LMFDB - Embedded newform 75.4.b.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.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:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.4.b.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} +20.0000i q^{7} +21.0000i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} +20.0000i q^{7} +21.0000i q^{8} -9.00000 q^{9} -24.0000 q^{11} -3.00000i q^{12} -74.0000i q^{13} -60.0000 q^{14} -71.0000 q^{16} +54.0000i q^{17} -27.0000i q^{18} +124.000 q^{19} -60.0000 q^{21} -72.0000i q^{22} +120.000i q^{23} -63.0000 q^{24} +222.000 q^{26} -27.0000i q^{27} -20.0000i q^{28} +78.0000 q^{29} +200.000 q^{31} -45.0000i q^{32} -72.0000i q^{33} -162.000 q^{34} +9.00000 q^{36} -70.0000i q^{37} +372.000i q^{38} +222.000 q^{39} +330.000 q^{41} -180.000i q^{42} -92.0000i q^{43} +24.0000 q^{44} -360.000 q^{46} -24.0000i q^{47} -213.000i q^{48} -57.0000 q^{49} -162.000 q^{51} +74.0000i q^{52} -450.000i q^{53} +81.0000 q^{54} -420.000 q^{56} +372.000i q^{57} +234.000i q^{58} -24.0000 q^{59} -322.000 q^{61} +600.000i q^{62} -180.000i q^{63} -433.000 q^{64} +216.000 q^{66} -196.000i q^{67} -54.0000i q^{68} -360.000 q^{69} -288.000 q^{71} -189.000i q^{72} +430.000i q^{73} +210.000 q^{74} -124.000 q^{76} -480.000i q^{77} +666.000i q^{78} +520.000 q^{79} +81.0000 q^{81} +990.000i q^{82} -156.000i q^{83} +60.0000 q^{84} +276.000 q^{86} +234.000i q^{87} -504.000i q^{88} -1026.00 q^{89} +1480.00 q^{91} -120.000i q^{92} +600.000i q^{93} +72.0000 q^{94} +135.000 q^{96} -286.000i q^{97} -171.000i q^{98} +216.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 18 q^{6} - 18 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 18 * q^6 - 18 * q^9	$ 2 q - 2 q^{4} - 18 q^{6} - 18 q^{9} - 48 q^{11} - 120 q^{14} - 142 q^{16} + 248 q^{19} - 120 q^{21} - 126 q^{24} + 444 q^{26} + 156 q^{29} + 400 q^{31} - 324 q^{34} + 18 q^{36} + 444 q^{39} + 660 q^{41} + 48 q^{44} - 720 q^{46} - 114 q^{49} - 324 q^{51} + 162 q^{54} - 840 q^{56} - 48 q^{59} - 644 q^{61} - 866 q^{64} + 432 q^{66} - 720 q^{69} - 576 q^{71} + 420 q^{74} - 248 q^{76} + 1040 q^{79} + 162 q^{81} + 120 q^{84} + 552 q^{86} - 2052 q^{89} + 2960 q^{91} + 144 q^{94} + 270 q^{96} + 432 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 18 * q^6 - 18 * q^9 - 48 * q^11 - 120 * q^14 - 142 * q^16 + 248 * q^19 - 120 * q^21 - 126 * q^24 + 444 * q^26 + 156 * q^29 + 400 * q^31 - 324 * q^34 + 18 * q^36 + 444 * q^39 + 660 * q^41 + 48 * q^44 - 720 * q^46 - 114 * q^49 - 324 * q^51 + 162 * q^54 - 840 * q^56 - 48 * q^59 - 644 * q^61 - 866 * q^64 + 432 * q^66 - 720 * q^69 - 576 * q^71 + 420 * q^74 - 248 * q^76 + 1040 * q^79 + 162 * q^81 + 120 * q^84 + 552 * q^86 - 2052 * q^89 + 2960 * q^91 + 144 * q^94 + 270 * q^96 + 432 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	3.00000i	1.06066i	0.847791	+	0.530330i	$0.177932\pi$
$2$	3.00000i	1.06066i	−0.847791	+	0.530330i	$0.822068\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−1.00000	−0.125000
$5$	0	0
$5$	0	0
$6$	−9.00000	−0.612372
$7$	20.0000i	1.07990i	0.841698	+	0.539949i	$0.181557\pi$
$7$	20.0000i	1.07990i	−0.841698	+	0.539949i	$0.818443\pi$
$8$	21.0000i	0.928078i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−24.0000	−0.657843	−0.328921	−	0.944357i	$-0.606685\pi$
$11$	−24.0000	−0.657843	−0.328921	+	0.944357i	$0.606685\pi$
$12$	− 3.00000i	− 0.0721688i
$13$	− 74.0000i	− 1.57876i	−0.613904	−	0.789381i	$-0.710402\pi$
$13$	− 74.0000i	− 1.57876i	0.613904	−	0.789381i	$-0.289598\pi$
$14$	−60.0000	−1.14541
$15$	0	0
$16$	−71.0000	−1.10938
$17$	54.0000i	0.770407i	0.922832	+	0.385204i	$0.125869\pi$
$17$	54.0000i	0.770407i	−0.922832	+	0.385204i	$0.874131\pi$
$18$	− 27.0000i	− 0.353553i
$19$	124.000	1.49724	0.748620	−	0.663000i	$-0.230717\pi$
$19$	124.000	1.49724	0.748620	+	0.663000i	$0.230717\pi$
$20$	0	0
$21$	−60.0000	−0.623480
$22$	− 72.0000i	− 0.697748i
$23$	120.000i	1.08790i	0.839117	+	0.543951i	$0.183072\pi$
$23$	120.000i	1.08790i	−0.839117	+	0.543951i	$0.816928\pi$
$24$	−63.0000	−0.535826
$25$	0	0
$26$	222.000	1.67453
$27$	− 27.0000i	− 0.192450i
$28$	− 20.0000i	− 0.134987i
$29$	78.0000	0.499456	0.249728	−	0.968316i	$-0.419659\pi$
$29$	78.0000	0.499456	0.249728	+	0.968316i	$0.419659\pi$
$30$	0	0
$31$	200.000	1.15874	0.579372	−	0.815063i	$-0.303298\pi$
$31$	200.000	1.15874	0.579372	+	0.815063i	$0.303298\pi$
$32$	− 45.0000i	− 0.248592i
$33$	− 72.0000i	− 0.379806i
$34$	−162.000	−0.817140
$35$	0	0
$36$	9.00000	0.0416667
$37$	− 70.0000i	− 0.311025i	−0.987834	−	0.155513i	$-0.950297\pi$
$37$	− 70.0000i	− 0.311025i	0.987834	−	0.155513i	$-0.0497029\pi$
$38$	372.000i	1.58806i
$39$	222.000	0.911499
$40$	0	0
$41$	330.000	1.25701	0.628504	−	0.777806i	$-0.283668\pi$
$41$	330.000	1.25701	0.628504	+	0.777806i	$0.283668\pi$
$42$	− 180.000i	− 0.661300i
$43$	− 92.0000i	− 0.326276i	−0.986603	−	0.163138i	$-0.947838\pi$
$43$	− 92.0000i	− 0.326276i	0.986603	−	0.163138i	$-0.0521616\pi$
$44$	24.0000	0.0822304
$45$	0	0
$46$	−360.000	−1.15389
$47$	− 24.0000i	− 0.0744843i	−0.999306	−	0.0372421i	$-0.988143\pi$
$47$	− 24.0000i	− 0.0744843i	0.999306	−	0.0372421i	$-0.0118573\pi$
$48$	− 213.000i	− 0.640498i
$49$	−57.0000	−0.166181
$50$	0	0
$51$	−162.000	−0.444795
$52$	74.0000i	0.197345i
$53$	− 450.000i	− 1.16627i	−0.812376	−	0.583134i	$-0.801826\pi$
$53$	− 450.000i	− 1.16627i	0.812376	−	0.583134i	$-0.198174\pi$
$54$	81.0000	0.204124
$55$	0	0
$56$	−420.000	−1.00223
$57$	372.000i	0.864432i
$58$	234.000i	0.529754i
$59$	−24.0000	−0.0529582	−0.0264791	−	0.999649i	$-0.508430\pi$
$59$	−24.0000	−0.0529582	−0.0264791	+	0.999649i	$0.508430\pi$
$60$	0	0
$61$	−322.000	−0.675867	−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000	−0.675867	−0.337933	+	0.941170i	$0.609728\pi$
