LMFDB - Embedded newform 7800.2.a.bv.1.1

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.16053$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} -6.46967 q^{11} +1.00000 q^{13} -3.49264 q^{17} -2.21016 q^{19} +2.82843 q^{21} +7.14949 q^{23} -1.00000 q^{27} +4.82843 q^{29} -9.65685 q^{31} +6.46967 q^{33} -6.93933 q^{37} -1.00000 q^{39} -0.148604 q^{41} -3.03858 q^{43} -6.79073 q^{47} +1.00000 q^{49} +3.49264 q^{51} -1.70279 q^{53} +2.21016 q^{57} -6.46967 q^{59} +2.66421 q^{61} -2.82843 q^{63} +7.70279 q^{67} -7.14949 q^{69} -11.0879 q^{71} -5.76776 q^{73} +18.2990 q^{77} -10.0239 q^{79} +1.00000 q^{81} +2.27171 q^{83} -4.82843 q^{87} -9.21104 q^{89} -2.82843 q^{91} +9.65685 q^{93} +8.11091 q^{97} -6.46967 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} + 4 q^{13} - 4 q^{17} - 12 q^{19} - 4 q^{23} - 4 q^{27} + 8 q^{29} - 16 q^{31} + 8 q^{33} + 8 q^{37} - 4 q^{39} - 4 q^{41} - 4 q^{43} + 12 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{57} - 8 q^{59} + 12 q^{61} + 24 q^{67} + 4 q^{69} - 12 q^{71} + 24 q^{73} + 8 q^{77} - 12 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{87} - 4 q^{89} + 16 q^{93} + 8 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 + 4 * q^13 - 4 * q^17 - 12 * q^19 - 4 * q^23 - 4 * q^27 + 8 * q^29 - 16 * q^31 + 8 * q^33 + 8 * q^37 - 4 * q^39 - 4 * q^41 - 4 * q^43 + 12 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^57 - 8 * q^59 + 12 * q^61 + 24 * q^67 + 4 * q^69 - 12 * q^71 + 24 * q^73 + 8 * q^77 - 12 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^87 - 4 * q^89 + 16 * q^93 + 8 * q^97 - 8 * 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
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.46967	−1.95068	−0.975339	−	0.220713i	$-0.929162\pi$
$11$	−6.46967	−1.95068	−0.975339	+	0.220713i	$0.929162\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.49264	−0.847089	−0.423544	−	0.905875i	$-0.639214\pi$
$17$	−3.49264	−0.847089	−0.423544	+	0.905875i	$0.639214\pi$
$18$	0	0
$19$	−2.21016	−0.507045	−0.253522	−	0.967330i	$-0.581589\pi$
$19$	−2.21016	−0.507045	−0.253522	+	0.967330i	$0.581589\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	7.14949	1.49077	0.745386	−	0.666633i	$-0.232265\pi$
$23$	7.14949	1.49077	0.745386	+	0.666633i	$0.232265\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	0	0
$33$	6.46967	1.12622
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.93933	−1.14082	−0.570410	−	0.821360i	$-0.693216\pi$
$37$	−6.93933	−1.14082	−0.570410	+	0.821360i	$0.693216\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−0.148604	−0.0232080	−0.0116040	−	0.999933i	$-0.503694\pi$
$41$	−0.148604	−0.0232080	−0.0116040	+	0.999933i	$0.503694\pi$
$42$	0	0
$43$	−3.03858	−0.463380	−0.231690	−	0.972790i	$-0.574425\pi$
$43$	−3.03858	−0.463380	−0.231690	+	0.972790i	$0.574425\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.79073	−0.990530	−0.495265	−	0.868742i	$-0.664929\pi$
$47$	−6.79073	−0.990530	−0.495265	+	0.868742i	$0.664929\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.49264	0.489067
$52$	0	0
$53$	−1.70279	−0.233897	−0.116948	−	0.993138i	$-0.537311\pi$
$53$	−1.70279	−0.233897	−0.116948	+	0.993138i	$0.537311\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.21016	0.292742
$58$	0	0
$59$	−6.46967	−0.842279	−0.421139	−	0.906996i	$-0.638370\pi$
$59$	−6.46967	−0.842279	−0.421139	+	0.906996i	$0.638370\pi$
$60$	0	0
$61$	2.66421	0.341117	0.170558	−	0.985348i	$-0.445443\pi$
$61$	2.66421	0.341117	0.170558	+	0.985348i	$0.445443\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.70279	0.941046	0.470523	−	0.882388i	$-0.344065\pi$
$67$	7.70279	0.941046	0.470523	+	0.882388i	$0.344065\pi$
$68$	0	0
$69$	−7.14949	−0.860697
$70$	0	0
$71$	−11.0879	−1.31590	−0.657948	−	0.753063i	$-0.728575\pi$
$71$	−11.0879	−1.31590	−0.657948	+	0.753063i	$0.728575\pi$
$72$	0	0
$73$	−5.76776	−0.675065	−0.337533	−	0.941314i	$-0.609592\pi$
$73$	−5.76776	−0.675065	−0.337533	+	0.941314i	$0.609592\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.2990	2.08536
$78$	0	0
$79$	−10.0239	−1.12777	−0.563886	−	0.825853i	$-0.690694\pi$
$79$	−10.0239	−1.12777	−0.563886	+	0.825853i	$0.690694\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.27171	0.249353	0.124676	−	0.992197i	$-0.460211\pi$
$83$	2.27171	0.249353	0.124676	+	0.992197i	$0.460211\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.82843	−0.517662
$88$	0	0
$89$	−9.21104	−0.976369	−0.488184	−	0.872741i	$-0.662341\pi$
