LMFDB - Embedded newform 2178.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,2,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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:	$ 1 $
Twist minimal:	no (minimal twist has level 242)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} -4.73205 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} -4.73205 q^{7} +1.00000 q^{8} +1.73205 q^{10} -3.00000 q^{13} -4.73205 q^{14} +1.00000 q^{16} -5.19615 q^{17} -1.26795 q^{19} +1.73205 q^{20} +1.26795 q^{23} -2.00000 q^{25} -3.00000 q^{26} -4.73205 q^{28} -3.00000 q^{29} +0.196152 q^{31} +1.00000 q^{32} -5.19615 q^{34} -8.19615 q^{35} -7.19615 q^{37} -1.26795 q^{38} +1.73205 q^{40} -0.803848 q^{41} +1.26795 q^{46} +8.19615 q^{47} +15.3923 q^{49} -2.00000 q^{50} -3.00000 q^{52} -11.1962 q^{53} -4.73205 q^{56} -3.00000 q^{58} -9.46410 q^{59} -2.53590 q^{61} +0.196152 q^{62} +1.00000 q^{64} -5.19615 q^{65} -10.1962 q^{67} -5.19615 q^{68} -8.19615 q^{70} +9.46410 q^{71} +0.928203 q^{73} -7.19615 q^{74} -1.26795 q^{76} +4.73205 q^{79} +1.73205 q^{80} -0.803848 q^{82} +8.19615 q^{83} -9.00000 q^{85} -6.46410 q^{89} +14.1962 q^{91} +1.26795 q^{92} +8.19615 q^{94} -2.19615 q^{95} -1.00000 q^{97} +15.3923 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 6 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 6 q^{7} + 2 q^{8} - 6 q^{13} - 6 q^{14} + 2 q^{16} - 6 q^{19} + 6 q^{23} - 4 q^{25} - 6 q^{26} - 6 q^{28} - 6 q^{29} - 10 q^{31} + 2 q^{32} - 6 q^{35} - 4 q^{37} - 6 q^{38} - 12 q^{41} + 6 q^{46} + 6 q^{47} + 10 q^{49} - 4 q^{50} - 6 q^{52} - 12 q^{53} - 6 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} - 10 q^{62} + 2 q^{64} - 10 q^{67} - 6 q^{70} + 12 q^{71} - 12 q^{73} - 4 q^{74} - 6 q^{76} + 6 q^{79} - 12 q^{82} + 6 q^{83} - 18 q^{85} - 6 q^{89} + 18 q^{91} + 6 q^{92} + 6 q^{94} + 6 q^{95} - 2 q^{97} + 10 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^7 + 2 * q^8 - 6 * q^13 - 6 * q^14 + 2 * q^16 - 6 * q^19 + 6 * q^23 - 4 * q^25 - 6 * q^26 - 6 * q^28 - 6 * q^29 - 10 * q^31 + 2 * q^32 - 6 * q^35 - 4 * q^37 - 6 * q^38 - 12 * q^41 + 6 * q^46 + 6 * q^47 + 10 * q^49 - 4 * q^50 - 6 * q^52 - 12 * q^53 - 6 * q^56 - 6 * q^58 - 12 * q^59 - 12 * q^61 - 10 * q^62 + 2 * q^64 - 10 * q^67 - 6 * q^70 + 12 * q^71 - 12 * q^73 - 4 * q^74 - 6 * q^76 + 6 * q^79 - 12 * q^82 + 6 * q^83 - 18 * q^85 - 6 * q^89 + 18 * q^91 + 6 * q^92 + 6 * q^94 + 6 * q^95 - 2 * q^97 + 10 * q^98

Display 100 coefficients 10 coefficients

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
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	−4.73205	−1.78855	−0.894274	−	0.447521i	$-0.852307\pi$
$7$	−4.73205	−1.78855	−0.894274	+	0.447521i	$0.852307\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.73205	0.547723
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−4.73205	−1.26469
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.19615	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−5.19615	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	0	0
$19$	−1.26795	−0.290887	−0.145444	−	0.989367i	$-0.546461\pi$
$19$	−1.26795	−0.290887	−0.145444	+	0.989367i	$0.546461\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	0	0
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	−3.00000	−0.588348
$27$	0	0
$28$	−4.73205	−0.894274
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	0.196152	0.0352300	0.0176150	−	0.999845i	$-0.494393\pi$
$31$	0.196152	0.0352300	0.0176150	+	0.999845i	$0.494393\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.19615	−0.891133
$35$	−8.19615	−1.38540
$36$	0	0
$37$	−7.19615	−1.18304	−0.591520	−	0.806290i	$-0.701472\pi$
$37$	−7.19615	−1.18304	−0.591520	+	0.806290i	$0.701472\pi$
$38$	−1.26795	−0.205689
$39$	0	0
$40$	1.73205	0.273861
$41$	−0.803848	−0.125540	−0.0627700	−	0.998028i	$-0.519993\pi$
$41$	−0.803848	−0.125540	−0.0627700	+	0.998028i	$0.519993\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	1.26795	0.186949
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	−2.00000	−0.282843
$51$	0	0
$52$	−3.00000	−0.416025
$53$	−11.1962	−1.53791	−0.768955	−	0.639303i	$-0.779223\pi$
$53$	−11.1962	−1.53791	−0.768955	+	0.639303i	$0.779223\pi$
$54$	0	0
$55$	0	0
$56$	−4.73205	−0.632347
$57$	0	0
$58$	−3.00000	−0.393919
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	0	0
$61$	−2.53590	−0.324689	−0.162344	−	0.986734i	$-0.551906\pi$
$61$	−2.53590	−0.324689	−0.162344	+	0.986734i	$0.551906\pi$
$62$	0.196152	0.0249114
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.19615	−0.644503
$66$	0	0
$67$	−10.1962	−1.24566	−0.622829	−	0.782358i	$-0.714017\pi$
$67$	−10.1962	−1.24566	−0.622829	+	0.782358i	$0.714017\pi$
$68$	−5.19615	−0.630126
$69$	0	0
$70$	−8.19615	−0.979628
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	0	0
