LMFDB - Embedded newform 6027.2.a.bm.1.14

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			1.14
Root			$-1.87592$ 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)

$f(q)$	$=$	$q+1.87592 q^{2} +1.00000 q^{3} +1.51907 q^{4} -1.33107 q^{5} +1.87592 q^{6} -0.902182 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.87592 q^{2} +1.00000 q^{3} +1.51907 q^{4} -1.33107 q^{5} +1.87592 q^{6} -0.902182 q^{8} +1.00000 q^{9} -2.49697 q^{10} +1.19223 q^{11} +1.51907 q^{12} +0.956777 q^{13} -1.33107 q^{15} -4.73056 q^{16} +3.26275 q^{17} +1.87592 q^{18} -6.16735 q^{19} -2.02199 q^{20} +2.23653 q^{22} -6.99354 q^{23} -0.902182 q^{24} -3.22826 q^{25} +1.79484 q^{26} +1.00000 q^{27} -2.28683 q^{29} -2.49697 q^{30} -1.84740 q^{31} -7.06979 q^{32} +1.19223 q^{33} +6.12065 q^{34} +1.51907 q^{36} -6.15612 q^{37} -11.5695 q^{38} +0.956777 q^{39} +1.20086 q^{40} +1.00000 q^{41} -5.98160 q^{43} +1.81108 q^{44} -1.33107 q^{45} -13.1193 q^{46} +6.24381 q^{47} -4.73056 q^{48} -6.05596 q^{50} +3.26275 q^{51} +1.45341 q^{52} +11.8675 q^{53} +1.87592 q^{54} -1.58694 q^{55} -6.16735 q^{57} -4.28990 q^{58} -4.26878 q^{59} -2.02199 q^{60} +4.85206 q^{61} -3.46558 q^{62} -3.80123 q^{64} -1.27353 q^{65} +2.23653 q^{66} +3.76060 q^{67} +4.95635 q^{68} -6.99354 q^{69} -11.3437 q^{71} -0.902182 q^{72} -6.06858 q^{73} -11.5484 q^{74} -3.22826 q^{75} -9.36866 q^{76} +1.79484 q^{78} -14.1656 q^{79} +6.29670 q^{80} +1.00000 q^{81} +1.87592 q^{82} +2.10474 q^{83} -4.34294 q^{85} -11.2210 q^{86} -2.28683 q^{87} -1.07561 q^{88} -9.71613 q^{89} -2.49697 q^{90} -10.6237 q^{92} -1.84740 q^{93} +11.7129 q^{94} +8.20916 q^{95} -7.06979 q^{96} +0.622368 q^{97} +1.19223 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.87592	1.32648	0.663238	−	0.748409i	$-0.269182\pi$
$2$	1.87592	1.32648	0.663238	+	0.748409i	$0.269182\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.51907	0.759536
$5$	−1.33107	−0.595271	−0.297636	−	0.954680i	$-0.596198\pi$
$5$	−1.33107	−0.595271	−0.297636	+	0.954680i	$0.596198\pi$
$6$	1.87592	0.765841
$7$	0	0
$7$	0	0
$8$	−0.902182	−0.318969
$9$	1.00000	0.333333
$10$	−2.49697	−0.789612
$11$	1.19223	0.359471	0.179736	−	0.983715i	$-0.442476\pi$
$11$	1.19223	0.359471	0.179736	+	0.983715i	$0.442476\pi$
$12$	1.51907	0.438518
$13$	0.956777	0.265362	0.132681	−	0.991159i	$-0.457641\pi$
$13$	0.956777	0.265362	0.132681	+	0.991159i	$0.457641\pi$
$14$	0	0
$15$	−1.33107	−0.343680
$16$	−4.73056	−1.18264
$17$	3.26275	0.791333	0.395666	−	0.918394i	$-0.370514\pi$
$17$	3.26275	0.791333	0.395666	+	0.918394i	$0.370514\pi$
$18$	1.87592	0.442158
$19$	−6.16735	−1.41489	−0.707444	−	0.706769i	$-0.750152\pi$
$19$	−6.16735	−1.41489	−0.707444	+	0.706769i	$0.750152\pi$
$20$	−2.02199	−0.452130
$21$	0	0
$22$	2.23653	0.476830
$23$	−6.99354	−1.45825	−0.729127	−	0.684378i	$-0.760074\pi$
$23$	−6.99354	−1.45825	−0.729127	+	0.684378i	$0.760074\pi$
$24$	−0.902182	−0.184157
$25$	−3.22826	−0.645652
$26$	1.79484	0.351996
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.28683	−0.424653	−0.212327	−	0.977199i	$-0.568104\pi$
$29$	−2.28683	−0.424653	−0.212327	+	0.977199i	$0.568104\pi$
$30$	−2.49697	−0.455883
$31$	−1.84740	−0.331804	−0.165902	−	0.986142i	$-0.553053\pi$
$31$	−1.84740	−0.331804	−0.165902	+	0.986142i	$0.553053\pi$
$32$	−7.06979	−1.24977
$33$	1.19223	0.207541
$34$	6.12065	1.04968
$35$	0	0
$36$	1.51907	0.253179
$37$	−6.15612	−1.01206	−0.506030	−	0.862516i	$-0.668888\pi$
$37$	−6.15612	−1.01206	−0.506030	+	0.862516i	$0.668888\pi$
$38$	−11.5695	−1.87681
$39$	0.956777	0.153207
$40$	1.20086	0.189873
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−5.98160	−0.912186	−0.456093	−	0.889932i	$-0.650752\pi$
$43$	−5.98160	−0.912186	−0.456093	+	0.889932i	$0.650752\pi$
$44$	1.81108	0.273031
$45$	−1.33107	−0.198424
$46$	−13.1193	−1.93434
$47$	6.24381	0.910754	0.455377	−	0.890299i	$-0.349505\pi$
$47$	6.24381	0.910754	0.455377	+	0.890299i	$0.349505\pi$
$48$	−4.73056	−0.682798
$49$	0	0
$50$	−6.05596	−0.856442
$51$	3.26275	0.456876
$52$	1.45341	0.201552
$53$	11.8675	1.63012	0.815060	−	0.579377i	$-0.196704\pi$
$53$	11.8675	1.63012	0.815060	+	0.579377i	$0.196704\pi$
$54$	1.87592	0.255280
$55$	−1.58694	−0.213983
$56$	0	0
$57$	−6.16735	−0.816886
$58$	−4.28990	−0.563292
$59$	−4.26878	−0.555748	−0.277874	−	0.960618i	$-0.589630\pi$
$59$	−4.26878	−0.555748	−0.277874	+	0.960618i	$0.589630\pi$
$60$	−2.02199	−0.261037
$61$	4.85206	0.621242	0.310621	−	0.950534i	$-0.399463\pi$
$61$	4.85206	0.621242	0.310621	+	0.950534i	$0.399463\pi$
$62$	−3.46558	−0.440129
$63$	0	0
$64$	−3.80123	−0.475153
$65$	−1.27353	−0.157962
$66$	2.23653	0.275298
$67$	3.76060	0.459430	0.229715	−	0.973258i	$-0.426221\pi$
$67$	3.76060	0.459430	0.229715	+	0.973258i	$0.426221\pi$
$68$	4.95635	0.601046
$69$	−6.99354	−0.841923
$70$	0	0
$71$	−11.3437	−1.34625	−0.673124	−	0.739530i	$-0.735048\pi$
$71$	−11.3437	−1.34625	−0.673124	+	0.739530i	$0.735048\pi$
$72$	−0.902182	−0.106323
$73$	−6.06858	−0.710273	−0.355137	−	0.934814i	$-0.615566\pi$
$73$	−6.06858	−0.710273	−0.355137	+	0.934814i	$0.615566\pi$
$74$	−11.5484	−1.34247
$75$	−3.22826	−0.372768
$76$	−9.36866	−1.07466
$77$	0	0
$78$	1.79484	0.203225
$79$	−14.1656	−1.59376	−0.796878	−	0.604140i	$-0.793517\pi$
