LMFDB - Embedded newform 2175.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.57789$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.57789 q^{2} -1.00000 q^{3} +4.64553 q^{4} +2.57789 q^{6} -4.69867 q^{7} -6.81989 q^{8} +1.00000 q^{9} -3.11503 q^{11} -4.64553 q^{12} +5.07945 q^{13} +12.1127 q^{14} +8.28989 q^{16} -1.40020 q^{17} -2.57789 q^{18} -3.76514 q^{19} +4.69867 q^{21} +8.03022 q^{22} -5.71483 q^{23} +6.81989 q^{24} -13.0943 q^{26} -1.00000 q^{27} -21.8278 q^{28} +1.00000 q^{29} -2.23218 q^{31} -7.73067 q^{32} +3.11503 q^{33} +3.60957 q^{34} +4.64553 q^{36} -5.79088 q^{37} +9.70614 q^{38} -5.07945 q^{39} -10.6968 q^{41} -12.1127 q^{42} +8.89527 q^{43} -14.4710 q^{44} +14.7322 q^{46} -3.62785 q^{47} -8.28989 q^{48} +15.0775 q^{49} +1.40020 q^{51} +23.5967 q^{52} +0.948260 q^{53} +2.57789 q^{54} +32.0445 q^{56} +3.76514 q^{57} -2.57789 q^{58} -8.53886 q^{59} -6.21467 q^{61} +5.75432 q^{62} -4.69867 q^{63} +3.34905 q^{64} -8.03022 q^{66} +13.9099 q^{67} -6.50468 q^{68} +5.71483 q^{69} +5.88726 q^{71} -6.81989 q^{72} -7.08398 q^{73} +14.9283 q^{74} -17.4911 q^{76} +14.6365 q^{77} +13.0943 q^{78} +7.31672 q^{79} +1.00000 q^{81} +27.5751 q^{82} -13.6579 q^{83} +21.8278 q^{84} -22.9311 q^{86} -1.00000 q^{87} +21.2442 q^{88} -7.98017 q^{89} -23.8667 q^{91} -26.5484 q^{92} +2.23218 q^{93} +9.35221 q^{94} +7.73067 q^{96} +13.8343 q^{97} -38.8683 q^{98} -3.11503 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 - 8 * q^3 + 12 * q^4 + 2 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 6 * q^11 - 12 * q^12 + 6 * q^13 + 9 * q^14 + 32 * q^16 - 12 * q^17 - 2 * q^18 - 2 * q^21 + 3 * q^22 - 14 * q^23 + 3 * q^24 + 18 * q^26 - 8 * q^27 + 14 * q^28 + 8 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^33 - 13 * q^34 + 12 * q^36 + 4 * q^37 - 26 * q^38 - 6 * q^39 + 2 * q^41 - 9 * q^42 + 2 * q^43 - 15 * q^44 + 24 * q^46 - 12 * q^47 - 32 * q^48 + 38 * q^49 + 12 * q^51 + 49 * q^52 - 4 * q^53 + 2 * q^54 + 58 * q^56 - 2 * q^58 + 18 * q^59 + 12 * q^61 + 4 * q^62 + 2 * q^63 + 21 * q^64 - 3 * q^66 + 26 * q^67 - 81 * q^68 + 14 * q^69 + 24 * q^71 - 3 * q^72 - 14 * q^73 - 22 * q^74 + 26 * q^77 - 18 * q^78 + 10 * q^79 + 8 * q^81 + 48 * q^82 - 40 * q^83 - 14 * q^84 + 8 * q^86 - 8 * q^87 - 10 * q^88 + 34 * q^89 + 26 * q^91 + 18 * q^92 - 8 * q^93 - 43 * q^94 - 2 * q^96 + 30 * q^97 - 29 * q^98 + 6 * q^99

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	$0.865015\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.64553	2.32277
$5$	0	0
$5$	0	0
$6$	2.57789	1.05242
$7$	−4.69867	−1.77593	−0.887966	−	0.459909i	$-0.847882\pi$
$7$	−4.69867	−1.77593	−0.887966	+	0.459909i	$0.847882\pi$
$8$	−6.81989	−2.41120
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.11503	−0.939218	−0.469609	−	0.882875i	$-0.655605\pi$
$11$	−3.11503	−0.939218	−0.469609	+	0.882875i	$0.655605\pi$
$12$	−4.64553	−1.34105
$13$	5.07945	1.40879	0.704393	−	0.709810i	$-0.251219\pi$
$13$	5.07945	1.40879	0.704393	+	0.709810i	$0.251219\pi$
$14$	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	$0.554312\pi$
$18$	−2.57789	−0.607615
$19$	−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
$21$	4.69867	1.02533
$22$	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	$0.703170\pi$
$24$	6.81989	1.39211
$25$	0	0
$26$	−13.0943	−2.56800
$27$	−1.00000	−0.192450
$28$	−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
$33$	3.11503	0.542258
$34$	3.60957	0.619036
$35$	0	0
$36$	4.64553	0.774255
$37$	−5.79088	−0.952015	−0.476008	−	0.879441i	$-0.657917\pi$
$37$	−5.79088	−0.952015	−0.476008	+	0.879441i	$0.657917\pi$
$38$	9.70614	1.57454
$39$	−5.07945	−0.813363
$40$	0	0
$41$	−10.6968	−1.67055	−0.835276	−	0.549830i	$-0.814692\pi$
$41$	−10.6968	−1.67055	−0.835276	+	0.549830i	$0.814692\pi$
$42$	−12.1127	−1.86903
$43$	8.89527	1.35652	0.678258	−	0.734824i	$-0.262735\pi$
$43$	8.89527	1.35652	0.678258	+	0.734824i	$0.262735\pi$
$44$	−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	$0.585236\pi$
$48$	−8.28989	−1.19654
$49$	15.0775	2.15393
$50$	0	0
$51$	1.40020	0.196067
$52$	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	$0.479255\pi$
$54$	2.57789	0.350807
$55$	0	0
$56$	32.0445	4.28212
$57$	3.76514	0.498706
$58$	−2.57789	−0.338494
$59$	−8.53886	−1.11166	−0.555832	−	0.831294i	$-0.687600\pi$
$59$	−8.53886	−1.11166	−0.555832	+	0.831294i	$0.687600\pi$
