LMFDB - Embedded newform 6525.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 $ x^8 - 2x^7 - 12x^6 + 23x^5 + 36x^4 - 62x^3 - 15x^2 + 14*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.57789$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57789 q^{2} +4.64553 q^{4} +4.69867 q^{7} -6.81989 q^{8} +O(q^{10})$	$q-2.57789 q^{2} +4.64553 q^{4} +4.69867 q^{7} -6.81989 q^{8} +3.11503 q^{11} -5.07945 q^{13} -12.1127 q^{14} +8.28989 q^{16} -1.40020 q^{17} -3.76514 q^{19} -8.03022 q^{22} -5.71483 q^{23} +13.0943 q^{26} +21.8278 q^{28} -1.00000 q^{29} -2.23218 q^{31} -7.73067 q^{32} +3.60957 q^{34} +5.79088 q^{37} +9.70614 q^{38} +10.6968 q^{41} -8.89527 q^{43} +14.4710 q^{44} +14.7322 q^{46} -3.62785 q^{47} +15.0775 q^{49} -23.5967 q^{52} +0.948260 q^{53} -32.0445 q^{56} +2.57789 q^{58} +8.53886 q^{59} -6.21467 q^{61} +5.75432 q^{62} +3.34905 q^{64} -13.9099 q^{67} -6.50468 q^{68} -5.88726 q^{71} +7.08398 q^{73} -14.9283 q^{74} -17.4911 q^{76} +14.6365 q^{77} +7.31672 q^{79} -27.5751 q^{82} -13.6579 q^{83} +22.9311 q^{86} -21.2442 q^{88} +7.98017 q^{89} -23.8667 q^{91} -26.5484 q^{92} +9.35221 q^{94} -13.8343 q^{97} -38.8683 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8	$ 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8} - 6 q^{11} - 6 q^{13} - 9 q^{14} + 32 q^{16} - 12 q^{17} - 3 q^{22} - 14 q^{23} - 18 q^{26} - 14 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 13 q^{34} - 4 q^{37} - 26 q^{38} - 2 q^{41} - 2 q^{43} + 15 q^{44} + 24 q^{46} - 12 q^{47} + 38 q^{49} - 49 q^{52} - 4 q^{53} - 58 q^{56} + 2 q^{58} - 18 q^{59} + 12 q^{61} + 4 q^{62} + 21 q^{64} - 26 q^{67} - 81 q^{68} - 24 q^{71} + 14 q^{73} + 22 q^{74} + 26 q^{77} + 10 q^{79} - 48 q^{82} - 40 q^{83} - 8 q^{86} + 10 q^{88} - 34 q^{89} + 26 q^{91} + 18 q^{92} - 43 q^{94} - 30 q^{97} - 29 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8 - 6 * q^11 - 6 * q^13 - 9 * q^14 + 32 * q^16 - 12 * q^17 - 3 * q^22 - 14 * q^23 - 18 * q^26 - 14 * q^28 - 8 * q^29 + 8 * q^31 + 2 * q^32 - 13 * q^34 - 4 * q^37 - 26 * q^38 - 2 * q^41 - 2 * q^43 + 15 * q^44 + 24 * q^46 - 12 * q^47 + 38 * q^49 - 49 * q^52 - 4 * q^53 - 58 * q^56 + 2 * q^58 - 18 * q^59 + 12 * q^61 + 4 * q^62 + 21 * q^64 - 26 * q^67 - 81 * q^68 - 24 * q^71 + 14 * q^73 + 22 * q^74 + 26 * q^77 + 10 * q^79 - 48 * q^82 - 40 * q^83 - 8 * q^86 + 10 * q^88 - 34 * q^89 + 26 * q^91 + 18 * q^92 - 43 * q^94 - 30 * q^97 - 29 * 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 $

$2$	−2.57789	−1.82285	−0.911423	−	0.411471i	$-0.865015\pi$
$2$	−2.57789	−1.82285	−0.911423	+	0.411471i	$0.865015\pi$
$3$	0	0
$3$	0	0
$4$	4.64553	2.32277
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.69867	1.77593	0.887966	−	0.459909i	$-0.152118\pi$
$7$	4.69867	1.77593	0.887966	+	0.459909i	$0.152118\pi$
$8$	−6.81989	−2.41120
$9$	0	0
$10$	0	0
$11$	3.11503	0.939218	0.469609	−	0.882875i	$-0.344395\pi$
$11$	3.11503	0.939218	0.469609	+	0.882875i	$0.344395\pi$
$12$	0	0
$13$	−5.07945	−1.40879	−0.704393	−	0.709810i	$-0.748781\pi$
$13$	−5.07945	−1.40879	−0.704393	+	0.709810i	$0.748781\pi$
$14$	−12.1127	−3.23725
$15$	0	0
$16$	8.28989	2.07247
$17$	−1.40020	−0.339599	−0.169799	−	0.985479i	$-0.554312\pi$
$17$	−1.40020	−0.339599	−0.169799	+	0.985479i	$0.554312\pi$
$18$	0	0
$19$	−3.76514	−0.863783	−0.431892	−	0.901925i	$-0.642154\pi$
$19$	−3.76514	−0.863783	−0.431892	+	0.901925i	$0.642154\pi$
$20$	0	0
$21$	0	0
$22$	−8.03022	−1.71205
$23$	−5.71483	−1.19162	−0.595812	−	0.803124i	$-0.703170\pi$
$23$	−5.71483	−1.19162	−0.595812	+	0.803124i	$0.703170\pi$
$24$	0	0
$25$	0	0
$26$	13.0943	2.56800
$27$	0	0
$28$	21.8278	4.12507
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.23218	−0.400911	−0.200456	−	0.979703i	$-0.564242\pi$
$31$	−2.23218	−0.400911	−0.200456	+	0.979703i	$0.564242\pi$
$32$	−7.73067	−1.36660
$33$	0	0
$34$	3.60957	0.619036
$35$	0	0
$36$	0	0
$37$	5.79088	0.952015	0.476008	−	0.879441i	$-0.342083\pi$
$37$	5.79088	0.952015	0.476008	+	0.879441i	$0.342083\pi$
$38$	9.70614	1.57454
$39$	0	0
$40$	0	0
$41$	10.6968	1.67055	0.835276	−	0.549830i	$-0.185308\pi$
$41$	10.6968	1.67055	0.835276	+	0.549830i	$0.185308\pi$
$42$	0	0
$43$	−8.89527	−1.35652	−0.678258	−	0.734824i	$-0.737265\pi$
$43$	−8.89527	−1.35652	−0.678258	+	0.734824i	$0.737265\pi$
$44$	14.4710	2.18158
$45$	0	0
$46$	14.7322	2.17215
$47$	−3.62785	−0.529176	−0.264588	−	0.964362i	$-0.585236\pi$
$47$	−3.62785	−0.529176	−0.264588	+	0.964362i	$0.585236\pi$
$48$	0	0
$49$	15.0775	2.15393
$50$	0	0
$51$	0	0
$52$	−23.5967	−3.27228
$53$	0.948260	0.130254	0.0651268	−	0.997877i	$-0.479255\pi$
$53$	0.948260	0.130254	0.0651268	+	0.997877i	$0.479255\pi$
$54$	0	0
$55$	0	0
$56$	−32.0445	−4.28212
$57$	0	0
$58$	2.57789	0.338494
$59$	8.53886	1.11166	0.555832	−	0.831294i	$-0.312400\pi$
$59$	8.53886	1.11166	0.555832	+	0.831294i	$0.312400\pi$
$60$	0	0
