LMFDB - Embedded newform 9075.2.a.dn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.9444552.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 7x^{2} + 9x - 6 $ x^5 - x^4 - 8x^3 + 7x^2 + 9*x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.583819$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.583819 q^{2} +1.00000 q^{3} -1.65916 q^{4} +0.583819 q^{6} +4.38942 q^{7} -2.13628 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.583819 q^{2} +1.00000 q^{3} -1.65916 q^{4} +0.583819 q^{6} +4.38942 q^{7} -2.13628 q^{8} +1.00000 q^{9} -1.65916 q^{12} -1.72010 q^{13} +2.56263 q^{14} +2.07111 q^{16} +3.73026 q^{17} +0.583819 q^{18} -2.79121 q^{19} +4.38942 q^{21} -6.45037 q^{23} -2.13628 q^{24} -1.00423 q^{26} +1.00000 q^{27} -7.28273 q^{28} -7.88094 q^{29} +3.05414 q^{31} +5.48172 q^{32} +2.17780 q^{34} -1.65916 q^{36} +9.21621 q^{37} -1.62956 q^{38} -1.72010 q^{39} +9.20605 q^{41} +2.56263 q^{42} +8.88094 q^{43} -3.76585 q^{46} -6.29289 q^{47} +2.07111 q^{48} +12.2670 q^{49} +3.73026 q^{51} +2.85392 q^{52} +6.32847 q^{53} +0.583819 q^{54} -9.37705 q^{56} -2.79121 q^{57} -4.60104 q^{58} -0.0655391 q^{59} +5.21162 q^{61} +1.78307 q^{62} +4.38942 q^{63} -0.941884 q^{64} -6.67215 q^{67} -6.18909 q^{68} -6.45037 q^{69} -15.9493 q^{71} -2.13628 q^{72} +3.40515 q^{73} +5.38060 q^{74} +4.63105 q^{76} -1.00423 q^{78} +3.56263 q^{79} +1.00000 q^{81} +5.37467 q^{82} +7.22920 q^{83} -7.28273 q^{84} +5.18486 q^{86} -7.88094 q^{87} +9.20605 q^{89} -7.55025 q^{91} +10.7022 q^{92} +3.05414 q^{93} -3.67391 q^{94} +5.48172 q^{96} +3.33404 q^{97} +7.16171 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 5 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + q^2 + 5 * q^3 + 7 * q^4 + q^6 - 4 * q^7 + 5 * q^9	$ 5 q + q^{2} + 5 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} + 5 q^{9} + 7 q^{12} + 4 q^{13} + 6 q^{14} + 15 q^{16} + 8 q^{17} + q^{18} - 6 q^{19} - 4 q^{21} - 9 q^{23} + 13 q^{26} + 5 q^{27} - 17 q^{28} - 2 q^{29} - 3 q^{31} + 11 q^{32} + 9 q^{34} + 7 q^{36} + q^{37} + 3 q^{38} + 4 q^{39} - q^{41} + 6 q^{42} + 7 q^{43} + 3 q^{46} - 14 q^{47} + 15 q^{48} + 17 q^{49} + 8 q^{51} + 20 q^{52} + 3 q^{53} + q^{54} + 23 q^{56} - 6 q^{57} + 27 q^{58} + 18 q^{59} + 2 q^{61} - 61 q^{62} - 4 q^{63} + 30 q^{64} + 12 q^{67} + 39 q^{68} - 9 q^{69} + 8 q^{71} + 16 q^{73} + 40 q^{74} - 61 q^{76} + 13 q^{78} + 11 q^{79} + 5 q^{81} + 20 q^{82} - 39 q^{83} - 17 q^{84} - 26 q^{86} - 2 q^{87} - q^{89} + 13 q^{91} - 26 q^{92} - 3 q^{93} + 16 q^{94} + 11 q^{96} + 11 q^{97} + 12 q^{98}+O(q^{100}) $ 5 * q + q^2 + 5 * q^3 + 7 * q^4 + q^6 - 4 * q^7 + 5 * q^9 + 7 * q^12 + 4 * q^13 + 6 * q^14 + 15 * q^16 + 8 * q^17 + q^18 - 6 * q^19 - 4 * q^21 - 9 * q^23 + 13 * q^26 + 5 * q^27 - 17 * q^28 - 2 * q^29 - 3 * q^31 + 11 * q^32 + 9 * q^34 + 7 * q^36 + q^37 + 3 * q^38 + 4 * q^39 - q^41 + 6 * q^42 + 7 * q^43 + 3 * q^46 - 14 * q^47 + 15 * q^48 + 17 * q^49 + 8 * q^51 + 20 * q^52 + 3 * q^53 + q^54 + 23 * q^56 - 6 * q^57 + 27 * q^58 + 18 * q^59 + 2 * q^61 - 61 * q^62 - 4 * q^63 + 30 * q^64 + 12 * q^67 + 39 * q^68 - 9 * q^69 + 8 * q^71 + 16 * q^73 + 40 * q^74 - 61 * q^76 + 13 * q^78 + 11 * q^79 + 5 * q^81 + 20 * q^82 - 39 * q^83 - 17 * q^84 - 26 * q^86 - 2 * q^87 - q^89 + 13 * q^91 - 26 * q^92 - 3 * q^93 + 16 * q^94 + 11 * q^96 + 11 * q^97 + 12 * 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 $

$2$	0.583819	0.412822	0.206411	−	0.978465i	$-0.433822\pi$
$2$	0.583819	0.412822	0.206411	+	0.978465i	$0.433822\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.65916	−0.829578
$5$	0	0
$5$	0	0
$6$	0.583819	0.238343
$7$	4.38942	1.65904	0.829522	−	0.558474i	$-0.188613\pi$
$7$	4.38942	1.65904	0.829522	+	0.558474i	$0.188613\pi$
$8$	−2.13628	−0.755290
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.65916	−0.478957
$13$	−1.72010	−0.477071	−0.238535	−	0.971134i	$-0.576667\pi$
$13$	−1.72010	−0.477071	−0.238535	+	0.971134i	$0.576667\pi$
$14$	2.56263	0.684891
$15$	0	0
$16$	2.07111	0.517777
$17$	3.73026	0.904722	0.452361	−	0.891835i	$-0.350582\pi$
$17$	3.73026	0.904722	0.452361	+	0.891835i	$0.350582\pi$
$18$	0.583819	0.137607
$19$	−2.79121	−0.640348	−0.320174	−	0.947359i	$-0.603741\pi$
$19$	−2.79121	−0.640348	−0.320174	+	0.947359i	$0.603741\pi$
$20$	0	0
$21$	4.38942	0.957850
$22$	0	0
$23$	−6.45037	−1.34499	−0.672497	−	0.740100i	$-0.734778\pi$
$23$	−6.45037	−1.34499	−0.672497	+	0.740100i	$0.734778\pi$
$24$	−2.13628	−0.436067
$25$	0	0
$26$	−1.00423	−0.196945
$27$	1.00000	0.192450
$28$	−7.28273	−1.37631
$29$	−7.88094	−1.46345	−0.731727	−	0.681598i	$-0.761285\pi$
$29$	−7.88094	−1.46345	−0.731727	+	0.681598i	$0.761285\pi$
$30$	0	0
$31$	3.05414	0.548540	0.274270	−	0.961653i	$-0.411564\pi$
$31$	3.05414	0.548540	0.274270	+	0.961653i	$0.411564\pi$
$32$	5.48172	0.969040
$33$	0	0
$34$	2.17780	0.373489
$35$	0	0
$36$	−1.65916	−0.276526
$37$	9.21621	1.51514	0.757568	−	0.652756i	$-0.226387\pi$
$37$	9.21621	1.51514	0.757568	+	0.652756i	$0.226387\pi$
$38$	−1.62956	−0.264350
$39$	−1.72010	−0.275437
$40$	0	0
$41$	9.20605	1.43774	0.718872	−	0.695143i	$-0.244659\pi$
$41$	9.20605	1.43774	0.718872	+	0.695143i	$0.244659\pi$
$42$	2.56263	0.395422
$43$	8.88094	1.35433	0.677165	−	0.735831i	$-0.263208\pi$
$43$	8.88094	1.35433	0.677165	+	0.735831i	$0.263208\pi$
$44$	0	0
$45$	0	0
$46$	−3.76585	−0.555244
$47$	−6.29289	−0.917912	−0.458956	−	0.888459i	$-0.651776\pi$
