LMFDB - Embedded newform 475.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.a (trivial)

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:	yes
Analytic conductor:	$3.79289409601$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} +0.903212 q^{6} +4.42864 q^{7} +1.21432 q^{8} +5.42864 q^{9} -2.62222 q^{11} +5.52543 q^{12} -0.474572 q^{13} -1.37778 q^{14} +3.42864 q^{16} -5.05086 q^{17} -1.68889 q^{18} -1.00000 q^{19} -12.8573 q^{21} +0.815792 q^{22} +1.37778 q^{23} -3.52543 q^{24} +0.147643 q^{26} -7.05086 q^{27} -8.42864 q^{28} -7.80642 q^{29} +1.24443 q^{31} -3.49532 q^{32} +7.61285 q^{33} +1.57136 q^{34} -10.3319 q^{36} -4.47457 q^{37} +0.311108 q^{38} +1.37778 q^{39} -5.05086 q^{41} +4.00000 q^{42} -12.0415 q^{43} +4.99063 q^{44} -0.428639 q^{46} +4.42864 q^{47} -9.95407 q^{48} +12.6128 q^{49} +14.6637 q^{51} +0.903212 q^{52} -7.52543 q^{53} +2.19358 q^{54} +5.37778 q^{56} +2.90321 q^{57} +2.42864 q^{58} -2.19358 q^{59} +3.67307 q^{61} -0.387152 q^{62} +24.0415 q^{63} -5.76986 q^{64} -2.36842 q^{66} +1.65878 q^{67} +9.61285 q^{68} -4.00000 q^{69} +7.61285 q^{71} +6.59210 q^{72} +3.80642 q^{73} +1.39207 q^{74} +1.90321 q^{76} -11.6128 q^{77} -0.428639 q^{78} -13.4193 q^{79} +4.18421 q^{81} +1.57136 q^{82} +10.6222 q^{83} +24.4701 q^{84} +3.74620 q^{86} +22.6637 q^{87} -3.18421 q^{88} -12.6637 q^{89} -2.10171 q^{91} -2.62222 q^{92} -3.61285 q^{93} -1.37778 q^{94} +10.1476 q^{96} -17.8938 q^{97} -3.92396 q^{98} -14.2351 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 3 q^{8} + 3 q^{9} - 8 q^{11} + 10 q^{12} - 8 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} - 5 q^{18} - 3 q^{19} - 12 q^{21} + 16 q^{22} + 4 q^{23} - 4 q^{24} - 6 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) $ 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 3 * q^8 + 3 * q^9 - 8 * q^11 + 10 * q^12 - 8 * q^13 - 4 * q^14 - 3 * q^16 - 2 * q^17 - 5 * q^18 - 3 * q^19 - 12 * q^21 + 16 * q^22 + 4 * q^23 - 4 * q^24 - 6 * q^26 - 8 * q^27 - 12 * q^28 - 10 * q^29 + 4 * q^31 + 3 * q^32 - 4 * q^33 + 18 * q^34 - 11 * q^36 - 20 * q^37 + q^38 + 4 * q^39 - 2 * q^41 + 12 * q^42 + 4 * q^43 - 12 * q^44 + 12 * q^46 - 10 * q^48 + 11 * q^49 + 4 * q^51 - 4 * q^52 - 16 * q^53 + 20 * q^54 + 16 * q^56 + 2 * q^57 - 6 * q^58 - 20 * q^59 - 2 * q^61 - 28 * q^62 + 32 * q^63 - 11 * q^64 + 20 * q^66 - 2 * q^67 + 2 * q^68 - 12 * q^69 - 4 * q^71 + 13 * q^72 - 2 * q^73 - 2 * q^74 - q^76 - 8 * q^77 + 12 * q^78 - q^81 + 18 * q^82 + 32 * q^83 + 20 * q^84 - 16 * q^86 + 28 * q^87 + 4 * q^88 + 2 * q^89 + 20 * q^91 - 8 * q^92 + 16 * q^93 - 4 * q^94 + 24 * q^96 - 20 * q^97 + 15 * q^98 - 16 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	−2.90321	−1.67617	−0.838085	−	0.545540i	$-0.816325\pi$
$3$	−2.90321	−1.67617	−0.838085	+	0.545540i	$0.816325\pi$
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	0.903212	0.368735
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	1.21432	0.429327
$9$	5.42864	1.80955
$10$	0	0
$11$	−2.62222	−0.790628	−0.395314	−	0.918546i	$-0.629364\pi$
$11$	−2.62222	−0.790628	−0.395314	+	0.918546i	$0.629364\pi$
$12$	5.52543	1.59505
$13$	−0.474572	−0.131623	−0.0658114	−	0.997832i	$-0.520964\pi$
$13$	−0.474572	−0.131623	−0.0658114	+	0.997832i	$0.520964\pi$
$14$	−1.37778	−0.368228
$15$	0	0
$16$	3.42864	0.857160
$17$	−5.05086	−1.22501	−0.612506	−	0.790466i	$-0.709839\pi$
$17$	−5.05086	−1.22501	−0.612506	+	0.790466i	$0.709839\pi$
$18$	−1.68889	−0.398076
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−12.8573	−2.80569
$22$	0.815792	0.173927
$23$	1.37778	0.287288	0.143644	−	0.989629i	$-0.454118\pi$
$23$	1.37778	0.287288	0.143644	+	0.989629i	$0.454118\pi$
$24$	−3.52543	−0.719625
$25$	0	0
$26$	0.147643	0.0289552
$27$	−7.05086	−1.35694
$28$	−8.42864	−1.59286
$29$	−7.80642	−1.44962	−0.724808	−	0.688951i	$-0.758072\pi$
$29$	−7.80642	−1.44962	−0.724808	+	0.688951i	$0.758072\pi$
$30$	0	0
$31$	1.24443	0.223506	0.111753	−	0.993736i	$-0.464353\pi$
$31$	1.24443	0.223506	0.111753	+	0.993736i	$0.464353\pi$
$32$	−3.49532	−0.617890
$33$	7.61285	1.32523
$34$	1.57136	0.269486
$35$	0	0
$36$	−10.3319	−1.72198
$37$	−4.47457	−0.735615	−0.367808	−	0.929902i	$-0.619891\pi$
$37$	−4.47457	−0.735615	−0.367808	+	0.929902i	$0.619891\pi$
$38$	0.311108	0.0504684
$39$	1.37778	0.220622
$40$	0	0
$41$	−5.05086	−0.788811	−0.394406	−	0.918936i	$-0.629049\pi$
$41$	−5.05086	−0.788811	−0.394406	+	0.918936i	$0.629049\pi$
$42$	4.00000	0.617213
$43$	−12.0415	−1.83631	−0.918155	−	0.396222i	$-0.870321\pi$
$43$	−12.0415	−1.83631	−0.918155	+	0.396222i	$0.870321\pi$
$44$	4.99063	0.752366
$45$	0	0
$46$	−0.428639	−0.0631994
$47$	4.42864	0.645983	0.322992	−	0.946402i	$-0.395311\pi$
$47$	4.42864	0.645983	0.322992	+	0.946402i	$0.395311\pi$
$48$	−9.95407	−1.43675
$49$	12.6128	1.80184
$50$	0	0
$51$	14.6637	2.05333
$52$	0.903212	0.125253
$53$	−7.52543	−1.03370	−0.516848	−	0.856077i	$-0.672895\pi$
$53$	−7.52543	−1.03370	−0.516848	+	0.856077i	$0.672895\pi$
$54$	2.19358	0.298508
$55$	0	0
$56$	5.37778	0.718637
$57$	2.90321	0.384540
$58$	2.42864	0.318896
$59$	−2.19358	−0.285579	−0.142790	−	0.989753i	$-0.545607\pi$
$59$	−2.19358	−0.285579	−0.142790	+	0.989753i	$0.545607\pi$
$60$	0	0
$61$	3.67307	0.470289	0.235144	−	0.971960i	$-0.424444\pi$
$61$	3.67307	0.470289	0.235144	+	0.971960i	$0.424444\pi$
$62$	−0.387152	−0.0491684
$63$	24.0415	3.02894
$64$	−5.76986	−0.721232
$65$	0	0
$66$	−2.36842	−0.291532
