LMFDB - Embedded newform 175.4.b.f.99.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(99,175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 64x^{6} + 1264x^{4} + 8905x^{2} + 14400 $ x^8 + 64x^6 + 1264x^4 + 8905*x^2 + 14400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.6
Root			$3.53510i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.f.99.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.53510i q^{2} -6.46622i q^{3} -12.5671 q^{4} +29.3249 q^{6} +7.00000i q^{7} -20.7124i q^{8} -14.8119 q^{9} -54.0684 q^{11} +81.2617i q^{12} -75.2159i q^{13} -31.7457 q^{14} -6.60441 q^{16} -71.2538i q^{17} -67.1736i q^{18} +65.5100 q^{19} +45.2635 q^{21} -245.206i q^{22} -125.688i q^{23} -133.931 q^{24} +341.111 q^{26} -78.8106i q^{27} -87.9699i q^{28} -190.405 q^{29} -193.105 q^{31} -195.650i q^{32} +349.618i q^{33} +323.143 q^{34} +186.144 q^{36} +114.673i q^{37} +297.094i q^{38} -486.362 q^{39} +216.896 q^{41} +205.275i q^{42} +413.032i q^{43} +679.484 q^{44} +570.007 q^{46} -113.555i q^{47} +42.7055i q^{48} -49.0000 q^{49} -460.743 q^{51} +945.247i q^{52} -584.366i q^{53} +357.414 q^{54} +144.986 q^{56} -423.602i q^{57} -863.504i q^{58} -203.748 q^{59} -162.539 q^{61} -875.749i q^{62} -103.684i q^{63} +834.459 q^{64} -1585.55 q^{66} +477.534i q^{67} +895.456i q^{68} -812.725 q^{69} +822.294 q^{71} +306.790i q^{72} +798.993i q^{73} -520.052 q^{74} -823.273 q^{76} -378.479i q^{77} -2205.70i q^{78} +468.087 q^{79} -909.529 q^{81} +983.643i q^{82} +310.333i q^{83} -568.832 q^{84} -1873.14 q^{86} +1231.20i q^{87} +1119.88i q^{88} -1314.90 q^{89} +526.511 q^{91} +1579.53i q^{92} +1248.66i q^{93} +514.985 q^{94} -1265.12 q^{96} -1314.66i q^{97} -222.220i q^{98} +800.858 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9} + 200 q^{11} - 56 q^{14} + 320 q^{16} + 58 q^{19} - 42 q^{21} + 42 q^{24} + 1400 q^{26} - 258 q^{29} + 228 q^{31} - 406 q^{34} + 2202 q^{36} - 1348 q^{39} + 1342 q^{41}+ \cdots - 4766 q^{99}+O(q^{100}) $ 8 * q - 72 * q^4 + 2 * q^6 - 122 * q^9 + 200 * q^11 - 56 * q^14 + 320 * q^16 + 58 * q^19 - 42 * q^21 + 42 * q^24 + 1400 * q^26 - 258 * q^29 + 228 * q^31 - 406 * q^34 + 2202 * q^36 - 1348 * q^39 + 1342 * q^41 - 876 * q^44 - 1994 * q^46 - 392 * q^49 - 1770 * q^51 + 5554 * q^54 + 378 * q^56 - 2036 * q^59 + 100 * q^61 + 4842 * q^64 - 7682 * q^66 - 2160 * q^69 + 430 * q^71 - 1246 * q^74 - 6514 * q^76 + 1902 * q^79 + 56 * q^81 + 2310 * q^84 - 198 * q^86 - 5638 * q^89 + 616 * q^91 + 6112 * q^94 - 2690 * q^96 - 4766 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.53510i	1.60340i	0.597727	+	0.801700i	$0.296071\pi$
$2$	4.53510i	1.60340i	−0.597727	+	0.801700i	$0.703929\pi$
$3$	− 6.46622i	− 1.24442i	−0.782849	−	0.622212i	$-0.786234\pi$
$3$	− 6.46622i	− 1.24442i	0.782849	−	0.622212i	$-0.213766\pi$
$4$	−12.5671	−1.57089
$5$	0	0
$5$	0	0
$6$	29.3249	1.99531
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	− 20.7124i	− 0.915365i
$9$	−14.8119	−0.548591
$10$	0	0
$11$	−54.0684	−1.48202	−0.741011	−	0.671493i	$-0.765653\pi$
$11$	−54.0684	−1.48202	−0.741011	+	0.671493i	$0.765653\pi$
$12$	81.2617i	1.95485i
$13$	− 75.2159i	− 1.60470i	−0.596853	−	0.802351i	$-0.703582\pi$
$13$	− 75.2159i	− 1.60470i	0.596853	−	0.802351i	$-0.296418\pi$
$14$	−31.7457	−0.606028
$15$	0	0
$16$	−6.60441	−0.103194
$17$	− 71.2538i	− 1.01656i	−0.861191	−	0.508282i	$-0.830281\pi$
$17$	− 71.2538i	− 1.01656i	0.861191	−	0.508282i	$-0.169719\pi$
$18$	− 67.1736i	− 0.879610i
$19$	65.5100	0.791002	0.395501	−	0.918466i	$-0.370571\pi$
$19$	65.5100	0.791002	0.395501	+	0.918466i	$0.370571\pi$
$20$	0	0
$21$	45.2635	0.470348
$22$	− 245.206i	− 2.37627i
$23$	− 125.688i	− 1.13947i	−0.821830	−	0.569733i	$-0.807047\pi$
$23$	− 125.688i	− 1.13947i	0.821830	−	0.569733i	$-0.192953\pi$
$24$	−133.931	−1.13910
$25$	0	0
$26$	341.111	2.57298
$27$	− 78.8106i	− 0.561745i
$28$	− 87.9699i	− 0.593741i
$29$	−190.405	−1.21922	−0.609608	−	0.792703i	$-0.708673\pi$
$29$	−190.405	−1.21922	−0.609608	+	0.792703i	$0.708673\pi$
$30$	0	0
$31$	−193.105	−1.11879	−0.559397	−	0.828900i	$-0.688967\pi$
$31$	−193.105	−1.11879	−0.559397	+	0.828900i	$0.688967\pi$
$32$	− 195.650i	− 1.08083i
$33$	349.618i	1.84426i
$34$	323.143	1.62996
$35$	0	0
$36$	186.144	0.861776
$37$	114.673i	0.509516i	0.967005	+	0.254758i	$0.0819957\pi$
$37$	114.673i	0.509516i	−0.967005	+	0.254758i	$0.918004\pi$
$38$	297.094i	1.26829i
$39$	−486.362	−1.99693
$40$	0	0
$41$	216.896	0.826180	0.413090	−	0.910690i	$-0.364449\pi$
$41$	216.896	0.826180	0.413090	+	0.910690i	$0.364449\pi$
$42$	205.275i	0.754156i
$43$	413.032i	1.46481i	0.680870	+	0.732405i	$0.261602\pi$
$43$	413.032i	1.46481i	−0.680870	+	0.732405i	$0.738398\pi$
$44$	679.484	2.32809
$45$	0	0
$46$	570.007	1.82702
$47$	− 113.555i	− 0.352421i	−0.984353	−	0.176210i	$-0.943616\pi$
$47$	− 113.555i	− 0.352421i	0.984353	−	0.176210i	$-0.0563839\pi$
$48$	42.7055i	0.128417i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−460.743	−1.26504
$52$	945.247i	2.52081i
$53$	− 584.366i	− 1.51451i	−0.653122	−	0.757253i	$-0.726541\pi$
$53$	− 584.366i	− 1.51451i	0.653122	−	0.757253i	$-0.273459\pi$
$54$	357.414	0.900701
$55$	0	0
$56$	144.986	0.345976
$57$	− 423.602i	− 0.984341i
$58$	− 863.504i	− 1.95489i
$59$	−203.748	−0.449589	−0.224795	−	0.974406i	$-0.572171\pi$
$59$	−203.748	−0.449589	−0.224795	+	0.974406i	$0.572171\pi$
$60$	0	0
$61$	−162.539	−0.341163	−0.170581	−	0.985344i	$-0.554565\pi$
$61$	−162.539	−0.341163	−0.170581	+	0.985344i	$0.554565\pi$
$62$	− 875.749i	− 1.79387i
$63$	− 103.684i	− 0.207348i
