LMFDB - Embedded newform 9248.2.a.bx.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9248,2,Mod(1,9248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9248, 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("9248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.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:	$73.8456517893$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 28x^{14} + 290x^{12} - 1436x^{10} + 3633x^{8} - 4632x^{6} + 2744x^{4} - 704x^{2} + 64 $ x^16 - 28x^14 + 290x^12 - 1436x^10 + 3633x^8 - 4632x^6 + 2744x^4 - 704*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.10698$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13196 q^{3} +2.34695 q^{5} +3.47109 q^{7} -1.71866 q^{9} +5.37774 q^{11} -0.828427 q^{13} -2.65666 q^{15} -1.43143 q^{19} -3.92915 q^{21} -0.826153 q^{23} +0.508169 q^{25} +5.34135 q^{27} +3.75887 q^{29} +8.37996 q^{31} -6.08741 q^{33} +8.14648 q^{35} -3.60295 q^{37} +0.937749 q^{39} +10.9431 q^{41} -12.8947 q^{43} -4.03361 q^{45} -3.75708 q^{47} +5.04849 q^{49} +7.32026 q^{53} +12.6213 q^{55} +1.62033 q^{57} -8.29914 q^{59} +2.46576 q^{61} -5.96563 q^{63} -1.94428 q^{65} +8.93145 q^{67} +0.935174 q^{69} -13.8456 q^{71} -2.02137 q^{73} -0.575228 q^{75} +18.6666 q^{77} +9.31771 q^{79} -0.890232 q^{81} +4.52835 q^{83} -4.25490 q^{87} -0.313491 q^{89} -2.87555 q^{91} -9.48581 q^{93} -3.35950 q^{95} +12.5458 q^{97} -9.24251 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{9} + 32 q^{13} + 48 q^{21} + 32 q^{33} + 48 q^{49} + 80 q^{53} + 64 q^{69} + 48 q^{77} - 48 q^{89}+O(q^{100}) $ 16 * q + 32 * q^9 + 32 * q^13 + 48 * q^21 + 32 * q^33 + 48 * q^49 + 80 * q^53 + 64 * q^69 + 48 * q^77 - 48 * q^89

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.13196	−0.653539	−0.326770	−	0.945104i	$-0.605960\pi$
$3$	−1.13196	−0.653539	−0.326770	+	0.945104i	$0.605960\pi$
$4$	0	0
$5$	2.34695	1.04959	0.524794	−	0.851229i	$-0.324142\pi$
$5$	2.34695	1.04959	0.524794	+	0.851229i	$0.324142\pi$
$6$	0	0
$7$	3.47109	1.31195	0.655975	−	0.754783i	$-0.272258\pi$
$7$	3.47109	1.31195	0.655975	+	0.754783i	$0.272258\pi$
$8$	0	0
$9$	−1.71866	−0.572886
$10$	0	0
$11$	5.37774	1.62145	0.810725	−	0.585427i	$-0.199073\pi$
$11$	5.37774	1.62145	0.810725	+	0.585427i	$0.199073\pi$
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	−2.65666	−0.685947
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.43143	−0.328393	−0.164196	−	0.986428i	$-0.552503\pi$
$19$	−1.43143	−0.328393	−0.164196	+	0.986428i	$0.552503\pi$
$20$	0	0
$21$	−3.92915	−0.857411
$22$	0	0
$23$	−0.826153	−0.172265	−0.0861324	−	0.996284i	$-0.527451\pi$
$23$	−0.826153	−0.172265	−0.0861324	+	0.996284i	$0.527451\pi$
$24$	0	0
$25$	0.508169	0.101634
$26$	0	0
$27$	5.34135	1.02794
$28$	0	0
$29$	3.75887	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$29$	3.75887	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$30$	0	0
$31$	8.37996	1.50509	0.752543	−	0.658544i	$-0.228827\pi$
$31$	8.37996	1.50509	0.752543	+	0.658544i	$0.228827\pi$
$32$	0	0
$33$	−6.08741	−1.05968
$34$	0	0
$35$	8.14648	1.37701
$36$	0	0
$37$	−3.60295	−0.592321	−0.296160	−	0.955138i	$-0.595706\pi$
$37$	−3.60295	−0.592321	−0.296160	+	0.955138i	$0.595706\pi$
$38$	0	0
$39$	0.937749	0.150160
$40$	0	0
$41$	10.9431	1.70903	0.854515	−	0.519426i	$-0.173854\pi$
$41$	10.9431	1.70903	0.854515	+	0.519426i	$0.173854\pi$
$42$	0	0
$43$	−12.8947	−1.96643	−0.983214	−	0.182455i	$-0.941596\pi$
$43$	−12.8947	−1.96643	−0.983214	+	0.182455i	$0.941596\pi$
$44$	0	0
$45$	−4.03361	−0.601294
$46$	0	0
$47$	−3.75708	−0.548027	−0.274014	−	0.961726i	$-0.588351\pi$
$47$	−3.75708	−0.548027	−0.274014	+	0.961726i	$0.588351\pi$
$48$	0	0
$49$	5.04849	0.721213
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.32026	1.00551	0.502757	−	0.864428i	$-0.332319\pi$
$53$	7.32026	1.00551	0.502757	+	0.864428i	$0.332319\pi$
$54$	0	0
$55$	12.6213	1.70185
$56$	0	0
$57$	1.62033	0.214618
$58$	0	0
$59$	−8.29914	−1.08046	−0.540228	−	0.841519i	$-0.681662\pi$
$59$	−8.29914	−1.08046	−0.540228	+	0.841519i	$0.681662\pi$
$60$	0	0
$61$	2.46576	0.315708	0.157854	−	0.987462i	$-0.449542\pi$
$61$	2.46576	0.315708	0.157854	+	0.987462i	$0.449542\pi$
$62$	0	0
$63$	−5.96563	−0.751598
$64$	0	0
$65$	−1.94428	−0.241158
$66$	0	0
$67$	8.93145	1.09115	0.545575	−	0.838062i	$-0.316311\pi$
$67$	8.93145	1.09115	0.545575	+	0.838062i	$0.316311\pi$
$68$	0	0
$69$	0.935174	0.112582
$70$	0	0
$71$	−13.8456	−1.64317	−0.821583	−	0.570089i	$-0.806909\pi$
$71$	−13.8456	−1.64317	−0.821583	+	0.570089i	$0.806909\pi$
$72$	0	0
