LMFDB - Embedded newform 756.2.b.b.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(55,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("756.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.b (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:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			55.1
Root			$-1.16372i$ of defining polynomial
Character	$\chi$	$=$	756.55
Dual form			756.2.b.b.55.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +0.250492i q^{5} +2.64575 q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +0.250492i q^{5} +2.64575 q^{7} +2.82843i q^{8} +0.354249 q^{10} +4.90538i q^{11} -3.74166i q^{14} +4.00000 q^{16} +3.74166i q^{17} -8.64575 q^{19} -0.500983i q^{20} +6.93725 q^{22} +9.14802i q^{23} +4.93725 q^{25} -5.29150 q^{28} +8.29150 q^{31} -5.65685i q^{32} +5.29150 q^{34} +0.662739i q^{35} +3.93725 q^{37} +12.2269i q^{38} -0.708497 q^{40} -12.4774i q^{41} -9.81076i q^{44} +12.9373 q^{46} +7.00000 q^{49} -6.98233i q^{50} -1.22876 q^{55} +7.48331i q^{56} -11.7260i q^{62} -8.00000 q^{64} -7.48331i q^{68} +0.937254 q^{70} +13.3907i q^{71} -5.56812i q^{74} +17.2915 q^{76} +12.9784i q^{77} +1.00197i q^{80} -17.6458 q^{82} -0.937254 q^{85} -13.8745 q^{88} +11.4755i q^{89} -18.2960i q^{92} -2.16569i q^{95} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 12 q^{10} + 16 q^{16} - 24 q^{19} - 4 q^{22} - 12 q^{25} + 12 q^{31} - 16 q^{37} - 24 q^{40} + 20 q^{46} + 28 q^{49} + 48 q^{55} - 32 q^{64} - 28 q^{70} + 48 q^{76} - 60 q^{82} + 28 q^{85} + 8 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 12 * q^10 + 16 * q^16 - 24 * q^19 - 4 * q^22 - 12 * q^25 + 12 * q^31 - 16 * q^37 - 24 * q^40 + 20 * q^46 + 28 * q^49 + 48 * q^55 - 32 * q^64 - 28 * q^70 + 48 * q^76 - 60 * q^82 + 28 * q^85 + 8 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0.250492i	0.112023i	0.998430	+	0.0560116i	$0.0178384\pi$
$5$	0.250492i	0.112023i	−0.998430	+	0.0560116i	$0.982162\pi$
$6$	0	0
$7$	2.64575	1.00000
$7$	2.64575	1.00000
$8$	2.82843i	1.00000i
$9$	0	0
$10$	0.354249	0.112023
$11$	4.90538i	1.47903i	0.673141	+	0.739514i	$0.264945\pi$
$11$	4.90538i	1.47903i	−0.673141	+	0.739514i	$0.735055\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	− 3.74166i	− 1.00000i
$15$	0	0
$16$	4.00000	1.00000
$17$	3.74166i	0.907485i	0.891133	+	0.453743i	$0.149911\pi$
$17$	3.74166i	0.907485i	−0.891133	+	0.453743i	$0.850089\pi$
$18$	0	0
$19$	−8.64575	−1.98347	−0.991736	−	0.128298i	$-0.959049\pi$
$19$	−8.64575	−1.98347	−0.991736	+	0.128298i	$0.959049\pi$
$20$	− 0.500983i	− 0.112023i
$21$	0	0
$22$	6.93725	1.47903
$23$	9.14802i	1.90749i	0.300610	+	0.953747i	$0.402810\pi$
$23$	9.14802i	1.90749i	−0.300610	+	0.953747i	$0.597190\pi$
$24$	0	0
$25$	4.93725	0.987451
$26$	0	0
$27$	0	0
$28$	−5.29150	−1.00000
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.29150	1.48920	0.744599	−	0.667512i	$-0.232641\pi$
$31$	8.29150	1.48920	0.744599	+	0.667512i	$0.232641\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	5.29150	0.907485
$35$	0.662739i	0.112023i
$36$	0	0
$37$	3.93725	0.647281	0.323640	−	0.946180i	$-0.395093\pi$
$37$	3.93725	0.647281	0.323640	+	0.946180i	$0.395093\pi$
$38$	12.2269i	1.98347i
$39$	0	0
$40$	−0.708497	−0.112023
$41$	− 12.4774i	− 1.94865i	−0.225152	−	0.974324i	$-0.572288\pi$
$41$	− 12.4774i	− 1.94865i	0.225152	−	0.974324i	$-0.427712\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	− 9.81076i	− 1.47903i