$60$	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
$63$	−4.69867	−0.591977
$64$	3.34905	0.418631
$65$	0	0
$66$	−8.03022	−0.988452
$67$	13.9099	1.69937	0.849685	−	0.527291i	$-0.176792\pi$
$67$	13.9099	1.69937	0.849685	+	0.527291i	$0.176792\pi$
$68$	−6.50468	−0.788808
$69$	5.71483	0.687985
$70$	0	0
$71$	5.88726	0.698690	0.349345	−	0.936994i	$-0.386404\pi$
$71$	5.88726	0.698690	0.349345	+	0.936994i	$0.386404\pi$
$72$	−6.81989	−0.803732
$73$	−7.08398	−0.829117	−0.414559	−	0.910023i	$-0.636064\pi$
$73$	−7.08398	−0.829117	−0.414559	+	0.910023i	$0.636064\pi$
$74$	14.9283	1.73538
$75$	0	0
$76$	−17.4911	−2.00637
$77$	14.6365	1.66799
$78$	13.0943	1.48263
$79$	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
$81$	1.00000	0.111111
$82$	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	$0.769743\pi$
$84$	21.8278	2.38161
$85$	0	0
$86$	−22.9311	−2.47272
$87$	−1.00000	−0.107211
$88$	21.2442	2.26464
$89$	−7.98017	−0.845897	−0.422948	−	0.906154i	$-0.639005\pi$
$89$	−7.98017	−0.845897	−0.422948	+	0.906154i	$0.639005\pi$
$90$	0	0
$91$	−23.8667	−2.50191
$92$	−26.5484	−2.76786
$93$	2.23218	0.231466
$94$	9.35221	0.964607
$95$	0	0
$96$	7.73067	0.789008
$97$	13.8343	1.40466	0.702329	−	0.711853i	$-0.252144\pi$
$97$	13.8343	1.40466	0.702329	+	0.711853i	$0.252144\pi$
$98$	−38.8683	−3.92629
$99$	−3.11503	−0.313073
$100$	0	0
$101$	12.7461	1.26829	0.634144	−	0.773215i	$-0.281353\pi$
$101$	12.7461	1.26829	0.634144	+	0.773215i	$0.281353\pi$
$102$	−3.60957	−0.357401
$103$	−8.20406	−0.808370	−0.404185	−	0.914677i	$-0.632445\pi$
$103$	−8.20406	−0.808370	−0.404185	+	0.914677i	$0.632445\pi$
$104$	−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	$0.550816\pi$
$108$	−4.64553	−0.447016
$109$	−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
$111$	5.79088	0.549646
$112$	−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	$0.737414\pi$
$114$	−9.70614	−0.909063
$115$	0	0
$116$	4.64553	0.431327
$117$	5.07945	0.469595
$118$	22.0123	2.02639
$119$	6.57909	0.603104
$120$	0	0
$121$	−1.29657	−0.117870
$122$	16.0207	1.45045
$123$	10.6968	0.964494
$124$	−10.3697	−0.931223
$125$	0	0
$126$	12.1127	1.07908
$127$	−21.1707	−1.87860	−0.939299	−	0.343101i	$-0.888523\pi$
$127$	−21.1707	−1.87860	−0.939299	+	0.343101i	$0.888523\pi$
$128$	6.82785	0.603503
$129$	−8.89527	−0.783185
$130$	0	0
$131$	−8.18994	−0.715559	−0.357779	−	0.933806i	$-0.616466\pi$
$131$	−8.18994	−0.715559	−0.357779	+	0.933806i	$0.616466\pi$
$132$	14.4710	1.25954
$133$	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	$0.233155\pi$
$138$	−14.7322	−1.25409
$139$	−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
$141$	3.62785	0.305520
$142$	−15.1767	−1.27360
$143$	−15.8226	−1.32316
$144$	8.28989	0.690825
$145$	0	0
$146$	18.2617	1.51135
$147$	−15.0775	−1.24357
$148$	−26.9017	−2.21131
$149$	23.7335	1.94432	0.972162	−	0.234311i	$-0.0752833\pi$
$149$	23.7335	1.94432	0.972162	+	0.234311i	$0.0752833\pi$
$150$	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
$153$	−1.40020	−0.113200
$154$	−37.7314	−3.04048
$155$	0	0
$156$	−23.5967	−1.88925
$157$	10.0537	0.802375	0.401188	−	0.915996i	$-0.368598\pi$
$157$	10.0537	0.802375	0.401188	+	0.915996i	$0.368598\pi$
$158$	−18.8617	−1.50056
$159$	−0.948260	−0.0752019
$160$	0	0
$161$	26.8521	2.11624
$162$	−2.57789	−0.202538
$163$	12.2602	0.960291	0.480145	−	0.877189i	$-0.340584\pi$
$163$	12.2602	0.960291	0.480145	+	0.877189i	$0.340584\pi$
$164$	−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	$0.676369\pi$
$168$	−32.0445	−2.47228
$169$	12.8008	0.984677
$170$	0	0
$171$	−3.76514	−0.287928
$172$	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	$0.397155\pi$
$174$	2.57789	0.195430
$175$	0	0
$176$	−25.8233	−1.94650
$177$	8.53886	0.641820
$178$	20.5720	1.54194
$179$	26.5865	1.98717	0.993583	−	0.113109i	$-0.0360810\pi$
$179$	26.5865	1.98717	0.993583	+	0.113109i	$0.0360810\pi$
$180$	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
$183$	6.21467	0.459401
$184$	38.9745	2.87324
$185$	0	0
$186$	−5.75432	−0.421927
$187$	4.36167	0.318957
$188$	−16.8533	−1.22915
$189$	4.69867	0.341778
$190$	0	0
$191$	6.76433	0.489450	0.244725	−	0.969593i	$-0.421302\pi$
$191$	6.76433	0.489450	0.244725	+	0.969593i	$0.421302\pi$
$192$	−3.34905	−0.241697
$193$	0.140715	0.0101289	0.00506444	−	0.999987i	$-0.498388\pi$
$193$	0.140715	0.0101289	0.00506444	+	0.999987i	$0.498388\pi$
