LMFDB - Embedded newform 9196.2.a.x.1.8

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			1.8
Root			$-0.593598$ 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)

$f(q)$	$=$	$q-0.593598 q^{3} -3.09847 q^{5} +1.91364 q^{7} -2.64764 q^{9} -3.47441 q^{13} +1.83925 q^{15} -4.59072 q^{17} -1.00000 q^{19} -1.13593 q^{21} -6.59650 q^{23} +4.60052 q^{25} +3.35243 q^{27} -2.21269 q^{29} -1.65053 q^{31} -5.92936 q^{35} -8.86862 q^{37} +2.06241 q^{39} +12.2832 q^{41} +0.943326 q^{43} +8.20364 q^{45} -5.03322 q^{47} -3.33798 q^{49} +2.72504 q^{51} -9.93586 q^{53} +0.593598 q^{57} +13.0150 q^{59} -14.4026 q^{61} -5.06664 q^{63} +10.7654 q^{65} -11.6489 q^{67} +3.91567 q^{69} -7.03788 q^{71} +5.63686 q^{73} -2.73086 q^{75} -1.35110 q^{79} +5.95293 q^{81} -16.6477 q^{83} +14.2242 q^{85} +1.31345 q^{87} +10.4695 q^{89} -6.64878 q^{91} +0.979750 q^{93} +3.09847 q^{95} -7.21671 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.593598	−0.342714	−0.171357	−	0.985209i	$-0.554815\pi$
$3$	−0.593598	−0.342714	−0.171357	+	0.985209i	$0.554815\pi$
$4$	0	0
$5$	−3.09847	−1.38568	−0.692839	−	0.721092i	$-0.743640\pi$
$5$	−3.09847	−1.38568	−0.692839	+	0.721092i	$0.743640\pi$
$6$	0	0
$7$	1.91364	0.723288	0.361644	−	0.932316i	$-0.382216\pi$
$7$	1.91364	0.723288	0.361644	+	0.932316i	$0.382216\pi$
$8$	0	0
$9$	−2.64764	−0.882547
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.47441	−0.963629	−0.481815	−	0.876273i	$-0.660022\pi$
$13$	−3.47441	−0.963629	−0.481815	+	0.876273i	$0.660022\pi$
$14$	0	0
$15$	1.83925	0.474891
$16$	0	0
$17$	−4.59072	−1.11341	−0.556706	−	0.830709i	$-0.687935\pi$
$17$	−4.59072	−1.11341	−0.556706	+	0.830709i	$0.687935\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.13593	−0.247881
$22$	0	0
$23$	−6.59650	−1.37547	−0.687733	−	0.725964i	$-0.741394\pi$
$23$	−6.59650	−1.37547	−0.687733	+	0.725964i	$0.741394\pi$
$24$	0	0
$25$	4.60052	0.920104
$26$	0	0
$27$	3.35243	0.645175
$28$	0	0
$29$	−2.21269	−0.410886	−0.205443	−	0.978669i	$-0.565863\pi$
$29$	−2.21269	−0.410886	−0.205443	+	0.978669i	$0.565863\pi$
$30$	0	0
$31$	−1.65053	−0.296444	−0.148222	−	0.988954i	$-0.547355\pi$
$31$	−1.65053	−0.296444	−0.148222	+	0.988954i	$0.547355\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.92936	−1.00225
$36$	0	0
$37$	−8.86862	−1.45799	−0.728996	−	0.684518i	$-0.760013\pi$
$37$	−8.86862	−1.45799	−0.728996	+	0.684518i	$0.760013\pi$
$38$	0	0
$39$	2.06241	0.330249
$40$	0	0
$41$	12.2832	1.91831	0.959156	−	0.282878i	$-0.0912890\pi$
$41$	12.2832	1.91831	0.959156	+	0.282878i	$0.0912890\pi$
$42$	0	0
$43$	0.943326	0.143856	0.0719279	−	0.997410i	$-0.477085\pi$
$43$	0.943326	0.143856	0.0719279	+	0.997410i	$0.477085\pi$
$44$	0	0
$45$	8.20364	1.22293
$46$	0	0
$47$	−5.03322	−0.734170	−0.367085	−	0.930187i	$-0.619644\pi$
$47$	−5.03322	−0.734170	−0.367085	+	0.930187i	$0.619644\pi$
$48$	0	0
$49$	−3.33798	−0.476854
$50$	0	0
$51$	2.72504	0.381582
$52$	0	0
$53$	−9.93586	−1.36480	−0.682398	−	0.730981i	$-0.739063\pi$
$53$	−9.93586	−1.36480	−0.682398	+	0.730981i	$0.739063\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.593598	0.0786240
$58$	0	0
$59$	13.0150	1.69441	0.847206	−	0.531265i	$-0.178283\pi$
$59$	13.0150	1.69441	0.847206	+	0.531265i	$0.178283\pi$
$60$	0	0
$61$	−14.4026	−1.84406	−0.922029	−	0.387120i	$-0.873470\pi$
$61$	−14.4026	−1.84406	−0.922029	+	0.387120i	$0.873470\pi$
$62$	0	0
$63$	−5.06664	−0.638336
$64$	0	0
$65$	10.7654	1.33528
$66$	0	0
$67$	−11.6489	−1.42313	−0.711567	−	0.702618i	$-0.752014\pi$
$67$	−11.6489	−1.42313	−0.711567	+	0.702618i	$0.752014\pi$
$68$	0	0
$69$	3.91567	0.471391
$70$	0	0
$71$	−7.03788	−0.835242	−0.417621	−	0.908621i	$-0.637136\pi$
$71$	−7.03788	−0.835242	−0.417621	+	0.908621i	$0.637136\pi$
$72$	0	0
$73$	5.63686	0.659745	0.329872	−	0.944026i	$-0.392994\pi$
$73$	5.63686	0.659745	0.329872	+	0.944026i	$0.392994\pi$
$74$	0	0
$75$	−2.73086	−0.315333
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.35110	−0.152011	−0.0760054	−	0.997107i	$-0.524217\pi$
$79$	−1.35110	−0.152011	−0.0760054	+	0.997107i	$0.524217\pi$
$80$	0	0
$81$	5.95293	0.661436
$82$	0	0
$83$	−16.6477	−1.82732	−0.913662	−	0.406476i	$-0.866758\pi$
$83$	−16.6477	−1.82732	−0.913662	+	0.406476i	$0.866758\pi$
$84$	0	0
$85$	14.2242	1.54283
$86$	0	0
$87$	1.31345	0.140816
$88$	0	0
$89$	10.4695	1.10977	0.554883	−	0.831929i	$-0.312763\pi$
$89$	10.4695	1.10977	0.554883	+	0.831929i	$0.312763\pi$
