LMFDB - Embedded newform 7360.2.a.cu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.255601784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 5x^{3} + 25x^{2} - 4 $ x^6 - x^5 - 10x^4 + 5x^3 + 25*x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.30036$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30036 q^{3} -1.00000 q^{5} +2.94683 q^{7} +7.89234 q^{9} +O(q^{10})$	$q-3.30036 q^{3} -1.00000 q^{5} +2.94683 q^{7} +7.89234 q^{9} +0.430924 q^{11} +2.53157 q^{13} +3.30036 q^{15} -1.14623 q^{17} +1.90207 q^{19} -9.72558 q^{21} +1.00000 q^{23} +1.00000 q^{25} -16.1465 q^{27} -0.895132 q^{29} -2.05620 q^{31} -1.42220 q^{33} -2.94683 q^{35} -0.715309 q^{37} -8.35508 q^{39} +4.25560 q^{41} -7.79139 q^{43} -7.89234 q^{45} -11.7071 q^{47} +1.68380 q^{49} +3.78298 q^{51} -2.36903 q^{53} -0.430924 q^{55} -6.27752 q^{57} -11.4496 q^{59} -3.84471 q^{61} +23.2574 q^{63} -2.53157 q^{65} +4.46867 q^{67} -3.30036 q^{69} -0.983853 q^{71} +8.60535 q^{73} -3.30036 q^{75} +1.26986 q^{77} -15.0564 q^{79} +29.6121 q^{81} -0.182907 q^{83} +1.14623 q^{85} +2.95425 q^{87} +5.54770 q^{89} +7.46010 q^{91} +6.78618 q^{93} -1.90207 q^{95} +12.6642 q^{97} +3.40100 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9	$ 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9} - 7 q^{11} - q^{13} + 5 q^{15} - 5 q^{19} + 6 q^{23} + 6 q^{25} - 20 q^{27} + 3 q^{29} + 12 q^{31} - 3 q^{33} - 4 q^{35} - 7 q^{37} + 8 q^{39} + 8 q^{41} - 28 q^{43} - 7 q^{45} + 8 q^{47} + 6 q^{49} - 7 q^{51} + 5 q^{53} + 7 q^{55} + 18 q^{57} - 9 q^{59} + 3 q^{61} - 5 q^{63} + q^{65} - 27 q^{67} - 5 q^{69} - 2 q^{71} - 2 q^{73} - 5 q^{75} + 14 q^{77} + 4 q^{79} + 14 q^{81} - 23 q^{83} + 12 q^{87} - 4 q^{89} + q^{91} + 4 q^{93} + 5 q^{95} - q^{97} - q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9 - 7 * q^11 - q^13 + 5 * q^15 - 5 * q^19 + 6 * q^23 + 6 * q^25 - 20 * q^27 + 3 * q^29 + 12 * q^31 - 3 * q^33 - 4 * q^35 - 7 * q^37 + 8 * q^39 + 8 * q^41 - 28 * q^43 - 7 * q^45 + 8 * q^47 + 6 * q^49 - 7 * q^51 + 5 * q^53 + 7 * q^55 + 18 * q^57 - 9 * q^59 + 3 * q^61 - 5 * q^63 + q^65 - 27 * q^67 - 5 * q^69 - 2 * q^71 - 2 * q^73 - 5 * q^75 + 14 * q^77 + 4 * q^79 + 14 * q^81 - 23 * q^83 + 12 * q^87 - 4 * q^89 + q^91 + 4 * q^93 + 5 * q^95 - q^97 - 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
$3$	−3.30036	−1.90546	−0.952730	−	0.303817i	$-0.901739\pi$
$3$	−3.30036	−1.90546	−0.952730	+	0.303817i	$0.901739\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.94683	1.11380	0.556898	−	0.830581i	$-0.311991\pi$
$7$	2.94683	1.11380	0.556898	+	0.830581i	$0.311991\pi$
$8$	0	0
$9$	7.89234	2.63078
$10$	0	0
$11$	0.430924	0.129928	0.0649642	−	0.997888i	$-0.479307\pi$
$11$	0.430924	0.129928	0.0649642	+	0.997888i	$0.479307\pi$
$12$	0	0
$13$	2.53157	0.702131	0.351066	−	0.936351i	$-0.385819\pi$
$13$	2.53157	0.702131	0.351066	+	0.936351i	$0.385819\pi$
$14$	0	0
$15$	3.30036	0.852148
$16$	0	0
$17$	−1.14623	−0.278002	−0.139001	−	0.990292i	$-0.544389\pi$
$17$	−1.14623	−0.278002	−0.139001	+	0.990292i	$0.544389\pi$
$18$	0	0
$19$	1.90207	0.436366	0.218183	−	0.975908i	$-0.429987\pi$
$19$	1.90207	0.436366	0.218183	+	0.975908i	$0.429987\pi$
$20$	0	0
$21$	−9.72558	−2.12230
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−16.1465	−3.10739
$28$	0	0
$29$	−0.895132	−0.166222	−0.0831110	−	0.996540i	$-0.526486\pi$
$29$	−0.895132	−0.166222	−0.0831110	+	0.996540i	$0.526486\pi$
$30$	0	0
$31$	−2.05620	−0.369304	−0.184652	−	0.982804i	$-0.559116\pi$
$31$	−2.05620	−0.369304	−0.184652	+	0.982804i	$0.559116\pi$
$32$	0	0
$33$	−1.42220	−0.247574
$34$	0	0
$35$	−2.94683	−0.498105
$36$	0	0
$37$	−0.715309	−0.117596	−0.0587980	−	0.998270i	$-0.518727\pi$
$37$	−0.715309	−0.117596	−0.0587980	+	0.998270i	$0.518727\pi$
$38$	0	0
$39$	−8.35508	−1.33788
$40$	0	0
$41$	4.25560	0.664613	0.332307	−	0.943171i	$-0.392173\pi$
$41$	4.25560	0.664613	0.332307	+	0.943171i	$0.392173\pi$
$42$	0	0
$43$	−7.79139	−1.18818	−0.594088	−	0.804400i	$-0.702487\pi$
$43$	−7.79139	−1.18818	−0.594088	+	0.804400i	$0.702487\pi$
$44$	0	0
$45$	−7.89234	−1.17652
$46$	0	0
$47$	−11.7071	−1.70766	−0.853830	−	0.520552i	$-0.825726\pi$
$47$	−11.7071	−1.70766	−0.853830	+	0.520552i	$0.825726\pi$
$48$	0	0
$49$	1.68380	0.240543
$50$	0	0
$51$	3.78298	0.529723
$52$	0	0
$53$	−2.36903	−0.325412	−0.162706	−	0.986675i	$-0.552022\pi$
$53$	−2.36903	−0.325412	−0.162706	+	0.986675i	$0.552022\pi$
$54$	0	0
$55$	−0.430924	−0.0581058
$56$	0	0
$57$	−6.27752	−0.831478
$58$	0	0
$59$	−11.4496	−1.49061	−0.745306	−	0.666722i	$-0.767697\pi$
$59$	−11.4496	−1.49061	−0.745306	+	0.666722i	$0.767697\pi$
$60$	0	0
$61$	−3.84471	−0.492265	−0.246132	−	0.969236i	$-0.579160\pi$
$61$	−3.84471	−0.492265	−0.246132	+	0.969236i	$0.579160\pi$
