LMFDB - Embedded newform 2175.4.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,4,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-1,-18] 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:	$128.329154262$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 $ x^6 - x^5 - 49x^4 + 27x^3 + 692x^2 - 82x - 2588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.35472$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.35472 q^{2} -3.00000 q^{3} -2.45528 q^{4} +7.06417 q^{6} +3.94392 q^{7} +24.6193 q^{8} +9.00000 q^{9} +42.8507 q^{11} +7.36583 q^{12} -36.4288 q^{13} -9.28685 q^{14} -38.3294 q^{16} -17.8279 q^{17} -21.1925 q^{18} +84.6809 q^{19} -11.8318 q^{21} -100.902 q^{22} -93.9097 q^{23} -73.8579 q^{24} +85.7797 q^{26} -27.0000 q^{27} -9.68343 q^{28} +29.0000 q^{29} +122.782 q^{31} -106.699 q^{32} -128.552 q^{33} +41.9798 q^{34} -22.0975 q^{36} +61.2323 q^{37} -199.400 q^{38} +109.286 q^{39} +13.8437 q^{41} +27.8605 q^{42} -397.388 q^{43} -105.210 q^{44} +221.131 q^{46} -235.310 q^{47} +114.988 q^{48} -327.445 q^{49} +53.4837 q^{51} +89.4428 q^{52} +50.4554 q^{53} +63.5775 q^{54} +97.0966 q^{56} -254.043 q^{57} -68.2870 q^{58} -476.712 q^{59} -75.4071 q^{61} -289.119 q^{62} +35.4953 q^{63} +557.882 q^{64} +302.705 q^{66} +698.609 q^{67} +43.7725 q^{68} +281.729 q^{69} +44.6799 q^{71} +221.574 q^{72} -669.611 q^{73} -144.185 q^{74} -207.915 q^{76} +169.000 q^{77} -257.339 q^{78} +269.236 q^{79} +81.0000 q^{81} -32.5981 q^{82} -869.721 q^{83} +29.0503 q^{84} +935.739 q^{86} -87.0000 q^{87} +1054.95 q^{88} -9.85326 q^{89} -143.672 q^{91} +230.574 q^{92} -368.347 q^{93} +554.089 q^{94} +320.098 q^{96} +1524.47 q^{97} +771.043 q^{98} +385.656 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) $ 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * 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 $

$2$	−2.35472	−0.832520	−0.416260	−	0.909246i	$-0.636659\pi$
$2$	−2.35472	−0.832520	−0.416260	+	0.909246i	$0.636659\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.45528	−0.306910
$5$	0	0
$5$	0	0
$6$	7.06417	0.480656
$7$	3.94392	0.212952	0.106476	−	0.994315i	$-0.466043\pi$
$7$	3.94392	0.212952	0.106476	+	0.994315i	$0.466043\pi$
$8$	24.6193	1.08803
$9$	9.00000	0.333333
$10$	0	0
$11$	42.8507	1.17454	0.587271	−	0.809390i	$-0.300202\pi$
$11$	42.8507	1.17454	0.587271	+	0.809390i	$0.300202\pi$
$12$	7.36583	0.177194
$13$	−36.4288	−0.777194	−0.388597	−	0.921408i	$-0.627040\pi$
$13$	−36.4288	−0.777194	−0.388597	+	0.921408i	$0.627040\pi$
$14$	−9.28685	−0.177287
$15$	0	0
$16$	−38.3294	−0.598897
$17$	−17.8279	−0.254347	−0.127174	−	0.991880i	$-0.540591\pi$
$17$	−17.8279	−0.254347	−0.127174	+	0.991880i	$0.540591\pi$
$18$	−21.1925	−0.277507
$19$	84.6809	1.02248	0.511241	−	0.859438i	$-0.329186\pi$
$19$	84.6809	1.02248	0.511241	+	0.859438i	$0.329186\pi$
$20$	0	0
$21$	−11.8318	−0.122948
$22$	−100.902	−0.977831
$23$	−93.9097	−0.851371	−0.425685	−	0.904871i	$-0.639967\pi$
$23$	−93.9097	−0.851371	−0.425685	+	0.904871i	$0.639967\pi$
$24$	−73.8579	−0.628174
$25$	0	0
$26$	85.7797	0.647030
$27$	−27.0000	−0.192450
$28$	−9.68343	−0.0653570
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	122.782	0.711367	0.355683	−	0.934607i	$-0.384248\pi$
$31$	122.782	0.711367	0.355683	+	0.934607i	$0.384248\pi$
$32$	−106.699	−0.589435
$33$	−128.552	−0.678123
$34$	41.9798	0.211749
$35$	0	0
$36$	−22.0975	−0.102303
$37$	61.2323	0.272069	0.136034	−	0.990704i	$-0.456564\pi$
$37$	61.2323	0.272069	0.136034	+	0.990704i	$0.456564\pi$
$38$	−199.400	−0.851236
$39$	109.286	0.448713
$40$	0	0
$41$	13.8437	0.0527323	0.0263662	−	0.999652i	$-0.491606\pi$
$41$	13.8437	0.0527323	0.0263662	+	0.999652i	$0.491606\pi$
$42$	27.8605	0.102357
$43$	−397.388	−1.40933	−0.704664	−	0.709541i	$-0.748902\pi$
$43$	−397.388	−1.40933	−0.704664	+	0.709541i	$0.748902\pi$
$44$	−105.210	−0.360479
$45$	0	0
$46$	221.131	0.708783
$47$	−235.310	−0.730286	−0.365143	−	0.930952i	$-0.618980\pi$
$47$	−235.310	−0.730286	−0.365143	+	0.930952i	$0.618980\pi$
$48$	114.988	0.345773
$49$	−327.445	−0.954652
$50$	0	0
$51$	53.4837	0.146847
$52$	89.4428	0.238529
$53$	50.4554	0.130766	0.0653829	−	0.997860i	$-0.479173\pi$
$53$	50.4554	0.130766	0.0653829	+	0.997860i	$0.479173\pi$
$54$	63.5775	0.160219
$55$	0	0
$56$	97.0966	0.231698
$57$	−254.043	−0.590330
$58$	−68.2870	−0.154595
$59$	−476.712	−1.05191	−0.525955	−	0.850513i	$-0.676292\pi$
$59$	−476.712	−1.05191	−0.525955	+	0.850513i	$0.676292\pi$
$60$	0	0
$61$	−75.4071	−0.158277	−0.0791384	−	0.996864i	$-0.525217\pi$
$61$	−75.4071	−0.158277	−0.0791384	+	0.996864i	$0.525217\pi$
$62$	−289.119	−0.592227
$63$	35.4953	0.0709839
$64$	557.882	1.08961
$65$	0	0
$66$	302.705	0.564551
$67$	698.609	1.27386	0.636931	−	0.770921i	$-0.280204\pi$
$67$	698.609	1.27386	0.636931	+	0.770921i	$0.280204\pi$
$68$	43.7725	0.0780616
$69$	281.729	0.491539
$70$	0	0
$71$	44.6799	0.0746835	0.0373417	−	0.999303i	$-0.488111\pi$
$71$	44.6799	0.0746835	0.0373417	+	0.999303i	$0.488111\pi$
$72$	221.574	0.362676
$73$	−669.611	−1.07359	−0.536795	−	0.843713i	$-0.680365\pi$
$73$	−669.611	−1.07359	−0.536795	+	0.843713i	$0.680365\pi$
$74$	−144.185	−0.226503
