LMFDB - Embedded newform 1440.2.f.h.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(289,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1440.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.f (of order $2$, degree $1$, 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1440.289
Dual form			1440.2.f.h.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +2.00000i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +2.00000i q^{7} +4.47214 q^{11} +4.47214i q^{13} -4.47214i q^{17} +4.00000i q^{23} -5.00000 q^{25} +4.00000 q^{29} +8.94427 q^{31} +4.47214 q^{35} +4.47214i q^{37} -10.0000 q^{41} +4.00000i q^{43} -8.00000i q^{47} +3.00000 q^{49} +4.47214i q^{53} -10.0000i q^{55} +13.4164 q^{59} +10.0000 q^{61} +10.0000 q^{65} -8.00000i q^{67} +8.94427 q^{71} -8.94427i q^{73} +8.94427i q^{77} +8.94427 q^{79} -4.00000i q^{83} -10.0000 q^{85} +6.00000 q^{89} -8.94427 q^{91} -17.8885i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 20 q^{25} + 16 q^{29} - 40 q^{41} + 12 q^{49} + 40 q^{61} + 40 q^{65} - 40 q^{85} + 24 q^{89}+O(q^{100}) $ 4 * q - 20 * q^25 + 16 * q^29 - 40 * q^41 + 12 * q^49 + 40 * q^61 + 40 * q^65 - 40 * q^85 + 24 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.47214i	− 1.08465i	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	− 4.47214i	− 1.08465i	0.840168	−	0.542326i	$-0.182456\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	8.94427	1.60644	0.803219	−	0.595683i	$-0.203119\pi$
$31$	8.94427	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.47214	0.755929
$36$	0	0
$37$	4.47214i	0.735215i	0.929981	+	0.367607i	$0.119823\pi$
$37$	4.47214i	0.735215i	−0.929981	+	0.367607i	$0.880177\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.47214i	0.614295i	0.951662	+	0.307148i	$0.0993745\pi$
$53$	4.47214i	0.614295i	−0.951662	+	0.307148i	$0.900625\pi$
$54$	0	0
$55$	− 10.0000i	− 1.34840i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.4164	1.74667	0.873334	−	0.487122i	$-0.161953\pi$
$59$	13.4164	1.74667	0.873334	+	0.487122i	$0.161953\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.0000	1.24035
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	− 8.94427i	− 1.04685i	−0.852072	−	0.523424i	$-0.824654\pi$
$73$	− 8.94427i	− 1.04685i	0.852072	−	0.523424i	$-0.175346\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.94427i	1.01929i
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−8.94427	−0.937614
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 17.8885i	− 1.81631i	−0.418638	−	0.908153i	$-0.637492\pi$
$97$	− 17.8885i	− 1.81631i	0.418638	−	0.908153i	$-0.362508\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.47214i	− 0.420703i	−0.977626	−	0.210352i	$-0.932539\pi$
$113$	− 4.47214i	− 0.420703i	0.977626	−	0.210352i	$-0.0674609\pi$
$114$	0	0
$115$	8.94427	0.834058
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.94427	0.819920
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.47214i	0.382080i	0.981582	+	0.191040i	$0.0611861\pi$
$137$	4.47214i	0.382080i	−0.981582	+	0.191040i	$0.938814\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.0000i	1.67248i
$144$	0	0
$145$	− 8.94427i	− 0.742781i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−8.94427	−0.727875	−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427	−0.727875	−0.363937	+	0.931423i	$0.618568\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 20.0000i	− 1.60644i
$156$	0	0
$157$	13.4164i	1.07075i	0.844616	+	0.535373i	$0.179829\pi$
$157$	13.4164i	1.07075i	−0.844616	+	0.535373i	$0.820171\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	24.0000i	1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$	24.0000i	1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.47214i	− 0.340010i	−0.985443	−	0.170005i	$-0.945622\pi$
$173$	− 4.47214i	− 0.340010i	0.985443	−	0.170005i	$-0.0543784\pi$
$174$	0	0
$175$	− 10.0000i	− 0.755929i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.3607	−1.67132	−0.835658	−	0.549250i	$-0.814913\pi$
