LMFDB - Embedded newform 3800.2.d.q.3649.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9 $ x^12 + 24x^10 + 194x^8 + 618x^6 + 733x^4 + 286*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.7
Root			$0.185519i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.q.3649.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.185519i q^{3} -4.45651i q^{7} +2.96558 q^{9} +2.64623 q^{11} -1.30142i q^{13} -3.51716i q^{17} +1.00000 q^{19} +0.826767 q^{21} +6.52882i q^{23} +1.10673i q^{27} +5.20946 q^{29} +10.8219 q^{31} +0.490925i q^{33} +2.04607i q^{37} +0.241439 q^{39} -3.80044 q^{41} -4.77089i q^{43} +1.49093i q^{47} -12.8605 q^{49} +0.652501 q^{51} -0.225583i q^{53} +0.185519i q^{57} +2.86056 q^{59} -6.31449 q^{61} -13.2161i q^{63} -13.1831i q^{67} -1.21122 q^{69} +12.3310 q^{71} +5.42276i q^{73} -11.7929i q^{77} -14.9688 q^{79} +8.69143 q^{81} -3.84470i q^{83} +0.966455i q^{87} -1.67666 q^{89} -5.79981 q^{91} +2.00768i q^{93} +9.48523i q^{97} +7.84760 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0.185519i	0.107109i	0.998565	+	0.0535547i	$0.0170552\pi$
$3$	0.185519i	0.107109i	−0.998565	+	0.0535547i	$0.982945\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.45651i	− 1.68440i	−0.539164	−	0.842201i	$-0.681260\pi$
$7$	− 4.45651i	− 1.68440i	0.539164	−	0.842201i	$-0.318740\pi$
$8$	0	0
$9$	2.96558	0.988528
$10$	0	0
$11$	2.64623	0.797867	0.398934	−	0.916980i	$-0.369380\pi$
$11$	2.64623	0.797867	0.398934	+	0.916980i	$0.369380\pi$
$12$	0	0
$13$	− 1.30142i	− 0.360950i	−0.983580	−	0.180475i	$-0.942236\pi$
$13$	− 1.30142i	− 0.360950i	0.983580	−	0.180475i	$-0.0577635\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.51716i	− 0.853037i	−0.904479	−	0.426519i	$-0.859740\pi$
$17$	− 3.51716i	− 0.853037i	0.904479	−	0.426519i	$-0.140260\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.826767	0.180415
$22$	0	0
$23$	6.52882i	1.36135i	0.732584	+	0.680677i	$0.238314\pi$
$23$	6.52882i	1.36135i	−0.732584	+	0.680677i	$0.761686\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.10673i	0.212990i
$28$	0	0
$29$	5.20946	0.967373	0.483686	−	0.875241i	$-0.339298\pi$
$29$	5.20946	0.967373	0.483686	+	0.875241i	$0.339298\pi$
$30$	0	0
$31$	10.8219	1.94368	0.971839	−	0.235646i	$-0.0757207\pi$
$31$	10.8219	1.94368	0.971839	+	0.235646i	$0.0757207\pi$
$32$	0	0
$33$	0.490925i	0.0854591i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.04607i	0.336373i	0.985755	+	0.168186i	$0.0537910\pi$
$37$	2.04607i	0.336373i	−0.985755	+	0.168186i	$0.946209\pi$
$38$	0	0
$39$	0.241439	0.0386612
$40$	0	0
$41$	−3.80044	−0.593529	−0.296764	−	0.954951i	$-0.595908\pi$
$41$	−3.80044	−0.593529	−0.296764	+	0.954951i	$0.595908\pi$
$42$	0	0
$43$	− 4.77089i	− 0.727554i	−0.931486	−	0.363777i	$-0.881487\pi$
$43$	− 4.77089i	− 0.727554i	0.931486	−	0.363777i	$-0.118513\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.49093i	0.217474i	0.994071	+	0.108737i	$0.0346806\pi$
$47$	1.49093i	0.217474i	−0.994071	+	0.108737i	$0.965319\pi$
$48$	0	0
$49$	−12.8605	−1.83721
$50$	0	0
$51$	0.652501	0.0913684
$52$	0	0
$53$	− 0.225583i	− 0.0309862i	−0.999880	−	0.0154931i	$-0.995068\pi$
$53$	− 0.225583i	− 0.0309862i	0.999880	−	0.0154931i	$-0.00493180\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.185519i	0.0245726i
$58$	0	0
$59$	2.86056	0.372413	0.186206	−	0.982511i	$-0.440381\pi$
$59$	2.86056	0.372413	0.186206	+	0.982511i	$0.440381\pi$
$60$	0	0
$61$	−6.31449	−0.808488	−0.404244	−	0.914651i	$-0.632465\pi$
$61$	−6.31449	−0.808488	−0.404244	+	0.914651i	$0.632465\pi$
$62$	0	0
$63$	− 13.2161i	− 1.66508i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.1831i	− 1.61058i	−0.592884	−	0.805288i	$-0.702011\pi$
$67$	− 13.1831i	− 1.61058i	0.592884	−	0.805288i	$-0.297989\pi$
$68$	0	0
$69$	−1.21122	−0.145814
$70$	0	0
$71$	12.3310	1.46342	0.731711	−	0.681615i	$-0.238722\pi$
$71$	12.3310	1.46342	0.731711	+	0.681615i	$0.238722\pi$
$72$	0	0
$73$	5.42276i	0.634686i	0.948311	+	0.317343i	$0.102791\pi$
$73$	5.42276i	0.634686i	−0.948311	+	0.317343i	$0.897209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11.7929i	− 1.34393i
$78$	0	0
$79$	−14.9688	−1.68413	−0.842063	−	0.539379i	$-0.818659\pi$
$79$	−14.9688	−1.68413	−0.842063	+	0.539379i	$0.818659\pi$
$80$	0	0
$81$	8.69143	0.965714
$82$	0	0
$83$	− 3.84470i	− 0.422011i	−0.977485	−	0.211005i	$-0.932326\pi$
$83$	− 3.84470i	− 0.422011i	0.977485	−	0.211005i	$-0.0676737\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.966455i	0.103615i
$88$	0	0
$89$	−1.67666	−0.177726	−0.0888629	−	0.996044i	$-0.528323\pi$
