LMFDB - Embedded newform 1600.3.h.i.1599.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1599,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1600.1599");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.h (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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 16 $ x^4 - 7*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1599.1
Root			$-1.93649 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1600.1599
Dual form			1600.3.h.i.1599.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.87298 q^{3} +7.74597 q^{7} +6.00000 q^{9} +O(q^{10})$	$q-3.87298 q^{3} +7.74597 q^{7} +6.00000 q^{9} -19.3649i q^{11} -20.0000i q^{13} -15.0000i q^{17} +19.3649i q^{19} -30.0000 q^{21} +7.74597 q^{23} +11.6190 q^{27} -48.0000 q^{29} -38.7298i q^{31} +75.0000i q^{33} +10.0000i q^{37} +77.4597i q^{39} +33.0000 q^{41} +61.9677 q^{43} -15.4919 q^{47} +11.0000 q^{49} +58.0948i q^{51} -30.0000i q^{53} -75.0000i q^{57} +77.4597i q^{59} -38.0000 q^{61} +46.4758 q^{63} +58.0948 q^{67} -30.0000 q^{69} +77.4597i q^{71} +5.00000i q^{73} -150.000i q^{77} -38.7298i q^{79} -99.0000 q^{81} -11.6190 q^{83} +185.903 q^{87} -87.0000 q^{89} -154.919i q^{91} +150.000i q^{93} -110.000i q^{97} -116.190i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{9}+O(q^{10}) $ 4 * q + 24 * q^9	$ 4 q + 24 q^{9} - 120 q^{21} - 192 q^{29} + 132 q^{41} + 44 q^{49} - 152 q^{61} - 120 q^{69} - 396 q^{81} - 348 q^{89}+O(q^{100}) $ 4 * q + 24 * q^9 - 120 * q^21 - 192 * q^29 + 132 * q^41 + 44 * q^49 - 152 * q^61 - 120 * q^69 - 396 * q^81 - 348 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−3.87298	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−3.87298	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.74597	1.10657	0.553283	−	0.832993i	$-0.313375\pi$
$7$	7.74597	1.10657	0.553283	+	0.832993i	$0.313375\pi$
$8$	0	0
$9$	6.00000	0.666667
$10$	0	0
$11$	− 19.3649i	− 1.76045i	−0.474559	−	0.880223i	$-0.657393\pi$
$11$	− 19.3649i	− 1.76045i	0.474559	−	0.880223i	$-0.342607\pi$
$12$	0	0
$13$	− 20.0000i	− 1.53846i	−0.638971	−	0.769231i	$-0.720640\pi$
$13$	− 20.0000i	− 1.53846i	0.638971	−	0.769231i	$-0.279360\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 15.0000i	− 0.882353i	−0.897420	−	0.441176i	$-0.854561\pi$
$17$	− 15.0000i	− 0.882353i	0.897420	−	0.441176i	$-0.145439\pi$
$18$	0	0
$19$	19.3649i	1.01921i	0.860410	+	0.509603i	$0.170208\pi$
$19$	19.3649i	1.01921i	−0.860410	+	0.509603i	$0.829792\pi$
$20$	0	0
$21$	−30.0000	−1.42857
$22$	0	0
$23$	7.74597	0.336781	0.168391	−	0.985720i	$-0.446143\pi$
$23$	7.74597	0.336781	0.168391	+	0.985720i	$0.446143\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	11.6190	0.430331
$28$	0	0
$29$	−48.0000	−1.65517	−0.827586	−	0.561339i	$-0.810287\pi$
$29$	−48.0000	−1.65517	−0.827586	+	0.561339i	$0.810287\pi$
$30$	0	0
$31$	− 38.7298i	− 1.24935i	−0.780885	−	0.624675i	$-0.785232\pi$
$31$	− 38.7298i	− 1.24935i	0.780885	−	0.624675i	$-0.214768\pi$
$32$	0	0
$33$	75.0000i	2.27273i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000i	0.270270i	0.990827	+	0.135135i	$0.0431469\pi$
$37$	10.0000i	0.270270i	−0.990827	+	0.135135i	$0.956853\pi$
$38$	0	0
$39$	77.4597i	1.98615i
$40$	0	0
$41$	33.0000	0.804878	0.402439	−	0.915447i	$-0.368163\pi$
$41$	33.0000	0.804878	0.402439	+	0.915447i	$0.368163\pi$
$42$	0	0
$43$	61.9677	1.44111	0.720555	−	0.693398i	$-0.243887\pi$
$43$	61.9677	1.44111	0.720555	+	0.693398i	$0.243887\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−15.4919	−0.329616	−0.164808	−	0.986326i	$-0.552700\pi$
$47$	−15.4919	−0.329616	−0.164808	+	0.986326i	$0.552700\pi$
$48$	0	0
$49$	11.0000	0.224490
$50$	0	0
$51$	58.0948i	1.13911i
$52$	0	0
$53$	− 30.0000i	− 0.566038i	−0.959114	−	0.283019i	$-0.908664\pi$
$53$	− 30.0000i	− 0.566038i	0.959114	−	0.283019i	$-0.0913359\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 75.0000i	− 1.31579i
$58$	0	0
$59$	77.4597i	1.31288i	0.754380	+	0.656438i	$0.227938\pi$
$59$	77.4597i	1.31288i	−0.754380	+	0.656438i	$0.772062\pi$
$60$	0	0
$61$	−38.0000	−0.622951	−0.311475	−	0.950254i	$-0.600823\pi$
$61$	−38.0000	−0.622951	−0.311475	+	0.950254i	$0.600823\pi$
$62$	0	0
$63$	46.4758	0.737711
$64$	0	0
$65$	0	0
$66$	0	0
$67$	58.0948	0.867086	0.433543	−	0.901133i	$-0.357263\pi$
$67$	58.0948	0.867086	0.433543	+	0.901133i	$0.357263\pi$
$68$	0	0
$69$	−30.0000	−0.434783
$70$	0	0
$71$	77.4597i	1.09098i	0.838117	+	0.545491i	$0.183657\pi$
$71$	77.4597i	1.09098i	−0.838117	+	0.545491i	$0.816343\pi$
$72$	0	0
$73$	5.00000i	0.0684932i	0.999413	+	0.0342466i	$0.0109032\pi$
$73$	5.00000i	0.0684932i	−0.999413	+	0.0342466i	$0.989097\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 150.000i	− 1.94805i
$78$	0	0
$79$	− 38.7298i	− 0.490251i	−0.969491	−	0.245126i	$-0.921171\pi$