$89$	−9.21104	−0.976369	−0.488184	+	0.872741i	$0.662341\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	9.65685	1.00137
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.11091	0.823538	0.411769	−	0.911288i	$-0.364911\pi$
$97$	8.11091	0.823538	0.411769	+	0.911288i	$0.364911\pi$
$98$	0	0
$99$	−6.46967	−0.650226
$100$	0	0
$101$	−16.1421	−1.60620	−0.803101	−	0.595843i	$-0.796818\pi$
$101$	−16.1421	−1.60620	−0.803101	+	0.595843i	$0.796818\pi$
$102$	0	0
$103$	19.5815	1.92942	0.964709	−	0.263318i	$-0.0848167\pi$
$103$	19.5815	1.92942	0.964709	+	0.263318i	$0.0848167\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.35965	−0.131442	−0.0657210	−	0.997838i	$-0.520935\pi$
$107$	−1.35965	−0.131442	−0.0657210	+	0.997838i	$0.520935\pi$
$108$	0	0
$109$	3.49264	0.334534	0.167267	−	0.985912i	$-0.446506\pi$
$109$	3.49264	0.334534	0.167267	+	0.985912i	$0.446506\pi$
$110$	0	0
$111$	6.93933	0.658652
$112$	0	0
$113$	12.4320	1.16950	0.584751	−	0.811213i	$-0.301192\pi$
$113$	12.4320	1.16950	0.584751	+	0.811213i	$0.301192\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	9.87867	0.905576
$120$	0	0
$121$	30.8566	2.80514
$122$	0	0
$123$	0.148604	0.0133991
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.6200	−1.65226	−0.826131	−	0.563478i	$-0.809463\pi$
$127$	−18.6200	−1.65226	−0.826131	+	0.563478i	$0.809463\pi$
$128$	0	0
$129$	3.03858	0.267532
$130$	0	0
$131$	−0.751258	−0.0656378	−0.0328189	−	0.999461i	$-0.510448\pi$
$131$	−0.751258	−0.0656378	−0.0328189	+	0.999461i	$0.510448\pi$
$132$	0	0
$133$	6.25127	0.542054
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.08794	−0.0929487	−0.0464743	−	0.998919i	$-0.514799\pi$
$137$	−1.08794	−0.0929487	−0.0464743	+	0.998919i	$0.514799\pi$
$138$	0	0
$139$	0.420314	0.0356506	0.0178253	−	0.999841i	$-0.494326\pi$
$139$	0.420314	0.0356506	0.0178253	+	0.999841i	$0.494326\pi$
$140$	0	0
$141$	6.79073	0.571883
$142$	0	0
$143$	−6.46967	−0.541021
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−14.1265	−1.15729	−0.578645	−	0.815580i	$-0.696418\pi$
$149$	−14.1265	−1.15729	−0.578645	+	0.815580i	$0.696418\pi$
$150$	0	0
$151$	6.07717	0.494553	0.247276	−	0.968945i	$-0.420464\pi$
$151$	6.07717	0.494553	0.247276	+	0.968945i	$0.420464\pi$
$152$	0	0
$153$	−3.49264	−0.282363
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.7266	0.935888	0.467944	−	0.883758i	$-0.344995\pi$
$157$	11.7266	0.935888	0.467944	+	0.883758i	$0.344995\pi$
$158$	0	0
$159$	1.70279	0.135040
$160$	0	0
$161$	−20.2218	−1.59370
$162$	0	0
$163$	−7.31371	−0.572854	−0.286427	−	0.958102i	$-0.592468\pi$
$163$	−7.31371	−0.572854	−0.286427	+	0.958102i	$0.592468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.0420	1.62828	0.814139	−	0.580670i	$-0.197209\pi$
$167$	21.0420	1.62828	0.814139	+	0.580670i	$0.197209\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.21016	−0.169015
$172$	0	0
$173$	18.9706	1.44231	0.721153	−	0.692776i	$-0.243613\pi$
$173$	18.9706	1.44231	0.721153	+	0.692776i	$0.243613\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.46967	0.486290
$178$	0	0
$179$	20.4099	1.52551	0.762753	−	0.646690i	$-0.223847\pi$
$179$	20.4099	1.52551	0.762753	+	0.646690i	$0.223847\pi$
$180$	0	0
$181$	24.8934	1.85031	0.925156	−	0.379588i	$-0.123934\pi$
$181$	24.8934	1.85031	0.925156	+	0.379588i	$0.123934\pi$
$182$	0	0
$183$	−2.66421	−0.196944
$184$	0	0
$185$	0	0
$186$	0	0
$187$	22.5962	1.65240
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	−14.9853	−1.08430	−0.542148	−	0.840283i	$-0.682389\pi$
$191$	−14.9853	−1.08430	−0.542148	+	0.840283i	$0.682389\pi$
$192$	0	0
$193$	18.7990	1.35318	0.676590	−	0.736360i	$-0.263457\pi$
$193$	18.7990	1.35318	0.676590	+	0.736360i	$0.263457\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.92857	−0.137405	−0.0687023	−	0.997637i	$-0.521886\pi$
$197$	−1.92857	−0.137405	−0.0687023	+	0.997637i	$0.521886\pi$
$198$	0	0
$199$	−10.0239	−0.710572	−0.355286	−	0.934758i	$-0.615617\pi$
$199$	−10.0239	−0.710572	−0.355286	+	0.934758i	$0.615617\pi$
$200$	0	0
$201$	−7.70279	−0.543313
$202$	0	0
$203$	−13.6569	−0.958523
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.14949	0.496924
$208$	0	0
$209$	14.2990	0.989081
$210$	0	0
$211$	20.2751	1.39580	0.697899	−	0.716197i	$-0.254119\pi$
$211$	20.2751	1.39580	0.697899	+	0.716197i	$0.254119\pi$
$212$	0	0
$213$	11.0879	0.759733
$214$	0	0
$215$	0	0
$216$	0	0
$217$	27.3137	1.85418
$218$	0	0
$219$	5.76776	0.389749
$220$	0	0
$221$	−3.49264	−0.234940
$222$	0	0
$223$	−16.1881	−1.08403	−0.542017	−	0.840368i	$-0.682339\pi$
