LMFDB - Embedded newform 7800.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{9} +1.50466 q^{11} +1.00000 q^{13} -2.72666 q^{17} +0.726656 q^{19} -4.72666 q^{23} +1.00000 q^{27} -7.55602 q^{29} -3.00933 q^{31} +1.50466 q^{33} -5.00933 q^{37} +1.00000 q^{39} +5.78734 q^{41} +2.72666 q^{43} -10.2313 q^{47} -7.00000 q^{49} -2.72666 q^{51} -7.55602 q^{53} +0.726656 q^{57} -12.5140 q^{59} +6.28267 q^{61} -12.5653 q^{67} -4.72666 q^{69} +4.77801 q^{71} +12.0187 q^{73} +5.27334 q^{79} +1.00000 q^{81} +7.78734 q^{83} -7.55602 q^{87} +1.78734 q^{89} -3.00933 q^{93} -6.00000 q^{97} +1.50466 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 4 q^{17} - 2 q^{19} - 10 q^{23} + 3 q^{27} - 10 q^{29} + 12 q^{31} - 6 q^{33} + 6 q^{37} + 3 q^{39} - 10 q^{41} + 4 q^{43} - 16 q^{47} - 21 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{57} - 6 q^{59} + 2 q^{61} - 4 q^{67} - 10 q^{69} + 8 q^{71} - 6 q^{73} + 20 q^{79} + 3 q^{81} - 4 q^{83} - 10 q^{87} - 22 q^{89} + 12 q^{93} - 18 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 4 * q^17 - 2 * q^19 - 10 * q^23 + 3 * q^27 - 10 * q^29 + 12 * q^31 - 6 * q^33 + 6 * q^37 + 3 * q^39 - 10 * q^41 + 4 * q^43 - 16 * q^47 - 21 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^57 - 6 * q^59 + 2 * q^61 - 4 * q^67 - 10 * q^69 + 8 * q^71 - 6 * q^73 + 20 * q^79 + 3 * q^81 - 4 * q^83 - 10 * q^87 - 22 * q^89 + 12 * q^93 - 18 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.50466	0.453673	0.226837	−	0.973933i	$-0.427162\pi$
$11$	1.50466	0.453673	0.226837	+	0.973933i	$0.427162\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.72666	−0.661311	−0.330656	−	0.943751i	$-0.607270\pi$
$17$	−2.72666	−0.661311	−0.330656	+	0.943751i	$0.607270\pi$
$18$	0	0
$19$	0.726656	0.166706	0.0833532	−	0.996520i	$-0.473437\pi$
$19$	0.726656	0.166706	0.0833532	+	0.996520i	$0.473437\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.72666	−0.985576	−0.492788	−	0.870149i	$-0.664022\pi$
$23$	−4.72666	−0.985576	−0.492788	+	0.870149i	$0.664022\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.55602	−1.40312	−0.701558	−	0.712612i	$-0.747512\pi$
$29$	−7.55602	−1.40312	−0.701558	+	0.712612i	$0.747512\pi$
$30$	0	0
$31$	−3.00933	−0.540491	−0.270246	−	0.962791i	$-0.587105\pi$
$31$	−3.00933	−0.540491	−0.270246	+	0.962791i	$0.587105\pi$
$32$	0	0
$33$	1.50466	0.261928
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.00933	−0.823529	−0.411764	−	0.911290i	$-0.635087\pi$
$37$	−5.00933	−0.823529	−0.411764	+	0.911290i	$0.635087\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	5.78734	0.903830	0.451915	−	0.892061i	$-0.350741\pi$
$41$	5.78734	0.903830	0.451915	+	0.892061i	$0.350741\pi$
$42$	0	0
$43$	2.72666	0.415811	0.207906	−	0.978149i	$-0.433335\pi$
$43$	2.72666	0.415811	0.207906	+	0.978149i	$0.433335\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.2313	−1.49239	−0.746196	−	0.665727i	$-0.768122\pi$
$47$	−10.2313	−1.49239	−0.746196	+	0.665727i	$0.768122\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−2.72666	−0.381808
$52$	0	0
$53$	−7.55602	−1.03790	−0.518949	−	0.854805i	$-0.673677\pi$
$53$	−7.55602	−1.03790	−0.518949	+	0.854805i	$0.673677\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.726656	0.0962480
$58$	0	0
$59$	−12.5140	−1.62918	−0.814592	−	0.580035i	$-0.803039\pi$
$59$	−12.5140	−1.62918	−0.814592	+	0.580035i	$0.803039\pi$
$60$	0	0
$61$	6.28267	0.804414	0.402207	−	0.915549i	$-0.368243\pi$
$61$	6.28267	0.804414	0.402207	+	0.915549i	$0.368243\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.5653	−1.53510	−0.767551	−	0.640988i	$-0.778525\pi$
$67$	−12.5653	−1.53510	−0.767551	+	0.640988i	$0.778525\pi$
$68$	0	0
$69$	−4.72666	−0.569023
$70$	0	0
$71$	4.77801	0.567045	0.283523	−	0.958966i	$-0.408497\pi$
$71$	4.77801	0.567045	0.283523	+	0.958966i	$0.408497\pi$
$72$	0	0
$73$	12.0187	1.40668	0.703339	−	0.710855i	$-0.251692\pi$
$73$	12.0187	1.40668	0.703339	+	0.710855i	$0.251692\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.27334	0.593297	0.296649	−	0.954987i	$-0.404131\pi$
$79$	5.27334	0.593297	0.296649	+	0.954987i	$0.404131\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.78734	0.854771	0.427386	−	0.904069i	$-0.359435\pi$
$83$	7.78734	0.854771	0.427386	+	0.904069i	$0.359435\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.55602	−0.810090
