.49853	0.636069	0.318034	+	0.948079i	$0.396977\pi$
$224$	0	0
$225$	−1.70367	−0.113578
$226$	0	0
$227$	−8.46076	−0.561560	−0.280780	−	0.959772i	$-0.590593\pi$
$227$	−8.46076	−0.561560	−0.280780	+	0.959772i	$0.590593\pi$
$228$	0	0
$229$	20.6743	1.36620	0.683099	−	0.730326i	$-0.260632\pi$
$229$	20.6743	1.36620	0.683099	+	0.730326i	$0.260632\pi$
$230$	0	0
$231$	3.68447	0.242420
$232$	0	0
$233$	−1.50991	−0.0989174	−0.0494587	−	0.998776i	$-0.515750\pi$
$233$	−1.50991	−0.0989174	−0.0494587	+	0.998776i	$0.515750\pi$
$234$	0	0
$235$	−2.59266	−0.169126
$236$	0	0
$237$	−8.45488	−0.549204
$238$	0	0
$239$	−1.17065	−0.0757228	−0.0378614	−	0.999283i	$-0.512055\pi$
$239$	−1.17065	−0.0757228	−0.0378614	+	0.999283i	$0.512055\pi$
$240$	0	0
$241$	6.87698	0.442985	0.221493	−	0.975162i	$-0.428907\pi$
$241$	6.87698	0.442985	0.221493	+	0.975162i	$0.428907\pi$
$242$	0	0
$243$	14.9431	0.958601
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−0.532397	−0.0338756
$248$	0	0
$249$	−0.222028	−0.0140705
$250$	0	0
$251$	−20.2657	−1.27916	−0.639581	−	0.768723i	$-0.720892\pi$
$251$	−20.2657	−1.27916	−0.639581	+	0.768723i	$0.720892\pi$
$252$	0	0
$253$	15.0264	0.944701
$254$	0	0
$255$	−4.99618	−0.312873
$256$	0	0
$257$	3.70617	0.231185	0.115592	−	0.993297i	$-0.463123\pi$
$257$	3.70617	0.231185	0.115592	+	0.993297i	$0.463123\pi$
$258$	0	0
$259$	−6.37463	−0.396100
$260$	0	0
$261$	−6.11101	−0.378262
$262$	0	0
$263$	−16.3009	−1.00515	−0.502577	−	0.864532i	$-0.667615\pi$
$263$	−16.3009	−1.00515	−0.502577	+	0.864532i	$0.667615\pi$
$264$	0	0
$265$	−9.53693	−0.585849
$266$	0	0
$267$	16.2292	0.993212
$268$	0	0
$269$	20.6197	1.25720	0.628602	−	0.777727i	$-0.283627\pi$
$269$	20.6197	1.25720	0.628602	+	0.777727i	$0.283627\pi$
$270$	0	0
$271$	13.7994	0.838253	0.419127	−	0.907928i	$-0.362336\pi$
$271$	13.7994	0.838253	0.419127	+	0.907928i	$0.362336\pi$
$272$	0	0
$273$	1.13856	0.0689090
$274$	0	0
$275$	−3.23607	−0.195142
$276$	0	0
$277$	2.91725	0.175281	0.0876403	−	0.996152i	$-0.472067\pi$
$277$	2.91725	0.175281	0.0876403	+	0.996152i	$0.472067\pi$
$278$	0	0
$279$	−12.0963	−0.724189
$280$	0	0
$281$	−26.8945	−1.60439	−0.802195	−	0.597062i	$-0.796335\pi$
$281$	−26.8945	−1.60439	−0.802195	+	0.597062i	$0.796335\pi$
$282$	0	0
$283$	−14.4005	−0.856021	−0.428010	−	0.903774i	$-0.640785\pi$
$283$	−14.4005	−0.856021	−0.428010	+	0.903774i	$0.640785\pi$
$284$	0	0
$285$	−0.606168	−0.0359063
$286$	0	0
$287$	−0.361122	−0.0213163
$288$	0	0
$289$	2.25580	0.132694
$290$	0	0
$291$	−6.08212	−0.356540
$292$	0	0
$293$	6.31553	0.368957	0.184479	−	0.982837i	$-0.440940\pi$
$293$	6.31553	0.368957	0.184479	+	0.982837i	$0.440940\pi$
$294$	0	0
$295$	−6.35090	−0.369764
$296$	0	0
$297$	17.3305	1.00562
$298$	0	0
$299$	4.64341	0.268535
$300$	0	0
$301$	−6.06745	−0.349722
$302$	0	0
$303$	7.27544	0.417963
$304$	0	0
$305$	10.6287	0.608600
$306$	0	0
$307$	18.4825	1.05485	0.527425	−	0.849602i	$-0.323158\pi$
$307$	18.4825	1.05485	0.527425	+	0.849602i	$0.323158\pi$
$308$	0	0
$309$	−1.84927	−0.105201
$310$	0	0
$311$	14.3041	0.811114	0.405557	−	0.914070i	$-0.367078\pi$
$311$	14.3041	0.811114	0.405557	+	0.914070i	$0.367078\pi$
$312$	0	0
$313$	12.7058	0.718174	0.359087	−	0.933304i	$-0.383088\pi$
$313$	12.7058	0.718174	0.359087	+	0.933304i	$0.383088\pi$
$314$	0	0
$315$	−1.70367	−0.0959910
$316$	0	0
$317$	0.548936	0.0308313	0.0154157	−	0.999881i	$-0.495093\pi$
$317$	0.548936	0.0308313	0.0154157	+	0.999881i	$0.495093\pi$
$318$	0	0