$67$	1.65878	0.202652	0.101326	−	0.994853i	$-0.467691\pi$
$67$	1.65878	0.202652	0.101326	+	0.994853i	$0.467691\pi$
$68$	9.61285	1.16573
$69$	−4.00000	−0.481543
$70$	0	0
$71$	7.61285	0.903479	0.451739	−	0.892150i	$-0.350804\pi$
$71$	7.61285	0.903479	0.451739	+	0.892150i	$0.350804\pi$
$72$	6.59210	0.776887
$73$	3.80642	0.445508	0.222754	−	0.974875i	$-0.428495\pi$
$73$	3.80642	0.445508	0.222754	+	0.974875i	$0.428495\pi$
$74$	1.39207	0.161825
$75$	0	0
$76$	1.90321	0.218313
$77$	−11.6128	−1.32341
$78$	−0.428639	−0.0485339
$79$	−13.4193	−1.50979	−0.754893	−	0.655848i	$-0.772311\pi$
$79$	−13.4193	−1.50979	−0.754893	+	0.655848i	$0.772311\pi$
$80$	0	0
$81$	4.18421	0.464912
$82$	1.57136	0.173528
$83$	10.6222	1.16594	0.582970	−	0.812494i	$-0.301891\pi$
$83$	10.6222	1.16594	0.582970	+	0.812494i	$0.301891\pi$
$84$	24.4701	2.66991
$85$	0	0
$86$	3.74620	0.403963
$87$	22.6637	2.42980
$88$	−3.18421	−0.339438
$89$	−12.6637	−1.34235	−0.671175	−	0.741299i	$-0.734210\pi$
$89$	−12.6637	−1.34235	−0.671175	+	0.741299i	$0.734210\pi$
$90$	0	0
$91$	−2.10171	−0.220319
$92$	−2.62222	−0.273385
$93$	−3.61285	−0.374635
$94$	−1.37778	−0.142108
$95$	0	0
$96$	10.1476	1.03569
$97$	−17.8938	−1.81684	−0.908422	−	0.418054i	$-0.862712\pi$
$97$	−17.8938	−1.81684	−0.908422	+	0.418054i	$0.862712\pi$
$98$	−3.92396	−0.396379
$99$	−14.2351	−1.43068
$100$	0	0
$101$	−10.4286	−1.03769	−0.518844	−	0.854869i	$-0.673638\pi$
$101$	−10.4286	−1.03769	−0.518844	+	0.854869i	$0.673638\pi$
$102$	−4.56199	−0.451705
$103$	5.65878	0.557576	0.278788	−	0.960353i	$-0.410067\pi$
$103$	5.65878	0.557576	0.278788	+	0.960353i	$0.410067\pi$
$104$	−0.576283	−0.0565092
$105$	0	0
$106$	2.34122	0.227399
$107$	−6.90321	−0.667359	−0.333679	−	0.942687i	$-0.608290\pi$
$107$	−6.90321	−0.667359	−0.333679	+	0.942687i	$0.608290\pi$
$108$	13.4193	1.29127
$109$	5.61285	0.537613	0.268807	−	0.963194i	$-0.413371\pi$
$109$	5.61285	0.537613	0.268807	+	0.963194i	$0.413371\pi$
$110$	0	0
$111$	12.9906	1.23302
$112$	15.1842	1.43477
$113$	−13.8938	−1.30702	−0.653511	−	0.756917i	$-0.726705\pi$
$113$	−13.8938	−1.30702	−0.653511	+	0.756917i	$0.726705\pi$
$114$	−0.903212	−0.0845935
$115$	0	0
$116$	14.8573	1.37946
$117$	−2.57628	−0.238177
$118$	0.682439	0.0628236
$119$	−22.3684	−2.05051
$120$	0	0
$121$	−4.12399	−0.374908
$122$	−1.14272	−0.103457
$123$	14.6637	1.32218
$124$	−2.36842	−0.212690
$125$	0	0
$126$	−7.47949	−0.666326
$127$	7.19850	0.638763	0.319382	−	0.947626i	$-0.396525\pi$
$127$	7.19850	0.638763	0.319382	+	0.947626i	$0.396525\pi$
$128$	8.78568	0.776552
$129$	34.9590	3.07797
$130$	0	0
$131$	−2.10171	−0.183627	−0.0918136	−	0.995776i	$-0.529266\pi$
$131$	−2.10171	−0.183627	−0.0918136	+	0.995776i	$0.529266\pi$
$132$	−14.4889	−1.26109
$133$	−4.42864	−0.384012
$134$	−0.516060	−0.0445808
$135$	0	0
$136$	−6.13335	−0.525931
$137$	−1.70471	−0.145644	−0.0728218	−	0.997345i	$-0.523200\pi$
$137$	−1.70471	−0.145644	−0.0728218	+	0.997345i	$0.523200\pi$
$138$	1.24443	0.105933
$139$	8.72393	0.739954	0.369977	−	0.929041i	$-0.379366\pi$
$139$	8.72393	0.739954	0.369977	+	0.929041i	$0.379366\pi$
$140$	0	0
$141$	−12.8573	−1.08278
$142$	−2.36842	−0.198753
$143$	1.24443	0.104065
$144$	18.6128	1.55107
$145$	0	0
$146$	−1.18421	−0.0980058
$147$	−36.6178	−3.02018
$148$	8.51606	0.700016
$149$	−6.81579	−0.558371	−0.279186	−	0.960237i	$-0.590065\pi$
$149$	−6.81579	−0.558371	−0.279186	+	0.960237i	$0.590065\pi$
$150$	0	0
$151$	−5.80642	−0.472520	−0.236260	−	0.971690i	$-0.575922\pi$
$151$	−5.80642	−0.472520	−0.236260	+	0.971690i	$0.575922\pi$
$152$	−1.21432	−0.0984943
$153$	−27.4193	−2.21672
$154$	3.61285	0.291132
$155$	0	0
$156$	−2.62222	−0.209945
$157$	−0.193576	−0.0154491	−0.00772453	−	0.999970i	$-0.502459\pi$
$157$	−0.193576	−0.0154491	−0.00772453	+	0.999970i	$0.502459\pi$
$158$	4.17484	0.332132
$159$	21.8479	1.73265
$160$	0	0
$161$	6.10171	0.480882
$162$	−1.30174	−0.102274
$163$	8.42864	0.660182	0.330091	−	0.943949i	$-0.392921\pi$
$163$	8.42864	0.660182	0.330091	+	0.943949i	$0.392921\pi$
$164$	9.61285	0.750637
$165$	0	0
$166$	−3.30465	−0.256491
$167$	−12.4429	−0.962863	−0.481431	−	0.876484i	$-0.659883\pi$
$167$	−12.4429	−0.962863	−0.481431	+	0.876484i	$0.659883\pi$
$168$	−15.6128	−1.20456
$169$	−12.7748	−0.982675
$170$	0	0
$171$	−5.42864	−0.415138
$172$	22.9175	1.74744
$173$	22.1891	1.68701	0.843504	−	0.537123i	$-0.180489\pi$
$173$	22.1891	1.68701	0.843504	+	0.537123i	$0.180489\pi$
$174$	−7.05086	−0.534524
$175$	0	0
$176$	−8.99063	−0.677694
$177$	6.36842	0.478679
$178$	3.93978	0.295299
$179$	−11.9081	−0.890056	−0.445028	−	0.895517i	$-0.646806\pi$
$179$	−11.9081	−0.890056	−0.445028	+	0.895517i	$0.646806\pi$
$180$	0	0
$181$	17.6128	1.30915	0.654576	−	0.755996i	$-0.272847\pi$
$181$	17.6128	1.30915	0.654576	+	0.755996i	$0.272847\pi$
$182$	0.653858	0.0484672
$183$	−10.6637	−0.788284
$184$	1.67307	0.123340
$185$	0	0
$186$	1.12399	0.0824146
$187$	13.2444	0.968529
$188$	−8.42864	−0.614722
$189$	−31.2257	−2.27134
$190$	0	0
$191$	−0.266706	−0.0192982	−0.00964909	−	0.999953i	$-0.503071\pi$
$191$	−0.266706	−0.0192982	−0.00964909	+	0.999953i	$0.503071\pi$
$192$	16.7511	1.20891
$193$	−2.66815	−0.192058	−0.0960288	−	0.995379i	$-0.530614\pi$
$193$	−2.66815	−0.192058	−0.0960288	+	0.995379i	$0.530614\pi$
$194$	5.56691	0.399681
$195$	0	0
$196$	−24.0049	−1.71464
$197$	5.34614	0.380897	0.190448	−	0.981697i	$-0.439006\pi$
