LMFDB - Embedded newform 7800.2.a.by.1.4

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:	4.4.30056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 26 $ x^4 - 11*x^2 + 26
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			1.4
Root			$2.74983$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.92778 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.92778 q^{7} +1.00000 q^{9} -1.38360 q^{11} -1.00000 q^{13} -0.195329 q^{17} +4.92778 q^{21} +2.19533 q^{23} +1.00000 q^{27} +7.49966 q^{29} -1.38360 q^{33} -6.05088 q^{37} -1.00000 q^{39} -2.11605 q^{41} -3.49966 q^{43} -2.31138 q^{47} +17.2830 q^{49} -0.195329 q^{51} +6.05088 q^{53} +9.43449 q^{59} -3.69498 q^{61} +4.92778 q^{63} +12.6228 q^{67} +2.19533 q^{69} +13.9716 q^{71} -4.73245 q^{73} -6.81809 q^{77} -8.07153 q^{79} +1.00000 q^{81} +13.3997 q^{83} +7.49966 q^{87} +3.02770 q^{89} -4.92778 q^{91} -6.01612 q^{97} -1.38360 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9} + q^{11} - 4 q^{13} + 3 q^{17} + 7 q^{21} + 5 q^{23} + 4 q^{27} + 8 q^{29} + q^{33} + 5 q^{37} - 4 q^{39} + 7 q^{41} + 8 q^{43} + 10 q^{47} + 9 q^{49} + 3 q^{51} - 5 q^{53} + 2 q^{59} + 11 q^{61} + 7 q^{63} + 12 q^{67} + 5 q^{69} + 15 q^{71} - 10 q^{73} + 15 q^{77} - q^{79} + 4 q^{81} + 22 q^{83} + 8 q^{87} + 9 q^{89} - 7 q^{91} + q^{97} + q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9 + q^11 - 4 * q^13 + 3 * q^17 + 7 * q^21 + 5 * q^23 + 4 * q^27 + 8 * q^29 + q^33 + 5 * q^37 - 4 * q^39 + 7 * q^41 + 8 * q^43 + 10 * q^47 + 9 * q^49 + 3 * q^51 - 5 * q^53 + 2 * q^59 + 11 * q^61 + 7 * q^63 + 12 * q^67 + 5 * q^69 + 15 * q^71 - 10 * q^73 + 15 * q^77 - q^79 + 4 * q^81 + 22 * q^83 + 8 * q^87 + 9 * q^89 - 7 * q^91 + q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.92778	1.86252	0.931262	−	0.364349i	$-0.118709\pi$
$7$	4.92778	1.86252	0.931262	+	0.364349i	$0.118709\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.38360	−0.417172	−0.208586	−	0.978004i	$-0.566886\pi$
$11$	−1.38360	−0.417172	−0.208586	+	0.978004i	$0.566886\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.195329	−0.0473741	−0.0236871	−	0.999719i	$-0.507541\pi$
$17$	−0.195329	−0.0473741	−0.0236871	+	0.999719i	$0.507541\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	4.92778	1.07533
$22$	0	0
$23$	2.19533	0.457758	0.228879	−	0.973455i	$-0.426494\pi$
$23$	2.19533	0.457758	0.228879	+	0.973455i	$0.426494\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.49966	1.39265	0.696326	−	0.717726i	$-0.254817\pi$
$29$	7.49966	1.39265	0.696326	+	0.717726i	$0.254817\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.38360	−0.240854
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.05088	−0.994759	−0.497379	−	0.867533i	$-0.665704\pi$
$37$	−6.05088	−0.994759	−0.497379	+	0.867533i	$0.665704\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.11605	−0.330472	−0.165236	−	0.986254i	$-0.552839\pi$
$41$	−2.11605	−0.330472	−0.165236	+	0.986254i	$0.552839\pi$
$42$	0	0
$43$	−3.49966	−0.533692	−0.266846	−	0.963739i	$-0.585982\pi$
$43$	−3.49966	−0.533692	−0.266846	+	0.963739i	$0.585982\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.31138	−0.337150	−0.168575	−	0.985689i	$-0.553916\pi$
$47$	−2.31138	−0.337150	−0.168575	+	0.985689i	$0.553916\pi$
$48$	0	0
$49$	17.2830	2.46900
$50$	0	0
$51$	−0.195329	−0.0273515
$52$	0	0
$53$	6.05088	0.831153	0.415576	−	0.909558i	$-0.363580\pi$
$53$	6.05088	0.831153	0.415576	+	0.909558i	$0.363580\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.43449	1.22827	0.614133	−	0.789203i	$-0.289506\pi$
$59$	9.43449	1.22827	0.614133	+	0.789203i	$0.289506\pi$
$60$	0	0
$61$	−3.69498	−0.473094	−0.236547	−	0.971620i	$-0.576016\pi$
$61$	−3.69498	−0.473094	−0.236547	+	0.971620i	$0.576016\pi$
$62$	0	0
$63$	4.92778	0.620842
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.6228	1.54212	0.771058	−	0.636765i	$-0.219728\pi$
$67$	12.6228	1.54212	0.771058	+	0.636765i	$0.219728\pi$
$68$	0	0
$69$	2.19533	0.264287
$70$	0	0
$71$	13.9716	1.65812	0.829062	−	0.559156i	$-0.188875\pi$
$71$	13.9716	1.65812	0.829062	+	0.559156i	$0.188875\pi$
$72$	0	0
$73$	−4.73245	−0.553891	−0.276946	−	0.960886i	$-0.589322\pi$
$73$	−4.73245	−0.553891	−0.276946	+	0.960886i	$0.589322\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.81809	−0.776993
$78$	0	0
$79$	−8.07153	−0.908119	−0.454059	−	0.890971i	$-0.650025\pi$
$79$	−8.07153	−0.908119	−0.454059	+	0.890971i	$0.650025\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.3997	1.47081	0.735406	−	0.677627i	$-0.236992\pi$
