LMFDB - Embedded newform 1521.2.b.k.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.k.1351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35690i q^{2} -3.55496 q^{4} -3.69202i q^{5} +0.801938i q^{7} +3.66487i q^{8} +O(q^{10})$	$q-2.35690i q^{2} -3.55496 q^{4} -3.69202i q^{5} +0.801938i q^{7} +3.66487i q^{8} -8.70171 q^{10} -2.85086i q^{11} +1.89008 q^{14} +1.52781 q^{16} +2.93900 q^{17} -2.44504i q^{19} +13.1250i q^{20} -6.71917 q^{22} -7.78986 q^{23} -8.63102 q^{25} -2.85086i q^{28} -3.85086 q^{29} +2.34481i q^{31} +3.72886i q^{32} -6.92692i q^{34} +2.96077 q^{35} -7.44504i q^{37} -5.76271 q^{38} +13.5308 q^{40} +0.850855i q^{41} +1.61596 q^{43} +10.1347i q^{44} +18.3599i q^{46} +2.44504i q^{47} +6.35690 q^{49} +20.3424i q^{50} +9.96077 q^{53} -10.5254 q^{55} -2.93900 q^{56} +9.07606i q^{58} -5.38404i q^{59} -13.2567 q^{61} +5.52648 q^{62} +11.8442 q^{64} +14.3937i q^{67} -10.4480 q^{68} -6.97823i q^{70} -8.12498i q^{71} +11.8877i q^{73} -17.5472 q^{74} +8.69202i q^{76} +2.28621 q^{77} +5.40581 q^{79} -5.64071i q^{80} +2.00538 q^{82} -7.04892i q^{83} -10.8509i q^{85} -3.80864i q^{86} +10.4480 q^{88} +1.13169i q^{89} +27.6926 q^{92} +5.76271 q^{94} -9.02715 q^{95} -5.94438i q^{97} -14.9825i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 22 q^{4}+O(q^{10}) $ 6 * q - 22 * q^4	$ 6 q - 22 q^{4} + 2 q^{10} + 10 q^{14} + 22 q^{16} - 2 q^{17} - 18 q^{22} - 22 q^{25} + 4 q^{29} - 8 q^{35} + 6 q^{40} + 30 q^{43} + 30 q^{49} + 34 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 4 q^{62} + 26 q^{68} - 30 q^{74} + 30 q^{77} + 6 q^{79} + 6 q^{82} - 26 q^{88} - 14 q^{92} - 42 q^{95}+O(q^{100}) $ 6 * q - 22 * q^4 + 2 * q^10 + 10 * q^14 + 22 * q^16 - 2 * q^17 - 18 * q^22 - 22 * q^25 + 4 * q^29 - 8 * q^35 + 6 * q^40 + 30 * q^43 + 30 * q^49 + 34 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 4 * q^62 + 26 * q^68 - 30 * q^74 + 30 * q^77 + 6 * q^79 + 6 * q^82 - 26 * q^88 - 14 * q^92 - 42 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.35690i	− 1.66658i	−0.552838	−	0.833289i	$-0.686455\pi$
$2$	− 2.35690i	− 1.66658i	0.552838	−	0.833289i	$-0.313545\pi$
$3$	0	0
$3$	0	0
$4$	−3.55496	−1.77748
$5$	− 3.69202i	− 1.65112i	−0.564313	−	0.825561i	$-0.690859\pi$
$5$	− 3.69202i	− 1.65112i	0.564313	−	0.825561i	$-0.309141\pi$
$6$	0	0
$7$	0.801938i	0.303104i	0.988449	+	0.151552i	$0.0484271\pi$
$7$	0.801938i	0.303104i	−0.988449	+	0.151552i	$0.951573\pi$
$8$	3.66487i	1.29573i
$9$	0	0
$10$	−8.70171	−2.75172
$11$	− 2.85086i	− 0.859565i	−0.902932	−	0.429783i	$-0.858590\pi$
$11$	− 2.85086i	− 0.859565i	0.902932	−	0.429783i	$-0.141410\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.89008	0.505146
$15$	0	0
$16$	1.52781	0.381953
$17$	2.93900	0.712812	0.356406	−	0.934331i	$-0.384002\pi$
$17$	2.93900	0.712812	0.356406	+	0.934331i	$0.384002\pi$
$18$	0	0
$19$	− 2.44504i	− 0.560931i	−0.959864	−	0.280466i	$-0.909511\pi$
$19$	− 2.44504i	− 0.560931i	0.959864	−	0.280466i	$-0.0904888\pi$
$20$	13.1250i	2.93484i
$21$	0	0
$22$	−6.71917	−1.43253
$23$	−7.78986	−1.62430	−0.812149	−	0.583451i	$-0.801702\pi$
$23$	−7.78986	−1.62430	−0.812149	+	0.583451i	$0.801702\pi$
$24$	0	0
$25$	−8.63102	−1.72620
$26$	0	0
$27$	0	0
$28$	− 2.85086i	− 0.538761i
$29$	−3.85086	−0.715086	−0.357543	−	0.933897i	$-0.616385\pi$
$29$	−3.85086	−0.715086	−0.357543	+	0.933897i	$0.616385\pi$
$30$	0	0
$31$	2.34481i	0.421141i	0.977579	+	0.210571i	$0.0675322\pi$
$31$	2.34481i	0.421141i	−0.977579	+	0.210571i	$0.932468\pi$
$32$	3.72886i	0.659175i
$33$	0	0
$34$	− 6.92692i	− 1.18796i
$35$	2.96077	0.500462
$36$	0	0
$37$	− 7.44504i	− 1.22396i	−0.790874	−	0.611979i	$-0.790374\pi$
$37$	− 7.44504i	− 1.22396i	0.790874	−	0.611979i	$-0.209626\pi$
$38$	−5.76271	−0.934835
$39$	0	0
$40$	13.5308	2.13941
$41$	0.850855i	0.132881i	0.997790	+	0.0664406i	$0.0211643\pi$
$41$	0.850855i	0.132881i	−0.997790	+	0.0664406i	$0.978836\pi$
$42$	0	0
$43$	1.61596	0.246431	0.123216	−	0.992380i	$-0.460679\pi$
$43$	1.61596	0.246431	0.123216	+	0.992380i	$0.460679\pi$
$44$	10.1347i	1.52786i
$45$	0	0
$46$	18.3599i	2.70702i
$47$	2.44504i	0.356646i	0.983972	+	0.178323i	$0.0570672\pi$
$47$	2.44504i	0.356646i	−0.983972	+	0.178323i	$0.942933\pi$
$48$	0	0
$49$	6.35690	0.908128
$50$	20.3424i	2.87685i
$51$	0	0
$52$	0	0
$53$	9.96077	1.36822	0.684109	−	0.729380i	$-0.260191\pi$
$53$	9.96077	1.36822	0.684109	+	0.729380i	$0.260191\pi$
$54$	0	0
$55$	−10.5254	−1.41925
$56$	−2.93900	−0.392741
$57$	0	0
$58$	9.07606i	1.19175i
$59$	− 5.38404i	− 0.700943i	−0.936573	−	0.350471i	$-0.886021\pi$
$59$	− 5.38404i	− 0.700943i	0.936573	−	0.350471i	$-0.113979\pi$
$60$	0	0
$61$	−13.2567	−1.69734	−0.848671	−	0.528921i	$-0.822597\pi$
$61$	−13.2567	−1.69734	−0.848671	+	0.528921i	$0.822597\pi$
$62$	5.52648	0.701864
$63$	0	0
$64$	11.8442	1.48052
$65$	0	0
$66$	0	0
$67$	14.3937i	1.75847i	0.476384	+	0.879237i	$0.341947\pi$
$67$	14.3937i	1.75847i	−0.476384	+	0.879237i	$0.658053\pi$
$68$	−10.4480	−1.26701
$69$	0	0
$70$	− 6.97823i	− 0.834058i
$71$	− 8.12498i	− 0.964258i	−0.876100	−	0.482129i	$-0.839864\pi$
$71$	− 8.12498i	− 0.964258i	0.876100	−	0.482129i	$-0.160136\pi$
$72$	0	0