$61$	−6.21467	−0.795707	−0.397853	−	0.917449i	$-0.630245\pi$
$61$	−6.21467	−0.795707	−0.397853	+	0.917449i	$0.630245\pi$
$62$	5.75432	0.730799
$63$	0	0
$64$	3.34905	0.418631
$65$	0	0
$66$	0	0
$67$	−13.9099	−1.69937	−0.849685	−	0.527291i	$-0.823208\pi$
$67$	−13.9099	−1.69937	−0.849685	+	0.527291i	$0.823208\pi$
$68$	−6.50468	−0.788808
$69$	0	0
$70$	0	0
$71$	−5.88726	−0.698690	−0.349345	−	0.936994i	$-0.613596\pi$
$71$	−5.88726	−0.698690	−0.349345	+	0.936994i	$0.613596\pi$
$72$	0	0
$73$	7.08398	0.829117	0.414559	−	0.910023i	$-0.363936\pi$
$73$	7.08398	0.829117	0.414559	+	0.910023i	$0.363936\pi$
$74$	−14.9283	−1.73538
$75$	0	0
$76$	−17.4911	−2.00637
$77$	14.6365	1.66799
$78$	0	0
$79$	7.31672	0.823195	0.411598	−	0.911366i	$-0.364971\pi$
$79$	7.31672	0.823195	0.411598	+	0.911366i	$0.364971\pi$
$80$	0	0
$81$	0	0
$82$	−27.5751	−3.04516
$83$	−13.6579	−1.49915	−0.749577	−	0.661917i	$-0.769743\pi$
$83$	−13.6579	−1.49915	−0.749577	+	0.661917i	$0.769743\pi$
$84$	0	0
$85$	0	0
$86$	22.9311	2.47272
$87$	0	0
$88$	−21.2442	−2.26464
$89$	7.98017	0.845897	0.422948	−	0.906154i	$-0.360995\pi$
$89$	7.98017	0.845897	0.422948	+	0.906154i	$0.360995\pi$
$90$	0	0
$91$	−23.8667	−2.50191
$92$	−26.5484	−2.76786
$93$	0	0
$94$	9.35221	0.964607
$95$	0	0
$96$	0	0
$97$	−13.8343	−1.40466	−0.702329	−	0.711853i	$-0.747856\pi$
$97$	−13.8343	−1.40466	−0.702329	+	0.711853i	$0.747856\pi$
$98$	−38.8683	−3.92629
$99$	0	0
$100$	0	0
$101$	−12.7461	−1.26829	−0.634144	−	0.773215i	$-0.718647\pi$
$101$	−12.7461	−1.26829	−0.634144	+	0.773215i	$0.718647\pi$
$102$	0	0
$103$	8.20406	0.808370	0.404185	−	0.914677i	$-0.367555\pi$
$103$	8.20406	0.808370	0.404185	+	0.914677i	$0.367555\pi$
$104$	34.6413	3.39686
$105$	0	0
$106$	−2.44451	−0.237432
$107$	−3.28873	−0.317933	−0.158967	−	0.987284i	$-0.550816\pi$
$107$	−3.28873	−0.317933	−0.158967	+	0.987284i	$0.550816\pi$
$108$	0	0
$109$	−0.246732	−0.0236326	−0.0118163	−	0.999930i	$-0.503761\pi$
$109$	−0.246732	−0.0236326	−0.0118163	+	0.999930i	$0.503761\pi$
$110$	0	0
$111$	0	0
$112$	38.9515	3.68057
$113$	−14.4273	−1.35721	−0.678603	−	0.734505i	$-0.737414\pi$
$113$	−14.4273	−1.35721	−0.678603	+	0.734505i	$0.737414\pi$
$114$	0	0
$115$	0	0
$116$	−4.64553	−0.431327
$117$	0	0
$118$	−22.0123	−2.02639
$119$	−6.57909	−0.603104
$120$	0	0
$121$	−1.29657	−0.117870
$122$	16.0207	1.45045
$123$	0	0
$124$	−10.3697	−0.931223
$125$	0	0
$126$	0	0
$127$	21.1707	1.87860	0.939299	−	0.343101i	$-0.111477\pi$
$127$	21.1707	1.87860	0.939299	+	0.343101i	$0.111477\pi$
$128$	6.82785	0.603503
$129$	0	0
$130$	0	0
$131$	8.18994	0.715559	0.357779	−	0.933806i	$-0.383534\pi$
$131$	8.18994	0.715559	0.357779	+	0.933806i	$0.383534\pi$
$132$	0	0
$133$	−17.6912	−1.53402
$134$	35.8583	3.09769
$135$	0	0
$136$	9.54923	0.818840
$137$	17.4054	1.48704	0.743520	−	0.668714i	$-0.233155\pi$
$137$	17.4054	1.48704	0.743520	+	0.668714i	$0.233155\pi$
$138$	0	0
$139$	−6.81900	−0.578380	−0.289190	−	0.957272i	$-0.593386\pi$
$139$	−6.81900	−0.578380	−0.289190	+	0.957272i	$0.593386\pi$
$140$	0	0
$141$	0	0
$142$	15.1767	1.27360
$143$	−15.8226	−1.32316
$144$	0	0
$145$	0	0
$146$	−18.2617	−1.51135
$147$	0	0
$148$	26.9017	2.21131
$149$	−23.7335	−1.94432	−0.972162	−	0.234311i	$-0.924717\pi$
$149$	−23.7335	−1.94432	−0.972162	+	0.234311i	$0.924717\pi$
$150$	0	0
$151$	−17.3095	−1.40863	−0.704315	−	0.709888i	$-0.748746\pi$
$151$	−17.3095	−1.40863	−0.704315	+	0.709888i	$0.748746\pi$
$152$	25.6779	2.08275
$153$	0	0
$154$	−37.7314	−3.04048
$155$	0	0
$156$	0	0
$157$	−10.0537	−0.802375	−0.401188	−	0.915996i	$-0.631402\pi$
$157$	−10.0537	−0.802375	−0.401188	+	0.915996i	$0.631402\pi$
$158$	−18.8617	−1.50056
$159$	0	0
$160$	0	0
$161$	−26.8521	−2.11624
$162$	0	0
$163$	−12.2602	−0.960291	−0.480145	−	0.877189i	$-0.659416\pi$
$163$	−12.2602	−0.960291	−0.480145	+	0.877189i	$0.659416\pi$
$164$	49.6921	3.88030
$165$	0	0
$166$	35.2087	2.73273
$167$	−13.5990	−1.05232	−0.526160	−	0.850386i	$-0.676369\pi$
$167$	−13.5990	−1.05232	−0.526160	+	0.850386i	$0.676369\pi$
$168$	0	0
$169$	12.8008	0.984677
$170$	0	0
$171$	0	0
$172$	−41.3233	−3.15087
$173$	8.35226	0.635011	0.317505	−	0.948256i	$-0.397155\pi$
$173$	8.35226	0.635011	0.317505	+	0.948256i	$0.397155\pi$
$174$	0	0
$175$	0	0
$176$	25.8233	1.94650
$177$	0	0
$178$	−20.5720	−1.54194
$179$	−26.5865	−1.98717	−0.993583	−	0.113109i	$-0.963919\pi$
$179$	−26.5865	−1.98717	−0.993583	+	0.113109i	$0.963919\pi$
$180$	0	0
$181$	3.72547	0.276912	0.138456	−	0.990369i	$-0.455786\pi$
$181$	3.72547	0.276912	0.138456	+	0.990369i	$0.455786\pi$
$182$	61.5257	4.56059
$183$	0	0
$184$	38.9745	2.87324
$185$	0	0
$186$	0	0
$187$	−4.36167	−0.318957
$188$	−16.8533	−1.22915
$189$	0	0
$190$	0	0
$191$	−6.76433	−0.489450	−0.244725	−	0.969593i	$-0.578698\pi$
$191$	−6.76433	−0.489450	−0.244725	+	0.969593i	$0.578698\pi$
$192$	0	0
