LMFDB - Embedded newform 9225.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9225,2,Mod(1,9225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9225 = 3^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9225.a (trivial)

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:	yes
Analytic conductor:	$73.6619958646$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	9225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81361 q^{2} +1.28917 q^{4} -2.52444 q^{7} +1.28917 q^{8} +O(q^{10})$	$q-1.81361 q^{2} +1.28917 q^{4} -2.52444 q^{7} +1.28917 q^{8} -0.813607 q^{11} -5.10278 q^{13} +4.57834 q^{14} -4.91638 q^{16} +3.39194 q^{17} +3.10278 q^{19} +1.47556 q^{22} -0.897225 q^{23} +9.25443 q^{26} -3.25443 q^{28} -4.44082 q^{29} -8.96526 q^{31} +6.33804 q^{32} -6.15165 q^{34} -2.08362 q^{37} -5.62721 q^{38} -1.00000 q^{41} -9.07306 q^{43} -1.04888 q^{44} +1.62721 q^{46} -0.235269 q^{47} -0.627213 q^{49} -6.57834 q^{52} +13.8328 q^{53} -3.25443 q^{56} +8.05390 q^{58} -1.04888 q^{59} +1.91638 q^{61} +16.2594 q^{62} -1.66196 q^{64} +10.0383 q^{67} +4.37279 q^{68} +12.8136 q^{71} -4.75971 q^{73} +3.77886 q^{74} +4.00000 q^{76} +2.05390 q^{77} -15.8328 q^{79} +1.81361 q^{82} -7.68111 q^{83} +16.4550 q^{86} -1.04888 q^{88} -13.2544 q^{89} +12.8816 q^{91} -1.15667 q^{92} +0.426686 q^{94} -2.72999 q^{97} +1.13752 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + 4 q^{11} - 8 q^{13} + 12 q^{14} - q^{16} + 2 q^{17} + 2 q^{19} + 10 q^{22} - 10 q^{23} + 2 q^{26} + 16 q^{28} + 6 q^{29} - 2 q^{31} + 7 q^{32} - 20 q^{37} - 4 q^{38} - 3 q^{41} - 10 q^{43} + 8 q^{44} - 8 q^{46} + 4 q^{47} + 11 q^{49} - 18 q^{52} + 14 q^{53} + 16 q^{56} + 28 q^{58} + 8 q^{59} - 8 q^{61} + 38 q^{62} - 17 q^{64} - 12 q^{67} + 26 q^{68} + 32 q^{71} - 4 q^{73} - 20 q^{74} + 12 q^{76} + 10 q^{77} - 20 q^{79} - q^{82} - 14 q^{83} - 6 q^{86} + 8 q^{88} - 14 q^{89} + 18 q^{94} + 12 q^{97} + 21 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 + 4 * q^11 - 8 * q^13 + 12 * q^14 - q^16 + 2 * q^17 + 2 * q^19 + 10 * q^22 - 10 * q^23 + 2 * q^26 + 16 * q^28 + 6 * q^29 - 2 * q^31 + 7 * q^32 - 20 * q^37 - 4 * q^38 - 3 * q^41 - 10 * q^43 + 8 * q^44 - 8 * q^46 + 4 * q^47 + 11 * q^49 - 18 * q^52 + 14 * q^53 + 16 * q^56 + 28 * q^58 + 8 * q^59 - 8 * q^61 + 38 * q^62 - 17 * q^64 - 12 * q^67 + 26 * q^68 + 32 * q^71 - 4 * q^73 - 20 * q^74 + 12 * q^76 + 10 * q^77 - 20 * q^79 - q^82 - 14 * q^83 - 6 * q^86 + 8 * q^88 - 14 * q^89 + 18 * q^94 + 12 * q^97 + 21 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.81361	−1.28241	−0.641207	−	0.767368i	$-0.721566\pi$
$2$	−1.81361	−1.28241	−0.641207	+	0.767368i	$0.721566\pi$
$3$	0	0
$3$	0	0
$4$	1.28917	0.644584
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.52444	−0.954148	−0.477074	−	0.878863i	$-0.658303\pi$
$7$	−2.52444	−0.954148	−0.477074	+	0.878863i	$0.658303\pi$
$8$	1.28917	0.455790
$9$	0	0
$10$	0	0
$11$	−0.813607	−0.245312	−0.122656	−	0.992449i	$-0.539141\pi$
$11$	−0.813607	−0.245312	−0.122656	+	0.992449i	$0.539141\pi$
$12$	0	0
$13$	−5.10278	−1.41526	−0.707628	−	0.706586i	$-0.750235\pi$
$13$	−5.10278	−1.41526	−0.707628	+	0.706586i	$0.750235\pi$
$14$	4.57834	1.22361
$15$	0	0
$16$	−4.91638	−1.22910
$17$	3.39194	0.822667	0.411334	−	0.911485i	$-0.365063\pi$
$17$	3.39194	0.822667	0.411334	+	0.911485i	$0.365063\pi$
$18$	0	0
$19$	3.10278	0.711825	0.355913	−	0.934519i	$-0.384170\pi$
$19$	3.10278	0.711825	0.355913	+	0.934519i	$0.384170\pi$
$20$	0	0
$21$	0	0
$22$	1.47556	0.314591
$23$	−0.897225	−0.187084	−0.0935422	−	0.995615i	$-0.529819\pi$
$23$	−0.897225	−0.187084	−0.0935422	+	0.995615i	$0.529819\pi$
$24$	0	0
$25$	0	0
$26$	9.25443	1.81494
$27$	0	0
$28$	−3.25443	−0.615029
$29$	−4.44082	−0.824639	−0.412320	−	0.911039i	$-0.635281\pi$
$29$	−4.44082	−0.824639	−0.412320	+	0.911039i	$0.635281\pi$
$30$	0	0
$31$	−8.96526	−1.61021	−0.805104	−	0.593134i	$-0.797890\pi$
$31$	−8.96526	−1.61021	−0.805104	+	0.593134i	$0.797890\pi$
$32$	6.33804	1.12042
$33$	0	0
$34$	−6.15165	−1.05500
$35$	0	0
$36$	0	0
$37$	−2.08362	−0.342545	−0.171272	−	0.985224i	$-0.554788\pi$
$37$	−2.08362	−0.342545	−0.171272	+	0.985224i	$0.554788\pi$
$38$	−5.62721	−0.912854
$39$	0	0
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−9.07306	−1.38363	−0.691814	−	0.722076i	$-0.743188\pi$
$43$	−9.07306	−1.38363	−0.691814	+	0.722076i	$0.743188\pi$
$44$	−1.04888	−0.158124
$45$	0	0
$46$	1.62721	0.239919
$47$	−0.235269	−0.0343176	−0.0171588	−	0.999853i	$-0.505462\pi$
$47$	−0.235269	−0.0343176	−0.0171588	+	0.999853i	$0.505462\pi$
$48$	0	0
$49$	−0.627213	−0.0896019
$50$	0	0
$51$	0	0
$52$	−6.57834	−0.912251
$53$	13.8328	1.90008	0.950038	−	0.312134i	$-0.101044\pi$
$53$	13.8328	1.90008	0.950038	+	0.312134i	$0.101044\pi$
$54$	0	0
$55$	0	0
$56$	−3.25443	−0.434891
$57$	0	0
$58$	8.05390	1.05753
$59$	−1.04888	−0.136552	−0.0682760	−	0.997666i	$-0.521750\pi$
