LMFDB - Embedded newform 2178.2.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2178,2,Mod(1,2178)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2178, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2178.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2178.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,0,0,0,-2,0,0,0,0,0,0,0,2,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.3914175602$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} +3.46410 q^{7} -1.00000 q^{8} +3.46410 q^{10} +3.46410 q^{13} -3.46410 q^{14} +1.00000 q^{16} -6.00000 q^{17} -6.92820 q^{19} -3.46410 q^{20} +3.46410 q^{23} +7.00000 q^{25} -3.46410 q^{26} +3.46410 q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -12.0000 q^{35} -2.00000 q^{37} +6.92820 q^{38} +3.46410 q^{40} -6.00000 q^{41} -3.46410 q^{46} +10.3923 q^{47} +5.00000 q^{49} -7.00000 q^{50} +3.46410 q^{52} +10.3923 q^{53} -3.46410 q^{56} -6.00000 q^{58} +6.92820 q^{59} -3.46410 q^{61} +8.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +4.00000 q^{67} -6.00000 q^{68} +12.0000 q^{70} +3.46410 q^{71} -13.8564 q^{73} +2.00000 q^{74} -6.92820 q^{76} -3.46410 q^{79} -3.46410 q^{80} +6.00000 q^{82} -12.0000 q^{83} +20.7846 q^{85} -13.8564 q^{89} +12.0000 q^{91} +3.46410 q^{92} -10.3923 q^{94} +24.0000 q^{95} -10.0000 q^{97} -5.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{16} - 12 q^{17} + 14 q^{25} + 12 q^{29} - 16 q^{31} - 2 q^{32} + 12 q^{34} - 24 q^{35} - 4 q^{37} - 12 q^{41} + 10 q^{49} - 14 q^{50} - 12 q^{58} + 16 q^{62} + 2 q^{64}+ \cdots - 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^16 - 12 * q^17 + 14 * q^25 + 12 * q^29 - 16 * q^31 - 2 * q^32 + 12 * q^34 - 24 * q^35 - 4 * q^37 - 12 * q^41 + 10 * q^49 - 14 * q^50 - 12 * q^58 + 16 * q^62 + 2 * q^64 - 24 * q^65 + 8 * q^67 - 12 * q^68 + 24 * q^70 + 4 * q^74 + 12 * q^82 - 24 * q^83 + 24 * q^91 + 48 * q^95 - 20 * q^97 - 10 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.46410	1.09545
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−6.92820	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$20$	−3.46410	−0.774597
$21$	0	0
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	−3.46410	−0.679366
$27$	0	0
$28$	3.46410	0.654654
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	−12.0000	−2.02837
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	6.92820	1.12390
$39$	0	0
$40$	3.46410	0.547723
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−3.46410	−0.510754
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	−7.00000	−0.989949
$51$	0	0
$52$	3.46410	0.480384
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	0	0
$56$	−3.46410	−0.462910
$57$	0	0
$58$	−6.00000	−0.787839
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	−3.46410	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	12.0000	1.43427
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	−13.8564	−1.62177	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	−13.8564	−1.62177	−0.810885	+	0.585206i	$0.801014\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.92820	−0.794719
$77$	0	0
$78$	0	0
$79$	−3.46410	−0.389742	−0.194871	−	0.980829i	$-0.562429\pi$
$79$	−3.46410	−0.389742	−0.194871	+	0.980829i	$0.562429\pi$
$80$	−3.46410	−0.387298
$81$	0	0
$82$	6.00000	0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	20.7846	2.25441
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.8564	−1.46878	−0.734388	−	0.678730i	$-0.762531\pi$
