LMFDB - Embedded newform 9196.2.a.l.1.6

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.114134848.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} - 6x^{3} + 20x^{2} + 14x - 5 $ x^6 - 10x^4 - 6x^3 + 20x^2 + 14x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.06973$ 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)

$f(q)$	$=$	$q+2.06973 q^{3} -0.437868 q^{5} -0.692506 q^{7} +1.28379 q^{9} -4.95486 q^{13} -0.906270 q^{15} +4.83743 q^{17} +1.00000 q^{19} -1.43330 q^{21} -3.86113 q^{23} -4.80827 q^{25} -3.55209 q^{27} +0.0211448 q^{29} +5.63341 q^{31} +0.303226 q^{35} +9.51598 q^{37} -10.2552 q^{39} -9.14630 q^{41} +3.99216 q^{43} -0.562132 q^{45} +7.90972 q^{47} -6.52044 q^{49} +10.0122 q^{51} +7.29473 q^{53} +2.06973 q^{57} +2.40871 q^{59} -5.72103 q^{61} -0.889035 q^{63} +2.16957 q^{65} +4.61821 q^{67} -7.99150 q^{69} +1.43977 q^{71} -0.736270 q^{73} -9.95184 q^{75} -16.3190 q^{79} -11.2033 q^{81} -16.0623 q^{83} -2.11815 q^{85} +0.0437641 q^{87} -4.22636 q^{89} +3.43127 q^{91} +11.6597 q^{93} -0.437868 q^{95} -11.4257 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 6 q^{17} + 6 q^{19} + 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} + 2 q^{29} - 6 q^{31} - 14 q^{35} + 4 q^{37} - 6 q^{39} - 14 q^{41} + 10 q^{43}+ \cdots + 28 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 6 * q^17 + 6 * q^19 + 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 + 2 * q^29 - 6 * q^31 - 14 * q^35 + 4 * q^37 - 6 * q^39 - 14 * q^41 + 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 - 18 * q^51 + 20 * q^59 + 2 * q^61 - 12 * q^63 + 12 * q^65 + 10 * q^67 - 20 * q^69 - 32 * q^73 + 18 * q^75 - 6 * q^79 - 6 * q^81 - 36 * q^83 - 6 * q^85 + 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 - 4 * q^95 + 28 * 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
$3$	2.06973	1.19496	0.597480	−	0.801884i	$-0.296169\pi$
$3$	2.06973	1.19496	0.597480	+	0.801884i	$0.296169\pi$
$4$	0	0
$5$	−0.437868	−0.195820	−0.0979102	−	0.995195i	$-0.531216\pi$
$5$	−0.437868	−0.195820	−0.0979102	+	0.995195i	$0.531216\pi$
$6$	0	0
$7$	−0.692506	−0.261743	−0.130871	−	0.991399i	$-0.541777\pi$
$7$	−0.692506	−0.261743	−0.130871	+	0.991399i	$0.541777\pi$
$8$	0	0
$9$	1.28379	0.427931
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.95486	−1.37423	−0.687115	−	0.726549i	$-0.741123\pi$
$13$	−4.95486	−1.37423	−0.687115	+	0.726549i	$0.741123\pi$
$14$	0	0
$15$	−0.906270	−0.233998
$16$	0	0
$17$	4.83743	1.17325	0.586624	−	0.809859i	$-0.300457\pi$
$17$	4.83743	1.17325	0.586624	+	0.809859i	$0.300457\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.43330	−0.312772
$22$	0	0
$23$	−3.86113	−0.805101	−0.402550	−	0.915398i	$-0.631876\pi$
$23$	−3.86113	−0.805101	−0.402550	+	0.915398i	$0.631876\pi$
$24$	0	0
$25$	−4.80827	−0.961654
$26$	0	0
$27$	−3.55209	−0.683600
$28$	0	0
$29$	0.0211448	0.00392649	0.00196325	−	0.999998i	$-0.499375\pi$
$29$	0.0211448	0.00392649	0.00196325	+	0.999998i	$0.499375\pi$
$30$	0	0
$31$	5.63341	1.01179	0.505895	−	0.862595i	$-0.331162\pi$
$31$	5.63341	1.01179	0.505895	+	0.862595i	$0.331162\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.303226	0.0512546
$36$	0	0
$37$	9.51598	1.56442	0.782209	−	0.623017i	$-0.214093\pi$
$37$	9.51598	1.56442	0.782209	+	0.623017i	$0.214093\pi$
$38$	0	0
$39$	−10.2552	−1.64215
$40$	0	0
$41$	−9.14630	−1.42841	−0.714206	−	0.699936i	$-0.753212\pi$
$41$	−9.14630	−1.42841	−0.714206	+	0.699936i	$0.753212\pi$
$42$	0	0
$43$	3.99216	0.608798	0.304399	−	0.952545i	$-0.401544\pi$
$43$	3.99216	0.608798	0.304399	+	0.952545i	$0.401544\pi$
$44$	0	0
$45$	−0.562132	−0.0837977
$46$	0	0
$47$	7.90972	1.15375	0.576875	−	0.816832i	$-0.304272\pi$
$47$	7.90972	1.15375	0.576875	+	0.816832i	$0.304272\pi$
$48$	0	0
$49$	−6.52044	−0.931491
$50$	0	0
$51$	10.0122	1.40199
$52$	0	0
$53$	7.29473	1.00201	0.501004	−	0.865445i	$-0.332964\pi$
$53$	7.29473	1.00201	0.501004	+	0.865445i	$0.332964\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.06973	0.274143
$58$	0	0
$59$	2.40871	0.313588	0.156794	−	0.987631i	$-0.449884\pi$
$59$	2.40871	0.313588	0.156794	+	0.987631i	$0.449884\pi$
$60$	0	0
$61$	−5.72103	−0.732503	−0.366251	−	0.930516i	$-0.619359\pi$
$61$	−5.72103	−0.732503	−0.366251	+	0.930516i	$0.619359\pi$
$62$	0	0
$63$	−0.889035	−0.112008
$64$	0	0
$65$	2.16957	0.269102
$66$	0	0
$67$	4.61821	0.564204	0.282102	−	0.959384i	$-0.408968\pi$
$67$	4.61821	0.564204	0.282102	+	0.959384i	$0.408968\pi$
$68$	0	0
$69$	−7.99150	−0.962064
$70$	0	0
$71$	1.43977	0.170869	0.0854345	−	0.996344i	$-0.472772\pi$
$71$	1.43977	0.170869	0.0854345	+	0.996344i	$0.472772\pi$