$64$	834.459	1.62980
$65$	0	0
$66$	−1585.55	−2.95709
$67$	477.534i	0.870747i	0.900250	+	0.435374i	$0.143384\pi$
$67$	477.534i	0.870747i	−0.900250	+	0.435374i	$0.856616\pi$
$68$	895.456i	1.59691i
$69$	−812.725	−1.41798
$70$	0	0
$71$	822.294	1.37448	0.687242	−	0.726429i	$-0.258821\pi$
$71$	822.294	1.37448	0.687242	+	0.726429i	$0.258821\pi$
$72$	306.790i	0.502161i
$73$	798.993i	1.28103i	0.767947	+	0.640514i	$0.221279\pi$
$73$	798.993i	1.28103i	−0.767947	+	0.640514i	$0.778721\pi$
$74$	−520.052	−0.816957
$75$	0	0
$76$	−823.273	−1.24258
$77$	− 378.479i	− 0.560152i
$78$	− 2205.70i	− 3.20188i
$79$	468.087	0.666632	0.333316	−	0.942815i	$-0.391832\pi$
$79$	468.087	0.666632	0.333316	+	0.942815i	$0.391832\pi$
$80$	0	0
$81$	−909.529	−1.24764
$82$	983.643i	1.32470i
$83$	310.333i	0.410403i	0.978720	+	0.205202i	$0.0657850\pi$
$83$	310.333i	0.410403i	−0.978720	+	0.205202i	$0.934215\pi$
$84$	−568.832	−0.738865
$85$	0	0
$86$	−1873.14	−2.34867
$87$	1231.20i	1.51722i
$88$	1119.88i	1.35659i
$89$	−1314.90	−1.56606	−0.783029	−	0.621985i	$-0.786326\pi$
$89$	−1314.90	−1.56606	−0.783029	+	0.621985i	$0.786326\pi$
$90$	0	0
$91$	526.511	0.606520
$92$	1579.53i	1.78998i
$93$	1248.66i	1.39225i
$94$	514.985	0.565071
$95$	0	0
$96$	−1265.12	−1.34501
$97$	− 1314.66i	− 1.37612i	−0.725656	−	0.688058i	$-0.758463\pi$
$97$	− 1314.66i	− 1.37612i	0.725656	−	0.688058i	$-0.241537\pi$
$98$	− 222.220i	− 0.229057i
$99$	800.858	0.813023
$100$	0	0
$101$	−401.240	−0.395296	−0.197648	−	0.980273i	$-0.563330\pi$
$101$	−401.240	−0.395296	−0.197648	+	0.980273i	$0.563330\pi$
$102$	− 2089.51i	− 2.02836i
$103$	− 291.844i	− 0.279187i	−0.990209	−	0.139594i	$-0.955420\pi$
$103$	− 291.844i	− 0.279187i	0.990209	−	0.139594i	$-0.0445796\pi$
$104$	−1557.90	−1.46889
$105$	0	0
$106$	2650.16	2.42836
$107$	− 21.9594i	− 0.0198402i	−0.999951	−	0.00992009i	$-0.996842\pi$
$107$	− 21.9594i	− 0.0198402i	0.999951	−	0.00992009i	$-0.00315772\pi$
$108$	990.422i	0.882439i
$109$	17.3019	0.0152039	0.00760194	−	0.999971i	$-0.497580\pi$
$109$	17.3019	0.0152039	0.00760194	+	0.999971i	$0.497580\pi$
$110$	0	0
$111$	741.498	0.634053
$112$	− 46.2308i	− 0.0390036i
$113$	− 58.8182i	− 0.0489660i	−0.999700	−	0.0244830i	$-0.992206\pi$
$113$	− 58.8182i	− 0.0489660i	0.999700	−	0.0244830i	$-0.00779396\pi$
$114$	1921.08	1.57829
$115$	0	0
$116$	2392.84	1.91526
$117$	1114.09i	0.880324i
$118$	− 924.018i	− 0.720871i
$119$	498.777	0.384225
$120$	0	0
$121$	1592.39	1.19639
$122$	− 737.129i	− 0.547020i
$123$	− 1402.49i	− 1.02812i
$124$	2426.77	1.75750
$125$	0	0
$126$	470.216	0.332461
$127$	1862.03i	1.30101i	0.759501	+	0.650507i	$0.225443\pi$
$127$	1862.03i	1.30101i	−0.759501	+	0.650507i	$0.774557\pi$
$128$	2219.15i	1.53240i
$129$	2670.75	1.82284
$130$	0	0
$131$	775.161	0.516994	0.258497	−	0.966012i	$-0.416773\pi$
$131$	775.161	0.516994	0.258497	+	0.966012i	$0.416773\pi$
$132$	− 4393.69i	− 2.89714i
$133$	458.570i	0.298971i
$134$	−2165.66	−1.39616
$135$	0	0
$136$	−1475.83	−0.930528
$137$	− 1027.95i	− 0.641051i	−0.947240	−	0.320525i	$-0.896141\pi$
$137$	− 1027.95i	− 0.641051i	0.947240	−	0.320525i	$-0.103859\pi$
$138$	− 3685.79i	− 2.27359i
$139$	1029.66	0.628309	0.314154	−	0.949372i	$-0.398279\pi$
$139$	1029.66	0.628309	0.314154	+	0.949372i	$0.398279\pi$
$140$	0	0
$141$	−734.274	−0.438561
$142$	3729.18i	2.20385i
$143$	4066.80i	2.37820i
$144$	97.8241	0.0566112
$145$	0	0
$146$	−3623.51	−2.05400
$147$	316.845i	0.177775i
$148$	− 1441.11i	− 0.800393i
$149$	−1414.67	−0.777814	−0.388907	−	0.921277i	$-0.627147\pi$
$149$	−1414.67	−0.777814	−0.388907	+	0.921277i	$0.627147\pi$
$150$	0	0
$151$	2094.96	1.12904	0.564522	−	0.825418i	$-0.309060\pi$
$151$	2094.96	1.12904	0.564522	+	0.825418i	$0.309060\pi$
$152$	− 1356.87i	− 0.724056i
$153$	1055.41i	0.557678i
$154$	1716.44	0.898147
$155$	0	0
$156$	6112.17	3.13696
$157$	− 709.495i	− 0.360662i	−0.983606	−	0.180331i	$-0.942283\pi$
$157$	− 709.495i	− 0.360662i	0.983606	−	0.180331i	$-0.0577168\pi$
$158$	2122.82i	1.06888i
$159$	−3778.64	−1.88469
$160$	0	0
$161$	879.815	0.430678
$162$	− 4124.80i	− 2.00046i
$163$	− 3276.23i	− 1.57432i	−0.616749	−	0.787160i	$-0.711551\pi$
$163$	− 3276.23i	− 1.57432i	0.616749	−	0.787160i	$-0.288449\pi$
$164$	−2725.75	−1.29784
$165$	0	0
$166$	−1407.39	−0.658040
$167$	2860.69i	1.32555i	0.748819	+	0.662775i	$0.230621\pi$
$167$	2860.69i	1.32555i	−0.748819	+	0.662775i	$0.769379\pi$
$168$	− 937.514i	− 0.430540i
$169$	−3460.42	−1.57507
$170$	0	0
$171$	−970.331	−0.433936
$172$	− 5190.62i	− 2.30105i
$173$	− 2803.44i	− 1.23203i	−0.787734	−	0.616015i	$-0.788746\pi$
$173$	− 2803.44i	− 1.23203i	0.787734	−	0.616015i	$-0.211254\pi$
$174$	−5583.61	−2.43271
$175$	0	0
$176$	357.090	0.152936
$177$	1317.48i	0.559479i
$178$	− 5963.20i	− 2.51102i
$179$	−248.617	−0.103813	−0.0519064	−	0.998652i	$-0.516530\pi$
$179$	−248.617	−0.103813	−0.0519064	+	0.998652i	$0.516530\pi$
$180$	0	0
$181$	2075.56	0.852347	0.426174	−	0.904641i	$-0.359861\pi$
$181$	2075.56	0.852347	0.426174	+	0.904641i	$0.359861\pi$
$182$	2387.78i	0.972494i
$183$	1051.01i	0.424551i
$184$	−2603.29	−1.04303
$185$	0	0
$186$	−5662.78	−2.23234
$187$	3852.58i	1.50657i
$188$	1427.07i	0.553614i
$189$	551.674	0.212319
$190$	0	0
$191$	−1252.95	−0.474663	−0.237331	−	0.971429i	$-0.576273\pi$
$191$	−1252.95	−0.474663	−0.237331	+	0.971429i	$0.576273\pi$
$192$	− 5395.79i	− 2.02817i
$193$	− 1602.12i	− 0.597527i	−0.954327	−	0.298764i	$-0.903426\pi$
$193$	− 1602.12i	− 0.597527i	0.954327	−	0.298764i	$-0.0965743\pi$
$194$	5962.10	2.20646
$195$	0	0
$196$	615.789	0.224413