$193$	−0.140715	−0.0101289	−0.00506444	−	0.999987i	$-0.501612\pi$
$193$	−0.140715	−0.0101289	−0.00506444	+	0.999987i	$0.501612\pi$
$194$	35.6633	2.56047
$195$	0	0
$196$	70.0431	5.00308
$197$	−1.48273	−0.105640	−0.0528200	−	0.998604i	$-0.516821\pi$
$197$	−1.48273	−0.105640	−0.0528200	+	0.998604i	$0.516821\pi$
$198$	0	0
$199$	25.8626	1.83335	0.916677	−	0.399628i	$-0.130861\pi$
$199$	25.8626	1.83335	0.916677	+	0.399628i	$0.130861\pi$
$200$	0	0
$201$	0	0
$202$	32.8582	2.31189
$203$	−4.69867	−0.329782
$204$	0	0
$205$	0	0
$206$	−21.1492	−1.47353
$207$	0	0
$208$	−42.1081	−2.91967
$209$	−11.7285	−0.811281
$210$	0	0
$211$	23.3428	1.60698	0.803491	−	0.595317i	$-0.202974\pi$
$211$	23.3428	1.60698	0.803491	+	0.595317i	$0.202974\pi$
$212$	4.40517	0.302548
$213$	0	0
$214$	8.47799	0.579543
$215$	0	0
$216$	0	0
$217$	−10.4883	−0.711991
$218$	0.636049	0.0430787
$219$	0	0
$220$	0	0
$221$	7.11225	0.478422
$222$	0	0
$223$	−22.4256	−1.50173	−0.750865	−	0.660455i	$-0.770363\pi$
$223$	−22.4256	−1.50173	−0.750865	+	0.660455i	$0.770363\pi$
$224$	−36.3239	−2.42699
$225$	0	0
$226$	37.1920	2.47398
$227$	−4.07327	−0.270353	−0.135176	−	0.990822i	$-0.543160\pi$
$227$	−4.07327	−0.270353	−0.135176	+	0.990822i	$0.543160\pi$
$228$	0	0
$229$	−15.7170	−1.03861	−0.519303	−	0.854590i	$-0.673808\pi$
$229$	−15.7170	−1.03861	−0.519303	+	0.854590i	$0.673808\pi$
$230$	0	0
$231$	0	0
$232$	6.81989	0.447748
$233$	16.4568	1.07812	0.539059	−	0.842268i	$-0.318780\pi$
$233$	16.4568	1.07812	0.539059	+	0.842268i	$0.318780\pi$
$234$	0	0
$235$	0	0
$236$	39.6675	2.58214
$237$	0	0
$238$	16.9602	1.09937
$239$	−3.18702	−0.206151	−0.103075	−	0.994674i	$-0.532868\pi$
$239$	−3.18702	−0.206151	−0.103075	+	0.994674i	$0.532868\pi$
$240$	0	0
$241$	11.1482	0.718118	0.359059	−	0.933315i	$-0.383098\pi$
$241$	11.1482	0.718118	0.359059	+	0.933315i	$0.383098\pi$
$242$	3.34243	0.214859
$243$	0	0
$244$	−28.8704	−1.84824
$245$	0	0
$246$	0	0
$247$	19.1249	1.21689
$248$	15.2232	0.966676
$249$	0	0
$250$	0	0
$251$	3.23388	0.204121	0.102060	−	0.994778i	$-0.467457\pi$
$251$	3.23388	0.204121	0.102060	+	0.994778i	$0.467457\pi$
$252$	0	0
$253$	−17.8019	−1.11919
$254$	−54.5758	−3.42439
$255$	0	0
$256$	−24.2996	−1.51872
$257$	3.96761	0.247493	0.123746	−	0.992314i	$-0.460509\pi$
$257$	3.96761	0.247493	0.123746	+	0.992314i	$0.460509\pi$
$258$	0	0
$259$	27.2095	1.69071
$260$	0	0
$261$	0	0
$262$	−21.1128	−1.30435
$263$	16.6865	1.02893	0.514467	−	0.857510i	$-0.327990\pi$
$263$	16.6865	1.02893	0.514467	+	0.857510i	$0.327990\pi$
$264$	0	0
$265$	0	0
$266$	45.6060	2.79628
$267$	0	0
$268$	−64.6191	−3.94724
$269$	−7.94329	−0.484311	−0.242155	−	0.970237i	$-0.577854\pi$
$269$	−7.94329	−0.484311	−0.242155	+	0.970237i	$0.577854\pi$
$270$	0	0
$271$	−17.3087	−1.05143	−0.525714	−	0.850662i	$-0.676202\pi$
$271$	−17.3087	−1.05143	−0.525714	+	0.850662i	$0.676202\pi$
$272$	−11.6075	−0.703810
$273$	0	0
$274$	−44.8691	−2.71064
$275$	0	0
$276$	0	0
$277$	−10.2791	−0.617612	−0.308806	−	0.951125i	$-0.599929\pi$
$277$	−10.2791	−0.617612	−0.308806	+	0.951125i	$0.599929\pi$
$278$	17.5786	1.05430
$279$	0	0
$280$	0	0
$281$	−4.57869	−0.273142	−0.136571	−	0.990630i	$-0.543608\pi$
$281$	−4.57869	−0.273142	−0.136571	+	0.990630i	$0.543608\pi$
$282$	0	0
$283$	−19.2431	−1.14388	−0.571941	−	0.820294i	$-0.693810\pi$
$283$	−19.2431	−1.14388	−0.571941	+	0.820294i	$0.693810\pi$
$284$	−27.3495	−1.62289
$285$	0	0
$286$	40.7891	2.41191
$287$	50.2606	2.96679
$288$	0	0
$289$	−15.0394	−0.884673
$290$	0	0
$291$	0	0
$292$	32.9089	1.92584
$293$	14.6045	0.853203	0.426602	−	0.904440i	$-0.359711\pi$
$293$	14.6045	0.853203	0.426602	+	0.904440i	$0.359711\pi$
$294$	0	0
$295$	0	0
$296$	−39.4932	−2.29550
$297$	0	0
$298$	61.1824	3.54420
$299$	29.0282	1.67874
$300$	0	0
$301$	−41.7960	−2.40908
$302$	44.6221	2.56771
$303$	0	0
$304$	−31.2127	−1.79017
$305$	0	0
$306$	0	0
$307$	−12.3324	−0.703850	−0.351925	−	0.936028i	$-0.614473\pi$
$307$	−12.3324	−0.703850	−0.351925	+	0.936028i	$0.614473\pi$
$308$	67.9944	3.87434
$309$	0	0
$310$	0	0
$311$	−12.3084	−0.697943	−0.348971	−	0.937133i	$-0.613469\pi$
$311$	−12.3084	−0.697943	−0.348971	+	0.937133i	$0.613469\pi$
$312$	0	0
$313$	−16.7306	−0.945672	−0.472836	−	0.881150i	$-0.656770\pi$
$313$	−16.7306	−0.945672	−0.472836	+	0.881150i	$0.656770\pi$
$314$	25.9174	1.46261
$315$	0	0
$316$	33.9901	1.91209
$317$	−10.9429	−0.614617	−0.307308	−	0.951610i	$-0.599428\pi$
$317$	−10.9429	−0.614617	−0.307308	+	0.951610i	$0.599428\pi$
$318$	0	0
$319$	−3.11503	−0.174408
$320$	0	0
$321$	0	0
$322$	69.2219	3.85759
$323$	5.27196	0.293340
$324$	0	0
$325$	0	0
$326$	31.6054	1.75046
$327$	0	0
$328$	−72.9508	−4.02803
$329$	−17.0461	−0.939781
$330$	0	0
$331$	−19.1431	−1.05220	−0.526099	−	0.850423i	$-0.676346\pi$
$331$	−19.1431	−1.05220	−0.526099	+	0.850423i	$0.676346\pi$
$332$	−63.4484	−3.48218
$333$	0	0
$334$	35.0567	1.91822
$335$	0	0
$336$	0	0
