LMFDB - Embedded newform 7800.2.a.bv.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:	4.4.10304.1
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			1.4
Root			$1.69230$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.82843 q^{7} +1.00000 q^{9} +0.393270 q^{11} +1.00000 q^{13} -6.21302 q^{17} -7.34271 q^{19} -2.82843 q^{21} -1.44383 q^{23} -1.00000 q^{27} -0.828427 q^{29} +1.65685 q^{31} -0.393270 q^{33} +6.78654 q^{37} -1.00000 q^{39} +3.77786 q^{41} -2.51428 q^{43} +3.00868 q^{47} +1.00000 q^{49} +6.21302 q^{51} -9.55573 q^{53} +7.34271 q^{57} +0.393270 q^{59} +11.0414 q^{61} +2.82843 q^{63} +15.5557 q^{67} +1.44383 q^{69} +6.56440 q^{71} +13.6150 q^{73} +1.11233 q^{77} -14.9403 q^{79} +1.00000 q^{81} +16.4633 q^{83} +0.828427 q^{87} -9.67674 q^{89} +2.82843 q^{91} -1.65685 q^{93} +0.0418875 q^{97} +0.393270 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.393270	0.118575	0.0592877	−	0.998241i	$-0.481117\pi$
$11$	0.393270	0.118575	0.0592877	+	0.998241i	$0.481117\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.21302	−1.50688	−0.753440	−	0.657517i	$-0.771607\pi$
$17$	−6.21302	−1.50688	−0.753440	+	0.657517i	$0.771607\pi$
$18$	0	0
$19$	−7.34271	−1.68453	−0.842266	−	0.539062i	$-0.818779\pi$
$19$	−7.34271	−1.68453	−0.842266	+	0.539062i	$0.818779\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	−1.44383	−0.301060	−0.150530	−	0.988605i	$-0.548098\pi$
$23$	−1.44383	−0.301060	−0.150530	+	0.988605i	$0.548098\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	1.65685	0.297580	0.148790	−	0.988869i	$-0.452462\pi$
$31$	1.65685	0.297580	0.148790	+	0.988869i	$0.452462\pi$
$32$	0	0
$33$	−0.393270	−0.0684595
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.78654	1.11570	0.557850	−	0.829942i	$-0.311626\pi$
$37$	6.78654	1.11570	0.557850	+	0.829942i	$0.311626\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.77786	0.590003	0.295002	−	0.955497i	$-0.404680\pi$
$41$	3.77786	0.590003	0.295002	+	0.955497i	$0.404680\pi$
$42$	0	0
$43$	−2.51428	−0.383424	−0.191712	−	0.981451i	$-0.561404\pi$
$43$	−2.51428	−0.383424	−0.191712	+	0.981451i	$0.561404\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.00868	0.438860	0.219430	−	0.975628i	$-0.429580\pi$
$47$	3.00868	0.438860	0.219430	+	0.975628i	$0.429580\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.21302	0.869997
$52$	0	0
$53$	−9.55573	−1.31258	−0.656290	−	0.754509i	$-0.727875\pi$
$53$	−9.55573	−1.31258	−0.656290	+	0.754509i	$0.727875\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.34271	0.972565
$58$	0	0
$59$	0.393270	0.0511994	0.0255997	−	0.999672i	$-0.491850\pi$
$59$	0.393270	0.0511994	0.0255997	+	0.999672i	$0.491850\pi$
$60$	0	0
$61$	11.0414	1.41371	0.706856	−	0.707357i	$-0.250113\pi$
$61$	11.0414	1.41371	0.706856	+	0.707357i	$0.250113\pi$
$62$	0	0
$63$	2.82843	0.356348
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.5557	1.90043	0.950217	−	0.311588i	$-0.100861\pi$
$67$	15.5557	1.90043	0.950217	+	0.311588i	$0.100861\pi$
$68$	0	0
$69$	1.44383	0.173817
$70$	0	0
$71$	6.56440	0.779051	0.389526	−	0.921016i	$-0.372639\pi$
$71$	6.56440	0.779051	0.389526	+	0.921016i	$0.372639\pi$
$72$	0	0
$73$	13.6150	1.59351	0.796756	−	0.604302i	$-0.206548\pi$
$73$	13.6150	1.59351	0.796756	+	0.604302i	$0.206548\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.11233	0.126762
$78$	0	0
$79$	−14.9403	−1.68092	−0.840459	−	0.541875i	$-0.817714\pi$
$79$	−14.9403	−1.68092	−0.840459	+	0.541875i	$0.817714\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.4633	1.80708	0.903540	−	0.428504i	$-0.140959\pi$
$83$	16.4633	1.80708	0.903540	+	0.428504i	$0.140959\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.828427	0.0888167
$88$	0	0
$89$	−9.67674	−1.02573	−0.512866	−	0.858469i	$-0.671416\pi$
$89$	−9.67674	−1.02573	−0.512866	+	0.858469i	$0.671416\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	−1.65685	−0.171808
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.0418875	0.00425303	0.00212652	−	0.999998i	$-0.499323\pi$
$97$	0.0418875	0.00425303	0.00212652	+	0.999998i	$0.499323\pi$
$98$	0	0
$99$	0.393270	0.0395251
$100$	0	0
$101$	12.1421	1.20819	0.604094	−	0.796913i	$-0.293535\pi$
$101$	12.1421	1.20819	0.604094	+	0.796913i	$0.293535\pi$
$102$	0	0
$103$	−0.0173504	−0.00170958	−0.000854792	−	1.00000i	$-0.500272\pi$