$319$	−11.6077	−0.649905
$320$	0	0
$321$	11.2754	0.629334
$322$	0	0
$323$	2.33623	0.129991
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	21.6000	1.19448
$328$	0	0
$329$	−2.59266	−0.142938
$330$	0	0
$331$	−24.7513	−1.36045	−0.680227	−	0.733001i	$-0.738119\pi$
$331$	−24.7513	−1.36045	−0.680227	+	0.733001i	$0.738119\pi$
$332$	0	0
$333$	−10.8603	−0.595140
$334$	0	0
$335$	0.0839932	0.00458904
$336$	0	0
$337$	29.3871	1.60082	0.800408	−	0.599456i	$-0.204616\pi$
$337$	29.3871	1.60082	0.800408	+	0.599456i	$0.204616\pi$
$338$	0	0
$339$	14.0631	0.763803
$340$	0	0
$341$	−22.9766	−1.24425
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.28682	0.284633
$346$	0	0
$347$	25.0003	1.34209	0.671044	−	0.741417i	$-0.265846\pi$
$347$	25.0003	1.34209	0.671044	+	0.741417i	$0.265846\pi$
$348$	0	0
$349$	−0.873789	−0.0467729	−0.0233864	−	0.999726i	$-0.507445\pi$
$349$	−0.873789	−0.0467729	−0.0233864	+	0.999726i	$0.507445\pi$
$350$	0	0
$351$	5.35543	0.285852
$352$	0	0
$353$	33.3396	1.77449	0.887244	−	0.461300i	$-0.152617\pi$
$353$	33.3396	1.77449	0.887244	+	0.461300i	$0.152617\pi$
$354$	0	0
$355$	−13.9469	−0.740226
$356$	0	0
$357$	−4.99618	−0.264426
$358$	0	0
$359$	5.94693	0.313867	0.156934	−	0.987609i	$-0.449839\pi$
$359$	5.94693	0.313867	0.156934	+	0.987609i	$0.449839\pi$
$360$	0	0
$361$	−18.7166	−0.985082
$362$	0	0
$363$	−0.601007	−0.0315447
$364$	0	0
$365$	5.22140	0.273301
$366$	0	0
$367$	−5.03343	−0.262743	−0.131371	−	0.991333i	$-0.541938\pi$
$367$	−5.03343	−0.262743	−0.131371	+	0.991333i	$0.541938\pi$
$368$	0	0
$369$	−0.615233	−0.0320277
$370$	0	0
$371$	−9.53693	−0.495133
$372$	0	0
$373$	−2.31321	−0.119774	−0.0598869	−	0.998205i	$-0.519074\pi$
$373$	−2.31321	−0.119774	−0.0598869	+	0.998205i	$0.519074\pi$
$374$	0	0
$375$	−1.13856	−0.0587952
$376$	0	0
$377$	−3.58697	−0.184738
$378$	0	0
$379$	−26.9643	−1.38506	−0.692531	−	0.721389i	$-0.743504\pi$
$379$	−26.9643	−1.38506	−0.692531	+	0.721389i	$0.743504\pi$
$380$	0	0
$381$	−0.869786	−0.0445605
$382$	0	0
$383$	24.6995	1.26208	0.631042	−	0.775748i	$-0.282627\pi$
$383$	24.6995	1.26208	0.631042	+	0.775748i	$0.282627\pi$
$384$	0	0
$385$	−3.23607	−0.164925
$386$	0	0
$387$	−10.3369	−0.525457
$388$	0	0
$389$	39.0222	1.97850	0.989252	−	0.146219i	$-0.0467105\pi$
$389$	39.0222	1.97850	0.989252	+	0.146219i	$0.0467105\pi$
$390$	0	0
$391$	−20.3759	−1.03046
$392$	0	0
$393$	−12.9443	−0.652952
$394$	0	0
$395$	7.42591	0.373638
$396$	0	0
$397$	3.33286	0.167271	0.0836356	−	0.996496i	$-0.473347\pi$
$397$	3.33286	0.167271	0.0836356	+	0.996496i	$0.473347\pi$
$398$	0	0
$399$	−0.606168	−0.0303463
$400$	0	0
$401$	19.6898	0.983261	0.491631	−	0.870804i	$-0.336401\pi$
$401$	19.6898	0.983261	0.491631	+	0.870804i	$0.336401\pi$
$402$	0	0
$403$	−7.10016	−0.353684
$404$	0	0
$405$	0.986489	0.0490191
$406$	0	0
$407$	−20.6287	−1.02253
$408$	0	0
$409$	−11.6294	−0.575035	−0.287518	−	0.957775i	$-0.592830\pi$
$409$	−11.6294	−0.575035	−0.287518	+	0.957775i	$0.592830\pi$
$410$	0	0
$411$	−24.0920	−1.18837
$412$	0	0
$413$	−6.35090	−0.312507
$414$	0	0
$415$	0.195007	0.00957254
$416$	0	0
$417$	−4.91351	−0.240615
$418$	0	0
$419$	−0.765964	−0.0374198	−0.0187099	−	0.999825i	$-0.505956\pi$
$419$	−0.765964	−0.0374198	−0.0187099	+	0.999825i	$0.505956\pi$
$420$	0	0
$421$	5.23810	0.255289	0.127645	−	0.991820i	$-0.459258\pi$
$421$	5.23810	0.255289	0.127645	+	0.991820i	$0.459258\pi$
$422$	0	0
$423$	−4.41704	−0.214764
$424$	0	0
$425$	4.38814	0.212856
$426$	0	0
$427$	10.6287	0.514361
