LMFDB - Embedded newform 575.4.b.j.24.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(24,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, 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("575.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.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:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 $ x^14 + 83x^12 + 2715x^10 + 44273x^8 + 372280x^6 + 1482448x^4 + 2136384x^2 + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.7
Root			$-0.711047i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.4.b.j.24.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.711047i q^{2} +0.689108i q^{3} +7.49441 q^{4} +0.489988 q^{6} -19.0526i q^{7} -11.0173i q^{8} +26.5251 q^{9} +50.7019 q^{11} +5.16446i q^{12} -45.4342i q^{13} -13.5473 q^{14} +52.1215 q^{16} +53.0220i q^{17} -18.8606i q^{18} -58.8778 q^{19} +13.1293 q^{21} -36.0514i q^{22} +23.0000i q^{23} +7.59208 q^{24} -32.3058 q^{26} +36.8846i q^{27} -142.788i q^{28} +70.7345 q^{29} -286.748 q^{31} -125.199i q^{32} +34.9391i q^{33} +37.7011 q^{34} +198.790 q^{36} -175.393i q^{37} +41.8649i q^{38} +31.3091 q^{39} -95.0049 q^{41} -9.33557i q^{42} -148.383i q^{43} +379.981 q^{44} +16.3541 q^{46} +23.5469i q^{47} +35.9174i q^{48} -20.0033 q^{49} -36.5379 q^{51} -340.503i q^{52} -379.219i q^{53} +26.2267 q^{54} -209.908 q^{56} -40.5731i q^{57} -50.2956i q^{58} +741.332 q^{59} +5.17078 q^{61} +203.891i q^{62} -505.374i q^{63} +327.950 q^{64} +24.8433 q^{66} +974.454i q^{67} +397.368i q^{68} -15.8495 q^{69} -336.469 q^{71} -292.234i q^{72} +317.142i q^{73} -124.713 q^{74} -441.254 q^{76} -966.005i q^{77} -22.2622i q^{78} +577.050 q^{79} +690.761 q^{81} +67.5530i q^{82} +225.514i q^{83} +98.3966 q^{84} -105.507 q^{86} +48.7438i q^{87} -558.596i q^{88} +1141.92 q^{89} -865.642 q^{91} +172.371i q^{92} -197.600i q^{93} +16.7429 q^{94} +86.2756 q^{96} -147.766i q^{97} +14.2233i q^{98} +1344.87 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) $ 14 * q - 54 * q^4 - 82 * q^6 - 20 * q^9 - 104 * q^11 + 84 * q^14 - 170 * q^16 + 20 * q^19 - 404 * q^21 + 606 * q^24 - 52 * q^26 + 910 * q^29 - 1380 * q^31 + 1314 * q^34 + 408 * q^36 + 554 * q^39 - 460 * q^41 + 2006 * q^44 - 138 * q^46 + 1604 * q^49 - 1390 * q^51 + 2494 * q^54 - 772 * q^56 + 1102 * q^59 - 2484 * q^61 + 3002 * q^64 + 1974 * q^66 + 46 * q^69 + 372 * q^71 + 3060 * q^74 - 3550 * q^76 + 4006 * q^79 - 730 * q^81 + 2704 * q^84 + 3788 * q^86 + 5258 * q^89 - 2590 * q^91 + 6936 * q^94 - 270 * q^96 + 4286 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 0.711047i	− 0.251393i	−0.992069	−	0.125697i	$-0.959883\pi$
$2$	− 0.711047i	− 0.251393i	0.992069	−	0.125697i	$-0.0401166\pi$
$3$	0.689108i	0.132619i	0.997799	+	0.0663095i	$0.0211225\pi$
$3$	0.689108i	0.132619i	−0.997799	+	0.0663095i	$0.978878\pi$
$4$	7.49441	0.936802
$5$	0	0
$5$	0	0
$6$	0.489988	0.0333395
$7$	− 19.0526i	− 1.02875i	−0.857566	−	0.514373i	$-0.828025\pi$
$7$	− 19.0526i	− 1.02875i	0.857566	−	0.514373i	$-0.171975\pi$
$8$	− 11.0173i	− 0.486899i
$9$	26.5251	0.982412
$10$	0	0
$11$	50.7019	1.38974	0.694872	−	0.719133i	$-0.255461\pi$
$11$	50.7019	1.38974	0.694872	+	0.719133i	$0.255461\pi$
$12$	5.16446i	0.124238i
$13$	− 45.4342i	− 0.969321i	−0.874702	−	0.484661i	$-0.838943\pi$
$13$	− 45.4342i	− 0.969321i	0.874702	−	0.484661i	$-0.161057\pi$
$14$	−13.5473	−0.258620
$15$	0	0
$16$	52.1215	0.814399
$17$	53.0220i	0.756454i	0.925713	+	0.378227i	$0.123466\pi$
$17$	53.0220i	0.756454i	−0.925713	+	0.378227i	$0.876534\pi$
$18$	− 18.8606i	− 0.246972i
$19$	−58.8778	−0.710920	−0.355460	−	0.934691i	$-0.615676\pi$
$19$	−58.8778	−0.710920	−0.355460	+	0.934691i	$0.615676\pi$
$20$	0	0
$21$	13.1293	0.136431
$22$	− 36.0514i	− 0.349372i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	7.59208	0.0645720
$25$	0	0
$26$	−32.3058	−0.243681
$27$	36.8846i	0.262905i
$28$	− 142.788i	− 0.963731i
$29$	70.7345	0.452934	0.226467	−	0.974019i	$-0.427283\pi$
$29$	70.7345	0.452934	0.226467	+	0.974019i	$0.427283\pi$
$30$	0	0
$31$	−286.748	−1.66134	−0.830669	−	0.556766i	$-0.812042\pi$
$31$	−286.748	−1.66134	−0.830669	+	0.556766i	$0.812042\pi$
$32$	− 125.199i	− 0.691633i
$33$	34.9391i	0.184306i
$34$	37.7011	0.190167
$35$	0	0
$36$	198.790	0.920325
$37$	− 175.393i	− 0.779310i	−0.920961	−	0.389655i	$-0.872594\pi$
$37$	− 175.393i	− 0.779310i	0.920961	−	0.389655i	$-0.127406\pi$
$38$	41.8649i	0.178720i
$39$	31.3091	0.128550
$40$	0	0
$41$	−95.0049	−0.361885	−0.180942	−	0.983494i	$-0.557915\pi$
$41$	−95.0049	−0.361885	−0.180942	+	0.983494i	$0.557915\pi$
$42$	− 9.33557i	− 0.0342979i
$43$	− 148.383i	− 0.526237i	−0.964764	−	0.263118i	$-0.915249\pi$
$43$	− 148.383i	− 0.526237i	0.964764	−	0.263118i	$-0.0847510\pi$
$44$	379.981	1.30191
$45$	0	0
$46$	16.3541	0.0524191
$47$	23.5469i	0.0730779i	0.999332	+	0.0365390i	$0.0116333\pi$
$47$	23.5469i	0.0730779i	−0.999332	+	0.0365390i	$0.988367\pi$
$48$	35.9174i	0.108005i
$49$	−20.0033	−0.0583188
$50$	0	0
$51$	−36.5379	−0.100320
$52$	− 340.503i	− 0.908062i
$53$	− 379.219i	− 0.982825i	−0.870927	−	0.491413i	$-0.836481\pi$
$53$	− 379.219i	− 0.982825i	0.870927	−	0.491413i	$-0.163519\pi$
$54$	26.2267	0.0660926
$55$	0	0
$56$	−209.908	−0.500895
$57$	− 40.5731i	− 0.0942815i
$58$	− 50.2956i	− 0.113864i
$59$	741.332	1.63582	0.817908	−	0.575349i	$-0.195134\pi$
$59$	741.332	1.63582	0.817908	+	0.575349i	$0.195134\pi$
$60$	0	0
$61$	5.17078	0.0108533	0.00542664	−	0.999985i	$-0.498273\pi$
$61$	5.17078	0.0108533	0.00542664	+	0.999985i	$0.498273\pi$
$62$	203.891i	0.417649i
