LMFDB - Embedded newform 9522.2.a.cd.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-8,0,8,0,0,0,-8,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,8,-12,0,0, 12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.546984493056.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 129x^{4} - 320x^{2} + 256 $ x^8 - 20x^6 + 129x^4 - 320*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.18696$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.77275 q^{5} -1.25511 q^{7} -1.00000 q^{8} -1.77275 q^{10} +2.17392 q^{11} +0.350306 q^{13} +1.25511 q^{14} +1.00000 q^{16} +0.495407 q^{17} -2.51023 q^{19} +1.77275 q^{20} -2.17392 q^{22} -1.85735 q^{25} -0.350306 q^{26} -1.25511 q^{28} -0.774998 q^{29} +4.19969 q^{31} -1.00000 q^{32} -0.495407 q^{34} -2.22500 q^{35} +2.77269 q^{37} +2.51023 q^{38} -1.77275 q^{40} -8.15674 q^{41} -10.0047 q^{43} +2.17392 q^{44} +7.77759 q^{47} -5.42469 q^{49} +1.85735 q^{50} +0.350306 q^{52} -5.99729 q^{53} +3.85382 q^{55} +1.25511 q^{56} +0.774998 q^{58} -3.35031 q^{59} +8.58135 q^{61} -4.19969 q^{62} +1.00000 q^{64} +0.621005 q^{65} -4.58078 q^{67} +0.495407 q^{68} +2.22500 q^{70} -14.4782 q^{71} +10.1058 q^{73} -2.77269 q^{74} -2.51023 q^{76} -2.72851 q^{77} -11.0531 q^{79} +1.77275 q^{80} +8.15674 q^{82} -4.91708 q^{83} +0.878234 q^{85} +10.0047 q^{86} -2.17392 q^{88} +10.4747 q^{89} -0.439673 q^{91} -7.77759 q^{94} -4.45000 q^{95} +0.674908 q^{97} +5.42469 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 8 * q^25 - 12 * q^26 + 12 * q^29 - 12 * q^31 - 8 * q^32 - 36 * q^35 - 24 * q^41 - 48 * q^47 - 16 * q^49 - 8 * q^50 + 12 * q^52 + 12 * q^55 - 12 * q^58 - 36 * q^59 + 12 * q^62 + 8 * q^64 + 36 * q^70 - 24 * q^71 + 12 * q^73 - 12 * q^77 + 24 * q^82 + 12 * q^85 + 48 * q^94 - 72 * q^95 + 16 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.77275	0.792798	0.396399	−	0.918078i	$-0.370260\pi$
$5$	1.77275	0.792798	0.396399	+	0.918078i	$0.370260\pi$
$6$	0	0
$7$	−1.25511	−0.474388	−0.237194	−	0.971462i	$-0.576228\pi$
$7$	−1.25511	−0.474388	−0.237194	+	0.971462i	$0.576228\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.77275	−0.560593
$11$	2.17392	0.655461	0.327731	−	0.944771i	$-0.393716\pi$
$11$	2.17392	0.655461	0.327731	+	0.944771i	$0.393716\pi$
$12$	0	0
$13$	0.350306	0.0971573	0.0485787	−	0.998819i	$-0.484531\pi$
$13$	0.350306	0.0971573	0.0485787	+	0.998819i	$0.484531\pi$
$14$	1.25511	0.335443
$15$	0	0
$16$	1.00000	0.250000
$17$	0.495407	0.120154	0.0600769	−	0.998194i	$-0.480865\pi$
$17$	0.495407	0.120154	0.0600769	+	0.998194i	$0.480865\pi$
$18$	0	0
$19$	−2.51023	−0.575885	−0.287943	−	0.957648i	$-0.592971\pi$
$19$	−2.51023	−0.575885	−0.287943	+	0.957648i	$0.592971\pi$
$20$	1.77275	0.396399
$21$	0	0
$22$	−2.17392	−0.463481
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.85735	−0.371471
$26$	−0.350306	−0.0687006
$27$	0	0
$28$	−1.25511	−0.237194
$29$	−0.774998	−0.143913	−0.0719567	−	0.997408i	$-0.522924\pi$
$29$	−0.774998	−0.143913	−0.0719567	+	0.997408i	$0.522924\pi$
$30$	0	0
$31$	4.19969	0.754286	0.377143	−	0.926155i	$-0.376906\pi$
$31$	4.19969	0.754286	0.377143	+	0.926155i	$0.376906\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.495407	−0.0849616
$35$	−2.22500	−0.376094
$36$	0	0
$37$	2.77269	0.455828	0.227914	−	0.973681i	$-0.426810\pi$
$37$	2.77269	0.455828	0.227914	+	0.973681i	$0.426810\pi$
$38$	2.51023	0.407212
$39$	0	0
$40$	−1.77275	−0.280297
$41$	−8.15674	−1.27387	−0.636935	−	0.770918i	$-0.719798\pi$
$41$	−8.15674	−1.27387	−0.636935	+	0.770918i	$0.719798\pi$
$42$	0	0
$43$	−10.0047	−1.52570	−0.762851	−	0.646575i	$-0.776201\pi$
$43$	−10.0047	−1.52570	−0.762851	+	0.646575i	$0.776201\pi$
$44$	2.17392	0.327731
$45$	0	0
$46$	0	0
$47$	7.77759	1.13448	0.567239	−	0.823553i	$-0.308012\pi$
$47$	7.77759	1.13448	0.567239	+	0.823553i	$0.308012\pi$
$48$	0	0
$49$	−5.42469	−0.774956
$50$	1.85735	0.262670
$51$	0	0
$52$	0.350306	0.0485787
$53$	−5.99729	−0.823791	−0.411895	−	0.911231i	$-0.635133\pi$
$53$	−5.99729	−0.823791	−0.411895	+	0.911231i	$0.635133\pi$
$54$	0	0
$55$	3.85382	0.519649
$56$	1.25511	0.167722
$57$	0	0
$58$	0.774998	0.101762
$59$	−3.35031	−0.436173	−0.218086	−	0.975929i	$-0.569981\pi$
$59$	−3.35031	−0.436173	−0.218086	+	0.975929i	$0.569981\pi$
$60$	0	0
$61$	8.58135	1.09873	0.549365	−	0.835583i	$-0.314870\pi$
$61$	8.58135	1.09873	0.549365	+	0.835583i	$0.314870\pi$
$62$	−4.19969	−0.533361
$63$	0	0
$64$	1.00000	0.125000
$65$	0.621005	0.0770262
$66$	0	0
$67$	−4.58078	−0.559631	−0.279816	−	0.960054i	$-0.590273\pi$
$67$	−4.58078	−0.559631	−0.279816	+	0.960054i	$0.590273\pi$
$68$	0.495407	0.0600769
$69$	0	0
$70$	2.22500	0.265939
$71$	−14.4782	−1.71825	−0.859123	−	0.511769i	$-0.828990\pi$
$71$	−14.4782	−1.71825	−0.859123	+	0.511769i	$0.828990\pi$
$72$	0	0
$73$	10.1058	1.18280	0.591399	−	0.806379i	$-0.298576\pi$
$73$	10.1058	1.18280	0.591399	+	0.806379i	$0.298576\pi$
