LMFDB - Embedded newform 2275.2.a.y.1.4

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			1.4
Root			$-0.340436$ 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)

$f(q)$	$=$	\(q-0.340436 q^{2} +2.66843 q^{3} -1.88410 q^{4} -0.908430 q^{6} -1.00000 q^{7} +1.32229 q^{8} +4.12053 q^{9} -5.19395 q^{11} -5.02760 q^{12} +1.00000 q^{13} +0.340436 q^{14} +3.31805 q^{16} -5.41029 q^{17} -1.40278 q^{18} +3.23870 q^{19} -2.66843 q^{21} +1.76821 q^{22} -0.617552 q^{23} +3.52844 q^{24} -0.340436 q^{26} +2.99006 q^{27} +1.88410 q^{28} -6.06550 q^{29} +6.67058 q^{31} -3.77416 q^{32} -13.8597 q^{33} +1.84186 q^{34} -7.76350 q^{36} +3.82885 q^{37} -1.10257 q^{38} +2.66843 q^{39} -9.06961 q^{41} +0.908430 q^{42} -5.70574 q^{43} +9.78594 q^{44} +0.210237 q^{46} -9.40633 q^{47} +8.85400 q^{48} +1.00000 q^{49} -14.4370 q^{51} -1.88410 q^{52} -7.97998 q^{53} -1.01792 q^{54} -1.32229 q^{56} +8.64224 q^{57} +2.06491 q^{58} -11.4360 q^{59} +3.03775 q^{61} -2.27090 q^{62} -4.12053 q^{63} -5.35125 q^{64} +4.71834 q^{66} -12.8492 q^{67} +10.1935 q^{68} -1.64790 q^{69} +8.06136 q^{71} +5.44853 q^{72} +8.32815 q^{73} -1.30348 q^{74} -6.10204 q^{76} +5.19395 q^{77} -0.908430 q^{78} -4.40268 q^{79} -4.38283 q^{81} +3.08762 q^{82} +9.35502 q^{83} +5.02760 q^{84} +1.94244 q^{86} -16.1854 q^{87} -6.86790 q^{88} -13.7193 q^{89} -1.00000 q^{91} +1.16353 q^{92} +17.8000 q^{93} +3.20225 q^{94} -10.0711 q^{96} +17.3085 q^{97} -0.340436 q^{98} -21.4018 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.340436	−0.240724	−0.120362	−	0.992730i	$-0.538406\pi$
$2$	−0.340436	−0.240724	−0.120362	+	0.992730i	$0.538406\pi$
$3$	2.66843	1.54062	0.770310	−	0.637670i	$-0.220101\pi$
$3$	2.66843	1.54062	0.770310	+	0.637670i	$0.220101\pi$
$4$	−1.88410	−0.942052
$5$	0	0
$5$	0	0
$6$	−0.908430	−0.370865
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.32229	0.467499
$9$	4.12053	1.37351
$10$	0	0
$11$	−5.19395	−1.56604	−0.783018	−	0.621999i	$-0.786321\pi$
$11$	−5.19395	−1.56604	−0.783018	+	0.621999i	$0.786321\pi$
$12$	−5.02760	−1.45134
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.340436	0.0909853
$15$	0	0
$16$	3.31805	0.829513
$17$	−5.41029	−1.31219	−0.656094	−	0.754680i	$-0.727792\pi$
$17$	−5.41029	−1.31219	−0.656094	+	0.754680i	$0.727792\pi$
$18$	−1.40278	−0.330637
$19$	3.23870	0.743008	0.371504	−	0.928431i	$-0.378842\pi$
$19$	3.23870	0.743008	0.371504	+	0.928431i	$0.378842\pi$
$20$	0	0
$21$	−2.66843	−0.582300
$22$	1.76821	0.376983
$23$	−0.617552	−0.128768	−0.0643842	−	0.997925i	$-0.520508\pi$
$23$	−0.617552	−0.128768	−0.0643842	+	0.997925i	$0.520508\pi$
$24$	3.52844	0.720239
$25$	0	0
$26$	−0.340436	−0.0667649
$27$	2.99006	0.575437
$28$	1.88410	0.356062
$29$	−6.06550	−1.12633	−0.563167	−	0.826343i	$-0.690417\pi$
$29$	−6.06550	−1.12633	−0.563167	+	0.826343i	$0.690417\pi$
$30$	0	0
$31$	6.67058	1.19807	0.599035	−	0.800723i	$-0.295551\pi$
$31$	6.67058	1.19807	0.599035	+	0.800723i	$0.295551\pi$
$32$	−3.77416	−0.667183
$33$	−13.8597	−2.41267
$34$	1.84186	0.315876
$35$	0	0
$36$	−7.76350	−1.29392
$37$	3.82885	0.629459	0.314729	−	0.949181i	$-0.398086\pi$
$37$	3.82885	0.629459	0.314729	+	0.949181i	$0.398086\pi$
$38$	−1.10257	−0.178860
$39$	2.66843	0.427291
$40$	0	0
$41$	−9.06961	−1.41644	−0.708218	−	0.705994i	$-0.750501\pi$
$41$	−9.06961	−1.41644	−0.708218	+	0.705994i	$0.750501\pi$
$42$	0.908430	0.140174
$43$	−5.70574	−0.870117	−0.435059	−	0.900402i	$-0.643272\pi$
$43$	−5.70574	−0.870117	−0.435059	+	0.900402i	$0.643272\pi$
$44$	9.78594	1.47529
$45$	0	0
$46$	0.210237	0.0309977
$47$	−9.40633	−1.37205	−0.686027	−	0.727576i	$-0.740647\pi$
$47$	−9.40633	−1.37205	−0.686027	+	0.727576i	$0.740647\pi$
$48$	8.85400	1.27796
$49$	1.00000	0.142857
$50$	0	0
$51$	−14.4370	−2.02158
$52$	−1.88410	−0.261278
$53$	−7.97998	−1.09613	−0.548067	−	0.836434i	$-0.684636\pi$
$53$	−7.97998	−1.09613	−0.548067	+	0.836434i	$0.684636\pi$
$54$	−1.01792	−0.138522
$55$	0	0
$56$	−1.32229	−0.176698
$57$	8.64224	1.14469
$58$	2.06491	0.271136
$59$	−11.4360	−1.48884	−0.744422	−	0.667709i	$-0.767275\pi$
$59$	−11.4360	−1.48884	−0.744422	+	0.667709i	$0.767275\pi$
$60$	0	0
$61$	3.03775	0.388944	0.194472	−	0.980908i	$-0.437701\pi$
$61$	3.03775	0.388944	0.194472	+	0.980908i	$0.437701\pi$
$62$	−2.27090	−0.288405
$63$	−4.12053	−0.519138
$64$	−5.35125	−0.668906
$65$	0	0
$66$	4.71834	0.580788
$67$	−12.8492	−1.56978	−0.784892	−	0.619632i	$-0.787282\pi$
$67$	−12.8492	−1.56978	−0.784892	+	0.619632i	$0.787282\pi$
$68$	10.1935	1.23615
$69$	−1.64790	−0.198383
$70$	0	0
$71$	8.06136	0.956708	0.478354	−	0.878167i	$-0.341234\pi$
$71$	8.06136	0.956708	0.478354	+	0.878167i	$0.341234\pi$
$72$	5.44853	0.642115
$73$	8.32815	0.974736	0.487368	−	0.873197i	$-0.337957\pi$
$73$	8.32815	0.974736	0.487368	+	0.873197i	$0.337957\pi$
$74$	−1.30348	−0.151526
$75$	0	0
$76$	−6.10204	−0.699952
$77$	5.19395	0.591906
$78$	−0.908430	−0.102859
$79$	−4.40268	−0.495340	−0.247670	−	0.968844i	$-0.579665\pi$
$79$	−4.40268	−0.495340	−0.247670	+	0.968844i	$0.579665\pi$
$80$	0	0
$81$	−4.38283	−0.486981
$82$	3.08762	0.340971
$83$	9.35502	1.02685	0.513423	−	0.858136i	$-0.328377\pi$