$62$	600.000i	1.22903i
$63$	− 180.000i	− 0.359966i
$64$	−433.000	−0.845703
$65$	0	0
$66$	216.000	0.402845
$67$	− 196.000i	− 0.357391i	−0.983904	−	0.178696i	$-0.942812\pi$
$67$	− 196.000i	− 0.357391i	0.983904	−	0.178696i	$-0.0571877\pi$
$68$	− 54.0000i	− 0.0963009i
$69$	−360.000	−0.628100
$70$	0	0
$71$	−288.000	−0.481399	−0.240699	−	0.970600i	$-0.577377\pi$
$71$	−288.000	−0.481399	−0.240699	+	0.970600i	$0.577377\pi$
$72$	− 189.000i	− 0.309359i
$73$	430.000i	0.689420i	0.938709	+	0.344710i	$0.112023\pi$
$73$	430.000i	0.689420i	−0.938709	+	0.344710i	$0.887977\pi$
$74$	210.000	0.329892
$75$	0	0
$76$	−124.000	−0.187155
$77$	− 480.000i	− 0.710404i
$78$	666.000i	0.966790i
$79$	520.000	0.740564	0.370282	−	0.928919i	$-0.379261\pi$
$79$	520.000	0.740564	0.370282	+	0.928919i	$0.379261\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	990.000i	1.33326i
$83$	− 156.000i	− 0.206304i	−0.994666	−	0.103152i	$-0.967107\pi$
$83$	− 156.000i	− 0.206304i	0.994666	−	0.103152i	$-0.0328928\pi$
$84$	60.0000	0.0779350
$85$	0	0
$86$	276.000	0.346068
$87$	234.000i	0.288361i
$88$	− 504.000i	− 0.610529i
$89$	−1026.00	−1.22198	−0.610988	−	0.791640i	$-0.709227\pi$
$89$	−1026.00	−1.22198	−0.610988	+	0.791640i	$0.709227\pi$
$90$	0	0
$91$	1480.00	1.70490
$92$	− 120.000i	− 0.135988i
$93$	600.000i	0.669001i
$94$	72.0000	0.0790025
$95$	0	0
$96$	135.000	0.143525
$97$	− 286.000i	− 0.299370i	−0.988734	−	0.149685i	$-0.952174\pi$
$97$	− 286.000i	− 0.299370i	0.988734	−	0.149685i	$-0.0478260\pi$
$98$	− 171.000i	− 0.176261i
$99$	216.000	0.219281
$100$	0	0
$101$	−1734.00	−1.70831	−0.854156	−	0.520017i	$-0.825925\pi$
$101$	−1734.00	−1.70831	−0.854156	+	0.520017i	$0.825925\pi$
$102$	− 486.000i	− 0.471776i
$103$	− 452.000i	− 0.432397i	−0.976349	−	0.216198i	$-0.930634\pi$
$103$	− 452.000i	− 0.432397i	0.976349	−	0.216198i	$-0.0693658\pi$
$104$	1554.00	1.46521
$105$	0	0
$106$	1350.00	1.23702
$107$	− 1404.00i	− 1.26850i	−0.773127	−	0.634251i	$-0.781308\pi$
$107$	− 1404.00i	− 1.26850i	0.773127	−	0.634251i	$-0.218692\pi$
$108$	27.0000i	0.0240563i
$109$	1474.00	1.29526	0.647631	−	0.761954i	$-0.275760\pi$
$109$	1474.00	1.29526	0.647631	+	0.761954i	$0.275760\pi$
$110$	0	0
$111$	210.000	0.179570
$112$	− 1420.00i	− 1.19801i
$113$	− 1086.00i	− 0.904091i	−0.891995	−	0.452046i	$-0.850694\pi$
$113$	− 1086.00i	− 0.904091i	0.891995	−	0.452046i	$-0.149306\pi$
$114$	−1116.00	−0.916868
$115$	0	0
$116$	−78.0000	−0.0624321
$117$	666.000i	0.526254i
$118$	− 72.0000i	− 0.0561707i
$119$	−1080.00	−0.831962
$120$	0	0
$121$	−755.000	−0.567243
$122$	− 966.000i	− 0.716865i
$123$	990.000i	0.725734i
$124$	−200.000	−0.144843
$125$	0	0
$126$	540.000	0.381802
$127$	1244.00i	0.869190i	0.900626	+	0.434595i	$0.143109\pi$
$127$	1244.00i	0.869190i	−0.900626	+	0.434595i	$0.856891\pi$
$128$	− 1659.00i	− 1.14560i
$129$	276.000	0.188376
$130$	0	0
$131$	2328.00	1.55266	0.776329	−	0.630327i	$-0.217079\pi$
$131$	2328.00	1.55266	0.776329	+	0.630327i	$0.217079\pi$
$132$	72.0000i	0.0474757i
$133$	2480.00i	1.61687i
$134$	588.000	0.379071
$135$	0	0
$136$	−1134.00	−0.714998
$137$	2118.00i	1.32082i	0.750903	+	0.660412i	$0.229618\pi$
$137$	2118.00i	1.32082i	−0.750903	+	0.660412i	$0.770382\pi$
$138$	− 1080.00i	− 0.666201i
$139$	−2324.00	−1.41812	−0.709062	−	0.705147i	$-0.750881\pi$
$139$	−2324.00	−1.41812	−0.709062	+	0.705147i	$0.750881\pi$
$140$	0	0
$141$	72.0000	0.0430035
$142$	− 864.000i	− 0.510600i
$143$	1776.00i	1.03858i
$144$	639.000	0.369792
$145$	0	0
$146$	−1290.00	−0.731241
$147$	− 171.000i	− 0.0959445i
$148$	70.0000i	0.0388781i
$149$	−258.000	−0.141854	−0.0709268	−	0.997482i	$-0.522596\pi$
$149$	−258.000	−0.141854	−0.0709268	+	0.997482i	$0.522596\pi$
$150$	0	0
$151$	−808.000	−0.435458	−0.217729	−	0.976009i	$-0.569865\pi$
$151$	−808.000	−0.435458	−0.217729	+	0.976009i	$0.569865\pi$
$152$	2604.00i	1.38955i
$153$	− 486.000i	− 0.256802i
$154$	1440.00	0.753497
$155$	0	0
$156$	−222.000	−0.113937
$157$	2378.00i	1.20882i	0.796673	+	0.604411i	$0.206592\pi$
$157$	2378.00i	1.20882i	−0.796673	+	0.604411i	$0.793408\pi$
$158$	1560.00i	0.785487i
$159$	1350.00	0.673346
$160$	0	0
$161$	−2400.00	−1.17482
$162$	243.000i	0.117851i
$163$	52.0000i	0.0249874i	0.999922	+	0.0124937i	$0.00397698\pi$
$163$	52.0000i	0.0249874i	−0.999922	+	0.0124937i	$0.996023\pi$
$164$	−330.000	−0.157126
$165$	0	0
$166$	468.000	0.218818
$167$	− 3720.00i	− 1.72373i	−0.507141	−	0.861863i	$-0.669298\pi$
$167$	− 3720.00i	− 1.72373i	0.507141	−	0.861863i	$-0.330702\pi$
$168$	− 1260.00i	− 0.578638i
$169$	−3279.00	−1.49249
$170$	0	0
$171$	−1116.00	−0.499080
$172$	92.0000i	0.0407845i
$173$	− 426.000i	− 0.187215i	−0.995609	−	0.0936075i	$-0.970160\pi$
$173$	− 426.000i	− 0.187215i	0.995609	−	0.0936075i	$-0.0298399\pi$
$174$	−702.000	−0.305853
$175$	0	0
$176$	1704.00	0.729795
$177$	− 72.0000i	− 0.0305754i
$178$	− 3078.00i	− 1.29610i
$179$	1440.00	0.601289	0.300644	−	0.953736i	$-0.402798\pi$
$179$	1440.00	0.601289	0.300644	+	0.953736i	$0.402798\pi$
$180$	0	0
$181$	−3130.00	−1.28537	−0.642683	−	0.766133i	$-0.722179\pi$
$181$	−3130.00	−1.28537	−0.642683	+	0.766133i	$0.722179\pi$
$182$	4440.00i	1.80832i
$183$	− 966.000i	− 0.390212i
$184$	−2520.00	−1.00966
$185$	0	0
$186$	−1800.00	−0.709583
$187$	− 1296.00i	− 0.506807i
$188$	24.0000i	0.00931053i
$189$	540.000	0.207827
$190$	0	0
$191$	3576.00	1.35471	0.677357	−	0.735655i	$-0.263125\pi$
$191$	3576.00	1.35471	0.677357	+	0.735655i	$0.263125\pi$
$192$	− 1299.00i	− 0.488267i
$193$	− 2666.00i	− 0.994315i	−0.867660	−	0.497158i	$-0.834377\pi$
$193$	− 2666.00i	− 0.994315i	0.867660	−	0.497158i	$-0.165623\pi$
$194$	858.000	0.317530
$195$	0	0
$196$	57.0000	0.0207726
$197$	− 2718.00i	− 0.982992i	−0.870880	−	0.491496i	$-0.836450\pi$
$197$	− 2718.00i	− 0.982992i	0.870880	−	0.491496i	$-0.163550\pi$
