LMFDB - Embedded newform 575.4.b.g.24.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 36x^{6} + 244x^{4} + 153x^{2} + 16 $ x^8 + 36x^6 + 244x^4 + 153*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.6
Root			$-2.83969i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.4.b.g.24.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.86845i q^{2} -3.43737i q^{3} -0.228032 q^{4} +9.85995 q^{6} +32.7301i q^{7} +22.2935i q^{8} +15.1845 q^{9} +26.7049 q^{11} +0.783832i q^{12} +14.4956i q^{13} -93.8848 q^{14} -65.7723 q^{16} +24.7016i q^{17} +43.5560i q^{18} -94.6224 q^{19} +112.505 q^{21} +76.6018i q^{22} +23.0000i q^{23} +76.6312 q^{24} -41.5801 q^{26} -145.004i q^{27} -7.46352i q^{28} +57.5965 q^{29} +88.8691 q^{31} -10.3165i q^{32} -91.7948i q^{33} -70.8553 q^{34} -3.46255 q^{36} -305.467i q^{37} -271.420i q^{38} +49.8269 q^{39} -179.205 q^{41} +322.717i q^{42} +96.5826i q^{43} -6.08959 q^{44} -65.9745 q^{46} +218.484i q^{47} +226.084i q^{48} -728.258 q^{49} +84.9085 q^{51} -3.30548i q^{52} +519.174i q^{53} +415.937 q^{54} -729.669 q^{56} +325.252i q^{57} +165.213i q^{58} +37.2884 q^{59} +96.3052 q^{61} +254.917i q^{62} +496.989i q^{63} -496.586 q^{64} +263.309 q^{66} +497.552i q^{67} -5.63276i q^{68} +79.0596 q^{69} -19.6235 q^{71} +338.515i q^{72} +208.235i q^{73} +876.219 q^{74} +21.5770 q^{76} +874.054i q^{77} +142.926i q^{78} +446.200 q^{79} -88.4513 q^{81} -514.042i q^{82} -501.151i q^{83} -25.6549 q^{84} -277.043 q^{86} -197.981i q^{87} +595.347i q^{88} -1102.82 q^{89} -474.444 q^{91} -5.24475i q^{92} -305.476i q^{93} -626.711 q^{94} -35.4615 q^{96} +1814.37i q^{97} -2088.98i q^{98} +405.500 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{4} - 34 q^{6} + 66 q^{9} + 16 q^{11} + 288 q^{14} + 128 q^{16} - 192 q^{19} + 360 q^{21} + 376 q^{24} - 458 q^{26} - 42 q^{29} - 386 q^{31} - 1332 q^{34} - 1258 q^{36} + 582 q^{39} - 250 q^{41}+ \cdots + 2996 q^{99}+O(q^{100}) $ 8 * q - 40 * q^4 - 34 * q^6 + 66 * q^9 + 16 * q^11 + 288 * q^14 + 128 * q^16 - 192 * q^19 + 360 * q^21 + 376 * q^24 - 458 * q^26 - 42 * q^29 - 386 * q^31 - 1332 * q^34 - 1258 * q^36 + 582 * q^39 - 250 * q^41 - 1660 * q^44 - 92 * q^46 - 2440 * q^49 - 680 * q^51 - 1282 * q^54 - 4348 * q^56 + 2280 * q^59 + 1508 * q^61 + 1610 * q^64 - 796 * q^66 + 322 * q^69 - 802 * q^71 - 2732 * q^74 - 7664 * q^76 + 1676 * q^79 - 1864 * q^81 - 5228 * q^84 + 1836 * q^86 - 4684 * q^89 + 584 * q^91 - 3134 * q^94 + 1598 * q^96 + 2996 * 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 $

$2$	2.86845i	1.01415i	0.861901	+	0.507076i	$0.169274\pi$
$2$	2.86845i	1.01415i	−0.861901	+	0.507076i	$0.830726\pi$
$3$	− 3.43737i	− 0.661523i	−0.943714	−	0.330761i	$-0.892694\pi$
$3$	− 3.43737i	− 0.661523i	0.943714	−	0.330761i	$-0.107306\pi$
$4$	−0.228032	−0.0285041
$5$	0	0
$5$	0	0
$6$	9.85995	0.670885
$7$	32.7301i	1.76726i	0.468187	+	0.883629i	$0.344907\pi$
$7$	32.7301i	1.76726i	−0.468187	+	0.883629i	$0.655093\pi$
$8$	22.2935i	0.985244i
$9$	15.1845	0.562388
$10$	0	0
$11$	26.7049	0.731985	0.365993	−	0.930618i	$-0.380730\pi$
$11$	26.7049	0.731985	0.365993	+	0.930618i	$0.380730\pi$
$12$	0.783832i	0.0188561i
$13$	14.4956i	0.309259i	0.987973	+	0.154630i	$0.0494184\pi$
$13$	14.4956i	0.309259i	−0.987973	+	0.154630i	$0.950582\pi$
$14$	−93.8848	−1.79227
$15$	0	0
$16$	−65.7723	−1.02769
$17$	24.7016i	0.352412i	0.984353	+	0.176206i	$0.0563825\pi$
$17$	24.7016i	0.352412i	−0.984353	+	0.176206i	$0.943617\pi$
$18$	43.5560i	0.570347i
$19$	−94.6224	−1.14252	−0.571259	−	0.820770i	$-0.693545\pi$
$19$	−94.6224	−1.14252	−0.571259	+	0.820770i	$0.693545\pi$
$20$	0	0
$21$	112.505	1.16908
$22$	76.6018i	0.742344i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	76.6312	0.651762
$25$	0	0
$26$	−41.5801	−0.313636
$27$	− 145.004i	− 1.03355i
$28$	− 7.46352i	− 0.0503740i
$29$	57.5965	0.368807	0.184403	−	0.982851i	$-0.440965\pi$
$29$	57.5965	0.368807	0.184403	+	0.982851i	$0.440965\pi$
$30$	0	0
$31$	88.8691	0.514882	0.257441	−	0.966294i	$-0.417121\pi$
$31$	88.8691	0.514882	0.257441	+	0.966294i	$0.417121\pi$
$32$	− 10.3165i	− 0.0569909i
$33$	− 91.7948i	− 0.484225i
$34$	−70.8553	−0.357400
$35$	0	0
$36$	−3.46255	−0.0160303
$37$	− 305.467i	− 1.35726i	−0.734482	−	0.678629i	$-0.762575\pi$
$37$	− 305.467i	− 1.35726i	0.734482	−	0.678629i	$-0.237425\pi$
$38$	− 271.420i	− 1.15869i
$39$	49.8269	0.204582
$40$	0	0
$41$	−179.205	−0.682613	−0.341306	−	0.939952i	$-0.610869\pi$
$41$	−179.205	−0.682613	−0.341306	+	0.939952i	$0.610869\pi$
$42$	322.717i	1.18563i
$43$	96.5826i	0.342528i	0.985225	+	0.171264i	$0.0547851\pi$
$43$	96.5826i	0.342528i	−0.985225	+	0.171264i	$0.945215\pi$
$44$	−6.08959	−0.0208645
$45$	0	0
$46$	−65.9745	−0.211465
$47$	218.484i	0.678067i	0.940774	+	0.339034i	$0.110100\pi$
$47$	218.484i	0.678067i	−0.940774	+	0.339034i	$0.889900\pi$
$48$	226.084i	0.679841i
$49$	−728.258	−2.12320
$50$	0	0
$51$	84.9085	0.233129
$52$	− 3.30548i	− 0.00881514i
$53$	519.174i	1.34555i	0.739848	+	0.672774i	$0.234897\pi$
$53$	519.174i	1.34555i	−0.739848	+	0.672774i	$0.765103\pi$
$54$	415.937	1.04818
$55$	0	0
$56$	−729.669	−1.74118
$57$	325.252i	0.755802i
$58$	165.213i	0.374026i
$59$	37.2884	0.0822803	0.0411401	−	0.999153i	$-0.486901\pi$
$59$	37.2884	0.0822803	0.0411401	+	0.999153i	$0.486901\pi$
$60$	0	0
$61$	96.3052	0.202141	0.101071	−	0.994879i	$-0.467773\pi$
$61$	96.3052	0.202141	0.101071	+	0.994879i	$0.467773\pi$
$62$	254.917i	0.522169i
$63$	496.989i	0.993884i
$64$	−496.586	−0.969894
$65$	0	0
$66$	263.309	0.491077
$67$	497.552i	0.907248i	0.891193	+	0.453624i	$0.149869\pi$
