LMFDB - Embedded newform 1134.2.a.o.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:	$0$
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.73205 q^{10} +1.26795 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.464102 q^{17} +4.19615 q^{19} +1.73205 q^{20} +1.26795 q^{22} +4.73205 q^{23} -2.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} -0.464102 q^{29} -6.19615 q^{31} +1.00000 q^{32} -0.464102 q^{34} +1.73205 q^{35} +7.19615 q^{37} +4.19615 q^{38} +1.73205 q^{40} +9.46410 q^{41} -8.39230 q^{43} +1.26795 q^{44} +4.73205 q^{46} -8.19615 q^{47} +1.00000 q^{49} -2.00000 q^{50} -1.00000 q^{52} +2.53590 q^{53} +2.19615 q^{55} +1.00000 q^{56} -0.464102 q^{58} +2.19615 q^{59} -11.3923 q^{61} -6.19615 q^{62} +1.00000 q^{64} -1.73205 q^{65} -6.19615 q^{67} -0.464102 q^{68} +1.73205 q^{70} +16.3923 q^{71} +1.19615 q^{73} +7.19615 q^{74} +4.19615 q^{76} +1.26795 q^{77} +4.19615 q^{79} +1.73205 q^{80} +9.46410 q^{82} +4.73205 q^{83} -0.803848 q^{85} -8.39230 q^{86} +1.26795 q^{88} -5.53590 q^{89} -1.00000 q^{91} +4.73205 q^{92} -8.19615 q^{94} +7.26795 q^{95} -16.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{19} + 6 q^{22} + 6 q^{23} - 4 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 2 q^{31} + 2 q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 12 q^{41} + 4 q^{43} + 6 q^{44} + 6 q^{46} - 6 q^{47} + 2 q^{49} - 4 q^{50} - 2 q^{52} + 12 q^{53} - 6 q^{55} + 2 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{67} + 6 q^{68} + 12 q^{71} - 8 q^{73} + 4 q^{74} - 2 q^{76} + 6 q^{77} - 2 q^{79} + 12 q^{82} + 6 q^{83} - 12 q^{85} + 4 q^{86} + 6 q^{88} - 18 q^{89} - 2 q^{91} + 6 q^{92} - 6 q^{94} + 18 q^{95} - 32 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 6 * q^17 - 2 * q^19 + 6 * q^22 + 6 * q^23 - 4 * q^25 - 2 * q^26 + 2 * q^28 + 6 * q^29 - 2 * q^31 + 2 * q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 12 * q^41 + 4 * q^43 + 6 * q^44 + 6 * q^46 - 6 * q^47 + 2 * q^49 - 4 * q^50 - 2 * q^52 + 12 * q^53 - 6 * q^55 + 2 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^67 + 6 * q^68 + 12 * q^71 - 8 * q^73 + 4 * q^74 - 2 * q^76 + 6 * q^77 - 2 * q^79 + 12 * q^82 + 6 * q^83 - 12 * q^85 + 4 * q^86 + 6 * q^88 - 18 * q^89 - 2 * q^91 + 6 * q^92 - 6 * q^94 + 18 * q^95 - 32 * 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
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	1.73205	0.547723
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.464102	−0.112561	−0.0562806	−	0.998415i	$-0.517924\pi$
$17$	−0.464102	−0.112561	−0.0562806	+	0.998415i	$0.517924\pi$
$18$	0	0
$19$	4.19615	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$19$	4.19615	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	1.26795	0.270328
$23$	4.73205	0.986701	0.493350	−	0.869831i	$-0.335772\pi$
$23$	4.73205	0.986701	0.493350	+	0.869831i	$0.335772\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−0.464102	−0.0861815	−0.0430908	−	0.999071i	$-0.513720\pi$
$29$	−0.464102	−0.0861815	−0.0430908	+	0.999071i	$0.513720\pi$
$30$	0	0
$31$	−6.19615	−1.11286	−0.556431	−	0.830894i	$-0.687830\pi$
$31$	−6.19615	−1.11286	−0.556431	+	0.830894i	$0.687830\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.464102	−0.0795928
$35$	1.73205	0.292770
$36$	0	0
$37$	7.19615	1.18304	0.591520	−	0.806290i	$-0.298528\pi$
$37$	7.19615	1.18304	0.591520	+	0.806290i	$0.298528\pi$
$38$	4.19615	0.680706
$39$	0	0
$40$	1.73205	0.273861
$41$	9.46410	1.47804	0.739022	−	0.673681i	$-0.235288\pi$
$41$	9.46410	1.47804	0.739022	+	0.673681i	$0.235288\pi$
$42$	0	0
$43$	−8.39230	−1.27981	−0.639907	−	0.768452i	$-0.721027\pi$
$43$	−8.39230	−1.27981	−0.639907	+	0.768452i	$0.721027\pi$
$44$	1.26795	0.191151
$45$	0	0
$46$	4.73205	0.697703
$47$	−8.19615	−1.19553	−0.597766	−	0.801671i	$-0.703945\pi$
$47$	−8.19615	−1.19553	−0.597766	+	0.801671i	$0.703945\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.00000	−0.282843
$51$	0	0
$52$	−1.00000	−0.138675
$53$	2.53590	0.348332	0.174166	−	0.984716i	$-0.444277\pi$
$53$	2.53590	0.348332	0.174166	+	0.984716i	$0.444277\pi$
$54$	0	0
$55$	2.19615	0.296129
$56$	1.00000	0.133631
$57$	0	0
$58$	−0.464102	−0.0609395
$59$	2.19615	0.285915	0.142957	−	0.989729i	$-0.454339\pi$
$59$	2.19615	0.285915	0.142957	+	0.989729i	$0.454339\pi$
$60$	0	0
$61$	−11.3923	−1.45864	−0.729318	−	0.684175i	$-0.760162\pi$
$61$	−11.3923	−1.45864	−0.729318	+	0.684175i	$0.760162\pi$
$62$	−6.19615	−0.786912
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.73205	−0.214834
$66$	0	0
$67$	−6.19615	−0.756980	−0.378490	−	0.925605i	$-0.623557\pi$
$67$	−6.19615	−0.756980	−0.378490	+	0.925605i	$0.623557\pi$
$68$	−0.464102	−0.0562806
$69$	0	0
$70$	1.73205	0.207020
$71$	16.3923	1.94541	0.972704	−	0.232048i	$-0.0745426\pi$
$71$	16.3923	1.94541	0.972704	+	0.232048i	$0.0745426\pi$
$72$	0	0
$73$	1.19615	0.139999	0.0699995	−	0.997547i	$-0.477700\pi$
