LMFDB - Embedded newform 7440.2.a.cd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2294036.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 8x^{2} + 6x - 4 $ x^5 - x^4 - 8x^3 + 8x^2 + 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3720)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.817478$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -3.33173 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -3.33173 q^{7} +1.00000 q^{9} -6.35857 q^{11} +3.86626 q^{13} +1.00000 q^{15} +5.25814 q^{17} -2.53453 q^{19} +3.33173 q^{21} +7.99353 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.12440 q^{29} -1.00000 q^{31} +6.35857 q^{33} +3.33173 q^{35} +2.79720 q^{37} -3.86626 q^{39} +4.20085 q^{41} -4.73538 q^{43} -1.00000 q^{45} -1.63496 q^{47} +4.10043 q^{49} -5.25814 q^{51} -4.37034 q^{53} +6.35857 q^{55} +2.53453 q^{57} +0.231304 q^{59} -2.89310 q^{61} -3.33173 q^{63} -3.86626 q^{65} +13.8533 q^{67} -7.99353 q^{69} +9.42929 q^{71} -2.95492 q^{73} -1.00000 q^{75} +21.1850 q^{77} +14.4810 q^{79} +1.00000 q^{81} +14.3521 q^{83} -5.25814 q^{85} +7.12440 q^{87} +10.8195 q^{89} -12.8813 q^{91} +1.00000 q^{93} +2.53453 q^{95} +6.46261 q^{97} -6.35857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 5 q^{5} - 3 q^{7} + 5 q^{9} - 5 q^{11} + 2 q^{13} + 5 q^{15} + 6 q^{17} - 9 q^{19} + 3 q^{21} + 3 q^{23} + 5 q^{25} - 5 q^{27} + 2 q^{29} - 5 q^{31} + 5 q^{33} + 3 q^{35} + 4 q^{37} - 2 q^{39} + 8 q^{41} - 7 q^{43} - 5 q^{45} + 2 q^{47} + 14 q^{49} - 6 q^{51} + 5 q^{53} + 5 q^{55} + 9 q^{57} - 6 q^{59} + 16 q^{61} - 3 q^{63} - 2 q^{65} - 22 q^{67} - 3 q^{69} + 9 q^{71} + 9 q^{73} - 5 q^{75} + q^{77} - 15 q^{79} + 5 q^{81} + 8 q^{83} - 6 q^{85} - 2 q^{87} + 17 q^{89} - 34 q^{91} + 5 q^{93} + 9 q^{95} + 18 q^{97} - 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 5 * q^5 - 3 * q^7 + 5 * q^9 - 5 * q^11 + 2 * q^13 + 5 * q^15 + 6 * q^17 - 9 * q^19 + 3 * q^21 + 3 * q^23 + 5 * q^25 - 5 * q^27 + 2 * q^29 - 5 * q^31 + 5 * q^33 + 3 * q^35 + 4 * q^37 - 2 * q^39 + 8 * q^41 - 7 * q^43 - 5 * q^45 + 2 * q^47 + 14 * q^49 - 6 * q^51 + 5 * q^53 + 5 * q^55 + 9 * q^57 - 6 * q^59 + 16 * q^61 - 3 * q^63 - 2 * q^65 - 22 * q^67 - 3 * q^69 + 9 * q^71 + 9 * q^73 - 5 * q^75 + q^77 - 15 * q^79 + 5 * q^81 + 8 * q^83 - 6 * q^85 - 2 * q^87 + 17 * q^89 - 34 * q^91 + 5 * q^93 + 9 * q^95 + 18 * q^97 - 5 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.33173	−1.25928	−0.629638	−	0.776889i	$-0.716797\pi$
$7$	−3.33173	−1.25928	−0.629638	+	0.776889i	$0.716797\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.35857	−1.91718	−0.958591	−	0.284788i	$-0.908077\pi$
$11$	−6.35857	−1.91718	−0.958591	+	0.284788i	$0.908077\pi$
$12$	0	0
$13$	3.86626	1.07231	0.536154	−	0.844120i	$-0.319877\pi$
$13$	3.86626	1.07231	0.536154	+	0.844120i	$0.319877\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.25814	1.27529	0.637644	−	0.770331i	$-0.279909\pi$
$17$	5.25814	1.27529	0.637644	+	0.770331i	$0.279909\pi$
$18$	0	0
$19$	−2.53453	−0.581461	−0.290730	−	0.956805i	$-0.593898\pi$
$19$	−2.53453	−0.581461	−0.290730	+	0.956805i	$0.593898\pi$
$20$	0	0
$21$	3.33173	0.727043
$22$	0	0
$23$	7.99353	1.66677	0.833383	−	0.552696i	$-0.186401\pi$
$23$	7.99353	1.66677	0.833383	+	0.552696i	$0.186401\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.12440	−1.32297	−0.661484	−	0.749959i	$-0.730073\pi$
$29$	−7.12440	−1.32297	−0.661484	+	0.749959i	$0.730073\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	6.35857	1.10689
$34$	0	0
$35$	3.33173	0.563165
$36$	0	0
$37$	2.79720	0.459857	0.229929	−	0.973208i	$-0.426151\pi$
$37$	2.79720	0.459857	0.229929	+	0.973208i	$0.426151\pi$
$38$	0	0
$39$	−3.86626	−0.619097
$40$	0	0
$41$	4.20085	0.656063	0.328031	−	0.944667i	$-0.393615\pi$
$41$	4.20085	0.656063	0.328031	+	0.944667i	$0.393615\pi$
$42$	0	0
$43$	−4.73538	−0.722139	−0.361069	−	0.932539i	$-0.617588\pi$
$43$	−4.73538	−0.722139	−0.361069	+	0.932539i	$0.617588\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.63496	−0.238483	−0.119241	−	0.992865i	$-0.538046\pi$
$47$	−1.63496	−0.238483	−0.119241	+	0.992865i	$0.538046\pi$
$48$	0	0
$49$	4.10043	0.585775
$50$	0	0
$51$	−5.25814	−0.736287
$52$	0	0
$53$	−4.37034	−0.600312	−0.300156	−	0.953890i	$-0.597039\pi$
$53$	−4.37034	−0.600312	−0.300156	+	0.953890i	$0.597039\pi$
$54$	0	0
$55$	6.35857	0.857389
$56$	0	0
$57$	2.53453	0.335707
$58$	0	0
$59$	0.231304	0.0301132	0.0150566	−	0.999887i	$-0.495207\pi$
$59$	0.231304	0.0301132	0.0150566	+	0.999887i	$0.495207\pi$
$60$	0	0
$61$	−2.89310	−0.370423	−0.185212	−	0.982699i	$-0.559297\pi$
$61$	−2.89310	−0.370423	−0.185212	+	0.982699i	$0.559297\pi$
$62$	0	0
$63$	−3.33173	−0.419759
$64$	0	0
$65$	−3.86626	−0.479550
$66$	0	0
