LMFDB - Embedded newform 1600.3.e.a.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(799,1600)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1600.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.e (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:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1600.799
Dual form			1600.3.e.a.799.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410i q^{3} -8.00000 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-3.46410i q^{3} -8.00000 q^{7} -3.00000 q^{9} -3.46410 q^{11} -6.92820 q^{13} +6.00000i q^{17} -24.2487 q^{19} +27.7128i q^{21} +24.0000 q^{23} -20.7846i q^{27} +20.7846i q^{29} +32.0000i q^{31} +12.0000i q^{33} -6.92820 q^{37} +24.0000i q^{39} +66.0000 q^{41} -31.1769i q^{43} -48.0000 q^{47} +15.0000 q^{49} +20.7846 q^{51} +90.0666 q^{53} +84.0000i q^{57} +31.1769 q^{59} +90.0666i q^{61} +24.0000 q^{63} -79.6743i q^{67} -83.1384i q^{69} +120.000i q^{71} +58.0000i q^{73} +27.7128 q^{77} +16.0000i q^{79} -99.0000 q^{81} -58.8897i q^{83} +72.0000 q^{87} +102.000 q^{89} +55.4256 q^{91} +110.851 q^{93} -26.0000i q^{97} +10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q - 32 * q^7 - 12 * q^9	$ 4 q - 32 q^{7} - 12 q^{9} + 96 q^{23} + 264 q^{41} - 192 q^{47} + 60 q^{49} + 96 q^{63} - 396 q^{81} + 288 q^{87} + 408 q^{89}+O(q^{100}) $ 4 * q - 32 * q^7 - 12 * q^9 + 96 * q^23 + 264 * q^41 - 192 * q^47 + 60 * q^49 + 96 * q^63 - 396 * q^81 + 288 * q^87 + 408 * 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
$3$	− 3.46410i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 3.46410i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−8.00000	−1.14286	−0.571429	−	0.820652i	$-0.693611\pi$
$7$	−8.00000	−1.14286	−0.571429	+	0.820652i	$0.693611\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−3.46410	−0.314918	−0.157459	−	0.987525i	$-0.550330\pi$
$11$	−3.46410	−0.314918	−0.157459	+	0.987525i	$0.550330\pi$
$12$	0	0
$13$	−6.92820	−0.532939	−0.266469	−	0.963843i	$-0.585857\pi$
$13$	−6.92820	−0.532939	−0.266469	+	0.963843i	$0.585857\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	0.352941i	0.984306	+	0.176471i	$0.0564680\pi$
$17$	6.00000i	0.352941i	−0.984306	+	0.176471i	$0.943532\pi$
$18$	0	0
$19$	−24.2487	−1.27625	−0.638124	−	0.769934i	$-0.720289\pi$
$19$	−24.2487	−1.27625	−0.638124	+	0.769934i	$0.720289\pi$
$20$	0	0
$21$	27.7128i	1.31966i
$22$	0	0
$23$	24.0000	1.04348	0.521739	−	0.853105i	$-0.325283\pi$
$23$	24.0000	1.04348	0.521739	+	0.853105i	$0.325283\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 20.7846i	− 0.769800i
$28$	0	0
$29$	20.7846i	0.716711i	0.933585	+	0.358355i	$0.116662\pi$
$29$	20.7846i	0.716711i	−0.933585	+	0.358355i	$0.883338\pi$
$30$	0	0
$31$	32.0000i	1.03226i	0.856511	+	0.516129i	$0.172628\pi$
$31$	32.0000i	1.03226i	−0.856511	+	0.516129i	$0.827372\pi$
$32$	0	0
$33$	12.0000i	0.363636i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.92820	−0.187249	−0.0936244	−	0.995608i	$-0.529845\pi$
$37$	−6.92820	−0.187249	−0.0936244	+	0.995608i	$0.529845\pi$
$38$	0	0
$39$	24.0000i	0.615385i
$40$	0	0
$41$	66.0000	1.60976	0.804878	−	0.593440i	$-0.202231\pi$
$41$	66.0000	1.60976	0.804878	+	0.593440i	$0.202231\pi$
$42$	0	0
$43$	− 31.1769i	− 0.725045i	−0.931975	−	0.362522i	$-0.881916\pi$
$43$	− 31.1769i	− 0.725045i	0.931975	−	0.362522i	$-0.118084\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−48.0000	−1.02128	−0.510638	−	0.859796i	$-0.670591\pi$
$47$	−48.0000	−1.02128	−0.510638	+	0.859796i	$0.670591\pi$
$48$	0	0
$49$	15.0000	0.306122
$50$	0	0
$51$	20.7846	0.407541
$52$	0	0
$53$	90.0666	1.69937	0.849685	−	0.527290i	$-0.176792\pi$
$53$	90.0666	1.69937	0.849685	+	0.527290i	$0.176792\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	84.0000i	1.47368i
$58$	0	0
$59$	31.1769	0.528422	0.264211	−	0.964465i	$-0.414888\pi$
$59$	31.1769	0.528422	0.264211	+	0.964465i	$0.414888\pi$
$60$	0	0
$61$	90.0666i	1.47650i	0.674526	+	0.738251i	$0.264348\pi$
$61$	90.0666i	1.47650i	−0.674526	+	0.738251i	$0.735652\pi$
$62$	0	0
$63$	24.0000	0.380952
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 79.6743i	− 1.18917i	−0.804033	−	0.594585i	$-0.797317\pi$
$67$	− 79.6743i	− 1.18917i	0.804033	−	0.594585i	$-0.202683\pi$
$68$	0	0
$69$	− 83.1384i	− 1.20490i
$70$	0	0
$71$	120.000i	1.69014i	0.534655	+	0.845070i	$0.320442\pi$
$71$	120.000i	1.69014i	−0.534655	+	0.845070i	$0.679558\pi$
$72$	0	0
$73$	58.0000i	0.794521i	0.917706	+	0.397260i	$0.130039\pi$
$73$	58.0000i	0.794521i	−0.917706	+	0.397260i	$0.869961\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	27.7128	0.359907
$78$	0	0
$79$	16.0000i	0.202532i	0.994859	+	0.101266i	$0.0322893\pi$
$79$	16.0000i	0.202532i	−0.994859	+	0.101266i	$0.967711\pi$
$80$	0	0
$81$	−99.0000	−1.22222
$82$	0	0
$83$	− 58.8897i	− 0.709515i	−0.934958	−	0.354757i	$-0.884563\pi$
