LMFDB - Embedded newform 475.4.b.f.324.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [475,4,Mod(324,475)] 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("475.324"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	475.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-42,0,-130] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$28.0259072527$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.158155776.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 $ x^6 - 2x^5 + 2x^4 + 24x^3 + 81x^2 + 54*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$-0.376763 - 0.376763i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.4.b.f.324.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.96257i q^{2} -6.71610i q^{3} -7.70200 q^{4} -26.6130 q^{6} -25.8362i q^{7} -1.18085i q^{8} -18.1060 q^{9} -8.13420 q^{11} +51.7274i q^{12} +4.56640i q^{13} -102.378 q^{14} -66.2952 q^{16} -62.5850i q^{17} +71.7464i q^{18} +19.0000 q^{19} -173.518 q^{21} +32.2324i q^{22} +52.7502i q^{23} -7.93070 q^{24} +18.0947 q^{26} -59.7330i q^{27} +198.990i q^{28} -171.620 q^{29} +168.749 q^{31} +253.253i q^{32} +54.6301i q^{33} -247.998 q^{34} +139.452 q^{36} -147.534i q^{37} -75.2889i q^{38} +30.6684 q^{39} +308.774 q^{41} +687.580i q^{42} +448.950i q^{43} +62.6496 q^{44} +209.027 q^{46} +113.335i q^{47} +445.245i q^{48} -324.509 q^{49} -420.327 q^{51} -35.1704i q^{52} -155.402i q^{53} -236.696 q^{54} -30.5086 q^{56} -127.606i q^{57} +680.059i q^{58} -182.347 q^{59} +404.080 q^{61} -668.681i q^{62} +467.790i q^{63} +473.172 q^{64} +216.476 q^{66} -106.400i q^{67} +482.030i q^{68} +354.276 q^{69} +472.079 q^{71} +21.3805i q^{72} -843.821i q^{73} -584.616 q^{74} -146.338 q^{76} +210.157i q^{77} -121.526i q^{78} +591.036 q^{79} -890.035 q^{81} -1223.54i q^{82} -290.388i q^{83} +1336.44 q^{84} +1779.00 q^{86} +1152.62i q^{87} +9.60526i q^{88} +964.896 q^{89} +117.978 q^{91} -406.282i q^{92} -1133.34i q^{93} +449.099 q^{94} +1700.87 q^{96} -219.495i q^{97} +1285.89i q^{98} +147.278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 42 q^{4} - 130 q^{6} - 96 q^{9} + 32 q^{11} - 74 q^{14} + 66 q^{16} + 114 q^{19} - 50 q^{21} + 978 q^{24} + 598 q^{26} - 754 q^{29} - 280 q^{31} - 658 q^{34} + 2148 q^{36} + 742 q^{39} + 1912 q^{41}+ \cdots - 500 q^{99}+O(q^{100}) $ 6 * q - 42 * q^4 - 130 * q^6 - 96 * q^9 + 32 * q^11 - 74 * q^14 + 66 * q^16 + 114 * q^19 - 50 * q^21 + 978 * q^24 + 598 * q^26 - 754 * q^29 - 280 * q^31 - 658 * q^34 + 2148 * q^36 + 742 * q^39 + 1912 * q^41 - 220 * q^44 - 478 * q^46 + 196 * q^49 + 374 * q^51 + 1894 * q^54 - 1830 * q^56 - 530 * q^59 + 1976 * q^61 + 318 * q^64 - 548 * q^66 - 1614 * q^69 + 1692 * q^71 + 1540 * q^74 - 798 * q^76 - 764 * q^79 - 1002 * q^81 - 1814 * q^84 + 5184 * q^86 + 344 * q^89 - 914 * q^91 + 192 * q^94 + 1726 * q^96 - 500 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 3.96257i	− 1.40098i	−0.713661	−	0.700491i	$-0.752964\pi$
$2$	− 3.96257i	− 1.40098i	0.713661	−	0.700491i	$-0.247036\pi$
$3$	− 6.71610i	− 1.29251i	−0.763120	−	0.646257i	$-0.776333\pi$
$3$	− 6.71610i	− 1.29251i	0.763120	−	0.646257i	$-0.223667\pi$
$4$	−7.70200	−0.962750
$5$	0	0
$5$	0	0
$6$	−26.6130	−1.81079
$7$	− 25.8362i	− 1.39502i	−0.716573	−	0.697512i	$-0.754290\pi$
$7$	− 25.8362i	− 1.39502i	0.716573	−	0.697512i	$-0.245710\pi$
$8$	− 1.18085i	− 0.0521866i
$9$	−18.1060	−0.670593
$10$	0	0
$11$	−8.13420	−0.222959	−0.111480	−	0.993767i	$-0.535559\pi$
$11$	−8.13420	−0.222959	−0.111480	+	0.993767i	$0.535559\pi$
$12$	51.7274i	1.24437i
$13$	4.56640i	0.0974224i	0.998813	+	0.0487112i	$0.0155114\pi$
$13$	4.56640i	0.0974224i	−0.998813	+	0.0487112i	$0.984489\pi$
$14$	−102.378	−1.95440
$15$	0	0
$16$	−66.2952	−1.03586
$17$	− 62.5850i	− 0.892888i	−0.894812	−	0.446444i	$-0.852690\pi$
$17$	− 62.5850i	− 0.892888i	0.894812	−	0.446444i	$-0.147310\pi$
$18$	71.7464i	0.939488i
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−173.518	−1.80309
$22$	32.2324i	0.312362i
$23$	52.7502i	0.478225i	0.970992	+	0.239113i	$0.0768565\pi$
$23$	52.7502i	0.478225i	−0.970992	+	0.239113i	$0.923144\pi$
$24$	−7.93070	−0.0674520
$25$	0	0
$26$	18.0947	0.136487
$27$	− 59.7330i	− 0.425764i
$28$	198.990i	1.34306i
$29$	−171.620	−1.09894	−0.549468	−	0.835515i	$-0.685169\pi$
$29$	−171.620	−1.09894	−0.549468	+	0.835515i	$0.685169\pi$
$30$	0	0
$31$	168.749	0.977685	0.488842	−	0.872372i	$-0.337419\pi$
$31$	168.749	0.977685	0.488842	+	0.872372i	$0.337419\pi$
$32$	253.253i	1.39904i
$33$	54.6301i	0.288178i
$34$	−247.998	−1.25092
$35$	0	0
$36$	139.452	0.645613
$37$	− 147.534i	− 0.655528i	−0.944760	−	0.327764i	$-0.893705\pi$
$37$	− 147.534i	− 0.655528i	0.944760	−	0.327764i	$-0.106295\pi$
$38$	− 75.2889i	− 0.321407i
$39$	30.6684	0.125920
$40$	0	0
$41$	308.774	1.17616	0.588078	−	0.808804i	$-0.299885\pi$
$41$	308.774	1.17616	0.588078	+	0.808804i	$0.299885\pi$
$42$	687.580i	2.52609i
$43$	448.950i	1.59219i	0.605170	+	0.796096i	$0.293105\pi$
$43$	448.950i	1.59219i	−0.605170	+	0.796096i	$0.706895\pi$
$44$	62.6496	0.214654
$45$	0	0
$46$	209.027	0.669985
$47$	113.335i	0.351737i	0.984414	+	0.175868i	$0.0562733\pi$
$47$	113.335i	0.351737i	−0.984414	+	0.175868i	$0.943727\pi$
$48$	445.245i	1.33887i
$49$	−324.509	−0.946091
$50$	0	0
$51$	−420.327	−1.15407
$52$	− 35.1704i	− 0.0937934i
$53$	− 155.402i	− 0.402758i	−0.979513	−	0.201379i	$-0.935458\pi$
$53$	− 155.402i	− 0.402758i	0.979513	−	0.201379i	$-0.0645422\pi$
$54$	−236.696	−0.596487
$55$	0	0
$56$	−30.5086	−0.0728016
$57$	− 127.606i	− 0.296523i
$58$	680.059i	1.53959i
$59$	−182.347	−0.402365	−0.201183	−	0.979554i	$-0.564478\pi$
