LMFDB - Embedded newform 1521.2.b.j.1351.3

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			1351.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.j.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{5} +2.82843i q^{7} +1.58579i q^{8} +O(q^{10})$	$q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{5} +2.82843i q^{7} +1.58579i q^{8} +1.17157 q^{10} +2.00000i q^{11} -1.17157 q^{14} +3.00000 q^{16} +7.65685 q^{17} +2.82843i q^{19} -5.17157i q^{20} -0.828427 q^{22} -4.00000 q^{23} -3.00000 q^{25} +5.17157i q^{28} -2.00000 q^{29} +1.17157i q^{31} +4.41421i q^{32} +3.17157i q^{34} +8.00000 q^{35} -7.65685i q^{37} -1.17157 q^{38} +4.48528 q^{40} +5.17157i q^{41} +1.65685 q^{43} +3.65685i q^{44} -1.65685i q^{46} +11.6569i q^{47} -1.00000 q^{49} -1.24264i q^{50} +2.00000 q^{53} +5.65685 q^{55} -4.48528 q^{56} -0.828427i q^{58} -7.65685i q^{59} +13.3137 q^{61} -0.485281 q^{62} +4.17157 q^{64} -6.82843i q^{67} +14.0000 q^{68} +3.31371i q^{70} +2.00000i q^{71} +0.343146i q^{73} +3.17157 q^{74} +5.17157i q^{76} -5.65685 q^{77} -11.3137 q^{79} -8.48528i q^{80} -2.14214 q^{82} +3.65685i q^{83} -21.6569i q^{85} +0.686292i q^{86} -3.17157 q^{88} -14.8284i q^{89} -7.31371 q^{92} -4.82843 q^{94} +8.00000 q^{95} -3.65685i q^{97} -0.414214i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	0	0
$3$	0	0
$4$	1.82843	0.914214
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0	0
$7$	2.82843i	1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$	2.82843i	1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$	1.58579i	0.560660i
$9$	0	0
$10$	1.17157	0.370484
$11$	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
$14$	−1.17157	−0.313116
$15$	0	0
$16$	3.00000	0.750000
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	2.82843i	0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$	2.82843i	0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$	− 5.17157i	− 1.15640i
$21$	0	0
$22$	−0.828427	−0.176621
$23$	−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
$28$	5.17157i	0.977335i
$29$	−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
$31$	1.17157i	0.210421i	0.994450	+	0.105210i	$0.0335516\pi$
$31$	1.17157i	0.210421i	−0.994450	+	0.105210i	$0.966448\pi$
$32$	4.41421i	0.780330i
$33$	0	0
$34$	3.17157i	0.543920i
$35$	8.00000	1.35225
$36$	0	0
$37$	− 7.65685i	− 1.25878i	−0.777090	−	0.629390i	$-0.783305\pi$
$37$	− 7.65685i	− 1.25878i	0.777090	−	0.629390i	$-0.216695\pi$
$38$	−1.17157	−0.190054
$39$	0	0
$40$	4.48528	0.709185
$41$	5.17157i	0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$41$	5.17157i	0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	3.65685i	0.551292i
$45$	0	0
$46$	− 1.65685i	− 0.244290i
$47$	11.6569i	1.70033i	0.526519	+	0.850163i	$0.323497\pi$
$47$	11.6569i	1.70033i	−0.526519	+	0.850163i	$0.676503\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	− 1.24264i	− 0.175736i
$51$	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
$56$	−4.48528	−0.599371
$57$	0	0
$58$	− 0.828427i	− 0.108778i
$59$	− 7.65685i	− 0.996838i	−0.866936	−	0.498419i	$-0.833914\pi$
$59$	− 7.65685i	− 0.996838i	0.866936	−	0.498419i	$-0.166086\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	−0.485281	−0.0616308
$63$	0	0
$64$	4.17157	0.521447
$65$	0	0
$66$	0	0
$67$	− 6.82843i	− 0.834225i	−0.908855	−	0.417113i	$-0.863042\pi$
$67$	− 6.82843i	− 0.834225i	0.908855	−	0.417113i	$-0.136958\pi$
$68$	14.0000	1.69775
$69$	0	0
$70$	3.31371i	0.396064i
$71$	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
$73$	0.343146i	0.0401622i	0.999798	+	0.0200811i	$0.00639244\pi$
$73$	0.343146i	0.0401622i	−0.999798	+	0.0200811i	$0.993608\pi$
$74$	3.17157	0.368688
$75$	0	0
$76$	5.17157i	0.593220i
$77$	−5.65685	−0.644658
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	− 8.48528i	− 0.948683i
$81$	0	0
$82$	−2.14214	−0.236559
$83$	3.65685i	0.401392i	0.979654	+	0.200696i	$0.0643203\pi$
$83$	3.65685i	0.401392i	−0.979654	+	0.200696i	$0.935680\pi$
$84$	0	0
$85$	− 21.6569i	− 2.34902i
$86$	0.686292i	0.0740047i
$87$	0	0
$88$	−3.17157	−0.338091
$89$	− 14.8284i	− 1.57181i	−0.618347	−	0.785905i	$-0.712197\pi$
$89$	− 14.8284i	− 1.57181i	0.618347	−	0.785905i	$-0.287803\pi$
$90$	0	0
$91$	0	0
$92$	−7.31371	−0.762507
$93$	0	0
$94$	−4.82843	−0.498014
$95$	8.00000	0.820783
$96$	0	0
$97$	− 3.65685i	− 0.371297i	−0.982616	−	0.185649i	$-0.940561\pi$
$97$	− 3.65685i	− 0.371297i	0.982616	−	0.185649i	$-0.0594386\pi$
$98$	− 0.414214i	− 0.0418419i
$99$	0	0
$100$	−5.48528	−0.548528
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	0	0
$106$	0.828427i	0.0804640i
$107$	11.3137	1.09374	0.546869	−	0.837218i	$-0.315820\pi$
$107$	11.3137	1.09374	0.546869	+	0.837218i	$0.315820\pi$
$108$	0	0
