LMFDB - Embedded newform 525.4.d.i.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.d (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:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.4.d.i.274.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43845i q^{2} +3.00000i q^{3} +5.93087 q^{4} +4.31534 q^{6} -7.00000i q^{7} -20.0388i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-1.43845i q^{2} +3.00000i q^{3} +5.93087 q^{4} +4.31534 q^{6} -7.00000i q^{7} -20.0388i q^{8} -9.00000 q^{9} +7.61553 q^{11} +17.7926i q^{12} -52.3542i q^{13} -10.0691 q^{14} +18.6222 q^{16} -49.7235i q^{17} +12.9460i q^{18} -140.600 q^{19} +21.0000 q^{21} -10.9545i q^{22} +23.4470i q^{23} +60.1165 q^{24} -75.3087 q^{26} -27.0000i q^{27} -41.5161i q^{28} -157.170 q^{29} +127.892 q^{31} -187.098i q^{32} +22.8466i q^{33} -71.5246 q^{34} -53.3778 q^{36} -115.477i q^{37} +202.246i q^{38} +157.062 q^{39} -188.617 q^{41} -30.2074i q^{42} -322.186i q^{43} +45.1667 q^{44} +33.7272 q^{46} -76.6477i q^{47} +55.8665i q^{48} -49.0000 q^{49} +149.170 q^{51} -310.506i q^{52} +424.172i q^{53} -38.8381 q^{54} -140.272 q^{56} -421.801i q^{57} +226.081i q^{58} -107.784 q^{59} +915.511 q^{61} -183.966i q^{62} +63.0000i q^{63} -120.153 q^{64} +32.8636 q^{66} -451.723i q^{67} -294.903i q^{68} -70.3409 q^{69} +907.312 q^{71} +180.349i q^{72} -755.956i q^{73} -166.108 q^{74} -833.882 q^{76} -53.3087i q^{77} -225.926i q^{78} -22.5834 q^{79} +81.0000 q^{81} +271.316i q^{82} -1112.25i q^{83} +124.548 q^{84} -463.447 q^{86} -471.511i q^{87} -152.606i q^{88} -1518.81 q^{89} -366.479 q^{91} +139.061i q^{92} +383.676i q^{93} -110.254 q^{94} +561.293 q^{96} -549.987i q^{97} +70.4839i q^{98} -68.5398 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 34 q^{4} + 42 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 34 * q^4 + 42 * q^6 - 36 * q^9	$ 4 q - 34 q^{4} + 42 q^{6} - 36 q^{9} - 52 q^{11} - 98 q^{14} + 594 q^{16} - 348 q^{19} + 84 q^{21} - 378 q^{24} + 276 q^{26} + 64 q^{29} + 660 q^{31} + 588 q^{34} + 306 q^{36} + 84 q^{39} + 400 q^{41} + 1632 q^{44} - 2240 q^{46} - 196 q^{49} - 96 q^{51} - 378 q^{54} + 882 q^{56} - 728 q^{59} + 1584 q^{61} - 5618 q^{64} - 1056 q^{66} + 1104 q^{69} + 908 q^{71} - 516 q^{74} - 136 q^{76} - 816 q^{79} + 324 q^{81} - 714 q^{84} - 1392 q^{86} - 72 q^{89} - 196 q^{91} + 3880 q^{94} + 6426 q^{96} + 468 q^{99}+O(q^{100}) $ 4 * q - 34 * q^4 + 42 * q^6 - 36 * q^9 - 52 * q^11 - 98 * q^14 + 594 * q^16 - 348 * q^19 + 84 * q^21 - 378 * q^24 + 276 * q^26 + 64 * q^29 + 660 * q^31 + 588 * q^34 + 306 * q^36 + 84 * q^39 + 400 * q^41 + 1632 * q^44 - 2240 * q^46 - 196 * q^49 - 96 * q^51 - 378 * q^54 + 882 * q^56 - 728 * q^59 + 1584 * q^61 - 5618 * q^64 - 1056 * q^66 + 1104 * q^69 + 908 * q^71 - 516 * q^74 - 136 * q^76 - 816 * q^79 + 324 * q^81 - 714 * q^84 - 1392 * q^86 - 72 * q^89 - 196 * q^91 + 3880 * q^94 + 6426 * q^96 + 468 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.43845i	− 0.508568i	−0.967130	−	0.254284i	$-0.918160\pi$
$2$	− 1.43845i	− 0.508568i	0.967130	−	0.254284i	$-0.0818398\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	5.93087	0.741359
$5$	0	0
$5$	0	0
$6$	4.31534	0.293622
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	− 20.0388i	− 0.885599i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	7.61553	0.208743	0.104371	−	0.994538i	$-0.466717\pi$
$11$	7.61553	0.208743	0.104371	+	0.994538i	$0.466717\pi$
$12$	17.7926i	0.428024i
$13$	− 52.3542i	− 1.11696i	−0.829519	−	0.558478i	$-0.811385\pi$
$13$	− 52.3542i	− 1.11696i	0.829519	−	0.558478i	$-0.188615\pi$
$14$	−10.0691	−0.192221
$15$	0	0
$16$	18.6222	0.290971
$17$	− 49.7235i	− 0.709395i	−0.934981	−	0.354697i	$-0.884584\pi$
$17$	− 49.7235i	− 0.709395i	0.934981	−	0.354697i	$-0.115416\pi$
$18$	12.9460i	0.169523i
$19$	−140.600	−1.69768	−0.848840	−	0.528649i	$-0.822699\pi$
$19$	−140.600	−1.69768	−0.848840	+	0.528649i	$0.822699\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	− 10.9545i	− 0.106160i
$23$	23.4470i	0.212566i	0.994336	+	0.106283i	$0.0338950\pi$
$23$	23.4470i	0.212566i	−0.994336	+	0.106283i	$0.966105\pi$
$24$	60.1165	0.511301
$25$	0	0
$26$	−75.3087	−0.568048
$27$	− 27.0000i	− 0.192450i
$28$	− 41.5161i	− 0.280207i
$29$	−157.170	−1.00641	−0.503204	−	0.864168i	$-0.667845\pi$
$29$	−157.170	−1.00641	−0.503204	+	0.864168i	$0.667845\pi$
$30$	0	0
$31$	127.892	0.740971	0.370485	−	0.928838i	$-0.379191\pi$
$31$	127.892	0.740971	0.370485	+	0.928838i	$0.379191\pi$
$32$	− 187.098i	− 1.03358i
$33$	22.8466i	0.120518i
$34$	−71.5246	−0.360776
$35$	0	0
$36$	−53.3778	−0.247120
$37$	− 115.477i	− 0.513090i	−0.966532	−	0.256545i	$-0.917416\pi$
$37$	− 115.477i	− 0.513090i	0.966532	−	0.256545i	$-0.0825843\pi$
$38$	202.246i	0.863386i
$39$	157.062	0.644875
$40$	0	0
$41$	−188.617	−0.718466	−0.359233	−	0.933248i	$-0.616962\pi$
$41$	−188.617	−0.718466	−0.359233	+	0.933248i	$0.616962\pi$
$42$	− 30.2074i	− 0.110979i
$43$	− 322.186i	− 1.14262i	−0.820733	−	0.571312i	$-0.806435\pi$
$43$	− 322.186i	− 1.14262i	0.820733	−	0.571312i	$-0.193565\pi$
$44$	45.1667	0.154753
$45$	0	0
$46$	33.7272	0.108104
$47$	− 76.6477i	− 0.237877i	−0.992902	−	0.118938i	$-0.962051\pi$
$47$	− 76.6477i	− 0.237877i	0.992902	−	0.118938i	$-0.0379491\pi$
$48$	55.8665i	0.167992i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	149.170	0.409569
$52$	− 310.506i	− 0.828065i
$53$	424.172i	1.09933i	0.835385	+	0.549666i	$0.185245\pi$
$53$	424.172i	1.09933i	−0.835385	+	0.549666i	$0.814755\pi$
$54$	−38.8381	−0.0978739
$55$	0	0
$56$	−140.272	−0.334725
$57$	− 421.801i	− 0.980157i
$58$	226.081i	0.511827i
$59$	−107.784	−0.237835	−0.118918	−	0.992904i	$-0.537942\pi$
$59$	−107.784	−0.237835	−0.118918	+	0.992904i	$0.537942\pi$
