LMFDB - Embedded newform 6534.2.a.ci.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, 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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{8} +3.46410 q^{10} +5.19615 q^{13} +1.73205 q^{14} +1.00000 q^{16} +3.46410 q^{19} +3.46410 q^{20} +6.92820 q^{23} +7.00000 q^{25} +5.19615 q^{26} +1.73205 q^{28} +6.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} +6.00000 q^{35} -2.00000 q^{37} +3.46410 q^{38} +3.46410 q^{40} -6.00000 q^{41} -10.3923 q^{43} +6.92820 q^{46} -10.3923 q^{47} -4.00000 q^{49} +7.00000 q^{50} +5.19615 q^{52} +1.73205 q^{56} +6.00000 q^{58} +3.46410 q^{59} -13.8564 q^{61} -5.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} +13.0000 q^{67} +6.00000 q^{70} -3.46410 q^{71} -1.73205 q^{73} -2.00000 q^{74} +3.46410 q^{76} +8.66025 q^{79} +3.46410 q^{80} -6.00000 q^{82} -6.00000 q^{83} -10.3923 q^{86} -17.3205 q^{89} +9.00000 q^{91} +6.92820 q^{92} -10.3923 q^{94} +12.0000 q^{95} -19.0000 q^{97} -4.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} + 14 q^{25} + 12 q^{29} - 10 q^{31} + 2 q^{32} + 12 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{49} + 14 q^{50} + 12 q^{58} - 10 q^{62} + 2 q^{64} + 36 q^{65} + 26 q^{67} + 12 q^{70} - 4 q^{74} - 12 q^{82} - 12 q^{83} + 18 q^{91} + 24 q^{95} - 38 q^{97} - 8 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^16 + 14 * q^25 + 12 * q^29 - 10 * q^31 + 2 * q^32 + 12 * q^35 - 4 * q^37 - 12 * q^41 - 8 * q^49 + 14 * q^50 + 12 * q^58 - 10 * q^62 + 2 * q^64 + 36 * q^65 + 26 * q^67 + 12 * q^70 - 4 * q^74 - 12 * q^82 - 12 * q^83 + 18 * q^91 + 24 * q^95 - 38 * q^97 - 8 * 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
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.46410	1.09545
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.19615	1.44115	0.720577	−	0.693375i	$-0.243877\pi$
$13$	5.19615	1.44115	0.720577	+	0.693375i	$0.243877\pi$
$14$	1.73205	0.462910
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	3.46410	0.774597
$21$	0	0
$22$	0	0
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	5.19615	1.01905
$27$	0	0
$28$	1.73205	0.327327
$29$	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
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	3.46410	0.561951
$39$	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
$43$	−10.3923	−1.58481	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	−10.3923	−1.58481	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	6.92820	1.02151
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	7.00000	0.989949
$51$	0	0
$52$	5.19615	0.720577
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	1.73205	0.231455
$57$	0	0
$58$	6.00000	0.787839
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	−13.8564	−1.77413	−0.887066	−	0.461644i	$-0.847260\pi$
$61$	−13.8564	−1.77413	−0.887066	+	0.461644i	$0.847260\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	18.0000	2.23263
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	0	0
$69$	0	0
$70$	6.00000	0.717137
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	−1.73205	−0.202721	−0.101361	−	0.994850i	$-0.532320\pi$
$73$	−1.73205	−0.202721	−0.101361	+	0.994850i	$0.532320\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	3.46410	0.397360
$77$	0	0
$78$	0	0
$79$	8.66025	0.974355	0.487177	−	0.873303i	$-0.338027\pi$
$79$	8.66025	0.974355	0.487177	+	0.873303i	$0.338027\pi$
$80$	3.46410	0.387298
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−10.3923	−1.12063
$87$	0	0
$88$	0	0
$89$	−17.3205	−1.83597	−0.917985	−	0.396615i	$-0.870185\pi$
$89$	−17.3205	−1.83597	−0.917985	+	0.396615i	$0.870185\pi$