$63$	− 505.374i	− 1.01065i
$64$	327.950	0.640527
$65$	0	0
$66$	24.8433	0.0463334
$67$	974.454i	1.77684i	0.459028	+	0.888422i	$0.348198\pi$
$67$	974.454i	1.77684i	−0.459028	+	0.888422i	$0.651802\pi$
$68$	397.368i	0.708647i
$69$	−15.8495	−0.0276530
$70$	0	0
$71$	−336.469	−0.562416	−0.281208	−	0.959647i	$-0.590735\pi$
$71$	−336.469	−0.562416	−0.281208	+	0.959647i	$0.590735\pi$
$72$	− 292.234i	− 0.478335i
$73$	317.142i	0.508474i	0.967142	+	0.254237i	$0.0818244\pi$
$73$	317.142i	0.508474i	−0.967142	+	0.254237i	$0.918176\pi$
$74$	−124.713	−0.195913
$75$	0	0
$76$	−441.254	−0.665991
$77$	− 966.005i	− 1.42969i
$78$	− 22.2622i	− 0.0323167i
$79$	577.050	0.821812	0.410906	−	0.911678i	$-0.365212\pi$
$79$	577.050	0.821812	0.410906	+	0.911678i	$0.365212\pi$
$80$	0	0
$81$	690.761	0.947546
$82$	67.5530i	0.0909754i
$83$	225.514i	0.298234i	0.988820	+	0.149117i	$0.0476430\pi$
$83$	225.514i	0.298234i	−0.988820	+	0.149117i	$0.952357\pi$
$84$	98.3966	0.127809
$85$	0	0
$86$	−105.507	−0.132292
$87$	48.7438i	0.0600676i
$88$	− 558.596i	− 0.676665i
$89$	1141.92	1.36003	0.680016	−	0.733197i	$-0.261973\pi$
$89$	1141.92	1.36003	0.680016	+	0.733197i	$0.261973\pi$
$90$	0	0
$91$	−865.642	−0.997186
$92$	172.371i	0.195337i
$93$	− 197.600i	− 0.220325i
$94$	16.7429	0.0183713
$95$	0	0
$96$	86.2756	0.0917236
$97$	− 147.766i	− 0.154674i	−0.997005	−	0.0773370i	$-0.975358\pi$
$97$	− 147.766i	− 0.154674i	0.997005	−	0.0773370i	$-0.0246417\pi$
$98$	14.2233i	0.0146609i
$99$	1344.87	1.36530
$100$	0	0
$101$	−909.233	−0.895763	−0.447881	−	0.894093i	$-0.647821\pi$
$101$	−909.233	−0.895763	−0.447881	+	0.894093i	$0.647821\pi$
$102$	25.9801i	0.0252198i
$103$	− 1330.08i	− 1.27240i	−0.771526	−	0.636198i	$-0.780506\pi$
$103$	− 1330.08i	− 1.27240i	0.771526	−	0.636198i	$-0.219494\pi$
$104$	−500.560	−0.471961
$105$	0	0
$106$	−269.643	−0.247075
$107$	− 1209.82i	− 1.09306i	−0.837440	−	0.546529i	$-0.815949\pi$
$107$	− 1209.82i	− 1.09306i	0.837440	−	0.546529i	$-0.184051\pi$
$108$	276.428i	0.246290i
$109$	−507.933	−0.446341	−0.223171	−	0.974779i	$-0.571641\pi$
$109$	−507.933	−0.446341	−0.223171	+	0.974779i	$0.571641\pi$
$110$	0	0
$111$	120.865	0.103351
$112$	− 993.053i	− 0.837809i
$113$	− 2.39609i	− 0.00199474i	−1.00000		0.000997369i	$-0.999683\pi$
$113$	− 2.39609i	− 0.00199474i	1.00000		0.000997369i	$-0.000317472\pi$
$114$	−28.8494	−0.0237017
$115$	0	0
$116$	530.114	0.424309
$117$	− 1205.15i	− 0.952273i
$118$	− 527.122i	− 0.411233i
$119$	1010.21	0.778199
$120$	0	0
$121$	1239.68	0.931390
$122$	− 3.67666i	− 0.00272844i
$123$	− 65.4687i	− 0.0479928i
$124$	−2149.01	−1.55634
$125$	0	0
$126$	−359.345	−0.254071
$127$	− 1355.81i	− 0.947310i	−0.880710	−	0.473655i	$-0.842934\pi$
$127$	− 1355.81i	− 0.947310i	0.880710	−	0.473655i	$-0.157066\pi$
$128$	− 1234.78i	− 0.852657i
$129$	102.252	0.0697890
$130$	0	0
$131$	−1701.47	−1.13479	−0.567397	−	0.823444i	$-0.692050\pi$
$131$	−1701.47	−1.13479	−0.567397	+	0.823444i	$0.692050\pi$
$132$	261.848i	0.172659i
$133$	1121.78i	0.731356i
$134$	692.883	0.446686
$135$	0	0
$136$	584.157	0.368316
$137$	− 2505.35i	− 1.56238i	−0.624292	−	0.781191i	$-0.714613\pi$
$137$	− 2505.35i	− 1.56238i	0.624292	−	0.781191i	$-0.285387\pi$
$138$	11.2697i	0.00695176i
$139$	245.374	0.149729	0.0748646	−	0.997194i	$-0.476148\pi$
$139$	245.374	0.149729	0.0748646	+	0.997194i	$0.476148\pi$
$140$	0	0
$141$	−16.2263	−0.00969151
$142$	239.245i	0.141388i
$143$	− 2303.60i	− 1.34711i
$144$	1382.53	0.800075
$145$	0	0
$146$	225.503	0.127827
$147$	− 13.7845i	− 0.00773418i
$148$	− 1314.47i	− 0.730059i
$149$	−2542.90	−1.39814	−0.699068	−	0.715055i	$-0.746402\pi$
$149$	−2542.90	−1.39814	−0.699068	+	0.715055i	$0.746402\pi$
$150$	0	0
$151$	2976.66	1.60422	0.802110	−	0.597176i	$-0.203711\pi$
$151$	2976.66	1.60422	0.802110	+	0.597176i	$0.203711\pi$
$152$	648.671i	0.346146i
$153$	1406.41i	0.743150i
$154$	−686.875	−0.359415
$155$	0	0
$156$	234.643	0.120426
$157$	3622.21i	1.84130i	0.390393	+	0.920648i	$0.372339\pi$
$157$	3622.21i	1.84130i	−0.390393	+	0.920648i	$0.627661\pi$
$158$	− 410.310i	− 0.206598i
$159$	261.323	0.130341
$160$	0	0
$161$	438.211	0.214508
$162$	− 491.164i	− 0.238207i
$163$	3486.81i	1.67551i	0.546048	+	0.837754i	$0.316132\pi$
$163$	3486.81i	1.67551i	−0.546048	+	0.837754i	$0.683868\pi$
$164$	−712.006	−0.339014
$165$	0	0
$166$	160.351	0.0749738
$167$	2818.90i	1.30618i	0.757278	+	0.653092i	$0.226529\pi$
$167$	2818.90i	1.30618i	−0.757278	+	0.653092i	$0.773471\pi$
$168$	− 144.649i	− 0.0664282i
$169$	132.734	0.0604161
$170$	0	0
$171$	−1561.74	−0.698417
$172$	− 1112.04i	− 0.492980i
$173$	2919.22i	1.28291i	0.767159	+	0.641456i	$0.221670\pi$
$173$	2919.22i	1.28291i	−0.767159	+	0.641456i	$0.778330\pi$
$174$	34.6591	0.0151006
$175$	0	0
$176$	2642.66	1.13181
$177$	510.858i	0.216940i
$178$	− 811.956i	− 0.341903i
$179$	2992.42	1.24952	0.624761	−	0.780816i	$-0.285197\pi$
$179$	2992.42	1.24952	0.624761	+	0.780816i	$0.285197\pi$
$180$	0	0
$181$	−3665.87	−1.50542	−0.752712	−	0.658350i	$-0.771255\pi$
$181$	−3665.87	−1.50542	−0.752712	+	0.658350i	$0.771255\pi$
$182$	615.512i	0.250686i
$183$	3.56322i	0.00143935i
$184$	253.397	0.101525
$185$	0	0
$186$	−140.503	−0.0553882
$187$	2688.31i	1.05128i
$188$	176.470i	0.0684595i
$189$	702.749	0.270463
$190$	0	0
$191$	−2159.35	−0.818035	−0.409018	−	0.912526i	$-0.634129\pi$
$191$	−2159.35	−0.818035	−0.409018	+	0.912526i	$0.634129\pi$
$192$	225.993i	0.0849460i
$193$	1865.76i	0.695856i	0.937521	+	0.347928i	$0.113115\pi$
$193$	1865.76i	0.695856i	−0.937521	+	0.347928i	$0.886885\pi$
