LMFDB - Embedded newform 175.4.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,4,Mod(1,175)] 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("175.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.3253342510$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +17.2132 q^{6} +7.00000 q^{7} +24.0833 q^{8} +17.3137 q^{9} +38.2548 q^{11} +8.74517 q^{12} -19.3431 q^{13} -18.1005 q^{14} -51.7645 q^{16} +87.2254 q^{17} -44.7696 q^{18} -44.2254 q^{19} -46.5980 q^{21} -98.9188 q^{22} -218.167 q^{23} -160.319 q^{24} +50.0172 q^{26} +64.4802 q^{27} -9.19596 q^{28} -46.9411 q^{29} +194.558 q^{31} -58.8141 q^{32} -254.657 q^{33} -225.546 q^{34} -22.7452 q^{36} -366.853 q^{37} +114.357 q^{38} +128.765 q^{39} -339.362 q^{41} +120.492 q^{42} +226.167 q^{43} -50.2557 q^{44} +564.132 q^{46} -11.6762 q^{47} +344.589 q^{48} +49.0000 q^{49} -580.647 q^{51} +25.4113 q^{52} +209.019 q^{53} -166.732 q^{54} +168.583 q^{56} +294.402 q^{57} +121.380 q^{58} -616.000 q^{59} +320.735 q^{61} -503.087 q^{62} +121.196 q^{63} +566.197 q^{64} +658.488 q^{66} -14.5097 q^{67} -114.589 q^{68} +1452.30 q^{69} -952.000 q^{71} +416.971 q^{72} -824.489 q^{73} +948.603 q^{74} +58.0993 q^{76} +267.784 q^{77} -332.958 q^{78} +156.275 q^{79} -896.706 q^{81} +877.519 q^{82} +1036.53 q^{83} +61.2162 q^{84} -584.818 q^{86} +312.480 q^{87} +921.301 q^{88} -170.225 q^{89} -135.402 q^{91} +286.607 q^{92} -1295.15 q^{93} +30.1921 q^{94} +391.517 q^{96} -1059.87 q^{97} -126.704 q^{98} +662.333 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 56 q^{14} + 168 q^{16} + 50 q^{17} - 16 q^{18} + 36 q^{19} - 14 q^{21} + 184 q^{22} - 244 q^{23}+ \cdots + 940 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 - 2 * q^3 + 20 * q^4 - 8 * q^6 + 14 * q^7 - 48 * q^8 + 12 * q^9 - 14 * q^11 + 108 * q^12 - 50 * q^13 - 56 * q^14 + 168 * q^16 + 50 * q^17 - 16 * q^18 + 36 * q^19 - 14 * q^21 + 184 * q^22 - 244 * q^23 - 496 * q^24 + 216 * q^26 - 86 * q^27 + 140 * q^28 - 26 * q^29 - 120 * q^31 - 672 * q^32 - 498 * q^33 - 24 * q^34 - 136 * q^36 - 564 * q^37 - 320 * q^38 - 14 * q^39 - 328 * q^41 - 56 * q^42 + 260 * q^43 - 1164 * q^44 + 704 * q^46 + 350 * q^47 + 1368 * q^48 + 98 * q^49 - 754 * q^51 - 628 * q^52 + 56 * q^53 + 648 * q^54 - 336 * q^56 + 668 * q^57 + 8 * q^58 - 1232 * q^59 + 336 * q^61 + 1200 * q^62 + 84 * q^63 + 2128 * q^64 + 1976 * q^66 + 152 * q^67 - 908 * q^68 + 1332 * q^69 - 1904 * q^71 + 800 * q^72 - 676 * q^73 + 2016 * q^74 + 1768 * q^76 - 98 * q^77 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 + 816 * q^82 + 376 * q^83 + 756 * q^84 - 768 * q^86 + 410 * q^87 + 4688 * q^88 - 216 * q^89 - 350 * q^91 - 264 * q^92 - 2760 * q^93 - 1928 * q^94 - 2464 * q^96 - 2742 * q^97 - 392 * q^98 + 940 * 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 $

$2$	−2.58579	−0.914214	−0.457107	−	0.889412i	$-0.651114\pi$
$2$	−2.58579	−0.914214	−0.457107	+	0.889412i	$0.651114\pi$
$3$	−6.65685	−1.28111	−0.640556	−	0.767911i	$-0.721296\pi$
$3$	−6.65685	−1.28111	−0.640556	+	0.767911i	$0.721296\pi$
$4$	−1.31371	−0.164214
$5$	0	0
$5$	0	0
$6$	17.2132	1.17121
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	24.0833	1.06434
$9$	17.3137	0.641248
$10$	0	0
$11$	38.2548	1.04857	0.524285	−	0.851543i	$-0.324333\pi$
$11$	38.2548	1.04857	0.524285	+	0.851543i	$0.324333\pi$
$12$	8.74517	0.210376
$13$	−19.3431	−0.412679	−0.206339	−	0.978480i	$-0.566155\pi$
$13$	−19.3431	−0.412679	−0.206339	+	0.978480i	$0.566155\pi$
$14$	−18.1005	−0.345540
$15$	0	0
$16$	−51.7645	−0.808820
$17$	87.2254	1.24443	0.622214	−	0.782847i	$-0.286233\pi$
$17$	87.2254	1.24443	0.622214	+	0.782847i	$0.286233\pi$
$18$	−44.7696	−0.586238
$19$	−44.2254	−0.534000	−0.267000	−	0.963697i	$-0.586032\pi$
$19$	−44.2254	−0.534000	−0.267000	+	0.963697i	$0.586032\pi$
$20$	0	0
$21$	−46.5980	−0.484215
$22$	−98.9188	−0.958617
$23$	−218.167	−1.97786	−0.988932	−	0.148371i	$-0.952597\pi$
$23$	−218.167	−1.97786	−0.988932	+	0.148371i	$0.952597\pi$
$24$	−160.319	−1.36354
$25$	0	0
$26$	50.0172	0.377276
$27$	64.4802	0.459601
$28$	−9.19596	−0.0620669
$29$	−46.9411	−0.300578	−0.150289	−	0.988642i	$-0.548020\pi$
$29$	−46.9411	−0.300578	−0.150289	+	0.988642i	$0.548020\pi$
$30$	0	0
$31$	194.558	1.12722	0.563609	−	0.826042i	$-0.309413\pi$
$31$	194.558	1.12722	0.563609	+	0.826042i	$0.309413\pi$
$32$	−58.8141	−0.324905
$33$	−254.657	−1.34334
$34$	−225.546	−1.13767
$35$	0	0
$36$	−22.7452	−0.105302
$37$	−366.853	−1.63001	−0.815003	−	0.579457i	$-0.803265\pi$
$37$	−366.853	−1.63001	−0.815003	+	0.579457i	$0.803265\pi$
$38$	114.357	0.488190
$39$	128.765	0.528688
$40$	0	0
$41$	−339.362	−1.29267	−0.646336	−	0.763053i	$-0.723699\pi$
$41$	−339.362	−1.29267	−0.646336	+	0.763053i	$0.723699\pi$
$42$	120.492	0.442676
$43$	226.167	0.802095	0.401047	−	0.916057i	$-0.368646\pi$
$43$	226.167	0.802095	0.401047	+	0.916057i	$0.368646\pi$
$44$	−50.2557	−0.172189
$45$	0	0
$46$	564.132	1.80819
$47$	−11.6762	−0.0362372	−0.0181186	−	0.999836i	$-0.505768\pi$
$47$	−11.6762	−0.0362372	−0.0181186	+	0.999836i	$0.505768\pi$
$48$	344.589	1.03619
$49$	49.0000	0.142857
$50$	0	0
$51$	−580.647	−1.59425
$52$	25.4113	0.0677674
$53$	209.019	0.541717	0.270859	−	0.962619i	$-0.412692\pi$
$53$	209.019	0.541717	0.270859	+	0.962619i	$0.412692\pi$
$54$	−166.732	−0.420173
$55$	0	0
$56$	168.583	0.402283
$57$	294.402	0.684114
$58$	121.380	0.274792
$59$	−616.000	−1.35926	−0.679630	−	0.733555i	$-0.737860\pi$
$59$	−616.000	−1.35926	−0.679630	+	0.733555i	$0.737860\pi$
$60$	0	0