$90$	0	0
$91$	−6.64878	−0.696982
$92$	0	0
$93$	0.979750	0.101595
$94$	0	0
$95$	3.09847	0.317896
$96$	0	0
$97$	−7.21671	−0.732746	−0.366373	−	0.930468i	$-0.619401\pi$
$97$	−7.21671	−0.732746	−0.366373	+	0.930468i	$0.619401\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.5740	1.45016	0.725082	−	0.688663i	$-0.241802\pi$
$101$	14.5740	1.45016	0.725082	+	0.688663i	$0.241802\pi$
$102$	0	0
$103$	−18.1505	−1.78842	−0.894210	−	0.447647i	$-0.852262\pi$
$103$	−18.1505	−1.78842	−0.894210	+	0.447647i	$0.852262\pi$
$104$	0	0
$105$	3.51966	0.343484
$106$	0	0
$107$	12.2256	1.18189	0.590945	−	0.806712i	$-0.298755\pi$
$107$	12.2256	1.18189	0.590945	+	0.806712i	$0.298755\pi$
$108$	0	0
$109$	12.2814	1.17634	0.588171	−	0.808737i	$-0.299848\pi$
$109$	12.2814	1.17634	0.588171	+	0.808737i	$0.299848\pi$
$110$	0	0
$111$	5.26439	0.499674
$112$	0	0
$113$	−11.5448	−1.08604	−0.543020	−	0.839720i	$-0.682719\pi$
$113$	−11.5448	−1.08604	−0.543020	+	0.839720i	$0.682719\pi$
$114$	0	0
$115$	20.4391	1.90595
$116$	0	0
$117$	9.19900	0.850448
$118$	0	0
$119$	−8.78499	−0.805318
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−7.29128	−0.657433
$124$	0	0
$125$	1.23777	0.110710
$126$	0	0
$127$	−6.83917	−0.606878	−0.303439	−	0.952851i	$-0.598135\pi$
$127$	−6.83917	−0.606878	−0.303439	+	0.952851i	$0.598135\pi$
$128$	0	0
$129$	−0.559957	−0.0493014
$130$	0	0
$131$	−16.7548	−1.46388	−0.731938	−	0.681372i	$-0.761384\pi$
$131$	−16.7548	−1.46388	−0.731938	+	0.681372i	$0.761384\pi$
$132$	0	0
$133$	−1.91364	−0.165934
$134$	0	0
$135$	−10.3874	−0.894006
$136$	0	0
$137$	18.4928	1.57995	0.789975	−	0.613139i	$-0.210093\pi$
$137$	18.4928	1.57995	0.789975	+	0.613139i	$0.210093\pi$
$138$	0	0
$139$	16.1878	1.37303	0.686515	−	0.727116i	$-0.259140\pi$
$139$	16.1878	1.37303	0.686515	+	0.727116i	$0.259140\pi$
$140$	0	0
$141$	2.98771	0.251610
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.85595	0.569355
$146$	0	0
$147$	1.98142	0.163425
$148$	0	0
$149$	9.60160	0.786593	0.393297	−	0.919412i	$-0.371335\pi$
$149$	9.60160	0.786593	0.393297	+	0.919412i	$0.371335\pi$
$150$	0	0
$151$	4.24817	0.345711	0.172856	−	0.984947i	$-0.444701\pi$
$151$	4.24817	0.345711	0.172856	+	0.984947i	$0.444701\pi$
$152$	0	0
$153$	12.1546	0.982639
$154$	0	0
$155$	5.11411	0.410775
$156$	0	0
$157$	−5.42626	−0.433063	−0.216532	−	0.976276i	$-0.569474\pi$
$157$	−5.42626	−0.433063	−0.216532	+	0.976276i	$0.569474\pi$
$158$	0	0
$159$	5.89791	0.467735
$160$	0	0
$161$	−12.6233	−0.994858
$162$	0	0
$163$	−8.61379	−0.674684	−0.337342	−	0.941382i	$-0.609528\pi$
$163$	−8.61379	−0.674684	−0.337342	+	0.941382i	$0.609528\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.68308	−0.439770	−0.219885	−	0.975526i	$-0.570568\pi$
$167$	−5.68308	−0.439770	−0.219885	+	0.975526i	$0.570568\pi$
$168$	0	0
$169$	−0.928446	−0.0714189
$170$	0	0
$171$	2.64764	0.202470
$172$	0	0
$173$	−24.5113	−1.86356	−0.931779	−	0.363025i	$-0.881744\pi$
$173$	−24.5113	−1.86356	−0.931779	+	0.363025i	$0.881744\pi$
$174$	0	0
$175$	8.80375	0.665501
$176$	0	0
$177$	−7.72569	−0.580699
$178$	0	0
$179$	7.80787	0.583588	0.291794	−	0.956481i	$-0.405748\pi$
$179$	7.80787	0.583588	0.291794	+	0.956481i	$0.405748\pi$
$180$	0	0
$181$	−3.65014	−0.271313	−0.135656	−	0.990756i	$-0.543314\pi$
$181$	−3.65014	−0.271313	−0.135656	+	0.990756i	$0.543314\pi$
$182$	0	0
$183$	8.54933	0.631985
$184$	0	0
$185$	27.4791	2.02031
$186$	0	0
$187$	0	0
$188$	0	0
$189$	6.41535	0.466648
$190$	0	0
$191$	12.1463	0.878876	0.439438	−	0.898273i	$-0.355178\pi$
$191$	12.1463	0.878876	0.439438	+	0.898273i	$0.355178\pi$
$192$	0	0
$193$	−4.14553	−0.298402	−0.149201	−	0.988807i	$-0.547670\pi$
$193$	−4.14553	−0.298402	−0.149201	+	0.988807i	$0.547670\pi$
$194$	0	0
$195$	−6.39031	−0.457619
$196$	0	0
$197$	−12.3484	−0.879786	−0.439893	−	0.898050i	$-0.644984\pi$
$197$	−12.3484	−0.879786	−0.439893	+	0.898050i	$0.644984\pi$
$198$	0	0
$199$	8.39863	0.595363	0.297682	−	0.954665i	$-0.403787\pi$
$199$	8.39863	0.595363	0.297682	+	0.954665i	$0.403787\pi$
$200$	0	0
$201$	6.91474	0.487728
$202$	0	0
$203$	−4.23429	−0.297189
$204$	0	0
$205$	−38.0591	−2.65816
$206$	0	0
$207$	17.4652	1.21391
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−14.6112	−1.00587	−0.502937	−	0.864323i	$-0.667747\pi$
$211$	−14.6112	−1.00587	−0.502937	+	0.864323i	$0.667747\pi$
$212$	0	0
$213$	4.17767	0.286249
$214$	0	0
$215$	−2.92287	−0.199338
$216$	0	0