$198$	648.000i	0.232583i
$199$	3832.00	1.36504	0.682521	−	0.730866i	$-0.260884\pi$
$199$	3832.00	1.36504	0.682521	+	0.730866i	$0.260884\pi$
$200$	0	0
$201$	588.000	0.206340
$202$	− 5202.00i	− 1.81194i
$203$	1560.00i	0.539362i
$204$	162.000	0.0555994
$205$	0	0
$206$	1356.00	0.458626
$207$	− 1080.00i	− 0.362634i
$208$	5254.00i	1.75144i
$209$	−2976.00	−0.984948
$210$	0	0
$211$	1100.00	0.358896	0.179448	−	0.983767i	$-0.442569\pi$
$211$	1100.00	0.358896	0.179448	+	0.983767i	$0.442569\pi$
$212$	450.000i	0.145784i
$213$	− 864.000i	− 0.277936i
$214$	4212.00	1.34545
$215$	0	0
$216$	567.000	0.178609
$217$	4000.00i	1.25133i
$218$	4422.00i	1.37383i
$219$	−1290.00	−0.398037
$220$	0	0
$221$	3996.00	1.21629
$222$	630.000i	0.190463i
$223$	− 1964.00i	− 0.589772i	−0.955532	−	0.294886i	$-0.904718\pi$
$223$	− 1964.00i	− 0.589772i	0.955532	−	0.294886i	$-0.0952817\pi$
$224$	900.000	0.268454
$225$	0	0
$226$	3258.00	0.958933
$227$	660.000i	0.192977i	0.995334	+	0.0964884i	$0.0307611\pi$
$227$	660.000i	0.192977i	−0.995334	+	0.0964884i	$0.969239\pi$
$228$	− 372.000i	− 0.108054i
$229$	1906.00	0.550009	0.275004	−	0.961443i	$-0.411321\pi$
$229$	1906.00	0.550009	0.275004	+	0.961443i	$0.411321\pi$
$230$	0	0
$231$	1440.00	0.410152
$232$	1638.00i	0.463534i
$233$	1458.00i	0.409943i	0.978768	+	0.204972i	$0.0657102\pi$
$233$	1458.00i	0.409943i	−0.978768	+	0.204972i	$0.934290\pi$
$234$	−1998.00	−0.558177
$235$	0	0
$236$	24.0000	0.00661978
$237$	1560.00i	0.427565i
$238$	− 3240.00i	− 0.882429i
$239$	−1176.00	−0.318281	−0.159140	−	0.987256i	$-0.550872\pi$
$239$	−1176.00	−0.318281	−0.159140	+	0.987256i	$0.550872\pi$
$240$	0	0
$241$	866.000	0.231469	0.115734	−	0.993280i	$-0.463078\pi$
$241$	866.000	0.231469	0.115734	+	0.993280i	$0.463078\pi$
$242$	− 2265.00i	− 0.601652i
$243$	243.000i	0.0641500i
$244$	322.000	0.0844834
$245$	0	0
$246$	−2970.00	−0.769757
$247$	− 9176.00i	− 2.36379i
$248$	4200.00i	1.07540i
$249$	468.000	0.119110
$250$	0	0
$251$	432.000	0.108636	0.0543179	−	0.998524i	$-0.482702\pi$
$251$	432.000	0.108636	0.0543179	+	0.998524i	$0.482702\pi$
$252$	180.000i	0.0449958i
$253$	− 2880.00i	− 0.715668i
$254$	−3732.00	−0.921915
$255$	0	0
$256$	1513.00	0.369385
$257$	2526.00i	0.613103i	0.951854	+	0.306552i	$0.0991752\pi$
$257$	2526.00i	0.613103i	−0.951854	+	0.306552i	$0.900825\pi$
$258$	828.000i	0.199802i
$259$	1400.00	0.335876
$260$	0	0
$261$	−702.000	−0.166485
$262$	6984.00i	1.64684i
$263$	− 5448.00i	− 1.27733i	−0.769484	−	0.638666i	$-0.779487\pi$
$263$	− 5448.00i	− 1.27733i	0.769484	−	0.638666i	$-0.220513\pi$
$264$	1512.00	0.352489
$265$	0	0
$266$	−7440.00	−1.71495
$267$	− 3078.00i	− 0.705508i
$268$	196.000i	0.0446739i
$269$	2574.00	0.583418	0.291709	−	0.956507i	$-0.405776\pi$
$269$	2574.00	0.583418	0.291709	+	0.956507i	$0.405776\pi$
$270$	0	0
$271$	−3184.00	−0.713706	−0.356853	−	0.934161i	$-0.616150\pi$
$271$	−3184.00	−0.713706	−0.356853	+	0.934161i	$0.616150\pi$
$272$	− 3834.00i	− 0.854671i
$273$	4440.00i	0.984326i
$274$	−6354.00	−1.40095
$275$	0	0
$276$	360.000	0.0785125
$277$	3962.00i	0.859399i	0.902972	+	0.429699i	$0.141380\pi$
$277$	3962.00i	0.859399i	−0.902972	+	0.429699i	$0.858620\pi$
$278$	− 6972.00i	− 1.50415i
$279$	−1800.00	−0.386248
$280$	0	0
$281$	−8286.00	−1.75908	−0.879540	−	0.475825i	$-0.842149\pi$
$281$	−8286.00	−1.75908	−0.879540	+	0.475825i	$0.842149\pi$
$282$	216.000i	0.0456121i
$283$	2716.00i	0.570493i	0.958454	+	0.285246i	$0.0920754\pi$
$283$	2716.00i	0.570493i	−0.958454	+	0.285246i	$0.907925\pi$
$284$	288.000	0.0601748
$285$	0	0
$286$	−5328.00	−1.10158
$287$	6600.00i	1.35744i
$288$	405.000i	0.0828641i
$289$	1997.00	0.406473
$290$	0	0
$291$	858.000	0.172841
$292$	− 430.000i	− 0.0861776i
$293$	− 6018.00i	− 1.19992i	−0.800032	−	0.599958i	$-0.795184\pi$
$293$	− 6018.00i	− 1.19992i	0.800032	−	0.599958i	$-0.204816\pi$
$294$	513.000	0.101765
$295$	0	0
$296$	1470.00	0.288655
$297$	648.000i	0.126602i
$298$	− 774.000i	− 0.150458i
$299$	8880.00	1.71754
$300$	0	0
$301$	1840.00	0.352345
$302$	− 2424.00i	− 0.461873i
$303$	− 5202.00i	− 0.986294i
$304$	−8804.00	−1.66100
$305$	0	0
$306$	1458.00	0.272380
$307$	9236.00i	1.71702i	0.512793	+	0.858512i	$0.328611\pi$
$307$	9236.00i	1.71702i	−0.512793	+	0.858512i	$0.671389\pi$
$308$	480.000i	0.0888004i
$309$	1356.00	0.249644
$310$	0	0
$311$	1536.00	0.280060	0.140030	−	0.990147i	$-0.455280\pi$
$311$	1536.00	0.280060	0.140030	+	0.990147i	$0.455280\pi$
$312$	4662.00i	0.845942i
$313$	7342.00i	1.32586i	0.748681	+	0.662930i	$0.230687\pi$
$313$	7342.00i	1.32586i	−0.748681	+	0.662930i	$0.769313\pi$
$314$	−7134.00	−1.28215
$315$	0	0
$316$	−520.000	−0.0925705
$317$	− 3894.00i	− 0.689933i	−0.938615	−	0.344967i	$-0.887890\pi$
$317$	− 3894.00i	− 0.689933i	0.938615	−	0.344967i	$-0.112110\pi$
$318$	4050.00i	0.714191i
$319$	−1872.00	−0.328564
$320$	0	0
$321$	4212.00	0.732370
$322$	− 7200.00i	− 1.24609i
$323$	6696.00i	1.15348i
$324$	−81.0000	−0.0138889
$325$	0	0
$326$	−156.000	−0.0265032
$327$	4422.00i	0.747820i
$328$	6930.00i	1.16660i
$329$	480.000	0.0804354
$330$	0	0
$331$	3692.00	0.613084	0.306542	−	0.951857i	$-0.400828\pi$
$331$	3692.00	0.613084	0.306542	+	0.951857i	$0.400828\pi$
$332$	156.000i	0.0257880i
$333$	630.000i	0.103675i
$334$	11160.0	1.82829
$335$	0	0
$336$	4260.00	0.691673
$337$	− 8998.00i	− 1.45446i	−0.686395	−	0.727229i	$-0.740808\pi$
$337$	− 8998.00i	− 1.45446i	0.686395	−	0.727229i	$-0.259192\pi$
$338$	− 9837.00i	− 1.58302i
$339$	3258.00	0.521977
$340$	0	0
$341$	−4800.00	−0.762271
$342$	− 3348.00i	− 0.529354i
$343$	5720.00i	0.900440i
$344$	1932.00	0.302809
$345$	0	0
$346$	1278.00	0.198571
$347$	5244.00i	0.811276i	0.914034	+	0.405638i	$0.132951\pi$
$347$	5244.00i	0.811276i	−0.914034	+	0.405638i	$0.867049\pi$
$348$	− 234.000i	− 0.0360452i