$73$	0.928203	0.108638	0.0543190	−	0.998524i	$-0.482701\pi$
$73$	0.928203	0.108638	0.0543190	+	0.998524i	$0.482701\pi$
$74$	−7.19615	−0.836536
$75$	0	0
$76$	−1.26795	−0.145444
$77$	0	0
$78$	0	0
$79$	4.73205	0.532397	0.266199	−	0.963918i	$-0.414232\pi$
$79$	4.73205	0.532397	0.266199	+	0.963918i	$0.414232\pi$
$80$	1.73205	0.193649
$81$	0	0
$82$	−0.803848	−0.0887701
$83$	8.19615	0.899645	0.449822	−	0.893118i	$-0.351487\pi$
$83$	8.19615	0.899645	0.449822	+	0.893118i	$0.351487\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.46410	−0.685193	−0.342597	−	0.939483i	$-0.611306\pi$
$89$	−6.46410	−0.685193	−0.342597	+	0.939483i	$0.611306\pi$
$90$	0	0
$91$	14.1962	1.48816
$92$	1.26795	0.132193
$93$	0	0
$94$	8.19615	0.845369
$95$	−2.19615	−0.225320
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	15.3923	1.55486
$99$	0	0
$100$	−2.00000	−0.200000
$101$	16.3923	1.63110	0.815548	−	0.578690i	$-0.196436\pi$
$101$	16.3923	1.63110	0.815548	+	0.578690i	$0.196436\pi$
$102$	0	0
$103$	−8.39230	−0.826918	−0.413459	−	0.910523i	$-0.635680\pi$
$103$	−8.39230	−0.826918	−0.413459	+	0.910523i	$0.635680\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−11.1962	−1.08747
$107$	12.5885	1.21697	0.608486	−	0.793565i	$-0.291777\pi$
$107$	12.5885	1.21697	0.608486	+	0.793565i	$0.291777\pi$
$108$	0	0
$109$	2.07180	0.198442	0.0992211	−	0.995065i	$-0.468365\pi$
$109$	2.07180	0.198442	0.0992211	+	0.995065i	$0.468365\pi$
$110$	0	0
$111$	0	0
$112$	−4.73205	−0.447137
$113$	−10.8564	−1.02128	−0.510642	−	0.859793i	$-0.670592\pi$
$113$	−10.8564	−1.02128	−0.510642	+	0.859793i	$0.670592\pi$
$114$	0	0
$115$	2.19615	0.204792
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−9.46410	−0.871241
$119$	24.5885	2.25402
$120$	0	0
$121$	0	0
$122$	−2.53590	−0.229589
$123$	0	0
$124$	0.196152	0.0176150
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−13.8564	−1.22956	−0.614779	−	0.788700i	$-0.710755\pi$
$127$	−13.8564	−1.22956	−0.614779	+	0.788700i	$0.710755\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.19615	−0.455733
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−10.1962	−0.880813
$135$	0	0
$136$	−5.19615	−0.445566
$137$	9.46410	0.808573	0.404286	−	0.914632i	$-0.367520\pi$
$137$	9.46410	0.808573	0.404286	+	0.914632i	$0.367520\pi$
$138$	0	0
$139$	−20.1962	−1.71302	−0.856508	−	0.516134i	$-0.827371\pi$
$139$	−20.1962	−1.71302	−0.856508	+	0.516134i	$0.827371\pi$
$140$	−8.19615	−0.692701
$141$	0	0
$142$	9.46410	0.794210
$143$	0	0
$144$	0	0
$145$	−5.19615	−0.431517
$146$	0.928203	0.0768186
$147$	0	0
$148$	−7.19615	−0.591520
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	4.39230	0.357441	0.178720	−	0.983900i	$-0.442804\pi$
$151$	4.39230	0.357441	0.178720	+	0.983900i	$0.442804\pi$
$152$	−1.26795	−0.102844
$153$	0	0
$154$	0	0
$155$	0.339746	0.0272891
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	4.73205	0.376462
$159$	0	0
$160$	1.73205	0.136931
$161$	−6.00000	−0.472866
$162$	0	0
$163$	22.1962	1.73854	0.869268	−	0.494340i	$-0.164590\pi$
$163$	22.1962	1.73854	0.869268	+	0.494340i	$0.164590\pi$
$164$	−0.803848	−0.0627700
$165$	0	0
$166$	8.19615	0.636145
$167$	16.3923	1.26847	0.634237	−	0.773138i	$-0.281314\pi$
$167$	16.3923	1.26847	0.634237	+	0.773138i	$0.281314\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	−9.00000	−0.690268
$171$	0	0
$172$	0	0
$173$	−4.39230	−0.333941	−0.166970	−	0.985962i	$-0.553398\pi$
$173$	−4.39230	−0.333941	−0.166970	+	0.985962i	$0.553398\pi$
$174$	0	0
$175$	9.46410	0.715419
$176$	0	0
$177$	0	0
$178$	−6.46410	−0.484505
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	7.19615	0.534886	0.267443	−	0.963574i	$-0.413821\pi$
$181$	7.19615	0.534886	0.267443	+	0.963574i	$0.413821\pi$
$182$	14.1962	1.05229
$183$	0	0
$184$	1.26795	0.0934745
$185$	−12.4641	−0.916379
$186$	0	0
$187$	0	0
$188$	8.19615	0.597766
$189$	0	0
$190$	−2.19615	−0.159326
$191$	21.4641	1.55309	0.776544	−	0.630063i	$-0.216971\pi$
$191$	21.4641	1.55309	0.776544	+	0.630063i	$0.216971\pi$
$192$	0	0
$193$	−10.2679	−0.739103	−0.369552	−	0.929210i	$-0.620489\pi$
$193$	−10.2679	−0.739103	−0.369552	+	0.929210i	$0.620489\pi$
$194$	−1.00000	−0.0717958
$195$	0	0
$196$	15.3923	1.09945
$197$	−13.3923	−0.954162	−0.477081	−	0.878859i	$-0.658305\pi$
$197$	−13.3923	−0.954162	−0.477081	+	0.878859i	$0.658305\pi$
$198$	0	0
$199$	−0.392305	−0.0278098	−0.0139049	−	0.999903i	$-0.504426\pi$
$199$	−0.392305	−0.0278098	−0.0139049	+	0.999903i	$0.504426\pi$
$200$	−2.00000	−0.141421
$201$	0	0
$202$	16.3923	1.15336
$203$	14.1962	0.996375
$204$	0	0
$205$	−1.39230	−0.0972428
$206$	−8.39230	−0.584720
$207$	0	0
$208$	−3.00000	−0.208013
$209$	0	0
$210$	0	0
$211$	25.8564	1.78003	0.890014	−	0.455933i	$-0.150694\pi$
$211$	25.8564	1.78003	0.890014	+	0.455933i	$0.150694\pi$
$212$	−11.1962	−0.768955