$88$	0	0
$89$	1.78734	0.189457	0.0947286	−	0.995503i	$-0.469802\pi$
$89$	1.78734	0.189457	0.0947286	+	0.995503i	$0.469802\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.00933	−0.312053
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.50466	0.151224
$100$	0	0
$101$	−2.99067	−0.297583	−0.148791	−	0.988869i	$-0.547538\pi$
$101$	−2.99067	−0.297583	−0.148791	+	0.988869i	$0.547538\pi$
$102$	0	0
$103$	−0.443984	−0.0437471	−0.0218735	−	0.999761i	$-0.506963\pi$
$103$	−0.443984	−0.0437471	−0.0218735	+	0.999761i	$0.506963\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.00933	0.677617	0.338809	−	0.940855i	$-0.389976\pi$
$107$	7.00933	0.677617	0.338809	+	0.940855i	$0.389976\pi$
$108$	0	0
$109$	−13.8387	−1.32551	−0.662753	−	0.748838i	$-0.730612\pi$
$109$	−13.8387	−1.32551	−0.662753	+	0.748838i	$0.730612\pi$
$110$	0	0
$111$	−5.00933	−0.475464
$112$	0	0
$113$	−4.28267	−0.402880	−0.201440	−	0.979501i	$-0.564562\pi$
$113$	−4.28267	−0.402880	−0.201440	+	0.979501i	$0.564562\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.73599	−0.794180
$122$	0	0
$123$	5.78734	0.521827
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.71733	−0.507331	−0.253665	−	0.967292i	$-0.581636\pi$
$127$	−5.71733	−0.507331	−0.253665	+	0.967292i	$0.581636\pi$
$128$	0	0
$129$	2.72666	0.240069
$130$	0	0
$131$	5.55602	0.485431	0.242716	−	0.970097i	$-0.421962\pi$
$131$	5.55602	0.485431	0.242716	+	0.970097i	$0.421962\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.67531	−0.570310	−0.285155	−	0.958481i	$-0.592045\pi$
$137$	−6.67531	−0.570310	−0.285155	+	0.958481i	$0.592045\pi$
$138$	0	0
$139$	19.4720	1.65159	0.825795	−	0.563970i	$-0.190727\pi$
$139$	19.4720	1.65159	0.825795	+	0.563970i	$0.190727\pi$
$140$	0	0
$141$	−10.2313	−0.861633
$142$	0	0
$143$	1.50466	0.125826
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	−14.5140	−1.18903	−0.594516	−	0.804084i	$-0.702656\pi$
$149$	−14.5140	−1.18903	−0.594516	+	0.804084i	$0.702656\pi$
$150$	0	0
$151$	−4.46264	−0.363165	−0.181582	−	0.983376i	$-0.558122\pi$
$151$	−4.46264	−0.363165	−0.181582	+	0.983376i	$0.558122\pi$
$152$	0	0
$153$	−2.72666	−0.220437
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.30133	0.662518	0.331259	−	0.943540i	$-0.392527\pi$
$157$	8.30133	0.662518	0.331259	+	0.943540i	$0.392527\pi$
$158$	0	0
$159$	−7.55602	−0.599231
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.0093	−0.862317	−0.431159	−	0.902276i	$-0.641895\pi$
$163$	−11.0093	−0.862317	−0.431159	+	0.902276i	$0.641895\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.76868	−0.446394	−0.223197	−	0.974773i	$-0.571649\pi$
$167$	−5.76868	−0.446394	−0.223197	+	0.974773i	$0.571649\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.726656	0.0555688
$172$	0	0
$173$	4.90663	0.373044	0.186522	−	0.982451i	$-0.440278\pi$
$173$	4.90663	0.373044	0.186522	+	0.982451i	$0.440278\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.5140	−0.940609
$178$	0	0
$179$	9.45331	0.706574	0.353287	−	0.935515i	$-0.385064\pi$
$179$	9.45331	0.706574	0.353287	+	0.935515i	$0.385064\pi$
$180$	0	0
$181$	17.4720	1.29868	0.649341	−	0.760498i	$-0.275045\pi$
$181$	17.4720	1.29868	0.649341	+	0.760498i	$0.275045\pi$
$182$	0	0
$183$	6.28267	0.464428
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.10270	−0.300019
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.1120	−0.804038	−0.402019	−	0.915631i	$-0.631691\pi$
$191$	−11.1120	−0.804038	−0.402019	+	0.915631i	$0.631691\pi$
$192$	0	0
$193$	6.10270	0.439282	0.219641	−	0.975581i	$-0.429511\pi$
$193$	6.10270	0.439282	0.219641	+	0.975581i	$0.429511\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.3620	1.52198	0.760990	−	0.648764i	$-0.224714\pi$
$197$	21.3620	1.52198	0.760990	+	0.648764i	$0.224714\pi$
$198$	0	0
$199$	8.38538	0.594423	0.297212	−	0.954812i	$-0.403943\pi$
$199$	8.38538	0.594423	0.297212	+	0.954812i	$0.403943\pi$
$200$	0	0
$201$	−12.5653	−0.886291
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.72666	−0.328525
$208$	0	0
$209$	1.09337	0.0756303
$210$	0	0
$211$	−1.27334	−0.0876606	−0.0438303	−	0.999039i	$-0.513956\pi$
$211$	−1.27334	−0.0876606	−0.0438303	+	0.999039i	$0.513956\pi$
$212$	0	0
$213$	4.77801	0.327384
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.0187	0.812146
$220$	0	0
$221$	−2.72666	−0.183415