$61$	320.735	0.673212	0.336606	−	0.941646i	$-0.390721\pi$
$61$	320.735	0.673212	0.336606	+	0.941646i	$0.390721\pi$
$62$	−503.087	−1.03052
$63$	121.196	0.242369
$64$	566.197	1.10585
$65$	0	0
$66$	658.488	1.22810
$67$	−14.5097	−0.0264573	−0.0132286	−	0.999912i	$-0.504211\pi$
$67$	−14.5097	−0.0264573	−0.0132286	+	0.999912i	$0.504211\pi$
$68$	−114.589	−0.204352
$69$	1452.30	2.53387
$70$	0	0
$71$	−952.000	−1.59129	−0.795645	−	0.605763i	$-0.792868\pi$
$71$	−952.000	−1.59129	−0.795645	+	0.605763i	$0.792868\pi$
$72$	416.971	0.682506
$73$	−824.489	−1.32191	−0.660953	−	0.750427i	$-0.729848\pi$
$73$	−824.489	−1.32191	−0.660953	+	0.750427i	$0.729848\pi$
$74$	948.603	1.49017
$75$	0	0
$76$	58.0993	0.0876901
$77$	267.784	0.396322
$78$	−332.958	−0.483334
$79$	156.275	0.222561	0.111280	−	0.993789i	$-0.464505\pi$
$79$	156.275	0.222561	0.111280	+	0.993789i	$0.464505\pi$
$80$	0	0
$81$	−896.706	−1.23005
$82$	877.519	1.18178
$83$	1036.53	1.37077	0.685384	−	0.728182i	$-0.259634\pi$
$83$	1036.53	1.37077	0.685384	+	0.728182i	$0.259634\pi$
$84$	61.2162	0.0795147
$85$	0	0
$86$	−584.818	−0.733286
$87$	312.480	0.385074
$88$	921.301	1.11603
$89$	−170.225	−0.202740	−0.101370	−	0.994849i	$-0.532323\pi$
$89$	−170.225	−0.202740	−0.101370	+	0.994849i	$0.532323\pi$
$90$	0	0
$91$	−135.402	−0.155978
$92$	286.607	0.324792
$93$	−1295.15	−1.44409
$94$	30.1921	0.0331285
$95$	0	0
$96$	391.517	0.416240
$97$	−1059.87	−1.10942	−0.554710	−	0.832044i	$-0.687171\pi$
$97$	−1059.87	−1.10942	−0.554710	+	0.832044i	$0.687171\pi$
$98$	−126.704	−0.130602
$99$	662.333	0.672394
$100$	0	0
$101$	−241.833	−0.238251	−0.119125	−	0.992879i	$-0.538009\pi$
$101$	−241.833	−0.238251	−0.119125	+	0.992879i	$0.538009\pi$
$102$	1501.43	1.45749
$103$	1679.58	1.60673	0.803367	−	0.595484i	$-0.203040\pi$
$103$	1679.58	1.60673	0.803367	+	0.595484i	$0.203040\pi$
$104$	−465.846	−0.439230
$105$	0	0
$106$	−540.479	−0.495245
$107$	−1506.88	−1.36146	−0.680728	−	0.732537i	$-0.738336\pi$
$107$	−1506.88	−1.36146	−0.680728	+	0.732537i	$0.738336\pi$
$108$	−84.7082	−0.0754727
$109$	−1252.41	−1.10054	−0.550271	−	0.834986i	$-0.685476\pi$
$109$	−1252.41	−1.10054	−0.550271	+	0.834986i	$0.685476\pi$
$110$	0	0
$111$	2442.09	2.08822
$112$	−362.352	−0.305705
$113$	−1370.20	−1.14069	−0.570345	−	0.821405i	$-0.693190\pi$
$113$	−1370.20	−1.14069	−0.570345	+	0.821405i	$0.693190\pi$
$114$	−761.261	−0.625426
$115$	0	0
$116$	61.6670	0.0493589
$117$	−334.902	−0.264630
$118$	1592.84	1.24265
$119$	610.578	0.470349
$120$	0	0
$121$	132.432	0.0994984
$122$	−829.352	−0.615459
$123$	2259.09	1.65606
$124$	−255.593	−0.185104
$125$	0	0
$126$	−313.387	−0.221577
$127$	−1213.49	−0.847873	−0.423936	−	0.905692i	$-0.639352\pi$
$127$	−1213.49	−0.847873	−0.423936	+	0.905692i	$0.639352\pi$
$128$	−993.551	−0.686081
$129$	−1505.56	−1.02757
$130$	0	0
$131$	−1982.42	−1.32217	−0.661087	−	0.750309i	$-0.729904\pi$
$131$	−1982.42	−1.32217	−0.661087	+	0.750309i	$0.729904\pi$
$132$	334.545	0.220594
$133$	−309.578	−0.201833
$134$	37.5189	0.0241876
$135$	0	0
$136$	2100.67	1.32449
$137$	−2210.95	−1.37879	−0.689394	−	0.724386i	$-0.742123\pi$
$137$	−2210.95	−1.37879	−0.689394	+	0.724386i	$0.742123\pi$
$138$	−3755.34	−2.31649
$139$	528.039	0.322213	0.161107	−	0.986937i	$-0.448494\pi$
$139$	528.039	0.322213	0.161107	+	0.986937i	$0.448494\pi$
$140$	0	0
$141$	77.7267	0.0464239
$142$	2461.67	1.45478
$143$	−739.969	−0.432722
$144$	−896.235	−0.518655
$145$	0	0
$146$	2131.95	1.20851
$147$	−326.186	−0.183016
$148$	481.938	0.267669
$149$	−328.372	−0.180545	−0.0902727	−	0.995917i	$-0.528774\pi$
$149$	−328.372	−0.180545	−0.0902727	+	0.995917i	$0.528774\pi$
$150$	0	0
$151$	1029.43	0.554793	0.277396	−	0.960756i	$-0.410528\pi$
$151$	1029.43	0.554793	0.277396	+	0.960756i	$0.410528\pi$
$152$	−1065.09	−0.568358
$153$	1510.20	0.797987
$154$	−692.432	−0.362323
$155$	0	0
$156$	−169.159	−0.0868177
$157$	−525.098	−0.266926	−0.133463	−	0.991054i	$-0.542610\pi$
$157$	−525.098	−0.266926	−0.133463	+	0.991054i	$0.542610\pi$
$158$	−404.094	−0.203468
$159$	−1391.41	−0.694001
$160$	0	0
$161$	−1527.17	−0.747562
$162$	2318.69	1.12453
$163$	−1002.63	−0.481790	−0.240895	−	0.970551i	$-0.577441\pi$
$163$	−1002.63	−0.481790	−0.240895	+	0.970551i	$0.577441\pi$
$164$	445.823	0.212274
$165$	0	0
$166$	−2680.24	−1.25317
$167$	359.422	0.166544	0.0832722	−	0.996527i	$-0.473463\pi$
$167$	359.422	0.166544	0.0832722	+	0.996527i	$0.473463\pi$
$168$	−1122.23	−0.515369
$169$	−1822.84	−0.829696
$170$	0	0
$171$	−765.706	−0.342427
$172$	−297.117	−0.131715
$173$	−3293.65	−1.44747	−0.723733	−	0.690080i	$-0.757575\pi$
$173$	−3293.65	−1.44747	−0.723733	+	0.690080i	$0.757575\pi$
$174$	−808.007	−0.352039
$175$	0	0
$176$	−1980.24	−0.848104
$177$	4100.62	1.74137
$178$	440.167	0.185348
$179$	2978.82	1.24384	0.621921	−	0.783080i	$-0.286353\pi$
$179$	2978.82	1.24384	0.621921	+	0.783080i	$0.286353\pi$
$180$	0	0
$181$	1462.31	0.600514	0.300257	−	0.953858i	$-0.402928\pi$
$181$	1462.31	0.600514	0.300257	+	0.953858i	$0.402928\pi$
$182$	350.121	0.142597
$183$	−2135.09	−0.862460
$184$	−5254.16	−2.10512
$185$	0	0
$186$	3348.97	1.32021
$187$	3336.79	1.30487
$188$	15.3391	0.00595064
$189$	451.362	0.173713
$190$	0	0
$191$	−374.923	−0.142034	−0.0710169	−	0.997475i	$-0.522624\pi$
$191$	−374.923	−0.142034	−0.0710169	+	0.997475i	$0.522624\pi$
$192$	−3769.09	−1.41672
$193$	−733.028	−0.273391	−0.136696	−	0.990613i	$-0.543648\pi$
$193$	−733.028	−0.273391	−0.136696	+	0.990613i	$0.543648\pi$
$194$	2740.61	1.01425
$195$	0	0
$196$	−64.3717	−0.0234591