$79$	− 38.7298i	− 0.490251i	0.969491	−	0.245126i	$-0.0788292\pi$
$80$	0	0
$81$	−99.0000	−1.22222
$82$	0	0
$83$	−11.6190	−0.139987	−0.0699937	−	0.997547i	$-0.522298\pi$
$83$	−11.6190	−0.139987	−0.0699937	+	0.997547i	$0.522298\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	185.903	2.13682
$88$	0	0
$89$	−87.0000	−0.977528	−0.488764	−	0.872416i	$-0.662552\pi$
$89$	−87.0000	−0.977528	−0.488764	+	0.872416i	$0.662552\pi$
$90$	0	0
$91$	− 154.919i	− 1.70241i
$92$	0	0
$93$	150.000i	1.61290i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 110.000i	− 1.13402i	−0.823711	−	0.567010i	$-0.808100\pi$
$97$	− 110.000i	− 1.13402i	0.823711	−	0.567010i	$-0.191900\pi$
$98$	0	0
$99$	− 116.190i	− 1.17363i
$100$	0	0
$101$	−42.0000	−0.415842	−0.207921	−	0.978146i	$-0.566670\pi$
$101$	−42.0000	−0.415842	−0.207921	+	0.978146i	$0.566670\pi$
$102$	0	0
$103$	77.4597	0.752036	0.376018	−	0.926612i	$-0.377293\pi$
$103$	77.4597	0.752036	0.376018	+	0.926612i	$0.377293\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−42.6028	−0.398157	−0.199079	−	0.979984i	$-0.563795\pi$
$107$	−42.6028	−0.398157	−0.199079	+	0.979984i	$0.563795\pi$
$108$	0	0
$109$	58.0000	0.532110	0.266055	−	0.963958i	$-0.414280\pi$
$109$	58.0000	0.532110	0.266055	+	0.963958i	$0.414280\pi$
$110$	0	0
$111$	− 38.7298i	− 0.348917i
$112$	0	0
$113$	− 135.000i	− 1.19469i	−0.801984	−	0.597345i	$-0.796222\pi$
$113$	− 135.000i	− 1.19469i	0.801984	−	0.597345i	$-0.203778\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 120.000i	− 1.02564i
$118$	0	0
$119$	− 116.190i	− 0.976382i
$120$	0	0
$121$	−254.000	−2.09917
$122$	0	0
$123$	−127.808	−1.03909
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−209.141	−1.64678	−0.823390	−	0.567476i	$-0.807920\pi$
$127$	−209.141	−1.64678	−0.823390	+	0.567476i	$0.807920\pi$
$128$	0	0
$129$	−240.000	−1.86047
$130$	0	0
$131$	154.919i	1.18259i	0.806455	+	0.591295i	$0.201383\pi$
$131$	154.919i	1.18259i	−0.806455	+	0.591295i	$0.798617\pi$
$132$	0	0
$133$	150.000i	1.12782i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	105.000i	0.766423i	0.923661	+	0.383212i	$0.125182\pi$
$137$	105.000i	0.766423i	−0.923661	+	0.383212i	$0.874818\pi$
$138$	0	0
$139$	19.3649i	0.139316i	0.997571	+	0.0696580i	$0.0221908\pi$
$139$	19.3649i	0.139316i	−0.997571	+	0.0696580i	$0.977809\pi$
$140$	0	0
$141$	60.0000	0.425532
$142$	0	0
$143$	−387.298	−2.70838
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−42.6028	−0.289815
$148$	0	0
$149$	−192.000	−1.28859	−0.644295	−	0.764777i	$-0.722849\pi$
$149$	−192.000	−1.28859	−0.644295	+	0.764777i	$0.722849\pi$
$150$	0	0
$151$	− 38.7298i	− 0.256489i	−0.991743	−	0.128244i	$-0.959066\pi$
$151$	− 38.7298i	− 0.256489i	0.991743	−	0.128244i	$-0.0409342\pi$
$152$	0	0
$153$	− 90.0000i	− 0.588235i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 110.000i	− 0.700637i	−0.936631	−	0.350318i	$-0.886073\pi$
$157$	− 110.000i	− 0.700637i	0.936631	−	0.350318i	$-0.113927\pi$
$158$	0	0
$159$	116.190i	0.730752i
$160$	0	0
$161$	60.0000	0.372671
$162$	0	0
$163$	127.808	0.784101	0.392050	−	0.919944i	$-0.371766\pi$
$163$	127.808	0.784101	0.392050	+	0.919944i	$0.371766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−294.347	−1.76256	−0.881278	−	0.472599i	$-0.843316\pi$
$167$	−294.347	−1.76256	−0.881278	+	0.472599i	$0.843316\pi$
$168$	0	0
$169$	−231.000	−1.36686
$170$	0	0
$171$	116.190i	0.679471i
$172$	0	0
$173$	240.000i	1.38728i	0.720320	+	0.693642i	$0.243995\pi$
$173$	240.000i	1.38728i	−0.720320	+	0.693642i	$0.756005\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 300.000i	− 1.69492i
$178$	0	0
$179$	− 213.014i	− 1.19002i	−0.803717	−	0.595011i	$-0.797148\pi$
$179$	− 213.014i	− 1.19002i	0.803717	−	0.595011i	$-0.202852\pi$
$180$	0	0
$181$	−22.0000	−0.121547	−0.0607735	−	0.998152i	$-0.519357\pi$
$181$	−22.0000	−0.121547	−0.0607735	+	0.998152i	$0.519357\pi$
$182$	0	0
$183$	147.173	0.804226
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−290.474	−1.55334
$188$	0	0
$189$	90.0000	0.476190
$190$	0	0
$191$	− 116.190i	− 0.608322i	−0.952621	−	0.304161i	$-0.901624\pi$
$191$	− 116.190i	− 0.608322i	0.952621	−	0.304161i	$-0.0983761\pi$
$192$	0	0
$193$	295.000i	1.52850i	0.644922	+	0.764249i	$0.276890\pi$
$193$	295.000i	1.52850i	−0.644922	+	0.764249i	$0.723110\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 330.000i	− 1.67513i	−0.546340	−	0.837563i	$-0.683979\pi$
$197$	− 330.000i	− 1.67513i	0.546340	−	0.837563i	$-0.316021\pi$
$198$	0	0
$199$	− 232.379i	− 1.16773i	−0.811849	−	0.583867i	$-0.801539\pi$
$199$	− 232.379i	− 1.16773i	0.811849	−	0.583867i	$-0.198461\pi$
$200$	0	0
$201$	−225.000	−1.11940
$202$	0	0
$203$	−371.806	−1.83156
$204$	0	0
$205$	0	0
$206$	0	0
$207$	46.4758	0.224521
$208$	0	0
$209$	375.000	1.79426