$59$	−182.347	−0.402365	−0.201183	+	0.979554i	$0.564478\pi$
$60$	0	0
$61$	404.080	0.848149	0.424075	−	0.905627i	$-0.360599\pi$
$61$	404.080	0.848149	0.424075	+	0.905627i	$0.360599\pi$
$62$	− 668.681i	− 1.36972i
$63$	467.790i	0.935492i
$64$	473.172	0.924164
$65$	0	0
$66$	216.476	0.403732
$67$	− 106.400i	− 0.194013i	−0.995284	−	0.0970064i	$-0.969073\pi$
$67$	− 106.400i	− 0.194013i	0.995284	−	0.0970064i	$-0.0309267\pi$
$68$	482.030i	0.859628i
$69$	354.276	0.618113
$70$	0	0
$71$	472.079	0.789091	0.394546	−	0.918876i	$-0.370902\pi$
$71$	472.079	0.789091	0.394546	+	0.918876i	$0.370902\pi$
$72$	21.3805i	0.0349960i
$73$	− 843.821i	− 1.35290i	−0.736488	−	0.676451i	$-0.763517\pi$
$73$	− 843.821i	− 1.35290i	0.736488	−	0.676451i	$-0.236483\pi$
$74$	−584.616	−0.918382
$75$	0	0
$76$	−146.338	−0.220870
$77$	210.157i	0.311034i
$78$	− 121.526i	− 0.176411i
$79$	591.036	0.841731	0.420866	−	0.907123i	$-0.361726\pi$
$79$	591.036	0.841731	0.420866	+	0.907123i	$0.361726\pi$
$80$	0	0
$81$	−890.035	−1.22090
$82$	− 1223.54i	− 1.64777i
$83$	− 290.388i	− 0.384027i	−0.981392	−	0.192013i	$-0.938498\pi$
$83$	− 290.388i	− 0.384027i	0.981392	−	0.192013i	$-0.0615017\pi$
$84$	1336.44	1.73592
$85$	0	0
$86$	1779.00	2.23063
$87$	1152.62i	1.42039i
$88$	9.60526i	0.0116355i
$89$	964.896	1.14920	0.574600	−	0.818435i	$-0.305158\pi$
$89$	964.896	1.14920	0.574600	+	0.818435i	$0.305158\pi$
$90$	0	0
$91$	117.978	0.135907
$92$	− 406.282i	− 0.460411i
$93$	− 1133.34i	− 1.26367i
$94$	449.099	0.492777
$95$	0	0
$96$	1700.87	1.80828
$97$	− 219.495i	− 0.229756i	−0.993380	−	0.114878i	$-0.963352\pi$
$97$	− 219.495i	− 0.229756i	0.993380	−	0.114878i	$-0.0366477\pi$
$98$	1285.89i	1.32546i
$99$	147.278	0.149515
$100$	0	0
$101$	1447.94	1.42649	0.713247	−	0.700913i	$-0.247224\pi$
$101$	1447.94	1.42649	0.713247	+	0.700913i	$0.247224\pi$
$102$	1665.58i	1.61683i
$103$	− 883.567i	− 0.845247i	−0.906305	−	0.422623i	$-0.861109\pi$
$103$	− 883.567i	− 0.845247i	0.906305	−	0.422623i	$-0.138891\pi$
$104$	5.39223	0.00508415
$105$	0	0
$106$	−615.793	−0.564256
$107$	− 1307.82i	− 1.18160i	−0.806817	−	0.590801i	$-0.798812\pi$
$107$	− 1307.82i	− 1.18160i	0.806817	−	0.590801i	$-0.201188\pi$
$108$	460.064i	0.409904i
$109$	−870.507	−0.764949	−0.382475	−	0.923966i	$-0.624928\pi$
$109$	−870.507	−0.764949	−0.382475	+	0.923966i	$0.624928\pi$
$110$	0	0
$111$	−990.856	−0.847279
$112$	1712.82i	1.44505i
$113$	1181.41i	0.983521i	0.870730	+	0.491761i	$0.163646\pi$
$113$	1181.41i	0.983521i	−0.870730	+	0.491761i	$0.836354\pi$
$114$	−505.648	−0.415423
$115$	0	0
$116$	1321.82	1.05800
$117$	− 82.6792i	− 0.0653307i
$118$	722.564i	0.563707i
$119$	−1616.96	−1.24560
$120$	0	0
$121$	−1264.83	−0.950289
$122$	− 1601.20i	− 1.18824i
$123$	− 2073.76i	− 1.52020i
$124$	−1299.71	−0.941266
$125$	0	0
$126$	1853.65	1.31061
$127$	887.509i	0.620108i	0.950719	+	0.310054i	$0.100347\pi$
$127$	887.509i	0.620108i	−0.950719	+	0.310054i	$0.899653\pi$
$128$	151.044i	0.104301i
$129$	3015.19	2.05793
$130$	0	0
$131$	−2344.76	−1.56384	−0.781920	−	0.623379i	$-0.785759\pi$
$131$	−2344.76	−1.56384	−0.781920	+	0.623379i	$0.785759\pi$
$132$	− 420.761i	− 0.277444i
$133$	− 490.888i	− 0.320040i
$134$	−421.619	−0.271809
$135$	0	0
$136$	−73.9034	−0.0465968
$137$	2244.82i	1.39991i	0.714186	+	0.699956i	$0.246797\pi$
$137$	2244.82i	1.39991i	−0.714186	+	0.699956i	$0.753203\pi$
$138$	− 1403.84i	− 0.865964i
$139$	296.146	0.180711	0.0903554	−	0.995910i	$-0.471200\pi$
$139$	296.146	0.180711	0.0903554	+	0.995910i	$0.471200\pi$
$140$	0	0
$141$	761.170	0.454625
$142$	− 1870.65i	− 1.10550i
$143$	− 37.1440i	− 0.0217212i
$144$	1200.34	0.694642
$145$	0	0
$146$	−3343.70	−1.89539
$147$	2179.44i	1.22284i
$148$	1136.31i	0.631109i
$149$	−1791.09	−0.984780	−0.492390	−	0.870375i	$-0.663877\pi$
$149$	−1791.09	−0.984780	−0.492390	+	0.870375i	$0.663877\pi$
$150$	0	0
$151$	−2352.65	−1.26792	−0.633960	−	0.773366i	$-0.718571\pi$
$151$	−2352.65	−1.26792	−0.633960	+	0.773366i	$0.718571\pi$
$152$	− 22.4361i	− 0.0119724i
$153$	1133.16i	0.598764i
$154$	832.762	0.435752
$155$	0	0
$156$	−236.208	−0.121229
$157$	− 1438.26i	− 0.731118i	−0.930788	−	0.365559i	$-0.880878\pi$
$157$	− 1438.26i	− 0.731118i	0.930788	−	0.365559i	$-0.119122\pi$
$158$	− 2342.02i	− 1.17925i
$159$	−1043.70	−0.520570
$160$	0	0
$161$	1362.86	0.667135
$162$	3526.83i	1.71046i
$163$	− 127.493i	− 0.0612640i	−0.999531	−	0.0306320i	$-0.990248\pi$
$163$	− 127.493i	− 0.0612640i	0.999531	−	0.0306320i	$-0.00975199\pi$
$164$	−2378.18	−1.13234
$165$	0	0
$166$	−1150.68	−0.538014
$167$	3419.05i	1.58428i	0.610341	+	0.792139i	$0.291033\pi$
$167$	3419.05i	1.58428i	−0.610341	+	0.792139i	$0.708967\pi$
$168$	204.899i	0.0940971i
$169$	2176.15	0.990509
$170$	0	0
$171$	−344.014	−0.153844
$172$	− 3457.81i	− 1.53288i
$173$	362.598i	0.159352i	0.996821	+	0.0796758i	$0.0253885\pi$
$173$	362.598i	0.159352i	−0.996821	+	0.0796758i	$0.974611\pi$
$174$	4567.34	1.98994
$175$	0	0
$176$	539.258	0.230955
$177$	1224.66i	0.520063i
$178$	− 3823.47i	− 1.61001i
$179$	−2417.89	−1.00962	−0.504809	−	0.863231i	$-0.668437\pi$
$179$	−2417.89	−1.00962	−0.504809	+	0.863231i	$0.668437\pi$
$180$	0	0
$181$	−2444.64	−1.00391	−0.501957	−	0.864892i	$-0.667387\pi$
$181$	−2444.64	−1.00391	−0.501957	+	0.864892i	$0.667387\pi$
$182$	− 467.498i	− 0.190403i
$183$	− 2713.84i	− 1.09624i
$184$	62.2900	0.0249570
$185$	0	0
$186$	−4490.93	−1.77038
$187$	509.079i	0.199078i
$188$	− 872.908i	− 0.338635i
$189$	−1543.27	−0.593951
$190$	0	0
$191$	−1387.66	−0.525693	−0.262846	−	0.964838i	$-0.584661\pi$
$191$	−1387.66	−0.525693	−0.262846	+	0.964838i	$0.584661\pi$
$192$	− 3177.87i	− 1.19449i
$193$	3208.03i	1.19647i	0.801319	+	0.598237i	$0.204132\pi$
