LMFDB - Embedded newform 7800.2.a.ba.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.41421 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.41421 q^{7} +1.00000 q^{9} +0.414214 q^{11} +1.00000 q^{13} -3.82843 q^{17} -2.00000 q^{19} -2.41421 q^{21} +0.828427 q^{23} +1.00000 q^{27} -8.65685 q^{29} +4.41421 q^{31} +0.414214 q^{33} +7.65685 q^{37} +1.00000 q^{39} +5.65685 q^{41} +10.4853 q^{43} +2.07107 q^{47} -1.17157 q^{49} -3.82843 q^{51} +5.82843 q^{53} -2.00000 q^{57} -6.41421 q^{59} -1.82843 q^{61} -2.41421 q^{63} -0.757359 q^{67} +0.828427 q^{69} -7.65685 q^{71} -9.31371 q^{73} -1.00000 q^{77} -8.82843 q^{79} +1.00000 q^{81} -13.7279 q^{83} -8.65685 q^{87} -2.41421 q^{91} +4.41421 q^{93} -14.9706 q^{97} +0.414214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} - 2 q^{21} - 4 q^{23} + 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 4 q^{43} - 10 q^{47} - 8 q^{49} - 2 q^{51} + 6 q^{53} - 4 q^{57} - 10 q^{59} + 2 q^{61} - 2 q^{63} - 10 q^{67} - 4 q^{69} - 4 q^{71} + 4 q^{73} - 2 q^{77} - 12 q^{79} + 2 q^{81} - 2 q^{83} - 6 q^{87} - 2 q^{91} + 6 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 4 * q^19 - 2 * q^21 - 4 * q^23 + 2 * q^27 - 6 * q^29 + 6 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 + 4 * q^43 - 10 * q^47 - 8 * q^49 - 2 * q^51 + 6 * q^53 - 4 * q^57 - 10 * q^59 + 2 * q^61 - 2 * q^63 - 10 * q^67 - 4 * q^69 - 4 * q^71 + 4 * q^73 - 2 * q^77 - 12 * q^79 + 2 * q^81 - 2 * q^83 - 6 * q^87 - 2 * q^91 + 6 * q^93 + 4 * q^97 - 2 * 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
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.414214	0.124890	0.0624450	−	0.998048i	$-0.480110\pi$
$11$	0.414214	0.124890	0.0624450	+	0.998048i	$0.480110\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.82843	−0.928530	−0.464265	−	0.885696i	$-0.653681\pi$
$17$	−3.82843	−0.928530	−0.464265	+	0.885696i	$0.653681\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−2.41421	−0.526825
$22$	0	0
$23$	0.828427	0.172739	0.0863695	−	0.996263i	$-0.472473\pi$
$23$	0.828427	0.172739	0.0863695	+	0.996263i	$0.472473\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.65685	−1.60754	−0.803769	−	0.594942i	$-0.797175\pi$
$29$	−8.65685	−1.60754	−0.803769	+	0.594942i	$0.797175\pi$
$30$	0	0
$31$	4.41421	0.792816	0.396408	−	0.918074i	$-0.370257\pi$
$31$	4.41421	0.792816	0.396408	+	0.918074i	$0.370257\pi$
$32$	0	0
$33$	0.414214	0.0721053
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.65685	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	10.4853	1.59899	0.799495	−	0.600672i	$-0.205100\pi$
$43$	10.4853	1.59899	0.799495	+	0.600672i	$0.205100\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.07107	0.302096	0.151048	−	0.988526i	$-0.451735\pi$
$47$	2.07107	0.302096	0.151048	+	0.988526i	$0.451735\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	−3.82843	−0.536087
$52$	0	0
$53$	5.82843	0.800596	0.400298	−	0.916385i	$-0.368907\pi$
$53$	5.82843	0.800596	0.400298	+	0.916385i	$0.368907\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−6.41421	−0.835059	−0.417530	−	0.908663i	$-0.637104\pi$
$59$	−6.41421	−0.835059	−0.417530	+	0.908663i	$0.637104\pi$
$60$	0	0
$61$	−1.82843	−0.234106	−0.117053	−	0.993126i	$-0.537345\pi$
$61$	−1.82843	−0.234106	−0.117053	+	0.993126i	$0.537345\pi$
$62$	0	0
$63$	−2.41421	−0.304162
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.757359	−0.0925262	−0.0462631	−	0.998929i	$-0.514731\pi$
$67$	−0.757359	−0.0925262	−0.0462631	+	0.998929i	$0.514731\pi$
$68$	0	0
$69$	0.828427	0.0997309
$70$	0	0
$71$	−7.65685	−0.908701	−0.454351	−	0.890823i	$-0.650129\pi$
$71$	−7.65685	−0.908701	−0.454351	+	0.890823i	$0.650129\pi$
$72$	0	0
$73$	−9.31371	−1.09009	−0.545044	−	0.838408i	$-0.683487\pi$
$73$	−9.31371	−1.09009	−0.545044	+	0.838408i	$0.683487\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−8.82843	−0.993276	−0.496638	−	0.867958i	$-0.665432\pi$
$79$	−8.82843	−0.993276	−0.496638	+	0.867958i	$0.665432\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.7279	−1.50684	−0.753418	−	0.657542i	$-0.771596\pi$