$67$	497.552i	0.907248i	−0.891193	+	0.453624i	$0.850131\pi$
$68$	− 5.63276i	− 0.0100452i
$69$	79.0596	0.137937
$70$	0	0
$71$	−19.6235	−0.0328011	−0.0164005	−	0.999866i	$-0.505221\pi$
$71$	−19.6235	−0.0328011	−0.0164005	+	0.999866i	$0.505221\pi$
$72$	338.515i	0.554089i
$73$	208.235i	0.333865i	0.985968	+	0.166932i	$0.0533861\pi$
$73$	208.235i	0.333865i	−0.985968	+	0.166932i	$0.946614\pi$
$74$	876.219	1.37647
$75$	0	0
$76$	21.5770	0.0325664
$77$	874.054i	1.29361i
$78$	142.926i	0.207477i
$79$	446.200	0.635461	0.317730	−	0.948181i	$-0.397079\pi$
$79$	446.200	0.635461	0.317730	+	0.948181i	$0.397079\pi$
$80$	0	0
$81$	−88.4513	−0.121332
$82$	− 514.042i	− 0.692273i
$83$	− 501.151i	− 0.662752i	−0.943499	−	0.331376i	$-0.892487\pi$
$83$	− 501.151i	− 0.662752i	0.943499	−	0.331376i	$-0.107513\pi$
$84$	−25.6549	−0.0333236
$85$	0	0
$86$	−277.043	−0.347375
$87$	− 197.981i	− 0.243974i
$88$	595.347i	0.721184i
$89$	−1102.82	−1.31347	−0.656733	−	0.754124i	$-0.728062\pi$
$89$	−1102.82	−1.31347	−0.656733	+	0.754124i	$0.728062\pi$
$90$	0	0
$91$	−474.444	−0.546541
$92$	− 5.24475i	− 0.00594351i
$93$	− 305.476i	− 0.340606i
$94$	−626.711	−0.687663
$95$	0	0
$96$	−35.4615	−0.0377008
$97$	1814.37i	1.89919i	0.313478	+	0.949595i	$0.398506\pi$
$97$	1814.37i	1.89919i	−0.313478	+	0.949595i	$0.601494\pi$
$98$	− 2088.98i	− 2.15325i
$99$	405.500	0.411659
$100$	0	0
$101$	−1386.24	−1.36570	−0.682852	−	0.730557i	$-0.739261\pi$
$101$	−1386.24	−1.36570	−0.682852	+	0.730557i	$0.739261\pi$
$102$	243.556i	0.236428i
$103$	− 1372.33i	− 1.31281i	−0.754407	−	0.656407i	$-0.772075\pi$
$103$	− 1372.33i	− 1.31281i	0.754407	−	0.656407i	$-0.227925\pi$
$104$	−323.159	−0.304696
$105$	0	0
$106$	−1489.23	−1.36459
$107$	1657.83i	1.49784i	0.662662	+	0.748919i	$0.269427\pi$
$107$	1657.83i	1.49784i	−0.662662	+	0.748919i	$0.730573\pi$
$108$	33.0656i	0.0294605i
$109$	821.369	0.721770	0.360885	−	0.932610i	$-0.382475\pi$
$109$	821.369	0.721770	0.360885	+	0.932610i	$0.382475\pi$
$110$	0	0
$111$	−1050.01	−0.897857
$112$	− 2152.73i	− 1.81620i
$113$	267.142i	0.222395i	0.993798	+	0.111197i	$0.0354686\pi$
$113$	267.142i	0.222395i	−0.993798	+	0.111197i	$0.964531\pi$
$114$	−932.972	−0.766498
$115$	0	0
$116$	−13.1339	−0.0105125
$117$	220.109i	0.173924i
$118$	106.960i	0.0834447i
$119$	−808.484	−0.622803
$120$	0	0
$121$	−617.847	−0.464198
$122$	276.247i	0.205002i
$123$	615.995i	0.451564i
$124$	−20.2650	−0.0146762
$125$	0	0
$126$	−1425.59	−1.00795
$127$	− 1446.21i	− 1.01048i	−0.862979	−	0.505239i	$-0.831404\pi$
$127$	− 1446.21i	− 1.01048i	0.862979	−	0.505239i	$-0.168596\pi$
$128$	− 1506.97i	− 1.04061i
$129$	331.990	0.226590
$130$	0	0
$131$	1459.33	0.973297	0.486649	−	0.873598i	$-0.338219\pi$
$131$	1459.33	0.973297	0.486649	+	0.873598i	$0.338219\pi$
$132$	20.9322i	0.0138024i
$133$	− 3097.00i	− 2.01913i
$134$	−1427.20	−0.920088
$135$	0	0
$136$	−550.685	−0.347212
$137$	− 1889.31i	− 1.17821i	−0.808056	−	0.589105i	$-0.799480\pi$
$137$	− 1889.31i	− 1.17821i	0.808056	−	0.589105i	$-0.200520\pi$
$138$	226.779i	0.139889i
$139$	1506.31	0.919159	0.459580	−	0.888137i	$-0.348000\pi$
$139$	1506.31	0.919159	0.459580	+	0.888137i	$0.348000\pi$
$140$	0	0
$141$	751.011	0.448557
$142$	− 56.2890i	− 0.0332653i
$143$	387.105i	0.226373i
$144$	−998.717	−0.577961
$145$	0	0
$146$	−597.314	−0.338589
$147$	2503.30i	1.40455i
$148$	69.6565i	0.0386873i
$149$	2946.38	1.61998	0.809989	−	0.586444i	$-0.199473\pi$
$149$	2946.38	1.61998	0.809989	+	0.586444i	$0.199473\pi$
$150$	0	0
$151$	1979.37	1.06675	0.533374	−	0.845880i	$-0.320924\pi$
$151$	1979.37	1.06675	0.533374	+	0.845880i	$0.320924\pi$
$152$	− 2109.47i	− 1.12566i
$153$	375.080i	0.198192i
$154$	−2507.18	−1.31191
$155$	0	0
$156$	−11.3622	−0.00583142
$157$	505.403i	0.256915i	0.991715	+	0.128457i	$0.0410025\pi$
$157$	505.403i	0.256915i	−0.991715	+	0.128457i	$0.958997\pi$
$158$	1279.90i	0.644454i
$159$	1784.59	0.890110
$160$	0	0
$161$	−752.792	−0.368499
$162$	− 253.719i	− 0.123049i
$163$	2604.26i	1.25142i	0.780055	+	0.625711i	$0.215191\pi$
$163$	2604.26i	1.25142i	−0.780055	+	0.625711i	$0.784809\pi$
$164$	40.8646	0.0194572
$165$	0	0
$166$	1437.53	0.672131
$167$	− 845.304i	− 0.391686i	−0.980635	−	0.195843i	$-0.937256\pi$
$167$	− 845.304i	− 0.391686i	0.980635	−	0.195843i	$-0.0627443\pi$
$168$	2508.15i	1.15183i
$169$	1986.88	0.904359
$170$	0	0
$171$	−1436.79	−0.642539
$172$	− 22.0240i	− 0.00976343i
$173$	− 886.843i	− 0.389742i	−0.980829	−	0.194871i	$-0.937571\pi$
$173$	− 886.843i	− 0.389742i	0.980829	−	0.194871i	$-0.0624289\pi$
$174$	567.898	0.247427
$175$	0	0
$176$	−1756.44	−0.752255
$177$	− 128.174i	− 0.0544303i
$178$	− 3163.38i	− 1.33205i
$179$	−1103.01	−0.460575	−0.230288	−	0.973123i	$-0.573967\pi$
$179$	−1103.01	−0.460575	−0.230288	+	0.973123i	$0.573967\pi$
$180$	0	0
$181$	−1200.43	−0.492967	−0.246483	−	0.969147i	$-0.579275\pi$
$181$	−1200.43	−0.492967	−0.246483	+	0.969147i	$0.579275\pi$
$182$	− 1360.92i	− 0.554275i
$183$	− 331.037i	− 0.133721i
$184$	−512.751	−0.205438
$185$	0	0
$186$	876.244	0.345427
$187$	659.653i	0.257961i
$188$	− 49.8214i	− 0.0193277i
$189$	4745.98	1.82656
$190$	0	0
$191$	2415.93	0.915237	0.457618	−	0.889149i	$-0.348703\pi$
$191$	2415.93	0.915237	0.457618	+	0.889149i	$0.348703\pi$
$192$	1706.95i	0.641607i
$193$	− 232.884i	− 0.0868568i	−0.999057	−	0.0434284i	$-0.986172\pi$
$193$	− 232.884i	− 0.0868568i	0.999057	−	0.0434284i	$-0.0138280\pi$
$194$	−5204.44	−1.92607
$195$	0	0
$196$	166.067	0.0605199
$197$	− 1418.48i	− 0.513008i	−0.966543	−	0.256504i	$-0.917429\pi$
$197$	− 1418.48i	− 0.513008i	0.966543	−	0.256504i	$-0.0825707\pi$