$194$	−105.069	−0.0388840
$195$	0	0
$196$	−149.913	−0.0546331
$197$	− 2638.16i	− 0.954117i	−0.878872	−	0.477058i	$-0.841703\pi$
$197$	− 2638.16i	− 0.954117i	0.878872	−	0.477058i	$-0.158297\pi$
$198$	− 956.269i	− 0.343228i
$199$	−626.916	−0.223321	−0.111661	−	0.993746i	$-0.535617\pi$
$199$	−626.916	−0.223321	−0.111661	+	0.993746i	$0.535617\pi$
$200$	0	0
$201$	−671.504	−0.235643
$202$	646.507i	0.225189i
$203$	− 1347.68i	− 0.465954i
$204$	−273.830	−0.0939800
$205$	0	0
$206$	−945.751	−0.319872
$207$	610.078i	0.204847i
$208$	− 2368.10i	− 0.789414i
$209$	−2985.21	−0.987998
$210$	0	0
$211$	−3090.41	−1.00831	−0.504154	−	0.863614i	$-0.668196\pi$
$211$	−3090.41	−1.00831	−0.504154	+	0.863614i	$0.668196\pi$
$212$	− 2842.02i	− 0.920712i
$213$	− 231.864i	− 0.0745870i
$214$	−860.236	−0.274787
$215$	0	0
$216$	406.367	0.128008
$217$	5463.31i	1.70910i
$218$	361.165i	0.112207i
$219$	−218.545	−0.0674333
$220$	0	0
$221$	2409.01	0.733247
$222$	− 85.9407i	− 0.0259818i
$223$	5702.67i	1.71246i	0.516594	+	0.856230i	$0.327200\pi$
$223$	5702.67i	1.71246i	−0.516594	+	0.856230i	$0.672800\pi$
$224$	−2385.37	−0.711515
$225$	0	0
$226$	−1.70373	−0.000501463 0
$227$	3203.45i	0.936654i	0.883555	+	0.468327i	$0.155143\pi$
$227$	3203.45i	0.936654i	−0.883555	+	0.468327i	$0.844857\pi$
$228$	− 304.072i	− 0.0883230i
$229$	−648.466	−0.187126	−0.0935630	−	0.995613i	$-0.529826\pi$
$229$	−648.466	−0.187126	−0.0935630	+	0.995613i	$0.529826\pi$
$230$	0	0
$231$	665.682	0.189605
$232$	− 779.301i	− 0.220533i
$233$	− 2556.98i	− 0.718941i	−0.933156	−	0.359470i	$-0.882957\pi$
$233$	− 2556.98i	− 0.718941i	0.933156	−	0.359470i	$-0.117043\pi$
$234$	−856.917	−0.239395
$235$	0	0
$236$	5555.85	1.53244
$237$	397.650i	0.108988i
$238$	− 718.306i	− 0.195634i
$239$	−4424.46	−1.19747	−0.598733	−	0.800949i	$-0.704329\pi$
$239$	−4424.46	−1.19747	−0.598733	+	0.800949i	$0.704329\pi$
$240$	0	0
$241$	−3335.19	−0.891447	−0.445724	−	0.895171i	$-0.647054\pi$
$241$	−3335.19	−0.891447	−0.445724	+	0.895171i	$0.647054\pi$
$242$	− 881.471i	− 0.234145i
$243$	1471.89i	0.388568i
$244$	38.7519	0.0101674
$245$	0	0
$246$	−46.5513	−0.0120651
$247$	2675.06i	0.689110i
$248$	3159.18i	0.808903i
$249$	−155.404	−0.0395514
$250$	0	0
$251$	1952.47	0.490990	0.245495	−	0.969398i	$-0.421049\pi$
$251$	1952.47	0.490990	0.245495	+	0.969398i	$0.421049\pi$
$252$	− 3787.48i	− 0.946781i
$253$	1166.14i	0.289782i
$254$	−964.042	−0.238147
$255$	0	0
$256$	1745.61	0.426175
$257$	− 543.328i	− 0.131875i	−0.997824	−	0.0659374i	$-0.978996\pi$
$257$	− 543.328i	− 0.131875i	0.997824	−	0.0659374i	$-0.0210038\pi$
$258$	− 72.7059i	− 0.0175445i
$259$	−3341.71	−0.801713
$260$	0	0
$261$	1876.24	0.444968
$262$	1209.82i	0.285279i
$263$	3370.82i	0.790318i	0.918613	+	0.395159i	$0.129310\pi$
$263$	3370.82i	0.790318i	−0.918613	+	0.395159i	$0.870690\pi$
$264$	384.933	0.0897385
$265$	0	0
$266$	797.636	0.183858
$267$	786.904i	0.180366i
$268$	7302.96i	1.66455i
$269$	2778.40	0.629747	0.314873	−	0.949134i	$-0.398038\pi$
$269$	2778.40	0.629747	0.314873	+	0.949134i	$0.398038\pi$
$270$	0	0
$271$	6476.65	1.45177	0.725883	−	0.687819i	$-0.241432\pi$
$271$	6476.65	1.45177	0.725883	+	0.687819i	$0.241432\pi$
$272$	2763.59i	0.616055i
$273$	− 596.521i	− 0.132246i
$274$	−1781.42	−0.392772
$275$	0	0
$276$	−118.783	−0.0259053
$277$	1555.57i	0.337420i	0.985666	+	0.168710i	$0.0539602\pi$
$277$	1555.57i	0.337420i	−0.985666	+	0.168710i	$0.946040\pi$
$278$	− 174.473i	− 0.0376409i
$279$	−7606.03	−1.63212
$280$	0	0
$281$	4311.09	0.915225	0.457612	−	0.889152i	$-0.348705\pi$
$281$	4311.09	0.915225	0.457612	+	0.889152i	$0.348705\pi$
$282$	11.5377i	0.00243638i
$283$	6866.82i	1.44237i	0.692744	+	0.721183i	$0.256402\pi$
$283$	6866.82i	1.44237i	−0.692744	+	0.721183i	$0.743598\pi$
$284$	−2521.64	−0.526872
$285$	0	0
$286$	−1637.97	−0.338654
$287$	1810.10i	0.372288i
$288$	− 3320.92i	− 0.679468i
$289$	2101.67	0.427777
$290$	0	0
$291$	101.827	0.0205127
$292$	2376.79i	0.476339i
$293$	− 7215.56i	− 1.43870i	−0.694650	−	0.719348i	$-0.744441\pi$
$293$	− 7215.56i	− 1.43870i	0.694650	−	0.719348i	$-0.255559\pi$
$294$	−9.80141	−0.00194432
$295$	0	0
$296$	−1932.35	−0.379445
$297$	1870.12i	0.365371i
$298$	1808.12i	0.351482i
$299$	1044.99	0.202117
$300$	0	0
$301$	−2827.09	−0.541364
$302$	− 2116.55i	− 0.403290i
$303$	− 626.560i	− 0.118795i
$304$	−3068.80	−0.578972
$305$	0	0
$306$	1000.03	0.186823
$307$	− 3524.91i	− 0.655300i	−0.944799	−	0.327650i	$-0.893743\pi$
$307$	− 3524.91i	− 0.655300i	0.944799	−	0.327650i	$-0.106257\pi$
$308$	− 7239.64i	− 1.33934i
$309$	916.570	0.168744
$310$	0	0
$311$	3378.27	0.615962	0.307981	−	0.951393i	$-0.400347\pi$
$311$	3378.27	0.615962	0.307981	+	0.951393i	$0.400347\pi$
$312$	− 344.940i	− 0.0625910i
$313$	2962.83i	0.535045i	0.963552	+	0.267522i	$0.0862049\pi$
$313$	2962.83i	0.535045i	−0.963552	+	0.267522i	$0.913795\pi$
$314$	2575.56	0.462889
$315$	0	0
$316$	4324.65	0.769875
$317$	− 553.945i	− 0.0981472i	−0.998795	−	0.0490736i	$-0.984373\pi$
$317$	− 553.945i	− 0.0981472i	0.998795	−	0.0490736i	$-0.0156269\pi$
$318$	− 185.813i	− 0.0327669i
$319$	3586.37	0.629462
$320$	0	0
$321$	833.694	0.144960
$322$	− 311.589i	− 0.0539259i
$323$	− 3121.81i	− 0.537778i
$324$	5176.85	0.887663
$325$	0	0
$326$	2479.28	0.421211
$327$	− 350.021i	− 0.0591933i
$328$	1046.69i	0.176201i
$329$	448.630	0.0751786
$330$	0	0
$331$	−2236.93	−0.371458	−0.185729	−	0.982601i	$-0.559465\pi$
$331$	−2236.93	−0.371458	−0.185729	+	0.982601i	$0.559465\pi$
$332$	1690.10i	0.279386i
$333$	− 4652.33i	− 0.765604i
$334$	2004.37	0.328366
$335$	0	0