$90$	0	0
$91$	9.00000	0.943456
$92$	6.92820	0.722315
$93$	0	0
$94$	−10.3923	−1.07188
$95$	12.0000	1.23117
$96$	0	0
$97$	−19.0000	−1.92916	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−19.0000	−1.92916	−0.964579	+	0.263795i	$0.915026\pi$
$98$	−4.00000	−0.404061
$99$	0	0
$100$	7.00000	0.700000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	5.19615	0.509525
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	12.1244	1.16130	0.580651	−	0.814152i	$-0.302798\pi$
$109$	12.1244	1.16130	0.580651	+	0.814152i	$0.302798\pi$
$110$	0	0
$111$	0	0
$112$	1.73205	0.163663
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	6.00000	0.557086
$117$	0	0
$118$	3.46410	0.318896
$119$	0	0
$120$	0	0
$121$	0	0
$122$	−13.8564	−1.25450
$123$	0	0
$124$	−5.00000	−0.449013
$125$	6.92820	0.619677
$126$	0	0
$127$	8.66025	0.768473	0.384237	−	0.923235i	$-0.374465\pi$
$127$	8.66025	0.768473	0.384237	+	0.923235i	$0.374465\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	18.0000	1.57870
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	13.0000	1.12303
$135$	0	0
$136$	0	0
$137$	−3.46410	−0.295958	−0.147979	−	0.988990i	$-0.547277\pi$
$137$	−3.46410	−0.295958	−0.147979	+	0.988990i	$0.547277\pi$
$138$	0	0
$139$	5.19615	0.440732	0.220366	−	0.975417i	$-0.429275\pi$
$139$	5.19615	0.440732	0.220366	+	0.975417i	$0.429275\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	−3.46410	−0.290701
$143$	0	0
$144$	0	0
$145$	20.7846	1.72607
$146$	−1.73205	−0.143346
$147$	0	0
$148$	−2.00000	−0.164399
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−5.19615	−0.422857	−0.211428	−	0.977393i	$-0.567812\pi$
$151$	−5.19615	−0.422857	−0.211428	+	0.977393i	$0.567812\pi$
$152$	3.46410	0.280976
$153$	0	0
$154$	0	0
$155$	−17.3205	−1.39122
$156$	0	0
$157$	−23.0000	−1.83560	−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000	−1.83560	−0.917800	+	0.397043i	$0.870036\pi$
$158$	8.66025	0.688973
$159$	0	0
$160$	3.46410	0.273861
$161$	12.0000	0.945732
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	14.0000	1.07692
$170$	0	0
$171$	0	0
$172$	−10.3923	−0.792406
$173$	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
$175$	12.1244	0.916515
$176$	0	0
$177$	0	0
$178$	−17.3205	−1.29823
$179$	−3.46410	−0.258919	−0.129460	−	0.991585i	$-0.541324\pi$
$179$	−3.46410	−0.258919	−0.129460	+	0.991585i	$0.541324\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	9.00000	0.667124
$183$	0	0
$184$	6.92820	0.510754
$185$	−6.92820	−0.509372
$186$	0	0
$187$	0	0
$188$	−10.3923	−0.757937
$189$	0	0
$190$	12.0000	0.870572
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	−6.92820	−0.498703	−0.249351	−	0.968413i	$-0.580217\pi$
$193$	−6.92820	−0.498703	−0.249351	+	0.968413i	$0.580217\pi$
$194$	−19.0000	−1.36412
$195$	0	0
$196$	−4.00000	−0.285714
$197$	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
$202$	−12.0000	−0.844317
$203$	10.3923	0.729397
$204$	0	0
$205$	−20.7846	−1.45166
$206$	−7.00000	−0.487713
$207$	0	0
$208$	5.19615	0.360288
$209$	0	0
$210$	0	0
$211$	−3.46410	−0.238479	−0.119239	−	0.992866i	$-0.538046\pi$
$211$	−3.46410	−0.238479	−0.119239	+	0.992866i	$0.538046\pi$
$212$	0	0
$213$	0	0
$214$	6.00000	0.410152
$215$	−36.0000	−2.45518
$216$	0	0
$217$	−8.66025	−0.587896
$218$	12.1244	0.821165
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	1.73205	0.115728
$225$	0	0
$226$	−17.3205	−1.15214
$227$	−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	$0.392813\pi$
$230$	24.0000	1.58251
$231$	0	0
$232$	6.00000	0.393919
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−36.0000	−2.34838
$236$	3.46410	0.225494