$109$	− 5.31371i	− 0.508961i	−0.967078	−	0.254480i	$-0.918096\pi$
$109$	− 5.31371i	− 0.508961i	0.967078	−	0.254480i	$-0.0819045\pi$
$110$	2.34315i	0.223410i
$111$	0	0
$112$	8.48528i	0.801784i
$113$	5.31371	0.499872	0.249936	−	0.968262i	$-0.419590\pi$
$113$	5.31371	0.499872	0.249936	+	0.968262i	$0.419590\pi$
$114$	0	0
$115$	11.3137i	1.05501i
$116$	−3.65685	−0.339530
$117$	0	0
$118$	3.17157	0.291967
$119$	21.6569i	1.98528i
$120$	0	0
$121$	7.00000	0.636364
$122$	5.51472i	0.499279i
$123$	0	0
$124$	2.14214i	0.192369i
$125$	− 5.65685i	− 0.505964i
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	10.5563i	0.933058i
$129$	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
$136$	12.1421i	1.04118i
$137$	10.8284i	0.925135i	0.886584	+	0.462567i	$0.153072\pi$
$137$	10.8284i	0.925135i	−0.886584	+	0.462567i	$0.846928\pi$
$138$	0	0
$139$	−7.31371	−0.620341	−0.310170	−	0.950681i	$-0.600386\pi$
$139$	−7.31371	−0.620341	−0.310170	+	0.950681i	$0.600386\pi$
$140$	14.6274	1.23624
$141$	0	0
$142$	−0.828427	−0.0695201
$143$	0	0
$144$	0	0
$145$	5.65685i	0.469776i
$146$	−0.142136	−0.0117632
$147$	0	0
$148$	− 14.0000i	− 1.15079i
$149$	− 9.17157i	− 0.751365i	−0.926749	−	0.375682i	$-0.877408\pi$
$149$	− 9.17157i	− 0.751365i	0.926749	−	0.375682i	$-0.122592\pi$
$150$	0	0
$151$	− 3.51472i	− 0.286024i	−0.989721	−	0.143012i	$-0.954321\pi$
$151$	− 3.51472i	− 0.286024i	0.989721	−	0.143012i	$-0.0456787\pi$
$152$	−4.48528	−0.363804
$153$	0	0
$154$	− 2.34315i	− 0.188816i
$155$	3.31371	0.266163
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	− 4.68629i	− 0.372821i
$159$	0	0
$160$	12.4853	0.987048
$161$	− 11.3137i	− 0.891645i
$162$	0	0
$163$	18.8284i	1.47476i	0.675480	+	0.737378i	$0.263936\pi$
$163$	18.8284i	1.47476i	−0.675480	+	0.737378i	$0.736064\pi$
$164$	9.45584i	0.738377i
$165$	0	0
$166$	−1.51472	−0.117565
$167$	3.65685i	0.282976i	0.989940	+	0.141488i	$0.0451886\pi$
$167$	3.65685i	0.282976i	−0.989940	+	0.141488i	$0.954811\pi$
$168$	0	0
$169$	0	0
$170$	8.97056	0.688011
$171$	0	0
$172$	3.02944	0.230992
$173$	−11.6569	−0.886254	−0.443127	−	0.896459i	$-0.646131\pi$
$173$	−11.6569	−0.886254	−0.443127	+	0.896459i	$0.646131\pi$
$174$	0	0
$175$	− 8.48528i	− 0.641427i
$176$	6.00000i	0.452267i
$177$	0	0
$178$	6.14214	0.460373
$179$	−23.3137	−1.74255	−0.871274	−	0.490797i	$-0.836706\pi$
$179$	−23.3137	−1.74255	−0.871274	+	0.490797i	$0.836706\pi$
$180$	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
$184$	− 6.34315i	− 0.467623i
$185$	−21.6569	−1.59224
$186$	0	0
$187$	15.3137i	1.11985i
$188$	21.3137i	1.55446i
$189$	0	0
$190$	3.31371i	0.240402i
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	5.31371i	0.382489i	0.981542	+	0.191245i	$0.0612524\pi$
$193$	5.31371i	0.382489i	−0.981542	+	0.191245i	$0.938748\pi$
$194$	1.51472	0.108750
$195$	0	0
$196$	−1.82843	−0.130602
$197$	0.485281i	0.0345749i	0.999851	+	0.0172874i	$0.00550304\pi$
$197$	0.485281i	0.0345749i	−0.999851	+	0.0172874i	$0.994497\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	− 4.75736i	− 0.336396i
$201$	0	0
$202$	3.17157i	0.223151i
$203$	− 5.65685i	− 0.397033i
$204$	0	0
$205$	14.6274	1.02162
$206$	− 0.970563i	− 0.0676223i
$207$	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	$0.635539\pi$
$212$	3.65685	0.251154
$213$	0	0
$214$	4.68629i	0.320348i
$215$	− 4.68629i	− 0.319602i
$216$	0	0
$217$	−3.31371	−0.224949
$218$	2.20101	0.149071
$219$	0	0
$220$	10.3431	0.697335
$221$	0	0
$222$	0	0
$223$	12.4853i	0.836076i	0.908429	+	0.418038i	$0.137282\pi$
$223$	12.4853i	0.836076i	−0.908429	+	0.418038i	$0.862718\pi$
$224$	−12.4853	−0.834208
$225$	0	0
$226$	2.20101i	0.146409i
$227$	− 17.3137i	− 1.14915i	−0.818452	−	0.574576i	$-0.805167\pi$
$227$	− 17.3137i	− 1.14915i	0.818452	−	0.574576i	$-0.194833\pi$
$228$	0	0
$229$	− 1.31371i	− 0.0868123i	−0.999058	−	0.0434062i	$-0.986179\pi$
$229$	− 1.31371i	− 0.0868123i	0.999058	−	0.0434062i	$-0.0138209\pi$
$230$	−4.68629	−0.309005
$231$	0	0
$232$	− 3.17157i	− 0.208224i
$233$	6.97056	0.456657	0.228328	−	0.973584i	$-0.426674\pi$
$233$	6.97056	0.456657	0.228328	+	0.973584i	$0.426674\pi$
$234$	0	0
$235$	32.9706	2.15076
$236$	− 14.0000i	− 0.911322i
$237$	0	0
$238$	−8.97056	−0.581475
$239$	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
$241$	0.343146i	0.0221040i	0.999939	+	0.0110520i	$0.00351803\pi$
$241$	0.343146i	0.0221040i	−0.999939	+	0.0110520i	$0.996482\pi$
$242$	2.89949i	0.186387i
$243$	0	0
$244$	24.3431	1.55841
$245$	2.82843i	0.180702i
$246$	0	0
$247$	0	0
$248$	−1.85786	−0.117975
$249$	0	0
$250$	2.34315	0.148194
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	− 2.34315i	− 0.147022i
$255$	0	0
$256$	3.97056	0.248160
$257$	−4.34315	−0.270918	−0.135459	−	0.990783i	$-0.543251\pi$