$67$	13.8533	1.69245	0.846226	−	0.532825i	$-0.178869\pi$
$67$	13.8533	1.69245	0.846226	+	0.532825i	$0.178869\pi$
$68$	0	0
$69$	−7.99353	−0.962307
$70$	0	0
$71$	9.42929	1.11905	0.559526	−	0.828813i	$-0.310983\pi$
$71$	9.42929	1.11905	0.559526	+	0.828813i	$0.310983\pi$
$72$	0	0
$73$	−2.95492	−0.345847	−0.172924	−	0.984935i	$-0.555321\pi$
$73$	−2.95492	−0.345847	−0.172924	+	0.984935i	$0.555321\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	21.1850	2.41426
$78$	0	0
$79$	14.4810	1.62924	0.814621	−	0.579993i	$-0.196945\pi$
$79$	14.4810	1.62924	0.814621	+	0.579993i	$0.196945\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.3521	1.57535	0.787674	−	0.616093i	$-0.211285\pi$
$83$	14.3521	1.57535	0.787674	+	0.616093i	$0.211285\pi$
$84$	0	0
$85$	−5.25814	−0.570326
$86$	0	0
$87$	7.12440	0.763816
$88$	0	0
$89$	10.8195	1.14687	0.573433	−	0.819252i	$-0.305611\pi$
$89$	10.8195	1.14687	0.573433	+	0.819252i	$0.305611\pi$
$90$	0	0
$91$	−12.8813	−1.35033
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.53453	0.260037
$96$	0	0
$97$	6.46261	0.656178	0.328089	−	0.944647i	$-0.393595\pi$
$97$	6.46261	0.656178	0.328089	+	0.944647i	$0.393595\pi$
$98$	0	0
$99$	−6.35857	−0.639060
$100$	0	0
$101$	4.99714	0.497234	0.248617	−	0.968602i	$-0.420024\pi$
$101$	4.99714	0.497234	0.248617	+	0.968602i	$0.420024\pi$
$102$	0	0
$103$	−5.86626	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$103$	−5.86626	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$104$	0	0
$105$	−3.33173	−0.325144
$106$	0	0
$107$	3.96474	0.383286	0.191643	−	0.981465i	$-0.438618\pi$
$107$	3.96474	0.383286	0.191643	+	0.981465i	$0.438618\pi$
$108$	0	0
$109$	−15.0156	−1.43823	−0.719115	−	0.694891i	$-0.755452\pi$
$109$	−15.0156	−1.43823	−0.719115	+	0.694891i	$0.755452\pi$
$110$	0	0
$111$	−2.79720	−0.265499
$112$	0	0
$113$	−14.8461	−1.39660	−0.698301	−	0.715805i	$-0.746060\pi$
$113$	−14.8461	−1.39660	−0.698301	+	0.715805i	$0.746060\pi$
$114$	0	0
$115$	−7.99353	−0.745400
$116$	0	0
$117$	3.86626	0.357436
$118$	0	0
$119$	−17.5187	−1.60594
$120$	0	0
$121$	29.4314	2.67558
$122$	0	0
$123$	−4.20085	−0.378778
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.2948	−1.53467	−0.767333	−	0.641249i	$-0.778417\pi$
$127$	−17.2948	−1.53467	−0.767333	+	0.641249i	$0.778417\pi$
$128$	0	0
$129$	4.73538	0.416927
$130$	0	0
$131$	0.231304	0.0202091	0.0101046	−	0.999949i	$-0.496784\pi$
$131$	0.231304	0.0202091	0.0101046	+	0.999949i	$0.496784\pi$
$132$	0	0
$133$	8.44437	0.732220
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.92160	−0.676788	−0.338394	−	0.941004i	$-0.609884\pi$
$137$	−7.92160	−0.676788	−0.338394	+	0.941004i	$0.609884\pi$
$138$	0	0
$139$	−7.36143	−0.624389	−0.312194	−	0.950018i	$-0.601064\pi$
$139$	−7.36143	−0.624389	−0.312194	+	0.950018i	$0.601064\pi$
$140$	0	0
$141$	1.63496	0.137688
$142$	0	0
$143$	−24.5839	−2.05581
$144$	0	0
$145$	7.12440	0.591650
$146$	0	0
$147$	−4.10043	−0.338197
$148$	0	0
$149$	−0.997137	−0.0816887	−0.0408443	−	0.999166i	$-0.513005\pi$
$149$	−0.997137	−0.0816887	−0.0408443	+	0.999166i	$0.513005\pi$
$150$	0	0
$151$	5.49111	0.446860	0.223430	−	0.974720i	$-0.428275\pi$
$151$	5.49111	0.446860	0.223430	+	0.974720i	$0.428275\pi$
$152$	0	0
$153$	5.25814	0.425096
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−3.78091	−0.301749	−0.150875	−	0.988553i	$-0.548209\pi$
$157$	−3.78091	−0.301749	−0.150875	+	0.988553i	$0.548209\pi$
$158$	0	0
$159$	4.37034	0.346590
$160$	0	0
$161$	−26.6323	−2.09892
$162$	0	0
$163$	−23.7761	−1.86229	−0.931144	−	0.364652i	$-0.881188\pi$
$163$	−23.7761	−1.86229	−0.931144	+	0.364652i	$0.881188\pi$
$164$	0	0
$165$	−6.35857	−0.495014
$166$	0	0
$167$	−5.95687	−0.460956	−0.230478	−	0.973078i	$-0.574029\pi$
$167$	−5.95687	−0.460956	−0.230478	+	0.973078i	$0.574029\pi$
$168$	0	0
$169$	1.94796	0.149843
$170$	0	0
$171$	−2.53453	−0.193820
$172$	0	0
$173$	−12.5163	−0.951596	−0.475798	−	0.879555i	$-0.657841\pi$
$173$	−12.5163	−0.951596	−0.475798	+	0.879555i	$0.657841\pi$
$174$	0	0
$175$	−3.33173	−0.251855
$176$	0	0
$177$	−0.231304	−0.0173859
$178$	0	0
$179$	0.430492	0.0321765	0.0160882	−	0.999871i	$-0.494879\pi$
$179$	0.430492	0.0321765	0.0160882	+	0.999871i	$0.494879\pi$
$180$	0	0
$181$	−9.89024	−0.735136	−0.367568	−	0.929997i	$-0.619809\pi$
$181$	−9.89024	−0.735136	−0.367568	+	0.929997i	$0.619809\pi$
$182$	0	0
$183$	2.89310	0.213864
$184$	0	0
$185$	−2.79720	−0.205654
$186$	0	0
$187$	−33.4343	−2.44496
$188$	0	0
$189$	3.33173	0.242348
$190$	0	0
$191$	4.83775	0.350048	0.175024	−	0.984564i	$-0.444000\pi$
$191$	4.83775	0.350048	0.175024	+	0.984564i	$0.444000\pi$
$192$	0	0
$193$	2.86431	0.206178	0.103089	−	0.994672i	$-0.467127\pi$
$193$	2.86431	0.206178	0.103089	+	0.994672i	$0.467127\pi$