$60$	0	0
$61$	915.511	1.92163	0.960813	−	0.277197i	$-0.0894053\pi$
$61$	915.511	1.92163	0.960813	+	0.277197i	$0.0894053\pi$
$62$	− 183.966i	− 0.376834i
$63$	63.0000i	0.125988i
$64$	−120.153	−0.234673
$65$	0	0
$66$	32.8636	0.0612914
$67$	− 451.723i	− 0.823684i	−0.911255	−	0.411842i	$-0.864886\pi$
$67$	− 451.723i	− 0.823684i	0.911255	−	0.411842i	$-0.135114\pi$
$68$	− 294.903i	− 0.525916i
$69$	−70.3409	−0.122725
$70$	0	0
$71$	907.312	1.51659	0.758297	−	0.651909i	$-0.226032\pi$
$71$	907.312	1.51659	0.758297	+	0.651909i	$0.226032\pi$
$72$	180.349i	0.295200i
$73$	− 755.956i	− 1.21203i	−0.795454	−	0.606014i	$-0.792768\pi$
$73$	− 755.956i	− 1.21203i	0.795454	−	0.606014i	$-0.207232\pi$
$74$	−166.108	−0.260941
$75$	0	0
$76$	−833.882	−1.25859
$77$	− 53.3087i	− 0.0788973i
$78$	− 225.926i	− 0.327963i
$79$	−22.5834	−0.0321624	−0.0160812	−	0.999871i	$-0.505119\pi$
$79$	−22.5834	−0.0321624	−0.0160812	+	0.999871i	$0.505119\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	271.316i	0.365389i
$83$	− 1112.25i	− 1.47091i	−0.677575	−	0.735454i	$-0.736969\pi$
$83$	− 1112.25i	− 1.47091i	0.677575	−	0.735454i	$-0.263031\pi$
$84$	124.548	0.161778
$85$	0	0
$86$	−463.447	−0.581102
$87$	− 471.511i	− 0.581050i
$88$	− 152.606i	− 0.184862i
$89$	−1518.81	−1.80892	−0.904458	−	0.426562i	$-0.859725\pi$
$89$	−1518.81	−1.80892	−0.904458	+	0.426562i	$0.859725\pi$
$90$	0	0
$91$	−366.479	−0.422170
$92$	139.061i	0.157588i
$93$	383.676i	0.427800i
$94$	−110.254	−0.120977
$95$	0	0
$96$	561.293	0.596736
$97$	− 549.987i	− 0.575698i	−0.957676	−	0.287849i	$-0.907060\pi$
$97$	− 549.987i	− 0.575698i	0.957676	−	0.287849i	$-0.0929401\pi$
$98$	70.4839i	0.0726526i
$99$	−68.5398	−0.0695809
$100$	0	0
$101$	−533.299	−0.525398	−0.262699	−	0.964878i	$-0.584613\pi$
$101$	−533.299	−0.525398	−0.262699	+	0.964878i	$0.584613\pi$
$102$	− 214.574i	− 0.208294i
$103$	− 1357.79i	− 1.29890i	−0.760404	−	0.649451i	$-0.774999\pi$
$103$	− 1357.79i	− 1.29890i	0.760404	−	0.649451i	$-0.225001\pi$
$104$	−1049.12	−0.989176
$105$	0	0
$106$	610.149	0.559084
$107$	913.693i	0.825515i	0.910841	+	0.412757i	$0.135434\pi$
$107$	913.693i	0.825515i	−0.910841	+	0.412757i	$0.864566\pi$
$108$	− 160.133i	− 0.142675i
$109$	160.489	0.141028	0.0705139	−	0.997511i	$-0.477536\pi$
$109$	160.489	0.141028	0.0705139	+	0.997511i	$0.477536\pi$
$110$	0	0
$111$	346.432	0.296233
$112$	− 130.355i	− 0.109977i
$113$	1788.48i	1.48891i	0.667675	+	0.744453i	$0.267290\pi$
$113$	1788.48i	1.48891i	−0.667675	+	0.744453i	$0.732710\pi$
$114$	−606.739	−0.498476
$115$	0	0
$116$	−932.157	−0.746109
$117$	471.187i	0.372319i
$118$	155.042i	0.120955i
$119$	−348.064	−0.268126
$120$	0	0
$121$	−1273.00	−0.956427
$122$	− 1316.91i	− 0.977277i
$123$	− 565.852i	− 0.414806i
$124$	758.511	0.549325
$125$	0	0
$126$	90.6222	0.0640735
$127$	− 271.015i	− 0.189360i	−0.995508	−	0.0946799i	$-0.969817\pi$
$127$	− 271.015i	− 0.189360i	0.995508	−	0.0946799i	$-0.0301828\pi$
$128$	− 1323.95i	− 0.914231i
$129$	966.557	0.659694
$130$	0	0
$131$	−763.151	−0.508984	−0.254492	−	0.967075i	$-0.581908\pi$
$131$	−763.151	−0.508984	−0.254492	+	0.967075i	$0.581908\pi$
$132$	135.500i	0.0893468i
$133$	984.203i	0.641663i
$134$	−649.780	−0.418899
$135$	0	0
$136$	−996.400	−0.628240
$137$	− 240.934i	− 0.150251i	−0.997174	−	0.0751254i	$-0.976064\pi$
$137$	− 240.934i	− 0.150251i	0.997174	−	0.0751254i	$-0.0239357\pi$
$138$	101.182i	0.0624141i
$139$	103.150	0.0629427	0.0314714	−	0.999505i	$-0.489981\pi$
$139$	103.150	0.0629427	0.0314714	+	0.999505i	$0.489981\pi$
$140$	0	0
$141$	229.943	0.137338
$142$	− 1305.12i	− 0.771291i
$143$	− 398.705i	− 0.233156i
$144$	−167.600	−0.0969905
$145$	0	0
$146$	−1087.40	−0.616398
$147$	− 147.000i	− 0.0824786i
$148$	− 684.881i	− 0.380384i
$149$	3256.71	1.79061	0.895303	−	0.445458i	$-0.146959\pi$
$149$	3256.71	1.79061	0.895303	+	0.445458i	$0.146959\pi$
$150$	0	0
$151$	1471.04	0.792793	0.396396	−	0.918080i	$-0.370261\pi$
$151$	1471.04	0.792793	0.396396	+	0.918080i	$0.370261\pi$
$152$	2817.47i	1.50346i
$153$	447.511i	0.236465i
$154$	−76.6817	−0.0401246
$155$	0	0
$156$	931.517	0.478084
$157$	− 1394.22i	− 0.708730i	−0.935107	−	0.354365i	$-0.884697\pi$
$157$	− 1394.22i	− 0.708730i	0.935107	−	0.354365i	$-0.115303\pi$
$158$	32.4850i	0.0163567i
$159$	−1272.52	−0.634699
$160$	0	0
$161$	164.129	0.0803426
$162$	− 116.514i	− 0.0565075i
$163$	3674.27i	1.76559i	0.469760	+	0.882794i	$0.344341\pi$
$163$	3674.27i	1.76559i	−0.469760	+	0.882794i	$0.655659\pi$
$164$	−1118.67	−0.532641
$165$	0	0
$166$	−1599.91	−0.748056
$167$	4041.09i	1.87251i	0.351325	+	0.936254i	$0.385731\pi$
$167$	4041.09i	1.87251i	−0.351325	+	0.936254i	$0.614269\pi$
$168$	− 420.815i	− 0.193254i
$169$	−543.958	−0.247591
$170$	0	0
$171$	1265.40	0.565894
$172$	− 1910.84i	− 0.847094i
$173$	− 59.4582i	− 0.0261302i	−0.999915	−	0.0130651i	$-0.995841\pi$
$173$	− 59.4582i	− 0.0261302i	0.999915	−	0.0130651i	$-0.00415886\pi$
$174$	−678.244	−0.295503
$175$	0	0
$176$	141.818	0.0607381
$177$	− 323.352i	− 0.137314i
$178$	2184.73i	0.919957i
$179$	2973.32	1.24155	0.620773	−	0.783991i	$-0.286819\pi$
$179$	2973.32	1.24155	0.620773	+	0.783991i	$0.286819\pi$
$180$	0	0
$181$	−676.220	−0.277696	−0.138848	−	0.990314i	$-0.544340\pi$
$181$	−676.220	−0.277696	−0.138848	+	0.990314i	$0.544340\pi$
$182$	527.161i	0.214702i
$183$	2746.53i	1.10945i
$184$	469.849	0.188249
$185$	0	0
$186$	551.898	0.217565
$187$	− 378.671i	− 0.148081i
$188$	− 454.588i	− 0.176352i
$189$	−189.000	−0.0727393
$190$	0	0
$191$	16.5589	0.00627308	0.00313654	−	0.999995i	$-0.499002\pi$
$191$	16.5589	0.00627308	0.00313654	+	0.999995i	$0.499002\pi$
$192$	− 360.458i	− 0.135489i
$193$	− 2694.68i	− 1.00501i	−0.864574	−	0.502506i	$-0.832411\pi$