$73$	−2.02137	−0.236583	−0.118291	−	0.992979i	$-0.537742\pi$
$73$	−2.02137	−0.236583	−0.118291	+	0.992979i	$0.537742\pi$
$74$	0	0
$75$	−0.575228	−0.0664217
$76$	0	0
$77$	18.6666	2.12726
$78$	0	0
$79$	9.31771	1.04832	0.524162	−	0.851619i	$-0.324379\pi$
$79$	9.31771	1.04832	0.524162	+	0.851619i	$0.324379\pi$
$80$	0	0
$81$	−0.890232	−0.0989147
$82$	0	0
$83$	4.52835	0.497051	0.248526	−	0.968625i	$-0.420054\pi$
$83$	4.52835	0.497051	0.248526	+	0.968625i	$0.420054\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.25490	−0.456174
$88$	0	0
$89$	−0.313491	−0.0332300	−0.0166150	−	0.999862i	$-0.505289\pi$
$89$	−0.313491	−0.0332300	−0.0166150	+	0.999862i	$0.505289\pi$
$90$	0	0
$91$	−2.87555	−0.301439
$92$	0	0
$93$	−9.48581	−0.983632
$94$	0	0
$95$	−3.35950	−0.344677
$96$	0	0
$97$	12.5458	1.27383	0.636915	−	0.770934i	$-0.280210\pi$
$97$	12.5458	1.27383	0.636915	+	0.770934i	$0.280210\pi$
$98$	0	0
$99$	−9.24251	−0.928907
$100$	0	0
$101$	11.1492	1.10939	0.554694	−	0.832054i	$-0.312835\pi$
$101$	11.1492	1.10939	0.554694	+	0.832054i	$0.312835\pi$
$102$	0	0
$103$	−4.18319	−0.412182	−0.206091	−	0.978533i	$-0.566074\pi$
$103$	−4.18319	−0.412182	−0.206091	+	0.978533i	$0.566074\pi$
$104$	0	0
$105$	−9.22151	−0.899928
$106$	0	0
$107$	12.0453	1.16446	0.582230	−	0.813024i	$-0.302180\pi$
$107$	12.0453	1.16446	0.582230	+	0.813024i	$0.302180\pi$
$108$	0	0
$109$	2.34695	0.224797	0.112398	−	0.993663i	$-0.464147\pi$
$109$	2.34695	0.224797	0.112398	+	0.993663i	$0.464147\pi$
$110$	0	0
$111$	4.07840	0.387105
$112$	0	0
$113$	20.6605	1.94357	0.971786	−	0.235863i	$-0.0757916\pi$
$113$	20.6605	1.94357	0.971786	+	0.235863i	$0.0757916\pi$
$114$	0	0
$115$	−1.93894	−0.180807
$116$	0	0
$117$	1.42378	0.131629
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.9201	1.62910
$122$	0	0
$123$	−12.3872	−1.11692
$124$	0	0
$125$	−10.5421	−0.942914
$126$	0	0
$127$	7.54387	0.669410	0.334705	−	0.942323i	$-0.391363\pi$
$127$	7.54387	0.669410	0.334705	+	0.942323i	$0.391363\pi$
$128$	0	0
$129$	14.5964	1.28514
$130$	0	0
$131$	4.20939	0.367776	0.183888	−	0.982947i	$-0.441132\pi$
$131$	4.20939	0.367776	0.183888	+	0.982947i	$0.441132\pi$
$132$	0	0
$133$	−4.96863	−0.430835
$134$	0	0
$135$	12.5359	1.07892
$136$	0	0
$137$	6.60889	0.564636	0.282318	−	0.959321i	$-0.408897\pi$
$137$	6.60889	0.564636	0.282318	+	0.959321i	$0.408897\pi$
$138$	0	0
$139$	3.93473	0.333739	0.166870	−	0.985979i	$-0.446634\pi$
$139$	3.93473	0.333739	0.166870	+	0.985979i	$0.446634\pi$
$140$	0	0
$141$	4.25288	0.358157
$142$	0	0
$143$	−4.45507	−0.372551
$144$	0	0
$145$	8.82188	0.732617
$146$	0	0
$147$	−5.71470	−0.471341
$148$	0	0
$149$	−7.14921	−0.585686	−0.292843	−	0.956161i	$-0.594601\pi$
$149$	−7.14921	−0.585686	−0.292843	+	0.956161i	$0.594601\pi$
$150$	0	0
$151$	−15.7758	−1.28381	−0.641907	−	0.766783i	$-0.721856\pi$
$151$	−15.7758	−1.28381	−0.641907	+	0.766783i	$0.721856\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	19.6673	1.57972
$156$	0	0
$157$	−5.78973	−0.462071	−0.231035	−	0.972945i	$-0.574211\pi$
$157$	−5.78973	−0.462071	−0.231035	+	0.972945i	$0.574211\pi$
$158$	0	0
$159$	−8.28626	−0.657143
$160$	0	0
$161$	−2.86765	−0.226003
$162$	0	0
$163$	−9.24251	−0.723929	−0.361964	−	0.932192i	$-0.617894\pi$
$163$	−9.24251	−0.723929	−0.361964	+	0.932192i	$0.617894\pi$
$164$	0	0
$165$	−14.2868	−1.11223
$166$	0	0
$167$	−8.37996	−0.648461	−0.324230	−	0.945978i	$-0.605105\pi$
$167$	−8.37996	−0.648461	−0.324230	+	0.945978i	$0.605105\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	2.46014	0.188132
$172$	0	0
$173$	21.4544	1.63115	0.815573	−	0.578654i	$-0.196422\pi$
$173$	21.4544	1.63115	0.815573	+	0.578654i	$0.196422\pi$
$174$	0	0
$175$	1.76390	0.133338
$176$	0	0
$177$	9.39432	0.706120
$178$	0	0
$179$	6.74475	0.504126	0.252063	−	0.967711i	$-0.418891\pi$
$179$	6.74475	0.504126	0.252063	+	0.967711i	$0.418891\pi$
$180$	0	0
$181$	−14.7670	−1.09762	−0.548811	−	0.835946i	$-0.684919\pi$
$181$	−14.7670	−1.09762	−0.548811	+	0.835946i	$0.684919\pi$
$182$	0	0
$183$	−2.79115	−0.206328
$184$	0	0
$185$	−8.45593	−0.621693
$186$	0	0
$187$	0	0
$188$	0	0
$189$	18.5403	1.34861
$190$	0	0
$191$	−3.62259	−0.262122	−0.131061	−	0.991374i	$-0.541838\pi$
$191$	−3.62259	−0.262122	−0.131061	+	0.991374i	$0.541838\pi$
$192$	0	0
$193$	−18.0297	−1.29780	−0.648901	−	0.760873i	$-0.724771\pi$
$193$	−18.0297	−1.29780	−0.648901	+	0.760873i	$0.724771\pi$
$194$	0	0
$195$	2.20085	0.157606
$196$	0	0
$197$	14.6945	1.04694	0.523470	−	0.852044i	$-0.324637\pi$
$197$	14.6945	1.04694	0.523470	+	0.852044i	$0.324637\pi$
$198$	0	0
$199$	−20.4067	−1.44660	−0.723298	−	0.690537i	$-0.757374\pi$