$336$	684.321	0.111109
$337$	− 586.710i	− 0.0948372i	−0.998875	−	0.0474186i	$-0.984901\pi$
$337$	− 586.710i	− 0.0948372i	0.998875	−	0.0474186i	$-0.0150995\pi$
$338$	− 94.3802i	− 0.0151882i
$339$	1.65117	0.000264540 0
$340$	0	0
$341$	−14538.7	−2.30884
$342$	1110.47i	0.175577i
$343$	− 6153.94i	− 0.968751i
$344$	−1634.77	−0.256224
$345$	0	0
$346$	2075.70	0.322515
$347$	9174.08i	1.41928i	0.704564	+	0.709640i	$0.251143\pi$
$347$	9174.08i	1.41928i	−0.704564	+	0.709640i	$0.748857\pi$
$348$	365.306i	0.0562714i
$349$	−5325.68	−0.816840	−0.408420	−	0.912794i	$-0.633920\pi$
$349$	−5325.68	−0.816840	−0.408420	+	0.912794i	$0.633920\pi$
$350$	0	0
$351$	1675.82	0.254840
$352$	− 6347.82i	− 0.961193i
$353$	9803.34i	1.47813i	0.673635	+	0.739064i	$0.264732\pi$
$353$	9803.34i	1.47813i	−0.673635	+	0.739064i	$0.735268\pi$
$354$	363.244	0.0545373
$355$	0	0
$356$	8557.99	1.27408
$357$	696.143i	0.103204i
$358$	− 2127.75i	− 0.314121i
$359$	−10089.9	−1.48336	−0.741681	−	0.670753i	$-0.765971\pi$
$359$	−10089.9	−1.48336	−0.741681	+	0.670753i	$0.765971\pi$
$360$	0	0
$361$	−3392.41	−0.494593
$362$	2606.60i	0.378453i
$363$	854.274i	0.123520i
$364$	−6487.48	−0.934165
$365$	0	0
$366$	2.53362	0.000361843 0
$367$	8291.81i	1.17937i	0.807633	+	0.589685i	$0.200748\pi$
$367$	8291.81i	1.17937i	−0.807633	+	0.589685i	$0.799252\pi$
$368$	1198.79i	0.169814i
$369$	−2520.02	−0.355520
$370$	0	0
$371$	−7225.13	−1.01108
$372$	− 1480.90i	− 0.206401i
$373$	7493.58i	1.04022i	0.854098	+	0.520111i	$0.174110\pi$
$373$	7493.58i	1.04022i	−0.854098	+	0.520111i	$0.825890\pi$
$374$	1911.52	0.264284
$375$	0	0
$376$	259.422	0.0355815
$377$	− 3213.77i	− 0.439038i
$378$	− 499.688i	− 0.0679925i
$379$	3630.87	0.492099	0.246049	−	0.969257i	$-0.420867\pi$
$379$	3630.87	0.492099	0.246049	+	0.969257i	$0.420867\pi$
$380$	0	0
$381$	934.297	0.125631
$382$	1535.40i	0.205648i
$383$	952.474i	0.127074i	0.997979	+	0.0635368i	$0.0202380\pi$
$383$	952.474i	0.127074i	−0.997979	+	0.0635368i	$0.979762\pi$
$384$	850.896	0.113078
$385$	0	0
$386$	1326.64	0.174933
$387$	− 3935.88i	− 0.516982i
$388$	− 1107.42i	− 0.144899i
$389$	4541.34	0.591916	0.295958	−	0.955201i	$-0.404361\pi$
$389$	4541.34	0.591916	0.295958	+	0.955201i	$0.404361\pi$
$390$	0	0
$391$	−1219.51	−0.157732
$392$	220.382i	0.0283953i
$393$	− 1172.50i	− 0.150495i
$394$	−1875.86	−0.239858
$395$	0	0
$396$	10079.0	1.27902
$397$	− 14345.6i	− 1.81356i	−0.421604	−	0.906780i	$-0.638533\pi$
$397$	− 14345.6i	− 1.81356i	0.421604	−	0.906780i	$-0.361467\pi$
$398$	445.767i	0.0561414i
$399$	−773.026	−0.0969917
$400$	0	0
$401$	−6358.53	−0.791845	−0.395923	−	0.918284i	$-0.629575\pi$
$401$	−6358.53	−0.791845	−0.395923	+	0.918284i	$0.629575\pi$
$402$	477.471i	0.0592390i
$403$	13028.2i	1.61037i
$404$	−6814.17	−0.839152
$405$	0	0
$406$	−958.264	−0.117138
$407$	− 8892.77i	− 1.08304i
$408$	402.547i	0.0488457i
$409$	11543.2	1.39553	0.697767	−	0.716325i	$-0.254177\pi$
$409$	11543.2	1.39553	0.697767	+	0.716325i	$0.254177\pi$
$410$	0	0
$411$	1726.46	0.207201
$412$	− 9968.18i	− 1.19198i
$413$	− 14124.3i	− 1.68284i
$414$	433.794	0.0514971
$415$	0	0
$416$	−5688.31	−0.670414
$417$	169.089i	0.0198569i
$418$	2122.63i	0.248376i
$419$	3691.25	0.430380	0.215190	−	0.976572i	$-0.430963\pi$
$419$	3691.25	0.430380	0.215190	+	0.976572i	$0.430963\pi$
$420$	0	0
$421$	−7551.44	−0.874191	−0.437096	−	0.899415i	$-0.643993\pi$
$421$	−7551.44	−0.874191	−0.437096	+	0.899415i	$0.643993\pi$
$422$	2197.43i	0.253482i
$423$	624.583i	0.0717926i
$424$	−4177.95	−0.478536
$425$	0	0
$426$	−164.866	−0.0187507
$427$	− 98.5170i	− 0.0111653i
$428$	− 9066.86i	− 1.02398i
$429$	1587.43	0.178652
$430$	0	0
$431$	−16426.2	−1.83578	−0.917891	−	0.396833i	$-0.870109\pi$
$431$	−16426.2	−1.83578	−0.917891	+	0.396833i	$0.870109\pi$
$432$	1922.48i	0.214110i
$433$	6363.88i	0.706301i	0.935566	+	0.353151i	$0.114890\pi$
$433$	6363.88i	0.706301i	−0.935566	+	0.353151i	$0.885110\pi$
$434$	3884.67	0.429655
$435$	0	0
$436$	−3806.66	−0.418133
$437$	− 1354.19i	− 0.148237i
$438$	155.396i	0.0169523i
$439$	7513.99	0.816910	0.408455	−	0.912779i	$-0.366068\pi$
$439$	7513.99	0.816910	0.408455	+	0.912779i	$0.366068\pi$
$440$	0	0
$441$	−530.591	−0.0572931
$442$	− 1712.92i	− 0.184333i
$443$	4413.43i	0.473337i	0.971590	+	0.236669i	$0.0760556\pi$
$443$	4413.43i	0.473337i	−0.971590	+	0.236669i	$0.923944\pi$
$444$	905.812	0.0968196
$445$	0	0
$446$	4054.86	0.430501
$447$	− 1752.33i	− 0.185419i
$448$	− 6248.31i	− 0.658940i
$449$	10795.9	1.13472	0.567361	−	0.823469i	$-0.307964\pi$
$449$	10795.9	1.13472	0.567361	+	0.823469i	$0.307964\pi$
$450$	0	0
$451$	−4816.93	−0.502928
$452$	− 17.9573i	− 0.00186867i
$453$	2051.24i	0.212750i
$454$	2277.80	0.235468
$455$	0	0
$456$	−447.005	−0.0459055
$457$	15726.1i	1.60971i	0.593473	+	0.804854i	$0.297756\pi$
$457$	15726.1i	1.60971i	−0.593473	+	0.804854i	$0.702244\pi$
$458$	461.090i	0.0470422i
$459$	−1955.69	−0.198876
$460$	0	0
$461$	−997.530	−0.100780	−0.0503900	−	0.998730i	$-0.516046\pi$
$461$	−997.530	−0.100780	−0.0503900	+	0.998730i	$0.516046\pi$
$462$	− 473.331i	− 0.0476653i
$463$	− 13338.6i	− 1.33887i	−0.742872	−	0.669434i	$-0.766537\pi$
$463$	− 13338.6i	− 1.33887i	0.742872	−	0.669434i	$-0.233463\pi$
$464$	3686.79	0.368869
$465$	0	0
$466$	−1818.13	−0.180737
$467$	5767.52i	0.571496i	0.958305	+	0.285748i	$0.0922421\pi$
$467$	5767.52i	0.571496i	−0.958305	+	0.285748i	$0.907758\pi$
$468$	− 9031.87i	− 0.892091i
$469$	18565.9	1.82792
$470$	0	0
$471$	−2496.09	−0.244191
$472$	− 8167.44i	− 0.796477i
$473$	− 7523.29i	− 0.731335i
$474$	282.748	0.0273988
