LMFDB - Embedded newform 9196.2.a.w.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9196,2,Mod(1,9196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9196, 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("9196.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.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:	$73.4304296988$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 6 x^{19} - 26 x^{18} + 208 x^{17} + 185 x^{16} - 2910 x^{15} + 687 x^{14} + 21067 x^{13} + \cdots - 3520 $ x^20 - 6x^19 - 26x^18 + 208x^17 + 185x^16 - 2910x^15 + 687x^14 + 21067x^13 - 16530x^12 - 83962x^11 + 93139x^10 + 180932x^9 - 242362x^8 - 194923x^7 + 303963x^6 + 96332x^5 - 176108x^4 - 24984x^3 + 44976x^2 + 3840*x - 3520
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$2.74725$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74725 q^{3} +3.77127 q^{5} -2.22761 q^{7} +4.54740 q^{9} +0.380193 q^{13} +10.3606 q^{15} +3.15750 q^{17} +1.00000 q^{19} -6.11981 q^{21} +7.90530 q^{23} +9.22250 q^{25} +4.25111 q^{27} -5.59221 q^{29} -3.96209 q^{31} -8.40093 q^{35} +3.34935 q^{37} +1.04449 q^{39} +0.889784 q^{41} +8.46165 q^{43} +17.1495 q^{45} -3.67335 q^{47} -2.03775 q^{49} +8.67445 q^{51} -3.93524 q^{53} +2.74725 q^{57} +11.0570 q^{59} -3.65104 q^{61} -10.1298 q^{63} +1.43381 q^{65} +6.20406 q^{67} +21.7179 q^{69} +0.936963 q^{71} +6.15755 q^{73} +25.3366 q^{75} -16.9962 q^{79} -1.96333 q^{81} +13.2446 q^{83} +11.9078 q^{85} -15.3632 q^{87} -9.43253 q^{89} -0.846923 q^{91} -10.8849 q^{93} +3.77127 q^{95} -0.252982 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{3} + 2 q^{5} - 4 q^{7} + 28 q^{9} + 5 q^{13} + 14 q^{15} + 6 q^{17} + 20 q^{19} + q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} - q^{29} + 23 q^{31} + 36 q^{35} + q^{37} - 21 q^{39} - 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) $ 20 * q + 6 * q^3 + 2 * q^5 - 4 * q^7 + 28 * q^9 + 5 * q^13 + 14 * q^15 + 6 * q^17 + 20 * q^19 + q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 - q^29 + 23 * q^31 + 36 * q^35 + q^37 - 21 * q^39 - 6 * q^41 + 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 - 14 * q^51 - 10 * q^53 + 6 * q^57 + 52 * q^59 + 2 * q^61 - 59 * q^63 - 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 + 13 * q^73 + 15 * q^75 + 2 * q^79 + 48 * q^81 + 2 * q^83 - 11 * q^85 - 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 + 2 * q^95 + 9 * q^97

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
$3$	2.74725	1.58613	0.793064	−	0.609138i	$-0.208485\pi$
$3$	2.74725	1.58613	0.793064	+	0.609138i	$0.208485\pi$
$4$	0	0
$5$	3.77127	1.68656	0.843282	−	0.537471i	$-0.180620\pi$
$5$	3.77127	1.68656	0.843282	+	0.537471i	$0.180620\pi$
$6$	0	0
$7$	−2.22761	−0.841958	−0.420979	−	0.907070i	$-0.638313\pi$
$7$	−2.22761	−0.841958	−0.420979	+	0.907070i	$0.638313\pi$
$8$	0	0
$9$	4.54740	1.51580
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.380193	0.105447	0.0527233	−	0.998609i	$-0.483210\pi$
$13$	0.380193	0.105447	0.0527233	+	0.998609i	$0.483210\pi$
$14$	0	0
$15$	10.3606	2.67511
$16$	0	0
$17$	3.15750	0.765806	0.382903	−	0.923789i	$-0.374924\pi$
$17$	3.15750	0.765806	0.382903	+	0.923789i	$0.374924\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.11981	−1.33545
$22$	0	0
$23$	7.90530	1.64837	0.824185	−	0.566321i	$-0.191634\pi$
$23$	7.90530	1.64837	0.824185	+	0.566321i	$0.191634\pi$
$24$	0	0
$25$	9.22250	1.84450
$26$	0	0
$27$	4.25111	0.818127
$28$	0	0
$29$	−5.59221	−1.03845	−0.519223	−	0.854639i	$-0.673779\pi$
$29$	−5.59221	−1.03845	−0.519223	+	0.854639i	$0.673779\pi$
$30$	0	0
$31$	−3.96209	−0.711613	−0.355807	−	0.934560i	$-0.615794\pi$
$31$	−3.96209	−0.711613	−0.355807	+	0.934560i	$0.615794\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.40093	−1.42002
$36$	0	0
$37$	3.34935	0.550630	0.275315	−	0.961354i	$-0.411218\pi$
$37$	3.34935	0.550630	0.275315	+	0.961354i	$0.411218\pi$
$38$	0	0
$39$	1.04449	0.167252
$40$	0	0
$41$	0.889784	0.138961	0.0694804	−	0.997583i	$-0.477866\pi$
$41$	0.889784	0.138961	0.0694804	+	0.997583i	$0.477866\pi$
$42$	0	0
$43$	8.46165	1.29039	0.645195	−	0.764018i	$-0.276776\pi$
$43$	8.46165	1.29039	0.645195	+	0.764018i	$0.276776\pi$
$44$	0	0
$45$	17.1495	2.55650
$46$	0	0
$47$	−3.67335	−0.535814	−0.267907	−	0.963445i	$-0.586332\pi$
$47$	−3.67335	−0.535814	−0.267907	+	0.963445i	$0.586332\pi$
$48$	0	0
$49$	−2.03775	−0.291107
$50$	0	0
$51$	8.67445	1.21467
$52$	0	0
$53$	−3.93524	−0.540546	−0.270273	−	0.962784i	$-0.587114\pi$
$53$	−3.93524	−0.540546	−0.270273	+	0.962784i	$0.587114\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.74725	0.363883
$58$	0	0
$59$	11.0570	1.43950	0.719749	−	0.694234i	$-0.244257\pi$
