LMFDB - Embedded newform 1764.4.f.b.881.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(881,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.f (of order $2$, degree $1$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.19
Character	$\chi$	$=$	1764.881
Dual form			1764.4.f.b.881.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.5703 q^{5} +O(q^{10})$	$q+13.5703 q^{5} -11.1126i q^{11} -25.5525i q^{13} -23.3148 q^{17} -53.1670i q^{19} -122.037i q^{23} +59.1529 q^{25} -43.3847i q^{29} +131.731i q^{31} -357.062 q^{37} +91.5050 q^{41} -395.464 q^{43} +94.4712 q^{47} -29.1725i q^{53} -150.801i q^{55} -727.585 q^{59} -728.461i q^{61} -346.755i q^{65} -37.2642 q^{67} -179.807i q^{71} +97.9041i q^{73} -123.029 q^{79} +217.139 q^{83} -316.388 q^{85} -158.542 q^{89} -721.492i q^{95} +597.401i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 888 q^{25} + 864 q^{37} - 1248 q^{43} + 1056 q^{67} - 8064 q^{85}+O(q^{100}) $ 24 * q + 888 * q^25 + 864 * q^37 - 1248 * q^43 + 1056 * q^67 - 8064 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	13.5703	1.21376	0.606882	−	0.794792i	$-0.292420\pi$
$5$	13.5703	1.21376	0.606882	+	0.794792i	$0.292420\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 11.1126i	− 0.304597i	−0.988335	−	0.152299i	$-0.951332\pi$
$11$	− 11.1126i	− 0.304597i	0.988335	−	0.152299i	$-0.0486676\pi$
$12$	0	0
$13$	− 25.5525i	− 0.545154i	−0.962134	−	0.272577i	$-0.912124\pi$
$13$	− 25.5525i	− 0.545154i	0.962134	−	0.272577i	$-0.0878759\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−23.3148	−0.332627	−0.166313	−	0.986073i	$-0.553186\pi$
$17$	−23.3148	−0.332627	−0.166313	+	0.986073i	$0.553186\pi$
$18$	0	0
$19$	− 53.1670i	− 0.641966i	−0.947085	−	0.320983i	$-0.895987\pi$
$19$	− 53.1670i	− 0.641966i	0.947085	−	0.320983i	$-0.104013\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 122.037i	− 1.10637i	−0.833058	−	0.553185i	$-0.813412\pi$
$23$	− 122.037i	− 1.10637i	0.833058	−	0.553185i	$-0.186588\pi$
$24$	0	0
$25$	59.1529	0.473223
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 43.3847i	− 0.277804i	−0.990306	−	0.138902i	$-0.955643\pi$
$29$	− 43.3847i	− 0.277804i	0.990306	−	0.138902i	$-0.0443574\pi$
$30$	0	0
$31$	131.731i	0.763212i	0.924325	+	0.381606i	$0.124629\pi$
$31$	131.731i	0.763212i	−0.924325	+	0.381606i	$0.875371\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−357.062	−1.58650	−0.793252	−	0.608894i	$-0.791614\pi$
$37$	−357.062	−1.58650	−0.793252	+	0.608894i	$0.791614\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	91.5050	0.348553	0.174277	−	0.984697i	$-0.444241\pi$
$41$	91.5050	0.348553	0.174277	+	0.984697i	$0.444241\pi$
$42$	0	0
$43$	−395.464	−1.40250	−0.701252	−	0.712914i	$-0.747375\pi$
$43$	−395.464	−1.40250	−0.701252	+	0.712914i	$0.747375\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	94.4712	0.293192	0.146596	−	0.989196i	$-0.453168\pi$
$47$	94.4712	0.293192	0.146596	+	0.989196i	$0.453168\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 29.1725i	− 0.0756067i	−0.999285	−	0.0378033i	$-0.987964\pi$
$53$	− 29.1725i	− 0.0756067i	0.999285	−	0.0378033i	$-0.0120360\pi$
$54$	0	0
$55$	− 150.801i	− 0.369709i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−727.585	−1.60548	−0.802742	−	0.596327i	$-0.796626\pi$
$59$	−727.585	−1.60548	−0.802742	+	0.596327i	$0.796626\pi$
$60$	0	0
$61$	− 728.461i	− 1.52901i	−0.644616	−	0.764507i	$-0.722983\pi$
$61$	− 728.461i	− 1.52901i	0.644616	−	0.764507i	$-0.277017\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 346.755i	− 0.661688i
$66$	0	0
$67$	−37.2642	−0.0679485	−0.0339742	−	0.999423i	$-0.510816\pi$
$67$	−37.2642	−0.0679485	−0.0339742	+	0.999423i	$0.510816\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 179.807i	− 0.300552i	−0.988644	−	0.150276i	$-0.951984\pi$
$71$	− 179.807i	− 0.300552i	0.988644	−	0.150276i	$-0.0480162\pi$
$72$	0	0
$73$	97.9041i	0.156970i	0.996915	+	0.0784850i	$0.0250083\pi$
$73$	97.9041i	0.156970i	−0.996915	+	0.0784850i	$0.974992\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−123.029	−0.175213	−0.0876066	−	0.996155i	$-0.527922\pi$
$79$	−123.029	−0.175213	−0.0876066	+	0.996155i	$0.527922\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	217.139	0.287158	0.143579	−	0.989639i	$-0.454139\pi$
$83$	217.139	0.287158	0.143579	+	0.989639i	$0.454139\pi$
$84$	0	0
$85$	−316.388	−0.403731