$45$	0	0
$46$	12.9373	1.90749
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	− 6.98233i	− 0.987451i
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−1.22876	−0.165686
$56$	7.48331i	1.00000i
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	− 11.7260i	− 1.48920i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 7.48331i	− 0.907485i
$69$	0	0
$70$	0.937254	0.112023
$71$	13.3907i	1.58918i	0.607147	+	0.794590i	$0.292314\pi$
$71$	13.3907i	1.58918i	−0.607147	+	0.794590i	$0.707686\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	− 5.56812i	− 0.647281i
$75$	0	0
$76$	17.2915	1.98347
$77$	12.9784i	1.47903i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.00197i	0.112023i
$81$	0	0
$82$	−17.6458	−1.94865
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−0.937254	−0.101659
$86$	0	0
$87$	0	0
$88$	−13.8745	−1.47903
$89$	11.4755i	1.21640i	0.793785	+	0.608198i	$0.208107\pi$
$89$	11.4755i	1.21640i	−0.793785	+	0.608198i	$0.791893\pi$
$90$	0	0
$91$	0	0
$92$	− 18.2960i	− 1.90749i
$93$	0	0
$94$	0	0
$95$	− 2.16569i	− 0.222195i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 9.89949i	− 1.00000i
$99$	0	0
$100$	−9.87451	−0.987451
$101$	− 18.7083i	− 1.86154i	−0.365600	−	0.930772i	$-0.619136\pi$
$101$	− 18.7083i	− 1.86154i	0.365600	−	0.930772i	$-0.380864\pi$
$102$	0	0
$103$	−9.70850	−0.956607	−0.478303	−	0.878195i	$-0.658748\pi$
$103$	−9.70850	−0.956607	−0.478303	+	0.878195i	$0.658748\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.07107i	− 0.683586i	−0.939775	−	0.341793i	$-0.888966\pi$
$107$	− 7.07107i	− 0.683586i	0.939775	−	0.341793i	$-0.111034\pi$
$108$	0	0
$109$	−10.8745	−1.04159	−0.520794	−	0.853682i	$-0.674364\pi$
$109$	−10.8745	−1.04159	−0.520794	+	0.853682i	$0.674364\pi$
$110$	1.73772i	0.165686i
$111$	0	0
$112$	10.5830	1.00000
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−2.29150	−0.213684
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.89949i	0.907485i
$120$	0	0
$121$	−13.0627	−1.18752
$122$	0	0
$123$	0	0
$124$	−16.5830	−1.48920
$125$	2.48920i	0.222641i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−22.8745	−1.98347
$134$	0	0
$135$	0	0
$136$	−10.5830	−0.907485
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	21.1660	1.79528	0.897639	−	0.440732i	$-0.145281\pi$
$139$	21.1660	1.79528	0.897639	+	0.440732i	$0.145281\pi$
$140$	− 1.32548i	− 0.112023i
$141$	0	0
$142$	18.9373	1.58918
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−7.87451	−0.647281
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	− 24.4539i	− 1.98347i
$153$	0	0
$154$	18.3542	1.47903
$155$	2.07695i	0.166825i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	1.41699	0.112023
$161$	24.2034i	1.90749i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	24.9549i	1.94865i
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	1.32548i	0.101659i
$171$	0	0
$172$	0	0
$173$	− 10.9745i	− 0.834374i	−0.908821	−	0.417187i	$-0.863016\pi$
$173$	− 10.9745i	− 0.834374i	0.908821	−	0.417187i	$-0.136984\pi$
$174$	0	0
$175$	13.0627	0.987451
$176$	19.6215i	1.47903i
$177$	0	0
$178$	16.2288	1.21640
$179$	18.3848i	1.37414i	0.726590	+	0.687071i	$0.241104\pi$
$179$	18.3848i	1.37414i	−0.726590	+	0.687071i	$0.758896\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−25.8745	−1.90749