$103$	−0.0173504	−0.00170958	−0.000854792		1.00000i	$0.500272\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.10113	0.203123	0.101562	−	0.994829i	$-0.467616\pi$
$107$	2.10113	0.203123	0.101562	+	0.994829i	$0.467616\pi$
$108$	0	0
$109$	6.21302	0.595100	0.297550	−	0.954706i	$-0.403831\pi$
$109$	6.21302	0.595100	0.297550	+	0.954706i	$0.403831\pi$
$110$	0	0
$111$	−6.78654	−0.644150
$112$	0	0
$113$	1.42648	0.134192	0.0670961	−	0.997747i	$-0.478627\pi$
$113$	1.42648	0.134192	0.0670961	+	0.997747i	$0.478627\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−17.5731	−1.61092
$120$	0	0
$121$	−10.8453	−0.985940
$122$	0	0
$123$	−3.77786	−0.340639
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.50307	0.133376	0.0666880	−	0.997774i	$-0.478757\pi$
$127$	1.50307	0.133376	0.0666880	+	0.997774i	$0.478757\pi$
$128$	0	0
$129$	2.51428	0.221370
$130$	0	0
$131$	3.85699	0.336987	0.168493	−	0.985703i	$-0.446110\pi$
$131$	3.85699	0.336987	0.168493	+	0.985703i	$0.446110\pi$
$132$	0	0
$133$	−20.7683	−1.80084
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.5644	1.41519	0.707596	−	0.706617i	$-0.249780\pi$
$137$	16.5644	1.41519	0.707596	+	0.706617i	$0.249780\pi$
$138$	0	0
$139$	10.6854	0.906325	0.453163	−	0.891428i	$-0.350296\pi$
$139$	10.6854	0.906325	0.453163	+	0.891428i	$0.350296\pi$
$140$	0	0
$141$	−3.00868	−0.253376
$142$	0	0
$143$	0.393270	0.0328869
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	4.05012	0.331799	0.165900	−	0.986143i	$-0.446947\pi$
$149$	4.05012	0.331799	0.165900	+	0.986143i	$0.446947\pi$
$150$	0	0
$151$	5.02856	0.409218	0.204609	−	0.978844i	$-0.434408\pi$
$151$	5.02856	0.409218	0.204609	+	0.978844i	$0.434408\pi$
$152$	0	0
$153$	−6.21302	−0.502293
$154$	0	0
$155$	0	0
$156$	0	0
$157$	24.4961	1.95500	0.977499	−	0.210940i	$-0.0676526\pi$
$157$	24.4961	1.95500	0.977499	+	0.210940i	$0.0676526\pi$
$158$	0	0
$159$	9.55573	0.757819
$160$	0	0
$161$	−4.08378	−0.321847
$162$	0	0
$163$	15.3137	1.19946	0.599731	−	0.800202i	$-0.295274\pi$
$163$	15.3137	1.19946	0.599731	+	0.800202i	$0.295274\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.7770	−1.22086	−0.610430	−	0.792070i	$-0.709003\pi$
$167$	−15.7770	−1.22086	−0.610430	+	0.792070i	$0.709003\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.34271	−0.561511
$172$	0	0
$173$	−14.9706	−1.13819	−0.569095	−	0.822272i	$-0.692706\pi$
$173$	−14.9706	−1.13819	−0.569095	+	0.822272i	$0.692706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.393270	−0.0295600
$178$	0	0
$179$	−4.84578	−0.362190	−0.181095	−	0.983466i	$-0.557964\pi$
$179$	−4.84578	−0.362190	−0.181095	+	0.983466i	$0.557964\pi$
$180$	0	0
$181$	−7.99912	−0.594570	−0.297285	−	0.954789i	$-0.596081\pi$
$181$	−7.99912	−0.594570	−0.297285	+	0.954789i	$0.596081\pi$
$182$	0	0
$183$	−11.0414	−0.816207
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.44339	−0.178679
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−20.4260	−1.47798	−0.738988	−	0.673718i	$-0.764696\pi$
$191$	−20.4260	−1.47798	−0.738988	+	0.673718i	$0.764696\pi$
$192$	0	0
$193$	24.0237	1.72926	0.864630	−	0.502408i	$-0.167553\pi$
$193$	24.0237	1.72926	0.864630	+	0.502408i	$0.167553\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.80642	−0.342444	−0.171222	−	0.985233i	$-0.554771\pi$
$197$	−4.80642	−0.342444	−0.171222	+	0.985233i	$0.554771\pi$
$198$	0	0
$199$	−14.9403	−1.05909	−0.529546	−	0.848281i	$-0.677638\pi$
$199$	−14.9403	−1.05909	−0.529546	+	0.848281i	$0.677638\pi$
$200$	0	0
$201$	−15.5557	−1.09722
$202$	0	0
$203$	−2.34315	−0.164457
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.44383	−0.100353
$208$	0	0
$209$	−2.88767	−0.199744
$210$	0	0
$211$	−1.82799	−0.125844	−0.0629220	−	0.998018i	$-0.520042\pi$
$211$	−1.82799	−0.125844	−0.0629220	+	0.998018i	$0.520042\pi$
$212$	0	0
$213$	−6.56440	−0.449786
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.68629	0.318126
$218$	0	0
$219$	−13.6150	−0.920014
$220$	0	0
$221$	−6.21302	−0.417933
$222$	0	0
$223$	−7.07045	−0.473472	−0.236736	−	0.971574i	$-0.576078\pi$
$223$	−7.07045	−0.473472	−0.236736	+	0.971574i	$0.576078\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.43560	0.360773	0.180387	−	0.983596i	$-0.442265\pi$
$227$	5.43560	0.360773	0.180387	+	0.983596i	$0.442265\pi$
$228$	0	0
$229$	24.8147	1.63980	0.819899	−	0.572508i	$-0.194029\pi$