$194$	−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	$0.516821\pi$
$198$	8.03022	0.570683
$199$	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
$201$	−13.9099	−0.981132
$202$	−32.8582	−2.31189
$203$	−4.69867	−0.329782
$204$	6.50468	0.455419
$205$	0	0
$206$	21.1492	1.47353
$207$	−5.71483	−0.397208
$208$	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
$213$	−5.88726	−0.403389
$214$	8.47799	0.579543
$215$	0	0
$216$	6.81989	0.464035
$217$	10.4883	0.711991
$218$	0.636049	0.0430787
$219$	7.08398	0.478691
$220$	0	0
$221$	−7.11225	−0.478422
$222$	−14.9283	−1.00192
$223$	22.4256	1.50173	0.750865	−	0.660455i	$-0.229637\pi$
$223$	22.4256	1.50173	0.750865	+	0.660455i	$0.229637\pi$
$224$	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	$0.543160\pi$
$228$	17.4911	1.15838
$229$	−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
$231$	−14.6365	−0.963012
$232$	−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	$0.318780\pi$
$234$	−13.0943	−0.856000
$235$	0	0
$236$	−39.6675	−2.58214
$237$	−7.31672	−0.475272
$238$	−16.9602	−1.09937
$239$	3.18702	0.206151	0.103075	−	0.994674i	$-0.467132\pi$
$239$	3.18702	0.206151	0.103075	+	0.994674i	$0.467132\pi$
$240$	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
$243$	−1.00000	−0.0641500
$244$	−28.8704	−1.84824
$245$	0	0
$246$	−27.5751	−1.75812
$247$	−19.1249	−1.21689
$248$	15.2232	0.966676
$249$	13.6579	0.865537
$250$	0	0
$251$	−3.23388	−0.204121	−0.102060	−	0.994778i	$-0.532543\pi$
$251$	−3.23388	−0.204121	−0.102060	+	0.994778i	$0.532543\pi$
$252$	−21.8278	−1.37502
$253$	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	$0.460509\pi$
$258$	22.9311	1.42763
$259$	27.2095	1.69071
$260$	0	0
$261$	1.00000	0.0618984
$262$	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	$0.327990\pi$
$264$	−21.2442	−1.30749
$265$	0	0
$266$	−45.6060	−2.79628
$267$	7.98017	0.488379
$268$	64.6191	3.94724
$269$	7.94329	0.484311	0.242155	−	0.970237i	$-0.422146\pi$
$269$	7.94329	0.484311	0.242155	+	0.970237i	$0.422146\pi$
$270$	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
$273$	23.8667	1.44448
$274$	−44.8691	−2.71064
$275$	0	0
$276$	26.5484	1.59803
$277$	10.2791	0.617612	0.308806	−	0.951125i	$-0.400071\pi$
$277$	10.2791	0.617612	0.308806	+	0.951125i	$0.400071\pi$
$278$	17.5786	1.05430
$279$	−2.23218	−0.133637
$280$	0	0
$281$	4.57869	0.273142	0.136571	−	0.990630i	$-0.456392\pi$
$281$	4.57869	0.273142	0.136571	+	0.990630i	$0.456392\pi$
$282$	−9.35221	−0.556916
$283$	19.2431	1.14388	0.571941	−	0.820294i	$-0.306190\pi$
$283$	19.2431	1.14388	0.571941	+	0.820294i	$0.306190\pi$
$284$	27.3495	1.62289
$285$	0	0
$286$	40.7891	2.41191
$287$	50.2606	2.96679
$288$	−7.73067	−0.455534
$289$	−15.0394	−0.884673
$290$	0	0
$291$	−13.8343	−0.810979
$292$	−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	$0.359711\pi$
$294$	38.8683	2.26684
$295$	0	0
$296$	39.4932	2.29550
$297$	3.11503	0.180753
$298$	−61.1824	−3.54420
$299$	−29.0282	−1.67874
$300$	0	0
$301$	−41.7960	−2.40908
$302$	44.6221	2.56771
$303$	−12.7461	−0.732246
$304$	−31.2127	−1.79017
$305$	0	0
$306$	3.60957	0.206345
$307$	12.3324	0.703850	0.351925	−	0.936028i	$-0.385527\pi$
$307$	12.3324	0.703850	0.351925	+	0.936028i	$0.385527\pi$
$308$	67.9944	3.87434
$309$	8.20406	0.466713
$310$	0	0
$311$	12.3084	0.697943	0.348971	−	0.937133i	$-0.386531\pi$
$311$	12.3084	0.697943	0.348971	+	0.937133i	$0.386531\pi$
$312$	34.6413	1.96118
$313$	16.7306	0.945672	0.472836	−	0.881150i	$-0.343230\pi$
$313$	16.7306	0.945672	0.472836	+	0.881150i	$0.343230\pi$
$314$	−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	$0.599428\pi$
$318$	2.44451	0.137081
$319$	−3.11503	−0.174408
$320$	0	0
$321$	3.28873	0.183559
$322$	−69.2219	−3.85759
$323$	5.27196	0.293340
$324$	4.64553	0.258085
$325$	0	0
$326$	−31.6054	−1.75046
$327$	0.246732	0.0136443
$328$	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
$333$	−5.79088	−0.317338
$334$	35.0567	1.91822
$335$	0	0
$336$	38.9515	2.12498
$337$	−17.0294	−0.927650	−0.463825	−	0.885927i	$-0.653523\pi$
$337$	−17.0294	−0.927650	−0.463825	+	0.885927i	$0.653523\pi$
$338$	−32.9991	−1.79491
$339$	14.4273	0.783583
$340$	0	0
$341$	6.95331	0.376543
$342$	9.70614	0.524848
$343$	−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	$0.653294\pi$
$348$	−4.64553	−0.249027
$349$	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
