LMFDB - Embedded newform 756.2.b.a.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			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	756.55
Dual form			756.2.b.a.55.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -3.99215i q^{5} +2.64575 q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -3.99215i q^{5} +2.64575 q^{7} +2.82843i q^{8} -5.64575 q^{10} -6.31959i q^{11} -3.74166i q^{14} +4.00000 q^{16} +3.74166i q^{17} +3.35425 q^{19} +7.98430i q^{20} -8.93725 q^{22} -2.07695i q^{23} -10.9373 q^{25} -5.29150 q^{28} +2.29150 q^{31} -5.65685i q^{32} +5.29150 q^{34} -10.5622i q^{35} -11.9373 q^{37} -4.74362i q^{38} +11.2915 q^{40} +8.73577i q^{41} +12.6392i q^{44} -2.93725 q^{46} +7.00000 q^{49} +15.4676i q^{50} -25.2288 q^{55} +7.48331i q^{56} -3.24067i q^{62} -8.00000 q^{64} -7.48331i q^{68} -14.9373 q^{70} +2.16569i q^{71} +16.8818i q^{74} -6.70850 q^{76} -16.7201i q^{77} -15.9686i q^{80} +12.3542 q^{82} +14.9373 q^{85} +17.8745 q^{88} +7.23282i q^{89} +4.15390i q^{92} -13.3907i q^{95} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 3.99215i	− 1.78534i	−0.450708	−	0.892672i	$-0.648828\pi$
$5$	− 3.99215i	− 1.78534i	0.450708	−	0.892672i	$-0.351172\pi$
$6$	0	0
$7$	2.64575	1.00000
$7$	2.64575	1.00000
$8$	2.82843i	1.00000i
$9$	0	0
$10$	−5.64575	−1.78534
$11$	− 6.31959i	− 1.90543i	−0.303867	−	0.952714i	$-0.598278\pi$
$11$	− 6.31959i	− 1.90543i	0.303867	−	0.952714i	$-0.401722\pi$
$12$	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
$19$	3.35425	0.769517	0.384759	−	0.923017i	$-0.374285\pi$
$19$	3.35425	0.769517	0.384759	+	0.923017i	$0.374285\pi$
$20$	7.98430i	1.78534i
$21$	0	0
$22$	−8.93725	−1.90543
$23$	− 2.07695i	− 0.433074i	−0.976274	−	0.216537i	$-0.930524\pi$
$23$	− 2.07695i	− 0.433074i	0.976274	−	0.216537i	$-0.0694763\pi$
$24$	0	0
$25$	−10.9373	−2.18745
$26$	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
$31$	2.29150	0.411566	0.205783	−	0.978598i	$-0.434026\pi$
$31$	2.29150	0.411566	0.205783	+	0.978598i	$0.434026\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	5.29150	0.907485
$35$	− 10.5622i	− 1.78534i
$36$	0	0
$37$	−11.9373	−1.96247	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−11.9373	−1.96247	−0.981236	+	0.192809i	$0.938240\pi$
$38$	− 4.74362i	− 0.769517i
$39$	0	0
$40$	11.2915	1.78534
$41$	8.73577i	1.36430i	0.731213	+	0.682149i	$0.238955\pi$
$41$	8.73577i	1.36430i	−0.731213	+	0.682149i	$0.761045\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	12.6392i	1.90543i
$45$	0	0
$46$	−2.93725	−0.433074
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	15.4676i	2.18745i
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−25.2288	−3.40184
$56$	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			$\pi$
$62$	− 3.24067i	− 0.411566i
$63$	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
$70$	−14.9373	−1.78534
$71$	2.16569i	0.257020i	0.991708	+	0.128510i	$0.0410194\pi$
$71$	2.16569i	0.257020i	−0.991708	+	0.128510i	$0.958981\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	16.8818i	1.96247i
$75$	0	0
$76$	−6.70850	−0.769517
$77$	− 16.7201i	− 1.90543i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 15.9686i	− 1.78534i
$81$	0	0
$82$	12.3542	1.36430
