LMFDB - Embedded newform 275.4.a.f.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,4,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,-12] 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:	$16.2255252516$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 24x^{3} + 31x^{2} + 108x - 84 $ x^5 - 2x^4 - 24x^3 + 31x^2 + 108x - 84
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.95823$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.95823 q^{2} -0.384445 q^{3} +7.66762 q^{4} -1.52172 q^{6} -27.0299 q^{7} -1.31563 q^{8} -26.8522 q^{9} +11.0000 q^{11} -2.94778 q^{12} -82.9282 q^{13} -106.991 q^{14} -66.5486 q^{16} +100.512 q^{17} -106.287 q^{18} +38.6071 q^{19} +10.3915 q^{21} +43.5406 q^{22} -19.5133 q^{23} +0.505788 q^{24} -328.249 q^{26} +20.7032 q^{27} -207.255 q^{28} -100.230 q^{29} +121.874 q^{31} -252.890 q^{32} -4.22889 q^{33} +397.851 q^{34} -205.893 q^{36} +347.319 q^{37} +152.816 q^{38} +31.8813 q^{39} +108.386 q^{41} +41.1320 q^{42} -268.039 q^{43} +84.3438 q^{44} -77.2381 q^{46} -568.074 q^{47} +25.5843 q^{48} +387.616 q^{49} -38.6414 q^{51} -635.862 q^{52} -603.970 q^{53} +81.9481 q^{54} +35.5614 q^{56} -14.8423 q^{57} -396.734 q^{58} +32.7793 q^{59} +156.323 q^{61} +482.407 q^{62} +725.813 q^{63} -468.608 q^{64} -16.7390 q^{66} -745.522 q^{67} +770.689 q^{68} +7.50178 q^{69} -264.725 q^{71} +35.3276 q^{72} -334.805 q^{73} +1374.77 q^{74} +296.025 q^{76} -297.329 q^{77} +126.194 q^{78} +1287.57 q^{79} +717.050 q^{81} +429.018 q^{82} +127.129 q^{83} +79.6782 q^{84} -1060.96 q^{86} +38.5329 q^{87} -14.4720 q^{88} -615.817 q^{89} +2241.54 q^{91} -149.620 q^{92} -46.8539 q^{93} -2248.57 q^{94} +97.2222 q^{96} +579.630 q^{97} +1534.28 q^{98} -295.374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - 12 q^{3} + 12 q^{4} + 8 q^{6} - 24 q^{7} - 27 q^{8} + 31 q^{9} + 55 q^{11} - 3 q^{12} - 111 q^{13} + 47 q^{14} - 56 q^{16} - 40 q^{17} - 217 q^{18} + 205 q^{19} - 94 q^{21} - 22 q^{22} - 287 q^{23}+ \cdots + 341 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 - 12 * q^3 + 12 * q^4 + 8 * q^6 - 24 * q^7 - 27 * q^8 + 31 * q^9 + 55 * q^11 - 3 * q^12 - 111 * q^13 + 47 * q^14 - 56 * q^16 - 40 * q^17 - 217 * q^18 + 205 * q^19 - 94 * q^21 - 22 * q^22 - 287 * q^23 + 273 * q^24 - 354 * q^26 - 270 * q^27 - 460 * q^28 + 251 * q^29 - 289 * q^31 - 248 * q^32 - 132 * q^33 + 522 * q^34 - 722 * q^36 - 224 * q^37 - 540 * q^38 + 538 * q^39 - 462 * q^41 - 175 * q^42 - 593 * q^43 + 132 * q^44 - 972 * q^46 - 766 * q^47 - 992 * q^48 + 75 * q^49 - 820 * q^51 - 696 * q^53 + 281 * q^54 - 527 * q^56 - 1170 * q^57 - 1461 * q^58 - 22 * q^59 + 720 * q^61 + 998 * q^62 + 168 * q^63 - 317 * q^64 + 88 * q^66 - 1230 * q^67 + 109 * q^68 - 514 * q^69 + 951 * q^71 + 1590 * q^72 - 666 * q^73 + 873 * q^74 + 290 * q^76 - 264 * q^77 - 56 * q^78 - 588 * q^79 + 1885 * q^81 + 1807 * q^82 + 867 * q^83 + 1321 * q^84 - 411 * q^86 - 766 * q^87 - 297 * q^88 - 51 * q^89 + 2172 * q^91 + 4137 * q^92 + 1916 * q^93 - 865 * q^94 + 603 * q^96 - 2849 * q^97 + 4104 * q^98 + 341 * 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 $

$2$	3.95823	1.39945	0.699724	−	0.714414i	$-0.253306\pi$
$2$	3.95823	1.39945	0.699724	+	0.714414i	$0.253306\pi$
$3$	−0.384445	−0.0739865	−0.0369932	−	0.999316i	$-0.511778\pi$
$3$	−0.384445	−0.0739865	−0.0369932	+	0.999316i	$0.511778\pi$
$4$	7.66762	0.958453
$5$	0	0
$5$	0	0
$6$	−1.52172	−0.103540
$7$	−27.0299	−1.45948	−0.729739	−	0.683726i	$-0.760358\pi$
$7$	−27.0299	−1.45948	−0.729739	+	0.683726i	$0.760358\pi$
$8$	−1.31563	−0.0581433
$9$	−26.8522	−0.994526
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−2.94778	−0.0709125
$13$	−82.9282	−1.76924	−0.884621	−	0.466310i	$-0.845583\pi$
$13$	−82.9282	−1.76924	−0.884621	+	0.466310i	$0.845583\pi$
$14$	−106.991	−2.04246
$15$	0	0
$16$	−66.5486	−1.03982
$17$	100.512	1.43399	0.716993	−	0.697080i	$-0.245518\pi$
$17$	100.512	1.43399	0.716993	+	0.697080i	$0.245518\pi$
$18$	−106.287	−1.39179
$19$	38.6071	0.466162	0.233081	−	0.972457i	$-0.425119\pi$
$19$	38.6071	0.466162	0.233081	+	0.972457i	$0.425119\pi$
$20$	0	0
$21$	10.3915	0.107982
$22$	43.5406	0.421949
$23$	−19.5133	−0.176904	−0.0884522	−	0.996080i	$-0.528192\pi$
$23$	−19.5133	−0.176904	−0.0884522	+	0.996080i	$0.528192\pi$
$24$	0.505788	0.00430181
$25$	0	0
$26$	−328.249	−2.47596
$27$	20.7032	0.147568
$28$	−207.255	−1.39884
$29$	−100.230	−0.641801	−0.320900	−	0.947113i	$-0.603986\pi$
$29$	−100.230	−0.641801	−0.320900	+	0.947113i	$0.603986\pi$
$30$	0	0
$31$	121.874	0.706105	0.353053	−	0.935603i	$-0.385144\pi$
$31$	121.874	0.706105	0.353053	+	0.935603i	$0.385144\pi$
$32$	−252.890	−1.39703
$33$	−4.22889	−0.0223078
$34$	397.851	2.00679
$35$	0	0
$36$	−205.893	−0.953206
$37$	347.319	1.54321	0.771606	−	0.636101i	$-0.219454\pi$
$37$	347.319	1.54321	0.771606	+	0.636101i	$0.219454\pi$
$38$	152.816	0.652370
$39$	31.8813	0.130900
$40$	0	0
$41$	108.386	0.412856	0.206428	−	0.978462i	$-0.433816\pi$
$41$	108.386	0.412856	0.206428	+	0.978462i	$0.433816\pi$
$42$	41.1320	0.151115
$43$	−268.039	−0.950594	−0.475297	−	0.879825i	$-0.657659\pi$
$43$	−268.039	−0.950594	−0.475297	+	0.879825i	$0.657659\pi$
$44$	84.3438	0.288984
$45$	0	0
$46$	−77.2381	−0.247568
$47$	−568.074	−1.76302	−0.881512	−	0.472162i	$-0.843474\pi$
$47$	−568.074	−1.76302	−0.881512	+	0.472162i	$0.843474\pi$
$48$	25.5843	0.0769327
$49$	387.616	1.13008
$50$	0	0
$51$	−38.6414	−0.106096
$52$	−635.862	−1.69574
$53$	−603.970	−1.56531	−0.782657	−	0.622453i	$-0.786136\pi$