$83$	13.3997	1.47081	0.735406	+	0.677627i	$0.236992\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.49966	0.804047
$88$	0	0
$89$	3.02770	0.320936	0.160468	−	0.987041i	$-0.448700\pi$
$89$	3.02770	0.320936	0.160468	+	0.987041i	$0.448700\pi$
$90$	0	0
$91$	−4.92778	−0.516571
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.01612	−0.610845	−0.305422	−	0.952217i	$-0.598798\pi$
$97$	−6.01612	−0.610845	−0.305422	+	0.952217i	$0.598798\pi$
$98$	0	0
$99$	−1.38360	−0.139057
$100$	0	0
$101$	−3.10900	−0.309357	−0.154678	−	0.987965i	$-0.549434\pi$
$101$	−3.10900	−0.309357	−0.154678	+	0.987965i	$0.549434\pi$
$102$	0	0
$103$	−16.8690	−1.66215	−0.831075	−	0.556161i	$-0.812274\pi$
$103$	−16.8690	−1.66215	−0.831075	+	0.556161i	$0.812274\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.1946	1.66227	0.831134	−	0.556072i	$-0.187692\pi$
$107$	17.1946	1.66227	0.831134	+	0.556072i	$0.187692\pi$
$108$	0	0
$109$	8.76721	0.839746	0.419873	−	0.907583i	$-0.362075\pi$
$109$	8.76721	0.839746	0.419873	+	0.907583i	$0.362075\pi$
$110$	0	0
$111$	−6.05088	−0.574324
$112$	0	0
$113$	−7.46490	−0.702238	−0.351119	−	0.936331i	$-0.614199\pi$
$113$	−7.46490	−0.702238	−0.351119	+	0.936331i	$0.614199\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−0.962536	−0.0882355
$120$	0	0
$121$	−9.08564	−0.825967
$122$	0	0
$123$	−2.11605	−0.190798
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.1231	0.987016	0.493508	−	0.869741i	$-0.335714\pi$
$127$	11.1231	0.987016	0.493508	+	0.869741i	$0.335714\pi$
$128$	0	0
$129$	−3.49966	−0.308127
$130$	0	0
$131$	−1.82080	−0.159084	−0.0795418	−	0.996832i	$-0.525346\pi$
$131$	−1.82080	−0.159084	−0.0795418	+	0.996832i	$0.525346\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.70204	0.743465	0.371733	−	0.928340i	$-0.378764\pi$
$137$	8.70204	0.743465	0.371733	+	0.928340i	$0.378764\pi$
$138$	0	0
$139$	6.81809	0.578303	0.289151	−	0.957283i	$-0.406627\pi$
$139$	6.81809	0.578303	0.289151	+	0.957283i	$0.406627\pi$
$140$	0	0
$141$	−2.31138	−0.194653
$142$	0	0
$143$	1.38360	0.115703
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.2830	1.42548
$148$	0	0
$149$	−7.48537	−0.613225	−0.306613	−	0.951834i	$-0.599196\pi$
$149$	−7.48537	−0.613225	−0.306613	+	0.951834i	$0.599196\pi$
$150$	0	0
$151$	12.8549	1.04611	0.523057	−	0.852298i	$-0.324791\pi$
$151$	12.8549	1.04611	0.523057	+	0.852298i	$0.324791\pi$
$152$	0	0
$153$	−0.195329	−0.0157914
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.7318	1.41515	0.707574	−	0.706639i	$-0.249790\pi$
$157$	17.7318	1.41515	0.707574	+	0.706639i	$0.249790\pi$
$158$	0	0
$159$	6.05088	0.479866
$160$	0	0
$161$	10.8181	0.852585
$162$	0	0
$163$	−18.1385	−1.42072	−0.710360	−	0.703838i	$-0.751468\pi$
$163$	−18.1385	−1.42072	−0.710360	+	0.703838i	$0.751468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.06517	0.314572	0.157286	−	0.987553i	$-0.449726\pi$
$167$	4.06517	0.314572	0.157286	+	0.987553i	$0.449726\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.2455	1.76732	0.883662	−	0.468125i	$-0.155070\pi$
$173$	23.2455	1.76732	0.883662	+	0.468125i	$0.155070\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.43449	0.709139
$178$	0	0
$179$	14.1224	1.05556	0.527779	−	0.849381i	$-0.323025\pi$
$179$	14.1224	1.05556	0.527779	+	0.849381i	$0.323025\pi$
$180$	0	0
$181$	3.67433	0.273111	0.136556	−	0.990632i	$-0.456397\pi$
$181$	3.67433	0.273111	0.136556	+	0.990632i	$0.456397\pi$
$182$	0	0
$183$	−3.69498	−0.273141
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.270257	0.0197632
$188$	0	0
$189$	4.92778	0.358443
$190$	0	0
$191$	−16.8549	−1.21958	−0.609788	−	0.792565i	$-0.708745\pi$
$191$	−16.8549	−1.21958	−0.609788	+	0.792565i	$0.708745\pi$
$192$	0	0
$193$	−6.53712	−0.470552	−0.235276	−	0.971929i	$-0.575599\pi$
$193$	−6.53712	−0.470552	−0.235276	+	0.971929i	$0.575599\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.7890	−1.83739	−0.918695	−	0.394967i	$-0.870756\pi$
$197$	−25.7890	−1.83739	−0.918695	+	0.394967i	$0.870756\pi$
$198$	0	0
$199$	−16.9646	−1.20259	−0.601293	−	0.799029i	$-0.705347\pi$
$199$	−16.9646	−1.20259	−0.601293	+	0.799029i	$0.705347\pi$
$200$	0	0
$201$	12.6228	0.890341
$202$	0	0
$203$	36.9566	2.59385
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.19533	0.152586
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−26.4577	−1.82142	−0.910710	−	0.413046i	$-0.864465\pi$
$211$	−26.4577	−1.82142	−0.910710	+	0.413046i	$0.864465\pi$
$212$	0	0