$257$	−4.34315	−0.270918	−0.135459	+	0.990783i	$0.543251\pi$
$258$	0	0
$259$	21.6569	1.34569
$260$	0	0
$261$	0	0
$262$	3.31371i	0.204722i
$263$	−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
$266$	− 3.31371i	− 0.203177i
$267$	0	0
$268$	− 12.4853i	− 0.762660i
$269$	−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
$271$	− 27.7990i	− 1.68867i	−0.535817	−	0.844334i	$-0.679996\pi$
$271$	− 27.7990i	− 1.68867i	0.535817	−	0.844334i	$-0.320004\pi$
$272$	22.9706	1.39279
$273$	0	0
$274$	−4.48528	−0.270966
$275$	− 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	$0.480863\pi$
$278$	− 3.02944i	− 0.181694i
$279$	0	0
$280$	12.6863i	0.758151i
$281$	− 21.1716i	− 1.26299i	−0.775380	−	0.631495i	$-0.782442\pi$
$281$	− 21.1716i	− 1.26299i	0.775380	−	0.631495i	$-0.217558\pi$
$282$	0	0
$283$	−28.9706	−1.72212	−0.861061	−	0.508502i	$-0.830199\pi$
$283$	−28.9706	−1.72212	−0.861061	+	0.508502i	$0.830199\pi$
$284$	3.65685i	0.216994i
$285$	0	0
$286$	0	0
$287$	−14.6274	−0.863429
$288$	0	0
$289$	41.6274	2.44867
$290$	−2.34315	−0.137594
$291$	0	0
$292$	0.627417i	0.0367168i
$293$	− 2.14214i	− 0.125145i	−0.998040	−	0.0625724i	$-0.980070\pi$
$293$	− 2.14214i	− 0.125145i	0.998040	−	0.0625724i	$-0.0199304\pi$
$294$	0	0
$295$	−21.6569	−1.26091
$296$	12.1421	0.705747
$297$	0	0
$298$	3.79899	0.220070
$299$	0	0
$300$	0	0
$301$	4.68629i	0.270113i
$302$	1.45584	0.0837744
$303$	0	0
$304$	8.48528i	0.486664i
$305$	− 37.6569i	− 2.15623i
$306$	0	0
$307$	− 22.8284i	− 1.30289i	−0.758697	−	0.651444i	$-0.774164\pi$
$307$	− 22.8284i	− 1.30289i	0.758697	−	0.651444i	$-0.225836\pi$
$308$	−10.3431	−0.589355
$309$	0	0
$310$	1.37258i	0.0779575i
$311$	−10.6274	−0.602626	−0.301313	−	0.953525i	$-0.597425\pi$
$311$	−10.6274	−0.602626	−0.301313	+	0.953525i	$0.597425\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	− 4.14214i	− 0.233754i
$315$	0	0
$316$	−20.6863	−1.16369
$317$	8.48528i	0.476581i	0.971194	+	0.238290i	$0.0765870\pi$
$317$	8.48528i	0.476581i	−0.971194	+	0.238290i	$0.923413\pi$
$318$	0	0
$319$	− 4.00000i	− 0.223957i
$320$	− 11.7990i	− 0.659584i
$321$	0	0
$322$	4.68629	0.261157
$323$	21.6569i	1.20502i
$324$	0	0
$325$	0	0
$326$	−7.79899	−0.431946
$327$	0	0
$328$	−8.20101	−0.452825
$329$	−32.9706	−1.81773
$330$	0	0
$331$	− 26.1421i	− 1.43690i	−0.695578	−	0.718451i	$-0.744852\pi$
$331$	− 26.1421i	− 1.43690i	0.695578	−	0.718451i	$-0.255148\pi$
$332$	6.68629i	0.366958i
$333$	0	0
$334$	−1.51472	−0.0828817
$335$	−19.3137	−1.05522
$336$	0	0
$337$	−9.31371	−0.507350	−0.253675	−	0.967290i	$-0.581639\pi$
$337$	−9.31371	−0.507350	−0.253675	+	0.967290i	$0.581639\pi$
$338$	0	0
$339$	0	0
$340$	− 39.5980i	− 2.14750i
$341$	−2.34315	−0.126888
$342$	0	0
$343$	16.9706i	0.916324i
$344$	2.62742i	0.141661i
$345$	0	0
$346$	− 4.82843i	− 0.259578i
$347$	8.68629	0.466305	0.233152	−	0.972440i	$-0.425096\pi$
$347$	8.68629	0.466305	0.233152	+	0.972440i	$0.425096\pi$
$348$	0	0
$349$	3.65685i	0.195747i	0.995199	+	0.0978735i	$0.0312040\pi$
$349$	3.65685i	0.195747i	−0.995199	+	0.0978735i	$0.968796\pi$
$350$	3.51472	0.187870
$351$	0	0
$352$	−8.82843	−0.470557
$353$	− 33.4558i	− 1.78067i	−0.455301	−	0.890337i	$-0.650468\pi$
$353$	− 33.4558i	− 1.78067i	0.455301	−	0.890337i	$-0.349532\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	− 27.1127i	− 1.43697i
$357$	0	0
$358$	− 9.65685i	− 0.510381i
$359$	− 34.9706i	− 1.84568i	−0.385189	−	0.922838i	$-0.625864\pi$
$359$	− 34.9706i	− 1.84568i	0.385189	−	0.922838i	$-0.374136\pi$
$360$	0	0
$361$	11.0000	0.578947
$362$	− 5.79899i	− 0.304788i
$363$	0	0
$364$	0	0
$365$	0.970563	0.0508016
$366$	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
$370$	− 8.97056i	− 0.466357i
$371$	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	$0.416643\pi$
$374$	−6.34315	−0.327996
$375$	0	0
$376$	−18.4853	−0.953306
$377$	0	0
$378$	0	0
$379$	− 0.485281i	− 0.0249272i	−0.999922	−	0.0124636i	$-0.996033\pi$
$379$	− 0.485281i	− 0.0249272i	0.999922	−	0.0124636i	$-0.00396739\pi$
$380$	14.6274	0.750371
$381$	0	0
$382$	− 1.37258i	− 0.0702275i
$383$	30.9706i	1.58252i	0.611479	+	0.791261i	$0.290575\pi$
$383$	30.9706i	1.58252i	−0.611479	+	0.791261i	$0.709425\pi$
$384$	0	0
$385$	16.0000i	0.815436i
$386$	−2.20101	−0.112028
$387$	0	0
$388$	− 6.68629i	− 0.339445i
$389$	−26.9706	−1.36746	−0.683731	−	0.729734i	$-0.739644\pi$
$389$	−26.9706	−1.36746	−0.683731	+	0.729734i	$0.739644\pi$
$390$	0	0
$391$	−30.6274	−1.54890
$392$	− 1.58579i	− 0.0800943i
$393$	0	0
$394$	−0.201010	−0.0101267
$395$	32.0000i	1.61009i
$396$	0	0
$397$	30.9706i	1.55437i	0.629273	+	0.777184i	$0.283353\pi$
$397$	30.9706i	1.55437i	−0.629273	+	0.777184i	$0.716647\pi$
$398$	− 8.97056i	− 0.449654i
$399$	0	0
$400$	−9.00000	−0.450000