$210$	0	0
$211$	− 58.0948i	− 0.275331i	−0.990479	−	0.137665i	$-0.956040\pi$
$211$	− 58.0948i	− 0.275331i	0.990479	−	0.137665i	$-0.0439598\pi$
$212$	0	0
$213$	− 300.000i	− 1.40845i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 300.000i	− 1.38249i
$218$	0	0
$219$	− 19.3649i	− 0.0884243i
$220$	0	0
$221$	−300.000	−1.35747
$222$	0	0
$223$	−15.4919	−0.0694706	−0.0347353	−	0.999397i	$-0.511059\pi$
$223$	−15.4919	−0.0694706	−0.0347353	+	0.999397i	$0.511059\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	30.9839	0.136493	0.0682464	−	0.997668i	$-0.478260\pi$
$227$	30.9839	0.136493	0.0682464	+	0.997668i	$0.478260\pi$
$228$	0	0
$229$	−68.0000	−0.296943	−0.148472	−	0.988917i	$-0.547435\pi$
$229$	−68.0000	−0.296943	−0.148472	+	0.988917i	$0.547435\pi$
$230$	0	0
$231$	580.948i	2.51492i
$232$	0	0
$233$	− 30.0000i	− 0.128755i	−0.997926	−	0.0643777i	$-0.979494\pi$
$233$	− 30.0000i	− 0.128755i	0.997926	−	0.0643777i	$-0.0205062\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	150.000i	0.632911i
$238$	0	0
$239$	− 154.919i	− 0.648198i	−0.946023	−	0.324099i	$-0.894939\pi$
$239$	− 154.919i	− 0.648198i	0.946023	−	0.324099i	$-0.105061\pi$
$240$	0	0
$241$	253.000	1.04979	0.524896	−	0.851166i	$-0.324104\pi$
$241$	253.000	1.04979	0.524896	+	0.851166i	$0.324104\pi$
$242$	0	0
$243$	278.855	1.14755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	387.298	1.56801
$248$	0	0
$249$	45.0000	0.180723
$250$	0	0
$251$	− 367.933i	− 1.46587i	−0.680299	−	0.732935i	$-0.738150\pi$
$251$	− 367.933i	− 1.46587i	0.680299	−	0.732935i	$-0.261850\pi$
$252$	0	0
$253$	− 150.000i	− 0.592885i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	270.000i	1.05058i	0.850922	+	0.525292i	$0.176044\pi$
$257$	270.000i	1.05058i	−0.850922	+	0.525292i	$0.823956\pi$
$258$	0	0
$259$	77.4597i	0.299072i
$260$	0	0
$261$	−288.000	−1.10345
$262$	0	0
$263$	100.698	0.382880	0.191440	−	0.981504i	$-0.438684\pi$
$263$	100.698	0.382880	0.191440	+	0.981504i	$0.438684\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	336.950	1.26198
$268$	0	0
$269$	−312.000	−1.15985	−0.579926	−	0.814669i	$-0.696918\pi$
$269$	−312.000	−1.15985	−0.579926	+	0.814669i	$0.696918\pi$
$270$	0	0
$271$	− 193.649i	− 0.714573i	−0.933995	−	0.357286i	$-0.883702\pi$
$271$	− 193.649i	− 0.714573i	0.933995	−	0.357286i	$-0.116298\pi$
$272$	0	0
$273$	600.000i	2.19780i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	440.000i	1.58845i	0.607625	+	0.794224i	$0.292122\pi$
$277$	440.000i	1.58845i	−0.607625	+	0.794224i	$0.707878\pi$
$278$	0	0
$279$	− 232.379i	− 0.832900i
$280$	0	0
$281$	198.000	0.704626	0.352313	−	0.935882i	$-0.385395\pi$
$281$	198.000	0.704626	0.352313	+	0.935882i	$0.385395\pi$
$282$	0	0
$283$	−213.014	−0.752700	−0.376350	−	0.926478i	$-0.622821\pi$
$283$	−213.014	−0.752700	−0.376350	+	0.926478i	$0.622821\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	255.617	0.890651
$288$	0	0
$289$	64.0000	0.221453
$290$	0	0
$291$	426.028i	1.46401i
$292$	0	0
$293$	− 390.000i	− 1.33106i	−0.746372	−	0.665529i	$-0.768206\pi$
$293$	− 390.000i	− 1.33106i	0.746372	−	0.665529i	$-0.231794\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 225.000i	− 0.757576i
$298$	0	0
$299$	− 154.919i	− 0.518125i
$300$	0	0
$301$	480.000	1.59468
$302$	0	0
$303$	162.665	0.536849
$304$	0	0
$305$	0	0
$306$	0	0
$307$	375.679	1.22371	0.611856	−	0.790969i	$-0.290423\pi$
$307$	375.679	1.22371	0.611856	+	0.790969i	$0.290423\pi$
$308$	0	0
$309$	−300.000	−0.970874
$310$	0	0
$311$	− 271.109i	− 0.871733i	−0.900011	−	0.435866i	$-0.856442\pi$
$311$	− 271.109i	− 0.871733i	0.900011	−	0.435866i	$-0.143558\pi$
$312$	0	0
$313$	− 50.0000i	− 0.159744i	−0.996805	−	0.0798722i	$-0.974549\pi$
$313$	− 50.0000i	− 0.159744i	0.996805	−	0.0798722i	$-0.0254512\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 300.000i	− 0.946372i	−0.880962	−	0.473186i	$-0.843104\pi$
$317$	− 300.000i	− 0.946372i	0.880962	−	0.473186i	$-0.156896\pi$
$318$	0	0
$319$	929.516i	2.91384i
$320$	0	0
$321$	165.000	0.514019
$322$	0	0
$323$	290.474	0.899300
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−224.633	−0.686951
$328$	0	0
$329$	−120.000	−0.364742
$330$	0	0
$331$	213.014i	0.643547i	0.946817	+	0.321774i	$0.104279\pi$
$331$	213.014i	0.643547i	−0.946817	+	0.321774i	$0.895721\pi$
$332$	0	0
$333$	60.0000i	0.180180i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 125.000i	− 0.370920i	−0.982652	−	0.185460i	$-0.940623\pi$
$337$	− 125.000i	− 0.370920i	0.982652	−	0.185460i	$-0.0593775\pi$
$338$	0	0
$339$	522.853i	1.54234i
$340$	0	0
$341$	−750.000	−2.19941
$342$	0	0
$343$	−294.347	−0.858154
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−476.377	−1.37284	−0.686422	−	0.727203i	$-0.740820\pi$
$347$	−476.377	−1.37284	−0.686422	+	0.727203i	$0.740820\pi$