$217$	−3.15852	−0.214414
$218$	0	0
$219$	−3.34603	−0.226104
$220$	0	0
$221$	15.9501	1.07292
$222$	0	0
$223$	−1.29147	−0.0864831	−0.0432416	−	0.999065i	$-0.513769\pi$
$223$	−1.29147	−0.0864831	−0.0432416	+	0.999065i	$0.513769\pi$
$224$	0	0
$225$	−12.1805	−0.812035
$226$	0	0
$227$	−2.42053	−0.160656	−0.0803280	−	0.996768i	$-0.525597\pi$
$227$	−2.42053	−0.160656	−0.0803280	+	0.996768i	$0.525597\pi$
$228$	0	0
$229$	−10.1848	−0.673028	−0.336514	−	0.941678i	$-0.609248\pi$
$229$	−10.1848	−0.673028	−0.336514	+	0.941678i	$0.609248\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.3079	1.46144	0.730719	−	0.682678i	$-0.239185\pi$
$233$	22.3079	1.46144	0.730719	+	0.682678i	$0.239185\pi$
$234$	0	0
$235$	15.5953	1.01732
$236$	0	0
$237$	0.802011	0.0520962
$238$	0	0
$239$	−2.32949	−0.150682	−0.0753412	−	0.997158i	$-0.524005\pi$
$239$	−2.32949	−0.150682	−0.0753412	+	0.997158i	$0.524005\pi$
$240$	0	0
$241$	−5.96521	−0.384253	−0.192126	−	0.981370i	$-0.561538\pi$
$241$	−5.96521	−0.384253	−0.192126	+	0.981370i	$0.561538\pi$
$242$	0	0
$243$	−13.5909	−0.871859
$244$	0	0
$245$	10.3426	0.660766
$246$	0	0
$247$	3.47441	0.221072
$248$	0	0
$249$	9.88205	0.626249
$250$	0	0
$251$	19.1521	1.20887	0.604435	−	0.796655i	$-0.293399\pi$
$251$	19.1521	1.20887	0.604435	+	0.796655i	$0.293399\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−8.44346	−0.528750
$256$	0	0
$257$	4.50418	0.280963	0.140482	−	0.990083i	$-0.455135\pi$
$257$	4.50418	0.280963	0.140482	+	0.990083i	$0.455135\pi$
$258$	0	0
$259$	−16.9713	−1.05455
$260$	0	0
$261$	5.85840	0.362626
$262$	0	0
$263$	3.77447	0.232744	0.116372	−	0.993206i	$-0.462874\pi$
$263$	3.77447	0.232744	0.116372	+	0.993206i	$0.462874\pi$
$264$	0	0
$265$	30.7860	1.89117
$266$	0	0
$267$	−6.21468	−0.380332
$268$	0	0
$269$	8.56383	0.522146	0.261073	−	0.965319i	$-0.415924\pi$
$269$	8.56383	0.522146	0.261073	+	0.965319i	$0.415924\pi$
$270$	0	0
$271$	−6.72572	−0.408559	−0.204279	−	0.978913i	$-0.565485\pi$
$271$	−6.72572	−0.408559	−0.204279	+	0.978913i	$0.565485\pi$
$272$	0	0
$273$	3.94671	0.238866
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.0925	−1.02699	−0.513495	−	0.858093i	$-0.671649\pi$
$277$	−17.0925	−1.02699	−0.513495	+	0.858093i	$0.671649\pi$
$278$	0	0
$279$	4.37001	0.261625
$280$	0	0
$281$	10.7687	0.642407	0.321204	−	0.947010i	$-0.395913\pi$
$281$	10.7687	0.642407	0.321204	+	0.947010i	$0.395913\pi$
$282$	0	0
$283$	6.60204	0.392450	0.196225	−	0.980559i	$-0.437132\pi$
$283$	6.60204	0.392450	0.196225	+	0.980559i	$0.437132\pi$
$284$	0	0
$285$	−1.83925	−0.108948
$286$	0	0
$287$	23.5056	1.38749
$288$	0	0
$289$	4.07468	0.239687
$290$	0	0
$291$	4.28383	0.251122
$292$	0	0
$293$	7.36721	0.430397	0.215199	−	0.976570i	$-0.430960\pi$
$293$	7.36721	0.430397	0.215199	+	0.976570i	$0.430960\pi$
$294$	0	0
$295$	−40.3267	−2.34791
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.9190	1.32544
$300$	0	0
$301$	1.80519	0.104049
$302$	0	0
$303$	−8.65108	−0.496992
$304$	0	0
$305$	44.6259	2.55527
$306$	0	0
$307$	−29.7569	−1.69831	−0.849157	−	0.528141i	$-0.822889\pi$
$307$	−29.7569	−1.69831	−0.849157	+	0.528141i	$0.822889\pi$
$308$	0	0
$309$	10.7741	0.612917
$310$	0	0
$311$	−21.9138	−1.24262	−0.621309	−	0.783566i	$-0.713399\pi$
$311$	−21.9138	−1.24262	−0.621309	+	0.783566i	$0.713399\pi$
$312$	0	0
$313$	28.5668	1.61469	0.807347	−	0.590078i	$-0.200903\pi$
$313$	28.5668	1.61469	0.807347	+	0.590078i	$0.200903\pi$
$314$	0	0
$315$	15.6988	0.884528
$316$	0	0
$317$	−20.9633	−1.17742	−0.588709	−	0.808345i	$-0.700364\pi$
$317$	−20.9633	−1.17742	−0.588709	+	0.808345i	$0.700364\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7.25707	−0.405050
$322$	0	0
$323$	4.59072	0.255434
$324$	0	0
$325$	−15.9841	−0.886639
$326$	0	0
$327$	−7.29020	−0.403149
$328$	0	0
$329$	−9.63177	−0.531017
$330$	0	0
$331$	24.6717	1.35608	0.678040	−	0.735025i	$-0.262829\pi$
$331$	24.6717	1.35608	0.678040	+	0.735025i	$0.262829\pi$
$332$	0	0
$333$	23.4809	1.28675
$334$	0	0
$335$	36.0936	1.97201
$336$	0	0
$337$	7.25510	0.395210	0.197605	−	0.980282i	$-0.436684\pi$
$337$	7.25510	0.395210	0.197605	+	0.980282i	$0.436684\pi$
$338$	0	0
$339$	6.85295	0.372201
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.7832	−1.06819
$344$	0	0
$345$	−12.1326	−0.653197
$346$	0	0
$347$	−6.38462	−0.342745	−0.171372	−	0.985206i	$-0.554820\pi$
$347$	−6.38462	−0.342745	−0.171372	+	0.985206i	$0.554820\pi$
$348$	0	0