$351$	−5.07945	−0.271121
$352$	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	$0.482660\pi$
$354$	−22.0123	−1.16994
$355$	0	0
$356$	−37.0721	−1.96482
$357$	−6.57909	−0.348202
$358$	−68.5370	−3.62229
$359$	−25.2336	−1.33178	−0.665889	−	0.746051i	$-0.731947\pi$
$359$	−25.2336	−1.33178	−0.665889	+	0.746051i	$0.731947\pi$
$360$	0	0
$361$	−4.82368	−0.253878
$362$	−9.60387	−0.504768
$363$	1.29657	0.0680525
$364$	−110.873	−5.81134
$365$	0	0
$366$	−16.0207	−0.837418
$367$	−9.13285	−0.476731	−0.238366	−	0.971176i	$-0.576612\pi$
$367$	−9.13285	−0.476731	−0.238366	+	0.971176i	$0.576612\pi$
$368$	−47.3753	−2.46961
$369$	−10.6968	−0.556851
$370$	0	0
$371$	−4.45557	−0.231321
$372$	10.3697	0.537642
$373$	5.41148	0.280196	0.140098	−	0.990138i	$-0.455258\pi$
$373$	5.41148	0.280196	0.140098	+	0.990138i	$0.455258\pi$
$374$	−11.2439	−0.581410
$375$	0	0
$376$	24.7416	1.27595
$377$	5.07945	0.261605
$378$	−12.1127	−0.623009
$379$	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
$381$	21.1707	1.08461
$382$	−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	$0.375762\pi$
$384$	−6.82785	−0.348432
$385$	0	0
$386$	−0.362748	−0.0184634
$387$	8.89527	0.452172
$388$	64.2675	3.26269
$389$	−15.7680	−0.799470	−0.399735	−	0.916631i	$-0.630898\pi$
$389$	−15.7680	−0.799470	−0.399735	+	0.916631i	$0.630898\pi$
$390$	0	0
$391$	8.00192	0.404674
$392$	−102.827	−5.19356
$393$	8.18994	0.413128
$394$	3.82232	0.192566
$395$	0	0
$396$	−14.4710	−0.727194
$397$	−26.5318	−1.33159	−0.665796	−	0.746134i	$-0.731908\pi$
$397$	−26.5318	−1.33159	−0.665796	+	0.746134i	$0.731908\pi$
$398$	−66.6711	−3.34192
$399$	−17.6912	−0.885667
$400$	0	0
$401$	9.71205	0.484997	0.242498	−	0.970152i	$-0.422033\pi$
$401$	9.71205	0.484997	0.242498	+	0.970152i	$0.422033\pi$
$402$	35.8583	1.78845
$403$	−11.3382	−0.564798
$404$	59.2126	2.94593
$405$	0	0
$406$	12.1127	0.601142
$407$	18.0388	0.894150
$408$	−9.54923	−0.472757
$409$	−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
$411$	−17.4054	−0.858543
$412$	−38.1122	−1.87765
$413$	40.1213	1.97424
$414$	14.7322	0.724049
$415$	0	0
$416$	−39.2675	−1.92525
$417$	6.81900	0.333928
$418$	−30.2349	−1.47884
$419$	26.0588	1.27306	0.636529	−	0.771253i	$-0.280370\pi$
$419$	26.0588	1.27306	0.636529	+	0.771253i	$0.280370\pi$
$420$	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
$423$	−3.62785	−0.176392
$424$	−6.46704	−0.314067
$425$	0	0
$426$	15.1767	0.735315
$427$	29.2007	1.41312
$428$	−15.2779	−0.738485
$429$	15.8226	0.763925
$430$	0	0
$431$	−0.150532	−0.00725086	−0.00362543	−	0.999993i	$-0.501154\pi$
$431$	−0.150532	−0.00725086	−0.00362543	+	0.999993i	$0.501154\pi$
$432$	−8.28989	−0.398848
$433$	−33.2461	−1.59770	−0.798852	−	0.601527i	$-0.794559\pi$
$433$	−33.2461	−1.59770	−0.798852	+	0.601527i	$0.794559\pi$
$434$	−27.0377	−1.29785
$435$	0	0
$436$	−1.14620	−0.0548931
$437$	21.5172	1.02931
$438$	−18.2617	−0.872580
$439$	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
$441$	15.0775	0.717978
$442$	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	$0.189796\pi$
$444$	26.9017	1.27670
$445$	0	0
$446$	−57.8108	−2.73742
$447$	−23.7335	−1.12256
$448$	−15.7361	−0.743460
$449$	36.1516	1.70610	0.853049	−	0.521830i	$-0.174751\pi$
$449$	36.1516	1.70610	0.853049	+	0.521830i	$0.174751\pi$
$450$	0	0
$451$	33.3207	1.56901
$452$	−67.0225	−3.15247
$453$	17.3095	0.813273
$454$	10.5005	0.492811
$455$	0	0
$456$	−25.6779	−1.20248
$457$	−8.93775	−0.418090	−0.209045	−	0.977906i	$-0.567036\pi$
$457$	−8.93775	−0.418090	−0.209045	+	0.977906i	$0.567036\pi$
$458$	40.5166	1.89322
$459$	1.40020	0.0653558
$460$	0	0
$461$	34.6733	1.61490	0.807449	−	0.589937i	$-0.200848\pi$
$461$	34.6733	1.61490	0.807449	+	0.589937i	$0.200848\pi$
$462$	37.7314	1.75542
$463$	30.9412	1.43796	0.718981	−	0.695030i	$-0.244609\pi$
$463$	30.9412	1.43796	0.718981	+	0.695030i	$0.244609\pi$
$464$	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	$0.460057\pi$
$468$	23.5967	1.09076
$469$	−65.3583	−3.01797
$470$	0	0
$471$	−10.0537	−0.463251
$472$	58.2341	2.68044
$473$	−27.7091	−1.27406
$474$	18.8617	0.866348
$475$	0	0
$476$	30.5634	1.40087
$477$	0.948260	0.0434179
$478$	−8.21578	−0.375781
$479$	13.4270	0.613494	0.306747	−	0.951791i	$-0.400759\pi$
$479$	13.4270	0.613494	0.306747	+	0.951791i	$0.400759\pi$
$480$	0	0
$481$	−29.4145	−1.34119
