LMFDB - Embedded newform 1521.2.b.j.1351.1

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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.j.1351.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{2} -3.82843 q^{4} +2.82843i q^{5} -2.82843i q^{7} +4.41421i q^{8} +O(q^{10})$	$q-2.41421i q^{2} -3.82843 q^{4} +2.82843i q^{5} -2.82843i q^{7} +4.41421i q^{8} +6.82843 q^{10} +2.00000i q^{11} -6.82843 q^{14} +3.00000 q^{16} -3.65685 q^{17} -2.82843i q^{19} -10.8284i q^{20} +4.82843 q^{22} -4.00000 q^{23} -3.00000 q^{25} +10.8284i q^{28} -2.00000 q^{29} +6.82843i q^{31} +1.58579i q^{32} +8.82843i q^{34} +8.00000 q^{35} +3.65685i q^{37} -6.82843 q^{38} -12.4853 q^{40} +10.8284i q^{41} -9.65685 q^{43} -7.65685i q^{44} +9.65685i q^{46} +0.343146i q^{47} -1.00000 q^{49} +7.24264i q^{50} +2.00000 q^{53} -5.65685 q^{55} +12.4853 q^{56} +4.82843i q^{58} +3.65685i q^{59} -9.31371 q^{61} +16.4853 q^{62} +9.82843 q^{64} -1.17157i q^{67} +14.0000 q^{68} -19.3137i q^{70} +2.00000i q^{71} +11.6569i q^{73} +8.82843 q^{74} +10.8284i q^{76} +5.65685 q^{77} +11.3137 q^{79} +8.48528i q^{80} +26.1421 q^{82} -7.65685i q^{83} -10.3431i q^{85} +23.3137i q^{86} -8.82843 q^{88} -9.17157i q^{89} +15.3137 q^{92} +0.828427 q^{94} +8.00000 q^{95} +7.65685i q^{97} +2.41421i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 16 q^{10} - 16 q^{14} + 12 q^{16} + 8 q^{17} + 8 q^{22} - 16 q^{23} - 12 q^{25} - 8 q^{29} + 32 q^{35} - 16 q^{38} - 16 q^{40} - 16 q^{43} - 4 q^{49} + 8 q^{53} + 16 q^{56} + 8 q^{61} + 32 q^{62} + 28 q^{64} + 56 q^{68} + 24 q^{74} + 48 q^{82} - 24 q^{88} + 16 q^{92} - 8 q^{94} + 32 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 16 * q^10 - 16 * q^14 + 12 * q^16 + 8 * q^17 + 8 * q^22 - 16 * q^23 - 12 * q^25 - 8 * q^29 + 32 * q^35 - 16 * q^38 - 16 * q^40 - 16 * q^43 - 4 * q^49 + 8 * q^53 + 16 * q^56 + 8 * q^61 + 32 * q^62 + 28 * q^64 + 56 * q^68 + 24 * q^74 + 48 * q^82 - 24 * q^88 + 16 * q^92 - 8 * q^94 + 32 * 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 $

$2$	− 2.41421i	− 1.70711i	−0.521005	−	0.853553i	$-0.674443\pi$
$2$	− 2.41421i	− 1.70711i	0.521005	−	0.853553i	$-0.325557\pi$
$3$	0	0
$3$	0	0
$4$	−3.82843	−1.91421
$5$	2.82843i	1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$	2.82843i	1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	4.41421i	1.56066i
$9$	0	0
$10$	6.82843	2.15934
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−6.82843	−1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	− 2.82843i	− 0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$	− 2.82843i	− 0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$	− 10.8284i	− 2.42131i
$21$	0	0
$22$	4.82843	1.02942
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	10.8284i	2.04638i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.82843i	1.22642i	0.789919	+	0.613211i	$0.210122\pi$
$31$	6.82843i	1.22642i	−0.789919	+	0.613211i	$0.789878\pi$
$32$	1.58579i	0.280330i
$33$	0	0
$34$	8.82843i	1.51406i
$35$	8.00000	1.35225
$36$	0	0
$37$	3.65685i	0.601183i	0.953753	+	0.300592i	$0.0971841\pi$
$37$	3.65685i	0.601183i	−0.953753	+	0.300592i	$0.902816\pi$
$38$	−6.82843	−1.10772
$39$	0	0
$40$	−12.4853	−1.97410
$41$	10.8284i	1.69112i	0.533883	+	0.845558i	$0.320732\pi$
$41$	10.8284i	1.69112i	−0.533883	+	0.845558i	$0.679268\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	− 7.65685i	− 1.15431i
$45$	0	0
$46$	9.65685i	1.42383i
$47$	0.343146i	0.0500530i	0.999687	+	0.0250265i	$0.00796701\pi$
$47$	0.343146i	0.0500530i	−0.999687	+	0.0250265i	$0.992033\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	7.24264i	1.02426i
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	12.4853	1.66842
$57$	0	0
$58$	4.82843i	0.634004i
$59$	3.65685i	0.476082i	0.971255	+	0.238041i	$0.0765052\pi$
$59$	3.65685i	0.476082i	−0.971255	+	0.238041i	$0.923495\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	16.4853	2.09363
$63$	0	0
$64$	9.82843	1.22855
$65$	0	0
$66$	0	0
$67$	− 1.17157i	− 0.143130i	−0.997436	−	0.0715652i	$-0.977201\pi$
$67$	− 1.17157i	− 0.143130i	0.997436	−	0.0715652i	$-0.0227994\pi$
$68$	14.0000	1.69775
$69$	0	0
$70$	− 19.3137i	− 2.30843i
$71$	2.00000i	0.237356i	0.992933	+	0.118678i	$0.0378657\pi$
$71$	2.00000i	0.237356i	−0.992933	+	0.118678i	$0.962134\pi$
$72$	0	0
$73$	11.6569i	1.36433i	0.731198	+	0.682166i	$0.238962\pi$
$73$	11.6569i	1.36433i	−0.731198	+	0.682166i	$0.761038\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	10.8284i	1.24211i
$77$	5.65685	0.644658
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	8.48528i	0.948683i
$81$	0	0
$82$	26.1421	2.88692
$83$	− 7.65685i	− 0.840449i	−0.907420	−	0.420224i	$-0.861951\pi$
$83$	− 7.65685i	− 0.840449i	0.907420	−	0.420224i	$-0.138049\pi$
$84$	0	0
$85$	− 10.3431i	− 1.12187i
$86$	23.3137i	2.51398i
$87$	0	0
$88$	−8.82843	−0.941113
$89$	− 9.17157i	− 0.972185i	−0.873907	−	0.486092i	$-0.838422\pi$
