LMFDB - Embedded newform 1440.2.h.c.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(1151,1440)]

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([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("1440.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.h (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:	$\Q(\zeta_{8})$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1440.1151
Dual form			1440.2.h.c.1151.1

$q$-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 $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

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

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 0.585786i	− 0.221406i	−0.993854	−	0.110703i	$-0.964690\pi$
$7$	− 0.585786i	− 0.221406i	0.993854	−	0.110703i	$-0.0353103\pi$
$8$	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	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	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	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	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	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	3.17157i	0.569631i	0.958582	+	0.284816i	$0.0919324\pi$
$31$	3.17157i	0.569631i	−0.958582	+	0.284816i	$0.908068\pi$
$32$	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	$-0.376973\pi$
$37$	4.58579	0.753899	0.376949	+	0.926234i	$0.376973\pi$
$38$	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	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	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	$0.155378\pi$
$53$	6.82843i	0.937957i	−0.883210	+	0.468978i	$0.844622\pi$
$54$	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	$-0.565082\pi$
$61$	−3.17157	−0.406078	−0.203039	+	0.979171i	$0.565082\pi$
$62$	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	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	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	$0.118871\pi$
$79$	6.48528i	0.729651i	−0.931076	+	0.364826i	$0.881129\pi$
$80$	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	$0.187590\pi$
$101$	11.1716i	1.11161i	−0.831312	+	0.555807i	$0.812410\pi$
$102$	0	0
$103$	2.24264i	0.220974i	0.993878	+	0.110487i	$0.0352410\pi$
$103$	2.24264i	0.220974i	−0.993878	+	0.110487i	$0.964759\pi$
$104$	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	$-0.361046\pi$
$109$	8.82843	0.845610	0.422805	+	0.906221i	$0.361046\pi$
$110$	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	$-0.811414\pi$
$127$	− 12.5858i	− 1.11681i	0.829569	−	0.558404i	$-0.188586\pi$
$128$	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	$0.0108035\pi$
$149$	0.828427i	0.0678674i	−0.999424	+	0.0339337i	$0.989196\pi$
$150$	0	0
$151$	22.9706i	1.86932i	0.355546	+	0.934659i	$0.384295\pi$
$151$	22.9706i	1.86932i	−0.355546	+	0.934659i	$0.615705\pi$
$152$	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	$-0.207955\pi$
$157$	19.8995	1.58815	0.794076	+	0.607818i	$0.207955\pi$
$158$	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	$-0.469344\pi$
$167$	2.48528	0.192317	0.0961584	+	0.995366i	$0.469344\pi$
$168$	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	$-0.784415\pi$
$173$	− 16.4853i	− 1.25335i	0.779280	−	0.626676i	$-0.215585\pi$
$174$	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	$-0.752440\pi$
$181$	−19.1716	−1.42501	−0.712506	+	0.701666i	$0.752440\pi$
$182$	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	$-0.420541\pi$
$191$	6.82843	0.494088	0.247044	+	0.969004i	$0.420541\pi$
$192$	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	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	4.34315i	0.307877i	0.988080	+	0.153939i	$0.0491958\pi$
$199$	4.34315i	0.307877i	−0.988080	+	0.153939i	$0.950804\pi$
$200$	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	$0.295281\pi$
$223$	23.8995i	1.60043i	−0.599714	+	0.800214i	$0.704719\pi$
$224$	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	$-0.538554\pi$
$229$	−3.65685	−0.241652	−0.120826	+	0.992674i	$0.538554\pi$
$230$	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	$-0.754087\pi$
$239$	−22.1421	−1.43226	−0.716128	+	0.697969i	$0.754087\pi$
$240$	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	$-0.701300\pi$
$263$	−19.1716	−1.18217	−0.591085	+	0.806609i	$0.701300\pi$
$264$	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	$-0.753063\pi$
$269$	− 22.9706i	− 1.40054i	0.713878	−	0.700270i	$-0.246937\pi$
$270$	0	0
$271$	21.3137i	1.29472i	0.762186	+	0.647358i	$0.224126\pi$
$271$	21.3137i	1.29472i	−0.762186	+	0.647358i	$0.775874\pi$