$73$	11.8877i	1.39135i	0.718357	+	0.695674i	$0.244894\pi$
$73$	11.8877i	1.39135i	−0.718357	+	0.695674i	$0.755106\pi$
$74$	−17.5472	−2.03982
$75$	0	0
$76$	8.69202i	0.997043i
$77$	2.28621	0.260538
$78$	0	0
$79$	5.40581	0.608202	0.304101	−	0.952640i	$-0.401644\pi$
$79$	5.40581	0.608202	0.304101	+	0.952640i	$0.401644\pi$
$80$	− 5.64071i	− 0.630651i
$81$	0	0
$82$	2.00538	0.221457
$83$	− 7.04892i	− 0.773719i	−0.922139	−	0.386860i	$-0.873560\pi$
$83$	− 7.04892i	− 0.773719i	0.922139	−	0.386860i	$-0.126440\pi$
$84$	0	0
$85$	− 10.8509i	− 1.17694i
$86$	− 3.80864i	− 0.410696i
$87$	0	0
$88$	10.4480	1.11376
$89$	1.13169i	0.119959i	0.998200	+	0.0599793i	$0.0191035\pi$
$89$	1.13169i	0.119959i	−0.998200	+	0.0599793i	$0.980897\pi$
$90$	0	0
$91$	0	0
$92$	27.6926	2.88715
$93$	0	0
$94$	5.76271	0.594378
$95$	−9.02715	−0.926166
$96$	0	0
$97$	− 5.94438i	− 0.603560i	−0.953378	−	0.301780i	$-0.902419\pi$
$97$	− 5.94438i	− 0.603560i	0.953378	−	0.301780i	$-0.0975808\pi$
$98$	− 14.9825i	− 1.51347i
$99$	0	0
$100$	30.6829	3.06829
$101$	4.62565	0.460269	0.230134	−	0.973159i	$-0.426083\pi$
$101$	4.62565	0.460269	0.230134	+	0.973159i	$0.426083\pi$
$102$	0	0
$103$	−1.20775	−0.119003	−0.0595016	−	0.998228i	$-0.518951\pi$
$103$	−1.20775	−0.119003	−0.0595016	+	0.998228i	$0.518951\pi$
$104$	0	0
$105$	0	0
$106$	− 23.4765i	− 2.28024i
$107$	9.52111	0.920440	0.460220	−	0.887805i	$-0.347771\pi$
$107$	9.52111	0.920440	0.460220	+	0.887805i	$0.347771\pi$
$108$	0	0
$109$	− 1.78448i	− 0.170922i	−0.996342	−	0.0854611i	$-0.972764\pi$
$109$	− 1.78448i	− 0.170922i	0.996342	−	0.0854611i	$-0.0272363\pi$
$110$	24.8073i	2.36528i
$111$	0	0
$112$	1.22521i	0.115771i
$113$	−4.95108	−0.465759	−0.232879	−	0.972506i	$-0.574815\pi$
$113$	−4.95108	−0.465759	−0.232879	+	0.972506i	$0.574815\pi$
$114$	0	0
$115$	28.7603i	2.68191i
$116$	13.6896	1.27105
$117$	0	0
$118$	−12.6896	−1.16817
$119$	2.35690i	0.216056i
$120$	0	0
$121$	2.87263	0.261148
$122$	31.2446i	2.82875i
$123$	0	0
$124$	− 8.33572i	− 0.748569i
$125$	13.4058i	1.19905i
$126$	0	0
$127$	−5.67025	−0.503153	−0.251577	−	0.967837i	$-0.580949\pi$
$127$	−5.67025	−0.503153	−0.251577	+	0.967837i	$0.580949\pi$
$128$	− 20.4577i	− 1.80822i
$129$	0	0
$130$	0	0
$131$	−18.2228	−1.59213	−0.796067	−	0.605208i	$-0.793090\pi$
$131$	−18.2228	−1.59213	−0.796067	+	0.605208i	$0.793090\pi$
$132$	0	0
$133$	1.96077	0.170020
$134$	33.9245	2.93063
$135$	0	0
$136$	10.7711i	0.923612i
$137$	9.45042i	0.807404i	0.914891	+	0.403702i	$0.132277\pi$
$137$	9.45042i	0.807404i	−0.914891	+	0.403702i	$0.867723\pi$
$138$	0	0
$139$	4.01507	0.340553	0.170277	−	0.985396i	$-0.445534\pi$
$139$	4.01507	0.340553	0.170277	+	0.985396i	$0.445534\pi$
$140$	−10.5254	−0.889560
$141$	0	0
$142$	−19.1497	−1.60701
$143$	0	0
$144$	0	0
$145$	14.2174i	1.18069i
$146$	28.0180	2.31879
$147$	0	0
$148$	26.4668i	2.17556i
$149$	− 19.4058i	− 1.58979i	−0.606750	−	0.794893i	$-0.707527\pi$
$149$	− 19.4058i	− 1.58979i	0.606750	−	0.794893i	$-0.292473\pi$
$150$	0	0
$151$	12.3623i	1.00603i	0.864278	+	0.503014i	$0.167775\pi$
$151$	12.3623i	1.00603i	−0.864278	+	0.503014i	$0.832225\pi$
$152$	8.96077	0.726815
$153$	0	0
$154$	− 5.38835i	− 0.434206i
$155$	8.65710	0.695355
$156$	0	0
$157$	−18.6775	−1.49063	−0.745315	−	0.666712i	$-0.767701\pi$
$157$	−18.6775	−1.49063	−0.745315	+	0.666712i	$0.767701\pi$
$158$	− 12.7409i	− 1.01361i
$159$	0	0
$160$	13.7670	1.08838
$161$	− 6.24698i	− 0.492331i
$162$	0	0
$163$	− 12.3394i	− 0.966499i	−0.875483	−	0.483250i	$-0.839456\pi$
$163$	− 12.3394i	− 0.966499i	0.875483	−	0.483250i	$-0.160544\pi$
$164$	− 3.02475i	− 0.236194i
$165$	0	0
$166$	−16.6136	−1.28946
$167$	11.4940i	0.889429i	0.895672	+	0.444715i	$0.146695\pi$
$167$	11.4940i	0.889429i	−0.895672	+	0.444715i	$0.853305\pi$
$168$	0	0
$169$	0	0
$170$	−25.5743	−1.96146
$171$	0	0
$172$	−5.74466	−0.438026
$173$	12.1142	0.921028	0.460514	−	0.887653i	$-0.347665\pi$
$173$	12.1142	0.921028	0.460514	+	0.887653i	$0.347665\pi$
$174$	0	0
$175$	− 6.92154i	− 0.523219i
$176$	− 4.35557i	− 0.328313i
$177$	0	0
$178$	2.66727	0.199920
$179$	−0.538565	−0.0402542	−0.0201271	−	0.999797i	$-0.506407\pi$
$179$	−0.538565	−0.0402542	−0.0201271	+	0.999797i	$0.506407\pi$
$180$	0	0
$181$	23.2838	1.73067	0.865336	−	0.501192i	$-0.167105\pi$
$181$	23.2838	1.73067	0.865336	+	0.501192i	$0.167105\pi$
$182$	0	0
$183$	0	0
$184$	− 28.5488i	− 2.10465i
$185$	−27.4873	−2.02090
$186$	0	0
$187$	− 8.37867i	− 0.612709i
$188$	− 8.69202i	− 0.633931i
$189$	0	0
$190$	21.2760i	1.54353i
$191$	−16.7657	−1.21312	−0.606561	−	0.795037i	$-0.707452\pi$
$191$	−16.7657	−1.21312	−0.606561	+	0.795037i	$0.707452\pi$
$192$	0	0
$193$	− 25.7439i	− 1.85309i	−0.376186	−	0.926544i	$-0.622765\pi$
$193$	− 25.7439i	− 1.85309i	0.376186	−	0.926544i	$-0.377235\pi$
$194$	−14.0103	−1.00588
$195$	0	0
$196$	−22.5985	−1.61418
$197$	− 21.4209i	− 1.52617i	−0.646296	−	0.763087i	$-0.723683\pi$
$197$	− 21.4209i	− 1.52617i	0.646296	−	0.763087i	$-0.276317\pi$
$198$	0	0
$199$	−3.52781	−0.250080	−0.125040	−	0.992152i	$-0.539906\pi$
$199$	−3.52781	−0.250080	−0.125040	+	0.992152i	$0.539906\pi$
$200$	− 31.6316i	− 2.23669i
$201$	0	0
$202$	− 10.9022i	− 0.767074i
$203$	− 3.08815i	− 0.216745i
$204$	0	0