$75$	0	0
$76$	−207.915	−0.313809
$77$	169.000	0.250121
$78$	−257.339	−0.373563
$79$	269.236	0.383435	0.191718	−	0.981450i	$-0.438594\pi$
$79$	269.236	0.383435	0.191718	+	0.981450i	$0.438594\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−32.5981	−0.0439007
$83$	−869.721	−1.15017	−0.575086	−	0.818093i	$-0.695031\pi$
$83$	−869.721	−1.15017	−0.575086	+	0.818093i	$0.695031\pi$
$84$	29.0503	0.0377339
$85$	0	0
$86$	935.739	1.17329
$87$	−87.0000	−0.107211
$88$	1054.95	1.27794
$89$	−9.85326	−0.0117353	−0.00586766	−	0.999983i	$-0.501868\pi$
$89$	−9.85326	−0.0117353	−0.00586766	+	0.999983i	$0.501868\pi$
$90$	0	0
$91$	−143.672	−0.165505
$92$	230.574	0.261294
$93$	−368.347	−0.410708
$94$	554.089	0.607978
$95$	0	0
$96$	320.098	0.340311
$97$	1524.47	1.59574	0.797868	−	0.602832i	$-0.205961\pi$
$97$	1524.47	1.59574	0.797868	+	0.602832i	$0.205961\pi$
$98$	771.043	0.794767
$99$	385.656	0.391514
$100$	0	0
$101$	517.330	0.509666	0.254833	−	0.966985i	$-0.417980\pi$
$101$	517.330	0.509666	0.254833	+	0.966985i	$0.417980\pi$
$102$	−125.939	−0.122253
$103$	1447.99	1.38519	0.692595	−	0.721327i	$-0.256467\pi$
$103$	1447.99	1.38519	0.692595	+	0.721327i	$0.256467\pi$
$104$	−896.851	−0.845610
$105$	0	0
$106$	−118.809	−0.108865
$107$	332.747	0.300634	0.150317	−	0.988638i	$-0.451971\pi$
$107$	332.747	0.300634	0.150317	+	0.988638i	$0.451971\pi$
$108$	66.2925	0.0590648
$109$	890.530	0.782544	0.391272	−	0.920275i	$-0.372035\pi$
$109$	890.530	0.782544	0.391272	+	0.920275i	$0.372035\pi$
$110$	0	0
$111$	−183.697	−0.157079
$112$	−151.168	−0.127536
$113$	936.802	0.779884	0.389942	−	0.920839i	$-0.372495\pi$
$113$	936.802	0.779884	0.389942	+	0.920839i	$0.372495\pi$
$114$	598.201	0.491462
$115$	0	0
$116$	−71.2031	−0.0569917
$117$	−327.859	−0.259065
$118$	1122.53	0.875736
$119$	−70.3119	−0.0541637
$120$	0	0
$121$	505.182	0.379551
$122$	177.563	0.131769
$123$	−41.5312	−0.0304450
$124$	−301.465	−0.218325
$125$	0	0
$126$	−83.5816	−0.0590956
$127$	−2450.72	−1.71234	−0.856168	−	0.516698i	$-0.827161\pi$
$127$	−2450.72	−1.71234	−0.856168	+	0.516698i	$0.827161\pi$
$128$	−460.065	−0.317690
$129$	1192.16	0.813676
$130$	0	0
$131$	546.681	0.364609	0.182304	−	0.983242i	$-0.441644\pi$
$131$	546.681	0.364609	0.182304	+	0.983242i	$0.441644\pi$
$132$	315.631	0.208122
$133$	333.975	0.217739
$134$	−1645.03	−1.06052
$135$	0	0
$136$	−438.910	−0.276737
$137$	618.064	0.385436	0.192718	−	0.981254i	$-0.438270\pi$
$137$	618.064	0.385436	0.192718	+	0.981254i	$0.438270\pi$
$138$	−663.394	−0.409216
$139$	1285.51	0.784428	0.392214	−	0.919874i	$-0.371709\pi$
$139$	1285.51	0.784428	0.392214	+	0.919874i	$0.371709\pi$
$140$	0	0
$141$	705.929	0.421631
$142$	−105.209	−0.0621755
$143$	−1561.00	−0.912848
$144$	−344.964	−0.199632
$145$	0	0
$146$	1576.75	0.893786
$147$	982.336	0.551168
$148$	−150.342	−0.0835005
$149$	−125.043	−0.0687512	−0.0343756	−	0.999409i	$-0.510944\pi$
$149$	−125.043	−0.0687512	−0.0343756	+	0.999409i	$0.510944\pi$
$150$	0	0
$151$	1968.65	1.06097	0.530485	−	0.847694i	$-0.322010\pi$
$151$	1968.65	1.06097	0.530485	+	0.847694i	$0.322010\pi$
$152$	2084.78	1.11249
$153$	−160.451	−0.0847824
$154$	−397.948	−0.208231
$155$	0	0
$156$	−268.328	−0.137715
$157$	−1125.84	−0.572304	−0.286152	−	0.958184i	$-0.592376\pi$
$157$	−1125.84	−0.572304	−0.286152	+	0.958184i	$0.592376\pi$
$158$	−633.976	−0.319218
$159$	−151.366	−0.0754977
$160$	0	0
$161$	−370.373	−0.181301
$162$	−190.733	−0.0925023
$163$	−1773.87	−0.852394	−0.426197	−	0.904630i	$-0.640147\pi$
$163$	−1773.87	−0.852394	−0.426197	+	0.904630i	$0.640147\pi$
$164$	−33.9902	−0.0161841
$165$	0	0
$166$	2047.95	0.957542
$167$	1420.91	0.658403	0.329201	−	0.944260i	$-0.393221\pi$
$167$	1420.91	0.658403	0.329201	+	0.944260i	$0.393221\pi$
$168$	−291.290	−0.133771
$169$	−869.944	−0.395969
$170$	0	0
$171$	762.129	0.340827
$172$	975.698	0.432536
$173$	339.659	0.149270	0.0746352	−	0.997211i	$-0.476221\pi$
$173$	339.659	0.149270	0.0746352	+	0.997211i	$0.476221\pi$
$174$	204.861	0.0892556
$175$	0	0
$176$	−1642.44	−0.703430
$177$	1430.14	0.607320
$178$	23.2017	0.00976990
$179$	−2593.10	−1.08278	−0.541389	−	0.840772i	$-0.682101\pi$
$179$	−2593.10	−1.08278	−0.541389	+	0.840772i	$0.682101\pi$
$180$	0	0
$181$	−335.225	−0.137663	−0.0688317	−	0.997628i	$-0.521927\pi$
$181$	−335.225	−0.137663	−0.0688317	+	0.997628i	$0.521927\pi$
$182$	338.309	0.137786
$183$	226.221	0.0913812
$184$	−2311.99	−0.926316
$185$	0	0
$186$	867.356	0.341923
$187$	−763.938	−0.298742
$188$	577.750	0.224132
$189$	−106.486	−0.0409826
$190$	0	0
$191$	927.961	0.351544	0.175772	−	0.984431i	$-0.443758\pi$
$191$	927.961	0.351544	0.175772	+	0.984431i	$0.443758\pi$
$192$	−1673.65	−0.629089
$193$	−2184.47	−0.814725	−0.407362	−	0.913267i	$-0.633551\pi$
$193$	−2184.47	−0.814725	−0.407362	+	0.913267i	$0.633551\pi$
$194$	−3589.70	−1.32848
$195$	0	0
$196$	803.970	0.292992
$197$	3765.55	1.36185	0.680924	−	0.732354i	$-0.261578\pi$
$197$	3765.55	1.36185	0.680924	+	0.732354i	$0.261578\pi$
$198$	−908.114	−0.325944
$199$	−60.0727	−0.0213992	−0.0106996	−	0.999943i	$-0.503406\pi$
$199$	−60.0727	−0.0213992	−0.0106996	+	0.999943i	$0.503406\pi$
$200$	0	0
$201$	−2095.83	−0.735464