$83$	−13.7279	−1.50684	−0.753418	+	0.657542i	$0.771596\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.65685	−0.928112
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−2.41421	−0.253078
$92$	0	0
$93$	4.41421	0.457733
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.9706	−1.52003	−0.760015	−	0.649905i	$-0.774809\pi$
$97$	−14.9706	−1.52003	−0.760015	+	0.649905i	$0.774809\pi$
$98$	0	0
$99$	0.414214	0.0416300
$100$	0	0
$101$	−6.17157	−0.614094	−0.307047	−	0.951694i	$-0.599341\pi$
$101$	−6.17157	−0.614094	−0.307047	+	0.951694i	$0.599341\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.4853	−1.78704	−0.893520	−	0.449024i	$-0.851772\pi$
$107$	−18.4853	−1.78704	−0.893520	+	0.449024i	$0.851772\pi$
$108$	0	0
$109$	−19.3137	−1.84992	−0.924959	−	0.380067i	$-0.875901\pi$
$109$	−19.3137	−1.84992	−0.924959	+	0.380067i	$0.875901\pi$
$110$	0	0
$111$	7.65685	0.726756
$112$	0	0
$113$	13.3137	1.25245	0.626224	−	0.779643i	$-0.284599\pi$
$113$	13.3137	1.25245	0.626224	+	0.779643i	$0.284599\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	9.24264	0.847271
$120$	0	0
$121$	−10.8284	−0.984402
$122$	0	0
$123$	5.65685	0.510061
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.1716	−0.991317	−0.495658	−	0.868518i	$-0.665073\pi$
$127$	−11.1716	−0.991317	−0.495658	+	0.868518i	$0.665073\pi$
$128$	0	0
$129$	10.4853	0.923178
$130$	0	0
$131$	−17.6569	−1.54269	−0.771343	−	0.636419i	$-0.780415\pi$
$131$	−17.6569	−1.54269	−0.771343	+	0.636419i	$0.780415\pi$
$132$	0	0
$133$	4.82843	0.418678
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.9706	1.44989	0.724947	−	0.688805i	$-0.241864\pi$
$137$	16.9706	1.44989	0.724947	+	0.688805i	$0.241864\pi$
$138$	0	0
$139$	−3.17157	−0.269009	−0.134505	−	0.990913i	$-0.542944\pi$
$139$	−3.17157	−0.269009	−0.134505	+	0.990913i	$0.542944\pi$
$140$	0	0
$141$	2.07107	0.174415
$142$	0	0
$143$	0.414214	0.0346383
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.17157	−0.0966297
$148$	0	0
$149$	17.3137	1.41839	0.709197	−	0.705010i	$-0.249058\pi$
$149$	17.3137	1.41839	0.709197	+	0.705010i	$0.249058\pi$
$150$	0	0
$151$	−7.24264	−0.589398	−0.294699	−	0.955590i	$-0.595219\pi$
$151$	−7.24264	−0.589398	−0.294699	+	0.955590i	$0.595219\pi$
$152$	0	0
$153$	−3.82843	−0.309510
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.3137	1.46159	0.730797	−	0.682595i	$-0.239149\pi$
$157$	18.3137	1.46159	0.730797	+	0.682595i	$0.239149\pi$
$158$	0	0
$159$	5.82843	0.462224
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	2.68629	0.210407	0.105203	−	0.994451i	$-0.466451\pi$
$163$	2.68629	0.210407	0.105203	+	0.994451i	$0.466451\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−25.3137	−1.95883	−0.979417	−	0.201848i	$-0.935305\pi$
$167$	−25.3137	−1.95883	−0.979417	+	0.201848i	$0.935305\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−3.48528	−0.264981	−0.132491	−	0.991184i	$-0.542297\pi$
$173$	−3.48528	−0.264981	−0.132491	+	0.991184i	$0.542297\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.41421	−0.482122
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−3.34315	−0.248494	−0.124247	−	0.992251i	$-0.539652\pi$
$181$	−3.34315	−0.248494	−0.124247	+	0.992251i	$0.539652\pi$
$182$	0	0
$183$	−1.82843	−0.135161
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.58579	−0.115964
$188$	0	0
$189$	−2.41421	−0.175608
$190$	0	0
$191$	14.4853	1.04812	0.524059	−	0.851682i	$-0.324417\pi$
$191$	14.4853	1.04812	0.524059	+	0.851682i	$0.324417\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	−0.828427	−0.0587256	−0.0293628	−	0.999569i	$-0.509348\pi$
$199$	−0.828427	−0.0587256	−0.0293628	+	0.999569i	$0.509348\pi$
$200$	0	0
$201$	−0.757359	−0.0534200
$202$	0	0
$203$	20.8995	1.46686
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.828427	0.0575797
$208$	0	0
$209$	−0.828427	−0.0573035
$210$	0	0
$211$	10.4853	0.721837	0.360918	−	0.932597i	$-0.382463\pi$
$211$	10.4853	0.721837	0.360918	+	0.932597i	$0.382463\pi$
$212$	0	0
$213$	−7.65685	−0.524639
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.6569	−0.723434
$218$	0	0
$219$	−9.31371	−0.629362
$220$	0	0
