LMFDB - Embedded newform 1134.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(1,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1134.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.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:	$9.05503558921$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{7} +1.00000 q^{8} -0.267949 q^{10} -6.19615 q^{11} -6.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} +0.732051 q^{19} -0.267949 q^{20} -6.19615 q^{22} +4.19615 q^{23} -4.92820 q^{25} -6.46410 q^{26} -1.00000 q^{28} -1.53590 q^{29} +8.19615 q^{31} +1.00000 q^{32} -7.00000 q^{34} +0.267949 q^{35} +10.6603 q^{37} +0.732051 q^{38} -0.267949 q^{40} -2.53590 q^{41} -1.46410 q^{43} -6.19615 q^{44} +4.19615 q^{46} -4.73205 q^{47} +1.00000 q^{49} -4.92820 q^{50} -6.46410 q^{52} +9.46410 q^{53} +1.66025 q^{55} -1.00000 q^{56} -1.53590 q^{58} -4.19615 q^{59} +3.92820 q^{61} +8.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} -6.73205 q^{67} -7.00000 q^{68} +0.267949 q^{70} -6.53590 q^{71} +8.26795 q^{73} +10.6603 q^{74} +0.732051 q^{76} +6.19615 q^{77} -9.12436 q^{79} -0.267949 q^{80} -2.53590 q^{82} -16.5885 q^{83} +1.87564 q^{85} -1.46410 q^{86} -6.19615 q^{88} -9.92820 q^{89} +6.46410 q^{91} +4.19615 q^{92} -4.73205 q^{94} -0.196152 q^{95} +10.9282 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 6 q^{13} - 2 q^{14} + 2 q^{16} - 14 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{23} + 4 q^{25} - 6 q^{26} - 2 q^{28} - 10 q^{29} + 6 q^{31} + 2 q^{32} - 14 q^{34} + 4 q^{35} + 4 q^{37} - 2 q^{38} - 4 q^{40} - 12 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{46} - 6 q^{47} + 2 q^{49} + 4 q^{50} - 6 q^{52} + 12 q^{53} - 14 q^{55} - 2 q^{56} - 10 q^{58} + 2 q^{59} - 6 q^{61} + 6 q^{62} + 2 q^{64} - 10 q^{67} - 14 q^{68} + 4 q^{70} - 20 q^{71} + 20 q^{73} + 4 q^{74} - 2 q^{76} + 2 q^{77} + 6 q^{79} - 4 q^{80} - 12 q^{82} - 2 q^{83} + 28 q^{85} + 4 q^{86} - 2 q^{88} - 6 q^{89} + 6 q^{91} - 2 q^{92} - 6 q^{94} + 10 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8 - 4 * q^10 - 2 * q^11 - 6 * q^13 - 2 * q^14 + 2 * q^16 - 14 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - 2 * q^23 + 4 * q^25 - 6 * q^26 - 2 * q^28 - 10 * q^29 + 6 * q^31 + 2 * q^32 - 14 * q^34 + 4 * q^35 + 4 * q^37 - 2 * q^38 - 4 * q^40 - 12 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^46 - 6 * q^47 + 2 * q^49 + 4 * q^50 - 6 * q^52 + 12 * q^53 - 14 * q^55 - 2 * q^56 - 10 * q^58 + 2 * q^59 - 6 * q^61 + 6 * q^62 + 2 * q^64 - 10 * q^67 - 14 * q^68 + 4 * q^70 - 20 * q^71 + 20 * q^73 + 4 * q^74 - 2 * q^76 + 2 * q^77 + 6 * q^79 - 4 * q^80 - 12 * q^82 - 2 * q^83 + 28 * q^85 + 4 * q^86 - 2 * q^88 - 6 * q^89 + 6 * q^91 - 2 * q^92 - 6 * q^94 + 10 * q^95 + 8 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.267949	−0.119831	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	−0.267949	−0.119831	−0.0599153	+	0.998203i	$0.519083\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.267949	−0.0847330
$11$	−6.19615	−1.86821	−0.934105	−	0.356998i	$-0.883800\pi$
$11$	−6.19615	−1.86821	−0.934105	+	0.356998i	$0.883800\pi$
$12$	0	0
$13$	−6.46410	−1.79282	−0.896410	−	0.443227i	$-0.853834\pi$
$13$	−6.46410	−1.79282	−0.896410	+	0.443227i	$0.853834\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	0.732051	0.167944	0.0839720	−	0.996468i	$-0.473239\pi$
$19$	0.732051	0.167944	0.0839720	+	0.996468i	$0.473239\pi$
$20$	−0.267949	−0.0599153
$21$	0	0
$22$	−6.19615	−1.32102
$23$	4.19615	0.874958	0.437479	−	0.899229i	$-0.355871\pi$
$23$	4.19615	0.874958	0.437479	+	0.899229i	$0.355871\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	−6.46410	−1.26771
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.53590	−0.285209	−0.142605	−	0.989780i	$-0.545548\pi$
$29$	−1.53590	−0.285209	−0.142605	+	0.989780i	$0.545548\pi$
$30$	0	0
$31$	8.19615	1.47207	0.736036	−	0.676942i	$-0.236695\pi$
$31$	8.19615	1.47207	0.736036	+	0.676942i	$0.236695\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.00000	−1.20049
$35$	0.267949	0.0452917
$36$	0	0
$37$	10.6603	1.75253	0.876267	−	0.481825i	$-0.160026\pi$
$37$	10.6603	1.75253	0.876267	+	0.481825i	$0.160026\pi$
$38$	0.732051	0.118754
$39$	0	0
$40$	−0.267949	−0.0423665
$41$	−2.53590	−0.396041	−0.198020	−	0.980198i	$-0.563451\pi$
$41$	−2.53590	−0.396041	−0.198020	+	0.980198i	$0.563451\pi$
$42$	0	0
$43$	−1.46410	−0.223273	−0.111637	−	0.993749i	$-0.535609\pi$
$43$	−1.46410	−0.223273	−0.111637	+	0.993749i	$0.535609\pi$
$44$	−6.19615	−0.934105
$45$	0	0
$46$	4.19615	0.618689
$47$	−4.73205	−0.690241	−0.345120	−	0.938558i	$-0.612162\pi$
$47$	−4.73205	−0.690241	−0.345120	+	0.938558i	$0.612162\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.92820	−0.696953
$51$	0	0
$52$	−6.46410	−0.896410
$53$	9.46410	1.29999	0.649997	−	0.759937i	$-0.274770\pi$
$53$	9.46410	1.29999	0.649997	+	0.759937i	$0.274770\pi$
$54$	0	0
$55$	1.66025	0.223869
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−1.53590	−0.201673
$59$	−4.19615	−0.546293	−0.273146	−	0.961973i	$-0.588064\pi$
$59$	−4.19615	−0.546293	−0.273146	+	0.961973i	$0.588064\pi$
