LMFDB - Embedded newform 2275.2.a.y.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2275,2,Mod(1,2275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2275 = 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.1659664598$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 $ x^7 - 8x^5 - 2x^4 + 16x^3 + 5x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 455)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.44914$ of defining polynomial
Character	$\chi$	$=$	2275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.44914 q^{2} -1.71242 q^{3} +3.99829 q^{4} -4.19396 q^{6} -1.00000 q^{7} +4.89410 q^{8} -0.0676101 q^{9} -4.08167 q^{11} -6.84677 q^{12} +1.00000 q^{13} -2.44914 q^{14} +3.98976 q^{16} +0.275564 q^{17} -0.165587 q^{18} -1.52434 q^{19} +1.71242 q^{21} -9.99659 q^{22} -7.26280 q^{23} -8.38077 q^{24} +2.44914 q^{26} +5.25304 q^{27} -3.99829 q^{28} -5.61055 q^{29} +5.18470 q^{31} -0.0167124 q^{32} +6.98954 q^{33} +0.674896 q^{34} -0.270325 q^{36} -5.84234 q^{37} -3.73331 q^{38} -1.71242 q^{39} -7.03310 q^{41} +4.19396 q^{42} -1.74083 q^{43} -16.3197 q^{44} -17.7876 q^{46} +6.80884 q^{47} -6.83216 q^{48} +1.00000 q^{49} -0.471882 q^{51} +3.99829 q^{52} -7.82802 q^{53} +12.8654 q^{54} -4.89410 q^{56} +2.61031 q^{57} -13.7410 q^{58} -11.0754 q^{59} -7.14032 q^{61} +12.6981 q^{62} +0.0676101 q^{63} -8.02046 q^{64} +17.1184 q^{66} +11.2782 q^{67} +1.10179 q^{68} +12.4370 q^{69} -6.28358 q^{71} -0.330891 q^{72} +8.45993 q^{73} -14.3087 q^{74} -6.09474 q^{76} +4.08167 q^{77} -4.19396 q^{78} +15.5187 q^{79} -8.79260 q^{81} -17.2251 q^{82} +14.7600 q^{83} +6.84677 q^{84} -4.26353 q^{86} +9.60763 q^{87} -19.9761 q^{88} -11.1977 q^{89} -1.00000 q^{91} -29.0388 q^{92} -8.87840 q^{93} +16.6758 q^{94} +0.0286186 q^{96} -6.19096 q^{97} +2.44914 q^{98} +0.275962 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * 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 $

$2$	2.44914	1.73180	0.865902	−	0.500213i	$-0.166745\pi$
$2$	2.44914	1.73180	0.865902	+	0.500213i	$0.166745\pi$
$3$	−1.71242	−0.988667	−0.494334	−	0.869272i	$-0.664588\pi$
$3$	−1.71242	−0.988667	−0.494334	+	0.869272i	$0.664588\pi$
$4$	3.99829	1.99915
$5$	0	0
$5$	0	0
$6$	−4.19396	−1.71218
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	4.89410	1.73033
$9$	−0.0676101	−0.0225367
$10$	0	0
$11$	−4.08167	−1.23067	−0.615335	−	0.788266i	$-0.710979\pi$
$11$	−4.08167	−1.23067	−0.615335	+	0.788266i	$0.710979\pi$
$12$	−6.84677	−1.97649
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.44914	−0.654561
$15$	0	0
$16$	3.98976	0.997441
$17$	0.275564	0.0668342	0.0334171	−	0.999441i	$-0.489361\pi$
$17$	0.275564	0.0668342	0.0334171	+	0.999441i	$0.489361\pi$
$18$	−0.165587	−0.0390292
$19$	−1.52434	−0.349707	−0.174853	−	0.984594i	$-0.555945\pi$
$19$	−1.52434	−0.349707	−0.174853	+	0.984594i	$0.555945\pi$
$20$	0	0
$21$	1.71242	0.373681
$22$	−9.99659	−2.13128
$23$	−7.26280	−1.51440	−0.757200	−	0.653184i	$-0.773433\pi$
$23$	−7.26280	−1.51440	−0.757200	+	0.653184i	$0.773433\pi$
$24$	−8.38077	−1.71072
$25$	0	0
$26$	2.44914	0.480316
$27$	5.25304	1.01095
$28$	−3.99829	−0.755606
$29$	−5.61055	−1.04185	−0.520926	−	0.853602i	$-0.674413\pi$
$29$	−5.61055	−1.04185	−0.520926	+	0.853602i	$0.674413\pi$
$30$	0	0
$31$	5.18470	0.931200	0.465600	−	0.884995i	$-0.345839\pi$
$31$	5.18470	0.931200	0.465600	+	0.884995i	$0.345839\pi$
$32$	−0.0167124	−0.00295435
$33$	6.98954	1.21672
$34$	0.674896	0.115744
$35$	0	0
$36$	−0.270325	−0.0450542
$37$	−5.84234	−0.960474	−0.480237	−	0.877139i	$-0.659449\pi$
$37$	−5.84234	−0.960474	−0.480237	+	0.877139i	$0.659449\pi$
$38$	−3.73331	−0.605624
$39$	−1.71242	−0.274207
$40$	0	0
$41$	−7.03310	−1.09839	−0.549193	−	0.835695i	$-0.685065\pi$
$41$	−7.03310	−1.09839	−0.549193	+	0.835695i	$0.685065\pi$
$42$	4.19396	0.647143
$43$	−1.74083	−0.265474	−0.132737	−	0.991151i	$-0.542377\pi$
$43$	−1.74083	−0.265474	−0.132737	+	0.991151i	$0.542377\pi$
$44$	−16.3197	−2.46029
$45$	0	0
$46$	−17.7876	−2.62264
$47$	6.80884	0.993171	0.496585	−	0.867988i	$-0.334587\pi$
$47$	6.80884	0.993171	0.496585	+	0.867988i	$0.334587\pi$
$48$	−6.83216	−0.986137
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.471882	−0.0660768
$52$	3.99829	0.554464
$53$	−7.82802	−1.07526	−0.537631	−	0.843180i	$-0.680681\pi$
$53$	−7.82802	−1.07526	−0.537631	+	0.843180i	$0.680681\pi$
$54$	12.8654	1.75077
$55$	0	0
$56$	−4.89410	−0.654002
$57$	2.61031	0.345744
$58$	−13.7410	−1.80429
$59$	−11.0754	−1.44189	−0.720944	−	0.692993i	$-0.756292\pi$
$59$	−11.0754	−1.44189	−0.720944	+	0.692993i	$0.756292\pi$
$60$	0	0
$61$	−7.14032	−0.914224	−0.457112	−	0.889409i	$-0.651116\pi$
$61$	−7.14032	−0.914224	−0.457112	+	0.889409i	$0.651116\pi$
$62$	12.6981	1.61266
$63$	0.0676101	0.00851808
$64$	−8.02046	−1.00256
$65$	0	0
$66$	17.1184	2.10713
$67$	11.2782	1.37786	0.688929	−	0.724829i	$-0.258081\pi$
$67$	11.2782	1.37786	0.688929	+	0.724829i	$0.258081\pi$
$68$	1.10179	0.133611
$69$	12.4370	1.49724
$70$	0	0
$71$	−6.28358	−0.745724	−0.372862	−	0.927887i	$-0.621624\pi$
$71$	−6.28358	−0.745724	−0.372862	+	0.927887i	$0.621624\pi$
$72$	−0.330891	−0.0389959
$73$	8.45993	0.990160	0.495080	−	0.868847i	$-0.335139\pi$