$47$	−6.29289	−0.917912	−0.458956	+	0.888459i	$0.651776\pi$
$48$	2.07111	0.298939
$49$	12.2670	1.75243
$50$	0	0
$51$	3.73026	0.522341
$52$	2.85392	0.395767
$53$	6.32847	0.869282	0.434641	−	0.900604i	$-0.356875\pi$
$53$	6.32847	0.869282	0.434641	+	0.900604i	$0.356875\pi$
$54$	0.583819	0.0794477
$55$	0	0
$56$	−9.37705	−1.25306
$57$	−2.79121	−0.369705
$58$	−4.60104	−0.604146
$59$	−0.0655391	−0.00853246	−0.00426623	−	0.999991i	$-0.501358\pi$
$59$	−0.0655391	−0.00853246	−0.00426623	+	0.999991i	$0.501358\pi$
$60$	0	0
$61$	5.21162	0.667280	0.333640	−	0.942701i	$-0.391723\pi$
$61$	5.21162	0.667280	0.333640	+	0.942701i	$0.391723\pi$
$62$	1.78307	0.226450
$63$	4.38942	0.553015
$64$	−0.941884	−0.117736
$65$	0	0
$66$	0	0
$67$	−6.67215	−0.815133	−0.407566	−	0.913176i	$-0.633622\pi$
$67$	−6.67215	−0.815133	−0.407566	+	0.913176i	$0.633622\pi$
$68$	−6.18909	−0.750537
$69$	−6.45037	−0.776533
$70$	0	0
$71$	−15.9493	−1.89283	−0.946417	−	0.322946i	$-0.895327\pi$
$71$	−15.9493	−1.89283	−0.946417	+	0.322946i	$0.895327\pi$
$72$	−2.13628	−0.251763
$73$	3.40515	0.398542	0.199271	−	0.979944i	$-0.436143\pi$
$73$	3.40515	0.398542	0.199271	+	0.979944i	$0.436143\pi$
$74$	5.38060	0.625482
$75$	0	0
$76$	4.63105	0.531218
$77$	0	0
$78$	−1.00423	−0.113706
$79$	3.56263	0.400827	0.200413	−	0.979711i	$-0.435771\pi$
$79$	3.56263	0.400827	0.200413	+	0.979711i	$0.435771\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.37467	0.593533
$83$	7.22920	0.793508	0.396754	−	0.917925i	$-0.370137\pi$
$83$	7.22920	0.793508	0.396754	+	0.917925i	$0.370137\pi$
$84$	−7.28273	−0.794611
$85$	0	0
$86$	5.18486	0.559098
$87$	−7.88094	−0.844925
$88$	0	0
$89$	9.20605	0.975840	0.487920	−	0.872888i	$-0.337756\pi$
$89$	9.20605	0.975840	0.487920	+	0.872888i	$0.337756\pi$
$90$	0	0
$91$	−7.55025	−0.791482
$92$	10.7022	1.11578
$93$	3.05414	0.316700
$94$	−3.67391	−0.378935
$95$	0	0
$96$	5.48172	0.559476
$97$	3.33404	0.338521	0.169260	−	0.985571i	$-0.445862\pi$
$97$	3.33404	0.338521	0.169260	+	0.985571i	$0.445862\pi$
$98$	7.16171	0.723442
$99$	0	0
$100$	0	0
$101$	−9.31151	−0.926530	−0.463265	−	0.886220i	$-0.653322\pi$
$101$	−9.31151	−0.926530	−0.463265	+	0.886220i	$0.653322\pi$
$102$	2.17780	0.215634
$103$	−2.75904	−0.271856	−0.135928	−	0.990719i	$-0.543402\pi$
$103$	−2.75904	−0.271856	−0.135928	+	0.990719i	$0.543402\pi$
$104$	3.67463	0.360327
$105$	0	0
$106$	3.69468	0.358859
$107$	12.2230	1.18164	0.590822	−	0.806802i	$-0.298804\pi$
$107$	12.2230	1.18164	0.590822	+	0.806802i	$0.298804\pi$
$108$	−1.65916	−0.159652
$109$	2.21162	0.211835	0.105917	−	0.994375i	$-0.466222\pi$
$109$	2.21162	0.211835	0.105917	+	0.994375i	$0.466222\pi$
$110$	0	0
$111$	9.21621	0.874764
$112$	9.09096	0.859015
$113$	20.5543	1.93359	0.966794	−	0.255557i	$-0.0822590\pi$
$113$	20.5543	1.93359	0.966794	+	0.255557i	$0.0822590\pi$
$114$	−1.62956	−0.152622
$115$	0	0
$116$	13.0757	1.21405
$117$	−1.72010	−0.159024
$118$	−0.0382629	−0.00352239
$119$	16.3737	1.50097
$120$	0	0
$121$	0	0
$122$	3.04264	0.275468
$123$	9.20605	0.830082
$124$	−5.06730	−0.455057
$125$	0	0
$126$	2.56263	0.228297
$127$	14.0938	1.25062	0.625311	−	0.780376i	$-0.284972\pi$
$127$	14.0938	1.25062	0.625311	+	0.780376i	$0.284972\pi$
$128$	−11.5133	−1.01764
$129$	8.88094	0.781923
$130$	0	0
$131$	−14.6242	−1.27772	−0.638861	−	0.769322i	$-0.720594\pi$
$131$	−14.6242	−1.27772	−0.638861	+	0.769322i	$0.720594\pi$
$132$	0	0
$133$	−12.2518	−1.06237
$134$	−3.89533	−0.336505
$135$	0	0
$136$	−7.96890	−0.683328
$137$	11.4893	0.981598	0.490799	−	0.871273i	$-0.336705\pi$
$137$	11.4893	0.981598	0.490799	+	0.871273i	$0.336705\pi$
$138$	−3.76585	−0.320570
$139$	−14.0356	−1.19048	−0.595241	−	0.803547i	$-0.702944\pi$
$139$	−14.0356	−1.19048	−0.595241	+	0.803547i	$0.702944\pi$
$140$	0	0
$141$	−6.29289	−0.529957
$142$	−9.31151	−0.781404
$143$	0	0
$144$	2.07111	0.172592
$145$	0	0
$146$	1.98799	0.164527
$147$	12.2670	1.01177
$148$	−15.2911	−1.25692
$149$	10.4791	0.858485	0.429242	−	0.903189i	$-0.358781\pi$
$149$	10.4791	0.858485	0.429242	+	0.903189i	$0.358781\pi$
$150$	0	0
$151$	8.49611	0.691404	0.345702	−	0.938344i	$-0.387641\pi$
$151$	8.49611	0.691404	0.345702	+	0.938344i	$0.387641\pi$
$152$	5.96282	0.483649
$153$	3.73026	0.301574
$154$	0	0
$155$	0	0
$156$	2.85392	0.228496
$157$	23.3455	1.86318	0.931588	−	0.363516i	$-0.118424\pi$
$157$	23.3455	1.86318	0.931588	+	0.363516i	$0.118424\pi$
$158$	2.07993	0.165470
$159$	6.32847	0.501880
$160$	0	0
$161$	−28.3134	−2.23141
$162$	0.583819	0.0458691
$163$	−3.72469	−0.291741	−0.145870	−	0.989304i	$-0.546598\pi$
$163$	−3.72469	−0.291741	−0.145870	+	0.989304i	$0.546598\pi$
$164$	−15.2743	−1.19272
$165$	0	0
$166$	4.22055	0.327578
$167$	0.743256	0.0575149	0.0287574	−	0.999586i	$-0.490845\pi$
$167$	0.743256	0.0575149	0.0287574	+	0.999586i	$0.490845\pi$
$168$	−9.37705	−0.723455
$169$	−10.0412	−0.772404
$170$	0	0
$171$	−2.79121	−0.213449
$172$	−14.7349	−1.12352
$173$	−13.0774	−0.994253	−0.497126	−	0.867678i	$-0.665611\pi$
$173$	−13.0774	−0.994253	−0.497126	+	0.867678i	$0.665611\pi$
$174$	−4.60104	−0.348804
$175$	0	0
$176$	0	0
$177$	−0.0655391	−0.00492622
$178$	5.37467	0.402848
$179$	−1.22399	−0.0914856	−0.0457428	−	0.998953i	$-0.514565\pi$
$179$	−1.22399	−0.0914856	−0.0457428	+	0.998953i	$0.514565\pi$
$180$	0	0
$181$	24.4476	1.81718	0.908589	−	0.417691i	$-0.137161\pi$
$181$	24.4476	1.81718	0.908589	+	0.417691i	$0.137161\pi$
$182$	−4.40798	−0.326741
$183$	5.21162	0.385254
$184$	13.7798	1.01586