$349$	−4.36965	−0.233902	−0.116951	−	0.993138i	$-0.537312\pi$
$349$	−4.36965	−0.233902	−0.116951	+	0.993138i	$0.537312\pi$
$350$	0	0
$351$	−11.6477	−0.621710
$352$	0	0
$353$	29.1492	1.55145	0.775727	−	0.631069i	$-0.217383\pi$
$353$	29.1492	1.55145	0.775727	+	0.631069i	$0.217383\pi$
$354$	0	0
$355$	21.8067	1.15738
$356$	0	0
$357$	5.21475	0.275994
$358$	0	0
$359$	8.96245	0.473020	0.236510	−	0.971629i	$-0.423996\pi$
$359$	8.96245	0.473020	0.236510	+	0.971629i	$0.423996\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.4657	−0.914194
$366$	0	0
$367$	−12.7720	−0.666693	−0.333346	−	0.942804i	$-0.608178\pi$
$367$	−12.7720	−0.666693	−0.333346	+	0.942804i	$0.608178\pi$
$368$	0	0
$369$	−32.5215	−1.69300
$370$	0	0
$371$	−19.0137	−0.987141
$372$	0	0
$373$	2.58911	0.134059	0.0670296	−	0.997751i	$-0.478648\pi$
$373$	2.58911	0.134059	0.0670296	+	0.997751i	$0.478648\pi$
$374$	0	0
$375$	−0.734740	−0.0379418
$376$	0	0
$377$	7.68779	0.395941
$378$	0	0
$379$	−28.2889	−1.45310	−0.726552	−	0.687111i	$-0.758879\pi$
$379$	−28.2889	−1.45310	−0.726552	+	0.687111i	$0.758879\pi$
$380$	0	0
$381$	4.05972	0.207986
$382$	0	0
$383$	35.5264	1.81532	0.907658	−	0.419710i	$-0.137868\pi$
$383$	35.5264	1.81532	0.907658	+	0.419710i	$0.137868\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.49759	−0.126960
$388$	0	0
$389$	22.0319	1.11706	0.558530	−	0.829485i	$-0.311366\pi$
$389$	22.0319	1.11706	0.558530	+	0.829485i	$0.311366\pi$
$390$	0	0
$391$	30.2827	1.53146
$392$	0	0
$393$	9.94563	0.501691
$394$	0	0
$395$	4.18635	0.210638
$396$	0	0
$397$	−14.2831	−0.716847	−0.358424	−	0.933559i	$-0.616686\pi$
$397$	−14.2831	−0.716847	−0.358424	+	0.933559i	$0.616686\pi$
$398$	0	0
$399$	1.13593	0.0568678
$400$	0	0
$401$	6.17835	0.308532	0.154266	−	0.988029i	$-0.450699\pi$
$401$	6.17835	0.308532	0.154266	+	0.988029i	$0.450699\pi$
$402$	0	0
$403$	5.73462	0.285662
$404$	0	0
$405$	−18.4450	−0.916538
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−9.16144	−0.453004	−0.226502	−	0.974011i	$-0.572729\pi$
$409$	−9.16144	−0.453004	−0.226502	+	0.974011i	$0.572729\pi$
$410$	0	0
$411$	−10.9773	−0.541471
$412$	0	0
$413$	24.9061	1.22555
$414$	0	0
$415$	51.5824	2.53208
$416$	0	0
$417$	−9.60904	−0.470557
$418$	0	0
$419$	−10.7931	−0.527277	−0.263639	−	0.964622i	$-0.584923\pi$
$419$	−10.7931	−0.527277	−0.263639	+	0.964622i	$0.584923\pi$
$420$	0	0
$421$	34.1194	1.66288	0.831440	−	0.555615i	$-0.187517\pi$
$421$	34.1194	1.66288	0.831440	+	0.555615i	$0.187517\pi$
$422$	0	0
$423$	13.3262	0.647940
$424$	0	0
$425$	−21.1197	−1.02446
$426$	0	0
$427$	−27.5613	−1.33379
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.3169	−1.07497	−0.537485	−	0.843274i	$-0.680625\pi$
$431$	−22.3169	−1.07497	−0.537485	+	0.843274i	$0.680625\pi$
$432$	0	0
$433$	9.76468	0.469261	0.234630	−	0.972085i	$-0.424612\pi$
$433$	9.76468	0.469261	0.234630	+	0.972085i	$0.424612\pi$
$434$	0	0
$435$	−4.06968	−0.195126
$436$	0	0
$437$	6.59650	0.315553
$438$	0	0
$439$	23.0676	1.10096	0.550479	−	0.834849i	$-0.314445\pi$
$439$	23.0676	1.10096	0.550479	+	0.834849i	$0.314445\pi$
$440$	0	0
$441$	8.83776	0.420846
$442$	0	0
$443$	−9.60340	−0.456271	−0.228136	−	0.973629i	$-0.573263\pi$
$443$	−9.60340	−0.456271	−0.228136	+	0.973629i	$0.573263\pi$
$444$	0	0
$445$	−32.4395	−1.53778
$446$	0	0
$447$	−5.69949	−0.269577
$448$	0	0
$449$	13.2045	0.623161	0.311580	−	0.950220i	$-0.399142\pi$
$449$	13.2045	0.623161	0.311580	+	0.950220i	$0.399142\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.52171	−0.118480
$454$	0	0
$455$	20.6011	0.965793
$456$	0	0
$457$	−3.94606	−0.184589	−0.0922944	−	0.995732i	$-0.529420\pi$
$457$	−3.94606	−0.184589	−0.0922944	+	0.995732i	$0.529420\pi$
$458$	0	0
$459$	−15.3901	−0.718346
$460$	0	0
$461$	−15.1580	−0.705981	−0.352990	−	0.935627i	$-0.614835\pi$
$461$	−15.1580	−0.705981	−0.352990	+	0.935627i	$0.614835\pi$
$462$	0	0
$463$	−5.39984	−0.250952	−0.125476	−	0.992097i	$-0.540046\pi$
$463$	−5.39984	−0.250952	−0.125476	+	0.992097i	$0.540046\pi$
$464$	0	0
$465$	−3.03573	−0.140779
$466$	0	0
$467$	34.8648	1.61335	0.806675	−	0.590995i	$-0.201265\pi$
$467$	34.8648	1.61335	0.806675	+	0.590995i	$0.201265\pi$
$468$	0	0
$469$	−22.2917	−1.02934
$470$	0	0
$471$	3.22102	0.148417
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.60052	−0.211086
$476$	0	0
$477$	26.3066	1.20450
$478$	0	0
$479$	−10.0115	−0.457437	−0.228718	−	0.973493i	$-0.573453\pi$
