LMFDB - Embedded newform 3750.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3750.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.a (trivial)

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:	yes
Analytic conductor:	$29.9439007580$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.71684000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 18x^{6} + 10x^{5} + 101x^{4} + 40x^{3} - 132x^{2} - 96x - 4 $ x^8 - 2x^7 - 18x^6 + 10x^5 + 101x^4 + 40x^3 - 132x^2 - 96*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.37243$ of defining polynomial
Character	$\chi$	$=$	3750.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.61995 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.61995 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.285809 q^{11} -1.00000 q^{12} +2.80530 q^{13} +2.61995 q^{14} +1.00000 q^{16} +6.49242 q^{17} -1.00000 q^{18} -0.443312 q^{19} +2.61995 q^{21} +0.285809 q^{22} -6.52498 q^{23} +1.00000 q^{24} -2.80530 q^{26} -1.00000 q^{27} -2.61995 q^{28} +7.25636 q^{29} +3.62674 q^{31} -1.00000 q^{32} +0.285809 q^{33} -6.49242 q^{34} +1.00000 q^{36} -4.53906 q^{37} +0.443312 q^{38} -2.80530 q^{39} +5.32407 q^{41} -2.61995 q^{42} -8.05390 q^{43} -0.285809 q^{44} +6.52498 q^{46} -8.99646 q^{47} -1.00000 q^{48} -0.135854 q^{49} -6.49242 q^{51} +2.80530 q^{52} -13.9158 q^{53} +1.00000 q^{54} +2.61995 q^{56} +0.443312 q^{57} -7.25636 q^{58} -3.74685 q^{59} +13.0189 q^{61} -3.62674 q^{62} -2.61995 q^{63} +1.00000 q^{64} -0.285809 q^{66} +10.2325 q^{67} +6.49242 q^{68} +6.52498 q^{69} -4.81783 q^{71} -1.00000 q^{72} +11.7867 q^{73} +4.53906 q^{74} -0.443312 q^{76} +0.748806 q^{77} +2.80530 q^{78} -7.81462 q^{79} +1.00000 q^{81} -5.32407 q^{82} +5.16187 q^{83} +2.61995 q^{84} +8.05390 q^{86} -7.25636 q^{87} +0.285809 q^{88} -6.16560 q^{89} -7.34974 q^{91} -6.52498 q^{92} -3.62674 q^{93} +8.99646 q^{94} +1.00000 q^{96} -5.45040 q^{97} +0.135854 q^{98} -0.285809 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{6} - 4 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 + 8 * q^6 - 4 * q^7 - 8 * q^8 + 8 * q^9	$ 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{6} - 4 q^{7} - 8 q^{8} + 8 q^{9} + 6 q^{11} - 8 q^{12} - 2 q^{13} + 4 q^{14} + 8 q^{16} - 14 q^{17} - 8 q^{18} + 10 q^{19} + 4 q^{21} - 6 q^{22} - 12 q^{23} + 8 q^{24} + 2 q^{26} - 8 q^{27} - 4 q^{28} + 10 q^{29} + 16 q^{31} - 8 q^{32} - 6 q^{33} + 14 q^{34} + 8 q^{36} + 6 q^{37} - 10 q^{38} + 2 q^{39} + 6 q^{41} - 4 q^{42} - 2 q^{43} + 6 q^{44} + 12 q^{46} - 14 q^{47} - 8 q^{48} + 26 q^{49} + 14 q^{51} - 2 q^{52} - 12 q^{53} + 8 q^{54} + 4 q^{56} - 10 q^{57} - 10 q^{58} + 16 q^{61} - 16 q^{62} - 4 q^{63} + 8 q^{64} + 6 q^{66} + 6 q^{67} - 14 q^{68} + 12 q^{69} + 6 q^{71} - 8 q^{72} + 8 q^{73} - 6 q^{74} + 10 q^{76} - 8 q^{77} - 2 q^{78} + 10 q^{79} + 8 q^{81} - 6 q^{82} - 22 q^{83} + 4 q^{84} + 2 q^{86} - 10 q^{87} - 6 q^{88} + 20 q^{89} + 6 q^{91} - 12 q^{92} - 16 q^{93} + 14 q^{94} + 8 q^{96} + 16 q^{97} - 26 q^{98} + 6 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 + 8 * q^6 - 4 * q^7 - 8 * q^8 + 8 * q^9 + 6 * q^11 - 8 * q^12 - 2 * q^13 + 4 * q^14 + 8 * q^16 - 14 * q^17 - 8 * q^18 + 10 * q^19 + 4 * q^21 - 6 * q^22 - 12 * q^23 + 8 * q^24 + 2 * q^26 - 8 * q^27 - 4 * q^28 + 10 * q^29 + 16 * q^31 - 8 * q^32 - 6 * q^33 + 14 * q^34 + 8 * q^36 + 6 * q^37 - 10 * q^38 + 2 * q^39 + 6 * q^41 - 4 * q^42 - 2 * q^43 + 6 * q^44 + 12 * q^46 - 14 * q^47 - 8 * q^48 + 26 * q^49 + 14 * q^51 - 2 * q^52 - 12 * q^53 + 8 * q^54 + 4 * q^56 - 10 * q^57 - 10 * q^58 + 16 * q^61 - 16 * q^62 - 4 * q^63 + 8 * q^64 + 6 * q^66 + 6 * q^67 - 14 * q^68 + 12 * q^69 + 6 * q^71 - 8 * q^72 + 8 * q^73 - 6 * q^74 + 10 * q^76 - 8 * q^77 - 2 * q^78 + 10 * q^79 + 8 * q^81 - 6 * q^82 - 22 * q^83 + 4 * q^84 + 2 * q^86 - 10 * q^87 - 6 * q^88 + 20 * q^89 + 6 * q^91 - 12 * q^92 - 16 * q^93 + 14 * q^94 + 8 * q^96 + 16 * q^97 - 26 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−2.61995	−0.990249	−0.495124	−	0.868822i	$-0.664877\pi$
$7$	−2.61995	−0.990249	−0.495124	+	0.868822i	$0.664877\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.285809	−0.0861746	−0.0430873	−	0.999071i	$-0.513719\pi$
$11$	−0.285809	−0.0861746	−0.0430873	+	0.999071i	$0.513719\pi$
$12$	−1.00000	−0.288675
$13$	2.80530	0.778049	0.389024	−	0.921227i	$-0.372812\pi$
$13$	2.80530	0.778049	0.389024	+	0.921227i	$0.372812\pi$
$14$	2.61995	0.700212
$15$	0	0
$16$	1.00000	0.250000
$17$	6.49242	1.57464	0.787322	−	0.616542i	$-0.211467\pi$
$17$	6.49242	1.57464	0.787322	+	0.616542i	$0.211467\pi$
$18$	−1.00000	−0.235702
$19$	−0.443312	−0.101703	−0.0508514	−	0.998706i	$-0.516193\pi$
$19$	−0.443312	−0.101703	−0.0508514	+	0.998706i	$0.516193\pi$
$20$	0	0
$21$	2.61995	0.571720
$22$	0.285809	0.0609347
$23$	−6.52498	−1.36055	−0.680276	−	0.732956i	$-0.738140\pi$
$23$	−6.52498	−1.36055	−0.680276	+	0.732956i	$0.738140\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.80530	−0.550164
$27$	−1.00000	−0.192450
$28$	−2.61995	−0.495124
$29$	7.25636	1.34747	0.673736	−	0.738972i	$-0.264689\pi$
$29$	7.25636	1.34747	0.673736	+	0.738972i	$0.264689\pi$
$30$	0	0
$31$	3.62674	0.651381	0.325691	−	0.945476i	$-0.394403\pi$
$31$	3.62674	0.651381	0.325691	+	0.945476i	$0.394403\pi$
$32$	−1.00000	−0.176777
$33$	0.285809	0.0497529
$34$	−6.49242	−1.11344
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.53906	−0.746217	−0.373109	−	0.927788i	$-0.621708\pi$
$37$	−4.53906	−0.746217	−0.373109	+	0.927788i	$0.621708\pi$
$38$	0.443312	0.0719147
$39$	−2.80530	−0.449207
$40$	0	0
$41$	5.32407	0.831480	0.415740	−	0.909484i	$-0.363523\pi$
$41$	5.32407	0.831480	0.415740	+	0.909484i	$0.363523\pi$
$42$	−2.61995	−0.404267
$43$	−8.05390	−1.22821	−0.614104	−	0.789225i	$-0.710483\pi$