$89$	−13.8564	−1.46878	−0.734388	+	0.678730i	$0.762531\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	3.46410	0.361158
$93$	0	0
$94$	−10.3923	−1.07188
$95$	24.0000	2.46235
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−5.00000	−0.505076
$99$	0	0
$100$	7.00000	0.700000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	−10.3923	−1.00939
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−10.3923	−0.995402	−0.497701	−	0.867349i	$-0.665822\pi$
$109$	−10.3923	−0.995402	−0.497701	+	0.867349i	$0.665822\pi$
$110$	0	0
$111$	0	0
$112$	3.46410	0.327327
$113$	−13.8564	−1.30350	−0.651751	−	0.758433i	$-0.725965\pi$
$113$	−13.8564	−1.30350	−0.651751	+	0.758433i	$0.725965\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	6.00000	0.557086
$117$	0	0
$118$	−6.92820	−0.637793
$119$	−20.7846	−1.90532
$120$	0	0
$121$	0	0
$122$	3.46410	0.313625
$123$	0	0
$124$	−8.00000	−0.718421
$125$	−6.92820	−0.619677
$126$	0	0
$127$	10.3923	0.922168	0.461084	−	0.887357i	$-0.347461\pi$
$127$	10.3923	0.922168	0.461084	+	0.887357i	$0.347461\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.0000	1.05247
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	0	0
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	−12.0000	−1.01419
$141$	0	0
$142$	−3.46410	−0.290701
$143$	0	0
$144$	0	0
$145$	−20.7846	−1.72607
$146$	13.8564	1.14676
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−3.46410	−0.281905	−0.140952	−	0.990016i	$-0.545016\pi$
$151$	−3.46410	−0.281905	−0.140952	+	0.990016i	$0.545016\pi$
$152$	6.92820	0.561951
$153$	0	0
$154$	0	0
$155$	27.7128	2.22595
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	3.46410	0.275589
$159$	0	0
$160$	3.46410	0.273861
$161$	12.0000	0.945732
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	−20.7846	−1.59411
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	24.2487	1.83303
$176$	0	0
$177$	0	0
$178$	13.8564	1.03858
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−12.0000	−0.889499
$183$	0	0
$184$	−3.46410	−0.255377
$185$	6.92820	0.509372
$186$	0	0
$187$	0	0
$188$	10.3923	0.757937
$189$	0	0
$190$	−24.0000	−1.74114
$191$	−17.3205	−1.25327	−0.626634	−	0.779314i	$-0.715568\pi$
$191$	−17.3205	−1.25327	−0.626634	+	0.779314i	$0.715568\pi$
$192$	0	0
$193$	−6.92820	−0.498703	−0.249351	−	0.968413i	$-0.580217\pi$
$193$	−6.92820	−0.498703	−0.249351	+	0.968413i	$0.580217\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	5.00000	0.357143
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−7.00000	−0.494975
$201$	0	0
$202$	18.0000	1.26648
$203$	20.7846	1.45879
$204$	0	0
$205$	20.7846	1.45166
$206$	16.0000	1.11477
$207$	0	0
$208$	3.46410	0.240192
$209$	0	0
$210$	0	0
$211$	27.7128	1.90783	0.953914	−	0.300079i	$-0.0970130\pi$
$211$	27.7128	1.90783	0.953914	+	0.300079i	$0.0970130\pi$
$212$	10.3923	0.713746
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−27.7128	−1.88127
$218$	10.3923	0.703856
$219$	0	0
$220$	0	0
$221$	−20.7846	−1.39812
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−3.46410	−0.231455
$225$	0	0
$226$	13.8564	0.921714
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−36.0000	−2.34838
$236$	6.92820	0.450988
$237$	0	0
$238$	20.7846	1.34727
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	6.92820	0.446285	0.223142	−	0.974786i	$-0.428369\pi$
$241$	6.92820	0.446285	0.223142	+	0.974786i	$0.428369\pi$
$242$	0	0
$243$	0	0
$244$	−3.46410	−0.221766
$245$	−17.3205	−1.10657
$246$	0	0
$247$	−24.0000	−1.52708
$248$	8.00000	0.508001