$74$	−2.77269	−0.322319
$75$	0	0
$76$	−2.51023	−0.287943
$77$	−2.72851	−0.310943
$78$	0	0
$79$	−11.0531	−1.24357	−0.621784	−	0.783189i	$-0.713592\pi$
$79$	−11.0531	−1.24357	−0.621784	+	0.783189i	$0.713592\pi$
$80$	1.77275	0.198200
$81$	0	0
$82$	8.15674	0.900762
$83$	−4.91708	−0.539720	−0.269860	−	0.962900i	$-0.586977\pi$
$83$	−4.91708	−0.539720	−0.269860	+	0.962900i	$0.586977\pi$
$84$	0	0
$85$	0.878234	0.0952578
$86$	10.0047	1.07883
$87$	0	0
$88$	−2.17392	−0.231741
$89$	10.4747	1.11032	0.555158	−	0.831745i	$-0.312658\pi$
$89$	10.4747	1.11032	0.555158	+	0.831745i	$0.312658\pi$
$90$	0	0
$91$	−0.439673	−0.0460903
$92$	0	0
$93$	0	0
$94$	−7.77759	−0.802197
$95$	−4.45000	−0.456561
$96$	0	0
$97$	0.674908	0.0685265	0.0342633	−	0.999413i	$-0.489092\pi$
$97$	0.674908	0.0685265	0.0342633	+	0.999413i	$0.489092\pi$
$98$	5.42469	0.547977
$99$	0	0
$100$	−1.85735	−0.185735
$101$	−5.81794	−0.578907	−0.289454	−	0.957192i	$-0.593474\pi$
$101$	−5.81794	−0.578907	−0.289454	+	0.957192i	$0.593474\pi$
$102$	0	0
$103$	2.40686	0.237155	0.118577	−	0.992945i	$-0.462167\pi$
$103$	2.40686	0.237155	0.118577	+	0.992945i	$0.462167\pi$
$104$	−0.350306	−0.0343503
$105$	0	0
$106$	5.99729	0.582508
$107$	−5.49958	−0.531665	−0.265832	−	0.964019i	$-0.585647\pi$
$107$	−5.49958	−0.531665	−0.265832	+	0.964019i	$0.585647\pi$
$108$	0	0
$109$	11.8382	1.13389	0.566946	−	0.823755i	$-0.308125\pi$
$109$	11.8382	1.13389	0.566946	+	0.823755i	$0.308125\pi$
$110$	−3.85382	−0.367447
$111$	0	0
$112$	−1.25511	−0.118597
$113$	−14.7668	−1.38914	−0.694572	−	0.719423i	$-0.744406\pi$
$113$	−14.7668	−1.38914	−0.694572	+	0.719423i	$0.744406\pi$
$114$	0	0
$115$	0	0
$116$	−0.774998	−0.0719567
$117$	0	0
$118$	3.35031	0.308421
$119$	−0.621792	−0.0569996
$120$	0	0
$121$	−6.27408	−0.570370
$122$	−8.58135	−0.776919
$123$	0	0
$124$	4.19969	0.377143
$125$	−12.1564	−1.08730
$126$	0	0
$127$	−4.67760	−0.415070	−0.207535	−	0.978228i	$-0.566544\pi$
$127$	−4.67760	−0.415070	−0.207535	+	0.978228i	$0.566544\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.621005	−0.0544657
$131$	8.05351	0.703638	0.351819	−	0.936068i	$-0.385563\pi$
$131$	8.05351	0.703638	0.351819	+	0.936068i	$0.385563\pi$
$132$	0	0
$133$	3.15062	0.273193
$134$	4.58078	0.395719
$135$	0	0
$136$	−0.495407	−0.0424808
$137$	6.05619	0.517415	0.258708	−	0.965956i	$-0.416703\pi$
$137$	6.05619	0.517415	0.258708	+	0.965956i	$0.416703\pi$
$138$	0	0
$139$	1.74939	0.148382	0.0741908	−	0.997244i	$-0.476363\pi$
$139$	1.74939	0.148382	0.0741908	+	0.997244i	$0.476363\pi$
$140$	−2.22500	−0.188047
$141$	0	0
$142$	14.4782	1.21498
$143$	0.761536	0.0636829
$144$	0	0
$145$	−1.37388	−0.114094
$146$	−10.1058	−0.836364
$147$	0	0
$148$	2.77269	0.227914
$149$	3.62164	0.296696	0.148348	−	0.988935i	$-0.452604\pi$
$149$	3.62164	0.296696	0.148348	+	0.988935i	$0.452604\pi$
$150$	0	0
$151$	9.02347	0.734319	0.367160	−	0.930158i	$-0.380330\pi$
$151$	9.02347	0.734319	0.367160	+	0.930158i	$0.380330\pi$
$152$	2.51023	0.203606
$153$	0	0
$154$	2.72851	0.219870
$155$	7.44500	0.597997
$156$	0	0
$157$	3.62664	0.289437	0.144719	−	0.989473i	$-0.453772\pi$
$157$	3.62664	0.289437	0.144719	+	0.989473i	$0.453772\pi$
$158$	11.0531	0.879335
$159$	0	0
$160$	−1.77275	−0.140148
$161$	0	0
$162$	0	0
$163$	7.55000	0.591361	0.295681	−	0.955287i	$-0.404454\pi$
$163$	7.55000	0.591361	0.295681	+	0.955287i	$0.404454\pi$
$164$	−8.15674	−0.636935
$165$	0	0
$166$	4.91708	0.381640
$167$	1.40122	0.108430	0.0542150	−	0.998529i	$-0.482734\pi$
$167$	1.40122	0.108430	0.0542150	+	0.998529i	$0.482734\pi$
$168$	0	0
$169$	−12.8773	−0.990560
$170$	−0.878234	−0.0673574
$171$	0	0
$172$	−10.0047	−0.762851
$173$	−20.0491	−1.52430	−0.762151	−	0.647399i	$-0.775857\pi$
$173$	−20.0491	−1.52430	−0.762151	+	0.647399i	$0.775857\pi$
$174$	0	0
$175$	2.33119	0.176221
$176$	2.17392	0.163865
$177$	0	0
$178$	−10.4747	−0.785112
$179$	−16.6776	−1.24654	−0.623271	−	0.782006i	$-0.714197\pi$
$179$	−16.6776	−1.24654	−0.623271	+	0.782006i	$0.714197\pi$
$180$	0	0
$181$	−16.7081	−1.24190	−0.620951	−	0.783849i	$-0.713254\pi$
$181$	−16.7081	−1.24190	−0.620951	+	0.783849i	$0.713254\pi$
$182$	0.439673	0.0325908
$183$	0	0
$184$	0	0
$185$	4.91529	0.361380
$186$	0	0
$187$	1.07698	0.0787562
$188$	7.77759	0.567239
$189$	0	0
$190$	4.45000	0.322837
$191$	−26.3651	−1.90771	−0.953854	−	0.300270i	$-0.902923\pi$
$191$	−26.3651	−1.90771	−0.953854	+	0.300270i	$0.902923\pi$
$192$	0	0
$193$	21.4493	1.54396	0.771979	−	0.635648i	$-0.219267\pi$
$193$	21.4493	1.54396	0.771979	+	0.635648i	$0.219267\pi$
$194$	−0.674908	−0.0484556
$195$	0	0
$196$	−5.42469	−0.387478
$197$	−14.9212	−1.06309	−0.531545	−	0.847030i	$-0.678388\pi$
$197$	−14.9212	−1.06309	−0.531545	+	0.847030i	$0.678388\pi$
$198$	0	0
$199$	22.8954	1.62301	0.811505	−	0.584345i	$-0.198649\pi$
$199$	22.8954	1.62301	0.811505	+	0.584345i	$0.198649\pi$
$200$	1.85735	0.131335