$348$	0	0
$349$	242.000	0.693410	0.346705	−	0.937974i	$-0.387301\pi$
$349$	242.000	0.693410	0.346705	+	0.937974i	$0.387301\pi$
$350$	0	0
$351$	− 232.379i	− 0.662048i
$352$	0	0
$353$	− 150.000i	− 0.424929i	−0.977169	−	0.212465i	$-0.931851\pi$
$353$	− 150.000i	− 0.424929i	0.977169	−	0.212465i	$-0.0681490\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	450.000i	1.26050i
$358$	0	0
$359$	426.028i	1.18671i	0.804942	+	0.593354i	$0.202197\pi$
$359$	426.028i	1.18671i	−0.804942	+	0.593354i	$0.797803\pi$
$360$	0	0
$361$	−14.0000	−0.0387812
$362$	0	0
$363$	983.738	2.71002
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−511.234	−1.39301	−0.696504	−	0.717553i	$-0.745262\pi$
$367$	−511.234	−1.39301	−0.696504	+	0.717553i	$0.745262\pi$
$368$	0	0
$369$	198.000	0.536585
$370$	0	0
$371$	− 232.379i	− 0.626358i
$372$	0	0
$373$	− 250.000i	− 0.670241i	−0.942175	−	0.335121i	$-0.891223\pi$
$373$	− 250.000i	− 0.670241i	0.942175	−	0.335121i	$-0.108777\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	960.000i	2.54642i
$378$	0	0
$379$	290.474i	0.766422i	0.923661	+	0.383211i	$0.125182\pi$
$379$	290.474i	0.766422i	−0.923661	+	0.383211i	$0.874818\pi$
$380$	0	0
$381$	810.000	2.12598
$382$	0	0
$383$	596.439	1.55728	0.778642	−	0.627469i	$-0.215909\pi$
$383$	596.439	1.55728	0.778642	+	0.627469i	$0.215909\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	371.806	0.960740
$388$	0	0
$389$	18.0000	0.0462725	0.0231362	−	0.999732i	$-0.492635\pi$
$389$	18.0000	0.0462725	0.0231362	+	0.999732i	$0.492635\pi$
$390$	0	0
$391$	− 116.190i	− 0.297160i
$392$	0	0
$393$	− 600.000i	− 1.52672i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	100.000i	0.251889i	0.992037	+	0.125945i	$0.0401962\pi$
$397$	100.000i	0.251889i	−0.992037	+	0.125945i	$0.959804\pi$
$398$	0	0
$399$	− 580.948i	− 1.45601i
$400$	0	0
$401$	−183.000	−0.456359	−0.228180	−	0.973619i	$-0.573277\pi$
$401$	−183.000	−0.456359	−0.228180	+	0.973619i	$0.573277\pi$
$402$	0	0
$403$	−774.597	−1.92208
$404$	0	0
$405$	0	0
$406$	0	0
$407$	193.649	0.475796
$408$	0	0
$409$	47.0000	0.114914	0.0574572	−	0.998348i	$-0.481701\pi$
$409$	47.0000	0.114914	0.0574572	+	0.998348i	$0.481701\pi$
$410$	0	0
$411$	− 406.663i	− 0.989448i
$412$	0	0
$413$	600.000i	1.45278i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 75.0000i	− 0.179856i
$418$	0	0
$419$	406.663i	0.970557i	0.874360	+	0.485278i	$0.161282\pi$
$419$	406.663i	0.970557i	−0.874360	+	0.485278i	$0.838718\pi$
$420$	0	0
$421$	452.000	1.07363	0.536817	−	0.843699i	$-0.319627\pi$
$421$	452.000	1.07363	0.536817	+	0.843699i	$0.319627\pi$
$422$	0	0
$423$	−92.9516	−0.219744
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−294.347	−0.689337
$428$	0	0
$429$	1500.00	3.49650
$430$	0	0
$431$	38.7298i	0.0898604i	0.998990	+	0.0449302i	$0.0143065\pi$
$431$	38.7298i	0.0898604i	−0.998990	+	0.0449302i	$0.985693\pi$
$432$	0	0
$433$	55.0000i	0.127021i	0.997981	+	0.0635104i	$0.0202296\pi$
$433$	55.0000i	0.127021i	−0.997981	+	0.0635104i	$0.979770\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	150.000i	0.343249i
$438$	0	0
$439$	619.677i	1.41157i	0.708428	+	0.705783i	$0.249405\pi$
$439$	619.677i	1.41157i	−0.708428	+	0.705783i	$0.750595\pi$
$440$	0	0
$441$	66.0000	0.149660
$442$	0	0
$443$	−321.458	−0.725638	−0.362819	−	0.931860i	$-0.618186\pi$
$443$	−321.458	−0.725638	−0.362819	+	0.931860i	$0.618186\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	743.613	1.66356
$448$	0	0
$449$	627.000	1.39644	0.698218	−	0.715885i	$-0.253977\pi$
$449$	627.000	1.39644	0.698218	+	0.715885i	$0.253977\pi$
$450$	0	0
$451$	− 639.042i	− 1.41695i
$452$	0	0
$453$	150.000i	0.331126i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	325.000i	0.711160i	0.934646	+	0.355580i	$0.115717\pi$
$457$	325.000i	0.711160i	−0.934646	+	0.355580i	$0.884283\pi$
$458$	0	0
$459$	− 174.284i	− 0.379704i
$460$	0	0
$461$	−12.0000	−0.0260304	−0.0130152	−	0.999915i	$-0.504143\pi$
$461$	−12.0000	−0.0260304	−0.0130152	+	0.999915i	$0.504143\pi$
$462$	0	0
$463$	−108.444	−0.234219	−0.117110	−	0.993119i	$-0.537363\pi$
$463$	−108.444	−0.234219	−0.117110	+	0.993119i	$0.537363\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−340.823	−0.729813	−0.364906	−	0.931044i	$-0.618899\pi$
$467$	−340.823	−0.729813	−0.364906	+	0.931044i	$0.618899\pi$
$468$	0	0
$469$	450.000	0.959488
$470$	0	0
$471$	426.028i	0.904518i
$472$	0	0
$473$	− 1200.00i	− 2.53700i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 180.000i	− 0.377358i
$478$	0	0
$479$	116.190i	0.242567i	0.992618	+	0.121283i	$0.0387010\pi$
$479$	116.190i	0.242567i	−0.992618	+	0.121283i	$0.961299\pi$
$480$	0	0
$481$	200.000	0.415800
$482$	0	0
$483$	−232.379	−0.481116
$484$	0	0
$485$	0	0
$486$	0	0
$487$	441.520	0.906612	0.453306	−	0.891355i	$-0.350244\pi$