$185$	0.986249i	0.0725105i
$186$	0	0
$187$	−18.3542	−1.34220
$188$	0	0
$189$	0	0
$190$	−3.06275	−0.222195
$191$	− 7.82254i	− 0.566019i	−0.959117	−	0.283010i	$-0.908667\pi$
$191$	− 7.82254i	− 0.566019i	0.959117	−	0.283010i	$-0.0913329\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−26.6458	−1.88887	−0.944434	−	0.328702i	$-0.893389\pi$
$199$	−26.6458	−1.88887	−0.944434	+	0.328702i	$0.893389\pi$
$200$	13.9647i	0.987451i
$201$	0	0
$202$	−26.4575	−1.86154
$203$	0	0
$204$	0	0
$205$	3.12549	0.218294
$206$	13.7299i	0.956607i
$207$	0	0
$208$	0	0
$209$	− 42.4107i	− 2.93361i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	−10.0000	−0.683586
$215$	0	0
$216$	0	0
$217$	21.9373	1.48920
$218$	15.3789i	1.04159i
$219$	0	0
$220$	2.45751	0.165686
$221$	0	0
$222$	0	0
$223$	25.2288	1.68944	0.844721	−	0.535207i	$-0.179766\pi$
$223$	25.2288	1.68944	0.844721	+	0.535207i	$0.179766\pi$
$224$	− 14.9666i	− 1.00000i
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	3.24067i	0.213684i
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	14.0000	0.907485
$239$	− 24.0416i	− 1.55512i	−0.628806	−	0.777562i	$-0.716456\pi$
$239$	− 24.0416i	− 1.55512i	0.628806	−	0.777562i	$-0.283544\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	18.4735i	1.18752i
$243$	0	0
$244$	0	0
$245$	1.75344i	0.112023i
$246$	0	0
$247$	0	0
$248$	23.4519i	1.48920i
$249$	0	0
$250$	3.52026	0.222641
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−44.8745	−2.82124
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	25.7063i	1.60352i	0.597648	+	0.801759i	$0.296102\pi$
$257$	25.7063i	1.60352i	−0.597648	+	0.801759i	$0.703898\pi$
$258$	0	0
$259$	10.4170	0.647281
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 28.7695i	− 1.77401i	−0.461764	−	0.887003i	$-0.652783\pi$
$263$	− 28.7695i	− 1.77401i	0.461764	−	0.887003i	$-0.347217\pi$
$264$	0	0
$265$	0	0
$266$	32.3494i	1.98347i
$267$	0	0
$268$	0	0
$269$	− 13.9804i	− 0.852399i	−0.904629	−	0.426199i	$-0.859852\pi$
$269$	− 13.9804i	− 0.852399i	0.904629	−	0.426199i	$-0.140148\pi$
$270$	0	0
$271$	−10.5830	−0.642872	−0.321436	−	0.946931i	$-0.604165\pi$
$271$	−10.5830	−0.642872	−0.321436	+	0.946931i	$0.604165\pi$
$272$	14.9666i	0.907485i
$273$	0	0
$274$	0	0
$275$	24.2191i	1.46047i
$276$	0	0
$277$	−8.06275	−0.484443	−0.242222	−	0.970221i	$-0.577876\pi$
$277$	−8.06275	−0.484443	−0.242222	+	0.970221i	$0.577876\pi$
$278$	− 29.9333i	− 1.79528i
$279$	0	0
$280$	−1.87451	−0.112023
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−26.4575	−1.57274	−0.786368	−	0.617758i	$-0.788041\pi$
$283$	−26.4575	−1.57274	−0.786368	+	0.617758i	$0.788041\pi$
$284$	− 26.7813i	− 1.58918i
$285$	0	0
$286$	0	0
$287$	− 33.0122i	− 1.94865i
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.1916i	1.53013i	0.643953	+	0.765065i	$0.277293\pi$
$293$	26.1916i	1.53013i	−0.643953	+	0.765065i	$0.722707\pi$
$294$	0	0
$295$	0	0
$296$	11.1362i	0.647281i
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−34.5830	−1.98347
$305$	0	0
$306$	0	0
$307$	27.3542	1.56119	0.780595	−	0.625038i	$-0.214916\pi$
$307$	27.3542	1.56119	0.780595	+	0.625038i	$0.214916\pi$
$308$	− 25.9568i	− 1.47903i
$309$	0	0
$310$	2.93725	0.166825
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	− 2.00393i	− 0.112023i
$321$	0	0
$322$	34.2288	1.90749
$323$	− 32.3494i	− 1.79997i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	35.2915	1.94865