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	14.9373	1.62017
$86$	0	0
$87$	0	0
$88$	17.8745	1.90543
$89$	7.23282i	0.766678i	0.923608	+	0.383339i	$0.125226\pi$
$89$	7.23282i	0.766678i	−0.923608	+	0.383339i	$0.874774\pi$
$90$	0	0
$91$	0	0
$92$	4.15390i	0.433074i
$93$	0	0
$94$	0	0
$95$	− 13.3907i	− 1.37385i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 9.89949i	− 1.00000i
$99$	0	0
$100$	21.8745	2.18745
$101$	− 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
$103$	20.2915	1.99938	0.999691	−	0.0248745i	$-0.00791862\pi$
$103$	20.2915	1.99938	0.999691	+	0.0248745i	$0.00791862\pi$
$104$	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
$109$	20.8745	1.99942	0.999708	−	0.0241802i	$-0.00769755\pi$
$109$	20.8745	1.99942	0.999708	+	0.0241802i	$0.00769755\pi$
$110$	35.6788i	3.40184i
$111$	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
$115$	−8.29150	−0.773187
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.89949i	0.907485i
$120$	0	0
$121$	−28.9373	−2.63066
$122$	0	0
$123$	0	0
$124$	−4.58301	−0.411566
$125$	23.7024i	2.12001i
$126$	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
$133$	8.87451	0.769517
$134$	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	$0.145281\pi$
$140$	21.1245i	1.78534i
$141$	0	0
$142$	3.06275	0.257020
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	23.8745	1.96247
$149$	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			$\pi$
$152$	9.48725i	0.769517i
$153$	0	0
$154$	−23.6458	−1.90543
$155$	− 9.14802i	− 0.734787i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−22.5830	−1.78534
$161$	− 5.49510i	− 0.433074i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 17.4715i	− 1.36430i
$165$	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
$170$	− 21.1245i	− 1.62017i
$171$	0	0
$172$	0	0
$173$	− 15.2171i	− 1.15694i	−0.815705	−	0.578468i	$-0.803651\pi$
$173$	− 15.2171i	− 1.15694i	0.815705	−	0.578468i	$-0.196349\pi$
$174$	0	0
$175$	−28.9373	−2.18745
$176$	− 25.2784i	− 1.90543i
$177$	0	0
$178$	10.2288	0.766678
$179$	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
$184$	5.87451	0.433074
$185$	47.6553i	3.50369i
$186$	0	0
$187$	23.6458	1.72915
$188$	0	0
$189$	0	0
$190$	−18.9373	−1.37385
$191$	− 19.0475i	− 1.37823i	−0.724652	−	0.689115i	$-0.758000\pi$
$191$	− 19.0475i	− 1.37823i	0.724652	−	0.689115i	$-0.242000\pi$
$192$	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
$199$	21.3542	1.51376	0.756881	−	0.653552i	$-0.226722\pi$
$199$	21.3542	1.51376	0.756881	+	0.653552i	$0.226722\pi$
$200$	− 30.9352i	− 2.18745i
$201$	0	0
$202$	−26.4575	−1.86154
$203$	0	0
$204$	0	0
$205$	34.8745	2.43574
$206$	− 28.6965i	− 1.99938i
$207$	0	0
$208$	0	0
$209$	− 21.1975i	− 1.46626i
$210$	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
$217$	6.06275	0.411566
$218$	− 29.5210i	− 1.99942i
$219$	0	0
$220$	50.4575	3.40184
$221$	0	0
$222$	0	0
$223$	1.22876	0.0822836	0.0411418	−	0.999153i	$-0.486900\pi$
$223$	1.22876	0.0822836	0.0411418	+	0.999153i	$0.486900\pi$
$224$	− 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			$\pi$
$230$	11.7260i	0.773187i
$231$	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			$\pi$
$242$	40.9235i	2.63066i
$243$	0	0
$244$	0	0
$245$	− 27.9450i	− 1.78534i
$246$	0	0