$213$	0	0
$214$	12.5885	0.860529
$215$	0	0
$216$	0	0
$217$	−0.928203	−0.0630105
$218$	2.07180	0.140320
$219$	0	0
$220$	0	0
$221$	15.5885	1.04859
$222$	0	0
$223$	−20.3923	−1.36557	−0.682785	−	0.730619i	$-0.739231\pi$
$223$	−20.3923	−1.36557	−0.682785	+	0.730619i	$0.739231\pi$
$224$	−4.73205	−0.316173
$225$	0	0
$226$	−10.8564	−0.722157
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	0	0
$229$	−13.1962	−0.872026	−0.436013	−	0.899940i	$-0.643610\pi$
$229$	−13.1962	−0.872026	−0.436013	+	0.899940i	$0.643610\pi$
$230$	2.19615	0.144810
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−6.80385	−0.445735	−0.222867	−	0.974849i	$-0.571542\pi$
$233$	−6.80385	−0.445735	−0.222867	+	0.974849i	$0.571542\pi$
$234$	0	0
$235$	14.1962	0.926055
$236$	−9.46410	−0.616061
$237$	0	0
$238$	24.5885	1.59383
$239$	−14.1962	−0.918273	−0.459136	−	0.888366i	$-0.651841\pi$
$239$	−14.1962	−0.918273	−0.459136	+	0.888366i	$0.651841\pi$
$240$	0	0
$241$	−26.7846	−1.72535	−0.862674	−	0.505760i	$-0.831212\pi$
$241$	−26.7846	−1.72535	−0.862674	+	0.505760i	$0.831212\pi$
$242$	0	0
$243$	0	0
$244$	−2.53590	−0.162344
$245$	26.6603	1.70326
$246$	0	0
$247$	3.80385	0.242033
$248$	0.196152	0.0124557
$249$	0	0
$250$	−12.1244	−0.766812
$251$	−9.12436	−0.575924	−0.287962	−	0.957642i	$-0.592978\pi$
$251$	−9.12436	−0.575924	−0.287962	+	0.957642i	$0.592978\pi$
$252$	0	0
$253$	0	0
$254$	−13.8564	−0.869428
$255$	0	0
$256$	1.00000	0.0625000
$257$	19.3923	1.20966	0.604829	−	0.796355i	$-0.293241\pi$
$257$	19.3923	1.20966	0.604829	+	0.796355i	$0.293241\pi$
$258$	0	0
$259$	34.0526	2.11592
$260$	−5.19615	−0.322252
$261$	0	0
$262$	0	0
$263$	9.80385	0.604531	0.302266	−	0.953224i	$-0.402257\pi$
$263$	9.80385	0.604531	0.302266	+	0.953224i	$0.402257\pi$
$264$	0	0
$265$	−19.3923	−1.19126
$266$	6.00000	0.367884
$267$	0	0
$268$	−10.1962	−0.622829
$269$	−9.58846	−0.584619	−0.292309	−	0.956324i	$-0.594424\pi$
$269$	−9.58846	−0.584619	−0.292309	+	0.956324i	$0.594424\pi$
$270$	0	0
$271$	16.3923	0.995762	0.497881	−	0.867245i	$-0.334112\pi$
$271$	16.3923	0.995762	0.497881	+	0.867245i	$0.334112\pi$
$272$	−5.19615	−0.315063
$273$	0	0
$274$	9.46410	0.571747
$275$	0	0
$276$	0	0
$277$	8.32051	0.499931	0.249965	−	0.968255i	$-0.419581\pi$
$277$	8.32051	0.499931	0.249965	+	0.968255i	$0.419581\pi$
$278$	−20.1962	−1.21128
$279$	0	0
$280$	−8.19615	−0.489814
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−6.92820	−0.411839	−0.205919	−	0.978569i	$-0.566018\pi$
$283$	−6.92820	−0.411839	−0.205919	+	0.978569i	$0.566018\pi$
$284$	9.46410	0.561591
$285$	0	0
$286$	0	0
$287$	3.80385	0.224534
$288$	0	0
$289$	10.0000	0.588235
$290$	−5.19615	−0.305129
$291$	0	0
$292$	0.928203	0.0543190
$293$	23.7846	1.38951	0.694756	−	0.719246i	$-0.255512\pi$
$293$	23.7846	1.38951	0.694756	+	0.719246i	$0.255512\pi$
$294$	0	0
$295$	−16.3923	−0.954397
$296$	−7.19615	−0.418268
$297$	0	0
$298$	−15.0000	−0.868927
$299$	−3.80385	−0.219982
$300$	0	0
$301$	0	0
$302$	4.39230	0.252749
$303$	0	0
$304$	−1.26795	−0.0727219
$305$	−4.39230	−0.251503
$306$	0	0
$307$	−25.2679	−1.44212	−0.721059	−	0.692874i	$-0.756344\pi$
$307$	−25.2679	−1.44212	−0.721059	+	0.692874i	$0.756344\pi$
$308$	0	0
$309$	0	0
$310$	0.339746	0.0192963
$311$	−8.78461	−0.498130	−0.249065	−	0.968487i	$-0.580123\pi$
$311$	−8.78461	−0.498130	−0.249065	+	0.968487i	$0.580123\pi$
$312$	0	0
$313$	−31.7846	−1.79657	−0.898286	−	0.439411i	$-0.855187\pi$
$313$	−31.7846	−1.79657	−0.898286	+	0.439411i	$0.855187\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	4.73205	0.266199
$317$	7.85641	0.441260	0.220630	−	0.975358i	$-0.429189\pi$
$317$	7.85641	0.441260	0.220630	+	0.975358i	$0.429189\pi$
$318$	0	0
$319$	0	0
$320$	1.73205	0.0968246
$321$	0	0
$322$	−6.00000	−0.334367
$323$	6.58846	0.366592
$324$	0	0
$325$	6.00000	0.332820
$326$	22.1962	1.22933
$327$	0	0
$328$	−0.803848	−0.0443851
$329$	−38.7846	−2.13826
$330$	0	0
$331$	28.7846	1.58215	0.791073	−	0.611722i	$-0.209523\pi$
$331$	28.7846	1.58215	0.791073	+	0.611722i	$0.209523\pi$
$332$	8.19615	0.449822
$333$	0	0
$334$	16.3923	0.896947
$335$	−17.6603	−0.964883
$336$	0	0
$337$	−2.66025	−0.144913	−0.0724566	−	0.997372i	$-0.523084\pi$
$337$	−2.66025	−0.144913	−0.0724566	+	0.997372i	$0.523084\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	−9.00000	−0.488094
$341$	0	0
$342$	0	0
$343$	−39.7128	−2.14429
$344$	0	0
$345$	0	0
$346$	−4.39230	−0.236132
$347$	−28.3923	−1.52418	−0.762089	−	0.647472i	$-0.775826\pi$
$347$	−28.3923	−1.52418	−0.762089	+	0.647472i	$0.775826\pi$
$348$	0	0
$349$	21.9282	1.17379	0.586895	−	0.809663i	$-0.300350\pi$
$349$	21.9282	1.17379	0.586895	+	0.809663i	$0.300350\pi$
$350$	9.46410	0.505878
$351$	0	0
$352$	0	0
$353$	−21.9282	−1.16712	−0.583560	−	0.812070i	$-0.698341\pi$
$353$	−21.9282	−1.16712	−0.583560	+	0.812070i	$0.698341\pi$