$53$	−603.970	−1.56531	−0.782657	+	0.622453i	$0.786136\pi$
$54$	81.9481	0.206514
$55$	0	0
$56$	35.5614	0.0848588
$57$	−14.8423	−0.0344897
$58$	−396.734	−0.898167
$59$	32.7793	0.0723304	0.0361652	−	0.999346i	$-0.488486\pi$
$59$	32.7793	0.0723304	0.0361652	+	0.999346i	$0.488486\pi$
$60$	0	0
$61$	156.323	0.328117	0.164059	−	0.986451i	$-0.447541\pi$
$61$	156.323	0.328117	0.164059	+	0.986451i	$0.447541\pi$
$62$	482.407	0.988157
$63$	725.813	1.45149
$64$	−468.608	−0.915251
$65$	0	0
$66$	−16.7390	−0.0312185
$67$	−745.522	−1.35940	−0.679702	−	0.733489i	$-0.737891\pi$
$67$	−745.522	−1.35940	−0.679702	+	0.733489i	$0.737891\pi$
$68$	770.689	1.37441
$69$	7.50178	0.0130885
$70$	0	0
$71$	−264.725	−0.442494	−0.221247	−	0.975218i	$-0.571013\pi$
$71$	−264.725	−0.442494	−0.221247	+	0.975218i	$0.571013\pi$
$72$	35.3276	0.0578250
$73$	−334.805	−0.536794	−0.268397	−	0.963308i	$-0.586494\pi$
$73$	−334.805	−0.536794	−0.268397	+	0.963308i	$0.586494\pi$
$74$	1374.77	2.15964
$75$	0	0
$76$	296.025	0.446795
$77$	−297.329	−0.440049
$78$	126.194	0.183188
$79$	1287.57	1.83370	0.916851	−	0.399230i	$-0.130723\pi$
$79$	1287.57	1.83370	0.916851	+	0.399230i	$0.130723\pi$
$80$	0	0
$81$	717.050	0.983608
$82$	429.018	0.577770
$83$	127.129	0.168123	0.0840617	−	0.996461i	$-0.473211\pi$
$83$	127.129	0.168123	0.0840617	+	0.996461i	$0.473211\pi$
$84$	79.6782	0.103495
$85$	0	0
$86$	−1060.96	−1.33031
$87$	38.5329	0.0474846
$88$	−14.4720	−0.0175309
$89$	−615.817	−0.733443	−0.366722	−	0.930331i	$-0.619520\pi$
$89$	−615.817	−0.733443	−0.366722	+	0.930331i	$0.619520\pi$
$90$	0	0
$91$	2241.54	2.58217
$92$	−149.620	−0.169554
$93$	−46.8539	−0.0522422
$94$	−2248.57	−2.46726
$95$	0	0
$96$	97.2222	0.103361
$97$	579.630	0.606727	0.303363	−	0.952875i	$-0.401890\pi$
$97$	579.630	0.606727	0.303363	+	0.952875i	$0.401890\pi$
$98$	1534.28	1.58148
$99$	−295.374	−0.299861
$100$	0	0
$101$	−599.271	−0.590393	−0.295197	−	0.955437i	$-0.595385\pi$
$101$	−599.271	−0.590393	−0.295197	+	0.955437i	$0.595385\pi$
$102$	−152.952	−0.148475
$103$	−991.632	−0.948625	−0.474313	−	0.880356i	$-0.657303\pi$
$103$	−991.632	−0.948625	−0.474313	+	0.880356i	$0.657303\pi$
$104$	109.103	0.102870
$105$	0	0
$106$	−2390.65	−2.19057
$107$	867.633	0.783899	0.391950	−	0.919987i	$-0.371801\pi$
$107$	867.633	0.783899	0.391950	+	0.919987i	$0.371801\pi$
$108$	158.744	0.141437
$109$	−1312.40	−1.15326	−0.576630	−	0.817006i	$-0.695632\pi$
$109$	−1312.40	−1.15326	−0.576630	+	0.817006i	$0.695632\pi$
$110$	0	0
$111$	−133.525	−0.114177
$112$	1798.80	1.51760
$113$	909.383	0.757058	0.378529	−	0.925589i	$-0.376430\pi$
$113$	909.383	0.757058	0.378529	+	0.925589i	$0.376430\pi$
$114$	−58.7494	−0.0482665
$115$	0	0
$116$	−768.525	−0.615136
$117$	2226.81	1.75956
$118$	129.748	0.101223
$119$	−2716.83	−2.09287
$120$	0	0
$121$	121.000	0.0909091
$122$	618.764	0.459183
$123$	−41.6686	−0.0305458
$124$	934.486	0.676768
$125$	0	0
$126$	2872.94	2.03128
$127$	339.579	0.237266	0.118633	−	0.992938i	$-0.462149\pi$
$127$	339.579	0.237266	0.118633	+	0.992938i	$0.462149\pi$
$128$	168.256	0.116186
$129$	103.046	0.0703311
$130$	0	0
$131$	1502.96	1.00240	0.501199	−	0.865332i	$-0.332892\pi$
$131$	1502.96	1.00240	0.501199	+	0.865332i	$0.332892\pi$
$132$	−32.4256	−0.0213809
$133$	−1043.55	−0.680354
$134$	−2950.95	−1.90241
$135$	0	0
$136$	−132.237	−0.0833767
$137$	−2164.22	−1.34965	−0.674825	−	0.737978i	$-0.735781\pi$
$137$	−2164.22	−1.34965	−0.674825	+	0.737978i	$0.735781\pi$
$138$	29.6938	0.0183167
$139$	−390.133	−0.238062	−0.119031	−	0.992891i	$-0.537979\pi$
$139$	−390.133	−0.238062	−0.119031	+	0.992891i	$0.537979\pi$
$140$	0	0
$141$	218.393	0.130440
$142$	−1047.84	−0.619246
$143$	−912.211	−0.533447
$144$	1786.98	1.03413
$145$	0	0
$146$	−1325.24	−0.751214
$147$	−149.017	−0.0836103
$148$	2663.11	1.47910
$149$	−2735.95	−1.50428	−0.752141	−	0.659003i	$-0.770979\pi$
$149$	−2735.95	−1.50428	−0.752141	+	0.659003i	$0.770979\pi$
$150$	0	0
$151$	1831.35	0.986974	0.493487	−	0.869753i	$-0.335722\pi$
$151$	1831.35	0.986974	0.493487	+	0.869753i	$0.335722\pi$
$152$	−50.7928	−0.0271042
$153$	−2698.97	−1.42614
$154$	−1176.90	−0.615826
$155$	0	0
$156$	244.454	0.125461
$157$	−3816.59	−1.94011	−0.970055	−	0.242884i	$-0.921907\pi$
$157$	−3816.59	−1.94011	−0.970055	+	0.242884i	$0.921907\pi$
$158$	5096.48	2.56617
$159$	232.193	0.115812
$160$	0	0
$161$	527.442	0.258188
$162$	2838.25	1.37651
$163$	4.62217	0.00222108	0.00111054	−	0.999999i	$-0.499647\pi$
$163$	4.62217	0.00222108	0.00111054	+	0.999999i	$0.499647\pi$
$164$	831.065	0.395703
$165$	0	0
$166$	503.207	0.235280
$167$	−2057.83	−0.953533	−0.476767	−	0.879030i	$-0.658191\pi$
$167$	−2057.83	−0.953533	−0.476767	+	0.879030i	$0.658191\pi$
$168$	−13.6714	−0.00627840
$169$	4680.09	2.13022
$170$	0	0
$171$	−1036.69	−0.463611
$172$	−2055.22	−0.911100
$173$	4045.58	1.77792	0.888959	−	0.457987i	$-0.151429\pi$
$173$	4045.58	1.77792	0.888959	+	0.457987i	$0.151429\pi$
$174$	152.522	0.0664522
$175$	0	0
$176$	−732.034	−0.313518
$177$	−12.6018	−0.00535147
$178$	−2437.55	−1.02642
$179$	3607.09	1.50618	0.753092	−	0.657916i	$-0.228562\pi$
$179$	3607.09	1.50618	0.753092	+	0.657916i	$0.228562\pi$
$180$	0	0
$181$	−664.649	−0.272945	−0.136472	−	0.990644i	$-0.543576\pi$
$181$	−664.649	−0.272945	−0.136472	+	0.990644i	$0.543576\pi$
$182$	8872.55	3.61361
$183$	−60.0977	−0.0242762
$184$	25.6723	0.0102858
$185$	0	0
$186$	−185.459	−0.0731102
$187$	1105.63	0.432363
$188$	−4355.78	−1.68977
$189$	−559.606	−0.215372
$190$	0	0