$197$	2093.24	0.757043	0.378521	−	0.925593i	$-0.376433\pi$
$197$	2093.24	0.757043	0.378521	+	0.925593i	$0.376433\pi$
$198$	−1712.65	−0.614711
$199$	2865.04	1.02059	0.510295	−	0.860000i	$-0.329536\pi$
$199$	2865.04	1.02059	0.510295	+	0.860000i	$0.329536\pi$
$200$	0	0
$201$	96.5887	0.0338948
$202$	625.330	0.217812
$203$	−328.588	−0.113608
$204$	762.801	0.261798
$205$	0	0
$206$	−4343.03	−1.46890
$207$	−3777.27	−1.26830
$208$	1001.29	0.333783
$209$	−1691.84	−0.559936
$210$	0	0
$211$	5643.65	1.84135	0.920674	−	0.390331i	$-0.127640\pi$
$211$	5643.65	1.84135	0.920674	+	0.390331i	$0.127640\pi$
$212$	−274.590	−0.0889573
$213$	6337.33	2.03862
$214$	3896.47	1.24466
$215$	0	0
$216$	1552.89	0.489172
$217$	1361.91	0.426048
$218$	3238.46	1.00613
$219$	5488.51	1.69351
$220$	0	0
$221$	−1687.21	−0.513549
$222$	−6314.71	−1.90908
$223$	6369.16	1.91260	0.956302	−	0.292381i	$-0.0944477\pi$
$223$	6369.16	1.91260	0.956302	+	0.292381i	$0.0944477\pi$
$224$	−411.699	−0.122803
$225$	0	0
$226$	3543.05	1.04283
$227$	1015.67	0.296972	0.148486	−	0.988914i	$-0.452560\pi$
$227$	1015.67	0.296972	0.148486	+	0.988914i	$0.452560\pi$
$228$	−386.758	−0.112341
$229$	4108.35	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$229$	4108.35	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$230$	0	0
$231$	−1782.60	−0.507733
$232$	−1130.50	−0.319917
$233$	−608.431	−0.171071	−0.0855357	−	0.996335i	$-0.527260\pi$
$233$	−608.431	−0.171071	−0.0855357	+	0.996335i	$0.527260\pi$
$234$	865.984	0.241928
$235$	0	0
$236$	809.244	0.223209
$237$	−1040.30	−0.285126
$238$	−1578.82	−0.430000
$239$	−5054.44	−1.36797	−0.683985	−	0.729496i	$-0.739755\pi$
$239$	−5054.44	−1.36797	−0.683985	+	0.729496i	$0.739755\pi$
$240$	0	0
$241$	4.86782	0.00130109	0.000650547	−	1.00000i	$-0.499793\pi$
$241$	4.86782	0.00130109	0.000650547		1.00000i	$0.499793\pi$
$242$	−342.442	−0.0909628
$243$	4228.27	1.11623
$244$	−421.352	−0.110551
$245$	0	0
$246$	−5841.52	−1.51399
$247$	855.458	0.220370
$248$	4685.60	1.19974
$249$	−6900.02	−1.75611
$250$	0	0
$251$	−547.921	−0.137787	−0.0688934	−	0.997624i	$-0.521947\pi$
$251$	−547.921	−0.137787	−0.0688934	+	0.997624i	$0.521947\pi$
$252$	−159.216	−0.0398003
$253$	−8345.92	−2.07393
$254$	3137.83	0.775137
$255$	0	0
$256$	−1960.46	−0.478629
$257$	1774.61	0.430729	0.215364	−	0.976534i	$-0.430906\pi$
$257$	1774.61	0.430729	0.215364	+	0.976534i	$0.430906\pi$
$258$	3893.05	0.939421
$259$	−2567.97	−0.616084
$260$	0	0
$261$	−812.725	−0.192745
$262$	5126.11	1.20875
$263$	1199.09	0.281138	0.140569	−	0.990071i	$-0.455107\pi$
$263$	1199.09	0.281138	0.140569	+	0.990071i	$0.455107\pi$
$264$	−6132.97	−1.42977
$265$	0	0
$266$	800.502	0.184519
$267$	1133.17	0.259733
$268$	19.0615	0.00434464
$269$	3250.29	0.736706	0.368353	−	0.929686i	$-0.379922\pi$
$269$	3250.29	0.736706	0.368353	+	0.929686i	$0.379922\pi$
$270$	0	0
$271$	−896.143	−0.200874	−0.100437	−	0.994943i	$-0.532024\pi$
$271$	−896.143	−0.200874	−0.100437	+	0.994943i	$0.532024\pi$
$272$	−4515.18	−1.00652
$273$	901.352	0.199825
$274$	5717.04	1.26051
$275$	0	0
$276$	−1907.90	−0.416095
$277$	386.562	0.0838492	0.0419246	−	0.999121i	$-0.486651\pi$
$277$	386.562	0.0838492	0.0419246	+	0.999121i	$0.486651\pi$
$278$	−1365.40	−0.294572
$279$	3368.53	0.722826
$280$	0	0
$281$	−3335.10	−0.708025	−0.354013	−	0.935241i	$-0.615183\pi$
$281$	−3335.10	−0.708025	−0.354013	+	0.935241i	$0.615183\pi$
$282$	−200.985	−0.0424414
$283$	−5412.26	−1.13684	−0.568419	−	0.822739i	$-0.692445\pi$
$283$	−5412.26	−1.13684	−0.568419	+	0.822739i	$0.692445\pi$
$284$	1250.65	0.261311
$285$	0	0
$286$	1913.40	0.395601
$287$	−2375.54	−0.488584
$288$	−1018.29	−0.208345
$289$	2695.27	0.548600
$290$	0	0
$291$	7055.42	1.42129
$292$	1083.14	0.217075
$293$	−282.211	−0.0562695	−0.0281347	−	0.999604i	$-0.508957\pi$
$293$	−282.211	−0.0562695	−0.0281347	+	0.999604i	$0.508957\pi$
$294$	843.447	0.167316
$295$	0	0
$296$	−8835.01	−1.73488
$297$	2466.68	0.481924
$298$	849.099	0.165057
$299$	4220.03	0.816222
$300$	0	0
$301$	1583.17	0.303163
$302$	−2661.88	−0.507199
$303$	1609.85	0.305226
$304$	2289.31	0.431910
$305$	0	0
$306$	−3905.04	−0.729531
$307$	−1919.67	−0.356878	−0.178439	−	0.983951i	$-0.557105\pi$
$307$	−1919.67	−0.356878	−0.178439	+	0.983951i	$0.557105\pi$
$308$	−351.790	−0.0650815
$309$	−11180.7	−2.05841
$310$	0	0
$311$	1213.31	0.221223	0.110612	−	0.993864i	$-0.464719\pi$
$311$	1213.31	0.221223	0.110612	+	0.993864i	$0.464719\pi$
$312$	3101.07	0.562703
$313$	1434.00	0.258960	0.129480	−	0.991582i	$-0.458669\pi$
$313$	1434.00	0.258960	0.129480	+	0.991582i	$0.458669\pi$
$314$	1357.79	0.244028
$315$	0	0
$316$	−205.300	−0.0365475
$317$	−6496.95	−1.15112	−0.575560	−	0.817760i	$-0.695216\pi$
$317$	−6496.95	−1.15112	−0.575560	+	0.817760i	$0.695216\pi$
$318$	3597.89	0.634465
$319$	−1795.72	−0.315176
$320$	0	0
$321$	10031.1	1.74418
$322$	3948.92	0.683432
$323$	−3857.58	−0.664524
$324$	1178.01	0.201991
$325$	0	0
$326$	2592.58	0.440459
$327$	8337.11	1.40992
$328$	−8172.96	−1.37584
$329$	−81.7333	−0.0136964
$330$	0	0
$331$	−9683.88	−1.60808	−0.804039	−	0.594576i	$-0.797320\pi$
$331$	−9683.88	−1.60808	−0.804039	+	0.594576i	$0.797320\pi$
$332$	−1361.70	−0.225099
$333$	−6351.58	−1.04524
$334$	−929.389	−0.152257
$335$	0	0
$336$	2412.12	0.391643
$337$	−29.1319	−0.00470895	−0.00235447	−	0.999997i	$-0.500749\pi$
$337$	−29.1319	−0.00470895	−0.00235447	+	0.999997i	$0.500749\pi$
$338$	4713.48	0.758520
$339$	9121.24	1.46135
$340$	0	0
$341$	7442.80	1.18197
$342$	1979.95	0.313051
$343$	343.000	0.0539949
$344$	5446.83	0.853701