$199$	−20.4067	−1.44660	−0.723298	+	0.690537i	$0.757374\pi$
$200$	0	0
$201$	−10.1101	−0.713109
$202$	0	0
$203$	13.0474	0.915747
$204$	0	0
$205$	25.6830	1.79378
$206$	0	0
$207$	1.41987	0.0986881
$208$	0	0
$209$	−7.69787	−0.532473
$210$	0	0
$211$	−1.08050	−0.0743844	−0.0371922	−	0.999308i	$-0.511841\pi$
$211$	−1.08050	−0.0743844	−0.0371922	+	0.999308i	$0.511841\pi$
$212$	0	0
$213$	15.6727	1.07387
$214$	0	0
$215$	−30.2633	−2.06394
$216$	0	0
$217$	29.0876	1.97460
$218$	0	0
$219$	2.28811	0.154616
$220$	0	0
$221$	0	0
$222$	0	0
$223$	9.87921	0.661561	0.330780	−	0.943708i	$-0.392688\pi$
$223$	9.87921	0.661561	0.330780	+	0.943708i	$0.392688\pi$
$224$	0	0
$225$	−0.873369	−0.0582246
$226$	0	0
$227$	15.0891	1.00150	0.500750	−	0.865592i	$-0.333058\pi$
$227$	15.0891	1.00150	0.500750	+	0.865592i	$0.333058\pi$
$228$	0	0
$229$	22.0947	1.46006	0.730029	−	0.683416i	$-0.239506\pi$
$229$	22.0947	1.46006	0.730029	+	0.683416i	$0.239506\pi$
$230$	0	0
$231$	−21.1300	−1.39025
$232$	0	0
$233$	8.24042	0.539848	0.269924	−	0.962882i	$-0.413001\pi$
$233$	8.24042	0.539848	0.269924	+	0.962882i	$0.413001\pi$
$234$	0	0
$235$	−8.81768	−0.575202
$236$	0	0
$237$	−10.5473	−0.685121
$238$	0	0
$239$	−10.5115	−0.679934	−0.339967	−	0.940437i	$-0.610416\pi$
$239$	−10.5115	−0.679934	−0.339967	+	0.940437i	$0.610416\pi$
$240$	0	0
$241$	−18.1429	−1.16869	−0.584343	−	0.811507i	$-0.698648\pi$
$241$	−18.1429	−1.16869	−0.584343	+	0.811507i	$0.698648\pi$
$242$	0	0
$243$	−15.0163	−0.963298
$244$	0	0
$245$	11.8485	0.756976
$246$	0	0
$247$	1.18584	0.0754530
$248$	0	0
$249$	−5.12593	−0.324842
$250$	0	0
$251$	−1.87169	−0.118140	−0.0590701	−	0.998254i	$-0.518814\pi$
$251$	−1.87169	−0.118140	−0.0590701	+	0.998254i	$0.518814\pi$
$252$	0	0
$253$	−4.44283	−0.279319
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.6544	−0.664601	−0.332300	−	0.943174i	$-0.607825\pi$
$257$	−10.6544	−0.664601	−0.332300	+	0.943174i	$0.607825\pi$
$258$	0	0
$259$	−12.5062	−0.777095
$260$	0	0
$261$	−6.46022	−0.399877
$262$	0	0
$263$	−31.6006	−1.94858	−0.974288	−	0.225305i	$-0.927662\pi$
$263$	−31.6006	−1.94858	−0.974288	+	0.225305i	$0.927662\pi$
$264$	0	0
$265$	17.1803	1.05538
$266$	0	0
$267$	0.354860	0.0217171
$268$	0	0
$269$	−24.3891	−1.48703	−0.743516	−	0.668718i	$-0.766843\pi$
$269$	−24.3891	−1.48703	−0.743516	+	0.668718i	$0.766843\pi$
$270$	0	0
$271$	−27.6373	−1.67885	−0.839424	−	0.543478i	$-0.817107\pi$
$271$	−27.6373	−1.67885	−0.839424	+	0.543478i	$0.817107\pi$
$272$	0	0
$273$	3.25501	0.197002
$274$	0	0
$275$	2.73280	0.164794
$276$	0	0
$277$	23.2348	1.39605	0.698023	−	0.716075i	$-0.254063\pi$
$277$	23.2348	1.39605	0.698023	+	0.716075i	$0.254063\pi$
$278$	0	0
$279$	−14.4023	−0.862243
$280$	0	0
$281$	4.21726	0.251581	0.125790	−	0.992057i	$-0.459853\pi$
$281$	4.21726	0.251581	0.125790	+	0.992057i	$0.459853\pi$
$282$	0	0
$283$	3.90116	0.231900	0.115950	−	0.993255i	$-0.463009\pi$
$283$	3.90116	0.231900	0.115950	+	0.993255i	$0.463009\pi$
$284$	0	0
$285$	3.80283	0.225260
$286$	0	0
$287$	37.9846	2.24216
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−14.2014	−0.832499
$292$	0	0
$293$	2.47602	0.144651	0.0723253	−	0.997381i	$-0.476958\pi$
$293$	2.47602	0.144651	0.0723253	+	0.997381i	$0.476958\pi$
$294$	0	0
$295$	−19.4776	−1.13403
$296$	0	0
$297$	28.7244	1.66676
$298$	0	0
$299$	0.684407	0.0395803
$300$	0	0
$301$	−44.7588	−2.57986
$302$	0	0
$303$	−12.6205	−0.725029
$304$	0	0
$305$	5.78701	0.331363
$306$	0	0
$307$	−22.1895	−1.26642	−0.633211	−	0.773979i	$-0.718263\pi$
$307$	−22.1895	−1.26642	−0.633211	+	0.773979i	$0.718263\pi$
$308$	0	0
$309$	4.73522	0.269377
$310$	0	0
$311$	−25.2977	−1.43450	−0.717251	−	0.696815i	$-0.754600\pi$
$311$	−25.2977	−1.43450	−0.717251	+	0.696815i	$0.754600\pi$
$312$	0	0
$313$	12.0626	0.681821	0.340910	−	0.940096i	$-0.389265\pi$
$313$	12.0626	0.681821	0.340910	+	0.940096i	$0.389265\pi$
$314$	0	0
$315$	−14.0010	−0.788868
$316$	0	0
$317$	−2.74050	−0.153922	−0.0769608	−	0.997034i	$-0.524522\pi$
$317$	−2.74050	−0.153922	−0.0769608	+	0.997034i	$0.524522\pi$
$318$	0	0
$319$	20.2142	1.13178
$320$	0	0
$321$	−13.6348	−0.761020
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.420981	−0.0233518
$326$	0	0
$327$	−2.65666	−0.146914
$328$	0	0
$329$	−13.0412	−0.718984
$330$	0	0
$331$	18.0736	0.993413	0.496706	−	0.867919i	$-0.334543\pi$
$331$	18.0736	0.993413	0.496706	+	0.867919i	$0.334543\pi$
$332$	0	0
$333$	6.19224	0.339333
$334$	0	0
$335$	20.9616	1.14526
$336$	0	0
$337$	−20.2116	−1.10099	−0.550497	−	0.834837i	$-0.685562\pi$
$337$	−20.2116	−1.10099	−0.550497	+	0.834837i	$0.685562\pi$