$59$	11.0570	1.43950	0.719749	+	0.694234i	$0.244257\pi$
$60$	0	0
$61$	−3.65104	−0.467468	−0.233734	−	0.972301i	$-0.575095\pi$
$61$	−3.65104	−0.467468	−0.233734	+	0.972301i	$0.575095\pi$
$62$	0	0
$63$	−10.1298	−1.27624
$64$	0	0
$65$	1.43381	0.177843
$66$	0	0
$67$	6.20406	0.757947	0.378974	−	0.925408i	$-0.376277\pi$
$67$	6.20406	0.757947	0.378974	+	0.925408i	$0.376277\pi$
$68$	0	0
$69$	21.7179	2.61453
$70$	0	0
$71$	0.936963	0.111197	0.0555985	−	0.998453i	$-0.482293\pi$
$71$	0.936963	0.111197	0.0555985	+	0.998453i	$0.482293\pi$
$72$	0	0
$73$	6.15755	0.720687	0.360344	−	0.932820i	$-0.382659\pi$
$73$	6.15755	0.720687	0.360344	+	0.932820i	$0.382659\pi$
$74$	0	0
$75$	25.3366	2.92561
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.9962	−1.91222	−0.956112	−	0.293002i	$-0.905346\pi$
$79$	−16.9962	−1.91222	−0.956112	+	0.293002i	$0.905346\pi$
$80$	0	0
$81$	−1.96333	−0.218147
$82$	0	0
$83$	13.2446	1.45378	0.726890	−	0.686754i	$-0.240965\pi$
$83$	13.2446	1.45378	0.726890	+	0.686754i	$0.240965\pi$
$84$	0	0
$85$	11.9078	1.29158
$86$	0	0
$87$	−15.3632	−1.64711
$88$	0	0
$89$	−9.43253	−0.999846	−0.499923	−	0.866070i	$-0.666638\pi$
$89$	−9.43253	−0.999846	−0.499923	+	0.866070i	$0.666638\pi$
$90$	0	0
$91$	−0.846923	−0.0887816
$92$	0	0
$93$	−10.8849	−1.12871
$94$	0	0
$95$	3.77127	0.386924
$96$	0	0
$97$	−0.252982	−0.0256864	−0.0128432	−	0.999918i	$-0.504088\pi$
$97$	−0.252982	−0.0256864	−0.0128432	+	0.999918i	$0.504088\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.5548	1.64727	0.823633	−	0.567123i	$-0.191943\pi$
$101$	16.5548	1.64727	0.823633	+	0.567123i	$0.191943\pi$
$102$	0	0
$103$	−17.0948	−1.68440	−0.842198	−	0.539168i	$-0.818739\pi$
$103$	−17.0948	−1.68440	−0.842198	+	0.539168i	$0.818739\pi$
$104$	0	0
$105$	−23.0795	−2.25233
$106$	0	0
$107$	19.7912	1.91329	0.956646	−	0.291255i	$-0.0940727\pi$
$107$	19.7912	1.91329	0.956646	+	0.291255i	$0.0940727\pi$
$108$	0	0
$109$	−8.35139	−0.799919	−0.399959	−	0.916533i	$-0.630976\pi$
$109$	−8.35139	−0.799919	−0.399959	+	0.916533i	$0.630976\pi$
$110$	0	0
$111$	9.20151	0.873369
$112$	0	0
$113$	−6.31426	−0.593996	−0.296998	−	0.954878i	$-0.595985\pi$
$113$	−6.31426	−0.593996	−0.296998	+	0.954878i	$0.595985\pi$
$114$	0	0
$115$	29.8131	2.78008
$116$	0	0
$117$	1.72889	0.159836
$118$	0	0
$119$	−7.03368	−0.644776
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2.44446	0.220410
$124$	0	0
$125$	15.9242	1.42430
$126$	0	0
$127$	−5.10778	−0.453242	−0.226621	−	0.973983i	$-0.572768\pi$
$127$	−5.10778	−0.453242	−0.226621	+	0.973983i	$0.572768\pi$
$128$	0	0
$129$	23.2463	2.04672
$130$	0	0
$131$	−20.3046	−1.77402	−0.887010	−	0.461750i	$-0.847222\pi$
$131$	−20.3046	−1.77402	−0.887010	+	0.461750i	$0.847222\pi$
$132$	0	0
$133$	−2.22761	−0.193158
$134$	0	0
$135$	16.0321	1.37982
$136$	0	0
$137$	2.04324	0.174566	0.0872829	−	0.996184i	$-0.472182\pi$
$137$	2.04324	0.174566	0.0872829	+	0.996184i	$0.472182\pi$
$138$	0	0
$139$	4.95300	0.420108	0.210054	−	0.977690i	$-0.432636\pi$
$139$	4.95300	0.420108	0.210054	+	0.977690i	$0.432636\pi$
$140$	0	0
$141$	−10.0916	−0.849869
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−21.0897	−1.75141
$146$	0	0
$147$	−5.59822	−0.461734
$148$	0	0
$149$	−19.2738	−1.57897	−0.789484	−	0.613771i	$-0.789652\pi$
$149$	−19.2738	−1.57897	−0.789484	+	0.613771i	$0.789652\pi$
$150$	0	0
$151$	21.0512	1.71312	0.856560	−	0.516048i	$-0.172597\pi$
$151$	21.0512	1.71312	0.856560	+	0.516048i	$0.172597\pi$
$152$	0	0
$153$	14.3584	1.16081
$154$	0	0
$155$	−14.9421	−1.20018
$156$	0	0
$157$	18.2398	1.45569	0.727847	−	0.685739i	$-0.240521\pi$
$157$	18.2398	1.45569	0.727847	+	0.685739i	$0.240521\pi$
$158$	0	0
$159$	−10.8111	−0.857376
$160$	0	0
$161$	−17.6099	−1.38786
$162$	0	0
$163$	−13.9246	−1.09066	−0.545328	−	0.838223i	$-0.683595\pi$
$163$	−13.9246	−1.09066	−0.545328	+	0.838223i	$0.683595\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.95687	−0.460956	−0.230478	−	0.973078i	$-0.574029\pi$
$167$	−5.95687	−0.460956	−0.230478	+	0.973078i	$0.574029\pi$
$168$	0	0
$169$	−12.8555	−0.988881
$170$	0	0
$171$	4.54740	0.347749
$172$	0	0
$173$	−6.66190	−0.506495	−0.253247	−	0.967402i	$-0.581499\pi$
$173$	−6.66190	−0.506495	−0.253247	+	0.967402i	$0.581499\pi$
$174$	0	0
$175$	−20.5441	−1.55299
$176$	0	0
$177$	30.3764	2.28323
$178$	0	0
$179$	6.54563	0.489243	0.244622	−	0.969619i	$-0.421336\pi$
$179$	6.54563	0.489243	0.244622	+	0.969619i	$0.421336\pi$
$180$	0	0
$181$	15.5603	1.15659	0.578295	−	0.815828i	$-0.303718\pi$
$181$	15.5603	1.15659	0.578295	+	0.815828i	$0.303718\pi$
$182$	0	0
$183$	−10.0303	−0.741465
$184$	0	0
$185$	12.6313	0.928672
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−9.46982	−0.688828
$190$	0	0