$89$	− 9.17157i	− 0.972185i	0.873907	−	0.486092i	$-0.161578\pi$
$90$	0	0
$91$	0	0
$92$	15.3137	1.59656
$93$	0	0
$94$	0.828427	0.0854457
$95$	8.00000	0.820783
$96$	0	0
$97$	7.65685i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$97$	7.65685i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$98$	2.41421i	0.243872i
$99$	0	0
$100$	11.4853	1.14853
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	0	0
$106$	− 4.82843i	− 0.468978i
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	0	0
$109$	17.3137i	1.65835i	0.558987	+	0.829176i	$0.311190\pi$
$109$	17.3137i	1.65835i	−0.558987	+	0.829176i	$0.688810\pi$
$110$	13.6569i	1.30213i
$111$	0	0
$112$	− 8.48528i	− 0.801784i
$113$	−17.3137	−1.62874	−0.814368	−	0.580348i	$-0.802916\pi$
$113$	−17.3137	−1.62874	−0.814368	+	0.580348i	$0.802916\pi$
$114$	0	0
$115$	− 11.3137i	− 1.05501i
$116$	7.65685	0.710921
$117$	0	0
$118$	8.82843	0.812723
$119$	10.3431i	0.948155i
$120$	0	0
$121$	7.00000	0.636364
$122$	22.4853i	2.03572i
$123$	0	0
$124$	− 26.1421i	− 2.34763i
$125$	5.65685i	0.505964i
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	− 20.5563i	− 1.81694i
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−2.82843	−0.244339
$135$	0	0
$136$	− 16.1421i	− 1.38418i
$137$	5.17157i	0.441837i	0.975292	+	0.220919i	$0.0709055\pi$
$137$	5.17157i	0.441837i	−0.975292	+	0.220919i	$0.929094\pi$
$138$	0	0
$139$	15.3137	1.29889	0.649446	−	0.760408i	$-0.275001\pi$
$139$	15.3137	1.29889	0.649446	+	0.760408i	$0.275001\pi$
$140$	−30.6274	−2.58849
$141$	0	0
$142$	4.82843	0.405193
$143$	0	0
$144$	0	0
$145$	− 5.65685i	− 0.469776i
$146$	28.1421	2.32906
$147$	0	0
$148$	− 14.0000i	− 1.15079i
$149$	− 14.8284i	− 1.21479i	−0.794399	−	0.607396i	$-0.792214\pi$
$149$	− 14.8284i	− 1.21479i	0.794399	−	0.607396i	$-0.207786\pi$
$150$	0	0
$151$	− 20.4853i	− 1.66707i	−0.552468	−	0.833534i	$-0.686314\pi$
$151$	− 20.4853i	− 1.66707i	0.552468	−	0.833534i	$-0.313686\pi$
$152$	12.4853	1.01269
$153$	0	0
$154$	− 13.6569i	− 1.10050i
$155$	−19.3137	−1.55131
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	− 27.3137i	− 2.17296i
$159$	0	0
$160$	−4.48528	−0.354593
$161$	11.3137i	0.891645i
$162$	0	0
$163$	13.1716i	1.03168i	0.856686	+	0.515839i	$0.172520\pi$
$163$	13.1716i	1.03168i	−0.856686	+	0.515839i	$0.827480\pi$
$164$	− 41.4558i	− 3.23716i
$165$	0	0
$166$	−18.4853	−1.43474
$167$	− 7.65685i	− 0.592505i	−0.955110	−	0.296253i	$-0.904263\pi$
$167$	− 7.65685i	− 0.592505i	0.955110	−	0.296253i	$-0.0957370\pi$
$168$	0	0
$169$	0	0
$170$	−24.9706	−1.91515
$171$	0	0
$172$	36.9706	2.81898
$173$	−0.343146	−0.0260889	−0.0130444	−	0.999915i	$-0.504152\pi$
$173$	−0.343146	−0.0260889	−0.0130444	+	0.999915i	$0.504152\pi$
$174$	0	0
$175$	8.48528i	0.641427i
$176$	6.00000i	0.452267i
$177$	0	0
$178$	−22.1421	−1.65962
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	− 17.6569i	− 1.30168i
$185$	−10.3431	−0.760443
$186$	0	0
$187$	− 7.31371i	− 0.534831i
$188$	− 1.31371i	− 0.0958120i
$189$	0	0
$190$	− 19.3137i	− 1.40116i
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	− 17.3137i	− 1.24627i	−0.782115	−	0.623134i	$-0.785859\pi$
$193$	− 17.3137i	− 1.24627i	0.782115	−	0.623134i	$-0.214141\pi$
$194$	18.4853	1.32717
$195$	0	0
$196$	3.82843	0.273459
$197$	− 16.4853i	− 1.17453i	−0.809396	−	0.587264i	$-0.800205\pi$
$197$	− 16.4853i	− 1.17453i	0.809396	−	0.587264i	$-0.199795\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	− 13.2426i	− 0.936396i
$201$	0	0
$202$	8.82843i	0.621166i
$203$	5.65685i	0.397033i
$204$	0	0
$205$	−30.6274	−2.13911
$206$	32.9706i	2.29717i
$207$	0	0
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−7.65685	−0.525875
$213$	0	0
$214$	27.3137i	1.86713i
$215$	− 27.3137i	− 1.86278i
$216$	0	0
$217$	19.3137	1.31110
$218$	41.7990	2.83098
$219$	0	0
$220$	21.6569	1.46010
$221$	0	0
$222$	0	0
$223$	− 4.48528i	− 0.300357i	−0.988659	−	0.150178i	$-0.952015\pi$
$223$	− 4.48528i	− 0.300357i	0.988659	−	0.150178i	$-0.0479848\pi$
$224$	4.48528	0.299685
$225$	0	0
$226$	41.7990i	2.78043i
$227$	5.31371i	0.352683i	0.984329	+	0.176342i	$0.0564263\pi$
$227$	5.31371i	0.352683i	−0.984329	+	0.176342i	$0.943574\pi$
$228$	0	0
$229$	21.3137i	1.40845i	0.709977	+	0.704225i	$0.248705\pi$
$229$	21.3137i	1.40845i	−0.709977	+	0.704225i	$0.751295\pi$
$230$	−27.3137	−1.80101
$231$	0	0
$232$	− 8.82843i	− 0.579615i
$233$	−26.9706	−1.76690	−0.883450	−	0.468525i	$-0.844786\pi$
$233$	−26.9706	−1.76690	−0.883450	+	0.468525i	$0.844786\pi$
$234$	0	0
$235$	−0.970563	−0.0633125
$236$	− 14.0000i	− 0.911322i
$237$	0	0
$238$	24.9706	1.61860
$239$	2.00000i	0.129369i	0.997906	+	0.0646846i	$0.0206041\pi$
$239$	2.00000i	0.129369i	−0.997906	+	0.0646846i	$0.979396\pi$
$240$	0	0
$241$	11.6569i	0.750884i	0.926846	+	0.375442i	$0.122509\pi$
$241$	11.6569i	0.750884i	−0.926846	+	0.375442i	$0.877491\pi$
$242$	− 16.8995i	− 1.08634i
$243$	0	0
$244$	35.6569	2.28270
$245$	− 2.82843i	− 0.180702i
$246$	0	0
$247$	0	0