$345$	0	0
$346$	8516.68	1.32329
$347$	7848.58	1.21422	0.607110	−	0.794618i	$-0.292329\pi$
$347$	7848.58	1.21422	0.607110	+	0.794618i	$0.292329\pi$
$348$	−410.508	−0.0632343
$349$	−10269.6	−1.57513	−0.787567	−	0.616229i	$-0.788659\pi$
$349$	−10269.6	−1.57513	−0.787567	+	0.616229i	$0.788659\pi$
$350$	0	0
$351$	−1247.25	−0.189668
$352$	−2249.93	−0.340686
$353$	−2799.93	−0.422168	−0.211084	−	0.977468i	$-0.567699\pi$
$353$	−2799.93	−0.422168	−0.211084	+	0.977468i	$0.567699\pi$
$354$	−10603.3	−1.59198
$355$	0	0
$356$	223.627	0.0332927
$357$	−4064.53	−0.602570
$358$	−7702.60	−1.13714
$359$	−3163.29	−0.465048	−0.232524	−	0.972591i	$-0.574698\pi$
$359$	−3163.29	−0.465048	−0.232524	+	0.972591i	$0.574698\pi$
$360$	0	0
$361$	−4903.11	−0.714844
$362$	−3781.23	−0.548998
$363$	−881.583	−0.127469
$364$	177.879	0.0256137
$365$	0	0
$366$	5520.88	0.788472
$367$	−3182.85	−0.452706	−0.226353	−	0.974045i	$-0.572680\pi$
$367$	−3182.85	−0.452706	−0.226353	+	0.974045i	$0.572680\pi$
$368$	11293.3	1.59974
$369$	−5875.62	−0.828923
$370$	0	0
$371$	1463.14	0.204750
$372$	1701.45	0.237139
$373$	2615.14	0.363021	0.181510	−	0.983389i	$-0.441901\pi$
$373$	2615.14	0.363021	0.181510	+	0.983389i	$0.441901\pi$
$374$	−8628.23	−1.19293
$375$	0	0
$376$	−281.201	−0.0385687
$377$	907.989	0.124042
$378$	−1167.12	−0.158811
$379$	−672.434	−0.0911362	−0.0455681	−	0.998961i	$-0.514510\pi$
$379$	−672.434	−0.0911362	−0.0455681	+	0.998961i	$0.514510\pi$
$380$	0	0
$381$	8078.03	1.08622
$382$	969.470	0.129849
$383$	−1169.86	−0.156075	−0.0780377	−	0.996950i	$-0.524865\pi$
$383$	−1169.86	−0.156075	−0.0780377	+	0.996950i	$0.524865\pi$
$384$	6613.92	0.878946
$385$	0	0
$386$	1895.45	0.249938
$387$	3915.78	0.514342
$388$	1392.36	0.182182
$389$	−1122.22	−0.146269	−0.0731347	−	0.997322i	$-0.523300\pi$
$389$	−1122.22	−0.146269	−0.0731347	+	0.997322i	$0.523300\pi$
$390$	0	0
$391$	−19029.7	−2.46131
$392$	1180.08	0.152049
$393$	13196.7	1.69385
$394$	−5412.68	−0.692099
$395$	0	0
$396$	−870.113	−0.110416
$397$	1985.93	0.251060	0.125530	−	0.992090i	$-0.459937\pi$
$397$	1985.93	0.251060	0.125530	+	0.992090i	$0.459937\pi$
$398$	−7408.38	−0.933037
$399$	2060.81	0.258571
$400$	0	0
$401$	−4172.38	−0.519597	−0.259799	−	0.965663i	$-0.583656\pi$
$401$	−4172.38	−0.519597	−0.259799	+	0.965663i	$0.583656\pi$
$402$	−249.758	−0.0309870
$403$	−3763.37	−0.465178
$404$	317.699	0.0391240
$405$	0	0
$406$	849.658	0.103862
$407$	−14033.9	−1.70918
$408$	−13983.9	−1.69682
$409$	−11700.8	−1.41459	−0.707295	−	0.706919i	$-0.750085\pi$
$409$	−11700.8	−1.41459	−0.707295	+	0.706919i	$0.750085\pi$
$410$	0	0
$411$	14718.0	1.76638
$412$	−2206.47	−0.263848
$413$	−4312.00	−0.513752
$414$	9767.22	1.15950
$415$	0	0
$416$	1137.65	0.134082
$417$	−3515.08	−0.412791
$418$	4374.72	0.511901
$419$	2733.20	0.318677	0.159339	−	0.987224i	$-0.449064\pi$
$419$	2733.20	0.318677	0.159339	+	0.987224i	$0.449064\pi$
$420$	0	0
$421$	13549.4	1.56854	0.784272	−	0.620417i	$-0.213037\pi$
$421$	13549.4	1.56854	0.784272	+	0.620417i	$0.213037\pi$
$422$	−14593.3	−1.68339
$423$	−202.158	−0.0232370
$424$	5033.87	0.576571
$425$	0	0
$426$	−16387.0	−1.86374
$427$	2245.15	0.254450
$428$	1979.60	0.223569
$429$	4925.86	0.554366
$430$	0	0
$431$	−6429.25	−0.718530	−0.359265	−	0.933236i	$-0.616973\pi$
$431$	−6429.25	−0.718530	−0.359265	+	0.933236i	$0.616973\pi$
$432$	−3337.79	−0.371735
$433$	−8022.03	−0.890333	−0.445166	−	0.895448i	$-0.646855\pi$
$433$	−8022.03	−0.890333	−0.445166	+	0.895448i	$0.646855\pi$
$434$	−3521.61	−0.389499
$435$	0	0
$436$	1645.30	0.180724
$437$	9648.50	1.05618
$438$	−14192.1	−1.54823
$439$	−5569.88	−0.605549	−0.302774	−	0.953062i	$-0.597913\pi$
$439$	−5569.88	−0.605549	−0.302774	+	0.953062i	$0.597913\pi$
$440$	0	0
$441$	848.372	0.0916069
$442$	4362.77	0.469493
$443$	5486.21	0.588392	0.294196	−	0.955745i	$-0.404948\pi$
$443$	5486.21	0.588392	0.294196	+	0.955745i	$0.404948\pi$
$444$	−3208.19	−0.342914
$445$	0	0
$446$	−16469.3	−1.74853
$447$	2185.92	0.231299
$448$	3963.38	0.417973
$449$	−7232.67	−0.760203	−0.380101	−	0.924945i	$-0.624111\pi$
$449$	−7232.67	−0.760203	−0.380101	+	0.924945i	$0.624111\pi$
$450$	0	0
$451$	−12982.3	−1.35546
$452$	1800.05	0.187317
$453$	−6852.76	−0.710752
$454$	−2626.32	−0.271496
$455$	0	0
$456$	7090.16	0.728130
$457$	2900.51	0.296893	0.148446	−	0.988920i	$-0.452573\pi$
$457$	2900.51	0.296893	0.148446	+	0.988920i	$0.452573\pi$
$458$	−10623.3	−1.08383
$459$	5624.31	0.571940
$460$	0	0
$461$	6073.57	0.613611	0.306805	−	0.951772i	$-0.400740\pi$
$461$	6073.57	0.613611	0.306805	+	0.951772i	$0.400740\pi$
$462$	4609.42	0.464176
$463$	18922.8	1.89939	0.949693	−	0.313183i	$-0.101395\pi$
$463$	18922.8	1.89939	0.949693	+	0.313183i	$0.101395\pi$
$464$	2429.88	0.243113
$465$	0	0
$466$	1573.27	0.156396
$467$	6776.71	0.671496	0.335748	−	0.941952i	$-0.391011\pi$
$467$	6776.71	0.671496	0.335748	+	0.941952i	$0.391011\pi$
$468$	439.963	0.0434558
$469$	−101.568	−0.00999991
$470$	0	0
$471$	3495.50	0.341962
$472$	−14835.3	−1.44672
$473$	8651.96	0.841052
$474$	2689.99	0.260666
$475$	0	0
$476$	−802.121	−0.0772377
$477$	3618.90	0.347375
$478$	13069.7	1.25062
$479$	2397.32	0.228677	0.114338	−	0.993442i	$-0.463525\pi$
$479$	2397.32	0.228677	0.114338	+	0.993442i	$0.463525\pi$
$480$	0	0
$481$	7096.09	0.672669
$482$	−12.5871	−0.00118948
$483$	10166.1	0.957711
$484$	−173.977	−0.0163390
$485$	0	0
$486$	−10933.4	−1.02047
$487$	−5586.17	−0.519781	−0.259890	−	0.965638i	$-0.583686\pi$
$487$	−5586.17	−0.519781	−0.259890	+	0.965638i	$0.583686\pi$