$43$	−8.05390	−1.22821	−0.614104	+	0.789225i	$0.710483\pi$
$44$	−0.285809	−0.0430873
$45$	0	0
$46$	6.52498	0.962056
$47$	−8.99646	−1.31227	−0.656134	−	0.754644i	$-0.727809\pi$
$47$	−8.99646	−1.31227	−0.656134	+	0.754644i	$0.727809\pi$
$48$	−1.00000	−0.144338
$49$	−0.135854	−0.0194077
$50$	0	0
$51$	−6.49242	−0.909121
$52$	2.80530	0.389024
$53$	−13.9158	−1.91148	−0.955738	−	0.294219i	$-0.904940\pi$
$53$	−13.9158	−1.91148	−0.955738	+	0.294219i	$0.904940\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.61995	0.350106
$57$	0.443312	0.0587181
$58$	−7.25636	−0.952806
$59$	−3.74685	−0.487798	−0.243899	−	0.969801i	$-0.578427\pi$
$59$	−3.74685	−0.487798	−0.243899	+	0.969801i	$0.578427\pi$
$60$	0	0
$61$	13.0189	1.66691	0.833453	−	0.552591i	$-0.186361\pi$
$61$	13.0189	1.66691	0.833453	+	0.552591i	$0.186361\pi$
$62$	−3.62674	−0.460596
$63$	−2.61995	−0.330083
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.285809	−0.0351806
$67$	10.2325	1.25010	0.625051	−	0.780584i	$-0.285078\pi$
$67$	10.2325	1.25010	0.625051	+	0.780584i	$0.285078\pi$
$68$	6.49242	0.787322
$69$	6.52498	0.785515
$70$	0	0
$71$	−4.81783	−0.571772	−0.285886	−	0.958264i	$-0.592288\pi$
$71$	−4.81783	−0.571772	−0.285886	+	0.958264i	$0.592288\pi$
$72$	−1.00000	−0.117851
$73$	11.7867	1.37953	0.689765	−	0.724033i	$-0.257714\pi$
$73$	11.7867	1.37953	0.689765	+	0.724033i	$0.257714\pi$
$74$	4.53906	0.527655
$75$	0	0
$76$	−0.443312	−0.0508514
$77$	0.748806	0.0853343
$78$	2.80530	0.317637
$79$	−7.81462	−0.879214	−0.439607	−	0.898190i	$-0.644882\pi$
$79$	−7.81462	−0.879214	−0.439607	+	0.898190i	$0.644882\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.32407	−0.587945
$83$	5.16187	0.566588	0.283294	−	0.959033i	$-0.408573\pi$
$83$	5.16187	0.566588	0.283294	+	0.959033i	$0.408573\pi$
$84$	2.61995	0.285860
$85$	0	0
$86$	8.05390	0.868475
$87$	−7.25636	−0.777963
$88$	0.285809	0.0304673
$89$	−6.16560	−0.653552	−0.326776	−	0.945102i	$-0.605962\pi$
$89$	−6.16560	−0.653552	−0.326776	+	0.945102i	$0.605962\pi$
$90$	0	0
$91$	−7.34974	−0.770462
$92$	−6.52498	−0.680276
$93$	−3.62674	−0.376075
$94$	8.99646	0.927914
$95$	0	0
$96$	1.00000	0.102062
$97$	−5.45040	−0.553405	−0.276702	−	0.960956i	$-0.589242\pi$
$97$	−5.45040	−0.553405	−0.276702	+	0.960956i	$0.589242\pi$
$98$	0.135854	0.0137233
$99$	−0.285809	−0.0287249
$100$	0	0
$101$	10.7765	1.07230	0.536152	−	0.844121i	$-0.319877\pi$
$101$	10.7765	1.07230	0.536152	+	0.844121i	$0.319877\pi$
$102$	6.49242	0.642846
$103$	17.9161	1.76533	0.882664	−	0.470004i	$-0.155748\pi$
$103$	17.9161	1.76533	0.882664	+	0.470004i	$0.155748\pi$
$104$	−2.80530	−0.275082
$105$	0	0
$106$	13.9158	1.35162
$107$	−3.81291	−0.368608	−0.184304	−	0.982869i	$-0.559003\pi$
$107$	−3.81291	−0.368608	−0.184304	+	0.982869i	$0.559003\pi$
$108$	−1.00000	−0.0962250
$109$	7.96590	0.762995	0.381497	−	0.924370i	$-0.375409\pi$
$109$	7.96590	0.762995	0.381497	+	0.924370i	$0.375409\pi$
$110$	0	0
$111$	4.53906	0.430829
$112$	−2.61995	−0.247562
$113$	−5.25015	−0.493893	−0.246946	−	0.969029i	$-0.579427\pi$
$113$	−5.25015	−0.493893	−0.246946	+	0.969029i	$0.579427\pi$
$114$	−0.443312	−0.0415200
$115$	0	0
$116$	7.25636	0.673736
$117$	2.80530	0.259350
$118$	3.74685	0.344926
$119$	−17.0098	−1.55929
$120$	0	0
$121$	−10.9183	−0.992574
$122$	−13.0189	−1.17868
$123$	−5.32407	−0.480055
$124$	3.62674	0.325691
$125$	0	0
$126$	2.61995	0.233404
$127$	18.6990	1.65927	0.829636	−	0.558305i	$-0.188548\pi$
$127$	18.6990	1.65927	0.829636	+	0.558305i	$0.188548\pi$
$128$	−1.00000	−0.0883883
$129$	8.05390	0.709107
$130$	0	0
$131$	4.55143	0.397661	0.198830	−	0.980034i	$-0.436286\pi$
$131$	4.55143	0.397661	0.198830	+	0.980034i	$0.436286\pi$
$132$	0.285809	0.0248765
$133$	1.16146	0.100711
$134$	−10.2325	−0.883956
$135$	0	0
$136$	−6.49242	−0.556721
$137$	15.5439	1.32801	0.664004	−	0.747729i	$-0.268856\pi$
$137$	15.5439	1.32801	0.664004	+	0.747729i	$0.268856\pi$
$138$	−6.52498	−0.555443
$139$	−9.00095	−0.763451	−0.381725	−	0.924276i	$-0.624670\pi$
$139$	−9.00095	−0.763451	−0.381725	+	0.924276i	$0.624670\pi$
$140$	0	0
$141$	8.99646	0.757638
$142$	4.81783	0.404304
$143$	−0.801778	−0.0670481
$144$	1.00000	0.0833333
$145$	0	0
$146$	−11.7867	−0.975476
$147$	0.135854	0.0112050
$148$	−4.53906	−0.373109
$149$	−7.87441	−0.645097	−0.322548	−	0.946553i	$-0.604540\pi$
$149$	−7.87441	−0.645097	−0.322548	+	0.946553i	$0.604540\pi$
$150$	0	0
$151$	17.7660	1.44578	0.722888	−	0.690965i	$-0.242814\pi$
$151$	17.7660	1.44578	0.722888	+	0.690965i	$0.242814\pi$
$152$	0.443312	0.0359573
$153$	6.49242	0.524881
$154$	−0.748806	−0.0603405
$155$	0	0
$156$	−2.80530	−0.224603
$157$	20.1191	1.60568	0.802840	−	0.596194i	$-0.203321\pi$
$157$	20.1191	1.60568	0.802840	+	0.596194i	$0.203321\pi$
$158$	7.81462	0.621698
$159$	13.9158	1.10359
$160$	0	0
$161$	17.0951	1.34729
$162$	−1.00000	−0.0785674
$163$	−2.11433	−0.165607	−0.0828034	−	0.996566i	$-0.526387\pi$
$163$	−2.11433	−0.165607	−0.0828034	+	0.996566i	$0.526387\pi$
$164$	5.32407	0.415740
$165$	0	0
$166$	−5.16187	−0.400639
$167$	19.4759	1.50709	0.753544	−	0.657398i	$-0.228343\pi$
$167$	19.4759	1.50709	0.753544	+	0.657398i	$0.228343\pi$
$168$	−2.61995	−0.202134
$169$	−5.13032	−0.394640
$170$	0	0
$171$	−0.443312	−0.0339009
$172$	−8.05390	−0.614104
$173$	3.31078	0.251714	0.125857	−	0.992048i	$-0.459832\pi$
$173$	3.31078	0.251714	0.125857	+	0.992048i	$0.459832\pi$
$174$	7.25636	0.550103
$175$	0	0
$176$	−0.285809	−0.0215437
$177$	3.74685	0.281631
$178$	6.16560	0.462131
$179$	−12.4603	−0.931326	−0.465663	−	0.884962i	$-0.654184\pi$
$179$	−12.4603	−0.931326	−0.465663	+	0.884962i	$0.654184\pi$
$180$	0	0
$181$	6.05748	0.450249	0.225124	−	0.974330i	$-0.427721\pi$
$181$	6.05748	0.450249	0.225124	+	0.974330i	$0.427721\pi$