$354$	0	0
$355$	16.3923	0.870013
$356$	−6.46410	−0.342597
$357$	0	0
$358$	13.8564	0.732334
$359$	−6.58846	−0.347725	−0.173863	−	0.984770i	$-0.555625\pi$
$359$	−6.58846	−0.347725	−0.173863	+	0.984770i	$0.555625\pi$
$360$	0	0
$361$	−17.3923	−0.915384
$362$	7.19615	0.378221
$363$	0	0
$364$	14.1962	0.744081
$365$	1.60770	0.0841506
$366$	0	0
$367$	8.58846	0.448314	0.224157	−	0.974553i	$-0.428037\pi$
$367$	8.58846	0.448314	0.224157	+	0.974553i	$0.428037\pi$
$368$	1.26795	0.0660964
$369$	0	0
$370$	−12.4641	−0.647978
$371$	52.9808	2.75062
$372$	0	0
$373$	11.3205	0.586154	0.293077	−	0.956089i	$-0.405321\pi$
$373$	11.3205	0.586154	0.293077	+	0.956089i	$0.405321\pi$
$374$	0	0
$375$	0	0
$376$	8.19615	0.422684
$377$	9.00000	0.463524
$378$	0	0
$379$	3.60770	0.185315	0.0926574	−	0.995698i	$-0.470464\pi$
$379$	3.60770	0.185315	0.0926574	+	0.995698i	$0.470464\pi$
$380$	−2.19615	−0.112660
$381$	0	0
$382$	21.4641	1.09820
$383$	0.679492	0.0347204	0.0173602	−	0.999849i	$-0.494474\pi$
$383$	0.679492	0.0347204	0.0173602	+	0.999849i	$0.494474\pi$
$384$	0	0
$385$	0	0
$386$	−10.2679	−0.522625
$387$	0	0
$388$	−1.00000	−0.0507673
$389$	−32.6603	−1.65594	−0.827970	−	0.560772i	$-0.810504\pi$
$389$	−32.6603	−1.65594	−0.827970	+	0.560772i	$0.810504\pi$
$390$	0	0
$391$	−6.58846	−0.333193
$392$	15.3923	0.777429
$393$	0	0
$394$	−13.3923	−0.674695
$395$	8.19615	0.412393
$396$	0	0
$397$	5.58846	0.280477	0.140238	−	0.990118i	$-0.455213\pi$
$397$	5.58846	0.280477	0.140238	+	0.990118i	$0.455213\pi$
$398$	−0.392305	−0.0196645
$399$	0	0
$400$	−2.00000	−0.100000
$401$	7.39230	0.369154	0.184577	−	0.982818i	$-0.440908\pi$
$401$	7.39230	0.369154	0.184577	+	0.982818i	$0.440908\pi$
$402$	0	0
$403$	−0.588457	−0.0293131
$404$	16.3923	0.815548
$405$	0	0
$406$	14.1962	0.704543
$407$	0	0
$408$	0	0
$409$	2.66025	0.131541	0.0657705	−	0.997835i	$-0.479049\pi$
$409$	2.66025	0.131541	0.0657705	+	0.997835i	$0.479049\pi$
$410$	−1.39230	−0.0687610
$411$	0	0
$412$	−8.39230	−0.413459
$413$	44.7846	2.20371
$414$	0	0
$415$	14.1962	0.696862
$416$	−3.00000	−0.147087
$417$	0	0
$418$	0	0
$419$	39.3731	1.92350	0.961750	−	0.273928i	$-0.0883231\pi$
$419$	39.3731	1.92350	0.961750	+	0.273928i	$0.0883231\pi$
$420$	0	0
$421$	7.58846	0.369839	0.184919	−	0.982754i	$-0.440798\pi$
$421$	7.58846	0.369839	0.184919	+	0.982754i	$0.440798\pi$
$422$	25.8564	1.25867
$423$	0	0
$424$	−11.1962	−0.543733
$425$	10.3923	0.504101
$426$	0	0
$427$	12.0000	0.580721
$428$	12.5885	0.608486
$429$	0	0
$430$	0	0
$431$	16.3923	0.789590	0.394795	−	0.918769i	$-0.370816\pi$
$431$	16.3923	0.789590	0.394795	+	0.918769i	$0.370816\pi$
$432$	0	0
$433$	31.7846	1.52747	0.763735	−	0.645529i	$-0.223363\pi$
$433$	31.7846	1.52747	0.763735	+	0.645529i	$0.223363\pi$
$434$	−0.928203	−0.0445552
$435$	0	0
$436$	2.07180	0.0992211
$437$	−1.60770	−0.0769065
$438$	0	0
$439$	21.1244	1.00821	0.504105	−	0.863642i	$-0.331822\pi$
$439$	21.1244	1.00821	0.504105	+	0.863642i	$0.331822\pi$
$440$	0	0
$441$	0	0
$442$	15.5885	0.741467
$443$	−17.0718	−0.811106	−0.405553	−	0.914072i	$-0.632921\pi$
$443$	−17.0718	−0.811106	−0.405553	+	0.914072i	$0.632921\pi$
$444$	0	0
$445$	−11.1962	−0.530749
$446$	−20.3923	−0.965604
$447$	0	0
$448$	−4.73205	−0.223568
$449$	9.92820	0.468541	0.234270	−	0.972171i	$-0.424730\pi$
$449$	9.92820	0.468541	0.234270	+	0.972171i	$0.424730\pi$
$450$	0	0
$451$	0	0
$452$	−10.8564	−0.510642
$453$	0	0
$454$	−16.3923	−0.769329
$455$	24.5885	1.15272
$456$	0	0
$457$	5.19615	0.243066	0.121533	−	0.992587i	$-0.461219\pi$
$457$	5.19615	0.243066	0.121533	+	0.992587i	$0.461219\pi$
$458$	−13.1962	−0.616616
$459$	0	0
$460$	2.19615	0.102396
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	28.7846	1.33773	0.668867	−	0.743382i	$-0.266780\pi$
$463$	28.7846	1.33773	0.668867	+	0.743382i	$0.266780\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−6.80385	−0.315182
$467$	−21.4641	−0.993240	−0.496620	−	0.867968i	$-0.665426\pi$
$467$	−21.4641	−0.993240	−0.496620	+	0.867968i	$0.665426\pi$
$468$	0	0
$469$	48.2487	2.22792
$470$	14.1962	0.654820
$471$	0	0
$472$	−9.46410	−0.435621
$473$	0	0
$474$	0	0
$475$	2.53590	0.116355
$476$	24.5885	1.12701
$477$	0	0
$478$	−14.1962	−0.649317
$479$	−8.78461	−0.401379	−0.200690	−	0.979655i	$-0.564318\pi$
$479$	−8.78461	−0.401379	−0.200690	+	0.979655i	$0.564318\pi$
$480$	0	0
$481$	21.5885	0.984349
$482$	−26.7846	−1.22001
$483$	0	0
$484$	0	0
$485$	−1.73205	−0.0786484
$486$	0	0
$487$	−12.9808	−0.588214	−0.294107	−	0.955772i	$-0.595022\pi$
$487$	−12.9808	−0.588214	−0.294107	+	0.955772i	$0.595022\pi$
$488$	−2.53590	−0.114795
$489$	0	0
$490$	26.6603	1.20439
$491$	−33.3731	−1.50611	−0.753053	−	0.657960i	$-0.771419\pi$
$491$	−33.3731	−1.50611	−0.753053	+	0.657960i	$0.771419\pi$
$492$	0	0
$493$	15.5885	0.702069
$494$	3.80385	0.171143
$495$	0	0