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.7490	1.67501	0.837503	−	0.546433i	$-0.184015\pi$
$337$	30.7490	1.67501	0.837503	+	0.546433i	$0.184015\pi$
$338$	− 18.3848i	− 1.00000i
$339$	0	0
$340$	1.87451	0.101659
$341$	40.6730i	2.20256i
$342$	0	0
$343$	18.5203	1.00000
$344$	0	0
$345$	0	0
$346$	−15.5203	−0.834374
$347$	− 37.2548i	− 1.99994i	−0.00752011	−	0.999972i	$-0.502394\pi$
$347$	− 37.2548i	− 1.99994i	0.00752011	−	0.999972i	$-0.497606\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	− 18.4735i	− 0.987451i
$351$	0	0
$352$	27.7490	1.47903
$353$	− 36.4303i	− 1.93899i	−0.245110	−	0.969495i	$-0.578824\pi$
$353$	− 36.4303i	− 1.93899i	0.245110	−	0.969495i	$-0.421176\pi$
$354$	0	0
$355$	−3.35425	−0.178025
$356$	− 22.9509i	− 1.21640i
$357$	0	0
$358$	26.0000	1.37414
$359$	− 32.5269i	− 1.71670i	−0.513061	−	0.858352i	$-0.671488\pi$
$359$	− 32.5269i	− 1.71670i	0.513061	−	0.858352i	$-0.328512\pi$
$360$	0	0
$361$	55.7490	2.93416
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.7712	−0.562254	−0.281127	−	0.959671i	$-0.590708\pi$
$367$	−10.7712	−0.562254	−0.281127	+	0.959671i	$0.590708\pi$
$368$	36.5921i	1.90749i
$369$	0	0
$370$	1.39477	0.0725105
$371$	0	0
$372$	0	0
$373$	1.12549	0.0582758	0.0291379	−	0.999575i	$-0.490724\pi$
$373$	1.12549	0.0582758	0.0291379	+	0.999575i	$0.490724\pi$
$374$	25.9568i	1.34220i
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	4.33138i	0.222195i
$381$	0	0
$382$	−11.0627	−0.566019
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−3.25098	−0.165686
$386$	5.65685i	0.287926i
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−34.2288	−1.73102
$392$	19.7990i	1.00000i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	37.6828i	1.88887i
$399$	0	0
$400$	19.7490	0.987451
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	37.4166i	1.86154i
$405$	0	0
$406$	0	0
$407$	19.3137i	0.957346i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	− 4.42011i	− 0.218294i
$411$	0	0
$412$	19.4170	0.956607
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−59.9778	−2.93361
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	27.9373	1.36158	0.680789	−	0.732479i	$-0.261637\pi$
$421$	27.9373	1.36158	0.680789	+	0.732479i	$0.261637\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.4735i	0.896097i
$426$	0	0
$427$	0	0
$428$	14.1421i	0.683586i
$429$	0	0
$430$	0	0
$431$	26.1186i	1.25809i	0.777370	+	0.629044i	$0.216553\pi$
$431$	26.1186i	1.25809i	−0.777370	+	0.629044i	$0.783447\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	− 31.0240i	− 1.48920i
$435$	0	0
$436$	21.7490	1.04159
$437$	− 79.0915i	− 3.78346i
$438$	0	0
$439$	5.29150	0.252550	0.126275	−	0.991995i	$-0.459698\pi$
$439$	5.29150	0.252550	0.126275	+	0.991995i	$0.459698\pi$
$440$	− 3.47545i	− 0.165686i
$441$	0	0
$442$	0	0
$443$	− 41.4975i	− 1.97160i	−0.167913	−	0.985802i	$-0.553703\pi$
$443$	− 41.4975i	− 1.97160i	0.167913	−	0.985802i	$-0.446297\pi$
$444$	0	0
$445$	−2.87451	−0.136265
$446$	− 35.6788i	− 1.68944i
$447$	0	0
$448$	−21.1660	−1.00000
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	61.2065	2.88210
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.7490	1.99971	0.999857	−	0.0168929i	$-0.00537742\pi$
$457$	42.7490	1.99971	0.999857	+	0.0168929i	$0.00537742\pi$
$458$	0	0
$459$	0	0
$460$	4.58301	0.213684