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−158.542	−0.188825	−0.0944126	−	0.995533i	$-0.530097\pi$
$89$	−158.542	−0.188825	−0.0944126	+	0.995533i	$0.530097\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 721.492i	− 0.779195i
$96$	0	0
$97$	597.401i	0.625329i	0.949864	+	0.312664i	$0.101221\pi$
$97$	597.401i	0.625329i	−0.949864	+	0.312664i	$0.898779\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−697.188	−0.686860	−0.343430	−	0.939178i	$-0.611589\pi$
$101$	−697.188	−0.686860	−0.343430	+	0.939178i	$0.611589\pi$
$102$	0	0
$103$	351.001i	0.335779i	0.985806	+	0.167889i	$0.0536951\pi$
$103$	351.001i	0.335779i	−0.985806	+	0.167889i	$0.946305\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 420.132i	− 0.379586i	−0.981824	−	0.189793i	$-0.939218\pi$
$107$	− 420.132i	− 0.379586i	0.981824	−	0.189793i	$-0.0607817\pi$
$108$	0	0
$109$	1427.80	1.25467	0.627333	−	0.778751i	$-0.284147\pi$
$109$	1427.80	1.25467	0.627333	+	0.778751i	$0.284147\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2036.67i	1.69552i	0.530377	+	0.847762i	$0.322050\pi$
$113$	2036.67i	1.69552i	−0.530377	+	0.847762i	$0.677950\pi$
$114$	0	0
$115$	− 1656.08i	− 1.34287i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1207.51	0.907220
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−893.565	−0.639383
$126$	0	0
$127$	−509.829	−0.356220	−0.178110	−	0.984011i	$-0.556998\pi$
$127$	−509.829	−0.356220	−0.178110	+	0.984011i	$0.556998\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2922.27	−1.94901	−0.974505	−	0.224366i	$-0.927969\pi$
$131$	−2922.27	−1.94901	−0.974505	+	0.224366i	$0.927969\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1583.48i	− 0.987490i	−0.869607	−	0.493745i	$-0.835628\pi$
$137$	− 1583.48i	− 0.987490i	0.869607	−	0.493745i	$-0.164372\pi$
$138$	0	0
$139$	− 1130.29i	− 0.689714i	−0.938655	−	0.344857i	$-0.887927\pi$
$139$	− 1130.29i	− 0.689714i	0.938655	−	0.344857i	$-0.112073\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−283.955	−0.166052
$144$	0	0
$145$	− 588.743i	− 0.337189i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1622.15i	0.891892i	0.895060	+	0.445946i	$0.147133\pi$
$149$	1622.15i	0.891892i	−0.895060	+	0.445946i	$0.852867\pi$
$150$	0	0
$151$	−356.740	−0.192259	−0.0961294	−	0.995369i	$-0.530646\pi$
$151$	−356.740	−0.192259	−0.0961294	+	0.995369i	$0.530646\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1787.63i	0.926360i
$156$	0	0
$157$	− 3397.39i	− 1.72701i	−0.504336	−	0.863507i	$-0.668263\pi$
$157$	− 3397.39i	− 1.72701i	0.504336	−	0.863507i	$-0.331737\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2873.36	1.38073	0.690366	−	0.723461i	$-0.257450\pi$
$163$	2873.36	1.38073	0.690366	+	0.723461i	$0.257450\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1733.75	0.803364	0.401682	−	0.915779i	$-0.368426\pi$
$167$	1733.75	0.803364	0.401682	+	0.915779i	$0.368426\pi$
$168$	0	0
$169$	1544.07	0.702807
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1238.61	−0.544333	−0.272167	−	0.962250i	$-0.587740\pi$
$173$	−1238.61	−0.544333	−0.272167	+	0.962250i	$0.587740\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3113.43i	− 1.30005i	−0.759914	−	0.650023i	$-0.774759\pi$
$179$	− 3113.43i	− 1.30005i	0.759914	−	0.650023i	$-0.225241\pi$
$180$	0	0
$181$	2749.21i	1.12899i	0.825437	+	0.564495i	$0.190929\pi$
$181$	2749.21i	1.12899i	−0.825437	+	0.564495i	$0.809071\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4845.44	−1.92564
$186$	0	0
$187$	259.087i	0.101317i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 425.392i	− 0.161153i	−0.996748	−	0.0805767i	$-0.974324\pi$
$191$	− 425.392i	− 0.161153i	0.996748	−	0.0805767i	$-0.0256762\pi$
$192$	0	0
$193$	2826.75	1.05427	0.527135	−	0.849782i	$-0.323266\pi$
$193$	2826.75	1.05427	0.527135	+	0.849782i	$0.323266\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4763.08i	1.72262i	0.508083	+	0.861308i	$0.330354\pi$
$197$	4763.08i	1.72262i	−0.508083	+	0.861308i	$0.669646\pi$
$198$	0	0
$199$	− 4312.96i	− 1.53637i	−0.640228	−	0.768185i	$-0.721160\pi$
$199$	− 4312.96i	− 1.53637i	0.640228	−	0.768185i	$-0.278840\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1241.75	0.423061
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−590.823	−0.195541
$210$	0	0
$211$	−4948.35	−1.61450	−0.807248	−	0.590213i	$-0.799044\pi$
$211$	−4948.35	−1.61450	−0.807248	+	0.590213i	$0.799044\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5366.56	−1.70231
