LMFDB - Embedded newform 360.4.f.e.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,4,Mod(289,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("360.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	360.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:	$21.2406876021$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.1
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	360.289
Dual form			360.4.f.e.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.79796 - 6.89898i) q^{5} +16.6969i q^{7} +19.1918 q^{11} -61.7980i q^{13} +30.3837i q^{17} -59.1918 q^{19} +205.687i q^{23} +(29.8082 + 121.394i) q^{25} +8.38367 q^{29} +331.151 q^{31} +(115.192 - 146.899i) q^{35} +266.565i q^{37} +320.788 q^{41} -83.1214i q^{43} +276.434i q^{47} +64.2122 q^{49} +390.888i q^{53} +(-168.849 - 132.404i) q^{55} +779.110 q^{59} -483.171 q^{61} +(-426.343 + 543.696i) q^{65} +123.707i q^{67} -187.233 q^{71} +778.706i q^{73} +320.445i q^{77} +446.384 q^{79} -1054.05i q^{83} +(209.616 - 267.314i) q^{85} -94.8490 q^{89} +1031.84 q^{91} +(520.767 + 408.363i) q^{95} +252.041i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 80 q^{11} - 80 q^{19} + 276 q^{25} - 280 q^{29} + 384 q^{31} + 304 q^{35} + 1048 q^{41} + 492 q^{49} - 1616 q^{55} + 1392 q^{59} - 1384 q^{61} - 608 q^{65} - 1376 q^{71} + 1472 q^{79} + 1152 q^{85}+ \cdots + 1456 q^{95}+O(q^{100}) $ 4 * q + 4 * q^5 - 80 * q^11 - 80 * q^19 + 276 * q^25 - 280 * q^29 + 384 * q^31 + 304 * q^35 + 1048 * q^41 + 492 * q^49 - 1616 * q^55 + 1392 * q^59 - 1384 * q^61 - 608 * q^65 - 1376 * q^71 + 1472 * q^79 + 1152 * q^85 - 1320 * q^89 + 992 * q^91 + 1456 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−8.79796	−	6.89898i	−0.786913	−	0.617063i
$5$	−8.79796	−	6.89898i	−0.786913	−	0.617063i
$6$		0			0
$7$			16.6969i			0.901550i	0.892638	+	0.450775i	$0.148852\pi$
$7$			16.6969i			0.901550i	−0.892638	+	0.450775i	$0.851148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	19.1918			0.526051			0.263025	−	0.964789i	$-0.415280\pi$
$11$	19.1918			0.526051			0.263025	+	0.964789i	$0.415280\pi$
$12$		0			0
$13$		−	61.7980i		−	1.31844i	−0.751952	−	0.659218i	$-0.770887\pi$
$13$		−	61.7980i		−	1.31844i	0.751952	−	0.659218i	$-0.229113\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			30.3837i			0.433478i	0.976230	+	0.216739i	$0.0695420\pi$
$17$			30.3837i			0.433478i	−0.976230	+	0.216739i	$0.930458\pi$
$18$		0			0
$19$	−59.1918			−0.714713			−0.357356	−	0.933968i	$-0.616322\pi$
$19$	−59.1918			−0.714713			−0.357356	+	0.933968i	$0.616322\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			205.687i			1.86472i	0.361526	+	0.932362i	$0.382256\pi$
$23$			205.687i			1.86472i	−0.361526	+	0.932362i	$0.617744\pi$
$24$		0			0
$25$	29.8082	+	121.394i	0.238465	+	0.971151i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.38367			0.0536831			0.0268415	−	0.999640i	$-0.491455\pi$
$29$	8.38367			0.0536831			0.0268415	+	0.999640i	$0.491455\pi$
$30$		0			0
$31$	331.151			1.91860			0.959298	−	0.282396i	$-0.0911291\pi$
$31$	331.151			1.91860			0.959298	+	0.282396i	$0.0911291\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	115.192	−	146.899i	0.556314	−	0.709442i
$36$		0			0
$37$			266.565i			1.18441i	0.805788	+	0.592204i	$0.201742\pi$
$37$			266.565i			1.18441i	−0.805788	+	0.592204i	$0.798258\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	320.788			1.22192			0.610959	−	0.791662i	$-0.290784\pi$
$41$	320.788			1.22192			0.610959	+	0.791662i	$0.290784\pi$
$42$		0			0
$43$		−	83.1214i		−	0.294788i	−0.989078	−	0.147394i	$-0.952911\pi$
$43$		−	83.1214i		−	0.294788i	0.989078	−	0.147394i	$-0.0470886\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			276.434i			0.857915i	0.903325	+	0.428957i	$0.141119\pi$
$47$			276.434i			0.857915i	−0.903325	+	0.428957i	$0.858881\pi$
$48$		0			0
$49$	64.2122			0.187208
$50$		0			0
$51$		0			0
$52$		0			0
$53$			390.888i			1.01307i	0.862220	+	0.506534i	$0.169073\pi$
$53$			390.888i			1.01307i	−0.862220	+	0.506534i	$0.830927\pi$
$54$		0			0
$55$	−168.849	−	132.404i	−0.413956	−	0.324607i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	779.110			1.71918			0.859589	−	0.510986i	$-0.170720\pi$
$59$	779.110			1.71918			0.859589	+	0.510986i	$0.170720\pi$
$60$		0			0
$61$	−483.171			−1.01416			−0.507080	−	0.861899i	$-0.669275\pi$
$61$	−483.171			−1.01416			−0.507080	+	0.861899i	$0.669275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−426.343	+	543.696i	−0.813559	+	1.03750i
$66$		0			0
$67$			123.707i			0.225571i	0.993619	+	0.112785i	$0.0359772\pi$
$67$			123.707i			0.225571i	−0.993619	+	0.112785i	$0.964023\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−187.233			−0.312964			−0.156482	−	0.987681i	$-0.550015\pi$
$71$	−187.233			−0.312964			−0.156482	+	0.987681i	$0.550015\pi$
$72$		0			0
$73$			778.706i			1.24850i	0.781224	+	0.624251i	$0.214596\pi$
$73$			778.706i			1.24850i	−0.781224	+	0.624251i	$0.785404\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			320.445i			0.474261i
$78$		0			0
$79$	446.384			0.635723			0.317861	−	0.948137i	$-0.397035\pi$
$79$	446.384			0.635723			0.317861	+	0.948137i	$0.397035\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	1054.05i		−	1.39394i	−0.717100	−	0.696970i	$-0.754531\pi$