$221$	−3.82843	−0.257528
$222$	0	0
$223$	23.6569	1.58418	0.792090	−	0.610404i	$-0.208993\pi$
$223$	23.6569	1.58418	0.792090	+	0.610404i	$0.208993\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.5858	−0.768976	−0.384488	−	0.923130i	$-0.625622\pi$
$227$	−11.5858	−0.768976	−0.384488	+	0.923130i	$0.625622\pi$
$228$	0	0
$229$	−4.34315	−0.287003	−0.143502	−	0.989650i	$-0.545836\pi$
$229$	−4.34315	−0.287003	−0.143502	+	0.989650i	$0.545836\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.82843	−0.573468
$238$	0	0
$239$	−5.24264	−0.339118	−0.169559	−	0.985520i	$-0.554234\pi$
$239$	−5.24264	−0.339118	−0.169559	+	0.985520i	$0.554234\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−13.7279	−0.869972
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0.343146	0.0215734
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.00000	−0.0623783	−0.0311891	−	0.999514i	$-0.509929\pi$
$257$	−1.00000	−0.0623783	−0.0311891	+	0.999514i	$0.509929\pi$
$258$	0	0
$259$	−18.4853	−1.14862
$260$	0	0
$261$	−8.65685	−0.535846
$262$	0	0
$263$	3.31371	0.204332	0.102166	−	0.994767i	$-0.467423\pi$
$263$	3.31371	0.204332	0.102166	+	0.994767i	$0.467423\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.3137	0.750780	0.375390	−	0.926867i	$-0.377509\pi$
$269$	12.3137	0.750780	0.375390	+	0.926867i	$0.377509\pi$
$270$	0	0
$271$	−4.89949	−0.297623	−0.148812	−	0.988866i	$-0.547545\pi$
$271$	−4.89949	−0.297623	−0.148812	+	0.988866i	$0.547545\pi$
$272$	0	0
$273$	−2.41421	−0.146115
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.31371	0.559607	0.279803	−	0.960057i	$-0.409731\pi$
$277$	9.31371	0.559607	0.279803	+	0.960057i	$0.409731\pi$
$278$	0	0
$279$	4.41421	0.264272
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	−27.4558	−1.63208	−0.816040	−	0.577995i	$-0.803835\pi$
$283$	−27.4558	−1.63208	−0.816040	+	0.577995i	$0.803835\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.6569	−0.806139
$288$	0	0
$289$	−2.34315	−0.137832
$290$	0	0
$291$	−14.9706	−0.877590
$292$	0	0
$293$	−5.31371	−0.310430	−0.155215	−	0.987881i	$-0.549607\pi$
$293$	−5.31371	−0.310430	−0.155215	+	0.987881i	$0.549607\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.414214	0.0240351
$298$	0	0
$299$	0.828427	0.0479092
$300$	0	0
$301$	−25.3137	−1.45906
$302$	0	0
$303$	−6.17157	−0.354548
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.4853	−1.04820	−0.524102	−	0.851655i	$-0.675599\pi$
$311$	−18.4853	−1.04820	−0.524102	+	0.851655i	$0.675599\pi$
$312$	0	0
$313$	19.8284	1.12077	0.560384	−	0.828233i	$-0.310653\pi$
$313$	19.8284	1.12077	0.560384	+	0.828233i	$0.310653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.97056	−0.166843	−0.0834217	−	0.996514i	$-0.526585\pi$
$317$	−2.97056	−0.166843	−0.0834217	+	0.996514i	$0.526585\pi$
$318$	0	0
$319$	−3.58579	−0.200765
$320$	0	0
$321$	−18.4853	−1.03175
$322$	0	0
$323$	7.65685	0.426039
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−19.3137	−1.06805
$328$	0	0
$329$	−5.00000	−0.275659
$330$	0	0
$331$	18.9706	1.04272	0.521358	−	0.853338i	$-0.325426\pi$
$331$	18.9706	1.04272	0.521358	+	0.853338i	$0.325426\pi$
$332$	0	0
$333$	7.65685	0.419593
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.48528	0.407749	0.203875	−	0.978997i	$-0.434647\pi$
$337$	7.48528	0.407749	0.203875	+	0.978997i	$0.434647\pi$
$338$	0	0
$339$	13.3137	0.723101
$340$	0	0
$341$	1.82843	0.0990149
$342$	0	0
$343$	19.7279	1.06521
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.5147	−0.725508	−0.362754	−	0.931885i	$-0.618163\pi$
$347$	−13.5147	−0.725508	−0.362754	+	0.931885i	$0.618163\pi$
$348$	0	0
$349$	−3.65685	−0.195747	−0.0978735	−	0.995199i	$-0.531204\pi$
$349$	−3.65685	−0.195747	−0.0978735	+	0.995199i	$0.531204\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−0.686292	−0.0365276	−0.0182638	−	0.999833i	$-0.505814\pi$
$353$	−0.686292	−0.0365276	−0.0182638	+	0.999833i	$0.505814\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.24264	0.489172
$358$	0	0
$359$	−1.10051	−0.0580824	−0.0290412	−	0.999578i	$-0.509245\pi$