$202$	−1218.17	−0.424307
$203$	114.374	0.0395442
$204$	−131.317	−0.0450689
$205$	0	0
$206$	−3409.61	−1.15320
$207$	−845.187	−0.283790
$208$	1396.29	0.465459
$209$	3628.64	1.20095
$210$	0	0
$211$	−2276.29	−0.742682	−0.371341	−	0.928497i	$-0.621102\pi$
$211$	−2276.29	−0.742682	−0.371341	+	0.928497i	$0.621102\pi$
$212$	−123.882	−0.0401333
$213$	−134.040	−0.0431185
$214$	−783.527	−0.250284
$215$	0	0
$216$	−664.721	−0.209391
$217$	484.244	0.151487
$218$	−2096.95	−0.651484
$219$	2008.83	0.619838
$220$	0	0
$221$	649.449	0.197677
$222$	432.556	0.130771
$223$	−3889.04	−1.16784	−0.583922	−	0.811809i	$-0.698483\pi$
$223$	−3889.04	−1.16784	−0.583922	+	0.811809i	$0.698483\pi$
$224$	−420.813	−0.125521
$225$	0	0
$226$	−2205.91	−0.649269
$227$	5689.80	1.66363	0.831817	−	0.555050i	$-0.187301\pi$
$227$	5689.80	1.66363	0.831817	+	0.555050i	$0.187301\pi$
$228$	623.746	0.181178
$229$	−3253.91	−0.938972	−0.469486	−	0.882940i	$-0.655561\pi$
$229$	−3253.91	−0.938972	−0.469486	+	0.882940i	$0.655561\pi$
$230$	0	0
$231$	−507.000	−0.144407
$232$	713.959	0.202042
$233$	−1363.72	−0.383436	−0.191718	−	0.981450i	$-0.561406\pi$
$233$	−1363.72	−0.383436	−0.191718	+	0.981450i	$0.561406\pi$
$234$	772.017	0.215677
$235$	0	0
$236$	1170.46	0.322841
$237$	−807.707	−0.221376
$238$	165.565	0.0450924
$239$	−4788.65	−1.29603	−0.648017	−	0.761626i	$-0.724402\pi$
$239$	−4788.65	−1.29603	−0.648017	+	0.761626i	$0.724402\pi$
$240$	0	0
$241$	−632.898	−0.169164	−0.0845820	−	0.996417i	$-0.526956\pi$
$241$	−632.898	−0.169164	−0.0845820	+	0.996417i	$0.526956\pi$
$242$	−1189.56	−0.315984
$243$	−243.000	−0.0641500
$244$	185.145	0.0485767
$245$	0	0
$246$	97.7944	0.0253461
$247$	−3084.82	−0.794667
$248$	3022.81	0.773988
$249$	2609.16	0.664052
$250$	0	0
$251$	−7413.17	−1.86420	−0.932101	−	0.362198i	$-0.882027\pi$
$251$	−7413.17	−1.86420	−0.932101	+	0.362198i	$0.882027\pi$
$252$	−87.1509	−0.0217857
$253$	−4024.10	−0.999971
$254$	5770.78	1.42555
$255$	0	0
$256$	−3379.73	−0.825130
$257$	3920.20	0.951500	0.475750	−	0.879581i	$-0.342177\pi$
$257$	3920.20	0.951500	0.475750	+	0.879581i	$0.342177\pi$
$258$	−2807.22	−0.677402
$259$	241.496	0.0579375
$260$	0	0
$261$	261.000	0.0618984
$262$	−1287.28	−0.303544
$263$	−2505.55	−0.587448	−0.293724	−	0.955890i	$-0.594895\pi$
$263$	−2505.55	−0.587448	−0.293724	+	0.955890i	$0.594895\pi$
$264$	−3164.86	−0.737817
$265$	0	0
$266$	−786.419	−0.181272
$267$	29.5598	0.00677539
$268$	−1715.28	−0.390960
$269$	−3707.17	−0.840261	−0.420131	−	0.907464i	$-0.638016\pi$
$269$	−3707.17	−0.840261	−0.420131	+	0.907464i	$0.638016\pi$
$270$	0	0
$271$	4768.39	1.06885	0.534426	−	0.845215i	$-0.320528\pi$
$271$	4768.39	1.06885	0.534426	+	0.845215i	$0.320528\pi$
$272$	683.332	0.152328
$273$	431.017	0.0955543
$274$	−1455.37	−0.320883
$275$	0	0
$276$	−691.723	−0.150858
$277$	−2908.10	−0.630798	−0.315399	−	0.948959i	$-0.602138\pi$
$277$	−2908.10	−0.630798	−0.315399	+	0.948959i	$0.602138\pi$
$278$	−3027.02	−0.653053
$279$	1105.04	0.237122
$280$	0	0
$281$	495.963	0.105291	0.0526454	−	0.998613i	$-0.483235\pi$
$281$	495.963	0.105291	0.0526454	+	0.998613i	$0.483235\pi$
$282$	−1662.27	−0.351016
$283$	−7296.12	−1.53254	−0.766271	−	0.642517i	$-0.777890\pi$
$283$	−7296.12	−1.53254	−0.766271	+	0.642517i	$0.777890\pi$
$284$	−109.702	−0.0229211
$285$	0	0
$286$	3675.72	0.759964
$287$	54.5986	0.0112294
$288$	−960.293	−0.196478
$289$	−4595.17	−0.935308
$290$	0	0
$291$	−4573.41	−0.921299
$292$	1644.08	0.329495
$293$	−853.949	−0.170267	−0.0851335	−	0.996370i	$-0.527132\pi$
$293$	−853.949	−0.170267	−0.0851335	+	0.996370i	$0.527132\pi$
$294$	−2313.13	−0.458859
$295$	0	0
$296$	1507.50	0.296018
$297$	−1156.97	−0.226041
$298$	294.442	0.0572368
$299$	3421.02	0.661680
$300$	0	0
$301$	−1567.27	−0.300119
$302$	−4635.63	−0.883280
$303$	−1551.99	−0.294256
$304$	−3245.77	−0.612361
$305$	0	0
$306$	377.818	0.0705831
$307$	694.287	0.129072	0.0645359	−	0.997915i	$-0.479443\pi$
$307$	694.287	0.129072	0.0645359	+	0.997915i	$0.479443\pi$
$308$	−414.942	−0.0767646
$309$	−4343.96	−0.799740
$310$	0	0
$311$	2587.58	0.471794	0.235897	−	0.971778i	$-0.424197\pi$
$311$	2587.58	0.471794	0.235897	+	0.971778i	$0.424197\pi$
$312$	2690.55	0.488213
$313$	3467.68	0.626214	0.313107	−	0.949718i	$-0.398630\pi$
$313$	3467.68	0.626214	0.313107	+	0.949718i	$0.398630\pi$
$314$	2651.04	0.476455
$315$	0	0
$316$	−661.048	−0.117680
$317$	5158.66	0.914003	0.457001	−	0.889466i	$-0.348923\pi$
$317$	5158.66	0.914003	0.457001	+	0.889466i	$0.348923\pi$
$318$	356.426	0.0628533
$319$	1242.67	0.218107
$320$	0	0
$321$	−998.241	−0.173571
$322$	872.125	0.150937
$323$	−1509.68	−0.260065
$324$	−198.878	−0.0341011
$325$	0	0
$326$	4176.97	0.709635
$327$	−2671.59	−0.451802
$328$	340.822	0.0573743
$329$	−928.043	−0.155516
$330$	0	0
$331$	11056.9	1.83609	0.918043	−	0.396480i	$-0.129768\pi$
$331$	11056.9	1.83609	0.918043	+	0.396480i	$0.129768\pi$
$332$	2135.41	0.352999
$333$	551.091	0.0906895
$334$	−3345.85	−0.548134
$335$	0	0
$336$	453.504	0.0736330
$337$	−6128.18	−0.990574	−0.495287	−	0.868729i	$-0.664937\pi$
$337$	−6128.18	−0.990574	−0.495287	+	0.868729i	$0.664937\pi$
$338$	2048.48	0.329652