$496$	0.196152	0.00880750
$497$	−44.7846	−2.00886
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−12.1244	−0.542218
$501$	0	0
$502$	−9.12436	−0.407240
$503$	−14.1962	−0.632975	−0.316488	−	0.948597i	$-0.602504\pi$
$503$	−14.1962	−0.632975	−0.316488	+	0.948597i	$0.602504\pi$
$504$	0	0
$505$	28.3923	1.26344
$506$	0	0
$507$	0	0
$508$	−13.8564	−0.614779
$509$	−12.9282	−0.573033	−0.286516	−	0.958075i	$-0.592497\pi$
$509$	−12.9282	−0.573033	−0.286516	+	0.958075i	$0.592497\pi$
$510$	0	0
$511$	−4.39230	−0.194304
$512$	1.00000	0.0441942
$513$	0	0
$514$	19.3923	0.855358
$515$	−14.5359	−0.640528
$516$	0	0
$517$	0	0
$518$	34.0526	1.49618
$519$	0	0
$520$	−5.19615	−0.227866
$521$	−9.46410	−0.414630	−0.207315	−	0.978274i	$-0.566472\pi$
$521$	−9.46410	−0.414630	−0.207315	+	0.978274i	$0.566472\pi$
$522$	0	0
$523$	−2.53590	−0.110887	−0.0554435	−	0.998462i	$-0.517657\pi$
$523$	−2.53590	−0.110887	−0.0554435	+	0.998462i	$0.517657\pi$
$524$	0	0
$525$	0	0
$526$	9.80385	0.427468
$527$	−1.01924	−0.0443987
$528$	0	0
$529$	−21.3923	−0.930100
$530$	−19.3923	−0.842348
$531$	0	0
$532$	6.00000	0.260133
$533$	2.41154	0.104456
$534$	0	0
$535$	21.8038	0.942663
$536$	−10.1962	−0.440407
$537$	0	0
$538$	−9.58846	−0.413388
$539$	0	0
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	16.3923	0.704110
$543$	0	0
$544$	−5.19615	−0.222783
$545$	3.58846	0.153713
$546$	0	0
$547$	24.0000	1.02617	0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000	1.02617	0.513083	+	0.858339i	$0.328503\pi$
$548$	9.46410	0.404286
$549$	0	0
$550$	0	0
$551$	3.80385	0.162049
$552$	0	0
$553$	−22.3923	−0.952218
$554$	8.32051	0.353505
$555$	0	0
$556$	−20.1962	−0.856508
$557$	37.1769	1.57524	0.787618	−	0.616164i	$-0.211314\pi$
$557$	37.1769	1.57524	0.787618	+	0.616164i	$0.211314\pi$
$558$	0	0
$559$	0	0
$560$	−8.19615	−0.346351
$561$	0	0
$562$	18.0000	0.759284
$563$	−0.588457	−0.0248005	−0.0124003	−	0.999923i	$-0.503947\pi$
$563$	−0.588457	−0.0248005	−0.0124003	+	0.999923i	$0.503947\pi$
$564$	0	0
$565$	−18.8038	−0.791084
$566$	−6.92820	−0.291214
$567$	0	0
$568$	9.46410	0.397105
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−5.66025	−0.236874	−0.118437	−	0.992962i	$-0.537788\pi$
$571$	−5.66025	−0.236874	−0.118437	+	0.992962i	$0.537788\pi$
$572$	0	0
$573$	0	0
$574$	3.80385	0.158770
$575$	−2.53590	−0.105754
$576$	0	0
$577$	14.6077	0.608126	0.304063	−	0.952652i	$-0.401657\pi$
$577$	14.6077	0.608126	0.304063	+	0.952652i	$0.401657\pi$
$578$	10.0000	0.415945
$579$	0	0
$580$	−5.19615	−0.215758
$581$	−38.7846	−1.60906
$582$	0	0
$583$	0	0
$584$	0.928203	0.0384093
$585$	0	0
$586$	23.7846	0.982533
$587$	−30.5885	−1.26252	−0.631260	−	0.775571i	$-0.717462\pi$
$587$	−30.5885	−1.26252	−0.631260	+	0.775571i	$0.717462\pi$
$588$	0	0
$589$	−0.248711	−0.0102480
$590$	−16.3923	−0.674861
$591$	0	0
$592$	−7.19615	−0.295760
$593$	−23.1962	−0.952552	−0.476276	−	0.879296i	$-0.658014\pi$
$593$	−23.1962	−0.952552	−0.476276	+	0.879296i	$0.658014\pi$
$594$	0	0
$595$	42.5885	1.74596
$596$	−15.0000	−0.614424
$597$	0	0
$598$	−3.80385	−0.155551
$599$	3.80385	0.155421	0.0777105	−	0.996976i	$-0.475239\pi$
$599$	3.80385	0.155421	0.0777105	+	0.996976i	$0.475239\pi$
$600$	0	0
$601$	8.66025	0.353259	0.176630	−	0.984277i	$-0.443481\pi$
$601$	8.66025	0.353259	0.176630	+	0.984277i	$0.443481\pi$
$602$	0	0
$603$	0	0
$604$	4.39230	0.178720
$605$	0	0
$606$	0	0
$607$	−26.1962	−1.06327	−0.531635	−	0.846974i	$-0.678422\pi$
$607$	−26.1962	−1.06327	−0.531635	+	0.846974i	$0.678422\pi$
$608$	−1.26795	−0.0514221
$609$	0	0
$610$	−4.39230	−0.177839
$611$	−24.5885	−0.994743
$612$	0	0
$613$	−20.3205	−0.820738	−0.410369	−	0.911920i	$-0.634600\pi$
$613$	−20.3205	−0.820738	−0.410369	+	0.911920i	$0.634600\pi$
$614$	−25.2679	−1.01973
$615$	0	0
$616$	0	0
$617$	−10.6077	−0.427050	−0.213525	−	0.976938i	$-0.568494\pi$
$617$	−10.6077	−0.427050	−0.213525	+	0.976938i	$0.568494\pi$
$618$	0	0
$619$	−30.1962	−1.21369	−0.606843	−	0.794822i	$-0.707564\pi$
$619$	−30.1962	−1.21369	−0.606843	+	0.794822i	$0.707564\pi$
$620$	0.339746	0.0136445
$621$	0	0
$622$	−8.78461	−0.352231
$623$	30.5885	1.22550
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−31.7846	−1.27037
$627$	0	0
$628$	4.00000	0.159617
$629$	37.3923	1.49093
$630$	0	0
$631$	−20.5885	−0.819614	−0.409807	−	0.912172i	$-0.634404\pi$
$631$	−20.5885	−0.819614	−0.409807	+	0.912172i	$0.634404\pi$
$632$	4.73205	0.188231
$633$	0	0
$634$	7.85641	0.312018
$635$	−24.0000	−0.952411
$636$	0	0
$637$	−46.1769	−1.82960
$638$	0	0
$639$	0	0
$640$	1.73205	0.0684653
$641$	−3.67949	−0.145331	−0.0726656	−	0.997356i	$-0.523151\pi$
$641$	−3.67949	−0.145331	−0.0726656	+	0.997356i	$0.523151\pi$
$642$	0	0
$643$	−25.8038	−1.01760	−0.508802	−	0.860883i	$-0.669912\pi$
$643$	−25.8038	−1.01760	−0.508802	+	0.860883i	$0.669912\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	6.58846	0.259219