$249$	0	0
$250$	6.92820	0.438178
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	0	0
$254$	−10.3923	−0.652071
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−6.92820	−0.430498
$260$	−12.0000	−0.744208
$261$	0	0
$262$	12.0000	0.741362
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	24.0000	1.47153
$267$	0	0
$268$	4.00000	0.244339
$269$	10.3923	0.633630	0.316815	−	0.948487i	$-0.397387\pi$
$269$	10.3923	0.633630	0.316815	+	0.948487i	$0.397387\pi$
$270$	0	0
$271$	3.46410	0.210429	0.105215	−	0.994450i	$-0.466447\pi$
$271$	3.46410	0.210429	0.105215	+	0.994450i	$0.466447\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	6.92820	0.418548
$275$	0	0
$276$	0	0
$277$	10.3923	0.624413	0.312207	−	0.950014i	$-0.398932\pi$
$277$	10.3923	0.624413	0.312207	+	0.950014i	$0.398932\pi$
$278$	−6.92820	−0.415526
$279$	0	0
$280$	12.0000	0.717137
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	3.46410	0.205557
$285$	0	0
$286$	0	0
$287$	−20.7846	−1.22688
$288$	0	0
$289$	19.0000	1.11765
$290$	20.7846	1.22051
$291$	0	0
$292$	−13.8564	−0.810885
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	2.00000	0.116248
$297$	0	0
$298$	6.00000	0.347571
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	3.46410	0.199337
$303$	0	0
$304$	−6.92820	−0.397360
$305$	12.0000	0.687118
$306$	0	0
$307$	−20.7846	−1.18624	−0.593120	−	0.805114i	$-0.702104\pi$
$307$	−20.7846	−1.18624	−0.593120	+	0.805114i	$0.702104\pi$
$308$	0	0
$309$	0	0
$310$	−27.7128	−1.57398
$311$	31.1769	1.76788	0.883940	−	0.467600i	$-0.154881\pi$
$311$	31.1769	1.76788	0.883940	+	0.467600i	$0.154881\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−3.46410	−0.194871
$317$	3.46410	0.194563	0.0972817	−	0.995257i	$-0.468985\pi$
$317$	3.46410	0.194563	0.0972817	+	0.995257i	$0.468985\pi$
$318$	0	0
$319$	0	0
$320$	−3.46410	−0.193649
$321$	0	0
$322$	−12.0000	−0.668734
$323$	41.5692	2.31297
$324$	0	0
$325$	24.2487	1.34508
$326$	−8.00000	−0.443079
$327$	0	0
$328$	6.00000	0.331295
$329$	36.0000	1.98474
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	0	0
$335$	−13.8564	−0.757056
$336$	0	0
$337$	−27.7128	−1.50961	−0.754807	−	0.655947i	$-0.772269\pi$
$337$	−27.7128	−1.50961	−0.754807	+	0.655947i	$0.772269\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	20.7846	1.12720
$341$	0	0
$342$	0	0
$343$	−6.92820	−0.374088
$344$	0	0
$345$	0	0
$346$	−6.00000	−0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	3.46410	0.185429	0.0927146	−	0.995693i	$-0.470446\pi$
$349$	3.46410	0.185429	0.0927146	+	0.995693i	$0.470446\pi$
$350$	−24.2487	−1.29615
$351$	0	0
$352$	0	0
$353$	13.8564	0.737502	0.368751	−	0.929528i	$-0.379785\pi$
$353$	13.8564	0.737502	0.368751	+	0.929528i	$0.379785\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	−13.8564	−0.734388
$357$	0	0
$358$	6.92820	0.366167
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	−2.00000	−0.105118
$363$	0	0
$364$	12.0000	0.628971
$365$	48.0000	2.51243
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	3.46410	0.180579
$369$	0	0
$370$	−6.92820	−0.360180
$371$	36.0000	1.86903
$372$	0	0
$373$	−24.2487	−1.25555	−0.627775	−	0.778395i	$-0.716034\pi$
$373$	−24.2487	−1.25555	−0.627775	+	0.778395i	$0.716034\pi$
$374$	0	0
$375$	0	0
$376$	−10.3923	−0.535942
$377$	20.7846	1.07046
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	17.3205	0.886194
$383$	3.46410	0.177007	0.0885037	−	0.996076i	$-0.471792\pi$
$383$	3.46410	0.177007	0.0885037	+	0.996076i	$0.471792\pi$
$384$	0	0
$385$	0	0