$488$	7724.35	0.716526
$489$	6674.33	0.617227
$490$	0	0
$491$	537.392	0.0493934	0.0246967	−	0.999695i	$-0.492138\pi$
$491$	537.392	0.0493934	0.0246967	+	0.999695i	$0.492138\pi$
$492$	−2967.78	−0.271947
$493$	−4094.46	−0.374047
$494$	−2212.03	−0.201466
$495$	0	0
$496$	−10071.2	−0.911716
$497$	−6664.00	−0.601451
$498$	17842.0	1.60546
$499$	598.965	0.0537342	0.0268671	−	0.999639i	$-0.491447\pi$
$499$	598.965	0.0537342	0.0268671	+	0.999639i	$0.491447\pi$
$500$	0	0
$501$	−2392.62	−0.213362
$502$	1416.81	0.125966
$503$	−4426.76	−0.392405	−0.196202	−	0.980563i	$-0.562861\pi$
$503$	−4426.76	−0.392405	−0.196202	+	0.980563i	$0.562861\pi$
$504$	2918.79	0.257963
$505$	0	0
$506$	21580.8	1.89601
$507$	12134.4	1.06293
$508$	1594.17	0.139232
$509$	17727.7	1.54374	0.771872	−	0.635779i	$-0.219321\pi$
$509$	17727.7	1.54374	0.771872	+	0.635779i	$0.219321\pi$
$510$	0	0
$511$	−5771.43	−0.499634
$512$	13017.7	1.12365
$513$	−2851.66	−0.245427
$514$	−4588.77	−0.393778
$515$	0	0
$516$	1977.86	0.168741
$517$	−446.671	−0.0379972
$518$	6640.22	0.563233
$519$	21925.4	1.85437
$520$	0	0
$521$	8662.79	0.728453	0.364226	−	0.931310i	$-0.381333\pi$
$521$	8662.79	0.728453	0.364226	+	0.931310i	$0.381333\pi$
$522$	2101.53	0.176210
$523$	7770.40	0.649667	0.324833	−	0.945771i	$-0.394692\pi$
$523$	7770.40	0.649667	0.324833	+	0.945771i	$0.394692\pi$
$524$	2604.32	0.217119
$525$	0	0
$526$	−3100.60	−0.257020
$527$	16970.4	1.40274
$528$	13182.2	1.08652
$529$	35429.6	2.91194
$530$	0	0
$531$	−10665.2	−0.871624
$532$	406.695	0.0331437
$533$	6564.34	0.533458
$534$	−2930.12	−0.237451
$535$	0	0
$536$	−349.440	−0.0281595
$537$	−19829.6	−1.59350
$538$	−8404.56	−0.673506
$539$	1874.49	0.149796
$540$	0	0
$541$	21641.0	1.71981	0.859906	−	0.510453i	$-0.170522\pi$
$541$	21641.0	1.71981	0.859906	+	0.510453i	$0.170522\pi$
$542$	2317.23	0.183642
$543$	−9734.41	−0.769325
$544$	−5130.09	−0.404321
$545$	0	0
$546$	−2330.70	−0.182683
$547$	−7489.29	−0.585409	−0.292705	−	0.956203i	$-0.594555\pi$
$547$	−7489.29	−0.585409	−0.292705	+	0.956203i	$0.594555\pi$
$548$	2904.54	0.226416
$549$	5553.11	0.431696
$550$	0	0
$551$	2075.99	0.160508
$552$	34976.2	2.69689
$553$	1093.93	0.0841201
$554$	−999.566	−0.0766561
$555$	0	0
$556$	−693.689	−0.0529118
$557$	−25297.9	−1.92443	−0.962214	−	0.272295i	$-0.912217\pi$
$557$	−25297.9	−1.92443	−0.962214	+	0.272295i	$0.912217\pi$
$558$	−8710.29	−0.660817
$559$	−4374.77	−0.331007
$560$	0	0
$561$	−22212.5	−1.67168
$562$	8623.85	0.647286
$563$	−15661.3	−1.17237	−0.586186	−	0.810177i	$-0.699371\pi$
$563$	−15661.3	−1.17237	−0.586186	+	0.810177i	$0.699371\pi$
$564$	−102.110	−0.00762343
$565$	0	0
$566$	13994.9	1.03931
$567$	−6276.94	−0.464915
$568$	−22927.3	−1.69367
$569$	−9982.75	−0.735498	−0.367749	−	0.929925i	$-0.619871\pi$
$569$	−9982.75	−0.735498	−0.367749	+	0.929925i	$0.619871\pi$
$570$	0	0
$571$	−11583.6	−0.848966	−0.424483	−	0.905436i	$-0.639544\pi$
$571$	−11583.6	−0.848966	−0.424483	+	0.905436i	$0.639544\pi$
$572$	972.103	0.0710589
$573$	2495.81	0.181961
$574$	6142.63	0.446670
$575$	0	0
$576$	9802.97	0.709127
$577$	595.378	0.0429565	0.0214783	−	0.999769i	$-0.493163\pi$
$577$	595.378	0.0429565	0.0214783	+	0.999769i	$0.493163\pi$
$578$	−6969.39	−0.501537
$579$	4879.66	0.350245
$580$	0	0
$581$	7255.70	0.518102
$582$	−18243.8	−1.29936
$583$	7996.00	0.568028
$584$	−19856.4	−1.40696
$585$	0	0
$586$	729.738	0.0514423
$587$	15750.3	1.10747	0.553736	−	0.832693i	$-0.313202\pi$
$587$	15750.3	1.10747	0.553736	+	0.832693i	$0.313202\pi$
$588$	428.513	0.0300537
$589$	−8604.42	−0.601934
$590$	0	0
$591$	−13934.4	−0.969856
$592$	18990.0	1.31838
$593$	417.878	0.0289379	0.0144690	−	0.999895i	$-0.495394\pi$
$593$	417.878	0.0289379	0.0144690	+	0.999895i	$0.495394\pi$
$594$	−6378.31	−0.440581
$595$	0	0
$596$	431.385	0.0296480
$597$	−19072.2	−1.30749
$598$	−10912.1	−0.746201
$599$	−19997.3	−1.36406	−0.682028	−	0.731326i	$-0.738902\pi$
$599$	−19997.3	−1.36406	−0.682028	+	0.731326i	$0.738902\pi$
$600$	0	0
$601$	−15992.6	−1.08545	−0.542723	−	0.839912i	$-0.682607\pi$
$601$	−15992.6	−1.08545	−0.542723	+	0.839912i	$0.682607\pi$
$602$	−4093.73	−0.277156
$603$	−251.216	−0.0169657
$604$	−1352.37	−0.0911045
$605$	0	0
$606$	−4162.73	−0.279042
$607$	−14159.2	−0.946793	−0.473396	−	0.880850i	$-0.656972\pi$
$607$	−14159.2	−0.946793	−0.473396	+	0.880850i	$0.656972\pi$
$608$	2601.08	0.173499
$609$	2187.36	0.145544
$610$	0	0
$611$	225.854	0.0149543
$612$	−1983.96	−0.131040
$613$	4629.41	0.305025	0.152512	−	0.988302i	$-0.451264\pi$
$613$	4629.41	0.305025	0.152512	+	0.988302i	$0.451264\pi$
$614$	4963.87	0.326263
$615$	0	0
$616$	6449.11	0.421821
$617$	23165.3	1.51151	0.755753	−	0.654857i	$-0.227271\pi$
$617$	23165.3	1.51151	0.755753	+	0.654857i	$0.227271\pi$
$618$	28910.9	1.88182
$619$	12370.6	0.803258	0.401629	−	0.915803i	$-0.368444\pi$
$619$	12370.6	0.803258	0.401629	+	0.915803i	$0.368444\pi$
$620$	0	0
$621$	−14067.4	−0.909028
$622$	−3137.36	−0.202245
$623$	−1191.58	−0.0766285
$624$	−6665.43	−0.427613
$625$	0	0
$626$	−3708.02	−0.236745
$627$	11262.3	0.717341
$628$	689.826	0.0438329
$629$	−31998.9	−2.02842
$630$	0	0
$631$	13980.2	0.882002	0.441001	−	0.897507i	$-0.354623\pi$
$631$	13980.2	0.882002	0.441001	+	0.897507i	$0.354623\pi$
$632$	3763.61	0.236880
$633$	−37568.9	−2.35897
$634$	16799.7	1.05237
$635$	0	0
$636$	1827.91	0.113964
$637$	−947.814	−0.0589541
$638$	4643.36	0.288139
$639$	−16482.7	−1.02041
$640$	0	0
$641$	−16060.9	−0.989655	−0.494828	−	0.868991i	$-0.664769\pi$
