LMFDB - Embedded newform 9248.2.a.bx.1.2

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			1.2
Root			$0.781338$ 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)

$f(q)$	$=$	$q-2.95274 q^{3} +0.499190 q^{5} -1.92584 q^{7} +5.71866 q^{9} -1.50049 q^{11} -0.828427 q^{13} -1.47398 q^{15} -4.67000 q^{19} +5.68651 q^{21} -6.70319 q^{23} -4.75081 q^{25} -8.02749 q^{27} +7.13736 q^{29} -4.64940 q^{31} +4.43055 q^{33} -0.961361 q^{35} -8.96052 q^{37} +2.44613 q^{39} -1.22581 q^{41} +4.94333 q^{43} +2.85470 q^{45} -2.08452 q^{47} -3.29113 q^{49} +12.5792 q^{53} -0.749029 q^{55} +13.7893 q^{57} +9.76347 q^{59} -7.66972 q^{61} -11.0132 q^{63} -0.413542 q^{65} -14.8939 q^{67} +19.7927 q^{69} -9.60768 q^{71} -7.69596 q^{73} +14.0279 q^{75} +2.88971 q^{77} -2.20327 q^{79} +6.54709 q^{81} +16.8805 q^{83} -21.0748 q^{87} -9.92915 q^{89} +1.59542 q^{91} +13.7284 q^{93} -2.33121 q^{95} +1.19659 q^{97} -8.58079 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.95274	−1.70476	−0.852382	−	0.522920i	$-0.824843\pi$
$3$	−2.95274	−1.70476	−0.852382	+	0.522920i	$0.824843\pi$
$4$	0	0
$5$	0.499190	0.223244	0.111622	−	0.993751i	$-0.464395\pi$
$5$	0.499190	0.223244	0.111622	+	0.993751i	$0.464395\pi$
$6$	0	0
$7$	−1.92584	−0.727900	−0.363950	−	0.931418i	$-0.618572\pi$
$7$	−1.92584	−0.727900	−0.363950	+	0.931418i	$0.618572\pi$
$8$	0	0
$9$	5.71866	1.90622
$10$	0	0
$11$	−1.50049	−0.452415	−0.226207	−	0.974079i	$-0.572633\pi$
$11$	−1.50049	−0.452415	−0.226207	+	0.974079i	$0.572633\pi$
$12$	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
$15$	−1.47398	−0.380579
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−4.67000	−1.07137	−0.535685	−	0.844418i	$-0.679947\pi$
$19$	−4.67000	−1.07137	−0.535685	+	0.844418i	$0.679947\pi$
$20$	0	0
$21$	5.68651	1.24090
$22$	0	0
$23$	−6.70319	−1.39771	−0.698855	−	0.715263i	$-0.746307\pi$
$23$	−6.70319	−1.39771	−0.698855	+	0.715263i	$0.746307\pi$
$24$	0	0
$25$	−4.75081	−0.950162
$26$	0	0
$27$	−8.02749	−1.54489
$28$	0	0
$29$	7.13736	1.32538	0.662688	−	0.748896i	$-0.269416\pi$
$29$	7.13736	1.32538	0.662688	+	0.748896i	$0.269416\pi$
$30$	0	0
$31$	−4.64940	−0.835056	−0.417528	−	0.908664i	$-0.637103\pi$
$31$	−4.64940	−0.835056	−0.417528	+	0.908664i	$0.637103\pi$
$32$	0	0
$33$	4.43055	0.771260
$34$	0	0
$35$	−0.961361	−0.162500
$36$	0	0
$37$	−8.96052	−1.47310	−0.736550	−	0.676383i	$-0.763546\pi$
$37$	−8.96052	−1.47310	−0.736550	+	0.676383i	$0.763546\pi$
$38$	0	0
$39$	2.44613	0.391694
$40$	0	0
$41$	−1.22581	−0.191439	−0.0957197	−	0.995408i	$-0.530515\pi$
$41$	−1.22581	−0.191439	−0.0957197	+	0.995408i	$0.530515\pi$
$42$	0	0
$43$	4.94333	0.753851	0.376926	−	0.926244i	$-0.376981\pi$
$43$	4.94333	0.753851	0.376926	+	0.926244i	$0.376981\pi$
$44$	0	0
$45$	2.85470	0.425553
$46$	0	0
$47$	−2.08452	−0.304058	−0.152029	−	0.988376i	$-0.548581\pi$
$47$	−2.08452	−0.304058	−0.152029	+	0.988376i	$0.548581\pi$
$48$	0	0
$49$	−3.29113	−0.470161
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.5792	1.72789	0.863946	−	0.503585i	$-0.167986\pi$
$53$	12.5792	1.72789	0.863946	+	0.503585i	$0.167986\pi$
$54$	0	0
$55$	−0.749029	−0.100999
$56$	0	0
$57$	13.7893	1.82643
$58$	0	0
$59$	9.76347	1.27110	0.635548	−	0.772062i	$-0.280774\pi$
$59$	9.76347	1.27110	0.635548	+	0.772062i	$0.280774\pi$
$60$	0	0
$61$	−7.66972	−0.982007	−0.491003	−	0.871158i	$-0.663370\pi$
$61$	−7.66972	−0.982007	−0.491003	+	0.871158i	$0.663370\pi$
$62$	0	0
$63$	−11.0132	−1.38754
$64$	0	0
$65$	−0.413542	−0.0512936
$66$	0	0
$67$	−14.8939	−1.81958	−0.909788	−	0.415074i	$-0.863756\pi$
$67$	−14.8939	−1.81958	−0.909788	+	0.415074i	$0.863756\pi$
$68$	0	0
$69$	19.7927	2.38277
$70$	0	0
$71$	−9.60768	−1.14022	−0.570111	−	0.821568i	$-0.693100\pi$
$71$	−9.60768	−1.14022	−0.570111	+	0.821568i	$0.693100\pi$
$72$	0	0
$73$	−7.69596	−0.900744	−0.450372	−	0.892841i	$-0.648709\pi$
$73$	−7.69596	−0.900744	−0.450372	+	0.892841i	$0.648709\pi$
$74$	0	0
$75$	14.0279	1.61980
$76$	0	0
$77$	2.88971	0.329313
$78$	0	0
$79$	−2.20327	−0.247887	−0.123944	−	0.992289i	$-0.539554\pi$
$79$	−2.20327	−0.247887	−0.123944	+	0.992289i	$0.539554\pi$
$80$	0	0
$81$	6.54709	0.727454
$82$	0	0
$83$	16.8805	1.85287	0.926436	−	0.376454i	$-0.122857\pi$
$83$	16.8805	1.85287	0.926436	+	0.376454i	$0.122857\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−21.0748	−2.25945
$88$	0	0
$89$	−9.92915	−1.05249	−0.526244	−	0.850334i	$-0.676400\pi$
$89$	−9.92915	−1.05249	−0.526244	+	0.850334i	$0.676400\pi$
$90$	0	0