$72$	0	0
$73$	−0.736270	−0.0861739	−0.0430870	−	0.999071i	$-0.513719\pi$
$73$	−0.736270	−0.0861739	−0.0430870	+	0.999071i	$0.513719\pi$
$74$	0	0
$75$	−9.95184	−1.14914
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.3190	−1.83603	−0.918017	−	0.396542i	$-0.870210\pi$
$79$	−16.3190	−1.83603	−0.918017	+	0.396542i	$0.870210\pi$
$80$	0	0
$81$	−11.2033	−1.24481
$82$	0	0
$83$	−16.0623	−1.76307	−0.881534	−	0.472120i	$-0.843489\pi$
$83$	−16.0623	−1.76307	−0.881534	+	0.472120i	$0.843489\pi$
$84$	0	0
$85$	−2.11815	−0.229746
$86$	0	0
$87$	0.0437641	0.00469200
$88$	0	0
$89$	−4.22636	−0.447994	−0.223997	−	0.974590i	$-0.571911\pi$
$89$	−4.22636	−0.447994	−0.223997	+	0.974590i	$0.571911\pi$
$90$	0	0
$91$	3.43127	0.359695
$92$	0	0
$93$	11.6597	1.20905
$94$	0	0
$95$	−0.437868	−0.0449243
$96$	0	0
$97$	−11.4257	−1.16010	−0.580052	−	0.814580i	$-0.696968\pi$
$97$	−11.4257	−1.16010	−0.580052	+	0.814580i	$0.696968\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.3923	−1.83010	−0.915052	−	0.403337i	$-0.867850\pi$
$101$	−18.3923	−1.83010	−0.915052	+	0.403337i	$0.867850\pi$
$102$	0	0
$103$	−4.08132	−0.402145	−0.201072	−	0.979576i	$-0.564443\pi$
$103$	−4.08132	−0.402145	−0.201072	+	0.979576i	$0.564443\pi$
$104$	0	0
$105$	0.627597	0.0612472
$106$	0	0
$107$	−14.9399	−1.44429	−0.722147	−	0.691740i	$-0.756845\pi$
$107$	−14.9399	−1.44429	−0.722147	+	0.691740i	$0.756845\pi$
$108$	0	0
$109$	−4.29391	−0.411282	−0.205641	−	0.978628i	$-0.565928\pi$
$109$	−4.29391	−0.411282	−0.205641	+	0.978628i	$0.565928\pi$
$110$	0	0
$111$	19.6955	1.86942
$112$	0	0
$113$	2.87912	0.270845	0.135423	−	0.990788i	$-0.456761\pi$
$113$	2.87912	0.270845	0.135423	+	0.990788i	$0.456761\pi$
$114$	0	0
$115$	1.69066	0.157655
$116$	0	0
$117$	−6.36101	−0.588076
$118$	0	0
$119$	−3.34995	−0.307089
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−18.9304	−1.70690
$124$	0	0
$125$	4.29473	0.384132
$126$	0	0
$127$	−20.4080	−1.81092	−0.905458	−	0.424435i	$-0.860473\pi$
$127$	−20.4080	−1.81092	−0.905458	+	0.424435i	$0.860473\pi$
$128$	0	0
$129$	8.26270	0.727490
$130$	0	0
$131$	−0.542045	−0.0473587	−0.0236793	−	0.999720i	$-0.507538\pi$
$131$	−0.542045	−0.0473587	−0.0236793	+	0.999720i	$0.507538\pi$
$132$	0	0
$133$	−0.692506	−0.0600479
$134$	0	0
$135$	1.55535	0.133863
$136$	0	0
$137$	8.78118	0.750227	0.375114	−	0.926979i	$-0.377604\pi$
$137$	8.78118	0.750227	0.375114	+	0.926979i	$0.377604\pi$
$138$	0	0
$139$	4.72530	0.400795	0.200397	−	0.979715i	$-0.435777\pi$
$139$	4.72530	0.400795	0.200397	+	0.979715i	$0.435777\pi$
$140$	0	0
$141$	16.3710	1.37869
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.00925863	−0.000768887 0
$146$	0	0
$147$	−13.4956	−1.11309
$148$	0	0
$149$	−11.2947	−0.925300	−0.462650	−	0.886541i	$-0.653101\pi$
$149$	−11.2947	−0.925300	−0.462650	+	0.886541i	$0.653101\pi$
$150$	0	0
$151$	12.5415	1.02061	0.510305	−	0.859994i	$-0.329533\pi$
$151$	12.5415	1.02061	0.510305	+	0.859994i	$0.329533\pi$
$152$	0	0
$153$	6.21026	0.502070
$154$	0	0
$155$	−2.46669	−0.198129
$156$	0	0
$157$	−9.63922	−0.769294	−0.384647	−	0.923064i	$-0.625677\pi$
$157$	−9.63922	−0.769294	−0.384647	+	0.923064i	$0.625677\pi$
$158$	0	0
$159$	15.0981	1.19736
$160$	0	0
$161$	2.67385	0.210729
$162$	0	0
$163$	−20.7050	−1.62174	−0.810872	−	0.585224i	$-0.801007\pi$
$163$	−20.7050	−1.62174	−0.810872	+	0.585224i	$0.801007\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0419	1.24136	0.620679	−	0.784065i	$-0.286857\pi$
$167$	16.0419	1.24136	0.620679	+	0.784065i	$0.286857\pi$
$168$	0	0
$169$	11.5506	0.888509
$170$	0	0
$171$	1.28379	0.0981742
$172$	0	0
$173$	−23.3559	−1.77572	−0.887859	−	0.460116i	$-0.847808\pi$
$173$	−23.3559	−1.77572	−0.887859	+	0.460116i	$0.847808\pi$
$174$	0	0
$175$	3.32976	0.251706
$176$	0	0
$177$	4.98539	0.374725
$178$	0	0
$179$	−8.40008	−0.627852	−0.313926	−	0.949448i	$-0.601644\pi$
$179$	−8.40008	−0.627852	−0.313926	+	0.949448i	$0.601644\pi$
$180$	0	0
$181$	−14.7596	−1.09707	−0.548537	−	0.836126i	$-0.684815\pi$
$181$	−14.7596	−1.09707	−0.548537	+	0.836126i	$0.684815\pi$
$182$	0	0
$183$	−11.8410	−0.875312
$184$	0	0
$185$	−4.16674	−0.306345
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.45984	0.178927
$190$	0	0
$191$	9.20325	0.665924	0.332962	−	0.942940i	$-0.391952\pi$
$191$	9.20325	0.665924	0.332962	+	0.942940i	$0.391952\pi$
$192$	0	0
$193$	4.76829	0.343229	0.171614	−	0.985164i	$-0.445102\pi$
$193$	4.76829	0.343229	0.171614	+	0.985164i	$0.445102\pi$
$194$	0	0
$195$	4.49044	0.321567
$196$	0	0
$197$	14.7979	1.05431	0.527153	−	0.849770i	$-0.323259\pi$