$337$	17.0294	0.927650	0.463825	−	0.885927i	$-0.346477\pi$
$337$	17.0294	0.927650	0.463825	+	0.885927i	$0.346477\pi$
$338$	−32.9991	−1.79491
$339$	0	0
$340$	0	0
$341$	−6.95331	−0.376543
$342$	0	0
$343$	37.9537	2.04931
$344$	60.6648	3.27083
$345$	0	0
$346$	−21.5312	−1.15753
$347$	−17.2564	−0.926373	−0.463186	−	0.886261i	$-0.653294\pi$
$347$	−17.2564	−0.926373	−0.463186	+	0.886261i	$0.653294\pi$
$348$	0	0
$349$	13.3969	0.717118	0.358559	−	0.933507i	$-0.383268\pi$
$349$	13.3969	0.717118	0.358559	+	0.933507i	$0.383268\pi$
$350$	0	0
$351$	0	0
$352$	−24.0813	−1.28354
$353$	2.04594	0.108894	0.0544472	−	0.998517i	$-0.482660\pi$
$353$	2.04594	0.108894	0.0544472	+	0.998517i	$0.482660\pi$
$354$	0	0
$355$	0	0
$356$	37.0721	1.96482
$357$	0	0
$358$	68.5370	3.62229
$359$	25.2336	1.33178	0.665889	−	0.746051i	$-0.268053\pi$
$359$	25.2336	1.33178	0.665889	+	0.746051i	$0.268053\pi$
$360$	0	0
$361$	−4.82368	−0.253878
$362$	−9.60387	−0.504768
$363$	0	0
$364$	−110.873	−5.81134
$365$	0	0
$366$	0	0
$367$	9.13285	0.476731	0.238366	−	0.971176i	$-0.423388\pi$
$367$	9.13285	0.476731	0.238366	+	0.971176i	$0.423388\pi$
$368$	−47.3753	−2.46961
$369$	0	0
$370$	0	0
$371$	4.45557	0.231321
$372$	0	0
$373$	−5.41148	−0.280196	−0.140098	−	0.990138i	$-0.544742\pi$
$373$	−5.41148	−0.280196	−0.140098	+	0.990138i	$0.544742\pi$
$374$	11.2439	0.581410
$375$	0	0
$376$	24.7416	1.27595
$377$	5.07945	0.261605
$378$	0	0
$379$	19.8284	1.01852	0.509258	−	0.860614i	$-0.329920\pi$
$379$	19.8284	1.01852	0.509258	+	0.860614i	$0.329920\pi$
$380$	0	0
$381$	0	0
$382$	17.4377	0.892191
$383$	14.8919	0.760941	0.380470	−	0.924793i	$-0.375762\pi$
$383$	14.8919	0.760941	0.380470	+	0.924793i	$0.375762\pi$
$384$	0	0
$385$	0	0
$386$	0.362748	0.0184634
$387$	0	0
$388$	−64.2675	−3.26269
$389$	15.7680	0.799470	0.399735	−	0.916631i	$-0.369102\pi$
$389$	15.7680	0.799470	0.399735	+	0.916631i	$0.369102\pi$
$390$	0	0
$391$	8.00192	0.404674
$392$	−102.827	−5.19356
$393$	0	0
$394$	3.82232	0.192566
$395$	0	0
$396$	0	0
$397$	26.5318	1.33159	0.665796	−	0.746134i	$-0.268092\pi$
$397$	26.5318	1.33159	0.665796	+	0.746134i	$0.268092\pi$
$398$	−66.6711	−3.34192
$399$	0	0
$400$	0	0
$401$	−9.71205	−0.484997	−0.242498	−	0.970152i	$-0.577967\pi$
$401$	−9.71205	−0.484997	−0.242498	+	0.970152i	$0.577967\pi$
$402$	0	0
$403$	11.3382	0.564798
$404$	−59.2126	−2.94593
$405$	0	0
$406$	12.1127	0.601142
$407$	18.0388	0.894150
$408$	0	0
$409$	−17.4468	−0.862691	−0.431346	−	0.902187i	$-0.641961\pi$
$409$	−17.4468	−0.862691	−0.431346	+	0.902187i	$0.641961\pi$
$410$	0	0
$411$	0	0
$412$	38.1122	1.87765
$413$	40.1213	1.97424
$414$	0	0
$415$	0	0
$416$	39.2675	1.92525
$417$	0	0
$418$	30.2349	1.47884
$419$	−26.0588	−1.27306	−0.636529	−	0.771253i	$-0.719630\pi$
$419$	−26.0588	−1.27306	−0.636529	+	0.771253i	$0.719630\pi$
$420$	0	0
$421$	2.66465	0.129867	0.0649337	−	0.997890i	$-0.479316\pi$
$421$	2.66465	0.129867	0.0649337	+	0.997890i	$0.479316\pi$
$422$	−60.1751	−2.92928
$423$	0	0
$424$	−6.46704	−0.314067
$425$	0	0
$426$	0	0
$427$	−29.2007	−1.41312
$428$	−15.2779	−0.738485
$429$	0	0
$430$	0	0
$431$	0.150532	0.00725086	0.00362543	−	0.999993i	$-0.498846\pi$
$431$	0.150532	0.00725086	0.00362543	+	0.999993i	$0.498846\pi$
$432$	0	0
$433$	33.2461	1.59770	0.798852	−	0.601527i	$-0.205441\pi$
$433$	33.2461	1.59770	0.798852	+	0.601527i	$0.205441\pi$
$434$	27.0377	1.29785
$435$	0	0
$436$	−1.14620	−0.0548931
$437$	21.5172	1.02931
$438$	0	0
$439$	18.1320	0.865391	0.432695	−	0.901540i	$-0.357563\pi$
$439$	18.1320	0.865391	0.432695	+	0.901540i	$0.357563\pi$
$440$	0	0
$441$	0	0
$442$	−18.3346	−0.872089
$443$	34.8312	1.65488	0.827440	−	0.561554i	$-0.189796\pi$
$443$	34.8312	1.65488	0.827440	+	0.561554i	$0.189796\pi$
$444$	0	0
$445$	0	0
$446$	57.8108	2.73742
$447$	0	0
$448$	15.7361	0.743460
$449$	−36.1516	−1.70610	−0.853049	−	0.521830i	$-0.825249\pi$
$449$	−36.1516	−1.70610	−0.853049	+	0.521830i	$0.825249\pi$
$450$	0	0
$451$	33.3207	1.56901
$452$	−67.0225	−3.15247
$453$	0	0
$454$	10.5005	0.492811
$455$	0	0
$456$	0	0
$457$	8.93775	0.418090	0.209045	−	0.977906i	$-0.432964\pi$
$457$	8.93775	0.418090	0.209045	+	0.977906i	$0.432964\pi$
$458$	40.5166	1.89322
$459$	0	0
$460$	0	0
$461$	−34.6733	−1.61490	−0.807449	−	0.589937i	$-0.799152\pi$
$461$	−34.6733	−1.61490	−0.807449	+	0.589937i	$0.799152\pi$
$462$	0	0
$463$	−30.9412	−1.43796	−0.718981	−	0.695030i	$-0.755391\pi$
$463$	−30.9412	−1.43796	−0.718981	+	0.695030i	$0.755391\pi$
$464$	−8.28989	−0.384849
$465$	0	0
$466$	−42.4237	−1.96524
$467$	5.40920	0.250308	0.125154	−	0.992137i	$-0.460057\pi$
$467$	5.40920	0.250308	0.125154	+	0.992137i	$0.460057\pi$
$468$	0	0
$469$	−65.3583	−3.01797
$470$	0	0
$471$	0	0
$472$	−58.2341	−2.68044
$473$	−27.7091	−1.27406
$474$	0	0
$475$	0	0
$476$	−30.5634	−1.40087
$477$	0	0
$478$	8.21578	0.375781
$479$	−13.4270	−0.613494	−0.306747	−	0.951791i	$-0.599241\pi$