$487$	441.520	0.906612	0.453306	+	0.891355i	$0.350244\pi$
$488$	0	0
$489$	−495.000	−1.01227
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	720.000i	1.46045i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	600.000i	1.20724i
$498$	0	0
$499$	− 309.839i	− 0.620919i	−0.950587	−	0.310460i	$-0.899517\pi$
$499$	− 309.839i	− 0.620919i	0.950587	−	0.310460i	$-0.100483\pi$
$500$	0	0
$501$	1140.00	2.27545
$502$	0	0
$503$	−46.4758	−0.0923972	−0.0461986	−	0.998932i	$-0.514711\pi$
$503$	−46.4758	−0.0923972	−0.0461986	+	0.998932i	$0.514711\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	894.659	1.76461
$508$	0	0
$509$	−618.000	−1.21415	−0.607073	−	0.794646i	$-0.707656\pi$
$509$	−618.000	−1.21415	−0.607073	+	0.794646i	$0.707656\pi$
$510$	0	0
$511$	38.7298i	0.0757922i
$512$	0	0
$513$	225.000i	0.438596i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	300.000i	0.580271i
$518$	0	0
$519$	− 929.516i	− 1.79097i
$520$	0	0
$521$	−423.000	−0.811900	−0.405950	−	0.913895i	$-0.633059\pi$
$521$	−423.000	−0.811900	−0.405950	+	0.913895i	$0.633059\pi$
$522$	0	0
$523$	120.062	0.229565	0.114782	−	0.993391i	$-0.463383\pi$
$523$	120.062	0.229565	0.114782	+	0.993391i	$0.463383\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−580.948	−1.10237
$528$	0	0
$529$	−469.000	−0.886578
$530$	0	0
$531$	464.758i	0.875250i
$532$	0	0
$533$	− 660.000i	− 1.23827i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	825.000i	1.53631i
$538$	0	0
$539$	− 213.014i	− 0.395202i
$540$	0	0
$541$	8.00000	0.0147874	0.00739372	−	0.999973i	$-0.497646\pi$
$541$	8.00000	0.0147874	0.00739372	+	0.999973i	$0.497646\pi$
$542$	0	0
$543$	85.2056	0.156916
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−398.917	−0.729282	−0.364641	−	0.931148i	$-0.618808\pi$
$547$	−398.917	−0.729282	−0.364641	+	0.931148i	$0.618808\pi$
$548$	0	0
$549$	−228.000	−0.415301
$550$	0	0
$551$	− 929.516i	− 1.68696i
$552$	0	0
$553$	− 300.000i	− 0.542495i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	660.000i	1.18492i	0.805600	+	0.592460i	$0.201843\pi$
$557$	660.000i	1.18492i	−0.805600	+	0.592460i	$0.798157\pi$
$558$	0	0
$559$	− 1239.35i	− 2.21709i
$560$	0	0
$561$	1125.00	2.00535
$562$	0	0
$563$	542.218	0.963086	0.481543	−	0.876422i	$-0.340076\pi$
$563$	542.218	0.963086	0.481543	+	0.876422i	$0.340076\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−766.851	−1.35247
$568$	0	0
$569$	−453.000	−0.796134	−0.398067	−	0.917356i	$-0.630319\pi$
$569$	−453.000	−0.796134	−0.398067	+	0.917356i	$0.630319\pi$
$570$	0	0
$571$	1084.44i	1.89919i	0.313485	+	0.949593i	$0.398503\pi$
$571$	1084.44i	1.89919i	−0.313485	+	0.949593i	$0.601497\pi$
$572$	0	0
$573$	450.000i	0.785340i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 325.000i	− 0.563258i	−0.959523	−	0.281629i	$-0.909125\pi$
$577$	− 325.000i	− 0.563258i	0.959523	−	0.281629i	$-0.0908748\pi$
$578$	0	0
$579$	− 1142.53i	− 1.97328i
$580$	0	0
$581$	−90.0000	−0.154905
$582$	0	0
$583$	−580.948	−0.996479
$584$	0	0
$585$	0	0
$586$	0	0
$587$	755.232	1.28660	0.643298	−	0.765616i	$-0.277566\pi$
$587$	755.232	1.28660	0.643298	+	0.765616i	$0.277566\pi$
$588$	0	0
$589$	750.000	1.27334
$590$	0	0
$591$	1278.08i	2.16258i
$592$	0	0
$593$	− 15.0000i	− 0.0252951i	−0.999920	−	0.0126476i	$-0.995974\pi$
$593$	− 15.0000i	− 0.0252951i	0.999920	−	0.0126476i	$-0.00402595\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	900.000i	1.50754i
$598$	0	0
$599$	1045.71i	1.74575i	0.487942	+	0.872876i	$0.337748\pi$
$599$	1045.71i	1.74575i	−0.487942	+	0.872876i	$0.662252\pi$
$600$	0	0
$601$	−637.000	−1.05990	−0.529950	−	0.848029i	$-0.677789\pi$
$601$	−637.000	−1.05990	−0.529950	+	0.848029i	$0.677789\pi$
$602$	0	0
$603$	348.569	0.578057
$604$	0	0
$605$	0	0
$606$	0	0
$607$	154.919	0.255221	0.127611	−	0.991824i	$-0.459269\pi$
$607$	154.919	0.255221	0.127611	+	0.991824i	$0.459269\pi$
$608$	0	0
$609$	1440.00	2.36453
$610$	0	0
$611$	309.839i	0.507101i
$612$	0	0
$613$	− 910.000i	− 1.48450i	−0.670122	−	0.742251i	$-0.733758\pi$
$613$	− 910.000i	− 1.48450i	0.670122	−	0.742251i	$-0.266242\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	630.000i	1.02107i	0.859857	+	0.510535i	$0.170553\pi$
$617$	630.000i	1.02107i	−0.859857	+	0.510535i	$0.829447\pi$
$618$	0	0
$619$	− 154.919i	− 0.250274i	−0.992139	−	0.125137i	$-0.960063\pi$
$619$	− 154.919i	− 0.250274i	0.992139	−	0.125137i	$-0.0399370\pi$
$620$	0	0
$621$	90.0000	0.144928
$622$	0	0
$623$	−673.899	−1.08170
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1452.37	−2.31638
$628$	0	0
$629$	150.000	0.238474
$630$	0	0
$631$	− 348.569i	− 0.552406i	−0.961099	−	0.276203i	$-0.910924\pi$
$631$	− 348.569i	− 0.552406i	0.961099	−	0.276203i	$-0.0890763\pi$
$632$	0	0
$633$	225.000i	0.355450i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 220.000i	− 0.345369i
