LMFDB - Embedded newform 2880.2.h.b.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(1151,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2880.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.h (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:	$22.9969157821$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2880.1151
Dual form			2880.2.h.b.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +0.585786i q^{7} +O(q^{10})$	$q-1.00000i q^{5} +0.585786i q^{7} +1.41421 q^{11} -2.24264 q^{13} +1.17157i q^{17} -5.65685i q^{19} -3.17157 q^{23} -1.00000 q^{25} -2.00000i q^{29} -3.17157i q^{31} +0.585786 q^{35} -4.58579 q^{37} +8.24264i q^{41} -6.82843i q^{43} -8.00000 q^{47} +6.65685 q^{49} -6.82843i q^{53} -1.41421i q^{55} +5.41421 q^{59} +3.17157 q^{61} +2.24264i q^{65} -11.3137i q^{67} -13.6569 q^{71} +0.828427 q^{73} +0.828427i q^{77} -6.48528i q^{79} -1.17157 q^{83} +1.17157 q^{85} +0.928932i q^{89} -1.31371i q^{91} -5.65685 q^{95} +10.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 8 q^{13} - 24 q^{23} - 4 q^{25} + 8 q^{35} - 24 q^{37} - 32 q^{47} + 4 q^{49} + 16 q^{59} + 24 q^{61} - 32 q^{71} - 8 q^{73} - 16 q^{83} + 16 q^{85} + 8 q^{97}+O(q^{100}) $ 4 * q + 8 * q^13 - 24 * q^23 - 4 * q^25 + 8 * q^35 - 24 * q^37 - 32 * q^47 + 4 * q^49 + 16 * q^59 + 24 * q^61 - 32 * q^71 - 8 * q^73 - 16 * q^83 + 16 * q^85 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	0.585786i	0.221406i	0.993854	+	0.110703i	$0.0353103\pi$
$7$	0.585786i	0.221406i	−0.993854	+	0.110703i	$0.964690\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	−2.24264	−0.621997	−0.310998	−	0.950410i	$-0.600663\pi$
$13$	−2.24264	−0.621997	−0.310998	+	0.950410i	$0.600663\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.17157i	0.284148i	0.989856	+	0.142074i	$0.0453771\pi$
$17$	1.17157i	0.284148i	−0.989856	+	0.142074i	$0.954623\pi$
$18$	0	0
$19$	− 5.65685i	− 1.29777i	−0.760886	−	0.648886i	$-0.775235\pi$
$19$	− 5.65685i	− 1.29777i	0.760886	−	0.648886i	$-0.224765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	− 3.17157i	− 0.569631i	−0.958582	−	0.284816i	$-0.908068\pi$
$31$	− 3.17157i	− 0.569631i	0.958582	−	0.284816i	$-0.0919324\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.585786	0.0990160
$36$	0	0
$37$	−4.58579	−0.753899	−0.376949	−	0.926234i	$-0.623027\pi$
$37$	−4.58579	−0.753899	−0.376949	+	0.926234i	$0.623027\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.24264i	1.28728i	0.765327	+	0.643642i	$0.222577\pi$
$41$	8.24264i	1.28728i	−0.765327	+	0.643642i	$0.777423\pi$
$42$	0	0
$43$	− 6.82843i	− 1.04133i	−0.853762	−	0.520663i	$-0.825685\pi$
$43$	− 6.82843i	− 1.04133i	0.853762	−	0.520663i	$-0.174315\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	6.65685	0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.82843i	− 0.937957i	−0.883210	−	0.468978i	$-0.844622\pi$
$53$	− 6.82843i	− 0.937957i	0.883210	−	0.468978i	$-0.155378\pi$
$54$	0	0
$55$	− 1.41421i	− 0.190693i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.41421	0.704871	0.352435	−	0.935836i	$-0.385354\pi$
$59$	5.41421	0.704871	0.352435	+	0.935836i	$0.385354\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.24264i	0.278165i
$66$	0	0
$67$	− 11.3137i	− 1.38219i	−0.722764	−	0.691095i	$-0.757129\pi$
$67$	− 11.3137i	− 1.38219i	0.722764	−	0.691095i	$-0.242871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.6569	−1.62077	−0.810385	−	0.585897i	$-0.800742\pi$
$71$	−13.6569	−1.62077	−0.810385	+	0.585897i	$0.800742\pi$
$72$	0	0
$73$	0.828427	0.0969601	0.0484800	−	0.998824i	$-0.484562\pi$
$73$	0.828427	0.0969601	0.0484800	+	0.998824i	$0.484562\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.828427i	0.0944080i
$78$	0	0
$79$	− 6.48528i	− 0.729651i	−0.931076	−	0.364826i	$-0.881129\pi$
$79$	− 6.48528i	− 0.729651i	0.931076	−	0.364826i	$-0.118871\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.17157	−0.128597	−0.0642984	−	0.997931i	$-0.520481\pi$
$83$	−1.17157	−0.128597	−0.0642984	+	0.997931i	$0.520481\pi$
$84$	0	0
$85$	1.17157	0.127075
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.928932i	0.0984666i	0.998787	+	0.0492333i	$0.0156778\pi$
$89$	0.928932i	0.0984666i	−0.998787	+	0.0492333i	$0.984322\pi$
$90$	0	0
$91$	− 1.31371i	− 0.137714i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	10.4853	1.06462	0.532310	−	0.846550i	$-0.321324\pi$
$97$	10.4853	1.06462	0.532310	+	0.846550i	$0.321324\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 11.1716i	− 1.11161i	−0.831312	−	0.555807i	$-0.812410\pi$