$179$	−22.3607	−1.67132	−0.835658	+	0.549250i	$0.814913\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.0000	0.735215
$186$	0	0
$187$	− 20.0000i	− 1.46254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.94427	−0.647185	−0.323592	−	0.946197i	$-0.604891\pi$
$191$	−8.94427	−0.647185	−0.323592	+	0.946197i	$0.604891\pi$
$192$	0	0
$193$	− 8.94427i	− 0.643823i	−0.946770	−	0.321911i	$-0.895675\pi$
$193$	− 8.94427i	− 0.643823i	0.946770	−	0.321911i	$-0.104325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.47214i	− 0.318626i	−0.987228	−	0.159313i	$-0.949072\pi$
$197$	− 4.47214i	− 0.318626i	0.987228	−	0.159313i	$-0.0509280\pi$
$198$	0	0
$199$	−26.8328	−1.90213	−0.951064	−	0.308994i	$-0.900008\pi$
$199$	−26.8328	−1.90213	−0.951064	+	0.308994i	$0.900008\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	22.3607i	1.56174i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	26.8328	1.84725	0.923624	−	0.383301i	$-0.125213\pi$
$211$	26.8328	1.84725	0.923624	+	0.383301i	$0.125213\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.94427	0.609994
$216$	0	0
$217$	17.8885i	1.21435i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	6.00000i	0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$	6.00000i	0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 13.4164i	− 0.878938i	−0.898258	−	0.439469i	$-0.855167\pi$
$233$	− 13.4164i	− 0.878938i	0.898258	−	0.439469i	$-0.144833\pi$
$234$	0	0
$235$	−17.8885	−1.16692
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.94427	−0.578557	−0.289278	−	0.957245i	$-0.593415\pi$
$239$	−8.94427	−0.578557	−0.289278	+	0.957245i	$0.593415\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.70820i	− 0.428571i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.47214	−0.282279	−0.141139	−	0.989990i	$-0.545077\pi$
$251$	−4.47214	−0.282279	−0.141139	+	0.989990i	$0.545077\pi$
$252$	0	0
$253$	17.8885i	1.12464i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 13.4164i	− 0.836893i	−0.908242	−	0.418446i	$-0.862575\pi$
$257$	− 13.4164i	− 0.836893i	0.908242	−	0.418446i	$-0.137425\pi$
$258$	0	0
$259$	−8.94427	−0.555770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 4.00000i	− 0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$	− 4.00000i	− 0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	−26.8328	−1.62998	−0.814989	−	0.579477i	$-0.803257\pi$
$271$	−26.8328	−1.62998	−0.814989	+	0.579477i	$0.803257\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.3607	−1.34840
$276$	0	0
$277$	− 13.4164i	− 0.806114i	−0.915175	−	0.403057i	$-0.867948\pi$
$277$	− 13.4164i	− 0.806114i	0.915175	−	0.403057i	$-0.132052\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 20.0000i	− 1.18056i
$288$	0	0
$289$	−3.00000	−0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.47214i	0.261265i	0.991431	+	0.130632i	$0.0417008\pi$
$293$	4.47214i	0.261265i	−0.991431	+	0.130632i	$0.958299\pi$
$294$	0	0
$295$	− 30.0000i	− 1.74667i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.8885	−1.03452
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 22.3607i	− 1.28037i
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.94427	−0.507183	−0.253592	−	0.967311i	$-0.581612\pi$
$311$	−8.94427	−0.507183	−0.253592	+	0.967311i	$0.581612\pi$
$312$	0	0
$313$	26.8328i	1.51668i	0.651859	+	0.758340i	$0.273989\pi$
$313$	26.8328i	1.51668i	−0.651859	+	0.758340i	$0.726011\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.4164i	− 0.753541i	−0.926307	−	0.376770i	$-0.877035\pi$
$317$	− 13.4164i	− 0.753541i	0.926307	−	0.376770i	$-0.122965\pi$
$318$	0	0
$319$	17.8885	1.00157
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 22.3607i	− 1.24035i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.8885	−0.977356
$336$	0	0
$337$	− 8.94427i	− 0.487226i	−0.969873	−	0.243613i	$-0.921667\pi$
$337$	− 8.94427i	− 0.487226i	0.969873	−	0.243613i	$-0.0783326\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.3607i	1.19014i	0.803674	+	0.595069i	$0.202875\pi$
