LMFDB - Embedded newform 720.7.c.a.449.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,7,Mod(449,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.449"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	720.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4320] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$165.638940206$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 630x^{10} + 143853x^{8} - 14514820x^{6} + 700911828x^{4} - 15238290240x^{2} + 141093389376 $ x^12 - 630x^10 + 143853x^8 - 14514820x^6 + 700911828x^4 - 15238290240*x^2 + 141093389376
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{20}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.9
Root			$-8.37795 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	720.449
Dual form			720.7.c.a.449.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(60.3283 - 109.478i) q^{5} -270.176i q^{7} +1187.10i q^{11} +526.105i q^{13} -6589.09 q^{17} +9088.18 q^{19} +8922.54 q^{23} +(-8346.00 - 13209.3i) q^{25} -7731.97i q^{29} -3114.29 q^{31} +(-29578.4 - 16299.2i) q^{35} +75996.2i q^{37} +56742.3i q^{41} +126636. i q^{43} +182063. q^{47} +44654.2 q^{49} -31815.2 q^{53} +(129962. + 71615.8i) q^{55} +319642. i q^{59} -359529. q^{61} +(57597.1 + 31739.0i) q^{65} -487383. i q^{67} +370686. i q^{71} +323083. i q^{73} +320726. q^{77} +66000.9 q^{79} +7403.28 q^{83} +(-397509. + 721363. i) q^{85} -991440. i q^{89} +142141. q^{91} +(548274. - 994959. i) q^{95} -1.52751e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4320 q^{19} - 63384 q^{25} - 60192 q^{31} + 711516 q^{49} + 104112 q^{55} - 449784 q^{61} - 4324608 q^{79} - 3305772 q^{85} - 631152 q^{91}+O(q^{100}) $ 12 * q - 4320 * q^19 - 63384 * q^25 - 60192 * q^31 + 711516 * q^49 + 104112 * q^55 - 449784 * q^61 - 4324608 * q^79 - 3305772 * q^85 - 631152 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	60.3283	−	109.478i	0.482626	−	0.875826i
$5$	60.3283	−	109.478i	0.482626	−	0.875826i
$6$		0			0
$7$		−	270.176i		−	0.787684i	−0.919178	−	0.393842i	$-0.871146\pi$
$7$		−	270.176i		−	0.787684i	0.919178	−	0.393842i	$-0.128854\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1187.10i			0.891888i	0.895061	+	0.445944i	$0.147132\pi$
$11$			1187.10i			0.891888i	−0.895061	+	0.445944i	$0.852868\pi$
$12$		0			0
$13$			526.105i			0.239465i	0.992806	+	0.119733i	$0.0382037\pi$
$13$			526.105i			0.239465i	−0.992806	+	0.119733i	$0.961796\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6589.09			−1.34115			−0.670577	−	0.741840i	$-0.733954\pi$
$17$	−6589.09			−1.34115			−0.670577	+	0.741840i	$0.733954\pi$
$18$		0			0
$19$	9088.18			1.32500			0.662500	−	0.749062i	$-0.269495\pi$
$19$	9088.18			1.32500			0.662500	+	0.749062i	$0.269495\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8922.54			0.733339			0.366670	−	0.930351i	$-0.380498\pi$
$23$	8922.54			0.733339			0.366670	+	0.930351i	$0.380498\pi$
$24$		0			0
$25$	−8346.00	−	13209.3i	−0.534144	−	0.845394i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7731.97i		−	0.317027i	−0.987357	−	0.158514i	$-0.949330\pi$
$29$		−	7731.97i		−	0.317027i	0.987357	−	0.158514i	$-0.0506702\pi$
$30$		0			0
$31$	−3114.29			−0.104538			−0.0522689	−	0.998633i	$-0.516645\pi$
$31$	−3114.29			−0.104538			−0.0522689	+	0.998633i	$0.516645\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−29578.4	−	16299.2i	−0.689874	−	0.380157i
$36$		0			0
$37$			75996.2i			1.50033i	0.661251	+	0.750165i	$0.270026\pi$
$37$			75996.2i			1.50033i	−0.661251	+	0.750165i	$0.729974\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			56742.3i			0.823295i	0.911343	+	0.411648i	$0.135047\pi$
$41$			56742.3i			0.823295i	−0.911343	+	0.411648i	$0.864953\pi$
$42$		0			0
$43$			126636.i			1.59277i	0.604790	+	0.796385i	$0.293257\pi$
$43$			126636.i			1.59277i	−0.604790	+	0.796385i	$0.706743\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	182063.			1.75359			0.876797	−	0.480861i	$-0.159676\pi$
$47$	182063.			1.75359			0.876797	+	0.480861i	$0.159676\pi$
$48$		0			0
$49$	44654.2			0.379554
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−31815.2			−0.213701			−0.106851	−	0.994275i	$-0.534077\pi$
$53$	−31815.2			−0.213701			−0.106851	+	0.994275i	$0.534077\pi$
$54$		0			0
$55$	129962.	+	71615.8i	0.781139	+	0.430448i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			319642.i			1.55635i	0.628048	+	0.778175i	$0.283854\pi$
$59$			319642.i			1.55635i	−0.628048	+	0.778175i	$0.716146\pi$
$60$		0			0
$61$	−359529.			−1.58396			−0.791981	−	0.610546i	$-0.790950\pi$
$61$	−359529.			−1.58396			−0.791981	+	0.610546i	$0.790950\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	57597.1	+	31739.0i	0.209730	+	0.115572i
$66$		0			0
$67$		−	487383.i		−	1.62049i	−0.586093	−	0.810244i	$-0.699335\pi$
$67$		−	487383.i		−	1.62049i	0.586093	−	0.810244i	$-0.300665\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			370686.i			1.03569i	0.855473	+	0.517847i	$0.173266\pi$
$71$			370686.i			1.03569i	−0.855473	+	0.517847i	$0.826734\pi$
$72$		0			0
$73$			323083.i			0.830511i	0.909705	+	0.415255i	$0.136308\pi$
$73$			323083.i			0.830511i	−0.909705	+	0.415255i	$0.863692\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	320726.			0.702525
$78$		0			0