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−25.9808	−1.67357	−0.836784	−	0.547533i	$-0.815567\pi$
$241$	−25.9808	−1.67357	−0.836784	+	0.547533i	$0.815567\pi$
$242$	0	0
$243$	0	0
$244$	−13.8564	−0.887066
$245$	−13.8564	−0.885253
$246$	0	0
$247$	18.0000	1.14531
$248$	−5.00000	−0.317500
$249$	0	0
$250$	6.92820	0.438178
$251$	17.3205	1.09326	0.546630	−	0.837374i	$-0.315910\pi$
$251$	17.3205	1.09326	0.546630	+	0.837374i	$0.315910\pi$
$252$	0	0
$253$	0	0
$254$	8.66025	0.543393
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.7846	1.29651	0.648254	−	0.761424i	$-0.275499\pi$
$257$	20.7846	1.29651	0.648254	+	0.761424i	$0.275499\pi$
$258$	0	0
$259$	−3.46410	−0.215249
$260$	18.0000	1.11631
$261$	0	0
$262$	18.0000	1.11204
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	0	0
$266$	6.00000	0.367884
$267$	0	0
$268$	13.0000	0.794101
$269$	−10.3923	−0.633630	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923	−0.633630	−0.316815	+	0.948487i	$0.602613\pi$
$270$	0	0
$271$	−15.5885	−0.946931	−0.473466	−	0.880812i	$-0.656997\pi$
$271$	−15.5885	−0.946931	−0.473466	+	0.880812i	$0.656997\pi$
$272$	0	0
$273$	0	0
$274$	−3.46410	−0.209274
$275$	0	0
$276$	0	0
$277$	8.66025	0.520344	0.260172	−	0.965562i	$-0.416221\pi$
$277$	8.66025	0.520344	0.260172	+	0.965562i	$0.416221\pi$
$278$	5.19615	0.311645
$279$	0	0
$280$	6.00000	0.358569
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−19.0526	−1.13256	−0.566279	−	0.824214i	$-0.691617\pi$
$283$	−19.0526	−1.13256	−0.566279	+	0.824214i	$0.691617\pi$
$284$	−3.46410	−0.205557
$285$	0	0
$286$	0	0
$287$	−10.3923	−0.613438
$288$	0	0
$289$	−17.0000	−1.00000
$290$	20.7846	1.22051
$291$	0	0
$292$	−1.73205	−0.101361
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	−2.00000	−0.116248
$297$	0	0
$298$	12.0000	0.695141
$299$	36.0000	2.08193
$300$	0	0
$301$	−18.0000	−1.03750
$302$	−5.19615	−0.299005
$303$	0	0
$304$	3.46410	0.198680
$305$	−48.0000	−2.74847
$306$	0	0
$307$	−1.73205	−0.0988534	−0.0494267	−	0.998778i	$-0.515739\pi$
$307$	−1.73205	−0.0988534	−0.0494267	+	0.998778i	$0.515739\pi$
$308$	0	0
$309$	0	0
$310$	−17.3205	−0.983739
$311$	−31.1769	−1.76788	−0.883940	−	0.467600i	$-0.845119\pi$
$311$	−31.1769	−1.76788	−0.883940	+	0.467600i	$0.845119\pi$
$312$	0	0
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	−23.0000	−1.29797
$315$	0	0
$316$	8.66025	0.487177
$317$	6.92820	0.389127	0.194563	−	0.980890i	$-0.437671\pi$
$317$	6.92820	0.389127	0.194563	+	0.980890i	$0.437671\pi$
$318$	0	0
$319$	0	0
$320$	3.46410	0.193649
$321$	0	0
$322$	12.0000	0.668734
$323$	0	0
$324$	0	0
$325$	36.3731	2.01761
$326$	17.0000	0.941543
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−18.0000	−0.992372
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	24.0000	1.31322
$335$	45.0333	2.46043
$336$	0	0
$337$	−6.92820	−0.377403	−0.188702	−	0.982034i	$-0.560428\pi$
$337$	−6.92820	−0.377403	−0.188702	+	0.982034i	$0.560428\pi$
$338$	14.0000	0.761500
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.0526	−1.02874
$344$	−10.3923	−0.560316
$345$	0	0
$346$	6.00000	0.322562
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−6.92820	−0.370858	−0.185429	−	0.982658i	$-0.559368\pi$
$349$	−6.92820	−0.370858	−0.185429	+	0.982658i	$0.559368\pi$
$350$	12.1244	0.648074
$351$	0	0
$352$	0	0
$353$	−34.6410	−1.84376	−0.921878	−	0.387481i	$-0.873345\pi$
$353$	−34.6410	−1.84376	−0.921878	+	0.387481i	$0.873345\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	−17.3205	−0.917985
$357$	0	0
$358$	−3.46410	−0.183083
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	−7.00000	−0.367912
$363$	0	0
$364$	9.00000	0.471728
$365$	−6.00000	−0.314054