$647$	23.3205	0.916824	0.458412	−	0.888740i	$-0.348418\pi$
$647$	23.3205	0.916824	0.458412	+	0.888740i	$0.348418\pi$
$648$	0	0
$649$	0	0
$650$	6.00000	0.235339
$651$	0	0
$652$	22.1962	0.869268
$653$	1.85641	0.0726468	0.0363234	−	0.999340i	$-0.488435\pi$
$653$	1.85641	0.0726468	0.0363234	+	0.999340i	$0.488435\pi$
$654$	0	0
$655$	0	0
$656$	−0.803848	−0.0313850
$657$	0	0
$658$	−38.7846	−1.51198
$659$	16.9808	0.661477	0.330738	−	0.943723i	$-0.392702\pi$
$659$	16.9808	0.661477	0.330738	+	0.943723i	$0.392702\pi$
$660$	0	0
$661$	−16.4115	−0.638335	−0.319168	−	0.947698i	$-0.603403\pi$
$661$	−16.4115	−0.638335	−0.319168	+	0.947698i	$0.603403\pi$
$662$	28.7846	1.11875
$663$	0	0
$664$	8.19615	0.318072
$665$	10.3923	0.402996
$666$	0	0
$667$	−3.80385	−0.147286
$668$	16.3923	0.634237
$669$	0	0
$670$	−17.6603	−0.682275
$671$	0	0
$672$	0	0
$673$	−0.928203	−0.0357796	−0.0178898	−	0.999840i	$-0.505695\pi$
$673$	−0.928203	−0.0357796	−0.0178898	+	0.999840i	$0.505695\pi$
$674$	−2.66025	−0.102469
$675$	0	0
$676$	−4.00000	−0.153846
$677$	4.60770	0.177088	0.0885441	−	0.996072i	$-0.471779\pi$
$677$	4.60770	0.177088	0.0885441	+	0.996072i	$0.471779\pi$
$678$	0	0
$679$	4.73205	0.181599
$680$	−9.00000	−0.345134
$681$	0	0
$682$	0	0
$683$	5.41154	0.207067	0.103533	−	0.994626i	$-0.466985\pi$
$683$	5.41154	0.207067	0.103533	+	0.994626i	$0.466985\pi$
$684$	0	0
$685$	16.3923	0.626318
$686$	−39.7128	−1.51624
$687$	0	0
$688$	0	0
$689$	33.5885	1.27962
$690$	0	0
$691$	−41.1769	−1.56644	−0.783222	−	0.621742i	$-0.786425\pi$
$691$	−41.1769	−1.56644	−0.783222	+	0.621742i	$0.786425\pi$
$692$	−4.39230	−0.166970
$693$	0	0
$694$	−28.3923	−1.07776
$695$	−34.9808	−1.32690
$696$	0	0
$697$	4.17691	0.158212
$698$	21.9282	0.829995
$699$	0	0
$700$	9.46410	0.357709
$701$	−17.7846	−0.671715	−0.335858	−	0.941913i	$-0.609026\pi$
$701$	−17.7846	−0.671715	−0.335858	+	0.941913i	$0.609026\pi$
$702$	0	0
$703$	9.12436	0.344132
$704$	0	0
$705$	0	0
$706$	−21.9282	−0.825279
$707$	−77.5692	−2.91729
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	16.3923	0.615192
$711$	0	0
$712$	−6.46410	−0.242252
$713$	0.248711	0.00931431
$714$	0	0
$715$	0	0
$716$	13.8564	0.517838
$717$	0	0
$718$	−6.58846	−0.245879
$719$	46.7321	1.74281	0.871406	−	0.490563i	$-0.163209\pi$
$719$	46.7321	1.74281	0.871406	+	0.490563i	$0.163209\pi$
$720$	0	0
$721$	39.7128	1.47898
$722$	−17.3923	−0.647275
$723$	0	0
$724$	7.19615	0.267443
$725$	6.00000	0.222834
$726$	0	0
$727$	−36.9808	−1.37154	−0.685770	−	0.727818i	$-0.740535\pi$
$727$	−36.9808	−1.37154	−0.685770	+	0.727818i	$0.740535\pi$
$728$	14.1962	0.526144
$729$	0	0
$730$	1.60770	0.0595035
$731$	0	0
$732$	0	0
$733$	−16.6077	−0.613419	−0.306710	−	0.951803i	$-0.599228\pi$
$733$	−16.6077	−0.613419	−0.306710	+	0.951803i	$0.599228\pi$
$734$	8.58846	0.317006
$735$	0	0
$736$	1.26795	0.0467372
$737$	0	0
$738$	0	0
$739$	−10.7321	−0.394785	−0.197392	−	0.980325i	$-0.563247\pi$
$739$	−10.7321	−0.394785	−0.197392	+	0.980325i	$0.563247\pi$
$740$	−12.4641	−0.458189
$741$	0	0
$742$	52.9808	1.94498
$743$	−38.1962	−1.40128	−0.700640	−	0.713514i	$-0.747102\pi$
$743$	−38.1962	−1.40128	−0.700640	+	0.713514i	$0.747102\pi$
$744$	0	0
$745$	−25.9808	−0.951861
$746$	11.3205	0.414473
$747$	0	0
$748$	0	0
$749$	−59.5692	−2.17661
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	8.19615	0.298883
$753$	0	0
$754$	9.00000	0.327761
$755$	7.60770	0.276872
$756$	0	0
$757$	−13.1962	−0.479622	−0.239811	−	0.970820i	$-0.577086\pi$
$757$	−13.1962	−0.479622	−0.239811	+	0.970820i	$0.577086\pi$
$758$	3.60770	0.131037
$759$	0	0
$760$	−2.19615	−0.0796628
$761$	19.9808	0.724302	0.362151	−	0.932119i	$-0.382042\pi$
$761$	19.9808	0.724302	0.362151	+	0.932119i	$0.382042\pi$
$762$	0	0
$763$	−9.80385	−0.354923
$764$	21.4641	0.776544
$765$	0	0
$766$	0.679492	0.0245510
$767$	28.3923	1.02519
$768$	0	0
$769$	−34.5167	−1.24470	−0.622351	−	0.782738i	$-0.713822\pi$
$769$	−34.5167	−1.24470	−0.622351	+	0.782738i	$0.713822\pi$
$770$	0	0
$771$	0	0
$772$	−10.2679	−0.369552
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	−0.392305	−0.0140920
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	−32.6603	−1.17093
$779$	1.01924	0.0365180
$780$	0	0
$781$	0	0
$782$	−6.58846	−0.235603
$783$	0	0
$784$	15.3923	0.549725
$785$	6.92820	0.247278
$786$	0	0
$787$	−44.7846	−1.59640	−0.798199	−	0.602393i	$-0.794214\pi$
$787$	−44.7846	−1.59640	−0.798199	+	0.602393i	$0.794214\pi$
$788$	−13.3923	−0.477081
$789$	0	0
$790$	8.19615	0.291606
$791$	51.3731	1.82662
$792$	0	0
$793$	7.60770	0.270157
$794$	5.58846	0.198327
$795$	0	0
$796$	−0.392305	−0.0139049
$797$	−36.9282	−1.30806	−0.654032	−	0.756467i	$-0.726924\pi$
$797$	−36.9282	−1.30806	−0.654032	+	0.756467i	$0.726924\pi$
$798$	0	0
$799$	−42.5885	−1.50667