$182$	7.34974	0.544799
$183$	−13.0189	−0.962388
$184$	6.52498	0.481028
$185$	0	0
$186$	3.62674	0.265925
$187$	−1.85559	−0.135694
$188$	−8.99646	−0.656134
$189$	2.61995	0.190573
$190$	0	0
$191$	23.3684	1.69088	0.845439	−	0.534072i	$-0.179339\pi$
$191$	23.3684	1.69088	0.845439	+	0.534072i	$0.179339\pi$
$192$	−1.00000	−0.0721688
$193$	2.13291	0.153530	0.0767650	−	0.997049i	$-0.475541\pi$
$193$	2.13291	0.153530	0.0767650	+	0.997049i	$0.475541\pi$
$194$	5.45040	0.391316
$195$	0	0
$196$	−0.135854	−0.00970384
$197$	−16.4563	−1.17246	−0.586230	−	0.810145i	$-0.699389\pi$
$197$	−16.4563	−1.17246	−0.586230	+	0.810145i	$0.699389\pi$
$198$	0.285809	0.0203116
$199$	1.66552	0.118065	0.0590327	−	0.998256i	$-0.481198\pi$
$199$	1.66552	0.118065	0.0590327	+	0.998256i	$0.481198\pi$
$200$	0	0
$201$	−10.2325	−0.721747
$202$	−10.7765	−0.758233
$203$	−19.0113	−1.33433
$204$	−6.49242	−0.454561
$205$	0	0
$206$	−17.9161	−1.24828
$207$	−6.52498	−0.453517
$208$	2.80530	0.194512
$209$	0.126703	0.00876420
$210$	0	0
$211$	−11.7638	−0.809855	−0.404927	−	0.914349i	$-0.632703\pi$
$211$	−11.7638	−0.809855	−0.404927	+	0.914349i	$0.632703\pi$
$212$	−13.9158	−0.955738
$213$	4.81783	0.330113
$214$	3.81291	0.260645
$215$	0	0
$216$	1.00000	0.0680414
$217$	−9.50188	−0.645029
$218$	−7.96590	−0.539519
$219$	−11.7867	−0.796472
$220$	0	0
$221$	18.2132	1.22515
$222$	−4.53906	−0.304642
$223$	−9.52156	−0.637611	−0.318805	−	0.947820i	$-0.603282\pi$
$223$	−9.52156	−0.637611	−0.318805	+	0.947820i	$0.603282\pi$
$224$	2.61995	0.175053
$225$	0	0
$226$	5.25015	0.349235
$227$	16.3173	1.08302	0.541509	−	0.840695i	$-0.317853\pi$
$227$	16.3173	1.08302	0.541509	+	0.840695i	$0.317853\pi$
$228$	0.443312	0.0293591
$229$	−0.533684	−0.0352668	−0.0176334	−	0.999845i	$-0.505613\pi$
$229$	−0.533684	−0.0352668	−0.0176334	+	0.999845i	$0.505613\pi$
$230$	0	0
$231$	−0.748806	−0.0492678
$232$	−7.25636	−0.476403
$233$	−10.9418	−0.716819	−0.358410	−	0.933564i	$-0.616681\pi$
$233$	−10.9418	−0.716819	−0.358410	+	0.933564i	$0.616681\pi$
$234$	−2.80530	−0.183388
$235$	0	0
$236$	−3.74685	−0.243899
$237$	7.81462	0.507614
$238$	17.0098	1.10258
$239$	6.03821	0.390579	0.195290	−	0.980746i	$-0.437435\pi$
$239$	6.03821	0.390579	0.195290	+	0.980746i	$0.437435\pi$
$240$	0	0
$241$	2.20701	0.142166	0.0710830	−	0.997470i	$-0.477354\pi$
$241$	2.20701	0.142166	0.0710830	+	0.997470i	$0.477354\pi$
$242$	10.9183	0.701856
$243$	−1.00000	−0.0641500
$244$	13.0189	0.833453
$245$	0	0
$246$	5.32407	0.339450
$247$	−1.24362	−0.0791297
$248$	−3.62674	−0.230298
$249$	−5.16187	−0.327120
$250$	0	0
$251$	−0.151651	−0.00957211	−0.00478606	−	0.999989i	$-0.501523\pi$
$251$	−0.151651	−0.00957211	−0.00478606	+	0.999989i	$0.501523\pi$
$252$	−2.61995	−0.165041
$253$	1.86490	0.117245
$254$	−18.6990	−1.17328
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.4462	−0.651614	−0.325807	−	0.945436i	$-0.605636\pi$
$257$	−10.4462	−0.651614	−0.325807	+	0.945436i	$0.605636\pi$
$258$	−8.05390	−0.501414
$259$	11.8921	0.738941
$260$	0	0
$261$	7.25636	0.449157
$262$	−4.55143	−0.281188
$263$	−21.9602	−1.35413	−0.677063	−	0.735925i	$-0.736747\pi$
$263$	−21.9602	−1.35413	−0.677063	+	0.735925i	$0.736747\pi$
$264$	−0.285809	−0.0175903
$265$	0	0
$266$	−1.16146	−0.0712134
$267$	6.16560	0.377329
$268$	10.2325	0.625051
$269$	−14.1299	−0.861516	−0.430758	−	0.902468i	$-0.641754\pi$
$269$	−14.1299	−0.861516	−0.430758	+	0.902468i	$0.641754\pi$
$270$	0	0
$271$	27.5290	1.67227	0.836134	−	0.548526i	$-0.184811\pi$
$271$	27.5290	1.67227	0.836134	+	0.548526i	$0.184811\pi$
$272$	6.49242	0.393661
$273$	7.34974	0.444826
$274$	−15.5439	−0.939043
$275$	0	0
$276$	6.52498	0.392758
$277$	23.5959	1.41774	0.708871	−	0.705338i	$-0.249205\pi$
$277$	23.5959	1.41774	0.708871	+	0.705338i	$0.249205\pi$
$278$	9.00095	0.539841
$279$	3.62674	0.217127
$280$	0	0
$281$	7.17134	0.427806	0.213903	−	0.976855i	$-0.431382\pi$
$281$	7.17134	0.427806	0.213903	+	0.976855i	$0.431382\pi$
$282$	−8.99646	−0.535731
$283$	23.0321	1.36912	0.684559	−	0.728957i	$-0.259995\pi$
$283$	23.0321	1.36912	0.684559	+	0.728957i	$0.259995\pi$
$284$	−4.81783	−0.285886
$285$	0	0
$286$	0.801778	0.0474101
$287$	−13.9488	−0.823372
$288$	−1.00000	−0.0589256
$289$	25.1516	1.47950
$290$	0	0
$291$	5.45040	0.319508
$292$	11.7867	0.689765
$293$	16.3851	0.957230	0.478615	−	0.878025i	$-0.341139\pi$
$293$	16.3851	0.957230	0.478615	+	0.878025i	$0.341139\pi$
$294$	−0.135854	−0.00792315
$295$	0	0
$296$	4.53906	0.263828
$297$	0.285809	0.0165843
$298$	7.87441	0.456152
$299$	−18.3045	−1.05858
$300$	0	0
$301$	21.1008	1.21623
$302$	−17.7660	−1.02232
$303$	−10.7765	−0.619095
$304$	−0.443312	−0.0254257
$305$	0	0
$306$	−6.49242	−0.371147
$307$	18.6027	1.06171	0.530857	−	0.847461i	$-0.321870\pi$
$307$	18.6027	1.06171	0.530857	+	0.847461i	$0.321870\pi$
$308$	0.748806	0.0426672
$309$	−17.9161	−1.01921
$310$	0	0
$311$	−13.9175	−0.789186	−0.394593	−	0.918856i	$-0.629114\pi$
$311$	−13.9175	−0.789186	−0.394593	+	0.918856i	$0.629114\pi$
$312$	2.80530	0.158819
$313$	15.3804	0.869354	0.434677	−	0.900586i	$-0.356863\pi$
$313$	15.3804	0.869354	0.434677	+	0.900586i	$0.356863\pi$
$314$	−20.1191	−1.13539
$315$	0	0
$316$	−7.81462	−0.439607
$317$	−6.42445	−0.360833	−0.180416	−	0.983590i	$-0.557745\pi$
$317$	−6.42445	−0.360833	−0.180416	+	0.983590i	$0.557745\pi$
$318$	−13.9158	−0.780357
$319$	−2.07393	−0.116118
$320$	0	0
$321$	3.81291	0.212816
$322$	−17.0951	−0.952674
$323$	−2.87817	−0.160146
$324$	1.00000	0.0555556
$325$	0	0
$326$	2.11433	0.117102
$327$	−7.96590	−0.440515
$328$	−5.32407	−0.293973
$329$	23.5703	1.29947
$330$	0	0
$331$	18.1676	0.998580	0.499290	−	0.866435i	$-0.333594\pi$
$331$	18.1676	0.998580	0.499290	+	0.866435i	$0.333594\pi$
$332$	5.16187	0.283294
$333$	−4.53906	−0.248739