$338$	0	0
$339$	−23.3869	−1.27020
$340$	0	0
$341$	45.0653	2.44042
$342$	0	0
$343$	−6.77388	−0.365755
$344$	0	0
$345$	2.19481	0.118164
$346$	0	0
$347$	34.2372	1.83795	0.918975	−	0.394315i	$-0.129018\pi$
$347$	34.2372	1.83795	0.918975	+	0.394315i	$0.129018\pi$
$348$	0	0
$349$	25.7412	1.37790	0.688948	−	0.724810i	$-0.258073\pi$
$349$	25.7412	1.37790	0.688948	+	0.724810i	$0.258073\pi$
$350$	0	0
$351$	−4.42492	−0.236185
$352$	0	0
$353$	17.6506	0.939447	0.469724	−	0.882814i	$-0.344354\pi$
$353$	17.6506	0.939447	0.469724	+	0.882814i	$0.344354\pi$
$354$	0	0
$355$	−32.4948	−1.72465
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.5572	1.13775	0.568873	−	0.822426i	$-0.307380\pi$
$359$	21.5572	1.13775	0.568873	+	0.822426i	$0.307380\pi$
$360$	0	0
$361$	−16.9510	−0.892158
$362$	0	0
$363$	−20.2849	−1.06468
$364$	0	0
$365$	−4.74404	−0.248315
$366$	0	0
$367$	30.5506	1.59473	0.797364	−	0.603499i	$-0.206227\pi$
$367$	30.5506	1.59473	0.797364	+	0.603499i	$0.206227\pi$
$368$	0	0
$369$	−18.8075	−0.979081
$370$	0	0
$371$	25.4093	1.31919
$372$	0	0
$373$	6.56695	0.340024	0.170012	−	0.985442i	$-0.445619\pi$
$373$	6.56695	0.340024	0.170012	+	0.985442i	$0.445619\pi$
$374$	0	0
$375$	11.9333	0.616231
$376$	0	0
$377$	−3.11395	−0.160377
$378$	0	0
$379$	7.79949	0.400633	0.200316	−	0.979731i	$-0.435803\pi$
$379$	7.79949	0.400633	0.200316	+	0.979731i	$0.435803\pi$
$380$	0	0
$381$	−8.53939	−0.437486
$382$	0	0
$383$	−19.2412	−0.983181	−0.491591	−	0.870826i	$-0.663584\pi$
$383$	−19.2412	−0.983181	−0.491591	+	0.870826i	$0.663584\pi$
$384$	0	0
$385$	43.8097	2.23275
$386$	0	0
$387$	22.1617	1.12654
$388$	0	0
$389$	−19.8148	−1.00465	−0.502325	−	0.864679i	$-0.667522\pi$
$389$	−19.8148	−1.00465	−0.502325	+	0.864679i	$0.667522\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.76487	−0.240356
$394$	0	0
$395$	21.8682	1.10031
$396$	0	0
$397$	−32.2108	−1.61662	−0.808308	−	0.588760i	$-0.799616\pi$
$397$	−32.2108	−1.61662	−0.808308	+	0.588760i	$0.799616\pi$
$398$	0	0
$399$	5.62431	0.281568
$400$	0	0
$401$	17.0184	0.849857	0.424928	−	0.905227i	$-0.360299\pi$
$401$	17.0184	0.849857	0.424928	+	0.905227i	$0.360299\pi$
$402$	0	0
$403$	−6.94219	−0.345815
$404$	0	0
$405$	−2.08933	−0.103820
$406$	0	0
$407$	−19.3757	−0.960419
$408$	0	0
$409$	−28.6315	−1.41574	−0.707868	−	0.706345i	$-0.750343\pi$
$409$	−28.6315	−1.41574	−0.707868	+	0.706345i	$0.750343\pi$
$410$	0	0
$411$	−7.48102	−0.369012
$412$	0	0
$413$	−28.8071	−1.41750
$414$	0	0
$415$	10.6278	0.521699
$416$	0	0
$417$	−4.45396	−0.218112
$418$	0	0
$419$	4.15792	0.203128	0.101564	−	0.994829i	$-0.467615\pi$
$419$	4.15792	0.203128	0.101564	+	0.994829i	$0.467615\pi$
$420$	0	0
$421$	34.1240	1.66310	0.831552	−	0.555447i	$-0.187453\pi$
$421$	34.1240	1.66310	0.831552	+	0.555447i	$0.187453\pi$
$422$	0	0
$423$	6.45715	0.313957
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.55889	0.414193
$428$	0	0
$429$	5.04297	0.243477
$430$	0	0
$431$	18.9123	0.910971	0.455486	−	0.890243i	$-0.349466\pi$
$431$	18.9123	0.910971	0.455486	+	0.890243i	$0.349466\pi$
$432$	0	0
$433$	−0.790252	−0.0379771	−0.0189886	−	0.999820i	$-0.506045\pi$
$433$	−0.790252	−0.0379771	−0.0189886	+	0.999820i	$0.506045\pi$
$434$	0	0
$435$	−9.98604	−0.478794
$436$	0	0
$437$	1.18258	0.0565705
$438$	0	0
$439$	14.6076	0.697182	0.348591	−	0.937275i	$-0.386660\pi$
$439$	14.6076	0.697182	0.348591	+	0.937275i	$0.386660\pi$
$440$	0	0
$441$	−8.67663	−0.413173
$442$	0	0
$443$	18.7059	0.888743	0.444371	−	0.895843i	$-0.353427\pi$
$443$	18.7059	0.888743	0.444371	+	0.895843i	$0.353427\pi$
$444$	0	0
$445$	−0.735747	−0.0348778
$446$	0	0
$447$	8.09264	0.382769
$448$	0	0
$449$	20.8164	0.982386	0.491193	−	0.871051i	$-0.336561\pi$
$449$	20.8164	0.982386	0.491193	+	0.871051i	$0.336561\pi$
$450$	0	0
$451$	58.8494	2.77111
$452$	0	0
$453$	17.8576	0.839022
$454$	0	0
$455$	−6.74876	−0.316387
$456$	0	0
$457$	9.60514	0.449310	0.224655	−	0.974438i	$-0.427875\pi$
$457$	9.60514	0.449310	0.224655	+	0.974438i	$0.427875\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.8546	1.53019	0.765095	−	0.643918i	$-0.222692\pi$
$461$	32.8546	1.53019	0.765095	+	0.643918i	$0.222692\pi$
$462$	0	0
$463$	14.5739	0.677308	0.338654	−	0.940911i	$-0.390028\pi$
$463$	14.5739	0.677308	0.338654	+	0.940911i	$0.390028\pi$
$464$	0	0
$465$	−22.2627	−1.03241
$466$	0	0
$467$	−6.08274	−0.281475	−0.140738	−	0.990047i	$-0.544947\pi$
$467$	−6.08274	−0.281475	−0.140738	+	0.990047i	$0.544947\pi$
$468$	0	0
$469$	31.0019	1.43153
$470$	0	0
$471$	6.55376	0.301981
$472$	0	0
$473$	−69.3445	−3.18847
$474$	0	0
$475$	−0.727409	−0.0333758