$89$	−1.67666	−0.177726	−0.0888629	+	0.996044i	$0.528323\pi$
$90$	0	0
$91$	−5.79981	−0.607985
$92$	0	0
$93$	2.00768i	0.208186i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.48523i	0.963079i	0.876424	+	0.481539i	$0.159922\pi$
$97$	9.48523i	0.963079i	−0.876424	+	0.481539i	$0.840078\pi$
$98$	0	0
$99$	7.84760	0.788714
$100$	0	0
$101$	−6.39100	−0.635928	−0.317964	−	0.948103i	$-0.602999\pi$
$101$	−6.39100	−0.635928	−0.317964	+	0.948103i	$0.602999\pi$
$102$	0	0
$103$	− 4.14301i	− 0.408223i	−0.978948	−	0.204112i	$-0.934569\pi$
$103$	− 4.14301i	− 0.408223i	0.978948	−	0.204112i	$-0.0654306\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.27597i	0.703394i	0.936114	+	0.351697i	$0.114395\pi$
$107$	7.27597i	0.703394i	−0.936114	+	0.351697i	$0.885605\pi$
$108$	0	0
$109$	−14.1077	−1.35127	−0.675637	−	0.737235i	$-0.736131\pi$
$109$	−14.1077	−1.35127	−0.675637	+	0.737235i	$0.736131\pi$
$110$	0	0
$111$	−0.379586	−0.0360287
$112$	0	0
$113$	− 7.03290i	− 0.661600i	−0.943701	−	0.330800i	$-0.892681\pi$
$113$	− 7.03290i	− 0.661600i	0.943701	−	0.330800i	$-0.107319\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.85948i	− 0.356809i
$118$	0	0
$119$	−15.6743	−1.43686
$120$	0	0
$121$	−3.99749	−0.363408
$122$	0	0
$123$	− 0.705053i	− 0.0635725i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.6240i	1.20893i	0.796630	+	0.604467i	$0.206614\pi$
$127$	13.6240i	1.20893i	−0.796630	+	0.604467i	$0.793386\pi$
$128$	0	0
$129$	0.885090	0.0779279
$130$	0	0
$131$	22.3345	1.95137	0.975685	−	0.219177i	$-0.0703373\pi$
$131$	22.3345	1.95137	0.975685	+	0.219177i	$0.0703373\pi$
$132$	0	0
$133$	− 4.45651i	− 0.386428i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.55578i	− 0.474662i	−0.971429	−	0.237331i	$-0.923727\pi$
$137$	− 5.55578i	− 0.474662i	0.971429	−	0.237331i	$-0.0762726\pi$
$138$	0	0
$139$	11.0452	0.936841	0.468420	−	0.883506i	$-0.344823\pi$
$139$	11.0452	0.936841	0.468420	+	0.883506i	$0.344823\pi$
$140$	0	0
$141$	−0.276595	−0.0232935
$142$	0	0
$143$	− 3.44386i	− 0.287990i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.38586i	− 0.196782i
$148$	0	0
$149$	−17.6003	−1.44188	−0.720938	−	0.693000i	$-0.756289\pi$
$149$	−17.6003	−1.44188	−0.720938	+	0.693000i	$0.756289\pi$
$150$	0	0
$151$	5.24140	0.426539	0.213269	−	0.976993i	$-0.431589\pi$
$151$	5.24140	0.426539	0.213269	+	0.976993i	$0.431589\pi$
$152$	0	0
$153$	− 10.4304i	− 0.843251i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.7965i	− 1.50012i	−0.661368	−	0.750061i	$-0.730024\pi$
$157$	− 18.7965i	− 1.50012i	0.661368	−	0.750061i	$-0.269976\pi$
$158$	0	0
$159$	0.0418499	0.00331891
$160$	0	0
$161$	29.0957	2.29307
$162$	0	0
$163$	− 12.1669i	− 0.952982i	−0.879179	−	0.476491i	$-0.841908\pi$
$163$	− 12.1669i	− 0.952982i	0.879179	−	0.476491i	$-0.158092\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.46393i	0.577576i	0.957393	+	0.288788i	$0.0932523\pi$
$167$	7.46393i	0.577576i	−0.957393	+	0.288788i	$0.906748\pi$
$168$	0	0
$169$	11.3063	0.869715
$170$	0	0
$171$	2.96558	0.226784
$172$	0	0
$173$	− 17.8070i	− 1.35384i	−0.736055	−	0.676921i	$-0.763314\pi$
$173$	− 17.8070i	− 1.35384i	0.736055	−	0.676921i	$-0.236686\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.530688i	0.0398889i
$178$	0	0
$179$	9.48154	0.708684	0.354342	−	0.935116i	$-0.384705\pi$
$179$	9.48154	0.708684	0.354342	+	0.935116i	$0.384705\pi$
$180$	0	0
$181$	4.32832	0.321721	0.160861	−	0.986977i	$-0.448573\pi$
$181$	4.32832	0.321721	0.160861	+	0.986977i	$0.448573\pi$
$182$	0	0
$183$	− 1.17146i	− 0.0865967i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.30720i	− 0.680610i
$188$	0	0
$189$	4.93215	0.358761
$190$	0	0
$191$	−8.21414	−0.594355	−0.297177	−	0.954822i	$-0.596045\pi$
$191$	−8.21414	−0.594355	−0.297177	+	0.954822i	$0.596045\pi$
$192$	0	0
$193$	− 2.89738i	− 0.208558i	−0.994548	−	0.104279i	$-0.966747\pi$
$193$	− 2.89738i	− 0.208558i	0.994548	−	0.104279i	$-0.0332534\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.4118i	− 1.59678i	−0.602143	−	0.798388i	$-0.705686\pi$
$197$	− 22.4118i	− 1.59678i	0.602143	−	0.798388i	$-0.294314\pi$
$198$	0	0
$199$	−2.68542	−0.190364	−0.0951821	−	0.995460i	$-0.530343\pi$
$199$	−2.68542	−0.190364	−0.0951821	+	0.995460i	$0.530343\pi$
$200$	0	0
$201$	2.44572	0.172508
$202$	0	0
$203$	− 23.2160i	− 1.62944i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	19.3618i	1.34573i
$208$	0	0
$209$	2.64623	0.183043
$210$	0	0
$211$	7.22869	0.497644	0.248822	−	0.968549i	$-0.419957\pi$
$211$	7.22869	0.497644	0.248822	+	0.968549i	$0.419957\pi$
$212$	0	0