$386$	6.92820	0.352636
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−3.46410	−0.175637	−0.0878185	−	0.996136i	$-0.527990\pi$
$389$	−3.46410	−0.175637	−0.0878185	+	0.996136i	$0.527990\pi$
$390$	0	0
$391$	−20.7846	−1.05112
$392$	−5.00000	−0.252538
$393$	0	0
$394$	−6.00000	−0.302276
$395$	12.0000	0.603786
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	7.00000	0.350000
$401$	−20.7846	−1.03793	−0.518967	−	0.854794i	$-0.673683\pi$
$401$	−20.7846	−1.03793	−0.518967	+	0.854794i	$0.673683\pi$
$402$	0	0
$403$	−27.7128	−1.38047
$404$	−18.0000	−0.895533
$405$	0	0
$406$	−20.7846	−1.03152
$407$	0	0
$408$	0	0
$409$	6.92820	0.342578	0.171289	−	0.985221i	$-0.445207\pi$
$409$	6.92820	0.342578	0.171289	+	0.985221i	$0.445207\pi$
$410$	−20.7846	−1.02648
$411$	0	0
$412$	−16.0000	−0.788263
$413$	24.0000	1.18096
$414$	0	0
$415$	41.5692	2.04055
$416$	−3.46410	−0.169842
$417$	0	0
$418$	0	0
$419$	20.7846	1.01539	0.507697	−	0.861536i	$-0.330497\pi$
$419$	20.7846	1.01539	0.507697	+	0.861536i	$0.330497\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−27.7128	−1.34904
$423$	0	0
$424$	−10.3923	−0.504695
$425$	−42.0000	−2.03730
$426$	0	0
$427$	−12.0000	−0.580721
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	27.7128	1.33026
$435$	0	0
$436$	−10.3923	−0.497701
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−31.1769	−1.48799	−0.743996	−	0.668184i	$-0.767072\pi$
$439$	−31.1769	−1.48799	−0.743996	+	0.668184i	$0.767072\pi$
$440$	0	0
$441$	0	0
$442$	20.7846	0.988623
$443$	−34.6410	−1.64584	−0.822922	−	0.568154i	$-0.807658\pi$
$443$	−34.6410	−1.64584	−0.822922	+	0.568154i	$0.807658\pi$
$444$	0	0
$445$	48.0000	2.27542
$446$	16.0000	0.757622
$447$	0	0
$448$	3.46410	0.163663
$449$	6.92820	0.326962	0.163481	−	0.986546i	$-0.447728\pi$
$449$	6.92820	0.326962	0.163481	+	0.986546i	$0.447728\pi$
$450$	0	0
$451$	0	0
$452$	−13.8564	−0.651751
$453$	0	0
$454$	24.0000	1.12638
$455$	−41.5692	−1.94880
$456$	0	0
$457$	−20.7846	−0.972263	−0.486132	−	0.873886i	$-0.661592\pi$
$457$	−20.7846	−0.972263	−0.486132	+	0.873886i	$0.661592\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	−12.0000	−0.559503
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	−13.8564	−0.641198	−0.320599	−	0.947215i	$-0.603884\pi$
$467$	−13.8564	−0.641198	−0.320599	+	0.947215i	$0.603884\pi$
$468$	0	0
$469$	13.8564	0.639829
$470$	36.0000	1.66056
$471$	0	0
$472$	−6.92820	−0.318896
$473$	0	0
$474$	0	0
$475$	−48.4974	−2.22521
$476$	−20.7846	−0.952661
$477$	0	0
$478$	24.0000	1.09773
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−6.92820	−0.315899
$482$	−6.92820	−0.315571
$483$	0	0
$484$	0	0
$485$	34.6410	1.57297
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	3.46410	0.156813
$489$	0	0
$490$	17.3205	0.782461
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	24.0000	1.07981
$495$	0	0
$496$	−8.00000	−0.359211
$497$	12.0000	0.538274
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−6.92820	−0.309839
$501$	0	0
$502$	6.92820	0.309221
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	62.3538	2.77471
$506$	0	0
$507$	0	0
$508$	10.3923	0.461084
$509$	24.2487	1.07481	0.537403	−	0.843326i	$-0.319406\pi$
$509$	24.2487	1.07481	0.537403	+	0.843326i	$0.319406\pi$
$510$	0	0
$511$	−48.0000	−2.12339
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0	0
$515$	55.4256	2.44234
$516$	0	0
$517$	0	0
$518$	6.92820	0.304408
$519$	0	0
$520$	12.0000	0.526235
$521$	6.92820	0.303530	0.151765	−	0.988417i	$-0.451504\pi$