$349$	−6302.00	−0.966585	−0.483293	−	0.875459i	$-0.660559\pi$
$349$	−6302.00	−0.966585	−0.483293	+	0.875459i	$0.660559\pi$
$350$	0	0
$351$	−1998.00	−0.303833
$352$	1080.00i	0.163535i
$353$	− 3414.00i	− 0.514756i	−0.966311	−	0.257378i	$-0.917141\pi$
$353$	− 3414.00i	− 0.514756i	0.966311	−	0.257378i	$-0.0828586\pi$
$354$	216.000	0.0324301
$355$	0	0
$356$	1026.00	0.152747
$357$	− 3240.00i	− 0.480333i
$358$	4320.00i	0.637763i
$359$	−4824.00	−0.709195	−0.354597	−	0.935019i	$-0.615382\pi$
$359$	−4824.00	−0.709195	−0.354597	+	0.935019i	$0.615382\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	− 9390.00i	− 1.36334i
$363$	− 2265.00i	− 0.327498i
$364$	−1480.00	−0.213113
$365$	0	0
$366$	2898.00	0.413882
$367$	− 3508.00i	− 0.498954i	−0.968381	−	0.249477i	$-0.919741\pi$
$367$	− 3508.00i	− 0.498954i	0.968381	−	0.249477i	$-0.0802587\pi$
$368$	− 8520.00i	− 1.20689i
$369$	−2970.00	−0.419003
$370$	0	0
$371$	9000.00	1.25945
$372$	− 600.000i	− 0.0836251i
$373$	− 10802.0i	− 1.49948i	−0.661732	−	0.749740i	$-0.730178\pi$
$373$	− 10802.0i	− 1.49948i	0.661732	−	0.749740i	$-0.269822\pi$
$374$	3888.00	0.537550
$375$	0	0
$376$	504.000	0.0691272
$377$	− 5772.00i	− 0.788523i
$378$	1620.00i	0.220433i
$379$	−1460.00	−0.197876	−0.0989382	−	0.995094i	$-0.531545\pi$
$379$	−1460.00	−0.197876	−0.0989382	+	0.995094i	$0.531545\pi$
$380$	0	0
$381$	−3732.00	−0.501827
$382$	10728.0i	1.43689i
$383$	4872.00i	0.649994i	0.945715	+	0.324997i	$0.105363\pi$
$383$	4872.00i	0.649994i	−0.945715	+	0.324997i	$0.894637\pi$
$384$	4977.00	0.661410
$385$	0	0
$386$	7998.00	1.05463
$387$	828.000i	0.108759i
$388$	286.000i	0.0374213i
$389$	14046.0	1.83075	0.915373	−	0.402606i	$-0.131896\pi$
$389$	14046.0	1.83075	0.915373	+	0.402606i	$0.131896\pi$
$390$	0	0
$391$	−6480.00	−0.838127
$392$	− 1197.00i	− 0.154229i
$393$	6984.00i	0.896428i
$394$	8154.00	1.04262
$395$	0	0
$396$	−216.000	−0.0274101
$397$	− 2734.00i	− 0.345631i	−0.984954	−	0.172816i	$-0.944714\pi$
$397$	− 2734.00i	− 0.345631i	0.984954	−	0.172816i	$-0.0552864\pi$
$398$	11496.0i	1.44785i
$399$	−7440.00	−0.933498
$400$	0	0
$401$	−15942.0	−1.98530	−0.992650	−	0.121019i	$-0.961384\pi$
$401$	−15942.0	−1.98530	−0.992650	+	0.121019i	$0.961384\pi$
$402$	1764.00i	0.218857i
$403$	− 14800.0i	− 1.82938i
$404$	1734.00	0.213539
$405$	0	0
$406$	−4680.00	−0.572080
$407$	1680.00i	0.204606i
$408$	− 3402.00i	− 0.412804i
$409$	−8714.00	−1.05350	−0.526748	−	0.850022i	$-0.676589\pi$
$409$	−8714.00	−1.05350	−0.526748	+	0.850022i	$0.676589\pi$
$410$	0	0
$411$	−6354.00	−0.762578
$412$	452.000i	0.0540496i
$413$	− 480.000i	− 0.0571895i
$414$	3240.00	0.384631
$415$	0	0
$416$	−3330.00	−0.392468
$417$	− 6972.00i	− 0.818754i
$418$	− 8928.00i	− 1.04470i
$419$	−11976.0	−1.39634	−0.698169	−	0.715933i	$-0.746002\pi$
$419$	−11976.0	−1.39634	−0.698169	+	0.715933i	$0.746002\pi$
$420$	0	0
$421$	11054.0	1.27967	0.639833	−	0.768514i	$-0.279004\pi$
$421$	11054.0	1.27967	0.639833	+	0.768514i	$0.279004\pi$
$422$	3300.00i	0.380667i
$423$	216.000i	0.0248281i
$424$	9450.00	1.08239
$425$	0	0
$426$	2592.00	0.294795
$427$	− 6440.00i	− 0.729868i
$428$	1404.00i	0.158563i
$429$	−5328.00	−0.599623
$430$	0	0
$431$	720.000	0.0804668	0.0402334	−	0.999190i	$-0.487190\pi$
$431$	720.000	0.0804668	0.0402334	+	0.999190i	$0.487190\pi$
$432$	1917.00i	0.213499i
$433$	15622.0i	1.73382i	0.498462	+	0.866912i	$0.333898\pi$
$433$	15622.0i	1.73382i	−0.498462	+	0.866912i	$0.666102\pi$
$434$	−12000.0	−1.32723
$435$	0	0
$436$	−1474.00	−0.161908
$437$	14880.0i	1.62885i
$438$	− 3870.00i	− 0.422182i
$439$	9880.00	1.07414	0.537069	−	0.843538i	$-0.319531\pi$
$439$	9880.00	1.07414	0.537069	+	0.843538i	$0.319531\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	11988.0i	1.29007i
$443$	16116.0i	1.72843i	0.503123	+	0.864215i	$0.332184\pi$
$443$	16116.0i	1.72843i	−0.503123	+	0.864215i	$0.667816\pi$
$444$	−210.000	−0.0224463
$445$	0	0
$446$	5892.00	0.625548
$447$	− 774.000i	− 0.0818992i
$448$	− 8660.00i	− 0.913274i
$449$	−9018.00	−0.947852	−0.473926	−	0.880565i	$-0.657164\pi$
$449$	−9018.00	−0.947852	−0.473926	+	0.880565i	$0.657164\pi$
$450$	0	0
$451$	−7920.00	−0.826914
$452$	1086.00i	0.113011i
$453$	− 2424.00i	− 0.251412i
$454$	−1980.00	−0.204683
$455$	0	0
$456$	−7812.00	−0.802260
$457$	− 3670.00i	− 0.375657i	−0.982202	−	0.187829i	$-0.939855\pi$
$457$	− 3670.00i	− 0.375657i	0.982202	−	0.187829i	$-0.0601450\pi$
$458$	5718.00i	0.583372i
$459$	1458.00	0.148265
$460$	0	0
$461$	17562.0	1.77428	0.887141	−	0.461499i	$-0.152688\pi$
$461$	17562.0	1.77428	0.887141	+	0.461499i	$0.152688\pi$
$462$	4320.00i	0.435032i
$463$	− 1172.00i	− 0.117640i	−0.998269	−	0.0588202i	$-0.981266\pi$
$463$	− 1172.00i	− 0.117640i	0.998269	−	0.0588202i	$-0.0187338\pi$
$464$	−5538.00	−0.554084
$465$	0	0
$466$	−4374.00	−0.434810
$467$	6876.00i	0.681335i	0.940184	+	0.340667i	$0.110653\pi$
$467$	6876.00i	0.681335i	−0.940184	+	0.340667i	$0.889347\pi$
$468$	− 666.000i	− 0.0657818i
$469$	3920.00	0.385946
$470$	0	0
$471$	−7134.00	−0.697914
$472$	− 504.000i	− 0.0491493i
$473$	2208.00i	0.214638i
$474$	−4680.00	−0.453501
$475$	0	0
$476$	1080.00	0.103995
$477$	4050.00i	0.388756i
$478$	− 3528.00i	− 0.337588i
$479$	−2280.00	−0.217486	−0.108743	−	0.994070i	$-0.534683\pi$
$479$	−2280.00	−0.217486	−0.108743	+	0.994070i	$0.534683\pi$
$480$	0	0
$481$	−5180.00	−0.491035
$482$	2598.00i	0.245510i
$483$	− 7200.00i	− 0.678284i
$484$	755.000	0.0709053
$485$	0	0
$486$	−729.000	−0.0680414
$487$	− 3076.00i	− 0.286215i	−0.989707	−	0.143108i	$-0.954290\pi$
$487$	− 3076.00i	− 0.286215i	0.989707	−	0.143108i	$-0.0457095\pi$
$488$	− 6762.00i	− 0.627257i
$489$	−156.000	−0.0144265
$490$	0	0
$491$	−18912.0	−1.73826	−0.869131	−	0.494582i	$-0.835321\pi$
$491$	−18912.0	−1.73826	−0.869131	+	0.494582i	$0.835321\pi$