$79$	66000.9			0.133866			0.0669328	−	0.997757i	$-0.478679\pi$
$79$	66000.9			0.133866			0.0669328	+	0.997757i	$0.478679\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7403.28			0.0129476			0.00647381	−	0.999979i	$-0.497939\pi$
$83$	7403.28			0.0129476			0.00647381	+	0.999979i	$0.497939\pi$
$84$		0			0
$85$	−397509.	+	721363.i	−0.647276	+	1.17462i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	991440.i		−	1.40636i	−0.711012	−	0.703180i	$-0.751763\pi$
$89$		−	991440.i		−	1.40636i	0.711012	−	0.703180i	$-0.248237\pi$
$90$		0			0
$91$	142141.			0.188623
$92$		0			0
$93$		0			0
$94$		0			0
$95$	548274.	−	994959.i	0.639480	−	1.16047i
$96$		0			0
$97$		−	1.52751e6i		−	1.67367i	−0.547458	−	0.836833i	$-0.684404\pi$
$97$		−	1.52751e6i		−	1.67367i	0.547458	−	0.836833i	$-0.315596\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			452341.i			0.439037i	0.975608	+	0.219519i	$0.0704487\pi$
$101$			452341.i			0.439037i	−0.975608	+	0.219519i	$0.929551\pi$
$102$		0			0
$103$		−	104961.i		−	0.0960542i	−0.998846	−	0.0480271i	$-0.984707\pi$
$103$		−	104961.i		−	0.0960542i	0.998846	−	0.0480271i	$-0.0152934\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−657099.			−0.536388			−0.268194	−	0.963365i	$-0.586427\pi$
$107$	−657099.			−0.536388			−0.268194	+	0.963365i	$0.586427\pi$
$108$		0			0
$109$	1.65657e6			1.27918			0.639589	−	0.768717i	$-0.279105\pi$
$109$	1.65657e6			1.27918			0.639589	+	0.768717i	$0.279105\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.02621e6			0.711216			0.355608	−	0.934635i	$-0.384274\pi$
$113$	1.02621e6			0.711216			0.355608	+	0.934635i	$0.384274\pi$
$114$		0			0
$115$	538281.	−	976824.i	0.353929	−	0.642278i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			1.78021e6i			1.05641i
$120$		0			0
$121$	362349.			0.204536
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.94963e6	+	116813.i	−0.998210	+	0.0598083i
$126$		0			0
$127$		−	375925.i		−	0.183523i	−0.995781	−	0.0917615i	$-0.970750\pi$
$127$		−	375925.i		−	0.183523i	0.995781	−	0.0917615i	$-0.0292497\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	1.42739e6i		−	0.634934i	−0.948269	−	0.317467i	$-0.897168\pi$
$131$		−	1.42739e6i		−	0.634934i	0.948269	−	0.317467i	$-0.102832\pi$
$132$		0			0
$133$		−	2.45540e6i		−	1.04368i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.76280e6			−1.07445			−0.537226	−	0.843438i	$-0.680528\pi$
$137$	−2.76280e6			−1.07445			−0.537226	+	0.843438i	$0.680528\pi$
$138$		0			0
$139$	2.33540e6			0.869596			0.434798	−	0.900528i	$-0.356820\pi$
$139$	2.33540e6			0.869596			0.434798	+	0.900528i	$0.356820\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−624540.			−0.213576
$144$		0			0
$145$	−846484.	−	466457.i	−0.277661	−	0.153006i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.57895e6i			1.68653i	0.537499	+	0.843265i	$0.319369\pi$
$149$			5.57895e6i			1.68653i	−0.537499	+	0.843265i	$0.680631\pi$
$150$		0			0
$151$	187475.			0.0544520			0.0272260	−	0.999629i	$-0.491333\pi$
$151$	187475.			0.0544520			0.0272260	+	0.999629i	$0.491333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−187879.	+	340947.i	−0.0504527	+	0.0915570i
$156$		0			0
$157$			2.11041e6i			0.545340i	0.962108	+	0.272670i	$0.0879066\pi$
$157$			2.11041e6i			0.545340i	−0.962108	+	0.272670i	$0.912093\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.41065e6i		−	0.577639i
$162$		0			0
$163$			4.41451e6i			1.01934i	0.860370	+	0.509671i	$0.170233\pi$
$163$			4.41451e6i			1.01934i	−0.860370	+	0.509671i	$0.829767\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	279043.			0.0599131			0.0299565	−	0.999551i	$-0.490463\pi$
$167$	279043.			0.0599131			0.0299565	+	0.999551i	$0.490463\pi$
$168$		0			0
$169$	4.55002e6			0.942656
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.97675e6			0.961187			0.480594	−	0.876943i	$-0.340421\pi$
$173$	4.97675e6			0.961187			0.480594	+	0.876943i	$0.340421\pi$
$174$		0			0
$175$	−3.56882e6	+	2.25488e6i	−0.665903	+	0.420736i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	3.25828e6i		−	0.568106i	−0.958809	−	0.284053i	$-0.908321\pi$
$179$		−	3.25828e6i		−	0.568106i	0.958809	−	0.284053i	$-0.0916791\pi$
$180$		0			0
$181$	1.15053e7			1.94027			0.970137	−	0.242559i	$-0.0779869\pi$
$181$	1.15053e7			1.94027			0.970137	+	0.242559i	$0.0779869\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.31994e6	+	4.58472e6i	1.31403	+	0.724098i
$186$		0			0
$187$		−	7.82193e6i		−	1.19616i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			6.07903e6i			0.872437i	0.899841	+	0.436218i	$0.143683\pi$
$191$			6.07903e6i			0.872437i	−0.899841	+	0.436218i	$0.856317\pi$
$192$		0			0
$193$		−	1.12931e7i		−	1.57088i	−0.618941	−	0.785438i	$-0.712438\pi$
$193$		−	1.12931e7i		−	1.57088i	0.618941	−	0.785438i	$-0.287562\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.46877e6			−0.453708			−0.226854	−	0.973929i	$-0.572844\pi$
$197$	−3.46877e6			−0.453708			−0.226854	+	0.973929i	$0.572844\pi$
$198$		0			0
$199$	5.80303e6			0.736369			0.368185	−	0.929753i	$-0.379979\pi$
$199$	5.80303e6			0.736369			0.368185	+	0.929753i	$0.379979\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.08899e6			−0.249717
$204$		0			0
$205$	6.21205e6	+	3.42317e6i	0.721064	+	0.397344i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.07886e7i			1.18175i