$213$	2.28764i	0.156746i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 48.2281i	− 3.27393i
$218$	0	0
$219$	−1.00603	−0.0679809
$220$	0	0
$221$	−4.57732	−0.307904
$222$	0	0
$223$	− 6.86629i	− 0.459801i	−0.973214	−	0.229900i	$-0.926160\pi$
$223$	− 6.86629i	− 0.459801i	0.973214	−	0.229900i	$-0.0738400\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.2176i	1.27552i	0.770237	+	0.637758i	$0.220138\pi$
$227$	19.2176i	1.27552i	−0.770237	+	0.637758i	$0.779862\pi$
$228$	0	0
$229$	−25.2205	−1.66661	−0.833307	−	0.552810i	$-0.813555\pi$
$229$	−25.2205	−1.66661	−0.833307	+	0.552810i	$0.813555\pi$
$230$	0	0
$231$	2.18781	0.143947
$232$	0	0
$233$	24.1123i	1.57965i	0.613332	+	0.789825i	$0.289829\pi$
$233$	24.1123i	1.57965i	−0.613332	+	0.789825i	$0.710171\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2.77701i	− 0.180386i
$238$	0	0
$239$	−4.34082	−0.280784	−0.140392	−	0.990096i	$-0.544836\pi$
$239$	−4.34082	−0.280784	−0.140392	+	0.990096i	$0.544836\pi$
$240$	0	0
$241$	21.9630	1.41476	0.707380	−	0.706834i	$-0.249877\pi$
$241$	21.9630	1.41476	0.707380	+	0.706834i	$0.249877\pi$
$242$	0	0
$243$	4.93261i	0.316427i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.30142i	− 0.0828077i
$248$	0	0
$249$	0.713265	0.0452013
$250$	0	0
$251$	4.41214	0.278492	0.139246	−	0.990258i	$-0.455532\pi$
$251$	4.41214	0.278492	0.139246	+	0.990258i	$0.455532\pi$
$252$	0	0
$253$	17.2767i	1.08618i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.9112i	− 0.743000i	−0.928433	−	0.371500i	$-0.878844\pi$
$257$	− 11.9112i	− 0.743000i	0.928433	−	0.371500i	$-0.121156\pi$
$258$	0	0
$259$	9.11835	0.566587
$260$	0	0
$261$	15.4491	0.956275
$262$	0	0
$263$	13.7779i	0.849581i	0.905292	+	0.424790i	$0.139652\pi$
$263$	13.7779i	0.849581i	−0.905292	+	0.424790i	$0.860348\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 0.311053i	− 0.0190361i
$268$	0	0
$269$	2.30474	0.140522	0.0702612	−	0.997529i	$-0.477617\pi$
$269$	2.30474	0.140522	0.0702612	+	0.997529i	$0.477617\pi$
$270$	0	0
$271$	−20.1878	−1.22632	−0.613161	−	0.789958i	$-0.710102\pi$
$271$	−20.1878	−1.22632	−0.613161	+	0.789958i	$0.710102\pi$
$272$	0	0
$273$	− 1.07597i	− 0.0651210i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.7021i	− 0.643024i	−0.946905	−	0.321512i	$-0.895809\pi$
$277$	− 10.7021i	− 0.643024i	0.946905	−	0.321512i	$-0.104191\pi$
$278$	0	0
$279$	32.0934	1.92138
$280$	0	0
$281$	−2.28603	−0.136373	−0.0681865	−	0.997673i	$-0.521721\pi$
$281$	−2.28603	−0.136373	−0.0681865	+	0.997673i	$0.521721\pi$
$282$	0	0
$283$	25.6515i	1.52482i	0.647093	+	0.762411i	$0.275984\pi$
$283$	25.6515i	1.52482i	−0.647093	+	0.762411i	$0.724016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.9367i	0.999741i
$288$	0	0
$289$	4.62957	0.272328
$290$	0	0
$291$	−1.75969	−0.103155
$292$	0	0
$293$	− 19.6027i	− 1.14520i	−0.819833	−	0.572602i	$-0.805934\pi$
$293$	− 19.6027i	− 1.14520i	0.819833	−	0.572602i	$-0.194066\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.92866i	0.169938i
$298$	0	0
$299$	8.49677	0.491381
$300$	0	0
$301$	−21.2615	−1.22549
$302$	0	0
$303$	− 1.18565i	− 0.0681139i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 31.6679i	− 1.80738i	−0.428183	−	0.903692i	$-0.640846\pi$
$307$	− 31.6679i	− 1.80738i	0.428183	−	0.903692i	$-0.359154\pi$
$308$	0	0
$309$	0.768608	0.0437246
$310$	0	0
$311$	14.8957	0.844655	0.422328	−	0.906443i	$-0.361213\pi$
$311$	14.8957	0.844655	0.422328	+	0.906443i	$0.361213\pi$
$312$	0	0
$313$	0.288235i	0.0162920i	0.999967	+	0.00814601i	$0.00259299\pi$
$313$	0.288235i	0.0162920i	−0.999967	+	0.00814601i	$0.997407\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.9612i	1.23347i	0.787173	+	0.616733i	$0.211544\pi$
$317$	21.9612i	1.23347i	−0.787173	+	0.616733i	$0.788456\pi$
$318$	0	0
$319$	13.7854	0.771835
$320$	0	0
$321$	−1.34983	−0.0753402
$322$	0	0
$323$	− 3.51716i	− 0.195700i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.61725i	− 0.144734i
$328$	0	0
$329$	6.64432	0.366313
$330$	0	0
$331$	24.0905	1.32413	0.662066	−	0.749445i	$-0.269680\pi$
$331$	24.0905	1.32413	0.662066	+	0.749445i	$0.269680\pi$
$332$	0	0
$333$	6.06780i	0.332514i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.7090i	0.964673i	0.875986	+	0.482337i	$0.160212\pi$
$337$	17.7090i	0.964673i	−0.875986	+	0.482337i	$0.839788\pi$
$338$	0	0
$339$	1.30474	0.0708636
$340$	0	0
$341$	28.6373	1.55080
$342$	0	0
$343$	26.1172i	1.41020i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.158465i	0.00850687i	0.999991	+	0.00425344i	$0.00135391\pi$
$347$	0.158465i	0.00850687i	−0.999991	+	0.00425344i	$0.998646\pi$