$521$	6.92820	0.303530	0.151765	+	0.988417i	$0.451504\pi$
$522$	0	0
$523$	−13.8564	−0.605898	−0.302949	−	0.953007i	$-0.597971\pi$
$523$	−13.8564	−0.605898	−0.302949	+	0.953007i	$0.597971\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	−11.0000	−0.478261
$530$	36.0000	1.56374
$531$	0	0
$532$	−24.0000	−1.04053
$533$	−20.7846	−0.900281
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−10.3923	−0.448044
$539$	0	0
$540$	0	0
$541$	17.3205	0.744667	0.372333	−	0.928099i	$-0.378558\pi$
$541$	17.3205	0.744667	0.372333	+	0.928099i	$0.378558\pi$
$542$	−3.46410	−0.148796
$543$	0	0
$544$	6.00000	0.257248
$545$	36.0000	1.54207
$546$	0	0
$547$	−6.92820	−0.296229	−0.148114	−	0.988970i	$-0.547320\pi$
$547$	−6.92820	−0.296229	−0.148114	+	0.988970i	$0.547320\pi$
$548$	−6.92820	−0.295958
$549$	0	0
$550$	0	0
$551$	−41.5692	−1.77091
$552$	0	0
$553$	−12.0000	−0.510292
$554$	−10.3923	−0.441527
$555$	0	0
$556$	6.92820	0.293821
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	0	0
$560$	−12.0000	−0.507093
$561$	0	0
$562$	−6.00000	−0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	48.0000	2.01938
$566$	0	0
$567$	0	0
$568$	−3.46410	−0.145350
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	6.92820	0.289936	0.144968	−	0.989436i	$-0.453692\pi$
$571$	6.92820	0.289936	0.144968	+	0.989436i	$0.453692\pi$
$572$	0	0
$573$	0	0
$574$	20.7846	0.867533
$575$	24.2487	1.01124
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	−20.7846	−0.863034
$581$	−41.5692	−1.72458
$582$	0	0
$583$	0	0
$584$	13.8564	0.573382
$585$	0	0
$586$	6.00000	0.247858
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	0	0
$589$	55.4256	2.28377
$590$	24.0000	0.988064
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	72.0000	2.95171
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−12.0000	−0.490716
$599$	−10.3923	−0.424618	−0.212309	−	0.977203i	$-0.568098\pi$
$599$	−10.3923	−0.424618	−0.212309	+	0.977203i	$0.568098\pi$
$600$	0	0
$601$	41.5692	1.69564	0.847822	−	0.530281i	$-0.177914\pi$
$601$	41.5692	1.69564	0.847822	+	0.530281i	$0.177914\pi$
$602$	0	0
$603$	0	0
$604$	−3.46410	−0.140952
$605$	0	0
$606$	0	0
$607$	−17.3205	−0.703018	−0.351509	−	0.936185i	$-0.614331\pi$
$607$	−17.3205	−0.703018	−0.351509	+	0.936185i	$0.614331\pi$
$608$	6.92820	0.280976
$609$	0	0
$610$	−12.0000	−0.485866
$611$	36.0000	1.45640
$612$	0	0
$613$	−17.3205	−0.699569	−0.349784	−	0.936830i	$-0.613745\pi$
$613$	−17.3205	−0.699569	−0.349784	+	0.936830i	$0.613745\pi$
$614$	20.7846	0.838799
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	27.7128	1.11297
$621$	0	0
$622$	−31.1769	−1.25008
$623$	−48.0000	−1.92308
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	10.0000	0.399043
$629$	12.0000	0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	3.46410	0.137795
$633$	0	0
$634$	−3.46410	−0.137577
$635$	−36.0000	−1.42862
$636$	0	0
$637$	17.3205	0.686264
$638$	0	0
$639$	0	0
$640$	3.46410	0.136931
$641$	6.92820	0.273648	0.136824	−	0.990595i	$-0.456311\pi$
$641$	6.92820	0.273648	0.136824	+	0.990595i	$0.456311\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−41.5692	−1.63552
$647$	−24.2487	−0.953315	−0.476658	−	0.879089i	$-0.658152\pi$
$647$	−24.2487	−0.953315	−0.476658	+	0.879089i	$0.658152\pi$
$648$	0	0
$649$	0	0
$650$	−24.2487	−0.951113
$651$	0	0
$652$	8.00000	0.313304
$653$	45.0333	1.76229	0.881145	−	0.472846i	$-0.156773\pi$
$653$	45.0333	1.76229	0.881145	+	0.472846i	$0.156773\pi$
$654$	0	0
$655$	41.5692	1.62424