$73$	1.19615	0.139999	0.0699995	+	0.997547i	$0.477700\pi$
$74$	7.19615	0.836536
$75$	0	0
$76$	4.19615	0.481332
$77$	1.26795	0.144496
$78$	0	0
$79$	4.19615	0.472104	0.236052	−	0.971740i	$-0.424146\pi$
$79$	4.19615	0.472104	0.236052	+	0.971740i	$0.424146\pi$
$80$	1.73205	0.193649
$81$	0	0
$82$	9.46410	1.04514
$83$	4.73205	0.519410	0.259705	−	0.965688i	$-0.416375\pi$
$83$	4.73205	0.519410	0.259705	+	0.965688i	$0.416375\pi$
$84$	0	0
$85$	−0.803848	−0.0871895
$86$	−8.39230	−0.904966
$87$	0	0
$88$	1.26795	0.135164
$89$	−5.53590	−0.586804	−0.293402	−	0.955989i	$-0.594787\pi$
$89$	−5.53590	−0.586804	−0.293402	+	0.955989i	$0.594787\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	4.73205	0.493350
$93$	0	0
$94$	−8.19615	−0.845369
$95$	7.26795	0.745676
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−0.928203	−0.0923597	−0.0461798	−	0.998933i	$-0.514705\pi$
$101$	−0.928203	−0.0923597	−0.0461798	+	0.998933i	$0.514705\pi$
$102$	0	0
$103$	12.3923	1.22105	0.610525	−	0.791997i	$-0.290958\pi$
$103$	12.3923	1.22105	0.610525	+	0.791997i	$0.290958\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	2.53590	0.246308
$107$	13.8564	1.33955	0.669775	−	0.742564i	$-0.266391\pi$
$107$	13.8564	1.33955	0.669775	+	0.742564i	$0.266391\pi$
$108$	0	0
$109$	−15.1962	−1.45553	−0.727764	−	0.685828i	$-0.759440\pi$
$109$	−15.1962	−1.45553	−0.727764	+	0.685828i	$0.759440\pi$
$110$	2.19615	0.209395
$111$	0	0
$112$	1.00000	0.0944911
$113$	−13.7321	−1.29180	−0.645901	−	0.763421i	$-0.723518\pi$
$113$	−13.7321	−1.29180	−0.645901	+	0.763421i	$0.723518\pi$
$114$	0	0
$115$	8.19615	0.764295
$116$	−0.464102	−0.0430908
$117$	0	0
$118$	2.19615	0.202172
$119$	−0.464102	−0.0425441
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−11.3923	−1.03141
$123$	0	0
$124$	−6.19615	−0.556431
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.73205	−0.151911
$131$	−9.46410	−0.826882	−0.413441	−	0.910531i	$-0.635673\pi$
$131$	−9.46410	−0.826882	−0.413441	+	0.910531i	$0.635673\pi$
$132$	0	0
$133$	4.19615	0.363853
$134$	−6.19615	−0.535266
$135$	0	0
$136$	−0.464102	−0.0397964
$137$	14.6603	1.25251	0.626255	−	0.779618i	$-0.284587\pi$
$137$	14.6603	1.25251	0.626255	+	0.779618i	$0.284587\pi$
$138$	0	0
$139$	−7.80385	−0.661914	−0.330957	−	0.943646i	$-0.607371\pi$
$139$	−7.80385	−0.661914	−0.330957	+	0.943646i	$0.607371\pi$
$140$	1.73205	0.146385
$141$	0	0
$142$	16.3923	1.37561
$143$	−1.26795	−0.106031
$144$	0	0
$145$	−0.803848	−0.0667559
$146$	1.19615	0.0989943
$147$	0	0
$148$	7.19615	0.591520
$149$	9.92820	0.813350	0.406675	−	0.913573i	$-0.366688\pi$
$149$	9.92820	0.813350	0.406675	+	0.913573i	$0.366688\pi$
$150$	0	0
$151$	8.58846	0.698919	0.349459	−	0.936952i	$-0.386365\pi$
$151$	8.58846	0.698919	0.349459	+	0.936952i	$0.386365\pi$
$152$	4.19615	0.340353
$153$	0	0
$154$	1.26795	0.102174
$155$	−10.7321	−0.862019
$156$	0	0
$157$	−5.39230	−0.430353	−0.215176	−	0.976575i	$-0.569033\pi$
$157$	−5.39230	−0.430353	−0.215176	+	0.976575i	$0.569033\pi$
$158$	4.19615	0.333828
$159$	0	0
$160$	1.73205	0.136931
$161$	4.73205	0.372938
$162$	0	0
$163$	3.60770	0.282576	0.141288	−	0.989969i	$-0.454876\pi$
$163$	3.60770	0.282576	0.141288	+	0.989969i	$0.454876\pi$
$164$	9.46410	0.739022
$165$	0	0
$166$	4.73205	0.367278
$167$	−10.7321	−0.830471	−0.415236	−	0.909714i	$-0.636301\pi$
$167$	−10.7321	−0.830471	−0.415236	+	0.909714i	$0.636301\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−0.803848	−0.0616523
$171$	0	0
$172$	−8.39230	−0.639907
$173$	−23.1962	−1.76357	−0.881785	−	0.471651i	$-0.843658\pi$
$173$	−23.1962	−1.76357	−0.881785	+	0.471651i	$0.843658\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	1.26795	0.0955753
$177$	0	0
$178$	−5.53590	−0.414933
$179$	10.7321	0.802151	0.401076	−	0.916045i	$-0.368636\pi$
$179$	10.7321	0.802151	0.401076	+	0.916045i	$0.368636\pi$
$180$	0	0
$181$	−20.3923	−1.51575	−0.757874	−	0.652401i	$-0.773762\pi$
$181$	−20.3923	−1.51575	−0.757874	+	0.652401i	$0.773762\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	4.73205	0.348851
$185$	12.4641	0.916379
$186$	0	0
$187$	−0.588457	−0.0430323
$188$	−8.19615	−0.597766
$189$	0	0
$190$	7.26795	0.527272
$191$	−6.58846	−0.476724	−0.238362	−	0.971176i	$-0.576610\pi$
$191$	−6.58846	−0.476724	−0.238362	+	0.971176i	$0.576610\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	1.00000	0.0714286
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	−10.5885	−0.750596	−0.375298	−	0.926904i	$-0.622460\pi$
$199$	−10.5885	−0.750596	−0.375298	+	0.926904i	$0.622460\pi$
$200$	−2.00000	−0.141421
$201$	0	0
$202$	−0.928203	−0.0653082
$203$	−0.464102	−0.0325735
$204$	0	0
$205$	16.3923	1.14489
$206$	12.3923	0.863413
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	5.32051	0.368027
$210$	0	0
$211$	22.1962	1.52805	0.764023	−	0.645189i	$-0.223221\pi$
$211$	22.1962	1.52805	0.764023	+	0.645189i	$0.223221\pi$
$212$	2.53590	0.174166
$213$	0	0