$91$	1.59542	0.167246
$92$	0	0
$93$	13.7284	1.42357
$94$	0	0
$95$	−2.33121	−0.239178
$96$	0	0
$97$	1.19659	0.121496	0.0607478	−	0.998153i	$-0.480651\pi$
$97$	1.19659	0.121496	0.0607478	+	0.998153i	$0.480651\pi$
$98$	0	0
$99$	−8.58079	−0.862402
$100$	0	0
$101$	−6.80606	−0.677229	−0.338614	−	0.940925i	$-0.609958\pi$
$101$	−6.80606	−0.677229	−0.338614	+	0.940925i	$0.609958\pi$
$102$	0	0
$103$	10.9119	1.07518	0.537590	−	0.843206i	$-0.319335\pi$
$103$	10.9119	1.07518	0.537590	+	0.843206i	$0.319335\pi$
$104$	0	0
$105$	2.83865	0.277024
$106$	0	0
$107$	−6.06863	−0.586677	−0.293338	−	0.956009i	$-0.594766\pi$
$107$	−6.06863	−0.586677	−0.293338	+	0.956009i	$0.594766\pi$
$108$	0	0
$109$	0.499190	0.0478137	0.0239069	−	0.999714i	$-0.492389\pi$
$109$	0.499190	0.0478137	0.0239069	+	0.999714i	$0.492389\pi$
$110$	0	0
$111$	26.4581	2.51129
$112$	0	0
$113$	8.49151	0.798814	0.399407	−	0.916774i	$-0.369216\pi$
$113$	8.49151	0.798814	0.399407	+	0.916774i	$0.369216\pi$
$114$	0	0
$115$	−3.34616	−0.312031
$116$	0	0
$117$	−4.73749	−0.437981
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.74853	−0.795321
$122$	0	0
$123$	3.61950	0.326359
$124$	0	0
$125$	−4.86750	−0.435363
$126$	0	0
$127$	17.4183	1.54563	0.772814	−	0.634633i	$-0.218849\pi$
$127$	17.4183	1.54563	0.772814	+	0.634633i	$0.218849\pi$
$128$	0	0
$129$	−14.5964	−1.28514
$130$	0	0
$131$	−10.9802	−0.959347	−0.479673	−	0.877447i	$-0.659245\pi$
$131$	−10.9802	−0.959347	−0.479673	+	0.877447i	$0.659245\pi$
$132$	0	0
$133$	8.99368	0.779851
$134$	0	0
$135$	−4.00724	−0.344888
$136$	0	0
$137$	−8.26575	−0.706190	−0.353095	−	0.935587i	$-0.614871\pi$
$137$	−8.26575	−0.706190	−0.353095	+	0.935587i	$0.614871\pi$
$138$	0	0
$139$	−11.6967	−0.992099	−0.496050	−	0.868294i	$-0.665217\pi$
$139$	−11.6967	−0.992099	−0.496050	+	0.868294i	$0.665217\pi$
$140$	0	0
$141$	6.15503	0.518347
$142$	0	0
$143$	1.24305	0.103949
$144$	0	0
$145$	3.56290	0.295883
$146$	0	0
$147$	9.71784	0.801514
$148$	0	0
$149$	10.8061	0.885267	0.442634	−	0.896703i	$-0.354044\pi$
$149$	10.8061	0.885267	0.442634	+	0.896703i	$0.354044\pi$
$150$	0	0
$151$	−19.7152	−1.60440	−0.802200	−	0.597056i	$-0.796337\pi$
$151$	−19.7152	−1.60440	−0.802200	+	0.597056i	$0.796337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.32093	−0.186422
$156$	0	0
$157$	1.64759	0.131492	0.0657461	−	0.997836i	$-0.479057\pi$
$157$	1.64759	0.131492	0.0657461	+	0.997836i	$0.479057\pi$
$158$	0	0
$159$	−37.1432	−2.94565
$160$	0	0
$161$	12.9093	1.01739
$162$	0	0
$163$	−8.58079	−0.672099	−0.336050	−	0.941844i	$-0.609091\pi$
$163$	−8.58079	−0.672099	−0.336050	+	0.941844i	$0.609091\pi$
$164$	0	0
$165$	2.21169	0.172180
$166$	0	0
$167$	4.64940	0.359781	0.179891	−	0.983687i	$-0.442426\pi$
$167$	4.64940	0.359781	0.179891	+	0.983687i	$0.442426\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	−26.7061	−2.04227
$172$	0	0
$173$	20.9516	1.59292	0.796462	−	0.604688i	$-0.206702\pi$
$173$	20.9516	1.59292	0.796462	+	0.604688i	$0.206702\pi$
$174$	0	0
$175$	9.14931	0.691623
$176$	0	0
$177$	−28.8290	−2.16692
$178$	0	0
$179$	7.61795	0.569392	0.284696	−	0.958618i	$-0.408107\pi$
$179$	7.61795	0.569392	0.284696	+	0.958618i	$0.408107\pi$
$180$	0	0
$181$	−22.9234	−1.70388	−0.851941	−	0.523637i	$-0.824575\pi$
$181$	−22.9234	−1.70388	−0.851941	+	0.523637i	$0.824575\pi$
$182$	0	0
$183$	22.6467	1.67409
$184$	0	0
$185$	−4.47300	−0.328861
$186$	0	0
$187$	0	0
$188$	0	0
$189$	15.4597	1.12453
$190$	0	0
$191$	−26.2052	−1.89614	−0.948069	−	0.318064i	$-0.896967\pi$
$191$	−26.2052	−1.89614	−0.948069	+	0.318064i	$0.896967\pi$
$192$	0	0
$193$	0.262230	0.0188758	0.00943788	−	0.999955i	$-0.496996\pi$
$193$	0.262230	0.0188758	0.00943788	+	0.999955i	$0.496996\pi$
$194$	0	0
$195$	1.22108	0.0874435
$196$	0	0
$197$	−6.15610	−0.438604	−0.219302	−	0.975657i	$-0.570378\pi$
$197$	−6.15610	−0.438604	−0.219302	+	0.975657i	$0.570378\pi$
$198$	0	0
$199$	−20.2905	−1.43836	−0.719178	−	0.694826i	$-0.755481\pi$
$199$	−20.2905	−1.43836	−0.719178	+	0.694826i	$0.755481\pi$
$200$	0	0
$201$	43.9777	3.10195
$202$	0	0
$203$	−13.7454	−0.964741
$204$	0	0
$205$	−0.611912	−0.0427378
$206$	0	0
$207$	−38.3332	−2.66434
$208$	0	0
$209$	7.00728	0.484704
$210$	0	0
$211$	6.27783	0.432184	0.216092	−	0.976373i	$-0.430669\pi$
$211$	6.27783	0.432184	0.216092	+	0.976373i	$0.430669\pi$
$212$	0	0
$213$	28.3690	1.94381
$214$	0	0
$215$	2.46766	0.168293
$216$	0	0
$217$	8.95401	0.607838
$218$	0	0
$219$	22.7241	1.53556