$201$	0	0
$202$	5.81794	0.409349
$203$	0.972709	0.0682708
$204$	0	0
$205$	−14.4599	−1.00992
$206$	−2.40686	−0.167694
$207$	0	0
$208$	0.350306	0.0242893
$209$	−5.45703	−0.377470
$210$	0	0
$211$	1.62881	0.112132	0.0560661	−	0.998427i	$-0.482144\pi$
$211$	1.62881	0.112132	0.0560661	+	0.998427i	$0.482144\pi$
$212$	−5.99729	−0.411895
$213$	0	0
$214$	5.49958	0.375944
$215$	−17.7358	−1.20957
$216$	0	0
$217$	−5.27108	−0.357824
$218$	−11.8382	−0.801783
$219$	0	0
$220$	3.85382	0.259824
$221$	0.173544	0.0116738
$222$	0	0
$223$	−12.6776	−0.848955	−0.424477	−	0.905439i	$-0.639542\pi$
$223$	−12.6776	−0.848955	−0.424477	+	0.905439i	$0.639542\pi$
$224$	1.25511	0.0838608
$225$	0	0
$226$	14.7668	0.982273
$227$	23.7647	1.57732	0.788660	−	0.614830i	$-0.210775\pi$
$227$	23.7647	1.57732	0.788660	+	0.614830i	$0.210775\pi$
$228$	0	0
$229$	12.2525	0.809665	0.404832	−	0.914391i	$-0.367330\pi$
$229$	12.2525	0.809665	0.404832	+	0.914391i	$0.367330\pi$
$230$	0	0
$231$	0	0
$232$	0.774998	0.0508811
$233$	−26.7208	−1.75054	−0.875270	−	0.483635i	$-0.839316\pi$
$233$	−26.7208	−1.75054	−0.875270	+	0.483635i	$0.839316\pi$
$234$	0	0
$235$	13.7877	0.899412
$236$	−3.35031	−0.218086
$237$	0	0
$238$	0.621792	0.0403048
$239$	15.9494	1.03168	0.515840	−	0.856685i	$-0.327480\pi$
$239$	15.9494	1.03168	0.515840	+	0.856685i	$0.327480\pi$
$240$	0	0
$241$	13.1867	0.849427	0.424714	−	0.905328i	$-0.360375\pi$
$241$	13.1867	0.849427	0.424714	+	0.905328i	$0.360375\pi$
$242$	6.27408	0.403313
$243$	0	0
$244$	8.58135	0.549365
$245$	−9.61663	−0.614384
$246$	0	0
$247$	−0.879347	−0.0559515
$248$	−4.19969	−0.266681
$249$	0	0
$250$	12.1564	0.768837
$251$	−2.69370	−0.170025	−0.0850125	−	0.996380i	$-0.527093\pi$
$251$	−2.69370	−0.170025	−0.0850125	+	0.996380i	$0.527093\pi$
$252$	0	0
$253$	0	0
$254$	4.67760	0.293499
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00797	−0.125254	−0.0626269	−	0.998037i	$-0.519948\pi$
$257$	−2.00797	−0.125254	−0.0626269	+	0.998037i	$0.519948\pi$
$258$	0	0
$259$	−3.48004	−0.216239
$260$	0.621005	0.0385131
$261$	0	0
$262$	−8.05351	−0.497547
$263$	−22.8604	−1.40963	−0.704815	−	0.709391i	$-0.748970\pi$
$263$	−22.8604	−1.40963	−0.704815	+	0.709391i	$0.748970\pi$
$264$	0	0
$265$	−10.6317	−0.653100
$266$	−3.15062	−0.193177
$267$	0	0
$268$	−4.58078	−0.279816
$269$	32.6831	1.99272	0.996361	−	0.0852352i	$-0.0271642\pi$
$269$	32.6831	1.99272	0.996361	+	0.0852352i	$0.0271642\pi$
$270$	0	0
$271$	−1.35290	−0.0821825	−0.0410913	−	0.999155i	$-0.513083\pi$
$271$	−1.35290	−0.0821825	−0.0410913	+	0.999155i	$0.513083\pi$
$272$	0.495407	0.0300385
$273$	0	0
$274$	−6.05619	−0.365868
$275$	−4.03774	−0.243485
$276$	0	0
$277$	9.17881	0.551501	0.275751	−	0.961229i	$-0.411074\pi$
$277$	9.17881	0.551501	0.275751	+	0.961229i	$0.411074\pi$
$278$	−1.74939	−0.104922
$279$	0	0
$280$	2.22500	0.132969
$281$	−16.5114	−0.984985	−0.492493	−	0.870317i	$-0.663914\pi$
$281$	−16.5114	−0.984985	−0.492493	+	0.870317i	$0.663914\pi$
$282$	0	0
$283$	−9.24316	−0.549449	−0.274724	−	0.961523i	$-0.588587\pi$
$283$	−9.24316	−0.549449	−0.274724	+	0.961523i	$0.588587\pi$
$284$	−14.4782	−0.859123
$285$	0	0
$286$	−0.761536	−0.0450306
$287$	10.2376	0.604308
$288$	0	0
$289$	−16.7546	−0.985563
$290$	1.37388	0.0806769
$291$	0	0
$292$	10.1058	0.591399
$293$	26.4726	1.54654	0.773271	−	0.634075i	$-0.218619\pi$
$293$	26.4726	1.54654	0.773271	+	0.634075i	$0.218619\pi$
$294$	0	0
$295$	−5.93926	−0.345797
$296$	−2.77269	−0.161160
$297$	0	0
$298$	−3.62164	−0.209796
$299$	0	0
$300$	0	0
$301$	12.5570	0.723774
$302$	−9.02347	−0.519242
$303$	0	0
$304$	−2.51023	−0.143971
$305$	15.2126	0.871071
$306$	0	0
$307$	−22.6500	−1.29270	−0.646351	−	0.763040i	$-0.723706\pi$
$307$	−22.6500	−1.29270	−0.646351	+	0.763040i	$0.723706\pi$
$308$	−2.72851	−0.155472
$309$	0	0
$310$	−7.44500	−0.422848
$311$	−7.55000	−0.428121	−0.214060	−	0.976820i	$-0.568669\pi$
$311$	−7.55000	−0.428121	−0.214060	+	0.976820i	$0.568669\pi$
$312$	0	0
$313$	−18.4696	−1.04396	−0.521981	−	0.852957i	$-0.674807\pi$
$313$	−18.4696	−1.04396	−0.521981	+	0.852957i	$0.674807\pi$
$314$	−3.62664	−0.204663
$315$	0	0
$316$	−11.0531	−0.621784
$317$	−14.4826	−0.813426	−0.406713	−	0.913556i	$-0.633325\pi$
$317$	−14.4826	−0.813426	−0.406713	+	0.913556i	$0.633325\pi$
$318$	0	0
$319$	−1.68478	−0.0943297
$320$	1.77275	0.0990998
$321$	0	0
$322$	0	0
$323$	−1.24358	−0.0691949
$324$	0	0
$325$	−0.650642	−0.0360911
$326$	−7.55000	−0.418156
$327$	0	0
$328$	8.15674	0.450381
$329$	−9.76175	−0.538183
$330$	0	0
$331$	2.19940	0.120890	0.0604449	−	0.998172i	$-0.480748\pi$
$331$	2.19940	0.120890	0.0604449	+	0.998172i	$0.480748\pi$
$332$	−4.91708	−0.269860
$333$	0	0
$334$	−1.40122	−0.0766715
$335$	−8.12058	−0.443675
$336$	0	0
$337$	−3.23643	−0.176299	−0.0881497	−	0.996107i	$-0.528095\pi$
$337$	−3.23643	−0.176299	−0.0881497	+	0.996107i	$0.528095\pi$
$338$	12.8773	0.700432
$339$	0	0