$62$	0	0
$63$	23.2574	2.93016
$64$	0	0
$65$	−2.53157	−0.314003
$66$	0	0
$67$	4.46867	0.545935	0.272967	−	0.962023i	$-0.411995\pi$
$67$	4.46867	0.545935	0.272967	+	0.962023i	$0.411995\pi$
$68$	0	0
$69$	−3.30036	−0.397316
$70$	0	0
$71$	−0.983853	−0.116762	−0.0583809	−	0.998294i	$-0.518594\pi$
$71$	−0.983853	−0.116762	−0.0583809	+	0.998294i	$0.518594\pi$
$72$	0	0
$73$	8.60535	1.00718	0.503590	−	0.863943i	$-0.332012\pi$
$73$	8.60535	1.00718	0.503590	+	0.863943i	$0.332012\pi$
$74$	0	0
$75$	−3.30036	−0.381092
$76$	0	0
$77$	1.26986	0.144714
$78$	0	0
$79$	−15.0564	−1.69398	−0.846991	−	0.531608i	$-0.821588\pi$
$79$	−15.0564	−1.69398	−0.846991	+	0.531608i	$0.821588\pi$
$80$	0	0
$81$	29.6121	3.29023
$82$	0	0
$83$	−0.182907	−0.0200766	−0.0100383	−	0.999950i	$-0.503195\pi$
$83$	−0.182907	−0.0200766	−0.0100383	+	0.999950i	$0.503195\pi$
$84$	0	0
$85$	1.14623	0.124326
$86$	0	0
$87$	2.95425	0.316729
$88$	0	0
$89$	5.54770	0.588055	0.294027	−	0.955797i	$-0.405004\pi$
$89$	5.54770	0.588055	0.294027	+	0.955797i	$0.405004\pi$
$90$	0	0
$91$	7.46010	0.782031
$92$	0	0
$93$	6.78618	0.703694
$94$	0	0
$95$	−1.90207	−0.195149
$96$	0	0
$97$	12.6642	1.28585	0.642925	−	0.765929i	$-0.277721\pi$
$97$	12.6642	1.28585	0.642925	+	0.765929i	$0.277721\pi$
$98$	0	0
$99$	3.40100	0.341813
$100$	0	0
$101$	18.1712	1.80810	0.904049	−	0.427429i	$-0.140580\pi$
$101$	18.1712	1.80810	0.904049	+	0.427429i	$0.140580\pi$
$102$	0	0
$103$	−4.37545	−0.431126	−0.215563	−	0.976490i	$-0.569159\pi$
$103$	−4.37545	−0.431126	−0.215563	+	0.976490i	$0.569159\pi$
$104$	0	0
$105$	9.72558	0.949120
$106$	0	0
$107$	−11.5772	−1.11921	−0.559603	−	0.828761i	$-0.689046\pi$
$107$	−11.5772	−1.11921	−0.559603	+	0.828761i	$0.689046\pi$
$108$	0	0
$109$	10.3790	0.994128	0.497064	−	0.867714i	$-0.334411\pi$
$109$	10.3790	0.994128	0.497064	+	0.867714i	$0.334411\pi$
$110$	0	0
$111$	2.36077	0.224075
$112$	0	0
$113$	−19.9948	−1.88096	−0.940479	−	0.339853i	$-0.889623\pi$
$113$	−19.9948	−1.88096	−0.940479	+	0.339853i	$0.889623\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	19.9800	1.84715
$118$	0	0
$119$	−3.37775	−0.309638
$120$	0	0
$121$	−10.8143	−0.983119
$122$	0	0
$123$	−14.0450	−1.26639
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.95377	−0.173369	−0.0866846	−	0.996236i	$-0.527627\pi$
$127$	−1.95377	−0.173369	−0.0866846	+	0.996236i	$0.527627\pi$
$128$	0	0
$129$	25.7144	2.26402
$130$	0	0
$131$	18.0992	1.58134	0.790669	−	0.612244i	$-0.209733\pi$
$131$	18.0992	1.58134	0.790669	+	0.612244i	$0.209733\pi$
$132$	0	0
$133$	5.60509	0.486023
$134$	0	0
$135$	16.1465	1.38967
$136$	0	0
$137$	−10.8975	−0.931038	−0.465519	−	0.885038i	$-0.654132\pi$
$137$	−10.8975	−0.931038	−0.465519	+	0.885038i	$0.654132\pi$
$138$	0	0
$139$	−7.26813	−0.616474	−0.308237	−	0.951310i	$-0.599739\pi$
$139$	−7.26813	−0.616474	−0.308237	+	0.951310i	$0.599739\pi$
$140$	0	0
$141$	38.6377	3.25388
$142$	0	0
$143$	1.09091	0.0912268
$144$	0	0
$145$	0.895132	0.0743367
$146$	0	0
$147$	−5.55715	−0.458346
$148$	0	0
$149$	−9.27722	−0.760019	−0.380010	−	0.924983i	$-0.624079\pi$
$149$	−9.27722	−0.760019	−0.380010	+	0.924983i	$0.624079\pi$
$150$	0	0
$151$	8.02016	0.652672	0.326336	−	0.945254i	$-0.394186\pi$
$151$	8.02016	0.652672	0.326336	+	0.945254i	$0.394186\pi$
$152$	0	0
$153$	−9.04647	−0.731363
$154$	0	0
$155$	2.05620	0.165158
$156$	0	0
$157$	−12.4105	−0.990465	−0.495232	−	0.868761i	$-0.664917\pi$
$157$	−12.4105	−0.990465	−0.495232	+	0.868761i	$0.664917\pi$
$158$	0	0
$159$	7.81865	0.620059
$160$	0	0
$161$	2.94683	0.232243
$162$	0	0
$163$	9.84914	0.771444	0.385722	−	0.922615i	$-0.373952\pi$
$163$	9.84914	0.771444	0.385722	+	0.922615i	$0.373952\pi$
$164$	0	0
$165$	1.42220	0.110718
$166$	0	0
$167$	−17.5888	−1.36107	−0.680533	−	0.732718i	$-0.738251\pi$
$167$	−17.5888	−1.36107	−0.680533	+	0.732718i	$0.738251\pi$
$168$	0	0
$169$	−6.59116	−0.507012
$170$	0	0
$171$	15.0118	1.14798
$172$	0	0
$173$	−9.57626	−0.728069	−0.364035	−	0.931385i	$-0.618601\pi$
$173$	−9.57626	−0.728069	−0.364035	+	0.931385i	$0.618601\pi$
$174$	0	0
$175$	2.94683	0.222759
$176$	0	0
$177$	37.7878	2.84030
$178$	0	0
$179$	7.59098	0.567376	0.283688	−	0.958917i	$-0.408442\pi$
$179$	7.59098	0.567376	0.283688	+	0.958917i	$0.408442\pi$
$180$	0	0
$181$	20.1013	1.49412	0.747061	−	0.664756i	$-0.231464\pi$
$181$	20.1013	1.49412	0.747061	+	0.664756i	$0.231464\pi$
$182$	0	0
$183$	12.6889	0.937991
$184$	0	0
$185$	0.715309	0.0525906
$186$	0	0
$187$	−0.493939	−0.0361204
$188$	0	0
$189$	−47.5809	−3.46100
$190$	0	0
$191$	19.3849	1.40264	0.701322	−	0.712845i	$-0.252594\pi$
$191$	19.3849	1.40264	0.701322	+	0.712845i	$0.252594\pi$
$192$	0	0
$193$	−18.7924	−1.35271	−0.676353	−	0.736577i	$-0.736441\pi$
$193$	−18.7924	−1.35271	−0.676353	+	0.736577i	$0.736441\pi$