$222$	0	0
$223$	−16.4626	−1.10242	−0.551210	−	0.834367i	$-0.685834\pi$
$223$	−16.4626	−1.10242	−0.551210	+	0.834367i	$0.685834\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.2500	−0.813060	−0.406530	−	0.913638i	$-0.633261\pi$
$227$	−12.2500	−0.813060	−0.406530	+	0.913638i	$0.633261\pi$
$228$	0	0
$229$	−1.27334	−0.0841449	−0.0420725	−	0.999115i	$-0.513396\pi$
$229$	−1.27334	−0.0841449	−0.0420725	+	0.999115i	$0.513396\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.3013	−1.19896	−0.599480	−	0.800390i	$-0.704626\pi$
$233$	−18.3013	−1.19896	−0.599480	+	0.800390i	$0.704626\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.27334	0.342540
$238$	0	0
$239$	6.23132	0.403071	0.201535	−	0.979481i	$-0.435407\pi$
$239$	6.23132	0.403071	0.201535	+	0.979481i	$0.435407\pi$
$240$	0	0
$241$	19.5560	1.25971	0.629857	−	0.776711i	$-0.283114\pi$
$241$	19.5560	1.25971	0.629857	+	0.776711i	$0.283114\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.726656	0.0462360
$248$	0	0
$249$	7.78734	0.493502
$250$	0	0
$251$	26.4813	1.67148	0.835742	−	0.549122i	$-0.185038\pi$
$251$	26.4813	1.67148	0.835742	+	0.549122i	$0.185038\pi$
$252$	0	0
$253$	−7.11203	−0.447130
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.8294	−0.925030	−0.462515	−	0.886611i	$-0.653053\pi$
$257$	−14.8294	−0.925030	−0.462515	+	0.886611i	$0.653053\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7.55602	−0.467706
$262$	0	0
$263$	−22.9507	−1.41520	−0.707601	−	0.706612i	$-0.750223\pi$
$263$	−22.9507	−1.41520	−0.707601	+	0.706612i	$0.750223\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.78734	0.109383
$268$	0	0
$269$	−27.9160	−1.70207	−0.851033	−	0.525112i	$-0.824023\pi$
$269$	−27.9160	−1.70207	−0.851033	+	0.525112i	$0.824023\pi$
$270$	0	0
$271$	−5.65872	−0.343743	−0.171871	−	0.985119i	$-0.554981\pi$
$271$	−5.65872	−0.343743	−0.171871	+	0.985119i	$0.554981\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.73599	−0.464810	−0.232405	−	0.972619i	$-0.574660\pi$
$277$	−7.73599	−0.464810	−0.232405	+	0.972619i	$0.574660\pi$
$278$	0	0
$279$	−3.00933	−0.180164
$280$	0	0
$281$	5.11929	0.305391	0.152696	−	0.988273i	$-0.451205\pi$
$281$	5.11929	0.305391	0.152696	+	0.988273i	$0.451205\pi$
$282$	0	0
$283$	−6.90663	−0.410556	−0.205278	−	0.978704i	$-0.565810\pi$
$283$	−6.90663	−0.410556	−0.205278	+	0.978704i	$0.565810\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.56534	−0.562667
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	−26.3527	−1.53954	−0.769770	−	0.638321i	$-0.779629\pi$
$293$	−26.3527	−1.53954	−0.769770	+	0.638321i	$0.779629\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.50466	0.0873095
$298$	0	0
$299$	−4.72666	−0.273350
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.99067	−0.171810
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−0.443984	−0.0252574
$310$	0	0
$311$	−11.8973	−0.674634	−0.337317	−	0.941391i	$-0.609519\pi$
$311$	−11.8973	−0.674634	−0.337317	+	0.941391i	$0.609519\pi$
$312$	0	0
$313$	−17.7360	−1.00250	−0.501249	−	0.865303i	$-0.667126\pi$
$313$	−17.7360	−1.00250	−0.501249	+	0.865303i	$0.667126\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.32469	0.0744023	0.0372011	−	0.999308i	$-0.488156\pi$
$317$	1.32469	0.0744023	0.0372011	+	0.999308i	$0.488156\pi$
$318$	0	0
$319$	−11.3693	−0.636557
$320$	0	0
$321$	7.00933	0.391223
$322$	0	0
$323$	−1.98134	−0.110245
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.8387	−0.765281
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3.73599	−0.205348	−0.102674	−	0.994715i	$-0.532740\pi$
$331$	−3.73599	−0.205348	−0.102674	+	0.994715i	$0.532740\pi$
$332$	0	0
$333$	−5.00933	−0.274510
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−23.3947	−1.27439	−0.637195	−	0.770702i	$-0.719906\pi$
$337$	−23.3947	−1.27439	−0.637195	+	0.770702i	$0.719906\pi$
$338$	0	0
$339$	−4.28267	−0.232603
$340$	0	0
$341$	−4.52803	−0.245207
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.5840	1.64184	0.820918	−	0.571047i	$-0.193462\pi$
$347$	30.5840	1.64184	0.820918	+	0.571047i	$0.193462\pi$
$348$	0	0
$349$	1.37605	0.0736581	0.0368290	−	0.999322i	$-0.488274\pi$
$349$	1.37605	0.0736581	0.0368290	+	0.999322i	$0.488274\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	19.9087	1.05963	0.529817	−	0.848112i	$-0.322261\pi$