$638$	0	0
$639$	464.758i	0.727321i
$640$	0	0
$641$	858.000	1.33853	0.669267	−	0.743022i	$-0.266608\pi$
$641$	858.000	1.33853	0.669267	+	0.743022i	$0.266608\pi$
$642$	0	0
$643$	−650.661	−1.01191	−0.505957	−	0.862558i	$-0.668861\pi$
$643$	−650.661	−1.01191	−0.505957	+	0.862558i	$0.668861\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−852.056	−1.31693	−0.658467	−	0.752610i	$-0.728795\pi$
$647$	−852.056	−1.31693	−0.658467	+	0.752610i	$0.728795\pi$
$648$	0	0
$649$	1500.00	2.31125
$650$	0	0
$651$	1161.90i	1.78478i
$652$	0	0
$653$	− 990.000i	− 1.51608i	−0.652208	−	0.758040i	$-0.726157\pi$
$653$	− 990.000i	− 1.51608i	0.652208	−	0.758040i	$-0.273843\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.0000i	0.0456621i
$658$	0	0
$659$	290.474i	0.440780i	0.975412	+	0.220390i	$0.0707329\pi$
$659$	290.474i	0.440780i	−0.975412	+	0.220390i	$0.929267\pi$
$660$	0	0
$661$	−968.000	−1.46445	−0.732224	−	0.681064i	$-0.761518\pi$
$661$	−968.000	−1.46445	−0.732224	+	0.681064i	$0.761518\pi$
$662$	0	0
$663$	1161.90	1.75248
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−371.806	−0.557431
$668$	0	0
$669$	60.0000	0.0896861
$670$	0	0
$671$	735.867i	1.09667i
$672$	0	0
$673$	790.000i	1.17385i	0.809642	+	0.586924i	$0.199661\pi$
$673$	790.000i	1.17385i	−0.809642	+	0.586924i	$0.800339\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	− 852.056i	− 1.25487i
$680$	0	0
$681$	−120.000	−0.176211
$682$	0	0
$683$	−468.631	−0.686136	−0.343068	−	0.939311i	$-0.611466\pi$
$683$	−468.631	−0.686136	−0.343068	+	0.939311i	$0.611466\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	263.363	0.383352
$688$	0	0
$689$	−600.000	−0.870827
$690$	0	0
$691$	− 1336.18i	− 1.93369i	−0.255365	−	0.966845i	$-0.582196\pi$
$691$	− 1336.18i	− 1.93369i	0.255365	−	0.966845i	$-0.417804\pi$
$692$	0	0
$693$	− 900.000i	− 1.29870i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 495.000i	− 0.710187i
$698$	0	0
$699$	116.190i	0.166222i
$700$	0	0
$701$	252.000	0.359486	0.179743	−	0.983714i	$-0.442473\pi$
$701$	252.000	0.359486	0.179743	+	0.983714i	$0.442473\pi$
$702$	0	0
$703$	−193.649	−0.275461
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−325.331	−0.460156
$708$	0	0
$709$	−1028.00	−1.44993	−0.724965	−	0.688786i	$-0.758144\pi$
$709$	−1028.00	−1.44993	−0.724965	+	0.688786i	$0.758144\pi$
$710$	0	0
$711$	− 232.379i	− 0.326834i
$712$	0	0
$713$	− 300.000i	− 0.420757i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	600.000i	0.836820i
$718$	0	0
$719$	− 1200.62i	− 1.66985i	−0.550361	−	0.834927i	$-0.685510\pi$
$719$	− 1200.62i	− 1.66985i	0.550361	−	0.834927i	$-0.314490\pi$
$720$	0	0
$721$	600.000	0.832178
$722$	0	0
$723$	−979.865	−1.35528
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−681.645	−0.937614	−0.468807	−	0.883301i	$-0.655316\pi$
$727$	−681.645	−0.937614	−0.468807	+	0.883301i	$0.655316\pi$
$728$	0	0
$729$	−189.000	−0.259259
$730$	0	0
$731$	− 929.516i	− 1.27157i
$732$	0	0
$733$	700.000i	0.954980i	0.878637	+	0.477490i	$0.158453\pi$
$733$	700.000i	0.954980i	−0.878637	+	0.477490i	$0.841547\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1125.00i	− 1.52646i
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	−1500.00	−2.02429
$742$	0	0
$743$	−735.867	−0.990400	−0.495200	−	0.868779i	$-0.664905\pi$
$743$	−735.867	−0.990400	−0.495200	+	0.868779i	$0.664905\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−69.7137	−0.0933249
$748$	0	0
$749$	−330.000	−0.440587
$750$	0	0
$751$	1084.44i	1.44399i	0.691899	+	0.721994i	$0.256774\pi$
$751$	1084.44i	1.44399i	−0.691899	+	0.721994i	$0.743226\pi$
$752$	0	0
$753$	1425.00i	1.89243i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 280.000i	− 0.369881i	−0.982750	−	0.184941i	$-0.940791\pi$
$757$	− 280.000i	− 0.369881i	0.982750	−	0.184941i	$-0.0592093\pi$
$758$	0	0
$759$	580.948i	0.765412i
$760$	0	0
$761$	537.000	0.705650	0.352825	−	0.935689i	$-0.385221\pi$
$761$	537.000	0.705650	0.352825	+	0.935689i	$0.385221\pi$
$762$	0	0
$763$	449.266	0.588815
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1549.19	2.01981
$768$	0	0
$769$	703.000	0.914174	0.457087	−	0.889422i	$-0.348893\pi$
$769$	703.000	0.914174	0.457087	+	0.889422i	$0.348893\pi$
$770$	0	0
$771$	− 1045.71i	− 1.35630i
$772$	0	0
$773$	− 1320.00i	− 1.70763i	−0.520574	−	0.853816i	$-0.674282\pi$
$773$	− 1320.00i	− 1.70763i	0.520574	−	0.853816i	$-0.325718\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 300.000i	− 0.386100i
$778$	0	0
$779$	639.042i	0.820337i
$780$	0	0
$781$	1500.00	1.92061
$782$	0	0
$783$	−557.710	−0.712273
$784$	0	0
$785$	0	0
$786$	0	0
$787$	681.645	0.866131	0.433065	−	0.901362i	$-0.357432\pi$
$787$	681.645	0.866131	0.433065	+	0.901362i	$0.357432\pi$
$788$	0	0
$789$	−390.000	−0.494297
$790$	0	0
$791$	− 1045.71i	− 1.32200i