$194$	0	0
$195$	8.35508	0.598320
$196$	0	0
$197$	25.0487	1.78464	0.892322	−	0.451399i	$-0.149075\pi$
$197$	25.0487	1.78464	0.892322	+	0.451399i	$0.149075\pi$
$198$	0	0
$199$	23.9015	1.69433	0.847164	−	0.531331i	$-0.178308\pi$
$199$	23.9015	1.69433	0.847164	+	0.531331i	$0.178308\pi$
$200$	0	0
$201$	−14.7482	−1.04026
$202$	0	0
$203$	−2.63780	−0.185137
$204$	0	0
$205$	−4.25560	−0.297224
$206$	0	0
$207$	7.89234	0.548556
$208$	0	0
$209$	0.819650	0.0566964
$210$	0	0
$211$	−16.0753	−1.10667	−0.553334	−	0.832959i	$-0.686645\pi$
$211$	−16.0753	−1.10667	−0.553334	+	0.832959i	$0.686645\pi$
$212$	0	0
$213$	3.24706	0.222485
$214$	0	0
$215$	7.79139	0.531369
$216$	0	0
$217$	−6.05926	−0.411329
$218$	0	0
$219$	−28.4007	−1.91914
$220$	0	0
$221$	−2.90177	−0.195194
$222$	0	0
$223$	11.1471	0.746463	0.373232	−	0.927738i	$-0.378250\pi$
$223$	11.1471	0.746463	0.373232	+	0.927738i	$0.378250\pi$
$224$	0	0
$225$	7.89234	0.526156
$226$	0	0
$227$	−24.6570	−1.63654	−0.818272	−	0.574832i	$-0.805068\pi$
$227$	−24.6570	−1.63654	−0.818272	+	0.574832i	$0.805068\pi$
$228$	0	0
$229$	11.7974	0.779597	0.389799	−	0.920900i	$-0.372545\pi$
$229$	11.7974	0.779597	0.389799	+	0.920900i	$0.372545\pi$
$230$	0	0
$231$	−4.19099	−0.275747
$232$	0	0
$233$	−26.8438	−1.75860	−0.879298	−	0.476272i	$-0.841988\pi$
$233$	−26.8438	−1.75860	−0.879298	+	0.476272i	$0.841988\pi$
$234$	0	0
$235$	11.7071	0.763689
$236$	0	0
$237$	49.6916	3.22781
$238$	0	0
$239$	0.301602	0.0195090	0.00975451	−	0.999952i	$-0.496895\pi$
$239$	0.301602	0.0195090	0.00975451	+	0.999952i	$0.496895\pi$
$240$	0	0
$241$	3.07421	0.198028	0.0990138	−	0.995086i	$-0.468431\pi$
$241$	3.07421	0.198028	0.0990138	+	0.995086i	$0.468431\pi$
$242$	0	0
$243$	−49.2909	−3.16201
$244$	0	0
$245$	−1.68380	−0.107574
$246$	0	0
$247$	4.81523	0.306386
$248$	0	0
$249$	0.603657	0.0382552
$250$	0	0
$251$	−11.2772	−0.711812	−0.355906	−	0.934522i	$-0.615828\pi$
$251$	−11.2772	−0.711812	−0.355906	+	0.934522i	$0.615828\pi$
$252$	0	0
$253$	0.430924	0.0270920
$254$	0	0
$255$	−3.78298	−0.236899
$256$	0	0
$257$	16.7453	1.04454	0.522271	−	0.852779i	$-0.325085\pi$
$257$	16.7453	1.04454	0.522271	+	0.852779i	$0.325085\pi$
$258$	0	0
$259$	−2.10789	−0.130978
$260$	0	0
$261$	−7.06469	−0.437293
$262$	0	0
$263$	−18.7253	−1.15465	−0.577324	−	0.816515i	$-0.695903\pi$
$263$	−18.7253	−1.15465	−0.577324	+	0.816515i	$0.695903\pi$
$264$	0	0
$265$	2.36903	0.145528
$266$	0	0
$267$	−18.3094	−1.12052
$268$	0	0
$269$	−16.5293	−1.00781	−0.503905	−	0.863759i	$-0.668104\pi$
$269$	−16.5293	−1.00781	−0.503905	+	0.863759i	$0.668104\pi$
$270$	0	0
$271$	10.6144	0.644781	0.322391	−	0.946607i	$-0.395514\pi$
$271$	10.6144	0.644781	0.322391	+	0.946607i	$0.395514\pi$
$272$	0	0
$273$	−24.6210	−1.49013
$274$	0	0
$275$	0.430924	0.0259857
$276$	0	0
$277$	−11.7428	−0.705558	−0.352779	−	0.935707i	$-0.614763\pi$
$277$	−11.7428	−0.705558	−0.352779	+	0.935707i	$0.614763\pi$
$278$	0	0
$279$	−16.2282	−0.971558
$280$	0	0
$281$	−15.8105	−0.943175	−0.471588	−	0.881819i	$-0.656319\pi$
$281$	−15.8105	−0.943175	−0.471588	+	0.881819i	$0.656319\pi$
$282$	0	0
$283$	22.0134	1.30856	0.654281	−	0.756252i	$-0.272971\pi$
$283$	22.0134	1.30856	0.654281	+	0.756252i	$0.272971\pi$
$284$	0	0
$285$	6.27752	0.371848
$286$	0	0
$287$	12.5405	0.740244
$288$	0	0
$289$	−15.6861	−0.922715
$290$	0	0
$291$	−41.7962	−2.45014
$292$	0	0
$293$	27.4356	1.60280	0.801402	−	0.598126i	$-0.204088\pi$
$293$	27.4356	1.60280	0.801402	+	0.598126i	$0.204088\pi$
$294$	0	0
$295$	11.4496	0.666622
$296$	0	0
$297$	−6.95790	−0.403738
$298$	0	0
$299$	2.53157	0.146404
$300$	0	0
$301$	−22.9599	−1.32339
$302$	0	0
$303$	−59.9713	−3.44526
$304$	0	0
$305$	3.84471	0.220147
$306$	0	0
$307$	−9.39799	−0.536371	−0.268186	−	0.963367i	$-0.586424\pi$
$307$	−9.39799	−0.536371	−0.268186	+	0.963367i	$0.586424\pi$
$308$	0	0
$309$	14.4405	0.821493
$310$	0	0
$311$	2.51267	0.142481	0.0712403	−	0.997459i	$-0.477304\pi$
$311$	2.51267	0.142481	0.0712403	+	0.997459i	$0.477304\pi$
$312$	0	0
$313$	12.7989	0.723436	0.361718	−	0.932287i	$-0.382190\pi$
$313$	12.7989	0.723436	0.361718	+	0.932287i	$0.382190\pi$
$314$	0	0
$315$	−23.2574	−1.31041
$316$	0	0
$317$	16.6739	0.936498	0.468249	−	0.883597i	$-0.344885\pi$
$317$	16.6739	0.936498	0.468249	+	0.883597i	$0.344885\pi$
$318$	0	0
$319$	−0.385734	−0.0215970
$320$	0	0
$321$	38.2087	2.13260
$322$	0	0
$323$	−2.18022	−0.121311
$324$	0	0
$325$	2.53157	0.140426
$326$	0	0
$327$	−34.2544	−1.89427
$328$	0	0
$329$	−34.4989	−1.90199
$330$	0	0
$331$	−17.4203	−0.957506	−0.478753	−	0.877950i	$-0.658911\pi$
$331$	−17.4203	−0.957506	−0.478753	+	0.877950i	$0.658911\pi$
$332$	0	0
$333$	−5.64546	−0.309370
$334$	0	0
$335$	−4.46867	−0.244149
$336$	0	0