$359$	−1.10051	−0.0580824	−0.0290412	+	0.999578i	$0.509245\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−10.8284	−0.568345
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.65685	0.504084	0.252042	−	0.967716i	$-0.418898\pi$
$367$	9.65685	0.504084	0.252042	+	0.967716i	$0.418898\pi$
$368$	0	0
$369$	5.65685	0.294484
$370$	0	0
$371$	−14.0711	−0.730533
$372$	0	0
$373$	32.6569	1.69091	0.845454	−	0.534048i	$-0.179330\pi$
$373$	32.6569	1.69091	0.845454	+	0.534048i	$0.179330\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.65685	−0.445851
$378$	0	0
$379$	0.0710678	0.00365051	0.00182525	−	0.999998i	$-0.499419\pi$
$379$	0.0710678	0.00365051	0.00182525	+	0.999998i	$0.499419\pi$
$380$	0	0
$381$	−11.1716	−0.572337
$382$	0	0
$383$	−3.65685	−0.186857	−0.0934283	−	0.995626i	$-0.529783\pi$
$383$	−3.65685	−0.186857	−0.0934283	+	0.995626i	$0.529783\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.4853	0.532997
$388$	0	0
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	0	0
$391$	−3.17157	−0.160393
$392$	0	0
$393$	−17.6569	−0.890670
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−34.2843	−1.72068	−0.860339	−	0.509722i	$-0.829748\pi$
$397$	−34.2843	−1.72068	−0.860339	+	0.509722i	$0.829748\pi$
$398$	0	0
$399$	4.82843	0.241724
$400$	0	0
$401$	−29.3137	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$401$	−29.3137	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$402$	0	0
$403$	4.41421	0.219888
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.17157	0.157209
$408$	0	0
$409$	19.3137	0.955001	0.477501	−	0.878631i	$-0.341543\pi$
$409$	19.3137	0.955001	0.477501	+	0.878631i	$0.341543\pi$
$410$	0	0
$411$	16.9706	0.837096
$412$	0	0
$413$	15.4853	0.761981
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.17157	−0.155313
$418$	0	0
$419$	2.34315	0.114470	0.0572351	−	0.998361i	$-0.481772\pi$
$419$	2.34315	0.114470	0.0572351	+	0.998361i	$0.481772\pi$
$420$	0	0
$421$	−11.3137	−0.551396	−0.275698	−	0.961244i	$-0.588909\pi$
$421$	−11.3137	−0.551396	−0.275698	+	0.961244i	$0.588909\pi$
$422$	0	0
$423$	2.07107	0.100699
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.41421	0.213619
$428$	0	0
$429$	0.414214	0.0199984
$430$	0	0
$431$	6.97056	0.335760	0.167880	−	0.985807i	$-0.446308\pi$
$431$	6.97056	0.335760	0.167880	+	0.985807i	$0.446308\pi$
$432$	0	0
$433$	25.3137	1.21650	0.608250	−	0.793746i	$-0.291872\pi$
$433$	25.3137	1.21650	0.608250	+	0.793746i	$0.291872\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.65685	−0.0792581
$438$	0	0
$439$	−18.4853	−0.882254	−0.441127	−	0.897445i	$-0.645421\pi$
$439$	−18.4853	−0.882254	−0.441127	+	0.897445i	$0.645421\pi$
$440$	0	0
$441$	−1.17157	−0.0557892
$442$	0	0
$443$	−23.4558	−1.11442	−0.557210	−	0.830371i	$-0.688128\pi$
$443$	−23.4558	−1.11442	−0.557210	+	0.830371i	$0.688128\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.3137	0.818910
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	2.34315	0.110334
$452$	0	0
$453$	−7.24264	−0.340289
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.6569	1.38729	0.693645	−	0.720317i	$-0.256004\pi$
$457$	29.6569	1.38729	0.693645	+	0.720317i	$0.256004\pi$
$458$	0	0
$459$	−3.82843	−0.178696
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	23.7279	1.10273	0.551365	−	0.834264i	$-0.314107\pi$
$463$	23.7279	1.10273	0.551365	+	0.834264i	$0.314107\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	1.82843	0.0844289
$470$	0	0
$471$	18.3137	0.843851
$472$	0	0
$473$	4.34315	0.199698
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.82843	0.266865
$478$	0	0
$479$	21.0416	0.961417	0.480708	−	0.876881i	$-0.340380\pi$
$479$	21.0416	0.961417	0.480708	+	0.876881i	$0.340380\pi$
$480$	0	0
$481$	7.65685	0.349123
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.24264	0.0563094	0.0281547	−	0.999604i	$-0.491037\pi$
$487$	1.24264	0.0563094	0.0281547	+	0.999604i	$0.491037\pi$
$488$	0	0
$489$	2.68629	0.121478
$490$	0	0
$491$	−22.4853	−1.01475	−0.507373	−	0.861726i	$-0.669383\pi$
$491$	−22.4853	−1.01475	−0.507373	+	0.861726i	$0.669383\pi$
$492$	0	0
$493$	33.1421	1.49265
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.4853	0.829178