$247$	0	0
$248$	6.48135i	0.411566i
$249$	0	0
$250$	33.5203	2.12001
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−13.1255	−0.825193
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 29.4480i	− 1.83692i	−0.395519	−	0.918458i	$-0.629435\pi$
$257$	− 29.4480i	− 1.83692i	0.395519	−	0.918458i	$-0.370565\pi$
$258$	0	0
$259$	−31.5830	−1.96247
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.3553i	1.68680i	0.537285	+	0.843401i	$0.319450\pi$
$263$	27.3553i	1.68680i	−0.537285	+	0.843401i	$0.680550\pi$
$264$	0	0
$265$	0	0
$266$	− 12.5504i	− 0.769517i
$267$	0	0
$268$	0	0
$269$	32.6887i	1.99306i	0.0832151	+	0.996532i	$0.473481\pi$
$269$	32.6887i	1.99306i	−0.0832151	+	0.996532i	$0.526519\pi$
$270$	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
$275$	69.1190i	4.16803i
$276$	0	0
$277$	−23.9373	−1.43825	−0.719125	−	0.694881i	$-0.755457\pi$
$277$	−23.9373	−1.43825	−0.719125	+	0.694881i	$0.755457\pi$
$278$	− 29.9333i	− 1.79528i
$279$	0	0
$280$	29.8745	1.78534
$281$	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	$0.788041\pi$
$284$	− 4.33138i	− 0.257020i
$285$	0	0
$286$	0	0
$287$	23.1127i	1.36430i
$288$	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
$296$	− 33.7637i	− 1.96247i
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	13.4170	0.769517
$305$	0	0
$306$	0	0
$307$	−32.6458	−1.86319	−0.931596	−	0.363496i	$-0.881583\pi$
$307$	−32.6458	−1.86319	−0.931596	+	0.363496i	$0.881583\pi$
$308$	33.4401i	1.90543i
$309$	0	0
$310$	−12.9373	−0.734787
$311$	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
$320$	31.9372i	1.78534i
$321$	0	0
$322$	−7.77124	−0.433074
$323$	12.5504i	0.698326i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−24.7085	−1.36430
$329$	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
$337$	−32.7490	−1.78395	−0.891976	−	0.452082i	$-0.850681\pi$
$337$	−32.7490	−1.78395	−0.891976	+	0.452082i	$0.850681\pi$
$338$	− 18.3848i	− 1.00000i
$339$	0	0
$340$	−29.8745	−1.62017
$341$	− 14.4814i	− 0.784210i
$342$	0	0
$343$	18.5203	1.00000
$344$	0	0
$345$	0	0
$346$	−21.5203	−1.15694
$347$	18.8700i	1.01300i	0.862241	+	0.506498i	$0.169060\pi$
$347$	18.8700i	1.01300i	−0.862241	+	0.506498i	$0.830940\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	40.9235i	2.18745i
$351$	0	0
$352$	−35.7490	−1.90543
$353$	10.2387i	0.544952i	0.962163	+	0.272476i	$0.0878426\pi$
$353$	10.2387i	0.544952i	−0.962163	+	0.272476i	$0.912157\pi$
$354$	0	0
$355$	8.64575	0.458869
$356$	− 14.4656i	− 0.766678i
$357$	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
$361$	−7.74902	−0.407843
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	37.2288	1.94333	0.971663	−	0.236372i	$-0.0759584\pi$
$367$	37.2288	1.94333	0.971663	+	0.236372i	$0.0759584\pi$
$368$	− 8.30781i	− 0.433074i
$369$	0	0
$370$	67.3948	3.50369
$371$	0	0
$372$	0	0
$373$	32.8745	1.70218	0.851089	−	0.525022i	$-0.175943\pi$
$373$	32.8745	1.70218	0.851089	+	0.525022i	$0.175943\pi$
$374$	− 33.4401i	− 1.72915i
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	26.7813i	1.37385i
$381$	0	0
$382$	−26.9373	−1.37823
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−66.7490	−3.40184
$386$	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
$391$	7.77124	0.393009