$197$	5.34614	0.380897	0.190448	+	0.981697i	$0.439006\pi$
$198$	4.42864	0.314730
$199$	17.1240	1.21389	0.606944	−	0.794745i	$-0.292395\pi$
$199$	17.1240	1.21389	0.606944	+	0.794745i	$0.292395\pi$
$200$	0	0
$201$	−4.81579	−0.339680
$202$	3.24443	0.228277
$203$	−34.5718	−2.42647
$204$	−27.9081	−1.95396
$205$	0	0
$206$	−1.76049	−0.122659
$207$	7.47949	0.519861
$208$	−1.62714	−0.112822
$209$	2.62222	0.181382
$210$	0	0
$211$	13.1526	0.905460	0.452730	−	0.891648i	$-0.350450\pi$
$211$	13.1526	0.905460	0.452730	+	0.891648i	$0.350450\pi$
$212$	14.3225	0.983672
$213$	−22.1017	−1.51438
$214$	2.14764	0.146810
$215$	0	0
$216$	−8.56199	−0.582570
$217$	5.51114	0.374120
$218$	−1.74620	−0.118268
$219$	−11.0509	−0.746748
$220$	0	0
$221$	2.39700	0.161239
$222$	−4.04149	−0.271247
$223$	10.5161	0.704207	0.352104	−	0.935961i	$-0.385466\pi$
$223$	10.5161	0.704207	0.352104	+	0.935961i	$0.385466\pi$
$224$	−15.4795	−1.03427
$225$	0	0
$226$	4.32248	0.287527
$227$	−6.90321	−0.458182	−0.229091	−	0.973405i	$-0.573575\pi$
$227$	−6.90321	−0.458182	−0.229091	+	0.973405i	$0.573575\pi$
$228$	−5.52543	−0.365930
$229$	18.0415	1.19222	0.596108	−	0.802905i	$-0.296713\pi$
$229$	18.0415	1.19222	0.596108	+	0.802905i	$0.296713\pi$
$230$	0	0
$231$	33.7146	2.21826
$232$	−9.47949	−0.622359
$233$	12.3684	0.810282	0.405141	−	0.914254i	$-0.367222\pi$
$233$	12.3684	0.810282	0.405141	+	0.914254i	$0.367222\pi$
$234$	0.801502	0.0523958
$235$	0	0
$236$	4.17484	0.271759
$237$	38.9590	2.53066
$238$	6.95899	0.451084
$239$	24.8573	1.60788	0.803942	−	0.594708i	$-0.202732\pi$
$239$	24.8573	1.60788	0.803942	+	0.594708i	$0.202732\pi$
$240$	0	0
$241$	9.05086	0.583017	0.291508	−	0.956568i	$-0.405843\pi$
$241$	9.05086	0.583017	0.291508	+	0.956568i	$0.405843\pi$
$242$	1.28300	0.0824746
$243$	9.00492	0.577666
$244$	−6.99063	−0.447529
$245$	0	0
$246$	−4.56199	−0.290862
$247$	0.474572	0.0301963
$248$	1.51114	0.0959573
$249$	−30.8385	−1.95431
$250$	0	0
$251$	−22.5718	−1.42472	−0.712361	−	0.701813i	$-0.752374\pi$
$251$	−22.5718	−1.42472	−0.712361	+	0.701813i	$0.752374\pi$
$252$	−45.7560	−2.88236
$253$	−3.61285	−0.227138
$254$	−2.23951	−0.140519
$255$	0	0
$256$	8.80642	0.550401
$257$	−4.94470	−0.308442	−0.154221	−	0.988036i	$-0.549287\pi$
$257$	−4.94470	−0.308442	−0.154221	+	0.988036i	$0.549287\pi$
$258$	−10.8760	−0.677111
$259$	−19.8163	−1.23132
$260$	0	0
$261$	−42.3783	−2.62315
$262$	0.653858	0.0403955
$263$	−9.37778	−0.578259	−0.289129	−	0.957290i	$-0.593366\pi$
$263$	−9.37778	−0.578259	−0.289129	+	0.957290i	$0.593366\pi$
$264$	9.24443	0.568955
$265$	0	0
$266$	1.37778	0.0844774
$267$	36.7654	2.25001
$268$	−3.15701	−0.192845
$269$	19.7146	1.20202	0.601009	−	0.799242i	$-0.294766\pi$
$269$	19.7146	1.20202	0.601009	+	0.799242i	$0.294766\pi$
$270$	0	0
$271$	−1.11108	−0.0674932	−0.0337466	−	0.999430i	$-0.510744\pi$
$271$	−1.11108	−0.0674932	−0.0337466	+	0.999430i	$0.510744\pi$
$272$	−17.3176	−1.05003
$273$	6.10171	0.369292
$274$	0.530350	0.0320396
$275$	0	0
$276$	7.61285	0.458240
$277$	−5.52098	−0.331724	−0.165862	−	0.986149i	$-0.553041\pi$
$277$	−5.52098	−0.331724	−0.165862	+	0.986149i	$0.553041\pi$
$278$	−2.71408	−0.162780
$279$	6.75557	0.404445
$280$	0	0
$281$	−15.8064	−0.942932	−0.471466	−	0.881884i	$-0.656275\pi$
$281$	−15.8064	−0.942932	−0.471466	+	0.881884i	$0.656275\pi$
$282$	4.00000	0.238197
$283$	−14.2351	−0.846187	−0.423093	−	0.906086i	$-0.639056\pi$
$283$	−14.2351	−0.846187	−0.423093	+	0.906086i	$0.639056\pi$
$284$	−14.4889	−0.859756
$285$	0	0
$286$	−0.387152	−0.0228928
$287$	−22.3684	−1.32037
$288$	−18.9748	−1.11810
$289$	8.51114	0.500655
$290$	0	0
$291$	51.9496	3.04534
$292$	−7.24443	−0.423948
$293$	−7.52543	−0.439640	−0.219820	−	0.975540i	$-0.570547\pi$
$293$	−7.52543	−0.439640	−0.219820	+	0.975540i	$0.570547\pi$
$294$	11.3921	0.664399
$295$	0	0
$296$	−5.43356	−0.315819
$297$	18.4889	1.07283
$298$	2.12045	0.122834
$299$	−0.653858	−0.0378136
$300$	0	0
$301$	−53.3274	−3.07374
$302$	1.80642	0.103948
$303$	30.2766	1.73934
$304$	−3.42864	−0.196646
$305$	0	0
$306$	8.53035	0.487648
$307$	2.81135	0.160452	0.0802260	−	0.996777i	$-0.474436\pi$
$307$	2.81135	0.160452	0.0802260	+	0.996777i	$0.474436\pi$
$308$	22.1017	1.25936
$309$	−16.4286	−0.934593
$310$	0	0
$311$	−13.8479	−0.785243	−0.392621	−	0.919700i	$-0.628432\pi$
$311$	−13.8479	−0.785243	−0.392621	+	0.919700i	$0.628432\pi$
$312$	1.67307	0.0947190
$313$	−23.2444	−1.31385	−0.656926	−	0.753955i	$-0.728144\pi$
$313$	−23.2444	−1.31385	−0.656926	+	0.753955i	$0.728144\pi$
$314$	0.0602231	0.00339858
$315$	0	0
$316$	25.5397	1.43672
$317$	−2.96343	−0.166443	−0.0832215	−	0.996531i	$-0.526521\pi$
$317$	−2.96343	−0.166443	−0.0832215	+	0.996531i	$0.526521\pi$
$318$	−6.79706	−0.381160
$319$	20.4701	1.14611
$320$	0	0
$321$	20.0415	1.11861
$322$	−1.89829	−0.105788
$323$	5.05086	0.281037
$324$	−7.96343	−0.442413
$325$	0	0
$326$	−2.62222	−0.145231
$327$	−16.2953	−0.901131
$328$	−6.13335	−0.338658
$329$	19.6128	1.08129
$330$	0	0
$331$	−0.949145	−0.0521697	−0.0260849	−	0.999660i	$-0.508304\pi$
$331$	−0.949145	−0.0521697	−0.0260849	+	0.999660i	$0.508304\pi$
$332$	−20.2163	−1.10952
$333$	−24.2908	−1.33113
$334$	3.87109	0.211817
$335$	0	0
$336$	−44.0830	−2.40492
$337$	−2.28100	−0.124254	−0.0621269	−	0.998068i	$-0.519788\pi$
$337$	−2.28100	−0.124254	−0.0621269	+	0.998068i	$0.519788\pi$
$338$	3.97433	0.216175
$339$	40.3368	2.19079
$340$	0	0
$341$	−3.26317	−0.176710
$342$	1.68889	0.0913248