$475$	0	0
$476$	7570.92	0.729018
$477$	− 10058.8i	− 0.965540i
$478$	3146.00i	0.301035i
$479$	−9581.36	−0.913953	−0.456977	−	0.889479i	$-0.651068\pi$
$479$	−9581.36	−0.913953	−0.456977	+	0.889479i	$0.651068\pi$
$480$	0	0
$481$	−7968.85	−0.755402
$482$	2371.48i	0.224104i
$483$	301.975i	0.0284479i
$484$	9290.68	0.872528
$485$	0	0
$486$	1046.59	0.0976833
$487$	4204.25i	0.391197i	0.980684	+	0.195598i	$0.0626649\pi$
$487$	4204.25i	0.391197i	−0.980684	+	0.195598i	$0.937335\pi$
$488$	− 56.9678i	− 0.00528444i
$489$	−2402.79	−0.222204
$490$	0	0
$491$	5650.56	0.519361	0.259681	−	0.965695i	$-0.416383\pi$
$491$	5650.56	0.519361	0.259681	+	0.965695i	$0.416383\pi$
$492$	− 490.649i	− 0.0449597i
$493$	3750.48i	0.342623i
$494$	1902.10	0.173238
$495$	0	0
$496$	−14945.7	−1.35299
$497$	6410.63i	0.578583i
$498$	110.499i	0.00994295i
$499$	19210.6	1.72341	0.861706	−	0.507408i	$-0.169396\pi$
$499$	19210.6	1.72341	0.861706	+	0.507408i	$0.169396\pi$
$500$	0	0
$501$	−1942.53	−0.173225
$502$	− 1388.30i	− 0.123432i
$503$	10429.5i	0.924506i	0.886748	+	0.462253i	$0.152959\pi$
$503$	10429.5i	0.924506i	−0.886748	+	0.462253i	$0.847041\pi$
$504$	−5567.83	−0.492085
$505$	0	0
$506$	829.183	0.0728491
$507$	91.4682i	0.00801231i
$508$	− 10161.0i	− 0.887442i
$509$	10350.4	0.901324	0.450662	−	0.892695i	$-0.351188\pi$
$509$	10350.4	0.901324	0.450662	+	0.892695i	$0.351188\pi$
$510$	0	0
$511$	6042.39	0.523091
$512$	− 11119.4i	− 0.959794i
$513$	− 2171.68i	− 0.186905i
$514$	−386.331	−0.0331524
$515$	0	0
$516$	766.318	0.0653784
$517$	1193.87i	0.101560i
$518$	2376.11i	0.201545i
$519$	−2011.66	−0.170139
$520$	0	0
$521$	12477.6	1.04924	0.524620	−	0.851336i	$-0.324207\pi$
$521$	12477.6	1.04924	0.524620	+	0.851336i	$0.324207\pi$
$522$	− 1334.10i	− 0.111862i
$523$	− 11220.6i	− 0.938134i	−0.883163	−	0.469067i	$-0.844590\pi$
$523$	− 11220.6i	− 0.938134i	0.883163	−	0.469067i	$-0.155410\pi$
$524$	−12751.5	−1.06308
$525$	0	0
$526$	2396.81	0.198680
$527$	− 15204.0i	− 1.25673i
$528$	1821.08i	0.150099i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	19663.9	1.60705
$532$	8407.06i	0.685136i
$533$	4316.47i	0.350783i
$534$	559.525	0.0453428
$535$	0	0
$536$	10735.8	0.865143
$537$	2062.10i	0.165710i
$538$	− 1975.57i	− 0.158314i
$539$	−1014.21	−0.0810482
$540$	0	0
$541$	−4220.58	−0.335410	−0.167705	−	0.985837i	$-0.553636\pi$
$541$	−4220.58	−0.335410	−0.167705	+	0.985837i	$0.553636\pi$
$542$	− 4605.20i	− 0.364964i
$543$	− 2526.18i	− 0.199648i
$544$	6638.29	0.523188
$545$	0	0
$546$	−424.154	−0.0332457
$547$	− 298.329i	− 0.0233193i	−0.999932	−	0.0116596i	$-0.996289\pi$
$547$	− 298.329i	− 0.0233193i	0.999932	−	0.0116596i	$-0.00371146\pi$
$548$	− 18776.1i	− 1.46364i
$549$	137.155	0.0106624
$550$	0	0
$551$	−4164.69	−0.322000
$552$	174.618i	0.0134642i
$553$	− 10994.3i	− 0.845436i
$554$	1106.09	0.0848251
$555$	0	0
$556$	1838.93	0.140267
$557$	1097.07i	0.0834545i	0.999129	+	0.0417272i	$0.0132860\pi$
$557$	1097.07i	0.0834545i	−0.999129	+	0.0417272i	$0.986714\pi$
$558$	5408.25i	0.410303i
$559$	−6741.66	−0.510093
$560$	0	0
$561$	−1852.54	−0.139419
$562$	− 3065.39i	− 0.230081i
$563$	5409.44i	0.404939i	0.979289	+	0.202469i	$0.0648967\pi$
$563$	5409.44i	0.404939i	−0.979289	+	0.202469i	$0.935103\pi$
$564$	−121.607	−0.00907902
$565$	0	0
$566$	4882.63	0.362601
$567$	− 13160.8i	− 0.974784i
$568$	3706.97i	0.273840i
$569$	−10942.8	−0.806231	−0.403115	−	0.915149i	$-0.632073\pi$
$569$	−10942.8	−0.806231	−0.403115	+	0.915149i	$0.632073\pi$
$570$	0	0
$571$	−1144.71	−0.0838959	−0.0419480	−	0.999120i	$-0.513356\pi$
$571$	−1144.71	−0.0838959	−0.0419480	+	0.999120i	$0.513356\pi$
$572$	− 17264.1i	− 1.26197i
$573$	− 1488.02i	− 0.108487i
$574$	1287.06	0.0935906
$575$	0	0
$576$	8698.91	0.629261
$577$	9892.27i	0.713727i	0.934157	+	0.356864i	$0.116154\pi$
$577$	9892.27i	0.713727i	−0.934157	+	0.356864i	$0.883846\pi$
$578$	− 1494.39i	− 0.107540i
$579$	−1285.71	−0.0922836
$580$	0	0
$581$	4296.64	0.306807
$582$	− 72.4036i	− 0.00515675i
$583$	− 19227.1i	− 1.36588i
$584$	3494.03	0.247575
$585$	0	0
$586$	−5130.60	−0.361678
$587$	− 12528.8i	− 0.880950i	−0.897765	−	0.440475i	$-0.854810\pi$
$587$	− 12528.8i	− 0.880950i	0.897765	−	0.440475i	$-0.145190\pi$
$588$	− 103.307i	− 0.00724539i
$589$	16883.1	1.18108
$590$	0	0
$591$	1817.98	0.126534
$592$	− 9141.76i	− 0.634669i
$593$	− 10978.3i	− 0.760241i	−0.924937	−	0.380121i	$-0.875882\pi$
$593$	− 10978.3i	− 0.760241i	0.924937	−	0.380121i	$-0.124118\pi$
$594$	1329.74	0.0918518
$595$	0	0
$596$	−19057.5	−1.30978
$597$	− 432.013i	− 0.0296166i
$598$	− 743.035i	− 0.0508109i
$599$	583.723	0.0398168	0.0199084	−	0.999802i	$-0.493663\pi$
$599$	583.723	0.0398168	0.0199084	+	0.999802i	$0.493663\pi$
$600$	0	0
$601$	−4111.87	−0.279080	−0.139540	−	0.990216i	$-0.544562\pi$
$601$	−4111.87	−0.279080	−0.139540	+	0.990216i	$0.544562\pi$
$602$	2010.19i	0.136095i
$603$	25847.5i	1.74559i
$604$	22308.3	1.50284
$605$	0	0
$606$	−445.514	−0.0298643
$607$	15367.4i	1.02758i	0.857915	+	0.513792i	$0.171760\pi$
$607$	15367.4i	1.02758i	−0.857915	+	0.513792i	$0.828240\pi$
$608$	7371.43i	0.491696i
$609$	928.697	0.0617943
$610$	0	0
$611$	1069.83	0.0708360
$612$	10540.3i	0.696184i
$613$	− 14127.7i	− 0.930853i	−0.885087	−	0.465426i	$-0.845901\pi$
$613$	− 14127.7i	− 0.930853i	0.885087	−	0.465426i	$-0.154099\pi$
$614$	−2506.38	−0.164738
$615$	0	0
$616$	−10642.7	−0.696116
$617$	− 24427.9i	− 1.59389i	−0.604053	−	0.796944i	$-0.706448\pi$
$617$	− 24427.9i	− 1.59389i	0.604053	−	0.796944i	$-0.293552\pi$
$618$	− 651.724i	− 0.0424210i