$83$	9.35502	1.02685	0.513423	+	0.858136i	$0.328377\pi$
$84$	5.02760	0.548556
$85$	0	0
$86$	1.94244	0.209459
$87$	−16.1854	−1.73525
$88$	−6.86790	−0.732120
$89$	−13.7193	−1.45425	−0.727123	−	0.686507i	$-0.759143\pi$
$89$	−13.7193	−1.45425	−0.727123	+	0.686507i	$0.759143\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	1.16353	0.121307
$93$	17.8000	1.84577
$94$	3.20225	0.330287
$95$	0	0
$96$	−10.0711	−1.02788
$97$	17.3085	1.75742	0.878708	−	0.477359i	$-0.158406\pi$
$97$	17.3085	1.75742	0.878708	+	0.477359i	$0.158406\pi$
$98$	−0.340436	−0.0343892
$99$	−21.4018	−2.15096
$100$	0	0
$101$	−12.2530	−1.21922	−0.609608	−	0.792703i	$-0.708673\pi$
$101$	−12.2530	−1.21922	−0.609608	+	0.792703i	$0.708673\pi$
$102$	4.91486	0.486644
$103$	2.50999	0.247316	0.123658	−	0.992325i	$-0.460537\pi$
$103$	2.50999	0.247316	0.123658	+	0.992325i	$0.460537\pi$
$104$	1.32229	0.129661
$105$	0	0
$106$	2.71667	0.263866
$107$	4.70287	0.454643	0.227322	−	0.973820i	$-0.427003\pi$
$107$	4.70287	0.454643	0.227322	+	0.973820i	$0.427003\pi$
$108$	−5.63358	−0.542091
$109$	3.71611	0.355939	0.177969	−	0.984036i	$-0.443047\pi$
$109$	3.71611	0.355939	0.177969	+	0.984036i	$0.443047\pi$
$110$	0	0
$111$	10.2170	0.969757
$112$	−3.31805	−0.313527
$113$	−1.76380	−0.165924	−0.0829622	−	0.996553i	$-0.526438\pi$
$113$	−1.76380	−0.165924	−0.0829622	+	0.996553i	$0.526438\pi$
$114$	−2.94213	−0.275556
$115$	0	0
$116$	11.4280	1.06107
$117$	4.12053	0.380943
$118$	3.89323	0.358401
$119$	5.41029	0.495960
$120$	0	0
$121$	15.9771	1.45247
$122$	−1.03416	−0.0936282
$123$	−24.2016	−2.18219
$124$	−12.5681	−1.12864
$125$	0	0
$126$	1.40278	0.124969
$127$	−17.4542	−1.54881	−0.774404	−	0.632691i	$-0.781950\pi$
$127$	−17.4542	−1.54881	−0.774404	+	0.632691i	$0.781950\pi$
$128$	9.37007	0.828205
$129$	−15.2254	−1.34052
$130$	0	0
$131$	−1.28810	−0.112542	−0.0562709	−	0.998416i	$-0.517921\pi$
$131$	−1.28810	−0.112542	−0.0562709	+	0.998416i	$0.517921\pi$
$132$	26.1131	2.27286
$133$	−3.23870	−0.280831
$134$	4.37434	0.377885
$135$	0	0
$136$	−7.15396	−0.613447
$137$	16.3542	1.39723	0.698615	−	0.715498i	$-0.253800\pi$
$137$	16.3542	1.39723	0.698615	+	0.715498i	$0.253800\pi$
$138$	0.561003	0.0477557
$139$	−15.9181	−1.35016	−0.675079	−	0.737746i	$-0.735890\pi$
$139$	−15.9181	−1.35016	−0.675079	+	0.737746i	$0.735890\pi$
$140$	0	0
$141$	−25.1002	−2.11381
$142$	−2.74438	−0.230303
$143$	−5.19395	−0.434340
$144$	13.6721	1.13934
$145$	0	0
$146$	−2.83520	−0.234643
$147$	2.66843	0.220089
$148$	−7.21394	−0.592983
$149$	−10.6535	−0.872767	−0.436384	−	0.899761i	$-0.643741\pi$
$149$	−10.6535	−0.872767	−0.436384	+	0.899761i	$0.643741\pi$
$150$	0	0
$151$	−2.63255	−0.214234	−0.107117	−	0.994246i	$-0.534162\pi$
$151$	−2.63255	−0.214234	−0.107117	+	0.994246i	$0.534162\pi$
$152$	4.28249	0.347356
$153$	−22.2932	−1.80230
$154$	−1.76821	−0.142486
$155$	0	0
$156$	−5.02760	−0.402530
$157$	−4.43438	−0.353902	−0.176951	−	0.984220i	$-0.556623\pi$
$157$	−4.43438	−0.353902	−0.176951	+	0.984220i	$0.556623\pi$
$158$	1.49883	0.119240
$159$	−21.2940	−1.68873
$160$	0	0
$161$	0.617552	0.0486699
$162$	1.49207	0.117228
$163$	12.6193	0.988418	0.494209	−	0.869343i	$-0.335458\pi$
$163$	12.6193	0.988418	0.494209	+	0.869343i	$0.335458\pi$
$164$	17.0881	1.33436
$165$	0	0
$166$	−3.18478	−0.247187
$167$	6.41659	0.496531	0.248265	−	0.968692i	$-0.420140\pi$
$167$	6.41659	0.496531	0.248265	+	0.968692i	$0.420140\pi$
$168$	−3.52844	−0.272225
$169$	1.00000	0.0769231
$170$	0	0
$171$	13.3451	1.02053
$172$	10.7502	0.819696
$173$	0.317546	0.0241425	0.0120713	−	0.999927i	$-0.496158\pi$
$173$	0.317546	0.0241425	0.0120713	+	0.999927i	$0.496158\pi$
$174$	5.51008	0.417718
$175$	0	0
$176$	−17.2338	−1.29905
$177$	−30.5163	−2.29374
$178$	4.67055	0.350073
$179$	11.9160	0.890643	0.445322	−	0.895371i	$-0.353089\pi$
$179$	11.9160	0.890643	0.445322	+	0.895371i	$0.353089\pi$
$180$	0	0
$181$	−18.0907	−1.34467	−0.672336	−	0.740246i	$-0.734709\pi$
$181$	−18.0907	−1.34467	−0.672336	+	0.740246i	$0.734709\pi$
$182$	0.340436	0.0252348
$183$	8.10602	0.599214
$184$	−0.816581	−0.0601992
$185$	0	0
$186$	−6.05975	−0.444322
$187$	28.1008	2.05493
$188$	17.7225	1.29255
$189$	−2.99006	−0.217495
$190$	0	0
$191$	22.6671	1.64013	0.820067	−	0.572267i	$-0.193936\pi$
$191$	22.6671	1.64013	0.820067	+	0.572267i	$0.193936\pi$
$192$	−14.2794	−1.03053
$193$	7.31897	0.526831	0.263416	−	0.964682i	$-0.415151\pi$
$193$	7.31897	0.526831	0.263416	+	0.964682i	$0.415151\pi$
$194$	−5.89245	−0.423053
$195$	0	0
$196$	−1.88410	−0.134579
$197$	5.49337	0.391387	0.195693	−	0.980665i	$-0.437304\pi$
$197$	5.49337	0.391387	0.195693	+	0.980665i	$0.437304\pi$
$198$	7.28595	0.517790
$199$	17.2940	1.22594	0.612971	−	0.790106i	$-0.289974\pi$
$199$	17.2940	1.22594	0.612971	+	0.790106i	$0.289974\pi$
$200$	0	0
$201$	−34.2873	−2.41844
$202$	4.17135	0.293495
$203$	6.06550	0.425715
$204$	27.2008	1.90443
$205$	0	0
$206$	−0.854489	−0.0595351
$207$	−2.54464	−0.176865
$208$	3.31805	0.230066
$209$	−16.8216	−1.16358
$210$	0	0
$211$	−15.7992	−1.08766	−0.543830	−	0.839196i	$-0.683026\pi$