$73$	8.45993	0.990160	0.495080	+	0.868847i	$0.335139\pi$
$74$	−14.3087	−1.66335
$75$	0	0
$76$	−6.09474	−0.699115
$77$	4.08167	0.465149
$78$	−4.19396	−0.474873
$79$	15.5187	1.74600	0.872998	−	0.487724i	$-0.162173\pi$
$79$	15.5187	1.74600	0.872998	+	0.487724i	$0.162173\pi$
$80$	0	0
$81$	−8.79260	−0.976955
$82$	−17.2251	−1.90219
$83$	14.7600	1.62012	0.810058	−	0.586350i	$-0.199436\pi$
$83$	14.7600	1.62012	0.810058	+	0.586350i	$0.199436\pi$
$84$	6.84677	0.747043
$85$	0	0
$86$	−4.26353	−0.459749
$87$	9.60763	1.03005
$88$	−19.9761	−2.12946
$89$	−11.1977	−1.18695	−0.593477	−	0.804851i	$-0.702245\pi$
$89$	−11.1977	−1.18695	−0.593477	+	0.804851i	$0.702245\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−29.0388	−3.02751
$93$	−8.87840	−0.920647
$94$	16.6758	1.71998
$95$	0	0
$96$	0.0286186	0.00292087
$97$	−6.19096	−0.628596	−0.314298	−	0.949324i	$-0.601769\pi$
$97$	−6.19096	−0.628596	−0.314298	+	0.949324i	$0.601769\pi$
$98$	2.44914	0.247401
$99$	0.275962	0.0277353
$100$	0	0
$101$	2.86283	0.284862	0.142431	−	0.989805i	$-0.454508\pi$
$101$	2.86283	0.284862	0.142431	+	0.989805i	$0.454508\pi$
$102$	−1.15571	−0.114432
$103$	−1.05707	−0.104157	−0.0520783	−	0.998643i	$-0.516585\pi$
$103$	−1.05707	−0.104157	−0.0520783	+	0.998643i	$0.516585\pi$
$104$	4.89410	0.479906
$105$	0	0
$106$	−19.1719	−1.86214
$107$	10.9563	1.05918	0.529592	−	0.848253i	$-0.322345\pi$
$107$	10.9563	1.05918	0.529592	+	0.848253i	$0.322345\pi$
$108$	21.0032	2.02103
$109$	−13.8448	−1.32609	−0.663046	−	0.748578i	$-0.730737\pi$
$109$	−13.8448	−1.32609	−0.663046	+	0.748578i	$0.730737\pi$
$110$	0	0
$111$	10.0045	0.949590
$112$	−3.98976	−0.376997
$113$	−7.59152	−0.714150	−0.357075	−	0.934076i	$-0.616226\pi$
$113$	−7.59152	−0.714150	−0.357075	+	0.934076i	$0.616226\pi$
$114$	6.39301	0.598760
$115$	0	0
$116$	−22.4326	−2.08282
$117$	−0.0676101	−0.00625056
$118$	−27.1251	−2.49707
$119$	−0.275564	−0.0252609
$120$	0	0
$121$	5.66003	0.514548
$122$	−17.4876	−1.58326
$123$	12.0436	1.08594
$124$	20.7300	1.86161
$125$	0	0
$126$	0.165587	0.0147516
$127$	17.1320	1.52022	0.760111	−	0.649794i	$-0.225145\pi$
$127$	17.1320	1.52022	0.760111	+	0.649794i	$0.225145\pi$
$128$	−19.6098	−1.73328
$129$	2.98103	0.262465
$130$	0	0
$131$	−4.49291	−0.392547	−0.196274	−	0.980549i	$-0.562884\pi$
$131$	−4.49291	−0.392547	−0.196274	+	0.980549i	$0.562884\pi$
$132$	27.9462	2.43241
$133$	1.52434	0.132177
$134$	27.6220	2.38618
$135$	0	0
$136$	1.34864	0.115645
$137$	7.85643	0.671220	0.335610	−	0.942001i	$-0.391058\pi$
$137$	7.85643	0.671220	0.335610	+	0.942001i	$0.391058\pi$
$138$	30.4599	2.59292
$139$	3.14968	0.267152	0.133576	−	0.991039i	$-0.457354\pi$
$139$	3.14968	0.267152	0.133576	+	0.991039i	$0.457354\pi$
$140$	0	0
$141$	−11.6596	−0.981916
$142$	−15.3894	−1.29145
$143$	−4.08167	−0.341326
$144$	−0.269748	−0.0224790
$145$	0	0
$146$	20.7196	1.71476
$147$	−1.71242	−0.141238
$148$	−23.3594	−1.92013
$149$	−5.66723	−0.464277	−0.232139	−	0.972683i	$-0.574572\pi$
$149$	−5.66723	−0.464277	−0.232139	+	0.972683i	$0.574572\pi$
$150$	0	0
$151$	8.91019	0.725101	0.362550	−	0.931964i	$-0.381906\pi$
$151$	8.91019	0.725101	0.362550	+	0.931964i	$0.381906\pi$
$152$	−7.46026	−0.605107
$153$	−0.0186309	−0.00150622
$154$	9.99659	0.805548
$155$	0	0
$156$	−6.84677	−0.548180
$157$	−8.77622	−0.700419	−0.350209	−	0.936671i	$-0.613890\pi$
$157$	−8.77622	−0.700419	−0.350209	+	0.936671i	$0.613890\pi$
$158$	38.0076	3.02372
$159$	13.4049	1.06308
$160$	0	0
$161$	7.26280	0.572389
$162$	−21.5343	−1.69190
$163$	−13.6841	−1.07182	−0.535910	−	0.844275i	$-0.680032\pi$
$163$	−13.6841	−1.07182	−0.535910	+	0.844275i	$0.680032\pi$
$164$	−28.1204	−2.19584
$165$	0	0
$166$	36.1492	2.80572
$167$	−11.8356	−0.915865	−0.457932	−	0.888987i	$-0.651410\pi$
$167$	−11.8356	−0.915865	−0.457932	+	0.888987i	$0.651410\pi$
$168$	8.38077	0.646590
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.103061	0.00788124
$172$	−6.96034	−0.530721
$173$	1.09649	0.0833648	0.0416824	−	0.999131i	$-0.486728\pi$
$173$	1.09649	0.0833648	0.0416824	+	0.999131i	$0.486728\pi$
$174$	23.5304	1.78384
$175$	0	0
$176$	−16.2849	−1.22752
$177$	18.9657	1.42555
$178$	−27.4247	−2.05557
$179$	−20.2385	−1.51270	−0.756349	−	0.654168i	$-0.773019\pi$
$179$	−20.2385	−1.51270	−0.756349	+	0.654168i	$0.773019\pi$
$180$	0	0
$181$	18.4100	1.36840	0.684201	−	0.729293i	$-0.260151\pi$
$181$	18.4100	1.36840	0.684201	+	0.729293i	$0.260151\pi$
$182$	−2.44914	−0.181542
$183$	12.2272	0.903863
$184$	−35.5449	−2.62041
$185$	0	0
$186$	−21.7445	−1.59438
$187$	−1.12476	−0.0822508
$188$	27.2237	1.98549
$189$	−5.25304	−0.382103
$190$	0	0
$191$	−1.25934	−0.0911230	−0.0455615	−	0.998962i	$-0.514508\pi$
$191$	−1.25934	−0.0911230	−0.0455615	+	0.998962i	$0.514508\pi$
$192$	13.7344	0.991196
$193$	11.4920	0.827214	0.413607	−	0.910456i	$-0.364269\pi$
$193$	11.4920	0.827214	0.413607	+	0.910456i	$0.364269\pi$
$194$	−15.1625	−1.08861
$195$	0	0
$196$	3.99829	0.285592
$197$	−9.47449	−0.675030	−0.337515	−	0.941320i	$-0.609586\pi$
$197$	−9.47449	−0.675030	−0.337515	+	0.941320i	$0.609586\pi$
$198$	0.675871	0.0480320
$199$	−17.4049	−1.23380	−0.616899	−	0.787042i	$-0.711611\pi$
