LMFDB - Embedded newform 575.4.b.j.24.4

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			24.4
Root			$-3.74805i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.4.b.j.24.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.74805i q^{2} +2.78570i q^{3} -6.04789 q^{4} +10.4409 q^{6} +9.96305i q^{7} -7.31660i q^{8} +19.2399 q^{9} -9.93900 q^{11} -16.8476i q^{12} -12.4060i q^{13} +37.3420 q^{14} -75.8061 q^{16} +32.9389i q^{17} -72.1121i q^{18} +51.1489 q^{19} -27.7540 q^{21} +37.2519i q^{22} +23.0000i q^{23} +20.3818 q^{24} -46.4982 q^{26} +128.810i q^{27} -60.2554i q^{28} +101.451 q^{29} +149.814 q^{31} +225.592i q^{32} -27.6871i q^{33} +123.457 q^{34} -116.361 q^{36} +57.2805i q^{37} -191.709i q^{38} +34.5593 q^{39} +297.967 q^{41} +104.024i q^{42} -188.213i q^{43} +60.1100 q^{44} +86.2052 q^{46} +108.166i q^{47} -211.173i q^{48} +243.738 q^{49} -91.7579 q^{51} +75.0300i q^{52} -113.422i q^{53} +482.788 q^{54} +72.8957 q^{56} +142.485i q^{57} -380.244i q^{58} +76.7575 q^{59} +563.750 q^{61} -561.511i q^{62} +191.688i q^{63} +239.083 q^{64} -103.772 q^{66} -690.928i q^{67} -199.211i q^{68} -64.0710 q^{69} +41.4554 q^{71} -140.771i q^{72} +112.320i q^{73} +214.690 q^{74} -309.343 q^{76} -99.0228i q^{77} -129.530i q^{78} +1085.16 q^{79} +160.651 q^{81} -1116.80i q^{82} -432.763i q^{83} +167.853 q^{84} -705.433 q^{86} +282.612i q^{87} +72.7198i q^{88} +278.456 q^{89} +123.601 q^{91} -139.101i q^{92} +417.337i q^{93} +405.414 q^{94} -628.432 q^{96} -447.096i q^{97} -913.542i q^{98} -191.225 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.74805i	− 1.32514i	−0.749002	−	0.662568i	$-0.769466\pi$
$2$	− 3.74805i	− 1.32514i	0.749002	−	0.662568i	$-0.230534\pi$
$3$	2.78570i	0.536108i	0.963404	+	0.268054i	$0.0863805\pi$
$3$	2.78570i	0.536108i	−0.963404	+	0.268054i	$0.913620\pi$
$4$	−6.04789	−0.755986
$5$	0	0
$5$	0	0
$6$	10.4409	0.710416
$7$	9.96305i	0.537954i	0.963147	+	0.268977i	$0.0866856\pi$
$7$	9.96305i	0.537954i	−0.963147	+	0.268977i	$0.913314\pi$
$8$	− 7.31660i	− 0.323351i
$9$	19.2399	0.712589
$10$	0	0
$11$	−9.93900	−0.272429	−0.136215	−	0.990679i	$-0.543494\pi$
$11$	−9.93900	−0.272429	−0.136215	+	0.990679i	$0.543494\pi$
$12$	− 16.8476i	− 0.405290i
$13$	− 12.4060i	− 0.264677i	−0.991205	−	0.132338i	$-0.957751\pi$
$13$	− 12.4060i	− 0.264677i	0.991205	−	0.132338i	$-0.0422486\pi$
$14$	37.3420	0.712862
$15$	0	0
$16$	−75.8061	−1.18447
$17$	32.9389i	0.469933i	0.972003	+	0.234967i	$0.0754981\pi$
$17$	32.9389i	0.469933i	−0.972003	+	0.234967i	$0.924502\pi$
$18$	− 72.1121i	− 0.944277i
$19$	51.1489	0.617598	0.308799	−	0.951127i	$-0.400073\pi$
$19$	51.1489	0.617598	0.308799	+	0.951127i	$0.400073\pi$
$20$	0	0
$21$	−27.7540	−0.288401
$22$	37.2519i	0.361006i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	20.3818	0.173351
$25$	0	0
$26$	−46.4982	−0.350733
$27$	128.810i	0.918132i
$28$	− 60.2554i	− 0.406686i
$29$	101.451	0.649621	0.324810	−	0.945779i	$-0.394699\pi$
$29$	101.451	0.649621	0.324810	+	0.945779i	$0.394699\pi$
$30$	0	0
$31$	149.814	0.867981	0.433991	−	0.900917i	$-0.357105\pi$
$31$	149.814	0.867981	0.433991	+	0.900917i	$0.357105\pi$
$32$	225.592i	1.24623i
$33$	− 27.6871i	− 0.146051i
$34$	123.457	0.622726
$35$	0	0
$36$	−116.361	−0.538707
$37$	57.2805i	0.254510i	0.991870	+	0.127255i	$0.0406166\pi$
$37$	57.2805i	0.254510i	−0.991870	+	0.127255i	$0.959383\pi$
$38$	− 191.709i	− 0.818401i
$39$	34.5593	0.141895
$40$	0	0
$41$	297.967	1.13499	0.567496	−	0.823376i	$-0.307912\pi$
$41$	297.967	1.13499	0.567496	+	0.823376i	$0.307912\pi$
$42$	104.024i	0.382171i
$43$	− 188.213i	− 0.667494i	−0.942663	−	0.333747i	$-0.891687\pi$
$43$	− 188.213i	− 0.667494i	0.942663	−	0.333747i	$-0.108313\pi$
$44$	60.1100	0.205953
$45$	0	0
$46$	86.2052	0.276310
$47$	108.166i	0.335696i	0.985813	+	0.167848i	$0.0536818\pi$
$47$	108.166i	0.335696i	−0.985813	+	0.167848i	$0.946318\pi$
$48$	− 211.173i	− 0.635004i
$49$	243.738	0.710606
$50$	0	0
$51$	−91.7579	−0.251935
$52$	75.0300i	0.200092i
$53$	− 113.422i	− 0.293958i	−0.989140	−	0.146979i	$-0.953045\pi$
$53$	− 113.422i	− 0.293958i	0.989140	−	0.146979i	$-0.0469550\pi$
$54$	482.788	1.21665
$55$	0	0
$56$	72.8957	0.173948
$57$	142.485i	0.331099i
$58$	− 380.244i	− 0.860836i
$59$	76.7575	0.169372	0.0846862	−	0.996408i	$-0.473011\pi$
$59$	76.7575	0.169372	0.0846862	+	0.996408i	$0.473011\pi$
$60$	0	0
$61$	563.750	1.18329	0.591646	−	0.806198i	$-0.298478\pi$
$61$	563.750	1.18329	0.591646	+	0.806198i	$0.298478\pi$
$62$	− 561.511i	− 1.15019i
$63$	191.688i	0.383340i
$64$	239.083	0.466959
$65$	0	0
$66$	−103.772	−0.193538
$67$	− 690.928i	− 1.25985i	−0.776654	−	0.629927i	$-0.783085\pi$
$67$	− 690.928i	− 1.25985i	0.776654	−	0.629927i	$-0.216915\pi$
$68$	− 199.211i	− 0.355263i
$69$	−64.0710	−0.111786
$70$	0	0
$71$	41.4554	0.0692937	0.0346468	−	0.999400i	$-0.488969\pi$
$71$	41.4554	0.0692937	0.0346468	+	0.999400i	$0.488969\pi$
$72$	− 140.771i	− 0.230416i
$73$	112.320i	0.180083i	0.995938	+	0.0900417i	$0.0287000\pi$
$73$	112.320i	0.180083i	−0.995938	+	0.0900417i	$0.971300\pi$
$74$	214.690	0.337260
$75$	0	0
$76$	−309.343	−0.466896
$77$	− 99.0228i	− 0.146554i
$78$	− 129.530i	− 0.188031i
$79$	1085.16	1.54545	0.772724	−	0.634742i	$-0.218894\pi$
$79$	1085.16	1.54545	0.772724	+	0.634742i	$0.218894\pi$
$80$	0	0
$81$	160.651	0.220371
$82$	− 1116.80i	− 1.50402i