$205$	3.14138	0.219403
$206$	2.84654i	0.198328i
$207$	0	0
$208$	0	0
$209$	−6.97046	−0.482157
$210$	0	0
$211$	1.21552	0.0836799	0.0418399	−	0.999124i	$-0.486678\pi$
$211$	1.21552	0.0836799	0.0418399	+	0.999124i	$0.486678\pi$
$212$	−35.4101	−2.43198
$213$	0	0
$214$	− 22.4403i	− 1.53398i
$215$	− 5.96615i	− 0.406888i
$216$	0	0
$217$	−1.88040	−0.127650
$218$	−4.20583	−0.284855
$219$	0	0
$220$	37.4174	2.52268
$221$	0	0
$222$	0	0
$223$	− 17.3884i	− 1.16441i	−0.813042	−	0.582205i	$-0.802190\pi$
$223$	− 17.3884i	− 1.16441i	0.813042	−	0.582205i	$-0.197810\pi$
$224$	−2.99031	−0.199799
$225$	0	0
$226$	11.6692i	0.776223i
$227$	− 17.4155i	− 1.15591i	−0.816070	−	0.577954i	$-0.803851\pi$
$227$	− 17.4155i	− 1.15591i	0.816070	−	0.577954i	$-0.196149\pi$
$228$	0	0
$229$	− 18.7603i	− 1.23972i	−0.784714	−	0.619858i	$-0.787190\pi$
$229$	− 18.7603i	− 1.23972i	0.784714	−	0.619858i	$-0.212810\pi$
$230$	67.7851	4.46962
$231$	0	0
$232$	− 14.1129i	− 0.926557i
$233$	3.95108	0.258844	0.129422	−	0.991590i	$-0.458688\pi$
$233$	3.95108	0.258844	0.129422	+	0.991590i	$0.458688\pi$
$234$	0	0
$235$	9.02715	0.588866
$236$	19.1400i	1.24591i
$237$	0	0
$238$	5.55496	0.360074
$239$	− 0.818331i	− 0.0529334i	−0.999650	−	0.0264667i	$-0.991574\pi$
$239$	− 0.818331i	− 0.0529334i	0.999650	−	0.0264667i	$-0.00842560\pi$
$240$	0	0
$241$	− 6.03252i	− 0.388589i	−0.980943	−	0.194295i	$-0.937758\pi$
$241$	− 6.03252i	− 0.388589i	0.980943	−	0.194295i	$-0.0622417\pi$
$242$	− 6.77048i	− 0.435223i
$243$	0	0
$244$	47.1269	3.01699
$245$	− 23.4698i	− 1.49943i
$246$	0	0
$247$	0	0
$248$	−8.59345	−0.545685
$249$	0	0
$250$	31.5961	1.99831
$251$	−26.8799	−1.69665	−0.848323	−	0.529479i	$-0.822387\pi$
$251$	−26.8799	−1.69665	−0.848323	+	0.529479i	$0.822387\pi$
$252$	0	0
$253$	22.2078i	1.39619i
$254$	13.3642i	0.838544i
$255$	0	0
$256$	−24.5284	−1.53303
$257$	−9.05323	−0.564725	−0.282362	−	0.959308i	$-0.591118\pi$
$257$	−9.05323	−0.564725	−0.282362	+	0.959308i	$0.591118\pi$
$258$	0	0
$259$	5.97046	0.370986
$260$	0	0
$261$	0	0
$262$	42.9493i	2.65342i
$263$	−23.1511	−1.42756	−0.713778	−	0.700372i	$-0.753017\pi$
$263$	−23.1511	−1.42756	−0.713778	+	0.700372i	$0.753017\pi$
$264$	0	0
$265$	− 36.7754i	− 2.25909i
$266$	− 4.62133i	− 0.283352i
$267$	0	0
$268$	− 51.1691i	− 3.12565i
$269$	2.42088	0.147604	0.0738018	−	0.997273i	$-0.476487\pi$
$269$	2.42088	0.147604	0.0738018	+	0.997273i	$0.476487\pi$
$270$	0	0
$271$	21.4450i	1.30269i	0.758780	+	0.651347i	$0.225796\pi$
$271$	21.4450i	1.30269i	−0.758780	+	0.651347i	$0.774204\pi$
$272$	4.49024	0.272261
$273$	0	0
$274$	22.2737	1.34560
$275$	24.6058i	1.48379i
$276$	0	0
$277$	14.8073	0.889685	0.444843	−	0.895609i	$-0.353260\pi$
$277$	14.8073	0.889685	0.444843	+	0.895609i	$0.353260\pi$
$278$	− 9.46309i	− 0.567558i
$279$	0	0
$280$	10.8509i	0.648463i
$281$	14.5036i	0.865215i	0.901582	+	0.432608i	$0.142406\pi$
$281$	14.5036i	0.865215i	−0.901582	+	0.432608i	$0.857594\pi$
$282$	0	0
$283$	−25.6722	−1.52605	−0.763026	−	0.646368i	$-0.776287\pi$
$283$	−25.6722	−1.52605	−0.763026	+	0.646368i	$0.776287\pi$
$284$	28.8840i	1.71395i
$285$	0	0
$286$	0	0
$287$	−0.682333	−0.0402768
$288$	0	0
$289$	−8.36227	−0.491898
$290$	33.5090	1.96772
$291$	0	0
$292$	− 42.2602i	− 2.47309i
$293$	− 26.5230i	− 1.54949i	−0.632273	−	0.774746i	$-0.717878\pi$
$293$	− 26.5230i	− 1.54949i	0.632273	−	0.774746i	$-0.282122\pi$
$294$	0	0
$295$	−19.8780	−1.15734
$296$	27.2851	1.58592
$297$	0	0
$298$	−45.7375	−2.64950
$299$	0	0
$300$	0	0
$301$	1.29590i	0.0746943i
$302$	29.1366	1.67662
$303$	0	0
$304$	− 3.73556i	− 0.214249i
$305$	48.9439i	2.80252i
$306$	0	0
$307$	8.24698i	0.470680i	0.971913	+	0.235340i	$0.0756204\pi$
$307$	8.24698i	0.470680i	−0.971913	+	0.235340i	$0.924380\pi$
$308$	−8.12737	−0.463100
$309$	0	0
$310$	− 20.4039i	− 1.15886i
$311$	−14.4179	−0.817564	−0.408782	−	0.912632i	$-0.634046\pi$
$311$	−14.4179	−0.817564	−0.408782	+	0.912632i	$0.634046\pi$
$312$	0	0
$313$	14.2338	0.804544	0.402272	−	0.915520i	$-0.368221\pi$
$313$	14.2338	0.804544	0.402272	+	0.915520i	$0.368221\pi$
$314$	44.0210i	2.48425i
$315$	0	0
$316$	−19.2174	−1.08107
$317$	6.84415i	0.384406i	0.981355	+	0.192203i	$0.0615632\pi$
$317$	6.84415i	0.384406i	−0.981355	+	0.192203i	$0.938437\pi$
$318$	0	0
$319$	10.9782i	0.614663i
$320$	− 43.7289i	− 2.44452i
$321$	0	0
$322$	−14.7235	−0.820507
$323$	− 7.18598i	− 0.399839i
$324$	0	0
$325$	0	0
$326$	−29.0828	−1.61075
$327$	0	0
$328$	−3.11828	−0.172178
$329$	−1.96077	−0.108101
$330$	0	0
$331$	9.44265i	0.519015i	0.965741	+	0.259507i	$0.0835602\pi$
$331$	9.44265i	0.519015i	−0.965741	+	0.259507i	$0.916440\pi$
$332$	25.0586i	1.37527i
$333$	0	0
$334$	27.0901	1.48230
$335$	53.1420	2.90346
$336$	0	0
$337$	−2.64310	−0.143979	−0.0719895	−	0.997405i	$-0.522935\pi$
$337$	−2.64310	−0.143979	−0.0719895	+	0.997405i	$0.522935\pi$
$338$	0	0
$339$	0	0
$340$	38.5743i	2.09199i
$341$	6.68473	0.361998
$342$	0	0
$343$	10.7114i	0.578361i
$344$	5.92228i	0.319308i
$345$	0	0
$346$	− 28.5520i	− 1.53496i
$347$	10.1588	0.545355	0.272677	−	0.962106i	$-0.412091\pi$
$347$	10.1588	0.545355	0.272677	+	0.962106i	$0.412091\pi$
$348$	0	0
$349$	− 10.4397i	− 0.558822i	−0.960172	−	0.279411i	$-0.909861\pi$