$461$	9.97251i	0.464466i	0.972660	+	0.232233i	$0.0746032\pi$
$461$	9.97251i	0.464466i	−0.972660	+	0.232233i	$0.925397\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−42.6863	−1.95858
$476$	− 19.7990i	− 0.907485i
$477$	0	0
$478$	−34.0000	−1.55512
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	26.1255	1.18752
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	2.47974	0.112023
$491$	− 16.3078i	− 0.735962i	−0.929833	−	0.367981i	$-0.880049\pi$
$491$	− 16.3078i	− 0.735962i	0.929833	−	0.367981i	$-0.119951\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	33.1660	1.48920
$497$	35.4284i	1.58918i
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 4.97840i	− 0.222641i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	4.68627	0.208536
$506$	63.4621i	2.82124i
$507$	0	0
$508$	0	0
$509$	26.1916i	1.16092i	0.814288	+	0.580461i	$0.197128\pi$
$509$	26.1916i	1.16092i	−0.814288	+	0.580461i	$0.802872\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	36.3542	1.60352
$515$	− 2.43190i	− 0.107162i
$516$	0	0
$517$	0	0
$518$	− 14.7319i	− 0.647281i
$519$	0	0
$520$	0	0
$521$	− 26.7083i	− 1.17011i	−0.810993	−	0.585056i	$-0.801073\pi$
$521$	− 26.7083i	− 1.17011i	0.810993	−	0.585056i	$-0.198927\pi$
$522$	0	0
$523$	6.16601	0.269621	0.134810	−	0.990871i	$-0.456957\pi$
$523$	6.16601	0.269621	0.134810	+	0.990871i	$0.456957\pi$
$524$	0	0
$525$	0	0
$526$	−40.6863	−1.77401
$527$	31.0240i	1.35143i
$528$	0	0
$529$	−60.6863	−2.63853
$530$	0	0
$531$	0	0
$532$	45.7490	1.98347
$533$	0	0
$534$	0	0
$535$	1.77124	0.0765775
$536$	0	0
$537$	0	0
$538$	−19.7712	−0.852399
$539$	34.3377i	1.47903i
$540$	0	0
$541$	−43.6863	−1.87822	−0.939110	−	0.343617i	$-0.888348\pi$
$541$	−43.6863	−1.87822	−0.939110	+	0.343617i	$0.888348\pi$
$542$	14.9666i	0.642872i
$543$	0	0
$544$	21.1660	0.907485
$545$	− 2.72397i	− 0.116682i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	34.2510	1.46047
$551$	0	0
$552$	0	0
$553$	0	0
$554$	11.4024i	0.484443i
$555$	0	0
$556$	−42.3320	−1.79528
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	2.65095i	0.112023i
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	37.4166i	1.57274i
$567$	0	0
$568$	−37.8745	−1.58918
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	−46.6863	−1.94865
$575$	45.1661i	1.88356i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 4.24264i	− 0.176471i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	37.0405	1.53013
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−71.6863	−2.95378
$590$	0	0
$591$	0	0
$592$	15.7490	0.647281
$593$	− 22.1995i	− 0.911622i	−0.890077	−	0.455811i	$-0.849349\pi$
$593$	− 22.1995i	− 0.911622i	0.890077	−	0.455811i	$-0.150651\pi$
$594$	0	0
$595$	−2.47974	−0.101659
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.0950i	1.22965i	0.788664	+	0.614824i	$0.210773\pi$
$599$	30.0950i	1.22965i	−0.788664	+	0.614824i	$0.789227\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.27211i	− 0.133030i
$606$	0	0
$607$	−26.4575	−1.07388	−0.536939	−	0.843621i	$-0.680419\pi$
$607$	−26.4575	−1.07388	−0.536939	+	0.843621i	$0.680419\pi$
$608$	48.9078i	1.98347i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	7.12549	0.287796	0.143898	−	0.989593i	$-0.454036\pi$
$613$	7.12549	0.287796	0.143898	+	0.989593i	$0.454036\pi$
$614$	− 38.6847i	− 1.56119i
$615$	0	0
$616$	−36.7085	−1.47903