$197$	14.7979	1.05431	0.527153	+	0.849770i	$0.323259\pi$
$198$	0	0
$199$	−11.2890	−0.800258	−0.400129	−	0.916459i	$-0.631035\pi$
$199$	−11.2890	−0.800258	−0.400129	+	0.916459i	$0.631035\pi$
$200$	0	0
$201$	9.55846	0.674202
$202$	0	0
$203$	−0.0146429	−0.00102773
$204$	0	0
$205$	4.00487	0.279712
$206$	0	0
$207$	−4.95689	−0.344528
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.97358	−0.135867	−0.0679334	−	0.997690i	$-0.521641\pi$
$211$	−1.97358	−0.135867	−0.0679334	+	0.997690i	$0.521641\pi$
$212$	0	0
$213$	2.97994	0.204182
$214$	0	0
$215$	−1.74804	−0.119215
$216$	0	0
$217$	−3.90117	−0.264829
$218$	0	0
$219$	−1.52388	−0.102974
$220$	0	0
$221$	−23.9688	−1.61231
$222$	0	0
$223$	−17.7936	−1.19155	−0.595774	−	0.803152i	$-0.703155\pi$
$223$	−17.7936	−1.19155	−0.595774	+	0.803152i	$0.703155\pi$
$224$	0	0
$225$	−6.17283	−0.411522
$226$	0	0
$227$	8.22967	0.546222	0.273111	−	0.961983i	$-0.411947\pi$
$227$	8.22967	0.546222	0.273111	+	0.961983i	$0.411947\pi$
$228$	0	0
$229$	5.22880	0.345529	0.172764	−	0.984963i	$-0.444730\pi$
$229$	5.22880	0.345529	0.172764	+	0.984963i	$0.444730\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.2336	−1.26004	−0.630018	−	0.776580i	$-0.716953\pi$
$233$	−19.2336	−1.26004	−0.630018	+	0.776580i	$0.716953\pi$
$234$	0	0
$235$	−3.46341	−0.225928
$236$	0	0
$237$	−33.7760	−2.19399
$238$	0	0
$239$	−2.51056	−0.162395	−0.0811974	−	0.996698i	$-0.525874\pi$
$239$	−2.51056	−0.162395	−0.0811974	+	0.996698i	$0.525874\pi$
$240$	0	0
$241$	22.7131	1.46308	0.731539	−	0.681800i	$-0.238803\pi$
$241$	22.7131	1.46308	0.731539	+	0.681800i	$0.238803\pi$
$242$	0	0
$243$	−12.5315	−0.803895
$244$	0	0
$245$	2.85509	0.182405
$246$	0	0
$247$	−4.95486	−0.315270
$248$	0	0
$249$	−33.2447	−2.10680
$250$	0	0
$251$	22.1646	1.39902	0.699510	−	0.714623i	$-0.253402\pi$
$251$	22.1646	1.39902	0.699510	+	0.714623i	$0.253402\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.38401	−0.274538
$256$	0	0
$257$	−28.0963	−1.75260	−0.876301	−	0.481764i	$-0.839996\pi$
$257$	−28.0963	−1.75260	−0.876301	+	0.481764i	$0.839996\pi$
$258$	0	0
$259$	−6.58988	−0.409475
$260$	0	0
$261$	0.0271456	0.00168027
$262$	0	0
$263$	4.57484	0.282096	0.141048	−	0.990003i	$-0.454953\pi$
$263$	4.57484	0.282096	0.141048	+	0.990003i	$0.454953\pi$
$264$	0	0
$265$	−3.19413	−0.196214
$266$	0	0
$267$	−8.74744	−0.535335
$268$	0	0
$269$	11.2467	0.685725	0.342863	−	0.939386i	$-0.388603\pi$
$269$	11.2467	0.685725	0.342863	+	0.939386i	$0.388603\pi$
$270$	0	0
$271$	28.2160	1.71400	0.856999	−	0.515318i	$-0.172326\pi$
$271$	28.2160	1.71400	0.856999	+	0.515318i	$0.172326\pi$
$272$	0	0
$273$	7.10181	0.429821
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.16357	0.129996	0.0649982	−	0.997885i	$-0.479296\pi$
$277$	2.16357	0.129996	0.0649982	+	0.997885i	$0.479296\pi$
$278$	0	0
$279$	7.23214	0.432977
$280$	0	0
$281$	−6.24083	−0.372297	−0.186148	−	0.982522i	$-0.559600\pi$
$281$	−6.24083	−0.372297	−0.186148	+	0.982522i	$0.559600\pi$
$282$	0	0
$283$	−22.5372	−1.33970	−0.669850	−	0.742497i	$-0.733641\pi$
$283$	−22.5372	−1.33970	−0.669850	+	0.742497i	$0.733641\pi$
$284$	0	0
$285$	−0.906270	−0.0536828
$286$	0	0
$287$	6.33387	0.373876
$288$	0	0
$289$	6.40070	0.376512
$290$	0	0
$291$	−23.6481	−1.38628
$292$	0	0
$293$	11.1895	0.653696	0.326848	−	0.945077i	$-0.394013\pi$
$293$	11.1895	0.653696	0.326848	+	0.945077i	$0.394013\pi$
$294$	0	0
$295$	−1.05470	−0.0614069
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.1313	1.10639
$300$	0	0
$301$	−2.76459	−0.159348
$302$	0	0
$303$	−38.0672	−2.18690
$304$	0	0
$305$	2.50506	0.143439
$306$	0	0
$307$	−8.94447	−0.510488	−0.255244	−	0.966877i	$-0.582156\pi$
$307$	−8.94447	−0.510488	−0.255244	+	0.966877i	$0.582156\pi$
$308$	0	0
$309$	−8.44724	−0.480547
$310$	0	0
$311$	30.5837	1.73425	0.867123	−	0.498095i	$-0.165967\pi$
$311$	30.5837	1.73425	0.867123	+	0.498095i	$0.165967\pi$
$312$	0	0
$313$	−4.81524	−0.272173	−0.136087	−	0.990697i	$-0.543453\pi$
$313$	−4.81524	−0.272173	−0.136087	+	0.990697i	$0.543453\pi$
$314$	0	0
$315$	0.389280	0.0219334
$316$	0	0
$317$	−12.0570	−0.677187	−0.338593	−	0.940933i	$-0.609951\pi$
$317$	−12.0570	−0.677187	−0.338593	+	0.940933i	$0.609951\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−30.9216	−1.72587
$322$	0	0
$323$	4.83743	0.269162
$324$	0	0
$325$	23.8243	1.32153
$326$	0	0
$327$	−8.88724	−0.491466
$328$	0	0
$329$	−5.47753	−0.301986
$330$	0	0
$331$	13.4580	0.739716	0.369858	−	0.929088i	$-0.379406\pi$
$331$	13.4580	0.739716	0.369858	+	0.929088i	$0.379406\pi$
$332$	0	0
$333$	12.2166	0.669463
$334$	0	0
$335$	−2.02217	−0.110483
$336$	0	0