$213$	13.9716	0.957319
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.73245	−0.319789
$220$	0	0
$221$	0.195329	0.0131392
$222$	0	0
$223$	2.02065	0.135313	0.0676564	−	0.997709i	$-0.478448\pi$
$223$	2.02065	0.135313	0.0676564	+	0.997709i	$0.478448\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.7904	1.18079	0.590395	−	0.807115i	$-0.298972\pi$
$227$	17.7904	1.18079	0.590395	+	0.807115i	$0.298972\pi$
$228$	0	0
$229$	−18.3339	−1.21154	−0.605768	−	0.795641i	$-0.707134\pi$
$229$	−18.3339	−1.21154	−0.605768	+	0.795641i	$0.707134\pi$
$230$	0	0
$231$	−6.81809	−0.448597
$232$	0	0
$233$	23.0361	1.50914	0.754572	−	0.656217i	$-0.227844\pi$
$233$	23.0361	1.50914	0.754572	+	0.656217i	$0.227844\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.07153	−0.524302
$238$	0	0
$239$	−0.986402	−0.0638051	−0.0319025	−	0.999491i	$-0.510157\pi$
$239$	−0.986402	−0.0638051	−0.0319025	+	0.999491i	$0.510157\pi$
$240$	0	0
$241$	1.52031	0.0979316	0.0489658	−	0.998800i	$-0.484407\pi$
$241$	1.52031	0.0979316	0.0489658	+	0.998800i	$0.484407\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	13.3997	0.849173
$250$	0	0
$251$	11.6763	0.737005	0.368502	−	0.929627i	$-0.379871\pi$
$251$	11.6763	0.737005	0.368502	+	0.929627i	$0.379871\pi$
$252$	0	0
$253$	−3.03746	−0.190964
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.00069	0.187178	0.0935889	−	0.995611i	$-0.470166\pi$
$257$	3.00069	0.187178	0.0935889	+	0.995611i	$0.470166\pi$
$258$	0	0
$259$	−29.8174	−1.85276
$260$	0	0
$261$	7.49966	0.464217
$262$	0	0
$263$	−25.8683	−1.59511	−0.797553	−	0.603248i	$-0.793873\pi$
$263$	−25.8683	−1.59511	−0.797553	+	0.603248i	$0.793873\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.02770	0.185293
$268$	0	0
$269$	11.1090	0.677328	0.338664	−	0.940907i	$-0.390025\pi$
$269$	11.1090	0.677328	0.338664	+	0.940907i	$0.390025\pi$
$270$	0	0
$271$	−13.9859	−0.849582	−0.424791	−	0.905291i	$-0.639653\pi$
$271$	−13.9859	−0.849582	−0.424791	+	0.905291i	$0.639653\pi$
$272$	0	0
$273$	−4.92778	−0.298243
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.0529	−0.604020	−0.302010	−	0.953305i	$-0.597658\pi$
$277$	−10.0529	−0.604020	−0.302010	+	0.953305i	$0.597658\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.6318	−1.17114	−0.585568	−	0.810623i	$-0.699129\pi$
$281$	−19.6318	−1.17114	−0.585568	+	0.810623i	$0.699129\pi$
$282$	0	0
$283$	25.5667	1.51978	0.759890	−	0.650051i	$-0.225253\pi$
$283$	25.5667	1.51978	0.759890	+	0.650051i	$0.225253\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.4274	−0.615512
$288$	0	0
$289$	−16.9618	−0.997756
$290$	0	0
$291$	−6.01612	−0.352671
$292$	0	0
$293$	−6.71614	−0.392361	−0.196181	−	0.980568i	$-0.562854\pi$
$293$	−6.71614	−0.392361	−0.196181	+	0.980568i	$0.562854\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.38360	−0.0802848
$298$	0	0
$299$	−2.19533	−0.126959
$300$	0	0
$301$	−17.2455	−0.994015
$302$	0	0
$303$	−3.10900	−0.178607
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.2689	−0.814368	−0.407184	−	0.913346i	$-0.633489\pi$
$307$	−14.2689	−0.814368	−0.407184	+	0.913346i	$0.633489\pi$
$308$	0	0
$309$	−16.8690	−0.959642
$310$	0	0
$311$	−7.15786	−0.405885	−0.202943	−	0.979191i	$-0.565050\pi$
$311$	−7.15786	−0.405885	−0.202943	+	0.979191i	$0.565050\pi$
$312$	0	0
$313$	−21.6568	−1.22412	−0.612058	−	0.790813i	$-0.709658\pi$
$313$	−21.6568	−1.22412	−0.612058	+	0.790813i	$0.709658\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.0232	1.63010	0.815052	−	0.579388i	$-0.196708\pi$
$317$	29.0232	1.63010	0.815052	+	0.579388i	$0.196708\pi$
$318$	0	0
$319$	−10.3765	−0.580975
$320$	0	0
$321$	17.1946	0.959711
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.76721	0.484828
$328$	0	0
$329$	−11.3900	−0.627949
$330$	0	0
$331$	21.8683	1.20199	0.600995	−	0.799253i	$-0.294771\pi$
$331$	21.8683	1.20199	0.600995	+	0.799253i	$0.294771\pi$
$332$	0	0
$333$	−6.05088	−0.331586
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.2991	1.21471	0.607355	−	0.794431i	$-0.292231\pi$
$337$	22.2991	1.21471	0.607355	+	0.794431i	$0.292231\pi$
$338$	0	0
$339$	−7.46490	−0.405438
$340$	0	0
$341$	0	0
$342$	0	0
$343$	50.6723	2.73605
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.9199	−1.23040	−0.615201	−	0.788370i	$-0.710925\pi$
$347$	−22.9199	−1.23040	−0.615201	+	0.788370i	$0.710925\pi$
$348$	0	0
$349$	6.75310	0.361485	0.180743	−	0.983530i	$-0.442150\pi$
$349$	6.75310	0.361485	0.180743	+	0.983530i	$0.442150\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−3.07859	−0.163857	−0.0819283	−	0.996638i	$-0.526108\pi$