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	6.92820	0.361158
$369$	0	0
$370$	−6.92820	−0.360180
$371$	0	0
$372$	0	0
$373$	8.66025	0.448411	0.224205	−	0.974542i	$-0.428021\pi$
$373$	8.66025	0.448411	0.224205	+	0.974542i	$0.428021\pi$
$374$	0	0
$375$	0	0
$376$	−10.3923	−0.535942
$377$	31.1769	1.60569
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	6.92820	0.354478
$383$	27.7128	1.41606	0.708029	−	0.706183i	$-0.249584\pi$
$383$	27.7128	1.41606	0.708029	+	0.706183i	$0.249584\pi$
$384$	0	0
$385$	0	0
$386$	−6.92820	−0.352636
$387$	0	0
$388$	−19.0000	−0.964579
$389$	24.2487	1.22946	0.614729	−	0.788738i	$-0.289265\pi$
$389$	24.2487	1.22946	0.614729	+	0.788738i	$0.289265\pi$
$390$	0	0
$391$	0	0
$392$	−4.00000	−0.202031
$393$	0	0
$394$	6.00000	0.302276
$395$	30.0000	1.50946
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−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
$403$	−25.9808	−1.29419
$404$	−12.0000	−0.597022
$405$	0	0
$406$	10.3923	0.515761
$407$	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
$412$	−7.00000	−0.344865
$413$	6.00000	0.295241
$414$	0	0
$415$	−20.7846	−1.02028
$416$	5.19615	0.254762
$417$	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
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	−3.46410	−0.168630
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−24.0000	−1.16144
$428$	6.00000	0.290021
$429$	0	0
$430$	−36.0000	−1.73607
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	−8.66025	−0.415705
$435$	0	0
$436$	12.1244	0.580651
$437$	24.0000	1.14808
$438$	0	0
$439$	19.0526	0.909329	0.454665	−	0.890663i	$-0.349759\pi$
$439$	19.0526	0.909329	0.454665	+	0.890663i	$0.349759\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.2487	1.15209	0.576046	−	0.817418i	$-0.304595\pi$
$443$	24.2487	1.15209	0.576046	+	0.817418i	$0.304595\pi$
$444$	0	0
$445$	−60.0000	−2.84427
$446$	−19.0000	−0.899676
$447$	0	0
$448$	1.73205	0.0818317
$449$	−6.92820	−0.326962	−0.163481	−	0.986546i	$-0.552272\pi$
$449$	−6.92820	−0.326962	−0.163481	+	0.986546i	$0.552272\pi$
$450$	0	0
$451$	0	0
$452$	−17.3205	−0.814688
$453$	0	0
$454$	−24.0000	−1.12638
$455$	31.1769	1.46160
$456$	0	0
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	24.0000	1.11901
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	24.0000	1.11178
$467$	24.2487	1.12210	0.561048	−	0.827783i	$-0.310398\pi$
$467$	24.2487	1.12210	0.561048	+	0.827783i	$0.310398\pi$
$468$	0	0
$469$	22.5167	1.03972
$470$	−36.0000	−1.66056
$471$	0	0
$472$	3.46410	0.159448
$473$	0	0
$474$	0	0
$475$	24.2487	1.11261
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	−10.3923	−0.473848
$482$	−25.9808	−1.18339
$483$	0	0
$484$	0	0
$485$	−65.8179	−2.98864
$486$	0	0
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	−13.8564	−0.627250
$489$	0	0
$490$	−13.8564	−0.625969
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	0	0
$494$	18.0000	0.809858
$495$	0	0
$496$	−5.00000	−0.224507
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−29.0000	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$500$	6.92820	0.309839
$501$	0	0
$502$	17.3205	0.773052
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	−41.5692	−1.84981
$506$	0	0
$507$	0	0
$508$	8.66025	0.384237
$509$	−3.46410	−0.153544	−0.0767718	−	0.997049i	$-0.524461\pi$
$509$	−3.46410	−0.153544	−0.0767718	+	0.997049i	$0.524461\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	1.00000	0.0441942
$513$	0	0
$514$	20.7846	0.916770
$515$	−24.2487	−1.06853
$516$	0	0
$517$	0	0
$518$	−3.46410	−0.152204
$519$	0	0
$520$	18.0000	0.789352
$521$	3.46410	0.151765	0.0758825	−	0.997117i	$-0.475823\pi$