$353$	19.9087	1.05963	0.529817	+	0.848112i	$0.322261\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.6940	1.40885	0.704427	−	0.709777i	$-0.251204\pi$
$359$	26.6940	1.40885	0.704427	+	0.709777i	$0.251204\pi$
$360$	0	0
$361$	−18.4720	−0.972209
$362$	0	0
$363$	−8.73599	−0.458520
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.12136	0.423932	0.211966	−	0.977277i	$-0.432013\pi$
$367$	8.12136	0.423932	0.211966	+	0.977277i	$0.432013\pi$
$368$	0	0
$369$	5.78734	0.301277
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9.47197	0.490440	0.245220	−	0.969467i	$-0.421140\pi$
$373$	9.47197	0.490440	0.245220	+	0.969467i	$0.421140\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.55602	−0.389155
$378$	0	0
$379$	16.7267	0.859191	0.429595	−	0.903022i	$-0.358656\pi$
$379$	16.7267	0.859191	0.429595	+	0.903022i	$0.358656\pi$
$380$	0	0
$381$	−5.71733	−0.292908
$382$	0	0
$383$	19.6846	1.00584	0.502919	−	0.864334i	$-0.332259\pi$
$383$	19.6846	1.00584	0.502919	+	0.864334i	$0.332259\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.72666	0.138604
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	12.8880	0.651773
$392$	0	0
$393$	5.55602	0.280264
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.35061	−0.168162	−0.0840812	−	0.996459i	$-0.526795\pi$
$397$	−3.35061	−0.168162	−0.0840812	+	0.996459i	$0.526795\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−36.3713	−1.81630	−0.908149	−	0.418647i	$-0.862504\pi$
$401$	−36.3713	−1.81630	−0.908149	+	0.418647i	$0.862504\pi$
$402$	0	0
$403$	−3.00933	−0.149905
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.53736	−0.373613
$408$	0	0
$409$	35.5933	1.75998	0.879988	−	0.474995i	$-0.157550\pi$
$409$	35.5933	1.75998	0.879988	+	0.474995i	$0.157550\pi$
$410$	0	0
$411$	−6.67531	−0.329269
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.4720	0.953546
$418$	0	0
$419$	−13.9160	−0.679839	−0.339919	−	0.940455i	$-0.610400\pi$
$419$	−13.9160	−0.679839	−0.339919	+	0.940455i	$0.610400\pi$
$420$	0	0
$421$	4.70800	0.229454	0.114727	−	0.993397i	$-0.463401\pi$
$421$	4.70800	0.229454	0.114727	+	0.993397i	$0.463401\pi$
$422$	0	0
$423$	−10.2313	−0.497464
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.50466	0.0726459
$430$	0	0
$431$	7.89004	0.380050	0.190025	−	0.981779i	$-0.439143\pi$
$431$	7.89004	0.380050	0.190025	+	0.981779i	$0.439143\pi$
$432$	0	0
$433$	−2.30133	−0.110595	−0.0552974	−	0.998470i	$-0.517611\pi$
$433$	−2.30133	−0.110595	−0.0552974	+	0.998470i	$0.517611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.43466	−0.164302
$438$	0	0
$439$	41.1307	1.96306	0.981530	−	0.191307i	$-0.0612725\pi$
$439$	41.1307	1.96306	0.981530	+	0.191307i	$0.0612725\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−29.5933	−1.40602	−0.703011	−	0.711179i	$-0.748161\pi$
$443$	−29.5933	−1.40602	−0.703011	+	0.711179i	$0.748161\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.5140	−0.686488
$448$	0	0
$449$	−11.4461	−0.540173	−0.270086	−	0.962836i	$-0.587052\pi$
$449$	−11.4461	−0.540173	−0.270086	+	0.962836i	$0.587052\pi$
$450$	0	0
$451$	8.70800	0.410044
$452$	0	0
$453$	−4.46264	−0.209673
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.5467	−0.774021	−0.387011	−	0.922075i	$-0.626492\pi$
$457$	−16.5467	−0.774021	−0.387011	+	0.922075i	$0.626492\pi$
$458$	0	0
$459$	−2.72666	−0.127269
$460$	0	0
$461$	3.50466	0.163228	0.0816142	−	0.996664i	$-0.473992\pi$
$461$	3.50466	0.163228	0.0816142	+	0.996664i	$0.473992\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.44398	−0.298192	−0.149096	−	0.988823i	$-0.547636\pi$
$467$	−6.44398	−0.298192	−0.149096	+	0.988823i	$0.547636\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.30133	0.382505
$472$	0	0
$473$	4.10270	0.188642
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.55602	−0.345966
$478$	0	0
$479$	22.2313	1.01577	0.507887	−	0.861423i	$-0.330427\pi$
$479$	22.2313	1.01577	0.507887	+	0.861423i	$0.330427\pi$
$480$	0	0
$481$	−5.00933	−0.228406
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.9160	0.630592	0.315296	−	0.948993i	$-0.397896\pi$
$487$	13.9160	0.630592	0.315296	+	0.948993i	$0.397896\pi$
$488$	0	0
$489$	−11.0093	−0.497859
$490$	0	0
$491$	−28.1400	−1.26994	−0.634971	−	0.772536i	$-0.718988\pi$
$491$	−28.1400	−1.26994	−0.634971	+	0.772536i	$0.718988\pi$
$492$	0	0
$493$	20.6027	0.927897