$60$	0	0
$61$	3.92820	0.502955	0.251477	−	0.967863i	$-0.419084\pi$
$61$	3.92820	0.502955	0.251477	+	0.967863i	$0.419084\pi$
$62$	8.19615	1.04091
$63$	0	0
$64$	1.00000	0.125000
$65$	1.73205	0.214834
$66$	0	0
$67$	−6.73205	−0.822451	−0.411225	−	0.911534i	$-0.634899\pi$
$67$	−6.73205	−0.822451	−0.411225	+	0.911534i	$0.634899\pi$
$68$	−7.00000	−0.848875
$69$	0	0
$70$	0.267949	0.0320261
$71$	−6.53590	−0.775668	−0.387834	−	0.921729i	$-0.626777\pi$
$71$	−6.53590	−0.775668	−0.387834	+	0.921729i	$0.626777\pi$
$72$	0	0
$73$	8.26795	0.967690	0.483845	−	0.875154i	$-0.339240\pi$
$73$	8.26795	0.967690	0.483845	+	0.875154i	$0.339240\pi$
$74$	10.6603	1.23923
$75$	0	0
$76$	0.732051	0.0839720
$77$	6.19615	0.706117
$78$	0	0
$79$	−9.12436	−1.02657	−0.513285	−	0.858218i	$-0.671572\pi$
$79$	−9.12436	−1.02657	−0.513285	+	0.858218i	$0.671572\pi$
$80$	−0.267949	−0.0299576
$81$	0	0
$82$	−2.53590	−0.280043
$83$	−16.5885	−1.82082	−0.910410	−	0.413707i	$-0.864234\pi$
$83$	−16.5885	−1.82082	−0.910410	+	0.413707i	$0.864234\pi$
$84$	0	0
$85$	1.87564	0.203442
$86$	−1.46410	−0.157878
$87$	0	0
$88$	−6.19615	−0.660512
$89$	−9.92820	−1.05239	−0.526194	−	0.850365i	$-0.676381\pi$
$89$	−9.92820	−1.05239	−0.526194	+	0.850365i	$0.676381\pi$
$90$	0	0
$91$	6.46410	0.677622
$92$	4.19615	0.437479
$93$	0	0
$94$	−4.73205	−0.488074
$95$	−0.196152	−0.0201248
$96$	0	0
$97$	10.9282	1.10959	0.554795	−	0.831987i	$-0.312797\pi$
$97$	10.9282	1.10959	0.554795	+	0.831987i	$0.312797\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−4.92820	−0.492820
$101$	−8.92820	−0.888389	−0.444195	−	0.895930i	$-0.646510\pi$
$101$	−8.92820	−0.888389	−0.444195	+	0.895930i	$0.646510\pi$
$102$	0	0
$103$	−8.39230	−0.826918	−0.413459	−	0.910523i	$-0.635680\pi$
$103$	−8.39230	−0.826918	−0.413459	+	0.910523i	$0.635680\pi$
$104$	−6.46410	−0.633857
$105$	0	0
$106$	9.46410	0.919235
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	3.19615	0.306136	0.153068	−	0.988216i	$-0.451085\pi$
$109$	3.19615	0.306136	0.153068	+	0.988216i	$0.451085\pi$
$110$	1.66025	0.158299
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	5.73205	0.539226	0.269613	−	0.962969i	$-0.413104\pi$
$113$	5.73205	0.539226	0.269613	+	0.962969i	$0.413104\pi$
$114$	0	0
$115$	−1.12436	−0.104847
$116$	−1.53590	−0.142605
$117$	0	0
$118$	−4.19615	−0.386287
$119$	7.00000	0.641689
$120$	0	0
$121$	27.3923	2.49021
$122$	3.92820	0.355643
$123$	0	0
$124$	8.19615	0.736036
$125$	2.66025	0.237940
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.73205	0.151911
$131$	−10.5359	−0.920526	−0.460263	−	0.887783i	$-0.652245\pi$
$131$	−10.5359	−0.920526	−0.460263	+	0.887783i	$0.652245\pi$
$132$	0	0
$133$	−0.732051	−0.0634769
$134$	−6.73205	−0.581561
$135$	0	0
$136$	−7.00000	−0.600245
$137$	8.26795	0.706379	0.353189	−	0.935552i	$-0.385097\pi$
$137$	8.26795	0.706379	0.353189	+	0.935552i	$0.385097\pi$
$138$	0	0
$139$	−3.26795	−0.277184	−0.138592	−	0.990350i	$-0.544258\pi$
$139$	−3.26795	−0.277184	−0.138592	+	0.990350i	$0.544258\pi$
$140$	0.267949	0.0226458
$141$	0	0
$142$	−6.53590	−0.548480
$143$	40.0526	3.34936
$144$	0	0
$145$	0.411543	0.0341768
$146$	8.26795	0.684260
$147$	0	0
$148$	10.6603	0.876267
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	−5.80385	−0.472310	−0.236155	−	0.971715i	$-0.575887\pi$
$151$	−5.80385	−0.472310	−0.236155	+	0.971715i	$0.575887\pi$
$152$	0.732051	0.0593772
$153$	0	0
$154$	6.19615	0.499300
$155$	−2.19615	−0.176399
$156$	0	0
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	−9.12436	−0.725895
$159$	0	0
$160$	−0.267949	−0.0211832
$161$	−4.19615	−0.330703
$162$	0	0
$163$	−13.4641	−1.05459	−0.527295	−	0.849682i	$-0.676794\pi$
$163$	−13.4641	−1.05459	−0.527295	+	0.849682i	$0.676794\pi$
$164$	−2.53590	−0.198020
$165$	0	0
$166$	−16.5885	−1.28751
$167$	1.80385	0.139586	0.0697930	−	0.997561i	$-0.477766\pi$
$167$	1.80385	0.139586	0.0697930	+	0.997561i	$0.477766\pi$
$168$	0	0
$169$	28.7846	2.21420
$170$	1.87564	0.143855
$171$	0	0
$172$	−1.46410	−0.111637
$173$	−6.26795	−0.476543	−0.238272	−	0.971199i	$-0.576581\pi$
$173$	−6.26795	−0.476543	−0.238272	+	0.971199i	$0.576581\pi$
$174$	0	0
$175$	4.92820	0.372537
$176$	−6.19615	−0.467053
$177$	0	0
$178$	−9.92820	−0.744150
$179$	2.19615	0.164148	0.0820741	−	0.996626i	$-0.473846\pi$
$179$	2.19615	0.164148	0.0820741	+	0.996626i	$0.473846\pi$
$180$	0	0
$181$	−16.3923	−1.21843	−0.609215	−	0.793005i	$-0.708515\pi$
$181$	−16.3923	−1.21843	−0.609215	+	0.793005i	$0.708515\pi$
$182$	6.46410	0.479151
$183$	0	0
$184$	4.19615	0.309344
$185$	−2.85641	−0.210007
$186$	0	0
$187$	43.3731	3.17175
$188$	−4.73205	−0.345120
$189$	0	0
$190$	−0.196152	−0.0142304
$191$	5.66025	0.409562	0.204781	−	0.978808i	$-0.434352\pi$
$191$	5.66025	0.409562	0.204781	+	0.978808i	$0.434352\pi$
$192$	0	0
$193$	18.8564	1.35731	0.678657	−	0.734455i	$-0.262562\pi$
$193$	18.8564	1.35731	0.678657	+	0.734455i	$0.262562\pi$
$194$	10.9282	0.784599
$195$	0	0
$196$	1.00000	0.0714286
$197$	−15.7846	−1.12461	−0.562303	−	0.826931i	$-0.690085\pi$
$197$	−15.7846	−1.12461	−0.562303	+	0.826931i	$0.690085\pi$
$198$	0	0