$197$	2346.38i	0.848593i	0.905523	+	0.424296i	$0.139479\pi$
$197$	2346.38i	0.848593i	−0.905523	+	0.424296i	$0.860521\pi$
$198$	3631.97i	1.30360i
$199$	1993.53	0.710140	0.355070	−	0.934840i	$-0.384457\pi$
$199$	1993.53	0.710140	0.355070	+	0.934840i	$0.384457\pi$
$200$	0	0
$201$	3087.84	1.08358
$202$	− 1819.67i	− 0.633818i
$203$	− 1332.83i	− 0.460820i
$204$	5790.21	1.98723
$205$	0	0
$206$	1323.54	0.447649
$207$	1861.68i	0.625101i
$208$	496.756i	0.165595i
$209$	−3542.02	−1.17228
$210$	0	0
$211$	−5852.55	−1.90951	−0.954754	−	0.297396i	$-0.903882\pi$
$211$	−5852.55	−1.90951	−0.954754	+	0.297396i	$0.903882\pi$
$212$	7343.80i	2.37912i
$213$	− 5317.13i	− 1.71044i
$214$	99.5883	0.0318117
$215$	0	0
$216$	−1632.35	−0.514202
$217$	− 1351.73i	− 0.422864i
$218$	78.4659i	0.0243779i
$219$	5166.46	1.59414
$220$	0	0
$221$	−5359.42	−1.63128
$222$	3362.77i	1.01664i
$223$	1719.58i	0.516374i	0.966095	+	0.258187i	$0.0831251\pi$
$223$	1719.58i	0.516374i	−0.966095	+	0.258187i	$0.916875\pi$
$224$	1369.55	0.408514
$225$	0	0
$226$	266.747	0.0785120
$227$	− 5686.00i	− 1.66252i	−0.555881	−	0.831262i	$-0.687619\pi$
$227$	− 5686.00i	− 1.66252i	0.555881	−	0.831262i	$-0.312381\pi$
$228$	5323.46i	1.54629i
$229$	3087.40	0.890923	0.445461	−	0.895301i	$-0.353040\pi$
$229$	3087.40	0.890923	0.445461	+	0.895301i	$0.353040\pi$
$230$	0	0
$231$	−2447.33	−0.697066
$232$	3943.73i	1.11603i
$233$	− 3997.55i	− 1.12398i	−0.827143	−	0.561991i	$-0.810035\pi$
$233$	− 3997.55i	− 1.12398i	0.827143	−	0.561991i	$-0.189965\pi$
$234$	−5052.52	−1.41151
$235$	0	0
$236$	2560.53	0.706255
$237$	− 3026.75i	− 0.829573i
$238$	2262.00i	0.616067i
$239$	4499.79	1.21786	0.608928	−	0.793226i	$-0.291600\pi$
$239$	4499.79	1.21786	0.608928	+	0.793226i	$0.291600\pi$
$240$	0	0
$241$	3633.26	0.971115	0.485557	−	0.874205i	$-0.338617\pi$
$241$	3633.26	0.971115	0.485557	+	0.874205i	$0.338617\pi$
$242$	7221.66i	1.91829i
$243$	3753.32i	0.990847i
$244$	2042.64	0.535929
$245$	0	0
$246$	6360.45	1.64849
$247$	− 4927.39i	− 1.26932i
$248$	3999.65i	1.02411i
$249$	2006.68	0.510715
$250$	0	0
$251$	−1211.21	−0.304585	−0.152292	−	0.988335i	$-0.548666\pi$
$251$	−1211.21	−0.304585	−0.152292	+	0.988335i	$0.548666\pi$
$252$	1303.00i	0.325721i
$253$	6795.74i	1.68871i
$254$	−8444.50	−2.08604
$255$	0	0
$256$	−3388.39	−0.827245
$257$	− 6225.81i	− 1.51111i	−0.655085	−	0.755556i	$-0.727367\pi$
$257$	− 6225.81i	− 1.51111i	0.655085	−	0.755556i	$-0.272633\pi$
$258$	12112.1i	2.92275i
$259$	−802.709	−0.192579
$260$	0	0
$261$	2820.27	0.668851
$262$	3515.43i	0.828947i
$263$	757.377i	0.177574i	0.996051	+	0.0887869i	$0.0282990\pi$
$263$	757.377i	0.177574i	−0.996051	+	0.0887869i	$0.971701\pi$
$264$	7241.41	1.68817
$265$	0	0
$266$	−2079.66	−0.479369
$267$	8502.43i	1.94884i
$268$	− 6001.23i	− 1.36785i
$269$	−1750.59	−0.396786	−0.198393	−	0.980123i	$-0.563572\pi$
$269$	−1750.59	−0.396786	−0.198393	+	0.980123i	$0.563572\pi$
$270$	0	0
$271$	4451.86	0.997902	0.498951	−	0.866630i	$-0.333719\pi$
$271$	4451.86	0.997902	0.498951	+	0.866630i	$0.333719\pi$
$272$	470.589i	0.104903i
$273$	− 3404.53i	− 0.754768i
$274$	4661.87	1.02786
$275$	0	0
$276$	10213.6	2.22749
$277$	− 7691.69i	− 1.66841i	−0.551456	−	0.834204i	$-0.685928\pi$
$277$	− 7691.69i	− 1.66841i	0.551456	−	0.834204i	$-0.314072\pi$
$278$	4669.62i	1.00743i
$279$	2860.26	0.613760
$280$	0	0
$281$	−199.034	−0.0422540	−0.0211270	−	0.999777i	$-0.506725\pi$
$281$	−199.034	−0.0422540	−0.0211270	+	0.999777i	$0.506725\pi$
$282$	− 3330.01i	− 0.703188i
$283$	799.307i	0.167893i	0.996470	+	0.0839467i	$0.0267526\pi$
$283$	799.307i	0.167893i	−0.996470	+	0.0839467i	$0.973247\pi$
$284$	−10333.9	−2.15916
$285$	0	0
$286$	−18443.3	−3.81321
$287$	1518.27i	0.312267i
$288$	2897.96i	0.592931i
$289$	−164.110	−0.0334031
$290$	0	0
$291$	−8500.86	−1.71247
$292$	− 10041.0i	− 2.01235i
$293$	− 3302.75i	− 0.658528i	−0.944238	−	0.329264i	$-0.893199\pi$
$293$	− 3302.75i	− 0.658528i	0.944238	−	0.329264i	$-0.106801\pi$
$294$	−1436.92	−0.285044
$295$	0	0
$296$	2375.14	0.466393
$297$	4261.16i	0.832518i
$298$	− 6415.67i	− 1.24715i
$299$	−9453.72	−1.82850
$300$	0	0
$301$	−2891.22	−0.553646
$302$	9500.87i	1.81031i
$303$	2594.51i	0.491916i
$304$	−432.655	−0.0816265
$305$	0	0
$306$	−4786.38	−0.894180
$307$	− 8628.17i	− 1.60402i	−0.597308	−	0.802012i	$-0.703763\pi$
$307$	− 8628.17i	− 1.60402i	0.597308	−	0.802012i	$-0.296237\pi$
$308$	4756.39i	0.879937i
$309$	−1887.13	−0.347427
$310$	0	0
$311$	−4900.60	−0.893530	−0.446765	−	0.894651i	$-0.647424\pi$
$311$	−4900.60	−0.893530	−0.446765	+	0.894651i	$0.647424\pi$
$312$	10073.7i	1.82792i
$313$	9114.26i	1.64590i	0.568110	+	0.822952i	$0.307675\pi$
$313$	9114.26i	1.64590i	−0.568110	+	0.822952i	$0.692325\pi$
$314$	3217.63	0.578285
$315$	0	0
$316$	−5882.51	−1.04721
$317$	− 8022.82i	− 1.42147i	−0.703459	−	0.710736i	$-0.748362\pi$
$317$	− 8022.82i	− 1.42147i	0.703459	−	0.710736i	$-0.251638\pi$
$318$	− 17136.5i	− 3.02191i
$319$	10294.9	1.80691
$320$	0	0
$321$	−141.995	−0.0246896
$322$	3990.05i	0.690549i
$323$	− 4667.84i	− 0.804104i
$324$	11430.2	1.95990
$325$	0	0
$326$	14858.0	2.52426
$327$	− 111.878i	− 0.0189201i
$328$	− 4492.42i	− 0.756257i
$329$	794.888	0.133202
$330$	0	0
$331$	−1333.58	−0.221451	−0.110725	−	0.993851i	$-0.535317\pi$
$331$	−1333.58	−0.221451	−0.110725	+	0.993851i	$0.535317\pi$
$332$	− 3899.99i	− 0.644698i
$333$	− 1698.53i	− 0.279515i
$334$	−12973.5	−2.12539
$335$	0	0
$336$	−298.939	−0.0485370
$337$	− 1687.33i	− 0.272744i	−0.990658	−	0.136372i	$-0.956456\pi$
$337$	− 1687.33i	− 0.272744i	0.990658	−	0.136372i	$-0.0435442\pi$