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	38.8480	1.73908	0.869538	−	0.493866i	$-0.164417\pi$
$499$	38.8480	1.73908	0.869538	+	0.493866i	$0.164417\pi$
$500$	0	0
$501$	−5.76868	−0.257726
$502$	0	0
$503$	−8.19863	−0.365559	−0.182779	−	0.983154i	$-0.558509\pi$
$503$	−8.19863	−0.365559	−0.182779	+	0.983154i	$0.558509\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	16.0700	0.712291	0.356145	−	0.934431i	$-0.384091\pi$
$509$	16.0700	0.712291	0.356145	+	0.934431i	$0.384091\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.726656	0.0320827
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−15.3947	−0.677058
$518$	0	0
$519$	4.90663	0.215377
$520$	0	0
$521$	−23.0280	−1.00887	−0.504437	−	0.863448i	$-0.668300\pi$
$521$	−23.0280	−1.00887	−0.504437	+	0.863448i	$0.668300\pi$
$522$	0	0
$523$	−5.47875	−0.239569	−0.119784	−	0.992800i	$-0.538220\pi$
$523$	−5.47875	−0.239569	−0.119784	+	0.992800i	$0.538220\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.20541	0.357433
$528$	0	0
$529$	−0.658719	−0.0286399
$530$	0	0
$531$	−12.5140	−0.543061
$532$	0	0
$533$	5.78734	0.250677
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.45331	0.407941
$538$	0	0
$539$	−10.5327	−0.453673
$540$	0	0
$541$	17.8387	0.766945	0.383473	−	0.923552i	$-0.374728\pi$
$541$	17.8387	0.766945	0.383473	+	0.923552i	$0.374728\pi$
$542$	0	0
$543$	17.4720	0.749794
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−29.1053	−1.24445	−0.622225	−	0.782838i	$-0.713771\pi$
$547$	−29.1053	−1.24445	−0.622225	+	0.782838i	$0.713771\pi$
$548$	0	0
$549$	6.28267	0.268138
$550$	0	0
$551$	−5.49063	−0.233909
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.0114	−1.86482	−0.932411	−	0.361399i	$-0.882299\pi$
$557$	−44.0114	−1.86482	−0.932411	+	0.361399i	$0.882299\pi$
$558$	0	0
$559$	2.72666	0.115325
$560$	0	0
$561$	−4.10270	−0.173216
$562$	0	0
$563$	−7.43466	−0.313333	−0.156667	−	0.987652i	$-0.550075\pi$
$563$	−7.43466	−0.313333	−0.156667	+	0.987652i	$0.550075\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.48130	0.355555	0.177777	−	0.984071i	$-0.443109\pi$
$569$	8.48130	0.355555	0.177777	+	0.984071i	$0.443109\pi$
$570$	0	0
$571$	40.7826	1.70670	0.853350	−	0.521339i	$-0.174567\pi$
$571$	40.7826	1.70670	0.853350	+	0.521339i	$0.174567\pi$
$572$	0	0
$573$	−11.1120	−0.464212
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.57467	−0.0655545	−0.0327773	−	0.999463i	$-0.510435\pi$
$577$	−1.57467	−0.0655545	−0.0327773	+	0.999463i	$0.510435\pi$
$578$	0	0
$579$	6.10270	0.253620
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.3693	−0.470867
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.8247	−1.14845	−0.574223	−	0.818699i	$-0.694696\pi$
$587$	−27.8247	−1.14845	−0.574223	+	0.818699i	$0.694696\pi$
$588$	0	0
$589$	−2.18675	−0.0901034
$590$	0	0
$591$	21.3620	0.878716
$592$	0	0
$593$	−32.6940	−1.34258	−0.671290	−	0.741195i	$-0.734260\pi$
$593$	−32.6940	−1.34258	−0.671290	+	0.741195i	$0.734260\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.38538	0.343191
$598$	0	0
$599$	−32.1400	−1.31321	−0.656603	−	0.754237i	$-0.728007\pi$
$599$	−32.1400	−1.31321	−0.656603	+	0.754237i	$0.728007\pi$
$600$	0	0
$601$	40.8667	1.66699	0.833493	−	0.552530i	$-0.186337\pi$
$601$	40.8667	1.66699	0.833493	+	0.552530i	$0.186337\pi$
$602$	0	0
$603$	−12.5653	−0.511700
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.9907	−0.608453	−0.304226	−	0.952600i	$-0.598398\pi$
$607$	−14.9907	−0.608453	−0.304226	+	0.952600i	$0.598398\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.2313	−0.413915
$612$	0	0
$613$	−13.5747	−0.548276	−0.274138	−	0.961690i	$-0.588392\pi$
$613$	−13.5747	−0.548276	−0.274138	+	0.961690i	$0.588392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.69396	0.188972	0.0944859	−	0.995526i	$-0.469879\pi$
$617$	4.69396	0.188972	0.0944859	+	0.995526i	$0.469879\pi$
$618$	0	0
$619$	−20.8667	−0.838702	−0.419351	−	0.907824i	$-0.637742\pi$
$619$	−20.8667	−0.838702	−0.419351	+	0.907824i	$0.637742\pi$
$620$	0	0
$621$	−4.72666	−0.189674
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.09337	0.0436652
$628$	0	0
$629$	13.6587	0.544609
$630$	0	0
$631$	45.9533	1.82937	0.914685	−	0.404167i	$-0.132438\pi$
$631$	45.9533	1.82937	0.914685	+	0.404167i	$0.132438\pi$
$632$	0	0
$633$	−1.27334	−0.0506109