$476$	0	0
$477$	−12.5810	−0.576046
$478$	0	0
$479$	30.0666	1.37378	0.686890	−	0.726761i	$-0.258975\pi$
$479$	30.0666	1.37378	0.686890	+	0.726761i	$0.258975\pi$
$480$	0	0
$481$	2.98478	0.136094
$482$	0	0
$483$	3.24608	0.147702
$484$	0	0
$485$	29.4443	1.33700
$486$	0	0
$487$	33.0093	1.49580	0.747898	−	0.663814i	$-0.231063\pi$
$487$	33.0093	1.49580	0.747898	+	0.663814i	$0.231063\pi$
$488$	0	0
$489$	10.4622	0.473116
$490$	0	0
$491$	−16.7094	−0.754084	−0.377042	−	0.926196i	$-0.623059\pi$
$491$	−16.7094	−0.754084	−0.377042	+	0.926196i	$0.623059\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−21.6917	−0.974969
$496$	0	0
$497$	−48.0592	−2.15575
$498$	0	0
$499$	26.3800	1.18093	0.590466	−	0.807063i	$-0.298944\pi$
$499$	26.3800	1.18093	0.590466	+	0.807063i	$0.298944\pi$
$500$	0	0
$501$	9.48581	0.423795
$502$	0	0
$503$	8.07173	0.359901	0.179950	−	0.983676i	$-0.442406\pi$
$503$	8.07173	0.359901	0.179950	+	0.983676i	$0.442406\pi$
$504$	0	0
$505$	26.1666	1.16440
$506$	0	0
$507$	13.9387	0.619038
$508$	0	0
$509$	0.0643005	0.00285007	0.00142503	−	0.999999i	$-0.499546\pi$
$509$	0.0643005	0.00285007	0.00142503	+	0.999999i	$0.499546\pi$
$510$	0	0
$511$	−7.01635	−0.310385
$512$	0	0
$513$	−7.64578	−0.337569
$514$	0	0
$515$	−9.81773	−0.432621
$516$	0	0
$517$	−20.2046	−0.888599
$518$	0	0
$519$	−24.2856	−1.06602
$520$	0	0
$521$	8.20562	0.359495	0.179747	−	0.983713i	$-0.442472\pi$
$521$	8.20562	0.359495	0.179747	+	0.983713i	$0.442472\pi$
$522$	0	0
$523$	9.10794	0.398263	0.199131	−	0.979973i	$-0.436188\pi$
$523$	9.10794	0.398263	0.199131	+	0.979973i	$0.436188\pi$
$524$	0	0
$525$	−1.99667	−0.0871419
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−22.3175	−0.970325
$530$	0	0
$531$	14.2634	0.618978
$532$	0	0
$533$	−9.06559	−0.392674
$534$	0	0
$535$	28.2696	1.22220
$536$	0	0
$537$	−7.63481	−0.329466
$538$	0	0
$539$	27.1495	1.16941
$540$	0	0
$541$	13.4948	0.580188	0.290094	−	0.956998i	$-0.406313\pi$
$541$	13.4948	0.580188	0.290094	+	0.956998i	$0.406313\pi$
$542$	0	0
$543$	16.7157	0.717339
$544$	0	0
$545$	5.50817	0.235944
$546$	0	0
$547$	29.4347	1.25854	0.629269	−	0.777188i	$-0.283355\pi$
$547$	29.4347	1.25854	0.629269	+	0.777188i	$0.283355\pi$
$548$	0	0
$549$	−4.23780	−0.180865
$550$	0	0
$551$	−5.38057	−0.229220
$552$	0	0
$553$	32.3426	1.37535
$554$	0	0
$555$	9.57181	0.406301
$556$	0	0
$557$	18.6792	0.791465	0.395732	−	0.918366i	$-0.370491\pi$
$557$	18.6792	0.791465	0.395732	+	0.918366i	$0.370491\pi$
$558$	0	0
$559$	10.6823	0.451815
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.1877	−1.23011	−0.615057	−	0.788482i	$-0.710867\pi$
$563$	−29.1877	−1.23011	−0.615057	+	0.788482i	$0.710867\pi$
$564$	0	0
$565$	48.4890	2.03995
$566$	0	0
$567$	−3.09008	−0.129771
$568$	0	0
$569$	23.6217	0.990273	0.495137	−	0.868815i	$-0.335118\pi$
$569$	23.6217	0.990273	0.495137	+	0.868815i	$0.335118\pi$
$570$	0	0
$571$	−15.2504	−0.638208	−0.319104	−	0.947720i	$-0.603382\pi$
$571$	−15.2504	−0.638208	−0.319104	+	0.947720i	$0.603382\pi$
$572$	0	0
$573$	4.10064	0.171307
$574$	0	0
$575$	−0.419825	−0.0175079
$576$	0	0
$577$	−7.85403	−0.326967	−0.163484	−	0.986546i	$-0.552273\pi$
$577$	−7.85403	−0.326967	−0.163484	+	0.986546i	$0.552273\pi$
$578$	0	0
$579$	20.4089	0.848165
$580$	0	0
$581$	15.7183	0.652106
$582$	0	0
$583$	39.3665	1.63039
$584$	0	0
$585$	3.34155	0.138156
$586$	0	0
$587$	−28.0757	−1.15881	−0.579405	−	0.815040i	$-0.696715\pi$
$587$	−28.0757	−1.15881	−0.579405	+	0.815040i	$0.696715\pi$
$588$	0	0
$589$	−11.9953	−0.494259
$590$	0	0
$591$	−16.6337	−0.684217
$592$	0	0
$593$	−15.3961	−0.632243	−0.316122	−	0.948719i	$-0.602381\pi$
$593$	−15.3961	−0.632243	−0.316122	+	0.948719i	$0.602381\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.0997	0.945407
$598$	0	0
$599$	−18.0394	−0.737069	−0.368535	−	0.929614i	$-0.620140\pi$
$599$	−18.0394	−0.737069	−0.368535	+	0.929614i	$0.620140\pi$
$600$	0	0
$601$	−16.7936	−0.685027	−0.342514	−	0.939513i	$-0.611278\pi$
$601$	−16.7936	−0.685027	−0.342514	+	0.939513i	$0.611278\pi$
$602$	0	0
$603$	−15.3501	−0.625105
$604$	0	0
$605$	42.0576	1.70988
$606$	0	0
$607$	21.4474	0.870524	0.435262	−	0.900304i	$-0.356656\pi$
$607$	21.4474	0.870524	0.435262	+	0.900304i	$0.356656\pi$
$608$	0	0
$609$	−14.7692	−0.598477
$610$	0	0
$611$	3.11247	0.125917
$612$	0	0
$613$	30.7109	1.24040	0.620200	−	0.784444i	$-0.287051\pi$
$613$	30.7109	1.24040	0.620200	+	0.784444i	$0.287051\pi$
$614$	0	0
$615$	−29.0722	−1.17230
$616$	0	0
$617$	−13.5636	−0.546049	−0.273025	−	0.962007i	$-0.588024\pi$
$617$	−13.5636	−0.546049	−0.273025	+	0.962007i	$0.588024\pi$