$339$	−2810.41	−0.450266
$340$	0	0
$341$	5261.31	0.835531
$342$	−1794.60	−0.283745
$343$	−2644.19	−0.416247
$344$	−9783.41	−1.53339
$345$	0	0
$346$	−799.803	−0.124271
$347$	9277.85	1.43533	0.717667	−	0.696386i	$-0.245210\pi$
$347$	9277.85	1.43533	0.717667	+	0.696386i	$0.245210\pi$
$348$	213.609	0.0329042
$349$	4375.41	0.671090	0.335545	−	0.942024i	$-0.391079\pi$
$349$	4375.41	0.671090	0.335545	+	0.942024i	$0.391079\pi$
$350$	0	0
$351$	983.577	0.149571
$352$	−4572.13	−0.692317
$353$	3256.89	0.491067	0.245534	−	0.969388i	$-0.421037\pi$
$353$	3256.89	0.491067	0.245534	+	0.969388i	$0.421037\pi$
$354$	−3367.58	−0.505606
$355$	0	0
$356$	24.1925	0.00360169
$357$	210.936	0.0312714
$358$	6106.03	0.901435
$359$	−6219.24	−0.914315	−0.457158	−	0.889386i	$-0.651132\pi$
$359$	−6219.24	−0.914315	−0.457158	+	0.889386i	$0.651132\pi$
$360$	0	0
$361$	311.863	0.0454677
$362$	789.362	0.114608
$363$	−1515.55	−0.219134
$364$	352.755	0.0507951
$365$	0	0
$366$	−532.688	−0.0760767
$367$	−10388.3	−1.47756	−0.738780	−	0.673947i	$-0.764598\pi$
$367$	−10388.3	−1.47756	−0.738780	+	0.673947i	$0.764598\pi$
$368$	3599.50	0.509883
$369$	124.593	0.0175774
$370$	0	0
$371$	198.992	0.0278468
$372$	904.395	0.126050
$373$	−13200.4	−1.83241	−0.916204	−	0.400712i	$-0.868763\pi$
$373$	−13200.4	−1.83241	−0.916204	+	0.400712i	$0.868763\pi$
$374$	1798.86	0.248708
$375$	0	0
$376$	−5793.15	−0.794572
$377$	−1056.43	−0.144321
$378$	250.745	0.0341189
$379$	−5927.59	−0.803376	−0.401688	−	0.915777i	$-0.631576\pi$
$379$	−5927.59	−0.803376	−0.401688	+	0.915777i	$0.631576\pi$
$380$	0	0
$381$	7352.17	0.988617
$382$	−2185.09	−0.292667
$383$	−7463.33	−0.995713	−0.497857	−	0.867259i	$-0.665879\pi$
$383$	−7463.33	−0.995713	−0.497857	+	0.867259i	$0.665879\pi$
$384$	1380.19	0.183419
$385$	0	0
$386$	5143.83	0.678275
$387$	−3576.49	−0.469776
$388$	−3743.00	−0.489747
$389$	−409.685	−0.0533981	−0.0266991	−	0.999644i	$-0.508500\pi$
$389$	−409.685	−0.0533981	−0.0266991	+	0.999644i	$0.508500\pi$
$390$	0	0
$391$	1674.21	0.216544
$392$	−8061.47	−1.03869
$393$	−1640.04	−0.210507
$394$	−8866.82	−1.13377
$395$	0	0
$396$	−946.893	−0.120160
$397$	−1112.31	−0.140618	−0.0703091	−	0.997525i	$-0.522399\pi$
$397$	−1112.31	−0.140618	−0.0703091	+	0.997525i	$0.522399\pi$
$398$	141.455	0.0178153
$399$	−1001.93	−0.125712
$400$	0	0
$401$	14999.2	1.86789	0.933947	−	0.357411i	$-0.116340\pi$
$401$	14999.2	1.86789	0.933947	+	0.357411i	$0.116340\pi$
$402$	4935.09	0.612289
$403$	−4472.81	−0.552870
$404$	−1270.19	−0.156421
$405$	0	0
$406$	−269.319	−0.0329213
$407$	2623.85	0.319556
$408$	1316.73	0.159774
$409$	7595.21	0.918238	0.459119	−	0.888375i	$-0.348165\pi$
$409$	7595.21	0.918238	0.459119	+	0.888375i	$0.348165\pi$
$410$	0	0
$411$	−1854.19	−0.222532
$412$	−3555.21	−0.425128
$413$	−1880.12	−0.224006
$414$	1990.18	0.236261
$415$	0	0
$416$	3886.92	0.458106
$417$	−3856.53	−0.452890
$418$	−8544.44	−0.999814
$419$	−6501.56	−0.758047	−0.379024	−	0.925387i	$-0.623740\pi$
$419$	−6501.56	−0.758047	−0.379024	+	0.925387i	$0.623740\pi$
$420$	0	0
$421$	−4381.60	−0.507235	−0.253617	−	0.967305i	$-0.581620\pi$
$421$	−4381.60	−0.507235	−0.253617	+	0.967305i	$0.581620\pi$
$422$	5360.02	0.618298
$423$	−2117.79	−0.243429
$424$	1242.18	0.142277
$425$	0	0
$426$	315.626	0.0358971
$427$	−297.400	−0.0337053
$428$	−816.986	−0.0922676
$429$	4683.00	0.527033
$430$	0	0
$431$	−698.269	−0.0780382	−0.0390191	−	0.999238i	$-0.512423\pi$
$431$	−698.269	−0.0780382	−0.0390191	+	0.999238i	$0.512423\pi$
$432$	1034.89	0.115258
$433$	6899.14	0.765707	0.382854	−	0.923809i	$-0.374941\pi$
$433$	6899.14	0.765707	0.382854	+	0.923809i	$0.374941\pi$
$434$	−1140.26	−0.126116
$435$	0	0
$436$	−2186.50	−0.240170
$437$	−7952.36	−0.870510
$438$	−4730.25	−0.516027
$439$	−12502.8	−1.35929	−0.679644	−	0.733542i	$-0.737866\pi$
$439$	−12502.8	−1.35929	−0.679644	+	0.733542i	$0.737866\pi$
$440$	0	0
$441$	−2947.01	−0.318217
$442$	−1529.27	−0.164570
$443$	−7684.41	−0.824148	−0.412074	−	0.911150i	$-0.635195\pi$
$443$	−7684.41	−0.824148	−0.412074	+	0.911150i	$0.635195\pi$
$444$	451.027	0.0482090
$445$	0	0
$446$	9157.61	0.972255
$447$	375.129	0.0396935
$448$	2200.24	0.232035
$449$	6931.63	0.728561	0.364280	−	0.931289i	$-0.381315\pi$
$449$	6931.63	0.728561	0.364280	+	0.931289i	$0.381315\pi$
$450$	0	0
$451$	593.213	0.0619364
$452$	−2300.11	−0.239354
$453$	−5905.95	−0.612552
$454$	−13397.9	−1.38501
$455$	0	0
$456$	−6254.35	−0.642296
$457$	669.132	0.0684916	0.0342458	−	0.999413i	$-0.489097\pi$
$457$	669.132	0.0684916	0.0342458	+	0.999413i	$0.489097\pi$
$458$	7662.06	0.781713
$459$	481.353	0.0489491
$460$	0	0
$461$	1369.33	0.138343	0.0691715	−	0.997605i	$-0.477964\pi$
$461$	1369.33	0.138343	0.0691715	+	0.997605i	$0.477964\pi$
$462$	1193.84	0.120222
$463$	8213.65	0.824451	0.412225	−	0.911082i	$-0.364752\pi$
$463$	8213.65	0.824451	0.412225	+	0.911082i	$0.364752\pi$
$464$	−1111.55	−0.111212
$465$	0	0
$466$	3211.19	0.319218
$467$	−19762.1	−1.95821	−0.979103	−	0.203366i	$-0.934812\pi$
$467$	−19762.1	−1.95821	−0.979103	+	0.203366i	$0.934812\pi$
$468$	804.985	0.0795095
$469$	2755.26	0.271271
$470$	0	0
$471$	3377.52	0.330420
$472$	−11736.3	−1.14451
$473$	−17028.3	−1.65532
