LMFDB - Embedded newform 3750.2.c.k.1249.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1249,3750)]

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, 1]))

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

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

chi := DirichletCharacter("3750.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.c (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:	$29.9439007580$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + \cdots + 11105 $ x^16 - 4x^15 - 24x^14 + 94x^13 + 262x^12 - 936x^11 - 1584x^10 + 4642x^9 + 6259x^8 - 11958x^7 - 15752x^6 + 14670x^5 + 18271x^4 - 10440x^3 + 1135x^2 + 21080*x + 11105
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.16
Root			$-2.79002 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	3750.1249
Dual form			3750.2.c.k.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.63137i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.63137i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.53967 q^{11} +1.00000i q^{12} +0.143974i q^{13} -4.63137 q^{14} +1.00000 q^{16} +7.49100i q^{17} -1.00000i q^{18} +6.74065 q^{19} +4.63137 q^{21} -2.53967i q^{22} -1.67733i q^{23} -1.00000 q^{24} -0.143974 q^{26} +1.00000i q^{27} -4.63137i q^{28} +2.25493 q^{29} +1.00385 q^{31} +1.00000i q^{32} +2.53967i q^{33} -7.49100 q^{34} +1.00000 q^{36} -0.0889810i q^{37} +6.74065i q^{38} +0.143974 q^{39} +3.07386 q^{41} +4.63137i q^{42} -9.02860i q^{43} +2.53967 q^{44} +1.67733 q^{46} +10.8854i q^{47} -1.00000i q^{48} -14.4496 q^{49} +7.49100 q^{51} -0.143974i q^{52} +4.96506i q^{53} -1.00000 q^{54} +4.63137 q^{56} -6.74065i q^{57} +2.25493i q^{58} -5.25365 q^{59} -13.7007 q^{61} +1.00385i q^{62} -4.63137i q^{63} -1.00000 q^{64} -2.53967 q^{66} -7.64929i q^{67} -7.49100i q^{68} -1.67733 q^{69} -10.2647 q^{71} +1.00000i q^{72} -1.96580i q^{73} +0.0889810 q^{74} -6.74065 q^{76} -11.7622i q^{77} +0.143974i q^{78} -0.747233 q^{79} +1.00000 q^{81} +3.07386i q^{82} +3.10257i q^{83} -4.63137 q^{84} +9.02860 q^{86} -2.25493i q^{87} +2.53967i q^{88} -0.733865 q^{89} -0.666798 q^{91} +1.67733i q^{92} -1.00385i q^{93} -10.8854 q^{94} +1.00000 q^{96} -9.12044i q^{97} -14.4496i q^{98} +2.53967 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9	$ 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 12 q^{11} - 8 q^{14} + 16 q^{16} - 20 q^{19} + 8 q^{21} - 16 q^{24} + 4 q^{26} - 20 q^{29} + 32 q^{31} - 28 q^{34} + 16 q^{36} - 4 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} - 52 q^{49} + 28 q^{51} - 16 q^{54} + 8 q^{56} + 32 q^{61} - 16 q^{64} + 12 q^{66} - 24 q^{69} + 12 q^{71} + 12 q^{74} + 20 q^{76} - 20 q^{79} + 16 q^{81} - 8 q^{84} + 4 q^{86} - 40 q^{89} + 12 q^{91} - 28 q^{94} + 16 q^{96} - 12 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 + 12 * q^11 - 8 * q^14 + 16 * q^16 - 20 * q^19 + 8 * q^21 - 16 * q^24 + 4 * q^26 - 20 * q^29 + 32 * q^31 - 28 * q^34 + 16 * q^36 - 4 * q^39 + 12 * q^41 - 12 * q^44 + 24 * q^46 - 52 * q^49 + 28 * q^51 - 16 * q^54 + 8 * q^56 + 32 * q^61 - 16 * q^64 + 12 * q^66 - 24 * q^69 + 12 * q^71 + 12 * q^74 + 20 * q^76 - 20 * q^79 + 16 * q^81 - 8 * q^84 + 4 * q^86 - 40 * q^89 + 12 * q^91 - 28 * q^94 + 16 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$.

$n$	$2501$	$3127$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	4.63137i	1.75049i	0.483677	+	0.875247i	$0.339301\pi$
$7$	4.63137i	1.75049i	−0.483677	+	0.875247i	$0.660699\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.53967	−0.765741	−0.382870	−	0.923802i	$-0.625064\pi$
$11$	−2.53967	−0.765741	−0.382870	+	0.923802i	$0.625064\pi$
$12$	1.00000i	0.288675i
$13$	0.143974i	0.0399313i	0.999801	+	0.0199656i	$0.00635568\pi$
$13$	0.143974i	0.0399313i	−0.999801	+	0.0199656i	$0.993644\pi$
$14$	−4.63137	−1.23779
$15$	0	0
$16$	1.00000	0.250000
$17$	7.49100i	1.81683i	0.418065	+	0.908417i	$0.362708\pi$
$17$	7.49100i	1.81683i	−0.418065	+	0.908417i	$0.637292\pi$
$18$	− 1.00000i	− 0.235702i
$19$	6.74065	1.54641	0.773206	−	0.634155i	$-0.218652\pi$
$19$	6.74065	1.54641	0.773206	+	0.634155i	$0.218652\pi$
$20$	0	0
$21$	4.63137	1.01065
$22$	− 2.53967i	− 0.541460i
$23$	− 1.67733i	− 0.349748i	−0.984591	−	0.174874i	$-0.944048\pi$
$23$	− 1.67733i	− 0.349748i	0.984591	−	0.174874i	$-0.0559517\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−0.143974	−0.0282357
$27$	1.00000i	0.192450i
$28$	− 4.63137i	− 0.875247i
$29$	2.25493	0.418730	0.209365	−	0.977838i	$-0.432860\pi$
$29$	2.25493	0.418730	0.209365	+	0.977838i	$0.432860\pi$
$30$	0	0
$31$	1.00385	0.180297	0.0901484	−	0.995928i	$-0.471266\pi$
$31$	1.00385	0.180297	0.0901484	+	0.995928i	$0.471266\pi$
$32$	1.00000i	0.176777i
$33$	2.53967i	0.442101i
$34$	−7.49100	−1.28470
$35$	0	0
$36$	1.00000	0.166667
$37$	− 0.0889810i	− 0.0146284i	−0.999973	−	0.00731419i	$-0.997672\pi$
$37$	− 0.0889810i	− 0.0146284i	0.999973	−	0.00731419i	$-0.00232820\pi$
$38$	6.74065i	1.09348i
$39$	0.143974	0.0230543
$40$	0	0
$41$	3.07386	0.480056	0.240028	−	0.970766i	$-0.422843\pi$
$41$	3.07386	0.480056	0.240028	+	0.970766i	$0.422843\pi$
$42$	4.63137i	0.714636i
$43$	− 9.02860i	− 1.37685i	−0.725308	−	0.688424i	$-0.758303\pi$
$43$	− 9.02860i	− 1.37685i	0.725308	−	0.688424i	$-0.241697\pi$
$44$	2.53967	0.382870
$45$	0	0
$46$	1.67733	0.247309
$47$	10.8854i	1.58779i	0.608053	+	0.793896i	$0.291951\pi$
$47$	10.8854i	1.58779i	−0.608053	+	0.793896i	$0.708049\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−14.4496	−2.06423
$50$	0	0
$51$	7.49100	1.04895
$52$	− 0.143974i	− 0.0199656i
$53$	4.96506i	0.682003i	0.940063	+	0.341002i	$0.110766\pi$
$53$	4.96506i	0.682003i	−0.940063	+	0.341002i	$0.889234\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	4.63137	0.618893
$57$	− 6.74065i	− 0.892821i