$392$	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			$\pi$
$398$	− 30.1995i	− 1.51376i
$399$	0	0
$400$	−43.7490	−2.18745
$401$	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
$407$	75.4386i	3.73935i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	− 49.3200i	− 2.43574i
$411$	0	0
$412$	−40.5830	−1.99938
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−29.9778	−1.46626
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	12.0627	0.587902	0.293951	−	0.955820i	$-0.405030\pi$
$421$	12.0627	0.587902	0.293951	+	0.955820i	$0.405030\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 40.9235i	− 1.98508i
$426$	0	0
$427$	0	0
$428$	14.1421i	0.683586i
$429$	0	0
$430$	0	0
$431$	14.8936i	0.717400i	0.933453	+	0.358700i	$0.116780\pi$
$431$	14.8936i	0.717400i	−0.933453	+	0.358700i	$0.883220\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	− 8.57402i	− 0.411566i
$435$	0	0
$436$	−41.7490	−1.99942
$437$	− 6.96661i	− 0.333258i
$438$	0	0
$439$	5.29150	0.252550	0.126275	−	0.991995i	$-0.459698\pi$
$439$	5.29150	0.252550	0.126275	+	0.991995i	$0.459698\pi$
$440$	− 71.3577i	− 3.40184i
$441$	0	0
$442$	0	0
$443$	14.6274i	0.694969i	0.937686	+	0.347484i	$0.112964\pi$
$443$	14.6274i	0.694969i	−0.937686	+	0.347484i	$0.887036\pi$
$444$	0	0
$445$	28.8745	1.36878
$446$	− 1.73772i	− 0.0822836i
$447$	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
$451$	55.2065	2.59957
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.7490	−0.970598	−0.485299	−	0.874348i	$-0.661289\pi$
$457$	−20.7490	−0.970598	−0.485299	+	0.874348i	$0.661289\pi$
$458$	0	0
$459$	0	0
$460$	16.5830	0.773187
$461$	31.1857i	1.45246i	0.687450	+	0.726232i	$0.258730\pi$
$461$	31.1857i	1.45246i	−0.687450	+	0.726232i	$0.741270\pi$
$462$	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
$475$	−36.6863	−1.68328
$476$	− 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
$484$	57.8745	2.63066
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−39.5203	−1.78534
$491$	− 27.5328i	− 1.24254i	−0.783597	−	0.621269i	$-0.786617\pi$
$491$	− 27.5328i	− 1.24254i	0.783597	−	0.621269i	$-0.213383\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	9.16601	0.411566
$497$	5.72987i	0.257020i
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 47.4048i	− 2.12001i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−74.6863	−3.32350
$506$	18.5622i	0.825193i
$507$	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
$514$	−41.6458	−1.83692
$515$	− 81.0067i	− 3.56958i
$516$	0	0
$517$	0	0
$518$	44.6651i	1.96247i
$519$	0	0
$520$	0	0
$521$	45.4166i	1.98974i	0.101177	+	0.994868i	$0.467739\pi$
$521$	45.4166i	1.98974i	−0.101177	+	0.994868i	$0.532261\pi$
$522$	0	0
$523$	36.1660	1.58143	0.790715	−	0.612185i	$-0.209709\pi$
$523$	36.1660	1.58143	0.790715	+	0.612185i	$0.209709\pi$
$524$	0	0
$525$	0	0
$526$	38.6863	1.68680
$527$	8.57402i	0.373490i
$528$	0	0
$529$	18.6863	0.812447
$530$	0	0
$531$	0	0
$532$	−17.7490	−0.769517
$533$	0	0
$534$	0	0
$535$	−28.2288	−1.22044
$536$	0	0
$537$	0	0
$538$	46.2288	1.99306
$539$	− 44.2372i	− 1.90543i
$540$	0	0
$541$	35.6863	1.53427	0.767136	−	0.641484i	$-0.221681\pi$
$541$	35.6863	1.53427	0.767136	+	0.641484i	$0.221681\pi$
$542$	14.9666i	0.642872i
$543$	0	0
