LMFDB - Embedded newform 6534.2.a.cf.1.1

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:	$1$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{8} -1.73205 q^{10} +6.92820 q^{13} +1.00000 q^{16} -6.00000 q^{17} -1.73205 q^{19} -1.73205 q^{20} -3.46410 q^{23} -2.00000 q^{25} +6.92820 q^{26} -9.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -2.00000 q^{37} -1.73205 q^{38} -1.73205 q^{40} +5.19615 q^{43} -3.46410 q^{46} -10.3923 q^{47} -7.00000 q^{49} -2.00000 q^{50} +6.92820 q^{52} -9.00000 q^{58} +3.46410 q^{59} -3.46410 q^{61} +4.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} -8.00000 q^{67} -6.00000 q^{68} +12.1244 q^{71} +5.19615 q^{73} -2.00000 q^{74} -1.73205 q^{76} +10.3923 q^{79} -1.73205 q^{80} +12.0000 q^{83} +10.3923 q^{85} +5.19615 q^{86} +3.46410 q^{89} -3.46410 q^{92} -10.3923 q^{94} +3.00000 q^{95} -7.00000 q^{97} -7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 12 q^{17} - 4 q^{25} - 18 q^{29} + 8 q^{31} + 2 q^{32} - 12 q^{34} - 4 q^{37} - 14 q^{49} - 4 q^{50} - 18 q^{58} + 8 q^{62} + 2 q^{64} - 24 q^{65} - 16 q^{67} - 12 q^{68} - 4 q^{74} + 24 q^{83} + 6 q^{95} - 14 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^16 - 12 * q^17 - 4 * q^25 - 18 * q^29 + 8 * q^31 + 2 * q^32 - 12 * q^34 - 4 * q^37 - 14 * q^49 - 4 * q^50 - 18 * q^58 + 8 * q^62 + 2 * q^64 - 24 * q^65 - 16 * q^67 - 12 * q^68 - 4 * q^74 + 24 * q^83 + 6 * q^95 - 14 * q^97 - 14 * 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	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.73205	−0.547723
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.92820	1.92154	0.960769	−	0.277350i	$-0.0894562\pi$
$13$	6.92820	1.92154	0.960769	+	0.277350i	$0.0894562\pi$
$14$	0	0
$15$	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
$19$	−1.73205	−0.397360	−0.198680	−	0.980064i	$-0.563665\pi$
$19$	−1.73205	−0.397360	−0.198680	+	0.980064i	$0.563665\pi$
$20$	−1.73205	−0.387298
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	6.92820	1.35873
$27$	0	0
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.73205	−0.280976
$39$	0	0
$40$	−1.73205	−0.273861
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	5.19615	0.792406	0.396203	−	0.918163i	$-0.370328\pi$
$43$	5.19615	0.792406	0.396203	+	0.918163i	$0.370328\pi$
$44$	0	0
$45$	0	0
$46$	−3.46410	−0.510754
$47$	−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
$49$	−7.00000	−1.00000
$50$	−2.00000	−0.282843
$51$	0	0
$52$	6.92820	0.960769
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−9.00000	−1.18176
$59$	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
$61$	−3.46410	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	12.1244	1.43890	0.719448	−	0.694546i	$-0.244395\pi$
$71$	12.1244	1.43890	0.719448	+	0.694546i	$0.244395\pi$
$72$	0	0
$73$	5.19615	0.608164	0.304082	−	0.952646i	$-0.401650\pi$
$73$	5.19615	0.608164	0.304082	+	0.952646i	$0.401650\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−1.73205	−0.198680
$77$	0	0
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	−1.73205	−0.193649
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	10.3923	1.12720
$86$	5.19615	0.560316
$87$	0	0
$88$	0	0
$89$	3.46410	0.367194	0.183597	−	0.983002i	$-0.441226\pi$
$89$	3.46410	0.367194	0.183597	+	0.983002i	$0.441226\pi$
$90$	0	0
$91$	0	0
$92$	−3.46410	−0.361158