$353$	22.3607i	1.19014i	−0.803674	+	0.595069i	$0.797125\pi$
$354$	0	0
$355$	− 20.0000i	− 1.06149i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.8885	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$359$	−17.8885	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−20.0000	−1.04685
$366$	0	0
$367$	18.0000i	0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$	18.0000i	0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.94427	−0.464363
$372$	0	0
$373$	− 13.4164i	− 0.694675i	−0.937740	−	0.347338i	$-0.887086\pi$
$373$	− 13.4164i	− 0.694675i	0.937740	−	0.347338i	$-0.112914\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.8885i	0.921307i
$378$	0	0
$379$	−26.8328	−1.37831	−0.689155	−	0.724614i	$-0.742018\pi$
$379$	−26.8328	−1.37831	−0.689155	+	0.724614i	$0.742018\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	20.0000	1.01929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	17.8885	0.904663
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 20.0000i	− 1.00631i
$396$	0	0
$397$	4.47214i	0.224450i	0.993683	+	0.112225i	$0.0357978\pi$
$397$	4.47214i	0.224450i	−0.993683	+	0.112225i	$0.964202\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	40.0000i	1.99254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000i	0.991363i
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	26.8328i	1.32036i
$414$	0	0
$415$	−8.94427	−0.439057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.3607	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$419$	22.3607	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	22.3607i	1.08465i
$426$	0	0
$427$	20.0000i	0.967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.8885	−0.861661	−0.430830	−	0.902433i	$-0.641779\pi$
$431$	−17.8885	−0.861661	−0.430830	+	0.902433i	$0.641779\pi$
$432$	0	0
$433$	− 26.8328i	− 1.28950i	−0.764392	−	0.644751i	$-0.776961\pi$
$433$	− 26.8328i	− 1.28950i	0.764392	−	0.644751i	$-0.223039\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.94427	0.426887	0.213443	−	0.976955i	$-0.431532\pi$
$439$	8.94427	0.426887	0.213443	+	0.976955i	$0.431532\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	− 13.4164i	− 0.635999i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−44.7214	−2.10585
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20.0000i	0.937614i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	− 6.00000i	− 0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$	− 6.00000i	− 0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17.8885i	0.822516i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.94427	0.408674	0.204337	−	0.978901i	$-0.434496\pi$
$479$	8.94427	0.408674	0.204337	+	0.978901i	$0.434496\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−40.0000	−1.81631
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.47214	−0.201825	−0.100912	−	0.994895i	$-0.532176\pi$
$491$	−4.47214	−0.201825	−0.100912	+	0.994895i	$0.532176\pi$
$492$	0	0
$493$	− 17.8885i	− 0.805659i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.8885i	0.802411i
$498$	0	0
$499$	17.8885	0.800801	0.400401	−	0.916340i	$-0.368871\pi$
$499$	17.8885	0.800801	0.400401	+	0.916340i	$0.368871\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	0	0
$505$	26.8328i	1.19404i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	17.8885	0.791343
$512$	0	0
$513$	0	0
$514$	0	0
$515$	31.3050	1.37946
$516$	0	0
$517$	− 35.7771i	− 1.57347i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	− 24.0000i	− 1.04945i	−0.851273	−	0.524723i	$-0.824169\pi$
$523$	− 24.0000i	− 1.04945i	0.851273	−	0.524723i	$-0.175831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 40.0000i	− 1.74243i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 44.7214i	− 1.93710i
$534$	0	0
$535$	26.8328	1.16008
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.4164	0.577886
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.3607i	0.957826i
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	17.8885i	0.760698i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.3050i	1.32643i	0.748427	+	0.663217i	$0.230809\pi$