$79$	−14.1656	−1.59376	−0.796878	+	0.604140i	$0.793517\pi$
$80$	6.29670	0.703992
$81$	1.00000	0.111111
$82$	1.87592	0.207161
$83$	2.10474	0.231025	0.115512	−	0.993306i	$-0.463149\pi$
$83$	2.10474	0.231025	0.115512	+	0.993306i	$0.463149\pi$
$84$	0	0
$85$	−4.34294	−0.471058
$86$	−11.2210	−1.20999
$87$	−2.28683	−0.245174
$88$	−1.07561	−0.114660
$89$	−9.71613	−1.02991	−0.514954	−	0.857218i	$-0.672191\pi$
$89$	−9.71613	−1.02991	−0.514954	+	0.857218i	$0.672191\pi$
$90$	−2.49697	−0.263204
$91$	0	0
$92$	−10.6237	−1.10760
$93$	−1.84740	−0.191567
$94$	11.7129	1.20809
$95$	8.20916	0.842242
$96$	−7.06979	−0.721558
$97$	0.622368	0.0631919	0.0315960	−	0.999501i	$-0.489941\pi$
$97$	0.622368	0.0631919	0.0315960	+	0.999501i	$0.489941\pi$
$98$	0	0
$99$	1.19223	0.119824
$100$	−4.90396	−0.490396
$101$	−7.06334	−0.702829	−0.351415	−	0.936220i	$-0.614299\pi$
$101$	−7.06334	−0.702829	−0.351415	+	0.936220i	$0.614299\pi$
$102$	6.12065	0.606035
$103$	17.8241	1.75626	0.878132	−	0.478418i	$-0.158790\pi$
$103$	17.8241	1.75626	0.878132	+	0.478418i	$0.158790\pi$
$104$	−0.863187	−0.0846425
$105$	0	0
$106$	22.2624	2.16231
$107$	10.7720	1.04137	0.520684	−	0.853749i	$-0.325677\pi$
$107$	10.7720	1.04137	0.520684	+	0.853749i	$0.325677\pi$
$108$	1.51907	0.146173
$109$	−9.04356	−0.866216	−0.433108	−	0.901342i	$-0.642583\pi$
$109$	−9.04356	−0.866216	−0.433108	+	0.901342i	$0.642583\pi$
$110$	−2.97697	−0.283843
$111$	−6.15612	−0.584313
$112$	0	0
$113$	−5.09784	−0.479565	−0.239782	−	0.970827i	$-0.577076\pi$
$113$	−5.09784	−0.479565	−0.239782	+	0.970827i	$0.577076\pi$
$114$	−11.5695	−1.08358
$115$	9.30887	0.868057
$116$	−3.47386	−0.322540
$117$	0.956777	0.0884541
$118$	−8.00789	−0.737186
$119$	0	0
$120$	1.20086	0.109623
$121$	−9.57858	−0.870780
$122$	9.10207	0.824062
$123$	1.00000	0.0901670
$124$	−2.80634	−0.252017
$125$	10.9524	0.979609
$126$	0	0
$127$	9.16734	0.813470	0.406735	−	0.913546i	$-0.366667\pi$
$127$	9.16734	0.813470	0.406735	+	0.913546i	$0.366667\pi$
$128$	7.00879	0.619495
$129$	−5.98160	−0.526651
$130$	−2.38905	−0.209533
$131$	4.49020	0.392311	0.196155	−	0.980573i	$-0.437154\pi$
$131$	4.49020	0.392311	0.196155	+	0.980573i	$0.437154\pi$
$132$	1.81108	0.157635
$133$	0	0
$134$	7.05457	0.609422
$135$	−1.33107	−0.114560
$136$	−2.94359	−0.252411
$137$	−7.16553	−0.612192	−0.306096	−	0.952001i	$-0.599023\pi$
$137$	−7.16553	−0.612192	−0.306096	+	0.952001i	$0.599023\pi$
$138$	−13.1193	−1.11679
$139$	−10.8857	−0.923310	−0.461655	−	0.887060i	$-0.652744\pi$
$139$	−10.8857	−0.923310	−0.461655	+	0.887060i	$0.652744\pi$
$140$	0	0
$141$	6.24381	0.525824
$142$	−21.2798	−1.78576
$143$	1.14070	0.0953901
$144$	−4.73056	−0.394214
$145$	3.04392	0.252784
$146$	−11.3842	−0.942160
$147$	0	0
$148$	−9.35160	−0.768696
$149$	−10.1880	−0.834633	−0.417317	−	0.908761i	$-0.637029\pi$
$149$	−10.1880	−0.834633	−0.417317	+	0.908761i	$0.637029\pi$
$150$	−6.05596	−0.494467
$151$	−17.1572	−1.39623	−0.698116	−	0.715985i	$-0.745978\pi$
$151$	−17.1572	−1.39623	−0.698116	+	0.715985i	$0.745978\pi$
$152$	5.56408	0.451306
$153$	3.26275	0.263778
$154$	0	0
$155$	2.45902	0.197513
$156$	1.45341	0.116366
$157$	17.1453	1.36834	0.684172	−	0.729320i	$-0.260164\pi$
$157$	17.1453	1.36834	0.684172	+	0.729320i	$0.260164\pi$
$158$	−26.5735	−2.11408
$159$	11.8675	0.941150
$160$	9.41036	0.743955
$161$	0	0
$162$	1.87592	0.147386
$163$	9.69691	0.759520	0.379760	−	0.925085i	$-0.376007\pi$
$163$	9.69691	0.759520	0.379760	+	0.925085i	$0.376007\pi$
$164$	1.51907	0.118620
$165$	−1.58694	−0.123543
$166$	3.94832	0.306449
$167$	1.78167	0.137869	0.0689347	−	0.997621i	$-0.478040\pi$
$167$	1.78167	0.137869	0.0689347	+	0.997621i	$0.478040\pi$
$168$	0	0
$169$	−12.0846	−0.929583
$170$	−8.14700	−0.624846
$171$	−6.16735	−0.471629
$172$	−9.08649	−0.692838
$173$	−7.95559	−0.604853	−0.302426	−	0.953173i	$-0.597797\pi$
$173$	−7.95559	−0.604853	−0.302426	+	0.953173i	$0.597797\pi$
$174$	−4.28990	−0.325217
$175$	0	0
$176$	−5.63993	−0.425125
$177$	−4.26878	−0.320861
$178$	−18.2267	−1.36615
$179$	6.55130	0.489667	0.244833	−	0.969565i	$-0.421267\pi$
$179$	6.55130	0.489667	0.244833	+	0.969565i	$0.421267\pi$
$180$	−2.02199	−0.150710
$181$	−16.4764	−1.22468	−0.612341	−	0.790593i	$-0.709772\pi$
$181$	−16.4764	−1.22468	−0.612341	+	0.790593i	$0.709772\pi$
$182$	0	0
$183$	4.85206	0.358674
$184$	6.30945	0.465139
$185$	8.19421	0.602450
$186$	−3.46558	−0.254109
$187$	3.88995	0.284461
$188$	9.48480	0.691750
$189$	0	0
$190$	15.3997	1.11721
$191$	−22.5981	−1.63514	−0.817570	−	0.575829i	$-0.804679\pi$
$191$	−22.5981	−1.63514	−0.817570	+	0.575829i	$0.804679\pi$
$192$	−3.80123	−0.274330
$193$	−2.60607	−0.187589	−0.0937943	−	0.995592i	$-0.529900\pi$
$193$	−2.60607	−0.187589	−0.0937943	+	0.995592i	$0.529900\pi$
$194$	1.16751	0.0838225
$195$	−1.27353	−0.0911997
$196$	0	0
$197$	−16.3863	−1.16747	−0.583737	−	0.811942i	$-0.698410\pi$
$197$	−16.3863	−1.16747	−0.583737	+	0.811942i	$0.698410\pi$
$198$	2.23653	0.158943
$199$	−8.52046	−0.604000	−0.302000	−	0.953308i	$-0.597654\pi$
$199$	−8.52046	−0.604000	−0.302000	+	0.953308i	$0.597654\pi$
$200$	2.91248	0.205943
$201$	3.76060	0.265252
$202$	−13.2503	−0.932285
$203$	0	0
$204$	4.95635	0.347014
$205$	−1.33107	−0.0929657
$206$	33.4366	2.32964
$207$	−6.99354	−0.486085