$492$	− 990.000i	− 0.0907168i
$493$	4212.00i	0.384785i
$494$	27528.0	2.50717
$495$	0	0
$496$	−14200.0	−1.28548
$497$	− 5760.00i	− 0.519862i
$498$	1404.00i	0.126335i
$499$	−9956.00	−0.893170	−0.446585	−	0.894741i	$-0.647360\pi$
$499$	−9956.00	−0.893170	−0.446585	+	0.894741i	$0.647360\pi$
$500$	0	0
$501$	11160.0	0.995194
$502$	1296.00i	0.115226i
$503$	10656.0i	0.944588i	0.881441	+	0.472294i	$0.156574\pi$
$503$	10656.0i	0.944588i	−0.881441	+	0.472294i	$0.843426\pi$
$504$	3780.00	0.334077
$505$	0	0
$506$	8640.00	0.759081
$507$	− 9837.00i	− 0.861689i
$508$	− 1244.00i	− 0.108649i
$509$	2766.00	0.240866	0.120433	−	0.992721i	$-0.461572\pi$
$509$	2766.00	0.240866	0.120433	+	0.992721i	$0.461572\pi$
$510$	0	0
$511$	−8600.00	−0.744504
$512$	− 8733.00i	− 0.753804i
$513$	− 3348.00i	− 0.288144i
$514$	−7578.00	−0.650294
$515$	0	0
$516$	−276.000	−0.0235469
$517$	576.000i	0.0489989i
$518$	4200.00i	0.356250i
$519$	1278.00	0.108089
$520$	0	0
$521$	10530.0	0.885466	0.442733	−	0.896654i	$-0.354009\pi$
$521$	10530.0	0.885466	0.442733	+	0.896654i	$0.354009\pi$
$522$	− 2106.00i	− 0.176585i
$523$	− 12692.0i	− 1.06115i	−0.847637	−	0.530576i	$-0.821976\pi$
$523$	− 12692.0i	− 1.06115i	0.847637	−	0.530576i	$-0.178024\pi$
$524$	−2328.00	−0.194082
$525$	0	0
$526$	16344.0	1.35481
$527$	10800.0i	0.892705i
$528$	5112.00i	0.421347i
$529$	−2233.00	−0.183529
$530$	0	0
$531$	216.000	0.0176527
$532$	− 2480.00i	− 0.202108i
$533$	− 24420.0i	− 1.98452i
$534$	9234.00	0.748304
$535$	0	0
$536$	4116.00	0.331687
$537$	4320.00i	0.347154i
$538$	7722.00i	0.618809i
$539$	1368.00	0.109321
$540$	0	0
$541$	18110.0	1.43920	0.719602	−	0.694386i	$-0.244324\pi$
$541$	18110.0	1.43920	0.719602	+	0.694386i	$0.244324\pi$
$542$	− 9552.00i	− 0.756999i
$543$	− 9390.00i	− 0.742106i
$544$	2430.00	0.191517
$545$	0	0
$546$	−13320.0	−1.04404
$547$	3620.00i	0.282962i	0.989941	+	0.141481i	$0.0451864\pi$
$547$	3620.00i	0.282962i	−0.989941	+	0.141481i	$0.954814\pi$
$548$	− 2118.00i	− 0.165103i
$549$	2898.00	0.225289
$550$	0	0
$551$	9672.00	0.747806
$552$	− 7560.00i	− 0.582926i
$553$	10400.0i	0.799734i
$554$	−11886.0	−0.911530
$555$	0	0
$556$	2324.00	0.177265
$557$	− 14166.0i	− 1.07762i	−0.842428	−	0.538809i	$-0.818875\pi$
$557$	− 14166.0i	− 1.07762i	0.842428	−	0.538809i	$-0.181125\pi$
$558$	− 5400.00i	− 0.409678i
$559$	−6808.00	−0.515112
$560$	0	0
$561$	3888.00	0.292605
$562$	− 24858.0i	− 1.86579i
$563$	13404.0i	1.00339i	0.865043	+	0.501697i	$0.167291\pi$
$563$	13404.0i	1.00339i	−0.865043	+	0.501697i	$0.832709\pi$
$564$	−72.0000	−0.00537544
$565$	0	0
$566$	−8148.00	−0.605099
$567$	1620.00i	0.119989i
$568$	− 6048.00i	− 0.446775i
$569$	18654.0	1.37437	0.687185	−	0.726483i	$-0.258846\pi$
$569$	18654.0	1.37437	0.687185	+	0.726483i	$0.258846\pi$
$570$	0	0
$571$	−7684.00	−0.563162	−0.281581	−	0.959537i	$-0.590859\pi$
$571$	−7684.00	−0.563162	−0.281581	+	0.959537i	$0.590859\pi$
$572$	− 1776.00i	− 0.129822i
$573$	10728.0i	0.782144i
$574$	−19800.0	−1.43978
$575$	0	0
$576$	3897.00	0.281901
$577$	− 1726.00i	− 0.124531i	−0.998060	−	0.0622654i	$-0.980167\pi$
$577$	− 1726.00i	− 0.124531i	0.998060	−	0.0622654i	$-0.0198325\pi$
$578$	5991.00i	0.431129i
$579$	7998.00	0.574068
$580$	0	0
$581$	3120.00	0.222787
$582$	2574.00i	0.183326i
$583$	10800.0i	0.767222i
$584$	−9030.00	−0.639836
$585$	0	0
$586$	18054.0	1.27270
$587$	10596.0i	0.745049i	0.928022	+	0.372524i	$0.121508\pi$
$587$	10596.0i	0.745049i	−0.928022	+	0.372524i	$0.878492\pi$
$588$	171.000i	0.0119931i
$589$	24800.0	1.73492
$590$	0	0
$591$	8154.00	0.567531
$592$	4970.00i	0.345043i
$593$	− 2862.00i	− 0.198193i	−0.995078	−	0.0990963i	$-0.968405\pi$
$593$	− 2862.00i	− 0.198193i	0.995078	−	0.0990963i	$-0.0315952\pi$
$594$	−1944.00	−0.134282
$595$	0	0
$596$	258.000	0.0177317
$597$	11496.0i	0.788107i
$598$	26640.0i	1.82172i
$599$	23592.0	1.60925	0.804627	−	0.593781i	$-0.202365\pi$
$599$	23592.0	1.60925	0.804627	+	0.593781i	$0.202365\pi$
$600$	0	0
$601$	−9574.00	−0.649803	−0.324902	−	0.945748i	$-0.605331\pi$
$601$	−9574.00	−0.649803	−0.324902	+	0.945748i	$0.605331\pi$
$602$	5520.00i	0.373718i
$603$	1764.00i	0.119130i
$604$	808.000	0.0544322
$605$	0	0
$606$	15606.0	1.04612
$607$	17444.0i	1.16644i	0.812314	+	0.583221i	$0.198208\pi$
$607$	17444.0i	1.16644i	−0.812314	+	0.583221i	$0.801792\pi$
$608$	− 5580.00i	− 0.372202i
$609$	−4680.00	−0.311401
$610$	0	0
$611$	−1776.00	−0.117593
$612$	486.000i	0.0321003i
$613$	2374.00i	0.156419i	0.996937	+	0.0782096i	$0.0249203\pi$
$613$	2374.00i	0.156419i	−0.996937	+	0.0782096i	$0.975080\pi$
$614$	−27708.0	−1.82118
$615$	0	0
$616$	10080.0	0.659310
$617$	− 12162.0i	− 0.793555i	−0.917915	−	0.396778i	$-0.870128\pi$
$617$	− 12162.0i	− 0.793555i	0.917915	−	0.396778i	$-0.129872\pi$
$618$	4068.00i	0.264788i
$619$	−8804.00	−0.571668	−0.285834	−	0.958279i	$-0.592271\pi$
$619$	−8804.00	−0.571668	−0.285834	+	0.958279i	$0.592271\pi$
$620$	0	0
$621$	3240.00	0.209367
$622$	4608.00i	0.297048i
$623$	− 20520.0i	− 1.31961i
$624$	−15762.0	−1.01119
$625$	0	0
$626$	−22026.0	−1.40629
$627$	− 8928.00i	− 0.568660i
$628$	− 2378.00i	− 0.151103i
$629$	3780.00	0.239616
$630$	0	0
$631$	−12688.0	−0.800478	−0.400239	−	0.916411i	$-0.631073\pi$
$631$	−12688.0	−0.800478	−0.400239	+	0.916411i	$0.631073\pi$
$632$	10920.0i	0.687301i
$633$	3300.00i	0.207209i
$634$	11682.0	0.731785
$635$	0	0
$636$	−1350.00	−0.0841682
$637$	4218.00i	0.262360i
$638$	− 5616.00i	− 0.348495i
$639$	2592.00	0.160466
$640$	0	0
$641$	−9150.00	−0.563812	−0.281906	−	0.959442i	$-0.590967\pi$
$641$	−9150.00	−0.563812	−0.281906	+	0.959442i	$0.590967\pi$
$642$	12636.0i	0.776796i
$643$	− 25292.0i	− 1.55120i	−0.631227	−	0.775598i	$-0.717448\pi$
$643$	− 25292.0i	− 1.55120i	0.631227	−	0.775598i	$-0.282552\pi$
$644$	2400.00	0.146853
$645$	0	0