$349$	− 10.4397i	− 0.558822i	0.960172	−	0.279411i	$-0.0901393\pi$
$350$	−16.3134	−0.871986
$351$	0	0
$352$	10.6304	0.566604
$353$	− 18.2911i	− 0.973538i	−0.873531	−	0.486769i	$-0.838175\pi$
$353$	− 18.2911i	− 0.973538i	0.873531	−	0.486769i	$-0.161825\pi$
$354$	0	0
$355$	−29.9976	−1.59211
$356$	− 4.02310i	− 0.213224i
$357$	0	0
$358$	1.26934i	0.0670867i
$359$	− 15.2731i	− 0.806081i	−0.915182	−	0.403041i	$-0.867953\pi$
$359$	− 15.2731i	− 0.806081i	0.915182	−	0.403041i	$-0.132047\pi$
$360$	0	0
$361$	13.0218	0.685356
$362$	− 54.8775i	− 2.88430i
$363$	0	0
$364$	0	0
$365$	43.8896	2.29729
$366$	0	0
$367$	−22.2717	−1.16258	−0.581288	−	0.813698i	$-0.697451\pi$
$367$	−22.2717	−1.16258	−0.581288	+	0.813698i	$0.697451\pi$
$368$	−11.9014	−0.620405
$369$	0	0
$370$	64.7846i	3.36799i
$371$	7.98792i	0.414712i
$372$	0	0
$373$	4.12631	0.213652	0.106826	−	0.994278i	$-0.465931\pi$
$373$	4.12631	0.213652	0.106826	+	0.994278i	$0.465931\pi$
$374$	−19.7476	−1.02113
$375$	0	0
$376$	−8.96077	−0.462116
$377$	0	0
$378$	0	0
$379$	− 10.7071i	− 0.549986i	−0.961446	−	0.274993i	$-0.911324\pi$
$379$	− 10.7071i	− 0.549986i	0.961446	−	0.274993i	$-0.0886756\pi$
$380$	32.0911	1.64624
$381$	0	0
$382$	39.5150i	2.02176i
$383$	− 6.52648i	− 0.333488i	−0.986000	−	0.166744i	$-0.946675\pi$
$383$	− 6.52648i	− 0.333488i	0.986000	−	0.166744i	$-0.0533253\pi$
$384$	0	0
$385$	− 8.44073i	− 0.430179i
$386$	−60.6757	−3.08831
$387$	0	0
$388$	21.1320i	1.07282i
$389$	−11.7922	−0.597891	−0.298945	−	0.954270i	$-0.596635\pi$
$389$	−11.7922	−0.597891	−0.298945	+	0.954270i	$0.596635\pi$
$390$	0	0
$391$	−22.8944	−1.15782
$392$	23.2972i	1.17669i
$393$	0	0
$394$	−50.4868	−2.54349
$395$	− 19.9584i	− 1.00422i
$396$	0	0
$397$	12.5429i	0.629509i	0.949173	+	0.314754i	$0.101922\pi$
$397$	12.5429i	0.629509i	−0.949173	+	0.314754i	$0.898078\pi$
$398$	8.31468i	0.416777i
$399$	0	0
$400$	−13.1866	−0.659329
$401$	17.8702i	0.892397i	0.894934	+	0.446198i	$0.147222\pi$
$401$	17.8702i	0.892397i	−0.894934	+	0.446198i	$0.852778\pi$
$402$	0	0
$403$	0	0
$404$	−16.4440	−0.818118
$405$	0	0
$406$	−7.27844	−0.361223
$407$	−21.2247	−1.05207
$408$	0	0
$409$	− 27.9119i	− 1.38015i	−0.723737	−	0.690076i	$-0.757577\pi$
$409$	− 27.9119i	− 1.38015i	0.723737	−	0.690076i	$-0.242423\pi$
$410$	− 7.40389i	− 0.365652i
$411$	0	0
$412$	4.29350	0.211526
$413$	4.31767	0.212459
$414$	0	0
$415$	−26.0248	−1.27750
$416$	0	0
$417$	0	0
$418$	16.4286i	0.803551i
$419$	16.4034	0.801360	0.400680	−	0.916218i	$-0.368774\pi$
$419$	16.4034	0.801360	0.400680	+	0.916218i	$0.368774\pi$
$420$	0	0
$421$	− 3.03684i	− 0.148006i	−0.997258	−	0.0740032i	$-0.976423\pi$
$421$	− 3.03684i	− 0.148006i	0.997258	−	0.0740032i	$-0.0235775\pi$
$422$	− 2.86486i	− 0.139459i
$423$	0	0
$424$	36.5050i	1.77284i
$425$	−25.3666	−1.23046
$426$	0	0
$427$	− 10.6310i	− 0.514471i
$428$	−33.8471	−1.63606
$429$	0	0
$430$	−14.0616	−0.678110
$431$	3.33811i	0.160791i	0.996763	+	0.0803955i	$0.0256183\pi$
$431$	3.33811i	0.160791i	−0.996763	+	0.0803955i	$0.974382\pi$
$432$	0	0
$433$	−11.9028	−0.572010	−0.286005	−	0.958228i	$-0.592327\pi$
$433$	−11.9028	−0.572010	−0.286005	+	0.958228i	$0.592327\pi$
$434$	4.43190i	0.212738i
$435$	0	0
$436$	6.34375i	0.303810i
$437$	19.0465i	0.911119i
$438$	0	0
$439$	−3.71810	−0.177455	−0.0887277	−	0.996056i	$-0.528280\pi$
$439$	−3.71810	−0.177455	−0.0887277	+	0.996056i	$0.528280\pi$
$440$	− 38.5743i	− 1.83896i
$441$	0	0
$442$	0	0
$443$	−1.45712	−0.0692300	−0.0346150	−	0.999401i	$-0.511021\pi$
$443$	−1.45712	−0.0692300	−0.0346150	+	0.999401i	$0.511021\pi$
$444$	0	0
$445$	4.17821	0.198066
$446$	−40.9825	−1.94058
$447$	0	0
$448$	9.49827i	0.448751i
$449$	− 12.1274i	− 0.572326i	−0.958181	−	0.286163i	$-0.907620\pi$
$449$	− 12.1274i	− 0.572326i	0.958181	−	0.286163i	$-0.0923799\pi$
$450$	0	0
$451$	2.42566	0.114220
$452$	17.6009	0.827876
$453$	0	0
$454$	−41.0465	−1.92641
$455$	0	0
$456$	0	0
$457$	− 3.44803i	− 0.161292i	−0.996743	−	0.0806459i	$-0.974302\pi$
$457$	− 3.44803i	− 0.161292i	0.996743	−	0.0806459i	$-0.0256983\pi$
$458$	−44.2161	−2.06608
$459$	0	0
$460$	− 102.242i	− 4.76704i
$461$	6.75600i	0.314658i	0.987546	+	0.157329i	$0.0502884\pi$
$461$	6.75600i	0.314658i	−0.987546	+	0.157329i	$0.949712\pi$
$462$	0	0
$463$	7.45175i	0.346312i	0.984894	+	0.173156i	$0.0553965\pi$
$463$	7.45175i	0.346312i	−0.984894	+	0.173156i	$0.944604\pi$
$464$	−5.88338	−0.273129
$465$	0	0
$466$	− 9.31229i	− 0.431384i
$467$	32.6098	1.50900	0.754502	−	0.656298i	$-0.227879\pi$
$467$	32.6098	1.50900	0.754502	+	0.656298i	$0.227879\pi$
$468$	0	0
$469$	−11.5429	−0.533001
$470$	− 21.2760i	− 0.981391i
$471$	0	0
$472$	19.7318	0.908232
$473$	− 4.60686i	− 0.211824i
$474$	0	0
$475$	21.1032i	0.968282i
$476$	− 8.37867i	− 0.384036i
$477$	0	0
$478$	−1.92872	−0.0882177
$479$	2.82908i	0.129264i	0.997909	+	0.0646321i	$0.0205874\pi$
$479$	2.82908i	0.129264i	−0.997909	+	0.0646321i	$0.979413\pi$
$480$	0	0
$481$	0	0
$482$	−14.2180	−0.647614
$483$	0	0
$484$	−10.2121	−0.464185
$485$	−21.9468	−0.996552
$486$	0	0
$487$	− 41.2935i	− 1.87119i	−0.353079	−	0.935594i	$-0.614865\pi$
$487$	− 41.2935i	− 1.87119i	0.353079	−	0.935594i	$-0.385135\pi$