$83$	− 58.8897i	− 0.709515i	0.934958	−	0.354757i	$-0.115437\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	72.0000	0.827586
$88$	0	0
$89$	102.000	1.14607	0.573034	−	0.819532i	$-0.305766\pi$
$89$	102.000	1.14607	0.573034	+	0.819532i	$0.305766\pi$
$90$	0	0
$91$	55.4256	0.609073
$92$	0	0
$93$	110.851	1.19195
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 26.0000i	− 0.268041i	−0.990979	−	0.134021i	$-0.957211\pi$
$97$	− 26.0000i	− 0.268041i	0.990979	−	0.134021i	$-0.0427888\pi$
$98$	0	0
$99$	10.3923	0.104973
$100$	0	0
$101$	− 6.92820i	− 0.0685961i	−0.999412	−	0.0342980i	$-0.989080\pi$
$101$	− 6.92820i	− 0.0685961i	0.999412	−	0.0342980i	$-0.0109195\pi$
$102$	0	0
$103$	40.0000	0.388350	0.194175	−	0.980967i	$-0.437797\pi$
$103$	40.0000	0.388350	0.194175	+	0.980967i	$0.437797\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	58.8897i	0.550371i	0.961391	+	0.275186i	$0.0887393\pi$
$107$	58.8897i	0.550371i	−0.961391	+	0.275186i	$0.911261\pi$
$108$	0	0
$109$	103.923i	0.953422i	0.879060	+	0.476711i	$0.158171\pi$
$109$	103.923i	0.953422i	−0.879060	+	0.476711i	$0.841829\pi$
$110$	0	0
$111$	24.0000i	0.216216i
$112$	0	0
$113$	66.0000i	0.584071i	0.956408	+	0.292035i	$0.0943325\pi$
$113$	66.0000i	0.584071i	−0.956408	+	0.292035i	$0.905667\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	20.7846	0.177646
$118$	0	0
$119$	− 48.0000i	− 0.403361i
$120$	0	0
$121$	−109.000	−0.900826
$122$	0	0
$123$	− 228.631i	− 1.85879i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	64.0000	0.503937	0.251969	−	0.967735i	$-0.418922\pi$
$127$	64.0000	0.503937	0.251969	+	0.967735i	$0.418922\pi$
$128$	0	0
$129$	−108.000	−0.837209
$130$	0	0
$131$	245.951	1.87749	0.938745	−	0.344612i	$-0.111990\pi$
$131$	245.951	1.87749	0.938745	+	0.344612i	$0.111990\pi$
$132$	0	0
$133$	193.990	1.45857
$134$	0	0
$135$	0	0
$136$	0	0
$137$	30.0000i	0.218978i	0.993988	+	0.109489i	$0.0349215\pi$
$137$	30.0000i	0.218978i	−0.993988	+	0.109489i	$0.965079\pi$
$138$	0	0
$139$	−79.6743	−0.573197	−0.286598	−	0.958051i	$-0.592525\pi$
$139$	−79.6743	−0.573197	−0.286598	+	0.958051i	$0.592525\pi$
$140$	0	0
$141$	166.277i	1.17927i
$142$	0	0
$143$	24.0000	0.167832
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 51.9615i	− 0.353480i
$148$	0	0
$149$	− 131.636i	− 0.883462i	−0.897148	−	0.441731i	$-0.854365\pi$
$149$	− 131.636i	− 0.883462i	0.897148	−	0.441731i	$-0.145635\pi$
$150$	0	0
$151$	− 152.000i	− 1.00662i	−0.864105	−	0.503311i	$-0.832115\pi$
$151$	− 152.000i	− 1.00662i	0.864105	−	0.503311i	$-0.167885\pi$
$152$	0	0
$153$	− 18.0000i	− 0.117647i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	145.492	0.926702	0.463351	−	0.886175i	$-0.346647\pi$
$157$	145.492	0.926702	0.463351	+	0.886175i	$0.346647\pi$
$158$	0	0
$159$	− 312.000i	− 1.96226i
$160$	0	0
$161$	−192.000	−1.19255
$162$	0	0
$163$	− 169.741i	− 1.04136i	−0.853753	−	0.520678i	$-0.825679\pi$
$163$	− 169.741i	− 1.04136i	0.853753	−	0.520678i	$-0.174321\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	216.000	1.29341	0.646707	−	0.762739i	$-0.276146\pi$
$167$	216.000	1.29341	0.646707	+	0.762739i	$0.276146\pi$
$168$	0	0
$169$	−121.000	−0.715976
$170$	0	0
$171$	72.7461	0.425416
$172$	0	0
$173$	−173.205	−1.00119	−0.500593	−	0.865683i	$-0.666885\pi$
$173$	−173.205	−1.00119	−0.500593	+	0.865683i	$0.666885\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 108.000i	− 0.610169i
$178$	0	0
$179$	−107.387	−0.599928	−0.299964	−	0.953950i	$-0.596975\pi$
$179$	−107.387	−0.599928	−0.299964	+	0.953950i	$0.596975\pi$
$180$	0	0
$181$	76.2102i	0.421051i	0.977588	+	0.210526i	$0.0675175\pi$
$181$	76.2102i	0.421051i	−0.977588	+	0.210526i	$0.932482\pi$
$182$	0	0
$183$	312.000	1.70492
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.7846i	− 0.111148i
$188$	0	0
$189$	166.277i	0.879772i
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	218.000i	1.12953i	0.825251	+	0.564767i	$0.191034\pi$
$193$	218.000i	1.12953i	−0.825251	+	0.564767i	$0.808966\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	214.774	1.09022	0.545112	−	0.838363i	$-0.316487\pi$
$197$	214.774	1.09022	0.545112	+	0.838363i	$0.316487\pi$
$198$	0	0
$199$	328.000i	1.64824i	0.566414	+	0.824121i	$0.308330\pi$
$199$	328.000i	1.64824i	−0.566414	+	0.824121i	$0.691670\pi$
$200$	0	0
$201$	−276.000	−1.37313
$202$	0	0
$203$	− 166.277i	− 0.819098i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−72.0000	−0.347826
$208$	0	0
$209$	84.0000	0.401914
$210$	0	0
$211$	107.387	0.508944	0.254472	−	0.967080i	$-0.418098\pi$
$211$	107.387	0.508944	0.254472	+	0.967080i	$0.418098\pi$
$212$	0	0
$213$	415.692	1.95161
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 256.000i	− 1.17972i
$218$	0	0
$219$	200.918	0.917433
$220$	0	0
$221$	− 41.5692i	− 0.188096i
$222$	0	0