$193$	− 2694.68i	− 1.00501i	0.864574	−	0.502506i	$-0.167589\pi$
$194$	−791.127	−0.292781
$195$	0	0
$196$	−290.613	−0.105908
$197$	1027.38i	0.371561i	0.982591	+	0.185781i	$0.0594814\pi$
$197$	1027.38i	0.371561i	−0.982591	+	0.185781i	$0.940519\pi$
$198$	98.5908i	0.0353866i
$199$	−2823.77	−1.00589	−0.502944	−	0.864319i	$-0.667750\pi$
$199$	−2823.77	−1.00589	−0.502944	+	0.864319i	$0.667750\pi$
$200$	0	0
$201$	1355.17	0.475554
$202$	767.123i	0.267201i
$203$	1100.19i	0.380386i
$204$	884.710	0.303638
$205$	0	0
$206$	−1953.11	−0.660579
$207$	− 211.023i	− 0.0708555i
$208$	− 974.948i	− 0.325002i
$209$	−1070.75	−0.354378
$210$	0	0
$211$	5151.16	1.68067	0.840333	−	0.542071i	$-0.182360\pi$
$211$	5151.16	1.68067	0.840333	+	0.542071i	$0.182360\pi$
$212$	2515.71i	0.814999i
$213$	2721.94i	0.875606i
$214$	1314.30	0.419830
$215$	0	0
$216$	−541.048	−0.170434
$217$	− 895.244i	− 0.280061i
$218$	− 230.855i	− 0.0717222i
$219$	2267.87	0.699764
$220$	0	0
$221$	−2603.23	−0.792363
$222$	− 498.324i	− 0.150655i
$223$	114.496i	0.0343822i	0.999852	+	0.0171911i	$0.00547237\pi$
$223$	114.496i	0.0343822i	−0.999852	+	0.0171911i	$0.994528\pi$
$224$	−1309.68	−0.390656
$225$	0	0
$226$	2572.64	0.757209
$227$	4744.75i	1.38731i	0.720306	+	0.693657i	$0.244001\pi$
$227$	4744.75i	1.38731i	−0.720306	+	0.693657i	$0.755999\pi$
$228$	− 2501.65i	− 0.726648i
$229$	5384.47	1.55378	0.776891	−	0.629635i	$-0.216796\pi$
$229$	5384.47	1.55378	0.776891	+	0.629635i	$0.216796\pi$
$230$	0	0
$231$	159.926	0.0455514
$232$	3149.51i	0.891274i
$233$	1608.91i	0.452373i	0.974084	+	0.226187i	$0.0726260\pi$
$233$	1608.91i	0.452373i	−0.974084	+	0.226187i	$0.927374\pi$
$234$	677.778	0.189349
$235$	0	0
$236$	−639.253	−0.176321
$237$	− 67.7501i	− 0.0185689i
$238$	500.672i	0.136360i
$239$	3113.11	0.842554	0.421277	−	0.906932i	$-0.361582\pi$
$239$	3113.11	0.842554	0.421277	+	0.906932i	$0.361582\pi$
$240$	0	0
$241$	7136.38	1.90745	0.953724	−	0.300685i	$-0.0972151\pi$
$241$	7136.38	1.90745	0.953724	+	0.300685i	$0.0972151\pi$
$242$	1831.15i	0.486408i
$243$	243.000i	0.0641500i
$244$	5429.78	1.42461
$245$	0	0
$246$	−813.948	−0.210957
$247$	7361.01i	1.89624i
$248$	− 2562.81i	− 0.656203i
$249$	3336.75	0.849229
$250$	0	0
$251$	−225.504	−0.0567079	−0.0283539	−	0.999598i	$-0.509027\pi$
$251$	−225.504	−0.0567079	−0.0283539	+	0.999598i	$0.509027\pi$
$252$	373.645i	0.0934024i
$253$	178.561i	0.0443717i
$254$	−389.841	−0.0963024
$255$	0	0
$256$	−2865.65	−0.699621
$257$	− 4423.05i	− 1.07355i	−0.843725	−	0.536775i	$-0.819642\pi$
$257$	− 4423.05i	− 1.07355i	0.843725	−	0.536775i	$-0.180358\pi$
$258$	− 1390.34i	− 0.335499i
$259$	−808.341	−0.193930
$260$	0	0
$261$	1414.53	0.335469
$262$	1097.75i	0.258853i
$263$	− 6540.56i	− 1.53349i	−0.641950	−	0.766746i	$-0.721875\pi$
$263$	− 6540.56i	− 1.53349i	0.641950	−	0.766746i	$-0.278125\pi$
$264$	457.819	0.106730
$265$	0	0
$266$	1415.72	0.326329
$267$	− 4556.43i	− 1.04438i
$268$	− 2679.11i	− 0.610645i
$269$	−2262.97	−0.512920	−0.256460	−	0.966555i	$-0.582556\pi$
$269$	−2262.97	−0.512920	−0.256460	+	0.966555i	$0.582556\pi$
$270$	0	0
$271$	1615.68	0.362160	0.181080	−	0.983468i	$-0.442041\pi$
$271$	1615.68	0.362160	0.181080	+	0.983468i	$0.442041\pi$
$272$	− 925.959i	− 0.206414i
$273$	− 1099.44i	− 0.243740i
$274$	−346.571	−0.0764127
$275$	0	0
$276$	−417.183	−0.0909835
$277$	4691.55i	1.01765i	0.860871	+	0.508823i	$0.169919\pi$
$277$	4691.55i	1.01765i	−0.860871	+	0.508823i	$0.830081\pi$
$278$	− 148.375i	− 0.0320107i
$279$	−1151.03	−0.246990
$280$	0	0
$281$	−8119.00	−1.72363	−0.861813	−	0.507226i	$-0.830671\pi$
$281$	−8119.00	−1.72363	−0.861813	+	0.507226i	$0.830671\pi$
$282$	− 330.761i	− 0.0698459i
$283$	3633.75i	0.763265i	0.924314	+	0.381632i	$0.124638\pi$
$283$	3633.75i	0.763265i	−0.924314	+	0.381632i	$0.875362\pi$
$284$	5381.15	1.12434
$285$	0	0
$286$	−573.515	−0.118576
$287$	1320.32i	0.271554i
$288$	1683.88i	0.344526i
$289$	2440.58	0.496759
$290$	0	0
$291$	1649.96	0.332379
$292$	− 4483.48i	− 0.898547i
$293$	7981.99i	1.59151i	0.605618	+	0.795756i	$0.292926\pi$
$293$	7981.99i	1.59151i	−0.605618	+	0.795756i	$0.707074\pi$
$294$	−211.452	−0.0419460
$295$	0	0
$296$	−2314.03	−0.454392
$297$	− 205.619i	− 0.0401725i
$298$	− 4684.61i	− 0.910645i
$299$	1227.55	0.237427
$300$	0	0
$301$	−2255.30	−0.431871
$302$	− 2116.02i	− 0.403189i
$303$	− 1599.90i	− 0.303339i
$304$	−2618.28	−0.493977
$305$	0	0
$306$	643.721	0.120259
$307$	− 7118.15i	− 1.32330i	−0.749811	−	0.661652i	$-0.769856\pi$
$307$	− 7118.15i	− 1.32330i	0.749811	−	0.661652i	$-0.230144\pi$
$308$	− 316.167i	− 0.0584912i
$309$	4073.36	0.749921
$310$	0	0
$311$	−9155.92	−1.66940	−0.834702	−	0.550703i	$-0.814360\pi$
$311$	−9155.92	−1.66940	−0.834702	+	0.550703i	$0.814360\pi$
$312$	− 3147.35i	− 0.571101i
$313$	6163.44i	1.11303i	0.830838	+	0.556515i	$0.187862\pi$
$313$	6163.44i	1.11303i	−0.830838	+	0.556515i	$0.812138\pi$
$314$	−2005.51	−0.360438
$315$	0	0
$316$	−133.939	−0.0238438
$317$	8658.37i	1.53408i	0.641601	+	0.767038i	$0.278270\pi$
$317$	8658.37i	1.53408i	−0.641601	+	0.767038i	$0.721730\pi$
$318$	1830.45i	0.322788i
$319$	−1196.94	−0.210080
$320$	0	0
$321$	−2741.08	−0.476611
$322$	− 236.090i	− 0.0408597i
$323$	6991.14i	1.20433i
$324$	480.400	0.0823732
$325$	0	0
$326$	5285.24	0.897922
$327$	481.466i	0.0814224i
$328$	3779.67i	0.636272i
$329$	−536.534	−0.0899090
$330$	0	0
$331$	−128.477	−0.0213346	−0.0106673	−	0.999943i	$-0.503396\pi$
$331$	−128.477	−0.0213346	−0.0106673	+	0.999943i	$0.503396\pi$
$332$	− 6596.61i	− 1.09047i
$333$	1039.30i	0.171030i
$334$	5812.89	0.952297
$335$	0	0
$336$	391.066	0.0634952
$337$	7784.57i	1.25832i	0.777277	+	0.629158i	$0.216600\pi$
$337$	7784.57i	1.25832i	−0.777277	+	0.629158i	$0.783400\pi$