$83$	− 432.763i	− 0.572313i	−0.958183	−	0.286156i	$-0.907622\pi$
$83$	− 432.763i	− 0.572313i	0.958183	−	0.286156i	$-0.0923777\pi$
$84$	167.853	0.218027
$85$	0	0
$86$	−705.433	−0.884521
$87$	282.612i	0.348267i
$88$	72.7198i	0.0880904i
$89$	278.456	0.331643	0.165822	−	0.986156i	$-0.446972\pi$
$89$	278.456	0.331643	0.165822	+	0.986156i	$0.446972\pi$
$90$	0	0
$91$	123.601	0.142384
$92$	− 139.101i	− 0.157634i
$93$	417.337i	0.465331i
$94$	405.414	0.444843
$95$	0	0
$96$	−628.432	−0.668116
$97$	− 447.096i	− 0.467997i	−0.972237	−	0.233999i	$-0.924819\pi$
$97$	− 447.096i	− 0.467997i	0.972237	−	0.233999i	$-0.0751811\pi$
$98$	− 913.542i	− 0.941649i
$99$	−191.225	−0.194130
$100$	0	0
$101$	−457.028	−0.450258	−0.225129	−	0.974329i	$-0.572280\pi$
$101$	−457.028	−0.450258	−0.225129	+	0.974329i	$0.572280\pi$
$102$	343.913i	0.333848i
$103$	374.787i	0.358532i	0.983801	+	0.179266i	$0.0573723\pi$
$103$	374.787i	0.358532i	−0.983801	+	0.179266i	$0.942628\pi$
$104$	−90.7696	−0.0855836
$105$	0	0
$106$	−425.113	−0.389534
$107$	− 558.564i	− 0.504658i	−0.967642	−	0.252329i	$-0.918803\pi$
$107$	− 558.564i	− 0.504658i	0.967642	−	0.252329i	$-0.0811965\pi$
$108$	− 779.031i	− 0.694095i
$109$	1540.07	1.35332	0.676662	−	0.736294i	$-0.263426\pi$
$109$	1540.07	1.35332	0.676662	+	0.736294i	$0.263426\pi$
$110$	0	0
$111$	−159.566	−0.136444
$112$	− 755.260i	− 0.637191i
$113$	793.720i	0.660769i	0.943846	+	0.330385i	$0.107178\pi$
$113$	793.720i	0.660769i	−0.943846	+	0.330385i	$0.892822\pi$
$114$	534.042	0.438751
$115$	0	0
$116$	−613.566	−0.491105
$117$	− 238.690i	− 0.188606i
$118$	− 287.691i	− 0.224442i
$119$	−328.172	−0.252802
$120$	0	0
$121$	−1232.22	−0.925782
$122$	− 2112.96i	− 1.56802i
$123$	830.046i	0.608478i
$124$	−906.060	−0.656182
$125$	0	0
$126$	718.456	0.507978
$127$	680.815i	0.475690i	0.971303	+	0.237845i	$0.0764410\pi$
$127$	680.815i	0.475690i	−0.971303	+	0.237845i	$0.923559\pi$
$128$	908.644i	0.627449i
$129$	524.305	0.357849
$130$	0	0
$131$	−831.977	−0.554887	−0.277443	−	0.960742i	$-0.589487\pi$
$131$	−831.977	−0.554887	−0.277443	+	0.960742i	$0.589487\pi$
$132$	167.448i	0.110413i
$133$	509.599i	0.332239i
$134$	−2589.63	−1.66948
$135$	0	0
$136$	241.001	0.151953
$137$	− 77.4758i	− 0.0483153i	−0.999708	−	0.0241577i	$-0.992310\pi$
$137$	− 77.4758i	− 0.0483153i	0.999708	−	0.0241577i	$-0.00769037\pi$
$138$	240.142i	0.148132i
$139$	11.3426	0.00692137	0.00346069	−	0.999994i	$-0.498898\pi$
$139$	11.3426	0.00692137	0.00346069	+	0.999994i	$0.498898\pi$
$140$	0	0
$141$	−301.319	−0.179969
$142$	− 155.377i	− 0.0918236i
$143$	123.303i	0.0721057i
$144$	−1458.50	−0.844041
$145$	0	0
$146$	420.982	0.238635
$147$	678.979i	0.380961i
$148$	− 346.426i	− 0.192406i
$149$	−235.762	−0.129627	−0.0648135	−	0.997897i	$-0.520645\pi$
$149$	−235.762	−0.129627	−0.0648135	+	0.997897i	$0.520645\pi$
$150$	0	0
$151$	−2306.45	−1.24302	−0.621510	−	0.783406i	$-0.713481\pi$
$151$	−2306.45	−1.24302	−0.621510	+	0.783406i	$0.713481\pi$
$152$	− 374.236i	− 0.199701i
$153$	633.742i	0.334869i
$154$	−371.142	−0.194205
$155$	0	0
$156$	−209.011	−0.107271
$157$	− 3211.78i	− 1.63266i	−0.577584	−	0.816332i	$-0.696004\pi$
$157$	− 3211.78i	− 1.63266i	0.577584	−	0.816332i	$-0.303996\pi$
$158$	− 4067.25i	− 2.04793i
$159$	315.960	0.157593
$160$	0	0
$161$	−229.150	−0.112171
$162$	− 602.127i	− 0.292022i
$163$	2763.11i	1.32775i	0.747844	+	0.663875i	$0.231089\pi$
$163$	2763.11i	1.32775i	−0.747844	+	0.663875i	$0.768911\pi$
$164$	−1802.07	−0.858038
$165$	0	0
$166$	−1622.02	−0.758393
$167$	334.205i	0.154860i	0.996998	+	0.0774299i	$0.0246714\pi$
$167$	334.205i	0.154860i	−0.996998	+	0.0774299i	$0.975329\pi$
$168$	203.065i	0.0932549i
$169$	2043.09	0.929946
$170$	0	0
$171$	984.099	0.440093
$172$	1138.29i	0.504617i
$173$	− 1610.42i	− 0.707735i	−0.935296	−	0.353868i	$-0.884866\pi$
$173$	− 1610.42i	− 0.707735i	0.935296	−	0.353868i	$-0.115134\pi$
$174$	1059.24	0.461501
$175$	0	0
$176$	753.438	0.322685
$177$	213.823i	0.0908019i
$178$	− 1043.67i	− 0.439472i
$179$	−797.514	−0.333011	−0.166506	−	0.986041i	$-0.553248\pi$
$179$	−797.514	−0.333011	−0.166506	+	0.986041i	$0.553248\pi$
$180$	0	0
$181$	1686.93	0.692753	0.346377	−	0.938096i	$-0.387412\pi$
$181$	1686.93	0.692753	0.346377	+	0.938096i	$0.387412\pi$
$182$	− 463.264i	− 0.188678i
$183$	1570.44i	0.634371i
$184$	168.282	0.0674234
$185$	0	0
$186$	1564.20	0.616628
$187$	− 327.380i	− 0.128024i
$188$	− 654.179i	− 0.253781i
$189$	−1283.34	−0.493913
$190$	0	0
$191$	−2026.48	−0.767701	−0.383851	−	0.923395i	$-0.625402\pi$
$191$	−2026.48	−0.767701	−0.383851	+	0.923395i	$0.625402\pi$
$192$	666.013i	0.250340i
$193$	832.542i	0.310506i	0.987875	+	0.155253i	$0.0496193\pi$
$193$	832.542i	0.310506i	−0.987875	+	0.155253i	$0.950381\pi$
$194$	−1675.74	−0.620160
$195$	0	0
$196$	−1474.10	−0.537208
$197$	2133.95i	0.771764i	0.922548	+	0.385882i	$0.126103\pi$
$197$	2133.95i	0.771764i	−0.922548	+	0.385882i	$0.873897\pi$
$198$	716.723i	0.257249i
$199$	343.656	0.122418	0.0612088	−	0.998125i	$-0.480504\pi$
$199$	343.656	0.122418	0.0612088	+	0.998125i	$0.480504\pi$
$200$	0	0
$201$	1924.71	0.675417
$202$	1712.97i	0.596653i
$203$	1010.76i	0.349466i
$204$	554.942	0.190459
$205$	0	0
$206$	1404.72	0.475104
$207$	442.518i	0.148585i
$208$	940.449i	0.313502i
$209$	−508.369	−0.168252
$210$	0	0