$58$	2.25493i	0.296087i
$59$	−5.25365	−0.683967	−0.341984	−	0.939706i	$-0.611099\pi$
$59$	−5.25365	−0.683967	−0.341984	+	0.939706i	$0.611099\pi$
$60$	0	0
$61$	−13.7007	−1.75419	−0.877096	−	0.480316i	$-0.840522\pi$
$61$	−13.7007	−1.75419	−0.877096	+	0.480316i	$0.840522\pi$
$62$	1.00385i	0.127489i
$63$	− 4.63137i	− 0.583498i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.53967	−0.312612
$67$	− 7.64929i	− 0.934509i	−0.884123	−	0.467255i	$-0.845243\pi$
$67$	− 7.64929i	− 0.934509i	0.884123	−	0.467255i	$-0.154757\pi$
$68$	− 7.49100i	− 0.908417i
$69$	−1.67733	−0.201927
$70$	0	0
$71$	−10.2647	−1.21819	−0.609096	−	0.793097i	$-0.708468\pi$
$71$	−10.2647	−1.21819	−0.609096	+	0.793097i	$0.708468\pi$
$72$	1.00000i	0.117851i
$73$	− 1.96580i	− 0.230079i	−0.993361	−	0.115040i	$-0.963301\pi$
$73$	− 1.96580i	− 0.230079i	0.993361	−	0.115040i	$-0.0366995\pi$
$74$	0.0889810	0.0103438
$75$	0	0
$76$	−6.74065	−0.773206
$77$	− 11.7622i	− 1.34042i
$78$	0.143974i	0.0163019i
$79$	−0.747233	−0.0840703	−0.0420352	−	0.999116i	$-0.513384\pi$
$79$	−0.747233	−0.0840703	−0.0420352	+	0.999116i	$0.513384\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.07386i	0.339451i
$83$	3.10257i	0.340551i	0.985397	+	0.170275i	$0.0544657\pi$
$83$	3.10257i	0.340551i	−0.985397	+	0.170275i	$0.945534\pi$
$84$	−4.63137	−0.505324
$85$	0	0
$86$	9.02860	0.973579
$87$	− 2.25493i	− 0.241754i
$88$	2.53967i	0.270730i
$89$	−0.733865	−0.0777896	−0.0388948	−	0.999243i	$-0.512384\pi$
$89$	−0.733865	−0.0777896	−0.0388948	+	0.999243i	$0.512384\pi$
$90$	0	0
$91$	−0.666798	−0.0698995
$92$	1.67733i	0.174874i
$93$	− 1.00385i	− 0.104094i
$94$	−10.8854	−1.12274
$95$	0	0
$96$	1.00000	0.102062
$97$	− 9.12044i	− 0.926041i	−0.886348	−	0.463020i	$-0.846766\pi$
$97$	− 9.12044i	− 0.926041i	0.886348	−	0.463020i	$-0.153234\pi$
$98$	− 14.4496i	− 1.45963i
$99$	2.53967	0.255247
$100$	0	0
$101$	−2.88013	−0.286583	−0.143292	−	0.989680i	$-0.545769\pi$
$101$	−2.88013	−0.286583	−0.143292	+	0.989680i	$0.545769\pi$
$102$	7.49100i	0.741720i
$103$	11.1713i	1.10075i	0.834919	+	0.550373i	$0.185514\pi$
$103$	11.1713i	1.10075i	−0.834919	+	0.550373i	$0.814486\pi$
$104$	0.143974	0.0141178
$105$	0	0
$106$	−4.96506	−0.482249
$107$	14.0538i	1.35863i	0.733846	+	0.679316i	$0.237723\pi$
$107$	14.0538i	1.35863i	−0.733846	+	0.679316i	$0.762277\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−6.71867	−0.643532	−0.321766	−	0.946819i	$-0.604276\pi$
$109$	−6.71867	−0.643532	−0.321766	+	0.946819i	$0.604276\pi$
$110$	0	0
$111$	−0.0889810	−0.00844570
$112$	4.63137i	0.437623i
$113$	− 0.997621i	− 0.0938483i	−0.998898	−	0.0469241i	$-0.985058\pi$
$113$	− 0.997621i	− 0.0938483i	0.998898	−	0.0469241i	$-0.0149419\pi$
$114$	6.74065	0.631320
$115$	0	0
$116$	−2.25493	−0.209365
$117$	− 0.143974i	− 0.0133104i
$118$	− 5.25365i	− 0.483638i
$119$	−34.6936	−3.18036
$120$	0	0
$121$	−4.55005	−0.413641
$122$	− 13.7007i	− 1.24040i
$123$	− 3.07386i	− 0.277160i
$124$	−1.00385	−0.0901484
$125$	0	0
$126$	4.63137	0.412595
$127$	11.5217i	1.02238i	0.859467	+	0.511192i	$0.170796\pi$
$127$	11.5217i	1.02238i	−0.859467	+	0.511192i	$0.829204\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−9.02860	−0.794924
$130$	0	0
$131$	−1.78409	−0.155877	−0.0779385	−	0.996958i	$-0.524834\pi$
$131$	−1.78409	−0.155877	−0.0779385	+	0.996958i	$0.524834\pi$
$132$	− 2.53967i	− 0.221050i
$133$	31.2184i	2.70698i
$134$	7.64929	0.660798
$135$	0	0
$136$	7.49100	0.642348
$137$	16.0233i	1.36897i	0.729029	+	0.684483i	$0.239972\pi$
$137$	16.0233i	1.36897i	−0.729029	+	0.684483i	$0.760028\pi$
$138$	− 1.67733i	− 0.142784i
$139$	3.70400	0.314169	0.157085	−	0.987585i	$-0.449790\pi$
$139$	3.70400	0.314169	0.157085	+	0.987585i	$0.449790\pi$
$140$	0	0
$141$	10.8854	0.916712
$142$	− 10.2647i	− 0.861392i
$143$	− 0.365648i	− 0.0305770i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	1.96580	0.162690
$147$	14.4496i	1.19178i
$148$	0.0889810i	0.00731419i
$149$	−1.88534	−0.154453	−0.0772267	−	0.997014i	$-0.524607\pi$
$149$	−1.88534	−0.154453	−0.0772267	+	0.997014i	$0.524607\pi$
$150$	0	0
$151$	−6.15090	−0.500553	−0.250276	−	0.968174i	$-0.580522\pi$
$151$	−6.15090	−0.500553	−0.250276	+	0.968174i	$0.580522\pi$
$152$	− 6.74065i	− 0.546739i
$153$	− 7.49100i	− 0.605611i
$154$	11.7622	0.947823
$155$	0	0
$156$	−0.143974	−0.0115272
$157$	− 23.4830i	− 1.87414i	−0.349137	−	0.937072i	$-0.613525\pi$
$157$	− 23.4830i	− 1.87414i	0.349137	−	0.937072i	$-0.386475\pi$
$158$	− 0.747233i	− 0.0594467i
$159$	4.96506	0.393755
$160$	0	0
$161$	7.76834	0.612231
$162$	1.00000i	0.0785674i
$163$	15.7773i	1.23577i	0.786267	+	0.617887i	$0.212011\pi$
$163$	15.7773i	1.23577i	−0.786267	+	0.617887i	$0.787989\pi$
$164$	−3.07386	−0.240028
$165$	0	0
$166$	−3.10257	−0.240806
$167$	− 4.03980i	− 0.312609i	−0.987709	−	0.156305i	$-0.950042\pi$
$167$	− 4.03980i	− 0.312609i	0.987709	−	0.156305i	$-0.0499582\pi$
$168$	− 4.63137i	− 0.357318i
$169$	12.9793	0.998405
$170$	0	0
$171$	−6.74065	−0.515470
$172$	9.02860i	0.688424i
$173$	− 20.1595i	− 1.53270i	−0.642422	−	0.766351i	$-0.722070\pi$
$173$	− 20.1595i	− 1.53270i	0.642422	−	0.766351i	$-0.277930\pi$
$174$	2.25493	0.170946
$175$	0	0
$176$	−2.53967	−0.191435
$177$	5.25365i	0.394889i
$178$	− 0.733865i	− 0.0550055i
$179$	7.76067	0.580060	0.290030	−	0.957018i	$-0.406335\pi$
$179$	7.76067	0.580060	0.290030	+	0.957018i	$0.406335\pi$
$180$	0	0
$181$	11.7909	0.876409	0.438205	−	0.898875i	$-0.355615\pi$
$181$	11.7909	0.876409	0.438205	+	0.898875i	$0.355615\pi$
$182$	− 0.666798i	− 0.0494264i
$183$	13.7007i	1.01278i
$184$	−1.67733	−0.123654
$185$	0	0
$186$	1.00385	0.0736058
$187$	− 19.0247i	− 1.39122i
$188$	− 10.8854i	− 0.793896i
$189$	−4.63137	−0.336883
$190$	0	0
$191$	4.88438	0.353422	0.176711	−	0.984263i	$-0.443454\pi$