$223$	32.0000	0.143498	0.0717489	−	0.997423i	$-0.477142\pi$
$223$	32.0000	0.143498	0.0717489	+	0.997423i	$0.477142\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	114.315i	0.503592i	0.967780	+	0.251796i	$0.0810212\pi$
$227$	114.315i	0.503592i	−0.967780	+	0.251796i	$0.918979\pi$
$228$	0	0
$229$	− 270.200i	− 1.17991i	−0.807435	−	0.589956i	$-0.799145\pi$
$229$	− 270.200i	− 1.17991i	0.807435	−	0.589956i	$-0.200855\pi$
$230$	0	0
$231$	− 96.0000i	− 0.415584i
$232$	0	0
$233$	186.000i	0.798283i	0.916889	+	0.399142i	$0.130692\pi$
$233$	186.000i	0.798283i	−0.916889	+	0.399142i	$0.869308\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	55.4256	0.233863
$238$	0	0
$239$	48.0000i	0.200837i	0.994945	+	0.100418i	$0.0320181\pi$
$239$	48.0000i	0.200837i	−0.994945	+	0.100418i	$0.967982\pi$
$240$	0	0
$241$	250.000	1.03734	0.518672	−	0.854973i	$-0.326427\pi$
$241$	250.000	1.03734	0.518672	+	0.854973i	$0.326427\pi$
$242$	0	0
$243$	155.885i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	168.000	0.680162
$248$	0	0
$249$	−204.000	−0.819277
$250$	0	0
$251$	301.377	1.20070	0.600352	−	0.799736i	$-0.295027\pi$
$251$	301.377	1.20070	0.600352	+	0.799736i	$0.295027\pi$
$252$	0	0
$253$	−83.1384	−0.328610
$254$	0	0
$255$	0	0
$256$	0	0
$257$	222.000i	0.863813i	0.901918	+	0.431907i	$0.142159\pi$
$257$	222.000i	0.863813i	−0.901918	+	0.431907i	$0.857841\pi$
$258$	0	0
$259$	55.4256	0.213999
$260$	0	0
$261$	− 62.3538i	− 0.238904i
$262$	0	0
$263$	−312.000	−1.18631	−0.593156	−	0.805088i	$-0.702118\pi$
$263$	−312.000	−1.18631	−0.593156	+	0.805088i	$0.702118\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 353.338i	− 1.32336i
$268$	0	0
$269$	− 62.3538i	− 0.231799i	−0.993261	−	0.115899i	$-0.963025\pi$
$269$	− 62.3538i	− 0.231799i	0.993261	−	0.115899i	$-0.0369750\pi$
$270$	0	0
$271$	112.000i	0.413284i	0.978417	+	0.206642i	$0.0662536\pi$
$271$	112.000i	0.413284i	−0.978417	+	0.206642i	$0.933746\pi$
$272$	0	0
$273$	− 192.000i	− 0.703297i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−145.492	−0.525243	−0.262621	−	0.964899i	$-0.584587\pi$
$277$	−145.492	−0.525243	−0.262621	+	0.964899i	$0.584587\pi$
$278$	0	0
$279$	− 96.0000i	− 0.344086i
$280$	0	0
$281$	90.0000	0.320285	0.160142	−	0.987094i	$-0.448805\pi$
$281$	90.0000	0.320285	0.160142	+	0.987094i	$0.448805\pi$
$282$	0	0
$283$	218.238i	0.771160i	0.922674	+	0.385580i	$0.125999\pi$
$283$	218.238i	0.771160i	−0.922674	+	0.385580i	$0.874001\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−528.000	−1.83972
$288$	0	0
$289$	253.000	0.875433
$290$	0	0
$291$	−90.0666	−0.309507
$292$	0	0
$293$	62.3538	0.212812	0.106406	−	0.994323i	$-0.466066\pi$
$293$	62.3538	0.212812	0.106406	+	0.994323i	$0.466066\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	72.0000i	0.242424i
$298$	0	0
$299$	−166.277	−0.556110
$300$	0	0
$301$	249.415i	0.828622i
$302$	0	0
$303$	−24.0000	−0.0792079
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 135.100i	− 0.440065i	−0.975493	−	0.220033i	$-0.929384\pi$
$307$	− 135.100i	− 0.440065i	0.975493	−	0.220033i	$-0.0706163\pi$
$308$	0	0
$309$	− 138.564i	− 0.448427i
$310$	0	0
$311$	264.000i	0.848875i	0.905457	+	0.424437i	$0.139528\pi$
$311$	264.000i	0.848875i	−0.905457	+	0.424437i	$0.860472\pi$
$312$	0	0
$313$	226.000i	0.722045i	0.932557	+	0.361022i	$0.117572\pi$
$313$	226.000i	0.722045i	−0.932557	+	0.361022i	$0.882428\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−408.764	−1.28948	−0.644738	−	0.764404i	$-0.723034\pi$
$317$	−408.764	−1.28948	−0.644738	+	0.764404i	$0.723034\pi$
$318$	0	0
$319$	− 72.0000i	− 0.225705i
$320$	0	0
$321$	204.000	0.635514
$322$	0	0
$323$	− 145.492i	− 0.450440i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	360.000	1.10092
$328$	0	0
$329$	384.000	1.16717
$330$	0	0
$331$	−502.295	−1.51751	−0.758753	−	0.651378i	$-0.774191\pi$
$331$	−502.295	−1.51751	−0.758753	+	0.651378i	$0.774191\pi$
$332$	0	0
$333$	20.7846	0.0624162
$334$	0	0
$335$	0	0
$336$	0	0
$337$	382.000i	1.13353i	0.823879	+	0.566766i	$0.191805\pi$
$337$	382.000i	1.13353i	−0.823879	+	0.566766i	$0.808195\pi$
$338$	0	0
$339$	228.631	0.674427
$340$	0	0
$341$	− 110.851i	− 0.325077i
$342$	0	0
$343$	272.000	0.793003
$344$	0	0
$345$	0	0
$346$	0	0
$347$	640.859i	1.84686i	0.383773	+	0.923428i	$0.374625\pi$
$347$	640.859i	1.84686i	−0.383773	+	0.923428i	$0.625375\pi$
$348$	0	0
$349$	464.190i	1.33006i	0.746818	+	0.665028i	$0.231580\pi$
$349$	464.190i	1.33006i	−0.746818	+	0.665028i	$0.768420\pi$
$350$	0	0
$351$	144.000i	0.410256i
$352$	0	0
$353$	− 318.000i	− 0.900850i	−0.892814	−	0.450425i	$-0.851272\pi$
$353$	− 318.000i	− 0.900850i	0.892814	−	0.450425i	$-0.148728\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−166.277	−0.465762
$358$	0	0
$359$	552.000i	1.53760i	0.639487	+	0.768802i	$0.279147\pi$