$211$	−15.7992	−1.08766	−0.543830	+	0.839196i	$0.683026\pi$
$212$	15.0351	1.03262
$213$	21.5112	1.47392
$214$	−1.60102	−0.109444
$215$	0	0
$216$	3.95372	0.269016
$217$	−6.67058	−0.452828
$218$	−1.26510	−0.0856832
$219$	22.2231	1.50170
$220$	0	0
$221$	−5.41029	−0.363935
$222$	−3.47824	−0.233444
$223$	−10.9332	−0.732143	−0.366072	−	0.930587i	$-0.619297\pi$
$223$	−10.9332	−0.732143	−0.366072	+	0.930587i	$0.619297\pi$
$224$	3.77416	0.252172
$225$	0	0
$226$	0.600461	0.0399420
$227$	18.9681	1.25896	0.629479	−	0.777018i	$-0.283269\pi$
$227$	18.9681	1.25896	0.629479	+	0.777018i	$0.283269\pi$
$228$	−16.2829	−1.07836
$229$	2.42775	0.160430	0.0802152	−	0.996778i	$-0.474439\pi$
$229$	2.42775	0.160430	0.0802152	+	0.996778i	$0.474439\pi$
$230$	0	0
$231$	13.8597	0.911902
$232$	−8.02034	−0.526561
$233$	−7.37746	−0.483313	−0.241657	−	0.970362i	$-0.577691\pi$
$233$	−7.37746	−0.483313	−0.241657	+	0.970362i	$0.577691\pi$
$234$	−1.40278	−0.0917023
$235$	0	0
$236$	21.5467	1.40257
$237$	−11.7483	−0.763131
$238$	−1.84186	−0.119390
$239$	−5.80498	−0.375493	−0.187746	−	0.982218i	$-0.560118\pi$
$239$	−5.80498	−0.375493	−0.187746	+	0.982218i	$0.560118\pi$
$240$	0	0
$241$	−2.46503	−0.158786	−0.0793932	−	0.996843i	$-0.525298\pi$
$241$	−2.46503	−0.158786	−0.0793932	+	0.996843i	$0.525298\pi$
$242$	−5.43919	−0.349644
$243$	−20.6654	−1.32569
$244$	−5.72343	−0.366405
$245$	0	0
$246$	8.23911	0.525306
$247$	3.23870	0.206073
$248$	8.82042	0.560097
$249$	24.9632	1.58198
$250$	0	0
$251$	−2.89196	−0.182539	−0.0912695	−	0.995826i	$-0.529092\pi$
$251$	−2.89196	−0.182539	−0.0912695	+	0.995826i	$0.529092\pi$
$252$	7.76350	0.489055
$253$	3.20754	0.201656
$254$	5.94203	0.372836
$255$	0	0
$256$	7.51259	0.469537
$257$	−8.67330	−0.541026	−0.270513	−	0.962716i	$-0.587193\pi$
$257$	−8.67330	−0.541026	−0.270513	+	0.962716i	$0.587193\pi$
$258$	5.18326	0.322696
$259$	−3.82885	−0.237913
$260$	0	0
$261$	−24.9931	−1.54703
$262$	0.438515	0.0270916
$263$	25.2854	1.55916	0.779582	−	0.626300i	$-0.215431\pi$
$263$	25.2854	1.55916	0.779582	+	0.626300i	$0.215431\pi$
$264$	−18.3265	−1.12792
$265$	0	0
$266$	1.10257	0.0676028
$267$	−36.6091	−2.24044
$268$	24.2093	1.47882
$269$	19.7623	1.20493	0.602465	−	0.798146i	$-0.294185\pi$
$269$	19.7623	1.20493	0.602465	+	0.798146i	$0.294185\pi$
$270$	0	0
$271$	9.64097	0.585647	0.292824	−	0.956166i	$-0.405405\pi$
$271$	9.64097	0.585647	0.292824	+	0.956166i	$0.405405\pi$
$272$	−17.9516	−1.08848
$273$	−2.66843	−0.161501
$274$	−5.56754	−0.336347
$275$	0	0
$276$	3.10481	0.186887
$277$	20.0897	1.20707	0.603537	−	0.797335i	$-0.293758\pi$
$277$	20.0897	1.20707	0.603537	+	0.797335i	$0.293758\pi$
$278$	5.41910	0.325016
$279$	27.4863	1.64556
$280$	0	0
$281$	9.64647	0.575460	0.287730	−	0.957712i	$-0.407099\pi$
$281$	9.64647	0.575460	0.287730	+	0.957712i	$0.407099\pi$
$282$	8.54499	0.508847
$283$	−18.1449	−1.07860	−0.539302	−	0.842113i	$-0.681312\pi$
$283$	−18.1449	−1.07860	−0.539302	+	0.842113i	$0.681312\pi$
$284$	−15.1884	−0.901268
$285$	0	0
$286$	1.76821	0.104556
$287$	9.06961	0.535362
$288$	−15.5515	−0.916383
$289$	12.2712	0.721835
$290$	0	0
$291$	46.1867	2.70751
$292$	−15.6911	−0.918252
$293$	−25.2982	−1.47794	−0.738968	−	0.673741i	$-0.764686\pi$
$293$	−25.2982	−1.47794	−0.738968	+	0.673741i	$0.764686\pi$
$294$	−0.908430	−0.0529807
$295$	0	0
$296$	5.06284	0.294271
$297$	−15.5302	−0.901154
$298$	3.62683	0.210096
$299$	−0.617552	−0.0357140
$300$	0	0
$301$	5.70574	0.328873
$302$	0.896213	0.0515713
$303$	−32.6962	−1.87835
$304$	10.7462	0.616335
$305$	0	0
$306$	7.58942	0.433858
$307$	10.4717	0.597649	0.298825	−	0.954308i	$-0.403405\pi$
$307$	10.4717	0.597649	0.298825	+	0.954308i	$0.403405\pi$
$308$	−9.78594	−0.557606
$309$	6.69773	0.381021
$310$	0	0
$311$	−11.5147	−0.652938	−0.326469	−	0.945208i	$-0.605859\pi$
$311$	−11.5147	−0.652938	−0.326469	+	0.945208i	$0.605859\pi$
$312$	3.52844	0.199758
$313$	−0.211335	−0.0119454	−0.00597269	−	0.999982i	$-0.501901\pi$
$313$	−0.211335	−0.0119454	−0.00597269	+	0.999982i	$0.501901\pi$
$314$	1.50962	0.0851928
$315$	0	0
$316$	8.29510	0.466636
$317$	5.97546	0.335615	0.167808	−	0.985820i	$-0.446331\pi$
$317$	5.97546	0.335615	0.167808	+	0.985820i	$0.446331\pi$
$318$	7.24925	0.406518
$319$	31.5039	1.76388
$320$	0	0
$321$	12.5493	0.700433
$322$	−0.210237	−0.0117160
$323$	−17.5223	−0.974965
$324$	8.25770	0.458761
$325$	0	0
$326$	−4.29606	−0.237936
$327$	9.91619	0.548366
$328$	−11.9926	−0.662183
$329$	9.40633	0.518588
$330$	0	0
$331$	13.1245	0.721388	0.360694	−	0.932684i	$-0.382540\pi$
$331$	13.1245	0.721388	0.360694	+	0.932684i	$0.382540\pi$
$332$	−17.6258	−0.967342
$333$	15.7769	0.864568
$334$	−2.18444	−0.119527
$335$	0	0
$336$	−8.85400	−0.483025
$337$	10.0655	0.548304	0.274152	−	0.961686i	$-0.411603\pi$
$337$	10.0655	0.548304	0.274152	+	0.961686i	$0.411603\pi$
$338$	−0.340436	−0.0185173
$339$	−4.70658	−0.255626
$340$	0	0
$341$	−34.6467	−1.87622
$342$	−4.54316	−0.245666
$343$	−1.00000	−0.0539949
$344$	−7.54463	−0.406779
$345$	0	0
$346$	−0.108104	−0.00581170
$347$	−5.21700	−0.280063	−0.140031	−	0.990147i	$-0.544720\pi$
$347$	−5.21700	−0.280063	−0.140031	+	0.990147i	$0.544720\pi$