$101$	− 11.1716i	− 1.11161i	0.831312	−	0.555807i	$-0.187590\pi$
$102$	0	0
$103$	− 2.24264i	− 0.220974i	−0.993878	−	0.110487i	$-0.964759\pi$
$103$	− 2.24264i	− 0.220974i	0.993878	−	0.110487i	$-0.0352410\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.1421	−1.75387	−0.876933	−	0.480612i	$-0.840414\pi$
$107$	−18.1421	−1.75387	−0.876933	+	0.480612i	$0.840414\pi$
$108$	0	0
$109$	−8.82843	−0.845610	−0.422805	−	0.906221i	$-0.638954\pi$
$109$	−8.82843	−0.845610	−0.422805	+	0.906221i	$0.638954\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	3.17157i	0.295751i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.686292	−0.0629122
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	12.5858i	1.11681i	0.829569	+	0.558404i	$0.188586\pi$
$127$	12.5858i	1.11681i	−0.829569	+	0.558404i	$0.811414\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.89949	0.515441	0.257721	−	0.966219i	$-0.417029\pi$
$131$	5.89949	0.515441	0.257721	+	0.966219i	$0.417029\pi$
$132$	0	0
$133$	3.31371	0.287335
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.65685i	− 0.483298i	−0.970364	−	0.241649i	$-0.922312\pi$
$137$	− 5.65685i	− 0.483298i	0.970364	−	0.241649i	$-0.0776882\pi$
$138$	0	0
$139$	− 19.6569i	− 1.66727i	−0.552314	−	0.833636i	$-0.686255\pi$
$139$	− 19.6569i	− 1.66727i	0.552314	−	0.833636i	$-0.313745\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.17157	−0.265220
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 0.828427i	− 0.0678674i	−0.999424	−	0.0339337i	$-0.989196\pi$
$149$	− 0.828427i	− 0.0678674i	0.999424	−	0.0339337i	$-0.0108035\pi$
$150$	0	0
$151$	− 22.9706i	− 1.86932i	−0.355546	−	0.934659i	$-0.615705\pi$
$151$	− 22.9706i	− 1.86932i	0.355546	−	0.934659i	$-0.384295\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.17157	−0.254747
$156$	0	0
$157$	−19.8995	−1.58815	−0.794076	−	0.607818i	$-0.792045\pi$
$157$	−19.8995	−1.58815	−0.794076	+	0.607818i	$0.792045\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.85786i	− 0.146420i
$162$	0	0
$163$	− 8.48528i	− 0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$	− 8.48528i	− 0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.48528	−0.192317	−0.0961584	−	0.995366i	$-0.530656\pi$
$167$	−2.48528	−0.192317	−0.0961584	+	0.995366i	$0.530656\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.4853i	1.25335i	0.779280	+	0.626676i	$0.215585\pi$
$173$	16.4853i	1.25335i	−0.779280	+	0.626676i	$0.784415\pi$
$174$	0	0
$175$	− 0.585786i	− 0.0442813i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.5858	−0.791219	−0.395609	−	0.918419i	$-0.629467\pi$
$179$	−10.5858	−0.791219	−0.395609	+	0.918419i	$0.629467\pi$
$180$	0	0
$181$	19.1716	1.42501	0.712506	−	0.701666i	$-0.247560\pi$
$181$	19.1716	1.42501	0.712506	+	0.701666i	$0.247560\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.58579i	0.337154i
$186$	0	0
$187$	1.65685i	0.121161i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.82843	−0.494088	−0.247044	−	0.969004i	$-0.579459\pi$
$191$	−6.82843	−0.494088	−0.247044	+	0.969004i	$0.579459\pi$
$192$	0	0
$193$	15.6569	1.12701	0.563503	−	0.826114i	$-0.309454\pi$
$193$	15.6569	1.12701	0.563503	+	0.826114i	$0.309454\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	− 4.34315i	− 0.307877i	−0.988080	−	0.153939i	$-0.950804\pi$
$199$	− 4.34315i	− 0.307877i	0.988080	−	0.153939i	$-0.0491958\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.17157	0.0822283
$204$	0	0
$205$	8.24264	0.575691
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 8.00000i	− 0.553372i
$210$	0	0
$211$	6.00000i	0.413057i	0.978441	+	0.206529i	$0.0662166\pi$
$211$	6.00000i	0.413057i	−0.978441	+	0.206529i	$0.933783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.82843	−0.465695
$216$	0	0
$217$	1.85786	0.126120
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.62742i	− 0.176739i
$222$	0	0
$223$	− 23.8995i	− 1.60043i	−0.599714	−	0.800214i	$-0.704719\pi$
$223$	− 23.8995i	− 1.60043i	0.599714	−	0.800214i	$-0.295281\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28.2843	−1.87729	−0.938647	−	0.344881i	$-0.887919\pi$
$227$	−28.2843	−1.87729	−0.938647	+	0.344881i	$0.887919\pi$
$228$	0	0
$229$	3.65685	0.241652	0.120826	−	0.992674i	$-0.461446\pi$
$229$	3.65685	0.241652	0.120826	+	0.992674i	$0.461446\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.48528i	− 0.293841i	−0.989148	−	0.146920i	$-0.953064\pi$