$83$		−	1054.05i		−	1.39394i	0.717100	−	0.696970i	$-0.245469\pi$
$84$		0			0
$85$	209.616	−	267.314i	0.267483	−	0.341109i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−94.8490			−0.112966			−0.0564830	−	0.998404i	$-0.517989\pi$
$89$	−94.8490			−0.112966			−0.0564830	+	0.998404i	$0.517989\pi$
$90$		0			0
$91$	1031.84			1.18864
$92$		0			0
$93$		0			0
$94$		0			0
$95$	520.767	+	408.363i	0.562417	+	0.441023i
$96$		0			0
$97$			252.041i			0.263823i	0.991261	+	0.131912i	$0.0421115\pi$
$97$			252.041i			0.263823i	−0.991261	+	0.131912i	$0.957888\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−37.9184			−0.0373566			−0.0186783	−	0.999826i	$-0.505946\pi$
$101$	−37.9184			−0.0373566			−0.0186783	+	0.999826i	$0.505946\pi$
$102$		0			0
$103$		−	94.4133i		−	0.0903186i	−0.998980	−	0.0451593i	$-0.985620\pi$
$103$		−	94.4133i		−	0.0903186i	0.998980	−	0.0451593i	$-0.0143795\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			901.464i			0.814466i	0.913324	+	0.407233i	$0.133506\pi$
$107$			901.464i			0.814466i	−0.913324	+	0.407233i	$0.866494\pi$
$108$		0			0
$109$	−1415.69			−1.24403			−0.622013	−	0.783007i	$-0.713685\pi$
$109$	−1415.69			−1.24403			−0.622013	+	0.783007i	$0.713685\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	293.576i		−	0.244401i	−0.992505	−	0.122200i	$-0.961005\pi$
$113$		−	293.576i		−	0.244401i	0.992505	−	0.122200i	$-0.0389950\pi$
$114$		0			0
$115$	1419.03	−	1809.62i	1.15065	−	1.46738i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−507.314			−0.390802
$120$		0			0
$121$	−962.673			−0.723271
$122$		0			0
$123$		0			0
$124$		0			0
$125$	575.243	−	1273.66i	0.411610	−	0.911360i
$126$		0			0
$127$			774.717i			0.541300i	0.962678	+	0.270650i	$0.0872385\pi$
$127$			774.717i			0.541300i	−0.962678	+	0.270650i	$0.912761\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−334.343			−0.222990			−0.111495	−	0.993765i	$-0.535564\pi$
$131$	−334.343			−0.222990			−0.111495	+	0.993765i	$0.535564\pi$
$132$		0			0
$133$		−	988.322i		−	0.644349i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	323.514i		−	0.201750i	−0.994899	−	0.100875i	$-0.967836\pi$
$137$		−	323.514i		−	0.201750i	0.994899	−	0.100875i	$-0.0321641\pi$
$138$		0			0
$139$	396.482			0.241936			0.120968	−	0.992656i	$-0.461400\pi$
$139$	396.482			0.241936			0.120968	+	0.992656i	$0.461400\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	1186.02i		−	0.693564i
$144$		0			0
$145$	−73.7592	−	57.8388i	−0.0422439	−	0.0331259i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1682.89			−0.925284			−0.462642	−	0.886545i	$-0.653099\pi$
$149$	−1682.89			−0.925284			−0.462642	+	0.886545i	$0.653099\pi$
$150$		0			0
$151$	−2924.52			−1.57612			−0.788060	−	0.615598i	$-0.788915\pi$
$151$	−2924.52			−1.57612			−0.788060	+	0.615598i	$0.788915\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2913.45	−	2284.60i	−1.50977	−	1.18390i
$156$		0			0
$157$			2768.42i			1.40729i	0.710553	+	0.703644i	$0.248445\pi$
$157$			2768.42i			1.40729i	−0.710553	+	0.703644i	$0.751555\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3434.34			−1.68114
$162$		0			0
$163$		−	2816.33i		−	1.35333i	−0.736292	−	0.676663i	$-0.763425\pi$
$163$		−	2816.33i		−	1.35333i	0.736292	−	0.676663i	$-0.236575\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1836.41i		−	0.850933i	−0.904974	−	0.425466i	$-0.860110\pi$
$167$		−	1836.41i		−	0.850933i	0.904974	−	0.425466i	$-0.139890\pi$
$168$		0			0
$169$	−1621.99			−0.738274
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1224.22i			0.538011i	0.963139	+	0.269006i	$0.0866950\pi$
$173$			1224.22i			0.538011i	−0.963139	+	0.269006i	$0.913305\pi$
$174$		0			0
$175$	−2026.91	+	497.705i	−0.875541	+	0.214988i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2729.58			1.13977			0.569883	−	0.821726i	$-0.306989\pi$
$179$	2729.58			1.13977			0.569883	+	0.821726i	$0.306989\pi$
$180$		0			0
$181$	2642.36			1.08511			0.542555	−	0.840020i	$-0.317457\pi$
$181$	2642.36			1.08511			0.542555	+	0.840020i	$0.317457\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1839.03	−	2345.23i	0.730854	−	0.932026i
$186$		0			0
$187$			583.118i			0.228031i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2339.23			0.886183			0.443091	−	0.896476i	$-0.353882\pi$
$191$	2339.23			0.886183			0.443091	+	0.896476i	$0.353882\pi$
$192$		0			0
$193$		−	4601.61i		−	1.71622i	−0.513464	−	0.858111i	$-0.671638\pi$
$193$		−	4601.61i		−	1.71622i	0.513464	−	0.858111i	$-0.328362\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			823.941i			0.297987i	0.988838	+	0.148993i	$0.0476033\pi$
$197$			823.941i			0.297987i	−0.988838	+	0.148993i	$0.952397\pi$
$198$		0			0
$199$	3329.70			1.18611			0.593055	−	0.805162i	$-0.297922\pi$
$199$	3329.70			1.18611			0.593055	+	0.805162i	$0.297922\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			139.982i			0.0483980i
$204$		0			0
$205$	−2822.28	−	2213.11i	−0.961543	−	0.754001i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1136.00			−0.375975
$210$		0			0
$211$	1018.78			0.332398			0.166199	−	0.986092i	$-0.446851\pi$
$211$	1018.78			0.332398			0.166199	+	0.986092i	$0.446851\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−573.453	+	731.299i	−0.181903	+	0.231973i