$474$	1901.93	0.184300
$475$	0	0
$476$	172.635	0.0166234
$477$	454.099	0.0435886
$478$	11276.0	1.07898
$479$	1437.29	0.137101	0.0685506	−	0.997648i	$-0.478163\pi$
$479$	1437.29	0.137101	0.0685506	+	0.997648i	$0.478163\pi$
$480$	0	0
$481$	−2230.62	−0.211450
$482$	1490.30	0.140833
$483$	1111.12	0.104674
$484$	−1240.36	−0.116488
$485$	0	0
$486$	572.198	0.0534062
$487$	−12731.4	−1.18463	−0.592313	−	0.805708i	$-0.701785\pi$
$487$	−12731.4	−1.18463	−0.592313	+	0.805708i	$0.701785\pi$
$488$	−1856.47	−0.172210
$489$	5321.61	0.492130
$490$	0	0
$491$	−15590.5	−1.43298	−0.716488	−	0.697600i	$-0.754251\pi$
$491$	−15590.5	−1.43298	−0.716488	+	0.697600i	$0.754251\pi$
$492$	101.971	0.00934387
$493$	−517.009	−0.0472311
$494$	7263.91	0.661576
$495$	0	0
$496$	−4706.17	−0.426035
$497$	176.214	0.0159040
$498$	−6143.86	−0.552837
$499$	−7309.53	−0.655750	−0.327875	−	0.944721i	$-0.606333\pi$
$499$	−7309.53	−0.655750	−0.327875	+	0.944721i	$0.606333\pi$
$500$	0	0
$501$	−4262.73	−0.380129
$502$	17456.0	1.55199
$503$	−11671.8	−1.03463	−0.517315	−	0.855795i	$-0.673068\pi$
$503$	−11671.8	−1.03463	−0.517315	+	0.855795i	$0.673068\pi$
$504$	873.869	0.0772326
$505$	0	0
$506$	9475.63	0.832496
$507$	2609.83	0.228613
$508$	6017.21	0.525532
$509$	−9056.40	−0.788640	−0.394320	−	0.918973i	$-0.629020\pi$
$509$	−9056.40	−0.788640	−0.394320	+	0.918973i	$0.629020\pi$
$510$	0	0
$511$	−2640.90	−0.228623
$512$	11638.9	1.00463
$513$	−2286.39	−0.196777
$514$	−9230.99	−0.792143
$515$	0	0
$516$	−2927.09	−0.249725
$517$	−10083.2	−0.857752
$518$	−568.655	−0.0482341
$519$	−1018.98	−0.0861813
$520$	0	0
$521$	−13061.2	−1.09831	−0.549156	−	0.835720i	$-0.685051\pi$
$521$	−13061.2	−1.09831	−0.549156	+	0.835720i	$0.685051\pi$
$522$	−614.583	−0.0515317
$523$	12711.3	1.06277	0.531383	−	0.847132i	$-0.321672\pi$
$523$	12711.3	1.06277	0.531383	+	0.847132i	$0.321672\pi$
$524$	−1342.25	−0.111902
$525$	0	0
$526$	5899.88	0.489063
$527$	−2188.95	−0.180934
$528$	4927.32	0.406125
$529$	−3347.97	−0.275168
$530$	0	0
$531$	−4290.41	−0.350636
$532$	−820.002	−0.0668263
$533$	−504.310	−0.0409833
$534$	−69.6051	−0.00564065
$535$	0	0
$536$	17199.3	1.38600
$537$	7779.30	0.625143
$538$	8729.36	0.699535
$539$	−14031.3	−1.12128
$540$	0	0
$541$	−16338.9	−1.29845	−0.649227	−	0.760595i	$-0.724908\pi$
$541$	−16338.9	−1.29845	−0.649227	+	0.760595i	$0.724908\pi$
$542$	−11228.2	−0.889841
$543$	1005.67	0.0794800
$544$	1902.22	0.149921
$545$	0	0
$546$	−1014.93	−0.0795509
$547$	9583.07	0.749072	0.374536	−	0.927212i	$-0.377802\pi$
$547$	9583.07	0.749072	0.374536	+	0.927212i	$0.377802\pi$
$548$	−1517.52	−0.118294
$549$	−678.664	−0.0527589
$550$	0	0
$551$	2455.75	0.189870
$552$	6935.97	0.534809
$553$	1061.84	0.0816532
$554$	6847.78	0.525152
$555$	0	0
$556$	−3156.28	−0.240749
$557$	1288.54	0.0980198	0.0490099	−	0.998798i	$-0.484393\pi$
$557$	1288.54	0.0980198	0.0490099	+	0.998798i	$0.484393\pi$
$558$	−2602.07	−0.197409
$559$	14476.4	1.09532
$560$	0	0
$561$	2291.81	0.172479
$562$	−1167.86	−0.0876567
$563$	13901.3	1.04062	0.520312	−	0.853976i	$-0.325816\pi$
$563$	13901.3	1.04062	0.520312	+	0.853976i	$0.325816\pi$
$564$	−1733.25	−0.129403
$565$	0	0
$566$	17180.4	1.27587
$567$	319.458	0.0236613
$568$	1099.99	0.0812578
$569$	12631.6	0.930654	0.465327	−	0.885139i	$-0.345937\pi$
$569$	12631.6	0.930654	0.465327	+	0.885139i	$0.345937\pi$
$570$	0	0
$571$	836.256	0.0612894	0.0306447	−	0.999530i	$-0.490244\pi$
$571$	836.256	0.0612894	0.0306447	+	0.999530i	$0.490244\pi$
$572$	3832.69	0.280162
$573$	−2783.88	−0.202964
$574$	−128.564	−0.00934874
$575$	0	0
$576$	5020.94	0.363205
$577$	−55.7407	−0.00402169	−0.00201085	−	0.999998i	$-0.500640\pi$
$577$	−55.7407	−0.00402169	−0.00201085	+	0.999998i	$0.500640\pi$
$578$	10820.3	0.778663
$579$	6553.42	0.470382
$580$	0	0
$581$	−3430.11	−0.244931
$582$	10769.1	0.767000
$583$	2162.05	0.153590
$584$	−16485.4	−1.16810
$585$	0	0
$586$	2010.81	0.141751
$587$	24044.3	1.69065	0.845327	−	0.534250i	$-0.179406\pi$
$587$	24044.3	1.69065	0.845327	+	0.534250i	$0.179406\pi$
$588$	−2411.91	−0.169159
$589$	10397.3	0.727359
$590$	0	0
$591$	−11296.6	−0.786264
$592$	−2347.00	−0.162941
$593$	15066.0	1.04332	0.521659	−	0.853154i	$-0.325313\pi$
$593$	15066.0	1.04332	0.521659	+	0.853154i	$0.325313\pi$
$594$	2724.34	0.188184
$595$	0	0
$596$	307.016	0.0211004
$597$	180.218	0.0123548
$598$	−8055.55	−0.550862
$599$	8354.77	0.569894	0.284947	−	0.958543i	$-0.408024\pi$
$599$	8354.77	0.569894	0.284947	+	0.958543i	$0.408024\pi$
$600$	0	0
$601$	5059.15	0.343372	0.171686	−	0.985152i	$-0.445078\pi$
$601$	5059.15	0.343372	0.171686	+	0.985152i	$0.445078\pi$
$602$	3690.48	0.249855
$603$	6287.48	0.424620
$604$	−4833.59	−0.325622
$605$	0	0
$606$	3654.51	0.244974
$607$	−15350.8	−1.02647	−0.513236	−	0.858248i	$-0.671553\pi$
$607$	−15350.8	−1.02647	−0.513236	+	0.858248i	$0.671553\pi$
$608$	−9035.39	−0.602687
$609$	−343.121	−0.0228308
$610$	0	0
$611$	8572.04	0.567574
$612$	393.952	0.0260205
$613$	17964.0	1.18362	0.591811	−	0.806077i	$-0.298413\pi$
$613$	17964.0	1.18362	0.591811	+	0.806077i	$0.298413\pi$
$614$	−1634.85	−0.107455
$615$	0	0
$616$	4160.66	0.272139