$338$	782.455i	0.125917i
$339$	−5365.45	−0.859620
$340$	0	0
$341$	973.965	0.154672
$342$	− 1820.22i	− 0.287795i
$343$	343.000i	0.0539949i
$344$	−6456.22	−1.01191
$345$	0	0
$346$	−85.5274	−0.0132890
$347$	3740.26i	0.578638i	0.957233	+	0.289319i	$0.0934289\pi$
$347$	3740.26i	0.578638i	−0.957233	+	0.289319i	$0.906571\pi$
$348$	− 2796.47i	− 0.430766i
$349$	−5676.86	−0.870703	−0.435351	−	0.900261i	$-0.643376\pi$
$349$	−5676.86	−0.870703	−0.435351	+	0.900261i	$0.643376\pi$
$350$	0	0
$351$	−1413.56	−0.214958
$352$	− 1424.85i	− 0.215752i
$353$	− 909.564i	− 0.137142i	−0.997646	−	0.0685711i	$-0.978156\pi$
$353$	− 909.564i	− 0.137142i	0.997646	−	0.0685711i	$-0.0218440\pi$
$354$	−465.125	−0.0698337
$355$	0	0
$356$	−9007.87	−1.34106
$357$	− 1044.19i	− 0.154803i
$358$	− 4276.97i	− 0.631410i
$359$	−2678.57	−0.393788	−0.196894	−	0.980425i	$-0.563085\pi$
$359$	−2678.57	−0.393788	−0.196894	+	0.980425i	$0.563085\pi$
$360$	0	0
$361$	12909.5	1.88212
$362$	972.706i	0.141227i
$363$	− 3819.01i	− 0.552193i
$364$	−2173.54	−0.312979
$365$	0	0
$366$	3950.74	0.564231
$367$	− 716.898i	− 0.101967i	−0.998700	−	0.0509833i	$-0.983764\pi$
$367$	− 716.898i	− 0.101967i	0.998700	−	0.0509833i	$-0.0162355\pi$
$368$	436.633i	0.0618508i
$369$	1697.56	0.239489
$370$	0	0
$371$	2969.21	0.415508
$372$	2275.53i	0.317153i
$373$	2006.15i	0.278484i	0.990258	+	0.139242i	$0.0444665\pi$
$373$	2006.15i	0.278484i	−0.990258	+	0.139242i	$0.955533\pi$
$374$	−544.698	−0.0753092
$375$	0	0
$376$	−1535.93	−0.210664
$377$	8228.53i	1.12411i
$378$	271.867i	0.0369929i
$379$	−7277.53	−0.986336	−0.493168	−	0.869934i	$-0.664161\pi$
$379$	−7277.53	−0.986336	−0.493168	+	0.869934i	$0.664161\pi$
$380$	0	0
$381$	813.045	0.109327
$382$	− 23.8191i	− 0.00319029i
$383$	− 5953.94i	− 0.794339i	−0.917745	−	0.397170i	$-0.869992\pi$
$383$	− 5953.94i	− 0.794339i	0.917745	−	0.397170i	$-0.130008\pi$
$384$	3971.84	0.527831
$385$	0	0
$386$	−3876.16	−0.511117
$387$	2899.67i	0.380875i
$388$	− 3261.90i	− 0.426799i
$389$	−1867.98	−0.243472	−0.121736	−	0.992563i	$-0.538846\pi$
$389$	−1867.98	−0.243472	−0.121736	+	0.992563i	$0.538846\pi$
$390$	0	0
$391$	1165.86	0.150794
$392$	981.902i	0.126514i
$393$	− 2289.45i	− 0.293862i
$394$	1477.83	0.188964
$395$	0	0
$396$	−406.500	−0.0515844
$397$	− 2160.07i	− 0.273075i	−0.990635	−	0.136538i	$-0.956403\pi$
$397$	− 2160.07i	− 0.273075i	0.990635	−	0.136538i	$-0.0435974\pi$
$398$	4061.85i	0.511563i
$399$	−2952.61	−0.370464
$400$	0	0
$401$	1954.81	0.243438	0.121719	−	0.992565i	$-0.461159\pi$
$401$	1954.81	0.243438	0.121719	+	0.992565i	$0.461159\pi$
$402$	− 1949.34i	− 0.241851i
$403$	− 6695.68i	− 0.827632i
$404$	−3162.93	−0.389509
$405$	0	0
$406$	1582.57	0.193452
$407$	− 879.420i	− 0.107104i
$408$	− 2989.20i	− 0.362714i
$409$	−14895.1	−1.80077	−0.900384	−	0.435096i	$-0.856714\pi$
$409$	−14895.1	−1.80077	−0.900384	+	0.435096i	$0.856714\pi$
$410$	0	0
$411$	722.801	0.0867474
$412$	− 8052.86i	− 0.962952i
$413$	754.489i	0.0898934i
$414$	−303.545	−0.0360348
$415$	0	0
$416$	−9795.34	−1.15446
$417$	309.449i	0.0363400i
$418$	1540.21i	0.180225i
$419$	−12608.9	−1.47013	−0.735067	−	0.677994i	$-0.762849\pi$
$419$	−12608.9	−1.47013	−0.735067	+	0.677994i	$0.762849\pi$
$420$	0	0
$421$	−7862.86	−0.910243	−0.455122	−	0.890429i	$-0.650404\pi$
$421$	−7862.86	−0.910243	−0.455122	+	0.890429i	$0.650404\pi$
$422$	− 7409.67i	− 0.854732i
$423$	689.829i	0.0792923i
$424$	8499.91	0.973567
$425$	0	0
$426$	3915.36	0.445305
$427$	− 6408.58i	− 0.726307i
$428$	5419.00i	0.612002i
$429$	1196.11	0.134613
$430$	0	0
$431$	14291.0	1.59715	0.798575	−	0.601896i	$-0.205588\pi$
$431$	14291.0	1.59715	0.798575	+	0.601896i	$0.205588\pi$
$432$	− 502.799i	− 0.0559975i
$433$	13759.0i	1.52705i	0.645776	+	0.763527i	$0.276534\pi$
$433$	13759.0i	1.52705i	−0.645776	+	0.763527i	$0.723466\pi$
$434$	−1287.76	−0.142430
$435$	0	0
$436$	951.838	0.104552
$437$	− 3296.65i	− 0.360870i
$438$	− 3262.21i	− 0.355878i
$439$	6093.13	0.662436	0.331218	−	0.943554i	$-0.392541\pi$
$439$	6093.13	0.662436	0.331218	+	0.943554i	$0.392541\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	3744.61i	0.402970i
$443$	− 13449.5i	− 1.44244i	−0.692704	−	0.721222i	$-0.743581\pi$
$443$	− 13449.5i	− 1.44244i	0.692704	−	0.721222i	$-0.256419\pi$
$444$	2054.64	0.219615
$445$	0	0
$446$	164.697	0.0174857
$447$	9770.14i	1.03381i
$448$	841.068i	0.0886981i
$449$	−6893.83	−0.724588	−0.362294	−	0.932064i	$-0.618006\pi$
$449$	−6893.83	−0.724588	−0.362294	+	0.932064i	$0.618006\pi$
$450$	0	0
$451$	−1436.42	−0.149974
$452$	10607.3i	1.10381i
$453$	4413.13i	0.457719i
$454$	6825.07	0.705543
$455$	0	0
$456$	−8452.40	−0.868026
$457$	− 11820.1i	− 1.20990i	−0.796265	−	0.604948i	$-0.793194\pi$
$457$	− 11820.1i	− 1.20990i	0.796265	−	0.604948i	$-0.206806\pi$
$458$	− 7745.28i	− 0.790203i
$459$	−1342.53	−0.136523
$460$	0	0
$461$	8443.38	0.853031	0.426516	−	0.904480i	$-0.359741\pi$
$461$	8443.38	0.853031	0.426516	+	0.904480i	$0.359741\pi$
$462$	− 230.045i	− 0.0231660i
$463$	1269.74i	0.127451i	0.997967	+	0.0637257i	$0.0202983\pi$
$463$	1269.74i	0.127451i	−0.997967	+	0.0637257i	$0.979702\pi$
$464$	−2926.86	−0.292836
$465$	0	0
$466$	2314.33	0.230063
$467$	− 16481.8i	− 1.63316i	−0.577233	−	0.816579i	$-0.695868\pi$
$467$	− 16481.8i	− 1.63316i	0.577233	−	0.816579i	$-0.304132\pi$
$468$	2794.55i	0.276022i
$469$	−3162.06	−0.311323
$470$	0	0
$471$	4182.65	0.409186
$472$	2159.87i	0.210627i
$473$	− 2453.61i	− 0.238514i
$474$	−97.4549	−0.00944357
$475$	0	0
$476$	−2064.32	−0.198778
$477$	− 3817.55i	− 0.366444i
$478$	− 4478.05i	− 0.428496i
$479$	1400.21	0.133564	0.0667822	−	0.997768i	$-0.478727\pi$
$479$	1400.21	0.133564	0.0667822	+	0.997768i	$0.478727\pi$
$480$	0	0
$481$	−6045.72	−0.573100