$498$	0	0
$499$	−24.4142	−1.09293	−0.546465	−	0.837482i	$-0.684027\pi$
$499$	−24.4142	−1.09293	−0.546465	+	0.837482i	$0.684027\pi$
$500$	0	0
$501$	−25.3137	−1.13093
$502$	0	0
$503$	−20.2843	−0.904431	−0.452215	−	0.891909i	$-0.649366\pi$
$503$	−20.2843	−0.904431	−0.452215	+	0.891909i	$0.649366\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	22.4853	0.994690
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.857864	0.0377288
$518$	0	0
$519$	−3.48528	−0.152987
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−10.4853	−0.458489	−0.229245	−	0.973369i	$-0.573626\pi$
$523$	−10.4853	−0.458489	−0.229245	+	0.973369i	$0.573626\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.8995	−0.736154
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	−6.41421	−0.278353
$532$	0	0
$533$	5.65685	0.245026
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	−0.485281	−0.0209025
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−3.34315	−0.143468
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−22.6274	−0.967478	−0.483739	−	0.875212i	$-0.660722\pi$
$547$	−22.6274	−0.967478	−0.483739	+	0.875212i	$0.660722\pi$
$548$	0	0
$549$	−1.82843	−0.0780354
$550$	0	0
$551$	17.3137	0.737589
$552$	0	0
$553$	21.3137	0.906351
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	10.4853	0.443480
$560$	0	0
$561$	−1.58579	−0.0669520
$562$	0	0
$563$	−10.4853	−0.441902	−0.220951	−	0.975285i	$-0.570916\pi$
$563$	−10.4853	−0.441902	−0.220951	+	0.975285i	$0.570916\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.41421	−0.101387
$568$	0	0
$569$	21.6863	0.909137	0.454568	−	0.890712i	$-0.349794\pi$
$569$	21.6863	0.909137	0.454568	+	0.890712i	$0.349794\pi$
$570$	0	0
$571$	−18.4853	−0.773585	−0.386792	−	0.922167i	$-0.626417\pi$
$571$	−18.4853	−0.773585	−0.386792	+	0.922167i	$0.626417\pi$
$572$	0	0
$573$	14.4853	0.605131
$574$	0	0
$575$	0	0
$576$	0	0
$577$	42.2843	1.76032	0.880159	−	0.474680i	$-0.157436\pi$
$577$	42.2843	1.76032	0.880159	+	0.474680i	$0.157436\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	33.1421	1.37497
$582$	0	0
$583$	2.41421	0.0999865
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.7279	0.484063	0.242032	−	0.970268i	$-0.422186\pi$
$587$	11.7279	0.484063	0.242032	+	0.970268i	$0.422186\pi$
$588$	0	0
$589$	−8.82843	−0.363769
$590$	0	0
$591$	4.00000	0.164538
$592$	0	0
$593$	−16.6274	−0.682806	−0.341403	−	0.939917i	$-0.610902\pi$
$593$	−16.6274	−0.682806	−0.341403	+	0.939917i	$0.610902\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.828427	−0.0339053
$598$	0	0
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	−27.1421	−1.10715	−0.553575	−	0.832799i	$-0.686737\pi$
$601$	−27.1421	−1.10715	−0.553575	+	0.832799i	$0.686737\pi$
$602$	0	0
$603$	−0.757359	−0.0308421
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.1421	−0.655189	−0.327595	−	0.944818i	$-0.606238\pi$
$607$	−16.1421	−0.655189	−0.327595	+	0.944818i	$0.606238\pi$
$608$	0	0
$609$	20.8995	0.846890
$610$	0	0
$611$	2.07107	0.0837864
$612$	0	0
$613$	24.3431	0.983210	0.491605	−	0.870818i	$-0.336410\pi$
$613$	24.3431	0.983210	0.491605	+	0.870818i	$0.336410\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.9706	1.56890	0.784448	−	0.620195i	$-0.212946\pi$
$617$	38.9706	1.56890	0.784448	+	0.620195i	$0.212946\pi$
$618$	0	0
$619$	−44.6274	−1.79373	−0.896864	−	0.442307i	$-0.854160\pi$
$619$	−44.6274	−1.79373	−0.896864	+	0.442307i	$0.854160\pi$
$620$	0	0
$621$	0.828427	0.0332436
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.828427	−0.0330842
$628$	0	0
$629$	−29.3137	−1.16881
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	10.4853	0.416753
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.17157	−0.0464194
$638$	0	0
$639$	−7.65685	−0.302900
$640$	0	0
$641$	−18.5147	−0.731287	−0.365644	−	0.930755i	$-0.619151\pi$
$641$	−18.5147	−0.731287	−0.365644	+	0.930755i	$0.619151\pi$
$642$	0	0
$643$	40.6274	1.60219	0.801094	−	0.598538i	$-0.204251\pi$
$643$	40.6274	1.60219	0.801094	+	0.598538i	$0.204251\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.9411	0.862595	0.431297	−	0.902210i	$-0.358056\pi$