$521$	3.46410	0.151765	0.0758825	+	0.997117i	$0.475823\pi$
$522$	0	0
$523$	17.3205	0.757373	0.378686	−	0.925525i	$-0.376376\pi$
$523$	17.3205	0.757373	0.378686	+	0.925525i	$0.376376\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	6.00000	0.261612
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	6.00000	0.260133
$533$	−31.1769	−1.35042
$534$	0	0
$535$	20.7846	0.898597
$536$	13.0000	0.561514
$537$	0	0
$538$	−10.3923	−0.448044
$539$	0	0
$540$	0	0
$541$	15.5885	0.670200	0.335100	−	0.942183i	$-0.391230\pi$
$541$	15.5885	0.670200	0.335100	+	0.942183i	$0.391230\pi$
$542$	−15.5885	−0.669582
$543$	0	0
$544$	0	0
$545$	42.0000	1.79908
$546$	0	0
$547$	−5.19615	−0.222171	−0.111086	−	0.993811i	$-0.535433\pi$
$547$	−5.19615	−0.222171	−0.111086	+	0.993811i	$0.535433\pi$
$548$	−3.46410	−0.147979
$549$	0	0
$550$	0	0
$551$	20.7846	0.885454
$552$	0	0
$553$	15.0000	0.637865
$554$	8.66025	0.367939
$555$	0	0
$556$	5.19615	0.220366
$557$	−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
$559$	−54.0000	−2.28396
$560$	6.00000	0.253546
$561$	0	0
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−60.0000	−2.52422
$566$	−19.0526	−0.800839
$567$	0	0
$568$	−3.46410	−0.145350
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	8.66025	0.362420	0.181210	−	0.983444i	$-0.441999\pi$
$571$	8.66025	0.362420	0.181210	+	0.983444i	$0.441999\pi$
$572$	0	0
$573$	0	0
$574$	−10.3923	−0.433766
$575$	48.4974	2.02248
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	20.7846	0.863034
$581$	−10.3923	−0.431145
$582$	0	0
$583$	0	0
$584$	−1.73205	−0.0716728
$585$	0	0
$586$	30.0000	1.23929
$587$	10.3923	0.428936	0.214468	−	0.976731i	$-0.431198\pi$
$587$	10.3923	0.428936	0.214468	+	0.976731i	$0.431198\pi$
$588$	0	0
$589$	−17.3205	−0.713679
$590$	12.0000	0.494032
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	0	0
$598$	36.0000	1.47215
$599$	20.7846	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$600$	0	0
$601$	1.73205	0.0706518	0.0353259	−	0.999376i	$-0.488753\pi$
$601$	1.73205	0.0706518	0.0353259	+	0.999376i	$0.488753\pi$
$602$	−18.0000	−0.733625
$603$	0	0
$604$	−5.19615	−0.211428
$605$	0	0
$606$	0	0
$607$	−25.9808	−1.05453	−0.527263	−	0.849702i	$-0.676782\pi$
$607$	−25.9808	−1.05453	−0.527263	+	0.849702i	$0.676782\pi$
$608$	3.46410	0.140488
$609$	0	0
$610$	−48.0000	−1.94346
$611$	−54.0000	−2.18461
$612$	0	0
$613$	12.1244	0.489698	0.244849	−	0.969561i	$-0.421262\pi$
$613$	12.1244	0.489698	0.244849	+	0.969561i	$0.421262\pi$
$614$	−1.73205	−0.0698999
$615$	0	0
$616$	0	0
$617$	10.3923	0.418378	0.209189	−	0.977875i	$-0.432918\pi$
$617$	10.3923	0.418378	0.209189	+	0.977875i	$0.432918\pi$
$618$	0	0
$619$	29.0000	1.16561	0.582804	−	0.812613i	$-0.301955\pi$
$619$	29.0000	1.16561	0.582804	+	0.812613i	$0.301955\pi$
$620$	−17.3205	−0.695608
$621$	0	0
$622$	−31.1769	−1.25008
$623$	−30.0000	−1.20192
$624$	0	0
$625$	−11.0000	−0.440000
$626$	25.0000	0.999201
$627$	0	0
$628$	−23.0000	−0.917800
$629$	0	0
$630$	0	0
$631$	31.0000	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$632$	8.66025	0.344486
$633$	0	0
$634$	6.92820	0.275154
$635$	30.0000	1.19051
$636$	0	0
$637$	−20.7846	−0.823516
$638$	0	0
$639$	0	0
$640$	3.46410	0.136931
$641$	−38.1051	−1.50506	−0.752531	−	0.658557i	$-0.771167\pi$
$641$	−38.1051	−1.50506	−0.752531	+	0.658557i	$0.771167\pi$
$642$	0	0
$643$	5.00000	0.197181	0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000	0.197181	0.0985904	+	0.995128i	$0.468567\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	0	0
$647$	24.2487	0.953315	0.476658	−	0.879089i	$-0.341848\pi$
$647$	24.2487	0.953315	0.476658	+	0.879089i	$0.341848\pi$
$648$	0	0