$211$	406.997	0.132791	0.0663953	−	0.997793i	$-0.478850\pi$
$211$	406.997	0.132791	0.0663953	+	0.997793i	$0.478850\pi$
$212$	685.966i	0.222228i
$213$	115.482i	0.0371489i
$214$	−2093.53	−0.668741
$215$	0	0
$216$	942.454	0.296879
$217$	1492.61i	0.466934i
$218$	− 5772.28i	− 1.79334i
$219$	−312.890	−0.0965440
$220$	0	0
$221$	408.639	0.124380
$222$	598.062i	0.180808i
$223$	− 1248.45i	− 0.374898i	−0.982274	−	0.187449i	$-0.939978\pi$
$223$	− 1248.45i	− 0.374898i	0.982274	−	0.187449i	$-0.0600220\pi$
$224$	−2247.59	−0.670417
$225$	0	0
$226$	2974.90	0.875609
$227$	2717.71i	0.794628i	0.917683	+	0.397314i	$0.130058\pi$
$227$	2717.71i	0.794628i	−0.917683	+	0.397314i	$0.869942\pi$
$228$	− 861.736i	− 0.250306i
$229$	3805.44	1.09812	0.549062	−	0.835782i	$-0.314985\pi$
$229$	3805.44	1.09812	0.549062	+	0.835782i	$0.314985\pi$
$230$	0	0
$231$	275.847	0.0785689
$232$	− 742.278i	− 0.210056i
$233$	1650.85i	0.464167i	0.972696	+	0.232083i	$0.0745542\pi$
$233$	1650.85i	0.464167i	−0.972696	+	0.232083i	$0.925446\pi$
$234$	−894.621	−0.249928
$235$	0	0
$236$	−464.221	−0.128043
$237$	3022.94i	0.828527i
$238$	1230.01i	0.334998i
$239$	−940.473	−0.254536	−0.127268	−	0.991868i	$-0.540621\pi$
$239$	−940.473	−0.254536	−0.127268	+	0.991868i	$0.540621\pi$
$240$	0	0
$241$	3962.35	1.05908	0.529538	−	0.848286i	$-0.322365\pi$
$241$	3962.35	1.05908	0.529538	+	0.848286i	$0.322365\pi$
$242$	4618.41i	1.22679i
$243$	3925.40i	1.03627i
$244$	−3409.50	−0.894552
$245$	0	0
$246$	3111.06	0.806316
$247$	− 634.552i	− 0.163464i
$248$	− 1096.13i	− 0.280663i
$249$	1205.55	0.306821
$250$	0	0
$251$	3048.43	0.766595	0.383298	−	0.923625i	$-0.374788\pi$
$251$	3048.43	0.766595	0.383298	+	0.923625i	$0.374788\pi$
$252$	− 1159.31i	− 0.289800i
$253$	− 228.597i	− 0.0568054i
$254$	2551.73	0.630354
$255$	0	0
$256$	5318.31	1.29842
$257$	4307.07i	1.04540i	0.852517	+	0.522699i	$0.175075\pi$
$257$	4307.07i	1.04540i	−0.852517	+	0.522699i	$0.824925\pi$
$258$	− 1965.12i	− 0.474198i
$259$	−570.688	−0.136914
$260$	0	0
$261$	1951.91	0.462913
$262$	3118.29i	0.735301i
$263$	2477.52i	0.580876i	0.956894	+	0.290438i	$0.0938010\pi$
$263$	2477.52i	0.580876i	−0.956894	+	0.290438i	$0.906199\pi$
$264$	−202.575	−0.0472259
$265$	0	0
$266$	1910.00	0.440262
$267$	775.693i	0.177796i
$268$	4178.66i	0.952433i
$269$	−2712.02	−0.614701	−0.307350	−	0.951596i	$-0.599442\pi$
$269$	−2712.02	−0.614701	−0.307350	+	0.951596i	$0.599442\pi$
$270$	0	0
$271$	−532.068	−0.119265	−0.0596326	−	0.998220i	$-0.518993\pi$
$271$	−532.068	−0.119265	−0.0596326	+	0.998220i	$0.518993\pi$
$272$	− 2496.97i	− 0.556622i
$273$	344.316i	0.0763331i
$274$	−290.383	−0.0640244
$275$	0	0
$276$	387.495	0.0845088
$277$	3450.11i	0.748364i	0.927355	+	0.374182i	$0.122076\pi$
$277$	3450.11i	0.748364i	−0.927355	+	0.374182i	$0.877924\pi$
$278$	− 42.5128i	− 0.00917176i
$279$	2882.41	0.618514
$280$	0	0
$281$	−5906.65	−1.25396	−0.626978	−	0.779037i	$-0.715708\pi$
$281$	−5906.65	−1.25396	−0.626978	+	0.779037i	$0.715708\pi$
$282$	1129.36i	0.238484i
$283$	− 732.420i	− 0.153844i	−0.997037	−	0.0769220i	$-0.975491\pi$
$283$	− 732.420i	− 0.153844i	0.997037	−	0.0769220i	$-0.0245092\pi$
$284$	−250.718	−0.0523851
$285$	0	0
$286$	462.146	0.0955499
$287$	2968.66i	0.610573i
$288$	4340.38i	0.888052i
$289$	3828.03	0.779163
$290$	0	0
$291$	1245.47	0.250897
$292$	− 679.300i	− 0.136141i
$293$	7832.49i	1.56170i	0.624716	+	0.780852i	$0.285215\pi$
$293$	7832.49i	1.56170i	−0.624716	+	0.780852i	$0.714785\pi$
$294$	2544.85	0.504825
$295$	0	0
$296$	419.099	0.0822960
$297$	− 1280.25i	− 0.250126i
$298$	883.650i	0.171773i
$299$	285.337	0.0551889
$300$	0	0
$301$	1875.18	0.359081
$302$	8644.69i	1.64717i
$303$	− 1273.14i	− 0.241387i
$304$	−3877.40	−0.731527
$305$	0	0
$306$	2375.30	0.443747
$307$	4123.87i	0.766651i	0.923613	+	0.383326i	$0.125221\pi$
$307$	4123.87i	0.766651i	−0.923613	+	0.383326i	$0.874779\pi$
$308$	598.879i	0.110793i
$309$	−1044.04	−0.192212
$310$	0	0
$311$	−9283.00	−1.69257	−0.846287	−	0.532727i	$-0.821167\pi$
$311$	−9283.00	−1.69257	−0.846287	+	0.532727i	$0.821167\pi$
$312$	− 252.857i	− 0.0458820i
$313$	8609.88i	1.55482i	0.628993	+	0.777411i	$0.283467\pi$
$313$	8609.88i	1.55482i	−0.628993	+	0.777411i	$0.716533\pi$
$314$	−12037.9	−2.16350
$315$	0	0
$316$	−6562.95	−1.16834
$317$	− 2840.16i	− 0.503215i	−0.967829	−	0.251607i	$-0.919041\pi$
$317$	− 2840.16i	− 0.503215i	0.967829	−	0.251607i	$-0.0809592\pi$
$318$	− 1184.24i	− 0.208832i
$319$	−1008.32	−0.176976
$320$	0	0
$321$	1555.99	0.270551
$322$	858.866i	0.148642i
$323$	1684.79i	0.290230i
$324$	−971.598	−0.166598
$325$	0	0
$326$	10356.3	1.75945
$327$	4290.18i	0.725527i
$328$	− 2180.11i	− 0.367001i
$329$	−1077.67	−0.180589
$330$	0	0
$331$	−10623.8	−1.76416	−0.882080	−	0.471099i	$-0.843857\pi$
$331$	−10623.8	−1.76416	−0.882080	+	0.471099i	$0.843857\pi$
$332$	2617.31i	0.432661i
$333$	1102.07i	0.181361i
$334$	1252.62	0.205210
$335$	0	0
$336$	2103.93	0.341603
$337$	− 1759.84i	− 0.284465i	−0.989833	−	0.142232i	$-0.954572\pi$
$337$	− 1759.84i	− 0.284465i	0.989833	−	0.142232i	$-0.0454281\pi$
$338$	− 7657.61i	− 1.23231i
$339$	−2211.06	−0.354243
$340$	0	0
$341$	−1489.00	−0.236464
$342$	− 3688.45i	− 0.583184i
$343$	5845.69i	0.920227i
$344$	−1377.08	−0.215835
$345$	0	0
$346$	−6035.95	−0.937846
$347$	− 6665.77i	− 1.03123i	−0.856820	−	0.515616i	$-0.827563\pi$