$233$	− 4.48528i	− 0.293841i	0.989148	−	0.146920i	$-0.0469361\pi$
$234$	0	0
$235$	8.00000i	0.521862i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.1421	1.43226	0.716128	−	0.697969i	$-0.245913\pi$
$239$	22.1421	1.43226	0.716128	+	0.697969i	$0.245913\pi$
$240$	0	0
$241$	−25.6569	−1.65270	−0.826352	−	0.563154i	$-0.809588\pi$
$241$	−25.6569	−1.65270	−0.826352	+	0.563154i	$0.809588\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.65685i	− 0.425291i
$246$	0	0
$247$	12.6863i	0.807209i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.72792	0.298424	0.149212	−	0.988805i	$-0.452326\pi$
$251$	4.72792	0.298424	0.149212	+	0.988805i	$0.452326\pi$
$252$	0	0
$253$	−4.48528	−0.281987
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 24.9706i	− 1.55762i	−0.627259	−	0.778810i	$-0.715823\pi$
$257$	− 24.9706i	− 1.55762i	0.627259	−	0.778810i	$-0.284177\pi$
$258$	0	0
$259$	− 2.68629i	− 0.166918i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.1716	1.18217	0.591085	−	0.806609i	$-0.298700\pi$
$263$	19.1716	1.18217	0.591085	+	0.806609i	$0.298700\pi$
$264$	0	0
$265$	−6.82843	−0.419467
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.9706i	1.40054i	0.713878	+	0.700270i	$0.246937\pi$
$269$	22.9706i	1.40054i	−0.713878	+	0.700270i	$0.753063\pi$
$270$	0	0
$271$	− 21.3137i	− 1.29472i	−0.762186	−	0.647358i	$-0.775874\pi$
$271$	− 21.3137i	− 1.29472i	0.762186	−	0.647358i	$-0.224126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	20.8701	1.25396	0.626980	−	0.779035i	$-0.284291\pi$
$277$	20.8701	1.25396	0.626980	+	0.779035i	$0.284291\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.8995i	1.06779i	0.845549	+	0.533897i	$0.179273\pi$
$281$	17.8995i	1.06779i	−0.845549	+	0.533897i	$0.820727\pi$
$282$	0	0
$283$	− 7.31371i	− 0.434755i	−0.976088	−	0.217377i	$-0.930250\pi$
$283$	− 7.31371i	− 0.434755i	0.976088	−	0.217377i	$-0.0697502\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.82843	−0.285013
$288$	0	0
$289$	15.6274	0.919260
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	0	0
$295$	− 5.41421i	− 0.315228i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.11270	0.411338
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.17157i	− 0.181604i
$306$	0	0
$307$	3.31371i	0.189123i	0.995519	+	0.0945617i	$0.0301450\pi$
$307$	3.31371i	0.189123i	−0.995519	+	0.0945617i	$0.969855\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.5147	0.652940	0.326470	−	0.945208i	$-0.394141\pi$
$311$	11.5147	0.652940	0.326470	+	0.945208i	$0.394141\pi$
$312$	0	0
$313$	−34.2843	−1.93786	−0.968931	−	0.247332i	$-0.920446\pi$
$313$	−34.2843	−1.93786	−0.968931	+	0.247332i	$0.920446\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.3137i	0.972435i	0.873838	+	0.486217i	$0.161624\pi$
$317$	17.3137i	0.972435i	−0.873838	+	0.486217i	$0.838376\pi$
$318$	0	0
$319$	− 2.82843i	− 0.158362i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.62742	0.368759
$324$	0	0
$325$	2.24264	0.124399
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.68629i	− 0.258364i
$330$	0	0
$331$	31.3137i	1.72116i	0.509318	+	0.860579i	$0.329898\pi$
$331$	31.3137i	1.72116i	−0.509318	+	0.860579i	$0.670102\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	16.8284	0.916703	0.458351	−	0.888771i	$-0.348440\pi$
$337$	16.8284	0.916703	0.458351	+	0.888771i	$0.348440\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4.48528i	− 0.242892i
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.9706	−1.12576	−0.562879	−	0.826539i	$-0.690306\pi$
$347$	−20.9706	−1.12576	−0.562879	+	0.826539i	$0.690306\pi$
$348$	0	0
$349$	−8.82843	−0.472575	−0.236287	−	0.971683i	$-0.575931\pi$
$349$	−8.82843	−0.472575	−0.236287	+	0.971683i	$0.575931\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 15.5147i	− 0.825765i	−0.910784	−	0.412883i	$-0.864522\pi$
$353$	− 15.5147i	− 0.825765i	0.910784	−	0.412883i	$-0.135478\pi$
$354$	0	0
$355$	13.6569i	0.724831i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.1127	1.64207	0.821033	−	0.570881i	$-0.193398\pi$
$359$	31.1127	1.64207	0.821033	+	0.570881i	$0.193398\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.828427i	− 0.0433619i
$366$	0	0
$367$	18.9289i	0.988082i	0.869439	+	0.494041i	$0.164481\pi$
$367$	18.9289i	0.988082i	−0.869439	+	0.494041i	$0.835519\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−2.92893	−0.151654	−0.0758272	−	0.997121i	$-0.524160\pi$