$482$	− 10265.3i	− 0.970066i
$483$	492.386i	0.0463858i
$484$	−7550.02	−0.709055
$485$	0	0
$486$	349.543	0.0326246
$487$	− 14165.9i	− 1.31811i	−0.752094	−	0.659055i	$-0.770956\pi$
$487$	− 14165.9i	− 1.31811i	0.752094	−	0.659055i	$-0.229044\pi$
$488$	− 18345.8i	− 1.70179i
$489$	−11022.8	−1.01936
$490$	0	0
$491$	4739.28	0.435603	0.217801	−	0.975993i	$-0.430112\pi$
$491$	4739.28	0.435603	0.217801	+	0.975993i	$0.430112\pi$
$492$	− 3356.00i	− 0.307520i
$493$	7815.06i	0.713940i
$494$	10588.4	0.964364
$495$	0	0
$496$	2381.63	0.215601
$497$	− 6351.19i	− 0.573219i
$498$	− 4799.74i	− 0.431890i
$499$	11370.0	1.02003	0.510013	−	0.860167i	$-0.329640\pi$
$499$	11370.0	1.02003	0.510013	+	0.860167i	$0.329640\pi$
$500$	0	0
$501$	−12123.3	−1.08109
$502$	324.375i	0.0288398i
$503$	− 9212.48i	− 0.816629i	−0.912841	−	0.408314i	$-0.866117\pi$
$503$	− 9212.48i	− 0.816629i	0.912841	−	0.408314i	$-0.133883\pi$
$504$	1262.45	0.111575
$505$	0	0
$506$	256.851	0.0225660
$507$	− 1631.87i	− 0.142947i
$508$	− 1607.36i	− 0.140384i
$509$	15938.3	1.38792	0.693960	−	0.720014i	$-0.255864\pi$
$509$	15938.3	1.38792	0.693960	+	0.720014i	$0.255864\pi$
$510$	0	0
$511$	−5291.69	−0.458103
$512$	− 6469.49i	− 0.558426i
$513$	3796.21i	0.326719i
$514$	−6362.33	−0.545973
$515$	0	0
$516$	5732.52	0.489070
$517$	− 583.713i	− 0.0496550i
$518$	1162.76i	0.0986265i
$519$	178.374	0.0150863
$520$	0	0
$521$	6442.99	0.541790	0.270895	−	0.962609i	$-0.412680\pi$
$521$	6442.99	0.541790	0.270895	+	0.962609i	$0.412680\pi$
$522$	− 2034.73i	− 0.170609i
$523$	− 986.655i	− 0.0824922i	−0.999149	−	0.0412461i	$-0.986867\pi$
$523$	− 986.655i	− 0.0824922i	0.999149	−	0.0412461i	$-0.0131328\pi$
$524$	−4526.15	−0.377339
$525$	0	0
$526$	−9408.26	−0.779885
$527$	− 6359.24i	− 0.525641i
$528$	425.453i	0.0350672i
$529$	11617.2	0.954815
$530$	0	0
$531$	970.057	0.0792785
$532$	5837.18i	0.475703i
$533$	9874.91i	0.802495i
$534$	−6554.19	−0.531137
$535$	0	0
$536$	−9052.01	−0.729454
$537$	8919.97i	0.716807i
$538$	3255.16i	0.260855i
$539$	−373.161	−0.0298204
$540$	0	0
$541$	−12681.1	−1.00777	−0.503885	−	0.863771i	$-0.668097\pi$
$541$	−12681.1	−1.00777	−0.503885	+	0.863771i	$0.668097\pi$
$542$	− 2324.06i	− 0.184183i
$543$	− 2028.66i	− 0.160328i
$544$	−9303.14	−0.733215
$545$	0	0
$546$	−1581.48	−0.123958
$547$	− 14826.4i	− 1.15892i	−0.814999	−	0.579462i	$-0.803263\pi$
$547$	− 14826.4i	− 1.15892i	0.814999	−	0.579462i	$-0.196737\pi$
$548$	− 1428.95i	− 0.111390i
$549$	−8239.60	−0.640542
$550$	0	0
$551$	22098.2	1.70856
$552$	1409.55i	0.108685i
$553$	158.083i	0.0121562i
$554$	6748.55	0.517542
$555$	0	0
$556$	611.767	0.0466632
$557$	1926.46i	0.146547i	0.997312	+	0.0732737i	$0.0233447\pi$
$557$	1926.46i	0.146547i	−0.997312	+	0.0732737i	$0.976655\pi$
$558$	1655.69i	0.125611i
$559$	−16867.8	−1.27626
$560$	0	0
$561$	1136.01	0.0854946
$562$	11678.8i	0.876581i
$563$	18624.8i	1.39422i	0.716966	+	0.697108i	$0.245530\pi$
$563$	18624.8i	1.39422i	−0.716966	+	0.697108i	$0.754470\pi$
$564$	1363.76	0.101817
$565$	0	0
$566$	5226.96	0.388172
$567$	− 567.000i	− 0.0419961i
$568$	− 18181.5i	− 1.34309i
$569$	20093.9	1.48045	0.740227	−	0.672357i	$-0.234718\pi$
$569$	20093.9	1.48045	0.740227	+	0.672357i	$0.234718\pi$
$570$	0	0
$571$	4535.25	0.332389	0.166195	−	0.986093i	$-0.446852\pi$
$571$	4535.25	0.332389	0.166195	+	0.986093i	$0.446852\pi$
$572$	− 2364.66i	− 0.172852i
$573$	49.6766i	0.00362176i
$574$	1899.21	0.138104
$575$	0	0
$576$	1081.37	0.0782243
$577$	10034.6i	0.723994i	0.932179	+	0.361997i	$0.117905\pi$
$577$	10034.6i	0.723994i	−0.932179	+	0.361997i	$0.882095\pi$
$578$	− 3510.64i	− 0.252636i
$579$	8084.05	0.580244
$580$	0	0
$581$	−7785.75	−0.555951
$582$	− 2373.38i	− 0.169037i
$583$	3230.30i	0.229477i
$584$	−15148.5	−1.07337
$585$	0	0
$586$	11481.7	0.809391
$587$	− 11192.6i	− 0.786999i	−0.919325	−	0.393499i	$-0.871264\pi$
$587$	− 11192.6i	− 0.786999i	0.919325	−	0.393499i	$-0.128736\pi$
$588$	− 871.838i	− 0.0611462i
$589$	−17981.7	−1.25793
$590$	0	0
$591$	−3082.13	−0.214521
$592$	− 2150.44i	− 0.149295i
$593$	− 20317.6i	− 1.40699i	−0.710703	−	0.703493i	$-0.751623\pi$
$593$	− 20317.6i	− 1.40699i	0.710703	−	0.703493i	$-0.248377\pi$
$594$	−295.772	−0.0204305
$595$	0	0
$596$	19315.1	1.32748
$597$	− 8471.31i	− 0.580750i
$598$	− 1765.76i	− 0.120748i
$599$	26376.5	1.79919	0.899594	−	0.436727i	$-0.143862\pi$
$599$	26376.5	1.79919	0.899594	+	0.436727i	$0.143862\pi$
$600$	0	0
$601$	9266.13	0.628907	0.314454	−	0.949273i	$-0.398179\pi$
$601$	9266.13	0.628907	0.314454	+	0.949273i	$0.398179\pi$
$602$	3244.13i	0.219636i
$603$	4065.51i	0.274561i
$604$	8724.56	0.587744
$605$	0	0
$606$	−2301.37	−0.154268
$607$	11338.0i	0.758149i	0.925366	+	0.379075i	$0.123758\pi$
$607$	11338.0i	0.758149i	−0.925366	+	0.379075i	$0.876242\pi$
$608$	26306.0i	1.75469i
$609$	−3300.58	−0.219616
$610$	0	0
$611$	−4012.83	−0.265698
$612$	2654.13i	0.175305i
$613$	− 25712.5i	− 1.69416i	−0.531469	−	0.847078i	$-0.678360\pi$
$613$	− 25712.5i	− 1.69416i	0.531469	−	0.847078i	$-0.321640\pi$
$614$	−10239.1	−0.672990
$615$	0	0
$616$	−1068.24	−0.0698714
$617$	663.465i	0.0432903i	0.999766	+	0.0216451i	$0.00689040\pi$
$617$	663.465i	0.0432903i	−0.999766	+	0.0216451i	$0.993110\pi$
$618$	− 5859.32i	− 0.381386i
$619$	12768.7	0.829108	0.414554	−	0.910025i	$-0.363938\pi$
$619$	12768.7	0.829108	0.414554	+	0.910025i	$0.363938\pi$
$620$	0	0
$621$	633.068	0.0409084
$622$	13170.3i	0.849005i
$623$	10631.7i	0.683706i
$624$	2924.84	0.187640
$625$	0	0
$626$	8865.78	0.566051
$627$	− 3212.24i	− 0.204600i
$628$	− 8268.92i	− 0.525423i
$629$	−5741.93	−0.363984
$630$	0	0
$631$	−14937.6	−0.942400	−0.471200	−	0.882026i	$-0.656179\pi$
$631$	−14937.6	−0.942400	−0.471200	+	0.882026i	$0.656179\pi$
$632$	452.544i	0.0284829i