$479$	−10.0115	−0.457437	−0.228718	+	0.973493i	$0.573453\pi$
$480$	0	0
$481$	30.8132	1.40496
$482$	0	0
$483$	7.49319	0.340952
$484$	0	0
$485$	22.3608	1.01535
$486$	0	0
$487$	19.3567	0.877135	0.438568	−	0.898698i	$-0.355486\pi$
$487$	19.3567	0.877135	0.438568	+	0.898698i	$0.355486\pi$
$488$	0	0
$489$	5.11313	0.231224
$490$	0	0
$491$	17.3728	0.784021	0.392011	−	0.919961i	$-0.371780\pi$
$491$	17.3728	0.784021	0.392011	+	0.919961i	$0.371780\pi$
$492$	0	0
$493$	10.1578	0.457485
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.4680	−0.604121
$498$	0	0
$499$	−0.802258	−0.0359140	−0.0179570	−	0.999839i	$-0.505716\pi$
$499$	−0.802258	−0.0359140	−0.0179570	+	0.999839i	$0.505716\pi$
$500$	0	0
$501$	3.37346	0.150715
$502$	0	0
$503$	32.9322	1.46838	0.734188	−	0.678946i	$-0.237563\pi$
$503$	32.9322	1.46838	0.734188	+	0.678946i	$0.237563\pi$
$504$	0	0
$505$	−45.1570	−2.00946
$506$	0	0
$507$	0.551124	0.0244763
$508$	0	0
$509$	7.41274	0.328564	0.164282	−	0.986413i	$-0.447469\pi$
$509$	7.41274	0.328564	0.164282	+	0.986413i	$0.447469\pi$
$510$	0	0
$511$	10.7869	0.477186
$512$	0	0
$513$	−3.35243	−0.148013
$514$	0	0
$515$	56.2387	2.47818
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.5499	0.638668
$520$	0	0
$521$	−0.622146	−0.0272567	−0.0136284	−	0.999907i	$-0.504338\pi$
$521$	−0.622146	−0.0272567	−0.0136284	+	0.999907i	$0.504338\pi$
$522$	0	0
$523$	−6.71374	−0.293571	−0.146786	−	0.989168i	$-0.546893\pi$
$523$	−6.71374	−0.293571	−0.146786	+	0.989168i	$0.546893\pi$
$524$	0	0
$525$	−5.22589	−0.228077
$526$	0	0
$527$	7.57711	0.330064
$528$	0	0
$529$	20.5138	0.891906
$530$	0	0
$531$	−34.4591	−1.49540
$532$	0	0
$533$	−42.6769	−1.84854
$534$	0	0
$535$	−37.8805	−1.63772
$536$	0	0
$537$	−4.63474	−0.200004
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.7772	1.83914	0.919568	−	0.392932i	$-0.128539\pi$
$541$	42.7772	1.83914	0.919568	+	0.392932i	$0.128539\pi$
$542$	0	0
$543$	2.16672	0.0929827
$544$	0	0
$545$	−38.0534	−1.63003
$546$	0	0
$547$	30.8980	1.32110	0.660552	−	0.750780i	$-0.270322\pi$
$547$	30.8980	1.32110	0.660552	+	0.750780i	$0.270322\pi$
$548$	0	0
$549$	38.1328	1.62747
$550$	0	0
$551$	2.21269	0.0942636
$552$	0	0
$553$	−2.58552	−0.109948
$554$	0	0
$555$	−16.3116	−0.692388
$556$	0	0
$557$	10.4039	0.440828	0.220414	−	0.975406i	$-0.429259\pi$
$557$	10.4039	0.440828	0.220414	+	0.975406i	$0.429259\pi$
$558$	0	0
$559$	−3.27750	−0.138624
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.2829	−0.896966	−0.448483	−	0.893791i	$-0.648036\pi$
$563$	−21.2829	−0.896966	−0.448483	+	0.893791i	$0.648036\pi$
$564$	0	0
$565$	35.7711	1.50490
$566$	0	0
$567$	11.3918	0.478409
$568$	0	0
$569$	1.61347	0.0676401	0.0338200	−	0.999428i	$-0.489233\pi$
$569$	1.61347	0.0676401	0.0338200	+	0.999428i	$0.489233\pi$
$570$	0	0
$571$	6.17441	0.258391	0.129195	−	0.991619i	$-0.458761\pi$
$571$	6.17441	0.258391	0.129195	+	0.991619i	$0.458761\pi$
$572$	0	0
$573$	−7.21003	−0.301203
$574$	0	0
$575$	−30.3473	−1.26557
$576$	0	0
$577$	29.0783	1.21055	0.605273	−	0.796018i	$-0.293064\pi$
$577$	29.0783	1.21055	0.605273	+	0.796018i	$0.293064\pi$
$578$	0	0
$579$	2.46078	0.102266
$580$	0	0
$581$	−31.8577	−1.32168
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−28.5028	−1.17845
$586$	0	0
$587$	−27.0388	−1.11601	−0.558005	−	0.829838i	$-0.688433\pi$
$587$	−27.0388	−1.11601	−0.558005	+	0.829838i	$0.688433\pi$
$588$	0	0
$589$	1.65053	0.0680088
$590$	0	0
$591$	7.32998	0.301515
$592$	0	0
$593$	−0.943627	−0.0387501	−0.0193751	−	0.999812i	$-0.506168\pi$
$593$	−0.943627	−0.0387501	−0.0193751	+	0.999812i	$0.506168\pi$
$594$	0	0
$595$	27.2200	1.11591
$596$	0	0
$597$	−4.98541	−0.204039
$598$	0	0
$599$	43.6389	1.78304	0.891519	−	0.452983i	$-0.149640\pi$
$599$	43.6389	1.78304	0.891519	+	0.452983i	$0.149640\pi$
$600$	0	0
$601$	19.6645	0.802132	0.401066	−	0.916049i	$-0.368640\pi$
$601$	19.6645	0.802132	0.401066	+	0.916049i	$0.368640\pi$
$602$	0	0
$603$	30.8420	1.25598
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.717308	−0.0291146	−0.0145573	−	0.999894i	$-0.504634\pi$
$607$	−0.717308	−0.0291146	−0.0145573	+	0.999894i	$0.504634\pi$
$608$	0	0
$609$	2.51347	0.101851
$610$	0	0
$611$	17.4875	0.707468
$612$	0	0
$613$	−9.68406	−0.391135	−0.195568	−	0.980690i	$-0.562655\pi$
$613$	−9.68406	−0.391135	−0.195568	+	0.980690i	$0.562655\pi$
$614$	0	0
$615$	22.5918	0.910990
$616$	0	0
$617$	−16.2192	−0.652959	−0.326479	−	0.945204i	$-0.605862\pi$