$199$	−19.1244	−1.35569	−0.677845	−	0.735205i	$-0.737086\pi$
$199$	−19.1244	−1.35569	−0.677845	+	0.735205i	$0.737086\pi$
$200$	−4.92820	−0.348477
$201$	0	0
$202$	−8.92820	−0.628186
$203$	1.53590	0.107799
$204$	0	0
$205$	0.679492	0.0474578
$206$	−8.39230	−0.584720
$207$	0	0
$208$	−6.46410	−0.448205
$209$	−4.53590	−0.313755
$210$	0	0
$211$	−17.2679	−1.18877	−0.594387	−	0.804179i	$-0.702605\pi$
$211$	−17.2679	−1.18877	−0.594387	+	0.804179i	$0.702605\pi$
$212$	9.46410	0.649997
$213$	0	0
$214$	0	0
$215$	0.392305	0.0267550
$216$	0	0
$217$	−8.19615	−0.556391
$218$	3.19615	0.216471
$219$	0	0
$220$	1.66025	0.111934
$221$	45.2487	3.04376
$222$	0	0
$223$	−25.4641	−1.70520	−0.852601	−	0.522562i	$-0.824976\pi$
$223$	−25.4641	−1.70520	−0.852601	+	0.522562i	$0.824976\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	5.73205	0.381290
$227$	18.9282	1.25631	0.628154	−	0.778089i	$-0.283811\pi$
$227$	18.9282	1.25631	0.628154	+	0.778089i	$0.283811\pi$
$228$	0	0
$229$	−2.46410	−0.162832	−0.0814162	−	0.996680i	$-0.525944\pi$
$229$	−2.46410	−0.162832	−0.0814162	+	0.996680i	$0.525944\pi$
$230$	−1.12436	−0.0741378
$231$	0	0
$232$	−1.53590	−0.100837
$233$	2.80385	0.183686	0.0918431	−	0.995773i	$-0.470724\pi$
$233$	2.80385	0.183686	0.0918431	+	0.995773i	$0.470724\pi$
$234$	0	0
$235$	1.26795	0.0827119
$236$	−4.19615	−0.273146
$237$	0	0
$238$	7.00000	0.453743
$239$	10.0526	0.650246	0.325123	−	0.945672i	$-0.394594\pi$
$239$	10.0526	0.650246	0.325123	+	0.945672i	$0.394594\pi$
$240$	0	0
$241$	−14.2679	−0.919079	−0.459540	−	0.888157i	$-0.651986\pi$
$241$	−14.2679	−0.919079	−0.459540	+	0.888157i	$0.651986\pi$
$242$	27.3923	1.76084
$243$	0	0
$244$	3.92820	0.251477
$245$	−0.267949	−0.0171186
$246$	0	0
$247$	−4.73205	−0.301093
$248$	8.19615	0.520456
$249$	0	0
$250$	2.66025	0.168249
$251$	22.0526	1.39195	0.695973	−	0.718068i	$-0.254973\pi$
$251$	22.0526	1.39195	0.695973	+	0.718068i	$0.254973\pi$
$252$	0	0
$253$	−26.0000	−1.63461
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.46410	−0.403220	−0.201610	−	0.979466i	$-0.564617\pi$
$257$	−6.46410	−0.403220	−0.201610	+	0.979466i	$0.564617\pi$
$258$	0	0
$259$	−10.6603	−0.662396
$260$	1.73205	0.107417
$261$	0	0
$262$	−10.5359	−0.650910
$263$	6.33975	0.390925	0.195463	−	0.980711i	$-0.437379\pi$
$263$	6.33975	0.390925	0.195463	+	0.980711i	$0.437379\pi$
$264$	0	0
$265$	−2.53590	−0.155779
$266$	−0.732051	−0.0448849
$267$	0	0
$268$	−6.73205	−0.411225
$269$	−5.58846	−0.340734	−0.170367	−	0.985381i	$-0.554495\pi$
$269$	−5.58846	−0.340734	−0.170367	+	0.985381i	$0.554495\pi$
$270$	0	0
$271$	−19.5167	−1.18555	−0.592776	−	0.805367i	$-0.701968\pi$
$271$	−19.5167	−1.18555	−0.592776	+	0.805367i	$0.701968\pi$
$272$	−7.00000	−0.424437
$273$	0	0
$274$	8.26795	0.499485
$275$	30.5359	1.84138
$276$	0	0
$277$	−18.7846	−1.12866	−0.564329	−	0.825550i	$-0.690865\pi$
$277$	−18.7846	−1.12866	−0.564329	+	0.825550i	$0.690865\pi$
$278$	−3.26795	−0.195999
$279$	0	0
$280$	0.267949	0.0160130
$281$	13.1962	0.787216	0.393608	−	0.919278i	$-0.371227\pi$
$281$	13.1962	0.787216	0.393608	+	0.919278i	$0.371227\pi$
$282$	0	0
$283$	15.3205	0.910710	0.455355	−	0.890310i	$-0.349512\pi$
$283$	15.3205	0.910710	0.455355	+	0.890310i	$0.349512\pi$
$284$	−6.53590	−0.387834
$285$	0	0
$286$	40.0526	2.36836
$287$	2.53590	0.149689
$288$	0	0
$289$	32.0000	1.88235
$290$	0.411543	0.0241666
$291$	0	0
$292$	8.26795	0.483845
$293$	−3.33975	−0.195110	−0.0975550	−	0.995230i	$-0.531102\pi$
$293$	−3.33975	−0.195110	−0.0975550	+	0.995230i	$0.531102\pi$
$294$	0	0
$295$	1.12436	0.0654625
$296$	10.6603	0.619615
$297$	0	0
$298$	−9.00000	−0.521356
$299$	−27.1244	−1.56864
$300$	0	0
$301$	1.46410	0.0843894
$302$	−5.80385	−0.333974
$303$	0	0
$304$	0.732051	0.0419860
$305$	−1.05256	−0.0602693
$306$	0	0
$307$	−21.8564	−1.24741	−0.623706	−	0.781659i	$-0.714374\pi$
$307$	−21.8564	−1.24741	−0.623706	+	0.781659i	$0.714374\pi$
$308$	6.19615	0.353059
$309$	0	0
$310$	−2.19615	−0.124733
$311$	−10.1962	−0.578171	−0.289085	−	0.957303i	$-0.593351\pi$
$311$	−10.1962	−0.578171	−0.289085	+	0.957303i	$0.593351\pi$
$312$	0	0
$313$	25.5885	1.44635	0.723173	−	0.690667i	$-0.242683\pi$
$313$	25.5885	1.44635	0.723173	+	0.690667i	$0.242683\pi$
$314$	−1.00000	−0.0564333
$315$	0	0
$316$	−9.12436	−0.513285
$317$	31.3923	1.76317	0.881584	−	0.472028i	$-0.156478\pi$
$317$	31.3923	1.76317	0.881584	+	0.472028i	$0.156478\pi$
$318$	0	0
$319$	9.51666	0.532831
$320$	−0.267949	−0.0149788
$321$	0	0
$322$	−4.19615	−0.233842
$323$	−5.12436	−0.285127
$324$	0	0
$325$	31.8564	1.76708
$326$	−13.4641	−0.745708
$327$	0	0
$328$	−2.53590	−0.140022
$329$	4.73205	0.260886
$330$	0	0
$331$	12.3923	0.681143	0.340571	−	0.940219i	$-0.389380\pi$
$331$	12.3923	0.681143	0.340571	+	0.940219i	$0.389380\pi$
$332$	−16.5885	−0.910410
$333$	0	0
$334$	1.80385	0.0987021
$335$	1.80385	0.0985547
$336$	0	0
$337$	−16.3923	−0.892946	−0.446473	−	0.894797i	$-0.647320\pi$
$337$	−16.3923	−0.892946	−0.446473	+	0.894797i	$0.647320\pi$
$338$	28.7846	1.56568
$339$	0	0
$340$	1.87564	0.101721
$341$	−50.7846	−2.75014
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.46410	−0.0789391
