LMFDB - Embedded newform 6027.2.a.bl.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
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} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} + \cdots - 49 $ x^16 - 4x^15 - 14x^14 + 68x^13 + 64x^12 - 456x^11 - 54x^10 + 1532x^9 - 400x^8 - 2708x^7 + 1218x^6 + 2424x^5 - 1276x^4 - 960x^3 + 500x^2 + 112*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.04648$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.04648 q^{2} -1.00000 q^{3} -0.904885 q^{4} -1.81328 q^{5} +1.04648 q^{6} +3.03990 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.04648 q^{2} -1.00000 q^{3} -0.904885 q^{4} -1.81328 q^{5} +1.04648 q^{6} +3.03990 q^{8} +1.00000 q^{9} +1.89755 q^{10} -0.445995 q^{11} +0.904885 q^{12} +2.67032 q^{13} +1.81328 q^{15} -1.37141 q^{16} -2.86907 q^{17} -1.04648 q^{18} +4.73993 q^{19} +1.64081 q^{20} +0.466723 q^{22} +3.59833 q^{23} -3.03990 q^{24} -1.71203 q^{25} -2.79443 q^{26} -1.00000 q^{27} -4.22128 q^{29} -1.89755 q^{30} -6.94583 q^{31} -4.64464 q^{32} +0.445995 q^{33} +3.00242 q^{34} -0.904885 q^{36} -9.17863 q^{37} -4.96023 q^{38} -2.67032 q^{39} -5.51217 q^{40} -1.00000 q^{41} -7.79959 q^{43} +0.403574 q^{44} -1.81328 q^{45} -3.76557 q^{46} +11.4891 q^{47} +1.37141 q^{48} +1.79160 q^{50} +2.86907 q^{51} -2.41633 q^{52} +10.4400 q^{53} +1.04648 q^{54} +0.808712 q^{55} -4.73993 q^{57} +4.41747 q^{58} -5.38717 q^{59} -1.64081 q^{60} +7.26684 q^{61} +7.26865 q^{62} +7.60334 q^{64} -4.84203 q^{65} -0.466723 q^{66} -14.2182 q^{67} +2.59618 q^{68} -3.59833 q^{69} +15.4800 q^{71} +3.03990 q^{72} +12.6484 q^{73} +9.60523 q^{74} +1.71203 q^{75} -4.28909 q^{76} +2.79443 q^{78} +5.02198 q^{79} +2.48675 q^{80} +1.00000 q^{81} +1.04648 q^{82} +7.60824 q^{83} +5.20242 q^{85} +8.16210 q^{86} +4.22128 q^{87} -1.35578 q^{88} -2.20300 q^{89} +1.89755 q^{90} -3.25607 q^{92} +6.94583 q^{93} -12.0230 q^{94} -8.59480 q^{95} +4.64464 q^{96} -1.75062 q^{97} -0.445995 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} - 16 q^{3} + 12 q^{4} + 12 q^{5} + 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 - 16 * q^3 + 12 * q^4 + 12 * q^5 + 4 * q^6 - 12 * q^8 + 16 * q^9	$ 16 q - 4 q^{2} - 16 q^{3} + 12 q^{4} + 12 q^{5} + 4 q^{6} - 12 q^{8} + 16 q^{9} + 4 q^{10} - 4 q^{11} - 12 q^{12} - 12 q^{15} + 8 q^{17} - 4 q^{18} - 4 q^{19} + 20 q^{20} - 16 q^{22} - 12 q^{23} + 12 q^{24} - 8 q^{25} + 8 q^{26} - 16 q^{27} - 16 q^{29} - 4 q^{30} + 4 q^{31} - 48 q^{32} + 4 q^{33} - 16 q^{34} + 12 q^{36} - 48 q^{37} + 4 q^{38} - 56 q^{40} - 16 q^{41} - 16 q^{43} + 12 q^{45} - 4 q^{46} + 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} + 4 q^{54} - 8 q^{55} + 4 q^{57} - 36 q^{58} + 36 q^{59} - 20 q^{60} + 4 q^{61} + 12 q^{62} + 52 q^{64} - 36 q^{65} + 16 q^{66} - 52 q^{67} + 8 q^{68} + 12 q^{69} - 12 q^{71} - 12 q^{72} + 16 q^{73} + 4 q^{74} + 8 q^{75} - 16 q^{76} - 8 q^{78} - 36 q^{79} + 68 q^{80} + 16 q^{81} + 4 q^{82} + 32 q^{83} - 28 q^{85} - 8 q^{86} + 16 q^{87} - 36 q^{88} + 12 q^{89} + 4 q^{90} - 36 q^{92} - 4 q^{93} - 24 q^{94} - 20 q^{95} + 48 q^{96} - 48 q^{97} - 4 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 - 16 * q^3 + 12 * q^4 + 12 * q^5 + 4 * q^6 - 12 * q^8 + 16 * q^9 + 4 * q^10 - 4 * q^11 - 12 * q^12 - 12 * q^15 + 8 * q^17 - 4 * q^18 - 4 * q^19 + 20 * q^20 - 16 * q^22 - 12 * q^23 + 12 * q^24 - 8 * q^25 + 8 * q^26 - 16 * q^27 - 16 * q^29 - 4 * q^30 + 4 * q^31 - 48 * q^32 + 4 * q^33 - 16 * q^34 + 12 * q^36 - 48 * q^37 + 4 * q^38 - 56 * q^40 - 16 * q^41 - 16 * q^43 + 12 * q^45 - 4 * q^46 + 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 + 4 * q^54 - 8 * q^55 + 4 * q^57 - 36 * q^58 + 36 * q^59 - 20 * q^60 + 4 * q^61 + 12 * q^62 + 52 * q^64 - 36 * q^65 + 16 * q^66 - 52 * q^67 + 8 * q^68 + 12 * q^69 - 12 * q^71 - 12 * q^72 + 16 * q^73 + 4 * q^74 + 8 * q^75 - 16 * q^76 - 8 * q^78 - 36 * q^79 + 68 * q^80 + 16 * q^81 + 4 * q^82 + 32 * q^83 - 28 * q^85 - 8 * q^86 + 16 * q^87 - 36 * q^88 + 12 * q^89 + 4 * q^90 - 36 * q^92 - 4 * q^93 - 24 * q^94 - 20 * q^95 + 48 * q^96 - 48 * q^97 - 4 * 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 $

$2$	−1.04648	−0.739971	−0.369986	−	0.929037i	$-0.620637\pi$
$2$	−1.04648	−0.739971	−0.369986	+	0.929037i	$0.620637\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.904885	−0.452442
$5$	−1.81328	−0.810922	−0.405461	−	0.914112i	$-0.632889\pi$
$5$	−1.81328	−0.810922	−0.405461	+	0.914112i	$0.632889\pi$
$6$	1.04648	0.427223
$7$	0	0
$7$	0	0
$8$	3.03990	1.07477
$9$	1.00000	0.333333
$10$	1.89755	0.600059
$11$	−0.445995	−0.134472	−0.0672362	−	0.997737i	$-0.521418\pi$
$11$	−0.445995	−0.134472	−0.0672362	+	0.997737i	$0.521418\pi$
$12$	0.904885	0.261218
$13$	2.67032	0.740613	0.370307	−	0.928910i	$-0.379253\pi$
$13$	2.67032	0.740613	0.370307	+	0.928910i	$0.379253\pi$
$14$	0	0
$15$	1.81328	0.468186
$16$	−1.37141	−0.342854
$17$	−2.86907	−0.695853	−0.347926	−	0.937522i	$-0.613114\pi$
$17$	−2.86907	−0.695853	−0.347926	+	0.937522i	$0.613114\pi$
$18$	−1.04648	−0.246657
$19$	4.73993	1.08741	0.543707	−	0.839275i	$-0.317020\pi$
$19$	4.73993	1.08741	0.543707	+	0.839275i	$0.317020\pi$
$20$	1.64081	0.366895
$21$	0	0
$22$	0.466723	0.0995058
$23$	3.59833	0.750303	0.375152	−	0.926963i	$-0.377591\pi$
$23$	3.59833	0.750303	0.375152	+	0.926963i	$0.377591\pi$
$24$	−3.03990	−0.620516
$25$	−1.71203	−0.342406
$26$	−2.79443	−0.548033
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.22128	−0.783871	−0.391936	−	0.919993i	$-0.628194\pi$
$29$	−4.22128	−0.783871	−0.391936	+	0.919993i	$0.628194\pi$
$30$	−1.89755	−0.346444
$31$	−6.94583	−1.24751	−0.623754	−	0.781621i	$-0.714393\pi$
$31$	−6.94583	−1.24751	−0.623754	+	0.781621i	$0.714393\pi$
$32$	−4.64464	−0.821064
$33$	0.445995	0.0776377
$34$	3.00242	0.514911
$35$	0	0
$36$	−0.904885	−0.150814