$223$	−16.1881	−1.08403	−0.542017	+	0.840368i	$0.682339\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.0879	1.53240	0.766200	−	0.642602i	$-0.222145\pi$
$227$	23.0879	1.53240	0.766200	+	0.642602i	$0.222145\pi$
$228$	0	0
$229$	−4.30886	−0.284738	−0.142369	−	0.989814i	$-0.545472\pi$
$229$	−4.30886	−0.284738	−0.142369	+	0.989814i	$0.545472\pi$
$230$	0	0
$231$	−18.2990	−1.20398
$232$	0	0
$233$	1.53857	0.100795	0.0503977	−	0.998729i	$-0.483951\pi$
$233$	1.53857	0.100795	0.0503977	+	0.998729i	$0.483951\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.0239	0.651119
$238$	0	0
$239$	−17.8514	−1.15471	−0.577355	−	0.816493i	$-0.695915\pi$
$239$	−17.8514	−1.15471	−0.577355	+	0.816493i	$0.695915\pi$
$240$	0	0
$241$	3.95406	0.254703	0.127352	−	0.991858i	$-0.459352\pi$
$241$	3.95406	0.254703	0.127352	+	0.991858i	$0.459352\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.21016	−0.140629
$248$	0	0
$249$	−2.27171	−0.143964
$250$	0	0
$251$	3.42284	0.216048	0.108024	−	0.994148i	$-0.465548\pi$
$251$	3.42284	0.216048	0.108024	+	0.994148i	$0.465548\pi$
$252$	0	0
$253$	−46.2548	−2.90802
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.14772	0.196349	0.0981746	−	0.995169i	$-0.468700\pi$
$257$	3.14772	0.196349	0.0981746	+	0.995169i	$0.468700\pi$
$258$	0	0
$259$	19.6274	1.21959
$260$	0	0
$261$	4.82843	0.298872
$262$	0	0
$263$	−13.7144	−0.845669	−0.422835	−	0.906207i	$-0.638965\pi$
$263$	−13.7144	−0.845669	−0.422835	+	0.906207i	$0.638965\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.21104	0.563707
$268$	0	0
$269$	9.84316	0.600148	0.300074	−	0.953916i	$-0.402989\pi$
$269$	9.84316	0.600148	0.300074	+	0.953916i	$0.402989\pi$
$270$	0	0
$271$	−2.86216	−0.173864	−0.0869320	−	0.996214i	$-0.527706\pi$
$271$	−2.86216	−0.173864	−0.0869320	+	0.996214i	$0.527706\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.24389	−0.375159	−0.187580	−	0.982249i	$-0.560064\pi$
$277$	−6.24389	−0.375159	−0.187580	+	0.982249i	$0.560064\pi$
$278$	0	0
$279$	−9.65685	−0.578141
$280$	0	0
$281$	−31.3869	−1.87239	−0.936193	−	0.351486i	$-0.885677\pi$
$281$	−31.3869	−1.87239	−0.936193	+	0.351486i	$0.885677\pi$
$282$	0	0
$283$	−2.56496	−0.152471	−0.0762354	−	0.997090i	$-0.524290\pi$
$283$	−2.56496	−0.152471	−0.0762354	+	0.997090i	$0.524290\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.420314	0.0248104
$288$	0	0
$289$	−4.80150	−0.282441
$290$	0	0
$291$	−8.11091	−0.475470
$292$	0	0
$293$	0.834895	0.0487751	0.0243875	−	0.999703i	$-0.492236\pi$
$293$	0.834895	0.0487751	0.0243875	+	0.999703i	$0.492236\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.46967	0.375408
$298$	0	0
$299$	7.14949	0.413466
$300$	0	0
$301$	8.59441	0.495374
$302$	0	0
$303$	16.1421	0.927341
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.7556	1.01336	0.506682	−	0.862133i	$-0.330872\pi$
$307$	17.7556	1.01336	0.506682	+	0.862133i	$0.330872\pi$
$308$	0	0
$309$	−19.5815	−1.11395
$310$	0	0
$311$	−1.35965	−0.0770985	−0.0385493	−	0.999257i	$-0.512274\pi$
$311$	−1.35965	−0.0770985	−0.0385493	+	0.999257i	$0.512274\pi$
$312$	0	0
$313$	7.75611	0.438401	0.219201	−	0.975680i	$-0.429655\pi$
$313$	7.75611	0.438401	0.219201	+	0.975680i	$0.429655\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.38514	0.527122	0.263561	−	0.964643i	$-0.415103\pi$
$317$	9.38514	0.527122	0.263561	+	0.964643i	$0.415103\pi$
$318$	0	0
$319$	−31.2383	−1.74901
$320$	0	0
$321$	1.35965	0.0758881
$322$	0	0
$323$	7.71927	0.429512
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.49264	−0.193143
$328$	0	0
$329$	19.2071	1.05892
$330$	0	0
$331$	16.0576	0.882605	0.441303	−	0.897358i	$-0.354517\pi$
$331$	16.0576	0.882605	0.441303	+	0.897358i	$0.354517\pi$
$332$	0	0
$333$	−6.93933	−0.380273
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.0698	−0.603011	−0.301506	−	0.953464i	$-0.597489\pi$
$337$	−11.0698	−0.603011	−0.301506	+	0.953464i	$0.597489\pi$
$338$	0	0
$339$	−12.4320	−0.675212
$340$	0	0
$341$	62.4766	3.38330
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.6127	−1.58969	−0.794846	−	0.606811i	$-0.792449\pi$
$347$	−29.6127	−1.58969	−0.794846	+	0.606811i	$0.792449\pi$
$348$	0	0
$349$	17.1183	0.916319	0.458160	−	0.888870i	$-0.348509\pi$
$349$	17.1183	0.916319	0.458160	+	0.888870i	$0.348509\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−18.3245	−0.975313	−0.487657	−	0.873035i	$-0.662148\pi$
$353$	−18.3245	−0.975313	−0.487657	+	0.873035i	$0.662148\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.87867	−0.522834