$193$	3208.03i	1.19647i	−0.801319	+	0.598237i	$0.795868\pi$
$194$	−869.764	−0.321884
$195$	0	0
$196$	2499.37	0.910849
$197$	− 3445.36i	− 1.24605i	−0.782202	−	0.623025i	$-0.785903\pi$
$197$	− 3445.36i	− 1.24605i	0.782202	−	0.623025i	$-0.214097\pi$
$198$	− 583.599i	− 0.209468i
$199$	−2025.71	−0.721602	−0.360801	−	0.932643i	$-0.617497\pi$
$199$	−2025.71	−0.721602	−0.360801	+	0.932643i	$0.617497\pi$
$200$	0	0
$201$	−714.595	−0.250764
$202$	− 5737.59i	− 1.99849i
$203$	4434.02i	1.53304i
$204$	3237.36	1.11108
$205$	0	0
$206$	−3501.20	−1.18418
$207$	− 955.095i	− 0.320694i
$208$	− 302.730i	− 0.100916i
$209$	−154.550	−0.0511504
$210$	0	0
$211$	4309.54	1.40607	0.703036	−	0.711155i	$-0.251827\pi$
$211$	4309.54	1.40607	0.703036	+	0.711155i	$0.251827\pi$
$212$	1196.91i	0.387755i
$213$	− 3170.53i	− 1.01991i
$214$	−5182.33	−1.65540
$215$	0	0
$216$	−70.5357	−0.0222192
$217$	− 4359.84i	− 1.36389i
$218$	3449.45i	1.07168i
$219$	−5667.19	−1.74864
$220$	0	0
$221$	285.788	0.0869873
$222$	3926.34i	1.18702i
$223$	− 825.648i	− 0.247935i	−0.992286	−	0.123968i	$-0.960438\pi$
$223$	− 825.648i	− 0.247935i	0.992286	−	0.123968i	$-0.0395619\pi$
$224$	6543.09	1.95169
$225$	0	0
$226$	4681.43	1.37790
$227$	− 1501.19i	− 0.438931i	−0.975620	−	0.219466i	$-0.929569\pi$
$227$	− 1501.19i	− 0.438931i	0.975620	−	0.219466i	$-0.0704314\pi$
$228$	982.821i	0.285478i
$229$	5250.40	1.51509	0.757547	−	0.652781i	$-0.226398\pi$
$229$	5250.40	1.51509	0.757547	+	0.652781i	$0.226398\pi$
$230$	0	0
$231$	1411.43	0.402015
$232$	202.658i	0.0573497i
$233$	− 2139.06i	− 0.601435i	−0.953713	−	0.300717i	$-0.902774\pi$
$233$	− 2139.06i	− 0.601435i	0.953713	−	0.300717i	$-0.0972261\pi$
$234$	−327.623	−0.0915272
$235$	0	0
$236$	1404.44	0.387377
$237$	− 3969.46i	− 1.08795i
$238$	6407.32i	1.74506i
$239$	−3772.70	−1.02107	−0.510534	−	0.859857i	$-0.670552\pi$
$239$	−3772.70	−1.02107	−0.510534	+	0.859857i	$0.670552\pi$
$240$	0	0
$241$	6415.39	1.71474	0.857369	−	0.514702i	$-0.172097\pi$
$241$	6415.39	1.71474	0.857369	+	0.514702i	$0.172097\pi$
$242$	5012.00i	1.33134i
$243$	4364.77i	1.15226i
$244$	−3112.22	−0.816555
$245$	0	0
$246$	−8217.41	−2.12977
$247$	86.7616i	0.0223502i
$248$	− 199.267i	− 0.0510221i
$249$	−1950.27	−0.496360
$250$	0	0
$251$	−6277.31	−1.57857	−0.789283	−	0.614029i	$-0.789548\pi$
$251$	−6277.31	−1.57857	−0.789283	+	0.614029i	$0.789548\pi$
$252$	− 3602.92i	− 0.900645i
$253$	− 429.081i	− 0.106625i
$254$	3516.82	0.868760
$255$	0	0
$256$	4383.90	1.07029
$257$	− 3183.98i	− 0.772807i	−0.922330	−	0.386404i	$-0.873717\pi$
$257$	− 3183.98i	− 0.772807i	0.922330	−	0.386404i	$-0.126283\pi$
$258$	− 11947.9i	− 2.88312i
$259$	−3811.73	−0.914476
$260$	0	0
$261$	3107.36	0.736938
$262$	9291.31i	2.19091i
$263$	2624.18i	0.615261i	0.951506	+	0.307630i	$0.0995360\pi$
$263$	2624.18i	0.615261i	−0.951506	+	0.307630i	$0.900464\pi$
$264$	64.5099	0.0150391
$265$	0	0
$266$	−1945.18	−0.448371
$267$	− 6480.34i	− 1.48536i
$268$	819.495i	0.186786i
$269$	−7444.76	−1.68742	−0.843708	−	0.536803i	$-0.819632\pi$
$269$	−7444.76	−1.68742	−0.843708	+	0.536803i	$0.819632\pi$
$270$	0	0
$271$	−4004.49	−0.897621	−0.448810	−	0.893627i	$-0.648152\pi$
$271$	−4004.49	−0.897621	−0.448810	+	0.893627i	$0.648152\pi$
$272$	4149.08i	0.924909i
$273$	− 792.355i	− 0.175661i
$274$	8895.27	1.96125
$275$	0	0
$276$	−2728.63	−0.595088
$277$	− 5830.66i	− 1.26473i	−0.774671	−	0.632365i	$-0.782084\pi$
$277$	− 5830.66i	− 1.26473i	0.774671	−	0.632365i	$-0.217916\pi$
$278$	− 1173.50i	− 0.253172i
$279$	−3055.37	−0.655628
$280$	0	0
$281$	−7504.37	−1.59314	−0.796572	−	0.604544i	$-0.793355\pi$
$281$	−7504.37	−1.59314	−0.796572	+	0.604544i	$0.793355\pi$
$282$	− 3016.19i	− 0.636921i
$283$	− 5910.87i	− 1.24157i	−0.783980	−	0.620785i	$-0.786814\pi$
$283$	− 5910.87i	− 1.24157i	0.783980	−	0.620785i	$-0.213186\pi$
$284$	−3635.95	−0.759697
$285$	0	0
$286$	−147.186	−0.0304311
$287$	− 7977.54i	− 1.64076i
$288$	− 4585.40i	− 0.938184i
$289$	996.118	0.202752
$290$	0	0
$291$	−1474.15	−0.296962
$292$	6499.11i	1.30251i
$293$	− 3245.59i	− 0.647131i	−0.946206	−	0.323566i	$-0.895118\pi$
$293$	− 3245.59i	− 0.647131i	0.946206	−	0.323566i	$-0.104882\pi$
$294$	8636.18	1.71317
$295$	0	0
$296$	−174.216	−0.0342098
$297$	485.880i	0.0949280i
$298$	7097.35i	1.37966i
$299$	−240.878	−0.0465898
$300$	0	0
$301$	11599.2	2.22115
$302$	9322.54i	1.77633i
$303$	− 9724.54i	− 1.84376i
$304$	−1259.61	−0.237643
$305$	0	0
$306$	4490.25	0.838857
$307$	7489.14i	1.39227i	0.717909	+	0.696137i	$0.245099\pi$
$307$	7489.14i	1.39227i	−0.717909	+	0.696137i	$0.754901\pi$
$308$	− 1618.63i	− 0.299448i
$309$	−5934.12	−1.09249
$310$	0	0
$311$	2136.71	0.389588	0.194794	−	0.980844i	$-0.437596\pi$
$311$	2136.71	0.389588	0.194794	+	0.980844i	$0.437596\pi$
$312$	− 36.2147i	− 0.00657133i
$313$	− 2212.15i	− 0.399483i	−0.979849	−	0.199742i	$-0.935990\pi$
$313$	− 2212.15i	− 0.399483i	0.979849	−	0.199742i	$-0.0640102\pi$
$314$	−5699.21	−1.02428
$315$	0	0
$316$	−4552.16	−0.810377
$317$	429.326i	0.0760674i	0.999276	+	0.0380337i	$0.0121094\pi$
$317$	429.326i	0.0760674i	−0.999276	+	0.0380337i	$0.987891\pi$
$318$	4135.73i	0.729309i
$319$	1396.00	0.245018
$320$	0	0
$321$	−8783.43	−1.52724
$322$	− 5400.45i	− 0.934644i
$323$	− 1189.11i	− 0.204842i
$324$	6855.05	1.17542
$325$	0	0
$326$	−505.201	−0.0858297
$327$	5846.42i	0.988708i
$328$	− 364.615i	− 0.0613796i
$329$	2928.15	0.490681
$330$	0	0
$331$	−765.454	−0.127109	−0.0635546	−	0.997978i	$-0.520244\pi$
$331$	−765.454	−0.127109	−0.0635546	+	0.997978i	$0.520244\pi$
$332$	2236.57i	0.369722i
$333$	2671.26i	0.439592i
$334$	13548.3	2.21954
$335$	0	0
$336$	11503.4	1.86775