$340$	0.878234	0.0476289
$341$	9.12979	0.494406
$342$	0	0
$343$	15.5944	0.842018
$344$	10.0047	0.539417
$345$	0	0
$346$	20.0491	1.07784
$347$	−29.0747	−1.56081	−0.780405	−	0.625274i	$-0.784987\pi$
$347$	−29.0747	−1.56081	−0.780405	+	0.625274i	$0.784987\pi$
$348$	0	0
$349$	−13.3693	−0.715644	−0.357822	−	0.933790i	$-0.616481\pi$
$349$	−13.3693	−0.715644	−0.357822	+	0.933790i	$0.616481\pi$
$350$	−2.33119	−0.124607
$351$	0	0
$352$	−2.17392	−0.115870
$353$	−17.7285	−0.943594	−0.471797	−	0.881707i	$-0.656394\pi$
$353$	−17.7285	−0.943594	−0.471797	+	0.881707i	$0.656394\pi$
$354$	0	0
$355$	−25.6662	−1.36222
$356$	10.4747	0.555158
$357$	0	0
$358$	16.6776	0.881438
$359$	22.1977	1.17155	0.585776	−	0.810473i	$-0.300790\pi$
$359$	22.1977	1.17155	0.585776	+	0.810473i	$0.300790\pi$
$360$	0	0
$361$	−12.6988	−0.668356
$362$	16.7081	0.878158
$363$	0	0
$364$	−0.439673	−0.0230451
$365$	17.9151	0.937720
$366$	0	0
$367$	15.7911	0.824290	0.412145	−	0.911118i	$-0.364780\pi$
$367$	15.7911	0.824290	0.412145	+	0.911118i	$0.364780\pi$
$368$	0	0
$369$	0	0
$370$	−4.91529	−0.255534
$371$	7.52727	0.390796
$372$	0	0
$373$	30.4604	1.57718	0.788591	−	0.614918i	$-0.210811\pi$
$373$	30.4604	1.57718	0.788591	+	0.614918i	$0.210811\pi$
$374$	−1.07698	−0.0556891
$375$	0	0
$376$	−7.77759	−0.401098
$377$	−0.271486	−0.0139822
$378$	0	0
$379$	0.0889235	0.00456770	0.00228385	−	0.999997i	$-0.499273\pi$
$379$	0.0889235	0.00456770	0.00228385	+	0.999997i	$0.499273\pi$
$380$	−4.45000	−0.228280
$381$	0	0
$382$	26.3651	1.34895
$383$	−14.2636	−0.728836	−0.364418	−	0.931235i	$-0.618732\pi$
$383$	−14.2636	−0.728836	−0.364418	+	0.931235i	$0.618732\pi$
$384$	0	0
$385$	−4.83698	−0.246515
$386$	−21.4493	−1.09174
$387$	0	0
$388$	0.674908	0.0342633
$389$	−8.21837	−0.416688	−0.208344	−	0.978056i	$-0.566807\pi$
$389$	−8.21837	−0.416688	−0.208344	+	0.978056i	$0.566807\pi$
$390$	0	0
$391$	0	0
$392$	5.42469	0.273988
$393$	0	0
$394$	14.9212	0.751718
$395$	−19.5943	−0.985898
$396$	0	0
$397$	19.8497	0.996227	0.498113	−	0.867112i	$-0.334026\pi$
$397$	19.8497	0.996227	0.498113	+	0.867112i	$0.334026\pi$
$398$	−22.8954	−1.14764
$399$	0	0
$400$	−1.85735	−0.0928677
$401$	−13.2995	−0.664143	−0.332072	−	0.943254i	$-0.607748\pi$
$401$	−13.2995	−0.664143	−0.332072	+	0.943254i	$0.607748\pi$
$402$	0	0
$403$	1.47118	0.0732845
$404$	−5.81794	−0.289454
$405$	0	0
$406$	−0.972709	−0.0482748
$407$	6.02761	0.298778
$408$	0	0
$409$	33.6760	1.66517	0.832587	−	0.553895i	$-0.186859\pi$
$409$	33.6760	1.66517	0.832587	+	0.553895i	$0.186859\pi$
$410$	14.4599	0.714122
$411$	0	0
$412$	2.40686	0.118577
$413$	4.20501	0.206915
$414$	0	0
$415$	−8.71677	−0.427889
$416$	−0.350306	−0.0171752
$417$	0	0
$418$	5.45703	0.266912
$419$	−28.2304	−1.37914	−0.689572	−	0.724217i	$-0.742201\pi$
$419$	−28.2304	−1.37914	−0.689572	+	0.724217i	$0.742201\pi$
$420$	0	0
$421$	2.56596	0.125057	0.0625286	−	0.998043i	$-0.480084\pi$
$421$	2.56596	0.125057	0.0625286	+	0.998043i	$0.480084\pi$
$422$	−1.62881	−0.0792895
$423$	0	0
$424$	5.99729	0.291254
$425$	−0.920147	−0.0446337
$426$	0	0
$427$	−10.7706	−0.521224
$428$	−5.49958	−0.265832
$429$	0	0
$430$	17.7358	0.855298
$431$	2.70696	0.130389	0.0651947	−	0.997873i	$-0.479233\pi$
$431$	2.70696	0.130389	0.0651947	+	0.997873i	$0.479233\pi$
$432$	0	0
$433$	7.77274	0.373534	0.186767	−	0.982404i	$-0.440199\pi$
$433$	7.77274	0.373534	0.186767	+	0.982404i	$0.440199\pi$
$434$	5.27108	0.253020
$435$	0	0
$436$	11.8382	0.566946
$437$	0	0
$438$	0	0
$439$	14.0561	0.670861	0.335430	−	0.942065i	$-0.391118\pi$
$439$	14.0561	0.670861	0.335430	+	0.942065i	$0.391118\pi$
$440$	−3.85382	−0.183724
$441$	0	0
$442$	−0.173544	−0.00825465
$443$	−17.4343	−0.828329	−0.414164	−	0.910202i	$-0.635926\pi$
$443$	−17.4343	−0.828329	−0.414164	+	0.910202i	$0.635926\pi$
$444$	0	0
$445$	18.5690	0.880256
$446$	12.6776	0.600302
$447$	0	0
$448$	−1.25511	−0.0592985
$449$	−2.21444	−0.104506	−0.0522530	−	0.998634i	$-0.516640\pi$
$449$	−2.21444	−0.104506	−0.0522530	+	0.998634i	$0.516640\pi$
$450$	0	0
$451$	−17.7321	−0.834972
$452$	−14.7668	−0.694572
$453$	0	0
$454$	−23.7647	−1.11533
$455$	−0.779431	−0.0365403
$456$	0	0
$457$	−5.24075	−0.245152	−0.122576	−	0.992459i	$-0.539116\pi$
$457$	−5.24075	−0.245152	−0.122576	+	0.992459i	$0.539116\pi$
$458$	−12.2525	−0.572519
$459$	0	0
$460$	0	0
$461$	24.6340	1.14732	0.573660	−	0.819094i	$-0.305523\pi$
$461$	24.6340	1.14732	0.573660	+	0.819094i	$0.305523\pi$
$462$	0	0
$463$	−23.0261	−1.07011	−0.535056	−	0.844817i	$-0.679709\pi$
$463$	−23.0261	−1.07011	−0.535056	+	0.844817i	$0.679709\pi$
$464$	−0.774998	−0.0359784
$465$	0	0
$466$	26.7208	1.23782
$467$	−35.2398	−1.63070	−0.815351	−	0.578967i	$-0.803456\pi$
$467$	−35.2398	−1.63070	−0.815351	+	0.578967i	$0.803456\pi$
$468$	0	0
$469$	5.74939	0.265482
$470$	−13.7877	−0.635980
$471$	0	0
$472$	3.35031	0.154210
$473$	−21.7494	−1.00004
$474$	0	0