$338$	− 15693.4i	− 2.52546i
$339$	−380.331	−0.0609344
$340$	0	0
$341$	10440.9	1.65808
$342$	− 4400.55i	− 0.695773i
$343$	− 343.000i	− 0.0539949i
$344$	8554.87	1.34084
$345$	0	0
$346$	12713.9	1.97544
$347$	5955.18i	0.921299i	0.887582	+	0.460649i	$0.152383\pi$
$347$	5955.18i	0.921299i	−0.887582	+	0.460649i	$0.847617\pi$
$348$	− 15472.6i	− 2.38339i
$349$	−6876.59	−1.05471	−0.527357	−	0.849644i	$-0.676817\pi$
$349$	−6876.59	−1.05471	−0.527357	+	0.849644i	$0.676817\pi$
$350$	0	0
$351$	−5927.81	−0.901433
$352$	10578.5i	1.60181i
$353$	− 1395.95i	− 0.210479i	−0.994447	−	0.105239i	$-0.966439\pi$
$353$	− 1395.95i	− 0.210479i	0.994447	−	0.105239i	$-0.0335609\pi$
$354$	−5974.90	−0.897069
$355$	0	0
$356$	16524.5	2.46011
$357$	− 3225.20i	− 0.478139i
$358$	− 1127.50i	− 0.166453i
$359$	382.988	0.0563046	0.0281523	−	0.999604i	$-0.491038\pi$
$359$	382.988	0.0563046	0.0281523	+	0.999604i	$0.491038\pi$
$360$	0	0
$361$	−2567.44	−0.374316
$362$	9412.85i	1.36665i
$363$	− 10296.8i	− 1.48881i
$364$	−6616.73	−0.952777
$365$	0	0
$366$	−4766.43	−0.680725
$367$	− 10178.4i	− 1.44771i	−0.689951	−	0.723856i	$-0.742368\pi$
$367$	− 10178.4i	− 1.44771i	0.689951	−	0.723856i	$-0.257632\pi$
$368$	830.094i	0.117586i
$369$	−3212.65	−0.453235
$370$	0	0
$371$	4090.56	0.572430
$372$	− 15692.0i	− 2.18708i
$373$	2991.00i	0.415196i	0.978214	+	0.207598i	$0.0665645\pi$
$373$	2991.00i	0.415196i	−0.978214	+	0.207598i	$0.933435\pi$
$374$	−17471.8	−2.41563
$375$	0	0
$376$	−2352.00	−0.322594
$377$	14321.5i	1.95648i
$378$	2501.90i	0.340433i
$379$	−7400.66	−1.00302	−0.501512	−	0.865150i	$-0.667223\pi$
$379$	−7400.66	−1.00302	−0.501512	+	0.865150i	$0.667223\pi$
$380$	0	0
$381$	12040.3	1.61901
$382$	− 5682.27i	− 0.761074i
$383$	10769.6i	1.43682i	0.695622	+	0.718408i	$0.255129\pi$
$383$	10769.6i	1.43682i	−0.695622	+	0.718408i	$0.744871\pi$
$384$	14349.5	1.90695
$385$	0	0
$386$	7265.75	0.958075
$387$	− 6117.81i	− 0.803580i
$388$	16521.5i	2.16173i
$389$	11193.3	1.45893	0.729466	−	0.684017i	$-0.239769\pi$
$389$	11193.3	1.45893	0.729466	+	0.684017i	$0.239769\pi$
$390$	0	0
$391$	−8955.74	−1.15834
$392$	1014.91i	0.130766i
$393$	− 5012.36i	− 0.643359i
$394$	−10641.1	−1.36063
$395$	0	0
$396$	−10064.5	−1.27717
$397$	10371.9i	1.31121i	0.755104	+	0.655605i	$0.227586\pi$
$397$	10371.9i	1.31121i	−0.755104	+	0.655605i	$0.772414\pi$
$398$	9040.88i	1.13864i
$399$	2965.21	0.372046
$400$	0	0
$401$	5402.20	0.672751	0.336375	−	0.941728i	$-0.390799\pi$
$401$	5402.20	0.672751	0.336375	+	0.941728i	$0.390799\pi$
$402$	14003.7i	1.73741i
$403$	14524.5i	1.79533i
$404$	5042.44	0.620967
$405$	0	0
$406$	6044.53	0.738879
$407$	− 6200.17i	− 0.755113i
$408$	9543.07i	1.15797i
$409$	12829.9	1.55110	0.775549	−	0.631288i	$-0.217473\pi$
$409$	12829.9	1.55110	0.775549	+	0.631288i	$0.217473\pi$
$410$	0	0
$411$	−6646.97	−0.797739
$412$	3667.64i	0.438572i
$413$	− 1426.24i	− 0.169929i
$414$	−8442.91	−1.00229
$415$	0	0
$416$	−14716.0	−1.73440
$417$	− 6658.02i	− 0.781882i
$418$	− 16063.4i	− 1.87964i
$419$	−5620.85	−0.655362	−0.327681	−	0.944788i	$-0.606267\pi$
$419$	−5620.85	−0.655362	−0.327681	+	0.944788i	$0.606267\pi$
$420$	0	0
$421$	−2177.11	−0.252033	−0.126017	−	0.992028i	$-0.540219\pi$
$421$	−2177.11	−0.252033	−0.126017	+	0.992028i	$0.540219\pi$
$422$	− 26541.9i	− 3.06171i
$423$	1681.98i	0.193335i
$424$	−12103.6	−1.38633
$425$	0	0
$426$	24113.7	2.74252
$427$	− 1137.77i	− 0.128947i
$428$	275.967i	0.0311668i
$429$	26296.8	2.95949
$430$	0	0
$431$	10396.4	1.16190	0.580950	−	0.813939i	$-0.302681\pi$
$431$	10396.4	1.16190	0.580950	+	0.813939i	$0.302681\pi$
$432$	520.497i	0.0579686i
$433$	− 11875.6i	− 1.31803i	−0.752129	−	0.659015i	$-0.770973\pi$
$433$	− 11875.6i	− 1.31803i	0.752129	−	0.659015i	$-0.229027\pi$
$434$	6130.24	0.678021
$435$	0	0
$436$	−217.435	−0.0238836
$437$	− 8233.81i	− 0.901320i
$438$	23430.4i	2.55605i
$439$	−640.369	−0.0696200	−0.0348100	−	0.999394i	$-0.511083\pi$
$439$	−640.369	−0.0696200	−0.0348100	+	0.999394i	$0.511083\pi$
$440$	0	0
$441$	725.785	0.0783701
$442$	− 24305.5i	− 2.61560i
$443$	854.852i	0.0916823i	0.998949	+	0.0458411i	$0.0145968\pi$
$443$	854.852i	0.0916823i	−0.998949	+	0.0458411i	$0.985403\pi$
$444$	−9318.50	−0.996028
$445$	0	0
$446$	−7798.46	−0.827954
$447$	9147.56i	0.967930i
$448$	5841.21i	0.616008i
$449$	−3427.44	−0.360247	−0.180123	−	0.983644i	$-0.557650\pi$
$449$	−3427.44	−0.360247	−0.180123	+	0.983644i	$0.557650\pi$
$450$	0	0
$451$	−11727.2	−1.22442
$452$	739.176i	0.0769202i
$453$	− 13546.5i	− 1.40501i
$454$	25786.6	2.66569
$455$	0	0
$456$	−8773.79	−0.901032
$457$	1709.31i	0.174963i	0.996166	+	0.0874814i	$0.0278818\pi$
$457$	1709.31i	0.174963i	−0.996166	+	0.0874814i	$0.972118\pi$
$458$	14001.7i	1.42851i
$459$	−5615.56	−0.571050
$460$	0	0
$461$	2251.18	0.227436	0.113718	−	0.993513i	$-0.463724\pi$
$461$	2251.18	0.227436	0.113718	+	0.993513i	$0.463724\pi$
$462$	− 11098.9i	− 1.11768i
$463$	3200.80i	0.321282i	0.987013	+	0.160641i	$0.0513562\pi$
$463$	3200.80i	0.321282i	−0.987013	+	0.160641i	$0.948644\pi$
$464$	1257.51	0.125816
$465$	0	0
$466$	18129.3	1.80219
$467$	5477.39i	0.542748i	0.962474	+	0.271374i	$0.0874780\pi$
$467$	5477.39i	0.542748i	−0.962474	+	0.271374i	$0.912522\pi$
$468$	− 14000.9i	− 1.38289i
$469$	−3342.74	−0.329112
$470$	0	0
$471$	−4587.75	−0.448816
$472$	4220.11i	0.411538i
$473$	− 22332.0i	− 2.17088i
$474$	13726.6	1.33014
$475$	0	0
$476$	−6268.19	−0.603576
$477$	8655.60i	0.830844i
$478$	20407.0i	1.95271i
$479$	14182.0	1.35280	0.676399	−	0.736535i	$-0.263539\pi$