$59$	−1.04888	−0.136552	−0.0682760	+	0.997666i	$0.521750\pi$
$60$	0	0
$61$	1.91638	0.245368	0.122684	−	0.992446i	$-0.460850\pi$
$61$	1.91638	0.245368	0.122684	+	0.992446i	$0.460850\pi$
$62$	16.2594	2.06495
$63$	0	0
$64$	−1.66196	−0.207744
$65$	0	0
$66$	0	0
$67$	10.0383	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$67$	10.0383	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$68$	4.37279	0.530278
$69$	0	0
$70$	0	0
$71$	12.8136	1.52070	0.760348	−	0.649516i	$-0.225029\pi$
$71$	12.8136	1.52070	0.760348	+	0.649516i	$0.225029\pi$
$72$	0	0
$73$	−4.75971	−0.557082	−0.278541	−	0.960424i	$-0.589851\pi$
$73$	−4.75971	−0.557082	−0.278541	+	0.960424i	$0.589851\pi$
$74$	3.77886	0.439284
$75$	0	0
$76$	4.00000	0.458831
$77$	2.05390	0.234064
$78$	0	0
$79$	−15.8328	−1.78133	−0.890663	−	0.454665i	$-0.849759\pi$
$79$	−15.8328	−1.78133	−0.890663	+	0.454665i	$0.849759\pi$
$80$	0	0
$81$	0	0
$82$	1.81361	0.200279
$83$	−7.68111	−0.843112	−0.421556	−	0.906802i	$-0.638516\pi$
$83$	−7.68111	−0.843112	−0.421556	+	0.906802i	$0.638516\pi$
$84$	0	0
$85$	0	0
$86$	16.4550	1.77438
$87$	0	0
$88$	−1.04888	−0.111811
$89$	−13.2544	−1.40497	−0.702483	−	0.711700i	$-0.747925\pi$
$89$	−13.2544	−1.40497	−0.702483	+	0.711700i	$0.747925\pi$
$90$	0	0
$91$	12.8816	1.35036
$92$	−1.15667	−0.120592
$93$	0	0
$94$	0.426686	0.0440093
$95$	0	0
$96$	0	0
$97$	−2.72999	−0.277188	−0.138594	−	0.990349i	$-0.544258\pi$
$97$	−2.72999	−0.277188	−0.138594	+	0.990349i	$0.544258\pi$
$98$	1.13752	0.114907
$99$	0	0
$100$	0	0
$101$	11.8625	1.18036	0.590181	−	0.807271i	$-0.299057\pi$
$101$	11.8625	1.18036	0.590181	+	0.807271i	$0.299057\pi$
$102$	0	0
$103$	0.494719	0.0487461	0.0243730	−	0.999703i	$-0.492241\pi$
$103$	0.494719	0.0487461	0.0243730	+	0.999703i	$0.492241\pi$
$104$	−6.57834	−0.645059
$105$	0	0
$106$	−25.0872	−2.43668
$107$	−15.0872	−1.45853	−0.729267	−	0.684229i	$-0.760139\pi$
$107$	−15.0872	−1.45853	−0.729267	+	0.684229i	$0.760139\pi$
$108$	0	0
$109$	5.10278	0.488757	0.244379	−	0.969680i	$-0.421416\pi$
$109$	5.10278	0.488757	0.244379	+	0.969680i	$0.421416\pi$
$110$	0	0
$111$	0	0
$112$	12.4111	1.17274
$113$	−17.5139	−1.64757	−0.823783	−	0.566905i	$-0.808141\pi$
$113$	−17.5139	−1.64757	−0.823783	+	0.566905i	$0.808141\pi$
$114$	0	0
$115$	0	0
$116$	−5.72496	−0.531550
$117$	0	0
$118$	1.90225	0.175116
$119$	−8.56275	−0.784946
$120$	0	0
$121$	−10.3380	−0.939822
$122$	−3.47556	−0.314663
$123$	0	0
$124$	−11.5577	−1.03791
$125$	0	0
$126$	0	0
$127$	−12.8816	−1.14306	−0.571530	−	0.820581i	$-0.693650\pi$
$127$	−12.8816	−1.14306	−0.571530	+	0.820581i	$0.693650\pi$
$128$	−9.66196	−0.854004
$129$	0	0
$130$	0	0
$131$	6.35720	0.555431	0.277716	−	0.960663i	$-0.410423\pi$
$131$	6.35720	0.555431	0.277716	+	0.960663i	$0.410423\pi$
$132$	0	0
$133$	−7.83276	−0.679187
$134$	−18.2056	−1.57272
$135$	0	0
$136$	4.37279	0.374963
$137$	−17.4897	−1.49425	−0.747123	−	0.664686i	$-0.768565\pi$
$137$	−17.4897	−1.49425	−0.747123	+	0.664686i	$0.768565\pi$
$138$	0	0
$139$	15.7250	1.33377	0.666887	−	0.745159i	$-0.267626\pi$
$139$	15.7250	1.33377	0.666887	+	0.745159i	$0.267626\pi$
$140$	0	0
$141$	0	0
$142$	−23.2388	−1.95016
$143$	4.15165	0.347178
$144$	0	0
$145$	0	0
$146$	8.63224	0.714409
$147$	0	0
$148$	−2.68614	−0.220799
$149$	−11.5678	−0.947669	−0.473835	−	0.880614i	$-0.657131\pi$
$149$	−11.5678	−0.947669	−0.473835	+	0.880614i	$0.657131\pi$
$150$	0	0
$151$	19.2544	1.56690	0.783451	−	0.621453i	$-0.213457\pi$
$151$	19.2544	1.56690	0.783451	+	0.621453i	$0.213457\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	−3.72496	−0.300166
$155$	0	0
$156$	0	0
$157$	20.9200	1.66959	0.834797	−	0.550558i	$-0.185585\pi$
$157$	20.9200	1.66959	0.834797	+	0.550558i	$0.185585\pi$
$158$	28.7144	2.28440
$159$	0	0
$160$	0	0
$161$	2.26499	0.178506
$162$	0	0
$163$	12.7980	1.00242	0.501209	−	0.865326i	$-0.332889\pi$
$163$	12.7980	1.00242	0.501209	+	0.865326i	$0.332889\pi$
$164$	−1.28917	−0.100667
$165$	0	0
$166$	13.9305	1.08122
$167$	9.62721	0.744976	0.372488	−	0.928037i	$-0.378505\pi$
$167$	9.62721	0.744976	0.372488	+	0.928037i	$0.378505\pi$
$168$	0	0
$169$	13.0383	1.00295
$170$	0	0
$171$	0	0
$172$	−11.6967	−0.891865
$173$	11.9461	0.908245	0.454123	−	0.890939i	$-0.349953\pi$
$173$	11.9461	0.908245	0.454123	+	0.890939i	$0.349953\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	24.0383	1.80175
$179$	2.13752	0.159766	0.0798828	−	0.996804i	$-0.474545\pi$
$179$	2.13752	0.159766	0.0798828	+	0.996804i	$0.474545\pi$
$180$	0	0
$181$	−1.83276	−0.136228	−0.0681141	−	0.997678i	$-0.521698\pi$
$181$	−1.83276	−0.136228	−0.0681141	+	0.997678i	$0.521698\pi$
$182$	−23.3622	−1.73172
$183$	0	0
$184$	−1.15667	−0.0852712
$185$	0	0
$186$	0	0
$187$	−2.75971	−0.201810
$188$	−0.303302	−0.0221206
$189$	0	0
$190$	0	0
$191$	18.6167	1.34705	0.673527	−	0.739163i	$-0.264779\pi$
$191$	18.6167	1.34705	0.673527	+	0.739163i	$0.264779\pi$
$192$	0	0
$193$	−15.6116	−1.12375	−0.561875	−	0.827222i	$-0.689920\pi$