$641$	−16060.9	−0.989655	−0.494828	+	0.868991i	$0.664769\pi$
$642$	−25938.3	−1.59455
$643$	4502.17	0.276125	0.138063	−	0.990424i	$-0.455913\pi$
$643$	4502.17	0.276125	0.138063	+	0.990424i	$0.455913\pi$
$644$	2006.25	0.122760
$645$	0	0
$646$	9974.87	0.607517
$647$	−29414.8	−1.78735	−0.893675	−	0.448715i	$-0.851882\pi$
$647$	−29414.8	−1.78735	−0.893675	+	0.448715i	$0.851882\pi$
$648$	−21595.6	−1.30919
$649$	−23565.0	−1.42528
$650$	0	0
$651$	−9066.03	−0.545815
$652$	1317.16	0.0791164
$653$	−13013.6	−0.779882	−0.389941	−	0.920840i	$-0.627505\pi$
$653$	−13013.6	−0.779882	−0.389941	+	0.920840i	$0.627505\pi$
$654$	−21558.0	−1.28897
$655$	0	0
$656$	17566.9	1.04554
$657$	−14275.0	−0.847671
$658$	211.345	0.0125214
$659$	23474.2	1.38759	0.693797	−	0.720171i	$-0.255937\pi$
$659$	23474.2	1.38759	0.693797	+	0.720171i	$0.255937\pi$
$660$	0	0
$661$	−9266.36	−0.545264	−0.272632	−	0.962118i	$-0.587894\pi$
$661$	−9266.36	−0.545264	−0.272632	+	0.962118i	$0.587894\pi$
$662$	25040.4	1.47013
$663$	11231.5	0.657914
$664$	24963.0	1.45896
$665$	0	0
$666$	16423.8	0.955572
$667$	10241.0	0.594501
$668$	−472.176	−0.0273489
$669$	−42398.6	−2.45026
$670$	0	0
$671$	12269.7	0.705909
$672$	2740.62	0.157324
$673$	25067.2	1.43576	0.717882	−	0.696164i	$-0.245112\pi$
$673$	25067.2	1.43576	0.717882	+	0.696164i	$0.245112\pi$
$674$	75.3288	0.00430498
$675$	0	0
$676$	2394.68	0.136247
$677$	22409.6	1.27219	0.636093	−	0.771613i	$-0.280550\pi$
$677$	22409.6	1.27219	0.636093	+	0.771613i	$0.280550\pi$
$678$	−23585.6	−1.33599
$679$	−7419.11	−0.419322
$680$	0	0
$681$	−6761.20	−0.380455
$682$	−19245.5	−1.08057
$683$	8757.53	0.490626	0.245313	−	0.969444i	$-0.421109\pi$
$683$	8757.53	0.490626	0.245313	+	0.969444i	$0.421109\pi$
$684$	1005.91	0.0562311
$685$	0	0
$686$	−886.925	−0.0493629
$687$	−27348.7	−1.51880
$688$	−11707.4	−0.648750
$689$	−4043.09	−0.223555
$690$	0	0
$691$	8468.42	0.466214	0.233107	−	0.972451i	$-0.425111\pi$
$691$	8468.42	0.466214	0.233107	+	0.972451i	$0.425111\pi$
$692$	4326.90	0.237694
$693$	4636.33	0.254141
$694$	−20294.8	−1.11006
$695$	0	0
$696$	7525.54	0.409849
$697$	−29601.0	−1.60864
$698$	26555.1	1.44001
$699$	4050.23	0.219162
$700$	0	0
$701$	15996.9	0.861906	0.430953	−	0.902374i	$-0.358177\pi$
$701$	15996.9	0.861906	0.430953	+	0.902374i	$0.358177\pi$
$702$	3225.12	0.173397
$703$	16224.2	0.870423
$704$	21659.8	1.15956
$705$	0	0
$706$	7240.03	0.385952
$707$	−1692.83	−0.0900503
$708$	−5387.02	−0.285956
$709$	19903.0	1.05426	0.527131	−	0.849784i	$-0.323268\pi$
$709$	19903.0	1.05426	0.527131	+	0.849784i	$0.323268\pi$
$710$	0	0
$711$	2705.70	0.142717
$712$	−4099.58	−0.215784
$713$	−42446.1	−2.22948
$714$	10510.0	0.550878
$715$	0	0
$716$	−3913.30	−0.204256
$717$	33646.7	1.75252
$718$	8179.59	0.425153
$719$	−11073.1	−0.574347	−0.287174	−	0.957879i	$-0.592716\pi$
$719$	−11073.1	−0.574347	−0.287174	+	0.957879i	$0.592716\pi$
$720$	0	0
$721$	11757.0	0.607288
$722$	12678.4	0.653520
$723$	−32.4043	−0.00166685
$724$	−1921.06	−0.0986125
$725$	0	0
$726$	2279.58	0.116533
$727$	−31652.7	−1.61476	−0.807382	−	0.590029i	$-0.799116\pi$
$727$	−31652.7	−1.61476	−0.807382	+	0.590029i	$0.799116\pi$
$728$	−3260.92	−0.166013
$729$	−3935.94	−0.199967
$730$	0	0
$731$	19727.5	0.998149
$732$	2804.88	0.141628
$733$	−16958.3	−0.854528	−0.427264	−	0.904127i	$-0.640523\pi$
$733$	−16958.3	−0.854528	−0.427264	+	0.904127i	$0.640523\pi$
$734$	8230.16	0.413870
$735$	0	0
$736$	12831.3	0.642618
$737$	−555.065	−0.0277423
$738$	15193.1	0.757813
$739$	−11616.6	−0.578245	−0.289123	−	0.957292i	$-0.593364\pi$
$739$	−11616.6	−0.578245	−0.289123	+	0.957292i	$0.593364\pi$
$740$	0	0
$741$	−5694.66	−0.282319
$742$	−3783.36	−0.187185
$743$	−15928.0	−0.786464	−0.393232	−	0.919439i	$-0.628643\pi$
$743$	−15928.0	−0.786464	−0.393232	+	0.919439i	$0.628643\pi$
$744$	−31191.4	−1.53700
$745$	0	0
$746$	−6762.19	−0.331879
$747$	17946.1	0.879003
$748$	−4383.57	−0.214277
$749$	−10548.2	−0.514582
$750$	0	0
$751$	25571.9	1.24252	0.621260	−	0.783604i	$-0.286621\pi$
$751$	25571.9	1.24252	0.621260	+	0.783604i	$0.286621\pi$
$752$	604.412	0.0293094
$753$	3647.43	0.176520
$754$	−2347.87	−0.113401
$755$	0	0
$756$	−592.958	−0.0285260
$757$	−6202.41	−0.297794	−0.148897	−	0.988853i	$-0.547572\pi$
$757$	−6202.41	−0.297794	−0.148897	+	0.988853i	$0.547572\pi$
$758$	1738.77	0.0833179
$759$	55557.6	2.65693
$760$	0	0
$761$	−29199.1	−1.39089	−0.695444	−	0.718580i	$-0.744792\pi$
$761$	−29199.1	−1.39089	−0.695444	+	0.718580i	$0.744792\pi$
$762$	−20888.1	−0.993037
$763$	−8766.87	−0.415966
$764$	492.539	0.0233239
$765$	0	0
$766$	3025.00	0.142686
$767$	11915.4	0.560938
$768$	13050.5	0.613177
$769$	21838.2	1.02407	0.512033	−	0.858966i	$-0.328892\pi$
$769$	21838.2	1.02407	0.512033	+	0.858966i	$0.328892\pi$
$770$	0	0
$771$	−11813.3	−0.551812
$772$	962.985	0.0448945
$773$	25544.8	1.18859	0.594296	−	0.804246i	$-0.297431\pi$
$773$	25544.8	1.18859	0.594296	+	0.804246i	$0.297431\pi$
$774$	−10125.4	−0.470218
$775$	0	0
$776$	−25525.2	−1.18080
$777$	17094.6	0.789273
$778$	2901.82	0.133721
$779$	15008.4	0.690286
$780$	0	0
$781$	−36418.6	−1.66858
$782$	49206.6	2.25016
$783$	−3026.77	−0.138146
$784$	−2536.46	−0.115546
$785$	0	0
$786$	−34123.8	−1.54854
$787$	37223.2	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$787$	37223.2	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$788$	−2749.91	−0.124317
$789$	−7982.19	−0.360169
$790$	0	0
$791$	−9591.42	−0.431140
$792$	15951.1	0.715655
$793$	−6204.03	−0.277820
$794$	−5135.18	−0.229522
$795$	0	0
$796$	−3763.83	−0.167595