$373$	−2.92893	−0.151654	−0.0758272	+	0.997121i	$0.524160\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.48528i	0.231004i
$378$	0	0
$379$	− 6.68629i	− 0.343452i	−0.985145	−	0.171726i	$-0.945066\pi$
$379$	− 6.68629i	− 0.343452i	0.985145	−	0.171726i	$-0.0549343\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.6274	1.56499	0.782494	−	0.622658i	$-0.213947\pi$
$383$	30.6274	1.56499	0.782494	+	0.622658i	$0.213947\pi$
$384$	0	0
$385$	0.828427	0.0422206
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.65685i	0.185410i	0.995694	+	0.0927049i	$0.0295513\pi$
$389$	3.65685i	0.185410i	−0.995694	+	0.0927049i	$0.970449\pi$
$390$	0	0
$391$	− 3.71573i	− 0.187912i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.48528	−0.326310
$396$	0	0
$397$	−20.3848	−1.02308	−0.511541	−	0.859259i	$-0.670925\pi$
$397$	−20.3848	−1.02308	−0.511541	+	0.859259i	$0.670925\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 8.72792i	− 0.435852i	−0.975965	−	0.217926i	$-0.930071\pi$
$401$	− 8.72792i	− 0.435852i	0.975965	−	0.217926i	$-0.0699291\pi$
$402$	0	0
$403$	7.11270i	0.354309i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.48528	−0.321463
$408$	0	0
$409$	0.343146	0.0169675	0.00848373	−	0.999964i	$-0.497300\pi$
$409$	0.343146	0.0169675	0.00848373	+	0.999964i	$0.497300\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.17157i	0.156063i
$414$	0	0
$415$	1.17157i	0.0575103i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	27.7574	1.35604	0.678018	−	0.735045i	$-0.262839\pi$
$419$	27.7574	1.35604	0.678018	+	0.735045i	$0.262839\pi$
$420$	0	0
$421$	39.9411	1.94661	0.973306	−	0.229513i	$-0.0737132\pi$
$421$	39.9411	1.94661	0.973306	+	0.229513i	$0.0737132\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.17157i	− 0.0568296i
$426$	0	0
$427$	1.85786i	0.0899084i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.48528	−0.216048	−0.108024	−	0.994148i	$-0.534452\pi$
$431$	−4.48528	−0.216048	−0.108024	+	0.994148i	$0.534452\pi$
$432$	0	0
$433$	12.8284	0.616495	0.308247	−	0.951306i	$-0.400258\pi$
$433$	12.8284	0.616495	0.308247	+	0.951306i	$0.400258\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.9411i	0.858240i
$438$	0	0
$439$	− 15.6569i	− 0.747261i	−0.927578	−	0.373630i	$-0.878113\pi$
$439$	− 15.6569i	− 0.747261i	0.927578	−	0.373630i	$-0.121887\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.31371	−0.347485	−0.173742	−	0.984791i	$-0.555586\pi$
$443$	−7.31371	−0.347485	−0.173742	+	0.984791i	$0.555586\pi$
$444$	0	0
$445$	0.928932	0.0440356
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 25.2132i	− 1.18988i	−0.803768	−	0.594942i	$-0.797175\pi$
$449$	− 25.2132i	− 1.18988i	0.803768	−	0.594942i	$-0.202825\pi$
$450$	0	0
$451$	11.6569i	0.548900i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.31371	−0.0615876
$456$	0	0
$457$	3.85786	0.180463	0.0902316	−	0.995921i	$-0.471239\pi$
$457$	3.85786	0.180463	0.0902316	+	0.995921i	$0.471239\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 5.79899i	− 0.270086i	−0.990840	−	0.135043i	$-0.956883\pi$
$461$	− 5.79899i	− 0.270086i	0.990840	−	0.135043i	$-0.0431172\pi$
$462$	0	0
$463$	16.3848i	0.761465i	0.924685	+	0.380733i	$0.124328\pi$
$463$	16.3848i	0.761465i	−0.924685	+	0.380733i	$0.875672\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.7990	−1.28638	−0.643192	−	0.765705i	$-0.722390\pi$
$467$	−27.7990	−1.28638	−0.643192	+	0.765705i	$0.722390\pi$
$468$	0	0
$469$	6.62742	0.306026
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 9.65685i	− 0.444023i
$474$	0	0
$475$	5.65685i	0.259554i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	10.2843	0.468922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.4853i	− 0.476112i
$486$	0	0
$487$	42.0416i	1.90509i	0.304401	+	0.952544i	$0.401544\pi$
$487$	42.0416i	1.90509i	−0.304401	+	0.952544i	$0.598456\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.0711	1.22170	0.610850	−	0.791746i	$-0.290828\pi$
$491$	27.0711	1.22170	0.610850	+	0.791746i	$0.290828\pi$
$492$	0	0
$493$	2.34315	0.105530
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	− 20.9706i	− 0.938771i	−0.882993	−	0.469386i	$-0.844475\pi$
$499$	− 20.9706i	− 0.938771i	0.882993	−	0.469386i	$-0.155525\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−11.1716	−0.497128
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 8.14214i	− 0.360894i	−0.983585	−	0.180447i	$-0.942246\pi$
$509$	− 8.14214i	− 0.360894i	0.983585	−	0.180447i	$-0.0577544\pi$
$510$	0	0