$185$	0	0
$186$	1.78307	0.130741
$187$	0	0
$188$	10.4409	0.761480
$189$	4.38942	0.319283
$190$	0	0
$191$	−11.6468	−0.842732	−0.421366	−	0.906891i	$-0.638449\pi$
$191$	−11.6468	−0.842732	−0.421366	+	0.906891i	$0.638449\pi$
$192$	−0.941884	−0.0679746
$193$	−20.5299	−1.47778	−0.738888	−	0.673828i	$-0.764649\pi$
$193$	−20.5299	−1.47778	−0.738888	+	0.673828i	$0.764649\pi$
$194$	1.94648	0.139749
$195$	0	0
$196$	−20.3529	−1.45378
$197$	16.5447	1.17876	0.589380	−	0.807856i	$-0.299372\pi$
$197$	16.5447	1.17876	0.589380	+	0.807856i	$0.299372\pi$
$198$	0	0
$199$	−16.2236	−1.15006	−0.575032	−	0.818131i	$-0.695010\pi$
$199$	−16.2236	−1.15006	−0.575032	+	0.818131i	$0.695010\pi$
$200$	0	0
$201$	−6.67215	−0.470617
$202$	−5.43623	−0.382492
$203$	−34.5927	−2.42793
$204$	−6.18909	−0.433323
$205$	0	0
$206$	−1.61078	−0.112228
$207$	−6.45037	−0.448331
$208$	−3.56252	−0.247016
$209$	0	0
$210$	0	0
$211$	−5.00407	−0.344494	−0.172247	−	0.985054i	$-0.555103\pi$
$211$	−5.00407	−0.344494	−0.172247	+	0.985054i	$0.555103\pi$
$212$	−10.4999	−0.721137
$213$	−15.9493	−1.09283
$214$	7.13603	0.487809
$215$	0	0
$216$	−2.13628	−0.145356
$217$	13.4059	0.910053
$218$	1.29119	0.0874501
$219$	3.40515	0.230099
$220$	0	0
$221$	−6.41644	−0.431616
$222$	5.38060	0.361122
$223$	11.0046	0.736922	0.368461	−	0.929643i	$-0.379885\pi$
$223$	11.0046	0.736922	0.368461	+	0.929643i	$0.379885\pi$
$224$	24.0616	1.60768
$225$	0	0
$226$	12.0000	0.798228
$227$	−26.8528	−1.78229	−0.891143	−	0.453723i	$-0.850095\pi$
$227$	−26.8528	−1.78229	−0.891143	+	0.453723i	$0.850095\pi$
$228$	4.63105	0.306699
$229$	23.9363	1.58176	0.790878	−	0.611974i	$-0.209624\pi$
$229$	23.9363	1.58176	0.790878	+	0.611974i	$0.209624\pi$
$230$	0	0
$231$	0	0
$232$	16.8359	1.10533
$233$	5.11173	0.334881	0.167440	−	0.985882i	$-0.446450\pi$
$233$	5.11173	0.334881	0.167440	+	0.985882i	$0.446450\pi$
$234$	−1.00423	−0.0656485
$235$	0	0
$236$	0.108739	0.00707834
$237$	3.56263	0.231417
$238$	9.55927	0.619635
$239$	−17.7423	−1.14765	−0.573826	−	0.818977i	$-0.694542\pi$
$239$	−17.7423	−1.14765	−0.573826	+	0.818977i	$0.694542\pi$
$240$	0	0
$241$	17.6965	1.13993	0.569967	−	0.821668i	$-0.306956\pi$
$241$	17.6965	1.13993	0.569967	+	0.821668i	$0.306956\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−8.64689	−0.553560
$245$	0	0
$246$	5.37467	0.342676
$247$	4.80117	0.305491
$248$	−6.52452	−0.414307
$249$	7.22920	0.458132
$250$	0	0
$251$	7.95478	0.502101	0.251051	−	0.967974i	$-0.419224\pi$
$251$	7.95478	0.502101	0.251051	+	0.967974i	$0.419224\pi$
$252$	−7.28273	−0.458769
$253$	0	0
$254$	8.22822	0.516285
$255$	0	0
$256$	−4.83793	−0.302371
$257$	−24.1670	−1.50750	−0.753749	−	0.657163i	$-0.771756\pi$
$257$	−24.1670	−1.50750	−0.753749	+	0.657163i	$0.771756\pi$
$258$	5.18486	0.322795
$259$	40.4538	2.51368
$260$	0	0
$261$	−7.88094	−0.487818
$262$	−8.53788	−0.527472
$263$	21.8894	1.34976	0.674879	−	0.737928i	$-0.264196\pi$
$263$	21.8894	1.34976	0.674879	+	0.737928i	$0.264196\pi$
$264$	0	0
$265$	0	0
$266$	−7.15283	−0.438568
$267$	9.20605	0.563401
$268$	11.0701	0.676216
$269$	21.0186	1.28153	0.640764	−	0.767738i	$-0.278618\pi$
$269$	21.0186	1.28153	0.640764	+	0.767738i	$0.278618\pi$
$270$	0	0
$271$	−19.5357	−1.18671	−0.593355	−	0.804941i	$-0.702197\pi$
$271$	−19.5357	−1.18671	−0.593355	+	0.804941i	$0.702197\pi$
$272$	7.72578	0.468444
$273$	−7.55025	−0.456962
$274$	6.70767	0.405225
$275$	0	0
$276$	10.7022	0.644194
$277$	11.8850	0.714101	0.357050	−	0.934085i	$-0.383782\pi$
$277$	11.8850	0.714101	0.357050	+	0.934085i	$0.383782\pi$
$278$	−8.19424	−0.491458
$279$	3.05414	0.182847
$280$	0	0
$281$	−11.9397	−0.712261	−0.356131	−	0.934436i	$-0.615904\pi$
$281$	−11.9397	−0.712261	−0.356131	+	0.934436i	$0.615904\pi$
$282$	−3.67391	−0.218778
$283$	−3.28113	−0.195043	−0.0975215	−	0.995233i	$-0.531091\pi$
$283$	−3.28113	−0.195043	−0.0975215	+	0.995233i	$0.531091\pi$
$284$	26.4624	1.57025
$285$	0	0
$286$	0	0
$287$	40.4092	2.38528
$288$	5.48172	0.323013
$289$	−3.08513	−0.181478
$290$	0	0
$291$	3.33404	0.195445
$292$	−5.64967	−0.330622
$293$	8.98304	0.524795	0.262397	−	0.964960i	$-0.415487\pi$
$293$	8.98304	0.524795	0.262397	+	0.964960i	$0.415487\pi$
$294$	7.16171	0.417679
$295$	0	0
$296$	−19.6884	−1.14437
$297$	0	0
$298$	6.11792	0.354402
$299$	11.0953	0.641657
$300$	0	0
$301$	38.9822	2.24689
$302$	4.96019	0.285427
$303$	−9.31151	−0.534932
$304$	−5.78090	−0.331557
$305$	0	0
$306$	2.17780	0.124496
$307$	−22.7433	−1.29803	−0.649015	−	0.760776i	$-0.724819\pi$
$307$	−22.7433	−1.29803	−0.649015	+	0.760776i	$0.724819\pi$
$308$	0	0
$309$	−2.75904	−0.156956
$310$	0	0
$311$	21.3539	1.21087	0.605435	−	0.795895i	$-0.292999\pi$
$311$	21.3539	1.21087	0.605435	+	0.795895i	$0.292999\pi$
$312$	3.67463	0.208035
$313$	22.2823	1.25947	0.629735	−	0.776810i	$-0.283163\pi$
$313$	22.2823	1.25947	0.629735	+	0.776810i	$0.283163\pi$
$314$	13.6296	0.769161
$315$	0	0
$316$	−5.91095	−0.332517
$317$	−10.8020	−0.606700	−0.303350	−	0.952879i	$-0.598105\pi$
$317$	−10.8020	−0.606700	−0.303350	+	0.952879i	$0.598105\pi$
$318$	3.69468	0.207187
$319$	0	0
$320$	0	0
$321$	12.2230	0.682222
$322$	−16.5299	−0.921174
$323$	−10.4120	−0.579337
$324$	−1.65916	−0.0921753
$325$	0	0
$326$	−2.17455	−0.120437
$327$	2.21162	0.122303
$328$	−19.6667	−1.08591
$329$	−27.6221	−1.52286
$330$	0	0
$331$	10.9753	0.603255	0.301627	−	0.953426i	$-0.402470\pi$
$331$	10.9753	0.603255	0.301627	+	0.953426i	$0.402470\pi$