$488$	− 48.5840i	− 2.19930i
$489$	0	0
$490$	−55.3159	−2.49892
$491$	−34.6698	−1.56463	−0.782313	−	0.622886i	$-0.785960\pi$
$491$	−34.6698	−1.56463	−0.782313	+	0.622886i	$0.785960\pi$
$492$	0	0
$493$	−11.3177	−0.509722
$494$	0	0
$495$	0	0
$496$	3.58243i	0.160856i
$497$	6.51573	0.292270
$498$	0	0
$499$	17.9409i	0.803146i	0.915827	+	0.401573i	$0.131536\pi$
$499$	17.9409i	0.803146i	−0.915827	+	0.401573i	$0.868464\pi$
$500$	− 47.6571i	− 2.13129i
$501$	0	0
$502$	63.3532i	2.82759i
$503$	26.1812	1.16736	0.583681	−	0.811983i	$-0.301612\pi$
$503$	26.1812	1.16736	0.583681	+	0.811983i	$0.301612\pi$
$504$	0	0
$505$	− 17.0780i	− 0.759960i
$506$	52.3414	2.32686
$507$	0	0
$508$	20.1575	0.894345
$509$	5.50604i	0.244051i	0.992527	+	0.122025i	$0.0389390\pi$
$509$	5.50604i	0.244051i	−0.992527	+	0.122025i	$0.961061\pi$
$510$	0	0
$511$	−9.53319	−0.421723
$512$	16.8955i	0.746681i
$513$	0	0
$514$	21.3375i	0.941158i
$515$	4.45904i	0.196489i
$516$	0	0
$517$	6.97046	0.306560
$518$	− 14.0718i	− 0.618277i
$519$	0	0
$520$	0	0
$521$	26.7211	1.17067	0.585336	−	0.810791i	$-0.300963\pi$
$521$	26.7211	1.17067	0.585336	+	0.810791i	$0.300963\pi$
$522$	0	0
$523$	36.5230	1.59704	0.798520	−	0.601968i	$-0.205617\pi$
$523$	36.5230	1.59704	0.798520	+	0.601968i	$0.205617\pi$
$524$	64.7813	2.82999
$525$	0	0
$526$	54.5646i	2.37913i
$527$	6.89141i	0.300195i
$528$	0	0
$529$	37.6819	1.63834
$530$	−86.6757	−3.76495
$531$	0	0
$532$	−6.97046	−0.302208
$533$	0	0
$534$	0	0
$535$	− 35.1521i	− 1.51976i
$536$	−52.7512	−2.27851
$537$	0	0
$538$	− 5.70576i	− 0.245993i
$539$	− 18.1226i	− 0.780595i
$540$	0	0
$541$	− 18.4655i	− 0.793893i	−0.917842	−	0.396947i	$-0.870070\pi$
$541$	− 18.4655i	− 0.793893i	0.917842	−	0.396947i	$-0.129930\pi$
$542$	50.5437	2.17104
$543$	0	0
$544$	10.9591i	0.469868i
$545$	−6.58834	−0.282213
$546$	0	0
$547$	39.8471	1.70374	0.851870	−	0.523753i	$-0.175468\pi$
$547$	39.8471	1.70374	0.851870	+	0.523753i	$0.175468\pi$
$548$	− 33.5958i	− 1.43514i
$549$	0	0
$550$	57.9933	2.47284
$551$	9.41550i	0.401114i
$552$	0	0
$553$	4.33513i	0.184348i
$554$	− 34.8993i	− 1.48273i
$555$	0	0
$556$	−14.2734	−0.605327
$557$	9.20477i	0.390018i	0.980801	+	0.195009i	$0.0624737\pi$
$557$	9.20477i	0.390018i	−0.980801	+	0.195009i	$0.937526\pi$
$558$	0	0
$559$	0	0
$560$	4.52350	0.191153
$561$	0	0
$562$	34.1836	1.44195
$563$	−0.975246	−0.0411017	−0.0205509	−	0.999789i	$-0.506542\pi$
$563$	−0.975246	−0.0411017	−0.0205509	+	0.999789i	$0.506542\pi$
$564$	0	0
$565$	18.2795i	0.769024i
$566$	60.5066i	2.54328i
$567$	0	0
$568$	29.7770	1.24942
$569$	−16.8944	−0.708250	−0.354125	−	0.935198i	$-0.615221\pi$
$569$	−16.8944	−0.708250	−0.354125	+	0.935198i	$0.615221\pi$
$570$	0	0
$571$	44.3226	1.85484	0.927421	−	0.374019i	$-0.122021\pi$
$571$	44.3226	1.85484	0.927421	+	0.374019i	$0.122021\pi$
$572$	0	0
$573$	0	0
$574$	1.60819i	0.0671244i
$575$	67.2344	2.80387
$576$	0	0
$577$	3.56704i	0.148498i	0.997240	+	0.0742489i	$0.0236559\pi$
$577$	3.56704i	0.148498i	−0.997240	+	0.0742489i	$0.976344\pi$
$578$	19.7090i	0.819787i
$579$	0	0
$580$	− 50.5424i	− 2.09866i
$581$	5.65279	0.234517
$582$	0	0
$583$	− 28.3967i	− 1.17607i
$584$	−43.5669	−1.80281
$585$	0	0
$586$	−62.5120	−2.58235
$587$	16.1172i	0.665229i	0.943063	+	0.332614i	$0.107931\pi$
$587$	16.1172i	0.665229i	−0.943063	+	0.332614i	$0.892069\pi$
$588$	0	0
$589$	5.73317	0.236231
$590$	46.8504i	1.92880i
$591$	0	0
$592$	− 11.3746i	− 0.467494i
$593$	− 42.8611i	− 1.76010i	−0.474885	−	0.880048i	$-0.657510\pi$
$593$	− 42.8611i	− 1.76010i	0.474885	−	0.880048i	$-0.342490\pi$
$594$	0	0
$595$	8.70171	0.356735
$596$	68.9869i	2.82581i
$597$	0	0
$598$	0	0
$599$	−40.9420	−1.67284	−0.836422	−	0.548086i	$-0.815357\pi$
$599$	−40.9420	−1.67284	−0.836422	+	0.548086i	$0.815357\pi$
$600$	0	0
$601$	1.18705	0.0484206	0.0242103	−	0.999707i	$-0.492293\pi$
$601$	1.18705	0.0484206	0.0242103	+	0.999707i	$0.492293\pi$
$602$	3.05429	0.124484
$603$	0	0
$604$	− 43.9474i	− 1.78819i
$605$	− 10.6058i	− 0.431187i
$606$	0	0
$607$	19.9922	0.811460	0.405730	−	0.913993i	$-0.367017\pi$
$607$	19.9922	0.811460	0.405730	+	0.913993i	$0.367017\pi$
$608$	9.11721	0.369752
$609$	0	0
$610$	115.356	4.67062
$611$	0	0
$612$	0	0
$613$	− 33.3618i	− 1.34747i	−0.738973	−	0.673735i	$-0.764689\pi$
$613$	− 33.3618i	− 1.34747i	0.738973	−	0.673735i	$-0.235311\pi$
$614$	19.4373	0.784424
$615$	0	0
$616$	8.37867i	0.337586i
$617$	− 11.6233i	− 0.467935i	−0.972244	−	0.233967i	$-0.924829\pi$
$617$	− 11.6233i	− 0.467935i	0.972244	−	0.233967i	$-0.0751709\pi$
$618$	0	0
$619$	16.5381i	0.664722i	0.943152	+	0.332361i	$0.107845\pi$
$619$	16.5381i	0.664722i	−0.943152	+	0.332361i	$0.892155\pi$
$620$	−30.7756	−1.23598
$621$	0	0
$622$	33.9815i	1.36253i
$623$	−0.907542	−0.0363599
$624$	0	0
$625$	6.33944	0.253577
$626$	− 33.5477i	− 1.34083i
$627$	0	0
$628$	66.3979	2.64956
$629$	− 21.8810i	− 0.872452i
$630$	0	0
$631$	36.4416i	1.45072i	0.688372	+	0.725358i	$0.258326\pi$
$631$	36.4416i	1.45072i	−0.688372	+	0.725358i	$0.741674\pi$
$632$	19.8116i	0.788064i
$633$	0	0
$634$	16.1309	0.640642
$635$	20.9347i	0.830768i
$636$	0	0
$637$	0	0
$638$	25.8745	1.02438
$639$	0	0
$640$	−75.5303	−2.98560