$511$	0.485281i	0.0214676i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.24264	−0.0988226
$516$	0	0
$517$	−11.3137	−0.497576
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.7279i	1.43384i	0.697157	+	0.716918i	$0.254448\pi$
$521$	32.7279i	1.43384i	−0.697157	+	0.716918i	$0.745552\pi$
$522$	0	0
$523$	11.7990i	0.515934i	0.966154	+	0.257967i	$0.0830526\pi$
$523$	11.7990i	0.515934i	−0.966154	+	0.257967i	$0.916947\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.71573	0.161860
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 18.4853i	− 0.800686i
$534$	0	0
$535$	18.1421i	0.784353i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.41421	0.405499
$540$	0	0
$541$	−24.1421	−1.03795	−0.518976	−	0.854789i	$-0.673687\pi$
$541$	−24.1421	−1.03795	−0.518976	+	0.854789i	$0.673687\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.82843i	0.378168i
$546$	0	0
$547$	43.7990i	1.87271i	0.351055	+	0.936355i	$0.385823\pi$
$547$	43.7990i	1.87271i	−0.351055	+	0.936355i	$0.614177\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	3.79899	0.161549
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.51472i	0.148923i	0.997224	+	0.0744617i	$0.0237239\pi$
$557$	3.51472i	0.148923i	−0.997224	+	0.0744617i	$0.976276\pi$
$558$	0	0
$559$	15.3137i	0.647701i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.6569	1.41847	0.709234	−	0.704974i	$-0.249041\pi$
$563$	33.6569	1.41847	0.709234	+	0.704974i	$0.249041\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.4142i	1.56849i	0.620454	+	0.784243i	$0.286948\pi$
$569$	37.4142i	1.56849i	−0.620454	+	0.784243i	$0.713052\pi$
$570$	0	0
$571$	10.6274i	0.444744i	0.974962	+	0.222372i	$0.0713799\pi$
$571$	10.6274i	0.444744i	−0.974962	+	0.222372i	$0.928620\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.17157	0.132264
$576$	0	0
$577$	23.4558	0.976480	0.488240	−	0.872710i	$-0.337639\pi$
$577$	23.4558	0.976480	0.488240	+	0.872710i	$0.337639\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.686292i	− 0.0284722i
$582$	0	0
$583$	− 9.65685i	− 0.399946i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.4558	−0.555382	−0.277691	−	0.960670i	$-0.589569\pi$
$587$	−13.4558	−0.555382	−0.277691	+	0.960670i	$0.589569\pi$
$588$	0	0
$589$	−17.9411	−0.739251
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.6274i	1.25772i	0.777520	+	0.628859i	$0.216478\pi$
$593$	30.6274i	1.25772i	−0.777520	+	0.628859i	$0.783522\pi$
$594$	0	0
$595$	0.686292i	0.0281352i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	42.1421	1.72188	0.860940	−	0.508706i	$-0.169876\pi$
$599$	42.1421	1.72188	0.860940	+	0.508706i	$0.169876\pi$
$600$	0	0
$601$	−21.9411	−0.894997	−0.447499	−	0.894285i	$-0.647685\pi$
$601$	−21.9411	−0.894997	−0.447499	+	0.894285i	$0.647685\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.00000i	0.365902i
$606$	0	0
$607$	38.0416i	1.54406i	0.635585	+	0.772031i	$0.280759\pi$
$607$	38.0416i	1.54406i	−0.635585	+	0.772031i	$0.719241\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17.9411	0.725820
$612$	0	0
$613$	6.92893	0.279857	0.139928	−	0.990162i	$-0.455313\pi$
$613$	6.92893	0.279857	0.139928	+	0.990162i	$0.455313\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.8284i	0.758004i	0.925396	+	0.379002i	$0.123733\pi$
$617$	18.8284i	0.758004i	−0.925396	+	0.379002i	$0.876267\pi$
$618$	0	0
$619$	6.97056i	0.280171i	0.990139	+	0.140085i	$0.0447377\pi$
$619$	6.97056i	0.280171i	−0.990139	+	0.140085i	$0.955262\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.544156	−0.0218011
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 5.37258i	− 0.214219i
$630$	0	0
$631$	12.1421i	0.483371i	0.970355	+	0.241685i	$0.0777002\pi$
$631$	12.1421i	0.483371i	−0.970355	+	0.241685i	$0.922300\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.5858	0.499452
$636$	0	0
$637$	−14.9289	−0.591506
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 38.8701i	− 1.53527i	−0.640884	−	0.767637i	$-0.721432\pi$
$641$	− 38.8701i	− 1.53527i	0.640884	−	0.767637i	$-0.278568\pi$
$642$	0	0
$643$	− 6.82843i	− 0.269287i	−0.990894	−	0.134643i	$-0.957011\pi$
$643$	− 6.82843i	− 0.269287i	0.990894	−	0.134643i	$-0.0429889\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.9706	−1.92523	−0.962616	−	0.270871i	$-0.912688\pi$
$647$	−48.9706	−1.92523	−0.962616	+	0.270871i	$0.912688\pi$
$648$	0	0
$649$	7.65685	0.300558
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.2843i	1.02858i	0.857615	+	0.514292i	$0.171945\pi$