$191$	89.6468	0.0339613	0.0169807	−	0.999856i	$-0.494595\pi$
$191$	89.6468	0.0339613	0.0169807	+	0.999856i	$0.494595\pi$
$192$	180.154	0.0677162
$193$	−2119.54	−0.790505	−0.395253	−	0.918572i	$-0.629343\pi$
$193$	−2119.54	−0.790505	−0.395253	+	0.918572i	$0.629343\pi$
$194$	2294.31	0.849082
$195$	0	0
$196$	2972.09	1.08312
$197$	−2736.33	−0.989623	−0.494812	−	0.869000i	$-0.664763\pi$
$197$	−2736.33	−0.989623	−0.494812	+	0.869000i	$0.664763\pi$
$198$	−1169.16	−0.419639
$199$	−1249.59	−0.445130	−0.222565	−	0.974918i	$-0.571443\pi$
$199$	−1249.59	−0.445130	−0.222565	+	0.974918i	$0.571443\pi$
$200$	0	0
$201$	286.612	0.100577
$202$	−2372.06	−0.826224
$203$	2709.21	0.936694
$204$	−296.287	−0.101688
$205$	0	0
$206$	−3925.11	−1.32755
$207$	523.975	0.175936
$208$	5518.75	1.83970
$209$	424.678	0.140553
$210$	0	0
$211$	944.864	0.308280	0.154140	−	0.988049i	$-0.450739\pi$
$211$	944.864	0.308280	0.154140	+	0.988049i	$0.450739\pi$
$212$	−4631.01	−1.50028
$213$	101.772	0.0327385
$214$	3434.29	1.09703
$215$	0	0
$216$	−27.2378	−0.00858008
$217$	−3294.25	−1.03054
$218$	−5194.79	−1.61393
$219$	128.714	0.0397155
$220$	0	0
$221$	−8335.29	−2.53707
$222$	−528.523	−0.159784
$223$	5252.19	1.57719	0.788593	−	0.614916i	$-0.210810\pi$
$223$	5252.19	1.57719	0.788593	+	0.614916i	$0.210810\pi$
$224$	6835.59	2.03894
$225$	0	0
$226$	3599.55	1.05946
$227$	−576.634	−0.168602	−0.0843008	−	0.996440i	$-0.526866\pi$
$227$	−576.634	−0.168602	−0.0843008	+	0.996440i	$0.526866\pi$
$228$	−113.805	−0.0330567
$229$	−4617.17	−1.33236	−0.666182	−	0.745789i	$-0.732073\pi$
$229$	−4617.17	−1.33236	−0.666182	+	0.745789i	$0.732073\pi$
$230$	0	0
$231$	114.307	0.0325577
$232$	131.866	0.0373164
$233$	−2916.94	−0.820150	−0.410075	−	0.912052i	$-0.634498\pi$
$233$	−2916.94	−0.820150	−0.410075	+	0.912052i	$0.634498\pi$
$234$	8814.22	2.46241
$235$	0	0
$236$	251.339	0.0693253
$237$	−494.998	−0.135669
$238$	−10753.9	−2.92886
$239$	−649.560	−0.175802	−0.0879008	−	0.996129i	$-0.528016\pi$
$239$	−649.560	−0.175802	−0.0879008	+	0.996129i	$0.528016\pi$
$240$	0	0
$241$	1206.97	0.322605	0.161302	−	0.986905i	$-0.448431\pi$
$241$	1206.97	0.322605	0.161302	+	0.986905i	$0.448431\pi$
$242$	478.946	0.127222
$243$	−834.653	−0.220342
$244$	1198.63	0.314485
$245$	0	0
$246$	−164.934	−0.0427472
$247$	−3201.62	−0.824754
$248$	−160.342	−0.0410553
$249$	−48.8742	−0.0124389
$250$	0	0
$251$	−2729.80	−0.686468	−0.343234	−	0.939250i	$-0.611522\pi$
$251$	−2729.80	−0.686468	−0.343234	+	0.939250i	$0.611522\pi$
$252$	5565.26	1.39118
$253$	−214.646	−0.0533387
$254$	1344.13	0.332041
$255$	0	0
$256$	4414.86	1.07785
$257$	−3924.17	−0.952464	−0.476232	−	0.879320i	$-0.657998\pi$
$257$	−3924.17	−0.952464	−0.476232	+	0.879320i	$0.657998\pi$
$258$	407.881	0.0984247
$259$	−9387.99	−2.25228
$260$	0	0
$261$	2691.39	0.638288
$262$	5949.07	1.40280
$263$	1503.06	0.352406	0.176203	−	0.984354i	$-0.443618\pi$
$263$	1503.06	0.352406	0.176203	+	0.984354i	$0.443618\pi$
$264$	5.56367	0.00129705
$265$	0	0
$266$	−4130.61	−0.952119
$267$	236.748	0.0542649
$268$	−5716.38	−1.30292
$269$	−2116.10	−0.479631	−0.239815	−	0.970819i	$-0.577087\pi$
$269$	−2116.10	−0.479631	−0.239815	+	0.970819i	$0.577087\pi$
$270$	0	0
$271$	−6259.37	−1.40306	−0.701531	−	0.712639i	$-0.747500\pi$
$271$	−6259.37	−1.40306	−0.701531	+	0.712639i	$0.747500\pi$
$272$	−6688.94	−1.49109
$273$	−861.750	−0.191046
$274$	−8566.50	−1.88876
$275$	0	0
$276$	57.5208	0.0125447
$277$	2873.23	0.623232	0.311616	−	0.950208i	$-0.399130\pi$
$277$	2873.23	0.623232	0.311616	+	0.950208i	$0.399130\pi$
$278$	−1544.24	−0.333155
$279$	−3272.59	−0.702240
$280$	0	0
$281$	−766.809	−0.162790	−0.0813949	−	0.996682i	$-0.525938\pi$
$281$	−766.809	−0.162790	−0.0813949	+	0.996682i	$0.525938\pi$
$282$	864.451	0.182544
$283$	−5160.63	−1.08399	−0.541993	−	0.840383i	$-0.682330\pi$
$283$	−5160.63	−1.08399	−0.541993	+	0.840383i	$0.682330\pi$
$284$	−2029.81	−0.424109
$285$	0	0
$286$	−3610.74	−0.746531
$287$	−2929.67	−0.602554
$288$	6790.65	1.38938
$289$	5189.69	1.05632
$290$	0	0
$291$	−222.836	−0.0448896
$292$	−2567.16	−0.514491
$293$	−3915.69	−0.780740	−0.390370	−	0.920658i	$-0.627653\pi$
$293$	−3915.69	−0.780740	−0.390370	+	0.920658i	$0.627653\pi$
$294$	−589.845	−0.117008
$295$	0	0
$296$	−456.944	−0.0897274
$297$	227.735	0.0444934
$298$	−10829.5	−2.10516
$299$	1618.20	0.312987
$300$	0	0
$301$	7245.07	1.38737
$302$	7248.91	1.38122
$303$	230.387	0.0436811
$304$	−2569.25	−0.484725
$305$	0	0
$306$	−10683.2	−1.99580
$307$	8255.96	1.53483	0.767415	−	0.641151i	$-0.221543\pi$
$307$	8255.96	1.53483	0.767415	+	0.641151i	$0.221543\pi$
$308$	−2279.81	−0.421766
$309$	381.228	0.0701854
$310$	0	0
$311$	−9181.60	−1.67409	−0.837043	−	0.547138i	$-0.815717\pi$
$311$	−9181.60	−1.67409	−0.837043	+	0.547138i	$0.815717\pi$
$312$	−41.9441	−0.00761095
$313$	7061.96	1.27529	0.637645	−	0.770330i	$-0.279909\pi$
$313$	7061.96	1.27529	0.637645	+	0.770330i	$0.279909\pi$
$314$	−15107.0	−2.71508
$315$	0	0
$316$	9872.56	1.75752
$317$	6122.11	1.08471	0.542353	−	0.840151i	$-0.317533\pi$
$317$	6122.11	1.08471	0.542353	+	0.840151i	$0.317533\pi$
$318$	919.075	0.162073
$319$	−1102.53	−0.193510
$320$	0	0
$321$	−333.557	−0.0579979
$322$	2087.74	0.361321
$323$	3880.48	0.668470
$324$	5498.07	0.942742
$325$	0	0
$326$	18.2956	0.00310828
$327$	504.546	0.0853256
$328$	−142.596	−0.0240048
$329$	15355.0	2.57309
$330$	0	0
$331$	4715.92	0.783113	0.391557	−	0.920154i	$-0.371937\pi$
$331$	4715.92	0.783113	0.391557	+	0.920154i	$0.371937\pi$
$332$	974.778	0.161138