$93$	0	0
$94$	−10.3923	−1.07188
$95$	3.00000	0.307794
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−6.92820	−0.663602	−0.331801	−	0.943349i	$-0.607656\pi$
$109$	−6.92820	−0.663602	−0.331801	+	0.943349i	$0.607656\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.92820	−0.651751	−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820	−0.651751	−0.325875	+	0.945413i	$0.605659\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	−9.00000	−0.835629
$117$	0	0
$118$	3.46410	0.318896
$119$	0	0
$120$	0	0
$121$	0	0
$122$	−3.46410	−0.313625
$123$	0	0
$124$	4.00000	0.359211
$125$	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
$130$	−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
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−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
$139$	−17.3205	−1.46911	−0.734553	−	0.678551i	$-0.762608\pi$
$139$	−17.3205	−1.46911	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	0	0
$142$	12.1244	1.01745
$143$	0	0
$144$	0	0
$145$	15.5885	1.29455
$146$	5.19615	0.430037
$147$	0	0
$148$	−2.00000	−0.164399
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−17.3205	−1.40952	−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205	−1.40952	−0.704761	+	0.709444i	$0.748946\pi$
$152$	−1.73205	−0.140488
$153$	0	0
$154$	0	0
$155$	−6.92820	−0.556487
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	10.3923	0.826767
$159$	0	0
$160$	−1.73205	−0.136931
$161$	0	0
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	0	0
$169$	35.0000	2.69231
$170$	10.3923	0.797053
$171$	0	0
$172$	5.19615	0.396203
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	3.46410	0.259645
$179$	−24.2487	−1.81243	−0.906217	−	0.422813i	$-0.861043\pi$
$179$	−24.2487	−1.81243	−0.906217	+	0.422813i	$0.861043\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	−3.46410	−0.255377
$185$	3.46410	0.254686
$186$	0	0
$187$	0	0
$188$	−10.3923	−0.757937
$189$	0	0
$190$	3.00000	0.217643
$191$	22.5167	1.62925	0.814624	−	0.579989i	$-0.196943\pi$
$191$	22.5167	1.62925	0.814624	+	0.579989i	$0.196943\pi$
$192$	0	0
$193$	19.0526	1.37143	0.685717	−	0.727869i	$-0.259489\pi$
$193$	19.0526	1.37143	0.685717	+	0.727869i	$0.259489\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	0	0
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	−2.00000	−0.141421
$201$	0	0
$202$	−15.0000	−1.05540
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	6.92820	0.480384
$209$	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
$215$	−9.00000	−0.613795
$216$	0	0
$217$	0	0
$218$	−6.92820	−0.469237
$219$	0	0
$220$	0	0
$221$	−41.5692	−2.79625
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	−6.92820	−0.460857
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	−9.00000	−0.590879
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	3.46410	0.225494
$237$	0	0
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	−1.73205	−0.111571	−0.0557856	−	0.998443i	$-0.517766\pi$
$241$	−1.73205	−0.111571	−0.0557856	+	0.998443i	$0.517766\pi$
$242$	0	0
$243$	0	0
$244$	−3.46410	−0.221766
$245$	12.1244	0.774597
$246$	0	0
$247$	−12.0000	−0.763542
$248$	4.00000	0.254000
$249$	0	0
$250$	12.1244	0.766812
$251$	−3.46410	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$251$	−3.46410	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$252$	0	0