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	595.751i	0.181333i
$222$	0	0
$223$	314.818i	0.0945370i	0.998882	+	0.0472685i	$0.0150516\pi$
$223$	314.818i	0.0945370i	−0.998882	+	0.0472685i	$0.984948\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4321.88	−1.26367	−0.631836	−	0.775102i	$-0.717698\pi$
$227$	−4321.88	−1.26367	−0.631836	+	0.775102i	$0.717698\pi$
$228$	0	0
$229$	− 3079.54i	− 0.888653i	−0.895865	−	0.444327i	$-0.853443\pi$
$229$	− 3079.54i	− 0.888653i	0.895865	−	0.444327i	$-0.146557\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3599.12i	− 1.01196i	−0.862546	−	0.505979i	$-0.831131\pi$
$233$	− 3599.12i	− 1.01196i	0.862546	−	0.505979i	$-0.168869\pi$
$234$	0	0
$235$	1282.00	0.355866
$236$	0	0
$237$	0	0
$238$	0	0
$239$	242.461i	0.0656213i	0.999462	+	0.0328107i	$0.0104458\pi$
$239$	242.461i	0.0656213i	−0.999462	+	0.0328107i	$0.989554\pi$
$240$	0	0
$241$	− 311.346i	− 0.0832180i	−0.999134	−	0.0416090i	$-0.986752\pi$
$241$	− 311.346i	− 0.0832180i	0.999134	−	0.0416090i	$-0.0132484\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1358.55	−0.349970
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1863.72	−0.468672	−0.234336	−	0.972156i	$-0.575292\pi$
$251$	−1863.72	−0.468672	−0.234336	+	0.972156i	$0.575292\pi$
$252$	0	0
$253$	−1356.15	−0.336998
$254$	0	0
$255$	0	0
$256$	0	0
$257$	333.924	0.0810490	0.0405245	−	0.999179i	$-0.487097\pi$
$257$	333.924	0.0810490	0.0405245	+	0.999179i	$0.487097\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 7255.52i	− 1.70112i	−0.525878	−	0.850560i	$-0.676263\pi$
$263$	− 7255.52i	− 1.70112i	0.525878	−	0.850560i	$-0.323737\pi$
$264$	0	0
$265$	− 395.880i	− 0.0917687i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5713.44	−1.29500	−0.647499	−	0.762066i	$-0.724185\pi$
$269$	−5713.44	−1.29500	−0.647499	+	0.762066i	$0.724185\pi$
$270$	0	0
$271$	− 6508.50i	− 1.45891i	−0.684031	−	0.729453i	$-0.739775\pi$
$271$	− 6508.50i	− 1.45891i	0.684031	−	0.729453i	$-0.260225\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 657.342i	− 0.144143i
$276$	0	0
$277$	5970.52	1.29507	0.647533	−	0.762037i	$-0.275801\pi$
$277$	5970.52	1.29507	0.647533	+	0.762037i	$0.275801\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4469.94i	0.948947i	0.880270	+	0.474474i	$0.157362\pi$
$281$	4469.94i	0.948947i	−0.880270	+	0.474474i	$0.842638\pi$
$282$	0	0
$283$	− 552.245i	− 0.115998i	−0.998317	−	0.0579992i	$-0.981528\pi$
$283$	− 552.245i	− 0.115998i	0.998317	−	0.0579992i	$-0.0184721\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4369.42	−0.889359
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5834.87	1.16340	0.581701	−	0.813403i	$-0.302387\pi$
$293$	5834.87	1.16340	0.581701	+	0.813403i	$0.302387\pi$
$294$	0	0
$295$	−9873.54	−1.94868
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3118.36	−0.603142
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 9885.43i	− 1.85586i
$306$	0	0
$307$	− 6605.19i	− 1.22794i	−0.789329	−	0.613971i	$-0.789571\pi$
$307$	− 6605.19i	− 1.22794i	0.789329	−	0.613971i	$-0.210429\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8914.19	−1.62533	−0.812665	−	0.582731i	$-0.801984\pi$
$311$	−8914.19	−1.62533	−0.812665	+	0.582731i	$0.801984\pi$
$312$	0	0
$313$	− 5495.12i	− 0.992341i	−0.868225	−	0.496170i	$-0.834739\pi$
$313$	− 5495.12i	− 0.992341i	0.868225	−	0.496170i	$-0.165261\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6462.99i	− 1.14510i	−0.819869	−	0.572552i	$-0.805954\pi$
$317$	− 6462.99i	− 1.14510i	0.819869	−	0.572552i	$-0.194046\pi$
$318$	0	0
$319$	−482.116	−0.0846185
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1239.58i	0.213535i
$324$	0	0
$325$	− 1511.51i	− 0.257979i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	714.857	0.118707	0.0593536	−	0.998237i	$-0.481096\pi$
$331$	714.857	0.118707	0.0593536	+	0.998237i	$0.481096\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−505.686	−0.0824734
$336$	0	0
$337$	−2098.15	−0.339151	−0.169575	−	0.985517i	$-0.554240\pi$
$337$	−2098.15	−0.339151	−0.169575	+	0.985517i	$0.554240\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1463.87	0.232473
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10802.4i	1.67119i	0.549345	+	0.835596i	$0.314877\pi$
$347$	10802.4i	1.67119i	−0.549345	+	0.835596i	$0.685123\pi$
$348$	0	0