$208$	−4.52610	−0.313828
$209$	−7.35291	−0.508612
$210$	0	0
$211$	23.0347	1.58577	0.792886	−	0.609369i	$-0.208577\pi$
$211$	23.0347	1.58577	0.792886	+	0.609369i	$0.208577\pi$
$212$	18.0275	1.23813
$213$	−11.3437	−0.777256
$214$	20.2074	1.38135
$215$	7.96191	0.542998
$216$	−0.902182	−0.0613857
$217$	0	0
$218$	−16.9650	−1.14901
$219$	−6.06858	−0.410076
$220$	−2.41067	−0.162528
$221$	3.12172	0.209990
$222$	−11.5484	−0.775077
$223$	9.56418	0.640465	0.320232	−	0.947339i	$-0.396239\pi$
$223$	9.56418	0.640465	0.320232	+	0.947339i	$0.396239\pi$
$224$	0	0
$225$	−3.22826	−0.215217
$226$	−9.56314	−0.636131
$227$	24.7536	1.64296	0.821478	−	0.570241i	$-0.193150\pi$
$227$	24.7536	1.64296	0.821478	+	0.570241i	$0.193150\pi$
$228$	−9.36866	−0.620454
$229$	3.35230	0.221526	0.110763	−	0.993847i	$-0.464671\pi$
$229$	3.35230	0.221526	0.110763	+	0.993847i	$0.464671\pi$
$230$	17.4627	1.15146
$231$	0	0
$232$	2.06314	0.135451
$233$	−1.02338	−0.0670440	−0.0335220	−	0.999438i	$-0.510672\pi$
$233$	−1.02338	−0.0670440	−0.0335220	+	0.999438i	$0.510672\pi$
$234$	1.79484	0.117332
$235$	−8.31093	−0.542145
$236$	−6.48459	−0.422111
$237$	−14.1656	−0.920155
$238$	0	0
$239$	−26.4140	−1.70858	−0.854290	−	0.519796i	$-0.826008\pi$
$239$	−26.4140	−1.70858	−0.854290	+	0.519796i	$0.826008\pi$
$240$	6.29670	0.406450
$241$	22.1097	1.42421	0.712105	−	0.702073i	$-0.247742\pi$
$241$	22.1097	1.42421	0.712105	+	0.702073i	$0.247742\pi$
$242$	−17.9686	−1.15507
$243$	1.00000	0.0641500
$244$	7.37062	0.471856
$245$	0	0
$246$	1.87592	0.119604
$247$	−5.90078	−0.375458
$248$	1.66669	0.105835
$249$	2.10474	0.133382
$250$	20.5457	1.29943
$251$	−19.7611	−1.24731	−0.623656	−	0.781699i	$-0.714353\pi$
$251$	−19.7611	−1.24731	−0.623656	+	0.781699i	$0.714353\pi$
$252$	0	0
$253$	−8.33792	−0.524200
$254$	17.1972	1.07905
$255$	−4.34294	−0.271965
$256$	20.7504	1.29690
$257$	−17.4492	−1.08845	−0.544224	−	0.838940i	$-0.683176\pi$
$257$	−17.4492	−1.08845	−0.544224	+	0.838940i	$0.683176\pi$
$258$	−11.2210	−0.698589
$259$	0	0
$260$	−1.93459	−0.119978
$261$	−2.28683	−0.141551
$262$	8.42325	0.520390
$263$	21.4392	1.32200	0.660998	−	0.750387i	$-0.270133\pi$
$263$	21.4392	1.32200	0.660998	+	0.750387i	$0.270133\pi$
$264$	−1.07561	−0.0661992
$265$	−15.7964	−0.970363
$266$	0	0
$267$	−9.71613	−0.594618
$268$	5.71262	0.348954
$269$	2.02049	0.123192	0.0615959	−	0.998101i	$-0.480381\pi$
$269$	2.02049	0.123192	0.0615959	+	0.998101i	$0.480381\pi$
$270$	−2.49697	−0.151961
$271$	11.0931	0.673861	0.336930	−	0.941530i	$-0.390611\pi$
$271$	11.0931	0.673861	0.336930	+	0.941530i	$0.390611\pi$
$272$	−15.4346	−0.935863
$273$	0	0
$274$	−13.4420	−0.812058
$275$	−3.84883	−0.232093
$276$	−10.6237	−0.639471
$277$	13.9305	0.837002	0.418501	−	0.908216i	$-0.362556\pi$
$277$	13.9305	0.837002	0.418501	+	0.908216i	$0.362556\pi$
$278$	−20.4206	−1.22475
$279$	−1.84740	−0.110601
$280$	0	0
$281$	−1.20264	−0.0717434	−0.0358717	−	0.999356i	$-0.511421\pi$
$281$	−1.20264	−0.0717434	−0.0358717	+	0.999356i	$0.511421\pi$
$282$	11.7129	0.697492
$283$	−5.10533	−0.303480	−0.151740	−	0.988420i	$-0.548488\pi$
$283$	−5.10533	−0.303480	−0.151740	+	0.988420i	$0.548488\pi$
$284$	−17.2319	−1.02252
$285$	8.20916	0.486269
$286$	2.13986	0.126533
$287$	0	0
$288$	−7.06979	−0.416591
$289$	−6.35447	−0.373792
$290$	5.71015	0.335311
$291$	0.622368	0.0364839
$292$	−9.21860	−0.539478
$293$	−28.2532	−1.65057	−0.825286	−	0.564715i	$-0.808986\pi$
$293$	−28.2532	−1.65057	−0.825286	+	0.564715i	$0.808986\pi$
$294$	0	0
$295$	5.68203	0.330821
$296$	5.55394	0.322816
$297$	1.19223	0.0691803
$298$	−19.1119	−1.10712
$299$	−6.69126	−0.386966
$300$	−4.90396	−0.283130
$301$	0	0
$302$	−32.1855	−1.85207
$303$	−7.06334	−0.405779
$304$	29.1751	1.67330
$305$	−6.45841	−0.369808
$306$	6.12065	0.349894
$307$	1.22364	0.0698370	0.0349185	−	0.999390i	$-0.488883\pi$
$307$	1.22364	0.0698370	0.0349185	+	0.999390i	$0.488883\pi$
$308$	0	0
$309$	17.8241	1.01398
$310$	4.61292	0.261996
$311$	−7.70437	−0.436875	−0.218437	−	0.975851i	$-0.570096\pi$
$311$	−7.70437	−0.436875	−0.218437	+	0.975851i	$0.570096\pi$
$312$	−0.863187	−0.0488683
$313$	2.06116	0.116504	0.0582518	−	0.998302i	$-0.481447\pi$
$313$	2.06116	0.116504	0.0582518	+	0.998302i	$0.481447\pi$
$314$	32.1632	1.81508
$315$	0	0
$316$	−21.5186	−1.21051
$317$	−10.8641	−0.610187	−0.305093	−	0.952322i	$-0.598688\pi$
$317$	−10.8641	−0.610187	−0.305093	+	0.952322i	$0.598688\pi$
$318$	22.2624	1.24841
$319$	−2.72643	−0.152651
$320$	5.05969	0.282845
$321$	10.7720	0.601234
$322$	0	0
$323$	−20.1225	−1.11965
$324$	1.51907	0.0843929
$325$	−3.08873	−0.171332
$326$	18.1906	1.00748
$327$	−9.04356	−0.500110
$328$	−0.902182	−0.0498147
$329$	0	0
$330$	−2.97697	−0.163877
$331$	28.8910	1.58799	0.793997	−	0.607921i	$-0.207996\pi$
$331$	28.8910	1.58799	0.793997	+	0.607921i	$0.207996\pi$
$332$	3.19725	0.175472
$333$	−6.15612	−0.337353
$334$	3.34226	0.182880
$335$	−5.00560	−0.273485
$336$	0	0
$337$	20.3472	1.10838	0.554192	−	0.832389i	$-0.313027\pi$
$337$	20.3472	1.10838	0.554192	+	0.832389i	$0.313027\pi$
$338$	−22.6697	−1.23307
$339$	−5.09784	−0.276877
$340$	−6.59723	−0.357785
$341$	−2.20253	−0.119274
$342$	−11.5695	−0.625605
$343$	0	0
$344$	5.39649	0.290959
$345$	9.30887	0.501173
$346$	−14.9241	−0.802322