$647$	21.9411	0.862595	0.431297	+	0.902210i	$0.358056\pi$
$648$	0	0
$649$	−2.65685	−0.104291
$650$	0	0
$651$	−10.6569	−0.417675
$652$	0	0
$653$	−23.3431	−0.913488	−0.456744	−	0.889598i	$-0.650984\pi$
$653$	−23.3431	−0.913488	−0.456744	+	0.889598i	$0.650984\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.31371	−0.363362
$658$	0	0
$659$	−3.85786	−0.150281	−0.0751405	−	0.997173i	$-0.523941\pi$
$659$	−3.85786	−0.150281	−0.0751405	+	0.997173i	$0.523941\pi$
$660$	0	0
$661$	45.9411	1.78690	0.893451	−	0.449160i	$-0.148277\pi$
$661$	45.9411	1.78690	0.893451	+	0.449160i	$0.148277\pi$
$662$	0	0
$663$	−3.82843	−0.148684
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.17157	−0.277684
$668$	0	0
$669$	23.6569	0.914627
$670$	0	0
$671$	−0.757359	−0.0292375
$672$	0	0
$673$	−7.14214	−0.275309	−0.137655	−	0.990480i	$-0.543956\pi$
$673$	−7.14214	−0.275309	−0.137655	+	0.990480i	$0.543956\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	36.1421	1.38701
$680$	0	0
$681$	−11.5858	−0.443968
$682$	0	0
$683$	43.1838	1.65238	0.826190	−	0.563391i	$-0.190503\pi$
$683$	43.1838	1.65238	0.826190	+	0.563391i	$0.190503\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.34315	−0.165701
$688$	0	0
$689$	5.82843	0.222045
$690$	0	0
$691$	9.24264	0.351607	0.175803	−	0.984425i	$-0.443748\pi$
$691$	9.24264	0.351607	0.175803	+	0.984425i	$0.443748\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−21.6569	−0.820312
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−32.3137	−1.22047	−0.610236	−	0.792220i	$-0.708925\pi$
$701$	−32.3137	−1.22047	−0.610236	+	0.792220i	$0.708925\pi$
$702$	0	0
$703$	−15.3137	−0.577567
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.8995	0.560353
$708$	0	0
$709$	34.6274	1.30046	0.650230	−	0.759737i	$-0.274673\pi$
$709$	34.6274	1.30046	0.650230	+	0.759737i	$0.274673\pi$
$710$	0	0
$711$	−8.82843	−0.331092
$712$	0	0
$713$	3.65685	0.136950
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.24264	−0.195790
$718$	0	0
$719$	−42.6274	−1.58973	−0.794867	−	0.606783i	$-0.792460\pi$
$719$	−42.6274	−1.58973	−0.794867	+	0.606783i	$0.792460\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.00000	−0.148762
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.686292	−0.0254531	−0.0127266	−	0.999919i	$-0.504051\pi$
$727$	−0.686292	−0.0254531	−0.0127266	+	0.999919i	$0.504051\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.1421	−1.48471
$732$	0	0
$733$	28.6863	1.05955	0.529776	−	0.848137i	$-0.322276\pi$
$733$	28.6863	1.05955	0.529776	+	0.848137i	$0.322276\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.313708	−0.0115556
$738$	0	0
$739$	34.5563	1.27118	0.635588	−	0.772028i	$-0.280758\pi$
$739$	34.5563	1.27118	0.635588	+	0.772028i	$0.280758\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	18.4142	0.675552	0.337776	−	0.941227i	$-0.390325\pi$
$743$	18.4142	0.675552	0.337776	+	0.941227i	$0.390325\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−13.7279	−0.502278
$748$	0	0
$749$	44.6274	1.63065
$750$	0	0
$751$	50.4853	1.84223	0.921117	−	0.389286i	$-0.127278\pi$
$751$	50.4853	1.84223	0.921117	+	0.389286i	$0.127278\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.7990	1.11941	0.559704	−	0.828692i	$-0.310915\pi$
$757$	30.7990	1.11941	0.559704	+	0.828692i	$0.310915\pi$
$758$	0	0
$759$	0.343146	0.0124554
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	46.6274	1.68803
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.41421	−0.231604
$768$	0	0
$769$	−17.6569	−0.636722	−0.318361	−	0.947969i	$-0.603132\pi$
$769$	−17.6569	−0.636722	−0.318361	+	0.947969i	$0.603132\pi$
$770$	0	0
$771$	−1.00000	−0.0360141
$772$	0	0
$773$	28.6274	1.02966	0.514828	−	0.857293i	$-0.327856\pi$
$773$	28.6274	1.02966	0.514828	+	0.857293i	$0.327856\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−18.4853	−0.663156
$778$	0	0
$779$	−11.3137	−0.405356
$780$	0	0
$781$	−3.17157	−0.113488
$782$	0	0
$783$	−8.65685	−0.309371
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.38478	0.120654	0.0603271	−	0.998179i	$-0.480786\pi$