$349$	4256.02i	0.652778i	0.945236	+	0.326389i	$0.105832\pi$
$349$	4256.02i	0.652778i	−0.945236	+	0.326389i	$0.894168\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5704.48	−0.860110	−0.430055	−	0.902803i	$-0.641506\pi$
$353$	−5704.48	−0.860110	−0.430055	+	0.902803i	$0.641506\pi$
$354$	0	0
$355$	− 2440.03i	− 0.364799i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 11701.3i	− 1.72026i	−0.510075	−	0.860130i	$-0.670382\pi$
$359$	− 11701.3i	− 1.72026i	0.510075	−	0.860130i	$-0.329618\pi$
$360$	0	0
$361$	4032.27	0.587880
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1328.59i	0.190524i
$366$	0	0
$367$	− 3695.78i	− 0.525663i	−0.964842	−	0.262831i	$-0.915344\pi$
$367$	− 3695.78i	− 0.525663i	0.964842	−	0.262831i	$-0.0846563\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4977.24	−0.690917	−0.345458	−	0.938434i	$-0.612277\pi$
$373$	−4977.24	−0.690917	−0.345458	+	0.938434i	$0.612277\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1108.59	−0.151446
$378$	0	0
$379$	263.270	0.0356815	0.0178408	−	0.999841i	$-0.494321\pi$
$379$	263.270	0.0356815	0.0178408	+	0.999841i	$0.494321\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7559.77	−1.00858	−0.504290	−	0.863535i	$-0.668246\pi$
$383$	−7559.77	−1.00858	−0.504290	+	0.863535i	$0.668246\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 7164.72i	− 0.933845i	−0.884298	−	0.466923i	$-0.845363\pi$
$389$	− 7164.72i	− 0.933845i	0.884298	−	0.466923i	$-0.154637\pi$
$390$	0	0
$391$	2845.27i	0.368008i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1669.54	−0.212668
$396$	0	0
$397$	2927.33i	0.370072i	0.982732	+	0.185036i	$0.0592402\pi$
$397$	2927.33i	0.370072i	−0.982732	+	0.185036i	$0.940760\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13841.7i	1.72374i	0.507128	+	0.861871i	$0.330707\pi$
$401$	13841.7i	1.72374i	−0.507128	+	0.861871i	$0.669293\pi$
$402$	0	0
$403$	3366.06	0.416068
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3967.88i	0.483245i
$408$	0	0
$409$	− 6284.10i	− 0.759728i	−0.925042	−	0.379864i	$-0.875971\pi$
$409$	− 6284.10i	− 0.759728i	0.925042	−	0.379864i	$-0.124029\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2946.64	0.348542
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4835.26	0.563766	0.281883	−	0.959449i	$-0.409041\pi$
$419$	4835.26	0.563766	0.281883	+	0.959449i	$0.409041\pi$
$420$	0	0
$421$	−4348.01	−0.503346	−0.251673	−	0.967812i	$-0.580981\pi$
$421$	−4348.01	−0.503346	−0.251673	+	0.967812i	$0.580981\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1379.13	−0.157407
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 6991.36i	− 0.781350i	−0.920529	−	0.390675i	$-0.872242\pi$
$431$	− 6991.36i	− 0.781350i	0.920529	−	0.390675i	$-0.127758\pi$
$432$	0	0
$433$	2274.87i	0.252479i	0.992000	+	0.126240i	$0.0402908\pi$
$433$	2274.87i	0.252479i	−0.992000	+	0.126240i	$0.959709\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6488.35	−0.710251
$438$	0	0
$439$	− 4831.70i	− 0.525295i	−0.964892	−	0.262648i	$-0.915404\pi$
$439$	− 4831.70i	− 0.525295i	0.964892	−	0.262648i	$-0.0845957\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3370.27i	0.361459i	0.983533	+	0.180729i	$0.0578458\pi$
$443$	3370.27i	0.361459i	−0.983533	+	0.180729i	$0.942154\pi$
$444$	0	0
$445$	−2151.46	−0.229189
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4674.68i	0.491340i	0.969353	+	0.245670i	$0.0790080\pi$
$449$	4674.68i	0.491340i	−0.969353	+	0.245670i	$0.920992\pi$
$450$	0	0
$451$	− 1016.86i	− 0.106168i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11140.1	1.14029	0.570146	−	0.821544i	$-0.306887\pi$
$457$	11140.1	1.14029	0.570146	+	0.821544i	$0.306887\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3295.95	−0.332989	−0.166494	−	0.986042i	$-0.553245\pi$
$461$	−3295.95	−0.332989	−0.166494	+	0.986042i	$0.553245\pi$
$462$	0	0
$463$	16451.6	1.65134	0.825670	−	0.564154i	$-0.190797\pi$
$463$	16451.6	1.65134	0.825670	+	0.564154i	$0.190797\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2461.96	−0.243953	−0.121976	−	0.992533i	$-0.538923\pi$
$467$	−2461.96	−0.243953	−0.121976	+	0.992533i	$0.538923\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4394.63i	0.427199i
$474$	0	0
$475$	− 3144.98i	− 0.303793i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14271.7	1.36136	0.680680	−	0.732581i	$-0.261684\pi$
$479$	14271.7	1.36136	0.680680	+	0.732581i	$0.261684\pi$
$480$	0	0