$198$	1163.16i	0.417485i
$199$	−1068.21	−0.380520	−0.190260	−	0.981734i	$-0.560933\pi$
$199$	−1068.21	−0.380520	−0.190260	+	0.981734i	$0.560933\pi$
$200$	0	0
$201$	1710.27	0.600165
$202$	− 3976.37i	− 1.38503i
$203$	1885.14i	0.651777i
$204$	−19.3619	−0.00664511
$205$	0	0
$206$	3936.47	1.33139
$207$	349.243i	0.117266i
$208$	− 953.412i	− 0.317823i
$209$	−2526.88	−0.836307
$210$	0	0
$211$	−1537.74	−0.501717	−0.250859	−	0.968024i	$-0.580713\pi$
$211$	−1537.74	−0.501717	−0.250859	+	0.968024i	$0.580713\pi$
$212$	− 118.388i	− 0.0383535i
$213$	67.4532i	0.0216987i
$214$	−4755.41	−1.51903
$215$	0	0
$216$	3232.65	1.01830
$217$	2908.69i	0.909930i
$218$	2356.06i	0.731984i
$219$	715.783	0.220859
$220$	0	0
$221$	−358.065	−0.108987
$222$	− 3011.89i	− 0.910563i
$223$	5359.68i	1.60946i	0.593638	+	0.804732i	$0.297691\pi$
$223$	5359.68i	1.60946i	−0.593638	+	0.804732i	$0.702309\pi$
$224$	337.658	0.100718
$225$	0	0
$226$	−766.285	−0.225542
$227$	1108.36i	0.324072i	0.986785	+	0.162036i	$0.0518061\pi$
$227$	1108.36i	0.324072i	−0.986785	+	0.162036i	$0.948194\pi$
$228$	− 74.1681i	− 0.0215434i
$229$	5046.34	1.45621	0.728104	−	0.685467i	$-0.240402\pi$
$229$	5046.34	1.45621	0.728104	+	0.685467i	$0.240402\pi$
$230$	0	0
$231$	3004.45	0.855750
$232$	1284.03i	0.363365i
$233$	− 7102.11i	− 1.99689i	−0.0557635	−	0.998444i	$-0.517759\pi$
$233$	− 7102.11i	− 1.99689i	0.0557635	−	0.998444i	$-0.482241\pi$
$234$	−631.372	−0.176385
$235$	0	0
$236$	−8.50296	−0.00234532
$237$	− 1533.75i	− 0.420372i
$238$	− 2319.10i	− 0.631617i
$239$	−1556.22	−0.421187	−0.210594	−	0.977574i	$-0.567540\pi$
$239$	−1556.22	−0.421187	−0.210594	+	0.977574i	$0.567540\pi$
$240$	0	0
$241$	3028.88	0.809573	0.404787	−	0.914411i	$-0.367346\pi$
$241$	3028.88	0.809573	0.404787	+	0.914411i	$0.367346\pi$
$242$	− 1772.27i	− 0.470767i
$243$	− 3611.06i	− 0.953291i
$244$	−21.9607	−0.00576184
$245$	0	0
$246$	−1766.95	−0.457954
$247$	− 1371.61i	− 0.353334i
$248$	1981.21i	0.507285i
$249$	−1722.64	−0.438426
$250$	0	0
$251$	1449.39	0.364480	0.182240	−	0.983254i	$-0.441665\pi$
$251$	1449.39	0.364480	0.182240	+	0.983254i	$0.441665\pi$
$252$	− 113.330i	− 0.0283297i
$253$	614.213i	0.152629i
$254$	4148.40	1.02478
$255$	0	0
$256$	349.976	0.0854433
$257$	− 3099.62i	− 0.752332i	−0.926552	−	0.376166i	$-0.877242\pi$
$257$	− 3099.62i	− 0.752332i	0.926552	−	0.376166i	$-0.122758\pi$
$258$	952.299i	0.229797i
$259$	9997.97	2.39862
$260$	0	0
$261$	874.572	0.207412
$262$	4186.01i	0.987071i
$263$	1087.92i	0.255073i	0.991834	+	0.127537i	$0.0407070\pi$
$263$	1087.92i	0.255073i	−0.991834	+	0.127537i	$0.959293\pi$
$264$	2046.43	0.477080
$265$	0	0
$266$	8883.60	2.04770
$267$	3790.79i	0.868887i
$268$	− 113.458i	− 0.0258603i
$269$	−4721.01	−1.07006	−0.535028	−	0.844834i	$-0.679699\pi$
$269$	−4721.01	−1.07006	−0.535028	+	0.844834i	$0.679699\pi$
$270$	0	0
$271$	1401.15	0.314074	0.157037	−	0.987593i	$-0.449806\pi$
$271$	1401.15	0.314074	0.157037	+	0.987593i	$0.449806\pi$
$272$	− 1624.68i	− 0.362171i
$273$	1630.84i	0.361549i
$274$	5419.41	1.19488
$275$	0	0
$276$	−18.0281	−0.00393176
$277$	4122.82i	0.894283i	0.894463	+	0.447142i	$0.147558\pi$
$277$	4122.82i	0.894283i	−0.894463	+	0.447142i	$0.852442\pi$
$278$	4320.77i	0.932167i
$279$	1349.43	0.289564
$280$	0	0
$281$	803.897	0.170664	0.0853318	−	0.996353i	$-0.472805\pi$
$281$	803.897	0.170664	0.0853318	+	0.996353i	$0.472805\pi$
$282$	2154.24i	0.454905i
$283$	3147.53i	0.661134i	0.943782	+	0.330567i	$0.107240\pi$
$283$	3147.53i	0.661134i	−0.943782	+	0.330567i	$0.892760\pi$
$284$	4.47479	0.000934964 0
$285$	0	0
$286$	−1110.39	−0.229577
$287$	− 5865.40i	− 1.20635i
$288$	− 156.650i	− 0.0320510i
$289$	4302.83	0.875806
$290$	0	0
$291$	6236.67	1.25636
$292$	− 47.4844i	− 0.00951649i
$293$	2812.88i	0.560854i	0.959875	+	0.280427i	$0.0904761\pi$
$293$	2812.88i	0.560854i	−0.959875	+	0.280427i	$0.909524\pi$
$294$	−7180.59	−1.42442
$295$	0	0
$296$	6809.95	1.33723
$297$	− 3872.31i	− 0.756547i
$298$	8451.56i	1.64290i
$299$	−333.400	−0.0644850
$300$	0	0
$301$	−3161.15	−0.605335
$302$	5677.73i	1.08184i
$303$	4765.02i	0.903444i
$304$	6223.53	1.17416
$305$	0	0
$306$	−1075.90	−0.200997
$307$	− 7885.52i	− 1.46596i	−0.680249	−	0.732981i	$-0.738128\pi$
$307$	− 7885.52i	− 1.46596i	0.680249	−	0.732981i	$-0.261872\pi$
$308$	− 199.313i	− 0.0368730i
$309$	−4717.22	−0.868457
$310$	0	0
$311$	−1904.46	−0.347240	−0.173620	−	0.984813i	$-0.555547\pi$
$311$	−1904.46	−0.347240	−0.173620	+	0.984813i	$0.555547\pi$
$312$	1110.82i	0.201563i
$313$	3854.89i	0.696137i	0.937469	+	0.348069i	$0.113162\pi$
$313$	3854.89i	0.696137i	−0.937469	+	0.348069i	$0.886838\pi$
$314$	−1449.73	−0.260550
$315$	0	0
$316$	−101.748	−0.0181132
$317$	1878.70i	0.332865i	0.986053	+	0.166432i	$0.0532247\pi$
$317$	1878.70i	0.332865i	−0.986053	+	0.166432i	$0.946775\pi$
$318$	5119.03i	0.902707i
$319$	1538.11	0.269961
$320$	0	0
$321$	5698.58	0.990853
$322$	− 2159.35i	− 0.373714i
$323$	− 2337.32i	− 0.402638i
$324$	20.1698	0.00345846
$325$	0	0
$326$	−7470.21	−1.26913
$327$	− 2823.35i	− 0.477467i
$328$	− 3995.11i	− 0.672541i
$329$	−7151.00	−1.19832
$330$	0	0
$331$	6829.96	1.13417	0.567083	−	0.823661i	$-0.308072\pi$
$331$	6829.96	1.13417	0.567083	+	0.823661i	$0.308072\pi$
$332$	114.279i	0.0188911i
$333$	− 4638.36i	− 0.763305i
$334$	2424.72	0.397229
$335$	0	0
$336$	−7399.74	−1.20146
$337$	11048.9i	1.78597i	0.450087	+	0.892985i	$0.351393\pi$
$337$	11048.9i	1.78597i	−0.450087	+	0.892985i	$0.648607\pi$
$338$	5699.26i	0.917157i
$339$	918.267	0.147119
$340$	0	0
$341$	2373.24	0.376886
$342$	− 4121.37i	− 0.651632i
$343$	− 12609.5i	− 1.98499i