$347$	8.70651	0.467390	0.233695	−	0.972310i	$-0.424918\pi$
$347$	8.70651	0.467390	0.233695	+	0.972310i	$0.424918\pi$
$348$	−3.47386	−0.186218
$349$	0.668717	0.0357956	0.0178978	−	0.999840i	$-0.494303\pi$
$349$	0.668717	0.0357956	0.0178978	+	0.999840i	$0.494303\pi$
$350$	0	0
$351$	0.956777	0.0510690
$352$	−8.42883	−0.449258
$353$	−11.0813	−0.589798	−0.294899	−	0.955528i	$-0.595286\pi$
$353$	−11.0813	−0.589798	−0.294899	+	0.955528i	$0.595286\pi$
$354$	−8.00789	−0.425615
$355$	15.0992	0.801382
$356$	−14.7595	−0.782252
$357$	0	0
$358$	12.2897	0.649531
$359$	−9.64446	−0.509015	−0.254507	−	0.967071i	$-0.581913\pi$
$359$	−9.64446	−0.509015	−0.254507	+	0.967071i	$0.581913\pi$
$360$	1.20086	0.0632911
$361$	19.0363	1.00191
$362$	−30.9084	−1.62451
$363$	−9.57858	−0.502745
$364$	0	0
$365$	8.07768	0.422805
$366$	9.10207	0.475773
$367$	35.5344	1.85488	0.927440	−	0.373971i	$-0.122004\pi$
$367$	35.5344	1.85488	0.927440	+	0.373971i	$0.122004\pi$
$368$	33.0834	1.72459
$369$	1.00000	0.0520579
$370$	15.3717	0.799135
$371$	0	0
$372$	−2.80634	−0.145502
$373$	0.914929	0.0473732	0.0236866	−	0.999719i	$-0.492460\pi$
$373$	0.914929	0.0473732	0.0236866	+	0.999719i	$0.492460\pi$
$374$	7.29723	0.377331
$375$	10.9524	0.565578
$376$	−5.63305	−0.290503
$377$	−2.18799	−0.112687
$378$	0	0
$379$	−1.54979	−0.0796071	−0.0398036	−	0.999208i	$-0.512673\pi$
$379$	−1.54979	−0.0796071	−0.0398036	+	0.999208i	$0.512673\pi$
$380$	12.4703	0.639713
$381$	9.16734	0.469657
$382$	−42.3922	−2.16897
$383$	38.3803	1.96114	0.980570	−	0.196167i	$-0.0628496\pi$
$383$	38.3803	1.96114	0.980570	+	0.196167i	$0.0628496\pi$
$384$	7.00879	0.357666
$385$	0	0
$386$	−4.88877	−0.248832
$387$	−5.98160	−0.304062
$388$	0.945422	0.0479965
$389$	4.21777	0.213849	0.106925	−	0.994267i	$-0.465900\pi$
$389$	4.21777	0.213849	0.106925	+	0.994267i	$0.465900\pi$
$390$	−2.38905	−0.120974
$391$	−22.8182	−1.15396
$392$	0	0
$393$	4.49020	0.226501
$394$	−30.7394	−1.54863
$395$	18.8554	0.948717
$396$	1.81108	0.0910104
$397$	29.8860	1.49993	0.749967	−	0.661475i	$-0.230069\pi$
$397$	29.8860	1.49993	0.749967	+	0.661475i	$0.230069\pi$
$398$	−15.9837	−0.801190
$399$	0	0
$400$	15.2715	0.763575
$401$	17.2699	0.862417	0.431209	−	0.902252i	$-0.358087\pi$
$401$	17.2699	0.862417	0.431209	+	0.902252i	$0.358087\pi$
$402$	7.05457	0.351850
$403$	−1.76755	−0.0880481
$404$	−10.7297	−0.533824
$405$	−1.33107	−0.0661412
$406$	0	0
$407$	−7.33952	−0.363807
$408$	−2.94359	−0.145730
$409$	0.572746	0.0283205	0.0141602	−	0.999900i	$-0.495493\pi$
$409$	0.572746	0.0283205	0.0141602	+	0.999900i	$0.495493\pi$
$410$	−2.49697	−0.123317
$411$	−7.16553	−0.353450
$412$	27.0761	1.33395
$413$	0	0
$414$	−13.1193	−0.644779
$415$	−2.80155	−0.137522
$416$	−6.76421	−0.331643
$417$	−10.8857	−0.533073
$418$	−13.7935	−0.674661
$419$	−9.37586	−0.458041	−0.229020	−	0.973422i	$-0.573552\pi$
$419$	−9.37586	−0.458041	−0.229020	+	0.973422i	$0.573552\pi$
$420$	0	0
$421$	0.621048	0.0302680	0.0151340	−	0.999885i	$-0.495183\pi$
$421$	0.621048	0.0302680	0.0151340	+	0.999885i	$0.495183\pi$
$422$	43.2112	2.10349
$423$	6.24381	0.303585
$424$	−10.7066	−0.519958
$425$	−10.5330	−0.510926
$426$	−21.2798	−1.03101
$427$	0	0
$428$	16.3634	0.790957
$429$	1.14070	0.0550735
$430$	14.9359	0.720273
$431$	39.2853	1.89231	0.946154	−	0.323718i	$-0.104933\pi$
$431$	39.2853	1.89231	0.946154	+	0.323718i	$0.104933\pi$
$432$	−4.73056	−0.227599
$433$	6.54587	0.314575	0.157287	−	0.987553i	$-0.449725\pi$
$433$	6.54587	0.314575	0.157287	+	0.987553i	$0.449725\pi$
$434$	0	0
$435$	3.04392	0.145945
$436$	−13.7378	−0.657922
$437$	43.1317	2.06327
$438$	−11.3842	−0.543956
$439$	−11.9986	−0.572664	−0.286332	−	0.958130i	$-0.592436\pi$
$439$	−11.9986	−0.572664	−0.286332	+	0.958130i	$0.592436\pi$
$440$	1.43171	0.0682540
$441$	0	0
$442$	5.85610	0.278546
$443$	−4.85767	−0.230795	−0.115397	−	0.993319i	$-0.536814\pi$
$443$	−4.85767	−0.230795	−0.115397	+	0.993319i	$0.536814\pi$
$444$	−9.35160	−0.443807
$445$	12.9328	0.613074
$446$	17.9416	0.849561
$447$	−10.1880	−0.481876
$448$	0	0
$449$	8.35529	0.394311	0.197155	−	0.980372i	$-0.436830\pi$
$449$	8.35529	0.394311	0.197155	+	0.980372i	$0.436830\pi$
$450$	−6.05596	−0.285481
$451$	1.19223	0.0561400
$452$	−7.74399	−0.364247
$453$	−17.1572	−0.806114
$454$	46.4358	2.17934
$455$	0	0
$456$	5.56408	0.260562
$457$	−38.9673	−1.82281	−0.911407	−	0.411506i	$-0.865003\pi$
$457$	−38.9673	−1.82281	−0.911407	+	0.411506i	$0.865003\pi$
$458$	6.28864	0.293849
$459$	3.26275	0.152292
$460$	14.1408	0.659320
$461$	23.6532	1.10164	0.550819	−	0.834625i	$-0.314315\pi$
$461$	23.6532	1.10164	0.550819	+	0.834625i	$0.314315\pi$
$462$	0	0
$463$	−1.86553	−0.0866985	−0.0433492	−	0.999060i	$-0.513803\pi$
$463$	−1.86553	−0.0866985	−0.0433492	+	0.999060i	$0.513803\pi$
$464$	10.8180	0.502213
$465$	2.45902	0.114034
$466$	−1.91978	−0.0889322
$467$	5.20134	0.240689	0.120345	−	0.992732i	$-0.461600\pi$
$467$	5.20134	0.240689	0.120345	+	0.992732i	$0.461600\pi$
$468$	1.45341	0.0671841
$469$	0	0
$470$	−15.5906	−0.719142
$471$	17.1453	0.790014
$472$	3.85122	0.177267
$473$	−7.13145	−0.327905
$474$	−26.5735	−1.22056
$475$	19.9098	0.913526
$476$	0	0
$477$	11.8675	0.543373
$478$	−49.5506	−2.26639
$479$	−2.97785	−0.136062	−0.0680308	−	0.997683i	$-0.521672\pi$