$272$	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	$-0.715709\pi$
$277$	−20.8701	−1.25396	−0.626980	+	0.779035i	$0.715709\pi$
$278$	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	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	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	$-0.605859\pi$
$311$	−11.5147	−0.652940	−0.326470	+	0.945208i	$0.605859\pi$
$312$	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	$-0.838376\pi$
$317$	− 17.3137i	− 0.972435i	0.873838	−	0.486217i	$-0.161624\pi$
$318$	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	$-0.424069\pi$
$349$	8.82843	0.472575	0.236287	+	0.971683i	$0.424069\pi$
$350$	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	$-0.806602\pi$
$359$	−31.1127	−1.64207	−0.821033	+	0.570881i	$0.806602\pi$
$360$	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	$-0.835519\pi$
$367$	− 18.9289i	− 0.988082i	0.869439	−	0.494041i	$-0.164481\pi$
$368$	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	$-0.475840\pi$
$373$	2.92893	0.151654	0.0758272	+	0.997121i	$0.475840\pi$
$374$	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	$-0.786053\pi$
$383$	−30.6274	−1.56499	−0.782494	+	0.622658i	$0.786053\pi$
$384$	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	$-0.970449\pi$
$389$	− 3.65685i	− 0.185410i	0.995694	−	0.0927049i	$-0.0295513\pi$
$390$	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	$-0.329075\pi$
$397$	20.3848	1.02308	0.511541	+	0.859259i	$0.329075\pi$
$398$	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	$-0.926287\pi$
$421$	−39.9411	−1.94661	−0.973306	+	0.229513i	$0.926287\pi$
$422$	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	$-0.465548\pi$
$431$	4.48528	0.216048	0.108024	+	0.994148i	$0.465548\pi$
$432$	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	$0.121887\pi$
$439$	15.6569i	0.747261i	−0.927578	+	0.373630i	$0.878113\pi$
$440$	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	$0.0431172\pi$
$461$	5.79899i	0.270086i	−0.990840	+	0.135043i	$0.956883\pi$
$462$	0	0
$463$	− 16.3848i	− 0.761465i	−0.924685	−	0.380733i	$-0.875672\pi$
$463$	− 16.3848i	− 0.761465i	0.924685	−	0.380733i	$-0.124328\pi$
$464$	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	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	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	$-0.598456\pi$
$487$	− 42.0416i	− 1.90509i	0.304401	−	0.952544i	$-0.401544\pi$
$488$	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	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	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	$0.0577544\pi$
$509$	8.14214i	0.360894i	−0.983585	+	0.180447i	$0.942246\pi$
$510$	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	$-0.326313\pi$
$541$	24.1421	1.03795	0.518976	+	0.854789i	$0.326313\pi$
$542$	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	$-0.976276\pi$
$557$	− 3.51472i	− 0.148923i	0.997224	−	0.0744617i	$-0.0237239\pi$
$558$	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	$-0.830124\pi$
$599$	−42.1421	−1.72188	−0.860940	+	0.508706i	$0.830124\pi$
$600$	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	$-0.719241\pi$
$607$	− 38.0416i	− 1.54406i	0.635585	−	0.772031i	$-0.280759\pi$
$608$	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	$-0.544687\pi$
$613$	−6.92893	−0.279857	−0.139928	+	0.990162i	$0.544687\pi$
$614$	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	$-0.922300\pi$
$631$	− 12.1421i	− 0.483371i	0.970355	−	0.241685i	$-0.0777002\pi$
$632$	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	$-0.0873116\pi$
$647$	48.9706	1.92523	0.962616	+	0.270871i	$0.0873116\pi$
$648$	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	$-0.828055\pi$
$653$	− 26.2843i	− 1.02858i	0.857615	−	0.514292i	$-0.171945\pi$
$654$	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	$-0.499120\pi$
$661$	0.142136	0.00552844	0.00276422	+	0.999996i	$0.499120\pi$
$662$	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	$0.190472\pi$
$677$	29.3137i	1.12662i	−0.826247	+	0.563309i	$0.809528\pi$
$678$	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	$0.258986\pi$
$701$	38.4853i	1.45357i	−0.686866	+	0.726785i	$0.741014\pi$
$702$	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	$-0.624765\pi$
$709$	−20.3431	−0.764003	−0.382001	+	0.924162i	$0.624765\pi$
$710$	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	$-0.570610\pi$
$719$	−11.7990	−0.440028	−0.220014	+	0.975497i	$0.570610\pi$
$720$	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	$-0.890151\pi$
$727$	− 18.2426i	− 0.676582i	0.941042	−	0.338291i	$-0.109849\pi$