$348$	0	0
$349$	−27.3776	−1.46549	−0.732745	−	0.680503i	$-0.761761\pi$
$349$	−27.3776	−1.46549	−0.732745	+	0.680503i	$0.761761\pi$
$350$	0	0
$351$	1.44032	0.0768788
$352$	0	0
$353$	18.3221i	0.975185i	0.873071	+	0.487592i	$0.162125\pi$
$353$	18.3221i	0.975185i	−0.873071	+	0.487592i	$0.837875\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.90787i	− 0.153901i
$358$	0	0
$359$	−35.6265	−1.88030	−0.940148	−	0.340767i	$-0.889313\pi$
$359$	−35.6265	−1.88030	−0.940148	+	0.340767i	$0.889313\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 0.741610i	− 0.0389245i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.03379i	0.158362i	0.996860	+	0.0791812i	$0.0252306\pi$
$367$	3.03379i	0.158362i	−0.996860	+	0.0791812i	$0.974769\pi$
$368$	0	0
$369$	−11.2705	−0.586719
$370$	0	0
$371$	−1.00531	−0.0521932
$372$	0	0
$373$	− 9.43451i	− 0.488500i	−0.969712	−	0.244250i	$-0.921458\pi$
$373$	− 9.43451i	− 0.488500i	0.969712	−	0.244250i	$-0.0785418\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.77972i	− 0.349173i
$378$	0	0
$379$	10.1522	0.521486	0.260743	−	0.965408i	$-0.416033\pi$
$379$	10.1522	0.521486	0.260743	+	0.965408i	$0.416033\pi$
$380$	0	0
$381$	−2.52751	−0.129488
$382$	0	0
$383$	− 19.0699i	− 0.974428i	−0.873283	−	0.487214i	$-0.838013\pi$
$383$	− 19.0699i	− 0.974428i	0.873283	−	0.487214i	$-0.161987\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 14.1485i	− 0.719207i
$388$	0	0
$389$	38.6448	1.95937	0.979685	−	0.200544i	$-0.0642709\pi$
$389$	38.6448	1.95937	0.979685	+	0.200544i	$0.0642709\pi$
$390$	0	0
$391$	22.9629	1.16128
$392$	0	0
$393$	4.14347i	0.209010i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.5339i	0.528679i	0.964430	+	0.264339i	$0.0851539\pi$
$397$	10.5339i	0.528679i	−0.964430	+	0.264339i	$0.914846\pi$
$398$	0	0
$399$	0.826767	0.0413901
$400$	0	0
$401$	−10.6994	−0.534304	−0.267152	−	0.963654i	$-0.586083\pi$
$401$	−10.6994	−0.534304	−0.267152	+	0.963654i	$0.586083\pi$
$402$	0	0
$403$	− 14.0839i	− 0.701571i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.41438i	0.268381i
$408$	0	0
$409$	−39.1036	−1.93355	−0.966775	−	0.255627i	$-0.917718\pi$
$409$	−39.1036	−1.93355	−0.966775	+	0.255627i	$0.917718\pi$
$410$	0	0
$411$	1.03070	0.0508408
$412$	0	0
$413$	− 12.7481i	− 0.627292i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.04909i	0.100345i
$418$	0	0
$419$	−5.00833	−0.244673	−0.122336	−	0.992489i	$-0.539039\pi$
$419$	−5.00833	−0.244673	−0.122336	+	0.992489i	$0.539039\pi$
$420$	0	0
$421$	−12.7557	−0.621674	−0.310837	−	0.950463i	$-0.600609\pi$
$421$	−12.7557	−0.621674	−0.310837	+	0.950463i	$0.600609\pi$
$422$	0	0
$423$	4.42146i	0.214979i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.1406i	1.36182i
$428$	0	0
$429$	0.638902	0.0308465
$430$	0	0
$431$	13.7140	0.660580	0.330290	−	0.943879i	$-0.392853\pi$
$431$	13.7140	0.660580	0.330290	+	0.943879i	$0.392853\pi$
$432$	0	0
$433$	− 1.49408i	− 0.0718007i	−0.999355	−	0.0359003i	$-0.988570\pi$
$433$	− 1.49408i	− 0.0718007i	0.999355	−	0.0359003i	$-0.0114299\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.52882i	0.312316i
$438$	0	0
$439$	−5.78568	−0.276136	−0.138068	−	0.990423i	$-0.544089\pi$
$439$	−5.78568	−0.276136	−0.138068	+	0.990423i	$0.544089\pi$
$440$	0	0
$441$	−38.1388	−1.81613
$442$	0	0
$443$	− 9.78426i	− 0.464864i	−0.972613	−	0.232432i	$-0.925332\pi$
$443$	− 9.78426i	− 0.464864i	0.972613	−	0.232432i	$-0.0746684\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.26520i	− 0.154439i
$448$	0	0
$449$	38.0631	1.79631	0.898154	−	0.439682i	$-0.144909\pi$
$449$	38.0631	1.79631	0.898154	+	0.439682i	$0.144909\pi$
$450$	0	0
$451$	−10.0568	−0.473557
$452$	0	0
$453$	0.972379i	0.0456864i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.3179i	0.856874i	0.903572	+	0.428437i	$0.140936\pi$
$457$	18.3179i	0.856874i	−0.903572	+	0.428437i	$0.859064\pi$
$458$	0	0
$459$	3.89255	0.181688
$460$	0	0
$461$	−35.8641	−1.67036	−0.835179	−	0.549977i	$-0.814636\pi$
$461$	−35.8641	−1.67036	−0.835179	+	0.549977i	$0.814636\pi$
$462$	0	0
$463$	35.3540i	1.64304i	0.570181	+	0.821519i	$0.306873\pi$
$463$	35.3540i	1.64304i	−0.570181	+	0.821519i	$0.693127\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.5118i	1.55074i	0.631506	+	0.775371i	$0.282437\pi$
$467$	33.5118i	1.55074i	−0.631506	+	0.775371i	$0.717563\pi$
$468$	0	0
$469$	−58.7507	−2.71286
$470$	0	0
$471$	3.48711	0.160677
$472$	0	0
$473$	− 12.6248i	− 0.580491i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.668984i	− 0.0306307i
$478$	0	0