$191$	4.88438	0.353422	0.176711	+	0.984263i	$0.443454\pi$
$192$	1.00000i	0.0721688i
$193$	18.7342i	1.34852i	0.738496	+	0.674258i	$0.235536\pi$
$193$	18.7342i	1.34852i	−0.738496	+	0.674258i	$0.764464\pi$
$194$	9.12044	0.654810
$195$	0	0
$196$	14.4496	1.03211
$197$	− 1.25454i	− 0.0893822i	−0.999001	−	0.0446911i	$-0.985770\pi$
$197$	− 1.25454i	− 0.0893822i	0.999001	−	0.0446911i	$-0.0142304\pi$
$198$	2.53967i	0.180487i
$199$	−19.3703	−1.37312	−0.686562	−	0.727071i	$-0.740881\pi$
$199$	−19.3703	−1.37312	−0.686562	+	0.727071i	$0.740881\pi$
$200$	0	0
$201$	−7.64929	−0.539539
$202$	− 2.88013i	− 0.202645i
$203$	10.4434i	0.732985i
$204$	−7.49100	−0.524475
$205$	0	0
$206$	−11.1713	−0.778345
$207$	1.67733i	0.116583i
$208$	0.143974i	0.00998282i
$209$	−17.1191	−1.18415
$210$	0	0
$211$	−1.24920	−0.0859983	−0.0429991	−	0.999075i	$-0.513691\pi$
$211$	−1.24920	−0.0859983	−0.0429991	+	0.999075i	$0.513691\pi$
$212$	− 4.96506i	− 0.341002i
$213$	10.2647i	0.703323i
$214$	−14.0538	−0.960698
$215$	0	0
$216$	1.00000	0.0680414
$217$	4.64920i	0.315608i
$218$	− 6.71867i	− 0.455046i
$219$	−1.96580	−0.132836
$220$	0	0
$221$	−1.07851	−0.0725485
$222$	− 0.0889810i	− 0.00597201i
$223$	− 21.8855i	− 1.46556i	−0.680463	−	0.732782i	$-0.738221\pi$
$223$	− 21.8855i	− 1.46556i	0.680463	−	0.732782i	$-0.261779\pi$
$224$	−4.63137	−0.309446
$225$	0	0
$226$	0.997621	0.0663607
$227$	14.5034i	0.962621i	0.876550	+	0.481311i	$0.159839\pi$
$227$	14.5034i	0.962621i	−0.876550	+	0.481311i	$0.840161\pi$
$228$	6.74065i	0.446410i
$229$	−4.70079	−0.310637	−0.155318	−	0.987864i	$-0.549640\pi$
$229$	−4.70079	−0.310637	−0.155318	+	0.987864i	$0.549640\pi$
$230$	0	0
$231$	−11.7622	−0.773894
$232$	− 2.25493i	− 0.148044i
$233$	11.8187i	0.774269i	0.922023	+	0.387135i	$0.126535\pi$
$233$	11.8187i	0.774269i	−0.922023	+	0.387135i	$0.873465\pi$
$234$	0.143974	0.00941189
$235$	0	0
$236$	5.25365	0.341984
$237$	0.747233i	0.0485380i
$238$	− 34.6936i	− 2.24885i
$239$	−9.71363	−0.628323	−0.314161	−	0.949370i	$-0.601723\pi$
$239$	−9.71363	−0.628323	−0.314161	+	0.949370i	$0.601723\pi$
$240$	0	0
$241$	6.68035	0.430319	0.215159	−	0.976579i	$-0.430973\pi$
$241$	6.68035	0.430319	0.215159	+	0.976579i	$0.430973\pi$
$242$	− 4.55005i	− 0.292488i
$243$	− 1.00000i	− 0.0641500i
$244$	13.7007	0.877096
$245$	0	0
$246$	3.07386	0.195982
$247$	0.970480i	0.0617502i
$248$	− 1.00385i	− 0.0637445i
$249$	3.10257	0.196617
$250$	0	0
$251$	−19.6023	−1.23729	−0.618644	−	0.785672i	$-0.712317\pi$
$251$	−19.6023	−1.23729	−0.618644	+	0.785672i	$0.712317\pi$
$252$	4.63137i	0.291749i
$253$	4.25987i	0.267816i
$254$	−11.5217	−0.722934
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.46931i	0.590679i	0.955392	+	0.295340i	$0.0954328\pi$
$257$	9.46931i	0.590679i	−0.955392	+	0.295340i	$0.904567\pi$
$258$	− 9.02860i	− 0.562096i
$259$	0.412104	0.0256069
$260$	0	0
$261$	−2.25493	−0.139577
$262$	− 1.78409i	− 0.110222i
$263$	21.1465i	1.30395i	0.758241	+	0.651974i	$0.226059\pi$
$263$	21.1465i	1.30395i	−0.758241	+	0.651974i	$0.773941\pi$
$264$	2.53967	0.156306
$265$	0	0
$266$	−31.2184	−1.91413
$267$	0.733865i	0.0449118i
$268$	7.64929i	0.467255i
$269$	1.95250	0.119046	0.0595230	−	0.998227i	$-0.481042\pi$
$269$	1.95250	0.119046	0.0595230	+	0.998227i	$0.481042\pi$
$270$	0	0
$271$	−15.9059	−0.966217	−0.483109	−	0.875560i	$-0.660492\pi$
$271$	−15.9059	−0.966217	−0.483109	+	0.875560i	$0.660492\pi$
$272$	7.49100i	0.454209i
$273$	0.666798i	0.0403565i
$274$	−16.0233	−0.968006
$275$	0	0
$276$	1.67733	0.100963
$277$	− 1.99048i	− 0.119596i	−0.998210	−	0.0597982i	$-0.980954\pi$
$277$	− 1.99048i	− 0.119596i	0.998210	−	0.0597982i	$-0.0190457\pi$
$278$	3.70400i	0.222151i
$279$	−1.00385	−0.0600989
$280$	0	0
$281$	−25.1864	−1.50249	−0.751247	−	0.660021i	$-0.770547\pi$
$281$	−25.1864	−1.50249	−0.751247	+	0.660021i	$0.770547\pi$
$282$	10.8854i	0.648214i
$283$	− 11.2601i	− 0.669346i	−0.942334	−	0.334673i	$-0.891374\pi$
$283$	− 11.2601i	− 0.669346i	0.942334	−	0.334673i	$-0.108626\pi$
$284$	10.2647	0.609096
$285$	0	0
$286$	0.365648	0.0216212
$287$	14.2362i	0.840335i
$288$	− 1.00000i	− 0.0589256i
$289$	−39.1151	−2.30089
$290$	0	0
$291$	−9.12044	−0.534650
$292$	1.96580i	0.115040i
$293$	− 20.0482i	− 1.17123i	−0.810591	−	0.585613i	$-0.800854\pi$
$293$	− 20.0482i	− 1.17123i	0.810591	−	0.585613i	$-0.199146\pi$
$294$	−14.4496	−0.842717
$295$	0	0
$296$	−0.0889810	−0.00517192
$297$	− 2.53967i	− 0.147367i
$298$	− 1.88534i	− 0.109215i
$299$	0.241492	0.0139659
$300$	0	0
$301$	41.8148	2.41016
$302$	− 6.15090i	− 0.353944i
$303$	2.88013i	0.165459i
$304$	6.74065	0.386603
$305$	0	0
$306$	7.49100	0.428232
$307$	− 14.4493i	− 0.824668i	−0.911033	−	0.412334i	$-0.864714\pi$
$307$	− 14.4493i	− 0.824668i	0.911033	−	0.412334i	$-0.135286\pi$
$308$	11.7622i	0.670212i
$309$	11.1713	0.635516
$310$	0	0
$311$	−23.5815	−1.33718	−0.668592	−	0.743630i	$-0.733103\pi$
$311$	−23.5815	−1.33718	−0.668592	+	0.743630i	$0.733103\pi$
$312$	− 0.143974i	− 0.00815094i
$313$	27.4927i	1.55398i	0.629515	+	0.776988i	$0.283254\pi$
$313$	27.4927i	1.55398i	−0.629515	+	0.776988i	$0.716746\pi$
$314$	23.4830	1.32522
$315$	0	0
$316$	0.747233	0.0420352
$317$	6.04575i	0.339563i	0.985482	+	0.169782i	$0.0543062\pi$
$317$	6.04575i	0.339563i	−0.985482	+	0.169782i	$0.945694\pi$
$318$	4.96506i	0.278427i
$319$	−5.72679	−0.320639
$320$	0	0
$321$	14.0538	0.784406
$322$	7.76834i	0.432913i
$323$	50.4942i	2.80957i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−15.7773	−0.873824
$327$	6.71867i	0.371543i
$328$	− 3.07386i	− 0.169725i
$329$	−50.4141	−2.77942
$330$	0	0
$331$	16.9315	0.930637	0.465319	−	0.885143i	$-0.345940\pi$
$331$	16.9315	0.930637	0.465319	+	0.885143i	$0.345940\pi$
$332$	− 3.10257i	− 0.170275i
$333$	0.0889810i	0.00487613i
$334$	4.03980	0.221048
$335$	0	0
$336$	4.63137	0.252662