$343$	24.8573	1.34217
$344$	−14.6222	−0.788377
$345$	0	0
$346$	−6.90321	−0.371119
$347$	−12.3368	−0.662273	−0.331136	−	0.943583i	$-0.607432\pi$
$347$	−12.3368	−0.662273	−0.331136	+	0.943583i	$0.607432\pi$
$348$	−43.1338	−2.31222
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	3.34614	0.178604
$352$	9.16547	0.488521
$353$	−2.56199	−0.136361	−0.0681806	−	0.997673i	$-0.521719\pi$
$353$	−2.56199	−0.136361	−0.0681806	+	0.997673i	$0.521719\pi$
$354$	−1.98126	−0.105303
$355$	0	0
$356$	24.1017	1.27739
$357$	64.9403	3.43700
$358$	3.70471	0.195800
$359$	24.3368	1.28445	0.642223	−	0.766518i	$-0.278012\pi$
$359$	24.3368	1.28445	0.642223	+	0.766518i	$0.278012\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.47949	−0.287996
$363$	11.9728	0.628409
$364$	4.00000	0.209657
$365$	0	0
$366$	3.31756	0.173412
$367$	4.42864	0.231173	0.115587	−	0.993297i	$-0.463125\pi$
$367$	4.42864	0.231173	0.115587	+	0.993297i	$0.463125\pi$
$368$	4.72393	0.246252
$369$	−27.4193	−1.42739
$370$	0	0
$371$	−33.3274	−1.73027
$372$	6.87601	0.356505
$373$	23.7003	1.22715	0.613577	−	0.789635i	$-0.289730\pi$
$373$	23.7003	1.22715	0.613577	+	0.789635i	$0.289730\pi$
$374$	−4.12045	−0.213063
$375$	0	0
$376$	5.37778	0.277338
$377$	3.70471	0.190802
$378$	9.71456	0.499663
$379$	8.20342	0.421381	0.210691	−	0.977553i	$-0.432429\pi$
$379$	8.20342	0.421381	0.210691	+	0.977553i	$0.432429\pi$
$380$	0	0
$381$	−20.8988	−1.07068
$382$	0.0829744	0.00424534
$383$	20.9131	1.06861	0.534304	−	0.845293i	$-0.320574\pi$
$383$	20.9131	1.06861	0.534304	+	0.845293i	$0.320574\pi$
$384$	−25.5067	−1.30163
$385$	0	0
$386$	0.830082	0.0422501
$387$	−65.3689	−3.32289
$388$	34.0558	1.72892
$389$	−24.1017	−1.22201	−0.611003	−	0.791629i	$-0.709234\pi$
$389$	−24.1017	−1.22201	−0.611003	+	0.791629i	$0.709234\pi$
$390$	0	0
$391$	−6.95899	−0.351931
$392$	15.3160	0.773576
$393$	6.10171	0.307791
$394$	−1.66323	−0.0837921
$395$	0	0
$396$	27.0923	1.36144
$397$	−7.92687	−0.397838	−0.198919	−	0.980016i	$-0.563743\pi$
$397$	−7.92687	−0.397838	−0.198919	+	0.980016i	$0.563743\pi$
$398$	−5.32741	−0.267039
$399$	12.8573	0.643669
$400$	0	0
$401$	−32.5718	−1.62656	−0.813280	−	0.581873i	$-0.802320\pi$
$401$	−32.5718	−1.62656	−0.813280	+	0.581873i	$0.802320\pi$
$402$	1.49823	0.0747249
$403$	−0.590573	−0.0294185
$404$	19.8479	0.987470
$405$	0	0
$406$	10.7556	0.533790
$407$	11.7333	0.581598
$408$	17.8064	0.881549
$409$	36.3684	1.79830	0.899151	−	0.437638i	$-0.144185\pi$
$409$	36.3684	1.79830	0.899151	+	0.437638i	$0.144185\pi$
$410$	0	0
$411$	4.94914	0.244123
$412$	−10.7699	−0.530593
$413$	−9.71456	−0.478022
$414$	−2.32693	−0.114362
$415$	0	0
$416$	1.65878	0.0813284
$417$	−25.3274	−1.24029
$418$	−0.815792	−0.0399017
$419$	−31.6958	−1.54844	−0.774221	−	0.632915i	$-0.781858\pi$
$419$	−31.6958	−1.54844	−0.774221	+	0.632915i	$0.781858\pi$
$420$	0	0
$421$	37.4005	1.82279	0.911395	−	0.411532i	$-0.135006\pi$
$421$	37.4005	1.82279	0.911395	+	0.411532i	$0.135006\pi$
$422$	−4.09187	−0.199189
$423$	24.0415	1.16894
$424$	−9.13828	−0.443794
$425$	0	0
$426$	6.87601	0.333144
$427$	16.2667	0.787201
$428$	13.1383	0.635063
$429$	−3.61285	−0.174430
$430$	0	0
$431$	4.94914	0.238392	0.119196	−	0.992871i	$-0.461968\pi$
$431$	4.94914	0.238392	0.119196	+	0.992871i	$0.461968\pi$
$432$	−24.1748	−1.16311
$433$	−32.3827	−1.55621	−0.778107	−	0.628132i	$-0.783820\pi$
$433$	−32.3827	−1.55621	−0.778107	+	0.628132i	$0.783820\pi$
$434$	−1.71456	−0.0823014
$435$	0	0
$436$	−10.6824	−0.511596
$437$	−1.37778	−0.0659084
$438$	3.43801	0.164274
$439$	10.0731	0.480764	0.240382	−	0.970678i	$-0.422727\pi$
$439$	10.0731	0.480764	0.240382	+	0.970678i	$0.422727\pi$
$440$	0	0
$441$	68.4706	3.26050
$442$	−0.745724	−0.0354705
$443$	13.9684	0.663657	0.331828	−	0.943340i	$-0.392335\pi$
$443$	13.9684	0.663657	0.331828	+	0.943340i	$0.392335\pi$
$444$	−24.7239	−1.17335
$445$	0	0
$446$	−3.27163	−0.154916
$447$	19.7877	0.935926
$448$	−25.5526	−1.20725
$449$	24.5718	1.15962	0.579808	−	0.814753i	$-0.303127\pi$
$449$	24.5718	1.15962	0.579808	+	0.814753i	$0.303127\pi$
$450$	0	0
$451$	13.2444	0.623656
$452$	26.4429	1.24377
$453$	16.8573	0.792024
$454$	2.14764	0.100794
$455$	0	0
$456$	3.52543	0.165093
$457$	3.51114	0.164244	0.0821220	−	0.996622i	$-0.473830\pi$
$457$	3.51114	0.164244	0.0821220	+	0.996622i	$0.473830\pi$
$458$	−5.61285	−0.262271
$459$	35.6128	1.66227
$460$	0	0
$461$	10.2034	0.475221	0.237610	−	0.971361i	$-0.423636\pi$
$461$	10.2034	0.475221	0.237610	+	0.971361i	$0.423636\pi$
$462$	−10.4889	−0.487986
$463$	−8.33677	−0.387443	−0.193721	−	0.981057i	$-0.562056\pi$
$463$	−8.33677	−0.387443	−0.193721	+	0.981057i	$0.562056\pi$
$464$	−26.7654	−1.24255
$465$	0	0
$466$	−3.84791	−0.178251
$467$	42.7052	1.97616	0.988080	−	0.153940i	$-0.0491961\pi$
$467$	42.7052	1.97616	0.988080	+	0.153940i	$0.0491961\pi$
$468$	4.90321	0.226651
$469$	7.34614	0.339213
$470$	0	0
$471$	0.561993	0.0258953
$472$	−2.66370	−0.122607
$473$	31.5754	1.45184
$474$	−12.1204	−0.556711
$475$	0	0
$476$	42.5718	1.95128
$477$	−40.8528	−1.87052
$478$	−7.73329	−0.353713
$479$	41.4608	1.89439	0.947195	−	0.320658i	$-0.103904\pi$
$479$	41.4608	1.89439	0.947195	+	0.320658i	$0.103904\pi$
$480$	0	0
$481$	2.12351	0.0968237
$482$	−2.81579	−0.128256
$483$	−17.7146	−0.806040
$484$	7.84882	0.356764
$485$	0	0
$486$	−2.80150	−0.127079
$487$	−30.0370	−1.36111	−0.680554	−	0.732698i	$-0.738261\pi$