$358$	0	0
$359$	27.1356	1.43216	0.716082	−	0.698016i	$-0.245933\pi$
$359$	27.1356	1.43216	0.716082	+	0.698016i	$0.245933\pi$
$360$	0	0
$361$	−14.1152	−0.742906
$362$	0	0
$363$	−30.8566	−1.61955
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.1855	0.792679	0.396340	−	0.918104i	$-0.370280\pi$
$367$	15.1855	0.792679	0.396340	+	0.918104i	$0.370280\pi$
$368$	0	0
$369$	−0.148604	−0.00773599
$370$	0	0
$371$	4.81623	0.250046
$372$	0	0
$373$	−22.1776	−1.14831	−0.574157	−	0.818745i	$-0.694670\pi$
$373$	−22.1776	−1.14831	−0.574157	+	0.818745i	$0.694670\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.82843	0.248677
$378$	0	0
$379$	14.7604	0.758191	0.379096	−	0.925358i	$-0.376235\pi$
$379$	14.7604	0.758191	0.379096	+	0.925358i	$0.376235\pi$
$380$	0	0
$381$	18.6200	0.953934
$382$	0	0
$383$	9.33238	0.476862	0.238431	−	0.971159i	$-0.423367\pi$
$383$	9.33238	0.476862	0.238431	+	0.971159i	$0.423367\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.03858	−0.154460
$388$	0	0
$389$	−3.09618	−0.156982	−0.0784912	−	0.996915i	$-0.525010\pi$
$389$	−3.09618	−0.156982	−0.0784912	+	0.996915i	$0.525010\pi$
$390$	0	0
$391$	−24.9706	−1.26282
$392$	0	0
$393$	0.751258	0.0378960
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.26777	−0.364759	−0.182379	−	0.983228i	$-0.558380\pi$
$397$	−7.26777	−0.364759	−0.182379	+	0.983228i	$0.558380\pi$
$398$	0	0
$399$	−6.25127	−0.312955
$400$	0	0
$401$	−8.40164	−0.419558	−0.209779	−	0.977749i	$-0.567274\pi$
$401$	−8.40164	−0.419558	−0.209779	+	0.977749i	$0.567274\pi$
$402$	0	0
$403$	−9.65685	−0.481042
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.8952	2.22537
$408$	0	0
$409$	11.5815	0.572666	0.286333	−	0.958130i	$-0.407564\pi$
$409$	11.5815	0.572666	0.286333	+	0.958130i	$0.407564\pi$
$410$	0	0
$411$	1.08794	0.0536639
$412$	0	0
$413$	18.2990	0.900434
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.420314	−0.0205829
$418$	0	0
$419$	−24.7071	−1.20702	−0.603510	−	0.797355i	$-0.706232\pi$
$419$	−24.7071	−1.20702	−0.603510	+	0.797355i	$0.706232\pi$
$420$	0	0
$421$	−20.7769	−1.01260	−0.506302	−	0.862356i	$-0.668988\pi$
$421$	−20.7769	−1.01260	−0.506302	+	0.862356i	$0.668988\pi$
$422$	0	0
$423$	−6.79073	−0.330177
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−7.53552	−0.364669
$428$	0	0
$429$	6.46967	0.312358
$430$	0	0
$431$	16.6694	0.802936	0.401468	−	0.915873i	$-0.368500\pi$
$431$	16.6694	0.802936	0.401468	+	0.915873i	$0.368500\pi$
$432$	0	0
$433$	−12.8860	−0.619263	−0.309631	−	0.950857i	$-0.600206\pi$
$433$	−12.8860	−0.619263	−0.309631	+	0.950857i	$0.600206\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.8015	−0.755888
$438$	0	0
$439$	37.1924	1.77510	0.887548	−	0.460716i	$-0.152407\pi$
$439$	37.1924	1.77510	0.887548	+	0.460716i	$0.152407\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	33.0693	1.57117	0.785584	−	0.618755i	$-0.212363\pi$
$443$	33.0693	1.57117	0.785584	+	0.618755i	$0.212363\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.1265	0.668161
$448$	0	0
$449$	−10.7448	−0.507078	−0.253539	−	0.967325i	$-0.581595\pi$
$449$	−10.7448	−0.507078	−0.253539	+	0.967325i	$0.581595\pi$
$450$	0	0
$451$	0.961416	0.0452713
$452$	0	0
$453$	−6.07717	−0.285530
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.9419	1.02640	0.513198	−	0.858270i	$-0.328460\pi$
$457$	21.9419	1.02640	0.513198	+	0.858270i	$0.328460\pi$
$458$	0	0
$459$	3.49264	0.163022
$460$	0	0
$461$	22.0034	1.02480	0.512401	−	0.858747i	$-0.328756\pi$
$461$	22.0034	1.02480	0.512401	+	0.858747i	$0.328756\pi$
$462$	0	0
$463$	32.0208	1.48813	0.744066	−	0.668106i	$-0.232895\pi$
$463$	32.0208	1.48813	0.744066	+	0.668106i	$0.232895\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.3596	1.17350	0.586752	−	0.809767i	$-0.300406\pi$
$467$	25.3596	1.17350	0.586752	+	0.809767i	$0.300406\pi$
$468$	0	0
$469$	−21.7868	−1.00602
$470$	0	0
$471$	−11.7266	−0.540335
$472$	0	0
$473$	19.6586	0.903905
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.70279	−0.0779655
$478$	0	0
$479$	9.46231	0.432344	0.216172	−	0.976355i	$-0.430643\pi$
$479$	9.46231	0.432344	0.216172	+	0.976355i	$0.430643\pi$
$480$	0	0
$481$	−6.93933	−0.316406
$482$	0	0
$483$	20.2218	0.920124
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−22.3327	−1.01199	−0.505996	−	0.862536i	$-0.668875\pi$
$487$	−22.3327	−1.01199	−0.505996	+	0.862536i	$0.668875\pi$
$488$	0	0
$489$	7.31371	0.330737
$490$	0	0
$491$	29.8762	1.34829	0.674146	−	0.738598i	$-0.264512\pi$