$333$	−9326.27	−1.53476
$334$	−8145.39	−1.33442
$335$	0	0
$336$	−691.540	−0.112282
$337$	−9490.05	−1.53399	−0.766997	−	0.641651i	$-0.778250\pi$
$337$	−9490.05	−1.53399	−0.766997	+	0.641651i	$0.778250\pi$
$338$	18524.9	2.98113
$339$	−349.608	−0.0560121
$340$	0	0
$341$	1340.62	0.212899
$342$	−4103.45	−0.648798
$343$	−1205.97	−0.189843
$344$	352.641	0.0552707
$345$	0	0
$346$	16013.4	2.48810
$347$	6489.79	1.00401	0.502003	−	0.864866i	$-0.332597\pi$
$347$	6489.79	1.00401	0.502003	+	0.864866i	$0.332597\pi$
$348$	295.456	0.0455117
$349$	7155.44	1.09748	0.548742	−	0.835992i	$-0.315107\pi$
$349$	7155.44	1.09748	0.548742	+	0.835992i	$0.315107\pi$
$350$	0	0
$351$	−1716.88	−0.261083
$352$	−2781.79	−0.421221
$353$	−929.258	−0.140112	−0.0700558	−	0.997543i	$-0.522318\pi$
$353$	−929.258	−0.140112	−0.0700558	+	0.997543i	$0.522318\pi$
$354$	−49.8810	−0.00748911
$355$	0	0
$356$	−4721.85	−0.702971
$357$	1044.47	0.154844
$358$	14277.7	2.10782
$359$	−8426.85	−1.23886	−0.619432	−	0.785050i	$-0.712637\pi$
$359$	−8426.85	−1.23886	−0.619432	+	0.785050i	$0.712637\pi$
$360$	0	0
$361$	−5368.49	−0.782693
$362$	−2630.84	−0.381972
$363$	−46.5178	−0.00672604
$364$	17187.3	2.47489
$365$	0	0
$366$	−237.881	−0.0339733
$367$	−5108.50	−0.726599	−0.363299	−	0.931672i	$-0.618350\pi$
$367$	−5108.50	−0.726599	−0.363299	+	0.931672i	$0.618350\pi$
$368$	1298.58	0.183949
$369$	−2910.41	−0.410596
$370$	0	0
$371$	16325.3	2.28454
$372$	−359.258	−0.0500717
$373$	12264.2	1.70246	0.851231	−	0.524791i	$-0.175856\pi$
$373$	12264.2	1.70246	0.851231	+	0.524791i	$0.175856\pi$
$374$	4376.36	0.605069
$375$	0	0
$376$	747.376	0.102508
$377$	8311.89	1.13550
$378$	−2215.05	−0.301402
$379$	−2724.96	−0.369318	−0.184659	−	0.982803i	$-0.559118\pi$
$379$	−2724.96	−0.369318	−0.184659	+	0.982803i	$0.559118\pi$
$380$	0	0
$381$	−130.549	−0.0175544
$382$	354.843	0.0475271
$383$	11606.0	1.54841	0.774205	−	0.632935i	$-0.218150\pi$
$383$	11606.0	1.54841	0.774205	+	0.632935i	$0.218150\pi$
$384$	−64.6850	−0.00859620
$385$	0	0
$386$	−8389.62	−1.10627
$387$	7197.44	0.945391
$388$	4444.38	0.581519
$389$	2655.45	0.346110	0.173055	−	0.984912i	$-0.444636\pi$
$389$	2655.45	0.346110	0.173055	+	0.984912i	$0.444636\pi$
$390$	0	0
$391$	−1961.32	−0.253678
$392$	−509.960	−0.0657063
$393$	−577.805	−0.0741639
$394$	−10831.1	−1.38493
$395$	0	0
$396$	−2264.82	−0.287402
$397$	−10994.8	−1.38996	−0.694980	−	0.719029i	$-0.744587\pi$
$397$	−10994.8	−1.38996	−0.694980	+	0.719029i	$0.744587\pi$
$398$	−4946.16	−0.622936
$399$	401.187	0.0503370
$400$	0	0
$401$	−4626.29	−0.576124	−0.288062	−	0.957612i	$-0.593011\pi$
$401$	−4626.29	−0.576124	−0.288062	+	0.957612i	$0.593011\pi$
$402$	1134.48	0.140753
$403$	−10106.8	−1.24927
$404$	−4594.98	−0.565864
$405$	0	0
$406$	10723.7	1.31085
$407$	3820.51	0.465296
$408$	50.8378	0.00616874
$409$	−7991.43	−0.966139	−0.483069	−	0.875582i	$-0.660478\pi$
$409$	−7991.43	−0.966139	−0.483069	+	0.875582i	$0.660478\pi$
$410$	0	0
$411$	832.024	0.0998558
$412$	−7603.46	−0.909212
$413$	−886.021	−0.105565
$414$	2074.01	0.246213
$415$	0	0
$416$	20971.7	2.47169
$417$	149.985	0.0176134
$418$	1680.98	0.196697
$419$	14032.5	1.63612	0.818059	−	0.575134i	$-0.195050\pi$
$419$	14032.5	1.63612	0.818059	+	0.575134i	$0.195050\pi$
$420$	0	0
$421$	5305.98	0.614247	0.307123	−	0.951670i	$-0.400634\pi$
$421$	5305.98	0.614247	0.307123	+	0.951670i	$0.400634\pi$
$422$	3739.99	0.431422
$423$	15254.0	1.75337
$424$	794.602	0.0910125
$425$	0	0
$426$	402.838	0.0458158
$427$	−4225.40	−0.478880
$428$	6652.68	0.751330
$429$	350.695	0.0394678
$430$	0	0
$431$	16459.2	1.83947	0.919737	−	0.392536i	$-0.128402\pi$
$431$	16459.2	1.83947	0.919737	+	0.392536i	$0.128402\pi$
$432$	−1377.77	−0.153444
$433$	−6138.61	−0.681300	−0.340650	−	0.940190i	$-0.610647\pi$
$433$	−6138.61	−0.681300	−0.340650	+	0.940190i	$0.610647\pi$
$434$	−13039.4	−1.44219
$435$	0	0
$436$	−10063.0	−1.10534
$437$	−753.352	−0.0824661
$438$	509.480	0.0555797
$439$	−13563.5	−1.47461	−0.737303	−	0.675562i	$-0.763901\pi$
$439$	−13563.5	−1.47461	−0.737303	+	0.675562i	$0.763901\pi$
$440$	0	0
$441$	−10408.3	−1.12389
$442$	−32993.0	−3.55050
$443$	2667.69	0.286108	0.143054	−	0.989715i	$-0.454308\pi$
$443$	2667.69	0.286108	0.143054	+	0.989715i	$0.454308\pi$
$444$	−1023.82	−0.109433
$445$	0	0
$446$	20789.4	2.20719
$447$	1051.82	0.111296
$448$	12666.4	1.33579
$449$	5625.14	0.591240	0.295620	−	0.955306i	$-0.404474\pi$
$449$	5625.14	0.591240	0.295620	+	0.955306i	$0.404474\pi$
$450$	0	0
$451$	1192.25	0.124481
$452$	6972.81	0.725604
$453$	−704.053	−0.0730227
$454$	−2282.45	−0.235949
$455$	0	0
$456$	19.5270	0.00200534
$457$	10988.9	1.12481	0.562404	−	0.826863i	$-0.309877\pi$
$457$	10988.9	1.12481	0.562404	+	0.826863i	$0.309877\pi$
$458$	−18275.9	−1.86457
$459$	2080.92	0.211610
$460$	0	0
$461$	1796.10	0.181459	0.0907295	−	0.995876i	$-0.471080\pi$
$461$	1796.10	0.181459	0.0907295	+	0.995876i	$0.471080\pi$
$462$	452.453	0.0455628
$463$	−720.304	−0.0723010	−0.0361505	−	0.999346i	$-0.511510\pi$
$463$	−720.304	−0.0723010	−0.0361505	+	0.999346i	$0.511510\pi$
$464$	6670.16	0.667358
$465$	0	0
$466$	−11545.9	−1.14776
$467$	−131.849	−0.0130647	−0.00653236	−	0.999979i	$-0.502079\pi$
$467$	−131.849	−0.0130647	−0.00653236	+	0.999979i	$0.502079\pi$
$468$	17074.3	1.68645
$469$	20151.4	1.98402
$470$	0	0
$471$	1467.27	0.143542
$472$	−43.1254	−0.00420553
$473$	−2948.43	−0.286615
$474$	−1959.32	−0.189862
$475$	0	0
$476$	−20831.7	−2.00592
$477$	16217.9	1.55675
$478$	−2571.11	−0.246025