$728$	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	$-0.463221\pi$
$733$	6.24264	0.230577	0.115289	+	0.993332i	$0.463221\pi$
$734$	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	$-0.770685\pi$
$743$	−40.9706	−1.50306	−0.751532	+	0.659697i	$0.770685\pi$
$744$	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	$-0.894721\pi$
$751$	− 17.7990i	− 0.649494i	0.945801	−	0.324747i	$-0.105279\pi$
$752$	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	$-0.503389\pi$
$757$	−0.585786	−0.0212908	−0.0106454	+	0.999943i	$0.503389\pi$
$758$	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	$0.323289\pi$
$773$	47.2548i	1.69964i	−0.527074	+	0.849819i	$0.676711\pi$
$774$	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	$0.0943105\pi$
$797$	16.4853i	0.583939i	−0.956428	+	0.291969i	$0.905689\pi$
$798$	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	$-0.902304\pi$
$821$	− 17.3137i	− 0.604253i	0.953268	−	0.302126i	$-0.0976964\pi$
$822$	0	0
$823$	− 16.8701i	− 0.588053i	−0.955797	−	0.294027i	$-0.905005\pi$
$823$	− 16.8701i	− 0.588053i	0.955797	−	0.294027i	$-0.0949954\pi$
$824$	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	$-0.322418\pi$
$829$	30.4853	1.05880	0.529399	+	0.848373i	$0.322418\pi$
$830$	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	$-0.458593\pi$
$839$	7.51472	0.259437	0.129718	+	0.991551i	$0.458593\pi$
$840$	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	$-0.473886\pi$
$853$	4.78680	0.163897	0.0819484	+	0.996637i	$0.473886\pi$
$854$	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	$-0.373324\pi$
$863$	22.7696	0.775085	0.387542	+	0.921852i	$0.373324\pi$
$864$	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	$-0.460049\pi$
$877$	7.41421	0.250360	0.125180	+	0.992134i	$0.460049\pi$
$878$	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	$-0.499240\pi$
$887$	0.142136	0.00477245	0.00238622	+	0.999997i	$0.499240\pi$
$888$	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	$-0.609082\pi$
$911$	−20.2843	−0.672048	−0.336024	+	0.941853i	$0.609082\pi$
$912$	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	$0.0501600\pi$
$919$	9.51472i	0.313862i	−0.987610	+	0.156931i	$0.949840\pi$
$920$	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	$0.0681194\pi$
$941$	13.0294i	0.424748i	−0.977189	+	0.212374i	$0.931881\pi$
$942$	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	$-0.999486\pi$
$967$	− 0.100505i	− 0.00323202i	0.999999	−	0.00161601i	$-0.000514393\pi$
$968$	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	$-0.516829\pi$
$983$	−3.31371	−0.105691	−0.0528454	+	0.998603i	$0.516829\pi$
$984$	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	$0.0592758\pi$
$991$	11.6569i	0.370292i	−0.982711	+	0.185146i	$0.940724\pi$
$992$	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	$-0.633729\pi$
$997$	−25.7574	−0.815744	−0.407872	+	0.913039i	$0.633729\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.h.c.1151.4	yes	4
3.2	odd	2		1440.2.h.b.1151.2	✓	4
4.3	odd	2		1440.2.h.b.1151.3	yes	4
5.2	odd	4		7200.2.o.k.7199.1		4
5.3	odd	4		7200.2.o.d.7199.4		4
5.4	even	2		7200.2.h.e.1151.3		4
8.3	odd	2		2880.2.h.b.1151.1		4
8.5	even	2		2880.2.h.c.1151.2		4
12.11	even	2	inner	1440.2.h.c.1151.1	yes	4
15.2	even	4		7200.2.o.l.7199.1		4
15.8	even	4		7200.2.o.c.7199.4		4
15.14	odd	2		7200.2.h.f.1151.3		4
20.3	even	4		7200.2.o.l.7199.2		4
20.7	even	4		7200.2.o.c.7199.3		4
20.19	odd	2		7200.2.h.f.1151.2		4
24.5	odd	2		2880.2.h.b.1151.4		4
24.11	even	2		2880.2.h.c.1151.3		4
60.23	odd	4		7200.2.o.k.7199.2		4
60.47	odd	4		7200.2.o.d.7199.3		4
60.59	even	2		7200.2.h.e.1151.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.h.b.1151.2	✓	4	3.2	odd	2
1440.2.h.b.1151.3	yes	4	4.3	odd	2
1440.2.h.c.1151.1	yes	4	12.11	even	2	inner
1440.2.h.c.1151.4	yes	4	1.1	even	1	trivial
2880.2.h.b.1151.1		4	8.3	odd	2
2880.2.h.b.1151.4		4	24.5	odd	2
2880.2.h.c.1151.2		4	8.5	even	2
2880.2.h.c.1151.3		4	24.11	even	2
7200.2.h.e.1151.2		4	60.59	even	2
7200.2.h.e.1151.3		4	5.4	even	2
7200.2.h.f.1151.2		4	20.19	odd	2
7200.2.h.f.1151.3		4	15.14	odd	2
7200.2.o.c.7199.3		4	20.7	even	4
7200.2.o.c.7199.4		4	15.8	even	4
7200.2.o.d.7199.3		4	60.47	odd	4
7200.2.o.d.7199.4		4	5.3	odd	4
7200.2.o.k.7199.1		4	5.2	odd	4
7200.2.o.k.7199.2		4	60.23	odd	4
7200.2.o.l.7199.1		4	15.2	even	4
7200.2.o.l.7199.2		4	20.3	even	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.h (of order \(2\), degree \(1\), 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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