$633$	15453.5i	0.970333i
$634$	12454.6	0.780182
$635$	0	0
$636$	−7547.13	−0.470540
$637$	2565.35i	0.159565i
$638$	1721.73i	0.106840i
$639$	−8165.81	−0.505531
$640$	0	0
$641$	10903.0	0.671829	0.335914	−	0.941893i	$-0.390955\pi$
$641$	10903.0	0.671829	0.335914	+	0.941893i	$0.390955\pi$
$642$	3942.90i	0.242389i
$643$	7623.47i	0.467559i	0.972290	+	0.233779i	$0.0751094\pi$
$643$	7623.47i	0.467559i	−0.972290	+	0.233779i	$0.924891\pi$
$644$	973.426	0.0595627
$645$	0	0
$646$	10056.4	0.612482
$647$	5384.84i	0.327202i	0.986527	+	0.163601i	$0.0523110\pi$
$647$	5384.84i	0.327202i	−0.986527	+	0.163601i	$0.947689\pi$
$648$	− 1623.14i	− 0.0983999i
$649$	−820.833	−0.0496464
$650$	0	0
$651$	2685.73	0.161693
$652$	21791.6i	1.30893i
$653$	− 297.318i	− 0.0178177i	−0.999960	−	0.00890883i	$-0.997164\pi$
$653$	− 297.318i	− 0.0178177i	0.999960	−	0.00890883i	$-0.00283581\pi$
$654$	692.564	0.0414088
$655$	0	0
$656$	−3512.47	−0.209053
$657$	6803.61i	0.404009i
$658$	771.776i	0.0457248i
$659$	−10324.7	−0.610309	−0.305155	−	0.952303i	$-0.598708\pi$
$659$	−10324.7	−0.610309	−0.305155	+	0.952303i	$0.598708\pi$
$660$	0	0
$661$	4272.98	0.251437	0.125718	−	0.992066i	$-0.459876\pi$
$661$	4272.98	0.251437	0.125718	+	0.992066i	$0.459876\pi$
$662$	184.808i	0.0108501i
$663$	− 7809.69i	− 0.457471i
$664$	−22288.2	−1.30263
$665$	0	0
$666$	1494.97	0.0869804
$667$	− 3685.17i	− 0.213928i
$668$	23967.2i	1.38820i
$669$	−343.488	−0.0198506
$670$	0	0
$671$	6972.10	0.401125
$672$	− 3929.05i	− 0.225545i
$673$	23033.4i	1.31927i	0.751584	+	0.659637i	$0.229290\pi$
$673$	23033.4i	1.31927i	−0.751584	+	0.659637i	$0.770710\pi$
$674$	11197.7	0.639940
$675$	0	0
$676$	−3226.15	−0.183554
$677$	5113.50i	0.290292i	0.989410	+	0.145146i	$0.0463652\pi$
$677$	5113.50i	0.290292i	−0.989410	+	0.145146i	$0.953635\pi$
$678$	7717.91i	0.437175i
$679$	−3849.91	−0.217593
$680$	0	0
$681$	−14234.2	−0.800966
$682$	− 1401.00i	− 0.0786613i
$683$	− 1341.02i	− 0.0751286i	−0.999294	−	0.0375643i	$-0.988040\pi$
$683$	− 1341.02i	− 0.0751286i	0.999294	−	0.0375643i	$-0.0119599\pi$
$684$	7504.94	0.419530
$685$	0	0
$686$	493.387	0.0274601
$687$	16153.4i	0.897076i
$688$	− 5999.80i	− 0.332471i
$689$	22207.2	1.22790
$690$	0	0
$691$	−16809.3	−0.925404	−0.462702	−	0.886514i	$-0.653120\pi$
$691$	−16809.3	−0.925404	−0.462702	+	0.886514i	$0.653120\pi$
$692$	− 352.639i	− 0.0193718i
$693$	479.778i	0.0262991i
$694$	5380.16	0.294277
$695$	0	0
$696$	−9448.53	−0.514577
$697$	9378.71i	0.509676i
$698$	8165.86i	0.442812i
$699$	−4826.72	−0.261178
$700$	0	0
$701$	13467.0	0.725592	0.362796	−	0.931869i	$-0.381822\pi$
$701$	13467.0	0.725592	0.362796	+	0.931869i	$0.381822\pi$
$702$	2033.33i	0.109321i
$703$	16236.1i	0.871064i
$704$	−915.025	−0.0489863
$705$	0	0
$706$	−1308.36	−0.0697461
$707$	3733.09i	0.198582i
$708$	− 1917.76i	− 0.101799i
$709$	−35514.3	−1.88119	−0.940597	−	0.339525i	$-0.889734\pi$
$709$	−35514.3	−1.88119	−0.940597	+	0.339525i	$0.889734\pi$
$710$	0	0
$711$	203.250	0.0107208
$712$	30435.2i	1.60198i
$713$	2998.68i	0.157506i
$714$	−1502.02	−0.0787277
$715$	0	0
$716$	17634.4	0.920431
$717$	9339.34i	0.486449i
$718$	3852.99i	0.200268i
$719$	−4993.61	−0.259013	−0.129506	−	0.991579i	$-0.541339\pi$
$719$	−4993.61	−0.259013	−0.129506	+	0.991579i	$0.541339\pi$
$720$	0	0
$721$	−9504.51	−0.490938
$722$	− 18569.6i	− 0.957186i
$723$	21409.1i	1.10127i
$724$	−4010.57	−0.205872
$725$	0	0
$726$	−5493.45	−0.280828
$727$	− 4223.35i	− 0.215454i	−0.994181	−	0.107727i	$-0.965643\pi$
$727$	− 4223.35i	− 0.215454i	0.994181	−	0.107727i	$-0.0343573\pi$
$728$	7343.81i	0.373873i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−16020.2	−0.810572
$732$	16289.3i	0.822502i
$733$	− 19030.9i	− 0.958968i	−0.877551	−	0.479484i	$-0.840824\pi$
$733$	− 19030.9i	− 0.958968i	0.877551	−	0.479484i	$-0.159176\pi$
$734$	−1031.22	−0.0518570
$735$	0	0
$736$	4386.87	0.219704
$737$	− 3440.11i	− 0.171938i
$738$	− 2441.85i	− 0.121796i
$739$	27772.5	1.38245	0.691224	−	0.722641i	$-0.257072\pi$
$739$	27772.5	1.38245	0.691224	+	0.722641i	$0.257072\pi$
$740$	0	0
$741$	−22083.0	−1.09479
$742$	− 4271.05i	− 0.211314i
$743$	− 26880.7i	− 1.32726i	−0.748060	−	0.663631i	$-0.769015\pi$
$743$	− 26880.7i	− 1.32726i	0.748060	−	0.663631i	$-0.230985\pi$
$744$	7688.42	0.378859
$745$	0	0
$746$	2885.74	0.141628
$747$	10010.2i	0.490302i
$748$	− 2245.85i	− 0.109781i
$749$	6395.85	0.312015
$750$	0	0
$751$	−35166.6	−1.70872	−0.854360	−	0.519681i	$-0.826051\pi$
$751$	−35166.6	−1.70872	−0.854360	+	0.519681i	$0.826051\pi$
$752$	− 1427.35i	− 0.0692154i
$753$	− 676.512i	− 0.0327403i
$754$	11836.3	0.571688
$755$	0	0
$756$	−1120.93	−0.0539259
$757$	14589.2i	0.700467i	0.936662	+	0.350233i	$0.113898\pi$
$757$	14589.2i	0.700467i	−0.936662	+	0.350233i	$0.886102\pi$
$758$	10468.3i	0.501619i
$759$	−535.683	−0.0256180
$760$	0	0
$761$	−782.826	−0.0372897	−0.0186448	−	0.999826i	$-0.505935\pi$
$761$	−782.826	−0.0372897	−0.0186448	+	0.999826i	$0.505935\pi$
$762$	− 1169.52i	− 0.0556002i
$763$	− 1123.42i	− 0.0533035i
$764$	98.2085	0.00465060
$765$	0	0
$766$	−8564.42	−0.403975
$767$	5642.95i	0.265652i
$768$	− 8596.95i	− 0.403927i
$769$	16548.4	0.776008	0.388004	−	0.921658i	$-0.373165\pi$
$769$	16548.4	0.776008	0.388004	+	0.921658i	$0.373165\pi$
$770$	0	0
$771$	13269.2	0.619815
$772$	− 15981.8i	− 0.745075i
$773$	− 5744.94i	− 0.267310i	−0.991028	−	0.133655i	$-0.957329\pi$
$773$	− 5744.94i	− 0.267310i	0.991028	−	0.133655i	$-0.0426715\pi$
$774$	4171.02	0.193701
$775$	0	0
$776$	−11021.1	−0.509837
$777$	− 2425.02i	− 0.111966i
$778$	2687.00i	0.123822i
$779$	26519.7	1.21973
$780$	0	0
$781$	6909.66	0.316578
$782$	− 1677.03i	− 0.0766888i
$783$	4243.60i	0.193683i
$784$	−912.486	−0.0415673
$785$	0	0