$337$	−9.98601	−0.543972	−0.271986	−	0.962301i	$-0.587681\pi$
$337$	−9.98601	−0.543972	−0.271986	+	0.962301i	$0.587681\pi$
$338$	0	0
$339$	65.9901	3.58409
$340$	0	0
$341$	−0.886065	−0.0479831
$342$	0	0
$343$	−15.6659	−0.845880
$344$	0	0
$345$	3.30036	0.177685
$346$	0	0
$347$	−21.4900	−1.15364	−0.576822	−	0.816870i	$-0.695707\pi$
$347$	−21.4900	−1.15364	−0.576822	+	0.816870i	$0.695707\pi$
$348$	0	0
$349$	−25.2213	−1.35006	−0.675032	−	0.737788i	$-0.735870\pi$
$349$	−25.2213	−1.35006	−0.675032	+	0.737788i	$0.735870\pi$
$350$	0	0
$351$	−40.8759	−2.18179
$352$	0	0
$353$	−34.5484	−1.83883	−0.919413	−	0.393294i	$-0.871335\pi$
$353$	−34.5484	−1.83883	−0.919413	+	0.393294i	$0.871335\pi$
$354$	0	0
$355$	0.983853	0.0522175
$356$	0	0
$357$	11.1478	0.590003
$358$	0	0
$359$	−1.59305	−0.0840778	−0.0420389	−	0.999116i	$-0.513385\pi$
$359$	−1.59305	−0.0840778	−0.0420389	+	0.999116i	$0.513385\pi$
$360$	0	0
$361$	−15.3821	−0.809585
$362$	0	0
$363$	35.6910	1.87329
$364$	0	0
$365$	−8.60535	−0.450424
$366$	0	0
$367$	5.04444	0.263318	0.131659	−	0.991295i	$-0.457970\pi$
$367$	5.04444	0.263318	0.131659	+	0.991295i	$0.457970\pi$
$368$	0	0
$369$	33.5867	1.74845
$370$	0	0
$371$	−6.98113	−0.362442
$372$	0	0
$373$	−15.3233	−0.793411	−0.396706	−	0.917946i	$-0.629847\pi$
$373$	−15.3233	−0.793411	−0.396706	+	0.917946i	$0.629847\pi$
$374$	0	0
$375$	3.30036	0.170430
$376$	0	0
$377$	−2.26609	−0.116710
$378$	0	0
$379$	−3.70004	−0.190058	−0.0950291	−	0.995474i	$-0.530294\pi$
$379$	−3.70004	−0.190058	−0.0950291	+	0.995474i	$0.530294\pi$
$380$	0	0
$381$	6.44814	0.330348
$382$	0	0
$383$	−20.0831	−1.02620	−0.513100	−	0.858329i	$-0.671503\pi$
$383$	−20.0831	−1.02620	−0.513100	+	0.858329i	$0.671503\pi$
$384$	0	0
$385$	−1.26986	−0.0647180
$386$	0	0
$387$	−61.4923	−3.12583
$388$	0	0
$389$	−8.30633	−0.421148	−0.210574	−	0.977578i	$-0.567533\pi$
$389$	−8.30633	−0.421148	−0.210574	+	0.977578i	$0.567533\pi$
$390$	0	0
$391$	−1.14623	−0.0579675
$392$	0	0
$393$	−59.7339	−3.01318
$394$	0	0
$395$	15.0564	0.757571
$396$	0	0
$397$	22.0236	1.10533	0.552666	−	0.833403i	$-0.313611\pi$
$397$	22.0236	1.10533	0.552666	+	0.833403i	$0.313611\pi$
$398$	0	0
$399$	−18.4988	−0.926098
$400$	0	0
$401$	−8.98102	−0.448491	−0.224245	−	0.974533i	$-0.571992\pi$
$401$	−8.98102	−0.448491	−0.224245	+	0.974533i	$0.571992\pi$
$402$	0	0
$403$	−5.20541	−0.259300
$404$	0	0
$405$	−29.6121	−1.47143
$406$	0	0
$407$	−0.308244	−0.0152791
$408$	0	0
$409$	38.5347	1.90542	0.952710	−	0.303882i	$-0.0982828\pi$
$409$	38.5347	1.90542	0.952710	+	0.303882i	$0.0982828\pi$
$410$	0	0
$411$	35.9657	1.77406
$412$	0	0
$413$	−33.7401	−1.66024
$414$	0	0
$415$	0.182907	0.00897853
$416$	0	0
$417$	23.9874	1.17467
$418$	0	0
$419$	−28.0123	−1.36849	−0.684244	−	0.729253i	$-0.739868\pi$
$419$	−28.0123	−1.36849	−0.684244	+	0.729253i	$0.739868\pi$
$420$	0	0
$421$	2.03264	0.0990647	0.0495323	−	0.998773i	$-0.484227\pi$
$421$	2.03264	0.0990647	0.0495323	+	0.998773i	$0.484227\pi$
$422$	0	0
$423$	−92.3967	−4.49248
$424$	0	0
$425$	−1.14623	−0.0556005
$426$	0	0
$427$	−11.3297	−0.548283
$428$	0	0
$429$	−3.60040	−0.173829
$430$	0	0
$431$	−18.4642	−0.889388	−0.444694	−	0.895683i	$-0.646688\pi$
$431$	−18.4642	−0.889388	−0.444694	+	0.895683i	$0.646688\pi$
$432$	0	0
$433$	16.1218	0.774765	0.387383	−	0.921919i	$-0.373379\pi$
$433$	16.1218	0.774765	0.387383	+	0.921919i	$0.373379\pi$
$434$	0	0
$435$	−2.95425	−0.141646
$436$	0	0
$437$	1.90207	0.0909886
$438$	0	0
$439$	−13.1528	−0.627750	−0.313875	−	0.949464i	$-0.601627\pi$
$439$	−13.1528	−0.627750	−0.313875	+	0.949464i	$0.601627\pi$
$440$	0	0
$441$	13.2892	0.632817
$442$	0	0
$443$	−14.0118	−0.665719	−0.332860	−	0.942976i	$-0.608014\pi$
$443$	−14.0118	−0.665719	−0.332860	+	0.942976i	$0.608014\pi$
$444$	0	0
$445$	−5.54770	−0.262986
$446$	0	0
$447$	30.6181	1.44819
$448$	0	0
$449$	17.6352	0.832255	0.416128	−	0.909306i	$-0.363387\pi$
$449$	17.6352	0.832255	0.416128	+	0.909306i	$0.363387\pi$
$450$	0	0
$451$	1.83384	0.0863522
$452$	0	0
$453$	−26.4694	−1.24364
$454$	0	0
$455$	−7.46010	−0.349735
$456$	0	0
$457$	12.9086	0.603839	0.301920	−	0.953333i	$-0.402373\pi$
$457$	12.9086	0.603839	0.301920	+	0.953333i	$0.402373\pi$
$458$	0	0
$459$	18.5076	0.863862
$460$	0	0
$461$	32.9570	1.53496	0.767480	−	0.641072i	$-0.221510\pi$
$461$	32.9570	1.53496	0.767480	+	0.641072i	$0.221510\pi$
$462$	0	0
$463$	13.4654	0.625789	0.312894	−	0.949788i	$-0.398701\pi$
$463$	13.4654	0.625789	0.312894	+	0.949788i	$0.398701\pi$
$464$	0	0
$465$	−6.78618	−0.314702
$466$	0	0
$467$	−15.4147	−0.713309	−0.356654	−	0.934236i	$-0.616083\pi$
$467$	−15.4147	−0.713309	−0.356654	+	0.934236i	$0.616083\pi$
$468$	0	0
$469$	13.1684	0.608060
$470$	0	0
$471$	40.9590	1.88729
$472$	0	0
$473$	−3.35750	−0.154378
$474$	0	0