$253$	0	0
$254$	−13.8564	−0.869428
$255$	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
$259$	0	0
$260$	−12.0000	−0.744208
$261$	0	0
$262$	−12.0000	−0.741362
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−15.5885	−0.950445	−0.475223	−	0.879866i	$-0.657632\pi$
$269$	−15.5885	−0.950445	−0.475223	+	0.879866i	$0.657632\pi$
$270$	0	0
$271$	−24.2487	−1.47300	−0.736502	−	0.676435i	$-0.763524\pi$
$271$	−24.2487	−1.47300	−0.736502	+	0.676435i	$0.763524\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−3.46410	−0.209274
$275$	0	0
$276$	0	0
$277$	−13.8564	−0.832551	−0.416275	−	0.909239i	$-0.636665\pi$
$277$	−13.8564	−0.832551	−0.416275	+	0.909239i	$0.636665\pi$
$278$	−17.3205	−1.03882
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−17.3205	−1.02960	−0.514799	−	0.857311i	$-0.672133\pi$
$283$	−17.3205	−1.02960	−0.514799	+	0.857311i	$0.672133\pi$
$284$	12.1244	0.719448
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	15.5885	0.915386
$291$	0	0
$292$	5.19615	0.304082
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	−2.00000	−0.116248
$297$	0	0
$298$	9.00000	0.521356
$299$	−24.0000	−1.38796
$300$	0	0
$301$	0	0
$302$	−17.3205	−0.996683
$303$	0	0
$304$	−1.73205	−0.0993399
$305$	6.00000	0.343559
$306$	0	0
$307$	−8.66025	−0.494267	−0.247133	−	0.968981i	$-0.579489\pi$
$307$	−8.66025	−0.494267	−0.247133	+	0.968981i	$0.579489\pi$
$308$	0	0
$309$	0	0
$310$	−6.92820	−0.393496
$311$	15.5885	0.883940	0.441970	−	0.897030i	$-0.354280\pi$
$311$	15.5885	0.883940	0.441970	+	0.897030i	$0.354280\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	10.3923	0.584613
$317$	12.1244	0.680972	0.340486	−	0.940250i	$-0.389408\pi$
$317$	12.1244	0.680972	0.340486	+	0.940250i	$0.389408\pi$
$318$	0	0
$319$	0	0
$320$	−1.73205	−0.0968246
$321$	0	0
$322$	0	0
$323$	10.3923	0.578243
$324$	0	0
$325$	−13.8564	−0.768615
$326$	11.0000	0.609234
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−15.0000	−0.820763
$335$	13.8564	0.757056
$336$	0	0
$337$	−12.1244	−0.660456	−0.330228	−	0.943901i	$-0.607126\pi$
$337$	−12.1244	−0.660456	−0.330228	+	0.943901i	$0.607126\pi$
$338$	35.0000	1.90375
$339$	0	0
$340$	10.3923	0.563602
$341$	0	0
$342$	0	0
$343$	0	0
$344$	5.19615	0.280158
$345$	0	0
$346$	9.00000	0.483843
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	3.46410	0.185429	0.0927146	−	0.995693i	$-0.470446\pi$
$349$	3.46410	0.185429	0.0927146	+	0.995693i	$0.470446\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.2487	−1.29063	−0.645314	−	0.763917i	$-0.723274\pi$
$353$	−24.2487	−1.29063	−0.645314	+	0.763917i	$0.723274\pi$
$354$	0	0
$355$	−21.0000	−1.11456
$356$	3.46410	0.183597
$357$	0	0
$358$	−24.2487	−1.28158
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	−22.0000	−1.15629
$363$	0	0
$364$	0	0
$365$	−9.00000	−0.471082
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−3.46410	−0.180579
$369$	0	0
$370$	3.46410	0.180090
$371$	0	0
$372$	0	0
$373$	−31.1769	−1.61428	−0.807140	−	0.590360i	$-0.798986\pi$
$373$	−31.1769	−1.61428	−0.807140	+	0.590360i	$0.798986\pi$
$374$	0	0
$375$	0	0
$376$	−10.3923	−0.535942
$377$	−62.3538	−3.21139
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	3.00000	0.153897
$381$	0	0
$382$	22.5167	1.15205
$383$	22.5167	1.15055	0.575274	−	0.817961i	$-0.304896\pi$