$194$	0	0
$195$	3.86626	0.276869
$196$	0	0
$197$	22.8684	1.62930	0.814652	−	0.579949i	$-0.196928\pi$
$197$	22.8684	1.62930	0.814652	+	0.579949i	$0.196928\pi$
$198$	0	0
$199$	−14.2232	−1.00825	−0.504127	−	0.863630i	$-0.668186\pi$
$199$	−14.2232	−1.00825	−0.504127	+	0.863630i	$0.668186\pi$
$200$	0	0
$201$	−13.8533	−0.977137
$202$	0	0
$203$	23.7366	1.66598
$204$	0	0
$205$	−4.20085	−0.293400
$206$	0	0
$207$	7.99353	0.555588
$208$	0	0
$209$	16.1160	1.11477
$210$	0	0
$211$	−0.748330	−0.0515171	−0.0257586	−	0.999668i	$-0.508200\pi$
$211$	−0.748330	−0.0515171	−0.0257586	+	0.999668i	$0.508200\pi$
$212$	0	0
$213$	−9.42929	−0.646085
$214$	0	0
$215$	4.73538	0.322950
$216$	0	0
$217$	3.33173	0.226173
$218$	0	0
$219$	2.95492	0.199675
$220$	0	0
$221$	20.3294	1.36750
$222$	0	0
$223$	−11.1549	−0.746984	−0.373492	−	0.927633i	$-0.621840\pi$
$223$	−11.1549	−0.746984	−0.373492	+	0.927633i	$0.621840\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.5347	1.09745	0.548724	−	0.836004i	$-0.315114\pi$
$227$	16.5347	1.09745	0.548724	+	0.836004i	$0.315114\pi$
$228$	0	0
$229$	14.8749	0.982958	0.491479	−	0.870889i	$-0.336456\pi$
$229$	14.8749	0.982958	0.491479	+	0.870889i	$0.336456\pi$
$230$	0	0
$231$	−21.1850	−1.39387
$232$	0	0
$233$	13.0875	0.857389	0.428695	−	0.903449i	$-0.358974\pi$
$233$	13.0875	0.857389	0.428695	+	0.903449i	$0.358974\pi$
$234$	0	0
$235$	1.63496	0.106653
$236$	0	0
$237$	−14.4810	−0.940644
$238$	0	0
$239$	−16.5124	−1.06810	−0.534049	−	0.845454i	$-0.679330\pi$
$239$	−16.5124	−1.06810	−0.534049	+	0.845454i	$0.679330\pi$
$240$	0	0
$241$	−20.1644	−1.29890	−0.649451	−	0.760404i	$-0.725001\pi$
$241$	−20.1644	−1.29890	−0.649451	+	0.760404i	$0.725001\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.10043	−0.261967
$246$	0	0
$247$	−9.79915	−0.623505
$248$	0	0
$249$	−14.3521	−0.909527
$250$	0	0
$251$	−25.9525	−1.63811	−0.819055	−	0.573715i	$-0.805502\pi$
$251$	−25.9525	−1.63811	−0.819055	+	0.573715i	$0.805502\pi$
$252$	0	0
$253$	−50.8274	−3.19549
$254$	0	0
$255$	5.25814	0.329278
$256$	0	0
$257$	−28.1508	−1.75600	−0.878000	−	0.478661i	$-0.841122\pi$
$257$	−28.1508	−1.75600	−0.878000	+	0.478661i	$0.841122\pi$
$258$	0	0
$259$	−9.31952	−0.579087
$260$	0	0
$261$	−7.12440	−0.440990
$262$	0	0
$263$	22.9468	1.41496	0.707480	−	0.706734i	$-0.249832\pi$
$263$	22.9468	1.41496	0.707480	+	0.706734i	$0.249832\pi$
$264$	0	0
$265$	4.37034	0.268468
$266$	0	0
$267$	−10.8195	−0.662144
$268$	0	0
$269$	−11.1871	−0.682092	−0.341046	−	0.940047i	$-0.610781\pi$
$269$	−11.1871	−0.682092	−0.341046	+	0.940047i	$0.610781\pi$
$270$	0	0
$271$	−13.5134	−0.820882	−0.410441	−	0.911887i	$-0.634625\pi$
$271$	−13.5134	−0.820882	−0.410441	+	0.911887i	$0.634625\pi$
$272$	0	0
$273$	12.8813	0.779614
$274$	0	0
$275$	−6.35857	−0.383436
$276$	0	0
$277$	14.2013	0.853276	0.426638	−	0.904423i	$-0.359698\pi$
$277$	14.2013	0.853276	0.426638	+	0.904423i	$0.359698\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−16.7944	−1.00187	−0.500934	−	0.865486i	$-0.667010\pi$
$281$	−16.7944	−1.00187	−0.500934	+	0.865486i	$0.667010\pi$
$282$	0	0
$283$	−28.8949	−1.71763	−0.858813	−	0.512289i	$-0.828798\pi$
$283$	−28.8949	−1.71763	−0.858813	+	0.512289i	$0.828798\pi$
$284$	0	0
$285$	−2.53453	−0.150133
$286$	0	0
$287$	−13.9961	−0.826164
$288$	0	0
$289$	10.6481	0.626358
$290$	0	0
$291$	−6.46261	−0.378845
$292$	0	0
$293$	22.6009	1.32036	0.660179	−	0.751108i	$-0.270480\pi$
$293$	22.6009	1.32036	0.660179	+	0.751108i	$0.270480\pi$
$294$	0	0
$295$	−0.231304	−0.0134670
$296$	0	0
$297$	6.35857	0.368962
$298$	0	0
$299$	30.9050	1.78728
$300$	0	0
$301$	15.7770	0.909372
$302$	0	0
$303$	−4.99714	−0.287078
$304$	0	0
$305$	2.89310	0.165658
$306$	0	0
$307$	−29.1362	−1.66289	−0.831445	−	0.555608i	$-0.812486\pi$
$307$	−29.1362	−1.66289	−0.831445	+	0.555608i	$0.812486\pi$
$308$	0	0
$309$	5.86626	0.333720
$310$	0	0
$311$	4.37848	0.248281	0.124140	−	0.992265i	$-0.460383\pi$
$311$	4.37848	0.248281	0.124140	+	0.992265i	$0.460383\pi$
$312$	0	0
$313$	−19.8139	−1.11995	−0.559975	−	0.828509i	$-0.689189\pi$
$313$	−19.8139	−1.11995	−0.559975	+	0.828509i	$0.689189\pi$
$314$	0	0
$315$	3.33173	0.187722
$316$	0	0
$317$	−25.3603	−1.42437	−0.712187	−	0.701990i	$-0.752295\pi$
$317$	−25.3603	−1.42437	−0.712187	+	0.701990i	$0.752295\pi$
$318$	0	0
$319$	45.3010	2.53637
$320$	0	0
$321$	−3.96474	−0.221290
$322$	0	0
$323$	−13.3269	−0.741530
$324$	0	0
$325$	3.86626	0.214461
$326$	0	0
$327$	15.0156	0.830362
$328$	0	0
$329$	5.44723	0.300316
$330$	0	0
$331$	27.6843	1.52167	0.760833	−	0.648948i	$-0.224791\pi$
$331$	27.6843	1.52167	0.760833	+	0.648948i	$0.224791\pi$
$332$	0	0
$333$	2.79720	0.153286
$334$	0	0
$335$	−13.8533	−0.756887
$336$	0	0