$479$	−13.4270	−0.613494	−0.306747	+	0.951791i	$0.599241\pi$
$480$	0	0
$481$	−29.4145	−1.34119
$482$	−28.7389	−1.30902
$483$	0	0
$484$	−6.02327	−0.273785
$485$	0	0
$486$	0	0
$487$	24.8746	1.12718	0.563589	−	0.826055i	$-0.309420\pi$
$487$	24.8746	1.12718	0.563589	+	0.826055i	$0.309420\pi$
$488$	42.3834	1.91861
$489$	0	0
$490$	0	0
$491$	−20.6717	−0.932902	−0.466451	−	0.884547i	$-0.654468\pi$
$491$	−20.6717	−0.932902	−0.466451	+	0.884547i	$0.654468\pi$
$492$	0	0
$493$	1.40020	0.0630619
$494$	−49.3018	−2.21819
$495$	0	0
$496$	−18.5045	−0.830878
$497$	−27.6623	−1.24083
$498$	0	0
$499$	21.2182	0.949856	0.474928	−	0.880025i	$-0.342474\pi$
$499$	21.2182	0.949856	0.474928	+	0.880025i	$0.342474\pi$
$500$	0	0
$501$	0	0
$502$	−8.33660	−0.372081
$503$	−35.9602	−1.60339	−0.801693	−	0.597736i	$-0.796067\pi$
$503$	−35.9602	−1.60339	−0.801693	+	0.597736i	$0.796067\pi$
$504$	0	0
$505$	0	0
$506$	45.8913	2.04012
$507$	0	0
$508$	98.3492	4.36354
$509$	−21.3861	−0.947924	−0.473962	−	0.880545i	$-0.657177\pi$
$509$	−21.3861	−0.947924	−0.473962	+	0.880545i	$0.657177\pi$
$510$	0	0
$511$	33.2853	1.47246
$512$	48.9860	2.16489
$513$	0	0
$514$	−10.2281	−0.451141
$515$	0	0
$516$	0	0
$517$	−11.3009	−0.497012
$518$	−70.1431	−3.08191
$519$	0	0
$520$	0	0
$521$	6.56775	0.287738	0.143869	−	0.989597i	$-0.454046\pi$
$521$	6.56775	0.287738	0.143869	+	0.989597i	$0.454046\pi$
$522$	0	0
$523$	21.1811	0.926184	0.463092	−	0.886310i	$-0.346740\pi$
$523$	21.1811	0.926184	0.463092	+	0.886310i	$0.346740\pi$
$524$	38.0466	1.66207
$525$	0	0
$526$	−43.0160	−1.87559
$527$	3.12550	0.136149
$528$	0	0
$529$	9.65929	0.419969
$530$	0	0
$531$	0	0
$532$	−82.1850	−3.56317
$533$	−54.3336	−2.35345
$534$	0	0
$535$	0	0
$536$	94.8643	4.09752
$537$	0	0
$538$	20.4769	0.882824
$539$	46.9670	2.02301
$540$	0	0
$541$	−9.98542	−0.429307	−0.214653	−	0.976690i	$-0.568862\pi$
$541$	−9.98542	−0.429307	−0.214653	+	0.976690i	$0.568862\pi$
$542$	44.6199	1.91659
$543$	0	0
$544$	10.8245	0.464097
$545$	0	0
$546$	0	0
$547$	1.01143	0.0432457	0.0216228	−	0.999766i	$-0.493117\pi$
$547$	1.01143	0.0432457	0.0216228	+	0.999766i	$0.493117\pi$
$548$	80.8571	3.45404
$549$	0	0
$550$	0	0
$551$	3.76514	0.160401
$552$	0	0
$553$	34.3789	1.46194
$554$	26.4985	1.12581
$555$	0	0
$556$	−31.6779	−1.34344
$557$	30.4618	1.29071	0.645353	−	0.763885i	$-0.276710\pi$
$557$	30.4618	1.29071	0.645353	+	0.763885i	$0.276710\pi$
$558$	0	0
$559$	45.1831	1.91104
$560$	0	0
$561$	0	0
$562$	11.8034	0.497895
$563$	2.49954	0.105343	0.0526716	−	0.998612i	$-0.483226\pi$
$563$	2.49954	0.105343	0.0526716	+	0.998612i	$0.483226\pi$
$564$	0	0
$565$	0	0
$566$	49.6066	2.08512
$567$	0	0
$568$	40.1505	1.68468
$569$	0.611166	0.0256214	0.0128107	−	0.999918i	$-0.495922\pi$
$569$	0.611166	0.0256214	0.0128107	+	0.999918i	$0.495922\pi$
$570$	0	0
$571$	−0.527376	−0.0220700	−0.0110350	−	0.999939i	$-0.503513\pi$
$571$	−0.527376	−0.0220700	−0.0110350	+	0.999939i	$0.503513\pi$
$572$	−73.5046	−3.07338
$573$	0	0
$574$	−129.566	−5.40800
$575$	0	0
$576$	0	0
$577$	−24.3938	−1.01553	−0.507764	−	0.861496i	$-0.669528\pi$
$577$	−24.3938	−1.01553	−0.507764	+	0.861496i	$0.669528\pi$
$578$	38.7700	1.61262
$579$	0	0
$580$	0	0
$581$	−64.1742	−2.66239
$582$	0	0
$583$	2.95386	0.122336
$584$	−48.3120	−1.99917
$585$	0	0
$586$	−37.6488	−1.55526
$587$	28.4529	1.17438	0.587188	−	0.809450i	$-0.300235\pi$
$587$	28.4529	1.17438	0.587188	+	0.809450i	$0.300235\pi$
$588$	0	0
$589$	8.40448	0.346301
$590$	0	0
$591$	0	0
$592$	48.0058	1.97303
$593$	−2.96753	−0.121862	−0.0609310	−	0.998142i	$-0.519407\pi$
$593$	−2.96753	−0.121862	−0.0609310	+	0.998142i	$0.519407\pi$
$594$	0	0
$595$	0	0
$596$	−110.255	−4.51621
$597$	0	0
$598$	−74.8316	−3.06009
$599$	−21.9546	−0.897040	−0.448520	−	0.893773i	$-0.648049\pi$
$599$	−21.9546	−0.897040	−0.448520	+	0.893773i	$0.648049\pi$
$600$	0	0
$601$	−21.4446	−0.874745	−0.437372	−	0.899280i	$-0.644091\pi$
$601$	−21.4446	−0.874745	−0.437372	+	0.899280i	$0.644091\pi$
$602$	107.746	4.39138
$603$	0	0
$604$	−80.4120	−3.27192
$605$	0	0
$606$	0	0
$607$	48.0114	1.94872	0.974361	−	0.224988i	$-0.0722344\pi$
$607$	48.0114	1.94872	0.974361	+	0.224988i	$0.0722344\pi$
$608$	29.1071	1.18045
$609$	0	0
$610$	0	0
$611$	18.4275	0.745496
$612$	0	0
$613$	37.9093	1.53114	0.765571	−	0.643351i	$-0.222456\pi$
$613$	37.9093	1.53114	0.765571	+	0.643351i	$0.222456\pi$
$614$	31.7917	1.28301
$615$	0	0
$616$	−99.8195	−4.02184
$617$	−8.29772	−0.334054	−0.167027	−	0.985952i	$-0.553417\pi$
$617$	−8.29772	−0.334054	−0.167027	+	0.985952i	$0.553417\pi$
$618$	0	0
$619$	−15.6135	−0.627559	−0.313779	−	0.949496i	$-0.601595\pi$
$619$	−15.6135	−0.627559	−0.313779	+	0.949496i	$0.601595\pi$
$620$	0	0
$621$	0	0
$622$	31.7296	1.27224
$623$	37.4962	1.50225
$624$	0	0
$625$	0	0
$626$	43.1298	1.72381
$627$	0	0
$628$	−46.7049	−1.86373