$344$	−2153.17	−0.337474
$345$	0	0
$346$	2543.87	0.395258
$347$	− 7093.14i	− 1.09735i	−0.836037	−	0.548674i	$-0.815133\pi$
$347$	− 7093.14i	− 1.09735i	0.836037	−	0.548674i	$-0.184867\pi$
$348$	45.1460i	0.00695425i
$349$	5363.16	0.822588	0.411294	−	0.911503i	$-0.365077\pi$
$349$	5363.16	0.822588	0.411294	+	0.911503i	$0.365077\pi$
$350$	0	0
$351$	2101.92	0.319636
$352$	− 275.500i	− 0.0417165i
$353$	− 11789.3i	− 1.77757i	−0.458320	−	0.888787i	$-0.651549\pi$
$353$	− 11789.3i	− 1.77757i	0.458320	−	0.888787i	$-0.348451\pi$
$354$	367.662	0.0552006
$355$	0	0
$356$	251.478	0.0374391
$357$	2779.06i	0.411999i
$358$	− 3163.94i	− 0.467093i
$359$	2645.66	0.388949	0.194474	−	0.980908i	$-0.437700\pi$
$359$	2645.66	0.388949	0.194474	+	0.980908i	$0.437700\pi$
$360$	0	0
$361$	2094.39	0.305349
$362$	− 3443.37i	− 0.499943i
$363$	2123.77i	0.307077i
$364$	108.189	0.0155786
$365$	0	0
$366$	949.564	0.135613
$367$	− 9775.68i	− 1.39043i	−0.718804	−	0.695213i	$-0.755310\pi$
$367$	− 9775.68i	− 1.39043i	0.718804	−	0.695213i	$-0.244690\pi$
$368$	− 1512.76i	− 0.214289i
$369$	−2721.13	−0.383893
$370$	0	0
$371$	−16992.6	−2.37793
$372$	69.6585i	0.00970866i
$373$	− 10981.1i	− 1.52435i	−0.647373	−	0.762174i	$-0.724132\pi$
$373$	− 10981.1i	− 1.52435i	0.647373	−	0.762174i	$-0.275868\pi$
$374$	−1892.19	−0.261611
$375$	0	0
$376$	−4870.78	−0.668062
$377$	834.899i	0.114057i
$378$	13613.6i	1.85241i
$379$	2313.77	0.313589	0.156795	−	0.987631i	$-0.449884\pi$
$379$	2313.77	0.313589	0.156795	+	0.987631i	$0.449884\pi$
$380$	0	0
$381$	−4971.18	−0.668455
$382$	6929.97i	0.928189i
$383$	− 9219.25i	− 1.22998i	−0.788535	−	0.614989i	$-0.789160\pi$
$383$	− 9219.25i	− 1.22998i	0.788535	−	0.614989i	$-0.210840\pi$
$384$	−5180.00	−0.688388
$385$	0	0
$386$	668.017	0.0880860
$387$	1466.55i	0.192633i
$388$	− 413.736i	− 0.0541346i
$389$	−5876.90	−0.765991	−0.382996	−	0.923750i	$-0.625108\pi$
$389$	−5876.90	−0.765991	−0.382996	+	0.923750i	$0.625108\pi$
$390$	0	0
$391$	−568.136	−0.0734830
$392$	− 16235.5i	− 2.09187i
$393$	− 5016.25i	− 0.643858i
$394$	4068.85	0.520268
$395$	0	0
$396$	−92.4671	−0.0117340
$397$	14268.6i	1.80383i	0.431911	+	0.901916i	$0.357840\pi$
$397$	14268.6i	1.80383i	−0.431911	+	0.901916i	$0.642160\pi$
$398$	− 3064.12i	− 0.385905i
$399$	−10645.5	−1.33570
$400$	0	0
$401$	−11556.1	−1.43911	−0.719557	−	0.694434i	$-0.755655\pi$
$401$	−11556.1	−1.43911	−0.719557	+	0.694434i	$0.755655\pi$
$402$	4905.84i	0.608659i
$403$	1288.21i	0.159232i
$404$	316.108	0.0389281
$405$	0	0
$406$	−5407.43	−0.661001
$407$	− 8157.48i	− 0.993492i
$408$	1892.91i	0.229689i
$409$	14148.3	1.71049	0.855245	−	0.518223i	$-0.173406\pi$
$409$	14148.3	1.71049	0.855245	+	0.518223i	$0.173406\pi$
$410$	0	0
$411$	−6494.27	−0.779413
$412$	312.936i	0.0374205i
$413$	1220.45i	0.145410i
$414$	−1001.79	−0.118925
$415$	0	0
$416$	149.544	0.0176250
$417$	− 5177.73i	− 0.608045i
$418$	− 7248.25i	− 0.848142i
$419$	2222.16	0.259092	0.129546	−	0.991573i	$-0.458648\pi$
$419$	2222.16	0.259092	0.129546	+	0.991573i	$0.458648\pi$
$420$	0	0
$421$	2518.51	0.291555	0.145777	−	0.989317i	$-0.453432\pi$
$421$	2518.51	0.291555	0.145777	+	0.989317i	$0.453432\pi$
$422$	− 4410.94i	− 0.508818i
$423$	3317.56i	0.381337i
$424$	−11574.2	−1.32569
$425$	0	0
$426$	−193.486	−0.0220057
$427$	3152.08i	0.357236i
$428$	− 378.039i	− 0.0426944i
$429$	1330.62	0.149751
$430$	0	0
$431$	9935.17	1.11035	0.555174	−	0.831734i	$-0.312652\pi$
$431$	9935.17	1.11035	0.555174	+	0.831734i	$0.312652\pi$
$432$	9537.22i	1.06218i
$433$	8427.47i	0.935331i	0.883906	+	0.467665i	$0.154905\pi$
$433$	8427.47i	0.935331i	−0.883906	+	0.467665i	$0.845095\pi$
$434$	−8343.45	−0.922807
$435$	0	0
$436$	−187.299	−0.0205734
$437$	− 2176.31i	− 0.238232i
$438$	2053.19i	0.223985i
$439$	−4886.04	−0.531203	−0.265601	−	0.964083i	$-0.585570\pi$
$439$	−4886.04	−0.531203	−0.265601	+	0.964083i	$0.585570\pi$
$440$	0	0
$441$	−11058.2	−1.19406
$442$	− 1027.09i	− 0.110529i
$443$	− 9724.04i	− 1.04290i	−0.853283	−	0.521448i	$-0.825392\pi$
$443$	− 9724.04i	− 1.04290i	0.853283	−	0.521448i	$-0.174608\pi$
$444$	239.435	0.0255926
$445$	0	0
$446$	−15374.0	−1.63224
$447$	− 10127.8i	− 1.07165i
$448$	− 16253.3i	− 1.71405i
$449$	10093.7	1.06091	0.530457	−	0.847712i	$-0.322020\pi$
$449$	10093.7	1.06091	0.530457	+	0.847712i	$0.322020\pi$
$450$	0	0
$451$	−4785.66	−0.499662
$452$	− 60.9171i	− 0.00633915i
$453$	− 6803.83i	− 0.705678i
$454$	−3179.28	−0.328658
$455$	0	0
$456$	−7251.02	−0.744650
$457$	3561.77i	0.364578i	0.983245	+	0.182289i	$0.0583507\pi$
$457$	3561.77i	0.364578i	−0.983245	+	0.182289i	$0.941649\pi$
$458$	14475.2i	1.47682i
$459$	3581.82	0.364237
$460$	0	0
$461$	−7492.00	−0.756914	−0.378457	−	0.925619i	$-0.623545\pi$
$461$	−7492.00	−0.756914	−0.378457	+	0.925619i	$0.623545\pi$
$462$	8618.13i	0.867861i
$463$	− 15427.0i	− 1.54849i	−0.632884	−	0.774247i	$-0.718129\pi$
$463$	− 15427.0i	− 1.54849i	0.632884	−	0.774247i	$-0.281871\pi$
$464$	−3788.25	−0.379020
$465$	0	0
$466$	20372.1	2.02515
$467$	11868.7i	1.17606i	0.808841	+	0.588028i	$0.200096\pi$
$467$	11868.7i	1.17606i	−0.808841	+	0.588028i	$0.799904\pi$
$468$	− 50.1919i	− 0.00495753i
$469$	−16284.9	−1.60334
$470$	0	0
$471$	1737.26	0.169955
$472$	831.290i	0.0810662i
$473$	2579.23i	0.250725i
$474$	4399.51	0.426321
$475$	0	0
$476$	184.361	0.0177524
$477$	7883.38i	0.756719i
$478$	− 4463.96i	− 0.427148i
$479$	−697.153	−0.0665005	−0.0332503	−	0.999447i	$-0.510586\pi$
$479$	−697.153	−0.0665005	−0.0332503	+	0.999447i	$0.510586\pi$
$480$	0	0
$481$	4427.95	0.419744
$482$	8688.20i	0.821030i
$483$	2587.63i	0.243770i