$359$	552.000i	1.53760i	−0.639487	+	0.768802i	$0.720853\pi$
$360$	0	0
$361$	227.000	0.628809
$362$	0	0
$363$	377.587i	1.04018i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−496.000	−1.35150	−0.675749	−	0.737132i	$-0.736180\pi$
$367$	−496.000	−1.35150	−0.675749	+	0.737132i	$0.736180\pi$
$368$	0	0
$369$	−198.000	−0.536585
$370$	0	0
$371$	−720.533	−1.94214
$372$	0	0
$373$	−575.041	−1.54166	−0.770832	−	0.637038i	$-0.780159\pi$
$373$	−575.041	−1.54166	−0.770832	+	0.637038i	$0.780159\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 144.000i	− 0.381963i
$378$	0	0
$379$	−245.951	−0.648948	−0.324474	−	0.945895i	$-0.605187\pi$
$379$	−245.951	−0.648948	−0.324474	+	0.945895i	$0.605187\pi$
$380$	0	0
$381$	− 221.703i	− 0.581896i
$382$	0	0
$383$	−384.000	−1.00261	−0.501305	−	0.865270i	$-0.667147\pi$
$383$	−384.000	−1.00261	−0.501305	+	0.865270i	$0.667147\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	93.5307i	0.241682i
$388$	0	0
$389$	284.056i	0.730222i	0.930964	+	0.365111i	$0.118969\pi$
$389$	284.056i	0.730222i	−0.930964	+	0.365111i	$0.881031\pi$
$390$	0	0
$391$	144.000i	0.368286i
$392$	0	0
$393$	− 852.000i	− 2.16794i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	394.908	0.994729	0.497365	−	0.867542i	$-0.334301\pi$
$397$	394.908	0.994729	0.497365	+	0.867542i	$0.334301\pi$
$398$	0	0
$399$	− 672.000i	− 1.68421i
$400$	0	0
$401$	378.000	0.942643	0.471322	−	0.881961i	$-0.343777\pi$
$401$	378.000	0.942643	0.471322	+	0.881961i	$0.343777\pi$
$402$	0	0
$403$	− 221.703i	− 0.550130i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.0000	0.0589681
$408$	0	0
$409$	−34.0000	−0.0831296	−0.0415648	−	0.999136i	$-0.513234\pi$
$409$	−34.0000	−0.0831296	−0.0415648	+	0.999136i	$0.513234\pi$
$410$	0	0
$411$	103.923	0.252854
$412$	0	0
$413$	−249.415	−0.603911
$414$	0	0
$415$	0	0
$416$	0	0
$417$	276.000i	0.661871i
$418$	0	0
$419$	−301.377	−0.719276	−0.359638	−	0.933092i	$-0.617100\pi$
$419$	−301.377	−0.719276	−0.359638	+	0.933092i	$0.617100\pi$
$420$	0	0
$421$	48.4974i	0.115196i	0.998340	+	0.0575979i	$0.0183441\pi$
$421$	48.4974i	0.115196i	−0.998340	+	0.0575979i	$0.981656\pi$
$422$	0	0
$423$	144.000	0.340426
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 720.533i	− 1.68743i
$428$	0	0
$429$	− 83.1384i	− 0.193796i
$430$	0	0
$431$	144.000i	0.334107i	0.985948	+	0.167053i	$0.0534252\pi$
$431$	144.000i	0.334107i	−0.985948	+	0.167053i	$0.946575\pi$
$432$	0	0
$433$	− 134.000i	− 0.309469i	−0.987956	−	0.154734i	$-0.950548\pi$
$433$	− 134.000i	− 0.309469i	0.987956	−	0.154734i	$-0.0494522\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−581.969	−1.33174
$438$	0	0
$439$	376.000i	0.856492i	0.903662	+	0.428246i	$0.140868\pi$
$439$	376.000i	0.856492i	−0.903662	+	0.428246i	$0.859132\pi$
$440$	0	0
$441$	−45.0000	−0.102041
$442$	0	0
$443$	384.515i	0.867980i	0.900918	+	0.433990i	$0.142895\pi$
$443$	384.515i	0.867980i	−0.900918	+	0.433990i	$0.857105\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−456.000	−1.02013
$448$	0	0
$449$	102.000	0.227171	0.113586	−	0.993528i	$-0.463766\pi$
$449$	102.000	0.227171	0.113586	+	0.993528i	$0.463766\pi$
$450$	0	0
$451$	−228.631	−0.506942
$452$	0	0
$453$	−526.543	−1.16235
$454$	0	0
$455$	0	0
$456$	0	0
$457$	254.000i	0.555799i	0.960610	+	0.277899i	$0.0896382\pi$
$457$	254.000i	0.555799i	−0.960610	+	0.277899i	$0.910362\pi$
$458$	0	0
$459$	124.708	0.271694
$460$	0	0
$461$	− 48.4974i	− 0.105200i	−0.998616	−	0.0526002i	$-0.983249\pi$
$461$	− 48.4974i	− 0.105200i	0.998616	−	0.0526002i	$-0.0167509\pi$
$462$	0	0
$463$	−880.000	−1.90065	−0.950324	−	0.311263i	$-0.899248\pi$
$463$	−880.000	−1.90065	−0.950324	+	0.311263i	$0.899248\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	142.028i	0.304129i	0.988371	+	0.152064i	$0.0485921\pi$
$467$	142.028i	0.304129i	−0.988371	+	0.152064i	$0.951408\pi$
$468$	0	0
$469$	637.395i	1.35905i
$470$	0	0
$471$	− 504.000i	− 1.07006i
$472$	0	0
$473$	108.000i	0.228330i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−270.200	−0.566457
$478$	0	0
$479$	− 480.000i	− 1.00209i	−0.865422	−	0.501044i	$-0.832950\pi$
$479$	− 480.000i	− 1.00209i	0.865422	−	0.501044i	$-0.167050\pi$
$480$	0	0
$481$	48.0000	0.0997921
$482$	0	0
$483$	665.108i	1.37703i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	280.000	0.574949	0.287474	−	0.957788i	$-0.407184\pi$
$487$	280.000	0.574949	0.287474	+	0.957788i	$0.407184\pi$
$488$	0	0
$489$	−588.000	−1.20245
$490$	0	0
$491$	−751.710	−1.53098	−0.765489	−	0.643449i	$-0.777503\pi$
$491$	−751.710	−1.53098	−0.765489	+	0.643449i	$0.777503\pi$
$492$	0	0
$493$	−124.708	−0.252957
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 960.000i	− 1.93159i
$498$	0	0
$499$	280.592	0.562309	0.281155	−	0.959663i	$-0.409283\pi$
$499$	280.592	0.562309	0.281155	+	0.959663i	$0.409283\pi$
$500$	0	0
$501$	− 748.246i	− 1.49350i