$353$	−3.07859	−0.163857	−0.0819283	+	0.996638i	$0.526108\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.962536	−0.0509428
$358$	0	0
$359$	−13.0091	−0.686592	−0.343296	−	0.939227i	$-0.611543\pi$
$359$	−13.0091	−0.686592	−0.343296	+	0.939227i	$0.611543\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−9.08564	−0.476872
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.4041	1.43048	0.715241	−	0.698878i	$-0.246317\pi$
$367$	27.4041	1.43048	0.715241	+	0.698878i	$0.246317\pi$
$368$	0	0
$369$	−2.11605	−0.110157
$370$	0	0
$371$	29.8174	1.54804
$372$	0	0
$373$	−19.9573	−1.03335	−0.516675	−	0.856181i	$-0.672831\pi$
$373$	−19.9573	−1.03335	−0.516675	+	0.856181i	$0.672831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.49966	−0.386252
$378$	0	0
$379$	−29.2042	−1.50012	−0.750060	−	0.661370i	$-0.769975\pi$
$379$	−29.2042	−1.50012	−0.750060	+	0.661370i	$0.769975\pi$
$380$	0	0
$381$	11.1231	0.569854
$382$	0	0
$383$	−15.5764	−0.795918	−0.397959	−	0.917403i	$-0.630281\pi$
$383$	−15.5764	−0.795918	−0.397959	+	0.917403i	$0.630281\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.49966	−0.177897
$388$	0	0
$389$	20.0529	1.01672	0.508361	−	0.861144i	$-0.330251\pi$
$389$	20.0529	1.01672	0.508361	+	0.861144i	$0.330251\pi$
$390$	0	0
$391$	−0.428810	−0.0216859
$392$	0	0
$393$	−1.82080	−0.0918470
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.9057	1.55112	0.775558	−	0.631277i	$-0.217469\pi$
$397$	30.9057	1.55112	0.775558	+	0.631277i	$0.217469\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.9902	−0.598764	−0.299382	−	0.954133i	$-0.596780\pi$
$401$	−11.9902	−0.598764	−0.299382	+	0.954133i	$0.596780\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.37202	0.414986
$408$	0	0
$409$	39.2169	1.93915	0.969577	−	0.244788i	$-0.0787183\pi$
$409$	39.2169	1.93915	0.969577	+	0.244788i	$0.0787183\pi$
$410$	0	0
$411$	8.70204	0.429240
$412$	0	0
$413$	46.4910	2.28767
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.81809	0.333883
$418$	0	0
$419$	−17.8621	−0.872621	−0.436310	−	0.899796i	$-0.643715\pi$
$419$	−17.8621	−0.872621	−0.436310	+	0.899796i	$0.643715\pi$
$420$	0	0
$421$	28.8690	1.40699	0.703494	−	0.710701i	$-0.251622\pi$
$421$	28.8690	1.40699	0.703494	+	0.710701i	$0.251622\pi$
$422$	0	0
$423$	−2.31138	−0.112383
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−18.2081	−0.881150
$428$	0	0
$429$	1.38360	0.0668010
$430$	0	0
$431$	17.0373	0.820657	0.410329	−	0.911938i	$-0.365414\pi$
$431$	17.0373	0.820657	0.410329	+	0.911938i	$0.365414\pi$
$432$	0	0
$433$	29.1554	1.40112	0.700558	−	0.713595i	$-0.252934\pi$
$433$	29.1554	1.40112	0.700558	+	0.713595i	$0.252934\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−23.2087	−1.10769	−0.553847	−	0.832619i	$-0.686841\pi$
$439$	−23.2087	−1.10769	−0.553847	+	0.832619i	$0.686841\pi$
$440$	0	0
$441$	17.2830	0.822999
$442$	0	0
$443$	14.9484	0.710221	0.355111	−	0.934824i	$-0.384443\pi$
$443$	14.9484	0.710221	0.355111	+	0.934824i	$0.384443\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.48537	−0.354046
$448$	0	0
$449$	−42.2023	−1.99165	−0.995826	−	0.0912763i	$-0.970905\pi$
$449$	−42.2023	−1.99165	−0.995826	+	0.0912763i	$0.970905\pi$
$450$	0	0
$451$	2.92778	0.137864
$452$	0	0
$453$	12.8549	0.603974
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.1599	0.709149	0.354575	−	0.935028i	$-0.384626\pi$
$457$	15.1599	0.709149	0.354575	+	0.935028i	$0.384626\pi$
$458$	0	0
$459$	−0.195329	−0.00911716
$460$	0	0
$461$	−0.848501	−0.0395186	−0.0197593	−	0.999805i	$-0.506290\pi$
$461$	−0.848501	−0.0395186	−0.0197593	+	0.999805i	$0.506290\pi$
$462$	0	0
$463$	−13.8109	−0.641845	−0.320922	−	0.947105i	$-0.603993\pi$
$463$	−13.8109	−0.641845	−0.320922	+	0.947105i	$0.603993\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.5165	−0.486644	−0.243322	−	0.969946i	$-0.578237\pi$
$467$	−10.5165	−0.486644	−0.243322	+	0.969946i	$0.578237\pi$
$468$	0	0
$469$	62.2022	2.87223
$470$	0	0
$471$	17.7318	0.817036
$472$	0	0
$473$	4.84214	0.222642
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.05088	0.277051
$478$	0	0
$479$	−6.41905	−0.293294	−0.146647	−	0.989189i	$-0.546848\pi$
$479$	−6.41905	−0.293294	−0.146647	+	0.989189i	$0.546848\pi$
$480$	0	0
$481$	6.05088	0.275897
$482$	0	0
$483$	10.8181	0.492240
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−23.5492	−1.06711	−0.533557	−	0.845764i	$-0.679145\pi$
$487$	−23.5492	−1.06711	−0.533557	+	0.845764i	$0.679145\pi$