$337$	− 3049.81i	− 0.492978i	−0.969146	−	0.246489i	$-0.920723\pi$
$337$	− 3049.81i	− 0.492978i	0.969146	−	0.246489i	$-0.0792769\pi$
$338$	− 8623.15i	− 1.38768i
$339$	7934.48	1.27121
$340$	0	0
$341$	−1372.64	−0.217984
$342$	1363.18i	0.215533i
$343$	− 477.732i	− 0.0752045i
$344$	530.142	0.0830912
$345$	0	0
$346$	1436.82	0.223249
$347$	− 5907.00i	− 0.913845i	−0.889507	−	0.456922i	$-0.848952\pi$
$347$	− 5907.00i	− 0.913845i	0.889507	−	0.456922i	$-0.151048\pi$
$348$	− 8877.48i	− 1.36748i
$349$	12107.4	1.85700	0.928502	−	0.371327i	$-0.121097\pi$
$349$	12107.4	1.85700	0.928502	+	0.371327i	$0.121097\pi$
$350$	0	0
$351$	272.765	0.0414789
$352$	− 2060.01i	− 0.311929i
$353$	2420.40i	0.364943i	0.983211	+	0.182471i	$0.0584097\pi$
$353$	2420.40i	0.364943i	−0.983211	+	0.182471i	$0.941590\pi$
$354$	4852.81	0.728599
$355$	0	0
$356$	−7431.63	−1.10639
$357$	10859.7i	1.60995i
$358$	9581.07i	1.41446i
$359$	1455.80	0.214023	0.107011	−	0.994258i	$-0.465872\pi$
$359$	1455.80	0.214023	0.107011	+	0.994258i	$0.465872\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	9687.06i	1.40647i
$363$	8494.76i	1.22826i
$364$	−908.670	−0.130844
$365$	0	0
$366$	−10753.8	−1.53582
$367$	8783.80i	1.24935i	0.780886	+	0.624674i	$0.214768\pi$
$367$	8783.80i	1.24935i	−0.780886	+	0.624674i	$0.785232\pi$
$368$	− 3497.08i	− 0.495375i
$369$	−5590.66	−0.788721
$370$	0	0
$371$	−4015.00	−0.561856
$372$	8728.95i	1.21660i
$373$	9199.84i	1.27708i	0.769590	+	0.638538i	$0.220461\pi$
$373$	9199.84i	1.27708i	−0.769590	+	0.638538i	$0.779539\pi$
$374$	2017.26	0.278904
$375$	0	0
$376$	133.832	0.0183560
$377$	− 783.688i	− 0.107061i
$378$	6115.34i	0.832114i
$379$	6161.38	0.835063	0.417531	−	0.908662i	$-0.362895\pi$
$379$	6161.38	0.835063	0.417531	+	0.908662i	$0.362895\pi$
$380$	0	0
$381$	5960.60	0.801498
$382$	5498.70i	0.736486i
$383$	− 2630.79i	− 0.350985i	−0.984481	−	0.175492i	$-0.943848\pi$
$383$	− 2630.79i	− 0.350985i	0.984481	−	0.175492i	$-0.0561518\pi$
$384$	1014.43	0.134810
$385$	0	0
$386$	12712.1	1.67624
$387$	− 8128.69i	− 1.06771i
$388$	1690.55i	0.221197i
$389$	5866.48	0.764633	0.382317	−	0.924031i	$-0.375126\pi$
$389$	5866.48	0.764633	0.382317	+	0.924031i	$0.375126\pi$
$390$	0	0
$391$	3301.37	0.427001
$392$	383.196i	0.0493733i
$393$	15747.7i	2.02129i
$394$	−13652.5	−1.74569
$395$	0	0
$396$	−1134.33	−0.143945
$397$	− 14254.0i	− 1.80199i	−0.433833	−	0.900993i	$-0.642839\pi$
$397$	− 14254.0i	− 1.80199i	0.433833	−	0.900993i	$-0.357161\pi$
$398$	8027.03i	1.01095i
$399$	−3296.85	−0.413657
$400$	0	0
$401$	9909.27	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$401$	9909.27	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$402$	2831.64i	0.351316i
$403$	770.576i	0.0952484i
$404$	−11152.1	−1.37336
$405$	0	0
$406$	17570.1	2.14776
$407$	1200.07i	0.146156i
$408$	496.343i	0.0602270i
$409$	−5805.87	−0.701912	−0.350956	−	0.936392i	$-0.614143\pi$
$409$	−5805.87	−0.701912	−0.350956	+	0.936392i	$0.614143\pi$
$410$	0	0
$411$	15076.4	1.80941
$412$	6805.23i	0.813761i
$413$	4711.15i	0.561309i
$414$	−3784.64	−0.449287
$415$	0	0
$416$	−1156.45	−0.136298
$417$	− 1988.95i	− 0.233571i
$418$	612.415i	0.0716608i
$419$	−12260.9	−1.42955	−0.714777	−	0.699353i	$-0.753472\pi$
$419$	−12260.9	−1.42955	−0.714777	+	0.699353i	$0.753472\pi$
$420$	0	0
$421$	5837.85	0.675818	0.337909	−	0.941179i	$-0.390280\pi$
$421$	5837.85	0.675818	0.337909	+	0.941179i	$0.390280\pi$
$422$	− 17076.9i	− 1.96988i
$423$	− 2052.05i	− 0.235872i
$424$	−183.507	−0.0210186
$425$	0	0
$426$	−12563.5	−1.42888
$427$	− 10439.9i	− 1.18319i
$428$	10072.8i	1.13759i
$429$	−249.463	−0.0280750
$430$	0	0
$431$	−2770.16	−0.309591	−0.154796	−	0.987946i	$-0.549472\pi$
$431$	−2770.16	−0.309591	−0.154796	+	0.987946i	$0.549472\pi$
$432$	3960.01i	0.441033i
$433$	− 5663.00i	− 0.628513i	−0.949338	−	0.314257i	$-0.898245\pi$
$433$	− 5663.00i	− 0.628513i	0.949338	−	0.314257i	$-0.101755\pi$
$434$	−17276.2	−1.91079
$435$	0	0
$436$	6704.65	0.736455
$437$	1002.25i	0.109712i
$438$	22456.6i	2.44982i
$439$	8399.20	0.913148	0.456574	−	0.889685i	$-0.349076\pi$
$439$	8399.20	0.913148	0.456574	+	0.889685i	$0.349076\pi$
$440$	0	0
$441$	5875.56	0.634442
$442$	− 1132.46i	− 0.121868i
$443$	− 6154.68i	− 0.660085i	−0.943966	−	0.330043i	$-0.892937\pi$
$443$	− 6154.68i	− 0.660085i	0.943966	−	0.330043i	$-0.107063\pi$
$444$	7631.58	0.815717
$445$	0	0
$446$	−3271.69	−0.347352
$447$	12029.2i	1.27284i
$448$	− 12225.0i	− 1.28923i
$449$	−3445.03	−0.362095	−0.181048	−	0.983474i	$-0.557949\pi$
$449$	−3445.03	−0.362095	−0.181048	+	0.983474i	$0.557949\pi$
$450$	0	0
$451$	−2511.63	−0.262235
$452$	− 9099.24i	− 0.946885i
$453$	15800.6i	1.63880i
$454$	−5948.57	−0.614935
$455$	0	0
$456$	−150.683	−0.0154745
$457$	− 502.346i	− 0.0514196i	−0.999669	−	0.0257098i	$-0.991815\pi$
$457$	− 502.346i	− 0.0514196i	0.999669	−	0.0257098i	$-0.00818459\pi$
$458$	− 20805.1i	− 2.12262i
$459$	−3738.39	−0.380159
$460$	0	0
$461$	546.259	0.0551883	0.0275942	−	0.999619i	$-0.491215\pi$
$461$	546.259	0.0551883	0.0275942	+	0.999619i	$0.491215\pi$
$462$	− 5592.91i	− 0.563216i
$463$	− 18540.2i	− 1.86098i	−0.366316	−	0.930490i	$-0.619381\pi$
$463$	− 18540.2i	− 1.86098i	0.366316	−	0.930490i	$-0.380619\pi$
$464$	11377.6	1.13835
$465$	0	0
$466$	−8476.18	−0.842599
$467$	12475.1i	1.23614i	0.786123	+	0.618070i	$0.212085\pi$
$467$	12475.1i	1.23614i	−0.786123	+	0.618070i	$0.787915\pi$
$468$	636.795i	0.0628972i
$469$	−2748.98	−0.270653
$470$	0	0
$471$	−9659.49	−0.944981
$472$	215.324i	0.0209981i
$473$	− 3651.85i	− 0.354994i
$474$	−15729.3	−1.52420
$475$	0	0
$476$	12453.8	1.19920
$477$	2813.71i	0.270086i
$478$	14949.6i	1.43050i