$484$	140.889	0.0132315
$485$	0	0
$486$	10358.2	0.966782
$487$	14414.0i	1.34119i	0.741822	+	0.670597i	$0.233962\pi$
$487$	14414.0i	1.34119i	−0.741822	+	0.670597i	$0.766038\pi$
$488$	2146.98i	0.199159i
$489$	8951.83	0.827844
$490$	0	0
$491$	17538.9	1.61206	0.806029	−	0.591876i	$-0.201612\pi$
$491$	17538.9	1.61206	0.806029	+	0.591876i	$0.201612\pi$
$492$	− 140.467i	− 0.0128714i
$493$	1422.72i	0.129972i
$494$	3934.41	0.358335
$495$	0	0
$496$	−5845.12	−0.529140
$497$	− 642.278i	− 0.0579680i
$498$	− 4941.32i	− 0.444630i
$499$	18798.6	1.68645	0.843226	−	0.537559i	$-0.180653\pi$
$499$	18798.6	1.68645	0.843226	+	0.537559i	$0.180653\pi$
$500$	0	0
$501$	−2905.62	−0.259109
$502$	4157.50i	0.369638i
$503$	− 4634.11i	− 0.410785i	−0.978680	−	0.205392i	$-0.934153\pi$
$503$	− 4634.11i	− 0.410785i	0.978680	−	0.205392i	$-0.0658470\pi$
$504$	−11079.6	−0.979219
$505$	0	0
$506$	−1761.84	−0.154789
$507$	− 6829.63i	− 0.598254i
$508$	329.784i	0.0288027i
$509$	2193.57	0.191018	0.0955092	−	0.995429i	$-0.469552\pi$
$509$	2193.57	0.191018	0.0955092	+	0.995429i	$0.469552\pi$
$510$	0	0
$511$	−6815.56	−0.590025
$512$	− 11051.8i	− 0.953958i
$513$	13720.6i	1.18086i
$514$	8891.13	0.762979
$515$	0	0
$516$	−75.7045	−0.00645873
$517$	5834.60i	0.496335i
$518$	28678.7i	2.43257i
$519$	−3048.41	−0.257823
$520$	0	0
$521$	−4401.81	−0.370147	−0.185074	−	0.982725i	$-0.559252\pi$
$521$	−4401.81	−0.370147	−0.185074	+	0.982725i	$0.559252\pi$
$522$	2508.67i	0.210348i
$523$	9974.09i	0.833913i	0.908926	+	0.416957i	$0.136903\pi$
$523$	9974.09i	0.833913i	−0.908926	+	0.416957i	$0.863097\pi$
$524$	−332.774	−0.0277429
$525$	0	0
$526$	−3120.66	−0.258683
$527$	2195.20i	0.181451i
$528$	6037.55i	0.497634i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	566.204	0.0462734
$532$	706.216i	0.0575533i
$533$	− 2597.69i	− 0.211104i
$534$	−10873.7	−0.881183
$535$	0	0
$536$	−11092.2	−0.893861
$537$	3791.46i	0.304681i
$538$	− 13542.0i	− 1.08520i
$539$	−19448.1	−1.55415
$540$	0	0
$541$	−9161.72	−0.728083	−0.364042	−	0.931383i	$-0.618603\pi$
$541$	−9161.72	−0.728083	−0.364042	+	0.931383i	$0.618603\pi$
$542$	4019.15i	0.318519i
$543$	4126.31i	0.326109i
$544$	254.833	0.0200843
$545$	0	0
$546$	−4677.99	−0.366666
$547$	1113.51i	0.0870390i	0.999053	+	0.0435195i	$0.0138571\pi$
$547$	1113.51i	0.0870390i	−0.999053	+	0.0435195i	$0.986143\pi$
$548$	430.825i	0.0335838i
$549$	1462.34	0.113682
$550$	0	0
$551$	−5449.92	−0.421369
$552$	1762.52i	0.135902i
$553$	14604.2i	1.12302i
$554$	−11826.1	−0.906939
$555$	0	0
$556$	−343.486	−0.0261998
$557$	− 7660.96i	− 0.582775i	−0.956605	−	0.291387i	$-0.905883\pi$
$557$	− 7660.96i	− 0.582775i	0.956605	−	0.291387i	$-0.0941168\pi$
$558$	3870.78i	0.293661i
$559$	−1400.03	−0.105930
$560$	0	0
$561$	2267.47	0.170647
$562$	2305.94i	0.173079i
$563$	− 17217.7i	− 1.28888i	−0.764654	−	0.644441i	$-0.777090\pi$
$563$	− 17217.7i	− 1.28888i	0.764654	−	0.644441i	$-0.222910\pi$
$564$	−171.255	−0.0127857
$565$	0	0
$566$	−9028.54	−0.670491
$567$	− 2895.02i	− 0.214426i
$568$	− 437.476i	− 0.0323171i
$569$	2385.63	0.175766	0.0878830	−	0.996131i	$-0.471990\pi$
$569$	2385.63	0.175766	0.0878830	+	0.996131i	$0.471990\pi$
$570$	0	0
$571$	16927.9	1.24065	0.620323	−	0.784347i	$-0.287002\pi$
$571$	16927.9	1.24065	0.620323	+	0.784347i	$0.287002\pi$
$572$	− 88.2725i	− 0.00645255i
$573$	− 8304.44i	− 0.605450i
$574$	16824.6	1.22343
$575$	0	0
$576$	−7540.39	−0.545457
$577$	2886.69i	0.208275i	0.994563	+	0.104137i	$0.0332081\pi$
$577$	2886.69i	0.208275i	−0.994563	+	0.104137i	$0.966792\pi$
$578$	12342.5i	0.888200i
$579$	−800.509	−0.0574578
$580$	0	0
$581$	16402.7	1.17125
$582$	17889.6i	1.27414i
$583$	13864.5i	0.984920i
$584$	−4642.30	−0.328938
$585$	0	0
$586$	−8068.62	−0.568791
$587$	503.810i	0.0354250i	0.999843	+	0.0177125i	$0.00563836\pi$
$587$	503.810i	0.0354250i	−0.999843	+	0.0177125i	$0.994362\pi$
$588$	− 570.832i	− 0.0400353i
$589$	−8409.00	−0.588263
$590$	0	0
$591$	−4875.85	−0.339366
$592$	20091.3i	1.39484i
$593$	7654.92i	0.530101i	0.964235	+	0.265050i	$0.0853886\pi$
$593$	7654.92i	0.530101i	−0.964235	+	0.265050i	$0.914611\pi$
$594$	11107.6	0.767253
$595$	0	0
$596$	−671.870	−0.0461760
$597$	3671.84i	0.251723i
$598$	− 956.343i	− 0.0653976i
$599$	−17366.5	−1.18460	−0.592301	−	0.805717i	$-0.701780\pi$
$599$	−17366.5	−1.18460	−0.592301	+	0.805717i	$0.701780\pi$
$600$	0	0
$601$	3872.81	0.262854	0.131427	−	0.991326i	$-0.458044\pi$
$601$	3872.81	0.262854	0.131427	+	0.991326i	$0.458044\pi$
$602$	− 9067.63i	− 0.613902i
$603$	7555.06i	0.510225i
$604$	−451.361	−0.0304066
$605$	0	0
$606$	−13668.3	−0.916229
$607$	− 12823.3i	− 0.857464i	−0.903432	−	0.428732i	$-0.858961\pi$
$607$	− 12823.3i	− 0.857464i	0.903432	−	0.428732i	$-0.141039\pi$
$608$	976.167i	0.0651132i
$609$	6479.92	0.431165
$610$	0	0
$611$	−3167.07	−0.209699
$612$	− 85.5304i	− 0.00564928i
$613$	17226.4i	1.13502i	0.823367	+	0.567509i	$0.192093\pi$
$613$	17226.4i	1.13502i	−0.823367	+	0.567509i	$0.807907\pi$
$614$	22619.3	1.48671
$615$	0	0
$616$	−19485.8	−1.27452
$617$	− 5892.12i	− 0.384454i	−0.981351	−	0.192227i	$-0.938429\pi$
$617$	− 5892.12i	− 0.384454i	0.981351	−	0.192227i	$-0.0615709\pi$
$618$	− 13531.1i	− 0.880747i
$619$	13036.6	0.846503	0.423251	−	0.906012i	$-0.360889\pi$
$619$	13036.6	0.846503	0.423251	+	0.906012i	$0.360889\pi$
$620$	0	0
$621$	3335.09	0.215511
$622$	− 5462.84i	− 0.352154i
$623$	− 36095.3i	− 2.32123i
$624$	−3277.23	−0.210247
$625$	0	0
$626$	−11057.6	−0.705989
$627$	8685.84i	0.553236i
$628$	− 115.248i	− 0.00732310i
$629$	7545.52	0.478314
$630$	0	0
$631$	−10044.6	−0.633707	−0.316853	−	0.948475i	$-0.602626\pi$