$337$	−7.72678	−0.420905	−0.210452	−	0.977604i	$-0.567494\pi$
$337$	−7.72678	−0.420905	−0.210452	+	0.977604i	$0.567494\pi$
$338$	0	0
$339$	14.8461	0.806328
$340$	0	0
$341$	6.35857	0.344336
$342$	0	0
$343$	9.66060	0.521623
$344$	0	0
$345$	7.99353	0.430357
$346$	0	0
$347$	−29.4578	−1.58138	−0.790689	−	0.612218i	$-0.790278\pi$
$347$	−29.4578	−1.58138	−0.790689	+	0.612218i	$0.790278\pi$
$348$	0	0
$349$	30.9399	1.65617	0.828087	−	0.560600i	$-0.189429\pi$
$349$	30.9399	1.65617	0.828087	+	0.560600i	$0.189429\pi$
$350$	0	0
$351$	−3.86626	−0.206366
$352$	0	0
$353$	−33.0622	−1.75972	−0.879860	−	0.475232i	$-0.842364\pi$
$353$	−33.0622	−1.75972	−0.879860	+	0.475232i	$0.842364\pi$
$354$	0	0
$355$	−9.42929	−0.500455
$356$	0	0
$357$	17.5187	0.927189
$358$	0	0
$359$	−6.87652	−0.362929	−0.181465	−	0.983397i	$-0.558084\pi$
$359$	−6.87652	−0.362929	−0.181465	+	0.983397i	$0.558084\pi$
$360$	0	0
$361$	−12.5762	−0.661903
$362$	0	0
$363$	−29.4314	−1.54475
$364$	0	0
$365$	2.95492	0.154668
$366$	0	0
$367$	22.4746	1.17316	0.586581	−	0.809891i	$-0.300474\pi$
$367$	22.4746	1.17316	0.586581	+	0.809891i	$0.300474\pi$
$368$	0	0
$369$	4.20085	0.218688
$370$	0	0
$371$	14.5608	0.755958
$372$	0	0
$373$	−10.7483	−0.556527	−0.278264	−	0.960505i	$-0.589759\pi$
$373$	−10.7483	−0.556527	−0.278264	+	0.960505i	$0.589759\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−27.5448	−1.41863
$378$	0	0
$379$	19.4338	0.998250	0.499125	−	0.866530i	$-0.333655\pi$
$379$	19.4338	0.998250	0.499125	+	0.866530i	$0.333655\pi$
$380$	0	0
$381$	17.2948	0.886040
$382$	0	0
$383$	30.9201	1.57995	0.789973	−	0.613142i	$-0.210095\pi$
$383$	30.9201	1.57995	0.789973	+	0.613142i	$0.210095\pi$
$384$	0	0
$385$	−21.1850	−1.07969
$386$	0	0
$387$	−4.73538	−0.240713
$388$	0	0
$389$	−16.1935	−0.821041	−0.410521	−	0.911851i	$-0.634653\pi$
$389$	−16.1935	−0.821041	−0.410521	+	0.911851i	$0.634653\pi$
$390$	0	0
$391$	42.0311	2.12560
$392$	0	0
$393$	−0.231304	−0.0116677
$394$	0	0
$395$	−14.4810	−0.728620
$396$	0	0
$397$	−2.61174	−0.131080	−0.0655398	−	0.997850i	$-0.520877\pi$
$397$	−2.61174	−0.131080	−0.0655398	+	0.997850i	$0.520877\pi$
$398$	0	0
$399$	−8.44437	−0.422747
$400$	0	0
$401$	24.2807	1.21252	0.606259	−	0.795267i	$-0.292669\pi$
$401$	24.2807	1.21252	0.606259	+	0.795267i	$0.292669\pi$
$402$	0	0
$403$	−3.86626	−0.192592
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−17.7862	−0.881629
$408$	0	0
$409$	−2.58292	−0.127717	−0.0638585	−	0.997959i	$-0.520341\pi$
$409$	−2.58292	−0.127717	−0.0638585	+	0.997959i	$0.520341\pi$
$410$	0	0
$411$	7.92160	0.390744
$412$	0	0
$413$	−0.770643	−0.0379208
$414$	0	0
$415$	−14.3521	−0.704517
$416$	0	0
$417$	7.36143	0.360491
$418$	0	0
$419$	−13.3501	−0.652197	−0.326099	−	0.945336i	$-0.605734\pi$
$419$	−13.3501	−0.652197	−0.326099	+	0.945336i	$0.605734\pi$
$420$	0	0
$421$	15.1359	0.737677	0.368838	−	0.929493i	$-0.379756\pi$
$421$	15.1359	0.737677	0.368838	+	0.929493i	$0.379756\pi$
$422$	0	0
$423$	−1.63496	−0.0794943
$424$	0	0
$425$	5.25814	0.255057
$426$	0	0
$427$	9.63903	0.466465
$428$	0	0
$429$	24.5839	1.18692
$430$	0	0
$431$	14.8191	0.713810	0.356905	−	0.934141i	$-0.383832\pi$
$431$	14.8191	0.713810	0.356905	+	0.934141i	$0.383832\pi$
$432$	0	0
$433$	−17.1558	−0.824454	−0.412227	−	0.911081i	$-0.635249\pi$
$433$	−17.1558	−0.824454	−0.412227	+	0.911081i	$0.635249\pi$
$434$	0	0
$435$	−7.12440	−0.341589
$436$	0	0
$437$	−20.2598	−0.969159
$438$	0	0
$439$	−1.47076	−0.0701957	−0.0350978	−	0.999384i	$-0.511174\pi$
$439$	−1.47076	−0.0701957	−0.0350978	+	0.999384i	$0.511174\pi$
$440$	0	0
$441$	4.10043	0.195258
$442$	0	0
$443$	35.5534	1.68919	0.844597	−	0.535403i	$-0.179840\pi$
$443$	35.5534	1.68919	0.844597	+	0.535403i	$0.179840\pi$
$444$	0	0
$445$	−10.8195	−0.512894
$446$	0	0
$447$	0.997137	0.0471630
$448$	0	0
$449$	−30.5965	−1.44394	−0.721970	−	0.691925i	$-0.756763\pi$
$449$	−30.5965	−1.44394	−0.721970	+	0.691925i	$0.756763\pi$
$450$	0	0
$451$	−26.7114	−1.25779
$452$	0	0
$453$	−5.49111	−0.257995
$454$	0	0
$455$	12.8813	0.603886
$456$	0	0
$457$	−19.9684	−0.934081	−0.467040	−	0.884236i	$-0.654680\pi$
$457$	−19.9684	−0.934081	−0.467040	+	0.884236i	$0.654680\pi$
$458$	0	0
$459$	−5.25814	−0.245429
$460$	0	0
$461$	24.1805	1.12620	0.563099	−	0.826389i	$-0.309609\pi$
$461$	24.1805	1.12620	0.563099	+	0.826389i	$0.309609\pi$
$462$	0	0
$463$	−7.43292	−0.345437	−0.172719	−	0.984971i	$-0.555255\pi$
$463$	−7.43292	−0.345437	−0.172719	+	0.984971i	$0.555255\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−29.6220	−1.37074	−0.685372	−	0.728194i	$-0.740360\pi$
$467$	−29.6220	−1.37074	−0.685372	+	0.728194i	$0.740360\pi$
$468$	0	0
$469$	−46.1555	−2.13126
$470$	0	0
$471$	3.78091	0.174215
$472$	0	0
$473$	30.1103	1.38447
$474$	0	0