$475$	1.90207	0.0872732
$476$	0	0
$477$	−18.6972	−0.856087
$478$	0	0
$479$	−21.9065	−1.00093	−0.500467	−	0.865756i	$-0.666838\pi$
$479$	−21.9065	−1.00093	−0.500467	+	0.865756i	$0.666838\pi$
$480$	0	0
$481$	−1.81085	−0.0825679
$482$	0	0
$483$	−9.72558	−0.442529
$484$	0	0
$485$	−12.6642	−0.575050
$486$	0	0
$487$	5.51458	0.249889	0.124945	−	0.992164i	$-0.460125\pi$
$487$	5.51458	0.249889	0.124945	+	0.992164i	$0.460125\pi$
$488$	0	0
$489$	−32.5057	−1.46996
$490$	0	0
$491$	32.8900	1.48430	0.742151	−	0.670232i	$-0.233805\pi$
$491$	32.8900	1.48430	0.742151	+	0.670232i	$0.233805\pi$
$492$	0	0
$493$	1.02603	0.0462101
$494$	0	0
$495$	−3.40100	−0.152864
$496$	0	0
$497$	−2.89925	−0.130049
$498$	0	0
$499$	9.65305	0.432130	0.216065	−	0.976379i	$-0.430678\pi$
$499$	9.65305	0.432130	0.216065	+	0.976379i	$0.430678\pi$
$500$	0	0
$501$	58.0494	2.59346
$502$	0	0
$503$	−3.12222	−0.139213	−0.0696064	−	0.997575i	$-0.522174\pi$
$503$	−3.12222	−0.139213	−0.0696064	+	0.997575i	$0.522174\pi$
$504$	0	0
$505$	−18.1712	−0.808606
$506$	0	0
$507$	21.7532	0.966091
$508$	0	0
$509$	−21.0864	−0.934637	−0.467318	−	0.884089i	$-0.654780\pi$
$509$	−21.0864	−0.934637	−0.467318	+	0.884089i	$0.654780\pi$
$510$	0	0
$511$	25.3585	1.12179
$512$	0	0
$513$	−30.7118	−1.35596
$514$	0	0
$515$	4.37545	0.192805
$516$	0	0
$517$	−5.04488	−0.221874
$518$	0	0
$519$	31.6050	1.38731
$520$	0	0
$521$	−23.1604	−1.01467	−0.507337	−	0.861748i	$-0.669370\pi$
$521$	−23.1604	−1.01467	−0.507337	+	0.861748i	$0.669370\pi$
$522$	0	0
$523$	−23.0299	−1.00703	−0.503513	−	0.863988i	$-0.667959\pi$
$523$	−23.0299	−1.00703	−0.503513	+	0.863988i	$0.667959\pi$
$524$	0	0
$525$	−9.72558	−0.424459
$526$	0	0
$527$	2.35688	0.102667
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−90.3643	−3.92148
$532$	0	0
$533$	10.7733	0.466646
$534$	0	0
$535$	11.5772	0.500524
$536$	0	0
$537$	−25.0529	−1.08111
$538$	0	0
$539$	0.725592	0.0312534
$540$	0	0
$541$	9.84743	0.423374	0.211687	−	0.977337i	$-0.432104\pi$
$541$	9.84743	0.423374	0.211687	+	0.977337i	$0.432104\pi$
$542$	0	0
$543$	−66.3416	−2.84699
$544$	0	0
$545$	−10.3790	−0.444588
$546$	0	0
$547$	12.5416	0.536242	0.268121	−	0.963385i	$-0.413597\pi$
$547$	12.5416	0.536242	0.268121	+	0.963385i	$0.413597\pi$
$548$	0	0
$549$	−30.3438	−1.29504
$550$	0	0
$551$	−1.70261	−0.0725336
$552$	0	0
$553$	−44.3687	−1.88675
$554$	0	0
$555$	−2.36077	−0.100209
$556$	0	0
$557$	−42.9496	−1.81983	−0.909917	−	0.414790i	$-0.863855\pi$
$557$	−42.9496	−1.81983	−0.909917	+	0.414790i	$0.863855\pi$
$558$	0	0
$559$	−19.7245	−0.834255
$560$	0	0
$561$	1.63018	0.0688261
$562$	0	0
$563$	30.3877	1.28069	0.640344	−	0.768088i	$-0.278792\pi$
$563$	30.3877	1.28069	0.640344	+	0.768088i	$0.278792\pi$
$564$	0	0
$565$	19.9948	0.841190
$566$	0	0
$567$	87.2617	3.66465
$568$	0	0
$569$	11.9305	0.500152	0.250076	−	0.968226i	$-0.419544\pi$
$569$	11.9305	0.500152	0.250076	+	0.968226i	$0.419544\pi$
$570$	0	0
$571$	−26.8597	−1.12404	−0.562021	−	0.827123i	$-0.689976\pi$
$571$	−26.8597	−1.12404	−0.562021	+	0.827123i	$0.689976\pi$
$572$	0	0
$573$	−63.9771	−2.67268
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−20.8228	−0.866864	−0.433432	−	0.901186i	$-0.642698\pi$
$577$	−20.8228	−0.866864	−0.433432	+	0.901186i	$0.642698\pi$
$578$	0	0
$579$	62.0216	2.57753
$580$	0	0
$581$	−0.538995	−0.0223613
$582$	0	0
$583$	−1.02087	−0.0422802
$584$	0	0
$585$	−19.9800	−0.826072
$586$	0	0
$587$	−6.09142	−0.251420	−0.125710	−	0.992067i	$-0.540121\pi$
$587$	−6.09142	−0.251420	−0.125710	+	0.992067i	$0.540121\pi$
$588$	0	0
$589$	−3.91104	−0.161152
$590$	0	0
$591$	−82.6695	−3.40057
$592$	0	0
$593$	−14.1034	−0.579157	−0.289578	−	0.957154i	$-0.593515\pi$
$593$	−14.1034	−0.579157	−0.289578	+	0.957154i	$0.593515\pi$
$594$	0	0
$595$	3.37775	0.138474
$596$	0	0
$597$	−78.8833	−3.22848
$598$	0	0
$599$	30.5418	1.24790	0.623951	−	0.781463i	$-0.285526\pi$
$599$	30.5418	1.24790	0.623951	+	0.781463i	$0.285526\pi$
$600$	0	0
$601$	−13.0180	−0.531017	−0.265508	−	0.964109i	$-0.585540\pi$
$601$	−13.0180	−0.531017	−0.265508	+	0.964109i	$0.585540\pi$
$602$	0	0
$603$	35.2683	1.43623
$604$	0	0
$605$	10.8143	0.439664
$606$	0	0
$607$	22.0597	0.895376	0.447688	−	0.894190i	$-0.352248\pi$
$607$	22.0597	0.895376	0.447688	+	0.894190i	$0.352248\pi$
$608$	0	0
$609$	8.70568	0.352772
$610$	0	0
$611$	−29.6374	−1.19900
$612$	0	0
$613$	−32.2053	−1.30076	−0.650380	−	0.759609i	$-0.725390\pi$
$613$	−32.2053	−1.30076	−0.650380	+	0.759609i	$0.725390\pi$
$614$	0	0
$615$	14.0450	0.566349
$616$	0	0
$617$	24.2947	0.978067	0.489033	−	0.872265i	$-0.337350\pi$
$617$	24.2947	0.978067	0.489033	+	0.872265i	$0.337350\pi$
$618$	0	0
$619$	10.1534	0.408100	0.204050	−	0.978961i	$-0.434590\pi$
$619$	10.1534	0.408100	0.204050	+	0.978961i	$0.434590\pi$