$199$	−17.4049	−1.23380	−0.616899	+	0.787042i	$0.711611\pi$
$200$	0	0
$201$	−19.3131	−1.36224
$202$	7.01148	0.493326
$203$	5.61055	0.393783
$204$	−1.88672	−0.132097
$205$	0	0
$206$	−2.58892	−0.180379
$207$	0.491039	0.0341296
$208$	3.98976	0.276640
$209$	6.22184	0.430373
$210$	0	0
$211$	19.9182	1.37123	0.685614	−	0.727965i	$-0.259534\pi$
$211$	19.9182	1.37123	0.685614	+	0.727965i	$0.259534\pi$
$212$	−31.2987	−2.14960
$213$	10.7601	0.737273
$214$	26.8335	1.83430
$215$	0	0
$216$	25.7089	1.74927
$217$	−5.18470	−0.351960
$218$	−33.9079	−2.29653
$219$	−14.4870	−0.978939
$220$	0	0
$221$	0.275564	0.0185365
$222$	24.5026	1.64450
$223$	−10.0341	−0.671932	−0.335966	−	0.941874i	$-0.609063\pi$
$223$	−10.0341	−0.671932	−0.335966	+	0.941874i	$0.609063\pi$
$224$	0.0167124	0.00111664
$225$	0	0
$226$	−18.5927	−1.23677
$227$	18.1092	1.20195	0.600975	−	0.799268i	$-0.294779\pi$
$227$	18.1092	1.20195	0.600975	+	0.799268i	$0.294779\pi$
$228$	10.4368	0.691192
$229$	−12.1959	−0.805929	−0.402965	−	0.915216i	$-0.632020\pi$
$229$	−12.1959	−0.805929	−0.402965	+	0.915216i	$0.632020\pi$
$230$	0	0
$231$	−6.98954	−0.459878
$232$	−27.4586	−1.80275
$233$	1.39717	0.0915316	0.0457658	−	0.998952i	$-0.485427\pi$
$233$	1.39717	0.0915316	0.0457658	+	0.998952i	$0.485427\pi$
$234$	−0.165587	−0.0108247
$235$	0	0
$236$	−44.2825	−2.88255
$237$	−26.5746	−1.72621
$238$	−0.674896	−0.0437470
$239$	20.4021	1.31970	0.659852	−	0.751396i	$-0.270619\pi$
$239$	20.4021	1.31970	0.659852	+	0.751396i	$0.270619\pi$
$240$	0	0
$241$	1.87381	0.120703	0.0603513	−	0.998177i	$-0.480778\pi$
$241$	1.87381	0.120703	0.0603513	+	0.998177i	$0.480778\pi$
$242$	13.8622	0.891097
$243$	−0.702490	−0.0450648
$244$	−28.5491	−1.82767
$245$	0	0
$246$	29.4966	1.88063
$247$	−1.52434	−0.0969912
$248$	25.3745	1.61128
$249$	−25.2753	−1.60176
$250$	0	0
$251$	25.1096	1.58490	0.792451	−	0.609935i	$-0.208804\pi$
$251$	25.1096	1.58490	0.792451	+	0.609935i	$0.208804\pi$
$252$	0.270325	0.0170289
$253$	29.6444	1.86373
$254$	41.9587	2.63273
$255$	0	0
$256$	−31.9863	−1.99914
$257$	18.5879	1.15948	0.579740	−	0.814801i	$-0.303154\pi$
$257$	18.5879	1.15948	0.579740	+	0.814801i	$0.303154\pi$
$258$	7.30097	0.454539
$259$	5.84234	0.363025
$260$	0	0
$261$	0.379330	0.0234799
$262$	−11.0038	−0.679815
$263$	19.6260	1.21019	0.605097	−	0.796152i	$-0.293134\pi$
$263$	19.6260	1.21019	0.605097	+	0.796152i	$0.293134\pi$
$264$	34.2075	2.10533
$265$	0	0
$266$	3.73331	0.228904
$267$	19.1752	1.17350
$268$	45.0937	2.75454
$269$	−14.8441	−0.905059	−0.452530	−	0.891749i	$-0.649478\pi$
$269$	−14.8441	−0.905059	−0.452530	+	0.891749i	$0.649478\pi$
$270$	0	0
$271$	4.42499	0.268799	0.134400	−	0.990927i	$-0.457089\pi$
$271$	4.42499	0.268799	0.134400	+	0.990927i	$0.457089\pi$
$272$	1.09944	0.0666631
$273$	1.71242	0.103641
$274$	19.2415	1.16242
$275$	0	0
$276$	49.7267	2.99320
$277$	22.8533	1.37312	0.686561	−	0.727072i	$-0.259119\pi$
$277$	22.8533	1.37312	0.686561	+	0.727072i	$0.259119\pi$
$278$	7.71401	0.462655
$279$	−0.350538	−0.0209862
$280$	0	0
$281$	1.40472	0.0837985	0.0418992	−	0.999122i	$-0.486659\pi$
$281$	1.40472	0.0837985	0.0418992	+	0.999122i	$0.486659\pi$
$282$	−28.5560	−1.70049
$283$	25.7114	1.52839	0.764193	−	0.644987i	$-0.223137\pi$
$283$	25.7114	1.52839	0.764193	+	0.644987i	$0.223137\pi$
$284$	−25.1236	−1.49081
$285$	0	0
$286$	−9.99659	−0.591111
$287$	7.03310	0.415151
$288$	0.00112992	6.65815e−5 0
$289$	−16.9241	−0.995533
$290$	0	0
$291$	10.6015	0.621473
$292$	33.8253	1.97947
$293$	32.5210	1.89989	0.949947	−	0.312410i	$-0.101136\pi$
$293$	32.5210	1.89989	0.949947	+	0.312410i	$0.101136\pi$
$294$	−4.19396	−0.244597
$295$	0	0
$296$	−28.5930	−1.66193
$297$	−21.4412	−1.24414
$298$	−13.8798	−0.804037
$299$	−7.26280	−0.420019
$300$	0	0
$301$	1.74083	0.100340
$302$	21.8223	1.25573
$303$	−4.90238	−0.281634
$304$	−6.08174	−0.348812
$305$	0	0
$306$	−0.0456298	−0.00260848
$307$	24.6822	1.40869	0.704343	−	0.709860i	$-0.251242\pi$
$307$	24.6822	1.40869	0.704343	+	0.709860i	$0.251242\pi$
$308$	16.3197	0.929902
$309$	1.81016	0.102976
$310$	0	0
$311$	−22.7363	−1.28926	−0.644630	−	0.764495i	$-0.722988\pi$
$311$	−22.7363	−1.28926	−0.644630	+	0.764495i	$0.722988\pi$
$312$	−8.38077	−0.474468
$313$	−22.4182	−1.26715	−0.633575	−	0.773681i	$-0.718413\pi$
$313$	−22.4182	−1.26715	−0.633575	+	0.773681i	$0.718413\pi$
$314$	−21.4942	−1.21299
$315$	0	0
$316$	62.0485	3.49050
$317$	−23.8714	−1.34075	−0.670375	−	0.742023i	$-0.733867\pi$
$317$	−23.8714	−1.34075	−0.670375	+	0.742023i	$0.733867\pi$
$318$	32.8304	1.84104
$319$	22.9004	1.28218
$320$	0	0
$321$	−18.7618	−1.04718
$322$	17.7876	0.991266
$323$	−0.420053	−0.0233724
$324$	−35.1554	−1.95308
$325$	0	0
$326$	−33.5143	−1.85618
$327$	23.7082	1.31106
$328$	−34.4207	−1.90057
$329$	−6.80884	−0.375383
$330$	0	0
$331$	−26.9719	−1.48251	−0.741256	−	0.671223i	$-0.765769\pi$
$331$	−26.9719	−1.48251	−0.741256	+	0.671223i	$0.765769\pi$
$332$	59.0146	3.23885
$333$	0.395001	0.0216459
$334$	−28.9870	−1.58610
$335$	0	0
$336$	6.83216	0.372725
$337$	11.5651	0.629993	0.314997	−	0.949093i	$-0.397997\pi$
$337$	11.5651	0.629993	0.314997	+	0.949093i	$0.397997\pi$
$338$	2.44914	0.133216
$339$	12.9999	0.706057