$193$	−15.6116	−1.12375	−0.561875	+	0.827222i	$0.689920\pi$
$194$	4.95112	0.355470
$195$	0	0
$196$	−0.808583	−0.0577559
$197$	−5.21057	−0.371238	−0.185619	−	0.982622i	$-0.559429\pi$
$197$	−5.21057	−0.371238	−0.185619	+	0.982622i	$0.559429\pi$
$198$	0	0
$199$	2.05390	0.145597	0.0727985	−	0.997347i	$-0.476807\pi$
$199$	2.05390	0.145597	0.0727985	+	0.997347i	$0.476807\pi$
$200$	0	0
$201$	0	0
$202$	−21.5139	−1.51371
$203$	11.2106	0.786828
$204$	0	0
$205$	0	0
$206$	−0.897225	−0.0625126
$207$	0	0
$208$	25.0872	1.73948
$209$	−2.52444	−0.174619
$210$	0	0
$211$	−9.04888	−0.622950	−0.311475	−	0.950254i	$-0.600823\pi$
$211$	−9.04888	−0.622950	−0.311475	+	0.950254i	$0.600823\pi$
$212$	17.8328	1.22476
$213$	0	0
$214$	27.3622	1.87044
$215$	0	0
$216$	0	0
$217$	22.6322	1.53638
$218$	−9.25443	−0.626789
$219$	0	0
$220$	0	0
$221$	−17.3083	−1.16428
$222$	0	0
$223$	−24.8222	−1.66222	−0.831109	−	0.556110i	$-0.812293\pi$
$223$	−24.8222	−1.66222	−0.831109	+	0.556110i	$0.812293\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	31.7633	2.11286
$227$	16.6464	1.10486	0.552429	−	0.833560i	$-0.313701\pi$
$227$	16.6464	1.10486	0.552429	+	0.833560i	$0.313701\pi$
$228$	0	0
$229$	−15.7789	−1.04270	−0.521348	−	0.853344i	$-0.674571\pi$
$229$	−15.7789	−1.04270	−0.521348	+	0.853344i	$0.674571\pi$
$230$	0	0
$231$	0	0
$232$	−5.72496	−0.375862
$233$	−24.9200	−1.63256	−0.816280	−	0.577656i	$-0.803967\pi$
$233$	−24.9200	−1.63256	−0.816280	+	0.577656i	$0.803967\pi$
$234$	0	0
$235$	0	0
$236$	−1.35218	−0.0880193
$237$	0	0
$238$	15.5295	1.00663
$239$	2.95112	0.190892	0.0954462	−	0.995435i	$-0.469572\pi$
$239$	2.95112	0.190892	0.0954462	+	0.995435i	$0.469572\pi$
$240$	0	0
$241$	−16.5925	−1.06881	−0.534407	−	0.845227i	$-0.679465\pi$
$241$	−16.5925	−1.06881	−0.534407	+	0.845227i	$0.679465\pi$
$242$	18.7491	1.20524
$243$	0	0
$244$	2.47054	0.158160
$245$	0	0
$246$	0	0
$247$	−15.8328	−1.00741
$248$	−11.5577	−0.733916
$249$	0	0
$250$	0	0
$251$	20.1361	1.27098	0.635489	−	0.772110i	$-0.280799\pi$
$251$	20.1361	1.27098	0.635489	+	0.772110i	$0.280799\pi$
$252$	0	0
$253$	0.729988	0.0458940
$254$	23.3622	1.46588
$255$	0	0
$256$	20.8469	1.30293
$257$	25.9008	1.61565	0.807824	−	0.589424i	$-0.200645\pi$
$257$	25.9008	1.61565	0.807824	+	0.589424i	$0.200645\pi$
$258$	0	0
$259$	5.25997	0.326838
$260$	0	0
$261$	0	0
$262$	−11.5295	−0.712292
$263$	−22.4408	−1.38376	−0.691880	−	0.722012i	$-0.743217\pi$
$263$	−22.4408	−1.38376	−0.691880	+	0.722012i	$0.743217\pi$
$264$	0	0
$265$	0	0
$266$	14.2056	0.870998
$267$	0	0
$268$	12.9411	0.790502
$269$	−7.62721	−0.465039	−0.232520	−	0.972592i	$-0.574697\pi$
$269$	−7.62721	−0.465039	−0.232520	+	0.972592i	$0.574697\pi$
$270$	0	0
$271$	19.9164	1.20983	0.604917	−	0.796289i	$-0.293206\pi$
$271$	19.9164	1.20983	0.604917	+	0.796289i	$0.293206\pi$
$272$	−16.6761	−1.01114
$273$	0	0
$274$	31.7194	1.91624
$275$	0	0
$276$	0	0
$277$	17.6413	1.05997	0.529983	−	0.848008i	$-0.322198\pi$
$277$	17.6413	1.05997	0.529983	+	0.848008i	$0.322198\pi$
$278$	−28.5189	−1.71045
$279$	0	0
$280$	0	0
$281$	0.0297193	0.00177291	0.000886453	−	1.00000i	$-0.499718\pi$
$281$	0.0297193	0.00177291	0.000886453		1.00000i	$0.499718\pi$
$282$	0	0
$283$	10.1814	0.605220	0.302610	−	0.953115i	$-0.402142\pi$
$283$	10.1814	0.605220	0.302610	+	0.953115i	$0.402142\pi$
$284$	16.5189	0.980216
$285$	0	0
$286$	−7.52946	−0.445226
$287$	2.52444	0.149013
$288$	0	0
$289$	−5.49472	−0.323219
$290$	0	0
$291$	0	0
$292$	−6.13607	−0.359086
$293$	3.14808	0.183913	0.0919564	−	0.995763i	$-0.470688\pi$
$293$	3.14808	0.183913	0.0919564	+	0.995763i	$0.470688\pi$
$294$	0	0
$295$	0	0
$296$	−2.68614	−0.156128
$297$	0	0
$298$	20.9794	1.21530
$299$	4.57834	0.264772
$300$	0	0
$301$	22.9044	1.32019
$302$	−34.9200	−2.00942
$303$	0	0
$304$	−15.2544	−0.874901
$305$	0	0
$306$	0	0
$307$	12.7980	0.730422	0.365211	−	0.930925i	$-0.380997\pi$
$307$	12.7980	0.730422	0.365211	+	0.930925i	$0.380997\pi$
$308$	2.64782	0.150874
$309$	0	0
$310$	0	0
$311$	−26.6167	−1.50929	−0.754646	−	0.656132i	$-0.772191\pi$
$311$	−26.6167	−1.50929	−0.754646	+	0.656132i	$0.772191\pi$
$312$	0	0
$313$	3.00502	0.169854	0.0849270	−	0.996387i	$-0.472934\pi$
$313$	3.00502	0.169854	0.0849270	+	0.996387i	$0.472934\pi$
$314$	−37.9406	−2.14111
$315$	0	0
$316$	−20.4111	−1.14821
$317$	−4.33302	−0.243367	−0.121683	−	0.992569i	$-0.538829\pi$
$317$	−4.33302	−0.243367	−0.121683	+	0.992569i	$0.538829\pi$
$318$	0	0
$319$	3.61308	0.202294
$320$	0	0
$321$	0	0
$322$	−4.10780	−0.228919
$323$	10.5244	0.585595
$324$	0	0
$325$	0	0
$326$	−23.2106	−1.28551
$327$	0	0
$328$	−1.28917	−0.0711824
$329$	0.593923	0.0327440
$330$	0	0
$331$	5.53500	0.304231	0.152116	−	0.988363i	$-0.451391\pi$
$331$	5.53500	0.304231	0.152116	+	0.988363i	$0.451391\pi$
$332$	−9.90225	−0.543456
$333$	0	0
$334$	−17.4600	−0.955367
$335$	0	0
$336$	0	0
$337$	3.71083	0.202142	0.101071	−	0.994879i	$-0.467773\pi$