$649$	0	0
$650$	36.3731	1.42667
$651$	0	0
$652$	17.0000	0.665771
$653$	48.4974	1.89785	0.948925	−	0.315501i	$-0.102172\pi$
$653$	48.4974	1.89785	0.948925	+	0.315501i	$0.102172\pi$
$654$	0	0
$655$	62.3538	2.43637
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−18.0000	−0.701713
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	−19.0000	−0.738456
$663$	0	0
$664$	−6.00000	−0.232845
$665$	20.7846	0.805993
$666$	0	0
$667$	41.5692	1.60957
$668$	24.0000	0.928588
$669$	0	0
$670$	45.0333	1.73979
$671$	0	0
$672$	0	0
$673$	12.1244	0.467360	0.233680	−	0.972314i	$-0.424923\pi$
$673$	12.1244	0.467360	0.233680	+	0.972314i	$0.424923\pi$
$674$	−6.92820	−0.266864
$675$	0	0
$676$	14.0000	0.538462
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	−32.9090	−1.26293
$680$	0	0
$681$	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
$685$	−12.0000	−0.458496
$686$	−19.0526	−0.727430
$687$	0	0
$688$	−10.3923	−0.396203
$689$	0	0
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	18.0000	0.683271
$695$	18.0000	0.682779
$696$	0	0
$697$	0	0
$698$	−6.92820	−0.262236
$699$	0	0
$700$	12.1244	0.458258
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−6.92820	−0.261302
$704$	0	0
$705$	0	0
$706$	−34.6410	−1.30373
$707$	−20.7846	−0.781686
$708$	0	0
$709$	−29.0000	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$709$	−29.0000	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	−17.3205	−0.649113
$713$	−34.6410	−1.29732
$714$	0	0
$715$	0	0
$716$	−3.46410	−0.129460
$717$	0	0
$718$	18.0000	0.671754
$719$	−27.7128	−1.03351	−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$720$	0	0
$721$	−12.1244	−0.451535
$722$	−7.00000	−0.260513
$723$	0	0
$724$	−7.00000	−0.260153
$725$	42.0000	1.55984
$726$	0	0
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	9.00000	0.333562
$729$	0	0
$730$	−6.00000	−0.222070
$731$	0	0
$732$	0	0
$733$	41.5692	1.53539	0.767697	−	0.640813i	$-0.221403\pi$
$733$	41.5692	1.53539	0.767697	+	0.640813i	$0.221403\pi$
$734$	11.0000	0.406017
$735$	0	0
$736$	6.92820	0.255377
$737$	0	0
$738$	0	0
$739$	−19.0526	−0.700860	−0.350430	−	0.936589i	$-0.613964\pi$
$739$	−19.0526	−0.700860	−0.350430	+	0.936589i	$0.613964\pi$
$740$	−6.92820	−0.254686
$741$	0	0
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	41.5692	1.52298
$746$	8.66025	0.317074
$747$	0	0
$748$	0	0
$749$	10.3923	0.379727
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−10.3923	−0.378968
$753$	0	0
$754$	31.1769	1.13540
$755$	−18.0000	−0.655087
$756$	0	0
$757$	−53.0000	−1.92632	−0.963159	−	0.268933i	$-0.913329\pi$
$757$	−53.0000	−1.92632	−0.963159	+	0.268933i	$0.913329\pi$
$758$	5.00000	0.181608
$759$	0	0
$760$	12.0000	0.435286
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	21.0000	0.760251
$764$	6.92820	0.250654
$765$	0	0
$766$	27.7128	1.00130
$767$	18.0000	0.649942
$768$	0	0
$769$	−50.2295	−1.81132	−0.905661	−	0.424003i	$-0.860624\pi$
$769$	−50.2295	−1.81132	−0.905661	+	0.424003i	$0.860624\pi$
$770$	0	0
$771$	0	0
$772$	−6.92820	−0.249351
$773$	31.1769	1.12136	0.560678	−	0.828034i	$-0.310541\pi$
$773$	31.1769	1.12136	0.560678	+	0.828034i	$0.310541\pi$
$774$	0	0
$775$	−35.0000	−1.25724
$776$	−19.0000	−0.682060
$777$	0	0
$778$	24.2487	0.869358
$779$	−20.7846	−0.744686
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−4.00000	−0.142857
$785$	−79.6743	−2.84370
$786$	0	0
$787$	−5.19615	−0.185223	−0.0926114	−	0.995702i	$-0.529521\pi$
$787$	−5.19615	−0.185223	−0.0926114	+	0.995702i	$0.529521\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	30.0000	1.06735
$791$	−30.0000	−1.06668
$792$	0	0
$793$	−72.0000	−2.55679
$794$	−1.00000	−0.0354887