$617$	−16.2192	−0.652959	−0.326479	+	0.945204i	$0.605862\pi$
$618$	0	0
$619$	30.4807	1.22512	0.612561	−	0.790423i	$-0.290139\pi$
$619$	30.4807	1.22512	0.612561	+	0.790423i	$0.290139\pi$
$620$	0	0
$621$	−22.1143	−0.887417
$622$	0	0
$623$	20.0349	0.802681
$624$	0	0
$625$	−26.8378	−1.07351
$626$	0	0
$627$	0	0
$628$	0	0
$629$	40.7133	1.62335
$630$	0	0
$631$	−25.8438	−1.02882	−0.514412	−	0.857543i	$-0.671990\pi$
$631$	−25.8438	−1.02882	−0.514412	+	0.857543i	$0.671990\pi$
$632$	0	0
$633$	8.67315	0.344727
$634$	0	0
$635$	21.1910	0.840938
$636$	0	0
$637$	11.5975	0.459510
$638$	0	0
$639$	18.6338	0.737141
$640$	0	0
$641$	−34.7158	−1.37119	−0.685595	−	0.727983i	$-0.740458\pi$
$641$	−34.7158	−1.37119	−0.685595	+	0.727983i	$0.740458\pi$
$642$	0	0
$643$	32.9515	1.29948	0.649741	−	0.760156i	$-0.274877\pi$
$643$	32.9515	1.29948	0.649741	+	0.760156i	$0.274877\pi$
$644$	0	0
$645$	1.73501	0.0683159
$646$	0	0
$647$	17.7250	0.696844	0.348422	−	0.937338i	$-0.386718\pi$
$647$	17.7250	0.696844	0.348422	+	0.937338i	$0.386718\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.87489	0.0734828
$652$	0	0
$653$	3.71122	0.145231	0.0726156	−	0.997360i	$-0.476865\pi$
$653$	3.71122	0.145231	0.0726156	+	0.997360i	$0.476865\pi$
$654$	0	0
$655$	51.9143	2.02846
$656$	0	0
$657$	−14.9244	−0.582256
$658$	0	0
$659$	21.9688	0.855782	0.427891	−	0.903830i	$-0.359257\pi$
$659$	21.9688	0.855782	0.427891	+	0.903830i	$0.359257\pi$
$660$	0	0
$661$	−36.3409	−1.41350	−0.706749	−	0.707465i	$-0.749839\pi$
$661$	−36.3409	−1.41350	−0.706749	+	0.707465i	$0.749839\pi$
$662$	0	0
$663$	−9.46792	−0.367704
$664$	0	0
$665$	5.92936	0.229931
$666$	0	0
$667$	14.5960	0.565159
$668$	0	0
$669$	0.766614	0.0296390
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8.77095	0.338095	0.169048	−	0.985608i	$-0.445931\pi$
$673$	8.77095	0.338095	0.169048	+	0.985608i	$0.445931\pi$
$674$	0	0
$675$	15.4229	0.593629
$676$	0	0
$677$	20.5001	0.787882	0.393941	−	0.919136i	$-0.371112\pi$
$677$	20.5001	0.787882	0.393941	+	0.919136i	$0.371112\pi$
$678$	0	0
$679$	−13.8102	−0.529987
$680$	0	0
$681$	1.43682	0.0550591
$682$	0	0
$683$	−21.8781	−0.837143	−0.418572	−	0.908184i	$-0.637469\pi$
$683$	−21.8781	−0.837143	−0.418572	+	0.908184i	$0.637469\pi$
$684$	0	0
$685$	−57.2995	−2.18930
$686$	0	0
$687$	6.04566	0.230656
$688$	0	0
$689$	34.5213	1.31516
$690$	0	0
$691$	15.2685	0.580841	0.290421	−	0.956899i	$-0.406205\pi$
$691$	15.2685	0.580841	0.290421	+	0.956899i	$0.406205\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−50.1574	−1.90258
$696$	0	0
$697$	−56.3886	−2.13587
$698$	0	0
$699$	−13.2419	−0.500855
$700$	0	0
$701$	35.3916	1.33672	0.668362	−	0.743836i	$-0.266996\pi$
$701$	35.3916	1.33672	0.668362	+	0.743836i	$0.266996\pi$
$702$	0	0
$703$	8.86862	0.334486
$704$	0	0
$705$	−9.25733	−0.348651
$706$	0	0
$707$	27.8893	1.04889
$708$	0	0
$709$	−6.88137	−0.258435	−0.129218	−	0.991616i	$-0.541247\pi$
$709$	−6.88137	−0.258435	−0.129218	+	0.991616i	$0.541247\pi$
$710$	0	0
$711$	3.57723	0.134157
$712$	0	0
$713$	10.8877	0.407748
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.38278	0.0516410
$718$	0	0
$719$	24.2631	0.904860	0.452430	−	0.891800i	$-0.350557\pi$
$719$	24.2631	0.904860	0.452430	+	0.891800i	$0.350557\pi$
$720$	0	0
$721$	−34.7335	−1.29354
$722$	0	0
$723$	3.54094	0.131689
$724$	0	0
$725$	−10.1795	−0.378058
$726$	0	0
$727$	23.2648	0.862844	0.431422	−	0.902150i	$-0.358012\pi$
$727$	23.2648	0.862844	0.431422	+	0.902150i	$0.358012\pi$
$728$	0	0
$729$	−9.79122	−0.362638
$730$	0	0
$731$	−4.33054	−0.160171
$732$	0	0
$733$	−28.6621	−1.05866	−0.529330	−	0.848416i	$-0.677557\pi$
$733$	−28.6621	−1.05866	−0.529330	+	0.848416i	$0.677557\pi$
$734$	0	0
$735$	−6.13936	−0.226454
$736$	0	0
$737$	0	0
$738$	0	0
$739$	14.4806	0.532676	0.266338	−	0.963880i	$-0.414186\pi$
$739$	14.4806	0.532676	0.266338	+	0.963880i	$0.414186\pi$
$740$	0	0
$741$	−2.06241	−0.0757644
$742$	0	0
$743$	6.38805	0.234355	0.117177	−	0.993111i	$-0.462615\pi$
$743$	6.38805	0.234355	0.117177	+	0.993111i	$0.462615\pi$
$744$	0	0
$745$	−29.7503	−1.08997
$746$	0	0
$747$	44.0772	1.61270
$748$	0	0
$749$	23.3953	0.854847
$750$	0	0
$751$	−8.26604	−0.301632	−0.150816	−	0.988562i	$-0.548190\pi$
$751$	−8.26604	−0.301632	−0.150816	+	0.988562i	$0.548190\pi$
$752$	0	0
$753$	−11.3686	−0.414297
$754$	0	0
$755$	−13.1628	−0.479045
$756$	0	0
$757$	−5.93299	−0.215638	−0.107819	−	0.994171i	$-0.534387\pi$