$334$	−19.4759	−1.06567
$335$	0	0
$336$	2.61995	0.142930
$337$	2.38677	0.130016	0.0650078	−	0.997885i	$-0.479293\pi$
$337$	2.38677	0.130016	0.0650078	+	0.997885i	$0.479293\pi$
$338$	5.13032	0.279053
$339$	5.25015	0.285149
$340$	0	0
$341$	−1.03655	−0.0561325
$342$	0.443312	0.0239716
$343$	18.6956	1.00947
$344$	8.05390	0.434237
$345$	0	0
$346$	−3.31078	−0.177989
$347$	25.9386	1.39246	0.696229	−	0.717820i	$-0.254860\pi$
$347$	25.9386	1.39246	0.696229	+	0.717820i	$0.254860\pi$
$348$	−7.25636	−0.388982
$349$	0.246770	0.0132093	0.00660463	−	0.999978i	$-0.497898\pi$
$349$	0.246770	0.0132093	0.00660463	+	0.999978i	$0.497898\pi$
$350$	0	0
$351$	−2.80530	−0.149736
$352$	0.285809	0.0152337
$353$	19.2023	1.02204	0.511018	−	0.859570i	$-0.329269\pi$
$353$	19.2023	1.02204	0.511018	+	0.859570i	$0.329269\pi$
$354$	−3.74685	−0.199143
$355$	0	0
$356$	−6.16560	−0.326776
$357$	17.0098	0.900256
$358$	12.4603	0.658547
$359$	15.4421	0.815005	0.407503	−	0.913204i	$-0.366400\pi$
$359$	15.4421	0.815005	0.407503	+	0.913204i	$0.366400\pi$
$360$	0	0
$361$	−18.8035	−0.989657
$362$	−6.05748	−0.318374
$363$	10.9183	0.573063
$364$	−7.34974	−0.385231
$365$	0	0
$366$	13.0189	0.680511
$367$	24.9709	1.30347	0.651735	−	0.758447i	$-0.274041\pi$
$367$	24.9709	1.30347	0.651735	+	0.758447i	$0.274041\pi$
$368$	−6.52498	−0.340138
$369$	5.32407	0.277160
$370$	0	0
$371$	36.4586	1.89284
$372$	−3.62674	−0.188038
$373$	18.6645	0.966409	0.483204	−	0.875508i	$-0.339473\pi$
$373$	18.6645	0.966409	0.483204	+	0.875508i	$0.339473\pi$
$374$	1.85559	0.0959504
$375$	0	0
$376$	8.99646	0.463957
$377$	20.3562	1.04840
$378$	−2.61995	−0.134756
$379$	29.2481	1.50237	0.751187	−	0.660089i	$-0.229482\pi$
$379$	29.2481	1.50237	0.751187	+	0.660089i	$0.229482\pi$
$380$	0	0
$381$	−18.6990	−0.957981
$382$	−23.3684	−1.19563
$383$	10.3716	0.529964	0.264982	−	0.964253i	$-0.414634\pi$
$383$	10.3716	0.529964	0.264982	+	0.964253i	$0.414634\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.13291	−0.108562
$387$	−8.05390	−0.409403
$388$	−5.45040	−0.276702
$389$	−9.15259	−0.464055	−0.232027	−	0.972709i	$-0.574536\pi$
$389$	−9.15259	−0.464055	−0.232027	+	0.972709i	$0.574536\pi$
$390$	0	0
$391$	−42.3629	−2.14239
$392$	0.135854	0.00686165
$393$	−4.55143	−0.229589
$394$	16.4563	0.829054
$395$	0	0
$396$	−0.285809	−0.0143624
$397$	16.1451	0.810297	0.405149	−	0.914251i	$-0.367220\pi$
$397$	16.1451	0.810297	0.405149	+	0.914251i	$0.367220\pi$
$398$	−1.66552	−0.0834849
$399$	−1.16146	−0.0581455
$400$	0	0
$401$	−25.9388	−1.29532	−0.647662	−	0.761928i	$-0.724253\pi$
$401$	−25.9388	−1.29532	−0.647662	+	0.761928i	$0.724253\pi$
$402$	10.2325	0.510352
$403$	10.1741	0.506807
$404$	10.7765	0.536152
$405$	0	0
$406$	19.0113	0.943515
$407$	1.29730	0.0643050
$408$	6.49242	0.321423
$409$	17.0650	0.843811	0.421906	−	0.906640i	$-0.361361\pi$
$409$	17.0650	0.843811	0.421906	+	0.906640i	$0.361361\pi$
$410$	0	0
$411$	−15.5439	−0.766725
$412$	17.9161	0.882664
$413$	9.81657	0.483042
$414$	6.52498	0.320685
$415$	0	0
$416$	−2.80530	−0.137541
$417$	9.00095	0.440778
$418$	−0.126703	−0.00619722
$419$	13.8128	0.674802	0.337401	−	0.941361i	$-0.390452\pi$
$419$	13.8128	0.674802	0.337401	+	0.941361i	$0.390452\pi$
$420$	0	0
$421$	−2.08499	−0.101616	−0.0508082	−	0.998708i	$-0.516180\pi$
$421$	−2.08499	−0.101616	−0.0508082	+	0.998708i	$0.516180\pi$
$422$	11.7638	0.572654
$423$	−8.99646	−0.437423
$424$	13.9158	0.675809
$425$	0	0
$426$	−4.81783	−0.233425
$427$	−34.1090	−1.65065
$428$	−3.81291	−0.184304
$429$	0.801778	0.0387102
$430$	0	0
$431$	9.33614	0.449706	0.224853	−	0.974393i	$-0.427810\pi$
$431$	9.33614	0.449706	0.224853	+	0.974393i	$0.427810\pi$
$432$	−1.00000	−0.0481125
$433$	−11.7140	−0.562941	−0.281470	−	0.959570i	$-0.590822\pi$
$433$	−11.7140	−0.562941	−0.281470	+	0.959570i	$0.590822\pi$
$434$	9.50188	0.456105
$435$	0	0
$436$	7.96590	0.381497
$437$	2.89260	0.138372
$438$	11.7867	0.563191
$439$	3.08200	0.147096	0.0735480	−	0.997292i	$-0.476568\pi$
$439$	3.08200	0.147096	0.0735480	+	0.997292i	$0.476568\pi$
$440$	0	0
$441$	−0.135854	−0.00646923
$442$	−18.2132	−0.866312
$443$	−10.6355	−0.505309	−0.252654	−	0.967557i	$-0.581304\pi$
$443$	−10.6355	−0.505309	−0.252654	+	0.967557i	$0.581304\pi$
$444$	4.53906	0.215414
$445$	0	0
$446$	9.52156	0.450859
$447$	7.87441	0.372447
$448$	−2.61995	−0.123781
$449$	−22.8276	−1.07730	−0.538651	−	0.842529i	$-0.681066\pi$
$449$	−22.8276	−1.07730	−0.538651	+	0.842529i	$0.681066\pi$
$450$	0	0
$451$	−1.52167	−0.0716525
$452$	−5.25015	−0.246946
$453$	−17.7660	−0.834719
$454$	−16.3173	−0.765809
$455$	0	0
$456$	−0.443312	−0.0207600
$457$	−25.7628	−1.20513	−0.602567	−	0.798068i	$-0.705855\pi$
$457$	−25.7628	−1.20513	−0.602567	+	0.798068i	$0.705855\pi$
$458$	0.533684	0.0249374
$459$	−6.49242	−0.303040
$460$	0	0
$461$	32.4218	1.51003	0.755016	−	0.655706i	$-0.227629\pi$
$461$	32.4218	1.51003	0.755016	+	0.655706i	$0.227629\pi$
$462$	0.748806	0.0348376
$463$	18.7775	0.872663	0.436331	−	0.899786i	$-0.356278\pi$
$463$	18.7775	0.872663	0.436331	+	0.899786i	$0.356278\pi$
$464$	7.25636	0.336868
$465$	0	0
$466$	10.9418	0.506868
$467$	1.96212	0.0907960	0.0453980	−	0.998969i	$-0.485544\pi$
$467$	1.96212	0.0907960	0.0453980	+	0.998969i	$0.485544\pi$
$468$	2.80530	0.129675
$469$	−26.8087	−1.23791
$470$	0	0
$471$	−20.1191	−0.927040
$472$	3.74685	0.172463
$473$	2.30188	0.105840
$474$	−7.81462	−0.358938
$475$	0	0
$476$	−17.0098	−0.779645
$477$	−13.9158	−0.637159
$478$	−6.03821	−0.276181
$479$	−6.74378	−0.308131	−0.154066	−	0.988061i	$-0.549237\pi$
$479$	−6.74378	−0.308131	−0.154066	+	0.988061i	$0.549237\pi$
$480$	0	0
$481$	−12.7334	−0.580594
$482$	−2.20701	−0.100526
$483$	−17.0951	−0.777855
$484$	−10.9183	−0.496287
$485$	0	0
$486$	1.00000	0.0453609