$345$	0	0
$346$	−6.26795	−0.336967
$347$	−21.4641	−1.15225	−0.576127	−	0.817360i	$-0.695437\pi$
$347$	−21.4641	−1.15225	−0.576127	+	0.817360i	$0.695437\pi$
$348$	0	0
$349$	1.46410	0.0783716	0.0391858	−	0.999232i	$-0.487524\pi$
$349$	1.46410	0.0783716	0.0391858	+	0.999232i	$0.487524\pi$
$350$	4.92820	0.263424
$351$	0	0
$352$	−6.19615	−0.330256
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	1.75129	0.0929488
$356$	−9.92820	−0.526194
$357$	0	0
$358$	2.19615	0.116070
$359$	10.9282	0.576769	0.288384	−	0.957515i	$-0.406882\pi$
$359$	10.9282	0.576769	0.288384	+	0.957515i	$0.406882\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	−16.3923	−0.861560
$363$	0	0
$364$	6.46410	0.338811
$365$	−2.21539	−0.115959
$366$	0	0
$367$	11.1244	0.580687	0.290343	−	0.956923i	$-0.406230\pi$
$367$	11.1244	0.580687	0.290343	+	0.956923i	$0.406230\pi$
$368$	4.19615	0.218740
$369$	0	0
$370$	−2.85641	−0.148498
$371$	−9.46410	−0.491352
$372$	0	0
$373$	−6.14359	−0.318103	−0.159052	−	0.987270i	$-0.550844\pi$
$373$	−6.14359	−0.318103	−0.159052	+	0.987270i	$0.550844\pi$
$374$	43.3731	2.24277
$375$	0	0
$376$	−4.73205	−0.244037
$377$	9.92820	0.511328
$378$	0	0
$379$	−27.5167	−1.41344	−0.706718	−	0.707495i	$-0.749825\pi$
$379$	−27.5167	−1.41344	−0.706718	+	0.707495i	$0.749825\pi$
$380$	−0.196152	−0.0100624
$381$	0	0
$382$	5.66025	0.289604
$383$	−19.7128	−1.00728	−0.503639	−	0.863914i	$-0.668006\pi$
$383$	−19.7128	−1.00728	−0.503639	+	0.863914i	$0.668006\pi$
$384$	0	0
$385$	−1.66025	−0.0846144
$386$	18.8564	0.959766
$387$	0	0
$388$	10.9282	0.554795
$389$	13.4641	0.682657	0.341329	−	0.939944i	$-0.389123\pi$
$389$	13.4641	0.682657	0.341329	+	0.939944i	$0.389123\pi$
$390$	0	0
$391$	−29.3731	−1.48546
$392$	1.00000	0.0505076
$393$	0	0
$394$	−15.7846	−0.795217
$395$	2.44486	0.123014
$396$	0	0
$397$	−21.0000	−1.05396	−0.526980	−	0.849878i	$-0.676676\pi$
$397$	−21.0000	−1.05396	−0.526980	+	0.849878i	$0.676676\pi$
$398$	−19.1244	−0.958617
$399$	0	0
$400$	−4.92820	−0.246410
$401$	−10.5167	−0.525177	−0.262588	−	0.964908i	$-0.584576\pi$
$401$	−10.5167	−0.525177	−0.262588	+	0.964908i	$0.584576\pi$
$402$	0	0
$403$	−52.9808	−2.63916
$404$	−8.92820	−0.444195
$405$	0	0
$406$	1.53590	0.0762254
$407$	−66.0526	−3.27410
$408$	0	0
$409$	17.3397	0.857395	0.428698	−	0.903448i	$-0.358973\pi$
$409$	17.3397	0.857395	0.428698	+	0.903448i	$0.358973\pi$
$410$	0.679492	0.0335577
$411$	0	0
$412$	−8.39230	−0.413459
$413$	4.19615	0.206479
$414$	0	0
$415$	4.44486	0.218190
$416$	−6.46410	−0.316929
$417$	0	0
$418$	−4.53590	−0.221858
$419$	9.46410	0.462352	0.231176	−	0.972912i	$-0.425743\pi$
$419$	9.46410	0.462352	0.231176	+	0.972912i	$0.425743\pi$
$420$	0	0
$421$	0.124356	0.00606072	0.00303036	−	0.999995i	$-0.499035\pi$
$421$	0.124356	0.00606072	0.00303036	+	0.999995i	$0.499035\pi$
$422$	−17.2679	−0.840591
$423$	0	0
$424$	9.46410	0.459617
$425$	34.4974	1.67337
$426$	0	0
$427$	−3.92820	−0.190099
$428$	0	0
$429$	0	0
$430$	0.392305	0.0189186
$431$	14.5359	0.700170	0.350085	−	0.936718i	$-0.386153\pi$
$431$	14.5359	0.700170	0.350085	+	0.936718i	$0.386153\pi$
$432$	0	0
$433$	15.7321	0.756034	0.378017	−	0.925799i	$-0.376606\pi$
$433$	15.7321	0.756034	0.378017	+	0.925799i	$0.376606\pi$
$434$	−8.19615	−0.393428
$435$	0	0
$436$	3.19615	0.153068
$437$	3.07180	0.146944
$438$	0	0
$439$	23.3205	1.11303	0.556514	−	0.830839i	$-0.312139\pi$
$439$	23.3205	1.11303	0.556514	+	0.830839i	$0.312139\pi$
$440$	1.66025	0.0791495
$441$	0	0
$442$	45.2487	2.15226
$443$	15.2679	0.725402	0.362701	−	0.931906i	$-0.381855\pi$
$443$	15.2679	0.725402	0.362701	+	0.931906i	$0.381855\pi$
$444$	0	0
$445$	2.66025	0.126108
$446$	−25.4641	−1.20576
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−15.8564	−0.748310	−0.374155	−	0.927366i	$-0.622067\pi$
$449$	−15.8564	−0.748310	−0.374155	+	0.927366i	$0.622067\pi$
$450$	0	0
$451$	15.7128	0.739887
$452$	5.73205	0.269613
$453$	0	0
$454$	18.9282	0.888345
$455$	−1.73205	−0.0811998
$456$	0	0
$457$	−6.85641	−0.320729	−0.160365	−	0.987058i	$-0.551267\pi$
$457$	−6.85641	−0.320729	−0.160365	+	0.987058i	$0.551267\pi$
$458$	−2.46410	−0.115140
$459$	0	0
$460$	−1.12436	−0.0524234
$461$	6.78461	0.315991	0.157995	−	0.987440i	$-0.449497\pi$
$461$	6.78461	0.315991	0.157995	+	0.987440i	$0.449497\pi$
$462$	0	0
$463$	−1.41154	−0.0656000	−0.0328000	−	0.999462i	$-0.510442\pi$
$463$	−1.41154	−0.0656000	−0.0328000	+	0.999462i	$0.510442\pi$
$464$	−1.53590	−0.0713023
$465$	0	0
$466$	2.80385	0.129886
$467$	16.5885	0.767622	0.383811	−	0.923412i	$-0.374611\pi$
$467$	16.5885	0.767622	0.383811	+	0.923412i	$0.374611\pi$
$468$	0	0
$469$	6.73205	0.310857
$470$	1.26795	0.0584861
$471$	0	0
$472$	−4.19615	−0.193144
$473$	9.07180	0.417122
$474$	0	0
$475$	−3.60770	−0.165532
$476$	7.00000	0.320844
$477$	0	0
$478$	10.0526	0.459793
$479$	21.5167	0.983121	0.491561	−	0.870843i	$-0.336427\pi$
$479$	21.5167	0.983121	0.491561	+	0.870843i	$0.336427\pi$
$480$	0	0
$481$	−68.9090	−3.14198
$482$	−14.2679	−0.649887
$483$	0	0
$484$	27.3923	1.24510
$485$	−2.92820	−0.132963
$486$	0	0