$629$	−8.10841	−0.323303
$630$	0	0
$631$	10.7848	0.429336	0.214668	−	0.976687i	$-0.431133\pi$
$631$	10.7848	0.429336	0.214668	+	0.976687i	$0.431133\pi$
$632$	−49.8993	−1.98489
$633$	0	0
$634$	28.2097	1.12035
$635$	0	0
$636$	0	0
$637$	−76.5856	−3.03443
$638$	8.03022	0.317919
$639$	0	0
$640$	0	0
$641$	−16.1226	−0.636805	−0.318403	−	0.947956i	$-0.603146\pi$
$641$	−16.1226	−0.636805	−0.318403	+	0.947956i	$0.603146\pi$
$642$	0	0
$643$	−25.0313	−0.987137	−0.493569	−	0.869707i	$-0.664308\pi$
$643$	−25.0313	−0.987137	−0.493569	+	0.869707i	$0.664308\pi$
$644$	−124.742	−4.91554
$645$	0	0
$646$	−13.5906	−0.534713
$647$	−34.9798	−1.37520	−0.687599	−	0.726091i	$-0.741335\pi$
$647$	−34.9798	−1.37520	−0.687599	+	0.726091i	$0.741335\pi$
$648$	0	0
$649$	26.5988	1.04409
$650$	0	0
$651$	0	0
$652$	−56.9550	−2.23053
$653$	−22.0250	−0.861904	−0.430952	−	0.902375i	$-0.641822\pi$
$653$	−22.0250	−0.861904	−0.430952	+	0.902375i	$0.641822\pi$
$654$	0	0
$655$	0	0
$656$	88.6750	3.46218
$657$	0	0
$658$	43.9430	1.71308
$659$	32.9279	1.28269	0.641345	−	0.767253i	$-0.278377\pi$
$659$	32.9279	1.28269	0.641345	+	0.767253i	$0.278377\pi$
$660$	0	0
$661$	9.78801	0.380710	0.190355	−	0.981715i	$-0.439036\pi$
$661$	9.78801	0.380710	0.190355	+	0.981715i	$0.439036\pi$
$662$	49.3488	1.91799
$663$	0	0
$664$	93.1457	3.61475
$665$	0	0
$666$	0	0
$667$	5.71483	0.221279
$668$	−63.1745	−2.44429
$669$	0	0
$670$	0	0
$671$	−19.3589	−0.747342
$672$	0	0
$673$	−25.3634	−0.977689	−0.488844	−	0.872371i	$-0.662581\pi$
$673$	−25.3634	−0.977689	−0.488844	+	0.872371i	$0.662581\pi$
$674$	−43.8999	−1.69096
$675$	0	0
$676$	59.4665	2.28717
$677$	19.7896	0.760578	0.380289	−	0.924868i	$-0.375825\pi$
$677$	19.7896	0.760578	0.380289	+	0.924868i	$0.375825\pi$
$678$	0	0
$679$	−65.0027	−2.49458
$680$	0	0
$681$	0	0
$682$	17.9249	0.686380
$683$	−44.9656	−1.72056	−0.860281	−	0.509820i	$-0.829712\pi$
$683$	−44.9656	−1.72056	−0.860281	+	0.509820i	$0.829712\pi$
$684$	0	0
$685$	0	0
$686$	−97.8406	−3.73557
$687$	0	0
$688$	−73.7409	−2.81134
$689$	−4.81664	−0.183499
$690$	0	0
$691$	−30.7758	−1.17077	−0.585384	−	0.810756i	$-0.699056\pi$
$691$	−30.7758	−1.17077	−0.585384	+	0.810756i	$0.699056\pi$
$692$	38.8007	1.47498
$693$	0	0
$694$	44.4852	1.68863
$695$	0	0
$696$	0	0
$697$	−14.9776	−0.567318
$698$	−34.5357	−1.30720
$699$	0	0
$700$	0	0
$701$	23.2459	0.877984	0.438992	−	0.898491i	$-0.355336\pi$
$701$	23.2459	0.877984	0.438992	+	0.898491i	$0.355336\pi$
$702$	0	0
$703$	−21.8035	−0.822335
$704$	10.4324	0.393186
$705$	0	0
$706$	−5.27422	−0.198498
$707$	−59.8899	−2.25239
$708$	0	0
$709$	7.86251	0.295283	0.147641	−	0.989041i	$-0.452832\pi$
$709$	7.86251	0.295283	0.147641	+	0.989041i	$0.452832\pi$
$710$	0	0
$711$	0	0
$712$	−54.4239	−2.03962
$713$	12.7565	0.477736
$714$	0	0
$715$	0	0
$716$	−123.508	−4.61572
$717$	0	0
$718$	−65.0495	−2.42762
$719$	−35.0267	−1.30628	−0.653138	−	0.757239i	$-0.726548\pi$
$719$	−35.0267	−1.30628	−0.653138	+	0.757239i	$0.726548\pi$
$720$	0	0
$721$	38.5482	1.43561
$722$	12.4349	0.462781
$723$	0	0
$724$	17.3068	0.643202
$725$	0	0
$726$	0	0
$727$	22.4518	0.832691	0.416346	−	0.909206i	$-0.363311\pi$
$727$	22.4518	0.832691	0.416346	+	0.909206i	$0.363311\pi$
$728$	162.768	6.03259
$729$	0	0
$730$	0	0
$731$	12.4552	0.460671
$732$	0	0
$733$	−15.0028	−0.554140	−0.277070	−	0.960850i	$-0.589363\pi$
$733$	−15.0028	−0.554140	−0.277070	+	0.960850i	$0.589363\pi$
$734$	−23.5435	−0.869007
$735$	0	0
$736$	44.1795	1.62848
$737$	−43.3299	−1.59608
$738$	0	0
$739$	−33.4801	−1.23159	−0.615793	−	0.787908i	$-0.711164\pi$
$739$	−33.4801	−1.23159	−0.615793	+	0.787908i	$0.711164\pi$
$740$	0	0
$741$	0	0
$742$	−11.4860	−0.421663
$743$	−6.75081	−0.247663	−0.123832	−	0.992303i	$-0.539518\pi$
$743$	−6.75081	−0.247663	−0.123832	+	0.992303i	$0.539518\pi$
$744$	0	0
$745$	0	0
$746$	13.9502	0.510754
$747$	0	0
$748$	−20.2623	−0.740863
$749$	−15.4527	−0.564628
$750$	0	0
$751$	28.0482	1.02349	0.511747	−	0.859136i	$-0.328999\pi$
$751$	28.0482	1.02349	0.511747	+	0.859136i	$0.328999\pi$
$752$	−30.0745	−1.09670
$753$	0	0
$754$	−13.0943	−0.476865
$755$	0	0
$756$	0	0
$757$	33.1698	1.20558	0.602789	−	0.797901i	$-0.294056\pi$
$757$	33.1698	1.20558	0.602789	+	0.797901i	$0.294056\pi$
$758$	−51.1155	−1.85660
$759$	0	0
$760$	0	0
$761$	12.0982	0.438560	0.219280	−	0.975662i	$-0.429629\pi$
$761$	12.0982	0.438560	0.219280	+	0.975662i	$0.429629\pi$
$762$	0	0
$763$	−1.15931	−0.0419700
$764$	−31.4239	−1.13688
$765$	0	0
$766$	−38.3897	−1.38708
$767$	−43.3727	−1.56610
$768$	0	0
$769$	−42.5811	−1.53551	−0.767756	−	0.640742i	$-0.778627\pi$
$769$	−42.5811	−1.53551	−0.767756	+	0.640742i	$0.778627\pi$
$770$	0	0
$771$	0	0
$772$	−0.653695	−0.0235270
$773$	−6.87342	−0.247220	−0.123610	−	0.992331i	$-0.539447\pi$
$773$	−6.87342	−0.247220	−0.123610	+	0.992331i	$0.539447\pi$
$774$	0	0
$775$	0	0
$776$	94.3483	3.38691