$401$	− 26.1421i	− 1.30548i	−0.757584	−	0.652738i	$-0.773620\pi$
$401$	− 26.1421i	− 1.30548i	0.757584	−	0.652738i	$-0.226380\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	2.34315	0.116288
$407$	15.3137	0.759072
$408$	0	0
$409$	34.9706i	1.72918i	0.502475	+	0.864592i	$0.332423\pi$
$409$	34.9706i	1.72918i	−0.502475	+	0.864592i	$0.667577\pi$
$410$	6.05887i	0.299226i
$411$	0	0
$412$	−4.28427	−0.211071
$413$	21.6569	1.06566
$414$	0	0
$415$	10.3431	0.507725
$416$	0	0
$417$	0	0
$418$	− 2.34315i	− 0.114607i
$419$	−14.6274	−0.714596	−0.357298	−	0.933990i	$-0.616302\pi$
$419$	−14.6274	−0.714596	−0.357298	+	0.933990i	$0.616302\pi$
$420$	0	0
$421$	− 37.3137i	− 1.81856i	−0.416186	−	0.909279i	$-0.636634\pi$
$421$	− 37.3137i	− 1.81856i	0.416186	−	0.909279i	$-0.363366\pi$
$422$	− 4.97056i	− 0.241963i
$423$	0	0
$424$	3.17157i	0.154025i
$425$	−22.9706	−1.11424
$426$	0	0
$427$	37.6569i	1.82234i
$428$	20.6863	0.999910
$429$	0	0
$430$	1.94113	0.0936094
$431$	8.34315i	0.401875i	0.979604	+	0.200938i	$0.0643988\pi$
$431$	8.34315i	0.401875i	−0.979604	+	0.200938i	$0.935601\pi$
$432$	0	0
$433$	21.3137	1.02427	0.512136	−	0.858905i	$-0.328854\pi$
$433$	21.3137	1.02427	0.512136	+	0.858905i	$0.328854\pi$
$434$	− 1.37258i	− 0.0658861i
$435$	0	0
$436$	− 9.71573i	− 0.465299i
$437$	− 11.3137i	− 0.541208i
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	8.97056i	0.427655i
$441$	0	0
$442$	0	0
$443$	25.9411	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$443$	25.9411	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$444$	0	0
$445$	−41.9411	−1.98820
$446$	−5.17157	−0.244881
$447$	0	0
$448$	11.7990i	0.557450i
$449$	31.7990i	1.50069i	0.661048	+	0.750344i	$0.270112\pi$
$449$	31.7990i	1.50069i	−0.661048	+	0.750344i	$0.729888\pi$
$450$	0	0
$451$	−10.3431	−0.487040
$452$	9.71573	0.456989
$453$	0	0
$454$	7.17157	0.336579
$455$	0	0
$456$	0	0
$457$	7.65685i	0.358173i	0.983833	+	0.179086i	$0.0573141\pi$
$457$	7.65685i	0.358173i	−0.983833	+	0.179086i	$0.942686\pi$
$458$	0.544156	0.0254267
$459$	0	0
$460$	20.6863i	0.964503i
$461$	5.17157i	0.240864i	0.992722	+	0.120432i	$0.0384280\pi$
$461$	5.17157i	0.240864i	−0.992722	+	0.120432i	$0.961572\pi$
$462$	0	0
$463$	− 24.4853i	− 1.13793i	−0.822363	−	0.568964i	$-0.807344\pi$
$463$	− 24.4853i	− 1.13793i	0.822363	−	0.568964i	$-0.192656\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	2.88730i	0.133752i
$467$	−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
$469$	19.3137	0.891824
$470$	13.6569i	0.629944i
$471$	0	0
$472$	12.1421	0.558887
$473$	3.31371i	0.152364i
$474$	0	0
$475$	− 8.48528i	− 0.389331i
$476$	39.5980i	1.81497i
$477$	0	0
$478$	−0.828427	−0.0378914
$479$	− 25.3137i	− 1.15661i	−0.815820	−	0.578306i	$-0.803714\pi$
$479$	− 25.3137i	− 1.15661i	0.815820	−	0.578306i	$-0.196286\pi$
$480$	0	0
$481$	0	0
$482$	−0.142136	−0.00647410
$483$	0	0
$484$	12.7990	0.581772
$485$	−10.3431	−0.469658
$486$	0	0
$487$	7.79899i	0.353406i	0.984264	+	0.176703i	$0.0565432\pi$
$487$	7.79899i	0.353406i	−0.984264	+	0.176703i	$0.943457\pi$
$488$	21.1127i	0.955727i
$489$	0	0
$490$	−1.17157	−0.0529263
$491$	30.6274	1.38220	0.691098	−	0.722761i	$-0.257127\pi$
$491$	30.6274	1.38220	0.691098	+	0.722761i	$0.257127\pi$
$492$	0	0
$493$	−15.3137	−0.689695
$494$	0	0
$495$	0	0
$496$	3.51472i	0.157816i
$497$	−5.65685	−0.253745
$498$	0	0
$499$	− 26.1421i	− 1.17028i	−0.810931	−	0.585141i	$-0.801039\pi$
$499$	− 26.1421i	− 1.17028i	0.810931	−	0.585141i	$-0.198961\pi$
$500$	− 10.3431i	− 0.462560i
$501$	0	0
$502$	0	0
$503$	7.31371	0.326102	0.163051	−	0.986618i	$-0.447866\pi$
$503$	7.31371	0.326102	0.163051	+	0.986618i	$0.447866\pi$
$504$	0	0
$505$	− 21.6569i	− 0.963717i
$506$	3.31371	0.147312
$507$	0	0
$508$	−10.3431	−0.458903
$509$	11.7990i	0.522981i	0.965206	+	0.261491i	$0.0842140\pi$
$509$	11.7990i	0.522981i	−0.965206	+	0.261491i	$0.915786\pi$
$510$	0	0
$511$	−0.970563	−0.0429352
$512$	22.7574i	1.00574i
$513$	0	0
$514$	− 1.79899i	− 0.0793500i
$515$	6.62742i	0.292039i
$516$	0	0
$517$	−23.3137	−1.02534
$518$	8.97056i	0.394144i
$519$	0	0
$520$	0	0
$521$	−25.3137	−1.10901	−0.554507	−	0.832179i	$-0.687093\pi$
$521$	−25.3137	−1.10901	−0.554507	+	0.832179i	$0.687093\pi$
$522$	0	0
$523$	−15.3137	−0.669622	−0.334811	−	0.942285i	$-0.608672\pi$
$523$	−15.3137	−0.669622	−0.334811	+	0.942285i	$0.608672\pi$
$524$	14.6274	0.639002
$525$	0	0
$526$	− 4.97056i	− 0.216727i
$527$	8.97056i	0.390764i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	2.34315	0.101780
$531$	0	0
$532$	−14.6274	−0.634179
$533$	0	0
$534$	0	0
$535$	− 32.0000i	− 1.38348i
$536$	10.8284	0.467717
$537$	0	0
$538$	− 7.45584i	− 0.321444i
$539$	− 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	$0.931036\pi$