$214$	13.8564	0.947204
$215$	−14.5359	−0.991340
$216$	0	0
$217$	−6.19615	−0.420622
$218$	−15.1962	−1.02921
$219$	0	0
$220$	2.19615	0.148065
$221$	0.464102	0.0312189
$222$	0	0
$223$	−8.39230	−0.561990	−0.280995	−	0.959709i	$-0.590664\pi$
$223$	−8.39230	−0.561990	−0.280995	+	0.959709i	$0.590664\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−13.7321	−0.913442
$227$	−18.9282	−1.25631	−0.628154	−	0.778089i	$-0.716189\pi$
$227$	−18.9282	−1.25631	−0.628154	+	0.778089i	$0.716189\pi$
$228$	0	0
$229$	19.7846	1.30740	0.653702	−	0.756752i	$-0.273215\pi$
$229$	19.7846	1.30740	0.653702	+	0.756752i	$0.273215\pi$
$230$	8.19615	0.540438
$231$	0	0
$232$	−0.464102	−0.0304698
$233$	10.2679	0.672676	0.336338	−	0.941741i	$-0.390812\pi$
$233$	10.2679	0.672676	0.336338	+	0.941741i	$0.390812\pi$
$234$	0	0
$235$	−14.1962	−0.926055
$236$	2.19615	0.142957
$237$	0	0
$238$	−0.464102	−0.0300832
$239$	−9.12436	−0.590206	−0.295103	−	0.955466i	$-0.595354\pi$
$239$	−9.12436	−0.590206	−0.295103	+	0.955466i	$0.595354\pi$
$240$	0	0
$241$	17.5885	1.13297	0.566486	−	0.824071i	$-0.308302\pi$
$241$	17.5885	1.13297	0.566486	+	0.824071i	$0.308302\pi$
$242$	−9.39230	−0.603760
$243$	0	0
$244$	−11.3923	−0.729318
$245$	1.73205	0.110657
$246$	0	0
$247$	−4.19615	−0.266995
$248$	−6.19615	−0.393456
$249$	0	0
$250$	−12.1244	−0.766812
$251$	−14.1962	−0.896053	−0.448027	−	0.894020i	$-0.647873\pi$
$251$	−14.1962	−0.896053	−0.448027	+	0.894020i	$0.647873\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.0718	−0.877775	−0.438887	−	0.898542i	$-0.644627\pi$
$257$	−14.0718	−0.877775	−0.438887	+	0.898542i	$0.644627\pi$
$258$	0	0
$259$	7.19615	0.447147
$260$	−1.73205	−0.107417
$261$	0	0
$262$	−9.46410	−0.584694
$263$	−9.12436	−0.562632	−0.281316	−	0.959615i	$-0.590771\pi$
$263$	−9.12436	−0.562632	−0.281316	+	0.959615i	$0.590771\pi$
$264$	0	0
$265$	4.39230	0.269817
$266$	4.19615	0.257283
$267$	0	0
$268$	−6.19615	−0.378490
$269$	−29.4449	−1.79529	−0.897643	−	0.440724i	$-0.854722\pi$
$269$	−29.4449	−1.79529	−0.897643	+	0.440724i	$0.854722\pi$
$270$	0	0
$271$	17.8038	1.08151	0.540753	−	0.841181i	$-0.318139\pi$
$271$	17.8038	1.08151	0.540753	+	0.841181i	$0.318139\pi$
$272$	−0.464102	−0.0281403
$273$	0	0
$274$	14.6603	0.885658
$275$	−2.53590	−0.152920
$276$	0	0
$277$	22.7846	1.36899	0.684497	−	0.729015i	$-0.260022\pi$
$277$	22.7846	1.36899	0.684497	+	0.729015i	$0.260022\pi$
$278$	−7.80385	−0.468044
$279$	0	0
$280$	1.73205	0.103510
$281$	−10.2679	−0.612534	−0.306267	−	0.951946i	$-0.599080\pi$
$281$	−10.2679	−0.612534	−0.306267	+	0.951946i	$0.599080\pi$
$282$	0	0
$283$	24.3923	1.44997	0.724986	−	0.688764i	$-0.241846\pi$
$283$	24.3923	1.44997	0.724986	+	0.688764i	$0.241846\pi$
$284$	16.3923	0.972704
$285$	0	0
$286$	−1.26795	−0.0749754
$287$	9.46410	0.558648
$288$	0	0
$289$	−16.7846	−0.987330
$290$	−0.803848	−0.0472036
$291$	0	0
$292$	1.19615	0.0699995
$293$	−9.33975	−0.545634	−0.272817	−	0.962066i	$-0.587955\pi$
$293$	−9.33975	−0.545634	−0.272817	+	0.962066i	$0.587955\pi$
$294$	0	0
$295$	3.80385	0.221469
$296$	7.19615	0.418268
$297$	0	0
$298$	9.92820	0.575125
$299$	−4.73205	−0.273662
$300$	0	0
$301$	−8.39230	−0.483724
$302$	8.58846	0.494210
$303$	0	0
$304$	4.19615	0.240666
$305$	−19.7321	−1.12985
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	1.26795	0.0722481
$309$	0	0
$310$	−10.7321	−0.609540
$311$	28.9808	1.64335	0.821674	−	0.569958i	$-0.193040\pi$
$311$	28.9808	1.64335	0.821674	+	0.569958i	$0.193040\pi$
$312$	0	0
$313$	−29.9808	−1.69461	−0.847306	−	0.531104i	$-0.821777\pi$
$313$	−29.9808	−1.69461	−0.847306	+	0.531104i	$0.821777\pi$
$314$	−5.39230	−0.304305
$315$	0	0
$316$	4.19615	0.236052
$317$	19.3923	1.08918	0.544590	−	0.838702i	$-0.316685\pi$
$317$	19.3923	1.08918	0.544590	+	0.838702i	$0.316685\pi$
$318$	0	0
$319$	−0.588457	−0.0329473
$320$	1.73205	0.0968246
$321$	0	0
$322$	4.73205	0.263707
$323$	−1.94744	−0.108359
$324$	0	0
$325$	2.00000	0.110940
$326$	3.60770	0.199812
$327$	0	0
$328$	9.46410	0.522568
$329$	−8.19615	−0.451869
$330$	0	0
$331$	33.1769	1.82357	0.911784	−	0.410670i	$-0.134705\pi$
$331$	33.1769	1.82357	0.911784	+	0.410670i	$0.134705\pi$
$332$	4.73205	0.259705
$333$	0	0
$334$	−10.7321	−0.587232
$335$	−10.7321	−0.586355
$336$	0	0
$337$	−11.6077	−0.632311	−0.316156	−	0.948707i	$-0.602392\pi$
$337$	−11.6077	−0.632311	−0.316156	+	0.948707i	$0.602392\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−0.803848	−0.0435948
$341$	−7.85641	−0.425448
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.39230	−0.452483
$345$	0	0
$346$	−23.1962	−1.24703
$347$	−4.39230	−0.235791	−0.117896	−	0.993026i	$-0.537615\pi$
$347$	−4.39230	−0.235791	−0.117896	+	0.993026i	$0.537615\pi$
$348$	0	0
$349$	−8.39230	−0.449230	−0.224615	−	0.974448i	$-0.572112\pi$
$349$	−8.39230	−0.449230	−0.224615	+	0.974448i	$0.572112\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	1.26795	0.0675819