$479$	21.1784	0.967667	0.483834	−	0.875160i	$-0.339244\pi$
$479$	21.1784	0.967667	0.483834	+	0.875160i	$0.339244\pi$
$480$	0	0
$481$	2.66281	0.121414
$482$	0	0
$483$	5.39781i	0.245609i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	35.2032i	1.59521i	0.603180	+	0.797605i	$0.293900\pi$
$487$	35.2032i	1.59521i	−0.603180	+	0.797605i	$0.706100\pi$
$488$	0	0
$489$	2.25719	0.102073
$490$	0	0
$491$	−42.7908	−1.93112	−0.965561	−	0.260177i	$-0.916219\pi$
$491$	−42.7908	−1.93112	−0.965561	+	0.260177i	$0.916219\pi$
$492$	0	0
$493$	− 18.3225i	− 0.825205i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 54.9533i	− 2.46499i
$498$	0	0
$499$	−18.3564	−0.821747	−0.410874	−	0.911692i	$-0.634776\pi$
$499$	−18.3564	−0.821747	−0.410874	+	0.911692i	$0.634776\pi$
$500$	0	0
$501$	−1.38470	−0.0618639
$502$	0	0
$503$	− 38.3650i	− 1.71061i	−0.518125	−	0.855305i	$-0.673370\pi$
$503$	− 38.3650i	− 1.71061i	0.518125	−	0.855305i	$-0.326630\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.09753i	0.0931547i
$508$	0	0
$509$	−1.73845	−0.0770556	−0.0385278	−	0.999258i	$-0.512267\pi$
$509$	−1.73845	−0.0770556	−0.0385278	+	0.999258i	$0.512267\pi$
$510$	0	0
$511$	24.1666	1.06907
$512$	0	0
$513$	1.10673i	0.0488633i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.94532i	0.173515i
$518$	0	0
$519$	3.30354	0.145009
$520$	0	0
$521$	−39.3764	−1.72511	−0.862556	−	0.505961i	$-0.831138\pi$
$521$	−39.3764	−1.72511	−0.862556	+	0.505961i	$0.831138\pi$
$522$	0	0
$523$	− 43.7296i	− 1.91216i	−0.293105	−	0.956080i	$-0.594689\pi$
$523$	− 43.7296i	− 1.91216i	0.293105	−	0.956080i	$-0.405311\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 38.0625i	− 1.65803i
$528$	0	0
$529$	−19.6255	−0.853282
$530$	0	0
$531$	8.48321	0.368140
$532$	0	0
$533$	4.94598i	0.214234i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.75901i	0.0759067i
$538$	0	0
$539$	−34.0317	−1.46585
$540$	0	0
$541$	−17.8936	−0.769305	−0.384652	−	0.923061i	$-0.625679\pi$
$541$	−17.8936	−0.769305	−0.384652	+	0.923061i	$0.625679\pi$
$542$	0	0
$543$	0.802985i	0.0344594i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.91310i	− 0.210069i	−0.994469	−	0.105034i	$-0.966505\pi$
$547$	− 4.91310i	− 0.210069i	0.994469	−	0.105034i	$-0.0334953\pi$
$548$	0	0
$549$	−18.7261	−0.799212
$550$	0	0
$551$	5.20946	0.221931
$552$	0	0
$553$	66.7088i	2.83675i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.6114i	1.63602i	0.575206	+	0.818008i	$0.304922\pi$
$557$	38.6114i	1.63602i	−0.575206	+	0.818008i	$0.695078\pi$
$558$	0	0
$559$	−6.20895	−0.262611
$560$	0	0
$561$	1.72666	0.0728998
$562$	0	0
$563$	− 12.2697i	− 0.517105i	−0.965997	−	0.258553i	$-0.916754\pi$
$563$	− 12.2697i	− 0.517105i	0.965997	−	0.258553i	$-0.0832455\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 38.7334i	− 1.62665i
$568$	0	0
$569$	32.0919	1.34536	0.672682	−	0.739932i	$-0.265142\pi$
$569$	32.0919	1.34536	0.672682	+	0.739932i	$0.265142\pi$
$570$	0	0
$571$	6.85246	0.286767	0.143383	−	0.989667i	$-0.454202\pi$
$571$	6.85246	0.286767	0.143383	+	0.989667i	$0.454202\pi$
$572$	0	0
$573$	− 1.52388i	− 0.0636610i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 20.8983i	− 0.870006i	−0.900429	−	0.435003i	$-0.856747\pi$
$577$	− 20.8983i	− 0.870006i	0.900429	−	0.435003i	$-0.143253\pi$
$578$	0	0
$579$	0.537519	0.0223385
$580$	0	0
$581$	−17.1339	−0.710835
$582$	0	0
$583$	− 0.596943i	− 0.0247228i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.5692i	0.683883i	0.939721	+	0.341942i	$0.111085\pi$
$587$	16.5692i	0.683883i	−0.939721	+	0.341942i	$0.888915\pi$
$588$	0	0
$589$	10.8219	0.445910
$590$	0	0
$591$	4.15782	0.171030
$592$	0	0
$593$	− 6.96253i	− 0.285917i	−0.989729	−	0.142958i	$-0.954338\pi$
$593$	− 6.96253i	− 0.285917i	0.989729	−	0.142958i	$-0.0456615\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 0.498196i	− 0.0203898i
$598$	0	0
$599$	23.8154	0.973072	0.486536	−	0.873660i	$-0.338260\pi$
$599$	23.8154	0.973072	0.486536	+	0.873660i	$0.338260\pi$
$600$	0	0
$601$	10.1339	0.413372	0.206686	−	0.978407i	$-0.433732\pi$
$601$	10.1339	0.413372	0.206686	+	0.978407i	$0.433732\pi$
$602$	0	0
$603$	− 39.0957i	− 1.59210i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.1179i	1.50657i	0.657694	+	0.753286i	$0.271532\pi$
$607$	37.1179i	1.50657i	−0.657694	+	0.753286i	$0.728468\pi$
$608$	0	0
$609$	4.30701	0.174529
$610$	0	0
$611$	1.94033	0.0784972
$612$	0	0
$613$	− 3.21621i	− 0.129901i	−0.997888	−	0.0649507i	$-0.979311\pi$
$613$	− 3.21621i	− 0.129901i	0.997888	−	0.0649507i	$-0.0206890\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.80395i	0.193400i	0.995314	+	0.0967000i	$0.0308287\pi$