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5.48122	−0.367049	−0.183525	−	0.983015i	$-0.558751\pi$
$223$	−5.48122	−0.367049	−0.183525	+	0.983015i	$0.558751\pi$
$224$	0	0
$225$	−27.1683	−1.81122
$226$	0	0
$227$	8.30336	0.551113	0.275557	−	0.961285i	$-0.411138\pi$
$227$	8.30336	0.551113	0.275557	+	0.961285i	$0.411138\pi$
$228$	0	0
$229$	−15.9942	−1.05693	−0.528463	−	0.848956i	$-0.677231\pi$
$229$	−15.9942	−1.05693	−0.528463	+	0.848956i	$0.677231\pi$
$230$	0	0
$231$	−8.53255	−0.561400
$232$	0	0
$233$	−13.9327	−0.912761	−0.456381	−	0.889785i	$-0.650854\pi$
$233$	−13.9327	−0.912761	−0.456381	+	0.889785i	$0.650854\pi$
$234$	0	0
$235$	−1.04057	−0.0678793
$236$	0	0
$237$	6.50567	0.422589
$238$	0	0
$239$	10.6116	0.686408	0.343204	−	0.939261i	$-0.388488\pi$
$239$	10.6116	0.686408	0.343204	+	0.939261i	$0.388488\pi$
$240$	0	0
$241$	29.5274	1.90203	0.951015	−	0.309146i	$-0.100043\pi$
$241$	29.5274	1.90203	0.951015	+	0.309146i	$0.100043\pi$
$242$	0	0
$243$	4.75063	0.304753
$244$	0	0
$245$	−1.64290	−0.104961
$246$	0	0
$247$	3.86875	0.246163
$248$	0	0
$249$	−49.8436	−3.15871
$250$	0	0
$251$	−15.4065	−0.972449	−0.486224	−	0.873834i	$-0.661626\pi$
$251$	−15.4065	−0.972449	−0.486224	+	0.873834i	$0.661626\pi$
$252$	0	0
$253$	10.0581	0.632345
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.13964	0.0710888	0.0355444	−	0.999368i	$-0.488683\pi$
$257$	1.13964	0.0710888	0.0355444	+	0.999368i	$0.488683\pi$
$258$	0	0
$259$	17.2565	1.07227
$260$	0	0
$261$	40.8162	2.52646
$262$	0	0
$263$	−12.0515	−0.743129	−0.371565	−	0.928407i	$-0.621179\pi$
$263$	−12.0515	−0.743129	−0.371565	+	0.928407i	$0.621179\pi$
$264$	0	0
$265$	6.27943	0.385742
$266$	0	0
$267$	29.3182	1.79424
$268$	0	0
$269$	3.77558	0.230201	0.115101	−	0.993354i	$-0.463281\pi$
$269$	3.77558	0.230201	0.115101	+	0.993354i	$0.463281\pi$
$270$	0	0
$271$	−2.10100	−0.127627	−0.0638133	−	0.997962i	$-0.520326\pi$
$271$	−2.10100	−0.127627	−0.0638133	+	0.997962i	$0.520326\pi$
$272$	0	0
$273$	−4.71086	−0.285114
$274$	0	0
$275$	7.12854	0.429867
$276$	0	0
$277$	−6.64633	−0.399339	−0.199670	−	0.979863i	$-0.563987\pi$
$277$	−6.64633	−0.399339	−0.199670	+	0.979863i	$0.563987\pi$
$278$	0	0
$279$	−26.5883	−1.59180
$280$	0	0
$281$	−2.31776	−0.138266	−0.0691331	−	0.997607i	$-0.522023\pi$
$281$	−2.31776	−0.138266	−0.0691331	+	0.997607i	$0.522023\pi$
$282$	0	0
$283$	16.6083	0.987259	0.493630	−	0.869672i	$-0.335670\pi$
$283$	16.6083	0.987259	0.493630	+	0.869672i	$0.335670\pi$
$284$	0	0
$285$	6.88346	0.407741
$286$	0	0
$287$	2.36072	0.139349
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−3.53323	−0.207121
$292$	0	0
$293$	−4.96130	−0.289842	−0.144921	−	0.989443i	$-0.546293\pi$
$293$	−4.96130	−0.289842	−0.144921	+	0.989443i	$0.546293\pi$
$294$	0	0
$295$	4.87382	0.283765
$296$	0	0
$297$	12.0452	0.698931
$298$	0	0
$299$	5.55310	0.321144
$300$	0	0
$301$	−9.52008	−0.548728
$302$	0	0
$303$	20.0965	1.15452
$304$	0	0
$305$	−3.82864	−0.219228
$306$	0	0
$307$	−23.2737	−1.32830	−0.664150	−	0.747600i	$-0.731206\pi$
$307$	−23.2737	−1.32830	−0.664150	+	0.747600i	$0.731206\pi$
$308$	0	0
$309$	−32.2200	−1.83293
$310$	0	0
$311$	19.9686	1.13231	0.566157	−	0.824297i	$-0.308430\pi$
$311$	19.9686	1.13231	0.566157	+	0.824297i	$0.308430\pi$
$312$	0	0
$313$	4.03755	0.228216	0.114108	−	0.993468i	$-0.463599\pi$
$313$	4.03755	0.228216	0.114108	+	0.993468i	$0.463599\pi$
$314$	0	0
$315$	−5.49770	−0.309760
$316$	0	0
$317$	17.6618	0.991984	0.495992	−	0.868327i	$-0.334804\pi$
$317$	17.6618	0.991984	0.495992	+	0.868327i	$0.334804\pi$
$318$	0	0
$319$	−10.7095	−0.599619
$320$	0	0
$321$	17.9191	1.00015
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.93570	0.218313
$326$	0	0
$327$	−1.47398	−0.0815111
$328$	0	0
$329$	4.01445	0.221324
$330$	0	0
$331$	22.1253	1.21611	0.608057	−	0.793893i	$-0.291949\pi$
$331$	22.1253	1.21611	0.608057	+	0.793893i	$0.291949\pi$
$332$	0	0
$333$	−51.2421	−2.80805
$334$	0	0
$335$	−7.43487	−0.406210
$336$	0	0
$337$	16.1865	0.881735	0.440868	−	0.897572i	$-0.354671\pi$
$337$	16.1865	0.881735	0.440868	+	0.897572i	$0.354671\pi$
$338$	0	0
$339$	−25.0732	−1.36179
$340$	0	0
$341$	6.97637	0.377792
$342$	0	0
$343$	19.8191	1.07013
$344$	0	0
$345$	9.88034	0.531939
$346$	0	0
$347$	27.1946	1.45988	0.729942	−	0.683510i	$-0.239547\pi$
$347$	27.1946	1.45988	0.729942	+	0.683510i	$0.239547\pi$
$348$	0	0
$349$	26.6435	1.42620	0.713098	−	0.701064i	$-0.247291\pi$
$349$	26.6435	1.42620	0.713098	+	0.701064i	$0.247291\pi$
$350$	0	0
$351$	6.65019	0.354961