$653$	26.2843i	1.02858i	−0.857615	+	0.514292i	$0.828055\pi$
$654$	0	0
$655$	− 5.89949i	− 0.230512i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.0416	1.71562	0.857809	−	0.513968i	$-0.171825\pi$
$659$	44.0416	1.71562	0.857809	+	0.513968i	$0.171825\pi$
$660$	0	0
$661$	−0.142136	−0.00552844	−0.00276422	−	0.999996i	$-0.500880\pi$
$661$	−0.142136	−0.00552844	−0.00276422	+	0.999996i	$0.500880\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.31371i	− 0.128500i
$666$	0	0
$667$	6.34315i	0.245608i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.48528	0.173152
$672$	0	0
$673$	30.7696	1.18608	0.593040	−	0.805173i	$-0.297928\pi$
$673$	30.7696	1.18608	0.593040	+	0.805173i	$0.297928\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 29.3137i	− 1.12662i	−0.826247	−	0.563309i	$-0.809528\pi$
$677$	− 29.3137i	− 1.12662i	0.826247	−	0.563309i	$-0.190472\pi$
$678$	0	0
$679$	6.14214i	0.235714i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.7990	0.757587	0.378794	−	0.925481i	$-0.376339\pi$
$683$	19.7990	0.757587	0.378794	+	0.925481i	$0.376339\pi$
$684$	0	0
$685$	−5.65685	−0.216137
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.3137i	0.583406i
$690$	0	0
$691$	18.3431i	0.697806i	0.937159	+	0.348903i	$0.113446\pi$
$691$	18.3431i	0.697806i	−0.937159	+	0.348903i	$0.886554\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.6569	−0.745627
$696$	0	0
$697$	−9.65685	−0.365779
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 38.4853i	− 1.45357i	−0.686866	−	0.726785i	$-0.741014\pi$
$701$	− 38.4853i	− 1.45357i	0.686866	−	0.726785i	$-0.258986\pi$
$702$	0	0
$703$	25.9411i	0.978388i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.54416	0.246118
$708$	0	0
$709$	20.3431	0.764003	0.382001	−	0.924162i	$-0.375235\pi$
$709$	20.3431	0.764003	0.382001	+	0.924162i	$0.375235\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.0589i	0.376708i
$714$	0	0
$715$	3.17157i	0.118610i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.7990	0.440028	0.220014	−	0.975497i	$-0.429390\pi$
$719$	11.7990	0.440028	0.220014	+	0.975497i	$0.429390\pi$
$720$	0	0
$721$	1.31371	0.0489251
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	18.2426i	0.676582i	0.941042	+	0.338291i	$0.109849\pi$
$727$	18.2426i	0.676582i	−0.941042	+	0.338291i	$0.890151\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−6.24264	−0.230577	−0.115289	−	0.993332i	$-0.536779\pi$
$733$	−6.24264	−0.230577	−0.115289	+	0.993332i	$0.536779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	− 12.0000i	− 0.441427i	−0.975339	−	0.220714i	$-0.929161\pi$
$739$	− 12.0000i	− 0.441427i	0.975339	−	0.220714i	$-0.0708386\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.9706	1.50306	0.751532	−	0.659697i	$-0.229315\pi$
$743$	40.9706	1.50306	0.751532	+	0.659697i	$0.229315\pi$
$744$	0	0
$745$	−0.828427	−0.0303512
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10.6274i	− 0.388317i
$750$	0	0
$751$	17.7990i	0.649494i	0.945801	+	0.324747i	$0.105279\pi$
$751$	17.7990i	0.649494i	−0.945801	+	0.324747i	$0.894721\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.9706	−0.835984
$756$	0	0
$757$	0.585786	0.0212908	0.0106454	−	0.999943i	$-0.496611\pi$
$757$	0.585786	0.0212908	0.0106454	+	0.999943i	$0.496611\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.8701i	0.539039i	0.962995	+	0.269520i	$0.0868649\pi$
$761$	14.8701i	0.539039i	−0.962995	+	0.269520i	$0.913135\pi$
$762$	0	0
$763$	− 5.17157i	− 0.187224i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.1421	−0.438427
$768$	0	0
$769$	47.5980	1.71643	0.858214	−	0.513293i	$-0.171575\pi$
$769$	47.5980	1.71643	0.858214	+	0.513293i	$0.171575\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 47.2548i	− 1.69964i	−0.527074	−	0.849819i	$-0.676711\pi$
$773$	− 47.2548i	− 1.69964i	0.527074	−	0.849819i	$-0.323289\pi$
$774$	0	0
$775$	3.17157i	0.113926i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	46.6274	1.67060
$780$	0	0
$781$	−19.3137	−0.691099
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.8995i	0.710243i
$786$	0	0
$787$	2.14214i	0.0763589i	0.999271	+	0.0381794i	$0.0121558\pi$
$787$	2.14214i	0.0763589i	−0.999271	+	0.0381794i	$0.987844\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.34315	−0.0833127
$792$	0	0
$793$	−7.11270	−0.252579
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.4853i	− 0.583939i	−0.956428	−	0.291969i	$-0.905689\pi$
$797$	− 16.4853i	− 0.583939i	0.956428	−	0.291969i	$-0.0943105\pi$