$475$	−2.53453	−0.116292
$476$	0	0
$477$	−4.37034	−0.200104
$478$	0	0
$479$	4.10237	0.187442	0.0937211	−	0.995598i	$-0.470124\pi$
$479$	4.10237	0.187442	0.0937211	+	0.995598i	$0.470124\pi$
$480$	0	0
$481$	10.8147	0.493108
$482$	0	0
$483$	26.6323	1.21181
$484$	0	0
$485$	−6.46261	−0.293452
$486$	0	0
$487$	20.0062	0.906568	0.453284	−	0.891366i	$-0.350252\pi$
$487$	20.0062	0.906568	0.453284	+	0.891366i	$0.350252\pi$
$488$	0	0
$489$	23.7761	1.07519
$490$	0	0
$491$	5.77565	0.260652	0.130326	−	0.991471i	$-0.458398\pi$
$491$	5.77565	0.260652	0.130326	+	0.991471i	$0.458398\pi$
$492$	0	0
$493$	−37.4611	−1.68717
$494$	0	0
$495$	6.35857	0.285796
$496$	0	0
$497$	−31.4159	−1.40919
$498$	0	0
$499$	−13.6265	−0.610007	−0.305003	−	0.952351i	$-0.598658\pi$
$499$	−13.6265	−0.610007	−0.305003	+	0.952351i	$0.598658\pi$
$500$	0	0
$501$	5.95687	0.266133
$502$	0	0
$503$	−33.9245	−1.51262	−0.756309	−	0.654214i	$-0.772999\pi$
$503$	−33.9245	−1.51262	−0.756309	+	0.654214i	$0.772999\pi$
$504$	0	0
$505$	−4.99714	−0.222370
$506$	0	0
$507$	−1.94796	−0.0865120
$508$	0	0
$509$	−19.2586	−0.853624	−0.426812	−	0.904340i	$-0.640363\pi$
$509$	−19.2586	−0.853624	−0.426812	+	0.904340i	$0.640363\pi$
$510$	0	0
$511$	9.84499	0.435517
$512$	0	0
$513$	2.53453	0.111902
$514$	0	0
$515$	5.86626	0.258498
$516$	0	0
$517$	10.3960	0.457215
$518$	0	0
$519$	12.5163	0.549404
$520$	0	0
$521$	16.1381	0.707024	0.353512	−	0.935430i	$-0.384987\pi$
$521$	16.1381	0.707024	0.353512	+	0.935430i	$0.384987\pi$
$522$	0	0
$523$	−22.9573	−1.00385	−0.501927	−	0.864910i	$-0.667375\pi$
$523$	−22.9573	−1.00385	−0.501927	+	0.864910i	$0.667375\pi$
$524$	0	0
$525$	3.33173	0.145409
$526$	0	0
$527$	−5.25814	−0.229048
$528$	0	0
$529$	40.8965	1.77811
$530$	0	0
$531$	0.231304	0.0100377
$532$	0	0
$533$	16.2416	0.703501
$534$	0	0
$535$	−3.96474	−0.171411
$536$	0	0
$537$	−0.430492	−0.0185771
$538$	0	0
$539$	−26.0728	−1.12304
$540$	0	0
$541$	39.4619	1.69660	0.848300	−	0.529517i	$-0.177627\pi$
$541$	39.4619	1.69660	0.848300	+	0.529517i	$0.177627\pi$
$542$	0	0
$543$	9.89024	0.424431
$544$	0	0
$545$	15.0156	0.643196
$546$	0	0
$547$	−37.1841	−1.58988	−0.794939	−	0.606689i	$-0.792497\pi$
$547$	−37.1841	−1.58988	−0.794939	+	0.606689i	$0.792497\pi$
$548$	0	0
$549$	−2.89310	−0.123474
$550$	0	0
$551$	18.0570	0.769254
$552$	0	0
$553$	−48.2469	−2.05167
$554$	0	0
$555$	2.79720	0.118735
$556$	0	0
$557$	−35.2769	−1.49473	−0.747365	−	0.664414i	$-0.768681\pi$
$557$	−35.2769	−1.49473	−0.747365	+	0.664414i	$0.768681\pi$
$558$	0	0
$559$	−18.3082	−0.774355
$560$	0	0
$561$	33.4343	1.41160
$562$	0	0
$563$	30.8416	1.29982	0.649909	−	0.760012i	$-0.274807\pi$
$563$	30.8416	1.29982	0.649909	+	0.760012i	$0.274807\pi$
$564$	0	0
$565$	14.8461	0.624579
$566$	0	0
$567$	−3.33173	−0.139920
$568$	0	0
$569$	14.2993	0.599459	0.299730	−	0.954024i	$-0.403104\pi$
$569$	14.2993	0.599459	0.299730	+	0.954024i	$0.403104\pi$
$570$	0	0
$571$	−13.9655	−0.584437	−0.292219	−	0.956352i	$-0.594394\pi$
$571$	−13.9655	−0.584437	−0.292219	+	0.956352i	$0.594394\pi$
$572$	0	0
$573$	−4.83775	−0.202100
$574$	0	0
$575$	7.99353	0.333353
$576$	0	0
$577$	−18.2343	−0.759105	−0.379553	−	0.925170i	$-0.623922\pi$
$577$	−18.2343	−0.759105	−0.379553	+	0.925170i	$0.623922\pi$
$578$	0	0
$579$	−2.86431	−0.119037
$580$	0	0
$581$	−47.8173	−1.98380
$582$	0	0
$583$	27.7891	1.15091
$584$	0	0
$585$	−3.86626	−0.159850
$586$	0	0
$587$	25.6529	1.05881	0.529405	−	0.848369i	$-0.322415\pi$
$587$	25.6529	1.05881	0.529405	+	0.848369i	$0.322415\pi$
$588$	0	0
$589$	2.53453	0.104433
$590$	0	0
$591$	−22.8684	−0.940680
$592$	0	0
$593$	−0.722866	−0.0296846	−0.0148423	−	0.999890i	$-0.504725\pi$
$593$	−0.722866	−0.0296846	−0.0148423	+	0.999890i	$0.504725\pi$
$594$	0	0
$595$	17.5187	0.718197
$596$	0	0
$597$	14.2232	0.582115
$598$	0	0
$599$	−27.9954	−1.14386	−0.571930	−	0.820302i	$-0.693805\pi$
$599$	−27.9954	−1.14386	−0.571930	+	0.820302i	$0.693805\pi$
$600$	0	0
$601$	−18.3058	−0.746710	−0.373355	−	0.927689i	$-0.621793\pi$
$601$	−18.3058	−0.746710	−0.373355	+	0.927689i	$0.621793\pi$
$602$	0	0
$603$	13.8533	0.564150
$604$	0	0
$605$	−29.4314	−1.19656
$606$	0	0
$607$	−15.7943	−0.641072	−0.320536	−	0.947236i	$-0.603863\pi$
$607$	−15.7943	−0.641072	−0.320536	+	0.947236i	$0.603863\pi$
$608$	0	0
$609$	−23.7366	−0.961855
$610$	0	0
$611$	−6.32116	−0.255727
$612$	0	0
$613$	−32.6461	−1.31857	−0.659283	−	0.751895i	$-0.729140\pi$
$613$	−32.6461	−1.31857	−0.659283	+	0.751895i	$0.729140\pi$
$614$	0	0
$615$	4.20085	0.169395
$616$	0	0
$617$	24.8639	1.00098	0.500491	−	0.865742i	$-0.333153\pi$
$617$	24.8639	1.00098	0.500491	+	0.865742i	$0.333153\pi$
$618$	0	0
$619$	−26.3181	−1.05782	−0.528908	−	0.848679i	$-0.677398\pi$