$337$	−19.9823	−1.08850	−0.544252	−	0.838922i	$-0.683186\pi$
$337$	−19.9823	−1.08850	−0.544252	+	0.838922i	$0.683186\pi$
$338$	0	0
$339$	5.95902	0.323649
$340$	0	0
$341$	0	0
$342$	0	0
$343$	9.36299	0.505554
$344$	0	0
$345$	3.49922	0.188392
$346$	0	0
$347$	−32.5836	−1.74918	−0.874589	−	0.484865i	$-0.838869\pi$
$347$	−32.5836	−1.74918	−0.874589	+	0.484865i	$0.838869\pi$
$348$	0	0
$349$	−3.49470	−0.187067	−0.0935336	−	0.995616i	$-0.529816\pi$
$349$	−3.49470	−0.187067	−0.0935336	+	0.995616i	$0.529816\pi$
$350$	0	0
$351$	17.6001	0.939423
$352$	0	0
$353$	23.4018	1.24555	0.622775	−	0.782401i	$-0.286005\pi$
$353$	23.4018	1.24555	0.622775	+	0.782401i	$0.286005\pi$
$354$	0	0
$355$	−0.630428	−0.0334597
$356$	0	0
$357$	−6.93350	−0.366960
$358$	0	0
$359$	−2.66333	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$359$	−2.66333	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.322389	0.0168746
$366$	0	0
$367$	1.68791	0.0881080	0.0440540	−	0.999029i	$-0.485973\pi$
$367$	1.68791	0.0881080	0.0440540	+	0.999029i	$0.485973\pi$
$368$	0	0
$369$	−11.7420	−0.611262
$370$	0	0
$371$	−5.05164	−0.262268
$372$	0	0
$373$	22.8819	1.18478	0.592389	−	0.805652i	$-0.298185\pi$
$373$	22.8819	1.18478	0.592389	+	0.805652i	$0.298185\pi$
$374$	0	0
$375$	8.88894	0.459023
$376$	0	0
$377$	−0.104769	−0.00539590
$378$	0	0
$379$	1.27786	0.0656393	0.0328197	−	0.999461i	$-0.489551\pi$
$379$	1.27786	0.0656393	0.0328197	+	0.999461i	$0.489551\pi$
$380$	0	0
$381$	−42.2391	−2.16397
$382$	0	0
$383$	28.0210	1.43181	0.715904	−	0.698199i	$-0.246015\pi$
$383$	28.0210	1.43181	0.715904	+	0.698199i	$0.246015\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.12510	0.260524
$388$	0	0
$389$	17.3045	0.877372	0.438686	−	0.898641i	$-0.355444\pi$
$389$	17.3045	0.877372	0.438686	+	0.898641i	$0.355444\pi$
$390$	0	0
$391$	−18.6779	−0.944583
$392$	0	0
$393$	−1.12189	−0.0565918
$394$	0	0
$395$	7.14558	0.359533
$396$	0	0
$397$	19.8924	0.998370	0.499185	−	0.866496i	$-0.333633\pi$
$397$	19.8924	0.998370	0.499185	+	0.866496i	$0.333633\pi$
$398$	0	0
$399$	−1.43330	−0.0717549
$400$	0	0
$401$	0.186866	0.00933163	0.00466581	−	0.999989i	$-0.498515\pi$
$401$	0.186866	0.00933163	0.00466581	+	0.999989i	$0.498515\pi$
$402$	0	0
$403$	−27.9127	−1.39043
$404$	0	0
$405$	4.90555	0.243759
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.79117	−0.138014	−0.0690072	−	0.997616i	$-0.521983\pi$
$409$	−2.79117	−0.138014	−0.0690072	+	0.997616i	$0.521983\pi$
$410$	0	0
$411$	18.1747	0.896492
$412$	0	0
$413$	−1.66805	−0.0820793
$414$	0	0
$415$	7.03317	0.345245
$416$	0	0
$417$	9.78010	0.478934
$418$	0	0
$419$	−1.89571	−0.0926117	−0.0463059	−	0.998927i	$-0.514745\pi$
$419$	−1.89571	−0.0926117	−0.0463059	+	0.998927i	$0.514745\pi$
$420$	0	0
$421$	15.0501	0.733497	0.366748	−	0.930320i	$-0.380471\pi$
$421$	15.0501	0.733497	0.366748	+	0.930320i	$0.380471\pi$
$422$	0	0
$423$	10.1544	0.493726
$424$	0	0
$425$	−23.2597	−1.12826
$426$	0	0
$427$	3.96185	0.191727
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.50689	0.168921	0.0844604	−	0.996427i	$-0.473083\pi$
$431$	3.50689	0.168921	0.0844604	+	0.996427i	$0.473083\pi$
$432$	0	0
$433$	−36.5590	−1.75691	−0.878456	−	0.477824i	$-0.841426\pi$
$433$	−36.5590	−1.75691	−0.878456	+	0.477824i	$0.841426\pi$
$434$	0	0
$435$	−0.0191629	−0.000918790 0
$436$	0	0
$437$	−3.86113	−0.184703
$438$	0	0
$439$	29.9247	1.42823	0.714113	−	0.700031i	$-0.246830\pi$
$439$	29.9247	1.42823	0.714113	+	0.700031i	$0.246830\pi$
$440$	0	0
$441$	−8.37089	−0.398614
$442$	0	0
$443$	−13.9182	−0.661272	−0.330636	−	0.943758i	$-0.607263\pi$
$443$	−13.9182	−0.661272	−0.330636	+	0.943758i	$0.607263\pi$
$444$	0	0
$445$	1.85059	0.0877263
$446$	0	0
$447$	−23.3771	−1.10570
$448$	0	0
$449$	−22.1501	−1.04533	−0.522663	−	0.852539i	$-0.675062\pi$
$449$	−22.1501	−1.04533	−0.522663	+	0.852539i	$0.675062\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	25.9575	1.21959
$454$	0	0
$455$	−1.50244	−0.0704356
$456$	0	0
$457$	17.2929	0.808928	0.404464	−	0.914554i	$-0.367458\pi$
$457$	17.2929	0.808928	0.404464	+	0.914554i	$0.367458\pi$
$458$	0	0
$459$	−17.1830	−0.802032
$460$	0	0
$461$	−18.4497	−0.859289	−0.429645	−	0.902998i	$-0.641361\pi$
$461$	−18.4497	−0.859289	−0.429645	+	0.902998i	$0.641361\pi$
$462$	0	0
$463$	−39.8532	−1.85213	−0.926067	−	0.377359i	$-0.876832\pi$
$463$	−39.8532	−1.85213	−0.926067	+	0.377359i	$0.876832\pi$
$464$	0	0
$465$	−5.10539	−0.236757
$466$	0	0
$467$	6.80337	0.314823	0.157411	−	0.987533i	$-0.449685\pi$
$467$	6.80337	0.314823	0.157411	+	0.987533i	$0.449685\pi$
$468$	0	0
$469$	−3.19814	−0.147676
$470$	0	0