$475$	4.66238	0.213925
$476$	−0.621792	−0.0284998
$477$	0	0
$478$	−15.9494	−0.729507
$479$	22.5196	1.02895	0.514474	−	0.857506i	$-0.327987\pi$
$479$	22.5196	1.02895	0.514474	+	0.857506i	$0.327987\pi$
$480$	0	0
$481$	0.971290	0.0442870
$482$	−13.1867	−0.600636
$483$	0	0
$484$	−6.27408	−0.285185
$485$	1.19644	0.0543277
$486$	0	0
$487$	−27.1279	−1.22928	−0.614641	−	0.788807i	$-0.710699\pi$
$487$	−27.1279	−1.22928	−0.614641	+	0.788807i	$0.710699\pi$
$488$	−8.58135	−0.388460
$489$	0	0
$490$	9.61663	0.434435
$491$	−41.1357	−1.85643	−0.928213	−	0.372049i	$-0.878655\pi$
$491$	−41.1357	−1.85643	−0.928213	+	0.372049i	$0.878655\pi$
$492$	0	0
$493$	−0.383939	−0.0172918
$494$	0.879347	0.0395637
$495$	0	0
$496$	4.19969	0.188572
$497$	18.1718	0.815115
$498$	0	0
$499$	−29.9634	−1.34135	−0.670673	−	0.741753i	$-0.733995\pi$
$499$	−29.9634	−1.34135	−0.670673	+	0.741753i	$0.733995\pi$
$500$	−12.1564	−0.543650
$501$	0	0
$502$	2.69370	0.120226
$503$	16.8528	0.751427	0.375714	−	0.926736i	$-0.377398\pi$
$503$	16.8528	0.751427	0.375714	+	0.926736i	$0.377398\pi$
$504$	0	0
$505$	−10.3138	−0.458957
$506$	0	0
$507$	0	0
$508$	−4.67760	−0.207535
$509$	−3.18324	−0.141095	−0.0705474	−	0.997508i	$-0.522475\pi$
$509$	−3.18324	−0.141095	−0.0705474	+	0.997508i	$0.522475\pi$
$510$	0	0
$511$	−12.6840	−0.561105
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00797	0.0885678
$515$	4.26676	0.188016
$516$	0	0
$517$	16.9078	0.743606
$518$	3.48004	0.152904
$519$	0	0
$520$	−0.621005	−0.0272329
$521$	26.5197	1.16185	0.580925	−	0.813957i	$-0.302691\pi$
$521$	26.5197	1.16185	0.580925	+	0.813957i	$0.302691\pi$
$522$	0	0
$523$	13.8501	0.605624	0.302812	−	0.953050i	$-0.402075\pi$
$523$	13.8501	0.605624	0.302812	+	0.953050i	$0.402075\pi$
$524$	8.05351	0.351819
$525$	0	0
$526$	22.8604	0.996759
$527$	2.08056	0.0906305
$528$	0	0
$529$	0	0
$530$	10.6317	0.461811
$531$	0	0
$532$	3.15062	0.136597
$533$	−2.85735	−0.123766
$534$	0	0
$535$	−9.74939	−0.421503
$536$	4.58078	0.197859
$537$	0	0
$538$	−32.6831	−1.40907
$539$	−11.7928	−0.507954
$540$	0	0
$541$	19.5491	0.840481	0.420241	−	0.907413i	$-0.361946\pi$
$541$	19.5491	0.840481	0.420241	+	0.907413i	$0.361946\pi$
$542$	1.35290	0.0581118
$543$	0	0
$544$	−0.495407	−0.0212404
$545$	20.9861	0.898948
$546$	0	0
$547$	−18.9794	−0.811501	−0.405751	−	0.913984i	$-0.632990\pi$
$547$	−18.9794	−0.811501	−0.405751	+	0.913984i	$0.632990\pi$
$548$	6.05619	0.258708
$549$	0	0
$550$	4.03774	0.172170
$551$	1.94542	0.0828776
$552$	0	0
$553$	13.8729	0.589934
$554$	−9.17881	−0.389970
$555$	0	0
$556$	1.74939	0.0741908
$557$	−19.1436	−0.811141	−0.405570	−	0.914064i	$-0.632927\pi$
$557$	−19.1436	−0.811141	−0.405570	+	0.914064i	$0.632927\pi$
$558$	0	0
$559$	−3.50470	−0.148233
$560$	−2.22500	−0.0940235
$561$	0	0
$562$	16.5114	0.696490
$563$	25.8091	1.08772	0.543861	−	0.839175i	$-0.316962\pi$
$563$	25.8091	1.08772	0.543861	+	0.839175i	$0.316962\pi$
$564$	0	0
$565$	−26.1779	−1.10131
$566$	9.24316	0.388519
$567$	0	0
$568$	14.4782	0.607492
$569$	43.5751	1.82676	0.913381	−	0.407105i	$-0.133462\pi$
$569$	43.5751	1.82676	0.913381	+	0.407105i	$0.133462\pi$
$570$	0	0
$571$	2.31350	0.0968168	0.0484084	−	0.998828i	$-0.484585\pi$
$571$	2.31350	0.0968168	0.0484084	+	0.998828i	$0.484585\pi$
$572$	0.761536	0.0318414
$573$	0	0
$574$	−10.2376	−0.427311
$575$	0	0
$576$	0	0
$577$	−5.01829	−0.208914	−0.104457	−	0.994529i	$-0.533310\pi$
$577$	−5.01829	−0.208914	−0.104457	+	0.994529i	$0.533310\pi$
$578$	16.7546	0.696898
$579$	0	0
$580$	−1.37388	−0.0570472
$581$	6.17150	0.256037
$582$	0	0
$583$	−13.0376	−0.539963
$584$	−10.1058	−0.418182
$585$	0	0
$586$	−26.4726	−1.09357
$587$	16.4046	0.677089	0.338544	−	0.940950i	$-0.390066\pi$
$587$	16.4046	0.677089	0.338544	+	0.940950i	$0.390066\pi$
$588$	0	0
$589$	−10.5422	−0.434382
$590$	5.93926	0.244515
$591$	0	0
$592$	2.77269	0.113957
$593$	−30.6040	−1.25675	−0.628377	−	0.777909i	$-0.716280\pi$
$593$	−30.6040	−1.25675	−0.628377	+	0.777909i	$0.716280\pi$
$594$	0	0
$595$	−1.10228	−0.0451892
$596$	3.62164	0.148348
$597$	0	0
$598$	0	0
$599$	19.7776	0.808090	0.404045	−	0.914739i	$-0.367604\pi$
$599$	19.7776	0.808090	0.404045	+	0.914739i	$0.367604\pi$
$600$	0	0
$601$	−10.9455	−0.446478	−0.223239	−	0.974764i	$-0.571663\pi$
$601$	−10.9455	−0.446478	−0.223239	+	0.974764i	$0.571663\pi$
$602$	−12.5570	−0.511786
$603$	0	0
$604$	9.02347	0.367160
$605$	−11.1224	−0.452189
$606$	0	0
$607$	26.3093	1.06786	0.533931	−	0.845528i	$-0.320714\pi$
$607$	26.3093	1.06786	0.533931	+	0.845528i	$0.320714\pi$
$608$	2.51023	0.101803
$609$	0	0
$610$	−15.2126	−0.615940
$611$	2.72453	0.110223
$612$	0	0
$613$	4.59618	0.185638	0.0928189	−	0.995683i	$-0.470412\pi$
$613$	4.59618	0.185638	0.0928189	+	0.995683i	$0.470412\pi$
$614$	22.6500	0.914079
$615$	0	0
$616$	2.72851	0.109935
$617$	28.6755	1.15443	0.577217	−	0.816591i	$-0.304139\pi$