$487$	2.58846	0.117294	0.0586471	−	0.998279i	$-0.481321\pi$
$487$	2.58846	0.117294	0.0586471	+	0.998279i	$0.481321\pi$
$488$	3.92820	0.177821
$489$	0	0
$490$	−0.267949	−0.0121047
$491$	−26.5359	−1.19755	−0.598774	−	0.800918i	$-0.704345\pi$
$491$	−26.5359	−1.19755	−0.598774	+	0.800918i	$0.704345\pi$
$492$	0	0
$493$	10.7513	0.484214
$494$	−4.73205	−0.212905
$495$	0	0
$496$	8.19615	0.368018
$497$	6.53590	0.293175
$498$	0	0
$499$	19.8038	0.886542	0.443271	−	0.896388i	$-0.353818\pi$
$499$	19.8038	0.886542	0.443271	+	0.896388i	$0.353818\pi$
$500$	2.66025	0.118970
$501$	0	0
$502$	22.0526	0.984254
$503$	−40.0526	−1.78586	−0.892928	−	0.450200i	$-0.851353\pi$
$503$	−40.0526	−1.78586	−0.892928	+	0.450200i	$0.851353\pi$
$504$	0	0
$505$	2.39230	0.106456
$506$	−26.0000	−1.15584
$507$	0	0
$508$	12.0000	0.532414
$509$	−31.8564	−1.41201	−0.706005	−	0.708207i	$-0.749504\pi$
$509$	−31.8564	−1.41201	−0.706005	+	0.708207i	$0.749504\pi$
$510$	0	0
$511$	−8.26795	−0.365753
$512$	1.00000	0.0441942
$513$	0	0
$514$	−6.46410	−0.285119
$515$	2.24871	0.0990901
$516$	0	0
$517$	29.3205	1.28951
$518$	−10.6603	−0.468385
$519$	0	0
$520$	1.73205	0.0759555
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	33.1769	1.45073	0.725363	−	0.688367i	$-0.241672\pi$
$523$	33.1769	1.45073	0.725363	+	0.688367i	$0.241672\pi$
$524$	−10.5359	−0.460263
$525$	0	0
$526$	6.33975	0.276426
$527$	−57.3731	−2.49921
$528$	0	0
$529$	−5.39230	−0.234448
$530$	−2.53590	−0.110152
$531$	0	0
$532$	−0.732051	−0.0317384
$533$	16.3923	0.710030
$534$	0	0
$535$	0	0
$536$	−6.73205	−0.290780
$537$	0	0
$538$	−5.58846	−0.240936
$539$	−6.19615	−0.266887
$540$	0	0
$541$	3.33975	0.143587	0.0717934	−	0.997420i	$-0.477128\pi$
$541$	3.33975	0.143587	0.0717934	+	0.997420i	$0.477128\pi$
$542$	−19.5167	−0.838312
$543$	0	0
$544$	−7.00000	−0.300123
$545$	−0.856406	−0.0366844
$546$	0	0
$547$	−22.7321	−0.971952	−0.485976	−	0.873972i	$-0.661536\pi$
$547$	−22.7321	−0.971952	−0.485976	+	0.873972i	$0.661536\pi$
$548$	8.26795	0.353189
$549$	0	0
$550$	30.5359	1.30206
$551$	−1.12436	−0.0478992
$552$	0	0
$553$	9.12436	0.388007
$554$	−18.7846	−0.798082
$555$	0	0
$556$	−3.26795	−0.138592
$557$	−23.9282	−1.01387	−0.506935	−	0.861984i	$-0.669222\pi$
$557$	−23.9282	−1.01387	−0.506935	+	0.861984i	$0.669222\pi$
$558$	0	0
$559$	9.46410	0.400289
$560$	0.267949	0.0113229
$561$	0	0
$562$	13.1962	0.556646
$563$	−19.7128	−0.830796	−0.415398	−	0.909640i	$-0.636358\pi$
$563$	−19.7128	−0.830796	−0.415398	+	0.909640i	$0.636358\pi$
$564$	0	0
$565$	−1.53590	−0.0646157
$566$	15.3205	0.643969
$567$	0	0
$568$	−6.53590	−0.274240
$569$	−23.1962	−0.972433	−0.486217	−	0.873838i	$-0.661623\pi$
$569$	−23.1962	−0.972433	−0.486217	+	0.873838i	$0.661623\pi$
$570$	0	0
$571$	−22.7321	−0.951307	−0.475653	−	0.879633i	$-0.657788\pi$
$571$	−22.7321	−0.951307	−0.475653	+	0.879633i	$0.657788\pi$
$572$	40.0526	1.67468
$573$	0	0
$574$	2.53590	0.105846
$575$	−20.6795	−0.862394
$576$	0	0
$577$	24.6603	1.02662	0.513310	−	0.858203i	$-0.328419\pi$
$577$	24.6603	1.02662	0.513310	+	0.858203i	$0.328419\pi$
$578$	32.0000	1.33102
$579$	0	0
$580$	0.411543	0.0170884
$581$	16.5885	0.688205
$582$	0	0
$583$	−58.6410	−2.42866
$584$	8.26795	0.342130
$585$	0	0
$586$	−3.33975	−0.137964
$587$	−14.7321	−0.608057	−0.304028	−	0.952663i	$-0.598332\pi$
$587$	−14.7321	−0.608057	−0.304028	+	0.952663i	$0.598332\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	1.12436	0.0462890
$591$	0	0
$592$	10.6603	0.438134
$593$	22.1769	0.910697	0.455348	−	0.890313i	$-0.349515\pi$
$593$	22.1769	0.910697	0.455348	+	0.890313i	$0.349515\pi$
$594$	0	0
$595$	−1.87564	−0.0768939
$596$	−9.00000	−0.368654
$597$	0	0
$598$	−27.1244	−1.10920
$599$	15.1244	0.617964	0.308982	−	0.951068i	$-0.400012\pi$
$599$	15.1244	0.617964	0.308982	+	0.951068i	$0.400012\pi$
$600$	0	0
$601$	19.1962	0.783027	0.391514	−	0.920172i	$-0.371952\pi$
$601$	19.1962	0.783027	0.391514	+	0.920172i	$0.371952\pi$
$602$	1.46410	0.0596723
$603$	0	0
$604$	−5.80385	−0.236155
$605$	−7.33975	−0.298403
$606$	0	0
$607$	−24.5885	−0.998015	−0.499007	−	0.866598i	$-0.666302\pi$
$607$	−24.5885	−0.998015	−0.499007	+	0.866598i	$0.666302\pi$
$608$	0.732051	0.0296886
$609$	0	0
$610$	−1.05256	−0.0426169
$611$	30.5885	1.23748
$612$	0	0
$613$	26.7846	1.08182	0.540910	−	0.841080i	$-0.318080\pi$
$613$	26.7846	1.08182	0.540910	+	0.841080i	$0.318080\pi$
$614$	−21.8564	−0.882053
$615$	0	0
$616$	6.19615	0.249650
$617$	−11.9808	−0.482327	−0.241164	−	0.970484i	$-0.577529\pi$
$617$	−11.9808	−0.482327	−0.241164	+	0.970484i	$0.577529\pi$
$618$	0	0
$619$	−23.7128	−0.953098	−0.476549	−	0.879148i	$-0.658113\pi$
$619$	−23.7128	−0.953098	−0.476549	+	0.879148i	$0.658113\pi$
$620$	−2.19615	−0.0881996
$621$	0	0
$622$	−10.1962	−0.408828
$623$	9.92820	0.397765
$624$	0	0
$625$	23.9282	0.957128
$626$	25.5885	1.02272
$627$	0	0
$628$	−1.00000	−0.0399043
$629$	−74.6218	−2.97537
$630$	0	0
$631$	−3.66025	−0.145712	−0.0728562	−	0.997342i	$-0.523211\pi$
$631$	−3.66025	−0.145712	−0.0728562	+	0.997342i	$0.523211\pi$
$632$	−9.12436	−0.362947
$633$	0	0
$634$	31.3923	1.24675
$635$	−3.21539	−0.127599
$636$	0	0