$646$	−20088.0	−1.22345
$647$	− 2736.00i	− 0.166249i	−0.996539	−	0.0831246i	$-0.973510\pi$
$647$	− 2736.00i	− 0.166249i	0.996539	−	0.0831246i	$-0.0264900\pi$
$648$	1701.00i	0.103120i
$649$	576.000	0.0348382
$650$	0	0
$651$	−12000.0	−0.722453
$652$	− 52.0000i	− 0.00312343i
$653$	− 22218.0i	− 1.33148i	−0.746183	−	0.665741i	$-0.768116\pi$
$653$	− 22218.0i	− 1.33148i	0.746183	−	0.665741i	$-0.231884\pi$
$654$	−13266.0	−0.793183
$655$	0	0
$656$	−23430.0	−1.39449
$657$	− 3870.00i	− 0.229807i
$658$	1440.00i	0.0853147i
$659$	−14520.0	−0.858299	−0.429149	−	0.903234i	$-0.641187\pi$
$659$	−14520.0	−0.858299	−0.429149	+	0.903234i	$0.641187\pi$
$660$	0	0
$661$	−10618.0	−0.624799	−0.312400	−	0.949951i	$-0.601133\pi$
$661$	−10618.0	−0.624799	−0.312400	+	0.949951i	$0.601133\pi$
$662$	11076.0i	0.650273i
$663$	11988.0i	0.702225i
$664$	3276.00	0.191466
$665$	0	0
$666$	−1890.00	−0.109964
$667$	9360.00i	0.543359i
$668$	3720.00i	0.215466i
$669$	5892.00	0.340505
$670$	0	0
$671$	7728.00	0.444614
$672$	2700.00i	0.154992i
$673$	− 1370.00i	− 0.0784690i	−0.999230	−	0.0392345i	$-0.987508\pi$
$673$	− 1370.00i	− 0.0784690i	0.999230	−	0.0392345i	$-0.0124919\pi$
$674$	26994.0	1.54269
$675$	0	0
$676$	3279.00	0.186561
$677$	− 13758.0i	− 0.781038i	−0.920595	−	0.390519i	$-0.872296\pi$
$677$	− 13758.0i	− 0.781038i	0.920595	−	0.390519i	$-0.127704\pi$
$678$	9774.00i	0.553640i
$679$	5720.00	0.323289
$680$	0	0
$681$	−1980.00	−0.111415
$682$	− 14400.0i	− 0.808511i
$683$	− 11988.0i	− 0.671608i	−0.941932	−	0.335804i	$-0.890992\pi$
$683$	− 11988.0i	− 0.671608i	0.941932	−	0.335804i	$-0.109008\pi$
$684$	1116.00	0.0623850
$685$	0	0
$686$	−17160.0	−0.955061
$687$	5718.00i	0.317548i
$688$	6532.00i	0.361962i
$689$	−33300.0	−1.84126
$690$	0	0
$691$	32996.0	1.81654	0.908268	−	0.418388i	$-0.137405\pi$
$691$	32996.0	1.81654	0.908268	+	0.418388i	$0.137405\pi$
$692$	426.000i	0.0234019i
$693$	4320.00i	0.236801i
$694$	−15732.0	−0.860488
$695$	0	0
$696$	−4914.00	−0.267622
$697$	17820.0i	0.968408i
$698$	− 18906.0i	− 1.02522i
$699$	−4374.00	−0.236681
$700$	0	0
$701$	−25902.0	−1.39558	−0.697792	−	0.716300i	$-0.745834\pi$
$701$	−25902.0	−1.39558	−0.697792	+	0.716300i	$0.745834\pi$
$702$	− 5994.00i	− 0.322263i
$703$	− 8680.00i	− 0.465679i
$704$	10392.0	0.556340
$705$	0	0
$706$	10242.0	0.545981
$707$	− 34680.0i	− 1.84480i
$708$	72.0000i	0.00382193i
$709$	27394.0	1.45106	0.725531	−	0.688189i	$-0.241594\pi$
$709$	27394.0	1.45106	0.725531	+	0.688189i	$0.241594\pi$
$710$	0	0
$711$	−4680.00	−0.246855
$712$	− 21546.0i	− 1.13409i
$713$	24000.0i	1.26060i
$714$	9720.00	0.509470
$715$	0	0
$716$	−1440.00	−0.0751611
$717$	− 3528.00i	− 0.183760i
$718$	− 14472.0i	− 0.752215i
$719$	−34848.0	−1.80753	−0.903763	−	0.428033i	$-0.859207\pi$
$719$	−34848.0	−1.80753	−0.903763	+	0.428033i	$0.859207\pi$
$720$	0	0
$721$	9040.00	0.466945
$722$	25551.0i	1.31705i
$723$	2598.00i	0.133639i
$724$	3130.00	0.160671
$725$	0	0
$726$	6795.00	0.347364
$727$	28028.0i	1.42985i	0.699201	+	0.714925i	$0.253539\pi$
$727$	28028.0i	1.42985i	−0.699201	+	0.714925i	$0.746461\pi$
$728$	31080.0i	1.58228i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	4968.00	0.251365
$732$	966.000i	0.0487765i
$733$	− 18002.0i	− 0.907120i	−0.891226	−	0.453560i	$-0.850154\pi$
$733$	− 18002.0i	− 0.907120i	0.891226	−	0.453560i	$-0.149846\pi$
$734$	10524.0	0.529221
$735$	0	0
$736$	5400.00	0.270444
$737$	4704.00i	0.235107i
$738$	− 8910.00i	− 0.444420i
$739$	−15284.0	−0.760800	−0.380400	−	0.924822i	$-0.624214\pi$
$739$	−15284.0	−0.760800	−0.380400	+	0.924822i	$0.624214\pi$
$740$	0	0
$741$	27528.0	1.36473
$742$	27000.0i	1.33585i
$743$	18768.0i	0.926691i	0.886178	+	0.463345i	$0.153351\pi$
$743$	18768.0i	0.926691i	−0.886178	+	0.463345i	$0.846649\pi$
$744$	−12600.0	−0.620885
$745$	0	0
$746$	32406.0	1.59044
$747$	1404.00i	0.0687680i
$748$	1296.00i	0.0633509i
$749$	28080.0	1.36985
$750$	0	0
$751$	8696.00	0.422532	0.211266	−	0.977429i	$-0.432241\pi$
$751$	8696.00	0.422532	0.211266	+	0.977429i	$0.432241\pi$
$752$	1704.00i	0.0826310i
$753$	1296.00i	0.0627209i
$754$	17316.0	0.836355
$755$	0	0
$756$	−540.000	−0.0259783
$757$	− 38662.0i	− 1.85627i	−0.372247	−	0.928134i	$-0.621413\pi$
$757$	− 38662.0i	− 1.85627i	0.372247	−	0.928134i	$-0.378587\pi$
$758$	− 4380.00i	− 0.209880i
$759$	8640.00	0.413191
$760$	0	0
$761$	23874.0	1.13723	0.568615	−	0.822604i	$-0.307479\pi$
$761$	23874.0	1.13723	0.568615	+	0.822604i	$0.307479\pi$
$762$	− 11196.0i	− 0.532268i
$763$	29480.0i	1.39875i
$764$	−3576.00	−0.169339
$765$	0	0
$766$	−14616.0	−0.689422
$767$	1776.00i	0.0836084i
$768$	4539.00i	0.213264i
$769$	−23618.0	−1.10753	−0.553763	−	0.832675i	$-0.686808\pi$
$769$	−23618.0	−1.10753	−0.553763	+	0.832675i	$0.686808\pi$
$770$	0	0
$771$	−7578.00	−0.353975
$772$	2666.00i	0.124289i
$773$	− 11538.0i	− 0.536860i	−0.963299	−	0.268430i	$-0.913495\pi$
$773$	− 11538.0i	− 0.536860i	0.963299	−	0.268430i	$-0.0865049\pi$
$774$	−2484.00	−0.115356
$775$	0	0
$776$	6006.00	0.277839
$777$	4200.00i	0.193918i
$778$	42138.0i	1.94180i
$779$	40920.0	1.88204
$780$	0	0
$781$	6912.00	0.316685
$782$	− 19440.0i	− 0.888968i
$783$	− 2106.00i	− 0.0961204i
$784$	4047.00	0.184357
$785$	0	0
$786$	−20952.0	−0.950805
$787$	− 14884.0i	− 0.674152i	−0.941478	−	0.337076i	$-0.890562\pi$
$787$	− 14884.0i	− 0.674152i	0.941478	−	0.337076i	$-0.109438\pi$
$788$	2718.00i	0.122874i
$789$	16344.0	0.737467
$790$	0	0
$791$	21720.0	0.976327
$792$	4536.00i	0.203510i
$793$	23828.0i	1.06703i
$794$	8202.00	0.366597
$795$	0	0
$796$	−3832.00	−0.170630
$797$	− 11334.0i	− 0.503728i	−0.967763	−	0.251864i	$-0.918957\pi$
$797$	− 11334.0i	− 0.503728i	0.967763	−	0.251864i	$-0.0810435\pi$
$798$	− 22320.0i	− 0.990125i
$799$	1296.00	0.0573832
$800$	0	0
$801$	9234.00	0.407325
$802$	− 47826.0i	− 2.10573i