$542$	11.5147	0.494600
$543$	0	0
$544$	33.7990i	1.44912i
$545$	−15.0294	−0.643790
$546$	0	0
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	19.7990i	0.845771i
$549$	0	0
$550$	2.48528	0.105973
$551$	− 5.65685i	− 0.240990i
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	0.828427i	0.0351965i
$555$	0	0
$556$	−13.3726	−0.567124
$557$	7.79899i	0.330454i	0.986256	+	0.165227i	$0.0528356\pi$
$557$	7.79899i	0.330454i	−0.986256	+	0.165227i	$0.947164\pi$
$558$	0	0
$559$	0	0
$560$	24.0000	1.01419
$561$	0	0
$562$	8.76955	0.369921
$563$	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
$565$	− 15.0294i	− 0.632293i
$566$	− 12.0000i	− 0.504398i
$567$	0	0
$568$	−3.17157	−0.133076
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	0	0
$571$	12.9706	0.542801	0.271401	−	0.962466i	$-0.412513\pi$
$571$	12.9706	0.542801	0.271401	+	0.962466i	$0.412513\pi$
$572$	0	0
$573$	0	0
$574$	− 6.05887i	− 0.252893i
$575$	12.0000	0.500435
$576$	0	0
$577$	31.9411i	1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$577$	31.9411i	1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$578$	17.2426i	0.717199i
$579$	0	0
$580$	10.3431i	0.429476i
$581$	−10.3431	−0.429106
$582$	0	0
$583$	4.00000i	0.165663i
$584$	−0.544156	−0.0225173
$585$	0	0
$586$	0.887302	0.0366541
$587$	− 10.9706i	− 0.452804i	−0.974034	−	0.226402i	$-0.927304\pi$
$587$	− 10.9706i	− 0.452804i	0.974034	−	0.226402i	$-0.0726962\pi$
$588$	0	0
$589$	−3.31371	−0.136539
$590$	− 8.97056i	− 0.369312i
$591$	0	0
$592$	− 22.9706i	− 0.944084i
$593$	20.4853i	0.841230i	0.907239	+	0.420615i	$0.138186\pi$
$593$	20.4853i	0.841230i	−0.907239	+	0.420615i	$0.861814\pi$
$594$	0	0
$595$	61.2548	2.51120
$596$	− 16.7696i	− 0.686908i
$597$	0	0
$598$	0	0
$599$	23.3137	0.952572	0.476286	−	0.879290i	$-0.341983\pi$
$599$	23.3137	0.952572	0.476286	+	0.879290i	$0.341983\pi$
$600$	0	0
$601$	−0.627417	−0.0255929	−0.0127964	−	0.999918i	$-0.504073\pi$
$601$	−0.627417	−0.0255929	−0.0127964	+	0.999918i	$0.504073\pi$
$602$	−1.94113	−0.0791144
$603$	0	0
$604$	− 6.42641i	− 0.261487i
$605$	− 19.7990i	− 0.804943i
$606$	0	0
$607$	41.9411	1.70234	0.851169	−	0.524892i	$-0.175894\pi$
$607$	41.9411	1.70234	0.851169	+	0.524892i	$0.175894\pi$
$608$	−12.4853	−0.506345
$609$	0	0
$610$	15.5980	0.631544
$611$	0	0
$612$	0	0
$613$	47.6569i	1.92484i	0.271561	+	0.962421i	$0.412460\pi$
$613$	47.6569i	1.92484i	−0.271561	+	0.962421i	$0.587540\pi$
$614$	9.45584	0.381607
$615$	0	0
$616$	− 8.97056i	− 0.361434i
$617$	− 34.8284i	− 1.40214i	−0.713093	−	0.701070i	$-0.752706\pi$
$617$	− 34.8284i	− 1.40214i	0.713093	−	0.701070i	$-0.247294\pi$
$618$	0	0
$619$	23.7990i	0.956562i	0.878207	+	0.478281i	$0.158740\pi$
$619$	23.7990i	0.956562i	−0.878207	+	0.478281i	$0.841260\pi$
$620$	6.05887	0.243330
$621$	0	0
$622$	− 4.40202i	− 0.176505i
$623$	41.9411	1.68034
$624$	0	0
$625$	−31.0000	−1.24000
$626$	2.48528i	0.0993318i
$627$	0	0
$628$	−18.2843	−0.729622
$629$	− 58.6274i	− 2.33763i
$630$	0	0
$631$	43.1127i	1.71629i	0.513408	+	0.858145i	$0.328383\pi$
$631$	43.1127i	1.71629i	−0.513408	+	0.858145i	$0.671617\pi$
$632$	− 17.9411i	− 0.713660i
$633$	0	0
$634$	−3.51472	−0.139587
$635$	16.0000i	0.634941i
$636$	0	0
$637$	0	0
$638$	1.65685	0.0655955
$639$	0	0
$640$	29.8579	1.18024
$641$	30.2843	1.19616	0.598078	−	0.801438i	$-0.295931\pi$
$641$	30.2843	1.19616	0.598078	+	0.801438i	$0.295931\pi$
$642$	0	0
$643$	− 22.8284i	− 0.900265i	−0.892962	−	0.450133i	$-0.851377\pi$
$643$	− 22.8284i	− 0.900265i	0.892962	−	0.450133i	$-0.148623\pi$
$644$	− 20.6863i	− 0.815154i
$645$	0	0
$646$	−8.97056	−0.352942
$647$	11.3137	0.444788	0.222394	−	0.974957i	$-0.428613\pi$
$647$	11.3137	0.444788	0.222394	+	0.974957i	$0.428613\pi$
$648$	0	0
$649$	15.3137	0.601116
$650$	0	0
$651$	0	0
$652$	34.4264i	1.34824i
$653$	25.3137	0.990602	0.495301	−	0.868721i	$-0.335058\pi$
$653$	25.3137	0.990602	0.495301	+	0.868721i	$0.335058\pi$
$654$	0	0
$655$	− 22.6274i	− 0.884126i
$656$	15.5147i	0.605748i
$657$	0	0
$658$	− 13.6569i	− 0.532400i
$659$	47.3137	1.84308	0.921540	−	0.388283i	$-0.126932\pi$
$659$	47.3137	1.84308	0.921540	+	0.388283i	$0.126932\pi$
$660$	0	0
$661$	− 34.9706i	− 1.36020i	−0.733121	−	0.680099i	$-0.761937\pi$
$661$	− 34.9706i	− 1.36020i	0.733121	−	0.680099i	$-0.238063\pi$
$662$	10.8284	0.420859
$663$	0	0
$664$	−5.79899	−0.225044
$665$	22.6274i	0.877454i
$666$	0	0
$667$	8.00000	0.309761
$668$	6.68629i	0.258700i
$669$	0	0
$670$	− 8.00000i	− 0.309067i
$671$	26.6274i	1.02794i
$672$	0	0
$673$	−16.6274	−0.640940	−0.320470	−	0.947259i	$-0.603841\pi$
$673$	−16.6274	−0.640940	−0.320470	+	0.947259i	$0.603841\pi$
$674$	− 3.85786i	− 0.148599i
$675$	0	0
$676$	0	0
$677$	−26.6863	−1.02564	−0.512819	−	0.858497i	$-0.671399\pi$