$488$	0	0
$489$	−18.1385	−0.820253
$490$	0	0
$491$	3.99346	0.180222	0.0901111	−	0.995932i	$-0.471278\pi$
$491$	3.99346	0.180222	0.0901111	+	0.995932i	$0.471278\pi$
$492$	0	0
$493$	−1.46490	−0.0659756
$494$	0	0
$495$	0	0
$496$	0	0
$497$	68.8490	3.08830
$498$	0	0
$499$	−5.30231	−0.237364	−0.118682	−	0.992932i	$-0.537867\pi$
$499$	−5.30231	−0.237364	−0.118682	+	0.992932i	$0.537867\pi$
$500$	0	0
$501$	4.06517	0.181618
$502$	0	0
$503$	−34.8408	−1.55347	−0.776736	−	0.629826i	$-0.783126\pi$
$503$	−34.8408	−1.55347	−0.776736	+	0.629826i	$0.783126\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	42.2526	1.87281	0.936406	−	0.350918i	$-0.114130\pi$
$509$	42.2526	1.87281	0.936406	+	0.350918i	$0.114130\pi$
$510$	0	0
$511$	−23.3205	−1.03164
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.19803	0.140649
$518$	0	0
$519$	23.2455	1.02037
$520$	0	0
$521$	−2.65098	−0.116141	−0.0580707	−	0.998312i	$-0.518495\pi$
$521$	−2.65098	−0.116141	−0.0580707	+	0.998312i	$0.518495\pi$
$522$	0	0
$523$	31.6014	1.38183	0.690917	−	0.722934i	$-0.257207\pi$
$523$	31.6014	1.38183	0.690917	+	0.722934i	$0.257207\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−18.1805	−0.790458
$530$	0	0
$531$	9.43449	0.409422
$532$	0	0
$533$	2.11605	0.0916564
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.1224	0.609427
$538$	0	0
$539$	−23.9128	−1.03000
$540$	0	0
$541$	1.54852	0.0665761	0.0332881	−	0.999446i	$-0.489402\pi$
$541$	1.54852	0.0665761	0.0332881	+	0.999446i	$0.489402\pi$
$542$	0	0
$543$	3.67433	0.157681
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.2376	1.42114	0.710569	−	0.703628i	$-0.248438\pi$
$547$	33.2376	1.42114	0.710569	+	0.703628i	$0.248438\pi$
$548$	0	0
$549$	−3.69498	−0.157698
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−39.7747	−1.69139
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.64594	−0.0697407	−0.0348703	−	0.999392i	$-0.511102\pi$
$557$	−1.64594	−0.0697407	−0.0348703	+	0.999392i	$0.511102\pi$
$558$	0	0
$559$	3.49966	0.148020
$560$	0	0
$561$	0.270257	0.0114103
$562$	0	0
$563$	18.7486	0.790158	0.395079	−	0.918647i	$-0.370717\pi$
$563$	18.7486	0.790158	0.395079	+	0.918647i	$0.370717\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.92778	0.206947
$568$	0	0
$569$	−41.6543	−1.74624	−0.873120	−	0.487505i	$-0.837907\pi$
$569$	−41.6543	−1.74624	−0.873120	+	0.487505i	$0.837907\pi$
$570$	0	0
$571$	−27.3184	−1.14324	−0.571620	−	0.820518i	$-0.693685\pi$
$571$	−27.3184	−1.14324	−0.571620	+	0.820518i	$0.693685\pi$
$572$	0	0
$573$	−16.8549	−0.704122
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.0715	1.00211	0.501056	−	0.865415i	$-0.332945\pi$
$577$	24.0715	1.00211	0.501056	+	0.865415i	$0.332945\pi$
$578$	0	0
$579$	−6.53712	−0.271673
$580$	0	0
$581$	66.0309	2.73942
$582$	0	0
$583$	−8.37202	−0.346734
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.6872	−0.977677	−0.488839	−	0.872374i	$-0.662579\pi$
$587$	−23.6872	−0.977677	−0.488839	+	0.872374i	$0.662579\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−25.7890	−1.06082
$592$	0	0
$593$	−38.8632	−1.59592	−0.797961	−	0.602709i	$-0.794088\pi$
$593$	−38.8632	−1.59592	−0.797961	+	0.602709i	$0.794088\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.9646	−0.694313
$598$	0	0
$599$	18.4783	0.755003	0.377502	−	0.926009i	$-0.376783\pi$
$599$	18.4783	0.755003	0.377502	+	0.926009i	$0.376783\pi$
$600$	0	0
$601$	−2.08702	−0.0851313	−0.0425656	−	0.999094i	$-0.513553\pi$
$601$	−2.08702	−0.0851313	−0.0425656	+	0.999094i	$0.513553\pi$
$602$	0	0
$603$	12.6228	0.514039
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.59120	0.145762	0.0728812	−	0.997341i	$-0.476781\pi$
$607$	3.59120	0.145762	0.0728812	+	0.997341i	$0.476781\pi$
$608$	0	0
$609$	36.9566	1.49756
$610$	0	0
$611$	2.31138	0.0935084
$612$	0	0
$613$	24.7895	1.00124	0.500620	−	0.865667i	$-0.333106\pi$
$613$	24.7895	1.00124	0.500620	+	0.865667i	$0.333106\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.7574	0.594112	0.297056	−	0.954860i	$-0.403995\pi$
$617$	14.7574	0.594112	0.297056	+	0.954860i	$0.403995\pi$
$618$	0	0
$619$	25.4081	1.02124	0.510619	−	0.859807i	$-0.329416\pi$
$619$	25.4081	1.02124	0.510619	+	0.859807i	$0.329416\pi$
$620$	0	0
$621$	2.19533	0.0880955
$622$	0	0
$623$	14.9199	0.597751
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.18191	0.0471258
$630$	0	0
$631$	9.46490	0.376792	0.188396	−	0.982093i	$-0.439671\pi$
$631$	9.46490	0.376792	0.188396	+	0.982093i	$0.439671\pi$