$482$	−28.7389	−1.30902
$483$	−26.8521	−1.22181
$484$	−6.02327	−0.273785
$485$	0	0
$486$	2.57789	0.116936
$487$	−24.8746	−1.12718	−0.563589	−	0.826055i	$-0.690580\pi$
$487$	−24.8746	−1.12718	−0.563589	+	0.826055i	$0.690580\pi$
$488$	42.3834	1.91861
$489$	−12.2602	−0.554424
$490$	0	0
$491$	20.6717	0.932902	0.466451	−	0.884547i	$-0.345532\pi$
$491$	20.6717	0.932902	0.466451	+	0.884547i	$0.345532\pi$
$492$	49.6921	2.24029
$493$	−1.40020	−0.0630619
$494$	49.3018	2.21819
$495$	0	0
$496$	−18.5045	−0.830878
$497$	−27.6623	−1.24083
$498$	−35.2087	−1.57774
$499$	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
$501$	13.5990	0.607557
$502$	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	$0.796067\pi$
$504$	32.0445	1.42737
$505$	0	0
$506$	−45.8913	−2.04012
$507$	−12.8008	−0.568504
$508$	−98.3492	−4.36354
$509$	21.3861	0.947924	0.473962	−	0.880545i	$-0.342823\pi$
$509$	21.3861	0.947924	0.473962	+	0.880545i	$0.342823\pi$
$510$	0	0
$511$	33.2853	1.47246
$512$	48.9860	2.16489
$513$	3.76514	0.166235
$514$	−10.2281	−0.451141
$515$	0	0
$516$	−41.3233	−1.81915
$517$	11.3009	0.497012
$518$	−70.1431	−3.08191
$519$	−8.35226	−0.366624
$520$	0	0
$521$	−6.56775	−0.287738	−0.143869	−	0.989597i	$-0.545954\pi$
$521$	−6.56775	−0.287738	−0.143869	+	0.989597i	$0.545954\pi$
$522$	−2.57789	−0.112831
$523$	−21.1811	−0.926184	−0.463092	−	0.886310i	$-0.653260\pi$
$523$	−21.1811	−0.926184	−0.463092	+	0.886310i	$0.653260\pi$
$524$	−38.0466	−1.66207
$525$	0	0
$526$	−43.0160	−1.87559
$527$	3.12550	0.136149
$528$	25.8233	1.12381
$529$	9.65929	0.419969
$530$	0	0
$531$	−8.53886	−0.370555
$532$	82.1850	3.56317
$533$	−54.3336	−2.35345
$534$	−20.5720	−0.890239
$535$	0	0
$536$	−94.8643	−4.09752
$537$	−26.5865	−1.14729
$538$	−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
$543$	−3.72547	−0.159875
$544$	10.8245	0.464097
$545$	0	0
$546$	−61.5257	−2.63306
$547$	−1.01143	−0.0432457	−0.0216228	−	0.999766i	$-0.506883\pi$
$547$	−1.01143	−0.0432457	−0.0216228	+	0.999766i	$0.506883\pi$
$548$	80.8571	3.45404
$549$	−6.21467	−0.265236
$550$	0	0
$551$	−3.76514	−0.160401
$552$	−38.9745	−1.65887
$553$	−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	$0.276710\pi$
$558$	5.75432	0.243600
$559$	45.1831	1.91104
$560$	0	0
$561$	−4.36167	−0.184150
$562$	−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	$0.483226\pi$
$564$	16.8533	0.709651
$565$	0	0
$566$	−49.6066	−2.08512
$567$	−4.69867	−0.197326
$568$	−40.1505	−1.68468
$569$	−0.611166	−0.0256214	−0.0128107	−	0.999918i	$-0.504078\pi$
$569$	−0.611166	−0.0256214	−0.0128107	+	0.999918i	$0.504078\pi$
$570$	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
$573$	−6.76433	−0.282584
$574$	−129.566	−5.40800
$575$	0	0
$576$	3.34905	0.139544
$577$	24.3938	1.01553	0.507764	−	0.861496i	$-0.330472\pi$
$577$	24.3938	1.01553	0.507764	+	0.861496i	$0.330472\pi$
$578$	38.7700	1.61262
$579$	−0.140715	−0.00584791
$580$	0	0
$581$	64.1742	2.66239
$582$	35.6633	1.47829
$583$	−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	$0.300235\pi$
$588$	−70.0431	−2.88853
$589$	8.40448	0.346301
$590$	0	0
$591$	1.48273	0.0609913
$592$	−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	$0.519407\pi$
$594$	−8.03022	−0.329484
$595$	0	0
$596$	110.255	4.51621
$597$	−25.8626	−1.05849
$598$	74.8316	3.06009
$599$	21.9546	0.897040	0.448520	−	0.893773i	$-0.351951\pi$
$599$	21.9546	0.897040	0.448520	+	0.893773i	$0.351951\pi$
$600$	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
$603$	13.9099	0.566457
$604$	−80.4120	−3.27192
$605$	0	0
$606$	32.8582	1.33477
$607$	−48.0114	−1.94872	−0.974361	−	0.224988i	$-0.927766\pi$
$607$	−48.0114	−1.94872	−0.974361	+	0.224988i	$0.927766\pi$
$608$	29.1071	1.18045
$609$	4.69867	0.190400
$610$	0	0
$611$	−18.4275	−0.745496
$612$	−6.50468	−0.262936
$613$	−37.9093	−1.53114	−0.765571	−	0.643351i	$-0.777544\pi$
$613$	−37.9093	−1.53114	−0.765571	+	0.643351i	$0.777544\pi$
$614$	−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	$0.553417\pi$
$618$	−21.1492	−0.850745
$619$	−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
$621$	5.71483	0.229328
$622$	−31.7296	−1.27224
$623$	37.4962	1.50225
$624$	−42.1081	−1.68567
$625$	0	0
$626$	−43.1298	−1.72381
$627$	−11.7285	−0.468393
$628$	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
$633$	−23.3428	−0.927791
$634$	28.2097	1.12035
$635$	0	0
$636$	−4.40517	−0.174676