$383$	22.5167	1.15055	0.575274	+	0.817961i	$0.304896\pi$
$384$	0	0
$385$	0	0
$386$	19.0526	0.969750
$387$	0	0
$388$	−7.00000	−0.355371
$389$	19.0526	0.966003	0.483002	−	0.875620i	$-0.339547\pi$
$389$	19.0526	0.966003	0.483002	+	0.875620i	$0.339547\pi$
$390$	0	0
$391$	20.7846	1.05112
$392$	−7.00000	−0.353553
$393$	0	0
$394$	−27.0000	−1.36024
$395$	−18.0000	−0.905678
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−26.0000	−1.30326
$399$	0	0
$400$	−2.00000	−0.100000
$401$	31.1769	1.55690	0.778450	−	0.627706i	$-0.216006\pi$
$401$	31.1769	1.55690	0.778450	+	0.627706i	$0.216006\pi$
$402$	0	0
$403$	27.7128	1.38047
$404$	−15.0000	−0.746278
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	32.9090	1.62724	0.813622	−	0.581394i	$-0.197493\pi$
$409$	32.9090	1.62724	0.813622	+	0.581394i	$0.197493\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	−20.7846	−1.02028
$416$	6.92820	0.339683
$417$	0	0
$418$	0	0
$419$	−20.7846	−1.01539	−0.507697	−	0.861536i	$-0.669503\pi$
$419$	−20.7846	−1.01539	−0.507697	+	0.861536i	$0.669503\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−3.46410	−0.168630
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−9.00000	−0.434019
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	−6.92820	−0.331801
$437$	6.00000	0.287019
$438$	0	0
$439$	27.7128	1.32266	0.661330	−	0.750095i	$-0.269992\pi$
$439$	27.7128	1.32266	0.661330	+	0.750095i	$0.269992\pi$
$440$	0	0
$441$	0	0
$442$	−41.5692	−1.97725
$443$	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
$445$	−6.00000	−0.284427
$446$	8.00000	0.378811
$447$	0	0
$448$	0	0
$449$	−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
$452$	−6.92820	−0.325875
$453$	0	0
$454$	−18.0000	−0.844782
$455$	0	0
$456$	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
$460$	6.00000	0.279751
$461$	27.0000	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$461$	27.0000	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	6.00000	0.277945
$467$	34.6410	1.60300	0.801498	−	0.597998i	$-0.204037\pi$
$467$	34.6410	1.60300	0.801498	+	0.597998i	$0.204037\pi$
$468$	0	0
$469$	0	0
$470$	18.0000	0.830278
$471$	0	0
$472$	3.46410	0.159448
$473$	0	0
$474$	0	0
$475$	3.46410	0.158944
$476$	0	0
$477$	0	0
$478$	21.0000	0.960518
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	−13.8564	−0.631798
$482$	−1.73205	−0.0788928
$483$	0	0
$484$	0	0
$485$	12.1244	0.550539
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−3.46410	−0.156813
$489$	0	0
$490$	12.1244	0.547723
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	54.0000	2.43204
$494$	−12.0000	−0.539906
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	12.1244	0.542218
$501$	0	0
$502$	−3.46410	−0.154610
$503$	−39.0000	−1.73892	−0.869462	−	0.494000i	$-0.835534\pi$
$503$	−39.0000	−1.73892	−0.869462	+	0.494000i	$0.835534\pi$
$504$	0	0
$505$	25.9808	1.15613
$506$	0	0
$507$	0	0
$508$	−13.8564	−0.614779
$509$	32.9090	1.45866	0.729332	−	0.684160i	$-0.239831\pi$
$509$	32.9090	1.45866	0.729332	+	0.684160i	$0.239831\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	0	0
$515$	6.92820	0.305293
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−12.0000	−0.526235
$521$	34.6410	1.51765	0.758825	−	0.651294i	$-0.225774\pi$
$521$	34.6410	1.51765	0.758825	+	0.651294i	$0.225774\pi$
$522$	0	0
$523$	38.1051	1.66622	0.833110	−	0.553107i	$-0.186558\pi$