$777$	0	0
$778$	−40.6482	−1.45731
$779$	−40.2748	−1.44300
$780$	0	0
$781$	−18.3390	−0.656222
$782$	−20.6281	−0.737659
$783$	0	0
$784$	124.991	4.46397
$785$	0	0
$786$	0	0
$787$	−4.51935	−0.161097	−0.0805487	−	0.996751i	$-0.525667\pi$
$787$	−4.51935	−0.161097	−0.0805487	+	0.996751i	$0.525667\pi$
$788$	−6.88806	−0.245377
$789$	0	0
$790$	0	0
$791$	−67.7892	−2.41031
$792$	0	0
$793$	31.5671	1.12098
$794$	−68.3961	−2.42729
$795$	0	0
$796$	120.146	4.25845
$797$	−13.9670	−0.494736	−0.247368	−	0.968922i	$-0.579566\pi$
$797$	−13.9670	−0.494736	−0.247368	+	0.968922i	$0.579566\pi$
$798$	0	0
$799$	5.07972	0.179708
$800$	0	0
$801$	0	0
$802$	25.0366	0.884074
$803$	22.0668	0.778722
$804$	0	0
$805$	0	0
$806$	−29.2288	−1.02954
$807$	0	0
$808$	86.9273	3.05809
$809$	−8.18148	−0.287646	−0.143823	−	0.989603i	$-0.545940\pi$
$809$	−8.18148	−0.287646	−0.143823	+	0.989603i	$0.545940\pi$
$810$	0	0
$811$	−44.4011	−1.55913	−0.779566	−	0.626320i	$-0.784560\pi$
$811$	−44.4011	−1.55913	−0.779566	+	0.626320i	$0.784560\pi$
$812$	−21.8278	−0.766007
$813$	0	0
$814$	−46.5021	−1.62990
$815$	0	0
$816$	0	0
$817$	33.4920	1.17174
$818$	44.9761	1.57255
$819$	0	0
$820$	0	0
$821$	−8.73119	−0.304721	−0.152360	−	0.988325i	$-0.548687\pi$
$821$	−8.73119	−0.304721	−0.152360	+	0.988325i	$0.548687\pi$
$822$	0	0
$823$	−39.1844	−1.36588	−0.682942	−	0.730473i	$-0.739300\pi$
$823$	−39.1844	−1.36588	−0.682942	+	0.730473i	$0.739300\pi$
$824$	−55.9508	−1.94914
$825$	0	0
$826$	−103.428	−3.59873
$827$	−6.33428	−0.220264	−0.110132	−	0.993917i	$-0.535127\pi$
$827$	−6.33428	−0.220264	−0.110132	+	0.993917i	$0.535127\pi$
$828$	0	0
$829$	1.36706	0.0474798	0.0237399	−	0.999718i	$-0.492443\pi$
$829$	1.36706	0.0474798	0.0237399	+	0.999718i	$0.492443\pi$
$830$	0	0
$831$	0	0
$832$	−17.0113	−0.589762
$833$	−21.1116	−0.731473
$834$	0	0
$835$	0	0
$836$	−54.4853	−1.88441
$837$	0	0
$838$	67.1769	2.32059
$839$	55.7902	1.92609	0.963046	−	0.269339i	$-0.0868051\pi$
$839$	55.7902	1.92609	0.963046	+	0.269339i	$0.0868051\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−6.86919	−0.236728
$843$	0	0
$844$	108.439	3.73264
$845$	0	0
$846$	0	0
$847$	−6.09218	−0.209330
$848$	7.86098	0.269947
$849$	0	0
$850$	0	0
$851$	−33.0939	−1.13444
$852$	0	0
$853$	6.42812	0.220095	0.110047	−	0.993926i	$-0.464900\pi$
$853$	6.42812	0.220095	0.110047	+	0.993926i	$0.464900\pi$
$854$	75.2762	2.57590
$855$	0	0
$856$	22.4288	0.766600
$857$	−35.0274	−1.19651	−0.598257	−	0.801304i	$-0.704140\pi$
$857$	−35.0274	−1.19651	−0.598257	+	0.801304i	$0.704140\pi$
$858$	0	0
$859$	18.4412	0.629207	0.314603	−	0.949223i	$-0.398128\pi$
$859$	18.4412	0.629207	0.314603	+	0.949223i	$0.398128\pi$
$860$	0	0
$861$	0	0
$862$	−0.388055	−0.0132172
$863$	−7.23911	−0.246422	−0.123211	−	0.992380i	$-0.539319\pi$
$863$	−7.23911	−0.246422	−0.123211	+	0.992380i	$0.539319\pi$
$864$	0	0
$865$	0	0
$866$	−85.7048	−2.91237
$867$	0	0
$868$	−48.7237	−1.65379
$869$	22.7918	0.773160
$870$	0	0
$871$	70.6549	2.39405
$872$	1.68269	0.0569830
$873$	0	0
$874$	−55.4689	−1.87626
$875$	0	0
$876$	0	0
$877$	24.6324	0.831777	0.415889	−	0.909416i	$-0.363471\pi$
$877$	24.6324	0.831777	0.415889	+	0.909416i	$0.363471\pi$
$878$	−46.7422	−1.57747
$879$	0	0
$880$	0	0
$881$	−26.2145	−0.883190	−0.441595	−	0.897214i	$-0.645587\pi$
$881$	−26.2145	−0.883190	−0.441595	+	0.897214i	$0.645587\pi$
$882$	0	0
$883$	10.7656	0.362290	0.181145	−	0.983456i	$-0.442020\pi$
$883$	10.7656	0.362290	0.181145	+	0.983456i	$0.442020\pi$
$884$	33.0402	1.11126
$885$	0	0
$886$	−89.7911	−3.01659
$887$	6.27796	0.210793	0.105397	−	0.994430i	$-0.466389\pi$
$887$	6.27796	0.210793	0.105397	+	0.994430i	$0.466389\pi$
$888$	0	0
$889$	99.4743	3.33626
$890$	0	0
$891$	0	0
$892$	−104.179	−3.48817
$893$	13.6594	0.457094
$894$	0	0
$895$	0	0
$896$	32.0818	1.07178
$897$	0	0
$898$	93.1949	3.10995
$899$	2.23218	0.0744474
$900$	0	0
$901$	−1.32776	−0.0442340
$902$	−85.8973	−2.86007
$903$	0	0
$904$	98.3927	3.27249
$905$	0	0
$906$	0	0
$907$	−21.7954	−0.723704	−0.361852	−	0.932235i	$-0.617855\pi$
$907$	−21.7954	−0.723704	−0.361852	+	0.932235i	$0.617855\pi$
$908$	−18.9225	−0.627966
$909$	0	0
$910$	0	0
$911$	5.56422	0.184351	0.0921753	−	0.995743i	$-0.470618\pi$
$911$	5.56422	0.184351	0.0921753	+	0.995743i	$0.470618\pi$
$912$	0	0
$913$	−42.5449	−1.40803
$914$	−23.0406	−0.762114
$915$	0	0
$916$	−73.0136	−2.41244
$917$	38.4819	1.27078
$918$	0	0
$919$	−7.76705	−0.256211	−0.128106	−	0.991761i	$-0.540890\pi$
$919$	−7.76705	−0.256211	−0.128106	+	0.991761i	$0.540890\pi$
$920$	0	0
$921$	0	0
$922$	89.3841	2.94371
$923$	29.9041	0.984304
$924$	0	0
$925$	0	0
$926$	79.7632	2.62118
$927$	0	0
$928$	7.73067	0.253772
$929$	11.3588	0.372672	0.186336	−	0.982486i	$-0.440339\pi$
$929$	11.3588	0.372672	0.186336	+	0.982486i	$0.440339\pi$
$930$	0	0
$931$	−56.7691	−1.86053
$932$	76.4503	2.50421
$933$	0	0
$934$	−13.9443	−0.456273