$619$	−26.3181	−1.05782	−0.528908	+	0.848679i	$0.677398\pi$
$620$	0	0
$621$	−7.99353	−0.320769
$622$	0	0
$623$	−36.0477	−1.44422
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.1160	−0.643610
$628$	0	0
$629$	14.7081	0.586450
$630$	0	0
$631$	19.1793	0.763517	0.381758	−	0.924262i	$-0.375319\pi$
$631$	19.1793	0.763517	0.381758	+	0.924262i	$0.375319\pi$
$632$	0	0
$633$	0.748330	0.0297434
$634$	0	0
$635$	17.2948	0.686323
$636$	0	0
$637$	15.8533	0.628131
$638$	0	0
$639$	9.42929	0.373017
$640$	0	0
$641$	−20.7346	−0.818969	−0.409485	−	0.912317i	$-0.634291\pi$
$641$	−20.7346	−0.818969	−0.409485	+	0.912317i	$0.634291\pi$
$642$	0	0
$643$	3.17835	0.125342	0.0626710	−	0.998034i	$-0.480038\pi$
$643$	3.17835	0.125342	0.0626710	+	0.998034i	$0.480038\pi$
$644$	0	0
$645$	−4.73538	−0.186455
$646$	0	0
$647$	46.5043	1.82827	0.914136	−	0.405408i	$-0.132871\pi$
$647$	46.5043	1.82827	0.914136	+	0.405408i	$0.132871\pi$
$648$	0	0
$649$	−1.47076	−0.0577325
$650$	0	0
$651$	−3.33173	−0.130581
$652$	0	0
$653$	9.23994	0.361587	0.180793	−	0.983521i	$-0.442133\pi$
$653$	9.23994	0.361587	0.180793	+	0.983521i	$0.442133\pi$
$654$	0	0
$655$	−0.231304	−0.00903780
$656$	0	0
$657$	−2.95492	−0.115282
$658$	0	0
$659$	−7.18578	−0.279918	−0.139959	−	0.990157i	$-0.544697\pi$
$659$	−7.18578	−0.279918	−0.139959	+	0.990157i	$0.544697\pi$
$660$	0	0
$661$	29.2962	1.13949	0.569744	−	0.821822i	$-0.307042\pi$
$661$	29.2962	1.13949	0.569744	+	0.821822i	$0.307042\pi$
$662$	0	0
$663$	−20.3294	−0.789527
$664$	0	0
$665$	−8.44437	−0.327459
$666$	0	0
$667$	−56.9491	−2.20508
$668$	0	0
$669$	11.1549	0.431272
$670$	0	0
$671$	18.3960	0.710169
$672$	0	0
$673$	−10.2771	−0.396154	−0.198077	−	0.980187i	$-0.563470\pi$
$673$	−10.2771	−0.396154	−0.198077	+	0.980187i	$0.563470\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−5.88809	−0.226298	−0.113149	−	0.993578i	$-0.536094\pi$
$677$	−5.88809	−0.226298	−0.113149	+	0.993578i	$0.536094\pi$
$678$	0	0
$679$	−21.5317	−0.826310
$680$	0	0
$681$	−16.5347	−0.633611
$682$	0	0
$683$	45.0242	1.72280	0.861401	−	0.507925i	$-0.169587\pi$
$683$	45.0242	1.72280	0.861401	+	0.507925i	$0.169587\pi$
$684$	0	0
$685$	7.92160	0.302669
$686$	0	0
$687$	−14.8749	−0.567511
$688$	0	0
$689$	−16.8969	−0.643719
$690$	0	0
$691$	−21.5867	−0.821199	−0.410599	−	0.911816i	$-0.634680\pi$
$691$	−21.5867	−0.821199	−0.410599	+	0.911816i	$0.634680\pi$
$692$	0	0
$693$	21.1850	0.804753
$694$	0	0
$695$	7.36143	0.279235
$696$	0	0
$697$	22.0887	0.836669
$698$	0	0
$699$	−13.0875	−0.495014
$700$	0	0
$701$	29.9453	1.13102	0.565509	−	0.824742i	$-0.308680\pi$
$701$	29.9453	1.13102	0.565509	+	0.824742i	$0.308680\pi$
$702$	0	0
$703$	−7.08959	−0.267389
$704$	0	0
$705$	−1.63496	−0.0615760
$706$	0	0
$707$	−16.6491	−0.626154
$708$	0	0
$709$	−45.5287	−1.70987	−0.854933	−	0.518738i	$-0.826402\pi$
$709$	−45.5287	−1.70987	−0.854933	+	0.518738i	$0.826402\pi$
$710$	0	0
$711$	14.4810	0.543081
$712$	0	0
$713$	−7.99353	−0.299360
$714$	0	0
$715$	24.5839	0.919385
$716$	0	0
$717$	16.5124	0.616667
$718$	0	0
$719$	11.1194	0.414684	0.207342	−	0.978269i	$-0.433519\pi$
$719$	11.1194	0.414684	0.207342	+	0.978269i	$0.433519\pi$
$720$	0	0
$721$	19.5448	0.727886
$722$	0	0
$723$	20.1644	0.749921
$724$	0	0
$725$	−7.12440	−0.264594
$726$	0	0
$727$	−5.19361	−0.192620	−0.0963102	−	0.995351i	$-0.530704\pi$
$727$	−5.19361	−0.192620	−0.0963102	+	0.995351i	$0.530704\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.8993	−0.920935
$732$	0	0
$733$	−12.8106	−0.473171	−0.236586	−	0.971611i	$-0.576028\pi$
$733$	−12.8106	−0.473171	−0.236586	+	0.971611i	$0.576028\pi$
$734$	0	0
$735$	4.10043	0.151246
$736$	0	0
$737$	−88.0873	−3.24474
$738$	0	0
$739$	7.16076	0.263413	0.131706	−	0.991289i	$-0.457954\pi$
$739$	7.16076	0.263413	0.131706	+	0.991289i	$0.457954\pi$
$740$	0	0
$741$	9.79915	0.359981
$742$	0	0
$743$	−7.40410	−0.271630	−0.135815	−	0.990734i	$-0.543365\pi$
$743$	−7.40410	−0.271630	−0.135815	+	0.990734i	$0.543365\pi$
$744$	0	0
$745$	0.997137	0.0365323
$746$	0	0
$747$	14.3521	0.525116
$748$	0	0
$749$	−13.2094	−0.482662
$750$	0	0
$751$	6.20085	0.226272	0.113136	−	0.993579i	$-0.463910\pi$
$751$	6.20085	0.226272	0.113136	+	0.993579i	$0.463910\pi$
$752$	0	0
$753$	25.9525	0.945763
$754$	0	0
$755$	−5.49111	−0.199842
$756$	0	0
$757$	−16.9833	−0.617268	−0.308634	−	0.951181i	$-0.599872\pi$
$757$	−16.9833	−0.617268	−0.308634	+	0.951181i	$0.599872\pi$
$758$	0	0
$759$	50.8274	1.84492
$760$	0	0
$761$	42.1190	1.52681	0.763406	−	0.645918i	$-0.223525\pi$
$761$	42.1190	1.52681	0.763406	+	0.645918i	$0.223525\pi$
$762$	0	0
$763$	50.0278	1.81113
$764$	0	0
$765$	−5.25814	−0.190109
$766$	0	0
$767$	0.894281	0.0322906
$768$	0	0
$769$	−1.75483	−0.0632809	−0.0316404	−	0.999499i	$-0.510073\pi$