$487$	−30.0370	−1.36111	−0.680554	+	0.732698i	$0.738261\pi$
$488$	4.46028	0.201907
$489$	−24.4701	−1.10658
$490$	0	0
$491$	15.3461	0.692562	0.346281	−	0.938131i	$-0.387444\pi$
$491$	15.3461	0.692562	0.346281	+	0.938131i	$0.387444\pi$
$492$	−27.9081	−1.25820
$493$	39.4291	1.77580
$494$	−0.147643	−0.00664278
$495$	0	0
$496$	4.26671	0.191581
$497$	33.7146	1.51230
$498$	9.59411	0.429922
$499$	−25.8479	−1.15711	−0.578556	−	0.815643i	$-0.696383\pi$
$499$	−25.8479	−1.15711	−0.578556	+	0.815643i	$0.696383\pi$
$500$	0	0
$501$	36.1245	1.61392
$502$	7.02227	0.313419
$503$	4.40006	0.196189	0.0980945	−	0.995177i	$-0.468725\pi$
$503$	4.40006	0.196189	0.0980945	+	0.995177i	$0.468725\pi$
$504$	29.1941	1.30041
$505$	0	0
$506$	1.12399	0.0499672
$507$	37.0879	1.64713
$508$	−13.7003	−0.607851
$509$	−27.2355	−1.20719	−0.603597	−	0.797290i	$-0.706266\pi$
$509$	−27.2355	−1.20719	−0.603597	+	0.797290i	$0.706266\pi$
$510$	0	0
$511$	16.8573	0.745722
$512$	−20.3111	−0.897633
$513$	7.05086	0.311303
$514$	1.53833	0.0678530
$515$	0	0
$516$	−66.5344	−2.92901
$517$	−11.6128	−0.510732
$518$	6.16500	0.270874
$519$	−64.4197	−2.82771
$520$	0	0
$521$	38.5531	1.68904	0.844521	−	0.535522i	$-0.179885\pi$
$521$	38.5531	1.68904	0.844521	+	0.535522i	$0.179885\pi$
$522$	13.1842	0.577057
$523$	−18.1575	−0.793971	−0.396986	−	0.917825i	$-0.629944\pi$
$523$	−18.1575	−0.793971	−0.396986	+	0.917825i	$0.629944\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	2.91750	0.127209
$527$	−6.28544	−0.273798
$528$	26.1017	1.13593
$529$	−21.1017	−0.917466
$530$	0	0
$531$	−11.9081	−0.516769
$532$	8.42864	0.365428
$533$	2.39700	0.103825
$534$	−11.4380	−0.494971
$535$	0	0
$536$	2.01429	0.0870041
$537$	34.5718	1.49188
$538$	−6.13335	−0.264428
$539$	−33.0736	−1.42458
$540$	0	0
$541$	−13.7748	−0.592224	−0.296112	−	0.955153i	$-0.595690\pi$
$541$	−13.7748	−0.592224	−0.296112	+	0.955153i	$0.595690\pi$
$542$	0.345665	0.0148476
$543$	−51.1338	−2.19436
$544$	17.6543	0.756923
$545$	0	0
$546$	−1.89829	−0.0812393
$547$	−42.9862	−1.83796	−0.918978	−	0.394308i	$-0.870984\pi$
$547$	−42.9862	−1.83796	−0.918978	+	0.394308i	$0.870984\pi$
$548$	3.24443	0.138595
$549$	19.9398	0.851009
$550$	0	0
$551$	7.80642	0.332565
$552$	−4.85728	−0.206740
$553$	−59.4291	−2.52718
$554$	1.71762	0.0729747
$555$	0	0
$556$	−16.6035	−0.704144
$557$	−14.2953	−0.605711	−0.302855	−	0.953037i	$-0.597940\pi$
$557$	−14.2953	−0.605711	−0.302855	+	0.953037i	$0.597940\pi$
$558$	−2.10171	−0.0889725
$559$	5.71456	0.241700
$560$	0	0
$561$	−38.4514	−1.62342
$562$	4.91750	0.207432
$563$	−29.9541	−1.26241	−0.631207	−	0.775615i	$-0.717440\pi$
$563$	−29.9541	−1.26241	−0.631207	+	0.775615i	$0.717440\pi$
$564$	24.4701	1.03038
$565$	0	0
$566$	4.42864	0.186150
$567$	18.5303	0.778202
$568$	9.24443	0.387888
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−38.2351	−1.60009	−0.800044	−	0.599942i	$-0.795190\pi$
$571$	−38.2351	−1.60009	−0.800044	+	0.599942i	$0.795190\pi$
$572$	−2.36842	−0.0990285
$573$	0.774305	0.0323470
$574$	6.95899	0.290463
$575$	0	0
$576$	−31.3225	−1.30510
$577$	16.5718	0.689895	0.344947	−	0.938622i	$-0.387897\pi$
$577$	16.5718	0.689895	0.344947	+	0.938622i	$0.387897\pi$
$578$	−2.64788	−0.110137
$579$	7.74620	0.321921
$580$	0	0
$581$	47.0420	1.95163
$582$	−16.1619	−0.669934
$583$	19.7333	0.817270
$584$	4.62222	0.191269
$585$	0	0
$586$	2.34122	0.0967149
$587$	−7.94962	−0.328116	−0.164058	−	0.986451i	$-0.552458\pi$
$587$	−7.94962	−0.328116	−0.164058	+	0.986451i	$0.552458\pi$
$588$	69.6914	2.87402
$589$	−1.24443	−0.0512759
$590$	0	0
$591$	−15.5210	−0.638448
$592$	−15.3417	−0.630540
$593$	17.0794	0.701368	0.350684	−	0.936494i	$-0.385949\pi$
$593$	17.0794	0.701368	0.350684	+	0.936494i	$0.385949\pi$
$594$	−5.75203	−0.236009
$595$	0	0
$596$	12.9719	0.531350
$597$	−49.7146	−2.03468
$598$	0.203420	0.00831848
$599$	5.68598	0.232323	0.116161	−	0.993230i	$-0.462941\pi$
$599$	5.68598	0.232323	0.116161	+	0.993230i	$0.462941\pi$
$600$	0	0
$601$	−33.2543	−1.35647	−0.678235	−	0.734845i	$-0.737255\pi$
$601$	−33.2543	−1.35647	−0.678235	+	0.734845i	$0.737255\pi$
$602$	16.5906	0.676181
$603$	9.00492	0.366709
$604$	11.0509	0.449653
$605$	0	0
$606$	−9.41927	−0.382632
$607$	10.9032	0.442548	0.221274	−	0.975212i	$-0.428979\pi$
$607$	10.9032	0.442548	0.221274	+	0.975212i	$0.428979\pi$
$608$	3.49532	0.141754
$609$	100.369	4.06717
$610$	0	0
$611$	−2.10171	−0.0850261
$612$	52.1847	2.10944
$613$	47.6227	1.92346	0.961731	−	0.273995i	$-0.0883451\pi$
$613$	47.6227	1.92346	0.961731	+	0.273995i	$0.0883451\pi$
$614$	−0.874632	−0.0352973
$615$	0	0
$616$	−14.1017	−0.568174
$617$	46.6450	1.87786	0.938928	−	0.344114i	$-0.111821\pi$
$617$	46.6450	1.87786	0.938928	+	0.344114i	$0.111821\pi$
$618$	5.11108	0.205598
$619$	−32.2163	−1.29488	−0.647442	−	0.762115i	$-0.724161\pi$
$619$	−32.2163	−1.29488	−0.647442	+	0.762115i	$0.724161\pi$
$620$	0	0
$621$	−9.71456	−0.389832
$622$	4.30819	0.172743
$623$	−56.0830	−2.24692
$624$	4.72393	0.189108
$625$	0	0
$626$	7.23152	0.289030
$627$	−7.61285	−0.304028
$628$	0.368416	0.0147014
$629$	22.6004	0.901138
$630$	0	0
$631$	−30.9719	−1.23297	−0.616486	−	0.787366i	$-0.711444\pi$
$631$	−30.9719	−1.23297	−0.616486	+	0.787366i	$0.711444\pi$
$632$	−16.2953	−0.648192
$633$	−38.1847	−1.51770
$634$	0.921948	0.0366152
$635$	0	0
$636$	−41.5812	−1.64880
$637$	−5.98571	−0.237162
$638$	−6.36842	−0.252128
$639$	41.3274	1.63489
$640$	0	0