$787$	3.38478	0.120654	0.0603271	+	0.998179i	$0.480786\pi$
$788$	0	0
$789$	3.31371	0.117971
$790$	0	0
$791$	−32.1421	−1.14284
$792$	0	0
$793$	−1.82843	−0.0649294
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.34315	−0.189264	−0.0946320	−	0.995512i	$-0.530167\pi$
$797$	−5.34315	−0.189264	−0.0946320	+	0.995512i	$0.530167\pi$
$798$	0	0
$799$	−7.92893	−0.280505
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.85786	−0.136141
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.3137	0.433463
$808$	0	0
$809$	41.3137	1.45251	0.726256	−	0.687424i	$-0.241259\pi$
$809$	41.3137	1.45251	0.726256	+	0.687424i	$0.241259\pi$
$810$	0	0
$811$	28.2721	0.992767	0.496383	−	0.868103i	$-0.334661\pi$
$811$	28.2721	0.992767	0.496383	+	0.868103i	$0.334661\pi$
$812$	0	0
$813$	−4.89949	−0.171833
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20.9706	−0.733667
$818$	0	0
$819$	−2.41421	−0.0843594
$820$	0	0
$821$	33.3137	1.16266	0.581328	−	0.813669i	$-0.302533\pi$
$821$	33.3137	1.16266	0.581328	+	0.813669i	$0.302533\pi$
$822$	0	0
$823$	−49.9411	−1.74084	−0.870419	−	0.492311i	$-0.836152\pi$
$823$	−49.9411	−1.74084	−0.870419	+	0.492311i	$0.836152\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.3848	−0.604528	−0.302264	−	0.953224i	$-0.597742\pi$
$827$	−17.3848	−0.604528	−0.302264	+	0.953224i	$0.597742\pi$
$828$	0	0
$829$	−3.82843	−0.132967	−0.0664834	−	0.997788i	$-0.521178\pi$
$829$	−3.82843	−0.132967	−0.0664834	+	0.997788i	$0.521178\pi$
$830$	0	0
$831$	9.31371	0.323089
$832$	0	0
$833$	4.48528	0.155406
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.41421	0.152578
$838$	0	0
$839$	7.65685	0.264344	0.132172	−	0.991227i	$-0.457805\pi$
$839$	7.65685	0.264344	0.132172	+	0.991227i	$0.457805\pi$
$840$	0	0
$841$	45.9411	1.58418
$842$	0	0
$843$	−14.0000	−0.482186
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.1421	0.898254
$848$	0	0
$849$	−27.4558	−0.942282
$850$	0	0
$851$	6.34315	0.217440
$852$	0	0
$853$	18.3431	0.628057	0.314029	−	0.949413i	$-0.398321\pi$
$853$	18.3431	0.628057	0.314029	+	0.949413i	$0.398321\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.6274	−0.431344	−0.215672	−	0.976466i	$-0.569194\pi$
$857$	−12.6274	−0.431344	−0.215672	+	0.976466i	$0.569194\pi$
$858$	0	0
$859$	−39.1716	−1.33652	−0.668258	−	0.743929i	$-0.732960\pi$
$859$	−39.1716	−1.33652	−0.668258	+	0.743929i	$0.732960\pi$
$860$	0	0
$861$	−13.6569	−0.465424
$862$	0	0
$863$	−29.5858	−1.00711	−0.503556	−	0.863963i	$-0.667975\pi$
$863$	−29.5858	−1.00711	−0.503556	+	0.863963i	$0.667975\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.34315	−0.0795774
$868$	0	0
$869$	−3.65685	−0.124050
$870$	0	0
$871$	−0.757359	−0.0256621
$872$	0	0
$873$	−14.9706	−0.506677
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.3431	0.484334	0.242167	−	0.970235i	$-0.422142\pi$
$877$	14.3431	0.484334	0.242167	+	0.970235i	$0.422142\pi$
$878$	0	0
$879$	−5.31371	−0.179227
$880$	0	0
$881$	−54.5980	−1.83945	−0.919726	−	0.392560i	$-0.871589\pi$
$881$	−54.5980	−1.83945	−0.919726	+	0.392560i	$0.871589\pi$
$882$	0	0
$883$	3.31371	0.111515	0.0557576	−	0.998444i	$-0.482243\pi$
$883$	3.31371	0.111515	0.0557576	+	0.998444i	$0.482243\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.17157	−0.106491	−0.0532455	−	0.998581i	$-0.516957\pi$
$887$	−3.17157	−0.106491	−0.0532455	+	0.998581i	$0.516957\pi$
$888$	0	0
$889$	26.9706	0.904564
$890$	0	0
$891$	0.414214	0.0138767
$892$	0	0
$893$	−4.14214	−0.138611
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.828427	0.0276604
$898$	0	0
$899$	−38.2132	−1.27448
$900$	0	0
$901$	−22.3137	−0.743377
$902$	0	0
$903$	−25.3137	−0.842387
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−33.9411	−1.12700	−0.563498	−	0.826117i	$-0.690545\pi$
$907$	−33.9411	−1.12700	−0.563498	+	0.826117i	$0.690545\pi$
$908$	0	0
$909$	−6.17157	−0.204698
$910$	0	0
$911$	−20.8284	−0.690077	−0.345038	−	0.938589i	$-0.612134\pi$
$911$	−20.8284	−0.690077	−0.345038	+	0.938589i	$0.612134\pi$
$912$	0	0
$913$	−5.68629	−0.188189
$914$	0	0
$915$	0	0
$916$	0	0
$917$	42.6274	1.40768
$918$	0	0