$479$	14182.0	1.35280	0.676399	+	0.736535i	$0.263539\pi$
$480$	0	0
$481$	8625.20	0.817620
$482$	16477.2i	1.55709i
$483$	− 5689.07i	− 0.535946i
$484$	−20011.8	−1.87940
$485$	0	0
$486$	−17021.7	−1.58872
$487$	− 5320.83i	− 0.495092i	−0.968876	−	0.247546i	$-0.920376\pi$
$487$	− 5320.83i	− 0.495092i	0.968876	−	0.247546i	$-0.0796242\pi$
$488$	3366.56i	0.312289i
$489$	−21184.8	−1.95912
$490$	0	0
$491$	−6574.80	−0.604311	−0.302155	−	0.953259i	$-0.597706\pi$
$491$	−6574.80	−0.604311	−0.302155	+	0.953259i	$0.597706\pi$
$492$	17625.3i	1.61506i
$493$	13567.1i	1.23941i
$494$	22346.2	2.03523
$495$	0	0
$496$	1275.34	0.115453
$497$	5756.06i	0.519506i
$498$	9100.49i	0.818881i
$499$	1507.68	0.135256	0.0676282	−	0.997711i	$-0.478457\pi$
$499$	1507.68	0.135256	0.0676282	+	0.997711i	$0.478457\pi$
$500$	0	0
$501$	18497.8	1.64955
$502$	− 5492.95i	− 0.488371i
$503$	8782.24i	0.778491i	0.921134	+	0.389245i	$0.127264\pi$
$503$	8782.24i	0.778491i	−0.921134	+	0.389245i	$0.872736\pi$
$504$	−2147.53	−0.189799
$505$	0	0
$506$	−30819.4	−2.70768
$507$	22375.9i	1.96005i
$508$	− 23400.4i	− 2.04375i
$509$	−12696.6	−1.10563	−0.552815	−	0.833304i	$-0.686447\pi$
$509$	−12696.6	−1.10563	−0.552815	+	0.833304i	$0.686447\pi$
$510$	0	0
$511$	−5592.95	−0.484183
$512$	2386.50i	0.205995i
$513$	− 5162.88i	− 0.444341i
$514$	28234.7	2.42291
$515$	0	0
$516$	−33563.7	−2.86349
$517$	6139.76i	0.522295i
$518$	− 3640.36i	− 0.308781i
$519$	−18127.6	−1.53317
$520$	0	0
$521$	−10434.7	−0.877455	−0.438727	−	0.898620i	$-0.644571\pi$
$521$	−10434.7	−0.877455	−0.438727	+	0.898620i	$0.644571\pi$
$522$	12790.2i	1.07243i
$523$	8403.03i	0.702560i	0.936270	+	0.351280i	$0.114254\pi$
$523$	8403.03i	0.702560i	−0.936270	+	0.351280i	$0.885746\pi$
$524$	−9741.55	−0.812140
$525$	0	0
$526$	−3434.78	−0.284722
$527$	13759.4i	1.13733i
$528$	− 2309.02i	− 0.190317i
$529$	−3630.43	−0.298384
$530$	0	0
$531$	3017.91	0.246640
$532$	− 5762.91i	− 0.469650i
$533$	− 16314.0i	− 1.32577i
$534$	−38559.4	−3.12477
$535$	0	0
$536$	9890.85	0.797052
$537$	1607.61i	0.129187i
$538$	− 7939.11i	− 0.636207i
$539$	2649.35	0.211717
$540$	0	0
$541$	8349.66	0.663549	0.331775	−	0.943359i	$-0.392353\pi$
$541$	8349.66	0.663549	0.331775	+	0.943359i	$0.392353\pi$
$542$	20189.6i	1.60004i
$543$	− 13421.0i	− 1.06068i
$544$	−13940.8	−1.09873
$545$	0	0
$546$	15439.9	1.21020
$547$	12185.7i	0.952509i	0.879308	+	0.476254i	$0.158006\pi$
$547$	12185.7i	0.952509i	−0.879308	+	0.476254i	$0.841994\pi$
$548$	12918.4i	1.00702i
$549$	2407.51	0.187159
$550$	0	0
$551$	−12473.4	−0.964402
$552$	16833.4i	1.29797i
$553$	3276.61i	0.251963i
$554$	34882.6	2.67512
$555$	0	0
$556$	−12939.9	−0.987004
$557$	− 540.209i	− 0.0410940i	−0.999789	−	0.0205470i	$-0.993459\pi$
$557$	− 540.209i	− 0.0410940i	0.999789	−	0.0205470i	$-0.00654078\pi$
$558$	12971.5i	0.984103i
$559$	31066.6	2.35058
$560$	0	0
$561$	24911.6	1.87481
$562$	− 902.639i	− 0.0677501i
$563$	1949.26i	0.145917i	0.997335	+	0.0729587i	$0.0232441\pi$
$563$	1949.26i	0.145917i	−0.997335	+	0.0729587i	$0.976756\pi$
$564$	9227.71	0.688930
$565$	0	0
$566$	−3624.94	−0.269200
$567$	− 6366.70i	− 0.471563i
$568$	− 17031.6i	− 1.25815i
$569$	−9487.28	−0.698994	−0.349497	−	0.936938i	$-0.613647\pi$
$569$	−9487.28	−0.698994	−0.349497	+	0.936938i	$0.613647\pi$
$570$	0	0
$571$	−20172.3	−1.47843	−0.739215	−	0.673470i	$-0.764803\pi$
$571$	−20172.3	−1.47843	−0.739215	+	0.673470i	$0.764803\pi$
$572$	− 51108.0i	− 3.73590i
$573$	8101.87i	0.590681i
$574$	−6885.50	−0.500689
$575$	0	0
$576$	−12360.0	−0.894094
$577$	12937.3i	0.933423i	0.884410	+	0.466712i	$0.154561\pi$
$577$	12937.3i	0.933423i	−0.884410	+	0.466712i	$0.845439\pi$
$578$	− 744.254i	− 0.0535586i
$579$	−10359.6	−0.743577
$580$	0	0
$581$	−2172.33	−0.155118
$582$	− 38552.2i	− 2.74578i
$583$	31595.7i	2.24453i
$584$	16549.0	1.17261
$585$	0	0
$586$	14978.3	1.05588
$587$	− 6489.43i	− 0.456299i	−0.973626	−	0.228149i	$-0.926733\pi$
$587$	− 6489.43i	− 0.456299i	0.973626	−	0.228149i	$-0.0732674\pi$
$588$	− 3981.82i	− 0.279265i
$589$	−12650.3	−0.884968
$590$	0	0
$591$	15172.2	1.05601
$592$	− 757.345i	− 0.0525789i
$593$	16803.6i	1.16364i	0.813317	+	0.581821i	$0.197660\pi$
$593$	16803.6i	1.16364i	−0.813317	+	0.581821i	$0.802340\pi$
$594$	−19324.8	−1.33486
$595$	0	0
$596$	17778.3	1.22186
$597$	− 12890.6i	− 0.883716i
$598$	− 42873.5i	− 2.93182i
$599$	−19139.6	−1.30554	−0.652772	−	0.757554i	$-0.726394\pi$
$599$	−19139.6	−1.30554	−0.652772	+	0.757554i	$0.726394\pi$
$600$	0	0
$601$	14304.0	0.970838	0.485419	−	0.874282i	$-0.338667\pi$
$601$	14304.0	0.970838	0.485419	+	0.874282i	$0.338667\pi$
$602$	− 13112.0i	− 0.887715i
$603$	− 7073.21i	− 0.477684i
$604$	−26327.7	−1.77361
$605$	0	0
$606$	−11766.3	−0.788738
$607$	− 2018.10i	− 0.134946i	−0.997721	−	0.0674730i	$-0.978506\pi$
$607$	− 2018.10i	− 0.134946i	0.997721	−	0.0674730i	$-0.0214936\pi$
$608$	− 12817.1i	− 0.854935i
$609$	−8618.39	−0.573456
$610$	0	0
$611$	−8541.17	−0.565530
$612$	− 13263.4i	− 0.876050i
$613$	− 21479.4i	− 1.41524i	−0.706592	−	0.707621i	$-0.749768\pi$
$613$	− 21479.4i	− 1.41524i	0.706592	−	0.707621i	$-0.250232\pi$
$614$	39129.6	2.57189
$615$	0	0
$616$	−7839.19	−0.512743
$617$	− 5856.83i	− 0.382151i	−0.981575	−	0.191075i	$-0.938803\pi$
$617$	− 5856.83i	− 0.382151i	0.981575	−	0.191075i	$-0.0611975\pi$
$618$	− 8558.31i	− 0.557065i
$619$	17619.3	1.14407	0.572035	−	0.820229i	$-0.306154\pi$
$619$	17619.3	1.14407	0.572035	+	0.820229i	$0.306154\pi$
$620$	0	0
$621$	−9905.53	−0.640089
$622$	− 22224.7i	− 1.43269i
$623$	− 9204.30i	− 0.591914i
$624$	3212.13	0.206071