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−36.0000	−1.40343
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	12.0000	0.465690
$665$	83.1384	3.22397
$666$	0	0
$667$	20.7846	0.804783
$668$	0	0
$669$	0	0
$670$	13.8564	0.535320
$671$	0	0
$672$	0	0
$673$	34.6410	1.33531	0.667657	−	0.744469i	$-0.267297\pi$
$673$	34.6410	1.33531	0.667657	+	0.744469i	$0.267297\pi$
$674$	27.7128	1.06746
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	−34.6410	−1.32940
$680$	−20.7846	−0.797053
$681$	0	0
$682$	0	0
$683$	−20.7846	−0.795301	−0.397650	−	0.917537i	$-0.630174\pi$
$683$	−20.7846	−0.795301	−0.397650	+	0.917537i	$0.630174\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	6.92820	0.264520
$687$	0	0
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−24.0000	−0.910372
$696$	0	0
$697$	36.0000	1.36360
$698$	−3.46410	−0.131118
$699$	0	0
$700$	24.2487	0.916515
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	13.8564	0.522604
$704$	0	0
$705$	0	0
$706$	−13.8564	−0.521493
$707$	−62.3538	−2.34506
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	13.8564	0.519291
$713$	−27.7128	−1.03785
$714$	0	0
$715$	0	0
$716$	−6.92820	−0.258919
$717$	0	0
$718$	24.0000	0.895672
$719$	−24.2487	−0.904324	−0.452162	−	0.891936i	$-0.649347\pi$
$719$	−24.2487	−0.904324	−0.452162	+	0.891936i	$0.649347\pi$
$720$	0	0
$721$	−55.4256	−2.06416
$722$	−29.0000	−1.07927
$723$	0	0
$724$	2.00000	0.0743294
$725$	42.0000	1.55984
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	−12.0000	−0.444750
$729$	0	0
$730$	−48.0000	−1.77656
$731$	0	0
$732$	0	0
$733$	31.1769	1.15155	0.575773	−	0.817610i	$-0.304701\pi$
$733$	31.1769	1.15155	0.575773	+	0.817610i	$0.304701\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−3.46410	−0.127688
$737$	0	0
$738$	0	0
$739$	41.5692	1.52915	0.764574	−	0.644536i	$-0.222949\pi$
$739$	41.5692	1.52915	0.764574	+	0.644536i	$0.222949\pi$
$740$	6.92820	0.254686
$741$	0	0
$742$	−36.0000	−1.32160
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	20.7846	0.761489
$746$	24.2487	0.887808
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	10.3923	0.378968
$753$	0	0
$754$	−20.7846	−0.756931
$755$	12.0000	0.436725
$756$	0	0
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	−24.0000	−0.870572
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	−17.3205	−0.626634
$765$	0	0
$766$	−3.46410	−0.125163
$767$	24.0000	0.866590
$768$	0	0
$769$	−6.92820	−0.249837	−0.124919	−	0.992167i	$-0.539867\pi$
$769$	−6.92820	−0.249837	−0.124919	+	0.992167i	$0.539867\pi$
$770$	0	0
$771$	0	0
$772$	−6.92820	−0.249351
$773$	10.3923	0.373785	0.186893	−	0.982380i	$-0.440158\pi$
$773$	10.3923	0.373785	0.186893	+	0.982380i	$0.440158\pi$
$774$	0	0
$775$	−56.0000	−2.01158
$776$	10.0000	0.358979
$777$	0	0
$778$	3.46410	0.124194
$779$	41.5692	1.48937
$780$	0	0
$781$	0	0
$782$	20.7846	0.743256
$783$	0	0
$784$	5.00000	0.178571
$785$	−34.6410	−1.23639
$786$	0	0
$787$	27.7128	0.987855	0.493928	−	0.869503i	$-0.335561\pi$
$787$	27.7128	0.987855	0.493928	+	0.869503i	$0.335561\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−48.0000	−1.70668
$792$	0	0
$793$	−12.0000	−0.426132
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−8.00000	−0.283552
$797$	24.2487	0.858933	0.429467	−	0.903083i	$-0.358702\pi$
$797$	24.2487	0.858933	0.429467	+	0.903083i	$0.358702\pi$
$798$	0	0
$799$	−62.3538	−2.20592
$800$	−7.00000	−0.247487
$801$	0	0
$802$	20.7846	0.733930
$803$	0	0
$804$	0	0
$805$	−41.5692	−1.46512
$806$	27.7128	0.976142