$792$	0	0
$793$	760.000i	0.958386i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 270.000i	− 0.338770i	−0.985550	−	0.169385i	$-0.945822\pi$
$797$	− 270.000i	− 0.338770i	0.985550	−	0.169385i	$-0.0541782\pi$
$798$	0	0
$799$	232.379i	0.290837i
$800$	0	0
$801$	−522.000	−0.651685
$802$	0	0
$803$	96.8246	0.120579
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1208.37	1.49736
$808$	0	0
$809$	−522.000	−0.645241	−0.322621	−	0.946528i	$-0.604564\pi$
$809$	−522.000	−0.645241	−0.322621	+	0.946528i	$0.604564\pi$
$810$	0	0
$811$	− 154.919i	− 0.191023i	−0.995428	−	0.0955113i	$-0.969551\pi$
$811$	− 154.919i	− 0.191023i	0.995428	−	0.0955113i	$-0.0304486\pi$
$812$	0	0
$813$	750.000i	0.922509i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1200.00i	1.46879i
$818$	0	0
$819$	− 929.516i	− 1.13494i
$820$	0	0
$821$	282.000	0.343484	0.171742	−	0.985142i	$-0.445061\pi$
$821$	282.000	0.343484	0.171742	+	0.985142i	$0.445061\pi$
$822$	0	0
$823$	697.137	0.847068	0.423534	−	0.905880i	$-0.360789\pi$
$823$	697.137	0.847068	0.423534	+	0.905880i	$0.360789\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	600.312	0.725892	0.362946	−	0.931810i	$-0.381771\pi$
$827$	600.312	0.725892	0.362946	+	0.931810i	$0.381771\pi$
$828$	0	0
$829$	748.000	0.902292	0.451146	−	0.892450i	$-0.351015\pi$
$829$	748.000	0.902292	0.451146	+	0.892450i	$0.351015\pi$
$830$	0	0
$831$	− 1704.11i	− 2.05068i
$832$	0	0
$833$	− 165.000i	− 0.198079i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 450.000i	− 0.537634i
$838$	0	0
$839$	− 154.919i	− 0.184648i	−0.995729	−	0.0923238i	$-0.970571\pi$
$839$	− 154.919i	− 0.184648i	0.995729	−	0.0923238i	$-0.0294295\pi$
$840$	0	0
$841$	1463.00	1.73960
$842$	0	0
$843$	−766.851	−0.909669
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1967.48	−2.32288
$848$	0	0
$849$	825.000	0.971731
$850$	0	0
$851$	77.4597i	0.0910219i
$852$	0	0
$853$	1430.00i	1.67644i	0.545335	+	0.838218i	$0.316402\pi$
$853$	1430.00i	1.67644i	−0.545335	+	0.838218i	$0.683598\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 975.000i	− 1.13769i	−0.822445	−	0.568845i	$-0.807391\pi$
$857$	− 975.000i	− 1.13769i	0.822445	−	0.568845i	$-0.192609\pi$
$858$	0	0
$859$	− 445.393i	− 0.518502i	−0.965810	−	0.259251i	$-0.916524\pi$
$859$	− 445.393i	− 0.518502i	0.965810	−	0.259251i	$-0.0834757\pi$
$860$	0	0
$861$	−990.000	−1.14983
$862$	0	0
$863$	1673.13	1.93874	0.969368	−	0.245614i	$-0.0789895\pi$
$863$	1673.13	1.93874	0.969368	+	0.245614i	$0.0789895\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−247.871	−0.285895
$868$	0	0
$869$	−750.000	−0.863061
$870$	0	0
$871$	− 1161.90i	− 1.33398i
$872$	0	0
$873$	− 660.000i	− 0.756014i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	220.000i	0.250855i	0.992103	+	0.125428i	$0.0400303\pi$
$877$	220.000i	0.250855i	−0.992103	+	0.125428i	$0.959970\pi$
$878$	0	0
$879$	1510.46i	1.71839i
$880$	0	0
$881$	−1602.00	−1.81839	−0.909194	−	0.416373i	$-0.863301\pi$
$881$	−1602.00	−1.81839	−0.909194	+	0.416373i	$0.863301\pi$
$882$	0	0
$883$	1576.30	1.78517	0.892584	−	0.450880i	$-0.148890\pi$
$883$	1576.30	1.78517	0.892584	+	0.450880i	$0.148890\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	108.444	0.122259	0.0611294	−	0.998130i	$-0.480530\pi$
$887$	108.444	0.122259	0.0611294	+	0.998130i	$0.480530\pi$
$888$	0	0
$889$	−1620.00	−1.82227
$890$	0	0
$891$	1917.13i	2.15166i
$892$	0	0
$893$	− 300.000i	− 0.335946i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	600.000i	0.668896i
$898$	0	0
$899$	1859.03i	2.06789i
$900$	0	0
$901$	−450.000	−0.499445
$902$	0	0
$903$	−1859.03	−2.05873
$904$	0	0
$905$	0	0
$906$	0	0
$907$	340.823	0.375769	0.187885	−	0.982191i	$-0.439837\pi$
$907$	340.823	0.375769	0.187885	+	0.982191i	$0.439837\pi$
$908$	0	0
$909$	−252.000	−0.277228
$910$	0	0
$911$	852.056i	0.935298i	0.883914	+	0.467649i	$0.154899\pi$
$911$	852.056i	0.935298i	−0.883914	+	0.467649i	$0.845101\pi$
$912$	0	0
$913$	225.000i	0.246440i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1200.00i	1.30862i
$918$	0	0
$919$	890.786i	0.969299i	0.874708	+	0.484650i	$0.161053\pi$
$919$	890.786i	0.969299i	−0.874708	+	0.484650i	$0.838947\pi$
$920$	0	0
$921$	−1455.00	−1.57980
$922$	0	0
$923$	1549.19	1.67843
$924$	0	0
$925$	0	0
$926$	0	0
$927$	464.758	0.501357
$928$	0	0
$929$	−1518.00	−1.63402	−0.817008	−	0.576627i	$-0.804369\pi$
$929$	−1518.00	−1.63402	−0.817008	+	0.576627i	$0.804369\pi$
$930$	0	0
$931$	213.014i	0.228801i
$932$	0	0
$933$	1050.00i	1.12540i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.0000i	0.0373533i	0.999826	+	0.0186766i	$0.00594530\pi$
$937$	35.0000i	0.0373533i	−0.999826	+	0.0186766i	$0.994055\pi$
$938$	0	0
$939$	193.649i	0.206229i
$940$	0	0
$941$	468.000	0.497343	0.248672	−	0.968588i	$-0.420006\pi$