$479$	15015.2	1.43228	0.716141	−	0.697956i	$-0.245907\pi$
$479$	15015.2	1.43228	0.716141	+	0.697956i	$0.245907\pi$
$480$	0	0
$481$	−28802.5	−2.73032
$482$	4777.47	0.451468
$483$	−202.772	−0.0191024
$484$	927.782	0.0871321
$485$	0	0
$486$	−3303.75	−0.308356
$487$	10065.4	0.936561	0.468281	−	0.883580i	$-0.344874\pi$
$487$	10065.4	0.936561	0.468281	+	0.883580i	$0.344874\pi$
$488$	−205.664	−0.0190778
$489$	−1.77697	−0.000164330 0
$490$	0	0
$491$	8049.15	0.739823	0.369912	−	0.929067i	$-0.379388\pi$
$491$	8049.15	0.739823	0.369912	+	0.929067i	$0.379388\pi$
$492$	−319.499	−0.0292767
$493$	−10074.3	−0.920334
$494$	−12672.8	−1.15420
$495$	0	0
$496$	−8110.55	−0.734223
$497$	7155.48	0.645810
$498$	−193.455	−0.0174075
$499$	−7688.18	−0.689720	−0.344860	−	0.938654i	$-0.612074\pi$
$499$	−7688.18	−0.689720	−0.344860	+	0.938654i	$0.612074\pi$
$500$	0	0
$501$	791.124	0.0705485
$502$	−10805.2	−0.960676
$503$	3158.78	0.280006	0.140003	−	0.990151i	$-0.455289\pi$
$503$	3158.78	0.280006	0.140003	+	0.990151i	$0.455289\pi$
$504$	−954.902	−0.0843943
$505$	0	0
$506$	−849.620	−0.0746447
$507$	−1799.24	−0.157607
$508$	2603.76	0.227408
$509$	−5722.71	−0.498340	−0.249170	−	0.968460i	$-0.580158\pi$
$509$	−5722.71	−0.498340	−0.249170	+	0.968460i	$0.580158\pi$
$510$	0	0
$511$	9049.74	0.783439
$512$	16129.0	1.39220
$513$	799.291	0.0687906
$514$	−15532.8	−1.33292
$515$	0	0
$516$	790.119	0.0674090
$517$	−6248.81	−0.531571
$518$	−37159.9	−3.15195
$519$	−1555.30	−0.131542
$520$	0	0
$521$	−3228.04	−0.271445	−0.135723	−	0.990747i	$-0.543336\pi$
$521$	−3228.04	−0.271445	−0.135723	+	0.990747i	$0.543336\pi$
$522$	10653.2	0.893250
$523$	8397.13	0.702067	0.351033	−	0.936363i	$-0.385830\pi$
$523$	8397.13	0.702067	0.351033	+	0.936363i	$0.385830\pi$
$524$	11524.1	0.960751
$525$	0	0
$526$	5949.48	0.493174
$527$	12249.8	1.01255
$528$	281.427	0.0231961
$529$	−11786.2	−0.968705
$530$	0	0
$531$	−880.195	−0.0719345
$532$	−8001.53	−0.652087
$533$	−8988.28	−0.730442
$534$	937.103	0.0759408
$535$	0	0
$536$	980.833	0.0790402
$537$	−1386.73	−0.111437
$538$	−8376.01	−0.671218
$539$	4263.78	0.340731
$540$	0	0
$541$	636.671	0.0505964	0.0252982	−	0.999680i	$-0.491946\pi$
$541$	636.671	0.0505964	0.0252982	+	0.999680i	$0.491946\pi$
$542$	−24776.1	−1.96351
$543$	255.521	0.0201942
$544$	−25418.5	−2.00332
$545$	0	0
$546$	−3411.01	−0.267358
$547$	16014.7	1.25181	0.625904	−	0.779900i	$-0.284730\pi$
$547$	16014.7	1.25181	0.625904	+	0.779900i	$0.284730\pi$
$548$	−16594.4	−1.29358
$549$	−4197.62	−0.326321
$550$	0	0
$551$	−3869.59	−0.299183
$552$	−9.86958	−0.000761010 0
$553$	−34802.8	−2.67625
$554$	11372.9	0.872181
$555$	0	0
$556$	−2991.39	−0.228171
$557$	17221.3	1.31004	0.655019	−	0.755612i	$-0.272660\pi$
$557$	17221.3	1.31004	0.655019	+	0.755612i	$0.272660\pi$
$558$	−12953.7	−0.982748
$559$	22228.0	1.68183
$560$	0	0
$561$	−425.055	−0.0319890
$562$	−3035.21	−0.227816
$563$	2137.21	0.159987	0.0799934	−	0.996795i	$-0.474510\pi$
$563$	2137.21	0.159987	0.0799934	+	0.996795i	$0.474510\pi$
$564$	1674.56	0.125020
$565$	0	0
$566$	−20427.0	−1.51698
$567$	−19381.8	−1.43555
$568$	348.280	0.0257280
$569$	−18820.1	−1.38661	−0.693304	−	0.720645i	$-0.743846\pi$
$569$	−18820.1	−1.38661	−0.693304	+	0.720645i	$0.743846\pi$
$570$	0	0
$571$	−12458.4	−0.913081	−0.456540	−	0.889703i	$-0.650912\pi$
$571$	−12458.4	−0.913081	−0.456540	+	0.889703i	$0.650912\pi$
$572$	−6994.49	−0.511283
$573$	−34.4642	−0.00251268
$574$	−11596.3	−0.843243
$575$	0	0
$576$	12583.2	0.910241
$577$	7596.79	0.548108	0.274054	−	0.961714i	$-0.411635\pi$
$577$	7596.79	0.548108	0.274054	+	0.961714i	$0.411635\pi$
$578$	20542.0	1.47826
$579$	814.845	0.0584867
$580$	0	0
$581$	−3436.29	−0.245372
$582$	−882.036	−0.0628206
$583$	−6643.67	−0.471960
$584$	440.480	0.0312109
$585$	0	0
$586$	−15499.2	−1.09260
$587$	−14595.8	−1.02629	−0.513146	−	0.858302i	$-0.671520\pi$
$587$	−14595.8	−1.02629	−0.513146	+	0.858302i	$0.671520\pi$
$588$	−1142.61	−0.0801366
$589$	4705.21	0.329160
$590$	0	0
$591$	1051.97	0.0732187
$592$	−23113.6	−1.60466
$593$	6816.78	0.472060	0.236030	−	0.971746i	$-0.424154\pi$
$593$	6816.78	0.472060	0.236030	+	0.971746i	$0.424154\pi$
$594$	901.430	0.0622662
$595$	0	0
$596$	−20978.2	−1.44178
$597$	480.398	0.0329336
$598$	6405.22	0.438008
$599$	−2023.88	−0.138053	−0.0690263	−	0.997615i	$-0.521989\pi$
$599$	−2023.88	−0.138053	−0.0690263	+	0.997615i	$0.521989\pi$
$600$	0	0
$601$	−14564.3	−0.988503	−0.494252	−	0.869319i	$-0.664558\pi$
$601$	−14564.3	−0.988503	−0.494252	+	0.869319i	$0.664558\pi$
$602$	28677.7	1.94155
$603$	20018.9	1.35196
$604$	14042.1	0.945968
$605$	0	0
$606$	911.925	0.0611294
$607$	29493.8	1.97218	0.986092	−	0.166199i	$-0.0531495\pi$
$607$	29493.8	1.97218	0.986092	+	0.166199i	$0.0531495\pi$
$608$	−9763.35	−0.651243
$609$	−1041.54	−0.0693027
$610$	0	0
$611$	47109.4	3.11922
$612$	−20694.7	−1.36688
$613$	−11895.7	−0.783788	−0.391894	−	0.920010i	$-0.628180\pi$
$613$	−11895.7	−0.783788	−0.391894	+	0.920010i	$0.628180\pi$
$614$	32679.0	2.14791
$615$	0	0
$616$	391.176	0.0255859
$617$	−4875.99	−0.318152	−0.159076	−	0.987266i	$-0.550852\pi$
$617$	−4875.99	−0.318152	−0.159076	+	0.987266i	$0.550852\pi$
$618$	1508.99	0.0982208
$619$	3651.22	0.237084	0.118542	−	0.992949i	$-0.462178\pi$
$619$	3651.22	0.237084	0.118542	+	0.992949i	$0.462178\pi$
$620$	0	0
$621$	−403.987	−0.0261054
$622$	−36342.9	−2.34279
$623$	16645.5	1.07044
$624$	−2121.66	−0.136113
$625$	0	0
$626$	27952.9	1.78470
$627$	−163.265	−0.0103990
$628$	−29264.2	−1.85950