$337$	− 22.6622i	− 1.23449i	−0.786771	−	0.617246i	$-0.788249\pi$
$337$	− 22.6622i	− 1.23449i	0.786771	−	0.617246i	$-0.211751\pi$
$338$	12.9793i	0.705979i
$339$	−0.997621	−0.0541833
$340$	0	0
$341$	−2.54945	−0.138061
$342$	− 6.74065i	− 0.364493i
$343$	− 34.5018i	− 1.86292i
$344$	−9.02860	−0.486789
$345$	0	0
$346$	20.1595	1.08378
$347$	− 10.2520i	− 0.550357i	−0.961393	−	0.275179i	$-0.911263\pi$
$347$	− 10.2520i	− 0.550357i	0.961393	−	0.275179i	$-0.0887370\pi$
$348$	2.25493i	0.120877i
$349$	−1.38746	−0.0742691	−0.0371346	−	0.999310i	$-0.511823\pi$
$349$	−1.38746	−0.0742691	−0.0371346	+	0.999310i	$0.511823\pi$
$350$	0	0
$351$	−0.143974	−0.00768478
$352$	− 2.53967i	− 0.135365i
$353$	− 12.9180i	− 0.687558i	−0.939051	−	0.343779i	$-0.888293\pi$
$353$	− 12.9180i	− 0.687558i	0.939051	−	0.343779i	$-0.111707\pi$
$354$	−5.25365	−0.279228
$355$	0	0
$356$	0.733865	0.0388948
$357$	34.6936i	1.83618i
$358$	7.76067i	0.410164i
$359$	−9.87465	−0.521164	−0.260582	−	0.965452i	$-0.583914\pi$
$359$	−9.87465	−0.521164	−0.260582	+	0.965452i	$0.583914\pi$
$360$	0	0
$361$	26.4364	1.39139
$362$	11.7909i	0.619715i
$363$	4.55005i	0.238816i
$364$	0.666798	0.0349497
$365$	0	0
$366$	−13.7007	−0.716146
$367$	5.31920i	0.277660i	0.990316	+	0.138830i	$0.0443341\pi$
$367$	5.31920i	0.277660i	−0.990316	+	0.138830i	$0.955666\pi$
$368$	− 1.67733i	− 0.0874369i
$369$	−3.07386	−0.160019
$370$	0	0
$371$	−22.9950	−1.19384
$372$	1.00385i	0.0520472i
$373$	9.68458i	0.501449i	0.968059	+	0.250724i	$0.0806688\pi$
$373$	9.68458i	0.501449i	−0.968059	+	0.250724i	$0.919331\pi$
$374$	19.0247	0.983744
$375$	0	0
$376$	10.8854	0.561369
$377$	0.324652i	0.0167204i
$378$	− 4.63137i	− 0.238212i
$379$	22.5220	1.15688	0.578439	−	0.815726i	$-0.303662\pi$
$379$	22.5220	1.15688	0.578439	+	0.815726i	$0.303662\pi$
$380$	0	0
$381$	11.5217	0.590273
$382$	4.88438i	0.249907i
$383$	7.21160i	0.368495i	0.982880	+	0.184248i	$0.0589849\pi$
$383$	7.21160i	0.368495i	−0.982880	+	0.184248i	$0.941015\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−18.7342	−0.953544
$387$	9.02860i	0.458949i
$388$	9.12044i	0.463020i
$389$	−13.2431	−0.671453	−0.335726	−	0.941960i	$-0.608982\pi$
$389$	−13.2431	−0.671453	−0.335726	+	0.941960i	$0.608982\pi$
$390$	0	0
$391$	12.5649	0.635433
$392$	14.4496i	0.729815i
$393$	1.78409i	0.0899957i
$394$	1.25454	0.0632027
$395$	0	0
$396$	−2.53967	−0.127623
$397$	− 29.2626i	− 1.46865i	−0.678798	−	0.734325i	$-0.737499\pi$
$397$	− 29.2626i	− 1.46865i	0.678798	−	0.734325i	$-0.262501\pi$
$398$	− 19.3703i	− 0.970945i
$399$	31.2184	1.56288
$400$	0	0
$401$	27.4005	1.36832	0.684158	−	0.729334i	$-0.260170\pi$
$401$	27.4005	1.36832	0.684158	+	0.729334i	$0.260170\pi$
$402$	− 7.64929i	− 0.381512i
$403$	0.144529i	0.00719948i
$404$	2.88013	0.143292
$405$	0	0
$406$	−10.4434	−0.518298
$407$	0.225983i	0.0112016i
$408$	− 7.49100i	− 0.370860i
$409$	−10.0026	−0.494596	−0.247298	−	0.968939i	$-0.579543\pi$
$409$	−10.0026	−0.494596	−0.247298	+	0.968939i	$0.579543\pi$
$410$	0	0
$411$	16.0233	0.790373
$412$	− 11.1713i	− 0.550373i
$413$	− 24.3316i	− 1.19728i
$414$	−1.67733	−0.0824363
$415$	0	0
$416$	−0.143974	−0.00705892
$417$	− 3.70400i	− 0.181386i
$418$	− 17.1191i	− 0.837320i
$419$	8.05118	0.393326	0.196663	−	0.980471i	$-0.436990\pi$
$419$	8.05118	0.393326	0.196663	+	0.980471i	$0.436990\pi$
$420$	0	0
$421$	10.0532	0.489965	0.244982	−	0.969528i	$-0.421218\pi$
$421$	10.0532	0.489965	0.244982	+	0.969528i	$0.421218\pi$
$422$	− 1.24920i	− 0.0608100i
$423$	− 10.8854i	− 0.529264i
$424$	4.96506	0.241125
$425$	0	0
$426$	−10.2647	−0.497325
$427$	− 63.4529i	− 3.07070i
$428$	− 14.0538i	− 0.679316i
$429$	−0.365648	−0.0176536
$430$	0	0
$431$	22.9694	1.10640	0.553198	−	0.833049i	$-0.313407\pi$
$431$	22.9694	1.10640	0.553198	+	0.833049i	$0.313407\pi$
$432$	1.00000i	0.0481125i
$433$	− 17.9964i	− 0.864854i	−0.901669	−	0.432427i	$-0.857657\pi$
$433$	− 17.9964i	− 0.864854i	0.901669	−	0.432427i	$-0.142343\pi$
$434$	−4.64920	−0.223169
$435$	0	0
$436$	6.71867	0.321766
$437$	− 11.3063i	− 0.540854i
$438$	− 1.96580i	− 0.0939294i
$439$	26.1564	1.24838	0.624188	−	0.781274i	$-0.285430\pi$
$439$	26.1564	1.24838	0.624188	+	0.781274i	$0.285430\pi$
$440$	0	0
$441$	14.4496	0.688076
$442$	− 1.07851i	− 0.0512996i
$443$	40.5689i	1.92749i	0.266833	+	0.963743i	$0.414023\pi$
$443$	40.5689i	1.92749i	−0.266833	+	0.963743i	$0.585977\pi$
$444$	0.0889810	0.00422285
$445$	0	0
$446$	21.8855	1.03631
$447$	1.88534i	0.0891737i
$448$	− 4.63137i	− 0.218812i
$449$	23.1589	1.09294	0.546468	−	0.837480i	$-0.315972\pi$
$449$	23.1589	1.09294	0.546468	+	0.837480i	$0.315972\pi$
$450$	0	0
$451$	−7.80660	−0.367598
$452$	0.997621i	0.0469241i
$453$	6.15090i	0.288994i
$454$	−14.5034	−0.680676
$455$	0	0
$456$	−6.74065	−0.315660
$457$	− 38.3997i	− 1.79626i	−0.439728	−	0.898131i	$-0.644925\pi$
$457$	− 38.3997i	− 1.79626i	0.439728	−	0.898131i	$-0.355075\pi$
$458$	− 4.70079i	− 0.219653i
$459$	−7.49100	−0.349650
$460$	0	0
$461$	−32.5913	−1.51793	−0.758965	−	0.651131i	$-0.774295\pi$
$461$	−32.5913	−1.51793	−0.758965	+	0.651131i	$0.774295\pi$
$462$	− 11.7622i	− 0.547226i
$463$	2.53778i	0.117941i	0.998260	+	0.0589703i	$0.0187817\pi$
$463$	2.53778i	0.117941i	−0.998260	+	0.0589703i	$0.981218\pi$
$464$	2.25493	0.104683
$465$	0	0
$466$	−11.8187	−0.547491
$467$	1.49654i	0.0692516i	0.999400	+	0.0346258i	$0.0110239\pi$
$467$	1.49654i	0.0692516i	−0.999400	+	0.0346258i	$0.988976\pi$
$468$	0.143974i	0.00665521i
$469$	35.4267	1.63585
$470$	0	0
$471$	−23.4830	−1.08204
$472$	5.25365i	0.241819i
$473$	22.9297i	1.05431i
$474$	−0.747233	−0.0343216
$475$	0	0
$476$	34.6936	1.59018
$477$	− 4.96506i	− 0.227334i
$478$	− 9.71363i	− 0.444291i
$479$	−24.2990	−1.11025	−0.555125	−	0.831767i	$-0.687330\pi$
$479$	−24.2990	−1.11025	−0.555125	+	0.831767i	$0.687330\pi$