$428$	0	0
$429$	3.68447	0.177888
$430$	0	0
$431$	−15.6428	−0.753486	−0.376743	−	0.926318i	$-0.622956\pi$
$431$	−15.6428	−0.753486	−0.376743	+	0.926318i	$0.622956\pi$
$432$	0	0
$433$	−12.6870	−0.609697	−0.304848	−	0.952401i	$-0.598606\pi$
$433$	−12.6870	−0.609697	−0.304848	+	0.952401i	$0.598606\pi$
$434$	0	0
$435$	−4.08399	−0.195812
$436$	0	0
$437$	−2.47214	−0.118258
$438$	0	0
$439$	−37.9490	−1.81121	−0.905603	−	0.424127i	$-0.860581\pi$
$439$	−37.9490	−1.81121	−0.905603	+	0.424127i	$0.860581\pi$
$440$	0	0
$441$	−1.70367	−0.0811272
$442$	0	0
$443$	−21.6584	−1.02902	−0.514511	−	0.857484i	$-0.672027\pi$
$443$	−21.6584	−1.02902	−0.514511	+	0.857484i	$0.672027\pi$
$444$	0	0
$445$	−14.2541	−0.675710
$446$	0	0
$447$	15.9707	0.755387
$448$	0	0
$449$	16.1258	0.761026	0.380513	−	0.924776i	$-0.375748\pi$
$449$	16.1258	0.761026	0.380513	+	0.924776i	$0.375748\pi$
$450$	0	0
$451$	−1.16861	−0.0550279
$452$	0	0
$453$	7.53924	0.354225
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	13.1726	0.616189	0.308095	−	0.951356i	$-0.400309\pi$
$457$	13.1726	0.616189	0.308095	+	0.951356i	$0.400309\pi$
$458$	0	0
$459$	−23.5004	−1.09690
$460$	0	0
$461$	10.9266	0.508901	0.254451	−	0.967086i	$-0.418105\pi$
$461$	10.9266	0.508901	0.254451	+	0.967086i	$0.418105\pi$
$462$	0	0
$463$	7.78516	0.361807	0.180904	−	0.983501i	$-0.442098\pi$
$463$	7.78516	0.361807	0.180904	+	0.983501i	$0.442098\pi$
$464$	0	0
$465$	−8.08399	−0.374886
$466$	0	0
$467$	0.366724	0.0169700	0.00848499	−	0.999964i	$-0.497299\pi$
$467$	0.366724	0.0169700	0.00848499	+	0.999964i	$0.497299\pi$
$468$	0	0
$469$	0.0839932	0.00387844
$470$	0	0
$471$	−1.59516	−0.0735010
$472$	0	0
$473$	−19.6347	−0.902804
$474$	0	0
$475$	0.532397	0.0244280
$476$	0	0
$477$	−16.2478	−0.743935
$478$	0	0
$479$	33.2298	1.51831	0.759153	−	0.650912i	$-0.225613\pi$
$479$	33.2298	1.51831	0.759153	+	0.650912i	$0.225613\pi$
$480$	0	0
$481$	−6.37463	−0.290658
$482$	0	0
$483$	5.28682	0.240559
$484$	0	0
$485$	5.34192	0.242564
$486$	0	0
$487$	−36.2481	−1.64256	−0.821278	−	0.570528i	$-0.806739\pi$
$487$	−36.2481	−1.64256	−0.821278	+	0.570528i	$0.806739\pi$
$488$	0	0
$489$	5.95209	0.269163
$490$	0	0
$491$	−6.54894	−0.295549	−0.147775	−	0.989021i	$-0.547211\pi$
$491$	−6.54894	−0.295549	−0.147775	+	0.989021i	$0.547211\pi$
$492$	0	0
$493$	15.7401	0.708900
$494$	0	0
$495$	−5.51320	−0.247800
$496$	0	0
$497$	−13.9469	−0.625605
$498$	0	0
$499$	−30.4100	−1.36134	−0.680669	−	0.732591i	$-0.738311\pi$
$499$	−30.4100	−1.36134	−0.680669	+	0.732591i	$0.738311\pi$
$500$	0	0
$501$	5.30352	0.236944
$502$	0	0
$503$	−0.931918	−0.0415522	−0.0207761	−	0.999784i	$-0.506614\pi$
$503$	−0.931918	−0.0415522	−0.0207761	+	0.999784i	$0.506614\pi$
$504$	0	0
$505$	−6.39001	−0.284352
$506$	0	0
$507$	1.13856	0.0505654
$508$	0	0
$509$	−11.9855	−0.531248	−0.265624	−	0.964077i	$-0.585578\pi$
$509$	−11.9855	−0.531248	−0.265624	+	0.964077i	$0.585578\pi$
$510$	0	0
$511$	5.22140	0.230981
$512$	0	0
$513$	−2.85121	−0.125884
$514$	0	0
$515$	1.62421	0.0715713
$516$	0	0
$517$	−8.39001	−0.368992
$518$	0	0
$519$	7.20389	0.316216
$520$	0	0
$521$	14.1475	0.619815	0.309908	−	0.950767i	$-0.399702\pi$
$521$	14.1475	0.619815	0.309908	+	0.950767i	$0.399702\pi$
$522$	0	0
$523$	−0.118489	−0.00518115	−0.00259058	−	0.999997i	$-0.500825\pi$
$523$	−0.118489	−0.00518115	−0.00259058	+	0.999997i	$0.500825\pi$
$524$	0	0
$525$	−1.13856	−0.0496910
$526$	0	0
$527$	31.1565	1.35720
$528$	0	0
$529$	−1.43874	−0.0625537