$216$		0			0
$217$			5529.21i			1.72971i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1877.65			0.571513
$222$		0			0
$223$			99.1581i			0.0297763i	0.999889	+	0.0148882i	$0.00473922\pi$
$223$			99.1581i			0.0297763i	−0.999889	+	0.0148882i	$0.995261\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1197.59i			0.350163i	0.984554	+	0.175081i	$0.0560188\pi$
$227$			1197.59i			0.350163i	−0.984554	+	0.175081i	$0.943981\pi$
$228$		0			0
$229$	453.592			0.130892			0.0654458	−	0.997856i	$-0.479153\pi$
$229$	453.592			0.130892			0.0654458	+	0.997856i	$0.479153\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3788.49i		−	1.06520i	−0.846367	−	0.532601i	$-0.821215\pi$
$233$		−	3788.49i		−	1.06520i	0.846367	−	0.532601i	$-0.178785\pi$
$234$		0			0
$235$	1907.11	−	2432.05i	0.529388	−	0.675105i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6000.47			−1.62401			−0.812005	−	0.583651i	$-0.801624\pi$
$239$	−6000.47			−1.62401			−0.812005	+	0.583651i	$0.801624\pi$
$240$		0			0
$241$	1842.53			0.492480			0.246240	−	0.969209i	$-0.420805\pi$
$241$	1842.53			0.492480			0.246240	+	0.969209i	$0.420805\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−564.937	−	442.999i	−0.147316	−	0.115519i
$246$		0			0
$247$			3657.93i			0.942303i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1149.46			−0.289057			−0.144529	−	0.989501i	$-0.546167\pi$
$251$	−1149.46			−0.289057			−0.144529	+	0.989501i	$0.546167\pi$
$252$		0			0
$253$			3947.51i			0.980939i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			5407.67i			1.31253i	0.754528	+	0.656267i	$0.227866\pi$
$257$			5407.67i			1.31253i	−0.754528	+	0.656267i	$0.772134\pi$
$258$		0			0
$259$	−4450.82			−1.06780
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2067.34i			0.484705i	0.970188	+	0.242352i	$0.0779190\pi$
$263$			2067.34i			0.484705i	−0.970188	+	0.242352i	$0.922081\pi$
$264$		0			0
$265$	2696.73	−	3439.01i	0.625127	−	0.797196i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−592.388			−0.134270			−0.0671348	−	0.997744i	$-0.521386\pi$
$269$	−592.388			−0.134270			−0.0671348	+	0.997744i	$0.521386\pi$
$270$		0			0
$271$	−2583.27			−0.579049			−0.289524	−	0.957171i	$-0.593497\pi$
$271$	−2583.27			−0.579049			−0.289524	+	0.957171i	$0.593497\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	572.073	+	2329.77i	0.125445	+	0.510875i
$276$		0			0
$277$		−	4488.09i		−	0.973513i	−0.873538	−	0.486756i	$-0.838180\pi$
$277$		−	4488.09i		−	0.973513i	0.873538	−	0.486756i	$-0.161820\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6280.54			1.33333			0.666665	−	0.745357i	$-0.267721\pi$
$281$	6280.54			1.33333			0.666665	+	0.745357i	$0.267721\pi$
$282$		0			0
$283$			5233.40i			1.09927i	0.835405	+	0.549635i	$0.185233\pi$
$283$			5233.40i			1.09927i	−0.835405	+	0.549635i	$0.814767\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5356.17i			1.10162i
$288$		0			0
$289$	3989.83			0.812097
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2438.21i		−	0.486149i	−0.970008	−	0.243074i	$-0.921844\pi$
$293$		−	2438.21i		−	0.486149i	0.970008	−	0.243074i	$-0.0781559\pi$
$294$		0			0
$295$	−6854.58	−	5375.07i	−1.35284	−	1.06084i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12711.0			2.45852
$300$		0			0
$301$	1387.87			0.265766
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4250.92	+	3333.39i	0.798056	+	0.625801i
$306$		0			0
$307$			7910.44i			1.47060i	0.677744	+	0.735298i	$0.262958\pi$
$307$			7910.44i			1.47060i	−0.677744	+	0.735298i	$0.737042\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5419.71			−0.988178			−0.494089	−	0.869411i	$-0.664498\pi$
$311$	−5419.71			−0.988178			−0.494089	+	0.869411i	$0.664498\pi$
$312$		0			0
$313$		−	5570.86i		−	1.00602i	−0.864281	−	0.503009i	$-0.832226\pi$
$313$		−	5570.86i		−	1.00602i	0.864281	−	0.503009i	$-0.167774\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			3724.09i			0.659829i	0.944011	+	0.329915i	$0.107020\pi$
$317$			3724.09i			0.659829i	−0.944011	+	0.329915i	$0.892980\pi$
$318$		0			0
$319$	160.898			0.0282400
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1798.47i		−	0.309812i
$324$		0			0
$325$	7501.89	−	1842.08i	1.28040	−	0.314401i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4615.60			−0.773453
$330$		0			0
$331$	3223.96			0.535362			0.267681	−	0.963508i	$-0.413743\pi$
$331$	3223.96			0.535362			0.267681	+	0.963508i	$0.413743\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	853.453	−	1088.37i	0.139191	−	0.177505i
$336$		0			0
$337$			9524.43i			1.53955i	0.638314	+	0.769776i	$0.279632\pi$
$337$			9524.43i			1.53955i	−0.638314	+	0.769776i	$0.720368\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6355.40			1.00928
$342$		0			0
$343$			6799.20i			1.07033i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6042.30i			0.934777i	0.884052	+	0.467388i	$0.154805\pi$
$347$			6042.30i			0.934777i	−0.884052	+	0.467388i	$0.845195\pi$
$348$		0			0
$349$	1626.20			0.249422			0.124711	−	0.992193i	$-0.460200\pi$
$349$	1626.20			0.249422			0.124711	+	0.992193i	$0.460200\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			886.955i			0.133733i	0.997762	+	0.0668667i	$0.0213002\pi$