$618$	0	0
$619$	−22.4425	−0.902039	−0.451020	−	0.892514i	$-0.648940\pi$
$619$	−22.4425	−0.902039	−0.451020	+	0.892514i	$0.648940\pi$
$620$	0	0
$621$	−4.41277	−0.177078
$622$	0	0
$623$	−1.08816	−0.0435961
$624$	0	0
$625$	−27.2826	−1.09130
$626$	0	0
$627$	8.71370	0.347992
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−35.9183	−1.42988	−0.714942	−	0.699184i	$-0.753547\pi$
$631$	−35.9183	−1.42988	−0.714942	+	0.699184i	$0.753547\pi$
$632$	0	0
$633$	1.22308	0.0486131
$634$	0	0
$635$	17.7051	0.702605
$636$	0	0
$637$	−4.18230	−0.165709
$638$	0	0
$639$	23.7958	0.941347
$640$	0	0
$641$	−10.8907	−0.430155	−0.215077	−	0.976597i	$-0.569000\pi$
$641$	−10.8907	−0.430155	−0.215077	+	0.976597i	$0.569000\pi$
$642$	0	0
$643$	−10.8433	−0.427620	−0.213810	−	0.976875i	$-0.568587\pi$
$643$	−10.8433	−0.427620	−0.213810	+	0.976875i	$0.568587\pi$
$644$	0	0
$645$	34.2569	1.34886
$646$	0	0
$647$	−8.06107	−0.316913	−0.158457	−	0.987366i	$-0.550652\pi$
$647$	−8.06107	−0.316913	−0.158457	+	0.987366i	$0.550652\pi$
$648$	0	0
$649$	−44.6306	−1.75190
$650$	0	0
$651$	−32.9261	−1.29048
$652$	0	0
$653$	41.2222	1.61315	0.806575	−	0.591132i	$-0.201319\pi$
$653$	41.2222	1.61315	0.806575	+	0.591132i	$0.201319\pi$
$654$	0	0
$655$	9.87921	0.386013
$656$	0	0
$657$	3.47404	0.135535
$658$	0	0
$659$	−33.2980	−1.29710	−0.648552	−	0.761170i	$-0.724625\pi$
$659$	−33.2980	−1.29710	−0.648552	+	0.761170i	$0.724625\pi$
$660$	0	0
$661$	−10.6084	−0.412618	−0.206309	−	0.978487i	$-0.566145\pi$
$661$	−10.6084	−0.412618	−0.206309	+	0.978487i	$0.566145\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.6611	−0.452199
$666$	0	0
$667$	−3.10540	−0.120242
$668$	0	0
$669$	−11.1829	−0.432356
$670$	0	0
$671$	13.2602	0.511905
$672$	0	0
$673$	19.3247	0.744911	0.372455	−	0.928050i	$-0.378516\pi$
$673$	19.3247	0.744911	0.372455	+	0.928050i	$0.378516\pi$
$674$	0	0
$675$	2.71431	0.104474
$676$	0	0
$677$	3.82346	0.146947	0.0734737	−	0.997297i	$-0.476592\pi$
$677$	3.82346	0.146947	0.0734737	+	0.997297i	$0.476592\pi$
$678$	0	0
$679$	43.5476	1.67120
$680$	0	0
$681$	−17.0803	−0.654520
$682$	0	0
$683$	−27.9664	−1.07010	−0.535052	−	0.844819i	$-0.679708\pi$
$683$	−27.9664	−1.07010	−0.535052	+	0.844819i	$0.679708\pi$
$684$	0	0
$685$	15.5107	0.592635
$686$	0	0
$687$	−25.0104	−0.954206
$688$	0	0
$689$	−6.06430	−0.231031
$690$	0	0
$691$	−26.8111	−1.01994	−0.509971	−	0.860192i	$-0.670344\pi$
$691$	−26.8111	−1.01994	−0.509971	+	0.860192i	$0.670344\pi$
$692$	0	0
$693$	−32.0816	−1.21868
$694$	0	0
$695$	9.23460	0.350288
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−9.32785	−0.352812
$700$	0	0
$701$	−29.7543	−1.12380	−0.561902	−	0.827204i	$-0.689930\pi$
$701$	−29.7543	−1.12380	−0.561902	+	0.827204i	$0.689930\pi$
$702$	0	0
$703$	5.15737	0.194514
$704$	0	0
$705$	9.98129	0.375917
$706$	0	0
$707$	38.6999	1.45546
$708$	0	0
$709$	−0.452690	−0.0170011	−0.00850057	−	0.999964i	$-0.502706\pi$
$709$	−0.452690	−0.0170011	−0.00850057	+	0.999964i	$0.502706\pi$
$710$	0	0
$711$	−16.0140	−0.600571
$712$	0	0
$713$	−6.92313	−0.259273
$714$	0	0
$715$	−10.4558	−0.391025
$716$	0	0
$717$	11.8987	0.444363
$718$	0	0
$719$	21.4068	0.798338	0.399169	−	0.916877i	$-0.369299\pi$
$719$	21.4068	0.798338	0.399169	+	0.916877i	$0.369299\pi$
$720$	0	0
$721$	−14.5202	−0.540762
$722$	0	0
$723$	20.5371	0.763782
$724$	0	0
$725$	1.91014	0.0709409
$726$	0	0
$727$	−22.4811	−0.833779	−0.416889	−	0.908957i	$-0.636880\pi$
$727$	−22.4811	−0.833779	−0.416889	+	0.908957i	$0.636880\pi$
$728$	0	0
$729$	19.6686	0.728468
$730$	0	0
$731$	0	0
$732$	0	0
$733$	21.1334	0.780580	0.390290	−	0.920692i	$-0.372375\pi$
$733$	21.1334	0.780580	0.390290	+	0.920692i	$0.372375\pi$
$734$	0	0
$735$	−13.4121	−0.494713
$736$	0	0
$737$	48.0310	1.76924
$738$	0	0
$739$	22.8218	0.839514	0.419757	−	0.907636i	$-0.362115\pi$
$739$	22.8218	0.839514	0.419757	+	0.907636i	$0.362115\pi$
$740$	0	0
$741$	−1.34232	−0.0493115
$742$	0	0
$743$	−37.5403	−1.37722	−0.688609	−	0.725133i	$-0.741778\pi$
$743$	−37.5403	−1.37722	−0.688609	+	0.725133i	$0.741778\pi$
$744$	0	0
$745$	−16.7788	−0.614729
$746$	0	0
$747$	−7.78269	−0.284754
$748$	0	0
$749$	41.8102	1.52771
$750$	0	0
$751$	41.8523	1.52721	0.763607	−	0.645682i	$-0.223427\pi$
$751$	41.8523	1.52721	0.763607	+	0.645682i	$0.223427\pi$
$752$	0	0
$753$	2.11869	0.0772092
$754$	0	0
$755$	−37.0249	−1.34747
$756$	0	0
$757$	−2.48300	−0.0902463	−0.0451231	−	0.998981i	$-0.514368\pi$
$757$	−2.48300	−0.0902463	−0.0451231	+	0.998981i	$0.514368\pi$
$758$	0	0
$759$	5.02913	0.182546
$760$	0	0
$761$	18.0881	0.655695	0.327847	−	0.944731i	$-0.393677\pi$