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.00000	−0.277350
$638$	0	0
$639$	4.77801	0.189015
$640$	0	0
$641$	20.0187	0.790689	0.395345	−	0.918533i	$-0.370625\pi$
$641$	20.0187	0.790689	0.395345	+	0.918533i	$0.370625\pi$
$642$	0	0
$643$	26.5840	1.04837	0.524185	−	0.851604i	$-0.324370\pi$
$643$	26.5840	1.04837	0.524185	+	0.851604i	$0.324370\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.39470	−0.369344	−0.184672	−	0.982800i	$-0.559122\pi$
$647$	−9.39470	−0.369344	−0.184672	+	0.982800i	$0.559122\pi$
$648$	0	0
$649$	−18.8294	−0.739117
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.0187	0.468892
$658$	0	0
$659$	47.6774	1.85725	0.928623	−	0.371024i	$-0.120993\pi$
$659$	47.6774	1.85725	0.928623	+	0.371024i	$0.120993\pi$
$660$	0	0
$661$	−22.0959	−0.859432	−0.429716	−	0.902964i	$-0.641386\pi$
$661$	−22.0959	−0.859432	−0.429716	+	0.902964i	$0.641386\pi$
$662$	0	0
$663$	−2.72666	−0.105895
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35.7147	1.38288
$668$	0	0
$669$	−16.4626	−0.636482
$670$	0	0
$671$	9.45331	0.364941
$672$	0	0
$673$	48.6027	1.87349	0.936747	−	0.350006i	$-0.113820\pi$
$673$	48.6027	1.87349	0.936747	+	0.350006i	$0.113820\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.4253	−0.400678	−0.200339	−	0.979727i	$-0.564204\pi$
$677$	−10.4253	−0.400678	−0.200339	+	0.979727i	$0.564204\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.2500	−0.469420
$682$	0	0
$683$	9.13795	0.349654	0.174827	−	0.984599i	$-0.444063\pi$
$683$	9.13795	0.349654	0.174827	+	0.984599i	$0.444063\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.27334	−0.0485811
$688$	0	0
$689$	−7.55602	−0.287861
$690$	0	0
$691$	33.7173	1.28267	0.641334	−	0.767262i	$-0.278381\pi$
$691$	33.7173	1.28267	0.641334	+	0.767262i	$0.278381\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.7801	−0.597713
$698$	0	0
$699$	−18.3013	−0.692220
$700$	0	0
$701$	−19.9813	−0.754685	−0.377342	−	0.926074i	$-0.623162\pi$
$701$	−19.9813	−0.754685	−0.377342	+	0.926074i	$0.623162\pi$
$702$	0	0
$703$	−3.64006	−0.137288
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−43.3293	−1.62727	−0.813633	−	0.581378i	$-0.802514\pi$
$709$	−43.3293	−1.62727	−0.813633	+	0.581378i	$0.802514\pi$
$710$	0	0
$711$	5.27334	0.197766
$712$	0	0
$713$	14.2241	0.532695
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.23132	0.232713
$718$	0	0
$719$	−34.6867	−1.29360	−0.646798	−	0.762661i	$-0.723892\pi$
$719$	−34.6867	−1.29360	−0.646798	+	0.762661i	$0.723892\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	19.5560	0.727296
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5.39470	0.200078	0.100039	−	0.994983i	$-0.468103\pi$
$727$	5.39470	0.200078	0.100039	+	0.994983i	$0.468103\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.43466	−0.274981
$732$	0	0
$733$	9.11203	0.336561	0.168280	−	0.985739i	$-0.446179\pi$
$733$	9.11203	0.336561	0.168280	+	0.985739i	$0.446179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.9066	−0.696435
$738$	0	0
$739$	−4.82936	−0.177651	−0.0888254	−	0.996047i	$-0.528311\pi$
$739$	−4.82936	−0.177651	−0.0888254	+	0.996047i	$0.528311\pi$
$740$	0	0
$741$	0.726656	0.0266944
$742$	0	0
$743$	50.4087	1.84931	0.924657	−	0.380801i	$-0.124352\pi$
$743$	50.4087	1.84931	0.924657	+	0.380801i	$0.124352\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.78734	0.284924
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.88797	0.178365	0.0891823	−	0.996015i	$-0.471575\pi$
$751$	4.88797	0.178365	0.0891823	+	0.996015i	$0.471575\pi$
$752$	0	0
$753$	26.4813	0.965032
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.2241	1.17120	0.585602	−	0.810599i	$-0.300858\pi$
$757$	32.2241	1.17120	0.585602	+	0.810599i	$0.300858\pi$
$758$	0	0
$759$	−7.11203	−0.258150
$760$	0	0
$761$	36.4740	1.32218	0.661091	−	0.750305i	$-0.270094\pi$
$761$	36.4740	1.32218	0.661091	+	0.750305i	$0.270094\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.5140	−0.451854
$768$	0	0
$769$	13.1120	0.472832	0.236416	−	0.971652i	$-0.424027\pi$
$769$	13.1120	0.472832	0.236416	+	0.971652i	$0.424027\pi$
$770$	0	0
$771$	−14.8294	−0.534066
$772$	0	0
$773$	−12.5913	−0.452876	−0.226438	−	0.974026i	$-0.572708\pi$
$773$	−12.5913	−0.452876	−0.226438	+	0.974026i	$0.572708\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.20541	0.150674
$780$	0	0