$502$	0	0
$503$	120.000	0.238569	0.119284	−	0.992860i	$-0.461940\pi$
$503$	120.000	0.238569	0.119284	+	0.992860i	$0.461940\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	419.156i	0.826738i
$508$	0	0
$509$	− 533.472i	− 1.04808i	−0.851694	−	0.524039i	$-0.824425\pi$
$509$	− 533.472i	− 1.04808i	0.851694	−	0.524039i	$-0.175575\pi$
$510$	0	0
$511$	− 464.000i	− 0.908023i
$512$	0	0
$513$	504.000i	0.982456i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	166.277	0.321619
$518$	0	0
$519$	600.000i	1.15607i
$520$	0	0
$521$	354.000	0.679463	0.339731	−	0.940523i	$-0.389664\pi$
$521$	354.000	0.679463	0.339731	+	0.940523i	$0.389664\pi$
$522$	0	0
$523$	162.813i	0.311305i	0.987812	+	0.155653i	$0.0497481\pi$
$523$	162.813i	0.311305i	−0.987812	+	0.155653i	$0.950252\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−192.000	−0.364326
$528$	0	0
$529$	47.0000	0.0888469
$530$	0	0
$531$	−93.5307	−0.176141
$532$	0	0
$533$	−457.261	−0.857901
$534$	0	0
$535$	0	0
$536$	0	0
$537$	372.000i	0.692737i
$538$	0	0
$539$	−51.9615	−0.0964036
$540$	0	0
$541$	− 1073.87i	− 1.98498i	−0.122346	−	0.992488i	$-0.539042\pi$
$541$	− 1073.87i	− 1.98498i	0.122346	−	0.992488i	$-0.460958\pi$
$542$	0	0
$543$	264.000	0.486188
$544$	0	0
$545$	0	0
$546$	0	0
$547$	917.987i	1.67822i	0.543961	+	0.839111i	$0.316924\pi$
$547$	917.987i	1.67822i	−0.543961	+	0.839111i	$0.683076\pi$
$548$	0	0
$549$	− 270.200i	− 0.492167i
$550$	0	0
$551$	− 504.000i	− 0.914701i
$552$	0	0
$553$	− 128.000i	− 0.231465i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−381.051	−0.684113	−0.342057	−	0.939679i	$-0.611123\pi$
$557$	−381.051	−0.684113	−0.342057	+	0.939679i	$0.611123\pi$
$558$	0	0
$559$	216.000i	0.386404i
$560$	0	0
$561$	−72.0000	−0.128342
$562$	0	0
$563$	911.059i	1.61822i	0.587656	+	0.809111i	$0.300051\pi$
$563$	911.059i	1.61822i	−0.587656	+	0.809111i	$0.699949\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	792.000	1.39683
$568$	0	0
$569$	702.000	1.23374	0.616872	−	0.787064i	$-0.288400\pi$
$569$	702.000	1.23374	0.616872	+	0.787064i	$0.288400\pi$
$570$	0	0
$571$	938.772	1.64408	0.822042	−	0.569427i	$-0.192835\pi$
$571$	938.772	1.64408	0.822042	+	0.569427i	$0.192835\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 802.000i	− 1.38995i	−0.719035	−	0.694974i	$-0.755416\pi$
$577$	− 802.000i	− 1.38995i	0.719035	−	0.694974i	$-0.244584\pi$
$578$	0	0
$579$	755.174	1.30427
$580$	0	0
$581$	471.118i	0.810874i
$582$	0	0
$583$	−312.000	−0.535163
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 107.387i	− 0.182942i	−0.995808	−	0.0914712i	$-0.970843\pi$
$587$	− 107.387i	− 0.182942i	0.995808	−	0.0914712i	$-0.0291569\pi$
$588$	0	0
$589$	− 775.959i	− 1.31742i
$590$	0	0
$591$	− 744.000i	− 1.25888i
$592$	0	0
$593$	− 798.000i	− 1.34570i	−0.739779	−	0.672850i	$-0.765070\pi$
$593$	− 798.000i	− 1.34570i	0.739779	−	0.672850i	$-0.234930\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1136.23	1.90323
$598$	0	0
$599$	24.0000i	0.0400668i	0.999799	+	0.0200334i	$0.00637725\pi$
$599$	24.0000i	0.0400668i	−0.999799	+	0.0200334i	$0.993623\pi$
$600$	0	0
$601$	218.000	0.362729	0.181364	−	0.983416i	$-0.441949\pi$
$601$	218.000	0.362729	0.181364	+	0.983416i	$0.441949\pi$
$602$	0	0
$603$	239.023i	0.396390i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−608.000	−1.00165	−0.500824	−	0.865549i	$-0.666970\pi$
$607$	−608.000	−1.00165	−0.500824	+	0.865549i	$0.666970\pi$
$608$	0	0
$609$	−576.000	−0.945813
$610$	0	0
$611$	332.554	0.544278
$612$	0	0
$613$	−381.051	−0.621617	−0.310808	−	0.950473i	$-0.600600\pi$
$613$	−381.051	−0.621617	−0.310808	+	0.950473i	$0.600600\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 954.000i	− 1.54619i	−0.634289	−	0.773096i	$-0.718707\pi$
$617$	− 954.000i	− 1.54619i	0.634289	−	0.773096i	$-0.281293\pi$
$618$	0	0
$619$	225.167	0.363759	0.181879	−	0.983321i	$-0.441782\pi$
$619$	225.167	0.363759	0.181879	+	0.983321i	$0.441782\pi$
$620$	0	0
$621$	− 498.831i	− 0.803270i
$622$	0	0
$623$	−816.000	−1.30979
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 290.985i	− 0.464090i
$628$	0	0
$629$	− 41.5692i	− 0.0660878i
$630$	0	0
$631$	− 248.000i	− 0.393027i	−0.980501	−	0.196513i	$-0.937038\pi$
$631$	− 248.000i	− 0.393027i	0.980501	−	0.196513i	$-0.0629619\pi$
$632$	0	0
$633$	− 372.000i	− 0.587678i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−103.923	−0.163145
$638$	0	0
$639$	− 360.000i	− 0.563380i
$640$	0	0
$641$	−1062.00	−1.65679	−0.828393	−	0.560147i	$-0.810745\pi$
$641$	−1062.00	−1.65679	−0.828393	+	0.560147i	$0.810745\pi$
$642$	0	0
$643$	301.377i	0.468704i	0.972152	+	0.234352i	$0.0752969\pi$
$643$	301.377i	0.468704i	−0.972152	+	0.234352i	$0.924703\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	696.000	1.07573	0.537867	−	0.843030i	$-0.319230\pi$