$37$	−9.17863	−1.50896	−0.754479	−	0.656324i	$-0.772110\pi$
$37$	−9.17863	−1.50896	−0.754479	+	0.656324i	$0.772110\pi$
$38$	−4.96023	−0.804656
$39$	−2.67032	−0.427593
$40$	−5.51217	−0.871551
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−7.79959	−1.18943	−0.594713	−	0.803938i	$-0.702735\pi$
$43$	−7.79959	−1.18943	−0.594713	+	0.803938i	$0.702735\pi$
$44$	0.403574	0.0608410
$45$	−1.81328	−0.270307
$46$	−3.76557	−0.555203
$47$	11.4891	1.67585	0.837926	−	0.545783i	$-0.183768\pi$
$47$	11.4891	1.67585	0.837926	+	0.545783i	$0.183768\pi$
$48$	1.37141	0.197947
$49$	0	0
$50$	1.79160	0.253370
$51$	2.86907	0.401751
$52$	−2.41633	−0.335085
$53$	10.4400	1.43405	0.717025	−	0.697048i	$-0.245503\pi$
$53$	10.4400	1.43405	0.717025	+	0.697048i	$0.245503\pi$
$54$	1.04648	0.142408
$55$	0.808712	0.109047
$56$	0	0
$57$	−4.73993	−0.627819
$58$	4.41747	0.580042
$59$	−5.38717	−0.701350	−0.350675	−	0.936497i	$-0.614048\pi$
$59$	−5.38717	−0.701350	−0.350675	+	0.936497i	$0.614048\pi$
$60$	−1.64081	−0.211827
$61$	7.26684	0.930423	0.465212	−	0.885199i	$-0.345978\pi$
$61$	7.26684	0.930423	0.465212	+	0.885199i	$0.345978\pi$
$62$	7.26865	0.923120
$63$	0	0
$64$	7.60334	0.950417
$65$	−4.84203	−0.600580
$66$	−0.466723	−0.0574497
$67$	−14.2182	−1.73702	−0.868512	−	0.495668i	$-0.834923\pi$
$67$	−14.2182	−1.73702	−0.868512	+	0.495668i	$0.834923\pi$
$68$	2.59618	0.314833
$69$	−3.59833	−0.433188
$70$	0	0
$71$	15.4800	1.83713	0.918567	−	0.395265i	$-0.129347\pi$
$71$	15.4800	1.83713	0.918567	+	0.395265i	$0.129347\pi$
$72$	3.03990	0.358255
$73$	12.6484	1.48038	0.740192	−	0.672396i	$-0.234735\pi$
$73$	12.6484	1.48038	0.740192	+	0.672396i	$0.234735\pi$
$74$	9.60523	1.11659
$75$	1.71203	0.197688
$76$	−4.28909	−0.491992
$77$	0	0
$78$	2.79443	0.316407
$79$	5.02198	0.565017	0.282508	−	0.959265i	$-0.408834\pi$
$79$	5.02198	0.565017	0.282508	+	0.959265i	$0.408834\pi$
$80$	2.48675	0.278027
$81$	1.00000	0.111111
$82$	1.04648	0.115564
$83$	7.60824	0.835113	0.417557	−	0.908651i	$-0.362887\pi$
$83$	7.60824	0.835113	0.417557	+	0.908651i	$0.362887\pi$
$84$	0	0
$85$	5.20242	0.564282
$86$	8.16210	0.880142
$87$	4.22128	0.452568
$88$	−1.35578	−0.144526
$89$	−2.20300	−0.233518	−0.116759	−	0.993160i	$-0.537250\pi$
$89$	−2.20300	−0.233518	−0.116759	+	0.993160i	$0.537250\pi$
$90$	1.89755	0.200020
$91$	0	0
$92$	−3.25607	−0.339469
$93$	6.94583	0.720249
$94$	−12.0230	−1.24008
$95$	−8.59480	−0.881808
$96$	4.64464	0.474042
$97$	−1.75062	−0.177748	−0.0888742	−	0.996043i	$-0.528327\pi$
$97$	−1.75062	−0.177748	−0.0888742	+	0.996043i	$0.528327\pi$
$98$	0	0
$99$	−0.445995	−0.0448242
$100$	1.54919	0.154919
$101$	10.0727	1.00227	0.501134	−	0.865370i	$-0.332916\pi$
$101$	10.0727	1.00227	0.501134	+	0.865370i	$0.332916\pi$
$102$	−3.00242	−0.297284
$103$	15.6394	1.54099	0.770496	−	0.637445i	$-0.220009\pi$
$103$	15.6394	1.54099	0.770496	+	0.637445i	$0.220009\pi$
$104$	8.11749	0.795986
$105$	0	0
$106$	−10.9253	−1.06116
$107$	−11.1705	−1.07990	−0.539948	−	0.841699i	$-0.681556\pi$
$107$	−11.1705	−1.07990	−0.539948	+	0.841699i	$0.681556\pi$
$108$	0.904885	0.0870726
$109$	14.9447	1.43145	0.715723	−	0.698384i	$-0.246097\pi$
$109$	14.9447	1.43145	0.715723	+	0.698384i	$0.246097\pi$
$110$	−0.846299	−0.0806914
$111$	9.17863	0.871197
$112$	0	0
$113$	11.9484	1.12401	0.562007	−	0.827133i	$-0.310030\pi$
$113$	11.9484	1.12401	0.562007	+	0.827133i	$0.310030\pi$
$114$	4.96023	0.464568
$115$	−6.52477	−0.608437
$116$	3.81977	0.354657
$117$	2.67032	0.246871
$118$	5.63755	0.518979
$119$	0	0
$120$	5.51217	0.503190
$121$	−10.8011	−0.981917
$122$	−7.60458	−0.688487
$123$	1.00000	0.0901670
$124$	6.28517	0.564425
$125$	12.1708	1.08859
$126$	0	0
$127$	−4.63833	−0.411586	−0.205793	−	0.978596i	$-0.565977\pi$
$127$	−4.63833	−0.411586	−0.205793	+	0.978596i	$0.565977\pi$
$128$	1.33256	0.117782
$129$	7.79959	0.686716
$130$	5.06707	0.444412
$131$	−7.50200	−0.655453	−0.327726	−	0.944773i	$-0.606282\pi$
$131$	−7.50200	−0.655453	−0.327726	+	0.944773i	$0.606282\pi$
$132$	−0.403574	−0.0351266
$133$	0	0
$134$	14.8790	1.28535
$135$	1.81328	0.156062
$136$	−8.72169	−0.747879
$137$	−18.9937	−1.62274	−0.811370	−	0.584533i	$-0.801278\pi$
$137$	−18.9937	−1.62274	−0.811370	+	0.584533i	$0.801278\pi$
$138$	3.76557	0.320547
$139$	−12.7746	−1.08353	−0.541766	−	0.840530i	$-0.682244\pi$
$139$	−12.7746	−1.08353	−0.541766	+	0.840530i	$0.682244\pi$
$140$	0	0
$141$	−11.4891	−0.967554
$142$	−16.1994	−1.35943
$143$	−1.19095	−0.0995921
$144$	−1.37141	−0.114285
$145$	7.65434	0.635658
$146$	−13.2363	−1.09544
$147$	0	0
$148$	8.30561	0.682717
$149$	−6.86535	−0.562431	−0.281216	−	0.959645i	$-0.590738\pi$
$149$	−6.86535	−0.562431	−0.281216	+	0.959645i	$0.590738\pi$
$150$	−1.79160	−0.146283
$151$	−6.76282	−0.550351	−0.275175	−	0.961394i	$-0.588736\pi$
$151$	−6.76282	−0.550351	−0.275175	+	0.961394i	$0.588736\pi$
$152$	14.4089	1.16872
$153$	−2.86907	−0.231951
$154$	0	0
$155$	12.5947	1.01163
$156$	2.41633	0.193461
$157$	−3.74193	−0.298639	−0.149319	−	0.988789i	$-0.547708\pi$
$157$	−3.74193	−0.298639	−0.149319	+	0.988789i	$0.547708\pi$
$158$	−5.25539	−0.418096
$159$	−10.4400	−0.827949
$160$	8.42202	0.665819
$161$	0	0
$162$	−1.04648	−0.0822190
$163$	−18.3200	−1.43494	−0.717468	−	0.696592i	$-0.754699\pi$
$163$	−18.3200	−1.43494	−0.717468	+	0.696592i	$0.754699\pi$
$164$	0.904885	0.0706596
$165$	−0.808712	−0.0629581
$166$	−7.96186	−0.617960
$167$	−12.9443	−1.00166	−0.500829	−	0.865546i	$-0.666971\pi$
$167$	−12.9443	−1.00166	−0.500829	+	0.865546i	$0.666971\pi$
$168$	0	0
$169$	−5.86940	−0.451492
$170$	−5.44422	−0.417553
$171$	4.73993	0.362472
$172$	7.05773	0.538147
$173$	−14.8684	−1.13042	−0.565212	−	0.824946i	$-0.691205\pi$
$173$	−14.8684	−1.13042	−0.565212	+	0.824946i	$0.691205\pi$
$174$	−4.41747	−0.334888
$175$	0	0
$176$	0.611643	0.0461044
$177$	5.38717	0.404925
$178$	2.30539	0.172796