$248$	−30.1421	−1.91403
$249$	0	0
$250$	13.6569	0.863735
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	− 13.6569i	− 0.856907i
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−15.6569	−0.976648	−0.488324	−	0.872662i	$-0.662392\pi$
$257$	−15.6569	−0.976648	−0.488324	+	0.872662i	$0.662392\pi$
$258$	0	0
$259$	10.3431	0.642692
$260$	0	0
$261$	0	0
$262$	− 19.3137i	− 1.19320i
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	5.65685i	0.347498i
$266$	19.3137i	1.18420i
$267$	0	0
$268$	4.48528i	0.273982i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	11.7990i	0.716738i	0.933580	+	0.358369i	$0.116667\pi$
$271$	11.7990i	0.716738i	−0.933580	+	0.358369i	$0.883333\pi$
$272$	−10.9706	−0.665188
$273$	0	0
$274$	12.4853	0.754263
$275$	− 6.00000i	− 0.361814i
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	− 36.9706i	− 2.21735i
$279$	0	0
$280$	35.3137i	2.11040i
$281$	− 26.8284i	− 1.60045i	−0.599700	−	0.800225i	$-0.704713\pi$
$281$	− 26.8284i	− 1.60045i	0.599700	−	0.800225i	$-0.295287\pi$
$282$	0	0
$283$	4.97056	0.295469	0.147735	−	0.989027i	$-0.452802\pi$
$283$	4.97056	0.295469	0.147735	+	0.989027i	$0.452802\pi$
$284$	− 7.65685i	− 0.454351i
$285$	0	0
$286$	0	0
$287$	30.6274	1.80788
$288$	0	0
$289$	−3.62742	−0.213377
$290$	−13.6569	−0.801958
$291$	0	0
$292$	− 44.6274i	− 2.61162i
$293$	26.1421i	1.52724i	0.645666	+	0.763620i	$0.276580\pi$
$293$	26.1421i	1.52724i	−0.645666	+	0.763620i	$0.723420\pi$
$294$	0	0
$295$	−10.3431	−0.602201
$296$	−16.1421	−0.938243
$297$	0	0
$298$	−35.7990	−2.07378
$299$	0	0
$300$	0	0
$301$	27.3137i	1.57434i
$302$	−49.4558	−2.84586
$303$	0	0
$304$	− 8.48528i	− 0.486664i
$305$	− 26.3431i	− 1.50840i
$306$	0	0
$307$	− 17.1716i	− 0.980033i	−0.871713	−	0.490017i	$-0.836991\pi$
$307$	− 17.1716i	− 0.980033i	0.871713	−	0.490017i	$-0.163009\pi$
$308$	−21.6569	−1.23401
$309$	0	0
$310$	46.6274i	2.64826i
$311$	34.6274	1.96354	0.981770	−	0.190071i	$-0.0608718\pi$
$311$	34.6274	1.96354	0.981770	+	0.190071i	$0.0608718\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	24.1421i	1.36242i
$315$	0	0
$316$	−43.3137	−2.43659
$317$	− 8.48528i	− 0.476581i	−0.971194	−	0.238290i	$-0.923413\pi$
$317$	− 8.48528i	− 0.476581i	0.971194	−	0.238290i	$-0.0765870\pi$
$318$	0	0
$319$	− 4.00000i	− 0.223957i
$320$	27.7990i	1.55401i
$321$	0	0
$322$	27.3137	1.52213
$323$	10.3431i	0.575508i
$324$	0	0
$325$	0	0
$326$	31.7990	1.76118
$327$	0	0
$328$	−47.7990	−2.63926
$329$	0.970563	0.0535089
$330$	0	0
$331$	2.14214i	0.117742i	0.998266	+	0.0588712i	$0.0187501\pi$
$331$	2.14214i	0.117742i	−0.998266	+	0.0588712i	$0.981250\pi$
$332$	29.3137i	1.60880i
$333$	0	0
$334$	−18.4853	−1.01147
$335$	3.31371	0.181047
$336$	0	0
$337$	13.3137	0.725244	0.362622	−	0.931936i	$-0.381882\pi$
$337$	13.3137	0.725244	0.362622	+	0.931936i	$0.381882\pi$
$338$	0	0
$339$	0	0
$340$	39.5980i	2.14750i
$341$	−13.6569	−0.739560
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	− 42.6274i	− 2.29832i
$345$	0	0
$346$	0.828427i	0.0445365i
$347$	31.3137	1.68101	0.840504	−	0.541805i	$-0.182259\pi$
$347$	31.3137	1.68101	0.840504	+	0.541805i	$0.182259\pi$
$348$	0	0
$349$	− 7.65685i	− 0.409862i	−0.978776	−	0.204931i	$-0.934303\pi$
$349$	− 7.65685i	− 0.409862i	0.978776	−	0.204931i	$-0.0656970\pi$
$350$	20.4853	1.09498
$351$	0	0
$352$	−3.17157	−0.169045
$353$	17.4558i	0.929081i	0.885552	+	0.464540i	$0.153780\pi$
$353$	17.4558i	0.929081i	−0.885552	+	0.464540i	$0.846220\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	35.1127i	1.86097i
$357$	0	0
$358$	1.65685i	0.0875675i
$359$	− 1.02944i	− 0.0543316i	−0.999631	−	0.0271658i	$-0.991352\pi$
$359$	− 1.02944i	− 0.0543316i	0.999631	−	0.0271658i	$-0.00864821\pi$
$360$	0	0
$361$	11.0000	0.578947
$362$	33.7990i	1.77644i
$363$	0	0
$364$	0	0
$365$	−32.9706	−1.72576
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	−12.0000	−0.625543
$369$	0	0
$370$	24.9706i	1.29816i
$371$	− 5.65685i	− 0.293689i
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−17.6569	−0.913014
$375$	0	0
$376$	−1.51472	−0.0781156
$377$	0	0
$378$	0	0
$379$	16.4853i	0.846792i	0.905945	+	0.423396i	$0.139162\pi$
$379$	16.4853i	0.846792i	−0.905945	+	0.423396i	$0.860838\pi$
$380$	−30.6274	−1.57115
$381$	0	0
$382$	− 46.6274i	− 2.38567i
$383$	− 2.97056i	− 0.151789i	−0.997116	−	0.0758943i	$-0.975819\pi$
$383$	− 2.97056i	− 0.151789i	0.997116	−	0.0758943i	$-0.0241812\pi$
$384$	0	0
$385$	16.0000i	0.815436i
$386$	−41.7990	−2.12751
$387$	0	0
$388$	− 29.3137i	− 1.48818i
$389$	6.97056	0.353422	0.176711	−	0.984263i	$-0.443454\pi$
$389$	6.97056	0.353422	0.176711	+	0.984263i	$0.443454\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	− 4.41421i	− 0.222951i
$393$	0	0
$394$	−39.7990	−2.00504
$395$	32.0000i	1.61009i
$396$	0	0
$397$	− 2.97056i	− 0.149088i	−0.997218	−	0.0745441i	$-0.976250\pi$
$397$	− 2.97056i	− 0.149088i	0.997218	−	0.0745441i	$-0.0237502\pi$
$398$	24.9706i	1.25166i
$399$	0	0