$795$	0	0
$796$	−8.00000	−0.283552
$797$	38.1051	1.34975	0.674876	−	0.737931i	$-0.264197\pi$
$797$	38.1051	1.34975	0.674876	+	0.737931i	$0.264197\pi$
$798$	0	0
$799$	0	0
$800$	7.00000	0.247487
$801$	0	0
$802$	−20.7846	−0.733930
$803$	0	0
$804$	0	0
$805$	41.5692	1.46512
$806$	−25.9808	−0.915133
$807$	0	0
$808$	−12.0000	−0.422159
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	−29.4449	−1.03395	−0.516975	−	0.856001i	$-0.672942\pi$
$811$	−29.4449	−1.03395	−0.516975	+	0.856001i	$0.672942\pi$
$812$	10.3923	0.364698
$813$	0	0
$814$	0	0
$815$	58.8897	2.06282
$816$	0	0
$817$	−36.0000	−1.25948
$818$	6.92820	0.242239
$819$	0	0
$820$	−20.7846	−0.725830
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−7.00000	−0.243857
$825$	0	0
$826$	6.00000	0.208767
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−20.7846	−0.721444
$831$	0	0
$832$	5.19615	0.180144
$833$	0	0
$834$	0	0
$835$	83.1384	2.87712
$836$	0	0
$837$	0	0
$838$	20.7846	0.717992
$839$	6.92820	0.239188	0.119594	−	0.992823i	$-0.461841\pi$
$839$	6.92820	0.239188	0.119594	+	0.992823i	$0.461841\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	19.0000	0.654783
$843$	0	0
$844$	−3.46410	−0.119239
$845$	48.4974	1.66836
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.8564	−0.474991
$852$	0	0
$853$	12.1244	0.415130	0.207565	−	0.978221i	$-0.433446\pi$
$853$	12.1244	0.415130	0.207565	+	0.978221i	$0.433446\pi$
$854$	−24.0000	−0.821263
$855$	0	0
$856$	6.00000	0.205076
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	−36.0000	−1.22759
$861$	0	0
$862$	6.00000	0.204361
$863$	−41.5692	−1.41503	−0.707516	−	0.706697i	$-0.750184\pi$
$863$	−41.5692	−1.41503	−0.707516	+	0.706697i	$0.750184\pi$
$864$	0	0
$865$	20.7846	0.706698
$866$	11.0000	0.373795
$867$	0	0
$868$	−8.66025	−0.293948
$869$	0	0
$870$	0	0
$871$	67.5500	2.28884
$872$	12.1244	0.410582
$873$	0	0
$874$	24.0000	0.811812
$875$	12.0000	0.405674
$876$	0	0
$877$	27.7128	0.935795	0.467898	−	0.883783i	$-0.345012\pi$
$877$	27.7128	0.935795	0.467898	+	0.883783i	$0.345012\pi$
$878$	19.0526	0.642993
$879$	0	0
$880$	0	0
$881$	6.92820	0.233417	0.116709	−	0.993166i	$-0.462766\pi$
$881$	6.92820	0.233417	0.116709	+	0.993166i	$0.462766\pi$
$882$	0	0
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	0	0
$885$	0	0
$886$	24.2487	0.814651
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	−60.0000	−2.01120
$891$	0	0
$892$	−19.0000	−0.636167
$893$	−36.0000	−1.20469
$894$	0	0
$895$	−12.0000	−0.401116
$896$	1.73205	0.0578638
$897$	0	0
$898$	−6.92820	−0.231197
$899$	−30.0000	−1.00056
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−17.3205	−0.576072
$905$	−24.2487	−0.806054
$906$	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
$910$	31.1769	1.03350
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	20.7846	0.687494
$915$	0	0
$916$	10.0000	0.330409
$917$	31.1769	1.02955
$918$	0	0
$919$	−29.4449	−0.971296	−0.485648	−	0.874154i	$-0.661416\pi$
$919$	−29.4449	−0.971296	−0.485648	+	0.874154i	$0.661416\pi$
$920$	24.0000	0.791257
$921$	0	0
$922$	24.0000	0.790398
$923$	−18.0000	−0.592477
$924$	0	0
$925$	−14.0000	−0.460317
$926$	−1.00000	−0.0328620
$927$	0	0
$928$	6.00000	0.196960
$929$	−10.3923	−0.340960	−0.170480	−	0.985361i	$-0.554532\pi$
$929$	−10.3923	−0.340960	−0.170480	+	0.985361i	$0.554532\pi$
$930$	0	0
$931$	−13.8564	−0.454125
$932$	24.0000	0.786146
$933$	0	0
$934$	24.2487	0.793442
$935$	0	0
$936$	0	0
$937$	−32.9090	−1.07509	−0.537545	−	0.843235i	$-0.680648\pi$
$937$	−32.9090	−1.07509	−0.537545	+	0.843235i	$0.680648\pi$
$938$	22.5167	0.735195
$939$	0	0
$940$	−36.0000	−1.17419