$803$	− 10320.0i	− 0.453530i
$804$	−588.000	−0.0257925
$805$	0	0
$806$	44400.0	1.94035
$807$	7722.00i	0.336837i
$808$	− 36414.0i	− 1.58545i
$809$	−44730.0	−1.94391	−0.971955	−	0.235167i	$-0.924436\pi$
$809$	−44730.0	−1.94391	−0.971955	+	0.235167i	$0.924436\pi$
$810$	0	0
$811$	−42748.0	−1.85091	−0.925453	−	0.378862i	$-0.876316\pi$
$811$	−42748.0	−1.85091	−0.925453	+	0.378862i	$0.876316\pi$
$812$	− 1560.00i	− 0.0674203i
$813$	− 9552.00i	− 0.412058i
$814$	−5040.00	−0.217017
$815$	0	0
$816$	11502.0	0.493444
$817$	− 11408.0i	− 0.488513i
$818$	− 26142.0i	− 1.11740i
$819$	−13320.0	−0.568301
$820$	0	0
$821$	−31686.0	−1.34695	−0.673477	−	0.739208i	$-0.735200\pi$
$821$	−31686.0	−1.34695	−0.673477	+	0.739208i	$0.735200\pi$
$822$	− 19062.0i	− 0.808836i
$823$	− 11036.0i	− 0.467425i	−0.972306	−	0.233713i	$-0.924913\pi$
$823$	− 11036.0i	− 0.467425i	0.972306	−	0.233713i	$-0.0750875\pi$
$824$	9492.00	0.401298
$825$	0	0
$826$	1440.00	0.0606586
$827$	25884.0i	1.08836i	0.838968	+	0.544181i	$0.183159\pi$
$827$	25884.0i	1.08836i	−0.838968	+	0.544181i	$0.816841\pi$
$828$	1080.00i	0.0453292i
$829$	−15950.0	−0.668234	−0.334117	−	0.942532i	$-0.608438\pi$
$829$	−15950.0	−0.668234	−0.334117	+	0.942532i	$0.608438\pi$
$830$	0	0
$831$	−11886.0	−0.496174
$832$	32042.0i	1.33516i
$833$	− 3078.00i	− 0.128027i
$834$	20916.0	0.868419
$835$	0	0
$836$	2976.00	0.123119
$837$	− 5400.00i	− 0.223000i
$838$	− 35928.0i	− 1.48104i
$839$	−13800.0	−0.567853	−0.283927	−	0.958846i	$-0.591637\pi$
$839$	−13800.0	−0.567853	−0.283927	+	0.958846i	$0.591637\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	33162.0i	1.35729i
$843$	− 24858.0i	− 1.01560i
$844$	−1100.00	−0.0448620
$845$	0	0
$846$	−648.000	−0.0263342
$847$	− 15100.0i	− 0.612565i
$848$	31950.0i	1.29383i
$849$	−8148.00	−0.329374
$850$	0	0
$851$	8400.00	0.338365
$852$	864.000i	0.0347420i
$853$	27862.0i	1.11838i	0.829040	+	0.559189i	$0.188887\pi$
$853$	27862.0i	1.11838i	−0.829040	+	0.559189i	$0.811113\pi$
$854$	19320.0	0.774141
$855$	0	0
$856$	29484.0	1.17727
$857$	− 7314.00i	− 0.291530i	−0.989319	−	0.145765i	$-0.953436\pi$
$857$	− 7314.00i	− 0.291530i	0.989319	−	0.145765i	$-0.0465644\pi$
$858$	− 15984.0i	− 0.635996i
$859$	28780.0	1.14314	0.571572	−	0.820552i	$-0.306334\pi$
$859$	28780.0	1.14314	0.571572	+	0.820552i	$0.306334\pi$
$860$	0	0
$861$	−19800.0	−0.783719
$862$	2160.00i	0.0853479i
$863$	32688.0i	1.28935i	0.764455	+	0.644677i	$0.223008\pi$
$863$	32688.0i	1.28935i	−0.764455	+	0.644677i	$0.776992\pi$
$864$	−1215.00	−0.0478416
$865$	0	0
$866$	−46866.0	−1.83900
$867$	5991.00i	0.234677i
$868$	− 4000.00i	− 0.156416i
$869$	−12480.0	−0.487175
$870$	0	0
$871$	−14504.0	−0.564236
$872$	30954.0i	1.20210i
$873$	2574.00i	0.0997900i
$874$	−44640.0	−1.72766
$875$	0	0
$876$	1290.00	0.0497546
$877$	36650.0i	1.41115i	0.708633	+	0.705577i	$0.249312\pi$
$877$	36650.0i	1.41115i	−0.708633	+	0.705577i	$0.750688\pi$
$878$	29640.0i	1.13930i
$879$	18054.0	0.692772
$880$	0	0
$881$	−2646.00	−0.101187	−0.0505936	−	0.998719i	$-0.516111\pi$
$881$	−2646.00	−0.101187	−0.0505936	+	0.998719i	$0.516111\pi$
$882$	1539.00i	0.0587538i
$883$	− 10892.0i	− 0.415113i	−0.978223	−	0.207557i	$-0.933449\pi$
$883$	− 10892.0i	− 0.415113i	0.978223	−	0.207557i	$-0.0665511\pi$
$884$	−3996.00	−0.152036
$885$	0	0
$886$	−48348.0	−1.83328
$887$	− 43464.0i	− 1.64530i	−0.568550	−	0.822648i	$-0.692496\pi$
$887$	− 43464.0i	− 1.64530i	0.568550	−	0.822648i	$-0.307504\pi$
$888$	4410.00i	0.166655i
$889$	−24880.0	−0.938637
$890$	0	0
$891$	−1944.00	−0.0730937
$892$	1964.00i	0.0737215i
$893$	− 2976.00i	− 0.111521i
$894$	2322.00	0.0868672
$895$	0	0
$896$	33180.0	1.23713
$897$	26640.0i	0.991621i
$898$	− 27054.0i	− 1.00535i
$899$	15600.0	0.578742
$900$	0	0
$901$	24300.0	0.898502
$902$	− 23760.0i	− 0.877075i
$903$	5520.00i	0.203426i
$904$	22806.0	0.839067
$905$	0	0
$906$	7272.00	0.266662
$907$	− 14884.0i	− 0.544890i	−0.962171	−	0.272445i	$-0.912168\pi$
$907$	− 14884.0i	− 0.544890i	0.962171	−	0.272445i	$-0.0878323\pi$
$908$	− 660.000i	− 0.0241221i
$909$	15606.0	0.569437
$910$	0	0
$911$	−1248.00	−0.0453876	−0.0226938	−	0.999742i	$-0.507224\pi$
$911$	−1248.00	−0.0453876	−0.0226938	+	0.999742i	$0.507224\pi$
$912$	− 26412.0i	− 0.958979i
$913$	3744.00i	0.135716i
$914$	11010.0	0.398445
$915$	0	0
$916$	−1906.00	−0.0687511
$917$	46560.0i	1.67671i
$918$	4374.00i	0.157259i
$919$	6640.00	0.238339	0.119169	−	0.992874i	$-0.461977\pi$
$919$	6640.00	0.238339	0.119169	+	0.992874i	$0.461977\pi$
$920$	0	0
$921$	−27708.0	−0.991324
$922$	52686.0i	1.88191i
$923$	21312.0i	0.760014i
$924$	−1440.00	−0.0512690
$925$	0	0
$926$	3516.00	0.124776
$927$	4068.00i	0.144132i
$928$	− 3510.00i	− 0.124161i
$929$	−29946.0	−1.05758	−0.528792	−	0.848751i	$-0.677355\pi$
$929$	−29946.0	−1.05758	−0.528792	+	0.848751i	$0.677355\pi$
$930$	0	0
$931$	−7068.00	−0.248812
$932$	− 1458.00i	− 0.0512429i
$933$	4608.00i	0.161693i
$934$	−20628.0	−0.722665
$935$	0	0
$936$	−13986.0	−0.488405
$937$	45002.0i	1.56900i	0.620130	+	0.784499i	$0.287080\pi$
$937$	45002.0i	1.56900i	−0.620130	+	0.784499i	$0.712920\pi$
$938$	11760.0i	0.409358i
$939$	−22026.0	−0.765486
$940$	0	0
$941$	6090.00	0.210976	0.105488	−	0.994421i	$-0.466360\pi$
$941$	6090.00	0.210976	0.105488	+	0.994421i	$0.466360\pi$
$942$	− 21402.0i	− 0.740249i
$943$	39600.0i	1.36750i
$944$	1704.00	0.0587505
$945$	0	0
$946$	−6624.00	−0.227658
$947$	56388.0i	1.93491i	0.253035	+	0.967457i	$0.418571\pi$
$947$	56388.0i	1.93491i	−0.253035	+	0.967457i	$0.581429\pi$
$948$	− 1560.00i	− 0.0534456i
$949$	31820.0	1.08843
$950$	0	0
$951$	11682.0	0.398333
$952$	− 22680.0i	− 0.772125i
$953$	− 10854.0i	− 0.368936i	−0.982839	−	0.184468i	$-0.940944\pi$
$953$	− 10854.0i	− 0.368936i	0.982839	−	0.184468i	$-0.0590561\pi$
$954$	−12150.0	−0.412338