$479$	10569.2	1.00818	0.504091	−	0.863651i	$-0.331828\pi$
$479$	10569.2	1.00818	0.504091	+	0.863651i	$0.331828\pi$
$480$	0	0
$481$	673.701	0.0638631
$482$	− 25421.5i	− 2.40232i
$483$	− 9153.13i	− 0.862282i
$484$	9741.76	0.914891
$485$	0	0
$486$	17295.7	1.61430
$487$	− 11227.9i	− 1.04473i	−0.852721	−	0.522366i	$-0.825049\pi$
$487$	− 11227.9i	− 1.04473i	0.852721	−	0.522366i	$-0.174951\pi$
$488$	− 477.157i	− 0.0442621i
$489$	−856.256	−0.0791846
$490$	0	0
$491$	−536.840	−0.0493427	−0.0246713	−	0.999696i	$-0.507854\pi$
$491$	−536.840	−0.0493427	−0.0246713	+	0.999696i	$0.507854\pi$
$492$	15972.1i	1.46357i
$493$	10740.9i	0.981226i
$494$	343.799	0.0313123
$495$	0	0
$496$	−11187.3	−1.01275
$497$	− 12196.7i	− 1.10080i
$498$	7728.11i	0.695391i
$499$	−1319.91	−0.118412	−0.0592058	−	0.998246i	$-0.518857\pi$
$499$	−1319.91	−0.118412	−0.0592058	+	0.998246i	$0.518857\pi$
$500$	0	0
$501$	22962.7	2.04770
$502$	24874.3i	2.21154i
$503$	− 1749.27i	− 0.155062i	−0.996990	−	0.0775310i	$-0.975296\pi$
$503$	− 1749.27i	− 0.155062i	0.996990	−	0.0775310i	$-0.0247037\pi$
$504$	552.390	0.0488202
$505$	0	0
$506$	−1700.26	−0.149379
$507$	− 14615.2i	− 1.28025i
$508$	− 6835.59i	− 0.597009i
$509$	−1882.19	−0.163903	−0.0819516	−	0.996636i	$-0.526115\pi$
$509$	−1882.19	−0.163903	−0.0819516	+	0.996636i	$0.526115\pi$
$510$	0	0
$511$	−21801.1	−1.88733
$512$	− 16163.2i	− 1.39515i
$513$	− 1134.93i	− 0.0976769i
$514$	−12616.8	−1.08269
$515$	0	0
$516$	−23223.0	−1.98127
$517$	− 921.891i	− 0.0784231i
$518$	15104.3i	1.28116i
$519$	2435.25	0.205964
$520$	0	0
$521$	−3238.50	−0.272325	−0.136163	−	0.990686i	$-0.543477\pi$
$521$	−3238.50	−0.272325	−0.136163	+	0.990686i	$0.543477\pi$
$522$	− 12313.1i	− 1.03244i
$523$	− 99.0144i	− 0.00827839i	−0.999991	−	0.00413919i	$-0.998682\pi$
$523$	− 99.0144i	− 0.00827839i	0.999991	−	0.00413919i	$-0.00131755\pi$
$524$	18059.4	1.50559
$525$	0	0
$526$	10398.5	0.861969
$527$	− 10561.2i	− 0.872963i
$528$	− 3621.71i	− 0.298513i
$529$	9384.42	0.771301
$530$	0	0
$531$	3301.57	0.269823
$532$	3780.82i	0.308119i
$533$	1409.98i	0.114584i
$534$	−25678.8	−2.08096
$535$	0	0
$536$	−125.643	−0.0101249
$537$	16238.8i	1.30495i
$538$	29500.4i	2.36404i
$539$	2639.62	0.210940
$540$	0	0
$541$	17183.7	1.36559	0.682794	−	0.730611i	$-0.260765\pi$
$541$	17183.7	1.36559	0.682794	+	0.730611i	$0.260765\pi$
$542$	15868.1i	1.25755i
$543$	16418.4i	1.29757i
$544$	15849.8	1.24918
$545$	0	0
$546$	−3139.77	−0.246098
$547$	− 1965.86i	− 0.153664i	−0.997044	−	0.0768319i	$-0.975520\pi$
$547$	− 1965.86i	− 0.153664i	0.997044	−	0.0768319i	$-0.0244805\pi$
$548$	− 17289.6i	− 1.34777i
$549$	−7316.27	−0.568762
$550$	0	0
$551$	−3260.79	−0.252113
$552$	− 418.346i	− 0.0322572i
$553$	− 15270.1i	− 1.17423i
$554$	−23104.4	−1.77186
$555$	0	0
$556$	−2280.92	−0.173979
$557$	6039.93i	0.459461i	0.973254	+	0.229731i	$0.0737845\pi$
$557$	6039.93i	0.459461i	−0.973254	+	0.229731i	$0.926215\pi$
$558$	12107.1i	0.918523i
$559$	−2050.09	−0.155115
$560$	0	0
$561$	3419.02	0.257311
$562$	29736.6i	2.23197i
$563$	− 5260.06i	− 0.393757i	−0.980428	−	0.196878i	$-0.936920\pi$
$563$	− 5260.06i	− 0.393757i	0.980428	−	0.196878i	$-0.0630804\pi$
$564$	−5862.53	−0.437690
$565$	0	0
$566$	−23422.3	−1.73942
$567$	22995.1i	1.70318i
$568$	− 557.454i	− 0.0411800i
$569$	−20567.4	−1.51534	−0.757672	−	0.652635i	$-0.773663\pi$
$569$	−20567.4	−1.51534	−0.757672	+	0.652635i	$0.773663\pi$
$570$	0	0
$571$	11462.4	0.840080	0.420040	−	0.907506i	$-0.362016\pi$
$571$	11462.4	0.840080	0.420040	+	0.907506i	$0.362016\pi$
$572$	286.083i	0.0209121i
$573$	9319.64i	0.679466i
$574$	−31611.6	−2.29868
$575$	0	0
$576$	−8567.25	−0.619737
$577$	− 27029.6i	− 1.95019i	−0.221790	−	0.975094i	$-0.571190\pi$
$577$	− 27029.6i	− 1.95019i	0.221790	−	0.975094i	$-0.428810\pi$
$578$	− 3947.19i	− 0.284051i
$579$	21545.5	1.54646
$580$	0	0
$581$	−7502.52	−0.535726
$582$	5841.42i	0.416039i
$583$	1264.07i	0.0897986i
$584$	−996.425	−0.0706034
$585$	0	0
$586$	−12860.9	−0.906619
$587$	15200.4i	1.06881i	0.845230	+	0.534403i	$0.179464\pi$
$587$	15200.4i	1.06881i	−0.845230	+	0.534403i	$0.820536\pi$
$588$	− 16786.0i	− 1.17729i
$589$	3206.23	0.224296
$590$	0	0
$591$	−23139.4	−1.61054
$592$	9780.83i	0.679036i
$593$	19026.7i	1.31759i	0.752322	+	0.658796i	$0.228934\pi$
$593$	19026.7i	1.31759i	−0.752322	+	0.658796i	$0.771066\pi$
$594$	1925.34	0.132992
$595$	0	0
$596$	13795.0	0.948096
$597$	13604.9i	0.932681i
$598$	954.499i	0.0652715i
$599$	−3927.31	−0.267889	−0.133945	−	0.990989i	$-0.542764\pi$
$599$	−3927.31	−0.267889	−0.133945	+	0.990989i	$0.542764\pi$
$600$	0	0
$601$	13718.1	0.931069	0.465534	−	0.885030i	$-0.345862\pi$
$601$	13718.1	0.931069	0.465534	+	0.885030i	$0.345862\pi$
$602$	− 45962.6i	− 3.11178i
$603$	1926.48i	0.130104i
$604$	18120.1	1.22069
$605$	0	0
$606$	−38534.2	−2.58308
$607$	26461.5i	1.76942i	0.466138	+	0.884712i	$0.345645\pi$
$607$	26461.5i	1.76942i	−0.466138	+	0.884712i	$0.654355\pi$
$608$	4811.81i	0.320961i
$609$	29779.3	1.98148
$610$	0	0
$611$	−517.534	−0.0342671
$612$	− 8727.63i	− 0.576460i
$613$	233.384i	0.0153773i	0.999970	+	0.00768865i	$0.00244740\pi$
$613$	233.384i	0.0153773i	−0.999970	+	0.00768865i	$0.997553\pi$
$614$	29676.3	1.95055
$615$	0	0
$616$	248.163	0.0162318
$617$	4202.77i	0.274225i	0.990555	+	0.137113i	$0.0437822\pi$
$617$	4202.77i	0.274225i	−0.990555	+	0.137113i	$0.956218\pi$
$618$	23514.4i	1.53056i
$619$	23009.4	1.49407	0.747033	−	0.664787i	$-0.231478\pi$
$619$	23009.4	1.49407	0.747033	+	0.664787i	$0.231478\pi$
$620$	0	0
$621$	3150.93	0.203611
$622$	− 8466.89i	− 0.545806i
$623$	− 24929.2i	− 1.60316i
$624$	−2033.17	−0.130436
$625$	0	0