$353$	31.8564	1.69555	0.847773	−	0.530360i	$-0.177943\pi$
$353$	31.8564	1.69555	0.847773	+	0.530360i	$0.177943\pi$
$354$	0	0
$355$	28.3923	1.50691
$356$	−5.53590	−0.293402
$357$	0	0
$358$	10.7321	0.567207
$359$	5.07180	0.267679	0.133840	−	0.991003i	$-0.457269\pi$
$359$	5.07180	0.267679	0.133840	+	0.991003i	$0.457269\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	−20.3923	−1.07180
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	2.07180	0.108443
$366$	0	0
$367$	26.5885	1.38791	0.693953	−	0.720020i	$-0.255867\pi$
$367$	26.5885	1.38791	0.693953	+	0.720020i	$0.255867\pi$
$368$	4.73205	0.246675
$369$	0	0
$370$	12.4641	0.647978
$371$	2.53590	0.131657
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−0.588457	−0.0304284
$375$	0	0
$376$	−8.19615	−0.422684
$377$	0.464102	0.0239024
$378$	0	0
$379$	14.5885	0.749359	0.374679	−	0.927154i	$-0.377753\pi$
$379$	14.5885	0.749359	0.374679	+	0.927154i	$0.377753\pi$
$380$	7.26795	0.372838
$381$	0	0
$382$	−6.58846	−0.337095
$383$	−37.8564	−1.93437	−0.967186	−	0.254069i	$-0.918231\pi$
$383$	−37.8564	−1.93437	−0.967186	+	0.254069i	$0.918231\pi$
$384$	0	0
$385$	2.19615	0.111926
$386$	−19.0000	−0.967075
$387$	0	0
$388$	−16.0000	−0.812277
$389$	18.2487	0.925246	0.462623	−	0.886555i	$-0.346908\pi$
$389$	18.2487	0.925246	0.462623	+	0.886555i	$0.346908\pi$
$390$	0	0
$391$	−2.19615	−0.111064
$392$	1.00000	0.0505076
$393$	0	0
$394$	−15.0000	−0.755689
$395$	7.26795	0.365690
$396$	0	0
$397$	−8.60770	−0.432008	−0.216004	−	0.976392i	$-0.569302\pi$
$397$	−8.60770	−0.432008	−0.216004	+	0.976392i	$0.569302\pi$
$398$	−10.5885	−0.530751
$399$	0	0
$400$	−2.00000	−0.100000
$401$	25.7321	1.28500	0.642499	−	0.766287i	$-0.277898\pi$
$401$	25.7321	1.28500	0.642499	+	0.766287i	$0.277898\pi$
$402$	0	0
$403$	6.19615	0.308652
$404$	−0.928203	−0.0461798
$405$	0	0
$406$	−0.464102	−0.0230330
$407$	9.12436	0.452278
$408$	0	0
$409$	−10.8038	−0.534216	−0.267108	−	0.963667i	$-0.586068\pi$
$409$	−10.8038	−0.534216	−0.267108	+	0.963667i	$0.586068\pi$
$410$	16.3923	0.809558
$411$	0	0
$412$	12.3923	0.610525
$413$	2.19615	0.108066
$414$	0	0
$415$	8.19615	0.402333
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	5.32051	0.260235
$419$	−11.3205	−0.553043	−0.276522	−	0.961008i	$-0.589182\pi$
$419$	−11.3205	−0.553043	−0.276522	+	0.961008i	$0.589182\pi$
$420$	0	0
$421$	2.80385	0.136651	0.0683256	−	0.997663i	$-0.478234\pi$
$421$	2.80385	0.136651	0.0683256	+	0.997663i	$0.478234\pi$
$422$	22.1962	1.08049
$423$	0	0
$424$	2.53590	0.123154
$425$	0.928203	0.0450245
$426$	0	0
$427$	−11.3923	−0.551312
$428$	13.8564	0.669775
$429$	0	0
$430$	−14.5359	−0.700983
$431$	35.3205	1.70133	0.850665	−	0.525709i	$-0.176200\pi$
$431$	35.3205	1.70133	0.850665	+	0.525709i	$0.176200\pi$
$432$	0	0
$433$	−0.411543	−0.0197775	−0.00988874	−	0.999951i	$-0.503148\pi$
$433$	−0.411543	−0.0197775	−0.00988874	+	0.999951i	$0.503148\pi$
$434$	−6.19615	−0.297425
$435$	0	0
$436$	−15.1962	−0.727764
$437$	19.8564	0.949861
$438$	0	0
$439$	−41.1769	−1.96527	−0.982633	−	0.185557i	$-0.940591\pi$
$439$	−41.1769	−1.96527	−0.982633	+	0.185557i	$0.940591\pi$
$440$	2.19615	0.104697
$441$	0	0
$442$	0.464102	0.0220751
$443$	−8.19615	−0.389411	−0.194705	−	0.980862i	$-0.562375\pi$
$443$	−8.19615	−0.389411	−0.194705	+	0.980862i	$0.562375\pi$
$444$	0	0
$445$	−9.58846	−0.454536
$446$	−8.39230	−0.397387
$447$	0	0
$448$	1.00000	0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−13.7321	−0.645901
$453$	0	0
$454$	−18.9282	−0.888345
$455$	−1.73205	−0.0811998
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	19.7846	0.924474
$459$	0	0
$460$	8.19615	0.382148
$461$	24.9282	1.16102	0.580511	−	0.814252i	$-0.302853\pi$
$461$	24.9282	1.16102	0.580511	+	0.814252i	$0.302853\pi$
$462$	0	0
$463$	16.1962	0.752699	0.376350	−	0.926478i	$-0.377179\pi$
$463$	16.1962	0.752699	0.376350	+	0.926478i	$0.377179\pi$
$464$	−0.464102	−0.0215454
$465$	0	0
$466$	10.2679	0.475654
$467$	−4.73205	−0.218973	−0.109487	−	0.993988i	$-0.534921\pi$
$467$	−4.73205	−0.218973	−0.109487	+	0.993988i	$0.534921\pi$
$468$	0	0
$469$	−6.19615	−0.286112
$470$	−14.1962	−0.654820
$471$	0	0
$472$	2.19615	0.101086
$473$	−10.6410	−0.489274
$474$	0	0
$475$	−8.39230	−0.385065
$476$	−0.464102	−0.0212721
$477$	0	0
$478$	−9.12436	−0.417338
$479$	13.2679	0.606228	0.303114	−	0.952954i	$-0.401974\pi$
$479$	13.2679	0.606228	0.303114	+	0.952954i	$0.401974\pi$
$480$	0	0
$481$	−7.19615	−0.328116
$482$	17.5885	0.801132
$483$	0	0
$484$	−9.39230	−0.426923
$485$	−27.7128	−1.25837
$486$	0	0
$487$	4.19615	0.190146	0.0950729	−	0.995470i	$-0.469692\pi$
$487$	4.19615	0.190146	0.0950729	+	0.995470i	$0.469692\pi$
$488$	−11.3923	−0.515705
$489$	0	0
$490$	1.73205	0.0782461
$491$	−23.3205	−1.05244	−0.526220	−	0.850349i	$-0.676391\pi$
$491$	−23.3205	−1.05244	−0.526220	+	0.850349i	$0.676391\pi$
$492$	0	0
$493$	0.215390	0.00970069
$494$	−4.19615	−0.188794