$617$	28.6755	1.15443	0.577217	+	0.816591i	$0.304139\pi$
$618$	0	0
$619$	26.3026	1.05719	0.528596	−	0.848873i	$-0.322719\pi$
$619$	26.3026	1.05719	0.528596	+	0.848873i	$0.322719\pi$
$620$	7.44500	0.298999
$621$	0	0
$622$	7.55000	0.302727
$623$	−13.1469	−0.526721
$624$	0	0
$625$	−12.2635	−0.490539
$626$	18.4696	0.738193
$627$	0	0
$628$	3.62664	0.144719
$629$	1.37361	0.0547695
$630$	0	0
$631$	−13.5415	−0.539080	−0.269540	−	0.962989i	$-0.586872\pi$
$631$	−13.5415	−0.539080	−0.269540	+	0.962989i	$0.586872\pi$
$632$	11.0531	0.439668
$633$	0	0
$634$	14.4826	0.575179
$635$	−8.29221	−0.329066
$636$	0	0
$637$	−1.90030	−0.0752927
$638$	1.68478	0.0667012
$639$	0	0
$640$	−1.77275	−0.0700741
$641$	16.2680	0.642546	0.321273	−	0.946987i	$-0.395889\pi$
$641$	16.2680	0.642546	0.321273	+	0.946987i	$0.395889\pi$
$642$	0	0
$643$	25.2528	0.995872	0.497936	−	0.867214i	$-0.334091\pi$
$643$	25.2528	0.995872	0.497936	+	0.867214i	$0.334091\pi$
$644$	0	0
$645$	0	0
$646$	1.24358	0.0489282
$647$	−36.5251	−1.43595	−0.717976	−	0.696068i	$-0.754931\pi$
$647$	−36.5251	−1.43595	−0.717976	+	0.696068i	$0.754931\pi$
$648$	0	0
$649$	−7.28329	−0.285894
$650$	0.650642	0.0255203
$651$	0	0
$652$	7.55000	0.295681
$653$	−6.68607	−0.261646	−0.130823	−	0.991406i	$-0.541762\pi$
$653$	−6.68607	−0.261646	−0.130823	+	0.991406i	$0.541762\pi$
$654$	0	0
$655$	14.2769	0.557843
$656$	−8.15674	−0.318467
$657$	0	0
$658$	9.76175	0.380553
$659$	−14.4998	−0.564832	−0.282416	−	0.959292i	$-0.591136\pi$
$659$	−14.4998	−0.564832	−0.282416	+	0.959292i	$0.591136\pi$
$660$	0	0
$661$	36.0706	1.40298	0.701492	−	0.712677i	$-0.252518\pi$
$661$	36.0706	1.40298	0.701492	+	0.712677i	$0.252518\pi$
$662$	−2.19940	−0.0854820
$663$	0	0
$664$	4.91708	0.190820
$665$	5.58526	0.216587
$666$	0	0
$667$	0	0
$668$	1.40122	0.0542150
$669$	0	0
$670$	8.12058	0.313725
$671$	18.6552	0.720175
$672$	0	0
$673$	−19.2209	−0.740910	−0.370455	−	0.928850i	$-0.620798\pi$
$673$	−19.2209	−0.740910	−0.370455	+	0.928850i	$0.620798\pi$
$674$	3.23643	0.124663
$675$	0	0
$676$	−12.8773	−0.495280
$677$	−27.5536	−1.05897	−0.529486	−	0.848319i	$-0.677615\pi$
$677$	−27.5536	−1.05897	−0.529486	+	0.848319i	$0.677615\pi$
$678$	0	0
$679$	−0.847086	−0.0325082
$680$	−0.878234	−0.0336787
$681$	0	0
$682$	−9.12979	−0.349598
$683$	−6.94695	−0.265818	−0.132909	−	0.991128i	$-0.542432\pi$
$683$	−6.94695	−0.265818	−0.132909	+	0.991128i	$0.542432\pi$
$684$	0	0
$685$	10.7361	0.410206
$686$	−15.5944	−0.595397
$687$	0	0
$688$	−10.0047	−0.381425
$689$	−2.10088	−0.0800373
$690$	0	0
$691$	−21.1788	−0.805680	−0.402840	−	0.915270i	$-0.631977\pi$
$691$	−21.1788	−0.805680	−0.402840	+	0.915270i	$0.631977\pi$
$692$	−20.0491	−0.762151
$693$	0	0
$694$	29.0747	1.10366
$695$	3.10124	0.117637
$696$	0	0
$697$	−4.04091	−0.153060
$698$	13.3693	0.506037
$699$	0	0
$700$	2.33119	0.0881107
$701$	11.9029	0.449566	0.224783	−	0.974409i	$-0.427833\pi$
$701$	11.9029	0.449566	0.224783	+	0.974409i	$0.427833\pi$
$702$	0	0
$703$	−6.96009	−0.262505
$704$	2.17392	0.0819327
$705$	0	0
$706$	17.7285	0.667221
$707$	7.30218	0.274627
$708$	0	0
$709$	−28.6940	−1.07762	−0.538812	−	0.842426i	$-0.681127\pi$
$709$	−28.6940	−1.07762	−0.538812	+	0.842426i	$0.681127\pi$
$710$	25.6662	0.963237
$711$	0	0
$712$	−10.4747	−0.392556
$713$	0	0
$714$	0	0
$715$	1.35001	0.0504877
$716$	−16.6776	−0.623271
$717$	0	0
$718$	−22.1977	−0.828413
$719$	3.82821	0.142768	0.0713841	−	0.997449i	$-0.477258\pi$
$719$	3.82821	0.142768	0.0713841	+	0.997449i	$0.477258\pi$
$720$	0	0
$721$	−3.02088	−0.112503
$722$	12.6988	0.472599
$723$	0	0
$724$	−16.7081	−0.620951
$725$	1.43945	0.0534596
$726$	0	0
$727$	−36.5081	−1.35401	−0.677006	−	0.735978i	$-0.736723\pi$
$727$	−36.5081	−1.35401	−0.677006	+	0.735978i	$0.736723\pi$
$728$	0.439673	0.0162954
$729$	0	0
$730$	−17.9151	−0.663068
$731$	−4.95640	−0.183319
$732$	0	0
$733$	−2.66837	−0.0985586	−0.0492793	−	0.998785i	$-0.515692\pi$
$733$	−2.66837	−0.0985586	−0.0492793	+	0.998785i	$0.515692\pi$
$734$	−15.7911	−0.582861
$735$	0	0
$736$	0	0
$737$	−9.95824	−0.366817
$738$	0	0
$739$	24.4046	0.897736	0.448868	−	0.893598i	$-0.351827\pi$
$739$	24.4046	0.897736	0.448868	+	0.893598i	$0.351827\pi$
$740$	4.91529	0.180690
$741$	0	0
$742$	−7.52727	−0.276335
$743$	6.83186	0.250637	0.125318	−	0.992117i	$-0.460005\pi$
$743$	6.83186	0.250637	0.125318	+	0.992117i	$0.460005\pi$
$744$	0	0
$745$	6.42026	0.235220
$746$	−30.4604	−1.11524
$747$	0	0
$748$	1.07698	0.0393781
$749$	6.90260	0.252215
$750$	0	0
$751$	−21.5369	−0.785893	−0.392946	−	0.919561i	$-0.628544\pi$
$751$	−21.5369	−0.785893	−0.392946	+	0.919561i	$0.628544\pi$
$752$	7.77759	0.283619
$753$	0	0
$754$	0.271486	0.00988694
$755$	15.9964	0.582167
$756$	0	0
$757$	10.3157	0.374932	0.187466	−	0.982271i	$-0.439973\pi$
$757$	10.3157	0.374932	0.187466	+	0.982271i	$0.439973\pi$
$758$	−0.0889235	−0.00322985
$759$	0	0
$760$	4.45000	0.161419