$761$	18.0881	0.655695	0.327847	+	0.944731i	$0.393677\pi$
$762$	0	0
$763$	8.14648	0.294922
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.87523	0.248250
$768$	0	0
$769$	−22.2047	−0.800721	−0.400360	−	0.916358i	$-0.631115\pi$
$769$	−22.2047	−0.800721	−0.400360	+	0.916358i	$0.631115\pi$
$770$	0	0
$771$	12.0603	0.434343
$772$	0	0
$773$	−19.3632	−0.696448	−0.348224	−	0.937411i	$-0.613215\pi$
$773$	−19.3632	−0.696448	−0.348224	+	0.937411i	$0.613215\pi$
$774$	0	0
$775$	4.25844	0.152968
$776$	0	0
$777$	14.1565	0.507862
$778$	0	0
$779$	−15.6644	−0.561234
$780$	0	0
$781$	−74.4579	−2.66431
$782$	0	0
$783$	20.0774	0.717509
$784$	0	0
$785$	−13.5882	−0.484983
$786$	0	0
$787$	24.7792	0.883282	0.441641	−	0.897192i	$-0.354396\pi$
$787$	24.7792	0.883282	0.441641	+	0.897192i	$0.354396\pi$
$788$	0	0
$789$	35.7707	1.27347
$790$	0	0
$791$	71.7144	2.54987
$792$	0	0
$793$	−2.04270	−0.0725385
$794$	0	0
$795$	−19.4474	−0.689729
$796$	0	0
$797$	−22.7455	−0.805688	−0.402844	−	0.915269i	$-0.631978\pi$
$797$	−22.7455	−0.805688	−0.402844	+	0.915269i	$0.631978\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.538784	0.0190370
$802$	0	0
$803$	−10.8704	−0.383607
$804$	0	0
$805$	−6.73023	−0.237210
$806$	0	0
$807$	27.6076	0.971834
$808$	0	0
$809$	−47.8685	−1.68297	−0.841484	−	0.540282i	$-0.818318\pi$
$809$	−47.8685	−1.68297	−0.841484	+	0.540282i	$0.818318\pi$
$810$	0	0
$811$	−42.9855	−1.50943	−0.754713	−	0.656055i	$-0.772224\pi$
$811$	−42.9855	−1.50943	−0.754713	+	0.656055i	$0.772224\pi$
$812$	0	0
$813$	31.2844	1.09719
$814$	0	0
$815$	−21.6917	−0.759827
$816$	0	0
$817$	18.4579	0.645761
$818$	0	0
$819$	4.94209	0.172690
$820$	0	0
$821$	33.3592	1.16424	0.582122	−	0.813102i	$-0.302223\pi$
$821$	33.3592	1.16424	0.582122	+	0.813102i	$0.302223\pi$
$822$	0	0
$823$	9.04647	0.315340	0.157670	−	0.987492i	$-0.449602\pi$
$823$	9.04647	0.315340	0.157670	+	0.987492i	$0.449602\pi$
$824$	0	0
$825$	−3.09343	−0.107699
$826$	0	0
$827$	14.6427	0.509178	0.254589	−	0.967049i	$-0.418060\pi$
$827$	14.6427	0.509178	0.254589	+	0.967049i	$0.418060\pi$
$828$	0	0
$829$	30.5539	1.06118	0.530590	−	0.847629i	$-0.321971\pi$
$829$	30.5539	1.06118	0.530590	+	0.847629i	$0.321971\pi$
$830$	0	0
$831$	−26.3010	−0.912371
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−19.6673	−0.680616
$836$	0	0
$837$	44.7603	1.54714
$838$	0	0
$839$	6.21413	0.214536	0.107268	−	0.994230i	$-0.465790\pi$
$839$	6.21413	0.214536	0.107268	+	0.994230i	$0.465790\pi$
$840$	0	0
$841$	−14.8709	−0.512789
$842$	0	0
$843$	−4.77378	−0.164418
$844$	0	0
$845$	−28.8996	−0.994178
$846$	0	0
$847$	62.2024	2.13730
$848$	0	0
$849$	−4.41597	−0.151556
$850$	0	0
$851$	2.97658	0.102036
$852$	0	0
$853$	25.0594	0.858016	0.429008	−	0.903301i	$-0.358863\pi$
$853$	25.0594	0.858016	0.429008	+	0.903301i	$0.358863\pi$
$854$	0	0
$855$	5.77383	0.197461
$856$	0	0
$857$	14.4424	0.493341	0.246671	−	0.969099i	$-0.420663\pi$
$857$	14.4424	0.493341	0.246671	+	0.969099i	$0.420663\pi$
$858$	0	0
$859$	20.9921	0.716242	0.358121	−	0.933675i	$-0.383417\pi$
$859$	20.9921	0.716242	0.358121	+	0.933675i	$0.383417\pi$
$860$	0	0
$861$	−42.9972	−1.46534
$862$	0	0
$863$	12.3137	0.419162	0.209581	−	0.977791i	$-0.432790\pi$
$863$	12.3137	0.419162	0.209581	+	0.977791i	$0.432790\pi$
$864$	0	0
$865$	50.3523	1.71203
$866$	0	0
$867$	0	0
$868$	0	0
$869$	50.1082	1.69981
$870$	0	0
$871$	−7.39905	−0.250707
$872$	0	0
$873$	−21.5619	−0.729760
$874$	0	0
$875$	−36.5926	−1.23706
$876$	0	0
$877$	−32.5217	−1.09818	−0.549091	−	0.835763i	$-0.685026\pi$
$877$	−32.5217	−1.09818	−0.549091	+	0.835763i	$0.685026\pi$
$878$	0	0
$879$	−2.80276	−0.0945348
$880$	0	0
$881$	36.3173	1.22356	0.611781	−	0.791027i	$-0.290453\pi$
$881$	36.3173	1.22356	0.611781	+	0.791027i	$0.290453\pi$
$882$	0	0
$883$	31.0615	1.04530	0.522652	−	0.852546i	$-0.324943\pi$
$883$	31.0615	1.04530	0.522652	+	0.852546i	$0.324943\pi$
$884$	0	0
$885$	22.0480	0.741135
$886$	0	0
$887$	48.5818	1.63122	0.815609	−	0.578604i	$-0.196402\pi$
$887$	48.5818	1.63122	0.815609	+	0.578604i	$0.196402\pi$
$888$	0	0
$889$	26.1855	0.878233
$890$	0	0
$891$	−4.78744	−0.160385
$892$	0	0
$893$	5.37801	0.179968
$894$	0	0
$895$	15.8296	0.529125
$896$	0	0
$897$	−0.774724	−0.0258673
$898$	0	0
$899$	31.4992	1.05056
$900$	0	0
$901$	0	0
$902$	0	0
$903$	50.6653	1.68604
$904$	0	0
$905$	−34.6574	−1.15205
$906$	0	0
$907$	−9.84329	−0.326841	−0.163421	−	0.986556i	$-0.552253\pi$
$907$	−9.84329	−0.326841	−0.163421	+	0.986556i	$0.552253\pi$
$908$	0	0
$909$	−19.1617	−0.635553
$910$	0	0
$911$	−6.32133	−0.209435	−0.104717	−	0.994502i	$-0.533394\pi$