$479$	−2.97785	−0.136062	−0.0680308	+	0.997683i	$0.521672\pi$
$480$	9.41036	0.429522
$481$	−5.89004	−0.268563
$482$	41.4760	1.88918
$483$	0	0
$484$	−14.5506	−0.661389
$485$	−0.828414	−0.0376163
$486$	1.87592	0.0850934
$487$	41.2011	1.86700	0.933500	−	0.358578i	$-0.116738\pi$
$487$	41.2011	1.86700	0.933500	+	0.358578i	$0.116738\pi$
$488$	−4.37744	−0.198157
$489$	9.69691	0.438509
$490$	0	0
$491$	−9.81187	−0.442803	−0.221402	−	0.975183i	$-0.571063\pi$
$491$	−9.81187	−0.442803	−0.221402	+	0.975183i	$0.571063\pi$
$492$	1.51907	0.0684851
$493$	−7.46135	−0.336042
$494$	−11.0694	−0.498036
$495$	−1.58694	−0.0713276
$496$	8.73926	0.392404
$497$	0	0
$498$	3.94832	0.176928
$499$	15.2282	0.681708	0.340854	−	0.940116i	$-0.389284\pi$
$499$	15.2282	0.681708	0.340854	+	0.940116i	$0.389284\pi$
$500$	16.6374	0.744048
$501$	1.78167	0.0795989
$502$	−37.0703	−1.65453
$503$	−7.15496	−0.319024	−0.159512	−	0.987196i	$-0.550992\pi$
$503$	−7.15496	−0.319024	−0.159512	+	0.987196i	$0.550992\pi$
$504$	0	0
$505$	9.40178	0.418374
$506$	−15.6413	−0.695339
$507$	−12.0846	−0.536695
$508$	13.9258	0.617860
$509$	−2.75329	−0.122037	−0.0610187	−	0.998137i	$-0.519435\pi$
$509$	−2.75329	−0.122037	−0.0610187	+	0.998137i	$0.519435\pi$
$510$	−8.14700	−0.360755
$511$	0	0
$512$	24.9084	1.10081
$513$	−6.16735	−0.272295
$514$	−32.7332	−1.44380
$515$	−23.7251	−1.04545
$516$	−9.08649	−0.400010
$517$	7.44407	0.327390
$518$	0	0
$519$	−7.95559	−0.349212
$520$	1.14896	0.0503852
$521$	−21.3386	−0.934861	−0.467431	−	0.884030i	$-0.654820\pi$
$521$	−21.3386	−0.934861	−0.467431	+	0.884030i	$0.654820\pi$
$522$	−4.28990	−0.187764
$523$	−5.54381	−0.242414	−0.121207	−	0.992627i	$-0.538676\pi$
$523$	−5.54381	−0.242414	−0.121207	+	0.992627i	$0.538676\pi$
$524$	6.82094	0.297974
$525$	0	0
$526$	40.2182	1.75360
$527$	−6.02761	−0.262567
$528$	−5.63993	−0.245446
$529$	25.9096	1.12651
$530$	−29.6327	−1.28716
$531$	−4.26878	−0.185249
$532$	0	0
$533$	0.956777	0.0414426
$534$	−18.2267	−0.788745
$535$	−14.3382	−0.619896
$536$	−3.39274	−0.146544
$537$	6.55130	0.282709
$538$	3.79028	0.163411
$539$	0	0
$540$	−2.02199	−0.0870124
$541$	9.42177	0.405073	0.202537	−	0.979275i	$-0.435081\pi$
$541$	9.42177	0.405073	0.202537	+	0.979275i	$0.435081\pi$
$542$	20.8098	0.893859
$543$	−16.4764	−0.707071
$544$	−23.0670	−0.988987
$545$	12.0376	0.515633
$546$	0	0
$547$	24.9252	1.06573	0.532863	−	0.846202i	$-0.321116\pi$
$547$	24.9252	1.06573	0.532863	+	0.846202i	$0.321116\pi$
$548$	−10.8850	−0.464982
$549$	4.85206	0.207081
$550$	−7.22010	−0.307866
$551$	14.1037	0.600837
$552$	6.30945	0.268548
$553$	0	0
$554$	26.1325	1.11026
$555$	8.19421	0.347825
$556$	−16.5361	−0.701287
$557$	7.44490	0.315451	0.157725	−	0.987483i	$-0.449584\pi$
$557$	7.44490	0.315451	0.157725	+	0.987483i	$0.449584\pi$
$558$	−3.46558	−0.146710
$559$	−5.72306	−0.242060
$560$	0	0
$561$	3.88995	0.164234
$562$	−2.25605	−0.0951658
$563$	33.3241	1.40444	0.702222	−	0.711958i	$-0.252191\pi$
$563$	33.3241	1.40444	0.702222	+	0.711958i	$0.252191\pi$
$564$	9.48480	0.399382
$565$	6.78557	0.285471
$566$	−9.57719	−0.402559
$567$	0	0
$568$	10.2341	0.429412
$569$	33.6371	1.41014	0.705071	−	0.709137i	$-0.250915\pi$
$569$	33.6371	1.41014	0.705071	+	0.709137i	$0.250915\pi$
$570$	15.3997	0.645023
$571$	16.5003	0.690518	0.345259	−	0.938507i	$-0.387791\pi$
$571$	16.5003	0.690518	0.345259	+	0.938507i	$0.387791\pi$
$572$	1.73280	0.0724522
$573$	−22.5981	−0.944049
$574$	0	0
$575$	22.5770	0.941525
$576$	−3.80123	−0.158384
$577$	16.8768	0.702592	0.351296	−	0.936265i	$-0.385741\pi$
$577$	16.8768	0.702592	0.351296	+	0.936265i	$0.385741\pi$
$578$	−11.9205	−0.495826
$579$	−2.60607	−0.108304
$580$	4.62393	0.191998
$581$	0	0
$582$	1.16751	0.0483949
$583$	14.1487	0.585981
$584$	5.47496	0.226555
$585$	−1.27353	−0.0526542
$586$	−53.0008	−2.18944
$587$	−25.8123	−1.06539	−0.532694	−	0.846308i	$-0.678820\pi$
$587$	−25.8123	−1.06539	−0.532694	+	0.846308i	$0.678820\pi$
$588$	0	0
$589$	11.3936	0.469465
$590$	10.6590	0.438826
$591$	−16.3863	−0.674042
$592$	29.1219	1.19690
$593$	17.2786	0.709547	0.354774	−	0.934952i	$-0.384558\pi$
$593$	17.2786	0.709547	0.354774	+	0.934952i	$0.384558\pi$
$594$	2.23653	0.0917659
$595$	0	0
$596$	−15.4763	−0.633934
$597$	−8.52046	−0.348719
$598$	−12.5523	−0.513300
$599$	25.0317	1.02277	0.511384	−	0.859352i	$-0.329133\pi$
$599$	25.0317	1.02277	0.511384	+	0.859352i	$0.329133\pi$
$600$	2.91248	0.118901
$601$	31.0336	1.26589	0.632944	−	0.774198i	$-0.281846\pi$
$601$	31.0336	1.26589	0.632944	+	0.774198i	$0.281846\pi$
$602$	0	0
$603$	3.76060	0.153143
$604$	−26.0630	−1.06049
$605$	12.7497	0.518350
$606$	−13.2503	−0.538255
$607$	−8.46352	−0.343524	−0.171762	−	0.985138i	$-0.554946\pi$
$607$	−8.46352	−0.343524	−0.171762	+	0.985138i	$0.554946\pi$
$608$	43.6019	1.76829
$609$	0	0
$610$	−12.1155	−0.490540
$611$	5.97394	0.241680
$612$	4.95635	0.200349
$613$	−40.4704	−1.63458	−0.817291	−	0.576225i	$-0.804525\pi$
$613$	−40.4704	−1.63458	−0.817291	+	0.576225i	$0.804525\pi$
$614$	2.29546	0.0926370
$615$	−1.33107	−0.0536738
$616$	0	0
$617$	−32.0283	−1.28941	−0.644706	−	0.764430i	$-0.723020\pi$
$617$	−32.0283	−1.28941	−0.644706	+	0.764430i	$0.723020\pi$
$618$	33.4366	1.34502
$619$	−40.1469	−1.61364	−0.806819	−	0.590798i	$-0.798813\pi$