$400$	−9.00000	−0.450000
$401$	2.14214i	0.106973i	0.998569	+	0.0534866i	$0.0170334\pi$
$401$	2.14214i	0.106973i	−0.998569	+	0.0534866i	$0.982967\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	13.6569	0.677778
$407$	−7.31371	−0.362527
$408$	0	0
$409$	1.02944i	0.0509024i	0.999676	+	0.0254512i	$0.00810224\pi$
$409$	1.02944i	0.0509024i	−0.999676	+	0.0254512i	$0.991898\pi$
$410$	73.9411i	3.65169i
$411$	0	0
$412$	52.2843	2.57586
$413$	10.3431	0.508953
$414$	0	0
$415$	21.6569	1.06309
$416$	0	0
$417$	0	0
$418$	− 13.6569i	− 0.667979i
$419$	30.6274	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$419$	30.6274	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$420$	0	0
$421$	− 14.6863i	− 0.715766i	−0.933766	−	0.357883i	$-0.883499\pi$
$421$	− 14.6863i	− 0.715766i	0.933766	−	0.357883i	$-0.116501\pi$
$422$	28.9706i	1.41026i
$423$	0	0
$424$	8.82843i	0.428746i
$425$	10.9706	0.532150
$426$	0	0
$427$	26.3431i	1.27483i
$428$	43.3137	2.09365
$429$	0	0
$430$	−65.9411	−3.17996
$431$	19.6569i	0.946837i	0.880838	+	0.473419i	$0.156980\pi$
$431$	19.6569i	0.946837i	−0.880838	+	0.473419i	$0.843020\pi$
$432$	0	0
$433$	−1.31371	−0.0631328	−0.0315664	−	0.999502i	$-0.510050\pi$
$433$	−1.31371	−0.0631328	−0.0315664	+	0.999502i	$0.510050\pi$
$434$	− 46.6274i	− 2.23819i
$435$	0	0
$436$	− 66.2843i	− 3.17444i
$437$	11.3137i	0.541208i
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	− 24.9706i	− 1.19042i
$441$	0	0
$442$	0	0
$443$	−41.9411	−1.99268	−0.996342	−	0.0854611i	$-0.972764\pi$
$443$	−41.9411	−1.99268	−0.996342	+	0.0854611i	$0.972764\pi$
$444$	0	0
$445$	25.9411	1.22973
$446$	−10.8284	−0.512741
$447$	0	0
$448$	− 27.7990i	− 1.31338i
$449$	− 7.79899i	− 0.368057i	−0.982921	−	0.184029i	$-0.941086\pi$
$449$	− 7.79899i	− 0.368057i	0.982921	−	0.184029i	$-0.0589139\pi$
$450$	0	0
$451$	−21.6569	−1.01978
$452$	66.2843	3.11775
$453$	0	0
$454$	12.8284	0.602068
$455$	0	0
$456$	0	0
$457$	− 3.65685i	− 0.171060i	−0.996336	−	0.0855302i	$-0.972742\pi$
$457$	− 3.65685i	− 0.171060i	0.996336	−	0.0855302i	$-0.0272584\pi$
$458$	51.4558	2.40437
$459$	0	0
$460$	43.3137i	2.01951i
$461$	10.8284i	0.504330i	0.967684	+	0.252165i	$0.0811426\pi$
$461$	10.8284i	0.504330i	−0.967684	+	0.252165i	$0.918857\pi$
$462$	0	0
$463$	− 7.51472i	− 0.349239i	−0.984636	−	0.174619i	$-0.944131\pi$
$463$	− 7.51472i	− 0.349239i	0.984636	−	0.174619i	$-0.0558695\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	65.1127i	3.01629i
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−3.31371	−0.153013
$470$	2.34315i	0.108081i
$471$	0	0
$472$	−16.1421	−0.743002
$473$	− 19.3137i	− 0.888045i
$474$	0	0
$475$	8.48528i	0.389331i
$476$	− 39.5980i	− 1.81497i
$477$	0	0
$478$	4.82843	0.220847
$479$	− 2.68629i	− 0.122740i	−0.998115	−	0.0613699i	$-0.980453\pi$
$479$	− 2.68629i	− 0.122740i	0.998115	−	0.0613699i	$-0.0195469\pi$
$480$	0	0
$481$	0	0
$482$	28.1421	1.28184
$483$	0	0
$484$	−26.7990	−1.21814
$485$	−21.6569	−0.983387
$486$	0	0
$487$	− 31.7990i	− 1.44095i	−0.693481	−	0.720475i	$-0.743924\pi$
$487$	− 31.7990i	− 1.44095i	0.693481	−	0.720475i	$-0.256076\pi$
$488$	− 41.1127i	− 1.86108i
$489$	0	0
$490$	−6.82843	−0.308477
$491$	−14.6274	−0.660126	−0.330063	−	0.943959i	$-0.607070\pi$
$491$	−14.6274	−0.660126	−0.330063	+	0.943959i	$0.607070\pi$
$492$	0	0
$493$	7.31371	0.329393
$494$	0	0
$495$	0	0
$496$	20.4853i	0.919816i
$497$	5.65685	0.253745
$498$	0	0
$499$	2.14214i	0.0958952i	0.998850	+	0.0479476i	$0.0152680\pi$
$499$	2.14214i	0.0958952i	−0.998850	+	0.0479476i	$0.984732\pi$
$500$	− 21.6569i	− 0.968524i
$501$	0	0
$502$	0	0
$503$	−15.3137	−0.682805	−0.341402	−	0.939917i	$-0.610902\pi$
$503$	−15.3137	−0.682805	−0.341402	+	0.939917i	$0.610902\pi$
$504$	0	0
$505$	− 10.3431i	− 0.460264i
$506$	−19.3137	−0.858599
$507$	0	0
$508$	−21.6569	−0.960868
$509$	− 27.7990i	− 1.23217i	−0.787680	−	0.616084i	$-0.788718\pi$
$509$	− 27.7990i	− 1.23217i	0.787680	−	0.616084i	$-0.211282\pi$
$510$	0	0
$511$	32.9706	1.45853
$512$	31.2426i	1.38074i
$513$	0	0
$514$	37.7990i	1.66724i
$515$	− 38.6274i	− 1.70213i
$516$	0	0
$517$	−0.686292	−0.0301831
$518$	− 24.9706i	− 1.09714i
$519$	0	0
$520$	0	0
$521$	−2.68629	−0.117689	−0.0588443	−	0.998267i	$-0.518742\pi$
$521$	−2.68629	−0.117689	−0.0588443	+	0.998267i	$0.518742\pi$
$522$	0	0
$523$	7.31371	0.319806	0.159903	−	0.987133i	$-0.448882\pi$
$523$	7.31371	0.319806	0.159903	+	0.987133i	$0.448882\pi$
$524$	−30.6274	−1.33796
$525$	0	0
$526$	28.9706i	1.26318i
$527$	− 24.9706i	− 1.08773i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	13.6569	0.593216
$531$	0	0
$532$	30.6274	1.32787
$533$	0	0
$534$	0	0
$535$	− 32.0000i	− 1.38348i
$536$	5.17157	0.223378
$537$	0	0
$538$	43.4558i	1.87351i
$539$	− 2.00000i	− 0.0861461i
$540$	0	0
$541$	10.0000i	0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$	10.0000i	0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	28.4853	1.22355
$543$	0	0
$544$	− 5.79899i	− 0.248630i