$625$	0	0
$626$	−41334.1	−2.63904
$627$	22903.5i	1.45882i
$628$	8916.31i	0.566560i
$629$	8170.87	0.517955
$630$	0	0
$631$	26395.7	1.66529	0.832643	−	0.553810i	$-0.186827\pi$
$631$	26395.7	1.66529	0.832643	+	0.553810i	$0.186827\pi$
$632$	− 9695.19i	− 0.610212i
$633$	37843.9i	2.37624i
$634$	36384.3	2.27919
$635$	0	0
$636$	47486.6	2.96064
$637$	3685.58i	0.229243i
$638$	46688.3i	2.89719i
$639$	−12179.8	−0.754029
$640$	0	0
$641$	3209.30	0.197753	0.0988764	−	0.995100i	$-0.468475\pi$
$641$	3209.30	0.197753	0.0988764	+	0.995100i	$0.468475\pi$
$642$	− 643.959i	− 0.0395873i
$643$	1762.48i	0.108096i	0.998538	+	0.0540478i	$0.0172123\pi$
$643$	1762.48i	0.108096i	−0.998538	+	0.0540478i	$0.982788\pi$
$644$	−11056.7	−0.676548
$645$	0	0
$646$	21169.1	1.28930
$647$	− 24372.7i	− 1.48098i	−0.672070	−	0.740488i	$-0.734595\pi$
$647$	− 24372.7i	− 1.48098i	0.672070	−	0.740488i	$-0.265405\pi$
$648$	18838.5i	1.14205i
$649$	11016.3	0.666301
$650$	0	0
$651$	−8740.60	−0.526223
$652$	41172.8i	2.47308i
$653$	− 6389.97i	− 0.382938i	−0.981499	−	0.191469i	$-0.938675\pi$
$653$	− 6389.97i	− 0.382938i	0.981499	−	0.191469i	$-0.0613252\pi$
$654$	507.377	0.0303364
$655$	0	0
$656$	−1432.47	−0.0852567
$657$	− 11834.6i	− 0.702760i
$658$	3604.90i	0.213577i
$659$	6111.68	0.361270	0.180635	−	0.983550i	$-0.442185\pi$
$659$	6111.68	0.361270	0.180635	+	0.983550i	$0.442185\pi$
$660$	0	0
$661$	16391.1	0.964510	0.482255	−	0.876031i	$-0.339818\pi$
$661$	16391.1	0.964510	0.482255	+	0.876031i	$0.339818\pi$
$662$	− 6047.92i	− 0.355074i
$663$	34655.2i	2.03001i
$664$	6427.72	0.375669
$665$	0	0
$666$	7702.98	0.448175
$667$	23931.6i	1.38926i
$668$	− 35950.6i	− 2.08229i
$669$	11119.2	0.642588
$670$	0	0
$671$	8788.20	0.505611
$672$	− 8855.83i	− 0.508365i
$673$	13055.9i	0.747799i	0.927469	+	0.373900i	$0.121980\pi$
$673$	13055.9i	0.747799i	−0.927469	+	0.373900i	$0.878020\pi$
$674$	7652.20	0.437317
$675$	0	0
$676$	43487.6	2.47426
$677$	− 3154.53i	− 0.179082i	−0.995983	−	0.0895410i	$-0.971460\pi$
$677$	− 3154.53i	− 0.179082i	0.995983	−	0.0895410i	$-0.0285400\pi$
$678$	− 1724.84i	− 0.0977022i
$679$	9202.60	0.520123
$680$	0	0
$681$	−36766.9	−2.06888
$682$	47350.4i	2.65856i
$683$	− 17282.7i	− 0.968233i	−0.875004	−	0.484117i	$-0.839141\pi$
$683$	− 17282.7i	− 0.968233i	0.875004	−	0.484117i	$-0.160859\pi$
$684$	12194.3	0.681666
$685$	0	0
$686$	1555.54	0.0865754
$687$	− 19963.8i	− 1.10869i
$688$	− 2727.83i	− 0.151159i
$689$	−43953.6	−2.43033
$690$	0	0
$691$	26838.3	1.47753	0.738766	−	0.673961i	$-0.235409\pi$
$691$	26838.3	1.47753	0.738766	+	0.673961i	$0.235409\pi$
$692$	35231.1i	1.93538i
$693$	5606.01i	0.307294i
$694$	−27007.3	−1.47721
$695$	0	0
$696$	25501.0	1.38881
$697$	− 15454.6i	− 0.839866i
$698$	− 31186.0i	− 1.69113i
$699$	−25849.0	−1.39871
$700$	0	0
$701$	−8374.52	−0.451214	−0.225607	−	0.974218i	$-0.572437\pi$
$701$	−8374.52	−0.451214	−0.225607	+	0.974218i	$0.572437\pi$
$702$	− 26883.2i	− 1.44536i
$703$	7512.21i	0.403028i
$704$	−45117.9	−2.41540
$705$	0	0
$706$	6330.79	0.337482
$707$	− 2808.68i	− 0.149408i
$708$	− 16556.9i	− 0.878881i
$709$	−11929.4	−0.631902	−0.315951	−	0.948775i	$-0.602324\pi$
$709$	−11929.4	−0.631902	−0.315951	+	0.948775i	$0.602324\pi$
$710$	0	0
$711$	−6933.28	−0.365708
$712$	27234.7i	1.43352i
$713$	24270.9i	1.27483i
$714$	14626.6	0.766648
$715$	0	0
$716$	3124.40	0.163078
$717$	− 29096.6i	− 1.51553i
$718$	1736.89i	0.0902788i
$719$	23392.3	1.21333	0.606666	−	0.794957i	$-0.292507\pi$
$719$	23392.3	1.21333	0.606666	+	0.794957i	$0.292507\pi$
$720$	0	0
$721$	2042.91	0.105523
$722$	− 11643.6i	− 0.600179i
$723$	− 23493.4i	− 1.20848i
$724$	−26083.8	−1.33894
$725$	0	0
$726$	46696.8	2.38717
$727$	10659.8i	0.543810i	0.962324	+	0.271905i	$0.0876536\pi$
$727$	10659.8i	0.543810i	−0.962324	+	0.271905i	$0.912346\pi$
$728$	− 10905.3i	− 0.555188i
$729$	−287.476	−0.0146053
$730$	0	0
$731$	29430.1	1.48907
$732$	− 13208.2i	− 0.666923i
$733$	− 23919.1i	− 1.20528i	−0.798011	−	0.602642i	$-0.794115\pi$
$733$	− 23919.1i	− 1.20528i	0.798011	−	0.602642i	$-0.205885\pi$
$734$	46160.2	2.32126
$735$	0	0
$736$	−24590.9	−1.23157
$737$	− 25819.5i	− 1.29047i
$738$	− 14569.7i	− 0.726717i
$739$	−34535.1	−1.71907	−0.859536	−	0.511074i	$-0.829248\pi$
$739$	−34535.1	−1.71907	−0.859536	+	0.511074i	$0.829248\pi$
$740$	0	0
$741$	−31861.6	−1.57957
$742$	18551.1i	0.917833i
$743$	− 17526.8i	− 0.865403i	−0.901537	−	0.432702i	$-0.857560\pi$
$743$	− 17526.8i	− 0.865403i	0.901537	−	0.432702i	$-0.142440\pi$
$744$	25862.6	1.27442
$745$	0	0
$746$	−13564.5	−0.665725
$747$	− 4596.63i	− 0.225143i
$748$	− 48415.9i	− 2.36666i
$749$	153.716	0.00749889
$750$	0	0
$751$	18534.3	0.900565	0.450283	−	0.892886i	$-0.351323\pi$
$751$	18534.3	0.900565	0.450283	+	0.892886i	$0.351323\pi$
$752$	749.966i	0.0363676i
$753$	7831.94i	0.379033i
$754$	−64949.2	−3.13702
$755$	0	0
$756$	−6932.96	−0.333531
$757$	− 9309.64i	− 0.446981i	−0.974706	−	0.223491i	$-0.928255\pi$
$757$	− 9309.64i	− 0.446981i	0.974706	−	0.223491i	$-0.0717452\pi$
$758$	− 33562.7i	− 1.60825i
$759$	43942.7	2.10148
$760$	0	0
$761$	−2968.41	−0.141399	−0.0706996	−	0.997498i	$-0.522523\pi$
$761$	−2968.41	−0.141399	−0.0706996	+	0.997498i	$0.522523\pi$
$762$	54604.0i	2.59592i
$763$	121.113i	0.00574653i
$764$	15746.0	0.745643
$765$	0	0
$766$	−48841.1	−2.30379
$767$	15325.1i	0.721457i
$768$	21910.1i	1.02944i
$769$	34932.9	1.63812	0.819060	−	0.573708i	$-0.194495\pi$
$769$	34932.9	1.63812	0.819060	+	0.573708i	$0.194495\pi$
$770$	0	0
$771$	−40257.5	−1.88046
$772$	20134.0i	0.938650i