$332$	−11.9944	−0.658277
$333$	9.21621	0.505045
$334$	0.433927	0.0237434
$335$	0	0
$336$	9.09096	0.495953
$337$	−14.4510	−0.787195	−0.393598	−	0.919283i	$-0.628770\pi$
$337$	−14.4510	−0.787195	−0.393598	+	0.919283i	$0.628770\pi$
$338$	−5.86227	−0.318865
$339$	20.5543	1.11636
$340$	0	0
$341$	0	0
$342$	−1.62956	−0.0881166
$343$	23.1191	1.24831
$344$	−18.9722	−1.02291
$345$	0	0
$346$	−7.63481	−0.410450
$347$	13.6236	0.731355	0.365677	−	0.930742i	$-0.380837\pi$
$347$	13.6236	0.731355	0.365677	+	0.930742i	$0.380837\pi$
$348$	13.0757	0.700931
$349$	35.3832	1.89402	0.947011	−	0.321202i	$-0.104087\pi$
$349$	35.3832	1.89402	0.947011	+	0.321202i	$0.104087\pi$
$350$	0	0
$351$	−1.72010	−0.0918123
$352$	0	0
$353$	−3.68221	−0.195984	−0.0979922	−	0.995187i	$-0.531242\pi$
$353$	−3.68221	−0.195984	−0.0979922	+	0.995187i	$0.531242\pi$
$354$	−0.0382629	−0.00203365
$355$	0	0
$356$	−15.2743	−0.809535
$357$	16.3737	0.866588
$358$	−0.714591	−0.0377673
$359$	−26.0120	−1.37286	−0.686431	−	0.727195i	$-0.740824\pi$
$359$	−26.0120	−1.37286	−0.686431	+	0.727195i	$0.740824\pi$
$360$	0	0
$361$	−11.2091	−0.589955
$362$	14.2730	0.750172
$363$	0	0
$364$	12.5270	0.656595
$365$	0	0
$366$	3.04264	0.159041
$367$	−1.09715	−0.0572707	−0.0286354	−	0.999590i	$-0.509116\pi$
$367$	−1.09715	−0.0572707	−0.0286354	+	0.999590i	$0.509116\pi$
$368$	−13.3594	−0.696407
$369$	9.20605	0.479248
$370$	0	0
$371$	27.7783	1.44218
$372$	−5.06730	−0.262727
$373$	−2.16207	−0.111948	−0.0559738	−	0.998432i	$-0.517826\pi$
$373$	−2.16207	−0.111948	−0.0559738	+	0.998432i	$0.517826\pi$
$374$	0	0
$375$	0	0
$376$	13.4434	0.693290
$377$	13.5560	0.698171
$378$	2.56263	0.131807
$379$	36.5593	1.87793	0.938963	−	0.344017i	$-0.111788\pi$
$379$	36.5593	1.87793	0.938963	+	0.344017i	$0.111788\pi$
$380$	0	0
$381$	14.0938	0.722047
$382$	−6.79961	−0.347898
$383$	10.3681	0.529783	0.264892	−	0.964278i	$-0.414664\pi$
$383$	10.3681	0.529783	0.264892	+	0.964278i	$0.414664\pi$
$384$	−11.5133	−0.587537
$385$	0	0
$386$	−11.9858	−0.610059
$387$	8.88094	0.451443
$388$	−5.53169	−0.280829
$389$	26.9092	1.36435	0.682175	−	0.731189i	$-0.261034\pi$
$389$	26.9092	1.36435	0.682175	+	0.731189i	$0.261034\pi$
$390$	0	0
$391$	−24.0616	−1.21685
$392$	−26.2058	−1.32359
$393$	−14.6242	−0.737693
$394$	9.65910	0.486618
$395$	0	0
$396$	0	0
$397$	3.29057	0.165149	0.0825744	−	0.996585i	$-0.473686\pi$
$397$	3.29057	0.165149	0.0825744	+	0.996585i	$0.473686\pi$
$398$	−9.47166	−0.474772
$399$	−12.2518	−0.613357
$400$	0	0
$401$	16.9470	0.846293	0.423146	−	0.906061i	$-0.360926\pi$
$401$	16.9470	0.846293	0.423146	+	0.906061i	$0.360926\pi$
$402$	−3.89533	−0.194281
$403$	−5.25344	−0.261693
$404$	15.4492	0.768628
$405$	0	0
$406$	−20.1959	−1.00231
$407$	0	0
$408$	−7.96890	−0.394519
$409$	−14.3894	−0.711511	−0.355755	−	0.934579i	$-0.615776\pi$
$409$	−14.3894	−0.711511	−0.355755	+	0.934579i	$0.615776\pi$
$410$	0	0
$411$	11.4893	0.566726
$412$	4.57768	0.225526
$413$	−0.287678	−0.0141557
$414$	−3.76585	−0.185081
$415$	0	0
$416$	−9.42912	−0.462301
$417$	−14.0356	−0.687325
$418$	0	0
$419$	15.6756	0.765801	0.382901	−	0.923790i	$-0.374925\pi$
$419$	15.6756	0.765801	0.382901	+	0.923790i	$0.374925\pi$
$420$	0	0
$421$	34.5379	1.68327	0.841637	−	0.540044i	$-0.181593\pi$
$421$	34.5379	1.68327	0.841637	+	0.540044i	$0.181593\pi$
$422$	−2.92147	−0.142215
$423$	−6.29289	−0.305971
$424$	−13.5194	−0.656561
$425$	0	0
$426$	−9.31151	−0.451144
$427$	22.8760	1.10705
$428$	−20.2799	−0.980265
$429$	0	0
$430$	0	0
$431$	28.1341	1.35517	0.677585	−	0.735444i	$-0.263027\pi$
$431$	28.1341	1.35517	0.677585	+	0.735444i	$0.263027\pi$
$432$	2.07111	0.0996462
$433$	−3.84659	−0.184855	−0.0924276	−	0.995719i	$-0.529463\pi$
$433$	−3.84659	−0.184855	−0.0924276	+	0.995719i	$0.529463\pi$
$434$	7.82663	0.375690
$435$	0	0
$436$	−3.66942	−0.175733
$437$	18.0043	0.861264
$438$	1.98799	0.0949898
$439$	−30.7891	−1.46948	−0.734742	−	0.678347i	$-0.762697\pi$
$439$	−30.7891	−1.46948	−0.734742	+	0.678347i	$0.762697\pi$
$440$	0	0
$441$	12.2670	0.584143
$442$	−3.74604	−0.178181
$443$	17.0346	0.809339	0.404669	−	0.914463i	$-0.367387\pi$
$443$	17.0346	0.809339	0.404669	+	0.914463i	$0.367387\pi$
$444$	−15.2911	−0.725685
$445$	0	0
$446$	6.42469	0.304218
$447$	10.4791	0.495647
$448$	−4.13432	−0.195328
$449$	−22.3641	−1.05543	−0.527713	−	0.849423i	$-0.676950\pi$
$449$	−22.3641	−1.05543	−0.527713	+	0.849423i	$0.676950\pi$
$450$	0	0
$451$	0	0
$452$	−34.1028	−1.60406
$453$	8.49611	0.399182
$454$	−15.6772	−0.735767
$455$	0	0
$456$	5.96282	0.279235
$457$	11.9860	0.560683	0.280341	−	0.959900i	$-0.409552\pi$
$457$	11.9860	0.560683	0.280341	+	0.959900i	$0.409552\pi$
$458$	13.9745	0.652984
$459$	3.73026	0.174114
$460$	0	0
$461$	15.6812	0.730345	0.365172	−	0.930940i	$-0.381010\pi$
$461$	15.6812	0.730345	0.365172	+	0.930940i	$0.381010\pi$
$462$	0	0
$463$	−2.77496	−0.128963	−0.0644816	−	0.997919i	$-0.520539\pi$
$463$	−2.77496	−0.128963	−0.0644816	+	0.997919i	$0.520539\pi$
$464$	−16.3223	−0.757742
$465$	0	0
$466$	2.98433	0.138246
$467$	−20.2229	−0.935805	−0.467903	−	0.883780i	$-0.654990\pi$
$467$	−20.2229	−0.935805	−0.467903	+	0.883780i	$0.654990\pi$
$468$	2.85392	0.131922
$469$	−29.2869	−1.35234
$470$	0	0
$471$	23.3455	1.07571
$472$	0.140010	0.00644449
$473$	0	0
$474$	2.07993	0.0955343
$475$	0	0
$476$	−27.1665	−1.24517
$477$	6.32847	0.289761
$478$	−10.3583	−0.473777
$479$	5.90017	0.269586	0.134793	−	0.990874i	$-0.456963\pi$