$481$	9123.84i	0.864888i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8106.91i	0.759001i
$486$	0	0
$487$	−7993.90	−0.743815	−0.371908	−	0.928270i	$-0.621296\pi$
$487$	−7993.90	−0.743815	−0.371908	+	0.928270i	$0.621296\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2326.48i	0.213834i	0.994268	+	0.106917i	$0.0340980\pi$
$491$	2326.48i	0.213834i	−0.994268	+	0.106917i	$0.965902\pi$
$492$	0	0
$493$	1011.50i	0.0924052i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15164.2	1.36041	0.680203	−	0.733023i	$-0.261891\pi$
$499$	15164.2	1.36041	0.680203	+	0.733023i	$0.261891\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17052.7	−1.51161	−0.755807	−	0.654794i	$-0.772755\pi$
$503$	−17052.7	−1.51161	−0.755807	+	0.654794i	$0.772755\pi$
$504$	0	0
$505$	−9461.05	−0.833686
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1886.95	−0.164317	−0.0821585	−	0.996619i	$-0.526181\pi$
$509$	−1886.95	−0.164317	−0.0821585	+	0.996619i	$0.526181\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4763.19i	0.407556i
$516$	0	0
$517$	− 1049.82i	− 0.0893056i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20873.7	1.75527	0.877635	−	0.479330i	$-0.159120\pi$
$521$	20873.7	1.75527	0.877635	+	0.479330i	$0.159120\pi$
$522$	0	0
$523$	6233.27i	0.521151i	0.965454	+	0.260575i	$0.0839122\pi$
$523$	6233.27i	0.521151i	−0.965454	+	0.260575i	$0.916088\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3071.28i	− 0.253865i
$528$	0	0
$529$	−2726.07	−0.224054
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2338.18i	− 0.190015i
$534$	0	0
$535$	− 5701.32i	− 0.460728i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−916.558	−0.0728390	−0.0364195	−	0.999337i	$-0.511595\pi$
$541$	−916.558	−0.0728390	−0.0364195	+	0.999337i	$0.511595\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19375.7	1.52287
$546$	0	0
$547$	14238.3	1.11295	0.556476	−	0.830864i	$-0.312153\pi$
$547$	14238.3	1.11295	0.556476	+	0.830864i	$0.312153\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2306.63	−0.178341
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11294.7i	− 0.859195i	−0.903021	−	0.429597i	$-0.858655\pi$
$557$	− 11294.7i	− 0.859195i	0.903021	−	0.429597i	$-0.141345\pi$
$558$	0	0
$559$	10105.1i	0.764580i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12400.1	−0.928249	−0.464124	−	0.885770i	$-0.653631\pi$
$563$	−12400.1	−0.928249	−0.464124	+	0.885770i	$0.653631\pi$
$564$	0	0
$565$	27638.3i	2.05797i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3444.61i	0.253788i	0.991916	+	0.126894i	$0.0405008\pi$
$569$	3444.61i	0.253788i	−0.991916	+	0.126894i	$0.959499\pi$
$570$	0	0
$571$	12742.4	0.933894	0.466947	−	0.884285i	$-0.345354\pi$
$571$	12742.4	0.933894	0.466947	+	0.884285i	$0.345354\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 7218.85i	− 0.523560i
$576$	0	0
$577$	− 4930.02i	− 0.355701i	−0.984058	−	0.177850i	$-0.943086\pi$
$577$	− 4930.02i	− 0.355701i	0.984058	−	0.177850i	$-0.0569143\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−324.182	−0.0230296
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13674.4	−0.961503	−0.480751	−	0.876857i	$-0.659636\pi$
$587$	−13674.4	−0.961503	−0.480751	+	0.876857i	$0.659636\pi$
$588$	0	0
$589$	7003.74	0.489956
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21506.8	1.48934	0.744671	−	0.667431i	$-0.232606\pi$
$593$	21506.8	1.48934	0.744671	+	0.667431i	$0.232606\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26621.2i	1.81588i	0.419097	+	0.907942i	$0.362347\pi$
$599$	26621.2i	1.81588i	−0.419097	+	0.907942i	$0.637653\pi$
$600$	0	0
$601$	1559.13i	0.105821i	0.998599	+	0.0529104i	$0.0168498\pi$
$601$	1559.13i	0.105821i	−0.998599	+	0.0529104i	$0.983150\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16386.3	1.10115
$606$	0	0
$607$	− 18118.9i	− 1.21157i	−0.795629	−	0.605784i	$-0.792860\pi$
$607$	− 18118.9i	− 1.21157i	0.795629	−	0.605784i	$-0.207140\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2413.98i	− 0.159835i
$612$	0	0
$613$	−3017.92	−0.198846	−0.0994230	−	0.995045i	$-0.531700\pi$
$613$	−3017.92	−0.198846	−0.0994230	+	0.995045i	$0.531700\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12242.8i	− 0.798830i	−0.916770	−	0.399415i	$-0.869213\pi$
$617$	− 12242.8i	− 0.798830i	0.916770	−	0.399415i	$-0.130787\pi$
$618$	0	0
$619$	25697.5i	1.66861i	0.551302	+	0.834306i	$0.314131\pi$