$471$	−19.9506	−0.919276
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.80827	−0.220619
$476$	0	0
$477$	9.36493	0.428790
$478$	0	0
$479$	7.50981	0.343132	0.171566	−	0.985173i	$-0.445117\pi$
$479$	7.50981	0.343132	0.171566	+	0.985173i	$0.445117\pi$
$480$	0	0
$481$	−47.1503	−2.14987
$482$	0	0
$483$	5.53416	0.251813
$484$	0	0
$485$	5.00295	0.227172
$486$	0	0
$487$	6.06847	0.274989	0.137494	−	0.990503i	$-0.456095\pi$
$487$	6.06847	0.274989	0.137494	+	0.990503i	$0.456095\pi$
$488$	0	0
$489$	−42.8539	−1.93792
$490$	0	0
$491$	−6.40136	−0.288889	−0.144445	−	0.989513i	$-0.546140\pi$
$491$	−6.40136	−0.288889	−0.144445	+	0.989513i	$0.546140\pi$
$492$	0	0
$493$	0.102286	0.00460675
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.997049	−0.0447237
$498$	0	0
$499$	8.06069	0.360846	0.180423	−	0.983589i	$-0.442253\pi$
$499$	8.06069	0.360846	0.180423	+	0.983589i	$0.442253\pi$
$500$	0	0
$501$	33.2024	1.48337
$502$	0	0
$503$	4.21016	0.187722	0.0938608	−	0.995585i	$-0.470079\pi$
$503$	4.21016	0.187722	0.0938608	+	0.995585i	$0.470079\pi$
$504$	0	0
$505$	8.05340	0.358372
$506$	0	0
$507$	23.9067	1.06173
$508$	0	0
$509$	−5.81571	−0.257777	−0.128888	−	0.991659i	$-0.541141\pi$
$509$	−5.81571	−0.257777	−0.128888	+	0.991659i	$0.541141\pi$
$510$	0	0
$511$	0.509872	0.0225554
$512$	0	0
$513$	−3.55209	−0.156829
$514$	0	0
$515$	1.78708	0.0787481
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−48.3405	−2.12191
$520$	0	0
$521$	33.7663	1.47933	0.739664	−	0.672976i	$-0.234984\pi$
$521$	33.7663	1.47933	0.739664	+	0.672976i	$0.234984\pi$
$522$	0	0
$523$	9.52752	0.416609	0.208305	−	0.978064i	$-0.433205\pi$
$523$	9.52752	0.416609	0.208305	+	0.978064i	$0.433205\pi$
$524$	0	0
$525$	6.89171	0.300779
$526$	0	0
$527$	27.2512	1.18708
$528$	0	0
$529$	−8.09170	−0.351813
$530$	0	0
$531$	3.09229	0.134194
$532$	0	0
$533$	45.3186	1.96297
$534$	0	0
$535$	6.54170	0.282822
$536$	0	0
$537$	−17.3859	−0.750258
$538$	0	0
$539$	0	0
$540$	0	0
$541$	26.2891	1.13026	0.565128	−	0.825003i	$-0.308827\pi$
$541$	26.2891	1.13026	0.565128	+	0.825003i	$0.308827\pi$
$542$	0	0
$543$	−30.5485	−1.31096
$544$	0	0
$545$	1.88016	0.0805374
$546$	0	0
$547$	13.0877	0.559591	0.279796	−	0.960060i	$-0.409733\pi$
$547$	13.0877	0.559591	0.279796	+	0.960060i	$0.409733\pi$
$548$	0	0
$549$	−7.34462	−0.313461
$550$	0	0
$551$	0.0211448	0.000900799 0
$552$	0	0
$553$	11.3010	0.480568
$554$	0	0
$555$	−8.62404	−0.366070
$556$	0	0
$557$	−13.5096	−0.572422	−0.286211	−	0.958167i	$-0.592396\pi$
$557$	−13.5096	−0.572422	−0.286211	+	0.958167i	$0.592396\pi$
$558$	0	0
$559$	−19.7806	−0.836629
$560$	0	0
$561$	0	0
$562$	0	0
$563$	35.4952	1.49595	0.747973	−	0.663730i	$-0.231027\pi$
$563$	35.4952	1.49595	0.747973	+	0.663730i	$0.231027\pi$
$564$	0	0
$565$	−1.26068	−0.0530370
$566$	0	0
$567$	7.75832	0.325819
$568$	0	0
$569$	1.71511	0.0719012	0.0359506	−	0.999354i	$-0.488554\pi$
$569$	1.71511	0.0719012	0.0359506	+	0.999354i	$0.488554\pi$
$570$	0	0
$571$	39.6069	1.65750	0.828748	−	0.559622i	$-0.189054\pi$
$571$	39.6069	1.65750	0.828748	+	0.559622i	$0.189054\pi$
$572$	0	0
$573$	19.0483	0.795753
$574$	0	0
$575$	18.5653	0.774229
$576$	0	0
$577$	24.5056	1.02018	0.510091	−	0.860121i	$-0.329612\pi$
$577$	24.5056	1.02018	0.510091	+	0.860121i	$0.329612\pi$
$578$	0	0
$579$	9.86908	0.410145
$580$	0	0
$581$	11.1233	0.461470
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.78528	0.115157
$586$	0	0
$587$	−16.3384	−0.674359	−0.337180	−	0.941440i	$-0.609473\pi$
$587$	−16.3384	−0.674359	−0.337180	+	0.941440i	$0.609473\pi$
$588$	0	0
$589$	5.63341	0.232121
$590$	0	0
$591$	30.6277	1.25985
$592$	0	0
$593$	17.6981	0.726776	0.363388	−	0.931638i	$-0.381620\pi$
$593$	17.6981	0.726776	0.363388	+	0.931638i	$0.381620\pi$
$594$	0	0
$595$	1.46683	0.0601344
$596$	0	0
$597$	−23.3653	−0.956276
$598$	0	0
$599$	−24.0307	−0.981868	−0.490934	−	0.871197i	$-0.663344\pi$
$599$	−24.0307	−0.981868	−0.490934	+	0.871197i	$0.663344\pi$
$600$	0	0
$601$	−39.0654	−1.59351	−0.796756	−	0.604301i	$-0.793453\pi$
$601$	−39.0654	−1.59351	−0.796756	+	0.604301i	$0.793453\pi$
$602$	0	0
$603$	5.92883	0.241440
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.47512	−0.0598732	−0.0299366	−	0.999552i	$-0.509531\pi$
$607$	−1.47512	−0.0598732	−0.0299366	+	0.999552i	$0.509531\pi$
$608$	0	0
$609$	−0.0303069	−0.00122810
$610$	0	0
$611$	−39.1915	−1.58552
$612$	0	0
$613$	−28.8847	−1.16664	−0.583321	−	0.812242i	$-0.698247\pi$
$613$	−28.8847	−1.16664	−0.583321	+	0.812242i	$0.698247\pi$
$614$	0	0
$615$	8.28901	0.334245
$616$	0	0
$617$	−43.5693	−1.75403	−0.877016	−	0.480461i	$-0.840469\pi$