$647$	696.000	1.07573	0.537867	+	0.843030i	$0.319230\pi$
$648$	0	0
$649$	−108.000	−0.166410
$650$	0	0
$651$	−886.810	−1.36223
$652$	0	0
$653$	−505.759	−0.774516	−0.387258	−	0.921971i	$-0.626578\pi$
$653$	−505.759	−0.774516	−0.387258	+	0.921971i	$0.626578\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 174.000i	− 0.264840i
$658$	0	0
$659$	−938.772	−1.42454	−0.712270	−	0.701906i	$-0.752333\pi$
$659$	−938.772	−1.42454	−0.712270	+	0.701906i	$0.752333\pi$
$660$	0	0
$661$	− 200.918i	− 0.303961i	−0.988384	−	0.151980i	$-0.951435\pi$
$661$	− 200.918i	− 0.303961i	0.988384	−	0.151980i	$-0.0485650\pi$
$662$	0	0
$663$	−144.000	−0.217195
$664$	0	0
$665$	0	0
$666$	0	0
$667$	498.831i	0.747872i
$668$	0	0
$669$	− 110.851i	− 0.165697i
$670$	0	0
$671$	− 312.000i	− 0.464978i
$672$	0	0
$673$	730.000i	1.08470i	0.840154	+	0.542348i	$0.182464\pi$
$673$	730.000i	1.08470i	−0.840154	+	0.542348i	$0.817536\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−893.738	−1.32015	−0.660073	−	0.751202i	$-0.729474\pi$
$677$	−893.738	−1.32015	−0.660073	+	0.751202i	$0.729474\pi$
$678$	0	0
$679$	208.000i	0.306333i
$680$	0	0
$681$	396.000	0.581498
$682$	0	0
$683$	467.654i	0.684705i	0.939572	+	0.342353i	$0.111224\pi$
$683$	467.654i	0.684705i	−0.939572	+	0.342353i	$0.888776\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−936.000	−1.36245
$688$	0	0
$689$	−624.000	−0.905660
$690$	0	0
$691$	−446.869	−0.646699	−0.323350	−	0.946280i	$-0.604809\pi$
$691$	−446.869	−0.646699	−0.323350	+	0.946280i	$0.604809\pi$
$692$	0	0
$693$	−83.1384	−0.119969
$694$	0	0
$695$	0	0
$696$	0	0
$697$	396.000i	0.568149i
$698$	0	0
$699$	644.323	0.921778
$700$	0	0
$701$	− 131.636i	− 0.187783i	−0.995582	−	0.0938915i	$-0.970069\pi$
$701$	− 131.636i	− 0.187783i	0.995582	−	0.0938915i	$-0.0299307\pi$
$702$	0	0
$703$	168.000	0.238976
$704$	0	0
$705$	0	0
$706$	0	0
$707$	55.4256i	0.0783955i
$708$	0	0
$709$	1281.72i	1.80778i	0.427763	+	0.903891i	$0.359302\pi$
$709$	1281.72i	1.80778i	−0.427763	+	0.903891i	$0.640698\pi$
$710$	0	0
$711$	− 48.0000i	− 0.0675105i
$712$	0	0
$713$	768.000i	1.07714i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	166.277	0.231906
$718$	0	0
$719$	− 1008.00i	− 1.40195i	−0.713187	−	0.700974i	$-0.752749\pi$
$719$	− 1008.00i	− 1.40195i	0.713187	−	0.700974i	$-0.247251\pi$
$720$	0	0
$721$	−320.000	−0.443828
$722$	0	0
$723$	− 866.025i	− 1.19782i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1384.00	1.90371	0.951857	−	0.306543i	$-0.0991722\pi$
$727$	1384.00	1.90371	0.951857	+	0.306543i	$0.0991722\pi$
$728$	0	0
$729$	−351.000	−0.481481
$730$	0	0
$731$	187.061	0.255898
$732$	0	0
$733$	1018.45	1.38942	0.694711	−	0.719289i	$-0.255532\pi$
$733$	1018.45	1.38942	0.694711	+	0.719289i	$0.255532\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	276.000i	0.374491i
$738$	0	0
$739$	446.869	0.604694	0.302347	−	0.953198i	$-0.402230\pi$
$739$	446.869	0.604694	0.302347	+	0.953198i	$0.402230\pi$
$740$	0	0
$741$	− 581.969i	− 0.785383i
$742$	0	0
$743$	744.000	1.00135	0.500673	−	0.865637i	$-0.333086\pi$
$743$	744.000	1.00135	0.500673	+	0.865637i	$0.333086\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	176.669i	0.236505i
$748$	0	0
$749$	− 471.118i	− 0.628996i
$750$	0	0
$751$	848.000i	1.12916i	0.825378	+	0.564581i	$0.190962\pi$
$751$	848.000i	1.12916i	−0.825378	+	0.564581i	$0.809038\pi$
$752$	0	0
$753$	− 1044.00i	− 1.38645i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	297.913	0.393544	0.196772	−	0.980449i	$-0.436954\pi$
$757$	297.913	0.393544	0.196772	+	0.980449i	$0.436954\pi$
$758$	0	0
$759$	288.000i	0.379447i
$760$	0	0
$761$	−414.000	−0.544021	−0.272011	−	0.962294i	$-0.587689\pi$
$761$	−414.000	−0.544021	−0.272011	+	0.962294i	$0.587689\pi$
$762$	0	0
$763$	− 831.384i	− 1.08963i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−216.000	−0.281617
$768$	0	0
$769$	358.000	0.465540	0.232770	−	0.972532i	$-0.425221\pi$
$769$	358.000	0.465540	0.232770	+	0.972532i	$0.425221\pi$
$770$	0	0
$771$	769.031	0.997446
$772$	0	0
$773$	−1267.86	−1.64018	−0.820091	−	0.572233i	$-0.806077\pi$
$773$	−1267.86	−1.64018	−0.820091	+	0.572233i	$0.806077\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 192.000i	− 0.247104i
$778$	0	0
$779$	−1600.41	−2.05445
$780$	0	0
$781$	− 415.692i	− 0.532256i
$782$	0	0
$783$	432.000	0.551724
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 772.495i	− 0.981569i	−0.871281	−	0.490784i	$-0.836710\pi$
$787$	− 772.495i	− 0.981569i	0.871281	−	0.490784i	$-0.163290\pi$
$788$	0	0
$789$	1080.80i	1.36983i
$790$	0	0
$791$	− 528.000i	− 0.667509i
$792$	0	0
$793$	− 624.000i	− 0.786885i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−76.2102	−0.0956214	−0.0478107	−	0.998856i	$-0.515224\pi$
$797$	−76.2102	−0.0956214	−0.0478107	+	0.998856i	$0.515224\pi$
$798$	0	0
$799$	− 288.000i	− 0.360451i
$800$	0	0