$617$	4.80395i	0.193400i	−0.995314	+	0.0967000i	$0.969171\pi$
$618$	0	0
$619$	10.3640	0.416564	0.208282	−	0.978069i	$-0.433213\pi$
$619$	10.3640	0.416564	0.208282	+	0.978069i	$0.433213\pi$
$620$	0	0
$621$	−7.22564	−0.289955
$622$	0	0
$623$	7.47205i	0.299362i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.490925i	0.0196057i
$628$	0	0
$629$	7.19638	0.286938
$630$	0	0
$631$	−0.213288	−0.00849084	−0.00424542	−	0.999991i	$-0.501351\pi$
$631$	−0.213288	−0.00849084	−0.00424542	+	0.999991i	$0.501351\pi$
$632$	0	0
$633$	1.34106i	0.0533024i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16.7369i	0.663141i
$638$	0	0
$639$	36.5686	1.44663
$640$	0	0
$641$	4.44567	0.175593	0.0877967	−	0.996138i	$-0.472017\pi$
$641$	4.44567	0.175593	0.0877967	+	0.996138i	$0.472017\pi$
$642$	0	0
$643$	20.0788i	0.791830i	0.918287	+	0.395915i	$0.129573\pi$
$643$	20.0788i	0.791830i	−0.918287	+	0.395915i	$0.870427\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26.5734i	− 1.04471i	−0.852729	−	0.522353i	$-0.825054\pi$
$647$	− 26.5734i	− 1.04471i	0.852729	−	0.522353i	$-0.174946\pi$
$648$	0	0
$649$	7.56968	0.297136
$650$	0	0
$651$	8.94722	0.350669
$652$	0	0
$653$	0.0209494i	0 0.000819814i	1.00000		0.000409907i	$0.000130477\pi$
$653$	0.0209494i	0 0.000819814i	−1.00000		0.000409907i	$0.999870\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	16.0816i	0.627405i
$658$	0	0
$659$	30.9236	1.20461	0.602307	−	0.798265i	$-0.294248\pi$
$659$	30.9236	1.20461	0.602307	+	0.798265i	$0.294248\pi$
$660$	0	0
$661$	−26.7194	−1.03927	−0.519633	−	0.854390i	$-0.673931\pi$
$661$	−26.7194	−1.03927	−0.519633	+	0.854390i	$0.673931\pi$
$662$	0	0
$663$	− 0.849180i	− 0.0329794i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	34.0116i	1.31694i
$668$	0	0
$669$	1.27383	0.0492490
$670$	0	0
$671$	−16.7096	−0.645066
$672$	0	0
$673$	17.3676i	0.669472i	0.942312	+	0.334736i	$0.108647\pi$
$673$	17.3676i	0.669472i	−0.942312	+	0.334736i	$0.891353\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 45.7712i	− 1.75913i	−0.475779	−	0.879565i	$-0.657834\pi$
$677$	− 45.7712i	− 1.75913i	0.475779	−	0.879565i	$-0.342166\pi$
$678$	0	0
$679$	42.2710	1.62221
$680$	0	0
$681$	−3.56523	−0.136620
$682$	0	0
$683$	36.4020i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$683$	36.4020i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 4.67887i	− 0.178510i
$688$	0	0
$689$	−0.293579	−0.0111845
$690$	0	0
$691$	−20.3322	−0.773472	−0.386736	−	0.922190i	$-0.626398\pi$
$691$	−20.3322	−0.773472	−0.386736	+	0.922190i	$0.626398\pi$
$692$	0	0
$693$	− 34.9729i	− 1.32851i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.3668i	0.506302i
$698$	0	0
$699$	−4.47329	−0.169196
$700$	0	0
$701$	−19.4294	−0.733840	−0.366920	−	0.930253i	$-0.619588\pi$
$701$	−19.4294	−0.733840	−0.366920	+	0.930253i	$0.619588\pi$
$702$	0	0
$703$	2.04607i	0.0771692i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.4815i	1.07116i
$708$	0	0
$709$	13.0978	0.491899	0.245950	−	0.969283i	$-0.420900\pi$
$709$	13.0978	0.491899	0.245950	+	0.969283i	$0.420900\pi$
$710$	0	0
$711$	−44.3913	−1.66481
$712$	0	0
$713$	70.6545i	2.64603i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.805305i	− 0.0300747i
$718$	0	0
$719$	−0.214142	−0.00798615	−0.00399308	−	0.999992i	$-0.501271\pi$
$719$	−0.214142	−0.00798615	−0.00399308	+	0.999992i	$0.501271\pi$
$720$	0	0
$721$	−18.4634	−0.687612
$722$	0	0
$723$	4.07455i	0.151534i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10.2004i	0.378310i	0.981947	+	0.189155i	$0.0605749\pi$
$727$	10.2004i	0.378310i	−0.981947	+	0.189155i	$0.939425\pi$
$728$	0	0
$729$	25.1592	0.931822
$730$	0	0
$731$	−16.7800	−0.620630
$732$	0	0
$733$	31.0210i	1.14579i	0.819630	+	0.572894i	$0.194179\pi$
$733$	31.0210i	1.14579i	−0.819630	+	0.572894i	$0.805821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 34.8855i	− 1.28503i
$738$	0	0
$739$	−35.2371	−1.29622	−0.648108	−	0.761548i	$-0.724440\pi$
$739$	−35.2371	−1.29622	−0.648108	+	0.761548i	$0.724440\pi$
$740$	0	0
$741$	0.241439	0.00886948
$742$	0	0
$743$	− 7.82541i	− 0.287086i	−0.989644	−	0.143543i	$-0.954150\pi$
$743$	− 7.82541i	− 0.287086i	0.989644	−	0.143543i	$-0.0458496\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 11.4018i	− 0.417169i
$748$	0	0
$749$	32.4254	1.18480
$750$	0	0
$751$	−24.2499	−0.884892	−0.442446	−	0.896795i	$-0.645889\pi$
$751$	−24.2499	−0.884892	−0.442446	+	0.896795i	$0.645889\pi$
$752$	0	0
$753$	0.818535i	0.0298291i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	36.4717i	1.32559i	0.748802	+	0.662793i	$0.230629\pi$