$352$	0	0
$353$	−1.20696	−0.0642401	−0.0321200	−	0.999484i	$-0.510226\pi$
$353$	−1.20696	−0.0642401	−0.0321200	+	0.999484i	$0.510226\pi$
$354$	0	0
$355$	−4.79606	−0.254548
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.4041	1.49911	0.749555	−	0.661942i	$-0.230268\pi$
$359$	28.4041	1.49911	0.749555	+	0.661942i	$0.230268\pi$
$360$	0	0
$361$	2.80887	0.147835
$362$	0	0
$363$	25.8321	1.35583
$364$	0	0
$365$	−3.84174	−0.201086
$366$	0	0
$367$	24.7904	1.29405	0.647025	−	0.762469i	$-0.276013\pi$
$367$	24.7904	1.29405	0.647025	+	0.762469i	$0.276013\pi$
$368$	0	0
$369$	−7.00999	−0.364926
$370$	0	0
$371$	−24.2256	−1.25773
$372$	0	0
$373$	5.29091	0.273953	0.136976	−	0.990574i	$-0.456262\pi$
$373$	5.29091	0.273953	0.136976	+	0.990574i	$0.456262\pi$
$374$	0	0
$375$	14.3725	0.742191
$376$	0	0
$377$	−5.91279	−0.304524
$378$	0	0
$379$	−1.61540	−0.0829777	−0.0414888	−	0.999139i	$-0.513210\pi$
$379$	−1.61540	−0.0829777	−0.0414888	+	0.999139i	$0.513210\pi$
$380$	0	0
$381$	−51.4318	−2.63493
$382$	0	0
$383$	−10.6755	−0.545492	−0.272746	−	0.962086i	$-0.587932\pi$
$383$	−10.6755	−0.545492	−0.272746	+	0.962086i	$0.587932\pi$
$384$	0	0
$385$	1.44251	0.0735172
$386$	0	0
$387$	28.2692	1.43701
$388$	0	0
$389$	−32.5111	−1.64838	−0.824189	−	0.566315i	$-0.808369\pi$
$389$	−32.5111	−1.64838	−0.824189	+	0.566315i	$0.808369\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	32.4217	1.63546
$394$	0	0
$395$	−1.09985	−0.0553394
$396$	0	0
$397$	22.9818	1.15343	0.576713	−	0.816947i	$-0.304335\pi$
$397$	22.9818	1.15343	0.576713	+	0.816947i	$0.304335\pi$
$398$	0	0
$399$	−26.5560	−1.32946
$400$	0	0
$401$	−30.7607	−1.53612	−0.768059	−	0.640379i	$-0.778777\pi$
$401$	−30.7607	−1.53612	−0.768059	+	0.640379i	$0.778777\pi$
$402$	0	0
$403$	3.85169	0.191866
$404$	0	0
$405$	3.26824	0.162400
$406$	0	0
$407$	13.4452	0.666452
$408$	0	0
$409$	29.2173	1.44470	0.722351	−	0.691527i	$-0.243062\pi$
$409$	29.2173	1.44470	0.722351	+	0.691527i	$0.243062\pi$
$410$	0	0
$411$	24.4066	1.20389
$412$	0	0
$413$	−18.8029	−0.925231
$414$	0	0
$415$	8.42655	0.413643
$416$	0	0
$417$	34.5372	1.69130
$418$	0	0
$419$	−20.2108	−0.987362	−0.493681	−	0.869643i	$-0.664349\pi$
$419$	−20.2108	−0.987362	−0.493681	+	0.869643i	$0.664349\pi$
$420$	0	0
$421$	0.0180903	0.000881669 0	0.000440835	−	1.00000i	$-0.499860\pi$
$421$	0.0180903	0.000881669 0	0.000440835		1.00000i	$0.499860\pi$
$422$	0	0
$423$	−11.9206	−0.579602
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14.7707	0.714803
$428$	0	0
$429$	−3.67039	−0.177208
$430$	0	0
$431$	0.863740	0.0416049	0.0208024	−	0.999784i	$-0.493378\pi$
$431$	0.863740	0.0416049	0.0208024	+	0.999784i	$0.493378\pi$
$432$	0	0
$433$	29.8613	1.43504	0.717522	−	0.696536i	$-0.245276\pi$
$433$	29.8613	1.43504	0.717522	+	0.696536i	$0.245276\pi$
$434$	0	0
$435$	−10.5203	−0.504410
$436$	0	0
$437$	31.3039	1.49747
$438$	0	0
$439$	−19.4613	−0.928838	−0.464419	−	0.885616i	$-0.653737\pi$
$439$	−19.4613	−0.928838	−0.464419	+	0.885616i	$0.653737\pi$
$440$	0	0
$441$	−18.8208	−0.896231
$442$	0	0
$443$	16.9949	0.807450	0.403725	−	0.914880i	$-0.367715\pi$
$443$	16.9949	0.807450	0.403725	+	0.914880i	$0.367715\pi$
$444$	0	0
$445$	−4.95653	−0.234962
$446$	0	0
$447$	−31.9075	−1.50917
$448$	0	0
$449$	6.66836	0.314699	0.157350	−	0.987543i	$-0.449705\pi$
$449$	6.66836	0.314699	0.157350	+	0.987543i	$0.449705\pi$
$450$	0	0
$451$	1.83932	0.0866100
$452$	0	0
$453$	58.2138	2.73512
$454$	0	0
$455$	0.796418	0.0373366
$456$	0	0
$457$	−12.3331	−0.576917	−0.288458	−	0.957492i	$-0.593143\pi$
$457$	−12.3331	−0.576917	−0.288458	+	0.957492i	$0.593143\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.55966	0.305514	0.152757	−	0.988264i	$-0.451185\pi$
$461$	6.55966	0.305514	0.152757	+	0.988264i	$0.451185\pi$
$462$	0	0
$463$	−26.1313	−1.21443	−0.607213	−	0.794539i	$-0.707712\pi$
$463$	−26.1313	−1.21443	−0.607213	+	0.794539i	$0.707712\pi$
$464$	0	0
$465$	6.85310	0.317805
$466$	0	0
$467$	0.500962	0.0231817	0.0115909	−	0.999933i	$-0.496310\pi$
$467$	0.500962	0.0231817	0.0115909	+	0.999933i	$0.496310\pi$
$468$	0	0
$469$	28.6832	1.32447
$470$	0	0
$471$	−4.86491	−0.224163
$472$	0	0
$473$	−7.41742	−0.341053
$474$	0	0
$475$	22.1863	1.01798
$476$	0	0
$477$	71.9364	3.29374
$478$	0	0
$479$	20.8638	0.953291	0.476645	−	0.879096i	$-0.341853\pi$
$479$	20.8638	0.953291	0.476645	+	0.879096i	$0.341853\pi$
$480$	0	0
$481$	7.42313	0.338466
$482$	0	0
$483$	−38.1177	−1.73442
$484$	0	0
$485$	0.597327	0.0271232
$486$	0	0