$757$	−5.93299	−0.215638	−0.107819	+	0.994171i	$0.534387\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−52.1838	−1.89166	−0.945831	−	0.324660i	$-0.894750\pi$
$761$	−52.1838	−1.89166	−0.945831	+	0.324660i	$0.894750\pi$
$762$	0	0
$763$	23.5021	0.850834
$764$	0	0
$765$	−37.6606	−1.36162
$766$	0	0
$767$	−45.2196	−1.63278
$768$	0	0
$769$	−42.4687	−1.53146	−0.765731	−	0.643161i	$-0.777622\pi$
$769$	−42.4687	−1.53146	−0.765731	+	0.643161i	$0.777622\pi$
$770$	0	0
$771$	−2.67368	−0.0962901
$772$	0	0
$773$	21.6781	0.779707	0.389853	−	0.920877i	$-0.372526\pi$
$773$	21.6781	0.779707	0.389853	+	0.920877i	$0.372526\pi$
$774$	0	0
$775$	−7.59329	−0.272759
$776$	0	0
$777$	10.0742	0.361409
$778$	0	0
$779$	−12.2832	−0.440091
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−7.41788	−0.265093
$784$	0	0
$785$	16.8131	0.600086
$786$	0	0
$787$	−53.3591	−1.90205	−0.951023	−	0.309121i	$-0.899965\pi$
$787$	−53.3591	−1.90205	−0.951023	+	0.309121i	$0.899965\pi$
$788$	0	0
$789$	−2.24052	−0.0797647
$790$	0	0
$791$	−22.0925	−0.785520
$792$	0	0
$793$	50.0405	1.77699
$794$	0	0
$795$	−18.2745	−0.648130
$796$	0	0
$797$	−36.6435	−1.29798	−0.648990	−	0.760797i	$-0.724808\pi$
$797$	−36.6435	−1.29798	−0.648990	+	0.760797i	$0.724808\pi$
$798$	0	0
$799$	23.1061	0.817434
$800$	0	0
$801$	−27.7195	−0.979420
$802$	0	0
$803$	0	0
$804$	0	0
$805$	39.1130	1.37855
$806$	0	0
$807$	−5.08348	−0.178947
$808$	0	0
$809$	−44.3884	−1.56061	−0.780307	−	0.625397i	$-0.784937\pi$
$809$	−44.3884	−1.56061	−0.780307	+	0.625397i	$0.784937\pi$
$810$	0	0
$811$	−36.4827	−1.28108	−0.640541	−	0.767924i	$-0.721290\pi$
$811$	−36.4827	−1.28108	−0.640541	+	0.767924i	$0.721290\pi$
$812$	0	0
$813$	3.99238	0.140019
$814$	0	0
$815$	26.6896	0.934895
$816$	0	0
$817$	−0.943326	−0.0330028
$818$	0	0
$819$	17.6036	0.615119
$820$	0	0
$821$	−28.5131	−0.995114	−0.497557	−	0.867431i	$-0.665769\pi$
$821$	−28.5131	−0.995114	−0.497557	+	0.867431i	$0.665769\pi$
$822$	0	0
$823$	23.9024	0.833185	0.416592	−	0.909093i	$-0.363224\pi$
$823$	23.9024	0.833185	0.416592	+	0.909093i	$0.363224\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.52622	−0.122619	−0.0613094	−	0.998119i	$-0.519528\pi$
$827$	−3.52622	−0.122619	−0.0613094	+	0.998119i	$0.519528\pi$
$828$	0	0
$829$	−11.0538	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$829$	−11.0538	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$830$	0	0
$831$	10.1461	0.351964
$832$	0	0
$833$	15.3237	0.530935
$834$	0	0
$835$	17.6088	0.609379
$836$	0	0
$837$	−5.53328	−0.191258
$838$	0	0
$839$	−8.73338	−0.301510	−0.150755	−	0.988571i	$-0.548170\pi$
$839$	−8.73338	−0.301510	−0.150755	+	0.988571i	$0.548170\pi$
$840$	0	0
$841$	−24.1040	−0.831173
$842$	0	0
$843$	−6.39229	−0.220162
$844$	0	0
$845$	2.87676	0.0989636
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.91896	−0.134498
$850$	0	0
$851$	58.5018	2.00542
$852$	0	0
$853$	−12.2466	−0.419316	−0.209658	−	0.977775i	$-0.567235\pi$
$853$	−12.2466	−0.419316	−0.209658	+	0.977775i	$0.567235\pi$
$854$	0	0
$855$	−8.20364	−0.280559
$856$	0	0
$857$	21.1665	0.723033	0.361517	−	0.932366i	$-0.382259\pi$
$857$	21.1665	0.723033	0.361517	+	0.932366i	$0.382259\pi$
$858$	0	0
$859$	24.0678	0.821184	0.410592	−	0.911819i	$-0.365322\pi$
$859$	24.0678	0.821184	0.410592	+	0.911819i	$0.365322\pi$
$860$	0	0
$861$	−13.9529	−0.475513
$862$	0	0
$863$	−7.20834	−0.245375	−0.122687	−	0.992445i	$-0.539151\pi$
$863$	−7.20834	−0.245375	−0.122687	+	0.992445i	$0.539151\pi$
$864$	0	0
$865$	75.9475	2.58229
$866$	0	0
$867$	−2.41872	−0.0821441
$868$	0	0
$869$	0	0
$870$	0	0
$871$	40.4729	1.37137
$872$	0	0
$873$	19.1073	0.646683
$874$	0	0
$875$	2.36865	0.0800751
$876$	0	0
$877$	51.9275	1.75347	0.876734	−	0.480975i	$-0.159717\pi$
$877$	51.9275	1.75347	0.876734	+	0.480975i	$0.159717\pi$
$878$	0	0
$879$	−4.37317	−0.147503
$880$	0	0
$881$	−43.4951	−1.46539	−0.732693	−	0.680559i	$-0.761737\pi$
$881$	−43.4951	−1.46539	−0.732693	+	0.680559i	$0.761737\pi$
$882$	0	0
$883$	−17.3622	−0.584285	−0.292142	−	0.956375i	$-0.594368\pi$
$883$	−17.3622	−0.584285	−0.292142	+	0.956375i	$0.594368\pi$
$884$	0	0
$885$	23.9378	0.804662
$886$	0	0
$887$	−1.37020	−0.0460068	−0.0230034	−	0.999735i	$-0.507323\pi$
$887$	−1.37020	−0.0460068	−0.0230034	+	0.999735i	$0.507323\pi$
$888$	0	0
$889$	−13.0877	−0.438948
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.03322	0.168430
$894$	0	0
$895$	−24.1925	−0.808665