$179$	−0.390628	−0.0291969	−0.0145984	−	0.999893i	$-0.504647\pi$
$179$	−0.390628	−0.0291969	−0.0145984	+	0.999893i	$0.504647\pi$
$180$	1.64081	0.122298
$181$	1.88861	0.140380	0.0701898	−	0.997534i	$-0.477640\pi$
$181$	1.88861	0.140380	0.0701898	+	0.997534i	$0.477640\pi$
$182$	0	0
$183$	−7.26684	−0.537180
$184$	10.9385	0.806400
$185$	16.6434	1.22365
$186$	−7.26865	−0.532963
$187$	1.27959	0.0935730
$188$	−10.3963	−0.758227
$189$	0	0
$190$	8.99427	0.652513
$191$	−8.98328	−0.650007	−0.325004	−	0.945713i	$-0.605366\pi$
$191$	−8.98328	−0.650007	−0.325004	+	0.945713i	$0.605366\pi$
$192$	−7.60334	−0.548724
$193$	16.2969	1.17307	0.586537	−	0.809922i	$-0.300491\pi$
$193$	16.2969	1.17307	0.586537	+	0.809922i	$0.300491\pi$
$194$	1.83198	0.131529
$195$	4.84203	0.346745
$196$	0	0
$197$	18.0411	1.28538	0.642688	−	0.766128i	$-0.277819\pi$
$197$	18.0411	1.28538	0.642688	+	0.766128i	$0.277819\pi$
$198$	0.466723	0.0331686
$199$	−3.32264	−0.235535	−0.117768	−	0.993041i	$-0.537574\pi$
$199$	−3.32264	−0.235535	−0.117768	+	0.993041i	$0.537574\pi$
$200$	−5.20439	−0.368006
$201$	14.2182	1.00287
$202$	−10.5408	−0.741650
$203$	0	0
$204$	−2.59618	−0.181769
$205$	1.81328	0.126645
$206$	−16.3662	−1.14029
$207$	3.59833	0.250101
$208$	−3.66211	−0.253922
$209$	−2.11398	−0.146227
$210$	0	0
$211$	18.3821	1.26547	0.632737	−	0.774367i	$-0.281932\pi$
$211$	18.3821	1.26547	0.632737	+	0.774367i	$0.281932\pi$
$212$	−9.44703	−0.648825
$213$	−15.4800	−1.06067
$214$	11.6897	0.799092
$215$	14.1428	0.964532
$216$	−3.03990	−0.206839
$217$	0	0
$218$	−15.6393	−1.05923
$219$	−12.6484	−0.854700
$220$	−0.731791	−0.0493373
$221$	−7.66134	−0.515358
$222$	−9.60523	−0.644661
$223$	8.28572	0.554853	0.277427	−	0.960747i	$-0.410518\pi$
$223$	8.28572	0.554853	0.277427	+	0.960747i	$0.410518\pi$
$224$	0	0
$225$	−1.71203	−0.114135
$226$	−12.5038	−0.831738
$227$	7.06231	0.468742	0.234371	−	0.972147i	$-0.424697\pi$
$227$	7.06231	0.468742	0.234371	+	0.972147i	$0.424697\pi$
$228$	4.28909	0.284052
$229$	20.2997	1.34144	0.670720	−	0.741711i	$-0.265985\pi$
$229$	20.2997	1.34144	0.670720	+	0.741711i	$0.265985\pi$
$230$	6.82802	0.450226
$231$	0	0
$232$	−12.8322	−0.842478
$233$	4.13151	0.270664	0.135332	−	0.990800i	$-0.456790\pi$
$233$	4.13151	0.270664	0.135332	+	0.990800i	$0.456790\pi$
$234$	−2.79443	−0.182678
$235$	−20.8329	−1.35899
$236$	4.87477	0.317320
$237$	−5.02198	−0.326213
$238$	0	0
$239$	−5.65744	−0.365950	−0.182975	−	0.983118i	$-0.558573\pi$
$239$	−5.65744	−0.365950	−0.182975	+	0.983118i	$0.558573\pi$
$240$	−2.48675	−0.160519
$241$	−18.0436	−1.16229	−0.581145	−	0.813800i	$-0.697395\pi$
$241$	−18.0436	−1.16229	−0.581145	+	0.813800i	$0.697395\pi$
$242$	11.3031	0.726591
$243$	−1.00000	−0.0641500
$244$	−6.57565	−0.420963
$245$	0	0
$246$	−1.04648	−0.0667210
$247$	12.6571	0.805354
$248$	−21.1146	−1.34078
$249$	−7.60824	−0.482153
$250$	−12.7364	−0.805523
$251$	16.5321	1.04349	0.521747	−	0.853100i	$-0.325280\pi$
$251$	16.5321	1.04349	0.521747	+	0.853100i	$0.325280\pi$
$252$	0	0
$253$	−1.60484	−0.100895
$254$	4.85391	0.304562
$255$	−5.20242	−0.325788
$256$	−16.6012	−1.03757
$257$	27.2214	1.69802	0.849012	−	0.528373i	$-0.177198\pi$
$257$	27.2214	1.69802	0.849012	+	0.528373i	$0.177198\pi$
$258$	−8.16210	−0.508150
$259$	0	0
$260$	4.38148	0.271728
$261$	−4.22128	−0.261290
$262$	7.85067	0.485016
$263$	0.195115	0.0120313	0.00601567	−	0.999982i	$-0.498085\pi$
$263$	0.195115	0.0120313	0.00601567	+	0.999982i	$0.498085\pi$
$264$	1.35578	0.0834424
$265$	−18.9307	−1.16290
$266$	0	0
$267$	2.20300	0.134822
$268$	12.8658	0.785904
$269$	−23.1477	−1.41134	−0.705670	−	0.708541i	$-0.749354\pi$
$269$	−23.1477	−1.41134	−0.705670	+	0.708541i	$0.749354\pi$
$270$	−1.89755	−0.115481
$271$	−6.82658	−0.414685	−0.207343	−	0.978268i	$-0.566481\pi$
$271$	−6.82658	−0.414685	−0.207343	+	0.978268i	$0.566481\pi$
$272$	3.93469	0.238575
$273$	0	0
$274$	19.8765	1.20078
$275$	0.763555	0.0460441
$276$	3.25607	0.195993
$277$	27.7102	1.66495	0.832474	−	0.554065i	$-0.186924\pi$
$277$	27.7102	1.66495	0.832474	+	0.554065i	$0.186924\pi$
$278$	13.3684	0.801782
$279$	−6.94583	−0.415836
$280$	0	0
$281$	−3.97578	−0.237175	−0.118587	−	0.992944i	$-0.537837\pi$
$281$	−3.97578	−0.237175	−0.118587	+	0.992944i	$0.537837\pi$
$282$	12.0230	0.715962
$283$	−12.5932	−0.748590	−0.374295	−	0.927310i	$-0.622115\pi$
$283$	−12.5932	−0.748590	−0.374295	+	0.927310i	$0.622115\pi$
$284$	−14.0076	−0.831197
$285$	8.59480	0.509112
$286$	1.24630	0.0736953
$287$	0	0
$288$	−4.64464	−0.273688
$289$	−8.76842	−0.515789
$290$	−8.01010	−0.470369
$291$	1.75062	0.102623
$292$	−11.4453	−0.669788
$293$	18.7975	1.09816	0.549081	−	0.835769i	$-0.314978\pi$
$293$	18.7975	1.09816	0.549081	+	0.835769i	$0.314978\pi$
$294$	0	0
$295$	9.76843	0.568740
$296$	−27.9021	−1.62178
$297$	0.445995	0.0258792
$298$	7.18443	0.416183
$299$	9.60868	0.555685
$300$	−1.54919	−0.0894424
$301$	0	0
$302$	7.07714	0.407244
$303$	−10.0727	−0.578660
$304$	−6.50041	−0.372824
$305$	−13.1768	−0.754501
$306$	3.00242	0.171637
$307$	−15.8413	−0.904109	−0.452055	−	0.891990i	$-0.649309\pi$
$307$	−15.8413	−0.904109	−0.452055	+	0.891990i	$0.649309\pi$
$308$	0	0
$309$	−15.6394	−0.889692
$310$	−13.1801	−0.748578
$311$	22.9872	1.30349	0.651743	−	0.758439i	$-0.274038\pi$
$311$	22.9872	1.30349	0.651743	+	0.758439i	$0.274038\pi$
$312$	−8.11749	−0.459563
$313$	−13.7375	−0.776489	−0.388244	−	0.921556i	$-0.626918\pi$
$313$	−13.7375	−0.776489	−0.388244	+	0.921556i	$0.626918\pi$
$314$	3.91585	0.220984
$315$	0	0
$316$	−4.54431	−0.255638
$317$	2.04989	0.115133	0.0575666	−	0.998342i	$-0.481666\pi$
$317$	2.04989	0.115133	0.0575666	+	0.998342i	$0.481666\pi$
$318$	10.9253	0.612658
$319$	1.88267	0.105409
$320$	−13.7870	−0.770714
$321$	11.1705	0.623478
$322$	0	0
$323$	−13.5992	−0.756680
$324$	−0.904885	−0.0502714