$353$			886.955i			0.133733i	−0.997762	+	0.0668667i	$0.978700\pi$
$354$		0			0
$355$	1647.27	+	1291.71i	0.246275	+	0.193119i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1722.27			0.253198			0.126599	−	0.991954i	$-0.459594\pi$
$359$	1722.27			0.253198			0.126599	+	0.991954i	$0.459594\pi$
$360$		0			0
$361$	−3355.33			−0.489186
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5372.28	−	6851.02i	0.770405	−	0.982463i
$366$		0			0
$367$			9271.77i			1.31875i	0.751813	+	0.659377i	$0.229180\pi$
$367$			9271.77i			1.31875i	−0.751813	+	0.659377i	$0.770820\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6526.63			−0.913331
$372$		0			0
$373$		−	5697.15i		−	0.790850i	−0.918498	−	0.395425i	$-0.870597\pi$
$373$		−	5697.15i		−	0.790850i	0.918498	−	0.395425i	$-0.129403\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	518.094i		−	0.0707777i
$378$		0			0
$379$	8526.24			1.15558			0.577789	−	0.816186i	$-0.303916\pi$
$379$	8526.24			1.15558			0.577789	+	0.816186i	$0.303916\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			4069.23i			0.542893i	0.962453	+	0.271447i	$0.0875020\pi$
$383$			4069.23i			0.542893i	−0.962453	+	0.271447i	$0.912498\pi$
$384$		0			0
$385$	2210.74	−	2819.26i	0.292649	−	0.373202i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2394.17			−0.312054			−0.156027	−	0.987753i	$-0.549869\pi$
$389$	−2394.17			−0.312054			−0.156027	+	0.987753i	$0.549869\pi$
$390$		0			0
$391$	−6249.52			−0.808316
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3927.27	−	3079.59i	−0.500259	−	0.392281i
$396$		0			0
$397$			3497.79i			0.442190i	0.975252	+	0.221095i	$0.0709630\pi$
$397$			3497.79i			0.442190i	−0.975252	+	0.221095i	$0.929037\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	8608.89			1.07209			0.536044	−	0.844190i	$-0.319918\pi$
$401$	8608.89			1.07209			0.536044	+	0.844190i	$0.319918\pi$
$402$		0			0
$403$		−	20464.5i		−	2.52955i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5115.88i			0.623058i
$408$		0			0
$409$	−1385.67			−0.167523			−0.0837615	−	0.996486i	$-0.526693\pi$
$409$	−1385.67			−0.167523			−0.0837615	+	0.996486i	$0.526693\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			13008.8i			1.54992i
$414$		0			0
$415$	−7271.87	+	9273.49i	−0.860149	+	1.09691i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3738.23			0.435858			0.217929	−	0.975965i	$-0.430070\pi$
$419$	3738.23			0.435858			0.217929	+	0.975965i	$0.430070\pi$
$420$		0			0
$421$	−8993.95			−1.04118			−0.520592	−	0.853806i	$-0.674289\pi$
$421$	−8993.95			−1.04118			−0.520592	+	0.853806i	$0.674289\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3688.39	+	905.681i	−0.420972	+	0.103369i
$426$		0			0
$427$		−	8067.48i		−	0.914316i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−462.547			−0.0516940			−0.0258470	−	0.999666i	$-0.508228\pi$
$431$	−462.547			−0.0516940			−0.0258470	+	0.999666i	$0.508228\pi$
$432$		0			0
$433$			5231.82i			0.580659i	0.956927	+	0.290329i	$0.0937649\pi$
$433$			5231.82i			0.580659i	−0.956927	+	0.290329i	$0.906235\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	12175.0i		−	1.33274i
$438$		0			0
$439$	−8995.77			−0.978006			−0.489003	−	0.872282i	$-0.662639\pi$
$439$	−8995.77			−0.978006			−0.489003	+	0.872282i	$0.662639\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5549.52i		−	0.595182i	−0.954694	−	0.297591i	$-0.903817\pi$
$443$		−	5549.52i		−	0.595182i	0.954694	−	0.297591i	$-0.0961831\pi$
$444$		0			0
$445$	834.477	+	654.361i	0.0888944	+	0.0697072i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5951.29			−0.625521			−0.312760	−	0.949832i	$-0.601254\pi$
$449$	−5951.29			−0.625521			−0.312760	+	0.949832i	$0.601254\pi$
$450$		0			0
$451$	6156.51			0.642791
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−9078.06	−	7118.62i	−0.935354	−	0.733464i
$456$		0			0
$457$		−	13912.7i		−	1.42409i	−0.702132	−	0.712047i	$-0.747768\pi$
$457$		−	13912.7i		−	1.42409i	0.702132	−	0.712047i	$-0.252232\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17467.6			−1.76475			−0.882374	−	0.470549i	$-0.844056\pi$
$461$	−17467.6			−1.76475			−0.882374	+	0.470549i	$0.844056\pi$
$462$		0			0
$463$		−	1575.36i		−	0.158127i	−0.996870	−	0.0790637i	$-0.974807\pi$
$463$		−	1575.36i		−	0.158127i	0.996870	−	0.0790637i	$-0.0251931\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	15618.3i		−	1.54760i	−0.633430	−	0.773800i	$-0.718354\pi$
$467$		−	15618.3i		−	1.54760i	0.633430	−	0.773800i	$-0.281646\pi$
$468$		0			0
$469$	−2065.53			−0.203363
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	1595.25i		−	0.155074i
$474$		0			0
$475$	−1764.40	−	7185.53i	−0.170434	−	0.694094i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9527.90			0.908854			0.454427	−	0.890784i	$-0.349844\pi$
$479$	9527.90			0.908854			0.454427	+	0.890784i	$0.349844\pi$
$480$		0			0
$481$	16473.2			1.56157
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1738.82	−	2217.44i	0.162796	−	0.207606i
$486$		0			0
$487$			15729.6i			1.46361i	0.681514	+	0.731805i	$0.261322\pi$
$487$			15729.6i			1.46361i	−0.681514	+	0.731805i	$0.738678\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2566.49			−0.235894			−0.117947	−	0.993020i	$-0.537631\pi$