$337$	3.71083	0.202142	0.101071	+	0.994879i	$0.467773\pi$
$338$	−23.6464	−1.28619
$339$	0	0
$340$	0	0
$341$	7.29419	0.395003
$342$	0	0
$343$	19.2544	1.03964
$344$	−11.6967	−0.630644
$345$	0	0
$346$	−21.6655	−1.16475
$347$	3.07860	0.165268	0.0826338	−	0.996580i	$-0.473667\pi$
$347$	3.07860	0.165268	0.0826338	+	0.996580i	$0.473667\pi$
$348$	0	0
$349$	14.7980	0.792120	0.396060	−	0.918225i	$-0.370377\pi$
$349$	14.7980	0.792120	0.396060	+	0.918225i	$0.370377\pi$
$350$	0	0
$351$	0	0
$352$	−5.15667	−0.274852
$353$	26.8222	1.42760	0.713801	−	0.700349i	$-0.246972\pi$
$353$	26.8222	1.42760	0.713801	+	0.700349i	$0.246972\pi$
$354$	0	0
$355$	0	0
$356$	−17.0872	−0.905619
$357$	0	0
$358$	−3.87662	−0.204886
$359$	−2.35720	−0.124408	−0.0622042	−	0.998063i	$-0.519813\pi$
$359$	−2.35720	−0.124408	−0.0622042	+	0.998063i	$0.519813\pi$
$360$	0	0
$361$	−9.37279	−0.493305
$362$	3.32391	0.174701
$363$	0	0
$364$	16.6066	0.870423
$365$	0	0
$366$	0	0
$367$	−23.1708	−1.20951	−0.604753	−	0.796413i	$-0.706728\pi$
$367$	−23.1708	−1.20951	−0.604753	+	0.796413i	$0.706728\pi$
$368$	4.41110	0.229944
$369$	0	0
$370$	0	0
$371$	−34.9200	−1.81295
$372$	0	0
$373$	−35.2091	−1.82306	−0.911530	−	0.411235i	$-0.865098\pi$
$373$	−35.2091	−1.82306	−0.911530	+	0.411235i	$0.865098\pi$
$374$	5.00502	0.258804
$375$	0	0
$376$	−0.303302	−0.0156416
$377$	22.6605	1.16708
$378$	0	0
$379$	−26.0383	−1.33750	−0.668749	−	0.743488i	$-0.733170\pi$
$379$	−26.0383	−1.33750	−0.668749	+	0.743488i	$0.733170\pi$
$380$	0	0
$381$	0	0
$382$	−33.7633	−1.72748
$383$	15.3819	0.785978	0.392989	−	0.919543i	$-0.371441\pi$
$383$	15.3819	0.785978	0.392989	+	0.919543i	$0.371441\pi$
$384$	0	0
$385$	0	0
$386$	28.3133	1.44111
$387$	0	0
$388$	−3.51941	−0.178671
$389$	−4.46500	−0.226384	−0.113192	−	0.993573i	$-0.536108\pi$
$389$	−4.46500	−0.226384	−0.113192	+	0.993573i	$0.536108\pi$
$390$	0	0
$391$	−3.04334	−0.153908
$392$	−0.808583	−0.0408396
$393$	0	0
$394$	9.44993	0.476081
$395$	0	0
$396$	0	0
$397$	10.5628	0.530129	0.265065	−	0.964231i	$-0.414607\pi$
$397$	10.5628	0.530129	0.265065	+	0.964231i	$0.414607\pi$
$398$	−3.72496	−0.186716
$399$	0	0
$400$	0	0
$401$	−13.0872	−0.653543	−0.326772	−	0.945103i	$-0.605961\pi$
$401$	−13.0872	−0.653543	−0.326772	+	0.945103i	$0.605961\pi$
$402$	0	0
$403$	45.7477	2.27885
$404$	15.2927	0.760842
$405$	0	0
$406$	−20.3316	−1.00904
$407$	1.69525	0.0840302
$408$	0	0
$409$	−18.5542	−0.917444	−0.458722	−	0.888580i	$-0.651693\pi$
$409$	−18.5542	−0.917444	−0.458722	+	0.888580i	$0.651693\pi$
$410$	0	0
$411$	0	0
$412$	0.637776	0.0314210
$413$	2.64782	0.130291
$414$	0	0
$415$	0	0
$416$	−32.3416	−1.58568
$417$	0	0
$418$	4.57834	0.223934
$419$	25.6116	1.25121	0.625605	−	0.780140i	$-0.284852\pi$
$419$	25.6116	1.25121	0.625605	+	0.780140i	$0.284852\pi$
$420$	0	0
$421$	−8.67609	−0.422847	−0.211423	−	0.977395i	$-0.567810\pi$
$421$	−8.67609	−0.422847	−0.211423	+	0.977395i	$0.567810\pi$
$422$	16.4111	0.798880
$423$	0	0
$424$	17.8328	0.866036
$425$	0	0
$426$	0	0
$427$	−4.83779	−0.234117
$428$	−19.4499	−0.940148
$429$	0	0
$430$	0	0
$431$	−7.10278	−0.342129	−0.171064	−	0.985260i	$-0.554721\pi$
$431$	−7.10278	−0.342129	−0.171064	+	0.985260i	$0.554721\pi$
$432$	0	0
$433$	11.0247	0.529813	0.264907	−	0.964274i	$-0.414659\pi$
$433$	11.0247	0.529813	0.264907	+	0.964274i	$0.414659\pi$
$434$	−41.0460	−1.97027
$435$	0	0
$436$	6.57834	0.315045
$437$	−2.78389	−0.133171
$438$	0	0
$439$	−1.32391	−0.0631868	−0.0315934	−	0.999501i	$-0.510058\pi$
$439$	−1.32391	−0.0631868	−0.0315934	+	0.999501i	$0.510058\pi$
$440$	0	0
$441$	0	0
$442$	31.3905	1.49309
$443$	15.5889	0.740651	0.370325	−	0.928902i	$-0.379246\pi$
$443$	15.5889	0.740651	0.370325	+	0.928902i	$0.379246\pi$
$444$	0	0
$445$	0	0
$446$	45.0177	2.13165
$447$	0	0
$448$	4.19550	0.198219
$449$	32.7839	1.54717	0.773584	−	0.633694i	$-0.218462\pi$
$449$	32.7839	1.54717	0.773584	+	0.633694i	$0.218462\pi$
$450$	0	0
$451$	0.813607	0.0383112
$452$	−22.5783	−1.06200
$453$	0	0
$454$	−30.1900	−1.41689
$455$	0	0
$456$	0	0
$457$	14.1672	0.662715	0.331358	−	0.943505i	$-0.392493\pi$
$457$	14.1672	0.662715	0.331358	+	0.943505i	$0.392493\pi$
$458$	28.6167	1.33717
$459$	0	0
$460$	0	0
$461$	30.0766	1.40081	0.700404	−	0.713747i	$-0.253003\pi$
$461$	30.0766	1.40081	0.700404	+	0.713747i	$0.253003\pi$
$462$	0	0
$463$	23.8766	1.10964	0.554820	−	0.831970i	$-0.312787\pi$
$463$	23.8766	1.10964	0.554820	+	0.831970i	$0.312787\pi$
$464$	21.8328	1.01356
$465$	0	0
$466$	45.1950	2.09362
$467$	−9.73501	−0.450483	−0.225241	−	0.974303i	$-0.572317\pi$
$467$	−9.73501	−0.450483	−0.225241	+	0.974303i	$0.572317\pi$
$468$	0	0
$469$	−25.3411	−1.17014
$470$	0	0
$471$	0	0
$472$	−1.35218	−0.0622390
$473$	7.38190	0.339420
$474$	0	0
$475$	0	0
$476$	−11.0388	−0.505964
$477$	0	0
$478$	−5.35218	−0.244803
$479$	1.17635	0.0537487	0.0268743	−	0.999639i	$-0.491445\pi$