$800$	−2.00000	−0.0707107
$801$	0	0
$802$	7.39230	0.261031
$803$	0	0
$804$	0	0
$805$	−10.3923	−0.366281
$806$	−0.588457	−0.0207275
$807$	0	0
$808$	16.3923	0.576679
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	14.1962	0.498187
$813$	0	0
$814$	0	0
$815$	38.4449	1.34666
$816$	0	0
$817$	0	0
$818$	2.66025	0.0930136
$819$	0	0
$820$	−1.39230	−0.0486214
$821$	−16.3923	−0.572095	−0.286048	−	0.958215i	$-0.592342\pi$
$821$	−16.3923	−0.572095	−0.286048	+	0.958215i	$0.592342\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−8.39230	−0.292360
$825$	0	0
$826$	44.7846	1.55826
$827$	−20.1962	−0.702289	−0.351145	−	0.936321i	$-0.614207\pi$
$827$	−20.1962	−0.702289	−0.351145	+	0.936321i	$0.614207\pi$
$828$	0	0
$829$	14.8038	0.514159	0.257079	−	0.966390i	$-0.417240\pi$
$829$	14.8038	0.514159	0.257079	+	0.966390i	$0.417240\pi$
$830$	14.1962	0.492756
$831$	0	0
$832$	−3.00000	−0.104006
$833$	−79.9808	−2.77117
$834$	0	0
$835$	28.3923	0.982556
$836$	0	0
$837$	0	0
$838$	39.3731	1.36012
$839$	50.4449	1.74155	0.870775	−	0.491682i	$-0.163618\pi$
$839$	50.4449	1.74155	0.870775	+	0.491682i	$0.163618\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	7.58846	0.261516
$843$	0	0
$844$	25.8564	0.890014
$845$	−6.92820	−0.238337
$846$	0	0
$847$	0	0
$848$	−11.1962	−0.384477
$849$	0	0
$850$	10.3923	0.356453
$851$	−9.12436	−0.312779
$852$	0	0
$853$	10.8564	0.371716	0.185858	−	0.982577i	$-0.440494\pi$
$853$	10.8564	0.371716	0.185858	+	0.982577i	$0.440494\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	12.5885	0.430265
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−24.7846	−0.845640	−0.422820	−	0.906214i	$-0.638960\pi$
$859$	−24.7846	−0.845640	−0.422820	+	0.906214i	$0.638960\pi$
$860$	0	0
$861$	0	0
$862$	16.3923	0.558324
$863$	3.80385	0.129484	0.0647422	−	0.997902i	$-0.479377\pi$
$863$	3.80385	0.129484	0.0647422	+	0.997902i	$0.479377\pi$
$864$	0	0
$865$	−7.60770	−0.258669
$866$	31.7846	1.08009
$867$	0	0
$868$	−0.928203	−0.0315053
$869$	0	0
$870$	0	0
$871$	30.5885	1.03645
$872$	2.07180	0.0701599
$873$	0	0
$874$	−1.60770	−0.0543811
$875$	57.3731	1.93956
$876$	0	0
$877$	6.46410	0.218277	0.109139	−	0.994027i	$-0.465191\pi$
$877$	6.46410	0.218277	0.109139	+	0.994027i	$0.465191\pi$
$878$	21.1244	0.712912
$879$	0	0
$880$	0	0
$881$	−41.5359	−1.39938	−0.699690	−	0.714447i	$-0.746679\pi$
$881$	−41.5359	−1.39938	−0.699690	+	0.714447i	$0.746679\pi$
$882$	0	0
$883$	−15.6077	−0.525241	−0.262620	−	0.964899i	$-0.584587\pi$
$883$	−15.6077	−0.525241	−0.262620	+	0.964899i	$0.584587\pi$
$884$	15.5885	0.524297
$885$	0	0
$886$	−17.0718	−0.573538
$887$	−45.8038	−1.53794	−0.768971	−	0.639283i	$-0.779231\pi$
$887$	−45.8038	−1.53794	−0.768971	+	0.639283i	$0.779231\pi$
$888$	0	0
$889$	65.5692	2.19912
$890$	−11.1962	−0.375296
$891$	0	0
$892$	−20.3923	−0.682785
$893$	−10.3923	−0.347765
$894$	0	0
$895$	24.0000	0.802232
$896$	−4.73205	−0.158087
$897$	0	0
$898$	9.92820	0.331308
$899$	−0.588457	−0.0196261
$900$	0	0
$901$	58.1769	1.93815
$902$	0	0
$903$	0	0
$904$	−10.8564	−0.361079
$905$	12.4641	0.414321
$906$	0	0
$907$	37.5692	1.24747	0.623733	−	0.781638i	$-0.285615\pi$
$907$	37.5692	1.24747	0.623733	+	0.781638i	$0.285615\pi$
$908$	−16.3923	−0.543998
$909$	0	0
$910$	24.5885	0.815099
$911$	31.6077	1.04721	0.523605	−	0.851961i	$-0.324587\pi$
$911$	31.6077	1.04721	0.523605	+	0.851961i	$0.324587\pi$
$912$	0	0
$913$	0	0
$914$	5.19615	0.171873
$915$	0	0
$916$	−13.1962	−0.436013
$917$	0	0
$918$	0	0
$919$	−22.1436	−0.730450	−0.365225	−	0.930919i	$-0.619008\pi$
$919$	−22.1436	−0.730450	−0.365225	+	0.930919i	$0.619008\pi$
$920$	2.19615	0.0724050
$921$	0	0
$922$	33.0000	1.08680
$923$	−28.3923	−0.934544
$924$	0	0
$925$	14.3923	0.473216
$926$	28.7846	0.945921
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	0	0
$931$	−19.5167	−0.639633
$932$	−6.80385	−0.222867
$933$	0	0
$934$	−21.4641	−0.702327
$935$	0	0
$936$	0	0
$937$	−35.4449	−1.15793	−0.578967	−	0.815351i	$-0.696544\pi$
$937$	−35.4449	−1.15793	−0.578967	+	0.815351i	$0.696544\pi$
$938$	48.2487	1.57538
$939$	0	0
$940$	14.1962	0.463027
$941$	−34.6077	−1.12818	−0.564089	−	0.825714i	$-0.690773\pi$
$941$	−34.6077	−1.12818	−0.564089	+	0.825714i	$0.690773\pi$
$942$	0	0
$943$	−1.01924	−0.0331910
$944$	−9.46410	−0.308030
$945$	0	0
$946$	0	0
$947$	−5.90897	−0.192016	−0.0960078	−	0.995381i	$-0.530607\pi$
$947$	−5.90897	−0.192016	−0.0960078	+	0.995381i	$0.530607\pi$
$948$	0	0
$949$	−2.78461	−0.0903923
$950$	2.53590	0.0822754
$951$	0	0
$952$	24.5885	0.796916
$953$	−39.5885	−1.28240	−0.641198	−	0.767376i	$-0.721562\pi$
$953$	−39.5885	−1.28240	−0.641198	+	0.767376i	$0.721562\pi$
$954$	0	0
$955$	37.1769	1.20302
$956$	−14.1962	−0.459136
$957$	0	0
$958$	−8.78461	−0.283818
$959$	−44.7846	−1.44617
$960$	0	0
$961$	−30.9615	−0.998759
$962$	21.5885	0.696040