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	28.4170	1.14218	0.571088	−	0.820889i	$-0.306522\pi$
$619$	28.4170	1.14218	0.571088	+	0.820889i	$0.306522\pi$
$620$	− 4.15390i	− 0.166825i
$621$	0	0
$622$	0	0
$623$	30.3612i	1.21640i
$624$	0	0
$625$	24.0627	0.962510
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.7319i	0.587398i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−2.83399	−0.112023
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	11.4797	0.452717	0.226358	−	0.974044i	$-0.427318\pi$
$643$	11.4797	0.452717	0.226358	+	0.974044i	$0.427318\pi$
$644$	− 48.4068i	− 1.90749i
$645$	0	0
$646$	−45.7490	−1.79997
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	− 49.9097i	− 1.94865i
$657$	0	0
$658$	0	0
$659$	51.3082i	1.99868i	0.0362652	+	0.999342i	$0.488454\pi$
$659$	51.3082i	1.99868i	−0.0362652	+	0.999342i	$0.511546\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.72987i	− 0.222195i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	− 43.4857i	− 1.67501i
$675$	0	0
$676$	−26.0000	−1.00000
$677$	48.1563i	1.85080i	0.378997	+	0.925398i	$0.376269\pi$
$677$	48.1563i	1.85080i	−0.378997	+	0.925398i	$0.623731\pi$
$678$	0	0
$679$	0	0
$680$	− 2.65095i	− 0.101659i
$681$	0	0
$682$	57.5203	2.20256
$683$	− 7.55633i	− 0.289135i	−0.989495	−	0.144568i	$-0.953821\pi$
$683$	− 7.55633i	− 0.289135i	0.989495	−	0.144568i	$-0.0461791\pi$
$684$	0	0
$685$	0	0
$686$	− 26.1916i	− 1.00000i
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−42.3320	−1.61039	−0.805193	−	0.593013i	$-0.797938\pi$
$691$	−42.3320	−1.61039	−0.805193	+	0.593013i	$0.797938\pi$
$692$	21.9490i	0.834374i
$693$	0	0
$694$	−52.6863	−1.99994
$695$	5.30191i	0.201113i
$696$	0	0
$697$	46.6863	1.76837
$698$	0	0
$699$	0	0
$700$	−26.1255	−0.987451
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−34.0405	−1.28386
$704$	− 39.2430i	− 1.47903i
$705$	0	0
$706$	−51.5203	−1.93899
$707$	− 49.4975i	− 1.86154i
$708$	0	0
$709$	−40.8745	−1.53507	−0.767537	−	0.641004i	$-0.778518\pi$
$709$	−40.8745	−1.53507	−0.767537	+	0.641004i	$0.778518\pi$
$710$	4.74362i	0.178025i
$711$	0	0
$712$	−32.4575	−1.21640
$713$	75.8508i	2.84064i
$714$	0	0
$715$	0	0
$716$	− 36.7696i	− 1.37414i
$717$	0	0
$718$	−46.0000	−1.71670
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−25.6863	−0.956607
$722$	− 78.8410i	− 2.93416i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	52.9150	1.96251	0.981255	−	0.192715i	$-0.0617292\pi$
$727$	52.9150	1.96251	0.981255	+	0.192715i	$0.0617292\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	15.2328i	0.562254i
$735$	0	0
$736$	51.7490	1.90749
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	− 1.97250i	− 0.0725105i
$741$	0	0
$742$	0	0
$743$	− 49.9827i	− 1.83369i	−0.399245	−	0.916844i	$-0.630728\pi$
$743$	− 49.9827i	− 1.83369i	0.399245	−	0.916844i	$-0.369272\pi$
$744$	0	0
$745$	0	0
$746$	− 1.59169i	− 0.0582758i
$747$	0	0
$748$	36.7085	1.34220
$749$	− 18.7083i	− 0.683586i
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	6.12549	0.222195
$761$	− 41.1582i	− 1.49198i	−0.665955	−	0.745992i	$-0.731976\pi$
$761$	− 41.1582i	− 1.49198i	0.665955	−	0.745992i	$-0.268024\pi$
$762$	0	0
$763$	−28.7712	−1.04159
$764$	15.6451i	0.566019i
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	4.59759i	0.165686i
$771$	0	0
$772$	8.00000	0.287926
$773$	− 47.6553i	− 1.71404i	−0.515282	−	0.857021i	$-0.672313\pi$