$626$	−8765.82	−0.559668
$627$	1037.97i	0.0661126i
$628$	11077.5i	0.703884i
$629$	−9233.45	−0.585313
$630$	0	0
$631$	18819.3	1.18730	0.593650	−	0.804723i	$-0.297686\pi$
$631$	18819.3	1.18730	0.593650	+	0.804723i	$0.297686\pi$
$632$	− 697.924i	− 0.0439271i
$633$	− 28943.3i	− 1.81737i
$634$	1701.24	0.106569
$635$	0	0
$636$	8038.56	0.501178
$637$	− 1481.84i	− 0.0921705i
$638$	− 5531.74i	− 0.343266i
$639$	−8547.46	−0.529159
$640$	0	0
$641$	25253.7	1.55610	0.778052	−	0.628200i	$-0.216208\pi$
$641$	25253.7	1.55610	0.778052	+	0.628200i	$0.216208\pi$
$642$	34805.0i	2.13963i
$643$	− 11712.1i	− 0.718324i	−0.933275	−	0.359162i	$-0.883063\pi$
$643$	− 11712.1i	− 0.718324i	0.933275	−	0.359162i	$-0.116937\pi$
$644$	−10496.8	−0.642284
$645$	0	0
$646$	−4711.96	−0.286981
$647$	− 26533.3i	− 1.61226i	−0.591737	−	0.806131i	$-0.701558\pi$
$647$	− 26533.3i	− 1.61226i	0.591737	−	0.806131i	$-0.298442\pi$
$648$	1051.00i	0.0637146i
$649$	1483.25	0.0897111
$650$	0	0
$651$	−29281.1	−1.76285
$652$	981.952i	0.0589819i
$653$	− 27898.9i	− 1.67193i	−0.548785	−	0.835964i	$-0.684909\pi$
$653$	− 27898.9i	− 1.67193i	0.548785	−	0.835964i	$-0.315091\pi$
$654$	23166.9	1.38516
$655$	0	0
$656$	−20470.2	−1.21834
$657$	15278.2i	0.907245i
$658$	− 11603.0i	− 0.687436i
$659$	1274.66	0.0753468	0.0376734	−	0.999290i	$-0.488005\pi$
$659$	1274.66	0.0753468	0.0376734	+	0.999290i	$0.488005\pi$
$660$	0	0
$661$	−5049.52	−0.297131	−0.148565	−	0.988903i	$-0.547466\pi$
$661$	−5049.52	−0.297131	−0.148565	+	0.988903i	$0.547466\pi$
$662$	3033.17i	0.178078i
$663$	− 1919.38i	− 0.112432i
$664$	−342.904	−0.0200411
$665$	0	0
$666$	10585.1	0.615860
$667$	− 9053.01i	− 0.525538i
$668$	− 26333.6i	− 1.52526i
$669$	−5545.14	−0.320460
$670$	0	0
$671$	−3286.86	−0.189103
$672$	− 43944.1i	− 2.52259i
$673$	− 8398.64i	− 0.481045i	−0.970644	−	0.240523i	$-0.922681\pi$
$673$	− 8398.64i	− 0.481045i	0.970644	−	0.240523i	$-0.0773189\pi$
$674$	−12085.1	−0.690653
$675$	0	0
$676$	−16760.7	−0.953612
$677$	9875.31i	0.560619i	0.959910	+	0.280309i	$0.0904371\pi$
$677$	9875.31i	0.560619i	−0.959910	+	0.280309i	$0.909563\pi$
$678$	− 31441.0i	− 1.78095i
$679$	−5670.91	−0.320515
$680$	0	0
$681$	−10082.1	−0.567325
$682$	5439.18i	0.305392i
$683$	8653.78i	0.484814i	0.970175	+	0.242407i	$0.0779369\pi$
$683$	8653.78i	0.484814i	−0.970175	+	0.242407i	$0.922063\pi$
$684$	2649.60	0.148114
$685$	0	0
$686$	−1893.05	−0.105360
$687$	− 35262.2i	− 1.95828i
$688$	− 29763.2i	− 1.64929i
$689$	709.629	0.0392376
$690$	0	0
$691$	2916.50	0.160563	0.0802813	−	0.996772i	$-0.474418\pi$
$691$	2916.50	0.160563	0.0802813	+	0.996772i	$0.474418\pi$
$692$	− 2792.73i	− 0.153416i
$693$	− 3805.10i	− 0.208577i
$694$	−23406.9	−1.28028
$695$	0	0
$696$	1361.07	0.0741254
$697$	− 19324.6i	− 1.05017i
$698$	− 47976.5i	− 2.60163i
$699$	−14366.1	−0.777363
$700$	0	0
$701$	−9070.78	−0.488729	−0.244364	−	0.969683i	$-0.578579\pi$
$701$	−9070.78	−0.488729	−0.244364	+	0.969683i	$0.578579\pi$
$702$	− 1080.85i	− 0.0581112i
$703$	− 2803.16i	− 0.150388i
$704$	−3848.88	−0.206051
$705$	0	0
$706$	9591.00	0.511278
$707$	− 37409.4i	− 1.98999i
$708$	− 9432.34i	− 0.500690i
$709$	5957.11	0.315549	0.157774	−	0.987475i	$-0.449568\pi$
$709$	5957.11	0.315549	0.157774	+	0.987475i	$0.449568\pi$
$710$	0	0
$711$	−10701.3	−0.564459
$712$	− 1139.40i	− 0.0599729i
$713$	8901.55i	0.467553i
$714$	43032.2	2.25552
$715$	0	0
$716$	18622.6	0.972010
$717$	25337.8i	1.31975i
$718$	− 5768.71i	− 0.299842i
$719$	31140.8	1.61524	0.807620	−	0.589703i	$-0.200755\pi$
$719$	31140.8	1.61524	0.807620	+	0.589703i	$0.200755\pi$
$720$	0	0
$721$	−22828.0	−1.17914
$722$	− 1430.49i	− 0.0737359i
$723$	− 43086.4i	− 2.21632i
$724$	18828.6	0.966519
$725$	0	0
$726$	33661.1	1.72077
$727$	14969.7i	0.763682i	0.924228	+	0.381841i	$0.124710\pi$
$727$	14969.7i	0.763682i	−0.924228	+	0.381841i	$0.875290\pi$
$728$	− 139.315i	− 0.00709251i
$729$	5283.30	0.268420
$730$	0	0
$731$	28097.5	1.42165
$732$	20902.0i	1.05541i
$733$	12414.1i	0.625545i	0.949828	+	0.312772i	$0.101258\pi$
$733$	12414.1i	0.625545i	−0.949828	+	0.312772i	$0.898742\pi$
$734$	34806.5	1.75031
$735$	0	0
$736$	−13359.1	−0.669055
$737$	865.481i	0.0432570i
$738$	22153.4i	1.10498i
$739$	−1324.11	−0.0659111	−0.0329555	−	0.999457i	$-0.510492\pi$
$739$	−1324.11	−0.0659111	−0.0329555	+	0.999457i	$0.510492\pi$
$740$	0	0
$741$	582.700	0.0288880
$742$	15909.8i	0.787150i
$743$	4391.55i	0.216838i	0.994105	+	0.108419i	$0.0345787\pi$
$743$	4391.55i	0.216838i	−0.994105	+	0.108419i	$0.965421\pi$
$744$	−1338.30	−0.0659468
$745$	0	0
$746$	36455.0	1.78916
$747$	5257.76i	0.257525i
$748$	− 3920.92i	− 0.191662i
$749$	−33789.0	−1.64836
$750$	0	0
$751$	−31947.5	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$751$	−31947.5	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$752$	− 7513.58i	− 0.364351i
$753$	42159.0i	2.04032i
$754$	−3105.42	−0.149990
$755$	0	0
$756$	11886.3	0.571826
$757$	18569.8i	0.891585i	0.895136	+	0.445793i	$0.147078\pi$
$757$	18569.8i	0.891585i	−0.895136	+	0.445793i	$0.852922\pi$
$758$	− 24414.9i	− 1.16991i
$759$	−2881.75	−0.137814
$760$	0	0
$761$	5507.32	0.262339	0.131170	−	0.991360i	$-0.458127\pi$
$761$	5507.32	0.262339	0.131170	+	0.991360i	$0.458127\pi$
$762$	− 23619.3i	− 1.12288i
$763$	22490.6i	1.06712i
$764$	10687.7	0.506111
$765$	0	0
$766$	−10424.7	−0.491723
$767$	− 832.669i	− 0.0391994i
$768$	− 29442.7i	− 1.38336i
$769$	14977.9	0.702362	0.351181	−	0.936308i	$-0.385780\pi$
$769$	14977.9	0.702362	0.351181	+	0.936308i	$0.385780\pi$
$770$	0	0
$771$	−21384.0	−0.998864
$772$	− 24708.3i	− 1.15190i
$773$	− 19545.6i	− 0.909450i	−0.890632	−	0.454725i	$-0.849738\pi$