$637$	76.5856	3.03443
$638$	8.03022	0.317919
$639$	5.88726	0.232897
$640$	0	0
$641$	16.1226	0.636805	0.318403	−	0.947956i	$-0.396854\pi$
$641$	16.1226	0.636805	0.318403	+	0.947956i	$0.396854\pi$
$642$	−8.47799	−0.334599
$643$	25.0313	0.987137	0.493569	−	0.869707i	$-0.335692\pi$
$643$	25.0313	0.987137	0.493569	+	0.869707i	$0.335692\pi$
$644$	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	$0.741335\pi$
$648$	−6.81989	−0.267911
$649$	26.5988	1.04409
$650$	0	0
$651$	−10.4883	−0.411068
$652$	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	$0.641822\pi$
$654$	−0.636049	−0.0248715
$655$	0	0
$656$	−88.6750	−3.46218
$657$	−7.08398	−0.276372
$658$	−43.9430	−1.71308
$659$	−32.9279	−1.28269	−0.641345	−	0.767253i	$-0.721623\pi$
$659$	−32.9279	−1.28269	−0.641345	+	0.767253i	$0.721623\pi$
$660$	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
$663$	7.11225	0.276217
$664$	93.1457	3.61475
$665$	0	0
$666$	14.9283	0.578459
$667$	−5.71483	−0.221279
$668$	−63.1745	−2.44429
$669$	−22.4256	−0.867025
$670$	0	0
$671$	19.3589	0.747342
$672$	−36.3239	−1.40122
$673$	25.3634	0.977689	0.488844	−	0.872371i	$-0.337419\pi$
$673$	25.3634	0.977689	0.488844	+	0.872371i	$0.337419\pi$
$674$	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	$0.375825\pi$
$678$	−37.1920	−1.42835
$679$	−65.0027	−2.49458
$680$	0	0
$681$	4.07327	0.156088
$682$	−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	$0.829712\pi$
$684$	−17.4911	−0.668789
$685$	0	0
$686$	97.8406	3.73557
$687$	15.7170	0.599640
$688$	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
$693$	14.6365	0.555995
$694$	44.4852	1.68863
$695$	0	0
$696$	6.81989	0.258507
$697$	14.9776	0.567318
$698$	−34.5357	−1.30720
$699$	−16.4568	−0.622452
$700$	0	0
$701$	−23.2459	−0.877984	−0.438992	−	0.898491i	$-0.644664\pi$
$701$	−23.2459	−0.877984	−0.438992	+	0.898491i	$0.644664\pi$
$702$	13.0943	0.494212
$703$	21.8035	0.822335
$704$	−10.4324	−0.393186
$705$	0	0
$706$	−5.27422	−0.198498
$707$	−59.8899	−2.25239
$708$	39.6675	1.49080
$709$	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
$711$	7.31672	0.274398
$712$	54.4239	2.03962
$713$	12.7565	0.477736
$714$	16.9602	0.634719
$715$	0	0
$716$	123.508	4.61572
$717$	−3.18702	−0.119021
$718$	65.0495	2.42762
$719$	35.0267	1.30628	0.653138	−	0.757239i	$-0.273452\pi$
$719$	35.0267	1.30628	0.653138	+	0.757239i	$0.273452\pi$
$720$	0	0
$721$	38.5482	1.43561
$722$	12.4349	0.462781
$723$	−11.1482	−0.414606
$724$	17.3068	0.643202
$725$	0	0
$726$	−3.34243	−0.124049
$727$	−22.4518	−0.832691	−0.416346	−	0.909206i	$-0.636689\pi$
$727$	−22.4518	−0.832691	−0.416346	+	0.909206i	$0.636689\pi$
$728$	162.768	6.03259
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.4552	−0.460671
$732$	28.8704	1.06708
$733$	15.0028	0.554140	0.277070	−	0.960850i	$-0.410637\pi$
$733$	15.0028	0.554140	0.277070	+	0.960850i	$0.410637\pi$
$734$	23.5435	0.869007
$735$	0	0
$736$	44.1795	1.62848
$737$	−43.3299	−1.59608
$738$	27.5751	1.01505
$739$	−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
$741$	19.1249	0.702569
$742$	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	$0.539518\pi$
$744$	−15.2232	−0.558111
$745$	0	0
$746$	−13.9502	−0.510754
$747$	−13.6579	−0.499718
$748$	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
$753$	3.23388	0.117849
$754$	−13.0943	−0.476865
$755$	0	0
$756$	21.8278	0.793871
$757$	−33.1698	−1.20558	−0.602789	−	0.797901i	$-0.705944\pi$
$757$	−33.1698	−1.20558	−0.602789	+	0.797901i	$0.705944\pi$
$758$	−51.1155	−1.85660
$759$	−17.8019	−0.646167
$760$	0	0
$761$	−12.0982	−0.438560	−0.219280	−	0.975662i	$-0.570371\pi$
$761$	−12.0982	−0.438560	−0.219280	+	0.975662i	$0.570371\pi$
$762$	−54.5758	−1.97707
$763$	1.15931	0.0419700
$764$	31.4239	1.13688
$765$	0	0
$766$	−38.3897	−1.38708
$767$	−43.3727	−1.56610
$768$	24.2996	0.876835
$769$	−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
$771$	−3.96761	−0.142890
$772$	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	$0.539447\pi$
$774$	−22.9311	−0.824240
$775$	0	0
$776$	−94.3483	−3.38691
$777$	−27.2095	−0.976134
$778$	40.6482	1.45731
$779$	40.2748	1.44300
$780$	0	0
$781$	−18.3390	−0.656222
$782$	−20.6281	−0.737659
$783$	−1.00000	−0.0357371
$784$	124.991	4.46397
$785$	0	0
$786$	−21.1128	−0.753068
$787$	4.51935	0.161097	0.0805487	−	0.996751i	$-0.474333\pi$