$620$	0	0
$621$	−16.1465	−0.647936
$622$	0	0
$623$	16.3481	0.654974
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.70514	−0.108033
$628$	0	0
$629$	0.819911	0.0326920
$630$	0	0
$631$	46.3806	1.84638	0.923191	−	0.384342i	$-0.125572\pi$
$631$	46.3806	1.84638	0.923191	+	0.384342i	$0.125572\pi$
$632$	0	0
$633$	53.0541	2.10871
$634$	0	0
$635$	1.95377	0.0775331
$636$	0	0
$637$	4.26267	0.168893
$638$	0	0
$639$	−7.76491	−0.307175
$640$	0	0
$641$	−27.8436	−1.09976	−0.549878	−	0.835245i	$-0.685326\pi$
$641$	−27.8436	−1.09976	−0.549878	+	0.835245i	$0.685326\pi$
$642$	0	0
$643$	−31.5956	−1.24601	−0.623004	−	0.782218i	$-0.714088\pi$
$643$	−31.5956	−1.24601	−0.623004	+	0.782218i	$0.714088\pi$
$644$	0	0
$645$	−25.7144	−1.01250
$646$	0	0
$647$	29.1616	1.14646	0.573230	−	0.819395i	$-0.305690\pi$
$647$	29.1616	1.14646	0.573230	+	0.819395i	$0.305690\pi$
$648$	0	0
$649$	−4.93391	−0.193673
$650$	0	0
$651$	19.9977	0.783772
$652$	0	0
$653$	−26.1065	−1.02163	−0.510814	−	0.859692i	$-0.670656\pi$
$653$	−26.1065	−1.02163	−0.510814	+	0.859692i	$0.670656\pi$
$654$	0	0
$655$	−18.0992	−0.707196
$656$	0	0
$657$	67.9163	2.64967
$658$	0	0
$659$	20.0278	0.780171	0.390086	−	0.920779i	$-0.372445\pi$
$659$	20.0278	0.780171	0.390086	+	0.920779i	$0.372445\pi$
$660$	0	0
$661$	−2.45494	−0.0954859	−0.0477430	−	0.998860i	$-0.515203\pi$
$661$	−2.45494	−0.0954859	−0.0477430	+	0.998860i	$0.515203\pi$
$662$	0	0
$663$	9.57687	0.371935
$664$	0	0
$665$	−5.60509	−0.217356
$666$	0	0
$667$	−0.895132	−0.0346597
$668$	0	0
$669$	−36.7893	−1.42236
$670$	0	0
$671$	−1.65678	−0.0639592
$672$	0	0
$673$	−17.1288	−0.660266	−0.330133	−	0.943934i	$-0.607094\pi$
$673$	−17.1288	−0.660266	−0.330133	+	0.943934i	$0.607094\pi$
$674$	0	0
$675$	−16.1465	−0.621478
$676$	0	0
$677$	−41.3054	−1.58750	−0.793748	−	0.608247i	$-0.791873\pi$
$677$	−41.3054	−1.58750	−0.793748	+	0.608247i	$0.791873\pi$
$678$	0	0
$679$	37.3191	1.43218
$680$	0	0
$681$	81.3769	3.11837
$682$	0	0
$683$	−11.7675	−0.450270	−0.225135	−	0.974328i	$-0.572282\pi$
$683$	−11.7675	−0.450270	−0.225135	+	0.974328i	$0.572282\pi$
$684$	0	0
$685$	10.8975	0.416373
$686$	0	0
$687$	−38.9358	−1.48549
$688$	0	0
$689$	−5.99737	−0.228482
$690$	0	0
$691$	39.9597	1.52014	0.760069	−	0.649842i	$-0.225165\pi$
$691$	39.9597	1.52014	0.760069	+	0.649842i	$0.225165\pi$
$692$	0	0
$693$	10.0222	0.380711
$694$	0	0
$695$	7.26813	0.275696
$696$	0	0
$697$	−4.87791	−0.184764
$698$	0	0
$699$	88.5941	3.35094
$700$	0	0
$701$	−12.8947	−0.487027	−0.243514	−	0.969898i	$-0.578300\pi$
$701$	−12.8947	−0.487027	−0.243514	+	0.969898i	$0.578300\pi$
$702$	0	0
$703$	−1.36057	−0.0513149
$704$	0	0
$705$	−38.6377	−1.45518
$706$	0	0
$707$	53.5473	2.01385
$708$	0	0
$709$	18.6787	0.701494	0.350747	−	0.936470i	$-0.385928\pi$
$709$	18.6787	0.701494	0.350747	+	0.936470i	$0.385928\pi$
$710$	0	0
$711$	−118.831	−4.45649
$712$	0	0
$713$	−2.05620	−0.0770052
$714$	0	0
$715$	−1.09091	−0.0407979
$716$	0	0
$717$	−0.995394	−0.0371737
$718$	0	0
$719$	−43.1158	−1.60795	−0.803974	−	0.594665i	$-0.797285\pi$
$719$	−43.1158	−1.60795	−0.803974	+	0.594665i	$0.797285\pi$
$720$	0	0
$721$	−12.8937	−0.480186
$722$	0	0
$723$	−10.1460	−0.377334
$724$	0	0
$725$	−0.895132	−0.0332444
$726$	0	0
$727$	9.51666	0.352953	0.176477	−	0.984305i	$-0.443530\pi$
$727$	9.51666	0.352953	0.176477	+	0.984305i	$0.443530\pi$
$728$	0	0
$729$	73.8413	2.73486
$730$	0	0
$731$	8.93075	0.330316
$732$	0	0
$733$	45.6498	1.68611	0.843057	−	0.537825i	$-0.180754\pi$
$733$	45.6498	1.68611	0.843057	+	0.537825i	$0.180754\pi$
$734$	0	0
$735$	5.55715	0.204979
$736$	0	0
$737$	1.92566	0.0709325
$738$	0	0
$739$	−15.2936	−0.562583	−0.281291	−	0.959622i	$-0.590763\pi$
$739$	−15.2936	−0.562583	−0.281291	+	0.959622i	$0.590763\pi$
$740$	0	0
$741$	−15.8920	−0.583807
$742$	0	0
$743$	−15.8735	−0.582340	−0.291170	−	0.956671i	$-0.594045\pi$
$743$	−15.8735	−0.582340	−0.291170	+	0.956671i	$0.594045\pi$
$744$	0	0
$745$	9.27722	0.339891
$746$	0	0
$747$	−1.44356	−0.0528172
$748$	0	0
$749$	−34.1159	−1.24657
$750$	0	0
$751$	36.1762	1.32009	0.660044	−	0.751227i	$-0.270538\pi$
$751$	36.1762	1.32009	0.660044	+	0.751227i	$0.270538\pi$
$752$	0	0
$753$	37.2188	1.35633
$754$	0	0
$755$	−8.02016	−0.291884
$756$	0	0
$757$	−28.1961	−1.02481	−0.512403	−	0.858745i	$-0.671245\pi$
$757$	−28.1961	−1.02481	−0.512403	+	0.858745i	$0.671245\pi$
$758$	0	0
$759$	−1.42220	−0.0516227
$760$	0	0
$761$	−25.4076	−0.921024	−0.460512	−	0.887653i	$-0.652334\pi$
$761$	−25.4076	−0.921024	−0.460512	+	0.887653i	$0.652334\pi$
$762$	0	0
$763$	30.5851	1.10726
$764$	0	0
$765$	9.04647	0.327076
$766$	0	0
$767$	−28.9855	−1.04661
$768$	0	0
$769$	−1.50361	−0.0542216	−0.0271108	−	0.999632i	$-0.508631\pi$
$769$	−1.50361	−0.0542216	−0.0271108	+	0.999632i	$0.508631\pi$