$557$	31.3050i	1.32643i	−0.748427	+	0.663217i	$0.769191\pi$
$558$	0	0
$559$	−17.8885	−0.756605
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 20.0000i	− 0.834058i
$576$	0	0
$577$	17.8885i	0.744710i	0.928090	+	0.372355i	$0.121450\pi$
$577$	17.8885i	0.744710i	−0.928090	+	0.372355i	$0.878550\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	20.0000i	0.828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 40.2492i	− 1.65284i	−0.563056	−	0.826419i	$-0.690374\pi$
$593$	− 40.2492i	− 1.65284i	0.563056	−	0.826419i	$-0.309626\pi$
$594$	0	0
$595$	− 20.0000i	− 0.819920i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.8885	−0.730906	−0.365453	−	0.930830i	$-0.619086\pi$
$599$	−17.8885	−0.730906	−0.365453	+	0.930830i	$0.619086\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 20.1246i	− 0.818182i
$606$	0	0
$607$	− 42.0000i	− 1.70473i	−0.522949	−	0.852364i	$-0.675168\pi$
$607$	− 42.0000i	− 1.70473i	0.522949	−	0.852364i	$-0.324832\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.7771	1.44739
$612$	0	0
$613$	− 22.3607i	− 0.903139i	−0.892236	−	0.451570i	$-0.850864\pi$
$613$	− 22.3607i	− 0.903139i	0.892236	−	0.451570i	$-0.149136\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 22.3607i	− 0.900207i	−0.892976	−	0.450104i	$-0.851387\pi$
$617$	− 22.3607i	− 0.900207i	0.892976	−	0.450104i	$-0.148613\pi$
$618$	0	0
$619$	26.8328	1.07850	0.539251	−	0.842145i	$-0.318707\pi$
$619$	26.8328	1.07850	0.539251	+	0.842145i	$0.318707\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000i	0.480770i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	−8.94427	−0.356066	−0.178033	−	0.984025i	$-0.556973\pi$
$631$	−8.94427	−0.356066	−0.178033	+	0.984025i	$0.556973\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.47214	0.177471
$636$	0	0
$637$	13.4164i	0.531577i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	0	0
$649$	60.0000	2.35521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 22.3607i	− 0.875041i	−0.899208	−	0.437521i	$-0.855857\pi$
$653$	− 22.3607i	− 0.875041i	0.899208	−	0.437521i	$-0.144143\pi$
$654$	0	0
$655$	10.0000i	0.390732i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.47214	0.174210	0.0871048	−	0.996199i	$-0.472238\pi$
$659$	4.47214	0.174210	0.0871048	+	0.996199i	$0.472238\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.7214	1.72645
$672$	0	0
$673$	26.8328i	1.03433i	0.855886	+	0.517165i	$0.173012\pi$
$673$	26.8328i	1.03433i	−0.855886	+	0.517165i	$0.826988\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 31.3050i	− 1.20315i	−0.798817	−	0.601574i	$-0.794541\pi$
$677$	− 31.3050i	− 1.20315i	0.798817	−	0.601574i	$-0.205459\pi$
$678$	0	0
$679$	35.7771	1.37300
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	17.8885	0.680512	0.340256	−	0.940333i	$-0.389486\pi$
$691$	17.8885	0.680512	0.340256	+	0.940333i	$0.389486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 20.0000i	− 0.758643i
$696$	0	0
$697$	44.7214i	1.69394i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	35.7771i	1.33986i
$714$	0	0
$715$	44.7214	1.67248
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.7771	1.33426	0.667130	−	0.744941i	$-0.267522\pi$
$719$	35.7771	1.33426	0.667130	+	0.744941i	$0.267522\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	− 18.0000i	− 0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$	− 18.0000i	− 0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.8885	0.661632
$732$	0	0
$733$	13.4164i	0.495546i	0.968818	+	0.247773i	$0.0796988\pi$
$733$	13.4164i	0.495546i	−0.968818	+	0.247773i	$0.920301\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 35.7771i	− 1.31787i
$738$	0	0
$739$	−35.7771	−1.31608	−0.658041	−	0.752982i	$-0.728615\pi$
$739$	−35.7771	−1.31608	−0.658041	+	0.752982i	$0.728615\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20.0000i	0.727875i
$756$	0	0
$757$	− 13.4164i	− 0.487628i	−0.969822	−	0.243814i	$-0.921601\pi$