$480$	0	0
$481$	0.0128110	0.000584130 0
$482$	6.68035i	0.304281i
$483$	− 7.76834i	− 0.353472i
$484$	4.55005	0.206821
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 10.0016i	− 0.453214i	−0.973986	−	0.226607i	$-0.927237\pi$
$487$	− 10.0016i	− 0.453214i	0.973986	−	0.226607i	$-0.0727632\pi$
$488$	13.7007i	0.620200i
$489$	15.7773	0.713474
$490$	0	0
$491$	4.17992	0.188637	0.0943186	−	0.995542i	$-0.469933\pi$
$491$	4.17992	0.188637	0.0943186	+	0.995542i	$0.469933\pi$
$492$	3.07386i	0.138580i
$493$	16.8917i	0.760764i
$494$	−0.970480	−0.0436640
$495$	0	0
$496$	1.00385	0.0450742
$497$	− 47.5395i	− 2.13244i
$498$	3.10257i	0.139029i
$499$	−1.70548	−0.0763476	−0.0381738	−	0.999271i	$-0.512154\pi$
$499$	−1.70548	−0.0763476	−0.0381738	+	0.999271i	$0.512154\pi$
$500$	0	0
$501$	−4.03980	−0.180485
$502$	− 19.6023i	− 0.874894i
$503$	30.9409i	1.37958i	0.724007	+	0.689792i	$0.242298\pi$
$503$	30.9409i	1.37958i	−0.724007	+	0.689792i	$0.757702\pi$
$504$	−4.63137	−0.206298
$505$	0	0
$506$	−4.25987	−0.189374
$507$	− 12.9793i	− 0.576430i
$508$	− 11.5217i	− 0.511192i
$509$	36.1157	1.60080	0.800399	−	0.599467i	$-0.204621\pi$
$509$	36.1157	1.60080	0.800399	+	0.599467i	$0.204621\pi$
$510$	0	0
$511$	9.10433	0.402752
$512$	1.00000i	0.0441942i
$513$	6.74065i	0.297607i
$514$	−9.46931	−0.417673
$515$	0	0
$516$	9.02860	0.397462
$517$	− 27.6453i	− 1.21584i
$518$	0.412104i	0.0181068i
$519$	−20.1595	−0.884906
$520$	0	0
$521$	4.47000	0.195834	0.0979171	−	0.995195i	$-0.468782\pi$
$521$	4.47000	0.195834	0.0979171	+	0.995195i	$0.468782\pi$
$522$	− 2.25493i	− 0.0986957i
$523$	30.6490i	1.34019i	0.742277	+	0.670093i	$0.233746\pi$
$523$	30.6490i	1.34019i	−0.742277	+	0.670093i	$0.766254\pi$
$524$	1.78409	0.0779385
$525$	0	0
$526$	−21.1465	−0.922030
$527$	7.51984i	0.327569i
$528$	2.53967i	0.110525i
$529$	20.1866	0.877677
$530$	0	0
$531$	5.25365	0.227989
$532$	− 31.2184i	− 1.35349i
$533$	0.442556i	0.0191692i
$534$	−0.733865	−0.0317575
$535$	0	0
$536$	−7.64929	−0.330399
$537$	− 7.76067i	− 0.334898i
$538$	1.95250i	0.0841782i
$539$	36.6973	1.58066
$540$	0	0
$541$	14.7172	0.632742	0.316371	−	0.948636i	$-0.397536\pi$
$541$	14.7172	0.632742	0.316371	+	0.948636i	$0.397536\pi$
$542$	− 15.9059i	− 0.683219i
$543$	− 11.7909i	− 0.505995i
$544$	−7.49100	−0.321174
$545$	0	0
$546$	−0.666798	−0.0285363
$547$	10.6678i	0.456123i	0.973647	+	0.228061i	$0.0732387\pi$
$547$	10.6678i	0.456123i	−0.973647	+	0.228061i	$0.926761\pi$
$548$	− 16.0233i	− 0.684483i
$549$	13.7007	0.584730
$550$	0	0
$551$	15.1997	0.647529
$552$	1.67733i	0.0713919i
$553$	− 3.46071i	− 0.147165i
$554$	1.99048	0.0845674
$555$	0	0
$556$	−3.70400	−0.157085
$557$	− 16.1652i	− 0.684942i	−0.939528	−	0.342471i	$-0.888736\pi$
$557$	− 16.1652i	− 0.684942i	0.939528	−	0.342471i	$-0.111264\pi$
$558$	− 1.00385i	− 0.0424963i
$559$	1.29989	0.0549793
$560$	0	0
$561$	−19.0247	−0.803224
$562$	− 25.1864i	− 1.06242i
$563$	− 43.4782i	− 1.83239i	−0.400733	−	0.916195i	$-0.631245\pi$
$563$	− 43.4782i	− 1.83239i	0.400733	−	0.916195i	$-0.368755\pi$
$564$	−10.8854	−0.458356
$565$	0	0
$566$	11.2601	0.473299
$567$	4.63137i	0.194499i
$568$	10.2647i	0.430696i
$569$	−12.0279	−0.504237	−0.252118	−	0.967696i	$-0.581127\pi$
$569$	−12.0279	−0.504237	−0.252118	+	0.967696i	$0.581127\pi$
$570$	0	0
$571$	18.7799	0.785915	0.392958	−	0.919557i	$-0.371452\pi$
$571$	18.7799	0.785915	0.392958	+	0.919557i	$0.371452\pi$
$572$	0.365648i	0.0152885i
$573$	− 4.88438i	− 0.204048i
$574$	−14.2362	−0.594206
$575$	0	0
$576$	1.00000	0.0416667
$577$	21.2351i	0.884029i	0.897008	+	0.442015i	$0.145736\pi$
$577$	21.2351i	0.884029i	−0.897008	+	0.442015i	$0.854264\pi$
$578$	− 39.1151i	− 1.62697i
$579$	18.7342	0.778566
$580$	0	0
$581$	−14.3691	−0.596132
$582$	− 9.12044i	− 0.378055i
$583$	− 12.6096i	− 0.522238i
$584$	−1.96580	−0.0813452
$585$	0	0
$586$	20.0482	0.828182
$587$	32.6278i	1.34669i	0.739327	+	0.673346i	$0.235144\pi$
$587$	32.6278i	1.34669i	−0.739327	+	0.673346i	$0.764856\pi$
$588$	− 14.4496i	− 0.595891i
$589$	6.76660	0.278813
$590$	0	0
$591$	−1.25454	−0.0516048
$592$	− 0.0889810i	− 0.00365710i
$593$	− 0.911868i	− 0.0374460i	−0.999825	−	0.0187230i	$-0.994040\pi$
$593$	− 0.911868i	− 0.0374460i	0.999825	−	0.0187230i	$-0.00596006\pi$
$594$	2.53967	0.104204
$595$	0	0
$596$	1.88534	0.0772267
$597$	19.3703i	0.792773i
$598$	0.241492i	0.00987536i
$599$	35.5516	1.45260	0.726300	−	0.687378i	$-0.241238\pi$
$599$	35.5516	1.45260	0.726300	+	0.687378i	$0.241238\pi$
$600$	0	0
$601$	−42.8608	−1.74833	−0.874164	−	0.485630i	$-0.838590\pi$
$601$	−42.8608	−1.74833	−0.874164	+	0.485630i	$0.838590\pi$
$602$	41.8148i	1.70424i
$603$	7.64929i	0.311503i
$604$	6.15090	0.250276
$605$	0	0
$606$	−2.88013	−0.116997
$607$	47.7389i	1.93766i	0.247724	+	0.968831i	$0.420317\pi$
$607$	47.7389i	1.93766i	−0.247724	+	0.968831i	$0.579683\pi$
$608$	6.74065i	0.273369i
$609$	10.4434	0.423189
$610$	0	0
$611$	−1.56721	−0.0634026
$612$	7.49100i	0.302806i
$613$	− 24.5031i	− 0.989670i	−0.868987	−	0.494835i	$-0.835229\pi$
$613$	− 24.5031i	− 0.989670i	0.868987	−	0.494835i	$-0.164771\pi$
$614$	14.4493	0.583128
$615$	0	0
$616$	−11.7622	−0.473911
$617$	2.62056i	0.105500i	0.998608	+	0.0527498i	$0.0167986\pi$
$617$	2.62056i	0.105500i	−0.998608	+	0.0527498i	$0.983201\pi$
$618$	11.1713i	0.449378i
$619$	6.75533	0.271519	0.135760	−	0.990742i	$-0.456652\pi$
$619$	6.75533	0.271519	0.135760	+	0.990742i	$0.456652\pi$
$620$	0	0
$621$	1.67733	0.0673089
$622$	− 23.5815i	− 0.945531i
$623$	− 3.39880i	− 0.136170i
$624$	0.143974	0.00576358
$625$	0	0
$626$	−27.4927	−1.09883
$627$	17.1191i	0.683669i
$628$	23.4830i	0.937072i
$629$	0.666557	0.0265774
$630$	0	0
$631$	−30.1555	−1.20047	−0.600235	−	0.799823i	$-0.704926\pi$
$631$	−30.1555	−1.20047	−0.600235	+	0.799823i	$0.704926\pi$