$348$	30.4949	1.63470
$349$	−27.2493	−1.45862	−0.729312	−	0.684181i	$-0.760160\pi$
$349$	−27.2493	−1.45862	−0.729312	+	0.684181i	$0.760160\pi$
$350$	0	0
$351$	2.99006	0.159597
$352$	19.6028	1.04483
$353$	−11.0785	−0.589651	−0.294825	−	0.955551i	$-0.595261\pi$
$353$	−11.0785	−0.589651	−0.294825	+	0.955551i	$0.595261\pi$
$354$	10.3888	0.552160
$355$	0	0
$356$	25.8487	1.36998
$357$	14.4370	0.764086
$358$	−4.05663	−0.214400
$359$	−4.18101	−0.220665	−0.110333	−	0.993895i	$-0.535192\pi$
$359$	−4.18101	−0.220665	−0.110333	+	0.993895i	$0.535192\pi$
$360$	0	0
$361$	−8.51085	−0.447939
$362$	6.15873	0.323696
$363$	42.6339	2.23770
$364$	1.88410	0.0987539
$365$	0	0
$366$	−2.75958	−0.144246
$367$	−18.7156	−0.976945	−0.488472	−	0.872579i	$-0.662446\pi$
$367$	−18.7156	−0.976945	−0.488472	+	0.872579i	$0.662446\pi$
$368$	−2.04907	−0.106815
$369$	−37.3716	−1.94549
$370$	0	0
$371$	7.97998	0.414300
$372$	−33.5370	−1.73881
$373$	−1.86241	−0.0964318	−0.0482159	−	0.998837i	$-0.515354\pi$
$373$	−1.86241	−0.0964318	−0.0482159	+	0.998837i	$0.515354\pi$
$374$	−9.56651	−0.494672
$375$	0	0
$376$	−12.4379	−0.641435
$377$	−6.06550	−0.312389
$378$	1.01792	0.0523563
$379$	−9.14419	−0.469706	−0.234853	−	0.972031i	$-0.575461\pi$
$379$	−9.14419	−0.469706	−0.234853	+	0.972031i	$0.575461\pi$
$380$	0	0
$381$	−46.5753	−2.38612
$382$	−7.71670	−0.394820
$383$	5.11756	0.261495	0.130748	−	0.991416i	$-0.458262\pi$
$383$	5.11756	0.261495	0.130748	+	0.991416i	$0.458262\pi$
$384$	25.0034	1.27595
$385$	0	0
$386$	−2.49164	−0.126821
$387$	−23.5107	−1.19511
$388$	−32.6111	−1.65558
$389$	−2.04130	−0.103498	−0.0517489	−	0.998660i	$-0.516480\pi$
$389$	−2.04130	−0.103498	−0.0517489	+	0.998660i	$0.516480\pi$
$390$	0	0
$391$	3.34113	0.168968
$392$	1.32229	0.0667856
$393$	−3.43721	−0.173384
$394$	−1.87014	−0.0942163
$395$	0	0
$396$	40.3233	2.02632
$397$	14.1198	0.708654	0.354327	−	0.935122i	$-0.384710\pi$
$397$	14.1198	0.708654	0.354327	+	0.935122i	$0.384710\pi$
$398$	−5.88751	−0.295114
$399$	−8.64224	−0.432653
$400$	0	0
$401$	25.4381	1.27032	0.635159	−	0.772381i	$-0.280935\pi$
$401$	25.4381	1.27032	0.635159	+	0.772381i	$0.280935\pi$
$402$	11.6726	0.582178
$403$	6.67058	0.332285
$404$	23.0859	1.14856
$405$	0	0
$406$	−2.06491	−0.102480
$407$	−19.8868	−0.985754
$408$	−19.0898	−0.945088
$409$	−24.5863	−1.21572	−0.607858	−	0.794046i	$-0.707971\pi$
$409$	−24.5863	−1.21572	−0.607858	+	0.794046i	$0.707971\pi$
$410$	0	0
$411$	43.6399	2.15260
$412$	−4.72908	−0.232985
$413$	11.4360	0.562730
$414$	0.866287	0.0425757
$415$	0	0
$416$	−3.77416	−0.185043
$417$	−42.4764	−2.08008
$418$	5.72669	0.280101
$419$	−29.8150	−1.45656	−0.728279	−	0.685281i	$-0.759680\pi$
$419$	−29.8150	−1.45656	−0.728279	+	0.685281i	$0.759680\pi$
$420$	0	0
$421$	10.7224	0.522579	0.261290	−	0.965260i	$-0.415852\pi$
$421$	10.7224	0.522579	0.261290	+	0.965260i	$0.415852\pi$
$422$	5.37860	0.261826
$423$	−38.7591	−1.88453
$424$	−10.5518	−0.512442
$425$	0	0
$426$	−7.32318	−0.354809
$427$	−3.03775	−0.147007
$428$	−8.86069	−0.428298
$429$	−13.8597	−0.669153
$430$	0	0
$431$	−8.48638	−0.408774	−0.204387	−	0.978890i	$-0.565520\pi$
$431$	−8.48638	−0.408774	−0.204387	+	0.978890i	$0.565520\pi$
$432$	9.92117	0.477332
$433$	−29.4000	−1.41287	−0.706437	−	0.707776i	$-0.749699\pi$
$433$	−29.4000	−1.41287	−0.706437	+	0.707776i	$0.749699\pi$
$434$	2.27090	0.109007
$435$	0	0
$436$	−7.00154	−0.335313
$437$	−2.00006	−0.0956760
$438$	−7.56554	−0.361496
$439$	−5.89751	−0.281473	−0.140736	−	0.990047i	$-0.544947\pi$
$439$	−5.89751	−0.281473	−0.140736	+	0.990047i	$0.544947\pi$
$440$	0	0
$441$	4.12053	0.196216
$442$	1.84186	0.0876081
$443$	10.1818	0.483754	0.241877	−	0.970307i	$-0.422237\pi$
$443$	10.1818	0.483754	0.241877	+	0.970307i	$0.422237\pi$
$444$	−19.2499	−0.913561
$445$	0	0
$446$	3.72206	0.176245
$447$	−28.4281	−1.34460
$448$	5.35125	0.252823
$449$	22.6117	1.06711	0.533556	−	0.845765i	$-0.320856\pi$
$449$	22.6117	1.06711	0.533556	+	0.845765i	$0.320856\pi$
$450$	0	0
$451$	47.1071	2.21819
$452$	3.32318	0.156309
$453$	−7.02477	−0.330053
$454$	−6.45742	−0.303062
$455$	0	0
$456$	11.4275	0.535143
$457$	−15.1884	−0.710483	−0.355241	−	0.934775i	$-0.615601\pi$
$457$	−15.1884	−0.710483	−0.355241	+	0.934775i	$0.615601\pi$
$458$	−0.826494	−0.0386195
$459$	−16.1771	−0.755081
$460$	0	0
$461$	−12.7941	−0.595881	−0.297941	−	0.954584i	$-0.596300\pi$
$461$	−12.7941	−0.595881	−0.297941	+	0.954584i	$0.596300\pi$
$462$	−4.71834	−0.219517
$463$	13.1270	0.610062	0.305031	−	0.952342i	$-0.401333\pi$
$463$	13.1270	0.610062	0.305031	+	0.952342i	$0.401333\pi$
$464$	−20.1256	−0.934310
$465$	0	0
$466$	2.51155	0.116345
$467$	39.4883	1.82730	0.913651	−	0.406500i	$-0.133251\pi$
$467$	39.4883	1.82730	0.913651	+	0.406500i	$0.133251\pi$
$468$	−7.76350	−0.358868
$469$	12.8492	0.593323
$470$	0	0
$471$	−11.8328	−0.545228
$472$	−15.1217	−0.696034
$473$	29.6353	1.36263
$474$	3.99953	0.183704
$475$	0	0
$476$	−10.1935	−0.467220
$477$	−32.8817	−1.50555
$478$	1.97622	0.0903903
$479$	−7.88009	−0.360051	−0.180025	−	0.983662i	$-0.557618\pi$
$479$	−7.88009	−0.360051	−0.180025	+	0.983662i	$0.557618\pi$