$191$	−12.8509	−0.929861	−0.464930	−	0.885347i	$-0.653921\pi$
$191$	−12.8509	−0.929861	−0.464930	+	0.885347i	$0.653921\pi$
$192$	0	0
$193$	6.48057	0.466482	0.233241	−	0.972419i	$-0.425067\pi$
$193$	6.48057	0.466482	0.233241	+	0.972419i	$0.425067\pi$
$194$	0	0
$195$	3.93905	0.282081
$196$	0	0
$197$	−1.15043	−0.0819645	−0.0409823	−	0.999160i	$-0.513049\pi$
$197$	−1.15043	−0.0819645	−0.0409823	+	0.999160i	$0.513049\pi$
$198$	0	0
$199$	14.3549	1.01759	0.508795	−	0.860888i	$-0.330091\pi$
$199$	14.3549	1.01759	0.508795	+	0.860888i	$0.330091\pi$
$200$	0	0
$201$	17.0441	1.20220
$202$	0	0
$203$	12.4573	0.874328
$204$	0	0
$205$	3.35562	0.234366
$206$	0	0
$207$	35.9486	2.49860
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−23.6599	−1.62881	−0.814406	−	0.580295i	$-0.802937\pi$
$211$	−23.6599	−1.62881	−0.814406	+	0.580295i	$0.802937\pi$
$212$	0	0
$213$	2.57408	0.176373
$214$	0	0
$215$	31.9112	2.17633
$216$	0	0
$217$	8.82600	0.599148
$218$	0	0
$219$	16.9164	1.14310
$220$	0	0
$221$	1.20046	0.0807517
$222$	0	0
$223$	13.4628	0.901533	0.450767	−	0.892642i	$-0.351151\pi$
$223$	13.4628	0.901533	0.450767	+	0.892642i	$0.351151\pi$
$224$	0	0
$225$	41.9384	2.79590
$226$	0	0
$227$	12.1878	0.808930	0.404465	−	0.914554i	$-0.367458\pi$
$227$	12.1878	0.808930	0.404465	+	0.914554i	$0.367458\pi$
$228$	0	0
$229$	17.1276	1.13183	0.565913	−	0.824465i	$-0.308524\pi$
$229$	17.1276	1.13183	0.565913	+	0.824465i	$0.308524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.43009	0.421249	0.210625	−	0.977567i	$-0.432450\pi$
$233$	6.43009	0.421249	0.210625	+	0.977567i	$0.432450\pi$
$234$	0	0
$235$	−13.8532	−0.903684
$236$	0	0
$237$	−46.6929	−3.03303
$238$	0	0
$239$	1.20787	0.0781309	0.0390654	−	0.999237i	$-0.487562\pi$
$239$	1.20787	0.0781309	0.0390654	+	0.999237i	$0.487562\pi$
$240$	0	0
$241$	−9.68622	−0.623945	−0.311972	−	0.950091i	$-0.600990\pi$
$241$	−9.68622	−0.623945	−0.311972	+	0.950091i	$0.600990\pi$
$242$	0	0
$243$	−18.1471	−1.16414
$244$	0	0
$245$	−7.68492	−0.490972
$246$	0	0
$247$	0.380193	0.0241911
$248$	0	0
$249$	36.3862	2.30588
$250$	0	0
$251$	29.4065	1.85612	0.928060	−	0.372430i	$-0.121475\pi$
$251$	29.4065	1.85612	0.928060	+	0.372430i	$0.121475\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	32.7137	2.04861
$256$	0	0
$257$	−27.7443	−1.73064	−0.865320	−	0.501221i	$-0.832885\pi$
$257$	−27.7443	−1.73064	−0.865320	+	0.501221i	$0.832885\pi$
$258$	0	0
$259$	−7.46104	−0.463607
$260$	0	0
$261$	−25.4300	−1.57408
$262$	0	0
$263$	7.54646	0.465335	0.232667	−	0.972556i	$-0.425255\pi$
$263$	7.54646	0.465335	0.232667	+	0.972556i	$0.425255\pi$
$264$	0	0
$265$	−14.8409	−0.911666
$266$	0	0
$267$	−25.9136	−1.58588
$268$	0	0
$269$	−13.0867	−0.797910	−0.398955	−	0.916970i	$-0.630627\pi$
$269$	−13.0867	−0.797910	−0.398955	+	0.916970i	$0.630627\pi$
$270$	0	0
$271$	2.09430	0.127220	0.0636099	−	0.997975i	$-0.479739\pi$
$271$	2.09430	0.127220	0.0636099	+	0.997975i	$0.479739\pi$
$272$	0	0
$273$	−2.32671	−0.140819
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.6426	0.939871	0.469935	−	0.882701i	$-0.344277\pi$
$277$	15.6426	0.939871	0.469935	+	0.882701i	$0.344277\pi$
$278$	0	0
$279$	−18.0172	−1.07866
$280$	0	0
$281$	2.89178	0.172509	0.0862545	−	0.996273i	$-0.472510\pi$
$281$	2.89178	0.172509	0.0862545	+	0.996273i	$0.472510\pi$
$282$	0	0
$283$	12.1380	0.721531	0.360765	−	0.932657i	$-0.382515\pi$
$283$	12.1380	0.721531	0.360765	+	0.932657i	$0.382515\pi$
$284$	0	0
$285$	10.3606	0.613712
$286$	0	0
$287$	−1.98209	−0.116999
$288$	0	0
$289$	−7.03020	−0.413541
$290$	0	0
$291$	−0.695005	−0.0407419
$292$	0	0
$293$	−1.89805	−0.110885	−0.0554426	−	0.998462i	$-0.517657\pi$
$293$	−1.89805	−0.110885	−0.0554426	+	0.998462i	$0.517657\pi$
$294$	0	0
$295$	41.6990	2.42781
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.00554	0.173815
$300$	0	0
$301$	−18.8493	−1.08645
$302$	0	0
$303$	45.4803	2.61278
$304$	0	0
$305$	−13.7691	−0.788416
$306$	0	0
$307$	26.1717	1.49370	0.746850	−	0.664992i	$-0.231565\pi$
$307$	26.1717	1.49370	0.746850	+	0.664992i	$0.231565\pi$
$308$	0	0
$309$	−46.9636	−2.67167
$310$	0	0
$311$	28.1884	1.59842	0.799210	−	0.601052i	$-0.205252\pi$
$311$	28.1884	1.59842	0.799210	+	0.601052i	$0.205252\pi$
$312$	0	0
$313$	−12.0996	−0.683912	−0.341956	−	0.939716i	$-0.611089\pi$
$313$	−12.0996	−0.683912	−0.341956	+	0.939716i	$0.611089\pi$
$314$	0	0
$315$	−38.2024	−2.15246
$316$	0	0
$317$	−27.8562	−1.56456	−0.782280	−	0.622928i	$-0.785943\pi$
$317$	−27.8562	−1.56456	−0.782280	+	0.622928i	$0.785943\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	54.3716	3.03472
$322$	0	0
$323$	3.15750	0.175688
$324$	0	0
$325$	3.50633	0.194496
$326$	0	0
$327$	−22.9434	−1.26877