$631$	−10044.6	−0.633707	−0.316853	+	0.948475i	$0.602626\pi$
$632$	9947.37i	0.626084i
$633$	5285.79i	0.331898i
$634$	−5388.95	−0.337575
$635$	0	0
$636$	−406.945	−0.0253717
$637$	− 10556.6i	− 0.656620i
$638$	4412.00i	0.273782i
$639$	−297.972	−0.0184469
$640$	0	0
$641$	4583.33	0.282419	0.141209	−	0.989980i	$-0.454901\pi$
$641$	4583.33	0.282419	0.141209	+	0.989980i	$0.454901\pi$
$642$	16346.1i	1.00488i
$643$	− 19260.0i	− 1.18125i	−0.806948	−	0.590623i	$-0.798882\pi$
$643$	− 19260.0i	− 1.18125i	0.806948	−	0.590623i	$-0.201118\pi$
$644$	171.661	0.0105037
$645$	0	0
$646$	6704.50	0.408336
$647$	− 1771.91i	− 0.107667i	−0.998550	−	0.0538337i	$-0.982856\pi$
$647$	− 1771.91i	− 0.107667i	0.998550	−	0.0538337i	$-0.0171441\pi$
$648$	− 1971.89i	− 0.119542i
$649$	995.784	0.0602279
$650$	0	0
$651$	9998.26	0.601940
$652$	− 593.857i	− 0.0356706i
$653$	28000.8i	1.67803i	0.544106	+	0.839017i	$0.316869\pi$
$653$	28000.8i	1.67803i	−0.544106	+	0.839017i	$0.683131\pi$
$654$	8098.66	0.484224
$655$	0	0
$656$	11786.7	0.701515
$657$	3161.94i	0.187761i
$658$	− 20512.3i	− 1.21528i
$659$	27664.0	1.63526	0.817629	−	0.575745i	$-0.195288\pi$
$659$	27664.0	1.63526	0.817629	+	0.575745i	$0.195288\pi$
$660$	0	0
$661$	23392.1	1.37647	0.688234	−	0.725489i	$-0.258386\pi$
$661$	23392.1	1.37647	0.688234	+	0.725489i	$0.258386\pi$
$662$	19591.4i	1.15022i
$663$	1230.80i	0.0720972i
$664$	11172.4	0.652973
$665$	0	0
$666$	13304.9	0.774107
$667$	1324.72i	0.0769016i
$668$	192.757i	0.0111646i
$669$	18423.2	1.06470
$670$	0	0
$671$	2571.82	0.147964
$672$	− 1160.66i	− 0.0666270i
$673$	− 4318.41i	− 0.247344i	−0.992323	−	0.123672i	$-0.960533\pi$
$673$	− 4318.41i	− 0.247344i	0.992323	−	0.123672i	$-0.0394670\pi$
$674$	−31693.3	−1.81124
$675$	0	0
$676$	−453.072	−0.0257779
$677$	18272.8i	1.03734i	0.854974	+	0.518671i	$0.173573\pi$
$677$	18272.8i	1.03734i	−0.854974	+	0.518671i	$0.826427\pi$
$678$	2634.01i	0.149201i
$679$	−59384.5	−3.35636
$680$	0	0
$681$	3809.84	0.214381
$682$	6807.53i	0.382220i
$683$	7245.38i	0.405910i	0.979188	+	0.202955i	$0.0650546\pi$
$683$	7245.38i	0.405910i	−0.979188	+	0.202955i	$0.934945\pi$
$684$	327.635	0.0183150
$685$	0	0
$686$	36169.9	2.01308
$687$	− 17346.2i	− 0.963314i
$688$	− 6352.45i	− 0.352013i
$689$	−7525.76	−0.416123
$690$	0	0
$691$	−12055.2	−0.663679	−0.331839	−	0.943336i	$-0.607669\pi$
$691$	−12055.2	−0.663679	−0.331839	+	0.943336i	$0.607669\pi$
$692$	202.229i	0.0111092i
$693$	13272.0i	0.727509i
$694$	20346.3	1.11288
$695$	0	0
$696$	4413.69	0.240374
$697$	− 4426.64i	− 0.240561i
$698$	15384.0i	0.834229i
$699$	−24412.6	−1.32099
$700$	0	0
$701$	−15301.9	−0.824455	−0.412228	−	0.911081i	$-0.635249\pi$
$701$	−15301.9	−0.824455	−0.412228	+	0.911081i	$0.635249\pi$
$702$	6029.27i	0.324160i
$703$	28904.0i	1.55069i
$704$	−13261.3	−0.709948
$705$	0	0
$706$	33817.2	1.80273
$707$	− 45371.8i	− 2.41355i
$708$	29.2279i	0.00155148i
$709$	12173.6	0.644835	0.322418	−	0.946597i	$-0.395504\pi$
$709$	12173.6	0.644835	0.322418	+	0.946597i	$0.395504\pi$
$710$	0	0
$711$	6775.30	0.357375
$712$	− 24585.7i	− 1.29408i
$713$	2043.99i	0.107360i
$714$	−7971.61	−0.417829
$715$	0	0
$716$	251.522	0.0131283
$717$	5349.32i	0.278625i
$718$	7588.95i	0.394453i
$719$	−2639.63	−0.136914	−0.0684572	−	0.997654i	$-0.521808\pi$
$719$	−2639.63	−0.136914	−0.0684572	+	0.997654i	$0.521808\pi$
$720$	0	0
$721$	44916.5	2.32008
$722$	6007.67i	0.309671i
$723$	− 10411.4i	− 0.535551i
$724$	273.736	0.0140515
$725$	0	0
$726$	−6091.94	−0.311423
$727$	− 32759.5i	− 1.67123i	−0.549319	−	0.835613i	$-0.685113\pi$
$727$	− 32759.5i	− 1.67123i	0.549319	−	0.835613i	$-0.314887\pi$
$728$	− 10577.0i	− 0.538476i
$729$	−14800.7	−0.751956
$730$	0	0
$731$	−2385.74	−0.120711
$732$	75.4871i	0.00381159i
$733$	− 29178.0i	− 1.47028i	−0.677917	−	0.735139i	$-0.737117\pi$
$733$	− 29178.0i	− 1.47028i	0.677917	−	0.735139i	$-0.262883\pi$
$734$	28041.1	1.41010
$735$	0	0
$736$	237.278	0.0118834
$737$	13287.1i	0.664092i
$738$	− 7805.45i	− 0.389326i
$739$	13704.9	0.682194	0.341097	−	0.940028i	$-0.389202\pi$
$739$	13704.9	0.682194	0.341097	+	0.940028i	$0.389202\pi$
$740$	0	0
$741$	−4714.74	−0.233739
$742$	− 48742.5i	− 2.41158i
$743$	991.593i	0.0489610i	0.999700	+	0.0244805i	$0.00779317\pi$
$743$	991.593i	0.0489610i	−0.999700	+	0.0244805i	$0.992207\pi$
$744$	6810.14	0.335581
$745$	0	0
$746$	31498.9	1.54592
$747$	− 7609.71i	− 0.372724i
$748$	− 150.422i	− 0.00735292i
$749$	−54260.9	−2.64706
$750$	0	0
$751$	9440.73	0.458718	0.229359	−	0.973342i	$-0.426337\pi$
$751$	9440.73	0.458718	0.229359	+	0.973342i	$0.426337\pi$
$752$	− 14370.2i	− 0.696844i
$753$	− 4982.09i	− 0.241112i
$754$	−2394.87	−0.115671
$755$	0	0
$756$	−1082.24	−0.0520643
$757$	10480.1i	0.503177i	0.967834	+	0.251589i	$0.0809530\pi$
$757$	10480.1i	0.503177i	−0.967834	+	0.251589i	$0.919047\pi$
$758$	6636.94i	0.318027i
$759$	2111.28	0.100968
$760$	0	0
$761$	−31314.9	−1.49168	−0.745838	−	0.666128i	$-0.767951\pi$
$761$	−31314.9	−1.49168	−0.745838	+	0.666128i	$0.767951\pi$
$762$	− 14259.6i	− 0.677915i
$763$	26883.5i	1.27555i
$764$	−550.909	−0.0260880
$765$	0	0
$766$	26445.0	1.24739
$767$	540.519i	0.0254459i
$768$	− 1203.00i	− 0.0565227i
$769$	20862.1	0.978292	0.489146	−	0.872202i	$-0.337309\pi$
$769$	20862.1	0.978292	0.489146	+	0.872202i	$0.337309\pi$
$770$	0	0
$771$	−10654.6	−0.497684
$772$	53.1051i	0.00247577i
$773$	20340.2i	0.946426i	0.880948	+	0.473213i	$0.156906\pi$
$773$	20340.2i	0.946426i	−0.880948	+	0.473213i	$0.843094\pi$
$774$	−4206.75	−0.195360
$775$	0	0
$776$	−40448.8	−1.87117
$777$	− 34366.8i	− 1.58674i
$778$	− 16857.6i	− 0.776832i