$479$	1.17635	0.0537487	0.0268743	+	0.999639i	$0.491445\pi$
$480$	0	0
$481$	10.6322	0.484788
$482$	30.0922	1.37066
$483$	0	0
$484$	−13.3275	−0.605795
$485$	0	0
$486$	0	0
$487$	−20.6025	−0.933589	−0.466795	−	0.884366i	$-0.654591\pi$
$487$	−20.6025	−0.933589	−0.466795	+	0.884366i	$0.654591\pi$
$488$	2.47054	0.111836
$489$	0	0
$490$	0	0
$491$	−14.1900	−0.640384	−0.320192	−	0.947353i	$-0.603747\pi$
$491$	−14.1900	−0.640384	−0.320192	+	0.947353i	$0.603747\pi$
$492$	0	0
$493$	−15.0630	−0.678404
$494$	28.7144	1.29192
$495$	0	0
$496$	44.0766	1.97910
$497$	−32.3472	−1.45097
$498$	0	0
$499$	−21.4600	−0.960680	−0.480340	−	0.877082i	$-0.659487\pi$
$499$	−21.4600	−0.960680	−0.480340	+	0.877082i	$0.659487\pi$
$500$	0	0
$501$	0	0
$502$	−36.5189	−1.62992
$503$	−15.5491	−0.693302	−0.346651	−	0.937994i	$-0.612681\pi$
$503$	−15.5491	−0.693302	−0.346651	+	0.937994i	$0.612681\pi$
$504$	0	0
$505$	0	0
$506$	−1.32391	−0.0588550
$507$	0	0
$508$	−16.6066	−0.736799
$509$	−22.7542	−1.00856	−0.504280	−	0.863540i	$-0.668242\pi$
$509$	−22.7542	−1.00856	−0.504280	+	0.863540i	$0.668242\pi$
$510$	0	0
$511$	12.0156	0.531538
$512$	−18.4842	−0.816892
$513$	0	0
$514$	−46.9739	−2.07193
$515$	0	0
$516$	0	0
$517$	0.191417	0.00841850
$518$	−9.53951	−0.419142
$519$	0	0
$520$	0	0
$521$	38.7230	1.69649	0.848243	−	0.529608i	$-0.177661\pi$
$521$	38.7230	1.69649	0.848243	+	0.529608i	$0.177661\pi$
$522$	0	0
$523$	−6.56829	−0.287211	−0.143606	−	0.989635i	$-0.545870\pi$
$523$	−6.56829	−0.287211	−0.143606	+	0.989635i	$0.545870\pi$
$524$	8.19550	0.358022
$525$	0	0
$526$	40.6988	1.77455
$527$	−30.4096	−1.32467
$528$	0	0
$529$	−22.1950	−0.964999
$530$	0	0
$531$	0	0
$532$	−10.0978	−0.437793
$533$	5.10278	0.221026
$534$	0	0
$535$	0	0
$536$	12.9411	0.558969
$537$	0	0
$538$	13.8328	0.596373
$539$	0.510305	0.0219804
$540$	0	0
$541$	42.0766	1.80902	0.904508	−	0.426457i	$-0.140239\pi$
$541$	42.0766	1.80902	0.904508	+	0.426457i	$0.140239\pi$
$542$	−36.1205	−1.55151
$543$	0	0
$544$	21.4983	0.921732
$545$	0	0
$546$	0	0
$547$	−19.9688	−0.853805	−0.426903	−	0.904298i	$-0.640395\pi$
$547$	−19.9688	−0.853805	−0.426903	+	0.904298i	$0.640395\pi$
$548$	−22.5472	−0.963167
$549$	0	0
$550$	0	0
$551$	−13.7789	−0.586999
$552$	0	0
$553$	39.9688	1.69965
$554$	−31.9945	−1.35931
$555$	0	0
$556$	20.2721	0.859730
$557$	13.5491	0.574095	0.287048	−	0.957916i	$-0.407326\pi$
$557$	13.5491	0.574095	0.287048	+	0.957916i	$0.407326\pi$
$558$	0	0
$559$	46.2978	1.95819
$560$	0	0
$561$	0	0
$562$	−0.0538991	−0.00227360
$563$	9.75468	0.411111	0.205555	−	0.978645i	$-0.434100\pi$
$563$	9.75468	0.411111	0.205555	+	0.978645i	$0.434100\pi$
$564$	0	0
$565$	0	0
$566$	−18.4650	−0.776142
$567$	0	0
$568$	16.5189	0.693118
$569$	21.4544	0.899417	0.449708	−	0.893175i	$-0.351528\pi$
$569$	21.4544	0.899417	0.449708	+	0.893175i	$0.351528\pi$
$570$	0	0
$571$	20.9411	0.876357	0.438178	−	0.898888i	$-0.355624\pi$
$571$	20.9411	0.876357	0.438178	+	0.898888i	$0.355624\pi$
$572$	5.35218	0.223786
$573$	0	0
$574$	−4.57834	−0.191096
$575$	0	0
$576$	0	0
$577$	−18.4705	−0.768939	−0.384469	−	0.923138i	$-0.625616\pi$
$577$	−18.4705	−0.768939	−0.384469	+	0.923138i	$0.625616\pi$
$578$	9.96526	0.414500
$579$	0	0
$580$	0	0
$581$	19.3905	0.804453
$582$	0	0
$583$	−11.2544	−0.466111
$584$	−6.13607	−0.253912
$585$	0	0
$586$	−5.70938	−0.235852
$587$	−0.127471	−0.00526130	−0.00263065	−	0.999997i	$-0.500837\pi$
$587$	−0.127471	−0.00526130	−0.00263065	+	0.999997i	$0.500837\pi$
$588$	0	0
$589$	−27.8172	−1.14619
$590$	0	0
$591$	0	0
$592$	10.2439	0.421020
$593$	−27.2841	−1.12043	−0.560213	−	0.828349i	$-0.689281\pi$
$593$	−27.2841	−1.12043	−0.560213	+	0.828349i	$0.689281\pi$
$594$	0	0
$595$	0	0
$596$	−14.9128	−0.610853
$597$	0	0
$598$	−8.30330	−0.339547
$599$	42.3260	1.72939	0.864697	−	0.502293i	$-0.167510\pi$
$599$	42.3260	1.72939	0.864697	+	0.502293i	$0.167510\pi$
$600$	0	0
$601$	−15.6756	−0.639420	−0.319710	−	0.947515i	$-0.603585\pi$
$601$	−15.6756	−0.639420	−0.319710	+	0.947515i	$0.603585\pi$
$602$	−41.5395	−1.69302
$603$	0	0
$604$	24.8222	1.01000
$605$	0	0
$606$	0	0
$607$	1.93051	0.0783572	0.0391786	−	0.999232i	$-0.487526\pi$
$607$	1.93051	0.0783572	0.0391786	+	0.999232i	$0.487526\pi$
$608$	19.6655	0.797542
$609$	0	0
$610$	0	0
$611$	1.20053	0.0485681
$612$	0	0
$613$	0.372787	0.0150567	0.00752836	−	0.999972i	$-0.497604\pi$
$613$	0.372787	0.0150567	0.00752836	+	0.999972i	$0.497604\pi$
$614$	−23.2106	−0.936703
$615$	0	0
$616$	2.64782	0.106684
$617$	31.3083	1.26043	0.630213	−	0.776422i	$-0.282968\pi$
$617$	31.3083	1.26043	0.630213	+	0.776422i	$0.282968\pi$
$618$	0	0
$619$	34.3658	1.38128	0.690639	−	0.723200i	$-0.257329\pi$
$619$	34.3658	1.38128	0.690639	+	0.723200i	$0.257329\pi$
$620$	0	0
$621$	0	0
$622$	48.2721	1.93554
$623$	33.4600	1.34055
$624$	0	0
$625$	0	0
$626$	−5.44993	−0.217823
$627$	0	0
$628$	26.9693	1.07619
$629$	−7.06752	−0.281800
$630$	0	0