$340$	0	0
$341$	−21.1622	−1.14600
$342$	0.252410	0.0136488
$343$	−1.00000	−0.0539949
$344$	−8.51979	−0.459356
$345$	0	0
$346$	2.68547	0.144372
$347$	−31.9334	−1.71427	−0.857137	−	0.515089i	$-0.827759\pi$
$347$	−31.9334	−1.71427	−0.857137	+	0.515089i	$0.827759\pi$
$348$	38.4141	2.05921
$349$	3.10981	0.166464	0.0832320	−	0.996530i	$-0.473476\pi$
$349$	3.10981	0.166464	0.0832320	+	0.996530i	$0.473476\pi$
$350$	0	0
$351$	5.25304	0.280387
$352$	0.0682143	0.00363584
$353$	−9.47462	−0.504283	−0.252141	−	0.967690i	$-0.581135\pi$
$353$	−9.47462	−0.504283	−0.252141	+	0.967690i	$0.581135\pi$
$354$	46.4496	2.46877
$355$	0	0
$356$	−44.7717	−2.37289
$357$	0.471882	0.0249747
$358$	−49.5670	−2.61970
$359$	6.89876	0.364102	0.182051	−	0.983289i	$-0.441726\pi$
$359$	6.89876	0.364102	0.182051	+	0.983289i	$0.441726\pi$
$360$	0	0
$361$	−16.6764	−0.877705
$362$	45.0886	2.36981
$363$	−9.69236	−0.508717
$364$	−3.99829	−0.209568
$365$	0	0
$366$	29.9462	1.56531
$367$	−19.9560	−1.04170	−0.520848	−	0.853649i	$-0.674384\pi$
$367$	−19.9560	−1.04170	−0.520848	+	0.853649i	$0.674384\pi$
$368$	−28.9769	−1.51052
$369$	0.475509	0.0247540
$370$	0	0
$371$	7.82802	0.406411
$372$	−35.4984	−1.84051
$373$	−24.0135	−1.24337	−0.621686	−	0.783267i	$-0.713552\pi$
$373$	−24.0135	−1.24337	−0.621686	+	0.783267i	$0.713552\pi$
$374$	−2.75470	−0.142442
$375$	0	0
$376$	33.3231	1.71851
$377$	−5.61055	−0.288958
$378$	−12.8654	−0.661727
$379$	−12.8251	−0.658782	−0.329391	−	0.944194i	$-0.606843\pi$
$379$	−12.8251	−0.658782	−0.329391	+	0.944194i	$0.606843\pi$
$380$	0	0
$381$	−29.3373	−1.50299
$382$	−3.08431	−0.157807
$383$	−28.4807	−1.45530	−0.727648	−	0.685950i	$-0.759387\pi$
$383$	−28.4807	−1.45530	−0.727648	+	0.685950i	$0.759387\pi$
$384$	33.5803	1.71364
$385$	0	0
$386$	28.1456	1.43257
$387$	0.117698	0.00598291
$388$	−24.7533	−1.25666
$389$	−3.11766	−0.158071	−0.0790357	−	0.996872i	$-0.525184\pi$
$389$	−3.11766	−0.158071	−0.0790357	+	0.996872i	$0.525184\pi$
$390$	0	0
$391$	−2.00137	−0.101214
$392$	4.89410	0.247190
$393$	7.69376	0.388099
$394$	−23.2044	−1.16902
$395$	0	0
$396$	1.10338	0.0554468
$397$	−28.6707	−1.43894	−0.719471	−	0.694523i	$-0.755615\pi$
$397$	−28.6707	−1.43894	−0.719471	+	0.694523i	$0.755615\pi$
$398$	−42.6270	−2.13670
$399$	−2.61031	−0.130679
$400$	0	0
$401$	30.5552	1.52585	0.762927	−	0.646484i	$-0.223761\pi$
$401$	30.5552	1.52585	0.762927	+	0.646484i	$0.223761\pi$
$402$	−47.3006	−2.35914
$403$	5.18470	0.258268
$404$	11.4464	0.569482
$405$	0	0
$406$	13.7410	0.681956
$407$	23.8465	1.18203
$408$	−2.30944	−0.114334
$409$	27.0116	1.33564	0.667818	−	0.744324i	$-0.267228\pi$
$409$	27.0116	1.33564	0.667818	+	0.744324i	$0.267228\pi$
$410$	0	0
$411$	−13.4535	−0.663613
$412$	−4.22649	−0.208224
$413$	11.0754	0.544983
$414$	1.20262	0.0591058
$415$	0	0
$416$	−0.0167124	−0.000819391 0
$417$	−5.39358	−0.264125
$418$	15.2382	0.745323
$419$	−11.7154	−0.572336	−0.286168	−	0.958179i	$-0.592382\pi$
$419$	−11.7154	−0.572336	−0.286168	+	0.958179i	$0.592382\pi$
$420$	0	0
$421$	0.0775773	0.00378088	0.00189044	−	0.999998i	$-0.499398\pi$
$421$	0.0775773	0.00378088	0.00189044	+	0.999998i	$0.499398\pi$
$422$	48.7826	2.37470
$423$	−0.460346	−0.0223828
$424$	−38.3111	−1.86055
$425$	0	0
$426$	26.3531	1.27681
$427$	7.14032	0.345544
$428$	43.8064	2.11746
$429$	6.98954	0.337458
$430$	0	0
$431$	−17.9072	−0.862562	−0.431281	−	0.902218i	$-0.641938\pi$
$431$	−17.9072	−0.862562	−0.431281	+	0.902218i	$0.641938\pi$
$432$	20.9584	1.00836
$433$	27.9725	1.34427	0.672137	−	0.740427i	$-0.265377\pi$
$433$	27.9725	1.34427	0.672137	+	0.740427i	$0.265377\pi$
$434$	−12.6981	−0.609527
$435$	0	0
$436$	−55.3556	−2.65105
$437$	11.0710	0.529595
$438$	−35.4806	−1.69533
$439$	7.35448	0.351010	0.175505	−	0.984479i	$-0.443844\pi$
$439$	7.35448	0.351010	0.175505	+	0.984479i	$0.443844\pi$
$440$	0	0
$441$	−0.0676101	−0.00321953
$442$	0.674896	0.0321015
$443$	−31.7691	−1.50940	−0.754699	−	0.656071i	$-0.772217\pi$
$443$	−31.7691	−1.50940	−0.754699	+	0.656071i	$0.772217\pi$
$444$	40.0011	1.89837
$445$	0	0
$446$	−24.5749	−1.16365
$447$	9.70468	0.459016
$448$	8.02046	0.378931
$449$	32.6539	1.54103	0.770515	−	0.637421i	$-0.219999\pi$
$449$	32.6539	1.54103	0.770515	+	0.637421i	$0.219999\pi$
$450$	0	0
$451$	28.7068	1.35175
$452$	−30.3531	−1.42769
$453$	−15.2580	−0.716884
$454$	44.3520	2.08154
$455$	0	0
$456$	12.7751	0.598249
$457$	−18.7555	−0.877346	−0.438673	−	0.898647i	$-0.644551\pi$
$457$	−18.7555	−0.877346	−0.438673	+	0.898647i	$0.644551\pi$
$458$	−29.8695	−1.39571
$459$	1.44755	0.0675659
$460$	0	0
$461$	6.99830	0.325943	0.162972	−	0.986631i	$-0.447892\pi$
$461$	6.99830	0.325943	0.162972	+	0.986631i	$0.447892\pi$
$462$	−17.1184	−0.796419
$463$	7.59583	0.353008	0.176504	−	0.984300i	$-0.443521\pi$
$463$	7.59583	0.353008	0.176504	+	0.984300i	$0.443521\pi$
$464$	−22.3848	−1.03919
$465$	0	0
$466$	3.42187	0.158515
$467$	−5.07306	−0.234753	−0.117377	−	0.993087i	$-0.537448\pi$
$467$	−5.07306	−0.234753	−0.117377	+	0.993087i	$0.537448\pi$
$468$	−0.270325	−0.0124958
$469$	−11.2782	−0.520781
$470$	0	0
$471$	15.0286	0.692481
$472$	−54.2039	−2.49494
$473$	7.10549	0.326711
$474$	−65.0851	−2.98946