$491$	−2566.49			−0.235894			−0.117947	+	0.993020i	$0.537631\pi$
$492$		0			0
$493$			254.727i			0.0232704i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	3126.21i		−	0.282152i
$498$		0			0
$499$	13560.4			1.21652			0.608261	−	0.793737i	$-0.291867\pi$
$499$	13560.4			1.21652			0.608261	+	0.793737i	$0.291867\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	5222.51i		−	0.462943i	−0.972842	−	0.231471i	$-0.925646\pi$
$503$		−	5222.51i		−	0.462943i	0.972842	−	0.231471i	$-0.0743540\pi$
$504$		0			0
$505$	333.604	+	261.598i	0.0293964	+	0.0230514i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11875.9			−1.03416			−0.517082	−	0.855936i	$-0.672982\pi$
$509$	−11875.9			−1.03416			−0.517082	+	0.855936i	$0.672982\pi$
$510$		0			0
$511$	−13002.0			−1.12559
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−651.355	+	830.644i	−0.0557323	+	0.0710729i
$516$		0			0
$517$			5305.27i			0.451307i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2456.04			−0.206528			−0.103264	−	0.994654i	$-0.532929\pi$
$521$	−2456.04			−0.206528			−0.103264	+	0.994654i	$0.532929\pi$
$522$		0			0
$523$		−	634.460i		−	0.0530459i	−0.999648	−	0.0265229i	$-0.991556\pi$
$523$		−	634.460i		−	0.0530459i	0.999648	−	0.0265229i	$-0.00844351\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			10061.6i			0.831669i
$528$		0			0
$529$	−30140.0			−2.47720
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	19824.0i		−	1.61102i
$534$		0			0
$535$	6219.18	−	7931.05i	0.502577	−	0.640914i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1232.35			0.0984807
$540$		0			0
$541$	18078.4			1.43669			0.718347	−	0.695685i	$-0.244899\pi$
$541$	18078.4			1.43669			0.718347	+	0.695685i	$0.244899\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12455.2	+	9766.84i	0.978941	+	0.767643i
$546$		0			0
$547$		−	24815.3i		−	1.93972i	−0.243661	−	0.969860i	$-0.578349\pi$
$547$		−	24815.3i		−	1.93972i	0.243661	−	0.969860i	$-0.421651\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−496.245			−0.0383680
$552$		0			0
$553$			7453.24i			0.573136i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			10073.7i			0.766310i	0.923684	+	0.383155i	$0.125162\pi$
$557$			10073.7i			0.766310i	−0.923684	+	0.383155i	$0.874838\pi$
$558$		0			0
$559$	−5136.73			−0.388660
$560$		0			0
$561$		0			0
$562$		0			0
$563$			7505.06i			0.561813i	0.959735	+	0.280906i	$0.0906350\pi$
$563$			7505.06i			0.561813i	−0.959735	+	0.280906i	$0.909365\pi$
$564$		0			0
$565$	−2025.37	+	2582.87i	−0.150811	+	0.192322i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15251.4			−1.12368			−0.561838	−	0.827247i	$-0.689906\pi$
$569$	−15251.4			−1.12368			−0.561838	+	0.827247i	$0.689906\pi$
$570$		0			0
$571$	−2683.78			−0.196695			−0.0983474	−	0.995152i	$-0.531356\pi$
$571$	−2683.78			−0.196695			−0.0983474	+	0.995152i	$0.531356\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24969.1	+	6131.14i	−1.81093	+	0.444672i
$576$		0			0
$577$		−	7369.88i		−	0.531737i	−0.964009	−	0.265868i	$-0.914341\pi$
$577$		−	7369.88i		−	0.531737i	0.964009	−	0.265868i	$-0.0856586\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	17599.4			1.25671
$582$		0			0
$583$			7501.85i			0.532925i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	20865.4i		−	1.46713i	−0.679618	−	0.733567i	$-0.737854\pi$
$587$		−	20865.4i		−	1.46713i	0.679618	−	0.733567i	$-0.262146\pi$
$588$		0			0
$589$	−19601.4			−1.37124
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	25894.0i		−	1.79316i	−0.442886	−	0.896578i	$-0.646046\pi$
$593$		−	25894.0i		−	1.79316i	0.442886	−	0.896578i	$-0.353954\pi$
$594$		0			0
$595$	4463.33	+	3499.95i	0.307527	+	0.241150i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5632.42			0.384198			0.192099	−	0.981376i	$-0.438471\pi$
$599$	5632.42			0.384198			0.192099	+	0.981376i	$0.438471\pi$
$600$		0			0
$601$	−13079.7			−0.887742			−0.443871	−	0.896091i	$-0.646395\pi$
$601$	−13079.7			−0.887742			−0.443871	+	0.896091i	$0.646395\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8469.56	+	6641.46i	0.569151	+	0.446304i
$606$		0			0
$607$		−	18890.2i		−	1.26314i	−0.775317	−	0.631572i	$-0.782410\pi$
$607$		−	18890.2i		−	1.26314i	0.775317	−	0.631572i	$-0.217590\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17083.0			1.13111
$612$		0			0
$613$			13631.5i			0.898155i	0.893493	+	0.449078i	$0.148247\pi$
$613$			13631.5i			0.898155i	−0.893493	+	0.449078i	$0.851753\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	14982.2i		−	0.977568i	−0.872405	−	0.488784i	$-0.837441\pi$
$617$		−	14982.2i		−	0.977568i	0.872405	−	0.488784i	$-0.162559\pi$
$618$		0			0
$619$	25049.9			1.62656			0.813279	−	0.581873i	$-0.197680\pi$
$619$	25049.9			1.62656			0.813279	+	0.581873i	$0.197680\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1583.69i		−	0.101844i
$624$		0			0
$625$	−13847.9	+	7237.06i	−0.886269	+	0.463172i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8099.23			−0.513414
$630$		0			0
$631$	18711.0			1.18046			0.590232	−	0.807233i	$-0.299036\pi$
$631$	18711.0			1.18046			0.590232	+	0.807233i	$0.299036\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5344.76	−	6815.93i	0.334016	−	0.425956i
$636$		0			0