$801$	−306.000	−0.382022
$802$	0	0
$803$	− 200.918i	− 0.250209i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−216.000	−0.267658
$808$	0	0
$809$	702.000	0.867738	0.433869	−	0.900976i	$-0.357148\pi$
$809$	702.000	0.867738	0.433869	+	0.900976i	$0.357148\pi$
$810$	0	0
$811$	−446.869	−0.551010	−0.275505	−	0.961300i	$-0.588845\pi$
$811$	−446.869	−0.551010	−0.275505	+	0.961300i	$0.588845\pi$
$812$	0	0
$813$	387.979	0.477219
$814$	0	0
$815$	0	0
$816$	0	0
$817$	756.000i	0.925337i
$818$	0	0
$819$	−166.277	−0.203024
$820$	0	0
$821$	1351.00i	1.64555i	0.568365	+	0.822777i	$0.307576\pi$
$821$	1351.00i	1.64555i	−0.568365	+	0.822777i	$0.692424\pi$
$822$	0	0
$823$	−520.000	−0.631835	−0.315917	−	0.948787i	$-0.602312\pi$
$823$	−520.000	−0.631835	−0.315917	+	0.948787i	$0.602312\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1111.98i	1.34459i	0.740283	+	0.672295i	$0.234692\pi$
$827$	1111.98i	1.34459i	−0.740283	+	0.672295i	$0.765308\pi$
$828$	0	0
$829$	− 200.918i	− 0.242362i	−0.992630	−	0.121181i	$-0.961332\pi$
$829$	− 200.918i	− 0.242362i	0.992630	−	0.121181i	$-0.0386681\pi$
$830$	0	0
$831$	504.000i	0.606498i
$832$	0	0
$833$	90.0000i	0.108043i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	665.108	0.794633
$838$	0	0
$839$	72.0000i	0.0858164i	0.999079	+	0.0429082i	$0.0136623\pi$
$839$	72.0000i	0.0858164i	−0.999079	+	0.0429082i	$0.986338\pi$
$840$	0	0
$841$	409.000	0.486326
$842$	0	0
$843$	− 311.769i	− 0.369833i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	872.000	1.02952
$848$	0	0
$849$	756.000	0.890459
$850$	0	0
$851$	−166.277	−0.195390
$852$	0	0
$853$	256.344	0.300520	0.150260	−	0.988647i	$-0.451989\pi$
$853$	256.344	0.300520	0.150260	+	0.988647i	$0.451989\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	510.000i	0.595099i	0.954706	+	0.297550i	$0.0961694\pi$
$857$	510.000i	0.595099i	−0.954706	+	0.297550i	$0.903831\pi$
$858$	0	0
$859$	973.413	1.13319	0.566596	−	0.823995i	$-0.308260\pi$
$859$	973.413	1.13319	0.566596	+	0.823995i	$0.308260\pi$
$860$	0	0
$861$	1829.05i	2.12433i
$862$	0	0
$863$	−1248.00	−1.44612	−0.723059	−	0.690786i	$-0.757265\pi$
$863$	−1248.00	−1.44612	−0.723059	+	0.690786i	$0.757265\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 876.418i	− 1.01086i
$868$	0	0
$869$	− 55.4256i	− 0.0637809i
$870$	0	0
$871$	552.000i	0.633754i
$872$	0	0
$873$	78.0000i	0.0893471i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	450.333	0.513493	0.256746	−	0.966479i	$-0.417349\pi$
$877$	450.333	0.513493	0.256746	+	0.966479i	$0.417349\pi$
$878$	0	0
$879$	− 216.000i	− 0.245734i
$880$	0	0
$881$	1218.00	1.38252	0.691260	−	0.722606i	$-0.257056\pi$
$881$	1218.00	1.38252	0.691260	+	0.722606i	$0.257056\pi$
$882$	0	0
$883$	− 252.879i	− 0.286387i	−0.989695	−	0.143193i	$-0.954263\pi$
$883$	− 252.879i	− 0.286387i	0.989695	−	0.143193i	$-0.0457371\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1464.00	−1.65051	−0.825254	−	0.564762i	$-0.808968\pi$
$887$	−1464.00	−1.65051	−0.825254	+	0.564762i	$0.808968\pi$
$888$	0	0
$889$	−512.000	−0.575928
$890$	0	0
$891$	342.946	0.384900
$892$	0	0
$893$	1163.94	1.30340
$894$	0	0
$895$	0	0
$896$	0	0
$897$	576.000i	0.642140i
$898$	0	0
$899$	−665.108	−0.739830
$900$	0	0
$901$	540.400i	0.599778i
$902$	0	0
$903$	864.000	0.956811
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1243.61i	− 1.37113i	−0.728013	−	0.685564i	$-0.759556\pi$
$907$	− 1243.61i	− 1.37113i	0.728013	−	0.685564i	$-0.240444\pi$
$908$	0	0
$909$	20.7846i	0.0228654i
$910$	0	0
$911$	624.000i	0.684962i	0.939525	+	0.342481i	$0.111267\pi$
$911$	624.000i	0.684962i	−0.939525	+	0.342481i	$0.888733\pi$
$912$	0	0
$913$	204.000i	0.223439i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1967.61	−2.14570
$918$	0	0
$919$	1304.00i	1.41893i	0.704739	+	0.709467i	$0.251064\pi$
$919$	1304.00i	1.41893i	−0.704739	+	0.709467i	$0.748936\pi$
$920$	0	0
$921$	−468.000	−0.508143
$922$	0	0
$923$	− 831.384i	− 0.900741i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−120.000	−0.129450
$928$	0	0
$929$	486.000	0.523143	0.261572	−	0.965184i	$-0.415759\pi$
$929$	486.000	0.523143	0.261572	+	0.965184i	$0.415759\pi$
$930$	0	0
$931$	−363.731	−0.390688
$932$	0	0
$933$	914.523	0.980196
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1094.00i	1.16756i	0.811913	+	0.583778i	$0.198426\pi$
$937$	1094.00i	1.16756i	−0.811913	+	0.583778i	$0.801574\pi$
$938$	0	0
$939$	782.887	0.833745
$940$	0	0
$941$	− 214.774i	− 0.228240i	−0.993467	−	0.114120i	$-0.963595\pi$
$941$	− 214.774i	− 0.228240i	0.993467	−	0.114120i	$-0.0364049\pi$
$942$	0	0
$943$	1584.00	1.67975
$944$	0	0
$945$	0	0
$946$	0	0
$947$	640.859i	0.676725i	0.941016	+	0.338363i	$0.109873\pi$
$947$	640.859i	0.676725i	−0.941016	+	0.338363i	$0.890127\pi$
$948$	0	0
$949$	− 401.836i	− 0.423431i
$950$	0	0
$951$	1416.00i	1.48896i