$328$	0	0
$329$	8.18280	0.451132
$330$	0	0
$331$	−6.91534	−0.380102	−0.190051	−	0.981774i	$-0.560865\pi$
$331$	−6.91534	−0.380102	−0.190051	+	0.981774i	$0.560865\pi$
$332$	0	0
$333$	15.2308	0.834645
$334$	0	0
$335$	23.3972	1.27833
$336$	0	0
$337$	3.06186	0.166790	0.0833950	−	0.996517i	$-0.473424\pi$
$337$	3.06186	0.166790	0.0833950	+	0.996517i	$0.473424\pi$
$338$	0	0
$339$	−17.3469	−0.942154
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.1326	1.08706
$344$	0	0
$345$	81.9040	4.40957
$346$	0	0
$347$	7.03695	0.377763	0.188882	−	0.982000i	$-0.439514\pi$
$347$	7.03695	0.377763	0.188882	+	0.982000i	$0.439514\pi$
$348$	0	0
$349$	−24.4280	−1.30760	−0.653801	−	0.756667i	$-0.726827\pi$
$349$	−24.4280	−1.30760	−0.653801	+	0.756667i	$0.726827\pi$
$350$	0	0
$351$	1.61625	0.0862688
$352$	0	0
$353$	−13.3929	−0.712829	−0.356415	−	0.934328i	$-0.616001\pi$
$353$	−13.3929	−0.712829	−0.356415	+	0.934328i	$0.616001\pi$
$354$	0	0
$355$	3.53354	0.187541
$356$	0	0
$357$	−19.3233	−1.02270
$358$	0	0
$359$	−27.7064	−1.46229	−0.731143	−	0.682224i	$-0.761013\pi$
$359$	−27.7064	−1.46229	−0.731143	+	0.682224i	$0.761013\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	23.2218	1.21549
$366$	0	0
$367$	1.46967	0.0767160	0.0383580	−	0.999264i	$-0.487787\pi$
$367$	1.46967	0.0767160	0.0383580	+	0.999264i	$0.487787\pi$
$368$	0	0
$369$	4.04621	0.210637
$370$	0	0
$371$	8.76617	0.455117
$372$	0	0
$373$	−18.5880	−0.962449	−0.481224	−	0.876597i	$-0.659808\pi$
$373$	−18.5880	−0.962449	−0.481224	+	0.876597i	$0.659808\pi$
$374$	0	0
$375$	43.7478	2.25913
$376$	0	0
$377$	−2.12612	−0.109501
$378$	0	0
$379$	−0.217768	−0.0111860	−0.00559301	−	0.999984i	$-0.501780\pi$
$379$	−0.217768	−0.0111860	−0.00559301	+	0.999984i	$0.501780\pi$
$380$	0	0
$381$	−14.0324	−0.718900
$382$	0	0
$383$	25.0035	1.27762	0.638809	−	0.769365i	$-0.279427\pi$
$383$	25.0035	1.27762	0.638809	+	0.769365i	$0.279427\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	38.4785	1.95597
$388$	0	0
$389$	−14.5749	−0.738979	−0.369489	−	0.929235i	$-0.620467\pi$
$389$	−14.5749	−0.738979	−0.369489	+	0.929235i	$0.620467\pi$
$390$	0	0
$391$	24.9610	1.26233
$392$	0	0
$393$	−55.7819	−2.81382
$394$	0	0
$395$	−64.0974	−3.22509
$396$	0	0
$397$	15.6107	0.783480	0.391740	−	0.920076i	$-0.371873\pi$
$397$	15.6107	0.783480	0.391740	+	0.920076i	$0.371873\pi$
$398$	0	0
$399$	−6.11981	−0.306374
$400$	0	0
$401$	−33.3928	−1.66756	−0.833778	−	0.552100i	$-0.813827\pi$
$401$	−33.3928	−1.66756	−0.833778	+	0.552100i	$0.813827\pi$
$402$	0	0
$403$	−1.50636	−0.0750372
$404$	0	0
$405$	−7.40424	−0.367920
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.9290	−0.738191	−0.369096	−	0.929391i	$-0.620333\pi$
$409$	−14.9290	−0.738191	−0.369096	+	0.929391i	$0.620333\pi$
$410$	0	0
$411$	5.61330	0.276884
$412$	0	0
$413$	−24.6307	−1.21200
$414$	0	0
$415$	49.9489	2.45190
$416$	0	0
$417$	13.6072	0.666345
$418$	0	0
$419$	−20.5457	−1.00372	−0.501861	−	0.864948i	$-0.667351\pi$
$419$	−20.5457	−1.00372	−0.501861	+	0.864948i	$0.667351\pi$
$420$	0	0
$421$	7.23866	0.352791	0.176395	−	0.984319i	$-0.443556\pi$
$421$	7.23866	0.352791	0.176395	+	0.984319i	$0.443556\pi$
$422$	0	0
$423$	−16.7042	−0.812187
$424$	0	0
$425$	29.1200	1.41253
$426$	0	0
$427$	8.13311	0.393589
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.4013	−1.80156	−0.900779	−	0.434279i	$-0.857003\pi$
$431$	−37.4013	−1.80156	−0.900779	+	0.434279i	$0.857003\pi$
$432$	0	0
$433$	25.8222	1.24094	0.620468	−	0.784232i	$-0.286943\pi$
$433$	25.8222	1.24094	0.620468	+	0.784232i	$0.286943\pi$
$434$	0	0
$435$	−57.9389	−2.77796
$436$	0	0
$437$	7.90530	0.378162
$438$	0	0
$439$	28.8669	1.37774	0.688871	−	0.724884i	$-0.258107\pi$
$439$	28.8669	1.37774	0.688871	+	0.724884i	$0.258107\pi$
$440$	0	0
$441$	−9.26648	−0.441261
$442$	0	0
$443$	−8.87429	−0.421630	−0.210815	−	0.977526i	$-0.567612\pi$
$443$	−8.87429	−0.421630	−0.210815	+	0.977526i	$0.567612\pi$
$444$	0	0
$445$	−35.5726	−1.68630
$446$	0	0
$447$	−52.9499	−2.50444
$448$	0	0
$449$	−28.0333	−1.32297	−0.661486	−	0.749957i	$-0.730074\pi$
$449$	−28.0333	−1.32297	−0.661486	+	0.749957i	$0.730074\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	57.8329	2.71723
$454$	0	0
$455$	−3.19398	−0.149736
$456$	0	0
$457$	17.0762	0.798792	0.399396	−	0.916778i	$-0.369220\pi$
$457$	17.0762	0.798792	0.399396	+	0.916778i	$0.369220\pi$
$458$	0	0
$459$	13.4229	0.626527
$460$	0	0
$461$	7.77013	0.361891	0.180946	−	0.983493i	$-0.442084\pi$
$461$	7.77013	0.361891	0.180946	+	0.983493i	$0.442084\pi$
$462$	0	0
$463$	−14.2038	−0.660105	−0.330052	−	0.943963i	$-0.607066\pi$
$463$	−14.2038	−0.660105	−0.330052	+	0.943963i	$0.607066\pi$
$464$	0	0
$465$	−41.0498	−1.90364
$466$	0	0