$963$	0	0
$964$	−26.7846	−0.862674
$965$	−17.7846	−0.572507
$966$	0	0
$967$	32.4449	1.04336	0.521678	−	0.853142i	$-0.325306\pi$
$967$	32.4449	1.04336	0.521678	+	0.853142i	$0.325306\pi$
$968$	0	0
$969$	0	0
$970$	−1.73205	−0.0556128
$971$	4.05256	0.130053	0.0650264	−	0.997884i	$-0.479287\pi$
$971$	4.05256	0.130053	0.0650264	+	0.997884i	$0.479287\pi$
$972$	0	0
$973$	95.5692	3.06381
$974$	−12.9808	−0.415930
$975$	0	0
$976$	−2.53590	−0.0811721
$977$	−32.3205	−1.03402	−0.517012	−	0.855978i	$-0.672956\pi$
$977$	−32.3205	−1.03402	−0.517012	+	0.855978i	$0.672956\pi$
$978$	0	0
$979$	0	0
$980$	26.6603	0.851631
$981$	0	0
$982$	−33.3731	−1.06498
$983$	−32.1051	−1.02399	−0.511997	−	0.858987i	$-0.671094\pi$
$983$	−32.1051	−1.02399	−0.511997	+	0.858987i	$0.671094\pi$
$984$	0	0
$985$	−23.1962	−0.739091
$986$	15.5885	0.496438
$987$	0	0
$988$	3.80385	0.121017
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0.196152	0.00622785
$993$	0	0
$994$	−44.7846	−1.42048
$995$	−0.679492	−0.0215413
$996$	0	0
$997$	−12.4641	−0.394742	−0.197371	−	0.980329i	$-0.563240\pi$
$997$	−12.4641	−0.394742	−0.197371	+	0.980329i	$0.563240\pi$
$998$	16.0000	0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.2.a.y.1.2		2
3.2	odd	2		242.2.a.c.1.2	✓	2
11.10	odd	2		2178.2.a.s.1.2		2
12.11	even	2		1936.2.a.y.1.1		2
15.14	odd	2		6050.2.a.cv.1.1		2
24.5	odd	2		7744.2.a.cs.1.1		2
24.11	even	2		7744.2.a.bt.1.2		2
33.2	even	10		242.2.c.f.81.2		8
33.5	odd	10		242.2.c.g.3.2		8
33.8	even	10		242.2.c.f.9.1		8
33.14	odd	10		242.2.c.g.9.1		8
33.17	even	10		242.2.c.f.3.2		8
33.20	odd	10		242.2.c.g.81.2		8
33.26	odd	10		242.2.c.g.27.1		8
33.29	even	10		242.2.c.f.27.1		8
33.32	even	2		242.2.a.e.1.2	yes	2
132.131	odd	2		1936.2.a.v.1.1		2
165.164	even	2		6050.2.a.cc.1.1		2
264.131	odd	2		7744.2.a.bq.1.2		2
264.197	even	2		7744.2.a.cv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.a.c.1.2	✓	2	3.2	odd	2
242.2.a.e.1.2	yes	2	33.32	even	2
242.2.c.f.3.2		8	33.17	even	10
242.2.c.f.9.1		8	33.8	even	10
242.2.c.f.27.1		8	33.29	even	10
242.2.c.f.81.2		8	33.2	even	10
242.2.c.g.3.2		8	33.5	odd	10
242.2.c.g.9.1		8	33.14	odd	10
242.2.c.g.27.1		8	33.26	odd	10
242.2.c.g.81.2		8	33.20	odd	10
1936.2.a.v.1.1		2	132.131	odd	2
1936.2.a.y.1.1		2	12.11	even	2
2178.2.a.s.1.2		2	11.10	odd	2
2178.2.a.y.1.2		2	1.1	even	1	trivial
6050.2.a.cc.1.1		2	165.164	even	2
6050.2.a.cv.1.1		2	15.14	odd	2
7744.2.a.bq.1.2		2	264.131	odd	2
7744.2.a.bt.1.2		2	24.11	even	2
7744.2.a.cs.1.1		2	24.5	odd	2
7744.2.a.cv.1.1		2	264.197	even	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} -4.73205 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} -4.73205 q^{7} +1.00000 q^{8} +1.73205 q^{10} -3.00000 q^{13} -4.73205 q^{14} +1.00000 q^{16} -5.19615 q^{17} -1.26795 q^{19} +1.73205 q^{20} +1.26795 q^{23} -2.00000 q^{25} -3.00000 q^{26} -4.73205 q^{28} -3.00000 q^{29} +0.196152 q^{31} +1.00000 q^{32} -5.19615 q^{34} -8.19615 q^{35} -7.19615 q^{37} -1.26795 q^{38} +1.73205 q^{40} -0.803848 q^{41} +1.26795 q^{46} +8.19615 q^{47} +15.3923 q^{49} -2.00000 q^{50} -3.00000 q^{52} -11.1962 q^{53} -4.73205 q^{56} -3.00000 q^{58} -9.46410 q^{59} -2.53590 q^{61} +0.196152 q^{62} +1.00000 q^{64} -5.19615 q^{65} -10.1962 q^{67} -5.19615 q^{68} -8.19615 q^{70} +9.46410 q^{71} +0.928203 q^{73} -7.19615 q^{74} -1.26795 q^{76} +4.73205 q^{79} +1.73205 q^{80} -0.803848 q^{82} +8.19615 q^{83} -9.00000 q^{85} -6.46410 q^{89} +14.1962 q^{91} +1.26795 q^{92} +8.19615 q^{94} -2.19615 q^{95} -1.00000 q^{97} +15.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 6 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 6 q^{7} + 2 q^{8} - 6 q^{13} - 6 q^{14} + 2 q^{16} - 6 q^{19} + 6 q^{23} - 4 q^{25} - 6 q^{26} - 6 q^{28} - 6 q^{29} - 10 q^{31} + 2 q^{32} - 6 q^{35} - 4 q^{37} - 6 q^{38} - 12 q^{41} + 6 q^{46} + 6 q^{47} + 10 q^{49} - 4 q^{50} - 6 q^{52} - 12 q^{53} - 6 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} - 10 q^{62} + 2 q^{64} - 10 q^{67} - 6 q^{70} + 12 q^{71} - 12 q^{73} - 4 q^{74} - 6 q^{76} + 6 q^{79} - 12 q^{82} + 6 q^{83} - 18 q^{85} - 6 q^{89} + 18 q^{91} + 6 q^{92} + 6 q^{94} + 6 q^{95} - 2 q^{97} + 10 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^7 + 2 * q^8 - 6 * q^13 - 6 * q^14 + 2 * q^16 - 6 * q^19 + 6 * q^23 - 4 * q^25 - 6 * q^26 - 6 * q^28 - 6 * q^29 - 10 * q^31 + 2 * q^32 - 6 * q^35 - 4 * q^37 - 6 * q^38 - 12 * q^41 + 6 * q^46 + 6 * q^47 + 10 * q^49 - 4 * q^50 - 6 * q^52 - 12 * q^53 - 6 * q^56 - 6 * q^58 - 12 * q^59 - 12 * q^61 - 10 * q^62 + 2 * q^64 - 10 * q^67 - 6 * q^70 + 12 * q^71 - 12 * q^73 - 4 * q^74 - 6 * q^76 + 6 * q^79 - 12 * q^82 + 6 * q^83 - 18 * q^85 - 6 * q^89 + 18 * q^91 + 6 * q^92 + 6 * q^94 + 6 * q^95 - 2 * q^97 + 10 * q^98

Introduction

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