$325$	−4.57166	−0.253590
$326$	19.1715	1.06181
$327$	−14.9447	−0.826446
$328$	−3.03990	−0.167850
$329$	0	0
$330$	0.846299	0.0465872
$331$	−29.0301	−1.59564	−0.797819	−	0.602897i	$-0.794013\pi$
$331$	−29.0301	−1.59564	−0.797819	+	0.602897i	$0.794013\pi$
$332$	−6.88458	−0.377841
$333$	−9.17863	−0.502986
$334$	13.5459	0.741198
$335$	25.7815	1.40859
$336$	0	0
$337$	−23.8798	−1.30082	−0.650408	−	0.759585i	$-0.725402\pi$
$337$	−23.8798	−1.30082	−0.650408	+	0.759585i	$0.725402\pi$
$338$	6.14219	0.334091
$339$	−11.9484	−0.648949
$340$	−4.70759	−0.255305
$341$	3.09780	0.167755
$342$	−4.96023	−0.268219
$343$	0	0
$344$	−23.7100	−1.27836
$345$	6.52477	0.351282
$346$	15.5594	0.836481
$347$	−6.57264	−0.352838	−0.176419	−	0.984315i	$-0.556451\pi$
$347$	−6.57264	−0.352838	−0.176419	+	0.984315i	$0.556451\pi$
$348$	−3.81977	−0.204761
$349$	−9.91409	−0.530689	−0.265345	−	0.964154i	$-0.585486\pi$
$349$	−9.91409	−0.530689	−0.265345	+	0.964154i	$0.585486\pi$
$350$	0	0
$351$	−2.67032	−0.142531
$352$	2.07148	0.110411
$353$	−6.00923	−0.319839	−0.159920	−	0.987130i	$-0.551123\pi$
$353$	−6.00923	−0.319839	−0.159920	+	0.987130i	$0.551123\pi$
$354$	−5.63755	−0.299633
$355$	−28.0695	−1.48977
$356$	1.99346	0.105653
$357$	0	0
$358$	0.408783	0.0216049
$359$	17.1440	0.904826	0.452413	−	0.891808i	$-0.350563\pi$
$359$	17.1440	0.904826	0.452413	+	0.891808i	$0.350563\pi$
$360$	−5.51217	−0.290517
$361$	3.46694	0.182470
$362$	−1.97639	−0.103877
$363$	10.8011	0.566910
$364$	0	0
$365$	−22.9351	−1.20048
$366$	7.60458	0.397498
$367$	−0.197446	−0.0103066	−0.00515330	−	0.999987i	$-0.501640\pi$
$367$	−0.197446	−0.0103066	−0.00515330	+	0.999987i	$0.501640\pi$
$368$	−4.93480	−0.257244
$369$	−1.00000	−0.0520579
$370$	−17.4169	−0.905464
$371$	0	0
$372$	−6.28517	−0.325871
$373$	−35.1529	−1.82015	−0.910074	−	0.414447i	$-0.863975\pi$
$373$	−35.1529	−1.82015	−0.910074	+	0.414447i	$0.863975\pi$
$374$	−1.33906	−0.0692414
$375$	−12.1708	−0.628496
$376$	34.9256	1.80115
$377$	−11.2722	−0.580545
$378$	0	0
$379$	−32.5491	−1.67193	−0.835967	−	0.548779i	$-0.815093\pi$
$379$	−32.5491	−1.67193	−0.835967	+	0.548779i	$0.815093\pi$
$380$	7.77731	0.398967
$381$	4.63833	0.237629
$382$	9.40080	0.480987
$383$	−16.1072	−0.823037	−0.411519	−	0.911401i	$-0.635001\pi$
$383$	−16.1072	−0.823037	−0.411519	+	0.911401i	$0.635001\pi$
$384$	−1.33256	−0.0680017
$385$	0	0
$386$	−17.0543	−0.868041
$387$	−7.79959	−0.396476
$388$	1.58411	0.0804209
$389$	−33.7726	−1.71234	−0.856168	−	0.516697i	$-0.827161\pi$
$389$	−33.7726	−1.71234	−0.856168	+	0.516697i	$0.827161\pi$
$390$	−5.06707	−0.256581
$391$	−10.3239	−0.522101
$392$	0	0
$393$	7.50200	0.378426
$394$	−18.8796	−0.951141
$395$	−9.10624	−0.458184
$396$	0.403574	0.0202803
$397$	18.8905	0.948087	0.474044	−	0.880501i	$-0.342794\pi$
$397$	18.8905	0.948087	0.474044	+	0.880501i	$0.342794\pi$
$398$	3.47706	0.174289
$399$	0	0
$400$	2.34790	0.117395
$401$	−15.2704	−0.762566	−0.381283	−	0.924458i	$-0.624518\pi$
$401$	−15.2704	−0.762566	−0.381283	+	0.924458i	$0.624518\pi$
$402$	−14.8790	−0.742096
$403$	−18.5476	−0.923920
$404$	−9.11461	−0.453469
$405$	−1.81328	−0.0901024
$406$	0	0
$407$	4.09362	0.202913
$408$	8.72169	0.431788
$409$	−32.9169	−1.62764	−0.813818	−	0.581119i	$-0.802615\pi$
$409$	−32.9169	−1.62764	−0.813818	+	0.581119i	$0.802615\pi$
$410$	−1.89755	−0.0937135
$411$	18.9937	0.936889
$412$	−14.1518	−0.697210
$413$	0	0
$414$	−3.76557	−0.185068
$415$	−13.7958	−0.677212
$416$	−12.4027	−0.608091
$417$	12.7746	0.625577
$418$	2.21224	0.108204
$419$	36.0486	1.76109	0.880544	−	0.473964i	$-0.157177\pi$
$419$	36.0486	1.76109	0.880544	+	0.473964i	$0.157177\pi$
$420$	0	0
$421$	−2.11308	−0.102985	−0.0514927	−	0.998673i	$-0.516398\pi$
$421$	−2.11308	−0.102985	−0.0514927	+	0.998673i	$0.516398\pi$
$422$	−19.2364	−0.936414
$423$	11.4891	0.558618
$424$	31.7366	1.54127
$425$	4.91193	0.238264
$426$	16.1994	0.784865
$427$	0	0
$428$	10.1080	0.488590
$429$	1.19095	0.0574995
$430$	−14.8001	−0.713726
$431$	25.2339	1.21547	0.607736	−	0.794139i	$-0.292078\pi$
$431$	25.2339	1.21547	0.607736	+	0.794139i	$0.292078\pi$
$432$	1.37141	0.0659822
$433$	−23.1074	−1.11047	−0.555235	−	0.831694i	$-0.687372\pi$
$433$	−23.1074	−1.11047	−0.555235	+	0.831694i	$0.687372\pi$
$434$	0	0
$435$	−7.65434	−0.366998
$436$	−13.5233	−0.647647
$437$	17.0558	0.815891
$438$	13.2363	0.632454
$439$	−21.0733	−1.00577	−0.502887	−	0.864352i	$-0.667729\pi$
$439$	−21.0733	−1.00577	−0.502887	+	0.864352i	$0.667729\pi$
$440$	2.45840	0.117200
$441$	0	0
$442$	8.01742	0.381350
$443$	10.7771	0.512035	0.256017	−	0.966672i	$-0.417590\pi$
$443$	10.7771	0.512035	0.256017	+	0.966672i	$0.417590\pi$
$444$	−8.30561	−0.394167
$445$	3.99465	0.189365
$446$	−8.67082	−0.410575
$447$	6.86535	0.324720
$448$	0	0
$449$	−1.96438	−0.0927048	−0.0463524	−	0.998925i	$-0.514760\pi$
$449$	−1.96438	−0.0927048	−0.0463524	+	0.998925i	$0.514760\pi$
$450$	1.79160	0.0844568
$451$	0.445995	0.0210011
$452$	−10.8119	−0.508551
$453$	6.76282	0.317745
$454$	−7.39055	−0.346856
$455$	0	0
$456$	−14.4089	−0.674758
$457$	10.4268	0.487744	0.243872	−	0.969807i	$-0.421582\pi$
$457$	10.4268	0.487744	0.243872	+	0.969807i	$0.421582\pi$
$458$	−21.2431	−0.992627
$459$	2.86907	0.133917
$460$	5.90416	0.275283
$461$	28.1751	1.31224	0.656122	−	0.754655i	$-0.272196\pi$
$461$	28.1751	1.31224	0.656122	+	0.754655i	$0.272196\pi$
$462$	0	0
$463$	−5.96875	−0.277392	−0.138696	−	0.990335i	$-0.544291\pi$
$463$	−5.96875	−0.277392	−0.138696	+	0.990335i	$0.544291\pi$
$464$	5.78912	0.268753
$465$	−12.5947	−0.584065
$466$	−4.32353	−0.200284
$467$	0.120608	0.00558107	0.00279053	−	0.999996i	$-0.499112\pi$
$467$	0.120608	0.00558107	0.00279053	+	0.999996i	$0.499112\pi$
$468$	−2.41633	−0.111695
$469$	0	0
$470$	21.8011	1.00561
$471$	3.74193	0.172419
$472$	−16.3764	−0.753787
$473$	3.47858	0.159945