$210$		0			0
$211$	1.31986e6			0.140502			0.0702509	−	0.997529i	$-0.477620\pi$
$211$	1.31986e6			0.140502			0.0702509	+	0.997529i	$0.477620\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.38639e7	+	7.63975e6i	1.39499	+	0.768712i
$216$		0			0
$217$			841404.i			0.0823427i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	3.46655e6i		−	0.321160i
$222$		0			0
$223$		−	1.33313e6i		−	0.120215i	−0.998192	−	0.0601076i	$-0.980856\pi$
$223$		−	1.33313e6i		−	0.120215i	0.998192	−	0.0601076i	$-0.0191444\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.97225e6			−0.681559			−0.340779	−	0.940143i	$-0.610691\pi$
$227$	−7.97225e6			−0.681559			−0.340779	+	0.940143i	$0.610691\pi$
$228$		0			0
$229$	−1.11916e7			−0.931938			−0.465969	−	0.884801i	$-0.654294\pi$
$229$	−1.11916e7			−0.931938			−0.465969	+	0.884801i	$0.654294\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.33843e7			1.05810			0.529052	−	0.848589i	$-0.322548\pi$
$233$	1.33843e7			1.05810			0.529052	+	0.848589i	$0.322548\pi$
$234$		0			0
$235$	1.09836e7	−	1.99320e7i	0.846330	−	1.53584i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			3.17061e6i			0.232247i	0.993235	+	0.116123i	$0.0370468\pi$
$239$			3.17061e6i			0.232247i	−0.993235	+	0.116123i	$0.962953\pi$
$240$		0			0
$241$	3.36319e6			0.240270			0.120135	−	0.992758i	$-0.461667\pi$
$241$	3.36319e6			0.240270			0.120135	+	0.992758i	$0.461667\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.69391e6	−	4.88867e6i	0.183183	−	0.332424i
$246$		0			0
$247$			4.78134e6i			0.317291i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			2.64563e7i			1.67305i	0.547932	+	0.836523i	$0.315415\pi$
$251$			2.64563e7i			1.67305i	−0.547932	+	0.836523i	$0.684585\pi$
$252$		0			0
$253$			1.05920e7i			0.654056i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.28701e6			0.252554			0.126277	−	0.991995i	$-0.459697\pi$
$257$	4.28701e6			0.252554			0.126277	+	0.991995i	$0.459697\pi$
$258$		0			0
$259$	2.05323e7			1.18179
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.94305e7			1.61782			0.808910	−	0.587933i	$-0.200058\pi$
$263$	2.94305e7			1.61782			0.808910	+	0.587933i	$0.200058\pi$
$264$		0			0
$265$	−1.91936e6	+	3.48308e6i	−0.103138	+	0.187165i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			1.40558e7i			0.722100i	0.932547	+	0.361050i	$0.117582\pi$
$269$			1.40558e7i			0.722100i	−0.932547	+	0.361050i	$0.882418\pi$
$270$		0			0
$271$	2.69073e6			0.135196			0.0675978	−	0.997713i	$-0.478467\pi$
$271$	2.69073e6			0.135196			0.0675978	+	0.997713i	$0.478467\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.56808e7	−	9.90756e6i	0.753996	−	0.476396i
$276$		0			0
$277$		−	4.16627e7i		−	1.96024i	−0.198414	−	0.980118i	$-0.563579\pi$
$277$		−	4.16627e7i		−	1.96024i	0.198414	−	0.980118i	$-0.436421\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.07292e7i			0.483556i	0.970332	+	0.241778i	$0.0777305\pi$
$281$			1.07292e7i			0.483556i	−0.970332	+	0.241778i	$0.922269\pi$
$282$		0			0
$283$		−	5.50312e6i		−	0.242800i	−0.992604	−	0.121400i	$-0.961262\pi$
$283$		−	5.50312e6i		−	0.242800i	0.992604	−	0.121400i	$-0.0387385\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.53304e7			0.648496
$288$		0			0
$289$	1.92786e7			0.798696
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.23547e7			−0.491169			−0.245584	−	0.969375i	$-0.578980\pi$
$293$	−1.23547e7			−0.491169			−0.245584	+	0.969375i	$0.578980\pi$
$294$		0			0
$295$	3.49938e7	+	1.92834e7i	1.36309	+	0.751135i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			4.69419e6i			0.175609i
$300$		0			0
$301$	3.42140e7			1.25460
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.16898e7	+	3.93606e7i	−0.764461	+	1.38728i
$306$		0			0
$307$		−	4.10198e7i		−	1.41768i	−0.705370	−	0.708840i	$-0.749219\pi$
$307$		−	4.10198e7i		−	1.41768i	0.705370	−	0.708840i	$-0.250781\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			4.55195e7i			1.51327i	0.653837	+	0.756635i	$0.273158\pi$
$311$			4.55195e7i			1.51327i	−0.653837	+	0.756635i	$0.726842\pi$
$312$		0			0
$313$		−	1.47679e7i		−	0.481598i	−0.970575	−	0.240799i	$-0.922591\pi$
$313$		−	1.47679e7i		−	0.481598i	0.970575	−	0.240799i	$-0.0774095\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.38900e6			0.231957			0.115979	−	0.993252i	$-0.463000\pi$
$317$	7.38900e6			0.231957			0.115979	+	0.993252i	$0.463000\pi$
$318$		0			0
$319$	9.17865e6			0.282753
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.98829e7			−1.77703
$324$		0			0
$325$	6.94946e6	−	4.39087e6i	0.202442	−	0.127909i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	4.91891e7i		−	1.38128i
$330$		0			0
$331$	4.89533e7			1.34989			0.674944	−	0.737869i	$-0.264168\pi$
$331$	4.89533e7			1.34989			0.674944	+	0.737869i	$0.264168\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.33579e7	−	2.94030e7i	−1.41927	−	0.782090i
$336$		0			0
$337$			2.52313e7i			0.659250i	0.944112	+	0.329625i	$0.106922\pi$
$337$			2.52313e7i			0.659250i	−0.944112	+	0.329625i	$0.893078\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	3.69698e6i		−	0.0932360i
$342$		0			0
$343$		−	4.38503e7i		−	1.08665i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.18476e7			1.00157			0.500787	−	0.865571i	$-0.333044\pi$