$347$	− 6665.77i	− 1.03123i	0.856820	−	0.515616i	$-0.172437\pi$
$348$	− 1709.21i	− 0.263285i
$349$	4447.06	0.682080	0.341040	−	0.940049i	$-0.389221\pi$
$349$	4447.06	0.682080	0.341040	+	0.940049i	$0.389221\pi$
$350$	0	0
$351$	1598.02	0.243008
$352$	− 2242.16i	− 0.339511i
$353$	− 7023.22i	− 1.05895i	−0.848326	−	0.529474i	$-0.822389\pi$
$353$	− 7023.22i	− 1.05895i	0.848326	−	0.529474i	$-0.177611\pi$
$354$	801.420	0.120325
$355$	0	0
$356$	−1684.07	−0.250718
$357$	− 914.188i	− 0.135529i
$358$	2989.12i	0.441285i
$359$	−3348.56	−0.492285	−0.246142	−	0.969234i	$-0.579163\pi$
$359$	−3348.56	−0.492285	−0.246142	+	0.969234i	$0.579163\pi$
$360$	0	0
$361$	−4242.79	−0.618573
$362$	− 6322.69i	− 0.917992i
$363$	− 3432.58i	− 0.496319i
$364$	−747.527	−0.107640
$365$	0	0
$366$	5886.08	0.840629
$367$	− 12057.2i	− 1.71494i	−0.514537	−	0.857468i	$-0.672036\pi$
$367$	− 12057.2i	− 1.71494i	0.514537	−	0.857468i	$-0.327964\pi$
$368$	− 1743.54i	− 0.246979i
$369$	5732.86	0.808782
$370$	0	0
$371$	1130.03	0.158136
$372$	− 2524.01i	− 0.351784i
$373$	− 6505.90i	− 0.903118i	−0.892241	−	0.451559i	$-0.850868\pi$
$373$	− 6505.90i	− 0.903118i	0.892241	−	0.451559i	$-0.149132\pi$
$374$	−1227.04	−0.169649
$375$	0	0
$376$	791.411	0.108548
$377$	− 1258.60i	− 0.171940i
$378$	4810.04i	0.654502i
$379$	2440.64	0.330785	0.165392	−	0.986228i	$-0.447111\pi$
$379$	2440.64	0.330785	0.165392	+	0.986228i	$0.447111\pi$
$380$	0	0
$381$	−1896.55	−0.255021
$382$	7595.35i	1.01731i
$383$	− 10873.0i	− 1.45061i	−0.688430	−	0.725303i	$-0.741700\pi$
$383$	− 10873.0i	− 1.45061i	0.688430	−	0.725303i	$-0.258300\pi$
$384$	−2531.21	−0.336380
$385$	0	0
$386$	3120.41	0.411463
$387$	− 3621.20i	− 0.475649i
$388$	2703.99i	0.353799i
$389$	−806.795	−0.105157	−0.0525786	−	0.998617i	$-0.516744\pi$
$389$	−806.795	−0.105157	−0.0525786	+	0.998617i	$0.516744\pi$
$390$	0	0
$391$	−757.595	−0.0979878
$392$	− 1783.33i	− 0.229775i
$393$	− 2317.64i	− 0.297479i
$394$	7998.15	1.02269
$395$	0	0
$396$	1156.51	0.146760
$397$	− 7589.48i	− 0.959459i	−0.877417	−	0.479729i	$-0.840735\pi$
$397$	− 7589.48i	− 0.959459i	0.877417	−	0.479729i	$-0.159265\pi$
$398$	− 1288.04i	− 0.162220i
$399$	−1419.59	−0.178116
$400$	0	0
$401$	−6643.66	−0.827353	−0.413677	−	0.910424i	$-0.635756\pi$
$401$	−6643.66	−0.827353	−0.413677	+	0.910424i	$0.635756\pi$
$402$	− 7213.93i	− 0.895020i
$403$	− 1858.59i	− 0.229735i
$404$	2764.06	0.340389
$405$	0	0
$406$	3788.39	0.463090
$407$	− 569.311i	− 0.0693359i
$408$	671.356i	0.0814634i
$409$	−7260.60	−0.877784	−0.438892	−	0.898540i	$-0.644629\pi$
$409$	−7260.60	−0.877784	−0.438892	+	0.898540i	$0.644629\pi$
$410$	0	0
$411$	215.824	0.0259022
$412$	− 2266.67i	− 0.271046i
$413$	764.739i	0.0911146i
$414$	1658.58	0.196895
$415$	0	0
$416$	2798.69	0.329849
$417$	31.5972i	0.00371060i
$418$	1905.39i	0.222957i
$419$	−12904.8	−1.50463	−0.752313	−	0.658805i	$-0.771062\pi$
$419$	−12904.8	−1.50463	−0.752313	+	0.658805i	$0.771062\pi$
$420$	0	0
$421$	−14736.4	−1.70596	−0.852980	−	0.521944i	$-0.825207\pi$
$421$	−14736.4	−1.70596	−0.852980	+	0.521944i	$0.825207\pi$
$422$	− 1525.45i	− 0.175966i
$423$	2081.11i	0.239213i
$424$	−829.867	−0.0950517
$425$	0	0
$426$	432.833	0.0492273
$427$	5616.67i	0.636556i
$428$	3378.13i	0.381515i
$429$	−343.485	−0.0386564
$430$	0	0
$431$	3289.60	0.367644	0.183822	−	0.982960i	$-0.441153\pi$
$431$	3289.60	0.367644	0.183822	+	0.982960i	$0.441153\pi$
$432$	− 9764.61i	− 1.08750i
$433$	7065.41i	0.784161i	0.919931	+	0.392081i	$0.128245\pi$
$433$	7065.41i	0.784161i	−0.919931	+	0.392081i	$0.871755\pi$
$434$	5594.36	0.618751
$435$	0	0
$436$	−9314.20	−1.02309
$437$	1176.42i	0.128778i
$438$	1172.73i	0.127934i
$439$	−8786.87	−0.955295	−0.477647	−	0.878552i	$-0.658510\pi$
$439$	−8786.87	−0.955295	−0.477647	+	0.878552i	$0.658510\pi$
$440$	0	0
$441$	4689.49	0.506369
$442$	− 1531.60i	− 0.164821i
$443$	− 15634.8i	− 1.67682i	−0.545036	−	0.838412i	$-0.683484\pi$
$443$	− 15634.8i	− 1.67682i	0.545036	−	0.838412i	$-0.316516\pi$
$444$	965.038	0.103150
$445$	0	0
$446$	−4679.25	−0.496791
$447$	− 656.763i	− 0.0694940i
$448$	2382.00i	0.251203i
$449$	−1326.49	−0.139423	−0.0697116	−	0.997567i	$-0.522208\pi$
$449$	−1326.49	−0.139423	−0.0697116	+	0.997567i	$0.522208\pi$
$450$	0	0
$451$	−2961.50	−0.309205
$452$	− 4800.33i	− 0.499533i
$453$	− 6425.07i	− 0.666393i
$454$	10186.1	1.05299
$455$	0	0
$456$	1042.51	0.107061
$457$	1577.23i	0.161444i	0.996737	+	0.0807219i	$0.0257226\pi$
$457$	1577.23i	0.161444i	−0.996737	+	0.0807219i	$0.974277\pi$
$458$	− 14263.0i	− 1.45516i
$459$	−4242.87	−0.431461
$460$	0	0
$461$	−18592.2	−1.87836	−0.939179	−	0.343427i	$-0.888412\pi$
$461$	−18592.2	−1.87836	−0.939179	+	0.343427i	$0.888412\pi$
$462$	− 1033.89i	− 0.104115i
$463$	16081.1i	1.61415i	0.590447	+	0.807076i	$0.298951\pi$
$463$	16081.1i	1.61415i	−0.590447	+	0.807076i	$0.701049\pi$
$464$	−7690.62	−0.769457
$465$	0	0
$466$	6187.47	0.615084
$467$	4803.27i	0.475951i	0.971271	+	0.237975i	$0.0764837\pi$
$467$	4803.27i	0.475951i	−0.971271	+	0.237975i	$0.923516\pi$
$468$	1443.57i	0.142583i
$469$	6883.74	0.677744
$470$	0	0
$471$	8947.05	0.875283
$472$	− 561.604i	− 0.0547668i
$473$	1870.65i	0.181845i
$474$	11330.1	1.09791
$475$	0	0
$476$	1984.75	0.191115
$477$	− 2182.24i	− 0.209471i
$478$	3524.94i	0.337295i
$479$	2632.36	0.251097	0.125548	−	0.992087i	$-0.459931\pi$