$474$	5.25539	0.241388
$475$	−8.11489	−0.372337
$476$	0	0
$477$	10.4400	0.478016
$478$	5.92039	0.270792
$479$	40.6769	1.85858	0.929288	−	0.369356i	$-0.120422\pi$
$479$	40.6769	1.85858	0.929288	+	0.369356i	$0.120422\pi$
$480$	−8.42202	−0.384411
$481$	−24.5099	−1.11755
$482$	18.8822	0.860061
$483$	0	0
$484$	9.77374	0.444261
$485$	3.17436	0.144140
$486$	1.04648	0.0474692
$487$	−28.6842	−1.29981	−0.649903	−	0.760017i	$-0.725190\pi$
$487$	−28.6842	−1.29981	−0.649903	+	0.760017i	$0.725190\pi$
$488$	22.0904	0.999987
$489$	18.3200	0.828460
$490$	0	0
$491$	2.32175	0.104779	0.0523896	−	0.998627i	$-0.483316\pi$
$491$	2.32175	0.104779	0.0523896	+	0.998627i	$0.483316\pi$
$492$	−0.904885	−0.0407954
$493$	12.1112	0.545459
$494$	−13.2454	−0.595939
$495$	0.808712	0.0363489
$496$	9.52560	0.427712
$497$	0	0
$498$	7.96186	0.356779
$499$	11.9822	0.536397	0.268198	−	0.963364i	$-0.413572\pi$
$499$	11.9822	0.536397	0.268198	+	0.963364i	$0.413572\pi$
$500$	−11.0131	−0.492523
$501$	12.9443	0.578307
$502$	−17.3004	−0.772156
$503$	−8.06028	−0.359390	−0.179695	−	0.983722i	$-0.557511\pi$
$503$	−8.06028	−0.359390	−0.179695	+	0.983722i	$0.557511\pi$
$504$	0	0
$505$	−18.2645	−0.812762
$506$	1.67942	0.0746595
$507$	5.86940	0.260669
$508$	4.19716	0.186219
$509$	−10.8090	−0.479101	−0.239550	−	0.970884i	$-0.577000\pi$
$509$	−10.8090	−0.479101	−0.239550	+	0.970884i	$0.577000\pi$
$510$	5.44422	0.241074
$511$	0	0
$512$	14.7076	0.649992
$513$	−4.73993	−0.209273
$514$	−28.4866	−1.25649
$515$	−28.3585	−1.24962
$516$	−7.05773	−0.310699
$517$	−5.12406	−0.225356
$518$	0	0
$519$	14.8684	0.652650
$520$	−14.7193	−0.645482
$521$	14.9072	0.653095	0.326547	−	0.945181i	$-0.394115\pi$
$521$	14.9072	0.653095	0.326547	+	0.945181i	$0.394115\pi$
$522$	4.41747	0.193347
$523$	24.1148	1.05447	0.527233	−	0.849721i	$-0.323230\pi$
$523$	24.1148	1.05447	0.527233	+	0.849721i	$0.323230\pi$
$524$	6.78844	0.296554
$525$	0	0
$526$	−0.204184	−0.00890284
$527$	19.9281	0.868081
$528$	−0.611643	−0.0266184
$529$	−10.0520	−0.437045
$530$	19.8105	0.860514
$531$	−5.38717	−0.233783
$532$	0	0
$533$	−2.67032	−0.115664
$534$	−2.30539	−0.0997641
$535$	20.2553	0.875711
$536$	−43.2217	−1.86689
$537$	0.390628	0.0168568
$538$	24.2235	1.04435
$539$	0	0
$540$	−1.64081	−0.0706091
$541$	−21.2142	−0.912072	−0.456036	−	0.889961i	$-0.650731\pi$
$541$	−21.2142	−0.912072	−0.456036	+	0.889961i	$0.650731\pi$
$542$	7.14386	0.306855
$543$	−1.88861	−0.0810482
$544$	13.3258	0.571339
$545$	−27.0989	−1.16079
$546$	0	0
$547$	−5.39628	−0.230728	−0.115364	−	0.993323i	$-0.536803\pi$
$547$	−5.39628	−0.230728	−0.115364	+	0.993323i	$0.536803\pi$
$548$	17.1871	0.734196
$549$	7.26684	0.310141
$550$	−0.799044	−0.0340713
$551$	−20.0086	−0.852393
$552$	−10.9385	−0.465575
$553$	0	0
$554$	−28.9981	−1.23201
$555$	−16.6434	−0.706473
$556$	11.5596	0.490236
$557$	−42.9811	−1.82117	−0.910584	−	0.413324i	$-0.864368\pi$
$557$	−42.9811	−1.82117	−0.910584	+	0.413324i	$0.864368\pi$
$558$	7.26865	0.307707
$559$	−20.8274	−0.880905
$560$	0	0
$561$	−1.27959	−0.0540244
$562$	4.16056	0.175503
$563$	−43.8925	−1.84985	−0.924924	−	0.380153i	$-0.875871\pi$
$563$	−43.8925	−1.84985	−0.924924	+	0.380153i	$0.875871\pi$
$564$	10.3963	0.437762
$565$	−21.6658	−0.911487
$566$	13.1785	0.553935
$567$	0	0
$568$	47.0575	1.97449
$569$	−10.8598	−0.455267	−0.227634	−	0.973747i	$-0.573099\pi$
$569$	−10.8598	−0.455267	−0.227634	+	0.973747i	$0.573099\pi$
$570$	−8.99427	−0.376728
$571$	14.8272	0.620498	0.310249	−	0.950655i	$-0.399588\pi$
$571$	14.8272	0.620498	0.310249	+	0.950655i	$0.399588\pi$
$572$	1.07767	0.0450597
$573$	8.98328	0.375282
$574$	0	0
$575$	−6.16044	−0.256908
$576$	7.60334	0.316806
$577$	17.5161	0.729204	0.364602	−	0.931164i	$-0.381205\pi$
$577$	17.5161	0.729204	0.364602	+	0.931164i	$0.381205\pi$
$578$	9.17595	0.381669
$579$	−16.2969	−0.677275
$580$	−6.92630	−0.287599
$581$	0	0
$582$	−1.83198	−0.0759381
$583$	−4.65620	−0.192840
$584$	38.4498	1.59107
$585$	−4.84203	−0.200193
$586$	−19.6712	−0.812608
$587$	−17.4318	−0.719488	−0.359744	−	0.933051i	$-0.617136\pi$
$587$	−17.4318	−0.719488	−0.359744	+	0.933051i	$0.617136\pi$
$588$	0	0
$589$	−32.9227	−1.35656
$590$	−10.2224	−0.420851
$591$	−18.0411	−0.742112
$592$	12.5877	0.517352
$593$	33.0499	1.35720	0.678598	−	0.734510i	$-0.262588\pi$
$593$	33.0499	1.35720	0.678598	+	0.734510i	$0.262588\pi$
$594$	−0.466723	−0.0191499
$595$	0	0
$596$	6.21235	0.254468
$597$	3.32264	0.135986
$598$	−10.0553	−0.411191
$599$	47.6320	1.94619	0.973095	−	0.230405i	$-0.0740052\pi$
$599$	47.6320	1.94619	0.973095	+	0.230405i	$0.0740052\pi$
$600$	5.20439	0.212468
$601$	14.1223	0.576060	0.288030	−	0.957621i	$-0.407000\pi$
$601$	14.1223	0.576060	0.288030	+	0.957621i	$0.407000\pi$
$602$	0	0
$603$	−14.2182	−0.579008
$604$	6.11957	0.249002
$605$	19.5854	0.796258
$606$	10.5408	0.428192
$607$	−25.8046	−1.04738	−0.523688	−	0.851910i	$-0.675444\pi$
$607$	−25.8046	−1.04738	−0.523688	+	0.851910i	$0.675444\pi$
$608$	−22.0153	−0.892837
$609$	0	0
$610$	13.7892	0.558309
$611$	30.6795	1.24116
$612$	2.59618	0.104944
$613$	−7.36270	−0.297377	−0.148688	−	0.988884i	$-0.547505\pi$
$613$	−7.36270	−0.297377	−0.148688	+	0.988884i	$0.547505\pi$
$614$	16.5775	0.669015
$615$	−1.81328	−0.0731184
$616$	0	0
$617$	−42.7294	−1.72022	−0.860110	−	0.510109i	$-0.829605\pi$
$617$	−42.7294	−1.72022	−0.860110	+	0.510109i	$0.829605\pi$
$618$	16.3662	0.658347
$619$	9.25040	0.371805	0.185902	−	0.982568i	$-0.440479\pi$
$619$	9.25040	0.371805	0.185902	+	0.982568i	$0.440479\pi$
$620$	−11.3968	−0.457705
$621$	−3.59833	−0.144396
$622$	−24.0556	−0.964543
$623$	0	0
$624$	3.66211	0.146602
$625$	−13.5088	−0.540353
$626$	14.3760	0.574580
$627$	2.11398	0.0844244
$628$	3.38602	0.135117
$629$	26.3342	1.05001
$630$	0	0
$631$	−2.08724	−0.0830918	−0.0415459	−	0.999137i	$-0.513228\pi$