$545$	−48.9706	−2.09767
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	− 19.7990i	− 0.845771i
$549$	0	0
$550$	−14.4853	−0.617654
$551$	5.65685i	0.240990i
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	− 4.82843i	− 0.205140i
$555$	0	0
$556$	−58.6274	−2.48636
$557$	− 31.7990i	− 1.34737i	−0.739020	−	0.673683i	$-0.764711\pi$
$557$	− 31.7990i	− 1.34737i	0.739020	−	0.673683i	$-0.235289\pi$
$558$	0	0
$559$	0	0
$560$	24.0000	1.01419
$561$	0	0
$562$	−64.7696	−2.73214
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	− 48.9706i	− 2.06021i
$566$	− 12.0000i	− 0.504398i
$567$	0	0
$568$	−8.82843	−0.370433
$569$	−9.02944	−0.378534	−0.189267	−	0.981926i	$-0.560611\pi$
$569$	−9.02944	−0.378534	−0.189267	+	0.981926i	$0.560611\pi$
$570$	0	0
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	0	0
$574$	− 73.9411i	− 3.08624i
$575$	12.0000	0.500435
$576$	0	0
$577$	− 35.9411i	− 1.49625i	−0.663559	−	0.748124i	$-0.730955\pi$
$577$	− 35.9411i	− 1.49625i	0.663559	−	0.748124i	$-0.269045\pi$
$578$	8.75736i	0.364258i
$579$	0	0
$580$	21.6569i	0.899252i
$581$	−21.6569	−0.898478
$582$	0	0
$583$	4.00000i	0.165663i
$584$	−51.4558	−2.12926
$585$	0	0
$586$	63.1127	2.60716
$587$	22.9706i	0.948097i	0.880499	+	0.474048i	$0.157208\pi$
$587$	22.9706i	0.948097i	−0.880499	+	0.474048i	$0.842792\pi$
$588$	0	0
$589$	19.3137	0.795807
$590$	24.9706i	1.02802i
$591$	0	0
$592$	10.9706i	0.450887i
$593$	3.51472i	0.144332i	0.997393	+	0.0721661i	$0.0229912\pi$
$593$	3.51472i	0.144332i	−0.997393	+	0.0721661i	$0.977009\pi$
$594$	0	0
$595$	−29.2548	−1.19933
$596$	56.7696i	2.32537i
$597$	0	0
$598$	0	0
$599$	0.686292	0.0280411	0.0140206	−	0.999902i	$-0.495537\pi$
$599$	0.686292	0.0280411	0.0140206	+	0.999902i	$0.495537\pi$
$600$	0	0
$601$	44.6274	1.82039	0.910195	−	0.414180i	$-0.135931\pi$
$601$	44.6274	1.82039	0.910195	+	0.414180i	$0.135931\pi$
$602$	65.9411	2.68756
$603$	0	0
$604$	78.4264i	3.19113i
$605$	19.7990i	0.804943i
$606$	0	0
$607$	−25.9411	−1.05292	−0.526459	−	0.850201i	$-0.676481\pi$
$607$	−25.9411	−1.05292	−0.526459	+	0.850201i	$0.676481\pi$
$608$	4.48528	0.181902
$609$	0	0
$610$	−63.5980	−2.57501
$611$	0	0
$612$	0	0
$613$	36.3431i	1.46789i	0.679211	+	0.733943i	$0.262322\pi$
$613$	36.3431i	1.46789i	−0.679211	+	0.733943i	$0.737678\pi$
$614$	−41.4558	−1.67302
$615$	0	0
$616$	24.9706i	1.00609i
$617$	− 29.1716i	− 1.17440i	−0.809441	−	0.587202i	$-0.800230\pi$
$617$	− 29.1716i	− 1.17440i	0.809441	−	0.587202i	$-0.199770\pi$
$618$	0	0
$619$	− 15.7990i	− 0.635015i	−0.948256	−	0.317508i	$-0.897154\pi$
$619$	− 15.7990i	− 0.635015i	0.948256	−	0.317508i	$-0.102846\pi$
$620$	73.9411	2.96955
$621$	0	0
$622$	− 83.5980i	− 3.35197i
$623$	−25.9411	−1.03931
$624$	0	0
$625$	−31.0000	−1.24000
$626$	− 14.4853i	− 0.578948i
$627$	0	0
$628$	38.2843	1.52771
$629$	− 13.3726i	− 0.533200i
$630$	0	0
$631$	− 19.1127i	− 0.760865i	−0.924809	−	0.380432i	$-0.875775\pi$
$631$	− 19.1127i	− 0.760865i	0.924809	−	0.380432i	$-0.124225\pi$
$632$	49.9411i	1.98655i
$633$	0	0
$634$	−20.4853	−0.813574
$635$	16.0000i	0.634941i
$636$	0	0
$637$	0	0
$638$	−9.65685	−0.382319
$639$	0	0
$640$	58.1421	2.29827
$641$	−26.2843	−1.03817	−0.519083	−	0.854724i	$-0.673727\pi$
$641$	−26.2843	−1.03817	−0.519083	+	0.854724i	$0.673727\pi$
$642$	0	0
$643$	− 17.1716i	− 0.677181i	−0.940934	−	0.338590i	$-0.890050\pi$
$643$	− 17.1716i	− 0.677181i	0.940934	−	0.338590i	$-0.109950\pi$
$644$	− 43.3137i	− 1.70680i
$645$	0	0
$646$	24.9706	0.982454
$647$	−11.3137	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$647$	−11.3137	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$648$	0	0
$649$	−7.31371	−0.287088
$650$	0	0
$651$	0	0
$652$	− 50.4264i	− 1.97485i
$653$	2.68629	0.105123	0.0525614	−	0.998618i	$-0.483261\pi$
$653$	2.68629	0.105123	0.0525614	+	0.998618i	$0.483261\pi$
$654$	0	0
$655$	22.6274i	0.884126i
$656$	32.4853i	1.26834i
$657$	0	0
$658$	− 2.34315i	− 0.0913453i
$659$	24.6863	0.961641	0.480821	−	0.876819i	$-0.340339\pi$
$659$	24.6863	0.961641	0.480821	+	0.876819i	$0.340339\pi$
$660$	0	0
$661$	− 1.02944i	− 0.0400405i	−0.999800	−	0.0200202i	$-0.993627\pi$
$661$	− 1.02944i	− 0.0400405i	0.999800	−	0.0200202i	$-0.00637306\pi$
$662$	5.17157	0.200999
$663$	0	0
$664$	33.7990	1.31166
$665$	− 22.6274i	− 0.877454i
$666$	0	0
$667$	8.00000	0.309761
$668$	29.3137i	1.13418i
$669$	0	0
$670$	− 8.00000i	− 0.309067i
$671$	− 18.6274i	− 0.719103i
$672$	0	0
$673$	28.6274	1.10351	0.551753	−	0.834008i	$-0.313959\pi$
$673$	28.6274	1.10351	0.551753	+	0.834008i	$0.313959\pi$
$674$	− 32.1421i	− 1.23807i
$675$	0	0
$676$	0	0
$677$	−49.3137	−1.89528	−0.947640	−	0.319341i	$-0.896538\pi$
$677$	−49.3137	−1.89528	−0.947640	+	0.319341i	$0.896538\pi$
$678$	0	0
$679$	21.6569	0.831114
$680$	45.6569	1.75086
$681$	0	0
$682$	32.9706i	1.26251i
$683$	19.9411i	0.763026i	0.924363	+	0.381513i	$0.124597\pi$
$683$	19.9411i	0.763026i	−0.924363	+	0.381513i	$0.875403\pi$
$684$	0	0