$641$	−22.1748	−0.875854	−0.437927	−	0.899011i	$-0.644287\pi$
$641$	−22.1748	−0.875854	−0.437927	+	0.899011i	$0.644287\pi$
$642$	−6.23506	−0.246078
$643$	6.23506	0.245887	0.122943	−	0.992414i	$-0.460767\pi$
$643$	6.23506	0.245887	0.122943	+	0.992414i	$0.460767\pi$
$644$	−11.6128	−0.457610
$645$	0	0
$646$	−1.57136	−0.0618244
$647$	1.29481	0.0509042	0.0254521	−	0.999676i	$-0.491897\pi$
$647$	1.29481	0.0509042	0.0254521	+	0.999676i	$0.491897\pi$
$648$	5.08097	0.199599
$649$	5.75203	0.225787
$650$	0	0
$651$	−16.0000	−0.627089
$652$	−16.0415	−0.628233
$653$	−30.2953	−1.18555	−0.592773	−	0.805370i	$-0.701967\pi$
$653$	−30.2953	−1.18555	−0.592773	+	0.805370i	$0.701967\pi$
$654$	5.06959	0.198237
$655$	0	0
$656$	−17.3176	−0.676137
$657$	20.6637	0.806168
$658$	−6.10171	−0.237869
$659$	−4.17484	−0.162629	−0.0813143	−	0.996689i	$-0.525912\pi$
$659$	−4.17484	−0.162629	−0.0813143	+	0.996689i	$0.525912\pi$
$660$	0	0
$661$	−6.56199	−0.255232	−0.127616	−	0.991824i	$-0.540732\pi$
$661$	−6.56199	−0.255232	−0.127616	+	0.991824i	$0.540732\pi$
$662$	0.295286	0.0114766
$663$	−6.95899	−0.270265
$664$	12.8988	0.500569
$665$	0	0
$666$	7.55707	0.292831
$667$	−10.7556	−0.416457
$668$	23.6815	0.916266
$669$	−30.5303	−1.18037
$670$	0	0
$671$	−9.63158	−0.371823
$672$	44.9403	1.73361
$673$	−12.7413	−0.491140	−0.245570	−	0.969379i	$-0.578975\pi$
$673$	−12.7413	−0.491140	−0.245570	+	0.969379i	$0.578975\pi$
$674$	0.709636	0.0273341
$675$	0	0
$676$	24.3131	0.935120
$677$	30.9260	1.18858	0.594291	−	0.804250i	$-0.297433\pi$
$677$	30.9260	1.18858	0.594291	+	0.804250i	$0.297433\pi$
$678$	−12.5491	−0.481945
$679$	−79.2454	−3.04116
$680$	0	0
$681$	20.0415	0.767991
$682$	1.01520	0.0388739
$683$	−34.3412	−1.31403	−0.657015	−	0.753877i	$-0.728181\pi$
$683$	−34.3412	−1.31403	−0.657015	+	0.753877i	$0.728181\pi$
$684$	10.3319	0.395048
$685$	0	0
$686$	−7.73329	−0.295259
$687$	−52.3783	−1.99836
$688$	−41.2859	−1.57401
$689$	3.57136	0.136058
$690$	0	0
$691$	24.2163	0.921233	0.460616	−	0.887599i	$-0.347628\pi$
$691$	24.2163	0.921233	0.460616	+	0.887599i	$0.347628\pi$
$692$	−42.2306	−1.60537
$693$	−63.0420	−2.39477
$694$	3.83807	0.145691
$695$	0	0
$696$	27.5210	1.04318
$697$	25.5111	0.966303
$698$	3.11108	0.117756
$699$	−35.9081	−1.35817
$700$	0	0
$701$	11.4064	0.430812	0.215406	−	0.976525i	$-0.430892\pi$
$701$	11.4064	0.430812	0.215406	+	0.976525i	$0.430892\pi$
$702$	−1.04101	−0.0392904
$703$	4.47457	0.168762
$704$	15.1298	0.570226
$705$	0	0
$706$	0.797056	0.0299976
$707$	−46.1847	−1.73695
$708$	−12.1204	−0.455514
$709$	−13.0223	−0.489062	−0.244531	−	0.969642i	$-0.578634\pi$
$709$	−13.0223	−0.489062	−0.244531	+	0.969642i	$0.578634\pi$
$710$	0	0
$711$	−72.8484	−2.73203
$712$	−15.3778	−0.576307
$713$	1.71456	0.0642107
$714$	−20.2034	−0.756094
$715$	0	0
$716$	22.6637	0.846982
$717$	−72.1659	−2.69509
$718$	−7.57136	−0.282561
$719$	52.2163	1.94734	0.973670	−	0.227961i	$-0.0732059\pi$
$719$	52.2163	1.94734	0.973670	+	0.227961i	$0.0732059\pi$
$720$	0	0
$721$	25.0607	0.933309
$722$	−0.311108	−0.0115782
$723$	−26.2766	−0.977235
$724$	−33.5210	−1.24580
$725$	0	0
$726$	−3.72483	−0.138242
$727$	−14.0602	−0.521465	−0.260732	−	0.965411i	$-0.583964\pi$
$727$	−14.0602	−0.521465	−0.260732	+	0.965411i	$0.583964\pi$
$728$	−2.55215	−0.0945889
$729$	−38.6958	−1.43318
$730$	0	0
$731$	60.8198	2.24950
$732$	20.2953	0.750135
$733$	4.48886	0.165800	0.0829000	−	0.996558i	$-0.473582\pi$
$733$	4.48886	0.165800	0.0829000	+	0.996558i	$0.473582\pi$
$734$	−1.37778	−0.0508549
$735$	0	0
$736$	−4.81579	−0.177512
$737$	−4.34968	−0.160223
$738$	8.53035	0.314007
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−1.37778	−0.0506142
$742$	10.3684	0.380637
$743$	37.5669	1.37820	0.689098	−	0.724668i	$-0.258007\pi$
$743$	37.5669	1.37820	0.689098	+	0.724668i	$0.258007\pi$
$744$	−4.38715	−0.160841
$745$	0	0
$746$	−7.37334	−0.269957
$747$	57.6642	2.10982
$748$	−25.2070	−0.921658
$749$	−30.5718	−1.11707
$750$	0	0
$751$	52.5817	1.91873	0.959366	−	0.282163i	$-0.0910520\pi$
$751$	52.5817	1.91873	0.959366	+	0.282163i	$0.0910520\pi$
$752$	15.1842	0.553711
$753$	65.5308	2.38808
$754$	−1.15257	−0.0419740
$755$	0	0
$756$	59.4291	2.16142
$757$	−5.73329	−0.208380	−0.104190	−	0.994557i	$-0.533225\pi$
$757$	−5.73329	−0.208380	−0.104190	+	0.994557i	$0.533225\pi$
$758$	−2.55215	−0.0926982
$759$	10.4889	0.380722
$760$	0	0
$761$	−32.7338	−1.18660	−0.593299	−	0.804982i	$-0.702175\pi$
$761$	−32.7338	−1.18660	−0.593299	+	0.804982i	$0.702175\pi$
$762$	6.50177	0.235534
$763$	24.8573	0.899894
$764$	0.507598	0.0183643
$765$	0	0
$766$	−6.50622	−0.235079
$767$	1.04101	0.0375887
$768$	−25.5669	−0.922567
$769$	10.1619	0.366449	0.183224	−	0.983071i	$-0.441347\pi$
$769$	10.1619	0.366449	0.183224	+	0.983071i	$0.441347\pi$
$770$	0	0
$771$	14.3555	0.517001
$772$	5.07805	0.182763
$773$	36.0785	1.29765	0.648827	−	0.760936i	$-0.275260\pi$
$773$	36.0785	1.29765	0.648827	+	0.760936i	$0.275260\pi$
$774$	20.3368	0.730990
$775$	0	0
$776$	−21.7288	−0.780020
$777$	57.5308	2.06391
$778$	7.49823	0.268825
$779$	5.05086	0.180966
$780$	0	0
$781$	−19.9625	−0.714315
$782$	2.16500	0.0774201
$783$	55.0420	1.96704
$784$	43.2449	1.54446
$785$	0	0
$786$	−1.89829	−0.0677098
$787$	30.5446	1.08880	0.544399	−	0.838826i	$-0.316758\pi$
$787$	30.5446	1.08880	0.544399	+	0.838826i	$0.316758\pi$
$788$	−10.1748	−0.362464
$789$	27.2257	0.969260
$790$	0	0
$791$	−61.5308	−2.18778
$792$	−17.2859	−0.614228