$629$	34909.7	2.21294
$630$	0	0
$631$	−10343.2	−0.652546	−0.326273	−	0.945276i	$-0.605793\pi$
$631$	−10343.2	−0.652546	−0.326273	+	0.945276i	$0.605793\pi$
$632$	−1693.96	−0.106617
$633$	−363.248	−0.0228086
$634$	24232.8	1.51799
$635$	0	0
$636$	1780.37	0.111000
$637$	−32144.3	−1.99938
$638$	−4364.07	−0.270807
$639$	7108.44	0.440071
$640$	0	0
$641$	28447.8	1.75292	0.876459	−	0.481477i	$-0.159899\pi$
$641$	28447.8	1.75292	0.876459	+	0.481477i	$0.159899\pi$
$642$	−1320.30	−0.0811650
$643$	−11854.4	−0.727048	−0.363524	−	0.931585i	$-0.618427\pi$
$643$	−11854.4	−0.727048	−0.363524	+	0.931585i	$0.618427\pi$
$644$	4044.23	0.247461
$645$	0	0
$646$	15359.9	0.935489
$647$	−16073.9	−0.976706	−0.488353	−	0.872646i	$-0.662402\pi$
$647$	−16073.9	−0.976706	−0.488353	+	0.872646i	$0.662402\pi$
$648$	−943.374	−0.0571902
$649$	360.572	0.0218084
$650$	0	0
$651$	1266.46	0.0762464
$652$	35.4410	0.00212880
$653$	−25950.1	−1.55514	−0.777569	−	0.628798i	$-0.783547\pi$
$653$	−25950.1	−1.55514	−0.777569	+	0.628798i	$0.783547\pi$
$654$	1997.11	0.119409
$655$	0	0
$656$	−7212.95	−0.429296
$657$	8990.25	0.533855
$658$	60778.6	3.60091
$659$	−23160.6	−1.36906	−0.684529	−	0.728986i	$-0.739992\pi$
$659$	−23160.6	−1.36906	−0.684529	+	0.728986i	$0.739992\pi$
$660$	0	0
$661$	−24160.6	−1.42169	−0.710847	−	0.703346i	$-0.751688\pi$
$661$	−24160.6	−1.42169	−0.710847	+	0.703346i	$0.751688\pi$
$662$	18666.7	1.09593
$663$	3204.46	0.187709
$664$	−167.255	−0.00977524
$665$	0	0
$666$	−36915.6	−2.14782
$667$	1955.81	0.113537
$668$	−15778.7	−0.913916
$669$	−2019.18	−0.116690
$670$	0	0
$671$	1719.56	0.0989310
$672$	−2627.91	−0.150854
$673$	−7079.68	−0.405500	−0.202750	−	0.979231i	$-0.564988\pi$
$673$	−7079.68	−0.405500	−0.202750	+	0.979231i	$0.564988\pi$
$674$	−37563.8	−2.14674
$675$	0	0
$676$	35885.2	2.04171
$677$	−26190.6	−1.48684	−0.743418	−	0.668827i	$-0.766797\pi$
$677$	−26190.6	−1.48684	−0.743418	+	0.668827i	$0.766797\pi$
$678$	−1383.83	−0.0783859
$679$	−15667.3	−0.885505
$680$	0	0
$681$	221.684	0.0124742
$682$	5306.47	0.297941
$683$	24378.6	1.36577	0.682887	−	0.730524i	$-0.260724\pi$
$683$	24378.6	1.36577	0.682887	+	0.730524i	$0.260724\pi$
$684$	−7948.92	−0.444349
$685$	0	0
$686$	−4773.51	−0.265676
$687$	1775.05	0.0985769
$688$	17837.6	0.988448
$689$	50086.2	2.76942
$690$	0	0
$691$	−20884.1	−1.14974	−0.574869	−	0.818245i	$-0.694947\pi$
$691$	−20884.1	−1.14974	−0.574869	+	0.818245i	$0.694947\pi$
$692$	31020.0	1.70405
$693$	7983.94	0.437640
$694$	25688.1	1.40505
$695$	0	0
$696$	−50.6951	−0.00276091
$697$	10894.1	0.592030
$698$	28322.9	1.53587
$699$	1121.40	0.0606800
$700$	0	0
$701$	−5279.66	−0.284465	−0.142232	−	0.989833i	$-0.545428\pi$
$701$	−5279.66	−0.284465	−0.142232	+	0.989833i	$0.545428\pi$
$702$	−6795.82	−0.365373
$703$	13409.0	0.719387
$704$	−5154.69	−0.275959
$705$	0	0
$706$	−3678.22	−0.196079
$707$	16198.2	0.861666
$708$	−96.6260	−0.00512913
$709$	9859.71	0.522270	0.261135	−	0.965302i	$-0.415903\pi$
$709$	9859.71	0.522270	0.261135	+	0.965302i	$0.415903\pi$
$710$	0	0
$711$	−34574.0	−1.82366
$712$	810.188	0.0426448
$713$	−2378.17	−0.124913
$714$	4134.27	0.216696
$715$	0	0
$716$	27657.8	1.44361
$717$	249.720	0.0130069
$718$	−33355.4	−1.73372
$719$	−11209.6	−0.581431	−0.290715	−	0.956810i	$-0.593893\pi$
$719$	−11209.6	−0.581431	−0.290715	+	0.956810i	$0.593893\pi$
$720$	0	0
$721$	26803.7	1.38450
$722$	−21249.7	−1.09534
$723$	−464.013	−0.0238684
$724$	−5096.28	−0.261605
$725$	0	0
$726$	−184.129	−0.00941274
$727$	−6297.20	−0.321252	−0.160626	−	0.987015i	$-0.551351\pi$
$727$	−6297.20	−0.321252	−0.160626	+	0.987015i	$0.551351\pi$
$728$	−2949.05	−0.150136
$729$	−19039.5	−0.967306
$730$	0	0
$731$	−26941.2	−1.36314
$732$	−460.806	−0.0232676
$733$	11939.8	0.601644	0.300822	−	0.953680i	$-0.402739\pi$
$733$	11939.8	0.601644	0.300822	+	0.953680i	$0.402739\pi$
$734$	−20220.7	−1.01684
$735$	0	0
$736$	4934.71	0.247141
$737$	−8200.75	−0.409876
$738$	−11520.1	−0.574607
$739$	24617.0	1.22537	0.612687	−	0.790325i	$-0.290089\pi$
$739$	24617.0	1.22537	0.612687	+	0.790325i	$0.290089\pi$
$740$	0	0
$741$	1230.85	0.0610206
$742$	64619.2	3.19710
$743$	34374.3	1.69727	0.848634	−	0.528980i	$-0.177425\pi$
$743$	34374.3	1.69727	0.848634	+	0.528980i	$0.177425\pi$
$744$	61.6425	0.00303753
$745$	0	0
$746$	48544.8	2.38251
$747$	−3413.70	−0.167203
$748$	8477.58	0.414400
$749$	−23452.0	−1.14408
$750$	0	0
$751$	23103.7	1.12259	0.561296	−	0.827615i	$-0.310303\pi$
$751$	23103.7	1.12259	0.561296	+	0.827615i	$0.310303\pi$
$752$	37804.5	1.83323
$753$	1049.46	0.0507894
$754$	32900.4	1.58907
$755$	0	0
$756$	−4290.85	−0.206424
$757$	−19530.2	−0.937696	−0.468848	−	0.883279i	$-0.655331\pi$
$757$	−19530.2	−0.937696	−0.468848	+	0.883279i	$0.655331\pi$
$758$	−10786.0	−0.516841
$759$	82.5196	0.00394634
$760$	0	0
$761$	−20257.0	−0.964934	−0.482467	−	0.875914i	$-0.660259\pi$
$761$	−20257.0	−0.964934	−0.482467	+	0.875914i	$0.660259\pi$
$762$	−516.745	−0.0245665
$763$	35474.1	1.68316
$764$	687.378	0.0325503
$765$	0	0
$766$	45939.4	2.16692
$767$	−2718.33	−0.127970
$768$	−1697.27	−0.0797461
$769$	−19356.1	−0.907669	−0.453834	−	0.891086i	$-0.649944\pi$
$769$	−19356.1	−0.907669	−0.453834	+	0.891086i	$0.649944\pi$
$770$	0	0
$771$	1508.63	0.0704694
$772$	−16251.8	−0.757662
$773$	4799.04	0.223298	0.111649	−	0.993748i	$-0.464387\pi$
$773$	4799.04	0.223298	0.111649	+	0.993748i	$0.464387\pi$
$774$	28489.1	1.32302
$775$	0	0
$776$	−762.580	−0.0352771
$777$	3609.17	0.166639
$778$	10510.9	0.484363
$779$	4184.48	0.192458