$495$	0	0
$496$	−6.19615	−0.278215
$497$	16.3923	0.735295
$498$	0	0
$499$	−10.5885	−0.474004	−0.237002	−	0.971509i	$-0.576165\pi$
$499$	−10.5885	−0.474004	−0.237002	+	0.971509i	$0.576165\pi$
$500$	−12.1244	−0.542218
$501$	0	0
$502$	−14.1962	−0.633605
$503$	28.9808	1.29219	0.646094	−	0.763258i	$-0.276401\pi$
$503$	28.9808	1.29219	0.646094	+	0.763258i	$0.276401\pi$
$504$	0	0
$505$	−1.60770	−0.0715415
$506$	6.00000	0.266733
$507$	0	0
$508$	−4.00000	−0.177471
$509$	33.7128	1.49429	0.747147	−	0.664659i	$-0.231423\pi$
$509$	33.7128	1.49429	0.747147	+	0.664659i	$0.231423\pi$
$510$	0	0
$511$	1.19615	0.0529147
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.0718	−0.620680
$515$	21.4641	0.945821
$516$	0	0
$517$	−10.3923	−0.457053
$518$	7.19615	0.316181
$519$	0	0
$520$	−1.73205	−0.0759555
$521$	16.1436	0.707264	0.353632	−	0.935385i	$-0.384947\pi$
$521$	16.1436	0.707264	0.353632	+	0.935385i	$0.384947\pi$
$522$	0	0
$523$	36.3923	1.59132	0.795662	−	0.605741i	$-0.207123\pi$
$523$	36.3923	1.59132	0.795662	+	0.605741i	$0.207123\pi$
$524$	−9.46410	−0.413441
$525$	0	0
$526$	−9.12436	−0.397841
$527$	2.87564	0.125265
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	4.39230	0.190790
$531$	0	0
$532$	4.19615	0.181926
$533$	−9.46410	−0.409936
$534$	0	0
$535$	24.0000	1.03761
$536$	−6.19615	−0.267633
$537$	0	0
$538$	−29.4449	−1.26946
$539$	1.26795	0.0546144
$540$	0	0
$541$	−0.411543	−0.0176936	−0.00884680	−	0.999961i	$-0.502816\pi$
$541$	−0.411543	−0.0176936	−0.00884680	+	0.999961i	$0.502816\pi$
$542$	17.8038	0.764741
$543$	0	0
$544$	−0.464102	−0.0198982
$545$	−26.3205	−1.12745
$546$	0	0
$547$	−30.1962	−1.29109	−0.645547	−	0.763720i	$-0.723371\pi$
$547$	−30.1962	−1.29109	−0.645547	+	0.763720i	$0.723371\pi$
$548$	14.6603	0.626255
$549$	0	0
$550$	−2.53590	−0.108131
$551$	−1.94744	−0.0829638
$552$	0	0
$553$	4.19615	0.178439
$554$	22.7846	0.968025
$555$	0	0
$556$	−7.80385	−0.330957
$557$	13.1436	0.556912	0.278456	−	0.960449i	$-0.410177\pi$
$557$	13.1436	0.556912	0.278456	+	0.960449i	$0.410177\pi$
$558$	0	0
$559$	8.39230	0.354957
$560$	1.73205	0.0731925
$561$	0	0
$562$	−10.2679	−0.433127
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−23.7846	−1.00063
$566$	24.3923	1.02529
$567$	0	0
$568$	16.3923	0.687806
$569$	43.9808	1.84377	0.921885	−	0.387464i	$-0.126649\pi$
$569$	43.9808	1.84377	0.921885	+	0.387464i	$0.126649\pi$
$570$	0	0
$571$	−30.1962	−1.26367	−0.631835	−	0.775103i	$-0.717698\pi$
$571$	−30.1962	−1.26367	−0.631835	+	0.775103i	$0.717698\pi$
$572$	−1.26795	−0.0530156
$573$	0	0
$574$	9.46410	0.395024
$575$	−9.46410	−0.394680
$576$	0	0
$577$	20.8038	0.866076	0.433038	−	0.901376i	$-0.357442\pi$
$577$	20.8038	0.866076	0.433038	+	0.901376i	$0.357442\pi$
$578$	−16.7846	−0.698148
$579$	0	0
$580$	−0.803848	−0.0333780
$581$	4.73205	0.196319
$582$	0	0
$583$	3.21539	0.133168
$584$	1.19615	0.0494971
$585$	0	0
$586$	−9.33975	−0.385821
$587$	−7.26795	−0.299980	−0.149990	−	0.988687i	$-0.547924\pi$
$587$	−7.26795	−0.299980	−0.149990	+	0.988687i	$0.547924\pi$
$588$	0	0
$589$	−26.0000	−1.07131
$590$	3.80385	0.156602
$591$	0	0
$592$	7.19615	0.295760
$593$	−13.1436	−0.539743	−0.269871	−	0.962896i	$-0.586981\pi$
$593$	−13.1436	−0.539743	−0.269871	+	0.962896i	$0.586981\pi$
$594$	0	0
$595$	−0.803848	−0.0329545
$596$	9.92820	0.406675
$597$	0	0
$598$	−4.73205	−0.193508
$599$	−14.1962	−0.580039	−0.290020	−	0.957021i	$-0.593662\pi$
$599$	−14.1962	−0.580039	−0.290020	+	0.957021i	$0.593662\pi$
$600$	0	0
$601$	1.19615	0.0487921	0.0243960	−	0.999702i	$-0.492234\pi$
$601$	1.19615	0.0487921	0.0243960	+	0.999702i	$0.492234\pi$
$602$	−8.39230	−0.342045
$603$	0	0
$604$	8.58846	0.349459
$605$	−16.2679	−0.661386
$606$	0	0
$607$	2.58846	0.105062	0.0525311	−	0.998619i	$-0.483271\pi$
$607$	2.58846	0.105062	0.0525311	+	0.998619i	$0.483271\pi$
$608$	4.19615	0.170176
$609$	0	0
$610$	−19.7321	−0.798927
$611$	8.19615	0.331581
$612$	0	0
$613$	34.7846	1.40494	0.702469	−	0.711715i	$-0.252081\pi$
$613$	34.7846	1.40494	0.702469	+	0.711715i	$0.252081\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	1.26795	0.0510871
$617$	−28.5167	−1.14804	−0.574019	−	0.818842i	$-0.694616\pi$
$617$	−28.5167	−1.14804	−0.574019	+	0.818842i	$0.694616\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−10.7321	−0.431010
$621$	0	0
$622$	28.9808	1.16202
$623$	−5.53590	−0.221791
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−29.9808	−1.19827
$627$	0	0
$628$	−5.39230	−0.215176
$629$	−3.33975	−0.133164
$630$	0	0
$631$	8.58846	0.341901	0.170951	−	0.985280i	$-0.445316\pi$
$631$	8.58846	0.341901	0.170951	+	0.985280i	$0.445316\pi$
$632$	4.19615	0.166914
$633$	0	0
$634$	19.3923	0.770167
$635$	−6.92820	−0.274937
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−0.588457	−0.0232972
$639$	0	0
$640$	1.73205	0.0684653
$641$	26.6603	1.05302	0.526508	−	0.850170i	$-0.323501\pi$
$641$	26.6603	1.05302	0.526508	+	0.850170i	$0.323501\pi$