$617$	−14791.0	−0.965093	−0.482546	−	0.875870i	$-0.660288\pi$
$617$	−14791.0	−0.965093	−0.482546	+	0.875870i	$0.660288\pi$
$618$	10228.8	0.665799
$619$	−12870.4	−0.835708	−0.417854	−	0.908514i	$-0.637218\pi$
$619$	−12870.4	−0.835708	−0.417854	+	0.908514i	$0.637218\pi$
$620$	0	0
$621$	2535.56	0.163846
$622$	−6093.03	−0.392779
$623$	−38.8605	−0.00249906
$624$	−4188.88	−0.268733
$625$	0	0
$626$	−8165.43	−0.521336
$627$	−10885.9	−0.693368
$628$	2764.25	0.175646
$629$	−1091.64	−0.0691998
$630$	0	0
$631$	−16299.2	−1.02831	−0.514154	−	0.857698i	$-0.671894\pi$
$631$	−16299.2	−1.02831	−0.514154	+	0.857698i	$0.671894\pi$
$632$	6628.39	0.417189
$633$	6828.86	0.428788
$634$	−12147.2	−0.760926
$635$	0	0
$636$	371.646	0.0231710
$637$	11928.4	0.741950
$638$	−2926.14	−0.181579
$639$	402.119	0.0248945
$640$	0	0
$641$	−9154.93	−0.564115	−0.282058	−	0.959397i	$-0.591017\pi$
$641$	−9154.93	−0.564115	−0.282058	+	0.959397i	$0.591017\pi$
$642$	2350.58	0.144502
$643$	−7509.43	−0.460564	−0.230282	−	0.973124i	$-0.573965\pi$
$643$	−7509.43	−0.460564	−0.230282	+	0.973124i	$0.573965\pi$
$644$	909.368	0.0556430
$645$	0	0
$646$	3554.89	0.216510
$647$	19945.4	1.21195	0.605977	−	0.795482i	$-0.292782\pi$
$647$	19945.4	1.21195	0.605977	+	0.795482i	$0.292782\pi$
$648$	1994.16	0.120892
$649$	−20427.5	−1.23551
$650$	0	0
$651$	−1452.73	−0.0874610
$652$	4355.34	0.261608
$653$	8028.91	0.481157	0.240579	−	0.970630i	$-0.422663\pi$
$653$	8028.91	0.481157	0.240579	+	0.970630i	$0.422663\pi$
$654$	6290.86	0.376134
$655$	0	0
$656$	−530.621	−0.0315812
$657$	−6026.50	−0.357863
$658$	2185.28	0.129470
$659$	−2943.00	−0.173965	−0.0869827	−	0.996210i	$-0.527722\pi$
$659$	−2943.00	−0.173965	−0.0869827	+	0.996210i	$0.527722\pi$
$660$	0	0
$661$	−30286.3	−1.78215	−0.891073	−	0.453859i	$-0.850047\pi$
$661$	−30286.3	−1.78215	−0.891073	+	0.453859i	$0.850047\pi$
$662$	−26036.0	−1.52858
$663$	−1948.35	−0.114129
$664$	−21411.9	−1.25142
$665$	0	0
$666$	−1297.67	−0.0755009
$667$	−2723.38	−0.158096
$668$	−3488.73	−0.202070
$669$	11667.1	0.674256
$670$	0	0
$671$	−3231.25	−0.185903
$672$	1262.44	0.0724698
$673$	−2201.07	−0.126070	−0.0630348	−	0.998011i	$-0.520078\pi$
$673$	−2201.07	−0.126070	−0.0630348	+	0.998011i	$0.520078\pi$
$674$	14430.2	0.824673
$675$	0	0
$676$	2135.95	0.121527
$677$	−22392.9	−1.27124	−0.635619	−	0.772003i	$-0.719255\pi$
$677$	−22392.9	−1.27124	−0.635619	+	0.772003i	$0.719255\pi$
$678$	6617.73	0.374856
$679$	6012.39	0.339815
$680$	0	0
$681$	−17069.4	−0.960500
$682$	−12388.9	−0.695596
$683$	31770.8	1.77991	0.889953	−	0.456053i	$-0.150737\pi$
$683$	31770.8	1.77991	0.889953	+	0.456053i	$0.150737\pi$
$684$	−1871.24	−0.104603
$685$	0	0
$686$	6226.33	0.346534
$687$	9761.73	0.542115
$688$	15231.6	0.844042
$689$	−1838.03	−0.101630
$690$	0	0
$691$	−19995.8	−1.10084	−0.550418	−	0.834889i	$-0.685532\pi$
$691$	−19995.8	−1.10084	−0.550418	+	0.834889i	$0.685532\pi$
$692$	−833.957	−0.0458126
$693$	1521.00	0.0833737
$694$	−21846.8	−1.19495
$695$	0	0
$696$	−2141.88	−0.116649
$697$	−246.804	−0.0134123
$698$	−10302.9	−0.558696
$699$	4091.17	0.221377
$700$	0	0
$701$	5940.97	0.320096	0.160048	−	0.987109i	$-0.448835\pi$
$701$	5940.97	0.320096	0.160048	+	0.987109i	$0.448835\pi$
$702$	−2316.05	−0.124521
$703$	5185.21	0.278185
$704$	23905.6	1.27980
$705$	0	0
$706$	−7669.08	−0.408824
$707$	2040.31	0.108534
$708$	−3511.38	−0.186393
$709$	−17955.3	−0.951094	−0.475547	−	0.879690i	$-0.657750\pi$
$709$	−17955.3	−0.951094	−0.475547	+	0.879690i	$0.657750\pi$
$710$	0	0
$711$	2423.12	0.127812
$712$	−242.580	−0.0127684
$713$	−11530.5	−0.605637
$714$	−496.695	−0.0260341
$715$	0	0
$716$	6366.78	0.332315
$717$	14366.0	0.748266
$718$	14644.6	0.761186
$719$	−22809.5	−1.18310	−0.591551	−	0.806268i	$-0.701484\pi$
$719$	−22809.5	−1.18310	−0.591551	+	0.806268i	$0.701484\pi$
$720$	0	0
$721$	5710.75	0.294979
$722$	−734.351	−0.0378528
$723$	1898.69	0.0976669
$724$	823.070	0.0422502
$725$	0	0
$726$	3568.69	0.182433
$727$	28267.1	1.44205	0.721023	−	0.692911i	$-0.243672\pi$
$727$	28267.1	1.44205	0.721023	+	0.692911i	$0.243672\pi$
$728$	−3537.11	−0.180074
$729$	729.000	0.0370370
$730$	0	0
$731$	7084.59	0.358458
$732$	−555.436	−0.0280458
$733$	−18679.6	−0.941264	−0.470632	−	0.882330i	$-0.655974\pi$
$733$	−18679.6	−0.941264	−0.470632	+	0.882330i	$0.655974\pi$
$734$	24461.6	1.23010
$735$	0	0
$736$	10020.1	0.501828
$737$	29935.9	1.49620
$738$	−293.383	−0.0146336
$739$	−8577.65	−0.426974	−0.213487	−	0.976946i	$-0.568482\pi$
$739$	−8577.65	−0.426974	−0.213487	+	0.976946i	$0.568482\pi$
$740$	0	0
$741$	9254.47	0.458801
$742$	−468.572	−0.0231830
$743$	−12442.0	−0.614337	−0.307169	−	0.951655i	$-0.599382\pi$
$743$	−12442.0	−0.614337	−0.307169	+	0.951655i	$0.599382\pi$
$744$	−9068.44	−0.446862
$745$	0	0
$746$	31083.2	1.52552
$747$	−7827.49	−0.383391
$748$	1875.68	0.0916867
$749$	1312.33	0.0640206
$750$	0	0
$751$	−28327.2	−1.37640	−0.688198	−	0.725523i	$-0.741598\pi$
$751$	−28327.2	−1.37640	−0.688198	+	0.725523i	$0.741598\pi$
$752$	9019.27	0.437366
$753$	22239.5	1.07630
$754$	2487.61	0.120150
$755$	0	0
$756$	261.453	0.0125780
$757$	−28073.3	−1.34787	−0.673937	−	0.738789i	$-0.735398\pi$