$632$	0.747233i	0.0297234i
$633$	1.24920i	0.0496511i
$634$	−6.04575	−0.240107
$635$	0	0
$636$	−4.96506	−0.196877
$637$	− 2.08037i	− 0.0824272i
$638$	− 5.72679i	− 0.226726i
$639$	10.2647	0.406064
$640$	0	0
$641$	20.5260	0.810727	0.405364	−	0.914156i	$-0.367145\pi$
$641$	20.5260	0.810727	0.405364	+	0.914156i	$0.367145\pi$
$642$	14.0538i	0.554659i
$643$	33.4413i	1.31880i	0.751793	+	0.659399i	$0.229189\pi$
$643$	33.4413i	1.31880i	−0.751793	+	0.659399i	$0.770811\pi$
$644$	−7.76834	−0.306115
$645$	0	0
$646$	−50.4942	−1.98667
$647$	18.0858i	0.711024i	0.934672	+	0.355512i	$0.115694\pi$
$647$	18.0858i	0.711024i	−0.934672	+	0.355512i	$0.884306\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	13.3426	0.523742
$650$	0	0
$651$	4.64920	0.182217
$652$	− 15.7773i	− 0.617887i
$653$	− 30.9940i	− 1.21289i	−0.795126	−	0.606444i	$-0.792596\pi$
$653$	− 30.9940i	− 1.21289i	0.795126	−	0.606444i	$-0.207404\pi$
$654$	−6.71867	−0.262721
$655$	0	0
$656$	3.07386	0.120014
$657$	1.96580i	0.0766930i
$658$	− 50.4141i	− 1.96535i
$659$	21.6253	0.842403	0.421202	−	0.906967i	$-0.361608\pi$
$659$	21.6253	0.842403	0.421202	+	0.906967i	$0.361608\pi$
$660$	0	0
$661$	−36.7601	−1.42980	−0.714902	−	0.699225i	$-0.753529\pi$
$661$	−36.7601	−1.42980	−0.714902	+	0.699225i	$0.753529\pi$
$662$	16.9315i	0.658060i
$663$	1.07851i	0.0418859i
$664$	3.10257	0.120403
$665$	0	0
$666$	−0.0889810	−0.00344794
$667$	− 3.78227i	− 0.146450i
$668$	4.03980i	0.156305i
$669$	−21.8855	−0.846144
$670$	0	0
$671$	34.7953	1.34326
$672$	4.63137i	0.178659i
$673$	− 30.8335i	− 1.18854i	−0.804264	−	0.594272i	$-0.797440\pi$
$673$	− 30.8335i	− 1.18854i	0.804264	−	0.594272i	$-0.202560\pi$
$674$	22.6622	0.872917
$675$	0	0
$676$	−12.9793	−0.499203
$677$	− 25.8101i	− 0.991963i	−0.868333	−	0.495981i	$-0.834808\pi$
$677$	− 25.8101i	− 0.991963i	0.868333	−	0.495981i	$-0.165192\pi$
$678$	− 0.997621i	− 0.0383134i
$679$	42.2402	1.62103
$680$	0	0
$681$	14.5034	0.555770
$682$	− 2.54945i	− 0.0976235i
$683$	− 20.8802i	− 0.798959i	−0.916742	−	0.399479i	$-0.869191\pi$
$683$	− 20.8802i	− 0.798959i	0.916742	−	0.399479i	$-0.130809\pi$
$684$	6.74065	0.257735
$685$	0	0
$686$	34.5018	1.31729
$687$	4.70079i	0.179346i
$688$	− 9.02860i	− 0.344212i
$689$	−0.714841	−0.0272333
$690$	0	0
$691$	20.6464	0.785427	0.392714	−	0.919661i	$-0.371536\pi$
$691$	20.6464	0.785427	0.392714	+	0.919661i	$0.371536\pi$
$692$	20.1595i	0.766351i
$693$	11.7622i	0.446808i
$694$	10.2520	0.389162
$695$	0	0
$696$	−2.25493	−0.0854730
$697$	23.0263i	0.872182i
$698$	− 1.38746i	− 0.0525162i
$699$	11.8187	0.447025
$700$	0	0
$701$	−12.3050	−0.464752	−0.232376	−	0.972626i	$-0.574650\pi$
$701$	−12.3050	−0.464752	−0.232376	+	0.972626i	$0.574650\pi$
$702$	− 0.143974i	− 0.00543396i
$703$	− 0.599790i	− 0.0226215i
$704$	2.53967	0.0957176
$705$	0	0
$706$	12.9180	0.486177
$707$	− 13.3389i	− 0.501662i
$708$	− 5.25365i	− 0.197444i
$709$	24.6984	0.927566	0.463783	−	0.885949i	$-0.346492\pi$
$709$	24.6984	0.927566	0.463783	+	0.885949i	$0.346492\pi$
$710$	0	0
$711$	0.747233	0.0280234
$712$	0.733865i	0.0275028i
$713$	− 1.68379i	− 0.0630583i
$714$	−34.6936	−1.29838
$715$	0	0
$716$	−7.76067	−0.290030
$717$	9.71363i	0.362762i
$718$	− 9.87465i	− 0.368519i
$719$	21.0231	0.784030	0.392015	−	0.919959i	$-0.371778\pi$
$719$	21.0231	0.784030	0.392015	+	0.919959i	$0.371778\pi$
$720$	0	0
$721$	−51.7387	−1.92685
$722$	26.4364i	0.983860i
$723$	− 6.68035i	− 0.248445i
$724$	−11.7909	−0.438205
$725$	0	0
$726$	−4.55005	−0.168868
$727$	8.67125i	0.321599i	0.986987	+	0.160799i	$0.0514072\pi$
$727$	8.67125i	0.321599i	−0.986987	+	0.160799i	$0.948593\pi$
$728$	0.666798i	0.0247132i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	67.6332	2.50151
$732$	− 13.7007i	− 0.506391i
$733$	33.9376i	1.25351i	0.779214	+	0.626757i	$0.215618\pi$
$733$	33.9376i	1.25351i	−0.779214	+	0.626757i	$0.784382\pi$
$734$	−5.31920	−0.196335
$735$	0	0
$736$	1.67733	0.0618272
$737$	19.4267i	0.715592i
$738$	− 3.07386i	− 0.113150i
$739$	16.4126	0.603747	0.301874	−	0.953348i	$-0.402388\pi$
$739$	16.4126	0.603747	0.301874	+	0.953348i	$0.402388\pi$
$740$	0	0
$741$	0.970480	0.0356515
$742$	− 22.9950i	− 0.844174i
$743$	3.98742i	0.146284i	0.997322	+	0.0731422i	$0.0233027\pi$
$743$	3.98742i	0.146284i	−0.997322	+	0.0731422i	$0.976697\pi$
$744$	−1.00385	−0.0368029
$745$	0	0
$746$	−9.68458	−0.354578
$747$	− 3.10257i	− 0.113517i
$748$	19.0247i	0.695612i
$749$	−65.0883	−2.37828
$750$	0	0
$751$	37.5827	1.37141	0.685706	−	0.727879i	$-0.259494\pi$
$751$	37.5827	1.37141	0.685706	+	0.727879i	$0.259494\pi$
$752$	10.8854i	0.396948i
$753$	19.6023i	0.714348i
$754$	−0.324652	−0.0118231
$755$	0	0
$756$	4.63137	0.168441
$757$	31.2749i	1.13671i	0.822785	+	0.568353i	$0.192419\pi$
$757$	31.2749i	1.13671i	−0.822785	+	0.568353i	$0.807581\pi$
$758$	22.5220i	0.818036i
$759$	4.25987	0.154624
$760$	0	0
$761$	29.9489	1.08565	0.542823	−	0.839847i	$-0.317355\pi$
$761$	29.9489	1.08565	0.542823	+	0.839847i	$0.317355\pi$
$762$	11.5217i	0.417386i
$763$	− 31.1167i	− 1.12650i
$764$	−4.88438	−0.176711
$765$	0	0
$766$	−7.21160	−0.260566
$767$	− 0.756391i	− 0.0273117i
$768$	− 1.00000i	− 0.0360844i
$769$	36.9037	1.33078	0.665391	−	0.746495i	$-0.268265\pi$
$769$	36.9037	1.33078	0.665391	+	0.746495i	$0.268265\pi$
$770$	0	0
$771$	9.46931	0.341029
$772$	− 18.7342i	− 0.674258i
$773$	44.2386i	1.59115i	0.605853	+	0.795576i	$0.292832\pi$
$773$	44.2386i	1.59115i	−0.605853	+	0.795576i	$0.707168\pi$
$774$	−9.02860	−0.324526
$775$	0	0
$776$	−9.12044	−0.327405
$777$	− 0.412104i	− 0.0147841i
$778$	− 13.2431i	− 0.474789i
$779$	20.7198	0.742364
$780$	0	0
$781$	26.0689	0.932819
$782$	12.5649i	0.449319i
$783$	2.25493i	0.0805847i
$784$	−14.4496	−0.516057
$785$	0	0
$786$	−1.78409	−0.0636365