$487$	−37.3415	−1.69211	−0.846054	−	0.533098i	$-0.821028\pi$
$487$	−37.3415	−1.69211	−0.846054	+	0.533098i	$0.821028\pi$
$488$	0	0
$489$	25.3368	1.14577
$490$	0	0
$491$	−5.39494	−0.243470	−0.121735	−	0.992563i	$-0.538846\pi$
$491$	−5.39494	−0.243470	−0.121735	+	0.992563i	$0.538846\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.28344	−0.192526
$496$	0	0
$497$	18.5029	0.829968
$498$	0	0
$499$	18.4596	0.826365	0.413182	−	0.910648i	$-0.364417\pi$
$499$	18.4596	0.826365	0.413182	+	0.910648i	$0.364417\pi$
$500$	0	0
$501$	−13.7284	−0.613342
$502$	0	0
$503$	22.9391	1.02280	0.511402	−	0.859342i	$-0.329126\pi$
$503$	22.9391	1.02280	0.511402	+	0.859342i	$0.329126\pi$
$504$	0	0
$505$	−3.39752	−0.151188
$506$	0	0
$507$	36.3592	1.61477
$508$	0	0
$509$	4.42098	0.195957	0.0979783	−	0.995189i	$-0.468762\pi$
$509$	4.42098	0.195957	0.0979783	+	0.995189i	$0.468762\pi$
$510$	0	0
$511$	14.8212	0.655652
$512$	0	0
$513$	37.4883	1.65515
$514$	0	0
$515$	5.44711	0.240028
$516$	0	0
$517$	3.12780	0.137560
$518$	0	0
$519$	−61.8647	−2.71556
$520$	0	0
$521$	−11.5401	−0.505580	−0.252790	−	0.967521i	$-0.581348\pi$
$521$	−11.5401	−0.505580	−0.252790	+	0.967521i	$0.581348\pi$
$522$	0	0
$523$	−20.2772	−0.886658	−0.443329	−	0.896359i	$-0.646203\pi$
$523$	−20.2772	−0.886658	−0.443329	+	0.896359i	$0.646203\pi$
$524$	0	0
$525$	−27.0155	−1.17905
$526$	0	0
$527$	0	0
$528$	0	0
$529$	21.9327	0.953595
$530$	0	0
$531$	55.8340	2.42299
$532$	0	0
$533$	1.01549	0.0439860
$534$	0	0
$535$	−3.02940	−0.130972
$536$	0	0
$537$	−22.4938	−0.970679
$538$	0	0
$539$	4.93830	0.212708
$540$	0	0
$541$	−23.7005	−1.01896	−0.509481	−	0.860482i	$-0.670163\pi$
$541$	−23.7005	−1.01896	−0.509481	+	0.860482i	$0.670163\pi$
$542$	0	0
$543$	67.6868	2.90472
$544$	0	0
$545$	0.249190	0.0106741
$546$	0	0
$547$	14.6672	0.627124	0.313562	−	0.949568i	$-0.398478\pi$
$547$	14.6672	0.627124	0.313562	+	0.949568i	$0.398478\pi$
$548$	0	0
$549$	−43.8605	−1.87192
$550$	0	0
$551$	−33.3315	−1.41997
$552$	0	0
$553$	4.24315	0.180437
$554$	0	0
$555$	13.2076	0.560631
$556$	0	0
$557$	−14.6792	−0.621979	−0.310990	−	0.950413i	$-0.600660\pi$
$557$	−14.6792	−0.621979	−0.310990	+	0.950413i	$0.600660\pi$
$558$	0	0
$559$	−4.09519	−0.173208
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.86605	0.289370	0.144685	−	0.989478i	$-0.453783\pi$
$563$	6.86605	0.289370	0.144685	+	0.989478i	$0.453783\pi$
$564$	0	0
$565$	4.23888	0.178331
$566$	0	0
$567$	−12.6087	−0.529514
$568$	0	0
$569$	−18.4501	−0.773470	−0.386735	−	0.922191i	$-0.626397\pi$
$569$	−18.4501	−0.773470	−0.386735	+	0.922191i	$0.626397\pi$
$570$	0	0
$571$	31.4291	1.31527	0.657634	−	0.753338i	$-0.271557\pi$
$571$	31.4291	1.31527	0.657634	+	0.753338i	$0.271557\pi$
$572$	0	0
$573$	77.3770	3.23247
$574$	0	0
$575$	31.8456	1.32805
$576$	0	0
$577$	−4.77339	−0.198719	−0.0993594	−	0.995052i	$-0.531679\pi$
$577$	−4.77339	−0.198719	−0.0993594	+	0.995052i	$0.531679\pi$
$578$	0	0
$579$	−0.774297	−0.0321787
$580$	0	0
$581$	−32.5091	−1.34871
$582$	0	0
$583$	−18.8750	−0.781723
$584$	0	0
$585$	−2.36491	−0.0977769
$586$	0	0
$587$	5.40737	0.223186	0.111593	−	0.993754i	$-0.464405\pi$
$587$	5.40737	0.223186	0.111593	+	0.993754i	$0.464405\pi$
$588$	0	0
$589$	21.7127	0.894655
$590$	0	0
$591$	18.1773	0.747716
$592$	0	0
$593$	18.7098	0.768321	0.384160	−	0.923266i	$-0.374491\pi$
$593$	18.7098	0.768321	0.384160	+	0.923266i	$0.374491\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	59.9125	2.45206
$598$	0	0
$599$	35.1710	1.43705	0.718525	−	0.695502i	$-0.244818\pi$
$599$	35.1710	1.43705	0.718525	+	0.695502i	$0.244818\pi$
$600$	0	0
$601$	−11.3817	−0.464269	−0.232134	−	0.972684i	$-0.574571\pi$
$601$	−11.3817	−0.464269	−0.232134	+	0.972684i	$0.574571\pi$
$602$	0	0
$603$	−85.1729	−3.46851
$604$	0	0
$605$	−4.36718	−0.177551
$606$	0	0
$607$	41.1978	1.67217	0.836083	−	0.548603i	$-0.184840\pi$
$607$	41.1978	1.67217	0.836083	+	0.548603i	$0.184840\pi$
$608$	0	0
$609$	40.5867	1.64466
$610$	0	0
$611$	1.72687	0.0698617
$612$	0	0
$613$	22.7450	0.918661	0.459331	−	0.888265i	$-0.348089\pi$
$613$	22.7450	0.918661	0.459331	+	0.888265i	$0.348089\pi$
$614$	0	0
$615$	1.80682	0.0728579
$616$	0	0
$617$	−0.869375	−0.0349997	−0.0174999	−	0.999847i	$-0.505571\pi$
$617$	−0.869375	−0.0349997	−0.0174999	+	0.999847i	$0.505571\pi$
$618$	0	0
$619$	4.67569	0.187932	0.0939658	−	0.995575i	$-0.470046\pi$
$619$	4.67569	0.187932	0.0939658	+	0.995575i	$0.470046\pi$
$620$	0	0
$621$	53.8097	2.15931
$622$	0	0