$487$	−1.16407	−0.0527489	−0.0263745	−	0.999652i	$-0.508396\pi$
$487$	−1.16407	−0.0527489	−0.0263745	+	0.999652i	$0.508396\pi$
$488$	−13.0189	−0.589340
$489$	2.11433	0.0956132
$490$	0	0
$491$	−8.38720	−0.378509	−0.189255	−	0.981928i	$-0.560607\pi$
$491$	−8.38720	−0.378509	−0.189255	+	0.981928i	$0.560607\pi$
$492$	−5.32407	−0.240028
$493$	47.1113	2.12179
$494$	1.24362	0.0559531
$495$	0	0
$496$	3.62674	0.162845
$497$	12.6225	0.566196
$498$	5.16187	0.231309
$499$	17.2590	0.772620	0.386310	−	0.922369i	$-0.373749\pi$
$499$	17.2590	0.772620	0.386310	+	0.922369i	$0.373749\pi$
$500$	0	0
$501$	−19.4759	−0.870117
$502$	0.151651	0.00676850
$503$	−8.82395	−0.393440	−0.196720	−	0.980460i	$-0.563029\pi$
$503$	−8.82395	−0.393440	−0.196720	+	0.980460i	$0.563029\pi$
$504$	2.61995	0.116702
$505$	0	0
$506$	−1.86490	−0.0829048
$507$	5.13032	0.227845
$508$	18.6990	0.829636
$509$	−13.6355	−0.604384	−0.302192	−	0.953247i	$-0.597718\pi$
$509$	−13.6355	−0.604384	−0.302192	+	0.953247i	$0.597718\pi$
$510$	0	0
$511$	−30.8806	−1.36608
$512$	−1.00000	−0.0441942
$513$	0.443312	0.0195727
$514$	10.4462	0.460761
$515$	0	0
$516$	8.05390	0.354553
$517$	2.57127	0.113084
$518$	−11.8921	−0.522510
$519$	−3.31078	−0.145327
$520$	0	0
$521$	−13.4897	−0.590995	−0.295497	−	0.955344i	$-0.595485\pi$
$521$	−13.4897	−0.590995	−0.295497	+	0.955344i	$0.595485\pi$
$522$	−7.25636	−0.317602
$523$	10.1795	0.445117	0.222559	−	0.974919i	$-0.428559\pi$
$523$	10.1795	0.445117	0.222559	+	0.974919i	$0.428559\pi$
$524$	4.55143	0.198830
$525$	0	0
$526$	21.9602	0.957511
$527$	23.5463	1.02569
$528$	0.285809	0.0124382
$529$	19.5754	0.851103
$530$	0	0
$531$	−3.74685	−0.162599
$532$	1.16146	0.0503555
$533$	14.9356	0.646932
$534$	−6.16560	−0.266812
$535$	0	0
$536$	−10.2325	−0.441978
$537$	12.4603	0.537701
$538$	14.1299	0.609184
$539$	0.0388282	0.00167245
$540$	0	0
$541$	−11.6124	−0.499254	−0.249627	−	0.968342i	$-0.580308\pi$
$541$	−11.6124	−0.499254	−0.249627	+	0.968342i	$0.580308\pi$
$542$	−27.5290	−1.18247
$543$	−6.05748	−0.259951
$544$	−6.49242	−0.278360
$545$	0	0
$546$	−7.34974	−0.314540
$547$	−20.1352	−0.860918	−0.430459	−	0.902610i	$-0.641648\pi$
$547$	−20.1352	−0.860918	−0.430459	+	0.902610i	$0.641648\pi$
$548$	15.5439	0.664004
$549$	13.0189	0.555635
$550$	0	0
$551$	−3.21683	−0.137042
$552$	−6.52498	−0.277722
$553$	20.4739	0.870640
$554$	−23.5959	−1.00250
$555$	0	0
$556$	−9.00095	−0.381725
$557$	8.99374	0.381077	0.190539	−	0.981680i	$-0.438977\pi$
$557$	8.99374	0.381077	0.190539	+	0.981680i	$0.438977\pi$
$558$	−3.62674	−0.153532
$559$	−22.5936	−0.955606
$560$	0	0
$561$	1.85559	0.0783432
$562$	−7.17134	−0.302505
$563$	−0.465203	−0.0196060	−0.00980299	−	0.999952i	$-0.503120\pi$
$563$	−0.465203	−0.0196060	−0.00980299	+	0.999952i	$0.503120\pi$
$564$	8.99646	0.378819
$565$	0	0
$566$	−23.0321	−0.968113
$567$	−2.61995	−0.110028
$568$	4.81783	0.202152
$569$	−5.10312	−0.213934	−0.106967	−	0.994263i	$-0.534114\pi$
$569$	−5.10312	−0.213934	−0.106967	+	0.994263i	$0.534114\pi$
$570$	0	0
$571$	−43.2082	−1.80821	−0.904103	−	0.427315i	$-0.859460\pi$
$571$	−43.2082	−1.80821	−0.904103	+	0.427315i	$0.859460\pi$
$572$	−0.801778	−0.0335240
$573$	−23.3684	−0.976229
$574$	13.9488	0.582212
$575$	0	0
$576$	1.00000	0.0416667
$577$	−15.4428	−0.642893	−0.321446	−	0.946928i	$-0.604169\pi$
$577$	−15.4428	−0.642893	−0.321446	+	0.946928i	$0.604169\pi$
$578$	−25.1516	−1.04617
$579$	−2.13291	−0.0886406
$580$	0	0
$581$	−13.5238	−0.561063
$582$	−5.45040	−0.225927
$583$	3.97725	0.164721
$584$	−11.7867	−0.487738
$585$	0	0
$586$	−16.3851	−0.676864
$587$	28.5643	1.17897	0.589487	−	0.807778i	$-0.299330\pi$
$587$	28.5643	1.17897	0.589487	+	0.807778i	$0.299330\pi$
$588$	0.135854	0.00560252
$589$	−1.60778	−0.0662473
$590$	0	0
$591$	16.4563	0.676920
$592$	−4.53906	−0.186554
$593$	−31.6123	−1.29816	−0.649080	−	0.760720i	$-0.724846\pi$
$593$	−31.6123	−1.29816	−0.649080	+	0.760720i	$0.724846\pi$
$594$	−0.285809	−0.0117269
$595$	0	0
$596$	−7.87441	−0.322548
$597$	−1.66552	−0.0681651
$598$	18.3045	0.748526
$599$	36.7361	1.50099	0.750497	−	0.660873i	$-0.229814\pi$
$599$	36.7361	1.50099	0.750497	+	0.660873i	$0.229814\pi$
$600$	0	0
$601$	45.3145	1.84842	0.924209	−	0.381887i	$-0.124726\pi$
$601$	45.3145	1.84842	0.924209	+	0.381887i	$0.124726\pi$
$602$	−21.1008	−0.860006
$603$	10.2325	0.416701
$604$	17.7660	0.722888
$605$	0	0
$606$	10.7765	0.437766
$607$	3.15730	0.128151	0.0640754	−	0.997945i	$-0.479590\pi$
$607$	3.15730	0.128151	0.0640754	+	0.997945i	$0.479590\pi$
$608$	0.443312	0.0179787
$609$	19.0113	0.770377
$610$	0	0
$611$	−25.2377	−1.02101
$612$	6.49242	0.262441
$613$	20.9594	0.846544	0.423272	−	0.906003i	$-0.360881\pi$
$613$	20.9594	0.846544	0.423272	+	0.906003i	$0.360881\pi$
$614$	−18.6027	−0.750746
$615$	0	0
$616$	−0.748806	−0.0301702
$617$	−36.7400	−1.47910	−0.739548	−	0.673103i	$-0.764961\pi$
$617$	−36.7400	−1.47910	−0.739548	+	0.673103i	$0.764961\pi$
$618$	17.9161	0.720692
$619$	−11.6388	−0.467801	−0.233901	−	0.972261i	$-0.575149\pi$
$619$	−11.6388	−0.467801	−0.233901	+	0.972261i	$0.575149\pi$
$620$	0	0
$621$	6.52498	0.261838
$622$	13.9175	0.558039
$623$	16.1536	0.647179
$624$	−2.80530	−0.112302
$625$	0	0
$626$	−15.3804	−0.614726
$627$	−0.126703	−0.00506001
$628$	20.1191	0.802840
$629$	−29.4695	−1.17503
$630$	0	0
$631$	4.62189	0.183995	0.0919973	−	0.995759i	$-0.470675\pi$
$631$	4.62189	0.183995	0.0919973	+	0.995759i	$0.470675\pi$
$632$	7.81462	0.310849
$633$	11.7638	0.467570
$634$	6.42445	0.255147
$635$	0	0
$636$	13.9158	0.551796
$637$	−0.381110	−0.0151001
$638$	2.07393	0.0821077
$639$	−4.81783	−0.190591
$640$	0	0
$641$	25.4057	1.00346	0.501732	−	0.865023i	$-0.332696\pi$
$641$	25.4057	1.00346	0.501732	+	0.865023i	$0.332696\pi$
$642$	−3.81291	−0.150483