$677$	−26.6863	−1.02564	−0.512819	+	0.858497i	$0.671399\pi$
$678$	0	0
$679$	10.3431	0.396934
$680$	34.3431	1.31700
$681$	0	0
$682$	− 0.970563i	− 0.0371648i
$683$	− 47.9411i	− 1.83442i	−0.398408	−	0.917208i	$-0.630437\pi$
$683$	− 47.9411i	− 1.83442i	0.398408	−	0.917208i	$-0.369563\pi$
$684$	0	0
$685$	30.6274	1.17021
$686$	−7.02944	−0.268385
$687$	0	0
$688$	4.97056	0.189501
$689$	0	0
$690$	0	0
$691$	5.85786i	0.222844i	0.993773	+	0.111422i	$0.0355405\pi$
$691$	5.85786i	0.222844i	−0.993773	+	0.111422i	$0.964460\pi$
$692$	−21.3137	−0.810226
$693$	0	0
$694$	3.59798i	0.136577i
$695$	20.6863i	0.784676i
$696$	0	0
$697$	39.5980i	1.49988i
$698$	−1.51472	−0.0573329
$699$	0	0
$700$	− 15.5147i	− 0.586401i
$701$	5.02944	0.189959	0.0949796	−	0.995479i	$-0.469721\pi$
$701$	5.02944	0.189959	0.0949796	+	0.995479i	$0.469721\pi$
$702$	0	0
$703$	21.6569	0.816804
$704$	8.34315i	0.314444i
$705$	0	0
$706$	13.8579	0.521548
$707$	21.6569i	0.814490i
$708$	0	0
$709$	− 4.62742i	− 0.173786i	−0.996218	−	0.0868931i	$-0.972306\pi$
$709$	− 4.62742i	− 0.173786i	0.996218	−	0.0868931i	$-0.0276939\pi$
$710$	2.34315i	0.0879367i
$711$	0	0
$712$	23.5147	0.881251
$713$	− 4.68629i	− 0.175503i
$714$	0	0
$715$	0	0
$716$	−42.6274	−1.59306
$717$	0	0
$718$	14.4853	0.540586
$719$	−29.9411	−1.11662	−0.558308	−	0.829634i	$-0.688549\pi$
$719$	−29.9411	−1.11662	−0.558308	+	0.829634i	$0.688549\pi$
$720$	0	0
$721$	− 6.62742i	− 0.246818i
$722$	4.55635i	0.169570i
$723$	0	0
$724$	−25.5980	−0.951341
$725$	6.00000	0.222834
$726$	0	0
$727$	10.3431	0.383606	0.191803	−	0.981433i	$-0.438567\pi$
$727$	10.3431	0.383606	0.191803	+	0.981433i	$0.438567\pi$
$728$	0	0
$729$	0	0
$730$	0.402020i	0.0148794i
$731$	12.6863	0.469219
$732$	0	0
$733$	36.6274i	1.35286i	0.736505	+	0.676432i	$0.236475\pi$
$733$	36.6274i	1.35286i	−0.736505	+	0.676432i	$0.763525\pi$
$734$	− 9.94113i	− 0.366934i
$735$	0	0
$736$	− 17.6569i	− 0.650840i
$737$	13.6569	0.503057
$738$	0	0
$739$	− 18.1421i	− 0.667369i	−0.942685	−	0.333685i	$-0.891708\pi$
$739$	− 18.1421i	− 0.667369i	0.942685	−	0.333685i	$-0.108292\pi$
$740$	−39.5980	−1.45565
$741$	0	0
$742$	−2.34315	−0.0860196
$743$	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
$745$	−25.9411	−0.950409
$746$	4.14214i	0.151654i
$747$	0	0
$748$	28.0000i	1.02378i
$749$	32.0000i	1.16925i
$750$	0	0
$751$	0.970563	0.0354163	0.0177082	−	0.999843i	$-0.494363\pi$
$751$	0.970563	0.0354163	0.0177082	+	0.999843i	$0.494363\pi$
$752$	34.9706i	1.27525i
$753$	0	0
$754$	0	0
$755$	−9.94113	−0.361795
$756$	0	0
$757$	51.9411	1.88783	0.943916	−	0.330185i	$-0.107111\pi$
$757$	51.9411	1.88783	0.943916	+	0.330185i	$0.107111\pi$
$758$	0.201010	0.00730102
$759$	0	0
$760$	12.6863i	0.460180i
$761$	− 32.4853i	− 1.17759i	−0.808282	−	0.588795i	$-0.799602\pi$
$761$	− 32.4853i	− 1.17759i	0.808282	−	0.588795i	$-0.200398\pi$
$762$	0	0
$763$	15.0294	0.544102
$764$	−6.05887	−0.219202
$765$	0	0
$766$	−12.8284	−0.463510
$767$	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	$-0.273472\pi$
$770$	−6.62742	−0.238836
$771$	0	0
$772$	9.71573i	0.349677i
$773$	34.1421i	1.22801i	0.789303	+	0.614004i	$0.210442\pi$
$773$	34.1421i	1.22801i	−0.789303	+	0.614004i	$0.789558\pi$
$774$	0	0
$775$	− 3.51472i	− 0.126252i
$776$	5.79899	0.208172
$777$	0	0
$778$	− 11.1716i	− 0.400520i
$779$	−14.6274	−0.524082
$780$	0	0
$781$	−4.00000	−0.143131
$782$	− 12.6863i	− 0.453661i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	28.2843i	1.00951i
$786$	0	0
$787$	− 40.7696i	− 1.45328i	−0.687020	−	0.726639i	$-0.741081\pi$
$787$	− 40.7696i	− 1.45328i	0.687020	−	0.726639i	$-0.258919\pi$
$788$	0.887302i	0.0316088i
$789$	0	0
$790$	−13.2548	−0.471586
$791$	15.0294i	0.534385i
$792$	0	0
$793$	0	0
$794$	−12.8284	−0.455264
$795$	0	0
$796$	−39.5980	−1.40351
$797$	24.3431	0.862278	0.431139	−	0.902285i	$-0.358112\pi$
$797$	24.3431	0.862278	0.431139	+	0.902285i	$0.358112\pi$
$798$	0	0
$799$	89.2548i	3.15761i
$800$	− 13.2426i	− 0.468198i
$801$	0	0
$802$	10.8284	0.382365
$803$	−0.686292	−0.0242187
$804$	0	0
$805$	−32.0000	−1.12785
$806$	0	0
$807$	0	0
$808$	12.1421i	0.427159i
$809$	−18.6863	−0.656975	−0.328488	−	0.944508i	$-0.606539\pi$
$809$	−18.6863	−0.656975	−0.328488	+	0.944508i	$0.606539\pi$
$810$	0	0
$811$	30.1421i	1.05843i	0.848487	+	0.529217i	$0.177514\pi$
$811$	30.1421i	1.05843i	−0.848487	+	0.529217i	$0.822486\pi$
$812$	− 10.3431i	− 0.362973i
$813$	0	0
$814$	6.34315i	0.222327i
$815$	53.2548	1.86544
$816$	0	0
$817$	4.68629i	0.163953i
$818$	−14.4853	−0.506466
$819$	0	0
$820$	26.7452	0.933982
$821$	− 23.7990i	− 0.830590i	−0.909687	−	0.415295i	$-0.863678\pi$
$821$	− 23.7990i	− 0.830590i	0.909687	−	0.415295i	$-0.136322\pi$
$822$	0	0
$823$	−15.0294	−0.523893	−0.261947	−	0.965082i	$-0.584364\pi$