$798$	0	0
$799$	− 9.37258i	− 0.331578i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.17157	0.0413439
$804$	0	0
$805$	−1.85786	−0.0654811
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 29.8995i	− 1.05121i	−0.850729	−	0.525605i	$-0.823839\pi$
$809$	− 29.8995i	− 1.05121i	0.850729	−	0.525605i	$-0.176161\pi$
$810$	0	0
$811$	− 32.6274i	− 1.14570i	−0.819659	−	0.572852i	$-0.805837\pi$
$811$	− 32.6274i	− 1.14570i	0.819659	−	0.572852i	$-0.194163\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.48528	−0.297226
$816$	0	0
$817$	−38.6274	−1.35140
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.3137i	0.604253i	0.953268	+	0.302126i	$0.0976964\pi$
$821$	17.3137i	0.604253i	−0.953268	+	0.302126i	$0.902304\pi$
$822$	0	0
$823$	16.8701i	0.588053i	0.955797	+	0.294027i	$0.0949954\pi$
$823$	16.8701i	0.588053i	−0.955797	+	0.294027i	$0.905005\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.37258	0.186823	0.0934115	−	0.995628i	$-0.470223\pi$
$827$	5.37258	0.186823	0.0934115	+	0.995628i	$0.470223\pi$
$828$	0	0
$829$	−30.4853	−1.05880	−0.529399	−	0.848373i	$-0.677582\pi$
$829$	−30.4853	−1.05880	−0.529399	+	0.848373i	$0.677582\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.79899i	0.270219i
$834$	0	0
$835$	2.48528i	0.0860067i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.51472	−0.259437	−0.129718	−	0.991551i	$-0.541407\pi$
$839$	−7.51472	−0.259437	−0.129718	+	0.991551i	$0.541407\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.97056i	0.274196i
$846$	0	0
$847$	− 5.27208i	− 0.181151i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.5442	0.498567
$852$	0	0
$853$	−4.78680	−0.163897	−0.0819484	−	0.996637i	$-0.526114\pi$
$853$	−4.78680	−0.163897	−0.0819484	+	0.996637i	$0.526114\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.5147i	1.07652i	0.842778	+	0.538261i	$0.180919\pi$
$857$	31.5147i	1.07652i	−0.842778	+	0.538261i	$0.819081\pi$
$858$	0	0
$859$	− 10.0000i	− 0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$859$	− 10.0000i	− 0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.7696	−0.775085	−0.387542	−	0.921852i	$-0.626676\pi$
$863$	−22.7696	−0.775085	−0.387542	+	0.921852i	$0.626676\pi$
$864$	0	0
$865$	16.4853	0.560516
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 9.17157i	− 0.311124i
$870$	0	0
$871$	25.3726i	0.859717i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.585786	−0.0198032
$876$	0	0
$877$	−7.41421	−0.250360	−0.125180	−	0.992134i	$-0.539951\pi$
$877$	−7.41421	−0.250360	−0.125180	+	0.992134i	$0.539951\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.5858i	0.760934i	0.924794	+	0.380467i	$0.124237\pi$
$881$	22.5858i	0.760934i	−0.924794	+	0.380467i	$0.875763\pi$
$882$	0	0
$883$	− 17.4558i	− 0.587436i	−0.955892	−	0.293718i	$-0.905107\pi$
$883$	− 17.4558i	− 0.587436i	0.955892	−	0.293718i	$-0.0948926\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.142136	−0.00477245	−0.00238622	−	0.999997i	$-0.500760\pi$
$887$	−0.142136	−0.00477245	−0.00238622	+	0.999997i	$0.500760\pi$
$888$	0	0
$889$	−7.37258	−0.247268
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.2548i	1.51440i
$894$	0	0
$895$	10.5858i	0.353844i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.34315	−0.211556
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.1716i	− 0.637285i
$906$	0	0
$907$	31.1127i	1.03308i	0.856263	+	0.516540i	$0.172780\pi$
$907$	31.1127i	1.03308i	−0.856263	+	0.516540i	$0.827220\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.2843	0.672048	0.336024	−	0.941853i	$-0.390918\pi$
$911$	20.2843	0.672048	0.336024	+	0.941853i	$0.390918\pi$
$912$	0	0
$913$	−1.65685	−0.0548339
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.45584i	0.114122i
$918$	0	0
$919$	− 9.51472i	− 0.313862i	−0.987610	−	0.156931i	$-0.949840\pi$
$919$	− 9.51472i	− 0.313862i	0.987610	−	0.156931i	$-0.0501600\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	30.6274	1.00811
$924$	0	0
$925$	4.58579	0.150780
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 7.27208i	− 0.238589i	−0.992859	−	0.119295i	$-0.961937\pi$
$929$	− 7.27208i	− 0.238589i	0.992859	−	0.119295i	$-0.0380633\pi$
$930$	0	0
$931$	− 37.6569i	− 1.23415i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.65685	0.0541849
$936$	0	0
$937$	14.2843	0.466647	0.233323	−	0.972399i	$-0.425040\pi$
$937$	14.2843	0.466647	0.233323	+	0.972399i	$0.425040\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 13.0294i	− 0.424748i	−0.977189	−	0.212374i	$-0.931881\pi$