$787$	4.51935	0.161097	0.0805487	+	0.996751i	$0.474333\pi$
$788$	−6.88806	−0.245377
$789$	−16.6865	−0.594056
$790$	0	0
$791$	67.7892	2.41031
$792$	21.2442	0.754879
$793$	−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	$0.579566\pi$
$798$	45.6060	1.61443
$799$	5.07972	0.179708
$800$	0	0
$801$	−7.98017	−0.281966
$802$	−25.0366	−0.884074
$803$	22.0668	0.778722
$804$	−64.6191	−2.27894
$805$	0	0
$806$	29.2288	1.02954
$807$	−7.94329	−0.279617
$808$	−86.9273	−3.05809
$809$	8.18148	0.287646	0.143823	−	0.989603i	$-0.454060\pi$
$809$	8.18148	0.287646	0.143823	+	0.989603i	$0.454060\pi$
$810$	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
$813$	17.3087	0.607042
$814$	−46.5021	−1.62990
$815$	0	0
$816$	11.6075	0.406345
$817$	−33.4920	−1.17174
$818$	44.9761	1.57255
$819$	−23.8667	−0.833969
$820$	0	0
$821$	8.73119	0.304721	0.152360	−	0.988325i	$-0.451313\pi$
$821$	8.73119	0.304721	0.152360	+	0.988325i	$0.451313\pi$
$822$	44.8691	1.56499
$823$	39.1844	1.36588	0.682942	−	0.730473i	$-0.260700\pi$
$823$	39.1844	1.36588	0.682942	+	0.730473i	$0.260700\pi$
$824$	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	$0.535127\pi$
$828$	−26.5484	−0.922621
$829$	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
$831$	−10.2791	−0.356579
$832$	17.0113	0.589762
$833$	−21.1116	−0.731473
$834$	−17.5786	−0.608699
$835$	0	0
$836$	54.4853	1.88441
$837$	2.23218	0.0771554
$838$	−67.1769	−2.32059
$839$	−55.7902	−1.92609	−0.963046	−	0.269339i	$-0.913195\pi$
$839$	−55.7902	−1.92609	−0.963046	+	0.269339i	$0.913195\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−6.86919	−0.236728
$843$	−4.57869	−0.157698
$844$	108.439	3.73264
$845$	0	0
$846$	9.35221	0.321536
$847$	6.09218	0.209330
$848$	7.86098	0.269947
$849$	−19.2431	−0.660421
$850$	0	0
$851$	33.0939	1.13444
$852$	−27.3495	−0.936977
$853$	−6.42812	−0.220095	−0.110047	−	0.993926i	$-0.535100\pi$
$853$	−6.42812	−0.220095	−0.110047	+	0.993926i	$0.535100\pi$
$854$	−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	$0.704140\pi$
$858$	−40.7891	−1.39252
$859$	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
$861$	−50.2606	−1.71288
$862$	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	$0.539319\pi$
$864$	7.73067	0.263003
$865$	0	0
$866$	85.7048	2.91237
$867$	15.0394	0.510766
$868$	48.7237	1.65379
$869$	−22.7918	−0.773160
$870$	0	0
$871$	70.6549	2.39405
$872$	1.68269	0.0569830
$873$	13.8343	0.468219
$874$	−55.4689	−1.87626
$875$	0	0
$876$	32.9089	1.11189
$877$	−24.6324	−0.831777	−0.415889	−	0.909416i	$-0.636529\pi$
$877$	−24.6324	−0.831777	−0.415889	+	0.909416i	$0.636529\pi$
$878$	−46.7422	−1.57747
$879$	−14.6045	−0.492597
$880$	0	0
$881$	26.2145	0.883190	0.441595	−	0.897214i	$-0.354413\pi$
$881$	26.2145	0.883190	0.441595	+	0.897214i	$0.354413\pi$
$882$	−38.8683	−1.30876
$883$	−10.7656	−0.362290	−0.181145	−	0.983456i	$-0.557980\pi$
$883$	−10.7656	−0.362290	−0.181145	+	0.983456i	$0.557980\pi$
$884$	−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	$0.466389\pi$
$888$	−39.4932	−1.32531
$889$	99.4743	3.33626
$890$	0	0
$891$	−3.11503	−0.104358
$892$	104.179	3.48817
$893$	13.6594	0.457094
$894$	61.1824	2.04625
$895$	0	0
$896$	−32.0818	−1.07178
$897$	29.0282	0.969223
$898$	−93.1949	−3.10995
$899$	−2.23218	−0.0744474
$900$	0	0
$901$	−1.32776	−0.0442340
$902$	−85.8973	−2.86007
$903$	41.7960	1.39088
$904$	98.3927	3.27249
$905$	0	0
$906$	−44.6221	−1.48247
$907$	21.7954	0.723704	0.361852	−	0.932235i	$-0.382145\pi$
$907$	21.7954	0.723704	0.361852	+	0.932235i	$0.382145\pi$
$908$	−18.9225	−0.627966
$909$	12.7461	0.422763
$910$	0	0
$911$	−5.56422	−0.184351	−0.0921753	−	0.995743i	$-0.529382\pi$
$911$	−5.56422	−0.184351	−0.0921753	+	0.995743i	$0.529382\pi$
$912$	31.2127	1.03355
$913$	42.5449	1.40803
$914$	23.0406	0.762114
$915$	0	0
$916$	−73.0136	−2.41244
$917$	38.4819	1.27078
$918$	−3.60957	−0.119134
$919$	−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
$921$	−12.3324	−0.406368
$922$	−89.3841	−2.94371
$923$	29.9041	0.984304
$924$	−67.9944	−2.23685
$925$	0	0
$926$	−79.7632	−2.62118
$927$	−8.20406	−0.269457
$928$	−7.73067	−0.253772
$929$	−11.3588	−0.372672	−0.186336	−	0.982486i	$-0.559661\pi$
$929$	−11.3588	−0.372672	−0.186336	+	0.982486i	$0.559661\pi$
$930$	0	0
$931$	−56.7691	−1.86053
$932$	76.4503	2.50421
$933$	−12.3084	−0.402957
$934$	−13.9443	−0.456273
$935$	0	0
$936$	−34.6413	−1.13229