$919$	17.7990	0.587135	0.293567	−	0.955938i	$-0.405158\pi$
$919$	17.7990	0.587135	0.293567	+	0.955938i	$0.405158\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	−7.65685	−0.252028
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	2.34315	0.0767935
$932$	0	0
$933$	−18.4853	−0.605181
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−46.5980	−1.52229	−0.761145	−	0.648582i	$-0.775362\pi$
$937$	−46.5980	−1.52229	−0.761145	+	0.648582i	$0.775362\pi$
$938$	0	0
$939$	19.8284	0.647076
$940$	0	0
$941$	45.3137	1.47718	0.738592	−	0.674152i	$-0.235491\pi$
$941$	45.3137	1.47718	0.738592	+	0.674152i	$0.235491\pi$
$942$	0	0
$943$	4.68629	0.152607
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.4142	−0.663373	−0.331686	−	0.943390i	$-0.607618\pi$
$947$	−20.4142	−0.663373	−0.331686	+	0.943390i	$0.607618\pi$
$948$	0	0
$949$	−9.31371	−0.302336
$950$	0	0
$951$	−2.97056	−0.0963271
$952$	0	0
$953$	−34.6569	−1.12265	−0.561323	−	0.827597i	$-0.689707\pi$
$953$	−34.6569	−1.12265	−0.561323	+	0.827597i	$0.689707\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.58579	−0.115912
$958$	0	0
$959$	−40.9706	−1.32301
$960$	0	0
$961$	−11.5147	−0.371443
$962$	0	0
$963$	−18.4853	−0.595680
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−24.0122	−0.772180	−0.386090	−	0.922461i	$-0.626175\pi$
$967$	−24.0122	−0.772180	−0.386090	+	0.922461i	$0.626175\pi$
$968$	0	0
$969$	7.65685	0.245974
$970$	0	0
$971$	−28.8284	−0.925148	−0.462574	−	0.886581i	$-0.653074\pi$
$971$	−28.8284	−0.925148	−0.462574	+	0.886581i	$0.653074\pi$
$972$	0	0
$973$	7.65685	0.245467
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.9706	−1.05482	−0.527411	−	0.849610i	$-0.676837\pi$
$977$	−32.9706	−1.05482	−0.527411	+	0.849610i	$0.676837\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−19.3137	−0.616639
$982$	0	0
$983$	−8.75736	−0.279316	−0.139658	−	0.990200i	$-0.544600\pi$
$983$	−8.75736	−0.279316	−0.139658	+	0.990200i	$0.544600\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.00000	−0.159152
$988$	0	0
$989$	8.68629	0.276208
$990$	0	0
$991$	15.8579	0.503742	0.251871	−	0.967761i	$-0.418954\pi$
$991$	15.8579	0.503742	0.251871	+	0.967761i	$0.418954\pi$
$992$	0	0
$993$	18.9706	0.602013
$994$	0	0
$995$	0	0
$996$	0	0
$997$	58.7990	1.86218	0.931091	−	0.364786i	$-0.118858\pi$
$997$	58.7990	1.86218	0.931091	+	0.364786i	$0.118858\pi$
$998$	0	0
$999$	7.65685	0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.ba.1.1	yes	2
5.4	even	2		7800.2.a.z.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.z.1.2	✓	2	5.4	even	2
7800.2.a.ba.1.1	yes	2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.41421 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.41421 q^{7} +1.00000 q^{9} +0.414214 q^{11} +1.00000 q^{13} -3.82843 q^{17} -2.00000 q^{19} -2.41421 q^{21} +0.828427 q^{23} +1.00000 q^{27} -8.65685 q^{29} +4.41421 q^{31} +0.414214 q^{33} +7.65685 q^{37} +1.00000 q^{39} +5.65685 q^{41} +10.4853 q^{43} +2.07107 q^{47} -1.17157 q^{49} -3.82843 q^{51} +5.82843 q^{53} -2.00000 q^{57} -6.41421 q^{59} -1.82843 q^{61} -2.41421 q^{63} -0.757359 q^{67} +0.828427 q^{69} -7.65685 q^{71} -9.31371 q^{73} -1.00000 q^{77} -8.82843 q^{79} +1.00000 q^{81} -13.7279 q^{83} -8.65685 q^{87} -2.41421 q^{91} +4.41421 q^{93} -14.9706 q^{97} +0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} - 2 q^{21} - 4 q^{23} + 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 4 q^{43} - 10 q^{47} - 8 q^{49} - 2 q^{51} + 6 q^{53} - 4 q^{57} - 10 q^{59} + 2 q^{61} - 2 q^{63} - 10 q^{67} - 4 q^{69} - 4 q^{71} + 4 q^{73} - 2 q^{77} - 12 q^{79} + 2 q^{81} - 2 q^{83} - 6 q^{87} - 2 q^{91} + 6 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 4 * q^19 - 2 * q^21 - 4 * q^23 + 2 * q^27 - 6 * q^29 + 6 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 + 4 * q^43 - 10 * q^47 - 8 * q^49 - 2 * q^51 + 6 * q^53 - 4 * q^57 - 10 * q^59 + 2 * q^61 - 2 * q^63 - 10 * q^67 - 4 * q^69 - 4 * q^71 + 4 * q^73 - 2 * q^77 - 12 * q^79 + 2 * q^81 - 2 * q^83 - 6 * q^87 - 2 * q^91 + 6 * q^93 + 4 * q^97 - 2 * q^99

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