$773$	13595.4i	0.632591i	0.948661	+	0.316295i	$0.102439\pi$
$773$	13595.4i	0.632591i	−0.948661	+	0.316295i	$0.897561\pi$
$774$	27744.9	1.28846
$775$	0	0
$776$	−27229.7	−1.25965
$777$	5190.49i	0.239650i
$778$	50762.9i	2.33925i
$779$	14208.8	0.653510
$780$	0	0
$781$	−44460.1	−2.03701
$782$	− 40615.2i	− 1.85728i
$783$	15005.9i	0.684888i
$784$	323.616	0.0147420
$785$	0	0
$786$	22731.6	1.03156
$787$	− 14228.1i	− 0.644444i	−0.946664	−	0.322222i	$-0.895570\pi$
$787$	− 14228.1i	− 0.644444i	0.946664	−	0.322222i	$-0.104430\pi$
$788$	− 29487.3i	− 1.33305i
$789$	4897.36	0.220977
$790$	0	0
$791$	411.728	0.0185074
$792$	− 16587.7i	− 0.744213i
$793$	12225.5i	0.547465i
$794$	−47037.5	−2.10239
$795$	0	0
$796$	−25053.0	−1.11555
$797$	11442.3i	0.508539i	0.967133	+	0.254270i	$0.0818351\pi$
$797$	11442.3i	0.508539i	−0.967133	+	0.254270i	$0.918165\pi$
$798$	13447.5i	0.596538i
$799$	−8091.26	−0.358258
$800$	0	0
$801$	19476.2	0.859125
$802$	24499.5i	1.07869i
$803$	− 43200.3i	− 1.89851i
$804$	−38805.2	−1.70218
$805$	0	0
$806$	−65870.2	−2.87863
$807$	11319.7i	0.493770i
$808$	8310.63i	0.361840i
$809$	−22732.9	−0.987944	−0.493972	−	0.869478i	$-0.664455\pi$
$809$	−22732.9	−0.987944	−0.493972	+	0.869478i	$0.664455\pi$
$810$	0	0
$811$	−29768.2	−1.28890	−0.644452	−	0.764644i	$-0.722915\pi$
$811$	−29768.2	−1.28890	−0.644452	+	0.764644i	$0.722915\pi$
$812$	16749.9i	0.723898i
$813$	− 28786.7i	− 1.24181i
$814$	28118.4	1.21075
$815$	0	0
$816$	3042.93	0.130544
$817$	27057.7i	1.15867i
$818$	58185.0i	2.48703i
$819$	−7798.65	−0.332731
$820$	0	0
$821$	6408.35	0.272416	0.136208	−	0.990680i	$-0.456509\pi$
$821$	6408.35	0.272416	0.136208	+	0.990680i	$0.456509\pi$
$822$	− 30144.6i	− 1.27909i
$823$	20876.0i	0.884194i	0.896967	+	0.442097i	$0.145765\pi$
$823$	20876.0i	0.884194i	−0.896967	+	0.442097i	$0.854235\pi$
$824$	−6044.78	−0.255558
$825$	0	0
$826$	6468.13	0.272464
$827$	− 41650.1i	− 1.75129i	−0.482955	−	0.875645i	$-0.660437\pi$
$827$	− 41650.1i	− 1.75129i	0.482955	−	0.875645i	$-0.339563\pi$
$828$	− 23396.0i	− 0.981964i
$829$	−17194.7	−0.720380	−0.360190	−	0.932879i	$-0.617288\pi$
$829$	−17194.7	−0.720380	−0.360190	+	0.932879i	$0.617288\pi$
$830$	0	0
$831$	−49736.1	−2.07621
$832$	− 62764.5i	− 2.61535i
$833$	3491.44i	0.145223i
$834$	30194.8	1.25367
$835$	0	0
$836$	44513.0	1.84153
$837$	15218.7i	0.628477i
$838$	− 25491.1i	− 1.05081i
$839$	−40583.7	−1.66997	−0.834985	−	0.550273i	$-0.814524\pi$
$839$	−40583.7	−1.66997	−0.834985	+	0.550273i	$0.814524\pi$
$840$	0	0
$841$	11865.0	0.486489
$842$	− 9873.42i	− 0.404110i
$843$	1287.00i	0.0525819i
$844$	73549.7	2.99963
$845$	0	0
$846$	−7627.93	−0.309993
$847$	11146.8i	0.452192i
$848$	3859.39i	0.156288i
$849$	5168.49	0.208931
$850$	0	0
$851$	14413.0	0.580576
$852$	66821.0i	2.68691i
$853$	− 23020.1i	− 0.924023i	−0.886874	−	0.462012i	$-0.847128\pi$
$853$	− 23020.1i	− 0.924023i	0.886874	−	0.462012i	$-0.152872\pi$
$854$	5159.90	0.206754
$855$	0	0
$856$	−454.832	−0.0181610
$857$	− 18075.5i	− 0.720476i	−0.932861	−	0.360238i	$-0.882696\pi$
$857$	− 18075.5i	− 0.720476i	0.932861	−	0.360238i	$-0.117304\pi$
$858$	119259.i	4.74525i
$859$	−20308.2	−0.806643	−0.403321	−	0.915058i	$-0.632144\pi$
$859$	−20308.2	−0.806643	−0.403321	+	0.915058i	$0.632144\pi$
$860$	0	0
$861$	9817.46	0.388592
$862$	47148.9i	1.86299i
$863$	31023.5i	1.22370i	0.790975	+	0.611849i	$0.209574\pi$
$863$	31023.5i	1.22370i	−0.790975	+	0.611849i	$0.790426\pi$
$864$	−15419.3	−0.607148
$865$	0	0
$866$	53857.2	2.11333
$867$	1061.17i	0.0415677i
$868$	16987.4i	0.664274i
$869$	−25308.7	−0.987963
$870$	0	0
$871$	35918.1	1.39729
$872$	− 358.363i	− 0.0139171i
$873$	19472.6i	0.754924i
$874$	37341.2	1.44518
$875$	0	0
$876$	−64927.5	−2.50422
$877$	− 28344.1i	− 1.09135i	−0.837997	−	0.545674i	$-0.816274\pi$
$877$	− 28344.1i	− 1.09135i	0.837997	−	0.545674i	$-0.183726\pi$
$878$	− 2904.14i	− 0.111629i
$879$	−21356.3	−0.819488
$880$	0	0
$881$	41264.5	1.57802	0.789010	−	0.614380i	$-0.210594\pi$
$881$	41264.5	1.57802	0.789010	+	0.614380i	$0.210594\pi$
$882$	3291.51i	0.125659i
$883$	7995.59i	0.304726i	0.988325	+	0.152363i	$0.0486883\pi$
$883$	7995.59i	0.304726i	−0.988325	+	0.152363i	$0.951312\pi$
$884$	67352.5	2.56257
$885$	0	0
$886$	−3876.84	−0.147003
$887$	24438.3i	0.925093i	0.886595	+	0.462547i	$0.153064\pi$
$887$	24438.3i	0.925093i	−0.886595	+	0.462547i	$0.846936\pi$
$888$	− 15358.2i	− 0.580390i
$889$	−13034.2	−0.491737
$890$	0	0
$891$	49176.8	1.84903
$892$	− 21610.1i	− 0.811167i
$893$	− 7439.02i	− 0.278765i
$894$	−41485.1	−1.55198
$895$	0	0
$896$	−15534.1	−0.579192
$897$	61129.8i	2.27543i
$898$	− 15543.8i	− 0.577620i
$899$	36768.0	1.36405
$900$	0	0
$901$	−41638.3	−1.53959
$902$	− 53184.0i	− 1.96323i
$903$	18695.3i	0.688970i
$904$	−1218.26	−0.0448218
$905$	0	0
$906$	61434.7	2.25279
$907$	17241.6i	0.631201i	0.948892	+	0.315601i	$0.102206\pi$
$907$	17241.6i	0.631201i	−0.948892	+	0.315601i	$0.897794\pi$
$908$	71456.6i	2.61164i
$909$	5943.15	0.216856
$910$	0	0
$911$	17407.8	0.633092	0.316546	−	0.948577i	$-0.397477\pi$
$911$	17407.8	0.633092	0.316546	+	0.948577i	$0.397477\pi$
$912$	2797.64i	0.101578i
$913$	− 16779.2i	− 0.608226i
$914$	−7751.87	−0.280535
$915$	0	0
$916$	−38799.8	−1.39954
$917$	5426.13i	0.195405i
$918$	− 25467.1i	− 0.915621i
$919$	−7704.64	−0.276553	−0.138277	−	0.990394i	$-0.544156\pi$
$919$	−7704.64	−0.276553	−0.138277	+	0.990394i	$0.544156\pi$
$920$	0	0
$921$	−55791.6	−1.99609
$922$	10209.3i	0.364671i
$923$	− 61849.5i	− 2.20564i
$924$	30755.9	1.09501
$925$	0	0
$926$	−14515.9	−0.515144
$927$	4322.78i	0.153159i
$928$	37252.8i	1.31776i