$479$	2632.36	0.251097	0.125548	+	0.992087i	$0.459931\pi$
$480$	0	0
$481$	710.620	0.0673627
$482$	− 14851.1i	− 1.40342i
$483$	− 638.343i	− 0.0601358i
$484$	7452.31	0.699879
$485$	0	0
$486$	14712.6	1.37320
$487$	7542.18i	0.701784i	0.936416	+	0.350892i	$0.114122\pi$
$487$	7542.18i	0.701784i	−0.936416	+	0.350892i	$0.885878\pi$
$488$	− 4124.73i	− 0.382619i
$489$	−7697.17	−0.711816
$490$	0	0
$491$	−9286.81	−0.853580	−0.426790	−	0.904351i	$-0.640356\pi$
$491$	−9286.81	−0.853580	−0.426790	+	0.904351i	$0.640356\pi$
$492$	− 5020.03i	− 0.460001i
$493$	3341.69i	0.305278i
$494$	−2378.33	−0.216612
$495$	0	0
$496$	−11356.8	−1.02810
$497$	413.022i	0.0372768i
$498$	− 4518.46i	− 0.406580i
$499$	7508.65	0.673614	0.336807	−	0.941574i	$-0.390653\pi$
$499$	7508.65	0.673614	0.336807	+	0.941574i	$0.390653\pi$
$500$	0	0
$501$	−930.994	−0.0830215
$502$	− 11425.7i	− 1.01584i
$503$	− 18833.3i	− 1.66946i	−0.550662	−	0.834728i	$-0.685625\pi$
$503$	− 18833.3i	− 1.66946i	0.550662	−	0.834728i	$-0.314375\pi$
$504$	1402.50	0.123953
$505$	0	0
$506$	−856.794	−0.0752750
$507$	5691.43i	0.498551i
$508$	− 4117.50i	− 0.359615i
$509$	8379.54	0.729698	0.364849	−	0.931067i	$-0.381121\pi$
$509$	8379.54	0.729698	0.364849	+	0.931067i	$0.381121\pi$
$510$	0	0
$511$	−1119.05	−0.0968765
$512$	− 12664.1i	− 1.09313i
$513$	6588.51i	0.567036i
$514$	16143.1	1.38530
$515$	0	0
$516$	−3170.94	−0.270529
$517$	− 1075.07i	− 0.0914534i
$518$	2138.97i	0.181430i
$519$	4486.15	0.379422
$520$	0	0
$521$	−3590.90	−0.301958	−0.150979	−	0.988537i	$-0.548243\pi$
$521$	−3590.90	−0.301958	−0.150979	+	0.988537i	$0.548243\pi$
$522$	− 7315.86i	− 0.613422i
$523$	5720.72i	0.478297i	0.970983	+	0.239149i	$0.0768683\pi$
$523$	5720.72i	0.478297i	−0.970983	+	0.239149i	$0.923132\pi$
$524$	5031.71	0.419487
$525$	0	0
$526$	9285.87	0.769740
$527$	4934.72i	0.407893i
$528$	2098.85i	0.172994i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	1476.81	0.120693
$532$	− 3082.00i	− 0.251168i
$533$	− 3696.57i	− 0.300406i
$534$	2907.34	0.235604
$535$	0	0
$536$	−5055.24	−0.407376
$537$	− 2221.63i	− 0.178530i
$538$	10164.8i	0.814562i
$539$	−2422.51	−0.193590
$540$	0	0
$541$	−21769.0	−1.72999	−0.864994	−	0.501783i	$-0.832678\pi$
$541$	−21769.0	−1.72999	−0.864994	+	0.501783i	$0.832678\pi$
$542$	1994.22i	0.158043i
$543$	4699.27i	0.371390i
$544$	−7430.78	−0.585647
$545$	0	0
$546$	1290.51	0.101152
$547$	20471.9i	1.60021i	0.599860	+	0.800105i	$0.295223\pi$
$547$	20471.9i	1.60021i	−0.599860	+	0.800105i	$0.704777\pi$
$548$	468.565i	0.0365257i
$549$	10846.5	0.843200
$550$	0	0
$551$	5189.11	0.401205
$552$	468.782i	0.0361462i
$553$	10811.5i	0.831380i
$554$	12931.2	0.991684
$555$	0	0
$556$	−68.5991	−0.00523246
$557$	− 4682.95i	− 0.356235i	−0.984009	−	0.178117i	$-0.942999\pi$
$557$	− 4682.95i	− 0.356235i	0.984009	−	0.178117i	$-0.0570007\pi$
$558$	− 10803.4i	− 0.819615i
$559$	−2334.97	−0.176670
$560$	0	0
$561$	911.982	0.0686344
$562$	22138.4i	1.66166i
$563$	− 12425.1i	− 0.930118i	−0.885280	−	0.465059i	$-0.846033\pi$
$563$	− 12425.1i	− 0.930118i	0.885280	−	0.465059i	$-0.153967\pi$
$564$	1822.34	0.136054
$565$	0	0
$566$	−2745.15	−0.203864
$567$	1600.57i	0.118550i
$568$	− 303.313i	− 0.0224062i
$569$	21359.2	1.57368	0.786842	−	0.617155i	$-0.211715\pi$
$569$	21359.2	1.57368	0.786842	+	0.617155i	$0.211715\pi$
$570$	0	0
$571$	6263.93	0.459085	0.229542	−	0.973299i	$-0.426277\pi$
$571$	6263.93	0.459085	0.229542	+	0.973299i	$0.426277\pi$
$572$	− 745.723i	− 0.0545109i
$573$	− 5645.16i	− 0.411570i
$574$	11126.7	0.809093
$575$	0	0
$576$	4599.94	0.332750
$577$	11385.6i	0.821470i	0.911755	+	0.410735i	$0.134728\pi$
$577$	11385.6i	0.821470i	−0.911755	+	0.410735i	$0.865272\pi$
$578$	− 14347.6i	− 1.03250i
$579$	−2319.21	−0.166465
$580$	0	0
$581$	4311.64	0.307878
$582$	− 4668.10i	− 0.332473i
$583$	1127.31i	0.0800827i
$584$	821.802	0.0582302
$585$	0	0
$586$	29356.6	2.06947
$587$	20252.6i	1.42405i	0.702155	+	0.712024i	$0.252221\pi$
$587$	20252.6i	1.42405i	−0.702155	+	0.712024i	$0.747779\pi$
$588$	− 4106.39i	− 0.288001i
$589$	7662.83	0.536064
$590$	0	0
$591$	−5944.53	−0.413749
$592$	− 4342.21i	− 0.301459i
$593$	− 13864.0i	− 0.960078i	−0.877247	−	0.480039i	$-0.840622\pi$
$593$	− 13864.0i	− 0.960078i	0.877247	−	0.480039i	$-0.159378\pi$
$594$	−4798.43	−0.331451
$595$	0	0
$596$	1425.87	0.0979962
$597$	957.320i	0.0656290i
$598$	− 1069.46i	− 0.0731328i
$599$	12357.3	0.842916	0.421458	−	0.906848i	$-0.361518\pi$
$599$	12357.3	0.842916	0.421458	+	0.906848i	$0.361518\pi$
$600$	0	0
$601$	8427.54	0.571991	0.285996	−	0.958231i	$-0.407676\pi$
$601$	8427.54	0.571991	0.285996	+	0.958231i	$0.407676\pi$
$602$	− 7028.26i	− 0.475831i
$603$	− 13293.4i	− 0.897758i
$604$	13949.2	0.939707
$605$	0	0
$606$	−4771.80	−0.319870
$607$	12024.5i	0.804049i	0.915629	+	0.402024i	$0.131693\pi$
$607$	12024.5i	0.804049i	−0.915629	+	0.402024i	$0.868307\pi$
$608$	11538.8i	0.769672i
$609$	−2815.68	−0.187351
$610$	0	0
$611$	1341.91	0.0888509
$612$	− 3832.80i	− 0.253156i
$613$	− 11845.0i	− 0.780447i	−0.920720	−	0.390224i	$-0.872398\pi$
$613$	− 11845.0i	− 0.780447i	0.920720	−	0.390224i	$-0.127602\pi$
$614$	15456.5	1.01592
$615$	0	0
$616$	−724.510	−0.0473886
$617$	− 6786.50i	− 0.442810i	−0.975182	−	0.221405i	$-0.928936\pi$
$617$	− 6786.50i	− 0.442810i	0.975182	−	0.221405i	$-0.0710643\pi$
$618$	3913.12i	0.254707i