$229$	24.8147	1.63980	0.819899	+	0.572508i	$0.194029\pi$
$230$	0	0
$231$	−1.11233	−0.0731863
$232$	0	0
$233$	23.4256	1.53466	0.767331	−	0.641251i	$-0.221584\pi$
$233$	23.4256	1.53466	0.767331	+	0.641251i	$0.221584\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.9403	0.970478
$238$	0	0
$239$	−21.7779	−1.40869	−0.704346	−	0.709856i	$-0.748760\pi$
$239$	−21.7779	−1.40869	−0.704346	+	0.709856i	$0.748760\pi$
$240$	0	0
$241$	−15.2126	−0.979929	−0.489964	−	0.871742i	$-0.662990\pi$
$241$	−15.2126	−0.979929	−0.489964	+	0.871742i	$0.662990\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.34271	−0.467205
$248$	0	0
$249$	−16.4633	−1.04332
$250$	0	0
$251$	−17.9399	−1.13236	−0.566178	−	0.824283i	$-0.691578\pi$
$251$	−17.9399	−1.13236	−0.566178	+	0.824283i	$0.691578\pi$
$252$	0	0
$253$	−0.567816	−0.0356983
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.88810	0.242533	0.121267	−	0.992620i	$-0.461304\pi$
$257$	3.88810	0.242533	0.121267	+	0.992620i	$0.461304\pi$
$258$	0	0
$259$	19.1952	1.19273
$260$	0	0
$261$	−0.828427	−0.0512784
$262$	0	0
$263$	−0.296797	−0.0183013	−0.00915064	−	0.999958i	$-0.502913\pi$
$263$	−0.296797	−0.0183013	−0.00915064	+	0.999958i	$0.502913\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.67674	0.592207
$268$	0	0
$269$	−1.25447	−0.0764864	−0.0382432	−	0.999268i	$-0.512176\pi$
$269$	−1.25447	−0.0764864	−0.0382432	+	0.999268i	$0.512176\pi$
$270$	0	0
$271$	9.81510	0.596225	0.298112	−	0.954531i	$-0.403643\pi$
$271$	9.81510	0.596225	0.298112	+	0.954531i	$0.403643\pi$
$272$	0	0
$273$	−2.82843	−0.171184
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.35603	−0.261729	−0.130864	−	0.991400i	$-0.541775\pi$
$277$	−4.35603	−0.261729	−0.130864	+	0.991400i	$0.541775\pi$
$278$	0	0
$279$	1.65685	0.0991933
$280$	0	0
$281$	3.45207	0.205933	0.102967	−	0.994685i	$-0.467167\pi$
$281$	3.45207	0.205933	0.102967	+	0.994685i	$0.467167\pi$
$282$	0	0
$283$	2.25937	0.134306	0.0671528	−	0.997743i	$-0.478608\pi$
$283$	2.25937	0.134306	0.0671528	+	0.997743i	$0.478608\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.6854	0.630740
$288$	0	0
$289$	21.6016	1.27068
$290$	0	0
$291$	−0.0418875	−0.00245549
$292$	0	0
$293$	19.5358	1.14130	0.570648	−	0.821195i	$-0.306692\pi$
$293$	19.5358	1.14130	0.570648	+	0.821195i	$0.306692\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.393270	−0.0228198
$298$	0	0
$299$	−1.44383	−0.0834990
$300$	0	0
$301$	−7.11146	−0.409898
$302$	0	0
$303$	−12.1421	−0.697547
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−27.8142	−1.58744	−0.793721	−	0.608282i	$-0.791859\pi$
$307$	−27.8142	−1.58744	−0.793721	+	0.608282i	$0.791859\pi$
$308$	0	0
$309$	0.0173504	0.000987028 0
$310$	0	0
$311$	2.10113	0.119144	0.0595719	−	0.998224i	$-0.481026\pi$
$311$	2.10113	0.119144	0.0595719	+	0.998224i	$0.481026\pi$
$312$	0	0
$313$	9.64397	0.545109	0.272555	−	0.962140i	$-0.412131\pi$
$313$	9.64397	0.545109	0.272555	+	0.962140i	$0.412131\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.1201	−0.905397	−0.452698	−	0.891664i	$-0.649539\pi$
$317$	−16.1201	−0.905397	−0.452698	+	0.891664i	$0.649539\pi$
$318$	0	0
$319$	−0.325795	−0.0182410
$320$	0	0
$321$	−2.10113	−0.117273
$322$	0	0
$323$	45.6204	2.53839
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.21302	−0.343581
$328$	0	0
$329$	8.50982	0.469161
$330$	0	0
$331$	13.9537	0.766962	0.383481	−	0.923549i	$-0.374725\pi$
$331$	13.9537	0.766962	0.383481	+	0.923549i	$0.374725\pi$
$332$	0	0
$333$	6.78654	0.371900
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.66974	0.526744	0.263372	−	0.964694i	$-0.415165\pi$
$337$	9.66974	0.526744	0.263372	+	0.964694i	$0.415165\pi$
$338$	0	0
$339$	−1.42648	−0.0774759
$340$	0	0
$341$	0.651591	0.0352856
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.2014	0.547638	0.273819	−	0.961781i	$-0.411713\pi$
$347$	10.2014	0.547638	0.273819	+	0.961781i	$0.411713\pi$
$348$	0	0
$349$	28.7402	1.53843	0.769214	−	0.638992i	$-0.220648\pi$
$349$	28.7402	1.53843	0.769214	+	0.638992i	$0.220648\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	20.9067	1.11275	0.556375	−	0.830931i	$-0.312192\pi$
$353$	20.9067	1.11275	0.556375	+	0.830931i	$0.312192\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	17.5731	0.930066
$358$	0	0
$359$	19.3162	1.01947	0.509736	−	0.860331i	$-0.329743\pi$
$359$	19.3162	1.01947	0.509736	+	0.860331i	$0.329743\pi$