$955$	0	0
$956$	1176.00	0.0397851
$957$	− 5616.00i	− 0.189696i
$958$	− 6840.00i	− 0.230679i
$959$	−42360.0	−1.42636
$960$	0	0
$961$	10209.0	0.342687
$962$	− 15540.0i	− 0.520821i
$963$	12636.0i	0.422834i
$964$	−866.000	−0.0289336
$965$	0	0
$966$	21600.0	0.719429
$967$	− 42316.0i	− 1.40723i	−0.710582	−	0.703615i	$-0.751568\pi$
$967$	− 42316.0i	− 1.40723i	0.710582	−	0.703615i	$-0.248432\pi$
$968$	− 15855.0i	− 0.526445i
$969$	−20088.0	−0.665964
$970$	0	0
$971$	24480.0	0.809063	0.404532	−	0.914524i	$-0.367435\pi$
$971$	24480.0	0.809063	0.404532	+	0.914524i	$0.367435\pi$
$972$	− 243.000i	− 0.00801875i
$973$	− 46480.0i	− 1.53143i
$974$	9228.00	0.303577
$975$	0	0
$976$	22862.0	0.749790
$977$	− 6906.00i	− 0.226144i	−0.993587	−	0.113072i	$-0.963931\pi$
$977$	− 6906.00i	− 0.226144i	0.993587	−	0.113072i	$-0.0360690\pi$
$978$	− 468.000i	− 0.0153016i
$979$	24624.0	0.803868
$980$	0	0
$981$	−13266.0	−0.431754
$982$	− 56736.0i	− 1.84371i
$983$	− 6960.00i	− 0.225829i	−0.993605	−	0.112914i	$-0.963981\pi$
$983$	− 6960.00i	− 0.225829i	0.993605	−	0.112914i	$-0.0360186\pi$
$984$	−20790.0	−0.673538
$985$	0	0
$986$	−12636.0	−0.408126
$987$	1440.00i	0.0464394i
$988$	9176.00i	0.295473i
$989$	11040.0	0.354956
$990$	0	0
$991$	47792.0	1.53195	0.765975	−	0.642870i	$-0.222256\pi$
$991$	47792.0	1.53195	0.765975	+	0.642870i	$0.222256\pi$
$992$	− 9000.00i	− 0.288055i
$993$	11076.0i	0.353964i
$994$	17280.0	0.551397
$995$	0	0
$996$	−468.000	−0.0148887
$997$	9938.00i	0.315687i	0.987464	+	0.157843i	$0.0504541\pi$
$997$	9938.00i	0.315687i	−0.987464	+	0.157843i	$0.949546\pi$
$998$	− 29868.0i	− 0.947350i
$999$	−1890.00	−0.0598568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.b.a.49.2		2
3.2	odd	2		225.4.b.d.199.1		2
4.3	odd	2		1200.4.f.m.49.1		2
5.2	odd	4		75.4.a.a.1.1		1
5.3	odd	4		15.4.a.b.1.1	✓	1
5.4	even	2	inner	75.4.b.a.49.1		2
15.2	even	4		225.4.a.g.1.1		1
15.8	even	4		45.4.a.b.1.1		1
15.14	odd	2		225.4.b.d.199.2		2
20.3	even	4		240.4.a.f.1.1		1
20.7	even	4		1200.4.a.o.1.1		1
20.19	odd	2		1200.4.f.m.49.2		2
35.13	even	4		735.4.a.i.1.1		1
40.3	even	4		960.4.a.l.1.1		1
40.13	odd	4		960.4.a.bi.1.1		1
45.13	odd	12		405.4.e.d.136.1		2
45.23	even	12		405.4.e.k.136.1		2
45.38	even	12		405.4.e.k.271.1		2
45.43	odd	12		405.4.e.d.271.1		2
55.43	even	4		1815.4.a.a.1.1		1
60.23	odd	4		720.4.a.r.1.1		1
105.83	odd	4		2205.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.b.1.1	✓	1	5.3	odd	4
45.4.a.b.1.1		1	15.8	even	4
75.4.a.a.1.1		1	5.2	odd	4
75.4.b.a.49.1		2	5.4	even	2	inner
75.4.b.a.49.2		2	1.1	even	1	trivial
225.4.a.g.1.1		1	15.2	even	4
225.4.b.d.199.1		2	3.2	odd	2
225.4.b.d.199.2		2	15.14	odd	2
240.4.a.f.1.1		1	20.3	even	4
405.4.e.d.136.1		2	45.13	odd	12
405.4.e.d.271.1		2	45.43	odd	12
405.4.e.k.136.1		2	45.23	even	12
405.4.e.k.271.1		2	45.38	even	12
720.4.a.r.1.1		1	60.23	odd	4
735.4.a.i.1.1		1	35.13	even	4
960.4.a.l.1.1		1	40.3	even	4
960.4.a.bi.1.1		1	40.13	odd	4
1200.4.a.o.1.1		1	20.7	even	4
1200.4.f.m.49.1		2	4.3	odd	2
1200.4.f.m.49.2		2	20.19	odd	2
1815.4.a.a.1.1		1	55.43	even	4
2205.4.a.c.1.1		1	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} +20.0000i q^{7} +21.0000i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} +20.0000i q^{7} +21.0000i q^{8} -9.00000 q^{9} -24.0000 q^{11} -3.00000i q^{12} -74.0000i q^{13} -60.0000 q^{14} -71.0000 q^{16} +54.0000i q^{17} -27.0000i q^{18} +124.000 q^{19} -60.0000 q^{21} -72.0000i q^{22} +120.000i q^{23} -63.0000 q^{24} +222.000 q^{26} -27.0000i q^{27} -20.0000i q^{28} +78.0000 q^{29} +200.000 q^{31} -45.0000i q^{32} -72.0000i q^{33} -162.000 q^{34} +9.00000 q^{36} -70.0000i q^{37} +372.000i q^{38} +222.000 q^{39} +330.000 q^{41} -180.000i q^{42} -92.0000i q^{43} +24.0000 q^{44} -360.000 q^{46} -24.0000i q^{47} -213.000i q^{48} -57.0000 q^{49} -162.000 q^{51} +74.0000i q^{52} -450.000i q^{53} +81.0000 q^{54} -420.000 q^{56} +372.000i q^{57} +234.000i q^{58} -24.0000 q^{59} -322.000 q^{61} +600.000i q^{62} -180.000i q^{63} -433.000 q^{64} +216.000 q^{66} -196.000i q^{67} -54.0000i q^{68} -360.000 q^{69} -288.000 q^{71} -189.000i q^{72} +430.000i q^{73} +210.000 q^{74} -124.000 q^{76} -480.000i q^{77} +666.000i q^{78} +520.000 q^{79} +81.0000 q^{81} +990.000i q^{82} -156.000i q^{83} +60.0000 q^{84} +276.000 q^{86} +234.000i q^{87} -504.000i q^{88} -1026.00 q^{89} +1480.00 q^{91} -120.000i q^{92} +600.000i q^{93} +72.0000 q^{94} +135.000 q^{96} -286.000i q^{97} -171.000i q^{98} +216.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 18 q^{6} - 18 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 18 * q^6 - 18 * q^9	\( 2 q - 2 q^{4} - 18 q^{6} - 18 q^{9} - 48 q^{11} - 120 q^{14} - 142 q^{16} + 248 q^{19} - 120 q^{21} - 126 q^{24} + 444 q^{26} + 156 q^{29} + 400 q^{31} - 324 q^{34} + 18 q^{36} + 444 q^{39} + 660 q^{41} + 48 q^{44} - 720 q^{46} - 114 q^{49} - 324 q^{51} + 162 q^{54} - 840 q^{56} - 48 q^{59} - 644 q^{61} - 866 q^{64} + 432 q^{66} - 720 q^{69} - 576 q^{71} + 420 q^{74} - 248 q^{76} + 1040 q^{79} + 162 q^{81} + 120 q^{84} + 552 q^{86} - 2052 q^{89} + 2960 q^{91} + 144 q^{94} + 270 q^{96} + 432 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 18 * q^6 - 18 * q^9 - 48 * q^11 - 120 * q^14 - 142 * q^16 + 248 * q^19 - 120 * q^21 - 126 * q^24 + 444 * q^26 + 156 * q^29 + 400 * q^31 - 324 * q^34 + 18 * q^36 + 444 * q^39 + 660 * q^41 + 48 * q^44 - 720 * q^46 - 114 * q^49 - 324 * q^51 + 162 * q^54 - 840 * q^56 - 48 * q^59 - 644 * q^61 - 866 * q^64 + 432 * q^66 - 720 * q^69 - 576 * q^71 + 420 * q^74 - 248 * q^76 + 1040 * q^79 + 162 * q^81 + 120 * q^84 + 552 * q^86 - 2052 * q^89 + 2960 * q^91 + 144 * q^94 + 270 * q^96 + 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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