$787$	40.0500i	1.42763i	0.700336	+	0.713813i	$0.253034\pi$
$787$	40.0500i	1.42763i	−0.700336	+	0.713813i	$0.746966\pi$
$788$	1.25454i	0.0446911i
$789$	21.1465	0.752834
$790$	0	0
$791$	4.62035	0.164281
$792$	− 2.53967i	− 0.0902434i
$793$	− 1.97254i	− 0.0700471i
$794$	29.2626	1.03849
$795$	0	0
$796$	19.3703	0.686562
$797$	44.4781i	1.57549i	0.615999	+	0.787747i	$0.288753\pi$
$797$	44.4781i	1.57549i	−0.615999	+	0.787747i	$0.711247\pi$
$798$	31.2184i	1.10512i
$799$	−81.5422	−2.88476
$800$	0	0
$801$	0.733865	0.0259299
$802$	27.4005i	0.967545i
$803$	4.99248i	0.176181i
$804$	7.64929	0.269770
$805$	0	0
$806$	−0.144529	−0.00509080
$807$	− 1.95250i	− 0.0687312i
$808$	2.88013i	0.101323i
$809$	34.2539	1.20430	0.602151	−	0.798382i	$-0.294311\pi$
$809$	34.2539	1.20430	0.602151	+	0.798382i	$0.294311\pi$
$810$	0	0
$811$	10.6488	0.373930	0.186965	−	0.982367i	$-0.440135\pi$
$811$	10.6488	0.373930	0.186965	+	0.982367i	$0.440135\pi$
$812$	− 10.4434i	− 0.366492i
$813$	15.9059i	0.557846i
$814$	−0.225983	−0.00792069
$815$	0	0
$816$	7.49100	0.262237
$817$	− 60.8586i	− 2.12917i
$818$	− 10.0026i	− 0.349732i
$819$	0.666798	0.0232998
$820$	0	0
$821$	−14.7387	−0.514384	−0.257192	−	0.966360i	$-0.582797\pi$
$821$	−14.7387	−0.514384	−0.257192	+	0.966360i	$0.582797\pi$
$822$	16.0233i	0.558878i
$823$	− 2.83258i	− 0.0987377i	−0.998781	−	0.0493688i	$-0.984279\pi$
$823$	− 2.83258i	− 0.0987377i	0.998781	−	0.0493688i	$-0.0157210\pi$
$824$	11.1713	0.389172
$825$	0	0
$826$	24.3316	0.846605
$827$	21.8116i	0.758463i	0.925302	+	0.379231i	$0.123812\pi$
$827$	21.8116i	0.758463i	−0.925302	+	0.379231i	$0.876188\pi$
$828$	− 1.67733i	− 0.0582913i
$829$	40.6037	1.41023	0.705113	−	0.709095i	$-0.250896\pi$
$829$	40.6037	1.41023	0.705113	+	0.709095i	$0.250896\pi$
$830$	0	0
$831$	−1.99048	−0.0690490
$832$	− 0.143974i	− 0.00499141i
$833$	− 108.242i	− 3.75036i
$834$	3.70400	0.128259
$835$	0	0
$836$	17.1191	0.592075
$837$	1.00385i	0.0346981i
$838$	8.05118i	0.278123i
$839$	−28.8888	−0.997353	−0.498676	−	0.866788i	$-0.666180\pi$
$839$	−28.8888	−0.997353	−0.498676	+	0.866788i	$0.666180\pi$
$840$	0	0
$841$	−23.9153	−0.824665
$842$	10.0532i	0.346458i
$843$	25.1864i	0.867465i
$844$	1.24920	0.0429991
$845$	0	0
$846$	10.8854	0.374246
$847$	− 21.0730i	− 0.724076i
$848$	4.96506i	0.170501i
$849$	−11.2601	−0.386447
$850$	0	0
$851$	−0.149251	−0.00511624
$852$	− 10.2647i	− 0.351662i
$853$	− 24.8092i	− 0.849450i	−0.905322	−	0.424725i	$-0.860371\pi$
$853$	− 24.8092i	− 0.849450i	0.905322	−	0.424725i	$-0.139629\pi$
$854$	63.4529	2.17131
$855$	0	0
$856$	14.0538	0.480349
$857$	− 20.0312i	− 0.684251i	−0.939654	−	0.342126i	$-0.888853\pi$
$857$	− 20.0312i	− 0.684251i	0.939654	−	0.342126i	$-0.111147\pi$
$858$	− 0.365648i	− 0.0124830i
$859$	31.7856	1.08451	0.542255	−	0.840214i	$-0.317571\pi$
$859$	31.7856	1.08451	0.542255	+	0.840214i	$0.317571\pi$
$860$	0	0
$861$	14.2362	0.485167
$862$	22.9694i	0.782341i
$863$	− 23.8998i	− 0.813560i	−0.913526	−	0.406780i	$-0.866652\pi$
$863$	− 23.8998i	− 0.813560i	0.913526	−	0.406780i	$-0.133348\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	17.9964	0.611544
$867$	39.1151i	1.32842i
$868$	− 4.64920i	− 0.157804i
$869$	1.89773	0.0643761
$870$	0	0
$871$	1.10130	0.0373162
$872$	6.71867i	0.227523i
$873$	9.12044i	0.308680i
$874$	11.3063	0.382441
$875$	0	0
$876$	1.96580	0.0664181
$877$	− 4.70430i	− 0.158853i	−0.996841	−	0.0794265i	$-0.974691\pi$
$877$	− 4.70430i	− 0.158853i	0.996841	−	0.0794265i	$-0.0253089\pi$
$878$	26.1564i	0.882735i
$879$	−20.0482	−0.676208
$880$	0	0
$881$	−44.6262	−1.50349	−0.751747	−	0.659452i	$-0.770789\pi$
$881$	−44.6262	−1.50349	−0.751747	+	0.659452i	$0.770789\pi$
$882$	14.4496i	0.486543i
$883$	− 32.2048i	− 1.08378i	−0.840450	−	0.541888i	$-0.817710\pi$
$883$	− 32.2048i	− 1.08378i	0.840450	−	0.541888i	$-0.182290\pi$
$884$	1.07851	0.0362743
$885$	0	0
$886$	−40.5689	−1.36294
$887$	− 43.0250i	− 1.44464i	−0.691560	−	0.722319i	$-0.743076\pi$
$887$	− 43.0250i	− 1.44464i	0.691560	−	0.722319i	$-0.256924\pi$
$888$	0.0889810i	0.00298601i
$889$	−53.3611	−1.78968
$890$	0	0
$891$	−2.53967	−0.0850823
$892$	21.8855i	0.732782i
$893$	73.3744i	2.45538i
$894$	−1.88534	−0.0630554
$895$	0	0
$896$	4.63137	0.154723
$897$	− 0.241492i	− 0.00806320i
$898$	23.1589i	0.772823i
$899$	2.26361	0.0754957
$900$	0	0
$901$	−37.1933	−1.23909
$902$	− 7.80660i	− 0.259931i
$903$	− 41.8148i	− 1.39151i
$904$	−0.997621	−0.0331804
$905$	0	0
$906$	−6.15090	−0.204350
$907$	39.3398i	1.30626i	0.757247	+	0.653129i	$0.226544\pi$
$907$	39.3398i	1.30626i	−0.757247	+	0.653129i	$0.773456\pi$
$908$	− 14.5034i	− 0.481311i
$909$	2.88013	0.0955278
$910$	0	0
$911$	−13.6023	−0.450664	−0.225332	−	0.974282i	$-0.572347\pi$
$911$	−13.6023	−0.450664	−0.225332	+	0.974282i	$0.572347\pi$
$912$	− 6.74065i	− 0.223205i
$913$	− 7.87951i	− 0.260774i
$914$	38.3997	1.27015
$915$	0	0
$916$	4.70079	0.155318
$917$	− 8.26280i	− 0.272862i
$918$	− 7.49100i	− 0.247240i
$919$	14.2048	0.468572	0.234286	−	0.972168i	$-0.424725\pi$
$919$	14.2048	0.468572	0.234286	+	0.972168i	$0.424725\pi$
$920$	0	0
$921$	−14.4493	−0.476122
$922$	− 32.5913i	− 1.07334i
$923$	− 1.47785i	− 0.0486440i
$924$	11.7622	0.386947
$925$	0	0
$926$	−2.53778	−0.0833966
$927$	− 11.1713i	− 0.366915i
$928$	2.25493i	0.0740218i
$929$	−8.10960	−0.266067	−0.133034	−	0.991112i	$-0.542472\pi$
$929$	−8.10960	−0.266067	−0.133034	+	0.991112i	$0.542472\pi$
$930$	0	0
$931$	−97.3996	−3.19214
$932$	− 11.8187i	− 0.387135i
$933$	23.5815i	0.772023i
$934$	−1.49654	−0.0489683
$935$	0	0
$936$	−0.143974	−0.00470595
$937$	10.5255i	0.343854i	0.985110	+	0.171927i	$0.0549993\pi$
$937$	10.5255i	0.343854i	−0.985110	+	0.171927i	$0.945001\pi$
$938$	35.4267i	1.15672i
$939$	27.4927	0.897189
$940$	0	0
$941$	−49.5406	−1.61498	−0.807489	−	0.589882i	$-0.799174\pi$