$757$	36.4717i	1.32559i	−0.748802	+	0.662793i	$0.769371\pi$
$758$	0	0
$759$	−3.20516	−0.116340
$760$	0	0
$761$	24.6495	0.893544	0.446772	−	0.894648i	$-0.352573\pi$
$761$	24.6495	0.893544	0.446772	+	0.894648i	$0.352573\pi$
$762$	0	0
$763$	62.8711i	2.27609i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.72280i	− 0.134422i
$768$	0	0
$769$	−14.3625	−0.517927	−0.258963	−	0.965887i	$-0.583381\pi$
$769$	−14.3625	−0.517927	−0.258963	+	0.965887i	$0.583381\pi$
$770$	0	0
$771$	2.20976	0.0795824
$772$	0	0
$773$	24.1630i	0.869081i	0.900652	+	0.434541i	$0.143089\pi$
$773$	24.1630i	0.869081i	−0.900652	+	0.434541i	$0.856911\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.69163i	0.0606868i
$778$	0	0
$779$	−3.80044	−0.136165
$780$	0	0
$781$	32.6306	1.16762
$782$	0	0
$783$	5.76546i	0.206041i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 29.6852i	− 1.05816i	−0.848571	−	0.529082i	$-0.822537\pi$
$787$	− 29.6852i	− 1.05816i	0.848571	−	0.529082i	$-0.177463\pi$
$788$	0	0
$789$	−2.55606	−0.0909981
$790$	0	0
$791$	−31.3422	−1.11440
$792$	0	0
$793$	8.21783i	0.291824i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0735i	1.49032i	0.666886	+	0.745160i	$0.267627\pi$
$797$	42.0735i	1.49032i	−0.666886	+	0.745160i	$0.732373\pi$
$798$	0	0
$799$	5.24383	0.185513
$800$	0	0
$801$	−4.97228	−0.175687
$802$	0	0
$803$	14.3498i	0.506395i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.427573i	0.0150513i
$808$	0	0
$809$	−27.8348	−0.978620	−0.489310	−	0.872110i	$-0.662751\pi$
$809$	−27.8348	−0.978620	−0.489310	+	0.872110i	$0.662751\pi$
$810$	0	0
$811$	14.9245	0.524069	0.262035	−	0.965059i	$-0.415607\pi$
$811$	14.9245	0.524069	0.262035	+	0.965059i	$0.415607\pi$
$812$	0	0
$813$	− 3.74522i	− 0.131351i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.77089i	− 0.166912i
$818$	0	0
$819$	−17.1998	−0.601010
$820$	0	0
$821$	0.503399	0.0175688	0.00878438	−	0.999961i	$-0.497204\pi$
$821$	0.503399	0.0175688	0.00878438	+	0.999961i	$0.497204\pi$
$822$	0	0
$823$	20.7905i	0.724713i	0.932040	+	0.362356i	$0.118028\pi$
$823$	20.7905i	0.724713i	−0.932040	+	0.362356i	$0.881972\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.3359i	1.29829i	0.760663	+	0.649147i	$0.224874\pi$
$827$	37.3359i	1.29829i	−0.760663	+	0.649147i	$0.775126\pi$
$828$	0	0
$829$	20.5586	0.714029	0.357014	−	0.934099i	$-0.383795\pi$
$829$	20.5586	0.714029	0.357014	+	0.934099i	$0.383795\pi$
$830$	0	0
$831$	1.98543	0.0688740
$832$	0	0
$833$	45.2323i	1.56721i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	11.9770i	0.413984i
$838$	0	0
$839$	8.13209	0.280751	0.140375	−	0.990098i	$-0.455169\pi$
$839$	8.13209	0.280751	0.140375	+	0.990098i	$0.455169\pi$
$840$	0	0
$841$	−1.86150	−0.0641896
$842$	0	0
$843$	− 0.424102i	− 0.0146068i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.8148i	0.612125i
$848$	0	0
$849$	−4.75884	−0.163323
$850$	0	0
$851$	−13.3585	−0.457922
$852$	0	0
$853$	− 32.7714i	− 1.12207i	−0.827792	−	0.561035i	$-0.810403\pi$
$853$	− 32.7714i	− 1.12207i	0.827792	−	0.561035i	$-0.189597\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.21619i	0.212341i	0.994348	+	0.106170i	$0.0338589\pi$
$857$	6.21619i	0.212341i	−0.994348	+	0.106170i	$0.966141\pi$
$858$	0	0
$859$	−21.2194	−0.723996	−0.361998	−	0.932179i	$-0.617905\pi$
$859$	−21.2194	−0.723996	−0.361998	+	0.932179i	$0.617905\pi$
$860$	0	0
$861$	−3.14208	−0.107082
$862$	0	0
$863$	− 19.4411i	− 0.661784i	−0.943669	−	0.330892i	$-0.892650\pi$
$863$	− 19.4411i	− 0.661784i	0.943669	−	0.330892i	$-0.107350\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.858873i	0.0291689i
$868$	0	0
$869$	−39.6109	−1.34371
$870$	0	0
$871$	−17.1569	−0.581338
$872$	0	0
$873$	28.1292i	0.952030i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.4953i	1.40120i	0.713555	+	0.700599i	$0.247084\pi$
$877$	41.4953i	1.40120i	−0.713555	+	0.700599i	$0.752916\pi$
$878$	0	0
$879$	3.63668	0.122662
$880$	0	0
$881$	−15.6473	−0.527171	−0.263585	−	0.964636i	$-0.584905\pi$
$881$	−15.6473	−0.527171	−0.263585	+	0.964636i	$0.584905\pi$
$882$	0	0
$883$	45.4821i	1.53059i	0.643677	+	0.765297i	$0.277408\pi$
$883$	45.4821i	1.53059i	−0.643677	+	0.765297i	$0.722592\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.9690i	− 0.435457i	−0.976009	−	0.217729i	$-0.930135\pi$
$887$	− 12.9690i	− 0.435457i	0.976009	−	0.217729i	$-0.0698648\pi$
$888$	0	0
$889$	60.7155	2.03633
$890$	0	0
$891$	22.9995	0.770512
$892$	0	0
$893$	1.49093i	0.0498919i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.57631i	0.0526315i