$360$	0	0
$361$	34.9153	1.83765
$362$	0	0
$363$	10.8453	0.569233
$364$	0	0
$365$	0	0
$366$	0	0
$367$	37.6957	1.96770	0.983851	−	0.178990i	$-0.0572828\pi$
$367$	37.6957	1.96770	0.983851	+	0.178990i	$0.0572828\pi$
$368$	0	0
$369$	3.77786	0.196668
$370$	0	0
$371$	−27.0277	−1.40321
$372$	0	0
$373$	22.4607	1.16297	0.581487	−	0.813556i	$-0.302471\pi$
$373$	22.4607	1.16297	0.581487	+	0.813556i	$0.302471\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.828427	−0.0426662
$378$	0	0
$379$	−24.3133	−1.24889	−0.624444	−	0.781069i	$-0.714675\pi$
$379$	−24.3133	−1.24889	−0.624444	+	0.781069i	$0.714675\pi$
$380$	0	0
$381$	−1.50307	−0.0770046
$382$	0	0
$383$	37.2498	1.90338	0.951688	−	0.307065i	$-0.0993471\pi$
$383$	37.2498	1.90338	0.951688	+	0.307065i	$0.0993471\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.51428	−0.127808
$388$	0	0
$389$	−0.467931	−0.0237250	−0.0118625	−	0.999930i	$-0.503776\pi$
$389$	−0.467931	−0.0237250	−0.0118625	+	0.999930i	$0.503776\pi$
$390$	0	0
$391$	8.97056	0.453661
$392$	0	0
$393$	−3.85699	−0.194559
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.5263	1.73282	0.866412	−	0.499329i	$-0.166420\pi$
$397$	34.5263	1.73282	0.866412	+	0.499329i	$0.166420\pi$
$398$	0	0
$399$	20.7683	1.03972
$400$	0	0
$401$	31.8781	1.59192	0.795958	−	0.605351i	$-0.206967\pi$
$401$	31.8781	1.59192	0.795958	+	0.605351i	$0.206967\pi$
$402$	0	0
$403$	1.65685	0.0825338
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.66894	0.132294
$408$	0	0
$409$	−8.01735	−0.396432	−0.198216	−	0.980158i	$-0.563515\pi$
$409$	−8.01735	−0.396432	−0.198216	+	0.980158i	$0.563515\pi$
$410$	0	0
$411$	−16.5644	−0.817062
$412$	0	0
$413$	1.11233	0.0547344
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.6854	−0.523267
$418$	0	0
$419$	8.40151	0.410440	0.205220	−	0.978716i	$-0.434209\pi$
$419$	8.40151	0.410440	0.205220	+	0.978716i	$0.434209\pi$
$420$	0	0
$421$	−11.7514	−0.572728	−0.286364	−	0.958121i	$-0.592447\pi$
$421$	−11.7514	−0.572728	−0.286364	+	0.958121i	$0.592447\pi$
$422$	0	0
$423$	3.00868	0.146287
$424$	0	0
$425$	0	0
$426$	0	0
$427$	31.2299	1.51132
$428$	0	0
$429$	−0.393270	−0.0189872
$430$	0	0
$431$	−20.5818	−0.991388	−0.495694	−	0.868497i	$-0.665086\pi$
$431$	−20.5818	−0.991388	−0.495694	+	0.868497i	$0.665086\pi$
$432$	0	0
$433$	−5.12522	−0.246303	−0.123151	−	0.992388i	$-0.539300\pi$
$433$	−5.12522	−0.246303	−0.123151	+	0.992388i	$0.539300\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.6016	0.507145
$438$	0	0
$439$	−12.8868	−0.615053	−0.307526	−	0.951540i	$-0.599501\pi$
$439$	−12.8868	−0.615053	−0.307526	+	0.951540i	$0.599501\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−35.1279	−1.66898	−0.834489	−	0.551024i	$-0.814237\pi$
$443$	−35.1279	−1.66898	−0.834489	+	0.551024i	$0.814237\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.05012	−0.191564
$448$	0	0
$449$	18.2213	0.859914	0.429957	−	0.902849i	$-0.358529\pi$
$449$	18.2213	0.859914	0.429957	+	0.902849i	$0.358529\pi$
$450$	0	0
$451$	1.48572	0.0699598
$452$	0	0
$453$	−5.02856	−0.236262
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.4118	−1.09516	−0.547580	−	0.836753i	$-0.684451\pi$
$457$	−23.4118	−1.09516	−0.547580	+	0.836753i	$0.684451\pi$
$458$	0	0
$459$	6.21302	0.289999
$460$	0	0
$461$	−14.2913	−0.665611	−0.332805	−	0.942996i	$-0.607995\pi$
$461$	−14.2913	−0.665611	−0.332805	+	0.942996i	$0.607995\pi$
$462$	0	0
$463$	−23.7152	−1.10214	−0.551070	−	0.834459i	$-0.685780\pi$
$463$	−23.7152	−1.10214	−0.551070	+	0.834459i	$0.685780\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.8989	1.01336	0.506680	−	0.862134i	$-0.330873\pi$
$467$	21.8989	1.01336	0.506680	+	0.862134i	$0.330873\pi$
$468$	0	0
$469$	43.9982	2.03165
$470$	0	0
$471$	−24.4961	−1.12872
$472$	0	0
$473$	−0.988790	−0.0454646
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.55573	−0.437527
$478$	0	0
$479$	−17.0916	−0.780934	−0.390467	−	0.920617i	$-0.627686\pi$
$479$	−17.0916	−0.780934	−0.390467	+	0.920617i	$0.627686\pi$
$480$	0	0
$481$	6.78654	0.309440
$482$	0	0
$483$	4.08378	0.185818
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.87434	0.0849343	0.0424672	−	0.999098i	$-0.486478\pi$
$487$	1.87434	0.0849343	0.0424672	+	0.999098i	$0.486478\pi$
$488$	0	0
$489$	−15.3137	−0.692510
$490$	0	0
$491$	−10.7704	−0.486063	−0.243031	−	0.970018i	$-0.578142\pi$