$637$	−6.46410	−0.256117
$638$	9.51666	0.376768
$639$	0	0
$640$	−0.267949	−0.0105916
$641$	−39.4449	−1.55798	−0.778989	−	0.627037i	$-0.784267\pi$
$641$	−39.4449	−1.55798	−0.778989	+	0.627037i	$0.784267\pi$
$642$	0	0
$643$	−9.41154	−0.371155	−0.185578	−	0.982630i	$-0.559416\pi$
$643$	−9.41154	−0.371155	−0.185578	+	0.982630i	$0.559416\pi$
$644$	−4.19615	−0.165352
$645$	0	0
$646$	−5.12436	−0.201615
$647$	4.39230	0.172679	0.0863397	−	0.996266i	$-0.472483\pi$
$647$	4.39230	0.172679	0.0863397	+	0.996266i	$0.472483\pi$
$648$	0	0
$649$	26.0000	1.02059
$650$	31.8564	1.24951
$651$	0	0
$652$	−13.4641	−0.527295
$653$	30.2487	1.18372	0.591862	−	0.806039i	$-0.298393\pi$
$653$	30.2487	1.18372	0.591862	+	0.806039i	$0.298393\pi$
$654$	0	0
$655$	2.82309	0.110307
$656$	−2.53590	−0.0990102
$657$	0	0
$658$	4.73205	0.184475
$659$	36.3923	1.41764	0.708821	−	0.705388i	$-0.249227\pi$
$659$	36.3923	1.41764	0.708821	+	0.705388i	$0.249227\pi$
$660$	0	0
$661$	−12.8564	−0.500056	−0.250028	−	0.968239i	$-0.580440\pi$
$661$	−12.8564	−0.500056	−0.250028	+	0.968239i	$0.580440\pi$
$662$	12.3923	0.481641
$663$	0	0
$664$	−16.5885	−0.643757
$665$	0.196152	0.00760646
$666$	0	0
$667$	−6.44486	−0.249546
$668$	1.80385	0.0697930
$669$	0	0
$670$	1.80385	0.0696887
$671$	−24.3397	−0.939625
$672$	0	0
$673$	−18.3205	−0.706204	−0.353102	−	0.935585i	$-0.614873\pi$
$673$	−18.3205	−0.706204	−0.353102	+	0.935585i	$0.614873\pi$
$674$	−16.3923	−0.631408
$675$	0	0
$676$	28.7846	1.10710
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	−10.9282	−0.419386
$680$	1.87564	0.0719277
$681$	0	0
$682$	−50.7846	−1.94464
$683$	1.85641	0.0710334	0.0355167	−	0.999369i	$-0.488692\pi$
$683$	1.85641	0.0710334	0.0355167	+	0.999369i	$0.488692\pi$
$684$	0	0
$685$	−2.21539	−0.0846457
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−1.46410	−0.0558184
$689$	−61.1769	−2.33065
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.26795	−0.238272
$693$	0	0
$694$	−21.4641	−0.814766
$695$	0.875644	0.0332151
$696$	0	0
$697$	17.7513	0.672378
$698$	1.46410	0.0554171
$699$	0	0
$700$	4.92820	0.186269
$701$	−27.3923	−1.03459	−0.517297	−	0.855806i	$-0.673062\pi$
$701$	−27.3923	−1.03459	−0.517297	+	0.855806i	$0.673062\pi$
$702$	0	0
$703$	7.80385	0.294328
$704$	−6.19615	−0.233526
$705$	0	0
$706$	18.0000	0.677439
$707$	8.92820	0.335780
$708$	0	0
$709$	−3.87564	−0.145553	−0.0727764	−	0.997348i	$-0.523186\pi$
$709$	−3.87564	−0.145553	−0.0727764	+	0.997348i	$0.523186\pi$
$710$	1.75129	0.0657247
$711$	0	0
$712$	−9.92820	−0.372075
$713$	34.3923	1.28800
$714$	0	0
$715$	−10.7321	−0.401356
$716$	2.19615	0.0820741
$717$	0	0
$718$	10.9282	0.407837
$719$	9.46410	0.352951	0.176476	−	0.984305i	$-0.443530\pi$
$719$	9.46410	0.352951	0.176476	+	0.984305i	$0.443530\pi$
$720$	0	0
$721$	8.39230	0.312546
$722$	−18.4641	−0.687163
$723$	0	0
$724$	−16.3923	−0.609215
$725$	7.56922	0.281114
$726$	0	0
$727$	51.3205	1.90337	0.951686	−	0.307072i	$-0.0993494\pi$
$727$	51.3205	1.90337	0.951686	+	0.307072i	$0.0993494\pi$
$728$	6.46410	0.239576
$729$	0	0
$730$	−2.21539	−0.0819953
$731$	10.2487	0.379062
$732$	0	0
$733$	47.3205	1.74782	0.873911	−	0.486085i	$-0.161576\pi$
$733$	47.3205	1.74782	0.873911	+	0.486085i	$0.161576\pi$
$734$	11.1244	0.410607
$735$	0	0
$736$	4.19615	0.154672
$737$	41.7128	1.53651
$738$	0	0
$739$	−13.2679	−0.488069	−0.244035	−	0.969767i	$-0.578471\pi$
$739$	−13.2679	−0.488069	−0.244035	+	0.969767i	$0.578471\pi$
$740$	−2.85641	−0.105004
$741$	0	0
$742$	−9.46410	−0.347438
$743$	40.3923	1.48185	0.740925	−	0.671588i	$-0.234387\pi$
$743$	40.3923	1.48185	0.740925	+	0.671588i	$0.234387\pi$
$744$	0	0
$745$	2.41154	0.0883521
$746$	−6.14359	−0.224933
$747$	0	0
$748$	43.3731	1.58588
$749$	0	0
$750$	0	0
$751$	−22.1436	−0.808031	−0.404016	−	0.914752i	$-0.632386\pi$
$751$	−22.1436	−0.808031	−0.404016	+	0.914752i	$0.632386\pi$
$752$	−4.73205	−0.172560
$753$	0	0
$754$	9.92820	0.361564
$755$	1.55514	0.0565972
$756$	0	0
$757$	20.7846	0.755429	0.377715	−	0.925922i	$-0.376710\pi$
$757$	20.7846	0.755429	0.377715	+	0.925922i	$0.376710\pi$
$758$	−27.5167	−0.999450
$759$	0	0
$760$	−0.196152	−0.00711520
$761$	−37.0000	−1.34125	−0.670624	−	0.741797i	$-0.733974\pi$
$761$	−37.0000	−1.34125	−0.670624	+	0.741797i	$0.733974\pi$
$762$	0	0
$763$	−3.19615	−0.115708
$764$	5.66025	0.204781
$765$	0	0
$766$	−19.7128	−0.712253
$767$	27.1244	0.979404
$768$	0	0
$769$	−4.41154	−0.159084	−0.0795421	−	0.996832i	$-0.525346\pi$
$769$	−4.41154	−0.159084	−0.0795421	+	0.996832i	$0.525346\pi$
$770$	−1.66025	−0.0598314
$771$	0	0
$772$	18.8564	0.678657
$773$	4.12436	0.148343	0.0741714	−	0.997246i	$-0.476369\pi$
$773$	4.12436	0.148343	0.0741714	+	0.997246i	$0.476369\pi$
$774$	0	0
$775$	−40.3923	−1.45093
$776$	10.9282	0.392300
$777$	0	0
$778$	13.4641	0.482711
$779$	−1.85641	−0.0665127
$780$	0	0
$781$	40.4974	1.44911
$782$	−29.3731	−1.05038
$783$	0	0
$784$	1.00000	0.0357143
$785$	0.267949	0.00956352
$786$	0	0
$787$	−28.3923	−1.01208	−0.506038	−	0.862511i	$-0.668891\pi$
$787$	−28.3923	−1.01208	−0.506038	+	0.862511i	$0.668891\pi$
$788$	−15.7846	−0.562303
$789$	0	0
$790$	2.44486	0.0869843