$619$	8051.54	0.522809	0.261405	−	0.965229i	$-0.415814\pi$
$619$	8051.54	0.522809	0.261405	+	0.965229i	$0.415814\pi$
$620$	0	0
$621$	−2962.64	−0.191444
$622$	34793.2i	2.24289i
$623$	2774.27i	0.178409i
$624$	−2619.81	−0.168071
$625$	0	0
$626$	32270.3	2.06035
$627$	− 1416.16i	− 0.0902011i
$628$	19424.5i	1.23427i
$629$	−1886.76	−0.119602
$630$	0	0
$631$	−17989.9	−1.13497	−0.567486	−	0.823383i	$-0.692084\pi$
$631$	−17989.9	−1.13497	−0.567486	+	0.823383i	$0.692084\pi$
$632$	− 7939.71i	− 0.499723i
$633$	1133.77i	0.0711901i
$634$	−10645.1	−0.666828
$635$	0	0
$636$	−1910.89	−0.119138
$637$	− 3023.80i	− 0.188081i
$638$	3779.25i	0.234517i
$639$	797.598	0.0493779
$640$	0	0
$641$	6793.04	0.418579	0.209289	−	0.977854i	$-0.432885\pi$
$641$	6793.04	0.418579	0.209289	+	0.977854i	$0.432885\pi$
$642$	− 5831.93i	− 0.358517i
$643$	− 5141.23i	− 0.315319i	−0.987494	−	0.157660i	$-0.949605\pi$
$643$	− 5141.23i	− 0.315319i	0.987494	−	0.157660i	$-0.0503949\pi$
$644$	1385.87	0.0847999
$645$	0	0
$646$	6314.68	0.384594
$647$	− 23855.9i	− 1.44957i	−0.688973	−	0.724787i	$-0.741938\pi$
$647$	− 23855.9i	− 1.44957i	0.688973	−	0.724787i	$-0.258062\pi$
$648$	− 1175.42i	− 0.0712573i
$649$	−762.893	−0.0461420
$650$	0	0
$651$	−4157.95	−0.250327
$652$	− 16711.0i	− 1.00376i
$653$	13524.4i	0.810490i	0.914208	+	0.405245i	$0.132814\pi$
$653$	13524.4i	0.810490i	−0.914208	+	0.405245i	$0.867186\pi$
$654$	16079.8	0.961423
$655$	0	0
$656$	−22587.7	−1.34436
$657$	2161.03i	0.128325i
$658$	4039.15i	0.239305i
$659$	−28598.1	−1.69047	−0.845237	−	0.534391i	$-0.820541\pi$
$659$	−28598.1	−1.69047	−0.845237	+	0.534391i	$0.820541\pi$
$660$	0	0
$661$	16103.2	0.947568	0.473784	−	0.880641i	$-0.342888\pi$
$661$	16103.2	0.947568	0.473784	+	0.880641i	$0.342888\pi$
$662$	39818.6i	2.33775i
$663$	1138.35i	0.0666813i
$664$	−3166.36	−0.185058
$665$	0	0
$666$	4130.62	0.240328
$667$	2333.38i	0.135455i
$668$	− 2021.24i	− 0.117072i
$669$	3477.80	0.200986
$670$	0	0
$671$	−5603.11	−0.322363
$672$	− 6261.10i	− 0.359415i
$673$	− 24118.9i	− 1.38145i	−0.723117	−	0.690726i	$-0.757291\pi$
$673$	− 24118.9i	− 1.38145i	0.723117	−	0.690726i	$-0.242709\pi$
$674$	−6595.98	−0.376955
$675$	0	0
$676$	−12356.4	−0.703027
$677$	2983.70i	0.169384i	0.996407	+	0.0846918i	$0.0269906\pi$
$677$	2983.70i	0.169384i	−0.996407	+	0.0846918i	$0.973009\pi$
$678$	8287.18i	0.469421i
$679$	4454.44	0.251761
$680$	0	0
$681$	−7570.70	−0.426006
$682$	5580.86i	0.313347i
$683$	− 9226.11i	− 0.516878i	−0.966028	−	0.258439i	$-0.916792\pi$
$683$	− 9226.11i	− 0.516878i	0.966028	−	0.258439i	$-0.0832080\pi$
$684$	−5951.73	−0.332705
$685$	0	0
$686$	21910.0	1.21943
$687$	10600.8i	0.588712i
$688$	14267.7i	0.790628i
$689$	−1407.12	−0.0778038
$690$	0	0
$691$	−2176.51	−0.119824	−0.0599120	−	0.998204i	$-0.519082\pi$
$691$	−2176.51	−0.119824	−0.0599120	+	0.998204i	$0.519082\pi$
$692$	9739.66i	0.535038i
$693$	− 1905.19i	− 0.104433i
$694$	−24983.7	−1.36652
$695$	0	0
$696$	2067.76	0.112612
$697$	9814.72i	0.533370i
$698$	− 16667.8i	− 0.903849i
$699$	−4598.77	−0.248843
$700$	0	0
$701$	34049.9	1.83459	0.917295	−	0.398208i	$-0.130368\pi$
$701$	34049.9	1.83459	0.917295	+	0.398208i	$0.130368\pi$
$702$	− 5989.45i	− 0.322019i
$703$	2929.83i	0.157185i
$704$	−2376.25	−0.127213
$705$	0	0
$706$	−26323.4	−1.40325
$707$	− 4553.40i	− 0.242218i
$708$	− 1293.18i	− 0.0686450i
$709$	1690.56	0.0895491	0.0447745	−	0.998997i	$-0.485743\pi$
$709$	1690.56	0.0895491	0.0447745	+	0.998997i	$0.485743\pi$
$710$	0	0
$711$	20878.4	1.10127
$712$	− 2037.35i	− 0.107237i
$713$	3445.73i	0.180987i
$714$	−3426.42	−0.179595
$715$	0	0
$716$	4823.28	0.251752
$717$	− 2619.87i	− 0.136459i
$718$	12550.6i	0.652344i
$719$	−24755.3	−1.28403	−0.642014	−	0.766693i	$-0.721901\pi$
$719$	−24755.3	−1.28403	−0.642014	+	0.766693i	$0.721901\pi$
$720$	0	0
$721$	−3734.02	−0.192874
$722$	15902.2i	0.819693i
$723$	11037.9i	0.567779i
$724$	−10202.4	−0.523712
$725$	0	0
$726$	−12865.5	−0.657690
$727$	11781.4i	0.601031i	0.953777	+	0.300515i	$0.0971586\pi$
$727$	11781.4i	0.601031i	−0.953777	+	0.300515i	$0.902841\pi$
$728$	− 904.342i	− 0.0460400i
$729$	−6597.41	−0.335183
$730$	0	0
$731$	6199.54	0.313678
$732$	− 9497.83i	− 0.479576i
$733$	− 11652.6i	− 0.587172i	−0.955933	−	0.293586i	$-0.905151\pi$
$733$	− 11652.6i	− 0.587172i	0.955933	−	0.293586i	$-0.0948488\pi$
$734$	−45191.1	−2.27252
$735$	0	0
$736$	−5188.63	−0.259858
$737$	6867.13i	0.343221i
$738$	− 21487.0i	− 1.07175i
$739$	−22216.1	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$739$	−22216.1	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$740$	0	0
$741$	1767.67	0.0876342
$742$	− 4235.42i	− 0.209551i
$743$	22539.9i	1.11293i	0.830871	+	0.556466i	$0.187843\pi$
$743$	22539.9i	1.11293i	−0.830871	+	0.556466i	$0.812157\pi$
$744$	3053.49	0.150466
$745$	0	0
$746$	−24384.5	−1.19675
$747$	− 8326.32i	− 0.407824i
$748$	1979.96i	0.0967841i
$749$	5565.00	0.271483
$750$	0	0
$751$	34513.6	1.67699	0.838494	−	0.544911i	$-0.183436\pi$
$751$	34513.6	1.67699	0.838494	+	0.544911i	$0.183436\pi$
$752$	− 8199.68i	− 0.397622i
$753$	8492.01i	0.410978i
$754$	−4717.30	−0.227843
$755$	0	0
$756$	7761.52	0.373391
$757$	− 35405.0i	− 1.69989i	−0.526870	−	0.849946i	$-0.676634\pi$
$757$	− 35405.0i	− 1.69989i	0.526870	−	0.849946i	$-0.323366\pi$
$758$	− 9147.65i	− 0.438335i
$759$	636.802	0.0304538
$760$	0	0