$479$	5.90017	0.269586	0.134793	+	0.990874i	$0.456963\pi$
$480$	0	0
$481$	−15.8528	−0.722827
$482$	10.3316	0.470590
$483$	−28.3134	−1.28830
$484$	0	0
$485$	0	0
$486$	0.583819	0.0264826
$487$	−18.5374	−0.840008	−0.420004	−	0.907522i	$-0.637971\pi$
$487$	−18.5374	−0.840008	−0.420004	+	0.907522i	$0.637971\pi$
$488$	−11.1335	−0.503990
$489$	−3.72469	−0.168437
$490$	0	0
$491$	0.752439	0.0339571	0.0169785	−	0.999856i	$-0.494595\pi$
$491$	0.752439	0.0339571	0.0169785	+	0.999856i	$0.494595\pi$
$492$	−15.2743	−0.688617
$493$	−29.3980	−1.32402
$494$	2.80301	0.126114
$495$	0	0
$496$	6.32546	0.284022
$497$	−70.0082	−3.14030
$498$	4.22055	0.189127
$499$	20.6605	0.924890	0.462445	−	0.886648i	$-0.346972\pi$
$499$	20.6605	0.924890	0.462445	+	0.886648i	$0.346972\pi$
$500$	0	0
$501$	0.743256	0.0332062
$502$	4.64415	0.207279
$503$	−22.7365	−1.01377	−0.506884	−	0.862014i	$-0.669203\pi$
$503$	−22.7365	−1.01377	−0.506884	+	0.862014i	$0.669203\pi$
$504$	−9.37705	−0.417687
$505$	0	0
$506$	0	0
$507$	−10.0412	−0.445947
$508$	−23.3838	−1.03749
$509$	37.2733	1.65211	0.826055	−	0.563590i	$-0.190580\pi$
$509$	37.2733	1.65211	0.826055	+	0.563590i	$0.190580\pi$
$510$	0	0
$511$	14.9466	0.661200
$512$	20.2022	0.892819
$513$	−2.79121	−0.123235
$514$	−14.1092	−0.622329
$515$	0	0
$516$	−14.7349	−0.648666
$517$	0	0
$518$	23.6177	1.03770
$519$	−13.0774	−0.574032
$520$	0	0
$521$	2.59424	0.113656	0.0568278	−	0.998384i	$-0.481901\pi$
$521$	2.59424	0.113656	0.0568278	+	0.998384i	$0.481901\pi$
$522$	−4.60104	−0.201382
$523$	−0.525447	−0.0229762	−0.0114881	−	0.999934i	$-0.503657\pi$
$523$	−0.525447	−0.0229762	−0.0114881	+	0.999934i	$0.503657\pi$
$524$	24.2638	1.05997
$525$	0	0
$526$	12.7794	0.557210
$527$	11.3928	0.496276
$528$	0	0
$529$	18.6072	0.809010
$530$	0	0
$531$	−0.0655391	−0.00284415
$532$	20.3276	0.881315
$533$	−15.8354	−0.685905
$534$	5.37467	0.232585
$535$	0	0
$536$	14.2536	0.615662
$537$	−1.22399	−0.0528192
$538$	12.2711	0.529043
$539$	0	0
$540$	0	0
$541$	−9.73212	−0.418416	−0.209208	−	0.977871i	$-0.567089\pi$
$541$	−9.73212	−0.418416	−0.209208	+	0.977871i	$0.567089\pi$
$542$	−11.4053	−0.489900
$543$	24.4476	1.04915
$544$	20.4483	0.876712
$545$	0	0
$546$	−4.40798	−0.188644
$547$	30.0380	1.28433	0.642166	−	0.766566i	$-0.278036\pi$
$547$	30.0380	1.28433	0.642166	+	0.766566i	$0.278036\pi$
$548$	−19.0625	−0.814312
$549$	5.21162	0.222427
$550$	0	0
$551$	21.9974	0.937119
$552$	13.7798	0.586508
$553$	15.6379	0.664989
$554$	6.93869	0.294797
$555$	0	0
$556$	23.2872	0.987598
$557$	−22.5984	−0.957523	−0.478762	−	0.877945i	$-0.658914\pi$
$557$	−22.5984	−0.957523	−0.478762	+	0.877945i	$0.658914\pi$
$558$	1.78307	0.0754832
$559$	−15.2761	−0.646111
$560$	0	0
$561$	0	0
$562$	−6.97061	−0.294037
$563$	−27.9757	−1.17904	−0.589518	−	0.807755i	$-0.700682\pi$
$563$	−27.9757	−1.17904	−0.589518	+	0.807755i	$0.700682\pi$
$564$	10.4409	0.439640
$565$	0	0
$566$	−1.91559	−0.0805181
$567$	4.38942	0.184338
$568$	34.0723	1.42964
$569$	5.57959	0.233909	0.116954	−	0.993137i	$-0.462687\pi$
$569$	5.57959	0.233909	0.116954	+	0.993137i	$0.462687\pi$
$570$	0	0
$571$	−24.5621	−1.02789	−0.513946	−	0.857823i	$-0.671817\pi$
$571$	−24.5621	−1.02789	−0.513946	+	0.857823i	$0.671817\pi$
$572$	0	0
$573$	−11.6468	−0.486551
$574$	23.5917	0.984697
$575$	0	0
$576$	−0.941884	−0.0392452
$577$	26.8363	1.11721	0.558606	−	0.829433i	$-0.311337\pi$
$577$	26.8363	1.11721	0.558606	+	0.829433i	$0.311337\pi$
$578$	−1.80116	−0.0749184
$579$	−20.5299	−0.853195
$580$	0	0
$581$	31.7320	1.31647
$582$	1.94648	0.0806840
$583$	0	0
$584$	−7.27436	−0.301015
$585$	0	0
$586$	5.24447	0.216647
$587$	−28.7941	−1.18846	−0.594230	−	0.804295i	$-0.702543\pi$
$587$	−28.7941	−1.18846	−0.594230	+	0.804295i	$0.702543\pi$
$588$	−20.3529	−0.839338
$589$	−8.52476	−0.351257
$590$	0	0
$591$	16.5447	0.680557
$592$	19.0878	0.784503
$593$	30.4424	1.25012	0.625060	−	0.780577i	$-0.285074\pi$
$593$	30.4424	1.25012	0.625060	+	0.780577i	$0.285074\pi$
$594$	0	0
$595$	0	0
$596$	−17.3865	−0.712180
$597$	−16.2236	−0.663989
$598$	6.47764	0.264891
$599$	48.3778	1.97666	0.988331	−	0.152323i	$-0.0486752\pi$
$599$	48.3778	1.97666	0.988331	+	0.152323i	$0.0486752\pi$
$600$	0	0
$601$	31.7738	1.29608	0.648041	−	0.761606i	$-0.275589\pi$
$601$	31.7738	1.29608	0.648041	+	0.761606i	$0.275589\pi$
$602$	22.7585	0.927568
$603$	−6.67215	−0.271711
$604$	−14.0964	−0.573573
$605$	0	0
$606$	−5.43623	−0.220832
$607$	15.1881	0.616466	0.308233	−	0.951311i	$-0.400262\pi$
$607$	15.1881	0.616466	0.308233	+	0.951311i	$0.400262\pi$
$608$	−15.3006	−0.620523
$609$	−34.5927	−1.40177
$610$	0	0
$611$	10.8244	0.437909
$612$	−6.18909	−0.250179
$613$	−24.7672	−1.00034	−0.500168	−	0.865928i	$-0.666729\pi$
$613$	−24.7672	−1.00034	−0.500168	+	0.865928i	$0.666729\pi$
$614$	−13.2780	−0.535855
$615$	0	0
$616$	0	0
$617$	−46.9607	−1.89057	−0.945283	−	0.326250i	$-0.894215\pi$
$617$	−46.9607	−1.89057	−0.945283	+	0.326250i	$0.894215\pi$
$618$	−1.61078	−0.0647951
$619$	−22.6308	−0.909608	−0.454804	−	0.890591i	$-0.650291\pi$
$619$	−22.6308	−0.909608	−0.454804	+	0.890591i	$0.650291\pi$
$620$	0	0
$621$	−6.45037	−0.258844
$622$	12.4668	0.499874
$623$	40.4092	1.61896
$624$	−3.56252	−0.142615
$625$	0	0
$626$	13.0088	0.519938
$627$	0	0
$628$	−38.7339	−1.54565
$629$	34.3789	1.37078
$630$	0	0
$631$	27.9087	1.11103	0.555515	−	0.831507i	$-0.312521\pi$
$631$	27.9087	1.11103	0.555515	+	0.831507i	$0.312521\pi$