$643$	−41.3338	−1.63005	−0.815023	−	0.579429i	$-0.803276\pi$
$643$	−41.3338	−1.63005	−0.815023	+	0.579429i	$0.803276\pi$
$644$	17.0951	0.673643
$645$	0	0
$646$	2.87817	0.113240
$647$	−5.45174	−0.214330	−0.107165	−	0.994241i	$-0.534177\pi$
$647$	−5.45174	−0.214330	−0.107165	+	0.994241i	$0.534177\pi$
$648$	−1.00000	−0.0392837
$649$	1.07088	0.0420358
$650$	0	0
$651$	9.50188	0.372408
$652$	−2.11433	−0.0828034
$653$	27.4805	1.07540	0.537698	−	0.843138i	$-0.319294\pi$
$653$	27.4805	1.07540	0.537698	+	0.843138i	$0.319294\pi$
$654$	7.96590	0.311491
$655$	0	0
$656$	5.32407	0.207870
$657$	11.7867	0.459844
$658$	−23.5703	−0.918865
$659$	42.2417	1.64550	0.822752	−	0.568400i	$-0.192437\pi$
$659$	42.2417	1.64550	0.822752	+	0.568400i	$0.192437\pi$
$660$	0	0
$661$	6.35815	0.247303	0.123652	−	0.992326i	$-0.460539\pi$
$661$	6.35815	0.247303	0.123652	+	0.992326i	$0.460539\pi$
$662$	−18.1676	−0.706103
$663$	−18.2132	−0.707341
$664$	−5.16187	−0.200319
$665$	0	0
$666$	4.53906	0.175885
$667$	−47.3476	−1.83331
$668$	19.4759	0.753544
$669$	9.52156	0.368125
$670$	0	0
$671$	−3.72093	−0.143645
$672$	−2.61995	−0.101067
$673$	20.3747	0.785386	0.392693	−	0.919670i	$-0.371543\pi$
$673$	20.3747	0.785386	0.392693	+	0.919670i	$0.371543\pi$
$674$	−2.38677	−0.0919349
$675$	0	0
$676$	−5.13032	−0.197320
$677$	−6.29870	−0.242079	−0.121039	−	0.992648i	$-0.538623\pi$
$677$	−6.29870	−0.242079	−0.121039	+	0.992648i	$0.538623\pi$
$678$	−5.25015	−0.201631
$679$	14.2798	0.548008
$680$	0	0
$681$	−16.3173	−0.625280
$682$	1.03655	0.0396917
$683$	−19.7006	−0.753822	−0.376911	−	0.926249i	$-0.623014\pi$
$683$	−19.7006	−0.753822	−0.376911	+	0.926249i	$0.623014\pi$
$684$	−0.443312	−0.0169505
$685$	0	0
$686$	−18.6956	−0.713801
$687$	0.533684	0.0203613
$688$	−8.05390	−0.307052
$689$	−39.0378	−1.48722
$690$	0	0
$691$	−48.7783	−1.85561	−0.927806	−	0.373062i	$-0.878308\pi$
$691$	−48.7783	−1.85561	−0.927806	+	0.373062i	$0.878308\pi$
$692$	3.31078	0.125857
$693$	0.748806	0.0284448
$694$	−25.9386	−0.984617
$695$	0	0
$696$	7.25636	0.275051
$697$	34.5661	1.30929
$698$	−0.246770	−0.00934036
$699$	10.9418	0.413856
$700$	0	0
$701$	−43.3213	−1.63622	−0.818111	−	0.575061i	$-0.804978\pi$
$701$	−43.3213	−1.63622	−0.818111	+	0.575061i	$0.804978\pi$
$702$	2.80530	0.105879
$703$	2.01222	0.0758923
$704$	−0.285809	−0.0107718
$705$	0	0
$706$	−19.2023	−0.722688
$707$	−28.2340	−1.06185
$708$	3.74685	0.140815
$709$	16.0229	0.601754	0.300877	−	0.953663i	$-0.402721\pi$
$709$	16.0229	0.601754	0.300877	+	0.953663i	$0.402721\pi$
$710$	0	0
$711$	−7.81462	−0.293071
$712$	6.16560	0.231066
$713$	−23.6644	−0.886238
$714$	−17.0098	−0.636577
$715$	0	0
$716$	−12.4603	−0.465663
$717$	−6.03821	−0.225501
$718$	−15.4421	−0.576296
$719$	27.7459	1.03475	0.517375	−	0.855759i	$-0.326909\pi$
$719$	27.7459	1.03475	0.517375	+	0.855759i	$0.326909\pi$
$720$	0	0
$721$	−46.9394	−1.74811
$722$	18.8035	0.699793
$723$	−2.20701	−0.0820795
$724$	6.05748	0.225124
$725$	0	0
$726$	−10.9183	−0.405217
$727$	−45.4681	−1.68632	−0.843159	−	0.537664i	$-0.819307\pi$
$727$	−45.4681	−1.68632	−0.843159	+	0.537664i	$0.819307\pi$
$728$	7.34974	0.272399
$729$	1.00000	0.0370370
$730$	0	0
$731$	−52.2894	−1.93399
$732$	−13.0189	−0.481194
$733$	31.4313	1.16094	0.580472	−	0.814281i	$-0.302868\pi$
$733$	31.4313	1.16094	0.580472	+	0.814281i	$0.302868\pi$
$734$	−24.9709	−0.921692
$735$	0	0
$736$	6.52498	0.240514
$737$	−2.92455	−0.107727
$738$	−5.32407	−0.195982
$739$	−17.0338	−0.626597	−0.313299	−	0.949655i	$-0.601434\pi$
$739$	−17.0338	−0.626597	−0.313299	+	0.949655i	$0.601434\pi$
$740$	0	0
$741$	1.24362	0.0456856
$742$	−36.4586	−1.33844
$743$	−1.44429	−0.0529858	−0.0264929	−	0.999649i	$-0.508434\pi$
$743$	−1.44429	−0.0529858	−0.0264929	+	0.999649i	$0.508434\pi$
$744$	3.62674	0.132963
$745$	0	0
$746$	−18.6645	−0.683354
$747$	5.16187	0.188863
$748$	−1.85559	−0.0678472
$749$	9.98963	0.365013
$750$	0	0
$751$	22.1167	0.807051	0.403526	−	0.914968i	$-0.367785\pi$
$751$	22.1167	0.807051	0.403526	+	0.914968i	$0.367785\pi$
$752$	−8.99646	−0.328067
$753$	0.151651	0.00552646
$754$	−20.3562	−0.741330
$755$	0	0
$756$	2.61995	0.0952867
$757$	18.0645	0.656565	0.328283	−	0.944580i	$-0.393530\pi$
$757$	18.0645	0.656565	0.328283	+	0.944580i	$0.393530\pi$
$758$	−29.2481	−1.06234
$759$	−1.86490	−0.0676915
$760$	0	0
$761$	14.3108	0.518765	0.259382	−	0.965775i	$-0.416481\pi$
$761$	14.3108	0.518765	0.259382	+	0.965775i	$0.416481\pi$
$762$	18.6990	0.677395
$763$	−20.8703	−0.755555
$764$	23.3684	0.845439
$765$	0	0
$766$	−10.3716	−0.374741
$767$	−10.5110	−0.379531
$768$	−1.00000	−0.0360844
$769$	34.4187	1.24117	0.620585	−	0.784140i	$-0.286895\pi$
$769$	34.4187	1.24117	0.620585	+	0.784140i	$0.286895\pi$
$770$	0	0
$771$	10.4462	0.376210
$772$	2.13291	0.0767650
$773$	−32.1629	−1.15682	−0.578410	−	0.815746i	$-0.696327\pi$
$773$	−32.1629	−1.15682	−0.578410	+	0.815746i	$0.696327\pi$
$774$	8.05390	0.289492
$775$	0	0
$776$	5.45040	0.195658
$777$	−11.8921	−0.426628
$778$	9.15259	0.328136
$779$	−2.36022	−0.0845638
$780$	0	0
$781$	1.37698	0.0492722
$782$	42.3629	1.51490
$783$	−7.25636	−0.259321
$784$	−0.135854	−0.00485192
$785$	0	0
$786$	4.55143	0.162344
$787$	31.5577	1.12491	0.562456	−	0.826827i	$-0.309857\pi$
$787$	31.5577	1.12491	0.562456	+	0.826827i	$0.309857\pi$
$788$	−16.4563	−0.586230
$789$	21.9602	0.781805
$790$	0	0
$791$	13.7551	0.489077
$792$	0.285809	0.0101558
$793$	36.5220	1.29693
$794$	−16.1451	−0.572967
$795$	0	0
$796$	1.66552	0.0590327
$797$	−27.2464	−0.965117	−0.482558	−	0.875864i	$-0.660292\pi$
$797$	−27.2464	−0.965117	−0.482558	+	0.875864i	$0.660292\pi$
$798$	1.16146	0.0411151
$799$	−58.4088	−2.06636
$800$	0	0
$801$	−6.16560	−0.217851
$802$	25.9388	0.915932
$803$	−3.36875	−0.118881
$804$	−10.2325	−0.360873