$937$	32.7813	1.07092	0.535460	−	0.844561i	$-0.320138\pi$
$937$	32.7813	1.07092	0.535460	+	0.844561i	$0.320138\pi$
$938$	168.487	5.50128
$939$	−16.7306	−0.545984
$940$	0	0
$941$	−13.3074	−0.433808	−0.216904	−	0.976193i	$-0.569596\pi$
$941$	−13.3074	−0.433808	−0.216904	+	0.976193i	$0.569596\pi$
$942$	25.9174	0.844436
$943$	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	$0.326283\pi$
$948$	−33.9901	−1.10395
$949$	−35.9827	−1.16805
$950$	0	0
$951$	10.9429	0.354849
$952$	−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	$0.833961\pi$
$954$	−2.44451	−0.0791440
$955$	0	0
$956$	14.8054	0.478840
$957$	3.11503	0.100695
$958$	−34.6133	−1.11831
$959$	−81.7821	−2.64088
$960$	0	0
$961$	−26.0174	−0.839270
$962$	75.8274	2.44477
$963$	−3.28873	−0.105978
$964$	51.7893	1.66802
$965$	0	0
$966$	69.2219	2.22718
$967$	16.9490	0.545044	0.272522	−	0.962150i	$-0.412142\pi$
$967$	16.9490	0.545044	0.272522	+	0.962150i	$0.412142\pi$
$968$	8.84250	0.284209
$969$	−5.27196	−0.169360
$970$	0	0
$971$	−28.5476	−0.916135	−0.458067	−	0.888917i	$-0.651458\pi$
$971$	−28.5476	−0.916135	−0.458067	+	0.888917i	$0.651458\pi$
$972$	−4.64553	−0.149005
$973$	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	$0.664290\pi$
$978$	31.6054	1.01063
$979$	24.8585	0.794481
$980$	0	0
$981$	−0.246732	−0.00787755
$982$	−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	$0.363956\pi$
$984$	−72.9508	−2.32559
$985$	0	0
$986$	3.60957	0.114952
$987$	−17.0461	−0.542583
$988$	−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
$993$	19.1431	0.607487
$994$	71.3105	2.26183
$995$	0	0
$996$	63.4484	2.01044
$997$	31.6331	1.00183	0.500916	−	0.865496i	$-0.332997\pi$
$997$	31.6331	1.00183	0.500916	+	0.865496i	$0.332997\pi$
$998$	−54.6982	−1.73144
$999$	5.79088	0.183215

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.2	✓	8	1.1	even	1	trivial
2175.2.a.bd.1.7	yes	8	5.4	even	2
2175.2.c.p.349.3		16	5.2	odd	4
2175.2.c.p.349.14		16	5.3	odd	4
6525.2.a.by.1.2		8	15.14	odd	2
6525.2.a.bz.1.7		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57789 q^{2} -1.00000 q^{3} +4.64553 q^{4} +2.57789 q^{6} -4.69867 q^{7} -6.81989 q^{8} +1.00000 q^{9} -3.11503 q^{11} -4.64553 q^{12} +5.07945 q^{13} +12.1127 q^{14} +8.28989 q^{16} -1.40020 q^{17} -2.57789 q^{18} -3.76514 q^{19} +4.69867 q^{21} +8.03022 q^{22} -5.71483 q^{23} +6.81989 q^{24} -13.0943 q^{26} -1.00000 q^{27} -21.8278 q^{28} +1.00000 q^{29} -2.23218 q^{31} -7.73067 q^{32} +3.11503 q^{33} +3.60957 q^{34} +4.64553 q^{36} -5.79088 q^{37} +9.70614 q^{38} -5.07945 q^{39} -10.6968 q^{41} -12.1127 q^{42} +8.89527 q^{43} -14.4710 q^{44} +14.7322 q^{46} -3.62785 q^{47} -8.28989 q^{48} +15.0775 q^{49} +1.40020 q^{51} +23.5967 q^{52} +0.948260 q^{53} +2.57789 q^{54} +32.0445 q^{56} +3.76514 q^{57} -2.57789 q^{58} -8.53886 q^{59} -6.21467 q^{61} +5.75432 q^{62} -4.69867 q^{63} +3.34905 q^{64} -8.03022 q^{66} +13.9099 q^{67} -6.50468 q^{68} +5.71483 q^{69} +5.88726 q^{71} -6.81989 q^{72} -7.08398 q^{73} +14.9283 q^{74} -17.4911 q^{76} +14.6365 q^{77} +13.0943 q^{78} +7.31672 q^{79} +1.00000 q^{81} +27.5751 q^{82} -13.6579 q^{83} +21.8278 q^{84} -22.9311 q^{86} -1.00000 q^{87} +21.2442 q^{88} -7.98017 q^{89} -23.8667 q^{91} -26.5484 q^{92} +2.23218 q^{93} +9.35221 q^{94} +7.73067 q^{96} +13.8343 q^{97} -38.8683 q^{98} -3.11503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^3 + 12 * q^4 + 2 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 6 * q^11 - 12 * q^12 + 6 * q^13 + 9 * q^14 + 32 * q^16 - 12 * q^17 - 2 * q^18 - 2 * q^21 + 3 * q^22 - 14 * q^23 + 3 * q^24 + 18 * q^26 - 8 * q^27 + 14 * q^28 + 8 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^33 - 13 * q^34 + 12 * q^36 + 4 * q^37 - 26 * q^38 - 6 * q^39 + 2 * q^41 - 9 * q^42 + 2 * q^43 - 15 * q^44 + 24 * q^46 - 12 * q^47 - 32 * q^48 + 38 * q^49 + 12 * q^51 + 49 * q^52 - 4 * q^53 + 2 * q^54 + 58 * q^56 - 2 * q^58 + 18 * q^59 + 12 * q^61 + 4 * q^62 + 2 * q^63 + 21 * q^64 - 3 * q^66 + 26 * q^67 - 81 * q^68 + 14 * q^69 + 24 * q^71 - 3 * q^72 - 14 * q^73 - 22 * q^74 + 26 * q^77 - 18 * q^78 + 10 * q^79 + 8 * q^81 + 48 * q^82 - 40 * q^83 - 14 * q^84 + 8 * q^86 - 8 * q^87 - 10 * q^88 + 34 * q^89 + 26 * q^91 + 18 * q^92 - 8 * q^93 - 43 * q^94 - 2 * q^96 + 30 * q^97 - 29 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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