$347$	4.18476e7			1.00157			0.500787	+	0.865571i	$0.333044\pi$
$348$		0			0
$349$	1.11598e7			0.262532			0.131266	−	0.991347i	$-0.458096\pi$
$349$	1.11598e7			0.262532			0.131266	+	0.991347i	$0.458096\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.13705e7			−0.485837			−0.242918	−	0.970047i	$-0.578105\pi$
$353$	−2.13705e7			−0.485837			−0.242918	+	0.970047i	$0.578105\pi$
$354$		0			0
$355$	4.05821e7	+	2.23629e7i	0.907088	+	0.499853i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.27851e7i			0.492455i	0.969212	+	0.246228i	$0.0791911\pi$
$359$			2.27851e7i			0.492455i	−0.969212	+	0.246228i	$0.920809\pi$
$360$		0			0
$361$	3.55491e7			0.755627
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.53706e7	+	1.94910e7i	0.727383	+	0.400826i
$366$		0			0
$367$		−	3.26916e7i		−	0.661360i	−0.943743	−	0.330680i	$-0.892722\pi$
$367$		−	3.26916e7i		−	0.661360i	0.943743	−	0.330680i	$-0.107278\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			8.59569e6i			0.168329i
$372$		0			0
$373$		−	2.20835e7i		−	0.425542i	−0.977102	−	0.212771i	$-0.931751\pi$
$373$		−	2.20835e7i		−	0.425542i	0.977102	−	0.212771i	$-0.0682488\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.06783e6			0.0759169
$378$		0			0
$379$	−2.12751e7			−0.390800			−0.195400	−	0.980724i	$-0.562601\pi$
$379$	−2.12751e7			−0.390800			−0.195400	+	0.980724i	$0.562601\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.76311e7			1.38178			0.690891	−	0.722959i	$-0.257219\pi$
$383$	7.76311e7			1.38178			0.690891	+	0.722959i	$0.257219\pi$
$384$		0			0
$385$	1.93488e7	−	3.51125e7i	0.339057	−	0.615290i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			7.43035e7i			1.26229i	0.775663	+	0.631147i	$0.217415\pi$
$389$			7.43035e7i			1.26229i	−0.775663	+	0.631147i	$0.782585\pi$
$390$		0			0
$391$	−5.87914e7			−0.983521
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.98172e6	−	7.22567e6i	0.0646070	−	0.117243i
$396$		0			0
$397$		−	2.25395e7i		−	0.360224i	−0.983646	−	0.180112i	$-0.942354\pi$
$397$		−	2.25395e7i		−	0.360224i	0.983646	−	0.180112i	$-0.0576459\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.85810e7i		−	0.753414i	−0.926332	−	0.376707i	$-0.877056\pi$
$401$		−	4.85810e7i		−	0.753414i	0.926332	−	0.376707i	$-0.122944\pi$
$402$		0			0
$403$		−	1.63844e6i		−	0.0250332i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.02153e7			−1.33813
$408$		0			0
$409$	6.71469e7			0.981423			0.490711	−	0.871322i	$-0.336737\pi$
$409$	6.71469e7			0.981423			0.490711	+	0.871322i	$0.336737\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.63593e7			1.22591
$414$		0			0
$415$	446627.	−	810499.i	0.00624886	−	0.0113399i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			4.06857e7i			0.553095i	0.961000	+	0.276548i	$0.0891904\pi$
$419$			4.06857e7i			0.553095i	−0.961000	+	0.276548i	$0.910810\pi$
$420$		0			0
$421$	9.91589e7			1.32888			0.664440	−	0.747342i	$-0.268670\pi$
$421$	9.91589e7			1.32888			0.664440	+	0.747342i	$0.268670\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.49926e7	+	8.70371e7i	0.716370	+	1.13380i
$426$		0			0
$427$			9.71360e7i			1.24766i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	2.37118e7i		−	0.296164i	−0.988975	−	0.148082i	$-0.952690\pi$
$431$		−	2.37118e7i		−	0.296164i	0.988975	−	0.148082i	$-0.0473099\pi$
$432$		0			0
$433$			6.09661e7i			0.750974i	0.926828	+	0.375487i	$0.122524\pi$
$433$			6.09661e7i			0.750974i	−0.926828	+	0.375487i	$0.877476\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.10896e7			0.971675
$438$		0			0
$439$	−1.53212e7			−0.181092			−0.0905458	−	0.995892i	$-0.528861\pi$
$439$	−1.53212e7			−0.181092			−0.0905458	+	0.995892i	$0.528861\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.12121e8			1.28966			0.644828	−	0.764327i	$-0.276929\pi$
$443$	1.12121e8			1.28966			0.644828	+	0.764327i	$0.276929\pi$
$444$		0			0
$445$	−1.08541e8	−	5.98119e7i	−1.23173	−	0.678746i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.27219e8i			1.40544i	0.711468	+	0.702719i	$0.248031\pi$
$449$			1.27219e8i			1.40544i	−0.711468	+	0.702719i	$0.751969\pi$
$450$		0			0
$451$	−6.73589e7			−0.734287
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.57510e6	−	1.55613e7i	0.0910343	−	0.165201i
$456$		0			0
$457$		−	7.74410e7i		−	0.811377i	−0.914011	−	0.405688i	$-0.867032\pi$
$457$		−	7.74410e7i		−	0.811377i	0.914011	−	0.405688i	$-0.132968\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	2.35645e7i		−	0.240523i	−0.992742	−	0.120261i	$-0.961627\pi$
$461$		−	2.35645e7i		−	0.240523i	0.992742	−	0.120261i	$-0.0383733\pi$
$462$		0			0
$463$		−	4.67479e7i		−	0.470998i	−0.971875	−	0.235499i	$-0.924328\pi$
$463$		−	4.67479e7i		−	0.470998i	0.971875	−	0.235499i	$-0.0756725\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.04192e8			1.02302			0.511511	−	0.859277i	$-0.329086\pi$
$467$	1.04192e8			1.02302			0.511511	+	0.859277i	$0.329086\pi$
$468$		0			0
$469$	−1.31679e8			−1.27643
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.50330e8			−1.42057
$474$		0			0
$475$	−7.58499e7	−	1.20048e8i	−0.707741	−	1.12015i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	9.95625e7i		−	0.905919i	−0.891531	−	0.452959i	$-0.850368\pi$
$479$		−	9.95625e7i		−	0.905919i	0.891531	−	0.452959i	$-0.149632\pi$