$480$	0	0
$481$	3.82885	0.174580
$482$	0.839183	0.0382238
$483$	1.64790	0.0749818
$484$	−30.1026	−1.36830
$485$	0	0
$486$	7.03526	0.319126
$487$	−11.1708	−0.506196	−0.253098	−	0.967441i	$-0.581450\pi$
$487$	−11.1708	−0.506196	−0.253098	+	0.967441i	$0.581450\pi$
$488$	4.01678	0.181831
$489$	33.6737	1.52278
$490$	0	0
$491$	−0.766090	−0.0345731	−0.0172866	−	0.999851i	$-0.505503\pi$
$491$	−0.766090	−0.0345731	−0.0172866	+	0.999851i	$0.505503\pi$
$492$	45.5984	2.05574
$493$	32.8161	1.47796
$494$	−1.10257	−0.0496069
$495$	0	0
$496$	22.1333	0.993816
$497$	−8.06136	−0.361602
$498$	−8.49837	−0.380821
$499$	22.7971	1.02054	0.510270	−	0.860014i	$-0.329545\pi$
$499$	22.7971	1.02054	0.510270	+	0.860014i	$0.329545\pi$
$500$	0	0
$501$	17.1222	0.764965
$502$	0.984527	0.0439416
$503$	34.1906	1.52448	0.762242	−	0.647292i	$-0.224099\pi$
$503$	34.1906	1.52448	0.762242	+	0.647292i	$0.224099\pi$
$504$	−5.44853	−0.242697
$505$	0	0
$506$	−1.09196	−0.0485435
$507$	2.66843	0.118509
$508$	32.8855	1.45906
$509$	9.99429	0.442989	0.221494	−	0.975162i	$-0.428907\pi$
$509$	9.99429	0.442989	0.221494	+	0.975162i	$0.428907\pi$
$510$	0	0
$511$	−8.32815	−0.368416
$512$	−21.2977	−0.941234
$513$	9.68389	0.427554
$514$	2.95270	0.130238
$515$	0	0
$516$	28.6862	1.26284
$517$	48.8560	2.14869
$518$	1.30348	0.0572715
$519$	0.847349	0.0371945
$520$	0	0
$521$	21.0167	0.920757	0.460379	−	0.887723i	$-0.347714\pi$
$521$	21.0167	0.920757	0.460379	+	0.887723i	$0.347714\pi$
$522$	8.50853	0.372408
$523$	−8.16277	−0.356933	−0.178466	−	0.983946i	$-0.557114\pi$
$523$	−8.16277	−0.356933	−0.178466	+	0.983946i	$0.557114\pi$
$524$	2.42691	0.106020
$525$	0	0
$526$	−8.60805	−0.375329
$527$	−36.0897	−1.57209
$528$	−45.9872	−2.00134
$529$	−22.6186	−0.983419
$530$	0	0
$531$	−47.1225	−2.04494
$532$	6.10204	0.264557
$533$	−9.06961	−0.392849
$534$	12.4631	0.539329
$535$	0	0
$536$	−16.9904	−0.733873
$537$	31.7970	1.37214
$538$	−6.72780	−0.290056
$539$	−5.19395	−0.223719
$540$	0	0
$541$	3.25598	0.139986	0.0699928	−	0.997547i	$-0.477702\pi$
$541$	3.25598	0.139986	0.0699928	+	0.997547i	$0.477702\pi$
$542$	−3.28213	−0.140980
$543$	−48.2739	−2.07163
$544$	20.4193	0.875470
$545$	0	0
$546$	0.908430	0.0388772
$547$	44.2346	1.89133	0.945667	−	0.325136i	$-0.105410\pi$
$547$	44.2346	1.89133	0.945667	+	0.325136i	$0.105410\pi$
$548$	−30.8129	−1.31626
$549$	12.5171	0.534218
$550$	0	0
$551$	−19.6443	−0.836876
$552$	−2.17899	−0.0927441
$553$	4.40268	0.187221
$554$	−6.83925	−0.290572
$555$	0	0
$556$	29.9914	1.27192
$557$	−30.7785	−1.30413	−0.652064	−	0.758164i	$-0.726097\pi$
$557$	−30.7785	−1.30413	−0.652064	+	0.758164i	$0.726097\pi$
$558$	−9.35732	−0.396127
$559$	−5.70574	−0.241327
$560$	0	0
$561$	74.9850	3.16587
$562$	−3.28400	−0.138527
$563$	22.9418	0.966881	0.483440	−	0.875377i	$-0.339387\pi$
$563$	22.9418	0.966881	0.483440	+	0.875377i	$0.339387\pi$
$564$	47.2913	1.99132
$565$	0	0
$566$	6.17718	0.259646
$567$	4.38283	0.184061
$568$	10.6594	0.447260
$569$	6.63172	0.278016	0.139008	−	0.990291i	$-0.455609\pi$
$569$	6.63172	0.278016	0.139008	+	0.990291i	$0.455609\pi$
$570$	0	0
$571$	−29.6156	−1.23937	−0.619686	−	0.784850i	$-0.712740\pi$
$571$	−29.6156	−1.23937	−0.619686	+	0.784850i	$0.712740\pi$
$572$	9.78594	0.409171
$573$	60.4856	2.52682
$574$	−3.08762	−0.128875
$575$	0	0
$576$	−22.0500	−0.918749
$577$	29.2894	1.21933	0.609667	−	0.792658i	$-0.291303\pi$
$577$	29.2894	1.21933	0.609667	+	0.792658i	$0.291303\pi$
$578$	−4.17755	−0.173763
$579$	19.5302	0.811647
$580$	0	0
$581$	−9.35502	−0.388111
$582$	−15.7236	−0.651764
$583$	41.4476	1.71659
$584$	11.0122	0.455689
$585$	0	0
$586$	8.61240	0.355775
$587$	4.60849	0.190213	0.0951064	−	0.995467i	$-0.469681\pi$
$587$	4.60849	0.190213	0.0951064	+	0.995467i	$0.469681\pi$
$588$	−5.02760	−0.207335
$589$	21.6040	0.890176
$590$	0	0
$591$	14.6587	0.602978
$592$	12.7043	0.522144
$593$	37.0023	1.51950	0.759751	−	0.650214i	$-0.225321\pi$
$593$	37.0023	1.51950	0.759751	+	0.650214i	$0.225321\pi$
$594$	5.28704	0.216930
$595$	0	0
$596$	20.0723	0.822192
$597$	46.1480	1.88871
$598$	0.210237	0.00859722
$599$	−33.8217	−1.38192	−0.690958	−	0.722895i	$-0.742811\pi$
$599$	−33.8217	−1.38192	−0.690958	+	0.722895i	$0.742811\pi$
$600$	0	0
$601$	−17.5180	−0.714573	−0.357286	−	0.933995i	$-0.616298\pi$
$601$	−17.5180	−0.714573	−0.357286	+	0.933995i	$0.616298\pi$
$602$	−1.94244	−0.0791679
$603$	−52.9457	−2.15611
$604$	4.95999	0.201819
$605$	0	0
$606$	11.1310	0.452165
$607$	−27.0434	−1.09766	−0.548828	−	0.835935i	$-0.684926\pi$
$607$	−27.0434	−1.09766	−0.548828	+	0.835935i	$0.684926\pi$
$608$	−12.2234	−0.495723
$609$	16.1854	0.655864
$610$	0	0
$611$	−9.40633	−0.380540
$612$	42.0028	1.69786
$613$	26.8767	1.08554	0.542771	−	0.839881i	$-0.317375\pi$
$613$	26.8767	1.08554	0.542771	+	0.839881i	$0.317375\pi$
$614$	−3.56493	−0.143869
$615$	0	0
$616$	6.86790	0.276716
$617$	0.400441	0.0161211	0.00806056	−	0.999968i	$-0.497434\pi$
$617$	0.400441	0.0161211	0.00806056	+	0.999968i	$0.497434\pi$
$618$	−2.28015	−0.0917210
$619$	35.9490	1.44491	0.722456	−	0.691417i	$-0.243013\pi$