$530$	0	0
$531$	−10.8198	−0.469541
$532$	0	0
$533$	−0.361122	−0.0156419
$534$	0	0
$535$	−9.90321	−0.428153
$536$	0	0
$537$	6.08425	0.262555
$538$	0	0
$539$	−3.23607	−0.139387
$540$	0	0
$541$	29.7860	1.28060	0.640300	−	0.768125i	$-0.278810\pi$
$541$	29.7860	1.28060	0.640300	+	0.768125i	$0.278810\pi$
$542$	0	0
$543$	−2.74020	−0.117593
$544$	0	0
$545$	−18.9713	−0.812641
$546$	0	0
$547$	8.64110	0.369467	0.184733	−	0.982789i	$-0.440858\pi$
$547$	8.64110	0.369467	0.184733	+	0.982789i	$0.440858\pi$
$548$	0	0
$549$	18.1079	0.772826
$550$	0	0
$551$	1.90969	0.0813555
$552$	0	0
$553$	7.42591	0.315782
$554$	0	0
$555$	−7.25793	−0.308082
$556$	0	0
$557$	0.122393	0.00518597	0.00259299	−	0.999997i	$-0.499175\pi$
$557$	0.122393	0.00518597	0.00259299	+	0.999997i	$0.499175\pi$
$558$	0	0
$559$	−6.06745	−0.256626
$560$	0	0
$561$	−16.1680	−0.682613
$562$	0	0
$563$	33.7220	1.42121	0.710606	−	0.703590i	$-0.248421\pi$
$563$	33.7220	1.42121	0.710606	+	0.703590i	$0.248421\pi$
$564$	0	0
$565$	−12.3516	−0.519636
$566$	0	0
$567$	0.986489	0.0414287
$568$	0	0
$569$	17.7807	0.745407	0.372703	−	0.927951i	$-0.378431\pi$
$569$	17.7807	0.745407	0.372703	+	0.927951i	$0.378431\pi$
$570$	0	0
$571$	34.5456	1.44569	0.722845	−	0.691011i	$-0.242834\pi$
$571$	34.5456	1.44569	0.722845	+	0.691011i	$0.242834\pi$
$572$	0	0
$573$	−26.1471	−1.09231
$574$	0	0
$575$	−4.64341	−0.193644
$576$	0	0
$577$	32.5903	1.35675	0.678377	−	0.734714i	$-0.262684\pi$
$577$	32.5903	1.35675	0.678377	+	0.734714i	$0.262684\pi$
$578$	0	0
$579$	4.36513	0.181408
$580$	0	0
$581$	0.195007	0.00809027
$582$	0	0
$583$	−30.8622	−1.27818
$584$	0	0
$585$	−1.70367	−0.0704381
$586$	0	0
$587$	0.855400	0.0353061	0.0176531	−	0.999844i	$-0.494381\pi$
$587$	0.855400	0.0353061	0.0176531	+	0.999844i	$0.494381\pi$
$588$	0	0
$589$	3.78010	0.155756
$590$	0	0
$591$	−5.51836	−0.226995
$592$	0	0
$593$	−2.46307	−0.101146	−0.0505731	−	0.998720i	$-0.516105\pi$
$593$	−2.46307	−0.101146	−0.0505731	+	0.998720i	$0.516105\pi$
$594$	0	0
$595$	4.38814	0.179896
$596$	0	0
$597$	−0.780030	−0.0319245
$598$	0	0
$599$	14.5596	0.594888	0.297444	−	0.954739i	$-0.403866\pi$
$599$	14.5596	0.594888	0.297444	+	0.954739i	$0.403866\pi$
$600$	0	0
$601$	−22.5272	−0.918905	−0.459453	−	0.888202i	$-0.651954\pi$
$601$	−22.5272	−0.918905	−0.459453	+	0.888202i	$0.651954\pi$
$602$	0	0
$603$	0.143097	0.00582735
$604$	0	0
$605$	0.527864	0.0214607
$606$	0	0
$607$	−12.0174	−0.487772	−0.243886	−	0.969804i	$-0.578422\pi$
$607$	−12.0174	−0.487772	−0.243886	+	0.969804i	$0.578422\pi$
$608$	0	0
$609$	−4.08399	−0.165492
$610$	0	0
$611$	−2.59266	−0.104888
$612$	0	0
$613$	32.7919	1.32445	0.662227	−	0.749304i	$-0.269612\pi$
$613$	32.7919	1.32445	0.662227	+	0.749304i	$0.269612\pi$
$614$	0	0
$615$	−0.411160	−0.0165796
$616$	0	0
$617$	2.38314	0.0959417	0.0479709	−	0.998849i	$-0.484725\pi$
$617$	2.38314	0.0959417	0.0479709	+	0.998849i	$0.484725\pi$
$618$	0	0
$619$	36.4577	1.46536	0.732680	−	0.680573i	$-0.238269\pi$
$619$	36.4577	1.46536	0.732680	+	0.680573i	$0.238269\pi$
$620$	0	0
$621$	24.8675	0.997897
$622$	0	0
$623$	−14.2541	−0.571079
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.96160	−0.0783388
$628$	0	0
$629$	27.9728	1.11535
$630$	0	0
$631$	−21.9469	−0.873694	−0.436847	−	0.899536i	$-0.643905\pi$
$631$	−21.9469	−0.873694	−0.436847	+	0.899536i	$0.643905\pi$
$632$	0	0
$633$	−16.4549	−0.654023
$634$	0	0
$635$	0.763932	0.0303157
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−23.7610	−0.939970