$642$	0	0
$643$	28.1962	1.11195	0.555974	−	0.831200i	$-0.312345\pi$
$643$	28.1962	1.11195	0.555974	+	0.831200i	$0.312345\pi$
$644$	4.73205	0.186469
$645$	0	0
$646$	−1.94744	−0.0766210
$647$	11.3205	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$647$	11.3205	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$648$	0	0
$649$	2.78461	0.109305
$650$	2.00000	0.0784465
$651$	0	0
$652$	3.60770	0.141288
$653$	−0.679492	−0.0265906	−0.0132953	−	0.999912i	$-0.504232\pi$
$653$	−0.679492	−0.0265906	−0.0132953	+	0.999912i	$0.504232\pi$
$654$	0	0
$655$	−16.3923	−0.640500
$656$	9.46410	0.369511
$657$	0	0
$658$	−8.19615	−0.319519
$659$	35.3205	1.37589	0.687946	−	0.725762i	$-0.258513\pi$
$659$	35.3205	1.37589	0.687946	+	0.725762i	$0.258513\pi$
$660$	0	0
$661$	12.6077	0.490383	0.245191	−	0.969475i	$-0.421149\pi$
$661$	12.6077	0.490383	0.245191	+	0.969475i	$0.421149\pi$
$662$	33.1769	1.28946
$663$	0	0
$664$	4.73205	0.183639
$665$	7.26795	0.281839
$666$	0	0
$667$	−2.19615	−0.0850354
$668$	−10.7321	−0.415236
$669$	0	0
$670$	−10.7321	−0.414615
$671$	−14.4449	−0.557638
$672$	0	0
$673$	−44.1769	−1.70289	−0.851447	−	0.524440i	$-0.824275\pi$
$673$	−44.1769	−1.70289	−0.851447	+	0.524440i	$0.824275\pi$
$674$	−11.6077	−0.447112
$675$	0	0
$676$	−12.0000	−0.461538
$677$	3.21539	0.123577	0.0617887	−	0.998089i	$-0.480319\pi$
$677$	3.21539	0.123577	0.0617887	+	0.998089i	$0.480319\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	−0.803848	−0.0308261
$681$	0	0
$682$	−7.85641	−0.300837
$683$	15.7128	0.601234	0.300617	−	0.953745i	$-0.402807\pi$
$683$	15.7128	0.601234	0.300617	+	0.953745i	$0.402807\pi$
$684$	0	0
$685$	25.3923	0.970190
$686$	1.00000	0.0381802
$687$	0	0
$688$	−8.39230	−0.319954
$689$	−2.53590	−0.0966100
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−23.1962	−0.881785
$693$	0	0
$694$	−4.39230	−0.166730
$695$	−13.5167	−0.512716
$696$	0	0
$697$	−4.39230	−0.166370
$698$	−8.39230	−0.317653
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−16.1769	−0.610994	−0.305497	−	0.952193i	$-0.598823\pi$
$701$	−16.1769	−0.610994	−0.305497	+	0.952193i	$0.598823\pi$
$702$	0	0
$703$	30.1962	1.13887
$704$	1.26795	0.0477876
$705$	0	0
$706$	31.8564	1.19893
$707$	−0.928203	−0.0349087
$708$	0	0
$709$	23.5885	0.885883	0.442942	−	0.896550i	$-0.353935\pi$
$709$	23.5885	0.885883	0.442942	+	0.896550i	$0.353935\pi$
$710$	28.3923	1.06554
$711$	0	0
$712$	−5.53590	−0.207467
$713$	−29.3205	−1.09806
$714$	0	0
$715$	−2.19615	−0.0821314
$716$	10.7321	0.401076
$717$	0	0
$718$	5.07180	0.189278
$719$	−11.3205	−0.422184	−0.211092	−	0.977466i	$-0.567702\pi$
$719$	−11.3205	−0.422184	−0.211092	+	0.977466i	$0.567702\pi$
$720$	0	0
$721$	12.3923	0.461514
$722$	−1.39230	−0.0518162
$723$	0	0
$724$	−20.3923	−0.757874
$725$	0.928203	0.0344726
$726$	0	0
$727$	36.3923	1.34972	0.674858	−	0.737948i	$-0.264205\pi$
$727$	36.3923	1.34972	0.674858	+	0.737948i	$0.264205\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	2.07180	0.0766806
$731$	3.89488	0.144057
$732$	0	0
$733$	33.1769	1.22542	0.612709	−	0.790309i	$-0.290080\pi$
$733$	33.1769	1.22542	0.612709	+	0.790309i	$0.290080\pi$
$734$	26.5885	0.981398
$735$	0	0
$736$	4.73205	0.174426
$737$	−7.85641	−0.289394
$738$	0	0
$739$	−13.8038	−0.507783	−0.253891	−	0.967233i	$-0.581711\pi$
$739$	−13.8038	−0.507783	−0.253891	+	0.967233i	$0.581711\pi$
$740$	12.4641	0.458189
$741$	0	0
$742$	2.53590	0.0930958
$743$	−14.5359	−0.533270	−0.266635	−	0.963798i	$-0.585912\pi$
$743$	−14.5359	−0.533270	−0.266635	+	0.963798i	$0.585912\pi$
$744$	0	0
$745$	17.1962	0.630018
$746$	20.0000	0.732252
$747$	0	0
$748$	−0.588457	−0.0215161
$749$	13.8564	0.506302
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−8.19615	−0.298883
$753$	0	0
$754$	0.464102	0.0169016
$755$	14.8756	0.541380
$756$	0	0
$757$	−36.7846	−1.33696	−0.668480	−	0.743730i	$-0.733055\pi$
$757$	−36.7846	−1.33696	−0.668480	+	0.743730i	$0.733055\pi$
$758$	14.5885	0.529877
$759$	0	0
$760$	7.26795	0.263636
$761$	26.3205	0.954118	0.477059	−	0.878871i	$-0.341703\pi$
$761$	26.3205	0.954118	0.477059	+	0.878871i	$0.341703\pi$
$762$	0	0
$763$	−15.1962	−0.550138
$764$	−6.58846	−0.238362
$765$	0	0
$766$	−37.8564	−1.36781
$767$	−2.19615	−0.0792985
$768$	0	0
$769$	−3.19615	−0.115256	−0.0576281	−	0.998338i	$-0.518354\pi$
$769$	−3.19615	−0.115256	−0.0576281	+	0.998338i	$0.518354\pi$
$770$	2.19615	0.0791438
$771$	0	0
$772$	−19.0000	−0.683825
$773$	−38.6603	−1.39051	−0.695256	−	0.718762i	$-0.744709\pi$
$773$	−38.6603	−1.39051	−0.695256	+	0.718762i	$0.744709\pi$
$774$	0	0
$775$	12.3923	0.445145
$776$	−16.0000	−0.574367
$777$	0	0
$778$	18.2487	0.654248
$779$	39.7128	1.42286
$780$	0	0
$781$	20.7846	0.743732
$782$	−2.19615	−0.0785343
$783$	0	0
$784$	1.00000	0.0357143
$785$	−9.33975	−0.333350
$786$	0	0
$787$	−17.1769	−0.612291	−0.306145	−	0.951985i	$-0.599039\pi$
$787$	−17.1769	−0.612291	−0.306145	+	0.951985i	$0.599039\pi$