$780$	0	0
$781$	−2911.97	−0.133417
$782$	−7763.37	−0.355010
$783$	−2075.08	−0.0947092
$784$	−25795.3	−1.17508
$785$	0	0
$786$	−2287.09	−0.103788
$787$	22195.1	1.00530	0.502648	−	0.864491i	$-0.332359\pi$
$787$	22195.1	1.00530	0.502648	+	0.864491i	$0.332359\pi$
$788$	−20981.2	−0.948507
$789$	−577.845	−0.0260733
$790$	0	0
$791$	−24580.5	−1.10491
$792$	388.604	0.0174349
$793$	−12963.6	−0.580519
$794$	−43520.0	−1.94517
$795$	0	0
$796$	−9581.36	−0.426636
$797$	−22630.2	−1.00578	−0.502888	−	0.864352i	$-0.667729\pi$
$797$	−22630.2	−1.00578	−0.502888	+	0.864352i	$0.667729\pi$
$798$	1587.99	0.0704439
$799$	−57098.3	−2.52815
$800$	0	0
$801$	16536.0	0.729428
$802$	−18311.9	−0.806255
$803$	−3682.85	−0.161849
$804$	2197.63	0.0963987
$805$	0	0
$806$	−40005.1	−1.74829
$807$	813.523	0.0354862
$808$	788.420	0.0343274
$809$	17563.6	0.763293	0.381647	−	0.924308i	$-0.375357\pi$
$809$	17563.6	0.763293	0.381647	+	0.924308i	$0.375357\pi$
$810$	0	0
$811$	6370.17	0.275816	0.137908	−	0.990445i	$-0.455962\pi$
$811$	6370.17	0.275816	0.137908	+	0.990445i	$0.455962\pi$
$812$	20773.2	0.897777
$813$	2406.38	0.103808
$814$	15122.5	0.651157
$815$	0	0
$816$	2571.53	0.110320
$817$	−10348.2	−0.443131
$818$	−31632.0	−1.35206
$819$	−60190.4	−2.56804
$820$	0	0
$821$	20761.6	0.882562	0.441281	−	0.897369i	$-0.354524\pi$
$821$	20761.6	0.882562	0.441281	+	0.897369i	$0.354524\pi$
$822$	3293.35	0.139743
$823$	666.345	0.0282228	0.0141114	−	0.999900i	$-0.495508\pi$
$823$	666.345	0.0282228	0.0141114	+	0.999900i	$0.495508\pi$
$824$	1304.62	0.0551562
$825$	0	0
$826$	−3507.08	−0.147732
$827$	−27888.5	−1.17265	−0.586323	−	0.810078i	$-0.699425\pi$
$827$	−27888.5	−1.17265	−0.586323	+	0.810078i	$0.699425\pi$
$828$	4017.64	0.168626
$829$	−24909.8	−1.04361	−0.521804	−	0.853065i	$-0.674741\pi$
$829$	−24909.8	−1.04361	−0.521804	+	0.853065i	$0.674741\pi$
$830$	0	0
$831$	−1104.60	−0.0461108
$832$	38860.9	1.61930
$833$	38960.1	1.62051
$834$	593.674	0.0246490
$835$	0	0
$836$	3256.27	0.134714
$837$	2523.19	0.104198
$838$	55544.0	2.28966
$839$	−7039.78	−0.289679	−0.144839	−	0.989455i	$-0.546267\pi$
$839$	−7039.78	−0.289679	−0.144839	+	0.989455i	$0.546267\pi$
$840$	0	0
$841$	−14343.0	−0.588092
$842$	21002.3	0.859606
$843$	294.796	0.0120442
$844$	7244.86	0.295472
$845$	0	0
$846$	60379.0	2.45375
$847$	−3270.62	−0.132680
$848$	40193.3	1.62765
$849$	1983.98	0.0802002
$850$	0	0
$851$	−6777.33	−0.273001
$852$	780.350	0.0313783
$853$	−15202.7	−0.610235	−0.305118	−	0.952315i	$-0.598696\pi$
$853$	−15202.7	−0.610235	−0.305118	+	0.952315i	$0.598696\pi$
$854$	−16725.1	−0.670167
$855$	0	0
$856$	−1141.49	−0.0455785
$857$	−6963.28	−0.277551	−0.138776	−	0.990324i	$-0.544317\pi$
$857$	−6963.28	−0.277551	−0.138776	+	0.990324i	$0.544317\pi$
$858$	1388.13	0.0552332
$859$	−25295.1	−1.00472	−0.502362	−	0.864658i	$-0.667535\pi$
$859$	−25295.1	−1.00472	−0.502362	+	0.864658i	$0.667535\pi$
$860$	0	0
$861$	1126.30	0.0445809
$862$	65149.5	2.57425
$863$	−40402.3	−1.59364	−0.796819	−	0.604219i	$-0.793485\pi$
$863$	−40402.3	−1.59364	−0.796819	+	0.604219i	$0.793485\pi$
$864$	−5235.63	−0.206157
$865$	0	0
$866$	−24298.1	−0.953444
$867$	−1995.15	−0.0781532
$868$	−25259.1	−0.987729
$869$	14163.2	0.552882
$870$	0	0
$871$	61824.8	2.40511
$872$	1726.64	0.0670543
$873$	−15564.3	−0.603406
$874$	−2981.94	−0.115407
$875$	0	0
$876$	986.930	0.0380654
$877$	−15316.6	−0.589744	−0.294872	−	0.955537i	$-0.595277\pi$
$877$	−15316.6	−0.589744	−0.294872	+	0.955537i	$0.595277\pi$
$878$	−53687.7	−2.06363
$879$	1505.37	0.0577642
$880$	0	0
$881$	−19661.6	−0.751892	−0.375946	−	0.926642i	$-0.622682\pi$
$881$	−19661.6	−0.751892	−0.375946	+	0.926642i	$0.622682\pi$
$882$	−41198.7	−1.57283
$883$	−44446.5	−1.69393	−0.846966	−	0.531646i	$-0.821574\pi$
$883$	−44446.5	−1.69393	−0.846966	+	0.531646i	$0.821574\pi$
$884$	−63911.9	−2.43166
$885$	0	0
$886$	10559.3	0.400392
$887$	−8495.24	−0.321581	−0.160790	−	0.986989i	$-0.551404\pi$
$887$	−8495.24	−0.321581	−0.160790	+	0.986989i	$0.551404\pi$
$888$	175.670	0.00663861
$889$	−9178.78	−0.346284
$890$	0	0
$891$	7887.55	0.296569
$892$	40271.8	1.51166
$893$	−21931.7	−0.821855
$894$	4163.36	0.155753
$895$	0	0
$896$	−4547.93	−0.169571
$897$	−622.110	−0.0231568
$898$	22265.6	0.827409
$899$	−12215.4	−0.453179
$900$	0	0
$901$	−60706.3	−2.24464
$902$	4719.20	0.174204
$903$	−2785.33	−0.102647
$904$	−1196.41	−0.0440178
$905$	0	0
$906$	−2786.81	−0.102191
$907$	27175.1	0.994857	0.497428	−	0.867505i	$-0.334278\pi$
$907$	27175.1	0.994857	0.497428	+	0.867505i	$0.334278\pi$
$908$	−4421.41	−0.161597
$909$	16091.7	0.587161
$910$	0	0
$911$	−15258.1	−0.554911	−0.277455	−	0.960738i	$-0.589491\pi$
$911$	−15258.1	−0.554911	−0.277455	+	0.960738i	$0.589491\pi$
$912$	987.735	0.0358631
$913$	1398.42	0.0506911
$914$	43496.5	1.57411
$915$	0	0
$916$	−35402.7	−1.27701
$917$	−40624.9	−1.46298
$918$	8236.78	0.296138
$919$	1227.88	0.0440739	0.0220370	−	0.999757i	$-0.492985\pi$
$919$	1227.88	0.0440739	0.0220370	+	0.999757i	$0.492985\pi$
$920$	0	0
$921$	−3173.96	−0.113557
$922$	7109.37	0.253942
$923$	21953.2	0.782878
$924$	876.460	0.0312050
$925$	0	0
$926$	−2851.13	−0.101181
$927$	26627.5	0.943432
$928$	25347.1	0.896616
$929$	27542.1	0.972687	0.486344	−	0.873768i	$-0.338330\pi$
$929$	27542.1	0.972687	0.486344	+	0.873768i	$0.338330\pi$
$930$	0	0
$931$	14964.7	0.526799
$932$	−22366.0	−0.786075
$933$	3529.82	0.123860
$934$	−521.888	−0.0182834
$935$	0	0
$936$	−2929.66	−0.102306
$937$	57180.2	1.99359	0.996796	−	0.0799827i	$-0.0254865\pi$