$637$		−	3968.19i		−	0.246821i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−25792.2			−1.58929			−0.794643	−	0.607077i	$-0.792342\pi$
$641$	−25792.2			−1.58929			−0.794643	+	0.607077i	$0.792342\pi$
$642$		0			0
$643$			20256.7i			1.24237i	0.783663	+	0.621186i	$0.213349\pi$
$643$			20256.7i			1.24237i	−0.783663	+	0.621186i	$0.786651\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			6655.43i			0.404408i	0.979343	+	0.202204i	$0.0648104\pi$
$647$			6655.43i			0.404408i	−0.979343	+	0.202204i	$0.935190\pi$
$648$		0			0
$649$	14952.6			0.904375
$650$		0			0
$651$		0			0
$652$		0			0
$653$			8490.91i			0.508844i	0.967093	+	0.254422i	$0.0818852\pi$
$653$			8490.91i			0.508844i	−0.967093	+	0.254422i	$0.918115\pi$
$654$		0			0
$655$	2941.53	+	2306.62i	0.175474	+	0.137599i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15543.8			0.918816			0.459408	−	0.888225i	$-0.348062\pi$
$659$	15543.8			0.918816			0.459408	+	0.888225i	$0.348062\pi$
$660$		0			0
$661$	−13519.8			−0.795553			−0.397777	−	0.917482i	$-0.630218\pi$
$661$	−13519.8			−0.795553			−0.397777	+	0.917482i	$0.630218\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6818.42	+	8695.22i	−0.397604	+	0.507047i
$666$		0			0
$667$			1724.41i			0.100104i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9272.95			−0.533499
$672$		0			0
$673$		−	11565.3i		−	0.662421i	−0.943557	−	0.331211i	$-0.892543\pi$
$673$		−	11565.3i		−	0.662421i	0.943557	−	0.331211i	$-0.107457\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			28227.5i			1.60247i	0.598350	+	0.801235i	$0.295823\pi$
$677$			28227.5i			1.60247i	−0.598350	+	0.801235i	$0.704177\pi$
$678$		0			0
$679$	−4208.31			−0.237850
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	6425.99i		−	0.360005i	−0.983666	−	0.180003i	$-0.942389\pi$
$683$		−	6425.99i		−	0.360005i	0.983666	−	0.180003i	$-0.0576106\pi$
$684$		0			0
$685$	−2231.92	+	2846.27i	−0.124492	+	0.158759i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24156.1			1.33566
$690$		0			0
$691$	−19066.9			−1.04969			−0.524846	−	0.851197i	$-0.675877\pi$
$691$	−19066.9			−1.04969			−0.524846	+	0.851197i	$0.675877\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3488.23	−	2735.32i	−0.190383	−	0.149290i
$696$		0			0
$697$			9746.71i			0.529674i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4796.00			−0.258406			−0.129203	−	0.991618i	$-0.541242\pi$
$701$	−4796.00			−0.258406			−0.129203	+	0.991618i	$0.541242\pi$
$702$		0			0
$703$		−	15778.5i		−	0.846511i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	633.121i		−	0.0336789i
$708$		0			0
$709$	9805.33			0.519389			0.259695	−	0.965691i	$-0.416378\pi$
$709$	9805.33			0.519389			0.259695	+	0.965691i	$0.416378\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			68113.4i			3.57765i
$714$		0			0
$715$	−8182.30	+	10434.5i	−0.427973	+	0.545775i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27539.7			1.42845			0.714227	−	0.699915i	$-0.246779\pi$
$719$	27539.7			1.42845			0.714227	+	0.699915i	$0.246779\pi$
$720$		0			0
$721$	1576.41			0.0814267
$722$		0			0
$723$		0			0
$724$		0			0
$725$	249.902	+	1017.73i	0.0128015	+	0.0521344i
$726$		0			0
$727$		−	16543.8i		−	0.843985i	−0.906599	−	0.421993i	$-0.861331\pi$
$727$		−	16543.8i		−	0.843985i	0.906599	−	0.421993i	$-0.138669\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2525.53			0.127784
$732$		0			0
$733$			16718.6i			0.842450i	0.906956	+	0.421225i	$0.138400\pi$
$733$			16718.6i			0.842450i	−0.906956	+	0.421225i	$0.861600\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2374.17i			0.118662i
$738$		0			0
$739$	24022.6			1.19579			0.597893	−	0.801576i	$-0.296005\pi$
$739$	24022.6			1.19579			0.597893	+	0.801576i	$0.296005\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			26201.8i			1.29374i	0.762600	+	0.646871i	$0.223923\pi$
$743$			26201.8i			1.29374i	−0.762600	+	0.646871i	$0.776077\pi$
$744$		0			0
$745$	14806.0	+	11610.2i	0.728119	+	0.570959i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15051.7			−0.734282
$750$		0			0
$751$	24451.7			1.18809			0.594045	−	0.804432i	$-0.297530\pi$
$751$	24451.7			1.18809			0.594045	+	0.804432i	$0.297530\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25729.8	+	20176.2i	1.24027	+	0.972567i
$756$		0			0
$757$		−	21403.8i		−	1.02765i	−0.857894	−	0.513827i	$-0.828227\pi$
$757$		−	21403.8i		−	1.02765i	0.857894	−	0.513827i	$-0.171773\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	28935.9			1.37835			0.689176	−	0.724594i	$-0.257973\pi$
$761$	28935.9			1.37835			0.689176	+	0.724594i	$0.257973\pi$
$762$		0			0
$763$		−	23637.8i		−	1.12155i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	48147.4i		−	2.26663i
$768$		0			0
$769$	11479.3			0.538301			0.269151	−	0.963098i	$-0.413257\pi$
$769$	11479.3			0.538301			0.269151	+	0.963098i	$0.413257\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	2512.27i		−	0.116895i	−0.998290	−	0.0584477i	$-0.981385\pi$
$773$		−	2512.27i		−	0.116895i	0.998290	−	0.0584477i	$-0.0186151\pi$
$774$		0			0
$775$	9871.00	+	40199.7i	0.457519	+	1.86325i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18988.0			−0.873320
$780$		0			0
$781$	−3593.34			−0.164635
$782$		0			0
$783$		0			0