$781$	7.18930	0.257253
$782$	0	0
$783$	−7.55602	−0.270030
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.9253	1.31624	0.658122	−	0.752911i	$-0.271351\pi$
$787$	36.9253	1.31624	0.658122	+	0.752911i	$0.271351\pi$
$788$	0	0
$789$	−22.9507	−0.817067
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.28267	0.223104
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.0466	−0.887198	−0.443599	−	0.896225i	$-0.646298\pi$
$797$	−25.0466	−0.887198	−0.443599	+	0.896225i	$0.646298\pi$
$798$	0	0
$799$	27.8973	0.986935
$800$	0	0
$801$	1.78734	0.0631524
$802$	0	0
$803$	18.0840	0.638172
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−27.9160	−0.982688
$808$	0	0
$809$	11.2334	0.394945	0.197473	−	0.980308i	$-0.436727\pi$
$809$	11.2334	0.394945	0.197473	+	0.980308i	$0.436727\pi$
$810$	0	0
$811$	31.8387	1.11801	0.559004	−	0.829165i	$-0.311184\pi$
$811$	31.8387	1.11801	0.559004	+	0.829165i	$0.311184\pi$
$812$	0	0
$813$	−5.65872	−0.198460
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.98134	0.0693184
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32.1073	−1.12055	−0.560277	−	0.828306i	$-0.689305\pi$
$821$	−32.1073	−1.12055	−0.560277	+	0.828306i	$0.689305\pi$
$822$	0	0
$823$	−46.7054	−1.62805	−0.814023	−	0.580832i	$-0.802727\pi$
$823$	−46.7054	−1.62805	−0.814023	+	0.580832i	$0.802727\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.6940	0.928240	0.464120	−	0.885772i	$-0.346371\pi$
$827$	26.6940	0.928240	0.464120	+	0.885772i	$0.346371\pi$
$828$	0	0
$829$	−28.8294	−1.00129	−0.500643	−	0.865654i	$-0.666903\pi$
$829$	−28.8294	−1.00129	−0.500643	+	0.865654i	$0.666903\pi$
$830$	0	0
$831$	−7.73599	−0.268358
$832$	0	0
$833$	19.0866	0.661311
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.00933	−0.104018
$838$	0	0
$839$	7.68463	0.265303	0.132652	−	0.991163i	$-0.457651\pi$
$839$	7.68463	0.265303	0.132652	+	0.991163i	$0.457651\pi$
$840$	0	0
$841$	28.0934	0.968737
$842$	0	0
$843$	5.11929	0.176318
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6.90663	−0.237035
$850$	0	0
$851$	23.6774	0.811650
$852$	0	0
$853$	18.4626	0.632149	0.316074	−	0.948734i	$-0.397635\pi$
$853$	18.4626	0.632149	0.316074	+	0.948734i	$0.397635\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32.2827	−1.10276	−0.551378	−	0.834256i	$-0.685898\pi$
$857$	−32.2827	−1.10276	−0.551378	+	0.834256i	$0.685898\pi$
$858$	0	0
$859$	−18.9694	−0.647227	−0.323613	−	0.946189i	$-0.604898\pi$
$859$	−18.9694	−0.647227	−0.323613	+	0.946189i	$0.604898\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.8807	0.574626	0.287313	−	0.957837i	$-0.407238\pi$
$863$	16.8807	0.574626	0.287313	+	0.957837i	$0.407238\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−9.56534	−0.324856
$868$	0	0
$869$	7.93461	0.269163
$870$	0	0
$871$	−12.5653	−0.425760
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.1214	0.949591	0.474795	−	0.880096i	$-0.342522\pi$
$877$	28.1214	0.949591	0.474795	+	0.880096i	$0.342522\pi$
$878$	0	0
$879$	−26.3527	−0.888854
$880$	0	0
$881$	−49.4066	−1.66455	−0.832275	−	0.554363i	$-0.812962\pi$
$881$	−49.4066	−1.66455	−0.832275	+	0.554363i	$0.812962\pi$
$882$	0	0
$883$	−5.65872	−0.190431	−0.0952155	−	0.995457i	$-0.530354\pi$
$883$	−5.65872	−0.190431	−0.0952155	+	0.995457i	$0.530354\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.2640	0.949013	0.474506	−	0.880252i	$-0.342627\pi$
$887$	28.2640	0.949013	0.474506	+	0.880252i	$0.342627\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.50466	0.0504082
$892$	0	0
$893$	−7.43466	−0.248791
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.72666	−0.157818
$898$	0	0
$899$	22.7385	0.758373
$900$	0	0
$901$	20.6027	0.686374
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.5840	−0.484254	−0.242127	−	0.970245i	$-0.577845\pi$
$907$	−14.5840	−0.484254	−0.242127	+	0.970245i	$0.577845\pi$
$908$	0	0
$909$	−2.99067	−0.0991943
$910$	0	0
$911$	29.1680	0.966379	0.483190	−	0.875516i	$-0.339478\pi$
$911$	29.1680	0.966379	0.483190	+	0.875516i	$0.339478\pi$
$912$	0	0
$913$	11.7173	0.387787
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.36672	0.0780708	0.0390354	−	0.999238i	$-0.487571\pi$
$919$	2.36672	0.0780708	0.0390354	+	0.999238i	$0.487571\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	4.77801	0.157270
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.443984	−0.0145824