$619$	25697.5i	1.66861i	−0.551302	+	0.834306i	$0.685869\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19520.0	−1.24928
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8324.81	0.527714
$630$	0	0
$631$	17745.4	1.11954	0.559772	−	0.828647i	$-0.310889\pi$
$631$	17745.4	1.11954	0.559772	+	0.828647i	$0.310889\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6918.53	−0.432367
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4490.03i	0.276670i	0.990386	+	0.138335i	$0.0441750\pi$
$641$	4490.03i	0.276670i	−0.990386	+	0.138335i	$0.955825\pi$
$642$	0	0
$643$	− 14156.1i	− 0.868217i	−0.900861	−	0.434109i	$-0.857063\pi$
$643$	− 14156.1i	− 0.868217i	0.900861	−	0.434109i	$-0.142937\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20881.4	1.26883	0.634414	−	0.772994i	$-0.281242\pi$
$647$	20881.4	1.26883	0.634414	+	0.772994i	$0.281242\pi$
$648$	0	0
$649$	8085.36i	0.489026i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18274.8i	1.09517i	0.836749	+	0.547587i	$0.184453\pi$
$653$	18274.8i	1.09517i	−0.836749	+	0.547587i	$0.815547\pi$
$654$	0	0
$655$	−39656.1	−2.36564
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 9657.98i	− 0.570897i	−0.958394	−	0.285449i	$-0.907857\pi$
$659$	− 9657.98i	− 0.570897i	0.958394	−	0.285449i	$-0.0921426\pi$
$660$	0	0
$661$	12736.0i	0.749427i	0.927141	+	0.374713i	$0.122259\pi$
$661$	12736.0i	0.749427i	−0.927141	+	0.374713i	$0.877741\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5294.54	−0.307354
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8095.09	−0.465734
$672$	0	0
$673$	−13482.5	−0.772232	−0.386116	−	0.922450i	$-0.626184\pi$
$673$	−13482.5	−0.772232	−0.386116	+	0.922450i	$0.626184\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28092.5	1.59480	0.797402	−	0.603449i	$-0.206207\pi$
$677$	28092.5	1.59480	0.797402	+	0.603449i	$0.206207\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26390.6i	1.47849i	0.673438	+	0.739244i	$0.264817\pi$
$683$	26390.6i	1.47849i	−0.673438	+	0.739244i	$0.735183\pi$
$684$	0	0
$685$	− 21488.3i	− 1.19858i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−745.432	−0.0412173
$690$	0	0
$691$	− 24456.3i	− 1.34640i	−0.739461	−	0.673200i	$-0.764919\pi$
$691$	− 24456.3i	− 1.34640i	0.739461	−	0.673200i	$-0.235081\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 15338.4i	− 0.837150i
$696$	0	0
$697$	−2133.42	−0.115938
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 16268.1i	− 0.876517i	−0.898849	−	0.438258i	$-0.855595\pi$
$701$	− 16268.1i	− 0.876517i	0.898849	−	0.438258i	$-0.144405\pi$
$702$	0	0
$703$	18983.9i	1.01848i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13388.8	−0.709207	−0.354603	−	0.935017i	$-0.615384\pi$
$709$	−13388.8	−0.709207	−0.354603	+	0.935017i	$0.615384\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16076.1	0.844395
$714$	0	0
$715$	−3853.35	−0.201548
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18431.0	0.955992	0.477996	−	0.878362i	$-0.341363\pi$
$719$	18431.0	0.955992	0.477996	+	0.878362i	$0.341363\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2566.33i	− 0.131463i
$726$	0	0
$727$	28217.0i	1.43949i	0.694238	+	0.719745i	$0.255741\pi$
$727$	28217.0i	1.43949i	−0.694238	+	0.719745i	$0.744259\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9220.14	0.466511
$732$	0	0
$733$	17062.6i	0.859782i	0.902881	+	0.429891i	$0.141448\pi$
$733$	17062.6i	0.859782i	−0.902881	+	0.429891i	$0.858552\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	414.102i	0.0206969i
$738$	0	0
$739$	10727.4	0.533982	0.266991	−	0.963699i	$-0.413971\pi$
$739$	10727.4	0.533982	0.266991	+	0.963699i	$0.413971\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9465.65i	− 0.467377i	−0.972312	−	0.233688i	$-0.924920\pi$
$743$	− 9465.65i	− 0.467377i	0.972312	−	0.233688i	$-0.0750796\pi$
$744$	0	0
$745$	22013.1i	1.08255i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−23783.2	−1.15561	−0.577804	−	0.816175i	$-0.696090\pi$
$751$	−23783.2	−1.15561	−0.577804	+	0.816175i	$0.696090\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4841.06	−0.233357
$756$	0	0
$757$	−1709.14	−0.0820605	−0.0410303	−	0.999158i	$-0.513064\pi$
$757$	−1709.14	−0.0820605	−0.0410303	+	0.999158i	$0.513064\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22364.2	1.06531	0.532656	−	0.846332i	$-0.321194\pi$