$791$	−5.73205	−0.203808
$792$	0	0
$793$	−25.3923	−0.901707
$794$	−21.0000	−0.745262
$795$	0	0
$796$	−19.1244	−0.677845
$797$	29.4449	1.04299	0.521495	−	0.853254i	$-0.325374\pi$
$797$	29.4449	1.04299	0.521495	+	0.853254i	$0.325374\pi$
$798$	0	0
$799$	33.1244	1.17186
$800$	−4.92820	−0.174238
$801$	0	0
$802$	−10.5167	−0.371356
$803$	−51.2295	−1.80785
$804$	0	0
$805$	1.12436	0.0396283
$806$	−52.9808	−1.86617
$807$	0	0
$808$	−8.92820	−0.314093
$809$	−32.1244	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$809$	−32.1244	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$810$	0	0
$811$	18.1962	0.638953	0.319477	−	0.947594i	$-0.396493\pi$
$811$	18.1962	0.638953	0.319477	+	0.947594i	$0.396493\pi$
$812$	1.53590	0.0538995
$813$	0	0
$814$	−66.0526	−2.31514
$815$	3.60770	0.126372
$816$	0	0
$817$	−1.07180	−0.0374974
$818$	17.3397	0.606270
$819$	0	0
$820$	0.679492	0.0237289
$821$	−25.9282	−0.904901	−0.452450	−	0.891790i	$-0.649450\pi$
$821$	−25.9282	−0.904901	−0.452450	+	0.891790i	$0.649450\pi$
$822$	0	0
$823$	0.784610	0.0273498	0.0136749	−	0.999906i	$-0.495647\pi$
$823$	0.784610	0.0273498	0.0136749	+	0.999906i	$0.495647\pi$
$824$	−8.39230	−0.292360
$825$	0	0
$826$	4.19615	0.146003
$827$	23.3205	0.810934	0.405467	−	0.914110i	$-0.367109\pi$
$827$	23.3205	0.810934	0.405467	+	0.914110i	$0.367109\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	4.44486	0.154283
$831$	0	0
$832$	−6.46410	−0.224102
$833$	−7.00000	−0.242536
$834$	0	0
$835$	−0.483340	−0.0167267
$836$	−4.53590	−0.156877
$837$	0	0
$838$	9.46410	0.326932
$839$	−1.46410	−0.0505464	−0.0252732	−	0.999681i	$-0.508046\pi$
$839$	−1.46410	−0.0505464	−0.0252732	+	0.999681i	$0.508046\pi$
$840$	0	0
$841$	−26.6410	−0.918656
$842$	0.124356	0.00428558
$843$	0	0
$844$	−17.2679	−0.594387
$845$	−7.71281	−0.265329
$846$	0	0
$847$	−27.3923	−0.941211
$848$	9.46410	0.324999
$849$	0	0
$850$	34.4974	1.18325
$851$	44.7321	1.53339
$852$	0	0
$853$	−49.7128	−1.70213	−0.851067	−	0.525057i	$-0.824044\pi$
$853$	−49.7128	−1.70213	−0.851067	+	0.525057i	$0.824044\pi$
$854$	−3.92820	−0.134420
$855$	0	0
$856$	0	0
$857$	16.8564	0.575804	0.287902	−	0.957660i	$-0.407042\pi$
$857$	16.8564	0.575804	0.287902	+	0.957660i	$0.407042\pi$
$858$	0	0
$859$	24.3923	0.832255	0.416127	−	0.909306i	$-0.363387\pi$
$859$	24.3923	0.832255	0.416127	+	0.909306i	$0.363387\pi$
$860$	0.392305	0.0133775
$861$	0	0
$862$	14.5359	0.495095
$863$	7.12436	0.242516	0.121258	−	0.992621i	$-0.461307\pi$
$863$	7.12436	0.242516	0.121258	+	0.992621i	$0.461307\pi$
$864$	0	0
$865$	1.67949	0.0571044
$866$	15.7321	0.534597
$867$	0	0
$868$	−8.19615	−0.278196
$869$	56.5359	1.91785
$870$	0	0
$871$	43.5167	1.47451
$872$	3.19615	0.108235
$873$	0	0
$874$	3.07180	0.103905
$875$	−2.66025	−0.0899330
$876$	0	0
$877$	−36.5167	−1.23308	−0.616540	−	0.787324i	$-0.711466\pi$
$877$	−36.5167	−1.23308	−0.616540	+	0.787324i	$0.711466\pi$
$878$	23.3205	0.787029
$879$	0	0
$880$	1.66025	0.0559672
$881$	−30.2487	−1.01910	−0.509552	−	0.860440i	$-0.670189\pi$
$881$	−30.2487	−1.01910	−0.509552	+	0.860440i	$0.670189\pi$
$882$	0	0
$883$	−9.66025	−0.325093	−0.162547	−	0.986701i	$-0.551971\pi$
$883$	−9.66025	−0.325093	−0.162547	+	0.986701i	$0.551971\pi$
$884$	45.2487	1.52188
$885$	0	0
$886$	15.2679	0.512937
$887$	56.4449	1.89523	0.947617	−	0.319410i	$-0.103485\pi$
$887$	56.4449	1.89523	0.947617	+	0.319410i	$0.103485\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	2.66025	0.0891719
$891$	0	0
$892$	−25.4641	−0.852601
$893$	−3.46410	−0.115922
$894$	0	0
$895$	−0.588457	−0.0196700
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−15.8564	−0.529135
$899$	−12.5885	−0.419849
$900$	0	0
$901$	−66.2487	−2.20706
$902$	15.7128	0.523179
$903$	0	0
$904$	5.73205	0.190645
$905$	4.39230	0.146005
$906$	0	0
$907$	−36.0000	−1.19536	−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000	−1.19536	−0.597680	+	0.801735i	$0.703911\pi$
$908$	18.9282	0.628154
$909$	0	0
$910$	−1.73205	−0.0574169
$911$	−42.2487	−1.39976	−0.699881	−	0.714259i	$-0.746764\pi$
$911$	−42.2487	−1.39976	−0.699881	+	0.714259i	$0.746764\pi$
$912$	0	0
$913$	102.785	3.40167
$914$	−6.85641	−0.226790
$915$	0	0
$916$	−2.46410	−0.0814162
$917$	10.5359	0.347926
$918$	0	0
$919$	26.9808	0.890013	0.445007	−	0.895527i	$-0.353201\pi$
$919$	26.9808	0.890013	0.445007	+	0.895527i	$0.353201\pi$
$920$	−1.12436	−0.0370689
$921$	0	0
$922$	6.78461	0.223439
$923$	42.2487	1.39063
$924$	0	0
$925$	−52.5359	−1.72737
$926$	−1.41154	−0.0463862
$927$	0	0
$928$	−1.53590	−0.0504183
$929$	51.4974	1.68958	0.844788	−	0.535101i	$-0.179727\pi$
$929$	51.4974	1.68958	0.844788	+	0.535101i	$0.179727\pi$
$930$	0	0
$931$	0.732051	0.0239920
$932$	2.80385	0.0918431
$933$	0	0
$934$	16.5885	0.542791
$935$	−11.6218	−0.380073
$936$	0	0
$937$	−25.8372	−0.844064	−0.422032	−	0.906581i	$-0.638683\pi$
$937$	−25.8372	−0.844064	−0.422032	+	0.906581i	$0.638683\pi$
$938$	6.73205	0.219809
$939$	0	0
$940$	1.26795	0.0413559
$941$	34.1244	1.11242	0.556211	−	0.831041i	$-0.312255\pi$
$941$	34.1244	1.11242	0.556211	+	0.831041i	$0.312255\pi$
$942$	0	0
$943$	−10.6410	−0.346519