$807$	0	0
$808$	18.0000	0.633238
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	41.5692	1.45969	0.729846	−	0.683611i	$-0.239592\pi$
$811$	41.5692	1.45969	0.729846	+	0.683611i	$0.239592\pi$
$812$	20.7846	0.729397
$813$	0	0
$814$	0	0
$815$	−27.7128	−0.970737
$816$	0	0
$817$	0	0
$818$	−6.92820	−0.242239
$819$	0	0
$820$	20.7846	0.725830
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−24.0000	−0.835067
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−41.5692	−1.44289
$831$	0	0
$832$	3.46410	0.120096
$833$	−30.0000	−1.03944
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−20.7846	−0.717992
$839$	−17.3205	−0.597970	−0.298985	−	0.954258i	$-0.596648\pi$
$839$	−17.3205	−0.597970	−0.298985	+	0.954258i	$0.596648\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−22.0000	−0.758170
$843$	0	0
$844$	27.7128	0.953914
$845$	3.46410	0.119169
$846$	0	0
$847$	0	0
$848$	10.3923	0.356873
$849$	0	0
$850$	42.0000	1.44059
$851$	−6.92820	−0.237496
$852$	0	0
$853$	−17.3205	−0.593043	−0.296521	−	0.955026i	$-0.595827\pi$
$853$	−17.3205	−0.593043	−0.296521	+	0.955026i	$0.595827\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−31.1769	−1.06127	−0.530637	−	0.847599i	$-0.678047\pi$
$863$	−31.1769	−1.06127	−0.530637	+	0.847599i	$0.678047\pi$
$864$	0	0
$865$	−20.7846	−0.706698
$866$	−14.0000	−0.475739
$867$	0	0
$868$	−27.7128	−0.940634
$869$	0	0
$870$	0	0
$871$	13.8564	0.469506
$872$	10.3923	0.351928
$873$	0	0
$874$	24.0000	0.811812
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−24.2487	−0.818821	−0.409410	−	0.912350i	$-0.634266\pi$
$877$	−24.2487	−0.818821	−0.409410	+	0.912350i	$0.634266\pi$
$878$	31.1769	1.05217
$879$	0	0
$880$	0	0
$881$	55.4256	1.86734	0.933668	−	0.358139i	$-0.116589\pi$
$881$	55.4256	1.86734	0.933668	+	0.358139i	$0.116589\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−20.7846	−0.699062
$885$	0	0
$886$	34.6410	1.16379
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	−48.0000	−1.60896
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−72.0000	−2.40939
$894$	0	0
$895$	24.0000	0.802232
$896$	−3.46410	−0.115728
$897$	0	0
$898$	−6.92820	−0.231197
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−62.3538	−2.07731
$902$	0	0
$903$	0	0
$904$	13.8564	0.460857
$905$	−6.92820	−0.230301
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−24.0000	−0.796468
$909$	0	0
$910$	41.5692	1.37801
$911$	−10.3923	−0.344312	−0.172156	−	0.985070i	$-0.555073\pi$
$911$	−10.3923	−0.344312	−0.172156	+	0.985070i	$0.555073\pi$
$912$	0	0
$913$	0	0
$914$	20.7846	0.687494
$915$	0	0
$916$	10.0000	0.330409
$917$	−41.5692	−1.37274
$918$	0	0
$919$	−51.9615	−1.71405	−0.857026	−	0.515273i	$-0.827691\pi$
$919$	−51.9615	−1.71405	−0.857026	+	0.515273i	$0.827691\pi$
$920$	12.0000	0.395628
$921$	0	0
$922$	30.0000	0.987997
$923$	12.0000	0.394985
$924$	0	0
$925$	−14.0000	−0.460317
$926$	−32.0000	−1.05159
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−20.7846	−0.681921	−0.340960	−	0.940078i	$-0.610752\pi$
$929$	−20.7846	−0.681921	−0.340960	+	0.940078i	$0.610752\pi$
$930$	0	0
$931$	−34.6410	−1.13531
$932$	−6.00000	−0.196537
$933$	0	0
$934$	13.8564	0.453395
$935$	0	0
$936$	0	0
$937$	−13.8564	−0.452669	−0.226335	−	0.974050i	$-0.572674\pi$
$937$	−13.8564	−0.452669	−0.226335	+	0.974050i	$0.572674\pi$
$938$	−13.8564	−0.452428
$939$	0	0
$940$	−36.0000	−1.17419
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	0	0
$943$	−20.7846	−0.676840
$944$	6.92820	0.225494
$945$	0	0