$475$	0	0
$476$	−1.10179	−0.0505003
$477$	0.529254	0.0242329
$478$	49.9677	2.28547
$479$	21.2446	0.970689	0.485345	−	0.874323i	$-0.338694\pi$
$479$	21.2446	0.970689	0.485345	+	0.874323i	$0.338694\pi$
$480$	0	0
$481$	−5.84234	−0.266388
$482$	4.58922	0.209033
$483$	−12.4370	−0.565902
$484$	22.6305	1.02866
$485$	0	0
$486$	−1.72050	−0.0780434
$487$	−25.9441	−1.17564	−0.587819	−	0.808992i	$-0.700013\pi$
$487$	−25.9441	−1.17564	−0.587819	+	0.808992i	$0.700013\pi$
$488$	−34.9454	−1.58191
$489$	23.4329	1.05967
$490$	0	0
$491$	−1.41408	−0.0638166	−0.0319083	−	0.999491i	$-0.510158\pi$
$491$	−1.41408	−0.0638166	−0.0319083	+	0.999491i	$0.510158\pi$
$492$	48.1540	2.17095
$493$	−1.54607	−0.0696314
$494$	−3.73331	−0.167970
$495$	0	0
$496$	20.6857	0.928817
$497$	6.28358	0.281857
$498$	−61.9027	−2.77393
$499$	−4.32253	−0.193503	−0.0967514	−	0.995309i	$-0.530845\pi$
$499$	−4.32253	−0.193503	−0.0967514	+	0.995309i	$0.530845\pi$
$500$	0	0
$501$	20.2675	0.905486
$502$	61.4969	2.74474
$503$	12.4066	0.553183	0.276592	−	0.960988i	$-0.410795\pi$
$503$	12.4066	0.553183	0.276592	+	0.960988i	$0.410795\pi$
$504$	0.330891	0.0147391
$505$	0	0
$506$	72.6032	3.22761
$507$	−1.71242	−0.0760513
$508$	68.4989	3.03915
$509$	−2.15053	−0.0953208	−0.0476604	−	0.998864i	$-0.515177\pi$
$509$	−2.15053	−0.0953208	−0.0476604	+	0.998864i	$0.515177\pi$
$510$	0	0
$511$	−8.45993	−0.374245
$512$	−39.1193	−1.72885
$513$	−8.00740	−0.353536
$514$	45.5243	2.00799
$515$	0	0
$516$	11.9190	0.524707
$517$	−27.7914	−1.22227
$518$	14.3087	0.628689
$519$	−1.87766	−0.0824201
$520$	0	0
$521$	33.7404	1.47819	0.739096	−	0.673600i	$-0.235253\pi$
$521$	33.7404	1.47819	0.739096	+	0.673600i	$0.235253\pi$
$522$	0.929033	0.0406627
$523$	20.2232	0.884299	0.442150	−	0.896941i	$-0.354216\pi$
$523$	20.2232	0.884299	0.442150	+	0.896941i	$0.354216\pi$
$524$	−17.9640	−0.784760
$525$	0	0
$526$	48.0670	2.09582
$527$	1.42872	0.0622360
$528$	27.8866	1.21361
$529$	29.7483	1.29340
$530$	0	0
$531$	0.748807	0.0324954
$532$	6.09474	0.264241
$533$	−7.03310	−0.304638
$534$	46.9627	2.03228
$535$	0	0
$536$	55.1969	2.38414
$537$	34.6569	1.49556
$538$	−36.3552	−1.56739
$539$	−4.08167	−0.175810
$540$	0	0
$541$	−17.4002	−0.748093	−0.374047	−	0.927410i	$-0.622030\pi$
$541$	−17.4002	−0.748093	−0.374047	+	0.927410i	$0.622030\pi$
$542$	10.8374	0.465508
$543$	−31.5256	−1.35289
$544$	−0.00460533	−0.000197452 0
$545$	0	0
$546$	4.19396	0.179485
$547$	−28.5667	−1.22142	−0.610711	−	0.791853i	$-0.709116\pi$
$547$	−28.5667	−1.22142	−0.610711	+	0.791853i	$0.709116\pi$
$548$	31.4123	1.34187
$549$	0.482758	0.0206036
$550$	0	0
$551$	8.55236	0.364343
$552$	60.8679	2.59071
$553$	−15.5187	−0.659924
$554$	55.9710	2.37798
$555$	0	0
$556$	12.5933	0.534076
$557$	−10.6941	−0.453125	−0.226563	−	0.973997i	$-0.572749\pi$
$557$	−10.6941	−0.453125	−0.226563	+	0.973997i	$0.572749\pi$
$558$	−0.858518	−0.0363440
$559$	−1.74083	−0.0736292
$560$	0	0
$561$	1.92607	0.0813187
$562$	3.44035	0.145123
$563$	−40.2438	−1.69607	−0.848037	−	0.529937i	$-0.822215\pi$
$563$	−40.2438	−1.69607	−0.848037	+	0.529937i	$0.822215\pi$
$564$	−46.6185	−1.96299
$565$	0	0
$566$	62.9710	2.64687
$567$	8.79260	0.369254
$568$	−30.7525	−1.29035
$569$	−13.5339	−0.567369	−0.283684	−	0.958918i	$-0.591557\pi$
$569$	−13.5339	−0.567369	−0.283684	+	0.958918i	$0.591557\pi$
$570$	0	0
$571$	−10.5069	−0.439700	−0.219850	−	0.975534i	$-0.570557\pi$
$571$	−10.5069	−0.439700	−0.219850	+	0.975534i	$0.570557\pi$
$572$	−16.3197	−0.682362
$573$	2.15653	0.0900903
$574$	17.2251	0.718960
$575$	0	0
$576$	0.542264	0.0225943
$577$	−15.5501	−0.647358	−0.323679	−	0.946167i	$-0.604920\pi$
$577$	−15.5501	−0.647358	−0.323679	+	0.946167i	$0.604920\pi$
$578$	−41.4494	−1.72407
$579$	−19.6792	−0.817839
$580$	0	0
$581$	−14.7600	−0.612346
$582$	25.9647	1.07627
$583$	31.9514	1.32329
$584$	41.4038	1.71330
$585$	0	0
$586$	79.6484	3.29025
$587$	33.9061	1.39946	0.699728	−	0.714410i	$-0.253305\pi$
$587$	33.9061	1.39946	0.699728	+	0.714410i	$0.253305\pi$
$588$	−6.84677	−0.282356
$589$	−7.90323	−0.325647
$590$	0	0
$591$	16.2243	0.667380
$592$	−23.3095	−0.958016
$593$	−27.5257	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$593$	−27.5257	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$594$	−52.5125	−2.15461
$595$	0	0
$596$	−22.6592	−0.928158
$597$	29.8045	1.21982
$598$	−17.7876	−0.727390
$599$	−26.0860	−1.06584	−0.532922	−	0.846165i	$-0.678906\pi$
$599$	−26.0860	−1.06584	−0.532922	+	0.846165i	$0.678906\pi$
$600$	0	0
$601$	−17.7962	−0.725922	−0.362961	−	0.931804i	$-0.618234\pi$
$601$	−17.7962	−0.725922	−0.362961	+	0.931804i	$0.618234\pi$
$602$	4.26353	0.173769
$603$	−0.762524	−0.0310524
$604$	35.6255	1.44958
$605$	0	0
$606$	−12.0066	−0.487735
$607$	−38.8492	−1.57684	−0.788421	−	0.615136i	$-0.789101\pi$
$607$	−38.8492	−1.57684	−0.788421	+	0.615136i	$0.789101\pi$
$608$	0.0254752	0.00103316
$609$	−9.60763	−0.389321
$610$	0	0
$611$	6.80884	0.275456
$612$	−0.0744920	−0.00301116
$613$	40.1447	1.62143	0.810715	−	0.585441i	$-0.199078\pi$
$613$	40.1447	1.62143	0.810715	+	0.585441i	$0.199078\pi$
$614$	60.4501	2.43957
$615$	0	0
$616$	19.9761	0.804861
$617$	−45.4711	−1.83060	−0.915299	−	0.402775i	$-0.868046\pi$