$952$	0	0
$953$	1218.00i	1.27807i	0.769178	+	0.639035i	$0.220666\pi$
$953$	1218.00i	1.27807i	−0.769178	+	0.639035i	$0.779334\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−249.415	−0.260622
$958$	0	0
$959$	− 240.000i	− 0.250261i
$960$	0	0
$961$	−63.0000	−0.0655567
$962$	0	0
$963$	− 176.669i	− 0.183457i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56.0000	0.0579111	0.0289555	−	0.999581i	$-0.490782\pi$
$967$	56.0000	0.0579111	0.0289555	+	0.999581i	$0.490782\pi$
$968$	0	0
$969$	−504.000	−0.520124
$970$	0	0
$971$	−86.6025	−0.0891890	−0.0445945	−	0.999005i	$-0.514200\pi$
$971$	−86.6025	−0.0891890	−0.0445945	+	0.999005i	$0.514200\pi$
$972$	0	0
$973$	637.395	0.655082
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 570.000i	− 0.583419i	−0.956507	−	0.291709i	$-0.905776\pi$
$977$	− 570.000i	− 0.583419i	0.956507	−	0.291709i	$-0.0942240\pi$
$978$	0	0
$979$	−353.338	−0.360918
$980$	0	0
$981$	− 311.769i	− 0.317807i
$982$	0	0
$983$	792.000	0.805697	0.402848	−	0.915267i	$-0.368020\pi$
$983$	792.000	0.805697	0.402848	+	0.915267i	$0.368020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1330.22i	− 1.34774i
$988$	0	0
$989$	− 748.246i	− 0.756568i
$990$	0	0
$991$	1184.00i	1.19475i	0.801961	+	0.597376i	$0.203790\pi$
$991$	1184.00i	1.19475i	−0.801961	+	0.597376i	$0.796210\pi$
$992$	0	0
$993$	1740.00i	1.75227i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	769.031	0.771345	0.385672	−	0.922636i	$-0.373970\pi$
$997$	769.031	0.771345	0.385672	+	0.922636i	$0.373970\pi$
$998$	0	0
$999$	144.000i	0.144144i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.e.a.799.1		4
4.3	odd	2		1600.3.e.f.799.4		4
5.2	odd	4		64.3.d.a.31.1	✓	4
5.3	odd	4		1600.3.g.c.351.4		4
5.4	even	2		1600.3.e.f.799.3		4
8.3	odd	2		1600.3.e.f.799.1		4
8.5	even	2	inner	1600.3.e.a.799.4		4
15.2	even	4		576.3.b.d.415.3		4
20.3	even	4		1600.3.g.c.351.1		4
20.7	even	4		64.3.d.a.31.3	yes	4
20.19	odd	2	inner	1600.3.e.a.799.2		4
40.3	even	4		1600.3.g.c.351.3		4
40.13	odd	4		1600.3.g.c.351.2		4
40.19	odd	2	inner	1600.3.e.a.799.3		4
40.27	even	4		64.3.d.a.31.2	yes	4
40.29	even	2		1600.3.e.f.799.2		4
40.37	odd	4		64.3.d.a.31.4	yes	4
60.47	odd	4		576.3.b.d.415.4		4
80.27	even	4		256.3.c.g.255.2		4
80.37	odd	4		256.3.c.g.255.4		4
80.67	even	4		256.3.c.g.255.3		4
80.77	odd	4		256.3.c.g.255.1		4
120.77	even	4		576.3.b.d.415.1		4
120.107	odd	4		576.3.b.d.415.2		4
240.77	even	4		2304.3.g.u.1279.4		4
240.107	odd	4		2304.3.g.u.1279.1		4
240.197	even	4		2304.3.g.u.1279.2		4
240.227	odd	4		2304.3.g.u.1279.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.3.d.a.31.1	✓	4	5.2	odd	4
64.3.d.a.31.2	yes	4	40.27	even	4
64.3.d.a.31.3	yes	4	20.7	even	4
64.3.d.a.31.4	yes	4	40.37	odd	4
256.3.c.g.255.1		4	80.77	odd	4
256.3.c.g.255.2		4	80.27	even	4
256.3.c.g.255.3		4	80.67	even	4
256.3.c.g.255.4		4	80.37	odd	4
576.3.b.d.415.1		4	120.77	even	4
576.3.b.d.415.2		4	120.107	odd	4
576.3.b.d.415.3		4	15.2	even	4
576.3.b.d.415.4		4	60.47	odd	4
1600.3.e.a.799.1		4	1.1	even	1	trivial
1600.3.e.a.799.2		4	20.19	odd	2	inner
1600.3.e.a.799.3		4	40.19	odd	2	inner
1600.3.e.a.799.4		4	8.5	even	2	inner
1600.3.e.f.799.1		4	8.3	odd	2
1600.3.e.f.799.2		4	40.29	even	2
1600.3.e.f.799.3		4	5.4	even	2
1600.3.e.f.799.4		4	4.3	odd	2
1600.3.g.c.351.1		4	20.3	even	4
1600.3.g.c.351.2		4	40.13	odd	4
1600.3.g.c.351.3		4	40.3	even	4
1600.3.g.c.351.4		4	5.3	odd	4
2304.3.g.u.1279.1		4	240.107	odd	4
2304.3.g.u.1279.2		4	240.197	even	4
2304.3.g.u.1279.3		4	240.227	odd	4
2304.3.g.u.1279.4		4	240.77	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.46410i q^{3} -8.00000 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-3.46410i q^{3} -8.00000 q^{7} -3.00000 q^{9} -3.46410 q^{11} -6.92820 q^{13} +6.00000i q^{17} -24.2487 q^{19} +27.7128i q^{21} +24.0000 q^{23} -20.7846i q^{27} +20.7846i q^{29} +32.0000i q^{31} +12.0000i q^{33} -6.92820 q^{37} +24.0000i q^{39} +66.0000 q^{41} -31.1769i q^{43} -48.0000 q^{47} +15.0000 q^{49} +20.7846 q^{51} +90.0666 q^{53} +84.0000i q^{57} +31.1769 q^{59} +90.0666i q^{61} +24.0000 q^{63} -79.6743i q^{67} -83.1384i q^{69} +120.000i q^{71} +58.0000i q^{73} +27.7128 q^{77} +16.0000i q^{79} -99.0000 q^{81} -58.8897i q^{83} +72.0000 q^{87} +102.000 q^{89} +55.4256 q^{91} +110.851 q^{93} -26.0000i q^{97} +10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q - 32 * q^7 - 12 * q^9	\( 4 q - 32 q^{7} - 12 q^{9} + 96 q^{23} + 264 q^{41} - 192 q^{47} + 60 q^{49} + 96 q^{63} - 396 q^{81} + 288 q^{87} + 408 q^{89}+O(q^{100}) \) 4 * q - 32 * q^7 - 12 * q^9 + 96 * q^23 + 264 * q^41 - 192 * q^47 + 60 * q^49 + 96 * q^63 - 396 * q^81 + 288 * q^87 + 408 * q^89

Introduction

L-functions

Modular forms

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Database

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