$632$	−7.61078	−0.302741
$633$	−5.00407	−0.198894
$634$	−6.30641	−0.250459
$635$	0	0
$636$	−10.4999	−0.416349
$637$	−21.1005	−0.836032
$638$	0	0
$639$	−15.9493	−0.630945
$640$	0	0
$641$	−42.6307	−1.68381	−0.841906	−	0.539624i	$-0.818567\pi$
$641$	−42.6307	−1.68381	−0.841906	+	0.539624i	$0.818567\pi$
$642$	7.13603	0.281637
$643$	2.06146	0.0812959	0.0406480	−	0.999174i	$-0.487058\pi$
$643$	2.06146	0.0812959	0.0406480	+	0.999174i	$0.487058\pi$
$644$	46.9763	1.85112
$645$	0	0
$646$	−6.07869	−0.239163
$647$	33.5358	1.31843	0.659213	−	0.751956i	$-0.270889\pi$
$647$	33.5358	1.31843	0.659213	+	0.751956i	$0.270889\pi$
$648$	−2.13628	−0.0839212
$649$	0	0
$650$	0	0
$651$	13.4059	0.525419
$652$	6.17985	0.242022
$653$	8.42680	0.329766	0.164883	−	0.986313i	$-0.447275\pi$
$653$	8.42680	0.329766	0.164883	+	0.986313i	$0.447275\pi$
$654$	1.29119	0.0504894
$655$	0	0
$656$	19.0667	0.744431
$657$	3.40515	0.132847
$658$	−16.1263	−0.628669
$659$	−5.24949	−0.204491	−0.102246	−	0.994759i	$-0.532603\pi$
$659$	−5.24949	−0.204491	−0.102246	+	0.994759i	$0.532603\pi$
$660$	0	0
$661$	−41.7498	−1.62388	−0.811939	−	0.583742i	$-0.801588\pi$
$661$	−41.7498	−1.62388	−0.811939	+	0.583742i	$0.801588\pi$
$662$	6.40756	0.249037
$663$	−6.41644	−0.249194
$664$	−15.4436	−0.599329
$665$	0	0
$666$	5.38060	0.208494
$667$	50.8349	1.96834
$668$	−1.23318	−0.0477130
$669$	11.0046	0.425462
$670$	0	0
$671$	0	0
$672$	24.0616	0.928195
$673$	−28.7639	−1.10877	−0.554384	−	0.832261i	$-0.687046\pi$
$673$	−28.7639	−1.10877	−0.554384	+	0.832261i	$0.687046\pi$
$674$	−8.43676	−0.324972
$675$	0	0
$676$	16.6600	0.640769
$677$	15.6208	0.600357	0.300179	−	0.953883i	$-0.402954\pi$
$677$	15.6208	0.600357	0.300179	+	0.953883i	$0.402954\pi$
$678$	12.0000	0.460857
$679$	14.6345	0.561621
$680$	0	0
$681$	−26.8528	−1.02900
$682$	0	0
$683$	−9.20172	−0.352094	−0.176047	−	0.984382i	$-0.556331\pi$
$683$	−9.20172	−0.352094	−0.176047	+	0.984382i	$0.556331\pi$
$684$	4.63105	0.177073
$685$	0	0
$686$	13.4973	0.515331
$687$	23.9363	0.913227
$688$	18.3934	0.701241
$689$	−10.8856	−0.414709
$690$	0	0
$691$	38.7015	1.47228	0.736138	−	0.676831i	$-0.236647\pi$
$691$	38.7015	1.47228	0.736138	+	0.676831i	$0.236647\pi$
$692$	21.6974	0.824810
$693$	0	0
$694$	7.95373	0.301920
$695$	0	0
$696$	16.8359	0.638164
$697$	34.3410	1.30076
$698$	20.6574	0.781894
$699$	5.11173	0.193344
$700$	0	0
$701$	17.0570	0.644235	0.322118	−	0.946700i	$-0.395605\pi$
$701$	17.0570	0.644235	0.322118	+	0.946700i	$0.395605\pi$
$702$	−1.00423	−0.0379022
$703$	−25.7244	−0.970214
$704$	0	0
$705$	0	0
$706$	−2.14975	−0.0809068
$707$	−40.8721	−1.53715
$708$	0.108739	0.00408668
$709$	−43.3869	−1.62943	−0.814715	−	0.579861i	$-0.803107\pi$
$709$	−43.3869	−1.62943	−0.814715	+	0.579861i	$0.803107\pi$
$710$	0	0
$711$	3.56263	0.133609
$712$	−19.6667	−0.737042
$713$	−19.7003	−0.737784
$714$	9.55927	0.357747
$715$	0	0
$716$	2.03080	0.0758944
$717$	−17.7423	−0.662598
$718$	−15.1863	−0.566748
$719$	−0.440769	−0.0164379	−0.00821895	−	0.999966i	$-0.502616\pi$
$719$	−0.440769	−0.0164379	−0.00821895	+	0.999966i	$0.502616\pi$
$720$	0	0
$721$	−12.1106	−0.451022
$722$	−6.54411	−0.243546
$723$	17.6965	0.658141
$724$	−40.5624	−1.50749
$725$	0	0
$726$	0	0
$727$	48.7576	1.80832	0.904159	−	0.427196i	$-0.140498\pi$
$727$	48.7576	1.80832	0.904159	+	0.427196i	$0.140498\pi$
$728$	16.1295	0.597798
$729$	1.00000	0.0370370
$730$	0	0
$731$	33.1282	1.22529
$732$	−8.64689	−0.319598
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	−0.640536	−0.0236426
$735$	0	0
$736$	−35.3591	−1.30335
$737$	0	0
$738$	5.37467	0.197844
$739$	−48.2429	−1.77464	−0.887322	−	0.461150i	$-0.847437\pi$
$739$	−48.2429	−1.77464	−0.887322	+	0.461150i	$0.847437\pi$
$740$	0	0
$741$	4.80117	0.176375
$742$	16.2175	0.595363
$743$	−0.336778	−0.0123552	−0.00617759	−	0.999981i	$-0.501966\pi$
$743$	−0.336778	−0.0123552	−0.00617759	+	0.999981i	$0.501966\pi$
$744$	−6.52452	−0.239200
$745$	0	0
$746$	−1.26226	−0.0462145
$747$	7.22920	0.264503
$748$	0	0
$749$	53.6519	1.96040
$750$	0	0
$751$	−13.1492	−0.479820	−0.239910	−	0.970795i	$-0.577118\pi$
$751$	−13.1492	−0.479820	−0.239910	+	0.970795i	$0.577118\pi$
$752$	−13.0333	−0.475274
$753$	7.95478	0.289888
$754$	7.91426	0.288220
$755$	0	0
$756$	−7.28273	−0.264870
$757$	−36.7715	−1.33648	−0.668241	−	0.743945i	$-0.732952\pi$
$757$	−36.7715	−1.33648	−0.668241	+	0.743945i	$0.732952\pi$
$758$	21.3440	0.775250
$759$	0	0
$760$	0	0
$761$	−3.36353	−0.121928	−0.0609639	−	0.998140i	$-0.519417\pi$
$761$	−3.36353	−0.121928	−0.0609639	+	0.998140i	$0.519417\pi$
$762$	8.22822	0.298077
$763$	9.70773	0.351443
$764$	19.3238	0.699111
$765$	0	0
$766$	6.05307	0.218706
$767$	0.112734	0.00407059
$768$	−4.83793	−0.174574
$769$	1.16042	0.0418457	0.0209228	−	0.999781i	$-0.493340\pi$
$769$	1.16042	0.0418457	0.0209228	+	0.999781i	$0.493340\pi$
$770$	0	0
$771$	−24.1670	−0.870354
$772$	34.0623	1.22593
$773$	−42.6763	−1.53496	−0.767479	−	0.641074i	$-0.778489\pi$
$773$	−42.6763	−1.53496	−0.767479	+	0.641074i	$0.778489\pi$
$774$	5.18486	0.186366
$775$	0	0
$776$	−7.12246	−0.255681
$777$	40.4538	1.45127
$778$	15.7101	0.563234
$779$	−25.6960	−0.920656
$780$	0	0
$781$	0	0
$782$	−14.0476	−0.502341
$783$	−7.88094	−0.281642
$784$	25.4063	0.907367
$785$	0	0
$786$	−8.53788	−0.304536
$787$	6.26464	0.223310	0.111655	−	0.993747i	$-0.464385\pi$
$787$	6.26464	0.223310	0.111655	+	0.993747i	$0.464385\pi$
$788$	−27.4502	−0.977873