$773$	− 47.6553i	− 1.71404i	0.515282	−	0.857021i	$-0.327687\pi$
$774$	0	0
$775$	40.9373	1.47051
$776$	0	0
$777$	0	0
$778$	0	0
$779$	107.877i	3.86509i
$780$	0	0
$781$	−65.6863	−2.35044
$782$	48.4068i	1.73102i
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	21.1660	0.754487	0.377243	−	0.926114i	$-0.376872\pi$
$787$	21.1660	0.754487	0.377243	+	0.926114i	$0.376872\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	53.2915	1.88887
$797$	− 50.6612i	− 1.79451i	−0.441511	−	0.897256i	$-0.645557\pi$
$797$	− 50.6612i	− 1.79451i	0.441511	−	0.897256i	$-0.354443\pi$
$798$	0	0
$799$	0	0
$800$	− 27.9293i	− 0.987451i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−6.06275	−0.213684
$806$	0	0
$807$	0	0
$808$	52.9150	1.86154
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−11.8340	−0.415548	−0.207774	−	0.978177i	$-0.566622\pi$
$811$	−11.8340	−0.415548	−0.207774	+	0.978177i	$0.566622\pi$
$812$	0	0
$813$	0	0
$814$	27.3137	0.957346
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−6.25098	−0.218294
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	− 27.4598i	− 0.956607i
$825$	0	0
$826$	0	0
$827$	21.6097i	0.751444i	0.926732	+	0.375722i	$0.122605\pi$
$827$	21.6097i	0.751444i	−0.926732	+	0.375722i	$0.877395\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.1916i	0.907485i
$834$	0	0
$835$	0	0
$836$	84.8214i	2.93361i
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 39.5092i	− 1.36158i
$843$	0	0
$844$	0	0
$845$	3.25639i	0.112023i
$846$	0	0
$847$	−34.5608	−1.18752
$848$	0	0
$849$	0	0
$850$	26.1255	0.896097
$851$	36.0181i	1.23468i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	20.0000	0.683586
$857$	− 15.4833i	− 0.528900i	−0.964399	−	0.264450i	$-0.914810\pi$
$857$	− 15.4833i	− 0.528900i	0.964399	−	0.264450i	$-0.0851905\pi$
$858$	0	0
$859$	23.1033	0.788273	0.394137	−	0.919052i	$-0.371044\pi$
$859$	23.1033	0.788273	0.394137	+	0.919052i	$0.371044\pi$
$860$	0	0
$861$	0	0
$862$	36.9373	1.25809
$863$	− 7.07107i	− 0.240702i	−0.992731	−	0.120351i	$-0.961598\pi$
$863$	− 7.07107i	− 0.240702i	0.992731	−	0.120351i	$-0.0384020\pi$
$864$	0	0
$865$	2.74902	0.0934693
$866$	0	0
$867$	0	0
$868$	−43.8745	−1.48920
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 30.7578i	− 1.04159i
$873$	0	0
$874$	−111.852	−3.78346
$875$	6.58580i	0.222641i
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	− 7.48331i	− 0.252550i
$879$	0	0
$880$	−4.91503	−0.165686
$881$	− 39.4362i	− 1.32864i	−0.747448	−	0.664320i	$-0.768721\pi$
$881$	− 39.4362i	− 1.32864i	0.747448	−	0.664320i	$-0.231279\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−58.6863	−1.97160
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	4.06517i	0.136265i
$891$	0	0
$892$	−50.4575	−1.68944
$893$	0	0
$894$	0	0
$895$	−4.60523	−0.153936
$896$	29.9333i	1.00000i
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 86.5591i	− 2.88210i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 15.5563i	− 0.515405i	−0.966224	−	0.257702i	$-0.917035\pi$
$911$	− 15.5563i	− 0.515405i	0.966224	−	0.257702i	$-0.0829654\pi$
$912$	0	0
$913$	0	0
$914$	− 60.4562i	− 1.99971i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	− 6.48135i	− 0.213684i
$921$	0	0
$922$	14.1033	0.464466
$923$	0	0
$924$	0	0
$925$	19.4392	0.639158
$926$	0	0
$927$	0	0
$928$	0	0
$929$	48.6415i	1.59588i	0.602739	+	0.797939i	$0.294076\pi$
$929$	48.6415i	1.59588i	−0.602739	+	0.797939i	$0.705924\pi$