$467$	39.6586	1.83518	0.917589	−	0.397529i	$-0.130132\pi$
$467$	39.6586	1.83518	0.917589	+	0.397529i	$0.130132\pi$
$468$	0	0
$469$	−13.8202	−0.638159
$470$	0	0
$471$	50.1094	2.30892
$472$	0	0
$473$	0	0
$474$	0	0
$475$	9.22250	0.423157
$476$	0	0
$477$	−17.8951	−0.819361
$478$	0	0
$479$	−28.6844	−1.31062	−0.655311	−	0.755359i	$-0.727462\pi$
$479$	−28.6844	−1.31062	−0.655311	+	0.755359i	$0.727462\pi$
$480$	0	0
$481$	1.27340	0.0580621
$482$	0	0
$483$	−48.3790	−2.20132
$484$	0	0
$485$	−0.954063	−0.0433217
$486$	0	0
$487$	−6.30786	−0.285837	−0.142918	−	0.989734i	$-0.545649\pi$
$487$	−6.30786	−0.285837	−0.142918	+	0.989734i	$0.545649\pi$
$488$	0	0
$489$	−38.2543	−1.72992
$490$	0	0
$491$	17.8721	0.806558	0.403279	−	0.915077i	$-0.367870\pi$
$491$	17.8721	0.806558	0.403279	+	0.915077i	$0.367870\pi$
$492$	0	0
$493$	−17.6574	−0.795249
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.08719	−0.0936232
$498$	0	0
$499$	−30.9538	−1.38568	−0.692841	−	0.721090i	$-0.743641\pi$
$499$	−30.9538	−1.38568	−0.692841	+	0.721090i	$0.743641\pi$
$500$	0	0
$501$	−16.3650	−0.731136
$502$	0	0
$503$	−21.8703	−0.975150	−0.487575	−	0.873081i	$-0.662119\pi$
$503$	−21.8703	−0.975150	−0.487575	+	0.873081i	$0.662119\pi$
$504$	0	0
$505$	62.4328	2.77822
$506$	0	0
$507$	−35.3172	−1.56849
$508$	0	0
$509$	−5.24221	−0.232357	−0.116178	−	0.993228i	$-0.537064\pi$
$509$	−5.24221	−0.232357	−0.116178	+	0.993228i	$0.537064\pi$
$510$	0	0
$511$	−13.7166	−0.606788
$512$	0	0
$513$	4.25111	0.187691
$514$	0	0
$515$	−64.4690	−2.84084
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.3019	−0.803365
$520$	0	0
$521$	−2.31697	−0.101508	−0.0507542	−	0.998711i	$-0.516163\pi$
$521$	−2.31697	−0.101508	−0.0507542	+	0.998711i	$0.516163\pi$
$522$	0	0
$523$	−28.4269	−1.24302	−0.621510	−	0.783406i	$-0.713480\pi$
$523$	−28.4269	−1.24302	−0.621510	+	0.783406i	$0.713480\pi$
$524$	0	0
$525$	−56.4400	−2.46324
$526$	0	0
$527$	−12.5103	−0.544958
$528$	0	0
$529$	39.4938	1.71712
$530$	0	0
$531$	50.2807	2.18199
$532$	0	0
$533$	0.338290	0.0146530
$534$	0	0
$535$	74.6382	3.22689
$536$	0	0
$537$	17.9825	0.776002
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.26206	0.0972538	0.0486269	−	0.998817i	$-0.484515\pi$
$541$	2.26206	0.0972538	0.0486269	+	0.998817i	$0.484515\pi$
$542$	0	0
$543$	42.7482	1.83450
$544$	0	0
$545$	−31.4954	−1.34911
$546$	0	0
$547$	18.2942	0.782205	0.391103	−	0.920347i	$-0.372094\pi$
$547$	18.2942	0.782205	0.391103	+	0.920347i	$0.372094\pi$
$548$	0	0
$549$	−16.6028	−0.708589
$550$	0	0
$551$	−5.59221	−0.238236
$552$	0	0
$553$	37.8609	1.61001
$554$	0	0
$555$	34.7014	1.47299
$556$	0	0
$557$	32.1716	1.36316	0.681578	−	0.731745i	$-0.261294\pi$
$557$	32.1716	1.36316	0.681578	+	0.731745i	$0.261294\pi$
$558$	0	0
$559$	3.21706	0.136067
$560$	0	0
$561$	0	0
$562$	0	0
$563$	40.1499	1.69212	0.846059	−	0.533089i	$-0.178969\pi$
$563$	40.1499	1.69212	0.846059	+	0.533089i	$0.178969\pi$
$564$	0	0
$565$	−23.8128	−1.00181
$566$	0	0
$567$	4.37352	0.183671
$568$	0	0
$569$	−40.5362	−1.69937	−0.849684	−	0.527292i	$-0.823207\pi$
$569$	−40.5362	−1.69937	−0.849684	+	0.527292i	$0.823207\pi$
$570$	0	0
$571$	5.14050	0.215123	0.107562	−	0.994198i	$-0.465696\pi$
$571$	5.14050	0.215123	0.107562	+	0.994198i	$0.465696\pi$
$572$	0	0
$573$	−35.3048	−1.47488
$574$	0	0
$575$	72.9067	3.04042
$576$	0	0
$577$	−19.5325	−0.813150	−0.406575	−	0.913618i	$-0.633277\pi$
$577$	−19.5325	−0.813150	−0.406575	+	0.913618i	$0.633277\pi$
$578$	0	0
$579$	17.8038	0.739900
$580$	0	0
$581$	−29.5037	−1.22402
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.52013	0.269574
$586$	0	0
$587$	1.68283	0.0694579	0.0347290	−	0.999397i	$-0.488943\pi$
$587$	1.68283	0.0694579	0.0347290	+	0.999397i	$0.488943\pi$
$588$	0	0
$589$	−3.96209	−0.163255
$590$	0	0
$591$	−3.16051	−0.130006
$592$	0	0
$593$	−40.6024	−1.66734	−0.833670	−	0.552263i	$-0.813765\pi$
$593$	−40.6024	−1.66734	−0.833670	+	0.552263i	$0.813765\pi$
$594$	0	0
$595$	−26.5259	−1.08746
$596$	0	0
$597$	39.4365	1.61403
$598$	0	0
$599$	−36.1681	−1.47779	−0.738893	−	0.673823i	$-0.764651\pi$
$599$	−36.1681	−1.47779	−0.738893	+	0.673823i	$0.764651\pi$
$600$	0	0
$601$	−11.6379	−0.474720	−0.237360	−	0.971422i	$-0.576282\pi$
$601$	−11.6379	−0.474720	−0.237360	+	0.971422i	$0.576282\pi$
$602$	0	0
$603$	28.2124	1.14890
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.5468	0.955734	0.477867	−	0.878432i	$-0.341410\pi$
$607$	23.5468	0.955734	0.477867	+	0.878432i	$0.341410\pi$
$608$	0	0
$609$	34.2232	1.38680
$610$	0	0
$611$	−1.39658	−0.0564998
$612$	0	0
$613$	29.5092	1.19187	0.595933	−	0.803034i	$-0.296783\pi$
$613$	29.5092	1.19187	0.595933	+	0.803034i	$0.296783\pi$
$614$	0	0
$615$	9.21873	0.371735