$623$	19.1220	0.766106
$624$	0	0
$625$	21.3242	0.852970
$626$	0	0
$627$	−20.6907	−0.826305
$628$	0	0
$629$	0	0
$630$	0	0
$631$	22.9810	0.914859	0.457430	−	0.889246i	$-0.348770\pi$
$631$	22.9810	0.914859	0.457430	+	0.889246i	$0.348770\pi$
$632$	0	0
$633$	−18.5368	−0.736771
$634$	0	0
$635$	8.69506	0.345053
$636$	0	0
$637$	2.72646	0.108026
$638$	0	0
$639$	−54.9431	−2.17351
$640$	0	0
$641$	10.2001	0.402878	0.201439	−	0.979501i	$-0.435438\pi$
$641$	10.2001	0.402878	0.201439	+	0.979501i	$0.435438\pi$
$642$	0	0
$643$	−12.7566	−0.503071	−0.251535	−	0.967848i	$-0.580935\pi$
$643$	−12.7566	−0.503071	−0.251535	+	0.967848i	$0.580935\pi$
$644$	0	0
$645$	−7.28636	−0.286900
$646$	0	0
$647$	4.21957	0.165888	0.0829442	−	0.996554i	$-0.473568\pi$
$647$	4.21957	0.165888	0.0829442	+	0.996554i	$0.473568\pi$
$648$	0	0
$649$	−14.6500	−0.575062
$650$	0	0
$651$	−26.4388	−1.03622
$652$	0	0
$653$	−16.5836	−0.648966	−0.324483	−	0.945891i	$-0.605190\pi$
$653$	−16.5836	−0.648966	−0.324483	+	0.945891i	$0.605190\pi$
$654$	0	0
$655$	−5.48122	−0.214169
$656$	0	0
$657$	−44.0106	−1.71702
$658$	0	0
$659$	−6.18208	−0.240820	−0.120410	−	0.992724i	$-0.538421\pi$
$659$	−6.18208	−0.240820	−0.120410	+	0.992724i	$0.538421\pi$
$660$	0	0
$661$	32.3657	1.25888	0.629441	−	0.777048i	$-0.283284\pi$
$661$	32.3657	1.25888	0.629441	+	0.777048i	$0.283284\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.48955	0.174097
$666$	0	0
$667$	−47.8431	−1.85249
$668$	0	0
$669$	16.1846	0.625733
$670$	0	0
$671$	11.5083	0.444274
$672$	0	0
$673$	−33.7576	−1.30126	−0.650630	−	0.759395i	$-0.725495\pi$
$673$	−33.7576	−1.30126	−0.650630	+	0.759395i	$0.725495\pi$
$674$	0	0
$675$	38.1371	1.46790
$676$	0	0
$677$	6.38219	0.245287	0.122644	−	0.992451i	$-0.460863\pi$
$677$	6.38219	0.245287	0.122644	+	0.992451i	$0.460863\pi$
$678$	0	0
$679$	−2.30445	−0.0884367
$680$	0	0
$681$	−24.5176	−0.939518
$682$	0	0
$683$	48.6122	1.86010	0.930048	−	0.367439i	$-0.119765\pi$
$683$	48.6122	1.86010	0.930048	+	0.367439i	$0.119765\pi$
$684$	0	0
$685$	−4.12618	−0.157653
$686$	0	0
$687$	47.2267	1.80181
$688$	0	0
$689$	−10.4210	−0.397008
$690$	0	0
$691$	−27.1197	−1.03168	−0.515840	−	0.856685i	$-0.672520\pi$
$691$	−27.1197	−1.03168	−0.515840	+	0.856685i	$0.672520\pi$
$692$	0	0
$693$	16.5252	0.627742
$694$	0	0
$695$	−5.83886	−0.221481
$696$	0	0
$697$	0	0
$698$	0	0
$699$	41.1396	1.55604
$700$	0	0
$701$	6.15629	0.232520	0.116260	−	0.993219i	$-0.462909\pi$
$701$	6.15629	0.232520	0.116260	+	0.993219i	$0.462909\pi$
$702$	0	0
$703$	41.8456	1.57824
$704$	0	0
$705$	3.07253	0.115718
$706$	0	0
$707$	13.1074	0.492955
$708$	0	0
$709$	−32.2360	−1.21065	−0.605324	−	0.795979i	$-0.706957\pi$
$709$	−32.2360	−1.21065	−0.605324	+	0.795979i	$0.706957\pi$
$710$	0	0
$711$	−12.5997	−0.472527
$712$	0	0
$713$	31.1658	1.16717
$714$	0	0
$715$	0.620516	0.0232060
$716$	0	0
$717$	−31.3333	−1.17016
$718$	0	0
$719$	13.8028	0.514758	0.257379	−	0.966310i	$-0.417141\pi$
$719$	13.8028	0.514758	0.257379	+	0.966310i	$0.417141\pi$
$720$	0	0
$721$	−21.0146	−0.782625
$722$	0	0
$723$	−87.1868	−3.24251
$724$	0	0
$725$	−33.9083	−1.25932
$726$	0	0
$727$	−34.3979	−1.27575	−0.637874	−	0.770141i	$-0.720186\pi$
$727$	−34.3979	−1.27575	−0.637874	+	0.770141i	$0.720186\pi$
$728$	0	0
$729$	−33.6686	−1.24699
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−9.51817	−0.351562	−0.175781	−	0.984429i	$-0.556245\pi$
$733$	−9.51817	−0.351562	−0.175781	+	0.984429i	$0.556245\pi$
$734$	0	0
$735$	4.85105	0.178934
$736$	0	0
$737$	22.3481	0.823202
$738$	0	0
$739$	18.1433	0.667412	0.333706	−	0.942677i	$-0.391701\pi$
$739$	18.1433	0.667412	0.333706	+	0.942677i	$0.391701\pi$
$740$	0	0
$741$	−11.4234	−0.419649
$742$	0	0
$743$	19.0905	0.700363	0.350182	−	0.936682i	$-0.386120\pi$
$743$	19.0905	0.700363	0.350182	+	0.936682i	$0.386120\pi$
$744$	0	0
$745$	5.39428	0.197631
$746$	0	0
$747$	96.5336	3.53198
$748$	0	0
$749$	11.6872	0.427042
$750$	0	0
$751$	16.7823	0.612394	0.306197	−	0.951968i	$-0.400943\pi$
$751$	16.7823	0.612394	0.306197	+	0.951968i	$0.400943\pi$
$752$	0	0
$753$	45.4913	1.65780
$754$	0	0
$755$	−9.84162	−0.358173
$756$	0	0
$757$	18.9267	0.687901	0.343950	−	0.938988i	$-0.388235\pi$
$757$	18.9267	0.687901	0.343950	+	0.938988i	$0.388235\pi$
$758$	0	0
$759$	−29.6988	−1.07800
$760$	0	0
$761$	−25.2597	−0.915664	−0.457832	−	0.889039i	$-0.651374\pi$
$761$	−25.2597	−0.915664	−0.457832	+	0.889039i	$0.651374\pi$
$762$	0	0
$763$	−0.961361	−0.0348036
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.08832	−0.292052