$640$	0	0
$641$	−38.2081	−1.50913	−0.754564	−	0.656226i	$-0.772152\pi$
$641$	−38.2081	−1.50913	−0.754564	+	0.656226i	$0.772152\pi$
$642$	0	0
$643$	−0.953163	−0.0375891	−0.0187945	−	0.999823i	$-0.505983\pi$
$643$	−0.953163	−0.0375891	−0.0187945	+	0.999823i	$0.505983\pi$
$644$	0	0
$645$	−6.90819	−0.272010
$646$	0	0
$647$	−10.0995	−0.397054	−0.198527	−	0.980095i	$-0.563616\pi$
$647$	−10.0995	−0.397054	−0.198527	+	0.980095i	$0.563616\pi$
$648$	0	0
$649$	−20.5519	−0.806734
$650$	0	0
$651$	−8.08399	−0.316837
$652$	0	0
$653$	−24.0258	−0.940201	−0.470100	−	0.882613i	$-0.655782\pi$
$653$	−24.0258	−0.940201	−0.470100	+	0.882613i	$0.655782\pi$
$654$	0	0
$655$	11.3689	0.444221
$656$	0	0
$657$	8.89555	0.347048
$658$	0	0
$659$	−21.1861	−0.825294	−0.412647	−	0.910891i	$-0.635396\pi$
$659$	−21.1861	−0.825294	−0.412647	+	0.910891i	$0.635396\pi$
$660$	0	0
$661$	18.7127	0.727841	0.363921	−	0.931430i	$-0.381438\pi$
$661$	18.7127	0.727841	0.363921	+	0.931430i	$0.381438\pi$
$662$	0	0
$663$	−4.99618	−0.194036
$664$	0	0
$665$	0.532397	0.0206455
$666$	0	0
$667$	−16.6558	−0.644914
$668$	0	0
$669$	10.8147	0.418120
$670$	0	0
$671$	34.3953	1.32782
$672$	0	0
$673$	−15.3803	−0.592868	−0.296434	−	0.955053i	$-0.595797\pi$
$673$	−15.3803	−0.592868	−0.296434	+	0.955053i	$0.595797\pi$
$674$	0	0
$675$	−5.35543	−0.206131
$676$	0	0
$677$	27.2034	1.04551	0.522757	−	0.852482i	$-0.324904\pi$
$677$	27.2034	1.04551	0.522757	+	0.852482i	$0.324904\pi$
$678$	0	0
$679$	5.34192	0.205004
$680$	0	0
$681$	−9.63312	−0.369142
$682$	0	0
$683$	29.6403	1.13415	0.567077	−	0.823664i	$-0.308074\pi$
$683$	29.6403	1.13415	0.567077	+	0.823664i	$0.308074\pi$
$684$	0	0
$685$	21.1600	0.808481
$686$	0	0
$687$	23.5391	0.898071
$688$	0	0
$689$	−9.53693	−0.363328
$690$	0	0
$691$	−41.1363	−1.56490	−0.782448	−	0.622715i	$-0.786029\pi$
$691$	−41.1363	−1.56490	−0.782448	+	0.622715i	$0.786029\pi$
$692$	0	0
$693$	−5.51320	−0.209429
$694$	0	0
$695$	4.31553	0.163697
$696$	0	0
$697$	1.58465	0.0600230
$698$	0	0
$699$	−1.71913	−0.0650234
$700$	0	0
$701$	13.2682	0.501135	0.250567	−	0.968099i	$-0.419383\pi$
$701$	13.2682	0.501135	0.250567	+	0.968099i	$0.419383\pi$
$702$	0	0
$703$	3.39383	0.128001
$704$	0	0
$705$	−2.95191	−0.111175
$706$	0	0
$707$	−6.39001	−0.240321
$708$	0	0
$709$	45.4834	1.70817	0.854083	−	0.520137i	$-0.174119\pi$
$709$	45.4834	1.70817	0.854083	+	0.520137i	$0.174119\pi$
$710$	0	0
$711$	12.6513	0.474462
$712$	0	0
$713$	−32.9690	−1.23470
$714$	0	0
$715$	−3.23607	−0.121022
$716$	0	0
$717$	−1.33286	−0.0497764
$718$	0	0
$719$	25.6475	0.956490	0.478245	−	0.878226i	$-0.341273\pi$
$719$	25.6475	0.956490	0.478245	+	0.878226i	$0.341273\pi$
$720$	0	0
$721$	1.62421	0.0604888
$722$	0	0
$723$	7.82988	0.291196
$724$	0	0
$725$	3.58697	0.133217
$726$	0	0
$727$	30.8321	1.14350	0.571750	−	0.820428i	$-0.306265\pi$
$727$	30.8321	1.14350	0.571750	+	0.820428i	$0.306265\pi$
$728$	0	0
$729$	19.9732	0.739747
$730$	0	0
$731$	26.6249	0.984756
$732$	0	0
$733$	−18.8945	−0.697884	−0.348942	−	0.937144i	$-0.613459\pi$
$733$	−18.8945	−0.697884	−0.348942	+	0.937144i	$0.613459\pi$
$734$	0	0
$735$	−1.13856	−0.0419966
$736$	0	0
$737$	0.271808	0.0100122
$738$	0	0
$739$	−39.5056	−1.45324	−0.726618	−	0.687042i	$-0.758909\pi$
$739$	−39.5056	−1.45324	−0.726618	+	0.687042i	$0.758909\pi$
$740$	0	0
$741$	−0.606168	−0.0222681
$742$	0	0
$743$	−46.3257	−1.69953	−0.849763	−	0.527165i	$-0.823255\pi$
$743$	−46.3257	−1.69953	−0.849763	+	0.527165i	$0.823255\pi$