$788$	−15.0000	−0.534353
$789$	0	0
$790$	7.26795	0.258582
$791$	−13.7321	−0.488256
$792$	0	0
$793$	11.3923	0.404553
$794$	−8.60770	−0.305476
$795$	0	0
$796$	−10.5885	−0.375298
$797$	38.6603	1.36942	0.684708	−	0.728817i	$-0.259930\pi$
$797$	38.6603	1.36942	0.684708	+	0.728817i	$0.259930\pi$
$798$	0	0
$799$	3.80385	0.134570
$800$	−2.00000	−0.0707107
$801$	0	0
$802$	25.7321	0.908630
$803$	1.51666	0.0535218
$804$	0	0
$805$	8.19615	0.288876
$806$	6.19615	0.218250
$807$	0	0
$808$	−0.928203	−0.0326541
$809$	24.1244	0.848167	0.424084	−	0.905623i	$-0.360596\pi$
$809$	24.1244	0.848167	0.424084	+	0.905623i	$0.360596\pi$
$810$	0	0
$811$	3.01924	0.106020	0.0530099	−	0.998594i	$-0.483119\pi$
$811$	3.01924	0.106020	0.0530099	+	0.998594i	$0.483119\pi$
$812$	−0.464102	−0.0162868
$813$	0	0
$814$	9.12436	0.319809
$815$	6.24871	0.218883
$816$	0	0
$817$	−35.2154	−1.23203
$818$	−10.8038	−0.377748
$819$	0	0
$820$	16.3923	0.572444
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	16.7846	0.585075	0.292537	−	0.956254i	$-0.405500\pi$
$823$	16.7846	0.585075	0.292537	+	0.956254i	$0.405500\pi$
$824$	12.3923	0.431706
$825$	0	0
$826$	2.19615	0.0764139
$827$	40.3923	1.40458	0.702289	−	0.711892i	$-0.252161\pi$
$827$	40.3923	1.40458	0.702289	+	0.711892i	$0.252161\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	8.19615	0.284493
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−0.464102	−0.0160802
$834$	0	0
$835$	−18.5885	−0.643280
$836$	5.32051	0.184014
$837$	0	0
$838$	−11.3205	−0.391060
$839$	−6.24871	−0.215729	−0.107865	−	0.994166i	$-0.534401\pi$
$839$	−6.24871	−0.215729	−0.107865	+	0.994166i	$0.534401\pi$
$840$	0	0
$841$	−28.7846	−0.992573
$842$	2.80385	0.0966270
$843$	0	0
$844$	22.1962	0.764023
$845$	−20.7846	−0.715012
$846$	0	0
$847$	−9.39230	−0.322723
$848$	2.53590	0.0870831
$849$	0	0
$850$	0.928203	0.0318371
$851$	34.0526	1.16731
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	−11.3923	−0.389837
$855$	0	0
$856$	13.8564	0.473602
$857$	−11.5359	−0.394059	−0.197029	−	0.980398i	$-0.563129\pi$
$857$	−11.5359	−0.394059	−0.197029	+	0.980398i	$0.563129\pi$
$858$	0	0
$859$	−20.3923	−0.695776	−0.347888	−	0.937536i	$-0.613101\pi$
$859$	−20.3923	−0.695776	−0.347888	+	0.937536i	$0.613101\pi$
$860$	−14.5359	−0.495670
$861$	0	0
$862$	35.3205	1.20302
$863$	−41.9090	−1.42660	−0.713299	−	0.700860i	$-0.752800\pi$
$863$	−41.9090	−1.42660	−0.713299	+	0.700860i	$0.752800\pi$
$864$	0	0
$865$	−40.1769	−1.36606
$866$	−0.411543	−0.0139848
$867$	0	0
$868$	−6.19615	−0.210311
$869$	5.32051	0.180486
$870$	0	0
$871$	6.19615	0.209949
$872$	−15.1962	−0.514607
$873$	0	0
$874$	19.8564	0.671653
$875$	−12.1244	−0.409878
$876$	0	0
$877$	20.8038	0.702496	0.351248	−	0.936282i	$-0.385757\pi$
$877$	20.8038	0.702496	0.351248	+	0.936282i	$0.385757\pi$
$878$	−41.1769	−1.38965
$879$	0	0
$880$	2.19615	0.0740323
$881$	37.1769	1.25252	0.626261	−	0.779613i	$-0.284584\pi$
$881$	37.1769	1.25252	0.626261	+	0.779613i	$0.284584\pi$
$882$	0	0
$883$	−38.9808	−1.31181	−0.655904	−	0.754845i	$-0.727712\pi$
$883$	−38.9808	−1.31181	−0.655904	+	0.754845i	$0.727712\pi$
$884$	0.464102	0.0156094
$885$	0	0
$886$	−8.19615	−0.275355
$887$	18.3397	0.615788	0.307894	−	0.951421i	$-0.400376\pi$
$887$	18.3397	0.615788	0.307894	+	0.951421i	$0.400376\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	−9.58846	−0.321406
$891$	0	0
$892$	−8.39230	−0.280995
$893$	−34.3923	−1.15089
$894$	0	0
$895$	18.5885	0.621344
$896$	1.00000	0.0334077
$897$	0	0
$898$	6.00000	0.200223
$899$	2.87564	0.0959081
$900$	0	0
$901$	−1.17691	−0.0392087
$902$	12.0000	0.399556
$903$	0	0
$904$	−13.7321	−0.456721
$905$	−35.3205	−1.17409
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−18.9282	−0.628154
$909$	0	0
$910$	−1.73205	−0.0574169
$911$	44.1051	1.46127	0.730634	−	0.682769i	$-0.239225\pi$
$911$	44.1051	1.46127	0.730634	+	0.682769i	$0.239225\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	−1.00000	−0.0330771
$915$	0	0
$916$	19.7846	0.653702
$917$	−9.46410	−0.312532
$918$	0	0
$919$	−48.1962	−1.58984	−0.794922	−	0.606711i	$-0.792488\pi$
$919$	−48.1962	−1.58984	−0.794922	+	0.606711i	$0.792488\pi$
$920$	8.19615	0.270219
$921$	0	0
$922$	24.9282	0.820967
$923$	−16.3923	−0.539559
$924$	0	0
$925$	−14.3923	−0.473216
$926$	16.1962	0.532239
$927$	0	0
$928$	−0.464102	−0.0152349
$929$	−13.3923	−0.439387	−0.219694	−	0.975569i	$-0.570506\pi$
$929$	−13.3923	−0.439387	−0.219694	+	0.975569i	$0.570506\pi$
$930$	0	0
$931$	4.19615	0.137523
$932$	10.2679	0.336338
$933$	0	0
$934$	−4.73205	−0.154837
$935$	−1.01924	−0.0333326
$936$	0	0
$937$	−33.1962	−1.08447	−0.542236	−	0.840227i	$-0.682422\pi$
$937$	−33.1962	−1.08447	−0.542236	+	0.840227i	$0.682422\pi$
$938$	−6.19615	−0.202312
$939$	0	0
$940$	−14.1962	−0.463027
$941$	46.7654	1.52451	0.762254	−	0.647278i	$-0.224093\pi$
$941$	46.7654	1.52451	0.762254	+	0.647278i	$0.224093\pi$