$480$		0			0
$481$	−3.99820e7			−0.359277
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.67229e8	−	9.21521e7i	−1.46584	−	0.807755i
$486$		0			0
$487$			1.90543e8i			1.64971i	0.565347	+	0.824853i	$0.308742\pi$
$487$			1.90543e8i			1.64971i	−0.565347	+	0.824853i	$0.691258\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			1.08462e8i			0.916287i	0.888878	+	0.458143i	$0.151485\pi$
$491$			1.08462e8i			0.916287i	−0.888878	+	0.458143i	$0.848515\pi$
$492$		0			0
$493$			5.09467e7i			0.425182i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.00150e8			0.815799
$498$		0			0
$499$	1.85918e8			1.49631			0.748153	−	0.663527i	$-0.230941\pi$
$499$	1.85918e8			1.49631			0.748153	+	0.663527i	$0.230941\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.39688e8			−1.09763			−0.548815	−	0.835944i	$-0.684921\pi$
$503$	−1.39688e8			−1.09763			−0.548815	+	0.835944i	$0.684921\pi$
$504$		0			0
$505$	4.95215e7	+	2.72889e7i	0.384521	+	0.211891i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			3.55184e7i			0.269340i	0.990891	+	0.134670i	$0.0429974\pi$
$509$			3.55184e7i			0.269340i	−0.990891	+	0.134670i	$0.957003\pi$
$510$		0			0
$511$	8.72890e7			0.654180
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.14910e7	−	6.33212e6i	−0.0841268	−	0.0463583i
$516$		0			0
$517$			2.16128e8i			1.56401i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.93385e8i		−	1.36745i	−0.729742	−	0.683723i	$-0.760360\pi$
$521$		−	1.93385e8i		−	1.36745i	0.729742	−	0.683723i	$-0.239640\pi$
$522$		0			0
$523$			9.39198e6i			0.0656527i	0.999461	+	0.0328263i	$0.0104508\pi$
$523$			9.39198e6i			0.0656527i	−0.999461	+	0.0328263i	$0.989549\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.05203e7			0.140201
$528$		0			0
$529$	−6.84242e7			−0.462214
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.98524e7			−0.197150
$534$		0			0
$535$	−3.96416e7	+	7.19381e7i	−0.258875	+	0.469783i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			5.30091e7i			0.338520i
$540$		0			0
$541$	2.07632e7			0.131130			0.0655651	−	0.997848i	$-0.479115\pi$
$541$	2.07632e7			0.131130			0.0655651	+	0.997848i	$0.479115\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.99382e7	−	1.81359e8i	0.617365	−	1.12034i
$546$		0			0
$547$		−	3.73970e7i		−	0.228494i	−0.993452	−	0.114247i	$-0.963554\pi$
$547$		−	3.73970e7i		−	0.228494i	0.993452	−	0.114247i	$-0.0364456\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	7.02696e7i		−	0.420061i
$552$		0			0
$553$		−	1.78318e7i		−	0.105444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.13408e8			−1.23494			−0.617468	−	0.786596i	$-0.711841\pi$
$557$	−2.13408e8			−1.23494			−0.617468	+	0.786596i	$0.711841\pi$
$558$		0			0
$559$	−6.66240e7			−0.381413
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.01902e8			−0.571029			−0.285515	−	0.958374i	$-0.592165\pi$
$563$	−1.01902e8			−0.571029			−0.285515	+	0.958374i	$0.592165\pi$
$564$		0			0
$565$	6.19095e7	−	1.12348e8i	0.343251	−	0.622902i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1.12478e8i			0.610566i	0.952262	+	0.305283i	$0.0987510\pi$
$569$			1.12478e8i			0.610566i	−0.952262	+	0.305283i	$0.901249\pi$
$570$		0			0
$571$	−1.75933e8			−0.945016			−0.472508	−	0.881326i	$-0.656651\pi$
$571$	−1.75933e8			−0.945016			−0.472508	+	0.881326i	$0.656651\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.44675e7	−	1.17860e8i	−0.391709	−	0.619960i
$576$		0			0
$577$			1.61356e8i			0.839957i	0.907534	+	0.419978i	$0.137962\pi$
$577$			1.61356e8i			0.839957i	−0.907534	+	0.419978i	$0.862038\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	2.00019e6i		−	0.0101986i
$582$		0			0
$583$		−	3.77679e7i		−	0.190598i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.39079e7			−0.217084			−0.108542	−	0.994092i	$-0.534618\pi$
$587$	−4.39079e7			−0.217084			−0.108542	+	0.994092i	$0.534618\pi$
$588$		0			0
$589$	−2.83032e7			−0.138513
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.94038e7			0.188962			0.0944808	−	0.995527i	$-0.469881\pi$
$593$	3.94038e7			0.188962			0.0944808	+	0.995527i	$0.469881\pi$
$594$		0			0
$595$	1.94895e8	+	1.07397e8i	0.925228	+	0.509849i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1.11994e8i		−	0.521092i	−0.965461	−	0.260546i	$-0.916097\pi$
$599$		−	1.11994e8i		−	0.521092i	0.965461	−	0.260546i	$-0.0839026\pi$
$600$		0			0
$601$	−3.96028e8			−1.82432			−0.912162	−	0.409830i	$-0.865588\pi$
$601$	−3.96028e8			−1.82432			−0.912162	+	0.409830i	$0.865588\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.18599e7	−	3.96693e7i	0.0987146	−	0.179138i
$606$		0			0
$607$		−	3.06455e8i		−	1.37025i	−0.728425	−	0.685125i	$-0.759747\pi$
$607$		−	3.06455e8i		−	1.37025i	0.728425	−	0.685125i	$-0.240253\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			9.57844e7i			0.419924i
$612$		0			0
$613$			1.80074e7i			0.0781751i	0.999236	+	0.0390875i	$0.0124451\pi$
$613$			1.80074e7i			0.0781751i	−0.999236	+	0.0390875i	$0.987555\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.66217e8			−0.707651			−0.353825	−	0.935311i	$-0.615119\pi$
$617$	−1.66217e8			−0.707651			−0.353825	+	0.935311i	$0.615119\pi$
$618$		0			0
$619$	−1.89841e8			−0.800422			−0.400211	−	0.916423i	$-0.631063\pi$