$744$	0	0
$745$	−14.0270	−0.513910
$746$	0	0
$747$	0.332229	0.0121556
$748$	0	0
$749$	−9.90321	−0.361855
$750$	0	0
$751$	−15.6486	−0.571027	−0.285514	−	0.958375i	$-0.592164\pi$
$751$	−15.6486	−0.571027	−0.285514	+	0.958375i	$0.592164\pi$
$752$	0	0
$753$	−23.0739	−0.840858
$754$	0	0
$755$	−6.62171	−0.240989
$756$	0	0
$757$	0.565636	0.0205584	0.0102792	−	0.999947i	$-0.496728\pi$
$757$	0.565636	0.0205584	0.0102792	+	0.999947i	$0.496728\pi$
$758$	0	0
$759$	17.1085	0.621000
$760$	0	0
$761$	−6.77760	−0.245688	−0.122844	−	0.992426i	$-0.539201\pi$
$761$	−6.77760	−0.245688	−0.122844	+	0.992426i	$0.539201\pi$
$762$	0	0
$763$	−18.9713	−0.686807
$764$	0	0
$765$	7.47595	0.270294
$766$	0	0
$767$	−6.35090	−0.229318
$768$	0	0
$769$	−19.6294	−0.707853	−0.353927	−	0.935273i	$-0.615154\pi$
$769$	−19.6294	−0.707853	−0.353927	+	0.935273i	$0.615154\pi$
$770$	0	0
$771$	4.21971	0.151969
$772$	0	0
$773$	−19.2138	−0.691071	−0.345536	−	0.938406i	$-0.612303\pi$
$773$	−19.2138	−0.691071	−0.345536	+	0.938406i	$0.612303\pi$
$774$	0	0
$775$	7.10016	0.255045
$776$	0	0
$777$	−7.25793	−0.260377
$778$	0	0
$779$	0.192260	0.00688843
$780$	0	0
$781$	−45.1332	−1.61499
$782$	0	0
$783$	−19.2098	−0.686501
$784$	0	0
$785$	1.40103	0.0500047
$786$	0	0
$787$	15.6868	0.559174	0.279587	−	0.960120i	$-0.409803\pi$
$787$	15.6868	0.559174	0.279587	+	0.960120i	$0.409803\pi$
$788$	0	0
$789$	−18.5596	−0.660738
$790$	0	0
$791$	−12.3516	−0.439173
$792$	0	0
$793$	10.6287	0.377438
$794$	0	0
$795$	−10.8584	−0.385108
$796$	0	0
$797$	44.6805	1.58267	0.791333	−	0.611386i	$-0.209388\pi$
$797$	44.6805	1.58267	0.791333	+	0.611386i	$0.209388\pi$
$798$	0	0
$799$	11.3769	0.402488
$800$	0	0
$801$	−24.2843	−0.858044
$802$	0	0
$803$	16.8968	0.596275
$804$	0	0
$805$	−4.64341	−0.163659
$806$	0	0
$807$	23.4768	0.826424
$808$	0	0
$809$	−0.000441937 0	−1.55377e−5 0	−7.76885e−6	−	1.00000i	$-0.500002\pi$
$809$	−0.000441937 0	−1.55377e−5 0	−7.76885e−6		1.00000i	$0.500002\pi$
$810$	0	0
$811$	−49.8716	−1.75123	−0.875614	−	0.483012i	$-0.839543\pi$
$811$	−49.8716	−1.75123	−0.875614	+	0.483012i	$0.839543\pi$
$812$	0	0
$813$	15.7115	0.551026
$814$	0	0
$815$	−5.22772	−0.183119
$816$	0	0
$817$	3.23029	0.113014
$818$	0	0
$819$	−1.70367	−0.0595311
$820$	0	0
$821$	0.737885	0.0257524	0.0128762	−	0.999917i	$-0.495901\pi$
$821$	0.737885	0.0257524	0.0128762	+	0.999917i	$0.495901\pi$
$822$	0	0
$823$	28.6285	0.997925	0.498963	−	0.866623i	$-0.333714\pi$
$823$	28.6285	0.997925	0.498963	+	0.866623i	$0.333714\pi$
$824$	0	0
$825$	−3.68447	−0.128277
$826$	0	0
$827$	−35.2228	−1.22482	−0.612409	−	0.790541i	$-0.709799\pi$
$827$	−35.2228	−1.22482	−0.612409	+	0.790541i	$0.709799\pi$
$828$	0	0
$829$	−26.4211	−0.917643	−0.458821	−	0.888529i	$-0.651728\pi$
$829$	−26.4211	−0.917643	−0.458821	+	0.888529i	$0.651728\pi$
$830$	0	0
$831$	3.32148	0.115221
$832$	0	0
$833$	4.38814	0.152040
$834$	0	0
$835$	−4.65808	−0.161199
$836$	0	0
$837$	−38.0244	−1.31432
$838$	0	0
$839$	−6.12787	−0.211558	−0.105779	−	0.994390i	$-0.533734\pi$
$839$	−6.12787	−0.211558	−0.105779	+	0.994390i	$0.533734\pi$
$840$	0	0
$841$	−16.1337	−0.556333
$842$	0	0
$843$	−30.6211	−1.05465
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	0.527864	0.0181376
$848$	0	0
$849$	−16.3959	−0.562706
$850$	0	0
$851$	−29.6000	−1.01468
$852$	0	0
$853$	31.0602	1.06348	0.531740	−	0.846908i	$-0.321538\pi$
$853$	31.0602	1.06348	0.531740	+	0.846908i	$0.321538\pi$
$854$	0	0
$855$	0.907029	0.0310197
$856$	0	0