$768$	0	0
$769$	−33.2512	−1.19907	−0.599534	−	0.800349i	$-0.704647\pi$
$769$	−33.2512	−1.19907	−0.599534	+	0.800349i	$0.704647\pi$
$770$	0	0
$771$	−3.36506	−0.121190
$772$	0	0
$773$	35.4049	1.27342	0.636712	−	0.771101i	$-0.280294\pi$
$773$	35.4049	1.27342	0.636712	+	0.771101i	$0.280294\pi$
$774$	0	0
$775$	22.0884	0.793439
$776$	0	0
$777$	−50.9541	−1.82797
$778$	0	0
$779$	5.72453	0.205103
$780$	0	0
$781$	14.4162	0.515853
$782$	0	0
$783$	−57.2951	−2.04756
$784$	0	0
$785$	0.822461	0.0293549
$786$	0	0
$787$	14.2838	0.509162	0.254581	−	0.967051i	$-0.418062\pi$
$787$	14.2838	0.509162	0.254581	+	0.967051i	$0.418062\pi$
$788$	0	0
$789$	35.5850	1.26686
$790$	0	0
$791$	−16.3533	−0.581457
$792$	0	0
$793$	6.35380	0.225630
$794$	0	0
$795$	−18.5415	−0.657599
$796$	0	0
$797$	43.8166	1.55206	0.776032	−	0.630694i	$-0.217230\pi$
$797$	43.8166	1.55206	0.776032	+	0.630694i	$0.217230\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−56.7814	−2.00627
$802$	0	0
$803$	11.5477	0.407510
$804$	0	0
$805$	6.44418	0.227128
$806$	0	0
$807$	−11.1483	−0.392439
$808$	0	0
$809$	39.8184	1.39994	0.699971	−	0.714171i	$-0.253196\pi$
$809$	39.8184	1.39994	0.699971	+	0.714171i	$0.253196\pi$
$810$	0	0
$811$	18.5309	0.650708	0.325354	−	0.945592i	$-0.394516\pi$
$811$	18.5309	0.650708	0.325354	+	0.945592i	$0.394516\pi$
$812$	0	0
$813$	6.20370	0.217573
$814$	0	0
$815$	−4.28344	−0.150042
$816$	0	0
$817$	−23.0853	−0.807654
$818$	0	0
$819$	9.12367	0.318807
$820$	0	0
$821$	35.8410	1.25086	0.625430	−	0.780281i	$-0.284924\pi$
$821$	35.8410	1.25086	0.625430	+	0.780281i	$0.284924\pi$
$822$	0	0
$823$	−36.6318	−1.27690	−0.638452	−	0.769661i	$-0.720425\pi$
$823$	−36.6318	−1.27690	−0.638452	+	0.769661i	$0.720425\pi$
$824$	0	0
$825$	−21.0487	−0.732822
$826$	0	0
$827$	25.3316	0.880866	0.440433	−	0.897786i	$-0.354825\pi$
$827$	25.3316	0.880866	0.440433	+	0.897786i	$0.354825\pi$
$828$	0	0
$829$	18.7599	0.651557	0.325778	−	0.945446i	$-0.394374\pi$
$829$	18.7599	0.651557	0.325778	+	0.945446i	$0.394374\pi$
$830$	0	0
$831$	19.6249	0.680779
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.32093	0.0803191
$836$	0	0
$837$	37.3230	1.29007
$838$	0	0
$839$	55.5824	1.91892	0.959458	−	0.281851i	$-0.0909483\pi$
$839$	55.5824	1.91892	0.959458	+	0.281851i	$0.0909483\pi$
$840$	0	0
$841$	21.9420	0.756619
$842$	0	0
$843$	6.84375	0.235711
$844$	0	0
$845$	−6.14688	−0.211459
$846$	0	0
$847$	16.8483	0.578914
$848$	0	0
$849$	−49.0399	−1.68304
$850$	0	0
$851$	60.0640	2.05897
$852$	0	0
$853$	53.8581	1.84407	0.922034	−	0.387109i	$-0.126526\pi$
$853$	53.8581	1.84407	0.922034	+	0.387109i	$0.126526\pi$
$854$	0	0
$855$	−13.3314	−0.455925
$856$	0	0
$857$	−21.8019	−0.744737	−0.372369	−	0.928085i	$-0.621454\pi$
$857$	−21.8019	−0.744737	−0.372369	+	0.928085i	$0.621454\pi$
$858$	0	0
$859$	21.4742	0.732689	0.366344	−	0.930479i	$-0.380609\pi$
$859$	21.4742	0.732689	0.366344	+	0.930479i	$0.380609\pi$
$860$	0	0
$861$	−6.97058	−0.237557
$862$	0	0
$863$	−53.8510	−1.83311	−0.916555	−	0.399909i	$-0.869042\pi$
$863$	−53.8510	−1.83311	−0.916555	+	0.399909i	$0.869042\pi$
$864$	0	0
$865$	10.4588	0.355611
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.30598	0.112148
$870$	0	0
$871$	12.3385	0.418074
$872$	0	0
$873$	6.84291	0.231597
$874$	0	0
$875$	9.37405	0.316901
$876$	0	0
$877$	−25.2939	−0.854114	−0.427057	−	0.904225i	$-0.640450\pi$
$877$	−25.2939	−0.854114	−0.427057	+	0.904225i	$0.640450\pi$
$878$	0	0
$879$	14.6494	0.494113
$880$	0	0
$881$	21.0100	0.707844	0.353922	−	0.935275i	$-0.384848\pi$
$881$	21.0100	0.707844	0.353922	+	0.935275i	$0.384848\pi$
$882$	0	0
$883$	−18.1188	−0.609746	−0.304873	−	0.952393i	$-0.598614\pi$
$883$	−18.1188	−0.609746	−0.304873	+	0.952393i	$0.598614\pi$
$884$	0	0
$885$	−14.3911	−0.483752
$886$	0	0
$887$	43.4325	1.45832	0.729160	−	0.684343i	$-0.239911\pi$
$887$	43.4325	1.45832	0.729160	+	0.684343i	$0.239911\pi$
$888$	0	0
$889$	−33.5450	−1.12506
$890$	0	0
$891$	−9.82383	−0.329111
$892$	0	0
$893$	9.73469	0.325759
$894$	0	0
$895$	3.80280	0.127114
$896$	0	0
$897$	−16.3968	−0.547475
$898$	0	0
$899$	−33.1844	−1.10676
$900$	0	0
$901$	0	0
$902$	0	0
$903$	28.1103	0.935452
$904$	0	0
$905$	−11.4431	−0.380382
$906$	0	0
$907$	−19.2443	−0.638995	−0.319498	−	0.947587i	$-0.603514\pi$
$907$	−19.2443	−0.638995	−0.319498	+	0.947587i	$0.603514\pi$
$908$	0	0
$909$	−38.9216	−1.29095
$910$	0	0
$911$	56.6045	1.87539	0.937696	−	0.347458i	$-0.112955\pi$