$619$	−23038.8	−1.49597	−0.747987	−	0.663713i	$-0.768980\pi$
$619$	−23038.8	−1.49597	−0.747987	+	0.663713i	$0.768980\pi$
$620$	0	0
$621$	−848.346	−0.0548196
$622$	− 2402.11i	− 0.154848i
$623$	− 21756.5i	− 1.39913i
$624$	1631.88	0.104691
$625$	0	0
$626$	2106.71	0.134507
$627$	− 2057.13i	− 0.131027i
$628$	27146.3i	1.72493i
$629$	9299.70	0.589512
$630$	0	0
$631$	−11336.7	−0.715223	−0.357611	−	0.933871i	$-0.616409\pi$
$631$	−11336.7	−0.715223	−0.357611	+	0.933871i	$0.616409\pi$
$632$	− 6357.50i	− 0.400139i
$633$	− 2129.63i	− 0.133721i
$634$	−393.881	−0.0246735
$635$	0	0
$636$	1958.46	0.122104
$637$	908.836i	0.0565297i
$638$	− 2550.08i	− 0.158242i
$639$	−8924.89	−0.552524
$640$	0	0
$641$	27030.1	1.66556	0.832782	−	0.553601i	$-0.186747\pi$
$641$	27030.1	1.66556	0.832782	+	0.553601i	$0.186747\pi$
$642$	− 592.795i	− 0.0364420i
$643$	− 6776.53i	− 0.415615i	−0.978170	−	0.207807i	$-0.933367\pi$
$643$	− 6776.53i	− 0.415615i	0.978170	−	0.207807i	$-0.0666327\pi$
$644$	3284.13	0.200952
$645$	0	0
$646$	−2219.76	−0.135194
$647$	25310.7i	1.53797i	0.639268	+	0.768984i	$0.279237\pi$
$647$	25310.7i	1.53797i	−0.639268	+	0.768984i	$0.720763\pi$
$648$	− 7610.29i	− 0.461359i
$649$	37586.9	2.27337
$650$	0	0
$651$	−3764.81	−0.226658
$652$	26131.6i	1.56962i
$653$	24830.6i	1.48805i	0.668153	+	0.744024i	$0.267085\pi$
$653$	24830.6i	1.48805i	−0.668153	+	0.744024i	$0.732915\pi$
$654$	−248.881	−0.0148808
$655$	0	0
$656$	−4951.80	−0.294719
$657$	8412.22i	0.499531i
$658$	− 318.997i	− 0.0188994i
$659$	−19514.4	−1.15353	−0.576764	−	0.816911i	$-0.695685\pi$
$659$	−19514.4	−1.15353	−0.576764	+	0.816911i	$0.695685\pi$
$660$	0	0
$661$	−20107.4	−1.18319	−0.591594	−	0.806236i	$-0.701501\pi$
$661$	−20107.4	−1.18319	−0.591594	+	0.806236i	$0.701501\pi$
$662$	1590.56i	0.0933820i
$663$	1660.07i	0.0972424i
$664$	2484.55	0.145209
$665$	0	0
$666$	−3308.03	−0.192468
$667$	1626.89i	0.0944432i
$668$	21126.0i	1.22364i
$669$	−3929.75	−0.227105
$670$	0	0
$671$	262.168	0.0150833
$672$	− 1643.78i	− 0.0943603i
$673$	34488.4i	1.97538i	0.156432	+	0.987689i	$0.450001\pi$
$673$	34488.4i	1.97538i	−0.156432	+	0.987689i	$0.549999\pi$
$674$	−417.179	−0.0238414
$675$	0	0
$676$	994.764	0.0565979
$677$	− 20506.3i	− 1.16414i	−0.813139	−	0.582070i	$-0.802243\pi$
$677$	− 20506.3i	− 1.16414i	0.813139	−	0.582070i	$-0.197757\pi$
$678$	− 1.17406i	0 6.65035e-5i
$679$	−2815.33	−0.159120
$680$	0	0
$681$	−2207.52	−0.124218
$682$	10337.7i	0.580425i
$683$	− 2269.13i	− 0.127124i	−0.997978	−	0.0635620i	$-0.979754\pi$
$683$	− 2269.13i	− 0.127124i	0.997978	−	0.0635620i	$-0.0202461\pi$
$684$	−11704.3	−0.654278
$685$	0	0
$686$	−4375.74	−0.243537
$687$	− 446.863i	− 0.0248164i
$688$	− 7733.94i	− 0.428567i
$689$	−17229.5	−0.952673
$690$	0	0
$691$	−27891.7	−1.53553	−0.767764	−	0.640733i	$-0.778631\pi$
$691$	−27891.7	−1.53553	−0.767764	+	0.640733i	$0.778631\pi$
$692$	21877.8i	1.20183i
$693$	− 25623.4i	− 1.40455i
$694$	6523.20	0.356797
$695$	0	0
$696$	537.022	0.0292468
$697$	− 5037.35i	− 0.273749i
$698$	3786.81i	0.205348i
$699$	1762.03	0.0953452
$700$	0	0
$701$	27181.1	1.46450	0.732251	−	0.681034i	$-0.238470\pi$
$701$	27181.1	1.46450	0.732251	+	0.681034i	$0.238470\pi$
$702$	− 1191.59i	− 0.0640650i
$703$	10326.8i	0.554027i
$704$	16627.7	0.890169
$705$	0	0
$706$	6970.64	0.371591
$707$	17323.3i	0.921513i
$708$	3828.58i	0.203230i
$709$	−2398.23	−0.127035	−0.0635173	−	0.997981i	$-0.520232\pi$
$709$	−2398.23	−0.127035	−0.0635173	+	0.997981i	$0.520232\pi$
$710$	0	0
$711$	15306.3	0.807358
$712$	− 12580.8i	− 0.662198i
$713$	− 6595.21i	− 0.346413i
$714$	494.991	0.0259448
$715$	0	0
$716$	22426.5	1.17055
$717$	− 3048.93i	− 0.158807i
$718$	7174.42i	0.372907i
$719$	−11387.8	−0.590674	−0.295337	−	0.955393i	$-0.595432\pi$
$719$	−11387.8	−0.590674	−0.295337	+	0.955393i	$0.595432\pi$
$720$	0	0
$721$	−25341.6	−1.30897
$722$	2412.16i	0.124337i
$723$	− 2298.31i	− 0.118223i
$724$	−27473.5	−1.41028
$725$	0	0
$726$	607.429	0.0310521
$727$	− 2762.84i	− 0.140946i	−0.997514	−	0.0704732i	$-0.977549\pi$
$727$	− 2762.84i	− 0.140946i	0.997514	−	0.0704732i	$-0.0224509\pi$
$728$	9537.00i	0.485528i
$729$	17636.3	0.896015
$730$	0	0
$731$	7867.56	0.398074
$732$	26.7043i	0.00134839i
$733$	− 1782.84i	− 0.0898370i	−0.998991	−	0.0449185i	$-0.985697\pi$
$733$	− 1782.84i	− 0.0898370i	0.998991	−	0.0449185i	$-0.0143028\pi$
$734$	5895.87	0.296486
$735$	0	0
$736$	2879.57	0.144215
$737$	49406.7i	2.46936i
$738$	1791.85i	0.0893753i
$739$	987.346	0.0491476	0.0245738	−	0.999698i	$-0.492177\pi$
$739$	987.346	0.0491476	0.0245738	+	0.999698i	$0.492177\pi$
$740$	0	0
$741$	−1843.41	−0.0913890
$742$	5137.40i	0.254178i
$743$	− 11744.8i	− 0.579910i	−0.957040	−	0.289955i	$-0.906360\pi$
$743$	− 11744.8i	− 0.579910i	0.957040	−	0.289955i	$-0.0936404\pi$
$744$	−2177.01	−0.107276
$745$	0	0
$746$	5328.29	0.261505
$747$	5981.79i	0.292988i
$748$	20147.3i	0.984839i
$749$	−23050.2	−1.12448
$750$	0	0
$751$	−14156.0	−0.687828	−0.343914	−	0.939001i	$-0.611753\pi$
$751$	−14156.0	−0.687828	−0.343914	+	0.939001i	$0.611753\pi$
$752$	1227.30i	0.0595145i
$753$	1345.46i	0.0651146i
$754$	−2285.14	−0.110371
$755$	0	0
$756$	5266.69	0.253370
$757$	35240.9i	1.69201i	0.533174	+	0.846005i	$0.320999\pi$
$757$	35240.9i	1.69201i	−0.533174	+	0.846005i	$0.679001\pi$
$758$	− 2581.72i	− 0.123710i
$759$	−803.599	−0.0384306
$760$	0	0
$761$	15253.8	0.726609	0.363305	−	0.931670i	$-0.381648\pi$
$761$	15253.8	0.726609	0.363305	+	0.931670i	$0.381648\pi$
$762$	− 664.329i	− 0.0315828i
$763$	9677.48i	0.459172i
$764$	−16183.0	−0.766337
$765$	0	0