$786$	−3293.26	−0.149449
$787$	− 34744.3i	− 1.57370i	−0.617146	−	0.786848i	$-0.711711\pi$
$787$	− 34744.3i	− 1.57370i	0.617146	−	0.786848i	$-0.288289\pi$
$788$	6093.24i	0.275460i
$789$	19621.7	0.885362
$790$	0	0
$791$	12519.4	0.562753
$792$	1373.46i	0.0616207i
$793$	− 47930.8i	− 2.14637i
$794$	−3107.15	−0.138877
$795$	0	0
$796$	−16747.4	−0.745724
$797$	21748.7i	0.966600i	0.875455	+	0.483300i	$0.160562\pi$
$797$	21748.7i	0.966600i	−0.875455	+	0.483300i	$0.839438\pi$
$798$	4247.17i	0.188406i
$799$	−3811.19	−0.168749
$800$	0	0
$801$	13669.3	0.602972
$802$	− 2811.89i	− 0.123805i
$803$	− 5757.01i	− 0.253002i
$804$	8037.34	0.352556
$805$	0	0
$806$	−9631.38	−0.420907
$807$	− 6788.90i	− 0.296134i
$808$	10686.7i	0.465292i
$809$	42350.6	1.84050	0.920252	−	0.391325i	$-0.127983\pi$
$809$	42350.6	1.84050	0.920252	+	0.391325i	$0.127983\pi$
$810$	0	0
$811$	−18910.7	−0.818796	−0.409398	−	0.912356i	$-0.634261\pi$
$811$	−18910.7	−0.818796	−0.409398	+	0.912356i	$0.634261\pi$
$812$	6525.10i	0.282003i
$813$	4847.03i	0.209093i
$814$	−1265.00	−0.0544696
$815$	0	0
$816$	2777.88	0.119173
$817$	45299.4i	1.93981i
$818$	21425.8i	0.915813i
$819$	3298.31	0.140723
$820$	0	0
$821$	6593.42	0.280283	0.140141	−	0.990132i	$-0.455244\pi$
$821$	6593.42	0.280283	0.140141	+	0.990132i	$0.455244\pi$
$822$	− 1039.71i	− 0.0441169i
$823$	− 26762.4i	− 1.13351i	−0.823886	−	0.566755i	$-0.808199\pi$
$823$	− 26762.4i	− 1.13351i	0.823886	−	0.566755i	$-0.191801\pi$
$824$	−27208.5	−1.15031
$825$	0	0
$826$	1085.29	0.0457169
$827$	− 24016.7i	− 1.00985i	−0.863164	−	0.504924i	$-0.831521\pi$
$827$	− 24016.7i	− 1.00985i	0.863164	−	0.504924i	$-0.168479\pi$
$828$	− 1251.55i	− 0.0525293i
$829$	28422.3	1.19077	0.595383	−	0.803442i	$-0.297000\pi$
$829$	28422.3	1.19077	0.595383	+	0.803442i	$0.297000\pi$
$830$	0	0
$831$	−14074.7	−0.587538
$832$	6290.49i	0.262120i
$833$	2436.45i	0.101342i
$834$	445.126	0.0184814
$835$	0	0
$836$	−6350.46	−0.262721
$837$	− 3453.09i	− 0.142600i
$838$	18137.3i	0.747663i
$839$	−6637.09	−0.273108	−0.136554	−	0.990633i	$-0.543603\pi$
$839$	−6637.09	−0.273108	−0.136554	+	0.990633i	$0.543603\pi$
$840$	0	0
$841$	313.546	0.0128560
$842$	11310.3i	0.462921i
$843$	− 24357.0i	− 0.995136i
$844$	30550.9	1.24598
$845$	0	0
$846$	992.283	0.0403255
$847$	8911.03i	0.361495i
$848$	7899.01i	0.319874i
$849$	−10901.2	−0.440671
$850$	0	0
$851$	2707.59	0.109066
$852$	16143.5i	0.649138i
$853$	1406.88i	0.0564720i	0.999601	+	0.0282360i	$0.00898899\pi$
$853$	1406.88i	0.0564720i	−0.999601	+	0.0282360i	$0.991011\pi$
$854$	−9218.40	−0.369376
$855$	0	0
$856$	18309.3	0.731075
$857$	27943.4i	1.11380i	0.830579	+	0.556901i	$0.188010\pi$
$857$	27943.4i	1.11380i	−0.830579	+	0.556901i	$0.811990\pi$
$858$	− 1720.55i	− 0.0684598i
$859$	−1936.63	−0.0769233	−0.0384616	−	0.999260i	$-0.512246\pi$
$859$	−1936.63	−0.0769233	−0.0384616	+	0.999260i	$0.512246\pi$
$860$	0	0
$861$	−3960.97	−0.156782
$862$	− 20556.8i	− 0.812259i
$863$	− 7947.70i	− 0.313491i	−0.987639	−	0.156746i	$-0.949900\pi$
$863$	− 7947.70i	− 0.313491i	0.987639	−	0.156746i	$-0.0501003\pi$
$864$	−5051.63	−0.198912
$865$	0	0
$866$	19791.6	0.776611
$867$	7321.73i	0.286804i
$868$	− 5309.58i	− 0.207625i
$869$	−171.984	−0.00671365
$870$	0	0
$871$	−23649.6	−0.920019
$872$	− 3216.00i	− 0.124894i
$873$	4949.88i	0.191899i
$874$	−4742.06	−0.183527
$875$	0	0
$876$	13450.4	0.518776
$877$	38655.0i	1.48835i	0.667983	+	0.744177i	$0.267158\pi$
$877$	38655.0i	1.48835i	−0.667983	+	0.744177i	$0.732842\pi$
$878$	− 8764.65i	− 0.336893i
$879$	−23946.0	−0.918859
$880$	0	0
$881$	−18879.4	−0.721978	−0.360989	−	0.932570i	$-0.617561\pi$
$881$	−18879.4	−0.721978	−0.360989	+	0.932570i	$0.617561\pi$
$882$	− 634.355i	− 0.0242175i
$883$	35098.1i	1.33765i	0.743420	+	0.668825i	$0.233202\pi$
$883$	35098.1i	1.33765i	−0.743420	+	0.668825i	$0.766798\pi$
$884$	−15439.4	−0.587425
$885$	0	0
$886$	−19346.3	−0.733581
$887$	− 48816.4i	− 1.84791i	−0.382504	−	0.923954i	$-0.624938\pi$
$887$	− 48816.4i	− 1.84791i	0.382504	−	0.923954i	$-0.375062\pi$
$888$	− 6942.08i	− 0.262344i
$889$	−1897.11	−0.0715713
$890$	0	0
$891$	616.858	0.0231936
$892$	679.062i	0.0254895i
$893$	10776.7i	0.403839i
$894$	14053.8	0.525761
$895$	0	0
$896$	−9267.63	−0.345547
$897$	3682.64i	0.137079i
$898$	9916.41i	0.368502i
$899$	−20100.8	−0.745718
$900$	0	0
$901$	21091.3	0.779860
$902$	2066.22i	0.0762721i
$903$	− 6765.90i	− 0.249341i
$904$	35839.1	1.31857
$905$	0	0
$906$	6348.05	0.232781
$907$	− 6010.83i	− 0.220051i	−0.993929	−	0.110026i	$-0.964907\pi$
$907$	− 6010.83i	− 0.220051i	0.993929	−	0.110026i	$-0.0350933\pi$
$908$	28140.5i	1.02850i
$909$	4799.69	0.175133
$910$	0	0
$911$	25780.1	0.937576	0.468788	−	0.883311i	$-0.344691\pi$
$911$	25780.1	0.937576	0.468788	+	0.883311i	$0.344691\pi$
$912$	− 7854.85i	− 0.285198i
$913$	− 8470.37i	− 0.307041i
$914$	−17002.6	−0.615314
$915$	0	0
$916$	31934.6	1.15191
$917$	5342.06i	0.192378i
$918$	1931.16i	0.0694313i
$919$	−26731.8	−0.959522	−0.479761	−	0.877399i	$-0.659277\pi$
$919$	−26731.8	−0.959522	−0.479761	+	0.877399i	$0.659277\pi$
$920$	0	0
$921$	21354.4	0.764010
$922$	− 12145.4i	− 0.433824i
$923$	− 47501.6i	− 1.69397i
$924$	948.501	0.0337699
$925$	0	0
$926$	1826.46	0.0648177
$927$	12220.1i	0.432967i
$928$	29406.2i	1.04020i
$929$	30464.6	1.07590	0.537949	−	0.842977i	$-0.319199\pi$
$929$	30464.6	1.07590	0.537949	+	0.842977i	$0.319199\pi$
$930$	0	0
$931$	6889.42	0.242526
$932$	9542.22i	0.335371i
$933$	− 27467.7i	− 0.963830i
$934$	−23708.1	−0.830572
$935$	0	0
$936$	9442.04	0.329725
$937$	− 28533.4i	− 0.994819i	−0.867516	−	0.497409i	$-0.834285\pi$
$937$	− 28533.4i	− 0.994819i	0.867516	−	0.497409i	$-0.165715\pi$
$938$	4548.46i	0.158329i
$939$	−18490.3	−0.642608
$940$	0	0