$616$	0	0
$617$	34.3045	1.38105	0.690524	−	0.723309i	$-0.257380\pi$
$617$	34.3045	1.38105	0.690524	+	0.723309i	$0.257380\pi$
$618$	0	0
$619$	19.1956	0.771536	0.385768	−	0.922596i	$-0.373937\pi$
$619$	19.1956	0.771536	0.385768	+	0.922596i	$0.373937\pi$
$620$	0	0
$621$	33.6063	1.34858
$622$	0	0
$623$	21.0120	0.841828
$624$	0	0
$625$	13.9420	0.557681
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.5756	0.421675
$630$	0	0
$631$	−20.5612	−0.818528	−0.409264	−	0.912416i	$-0.634215\pi$
$631$	−20.5612	−0.818528	−0.409264	+	0.912416i	$0.634215\pi$
$632$	0	0
$633$	−64.9997	−2.58351
$634$	0	0
$635$	−19.2628	−0.764422
$636$	0	0
$637$	−0.774740	−0.0306963
$638$	0	0
$639$	4.26075	0.168553
$640$	0	0
$641$	−23.7304	−0.937296	−0.468648	−	0.883385i	$-0.655259\pi$
$641$	−23.7304	−0.937296	−0.468648	+	0.883385i	$0.655259\pi$
$642$	0	0
$643$	−27.8525	−1.09840	−0.549198	−	0.835692i	$-0.685067\pi$
$643$	−27.8525	−1.09840	−0.549198	+	0.835692i	$0.685067\pi$
$644$	0	0
$645$	87.6682	3.45193
$646$	0	0
$647$	−45.4458	−1.78666	−0.893329	−	0.449402i	$-0.851637\pi$
$647$	−45.4458	−1.78666	−0.893329	+	0.449402i	$0.851637\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	24.2473	0.950325
$652$	0	0
$653$	−5.62046	−0.219946	−0.109973	−	0.993935i	$-0.535076\pi$
$653$	−5.62046	−0.219946	−0.109973	+	0.993935i	$0.535076\pi$
$654$	0	0
$655$	−76.5741	−2.99200
$656$	0	0
$657$	28.0009	1.09242
$658$	0	0
$659$	−22.0625	−0.859433	−0.429716	−	0.902964i	$-0.641386\pi$
$659$	−22.0625	−0.859433	−0.429716	+	0.902964i	$0.641386\pi$
$660$	0	0
$661$	−22.6999	−0.882925	−0.441463	−	0.897280i	$-0.645540\pi$
$661$	−22.6999	−0.882925	−0.441463	+	0.897280i	$0.645540\pi$
$662$	0	0
$663$	3.29797	0.128083
$664$	0	0
$665$	−8.40093	−0.325774
$666$	0	0
$667$	−44.2081	−1.71174
$668$	0	0
$669$	36.9856	1.42995
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.2053	−0.393386	−0.196693	−	0.980465i	$-0.563020\pi$
$673$	−10.2053	−0.393386	−0.196693	+	0.980465i	$0.563020\pi$
$674$	0	0
$675$	39.2059	1.50904
$676$	0	0
$677$	−44.7165	−1.71859	−0.859297	−	0.511477i	$-0.829099\pi$
$677$	−44.7165	−1.71859	−0.859297	+	0.511477i	$0.829099\pi$
$678$	0	0
$679$	0.563544	0.0216268
$680$	0	0
$681$	33.4829	1.28307
$682$	0	0
$683$	6.97812	0.267010	0.133505	−	0.991048i	$-0.457377\pi$
$683$	6.97812	0.267010	0.133505	+	0.991048i	$0.457377\pi$
$684$	0	0
$685$	7.70562	0.294416
$686$	0	0
$687$	47.0540	1.79522
$688$	0	0
$689$	−1.49615	−0.0569988
$690$	0	0
$691$	18.0354	0.686099	0.343050	−	0.939317i	$-0.388540\pi$
$691$	18.0354	0.686099	0.343050	+	0.939317i	$0.388540\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.6791	0.708539
$696$	0	0
$697$	2.80949	0.106417
$698$	0	0
$699$	17.6651	0.668155
$700$	0	0
$701$	27.0762	1.02265	0.511326	−	0.859387i	$-0.329154\pi$
$701$	27.0762	1.02265	0.511326	+	0.859387i	$0.329154\pi$
$702$	0	0
$703$	3.34935	0.126323
$704$	0	0
$705$	−38.0583	−1.43336
$706$	0	0
$707$	−36.8777	−1.38693
$708$	0	0
$709$	−11.7101	−0.439784	−0.219892	−	0.975524i	$-0.570570\pi$
$709$	−11.7101	−0.439784	−0.219892	+	0.975524i	$0.570570\pi$
$710$	0	0
$711$	−77.2887	−2.89855
$712$	0	0
$713$	−31.3216	−1.17300
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.31834	0.123926
$718$	0	0
$719$	−9.47250	−0.353265	−0.176632	−	0.984277i	$-0.556520\pi$
$719$	−9.47250	−0.353265	−0.176632	+	0.984277i	$0.556520\pi$
$720$	0	0
$721$	38.0804	1.41819
$722$	0	0
$723$	−26.6105	−0.989656
$724$	0	0
$725$	−51.5741	−1.91541
$726$	0	0
$727$	11.9864	0.444552	0.222276	−	0.974984i	$-0.428651\pi$
$727$	11.9864	0.444552	0.222276	+	0.974984i	$0.428651\pi$
$728$	0	0
$729$	−43.9647	−1.62832
$730$	0	0
$731$	26.7177	0.988188
$732$	0	0
$733$	15.7162	0.580492	0.290246	−	0.956952i	$-0.406263\pi$
$733$	15.7162	0.580492	0.290246	+	0.956952i	$0.406263\pi$
$734$	0	0
$735$	−21.1124	−0.778744
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−38.5663	−1.41868	−0.709342	−	0.704864i	$-0.751008\pi$
$739$	−38.5663	−1.41868	−0.709342	+	0.704864i	$0.751008\pi$
$740$	0	0
$741$	1.04449	0.0383702
$742$	0	0
$743$	−21.5759	−0.791544	−0.395772	−	0.918349i	$-0.629523\pi$
$743$	−21.5759	−0.791544	−0.395772	+	0.918349i	$0.629523\pi$
$744$	0	0
$745$	−72.6866	−2.66303
$746$	0	0
$747$	60.2284	2.20364
$748$	0	0
$749$	−44.0872	−1.61091
$750$	0	0
$751$	−34.8615	−1.27211	−0.636057	−	0.771642i	$-0.719436\pi$
$751$	−34.8615	−1.27211	−0.636057	+	0.771642i	$0.719436\pi$
$752$	0	0
$753$	80.7871	2.94404
$754$	0	0
$755$	79.3897	2.88929
$756$	0	0
$757$	19.3996	0.705092	0.352546	−	0.935795i	$-0.385316\pi$
$757$	19.3996	0.705092	0.352546	+	0.935795i	$0.385316\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.9535	−1.26706	−0.633531	−	0.773717i	$-0.718395\pi$