$523$	38.1051	1.66622	0.833110	+	0.553107i	$0.186558\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	3.00000	0.130806
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.3923	0.449299
$536$	−8.00000	−0.345547
$537$	0	0
$538$	−15.5885	−0.672066
$539$	0	0
$540$	0	0
$541$	24.2487	1.04253	0.521267	−	0.853394i	$-0.325460\pi$
$541$	24.2487	1.04253	0.521267	+	0.853394i	$0.325460\pi$
$542$	−24.2487	−1.04157
$543$	0	0
$544$	−6.00000	−0.257248
$545$	12.0000	0.514024
$546$	0	0
$547$	−24.2487	−1.03680	−0.518400	−	0.855138i	$-0.673472\pi$
$547$	−24.2487	−1.03680	−0.518400	+	0.855138i	$0.673472\pi$
$548$	−3.46410	−0.147979
$549$	0	0
$550$	0	0
$551$	15.5885	0.664091
$552$	0	0
$553$	0	0
$554$	−13.8564	−0.588702
$555$	0	0
$556$	−17.3205	−0.734553
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−17.3205	−0.728035
$567$	0	0
$568$	12.1244	0.508727
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−5.19615	−0.217452	−0.108726	−	0.994072i	$-0.534677\pi$
$571$	−5.19615	−0.217452	−0.108726	+	0.994072i	$0.534677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.92820	0.288926
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	15.5885	0.647275
$581$	0	0
$582$	0	0
$583$	0	0
$584$	5.19615	0.215018
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−20.7846	−0.857873	−0.428936	−	0.903335i	$-0.641112\pi$
$587$	−20.7846	−0.857873	−0.428936	+	0.903335i	$0.641112\pi$
$588$	0	0
$589$	−6.92820	−0.285472
$590$	−6.00000	−0.247016
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	9.00000	0.368654
$597$	0	0
$598$	−24.0000	−0.981433
$599$	−31.1769	−1.27385	−0.636927	−	0.770924i	$-0.719795\pi$
$599$	−31.1769	−1.27385	−0.636927	+	0.770924i	$0.719795\pi$
$600$	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
$604$	−17.3205	−0.704761
$605$	0	0
$606$	0	0
$607$	17.3205	0.703018	0.351509	−	0.936185i	$-0.385669\pi$
$607$	17.3205	0.703018	0.351509	+	0.936185i	$0.385669\pi$
$608$	−1.73205	−0.0702439
$609$	0	0
$610$	6.00000	0.242933
$611$	−72.0000	−2.91281
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−8.66025	−0.349499
$615$	0	0
$616$	0	0
$617$	−31.1769	−1.25514	−0.627568	−	0.778562i	$-0.715949\pi$
$617$	−31.1769	−1.25514	−0.627568	+	0.778562i	$0.715949\pi$
$618$	0	0
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	−6.92820	−0.278243
$621$	0	0
$622$	15.5885	0.625040
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	19.0000	0.759393
$627$	0	0
$628$	4.00000	0.159617
$629$	12.0000	0.478471
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	10.3923	0.413384
$633$	0	0
$634$	12.1244	0.481520
$635$	24.0000	0.952411
$636$	0	0
$637$	−48.4974	−1.92154
$638$	0	0
$639$	0	0
$640$	−1.73205	−0.0684653
$641$	−17.3205	−0.684119	−0.342059	−	0.939678i	$-0.611124\pi$
$641$	−17.3205	−0.684119	−0.342059	+	0.939678i	$0.611124\pi$
$642$	0	0
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	0	0
$645$	0	0
$646$	10.3923	0.408880
$647$	−12.1244	−0.476658	−0.238329	−	0.971185i	$-0.576600\pi$
$647$	−12.1244	−0.476658	−0.238329	+	0.971185i	$0.576600\pi$
$648$	0	0
$649$	0	0
$650$	−13.8564	−0.543493
$651$	0	0
$652$	11.0000	0.430793
$653$	1.73205	0.0677804	0.0338902	−	0.999426i	$-0.489210\pi$
$653$	1.73205	0.0677804	0.0338902	+	0.999426i	$0.489210\pi$
$654$	0	0
$655$	20.7846	0.812122