$941$	−49.5406	−1.61498	−0.807489	+	0.589882i	$0.799174\pi$
$942$	− 23.4830i	− 0.765116i
$943$	− 5.15587i	− 0.167898i
$944$	−5.25365	−0.170992
$945$	0	0
$946$	−22.9297	−0.745509
$947$	10.5814i	0.343849i	0.985110	+	0.171925i	$0.0549986\pi$
$947$	10.5814i	0.343849i	−0.985110	+	0.171925i	$0.945001\pi$
$948$	− 0.747233i	− 0.0242690i
$949$	0.283024	0.00918735
$950$	0	0
$951$	6.04575	0.196047
$952$	34.6936i	1.12443i
$953$	26.0143i	0.842686i	0.906901	+	0.421343i	$0.138441\pi$
$953$	26.0143i	0.842686i	−0.906901	+	0.421343i	$0.861559\pi$
$954$	4.96506	0.160750
$955$	0	0
$956$	9.71363	0.314161
$957$	5.72679i	0.185121i
$958$	− 24.2990i	− 0.785065i
$959$	−74.2100	−2.39637
$960$	0	0
$961$	−29.9923	−0.967493
$962$	0.0128110i	0 0.000413043i
$963$	− 14.0538i	− 0.452877i
$964$	−6.68035	−0.215159
$965$	0	0
$966$	7.76834	0.249942
$967$	29.5512i	0.950304i	0.879904	+	0.475152i	$0.157607\pi$
$967$	29.5512i	0.950304i	−0.879904	+	0.475152i	$0.842393\pi$
$968$	4.55005i	0.146244i
$969$	50.4942	1.62211
$970$	0	0
$971$	35.8718	1.15118	0.575590	−	0.817739i	$-0.304773\pi$
$971$	35.8718	1.15118	0.575590	+	0.817739i	$0.304773\pi$
$972$	1.00000i	0.0320750i
$973$	17.1546i	0.549952i
$974$	10.0016	0.320470
$975$	0	0
$976$	−13.7007	−0.438548
$977$	21.6261i	0.691881i	0.938256	+	0.345940i	$0.112440\pi$
$977$	21.6261i	0.691881i	−0.938256	+	0.345940i	$0.887560\pi$
$978$	15.7773i	0.504502i
$979$	1.86378	0.0595666
$980$	0	0
$981$	6.71867	0.214511
$982$	4.17992i	0.133387i
$983$	14.6180i	0.466243i	0.972448	+	0.233122i	$0.0748940\pi$
$983$	14.6180i	0.466243i	−0.972448	+	0.233122i	$0.925106\pi$
$984$	−3.07386	−0.0979910
$985$	0	0
$986$	−16.8917	−0.537941
$987$	50.4141i	1.60470i
$988$	− 0.970480i	− 0.0308751i
$989$	−15.1439	−0.481549
$990$	0	0
$991$	−13.0741	−0.415313	−0.207656	−	0.978202i	$-0.566584\pi$
$991$	−13.0741	−0.415313	−0.207656	+	0.978202i	$0.566584\pi$
$992$	1.00385i	0.0318723i
$993$	− 16.9315i	− 0.537304i
$994$	47.5395	1.50786
$995$	0	0
$996$	−3.10257	−0.0983086
$997$	− 12.1968i	− 0.386275i	−0.981172	−	0.193138i	$-0.938134\pi$
$997$	− 12.1968i	− 0.386275i	0.981172	−	0.193138i	$-0.0618664\pi$
$998$	− 1.70548i	− 0.0539859i
$999$	0.0889810	0.00281523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.c.k.1249.16		16
5.2	odd	4		3750.2.a.u.1.1		8
5.3	odd	4		3750.2.a.v.1.8		8
5.4	even	2	inner	3750.2.c.k.1249.1		16
25.3	odd	20		750.2.g.f.451.4		16
25.4	even	10		150.2.h.b.109.1	✓	16
25.6	even	5		150.2.h.b.139.1	yes	16
25.8	odd	20		750.2.g.f.301.4		16
25.17	odd	20		750.2.g.g.301.1		16
25.19	even	10		750.2.h.d.199.3		16
25.21	even	5		750.2.h.d.49.4		16
25.22	odd	20		750.2.g.g.451.1		16
75.29	odd	10		450.2.l.c.109.4		16
75.56	odd	10		450.2.l.c.289.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.h.b.109.1	✓	16	25.4	even	10
150.2.h.b.139.1	yes	16	25.6	even	5
450.2.l.c.109.4		16	75.29	odd	10
450.2.l.c.289.4		16	75.56	odd	10
750.2.g.f.301.4		16	25.8	odd	20
750.2.g.f.451.4		16	25.3	odd	20
750.2.g.g.301.1		16	25.17	odd	20
750.2.g.g.451.1		16	25.22	odd	20
750.2.h.d.49.4		16	25.21	even	5
750.2.h.d.199.3		16	25.19	even	10
3750.2.a.u.1.1		8	5.2	odd	4
3750.2.a.v.1.8		8	5.3	odd	4
3750.2.c.k.1249.1		16	5.4	even	2	inner
3750.2.c.k.1249.16		16	1.1	even	1	trivial

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.63137i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.63137i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.53967 q^{11} +1.00000i q^{12} +0.143974i q^{13} -4.63137 q^{14} +1.00000 q^{16} +7.49100i q^{17} -1.00000i q^{18} +6.74065 q^{19} +4.63137 q^{21} -2.53967i q^{22} -1.67733i q^{23} -1.00000 q^{24} -0.143974 q^{26} +1.00000i q^{27} -4.63137i q^{28} +2.25493 q^{29} +1.00385 q^{31} +1.00000i q^{32} +2.53967i q^{33} -7.49100 q^{34} +1.00000 q^{36} -0.0889810i q^{37} +6.74065i q^{38} +0.143974 q^{39} +3.07386 q^{41} +4.63137i q^{42} -9.02860i q^{43} +2.53967 q^{44} +1.67733 q^{46} +10.8854i q^{47} -1.00000i q^{48} -14.4496 q^{49} +7.49100 q^{51} -0.143974i q^{52} +4.96506i q^{53} -1.00000 q^{54} +4.63137 q^{56} -6.74065i q^{57} +2.25493i q^{58} -5.25365 q^{59} -13.7007 q^{61} +1.00385i q^{62} -4.63137i q^{63} -1.00000 q^{64} -2.53967 q^{66} -7.64929i q^{67} -7.49100i q^{68} -1.67733 q^{69} -10.2647 q^{71} +1.00000i q^{72} -1.96580i q^{73} +0.0889810 q^{74} -6.74065 q^{76} -11.7622i q^{77} +0.143974i q^{78} -0.747233 q^{79} +1.00000 q^{81} +3.07386i q^{82} +3.10257i q^{83} -4.63137 q^{84} +9.02860 q^{86} -2.25493i q^{87} +2.53967i q^{88} -0.733865 q^{89} -0.666798 q^{91} +1.67733i q^{92} -1.00385i q^{93} -10.8854 q^{94} +1.00000 q^{96} -9.12044i q^{97} -14.4496i q^{98} +2.53967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9	\( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 12 q^{11} - 8 q^{14} + 16 q^{16} - 20 q^{19} + 8 q^{21} - 16 q^{24} + 4 q^{26} - 20 q^{29} + 32 q^{31} - 28 q^{34} + 16 q^{36} - 4 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} - 52 q^{49} + 28 q^{51} - 16 q^{54} + 8 q^{56} + 32 q^{61} - 16 q^{64} + 12 q^{66} - 24 q^{69} + 12 q^{71} + 12 q^{74} + 20 q^{76} - 20 q^{79} + 16 q^{81} - 8 q^{84} + 4 q^{86} - 40 q^{89} + 12 q^{91} - 28 q^{94} + 16 q^{96} - 12 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 + 12 * q^11 - 8 * q^14 + 16 * q^16 - 20 * q^19 + 8 * q^21 - 16 * q^24 + 4 * q^26 - 20 * q^29 + 32 * q^31 - 28 * q^34 + 16 * q^36 - 4 * q^39 + 12 * q^41 - 12 * q^44 + 24 * q^46 - 52 * q^49 + 28 * q^51 - 16 * q^54 + 8 * q^56 + 32 * q^61 - 16 * q^64 + 12 * q^66 - 24 * q^69 + 12 * q^71 + 12 * q^74 + 20 * q^76 - 20 * q^79 + 16 * q^81 - 8 * q^84 + 4 * q^86 - 40 * q^89 + 12 * q^91 - 28 * q^94 + 16 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

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Database

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