$632$	0	0
$633$	−26.4577	−1.05160
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−17.2830	−0.684777
$638$	0	0
$639$	13.9716	0.552708
$640$	0	0
$641$	7.89686	0.311907	0.155954	−	0.987764i	$-0.450155\pi$
$641$	7.89686	0.311907	0.155954	+	0.987764i	$0.450155\pi$
$642$	0	0
$643$	−16.1204	−0.635727	−0.317863	−	0.948137i	$-0.602965\pi$
$643$	−16.1204	−0.635727	−0.317863	+	0.948137i	$0.602965\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.99616	−0.117791	−0.0588956	−	0.998264i	$-0.518758\pi$
$647$	−2.99616	−0.117791	−0.0588956	+	0.998264i	$0.518758\pi$
$648$	0	0
$649$	−13.0536	−0.512398
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21.5330	−0.842653	−0.421326	−	0.906909i	$-0.638435\pi$
$653$	−21.5330	−0.842653	−0.421326	+	0.906909i	$0.638435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.73245	−0.184630
$658$	0	0
$659$	17.4429	0.679478	0.339739	−	0.940520i	$-0.389661\pi$
$659$	17.4429	0.679478	0.339739	+	0.940520i	$0.389661\pi$
$660$	0	0
$661$	−22.4642	−0.873756	−0.436878	−	0.899521i	$-0.643916\pi$
$661$	−22.4642	−0.873756	−0.436878	+	0.899521i	$0.643916\pi$
$662$	0	0
$663$	0.195329	0.00758593
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.4642	0.637497
$668$	0	0
$669$	2.02065	0.0781229
$670$	0	0
$671$	5.11239	0.197362
$672$	0	0
$673$	46.8122	1.80448	0.902239	−	0.431237i	$-0.141923\pi$
$673$	46.8122	1.80448	0.902239	+	0.431237i	$0.141923\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.2682	1.20173	0.600867	−	0.799349i	$-0.294822\pi$
$677$	31.2682	1.20173	0.600867	+	0.799349i	$0.294822\pi$
$678$	0	0
$679$	−29.6461	−1.13771
$680$	0	0
$681$	17.7904	0.681729
$682$	0	0
$683$	−44.1797	−1.69049	−0.845244	−	0.534381i	$-0.820545\pi$
$683$	−44.1797	−1.69049	−0.845244	+	0.534381i	$0.820545\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18.3339	−0.699481
$688$	0	0
$689$	−6.05088	−0.230520
$690$	0	0
$691$	−25.8360	−0.982849	−0.491425	−	0.870920i	$-0.663524\pi$
$691$	−25.8360	−0.982849	−0.491425	+	0.870920i	$0.663524\pi$
$692$	0	0
$693$	−6.81809	−0.258998
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.413325	0.0156558
$698$	0	0
$699$	23.0361	0.871305
$700$	0	0
$701$	15.5887	0.588777	0.294388	−	0.955686i	$-0.404884\pi$
$701$	15.5887	0.588777	0.294388	+	0.955686i	$0.404884\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.3205	−0.576185
$708$	0	0
$709$	−4.05541	−0.152304	−0.0761521	−	0.997096i	$-0.524263\pi$
$709$	−4.05541	−0.152304	−0.0761521	+	0.997096i	$0.524263\pi$
$710$	0	0
$711$	−8.07153	−0.302706
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.986402	−0.0368379
$718$	0	0
$719$	13.4081	0.500038	0.250019	−	0.968241i	$-0.419563\pi$
$719$	13.4081	0.500038	0.250019	+	0.968241i	$0.419563\pi$
$720$	0	0
$721$	−83.1265	−3.09579
$722$	0	0
$723$	1.52031	0.0565408
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12.9787	0.481352	0.240676	−	0.970606i	$-0.422631\pi$
$727$	12.9787	0.481352	0.240676	+	0.970606i	$0.422631\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.683583	0.0252832
$732$	0	0
$733$	49.2410	1.81876	0.909379	−	0.415969i	$-0.136557\pi$
$733$	49.2410	1.81876	0.909379	+	0.415969i	$0.136557\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.4649	−0.643328
$738$	0	0
$739$	−11.3359	−0.416999	−0.208500	−	0.978022i	$-0.566858\pi$
$739$	−11.3359	−0.416999	−0.208500	+	0.978022i	$0.566858\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.07790	0.296349	0.148175	−	0.988961i	$-0.452660\pi$
$743$	8.07790	0.296349	0.148175	+	0.988961i	$0.452660\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.3997	0.490270
$748$	0	0
$749$	84.7314	3.09602
$750$	0	0
$751$	18.2403	0.665598	0.332799	−	0.942998i	$-0.392007\pi$
$751$	18.2403	0.665598	0.332799	+	0.942998i	$0.392007\pi$
$752$	0	0
$753$	11.6763	0.425510
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−5.71111	−0.207574	−0.103787	−	0.994600i	$-0.533096\pi$
$757$	−5.71111	−0.207574	−0.103787	+	0.994600i	$0.533096\pi$
$758$	0	0
$759$	−3.03746	−0.110253
$760$	0	0
$761$	12.6543	0.458718	0.229359	−	0.973342i	$-0.426337\pi$
$761$	12.6543	0.458718	0.229359	+	0.973342i	$0.426337\pi$
$762$	0	0
$763$	43.2028	1.56405
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.43449	−0.340660
$768$	0	0
$769$	−21.4222	−0.772505	−0.386252	−	0.922393i	$-0.626231\pi$
$769$	−21.4222	−0.772505	−0.386252	+	0.922393i	$0.626231\pi$
$770$	0	0
$771$	3.00069	0.108067
$772$	0	0
$773$	−36.0547	−1.29680	−0.648399	−	0.761300i	$-0.724561\pi$