$641$	27.2067	1.07460	0.537300	−	0.843391i	$-0.319444\pi$
$641$	27.2067	1.07460	0.537300	+	0.843391i	$0.319444\pi$
$642$	0	0
$643$	− 5.06962i	− 0.199926i	−0.994991	−	0.0999632i	$-0.968127\pi$
$643$	− 5.06962i	− 0.199926i	0.994991	−	0.0999632i	$-0.0318725\pi$
$644$	22.2078i	0.875108i
$645$	0	0
$646$	−16.9366	−0.666362
$647$	19.3207	0.759573	0.379787	−	0.925074i	$-0.375997\pi$
$647$	19.3207	0.759573	0.379787	+	0.925074i	$0.375997\pi$
$648$	0	0
$649$	−15.3491	−0.602506
$650$	0	0
$651$	0	0
$652$	43.8662i	1.71793i
$653$	−35.2355	−1.37887	−0.689436	−	0.724347i	$-0.742141\pi$
$653$	−35.2355	−1.37887	−0.689436	+	0.724347i	$0.742141\pi$
$654$	0	0
$655$	67.2790i	2.62881i
$656$	1.29995i	0.0507544i
$657$	0	0
$658$	4.62133i	0.180158i
$659$	4.36168	0.169907	0.0849535	−	0.996385i	$-0.472926\pi$
$659$	4.36168	0.169907	0.0849535	+	0.996385i	$0.472926\pi$
$660$	0	0
$661$	15.4709i	0.601747i	0.953664	+	0.300873i	$0.0972782\pi$
$661$	15.4709i	0.601747i	−0.953664	+	0.300873i	$0.902722\pi$
$662$	22.2553	0.864978
$663$	0	0
$664$	25.8334	1.00253
$665$	− 7.23921i	− 0.280725i
$666$	0	0
$667$	29.9976	1.16151
$668$	− 40.8605i	− 1.58094i
$669$	0	0
$670$	− 125.250i	− 4.83883i
$671$	37.7928i	1.45898i
$672$	0	0
$673$	11.7409	0.452580	0.226290	−	0.974060i	$-0.427340\pi$
$673$	11.7409	0.452580	0.226290	+	0.974060i	$0.427340\pi$
$674$	6.22952i	0.239952i
$675$	0	0
$676$	0	0
$677$	−3.44504	−0.132404	−0.0662019	−	0.997806i	$-0.521088\pi$
$677$	−3.44504	−0.132404	−0.0662019	+	0.997806i	$0.521088\pi$
$678$	0	0
$679$	4.76702	0.182941
$680$	39.7670	1.52500
$681$	0	0
$682$	− 15.7552i	− 0.603298i
$683$	20.4058i	0.780807i	0.920644	+	0.390403i	$0.127664\pi$
$683$	20.4058i	0.780807i	−0.920644	+	0.390403i	$0.872336\pi$
$684$	0	0
$685$	34.8911	1.33312
$686$	25.2457	0.963883
$687$	0	0
$688$	2.46888	0.0941251
$689$	0	0
$690$	0	0
$691$	27.8039i	1.05771i	0.848713	+	0.528854i	$0.177378\pi$
$691$	27.8039i	1.05771i	−0.848713	+	0.528854i	$0.822622\pi$
$692$	−43.0656	−1.63711
$693$	0	0
$694$	− 23.9433i	− 0.908876i
$695$	− 14.8237i	− 0.562295i
$696$	0	0
$697$	2.50066i	0.0947194i
$698$	−24.6052	−0.931321
$699$	0	0
$700$	24.6058i	0.930012i
$701$	11.9715	0.452158	0.226079	−	0.974109i	$-0.427409\pi$
$701$	11.9715	0.452158	0.226079	+	0.974109i	$0.427409\pi$
$702$	0	0
$703$	−18.2034	−0.686556
$704$	− 33.7660i	− 1.27260i
$705$	0	0
$706$	−43.1102	−1.62248
$707$	3.70948i	0.139509i
$708$	0	0
$709$	− 32.2664i	− 1.21179i	−0.795545	−	0.605894i	$-0.792815\pi$
$709$	− 32.2664i	− 1.21179i	0.795545	−	0.605894i	$-0.207185\pi$
$710$	70.7012i	2.65337i
$711$	0	0
$712$	−4.14749	−0.155434
$713$	− 18.2658i	− 0.684058i
$714$	0	0
$715$	0	0
$716$	1.91457	0.0715510
$717$	0	0
$718$	−35.9970	−1.34340
$719$	12.1086	0.451574	0.225787	−	0.974177i	$-0.427505\pi$
$719$	12.1086	0.451574	0.225787	+	0.974177i	$0.427505\pi$
$720$	0	0
$721$	− 0.968541i	− 0.0360704i
$722$	− 30.6910i	− 1.14220i
$723$	0	0
$724$	−82.7730	−3.07623
$725$	33.2368	1.23438
$726$	0	0
$727$	16.6200	0.616402	0.308201	−	0.951321i	$-0.400273\pi$
$727$	16.6200	0.616402	0.308201	+	0.951321i	$0.400273\pi$
$728$	0	0
$729$	0	0
$730$	− 103.443i	− 3.82861i
$731$	4.74930	0.175659
$732$	0	0
$733$	17.7912i	0.657132i	0.944481	+	0.328566i	$0.106565\pi$
$733$	17.7912i	0.657132i	−0.944481	+	0.328566i	$0.893435\pi$
$734$	52.4922i	1.93752i
$735$	0	0
$736$	− 29.0473i	− 1.07070i
$737$	41.0344	1.51152
$738$	0	0
$739$	27.3618i	1.00652i	0.864135	+	0.503260i	$0.167866\pi$
$739$	27.3618i	1.00652i	−0.864135	+	0.503260i	$0.832134\pi$
$740$	97.7160	3.59211
$741$	0	0
$742$	18.8267	0.691150
$743$	− 8.38596i	− 0.307651i	−0.988098	−	0.153826i	$-0.950841\pi$
$743$	− 8.38596i	− 0.307651i	0.988098	−	0.153826i	$-0.0491594\pi$
$744$	0	0
$745$	−71.6467	−2.62493
$746$	− 9.72528i	− 0.356068i
$747$	0	0
$748$	29.7858i	1.08908i
$749$	7.63533i	0.278989i
$750$	0	0
$751$	38.7778	1.41502	0.707511	−	0.706703i	$-0.249818\pi$
$751$	38.7778	1.41502	0.707511	+	0.706703i	$0.249818\pi$
$752$	3.73556i	0.136222i
$753$	0	0
$754$	0	0
$755$	45.6418	1.66107
$756$	0	0
$757$	12.9729	0.471506	0.235753	−	0.971813i	$-0.424244\pi$
$757$	12.9729	0.471506	0.235753	+	0.971813i	$0.424244\pi$
$758$	−25.2355	−0.916594
$759$	0	0
$760$	− 33.0834i	− 1.20006i
$761$	− 5.15585i	− 0.186899i	−0.995624	−	0.0934497i	$-0.970211\pi$
$761$	− 5.15585i	− 0.186899i	0.995624	−	0.0934497i	$-0.0297894\pi$
$762$	0	0
$763$	1.43104	0.0518072
$764$	59.6013	2.15630
$765$	0	0
$766$	−15.3822	−0.555783
$767$	0	0
$768$	0	0
$769$	− 35.5013i	− 1.28021i	−0.768288	−	0.640104i	$-0.778891\pi$
$769$	− 35.5013i	− 1.28021i	0.768288	−	0.640104i	$-0.221109\pi$
$770$	−19.8939	−0.716927
$771$	0	0
$772$	91.5186i	3.29383i
$773$	6.15585i	0.221411i	0.993853	+	0.110705i	$0.0353110\pi$
$773$	6.15585i	0.221411i	−0.993853	+	0.110705i	$0.964689\pi$
$774$	0	0
$775$	− 20.2381i	− 0.726976i
$776$	21.7854	0.782050
$777$	0	0
$778$	27.7931i	0.996431i
$779$	2.08038	0.0745372
$780$	0	0
$781$	−23.1631	−0.828843
$782$	53.9597i	1.92960i
$783$	0	0
$784$	9.71214	0.346862
$785$	68.9579i	2.46121i
$786$	0	0
$787$	− 14.2107i	− 0.506558i	−0.967393	−	0.253279i	$-0.918491\pi$
$787$	− 14.2107i	− 0.506558i	0.967393	−	0.253279i	$-0.0815091\pi$