$619$	−1.89841e8			−0.800422			−0.400211	+	0.916423i	$0.631063\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.67863e8			−1.10777
$624$		0			0
$625$	−1.04829e8	+	2.20489e8i	−0.429380	+	0.903124i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	5.00746e8i		−	2.01217i
$630$		0			0
$631$	−2.56171e8			−1.01963			−0.509814	−	0.860285i	$-0.670286\pi$
$631$	−2.56171e8			−1.01963			−0.509814	+	0.860285i	$0.670286\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.11557e7	−	2.26789e7i	−0.160734	−	0.0885730i
$636$		0			0
$637$			2.34928e7i			0.0908900i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	4.41284e7i		−	0.167550i	−0.996485	−	0.0837749i	$-0.973302\pi$
$641$		−	4.41284e7i		−	0.167550i	0.996485	−	0.0837749i	$-0.0266977\pi$
$642$		0			0
$643$			1.96340e8i			0.738545i	0.929321	+	0.369272i	$0.120393\pi$
$643$			1.96340e8i			0.738545i	−0.929321	+	0.369272i	$0.879607\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.99293e7			−0.368961			−0.184480	−	0.982836i	$-0.559060\pi$
$647$	−9.99293e7			−0.368961			−0.184480	+	0.982836i	$0.559060\pi$
$648$		0			0
$649$	−3.79447e8			−1.38809
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.49531e7			0.0896160			0.0448080	−	0.998996i	$-0.485732\pi$
$653$	2.49531e7			0.0896160			0.0448080	+	0.998996i	$0.485732\pi$
$654$		0			0
$655$	−1.56268e8	−	8.61120e7i	−0.556092	−	0.306436i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1.49890e8i			0.523739i	0.965103	+	0.261870i	$0.0843390\pi$
$659$			1.49890e8i			0.523739i	−0.965103	+	0.261870i	$0.915661\pi$
$660$		0			0
$661$	−4.68646e6			−0.0162271			−0.00811355	−	0.999967i	$-0.502583\pi$
$661$	−4.68646e6			−0.0162271			−0.00811355	+	0.999967i	$0.502583\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.68813e8	−	1.48130e8i	−0.914084	−	0.503708i
$666$		0			0
$667$		−	6.89888e7i		−	0.232488i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	4.26798e8i		−	1.41272i
$672$		0			0
$673$		−	3.58399e8i		−	1.17577i	−0.808945	−	0.587884i	$-0.799961\pi$
$673$		−	3.58399e8i		−	1.17577i	0.808945	−	0.587884i	$-0.200039\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.26026e7			−0.0728437			−0.0364218	−	0.999337i	$-0.511596\pi$
$677$	−2.26026e7			−0.0728437			−0.0364218	+	0.999337i	$0.511596\pi$
$678$		0			0
$679$	−4.12696e8			−1.31832
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.32455e8			1.67117			0.835585	−	0.549361i	$-0.185129\pi$
$683$	5.32455e8			1.67117			0.835585	+	0.549361i	$0.185129\pi$
$684$		0			0
$685$	−1.66675e8	+	3.02466e8i	−0.518559	+	0.941033i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1.67381e7i		−	0.0511740i
$690$		0			0
$691$	5.20725e8			1.57825			0.789123	−	0.614236i	$-0.210536\pi$
$691$	5.20725e8			1.57825			0.789123	+	0.614236i	$0.210536\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.40891e8	−	2.55676e8i	0.419690	−	0.761615i
$696$		0			0
$697$		−	3.73880e8i		−	1.10417i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			2.97763e8i			0.864404i	0.901777	+	0.432202i	$0.142263\pi$
$701$			2.97763e8i			0.864404i	−0.901777	+	0.432202i	$0.857737\pi$
$702$		0			0
$703$			6.90667e8i			1.98794i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.22211e8			0.345823
$708$		0			0
$709$	1.89039e8			0.530411			0.265205	−	0.964192i	$-0.414560\pi$
$709$	1.89039e8			0.530411			0.265205	+	0.964192i	$0.414560\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.77873e7			−0.0766617
$714$		0			0
$715$	−3.76774e7	+	6.83736e7i	−0.103077	+	0.187055i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1.30902e8i			0.352176i	0.984374	+	0.176088i	$0.0563443\pi$
$719$			1.30902e8i			0.352176i	−0.984374	+	0.176088i	$0.943656\pi$
$720$		0			0
$721$	−2.83579e7			−0.0756603
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.02134e8	+	6.45311e7i	−0.268013	+	0.169338i
$726$		0			0
$727$			4.44143e8i			1.15590i	0.816073	+	0.577949i	$0.196147\pi$
$727$			4.44143e8i			1.15590i	−0.816073	+	0.577949i	$0.803853\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	8.34419e8i		−	2.13615i
$732$		0			0
$733$			4.71631e8i			1.19754i	0.800921	+	0.598770i	$0.204344\pi$
$733$			4.71631e8i			1.19754i	−0.800921	+	0.598770i	$0.795656\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.78573e8			1.44529
$738$		0			0
$739$	−6.40610e8			−1.58731			−0.793653	−	0.608371i	$-0.791823\pi$
$739$	−6.40610e8			−1.58731			−0.793653	+	0.608371i	$0.791823\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.44941e8			−1.32857			−0.664283	−	0.747481i	$-0.731263\pi$
$743$	−5.44941e8			−1.32857			−0.664283	+	0.747481i	$0.731263\pi$
$744$		0			0
$745$	6.10774e8	+	3.36569e8i	1.47711	+	0.813963i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1.77532e8i			0.422504i
$750$		0			0
$751$	−5.56224e8			−1.31320			−0.656598	−	0.754241i	$-0.728005\pi$
$751$	−5.56224e8			−1.31320			−0.656598	+	0.754241i	$0.728005\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.13101e7	−	2.05245e7i	0.0262799	−	0.0476905i
$756$		0			0
$757$		−	4.02524e8i		−	0.927906i	−0.885860	−	0.463953i	$-0.846431\pi$
$757$		−	4.02524e8i		−	0.927906i	0.885860	−	0.463953i	$-0.153569\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.44544e8i		−	0.327979i	−0.986462	−	0.163989i	$-0.947564\pi$
$761$		−	1.44544e8i		−	0.327979i	0.986462	−	0.163989i	$-0.0524363\pi$
$762$		0			0