$942$	0	0
$943$	44.7846	1.45839
$944$	2.19615	0.0714787
$945$	0	0
$946$	−10.6410	−0.345969
$947$	−28.3923	−0.922626	−0.461313	−	0.887237i	$-0.652621\pi$
$947$	−28.3923	−0.922626	−0.461313	+	0.887237i	$0.652621\pi$
$948$	0	0
$949$	−1.19615	−0.0388288
$950$	−8.39230	−0.272282
$951$	0	0
$952$	−0.464102	−0.0150416
$953$	−4.01924	−0.130196	−0.0650979	−	0.997879i	$-0.520736\pi$
$953$	−4.01924	−0.130196	−0.0650979	+	0.997879i	$0.520736\pi$
$954$	0	0
$955$	−11.4115	−0.369269
$956$	−9.12436	−0.295103
$957$	0	0
$958$	13.2679	0.428668
$959$	14.6603	0.473404
$960$	0	0
$961$	7.39230	0.238461
$962$	−7.19615	−0.232013
$963$	0	0
$964$	17.5885	0.566486
$965$	−32.9090	−1.05938
$966$	0	0
$967$	8.58846	0.276186	0.138093	−	0.990419i	$-0.455903\pi$
$967$	8.58846	0.276186	0.138093	+	0.990419i	$0.455903\pi$
$968$	−9.39230	−0.301880
$969$	0	0
$970$	−27.7128	−0.889805
$971$	−21.1244	−0.677913	−0.338956	−	0.940802i	$-0.610074\pi$
$971$	−21.1244	−0.677913	−0.338956	+	0.940802i	$0.610074\pi$
$972$	0	0
$973$	−7.80385	−0.250180
$974$	4.19615	0.134453
$975$	0	0
$976$	−11.3923	−0.364659
$977$	−47.5692	−1.52187	−0.760937	−	0.648826i	$-0.775260\pi$
$977$	−47.5692	−1.52187	−0.760937	+	0.648826i	$0.775260\pi$
$978$	0	0
$979$	−7.01924	−0.224336
$980$	1.73205	0.0553283
$981$	0	0
$982$	−23.3205	−0.744187
$983$	−54.9282	−1.75194	−0.875969	−	0.482368i	$-0.839777\pi$
$983$	−54.9282	−1.75194	−0.875969	+	0.482368i	$0.839777\pi$
$984$	0	0
$985$	−25.9808	−0.827816
$986$	0.215390	0.00685942
$987$	0	0
$988$	−4.19615	−0.133497
$989$	−39.7128	−1.26279
$990$	0	0
$991$	−8.98076	−0.285283	−0.142642	−	0.989774i	$-0.545560\pi$
$991$	−8.98076	−0.285283	−0.142642	+	0.989774i	$0.545560\pi$
$992$	−6.19615	−0.196728
$993$	0	0
$994$	16.3923	0.519932
$995$	−18.3397	−0.581409
$996$	0	0
$997$	19.7846	0.626585	0.313292	−	0.949657i	$-0.398568\pi$
$997$	19.7846	0.626585	0.313292	+	0.949657i	$0.398568\pi$
$998$	−10.5885	−0.335172
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.2.a.o.1.2	yes	2
3.2	odd	2		1134.2.a.j.1.1	✓	2
4.3	odd	2		9072.2.a.bf.1.2		2
7.6	odd	2		7938.2.a.br.1.1		2
9.2	odd	6		1134.2.f.t.757.2		4
9.4	even	3		1134.2.f.q.379.1		4
9.5	odd	6		1134.2.f.t.379.2		4
9.7	even	3		1134.2.f.q.757.1		4
12.11	even	2		9072.2.a.bi.1.1		2
21.20	even	2		7938.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.a.j.1.1	✓	2	3.2	odd	2
1134.2.a.o.1.2	yes	2	1.1	even	1	trivial
1134.2.f.q.379.1		4	9.4	even	3
1134.2.f.q.757.1		4	9.7	even	3
1134.2.f.t.379.2		4	9.5	odd	6
1134.2.f.t.757.2		4	9.2	odd	6
7938.2.a.bi.1.2		2	21.20	even	2
7938.2.a.br.1.1		2	7.6	odd	2
9072.2.a.bf.1.2		2	4.3	odd	2
9072.2.a.bi.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.73205 q^{10} +1.26795 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.464102 q^{17} +4.19615 q^{19} +1.73205 q^{20} +1.26795 q^{22} +4.73205 q^{23} -2.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} -0.464102 q^{29} -6.19615 q^{31} +1.00000 q^{32} -0.464102 q^{34} +1.73205 q^{35} +7.19615 q^{37} +4.19615 q^{38} +1.73205 q^{40} +9.46410 q^{41} -8.39230 q^{43} +1.26795 q^{44} +4.73205 q^{46} -8.19615 q^{47} +1.00000 q^{49} -2.00000 q^{50} -1.00000 q^{52} +2.53590 q^{53} +2.19615 q^{55} +1.00000 q^{56} -0.464102 q^{58} +2.19615 q^{59} -11.3923 q^{61} -6.19615 q^{62} +1.00000 q^{64} -1.73205 q^{65} -6.19615 q^{67} -0.464102 q^{68} +1.73205 q^{70} +16.3923 q^{71} +1.19615 q^{73} +7.19615 q^{74} +4.19615 q^{76} +1.26795 q^{77} +4.19615 q^{79} +1.73205 q^{80} +9.46410 q^{82} +4.73205 q^{83} -0.803848 q^{85} -8.39230 q^{86} +1.26795 q^{88} -5.53590 q^{89} -1.00000 q^{91} +4.73205 q^{92} -8.19615 q^{94} +7.26795 q^{95} -16.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{19} + 6 q^{22} + 6 q^{23} - 4 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 2 q^{31} + 2 q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 12 q^{41} + 4 q^{43} + 6 q^{44} + 6 q^{46} - 6 q^{47} + 2 q^{49} - 4 q^{50} - 2 q^{52} + 12 q^{53} - 6 q^{55} + 2 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{67} + 6 q^{68} + 12 q^{71} - 8 q^{73} + 4 q^{74} - 2 q^{76} + 6 q^{77} - 2 q^{79} + 12 q^{82} + 6 q^{83} - 12 q^{85} + 4 q^{86} + 6 q^{88} - 18 q^{89} - 2 q^{91} + 6 q^{92} - 6 q^{94} + 18 q^{95} - 32 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 6 * q^17 - 2 * q^19 + 6 * q^22 + 6 * q^23 - 4 * q^25 - 2 * q^26 + 2 * q^28 + 6 * q^29 - 2 * q^31 + 2 * q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 12 * q^41 + 4 * q^43 + 6 * q^44 + 6 * q^46 - 6 * q^47 + 2 * q^49 - 4 * q^50 - 2 * q^52 + 12 * q^53 - 6 * q^55 + 2 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^67 + 6 * q^68 + 12 * q^71 - 8 * q^73 + 4 * q^74 - 2 * q^76 + 6 * q^77 - 2 * q^79 + 12 * q^82 + 6 * q^83 - 12 * q^85 + 4 * q^86 + 6 * q^88 - 18 * q^89 - 2 * q^91 + 6 * q^92 - 6 * q^94 + 18 * q^95 - 32 * q^97 + 2 * q^98

Introduction

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