$898$	0	0
$899$	56.3765	1.88026
$900$	0	0
$901$	−0.793411	−0.0264324
$902$	0	0
$903$	− 3.94441i	− 0.131262i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.07053i	0.0687508i	0.999409	+	0.0343754i	$0.0109442\pi$
$907$	2.07053i	0.0687508i	−0.999409	+	0.0343754i	$0.989056\pi$
$908$	0	0
$909$	−18.9530	−0.628633
$910$	0	0
$911$	25.5409	0.846207	0.423104	−	0.906081i	$-0.360941\pi$
$911$	25.5409	0.846207	0.423104	+	0.906081i	$0.360941\pi$
$912$	0	0
$913$	− 10.1739i	− 0.336708i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 99.5337i	− 3.28689i
$918$	0	0
$919$	−25.0300	−0.825664	−0.412832	−	0.910807i	$-0.635460\pi$
$919$	−25.0300	−0.825664	−0.412832	+	0.910807i	$0.635460\pi$
$920$	0	0
$921$	5.87500	0.193588
$922$	0	0
$923$	− 16.0479i	− 0.528223i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.2865i	− 0.403540i
$928$	0	0
$929$	16.4704	0.540377	0.270188	−	0.962807i	$-0.412914\pi$
$929$	16.4704	0.540377	0.270188	+	0.962807i	$0.412914\pi$
$930$	0	0
$931$	−12.8605	−0.421485
$932$	0	0
$933$	2.76343i	0.0904706i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.56119i	0.149008i	0.997221	+	0.0745039i	$0.0237373\pi$
$937$	4.56119i	0.149008i	−0.997221	+	0.0745039i	$0.976263\pi$
$938$	0	0
$939$	−0.0534732	−0.00174503
$940$	0	0
$941$	−10.3721	−0.338119	−0.169060	−	0.985606i	$-0.554073\pi$
$941$	−10.3721	−0.338119	−0.169060	+	0.985606i	$0.554073\pi$
$942$	0	0
$943$	− 24.8124i	− 0.808002i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.70431i	0.0553828i	0.999617	+	0.0276914i	$0.00881557\pi$
$947$	1.70431i	0.0553828i	−0.999617	+	0.0276914i	$0.991184\pi$
$948$	0	0
$949$	7.05731	0.229090
$950$	0	0
$951$	−4.07422	−0.132116
$952$	0	0
$953$	− 18.5324i	− 0.600322i	−0.953888	−	0.300161i	$-0.902960\pi$
$953$	− 18.5324i	− 0.600322i	0.953888	−	0.300161i	$-0.0970405\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.55746i	0.0826708i
$958$	0	0
$959$	−24.7594	−0.799522
$960$	0	0
$961$	86.1144	2.77788
$962$	0	0
$963$	21.5775i	0.695325i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	43.7127i	1.40570i	0.711336	+	0.702852i	$0.248091\pi$
$967$	43.7127i	1.40570i	−0.711336	+	0.702852i	$0.751909\pi$
$968$	0	0
$969$	0.652501	0.0209613
$970$	0	0
$971$	−32.5273	−1.04385	−0.521925	−	0.852992i	$-0.674786\pi$
$971$	−32.5273	−1.04385	−0.521925	+	0.852992i	$0.674786\pi$
$972$	0	0
$973$	− 49.2230i	− 1.57802i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.03542i	0.0651187i	0.999470	+	0.0325594i	$0.0103658\pi$
$977$	2.03542i	0.0651187i	−0.999470	+	0.0325594i	$0.989634\pi$
$978$	0	0
$979$	−4.43682	−0.141802
$980$	0	0
$981$	−41.8376	−1.33577
$982$	0	0
$983$	19.2872i	0.615166i	0.951521	+	0.307583i	$0.0995202\pi$
$983$	19.2872i	0.615166i	−0.951521	+	0.307583i	$0.900480\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.23265i	0.0392356i
$988$	0	0
$989$	31.1483	0.990457
$990$	0	0
$991$	−0.611835	−0.0194356	−0.00971778	−	0.999953i	$-0.503093\pi$
$991$	−0.611835	−0.0194356	−0.00971778	+	0.999953i	$0.503093\pi$
$992$	0	0
$993$	4.46924i	0.141827i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	58.8380i	1.86342i	0.363207	+	0.931709i	$0.381682\pi$
$997$	58.8380i	1.86342i	−0.363207	+	0.931709i	$0.618318\pi$
$998$	0	0
$999$	−2.26445	−0.0716441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.q.3649.7		12
5.2	odd	4		3800.2.a.ba.1.4	✓	6
5.3	odd	4		3800.2.a.bc.1.3	yes	6
5.4	even	2	inner	3800.2.d.q.3649.6		12
20.3	even	4		7600.2.a.ch.1.4		6
20.7	even	4		7600.2.a.cl.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.4	✓	6	5.2	odd	4
3800.2.a.bc.1.3	yes	6	5.3	odd	4
3800.2.d.q.3649.6		12	5.4	even	2	inner
3800.2.d.q.3649.7		12	1.1	even	1	trivial
7600.2.a.ch.1.4		6	20.3	even	4
7600.2.a.cl.1.3		6	20.7	even	4

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.185519i q^{3} -4.45651i q^{7} +2.96558 q^{9} +2.64623 q^{11} -1.30142i q^{13} -3.51716i q^{17} +1.00000 q^{19} +0.826767 q^{21} +6.52882i q^{23} +1.10673i q^{27} +5.20946 q^{29} +10.8219 q^{31} +0.490925i q^{33} +2.04607i q^{37} +0.241439 q^{39} -3.80044 q^{41} -4.77089i q^{43} +1.49093i q^{47} -12.8605 q^{49} +0.652501 q^{51} -0.225583i q^{53} +0.185519i q^{57} +2.86056 q^{59} -6.31449 q^{61} -13.2161i q^{63} -13.1831i q^{67} -1.21122 q^{69} +12.3310 q^{71} +5.42276i q^{73} -11.7929i q^{77} -14.9688 q^{79} +8.69143 q^{81} -3.84470i q^{83} +0.966455i q^{87} -1.67666 q^{89} -5.79981 q^{91} +2.00768i q^{93} +9.48523i q^{97} +7.84760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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