$761$	20280.8	0.966069	0.483035	−	0.875601i	$-0.339534\pi$
$761$	20280.8	0.966069	0.483035	+	0.875601i	$0.339534\pi$
$762$	7108.35i	0.337937i
$763$	15343.8i	0.728026i
$764$	12255.9	0.580372
$765$	0	0
$766$	−40752.4	−1.92225
$767$	− 952.252i	− 0.0448290i
$768$	14815.2i	0.696090i
$769$	8677.76	0.406928	0.203464	−	0.979082i	$-0.434780\pi$
$769$	8677.76	0.406928	0.203464	+	0.979082i	$0.434780\pi$
$770$	0	0
$771$	−11998.2	−0.560446
$772$	− 5035.12i	− 0.234738i
$773$	− 18442.0i	− 0.858103i	−0.903280	−	0.429051i	$-0.858848\pi$
$773$	− 18442.0i	− 0.858103i	0.903280	−	0.429051i	$-0.141152\pi$
$774$	−13572.5	−0.630300
$775$	0	0
$776$	−3271.22	−0.151327
$777$	− 1589.76i	− 0.0734008i
$778$	3023.91i	0.139348i
$779$	15240.7	0.700969
$780$	0	0
$781$	−412.026	−0.0188776
$782$	2839.51i	0.129847i
$783$	13068.0i	0.596438i
$784$	−18476.8	−0.841692
$785$	0	0
$786$	−8686.62	−0.394200
$787$	12802.8i	0.579885i	0.957044	+	0.289942i	$0.0936362\pi$
$787$	12802.8i	0.579885i	−0.957044	+	0.289942i	$0.906364\pi$
$788$	− 12905.9i	− 0.583443i
$789$	−6901.62	−0.311412
$790$	0	0
$791$	−7907.87	−0.355463
$792$	1399.12i	0.0627722i
$793$	− 6993.87i	− 0.313190i
$794$	−28445.8	−1.27141
$795$	0	0
$796$	−2078.39	−0.0925460
$797$	− 28955.6i	− 1.28690i	−0.765488	−	0.643450i	$-0.777502\pi$
$797$	− 28955.6i	− 1.28690i	0.765488	−	0.643450i	$-0.222498\pi$
$798$	5320.69i	0.236028i
$799$	−3562.89	−0.157755
$800$	0	0
$801$	5357.46	0.236325
$802$	24900.8i	1.09636i
$803$	− 1116.35i	− 0.0490600i
$804$	−11640.5	−0.510606
$805$	0	0
$806$	−6966.09	−0.304430
$807$	− 7554.85i	− 0.329546i
$808$	3343.90i	0.145591i
$809$	−20743.7	−0.901493	−0.450747	−	0.892652i	$-0.648842\pi$
$809$	−20743.7	−0.901493	−0.450747	+	0.892652i	$0.648842\pi$
$810$	0	0
$811$	10421.2	0.451220	0.225610	−	0.974218i	$-0.427563\pi$
$811$	10421.2	0.451220	0.225610	+	0.974218i	$0.427563\pi$
$812$	− 6112.98i	− 0.264192i
$813$	− 1482.18i	− 0.0639389i
$814$	−2133.81	−0.0918795
$815$	0	0
$816$	6955.81	0.298409
$817$	− 9626.90i	− 0.412243i
$818$	27213.1i	1.16318i
$819$	2378.08	0.101461
$820$	0	0
$821$	4432.04	0.188404	0.0942018	−	0.995553i	$-0.469970\pi$
$821$	4432.04	0.188404	0.0942018	+	0.995553i	$0.469970\pi$
$822$	− 808.919i	− 0.0343240i
$823$	− 6572.80i	− 0.278388i	−0.990265	−	0.139194i	$-0.955549\pi$
$823$	− 6572.80i	− 0.278388i	0.990265	−	0.139194i	$-0.0444512\pi$
$824$	2742.17	0.115932
$825$	0	0
$826$	2866.28	0.120739
$827$	− 14107.7i	− 0.593197i	−0.955002	−	0.296598i	$-0.904148\pi$
$827$	− 14107.7i	− 0.593197i	0.955002	−	0.296598i	$-0.0958523\pi$
$828$	− 2676.30i	− 0.112328i
$829$	46543.8	1.94998	0.974990	−	0.222250i	$-0.0713400\pi$
$829$	46543.8	1.94998	0.974990	+	0.222250i	$0.0713400\pi$
$830$	0	0
$831$	−9610.95	−0.401203
$832$	− 2966.06i	− 0.123593i
$833$	8028.46i	0.333937i
$834$	118.428	0.00491705
$835$	0	0
$836$	3074.56	0.127196
$837$	19297.6i	0.796921i
$838$	48367.7i	1.99384i
$839$	−38019.7	−1.56447	−0.782233	−	0.622986i	$-0.785919\pi$
$839$	−38019.7	−1.56447	−0.782233	+	0.622986i	$0.785919\pi$
$840$	0	0
$841$	−14096.7	−0.577993
$842$	55232.8i	2.26063i
$843$	− 16454.1i	− 0.672255i
$844$	−2461.47	−0.100388
$845$	0	0
$846$	7800.11	0.316990
$847$	− 12276.6i	− 0.498028i
$848$	8598.12i	0.348185i
$849$	2040.30	0.0824769
$850$	0	0
$851$	−1317.45	−0.0530689
$852$	− 698.424i	− 0.0280840i
$853$	156.574i	0.00628485i	0.999995	+	0.00314242i	$0.00100027\pi$
$853$	156.574i	0.00628485i	−0.999995	+	0.00314242i	$0.999000\pi$
$854$	21051.6	0.843524
$855$	0	0
$856$	−4086.79	−0.163182
$857$	− 35179.5i	− 1.40223i	−0.713050	−	0.701113i	$-0.752687\pi$
$857$	− 35179.5i	− 1.40223i	0.713050	−	0.701113i	$-0.247313\pi$
$858$	1287.40i	0.0512250i
$859$	−14183.8	−0.563383	−0.281692	−	0.959505i	$-0.590895\pi$
$859$	−14183.8	−0.563383	−0.281692	+	0.959505i	$0.590895\pi$
$860$	0	0
$861$	−8269.79	−0.327333
$862$	− 12329.6i	− 0.487179i
$863$	33263.0i	1.31203i	0.754746	+	0.656017i	$0.227760\pi$
$863$	33263.0i	1.31203i	−0.754746	+	0.656017i	$0.772240\pi$
$864$	−29058.6	−1.14421
$865$	0	0
$866$	26481.5	1.03912
$867$	10663.7i	0.417715i
$868$	− 9027.12i	− 0.352996i
$869$	−10785.4	−0.421026
$870$	0	0
$871$	−8571.63	−0.333454
$872$	− 11268.1i	− 0.437599i
$873$	− 8602.08i	− 0.333489i
$874$	4409.30	0.170649
$875$	0	0
$876$	1892.32	0.0729860
$877$	− 19178.7i	− 0.738449i	−0.929340	−	0.369225i	$-0.879623\pi$
$877$	− 19178.7i	− 0.738449i	0.929340	−	0.369225i	$-0.120377\pi$
$878$	32933.6i	1.26590i
$879$	−21819.0	−0.837241
$880$	0	0
$881$	−33525.0	−1.28205	−0.641026	−	0.767519i	$-0.721491\pi$
$881$	−33525.0	−1.28205	−0.641026	+	0.767519i	$0.721491\pi$
$882$	− 17576.4i	− 0.671009i
$883$	37557.9i	1.43140i	0.698409	+	0.715699i	$0.253892\pi$
$883$	37557.9i	1.43140i	−0.698409	+	0.715699i	$0.746108\pi$
$884$	−2471.41	−0.0940299
$885$	0	0
$886$	−58600.2	−2.22202
$887$	7495.92i	0.283753i	0.989884	+	0.141876i	$0.0453135\pi$
$887$	7495.92i	0.283753i	−0.989884	+	0.141876i	$0.954686\pi$
$888$	1167.48i	0.0441195i
$889$	−6783.00	−0.255899
$890$	0	0
$891$	−1596.71	−0.0600356
$892$	7550.48i	0.283418i
$893$	5532.60i	0.207325i
$894$	−2461.58	−0.0920890
$895$	0	0
$896$	−9052.86	−0.337539
$897$	794.863i	0.0295872i
$898$	4971.76i	0.184755i
$899$	15198.8	0.563859
$900$	0	0
$901$	3736.01	0.138141
$902$	11099.8i	0.409739i
$903$	5223.68i	0.192506i
$904$	5807.34	0.213661
$905$	0	0
$906$	−24081.5	−0.883061