$491$	29.8762	1.34829	0.674146	+	0.738598i	$0.264512\pi$
$492$	0	0
$493$	−16.8639	−0.759513
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.3614	1.40675
$498$	0	0
$499$	−24.7145	−1.10637	−0.553186	−	0.833058i	$-0.686588\pi$
$499$	−24.7145	−1.10637	−0.553186	+	0.833058i	$0.686588\pi$
$500$	0	0
$501$	−21.0420	−0.940087
$502$	0	0
$503$	−17.2267	−0.768099	−0.384049	−	0.923313i	$-0.625471\pi$
$503$	−17.2267	−0.768099	−0.384049	+	0.923313i	$0.625471\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	26.1430	1.15877	0.579384	−	0.815055i	$-0.303293\pi$
$509$	26.1430	1.15877	0.579384	+	0.815055i	$0.303293\pi$
$510$	0	0
$511$	16.3137	0.721675
$512$	0	0
$513$	2.21016	0.0975808
$514$	0	0
$515$	0	0
$516$	0	0
$517$	43.9338	1.93220
$518$	0	0
$519$	−18.9706	−0.832715
$520$	0	0
$521$	−33.0165	−1.44648	−0.723240	−	0.690597i	$-0.757348\pi$
$521$	−33.0165	−1.44648	−0.723240	+	0.690597i	$0.757348\pi$
$522$	0	0
$523$	32.2751	1.41129	0.705646	−	0.708564i	$-0.250657\pi$
$523$	32.2751	1.41129	0.705646	+	0.708564i	$0.250657\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.7279	1.46921
$528$	0	0
$529$	28.1152	1.22240
$530$	0	0
$531$	−6.46967	−0.280760
$532$	0	0
$533$	−0.148604	−0.00643674
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−20.4099	−0.880752
$538$	0	0
$539$	−6.46967	−0.278668
$540$	0	0
$541$	−2.21193	−0.0950983	−0.0475491	−	0.998869i	$-0.515141\pi$
$541$	−2.21193	−0.0950983	−0.0475491	+	0.998869i	$0.515141\pi$
$542$	0	0
$543$	−24.8934	−1.06828
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.4822	−1.34608	−0.673041	−	0.739605i	$-0.735012\pi$
$547$	−31.4822	−1.34608	−0.673041	+	0.739605i	$0.735012\pi$
$548$	0	0
$549$	2.66421	0.113706
$550$	0	0
$551$	−10.6716	−0.454625
$552$	0	0
$553$	28.3517	1.20564
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.1209	−1.48812	−0.744061	−	0.668112i	$-0.767103\pi$
$557$	−35.1209	−1.48812	−0.744061	+	0.668112i	$0.767103\pi$
$558$	0	0
$559$	−3.03858	−0.128518
$560$	0	0
$561$	−22.5962	−0.954012
$562$	0	0
$563$	−30.3467	−1.27896	−0.639480	−	0.768808i	$-0.720850\pi$
$563$	−30.3467	−1.27896	−0.639480	+	0.768808i	$0.720850\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	30.8327	1.29257	0.646287	−	0.763094i	$-0.276321\pi$
$569$	30.8327	1.29257	0.646287	+	0.763094i	$0.276321\pi$
$570$	0	0
$571$	21.1832	0.886490	0.443245	−	0.896400i	$-0.353827\pi$
$571$	21.1832	0.886490	0.443245	+	0.896400i	$0.353827\pi$
$572$	0	0
$573$	14.9853	0.626019
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.6594	0.943322	0.471661	−	0.881780i	$-0.343655\pi$
$577$	22.6594	0.943322	0.471661	+	0.881780i	$0.343655\pi$
$578$	0	0
$579$	−18.7990	−0.781259
$580$	0	0
$581$	−6.42537	−0.266569
$582$	0	0
$583$	11.0165	0.456257
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.6529	−0.563515	−0.281758	−	0.959486i	$-0.590917\pi$
$587$	−13.6529	−0.563515	−0.281758	+	0.959486i	$0.590917\pi$
$588$	0	0
$589$	21.3432	0.879430
$590$	0	0
$591$	1.92857	0.0793306
$592$	0	0
$593$	−18.7448	−0.769756	−0.384878	−	0.922967i	$-0.625757\pi$
$593$	−18.7448	−0.769756	−0.384878	+	0.922967i	$0.625757\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.0239	0.410249
$598$	0	0
$599$	−48.7885	−1.99345	−0.996723	−	0.0808922i	$-0.974223\pi$
$599$	−48.7885	−1.99345	−0.996723	+	0.0808922i	$0.974223\pi$
$600$	0	0
$601$	16.5870	0.676599	0.338300	−	0.941038i	$-0.390148\pi$
$601$	16.5870	0.676599	0.338300	+	0.941038i	$0.390148\pi$
$602$	0	0
$603$	7.70279	0.313682
$604$	0	0
$605$	0	0
$606$	0	0
$607$	29.0642	1.17968	0.589840	−	0.807520i	$-0.299191\pi$
$607$	29.0642	1.17968	0.589840	+	0.807520i	$0.299191\pi$
$608$	0	0
$609$	13.6569	0.553404
$610$	0	0
$611$	−6.79073	−0.274723
$612$	0	0
$613$	22.3926	0.904430	0.452215	−	0.891909i	$-0.350634\pi$
$613$	22.3926	0.904430	0.452215	+	0.891909i	$0.350634\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.63136	−0.0656760	−0.0328380	−	0.999461i	$-0.510455\pi$
$617$	−1.63136	−0.0656760	−0.0328380	+	0.999461i	$0.510455\pi$
$618$	0	0
$619$	−37.4485	−1.50518	−0.752591	−	0.658489i	$-0.771196\pi$
$619$	−37.4485	−1.50518	−0.752591	+	0.658489i	$0.771196\pi$
$620$	0	0
$621$	−7.14949	−0.286899
$622$	0	0
$623$	26.0528	1.04378
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−14.2990	−0.571046
$628$	0	0
$629$	24.2366	0.966375
$630$	0	0
$631$	10.7193	0.426728	0.213364	−	0.976973i	$-0.431558\pi$
$631$	10.7193	0.426728	0.213364	+	0.976973i	$0.431558\pi$
$632$	0	0
$633$	−20.2751	−0.805864
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−11.0879	−0.438632