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	28.0000	1.08907	0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000	1.08907	0.544537	+	0.838737i	$0.316705\pi$
$662$	11.0000	0.427527
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	31.1769	1.20717
$668$	−15.0000	−0.580367
$669$	0	0
$670$	13.8564	0.535320
$671$	0	0
$672$	0	0
$673$	−15.5885	−0.600891	−0.300445	−	0.953799i	$-0.597135\pi$
$673$	−15.5885	−0.600891	−0.300445	+	0.953799i	$0.597135\pi$
$674$	−12.1244	−0.467013
$675$	0	0
$676$	35.0000	1.34615
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	0	0
$680$	10.3923	0.398527
$681$	0	0
$682$	0	0
$683$	20.7846	0.795301	0.397650	−	0.917537i	$-0.369826\pi$
$683$	20.7846	0.795301	0.397650	+	0.917537i	$0.369826\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	0	0
$688$	5.19615	0.198101
$689$	0	0
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	9.00000	0.342129
$693$	0	0
$694$	30.0000	1.13878
$695$	30.0000	1.13796
$696$	0	0
$697$	0	0
$698$	3.46410	0.131118
$699$	0	0
$700$	0	0
$701$	−39.0000	−1.47301	−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000	−1.47301	−0.736505	+	0.676432i	$0.763525\pi$
$702$	0	0
$703$	3.46410	0.130651
$704$	0	0
$705$	0	0
$706$	−24.2487	−0.912612
$707$	0	0
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	−21.0000	−0.788116
$711$	0	0
$712$	3.46410	0.129823
$713$	−13.8564	−0.518927
$714$	0	0
$715$	0	0
$716$	−24.2487	−0.906217
$717$	0	0
$718$	−15.0000	−0.559795
$719$	19.0526	0.710541	0.355270	−	0.934764i	$-0.384389\pi$
$719$	19.0526	0.710541	0.355270	+	0.934764i	$0.384389\pi$
$720$	0	0
$721$	0	0
$722$	−16.0000	−0.595458
$723$	0	0
$724$	−22.0000	−0.817624
$725$	18.0000	0.668503
$726$	0	0
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	0	0
$729$	0	0
$730$	−9.00000	−0.333105
$731$	−31.1769	−1.15312
$732$	0	0
$733$	−31.1769	−1.15155	−0.575773	−	0.817610i	$-0.695299\pi$
$733$	−31.1769	−1.15155	−0.575773	+	0.817610i	$0.695299\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−3.46410	−0.127688
$737$	0	0
$738$	0	0
$739$	−1.73205	−0.0637145	−0.0318573	−	0.999492i	$-0.510142\pi$
$739$	−1.73205	−0.0637145	−0.0318573	+	0.999492i	$0.510142\pi$
$740$	3.46410	0.127343
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−15.5885	−0.571117
$746$	−31.1769	−1.14147
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	−10.3923	−0.378968
$753$	0	0
$754$	−62.3538	−2.27079
$755$	30.0000	1.09181
$756$	0	0
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	3.00000	0.108821
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	0	0
$764$	22.5167	0.814624
$765$	0	0
$766$	22.5167	0.813560
$767$	24.0000	0.866590
$768$	0	0
$769$	−5.19615	−0.187378	−0.0936890	−	0.995602i	$-0.529866\pi$
$769$	−5.19615	−0.187378	−0.0936890	+	0.995602i	$0.529866\pi$
$770$	0	0
$771$	0	0
$772$	19.0526	0.685717
$773$	5.19615	0.186893	0.0934463	−	0.995624i	$-0.470212\pi$
$773$	5.19615	0.186893	0.0934463	+	0.995624i	$0.470212\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	−7.00000	−0.251285
$777$	0	0
$778$	19.0526	0.683067
$779$	0	0
$780$	0	0
$781$	0	0
$782$	20.7846	0.743256
$783$	0	0
$784$	−7.00000	−0.250000
$785$	−6.92820	−0.247278
$786$	0	0
$787$	−1.73205	−0.0617409	−0.0308705	−	0.999523i	$-0.509828\pi$
$787$	−1.73205	−0.0617409	−0.0308705	+	0.999523i	$0.509828\pi$
$788$	−27.0000	−0.961835
$789$	0	0
$790$	−18.0000	−0.640411
$791$	0	0