$907$	25397.1i	0.929764i	0.885373	+	0.464882i	$0.153903\pi$
$907$	25397.1i	0.929764i	−0.885373	+	0.464882i	$0.846097\pi$
$908$	− 16436.4i	− 0.600728i
$909$	−8793.18	−0.320849
$910$	0	0
$911$	21732.9	0.790387	0.395193	−	0.918598i	$-0.370678\pi$
$911$	21732.9	0.790387	0.395193	+	0.918598i	$0.370678\pi$
$912$	− 10801.3i	− 0.392177i
$913$	4301.24i	0.155915i
$914$	5911.55	0.213935
$915$	0	0
$916$	−23014.9	−0.830166
$917$	− 8289.02i	− 0.298504i
$918$	15902.5i	0.571744i
$919$	−13684.9	−0.491211	−0.245605	−	0.969370i	$-0.578987\pi$
$919$	−13684.9	−0.491211	−0.245605	+	0.969370i	$0.578987\pi$
$920$	0	0
$921$	−11487.9	−0.411008
$922$	69684.4i	2.48908i
$923$	− 514.295i	− 0.0183404i
$924$	−1668.29	−0.0593970
$925$	0	0
$926$	60272.8	2.13897
$927$	7210.86i	0.255486i
$928$	22886.6i	0.809580i
$929$	46187.0	1.63116	0.815580	−	0.578645i	$-0.196418\pi$
$929$	46187.0	1.63116	0.815580	+	0.578645i	$0.196418\pi$
$930$	0	0
$931$	12466.9	0.438869
$932$	− 9984.17i	− 0.350904i
$933$	− 25859.6i	− 0.907402i
$934$	18002.9	0.630700
$935$	0	0
$936$	−1746.40	−0.0609859
$937$	37728.4i	1.31540i	0.753278	+	0.657702i	$0.228472\pi$
$937$	37728.4i	1.31540i	−0.753278	+	0.657702i	$0.771528\pi$
$938$	− 25800.6i	− 0.898103i
$939$	−23984.5	−0.833551
$940$	0	0
$941$	33366.7	1.15592	0.577962	−	0.816064i	$-0.303848\pi$
$941$	33366.7	1.15592	0.577962	+	0.816064i	$0.303848\pi$
$942$	− 33534.0i	− 1.15987i
$943$	6853.25i	0.236662i
$944$	−5818.69	−0.200617
$945$	0	0
$946$	7011.30	0.240969
$947$	− 38414.4i	− 1.31816i	−0.752072	−	0.659081i	$-0.770946\pi$
$947$	− 38414.4i	− 1.31816i	0.752072	−	0.659081i	$-0.229054\pi$
$948$	− 18282.4i	− 0.626355i
$949$	1393.44	0.0476639
$950$	0	0
$951$	7911.82	0.269777
$952$	2401.11i	0.0817440i
$953$	47050.2i	1.59927i	0.600487	+	0.799635i	$0.294974\pi$
$953$	47050.2i	1.59927i	−0.600487	+	0.799635i	$0.705026\pi$
$954$	−8179.13	−0.277578
$955$	0	0
$956$	5687.88	0.192426
$957$	− 2808.88i	− 0.0948781i
$958$	− 9866.20i	− 0.332738i
$959$	771.895	0.0259914
$960$	0	0
$961$	−7346.71	−0.246608
$962$	− 2663.44i	− 0.0892648i
$963$	− 10746.7i	− 0.359614i
$964$	−23963.8	−0.800647
$965$	0	0
$966$	−2392.54	−0.0796881
$967$	15040.7i	0.500182i	0.968222	+	0.250091i	$0.0804605\pi$
$967$	15040.7i	0.500182i	−0.968222	+	0.250091i	$0.919540\pi$
$968$	9015.64i	0.299353i
$969$	−4693.31	−0.155594
$970$	0	0
$971$	54555.9	1.80307	0.901536	−	0.432703i	$-0.142440\pi$
$971$	54555.9	1.80307	0.901536	+	0.432703i	$0.142440\pi$
$972$	− 23740.4i	− 0.783409i
$973$	113.007i	0.00372338i
$974$	28268.5	0.929959
$975$	0	0
$976$	−42735.7	−1.40157
$977$	− 43553.2i	− 1.42619i	−0.701066	−	0.713096i	$-0.747292\pi$
$977$	− 43553.2i	− 1.42619i	0.701066	−	0.713096i	$-0.252708\pi$
$978$	28849.4i	0.943254i
$979$	−2767.57	−0.0903493
$980$	0	0
$981$	29630.9	0.964363
$982$	34807.4i	1.13111i
$983$	38375.8i	1.24516i	0.782554	+	0.622582i	$0.213916\pi$
$983$	38375.8i	1.24516i	−0.782554	+	0.622582i	$0.786084\pi$
$984$	6073.12	0.196752
$985$	0	0
$986$	12524.8	0.404536
$987$	− 3002.06i	− 0.0968151i
$988$	3837.70i	0.123576i
$989$	4328.90	0.139182
$990$	0	0
$991$	28505.2	0.913722	0.456861	−	0.889538i	$-0.348974\pi$
$991$	28505.2	0.913722	0.456861	+	0.889538i	$0.348974\pi$
$992$	33797.0i	1.08171i
$993$	− 29594.7i	− 0.945780i
$994$	1548.03	0.0493969
$995$	0	0
$996$	−7291.02	−0.231953
$997$	− 35545.6i	− 1.12913i	−0.825389	−	0.564565i	$-0.809044\pi$
$997$	− 35545.6i	− 1.12913i	0.825389	−	0.564565i	$-0.190956\pi$
$998$	− 28142.8i	− 0.892630i
$999$	−7378.32	−0.233673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.b.j.24.4		14
5.2	odd	4		575.4.a.l.1.6	✓	7
5.3	odd	4		575.4.a.m.1.2	yes	7
5.4	even	2	inner	575.4.b.j.24.11		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.4.a.l.1.6	✓	7	5.2	odd	4
575.4.a.m.1.2	yes	7	5.3	odd	4
575.4.b.j.24.4		14	1.1	even	1	trivial
575.4.b.j.24.11		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)

\(f(q)\)	\(=\)	\(q-3.74805i q^{2} +2.78570i q^{3} -6.04789 q^{4} +10.4409 q^{6} +9.96305i q^{7} -7.31660i q^{8} +19.2399 q^{9} -9.93900 q^{11} -16.8476i q^{12} -12.4060i q^{13} +37.3420 q^{14} -75.8061 q^{16} +32.9389i q^{17} -72.1121i q^{18} +51.1489 q^{19} -27.7540 q^{21} +37.2519i q^{22} +23.0000i q^{23} +20.3818 q^{24} -46.4982 q^{26} +128.810i q^{27} -60.2554i q^{28} +101.451 q^{29} +149.814 q^{31} +225.592i q^{32} -27.6871i q^{33} +123.457 q^{34} -116.361 q^{36} +57.2805i q^{37} -191.709i q^{38} +34.5593 q^{39} +297.967 q^{41} +104.024i q^{42} -188.213i q^{43} +60.1100 q^{44} +86.2052 q^{46} +108.166i q^{47} -211.173i q^{48} +243.738 q^{49} -91.7579 q^{51} +75.0300i q^{52} -113.422i q^{53} +482.788 q^{54} +72.8957 q^{56} +142.485i q^{57} -380.244i q^{58} +76.7575 q^{59} +563.750 q^{61} -561.511i q^{62} +191.688i q^{63} +239.083 q^{64} -103.772 q^{66} -690.928i q^{67} -199.211i q^{68} -64.0710 q^{69} +41.4554 q^{71} -140.771i q^{72} +112.320i q^{73} +214.690 q^{74} -309.343 q^{76} -99.0228i q^{77} -129.530i q^{78} +1085.16 q^{79} +160.651 q^{81} -1116.80i q^{82} -432.763i q^{83} +167.853 q^{84} -705.433 q^{86} +282.612i q^{87} +72.7198i q^{88} +278.456 q^{89} +123.601 q^{91} -139.101i q^{92} +417.337i q^{93} +405.414 q^{94} -628.432 q^{96} -447.096i q^{97} -913.542i q^{98} -191.225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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