$544$	21.1660	0.907485
$545$	− 83.3341i	− 3.56964i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	97.7490	4.16803
$551$	0	0
$552$	0	0
$553$	0	0
$554$	33.8524i	1.43825i
$555$	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
$560$	− 42.2489i	− 1.78534i
$561$	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
$568$	−6.12549	−0.257020
$569$	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
$574$	32.6863	1.36430
$575$	22.7162i	0.947329i
$576$	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
$589$	7.68627	0.316707
$590$	0	0
$591$	0	0
$592$	−47.7490	−1.96247
$593$	− 26.4421i	− 1.08585i	−0.839782	−	0.542923i	$-0.817317\pi$
$593$	− 26.4421i	− 1.08585i	0.839782	−	0.542923i	$-0.182683\pi$
$594$	0	0
$595$	39.5203	1.62017
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 48.4798i	− 1.98083i	−0.138122	−	0.990415i	$-0.544106\pi$
$599$	− 48.4798i	− 1.98083i	0.138122	−	0.990415i	$-0.455894\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	115.522i	4.69663i
$606$	0	0
$607$	−26.4575	−1.07388	−0.536939	−	0.843621i	$-0.680419\pi$
$607$	−26.4575	−1.07388	−0.536939	+	0.843621i	$0.680419\pi$
$608$	− 18.9745i	− 0.769517i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	38.8745	1.57013	0.785063	−	0.619416i	$-0.212630\pi$
$613$	38.8745	1.57013	0.785063	+	0.619416i	$0.212630\pi$
$614$	46.1681i	1.86319i
$615$	0	0
$616$	47.2915	1.90543
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−49.5830	−1.99291	−0.996455	−	0.0841320i	$-0.973188\pi$
$619$	−49.5830	−1.99291	−0.996455	+	0.0841320i	$0.973188\pi$
$620$	18.2960i	0.734787i
$621$	0	0
$622$	0	0
$623$	19.1363i	0.766678i
$624$	0	0
$625$	39.9373	1.59749
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 44.6651i	− 1.78091i
$630$	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
$640$	45.1660	1.78534
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−48.5203	−1.91345	−0.956726	−	0.290990i	$-0.906015\pi$
$643$	−48.5203	−1.91345	−0.956726	+	0.290990i	$0.906015\pi$
$644$	10.9902i	0.433074i
$645$	0	0
$646$	17.7490	0.698326
$647$	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
$656$	34.9431i	1.36430i
$657$	0	0
$658$	0	0
$659$	− 27.2666i	− 1.06216i	−0.847323	−	0.531078i	$-0.821787\pi$
$659$	− 27.2666i	− 1.06216i	0.847323	−	0.531078i	$-0.178213\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 35.4284i	− 1.37385i
$666$	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	$0.177784\pi$
$674$	46.3141i	1.78395i
$675$	0	0
$676$	−26.0000	−1.00000
$677$	− 6.99805i	− 0.268957i	−0.990917	−	0.134478i	$-0.957064\pi$
$677$	− 6.99805i	− 0.268957i	0.990917	−	0.134478i	$-0.0429359\pi$
$678$	0	0
$679$	0	0
$680$	42.2489i	1.62017i
$681$	0	0
$682$	−20.4797	−0.784210
$683$	48.5685i	1.85842i	0.369548	+	0.929212i	$0.379512\pi$
$683$	48.5685i	1.85842i	−0.369548	+	0.929212i	$0.620488\pi$
$684$	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	$0.797938\pi$
$692$	30.4342i	1.15694i
$693$	0	0
$694$	26.6863	1.01300
$695$	− 84.4979i	− 3.20519i
$696$	0	0
$697$	−32.6863	−1.23808
$698$	0	0
$699$	0	0
$700$	57.8745	2.18745
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−40.0405	−1.51016
$704$	50.5567i	1.90543i
$705$	0	0
$706$	14.4797	0.544952
$707$	− 49.4975i	− 1.86154i
$708$	0	0