$941$	468.000	0.497343	0.248672	+	0.968588i	$0.420006\pi$
$942$	0	0
$943$	255.617	0.271068
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1533.70	1.61954	0.809768	−	0.586750i	$-0.199593\pi$
$947$	1533.70	1.61954	0.809768	+	0.586750i	$0.199593\pi$
$948$	0	0
$949$	100.000	0.105374
$950$	0	0
$951$	1161.90i	1.22176i
$952$	0	0
$953$	− 1485.00i	− 1.55824i	−0.626877	−	0.779119i	$-0.715667\pi$
$953$	− 1485.00i	− 1.55824i	0.626877	−	0.779119i	$-0.284333\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3600.00i	− 3.76176i
$958$	0	0
$959$	813.327i	0.848099i
$960$	0	0
$961$	−539.000	−0.560874
$962$	0	0
$963$	−255.617	−0.265438
$964$	0	0
$965$	0	0
$966$	0	0
$967$	402.790	0.416536	0.208268	−	0.978072i	$-0.433217\pi$
$967$	402.790	0.416536	0.208268	+	0.978072i	$0.433217\pi$
$968$	0	0
$969$	−1125.00	−1.16099
$970$	0	0
$971$	− 484.123i	− 0.498582i	−0.968429	−	0.249291i	$-0.919802\pi$
$971$	− 484.123i	− 0.498582i	0.968429	−	0.249291i	$-0.0801975\pi$
$972$	0	0
$973$	150.000i	0.154162i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1755.00i	− 1.79632i	−0.439674	−	0.898158i	$-0.644906\pi$
$977$	− 1755.00i	− 1.79632i	0.439674	−	0.898158i	$-0.355094\pi$
$978$	0	0
$979$	1684.75i	1.72089i
$980$	0	0
$981$	348.000	0.354740
$982$	0	0
$983$	−426.028	−0.433396	−0.216698	−	0.976239i	$-0.569529\pi$
$983$	−426.028	−0.433396	−0.216698	+	0.976239i	$0.569529\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	464.758	0.470879
$988$	0	0
$989$	480.000	0.485339
$990$	0	0
$991$	− 426.028i	− 0.429897i	−0.976625	−	0.214949i	$-0.931042\pi$
$991$	− 426.028i	− 0.429897i	0.976625	−	0.214949i	$-0.0689584\pi$
$992$	0	0
$993$	− 825.000i	− 0.830816i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1700.00i	− 1.70512i	−0.522633	−	0.852558i	$-0.675050\pi$
$997$	− 1700.00i	− 1.70512i	0.522633	−	0.852558i	$-0.324950\pi$
$998$	0	0
$999$	116.190i	0.116306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.h.i.1599.1		4
4.3	odd	2	inner	1600.3.h.i.1599.4		4
5.2	odd	4		1600.3.b.g.1151.2		2
5.3	odd	4		1600.3.b.h.1151.1		2
5.4	even	2	inner	1600.3.h.i.1599.3		4
8.3	odd	2		400.3.h.c.399.1		4
8.5	even	2		400.3.h.c.399.4		4
20.3	even	4		1600.3.b.h.1151.2		2
20.7	even	4		1600.3.b.g.1151.1		2
20.19	odd	2	inner	1600.3.h.i.1599.2		4
24.5	odd	2		3600.3.j.b.1999.3		4
24.11	even	2		3600.3.j.b.1999.2		4
40.3	even	4		400.3.b.c.351.1	✓	2
40.13	odd	4		400.3.b.c.351.2	yes	2
40.19	odd	2		400.3.h.c.399.3		4
40.27	even	4		400.3.b.d.351.2	yes	2
40.29	even	2		400.3.h.c.399.2		4
40.37	odd	4		400.3.b.d.351.1	yes	2
120.29	odd	2		3600.3.j.b.1999.1		4
120.53	even	4		3600.3.e.i.3151.1		2
120.59	even	2		3600.3.j.b.1999.4		4
120.77	even	4		3600.3.e.v.3151.2		2
120.83	odd	4		3600.3.e.i.3151.2		2
120.107	odd	4		3600.3.e.v.3151.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.3.b.c.351.1	✓	2	40.3	even	4
400.3.b.c.351.2	yes	2	40.13	odd	4
400.3.b.d.351.1	yes	2	40.37	odd	4
400.3.b.d.351.2	yes	2	40.27	even	4
400.3.h.c.399.1		4	8.3	odd	2
400.3.h.c.399.2		4	40.29	even	2
400.3.h.c.399.3		4	40.19	odd	2
400.3.h.c.399.4		4	8.5	even	2
1600.3.b.g.1151.1		2	20.7	even	4
1600.3.b.g.1151.2		2	5.2	odd	4
1600.3.b.h.1151.1		2	5.3	odd	4
1600.3.b.h.1151.2		2	20.3	even	4
1600.3.h.i.1599.1		4	1.1	even	1	trivial
1600.3.h.i.1599.2		4	20.19	odd	2	inner
1600.3.h.i.1599.3		4	5.4	even	2	inner
1600.3.h.i.1599.4		4	4.3	odd	2	inner
3600.3.e.i.3151.1		2	120.53	even	4
3600.3.e.i.3151.2		2	120.83	odd	4
3600.3.e.v.3151.1		2	120.107	odd	4
3600.3.e.v.3151.2		2	120.77	even	4
3600.3.j.b.1999.1		4	120.29	odd	2
3600.3.j.b.1999.2		4	24.11	even	2
3600.3.j.b.1999.3		4	24.5	odd	2
3600.3.j.b.1999.4		4	120.59	even	2

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.87298 q^{3} +7.74597 q^{7} +6.00000 q^{9} +O(q^{10})\)	\(q-3.87298 q^{3} +7.74597 q^{7} +6.00000 q^{9} -19.3649i q^{11} -20.0000i q^{13} -15.0000i q^{17} +19.3649i q^{19} -30.0000 q^{21} +7.74597 q^{23} +11.6190 q^{27} -48.0000 q^{29} -38.7298i q^{31} +75.0000i q^{33} +10.0000i q^{37} +77.4597i q^{39} +33.0000 q^{41} +61.9677 q^{43} -15.4919 q^{47} +11.0000 q^{49} +58.0948i q^{51} -30.0000i q^{53} -75.0000i q^{57} +77.4597i q^{59} -38.0000 q^{61} +46.4758 q^{63} +58.0948 q^{67} -30.0000 q^{69} +77.4597i q^{71} +5.00000i q^{73} -150.000i q^{77} -38.7298i q^{79} -99.0000 q^{81} -11.6190 q^{83} +185.903 q^{87} -87.0000 q^{89} -154.919i q^{91} +150.000i q^{93} -110.000i q^{97} -116.190i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{9}+O(q^{10}) \) 4 * q + 24 * q^9	\( 4 q + 24 q^{9} - 120 q^{21} - 192 q^{29} + 132 q^{41} + 44 q^{49} - 152 q^{61} - 120 q^{69} - 396 q^{81} - 348 q^{89}+O(q^{100}) \) 4 * q + 24 * q^9 - 120 * q^21 - 192 * q^29 + 132 * q^41 + 44 * q^49 - 152 * q^61 - 120 * q^69 - 396 * q^81 - 348 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more