$761$	19.0242	0.689627	0.344814	−	0.938671i	$-0.387942\pi$
$761$	19.0242	0.689627	0.344814	+	0.938671i	$0.387942\pi$
$762$	0	0
$763$	−14.8583	−0.537905
$764$	−26.3651	−0.953854
$765$	0	0
$766$	14.2636	0.515365
$767$	−1.17363	−0.0423774
$768$	0	0
$769$	54.4255	1.96264	0.981318	−	0.192395i	$-0.0616253\pi$
$769$	54.4255	1.96264	0.981318	+	0.192395i	$0.0616253\pi$
$770$	4.83698	0.174312
$771$	0	0
$772$	21.4493	0.771979
$773$	9.17298	0.329929	0.164965	−	0.986299i	$-0.447249\pi$
$773$	9.17298	0.329929	0.164965	+	0.986299i	$0.447249\pi$
$774$	0	0
$775$	−7.80031	−0.280195
$776$	−0.674908	−0.0242278
$777$	0	0
$778$	8.21837	0.294643
$779$	20.4753	0.733602
$780$	0	0
$781$	−31.4744	−1.12624
$782$	0	0
$783$	0	0
$784$	−5.42469	−0.193739
$785$	6.42913	0.229465
$786$	0	0
$787$	−29.9252	−1.06672	−0.533358	−	0.845889i	$-0.679070\pi$
$787$	−29.9252	−1.06672	−0.533358	+	0.845889i	$0.679070\pi$
$788$	−14.9212	−0.531545
$789$	0	0
$790$	19.5943	0.697135
$791$	18.5340	0.658993
$792$	0	0
$793$	3.00610	0.106750
$794$	−19.8497	−0.704439
$795$	0	0
$796$	22.8954	0.811505
$797$	−4.85650	−0.172026	−0.0860130	−	0.996294i	$-0.527413\pi$
$797$	−4.85650	−0.172026	−0.0860130	+	0.996294i	$0.527413\pi$
$798$	0	0
$799$	3.85307	0.136312
$800$	1.85735	0.0656674
$801$	0	0
$802$	13.2995	0.469620
$803$	21.9692	0.775278
$804$	0	0
$805$	0	0
$806$	−1.47118	−0.0518199
$807$	0	0
$808$	5.81794	0.204675
$809$	6.29754	0.221410	0.110705	−	0.993853i	$-0.464689\pi$
$809$	6.29754	0.221410	0.110705	+	0.993853i	$0.464689\pi$
$810$	0	0
$811$	−56.9058	−1.99823	−0.999116	−	0.0420398i	$-0.986614\pi$
$811$	−56.9058	−1.99823	−0.999116	+	0.0420398i	$0.986614\pi$
$812$	0.972709	0.0341354
$813$	0	0
$814$	−6.02761	−0.211268
$815$	13.3843	0.468830
$816$	0	0
$817$	25.1140	0.878629
$818$	−33.6760	−1.17746
$819$	0	0
$820$	−14.4599	−0.504961
$821$	−7.03144	−0.245399	−0.122699	−	0.992444i	$-0.539155\pi$
$821$	−7.03144	−0.245399	−0.122699	+	0.992444i	$0.539155\pi$
$822$	0	0
$823$	−36.4552	−1.27075	−0.635374	−	0.772205i	$-0.719154\pi$
$823$	−36.4552	−1.27075	−0.635374	+	0.772205i	$0.719154\pi$
$824$	−2.40686	−0.0838469
$825$	0	0
$826$	−4.20501	−0.146311
$827$	−40.7408	−1.41670	−0.708349	−	0.705862i	$-0.750560\pi$
$827$	−40.7408	−1.41670	−0.708349	+	0.705862i	$0.750560\pi$
$828$	0	0
$829$	−36.8349	−1.27933	−0.639665	−	0.768654i	$-0.720927\pi$
$829$	−36.8349	−1.27933	−0.639665	+	0.768654i	$0.720927\pi$
$830$	8.71677	0.302563
$831$	0	0
$832$	0.350306	0.0121447
$833$	−2.68743	−0.0931140
$834$	0	0
$835$	2.48402	0.0859630
$836$	−5.45703	−0.188735
$837$	0	0
$838$	28.2304	0.975202
$839$	−36.3073	−1.25347	−0.626734	−	0.779233i	$-0.715609\pi$
$839$	−36.3073	−1.25347	−0.626734	+	0.779233i	$0.715609\pi$
$840$	0	0
$841$	−28.3994	−0.979289
$842$	−2.56596	−0.0884288
$843$	0	0
$844$	1.62881	0.0560661
$845$	−22.8282	−0.785315
$846$	0	0
$847$	7.87467	0.270577
$848$	−5.99729	−0.205948
$849$	0	0
$850$	0.920147	0.0315608
$851$	0	0
$852$	0	0
$853$	−6.29237	−0.215446	−0.107723	−	0.994181i	$-0.534356\pi$
$853$	−6.29237	−0.215446	−0.107723	+	0.994181i	$0.534356\pi$
$854$	10.7706	0.368561
$855$	0	0
$856$	5.49958	0.187972
$857$	−26.8972	−0.918791	−0.459396	−	0.888232i	$-0.651934\pi$
$857$	−26.8972	−0.918791	−0.459396	+	0.888232i	$0.651934\pi$
$858$	0	0
$859$	53.2910	1.81827	0.909133	−	0.416506i	$-0.136746\pi$
$859$	53.2910	1.81827	0.909133	+	0.416506i	$0.136746\pi$
$860$	−17.7358	−0.604787
$861$	0	0
$862$	−2.70696	−0.0921993
$863$	12.9193	0.439779	0.219890	−	0.975525i	$-0.429430\pi$
$863$	12.9193	0.439779	0.219890	+	0.975525i	$0.429430\pi$
$864$	0	0
$865$	−35.5420	−1.20846
$866$	−7.77274	−0.264129
$867$	0	0
$868$	−5.27108	−0.178912
$869$	−24.0285	−0.815111
$870$	0	0
$871$	−1.60467	−0.0543723
$872$	−11.8382	−0.400891
$873$	0	0
$874$	0	0
$875$	15.2576	0.515802
$876$	0	0
$877$	−40.3858	−1.36373	−0.681866	−	0.731477i	$-0.738831\pi$
$877$	−40.3858	−1.36373	−0.681866	+	0.731477i	$0.738831\pi$
$878$	−14.0561	−0.474370
$879$	0	0
$880$	3.85382	0.129912
$881$	−3.16076	−0.106489	−0.0532443	−	0.998582i	$-0.516956\pi$
$881$	−3.16076	−0.106489	−0.0532443	+	0.998582i	$0.516956\pi$
$882$	0	0
$883$	−17.6236	−0.593083	−0.296541	−	0.955020i	$-0.595833\pi$
$883$	−17.6236	−0.593083	−0.296541	+	0.955020i	$0.595833\pi$
$884$	0.173544	0.00583692
$885$	0	0
$886$	17.4343	0.585717
$887$	−22.2558	−0.747276	−0.373638	−	0.927575i	$-0.621890\pi$
$887$	−22.2558	−0.747276	−0.373638	+	0.927575i	$0.621890\pi$
$888$	0	0
$889$	5.87091	0.196904
$890$	−18.5690	−0.622435
$891$	0	0
$892$	−12.6776	−0.424477
$893$	−19.5235	−0.653329
$894$	0	0
$895$	−29.5652	−0.988256
$896$	1.25511	0.0419304
$897$	0	0
$898$	2.21444	0.0738969
$899$	−3.25475	−0.108552
$900$	0	0
$901$	−2.97110	−0.0989817
$902$	17.7321	0.590414
$903$	0	0
$904$	14.7668	0.491137
$905$	−29.6193	−0.984579
$906$	0	0
$907$	−20.3863	−0.676917	−0.338459	−	0.940981i	$-0.609906\pi$
$907$	−20.3863	−0.676917	−0.338459	+	0.940981i	$0.609906\pi$