$823$	−15.0294	−0.523893	−0.261947	+	0.965082i	$0.584364\pi$
$824$	− 3.71573i	− 0.129444i
$825$	0	0
$826$	8.97056i	0.312126i
$827$	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
$829$	17.3137	0.601330	0.300665	−	0.953730i	$-0.402791\pi$
$829$	17.3137	0.601330	0.300665	+	0.953730i	$0.402791\pi$
$830$	4.28427i	0.148709i
$831$	0	0
$832$	0	0
$833$	−7.65685	−0.265294
$834$	0	0
$835$	10.3431	0.357939
$836$	−10.3431	−0.357725
$837$	0	0
$838$	− 6.05887i	− 0.209300i
$839$	− 43.2548i	− 1.49332i	−0.665204	−	0.746661i	$-0.731656\pi$
$839$	− 43.2548i	− 1.49332i	0.665204	−	0.746661i	$-0.268344\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	15.4558	0.532644
$843$	0	0
$844$	−21.9411	−0.755245
$845$	0	0
$846$	0	0
$847$	19.7990i	0.680301i
$848$	6.00000	0.206041
$849$	0	0
$850$	− 9.51472i	− 0.326352i
$851$	30.6274i	1.04989i
$852$	0	0
$853$	3.65685i	0.125208i	0.998038	+	0.0626042i	$0.0199406\pi$
$853$	3.65685i	0.125208i	−0.998038	+	0.0626042i	$0.980059\pi$
$854$	−15.5980	−0.533752
$855$	0	0
$856$	17.9411i	0.613215i
$857$	−49.5980	−1.69423	−0.847117	−	0.531406i	$-0.821664\pi$
$857$	−49.5980	−1.69423	−0.847117	+	0.531406i	$0.821664\pi$
$858$	0	0
$859$	−0.686292	−0.0234160	−0.0117080	−	0.999931i	$-0.503727\pi$
$859$	−0.686292	−0.0234160	−0.0117080	+	0.999931i	$0.503727\pi$
$860$	− 8.56854i	− 0.292185i
$861$	0	0
$862$	−3.45584	−0.117707
$863$	− 28.3431i	− 0.964812i	−0.875948	−	0.482406i	$-0.839763\pi$
$863$	− 28.3431i	− 0.964812i	0.875948	−	0.482406i	$-0.160237\pi$
$864$	0	0
$865$	32.9706i	1.12103i
$866$	8.82843i	0.300002i
$867$	0	0
$868$	−6.05887	−0.205652
$869$	− 22.6274i	− 0.767583i
$870$	0	0
$871$	0	0
$872$	8.42641	0.285354
$873$	0	0
$874$	4.68629	0.158516
$875$	16.0000	0.540899
$876$	0	0
$877$	− 42.2843i	− 1.42784i	−0.700228	−	0.713919i	$-0.746918\pi$
$877$	− 42.2843i	− 1.42784i	0.700228	−	0.713919i	$-0.253082\pi$
$878$	− 7.02944i	− 0.237232i
$879$	0	0
$880$	16.9706	0.572078
$881$	−25.5980	−0.862418	−0.431209	−	0.902252i	$-0.641913\pi$
$881$	−25.5980	−0.862418	−0.431209	+	0.902252i	$0.641913\pi$
$882$	0	0
$883$	−27.5980	−0.928746	−0.464373	−	0.885640i	$-0.653720\pi$
$883$	−27.5980	−0.928746	−0.464373	+	0.885640i	$0.653720\pi$
$884$	0	0
$885$	0	0
$886$	10.7452i	0.360991i
$887$	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
$890$	− 17.3726i	− 0.582330i
$891$	0	0
$892$	22.8284i	0.764352i
$893$	−32.9706	−1.10332
$894$	0	0
$895$	65.9411i	2.20417i
$896$	−29.8579	−0.997481
$897$	0	0
$898$	−13.1716	−0.439541
$899$	− 2.34315i	− 0.0781483i
$900$	0	0
$901$	15.3137	0.510174
$902$	− 4.28427i	− 0.142651i
$903$	0	0
$904$	8.42641i	0.280258i
$905$	39.5980i	1.31628i
$906$	0	0
$907$	12.9706	0.430680	0.215340	−	0.976539i	$-0.430914\pi$
$907$	12.9706	0.430680	0.215340	+	0.976539i	$0.430914\pi$
$908$	− 31.6569i	− 1.05057i
$909$	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
$913$	−7.31371	−0.242048
$914$	−3.17157	−0.104906
$915$	0	0
$916$	− 2.40202i	− 0.0793650i
$917$	22.6274i	0.747223i
$918$	0	0
$919$	3.31371	0.109309	0.0546546	−	0.998505i	$-0.482594\pi$
$919$	3.31371	0.109309	0.0546546	+	0.998505i	$0.482594\pi$
$920$	−17.9411	−0.591501
$921$	0	0
$922$	−2.14214	−0.0705475
$923$	0	0
$924$	0	0
$925$	22.9706i	0.755267i
$926$	10.1421	0.333291
$927$	0	0
$928$	− 8.82843i	− 0.289807i
$929$	11.7990i	0.387112i	0.981089	+	0.193556i	$0.0620022\pi$
$929$	11.7990i	0.387112i	−0.981089	+	0.193556i	$0.937998\pi$
$930$	0	0
$931$	− 2.82843i	− 0.0926980i
$932$	12.7452	0.417482
$933$	0	0
$934$	− 3.31371i	− 0.108428i
$935$	43.3137	1.41651
$936$	0	0
$937$	−21.3137	−0.696289	−0.348144	−	0.937441i	$-0.613188\pi$
$937$	−21.3137	−0.696289	−0.348144	+	0.937441i	$0.613188\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	60.2843	1.96626
$941$	34.1421i	1.11300i	0.830847	+	0.556501i	$0.187856\pi$
$941$	34.1421i	1.11300i	−0.830847	+	0.556501i	$0.812144\pi$
$942$	0	0
$943$	− 20.6863i	− 0.673638i
$944$	− 22.9706i	− 0.747628i
$945$	0	0
$946$	−1.37258	−0.0446265
$947$	21.0294i	0.683365i	0.939815	+	0.341682i	$0.110997\pi$
$947$	21.0294i	0.683365i	−0.939815	+	0.341682i	$0.889003\pi$
$948$	0	0
$949$	0	0
$950$	3.51472	0.114033
$951$	0	0
$952$	−34.3431	−1.11307
$953$	40.3431	1.30684	0.653421	−	0.756994i	$-0.273333\pi$
$953$	40.3431	1.30684	0.653421	+	0.756994i	$0.273333\pi$
$954$	0	0
$955$	9.37258i	0.303290i
$956$	3.65685i	0.118271i
$957$	0	0
$958$	10.4853	0.338764
$959$	−30.6274	−0.989011
$960$	0	0
$961$	29.6274	0.955723
$962$	0	0
$963$	0	0
$964$	0.627417i	0.0202077i
$965$	15.0294	0.483815
$966$	0	0
$967$	− 18.1421i	− 0.583412i	−0.956508	−	0.291706i	$-0.905777\pi$
$967$	− 18.1421i	− 0.583412i	0.956508	−	0.291706i	$-0.0942228\pi$
$968$	11.1005i	0.356784i
$969$	0	0
$970$	− 4.28427i	− 0.137560i