$941$	36.0000	1.17357	0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000	1.17357	0.586783	+	0.809744i	$0.300394\pi$
$942$	0	0
$943$	−41.5692	−1.35368
$944$	3.46410	0.112747
$945$	0	0
$946$	0	0
$947$	27.7128	0.900545	0.450273	−	0.892891i	$-0.351327\pi$
$947$	27.7128	0.900545	0.450273	+	0.892891i	$0.351327\pi$
$948$	0	0
$949$	−9.00000	−0.292152
$950$	24.2487	0.786732
$951$	0	0
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	0	0
$957$	0	0
$958$	30.0000	0.969256
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−10.3923	−0.335061
$963$	0	0
$964$	−25.9808	−0.836784
$965$	−24.0000	−0.772587
$966$	0	0
$967$	−24.2487	−0.779786	−0.389893	−	0.920860i	$-0.627488\pi$
$967$	−24.2487	−0.779786	−0.389893	+	0.920860i	$0.627488\pi$
$968$	0	0
$969$	0	0
$970$	−65.8179	−2.11329
$971$	−45.0333	−1.44519	−0.722594	−	0.691273i	$-0.757050\pi$
$971$	−45.0333	−1.44519	−0.722594	+	0.691273i	$0.757050\pi$
$972$	0	0
$973$	9.00000	0.288527
$974$	13.0000	0.416547
$975$	0	0
$976$	−13.8564	−0.443533
$977$	−24.2487	−0.775785	−0.387893	−	0.921705i	$-0.626797\pi$
$977$	−24.2487	−0.775785	−0.387893	+	0.921705i	$0.626797\pi$
$978$	0	0
$979$	0	0
$980$	−13.8564	−0.442627
$981$	0	0
$982$	−6.00000	−0.191468
$983$	38.1051	1.21536	0.607682	−	0.794180i	$-0.292099\pi$
$983$	38.1051	1.21536	0.607682	+	0.794180i	$0.292099\pi$
$984$	0	0
$985$	20.7846	0.662253
$986$	0	0
$987$	0	0
$988$	18.0000	0.572656
$989$	−72.0000	−2.28947
$990$	0	0
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	−5.00000	−0.158750
$993$	0	0
$994$	−6.00000	−0.190308
$995$	−27.7128	−0.878555
$996$	0	0
$997$	−43.3013	−1.37136	−0.685682	−	0.727901i	$-0.740496\pi$
$997$	−43.3013	−1.37136	−0.685682	+	0.727901i	$0.740496\pi$
$998$	−29.0000	−0.917979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.ci.1.2	yes	2
3.2	odd	2		6534.2.a.bm.1.1	✓	2
11.10	odd	2		6534.2.a.bm.1.2	yes	2
33.32	even	2	inner	6534.2.a.ci.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6534.2.a.bm.1.1	✓	2	3.2	odd	2
6534.2.a.bm.1.2	yes	2	11.10	odd	2
6534.2.a.ci.1.1	yes	2	33.32	even	2	inner
6534.2.a.ci.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{8} +3.46410 q^{10} +5.19615 q^{13} +1.73205 q^{14} +1.00000 q^{16} +3.46410 q^{19} +3.46410 q^{20} +6.92820 q^{23} +7.00000 q^{25} +5.19615 q^{26} +1.73205 q^{28} +6.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} +6.00000 q^{35} -2.00000 q^{37} +3.46410 q^{38} +3.46410 q^{40} -6.00000 q^{41} -10.3923 q^{43} +6.92820 q^{46} -10.3923 q^{47} -4.00000 q^{49} +7.00000 q^{50} +5.19615 q^{52} +1.73205 q^{56} +6.00000 q^{58} +3.46410 q^{59} -13.8564 q^{61} -5.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} +13.0000 q^{67} +6.00000 q^{70} -3.46410 q^{71} -1.73205 q^{73} -2.00000 q^{74} +3.46410 q^{76} +8.66025 q^{79} +3.46410 q^{80} -6.00000 q^{82} -6.00000 q^{83} -10.3923 q^{86} -17.3205 q^{89} +9.00000 q^{91} +6.92820 q^{92} -10.3923 q^{94} +12.0000 q^{95} -19.0000 q^{97} -4.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} + 14 q^{25} + 12 q^{29} - 10 q^{31} + 2 q^{32} + 12 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{49} + 14 q^{50} + 12 q^{58} - 10 q^{62} + 2 q^{64} + 36 q^{65} + 26 q^{67} + 12 q^{70} - 4 q^{74} - 12 q^{82} - 12 q^{83} + 18 q^{91} + 24 q^{95} - 38 q^{97} - 8 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^16 + 14 * q^25 + 12 * q^29 - 10 * q^31 + 2 * q^32 + 12 * q^35 - 4 * q^37 - 12 * q^41 - 8 * q^49 + 14 * q^50 + 12 * q^58 - 10 * q^62 + 2 * q^64 + 36 * q^65 + 26 * q^67 + 12 * q^70 - 4 * q^74 - 12 * q^82 - 12 * q^83 + 18 * q^91 + 24 * q^95 - 38 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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