$709$	−9.12549	−0.342715	−0.171358	−	0.985209i	$-0.554815\pi$
$709$	−9.12549	−0.342715	−0.171358	+	0.985209i	$0.554815\pi$
$710$	− 12.2269i	− 0.458869i
$711$	0	0
$712$	−20.4575	−0.766678
$713$	− 4.75934i	− 0.178239i
$714$	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
$721$	53.6863	1.99938
$722$	10.9588i	0.407843i
$723$	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			$\pi$
$734$	− 52.6494i	− 1.94333i
$735$	0	0
$736$	−11.7490	−0.433074
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	− 95.3106i	− 3.50369i
$741$	0	0
$742$	0	0
$743$	6.14212i	0.225333i	0.993633	+	0.112666i	$0.0359391\pi$
$743$	6.14212i	0.225333i	−0.993633	+	0.112666i	$0.964061\pi$
$744$	0	0
$745$	0	0
$746$	− 46.4916i	− 1.70218i
$747$	0	0
$748$	−47.2915	−1.72915
$749$	− 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
$760$	37.8745	1.37385
$761$	− 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
$763$	55.2288	1.99942
$764$	38.0950i	1.37823i
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	94.3974i	3.40184i
$771$	0	0
$772$	8.00000	0.287926
$773$	− 0.986249i	− 0.0354729i	−0.999843	−	0.0177365i	$-0.994354\pi$
$773$	− 0.986249i	− 0.0354729i	0.999843	−	0.0177365i	$-0.00564599\pi$
$774$	0	0
$775$	−25.0627	−0.900280
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.3020i	1.04985i
$780$	0	0
$781$	13.6863	0.489733
$782$	− 10.9902i	− 0.393009i
$783$	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
$796$	−42.7085	−1.51376
$797$	46.9195i	1.66198i	0.556291	+	0.830988i	$0.312224\pi$
$797$	46.9195i	1.66198i	−0.556291	+	0.830988i	$0.687776\pi$
$798$	0	0
$799$	0	0
$800$	61.8705i	2.18745i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−21.9373	−0.773187
$806$	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
$811$	54.1660	1.90203	0.951013	−	0.309151i	$-0.100045\pi$
$811$	54.1660	1.90203	0.951013	+	0.309151i	$0.100045\pi$
$812$	0	0
$813$	0	0
$814$	106.686	3.73935
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−69.7490	−2.43574
$821$	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			$\pi$
$824$	57.3930i	1.99938i
$825$	0	0
$826$	0	0
$827$	− 56.9651i	− 1.98087i	−0.137981	−	0.990435i	$-0.544061\pi$
$827$	− 56.9651i	− 1.98087i	0.137981	−	0.990435i	$-0.455939\pi$
$828$	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
$836$	42.3950i	1.46626i
$837$	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
$842$	− 17.0593i	− 0.587902i
$843$	0	0
$844$	0	0
$845$	− 51.8979i	− 1.78534i
$846$	0	0
$847$	−76.5608	−2.63066
$848$	0	0
$849$	0	0
$850$	−57.8745	−1.98508
$851$	24.7931i	0.849897i
$852$	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
$857$	56.6416i	1.93484i	0.253179	+	0.967419i	$0.418524\pi$
$857$	56.6416i	1.93484i	−0.253179	+	0.967419i	$0.581476\pi$
$858$	0	0
$859$	35.1033	1.19771	0.598854	−	0.800858i	$-0.295623\pi$
$859$	35.1033	1.19771	0.598854	+	0.800858i	$0.295623\pi$
$860$	0	0
$861$	0	0
$862$	21.0627	0.717400
$863$	− 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
$865$	−60.7490	−2.06553
$866$	0	0
$867$	0	0
$868$	−12.1255	−0.411566
$869$	0	0
$870$	0	0
$871$	0	0
$872$	59.0420i	1.99942i
$873$	0	0
$874$	−9.85228	−0.333258
$875$	62.7107i	2.12001i
$876$	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