$631$	−2.08724	−0.0830918	−0.0415459	+	0.999137i	$0.513228\pi$
$632$	15.2663	0.607261
$633$	−18.3821	−0.730621
$634$	−2.14516	−0.0851953
$635$	8.41058	0.333764
$636$	9.44703	0.374599
$637$	0	0
$638$	−1.97017	−0.0779997
$639$	15.4800	0.612378
$640$	−2.41629	−0.0955123
$641$	8.30068	0.327857	0.163929	−	0.986472i	$-0.447583\pi$
$641$	8.30068	0.327857	0.163929	+	0.986472i	$0.447583\pi$
$642$	−11.6897	−0.461356
$643$	39.5985	1.56161	0.780807	−	0.624772i	$-0.214808\pi$
$643$	39.5985	1.56161	0.780807	+	0.624772i	$0.214808\pi$
$644$	0	0
$645$	−14.1428	−0.556873
$646$	14.2313	0.559922
$647$	−7.17653	−0.282139	−0.141069	−	0.990000i	$-0.545054\pi$
$647$	−7.17653	−0.282139	−0.141069	+	0.990000i	$0.545054\pi$
$648$	3.03990	0.119418
$649$	2.40265	0.0943123
$650$	4.78414	0.187649
$651$	0	0
$652$	16.5775	0.649226
$653$	−25.7808	−1.00888	−0.504440	−	0.863447i	$-0.668301\pi$
$653$	−25.7808	−1.00888	−0.504440	+	0.863447i	$0.668301\pi$
$654$	15.6393	0.611546
$655$	13.6032	0.531521
$656$	1.37141	0.0535447
$657$	12.6484	0.493461
$658$	0	0
$659$	2.94995	0.114914	0.0574569	−	0.998348i	$-0.481701\pi$
$659$	2.94995	0.114914	0.0574569	+	0.998348i	$0.481701\pi$
$660$	0.731791	0.0284849
$661$	−17.7965	−0.692204	−0.346102	−	0.938197i	$-0.612495\pi$
$661$	−17.7965	−0.692204	−0.346102	+	0.938197i	$0.612495\pi$
$662$	30.3793	1.18073
$663$	7.66134	0.297542
$664$	23.1283	0.897551
$665$	0	0
$666$	9.60523	0.372195
$667$	−15.1895	−0.588141
$668$	11.7131	0.453193
$669$	−8.28572	−0.320345
$670$	−26.9797	−1.04232
$671$	−3.24097	−0.125116
$672$	0	0
$673$	35.0368	1.35057	0.675285	−	0.737557i	$-0.264021\pi$
$673$	35.0368	1.35057	0.675285	+	0.737557i	$0.264021\pi$
$674$	24.9897	0.962567
$675$	1.71203	0.0658960
$676$	5.31113	0.204274
$677$	−32.5996	−1.25291	−0.626453	−	0.779459i	$-0.715494\pi$
$677$	−32.5996	−1.25291	−0.626453	+	0.779459i	$0.715494\pi$
$678$	12.5038	0.480204
$679$	0	0
$680$	15.8148	0.606471
$681$	−7.06231	−0.270628
$682$	−3.24178	−0.124134
$683$	−36.4205	−1.39359	−0.696796	−	0.717269i	$-0.745392\pi$
$683$	−36.4205	−1.39359	−0.696796	+	0.717269i	$0.745392\pi$
$684$	−4.28909	−0.163997
$685$	34.4408	1.31592
$686$	0	0
$687$	−20.2997	−0.774480
$688$	10.6965	0.407799
$689$	27.8782	1.06208
$690$	−6.82802	−0.259938
$691$	−25.8214	−0.982293	−0.491147	−	0.871077i	$-0.663422\pi$
$691$	−25.8214	−0.982293	−0.491147	+	0.871077i	$0.663422\pi$
$692$	13.4542	0.511451
$693$	0	0
$694$	6.87812	0.261090
$695$	23.1640	0.878659
$696$	12.8322	0.486405
$697$	2.86907	0.108674
$698$	10.3749	0.392695
$699$	−4.13151	−0.156268
$700$	0	0
$701$	−7.64248	−0.288653	−0.144326	−	0.989530i	$-0.546101\pi$
$701$	−7.64248	−0.288653	−0.144326	+	0.989530i	$0.546101\pi$
$702$	2.79443	0.105469
$703$	−43.5061	−1.64086
$704$	−3.39105	−0.127805
$705$	20.8329	0.784611
$706$	6.28853	0.236672
$707$	0	0
$708$	−4.87477	−0.183205
$709$	−16.9791	−0.637664	−0.318832	−	0.947811i	$-0.603290\pi$
$709$	−16.9791	−0.637664	−0.318832	+	0.947811i	$0.603290\pi$
$710$	29.3741	1.10239
$711$	5.02198	0.188339
$712$	−6.69690	−0.250977
$713$	−24.9934	−0.936009
$714$	0	0
$715$	2.15952	0.0807614
$716$	0.353473	0.0132099
$717$	5.65744	0.211281
$718$	−17.9408	−0.669545
$719$	1.92341	0.0717310	0.0358655	−	0.999357i	$-0.488581\pi$
$719$	1.92341	0.0717310	0.0358655	+	0.999357i	$0.488581\pi$
$720$	2.48675	0.0926758
$721$	0	0
$722$	−3.62807	−0.135023
$723$	18.0436	0.671049
$724$	−1.70898	−0.0635137
$725$	7.22694	0.268402
$726$	−11.3031	−0.419497
$727$	5.46716	0.202766	0.101383	−	0.994847i	$-0.467673\pi$
$727$	5.46716	0.202766	0.101383	+	0.994847i	$0.467673\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	24.0010	0.888318
$731$	22.3776	0.827666
$732$	6.57565	0.243043
$733$	−19.6346	−0.725219	−0.362609	−	0.931941i	$-0.618114\pi$
$733$	−19.6346	−0.725219	−0.362609	+	0.931941i	$0.618114\pi$
$734$	0.206623	0.00762659
$735$	0	0
$736$	−16.7129	−0.616047
$737$	6.34122	0.233582
$738$	1.04648	0.0385214
$739$	33.1279	1.21863	0.609315	−	0.792929i	$-0.291445\pi$
$739$	33.1279	1.21863	0.609315	+	0.792929i	$0.291445\pi$
$740$	−15.0604	−0.553630
$741$	−12.6571	−0.464971
$742$	0	0
$743$	−14.1798	−0.520206	−0.260103	−	0.965581i	$-0.583756\pi$
$743$	−14.1798	−0.520206	−0.260103	+	0.965581i	$0.583756\pi$
$744$	21.1146	0.774099
$745$	12.4488	0.456088
$746$	36.7867	1.34686
$747$	7.60824	0.278371
$748$	−1.15788	−0.0423364
$749$	0	0
$750$	12.7364	0.465069
$751$	−3.84986	−0.140483	−0.0702416	−	0.997530i	$-0.522377\pi$
$751$	−3.84986	−0.140483	−0.0702416	+	0.997530i	$0.522377\pi$
$752$	−15.7563	−0.574572
$753$	−16.5321	−0.602462
$754$	11.7961	0.429587
$755$	12.2629	0.446291
$756$	0	0
$757$	−45.9774	−1.67108	−0.835539	−	0.549432i	$-0.814844\pi$
$757$	−45.9774	−1.67108	−0.835539	+	0.549432i	$0.814844\pi$
$758$	34.0619	1.23718
$759$	1.60484	0.0582518
$760$	−26.1273	−0.947737
$761$	−44.3346	−1.60713	−0.803564	−	0.595218i	$-0.797066\pi$
$761$	−44.3346	−1.60713	−0.803564	+	0.595218i	$0.797066\pi$
$762$	−4.85391	−0.175839
$763$	0	0
$764$	8.12883	0.294091
$765$	5.20242	0.188094
$766$	16.8558	0.609024
$767$	−14.3855	−0.519429
$768$	16.6012	0.599043
$769$	4.11151	0.148265	0.0741324	−	0.997248i	$-0.476381\pi$
$769$	4.11151	0.148265	0.0741324	+	0.997248i	$0.476381\pi$
$770$	0	0
$771$	−27.2214	−0.980355
$772$	−14.7468	−0.530749
$773$	19.1489	0.688737	0.344369	−	0.938835i	$-0.388093\pi$
$773$	19.1489	0.688737	0.344369	+	0.938835i	$0.388093\pi$
$774$	8.16210	0.293381
$775$	11.8914	0.427153
$776$	−5.32170	−0.191038
$777$	0	0
$778$	35.3422	1.26708
$779$	−4.73993	−0.169826
$780$	−4.38148	−0.156882
$781$	−6.90398	−0.247044
$782$	10.8037	0.386339
$783$	4.22128	0.150856
$784$	0	0
$785$	6.78516	0.242173
$786$	−7.85067	−0.280024
$787$	42.9937	1.53256	0.766280	−	0.642506i	$-0.222105\pi$
$787$	42.9937	1.53256	0.766280	+	0.642506i	$0.222105\pi$
$788$	−16.3251	−0.581558