$805$	0	0
$806$	−10.1741	−0.358366
$807$	14.1299	0.497396
$808$	−10.7765	−0.379117
$809$	15.6915	0.551683	0.275842	−	0.961203i	$-0.411043\pi$
$809$	15.6915	0.551683	0.275842	+	0.961203i	$0.411043\pi$
$810$	0	0
$811$	−23.0255	−0.808536	−0.404268	−	0.914641i	$-0.632474\pi$
$811$	−23.0255	−0.808536	−0.404268	+	0.914641i	$0.632474\pi$
$812$	−19.0113	−0.667166
$813$	−27.5290	−0.965484
$814$	−1.29730	−0.0454705
$815$	0	0
$816$	−6.49242	−0.227280
$817$	3.57039	0.124912
$818$	−17.0650	−0.596665
$819$	−7.34974	−0.256821
$820$	0	0
$821$	−11.0991	−0.387363	−0.193681	−	0.981065i	$-0.562043\pi$
$821$	−11.0991	−0.387363	−0.193681	+	0.981065i	$0.562043\pi$
$822$	15.5439	0.542157
$823$	−44.2965	−1.54408	−0.772039	−	0.635575i	$-0.780763\pi$
$823$	−44.2965	−1.54408	−0.772039	+	0.635575i	$0.780763\pi$
$824$	−17.9161	−0.624138
$825$	0	0
$826$	−9.81657	−0.341562
$827$	10.3291	0.359179	0.179589	−	0.983742i	$-0.442523\pi$
$827$	10.3291	0.359179	0.179589	+	0.983742i	$0.442523\pi$
$828$	−6.52498	−0.226759
$829$	−48.2004	−1.67407	−0.837034	−	0.547151i	$-0.815712\pi$
$829$	−48.2004	−1.67407	−0.837034	+	0.547151i	$0.815712\pi$
$830$	0	0
$831$	−23.5959	−0.818534
$832$	2.80530	0.0972561
$833$	−0.882021	−0.0305602
$834$	−9.00095	−0.311677
$835$	0	0
$836$	0.126703	0.00438210
$837$	−3.62674	−0.125358
$838$	−13.8128	−0.477157
$839$	2.61715	0.0903542	0.0451771	−	0.998979i	$-0.485615\pi$
$839$	2.61715	0.0903542	0.0451771	+	0.998979i	$0.485615\pi$
$840$	0	0
$841$	23.6547	0.815680
$842$	2.08499	0.0718536
$843$	−7.17134	−0.246994
$844$	−11.7638	−0.404927
$845$	0	0
$846$	8.99646	0.309305
$847$	28.6055	0.982895
$848$	−13.9158	−0.477869
$849$	−23.0321	−0.790461
$850$	0	0
$851$	29.6173	1.01527
$852$	4.81783	0.165056
$853$	−15.2058	−0.520638	−0.260319	−	0.965523i	$-0.583828\pi$
$853$	−15.2058	−0.520638	−0.260319	+	0.965523i	$0.583828\pi$
$854$	34.1090	1.16719
$855$	0	0
$856$	3.81291	0.130322
$857$	−26.1596	−0.893595	−0.446797	−	0.894635i	$-0.647436\pi$
$857$	−26.1596	−0.893595	−0.446797	+	0.894635i	$0.647436\pi$
$858$	−0.801778	−0.0273723
$859$	7.18205	0.245048	0.122524	−	0.992466i	$-0.460901\pi$
$859$	7.18205	0.245048	0.122524	+	0.992466i	$0.460901\pi$
$860$	0	0
$861$	13.9488	0.475374
$862$	−9.33614	−0.317990
$863$	−36.7109	−1.24965	−0.624827	−	0.780763i	$-0.714830\pi$
$863$	−36.7109	−1.24965	−0.624827	+	0.780763i	$0.714830\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	11.7140	0.398059
$867$	−25.1516	−0.854192
$868$	−9.50188	−0.322515
$869$	2.23349	0.0757659
$870$	0	0
$871$	28.7053	0.972641
$872$	−7.96590	−0.269759
$873$	−5.45040	−0.184468
$874$	−2.89260	−0.0978437
$875$	0	0
$876$	−11.7867	−0.398236
$877$	−18.0104	−0.608167	−0.304084	−	0.952645i	$-0.598350\pi$
$877$	−18.0104	−0.608167	−0.304084	+	0.952645i	$0.598350\pi$
$878$	−3.08200	−0.104013
$879$	−16.3851	−0.552657
$880$	0	0
$881$	−2.86351	−0.0964740	−0.0482370	−	0.998836i	$-0.515360\pi$
$881$	−2.86351	−0.0964740	−0.0482370	+	0.998836i	$0.515360\pi$
$882$	0.135854	0.00457444
$883$	−45.8301	−1.54231	−0.771153	−	0.636650i	$-0.780320\pi$
$883$	−45.8301	−1.54231	−0.771153	+	0.636650i	$0.780320\pi$
$884$	18.2132	0.612575
$885$	0	0
$886$	10.6355	0.357307
$887$	14.8534	0.498728	0.249364	−	0.968410i	$-0.419778\pi$
$887$	14.8534	0.498728	0.249364	+	0.968410i	$0.419778\pi$
$888$	−4.53906	−0.152321
$889$	−48.9906	−1.64309
$890$	0	0
$891$	−0.285809	−0.00957496
$892$	−9.52156	−0.318805
$893$	3.98824	0.133461
$894$	−7.87441	−0.263360
$895$	0	0
$896$	2.61995	0.0875264
$897$	18.3045	0.611169
$898$	22.8276	0.761767
$899$	26.3169	0.877718
$900$	0	0
$901$	−90.3470	−3.00989
$902$	1.52167	0.0506660
$903$	−21.1008	−0.702192
$904$	5.25015	0.174617
$905$	0	0
$906$	17.7660	0.590235
$907$	38.6577	1.28361	0.641804	−	0.766869i	$-0.278186\pi$
$907$	38.6577	1.28361	0.641804	+	0.766869i	$0.278186\pi$
$908$	16.3173	0.541509
$909$	10.7765	0.357435
$910$	0	0
$911$	48.5596	1.60885	0.804425	−	0.594054i	$-0.202474\pi$
$911$	48.5596	1.60885	0.804425	+	0.594054i	$0.202474\pi$
$912$	0.443312	0.0146795
$913$	−1.47531	−0.0488255
$914$	25.7628	0.852159
$915$	0	0
$916$	−0.533684	−0.0176334
$917$	−11.9245	−0.393783
$918$	6.49242	0.214282
$919$	−29.3888	−0.969446	−0.484723	−	0.874668i	$-0.661080\pi$
$919$	−29.3888	−0.969446	−0.484723	+	0.874668i	$0.661080\pi$
$920$	0	0
$921$	−18.6027	−0.612981
$922$	−32.4218	−1.06775
$923$	−13.5154	−0.444866
$924$	−0.748806	−0.0246339
$925$	0	0
$926$	−18.7775	−0.617066
$927$	17.9161	0.588443
$928$	−7.25636	−0.238202
$929$	47.3650	1.55400	0.776998	−	0.629503i	$-0.216741\pi$
$929$	47.3650	1.55400	0.776998	+	0.629503i	$0.216741\pi$
$930$	0	0
$931$	0.0602256	0.00197381
$932$	−10.9418	−0.358410
$933$	13.9175	0.455637
$934$	−1.96212	−0.0642025
$935$	0	0
$936$	−2.80530	−0.0916939
$937$	−24.3409	−0.795183	−0.397592	−	0.917562i	$-0.630154\pi$
$937$	−24.3409	−0.795183	−0.397592	+	0.917562i	$0.630154\pi$
$938$	26.8087	0.875336
$939$	−15.3804	−0.501922
$940$	0	0
$941$	19.5665	0.637850	0.318925	−	0.947780i	$-0.396678\pi$
$941$	19.5665	0.637850	0.318925	+	0.947780i	$0.396678\pi$
$942$	20.1191	0.655516
$943$	−34.7394	−1.13127
$944$	−3.74685	−0.121950
$945$	0	0
$946$	−2.30188	−0.0748405
$947$	45.9209	1.49223	0.746114	−	0.665818i	$-0.231917\pi$
$947$	45.9209	1.49223	0.746114	+	0.665818i	$0.231917\pi$
$948$	7.81462	0.253807
$949$	33.0652	1.07334
$950$	0	0
$951$	6.42445	0.208327
$952$	17.0098	0.551292
$953$	19.7792	0.640711	0.320355	−	0.947297i	$-0.396198\pi$
$953$	19.7792	0.640711	0.320355	+	0.947297i	$0.396198\pi$
$954$	13.9158	0.450539
$955$	0	0
$956$	6.03821	0.195290
$957$	2.07393	0.0670407
$958$	6.74378	0.217882
$959$	−40.7243	−1.31506
$960$	0	0
$961$	−17.8468	−0.575702
$962$	12.7334	0.410542
$963$	−3.81291	−0.122869
$964$	2.20701	0.0710830
$965$	0	0