$761$	22364.2	1.06531	0.532656	+	0.846332i	$0.321194\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18591.6i	0.875235i
$768$	0	0
$769$	266.623i	0.0125028i	0.999980	+	0.00625141i	$0.00198990\pi$
$769$	266.623i	0.0125028i	−0.999980	+	0.00625141i	$0.998010\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13319.8	0.619768	0.309884	−	0.950774i	$-0.399710\pi$
$773$	13319.8	0.619768	0.309884	+	0.950774i	$0.399710\pi$
$774$	0	0
$775$	7792.27i	0.361170i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4865.05i	− 0.223759i
$780$	0	0
$781$	−1998.12	−0.0915473
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 46103.6i	− 2.09619i
$786$	0	0
$787$	33820.2i	1.53184i	0.642936	+	0.765920i	$0.277716\pi$
$787$	33820.2i	1.53184i	−0.642936	+	0.765920i	$0.722284\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18614.0	−0.833547
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3193.62	0.141937	0.0709685	−	0.997479i	$-0.477391\pi$
$797$	3193.62	0.141937	0.0709685	+	0.997479i	$0.477391\pi$
$798$	0	0
$799$	−2202.57	−0.0975236
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1087.97	0.0478126
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 6329.00i	− 0.275050i	−0.990498	−	0.137525i	$-0.956085\pi$
$809$	− 6329.00i	− 0.275050i	0.990498	−	0.137525i	$-0.0439148\pi$
$810$	0	0
$811$	− 36542.9i	− 1.58224i	−0.611661	−	0.791120i	$-0.709498\pi$
$811$	− 36542.9i	− 1.58224i	0.611661	−	0.791120i	$-0.290502\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	38992.4	1.67588
$816$	0	0
$817$	21025.6i	0.900359i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 6739.92i	− 0.286510i	−0.989686	−	0.143255i	$-0.954243\pi$
$821$	− 6739.92i	− 0.286510i	0.989686	−	0.143255i	$-0.0457569\pi$
$822$	0	0
$823$	−21115.4	−0.894333	−0.447167	−	0.894451i	$-0.647567\pi$
$823$	−21115.4	−0.894333	−0.447167	+	0.894451i	$0.647567\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27049.4i	− 1.13736i	−0.822558	−	0.568681i	$-0.807454\pi$
$827$	− 27049.4i	− 1.13736i	0.822558	−	0.568681i	$-0.192546\pi$
$828$	0	0
$829$	14557.3i	0.609888i	0.952370	+	0.304944i	$0.0986378\pi$
$829$	14557.3i	0.609888i	−0.952370	+	0.304944i	$0.901362\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	23527.5	0.975095
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33136.3	−1.36352	−0.681759	−	0.731577i	$-0.738785\pi$
$839$	−33136.3	−1.36352	−0.681759	+	0.731577i	$0.738785\pi$
$840$	0	0
$841$	22506.8	0.922825
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20953.5	0.853042
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	43574.8i	1.75526i
$852$	0	0
$853$	38789.6i	1.55701i	0.627638	+	0.778505i	$0.284022\pi$
$853$	38789.6i	1.55701i	−0.627638	+	0.778505i	$0.715978\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−376.792	−0.0150186	−0.00750932	−	0.999972i	$-0.502390\pi$
$857$	−376.792	−0.0150186	−0.00750932	+	0.999972i	$0.502390\pi$
$858$	0	0
$859$	848.036i	0.0336841i	0.999858	+	0.0168420i	$0.00536124\pi$
$859$	848.036i	0.0336841i	−0.999858	+	0.0168420i	$0.994639\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 14500.0i	− 0.571944i	−0.958238	−	0.285972i	$-0.907684\pi$
$863$	− 14500.0i	− 0.571944i	0.958238	−	0.285972i	$-0.0923164\pi$
$864$	0	0
$865$	−16808.3	−0.660692
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1367.17i	0.0533695i
$870$	0	0
$871$	952.195i	0.0370424i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6492.50	−0.249984	−0.124992	−	0.992158i	$-0.539891\pi$
$877$	−6492.50	−0.249984	−0.124992	+	0.992158i	$0.539891\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5687.73	0.217508	0.108754	−	0.994069i	$-0.465314\pi$
$881$	5687.73	0.217508	0.108754	+	0.994069i	$0.465314\pi$
$882$	0	0
$883$	38104.3	1.45222	0.726111	−	0.687578i	$-0.241326\pi$
$883$	38104.3	1.45222	0.726111	+	0.687578i	$0.241326\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8130.39	0.307770	0.153885	−	0.988089i	$-0.450821\pi$
$887$	8130.39	0.307770	0.153885	+	0.988089i	$0.450821\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5022.75i	− 0.188219i
$894$	0	0
$895$	− 42250.1i	− 1.57795i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5715.10	0.212024
$900$	0	0
$901$	680.150i	0.0251488i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	37307.6i	1.37033i
$906$	0	0
$907$	2510.32	0.0919005	0.0459502	−	0.998944i	$-0.485368\pi$
$907$	2510.32	0.0919005	0.0459502	+	0.998944i	$0.485368\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	17140.3i	0.623361i	0.950187	+	0.311681i	$0.100892\pi$