$789$	−0.195115	−0.00694629
$790$	9.52947	0.339043
$791$	0	0
$792$	−1.35578	−0.0481755
$793$	19.4048	0.689084
$794$	−19.7685	−0.701557
$795$	18.9307	0.671402
$796$	3.00660	0.106566
$797$	−14.2882	−0.506116	−0.253058	−	0.967451i	$-0.581436\pi$
$797$	−14.2882	−0.506116	−0.253058	+	0.967451i	$0.581436\pi$
$798$	0	0
$799$	−32.9630	−1.16615
$800$	7.95175	0.281137
$801$	−2.20300	−0.0778392
$802$	15.9801	0.564277
$803$	−5.64112	−0.199071
$804$	−12.8658	−0.453742
$805$	0	0
$806$	19.4096	0.683675
$807$	23.1477	0.814837
$808$	30.6199	1.07720
$809$	−22.9124	−0.805555	−0.402778	−	0.915298i	$-0.631955\pi$
$809$	−22.9124	−0.805555	−0.402778	+	0.915298i	$0.631955\pi$
$810$	1.89755	0.0666732
$811$	−33.4405	−1.17425	−0.587127	−	0.809495i	$-0.699741\pi$
$811$	−33.4405	−1.17425	−0.587127	+	0.809495i	$0.699741\pi$
$812$	0	0
$813$	6.82658	0.239419
$814$	−4.28388	−0.150150
$815$	33.2193	1.16362
$816$	−3.93469	−0.137742
$817$	−36.9695	−1.29340
$818$	34.4468	1.20440
$819$	0	0
$820$	−1.64081	−0.0572994
$821$	−19.3510	−0.675355	−0.337677	−	0.941262i	$-0.609641\pi$
$821$	−19.3510	−0.675355	−0.337677	+	0.941262i	$0.609641\pi$
$822$	−19.8765	−0.693271
$823$	14.9217	0.520139	0.260070	−	0.965590i	$-0.416255\pi$
$823$	14.9217	0.520139	0.260070	+	0.965590i	$0.416255\pi$
$824$	47.5420	1.65621
$825$	−0.763555	−0.0265836
$826$	0	0
$827$	−10.1637	−0.353425	−0.176713	−	0.984262i	$-0.556546\pi$
$827$	−10.1637	−0.353425	−0.176713	+	0.984262i	$0.556546\pi$
$828$	−3.25607	−0.113156
$829$	−7.08509	−0.246075	−0.123038	−	0.992402i	$-0.539264\pi$
$829$	−7.08509	−0.246075	−0.123038	+	0.992402i	$0.539264\pi$
$830$	14.4370	0.501117
$831$	−27.7102	−0.961258
$832$	20.3033	0.703892
$833$	0	0
$834$	−13.3684	−0.462909
$835$	23.4715	0.812266
$836$	1.91291	0.0661594
$837$	6.94583	0.240083
$838$	−37.7240	−1.30315
$839$	−23.0788	−0.796770	−0.398385	−	0.917218i	$-0.630429\pi$
$839$	−23.0788	−0.796770	−0.398385	+	0.917218i	$0.630429\pi$
$840$	0	0
$841$	−11.1808	−0.385546
$842$	2.21129	0.0762062
$843$	3.97578	0.136933
$844$	−16.6337	−0.572554
$845$	10.6428	0.366125
$846$	−12.0230	−0.413361
$847$	0	0
$848$	−14.3176	−0.491669
$849$	12.5932	0.432199
$850$	−5.14023	−0.176308
$851$	−33.0277	−1.13218
$852$	14.0076	0.479892
$853$	−55.1310	−1.88765	−0.943825	−	0.330446i	$-0.892801\pi$
$853$	−55.1310	−1.88765	−0.943825	+	0.330446i	$0.892801\pi$
$854$	0	0
$855$	−8.59480	−0.293936
$856$	−33.9572	−1.16063
$857$	−21.0816	−0.720135	−0.360068	−	0.932926i	$-0.617246\pi$
$857$	−21.0816	−0.720135	−0.360068	+	0.932926i	$0.617246\pi$
$858$	−1.24630	−0.0425480
$859$	−35.7227	−1.21884	−0.609421	−	0.792847i	$-0.708598\pi$
$859$	−35.7227	−1.21884	−0.609421	+	0.792847i	$0.708598\pi$
$860$	−12.7976	−0.436395
$861$	0	0
$862$	−26.4067	−0.899415
$863$	−41.9838	−1.42914	−0.714572	−	0.699562i	$-0.753378\pi$
$863$	−41.9838	−1.42914	−0.714572	+	0.699562i	$0.753378\pi$
$864$	4.64464	0.158014
$865$	26.9605	0.916685
$866$	24.1814	0.821716
$867$	8.76842	0.297791
$868$	0	0
$869$	−2.23978	−0.0759792
$870$	8.01010	0.271568
$871$	−37.9670	−1.28646
$872$	45.4304	1.53847
$873$	−1.75062	−0.0592495
$874$	−17.8485	−0.603736
$875$	0	0
$876$	11.4453	0.386703
$877$	−54.7628	−1.84921	−0.924604	−	0.380931i	$-0.875604\pi$
$877$	−54.7628	−1.84921	−0.924604	+	0.380931i	$0.875604\pi$
$878$	22.0527	0.744244
$879$	−18.7975	−0.634024
$880$	−1.10908	−0.0373870
$881$	25.1750	0.848166	0.424083	−	0.905623i	$-0.360596\pi$
$881$	25.1750	0.848166	0.424083	+	0.905623i	$0.360596\pi$
$882$	0	0
$883$	55.2757	1.86018	0.930088	−	0.367336i	$-0.119730\pi$
$883$	55.2757	1.86018	0.930088	+	0.367336i	$0.119730\pi$
$884$	6.93263	0.233170
$885$	−9.76843	−0.328362
$886$	−11.2780	−0.378891
$887$	−10.4173	−0.349780	−0.174890	−	0.984588i	$-0.555957\pi$
$887$	−10.4173	−0.349780	−0.174890	+	0.984588i	$0.555957\pi$
$888$	27.9021	0.936333
$889$	0	0
$890$	−4.18031	−0.140124
$891$	−0.445995	−0.0149414
$892$	−7.49762	−0.251039
$893$	54.4574	1.82235
$894$	−7.18443	−0.240283
$895$	0.708316	0.0236764
$896$	0	0
$897$	−9.60868	−0.320825
$898$	2.05568	0.0685989
$899$	29.3202	0.977885
$900$	1.54919	0.0516396
$901$	−29.9532	−0.997887
$902$	−0.466723	−0.0155402
$903$	0	0
$904$	36.3220	1.20805
$905$	−3.42458	−0.113837
$906$	−7.07714	−0.235122
$907$	10.0366	0.333258	0.166629	−	0.986020i	$-0.446712\pi$
$907$	10.0366	0.333258	0.166629	+	0.986020i	$0.446712\pi$
$908$	−6.39058	−0.212079
$909$	10.0727	0.334089
$910$	0	0
$911$	17.5250	0.580628	0.290314	−	0.956931i	$-0.406240\pi$
$911$	17.5250	0.580628	0.290314	+	0.956931i	$0.406240\pi$
$912$	6.50041	0.215250
$913$	−3.39324	−0.112300
$914$	−10.9114	−0.360917
$915$	13.1768	0.435611
$916$	−18.3689	−0.606924
$917$	0	0
$918$	−3.00242	−0.0990947
$919$	−41.1561	−1.35761	−0.678807	−	0.734317i	$-0.737503\pi$
$919$	−41.1561	−1.35761	−0.678807	+	0.734317i	$0.737503\pi$
$920$	−19.8346	−0.653928
$921$	15.8413	0.521988
$922$	−29.4846	−0.971023
$923$	41.3365	1.36061
$924$	0	0
$925$	15.7141	0.516676
$926$	6.24617	0.205262
$927$	15.6394	0.513664
$928$	19.6063	0.643608
$929$	19.3546	0.635002	0.317501	−	0.948258i	$-0.397156\pi$
$929$	19.3546	0.635002	0.317501	+	0.948258i	$0.397156\pi$
$930$	13.1801	0.432192
$931$	0	0
$932$	−3.73854	−0.122460
$933$	−22.9872	−0.752569
$934$	−0.126213	−0.00412983
$935$	−2.32025	−0.0758804
$936$	8.11749	0.265329
$937$	−16.6445	−0.543753	−0.271876	−	0.962332i	$-0.587644\pi$
$937$	−16.6445	−0.543753	−0.271876	+	0.962332i	$0.587644\pi$
$938$	0	0
$939$	13.7375	0.448306
$940$	18.8513	0.614863
$941$	11.3034	0.368480	0.184240	−	0.982881i	$-0.441018\pi$
$941$	11.3034	0.368480	0.184240	+	0.982881i	$0.441018\pi$
$942$	−3.91585	−0.127585
$943$	−3.59833	−0.117178
$944$	7.38804	0.240460
$945$	0	0
$946$	−3.64025	−0.118355
$947$	4.08943	0.132889	0.0664443	−	0.997790i	$-0.478835\pi$
$947$	4.08943	0.132889	0.0664443	+	0.997790i	$0.478835\pi$