$640$	0	0
$641$	−31.0183	−1.22515	−0.612574	−	0.790413i	$-0.709866\pi$
$641$	−31.0183	−1.22515	−0.612574	+	0.790413i	$0.709866\pi$
$642$	0	0
$643$	−12.4515	−0.491041	−0.245520	−	0.969391i	$-0.578959\pi$
$643$	−12.4515	−0.491041	−0.245520	+	0.969391i	$0.578959\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.6538	1.63758	0.818790	−	0.574093i	$-0.194645\pi$
$647$	41.6538	1.63758	0.818790	+	0.574093i	$0.194645\pi$
$648$	0	0
$649$	41.8566	1.64301
$650$	0	0
$651$	−27.3137	−1.07051
$652$	0	0
$653$	3.83095	0.149917	0.0749584	−	0.997187i	$-0.476118\pi$
$653$	3.83095	0.149917	0.0749584	+	0.997187i	$0.476118\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.76776	−0.225022
$658$	0	0
$659$	−41.8926	−1.63191	−0.815953	−	0.578119i	$-0.803787\pi$
$659$	−41.8926	−1.63191	−0.815953	+	0.578119i	$0.803787\pi$
$660$	0	0
$661$	40.8523	1.58897	0.794485	−	0.607284i	$-0.207741\pi$
$661$	40.8523	1.58897	0.794485	+	0.607284i	$0.207741\pi$
$662$	0	0
$663$	3.49264	0.135643
$664$	0	0
$665$	0	0
$666$	0	0
$667$	34.5208	1.33665
$668$	0	0
$669$	16.1881	0.625867
$670$	0	0
$671$	−17.2365	−0.665409
$672$	0	0
$673$	45.2401	1.74388	0.871939	−	0.489615i	$-0.162863\pi$
$673$	45.2401	1.74388	0.871939	+	0.489615i	$0.162863\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.58146	0.137647	0.0688233	−	0.997629i	$-0.478076\pi$
$677$	3.58146	0.137647	0.0688233	+	0.997629i	$0.478076\pi$
$678$	0	0
$679$	−22.9411	−0.880399
$680$	0	0
$681$	−23.0879	−0.884732
$682$	0	0
$683$	−44.8402	−1.71576	−0.857882	−	0.513848i	$-0.828220\pi$
$683$	−44.8402	−1.71576	−0.857882	+	0.513848i	$0.828220\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.30886	0.164393
$688$	0	0
$689$	−1.70279	−0.0648712
$690$	0	0
$691$	18.2891	0.695750	0.347875	−	0.937541i	$-0.386903\pi$
$691$	18.2891	0.695750	0.347875	+	0.937541i	$0.386903\pi$
$692$	0	0
$693$	18.2990	0.695121
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.519018	0.0196592
$698$	0	0
$699$	−1.53857	−0.0581942
$700$	0	0
$701$	36.0667	1.36222	0.681111	−	0.732180i	$-0.261497\pi$
$701$	36.0667	1.36222	0.681111	+	0.732180i	$0.261497\pi$
$702$	0	0
$703$	15.3370	0.578447
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45.6569	1.71710
$708$	0	0
$709$	−11.7769	−0.442291	−0.221146	−	0.975241i	$-0.570980\pi$
$709$	−11.7769	−0.442291	−0.221146	+	0.975241i	$0.570980\pi$
$710$	0	0
$711$	−10.0239	−0.375924
$712$	0	0
$713$	−69.0416	−2.58563
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.8514	0.666673
$718$	0	0
$719$	−46.7261	−1.74259	−0.871295	−	0.490759i	$-0.836719\pi$
$719$	−46.7261	−1.74259	−0.871295	+	0.490759i	$0.836719\pi$
$720$	0	0
$721$	−55.3847	−2.06264
$722$	0	0
$723$	−3.95406	−0.147053
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.8038	−1.40207	−0.701033	−	0.713129i	$-0.747277\pi$
$727$	−37.8038	−1.40207	−0.701033	+	0.713129i	$0.747277\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.6127	0.392524
$732$	0	0
$733$	20.9411	0.773477	0.386739	−	0.922189i	$-0.373602\pi$
$733$	20.9411	0.773477	0.386739	+	0.922189i	$0.373602\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−49.8345	−1.83568
$738$	0	0
$739$	31.5012	1.15879	0.579396	−	0.815046i	$-0.303289\pi$
$739$	31.5012	1.15879	0.579396	+	0.815046i	$0.303289\pi$
$740$	0	0
$741$	2.21016	0.0811922
$742$	0	0
$743$	−2.96483	−0.108769	−0.0543845	−	0.998520i	$-0.517320\pi$
$743$	−2.96483	−0.108769	−0.0543845	+	0.998520i	$0.517320\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.27171	0.0831176
$748$	0	0
$749$	3.84566	0.140517
$750$	0	0
$751$	10.0295	0.365980	0.182990	−	0.983115i	$-0.441422\pi$
$751$	10.0295	0.365980	0.182990	+	0.983115i	$0.441422\pi$
$752$	0	0
$753$	−3.42284	−0.124735
$754$	0	0
$755$	0	0
$756$	0	0
$757$	36.4731	1.32564	0.662818	−	0.748780i	$-0.269360\pi$
$757$	36.4731	1.32564	0.662818	+	0.748780i	$0.269360\pi$
$758$	0	0
$759$	46.2548	1.67894
$760$	0	0
$761$	−9.88085	−0.358181	−0.179090	−	0.983833i	$-0.557315\pi$
$761$	−9.88085	−0.358181	−0.179090	+	0.983833i	$0.557315\pi$
$762$	0	0
$763$	−9.87867	−0.357632
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.46967	−0.233606
$768$	0	0
$769$	32.6274	1.17657	0.588287	−	0.808652i	$-0.299802\pi$
$769$	32.6274	1.17657	0.588287	+	0.808652i	$0.299802\pi$
$770$	0	0
$771$	−3.14772	−0.113362
$772$	0	0
$773$	44.2950	1.59318	0.796591	−	0.604519i	$-0.206635\pi$
$773$	44.2950	1.59318	0.796591	+	0.604519i	$0.206635\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−19.6274	−0.704129
$778$	0	0
$779$	0.328437	0.0117675