$792$	0	0
$793$	−24.0000	−0.852265
$794$	26.0000	0.922705
$795$	0	0
$796$	−26.0000	−0.921546
$797$	−39.8372	−1.41110	−0.705552	−	0.708658i	$-0.749301\pi$
$797$	−39.8372	−1.41110	−0.705552	+	0.708658i	$0.749301\pi$
$798$	0	0
$799$	62.3538	2.20592
$800$	−2.00000	−0.0707107
$801$	0	0
$802$	31.1769	1.10090
$803$	0	0
$804$	0	0
$805$	0	0
$806$	27.7128	0.976142
$807$	0	0
$808$	−15.0000	−0.527698
$809$	−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
$811$	8.66025	0.304103	0.152051	−	0.988373i	$-0.451412\pi$
$811$	8.66025	0.304103	0.152051	+	0.988373i	$0.451412\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.0526	−0.667382
$816$	0	0
$817$	−9.00000	−0.314870
$818$	32.9090	1.15063
$819$	0	0
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	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
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−20.7846	−0.721444
$831$	0	0
$832$	6.92820	0.240192
$833$	42.0000	1.45521
$834$	0	0
$835$	25.9808	0.899101
$836$	0	0
$837$	0	0
$838$	−20.7846	−0.717992
$839$	−50.2295	−1.73411	−0.867057	−	0.498209i	$-0.833991\pi$
$839$	−50.2295	−1.73411	−0.867057	+	0.498209i	$0.833991\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	22.0000	0.758170
$843$	0	0
$844$	−3.46410	−0.119239
$845$	−60.6218	−2.08545
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	12.0000	0.411597
$851$	6.92820	0.237496
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−6.00000	−0.205076
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	−9.00000	−0.306897
$861$	0	0
$862$	−36.0000	−1.22616
$863$	5.19615	0.176879	0.0884395	−	0.996082i	$-0.471812\pi$
$863$	5.19615	0.176879	0.0884395	+	0.996082i	$0.471812\pi$
$864$	0	0
$865$	−15.5885	−0.530023
$866$	2.00000	0.0679628
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−55.4256	−1.87803
$872$	−6.92820	−0.234619
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	−34.6410	−1.16974	−0.584872	−	0.811126i	$-0.698855\pi$
$877$	−34.6410	−1.16974	−0.584872	+	0.811126i	$0.698855\pi$
$878$	27.7128	0.935262
$879$	0	0
$880$	0	0
$881$	48.4974	1.63392	0.816960	−	0.576695i	$-0.195658\pi$
$881$	48.4974	1.63392	0.816960	+	0.576695i	$0.195658\pi$
$882$	0	0
$883$	−29.0000	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$884$	−41.5692	−1.39812
$885$	0	0
$886$	24.2487	0.814651
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	0	0
$890$	−6.00000	−0.201120
$891$	0	0
$892$	8.00000	0.267860
$893$	18.0000	0.602347
$894$	0	0
$895$	42.0000	1.40391
$896$	0	0
$897$	0	0
$898$	−6.92820	−0.231197
$899$	−36.0000	−1.20067
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−6.92820	−0.230429
$905$	38.1051	1.26666
$906$	0	0
$907$	−31.0000	−1.02934	−0.514669	−	0.857389i	$-0.672085\pi$
$907$	−31.0000	−1.02934	−0.514669	+	0.857389i	$0.672085\pi$
$908$	−18.0000	−0.597351
$909$	0	0
$910$	0	0
$911$	31.1769	1.03294	0.516469	−	0.856306i	$-0.327246\pi$
$911$	31.1769	1.03294	0.516469	+	0.856306i	$0.327246\pi$
$912$	0	0
$913$	0	0
$914$	20.7846	0.687494
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	3.46410	0.114270	0.0571351	−	0.998366i	$-0.481803\pi$
$919$	3.46410	0.114270	0.0571351	+	0.998366i	$0.481803\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	27.0000	0.889198
$923$	84.0000	2.76489
$924$	0	0
$925$	4.00000	0.131519
$926$	14.0000	0.460069
$927$	0	0