$966$	17.0951	0.550027
$967$	21.1310	0.679529	0.339764	−	0.940511i	$-0.389653\pi$
$967$	21.1310	0.679529	0.339764	+	0.940511i	$0.389653\pi$
$968$	10.9183	0.350928
$969$	2.87817	0.0924601
$970$	0	0
$971$	−48.1608	−1.54555	−0.772777	−	0.634678i	$-0.781133\pi$
$971$	−48.1608	−1.54555	−0.772777	+	0.634678i	$0.781133\pi$
$972$	−1.00000	−0.0320750
$973$	23.5821	0.756006
$974$	1.16407	0.0372991
$975$	0	0
$976$	13.0189	0.416726
$977$	−50.5593	−1.61753	−0.808767	−	0.588129i	$-0.799865\pi$
$977$	−50.5593	−1.61753	−0.808767	+	0.588129i	$0.799865\pi$
$978$	−2.11433	−0.0676087
$979$	1.76218	0.0563196
$980$	0	0
$981$	7.96590	0.254332
$982$	8.38720	0.267646
$983$	−33.9077	−1.08149	−0.540744	−	0.841187i	$-0.681857\pi$
$983$	−33.9077	−1.08149	−0.540744	+	0.841187i	$0.681857\pi$
$984$	5.32407	0.169725
$985$	0	0
$986$	−47.1113	−1.50033
$987$	−23.5703	−0.750250
$988$	−1.24362	−0.0395649
$989$	52.5515	1.67104
$990$	0	0
$991$	−51.5839	−1.63862	−0.819308	−	0.573353i	$-0.805642\pi$
$991$	−51.5839	−1.63862	−0.819308	+	0.573353i	$0.805642\pi$
$992$	−3.62674	−0.115149
$993$	−18.1676	−0.576530
$994$	−12.6225	−0.400361
$995$	0	0
$996$	−5.16187	−0.163560
$997$	34.2263	1.08396	0.541979	−	0.840392i	$-0.317675\pi$
$997$	34.2263	1.08396	0.541979	+	0.840392i	$0.317675\pi$
$998$	−17.2590	−0.546325
$999$	4.53906	0.143610

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.a.u.1.4		8
5.2	odd	4		3750.2.c.k.1249.4		16
5.3	odd	4		3750.2.c.k.1249.13		16
5.4	even	2		3750.2.a.v.1.5		8
25.2	odd	20		150.2.h.b.79.1	yes	16
25.9	even	10		750.2.g.f.151.3		16
25.11	even	5		750.2.g.g.601.2		16
25.12	odd	20		750.2.h.d.349.3		16
25.13	odd	20		150.2.h.b.19.1	✓	16
25.14	even	10		750.2.g.f.601.3		16
25.16	even	5		750.2.g.g.151.2		16
25.23	odd	20		750.2.h.d.649.4		16
75.2	even	20		450.2.l.c.379.4		16
75.38	even	20		450.2.l.c.19.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.h.b.19.1	✓	16	25.13	odd	20
150.2.h.b.79.1	yes	16	25.2	odd	20
450.2.l.c.19.4		16	75.38	even	20
450.2.l.c.379.4		16	75.2	even	20
750.2.g.f.151.3		16	25.9	even	10
750.2.g.f.601.3		16	25.14	even	10
750.2.g.g.151.2		16	25.16	even	5
750.2.g.g.601.2		16	25.11	even	5
750.2.h.d.349.3		16	25.12	odd	20
750.2.h.d.649.4		16	25.23	odd	20
3750.2.a.u.1.4		8	1.1	even	1	trivial
3750.2.a.v.1.5		8	5.4	even	2
3750.2.c.k.1249.4		16	5.2	odd	4
3750.2.c.k.1249.13		16	5.3	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:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.61995 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.61995 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.285809 q^{11} -1.00000 q^{12} +2.80530 q^{13} +2.61995 q^{14} +1.00000 q^{16} +6.49242 q^{17} -1.00000 q^{18} -0.443312 q^{19} +2.61995 q^{21} +0.285809 q^{22} -6.52498 q^{23} +1.00000 q^{24} -2.80530 q^{26} -1.00000 q^{27} -2.61995 q^{28} +7.25636 q^{29} +3.62674 q^{31} -1.00000 q^{32} +0.285809 q^{33} -6.49242 q^{34} +1.00000 q^{36} -4.53906 q^{37} +0.443312 q^{38} -2.80530 q^{39} +5.32407 q^{41} -2.61995 q^{42} -8.05390 q^{43} -0.285809 q^{44} +6.52498 q^{46} -8.99646 q^{47} -1.00000 q^{48} -0.135854 q^{49} -6.49242 q^{51} +2.80530 q^{52} -13.9158 q^{53} +1.00000 q^{54} +2.61995 q^{56} +0.443312 q^{57} -7.25636 q^{58} -3.74685 q^{59} +13.0189 q^{61} -3.62674 q^{62} -2.61995 q^{63} +1.00000 q^{64} -0.285809 q^{66} +10.2325 q^{67} +6.49242 q^{68} +6.52498 q^{69} -4.81783 q^{71} -1.00000 q^{72} +11.7867 q^{73} +4.53906 q^{74} -0.443312 q^{76} +0.748806 q^{77} +2.80530 q^{78} -7.81462 q^{79} +1.00000 q^{81} -5.32407 q^{82} +5.16187 q^{83} +2.61995 q^{84} +8.05390 q^{86} -7.25636 q^{87} +0.285809 q^{88} -6.16560 q^{89} -7.34974 q^{91} -6.52498 q^{92} -3.62674 q^{93} +8.99646 q^{94} +1.00000 q^{96} -5.45040 q^{97} +0.135854 q^{98} -0.285809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{6} - 4 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 + 8 * q^6 - 4 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} + 8 q^{6} - 4 q^{7} - 8 q^{8} + 8 q^{9} + 6 q^{11} - 8 q^{12} - 2 q^{13} + 4 q^{14} + 8 q^{16} - 14 q^{17} - 8 q^{18} + 10 q^{19} + 4 q^{21} - 6 q^{22} - 12 q^{23} + 8 q^{24} + 2 q^{26} - 8 q^{27} - 4 q^{28} + 10 q^{29} + 16 q^{31} - 8 q^{32} - 6 q^{33} + 14 q^{34} + 8 q^{36} + 6 q^{37} - 10 q^{38} + 2 q^{39} + 6 q^{41} - 4 q^{42} - 2 q^{43} + 6 q^{44} + 12 q^{46} - 14 q^{47} - 8 q^{48} + 26 q^{49} + 14 q^{51} - 2 q^{52} - 12 q^{53} + 8 q^{54} + 4 q^{56} - 10 q^{57} - 10 q^{58} + 16 q^{61} - 16 q^{62} - 4 q^{63} + 8 q^{64} + 6 q^{66} + 6 q^{67} - 14 q^{68} + 12 q^{69} + 6 q^{71} - 8 q^{72} + 8 q^{73} - 6 q^{74} + 10 q^{76} - 8 q^{77} - 2 q^{78} + 10 q^{79} + 8 q^{81} - 6 q^{82} - 22 q^{83} + 4 q^{84} + 2 q^{86} - 10 q^{87} - 6 q^{88} + 20 q^{89} + 6 q^{91} - 12 q^{92} - 16 q^{93} + 14 q^{94} + 8 q^{96} + 16 q^{97} - 26 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 + 8 * q^6 - 4 * q^7 - 8 * q^8 + 8 * q^9 + 6 * q^11 - 8 * q^12 - 2 * q^13 + 4 * q^14 + 8 * q^16 - 14 * q^17 - 8 * q^18 + 10 * q^19 + 4 * q^21 - 6 * q^22 - 12 * q^23 + 8 * q^24 + 2 * q^26 - 8 * q^27 - 4 * q^28 + 10 * q^29 + 16 * q^31 - 8 * q^32 - 6 * q^33 + 14 * q^34 + 8 * q^36 + 6 * q^37 - 10 * q^38 + 2 * q^39 + 6 * q^41 - 4 * q^42 - 2 * q^43 + 6 * q^44 + 12 * q^46 - 14 * q^47 - 8 * q^48 + 26 * q^49 + 14 * q^51 - 2 * q^52 - 12 * q^53 + 8 * q^54 + 4 * q^56 - 10 * q^57 - 10 * q^58 + 16 * q^61 - 16 * q^62 - 4 * q^63 + 8 * q^64 + 6 * q^66 + 6 * q^67 - 14 * q^68 + 12 * q^69 + 6 * q^71 - 8 * q^72 + 8 * q^73 - 6 * q^74 + 10 * q^76 - 8 * q^77 - 2 * q^78 + 10 * q^79 + 8 * q^81 - 6 * q^82 - 22 * q^83 + 4 * q^84 + 2 * q^86 - 10 * q^87 - 6 * q^88 + 20 * q^89 + 6 * q^91 - 12 * q^92 - 16 * q^93 + 14 * q^94 + 8 * q^96 + 16 * q^97 - 26 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more