$793$	−1.74314	−0.0619007
$794$	2.46611	0.0875190
$795$	0	0
$796$	−32.5906	−1.15514
$797$	27.9037	0.988399	0.494200	−	0.869348i	$-0.335461\pi$
$797$	27.9037	0.988399	0.494200	+	0.869348i	$0.335461\pi$
$798$	−4.00000	−0.141598
$799$	−22.3684	−0.791338
$800$	0	0
$801$	−68.7467	−2.42904
$802$	10.1334	0.357821
$803$	−9.98126	−0.352231
$804$	9.16547	0.323241
$805$	0	0
$806$	0.183732	0.00647168
$807$	−57.2355	−2.01479
$808$	−12.6637	−0.445508
$809$	−25.6128	−0.900500	−0.450250	−	0.892903i	$-0.648665\pi$
$809$	−25.6128	−0.900500	−0.450250	+	0.892903i	$0.648665\pi$
$810$	0	0
$811$	−6.01874	−0.211346	−0.105673	−	0.994401i	$-0.533700\pi$
$811$	−6.01874	−0.211346	−0.105673	+	0.994401i	$0.533700\pi$
$812$	65.7975	2.30904
$813$	3.22570	0.113130
$814$	−3.65032	−0.127944
$815$	0	0
$816$	50.2766	1.76003
$817$	12.0415	0.421278
$818$	−11.3145	−0.395602
$819$	−11.4094	−0.398678
$820$	0	0
$821$	−6.20342	−0.216501	−0.108250	−	0.994124i	$-0.534525\pi$
$821$	−6.20342	−0.216501	−0.108250	+	0.994124i	$0.534525\pi$
$822$	−1.53972	−0.0537038
$823$	−1.75605	−0.0612119	−0.0306059	−	0.999532i	$-0.509744\pi$
$823$	−1.75605	−0.0612119	−0.0306059	+	0.999532i	$0.509744\pi$
$824$	6.87157	0.239382
$825$	0	0
$826$	3.02227	0.105158
$827$	53.2083	1.85024	0.925118	−	0.379680i	$-0.123966\pi$
$827$	53.2083	1.85024	0.925118	+	0.379680i	$0.123966\pi$
$828$	−14.2351	−0.494703
$829$	−26.9777	−0.936975	−0.468488	−	0.883470i	$-0.655201\pi$
$829$	−26.9777	−0.936975	−0.468488	+	0.883470i	$0.655201\pi$
$830$	0	0
$831$	16.0286	0.556025
$832$	2.73822	0.0949306
$833$	−63.7057	−2.20727
$834$	7.87955	0.272847
$835$	0	0
$836$	−4.99063	−0.172605
$837$	−8.77430	−0.303284
$838$	9.86082	0.340636
$839$	46.9501	1.62090	0.810449	−	0.585810i	$-0.199223\pi$
$839$	46.9501	1.62090	0.810449	+	0.585810i	$0.199223\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	−11.6356	−0.400989
$843$	45.8894	1.58051
$844$	−25.0321	−0.861641
$845$	0	0
$846$	−7.47949	−0.257150
$847$	−18.2636	−0.627546
$848$	−25.8020	−0.886044
$849$	41.3274	1.41835
$850$	0	0
$851$	−6.16500	−0.211333
$852$	42.0642	1.44110
$853$	46.4701	1.59111	0.795553	−	0.605883i	$-0.207180\pi$
$853$	46.4701	1.59111	0.795553	+	0.605883i	$0.207180\pi$
$854$	−5.06070	−0.173174
$855$	0	0
$856$	−8.38271	−0.286515
$857$	7.79213	0.266174	0.133087	−	0.991104i	$-0.457511\pi$
$857$	7.79213	0.266174	0.133087	+	0.991104i	$0.457511\pi$
$858$	1.12399	0.0383722
$859$	−37.4479	−1.27770	−0.638852	−	0.769330i	$-0.720590\pi$
$859$	−37.4479	−1.27770	−0.638852	+	0.769330i	$0.720590\pi$
$860$	0	0
$861$	64.9403	2.21316
$862$	−1.53972	−0.0524430
$863$	−47.7605	−1.62579	−0.812893	−	0.582413i	$-0.802109\pi$
$863$	−47.7605	−1.62579	−0.812893	+	0.582413i	$0.802109\pi$
$864$	24.6450	0.838439
$865$	0	0
$866$	10.0745	0.342346
$867$	−24.7096	−0.839183
$868$	−10.4889	−0.356015
$869$	35.1882	1.19368
$870$	0	0
$871$	−0.787212	−0.0266736
$872$	6.81579	0.230812
$873$	−97.1392	−3.28766
$874$	0.428639	0.0144989
$875$	0	0
$876$	21.0321	0.710609
$877$	22.5763	0.762347	0.381173	−	0.924504i	$-0.375520\pi$
$877$	22.5763	0.762347	0.381173	+	0.924504i	$0.375520\pi$
$878$	−3.13383	−0.105762
$879$	21.8479	0.736912
$880$	0	0
$881$	−10.8988	−0.367189	−0.183594	−	0.983002i	$-0.558773\pi$
$881$	−10.8988	−0.367189	−0.183594	+	0.983002i	$0.558773\pi$
$882$	−21.3017	−0.717267
$883$	39.9782	1.34537	0.672687	−	0.739927i	$-0.265140\pi$
$883$	39.9782	1.34537	0.672687	+	0.739927i	$0.265140\pi$
$884$	−4.56199	−0.153436
$885$	0	0
$886$	−4.34567	−0.145995
$887$	−49.6785	−1.66804	−0.834020	−	0.551734i	$-0.813966\pi$
$887$	−49.6785	−1.66804	−0.834020	+	0.551734i	$0.813966\pi$
$888$	15.7748	0.529367
$889$	31.8796	1.06921
$890$	0	0
$891$	−10.9719	−0.367572
$892$	−20.0143	−0.670128
$893$	−4.42864	−0.148199
$894$	−6.15610	−0.205891
$895$	0	0
$896$	38.9086	1.29985
$897$	1.89829	0.0633821
$898$	−7.64449	−0.255100
$899$	−9.71456	−0.323999
$900$	0	0
$901$	38.0098	1.26629
$902$	−4.12045	−0.137196
$903$	154.821	5.15211
$904$	−16.8716	−0.561140
$905$	0	0
$906$	−5.24443	−0.174235
$907$	18.2779	0.606909	0.303454	−	0.952846i	$-0.401860\pi$
$907$	18.2779	0.606909	0.303454	+	0.952846i	$0.401860\pi$
$908$	13.1383	0.436009
$909$	−56.6133	−1.87775
$910$	0	0
$911$	−44.7654	−1.48314	−0.741572	−	0.670873i	$-0.765920\pi$
$911$	−44.7654	−1.48314	−0.741572	+	0.670873i	$0.765920\pi$
$912$	9.95407	0.329612
$913$	−27.8537	−0.921824
$914$	−1.09234	−0.0361315
$915$	0	0
$916$	−34.3368	−1.13452
$917$	−9.30772	−0.307368
$918$	−11.0794	−0.365676
$919$	−33.6316	−1.10940	−0.554702	−	0.832049i	$-0.687168\pi$
$919$	−33.6316	−1.10940	−0.554702	+	0.832049i	$0.687168\pi$
$920$	0	0
$921$	−8.16193	−0.268945
$922$	−3.17436	−0.104542
$923$	−3.61285	−0.118918
$924$	−64.1659	−2.11090
$925$	0	0
$926$	2.59364	0.0852321
$927$	30.7195	1.00896
$928$	27.2859	0.895704
$929$	13.0223	0.427247	0.213623	−	0.976916i	$-0.431473\pi$
$929$	13.0223	0.427247	0.213623	+	0.976916i	$0.431473\pi$
$930$	0	0
$931$	−12.6128	−0.413369
$932$	−23.5397	−0.771069
$933$	40.2034	1.31620
$934$	−13.2859	−0.434729
$935$	0	0
$936$	−3.12843	−0.102256
$937$	−28.5433	−0.932468	−0.466234	−	0.884662i	$-0.654389\pi$
$937$	−28.5433	−0.932468	−0.466234	+	0.884662i	$0.654389\pi$
$938$	−2.28544	−0.0746223
$939$	67.4835	2.20224
$940$	0	0
$941$	13.2257	0.431145	0.215573	−	0.976488i	$-0.430838\pi$
$941$	13.2257	0.431145	0.215573	+	0.976488i	$0.430838\pi$
$942$	−0.174840	−0.00569660
$943$	−6.95899	−0.226616