$773$	− 19545.6i	− 0.909450i	0.890632	−	0.454725i	$-0.150262\pi$
$774$	−32210.5	−1.49585
$775$	0	0
$776$	−259.190	−0.0119902
$777$	25600.0i	1.18197i
$778$	− 23246.4i	− 1.07124i
$779$	5866.70	0.269829
$780$	0	0
$781$	−3839.98	−0.175935
$782$	− 13081.9i	− 0.598221i
$783$	10251.4i	0.467887i
$784$	21513.4	0.980020
$785$	0	0
$786$	62401.3	2.83178
$787$	− 4274.62i	− 0.193613i	−0.995303	−	0.0968067i	$-0.969137\pi$
$787$	− 4274.62i	− 0.193613i	0.995303	−	0.0968067i	$-0.0308629\pi$
$788$	26536.2i	1.19963i
$789$	17624.2	0.795233
$790$	0	0
$791$	30523.2	1.37204
$792$	− 173.913i	− 0.00780268i
$793$	1845.19i	0.0826287i
$794$	−56482.6	−2.52455
$795$	0	0
$796$	15602.0	0.694722
$797$	− 25450.6i	− 1.13112i	−0.824706	−	0.565562i	$-0.808659\pi$
$797$	− 25450.6i	− 1.13112i	0.824706	−	0.565562i	$-0.191341\pi$
$798$	13064.0i	0.579525i
$799$	7093.08	0.314062
$800$	0	0
$801$	−17470.4	−0.770645
$802$	− 39266.2i	− 1.72885i
$803$	6863.81i	0.301642i
$804$	5503.81	0.241423
$805$	0	0
$806$	3053.46	0.133441
$807$	49999.7i	2.18101i
$808$	− 1709.80i	− 0.0744439i
$809$	−4002.04	−0.173924	−0.0869619	−	0.996212i	$-0.527716\pi$
$809$	−4002.04	−0.173924	−0.0869619	+	0.996212i	$0.527716\pi$
$810$	0	0
$811$	−37915.1	−1.64165	−0.820826	−	0.571179i	$-0.806486\pi$
$811$	−37915.1	−1.64165	−0.820826	+	0.571179i	$0.806486\pi$
$812$	− 34150.8i	− 1.47593i
$813$	26894.5i	1.16019i
$814$	4755.39	0.204762
$815$	0	0
$816$	27865.7	1.19546
$817$	8530.05i	0.365274i
$818$	23006.2i	0.983366i
$819$	−2136.12	−0.0911379
$820$	0	0
$821$	4739.43	0.201470	0.100735	−	0.994913i	$-0.467881\pi$
$821$	4739.43	0.201470	0.100735	+	0.994913i	$0.467881\pi$
$822$	− 59741.5i	− 2.53494i
$823$	− 20752.2i	− 0.878952i	−0.898254	−	0.439476i	$-0.855164\pi$
$823$	− 20752.2i	− 0.878952i	0.898254	−	0.439476i	$-0.144836\pi$
$824$	−1043.36	−0.0441106
$825$	0	0
$826$	18668.3	0.786384
$827$	34264.8i	1.44075i	0.693583	+	0.720377i	$0.256031\pi$
$827$	34264.8i	1.44075i	−0.693583	+	0.720377i	$0.743969\pi$
$828$	7356.14i	0.308748i
$829$	39707.5	1.66357	0.831784	−	0.555100i	$-0.187320\pi$
$829$	39707.5	1.66357	0.831784	+	0.555100i	$0.187320\pi$
$830$	0	0
$831$	−39159.3	−1.63468
$832$	2160.69i	0.0900343i
$833$	20309.4i	0.844753i
$834$	−7881.35	−0.327229
$835$	0	0
$836$	1190.34	0.0492450
$837$	− 10079.9i	− 0.416263i
$838$	48584.6i	2.00278i
$839$	4524.04	0.186159	0.0930794	−	0.995659i	$-0.470329\pi$
$839$	4524.04	0.186159	0.0930794	+	0.995659i	$0.470329\pi$
$840$	0	0
$841$	5064.59	0.207659
$842$	− 23132.9i	− 0.946809i
$843$	50400.1i	2.05916i
$844$	−33192.1	−1.35370
$845$	0	0
$846$	−8131.39	−0.330453
$847$	32678.5i	1.32568i
$848$	10302.4i	0.417201i
$849$	−39698.0	−1.60475
$850$	0	0
$851$	7782.47	0.313490
$852$	24419.4i	0.981920i
$853$	7595.54i	0.304884i	0.988312	+	0.152442i	$0.0487138\pi$
$853$	7595.54i	0.304884i	−0.988312	+	0.152442i	$0.951286\pi$
$854$	−41368.8	−1.65762
$855$	0	0
$856$	−1544.34	−0.0616639
$857$	− 19528.9i	− 0.778405i	−0.921152	−	0.389203i	$-0.872751\pi$
$857$	− 19528.9i	− 0.778405i	0.921152	−	0.389203i	$-0.127249\pi$
$858$	988.515i	0.0393326i
$859$	−25980.8	−1.03196	−0.515979	−	0.856601i	$-0.672572\pi$
$859$	−25980.8	−1.03196	−0.515979	+	0.856601i	$0.672572\pi$
$860$	0	0
$861$	−53578.0	−2.12071
$862$	10977.0i	0.433732i
$863$	48294.6i	1.90494i	0.304625	+	0.952472i	$0.401469\pi$
$863$	48294.6i	1.90494i	−0.304625	+	0.952472i	$0.598531\pi$
$864$	15127.6	0.595660
$865$	0	0
$866$	−22440.1	−0.880536
$867$	− 6690.03i	− 0.262059i
$868$	33579.4i	1.31309i
$869$	−4807.61	−0.187672
$870$	0	0
$871$	485.866	0.0189012
$872$	1027.94i	0.0399201i
$873$	3974.17i	0.154072i
$874$	3971.51	0.153705
$875$	0	0
$876$	43648.7	1.68351
$877$	44377.5i	1.70869i	0.519705	+	0.854346i	$0.326042\pi$
$877$	44377.5i	1.70869i	−0.519705	+	0.854346i	$0.673958\pi$
$878$	− 33282.5i	− 1.27930i
$879$	−21797.7	−0.836426
$880$	0	0
$881$	−12139.4	−0.464231	−0.232116	−	0.972688i	$-0.574565\pi$
$881$	−12139.4	−0.464231	−0.232116	+	0.972688i	$0.574565\pi$
$882$	− 23282.4i	− 0.888841i
$883$	− 5048.07i	− 0.192391i	−0.995362	−	0.0961954i	$-0.969333\pi$
$883$	− 5048.07i	− 0.192391i	0.995362	−	0.0961954i	$-0.0306674\pi$
$884$	−2201.14	−0.0837470
$885$	0	0
$886$	−24388.4	−0.924767
$887$	20373.4i	0.771221i	0.922662	+	0.385610i	$0.126009\pi$
$887$	20373.4i	0.771221i	−0.922662	+	0.385610i	$0.873991\pi$
$888$	1170.05i	0.0442166i
$889$	22929.9	0.865065
$890$	0	0
$891$	7239.72	0.272211
$892$	6359.14i	0.238699i
$893$	2153.37i	0.0806940i
$894$	47666.5	1.78323
$895$	0	0
$896$	3902.40	0.145502
$897$	1617.76i	0.0602180i
$898$	13651.2i	0.507289i
$899$	−28960.8	−1.07441
$900$	0	0
$901$	−9725.85	−0.359617
$902$	9952.52i	0.367386i
$903$	− 77901.2i	− 2.87086i
$904$	1395.07	0.0513267
$905$	0	0
$906$	62611.1	2.29593
$907$	− 7456.13i	− 0.272962i	−0.990643	−	0.136481i	$-0.956421\pi$
$907$	− 7456.13i	− 0.272962i	0.990643	−	0.136481i	$-0.0435793\pi$
$908$	11562.2i	0.422581i
$909$	−26216.5	−0.956596
$910$	0	0
$911$	−10653.2	−0.387440	−0.193720	−	0.981057i	$-0.562055\pi$
$911$	−10653.2	−0.387440	−0.193720	+	0.981057i	$0.562055\pi$
$912$	8459.66i	0.307157i
$913$	2362.07i	0.0856224i
$914$	−1990.59	−0.0720380
$915$	0	0
$916$	−40438.6	−1.45866
$917$	60579.8i	2.18159i
$918$	14813.6i	0.532596i
$919$	12569.7	0.451183	0.225591	−	0.974222i	$-0.427569\pi$
$919$	12569.7	0.451183	0.225591	+	0.974222i	$0.427569\pi$
$920$	0	0
$921$	50297.8	1.79953
$922$	− 2164.59i	− 0.0773179i
$923$	2155.70i	0.0768752i
$924$	−10870.9	−0.387040
$925$	0	0
$926$	−73466.8	−2.60720
$927$	15997.9i	0.566816i
$928$	− 43463.4i	− 1.53745i
$929$	−4920.06	−0.173759	−0.0868795	−	0.996219i	$-0.527689\pi$
$929$	−4920.06	−0.173759	−0.0868795	+	0.996219i	$0.527689\pi$