$491$	−10.7704	−0.486063	−0.243031	+	0.970018i	$0.578142\pi$
$492$	0	0
$493$	5.14704	0.231811
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.5669	0.832841
$498$	0	0
$499$	33.5259	1.50082	0.750412	−	0.660971i	$-0.229855\pi$
$499$	33.5259	1.50082	0.750412	+	0.660971i	$0.229855\pi$
$500$	0	0
$501$	15.7770	0.704864
$502$	0	0
$503$	−7.58473	−0.338186	−0.169093	−	0.985600i	$-0.554084\pi$
$503$	−7.58473	−0.338186	−0.169093	+	0.985600i	$0.554084\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	38.0145	1.68497	0.842483	−	0.538724i	$-0.181093\pi$
$509$	38.0145	1.68497	0.842483	+	0.538724i	$0.181093\pi$
$510$	0	0
$511$	38.5089	1.70354
$512$	0	0
$513$	7.34271	0.324188
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.18322	0.0520380
$518$	0	0
$519$	14.9706	0.657135
$520$	0	0
$521$	−18.2420	−0.799197	−0.399599	−	0.916690i	$-0.630851\pi$
$521$	−18.2420	−0.799197	−0.399599	+	0.916690i	$0.630851\pi$
$522$	0	0
$523$	10.1720	0.444791	0.222396	−	0.974957i	$-0.428612\pi$
$523$	10.1720	0.444791	0.222396	+	0.974957i	$0.428612\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.2941	−0.448417
$528$	0	0
$529$	−20.9153	−0.909363
$530$	0	0
$531$	0.393270	0.0170665
$532$	0	0
$533$	3.77786	0.163637
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.84578	0.209111
$538$	0	0
$539$	0.393270	0.0169393
$540$	0	0
$541$	1.98923	0.0855236	0.0427618	−	0.999085i	$-0.486384\pi$
$541$	1.98923	0.0855236	0.0427618	+	0.999085i	$0.486384\pi$
$542$	0	0
$543$	7.99912	0.343275
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.31817	0.0563609	0.0281804	−	0.999603i	$-0.491029\pi$
$547$	1.31817	0.0563609	0.0281804	+	0.999603i	$0.491029\pi$
$548$	0	0
$549$	11.0414	0.471238
$550$	0	0
$551$	6.08290	0.259140
$552$	0	0
$553$	−42.2576	−1.79698
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0804	0.511861	0.255931	−	0.966695i	$-0.417618\pi$
$557$	12.0804	0.511861	0.255931	+	0.966695i	$0.417618\pi$
$558$	0	0
$559$	−2.51428	−0.106343
$560$	0	0
$561$	2.44339	0.103160
$562$	0	0
$563$	−22.9930	−0.969039	−0.484519	−	0.874781i	$-0.661005\pi$
$563$	−22.9930	−0.969039	−0.484519	+	0.874781i	$0.661005\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	29.0370	1.21729	0.608647	−	0.793441i	$-0.291713\pi$
$569$	29.0370	1.21729	0.608647	+	0.793441i	$0.291713\pi$
$570$	0	0
$571$	5.56950	0.233076	0.116538	−	0.993186i	$-0.462820\pi$
$571$	5.56950	0.233076	0.116538	+	0.993186i	$0.462820\pi$
$572$	0	0
$573$	20.4260	0.853310
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20.2822	−0.844357	−0.422179	−	0.906513i	$-0.638734\pi$
$577$	−20.2822	−0.844357	−0.422179	+	0.906513i	$0.638734\pi$
$578$	0	0
$579$	−24.0237	−0.998389
$580$	0	0
$581$	46.5652	1.93185
$582$	0	0
$583$	−3.75798	−0.155640
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.82377	0.364196	0.182098	−	0.983280i	$-0.441711\pi$
$587$	8.82377	0.364196	0.182098	+	0.983280i	$0.441711\pi$
$588$	0	0
$589$	−12.1658	−0.501283
$590$	0	0
$591$	4.80642	0.197710
$592$	0	0
$593$	10.2213	0.419737	0.209868	−	0.977730i	$-0.432696\pi$
$593$	10.2213	0.419737	0.209868	+	0.977730i	$0.432696\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.9403	0.611467
$598$	0	0
$599$	−18.4925	−0.755582	−0.377791	−	0.925891i	$-0.623316\pi$
$599$	−18.4925	−0.755582	−0.377791	+	0.925891i	$0.623316\pi$
$600$	0	0
$601$	26.0129	1.06109	0.530544	−	0.847657i	$-0.321988\pi$
$601$	26.0129	1.06109	0.530544	+	0.847657i	$0.321988\pi$
$602$	0	0
$603$	15.5557	0.633478
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.1227	0.979109	0.489554	−	0.871973i	$-0.337159\pi$
$607$	24.1227	0.979109	0.489554	+	0.871973i	$0.337159\pi$
$608$	0	0
$609$	2.34315	0.0949491
$610$	0	0
$611$	3.00868	0.121718
$612$	0	0
$613$	34.2056	1.38155	0.690775	−	0.723070i	$-0.257270\pi$
$613$	34.2056	1.38155	0.690775	+	0.723070i	$0.257270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.3622	−0.497682	−0.248841	−	0.968544i	$-0.580050\pi$
$617$	−12.3622	−0.497682	−0.248841	+	0.968544i	$0.580050\pi$
$618$	0	0
$619$	−11.6685	−0.468997	−0.234498	−	0.972117i	$-0.575345\pi$
$619$	−11.6685	−0.468997	−0.234498	+	0.972117i	$0.575345\pi$
$620$	0	0
$621$	1.44383	0.0579390
$622$	0	0
$623$	−27.3700	−1.09655
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.88767	0.115322
$628$	0	0
$629$	−42.1649	−1.68123
$630$	0	0