$763$		−	4.47565e8i		−	1.00759i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.68165e8			−0.372691
$768$		0			0
$769$	−2.64281e8			−0.581148			−0.290574	−	0.956853i	$-0.593846\pi$
$769$	−2.64281e8			−0.581148			−0.290574	+	0.956853i	$0.593846\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.66034e8			0.359466			0.179733	−	0.983715i	$-0.442477\pi$
$773$	1.66034e8			0.359466			0.179733	+	0.983715i	$0.442477\pi$
$774$		0			0
$775$	2.59918e7	+	4.11374e7i	0.0558382	+	0.0883756i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			5.15684e8i			1.09087i
$780$		0			0
$781$	−4.40043e8			−0.923723
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.31044e8	+	1.27317e8i	0.477623	+	0.263195i
$786$		0			0
$787$			2.25184e8i			0.461970i	0.972957	+	0.230985i	$0.0741948\pi$
$787$			2.25184e8i			0.461970i	−0.972957	+	0.230985i	$0.925805\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2.77257e8i		−	0.560213i
$792$		0			0
$793$		−	1.89150e8i		−	0.379303i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−5.51423e8			−1.08921			−0.544603	−	0.838694i	$-0.683320\pi$
$797$	−5.51423e8			−1.08921			−0.544603	+	0.838694i	$0.683320\pi$
$798$		0			0
$799$	−1.19963e9			−2.35184
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.83532e8			−0.740722
$804$		0			0
$805$	−2.63914e8	−	1.45430e8i	−0.505912	−	0.278784i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			3.95120e7i			0.0746249i	0.999304	+	0.0373125i	$0.0118797\pi$
$809$			3.95120e7i			0.0746249i	−0.999304	+	0.0373125i	$0.988120\pi$
$810$		0			0
$811$	8.11405e8			1.52116			0.760580	−	0.649244i	$-0.224915\pi$
$811$	8.11405e8			1.52116			0.760580	+	0.649244i	$0.224915\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.83293e8	+	2.66320e8i	0.892766	+	0.491961i
$816$		0			0
$817$			1.15089e9i			2.11042i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1.22832e8i			0.221963i	0.993822	+	0.110982i	$0.0353994\pi$
$821$			1.22832e8i			0.221963i	−0.993822	+	0.110982i	$0.964601\pi$
$822$		0			0
$823$			8.46296e8i			1.51818i	0.650987	+	0.759089i	$0.274355\pi$
$823$			8.46296e8i			1.51818i	−0.650987	+	0.759089i	$0.725645\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.80388e8			−1.02613			−0.513065	−	0.858350i	$-0.671490\pi$
$827$	−5.80388e8			−1.02613			−0.513065	+	0.858350i	$0.671490\pi$
$828$		0			0
$829$	−6.30126e8			−1.10602			−0.553011	−	0.833174i	$-0.686521\pi$
$829$	−6.30126e8			−1.10602			−0.553011	+	0.833174i	$0.686521\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.94231e8			−0.509041
$834$		0			0
$835$	1.68342e7	−	3.05491e7i	0.0289156	−	0.0524735i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	2.51051e7i		−	0.0425085i	−0.999774	−	0.0212542i	$-0.993234\pi$
$839$		−	2.51051e7i		−	0.0425085i	0.999774	−	0.0212542i	$-0.00676595\pi$
$840$		0			0
$841$	5.35040e8			0.899494
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.74495e8	−	4.98129e8i	0.454951	−	0.825603i
$846$		0			0
$847$		−	9.78978e7i		−	0.161110i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			6.78079e8i			1.10025i
$852$		0			0
$853$			3.53908e8i			0.570221i	0.958495	+	0.285111i	$0.0920303\pi$
$853$			3.53908e8i			0.570221i	−0.958495	+	0.285111i	$0.907970\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.06395e8			−1.28117			−0.640583	−	0.767889i	$-0.721307\pi$
$857$	−8.06395e8			−1.28117			−0.640583	+	0.767889i	$0.721307\pi$
$858$		0			0
$859$	9.26709e8			1.46206			0.731028	−	0.682348i	$-0.239041\pi$
$859$	9.26709e8			1.46206			0.731028	+	0.682348i	$0.239041\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.89660e8			1.07301			0.536504	−	0.843898i	$-0.319745\pi$
$863$	6.89660e8			1.07301			0.536504	+	0.843898i	$0.319745\pi$
$864$		0			0
$865$	3.00239e8	−	5.44847e8i	0.463894	−	0.841833i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			7.83499e7i			0.119393i
$870$		0			0
$871$	2.56414e8			0.388050
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.15601e7	+	5.26742e8i	0.0471101	+	0.786274i
$876$		0			0
$877$			2.94275e8i			0.436269i	0.975919	+	0.218134i	$0.0699971\pi$
$877$			2.94275e8i			0.436269i	−0.975919	+	0.218134i	$0.930003\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			6.79591e7i			0.0993847i	0.998765	+	0.0496924i	$0.0158241\pi$
$881$			6.79591e7i			0.0993847i	−0.998765	+	0.0496924i	$0.984176\pi$
$882$		0			0
$883$		−	9.49523e8i		−	1.37919i	−0.724196	−	0.689594i	$-0.757789\pi$
$883$		−	9.49523e8i		−	1.37919i	0.724196	−	0.689594i	$-0.242211\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.87598e8			0.268817			0.134409	−	0.990926i	$-0.457086\pi$
$887$	1.87598e8			0.268817			0.134409	+	0.990926i	$0.457086\pi$
$888$		0			0
$889$	−1.01566e8			−0.144558
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.65462e9			2.32351
$894$		0			0
$895$	−3.56711e8	−	1.96566e8i	−0.497562	−	0.274183i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			2.40796e7i			0.0331413i
$900$		0			0
$901$	2.09633e8			0.286607
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.94096e8	−	1.25958e9i	0.936427	−	1.69934i
$906$		0			0
$907$		−	1.30184e9i		−	1.74476i	−0.488832	−	0.872378i	$-0.662577\pi$
$907$		−	1.30184e9i		−	1.74476i	0.488832	−	0.872378i	$-0.337423\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1.25808e9i		−	1.66400i	−0.554775	−	0.832000i	$-0.687196\pi$