$766$	677.254	0.0319454
$767$	− 33681.8i	− 1.58563i
$768$	1202.92i	0.0565189i
$769$	−33344.6	−1.56364	−0.781818	−	0.623507i	$-0.785707\pi$
$769$	−33344.6	−1.56364	−0.781818	+	0.623507i	$0.785707\pi$
$770$	0	0
$771$	374.411	0.0174891
$772$	13982.8i	0.651879i
$773$	− 14412.0i	− 0.670587i	−0.942114	−	0.335294i	$-0.891164\pi$
$773$	− 14412.0i	− 0.670587i	0.942114	−	0.335294i	$-0.108836\pi$
$774$	−2798.59	−0.129966
$775$	0	0
$776$	−1627.98	−0.0753105
$777$	− 2302.80i	− 0.106322i
$778$	− 3229.11i	− 0.148803i
$779$	5593.68	0.257271
$780$	0	0
$781$	−17059.6	−0.781615
$782$	867.126i	0.0396526i
$783$	2609.02i	0.119079i
$784$	−1042.60	−0.0474947
$785$	0	0
$786$	−833.700	−0.0378335
$787$	15474.1i	0.700880i	0.936585	+	0.350440i	$0.113968\pi$
$787$	15474.1i	0.700880i	−0.936585	+	0.350440i	$0.886032\pi$
$788$	− 19771.5i	− 0.893818i
$789$	−2322.86	−0.104811
$790$	0	0
$791$	−45.6519	−0.00205208
$792$	− 14816.8i	− 0.664764i
$793$	− 234.930i	− 0.0105203i
$794$	−10200.4	−0.455916
$795$	0	0
$796$	−4698.37	−0.209208
$797$	− 15561.2i	− 0.691603i	−0.938308	−	0.345802i	$-0.887607\pi$
$797$	− 15561.2i	− 0.691603i	0.938308	−	0.345802i	$-0.112393\pi$
$798$	549.658i	0.0243830i
$799$	−1248.50	−0.0552801
$800$	0	0
$801$	30289.5	1.33611
$802$	4521.21i	0.199064i
$803$	16079.7i	0.706649i
$804$	−5032.53	−0.220751
$805$	0	0
$806$	9263.64	0.404836
$807$	1914.62i	0.0835163i
$808$	10017.3i	0.436146i
$809$	−14327.9	−0.622674	−0.311337	−	0.950299i	$-0.600777\pi$
$809$	−14327.9	−0.622674	−0.311337	+	0.950299i	$0.600777\pi$
$810$	0	0
$811$	−8539.31	−0.369736	−0.184868	−	0.982763i	$-0.559186\pi$
$811$	−8539.31	−0.369736	−0.184868	+	0.982763i	$0.559186\pi$
$812$	− 10100.1i	− 0.436506i
$813$	4463.11i	0.192532i
$814$	−6323.18	−0.272269
$815$	0	0
$816$	−1904.41	−0.0817005
$817$	8736.45i	0.374112i
$818$	− 8207.74i	− 0.350828i
$819$	−22961.3	−0.979647
$820$	0	0
$821$	42664.3	1.81364	0.906818	−	0.421523i	$-0.138504\pi$
$821$	42664.3	1.81364	0.906818	+	0.421523i	$0.138504\pi$
$822$	− 1227.59i	− 0.0520890i
$823$	7687.86i	0.325616i	0.986658	+	0.162808i	$0.0520551\pi$
$823$	7687.86i	0.325616i	−0.986658	+	0.162808i	$0.947945\pi$
$824$	−14653.8	−0.619528
$825$	0	0
$826$	−10043.1	−0.423054
$827$	− 27712.7i	− 1.16525i	−0.812740	−	0.582627i	$-0.802025\pi$
$827$	− 27712.7i	− 1.16525i	0.812740	−	0.582627i	$-0.197975\pi$
$828$	4572.18i	0.191901i
$829$	13667.8	0.572619	0.286310	−	0.958137i	$-0.407571\pi$
$829$	13667.8	0.572619	0.286310	+	0.958137i	$0.407571\pi$
$830$	0	0
$831$	−1071.96	−0.0447483
$832$	− 14900.1i	− 0.620876i
$833$	− 1060.62i	− 0.0441155i
$834$	120.230	0.00499189
$835$	0	0
$836$	−22372.4	−0.925558
$837$	− 10576.6i	− 0.436775i
$838$	− 2624.65i	− 0.108195i
$839$	13429.4	0.552604	0.276302	−	0.961071i	$-0.410891\pi$
$839$	13429.4	0.552604	0.276302	+	0.961071i	$0.410891\pi$
$840$	0	0
$841$	−19385.6	−0.794851
$842$	5369.43i	0.219766i
$843$	2970.81i	0.121376i
$844$	−23160.8	−0.944584
$845$	0	0
$846$	444.108	0.0180482
$847$	− 23619.2i	− 0.958164i
$848$	− 19765.5i	− 0.800411i
$849$	−4731.98	−0.191285
$850$	0	0
$851$	4034.05	0.162497
$852$	− 1737.68i	− 0.0698732i
$853$	− 1540.70i	− 0.0618434i	−0.999522	−	0.0309217i	$-0.990156\pi$
$853$	− 1540.70i	− 0.0618434i	0.999522	−	0.0309217i	$-0.00984425\pi$
$854$	−70.0502	−0.00280687
$855$	0	0
$856$	−13328.8	−0.532209
$857$	− 33588.2i	− 1.33880i	−0.742903	−	0.669399i	$-0.766552\pi$
$857$	− 33588.2i	− 1.33880i	0.742903	−	0.669399i	$-0.233448\pi$
$858$	− 1128.74i	− 0.0449119i
$859$	−4955.28	−0.196824	−0.0984122	−	0.995146i	$-0.531376\pi$
$859$	−4955.28	−0.196824	−0.0984122	+	0.995146i	$0.531376\pi$
$860$	0	0
$861$	−1247.35	−0.0493724
$862$	11679.8i	0.461503i
$863$	41833.5i	1.65009i	0.565066	+	0.825046i	$0.308851\pi$
$863$	41833.5i	1.65009i	−0.565066	+	0.825046i	$0.691149\pi$
$864$	4617.91	0.181834
$865$	0	0
$866$	4525.02	0.177559
$867$	1448.28i	0.0567314i
$868$	40944.3i	1.60108i
$869$	29257.5	1.14211
$870$	0	0
$871$	44273.5	1.72233
$872$	5596.03i	0.217323i
$873$	− 3919.51i	− 0.151954i
$874$	−962.892	−0.0372658
$875$	0	0
$876$	−1637.87	−0.0631716
$877$	− 34708.9i	− 1.33642i	−0.743974	−	0.668208i	$-0.767062\pi$
$877$	− 34708.9i	− 1.33642i	0.743974	−	0.668208i	$-0.232938\pi$
$878$	− 5342.80i	− 0.205365i
$879$	4972.30	0.190798
$880$	0	0
$881$	−6072.43	−0.232219	−0.116110	−	0.993236i	$-0.537042\pi$
$881$	−6072.43	−0.232219	−0.116110	+	0.993236i	$0.537042\pi$
$882$	377.275i	0.0144031i
$883$	18884.3i	0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$883$	18884.3i	0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$884$	18054.1	0.686907
$885$	0	0
$886$	3138.16	0.118994
$887$	46338.3i	1.75410i	0.480397	+	0.877051i	$0.340493\pi$
$887$	46338.3i	1.75410i	−0.480397	+	0.877051i	$0.659507\pi$
$888$	− 1331.60i	− 0.0503216i
$889$	−25831.7	−0.974542
$890$	0	0
$891$	35022.9	1.31685
$892$	42738.1i	1.60424i
$893$	− 1386.39i	− 0.0519526i
$894$	−1245.99	−0.0466132
$895$	0	0
$896$	−23525.8	−0.877167
$897$	720.109i	0.0268046i
$898$	− 7676.40i	− 0.285261i
$899$	−20283.0	−0.752476
$900$	0	0
$901$	20106.9	0.743462
$902$	3425.06i	0.126433i
$903$	− 1948.17i	− 0.0717951i
$904$	−26.3984	−0.000971235 0
$905$	0	0
$906$	1458.53	0.0534839
$907$	22074.9i	0.808141i	0.914728	+	0.404070i	$0.132405\pi$
$907$	22074.9i	0.808141i	−0.914728	+	0.404070i	$0.867595\pi$
$908$	24008.0i	0.877459i
$909$	−24117.5	−0.880008
$910$	0	0
$911$	−16529.7	−0.601156	−0.300578	−	0.953757i	$-0.597180\pi$
$911$	−16529.7	−0.601156	−0.300578	+	0.953757i	$0.597180\pi$
$912$	− 2114.73i	− 0.0767827i
$913$	11434.0i	0.414468i