$780$	0	0
$781$	71.7352	2.56689
$782$	0	0
$783$	−4.82843	−0.172554
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−24.0068	−0.855751	−0.427875	−	0.903838i	$-0.640738\pi$
$787$	−24.0068	−0.855751	−0.427875	+	0.903838i	$0.640738\pi$
$788$	0	0
$789$	13.7144	0.488247
$790$	0	0
$791$	−35.1629	−1.25025
$792$	0	0
$793$	2.66421	0.0946088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.4271	−1.50285	−0.751423	−	0.659821i	$-0.770632\pi$
$797$	−42.4271	−1.50285	−0.751423	+	0.659821i	$0.770632\pi$
$798$	0	0
$799$	23.7175	0.839066
$800$	0	0
$801$	−9.21104	−0.325456
$802$	0	0
$803$	37.3155	1.31683
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.84316	−0.346495
$808$	0	0
$809$	−12.0495	−0.423637	−0.211819	−	0.977309i	$-0.567939\pi$
$809$	−12.0495	−0.423637	−0.211819	+	0.977309i	$0.567939\pi$
$810$	0	0
$811$	−34.5619	−1.21363	−0.606816	−	0.794842i	$-0.707554\pi$
$811$	−34.5619	−1.21363	−0.606816	+	0.794842i	$0.707554\pi$
$812$	0	0
$813$	2.86216	0.100380
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.71575	0.234954
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	−1.92119	−0.0670500	−0.0335250	−	0.999438i	$-0.510673\pi$
$821$	−1.92119	−0.0670500	−0.0335250	+	0.999438i	$0.510673\pi$
$822$	0	0
$823$	36.9246	1.28711	0.643556	−	0.765399i	$-0.277458\pi$
$823$	36.9246	1.28711	0.643556	+	0.765399i	$0.277458\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.9501	−0.728507	−0.364254	−	0.931300i	$-0.618676\pi$
$827$	−20.9501	−0.728507	−0.364254	+	0.931300i	$0.618676\pi$
$828$	0	0
$829$	−3.55255	−0.123385	−0.0616926	−	0.998095i	$-0.519650\pi$
$829$	−3.55255	−0.123385	−0.0616926	+	0.998095i	$0.519650\pi$
$830$	0	0
$831$	6.24389	0.216598
$832$	0	0
$833$	−3.49264	−0.121013
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.65685	0.333790
$838$	0	0
$839$	−12.2768	−0.423841	−0.211920	−	0.977287i	$-0.567972\pi$
$839$	−12.2768	−0.423841	−0.211920	+	0.977287i	$0.567972\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	31.3869	1.08102
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−87.2756	−2.99882
$848$	0	0
$849$	2.56496	0.0880291
$850$	0	0
$851$	−49.6127	−1.70070
$852$	0	0
$853$	10.6734	0.365449	0.182724	−	0.983164i	$-0.441508\pi$
$853$	10.6734	0.365449	0.182724	+	0.983164i	$0.441508\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−54.5324	−1.86279	−0.931396	−	0.364006i	$-0.881409\pi$
$857$	−54.5324	−1.86279	−0.931396	+	0.364006i	$0.881409\pi$
$858$	0	0
$859$	−45.6218	−1.55660	−0.778298	−	0.627894i	$-0.783917\pi$
$859$	−45.6218	−1.55660	−0.778298	+	0.627894i	$0.783917\pi$
$860$	0	0
$861$	−0.420314	−0.0143243
$862$	0	0
$863$	18.1798	0.618849	0.309424	−	0.950924i	$-0.399864\pi$
$863$	18.1798	0.618849	0.309424	+	0.950924i	$0.399864\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.80150	0.163067
$868$	0	0
$869$	64.8510	2.19992
$870$	0	0
$871$	7.70279	0.260999
$872$	0	0
$873$	8.11091	0.274513
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.1611	0.782096	0.391048	−	0.920370i	$-0.372113\pi$
$877$	23.1611	0.782096	0.391048	+	0.920370i	$0.372113\pi$
$878$	0	0
$879$	−0.834895	−0.0281603
$880$	0	0
$881$	16.7358	0.563843	0.281921	−	0.959438i	$-0.409028\pi$
$881$	16.7358	0.563843	0.281921	+	0.959438i	$0.409028\pi$
$882$	0	0
$883$	−20.3959	−0.686377	−0.343189	−	0.939267i	$-0.611507\pi$
$883$	−20.3959	−0.686377	−0.343189	+	0.939267i	$0.611507\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−58.7395	−1.97228	−0.986140	−	0.165912i	$-0.946943\pi$
$887$	−58.7395	−1.97228	−0.986140	+	0.165912i	$0.946943\pi$
$888$	0	0
$889$	52.6654	1.76634
$890$	0	0
$891$	−6.46967	−0.216742
$892$	0	0
$893$	15.0086	0.502243
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.14949	−0.238715
$898$	0	0
$899$	−46.6274	−1.55511
$900$	0	0
$901$	5.94723	0.198131
$902$	0	0
$903$	−8.59441	−0.286004
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17.9853	0.597192	0.298596	−	0.954380i	$-0.403482\pi$
$907$	17.9853	0.597192	0.298596	+	0.954380i	$0.403482\pi$
$908$	0	0
$909$	−16.1421	−0.535401
$910$	0	0
$911$	55.0269	1.82312	0.911561	−	0.411165i	$-0.134878\pi$
$911$	55.0269	1.82312	0.911561	+	0.411165i	$0.134878\pi$
$912$	0	0
$913$	−14.6972	−0.486407
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.12488	0.0701697
$918$	0	0
$919$	1.00913	0.0332880	0.0166440	−	0.999861i	$-0.494702\pi$
$919$	1.00913	0.0332880	0.0166440	+	0.999861i	$0.494702\pi$
$920$	0	0
$921$	−17.7556	−0.585066
$922$	0	0
$923$	−11.0879	−0.364964
$924$	0	0
$925$	0	0
$926$	0	0