$789$	21.8894	0.779283
$790$	0	0
$791$	90.2215	3.20791
$792$	0	0
$793$	−8.96452	−0.318340
$794$	1.92109	0.0681771
$795$	0	0
$796$	26.9175	0.954067
$797$	12.0740	0.427684	0.213842	−	0.976868i	$-0.431402\pi$
$797$	12.0740	0.427684	0.213842	+	0.976868i	$0.431402\pi$
$798$	−7.15283	−0.253207
$799$	−23.4741	−0.830455
$800$	0	0
$801$	9.20605	0.325280
$802$	9.89398	0.349369
$803$	0	0
$804$	11.0701	0.390413
$805$	0	0
$806$	−3.06706	−0.108033
$807$	21.0186	0.739890
$808$	19.8920	0.699799
$809$	−26.8838	−0.945186	−0.472593	−	0.881281i	$-0.656682\pi$
$809$	−26.8838	−0.945186	−0.472593	+	0.881281i	$0.656682\pi$
$810$	0	0
$811$	−21.0792	−0.740192	−0.370096	−	0.928993i	$-0.620675\pi$
$811$	−21.0792	−0.740192	−0.370096	+	0.928993i	$0.620675\pi$
$812$	57.3947	2.01416
$813$	−19.5357	−0.685147
$814$	0	0
$815$	0	0
$816$	7.72578	0.270456
$817$	−24.7886	−0.867242
$818$	−8.40081	−0.293728
$819$	−7.55025	−0.263827
$820$	0	0
$821$	−16.6584	−0.581384	−0.290692	−	0.956817i	$-0.593885\pi$
$821$	−16.6584	−0.581384	−0.290692	+	0.956817i	$0.593885\pi$
$822$	6.70767	0.233957
$823$	54.8859	1.91320	0.956601	−	0.291402i	$-0.0941216\pi$
$823$	54.8859	1.91320	0.956601	+	0.291402i	$0.0941216\pi$
$824$	5.89410	0.205331
$825$	0	0
$826$	−0.167952	−0.00584380
$827$	−36.8708	−1.28212	−0.641061	−	0.767490i	$-0.721506\pi$
$827$	−36.8708	−1.28212	−0.641061	+	0.767490i	$0.721506\pi$
$828$	10.7022	0.371926
$829$	38.7097	1.34444	0.672221	−	0.740350i	$-0.265340\pi$
$829$	38.7097	1.34444	0.672221	+	0.740350i	$0.265340\pi$
$830$	0	0
$831$	11.8850	0.412286
$832$	1.62014	0.0561682
$833$	45.7591	1.58546
$834$	−8.19424	−0.283743
$835$	0	0
$836$	0	0
$837$	3.05414	0.105567
$838$	9.15169	0.316140
$839$	−6.01809	−0.207768	−0.103884	−	0.994589i	$-0.533127\pi$
$839$	−6.01809	−0.207768	−0.103884	+	0.994589i	$0.533127\pi$
$840$	0	0
$841$	33.1092	1.14170
$842$	20.1639	0.694893
$843$	−11.9397	−0.411224
$844$	8.30252	0.285785
$845$	0	0
$846$	−3.67391	−0.126312
$847$	0	0
$848$	13.1069	0.450094
$849$	−3.28113	−0.112608
$850$	0	0
$851$	−59.4479	−2.03785
$852$	26.4624	0.906586
$853$	−6.43119	−0.220200	−0.110100	−	0.993921i	$-0.535117\pi$
$853$	−6.43119	−0.220200	−0.110100	+	0.993921i	$0.535117\pi$
$854$	13.3554	0.457014
$855$	0	0
$856$	−26.1118	−0.892484
$857$	−20.8597	−0.712553	−0.356276	−	0.934381i	$-0.615954\pi$
$857$	−20.8597	−0.712553	−0.356276	+	0.934381i	$0.615954\pi$
$858$	0	0
$859$	−28.1757	−0.961342	−0.480671	−	0.876901i	$-0.659607\pi$
$859$	−28.1757	−0.961342	−0.480671	+	0.876901i	$0.659607\pi$
$860$	0	0
$861$	40.4092	1.37714
$862$	16.4252	0.559444
$863$	−21.8890	−0.745111	−0.372556	−	0.928010i	$-0.621518\pi$
$863$	−21.8890	−0.745111	−0.372556	+	0.928010i	$0.621518\pi$
$864$	5.48172	0.186492
$865$	0	0
$866$	−2.24571	−0.0763124
$867$	−3.08513	−0.104777
$868$	−22.2425	−0.754960
$869$	0	0
$870$	0	0
$871$	11.4768	0.388876
$872$	−4.72465	−0.159997
$873$	3.33404	0.112840
$874$	10.5113	0.355549
$875$	0	0
$876$	−5.64967	−0.190885
$877$	7.17038	0.242126	0.121063	−	0.992645i	$-0.461370\pi$
$877$	7.17038	0.242126	0.121063	+	0.992645i	$0.461370\pi$
$878$	−17.9753	−0.606635
$879$	8.98304	0.302990
$880$	0	0
$881$	37.6149	1.26728	0.633639	−	0.773629i	$-0.281560\pi$
$881$	37.6149	1.26728	0.633639	+	0.773629i	$0.281560\pi$
$882$	7.16171	0.241147
$883$	25.6366	0.862739	0.431370	−	0.902175i	$-0.358031\pi$
$883$	25.6366	0.862739	0.431370	+	0.902175i	$0.358031\pi$
$884$	10.6459	0.358059
$885$	0	0
$886$	9.94512	0.334113
$887$	−11.2399	−0.377399	−0.188699	−	0.982035i	$-0.560427\pi$
$887$	−11.2399	−0.377399	−0.188699	+	0.982035i	$0.560427\pi$
$888$	−19.6884	−0.660701
$889$	61.8636	2.07484
$890$	0	0
$891$	0	0
$892$	−18.2583	−0.611334
$893$	17.5648	0.587783
$894$	6.11792	0.204614
$895$	0	0
$896$	−50.5368	−1.68832
$897$	11.0953	0.370461
$898$	−13.0566	−0.435704
$899$	−24.0695	−0.802763
$900$	0	0
$901$	23.6069	0.786459
$902$	0	0
$903$	38.9822	1.29724
$904$	−43.9099	−1.46042
$905$	0	0
$906$	4.96019	0.164791
$907$	−24.9657	−0.828971	−0.414486	−	0.910056i	$-0.636038\pi$
$907$	−24.9657	−0.828971	−0.414486	+	0.910056i	$0.636038\pi$
$908$	44.5530	1.47854
$909$	−9.31151	−0.308843
$910$	0	0
$911$	15.8054	0.523656	0.261828	−	0.965115i	$-0.415675\pi$
$911$	15.8054	0.523656	0.261828	+	0.965115i	$0.415675\pi$
$912$	−5.78090	−0.191425
$913$	0	0
$914$	6.99767	0.231462
$915$	0	0
$916$	−39.7141	−1.31219
$917$	−64.1917	−2.11980
$918$	2.17780	0.0718781
$919$	−10.8206	−0.356937	−0.178469	−	0.983946i	$-0.557114\pi$
$919$	−10.8206	−0.356937	−0.178469	+	0.983946i	$0.557114\pi$
$920$	0	0
$921$	−22.7433	−0.749418
$922$	9.15496	0.301503
$923$	27.4345	0.903016
$924$	0	0
$925$	0	0
$926$	−1.62007	−0.0532389
$927$	−2.75904	−0.0906188
$928$	−43.2011	−1.41815
$929$	−4.68302	−0.153645	−0.0768224	−	0.997045i	$-0.524477\pi$
$929$	−4.68302	−0.153645	−0.0768224	+	0.997045i	$0.524477\pi$
$930$	0	0
$931$	−34.2398	−1.12216
$932$	−8.48116	−0.277810
$933$	21.3539	0.699096
$934$	−11.8065	−0.386321
$935$	0	0
$936$	3.67463	0.120109
$937$	−54.2836	−1.77337	−0.886684	−	0.462376i	$-0.846997\pi$
$937$	−54.2836	−1.77337	−0.886684	+	0.462376i	$0.846997\pi$
$938$	−17.0982	−0.558277
$939$	22.2823	0.727156
$940$	0	0
$941$	−32.2173	−1.05025	−0.525127	−	0.851024i	$-0.675982\pi$
$941$	−32.2173	−1.05025	−0.525127	+	0.851024i	$0.675982\pi$
$942$	13.6296	0.444075
$943$	−59.3824	−1.93376
$944$	−0.135738	−0.00441791
$945$	0	0
$946$	0	0
$947$	0.238187	0.00774002	0.00387001	−	0.999993i	$-0.498768\pi$