$914$	11182.0	0.404670
$915$	0	0
$916$	−4859.87	−0.175300
$917$	32417.5i	1.16742i
$918$	1390.59i	0.0499960i
$919$	−37282.7	−1.33824	−0.669120	−	0.743154i	$-0.733329\pi$
$919$	−37282.7	−1.33824	−0.669120	+	0.743154i	$0.733329\pi$
$920$	0	0
$921$	2429.04	0.0869052
$922$	709.291i	0.0253354i
$923$	15287.2i	0.545162i
$924$	4988.89	0.177622
$925$	0	0
$926$	−9484.35	−0.336582
$927$	− 35280.6i	− 1.25002i
$928$	− 8855.89i	− 0.313264i
$929$	−37919.2	−1.33917	−0.669585	−	0.742736i	$-0.733528\pi$
$929$	−37919.2	−1.33917	−0.669585	+	0.742736i	$0.733528\pi$
$930$	0	0
$931$	1177.75	0.0414600
$932$	− 19163.0i	− 0.673505i
$933$	2327.99i	0.0816882i
$934$	4100.97	0.143670
$935$	0	0
$936$	−13277.4	−0.463660
$937$	− 7139.87i	− 0.248932i	−0.992224	−	0.124466i	$-0.960278\pi$
$937$	− 7139.87i	− 0.248932i	0.992224	−	0.124466i	$-0.0397218\pi$
$938$	− 13201.2i	− 0.459527i
$939$	−2041.71	−0.0709570
$940$	0	0
$941$	44548.2	1.54328	0.771642	−	0.636057i	$-0.219436\pi$
$941$	44548.2	1.54328	0.771642	+	0.636057i	$0.219436\pi$
$942$	1774.84i	0.0613879i
$943$	− 2185.11i	− 0.0754582i
$944$	38639.3	1.33221
$945$	0	0
$946$	−5349.42	−0.183853
$947$	11796.3i	0.404782i	0.979305	+	0.202391i	$0.0648712\pi$
$947$	11796.3i	0.404782i	−0.979305	+	0.202391i	$0.935129\pi$
$948$	2980.15i	0.102100i
$949$	14409.1	0.492875
$950$	0	0
$951$	381.728	0.0130162
$952$	− 11129.7i	− 0.378904i
$953$	− 16156.5i	− 0.549171i	−0.961563	−	0.274586i	$-0.911459\pi$
$953$	− 16156.5i	− 0.549171i	0.961563	−	0.274586i	$-0.0885407\pi$
$954$	−7152.30	−0.242730
$955$	0	0
$956$	−33158.7	−1.12179
$957$	2471.40i	0.0834786i
$958$	6812.80i	0.229762i
$959$	−47733.5	−1.60729
$960$	0	0
$961$	52433.5	1.76004
$962$	5666.23i	0.189903i
$963$	− 32090.5i	− 1.07383i
$964$	−24995.3	−0.835109
$965$	0	0
$966$	214.718	0.00715160
$967$	50550.0i	1.68105i	0.541770	+	0.840527i	$0.317754\pi$
$967$	50550.0i	1.68105i	−0.541770	+	0.840527i	$0.682246\pi$
$968$	− 13657.9i	− 0.453493i
$969$	2151.27	0.0713196
$970$	0	0
$971$	33282.9	1.10000	0.550000	−	0.835165i	$-0.314628\pi$
$971$	33282.9	1.10000	0.550000	+	0.835165i	$0.314628\pi$
$972$	11031.0i	0.364011i
$973$	− 4675.03i	− 0.154033i
$974$	2989.42	0.0983441
$975$	0	0
$976$	269.509	0.00883889
$977$	− 2729.36i	− 0.0893755i	−0.999001	−	0.0446877i	$-0.985771\pi$
$977$	− 2729.36i	− 0.0893755i	0.999001	−	0.0446877i	$-0.0142293\pi$
$978$	1708.49i	0.0558606i
$979$	57897.3	1.89010
$980$	0	0
$981$	−13473.0	−0.438491
$982$	− 4017.82i	− 0.130564i
$983$	− 14904.5i	− 0.483600i	−0.970326	−	0.241800i	$-0.922262\pi$
$983$	− 14904.5i	− 0.483600i	0.970326	−	0.241800i	$-0.0777378\pi$
$984$	−721.285	−0.0233676
$985$	0	0
$986$	2666.77	0.0861332
$987$	309.154i	0.00997011i
$988$	20048.0i	0.645559i
$989$	3412.81	0.109728
$990$	0	0
$991$	39748.4	1.27412	0.637058	−	0.770816i	$-0.280151\pi$
$991$	39748.4	1.27412	0.637058	+	0.770816i	$0.280151\pi$
$992$	35900.5i	1.14904i
$993$	− 1541.48i	− 0.0492624i
$994$	4558.26	0.145452
$995$	0	0
$996$	−1164.66	−0.0370518
$997$	− 47343.4i	− 1.50389i	−0.659225	−	0.751946i	$-0.729116\pi$
$997$	− 47343.4i	− 1.50389i	0.659225	−	0.751946i	$-0.270884\pi$
$998$	− 13659.6i	− 0.433254i
$999$	6469.31	0.204885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.b.j.24.7		14
5.2	odd	4		575.4.a.l.1.5	✓	7
5.3	odd	4		575.4.a.m.1.3	yes	7
5.4	even	2	inner	575.4.b.j.24.8		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.4.a.l.1.5	✓	7	5.2	odd	4
575.4.a.m.1.3	yes	7	5.3	odd	4
575.4.b.j.24.7		14	1.1	even	1	trivial
575.4.b.j.24.8		14	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.711047i q^{2} +0.689108i q^{3} +7.49441 q^{4} +0.489988 q^{6} -19.0526i q^{7} -11.0173i q^{8} +26.5251 q^{9} +50.7019 q^{11} +5.16446i q^{12} -45.4342i q^{13} -13.5473 q^{14} +52.1215 q^{16} +53.0220i q^{17} -18.8606i q^{18} -58.8778 q^{19} +13.1293 q^{21} -36.0514i q^{22} +23.0000i q^{23} +7.59208 q^{24} -32.3058 q^{26} +36.8846i q^{27} -142.788i q^{28} +70.7345 q^{29} -286.748 q^{31} -125.199i q^{32} +34.9391i q^{33} +37.7011 q^{34} +198.790 q^{36} -175.393i q^{37} +41.8649i q^{38} +31.3091 q^{39} -95.0049 q^{41} -9.33557i q^{42} -148.383i q^{43} +379.981 q^{44} +16.3541 q^{46} +23.5469i q^{47} +35.9174i q^{48} -20.0033 q^{49} -36.5379 q^{51} -340.503i q^{52} -379.219i q^{53} +26.2267 q^{54} -209.908 q^{56} -40.5731i q^{57} -50.2956i q^{58} +741.332 q^{59} +5.17078 q^{61} +203.891i q^{62} -505.374i q^{63} +327.950 q^{64} +24.8433 q^{66} +974.454i q^{67} +397.368i q^{68} -15.8495 q^{69} -336.469 q^{71} -292.234i q^{72} +317.142i q^{73} -124.713 q^{74} -441.254 q^{76} -966.005i q^{77} -22.2622i q^{78} +577.050 q^{79} +690.761 q^{81} +67.5530i q^{82} +225.514i q^{83} +98.3966 q^{84} -105.507 q^{86} +48.7438i q^{87} -558.596i q^{88} +1141.92 q^{89} -865.642 q^{91} +172.371i q^{92} -197.600i q^{93} +16.7429 q^{94} +86.2756 q^{96} -147.766i q^{97} +14.2233i q^{98} +1344.87 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) \) 14 * q - 54 * q^4 - 82 * q^6 - 20 * q^9 - 104 * q^11 + 84 * q^14 - 170 * q^16 + 20 * q^19 - 404 * q^21 + 606 * q^24 - 52 * q^26 + 910 * q^29 - 1380 * q^31 + 1314 * q^34 + 408 * q^36 + 554 * q^39 - 460 * q^41 + 2006 * q^44 - 138 * q^46 + 1604 * q^49 - 1390 * q^51 + 2494 * q^54 - 772 * q^56 + 1102 * q^59 - 2484 * q^61 + 3002 * q^64 + 1974 * q^66 + 46 * q^69 + 372 * q^71 + 3060 * q^74 - 3550 * q^76 + 4006 * q^79 - 730 * q^81 + 2704 * q^84 + 3788 * q^86 + 5258 * q^89 - 2590 * q^91 + 6936 * q^94 - 270 * q^96 + 4286 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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