$770$	0	0
$771$	−55.2654	−1.99034
$772$	0	0
$773$	6.32092	0.227348	0.113674	−	0.993518i	$-0.463738\pi$
$773$	6.32092	0.227348	0.113674	+	0.993518i	$0.463738\pi$
$774$	0	0
$775$	−2.05620	−0.0738608
$776$	0	0
$777$	6.95680	0.249574
$778$	0	0
$779$	8.09447	0.290014
$780$	0	0
$781$	−0.423966	−0.0151707
$782$	0	0
$783$	14.4532	0.516516
$784$	0	0
$785$	12.4105	0.442949
$786$	0	0
$787$	−40.1501	−1.43120	−0.715598	−	0.698512i	$-0.753846\pi$
$787$	−40.1501	−1.43120	−0.715598	+	0.698512i	$0.753846\pi$
$788$	0	0
$789$	61.8000	2.20014
$790$	0	0
$791$	−58.9214	−2.09500
$792$	0	0
$793$	−9.73315	−0.345634
$794$	0	0
$795$	−7.81865	−0.277299
$796$	0	0
$797$	−20.6554	−0.731652	−0.365826	−	0.930683i	$-0.619213\pi$
$797$	−20.6554	−0.731652	−0.365826	+	0.930683i	$0.619213\pi$
$798$	0	0
$799$	13.4191	0.474734
$800$	0	0
$801$	43.7843	1.54704
$802$	0	0
$803$	3.70825	0.130861
$804$	0	0
$805$	−2.94683	−0.103862
$806$	0	0
$807$	54.5526	1.92034
$808$	0	0
$809$	22.9413	0.806572	0.403286	−	0.915074i	$-0.367868\pi$
$809$	22.9413	0.806572	0.403286	+	0.915074i	$0.367868\pi$
$810$	0	0
$811$	48.3124	1.69648	0.848239	−	0.529613i	$-0.177663\pi$
$811$	48.3124	1.69648	0.848239	+	0.529613i	$0.177663\pi$
$812$	0	0
$813$	−35.0314	−1.22861
$814$	0	0
$815$	−9.84914	−0.345000
$816$	0	0
$817$	−14.8198	−0.518480
$818$	0	0
$819$	58.8777	2.05735
$820$	0	0
$821$	47.6009	1.66128	0.830642	−	0.556807i	$-0.187974\pi$
$821$	47.6009	1.66128	0.830642	+	0.556807i	$0.187974\pi$
$822$	0	0
$823$	−9.54587	−0.332748	−0.166374	−	0.986063i	$-0.553206\pi$
$823$	−9.54587	−0.332748	−0.166374	+	0.986063i	$0.553206\pi$
$824$	0	0
$825$	−1.42220	−0.0495147
$826$	0	0
$827$	43.7981	1.52301	0.761505	−	0.648159i	$-0.224461\pi$
$827$	43.7981	1.52301	0.761505	+	0.648159i	$0.224461\pi$
$828$	0	0
$829$	−27.6442	−0.960124	−0.480062	−	0.877235i	$-0.659386\pi$
$829$	−27.6442	−0.960124	−0.480062	+	0.877235i	$0.659386\pi$
$830$	0	0
$831$	38.7555	1.34441
$832$	0	0
$833$	−1.93003	−0.0668716
$834$	0	0
$835$	17.5888	0.608687
$836$	0	0
$837$	33.2003	1.14757
$838$	0	0
$839$	35.0541	1.21020	0.605100	−	0.796149i	$-0.293133\pi$
$839$	35.0541	1.21020	0.605100	+	0.796149i	$0.293133\pi$
$840$	0	0
$841$	−28.1987	−0.972370
$842$	0	0
$843$	52.1803	1.79718
$844$	0	0
$845$	6.59116	0.226743
$846$	0	0
$847$	−31.8679	−1.09499
$848$	0	0
$849$	−72.6521	−2.49341
$850$	0	0
$851$	−0.715309	−0.0245205
$852$	0	0
$853$	25.2255	0.863703	0.431852	−	0.901945i	$-0.357860\pi$
$853$	25.2255	0.863703	0.431852	+	0.901945i	$0.357860\pi$
$854$	0	0
$855$	−15.0118	−0.513394
$856$	0	0
$857$	−36.0303	−1.23077	−0.615385	−	0.788226i	$-0.711000\pi$
$857$	−36.0303	−1.23077	−0.615385	+	0.788226i	$0.711000\pi$
$858$	0	0
$859$	−7.58681	−0.258859	−0.129429	−	0.991589i	$-0.541315\pi$
$859$	−7.58681	−0.258859	−0.129429	+	0.991589i	$0.541315\pi$
$860$	0	0
$861$	−41.3882	−1.41051
$862$	0	0
$863$	21.4901	0.731532	0.365766	−	0.930707i	$-0.380807\pi$
$863$	21.4901	0.731532	0.365766	+	0.930707i	$0.380807\pi$
$864$	0	0
$865$	9.57626	0.325602
$866$	0	0
$867$	51.7699	1.75820
$868$	0	0
$869$	−6.48818	−0.220096
$870$	0	0
$871$	11.3127	0.383318
$872$	0	0
$873$	99.9499	3.38279
$874$	0	0
$875$	−2.94683	−0.0996210
$876$	0	0
$877$	33.4705	1.13022	0.565109	−	0.825016i	$-0.308834\pi$
$877$	33.4705	1.13022	0.565109	+	0.825016i	$0.308834\pi$
$878$	0	0
$879$	−90.5472	−3.05408
$880$	0	0
$881$	58.8645	1.98319	0.991597	−	0.129365i	$-0.0412938\pi$
$881$	58.8645	1.98319	0.991597	+	0.129365i	$0.0412938\pi$
$882$	0	0
$883$	17.2792	0.581491	0.290745	−	0.956800i	$-0.406097\pi$
$883$	17.2792	0.581491	0.290745	+	0.956800i	$0.406097\pi$
$884$	0	0
$885$	−37.7878	−1.27022
$886$	0	0
$887$	−25.2547	−0.847970	−0.423985	−	0.905669i	$-0.639369\pi$
$887$	−25.2547	−0.847970	−0.423985	+	0.905669i	$0.639369\pi$
$888$	0	0
$889$	−5.75743	−0.193098
$890$	0	0
$891$	12.7605	0.427494
$892$	0	0
$893$	−22.2678	−0.745165
$894$	0	0
$895$	−7.59098	−0.253738
$896$	0	0
$897$	−8.35508	−0.278968
$898$	0	0
$899$	1.84057	0.0613864
$900$	0	0
$901$	2.71546	0.0904652
$902$	0	0
$903$	75.7758	2.52166
$904$	0	0
$905$	−20.1013	−0.668191
$906$	0	0
$907$	1.16991	0.0388461	0.0194230	−	0.999811i	$-0.493817\pi$
$907$	1.16991	0.0388461	0.0194230	+	0.999811i	$0.493817\pi$
$908$	0	0
$909$	143.413	4.75671
$910$	0	0
$911$	−13.7927	−0.456972	−0.228486	−	0.973547i	$-0.573377\pi$
$911$	−13.7927	−0.456972	−0.228486	+	0.973547i	$0.573377\pi$
$912$	0	0
$913$	−0.0788189	−0.00260852
$914$	0	0
$915$	−12.6889	−0.419482
$916$	0	0
$917$	53.3354	1.76129
$918$	0	0
$919$	23.8592	0.787042	0.393521	−	0.919316i	$-0.371257\pi$
$919$	23.8592	0.787042	0.393521	+	0.919316i	$0.371257\pi$
$920$	0	0
$921$	31.0167	1.02203
$922$	0	0
$923$	−2.49069	−0.0819822
$924$	0	0
$925$	−0.715309	−0.0235192