$761$	−34.9535	−1.26706	−0.633531	+	0.773717i	$0.718395\pi$
$762$	0	0
$763$	18.6037	0.673497
$764$	0	0
$765$	54.1496	1.95778
$766$	0	0
$767$	4.20380	0.151790
$768$	0	0
$769$	−4.34543	−0.156700	−0.0783500	−	0.996926i	$-0.524965\pi$
$769$	−4.34543	−0.156700	−0.0783500	+	0.996926i	$0.524965\pi$
$770$	0	0
$771$	−76.2205	−2.74501
$772$	0	0
$773$	−9.20741	−0.331168	−0.165584	−	0.986196i	$-0.552951\pi$
$773$	−9.20741	−0.331168	−0.165584	+	0.986196i	$0.552951\pi$
$774$	0	0
$775$	−36.5404	−1.31257
$776$	0	0
$777$	−20.4974	−0.735340
$778$	0	0
$779$	0.889784	0.0318798
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−23.7731	−0.849581
$784$	0	0
$785$	68.7873	2.45512
$786$	0	0
$787$	24.9743	0.890237	0.445118	−	0.895472i	$-0.353162\pi$
$787$	24.9743	0.890237	0.445118	+	0.895472i	$0.353162\pi$
$788$	0	0
$789$	20.7320	0.738080
$790$	0	0
$791$	14.0657	0.500119
$792$	0	0
$793$	−1.38810	−0.0492930
$794$	0	0
$795$	−40.7716	−1.44602
$796$	0	0
$797$	2.21063	0.0783046	0.0391523	−	0.999233i	$-0.487534\pi$
$797$	2.21063	0.0783046	0.0391523	+	0.999233i	$0.487534\pi$
$798$	0	0
$799$	−11.5986	−0.410329
$800$	0	0
$801$	−42.8935	−1.51557
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−66.4119	−2.34071
$806$	0	0
$807$	−35.9525	−1.26559
$808$	0	0
$809$	43.4993	1.52935	0.764677	−	0.644413i	$-0.222898\pi$
$809$	43.4993	1.52935	0.764677	+	0.644413i	$0.222898\pi$
$810$	0	0
$811$	−43.3329	−1.52162	−0.760811	−	0.648973i	$-0.775199\pi$
$811$	−43.3329	−1.52162	−0.760811	+	0.648973i	$0.775199\pi$
$812$	0	0
$813$	5.75358	0.201787
$814$	0	0
$815$	−52.5133	−1.83946
$816$	0	0
$817$	8.46165	0.296036
$818$	0	0
$819$	−3.85130	−0.134575
$820$	0	0
$821$	−28.8759	−1.00778	−0.503888	−	0.863769i	$-0.668098\pi$
$821$	−28.8759	−1.00778	−0.503888	+	0.863769i	$0.668098\pi$
$822$	0	0
$823$	13.3431	0.465113	0.232556	−	0.972583i	$-0.425291\pi$
$823$	13.3431	0.465113	0.232556	+	0.972583i	$0.425291\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.0201	1.53073	0.765365	−	0.643596i	$-0.222558\pi$
$827$	44.0201	1.53073	0.765365	+	0.643596i	$0.222558\pi$
$828$	0	0
$829$	−21.0098	−0.729700	−0.364850	−	0.931066i	$-0.618880\pi$
$829$	−21.0098	−0.729700	−0.364850	+	0.931066i	$0.618880\pi$
$830$	0	0
$831$	42.9741	1.49075
$832$	0	0
$833$	−6.43420	−0.222932
$834$	0	0
$835$	−22.4650	−0.777433
$836$	0	0
$837$	−16.8433	−0.582190
$838$	0	0
$839$	−37.6681	−1.30045	−0.650224	−	0.759742i	$-0.725325\pi$
$839$	−37.6681	−1.30045	−0.650224	+	0.759742i	$0.725325\pi$
$840$	0	0
$841$	2.27276	0.0783711
$842$	0	0
$843$	7.94444	0.273621
$844$	0	0
$845$	−48.4814	−1.66781
$846$	0	0
$847$	0	0
$848$	0	0
$849$	33.3462	1.14444
$850$	0	0
$851$	26.4776	0.907641
$852$	0	0
$853$	−11.1160	−0.380605	−0.190302	−	0.981726i	$-0.560947\pi$
$853$	−11.1160	−0.380605	−0.190302	+	0.981726i	$0.560947\pi$
$854$	0	0
$855$	17.1495	0.586501
$856$	0	0
$857$	−9.61907	−0.328581	−0.164291	−	0.986412i	$-0.552533\pi$
$857$	−9.61907	−0.328581	−0.164291	+	0.986412i	$0.552533\pi$
$858$	0	0
$859$	42.3020	1.44333	0.721663	−	0.692245i	$-0.243378\pi$
$859$	42.3020	1.44333	0.721663	+	0.692245i	$0.243378\pi$
$860$	0	0
$861$	−5.44531	−0.185576
$862$	0	0
$863$	45.8473	1.56066	0.780331	−	0.625367i	$-0.215051\pi$
$863$	45.8473	1.56066	0.780331	+	0.625367i	$0.215051\pi$
$864$	0	0
$865$	−25.1238	−0.854236
$866$	0	0
$867$	−19.3137	−0.655929
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.35874	0.0799230
$872$	0	0
$873$	−1.15041	−0.0389355
$874$	0	0
$875$	−35.4729	−1.19920
$876$	0	0
$877$	−10.5638	−0.356713	−0.178356	−	0.983966i	$-0.557078\pi$
$877$	−10.5638	−0.356713	−0.178356	+	0.983966i	$0.557078\pi$
$878$	0	0
$879$	−5.21442	−0.175878
$880$	0	0
$881$	2.16642	0.0729885	0.0364943	−	0.999334i	$-0.488381\pi$
$881$	2.16642	0.0729885	0.0364943	+	0.999334i	$0.488381\pi$
$882$	0	0
$883$	−26.4998	−0.891789	−0.445895	−	0.895085i	$-0.647114\pi$
$883$	−26.4998	−0.891789	−0.445895	+	0.895085i	$0.647114\pi$
$884$	0	0
$885$	114.558	3.85081
$886$	0	0
$887$	−1.35840	−0.0456105	−0.0228052	−	0.999740i	$-0.507260\pi$
$887$	−1.35840	−0.0456105	−0.0228052	+	0.999740i	$0.507260\pi$
$888$	0	0
$889$	11.3781	0.381610
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.67335	−0.122924
$894$	0	0
$895$	24.6853	0.825140
$896$	0	0
$897$	8.25699	0.275693
$898$	0	0
$899$	22.1568	0.738972
$900$	0	0
$901$	−12.4255	−0.413954
$902$	0	0
$903$	−51.7837	−1.72325
$904$	0	0
$905$	58.6822	1.95066
$906$	0	0
$907$	24.5948	0.816658	0.408329	−	0.912835i	$-0.366112\pi$
$907$	24.5948	0.816658	0.408329	+	0.912835i	$0.366112\pi$
$908$	0	0
$909$	75.2815	2.49693
$910$	0	0
$911$	9.05155	0.299891	0.149946	−	0.988694i	$-0.452090\pi$