$947$	0.238187	0.00774002	0.00387001	+	0.999993i	$0.498768\pi$
$948$	−5.91095	−0.191979
$949$	−5.85721	−0.190133
$950$	0	0
$951$	−10.8020	−0.350279
$952$	−34.9789	−1.13367
$953$	−15.0130	−0.486319	−0.243159	−	0.969986i	$-0.578184\pi$
$953$	−15.0130	−0.486319	−0.243159	+	0.969986i	$0.578184\pi$
$954$	3.69468	0.119620
$955$	0	0
$956$	29.4372	0.952067
$957$	0	0
$958$	3.44463	0.111291
$959$	50.4314	1.62851
$960$	0	0
$961$	−21.6722	−0.699103
$962$	−9.25518	−0.298399
$963$	12.2230	0.393881
$964$	−29.3613	−0.945664
$965$	0	0
$966$	−16.5299	−0.531840
$967$	−37.6943	−1.21217	−0.606084	−	0.795401i	$-0.707260\pi$
$967$	−37.6943	−1.21217	−0.606084	+	0.795401i	$0.707260\pi$
$968$	0	0
$969$	−10.4120	−0.334480
$970$	0	0
$971$	−25.1102	−0.805826	−0.402913	−	0.915238i	$-0.632002\pi$
$971$	−25.1102	−0.805826	−0.402913	+	0.915238i	$0.632002\pi$
$972$	−1.65916	−0.0532174
$973$	−61.6081	−1.97506
$974$	−10.8225	−0.346774
$975$	0	0
$976$	10.7938	0.345502
$977$	−19.9204	−0.637311	−0.318656	−	0.947871i	$-0.603231\pi$
$977$	−19.9204	−0.637311	−0.318656	+	0.947871i	$0.603231\pi$
$978$	−2.17455	−0.0695343
$979$	0	0
$980$	0	0
$981$	2.21162	0.0706116
$982$	0.439288	0.0140182
$983$	−37.3102	−1.19001	−0.595005	−	0.803722i	$-0.702850\pi$
$983$	−37.3102	−1.19001	−0.595005	+	0.803722i	$0.702850\pi$
$984$	−19.6667	−0.626953
$985$	0	0
$986$	−17.1631	−0.546584
$987$	−27.6221	−0.879222
$988$	−7.96589	−0.253429
$989$	−57.2853	−1.82157
$990$	0	0
$991$	−2.19966	−0.0698746	−0.0349373	−	0.999390i	$-0.511123\pi$
$991$	−2.19966	−0.0698746	−0.0349373	+	0.999390i	$0.511123\pi$
$992$	16.7420	0.531558
$993$	10.9753	0.348289
$994$	−40.8721	−1.29638
$995$	0	0
$996$	−11.9944	−0.380056
$997$	3.62461	0.114792	0.0573962	−	0.998351i	$-0.481720\pi$
$997$	3.62461	0.114792	0.0573962	+	0.998351i	$0.481720\pi$
$998$	12.0620	0.381815
$999$	9.21621	0.291588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dn.1.3	yes	5
5.4	even	2		9075.2.a.dk.1.3	✓	5
11.10	odd	2		9075.2.a.dl.1.3	yes	5
55.54	odd	2		9075.2.a.dm.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.dk.1.3	✓	5	5.4	even	2
9075.2.a.dl.1.3	yes	5	11.10	odd	2
9075.2.a.dm.1.3	yes	5	55.54	odd	2
9075.2.a.dn.1.3	yes	5	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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+0.583819 q^{2} +1.00000 q^{3} -1.65916 q^{4} +0.583819 q^{6} +4.38942 q^{7} -2.13628 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.583819 q^{2} +1.00000 q^{3} -1.65916 q^{4} +0.583819 q^{6} +4.38942 q^{7} -2.13628 q^{8} +1.00000 q^{9} -1.65916 q^{12} -1.72010 q^{13} +2.56263 q^{14} +2.07111 q^{16} +3.73026 q^{17} +0.583819 q^{18} -2.79121 q^{19} +4.38942 q^{21} -6.45037 q^{23} -2.13628 q^{24} -1.00423 q^{26} +1.00000 q^{27} -7.28273 q^{28} -7.88094 q^{29} +3.05414 q^{31} +5.48172 q^{32} +2.17780 q^{34} -1.65916 q^{36} +9.21621 q^{37} -1.62956 q^{38} -1.72010 q^{39} +9.20605 q^{41} +2.56263 q^{42} +8.88094 q^{43} -3.76585 q^{46} -6.29289 q^{47} +2.07111 q^{48} +12.2670 q^{49} +3.73026 q^{51} +2.85392 q^{52} +6.32847 q^{53} +0.583819 q^{54} -9.37705 q^{56} -2.79121 q^{57} -4.60104 q^{58} -0.0655391 q^{59} +5.21162 q^{61} +1.78307 q^{62} +4.38942 q^{63} -0.941884 q^{64} -6.67215 q^{67} -6.18909 q^{68} -6.45037 q^{69} -15.9493 q^{71} -2.13628 q^{72} +3.40515 q^{73} +5.38060 q^{74} +4.63105 q^{76} -1.00423 q^{78} +3.56263 q^{79} +1.00000 q^{81} +5.37467 q^{82} +7.22920 q^{83} -7.28273 q^{84} +5.18486 q^{86} -7.88094 q^{87} +9.20605 q^{89} -7.55025 q^{91} +10.7022 q^{92} +3.05414 q^{93} -3.67391 q^{94} +5.48172 q^{96} +3.33404 q^{97} +7.16171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 5 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + q^2 + 5 * q^3 + 7 * q^4 + q^6 - 4 * q^7 + 5 * q^9	\( 5 q + q^{2} + 5 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} + 5 q^{9} + 7 q^{12} + 4 q^{13} + 6 q^{14} + 15 q^{16} + 8 q^{17} + q^{18} - 6 q^{19} - 4 q^{21} - 9 q^{23} + 13 q^{26} + 5 q^{27} - 17 q^{28} - 2 q^{29} - 3 q^{31} + 11 q^{32} + 9 q^{34} + 7 q^{36} + q^{37} + 3 q^{38} + 4 q^{39} - q^{41} + 6 q^{42} + 7 q^{43} + 3 q^{46} - 14 q^{47} + 15 q^{48} + 17 q^{49} + 8 q^{51} + 20 q^{52} + 3 q^{53} + q^{54} + 23 q^{56} - 6 q^{57} + 27 q^{58} + 18 q^{59} + 2 q^{61} - 61 q^{62} - 4 q^{63} + 30 q^{64} + 12 q^{67} + 39 q^{68} - 9 q^{69} + 8 q^{71} + 16 q^{73} + 40 q^{74} - 61 q^{76} + 13 q^{78} + 11 q^{79} + 5 q^{81} + 20 q^{82} - 39 q^{83} - 17 q^{84} - 26 q^{86} - 2 q^{87} - q^{89} + 13 q^{91} - 26 q^{92} - 3 q^{93} + 16 q^{94} + 11 q^{96} + 11 q^{97} + 12 q^{98}+O(q^{100}) \) 5 * q + q^2 + 5 * q^3 + 7 * q^4 + q^6 - 4 * q^7 + 5 * q^9 + 7 * q^12 + 4 * q^13 + 6 * q^14 + 15 * q^16 + 8 * q^17 + q^18 - 6 * q^19 - 4 * q^21 - 9 * q^23 + 13 * q^26 + 5 * q^27 - 17 * q^28 - 2 * q^29 - 3 * q^31 + 11 * q^32 + 9 * q^34 + 7 * q^36 + q^37 + 3 * q^38 + 4 * q^39 - q^41 + 6 * q^42 + 7 * q^43 + 3 * q^46 - 14 * q^47 + 15 * q^48 + 17 * q^49 + 8 * q^51 + 20 * q^52 + 3 * q^53 + q^54 + 23 * q^56 - 6 * q^57 + 27 * q^58 + 18 * q^59 + 2 * q^61 - 61 * q^62 - 4 * q^63 + 30 * q^64 + 12 * q^67 + 39 * q^68 - 9 * q^69 + 8 * q^71 + 16 * q^73 + 40 * q^74 - 61 * q^76 + 13 * q^78 + 11 * q^79 + 5 * q^81 + 20 * q^82 - 39 * q^83 - 17 * q^84 - 26 * q^86 - 2 * q^87 - q^89 + 13 * q^91 - 26 * q^92 - 3 * q^93 + 16 * q^94 + 11 * q^96 + 11 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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