$631$	3.79775	0.151186	0.0755930	−	0.997139i	$-0.475915\pi$
$631$	3.79775	0.151186	0.0755930	+	0.997139i	$0.475915\pi$
$632$	0	0
$633$	1.82799	0.0726560
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	6.56440	0.259684
$640$	0	0
$641$	−6.91008	−0.272932	−0.136466	−	0.990645i	$-0.543574\pi$
$641$	−6.91008	−0.272932	−0.136466	+	0.990645i	$0.543574\pi$
$642$	0	0
$643$	−2.50139	−0.0986452	−0.0493226	−	0.998783i	$-0.515706\pi$
$643$	−2.50139	−0.0986452	−0.0493226	+	0.998783i	$0.515706\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.3124	−1.19170	−0.595852	−	0.803095i	$-0.703185\pi$
$647$	−30.3124	−1.19170	−0.595852	+	0.803095i	$0.703185\pi$
$648$	0	0
$649$	0.154661	0.00607098
$650$	0	0
$651$	−4.68629	−0.183670
$652$	0	0
$653$	−33.4537	−1.30915	−0.654573	−	0.755999i	$-0.727151\pi$
$653$	−33.4537	−1.30915	−0.654573	+	0.755999i	$0.727151\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.6150	0.531170
$658$	0	0
$659$	−31.2942	−1.21905	−0.609525	−	0.792767i	$-0.708640\pi$
$659$	−31.2942	−1.21905	−0.609525	+	0.792767i	$0.708640\pi$
$660$	0	0
$661$	40.1119	1.56017	0.780086	−	0.625672i	$-0.215175\pi$
$661$	40.1119	1.56017	0.780086	+	0.625672i	$0.215175\pi$
$662$	0	0
$663$	6.21302	0.241294
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.19611	0.0463136
$668$	0	0
$669$	7.07045	0.273359
$670$	0	0
$671$	4.34227	0.167631
$672$	0	0
$673$	4.99386	0.192499	0.0962496	−	0.995357i	$-0.469315\pi$
$673$	4.99386	0.192499	0.0962496	+	0.995357i	$0.469315\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.0174	−0.615597	−0.307798	−	0.951452i	$-0.599592\pi$
$677$	−16.0174	−0.615597	−0.307798	+	0.951452i	$0.599592\pi$
$678$	0	0
$679$	0.118476	0.00454668
$680$	0	0
$681$	−5.43560	−0.208292
$682$	0	0
$683$	−35.5400	−1.35990	−0.679951	−	0.733258i	$-0.737999\pi$
$683$	−35.5400	−1.35990	−0.679951	+	0.733258i	$0.737999\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−24.8147	−0.946738
$688$	0	0
$689$	−9.55573	−0.364044
$690$	0	0
$691$	13.0393	0.496040	0.248020	−	0.968755i	$-0.420220\pi$
$691$	13.0393	0.496040	0.248020	+	0.968755i	$0.420220\pi$
$692$	0	0
$693$	1.11233	0.0422541
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23.4720	−0.889064
$698$	0	0
$699$	−23.4256	−0.886038
$700$	0	0
$701$	−0.502632	−0.0189841	−0.00949207	−	0.999955i	$-0.503021\pi$
$701$	−0.502632	−0.0189841	−0.00949207	+	0.999955i	$0.503021\pi$
$702$	0	0
$703$	−49.8316	−1.87943
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.3431	1.29161
$708$	0	0
$709$	42.0712	1.58002	0.790009	−	0.613095i	$-0.210076\pi$
$709$	42.0712	1.58002	0.790009	+	0.613095i	$0.210076\pi$
$710$	0	0
$711$	−14.9403	−0.560306
$712$	0	0
$713$	−2.39222	−0.0895893
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.7779	0.813309
$718$	0	0
$719$	32.7848	1.22267	0.611333	−	0.791373i	$-0.290634\pi$
$719$	32.7848	1.22267	0.611333	+	0.791373i	$0.290634\pi$
$720$	0	0
$721$	−0.0490743	−0.00182762
$722$	0	0
$723$	15.2126	0.565762
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−49.5246	−1.83677	−0.918383	−	0.395692i	$-0.870505\pi$
$727$	−49.5246	−1.83677	−0.918383	+	0.395692i	$0.870505\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.6213	0.577774
$732$	0	0
$733$	−2.11848	−0.0782477	−0.0391238	−	0.999234i	$-0.512457\pi$
$733$	−2.11848	−0.0782477	−0.0391238	+	0.999234i	$0.512457\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.11760	0.225345
$738$	0	0
$739$	−47.7014	−1.75473	−0.877363	−	0.479827i	$-0.840699\pi$
$739$	−47.7014	−1.75473	−0.877363	+	0.479827i	$0.840699\pi$
$740$	0	0
$741$	7.34271	0.269741
$742$	0	0
$743$	32.8055	1.20352	0.601759	−	0.798677i	$-0.294467\pi$
$743$	32.8055	1.20352	0.601759	+	0.798677i	$0.294467\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16.4633	0.602360
$748$	0	0
$749$	5.94288	0.217148
$750$	0	0
$751$	−0.852087	−0.0310931	−0.0155465	−	0.999879i	$-0.504949\pi$
$751$	−0.852087	−0.0310931	−0.0155465	+	0.999879i	$0.504949\pi$
$752$	0	0
$753$	17.9399	0.653766
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.68454	−0.242954	−0.121477	−	0.992594i	$-0.538763\pi$
$757$	−6.68454	−0.242954	−0.121477	+	0.992594i	$0.538763\pi$
$758$	0	0
$759$	0.567816	0.0206104
$760$	0	0
$761$	−2.92578	−0.106059	−0.0530297	−	0.998593i	$-0.516888\pi$
$761$	−2.92578	−0.106059	−0.0530297	+	0.998593i	$0.516888\pi$
$762$	0	0
$763$	17.5731	0.636188