$948$	4.54431	0.147592
$949$	33.7753	1.09639
$950$	8.49205	0.275519
$951$	−2.04989	−0.0664722
$952$	0	0
$953$	−14.0371	−0.454707	−0.227353	−	0.973812i	$-0.573007\pi$
$953$	−14.0371	−0.454707	−0.227353	+	0.973812i	$0.573007\pi$
$954$	−10.9253	−0.353718
$955$	16.2892	0.527105
$956$	5.11934	0.165571
$957$	−1.88267	−0.0608580
$958$	−42.5675	−1.37529
$959$	0	0
$960$	13.7870	0.444972
$961$	17.2445	0.556274
$962$	25.6490	0.826958
$963$	−11.1705	−0.359965
$964$	16.3274	0.525869
$965$	−29.5507	−0.951272
$966$	0	0
$967$	43.1943	1.38904	0.694518	−	0.719475i	$-0.255618\pi$
$967$	43.1943	1.38904	0.694518	+	0.719475i	$0.255618\pi$
$968$	−32.8342	−1.05533
$969$	13.5992	0.436869
$970$	−3.32189	−0.106660
$971$	−34.2336	−1.09861	−0.549304	−	0.835623i	$-0.685107\pi$
$971$	−34.2336	−1.09861	−0.549304	+	0.835623i	$0.685107\pi$
$972$	0.904885	0.0290242
$973$	0	0
$974$	30.0174	0.961819
$975$	4.57166	0.146410
$976$	−9.96585	−0.318999
$977$	−25.7841	−0.824908	−0.412454	−	0.910978i	$-0.635328\pi$
$977$	−25.7841	−0.824908	−0.412454	+	0.910978i	$0.635328\pi$
$978$	−19.1715	−0.613037
$979$	0.982527	0.0314017
$980$	0	0
$981$	14.9447	0.477149
$982$	−2.42966	−0.0775336
$983$	−15.7033	−0.500858	−0.250429	−	0.968135i	$-0.580572\pi$
$983$	−15.7033	−0.500858	−0.250429	+	0.968135i	$0.580572\pi$
$984$	3.03990	0.0969084
$985$	−32.7135	−1.04234
$986$	−12.6740	−0.403624
$987$	0	0
$988$	−11.4532	−0.364376
$989$	−28.0655	−0.892431
$990$	−0.846299	−0.0268971
$991$	47.9872	1.52436	0.762182	−	0.647363i	$-0.224128\pi$
$991$	47.9872	1.52436	0.762182	+	0.647363i	$0.224128\pi$
$992$	32.2609	1.02428
$993$	29.0301	0.921242
$994$	0	0
$995$	6.02486	0.191001
$996$	6.88458	0.218146
$997$	32.6743	1.03481	0.517403	−	0.855742i	$-0.326899\pi$
$997$	32.6743	1.03481	0.517403	+	0.855742i	$0.326899\pi$
$998$	−12.5391	−0.396918
$999$	9.17863	0.290399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bl.1.7	✓	16
7.6	odd	2		6027.2.a.bm.1.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.7	✓	16	1.1	even	1	trivial
6027.2.a.bm.1.7	yes	16	7.6	odd	2

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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-1.04648 q^{2} -1.00000 q^{3} -0.904885 q^{4} -1.81328 q^{5} +1.04648 q^{6} +3.03990 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.04648 q^{2} -1.00000 q^{3} -0.904885 q^{4} -1.81328 q^{5} +1.04648 q^{6} +3.03990 q^{8} +1.00000 q^{9} +1.89755 q^{10} -0.445995 q^{11} +0.904885 q^{12} +2.67032 q^{13} +1.81328 q^{15} -1.37141 q^{16} -2.86907 q^{17} -1.04648 q^{18} +4.73993 q^{19} +1.64081 q^{20} +0.466723 q^{22} +3.59833 q^{23} -3.03990 q^{24} -1.71203 q^{25} -2.79443 q^{26} -1.00000 q^{27} -4.22128 q^{29} -1.89755 q^{30} -6.94583 q^{31} -4.64464 q^{32} +0.445995 q^{33} +3.00242 q^{34} -0.904885 q^{36} -9.17863 q^{37} -4.96023 q^{38} -2.67032 q^{39} -5.51217 q^{40} -1.00000 q^{41} -7.79959 q^{43} +0.403574 q^{44} -1.81328 q^{45} -3.76557 q^{46} +11.4891 q^{47} +1.37141 q^{48} +1.79160 q^{50} +2.86907 q^{51} -2.41633 q^{52} +10.4400 q^{53} +1.04648 q^{54} +0.808712 q^{55} -4.73993 q^{57} +4.41747 q^{58} -5.38717 q^{59} -1.64081 q^{60} +7.26684 q^{61} +7.26865 q^{62} +7.60334 q^{64} -4.84203 q^{65} -0.466723 q^{66} -14.2182 q^{67} +2.59618 q^{68} -3.59833 q^{69} +15.4800 q^{71} +3.03990 q^{72} +12.6484 q^{73} +9.60523 q^{74} +1.71203 q^{75} -4.28909 q^{76} +2.79443 q^{78} +5.02198 q^{79} +2.48675 q^{80} +1.00000 q^{81} +1.04648 q^{82} +7.60824 q^{83} +5.20242 q^{85} +8.16210 q^{86} +4.22128 q^{87} -1.35578 q^{88} -2.20300 q^{89} +1.89755 q^{90} -3.25607 q^{92} +6.94583 q^{93} -12.0230 q^{94} -8.59480 q^{95} +4.64464 q^{96} -1.75062 q^{97} -0.445995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} - 16 q^{3} + 12 q^{4} + 12 q^{5} + 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 - 16 * q^3 + 12 * q^4 + 12 * q^5 + 4 * q^6 - 12 * q^8 + 16 * q^9	\( 16 q - 4 q^{2} - 16 q^{3} + 12 q^{4} + 12 q^{5} + 4 q^{6} - 12 q^{8} + 16 q^{9} + 4 q^{10} - 4 q^{11} - 12 q^{12} - 12 q^{15} + 8 q^{17} - 4 q^{18} - 4 q^{19} + 20 q^{20} - 16 q^{22} - 12 q^{23} + 12 q^{24} - 8 q^{25} + 8 q^{26} - 16 q^{27} - 16 q^{29} - 4 q^{30} + 4 q^{31} - 48 q^{32} + 4 q^{33} - 16 q^{34} + 12 q^{36} - 48 q^{37} + 4 q^{38} - 56 q^{40} - 16 q^{41} - 16 q^{43} + 12 q^{45} - 4 q^{46} + 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} + 4 q^{54} - 8 q^{55} + 4 q^{57} - 36 q^{58} + 36 q^{59} - 20 q^{60} + 4 q^{61} + 12 q^{62} + 52 q^{64} - 36 q^{65} + 16 q^{66} - 52 q^{67} + 8 q^{68} + 12 q^{69} - 12 q^{71} - 12 q^{72} + 16 q^{73} + 4 q^{74} + 8 q^{75} - 16 q^{76} - 8 q^{78} - 36 q^{79} + 68 q^{80} + 16 q^{81} + 4 q^{82} + 32 q^{83} - 28 q^{85} - 8 q^{86} + 16 q^{87} - 36 q^{88} + 12 q^{89} + 4 q^{90} - 36 q^{92} - 4 q^{93} - 24 q^{94} - 20 q^{95} + 48 q^{96} - 48 q^{97} - 4 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 - 16 * q^3 + 12 * q^4 + 12 * q^5 + 4 * q^6 - 12 * q^8 + 16 * q^9 + 4 * q^10 - 4 * q^11 - 12 * q^12 - 12 * q^15 + 8 * q^17 - 4 * q^18 - 4 * q^19 + 20 * q^20 - 16 * q^22 - 12 * q^23 + 12 * q^24 - 8 * q^25 + 8 * q^26 - 16 * q^27 - 16 * q^29 - 4 * q^30 + 4 * q^31 - 48 * q^32 + 4 * q^33 - 16 * q^34 + 12 * q^36 - 48 * q^37 + 4 * q^38 - 56 * q^40 - 16 * q^41 - 16 * q^43 + 12 * q^45 - 4 * q^46 + 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 + 4 * q^54 - 8 * q^55 + 4 * q^57 - 36 * q^58 + 36 * q^59 - 20 * q^60 + 4 * q^61 + 12 * q^62 + 52 * q^64 - 36 * q^65 + 16 * q^66 - 52 * q^67 + 8 * q^68 + 12 * q^69 - 12 * q^71 - 12 * q^72 + 16 * q^73 + 4 * q^74 + 8 * q^75 - 16 * q^76 - 8 * q^78 - 36 * q^79 + 68 * q^80 + 16 * q^81 + 4 * q^82 + 32 * q^83 - 28 * q^85 - 8 * q^86 + 16 * q^87 - 36 * q^88 + 12 * q^89 + 4 * q^90 - 36 * q^92 - 4 * q^93 - 24 * q^94 - 20 * q^95 + 48 * q^96 - 48 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more