$937$	57180.2	1.99359	0.996796	+	0.0799827i	$0.0254865\pi$
$938$	79764.0	2.77653
$939$	−2714.94	−0.0943542
$940$	0	0
$941$	23474.8	0.813237	0.406618	−	0.913598i	$-0.366708\pi$
$941$	23474.8	0.813237	0.406618	+	0.913598i	$0.366708\pi$
$942$	5807.80	0.200879
$943$	−2114.97	−0.0730360
$944$	−2181.41	−0.0752107
$945$	0	0
$946$	−11670.6	−0.401103
$947$	−13294.2	−0.456180	−0.228090	−	0.973640i	$-0.573248\pi$
$947$	−13294.2	−0.456180	−0.228090	+	0.973640i	$0.573248\pi$
$948$	−3795.46	−0.130032
$949$	27764.8	0.949718
$950$	0	0
$951$	−2353.61	−0.0802536
$952$	3574.35	0.121686
$953$	16727.6	0.568582	0.284291	−	0.958738i	$-0.408242\pi$
$953$	16727.6	0.568582	0.284291	+	0.958738i	$0.408242\pi$
$954$	64194.3	2.17858
$955$	0	0
$956$	−4980.58	−0.168497
$957$	423.862	0.0143171
$958$	59433.8	2.00440
$959$	58498.8	1.96978
$960$	0	0
$961$	−14937.7	−0.501416
$962$	−114007.	−3.82093
$963$	−23297.8	−0.779608
$964$	9254.59	0.309201
$965$	0	0
$966$	−802.621	−0.0267328
$967$	17648.1	0.586893	0.293446	−	0.955976i	$-0.405198\pi$
$967$	17648.1	0.586893	0.293446	+	0.955976i	$0.405198\pi$
$968$	−159.191	−0.00528575
$969$	−1491.83	−0.0494578
$970$	0	0
$971$	−1627.65	−0.0537939	−0.0268970	−	0.999638i	$-0.508563\pi$
$971$	−1627.65	−0.0537939	−0.0268970	+	0.999638i	$0.508563\pi$
$972$	−6399.80	−0.211187
$973$	10545.3	0.347447
$974$	39841.1	1.31067
$975$	0	0
$976$	−10403.1	−0.341183
$977$	10385.4	0.340080	0.170040	−	0.985437i	$-0.445610\pi$
$977$	10385.4	0.340080	0.170040	+	0.985437i	$0.445610\pi$
$978$	−7.03366	−0.000229971 0
$979$	−6773.98	−0.221141
$980$	0	0
$981$	35240.9	1.14695
$982$	31860.4	1.03534
$983$	−26696.5	−0.866211	−0.433105	−	0.901343i	$-0.642582\pi$
$983$	−26696.5	−0.866211	−0.433105	+	0.901343i	$0.642582\pi$
$984$	54.8205	0.00177603
$985$	0	0
$986$	−39876.5	−1.28796
$987$	−5903.15	−0.190374
$988$	−24548.8	−0.790488
$989$	5230.32	0.168164
$990$	0	0
$991$	56889.6	1.82357	0.911784	−	0.410670i	$-0.134705\pi$
$991$	56889.6	1.82357	0.911784	+	0.410670i	$0.134705\pi$
$992$	−30820.7	−0.986451
$993$	−1813.01	−0.0579398
$994$	28323.1	0.903776
$995$	0	0
$996$	−374.749	−0.0119221
$997$	−34587.2	−1.09868	−0.549341	−	0.835598i	$-0.685121\pi$
$997$	−34587.2	−1.09868	−0.549341	+	0.835598i	$0.685121\pi$
$998$	−30431.6	−0.965227
$999$	7190.61	0.227729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.a.f.1.5	✓	5
3.2	odd	2		2475.4.a.bk.1.1		5
5.2	odd	4		275.4.b.g.199.9		10
5.3	odd	4		275.4.b.g.199.2		10
5.4	even	2		275.4.a.i.1.1	yes	5
15.14	odd	2		2475.4.a.bg.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.f.1.5	✓	5	1.1	even	1	trivial
275.4.a.i.1.1	yes	5	5.4	even	2
275.4.b.g.199.2		10	5.3	odd	4
275.4.b.g.199.9		10	5.2	odd	4
2475.4.a.bg.1.5		5	15.14	odd	2
2475.4.a.bk.1.1		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q+3.95823 q^{2} -0.384445 q^{3} +7.66762 q^{4} -1.52172 q^{6} -27.0299 q^{7} -1.31563 q^{8} -26.8522 q^{9} +11.0000 q^{11} -2.94778 q^{12} -82.9282 q^{13} -106.991 q^{14} -66.5486 q^{16} +100.512 q^{17} -106.287 q^{18} +38.6071 q^{19} +10.3915 q^{21} +43.5406 q^{22} -19.5133 q^{23} +0.505788 q^{24} -328.249 q^{26} +20.7032 q^{27} -207.255 q^{28} -100.230 q^{29} +121.874 q^{31} -252.890 q^{32} -4.22889 q^{33} +397.851 q^{34} -205.893 q^{36} +347.319 q^{37} +152.816 q^{38} +31.8813 q^{39} +108.386 q^{41} +41.1320 q^{42} -268.039 q^{43} +84.3438 q^{44} -77.2381 q^{46} -568.074 q^{47} +25.5843 q^{48} +387.616 q^{49} -38.6414 q^{51} -635.862 q^{52} -603.970 q^{53} +81.9481 q^{54} +35.5614 q^{56} -14.8423 q^{57} -396.734 q^{58} +32.7793 q^{59} +156.323 q^{61} +482.407 q^{62} +725.813 q^{63} -468.608 q^{64} -16.7390 q^{66} -745.522 q^{67} +770.689 q^{68} +7.50178 q^{69} -264.725 q^{71} +35.3276 q^{72} -334.805 q^{73} +1374.77 q^{74} +296.025 q^{76} -297.329 q^{77} +126.194 q^{78} +1287.57 q^{79} +717.050 q^{81} +429.018 q^{82} +127.129 q^{83} +79.6782 q^{84} -1060.96 q^{86} +38.5329 q^{87} -14.4720 q^{88} -615.817 q^{89} +2241.54 q^{91} -149.620 q^{92} -46.8539 q^{93} -2248.57 q^{94} +97.2222 q^{96} +579.630 q^{97} +1534.28 q^{98} -295.374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - 12 q^{3} + 12 q^{4} + 8 q^{6} - 24 q^{7} - 27 q^{8} + 31 q^{9} + 55 q^{11} - 3 q^{12} - 111 q^{13} + 47 q^{14} - 56 q^{16} - 40 q^{17} - 217 q^{18} + 205 q^{19} - 94 q^{21} - 22 q^{22} - 287 q^{23}+ \cdots + 341 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 12 * q^3 + 12 * q^4 + 8 * q^6 - 24 * q^7 - 27 * q^8 + 31 * q^9 + 55 * q^11 - 3 * q^12 - 111 * q^13 + 47 * q^14 - 56 * q^16 - 40 * q^17 - 217 * q^18 + 205 * q^19 - 94 * q^21 - 22 * q^22 - 287 * q^23 + 273 * q^24 - 354 * q^26 - 270 * q^27 - 460 * q^28 + 251 * q^29 - 289 * q^31 - 248 * q^32 - 132 * q^33 + 522 * q^34 - 722 * q^36 - 224 * q^37 - 540 * q^38 + 538 * q^39 - 462 * q^41 - 175 * q^42 - 593 * q^43 + 132 * q^44 - 972 * q^46 - 766 * q^47 - 992 * q^48 + 75 * q^49 - 820 * q^51 - 696 * q^53 + 281 * q^54 - 527 * q^56 - 1170 * q^57 - 1461 * q^58 - 22 * q^59 + 720 * q^61 + 998 * q^62 + 168 * q^63 - 317 * q^64 + 88 * q^66 - 1230 * q^67 + 109 * q^68 - 514 * q^69 + 951 * q^71 + 1590 * q^72 - 666 * q^73 + 873 * q^74 + 290 * q^76 - 264 * q^77 - 56 * q^78 - 588 * q^79 + 1885 * q^81 + 1807 * q^82 + 867 * q^83 + 1321 * q^84 - 411 * q^86 - 766 * q^87 - 297 * q^88 - 51 * q^89 + 2172 * q^91 + 4137 * q^92 + 1916 * q^93 - 865 * q^94 + 603 * q^96 - 2849 * q^97 + 4104 * q^98 + 341 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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