$797$	40384.6	1.79485	0.897425	−	0.441168i	$-0.145436\pi$
$797$	40384.6	1.79485	0.897425	+	0.441168i	$0.145436\pi$
$798$	−5328.83	−0.236389
$799$	−1018.46	−0.0450945
$800$	0	0
$801$	−2947.23	−0.130007
$802$	10788.9	0.475023
$803$	−31540.7	−1.38611
$804$	−126.889	−0.00556598
$805$	0	0
$806$	9731.28	0.425272
$807$	−21636.7	−0.943803
$808$	−5824.14	−0.253580
$809$	−1955.76	−0.0849948	−0.0424974	−	0.999097i	$-0.513531\pi$
$809$	−1955.76	−0.0849948	−0.0424974	+	0.999097i	$0.513531\pi$
$810$	0	0
$811$	−34301.8	−1.48520	−0.742600	−	0.669735i	$-0.766408\pi$
$811$	−34301.8	−1.48520	−0.742600	+	0.669735i	$0.766408\pi$
$812$	431.669	0.0186559
$813$	5965.49	0.257342
$814$	36288.7	1.56255
$815$	0	0
$816$	30056.9	1.28946
$817$	−10002.3	−0.428319
$818$	30255.8	1.29324
$819$	−2344.31	−0.100021
$820$	0	0
$821$	−13665.6	−0.580918	−0.290459	−	0.956887i	$-0.593808\pi$
$821$	−13665.6	−0.580918	−0.290459	+	0.956887i	$0.593808\pi$
$822$	−38057.5	−1.61485
$823$	21519.5	0.911449	0.455724	−	0.890121i	$-0.349380\pi$
$823$	21519.5	0.911449	0.455724	+	0.890121i	$0.349380\pi$
$824$	40449.7	1.71011
$825$	0	0
$826$	11149.9	0.469679
$827$	−35220.6	−1.48094	−0.740471	−	0.672088i	$-0.765398\pi$
$827$	−35220.6	−1.48094	−0.740471	+	0.672088i	$0.765398\pi$
$828$	4962.23	0.208272
$829$	31365.5	1.31408	0.657039	−	0.753857i	$-0.271809\pi$
$829$	31365.5	1.31408	0.657039	+	0.753857i	$0.271809\pi$
$830$	0	0
$831$	−2573.28	−0.107420
$832$	−10952.0	−0.456362
$833$	4274.04	0.177775
$834$	9089.24	0.377380
$835$	0	0
$836$	2222.58	0.0919491
$837$	12545.2	0.518070
$838$	−7067.48	−0.291339
$839$	−28287.1	−1.16398	−0.581990	−	0.813196i	$-0.697726\pi$
$839$	−28287.1	−1.16398	−0.581990	+	0.813196i	$0.697726\pi$
$840$	0	0
$841$	−22185.5	−0.909653
$842$	−35035.8	−1.43398
$843$	22201.2	0.907060
$844$	−7414.11	−0.302374
$845$	0	0
$846$	522.738	0.0212436
$847$	927.026	0.0376068
$848$	−10819.8	−0.438152
$849$	36028.6	1.45642
$850$	0	0
$851$	80035.0	3.22393
$852$	−8325.40	−0.334769
$853$	−9405.41	−0.377533	−0.188766	−	0.982022i	$-0.560449\pi$
$853$	−9405.41	−0.377533	−0.188766	+	0.982022i	$0.560449\pi$
$854$	−5805.47	−0.232622
$855$	0	0
$856$	−36290.6	−1.44905
$857$	27966.9	1.11474	0.557369	−	0.830265i	$-0.311811\pi$
$857$	27966.9	1.11474	0.557369	+	0.830265i	$0.311811\pi$
$858$	−12737.2	−0.506809
$859$	6281.11	0.249486	0.124743	−	0.992189i	$-0.460189\pi$
$859$	6281.11	0.249486	0.124743	+	0.992189i	$0.460189\pi$
$860$	0	0
$861$	15813.6	0.625931
$862$	16624.7	0.656890
$863$	4757.13	0.187642	0.0938208	−	0.995589i	$-0.470092\pi$
$863$	4757.13	0.187642	0.0938208	+	0.995589i	$0.470092\pi$
$864$	−3792.35	−0.149327
$865$	0	0
$866$	20743.3	0.813954
$867$	−17942.0	−0.702818
$868$	−1789.15	−0.0699629
$869$	5978.28	0.233371
$870$	0	0
$871$	280.663	0.0109184
$872$	−30162.1	−1.17135
$873$	−18350.3	−0.711414
$874$	−24949.0	−0.965574
$875$	0	0
$876$	−7210.30	−0.278097
$877$	30240.5	1.16437	0.582184	−	0.813057i	$-0.302198\pi$
$877$	30240.5	1.16437	0.582184	+	0.813057i	$0.302198\pi$
$878$	14402.5	0.553601
$879$	1878.64	0.0720875
$880$	0	0
$881$	44875.5	1.71611	0.858056	−	0.513556i	$-0.171672\pi$
$881$	44875.5	1.71611	0.858056	+	0.513556i	$0.171672\pi$
$882$	−2193.71	−0.0837483
$883$	−4892.13	−0.186448	−0.0932238	−	0.995645i	$-0.529717\pi$
$883$	−4892.13	−0.186448	−0.0932238	+	0.995645i	$0.529717\pi$
$884$	2216.51	0.0843317
$885$	0	0
$886$	−14186.2	−0.537916
$887$	−1761.40	−0.0666765	−0.0333382	−	0.999444i	$-0.510614\pi$
$887$	−1761.40	−0.0666765	−0.0333382	+	0.999444i	$0.510614\pi$
$888$	58813.4	2.22258
$889$	−8494.43	−0.320466
$890$	0	0
$891$	−34303.3	−1.28979
$892$	−8367.22	−0.314075
$893$	516.384	0.0193507
$894$	−5652.33	−0.211457
$895$	0	0
$896$	−6954.86	−0.259314
$897$	−28092.1	−1.04567
$898$	18702.2	0.694988
$899$	−9132.79	−0.338816
$900$	0	0
$901$	18231.8	0.674128
$902$	33569.3	1.23918
$903$	−10538.9	−0.388386
$904$	−32999.0	−1.21408
$905$	0	0
$906$	17719.8	0.649779
$907$	−23689.1	−0.867238	−0.433619	−	0.901096i	$-0.642764\pi$
$907$	−23689.1	−0.867238	−0.433619	+	0.901096i	$0.642764\pi$
$908$	−1334.30	−0.0487669
$909$	−4187.03	−0.152778
$910$	0	0
$911$	−13877.3	−0.504692	−0.252346	−	0.967637i	$-0.581202\pi$
$911$	−13877.3	−0.504692	−0.252346	+	0.967637i	$0.581202\pi$
$912$	−15239.6	−0.553325
$913$	39652.2	1.43735
$914$	−7500.09	−0.271423
$915$	0	0
$916$	−5397.18	−0.194681
$917$	−13876.9	−0.499735
$918$	−14543.3	−0.522875
$919$	14331.6	0.514426	0.257213	−	0.966355i	$-0.417196\pi$
$919$	14331.6	0.514426	0.257213	+	0.966355i	$0.417196\pi$
$920$	0	0
$921$	12779.0	0.457201
$922$	−15705.0	−0.560971
$923$	18414.7	0.656692
$924$	2341.81	0.0833767
$925$	0	0
$926$	−48930.2	−1.73644
$927$	29079.7	1.03032
$928$	2760.80	0.0976592
$929$	16668.4	0.588668	0.294334	−	0.955703i	$-0.404902\pi$
$929$	16668.4	0.588668	0.294334	+	0.955703i	$0.404902\pi$
$930$	0	0
$931$	−2167.04	−0.0762857
$932$	799.300	0.0280922
$933$	−8076.82	−0.283412
$934$	−17523.1	−0.613891
$935$	0	0
$936$	−8065.52	−0.281656
$937$	−30384.9	−1.05937	−0.529685	−	0.848194i	$-0.677690\pi$
$937$	−30384.9	−1.05937	−0.529685	+	0.848194i	$0.677690\pi$
$938$	262.632	0.00914206
$939$	−9545.94	−0.331757
$940$	0	0
$941$	−1196.35	−0.0414452	−0.0207226	−	0.999785i	$-0.506597\pi$
$941$	−1196.35	−0.0414452	−0.0207226	+	0.999785i	$0.506597\pi$
$942$	−9038.63	−0.312627
$943$	74037.5	2.55673
$944$	31886.9	1.09940
$945$	0	0
$946$	−22372.1	−0.768901
$947$	−1788.41	−0.0613681	−0.0306840	−	0.999529i	$-0.509769\pi$
$947$	−1788.41	−0.0613681	−0.0306840	+	0.999529i	$0.509769\pi$