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.393270	0.0142001
$768$	0	0
$769$	−12.6274	−0.455356	−0.227678	−	0.973736i	$-0.573113\pi$
$769$	−12.6274	−0.455356	−0.227678	+	0.973736i	$0.573113\pi$
$770$	0	0
$771$	−3.88810	−0.140027
$772$	0	0
$773$	15.9454	0.573517	0.286758	−	0.958003i	$-0.407422\pi$
$773$	15.9454	0.573517	0.286758	+	0.958003i	$0.407422\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−19.1952	−0.688625
$778$	0	0
$779$	−27.7398	−0.993880
$780$	0	0
$781$	2.58158	0.0923763
$782$	0	0
$783$	0.828427	0.0296056
$784$	0	0
$785$	0	0
$786$	0	0
$787$	48.5825	1.73178	0.865890	−	0.500234i	$-0.166753\pi$
$787$	48.5825	1.73178	0.865890	+	0.500234i	$0.166753\pi$
$788$	0	0
$789$	0.296797	0.0105662
$790$	0	0
$791$	4.03470	0.143457
$792$	0	0
$793$	11.0414	0.392093
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.8971	0.704792	0.352396	−	0.935851i	$-0.385367\pi$
$797$	19.8971	0.704792	0.352396	+	0.935851i	$0.385367\pi$
$798$	0	0
$799$	−18.6930	−0.661310
$800$	0	0
$801$	−9.67674	−0.341911
$802$	0	0
$803$	5.35436	0.188951
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.25447	0.0441595
$808$	0	0
$809$	−12.5487	−0.441189	−0.220595	−	0.975366i	$-0.570800\pi$
$809$	−12.5487	−0.441189	−0.220595	+	0.975366i	$0.570800\pi$
$810$	0	0
$811$	30.9149	1.08557	0.542785	−	0.839872i	$-0.317370\pi$
$811$	30.9149	1.08557	0.542785	+	0.839872i	$0.317370\pi$
$812$	0	0
$813$	−9.81510	−0.344231
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.4616	0.645890
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	−29.9308	−1.04459	−0.522296	−	0.852765i	$-0.674924\pi$
$821$	−29.9308	−1.04459	−0.522296	+	0.852765i	$0.674924\pi$
$822$	0	0
$823$	−16.1831	−0.564109	−0.282055	−	0.959398i	$-0.591016\pi$
$823$	−16.1831	−0.564109	−0.282055	+	0.959398i	$0.591016\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.2022	1.88479	0.942397	−	0.334497i	$-0.108566\pi$
$827$	54.2022	1.88479	0.942397	+	0.334497i	$0.108566\pi$
$828$	0	0
$829$	−42.2929	−1.46889	−0.734447	−	0.678666i	$-0.762558\pi$
$829$	−42.2929	−1.46889	−0.734447	+	0.678666i	$0.762558\pi$
$830$	0	0
$831$	4.35603	0.151109
$832$	0	0
$833$	−6.21302	−0.215268
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.65685	−0.0572693
$838$	0	0
$839$	36.7873	1.27004	0.635020	−	0.772496i	$-0.280992\pi$
$839$	36.7873	1.27004	0.635020	+	0.772496i	$0.280992\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	−3.45207	−0.118896
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−30.6753	−1.05401
$848$	0	0
$849$	−2.25937	−0.0775414
$850$	0	0
$851$	−9.79863	−0.335893
$852$	0	0
$853$	−15.4148	−0.527794	−0.263897	−	0.964551i	$-0.585008\pi$
$853$	−15.4148	−0.527794	−0.263897	+	0.964551i	$0.585008\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.0628221	0.00214596	0.00107298	−	0.999999i	$-0.499658\pi$
$857$	0.0628221	0.00214596	0.00107298	+	0.999999i	$0.499658\pi$
$858$	0	0
$859$	28.6577	0.977787	0.488893	−	0.872344i	$-0.337401\pi$
$859$	28.6577	0.977787	0.488893	+	0.872344i	$0.337401\pi$
$860$	0	0
$861$	−10.6854	−0.364158
$862$	0	0
$863$	−5.96189	−0.202945	−0.101473	−	0.994838i	$-0.532355\pi$
$863$	−5.96189	−0.202945	−0.101473	+	0.994838i	$0.532355\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−21.6016	−0.733630
$868$	0	0
$869$	−5.87558	−0.199315
$870$	0	0
$871$	15.5557	0.527086
$872$	0	0
$873$	0.0418875	0.00141768
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.70276	−0.226336	−0.113168	−	0.993576i	$-0.536100\pi$
$877$	−6.70276	−0.226336	−0.113168	+	0.993576i	$0.536100\pi$
$878$	0	0
$879$	−19.5358	−0.658928
$880$	0	0
$881$	39.8624	1.34300	0.671500	−	0.741005i	$-0.265651\pi$
$881$	39.8624	1.34300	0.671500	+	0.741005i	$0.265651\pi$
$882$	0	0
$883$	21.7131	0.730704	0.365352	−	0.930869i	$-0.380949\pi$
$883$	21.7131	0.730704	0.365352	+	0.930869i	$0.380949\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	51.3757	1.72503	0.862513	−	0.506035i	$-0.168889\pi$
$887$	51.3757	1.72503	0.862513	+	0.506035i	$0.168889\pi$
$888$	0	0
$889$	4.25133	0.142585
$890$	0	0
$891$	0.393270	0.0131750
$892$	0	0
$893$	−22.0918	−0.739275
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.44383	0.0482082
$898$	0	0
$899$	−1.37258	−0.0457782
$900$	0	0
$901$	59.3700	1.97790
$902$	0	0
$903$	7.11146	0.236654
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−21.3966	−0.710463	−0.355231	−	0.934778i	$-0.615598\pi$