$857$	−46.7894	−1.59829	−0.799147	−	0.601136i	$-0.794715\pi$
$857$	−46.7894	−1.59829	−0.799147	+	0.601136i	$0.794715\pi$
$858$	0	0
$859$	−38.3953	−1.31003	−0.655016	−	0.755615i	$-0.727338\pi$
$859$	−38.3953	−1.31003	−0.655016	+	0.755615i	$0.727338\pi$
$860$	0	0
$861$	−0.411160	−0.0140123
$862$	0	0
$863$	3.18719	0.108493	0.0542465	−	0.998528i	$-0.482724\pi$
$863$	3.18719	0.108493	0.0542465	+	0.998528i	$0.482724\pi$
$864$	0	0
$865$	−6.32717	−0.215130
$866$	0	0
$867$	2.56837	0.0872264
$868$	0	0
$869$	24.0308	0.815188
$870$	0	0
$871$	0.0839932	0.00284600
$872$	0	0
$873$	9.10088	0.308018
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	52.6389	1.77749	0.888745	−	0.458401i	$-0.151578\pi$
$877$	52.6389	1.77749	0.888745	+	0.458401i	$0.151578\pi$
$878$	0	0
$879$	7.19064	0.242534
$880$	0	0
$881$	−31.9593	−1.07674	−0.538368	−	0.842710i	$-0.680959\pi$
$881$	−31.9593	−1.07674	−0.538368	+	0.842710i	$0.680959\pi$
$882$	0	0
$883$	21.3722	0.719233	0.359616	−	0.933100i	$-0.382908\pi$
$883$	21.3722	0.719233	0.359616	+	0.933100i	$0.382908\pi$
$884$	0	0
$885$	−7.23091	−0.243064
$886$	0	0
$887$	29.9586	1.00591	0.502955	−	0.864312i	$-0.332246\pi$
$887$	29.9586	1.00591	0.502955	+	0.864312i	$0.332246\pi$
$888$	0	0
$889$	0.763932	0.0256215
$890$	0	0
$891$	3.19235	0.106948
$892$	0	0
$893$	1.38032	0.0461907
$894$	0	0
$895$	−5.34379	−0.178623
$896$	0	0
$897$	5.28682	0.176522
$898$	0	0
$899$	25.4681	0.849407
$900$	0	0
$901$	41.8494	1.39421
$902$	0	0
$903$	−6.90819	−0.229890
$904$	0	0
$905$	2.40671	0.0800019
$906$	0	0
$907$	−5.12693	−0.170237	−0.0851184	−	0.996371i	$-0.527127\pi$
$907$	−5.12693	−0.170237	−0.0851184	+	0.996371i	$0.527127\pi$
$908$	0	0
$909$	−10.8865	−0.361082
$910$	0	0
$911$	−0.603420	−0.0199922	−0.00999610	−	0.999950i	$-0.503182\pi$
$911$	−0.603420	−0.0199922	−0.00999610	+	0.999950i	$0.503182\pi$
$912$	0	0
$913$	0.631057	0.0208849
$914$	0	0
$915$	12.1015	0.400063
$916$	0	0
$917$	11.3689	0.375436
$918$	0	0
$919$	−25.6600	−0.846446	−0.423223	−	0.906026i	$-0.639101\pi$
$919$	−25.6600	−0.846446	−0.423223	+	0.906026i	$0.639101\pi$
$920$	0	0
$921$	21.0435	0.693406
$922$	0	0
$923$	−13.9469	−0.459069
$924$	0	0
$925$	6.37463	0.209597
$926$	0	0
$927$	2.76712	0.0908842
$928$	0	0
$929$	−41.4053	−1.35846	−0.679232	−	0.733924i	$-0.737687\pi$
$929$	−41.4053	−1.35846	−0.679232	+	0.733924i	$0.737687\pi$
$930$	0	0
$931$	0.532397	0.0174486
$932$	0	0
$933$	16.2862	0.533186
$934$	0	0
$935$	14.2003	0.464400
$936$	0	0
$937$	9.33241	0.304877	0.152438	−	0.988313i	$-0.451287\pi$
$937$	9.33241	0.304877	0.152438	+	0.988313i	$0.451287\pi$
$938$	0	0
$939$	14.4664	0.472092
$940$	0	0
$941$	−44.5245	−1.45146	−0.725728	−	0.687981i	$-0.758497\pi$
$941$	−44.5245	−1.45146	−0.725728	+	0.687981i	$0.758497\pi$
$942$	0	0
$943$	−1.67684	−0.0546053
$944$	0	0
$945$	−5.35543	−0.174212
$946$	0	0
$947$	30.1327	0.979182	0.489591	−	0.871952i	$-0.337146\pi$
$947$	30.1327	0.979182	0.489591	+	0.871952i	$0.337146\pi$
$948$	0	0
$949$	5.22140	0.169494
$950$	0	0
$951$	0.624999	0.0202670
$952$	0	0
$953$	38.2736	1.23980	0.619901	−	0.784680i	$-0.287173\pi$
$953$	38.2736	1.23980	0.619901	+	0.784680i	$0.287173\pi$
$954$	0	0
$955$	22.9650	0.743129
$956$	0	0
$957$	−13.2161	−0.427215
$958$	0	0
$959$	21.1600	0.683291
$960$	0	0
$961$	19.4123	0.626204
$962$	0	0
$963$	−16.8718	−0.543687
$964$	0	0
$965$	−3.83389	−0.123417
$966$	0	0
$967$	−31.0395	−0.998162	−0.499081	−	0.866555i	$-0.666329\pi$