$619$	−40.1469	−1.61364	−0.806819	+	0.590798i	$0.798813\pi$
$620$	3.73542	0.150018
$621$	−6.99354	−0.280641
$622$	−14.4528	−0.579504
$623$	0	0
$624$	−4.52610	−0.181189
$625$	1.56298	0.0625194
$626$	3.86657	0.154539
$627$	−7.35291	−0.293647
$628$	26.0450	1.03931
$629$	−20.0859	−0.800877
$630$	0	0
$631$	8.26664	0.329090	0.164545	−	0.986370i	$-0.447384\pi$
$631$	8.26664	0.329090	0.164545	+	0.986370i	$0.447384\pi$
$632$	12.7800	0.508359
$633$	23.0347	0.915546
$634$	−20.3801	−0.809398
$635$	−12.2023	−0.484235
$636$	18.0275	0.714837
$637$	0	0
$638$	−5.11456	−0.202487
$639$	−11.3437	−0.448749
$640$	−9.32916	−0.368768
$641$	−10.8409	−0.428189	−0.214094	−	0.976813i	$-0.568680\pi$
$641$	−10.8409	−0.428189	−0.214094	+	0.976813i	$0.568680\pi$
$642$	20.2074	0.797522
$643$	−27.7902	−1.09594	−0.547969	−	0.836499i	$-0.684599\pi$
$643$	−27.7902	−1.09594	−0.547969	+	0.836499i	$0.684599\pi$
$644$	0	0
$645$	7.96191	0.313500
$646$	−37.7482	−1.48518
$647$	−34.1540	−1.34273	−0.671366	−	0.741126i	$-0.734292\pi$
$647$	−34.1540	−1.34273	−0.671366	+	0.741126i	$0.734292\pi$
$648$	−0.902182	−0.0354411
$649$	−5.08938	−0.199775
$650$	−5.79420	−0.227267
$651$	0	0
$652$	14.7303	0.576883
$653$	−0.920987	−0.0360410	−0.0180205	−	0.999838i	$-0.505736\pi$
$653$	−0.920987	−0.0360410	−0.0180205	+	0.999838i	$0.505736\pi$
$654$	−16.9650	−0.663383
$655$	−5.97675	−0.233531
$656$	−4.73056	−0.184698
$657$	−6.06858	−0.236758
$658$	0	0
$659$	5.06675	0.197372	0.0986862	−	0.995119i	$-0.468536\pi$
$659$	5.06675	0.197372	0.0986862	+	0.995119i	$0.468536\pi$
$660$	−2.41067	−0.0938354
$661$	−22.6196	−0.879801	−0.439901	−	0.898046i	$-0.644986\pi$
$661$	−22.6196	−0.879801	−0.439901	+	0.898046i	$0.644986\pi$
$662$	54.1972	2.10644
$663$	3.12172	0.121238
$664$	−1.89886	−0.0736899
$665$	0	0
$666$	−11.5484	−0.447491
$667$	15.9930	0.619253
$668$	2.70648	0.104717
$669$	9.56418	0.369773
$670$	−9.39011	−0.362772
$671$	5.78477	0.223319
$672$	0	0
$673$	13.7602	0.530418	0.265209	−	0.964191i	$-0.414559\pi$
$673$	13.7602	0.530418	0.265209	+	0.964191i	$0.414559\pi$
$674$	38.1697	1.47024
$675$	−3.22826	−0.124256
$676$	−18.3573	−0.706052
$677$	0.583084	0.0224097	0.0112049	−	0.999937i	$-0.496433\pi$
$677$	0.583084	0.0224097	0.0112049	+	0.999937i	$0.496433\pi$
$678$	−9.56314	−0.367270
$679$	0	0
$680$	3.91812	0.150253
$681$	24.7536	0.948561
$682$	−4.13177	−0.158214
$683$	27.6868	1.05941	0.529704	−	0.848183i	$-0.322303\pi$
$683$	27.6868	1.05941	0.529704	+	0.848183i	$0.322303\pi$
$684$	−9.36866	−0.358220
$685$	9.53780	0.364420
$686$	0	0
$687$	3.35230	0.127898
$688$	28.2964	1.07879
$689$	11.3545	0.432572
$690$	17.4627	0.664793
$691$	−14.8271	−0.564049	−0.282025	−	0.959407i	$-0.591006\pi$
$691$	−14.8271	−0.564049	−0.282025	+	0.959407i	$0.591006\pi$
$692$	−12.0851	−0.459407
$693$	0	0
$694$	16.3327	0.619981
$695$	14.4895	0.549620
$696$	2.06314	0.0782029
$697$	3.26275	0.123585
$698$	1.25446	0.0474820
$699$	−1.02338	−0.0387079
$700$	0	0
$701$	2.79286	0.105485	0.0527424	−	0.998608i	$-0.483204\pi$
$701$	2.79286	0.105485	0.0527424	+	0.998608i	$0.483204\pi$
$702$	1.79484	0.0677417
$703$	37.9670	1.43195
$704$	−4.53194	−0.170804
$705$	−8.31093	−0.313008
$706$	−20.7876	−0.782353
$707$	0	0
$708$	−6.48459	−0.243706
$709$	−42.6863	−1.60312	−0.801558	−	0.597917i	$-0.795995\pi$
$709$	−42.6863	−1.60312	−0.801558	+	0.597917i	$0.795995\pi$
$710$	28.3249	1.06301
$711$	−14.1656	−0.531252
$712$	8.76572	0.328509
$713$	12.9199	0.483854
$714$	0	0
$715$	−1.51835	−0.0567830
$716$	9.95189	0.371920
$717$	−26.4140	−0.986449
$718$	−18.0922	−0.675196
$719$	−44.1357	−1.64598	−0.822992	−	0.568053i	$-0.807697\pi$
$719$	−44.1357	−1.64598	−0.822992	+	0.568053i	$0.807697\pi$
$720$	6.29670	0.234664
$721$	0	0
$722$	35.7105	1.32901
$723$	22.1097	0.822268
$724$	−25.0289	−0.930191
$725$	7.38248	0.274178
$726$	−17.9686	−0.666879
$727$	13.6510	0.506288	0.253144	−	0.967429i	$-0.418535\pi$
$727$	13.6510	0.506288	0.253144	+	0.967429i	$0.418535\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	15.1531	0.560840
$731$	−19.5165	−0.721843
$732$	7.37062	0.272426
$733$	−15.2128	−0.561896	−0.280948	−	0.959723i	$-0.590649\pi$
$733$	−15.2128	−0.561896	−0.280948	+	0.959723i	$0.590649\pi$
$734$	66.6597	2.46045
$735$	0	0
$736$	49.4429	1.82249
$737$	4.48350	0.165152
$738$	1.87592	0.0690535
$739$	12.6595	0.465689	0.232844	−	0.972514i	$-0.425197\pi$
$739$	12.6595	0.465689	0.232844	+	0.972514i	$0.425197\pi$
$740$	12.4476	0.457583
$741$	−5.90078	−0.216771
$742$	0	0
$743$	−32.8862	−1.20648	−0.603238	−	0.797561i	$-0.706123\pi$
$743$	−32.8862	−1.20648	−0.603238	+	0.797561i	$0.706123\pi$
$744$	1.66669	0.0611040
$745$	13.5609	0.496833
$746$	1.71633	0.0628394
$747$	2.10474	0.0770083
$748$	5.90912	0.216059
$749$	0	0
$750$	20.5457	0.750225
$751$	5.59396	0.204127	0.102063	−	0.994778i	$-0.467456\pi$
$751$	5.59396	0.204127	0.102063	+	0.994778i	$0.467456\pi$
$752$	−29.5368	−1.07709
$753$	−19.7611	−0.720136
$754$	−4.10448	−0.149476
$755$	22.8373	0.831136
$756$	0	0
$757$	−49.2675	−1.79066	−0.895329	−	0.445406i	$-0.853059\pi$
$757$	−49.2675	−1.79066	−0.895329	+	0.445406i	$0.853059\pi$
$758$	−2.90727	−0.105597
$759$	−8.33792	−0.302647
$760$	−7.40616	−0.268649
$761$	−37.0827	−1.34425	−0.672123	−	0.740439i	$-0.734618\pi$