$773$	−36.0547	−1.29680	−0.648399	+	0.761300i	$0.724561\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−29.8174	−1.06969
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−19.3312	−0.691723
$782$	0	0
$783$	7.49966	0.268016
$784$	0	0
$785$	0	0
$786$	0	0
$787$	40.8394	1.45577	0.727883	−	0.685701i	$-0.240504\pi$
$787$	40.8394	1.45577	0.727883	+	0.685701i	$0.240504\pi$
$788$	0	0
$789$	−25.8683	−0.920935
$790$	0	0
$791$	−36.7853	−1.30794
$792$	0	0
$793$	3.69498	0.131213
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.4691	0.902161	0.451080	−	0.892483i	$-0.351039\pi$
$797$	25.4691	0.902161	0.451080	+	0.892483i	$0.351039\pi$
$798$	0	0
$799$	0.451479	0.0159722
$800$	0	0
$801$	3.02770	0.106979
$802$	0	0
$803$	6.54783	0.231068
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.1090	0.391055
$808$	0	0
$809$	15.7252	0.552869	0.276435	−	0.961033i	$-0.410847\pi$
$809$	15.7252	0.552869	0.276435	+	0.961033i	$0.410847\pi$
$810$	0	0
$811$	−49.6815	−1.74455	−0.872277	−	0.489012i	$-0.837357\pi$
$811$	−49.6815	−1.74455	−0.872277	+	0.489012i	$0.837357\pi$
$812$	0	0
$813$	−13.9859	−0.490507
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−4.92778	−0.172190
$820$	0	0
$821$	23.9936	0.837384	0.418692	−	0.908128i	$-0.362489\pi$
$821$	23.9936	0.837384	0.418692	+	0.908128i	$0.362489\pi$
$822$	0	0
$823$	23.0457	0.803321	0.401661	−	0.915789i	$-0.368433\pi$
$823$	23.0457	0.803321	0.401661	+	0.915789i	$0.368433\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−49.1195	−1.70805	−0.854027	−	0.520229i	$-0.825847\pi$
$827$	−49.1195	−1.70805	−0.854027	+	0.520229i	$0.825847\pi$
$828$	0	0
$829$	−5.54739	−0.192669	−0.0963344	−	0.995349i	$-0.530712\pi$
$829$	−5.54739	−0.192669	−0.0963344	+	0.995349i	$0.530712\pi$
$830$	0	0
$831$	−10.0529	−0.348731
$832$	0	0
$833$	−3.37586	−0.116967
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.7106	−0.991200	−0.495600	−	0.868551i	$-0.665052\pi$
$839$	−28.7106	−0.991200	−0.495600	+	0.868551i	$0.665052\pi$
$840$	0	0
$841$	27.2448	0.939477
$842$	0	0
$843$	−19.6318	−0.676156
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−44.7720	−1.53838
$848$	0	0
$849$	25.5667	0.877446
$850$	0	0
$851$	−13.2837	−0.455359
$852$	0	0
$853$	27.7606	0.950505	0.475253	−	0.879849i	$-0.342357\pi$
$853$	27.7606	0.950505	0.475253	+	0.879849i	$0.342357\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.8869	1.22587	0.612937	−	0.790132i	$-0.289988\pi$
$857$	35.8869	1.22587	0.612937	+	0.790132i	$0.289988\pi$
$858$	0	0
$859$	−26.6107	−0.907944	−0.453972	−	0.891016i	$-0.649993\pi$
$859$	−26.6107	−0.907944	−0.453972	+	0.891016i	$0.649993\pi$
$860$	0	0
$861$	−10.4274	−0.355366
$862$	0	0
$863$	20.0366	0.682054	0.341027	−	0.940054i	$-0.389225\pi$
$863$	20.0366	0.682054	0.341027	+	0.940054i	$0.389225\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.9618	−0.576055
$868$	0	0
$869$	11.1678	0.378842
$870$	0	0
$871$	−12.6228	−0.427706
$872$	0	0
$873$	−6.01612	−0.203615
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.15855	−0.0728892	−0.0364446	−	0.999336i	$-0.511603\pi$
$877$	−2.15855	−0.0728892	−0.0364446	+	0.999336i	$0.511603\pi$
$878$	0	0
$879$	−6.71614	−0.226530
$880$	0	0
$881$	−14.8421	−0.500044	−0.250022	−	0.968240i	$-0.580438\pi$
$881$	−14.8421	−0.500044	−0.250022	+	0.968240i	$0.580438\pi$
$882$	0	0
$883$	−13.0742	−0.439983	−0.219992	−	0.975502i	$-0.570603\pi$
$883$	−13.0742	−0.439983	−0.219992	+	0.975502i	$0.570603\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.4254	1.32377	0.661887	−	0.749604i	$-0.269756\pi$
$887$	39.4254	1.32377	0.661887	+	0.749604i	$0.269756\pi$
$888$	0	0
$889$	54.8122	1.83834
$890$	0	0
$891$	−1.38360	−0.0463525
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.19533	−0.0732999
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−1.18191	−0.0393751
$902$	0	0
$903$	−17.2455	−0.573895
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−49.8397	−1.65490	−0.827450	−	0.561539i	$-0.810209\pi$
$907$	−49.8397	−1.65490	−0.827450	+	0.561539i	$0.810209\pi$
$908$	0	0
$909$	−3.10900	−0.103119
$910$	0	0
$911$	8.92438	0.295678	0.147839	−	0.989011i	$-0.452768\pi$
$911$	8.92438	0.295678	0.147839	+	0.989011i	$0.452768\pi$
$912$	0	0
$913$	−18.5399	−0.613581
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.97247	−0.296297
$918$	0	0
$919$	−21.9876	−0.725302	−0.362651	−	0.931925i	$-0.618128\pi$
$919$	−21.9876	−0.725302	−0.362651	+	0.931925i	$0.618128\pi$
$920$	0	0
$921$	−14.2689	−0.470176
$922$	0	0
$923$	−13.9716	−0.459881