$928$	−9.00000	−0.295439
$929$	20.7846	0.681921	0.340960	−	0.940078i	$-0.389248\pi$
$929$	20.7846	0.681921	0.340960	+	0.940078i	$0.389248\pi$
$930$	0	0
$931$	12.1244	0.397360
$932$	6.00000	0.196537
$933$	0	0
$934$	34.6410	1.13349
$935$	0	0
$936$	0	0
$937$	25.9808	0.848755	0.424377	−	0.905485i	$-0.360493\pi$
$937$	25.9808	0.848755	0.424377	+	0.905485i	$0.360493\pi$
$938$	0	0
$939$	0	0
$940$	18.0000	0.587095
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	0	0
$944$	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
$949$	36.0000	1.16861
$950$	3.46410	0.112390
$951$	0	0
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−39.0000	−1.26201
$956$	21.0000	0.679189
$957$	0	0
$958$	21.0000	0.678479
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−13.8564	−0.446748
$963$	0	0
$964$	−1.73205	−0.0557856
$965$	−33.0000	−1.06231
$966$	0	0
$967$	−13.8564	−0.445592	−0.222796	−	0.974865i	$-0.571518\pi$
$967$	−13.8564	−0.445592	−0.222796	+	0.974865i	$0.571518\pi$
$968$	0	0
$969$	0	0
$970$	12.1244	0.389290
$971$	−3.46410	−0.111168	−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410	−0.111168	−0.0555842	+	0.998454i	$0.517702\pi$
$972$	0	0
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−3.46410	−0.110883
$977$	17.3205	0.554132	0.277066	−	0.960851i	$-0.410638\pi$
$977$	17.3205	0.554132	0.277066	+	0.960851i	$0.410638\pi$
$978$	0	0
$979$	0	0
$980$	12.1244	0.387298
$981$	0	0
$982$	30.0000	0.957338
$983$	−60.6218	−1.93353	−0.966767	−	0.255658i	$-0.917708\pi$
$983$	−60.6218	−1.93353	−0.966767	+	0.255658i	$0.917708\pi$
$984$	0	0
$985$	46.7654	1.49007
$986$	54.0000	1.71971
$987$	0	0
$988$	−12.0000	−0.381771
$989$	−18.0000	−0.572367
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	0	0
$995$	45.0333	1.42765
$996$	0	0
$997$	−13.8564	−0.438837	−0.219418	−	0.975631i	$-0.570416\pi$
$997$	−13.8564	−0.438837	−0.219418	+	0.975631i	$0.570416\pi$
$998$	−11.0000	−0.348199
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{8} -1.73205 q^{10} +6.92820 q^{13} +1.00000 q^{16} -6.00000 q^{17} -1.73205 q^{19} -1.73205 q^{20} -3.46410 q^{23} -2.00000 q^{25} +6.92820 q^{26} -9.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -2.00000 q^{37} -1.73205 q^{38} -1.73205 q^{40} +5.19615 q^{43} -3.46410 q^{46} -10.3923 q^{47} -7.00000 q^{49} -2.00000 q^{50} +6.92820 q^{52} -9.00000 q^{58} +3.46410 q^{59} -3.46410 q^{61} +4.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} -8.00000 q^{67} -6.00000 q^{68} +12.1244 q^{71} +5.19615 q^{73} -2.00000 q^{74} -1.73205 q^{76} +10.3923 q^{79} -1.73205 q^{80} +12.0000 q^{83} +10.3923 q^{85} +5.19615 q^{86} +3.46410 q^{89} -3.46410 q^{92} -10.3923 q^{94} +3.00000 q^{95} -7.00000 q^{97} -7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} - 12 q^{17} - 4 q^{25} - 18 q^{29} + 8 q^{31} + 2 q^{32} - 12 q^{34} - 4 q^{37} - 14 q^{49} - 4 q^{50} - 18 q^{58} + 8 q^{62} + 2 q^{64} - 24 q^{65} - 16 q^{67} - 12 q^{68} - 4 q^{74} + 24 q^{83} + 6 q^{95} - 14 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^16 - 12 * q^17 - 4 * q^25 - 18 * q^29 + 8 * q^31 + 2 * q^32 - 12 * q^34 - 4 * q^37 - 14 * q^49 - 4 * q^50 - 18 * q^58 + 8 * q^62 + 2 * q^64 - 24 * q^65 - 16 * q^67 - 12 * q^68 - 4 * q^74 + 24 * q^83 + 6 * q^95 - 14 * q^97 - 14 * q^98

Introduction

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