$941$	34455.8	1.19365	0.596827	−	0.802370i	$-0.296428\pi$
$941$	34455.8	1.19365	0.596827	+	0.802370i	$0.296428\pi$
$942$	− 6016.52i	− 0.208099i
$943$	− 4422.50i	− 0.152722i
$944$	−2007.17	−0.0692033
$945$	0	0
$946$	−3529.39	−0.121301
$947$	2477.68i	0.0850198i	0.999096	+	0.0425099i	$0.0135354\pi$
$947$	2477.68i	0.0850198i	−0.999096	+	0.0425099i	$0.986465\pi$
$948$	− 401.817i	− 0.0137662i
$949$	−39577.5	−1.35378
$950$	0	0
$951$	−25975.1	−0.885699
$952$	6974.80i	0.237452i
$953$	− 41690.6i	− 1.41709i	−0.705663	−	0.708547i	$-0.749351\pi$
$953$	− 41690.6i	− 1.41709i	0.705663	−	0.708547i	$-0.250649\pi$
$954$	−5491.35	−0.186361
$955$	0	0
$956$	18463.5	0.624635
$957$	− 3590.81i	− 0.121290i
$958$	− 2014.13i	− 0.0679266i
$959$	−1686.54	−0.0567895
$960$	0	0
$961$	−13434.6	−0.450962
$962$	8696.44i	0.291460i
$963$	− 8223.24i	− 0.275172i
$964$	42325.0	1.41410
$965$	0	0
$966$	708.271	0.0235903
$967$	− 20641.3i	− 0.686430i	−0.939257	−	0.343215i	$-0.888484\pi$
$967$	− 20641.3i	− 0.686430i	0.939257	−	0.343215i	$-0.111516\pi$
$968$	25509.5i	0.847010i
$969$	−20973.4	−0.695318
$970$	0	0
$971$	6626.17	0.218995	0.109497	−	0.993987i	$-0.465076\pi$
$971$	6626.17	0.218995	0.109497	+	0.993987i	$0.465076\pi$
$972$	1441.20i	0.0475582i
$973$	− 722.048i	− 0.0237901i
$974$	−20377.0	−0.670349
$975$	0	0
$976$	17048.8	0.559138
$977$	41961.0i	1.37405i	0.726632	+	0.687027i	$0.241085\pi$
$977$	41961.0i	1.37405i	−0.726632	+	0.687027i	$0.758915\pi$
$978$	15855.7i	0.518415i
$979$	−11566.5	−0.377598
$980$	0	0
$981$	−1444.40	−0.0470093
$982$	− 6817.21i	− 0.221533i
$983$	16781.7i	0.544510i	0.962225	+	0.272255i	$0.0877695\pi$
$983$	16781.7i	0.544510i	−0.962225	+	0.272255i	$0.912231\pi$
$984$	−11339.0	−0.367352
$985$	0	0
$986$	11241.6	0.363087
$987$	− 1609.60i	− 0.0519090i
$988$	43657.2i	1.40579i
$989$	7554.27	0.242884
$990$	0	0
$991$	50319.8	1.61298	0.806489	−	0.591250i	$-0.201365\pi$
$991$	50319.8	1.61298	0.806489	+	0.591250i	$0.201365\pi$
$992$	− 23928.3i	− 0.765851i
$993$	− 385.432i	− 0.0123176i
$994$	−9135.85	−0.291521
$995$	0	0
$996$	19789.8	0.629583
$997$	12949.5i	0.411348i	0.978621	+	0.205674i	$0.0659386\pi$
$997$	12949.5i	0.411348i	−0.978621	+	0.205674i	$0.934061\pi$
$998$	− 16355.2i	− 0.518753i
$999$	−3117.89	−0.0987443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.d.i.274.2		4
5.2	odd	4		525.4.a.p.1.1		2
5.3	odd	4		105.4.a.c.1.2	✓	2
5.4	even	2	inner	525.4.d.i.274.3		4
15.2	even	4		1575.4.a.m.1.2		2
15.8	even	4		315.4.a.m.1.1		2
20.3	even	4		1680.4.a.bk.1.1		2
35.13	even	4		735.4.a.k.1.2		2
105.83	odd	4		2205.4.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.c.1.2	✓	2	5.3	odd	4
315.4.a.m.1.1		2	15.8	even	4
525.4.a.p.1.1		2	5.2	odd	4
525.4.d.i.274.2		4	1.1	even	1	trivial
525.4.d.i.274.3		4	5.4	even	2	inner
735.4.a.k.1.2		2	35.13	even	4
1575.4.a.m.1.2		2	15.2	even	4
1680.4.a.bk.1.1		2	20.3	even	4
2205.4.a.bh.1.1		2	105.83	odd	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.43845i q^{2} +3.00000i q^{3} +5.93087 q^{4} +4.31534 q^{6} -7.00000i q^{7} -20.0388i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-1.43845i q^{2} +3.00000i q^{3} +5.93087 q^{4} +4.31534 q^{6} -7.00000i q^{7} -20.0388i q^{8} -9.00000 q^{9} +7.61553 q^{11} +17.7926i q^{12} -52.3542i q^{13} -10.0691 q^{14} +18.6222 q^{16} -49.7235i q^{17} +12.9460i q^{18} -140.600 q^{19} +21.0000 q^{21} -10.9545i q^{22} +23.4470i q^{23} +60.1165 q^{24} -75.3087 q^{26} -27.0000i q^{27} -41.5161i q^{28} -157.170 q^{29} +127.892 q^{31} -187.098i q^{32} +22.8466i q^{33} -71.5246 q^{34} -53.3778 q^{36} -115.477i q^{37} +202.246i q^{38} +157.062 q^{39} -188.617 q^{41} -30.2074i q^{42} -322.186i q^{43} +45.1667 q^{44} +33.7272 q^{46} -76.6477i q^{47} +55.8665i q^{48} -49.0000 q^{49} +149.170 q^{51} -310.506i q^{52} +424.172i q^{53} -38.8381 q^{54} -140.272 q^{56} -421.801i q^{57} +226.081i q^{58} -107.784 q^{59} +915.511 q^{61} -183.966i q^{62} +63.0000i q^{63} -120.153 q^{64} +32.8636 q^{66} -451.723i q^{67} -294.903i q^{68} -70.3409 q^{69} +907.312 q^{71} +180.349i q^{72} -755.956i q^{73} -166.108 q^{74} -833.882 q^{76} -53.3087i q^{77} -225.926i q^{78} -22.5834 q^{79} +81.0000 q^{81} +271.316i q^{82} -1112.25i q^{83} +124.548 q^{84} -463.447 q^{86} -471.511i q^{87} -152.606i q^{88} -1518.81 q^{89} -366.479 q^{91} +139.061i q^{92} +383.676i q^{93} -110.254 q^{94} +561.293 q^{96} -549.987i q^{97} +70.4839i q^{98} -68.5398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 34 q^{4} + 42 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 34 * q^4 + 42 * q^6 - 36 * q^9	\( 4 q - 34 q^{4} + 42 q^{6} - 36 q^{9} - 52 q^{11} - 98 q^{14} + 594 q^{16} - 348 q^{19} + 84 q^{21} - 378 q^{24} + 276 q^{26} + 64 q^{29} + 660 q^{31} + 588 q^{34} + 306 q^{36} + 84 q^{39} + 400 q^{41} + 1632 q^{44} - 2240 q^{46} - 196 q^{49} - 96 q^{51} - 378 q^{54} + 882 q^{56} - 728 q^{59} + 1584 q^{61} - 5618 q^{64} - 1056 q^{66} + 1104 q^{69} + 908 q^{71} - 516 q^{74} - 136 q^{76} - 816 q^{79} + 324 q^{81} - 714 q^{84} - 1392 q^{86} - 72 q^{89} - 196 q^{91} + 3880 q^{94} + 6426 q^{96} + 468 q^{99}+O(q^{100}) \) 4 * q - 34 * q^4 + 42 * q^6 - 36 * q^9 - 52 * q^11 - 98 * q^14 + 594 * q^16 - 348 * q^19 + 84 * q^21 - 378 * q^24 + 276 * q^26 + 64 * q^29 + 660 * q^31 + 588 * q^34 + 306 * q^36 + 84 * q^39 + 400 * q^41 + 1632 * q^44 - 2240 * q^46 - 196 * q^49 - 96 * q^51 - 378 * q^54 + 882 * q^56 - 728 * q^59 + 1584 * q^61 - 5618 * q^64 - 1056 * q^66 + 1104 * q^69 + 908 * q^71 - 516 * q^74 - 136 * q^76 - 816 * q^79 + 324 * q^81 - 714 * q^84 - 1392 * q^86 - 72 * q^89 - 196 * q^91 + 3880 * q^94 + 6426 * q^96 + 468 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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