$631$	7.33804	0.292123	0.146061	−	0.989276i	$-0.453340\pi$
$631$	7.33804	0.292123	0.146061	+	0.989276i	$0.453340\pi$
$632$	−20.4111	−0.811910
$633$	0	0
$634$	7.85840	0.312097
$635$	0	0
$636$	0	0
$637$	3.20053	0.126809
$638$	−6.55270	−0.259424
$639$	0	0
$640$	0	0
$641$	31.5764	1.24719	0.623596	−	0.781747i	$-0.285671\pi$
$641$	31.5764	1.24719	0.623596	+	0.781747i	$0.285671\pi$
$642$	0	0
$643$	4.51890	0.178208	0.0891040	−	0.996022i	$-0.471600\pi$
$643$	4.51890	0.178208	0.0891040	+	0.996022i	$0.471600\pi$
$644$	2.91995	0.115062
$645$	0	0
$646$	−19.0872	−0.750975
$647$	−20.6705	−0.812643	−0.406322	−	0.913730i	$-0.633189\pi$
$647$	−20.6705	−0.812643	−0.406322	+	0.913730i	$0.633189\pi$
$648$	0	0
$649$	0.853372	0.0334978
$650$	0	0
$651$	0	0
$652$	16.4988	0.646143
$653$	0.381381	0.0149246	0.00746229	−	0.999972i	$-0.497625\pi$
$653$	0.381381	0.0149246	0.00746229	+	0.999972i	$0.497625\pi$
$654$	0	0
$655$	0	0
$656$	4.91638	0.191952
$657$	0	0
$658$	−1.07714	−0.0419914
$659$	−13.4600	−0.524326	−0.262163	−	0.965024i	$-0.584436\pi$
$659$	−13.4600	−0.524326	−0.262163	+	0.965024i	$0.584436\pi$
$660$	0	0
$661$	30.7738	1.19696	0.598482	−	0.801136i	$-0.295771\pi$
$661$	30.7738	1.19696	0.598482	+	0.801136i	$0.295771\pi$
$662$	−10.0383	−0.390150
$663$	0	0
$664$	−9.90225	−0.384282
$665$	0	0
$666$	0	0
$667$	3.98441	0.154277
$668$	12.4111	0.480200
$669$	0	0
$670$	0	0
$671$	−1.55918	−0.0601915
$672$	0	0
$673$	−1.48110	−0.0570923	−0.0285461	−	0.999592i	$-0.509088\pi$
$673$	−1.48110	−0.0570923	−0.0285461	+	0.999592i	$0.509088\pi$
$674$	−6.72999	−0.259229
$675$	0	0
$676$	16.8086	0.646484
$677$	45.3749	1.74390	0.871950	−	0.489596i	$-0.162856\pi$
$677$	45.3749	1.74390	0.871950	+	0.489596i	$0.162856\pi$
$678$	0	0
$679$	6.89169	0.264479
$680$	0	0
$681$	0	0
$682$	−13.2288	−0.506557
$683$	−0.0594386	−0.00227436	−0.00113718	−	0.999999i	$-0.500362\pi$
$683$	−0.0594386	−0.00227436	−0.00113718	+	0.999999i	$0.500362\pi$
$684$	0	0
$685$	0	0
$686$	−34.9200	−1.33325
$687$	0	0
$688$	44.6066	1.70061
$689$	−70.5855	−2.68909
$690$	0	0
$691$	−18.9355	−0.720342	−0.360171	−	0.932886i	$-0.617282\pi$
$691$	−18.9355	−0.720342	−0.360171	+	0.932886i	$0.617282\pi$
$692$	15.4005	0.585441
$693$	0	0
$694$	−5.58336	−0.211941
$695$	0	0
$696$	0	0
$697$	−3.39194	−0.128479
$698$	−26.8378	−1.01583
$699$	0	0
$700$	0	0
$701$	0.540024	0.0203964	0.0101982	−	0.999948i	$-0.496754\pi$
$701$	0.540024	0.0203964	0.0101982	+	0.999948i	$0.496754\pi$
$702$	0	0
$703$	−6.46500	−0.243832
$704$	1.35218	0.0509621
$705$	0	0
$706$	−48.6449	−1.83078
$707$	−29.9461	−1.12624
$708$	0	0
$709$	28.0867	1.05482	0.527409	−	0.849612i	$-0.323164\pi$
$709$	28.0867	1.05482	0.527409	+	0.849612i	$0.323164\pi$
$710$	0	0
$711$	0	0
$712$	−17.0872	−0.640369
$713$	8.04385	0.301245
$714$	0	0
$715$	0	0
$716$	2.75562	0.102982
$717$	0	0
$718$	4.27504	0.159543
$719$	−9.86248	−0.367809	−0.183904	−	0.982944i	$-0.558874\pi$
$719$	−9.86248	−0.367809	−0.183904	+	0.982944i	$0.558874\pi$
$720$	0	0
$721$	−1.24889	−0.0465110
$722$	16.9985	0.632620
$723$	0	0
$724$	−2.36274	−0.0878106
$725$	0	0
$726$	0	0
$727$	30.4650	1.12988	0.564942	−	0.825131i	$-0.308898\pi$
$727$	30.4650	1.12988	0.564942	+	0.825131i	$0.308898\pi$
$728$	16.6066	0.615482
$729$	0	0
$730$	0	0
$731$	−30.7753	−1.13827
$732$	0	0
$733$	−27.5436	−1.01735	−0.508673	−	0.860960i	$-0.669864\pi$
$733$	−27.5436	−1.01735	−0.508673	+	0.860960i	$0.669864\pi$
$734$	42.0227	1.55109
$735$	0	0
$736$	−5.68665	−0.209613
$737$	−8.16724	−0.300844
$738$	0	0
$739$	13.1013	0.481940	0.240970	−	0.970533i	$-0.422534\pi$
$739$	13.1013	0.481940	0.240970	+	0.970533i	$0.422534\pi$
$740$	0	0
$741$	0	0
$742$	63.3311	2.32496
$743$	34.7527	1.27495	0.637477	−	0.770470i	$-0.279978\pi$
$743$	34.7527	1.27495	0.637477	+	0.770470i	$0.279978\pi$
$744$	0	0
$745$	0	0
$746$	63.8555	2.33792
$747$	0	0
$748$	−3.55773	−0.130083
$749$	38.0867	1.39166
$750$	0	0
$751$	32.7738	1.19593	0.597967	−	0.801521i	$-0.295975\pi$
$751$	32.7738	1.19593	0.597967	+	0.801521i	$0.295975\pi$
$752$	1.15667	0.0421796
$753$	0	0
$754$	−41.0972	−1.49667
$755$	0	0
$756$	0	0
$757$	32.2978	1.17388	0.586941	−	0.809630i	$-0.300332\pi$
$757$	32.2978	1.17388	0.586941	+	0.809630i	$0.300332\pi$
$758$	47.2233	1.71523
$759$	0	0
$760$	0	0
$761$	−44.4494	−1.61129	−0.805645	−	0.592399i	$-0.798181\pi$
$761$	−44.4494	−1.61129	−0.805645	+	0.592399i	$0.798181\pi$
$762$	0	0
$763$	−12.8816	−0.466347
$764$	24.0000	0.868290
$765$	0	0
$766$	−27.8967	−1.00795
$767$	5.35218	0.193256
$768$	0	0
$769$	−19.7108	−0.710791	−0.355395	−	0.934716i	$-0.615654\pi$
$769$	−19.7108	−0.710791	−0.355395	+	0.934716i	$0.615654\pi$
$770$	0	0
$771$	0	0
$772$	−20.1260	−0.724351
$773$	43.2530	1.55570	0.777851	−	0.628449i	$-0.216310\pi$
$773$	43.2530	1.55570	0.777851	+	0.628449i	$0.216310\pi$
$774$	0	0
$775$	0	0
$776$	−3.51941	−0.126340
$777$	0	0
$778$	8.09775	0.290318
$779$	−3.10278	−0.111168
$780$	0	0
$781$	−10.4252	−0.373044