$685$	−14.6274	−0.558885
$686$	−40.9706	−1.56426
$687$	0	0
$688$	−28.9706	−1.10449
$689$	0	0
$690$	0	0
$691$	34.1421i	1.29883i	0.760435	+	0.649414i	$0.224986\pi$
$691$	34.1421i	1.29883i	−0.760435	+	0.649414i	$0.775014\pi$
$692$	1.31371	0.0499397
$693$	0	0
$694$	− 75.5980i	− 2.86966i
$695$	43.3137i	1.64298i
$696$	0	0
$697$	− 39.5980i	− 1.49988i
$698$	−18.4853	−0.699678
$699$	0	0
$700$	− 32.4853i	− 1.22783i
$701$	38.9706	1.47190	0.735949	−	0.677037i	$-0.236736\pi$
$701$	38.9706	1.47190	0.735949	+	0.677037i	$0.236736\pi$
$702$	0	0
$703$	10.3431	0.390099
$704$	19.6569i	0.740846i
$705$	0	0
$706$	42.1421	1.58604
$707$	10.3431i	0.388994i
$708$	0	0
$709$	40.6274i	1.52579i	0.646520	+	0.762897i	$0.276224\pi$
$709$	40.6274i	1.52579i	−0.646520	+	0.762897i	$0.723776\pi$
$710$	13.6569i	0.512533i
$711$	0	0
$712$	40.4853	1.51725
$713$	− 27.3137i	− 1.02291i
$714$	0	0
$715$	0	0
$716$	2.62742	0.0981912
$717$	0	0
$718$	−2.48528	−0.0927499
$719$	37.9411	1.41497	0.707483	−	0.706731i	$-0.249831\pi$
$719$	37.9411	1.41497	0.707483	+	0.706731i	$0.249831\pi$
$720$	0	0
$721$	38.6274i	1.43856i
$722$	− 26.5563i	− 0.988325i
$723$	0	0
$724$	53.5980	1.99195
$725$	6.00000	0.222834
$726$	0	0
$727$	21.6569	0.803208	0.401604	−	0.915813i	$-0.368453\pi$
$727$	21.6569	0.803208	0.401604	+	0.915813i	$0.368453\pi$
$728$	0	0
$729$	0	0
$730$	79.5980i	2.94605i
$731$	35.3137	1.30612
$732$	0	0
$733$	− 8.62742i	− 0.318661i	−0.987225	−	0.159330i	$-0.949066\pi$
$733$	− 8.62742i	− 0.318661i	0.987225	−	0.159330i	$-0.0509335\pi$
$734$	57.9411i	2.13865i
$735$	0	0
$736$	− 6.34315i	− 0.233811i
$737$	2.34315	0.0863109
$738$	0	0
$739$	10.1421i	0.373084i	0.982447	+	0.186542i	$0.0597281\pi$
$739$	10.1421i	0.373084i	−0.982447	+	0.186542i	$0.940272\pi$
$740$	39.5980	1.45565
$741$	0	0
$742$	−13.6569	−0.501359
$743$	2.00000i	0.0733729i	0.999327	+	0.0366864i	$0.0116803\pi$
$743$	2.00000i	0.0733729i	−0.999327	+	0.0366864i	$0.988320\pi$
$744$	0	0
$745$	41.9411	1.53660
$746$	− 24.1421i	− 0.883906i
$747$	0	0
$748$	28.0000i	1.02378i
$749$	32.0000i	1.16925i
$750$	0	0
$751$	−32.9706	−1.20311	−0.601556	−	0.798830i	$-0.705453\pi$
$751$	−32.9706	−1.20311	−0.601556	+	0.798830i	$0.705453\pi$
$752$	1.02944i	0.0375397i
$753$	0	0
$754$	0	0
$755$	57.9411	2.10869
$756$	0	0
$757$	−15.9411	−0.579390	−0.289695	−	0.957119i	$-0.593554\pi$
$757$	−15.9411	−0.579390	−0.289695	+	0.957119i	$0.593554\pi$
$758$	39.7990	1.44556
$759$	0	0
$760$	35.3137i	1.28096i
$761$	− 15.5147i	− 0.562408i	−0.959648	−	0.281204i	$-0.909266\pi$
$761$	− 15.5147i	− 0.562408i	0.959648	−	0.281204i	$-0.0907338\pi$
$762$	0	0
$763$	48.9706	1.77285
$764$	−73.9411	−2.67510
$765$	0	0
$766$	−7.17157	−0.259119
$767$	0	0
$768$	0	0
$769$	− 42.0000i	− 1.51456i	−0.653091	−	0.757279i	$-0.726528\pi$
$769$	− 42.0000i	− 1.51456i	0.653091	−	0.757279i	$-0.273472\pi$
$770$	38.6274	1.39204
$771$	0	0
$772$	66.2843i	2.38562i
$773$	5.85786i	0.210693i	0.994436	+	0.105346i	$0.0335951\pi$
$773$	5.85786i	0.210693i	−0.994436	+	0.105346i	$0.966405\pi$
$774$	0	0
$775$	− 20.4853i	− 0.735853i
$776$	−33.7990	−1.21331
$777$	0	0
$778$	− 16.8284i	− 0.603328i
$779$	30.6274	1.09734
$780$	0	0
$781$	−4.00000	−0.143131
$782$	− 35.3137i	− 1.26282i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	− 28.2843i	− 1.00951i
$786$	0	0
$787$	32.7696i	1.16811i	0.811715	+	0.584054i	$0.198534\pi$
$787$	32.7696i	1.16811i	−0.811715	+	0.584054i	$0.801466\pi$
$788$	63.1127i	2.24830i
$789$	0	0
$790$	77.2548	2.74860
$791$	48.9706i	1.74119i
$792$	0	0
$793$	0	0
$794$	−7.17157	−0.254510
$795$	0	0
$796$	39.5980	1.40351
$797$	35.6569	1.26303	0.631515	−	0.775363i	$-0.282433\pi$
$797$	35.6569	1.26303	0.631515	+	0.775363i	$0.282433\pi$
$798$	0	0
$799$	− 1.25483i	− 0.0443928i
$800$	− 4.75736i	− 0.168198i
$801$	0	0
$802$	5.17157	0.182615
$803$	−23.3137	−0.822723
$804$	0	0
$805$	−32.0000	−1.12785
$806$	0	0
$807$	0	0
$808$	− 16.1421i	− 0.567878i
$809$	−41.3137	−1.45251	−0.726256	−	0.687424i	$-0.758741\pi$
$809$	−41.3137	−1.45251	−0.726256	+	0.687424i	$0.758741\pi$
$810$	0	0
$811$	1.85786i	0.0652384i	0.999468	+	0.0326192i	$0.0103849\pi$
$811$	1.85786i	0.0652384i	−0.999468	+	0.0326192i	$0.989615\pi$
$812$	− 21.6569i	− 0.760007i
$813$	0	0
$814$	17.6569i	0.618872i
$815$	−37.2548	−1.30498
$816$	0	0
$817$	27.3137i	0.955586i
$818$	2.48528	0.0868958
$819$	0	0
$820$	117.255	4.09472
$821$	15.7990i	0.551389i	0.961245	+	0.275694i	$0.0889077\pi$
$821$	15.7990i	0.551389i	−0.961245	+	0.275694i	$0.911092\pi$
$822$	0	0
$823$	−48.9706	−1.70701	−0.853503	−	0.521088i	$-0.825527\pi$
$823$	−48.9706	−1.70701	−0.853503	+	0.521088i	$0.825527\pi$
$824$	− 60.2843i	− 2.10010i
$825$	0	0
$826$	− 24.9706i	− 0.868837i
$827$	26.0000i	0.904109i	0.891990	+	0.452054i	$0.149309\pi$
$827$	26.0000i	0.904109i	−0.891990	+	0.452054i	$0.850691\pi$
$828$	0	0
$829$	−5.31371	−0.184553	−0.0922764	−	0.995733i	$-0.529414\pi$
$829$	−5.31371	−0.184553	−0.0922764	+	0.995733i	$0.529414\pi$