$929$	48184.3	1.70170	0.850849	−	0.525411i	$-0.176088\pi$
$929$	48184.3	1.70170	0.850849	+	0.525411i	$0.176088\pi$
$930$	0	0
$931$	−3209.99	−0.113000
$932$	50237.7i	1.76565i
$933$	31688.4i	1.11193i
$934$	−24840.5	−0.870242
$935$	0	0
$936$	23075.5	0.805818
$937$	23371.6i	0.814853i	0.913238	+	0.407427i	$0.133574\pi$
$937$	23371.6i	0.814853i	−0.913238	+	0.407427i	$0.866426\pi$
$938$	− 15159.7i	− 0.527697i
$939$	58934.7	2.04820
$940$	0	0
$941$	18048.6	0.625258	0.312629	−	0.949875i	$-0.398790\pi$
$941$	18048.6	0.625258	0.312629	+	0.949875i	$0.398790\pi$
$942$	− 20805.9i	− 0.719631i
$943$	− 27261.1i	− 0.941405i
$944$	1345.64	0.0463948
$945$	0	0
$946$	101278.	3.48079
$947$	10346.2i	0.355022i	0.984119	+	0.177511i	$0.0568045\pi$
$947$	10346.2i	0.355022i	−0.984119	+	0.177511i	$0.943195\pi$
$948$	38037.6i	1.30317i
$949$	60096.9	2.05567
$950$	0	0
$951$	−51877.3	−1.76891
$952$	− 10330.8i	− 0.351706i
$953$	− 8691.77i	− 0.295440i	−0.989029	−	0.147720i	$-0.952807\pi$
$953$	− 8691.77i	− 0.295440i	0.989029	−	0.147720i	$-0.0471934\pi$
$954$	−39254.0	−1.33217
$955$	0	0
$956$	−56549.5	−1.91312
$957$	− 66568.9i	− 2.24856i
$958$	64316.6i	2.16908i
$959$	7195.67	0.242294
$960$	0	0
$961$	7498.41	0.251701
$962$	39116.2i	1.31097i
$963$	325.262i	0.0108841i
$964$	−45659.6	−1.52552
$965$	0	0
$966$	25800.5	0.859335
$967$	8971.58i	0.298352i	0.988811	+	0.149176i	$0.0476621\pi$
$967$	8971.58i	0.298352i	−0.988811	+	0.149176i	$0.952338\pi$
$968$	− 32982.2i	− 1.09513i
$969$	−30183.3	−1.00065
$970$	0	0
$971$	29275.1	0.967541	0.483771	−	0.875195i	$-0.339267\pi$
$971$	29275.1	0.967541	0.483771	+	0.875195i	$0.339267\pi$
$972$	− 47168.5i	− 1.55651i
$973$	7207.64i	0.237478i
$974$	24130.5	0.793831
$975$	0	0
$976$	1073.47	0.0352059
$977$	− 16477.5i	− 0.539572i	−0.962920	−	0.269786i	$-0.913047\pi$
$977$	− 16477.5i	− 0.539572i	0.962920	−	0.269786i	$-0.0869529\pi$
$978$	− 96075.2i	− 3.14125i
$979$	71094.6	2.32093
$980$	0	0
$981$	−256.275	−0.00834071
$982$	− 29817.4i	− 0.968952i
$983$	− 45912.9i	− 1.48972i	−0.667222	−	0.744859i	$-0.732517\pi$
$983$	− 45912.9i	− 1.48972i	0.667222	−	0.744859i	$-0.267483\pi$
$984$	−29048.9	−0.941104
$985$	0	0
$986$	−61528.0	−1.98727
$987$	− 5139.92i	− 0.165760i
$988$	61923.2i	1.99397i
$989$	51913.1	1.66910
$990$	0	0
$991$	46124.8	1.47851	0.739255	−	0.673426i	$-0.235178\pi$
$991$	46124.8	1.47851	0.739255	+	0.673426i	$0.235178\pi$
$992$	37781.0i	1.20922i
$993$	8623.22i	0.275579i
$994$	−26104.3	−0.832975
$995$	0	0
$996$	−25218.2	−0.802278
$997$	− 30984.5i	− 0.984240i	−0.870527	−	0.492120i	$-0.836222\pi$
$997$	− 30984.5i	− 0.984240i	0.870527	−	0.492120i	$-0.163778\pi$
$998$	6837.47i	0.216870i
$999$	9037.42	0.286218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.f.99.6		8
5.2	odd	4		175.4.a.g.1.2	✓	4
5.3	odd	4		175.4.a.h.1.3	yes	4
5.4	even	2	inner	175.4.b.f.99.3		8
15.2	even	4		1575.4.a.bl.1.3		4
15.8	even	4		1575.4.a.bg.1.2		4
35.13	even	4		1225.4.a.bd.1.3		4
35.27	even	4		1225.4.a.z.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.2	✓	4	5.2	odd	4
175.4.a.h.1.3	yes	4	5.3	odd	4
175.4.b.f.99.3		8	5.4	even	2	inner
175.4.b.f.99.6		8	1.1	even	1	trivial
1225.4.a.z.1.2		4	35.27	even	4
1225.4.a.bd.1.3		4	35.13	even	4
1575.4.a.bg.1.2		4	15.8	even	4
1575.4.a.bl.1.3		4	15.2	even	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.53510i q^{2} -6.46622i q^{3} -12.5671 q^{4} +29.3249 q^{6} +7.00000i q^{7} -20.7124i q^{8} -14.8119 q^{9} -54.0684 q^{11} +81.2617i q^{12} -75.2159i q^{13} -31.7457 q^{14} -6.60441 q^{16} -71.2538i q^{17} -67.1736i q^{18} +65.5100 q^{19} +45.2635 q^{21} -245.206i q^{22} -125.688i q^{23} -133.931 q^{24} +341.111 q^{26} -78.8106i q^{27} -87.9699i q^{28} -190.405 q^{29} -193.105 q^{31} -195.650i q^{32} +349.618i q^{33} +323.143 q^{34} +186.144 q^{36} +114.673i q^{37} +297.094i q^{38} -486.362 q^{39} +216.896 q^{41} +205.275i q^{42} +413.032i q^{43} +679.484 q^{44} +570.007 q^{46} -113.555i q^{47} +42.7055i q^{48} -49.0000 q^{49} -460.743 q^{51} +945.247i q^{52} -584.366i q^{53} +357.414 q^{54} +144.986 q^{56} -423.602i q^{57} -863.504i q^{58} -203.748 q^{59} -162.539 q^{61} -875.749i q^{62} -103.684i q^{63} +834.459 q^{64} -1585.55 q^{66} +477.534i q^{67} +895.456i q^{68} -812.725 q^{69} +822.294 q^{71} +306.790i q^{72} +798.993i q^{73} -520.052 q^{74} -823.273 q^{76} -378.479i q^{77} -2205.70i q^{78} +468.087 q^{79} -909.529 q^{81} +983.643i q^{82} +310.333i q^{83} -568.832 q^{84} -1873.14 q^{86} +1231.20i q^{87} +1119.88i q^{88} -1314.90 q^{89} +526.511 q^{91} +1579.53i q^{92} +1248.66i q^{93} +514.985 q^{94} -1265.12 q^{96} -1314.66i q^{97} -222.220i q^{98} +800.858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9} + 200 q^{11} - 56 q^{14} + 320 q^{16} + 58 q^{19} - 42 q^{21} + 42 q^{24} + 1400 q^{26} - 258 q^{29} + 228 q^{31} - 406 q^{34} + 2202 q^{36} - 1348 q^{39} + 1342 q^{41}+ \cdots - 4766 q^{99}+O(q^{100}) \) 8 * q - 72 * q^4 + 2 * q^6 - 122 * q^9 + 200 * q^11 - 56 * q^14 + 320 * q^16 + 58 * q^19 - 42 * q^21 + 42 * q^24 + 1400 * q^26 - 258 * q^29 + 228 * q^31 - 406 * q^34 + 2202 * q^36 - 1348 * q^39 + 1342 * q^41 - 876 * q^44 - 1994 * q^46 - 392 * q^49 - 1770 * q^51 + 5554 * q^54 + 378 * q^56 - 2036 * q^59 + 100 * q^61 + 4842 * q^64 - 7682 * q^66 - 2160 * q^69 + 430 * q^71 - 1246 * q^74 - 6514 * q^76 + 1902 * q^79 + 56 * q^81 + 2310 * q^84 - 198 * q^86 - 5638 * q^89 + 616 * q^91 + 6112 * q^94 - 2690 * q^96 - 4766 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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