$788$	76.1503i	2.71274i
$789$	0	0
$790$	−47.0398	−1.67360
$791$	− 3.97046i	− 0.141173i
$792$	0	0
$793$	0	0
$794$	29.5623	1.04913
$795$	0	0
$796$	12.5412	0.444512
$797$	11.9022	0.421596	0.210798	−	0.977530i	$-0.432394\pi$
$797$	11.9022	0.421596	0.210798	+	0.977530i	$0.432394\pi$
$798$	0	0
$799$	7.18598i	0.254222i
$800$	− 32.1839i	− 1.13787i
$801$	0	0
$802$	42.1183	1.48725
$803$	33.8901	1.19596
$804$	0	0
$805$	−23.0640	−0.812899
$806$	0	0
$807$	0	0
$808$	16.9524i	0.596384i
$809$	−1.61596	−0.0568140	−0.0284070	−	0.999596i	$-0.509043\pi$
$809$	−1.61596	−0.0568140	−0.0284070	+	0.999596i	$0.509043\pi$
$810$	0	0
$811$	− 51.9657i	− 1.82476i	−0.409342	−	0.912381i	$-0.634242\pi$
$811$	− 51.9657i	− 1.82476i	0.409342	−	0.912381i	$-0.365758\pi$
$812$	10.9782i	0.385260i
$813$	0	0
$814$	50.0245i	1.75336i
$815$	−45.5575	−1.59581
$816$	0	0
$817$	− 3.95108i	− 0.138231i
$818$	−65.7853	−2.30013
$819$	0	0
$820$	−11.1675	−0.389985
$821$	− 54.8327i	− 1.91367i	−0.290627	−	0.956837i	$-0.593864\pi$
$821$	− 54.8327i	− 1.91367i	0.290627	−	0.956837i	$-0.406136\pi$
$822$	0	0
$823$	46.8514	1.63314	0.816569	−	0.577247i	$-0.195873\pi$
$823$	46.8514	1.63314	0.816569	+	0.577247i	$0.195873\pi$
$824$	− 4.42626i	− 0.154196i
$825$	0	0
$826$	− 10.1763i	− 0.354078i
$827$	21.2021i	0.737270i	0.929574	+	0.368635i	$0.120175\pi$
$827$	21.2021i	0.737270i	−0.929574	+	0.368635i	$0.879825\pi$
$828$	0	0
$829$	−18.2972	−0.635489	−0.317744	−	0.948176i	$-0.602925\pi$
$829$	−18.2972	−0.635489	−0.317744	+	0.948176i	$0.602925\pi$
$830$	61.3376i	2.12906i
$831$	0	0
$832$	0	0
$833$	18.6829	0.647325
$834$	0	0
$835$	42.4359	1.46856
$836$	24.7797	0.857024
$837$	0	0
$838$	− 38.6612i	− 1.33553i
$839$	21.1414i	0.729881i	0.931031	+	0.364941i	$0.118911\pi$
$839$	21.1414i	0.729881i	−0.931031	+	0.364941i	$0.881089\pi$
$840$	0	0
$841$	−14.1709	−0.488652
$842$	−7.15751	−0.246664
$843$	0	0
$844$	−4.32113	−0.148739
$845$	0	0
$846$	0	0
$847$	2.30367i	0.0791549i
$848$	15.2182	0.522594
$849$	0	0
$850$	59.7864i	2.05066i
$851$	57.9958i	1.98807i
$852$	0	0
$853$	7.13036i	0.244139i	0.992522	+	0.122069i	$0.0389531\pi$
$853$	7.13036i	0.244139i	−0.992522	+	0.122069i	$0.961047\pi$
$854$	−25.0562	−0.857406
$855$	0	0
$856$	34.8937i	1.19264i
$857$	−44.7741	−1.52945	−0.764726	−	0.644355i	$-0.777126\pi$
$857$	−44.7741	−1.52945	−0.764726	+	0.644355i	$0.777126\pi$
$858$	0	0
$859$	57.3782	1.95772	0.978859	−	0.204535i	$-0.0655681\pi$
$859$	57.3782	1.95772	0.978859	+	0.204535i	$0.0655681\pi$
$860$	21.2094i	0.723235i
$861$	0	0
$862$	7.86758	0.267971
$863$	6.67563i	0.227241i	0.993524	+	0.113621i	$0.0362448\pi$
$863$	6.67563i	0.227241i	−0.993524	+	0.113621i	$0.963755\pi$
$864$	0	0
$865$	− 44.7260i	− 1.52073i
$866$	28.0536i	0.953299i
$867$	0	0
$868$	6.68473	0.226894
$869$	− 15.4112i	− 0.522789i
$870$	0	0
$871$	0	0
$872$	6.53989	0.221469
$873$	0	0
$874$	44.8907	1.51845
$875$	−10.7506	−0.363438
$876$	0	0
$877$	25.2983i	0.854263i	0.904190	+	0.427131i	$0.140476\pi$
$877$	25.2983i	0.854263i	−0.904190	+	0.427131i	$0.859524\pi$
$878$	8.76318i	0.295743i
$879$	0	0
$880$	−16.0809	−0.542085
$881$	−35.3787	−1.19194	−0.595969	−	0.803008i	$-0.703232\pi$
$881$	−35.3787	−1.19194	−0.595969	+	0.803008i	$0.703232\pi$
$882$	0	0
$883$	11.5851	0.389869	0.194935	−	0.980816i	$-0.437551\pi$
$883$	11.5851	0.389869	0.194935	+	0.980816i	$0.437551\pi$
$884$	0	0
$885$	0	0
$886$	3.43429i	0.115377i
$887$	27.2892	0.916281	0.458141	−	0.888880i	$-0.348516\pi$
$887$	27.2892	0.916281	0.458141	+	0.888880i	$0.348516\pi$
$888$	0	0
$889$	− 4.54719i	− 0.152508i
$890$	− 9.84761i	− 0.330093i
$891$	0	0
$892$	61.8149i	2.06972i
$893$	5.97823	0.200054
$894$	0	0
$895$	1.98839i	0.0664646i
$896$	16.4058	0.548080
$897$	0	0
$898$	−28.5830	−0.953826
$899$	− 9.02954i	− 0.301152i
$900$	0	0
$901$	29.2747	0.975282
$902$	− 5.71704i	− 0.190357i
$903$	0	0
$904$	− 18.1451i	− 0.603497i
$905$	− 85.9643i	− 2.85755i
$906$	0	0
$907$	30.4219	1.01014	0.505072	−	0.863077i	$-0.331466\pi$
$907$	30.4219	1.01014	0.505072	+	0.863077i	$0.331466\pi$
$908$	61.9114i	2.05460i
$909$	0	0
$910$	0	0
$911$	−53.5719	−1.77492	−0.887459	−	0.460887i	$-0.847531\pi$
$911$	−53.5719	−1.77492	−0.887459	+	0.460887i	$0.847531\pi$
$912$	0	0
$913$	−20.0954	−0.665062
$914$	−8.12664	−0.268805
$915$	0	0
$916$	66.6921i	2.20357i
$917$	− 14.6136i	− 0.482582i
$918$	0	0
$919$	−36.1672	−1.19305	−0.596523	−	0.802596i	$-0.703451\pi$
$919$	−36.1672	−1.19305	−0.596523	+	0.802596i	$0.703451\pi$
$920$	−105.403	−3.47503
$921$	0	0
$922$	15.9232	0.524403
$923$	0	0
$924$	0	0
$925$	64.2583i	2.11280i
$926$	17.5630	0.577156
$927$	0	0
$928$	− 14.3593i	− 0.471367i
$929$	13.3478i	0.437927i	0.975733	+	0.218964i	$0.0702676\pi$
$929$	13.3478i	0.437927i	−0.975733	+	0.218964i	$0.929732\pi$
$930$	0	0
$931$	− 15.5429i	− 0.509397i
$932$	−14.0459	−0.460090
$933$	0	0
$934$	− 76.8580i	− 2.51487i
$935$	−30.9342	−1.01166
$936$	0	0
$937$	38.6872	1.26386	0.631928	−	0.775027i	$-0.282264\pi$
$937$	38.6872	1.26386	0.631928	+	0.775027i	$0.282264\pi$
$938$	27.2054i	0.888286i
$939$	0	0
$940$	−32.0911	−1.04670
$941$	− 35.7275i	− 1.16468i	−0.812944	−	0.582342i	$-0.802136\pi$