$617$	−43.5693	−1.75403	−0.877016	+	0.480461i	$0.840469\pi$
$618$	0	0
$619$	−10.6711	−0.428906	−0.214453	−	0.976734i	$-0.568797\pi$
$619$	−10.6711	−0.428906	−0.214453	+	0.976734i	$0.568797\pi$
$620$	0	0
$621$	13.7151	0.550367
$622$	0	0
$623$	2.92678	0.117259
$624$	0	0
$625$	22.1608	0.886433
$626$	0	0
$627$	0	0
$628$	0	0
$629$	46.0329	1.83545
$630$	0	0
$631$	−12.6253	−0.502605	−0.251303	−	0.967909i	$-0.580859\pi$
$631$	−12.6253	−0.502605	−0.251303	+	0.967909i	$0.580859\pi$
$632$	0	0
$633$	−4.08478	−0.162355
$634$	0	0
$635$	8.93601	0.354615
$636$	0	0
$637$	32.3078	1.28008
$638$	0	0
$639$	1.84837	0.0731202
$640$	0	0
$641$	−3.57802	−0.141323	−0.0706617	−	0.997500i	$-0.522511\pi$
$641$	−3.57802	−0.141323	−0.0706617	+	0.997500i	$0.522511\pi$
$642$	0	0
$643$	14.5338	0.573155	0.286578	−	0.958057i	$-0.407482\pi$
$643$	14.5338	0.573155	0.286578	+	0.958057i	$0.407482\pi$
$644$	0	0
$645$	−3.61797	−0.142457
$646$	0	0
$647$	14.5002	0.570063	0.285031	−	0.958518i	$-0.407996\pi$
$647$	14.5002	0.570063	0.285031	+	0.958518i	$0.407996\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.07438	−0.316460
$652$	0	0
$653$	20.9992	0.821762	0.410881	−	0.911689i	$-0.365221\pi$
$653$	20.9992	0.821762	0.410881	+	0.911689i	$0.365221\pi$
$654$	0	0
$655$	0.237344	0.00927380
$656$	0	0
$657$	−0.945219	−0.0368765
$658$	0	0
$659$	−43.7854	−1.70564	−0.852819	−	0.522206i	$-0.825109\pi$
$659$	−43.7854	−1.70564	−0.852819	+	0.522206i	$0.825109\pi$
$660$	0	0
$661$	44.4370	1.72840	0.864199	−	0.503150i	$-0.167826\pi$
$661$	44.4370	1.72840	0.864199	+	0.503150i	$0.167826\pi$
$662$	0	0
$663$	−49.6089	−1.92665
$664$	0	0
$665$	0.303226	0.0117586
$666$	0	0
$667$	−0.0816428	−0.00316122
$668$	0	0
$669$	−36.8280	−1.42385
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.6117	0.755976	0.377988	−	0.925810i	$-0.376616\pi$
$673$	19.6117	0.755976	0.377988	+	0.925810i	$0.376616\pi$
$674$	0	0
$675$	17.0794	0.657387
$676$	0	0
$677$	37.0347	1.42336	0.711680	−	0.702504i	$-0.247934\pi$
$677$	37.0347	1.42336	0.711680	+	0.702504i	$0.247934\pi$
$678$	0	0
$679$	7.91236	0.303649
$680$	0	0
$681$	17.0332	0.652714
$682$	0	0
$683$	17.9808	0.688015	0.344007	−	0.938967i	$-0.388215\pi$
$683$	17.9808	0.688015	0.344007	+	0.938967i	$0.388215\pi$
$684$	0	0
$685$	−3.84500	−0.146910
$686$	0	0
$687$	10.8222	0.412893
$688$	0	0
$689$	−36.1443	−1.37699
$690$	0	0
$691$	16.0334	0.609938	0.304969	−	0.952362i	$-0.401354\pi$
$691$	16.0334	0.609938	0.304969	+	0.952362i	$0.401354\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.06906	−0.0784838
$696$	0	0
$697$	−44.2446	−1.67588
$698$	0	0
$699$	−39.8085	−1.50569
$700$	0	0
$701$	−8.23224	−0.310927	−0.155464	−	0.987842i	$-0.549687\pi$
$701$	−8.23224	−0.310927	−0.155464	+	0.987842i	$0.549687\pi$
$702$	0	0
$703$	9.51598	0.358902
$704$	0	0
$705$	−7.16833	−0.269975
$706$	0	0
$707$	12.7368	0.479016
$708$	0	0
$709$	7.90414	0.296846	0.148423	−	0.988924i	$-0.452580\pi$
$709$	7.90414	0.296846	0.148423	+	0.988924i	$0.452580\pi$
$710$	0	0
$711$	−20.9503	−0.785696
$712$	0	0
$713$	−21.7513	−0.814593
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.19619	−0.194055
$718$	0	0
$719$	14.9502	0.557549	0.278774	−	0.960357i	$-0.410072\pi$
$719$	14.9502	0.557549	0.278774	+	0.960357i	$0.410072\pi$
$720$	0	0
$721$	2.82634	0.105258
$722$	0	0
$723$	47.0100	1.74832
$724$	0	0
$725$	−0.101670	−0.00377593
$726$	0	0
$727$	−8.92157	−0.330883	−0.165441	−	0.986220i	$-0.552905\pi$
$727$	−8.92157	−0.330883	−0.165441	+	0.986220i	$0.552905\pi$
$728$	0	0
$729$	7.67295	0.284183
$730$	0	0
$731$	19.3118	0.714271
$732$	0	0
$733$	−46.9378	−1.73369	−0.866844	−	0.498579i	$-0.833855\pi$
$733$	−46.9378	−1.73369	−0.866844	+	0.498579i	$0.833855\pi$
$734$	0	0
$735$	5.90927	0.217967
$736$	0	0
$737$	0	0
$738$	0	0
$739$	13.1963	0.485434	0.242717	−	0.970097i	$-0.421961\pi$
$739$	13.1963	0.485434	0.242717	+	0.970097i	$0.421961\pi$
$740$	0	0
$741$	−10.2552	−0.376735
$742$	0	0
$743$	4.99731	0.183334	0.0916668	−	0.995790i	$-0.470781\pi$
$743$	4.99731	0.183334	0.0916668	+	0.995790i	$0.470781\pi$
$744$	0	0
$745$	4.94560	0.181193
$746$	0	0
$747$	−20.6207	−0.754472
$748$	0	0
$749$	10.3460	0.378033
$750$	0	0
$751$	−37.4419	−1.36627	−0.683136	−	0.730291i	$-0.739385\pi$
$751$	−37.4419	−1.36627	−0.683136	+	0.730291i	$0.739385\pi$
$752$	0	0
$753$	45.8749	1.67177
$754$	0	0
$755$	−5.49150	−0.199856
$756$	0	0
$757$	2.19395	0.0797405	0.0398703	−	0.999205i	$-0.487306\pi$
$757$	2.19395	0.0797405	0.0398703	+	0.999205i	$0.487306\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26.7512	0.969732	0.484866	−	0.874589i	$-0.338868\pi$