$896$	0	0
$897$	−13.6047	−0.454247
$898$	0	0
$899$	3.65210	0.121804
$900$	0	0
$901$	45.6127	1.51958
$902$	0	0
$903$	−1.07156	−0.0356592
$904$	0	0
$905$	11.3099	0.375952
$906$	0	0
$907$	6.17010	0.204875	0.102437	−	0.994739i	$-0.467336\pi$
$907$	6.17010	0.204875	0.102437	+	0.994739i	$0.467336\pi$
$908$	0	0
$909$	−38.5866	−1.27984
$910$	0	0
$911$	31.1950	1.03354	0.516769	−	0.856125i	$-0.327135\pi$
$911$	31.1950	1.03354	0.516769	+	0.856125i	$0.327135\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−26.4899	−0.875728
$916$	0	0
$917$	−32.0627	−1.05880
$918$	0	0
$919$	−12.2119	−0.402833	−0.201416	−	0.979506i	$-0.564554\pi$
$919$	−12.2119	−0.402833	−0.201416	+	0.979506i	$0.564554\pi$
$920$	0	0
$921$	17.6636	0.582036
$922$	0	0
$923$	24.4525	0.804864
$924$	0	0
$925$	−40.8003	−1.34150
$926$	0	0
$927$	48.0560	1.57836
$928$	0	0
$929$	21.3907	0.701807	0.350904	−	0.936412i	$-0.385874\pi$
$929$	21.3907	0.701807	0.350904	+	0.936412i	$0.385874\pi$
$930$	0	0
$931$	3.33798	0.109398
$932$	0	0
$933$	13.0080	0.425863
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.4790	−0.603683	−0.301841	−	0.953358i	$-0.597601\pi$
$937$	−18.4790	−0.603683	−0.301841	+	0.953358i	$0.597601\pi$
$938$	0	0
$939$	−16.9572	−0.553378
$940$	0	0
$941$	−16.1725	−0.527208	−0.263604	−	0.964631i	$-0.584911\pi$
$941$	−16.1725	−0.527208	−0.263604	+	0.964631i	$0.584911\pi$
$942$	0	0
$943$	−81.0261	−2.63857
$944$	0	0
$945$	−19.8778	−0.646624
$946$	0	0
$947$	−22.0138	−0.715353	−0.357677	−	0.933846i	$-0.616431\pi$
$947$	−22.0138	−0.715353	−0.357677	+	0.933846i	$0.616431\pi$
$948$	0	0
$949$	−19.5848	−0.635749
$950$	0	0
$951$	12.4438	0.403518
$952$	0	0
$953$	0.628681	0.0203650	0.0101825	−	0.999948i	$-0.496759\pi$
$953$	0.628681	0.0203650	0.0101825	+	0.999948i	$0.496759\pi$
$954$	0	0
$955$	−37.6350	−1.21784
$956$	0	0
$957$	0	0
$958$	0	0
$959$	35.3887	1.14276
$960$	0	0
$961$	−28.2758	−0.912121
$962$	0	0
$963$	−32.3689	−1.04307
$964$	0	0
$965$	12.8448	0.413489
$966$	0	0
$967$	14.7814	0.475336	0.237668	−	0.971346i	$-0.423617\pi$
$967$	14.7814	0.475336	0.237668	+	0.971346i	$0.423617\pi$
$968$	0	0
$969$	−2.72504	−0.0875409
$970$	0	0
$971$	−28.4506	−0.913023	−0.456511	−	0.889718i	$-0.650901\pi$
$971$	−28.4506	−0.913023	−0.456511	+	0.889718i	$0.650901\pi$
$972$	0	0
$973$	30.9776	0.993096
$974$	0	0
$975$	9.48814	0.303864
$976$	0	0
$977$	−25.5690	−0.818024	−0.409012	−	0.912529i	$-0.634127\pi$
$977$	−25.5690	−0.818024	−0.409012	+	0.912529i	$0.634127\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−32.5166	−1.03818
$982$	0	0
$983$	60.1871	1.91967	0.959835	−	0.280565i	$-0.0905218\pi$
$983$	60.1871	1.91967	0.959835	+	0.280565i	$0.0905218\pi$
$984$	0	0
$985$	38.2611	1.21910
$986$	0	0
$987$	5.71740	0.181987
$988$	0	0
$989$	−6.22265	−0.197869
$990$	0	0
$991$	14.5621	0.462581	0.231290	−	0.972885i	$-0.425705\pi$
$991$	14.5621	0.462581	0.231290	+	0.972885i	$0.425705\pi$
$992$	0	0
$993$	−14.6451	−0.464748
$994$	0	0
$995$	−26.0229	−0.824982
$996$	0	0
$997$	5.28710	0.167444	0.0837221	−	0.996489i	$-0.473319\pi$
$997$	5.28710	0.167444	0.0837221	+	0.996489i	$0.473319\pi$
$998$	0	0
$999$	−29.7314	−0.940660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.x.1.8		20
11.3	even	5		836.2.j.c.229.7	✓	40
11.4	even	5		836.2.j.c.533.7	yes	40
11.10	odd	2		9196.2.a.w.1.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.c.229.7	✓	40	11.3	even	5
836.2.j.c.533.7	yes	40	11.4	even	5
9196.2.a.w.1.8		20	11.10	odd	2
9196.2.a.x.1.8		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.593598 q^{3} -3.09847 q^{5} +1.91364 q^{7} -2.64764 q^{9} -3.47441 q^{13} +1.83925 q^{15} -4.59072 q^{17} -1.00000 q^{19} -1.13593 q^{21} -6.59650 q^{23} +4.60052 q^{25} +3.35243 q^{27} -2.21269 q^{29} -1.65053 q^{31} -5.92936 q^{35} -8.86862 q^{37} +2.06241 q^{39} +12.2832 q^{41} +0.943326 q^{43} +8.20364 q^{45} -5.03322 q^{47} -3.33798 q^{49} +2.72504 q^{51} -9.93586 q^{53} +0.593598 q^{57} +13.0150 q^{59} -14.4026 q^{61} -5.06664 q^{63} +10.7654 q^{65} -11.6489 q^{67} +3.91567 q^{69} -7.03788 q^{71} +5.63686 q^{73} -2.73086 q^{75} -1.35110 q^{79} +5.95293 q^{81} -16.6477 q^{83} +14.2242 q^{85} +1.31345 q^{87} +10.4695 q^{89} -6.64878 q^{91} +0.979750 q^{93} +3.09847 q^{95} -7.21671 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) \) 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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