$911$	56.6045	1.87539	0.937696	+	0.347458i	$0.112955\pi$
$912$	0	0
$913$	−25.3289	−0.838266
$914$	0	0
$915$	11.3050	0.373731
$916$	0	0
$917$	21.1462	0.698309
$918$	0	0
$919$	27.0662	0.892832	0.446416	−	0.894825i	$-0.352700\pi$
$919$	27.0662	0.892832	0.446416	+	0.894825i	$0.352700\pi$
$920$	0	0
$921$	68.7211	2.26444
$922$	0	0
$923$	7.95926	0.261982
$924$	0	0
$925$	42.5697	1.39968
$926$	0	0
$927$	62.4014	2.04953
$928$	0	0
$929$	−36.2452	−1.18917	−0.594583	−	0.804034i	$-0.702683\pi$
$929$	−36.2452	−1.18917	−0.594583	+	0.804034i	$0.702683\pi$
$930$	0	0
$931$	15.3696	0.503717
$932$	0	0
$933$	−58.9620	−1.93033
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42.6134	−1.39212	−0.696060	−	0.717984i	$-0.745065\pi$
$937$	−42.6134	−1.39212	−0.696060	+	0.717984i	$0.745065\pi$
$938$	0	0
$939$	−11.9218	−0.389054
$940$	0	0
$941$	18.0865	0.589602	0.294801	−	0.955559i	$-0.404747\pi$
$941$	18.0865	0.589602	0.294801	+	0.955559i	$0.404747\pi$
$942$	0	0
$943$	8.21684	0.267577
$944$	0	0
$945$	7.71731	0.251044
$946$	0	0
$947$	−44.8948	−1.45888	−0.729442	−	0.684042i	$-0.760220\pi$
$947$	−44.8948	−1.45888	−0.729442	+	0.684042i	$0.760220\pi$
$948$	0	0
$949$	6.37554	0.206959
$950$	0	0
$951$	−52.1506	−1.69110
$952$	0	0
$953$	−48.6834	−1.57701	−0.788505	−	0.615028i	$-0.789145\pi$
$953$	−48.6834	−1.57701	−0.788505	+	0.615028i	$0.789145\pi$
$954$	0	0
$955$	−13.0813	−0.423302
$956$	0	0
$957$	31.6225	1.02221
$958$	0	0
$959$	15.9185	0.514036
$960$	0	0
$961$	−9.38311	−0.302681
$962$	0	0
$963$	−34.7044	−1.11833
$964$	0	0
$965$	0.130903	0.00421391
$966$	0	0
$967$	21.4829	0.690845	0.345422	−	0.938447i	$-0.387736\pi$
$967$	21.4829	0.690845	0.345422	+	0.938447i	$0.387736\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−59.0629	−1.89542	−0.947710	−	0.319133i	$-0.896608\pi$
$971$	−59.0629	−1.89542	−0.947710	+	0.319133i	$0.896608\pi$
$972$	0	0
$973$	22.5260	0.722149
$974$	0	0
$975$	−11.6211	−0.372173
$976$	0	0
$977$	−22.2246	−0.711026	−0.355513	−	0.934671i	$-0.615694\pi$
$977$	−22.2246	−0.711026	−0.355513	+	0.934671i	$0.615694\pi$
$978$	0	0
$979$	14.8986	0.476161
$980$	0	0
$981$	2.85470	0.0911434
$982$	0	0
$983$	−33.0671	−1.05468	−0.527338	−	0.849656i	$-0.676810\pi$
$983$	−33.0671	−1.05468	−0.527338	+	0.849656i	$0.676810\pi$
$984$	0	0
$985$	−3.07306	−0.0979159
$986$	0	0
$987$	−11.8536	−0.377305
$988$	0	0
$989$	−33.1361	−1.05367
$990$	0	0
$991$	−24.8964	−0.790861	−0.395430	−	0.918496i	$-0.629405\pi$
$991$	−24.8964	−0.790861	−0.395430	+	0.918496i	$0.629405\pi$
$992$	0	0
$993$	−65.3301	−2.07319
$994$	0	0
$995$	−10.1288	−0.321105
$996$	0	0
$997$	20.8524	0.660403	0.330201	−	0.943911i	$-0.392883\pi$
$997$	20.8524	0.660403	0.330201	+	0.943911i	$0.392883\pi$
$998$	0	0
$999$	71.9304	2.27578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.bx.1.2		16
4.3	odd	2	inner	9248.2.a.bx.1.16		16
17.5	odd	16		544.2.bb.d.161.4	yes	16
17.7	odd	16		544.2.bb.d.321.4	yes	16
17.16	even	2	inner	9248.2.a.bx.1.15		16
68.7	even	16		544.2.bb.d.321.1	yes	16
68.39	even	16		544.2.bb.d.161.1	✓	16
68.67	odd	2	inner	9248.2.a.bx.1.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.bb.d.161.1	✓	16	68.39	even	16
544.2.bb.d.161.4	yes	16	17.5	odd	16
544.2.bb.d.321.1	yes	16	68.7	even	16
544.2.bb.d.321.4	yes	16	17.7	odd	16
9248.2.a.bx.1.1		16	68.67	odd	2	inner
9248.2.a.bx.1.2		16	1.1	even	1	trivial
9248.2.a.bx.1.15		16	17.16	even	2	inner
9248.2.a.bx.1.16		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)

\(f(q)\)	\(=\)	\(q-2.95274 q^{3} +0.499190 q^{5} -1.92584 q^{7} +5.71866 q^{9} -1.50049 q^{11} -0.828427 q^{13} -1.47398 q^{15} -4.67000 q^{19} +5.68651 q^{21} -6.70319 q^{23} -4.75081 q^{25} -8.02749 q^{27} +7.13736 q^{29} -4.64940 q^{31} +4.43055 q^{33} -0.961361 q^{35} -8.96052 q^{37} +2.44613 q^{39} -1.22581 q^{41} +4.94333 q^{43} +2.85470 q^{45} -2.08452 q^{47} -3.29113 q^{49} +12.5792 q^{53} -0.749029 q^{55} +13.7893 q^{57} +9.76347 q^{59} -7.66972 q^{61} -11.0132 q^{63} -0.413542 q^{65} -14.8939 q^{67} +19.7927 q^{69} -9.60768 q^{71} -7.69596 q^{73} +14.0279 q^{75} +2.88971 q^{77} -2.20327 q^{79} +6.54709 q^{81} +16.8805 q^{83} -21.0748 q^{87} -9.92915 q^{89} +1.59542 q^{91} +13.7284 q^{93} -2.33121 q^{95} +1.19659 q^{97} -8.58079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{9} + 32 q^{13} + 48 q^{21} + 32 q^{33} + 48 q^{49} + 80 q^{53} + 64 q^{69} + 48 q^{77} - 48 q^{89}+O(q^{100}) \) 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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