$911$	9.05155	0.299891	0.149946	+	0.988694i	$0.452090\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−37.8272	−1.25053
$916$	0	0
$917$	45.2307	1.49365
$918$	0	0
$919$	−52.0002	−1.71533	−0.857665	−	0.514209i	$-0.828086\pi$
$919$	−52.0002	−1.71533	−0.857665	+	0.514209i	$0.828086\pi$
$920$	0	0
$921$	71.9004	2.36920
$922$	0	0
$923$	0.356227	0.0117254
$924$	0	0
$925$	30.8894	1.01564
$926$	0	0
$927$	−77.7368	−2.55321
$928$	0	0
$929$	−13.9116	−0.456424	−0.228212	−	0.973611i	$-0.573288\pi$
$929$	−13.9116	−0.456424	−0.228212	+	0.973611i	$0.573288\pi$
$930$	0	0
$931$	−2.03775	−0.0667846
$932$	0	0
$933$	77.4408	2.53530
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.9634	0.880857	0.440429	−	0.897788i	$-0.354827\pi$
$937$	26.9634	0.880857	0.440429	+	0.897788i	$0.354827\pi$
$938$	0	0
$939$	−33.2408	−1.08477
$940$	0	0
$941$	−4.72542	−0.154044	−0.0770221	−	0.997029i	$-0.524541\pi$
$941$	−4.72542	−0.154044	−0.0770221	+	0.997029i	$0.524541\pi$
$942$	0	0
$943$	7.03401	0.229059
$944$	0	0
$945$	−35.7133	−1.16175
$946$	0	0
$947$	5.90710	0.191955	0.0959774	−	0.995384i	$-0.469402\pi$
$947$	5.90710	0.191955	0.0959774	+	0.995384i	$0.469402\pi$
$948$	0	0
$949$	2.34106	0.0759941
$950$	0	0
$951$	−76.5280	−2.48159
$952$	0	0
$953$	−6.74458	−0.218478	−0.109239	−	0.994015i	$-0.534841\pi$
$953$	−6.74458	−0.218478	−0.109239	+	0.994015i	$0.534841\pi$
$954$	0	0
$955$	−48.4644	−1.56827
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.55154	−0.146977
$960$	0	0
$961$	−15.3018	−0.493607
$962$	0	0
$963$	89.9988	2.90017
$964$	0	0
$965$	24.4400	0.786752
$966$	0	0
$967$	−52.7615	−1.69670	−0.848348	−	0.529440i	$-0.822402\pi$
$967$	−52.7615	−1.69670	−0.848348	+	0.529440i	$0.822402\pi$
$968$	0	0
$969$	8.67445	0.278664
$970$	0	0
$971$	46.1658	1.48153	0.740766	−	0.671763i	$-0.234463\pi$
$971$	46.1658	1.48153	0.740766	+	0.671763i	$0.234463\pi$
$972$	0	0
$973$	−11.0334	−0.353713
$974$	0	0
$975$	9.63279	0.308496
$976$	0	0
$977$	49.4707	1.58271	0.791355	−	0.611357i	$-0.209376\pi$
$977$	49.4707	1.58271	0.791355	+	0.611357i	$0.209376\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−37.9772	−1.21252
$982$	0	0
$983$	−38.0001	−1.21201	−0.606007	−	0.795459i	$-0.707230\pi$
$983$	−38.0001	−1.21201	−0.606007	+	0.795459i	$0.707230\pi$
$984$	0	0
$985$	−4.33857	−0.138238
$986$	0	0
$987$	22.4802	0.715554
$988$	0	0
$989$	66.8919	2.12704
$990$	0	0
$991$	56.1034	1.78218	0.891091	−	0.453824i	$-0.149941\pi$
$991$	56.1034	1.78218	0.891091	+	0.453824i	$0.149941\pi$
$992$	0	0
$993$	−18.9982	−0.602890
$994$	0	0
$995$	54.1362	1.71623
$996$	0	0
$997$	34.8899	1.10498	0.552488	−	0.833521i	$-0.313679\pi$
$997$	34.8899	1.10498	0.552488	+	0.833521i	$0.313679\pi$
$998$	0	0
$999$	14.2385	0.450485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.w.1.17		20
11.2	odd	10		836.2.j.c.609.9	✓	40
11.6	odd	10		836.2.j.c.685.9	yes	40
11.10	odd	2		9196.2.a.x.1.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.c.609.9	✓	40	11.2	odd	10
836.2.j.c.685.9	yes	40	11.6	odd	10
9196.2.a.w.1.17		20	1.1	even	1	trivial
9196.2.a.x.1.17		20	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74725 q^{3} +3.77127 q^{5} -2.22761 q^{7} +4.54740 q^{9} +0.380193 q^{13} +10.3606 q^{15} +3.15750 q^{17} +1.00000 q^{19} -6.11981 q^{21} +7.90530 q^{23} +9.22250 q^{25} +4.25111 q^{27} -5.59221 q^{29} -3.96209 q^{31} -8.40093 q^{35} +3.34935 q^{37} +1.04449 q^{39} +0.889784 q^{41} +8.46165 q^{43} +17.1495 q^{45} -3.67335 q^{47} -2.03775 q^{49} +8.67445 q^{51} -3.93524 q^{53} +2.74725 q^{57} +11.0570 q^{59} -3.65104 q^{61} -10.1298 q^{63} +1.43381 q^{65} +6.20406 q^{67} +21.7179 q^{69} +0.936963 q^{71} +6.15755 q^{73} +25.3366 q^{75} -16.9962 q^{79} -1.96333 q^{81} +13.2446 q^{83} +11.9078 q^{85} -15.3632 q^{87} -9.43253 q^{89} -0.846923 q^{91} -10.8849 q^{93} +3.77127 q^{95} -0.252982 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 2 q^{5} - 4 q^{7} + 28 q^{9} + 5 q^{13} + 14 q^{15} + 6 q^{17} + 20 q^{19} + q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} - q^{29} + 23 q^{31} + 36 q^{35} + q^{37} - 21 q^{39} - 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) \) 20 * q + 6 * q^3 + 2 * q^5 - 4 * q^7 + 28 * q^9 + 5 * q^13 + 14 * q^15 + 6 * q^17 + 20 * q^19 + q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 - q^29 + 23 * q^31 + 36 * q^35 + q^37 - 21 * q^39 - 6 * q^41 + 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 - 14 * q^51 - 10 * q^53 + 6 * q^57 + 52 * q^59 + 2 * q^61 - 59 * q^63 - 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 + 13 * q^73 + 15 * q^75 + 2 * q^79 + 48 * q^81 + 2 * q^83 - 11 * q^85 - 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 + 2 * q^95 + 9 * q^97

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