$617$	−45.4711	−1.83060	−0.915299	+	0.402775i	$0.868046\pi$
$618$	4.43333	0.178335
$619$	−42.7407	−1.71790	−0.858948	−	0.512063i	$-0.828881\pi$
$619$	−42.7407	−1.71790	−0.858948	+	0.512063i	$0.828881\pi$
$620$	0	0
$621$	−38.1518	−1.53098
$622$	−55.6845	−2.23275
$623$	11.1977	0.448626
$624$	−6.83216	−0.273505
$625$	0	0
$626$	−54.9053	−2.19446
$627$	−10.6544	−0.425496
$628$	−35.0899	−1.40024
$629$	−1.60994	−0.0641925
$630$	0	0
$631$	−24.9039	−0.991410	−0.495705	−	0.868491i	$-0.665090\pi$
$631$	−24.9039	−0.991410	−0.495705	+	0.868491i	$0.665090\pi$
$632$	75.9503	3.02114
$633$	−34.1085	−1.35569
$634$	−58.4644	−2.32192
$635$	0	0
$636$	53.5966	2.12524
$637$	1.00000	0.0396214
$638$	56.0863	2.22048
$639$	0.424834	0.0168062
$640$	0	0
$641$	−28.0998	−1.10988	−0.554939	−	0.831891i	$-0.687258\pi$
$641$	−28.0998	−1.10988	−0.554939	+	0.831891i	$0.687258\pi$
$642$	−45.9502	−1.81351
$643$	33.8666	1.33557	0.667784	−	0.744355i	$-0.267243\pi$
$643$	33.8666	1.33557	0.667784	+	0.744355i	$0.267243\pi$
$644$	29.0388	1.14429
$645$	0	0
$646$	−1.02877	−0.0404763
$647$	26.7838	1.05298	0.526491	−	0.850181i	$-0.323507\pi$
$647$	26.7838	1.05298	0.526491	+	0.850181i	$0.323507\pi$
$648$	−43.0319	−1.69045
$649$	45.2060	1.77449
$650$	0	0
$651$	8.87840	0.347972
$652$	−54.7130	−2.14273
$653$	29.3612	1.14899	0.574496	−	0.818507i	$-0.305198\pi$
$653$	29.3612	1.14899	0.574496	+	0.818507i	$0.305198\pi$
$654$	58.0646	2.27051
$655$	0	0
$656$	−28.0604	−1.09558
$657$	−0.571977	−0.0223150
$658$	−16.6758	−0.650090
$659$	−31.2087	−1.21572	−0.607860	−	0.794045i	$-0.707972\pi$
$659$	−31.2087	−1.21572	−0.607860	+	0.794045i	$0.707972\pi$
$660$	0	0
$661$	36.3164	1.41254	0.706272	−	0.707941i	$-0.250376\pi$
$661$	36.3164	1.41254	0.706272	+	0.707941i	$0.250376\pi$
$662$	−66.0581	−2.56742
$663$	−0.471882	−0.0183264
$664$	72.2367	2.80333
$665$	0	0
$666$	0.967414	0.0374865
$667$	40.7483	1.57778
$668$	−47.3221	−1.83095
$669$	17.1826	0.664317
$670$	0	0
$671$	29.1444	1.12511
$672$	−0.0286186	−0.00110399
$673$	−1.43002	−0.0551232	−0.0275616	−	0.999620i	$-0.508774\pi$
$673$	−1.43002	−0.0551232	−0.0275616	+	0.999620i	$0.508774\pi$
$674$	28.3247	1.09103
$675$	0	0
$676$	3.99829	0.153781
$677$	−19.1740	−0.736915	−0.368458	−	0.929645i	$-0.620114\pi$
$677$	−19.1740	−0.736915	−0.368458	+	0.929645i	$0.620114\pi$
$678$	31.8386	1.22275
$679$	6.19096	0.237587
$680$	0	0
$681$	−31.0106	−1.18833
$682$	−51.8293	−1.98465
$683$	−29.2670	−1.11987	−0.559935	−	0.828536i	$-0.689174\pi$
$683$	−29.2670	−1.11987	−0.559935	+	0.828536i	$0.689174\pi$
$684$	0.412066	0.0157558
$685$	0	0
$686$	−2.44914	−0.0935087
$687$	20.8846	0.796796
$688$	−6.94549	−0.264794
$689$	−7.82802	−0.298224
$690$	0	0
$691$	−0.420810	−0.0160084	−0.00800419	−	0.999968i	$-0.502548\pi$
$691$	−0.420810	−0.0160084	−0.00800419	+	0.999968i	$0.502548\pi$
$692$	4.38410	0.166658
$693$	−0.275962	−0.0104829
$694$	−78.2093	−2.96879
$695$	0	0
$696$	47.0207	1.78232
$697$	−1.93807	−0.0734097
$698$	7.61635	0.288283
$699$	−2.39254	−0.0904943
$700$	0	0
$701$	−33.7661	−1.27533	−0.637664	−	0.770314i	$-0.720099\pi$
$701$	−33.7661	−1.27533	−0.637664	+	0.770314i	$0.720099\pi$
$702$	12.8654	0.485575
$703$	8.90569	0.335884
$704$	32.7369	1.23382
$705$	0	0
$706$	−23.2047	−0.873319
$707$	−2.86283	−0.107668
$708$	75.8304	2.84988
$709$	38.1704	1.43352	0.716760	−	0.697320i	$-0.245624\pi$
$709$	38.1704	1.43352	0.716760	+	0.697320i	$0.245624\pi$
$710$	0	0
$711$	−1.04922	−0.0393490
$712$	−54.8027	−2.05382
$713$	−37.6555	−1.41021
$714$	1.15571	0.0432512
$715$	0	0
$716$	−80.9195	−3.02411
$717$	−34.9371	−1.30475
$718$	16.8960	0.630554
$719$	25.4755	0.950077	0.475039	−	0.879965i	$-0.342434\pi$
$719$	25.4755	0.950077	0.475039	+	0.879965i	$0.342434\pi$
$720$	0	0
$721$	1.05707	0.0393675
$722$	−40.8429	−1.52001
$723$	−3.20875	−0.119335
$724$	73.6085	2.73564
$725$	0	0
$726$	−23.7380	−0.880998
$727$	−25.9144	−0.961111	−0.480555	−	0.876964i	$-0.659565\pi$
$727$	−25.9144	−0.961111	−0.480555	+	0.876964i	$0.659565\pi$
$728$	−4.89410	−0.181388
$729$	27.5808	1.02151
$730$	0	0
$731$	−0.479710	−0.0177427
$732$	48.8881	1.80696
$733$	7.72915	0.285483	0.142741	−	0.989760i	$-0.454408\pi$
$733$	7.72915	0.285483	0.142741	+	0.989760i	$0.454408\pi$
$734$	−48.8751	−1.80401
$735$	0	0
$736$	0.121379	0.00447407
$737$	−46.0341	−1.69569
$738$	1.16459	0.0428691
$739$	−19.7206	−0.725435	−0.362718	−	0.931899i	$-0.618151\pi$
$739$	−19.7206	−0.725435	−0.362718	+	0.931899i	$0.618151\pi$
$740$	0	0
$741$	2.61031	0.0958920
$742$	19.1719	0.703824
$743$	−3.44689	−0.126454	−0.0632271	−	0.997999i	$-0.520139\pi$
$743$	−3.44689	−0.126454	−0.0632271	+	0.997999i	$0.520139\pi$
$744$	−43.4518	−1.59302
$745$	0	0
$746$	−58.8124	−2.15328
$747$	−0.997923	−0.0365121
$748$	−4.49713	−0.164431
$749$	−10.9563	−0.400334
$750$	0	0
$751$	0.452308	0.0165050	0.00825248	−	0.999966i	$-0.497373\pi$
$751$	0.452308	0.0165050	0.00825248	+	0.999966i	$0.497373\pi$
$752$	27.1656	0.990629
$753$	−42.9982	−1.56694
$754$	−13.7410	−0.500419
$755$	0	0
$756$	−21.0032	−0.763879
$757$	6.02600	0.219019	0.109509	−	0.993986i	$-0.465072\pi$
$757$	6.02600	0.219019	0.109509	+	0.993986i	$0.465072\pi$