$619$	35.9490	1.44491	0.722456	+	0.691417i	$0.243013\pi$
$620$	0	0
$621$	−1.84652	−0.0740981
$622$	3.92001	0.157178
$623$	13.7193	0.549654
$624$	8.85400	0.354444
$625$	0	0
$626$	0.0719461	0.00287554
$627$	−44.8874	−1.79263
$628$	8.35482	0.333394
$629$	−20.7152	−0.825968
$630$	0	0
$631$	−0.685851	−0.0273033	−0.0136517	−	0.999907i	$-0.504346\pi$
$631$	−0.685851	−0.0273033	−0.0136517	+	0.999907i	$0.504346\pi$
$632$	−5.82161	−0.231571
$633$	−42.1590	−1.67567
$634$	−2.03426	−0.0807908
$635$	0	0
$636$	40.1202	1.59087
$637$	1.00000	0.0396214
$638$	−10.7251	−0.424609
$639$	33.2171	1.31405
$640$	0	0
$641$	39.1581	1.54665	0.773326	−	0.634009i	$-0.218592\pi$
$641$	39.1581	1.54665	0.773326	+	0.634009i	$0.218592\pi$
$642$	−4.27222	−0.168611
$643$	41.9100	1.65277	0.826384	−	0.563108i	$-0.190394\pi$
$643$	41.9100	1.65277	0.826384	+	0.563108i	$0.190394\pi$
$644$	−1.16353	−0.0458496
$645$	0	0
$646$	5.96521	0.234698
$647$	16.6139	0.653159	0.326580	−	0.945170i	$-0.394104\pi$
$647$	16.6139	0.653159	0.326580	+	0.945170i	$0.394104\pi$
$648$	−5.79536	−0.227663
$649$	59.3982	2.33158
$650$	0	0
$651$	−17.8000	−0.697636
$652$	−23.7760	−0.931141
$653$	−3.83448	−0.150055	−0.0750274	−	0.997181i	$-0.523904\pi$
$653$	−3.83448	−0.150055	−0.0750274	+	0.997181i	$0.523904\pi$
$654$	−3.37582	−0.132005
$655$	0	0
$656$	−30.0935	−1.17495
$657$	34.3164	1.33881
$658$	−3.20225	−0.124837
$659$	−29.8578	−1.16310	−0.581548	−	0.813512i	$-0.697553\pi$
$659$	−29.8578	−1.16310	−0.581548	+	0.813512i	$0.697553\pi$
$660$	0	0
$661$	−19.7018	−0.766312	−0.383156	−	0.923684i	$-0.625163\pi$
$661$	−19.7018	−0.766312	−0.383156	+	0.923684i	$0.625163\pi$
$662$	−4.46805	−0.173656
$663$	−14.4370	−0.560686
$664$	12.3700	0.480050
$665$	0	0
$666$	−5.37101	−0.208123
$667$	3.74576	0.145036
$668$	−12.0895	−0.467758
$669$	−29.1746	−1.12795
$670$	0	0
$671$	−15.7779	−0.609099
$672$	10.0711	0.388501
$673$	41.8355	1.61264	0.806320	−	0.591480i	$-0.201456\pi$
$673$	41.8355	1.61264	0.806320	+	0.591480i	$0.201456\pi$
$674$	−3.42667	−0.131990
$675$	0	0
$676$	−1.88410	−0.0724655
$677$	10.5402	0.405092	0.202546	−	0.979273i	$-0.435078\pi$
$677$	10.5402	0.405092	0.202546	+	0.979273i	$0.435078\pi$
$678$	1.60229	0.0615355
$679$	−17.3085	−0.664241
$680$	0	0
$681$	50.6151	1.93957
$682$	11.7950	0.451652
$683$	−8.62006	−0.329837	−0.164919	−	0.986307i	$-0.552736\pi$
$683$	−8.62006	−0.329837	−0.164919	+	0.986307i	$0.552736\pi$
$684$	−25.1436	−0.961391
$685$	0	0
$686$	0.340436	0.0129979
$687$	6.47829	0.247162
$688$	−18.9320	−0.721774
$689$	−7.97998	−0.304013
$690$	0	0
$691$	−7.28104	−0.276984	−0.138492	−	0.990364i	$-0.544225\pi$
$691$	−7.28104	−0.276984	−0.138492	+	0.990364i	$0.544225\pi$
$692$	−0.598289	−0.0227435
$693$	21.4018	0.812988
$694$	1.77605	0.0674180
$695$	0	0
$696$	−21.4017	−0.811230
$697$	49.0692	1.85863
$698$	9.27665	0.351126
$699$	−19.6862	−0.744602
$700$	0	0
$701$	−43.8526	−1.65629	−0.828146	−	0.560513i	$-0.810604\pi$
$701$	−43.8526	−1.65629	−0.828146	+	0.560513i	$0.810604\pi$
$702$	−1.01792	−0.0384190
$703$	12.4005	0.467693
$704$	27.7941	1.04753
$705$	0	0
$706$	3.77153	0.141943
$707$	12.2530	0.460820
$708$	57.4958	2.16083
$709$	−46.2343	−1.73637	−0.868183	−	0.496243i	$-0.834712\pi$
$709$	−46.2343	−1.73637	−0.868183	+	0.496243i	$0.834712\pi$
$710$	0	0
$711$	−18.1414	−0.680355
$712$	−18.1409	−0.679859
$713$	−4.11943	−0.154274
$714$	−4.91486	−0.183934
$715$	0	0
$716$	−22.4510	−0.839032
$717$	−15.4902	−0.578491
$718$	1.42337	0.0531196
$719$	−44.1163	−1.64526	−0.822631	−	0.568576i	$-0.807494\pi$
$719$	−44.1163	−1.64526	−0.822631	+	0.568576i	$0.807494\pi$
$720$	0	0
$721$	−2.50999	−0.0934768
$722$	2.89740	0.107830
$723$	−6.57776	−0.244629
$724$	34.0848	1.26675
$725$	0	0
$726$	−14.5141	−0.538669
$727$	−18.1347	−0.672577	−0.336289	−	0.941759i	$-0.609172\pi$
$727$	−18.1347	−0.672577	−0.336289	+	0.941759i	$0.609172\pi$
$728$	−1.32229	−0.0490072
$729$	−41.9958	−1.55540
$730$	0	0
$731$	30.8697	1.14176
$732$	−15.2726	−0.564491
$733$	0.317708	0.0117348	0.00586740	−	0.999983i	$-0.498132\pi$
$733$	0.317708	0.0117348	0.00586740	+	0.999983i	$0.498132\pi$
$734$	6.37145	0.235174
$735$	0	0
$736$	2.33074	0.0859122
$737$	66.7383	2.45834
$738$	12.7226	0.468327
$739$	−16.4727	−0.605957	−0.302978	−	0.952997i	$-0.597981\pi$
$739$	−16.4727	−0.605957	−0.302978	+	0.952997i	$0.597981\pi$
$740$	0	0
$741$	8.64224	0.317481
$742$	−2.71667	−0.0997321
$743$	21.5420	0.790299	0.395149	−	0.918617i	$-0.370693\pi$
$743$	21.5420	0.790299	0.395149	+	0.918617i	$0.370693\pi$
$744$	23.5367	0.862897
$745$	0	0
$746$	0.634030	0.0232135
$747$	38.5476	1.41038
$748$	−52.9448	−1.93585
$749$	−4.70287	−0.171839
$750$	0	0
$751$	2.25542	0.0823013	0.0411507	−	0.999153i	$-0.486898\pi$
$751$	2.25542	0.0823013	0.0411507	+	0.999153i	$0.486898\pi$
$752$	−31.2107	−1.13814
$753$	−7.71700	−0.281223
$754$	2.06491	0.0751997
$755$	0	0
$756$	5.63358	0.204891
$757$	−42.6874	−1.55150	−0.775749	−	0.631041i	$-0.782628\pi$
$757$	−42.6874	−1.55150	−0.775749	+	0.631041i	$0.782628\pi$
$758$	3.11301	0.113070
$759$	8.55909	0.310675
$760$	0	0
$761$	22.4626	0.814268	0.407134	−	0.913368i	$-0.366528\pi$