$944$	−7.52098	−0.244787
$945$	0	0
$946$	−9.82335	−0.319385
$947$	−3.66323	−0.119039	−0.0595194	−	0.998227i	$-0.518957\pi$
$947$	−3.66323	−0.119039	−0.0595194	+	0.998227i	$0.518957\pi$
$948$	−74.1472	−2.40819
$949$	−1.80642	−0.0586390
$950$	0	0
$951$	8.60348	0.278987
$952$	−27.1624	−0.880339
$953$	−12.1704	−0.394238	−0.197119	−	0.980380i	$-0.563158\pi$
$953$	−12.1704	−0.394238	−0.197119	+	0.980380i	$0.563158\pi$
$954$	12.7096	0.411490
$955$	0	0
$956$	−47.3087	−1.53007
$957$	−59.4291	−1.92107
$958$	−12.8988	−0.416740
$959$	−7.54956	−0.243788
$960$	0	0
$961$	−29.4514	−0.950045
$962$	−0.660640	−0.0212999
$963$	−37.4750	−1.20762
$964$	−17.2257	−0.554802
$965$	0	0
$966$	5.51114	0.177318
$967$	−8.52051	−0.274001	−0.137000	−	0.990571i	$-0.543746\pi$
$967$	−8.52051	−0.274001	−0.137000	+	0.990571i	$0.543746\pi$
$968$	−5.00784	−0.160958
$969$	−14.6637	−0.471066
$970$	0	0
$971$	−2.67259	−0.0857676	−0.0428838	−	0.999080i	$-0.513655\pi$
$971$	−2.67259	−0.0857676	−0.0428838	+	0.999080i	$0.513655\pi$
$972$	−17.1383	−0.549710
$973$	38.6351	1.23859
$974$	9.34476	0.299425
$975$	0	0
$976$	12.5936	0.403112
$977$	20.3827	0.652101	0.326050	−	0.945352i	$-0.394282\pi$
$977$	20.3827	0.652101	0.326050	+	0.945352i	$0.394282\pi$
$978$	7.61285	0.243432
$979$	33.2070	1.06130
$980$	0	0
$981$	30.4701	0.972836
$982$	−4.77430	−0.152354
$983$	−10.3126	−0.328922	−0.164461	−	0.986384i	$-0.552588\pi$
$983$	−10.3126	−0.328922	−0.164461	+	0.986384i	$0.552588\pi$
$984$	17.8064	0.567648
$985$	0	0
$986$	−12.2667	−0.390652
$987$	−56.9403	−1.81243
$988$	−0.903212	−0.0287350
$989$	−16.5906	−0.527550
$990$	0	0
$991$	20.0919	0.638239	0.319120	−	0.947714i	$-0.396613\pi$
$991$	20.0919	0.638239	0.319120	+	0.947714i	$0.396613\pi$
$992$	−4.34968	−0.138102
$993$	2.75557	0.0874453
$994$	−10.4889	−0.332687
$995$	0	0
$996$	58.6923	1.85974
$997$	−25.0509	−0.793369	−0.396684	−	0.917955i	$-0.629839\pi$
$997$	−25.0509	−0.793369	−0.396684	+	0.917955i	$0.629839\pi$
$998$	8.04149	0.254549
$999$	31.5496	0.998184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.f.1.2		3
3.2	odd	2		4275.2.a.bk.1.2		3
4.3	odd	2		7600.2.a.bx.1.3		3
5.2	odd	4		475.2.b.d.324.3		6
5.3	odd	4		475.2.b.d.324.4		6
5.4	even	2		95.2.a.a.1.2	✓	3
15.14	odd	2		855.2.a.i.1.2		3
19.18	odd	2		9025.2.a.bb.1.2		3
20.19	odd	2		1520.2.a.p.1.1		3
35.34	odd	2		4655.2.a.u.1.2		3
40.19	odd	2		6080.2.a.by.1.3		3
40.29	even	2		6080.2.a.bo.1.1		3
95.94	odd	2		1805.2.a.f.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.2	✓	3	5.4	even	2
475.2.a.f.1.2		3	1.1	even	1	trivial
475.2.b.d.324.3		6	5.2	odd	4
475.2.b.d.324.4		6	5.3	odd	4
855.2.a.i.1.2		3	15.14	odd	2
1520.2.a.p.1.1		3	20.19	odd	2
1805.2.a.f.1.2		3	95.94	odd	2
4275.2.a.bk.1.2		3	3.2	odd	2
4655.2.a.u.1.2		3	35.34	odd	2
6080.2.a.bo.1.1		3	40.29	even	2
6080.2.a.by.1.3		3	40.19	odd	2
7600.2.a.bx.1.3		3	4.3	odd	2
9025.2.a.bb.1.2		3	19.18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} +0.903212 q^{6} +4.42864 q^{7} +1.21432 q^{8} +5.42864 q^{9} -2.62222 q^{11} +5.52543 q^{12} -0.474572 q^{13} -1.37778 q^{14} +3.42864 q^{16} -5.05086 q^{17} -1.68889 q^{18} -1.00000 q^{19} -12.8573 q^{21} +0.815792 q^{22} +1.37778 q^{23} -3.52543 q^{24} +0.147643 q^{26} -7.05086 q^{27} -8.42864 q^{28} -7.80642 q^{29} +1.24443 q^{31} -3.49532 q^{32} +7.61285 q^{33} +1.57136 q^{34} -10.3319 q^{36} -4.47457 q^{37} +0.311108 q^{38} +1.37778 q^{39} -5.05086 q^{41} +4.00000 q^{42} -12.0415 q^{43} +4.99063 q^{44} -0.428639 q^{46} +4.42864 q^{47} -9.95407 q^{48} +12.6128 q^{49} +14.6637 q^{51} +0.903212 q^{52} -7.52543 q^{53} +2.19358 q^{54} +5.37778 q^{56} +2.90321 q^{57} +2.42864 q^{58} -2.19358 q^{59} +3.67307 q^{61} -0.387152 q^{62} +24.0415 q^{63} -5.76986 q^{64} -2.36842 q^{66} +1.65878 q^{67} +9.61285 q^{68} -4.00000 q^{69} +7.61285 q^{71} +6.59210 q^{72} +3.80642 q^{73} +1.39207 q^{74} +1.90321 q^{76} -11.6128 q^{77} -0.428639 q^{78} -13.4193 q^{79} +4.18421 q^{81} +1.57136 q^{82} +10.6222 q^{83} +24.4701 q^{84} +3.74620 q^{86} +22.6637 q^{87} -3.18421 q^{88} -12.6637 q^{89} -2.10171 q^{91} -2.62222 q^{92} -3.61285 q^{93} -1.37778 q^{94} +10.1476 q^{96} -17.8938 q^{97} -3.92396 q^{98} -14.2351 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 3 q^{8} + 3 q^{9} - 8 q^{11} + 10 q^{12} - 8 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} - 5 q^{18} - 3 q^{19} - 12 q^{21} + 16 q^{22} + 4 q^{23} - 4 q^{24} - 6 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) \) 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 3 * q^8 + 3 * q^9 - 8 * q^11 + 10 * q^12 - 8 * q^13 - 4 * q^14 - 3 * q^16 - 2 * q^17 - 5 * q^18 - 3 * q^19 - 12 * q^21 + 16 * q^22 + 4 * q^23 - 4 * q^24 - 6 * q^26 - 8 * q^27 - 12 * q^28 - 10 * q^29 + 4 * q^31 + 3 * q^32 - 4 * q^33 + 18 * q^34 - 11 * q^36 - 20 * q^37 + q^38 + 4 * q^39 - 2 * q^41 + 12 * q^42 + 4 * q^43 - 12 * q^44 + 12 * q^46 - 10 * q^48 + 11 * q^49 + 4 * q^51 - 4 * q^52 - 16 * q^53 + 20 * q^54 + 16 * q^56 + 2 * q^57 - 6 * q^58 - 20 * q^59 - 2 * q^61 - 28 * q^62 + 32 * q^63 - 11 * q^64 + 20 * q^66 - 2 * q^67 + 2 * q^68 - 12 * q^69 - 4 * q^71 + 13 * q^72 - 2 * q^73 - 2 * q^74 - q^76 - 8 * q^77 + 12 * q^78 - q^81 + 18 * q^82 + 32 * q^83 + 20 * q^84 - 16 * q^86 + 28 * q^87 + 4 * q^88 + 2 * q^89 + 20 * q^91 - 8 * q^92 + 16 * q^93 - 4 * q^94 + 24 * q^96 - 20 * q^97 + 15 * q^98 - 16 * q^99

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