$761$	−37.0827	−1.34425	−0.672123	+	0.740439i	$0.734618\pi$
$762$	17.1972	0.622988
$763$	0	0
$764$	−34.3281	−1.24195
$765$	−4.34294	−0.157019
$766$	71.9983	2.60140
$767$	−4.08427	−0.147475
$768$	20.7504	0.748765
$769$	46.4807	1.67614	0.838069	−	0.545564i	$-0.183684\pi$
$769$	46.4807	1.67614	0.838069	+	0.545564i	$0.183684\pi$
$770$	0	0
$771$	−17.4492	−0.628416
$772$	−3.95880	−0.142480
$773$	53.4446	1.92227	0.961134	−	0.276081i	$-0.0890360\pi$
$773$	53.4446	1.92227	0.961134	+	0.276081i	$0.0890360\pi$
$774$	−11.2210	−0.403331
$775$	5.96390	0.214230
$776$	−0.561489	−0.0201563
$777$	0	0
$778$	7.91219	0.283666
$779$	−6.16735	−0.220968
$780$	−1.93459	−0.0692694
$781$	−13.5243	−0.483937
$782$	−42.8050	−1.53071
$783$	−2.28683	−0.0817246
$784$	0	0
$785$	−22.8216	−0.814536
$786$	8.42325	0.300447
$787$	19.7132	0.702700	0.351350	−	0.936244i	$-0.385723\pi$
$787$	19.7132	0.702700	0.351350	+	0.936244i	$0.385723\pi$
$788$	−24.8920	−0.886739
$789$	21.4392	0.763255
$790$	35.3711	1.25845
$791$	0	0
$792$	−1.07561	−0.0382201
$793$	4.64234	0.164854
$794$	56.0637	1.98962
$795$	−15.7964	−0.560239
$796$	−12.9432	−0.458759
$797$	−42.7009	−1.51254	−0.756271	−	0.654259i	$-0.772981\pi$
$797$	−42.7009	−1.51254	−0.756271	+	0.654259i	$0.772981\pi$
$798$	0	0
$799$	20.3720	0.720709
$800$	22.8231	0.806920
$801$	−9.71613	−0.343303
$802$	32.3969	1.14397
$803$	−7.23515	−0.255323
$804$	5.71262	0.201468
$805$	0	0
$806$	−3.31579	−0.116794
$807$	2.02049	0.0711248
$808$	6.37242	0.224181
$809$	−6.30079	−0.221524	−0.110762	−	0.993847i	$-0.535329\pi$
$809$	−6.30079	−0.221524	−0.110762	+	0.993847i	$0.535329\pi$
$810$	−2.49697	−0.0877347
$811$	−21.0846	−0.740379	−0.370189	−	0.928956i	$-0.620707\pi$
$811$	−21.0846	−0.740379	−0.370189	+	0.928956i	$0.620707\pi$
$812$	0	0
$813$	11.0931	0.389054
$814$	−13.7683	−0.482580
$815$	−12.9072	−0.452121
$816$	−15.4346	−0.540321
$817$	36.8907	1.29064
$818$	1.07443	0.0375664
$819$	0	0
$820$	−2.02199	−0.0706108
$821$	−4.52453	−0.157907	−0.0789537	−	0.996878i	$-0.525158\pi$
$821$	−4.52453	−0.157907	−0.0789537	+	0.996878i	$0.525158\pi$
$822$	−13.4420	−0.468842
$823$	−40.2931	−1.40453	−0.702264	−	0.711916i	$-0.747827\pi$
$823$	−40.2931	−1.40453	−0.702264	+	0.711916i	$0.747827\pi$
$824$	−16.0806	−0.560195
$825$	−3.84883	−0.133999
$826$	0	0
$827$	−11.1628	−0.388169	−0.194085	−	0.980985i	$-0.562174\pi$
$827$	−11.1628	−0.388169	−0.194085	+	0.980985i	$0.562174\pi$
$828$	−10.6237	−0.369199
$829$	−38.5413	−1.33859	−0.669297	−	0.742995i	$-0.733404\pi$
$829$	−38.5413	−1.33859	−0.669297	+	0.742995i	$0.733404\pi$
$830$	−5.25547	−0.182420
$831$	13.9305	0.483243
$832$	−3.63693	−0.126088
$833$	0	0
$834$	−20.4206	−0.707108
$835$	−2.37151	−0.0820697
$836$	−11.1696	−0.386309
$837$	−1.84740	−0.0638556
$838$	−17.5883	−0.607579
$839$	0.657012	0.0226826	0.0113413	−	0.999936i	$-0.496390\pi$
$839$	0.657012	0.0226826	0.0113413	+	0.999936i	$0.496390\pi$
$840$	0	0
$841$	−23.7704	−0.819670
$842$	1.16504	0.0401498
$843$	−1.20264	−0.0414211
$844$	34.9913	1.20445
$845$	16.0854	0.553354
$846$	11.7129	0.402697
$847$	0	0
$848$	−56.1397	−1.92785
$849$	−5.10533	−0.175214
$850$	−19.7591	−0.677730
$851$	43.0531	1.47584
$852$	−17.2319	−0.590354
$853$	−2.29577	−0.0786058	−0.0393029	−	0.999227i	$-0.512514\pi$
$853$	−2.29577	−0.0786058	−0.0393029	+	0.999227i	$0.512514\pi$
$854$	0	0
$855$	8.20916	0.280747
$856$	−9.71830	−0.332165
$857$	−18.9064	−0.645830	−0.322915	−	0.946428i	$-0.604663\pi$
$857$	−18.9064	−0.645830	−0.322915	+	0.946428i	$0.604663\pi$
$858$	2.13986	0.0730536
$859$	−33.0093	−1.12626	−0.563131	−	0.826368i	$-0.690403\pi$
$859$	−33.0093	−1.12626	−0.563131	+	0.826368i	$0.690403\pi$
$860$	12.0947	0.412426
$861$	0	0
$862$	73.6961	2.51010
$863$	−32.3940	−1.10271	−0.551353	−	0.834272i	$-0.685888\pi$
$863$	−32.3940	−1.10271	−0.551353	+	0.834272i	$0.685888\pi$
$864$	−7.06979	−0.240519
$865$	10.5894	0.360051
$866$	12.2795	0.417275
$867$	−6.35447	−0.215809
$868$	0	0
$869$	−16.8887	−0.572909
$870$	5.71015	0.193592
$871$	3.59805	0.121915
$872$	8.15893	0.276296
$873$	0.622368	0.0210640
$874$	80.9115	2.73687
$875$	0	0
$876$	−9.21860	−0.311468
$877$	2.48394	0.0838768	0.0419384	−	0.999120i	$-0.486647\pi$
$877$	2.48394	0.0838768	0.0419384	+	0.999120i	$0.486647\pi$
$878$	−22.5085	−0.759625
$879$	−28.2532	−0.952958
$880$	7.50712	0.253065
$881$	−4.62881	−0.155949	−0.0779744	−	0.996955i	$-0.524845\pi$
$881$	−4.62881	−0.155949	−0.0779744	+	0.996955i	$0.524845\pi$
$882$	0	0
$883$	13.8165	0.464962	0.232481	−	0.972601i	$-0.425316\pi$
$883$	13.8165	0.464962	0.232481	+	0.972601i	$0.425316\pi$
$884$	4.74212	0.159495
$885$	5.68203	0.190999
$886$	−9.11260	−0.306144
$887$	13.3709	0.448950	0.224475	−	0.974480i	$-0.427933\pi$
$887$	13.3709	0.448950	0.224475	+	0.974480i	$0.427933\pi$
$888$	5.55394	0.186378
$889$	0	0
$890$	24.2609	0.813228
$891$	1.19223	0.0399412
$892$	14.5287	0.486456
$893$	−38.5078	−1.28861
$894$	−19.1119	−0.639196
$895$	−8.72021	−0.291485
$896$	0	0
$897$	−6.69126	−0.223415
$898$	15.6739	0.523043
$899$	4.22470	0.140901
$900$	−4.90396	−0.163465
$901$	38.7205	1.28997
$902$	2.23653	0.0744683
$903$	0	0
$904$	4.59918	0.152966
$905$	21.9312	0.729018
$906$	−32.1855	−1.06929
$907$	35.7979	1.18865	0.594325	−	0.804225i	$-0.297419\pi$