$930$	0	0
$931$	−60.5203	−1.98347
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 4.59759i	− 0.150357i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	51.1622i	1.66784i	0.551886	+	0.833920i	$0.313908\pi$
$941$	51.1622i	1.66784i	−0.551886	+	0.833920i	$0.686092\pi$
$942$	0	0
$943$	114.144	3.71703
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 54.2254i	− 1.76209i	−0.473034	−	0.881044i	$-0.656841\pi$
$947$	− 54.2254i	− 1.76209i	0.473034	−	0.881044i	$-0.343159\pi$
$948$	0	0
$949$	0	0
$950$	60.3675i	1.95858i
$951$	0	0
$952$	−28.0000	−0.907485
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	1.95948	0.0634073
$956$	48.0833i	1.55512i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	37.7490	1.21771
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.00197i	− 0.0322544i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 36.9470i	− 1.18752i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	56.0000	1.79528
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−56.2915	−1.79908
$980$	− 3.50688i	− 0.112023i
$981$	0	0
$982$	−23.0627	−0.735962
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	− 46.9038i	− 1.48920i
$993$	0	0
$994$	50.1033	1.58918
$995$	− 6.67454i	− 0.211597i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.b.b.55.1	yes	4
3.2	odd	2	inner	756.2.b.b.55.4	yes	4
4.3	odd	2		756.2.b.a.55.3	yes	4
7.6	odd	2		756.2.b.a.55.2	✓	4
12.11	even	2		756.2.b.a.55.2	✓	4
21.20	even	2		756.2.b.a.55.3	yes	4
28.27	even	2	inner	756.2.b.b.55.4	yes	4
84.83	odd	2	CM	756.2.b.b.55.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.b.a.55.2	✓	4	7.6	odd	2
756.2.b.a.55.2	✓	4	12.11	even	2
756.2.b.a.55.3	yes	4	4.3	odd	2
756.2.b.a.55.3	yes	4	21.20	even	2
756.2.b.b.55.1	yes	4	1.1	even	1	trivial
756.2.b.b.55.1	yes	4	84.83	odd	2	CM
756.2.b.b.55.4	yes	4	3.2	odd	2	inner
756.2.b.b.55.4	yes	4	28.27	even	2	inner

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +0.250492i q^{5} +2.64575 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +0.250492i q^{5} +2.64575 q^{7} +2.82843i q^{8} +0.354249 q^{10} +4.90538i q^{11} -3.74166i q^{14} +4.00000 q^{16} +3.74166i q^{17} -8.64575 q^{19} -0.500983i q^{20} +6.93725 q^{22} +9.14802i q^{23} +4.93725 q^{25} -5.29150 q^{28} +8.29150 q^{31} -5.65685i q^{32} +5.29150 q^{34} +0.662739i q^{35} +3.93725 q^{37} +12.2269i q^{38} -0.708497 q^{40} -12.4774i q^{41} -9.81076i q^{44} +12.9373 q^{46} +7.00000 q^{49} -6.98233i q^{50} -1.22876 q^{55} +7.48331i q^{56} -11.7260i q^{62} -8.00000 q^{64} -7.48331i q^{68} +0.937254 q^{70} +13.3907i q^{71} -5.56812i q^{74} +17.2915 q^{76} +12.9784i q^{77} +1.00197i q^{80} -17.6458 q^{82} -0.937254 q^{85} -13.8745 q^{88} +11.4755i q^{89} -18.2960i q^{92} -2.16569i q^{95} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 12 q^{10} + 16 q^{16} - 24 q^{19} - 4 q^{22} - 12 q^{25} + 12 q^{31} - 16 q^{37} - 24 q^{40} + 20 q^{46} + 28 q^{49} + 48 q^{55} - 32 q^{64} - 28 q^{70} + 48 q^{76} - 60 q^{82} + 28 q^{85} + 8 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 12 * q^10 + 16 * q^16 - 24 * q^19 - 4 * q^22 - 12 * q^25 + 12 * q^31 - 16 * q^37 - 24 * q^40 + 20 * q^46 + 28 * q^49 + 48 * q^55 - 32 * q^64 - 28 * q^70 + 48 * q^76 - 60 * q^82 + 28 * q^85 + 8 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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