$757$	−28073.3	−1.34787	−0.673937	+	0.738789i	$0.735398\pi$
$758$	13957.8	0.668827
$759$	12072.3	0.577334
$760$	0	0
$761$	−12715.7	−0.605708	−0.302854	−	0.953037i	$-0.597939\pi$
$761$	−12715.7	−0.605708	−0.302854	+	0.953037i	$0.597939\pi$
$762$	−17312.3	−0.823044
$763$	3512.18	0.166644
$764$	−2278.40	−0.107892
$765$	0	0
$766$	17574.1	0.828952
$767$	17366.0	0.817538
$768$	10139.2	0.476389
$769$	−2730.97	−0.128064	−0.0640320	−	0.997948i	$-0.520396\pi$
$769$	−2730.97	−0.128064	−0.0640320	+	0.997948i	$0.520396\pi$
$770$	0	0
$771$	−11760.6	−0.549349
$772$	5363.49	0.250047
$773$	−21816.6	−1.01512	−0.507561	−	0.861616i	$-0.669453\pi$
$773$	−21816.6	−1.01512	−0.507561	+	0.861616i	$0.669453\pi$
$774$	8421.65	0.391098
$775$	0	0
$776$	37531.3	1.73621
$777$	−724.487	−0.0334502
$778$	964.696	0.0444550
$779$	1172.30	0.0539178
$780$	0	0
$781$	1914.56	0.0877189
$782$	−3942.31	−0.180277
$783$	−783.000	−0.0357371
$784$	12550.8	0.571738
$785$	0	0
$786$	3861.85	0.175251
$787$	−16740.5	−0.758239	−0.379119	−	0.925348i	$-0.623773\pi$
$787$	−16740.5	−0.758239	−0.379119	+	0.925348i	$0.623773\pi$
$788$	−9245.47	−0.417965
$789$	7516.65	0.339163
$790$	0	0
$791$	3694.67	0.166078
$792$	9494.58	0.425979
$793$	2746.99	0.123012
$794$	2619.19	0.117068
$795$	0	0
$796$	147.495	0.00656762
$797$	−3731.13	−0.165826	−0.0829131	−	0.996557i	$-0.526422\pi$
$797$	−3731.13	−0.165826	−0.0829131	+	0.996557i	$0.526422\pi$
$798$	2359.26	0.104658
$799$	4195.07	0.185746
$800$	0	0
$801$	−88.6794	−0.00391177
$802$	−35319.0	−1.55506
$803$	−28693.3	−1.26098
$804$	5145.84	0.225721
$805$	0	0
$806$	10532.2	0.460276
$807$	11121.5	0.485125
$808$	12736.3	0.554531
$809$	−1318.79	−0.0573129	−0.0286564	−	0.999589i	$-0.509123\pi$
$809$	−1318.79	−0.0573129	−0.0286564	+	0.999589i	$0.509123\pi$
$810$	0	0
$811$	−6890.79	−0.298358	−0.149179	−	0.988810i	$-0.547663\pi$
$811$	−6890.79	−0.298358	−0.149179	+	0.988810i	$0.547663\pi$
$812$	−280.819	−0.0121365
$813$	−14305.2	−0.617102
$814$	−6178.44	−0.266037
$815$	0	0
$816$	−2050.00	−0.0879464
$817$	−33651.2	−1.44101
$818$	−17884.6	−0.764452
$819$	−1293.05	−0.0551683
$820$	0	0
$821$	25055.7	1.06510	0.532552	−	0.846397i	$-0.321233\pi$
$821$	25055.7	1.06510	0.532552	+	0.846397i	$0.321233\pi$
$822$	4366.11	0.185262
$823$	28476.2	1.20610	0.603049	−	0.797704i	$-0.293952\pi$
$823$	28476.2	1.20610	0.603049	+	0.797704i	$0.293952\pi$
$824$	35648.4	1.50713
$825$	0	0
$826$	4427.15	0.186490
$827$	6404.82	0.269308	0.134654	−	0.990893i	$-0.457008\pi$
$827$	6404.82	0.269308	0.134654	+	0.990893i	$0.457008\pi$
$828$	2075.17	0.0870980
$829$	42500.5	1.78058	0.890292	−	0.455390i	$-0.150500\pi$
$829$	42500.5	1.78058	0.890292	+	0.455390i	$0.150500\pi$
$830$	0	0
$831$	8724.31	0.364191
$832$	−20323.0	−0.846841
$833$	5837.66	0.242813
$834$	9081.06	0.377040
$835$	0	0
$836$	−8909.31	−0.368583
$837$	−3315.12	−0.136903
$838$	15309.4	0.631090
$839$	7738.21	0.318418	0.159209	−	0.987245i	$-0.449106\pi$
$839$	7738.21	0.318418	0.159209	+	0.987245i	$0.449106\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	10317.4	0.422283
$843$	−1487.89	−0.0607896
$844$	5588.91	0.227936
$845$	0	0
$846$	4986.80	0.202659
$847$	1992.40	0.0808260
$848$	−1933.93	−0.0783152
$849$	21888.4	0.884814
$850$	0	0
$851$	−5750.31	−0.231631
$852$	329.105	0.0132335
$853$	−31504.1	−1.26457	−0.632286	−	0.774735i	$-0.717883\pi$
$853$	−31504.1	−1.26457	−0.632286	+	0.774735i	$0.717883\pi$
$854$	700.294	0.0280604
$855$	0	0
$856$	8191.99	0.327099
$857$	33621.2	1.34011	0.670057	−	0.742310i	$-0.266270\pi$
$857$	33621.2	1.34011	0.670057	+	0.742310i	$0.266270\pi$
$858$	−11027.2	−0.438766
$859$	35655.1	1.41622	0.708111	−	0.706101i	$-0.249547\pi$
$859$	35655.1	1.41622	0.708111	+	0.706101i	$0.249547\pi$
$860$	0	0
$861$	−163.796	−0.00648332
$862$	1644.23	0.0649684
$863$	20500.2	0.808615	0.404307	−	0.914623i	$-0.367513\pi$
$863$	20500.2	0.808615	0.404307	+	0.914623i	$0.367513\pi$
$864$	2880.88	0.113437
$865$	0	0
$866$	−16245.6	−0.637467
$867$	13785.5	0.540000
$868$	−1188.95	−0.0464928
$869$	11536.9	0.450361
$870$	0	0
$871$	−25449.5	−0.990038
$872$	21924.2	0.851431
$873$	13720.2	0.531912
$874$	18725.6	0.724718
$875$	0	0
$876$	−4932.25	−0.190234
$877$	15240.7	0.586822	0.293411	−	0.955986i	$-0.405210\pi$
$877$	15240.7	0.586822	0.293411	+	0.955986i	$0.405210\pi$
$878$	29440.7	1.13164
$879$	2561.85	0.0983036
$880$	0	0
$881$	−17749.3	−0.678761	−0.339381	−	0.940649i	$-0.610217\pi$
$881$	−17749.3	−0.678761	−0.339381	+	0.940649i	$0.610217\pi$
$882$	6939.39	0.264922
$883$	2290.27	0.0872861	0.0436431	−	0.999047i	$-0.486104\pi$
$883$	2290.27	0.0872861	0.0436431	+	0.999047i	$0.486104\pi$
$884$	−1594.58	−0.0606690
$885$	0	0
$886$	18094.7	0.686120
$887$	45630.9	1.72732	0.863662	−	0.504072i	$-0.168165\pi$
$887$	45630.9	1.72732	0.863662	+	0.504072i	$0.168165\pi$
$888$	−4522.49	−0.170906
$889$	−9665.47	−0.364645
$890$	0	0
$891$	3470.91	0.130505
$892$	9548.68	0.358423
$893$	−19926.2	−0.746703
$894$	−883.326	−0.0330457
$895$	0	0
$896$	−1814.46	−0.0676527
$897$	−10263.0	−0.382021
$898$	−16322.1	−0.606542
$899$	3560.69	0.132097
$900$	0	0
$901$	−899.514	−0.0332599
$902$	−1396.85	−0.0515633
$903$	4701.80	0.173274
$904$	23063.4	0.848537