$967$	−31.0395	−0.998162	−0.499081	+	0.866555i	$0.666329\pi$
$968$	0	0
$969$	2.65995	0.0854499
$970$	0	0
$971$	−11.0595	−0.354915	−0.177458	−	0.984128i	$-0.556787\pi$
$971$	−11.0595	−0.354915	−0.177458	+	0.984128i	$0.556787\pi$
$972$	0	0
$973$	4.31553	0.138349
$974$	0	0
$975$	−1.13856	−0.0364632
$976$	0	0
$977$	−56.3398	−1.80247	−0.901235	−	0.433331i	$-0.857338\pi$
$977$	−56.3398	−1.80247	−0.901235	+	0.433331i	$0.857338\pi$
$978$	0	0
$979$	−46.1273	−1.47423
$980$	0	0
$981$	−32.3208	−1.03193
$982$	0	0
$983$	39.5119	1.26023	0.630116	−	0.776501i	$-0.283007\pi$
$983$	39.5119	1.26023	0.630116	+	0.776501i	$0.283007\pi$
$984$	0	0
$985$	4.84677	0.154431
$986$	0	0
$987$	−2.95191	−0.0939602
$988$	0	0
$989$	−28.1737	−0.895871
$990$	0	0
$991$	−46.4836	−1.47660	−0.738300	−	0.674472i	$-0.764371\pi$
$991$	−46.4836	−1.47660	−0.738300	+	0.674472i	$0.764371\pi$
$992$	0	0
$993$	−28.1809	−0.894295
$994$	0	0
$995$	0.685099	0.0217191
$996$	0	0
$997$	6.48433	0.205361	0.102680	−	0.994714i	$-0.467258\pi$
$997$	6.48433	0.205361	0.102680	+	0.994714i	$0.467258\pi$
$998$	0	0
$999$	−34.1389	−1.08011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.v.1.3	✓	4
4.3	odd	2		7280.2.a.bt.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.v.1.3	✓	4	1.1	even	1	trivial
7280.2.a.bt.1.2		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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.13856 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} +O(q^{10})\)	\(q+1.13856 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} -3.23607 q^{11} -1.00000 q^{13} -1.13856 q^{15} +4.38814 q^{17} +0.532397 q^{19} -1.13856 q^{21} -4.64341 q^{23} +1.00000 q^{25} -5.35543 q^{27} +3.58697 q^{29} +7.10016 q^{31} -3.68447 q^{33} +1.00000 q^{35} +6.37463 q^{37} -1.13856 q^{39} +0.361122 q^{41} +6.06745 q^{43} +1.70367 q^{45} +2.59266 q^{47} +1.00000 q^{49} +4.99618 q^{51} +9.53693 q^{53} +3.23607 q^{55} +0.606168 q^{57} +6.35090 q^{59} -10.6287 q^{61} +1.70367 q^{63} +1.00000 q^{65} -0.0839932 q^{67} -5.28682 q^{69} +13.9469 q^{71} -5.22140 q^{73} +1.13856 q^{75} +3.23607 q^{77} -7.42591 q^{79} -0.986489 q^{81} -0.195007 q^{83} -4.38814 q^{85} +4.08399 q^{87} +14.2541 q^{89} +1.00000 q^{91} +8.08399 q^{93} -0.532397 q^{95} -5.34192 q^{97} +5.51320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9	\( 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 7 q^{17} - q^{19} - q^{21} + 2 q^{23} + 4 q^{25} + 4 q^{27} + q^{29} - q^{31} + 4 q^{33} + 4 q^{35} + 13 q^{37} - q^{39} + q^{41} - 6 q^{43} + q^{45} + 22 q^{47} + 4 q^{49} - 2 q^{51} + 14 q^{53} + 4 q^{55} + 2 q^{57} + 21 q^{59} + 12 q^{61} + q^{63} + 4 q^{65} - 7 q^{67} + 20 q^{69} - 4 q^{71} + 22 q^{73} + q^{75} + 4 q^{77} - 23 q^{79} - 16 q^{81} + 10 q^{83} + 7 q^{85} + 23 q^{87} + 15 q^{89} + 4 q^{91} + 39 q^{93} + q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 - 4 * q^13 - q^15 - 7 * q^17 - q^19 - q^21 + 2 * q^23 + 4 * q^25 + 4 * q^27 + q^29 - q^31 + 4 * q^33 + 4 * q^35 + 13 * q^37 - q^39 + q^41 - 6 * q^43 + q^45 + 22 * q^47 + 4 * q^49 - 2 * q^51 + 14 * q^53 + 4 * q^55 + 2 * q^57 + 21 * q^59 + 12 * q^61 + q^63 + 4 * q^65 - 7 * q^67 + 20 * q^69 - 4 * q^71 + 22 * q^73 + q^75 + 4 * q^77 - 23 * q^79 - 16 * q^81 + 10 * q^83 + 7 * q^85 + 23 * q^87 + 15 * q^89 + 4 * q^91 + 39 * q^93 + q^95 - 8 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

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Database

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