$757$	− 13.4164i	− 0.487628i	0.969822	−	0.243814i	$-0.0783986\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	− 20.0000i	− 0.724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	60.0000i	2.16647i
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.47214i	− 0.160852i	−0.996761	−	0.0804258i	$-0.974372\pi$
$773$	− 4.47214i	− 0.160852i	0.996761	−	0.0804258i	$-0.0256280\pi$
$774$	0	0
$775$	−44.7214	−1.60644
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.94427	0.318022
$792$	0	0
$793$	44.7214i	1.58810i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.47214i	0.158411i	0.996858	+	0.0792056i	$0.0252384\pi$
$797$	4.47214i	0.158411i	−0.996858	+	0.0792056i	$0.974762\pi$
$798$	0	0
$799$	−35.7771	−1.26570
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	17.8885i	0.630488i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	26.8328	0.942228	0.471114	−	0.882072i	$-0.343852\pi$
$811$	26.8328	0.942228	0.471114	+	0.882072i	$0.343852\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	53.6656	1.87983
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	0	0
$823$	26.0000i	0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$	26.0000i	0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 13.4164i	− 0.464851i
$834$	0	0
$835$	−26.8328	−0.928588
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.7771	1.23516	0.617581	−	0.786507i	$-0.288113\pi$
$839$	35.7771	1.23516	0.617581	+	0.786507i	$0.288113\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.6525i	0.538462i
$846$	0	0
$847$	18.0000i	0.618487i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−17.8885	−0.613211
$852$	0	0
$853$	− 40.2492i	− 1.37811i	−0.724710	−	0.689054i	$-0.758026\pi$
$853$	− 40.2492i	− 1.37811i	0.724710	−	0.689054i	$-0.241974\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.1935i	1.68042i	0.542263	+	0.840209i	$0.317568\pi$
$857$	49.1935i	1.68042i	−0.542263	+	0.840209i	$0.682432\pi$
$858$	0	0
$859$	44.7214	1.52587	0.762937	−	0.646473i	$-0.223757\pi$
$859$	44.7214	1.52587	0.762937	+	0.646473i	$0.223757\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	35.7771	1.21226
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−22.3607	−0.755929
$876$	0	0
$877$	13.4164i	0.453040i	0.974007	+	0.226520i	$0.0727348\pi$
$877$	13.4164i	0.453040i	−0.974007	+	0.226520i	$0.927265\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	50.0000i	1.67132i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35.7771	1.19323
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 40.2492i	− 1.33793i
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.7771	1.18535	0.592674	−	0.805443i	$-0.298072\pi$
$911$	35.7771	1.18535	0.592674	+	0.805443i	$0.298072\pi$
$912$	0	0
$913$	− 17.8885i	− 0.592024i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.94427i	− 0.295366i
$918$	0	0
$919$	8.94427	0.295044	0.147522	−	0.989059i	$-0.452870\pi$
$919$	8.94427	0.295044	0.147522	+	0.989059i	$0.452870\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.0000i	1.31662i
$924$	0	0
$925$	− 22.3607i	− 0.735215i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−44.7214	−1.46254
$936$	0	0
$937$	− 53.6656i	− 1.75318i	−0.481238	−	0.876590i	$-0.659813\pi$
$937$	− 53.6656i	− 1.75318i	0.481238	−	0.876590i	$-0.340187\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	0	0
$943$	− 40.0000i	− 1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	0	0
$949$	40.0000	1.29845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31.3050i	1.01407i	0.861926	+	0.507033i	$0.169258\pi$
$953$	31.3050i	1.01407i	−0.861926	+	0.507033i	$0.830742\pi$
$954$	0	0
$955$	20.0000i	0.647185i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.94427	−0.288826
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.0000	−0.643823
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.2492	−1.29166	−0.645830	−	0.763482i	$-0.723488\pi$
$971$	−40.2492	−1.29166	−0.645830	+	0.763482i	$0.723488\pi$
$972$	0	0