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.8690	−0.554050
$928$	0	0
$929$	4.14289	0.135924	0.0679619	−	0.997688i	$-0.478350\pi$
$929$	4.14289	0.135924	0.0679619	+	0.997688i	$0.478350\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−7.15786	−0.234338
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−59.2235	−1.93475	−0.967374	−	0.253354i	$-0.918466\pi$
$937$	−59.2235	−1.93475	−0.967374	+	0.253354i	$0.918466\pi$
$938$	0	0
$939$	−21.6568	−0.706744
$940$	0	0
$941$	11.7615	0.383415	0.191707	−	0.981452i	$-0.438598\pi$
$941$	11.7615	0.383415	0.191707	+	0.981452i	$0.438598\pi$
$942$	0	0
$943$	−4.64543	−0.151276
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.51629	0.276742	0.138371	−	0.990380i	$-0.455813\pi$
$947$	8.51629	0.276742	0.138371	+	0.990380i	$0.455813\pi$
$948$	0	0
$949$	4.73245	0.153622
$950$	0	0
$951$	29.0232	0.941141
$952$	0	0
$953$	−35.4872	−1.14954	−0.574772	−	0.818314i	$-0.694909\pi$
$953$	−35.4872	−1.14954	−0.574772	+	0.818314i	$0.694909\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−10.3765	−0.335426
$958$	0	0
$959$	42.8817	1.38472
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	17.1946	0.554090
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.51445	0.273806	0.136903	−	0.990584i	$-0.456285\pi$
$967$	8.51445	0.273806	0.136903	+	0.990584i	$0.456285\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−50.5013	−1.62066	−0.810331	−	0.585972i	$-0.800713\pi$
$971$	−50.5013	−1.62066	−0.810331	+	0.585972i	$0.800713\pi$
$972$	0	0
$973$	33.5980	1.07710
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39.0540	1.24945	0.624725	−	0.780845i	$-0.285211\pi$
$977$	39.0540	1.24945	0.624725	+	0.780845i	$0.285211\pi$
$978$	0	0
$979$	−4.18914	−0.133886
$980$	0	0
$981$	8.76721	0.279915
$982$	0	0
$983$	−54.9882	−1.75385	−0.876925	−	0.480627i	$-0.840409\pi$
$983$	−54.9882	−1.75385	−0.876925	+	0.480627i	$0.840409\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−11.3900	−0.362547
$988$	0	0
$989$	−7.68289	−0.244302
$990$	0	0
$991$	−15.0052	−0.476656	−0.238328	−	0.971185i	$-0.576599\pi$
$991$	−15.0052	−0.476656	−0.238328	+	0.971185i	$0.576599\pi$
$992$	0	0
$993$	21.8683	0.693969
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.4113	−0.709773	−0.354887	−	0.934909i	$-0.615481\pi$
$997$	−22.4113	−0.709773	−0.354887	+	0.934909i	$0.615481\pi$
$998$	0	0
$999$	−6.05088	−0.191441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.by.1.4		4
5.2	odd	4		1560.2.l.d.1249.2	✓	8
5.3	odd	4		1560.2.l.d.1249.6	yes	8
5.4	even	2		7800.2.a.bt.1.1		4
15.2	even	4		4680.2.l.g.2809.5		8
15.8	even	4		4680.2.l.g.2809.6		8
20.3	even	4		3120.2.l.n.1249.2		8
20.7	even	4		3120.2.l.n.1249.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.d.1249.2	✓	8	5.2	odd	4
1560.2.l.d.1249.6	yes	8	5.3	odd	4
3120.2.l.n.1249.2		8	20.3	even	4
3120.2.l.n.1249.6		8	20.7	even	4
4680.2.l.g.2809.5		8	15.2	even	4
4680.2.l.g.2809.6		8	15.8	even	4
7800.2.a.bt.1.1		4	5.4	even	2
7800.2.a.by.1.4		4	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.92778 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.92778 q^{7} +1.00000 q^{9} -1.38360 q^{11} -1.00000 q^{13} -0.195329 q^{17} +4.92778 q^{21} +2.19533 q^{23} +1.00000 q^{27} +7.49966 q^{29} -1.38360 q^{33} -6.05088 q^{37} -1.00000 q^{39} -2.11605 q^{41} -3.49966 q^{43} -2.31138 q^{47} +17.2830 q^{49} -0.195329 q^{51} +6.05088 q^{53} +9.43449 q^{59} -3.69498 q^{61} +4.92778 q^{63} +12.6228 q^{67} +2.19533 q^{69} +13.9716 q^{71} -4.73245 q^{73} -6.81809 q^{77} -8.07153 q^{79} +1.00000 q^{81} +13.3997 q^{83} +7.49966 q^{87} +3.02770 q^{89} -4.92778 q^{91} -6.01612 q^{97} -1.38360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9} + q^{11} - 4 q^{13} + 3 q^{17} + 7 q^{21} + 5 q^{23} + 4 q^{27} + 8 q^{29} + q^{33} + 5 q^{37} - 4 q^{39} + 7 q^{41} + 8 q^{43} + 10 q^{47} + 9 q^{49} + 3 q^{51} - 5 q^{53} + 2 q^{59} + 11 q^{61} + 7 q^{63} + 12 q^{67} + 5 q^{69} + 15 q^{71} - 10 q^{73} + 15 q^{77} - q^{79} + 4 q^{81} + 22 q^{83} + 8 q^{87} + 9 q^{89} - 7 q^{91} + q^{97} + q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9 + q^11 - 4 * q^13 + 3 * q^17 + 7 * q^21 + 5 * q^23 + 4 * q^27 + 8 * q^29 + q^33 + 5 * q^37 - 4 * q^39 + 7 * q^41 + 8 * q^43 + 10 * q^47 + 9 * q^49 + 3 * q^51 - 5 * q^53 + 2 * q^59 + 11 * q^61 + 7 * q^63 + 12 * q^67 + 5 * q^69 + 15 * q^71 - 10 * q^73 + 15 * q^77 - q^79 + 4 * q^81 + 22 * q^83 + 8 * q^87 + 9 * q^89 - 7 * q^91 + q^97 + q^99

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