$908$	23.7647	0.788660
$909$	0	0
$910$	0.779431	0.0258379
$911$	−16.1901	−0.536403	−0.268202	−	0.963363i	$-0.586429\pi$
$911$	−16.1901	−0.536403	−0.268202	+	0.963363i	$0.586429\pi$
$912$	0	0
$913$	−10.6893	−0.353766
$914$	5.24075	0.173349
$915$	0	0
$916$	12.2525	0.404832
$917$	−10.1081	−0.333798
$918$	0	0
$919$	−18.9754	−0.625942	−0.312971	−	0.949763i	$-0.601324\pi$
$919$	−18.9754	−0.625942	−0.312971	+	0.949763i	$0.601324\pi$
$920$	0	0
$921$	0	0
$922$	−24.6340	−0.811277
$923$	−5.07180	−0.166940
$924$	0	0
$925$	−5.14987	−0.169327
$926$	23.0261	0.756683
$927$	0	0
$928$	0.774998	0.0254405
$929$	−51.8527	−1.70123	−0.850616	−	0.525787i	$-0.823771\pi$
$929$	−51.8527	−1.70123	−0.850616	+	0.525787i	$0.823771\pi$
$930$	0	0
$931$	13.6172	0.446286
$932$	−26.7208	−0.875270
$933$	0	0
$934$	35.2398	1.15308
$935$	1.90921	0.0624378
$936$	0	0
$937$	53.7061	1.75450	0.877252	−	0.480031i	$-0.159374\pi$
$937$	53.7061	1.75450	0.877252	+	0.480031i	$0.159374\pi$
$938$	−5.74939	−0.187724
$939$	0	0
$940$	13.7877	0.449706
$941$	−45.0998	−1.47021	−0.735106	−	0.677952i	$-0.762868\pi$
$941$	−45.0998	−1.47021	−0.735106	+	0.677952i	$0.762868\pi$
$942$	0	0
$943$	0	0
$944$	−3.35031	−0.109043
$945$	0	0
$946$	21.7494	0.707134
$947$	28.5041	0.926259	0.463129	−	0.886291i	$-0.346727\pi$
$947$	28.5041	0.926259	0.463129	+	0.886291i	$0.346727\pi$
$948$	0	0
$949$	3.54013	0.114917
$950$	−4.66238	−0.151268
$951$	0	0
$952$	0.621792	0.0201524
$953$	27.7838	0.900006	0.450003	−	0.893027i	$-0.351423\pi$
$953$	27.7838	0.900006	0.450003	+	0.893027i	$0.351423\pi$
$954$	0	0
$955$	−46.7387	−1.51243
$956$	15.9494	0.515840
$957$	0	0
$958$	−22.5196	−0.727576
$959$	−7.60120	−0.245456
$960$	0	0
$961$	−13.3626	−0.431052
$962$	−0.971290	−0.0313157
$963$	0	0
$964$	13.1867	0.424714
$965$	38.0244	1.22405
$966$	0	0
$967$	−42.5113	−1.36707	−0.683535	−	0.729917i	$-0.739558\pi$
$967$	−42.5113	−1.36707	−0.683535	+	0.729917i	$0.739558\pi$
$968$	6.27408	0.201656
$969$	0	0
$970$	−1.19644	−0.0384155
$971$	6.68347	0.214483	0.107241	−	0.994233i	$-0.465798\pi$
$971$	6.68347	0.214483	0.107241	+	0.994233i	$0.465798\pi$
$972$	0	0
$973$	−2.19569	−0.0703905
$974$	27.1279	0.869234
$975$	0	0
$976$	8.58135	0.274682
$977$	−19.9124	−0.637054	−0.318527	−	0.947914i	$-0.603188\pi$
$977$	−19.9124	−0.637054	−0.318527	+	0.947914i	$0.603188\pi$
$978$	0	0
$979$	22.7711	0.727769
$980$	−9.61663	−0.307192
$981$	0	0
$982$	41.1357	1.31269
$983$	34.6211	1.10424	0.552120	−	0.833765i	$-0.313819\pi$
$983$	34.6211	1.10424	0.552120	+	0.833765i	$0.313819\pi$
$984$	0	0
$985$	−26.4515	−0.842816
$986$	0.383939	0.0122271
$987$	0	0
$988$	−0.879347	−0.0279757
$989$	0	0
$990$	0	0
$991$	−12.1439	−0.385763	−0.192882	−	0.981222i	$-0.561783\pi$
$991$	−12.1439	−0.385763	−0.192882	+	0.981222i	$0.561783\pi$
$992$	−4.19969	−0.133340
$993$	0	0
$994$	−18.1718	−0.576374
$995$	40.5878	1.28672
$996$	0	0
$997$	−19.6368	−0.621903	−0.310952	−	0.950426i	$-0.600648\pi$
$997$	−19.6368	−0.621903	−0.310952	+	0.950426i	$0.600648\pi$
$998$	29.9634	0.948475
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cd.1.6	yes	8
3.2	odd	2		9522.2.a.cf.1.3	yes	8
23.22	odd	2	inner	9522.2.a.cd.1.3	✓	8
69.68	even	2		9522.2.a.cf.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.cd.1.3	✓	8	23.22	odd	2	inner
9522.2.a.cd.1.6	yes	8	1.1	even	1	trivial
9522.2.a.cf.1.3	yes	8	3.2	odd	2
9522.2.a.cf.1.6	yes	8	69.68	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.77275 q^{5} -1.25511 q^{7} -1.00000 q^{8} -1.77275 q^{10} +2.17392 q^{11} +0.350306 q^{13} +1.25511 q^{14} +1.00000 q^{16} +0.495407 q^{17} -2.51023 q^{19} +1.77275 q^{20} -2.17392 q^{22} -1.85735 q^{25} -0.350306 q^{26} -1.25511 q^{28} -0.774998 q^{29} +4.19969 q^{31} -1.00000 q^{32} -0.495407 q^{34} -2.22500 q^{35} +2.77269 q^{37} +2.51023 q^{38} -1.77275 q^{40} -8.15674 q^{41} -10.0047 q^{43} +2.17392 q^{44} +7.77759 q^{47} -5.42469 q^{49} +1.85735 q^{50} +0.350306 q^{52} -5.99729 q^{53} +3.85382 q^{55} +1.25511 q^{56} +0.774998 q^{58} -3.35031 q^{59} +8.58135 q^{61} -4.19969 q^{62} +1.00000 q^{64} +0.621005 q^{65} -4.58078 q^{67} +0.495407 q^{68} +2.22500 q^{70} -14.4782 q^{71} +10.1058 q^{73} -2.77269 q^{74} -2.51023 q^{76} -2.72851 q^{77} -11.0531 q^{79} +1.77275 q^{80} +8.15674 q^{82} -4.91708 q^{83} +0.878234 q^{85} +10.0047 q^{86} -2.17392 q^{88} +10.4747 q^{89} -0.439673 q^{91} -7.77759 q^{94} -4.45000 q^{95} +0.674908 q^{97} +5.42469 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 8 * q^25 - 12 * q^26 + 12 * q^29 - 12 * q^31 - 8 * q^32 - 36 * q^35 - 24 * q^41 - 48 * q^47 - 16 * q^49 - 8 * q^50 + 12 * q^52 + 12 * q^55 - 12 * q^58 - 36 * q^59 + 12 * q^62 + 8 * q^64 + 36 * q^70 - 24 * q^71 + 12 * q^73 - 12 * q^77 + 24 * q^82 + 12 * q^85 + 48 * q^94 - 72 * q^95 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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