$761$	26.7512	0.969732	0.484866	+	0.874589i	$0.338868\pi$
$762$	0	0
$763$	2.97356	0.107650
$764$	0	0
$765$	−2.71927	−0.0983155
$766$	0	0
$767$	−11.9348	−0.430942
$768$	0	0
$769$	−50.5117	−1.82150	−0.910749	−	0.412961i	$-0.864495\pi$
$769$	−50.5117	−1.82150	−0.910749	+	0.412961i	$0.864495\pi$
$770$	0	0
$771$	−58.1519	−2.09429
$772$	0	0
$773$	13.2270	0.475740	0.237870	−	0.971297i	$-0.423551\pi$
$773$	13.2270	0.475740	0.237870	+	0.971297i	$0.423551\pi$
$774$	0	0
$775$	−27.0870	−0.972992
$776$	0	0
$777$	−13.6393	−0.489306
$778$	0	0
$779$	−9.14630	−0.327700
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.0751082	−0.00268415
$784$	0	0
$785$	4.22071	0.150643
$786$	0	0
$787$	−33.2240	−1.18431	−0.592154	−	0.805825i	$-0.701722\pi$
$787$	−33.2240	−1.18431	−0.592154	+	0.805825i	$0.701722\pi$
$788$	0	0
$789$	9.46869	0.337094
$790$	0	0
$791$	−1.99381	−0.0708918
$792$	0	0
$793$	28.3469	1.00663
$794$	0	0
$795$	−6.61099	−0.234468
$796$	0	0
$797$	−28.8947	−1.02350	−0.511751	−	0.859134i	$-0.671003\pi$
$797$	−28.8947	−1.02350	−0.511751	+	0.859134i	$0.671003\pi$
$798$	0	0
$799$	38.2627	1.35364
$800$	0	0
$801$	−5.42578	−0.191710
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.17080	−0.0412651
$806$	0	0
$807$	23.2777	0.819415
$808$	0	0
$809$	31.4004	1.10398	0.551989	−	0.833851i	$-0.313869\pi$
$809$	31.4004	1.10398	0.551989	+	0.833851i	$0.313869\pi$
$810$	0	0
$811$	−32.3605	−1.13633	−0.568166	−	0.822914i	$-0.692347\pi$
$811$	−32.3605	−1.13633	−0.568166	+	0.822914i	$0.692347\pi$
$812$	0	0
$813$	58.3995	2.04816
$814$	0	0
$815$	9.06607	0.317571
$816$	0	0
$817$	3.99216	0.139668
$818$	0	0
$819$	4.40504	0.153925
$820$	0	0
$821$	−7.10604	−0.248003	−0.124001	−	0.992282i	$-0.539573\pi$
$821$	−7.10604	−0.248003	−0.124001	+	0.992282i	$0.539573\pi$
$822$	0	0
$823$	−14.7226	−0.513198	−0.256599	−	0.966518i	$-0.582602\pi$
$823$	−14.7226	−0.513198	−0.256599	+	0.966518i	$0.582602\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.45289	−0.189616	−0.0948078	−	0.995496i	$-0.530224\pi$
$827$	−5.45289	−0.189616	−0.0948078	+	0.995496i	$0.530224\pi$
$828$	0	0
$829$	41.3069	1.43465	0.717324	−	0.696740i	$-0.245367\pi$
$829$	41.3069	1.43465	0.717324	+	0.696740i	$0.245367\pi$
$830$	0	0
$831$	4.47802	0.155341
$832$	0	0
$833$	−31.5421	−1.09287
$834$	0	0
$835$	−7.02422	−0.243083
$836$	0	0
$837$	−20.0104	−0.691660
$838$	0	0
$839$	−37.6135	−1.29856	−0.649280	−	0.760549i	$-0.724930\pi$
$839$	−37.6135	−1.29856	−0.649280	+	0.760549i	$0.724930\pi$
$840$	0	0
$841$	−28.9996	−0.999985
$842$	0	0
$843$	−12.9168	−0.444880
$844$	0	0
$845$	−5.05764	−0.173988
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−46.6460	−1.60089
$850$	0	0
$851$	−36.7424	−1.25951
$852$	0	0
$853$	−9.52973	−0.326292	−0.163146	−	0.986602i	$-0.552164\pi$
$853$	−9.52973	−0.326292	−0.163146	+	0.986602i	$0.552164\pi$
$854$	0	0
$855$	−0.562132	−0.0192245
$856$	0	0
$857$	38.4495	1.31341	0.656705	−	0.754147i	$-0.271950\pi$
$857$	38.4495	1.31341	0.656705	+	0.754147i	$0.271950\pi$
$858$	0	0
$859$	−15.0710	−0.514216	−0.257108	−	0.966383i	$-0.582770\pi$
$859$	−15.0710	−0.514216	−0.257108	+	0.966383i	$0.582770\pi$
$860$	0	0
$861$	13.1094	0.446768
$862$	0	0
$863$	−9.98060	−0.339743	−0.169872	−	0.985466i	$-0.554335\pi$
$863$	−9.98060	−0.339743	−0.169872	+	0.985466i	$0.554335\pi$
$864$	0	0
$865$	10.2268	0.347722
$866$	0	0
$867$	13.2477	0.449917
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−22.8826	−0.775346
$872$	0	0
$873$	−14.6682	−0.496445
$874$	0	0
$875$	−2.97413	−0.100544
$876$	0	0
$877$	−8.76513	−0.295977	−0.147989	−	0.988989i	$-0.547280\pi$
$877$	−8.76513	−0.295977	−0.147989	+	0.988989i	$0.547280\pi$
$878$	0	0
$879$	23.1592	0.781141
$880$	0	0
$881$	−8.25318	−0.278057	−0.139028	−	0.990288i	$-0.544398\pi$
$881$	−8.25318	−0.278057	−0.139028	+	0.990288i	$0.544398\pi$
$882$	0	0
$883$	38.5899	1.29865	0.649327	−	0.760509i	$-0.275051\pi$
$883$	38.5899	1.29865	0.649327	+	0.760509i	$0.275051\pi$
$884$	0	0
$885$	−2.18294	−0.0733788
$886$	0	0
$887$	26.8505	0.901553	0.450776	−	0.892637i	$-0.351147\pi$
$887$	26.8505	0.901553	0.450776	+	0.892637i	$0.351147\pi$
$888$	0	0
$889$	14.1327	0.473994
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.90972	0.264689
$894$	0	0
$895$	3.67813	0.122946
$896$	0	0
$897$	39.5968	1.32210
$898$	0	0
$899$	0.119117	0.00397279
$900$	0	0
$901$	35.2877	1.17560
$902$	0	0
$903$	−5.72197	−0.190415
$904$	0	0
$905$	6.46276	0.214830
$906$	0	0
$907$	4.64221	0.154142	0.0770710	−	0.997026i	$-0.475443\pi$
$907$	4.64221	0.154142	0.0770710	+	0.997026i	$0.475443\pi$
$908$	0	0
$909$	−23.6119	−0.783158