$761$	22.4626	0.814268	0.407134	+	0.913368i	$0.366528\pi$
$762$	15.8559	0.574398
$763$	−3.71611	−0.134532
$764$	−42.7072	−1.54509
$765$	0	0
$766$	−1.74220	−0.0629483
$767$	−11.4360	−0.412931
$768$	20.0468	0.723377
$769$	11.0481	0.398404	0.199202	−	0.979958i	$-0.436165\pi$
$769$	11.0481	0.398404	0.199202	+	0.979958i	$0.436165\pi$
$770$	0	0
$771$	−23.1441	−0.833515
$772$	−13.7897	−0.496302
$773$	24.5304	0.882297	0.441148	−	0.897434i	$-0.354571\pi$
$773$	24.5304	0.882297	0.441148	+	0.897434i	$0.354571\pi$
$774$	8.00387	0.287693
$775$	0	0
$776$	22.8869	0.821591
$777$	−10.2170	−0.366534
$778$	0.694931	0.0249145
$779$	−29.3737	−1.05242
$780$	0	0
$781$	−41.8703	−1.49824
$782$	−1.13744	−0.0406748
$783$	−18.1362	−0.648134
$784$	3.31805	0.118502
$785$	0	0
$786$	1.17015	0.0417378
$787$	26.9676	0.961291	0.480646	−	0.876915i	$-0.340402\pi$
$787$	26.9676	0.961291	0.480646	+	0.876915i	$0.340402\pi$
$788$	−10.3501	−0.368706
$789$	67.4724	2.40208
$790$	0	0
$791$	1.76380	0.0627135
$792$	−28.2994	−1.00557
$793$	3.03775	0.107874
$794$	−4.80690	−0.170590
$795$	0	0
$796$	−32.5837	−1.15490
$797$	−14.8858	−0.527283	−0.263642	−	0.964621i	$-0.584924\pi$
$797$	−14.8858	−0.527283	−0.263642	+	0.964621i	$0.584924\pi$
$798$	2.94213	0.104150
$799$	50.8910	1.80039
$800$	0	0
$801$	−56.5309	−1.99742
$802$	−8.66004	−0.305797
$803$	−43.2560	−1.52647
$804$	64.6009	2.27830
$805$	0	0
$806$	−2.27090	−0.0799891
$807$	52.7344	1.85634
$808$	−16.2020	−0.569983
$809$	−10.8894	−0.382849	−0.191425	−	0.981507i	$-0.561311\pi$
$809$	−10.8894	−0.382849	−0.191425	+	0.981507i	$0.561311\pi$
$810$	0	0
$811$	−40.6296	−1.42670	−0.713349	−	0.700809i	$-0.752822\pi$
$811$	−40.6296	−1.42670	−0.713349	+	0.700809i	$0.752822\pi$
$812$	−11.4280	−0.401045
$813$	25.7263	0.902260
$814$	6.77019	0.237295
$815$	0	0
$816$	−47.9027	−1.67693
$817$	−18.4792	−0.646504
$818$	8.37007	0.292653
$819$	−4.12053	−0.143983
$820$	0	0
$821$	9.29079	0.324251	0.162125	−	0.986770i	$-0.448165\pi$
$821$	9.29079	0.324251	0.162125	+	0.986770i	$0.448165\pi$
$822$	−14.8566	−0.518183
$823$	−13.9665	−0.486840	−0.243420	−	0.969921i	$-0.578269\pi$
$823$	−13.9665	−0.486840	−0.243420	+	0.969921i	$0.578269\pi$
$824$	3.31893	0.115620
$825$	0	0
$826$	−3.89323	−0.135463
$827$	−34.8658	−1.21240	−0.606202	−	0.795311i	$-0.707308\pi$
$827$	−34.8658	−1.21240	−0.606202	+	0.795311i	$0.707308\pi$
$828$	4.79437	0.166616
$829$	41.9973	1.45863	0.729313	−	0.684180i	$-0.239840\pi$
$829$	41.9973	1.45863	0.729313	+	0.684180i	$0.239840\pi$
$830$	0	0
$831$	53.6080	1.85964
$832$	−5.35125	−0.185521
$833$	−5.41029	−0.187455
$834$	14.4605	0.500726
$835$	0	0
$836$	31.6937	1.09615
$837$	19.9454	0.689414
$838$	10.1501	0.350629
$839$	−30.1116	−1.03957	−0.519784	−	0.854298i	$-0.673987\pi$
$839$	−30.1116	−1.03957	−0.519784	+	0.854298i	$0.673987\pi$
$840$	0	0
$841$	7.79028	0.268630
$842$	−3.65030	−0.125798
$843$	25.7409	0.886565
$844$	29.7673	1.02463
$845$	0	0
$846$	13.1950	0.453653
$847$	−15.9771	−0.548981
$848$	−26.4780	−0.909258
$849$	−48.4185	−1.66172
$850$	0	0
$851$	−2.36451	−0.0810544
$852$	−40.5293	−1.38851
$853$	−46.3312	−1.58635	−0.793175	−	0.608994i	$-0.791573\pi$
$853$	−46.3312	−1.58635	−0.793175	+	0.608994i	$0.791573\pi$
$854$	1.03416	0.0353881
$855$	0	0
$856$	6.21854	0.212545
$857$	−24.6073	−0.840570	−0.420285	−	0.907392i	$-0.638070\pi$
$857$	−24.6073	−0.840570	−0.420285	+	0.907392i	$0.638070\pi$
$858$	4.71834	0.161081
$859$	16.7463	0.571375	0.285687	−	0.958323i	$-0.407778\pi$
$859$	16.7463	0.571375	0.285687	+	0.958323i	$0.407778\pi$
$860$	0	0
$861$	24.2016	0.824790
$862$	2.88907	0.0984020
$863$	−57.8952	−1.97077	−0.985387	−	0.170329i	$-0.945517\pi$
$863$	−57.8952	−1.97077	−0.985387	+	0.170329i	$0.945517\pi$
$864$	−11.2850	−0.383922
$865$	0	0
$866$	10.0088	0.340113
$867$	32.7449	1.11207
$868$	12.5681	0.426588
$869$	22.8673	0.775720
$870$	0	0
$871$	−12.8492	−0.435380
$872$	4.91377	0.166401
$873$	71.3204	2.41383
$874$	0.680893	0.0230316
$875$	0	0
$876$	−41.8706	−1.41468
$877$	−16.4044	−0.553937	−0.276968	−	0.960879i	$-0.589330\pi$
$877$	−16.4044	−0.553937	−0.276968	+	0.960879i	$0.589330\pi$
$878$	2.00772	0.0677574
$879$	−67.5065	−2.27694
$880$	0	0
$881$	−42.7051	−1.43877	−0.719385	−	0.694612i	$-0.755576\pi$
$881$	−42.7051	−1.43877	−0.719385	+	0.694612i	$0.755576\pi$
$882$	−1.40278	−0.0472339
$883$	−17.4905	−0.588601	−0.294301	−	0.955713i	$-0.595087\pi$
$883$	−17.4905	−0.588601	−0.294301	+	0.955713i	$0.595087\pi$
$884$	10.1935	0.342846
$885$	0	0
$886$	−3.46626	−0.116451
$887$	43.9942	1.47718	0.738591	−	0.674154i	$-0.235492\pi$
$887$	43.9942	1.47718	0.738591	+	0.674154i	$0.235492\pi$
$888$	13.5098	0.453361
$889$	17.4542	0.585394
$890$	0	0
$891$	22.7642	0.762629
$892$	20.5993	0.689717
$893$	−30.4643	−1.01945
$894$	9.67794	0.323679
$895$	0	0
$896$	−9.37007	−0.313032
$897$	−1.64790	−0.0550216
$898$	−7.69782	−0.256880
$899$	−40.4604	−1.34943
$900$	0	0
$901$	43.1740	1.43833
$902$	−16.0370	−0.533972
$903$	15.2254	0.506669
$904$	−2.33225	−0.0775695
$905$	0	0
$906$	2.39148	0.0794517
$907$	−42.6498	−1.41616	−0.708081	−	0.706131i	$-0.750439\pi$