$935$	0	0
$936$	0	0
$937$	−32.7813	−1.07092	−0.535460	−	0.844561i	$-0.679862\pi$
$937$	−32.7813	−1.07092	−0.535460	+	0.844561i	$0.679862\pi$
$938$	168.487	5.50128
$939$	0	0
$940$	0	0
$941$	13.3074	0.433808	0.216904	−	0.976193i	$-0.430404\pi$
$941$	13.3074	0.433808	0.216904	+	0.976193i	$0.430404\pi$
$942$	0	0
$943$	−61.1302	−1.99067
$944$	70.7862	2.30390
$945$	0	0
$946$	71.4310	2.32242
$947$	31.9463	1.03811	0.519057	−	0.854740i	$-0.326283\pi$
$947$	31.9463	1.03811	0.519057	+	0.854740i	$0.326283\pi$
$948$	0	0
$949$	−35.9827	−1.16805
$950$	0	0
$951$	0	0
$952$	44.8687	1.45420
$953$	−53.5304	−1.73402	−0.867009	−	0.498292i	$-0.833961\pi$
$953$	−53.5304	−1.73402	−0.867009	+	0.498292i	$0.833961\pi$
$954$	0	0
$955$	0	0
$956$	−14.8054	−0.478840
$957$	0	0
$958$	34.6133	1.11831
$959$	81.7821	2.64088
$960$	0	0
$961$	−26.0174	−0.839270
$962$	75.8274	2.44477
$963$	0	0
$964$	51.7893	1.66802
$965$	0	0
$966$	0	0
$967$	−16.9490	−0.545044	−0.272522	−	0.962150i	$-0.587858\pi$
$967$	−16.9490	−0.545044	−0.272522	+	0.962150i	$0.587858\pi$
$968$	8.84250	0.284209
$969$	0	0
$970$	0	0
$971$	28.5476	0.916135	0.458067	−	0.888917i	$-0.348542\pi$
$971$	28.5476	0.916135	0.458067	+	0.888917i	$0.348542\pi$
$972$	0	0
$973$	−32.0402	−1.02716
$974$	−64.1242	−2.05467
$975$	0	0
$976$	−51.5189	−1.64908
$977$	−30.8519	−0.987040	−0.493520	−	0.869734i	$-0.664290\pi$
$977$	−30.8519	−0.987040	−0.493520	+	0.869734i	$0.664290\pi$
$978$	0	0
$979$	24.8585	0.794481
$980$	0	0
$981$	0	0
$982$	53.2895	1.70054
$983$	25.9915	0.829000	0.414500	−	0.910049i	$-0.363956\pi$
$983$	25.9915	0.829000	0.414500	+	0.910049i	$0.363956\pi$
$984$	0	0
$985$	0	0
$986$	−3.60957	−0.114952
$987$	0	0
$988$	88.8451	2.82654
$989$	50.8350	1.61646
$990$	0	0
$991$	−3.17860	−0.100972	−0.0504858	−	0.998725i	$-0.516077\pi$
$991$	−3.17860	−0.100972	−0.0504858	+	0.998725i	$0.516077\pi$
$992$	17.2562	0.547886
$993$	0	0
$994$	71.3105	2.26183
$995$	0	0
$996$	0	0
$997$	−31.6331	−1.00183	−0.500916	−	0.865496i	$-0.667003\pi$
$997$	−31.6331	−1.00183	−0.500916	+	0.865496i	$0.667003\pi$
$998$	−54.6982	−1.73144
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.by.1.2		8
3.2	odd	2		2175.2.a.bd.1.7	yes	8
5.4	even	2		6525.2.a.bz.1.7		8
15.2	even	4		2175.2.c.p.349.14		16
15.8	even	4		2175.2.c.p.349.3		16
15.14	odd	2		2175.2.a.bc.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.2	✓	8	15.14	odd	2
2175.2.a.bd.1.7	yes	8	3.2	odd	2
2175.2.c.p.349.3		16	15.8	even	4
2175.2.c.p.349.14		16	15.2	even	4
6525.2.a.by.1.2		8	1.1	even	1	trivial
6525.2.a.bz.1.7		8	5.4	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57789 q^{2} +4.64553 q^{4} +4.69867 q^{7} -6.81989 q^{8} +O(q^{10})\)	\(q-2.57789 q^{2} +4.64553 q^{4} +4.69867 q^{7} -6.81989 q^{8} +3.11503 q^{11} -5.07945 q^{13} -12.1127 q^{14} +8.28989 q^{16} -1.40020 q^{17} -3.76514 q^{19} -8.03022 q^{22} -5.71483 q^{23} +13.0943 q^{26} +21.8278 q^{28} -1.00000 q^{29} -2.23218 q^{31} -7.73067 q^{32} +3.60957 q^{34} +5.79088 q^{37} +9.70614 q^{38} +10.6968 q^{41} -8.89527 q^{43} +14.4710 q^{44} +14.7322 q^{46} -3.62785 q^{47} +15.0775 q^{49} -23.5967 q^{52} +0.948260 q^{53} -32.0445 q^{56} +2.57789 q^{58} +8.53886 q^{59} -6.21467 q^{61} +5.75432 q^{62} +3.34905 q^{64} -13.9099 q^{67} -6.50468 q^{68} -5.88726 q^{71} +7.08398 q^{73} -14.9283 q^{74} -17.4911 q^{76} +14.6365 q^{77} +7.31672 q^{79} -27.5751 q^{82} -13.6579 q^{83} +22.9311 q^{86} -21.2442 q^{88} +7.98017 q^{89} -23.8667 q^{91} -26.5484 q^{92} +9.35221 q^{94} -13.8343 q^{97} -38.8683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8	\( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8} - 6 q^{11} - 6 q^{13} - 9 q^{14} + 32 q^{16} - 12 q^{17} - 3 q^{22} - 14 q^{23} - 18 q^{26} - 14 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 13 q^{34} - 4 q^{37} - 26 q^{38} - 2 q^{41} - 2 q^{43} + 15 q^{44} + 24 q^{46} - 12 q^{47} + 38 q^{49} - 49 q^{52} - 4 q^{53} - 58 q^{56} + 2 q^{58} - 18 q^{59} + 12 q^{61} + 4 q^{62} + 21 q^{64} - 26 q^{67} - 81 q^{68} - 24 q^{71} + 14 q^{73} + 22 q^{74} + 26 q^{77} + 10 q^{79} - 48 q^{82} - 40 q^{83} - 8 q^{86} + 10 q^{88} - 34 q^{89} + 26 q^{91} + 18 q^{92} - 43 q^{94} - 30 q^{97} - 29 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8 - 6 * q^11 - 6 * q^13 - 9 * q^14 + 32 * q^16 - 12 * q^17 - 3 * q^22 - 14 * q^23 - 18 * q^26 - 14 * q^28 - 8 * q^29 + 8 * q^31 + 2 * q^32 - 13 * q^34 - 4 * q^37 - 26 * q^38 - 2 * q^41 - 2 * q^43 + 15 * q^44 + 24 * q^46 - 12 * q^47 + 38 * q^49 - 49 * q^52 - 4 * q^53 - 58 * q^56 + 2 * q^58 - 18 * q^59 + 12 * q^61 + 4 * q^62 + 21 * q^64 - 26 * q^67 - 81 * q^68 - 24 * q^71 + 14 * q^73 + 22 * q^74 + 26 * q^77 + 10 * q^79 - 48 * q^82 - 40 * q^83 - 8 * q^86 + 10 * q^88 - 34 * q^89 + 26 * q^91 + 18 * q^92 - 43 * q^94 - 30 * q^97 - 29 * q^98

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