$779$	16956.8	0.779898
$780$	0	0
$781$	−524.043	−0.0240099
$782$	− 1629.67i	− 0.0745230i
$783$	− 8351.71i	− 0.381182i
$784$	47899.2	2.18200
$785$	0	0
$786$	14388.9	0.652970
$787$	31293.8i	1.41741i	0.705503	+	0.708707i	$0.250721\pi$
$787$	31293.8i	1.41741i	−0.705503	+	0.708707i	$0.749279\pi$
$788$	323.460i	0.0146228i
$789$	3739.60	0.168737
$790$	0	0
$791$	−8743.58	−0.393029
$792$	9040.03i	0.405585i
$793$	1396.01i	0.0625140i
$794$	−40928.9	−1.82936
$795$	0	0
$796$	243.587	0.0108464
$797$	32015.4i	1.42289i	0.702741	+	0.711446i	$0.251959\pi$
$797$	32015.4i	1.42289i	−0.702741	+	0.711446i	$0.748041\pi$
$798$	− 30536.2i	− 1.35460i
$799$	−5396.89	−0.238959
$800$	0	0
$801$	−16745.7	−0.738677
$802$	− 33148.2i	− 1.45948i
$803$	5560.91i	0.244384i
$804$	−389.997	−0.0171071
$805$	0	0
$806$	−3695.19	−0.161486
$807$	16227.9i	0.707867i
$808$	− 30904.2i	− 1.34555i
$809$	−5234.16	−0.227470	−0.113735	−	0.993511i	$-0.536281\pi$
$809$	−5234.16	−0.227470	−0.113735	+	0.993511i	$0.536281\pi$
$810$	0	0
$811$	−2377.40	−0.102937	−0.0514685	−	0.998675i	$-0.516390\pi$
$811$	−2377.40	−0.102937	−0.0514685	+	0.998675i	$0.516390\pi$
$812$	− 429.873i	− 0.0185783i
$813$	− 4816.29i	− 0.207767i
$814$	23399.4	1.00755
$815$	0	0
$816$	−5584.62	−0.239584
$817$	− 9138.87i	− 0.391345i
$818$	40583.9i	1.73470i
$819$	−7204.18	−0.307368
$820$	0	0
$821$	5174.70	0.219974	0.109987	−	0.993933i	$-0.464919\pi$
$821$	5174.70	0.219974	0.109987	+	0.993933i	$0.464919\pi$
$822$	− 18628.5i	− 0.790443i
$823$	− 44178.3i	− 1.87115i	−0.353126	−	0.935576i	$-0.614881\pi$
$823$	− 44178.3i	− 1.87115i	0.353126	−	0.935576i	$-0.385119\pi$
$824$	30594.1	1.29344
$825$	0	0
$826$	−3500.81	−0.147468
$827$	5766.56i	0.242470i	0.992624	+	0.121235i	$0.0386855\pi$
$827$	5766.56i	0.242470i	−0.992624	+	0.121235i	$0.961314\pi$
$828$	− 79.6387i	− 0.00334255i
$829$	−38465.6	−1.61154	−0.805770	−	0.592229i	$-0.798248\pi$
$829$	−38465.6	−1.61154	−0.805770	+	0.592229i	$0.798248\pi$
$830$	0	0
$831$	14171.7	0.591589
$832$	− 7198.33i	− 0.299949i
$833$	− 17989.1i	− 0.748242i
$834$	14852.1	0.616650
$835$	0	0
$836$	576.211	0.0238381
$837$	− 12886.3i	− 0.532159i
$838$	6374.17i	0.262759i
$839$	17261.8	0.710301	0.355151	−	0.934809i	$-0.384430\pi$
$839$	17261.8	0.710301	0.355151	+	0.934809i	$0.384430\pi$
$840$	0	0
$841$	−21071.6	−0.863981
$842$	7224.22i	0.295681i
$843$	− 2763.29i	− 0.112898i
$844$	350.655	0.0143010
$845$	0	0
$846$	−9516.28	−0.386733
$847$	− 20222.2i	− 0.820358i
$848$	− 34147.2i	− 1.38281i
$849$	10819.2	0.437355
$850$	0	0
$851$	7025.75	0.283008
$852$	− 15.3815i	0 0.000618500i
$853$	− 29038.7i	− 1.16561i	−0.812612	−	0.582805i	$-0.801955\pi$
$853$	− 29038.7i	− 1.16561i	0.812612	−	0.582805i	$-0.198045\pi$
$854$	−9041.59	−0.362291
$855$	0	0
$856$	−36958.9	−1.47574
$857$	− 9865.16i	− 0.393217i	−0.980482	−	0.196609i	$-0.937007\pi$
$857$	− 9865.16i	− 0.393217i	0.980482	−	0.196609i	$-0.0629929\pi$
$858$	3816.84i	0.151870i
$859$	24476.6	0.972214	0.486107	−	0.873899i	$-0.338416\pi$
$859$	24476.6	0.972214	0.486107	+	0.873899i	$0.338416\pi$
$860$	0	0
$861$	−20161.6	−0.798030
$862$	28498.6i	1.12606i
$863$	37353.6i	1.47339i	0.676227	+	0.736693i	$0.263614\pi$
$863$	37353.6i	1.47339i	−0.676227	+	0.736693i	$0.736386\pi$
$864$	−1495.92	−0.0589032
$865$	0	0
$866$	−24173.8	−0.948567
$867$	− 14790.4i	− 0.579365i
$868$	− 663.276i	− 0.0259367i
$869$	11915.7	0.465148
$870$	0	0
$871$	−7212.34	−0.280575
$872$	18311.2i	0.711120i
$873$	27550.3i	1.06808i
$874$	6242.66	0.241603
$875$	0	0
$876$	−163.222	−0.00629538
$877$	− 25920.7i	− 0.998040i	−0.866590	−	0.499020i	$-0.833693\pi$
$877$	− 25920.7i	− 0.998040i	0.866590	−	0.499020i	$-0.166307\pi$
$878$	− 14015.4i	− 0.538720i
$879$	9668.92	0.371018
$880$	0	0
$881$	−9337.06	−0.357064	−0.178532	−	0.983934i	$-0.557135\pi$
$881$	−9337.06	−0.357064	−0.178532	+	0.983934i	$0.557135\pi$
$882$	− 31720.0i	− 1.21096i
$883$	70.8656i	0.00270081i	0.999999	+	0.00135041i	$0.000429848\pi$
$883$	70.8656i	0.00270081i	−0.999999	+	0.00135041i	$0.999570\pi$
$884$	81.6505	0.00310656
$885$	0	0
$886$	27893.0	1.05765
$887$	− 1058.85i	− 0.0400819i	−0.999799	−	0.0200409i	$-0.993620\pi$
$887$	− 1058.85i	− 0.0400819i	0.999799	−	0.0200409i	$-0.00637965\pi$
$888$	− 23408.3i	− 0.884608i
$889$	47334.7	1.78578
$890$	0	0
$891$	−2362.08	−0.0888135
$892$	− 1222.18i	− 0.0458762i
$893$	− 20673.5i	− 0.774705i
$894$	29051.2	1.08682
$895$	0	0
$896$	49323.1	1.83903
$897$	1146.02i	0.0426583i
$898$	28953.3i	1.07593i
$899$	5118.55	0.189892
$900$	0	0
$901$	−12824.4	−0.474187
$902$	− 13727.4i	− 0.506734i
$903$	10866.1i	0.400443i
$904$	−5955.54	−0.219113
$905$	0	0
$906$	19516.5	0.715664
$907$	29050.2i	1.06350i	0.846900	+	0.531751i	$0.178466\pi$
$907$	29050.2i	1.06350i	−0.846900	+	0.531751i	$0.821534\pi$
$908$	− 252.742i	− 0.00923736i
$909$	−21049.3	−0.768055
$910$	0	0
$911$	16041.6	0.583404	0.291702	−	0.956509i	$-0.405778\pi$
$911$	16041.6	0.583404	0.291702	+	0.956509i	$0.405778\pi$
$912$	− 21392.6i	− 0.776732i
$913$	− 13383.2i	− 0.485125i
$914$	−10216.8	−0.369738
$915$	0	0
$916$	−1150.73	−0.0415078
$917$	47763.9i	1.72007i
$918$	10274.3i	0.369392i
$919$	30151.2	1.08226	0.541130	−	0.840939i	$-0.317997\pi$
$919$	30151.2	1.08226	0.541130	+	0.840939i	$0.317997\pi$
$920$	0	0
$921$	−27105.5	−0.969767
$922$	− 21490.5i	− 0.767626i
$923$	− 284.455i	− 0.0101440i
$924$	−685.112	−0.0243923
$925$	0	0
$926$	44251.6	1.57041
$927$	− 20838.1i	− 0.738311i
$928$	− 594.192i	− 0.0210186i
$929$	−43339.8	−1.53061	−0.765303	−	0.643670i	$-0.777411\pi$
$929$	−43339.8	−1.53061	−0.765303	+	0.643670i	$0.777411\pi$