$782$	5.51941	0.197374
$783$	0	0
$784$	3.08362	0.110129
$785$	0	0
$786$	0	0
$787$	21.4358	0.764104	0.382052	−	0.924141i	$-0.375218\pi$
$787$	21.4358	0.764104	0.382052	+	0.924141i	$0.375218\pi$
$788$	−6.71731	−0.239294
$789$	0	0
$790$	0	0
$791$	44.2127	1.57202
$792$	0	0
$793$	−9.77886	−0.347258
$794$	−19.1567	−0.679845
$795$	0	0
$796$	2.64782	0.0938496
$797$	−4.40105	−0.155893	−0.0779467	−	0.996958i	$-0.524836\pi$
$797$	−4.40105	−0.155893	−0.0779467	+	0.996958i	$0.524836\pi$
$798$	0	0
$799$	−0.798021	−0.0282319
$800$	0	0
$801$	0	0
$802$	23.7350	0.838112
$803$	3.87253	0.136659
$804$	0	0
$805$	0	0
$806$	−82.9683	−2.92243
$807$	0	0
$808$	15.2927	0.537997
$809$	26.3799	0.927469	0.463734	−	0.885974i	$-0.346509\pi$
$809$	26.3799	0.927469	0.463734	+	0.885974i	$0.346509\pi$
$810$	0	0
$811$	4.96526	0.174354	0.0871769	−	0.996193i	$-0.472215\pi$
$811$	4.96526	0.174354	0.0871769	+	0.996193i	$0.472215\pi$
$812$	14.4523	0.507177
$813$	0	0
$814$	−3.07451	−0.107761
$815$	0	0
$816$	0	0
$817$	−28.1517	−0.984902
$818$	33.6499	1.17654
$819$	0	0
$820$	0	0
$821$	15.7789	0.550686	0.275343	−	0.961346i	$-0.411209\pi$
$821$	15.7789	0.550686	0.275343	+	0.961346i	$0.411209\pi$
$822$	0	0
$823$	8.35166	0.291121	0.145560	−	0.989349i	$-0.453502\pi$
$823$	8.35166	0.291121	0.145560	+	0.989349i	$0.453502\pi$
$824$	0.637776	0.0222180
$825$	0	0
$826$	−4.80211	−0.167087
$827$	−18.3925	−0.639568	−0.319784	−	0.947490i	$-0.603610\pi$
$827$	−18.3925	−0.639568	−0.319784	+	0.947490i	$0.603610\pi$
$828$	0	0
$829$	−29.4147	−1.02161	−0.510807	−	0.859695i	$-0.670653\pi$
$829$	−29.4147	−1.02161	−0.510807	+	0.859695i	$0.670653\pi$
$830$	0	0
$831$	0	0
$832$	8.48059	0.294011
$833$	−2.12747	−0.0737125
$834$	0	0
$835$	0	0
$836$	−3.25443	−0.112557
$837$	0	0
$838$	−46.4494	−1.60457
$839$	33.4600	1.15517	0.577583	−	0.816332i	$-0.303996\pi$
$839$	33.4600	1.15517	0.577583	+	0.816332i	$0.303996\pi$
$840$	0	0
$841$	−9.27912	−0.319970
$842$	15.7350	0.542264
$843$	0	0
$844$	−11.6655	−0.401544
$845$	0	0
$846$	0	0
$847$	26.0978	0.896729
$848$	−68.0071	−2.33537
$849$	0	0
$850$	0	0
$851$	1.86947	0.0640848
$852$	0	0
$853$	−40.4494	−1.38496	−0.692481	−	0.721436i	$-0.743482\pi$
$853$	−40.4494	−1.38496	−0.692481	+	0.721436i	$0.743482\pi$
$854$	8.77384	0.300235
$855$	0	0
$856$	−19.4499	−0.664785
$857$	12.4494	0.425264	0.212632	−	0.977132i	$-0.431796\pi$
$857$	12.4494	0.425264	0.212632	+	0.977132i	$0.431796\pi$
$858$	0	0
$859$	−21.6514	−0.738736	−0.369368	−	0.929283i	$-0.620426\pi$
$859$	−21.6514	−0.738736	−0.369368	+	0.929283i	$0.620426\pi$
$860$	0	0
$861$	0	0
$862$	12.8816	0.438750
$863$	19.3778	0.659628	0.329814	−	0.944046i	$-0.393014\pi$
$863$	19.3778	0.659628	0.329814	+	0.944046i	$0.393014\pi$
$864$	0	0
$865$	0	0
$866$	−19.9945	−0.679439
$867$	0	0
$868$	29.1768	0.990324
$869$	12.8816	0.436980
$870$	0	0
$871$	−51.2233	−1.73563
$872$	6.57834	0.222771
$873$	0	0
$874$	5.04888	0.170781
$875$	0	0
$876$	0	0
$877$	3.08413	0.104144	0.0520719	−	0.998643i	$-0.483417\pi$
$877$	3.08413	0.104144	0.0520719	+	0.998643i	$0.483417\pi$
$878$	2.40105	0.0810316
$879$	0	0
$880$	0	0
$881$	34.5783	1.16497	0.582487	−	0.812840i	$-0.302080\pi$
$881$	34.5783	1.16497	0.582487	+	0.812840i	$0.302080\pi$
$882$	0	0
$883$	9.64280	0.324506	0.162253	−	0.986749i	$-0.448124\pi$
$883$	9.64280	0.324506	0.162253	+	0.986749i	$0.448124\pi$
$884$	−22.3133	−0.750479
$885$	0	0
$886$	−28.2721	−0.949821
$887$	32.2041	1.08131	0.540654	−	0.841245i	$-0.318177\pi$
$887$	32.2041	1.08131	0.540654	+	0.841245i	$0.318177\pi$
$888$	0	0
$889$	32.5189	1.09065
$890$	0	0
$891$	0	0
$892$	−32.0000	−1.07144
$893$	−0.729988	−0.0244281
$894$	0	0
$895$	0	0
$896$	24.3910	0.814846
$897$	0	0
$898$	−59.4571	−1.98411
$899$	39.8131	1.32784
$900$	0	0
$901$	46.9200	1.56313
$902$	−1.47556	−0.0491308
$903$	0	0
$904$	−22.5783	−0.750944
$905$	0	0
$906$	0	0
$907$	17.8227	0.591794	0.295897	−	0.955220i	$-0.404382\pi$
$907$	17.8227	0.591794	0.295897	+	0.955220i	$0.404382\pi$
$908$	21.4600	0.712174
$909$	0	0
$910$	0	0
$911$	20.7894	0.688784	0.344392	−	0.938826i	$-0.388085\pi$
$911$	20.7894	0.688784	0.344392	+	0.938826i	$0.388085\pi$
$912$	0	0
$913$	6.24940	0.206825
$914$	−25.6938	−0.849875
$915$	0	0
$916$	−20.3416	−0.672106
$917$	−16.0484	−0.529964
$918$	0	0
$919$	33.8555	1.11679	0.558395	−	0.829575i	$-0.311417\pi$
$919$	33.8555	1.11679	0.558395	+	0.829575i	$0.311417\pi$
$920$	0	0
$921$	0	0
$922$	−54.5472	−1.79642
$923$	−65.3850	−2.15217
$924$	0	0
$925$	0	0
$926$	−43.3028	−1.42302
$927$	0	0
$928$	−28.1461	−0.923941
$929$	4.10635	0.134725	0.0673624	−	0.997729i	$-0.478542\pi$
$929$	4.10635	0.134725	0.0673624	+	0.997729i	$0.478542\pi$
$930$	0	0
$931$	−1.94610	−0.0637809
$932$	−32.1260	−1.05232
$933$	0	0
$934$	17.6555	0.577705
$935$	0	0
$936$	0	0
$937$	−13.3028	−0.434583	−0.217292	−	0.976107i	$-0.569722\pi$
$937$	−13.3028	−0.434583	−0.217292	+	0.976107i	$0.569722\pi$