$758$	−31.4105	−1.14088
$759$	−50.7637	−1.84260
$760$	0	0
$761$	22.5566	0.817675	0.408838	−	0.912607i	$-0.365934\pi$
$761$	22.5566	0.817675	0.408838	+	0.912607i	$0.365934\pi$
$762$	−71.8511	−2.60289
$763$	13.8448	0.501216
$764$	−5.03523	−0.182168
$765$	0	0
$766$	−69.7533	−2.52029
$767$	−11.0754	−0.399908
$768$	54.7740	1.97649
$769$	−51.6908	−1.86402	−0.932008	−	0.362437i	$-0.881945\pi$
$769$	−51.6908	−1.86402	−0.932008	+	0.362437i	$0.881945\pi$
$770$	0	0
$771$	−31.8303	−1.14634
$772$	45.9485	1.65372
$773$	4.91863	0.176911	0.0884555	−	0.996080i	$-0.471807\pi$
$773$	4.91863	0.176911	0.0884555	+	0.996080i	$0.471807\pi$
$774$	0.288258	0.0103612
$775$	0	0
$776$	−30.2992	−1.08768
$777$	−10.0045	−0.358911
$778$	−7.63558	−0.273749
$779$	10.7208	0.384113
$780$	0	0
$781$	25.6475	0.917740
$782$	−4.90164	−0.175282
$783$	−29.4725	−1.05326
$784$	3.98976	0.142492
$785$	0	0
$786$	18.8431	0.672111
$787$	−7.15207	−0.254944	−0.127472	−	0.991842i	$-0.540686\pi$
$787$	−7.15207	−0.254944	−0.127472	+	0.991842i	$0.540686\pi$
$788$	−37.8818	−1.34948
$789$	−33.6081	−1.19648
$790$	0	0
$791$	7.59152	0.269923
$792$	1.35059	0.0479911
$793$	−7.14032	−0.253560
$794$	−70.2186	−2.49196
$795$	0	0
$796$	−69.5898	−2.46654
$797$	−1.90848	−0.0676019	−0.0338010	−	0.999429i	$-0.510761\pi$
$797$	−1.90848	−0.0676019	−0.0338010	+	0.999429i	$0.510761\pi$
$798$	−6.39301	−0.226310
$799$	1.87627	0.0663777
$800$	0	0
$801$	0.757078	0.0267500
$802$	74.8341	2.64248
$803$	−34.5306	−1.21856
$804$	−77.2195	−2.72332
$805$	0	0
$806$	12.6981	0.447270
$807$	25.4193	0.894803
$808$	14.0110	0.492905
$809$	−30.8689	−1.08529	−0.542646	−	0.839961i	$-0.682577\pi$
$809$	−30.8689	−1.08529	−0.542646	+	0.839961i	$0.682577\pi$
$810$	0	0
$811$	22.3766	0.785750	0.392875	−	0.919592i	$-0.371480\pi$
$811$	22.3766	0.785750	0.392875	+	0.919592i	$0.371480\pi$
$812$	22.4326	0.787231
$813$	−7.57746	−0.265753
$814$	58.4034	2.04704
$815$	0	0
$816$	−1.88270	−0.0659077
$817$	2.65361	0.0928380
$818$	66.1552	2.31306
$819$	0.0676101	0.00236249
$820$	0	0
$821$	2.63859	0.0920873	0.0460437	−	0.998939i	$-0.485339\pi$
$821$	2.63859	0.0920873	0.0460437	+	0.998939i	$0.485339\pi$
$822$	−32.9496	−1.14925
$823$	−32.5426	−1.13436	−0.567182	−	0.823592i	$-0.691967\pi$
$823$	−32.5426	−1.13436	−0.567182	+	0.823592i	$0.691967\pi$
$824$	−5.17343	−0.180225
$825$	0	0
$826$	27.1251	0.943803
$827$	26.6557	0.926909	0.463454	−	0.886121i	$-0.346610\pi$
$827$	26.6557	0.926909	0.463454	+	0.886121i	$0.346610\pi$
$828$	1.96332	0.0682300
$829$	−39.0751	−1.35714	−0.678568	−	0.734538i	$-0.737399\pi$
$829$	−39.0751	−1.35714	−0.678568	+	0.734538i	$0.737399\pi$
$830$	0	0
$831$	−39.1345	−1.35756
$832$	−8.02046	−0.278059
$833$	0.275564	0.00954774
$834$	−13.2096	−0.457412
$835$	0	0
$836$	24.8767	0.860380
$837$	27.2355	0.941395
$838$	−28.6928	−0.991175
$839$	−5.17263	−0.178579	−0.0892896	−	0.996006i	$-0.528460\pi$
$839$	−5.17263	−0.178579	−0.0892896	+	0.996006i	$0.528460\pi$
$840$	0	0
$841$	2.47825	0.0854570
$842$	0.189998	0.00654775
$843$	−2.40547	−0.0828488
$844$	79.6390	2.74129
$845$	0	0
$846$	−1.12745	−0.0387627
$847$	−5.66003	−0.194481
$848$	−31.2319	−1.07251
$849$	−44.0288	−1.51107
$850$	0	0
$851$	42.4317	1.45454
$852$	43.0222	1.47392
$853$	−13.8622	−0.474634	−0.237317	−	0.971432i	$-0.576268\pi$
$853$	−13.8622	−0.474634	−0.237317	+	0.971432i	$0.576268\pi$
$854$	17.4876	0.598415
$855$	0	0
$856$	53.6211	1.83273
$857$	−9.21635	−0.314825	−0.157412	−	0.987533i	$-0.550315\pi$
$857$	−9.21635	−0.314825	−0.157412	+	0.987533i	$0.550315\pi$
$858$	17.1184	0.584412
$859$	3.01479	0.102863	0.0514317	−	0.998677i	$-0.483622\pi$
$859$	3.01479	0.102863	0.0514317	+	0.998677i	$0.483622\pi$
$860$	0	0
$861$	−12.0436	−0.410446
$862$	−43.8574	−1.49379
$863$	12.0474	0.410097	0.205049	−	0.978752i	$-0.434265\pi$
$863$	12.0474	0.410097	0.205049	+	0.978752i	$0.434265\pi$
$864$	−0.0877907	−0.00298670
$865$	0	0
$866$	68.5087	2.32802
$867$	28.9811	0.984251
$868$	−20.7300	−0.703621
$869$	−63.3424	−2.14874
$870$	0	0
$871$	11.2782	0.382149
$872$	−67.7579	−2.29457
$873$	0.418572	0.0141665
$874$	27.1143	0.917156
$875$	0	0
$876$	−57.9232	−1.95704
$877$	−4.61405	−0.155805	−0.0779027	−	0.996961i	$-0.524822\pi$
$877$	−4.61405	−0.155805	−0.0779027	+	0.996961i	$0.524822\pi$
$878$	18.0122	0.607881
$879$	−55.6896	−1.87836
$880$	0	0
$881$	−10.7220	−0.361234	−0.180617	−	0.983553i	$-0.557809\pi$
$881$	−10.7220	−0.361234	−0.180617	+	0.983553i	$0.557809\pi$
$882$	−0.165587	−0.00557560
$883$	−25.9098	−0.871933	−0.435966	−	0.899963i	$-0.643593\pi$
$883$	−25.9098	−0.871933	−0.435966	+	0.899963i	$0.643593\pi$
$884$	1.10179	0.0370571
$885$	0	0
$886$	−77.8071	−2.61398
$887$	−36.4014	−1.22224	−0.611120	−	0.791538i	$-0.709281\pi$
$887$	−36.4014	−1.22224	−0.611120	+	0.791538i	$0.709281\pi$
$888$	48.9633	1.64310
$889$	−17.1320	−0.574590
$890$	0	0
$891$	35.8885	1.20231
$892$	−40.1192	−1.34329
$893$	−10.3790	−0.347318
$894$	23.7681	0.794926
$895$	0	0
$896$	19.6098	0.655118
$897$	12.4370	0.415259
$898$	79.9739	2.66876
$899$	−29.0890	−0.970173
$900$	0	0
$901$	−2.15712	−0.0718642
$902$	70.3070	2.34097
$903$	−2.98103	−0.0992026
$904$	−37.1537	−1.23571
$905$	0	0