$971$	15.3137	0.491440	0.245720	−	0.969341i	$-0.420976\pi$
$971$	15.3137	0.491440	0.245720	+	0.969341i	$0.420976\pi$
$972$	0	0
$973$	− 20.6863i	− 0.663172i
$974$	−3.23045	−0.103510
$975$	0	0
$976$	39.9411	1.27848
$977$	42.1421i	1.34825i	0.738619	+	0.674123i	$0.235478\pi$
$977$	42.1421i	1.34825i	−0.738619	+	0.674123i	$0.764522\pi$
$978$	0	0
$979$	29.6569	0.947837
$980$	5.17157i	0.165200i
$981$	0	0
$982$	12.6863i	0.404836i
$983$	− 25.3137i	− 0.807382i	−0.914895	−	0.403691i	$-0.867727\pi$
$983$	− 25.3137i	− 0.807382i	0.914895	−	0.403691i	$-0.132273\pi$
$984$	0	0
$985$	1.37258	0.0437341
$986$	− 6.34315i	− 0.202007i
$987$	0	0
$988$	0	0
$989$	−6.62742	−0.210740
$990$	0	0
$991$	4.68629	0.148865	0.0744325	−	0.997226i	$-0.476285\pi$
$991$	4.68629	0.148865	0.0744325	+	0.997226i	$0.476285\pi$
$992$	−5.17157	−0.164198
$993$	0	0
$994$	− 2.34315i	− 0.0743201i
$995$	61.2548i	1.94191i
$996$	0	0
$997$	−39.2548	−1.24321	−0.621607	−	0.783330i	$-0.713520\pi$
$997$	−39.2548	−1.24321	−0.621607	+	0.783330i	$0.713520\pi$
$998$	10.8284	0.342768
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.j.1351.3		4
3.2	odd	2		507.2.b.e.337.2		4
13.5	odd	4		1521.2.a.f.1.2		2
13.8	odd	4		117.2.a.c.1.1		2
13.12	even	2	inner	1521.2.b.j.1351.2		4
39.2	even	12		507.2.e.d.22.2		4
39.5	even	4		507.2.a.h.1.1		2
39.8	even	4		39.2.a.b.1.2	✓	2
39.11	even	12		507.2.e.h.22.1		4
39.17	odd	6		507.2.j.f.361.3		8
39.20	even	12		507.2.e.h.484.1		4
39.23	odd	6		507.2.j.f.316.2		8
39.29	odd	6		507.2.j.f.316.3		8
39.32	even	12		507.2.e.d.484.2		4
39.35	odd	6		507.2.j.f.361.2		8
39.38	odd	2		507.2.b.e.337.3		4
52.47	even	4		1872.2.a.w.1.2		2
65.8	even	4		2925.2.c.u.2224.3		4
65.34	odd	4		2925.2.a.v.1.2		2
65.47	even	4		2925.2.c.u.2224.2		4
91.34	even	4		5733.2.a.u.1.1		2
104.21	odd	4		7488.2.a.cl.1.1		2
104.99	even	4		7488.2.a.co.1.1		2
117.34	odd	12		1053.2.e.e.352.2		4
117.47	even	12		1053.2.e.m.352.1		4
117.86	even	12		1053.2.e.m.703.1		4
117.112	odd	12		1053.2.e.e.703.2		4
156.47	odd	4		624.2.a.k.1.1		2
156.83	odd	4		8112.2.a.bm.1.2		2
195.8	odd	4		975.2.c.h.274.2		4
195.47	odd	4		975.2.c.h.274.3		4
195.164	even	4		975.2.a.l.1.1		2
273.125	odd	4		1911.2.a.h.1.2		2
312.125	even	4		2496.2.a.bf.1.2		2
312.203	odd	4		2496.2.a.bi.1.2		2
429.164	odd	4		4719.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.2	✓	2	39.8	even	4
117.2.a.c.1.1		2	13.8	odd	4
507.2.a.h.1.1		2	39.5	even	4
507.2.b.e.337.2		4	3.2	odd	2
507.2.b.e.337.3		4	39.38	odd	2
507.2.e.d.22.2		4	39.2	even	12
507.2.e.d.484.2		4	39.32	even	12
507.2.e.h.22.1		4	39.11	even	12
507.2.e.h.484.1		4	39.20	even	12
507.2.j.f.316.2		8	39.23	odd	6
507.2.j.f.316.3		8	39.29	odd	6
507.2.j.f.361.2		8	39.35	odd	6
507.2.j.f.361.3		8	39.17	odd	6
624.2.a.k.1.1		2	156.47	odd	4
975.2.a.l.1.1		2	195.164	even	4
975.2.c.h.274.2		4	195.8	odd	4
975.2.c.h.274.3		4	195.47	odd	4
1053.2.e.e.352.2		4	117.34	odd	12
1053.2.e.e.703.2		4	117.112	odd	12
1053.2.e.m.352.1		4	117.47	even	12
1053.2.e.m.703.1		4	117.86	even	12
1521.2.a.f.1.2		2	13.5	odd	4
1521.2.b.j.1351.2		4	13.12	even	2	inner
1521.2.b.j.1351.3		4	1.1	even	1	trivial
1872.2.a.w.1.2		2	52.47	even	4
1911.2.a.h.1.2		2	273.125	odd	4
2496.2.a.bf.1.2		2	312.125	even	4
2496.2.a.bi.1.2		2	312.203	odd	4
2925.2.a.v.1.2		2	65.34	odd	4
2925.2.c.u.2224.2		4	65.47	even	4
2925.2.c.u.2224.3		4	65.8	even	4
4719.2.a.p.1.1		2	429.164	odd	4
5733.2.a.u.1.1		2	91.34	even	4
7488.2.a.cl.1.1		2	104.21	odd	4
7488.2.a.co.1.1		2	104.99	even	4
8112.2.a.bm.1.2		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)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{5} +2.82843i q^{7} +1.58579i q^{8} +O(q^{10})\)	\(q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{5} +2.82843i q^{7} +1.58579i q^{8} +1.17157 q^{10} +2.00000i q^{11} -1.17157 q^{14} +3.00000 q^{16} +7.65685 q^{17} +2.82843i q^{19} -5.17157i q^{20} -0.828427 q^{22} -4.00000 q^{23} -3.00000 q^{25} +5.17157i q^{28} -2.00000 q^{29} +1.17157i q^{31} +4.41421i q^{32} +3.17157i q^{34} +8.00000 q^{35} -7.65685i q^{37} -1.17157 q^{38} +4.48528 q^{40} +5.17157i q^{41} +1.65685 q^{43} +3.65685i q^{44} -1.65685i q^{46} +11.6569i q^{47} -1.00000 q^{49} -1.24264i q^{50} +2.00000 q^{53} +5.65685 q^{55} -4.48528 q^{56} -0.828427i q^{58} -7.65685i q^{59} +13.3137 q^{61} -0.485281 q^{62} +4.17157 q^{64} -6.82843i q^{67} +14.0000 q^{68} +3.31371i q^{70} +2.00000i q^{71} +0.343146i q^{73} +3.17157 q^{74} +5.17157i q^{76} -5.65685 q^{77} -11.3137 q^{79} -8.48528i q^{80} -2.14214 q^{82} +3.65685i q^{83} -21.6569i q^{85} +0.686292i q^{86} -3.17157 q^{88} -14.8284i q^{89} -7.31371 q^{92} -4.82843 q^{94} +8.00000 q^{95} -3.65685i q^{97} -0.414214i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more