$905$	0	0
$906$	13906.9	0.509962
$907$	−38959.7	−1.42628	−0.713140	−	0.701022i	$-0.752728\pi$
$907$	−38959.7	−1.42628	−0.713140	+	0.701022i	$0.752728\pi$
$908$	−13970.0	−0.510586
$909$	4655.97	0.169889
$910$	0	0
$911$	34221.7	1.24458	0.622291	−	0.782786i	$-0.286202\pi$
$911$	34221.7	1.24458	0.622291	+	0.782786i	$0.286202\pi$
$912$	9737.31	0.353547
$913$	−37268.1	−1.35093
$914$	−1575.62	−0.0570207
$915$	0	0
$916$	7989.26	0.288180
$917$	2156.07	0.0776441
$918$	−1133.45	−0.0407511
$919$	519.522	0.0186479	0.00932396	−	0.999957i	$-0.497032\pi$
$919$	519.522	0.0186479	0.00932396	+	0.999957i	$0.497032\pi$
$920$	0	0
$921$	−2082.86	−0.0745197
$922$	−3224.40	−0.115173
$923$	−1627.63	−0.0580436
$924$	1244.82	0.0443201
$925$	0	0
$926$	−19340.9	−0.686372
$927$	13031.9	0.461730
$928$	−3094.28	−0.109455
$929$	36926.5	1.30411	0.652056	−	0.758171i	$-0.273907\pi$
$929$	36926.5	1.30411	0.652056	+	0.758171i	$0.273907\pi$
$930$	0	0
$931$	−27728.4	−0.976113
$932$	3348.32	0.117680
$933$	−7762.73	−0.272391
$934$	46534.3	1.63025
$935$	0	0
$936$	−8071.66	−0.281870
$937$	−24916.0	−0.868697	−0.434348	−	0.900745i	$-0.643021\pi$
$937$	−24916.0	−0.868697	−0.434348	+	0.900745i	$0.643021\pi$
$938$	−6487.88	−0.225839
$939$	−10403.0	−0.361545
$940$	0	0
$941$	−41585.1	−1.44063	−0.720317	−	0.693645i	$-0.756004\pi$
$941$	−41585.1	−1.44063	−0.720317	+	0.693645i	$0.756004\pi$
$942$	−7953.12	−0.275081
$943$	−1300.06	−0.0448948
$944$	18272.1	0.629985
$945$	0	0
$946$	40097.1	1.37808
$947$	−173.666	−0.00595923	−0.00297962	−	0.999996i	$-0.500948\pi$
$947$	−173.666	−0.00595923	−0.00297962	+	0.999996i	$0.500948\pi$
$948$	1983.15	0.0679426
$949$	24393.1	0.834388
$950$	0	0
$951$	−15476.0	−0.527700
$952$	−1731.03	−0.0589317
$953$	22394.5	0.761207	0.380603	−	0.924738i	$-0.375716\pi$
$953$	22394.5	0.761207	0.380603	+	0.924738i	$0.375716\pi$
$954$	−1069.28	−0.0362884
$955$	0	0
$956$	11757.5	0.397766
$957$	−3728.01	−0.125924
$958$	−3384.42	−0.114140
$959$	2437.60	0.0820793
$960$	0	0
$961$	−14715.5	−0.493957
$962$	5252.49	0.176037
$963$	2994.72	0.100211
$964$	1553.94	0.0519181
$965$	0	0
$966$	−2616.38	−0.0871434
$967$	−35240.6	−1.17194	−0.585968	−	0.810334i	$-0.699285\pi$
$967$	−35240.6	−1.17194	−0.585968	+	0.810334i	$0.699285\pi$
$968$	12437.2	0.412962
$969$	4529.05	0.150149
$970$	0	0
$971$	−23385.4	−0.772888	−0.386444	−	0.922313i	$-0.626297\pi$
$971$	−23385.4	−0.772888	−0.386444	+	0.922313i	$0.626297\pi$
$972$	596.633	0.0196883
$973$	5069.95	0.167045
$974$	29978.8	0.986226
$975$	0	0
$976$	2890.31	0.0947915
$977$	−37442.2	−1.22608	−0.613041	−	0.790051i	$-0.710054\pi$
$977$	−37442.2	−1.22608	−0.613041	+	0.790051i	$0.710054\pi$
$978$	−12530.9	−0.409708
$979$	−422.219	−0.0137836
$980$	0	0
$981$	8014.77	0.260848
$982$	36711.4	1.19298
$983$	35213.6	1.14256	0.571282	−	0.820754i	$-0.306446\pi$
$983$	35213.6	1.14256	0.571282	+	0.820754i	$0.306446\pi$
$984$	−1022.47	−0.0331251
$985$	0	0
$986$	1217.41	0.0393208
$987$	2784.13	0.0897870
$988$	7574.10	0.243891
$989$	37318.6	1.19986
$990$	0	0
$991$	−43883.0	−1.40665	−0.703324	−	0.710869i	$-0.748302\pi$
$991$	−43883.0	−1.40665	−0.703324	+	0.710869i	$0.748302\pi$
$992$	−13100.8	−0.419305
$993$	−33170.8	−1.06007
$994$	−414.935	−0.0132404
$995$	0	0
$996$	−6406.22	−0.203804
$997$	8968.04	0.284875	0.142438	−	0.989804i	$-0.454506\pi$
$997$	8968.04	0.284875	0.142438	+	0.989804i	$0.454506\pi$
$998$	17211.9	0.545925
$999$	−1653.27	−0.0523596

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.k.1.3		6
5.4	even	2		435.4.a.h.1.4	✓	6
15.14	odd	2		1305.4.a.h.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.4	✓	6	5.4	even	2
1305.4.a.h.1.3		6	15.14	odd	2
2175.4.a.k.1.3		6	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35472 q^{2} -3.00000 q^{3} -2.45528 q^{4} +7.06417 q^{6} +3.94392 q^{7} +24.6193 q^{8} +9.00000 q^{9} +42.8507 q^{11} +7.36583 q^{12} -36.4288 q^{13} -9.28685 q^{14} -38.3294 q^{16} -17.8279 q^{17} -21.1925 q^{18} +84.6809 q^{19} -11.8318 q^{21} -100.902 q^{22} -93.9097 q^{23} -73.8579 q^{24} +85.7797 q^{26} -27.0000 q^{27} -9.68343 q^{28} +29.0000 q^{29} +122.782 q^{31} -106.699 q^{32} -128.552 q^{33} +41.9798 q^{34} -22.0975 q^{36} +61.2323 q^{37} -199.400 q^{38} +109.286 q^{39} +13.8437 q^{41} +27.8605 q^{42} -397.388 q^{43} -105.210 q^{44} +221.131 q^{46} -235.310 q^{47} +114.988 q^{48} -327.445 q^{49} +53.4837 q^{51} +89.4428 q^{52} +50.4554 q^{53} +63.5775 q^{54} +97.0966 q^{56} -254.043 q^{57} -68.2870 q^{58} -476.712 q^{59} -75.4071 q^{61} -289.119 q^{62} +35.4953 q^{63} +557.882 q^{64} +302.705 q^{66} +698.609 q^{67} +43.7725 q^{68} +281.729 q^{69} +44.6799 q^{71} +221.574 q^{72} -669.611 q^{73} -144.185 q^{74} -207.915 q^{76} +169.000 q^{77} -257.339 q^{78} +269.236 q^{79} +81.0000 q^{81} -32.5981 q^{82} -869.721 q^{83} +29.0503 q^{84} +935.739 q^{86} -87.0000 q^{87} +1054.95 q^{88} -9.85326 q^{89} -143.672 q^{91} +230.574 q^{92} -368.347 q^{93} +554.089 q^{94} +320.098 q^{96} +1524.47 q^{97} +771.043 q^{98} +385.656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) \) 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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