$910$	0	0
$911$	43.6674	1.44677	0.723383	−	0.690447i	$-0.242586\pi$
$911$	43.6674	1.44677	0.723383	+	0.690447i	$0.242586\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	5.18480	0.171404
$916$	0	0
$917$	0.375369	0.0123958
$918$	0	0
$919$	29.8684	0.985269	0.492634	−	0.870236i	$-0.336034\pi$
$919$	29.8684	0.985269	0.492634	+	0.870236i	$0.336034\pi$
$920$	0	0
$921$	−18.5127	−0.610013
$922$	0	0
$923$	−7.13385	−0.234813
$924$	0	0
$925$	−45.7554	−1.50443
$926$	0	0
$927$	−5.23957	−0.172090
$928$	0	0
$929$	−52.6120	−1.72614	−0.863071	−	0.505082i	$-0.831462\pi$
$929$	−52.6120	−1.72614	−0.863071	+	0.505082i	$0.831462\pi$
$930$	0	0
$931$	−6.52044	−0.213699
$932$	0	0
$933$	63.3002	2.07236
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.8852	0.910970	0.455485	−	0.890244i	$-0.349466\pi$
$937$	27.8852	0.910970	0.455485	+	0.890244i	$0.349466\pi$
$938$	0	0
$939$	−9.96626	−0.325236
$940$	0	0
$941$	−23.1273	−0.753927	−0.376964	−	0.926228i	$-0.623032\pi$
$941$	−23.1273	−0.753927	−0.376964	+	0.926228i	$0.623032\pi$
$942$	0	0
$943$	35.3150	1.15002
$944$	0	0
$945$	−1.07709	−0.0350376
$946$	0	0
$947$	−47.2334	−1.53488	−0.767440	−	0.641120i	$-0.778470\pi$
$947$	−47.2334	−1.53488	−0.767440	+	0.641120i	$0.778470\pi$
$948$	0	0
$949$	3.64811	0.118423
$950$	0	0
$951$	−24.9547	−0.809212
$952$	0	0
$953$	−7.64014	−0.247488	−0.123744	−	0.992314i	$-0.539490\pi$
$953$	−7.64014	−0.247488	−0.123744	+	0.992314i	$0.539490\pi$
$954$	0	0
$955$	−4.02981	−0.130402
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.08102	−0.196367
$960$	0	0
$961$	0.735306	0.0237195
$962$	0	0
$963$	−19.1797	−0.618059
$964$	0	0
$965$	−2.08788	−0.0672112
$966$	0	0
$967$	41.2560	1.32670	0.663351	−	0.748308i	$-0.269134\pi$
$967$	41.2560	1.32670	0.663351	+	0.748308i	$0.269134\pi$
$968$	0	0
$969$	10.0122	0.321638
$970$	0	0
$971$	−20.0087	−0.642109	−0.321054	−	0.947061i	$-0.604037\pi$
$971$	−20.0087	−0.642109	−0.321054	+	0.947061i	$0.604037\pi$
$972$	0	0
$973$	−3.27230	−0.104905
$974$	0	0
$975$	49.3099	1.57918
$976$	0	0
$977$	−1.57451	−0.0503730	−0.0251865	−	0.999683i	$-0.508018\pi$
$977$	−1.57451	−0.0503730	−0.0251865	+	0.999683i	$0.508018\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−5.51249	−0.176000
$982$	0	0
$983$	−32.5468	−1.03808	−0.519042	−	0.854749i	$-0.673711\pi$
$983$	−32.5468	−1.03808	−0.519042	+	0.854749i	$0.673711\pi$
$984$	0	0
$985$	−6.47952	−0.206455
$986$	0	0
$987$	−11.3370	−0.360861
$988$	0	0
$989$	−15.4142	−0.490144
$990$	0	0
$991$	40.7111	1.29323	0.646616	−	0.762816i	$-0.276184\pi$
$991$	40.7111	1.29323	0.646616	+	0.762816i	$0.276184\pi$
$992$	0	0
$993$	27.8544	0.883931
$994$	0	0
$995$	4.94310	0.156707
$996$	0	0
$997$	48.2060	1.52670	0.763350	−	0.645985i	$-0.223553\pi$
$997$	48.2060	1.52670	0.763350	+	0.645985i	$0.223553\pi$
$998$	0	0
$999$	−33.8016	−1.06944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.l.1.6	✓	6
11.10	odd	2		9196.2.a.m.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.l.1.6	✓	6	1.1	even	1	trivial
9196.2.a.m.1.6	yes	6	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)

\(f(q)\)	\(=\)	\(q+2.06973 q^{3} -0.437868 q^{5} -0.692506 q^{7} +1.28379 q^{9} -4.95486 q^{13} -0.906270 q^{15} +4.83743 q^{17} +1.00000 q^{19} -1.43330 q^{21} -3.86113 q^{23} -4.80827 q^{25} -3.55209 q^{27} +0.0211448 q^{29} +5.63341 q^{31} +0.303226 q^{35} +9.51598 q^{37} -10.2552 q^{39} -9.14630 q^{41} +3.99216 q^{43} -0.562132 q^{45} +7.90972 q^{47} -6.52044 q^{49} +10.0122 q^{51} +7.29473 q^{53} +2.06973 q^{57} +2.40871 q^{59} -5.72103 q^{61} -0.889035 q^{63} +2.16957 q^{65} +4.61821 q^{67} -7.99150 q^{69} +1.43977 q^{71} -0.736270 q^{73} -9.95184 q^{75} -16.3190 q^{79} -11.2033 q^{81} -16.0623 q^{83} -2.11815 q^{85} +0.0437641 q^{87} -4.22636 q^{89} +3.43127 q^{91} +11.6597 q^{93} -0.437868 q^{95} -11.4257 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 6 q^{17} + 6 q^{19} + 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} + 2 q^{29} - 6 q^{31} - 14 q^{35} + 4 q^{37} - 6 q^{39} - 14 q^{41} + 10 q^{43}+ \cdots + 28 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 6 * q^17 + 6 * q^19 + 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 + 2 * q^29 - 6 * q^31 - 14 * q^35 + 4 * q^37 - 6 * q^39 - 14 * q^41 + 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 - 18 * q^51 + 20 * q^59 + 2 * q^61 - 12 * q^63 + 12 * q^65 + 10 * q^67 - 20 * q^69 - 32 * q^73 + 18 * q^75 - 6 * q^79 - 6 * q^81 - 36 * q^83 - 6 * q^85 + 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 - 4 * q^95 + 28 * q^97

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