$769$	−1.75483	−0.0632809	−0.0316404	+	0.999499i	$0.510073\pi$
$770$	0	0
$771$	28.1508	1.01383
$772$	0	0
$773$	−53.5209	−1.92501	−0.962506	−	0.271261i	$-0.912559\pi$
$773$	−53.5209	−1.92501	−0.962506	+	0.271261i	$0.912559\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	9.31952	0.334336
$778$	0	0
$779$	−10.6472	−0.381475
$780$	0	0
$781$	−59.9568	−2.14542
$782$	0	0
$783$	7.12440	0.254605
$784$	0	0
$785$	3.78091	0.134946
$786$	0	0
$787$	−20.9338	−0.746211	−0.373105	−	0.927789i	$-0.621707\pi$
$787$	−20.9338	−0.746211	−0.373105	+	0.927789i	$0.621707\pi$
$788$	0	0
$789$	−22.9468	−0.816927
$790$	0	0
$791$	49.4631	1.75871
$792$	0	0
$793$	−11.1855	−0.397208
$794$	0	0
$795$	−4.37034	−0.155000
$796$	0	0
$797$	28.2188	0.999563	0.499781	−	0.866152i	$-0.333414\pi$
$797$	28.2188	0.999563	0.499781	+	0.866152i	$0.333414\pi$
$798$	0	0
$799$	−8.59683	−0.304134
$800$	0	0
$801$	10.8195	0.382289
$802$	0	0
$803$	18.7891	0.663052
$804$	0	0
$805$	26.6323	0.938664
$806$	0	0
$807$	11.1871	0.393806
$808$	0	0
$809$	35.1987	1.23752	0.618759	−	0.785581i	$-0.287636\pi$
$809$	35.1987	1.23752	0.618759	+	0.785581i	$0.287636\pi$
$810$	0	0
$811$	4.33368	0.152176	0.0760880	−	0.997101i	$-0.475757\pi$
$811$	4.33368	0.152176	0.0760880	+	0.997101i	$0.475757\pi$
$812$	0	0
$813$	13.5134	0.473937
$814$	0	0
$815$	23.7761	0.832840
$816$	0	0
$817$	12.0020	0.419896
$818$	0	0
$819$	−12.8813	−0.450110
$820$	0	0
$821$	−13.2390	−0.462044	−0.231022	−	0.972949i	$-0.574207\pi$
$821$	−13.2390	−0.462044	−0.231022	+	0.972949i	$0.574207\pi$
$822$	0	0
$823$	51.5038	1.79531	0.897655	−	0.440699i	$-0.145269\pi$
$823$	51.5038	1.79531	0.897655	+	0.440699i	$0.145269\pi$
$824$	0	0
$825$	6.35857	0.221377
$826$	0	0
$827$	−43.2252	−1.50309	−0.751543	−	0.659684i	$-0.770690\pi$
$827$	−43.2252	−1.50309	−0.751543	+	0.659684i	$0.770690\pi$
$828$	0	0
$829$	6.70659	0.232930	0.116465	−	0.993195i	$-0.462844\pi$
$829$	6.70659	0.232930	0.116465	+	0.993195i	$0.462844\pi$
$830$	0	0
$831$	−14.2013	−0.492639
$832$	0	0
$833$	21.5606	0.747032
$834$	0	0
$835$	5.95687	0.206146
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−15.5545	−0.537000	−0.268500	−	0.963280i	$-0.586528\pi$
$839$	−15.5545	−0.537000	−0.268500	+	0.963280i	$0.586528\pi$
$840$	0	0
$841$	21.7571	0.750246
$842$	0	0
$843$	16.7944	0.578428
$844$	0	0
$845$	−1.94796	−0.0670119
$846$	0	0
$847$	−98.0575	−3.36930
$848$	0	0
$849$	28.8949	0.991672
$850$	0	0
$851$	22.3595	0.766474
$852$	0	0
$853$	−3.96456	−0.135744	−0.0678720	−	0.997694i	$-0.521621\pi$
$853$	−3.96456	−0.135744	−0.0678720	+	0.997694i	$0.521621\pi$
$854$	0	0
$855$	2.53453	0.0866791
$856$	0	0
$857$	31.9056	1.08988	0.544938	−	0.838476i	$-0.316553\pi$
$857$	31.9056	1.08988	0.544938	+	0.838476i	$0.316553\pi$
$858$	0	0
$859$	−45.3063	−1.54583	−0.772915	−	0.634509i	$-0.781202\pi$
$859$	−45.3063	−1.54583	−0.772915	+	0.634509i	$0.781202\pi$
$860$	0	0
$861$	13.9961	0.476986
$862$	0	0
$863$	45.9074	1.56271	0.781354	−	0.624088i	$-0.214530\pi$
$863$	45.9074	1.56271	0.781354	+	0.624088i	$0.214530\pi$
$864$	0	0
$865$	12.5163	0.425567
$866$	0	0
$867$	−10.6481	−0.361628
$868$	0	0
$869$	−92.0786	−3.12355
$870$	0	0
$871$	53.5605	1.81483
$872$	0	0
$873$	6.46261	0.218726
$874$	0	0
$875$	3.33173	0.112633
$876$	0	0
$877$	21.7675	0.735037	0.367519	−	0.930016i	$-0.380207\pi$
$877$	21.7675	0.735037	0.367519	+	0.930016i	$0.380207\pi$
$878$	0	0
$879$	−22.6009	−0.762310
$880$	0	0
$881$	10.1686	0.342588	0.171294	−	0.985220i	$-0.445205\pi$
$881$	10.1686	0.342588	0.171294	+	0.985220i	$0.445205\pi$
$882$	0	0
$883$	−48.6999	−1.63888	−0.819441	−	0.573164i	$-0.805716\pi$
$883$	−48.6999	−1.63888	−0.819441	+	0.573164i	$0.805716\pi$
$884$	0	0
$885$	0.231304	0.00777520
$886$	0	0
$887$	−4.23568	−0.142220	−0.0711101	−	0.997468i	$-0.522654\pi$
$887$	−4.23568	−0.142220	−0.0711101	+	0.997468i	$0.522654\pi$
$888$	0	0
$889$	57.6216	1.93257
$890$	0	0
$891$	−6.35857	−0.213020
$892$	0	0
$893$	4.14384	0.138668
$894$	0	0
$895$	−0.430492	−0.0143898
$896$	0	0
$897$	−30.9050	−1.03189
$898$	0	0
$899$	7.12440	0.237612
$900$	0	0
$901$	−22.9799	−0.765570
$902$	0	0
$903$	−15.7770	−0.525026
$904$	0	0
$905$	9.89024	0.328763
$906$	0	0
$907$	31.9176	1.05981	0.529903	−	0.848058i	$-0.322228\pi$
$907$	31.9176	1.05981	0.529903	+	0.848058i	$0.322228\pi$
$908$	0	0
$909$	4.99714	0.165745
$910$	0	0
$911$	12.3667	0.409728	0.204864	−	0.978790i	$-0.434325\pi$
$911$	12.3667	0.409728	0.204864	+	0.978790i	$0.434325\pi$
$912$	0	0
$913$	−91.2588	−3.02023
$914$	0	0
$915$	−2.89310	−0.0956429
$916$	0	0
$917$	−0.770643	−0.0254489
$918$	0	0
$919$	−25.8456	−0.852568	−0.426284	−	0.904589i	$-0.640178\pi$
$919$	−25.8456	−0.852568	−0.426284	+	0.904589i	$0.640178\pi$
$920$	0	0
$921$	29.1362	0.960069