$784$		0			0
$785$	19099.3	−	24356.5i	0.868386	−	1.10741i
$786$		0			0
$787$		−	2650.18i		−	0.120036i	−0.998197	−	0.0600182i	$-0.980884\pi$
$787$		−	2650.18i		−	0.120036i	0.998197	−	0.0600182i	$-0.0191159\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4901.81			0.220339
$792$		0			0
$793$			29859.0i			1.33711i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	24516.7i		−	1.08962i	−0.838560	−	0.544810i	$-0.816602\pi$
$797$		−	24516.7i		−	1.08962i	0.838560	−	0.544810i	$-0.183398\pi$
$798$		0			0
$799$	−8399.07			−0.371887
$800$		0			0
$801$		0			0
$802$		0			0
$803$			14944.8i			0.656775i
$804$		0			0
$805$	30215.2	+	23693.4i	1.32291	+	1.03737i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−29725.5			−1.29183			−0.645917	−	0.763408i	$-0.723525\pi$
$809$	−29725.5			−1.29183			−0.645917	+	0.763408i	$0.723525\pi$
$810$		0			0
$811$	−2050.38			−0.0887773			−0.0443887	−	0.999014i	$-0.514134\pi$
$811$	−2050.38			−0.0887773			−0.0443887	+	0.999014i	$0.514134\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19429.8	+	24778.0i	−0.835089	+	1.06495i
$816$		0			0
$817$			4920.11i			0.210689i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33863.4			1.43952			0.719758	−	0.694225i	$-0.244253\pi$
$821$	33863.4			1.43952			0.719758	+	0.694225i	$0.244253\pi$
$822$		0			0
$823$			1216.52i			0.0515251i	0.999668	+	0.0257625i	$0.00820138\pi$
$823$			1216.52i			0.0515251i	−0.999668	+	0.0257625i	$0.991799\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			25654.4i			1.07871i	0.842079	+	0.539355i	$0.181332\pi$
$827$			25654.4i			1.07871i	−0.842079	+	0.539355i	$0.818668\pi$
$828$		0			0
$829$	−24544.6			−1.02831			−0.514154	−	0.857698i	$-0.671894\pi$
$829$	−24544.6			−1.02831			−0.514154	+	0.857698i	$0.671894\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1951.00i			0.0811504i
$834$		0			0
$835$	−12669.4	+	16156.7i	−0.525080	+	0.669610i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16548.6			0.680954			0.340477	−	0.940253i	$-0.389411\pi$
$839$	16548.6			0.680954			0.340477	+	0.940253i	$0.389411\pi$
$840$		0			0
$841$	−24318.7			−0.997118
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14270.2	+	11190.1i	0.580958	+	0.455562i
$846$		0			0
$847$		−	16073.7i		−	0.652065i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−54828.9			−2.20859
$852$		0			0
$853$		−	24363.5i		−	0.977950i	−0.872298	−	0.488975i	$-0.837371\pi$
$853$		−	24363.5i		−	0.977950i	0.872298	−	0.488975i	$-0.162629\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	575.718i		−	0.0229477i	−0.999934	−	0.0114738i	$-0.996348\pi$
$857$		−	575.718i		−	0.0229477i	0.999934	−	0.0114738i	$-0.00365232\pi$
$858$		0			0
$859$	−1531.31			−0.0608236			−0.0304118	−	0.999537i	$-0.509682\pi$
$859$	−1531.31			−0.0608236			−0.0304118	+	0.999537i	$0.509682\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	7706.51i		−	0.303978i	−0.988382	−	0.151989i	$-0.951432\pi$
$863$		−	7706.51i		−	0.303978i	0.988382	−	0.151989i	$-0.0485678\pi$
$864$		0			0
$865$	8445.89	−	10770.7i	0.331987	−	0.423368i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8566.92			0.334422
$870$		0			0
$871$	7644.85			0.297400
$872$		0			0
$873$		0			0
$874$		0			0
$875$	21266.3	+	9604.79i	0.821637	+	0.371087i
$876$		0			0
$877$		−	43618.9i		−	1.67948i	−0.542988	−	0.839741i	$-0.682707\pi$
$877$		−	43618.9i		−	1.67948i	0.542988	−	0.839741i	$-0.317293\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13416.4			0.513066			0.256533	−	0.966536i	$-0.417420\pi$
$881$	13416.4			0.513066			0.256533	+	0.966536i	$0.417420\pi$
$882$		0			0
$883$			29538.2i			1.12575i	0.826542	+	0.562875i	$0.190305\pi$
$883$			29538.2i			1.12575i	−0.826542	+	0.562875i	$0.809695\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1950.88i		−	0.0738490i	−0.999318	−	0.0369245i	$-0.988244\pi$
$887$		−	1950.88i		−	0.0738490i	0.999318	−	0.0369245i	$-0.0117561\pi$
$888$		0			0
$889$	−12935.4			−0.488009
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	16362.6i		−	0.613162i
$894$		0			0
$895$	−24014.7	−	18831.3i	−0.896897	−	0.703308i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2776.26			0.102996
$900$		0			0
$901$	−11876.6			−0.439142
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23247.4	−	18229.6i	−0.853888	−	0.669582i
$906$		0			0
$907$		−	3507.55i		−	0.128408i	−0.997937	−	0.0642042i	$-0.979549\pi$
$907$		−	3507.55i		−	0.128408i	0.997937	−	0.0642042i	$-0.0204509\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	33841.4			1.23075			0.615377	−	0.788233i	$-0.289004\pi$
$911$	33841.4			1.23075			0.615377	+	0.788233i	$0.289004\pi$
$912$		0			0
$913$		−	20229.2i		−	0.733283i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	5582.50i		−	0.201036i
$918$		0			0
$919$	−25440.7			−0.913178			−0.456589	−	0.889678i	$-0.650929\pi$
$919$	−25440.7			−0.913178			−0.456589	+	0.889678i	$0.650929\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			11570.6i			0.412623i
$924$		0			0
$925$	−32359.4	+	7945.82i	−1.15024	+	0.282440i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−26416.7			−0.932941			−0.466471	−	0.884537i	$-0.654475\pi$
$929$	−26416.7			−0.932941			−0.466471	+	0.884537i	$0.654475\pi$
$930$		0			0
$931$	−3800.84			−0.133800