$928$	0	0
$929$	31.1753	1.02283	0.511414	−	0.859335i	$-0.329122\pi$
$929$	31.1753	1.02283	0.511414	+	0.859335i	$0.329122\pi$
$930$	0	0
$931$	−5.08660	−0.166706
$932$	0	0
$933$	−11.8973	−0.389500
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.46942	0.211347	0.105673	−	0.994401i	$-0.466300\pi$
$937$	6.46942	0.211347	0.105673	+	0.994401i	$0.466300\pi$
$938$	0	0
$939$	−17.7360	−0.578792
$940$	0	0
$941$	−19.4020	−0.632486	−0.316243	−	0.948678i	$-0.602421\pi$
$941$	−19.4020	−0.632486	−0.316243	+	0.948678i	$0.602421\pi$
$942$	0	0
$943$	−27.3548	−0.890793
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.0632	1.62684	0.813418	−	0.581679i	$-0.197604\pi$
$947$	50.0632	1.62684	0.813418	+	0.581679i	$0.197604\pi$
$948$	0	0
$949$	12.0187	0.390142
$950$	0	0
$951$	1.32469	0.0429562
$952$	0	0
$953$	6.30133	0.204120	0.102060	−	0.994778i	$-0.467457\pi$
$953$	6.30133	0.204120	0.102060	+	0.994778i	$0.467457\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−11.3693	−0.367516
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21.9439	−0.707869
$962$	0	0
$963$	7.00933	0.225872
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37.2334	−1.19735	−0.598673	−	0.800994i	$-0.704305\pi$
$967$	−37.2334	−1.19735	−0.598673	+	0.800994i	$0.704305\pi$
$968$	0	0
$969$	−1.98134	−0.0636499
$970$	0	0
$971$	13.4533	0.431737	0.215869	−	0.976422i	$-0.430742\pi$
$971$	13.4533	0.431737	0.215869	+	0.976422i	$0.430742\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.4647	−1.19860	−0.599301	−	0.800524i	$-0.704555\pi$
$977$	−37.4647	−1.19860	−0.599301	+	0.800524i	$0.704555\pi$
$978$	0	0
$979$	2.68934	0.0859517
$980$	0	0
$981$	−13.8387	−0.441835
$982$	0	0
$983$	29.2406	0.932632	0.466316	−	0.884618i	$-0.345581\pi$
$983$	29.2406	0.932632	0.466316	+	0.884618i	$0.345581\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.8880	−0.409814
$990$	0	0
$991$	3.11203	0.0988569	0.0494285	−	0.998778i	$-0.484260\pi$
$991$	3.11203	0.0988569	0.0494285	+	0.998778i	$0.484260\pi$
$992$	0	0
$993$	−3.73599	−0.118558
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−42.3200	−1.34029	−0.670144	−	0.742231i	$-0.733768\pi$
$997$	−42.3200	−1.34029	−0.670144	+	0.742231i	$0.733768\pi$
$998$	0	0
$999$	−5.00933	−0.158488

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bp.1.3		3
5.2	odd	4		1560.2.l.c.1249.1	✓	6
5.3	odd	4		1560.2.l.c.1249.4	yes	6
5.4	even	2		7800.2.a.bj.1.3		3
15.2	even	4		4680.2.l.e.2809.5		6
15.8	even	4		4680.2.l.e.2809.6		6
20.3	even	4		3120.2.l.m.1249.1		6
20.7	even	4		3120.2.l.m.1249.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.c.1249.1	✓	6	5.2	odd	4
1560.2.l.c.1249.4	yes	6	5.3	odd	4
3120.2.l.m.1249.1		6	20.3	even	4
3120.2.l.m.1249.4		6	20.7	even	4
4680.2.l.e.2809.5		6	15.2	even	4
4680.2.l.e.2809.6		6	15.8	even	4
7800.2.a.bj.1.3		3	5.4	even	2
7800.2.a.bp.1.3		3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{9} +1.50466 q^{11} +1.00000 q^{13} -2.72666 q^{17} +0.726656 q^{19} -4.72666 q^{23} +1.00000 q^{27} -7.55602 q^{29} -3.00933 q^{31} +1.50466 q^{33} -5.00933 q^{37} +1.00000 q^{39} +5.78734 q^{41} +2.72666 q^{43} -10.2313 q^{47} -7.00000 q^{49} -2.72666 q^{51} -7.55602 q^{53} +0.726656 q^{57} -12.5140 q^{59} +6.28267 q^{61} -12.5653 q^{67} -4.72666 q^{69} +4.77801 q^{71} +12.0187 q^{73} +5.27334 q^{79} +1.00000 q^{81} +7.78734 q^{83} -7.55602 q^{87} +1.78734 q^{89} -3.00933 q^{93} -6.00000 q^{97} +1.50466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 4 q^{17} - 2 q^{19} - 10 q^{23} + 3 q^{27} - 10 q^{29} + 12 q^{31} - 6 q^{33} + 6 q^{37} + 3 q^{39} - 10 q^{41} + 4 q^{43} - 16 q^{47} - 21 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{57} - 6 q^{59} + 2 q^{61} - 4 q^{67} - 10 q^{69} + 8 q^{71} - 6 q^{73} + 20 q^{79} + 3 q^{81} - 4 q^{83} - 10 q^{87} - 22 q^{89} + 12 q^{93} - 18 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 4 * q^17 - 2 * q^19 - 10 * q^23 + 3 * q^27 - 10 * q^29 + 12 * q^31 - 6 * q^33 + 6 * q^37 + 3 * q^39 - 10 * q^41 + 4 * q^43 - 16 * q^47 - 21 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^57 - 6 * q^59 + 2 * q^61 - 4 * q^67 - 10 * q^69 + 8 * q^71 - 6 * q^73 + 20 * q^79 + 3 * q^81 - 4 * q^83 - 10 * q^87 - 22 * q^89 + 12 * q^93 - 18 * q^97 - 6 * q^99

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