$941$	− 13.0294i	− 0.424748i	0.977189	−	0.212374i	$-0.0681194\pi$
$942$	0	0
$943$	− 26.1421i	− 0.851305i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.0294	−0.358409	−0.179204	−	0.983812i	$-0.557352\pi$
$947$	−11.0294	−0.358409	−0.179204	+	0.983812i	$0.557352\pi$
$948$	0	0
$949$	−1.85786	−0.0603088
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 13.6569i	− 0.442389i	−0.975230	−	0.221194i	$-0.929004\pi$
$953$	− 13.6569i	− 0.442389i	0.975230	−	0.221194i	$-0.0709955\pi$
$954$	0	0
$955$	6.82843i	0.220963i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.31371	0.107005
$960$	0	0
$961$	20.9411	0.675520
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 15.6569i	− 0.504012i
$966$	0	0
$967$	0.100505i	0.00323202i	0.999999	+	0.00161601i	$0.000514393\pi$
$967$	0.100505i	0.00323202i	−0.999999	+	0.00161601i	$0.999486\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.6985	−0.953070	−0.476535	−	0.879156i	$-0.658107\pi$
$971$	−29.6985	−0.953070	−0.476535	+	0.879156i	$0.658107\pi$
$972$	0	0
$973$	11.5147	0.369145
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.51472i	0.240417i	0.992749	+	0.120209i	$0.0383563\pi$
$977$	7.51472i	0.240417i	−0.992749	+	0.120209i	$0.961644\pi$
$978$	0	0
$979$	1.31371i	0.0419863i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.31371	0.105691	0.0528454	−	0.998603i	$-0.483171\pi$
$983$	3.31371	0.105691	0.0528454	+	0.998603i	$0.483171\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.6569i	0.688648i
$990$	0	0
$991$	− 11.6569i	− 0.370292i	−0.982711	−	0.185146i	$-0.940724\pi$
$991$	− 11.6569i	− 0.370292i	0.982711	−	0.185146i	$-0.0592758\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.34315	−0.137687
$996$	0	0
$997$	25.7574	0.815744	0.407872	−	0.913039i	$-0.366271\pi$
$997$	25.7574	0.815744	0.407872	+	0.913039i	$0.366271\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.h.b.1151.1		4
3.2	odd	2		2880.2.h.c.1151.3		4
4.3	odd	2		2880.2.h.c.1151.2		4
8.3	odd	2		1440.2.h.c.1151.4	yes	4
8.5	even	2		1440.2.h.b.1151.3	yes	4
12.11	even	2	inner	2880.2.h.b.1151.4		4
24.5	odd	2		1440.2.h.c.1151.1	yes	4
24.11	even	2		1440.2.h.b.1151.2	✓	4
40.3	even	4		7200.2.o.d.7199.4		4
40.13	odd	4		7200.2.o.l.7199.2		4
40.19	odd	2		7200.2.h.e.1151.3		4
40.27	even	4		7200.2.o.k.7199.1		4
40.29	even	2		7200.2.h.f.1151.2		4
40.37	odd	4		7200.2.o.c.7199.3		4
120.29	odd	2		7200.2.h.e.1151.2		4
120.53	even	4		7200.2.o.k.7199.2		4
120.59	even	2		7200.2.h.f.1151.3		4
120.77	even	4		7200.2.o.d.7199.3		4
120.83	odd	4		7200.2.o.c.7199.4		4
120.107	odd	4		7200.2.o.l.7199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.h.b.1151.2	✓	4	24.11	even	2
1440.2.h.b.1151.3	yes	4	8.5	even	2
1440.2.h.c.1151.1	yes	4	24.5	odd	2
1440.2.h.c.1151.4	yes	4	8.3	odd	2
2880.2.h.b.1151.1		4	1.1	even	1	trivial
2880.2.h.b.1151.4		4	12.11	even	2	inner
2880.2.h.c.1151.2		4	4.3	odd	2
2880.2.h.c.1151.3		4	3.2	odd	2
7200.2.h.e.1151.2		4	120.29	odd	2
7200.2.h.e.1151.3		4	40.19	odd	2
7200.2.h.f.1151.2		4	40.29	even	2
7200.2.h.f.1151.3		4	120.59	even	2
7200.2.o.c.7199.3		4	40.37	odd	4
7200.2.o.c.7199.4		4	120.83	odd	4
7200.2.o.d.7199.3		4	120.77	even	4
7200.2.o.d.7199.4		4	40.3	even	4
7200.2.o.k.7199.1		4	40.27	even	4
7200.2.o.k.7199.2		4	120.53	even	4
7200.2.o.l.7199.1		4	120.107	odd	4
7200.2.o.l.7199.2		4	40.13	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +0.585786i q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} +0.585786i q^{7} +1.41421 q^{11} -2.24264 q^{13} +1.17157i q^{17} -5.65685i q^{19} -3.17157 q^{23} -1.00000 q^{25} -2.00000i q^{29} -3.17157i q^{31} +0.585786 q^{35} -4.58579 q^{37} +8.24264i q^{41} -6.82843i q^{43} -8.00000 q^{47} +6.65685 q^{49} -6.82843i q^{53} -1.41421i q^{55} +5.41421 q^{59} +3.17157 q^{61} +2.24264i q^{65} -11.3137i q^{67} -13.6569 q^{71} +0.828427 q^{73} +0.828427i q^{77} -6.48528i q^{79} -1.17157 q^{83} +1.17157 q^{85} +0.928932i q^{89} -1.31371i q^{91} -5.65685 q^{95} +10.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 8 q^{13} - 24 q^{23} - 4 q^{25} + 8 q^{35} - 24 q^{37} - 32 q^{47} + 4 q^{49} + 16 q^{59} + 24 q^{61} - 32 q^{71} - 8 q^{73} - 16 q^{83} + 16 q^{85} + 8 q^{97}+O(q^{100}) \) 4 * q + 8 * q^13 - 24 * q^23 - 4 * q^25 + 8 * q^35 - 24 * q^37 - 32 * q^47 + 4 * q^49 + 16 * q^59 + 24 * q^61 - 32 * q^71 - 8 * q^73 - 16 * q^83 + 16 * q^85 + 8 * q^97

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