$930$	0	0
$931$	68909.5	2.42580
$932$	1619.51i	0.0569194i
$933$	6546.32i	0.229707i
$934$	−34044.8	−1.19270
$935$	0	0
$936$	−4907.00	−0.171357
$937$	27291.9i	0.951533i	0.879572	+	0.475766i	$0.157829\pi$
$937$	27291.9i	0.951533i	−0.879572	+	0.475766i	$0.842171\pi$
$938$	− 46712.5i	− 1.62603i
$939$	13250.7	0.460511
$940$	0	0
$941$	−4358.15	−0.150980	−0.0754898	−	0.997147i	$-0.524052\pi$
$941$	−4358.15	−0.150980	−0.0754898	+	0.997147i	$0.524052\pi$
$942$	4983.25i	0.172360i
$943$	− 4121.72i	− 0.142335i
$944$	−2452.54	−0.0845587
$945$	0	0
$946$	−7398.40	−0.254274
$947$	39067.8i	1.34059i	0.742097	+	0.670293i	$0.233831\pi$
$947$	39067.8i	1.34059i	−0.742097	+	0.670293i	$0.766169\pi$
$948$	349.746i	0.0119823i
$949$	−3018.51	−0.103251
$950$	0	0
$951$	6457.78	0.220197
$952$	− 18024.0i	− 0.613614i
$953$	52563.8i	1.78668i	0.449379	+	0.893341i	$0.351645\pi$
$953$	52563.8i	1.78668i	−0.449379	+	0.893341i	$0.648355\pi$
$954$	−22613.1	−0.767428
$955$	0	0
$956$	354.869	0.0120055
$957$	− 5287.06i	− 0.178585i
$958$	− 1999.75i	− 0.0674416i
$959$	61837.4	2.08220
$960$	0	0
$961$	−21893.3	−0.734896
$962$	12701.4i	0.425684i
$963$	25173.3i	0.842365i
$964$	−690.682	−0.0230761
$965$	0	0
$966$	−7422.49	−0.247220
$967$	− 2440.46i	− 0.0811580i	−0.999176	−	0.0405790i	$-0.987080\pi$
$967$	− 2440.46i	− 0.0811580i	0.999176	−	0.0405790i	$-0.0129203\pi$
$968$	− 13774.0i	− 0.457348i
$969$	−8034.24	−0.266354
$970$	0	0
$971$	45490.7	1.50347	0.751733	−	0.659468i	$-0.229218\pi$
$971$	45490.7	1.50347	0.751733	+	0.659468i	$0.229218\pi$
$972$	823.439i	0.0271727i
$973$	49301.5i	1.62439i
$974$	−41345.9	−1.36017
$975$	0	0
$976$	−6334.21	−0.207739
$977$	− 33244.4i	− 1.08862i	−0.838884	−	0.544311i	$-0.816791\pi$
$977$	− 33244.4i	− 1.08862i	0.838884	−	0.544311i	$-0.183209\pi$
$978$	25677.9i	0.839559i
$979$	−29450.6	−0.961437
$980$	0	0
$981$	12472.1	0.405914
$982$	50309.6i	1.63487i
$983$	− 1171.32i	− 0.0380053i	−0.999819	−	0.0190027i	$-0.993951\pi$
$983$	− 1171.32i	− 0.0380053i	0.999819	−	0.0190027i	$-0.00604910\pi$
$984$	−13732.7	−0.444901
$985$	0	0
$986$	−4081.02	−0.131811
$987$	24580.6i	0.792716i
$988$	312.772i	0.0100715i
$989$	−2221.40	−0.0714220
$990$	0	0
$991$	−3714.32	−0.119061	−0.0595304	−	0.998226i	$-0.518960\pi$
$991$	−3714.32	−0.119061	−0.0595304	+	0.998226i	$0.518960\pi$
$992$	− 916.814i	− 0.0293436i
$993$	− 23477.1i	− 0.750276i
$994$	1842.34	0.0587883
$995$	0	0
$996$	392.818	0.0124969
$997$	19521.0i	0.620098i	0.950721	+	0.310049i	$0.100345\pi$
$997$	19521.0i	0.620098i	−0.950721	+	0.310049i	$0.899655\pi$
$998$	53922.9i	1.71032i
$999$	−44293.9	−1.40280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.b.g.24.6		8
5.2	odd	4		575.4.a.i.1.2		4
5.3	odd	4		23.4.a.b.1.3	✓	4
5.4	even	2	inner	575.4.b.g.24.3		8
15.8	even	4		207.4.a.e.1.2		4
20.3	even	4		368.4.a.l.1.2		4
35.13	even	4		1127.4.a.c.1.3		4
40.3	even	4		1472.4.a.bf.1.3		4
40.13	odd	4		1472.4.a.y.1.2		4
115.68	even	4		529.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.3	✓	4	5.3	odd	4
207.4.a.e.1.2		4	15.8	even	4
368.4.a.l.1.2		4	20.3	even	4
529.4.a.g.1.3		4	115.68	even	4
575.4.a.i.1.2		4	5.2	odd	4
575.4.b.g.24.3		8	5.4	even	2	inner
575.4.b.g.24.6		8	1.1	even	1	trivial
1127.4.a.c.1.3		4	35.13	even	4
1472.4.a.y.1.2		4	40.13	odd	4
1472.4.a.bf.1.3		4	40.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.86845i q^{2} -3.43737i q^{3} -0.228032 q^{4} +9.85995 q^{6} +32.7301i q^{7} +22.2935i q^{8} +15.1845 q^{9} +26.7049 q^{11} +0.783832i q^{12} +14.4956i q^{13} -93.8848 q^{14} -65.7723 q^{16} +24.7016i q^{17} +43.5560i q^{18} -94.6224 q^{19} +112.505 q^{21} +76.6018i q^{22} +23.0000i q^{23} +76.6312 q^{24} -41.5801 q^{26} -145.004i q^{27} -7.46352i q^{28} +57.5965 q^{29} +88.8691 q^{31} -10.3165i q^{32} -91.7948i q^{33} -70.8553 q^{34} -3.46255 q^{36} -305.467i q^{37} -271.420i q^{38} +49.8269 q^{39} -179.205 q^{41} +322.717i q^{42} +96.5826i q^{43} -6.08959 q^{44} -65.9745 q^{46} +218.484i q^{47} +226.084i q^{48} -728.258 q^{49} +84.9085 q^{51} -3.30548i q^{52} +519.174i q^{53} +415.937 q^{54} -729.669 q^{56} +325.252i q^{57} +165.213i q^{58} +37.2884 q^{59} +96.3052 q^{61} +254.917i q^{62} +496.989i q^{63} -496.586 q^{64} +263.309 q^{66} +497.552i q^{67} -5.63276i q^{68} +79.0596 q^{69} -19.6235 q^{71} +338.515i q^{72} +208.235i q^{73} +876.219 q^{74} +21.5770 q^{76} +874.054i q^{77} +142.926i q^{78} +446.200 q^{79} -88.4513 q^{81} -514.042i q^{82} -501.151i q^{83} -25.6549 q^{84} -277.043 q^{86} -197.981i q^{87} +595.347i q^{88} -1102.82 q^{89} -474.444 q^{91} -5.24475i q^{92} -305.476i q^{93} -626.711 q^{94} -35.4615 q^{96} +1814.37i q^{97} -2088.98i q^{98} +405.500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{4} - 34 q^{6} + 66 q^{9} + 16 q^{11} + 288 q^{14} + 128 q^{16} - 192 q^{19} + 360 q^{21} + 376 q^{24} - 458 q^{26} - 42 q^{29} - 386 q^{31} - 1332 q^{34} - 1258 q^{36} + 582 q^{39} - 250 q^{41}+ \cdots + 2996 q^{99}+O(q^{100}) \) 8 * q - 40 * q^4 - 34 * q^6 + 66 * q^9 + 16 * q^11 + 288 * q^14 + 128 * q^16 - 192 * q^19 + 360 * q^21 + 376 * q^24 - 458 * q^26 - 42 * q^29 - 386 * q^31 - 1332 * q^34 - 1258 * q^36 + 582 * q^39 - 250 * q^41 - 1660 * q^44 - 92 * q^46 - 2440 * q^49 - 680 * q^51 - 1282 * q^54 - 4348 * q^56 + 2280 * q^59 + 1508 * q^61 + 1610 * q^64 - 796 * q^66 + 322 * q^69 - 802 * q^71 - 2732 * q^74 - 7664 * q^76 + 1676 * q^79 - 1864 * q^81 - 5228 * q^84 + 1836 * q^86 - 4684 * q^89 + 584 * q^91 - 3134 * q^94 + 1598 * q^96 + 2996 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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