$907$	−21.3966	−0.710463	−0.355231	+	0.934778i	$0.615598\pi$
$908$	0	0
$909$	12.1421	0.402729
$910$	0	0
$911$	−51.0044	−1.68985	−0.844925	−	0.534884i	$-0.820355\pi$
$911$	−51.0044	−1.68985	−0.844925	+	0.534884i	$0.820355\pi$
$912$	0	0
$913$	6.47451	0.214275
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.9092	0.360254
$918$	0	0
$919$	11.3664	0.374942	0.187471	−	0.982270i	$-0.439971\pi$
$919$	11.3664	0.374942	0.187471	+	0.982270i	$0.439971\pi$
$920$	0	0
$921$	27.8142	0.916510
$922$	0	0
$923$	6.56440	0.216070
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.0173504	−0.000569861 0
$928$	0	0
$929$	47.9344	1.57268	0.786338	−	0.617797i	$-0.211975\pi$
$929$	47.9344	1.57268	0.786338	+	0.617797i	$0.211975\pi$
$930$	0	0
$931$	−7.34271	−0.240648
$932$	0	0
$933$	−2.10113	−0.0687878
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.3266	−0.762047	−0.381023	−	0.924565i	$-0.624428\pi$
$937$	−23.3266	−0.762047	−0.381023	+	0.924565i	$0.624428\pi$
$938$	0	0
$939$	−9.64397	−0.314719
$940$	0	0
$941$	−39.6316	−1.29195	−0.645977	−	0.763357i	$-0.723550\pi$
$941$	−39.6316	−1.29195	−0.645977	+	0.763357i	$0.723550\pi$
$942$	0	0
$943$	−5.45460	−0.177626
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.7770	−0.642666	−0.321333	−	0.946966i	$-0.604131\pi$
$947$	−19.7770	−0.642666	−0.321333	+	0.946966i	$0.604131\pi$
$948$	0	0
$949$	13.6150	0.441961
$950$	0	0
$951$	16.1201	0.522731
$952$	0	0
$953$	−45.2847	−1.46692	−0.733458	−	0.679735i	$-0.762095\pi$
$953$	−45.2847	−1.46692	−0.733458	+	0.679735i	$0.762095\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.325795	0.0105315
$958$	0	0
$959$	46.8512	1.51290
$960$	0	0
$961$	−28.2548	−0.911446
$962$	0	0
$963$	2.10113	0.0677078
$964$	0	0
$965$	0	0
$966$	0	0
$967$	13.3903	0.430603	0.215301	−	0.976548i	$-0.430927\pi$
$967$	13.3903	0.430603	0.215301	+	0.976548i	$0.430927\pi$
$968$	0	0
$969$	−45.6204	−1.46554
$970$	0	0
$971$	−39.6150	−1.27130	−0.635652	−	0.771975i	$-0.719269\pi$
$971$	−39.6150	−1.27130	−0.635652	+	0.771975i	$0.719269\pi$
$972$	0	0
$973$	30.2229	0.968902
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.2100	−0.614584	−0.307292	−	0.951615i	$-0.599423\pi$
$977$	−19.2100	−0.614584	−0.307292	+	0.951615i	$0.599423\pi$
$978$	0	0
$979$	−3.80557	−0.121627
$980$	0	0
$981$	6.21302	0.198367
$982$	0	0
$983$	−32.3960	−1.03327	−0.516636	−	0.856205i	$-0.672816\pi$
$983$	−32.3960	−1.03327	−0.516636	+	0.856205i	$0.672816\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.50982	−0.270871
$988$	0	0
$989$	3.63020	0.115434
$990$	0	0
$991$	10.5107	0.333883	0.166942	−	0.985967i	$-0.446611\pi$
$991$	10.5107	0.333883	0.166942	+	0.985967i	$0.446611\pi$
$992$	0	0
$993$	−13.9537	−0.442806
$994$	0	0
$995$	0	0
$996$	0	0
$997$	41.0753	1.30087	0.650433	−	0.759563i	$-0.274587\pi$
$997$	41.0753	1.30087	0.650433	+	0.759563i	$0.274587\pi$
$998$	0	0
$999$	−6.78654	−0.214717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bv.1.4		4
5.2	odd	4		1560.2.l.e.1249.6	yes	8
5.3	odd	4		1560.2.l.e.1249.2	✓	8
5.4	even	2		7800.2.a.bw.1.2		4
15.2	even	4		4680.2.l.f.2809.6		8
15.8	even	4		4680.2.l.f.2809.5		8
20.3	even	4		3120.2.l.o.1249.6		8
20.7	even	4		3120.2.l.o.1249.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.2	✓	8	5.3	odd	4
1560.2.l.e.1249.6	yes	8	5.2	odd	4
3120.2.l.o.1249.2		8	20.7	even	4
3120.2.l.o.1249.6		8	20.3	even	4
4680.2.l.f.2809.5		8	15.8	even	4
4680.2.l.f.2809.6		8	15.2	even	4
7800.2.a.bv.1.4		4	1.1	even	1	trivial
7800.2.a.bw.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.82843 q^{7} +1.00000 q^{9} +0.393270 q^{11} +1.00000 q^{13} -6.21302 q^{17} -7.34271 q^{19} -2.82843 q^{21} -1.44383 q^{23} -1.00000 q^{27} -0.828427 q^{29} +1.65685 q^{31} -0.393270 q^{33} +6.78654 q^{37} -1.00000 q^{39} +3.77786 q^{41} -2.51428 q^{43} +3.00868 q^{47} +1.00000 q^{49} +6.21302 q^{51} -9.55573 q^{53} +7.34271 q^{57} +0.393270 q^{59} +11.0414 q^{61} +2.82843 q^{63} +15.5557 q^{67} +1.44383 q^{69} +6.56440 q^{71} +13.6150 q^{73} +1.11233 q^{77} -14.9403 q^{79} +1.00000 q^{81} +16.4633 q^{83} +0.828427 q^{87} -9.67674 q^{89} +2.82843 q^{91} -1.65685 q^{93} +0.0418875 q^{97} +0.393270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 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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