$907$	−42.6498	−1.41616	−0.708081	+	0.706131i	$0.750439\pi$
$908$	−35.7379	−1.18600
$909$	−50.4887	−1.67461
$910$	0	0
$911$	21.3427	0.707115	0.353557	−	0.935413i	$-0.384972\pi$
$911$	21.3427	0.707115	0.353557	+	0.935413i	$0.384972\pi$
$912$	28.6754	0.949538
$913$	−48.5895	−1.60808
$914$	5.17067	0.171031
$915$	0	0
$916$	−4.57414	−0.151134
$917$	1.28810	0.0425368
$918$	5.50725	0.181766
$919$	54.0878	1.78419	0.892096	−	0.451845i	$-0.149234\pi$
$919$	54.0878	1.78419	0.892096	+	0.451845i	$0.149234\pi$
$920$	0	0
$921$	27.9429	0.920750
$922$	4.35557	0.143443
$923$	8.06136	0.265343
$924$	−26.1131	−0.859059
$925$	0	0
$926$	−4.46889	−0.146857
$927$	10.3425	0.339691
$928$	22.8922	0.751472
$929$	−15.4818	−0.507941	−0.253971	−	0.967212i	$-0.581737\pi$
$929$	−15.4818	−0.507941	−0.253971	+	0.967212i	$0.581737\pi$
$930$	0	0
$931$	3.23870	0.106144
$932$	13.8999	0.455306
$933$	−30.7262	−1.00593
$934$	−13.4432	−0.439876
$935$	0	0
$936$	5.44853	0.178091
$937$	44.3484	1.44880	0.724400	−	0.689380i	$-0.242117\pi$
$937$	44.3484	1.44880	0.724400	+	0.689380i	$0.242117\pi$
$938$	−4.37434	−0.142827
$939$	−0.563934	−0.0184033
$940$	0	0
$941$	−47.4472	−1.54673	−0.773367	−	0.633959i	$-0.781429\pi$
$941$	−47.4472	−1.54673	−0.773367	+	0.633959i	$0.781429\pi$
$942$	4.02832	0.131250
$943$	5.60096	0.182392
$944$	−37.9454	−1.23502
$945$	0	0
$946$	−10.0889	−0.328019
$947$	56.6670	1.84143	0.920714	−	0.390237i	$-0.127607\pi$
$947$	56.6670	1.84143	0.920714	+	0.390237i	$0.127607\pi$
$948$	22.1349	0.718909
$949$	8.32815	0.270343
$950$	0	0
$951$	15.9451	0.517056
$952$	7.15396	0.231861
$953$	12.8654	0.416750	0.208375	−	0.978049i	$-0.433183\pi$
$953$	12.8654	0.416750	0.208375	+	0.978049i	$0.433183\pi$
$954$	11.1941	0.362423
$955$	0	0
$956$	10.9372	0.353734
$957$	84.0660	2.71747
$958$	2.68267	0.0866730
$959$	−16.3542	−0.528103
$960$	0	0
$961$	13.4966	0.435374
$962$	−1.30348	−0.0420258
$963$	19.3783	0.624457
$964$	4.64437	0.149585
$965$	0	0
$966$	−0.561003	−0.0180500
$967$	−46.2209	−1.48636	−0.743181	−	0.669090i	$-0.766684\pi$
$967$	−46.2209	−1.48636	−0.743181	+	0.669090i	$0.766684\pi$
$968$	21.1264	0.679027
$969$	−46.7570	−1.50205
$970$	0	0
$971$	25.8952	0.831017	0.415509	−	0.909589i	$-0.363604\pi$
$971$	25.8952	0.831017	0.415509	+	0.909589i	$0.363604\pi$
$972$	38.9358	1.24887
$973$	15.9181	0.510312
$974$	3.80293	0.121854
$975$	0	0
$976$	10.0794	0.322634
$977$	51.4558	1.64622	0.823108	−	0.567885i	$-0.192238\pi$
$977$	51.4558	1.64622	0.823108	+	0.567885i	$0.192238\pi$
$978$	−11.4637	−0.366570
$979$	71.2576	2.27740
$980$	0	0
$981$	15.3123	0.488885
$982$	0.260804	0.00832260
$983$	36.8815	1.17634	0.588168	−	0.808738i	$-0.299849\pi$
$983$	36.8815	1.17634	0.588168	+	0.808738i	$0.299849\pi$
$984$	−32.0015	−1.02017
$985$	0	0
$986$	−11.1718	−0.355782
$987$	25.1002	0.798947
$988$	−6.10204	−0.194132
$989$	3.52359	0.112044
$990$	0	0
$991$	48.3732	1.53663	0.768313	−	0.640074i	$-0.221096\pi$
$991$	48.3732	1.53663	0.768313	+	0.640074i	$0.221096\pi$
$992$	−25.1758	−0.799333
$993$	35.0219	1.11138
$994$	2.74438	0.0870463
$995$	0	0
$996$	−47.0333	−1.49031
$997$	−0.0221861	−0.000702641 0	−0.000351320	−	1.00000i	$-0.500112\pi$
$997$	−0.0221861	−0.000702641 0	−0.000351320		1.00000i	$0.500112\pi$
$998$	−7.76096	−0.245669
$999$	11.4485	0.362214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2275.2.a.y.1.4		7
5.2	odd	4		455.2.c.b.274.7	✓	14
5.3	odd	4		455.2.c.b.274.8	yes	14
5.4	even	2		2275.2.a.w.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.c.b.274.7	✓	14	5.2	odd	4
455.2.c.b.274.8	yes	14	5.3	odd	4
2275.2.a.w.1.4		7	5.4	even	2
2275.2.a.y.1.4		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)

\(f(q)\)	\(=\)	\(q-0.340436 q^{2} +2.66843 q^{3} -1.88410 q^{4} -0.908430 q^{6} -1.00000 q^{7} +1.32229 q^{8} +4.12053 q^{9} -5.19395 q^{11} -5.02760 q^{12} +1.00000 q^{13} +0.340436 q^{14} +3.31805 q^{16} -5.41029 q^{17} -1.40278 q^{18} +3.23870 q^{19} -2.66843 q^{21} +1.76821 q^{22} -0.617552 q^{23} +3.52844 q^{24} -0.340436 q^{26} +2.99006 q^{27} +1.88410 q^{28} -6.06550 q^{29} +6.67058 q^{31} -3.77416 q^{32} -13.8597 q^{33} +1.84186 q^{34} -7.76350 q^{36} +3.82885 q^{37} -1.10257 q^{38} +2.66843 q^{39} -9.06961 q^{41} +0.908430 q^{42} -5.70574 q^{43} +9.78594 q^{44} +0.210237 q^{46} -9.40633 q^{47} +8.85400 q^{48} +1.00000 q^{49} -14.4370 q^{51} -1.88410 q^{52} -7.97998 q^{53} -1.01792 q^{54} -1.32229 q^{56} +8.64224 q^{57} +2.06491 q^{58} -11.4360 q^{59} +3.03775 q^{61} -2.27090 q^{62} -4.12053 q^{63} -5.35125 q^{64} +4.71834 q^{66} -12.8492 q^{67} +10.1935 q^{68} -1.64790 q^{69} +8.06136 q^{71} +5.44853 q^{72} +8.32815 q^{73} -1.30348 q^{74} -6.10204 q^{76} +5.19395 q^{77} -0.908430 q^{78} -4.40268 q^{79} -4.38283 q^{81} +3.08762 q^{82} +9.35502 q^{83} +5.02760 q^{84} +1.94244 q^{86} -16.1854 q^{87} -6.86790 q^{88} -13.7193 q^{89} -1.00000 q^{91} +1.16353 q^{92} +17.8000 q^{93} +3.20225 q^{94} -10.0711 q^{96} +17.3085 q^{97} -0.340436 q^{98} -21.4018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) \) 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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