$941$	− 35.7275i	− 1.16468i	0.812944	−	0.582342i	$-0.197864\pi$
$942$	0	0
$943$	− 6.62804i	− 0.215839i
$944$	− 8.22580i	− 0.267727i
$945$	0	0
$946$	−10.8579	−0.353020
$947$	− 17.8436i	− 0.579838i	−0.957051	−	0.289919i	$-0.906372\pi$
$947$	− 17.8436i	− 0.579838i	0.957051	−	0.289919i	$-0.0936283\pi$
$948$	0	0
$949$	0	0
$950$	49.7381	1.61372
$951$	0	0
$952$	−8.63773	−0.279950
$953$	20.1691	0.653342	0.326671	−	0.945138i	$-0.394073\pi$
$953$	20.1691	0.653342	0.326671	+	0.945138i	$0.394073\pi$
$954$	0	0
$955$	61.8993i	2.00301i
$956$	2.90913i	0.0940881i
$957$	0	0
$958$	6.66786	0.215429
$959$	−7.57865	−0.244727
$960$	0	0
$961$	25.5018	0.822640
$962$	0	0
$963$	0	0
$964$	21.4454i	0.690709i
$965$	−95.0471	−3.05967
$966$	0	0
$967$	− 32.7894i	− 1.05444i	−0.849730	−	0.527218i	$-0.823235\pi$
$967$	− 32.7894i	− 1.05444i	0.849730	−	0.527218i	$-0.176765\pi$
$968$	10.5278i	0.338377i
$969$	0	0
$970$	51.7263i	1.66083i
$971$	26.9845	0.865973	0.432986	−	0.901401i	$-0.357460\pi$
$971$	26.9845	0.865973	0.432986	+	0.901401i	$0.357460\pi$
$972$	0	0
$973$	3.21983i	0.103223i
$974$	−97.3245	−3.11848
$975$	0	0
$976$	−20.2537	−0.648305
$977$	16.9571i	0.542504i	0.962508	+	0.271252i	$0.0874377\pi$
$977$	16.9571i	0.542504i	−0.962508	+	0.271252i	$0.912562\pi$
$978$	0	0
$979$	3.22627	0.103112
$980$	83.4341i	2.66521i
$981$	0	0
$982$	81.7131i	2.60757i
$983$	− 32.6631i	− 1.04179i	−0.853621	−	0.520895i	$-0.825598\pi$
$983$	− 32.6631i	− 1.04179i	0.853621	−	0.520895i	$-0.174402\pi$
$984$	0	0
$985$	−79.0863	−2.51990
$986$	26.6746i	0.849491i
$987$	0	0
$988$	0	0
$989$	−12.5881	−0.400277
$990$	0	0
$991$	7.64310	0.242791	0.121396	−	0.992604i	$-0.461263\pi$
$991$	7.64310	0.242791	0.121396	+	0.992604i	$0.461263\pi$
$992$	−8.74348	−0.277606
$993$	0	0
$994$	− 15.3569i	− 0.487091i
$995$	13.0248i	0.412912i
$996$	0	0
$997$	36.1256	1.14411	0.572054	−	0.820216i	$-0.306147\pi$
$997$	36.1256	1.14411	0.572054	+	0.820216i	$0.306147\pi$
$998$	42.2849	1.33850
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.k.1351.2		6
3.2	odd	2		507.2.b.f.337.5		6
13.5	odd	4		1521.2.a.n.1.2		3
13.8	odd	4		1521.2.a.s.1.2		3
13.12	even	2	inner	1521.2.b.k.1351.5		6
39.2	even	12		507.2.e.i.22.2		6
39.5	even	4		507.2.a.l.1.2	yes	3
39.8	even	4		507.2.a.i.1.2	✓	3
39.11	even	12		507.2.e.l.22.2		6
39.17	odd	6		507.2.j.i.361.2		12
39.20	even	12		507.2.e.l.484.2		6
39.23	odd	6		507.2.j.i.316.5		12
39.29	odd	6		507.2.j.i.316.2		12
39.32	even	12		507.2.e.i.484.2		6
39.35	odd	6		507.2.j.i.361.5		12
39.38	odd	2		507.2.b.f.337.2		6
156.47	odd	4		8112.2.a.cg.1.1		3
156.83	odd	4		8112.2.a.cp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.2	✓	3	39.8	even	4
507.2.a.l.1.2	yes	3	39.5	even	4
507.2.b.f.337.2		6	39.38	odd	2
507.2.b.f.337.5		6	3.2	odd	2
507.2.e.i.22.2		6	39.2	even	12
507.2.e.i.484.2		6	39.32	even	12
507.2.e.l.22.2		6	39.11	even	12
507.2.e.l.484.2		6	39.20	even	12
507.2.j.i.316.2		12	39.29	odd	6
507.2.j.i.316.5		12	39.23	odd	6
507.2.j.i.361.2		12	39.17	odd	6
507.2.j.i.361.5		12	39.35	odd	6
1521.2.a.n.1.2		3	13.5	odd	4
1521.2.a.s.1.2		3	13.8	odd	4
1521.2.b.k.1351.2		6	1.1	even	1	trivial
1521.2.b.k.1351.5		6	13.12	even	2	inner
8112.2.a.cg.1.1		3	156.47	odd	4
8112.2.a.cp.1.3		3	156.83	odd	4

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.35690i q^{2} -3.55496 q^{4} -3.69202i q^{5} +0.801938i q^{7} +3.66487i q^{8} +O(q^{10})\)	\(q-2.35690i q^{2} -3.55496 q^{4} -3.69202i q^{5} +0.801938i q^{7} +3.66487i q^{8} -8.70171 q^{10} -2.85086i q^{11} +1.89008 q^{14} +1.52781 q^{16} +2.93900 q^{17} -2.44504i q^{19} +13.1250i q^{20} -6.71917 q^{22} -7.78986 q^{23} -8.63102 q^{25} -2.85086i q^{28} -3.85086 q^{29} +2.34481i q^{31} +3.72886i q^{32} -6.92692i q^{34} +2.96077 q^{35} -7.44504i q^{37} -5.76271 q^{38} +13.5308 q^{40} +0.850855i q^{41} +1.61596 q^{43} +10.1347i q^{44} +18.3599i q^{46} +2.44504i q^{47} +6.35690 q^{49} +20.3424i q^{50} +9.96077 q^{53} -10.5254 q^{55} -2.93900 q^{56} +9.07606i q^{58} -5.38404i q^{59} -13.2567 q^{61} +5.52648 q^{62} +11.8442 q^{64} +14.3937i q^{67} -10.4480 q^{68} -6.97823i q^{70} -8.12498i q^{71} +11.8877i q^{73} -17.5472 q^{74} +8.69202i q^{76} +2.28621 q^{77} +5.40581 q^{79} -5.64071i q^{80} +2.00538 q^{82} -7.04892i q^{83} -10.8509i q^{85} -3.80864i q^{86} +10.4480 q^{88} +1.13169i q^{89} +27.6926 q^{92} +5.76271 q^{94} -9.02715 q^{95} -5.94438i q^{97} -14.9825i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 22 q^{4}+O(q^{10}) \) 6 * q - 22 * q^4	\( 6 q - 22 q^{4} + 2 q^{10} + 10 q^{14} + 22 q^{16} - 2 q^{17} - 18 q^{22} - 22 q^{25} + 4 q^{29} - 8 q^{35} + 6 q^{40} + 30 q^{43} + 30 q^{49} + 34 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 4 q^{62} + 26 q^{68} - 30 q^{74} + 30 q^{77} + 6 q^{79} + 6 q^{82} - 26 q^{88} - 14 q^{92} - 42 q^{95}+O(q^{100}) \) 6 * q - 22 * q^4 + 2 * q^10 + 10 * q^14 + 22 * q^16 - 2 * q^17 - 18 * q^22 - 22 * q^25 + 4 * q^29 - 8 * q^35 + 6 * q^40 + 30 * q^43 + 30 * q^49 + 34 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 4 * q^62 + 26 * q^68 - 30 * q^74 + 30 * q^77 + 6 * q^79 + 6 * q^82 - 26 * q^88 - 14 * q^92 - 42 * q^95

Introduction

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