$930$	0	0
$931$	−6165.67	−0.217048
$932$	16475.0i	0.579031i
$933$	− 14350.4i	− 0.503548i
$934$	49433.4	1.73181
$935$	0	0
$936$	−97.6317	−0.00340939
$937$	1991.87i	0.0694465i	0.999397	+	0.0347233i	$0.0110550\pi$
$937$	1991.87i	0.0694465i	−0.999397	+	0.0347233i	$0.988945\pi$
$938$	10893.0i	0.379179i
$939$	−14857.0	−0.516337
$940$	0	0
$941$	−7640.33	−0.264684	−0.132342	−	0.991204i	$-0.542250\pi$
$941$	−7640.33	−0.264684	−0.132342	+	0.991204i	$0.542250\pi$
$942$	38276.5i	1.32390i
$943$	16287.9i	0.562467i
$944$	12088.7	0.416795
$945$	0	0
$946$	−14470.7	−0.497340
$947$	− 6521.15i	− 0.223769i	−0.993721	−	0.111884i	$-0.964311\pi$
$947$	− 6521.15i	− 0.223769i	0.993721	−	0.111884i	$-0.0356886\pi$
$948$	30572.8i	1.04742i
$949$	3853.22	0.131803
$950$	0	0
$951$	2883.40	0.0983182
$952$	1909.38i	0.0650037i
$953$	35757.9i	1.21544i	0.794152	+	0.607719i	$0.207915\pi$
$953$	35757.9i	1.21544i	−0.794152	+	0.607719i	$0.792085\pi$
$954$	11149.6	0.378386
$955$	0	0
$956$	29057.3	0.983034
$957$	− 9375.64i	− 0.316689i
$958$	− 41881.3i	− 1.41244i
$959$	57997.6	1.95291
$960$	0	0
$961$	−1314.75	−0.0441323
$962$	− 2669.59i	− 0.0894710i
$963$	23679.3i	0.792374i
$964$	−49411.4	−1.65086
$965$	0	0
$966$	−36270.0	−1.20804
$967$	− 6342.30i	− 0.210915i	−0.994424	−	0.105457i	$-0.966369\pi$
$967$	− 6342.30i	− 0.210915i	0.994424	−	0.105457i	$-0.0336307\pi$
$968$	1493.58i	0.0495924i
$969$	−7986.21	−0.264762
$970$	0	0
$971$	30351.2	1.00311	0.501553	−	0.865127i	$-0.332762\pi$
$971$	30351.2	1.00311	0.501553	+	0.865127i	$0.332762\pi$
$972$	− 33617.5i	− 1.10934i
$973$	− 7651.29i	− 0.252096i
$974$	−44491.4	−1.46365
$975$	0	0
$976$	−26788.5	−0.878566
$977$	39843.6i	1.30472i	0.757910	+	0.652359i	$0.226220\pi$
$977$	39843.6i	1.30472i	−0.757910	+	0.652359i	$0.773780\pi$
$978$	3392.98i	0.110936i
$979$	−7848.66	−0.256225
$980$	0	0
$981$	15761.4	0.512969
$982$	2127.27i	0.0691282i
$983$	− 24068.7i	− 0.780949i	−0.920614	−	0.390475i	$-0.872311\pi$
$983$	− 24068.7i	− 0.780949i	0.920614	−	0.390475i	$-0.127689\pi$
$984$	−2448.79	−0.0793340
$985$	0	0
$986$	42561.5	1.37468
$987$	− 19665.8i	− 0.634213i
$988$	− 668.238i	− 0.0215177i
$989$	−23682.2	−0.761426
$990$	0	0
$991$	3235.83	0.103723	0.0518615	−	0.998654i	$-0.483485\pi$
$991$	3235.83	0.103723	0.0518615	+	0.998654i	$0.483485\pi$
$992$	42736.2i	1.36782i
$993$	5140.86i	0.164290i
$994$	−48330.4	−1.54220
$995$	0	0
$996$	15021.0	0.477871
$997$	19444.5i	0.617665i	0.951116	+	0.308833i	$0.0999383\pi$
$997$	19444.5i	0.617665i	−0.951116	+	0.308833i	$0.900062\pi$
$998$	5230.25i	0.165892i
$999$	−8812.68	−0.279100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.b.f.324.2		6
5.2	odd	4		475.4.a.f.1.3		3
5.3	odd	4		19.4.a.b.1.1	✓	3
5.4	even	2	inner	475.4.b.f.324.5		6
15.8	even	4		171.4.a.f.1.3		3
20.3	even	4		304.4.a.i.1.1		3
35.13	even	4		931.4.a.c.1.1		3
40.3	even	4		1216.4.a.u.1.3		3
40.13	odd	4		1216.4.a.s.1.1		3
55.43	even	4		2299.4.a.h.1.3		3
95.18	even	4		361.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.1	✓	3	5.3	odd	4
171.4.a.f.1.3		3	15.8	even	4
304.4.a.i.1.1		3	20.3	even	4
361.4.a.i.1.3		3	95.18	even	4
475.4.a.f.1.3		3	5.2	odd	4
475.4.b.f.324.2		6	1.1	even	1	trivial
475.4.b.f.324.5		6	5.4	even	2	inner
931.4.a.c.1.1		3	35.13	even	4
1216.4.a.s.1.1		3	40.13	odd	4
1216.4.a.u.1.3		3	40.3	even	4
2299.4.a.h.1.3		3	55.43	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.96257i q^{2} -6.71610i q^{3} -7.70200 q^{4} -26.6130 q^{6} -25.8362i q^{7} -1.18085i q^{8} -18.1060 q^{9} -8.13420 q^{11} +51.7274i q^{12} +4.56640i q^{13} -102.378 q^{14} -66.2952 q^{16} -62.5850i q^{17} +71.7464i q^{18} +19.0000 q^{19} -173.518 q^{21} +32.2324i q^{22} +52.7502i q^{23} -7.93070 q^{24} +18.0947 q^{26} -59.7330i q^{27} +198.990i q^{28} -171.620 q^{29} +168.749 q^{31} +253.253i q^{32} +54.6301i q^{33} -247.998 q^{34} +139.452 q^{36} -147.534i q^{37} -75.2889i q^{38} +30.6684 q^{39} +308.774 q^{41} +687.580i q^{42} +448.950i q^{43} +62.6496 q^{44} +209.027 q^{46} +113.335i q^{47} +445.245i q^{48} -324.509 q^{49} -420.327 q^{51} -35.1704i q^{52} -155.402i q^{53} -236.696 q^{54} -30.5086 q^{56} -127.606i q^{57} +680.059i q^{58} -182.347 q^{59} +404.080 q^{61} -668.681i q^{62} +467.790i q^{63} +473.172 q^{64} +216.476 q^{66} -106.400i q^{67} +482.030i q^{68} +354.276 q^{69} +472.079 q^{71} +21.3805i q^{72} -843.821i q^{73} -584.616 q^{74} -146.338 q^{76} +210.157i q^{77} -121.526i q^{78} +591.036 q^{79} -890.035 q^{81} -1223.54i q^{82} -290.388i q^{83} +1336.44 q^{84} +1779.00 q^{86} +1152.62i q^{87} +9.60526i q^{88} +964.896 q^{89} +117.978 q^{91} -406.282i q^{92} -1133.34i q^{93} +449.099 q^{94} +1700.87 q^{96} -219.495i q^{97} +1285.89i q^{98} +147.278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 42 q^{4} - 130 q^{6} - 96 q^{9} + 32 q^{11} - 74 q^{14} + 66 q^{16} + 114 q^{19} - 50 q^{21} + 978 q^{24} + 598 q^{26} - 754 q^{29} - 280 q^{31} - 658 q^{34} + 2148 q^{36} + 742 q^{39} + 1912 q^{41}+ \cdots - 500 q^{99}+O(q^{100}) \) 6 * q - 42 * q^4 - 130 * q^6 - 96 * q^9 + 32 * q^11 - 74 * q^14 + 66 * q^16 + 114 * q^19 - 50 * q^21 + 978 * q^24 + 598 * q^26 - 754 * q^29 - 280 * q^31 - 658 * q^34 + 2148 * q^36 + 742 * q^39 + 1912 * q^41 - 220 * q^44 - 478 * q^46 + 196 * q^49 + 374 * q^51 + 1894 * q^54 - 1830 * q^56 - 530 * q^59 + 1976 * q^61 + 318 * q^64 - 548 * q^66 - 1614 * q^69 + 1692 * q^71 + 1540 * q^74 - 798 * q^76 - 764 * q^79 - 1002 * q^81 - 1814 * q^84 + 5184 * q^86 + 344 * q^89 - 914 * q^91 + 192 * q^94 + 1726 * q^96 - 500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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