$907$	35.7979	1.18865	0.594325	+	0.804225i	$0.297419\pi$
$908$	37.6025	1.24788
$909$	−7.06334	−0.234276
$910$	0	0
$911$	42.9779	1.42392	0.711960	−	0.702220i	$-0.247808\pi$
$911$	42.9779	1.42392	0.711960	+	0.702220i	$0.247808\pi$
$912$	29.1751	0.966083
$913$	2.50933	0.0830468
$914$	−73.0995	−2.41792
$915$	−6.45841	−0.213508
$916$	5.09238	0.168257
$917$	0	0
$918$	6.12065	0.202012
$919$	13.4263	0.442893	0.221447	−	0.975173i	$-0.428922\pi$
$919$	13.4263	0.442893	0.221447	+	0.975173i	$0.428922\pi$
$920$	−8.39829	−0.276884
$921$	1.22364	0.0403204
$922$	44.3715	1.46130
$923$	−10.8534	−0.357243
$924$	0	0
$925$	19.8736	0.653439
$926$	−3.49958	−0.115003
$927$	17.8241	0.585421
$928$	16.1674	0.530721
$929$	32.5053	1.06646	0.533232	−	0.845969i	$-0.320977\pi$
$929$	32.5053	1.06646	0.533232	+	0.845969i	$0.320977\pi$
$930$	4.61292	0.151264
$931$	0	0
$932$	−1.55459	−0.0509223
$933$	−7.70437	−0.252230
$934$	9.75729	0.319268
$935$	−5.17778	−0.169332
$936$	−0.863187	−0.0282142
$937$	10.0847	0.329454	0.164727	−	0.986339i	$-0.447326\pi$
$937$	10.0847	0.329454	0.164727	+	0.986339i	$0.447326\pi$
$938$	0	0
$939$	2.06116	0.0672634
$940$	−12.6249	−0.411779
$941$	−45.9898	−1.49923	−0.749613	−	0.661876i	$-0.769760\pi$
$941$	−45.9898	−1.49923	−0.749613	+	0.661876i	$0.769760\pi$
$942$	32.1632	1.04793
$943$	−6.99354	−0.227741
$944$	20.1938	0.657251
$945$	0	0
$946$	−13.3780	−0.434957
$947$	2.76566	0.0898719	0.0449359	−	0.998990i	$-0.485692\pi$
$947$	2.76566	0.0898719	0.0449359	+	0.998990i	$0.485692\pi$
$948$	−21.5186	−0.698891
$949$	−5.80628	−0.188480
$950$	37.3492	1.21177
$951$	−10.8641	−0.352292
$952$	0	0
$953$	9.52118	0.308421	0.154211	−	0.988038i	$-0.450717\pi$
$953$	9.52118	0.308421	0.154211	+	0.988038i	$0.450717\pi$
$954$	22.2624	0.720771
$955$	30.0796	0.973352
$956$	−40.1248	−1.29773
$957$	−2.72643	−0.0881329
$958$	−5.58621	−0.180482
$959$	0	0
$960$	5.05969	0.163301
$961$	−27.5871	−0.889906
$962$	−11.0492	−0.356242
$963$	10.7720	0.347123
$964$	33.5862	1.08174
$965$	3.46885	0.111666
$966$	0	0
$967$	−12.5417	−0.403314	−0.201657	−	0.979456i	$-0.564633\pi$
$967$	−12.5417	−0.403314	−0.201657	+	0.979456i	$0.564633\pi$
$968$	8.64163	0.277752
$969$	−20.1225	−0.646429
$970$	−1.55404	−0.0498971
$971$	1.59591	0.0512153	0.0256076	−	0.999672i	$-0.491848\pi$
$971$	1.59591	0.0512153	0.0256076	+	0.999672i	$0.491848\pi$
$972$	1.51907	0.0487243
$973$	0	0
$974$	77.2899	2.47653
$975$	−3.08873	−0.0989184
$976$	−22.9530	−0.734707
$977$	3.70916	0.118667	0.0593333	−	0.998238i	$-0.481103\pi$
$977$	3.70916	0.118667	0.0593333	+	0.998238i	$0.481103\pi$
$978$	18.1906	0.581672
$979$	−11.5839	−0.370222
$980$	0	0
$981$	−9.04356	−0.288739
$982$	−18.4063	−0.587368
$983$	14.2656	0.455001	0.227501	−	0.973778i	$-0.426945\pi$
$983$	14.2656	0.455001	0.227501	+	0.973778i	$0.426945\pi$
$984$	−0.902182	−0.0287605
$985$	21.8112	0.694964
$986$	−13.9969	−0.445752
$987$	0	0
$988$	−8.96372	−0.285174
$989$	41.8326	1.33020
$990$	−2.97697	−0.0946143
$991$	57.4385	1.82459	0.912297	−	0.409530i	$-0.134307\pi$
$991$	57.4385	1.82459	0.912297	+	0.409530i	$0.134307\pi$
$992$	13.0608	0.414680
$993$	28.8910	0.916829
$994$	0	0
$995$	11.3413	0.359543
$996$	3.19725	0.101309
$997$	−28.4978	−0.902534	−0.451267	−	0.892389i	$-0.649028\pi$
$997$	−28.4978	−0.902534	−0.451267	+	0.892389i	$0.649028\pi$
$998$	28.5669	0.904268
$999$	−6.15612	−0.194771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bm.1.14	yes	16
7.6	odd	2		6027.2.a.bl.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.14	✓	16	7.6	odd	2
6027.2.a.bm.1.14	yes	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+1.87592 q^{2} +1.00000 q^{3} +1.51907 q^{4} -1.33107 q^{5} +1.87592 q^{6} -0.902182 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.87592 q^{2} +1.00000 q^{3} +1.51907 q^{4} -1.33107 q^{5} +1.87592 q^{6} -0.902182 q^{8} +1.00000 q^{9} -2.49697 q^{10} +1.19223 q^{11} +1.51907 q^{12} +0.956777 q^{13} -1.33107 q^{15} -4.73056 q^{16} +3.26275 q^{17} +1.87592 q^{18} -6.16735 q^{19} -2.02199 q^{20} +2.23653 q^{22} -6.99354 q^{23} -0.902182 q^{24} -3.22826 q^{25} +1.79484 q^{26} +1.00000 q^{27} -2.28683 q^{29} -2.49697 q^{30} -1.84740 q^{31} -7.06979 q^{32} +1.19223 q^{33} +6.12065 q^{34} +1.51907 q^{36} -6.15612 q^{37} -11.5695 q^{38} +0.956777 q^{39} +1.20086 q^{40} +1.00000 q^{41} -5.98160 q^{43} +1.81108 q^{44} -1.33107 q^{45} -13.1193 q^{46} +6.24381 q^{47} -4.73056 q^{48} -6.05596 q^{50} +3.26275 q^{51} +1.45341 q^{52} +11.8675 q^{53} +1.87592 q^{54} -1.58694 q^{55} -6.16735 q^{57} -4.28990 q^{58} -4.26878 q^{59} -2.02199 q^{60} +4.85206 q^{61} -3.46558 q^{62} -3.80123 q^{64} -1.27353 q^{65} +2.23653 q^{66} +3.76060 q^{67} +4.95635 q^{68} -6.99354 q^{69} -11.3437 q^{71} -0.902182 q^{72} -6.06858 q^{73} -11.5484 q^{74} -3.22826 q^{75} -9.36866 q^{76} +1.79484 q^{78} -14.1656 q^{79} +6.29670 q^{80} +1.00000 q^{81} +1.87592 q^{82} +2.10474 q^{83} -4.34294 q^{85} -11.2210 q^{86} -2.28683 q^{87} -1.07561 q^{88} -9.71613 q^{89} -2.49697 q^{90} -10.6237 q^{92} -1.84740 q^{93} +11.7129 q^{94} +8.20916 q^{95} -7.06979 q^{96} +0.622368 q^{97} +1.19223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 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