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4022.92	−	5130.25i	0.140710	−	0.179441i
$936$		0			0
$937$			1936.70i			0.0675231i	0.999430	+	0.0337616i	$0.0107487\pi$
$937$			1936.70i			0.0675231i	−0.999430	+	0.0337616i	$0.989251\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23459.1			−0.812694			−0.406347	−	0.913719i	$-0.633198\pi$
$941$	−23459.1			−0.812694			−0.406347	+	0.913719i	$0.633198\pi$
$942$		0			0
$943$			65981.8i			2.27854i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			23606.8i			0.810049i	0.914306	+	0.405025i	$0.132737\pi$
$947$			23606.8i			0.810049i	−0.914306	+	0.405025i	$0.867263\pi$
$948$		0			0
$949$	48122.4			1.64607
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	30164.2i		−	1.02530i	−0.858596	−	0.512652i	$-0.828663\pi$
$953$		−	30164.2i		−	1.02530i	0.858596	−	0.512652i	$-0.171337\pi$
$954$		0			0
$955$	−20580.5	−	16138.3i	−0.697349	−	0.546831i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5401.70			0.181887
$960$		0			0
$961$	79870.0			2.68101
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−31746.4	+	40484.8i	−1.05902	+	1.35052i
$966$		0			0
$967$		−	9034.04i		−	0.300429i	−0.988653	−	0.150215i	$-0.952004\pi$
$967$		−	9034.04i		−	0.300429i	0.988653	−	0.150215i	$-0.0479965\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36159.0			−1.19506			−0.597528	−	0.801848i	$-0.703850\pi$
$971$	−36159.0			−1.19506			−0.597528	+	0.801848i	$0.703850\pi$
$972$		0			0
$973$			6620.03i			0.218118i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	38584.0i		−	1.26347i	−0.775184	−	0.631736i	$-0.782343\pi$
$977$		−	38584.0i		−	1.26347i	0.775184	−	0.631736i	$-0.217657\pi$
$978$		0			0
$979$	−1820.33			−0.0594258
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	53046.3i		−	1.72117i	−0.509303	−	0.860587i	$-0.670097\pi$
$983$		−	53046.3i		−	1.72117i	0.509303	−	0.860587i	$-0.329903\pi$
$984$		0			0
$985$	5684.35	−	7249.00i	0.183877	−	0.234490i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17097.0			0.549699
$990$		0			0
$991$	−16425.7			−0.526517			−0.263259	−	0.964725i	$-0.584797\pi$
$991$	−16425.7			−0.526517			−0.263259	+	0.964725i	$0.584797\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29294.5	−	22971.5i	−0.933366	−	0.731906i
$996$		0			0
$997$		−	20024.6i		−	0.636092i	−0.948075	−	0.318046i	$-0.896973\pi$
$997$		−	20024.6i		−	0.636092i	0.948075	−	0.318046i	$-0.103027\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.4.f.e.289.1		4
3.2	odd	2		40.4.c.a.9.3	yes	4
4.3	odd	2		720.4.f.m.289.1		4
5.2	odd	4		1800.4.a.bk.1.1		2
5.3	odd	4		1800.4.a.bp.1.2		2
5.4	even	2	inner	360.4.f.e.289.2		4
12.11	even	2		80.4.c.c.49.2		4
15.2	even	4		200.4.a.k.1.2		2
15.8	even	4		200.4.a.l.1.1		2
15.14	odd	2		40.4.c.a.9.2	✓	4
20.19	odd	2		720.4.f.m.289.2		4
24.5	odd	2		320.4.c.g.129.2		4
24.11	even	2		320.4.c.h.129.3		4
60.23	odd	4		400.4.a.v.1.2		2
60.47	odd	4		400.4.a.x.1.1		2
60.59	even	2		80.4.c.c.49.3		4
120.29	odd	2		320.4.c.g.129.3		4
120.53	even	4		1600.4.a.ce.1.2		2
120.59	even	2		320.4.c.h.129.2		4
120.77	even	4		1600.4.a.cl.1.1		2
120.83	odd	4		1600.4.a.cm.1.1		2
120.107	odd	4		1600.4.a.cf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.c.a.9.2	✓	4	15.14	odd	2
40.4.c.a.9.3	yes	4	3.2	odd	2
80.4.c.c.49.2		4	12.11	even	2
80.4.c.c.49.3		4	60.59	even	2
200.4.a.k.1.2		2	15.2	even	4
200.4.a.l.1.1		2	15.8	even	4
320.4.c.g.129.2		4	24.5	odd	2
320.4.c.g.129.3		4	120.29	odd	2
320.4.c.h.129.2		4	120.59	even	2
320.4.c.h.129.3		4	24.11	even	2
360.4.f.e.289.1		4	1.1	even	1	trivial
360.4.f.e.289.2		4	5.4	even	2	inner
400.4.a.v.1.2		2	60.23	odd	4
400.4.a.x.1.1		2	60.47	odd	4
720.4.f.m.289.1		4	4.3	odd	2
720.4.f.m.289.2		4	20.19	odd	2
1600.4.a.ce.1.2		2	120.53	even	4
1600.4.a.cf.1.2		2	120.107	odd	4
1600.4.a.cl.1.1		2	120.77	even	4
1600.4.a.cm.1.1		2	120.83	odd	4
1800.4.a.bk.1.1		2	5.2	odd	4
1800.4.a.bp.1.2		2	5.3	odd	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:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.79796 - 6.89898i) q^{5} +16.6969i q^{7} +19.1918 q^{11} -61.7980i q^{13} +30.3837i q^{17} -59.1918 q^{19} +205.687i q^{23} +(29.8082 + 121.394i) q^{25} +8.38367 q^{29} +331.151 q^{31} +(115.192 - 146.899i) q^{35} +266.565i q^{37} +320.788 q^{41} -83.1214i q^{43} +276.434i q^{47} +64.2122 q^{49} +390.888i q^{53} +(-168.849 - 132.404i) q^{55} +779.110 q^{59} -483.171 q^{61} +(-426.343 + 543.696i) q^{65} +123.707i q^{67} -187.233 q^{71} +778.706i q^{73} +320.445i q^{77} +446.384 q^{79} -1054.05i q^{83} +(209.616 - 267.314i) q^{85} -94.8490 q^{89} +1031.84 q^{91} +(520.767 + 408.363i) q^{95} +252.041i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 80 q^{11} - 80 q^{19} + 276 q^{25} - 280 q^{29} + 384 q^{31} + 304 q^{35} + 1048 q^{41} + 492 q^{49} - 1616 q^{55} + 1392 q^{59} - 1384 q^{61} - 608 q^{65} - 1376 q^{71} + 1472 q^{79} + 1152 q^{85}+ \cdots + 1456 q^{95}+O(q^{100}) \) 4 * q + 4 * q^5 - 80 * q^11 - 80 * q^19 + 276 * q^25 - 280 * q^29 + 384 * q^31 + 304 * q^35 + 1048 * q^41 + 492 * q^49 - 1616 * q^55 + 1392 * q^59 - 1384 * q^61 - 608 * q^65 - 1376 * q^71 + 1472 * q^79 + 1152 * q^85 - 1320 * q^89 + 992 * q^91 + 1456 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more