$911$	17140.3i	0.623361i	−0.950187	+	0.311681i	$0.899108\pi$
$912$	0	0
$913$	− 2412.98i	− 0.0874675i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5879.85	0.211054	0.105527	−	0.994416i	$-0.466347\pi$
$919$	5879.85	0.211054	0.105527	+	0.994416i	$0.466347\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4594.52	−0.163847
$924$	0	0
$925$	−21121.2	−0.750770
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32296.0	1.14058	0.570290	−	0.821443i	$-0.306831\pi$
$929$	32296.0	1.14058	0.570290	+	0.821443i	$0.306831\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3515.89i	0.122975i
$936$	0	0
$937$	24422.5i	0.851494i	0.904842	+	0.425747i	$0.139989\pi$
$937$	24422.5i	0.851494i	−0.904842	+	0.425747i	$0.860011\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	43797.0	1.51726	0.758629	−	0.651523i	$-0.225869\pi$
$941$	43797.0	1.51726	0.758629	+	0.651523i	$0.225869\pi$
$942$	0	0
$943$	− 11167.0i	− 0.385629i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32248.8i	1.10660i	0.832984	+	0.553298i	$0.186631\pi$
$947$	32248.8i	1.10660i	−0.832984	+	0.553298i	$0.813369\pi$
$948$	0	0
$949$	2501.70	0.0855727
$950$	0	0
$951$	0	0
$952$	0	0
$953$	37006.9i	1.25789i	0.777449	+	0.628946i	$0.216513\pi$
$953$	37006.9i	1.25789i	−0.777449	+	0.628946i	$0.783487\pi$
$954$	0	0
$955$	− 5772.70i	− 0.195602i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12437.9	0.417507
$962$	0	0
$963$	0	0
$964$	0	0
$965$	38359.8	1.27963
$966$	0	0
$967$	24660.8	0.820100	0.410050	−	0.912063i	$-0.365511\pi$
$967$	24660.8	0.820100	0.410050	+	0.912063i	$0.365511\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47325.9	−1.56412	−0.782059	−	0.623204i	$-0.785831\pi$
$971$	−47325.9	−1.56412	−0.782059	+	0.623204i	$0.785831\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4776.97i	− 0.156427i	−0.996937	−	0.0782133i	$-0.975078\pi$
$977$	− 4776.97i	− 0.156427i	0.996937	−	0.0782133i	$-0.0249215\pi$
$978$	0	0
$979$	1761.81i	0.0575157i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35153.0	1.14060	0.570299	−	0.821437i	$-0.306827\pi$
$983$	35153.0	1.14060	0.570299	+	0.821437i	$0.306827\pi$
$984$	0	0
$985$	64636.4i	2.09085i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48261.3i	1.55169i
$990$	0	0
$991$	−39456.4	−1.26476	−0.632378	−	0.774660i	$-0.717921\pi$
$991$	−39456.4	−1.26476	−0.632378	+	0.774660i	$0.717921\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 58528.1i	− 1.86479i
$996$	0	0
$997$	15166.4i	0.481769i	0.970554	+	0.240884i	$0.0774375\pi$
$997$	15166.4i	0.481769i	−0.970554	+	0.240884i	$0.922563\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.f.b.881.19	yes	24
3.2	odd	2	inner	1764.4.f.b.881.6	yes	24
7.2	even	3		1764.4.t.c.521.5		48
7.3	odd	6		1764.4.t.c.1097.20		48
7.4	even	3		1764.4.t.c.1097.6		48
7.5	odd	6		1764.4.t.c.521.19		48
7.6	odd	2	inner	1764.4.f.b.881.5	✓	24
21.2	odd	6		1764.4.t.c.521.20		48
21.5	even	6		1764.4.t.c.521.6		48
21.11	odd	6		1764.4.t.c.1097.19		48
21.17	even	6		1764.4.t.c.1097.5		48
21.20	even	2	inner	1764.4.f.b.881.20	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.f.b.881.5	✓	24	7.6	odd	2	inner
1764.4.f.b.881.6	yes	24	3.2	odd	2	inner
1764.4.f.b.881.19	yes	24	1.1	even	1	trivial
1764.4.f.b.881.20	yes	24	21.20	even	2	inner
1764.4.t.c.521.5		48	7.2	even	3
1764.4.t.c.521.6		48	21.5	even	6
1764.4.t.c.521.19		48	7.5	odd	6
1764.4.t.c.521.20		48	21.2	odd	6
1764.4.t.c.1097.5		48	21.17	even	6
1764.4.t.c.1097.6		48	7.4	even	3
1764.4.t.c.1097.19		48	21.11	odd	6
1764.4.t.c.1097.20		48	7.3	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+13.5703 q^{5} +O(q^{10})\)	\(q+13.5703 q^{5} -11.1126i q^{11} -25.5525i q^{13} -23.3148 q^{17} -53.1670i q^{19} -122.037i q^{23} +59.1529 q^{25} -43.3847i q^{29} +131.731i q^{31} -357.062 q^{37} +91.5050 q^{41} -395.464 q^{43} +94.4712 q^{47} -29.1725i q^{53} -150.801i q^{55} -727.585 q^{59} -728.461i q^{61} -346.755i q^{65} -37.2642 q^{67} -179.807i q^{71} +97.9041i q^{73} -123.029 q^{79} +217.139 q^{83} -316.388 q^{85} -158.542 q^{89} -721.492i q^{95} +597.401i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 888 q^{25} + 864 q^{37} - 1248 q^{43} + 1056 q^{67} - 8064 q^{85}+O(q^{100}) \) 24 * q + 888 * q^25 + 864 * q^37 - 1248 * q^43 + 1056 * q^67 - 8064 * q^85

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