$922$	0	0
$923$	36.4561	1.19997
$924$	0	0
$925$	2.79720	0.0919714
$926$	0	0
$927$	−5.86626	−0.192673
$928$	0	0
$929$	−12.4441	−0.408278	−0.204139	−	0.978942i	$-0.565439\pi$
$929$	−12.4441	−0.408278	−0.204139	+	0.978942i	$0.565439\pi$
$930$	0	0
$931$	−10.3927	−0.340605
$932$	0	0
$933$	−4.37848	−0.143345
$934$	0	0
$935$	33.4343	1.09342
$936$	0	0
$937$	−20.7708	−0.678553	−0.339277	−	0.940687i	$-0.610182\pi$
$937$	−20.7708	−0.678553	−0.339277	+	0.940687i	$0.610182\pi$
$938$	0	0
$939$	19.8139	0.646603
$940$	0	0
$941$	27.1211	0.884122	0.442061	−	0.896985i	$-0.354248\pi$
$941$	27.1211	0.884122	0.442061	+	0.896985i	$0.354248\pi$
$942$	0	0
$943$	33.5796	1.09350
$944$	0	0
$945$	−3.33173	−0.108381
$946$	0	0
$947$	−10.3913	−0.337672	−0.168836	−	0.985644i	$-0.554001\pi$
$947$	−10.3913	−0.337672	−0.168836	+	0.985644i	$0.554001\pi$
$948$	0	0
$949$	−11.4245	−0.370854
$950$	0	0
$951$	25.3603	0.822363
$952$	0	0
$953$	29.9657	0.970683	0.485342	−	0.874325i	$-0.338695\pi$
$953$	29.9657	0.970683	0.485342	+	0.874325i	$0.338695\pi$
$954$	0	0
$955$	−4.83775	−0.156546
$956$	0	0
$957$	−45.3010	−1.46437
$958$	0	0
$959$	26.3927	0.852263
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	3.96474	0.127762
$964$	0	0
$965$	−2.86431	−0.0922055
$966$	0	0
$967$	−32.6754	−1.05077	−0.525385	−	0.850865i	$-0.676079\pi$
$967$	−32.6754	−1.05077	−0.525385	+	0.850865i	$0.676079\pi$
$968$	0	0
$969$	13.3269	0.428122
$970$	0	0
$971$	50.8324	1.63129	0.815644	−	0.578553i	$-0.196383\pi$
$971$	50.8324	1.63129	0.815644	+	0.578553i	$0.196383\pi$
$972$	0	0
$973$	24.5263	0.786277
$974$	0	0
$975$	−3.86626	−0.123819
$976$	0	0
$977$	9.87725	0.316001	0.158001	−	0.987439i	$-0.449495\pi$
$977$	9.87725	0.316001	0.158001	+	0.987439i	$0.449495\pi$
$978$	0	0
$979$	−68.7966	−2.19875
$980$	0	0
$981$	−15.0156	−0.479410
$982$	0	0
$983$	27.9712	0.892143	0.446072	−	0.894997i	$-0.352823\pi$
$983$	27.9712	0.892143	0.446072	+	0.894997i	$0.352823\pi$
$984$	0	0
$985$	−22.8684	−0.728647
$986$	0	0
$987$	−5.44723	−0.173387
$988$	0	0
$989$	−37.8524	−1.20364
$990$	0	0
$991$	−55.8259	−1.77337	−0.886685	−	0.462375i	$-0.846998\pi$
$991$	−55.8259	−1.77337	−0.886685	+	0.462375i	$0.846998\pi$
$992$	0	0
$993$	−27.6843	−0.878534
$994$	0	0
$995$	14.2232	0.450905
$996$	0	0
$997$	−26.8643	−0.850801	−0.425401	−	0.905005i	$-0.639867\pi$
$997$	−26.8643	−0.850801	−0.425401	+	0.905005i	$0.639867\pi$
$998$	0	0
$999$	−2.79720	−0.0884995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.cd.1.2		5
4.3	odd	2		3720.2.a.v.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.v.1.4	✓	5	4.3	odd	2
7440.2.a.cd.1.2		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -3.33173 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -3.33173 q^{7} +1.00000 q^{9} -6.35857 q^{11} +3.86626 q^{13} +1.00000 q^{15} +5.25814 q^{17} -2.53453 q^{19} +3.33173 q^{21} +7.99353 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.12440 q^{29} -1.00000 q^{31} +6.35857 q^{33} +3.33173 q^{35} +2.79720 q^{37} -3.86626 q^{39} +4.20085 q^{41} -4.73538 q^{43} -1.00000 q^{45} -1.63496 q^{47} +4.10043 q^{49} -5.25814 q^{51} -4.37034 q^{53} +6.35857 q^{55} +2.53453 q^{57} +0.231304 q^{59} -2.89310 q^{61} -3.33173 q^{63} -3.86626 q^{65} +13.8533 q^{67} -7.99353 q^{69} +9.42929 q^{71} -2.95492 q^{73} -1.00000 q^{75} +21.1850 q^{77} +14.4810 q^{79} +1.00000 q^{81} +14.3521 q^{83} -5.25814 q^{85} +7.12440 q^{87} +10.8195 q^{89} -12.8813 q^{91} +1.00000 q^{93} +2.53453 q^{95} +6.46261 q^{97} -6.35857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 5 q^{5} - 3 q^{7} + 5 q^{9} - 5 q^{11} + 2 q^{13} + 5 q^{15} + 6 q^{17} - 9 q^{19} + 3 q^{21} + 3 q^{23} + 5 q^{25} - 5 q^{27} + 2 q^{29} - 5 q^{31} + 5 q^{33} + 3 q^{35} + 4 q^{37} - 2 q^{39} + 8 q^{41} - 7 q^{43} - 5 q^{45} + 2 q^{47} + 14 q^{49} - 6 q^{51} + 5 q^{53} + 5 q^{55} + 9 q^{57} - 6 q^{59} + 16 q^{61} - 3 q^{63} - 2 q^{65} - 22 q^{67} - 3 q^{69} + 9 q^{71} + 9 q^{73} - 5 q^{75} + q^{77} - 15 q^{79} + 5 q^{81} + 8 q^{83} - 6 q^{85} - 2 q^{87} + 17 q^{89} - 34 q^{91} + 5 q^{93} + 9 q^{95} + 18 q^{97} - 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 5 * q^5 - 3 * q^7 + 5 * q^9 - 5 * q^11 + 2 * q^13 + 5 * q^15 + 6 * q^17 - 9 * q^19 + 3 * q^21 + 3 * q^23 + 5 * q^25 - 5 * q^27 + 2 * q^29 - 5 * q^31 + 5 * q^33 + 3 * q^35 + 4 * q^37 - 2 * q^39 + 8 * q^41 - 7 * q^43 - 5 * q^45 + 2 * q^47 + 14 * q^49 - 6 * q^51 + 5 * q^53 + 5 * q^55 + 9 * q^57 - 6 * q^59 + 16 * q^61 - 3 * q^63 - 2 * q^65 - 22 * q^67 - 3 * q^69 + 9 * q^71 + 9 * q^73 - 5 * q^75 + q^77 - 15 * q^79 + 5 * q^81 + 8 * q^83 - 6 * q^85 - 2 * q^87 + 17 * q^89 - 34 * q^91 + 5 * q^93 + 9 * q^95 + 18 * q^97 - 5 * q^99

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