$830$	− 52.2843i	− 1.81481i
$831$	0	0
$832$	0	0
$833$	3.65685	0.126702
$834$	0	0
$835$	21.6569	0.749466
$836$	−21.6569	−0.749018
$837$	0	0
$838$	− 73.9411i	− 2.55425i
$839$	47.2548i	1.63142i	0.578462	+	0.815709i	$0.303653\pi$
$839$	47.2548i	1.63142i	−0.578462	+	0.815709i	$0.696347\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−35.4558	−1.22189
$843$	0	0
$844$	45.9411	1.58136
$845$	0	0
$846$	0	0
$847$	− 19.7990i	− 0.680301i
$848$	6.00000	0.206041
$849$	0	0
$850$	− 26.4853i	− 0.908438i
$851$	− 14.6274i	− 0.501421i
$852$	0	0
$853$	− 7.65685i	− 0.262166i	−0.991371	−	0.131083i	$-0.958155\pi$
$853$	− 7.65685i	− 0.262166i	0.991371	−	0.131083i	$-0.0418454\pi$
$854$	63.5980	2.17628
$855$	0	0
$856$	− 49.9411i	− 1.70695i
$857$	29.5980	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$857$	29.5980	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$858$	0	0
$859$	−23.3137	−0.795453	−0.397727	−	0.917504i	$-0.630201\pi$
$859$	−23.3137	−0.795453	−0.397727	+	0.917504i	$0.630201\pi$
$860$	104.569i	3.56576i
$861$	0	0
$862$	47.4558	1.61635
$863$	− 39.6569i	− 1.34994i	−0.737847	−	0.674968i	$-0.764158\pi$
$863$	− 39.6569i	− 1.34994i	0.737847	−	0.674968i	$-0.235842\pi$
$864$	0	0
$865$	− 0.970563i	− 0.0330001i
$866$	3.17157i	0.107774i
$867$	0	0
$868$	−73.9411	−2.50973
$869$	22.6274i	0.767583i
$870$	0	0
$871$	0	0
$872$	−76.4264	−2.58812
$873$	0	0
$874$	27.3137	0.923900
$875$	16.0000	0.540899
$876$	0	0
$877$	14.2843i	0.482346i	0.970482	+	0.241173i	$0.0775321\pi$
$877$	14.2843i	0.482346i	−0.970482	+	0.241173i	$0.922468\pi$
$878$	− 40.9706i	− 1.38269i
$879$	0	0
$880$	−16.9706	−0.572078
$881$	53.5980	1.80576	0.902881	−	0.429891i	$-0.141448\pi$
$881$	53.5980	1.80576	0.902881	+	0.429891i	$0.141448\pi$
$882$	0	0
$883$	51.5980	1.73641	0.868205	−	0.496205i	$-0.165274\pi$
$883$	51.5980	1.73641	0.868205	+	0.496205i	$0.165274\pi$
$884$	0	0
$885$	0	0
$886$	101.255i	3.40172i
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	− 16.0000i	− 0.536623i
$890$	− 62.6274i	− 2.09928i
$891$	0	0
$892$	17.1716i	0.574947i
$893$	0.970563	0.0324786
$894$	0	0
$895$	− 1.94113i	− 0.0648847i
$896$	−58.1421	−1.94239
$897$	0	0
$898$	−18.8284	−0.628313
$899$	− 13.6569i	− 0.455482i
$900$	0	0
$901$	−7.31371	−0.243655
$902$	52.2843i	1.74088i
$903$	0	0
$904$	− 76.4264i	− 2.54190i
$905$	− 39.5980i	− 1.31628i
$906$	0	0
$907$	−20.9706	−0.696316	−0.348158	−	0.937436i	$-0.613193\pi$
$907$	−20.9706	−0.696316	−0.348158	+	0.937436i	$0.613193\pi$
$908$	− 20.3431i	− 0.675111i
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	15.3137	0.506810
$914$	−8.82843	−0.292018
$915$	0	0
$916$	− 81.5980i	− 2.69607i
$917$	− 22.6274i	− 0.747223i
$918$	0	0
$919$	−19.3137	−0.637100	−0.318550	−	0.947906i	$-0.603196\pi$
$919$	−19.3137	−0.637100	−0.318550	+	0.947906i	$0.603196\pi$
$920$	49.9411	1.64651
$921$	0	0
$922$	26.1421	0.860945
$923$	0	0
$924$	0	0
$925$	− 10.9706i	− 0.360710i
$926$	−18.1421	−0.596188
$927$	0	0
$928$	− 3.17157i	− 0.104112i
$929$	− 27.7990i	− 0.912055i	−0.889966	−	0.456028i	$-0.849272\pi$
$929$	− 27.7990i	− 0.912055i	0.889966	−	0.456028i	$-0.150728\pi$
$930$	0	0
$931$	2.82843i	0.0926980i
$932$	103.255	3.38222
$933$	0	0
$934$	19.3137i	0.631964i
$935$	20.6863	0.676514
$936$	0	0
$937$	1.31371	0.0429170	0.0214585	−	0.999770i	$-0.493169\pi$
$937$	1.31371	0.0429170	0.0214585	+	0.999770i	$0.493169\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	3.71573	0.121194
$941$	5.85786i	0.190961i	0.995431	+	0.0954805i	$0.0304387\pi$
$941$	5.85786i	0.190961i	−0.995431	+	0.0954805i	$0.969561\pi$
$942$	0	0
$943$	− 43.3137i	− 1.41049i
$944$	10.9706i	0.357061i
$945$	0	0
$946$	−46.6274	−1.51599
$947$	54.9706i	1.78630i	0.449756	+	0.893152i	$0.351511\pi$
$947$	54.9706i	1.78630i	−0.449756	+	0.893152i	$0.648489\pi$
$948$	0	0
$949$	0	0
$950$	20.4853	0.664630
$951$	0	0
$952$	−45.6569	−1.47975
$953$	51.6569	1.67333	0.836665	−	0.547715i	$-0.184502\pi$
$953$	51.6569	1.67333	0.836665	+	0.547715i	$0.184502\pi$
$954$	0	0
$955$	54.6274i	1.76770i
$956$	− 7.65685i	− 0.247640i
$957$	0	0
$958$	−6.48528	−0.209530
$959$	14.6274	0.472344
$960$	0	0
$961$	−15.6274	−0.504110
$962$	0	0
$963$	0	0
$964$	− 44.6274i	− 1.43735i
$965$	48.9706	1.57642
$966$	0	0
$967$	10.1421i	0.326149i	0.986614	+	0.163075i	$0.0521411\pi$
$967$	10.1421i	0.326149i	−0.986614	+	0.163075i	$0.947859\pi$
$968$	30.8995i	0.993147i
$969$	0	0
$970$	52.2843i	1.67875i
$971$	−7.31371	−0.234708	−0.117354	−	0.993090i	$-0.537441\pi$
$971$	−7.31371	−0.234708	−0.117354	+	0.993090i	$0.537441\pi$
$972$	0	0
$973$	− 43.3137i	− 1.38857i
$974$	−76.7696	−2.45986
$975$	0	0
$976$	−27.9411	−0.894374
$977$	13.8579i	0.443352i	0.975120	+	0.221676i	$0.0711528\pi$
$977$	13.8579i	0.443352i	−0.975120	+	0.221676i	$0.928847\pi$
$978$	0	0
$979$	18.3431	0.586249
$980$	10.8284i	0.345901i
$981$	0	0
$982$	35.3137i	1.12691i
$983$	− 2.68629i	− 0.0856794i	−0.999082	−	0.0428397i	$-0.986360\pi$