$948$	1366.65	0.0468215
$949$	15948.2	0.545523
$950$	0	0
$951$	43249.2	1.47471
$952$	14704.7	0.500612
$953$	−8578.60	−0.291593	−0.145796	−	0.989315i	$-0.546574\pi$
$953$	−8578.60	−0.291593	−0.145796	+	0.989315i	$0.546574\pi$
$954$	−9357.70	−0.317575
$955$	0	0
$956$	6640.06	0.224639
$957$	11953.9	0.403776
$958$	−6198.95	−0.209059
$959$	−15476.6	−0.521133
$960$	0	0
$961$	8061.99	0.270618
$962$	−18349.0	−0.614963
$963$	−26089.7	−0.873031
$964$	−6.39489	−0.000213657 0
$965$	0	0
$966$	−26287.4	−0.875552
$967$	55459.3	1.84431	0.922156	−	0.386818i	$-0.126426\pi$
$967$	55459.3	1.84431	0.922156	+	0.386818i	$0.126426\pi$
$968$	3189.40	0.105900
$969$	25679.3	0.851330
$970$	0	0
$971$	22047.3	0.728662	0.364331	−	0.931270i	$-0.381298\pi$
$971$	22047.3	0.728662	0.364331	+	0.931270i	$0.381298\pi$
$972$	−5554.72	−0.183300
$973$	3696.27	0.121785
$974$	14444.6	0.475191
$975$	0	0
$976$	−16602.7	−0.544507
$977$	−14402.3	−0.471617	−0.235809	−	0.971800i	$-0.575774\pi$
$977$	−14402.3	−0.471617	−0.235809	+	0.971800i	$0.575774\pi$
$978$	−17258.4	−0.564277
$979$	−6511.94	−0.212587
$980$	0	0
$981$	−21683.9	−0.705721
$982$	−1389.58	−0.0451561
$983$	−7817.11	−0.253639	−0.126819	−	0.991926i	$-0.540477\pi$
$983$	−7817.11	−0.253639	−0.126819	+	0.991926i	$0.540477\pi$
$984$	54406.2	1.76261
$985$	0	0
$986$	10587.4	0.341959
$987$	544.087	0.0175466
$988$	−1123.82	−0.0361878
$989$	−49342.0	−1.58643
$990$	0	0
$991$	24501.6	0.785386	0.392693	−	0.919670i	$-0.371543\pi$
$991$	24501.6	0.785386	0.392693	+	0.919670i	$0.371543\pi$
$992$	−11442.8	−0.366239
$993$	64464.2	2.06013
$994$	17231.7	0.549855
$995$	0	0
$996$	9064.61	0.288377
$997$	50696.0	1.61039	0.805195	−	0.593010i	$-0.202060\pi$
$997$	50696.0	1.61039	0.805195	+	0.593010i	$0.202060\pi$
$998$	−1548.80	−0.0491245
$999$	−23654.8	−0.749152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.a.c.1.2		2
3.2	odd	2		1575.4.a.z.1.1		2
5.2	odd	4		175.4.b.c.99.2		4
5.3	odd	4		175.4.b.c.99.3		4
5.4	even	2		35.4.a.b.1.1	✓	2
7.6	odd	2		1225.4.a.m.1.2		2
15.14	odd	2		315.4.a.f.1.2		2
20.19	odd	2		560.4.a.r.1.1		2
35.4	even	6		245.4.e.h.226.2		4
35.9	even	6		245.4.e.h.116.2		4
35.19	odd	6		245.4.e.i.116.2		4
35.24	odd	6		245.4.e.i.226.2		4
35.34	odd	2		245.4.a.k.1.1		2
40.19	odd	2		2240.4.a.bo.1.2		2
40.29	even	2		2240.4.a.bn.1.1		2
105.104	even	2		2205.4.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.1	✓	2	5.4	even	2
175.4.a.c.1.2		2	1.1	even	1	trivial
175.4.b.c.99.2		4	5.2	odd	4
175.4.b.c.99.3		4	5.3	odd	4
245.4.a.k.1.1		2	35.34	odd	2
245.4.e.h.116.2		4	35.9	even	6
245.4.e.h.226.2		4	35.4	even	6
245.4.e.i.116.2		4	35.19	odd	6
245.4.e.i.226.2		4	35.24	odd	6
315.4.a.f.1.2		2	15.14	odd	2
560.4.a.r.1.1		2	20.19	odd	2
1225.4.a.m.1.2		2	7.6	odd	2
1575.4.a.z.1.1		2	3.2	odd	2
2205.4.a.u.1.2		2	105.104	even	2
2240.4.a.bn.1.1		2	40.29	even	2
2240.4.a.bo.1.2		2	40.19	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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +17.2132 q^{6} +7.00000 q^{7} +24.0833 q^{8} +17.3137 q^{9} +38.2548 q^{11} +8.74517 q^{12} -19.3431 q^{13} -18.1005 q^{14} -51.7645 q^{16} +87.2254 q^{17} -44.7696 q^{18} -44.2254 q^{19} -46.5980 q^{21} -98.9188 q^{22} -218.167 q^{23} -160.319 q^{24} +50.0172 q^{26} +64.4802 q^{27} -9.19596 q^{28} -46.9411 q^{29} +194.558 q^{31} -58.8141 q^{32} -254.657 q^{33} -225.546 q^{34} -22.7452 q^{36} -366.853 q^{37} +114.357 q^{38} +128.765 q^{39} -339.362 q^{41} +120.492 q^{42} +226.167 q^{43} -50.2557 q^{44} +564.132 q^{46} -11.6762 q^{47} +344.589 q^{48} +49.0000 q^{49} -580.647 q^{51} +25.4113 q^{52} +209.019 q^{53} -166.732 q^{54} +168.583 q^{56} +294.402 q^{57} +121.380 q^{58} -616.000 q^{59} +320.735 q^{61} -503.087 q^{62} +121.196 q^{63} +566.197 q^{64} +658.488 q^{66} -14.5097 q^{67} -114.589 q^{68} +1452.30 q^{69} -952.000 q^{71} +416.971 q^{72} -824.489 q^{73} +948.603 q^{74} +58.0993 q^{76} +267.784 q^{77} -332.958 q^{78} +156.275 q^{79} -896.706 q^{81} +877.519 q^{82} +1036.53 q^{83} +61.2162 q^{84} -584.818 q^{86} +312.480 q^{87} +921.301 q^{88} -170.225 q^{89} -135.402 q^{91} +286.607 q^{92} -1295.15 q^{93} +30.1921 q^{94} +391.517 q^{96} -1059.87 q^{97} -126.704 q^{98} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 56 q^{14} + 168 q^{16} + 50 q^{17} - 16 q^{18} + 36 q^{19} - 14 q^{21} + 184 q^{22} - 244 q^{23}+ \cdots + 940 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 - 2 * q^3 + 20 * q^4 - 8 * q^6 + 14 * q^7 - 48 * q^8 + 12 * q^9 - 14 * q^11 + 108 * q^12 - 50 * q^13 - 56 * q^14 + 168 * q^16 + 50 * q^17 - 16 * q^18 + 36 * q^19 - 14 * q^21 + 184 * q^22 - 244 * q^23 - 496 * q^24 + 216 * q^26 - 86 * q^27 + 140 * q^28 - 26 * q^29 - 120 * q^31 - 672 * q^32 - 498 * q^33 - 24 * q^34 - 136 * q^36 - 564 * q^37 - 320 * q^38 - 14 * q^39 - 328 * q^41 - 56 * q^42 + 260 * q^43 - 1164 * q^44 + 704 * q^46 + 350 * q^47 + 1368 * q^48 + 98 * q^49 - 754 * q^51 - 628 * q^52 + 56 * q^53 + 648 * q^54 - 336 * q^56 + 668 * q^57 + 8 * q^58 - 1232 * q^59 + 336 * q^61 + 1200 * q^62 + 84 * q^63 + 2128 * q^64 + 1976 * q^66 + 152 * q^67 - 908 * q^68 + 1332 * q^69 - 1904 * q^71 + 800 * q^72 - 676 * q^73 + 2016 * q^74 + 1768 * q^76 - 98 * q^77 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 + 816 * q^82 + 376 * q^83 + 756 * q^84 - 768 * q^86 + 410 * q^87 + 4688 * q^88 - 216 * q^89 - 350 * q^91 - 264 * q^92 - 2760 * q^93 - 1928 * q^94 - 2464 * q^96 - 2742 * q^97 - 392 * q^98 + 940 * q^99

Introduction

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