$911$	−6.32133	−0.209435	−0.104717	+	0.994502i	$0.533394\pi$
$912$	0	0
$913$	24.3523	0.805944
$914$	0	0
$915$	−6.55069	−0.216559
$916$	0	0
$917$	14.6112	0.482503
$918$	0	0
$919$	−35.0444	−1.15601	−0.578004	−	0.816034i	$-0.696168\pi$
$919$	−35.0444	−1.15601	−0.578004	+	0.816034i	$0.696168\pi$
$920$	0	0
$921$	25.1177	0.827657
$922$	0	0
$923$	11.4700	0.377541
$924$	0	0
$925$	−1.83091	−0.0601998
$926$	0	0
$927$	7.18948	0.236133
$928$	0	0
$929$	−6.64917	−0.218152	−0.109076	−	0.994033i	$-0.534789\pi$
$929$	−6.64917	−0.218152	−0.109076	+	0.994033i	$0.534789\pi$
$930$	0	0
$931$	−7.22657	−0.236841
$932$	0	0
$933$	28.6361	0.937503
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.3413	1.61191	0.805955	−	0.591976i	$-0.201652\pi$
$937$	49.3413	1.61191	0.805955	+	0.591976i	$0.201652\pi$
$938$	0	0
$939$	−13.6545	−0.445597
$940$	0	0
$941$	41.3964	1.34948	0.674742	−	0.738054i	$-0.264255\pi$
$941$	41.3964	1.34948	0.674742	+	0.738054i	$0.264255\pi$
$942$	0	0
$943$	−9.04070	−0.294406
$944$	0	0
$945$	43.5132	1.41548
$946$	0	0
$947$	17.4856	0.568204	0.284102	−	0.958794i	$-0.408305\pi$
$947$	17.4856	0.568204	0.284102	+	0.958794i	$0.408305\pi$
$948$	0	0
$949$	1.67455	0.0543583
$950$	0	0
$951$	3.10214	0.100594
$952$	0	0
$953$	−54.4709	−1.76449	−0.882243	−	0.470794i	$-0.843968\pi$
$953$	−54.4709	−1.76449	−0.882243	+	0.470794i	$0.843968\pi$
$954$	0	0
$955$	−8.50204	−0.275120
$956$	0	0
$957$	−22.8818	−0.739663
$958$	0	0
$959$	22.9401	0.740774
$960$	0	0
$961$	39.2237	1.26528
$962$	0	0
$963$	−20.7017	−0.667103
$964$	0	0
$965$	−42.3147	−1.36216
$966$	0	0
$967$	−10.6757	−0.343308	−0.171654	−	0.985157i	$-0.554911\pi$
$967$	−10.6757	−0.343308	−0.171654	+	0.985157i	$0.554911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−60.7986	−1.95112	−0.975560	−	0.219732i	$-0.929482\pi$
$971$	−60.7986	−1.95112	−0.975560	+	0.219732i	$0.929482\pi$
$972$	0	0
$973$	13.6578	0.437849
$974$	0	0
$975$	0.476535	0.0152613
$976$	0	0
$977$	11.8814	0.380120	0.190060	−	0.981772i	$-0.439132\pi$
$977$	11.8814	0.380120	0.190060	+	0.981772i	$0.439132\pi$
$978$	0	0
$979$	−1.68587	−0.0538807
$980$	0	0
$981$	−4.03361	−0.128783
$982$	0	0
$983$	−26.3261	−0.839673	−0.419837	−	0.907600i	$-0.637913\pi$
$983$	−26.3261	−0.839673	−0.419837	+	0.907600i	$0.637913\pi$
$984$	0	0
$985$	34.4873	1.09886
$986$	0	0
$987$	14.7621	0.469884
$988$	0	0
$989$	10.6530	0.338746
$990$	0	0
$991$	−41.0527	−1.30408	−0.652042	−	0.758183i	$-0.726087\pi$
$991$	−41.0527	−1.30408	−0.652042	+	0.758183i	$0.726087\pi$
$992$	0	0
$993$	−20.4586	−0.649234
$994$	0	0
$995$	−47.8936	−1.51833
$996$	0	0
$997$	−7.31231	−0.231583	−0.115792	−	0.993274i	$-0.536940\pi$
$997$	−7.31231	−0.231583	−0.115792	+	0.993274i	$0.536940\pi$
$998$	0	0
$999$	−19.2446	−0.608872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.bx.1.8		16
4.3	odd	2	inner	9248.2.a.bx.1.10		16
17.10	odd	16		544.2.bb.d.321.2	yes	16
17.12	odd	16		544.2.bb.d.161.2	✓	16
17.16	even	2	inner	9248.2.a.bx.1.9		16
68.27	even	16		544.2.bb.d.321.3	yes	16
68.63	even	16		544.2.bb.d.161.3	yes	16
68.67	odd	2	inner	9248.2.a.bx.1.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.bb.d.161.2	✓	16	17.12	odd	16
544.2.bb.d.161.3	yes	16	68.63	even	16
544.2.bb.d.321.2	yes	16	17.10	odd	16
544.2.bb.d.321.3	yes	16	68.27	even	16
9248.2.a.bx.1.7		16	68.67	odd	2	inner
9248.2.a.bx.1.8		16	1.1	even	1	trivial
9248.2.a.bx.1.9		16	17.16	even	2	inner
9248.2.a.bx.1.10		16	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13196 q^{3} +2.34695 q^{5} +3.47109 q^{7} -1.71866 q^{9} +5.37774 q^{11} -0.828427 q^{13} -2.65666 q^{15} -1.43143 q^{19} -3.92915 q^{21} -0.826153 q^{23} +0.508169 q^{25} +5.34135 q^{27} +3.75887 q^{29} +8.37996 q^{31} -6.08741 q^{33} +8.14648 q^{35} -3.60295 q^{37} +0.937749 q^{39} +10.9431 q^{41} -12.8947 q^{43} -4.03361 q^{45} -3.75708 q^{47} +5.04849 q^{49} +7.32026 q^{53} +12.6213 q^{55} +1.62033 q^{57} -8.29914 q^{59} +2.46576 q^{61} -5.96563 q^{63} -1.94428 q^{65} +8.93145 q^{67} +0.935174 q^{69} -13.8456 q^{71} -2.02137 q^{73} -0.575228 q^{75} +18.6666 q^{77} +9.31771 q^{79} -0.890232 q^{81} +4.52835 q^{83} -4.25490 q^{87} -0.313491 q^{89} -2.87555 q^{91} -9.48581 q^{93} -3.35950 q^{95} +12.5458 q^{97} -9.24251 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{9} + 32 q^{13} + 48 q^{21} + 32 q^{33} + 48 q^{49} + 80 q^{53} + 64 q^{69} + 48 q^{77} - 48 q^{89}+O(q^{100}) \) 16 * q + 32 * q^9 + 32 * q^13 + 48 * q^21 + 32 * q^33 + 48 * q^49 + 80 * q^53 + 64 * q^69 + 48 * q^77 - 48 * q^89

Introduction

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