$927$	19.5815	0.643139
$928$	0	0
$929$	−23.1407	−0.759222	−0.379611	−	0.925146i	$-0.623942\pi$
$929$	−23.1407	−0.759222	−0.379611	+	0.925146i	$0.623942\pi$
$930$	0	0
$931$	−2.21016	−0.0724350
$932$	0	0
$933$	1.35965	0.0445128
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.72667	0.285088	0.142544	−	0.989788i	$-0.454472\pi$
$937$	8.72667	0.285088	0.142544	+	0.989788i	$0.454472\pi$
$938$	0	0
$939$	−7.75611	−0.253111
$940$	0	0
$941$	5.24965	0.171134	0.0855668	−	0.996332i	$-0.472730\pi$
$941$	5.24965	0.171134	0.0855668	+	0.996332i	$0.472730\pi$
$942$	0	0
$943$	−1.06244	−0.0345978
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.0420	0.553791	0.276895	−	0.960900i	$-0.410694\pi$
$947$	17.0420	0.553791	0.276895	+	0.960900i	$0.410694\pi$
$948$	0	0
$949$	−5.76776	−0.187229
$950$	0	0
$951$	−9.38514	−0.304334
$952$	0	0
$953$	−5.16243	−0.167227	−0.0836137	−	0.996498i	$-0.526646\pi$
$953$	−5.16243	−0.167227	−0.0836137	+	0.996498i	$0.526646\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	31.2383	1.00979
$958$	0	0
$959$	3.07715	0.0993663
$960$	0	0
$961$	62.2548	2.00822
$962$	0	0
$963$	−1.35965	−0.0438140
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40.3657	−1.29807	−0.649037	−	0.760757i	$-0.724828\pi$
$967$	−40.3657	−1.29807	−0.649037	+	0.760757i	$0.724828\pi$
$968$	0	0
$969$	−7.71927	−0.247979
$970$	0	0
$971$	−20.2322	−0.649283	−0.324642	−	0.945837i	$-0.605244\pi$
$971$	−20.2322	−0.649283	−0.324642	+	0.945837i	$0.605244\pi$
$972$	0	0
$973$	−1.18883	−0.0381121
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.4034	0.972691	0.486346	−	0.873766i	$-0.338330\pi$
$977$	30.4034	0.972691	0.486346	+	0.873766i	$0.338330\pi$
$978$	0	0
$979$	59.5924	1.90458
$980$	0	0
$981$	3.49264	0.111511
$982$	0	0
$983$	50.4250	1.60831	0.804153	−	0.594422i	$-0.202619\pi$
$983$	50.4250	1.60831	0.804153	+	0.594422i	$0.202619\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−19.2071	−0.611368
$988$	0	0
$989$	−21.7243	−0.690793
$990$	0	0
$991$	54.1005	1.71856	0.859279	−	0.511507i	$-0.170913\pi$
$991$	54.1005	1.71856	0.859279	+	0.511507i	$0.170913\pi$
$992$	0	0
$993$	−16.0576	−0.509572
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.3921	−1.24756	−0.623780	−	0.781600i	$-0.714404\pi$
$997$	−39.3921	−1.24756	−0.623780	+	0.781600i	$0.714404\pi$
$998$	0	0
$999$	6.93933	0.219551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bv.1.1		4
5.2	odd	4		1560.2.l.e.1249.8	yes	8
5.3	odd	4		1560.2.l.e.1249.4	✓	8
5.4	even	2		7800.2.a.bw.1.3		4
15.2	even	4		4680.2.l.f.2809.1		8
15.8	even	4		4680.2.l.f.2809.2		8
20.3	even	4		3120.2.l.o.1249.8		8
20.7	even	4		3120.2.l.o.1249.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.4	✓	8	5.3	odd	4
1560.2.l.e.1249.8	yes	8	5.2	odd	4
3120.2.l.o.1249.4		8	20.7	even	4
3120.2.l.o.1249.8		8	20.3	even	4
4680.2.l.f.2809.1		8	15.2	even	4
4680.2.l.f.2809.2		8	15.8	even	4
7800.2.a.bv.1.1		4	1.1	even	1	trivial
7800.2.a.bw.1.3		4	5.4	even	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} -6.46967 q^{11} +1.00000 q^{13} -3.49264 q^{17} -2.21016 q^{19} +2.82843 q^{21} +7.14949 q^{23} -1.00000 q^{27} +4.82843 q^{29} -9.65685 q^{31} +6.46967 q^{33} -6.93933 q^{37} -1.00000 q^{39} -0.148604 q^{41} -3.03858 q^{43} -6.79073 q^{47} +1.00000 q^{49} +3.49264 q^{51} -1.70279 q^{53} +2.21016 q^{57} -6.46967 q^{59} +2.66421 q^{61} -2.82843 q^{63} +7.70279 q^{67} -7.14949 q^{69} -11.0879 q^{71} -5.76776 q^{73} +18.2990 q^{77} -10.0239 q^{79} +1.00000 q^{81} +2.27171 q^{83} -4.82843 q^{87} -9.21104 q^{89} -2.82843 q^{91} +9.65685 q^{93} +8.11091 q^{97} -6.46967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} + 4 q^{13} - 4 q^{17} - 12 q^{19} - 4 q^{23} - 4 q^{27} + 8 q^{29} - 16 q^{31} + 8 q^{33} + 8 q^{37} - 4 q^{39} - 4 q^{41} - 4 q^{43} + 12 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{57} - 8 q^{59} + 12 q^{61} + 24 q^{67} + 4 q^{69} - 12 q^{71} + 24 q^{73} + 8 q^{77} - 12 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{87} - 4 q^{89} + 16 q^{93} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 + 4 * q^13 - 4 * q^17 - 12 * q^19 - 4 * q^23 - 4 * q^27 + 8 * q^29 - 16 * q^31 + 8 * q^33 + 8 * q^37 - 4 * q^39 - 4 * q^41 - 4 * q^43 + 12 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^57 - 8 * q^59 + 12 * q^61 + 24 * q^67 + 4 * q^69 - 12 * q^71 + 24 * q^73 + 8 * q^77 - 12 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^87 - 4 * q^89 + 16 * q^93 + 8 * q^97 - 8 * q^99

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