$944$	−4.19615	−0.136573
$945$	0	0
$946$	9.07180	0.294950
$947$	46.2487	1.50288	0.751441	−	0.659801i	$-0.229359\pi$
$947$	46.2487	1.50288	0.751441	+	0.659801i	$0.229359\pi$
$948$	0	0
$949$	−53.4449	−1.73489
$950$	−3.60770	−0.117049
$951$	0	0
$952$	7.00000	0.226871
$953$	41.5885	1.34718	0.673591	−	0.739104i	$-0.264751\pi$
$953$	41.5885	1.34718	0.673591	+	0.739104i	$0.264751\pi$
$954$	0	0
$955$	−1.51666	−0.0490780
$956$	10.0526	0.325123
$957$	0	0
$958$	21.5167	0.695172
$959$	−8.26795	−0.266986
$960$	0	0
$961$	36.1769	1.16700
$962$	−68.9090	−2.22171
$963$	0	0
$964$	−14.2679	−0.459540
$965$	−5.05256	−0.162648
$966$	0	0
$967$	−3.66025	−0.117706	−0.0588529	−	0.998267i	$-0.518744\pi$
$967$	−3.66025	−0.117706	−0.0588529	+	0.998267i	$0.518744\pi$
$968$	27.3923	0.880422
$969$	0	0
$970$	−2.92820	−0.0940189
$971$	−8.87564	−0.284833	−0.142416	−	0.989807i	$-0.545487\pi$
$971$	−8.87564	−0.284833	−0.142416	+	0.989807i	$0.545487\pi$
$972$	0	0
$973$	3.26795	0.104766
$974$	2.58846	0.0829395
$975$	0	0
$976$	3.92820	0.125739
$977$	−57.7128	−1.84640	−0.923198	−	0.384324i	$-0.874435\pi$
$977$	−57.7128	−1.84640	−0.923198	+	0.384324i	$0.874435\pi$
$978$	0	0
$979$	61.5167	1.96608
$980$	−0.267949	−0.00855932
$981$	0	0
$982$	−26.5359	−0.846795
$983$	−58.6410	−1.87036	−0.935179	−	0.354176i	$-0.884762\pi$
$983$	−58.6410	−1.87036	−0.935179	+	0.354176i	$0.884762\pi$
$984$	0	0
$985$	4.22947	0.134762
$986$	10.7513	0.342391
$987$	0	0
$988$	−4.73205	−0.150547
$989$	−6.14359	−0.195355
$990$	0	0
$991$	−27.6603	−0.878657	−0.439328	−	0.898327i	$-0.644784\pi$
$991$	−27.6603	−0.878657	−0.439328	+	0.898327i	$0.644784\pi$
$992$	8.19615	0.260228
$993$	0	0
$994$	6.53590	0.207306
$995$	5.12436	0.162453
$996$	0	0
$997$	41.2487	1.30636	0.653180	−	0.757203i	$-0.273435\pi$
$997$	41.2487	1.30636	0.653180	+	0.757203i	$0.273435\pi$
$998$	19.8038	0.626880
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.2.a.m.1.2	yes	2
3.2	odd	2		1134.2.a.l.1.1	✓	2
4.3	odd	2		9072.2.a.y.1.2		2
7.6	odd	2		7938.2.a.bt.1.1		2
9.2	odd	6		1134.2.f.s.757.2		4
9.4	even	3		1134.2.f.r.379.1		4
9.5	odd	6		1134.2.f.s.379.2		4
9.7	even	3		1134.2.f.r.757.1		4
12.11	even	2		9072.2.a.bp.1.1		2
21.20	even	2		7938.2.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.a.l.1.1	✓	2	3.2	odd	2
1134.2.a.m.1.2	yes	2	1.1	even	1	trivial
1134.2.f.r.379.1		4	9.4	even	3
1134.2.f.r.757.1		4	9.7	even	3
1134.2.f.s.379.2		4	9.5	odd	6
1134.2.f.s.757.2		4	9.2	odd	6
7938.2.a.bg.1.2		2	21.20	even	2
7938.2.a.bt.1.1		2	7.6	odd	2
9072.2.a.y.1.2		2	4.3	odd	2
9072.2.a.bp.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{7} +1.00000 q^{8} -0.267949 q^{10} -6.19615 q^{11} -6.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} +0.732051 q^{19} -0.267949 q^{20} -6.19615 q^{22} +4.19615 q^{23} -4.92820 q^{25} -6.46410 q^{26} -1.00000 q^{28} -1.53590 q^{29} +8.19615 q^{31} +1.00000 q^{32} -7.00000 q^{34} +0.267949 q^{35} +10.6603 q^{37} +0.732051 q^{38} -0.267949 q^{40} -2.53590 q^{41} -1.46410 q^{43} -6.19615 q^{44} +4.19615 q^{46} -4.73205 q^{47} +1.00000 q^{49} -4.92820 q^{50} -6.46410 q^{52} +9.46410 q^{53} +1.66025 q^{55} -1.00000 q^{56} -1.53590 q^{58} -4.19615 q^{59} +3.92820 q^{61} +8.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} -6.73205 q^{67} -7.00000 q^{68} +0.267949 q^{70} -6.53590 q^{71} +8.26795 q^{73} +10.6603 q^{74} +0.732051 q^{76} +6.19615 q^{77} -9.12436 q^{79} -0.267949 q^{80} -2.53590 q^{82} -16.5885 q^{83} +1.87564 q^{85} -1.46410 q^{86} -6.19615 q^{88} -9.92820 q^{89} +6.46410 q^{91} +4.19615 q^{92} -4.73205 q^{94} -0.196152 q^{95} +10.9282 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 6 q^{13} - 2 q^{14} + 2 q^{16} - 14 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{23} + 4 q^{25} - 6 q^{26} - 2 q^{28} - 10 q^{29} + 6 q^{31} + 2 q^{32} - 14 q^{34} + 4 q^{35} + 4 q^{37} - 2 q^{38} - 4 q^{40} - 12 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{46} - 6 q^{47} + 2 q^{49} + 4 q^{50} - 6 q^{52} + 12 q^{53} - 14 q^{55} - 2 q^{56} - 10 q^{58} + 2 q^{59} - 6 q^{61} + 6 q^{62} + 2 q^{64} - 10 q^{67} - 14 q^{68} + 4 q^{70} - 20 q^{71} + 20 q^{73} + 4 q^{74} - 2 q^{76} + 2 q^{77} + 6 q^{79} - 4 q^{80} - 12 q^{82} - 2 q^{83} + 28 q^{85} + 4 q^{86} - 2 q^{88} - 6 q^{89} + 6 q^{91} - 2 q^{92} - 6 q^{94} + 10 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8 - 4 * q^10 - 2 * q^11 - 6 * q^13 - 2 * q^14 + 2 * q^16 - 14 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - 2 * q^23 + 4 * q^25 - 6 * q^26 - 2 * q^28 - 10 * q^29 + 6 * q^31 + 2 * q^32 - 14 * q^34 + 4 * q^35 + 4 * q^37 - 2 * q^38 - 4 * q^40 - 12 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^46 - 6 * q^47 + 2 * q^49 + 4 * q^50 - 6 * q^52 + 12 * q^53 - 14 * q^55 - 2 * q^56 - 10 * q^58 + 2 * q^59 - 6 * q^61 + 6 * q^62 + 2 * q^64 - 10 * q^67 - 14 * q^68 + 4 * q^70 - 20 * q^71 + 20 * q^73 + 4 * q^74 - 2 * q^76 + 2 * q^77 + 6 * q^79 - 4 * q^80 - 12 * q^82 - 2 * q^83 + 28 * q^85 + 4 * q^86 - 2 * q^88 - 6 * q^89 + 6 * q^91 - 2 * q^92 - 6 * q^94 + 10 * q^95 + 8 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more