$926$	0	0
$927$	−34.5325	−1.13420
$928$	0	0
$929$	−49.0047	−1.60779	−0.803896	−	0.594770i	$-0.797243\pi$
$929$	−49.0047	−1.60779	−0.803896	+	0.594770i	$0.797243\pi$
$930$	0	0
$931$	3.20272	0.104965
$932$	0	0
$933$	−8.29272	−0.271491
$934$	0	0
$935$	0.493939	0.0161535
$936$	0	0
$937$	3.46095	0.113064	0.0565322	−	0.998401i	$-0.481996\pi$
$937$	3.46095	0.113064	0.0565322	+	0.998401i	$0.481996\pi$
$938$	0	0
$939$	−42.2409	−1.37848
$940$	0	0
$941$	−1.03575	−0.0337644	−0.0168822	−	0.999857i	$-0.505374\pi$
$941$	−1.03575	−0.0337644	−0.0168822	+	0.999857i	$0.505374\pi$
$942$	0	0
$943$	4.25560	0.138581
$944$	0	0
$945$	47.5809	1.54781
$946$	0	0
$947$	36.4568	1.18469	0.592344	−	0.805685i	$-0.298203\pi$
$947$	36.4568	1.18469	0.592344	+	0.805685i	$0.298203\pi$
$948$	0	0
$949$	21.7850	0.707172
$950$	0	0
$951$	−55.0297	−1.78446
$952$	0	0
$953$	−14.0726	−0.455856	−0.227928	−	0.973678i	$-0.573195\pi$
$953$	−14.0726	−0.455856	−0.227928	+	0.973678i	$0.573195\pi$
$954$	0	0
$955$	−19.3849	−0.627281
$956$	0	0
$957$	1.27306	0.0411522
$958$	0	0
$959$	−32.1131	−1.03699
$960$	0	0
$961$	−26.7721	−0.863615
$962$	0	0
$963$	−91.3709	−2.94439
$964$	0	0
$965$	18.7924	0.604949
$966$	0	0
$967$	34.7709	1.11816	0.559078	−	0.829115i	$-0.311155\pi$
$967$	34.7709	1.11816	0.559078	+	0.829115i	$0.311155\pi$
$968$	0	0
$969$	7.19550	0.231153
$970$	0	0
$971$	26.8685	0.862252	0.431126	−	0.902292i	$-0.358116\pi$
$971$	26.8685	0.862252	0.431126	+	0.902292i	$0.358116\pi$
$972$	0	0
$973$	−21.4179	−0.686627
$974$	0	0
$975$	−8.35508	−0.267577
$976$	0	0
$977$	5.85008	0.187161	0.0935804	−	0.995612i	$-0.470169\pi$
$977$	5.85008	0.187161	0.0935804	+	0.995612i	$0.470169\pi$
$978$	0	0
$979$	2.39064	0.0764051
$980$	0	0
$981$	81.9146	2.61533
$982$	0	0
$983$	9.51976	0.303633	0.151817	−	0.988409i	$-0.451488\pi$
$983$	9.51976	0.303633	0.151817	+	0.988409i	$0.451488\pi$
$984$	0	0
$985$	−25.0487	−0.798117
$986$	0	0
$987$	113.859	3.62416
$988$	0	0
$989$	−7.79139	−0.247752
$990$	0	0
$991$	−27.5574	−0.875389	−0.437695	−	0.899124i	$-0.644205\pi$
$991$	−27.5574	−0.875389	−0.437695	+	0.899124i	$0.644205\pi$
$992$	0	0
$993$	57.4931	1.82449
$994$	0	0
$995$	−23.9015	−0.757727
$996$	0	0
$997$	16.8441	0.533458	0.266729	−	0.963772i	$-0.414057\pi$
$997$	16.8441	0.533458	0.266729	+	0.963772i	$0.414057\pi$
$998$	0	0
$999$	11.5497	0.365417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.cu.1.1		6
4.3	odd	2		7360.2.a.cv.1.6		6
8.3	odd	2		3680.2.a.bc.1.1	✓	6
8.5	even	2		3680.2.a.bd.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.bc.1.1	✓	6	8.3	odd	2
3680.2.a.bd.1.6	yes	6	8.5	even	2
7360.2.a.cu.1.1		6	1.1	even	1	trivial
7360.2.a.cv.1.6		6	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q-3.30036 q^{3} -1.00000 q^{5} +2.94683 q^{7} +7.89234 q^{9} +O(q^{10})\)	\(q-3.30036 q^{3} -1.00000 q^{5} +2.94683 q^{7} +7.89234 q^{9} +0.430924 q^{11} +2.53157 q^{13} +3.30036 q^{15} -1.14623 q^{17} +1.90207 q^{19} -9.72558 q^{21} +1.00000 q^{23} +1.00000 q^{25} -16.1465 q^{27} -0.895132 q^{29} -2.05620 q^{31} -1.42220 q^{33} -2.94683 q^{35} -0.715309 q^{37} -8.35508 q^{39} +4.25560 q^{41} -7.79139 q^{43} -7.89234 q^{45} -11.7071 q^{47} +1.68380 q^{49} +3.78298 q^{51} -2.36903 q^{53} -0.430924 q^{55} -6.27752 q^{57} -11.4496 q^{59} -3.84471 q^{61} +23.2574 q^{63} -2.53157 q^{65} +4.46867 q^{67} -3.30036 q^{69} -0.983853 q^{71} +8.60535 q^{73} -3.30036 q^{75} +1.26986 q^{77} -15.0564 q^{79} +29.6121 q^{81} -0.182907 q^{83} +1.14623 q^{85} +2.95425 q^{87} +5.54770 q^{89} +7.46010 q^{91} +6.78618 q^{93} -1.90207 q^{95} +12.6642 q^{97} +3.40100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9	\( 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9} - 7 q^{11} - q^{13} + 5 q^{15} - 5 q^{19} + 6 q^{23} + 6 q^{25} - 20 q^{27} + 3 q^{29} + 12 q^{31} - 3 q^{33} - 4 q^{35} - 7 q^{37} + 8 q^{39} + 8 q^{41} - 28 q^{43} - 7 q^{45} + 8 q^{47} + 6 q^{49} - 7 q^{51} + 5 q^{53} + 7 q^{55} + 18 q^{57} - 9 q^{59} + 3 q^{61} - 5 q^{63} + q^{65} - 27 q^{67} - 5 q^{69} - 2 q^{71} - 2 q^{73} - 5 q^{75} + 14 q^{77} + 4 q^{79} + 14 q^{81} - 23 q^{83} + 12 q^{87} - 4 q^{89} + q^{91} + 4 q^{93} + 5 q^{95} - q^{97} - q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9 - 7 * q^11 - q^13 + 5 * q^15 - 5 * q^19 + 6 * q^23 + 6 * q^25 - 20 * q^27 + 3 * q^29 + 12 * q^31 - 3 * q^33 - 4 * q^35 - 7 * q^37 + 8 * q^39 + 8 * q^41 - 28 * q^43 - 7 * q^45 + 8 * q^47 + 6 * q^49 - 7 * q^51 + 5 * q^53 + 7 * q^55 + 18 * q^57 - 9 * q^59 + 3 * q^61 - 5 * q^63 + q^65 - 27 * q^67 - 5 * q^69 - 2 * q^71 - 2 * q^73 - 5 * q^75 + 14 * q^77 + 4 * q^79 + 14 * q^81 - 23 * q^83 + 12 * q^87 - 4 * q^89 + q^91 + 4 * q^93 + 5 * q^95 - q^97 - q^99

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