$906$	−37.3690	−1.24150
$907$	−38.7724	−1.28742	−0.643708	−	0.765271i	$-0.722605\pi$
$907$	−38.7724	−1.28742	−0.643708	+	0.765271i	$0.722605\pi$
$908$	72.4060	2.40288
$909$	−0.193556	−0.00641986
$910$	0	0
$911$	9.72950	0.322353	0.161176	−	0.986926i	$-0.448471\pi$
$911$	9.72950	0.322353	0.161176	+	0.986926i	$0.448471\pi$
$912$	10.4145	0.344859
$913$	−60.2453	−1.99383
$914$	−45.9349	−1.51939
$915$	0	0
$916$	−48.7629	−1.61117
$917$	4.49291	0.148369
$918$	3.54526	0.117011
$919$	58.5560	1.93159	0.965793	−	0.259316i	$-0.0834971\pi$
$919$	58.5560	1.93159	0.965793	+	0.259316i	$0.0834971\pi$
$920$	0	0
$921$	−42.2663	−1.39272
$922$	17.1398	0.564470
$923$	−6.28358	−0.206827
$924$	−27.9462	−0.919364
$925$	0	0
$926$	18.6033	0.611341
$927$	0.0714689	0.00234735
$928$	0.0937655	0.00307800
$929$	−45.6799	−1.49871	−0.749354	−	0.662170i	$-0.769636\pi$
$929$	−45.6799	−1.49871	−0.749354	+	0.662170i	$0.769636\pi$
$930$	0	0
$931$	−1.52434	−0.0499581
$932$	5.58629	0.182985
$933$	38.9342	1.27465
$934$	−12.4246	−0.406547
$935$	0	0
$936$	−0.330891	−0.0108155
$937$	14.8314	0.484520	0.242260	−	0.970211i	$-0.422111\pi$
$937$	14.8314	0.484520	0.242260	+	0.970211i	$0.422111\pi$
$938$	−27.6220	−0.901891
$939$	38.3894	1.25279
$940$	0	0
$941$	−34.9206	−1.13838	−0.569189	−	0.822206i	$-0.692743\pi$
$941$	−34.9206	−1.13838	−0.569189	+	0.822206i	$0.692743\pi$
$942$	36.8072	1.19924
$943$	51.0801	1.66340
$944$	−44.1880	−1.43820
$945$	0	0
$946$	17.4023	0.565799
$947$	−16.6933	−0.542459	−0.271229	−	0.962515i	$-0.587430\pi$
$947$	−16.6933	−0.542459	−0.271229	+	0.962515i	$0.587430\pi$
$948$	−106.253	−3.45094
$949$	8.45993	0.274621
$950$	0	0
$951$	40.8779	1.32556
$952$	−1.34864	−0.0437097
$953$	30.1853	0.977798	0.488899	−	0.872341i	$-0.337399\pi$
$953$	30.1853	0.977798	0.488899	+	0.872341i	$0.337399\pi$
$954$	1.29622	0.0419666
$955$	0	0
$956$	81.5737	2.63828
$957$	−39.2152	−1.26765
$958$	52.0309	1.68104
$959$	−7.85643	−0.253697
$960$	0	0
$961$	−4.11887	−0.132867
$962$	−14.3087	−0.461331
$963$	−0.740755	−0.0238705
$964$	7.49203	0.241302
$965$	0	0
$966$	−30.4599	−0.980032
$967$	−17.2203	−0.553768	−0.276884	−	0.960903i	$-0.589302\pi$
$967$	−17.2203	−0.553768	−0.276884	+	0.960903i	$0.589302\pi$
$968$	27.7008	0.890336
$969$	0.719307	0.0231075
$970$	0	0
$971$	−44.8516	−1.43936	−0.719678	−	0.694308i	$-0.755711\pi$
$971$	−44.8516	−1.43936	−0.719678	+	0.694308i	$0.755711\pi$
$972$	−2.80876	−0.0900911
$973$	−3.14968	−0.100974
$974$	−63.5407	−2.03598
$975$	0	0
$976$	−28.4882	−0.911884
$977$	25.9934	0.831601	0.415800	−	0.909456i	$-0.363501\pi$
$977$	25.9934	0.831601	0.415800	+	0.909456i	$0.363501\pi$
$978$	57.3906	1.83515
$979$	45.7053	1.46075
$980$	0	0
$981$	0.936050	0.0298858
$982$	−3.46329	−0.110518
$983$	−1.87284	−0.0597345	−0.0298672	−	0.999554i	$-0.509508\pi$
$983$	−1.87284	−0.0597345	−0.0298672	+	0.999554i	$0.509508\pi$
$984$	58.9428	1.87903
$985$	0	0
$986$	−3.78654	−0.120588
$987$	11.6596	0.371129
$988$	−6.09474	−0.193900
$989$	12.6433	0.402033
$990$	0	0
$991$	52.3129	1.66177	0.830887	−	0.556441i	$-0.187834\pi$
$991$	52.3129	1.66177	0.830887	+	0.556441i	$0.187834\pi$
$992$	−0.0866486	−0.00275110
$993$	46.1873	1.46571
$994$	15.3894	0.488122
$995$	0	0
$996$	−101.058	−3.20214
$997$	12.9290	0.409465	0.204732	−	0.978818i	$-0.434368\pi$
$997$	12.9290	0.409465	0.204732	+	0.978818i	$0.434368\pi$
$998$	−10.5865	−0.335109
$999$	−30.6901	−0.970990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2275.2.a.y.1.7		7
5.2	odd	4		455.2.c.b.274.14	yes	14
5.3	odd	4		455.2.c.b.274.1	✓	14
5.4	even	2		2275.2.a.w.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.c.b.274.1	✓	14	5.3	odd	4
455.2.c.b.274.14	yes	14	5.2	odd	4
2275.2.a.w.1.1		7	5.4	even	2
2275.2.a.y.1.7		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44914 q^{2} -1.71242 q^{3} +3.99829 q^{4} -4.19396 q^{6} -1.00000 q^{7} +4.89410 q^{8} -0.0676101 q^{9} -4.08167 q^{11} -6.84677 q^{12} +1.00000 q^{13} -2.44914 q^{14} +3.98976 q^{16} +0.275564 q^{17} -0.165587 q^{18} -1.52434 q^{19} +1.71242 q^{21} -9.99659 q^{22} -7.26280 q^{23} -8.38077 q^{24} +2.44914 q^{26} +5.25304 q^{27} -3.99829 q^{28} -5.61055 q^{29} +5.18470 q^{31} -0.0167124 q^{32} +6.98954 q^{33} +0.674896 q^{34} -0.270325 q^{36} -5.84234 q^{37} -3.73331 q^{38} -1.71242 q^{39} -7.03310 q^{41} +4.19396 q^{42} -1.74083 q^{43} -16.3197 q^{44} -17.7876 q^{46} +6.80884 q^{47} -6.83216 q^{48} +1.00000 q^{49} -0.471882 q^{51} +3.99829 q^{52} -7.82802 q^{53} +12.8654 q^{54} -4.89410 q^{56} +2.61031 q^{57} -13.7410 q^{58} -11.0754 q^{59} -7.14032 q^{61} +12.6981 q^{62} +0.0676101 q^{63} -8.02046 q^{64} +17.1184 q^{66} +11.2782 q^{67} +1.10179 q^{68} +12.4370 q^{69} -6.28358 q^{71} -0.330891 q^{72} +8.45993 q^{73} -14.3087 q^{74} -6.09474 q^{76} +4.08167 q^{77} -4.19396 q^{78} +15.5187 q^{79} -8.79260 q^{81} -17.2251 q^{82} +14.7600 q^{83} +6.84677 q^{84} -4.26353 q^{86} +9.60763 q^{87} -19.9761 q^{88} -11.1977 q^{89} -1.00000 q^{91} -29.0388 q^{92} -8.87840 q^{93} +16.6758 q^{94} +0.0286186 q^{96} -6.19096 q^{97} +2.44914 q^{98} +0.275962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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