$911$		−	1.25808e9i		−	1.66400i	0.554775	−	0.832000i	$-0.312804\pi$
$912$		0			0
$913$			8.78846e6i			0.0115478i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.85646e8			−0.500127
$918$		0			0
$919$	2.32755e8			0.299883			0.149942	−	0.988695i	$-0.452091\pi$
$919$	2.32755e8			0.299883			0.149942	+	0.988695i	$0.452091\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.95020e8			−0.248013
$924$		0			0
$925$	1.00385e9	−	6.34264e8i	1.26837	−	0.801392i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.02421e9i			1.27745i	0.769436	+	0.638724i	$0.220538\pi$
$929$			1.02421e9i			1.27745i	−0.769436	+	0.638724i	$0.779462\pi$
$930$		0			0
$931$	4.05825e8			0.502910
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.56331e8	−	4.71883e8i	−1.04763	−	0.577298i
$936$		0			0
$937$		−	5.87353e8i		−	0.713971i	−0.934110	−	0.356985i	$-0.883805\pi$
$937$		−	5.87353e8i		−	0.713971i	0.934110	−	0.356985i	$-0.116195\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.44532e8i		−	0.173458i	−0.996232	−	0.0867292i	$-0.972359\pi$
$941$		−	1.44532e8i		−	0.173458i	0.996232	−	0.0867292i	$-0.0276415\pi$
$942$		0			0
$943$			5.06285e8i			0.603755i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.40595e8			−0.283294			−0.141647	−	0.989917i	$-0.545240\pi$
$947$	−2.40595e8			−0.283294			−0.141647	+	0.989917i	$0.545240\pi$
$948$		0			0
$949$	−1.69975e8			−0.198878
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.59931e9			−1.84779			−0.923896	−	0.382644i	$-0.875014\pi$
$953$	−1.59931e9			−1.84779			−0.923896	+	0.382644i	$0.875014\pi$
$954$		0			0
$955$	6.65522e8	+	3.66737e8i	0.764103	+	0.421061i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			7.46440e8i			0.846328i
$960$		0			0
$961$	−8.77805e8			−0.989072
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.23635e9	−	6.81294e8i	−1.37581	−	0.758145i
$966$		0			0
$967$			3.53519e8i			0.390960i	0.980708	+	0.195480i	$0.0626265\pi$
$967$			3.53519e8i			0.390960i	−0.980708	+	0.195480i	$0.937373\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.33953e8i		−	0.146317i	−0.997320	−	0.0731583i	$-0.976692\pi$
$971$		−	1.33953e8i		−	0.146317i	0.997320	−	0.0731583i	$-0.0233078\pi$
$972$		0			0
$973$		−	6.30969e8i		−	0.684967i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.99781e8			−0.321455			−0.160728	−	0.986999i	$-0.551384\pi$
$977$	−2.99781e8			−0.321455			−0.160728	+	0.986999i	$0.551384\pi$
$978$		0			0
$979$	1.17694e9			1.25431
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.04294e9			1.09799			0.548996	−	0.835825i	$-0.315010\pi$
$983$	1.04294e9			1.09799			0.548996	+	0.835825i	$0.315010\pi$
$984$		0			0
$985$	−2.09265e8	+	3.79755e8i	−0.218971	+	0.397369i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.12992e9i			1.16804i
$990$		0			0
$991$	−3.39676e8			−0.349014			−0.174507	−	0.984656i	$-0.555833\pi$
$991$	−3.39676e8			−0.349014			−0.174507	+	0.984656i	$0.555833\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.50087e8	−	6.35306e8i	0.355391	−	0.644931i
$996$		0			0
$997$		−	1.79345e9i		−	1.80969i	−0.425739	−	0.904846i	$-0.639986\pi$
$997$		−	1.79345e9i		−	1.80969i	0.425739	−	0.904846i	$-0.360014\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.7.c.a.449.9		12
3.2	odd	2	inner	720.7.c.a.449.4		12
4.3	odd	2		45.7.d.a.44.3	✓	12
5.4	even	2	inner	720.7.c.a.449.3		12
12.11	even	2		45.7.d.a.44.10	yes	12
15.14	odd	2	inner	720.7.c.a.449.10		12
20.3	even	4		225.7.c.e.26.10		12
20.7	even	4		225.7.c.e.26.3		12
20.19	odd	2		45.7.d.a.44.9	yes	12
60.23	odd	4		225.7.c.e.26.4		12
60.47	odd	4		225.7.c.e.26.9		12
60.59	even	2		45.7.d.a.44.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.7.d.a.44.3	✓	12	4.3	odd	2
45.7.d.a.44.4	yes	12	60.59	even	2
45.7.d.a.44.9	yes	12	20.19	odd	2
45.7.d.a.44.10	yes	12	12.11	even	2
225.7.c.e.26.3		12	20.7	even	4
225.7.c.e.26.4		12	60.23	odd	4
225.7.c.e.26.9		12	60.47	odd	4
225.7.c.e.26.10		12	20.3	even	4
720.7.c.a.449.3		12	5.4	even	2	inner
720.7.c.a.449.4		12	3.2	odd	2	inner
720.7.c.a.449.9		12	1.1	even	1	trivial
720.7.c.a.449.10		12	15.14	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	720.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(60.3283 - 109.478i) q^{5} -270.176i q^{7} +1187.10i q^{11} +526.105i q^{13} -6589.09 q^{17} +9088.18 q^{19} +8922.54 q^{23} +(-8346.00 - 13209.3i) q^{25} -7731.97i q^{29} -3114.29 q^{31} +(-29578.4 - 16299.2i) q^{35} +75996.2i q^{37} +56742.3i q^{41} +126636. i q^{43} +182063. q^{47} +44654.2 q^{49} -31815.2 q^{53} +(129962. + 71615.8i) q^{55} +319642. i q^{59} -359529. q^{61} +(57597.1 + 31739.0i) q^{65} -487383. i q^{67} +370686. i q^{71} +323083. i q^{73} +320726. q^{77} +66000.9 q^{79} +7403.28 q^{83} +(-397509. + 721363. i) q^{85} -991440. i q^{89} +142141. q^{91} +(548274. - 994959. i) q^{95} -1.52751e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4320 q^{19} - 63384 q^{25} - 60192 q^{31} + 711516 q^{49} + 104112 q^{55} - 449784 q^{61} - 4324608 q^{79} - 3305772 q^{85} - 631152 q^{91}+O(q^{100}) \) 12 * q - 4320 * q^19 - 63384 * q^25 - 60192 * q^31 + 711516 * q^49 + 104112 * q^55 - 449784 * q^61 - 4324608 * q^79 - 3305772 * q^85 - 631152 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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