$880$	−100.915	−3.40184
$881$	58.1445i	1.95894i	0.201594	+	0.979469i	$0.435388\pi$
$881$	58.1445i	1.95894i	−0.201594	+	0.979469i	$0.564612\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	20.6863	0.694969
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	− 40.8347i	− 1.36878i
$891$	0	0
$892$	−2.45751	−0.0822836
$893$	0	0
$894$	0	0
$895$	73.3948	2.45332
$896$	29.9333i	1.00000i
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 78.0738i	− 2.59957i
$903$	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
$914$	29.3435i	0.970598i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	− 23.4519i	− 0.773187i
$921$	0	0
$922$	44.1033	1.45246
$923$	0	0
$924$	0	0
$925$	130.561	4.29281
$926$	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
$931$	23.4797	0.769517
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 94.3974i	− 3.08712i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 54.9038i	− 1.78981i	−0.446253	−	0.894907i	$-0.647242\pi$
$941$	− 54.9038i	− 1.78981i	0.446253	−	0.894907i	$-0.352758\pi$
$942$	0	0
$943$	18.1438	0.590843
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.89948i	0.0617248i	0.999524	+	0.0308624i	$0.00982537\pi$
$947$	1.89948i	0.0617248i	−0.999524	+	0.0308624i	$0.990175\pi$
$948$	0	0
$949$	0	0
$950$	51.8822i	1.68328i
$951$	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
$955$	−76.0405	−2.46061
$956$	48.0833i	1.55512i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.7490	−0.830613
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.9686i	0.514047i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 81.8469i	− 2.63066i
$969$	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
$979$	45.7085	1.46085
$980$	55.8901i	1.78534i
$981$	0	0
$982$	−38.9373	−1.24254
$983$	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			$\pi$
$992$	− 12.9627i	− 0.411566i
$993$	0	0
$994$	8.10326	0.257020
$995$	− 85.2493i	− 2.70259i
$996$	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.a.55.1	✓	4
3.2	odd	2	inner	756.2.b.a.55.4	yes	4
4.3	odd	2		756.2.b.b.55.3	yes	4
7.6	odd	2		756.2.b.b.55.2	yes	4
12.11	even	2		756.2.b.b.55.2	yes	4
21.20	even	2		756.2.b.b.55.3	yes	4
28.27	even	2	inner	756.2.b.a.55.4	yes	4
84.83	odd	2	CM	756.2.b.a.55.1	✓	4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -3.99215i q^{5} +2.64575 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -3.99215i q^{5} +2.64575 q^{7} +2.82843i q^{8} -5.64575 q^{10} -6.31959i q^{11} -3.74166i q^{14} +4.00000 q^{16} +3.74166i q^{17} +3.35425 q^{19} +7.98430i q^{20} -8.93725 q^{22} -2.07695i q^{23} -10.9373 q^{25} -5.29150 q^{28} +2.29150 q^{31} -5.65685i q^{32} +5.29150 q^{34} -10.5622i q^{35} -11.9373 q^{37} -4.74362i q^{38} +11.2915 q^{40} +8.73577i q^{41} +12.6392i q^{44} -2.93725 q^{46} +7.00000 q^{49} +15.4676i q^{50} -25.2288 q^{55} +7.48331i q^{56} -3.24067i q^{62} -8.00000 q^{64} -7.48331i q^{68} -14.9373 q^{70} +2.16569i q^{71} +16.8818i q^{74} -6.70850 q^{76} -16.7201i q^{77} -15.9686i q^{80} +12.3542 q^{82} +14.9373 q^{85} +17.8745 q^{88} +7.23282i q^{89} +4.15390i q^{92} -13.3907i q^{95} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 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

Related objects

Downloads

Learn more