$973$	17.8885i	0.573480i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.1935i	1.57384i	0.617055	+	0.786920i	$0.288325\pi$
$977$	49.1935i	1.57384i	−0.617055	+	0.786920i	$0.711675\pi$
$978$	0	0
$979$	26.8328	0.857581
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−44.7214	−1.42062	−0.710310	−	0.703889i	$-0.751445\pi$
$991$	−44.7214	−1.42062	−0.710310	+	0.703889i	$0.751445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	60.0000i	1.90213i
$996$	0	0
$997$	− 4.47214i	− 0.141634i	−0.997489	−	0.0708170i	$-0.977439\pi$
$997$	− 4.47214i	− 0.141634i	0.997489	−	0.0708170i	$-0.0225606\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.f.h.289.2		4
3.2	odd	2		480.2.f.e.289.2	yes	4
4.3	odd	2	inner	1440.2.f.h.289.1		4
5.2	odd	4		7200.2.a.cc.1.2		2
5.3	odd	4		7200.2.a.cq.1.2		2
5.4	even	2	inner	1440.2.f.h.289.3		4
8.3	odd	2		2880.2.f.v.1729.3		4
8.5	even	2		2880.2.f.v.1729.4		4
12.11	even	2		480.2.f.e.289.4	yes	4
15.2	even	4		2400.2.a.bi.1.1		2
15.8	even	4		2400.2.a.bj.1.1		2
15.14	odd	2		480.2.f.e.289.3	yes	4
20.3	even	4		7200.2.a.cc.1.1		2
20.7	even	4		7200.2.a.cq.1.1		2
20.19	odd	2	inner	1440.2.f.h.289.4		4
24.5	odd	2		960.2.f.k.769.3		4
24.11	even	2		960.2.f.k.769.1		4
40.19	odd	2		2880.2.f.v.1729.2		4
40.29	even	2		2880.2.f.v.1729.1		4
48.5	odd	4		3840.2.d.bg.2689.1		4
48.11	even	4		3840.2.d.bh.2689.2		4
48.29	odd	4		3840.2.d.bh.2689.3		4
48.35	even	4		3840.2.d.bg.2689.4		4
60.23	odd	4		2400.2.a.bi.1.2		2
60.47	odd	4		2400.2.a.bj.1.2		2
60.59	even	2		480.2.f.e.289.1	✓	4
120.29	odd	2		960.2.f.k.769.2		4
120.53	even	4		4800.2.a.cu.1.2		2
120.59	even	2		960.2.f.k.769.4		4
120.77	even	4		4800.2.a.cv.1.2		2
120.83	odd	4		4800.2.a.cv.1.1		2
120.107	odd	4		4800.2.a.cu.1.1		2
240.29	odd	4		3840.2.d.bg.2689.2		4
240.59	even	4		3840.2.d.bg.2689.3		4
240.149	odd	4		3840.2.d.bh.2689.4		4
240.179	even	4		3840.2.d.bh.2689.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.f.e.289.1	✓	4	60.59	even	2
480.2.f.e.289.2	yes	4	3.2	odd	2
480.2.f.e.289.3	yes	4	15.14	odd	2
480.2.f.e.289.4	yes	4	12.11	even	2
960.2.f.k.769.1		4	24.11	even	2
960.2.f.k.769.2		4	120.29	odd	2
960.2.f.k.769.3		4	24.5	odd	2
960.2.f.k.769.4		4	120.59	even	2
1440.2.f.h.289.1		4	4.3	odd	2	inner
1440.2.f.h.289.2		4	1.1	even	1	trivial
1440.2.f.h.289.3		4	5.4	even	2	inner
1440.2.f.h.289.4		4	20.19	odd	2	inner
2400.2.a.bi.1.1		2	15.2	even	4
2400.2.a.bi.1.2		2	60.23	odd	4
2400.2.a.bj.1.1		2	15.8	even	4
2400.2.a.bj.1.2		2	60.47	odd	4
2880.2.f.v.1729.1		4	40.29	even	2
2880.2.f.v.1729.2		4	40.19	odd	2
2880.2.f.v.1729.3		4	8.3	odd	2
2880.2.f.v.1729.4		4	8.5	even	2
3840.2.d.bg.2689.1		4	48.5	odd	4
3840.2.d.bg.2689.2		4	240.29	odd	4
3840.2.d.bg.2689.3		4	240.59	even	4
3840.2.d.bg.2689.4		4	48.35	even	4
3840.2.d.bh.2689.1		4	240.179	even	4
3840.2.d.bh.2689.2		4	48.11	even	4
3840.2.d.bh.2689.3		4	48.29	odd	4
3840.2.d.bh.2689.4		4	240.149	odd	4
4800.2.a.cu.1.1		2	120.107	odd	4
4800.2.a.cu.1.2		2	120.53	even	4
4800.2.a.cv.1.1		2	120.83	odd	4
4800.2.a.cv.1.2		2	120.77	even	4
7200.2.a.cc.1.1		2	20.3	even	4
7200.2.a.cc.1.2		2	5.2	odd	4
7200.2.a.cq.1.1		2	20.7	even	4
7200.2.a.cq.1.2		2	5.3	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +2.00000i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +2.00000i q^{7} +4.47214 q^{11} +4.47214i q^{13} -4.47214i q^{17} +4.00000i q^{23} -5.00000 q^{25} +4.00000 q^{29} +8.94427 q^{31} +4.47214 q^{35} +4.47214i q^{37} -10.0000 q^{41} +4.00000i q^{43} -8.00000i q^{47} +3.00000 q^{49} +4.47214i q^{53} -10.0000i q^{55} +13.4164 q^{59} +10.0000 q^{61} +10.0000 q^{65} -8.00000i q^{67} +8.94427 q^{71} -8.94427i q^{73} +8.94427i q^{77} +8.94427 q^{79} -4.00000i q^{83} -10.0000 q^{85} +6.00000 q^{89} -8.94427 q^{91} -17.8885i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 20 q^{25} + 16 q^{29} - 40 q^{41} + 12 q^{49} + 40 q^{61} + 40 q^{65} - 40 q^{85} + 24 q^{89}+O(q^{100}) \) 4 * q - 20 * q^25 + 16 * q^29 - 40 * q^41 + 12 * q^49 + 40 * q^61 + 40 * q^65 - 40 * q^85 + 24 * q^89

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