$983$	− 2.68629i	− 0.0856794i	0.999082	−	0.0428397i	$-0.0136405\pi$
$984$	0	0
$985$	46.6274	1.48567
$986$	− 17.6569i	− 0.562309i
$987$	0	0
$988$	0	0
$989$	38.6274	1.22828
$990$	0	0
$991$	27.3137	0.867649	0.433824	−	0.900998i	$-0.357164\pi$
$991$	27.3137	0.867649	0.433824	+	0.900998i	$0.357164\pi$
$992$	−10.8284	−0.343803
$993$	0	0
$994$	− 13.6569i	− 0.433169i
$995$	− 29.2548i	− 0.927441i
$996$	0	0
$997$	51.2548	1.62326	0.811628	−	0.584174i	$-0.198581\pi$
$997$	51.2548	1.62326	0.811628	+	0.584174i	$0.198581\pi$
$998$	5.17157	0.163703
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.j.1351.1		4
3.2	odd	2		507.2.b.e.337.4		4
13.5	odd	4		1521.2.a.f.1.1		2
13.8	odd	4		117.2.a.c.1.2		2
13.12	even	2	inner	1521.2.b.j.1351.4		4
39.2	even	12		507.2.e.d.22.1		4
39.5	even	4		507.2.a.h.1.2		2
39.8	even	4		39.2.a.b.1.1	✓	2
39.11	even	12		507.2.e.h.22.2		4
39.17	odd	6		507.2.j.f.361.1		8
39.20	even	12		507.2.e.h.484.2		4
39.23	odd	6		507.2.j.f.316.4		8
39.29	odd	6		507.2.j.f.316.1		8
39.32	even	12		507.2.e.d.484.1		4
39.35	odd	6		507.2.j.f.361.4		8
39.38	odd	2		507.2.b.e.337.1		4
52.47	even	4		1872.2.a.w.1.1		2
65.8	even	4		2925.2.c.u.2224.1		4
65.34	odd	4		2925.2.a.v.1.1		2
65.47	even	4		2925.2.c.u.2224.4		4
91.34	even	4		5733.2.a.u.1.2		2
104.21	odd	4		7488.2.a.cl.1.2		2
104.99	even	4		7488.2.a.co.1.2		2
117.34	odd	12		1053.2.e.e.352.1		4
117.47	even	12		1053.2.e.m.352.2		4
117.86	even	12		1053.2.e.m.703.2		4
117.112	odd	12		1053.2.e.e.703.1		4
156.47	odd	4		624.2.a.k.1.2		2
156.83	odd	4		8112.2.a.bm.1.1		2
195.8	odd	4		975.2.c.h.274.4		4
195.47	odd	4		975.2.c.h.274.1		4
195.164	even	4		975.2.a.l.1.2		2
273.125	odd	4		1911.2.a.h.1.1		2
312.125	even	4		2496.2.a.bf.1.1		2
312.203	odd	4		2496.2.a.bi.1.1		2
429.164	odd	4		4719.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.1	✓	2	39.8	even	4
117.2.a.c.1.2		2	13.8	odd	4
507.2.a.h.1.2		2	39.5	even	4
507.2.b.e.337.1		4	39.38	odd	2
507.2.b.e.337.4		4	3.2	odd	2
507.2.e.d.22.1		4	39.2	even	12
507.2.e.d.484.1		4	39.32	even	12
507.2.e.h.22.2		4	39.11	even	12
507.2.e.h.484.2		4	39.20	even	12
507.2.j.f.316.1		8	39.29	odd	6
507.2.j.f.316.4		8	39.23	odd	6
507.2.j.f.361.1		8	39.17	odd	6
507.2.j.f.361.4		8	39.35	odd	6
624.2.a.k.1.2		2	156.47	odd	4
975.2.a.l.1.2		2	195.164	even	4
975.2.c.h.274.1		4	195.47	odd	4
975.2.c.h.274.4		4	195.8	odd	4
1053.2.e.e.352.1		4	117.34	odd	12
1053.2.e.e.703.1		4	117.112	odd	12
1053.2.e.m.352.2		4	117.47	even	12
1053.2.e.m.703.2		4	117.86	even	12
1521.2.a.f.1.1		2	13.5	odd	4
1521.2.b.j.1351.1		4	1.1	even	1	trivial
1521.2.b.j.1351.4		4	13.12	even	2	inner
1872.2.a.w.1.1		2	52.47	even	4
1911.2.a.h.1.1		2	273.125	odd	4
2496.2.a.bf.1.1		2	312.125	even	4
2496.2.a.bi.1.1		2	312.203	odd	4
2925.2.a.v.1.1		2	65.34	odd	4
2925.2.c.u.2224.1		4	65.8	even	4
2925.2.c.u.2224.4		4	65.47	even	4
4719.2.a.p.1.2		2	429.164	odd	4
5733.2.a.u.1.2		2	91.34	even	4
7488.2.a.cl.1.2		2	104.21	odd	4
7488.2.a.co.1.2		2	104.99	even	4
8112.2.a.bm.1.1		2	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)

\(f(q)\)	\(=\)	\(q-2.41421i q^{2} -3.82843 q^{4} +2.82843i q^{5} -2.82843i q^{7} +4.41421i q^{8} +O(q^{10})\)	\(q-2.41421i q^{2} -3.82843 q^{4} +2.82843i q^{5} -2.82843i q^{7} +4.41421i q^{8} +6.82843 q^{10} +2.00000i q^{11} -6.82843 q^{14} +3.00000 q^{16} -3.65685 q^{17} -2.82843i q^{19} -10.8284i q^{20} +4.82843 q^{22} -4.00000 q^{23} -3.00000 q^{25} +10.8284i q^{28} -2.00000 q^{29} +6.82843i q^{31} +1.58579i q^{32} +8.82843i q^{34} +8.00000 q^{35} +3.65685i q^{37} -6.82843 q^{38} -12.4853 q^{40} +10.8284i q^{41} -9.65685 q^{43} -7.65685i q^{44} +9.65685i q^{46} +0.343146i q^{47} -1.00000 q^{49} +7.24264i q^{50} +2.00000 q^{53} -5.65685 q^{55} +12.4853 q^{56} +4.82843i q^{58} +3.65685i q^{59} -9.31371 q^{61} +16.4853 q^{62} +9.82843 q^{64} -1.17157i q^{67} +14.0000 q^{68} -19.3137i q^{70} +2.00000i q^{71} +11.6569i q^{73} +8.82843 q^{74} +10.8284i q^{76} +5.65685 q^{77} +11.3137 q^{79} +8.48528i q^{80} +26.1421 q^{82} -7.65685i q^{83} -10.3431i q^{85} +23.3137i q^{86} -8.82843 q^{88} -9.17157i q^{89} +15.3137 q^{92} +0.828427 q^{94} +8.00000 q^{95} +7.65685i q^{97} +2.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 16 q^{10} - 16 q^{14} + 12 q^{16} + 8 q^{17} + 8 q^{22} - 16 q^{23} - 12 q^{25} - 8 q^{29} + 32 q^{35} - 16 q^{38} - 16 q^{40} - 16 q^{43} - 4 q^{49} + 8 q^{53} + 16 q^{56} + 8 q^{61} + 32 q^{62} + 28 q^{64} + 56 q^{68} + 24 q^{74} + 48 q^{82} - 24 q^{88} + 16 q^{92} - 8 q^{94} + 32 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 16 * q^10 - 16 * q^14 + 12 * q^16 + 8 * q^17 + 8 * q^22 - 16 * q^23 - 12 * q^25 - 8 * q^29 + 32 * q^35 - 16 * q^38 - 16 * q^40 - 16 * q^43 - 4 * q^49 + 8 * q^53 + 16 * q^56 + 8 * q^61 + 32 * q^62 + 28 * q^64 + 56 * q^68 + 24 * q^74 + 48 * q^82 - 24 * q^88 + 16 * q^92 - 8 * q^94 + 32 * q^95

Introduction

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