$946$	0	0
$947$	13.8564	0.450273	0.225136	−	0.974327i	$-0.427717\pi$
$947$	13.8564	0.450273	0.225136	+	0.974327i	$0.427717\pi$
$948$	0	0
$949$	−48.0000	−1.55815
$950$	48.4974	1.57346
$951$	0	0
$952$	20.7846	0.673633
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	60.0000	1.94155
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−24.0000	−0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	6.92820	0.223374
$963$	0	0
$964$	6.92820	0.223142
$965$	24.0000	0.772587
$966$	0	0
$967$	−45.0333	−1.44817	−0.724087	−	0.689709i	$-0.757739\pi$
$967$	−45.0333	−1.44817	−0.724087	+	0.689709i	$0.757739\pi$
$968$	0	0
$969$	0	0
$970$	−34.6410	−1.11226
$971$	−48.4974	−1.55636	−0.778178	−	0.628044i	$-0.783856\pi$
$971$	−48.4974	−1.55636	−0.778178	+	0.628044i	$0.783856\pi$
$972$	0	0
$973$	24.0000	0.769405
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−3.46410	−0.110883
$977$	−48.4974	−1.55157	−0.775785	−	0.630997i	$-0.782646\pi$
$977$	−48.4974	−1.55157	−0.775785	+	0.630997i	$0.782646\pi$
$978$	0	0
$979$	0	0
$980$	−17.3205	−0.553283
$981$	0	0
$982$	−24.0000	−0.765871
$983$	−38.1051	−1.21536	−0.607682	−	0.794180i	$-0.707901\pi$
$983$	−38.1051	−1.21536	−0.607682	+	0.794180i	$0.707901\pi$
$984$	0	0
$985$	−20.7846	−0.662253
$986$	36.0000	1.14647
$987$	0	0
$988$	−24.0000	−0.763542
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	−12.0000	−0.380617
$995$	27.7128	0.878555
$996$	0	0
$997$	−51.9615	−1.64564	−0.822819	−	0.568304i	$-0.807600\pi$
$997$	−51.9615	−1.64564	−0.822819	+	0.568304i	$0.807600\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.2.a.q.1.1	✓	2
3.2	odd	2		2178.2.a.ba.1.2	yes	2
11.10	odd	2		2178.2.a.ba.1.1	yes	2
33.32	even	2	inner	2178.2.a.q.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2178.2.a.q.1.1	✓	2	1.1	even	1	trivial
2178.2.a.q.1.2	yes	2	33.32	even	2	inner
2178.2.a.ba.1.1	yes	2	11.10	odd	2
2178.2.a.ba.1.2	yes	2	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} +3.46410 q^{7} -1.00000 q^{8} +3.46410 q^{10} +3.46410 q^{13} -3.46410 q^{14} +1.00000 q^{16} -6.00000 q^{17} -6.92820 q^{19} -3.46410 q^{20} +3.46410 q^{23} +7.00000 q^{25} -3.46410 q^{26} +3.46410 q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -12.0000 q^{35} -2.00000 q^{37} +6.92820 q^{38} +3.46410 q^{40} -6.00000 q^{41} -3.46410 q^{46} +10.3923 q^{47} +5.00000 q^{49} -7.00000 q^{50} +3.46410 q^{52} +10.3923 q^{53} -3.46410 q^{56} -6.00000 q^{58} +6.92820 q^{59} -3.46410 q^{61} +8.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +4.00000 q^{67} -6.00000 q^{68} +12.0000 q^{70} +3.46410 q^{71} -13.8564 q^{73} +2.00000 q^{74} -6.92820 q^{76} -3.46410 q^{79} -3.46410 q^{80} +6.00000 q^{82} -12.0000 q^{83} +20.7846 q^{85} -13.8564 q^{89} +12.0000 q^{91} +3.46410 q^{92} -10.3923 q^{94} +24.0000 q^{95} -10.0000 q^{97} -5.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{16} - 12 q^{17} + 14 q^{25} + 12 q^{29} - 16 q^{31} - 2 q^{32} + 12 q^{34} - 24 q^{35} - 4 q^{37} - 12 q^{41} + 10 q^{49} - 14 q^{50} - 12 q^{58} + 16 q^{62} + 2 q^{64}+ \cdots - 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^16 - 12 * q^17 + 14 * q^25 + 12 * q^29 - 16 * q^31 - 2 * q^32 + 12 * q^34 - 24 * q^35 - 4 * q^37 - 12 * q^41 + 10 * q^49 - 14 * q^50 - 12 * q^58 + 16 * q^62 + 2 * q^64 - 24 * q^65 + 8 * q^67 - 12 * q^68 + 24 * q^70 + 4 * q^74 + 12 * q^82 - 24 * q^83 + 24 * q^91 + 48 * q^95 - 20 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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