$938$	45.9588	1.50061
$939$	0	0
$940$	0	0
$941$	11.7633	0.383472	0.191736	−	0.981447i	$-0.438588\pi$
$941$	11.7633	0.383472	0.191736	+	0.981447i	$0.438588\pi$
$942$	0	0
$943$	0.897225	0.0292177
$944$	5.15667	0.167835
$945$	0	0
$946$	−13.3879	−0.435277
$947$	1.40054	0.0455114	0.0227557	−	0.999741i	$-0.492756\pi$
$947$	1.40054	0.0455114	0.0227557	+	0.999741i	$0.492756\pi$
$948$	0	0
$949$	24.2877	0.788413
$950$	0	0
$951$	0	0
$952$	−11.0388	−0.357771
$953$	33.7577	1.09352	0.546760	−	0.837289i	$-0.315861\pi$
$953$	33.7577	1.09352	0.546760	+	0.837289i	$0.315861\pi$
$954$	0	0
$955$	0	0
$956$	3.80450	0.123046
$957$	0	0
$958$	−2.13343	−0.0689280
$959$	44.1517	1.42573
$960$	0	0
$961$	49.3758	1.59277
$962$	−19.2827	−0.621699
$963$	0	0
$964$	−21.3905	−0.688941
$965$	0	0
$966$	0	0
$967$	10.4806	0.337033	0.168516	−	0.985699i	$-0.446102\pi$
$967$	10.4806	0.337033	0.168516	+	0.985699i	$0.446102\pi$
$968$	−13.3275	−0.428361
$969$	0	0
$970$	0	0
$971$	−57.6358	−1.84962	−0.924811	−	0.380428i	$-0.875777\pi$
$971$	−57.6358	−1.84962	−0.924811	+	0.380428i	$0.875777\pi$
$972$	0	0
$973$	−39.6967	−1.27262
$974$	37.3649	1.19725
$975$	0	0
$976$	−9.42166	−0.301580
$977$	41.9008	1.34053	0.670263	−	0.742124i	$-0.266181\pi$
$977$	41.9008	1.34053	0.670263	+	0.742124i	$0.266181\pi$
$978$	0	0
$979$	10.7839	0.344655
$980$	0	0
$981$	0	0
$982$	25.7350	0.821237
$983$	−18.4056	−0.587046	−0.293523	−	0.955952i	$-0.594828\pi$
$983$	−18.4056	−0.587046	−0.293523	+	0.955952i	$0.594828\pi$
$984$	0	0
$985$	0	0
$986$	27.3184	0.869994
$987$	0	0
$988$	−20.4111	−0.649364
$989$	8.14057	0.258855
$990$	0	0
$991$	26.0666	0.828032	0.414016	−	0.910270i	$-0.364126\pi$
$991$	26.0666	0.828032	0.414016	+	0.910270i	$0.364126\pi$
$992$	−56.8222	−1.80411
$993$	0	0
$994$	58.6650	1.86074
$995$	0	0
$996$	0	0
$997$	29.8483	0.945307	0.472653	−	0.881248i	$-0.343296\pi$
$997$	29.8483	0.945307	0.472653	+	0.881248i	$0.343296\pi$
$998$	38.9200	1.23199
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9225.2.a.bx.1.1		3
3.2	odd	2		3075.2.a.t.1.3		3
5.4	even	2		369.2.a.e.1.3		3
15.14	odd	2		123.2.a.d.1.1	✓	3
20.19	odd	2		5904.2.a.bd.1.3		3
60.59	even	2		1968.2.a.w.1.1		3
105.104	even	2		6027.2.a.s.1.1		3
120.29	odd	2		7872.2.a.bx.1.3		3
120.59	even	2		7872.2.a.bs.1.3		3
615.614	odd	2		5043.2.a.n.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.d.1.1	✓	3	15.14	odd	2
369.2.a.e.1.3		3	5.4	even	2
1968.2.a.w.1.1		3	60.59	even	2
3075.2.a.t.1.3		3	3.2	odd	2
5043.2.a.n.1.1		3	615.614	odd	2
5904.2.a.bd.1.3		3	20.19	odd	2
6027.2.a.s.1.1		3	105.104	even	2
7872.2.a.bs.1.3		3	120.59	even	2
7872.2.a.bx.1.3		3	120.29	odd	2
9225.2.a.bx.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9225 = 3^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.81361 q^{2} +1.28917 q^{4} -2.52444 q^{7} +1.28917 q^{8} +O(q^{10})\)	\(q-1.81361 q^{2} +1.28917 q^{4} -2.52444 q^{7} +1.28917 q^{8} -0.813607 q^{11} -5.10278 q^{13} +4.57834 q^{14} -4.91638 q^{16} +3.39194 q^{17} +3.10278 q^{19} +1.47556 q^{22} -0.897225 q^{23} +9.25443 q^{26} -3.25443 q^{28} -4.44082 q^{29} -8.96526 q^{31} +6.33804 q^{32} -6.15165 q^{34} -2.08362 q^{37} -5.62721 q^{38} -1.00000 q^{41} -9.07306 q^{43} -1.04888 q^{44} +1.62721 q^{46} -0.235269 q^{47} -0.627213 q^{49} -6.57834 q^{52} +13.8328 q^{53} -3.25443 q^{56} +8.05390 q^{58} -1.04888 q^{59} +1.91638 q^{61} +16.2594 q^{62} -1.66196 q^{64} +10.0383 q^{67} +4.37279 q^{68} +12.8136 q^{71} -4.75971 q^{73} +3.77886 q^{74} +4.00000 q^{76} +2.05390 q^{77} -15.8328 q^{79} +1.81361 q^{82} -7.68111 q^{83} +16.4550 q^{86} -1.04888 q^{88} -13.2544 q^{89} +12.8816 q^{91} -1.15667 q^{92} +0.426686 q^{94} -2.72999 q^{97} +1.13752 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + 4 q^{11} - 8 q^{13} + 12 q^{14} - q^{16} + 2 q^{17} + 2 q^{19} + 10 q^{22} - 10 q^{23} + 2 q^{26} + 16 q^{28} + 6 q^{29} - 2 q^{31} + 7 q^{32} - 20 q^{37} - 4 q^{38} - 3 q^{41} - 10 q^{43} + 8 q^{44} - 8 q^{46} + 4 q^{47} + 11 q^{49} - 18 q^{52} + 14 q^{53} + 16 q^{56} + 28 q^{58} + 8 q^{59} - 8 q^{61} + 38 q^{62} - 17 q^{64} - 12 q^{67} + 26 q^{68} + 32 q^{71} - 4 q^{73} - 20 q^{74} + 12 q^{76} + 10 q^{77} - 20 q^{79} - q^{82} - 14 q^{83} - 6 q^{86} + 8 q^{88} - 14 q^{89} + 18 q^{94} + 12 q^{97} + 21 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 + 4 * q^11 - 8 * q^13 + 12 * q^14 - q^16 + 2 * q^17 + 2 * q^19 + 10 * q^22 - 10 * q^23 + 2 * q^26 + 16 * q^28 + 6 * q^29 - 2 * q^31 + 7 * q^32 - 20 * q^37 - 4 * q^38 - 3 * q^41 - 10 * q^43 + 8 * q^44 - 8 * q^46 + 4 * q^47 + 11 * q^49 - 18 * q^52 + 14 * q^53 + 16 * q^56 + 28 * q^58 + 8 * q^59 - 8 * q^61 + 38 * q^62 - 17 * q^64 - 12 * q^67 + 26 * q^68 + 32 * q^71 - 4 * q^73 - 20 * q^74 + 12 * q^76 + 10 * q^77 - 20 * q^79 - q^82 - 14 * q^83 - 6 * q^86 + 8 * q^88 - 14 * q^89 + 18 * q^94 + 12 * q^97 + 21 * q^98

Introduction

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