LMFDB - Embedded newform 8280.2.p.a.1241.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 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("8280.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.p (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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.2
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.47

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -4.64793i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -4.64793i q^{7} +3.84841 q^{11} -6.83108 q^{13} -4.29829 q^{17} +6.20683i q^{19} +(2.23033 - 4.24566i) q^{23} +1.00000 q^{25} +1.90324i q^{29} +5.07055 q^{31} +4.64793i q^{35} -2.26790i q^{37} -10.8603i q^{41} +3.10929i q^{43} +3.10323i q^{47} -14.6033 q^{49} -8.56168 q^{53} -3.84841 q^{55} +8.43405i q^{59} +8.26761i q^{61} +6.83108 q^{65} -12.6821i q^{67} -10.5482i q^{71} -5.77129 q^{73} -17.8872i q^{77} -2.14676i q^{79} -14.9978 q^{83} +4.29829 q^{85} -2.95587 q^{89} +31.7504i q^{91} -6.20683i q^{95} +3.87584i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	4.64793i		−	1.75675i	−0.477969	−	0.878377i	$-0.658627\pi$
$7$		−	4.64793i		−	1.75675i	0.477969	−	0.878377i	$-0.341373\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.84841			1.16034			0.580170	−	0.814495i	$-0.302986\pi$
$11$	3.84841			1.16034			0.580170	+	0.814495i	$0.302986\pi$
$12$		0			0
$13$	−6.83108			−1.89460			−0.947300	−	0.320349i	$-0.896200\pi$
$13$	−6.83108			−1.89460			−0.947300	+	0.320349i	$0.896200\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.29829			−1.04249			−0.521244	−	0.853408i	$-0.674532\pi$
$17$	−4.29829			−1.04249			−0.521244	+	0.853408i	$0.674532\pi$
$18$		0			0
$19$			6.20683i			1.42394i	0.702208	+	0.711972i	$0.252198\pi$
$19$			6.20683i			1.42394i	−0.702208	+	0.711972i	$0.747802\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.23033	−	4.24566i	0.465056	−	0.885281i
$23$	2.23033	−	4.24566i	0.465056	−	0.885281i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.90324i			0.353423i	0.984263	+	0.176712i	$0.0565460\pi$
$29$			1.90324i			0.353423i	−0.984263	+	0.176712i	$0.943454\pi$
$30$		0			0
$31$	5.07055			0.910698			0.455349	−	0.890313i	$-0.349515\pi$
$31$	5.07055			0.910698			0.455349	+	0.890313i	$0.349515\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			4.64793i			0.785644i
$36$		0			0
$37$		−	2.26790i		−	0.372841i	−0.982470	−	0.186420i	$-0.940311\pi$
$37$		−	2.26790i		−	0.372841i	0.982470	−	0.186420i	$-0.0596887\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.8603i		−	1.69609i	−0.529927	−	0.848043i	$-0.677781\pi$
$41$		−	10.8603i		−	1.69609i	0.529927	−	0.848043i	$-0.322219\pi$
$42$		0			0
$43$			3.10929i			0.474162i	0.971490	+	0.237081i	$0.0761906\pi$
$43$			3.10929i			0.474162i	−0.971490	+	0.237081i	$0.923809\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.10323i			0.452652i	0.974052	+	0.226326i	$0.0726715\pi$
$47$			3.10323i			0.452652i	−0.974052	+	0.226326i	$0.927328\pi$
$48$		0			0
$49$	−14.6033			−2.08618
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.56168			−1.17604			−0.588018	−	0.808848i	$-0.700092\pi$
$53$	−8.56168			−1.17604			−0.588018	+	0.808848i	$0.700092\pi$
$54$		0			0
$55$	−3.84841			−0.518920
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.43405i			1.09802i	0.835816	+	0.549010i	$0.184995\pi$
$59$			8.43405i			1.09802i	−0.835816	+	0.549010i	$0.815005\pi$
$60$		0			0
$61$			8.26761i			1.05856i	0.848447	+	0.529280i	$0.177538\pi$
$61$			8.26761i			1.05856i	−0.848447	+	0.529280i	$0.822462\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.83108			0.847291
$66$		0			0
$67$		−	12.6821i		−	1.54936i	−0.632353	−	0.774680i	$-0.717911\pi$
$67$		−	12.6821i		−	1.54936i	0.632353	−	0.774680i	$-0.282089\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.5482i		−	1.25184i	−0.779886	−	0.625922i	$-0.784723\pi$
$71$		−	10.5482i		−	1.25184i	0.779886	−	0.625922i	$-0.215277\pi$
$72$		0			0
$73$	−5.77129			−0.675479			−0.337739	−	0.941240i	$-0.609662\pi$
$73$	−5.77129			−0.675479			−0.337739	+	0.941240i	$0.609662\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	17.8872i		−	2.03843i
$78$		0			0
$79$		−	2.14676i		−	0.241530i	−0.992681	−	0.120765i	$-0.961465\pi$
$79$		−	2.14676i		−	0.241530i	0.992681	−	0.120765i	$-0.0385347\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.9978			−1.64622			−0.823109	−	0.567883i	$-0.807762\pi$
$83$	−14.9978			−1.64622			−0.823109	+	0.567883i	$0.807762\pi$
$84$		0			0
$85$	4.29829			0.466215
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.95587			−0.313322			−0.156661	−	0.987652i	$-0.550073\pi$
$89$	−2.95587			−0.313322			−0.156661	+	0.987652i	$0.550073\pi$
$90$		0			0
$91$			31.7504i			3.32834i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	6.20683i		−	0.636807i
$96$		0			0
$97$			3.87584i			0.393532i	0.980450	+	0.196766i	$0.0630439\pi$
$97$			3.87584i			0.393532i	−0.980450	+	0.196766i	$0.936956\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	0.965924i		−	0.0961130i	−0.998845	−	0.0480565i	$-0.984697\pi$
$101$		−	0.965924i		−	0.0961130i	0.998845	−	0.0480565i	$-0.0153028\pi$
$102$		0			0
$103$			4.95069i			0.487806i	0.969800	+	0.243903i	$0.0784278\pi$
$103$			4.95069i			0.487806i	−0.969800	+	0.243903i	$0.921572\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.8916			1.34295			0.671475	−	0.741027i	$-0.265661\pi$
$107$	13.8916			1.34295			0.671475	+	0.741027i	$0.265661\pi$
$108$		0			0
$109$			5.90475i			0.565573i	0.959183	+	0.282786i	$0.0912588\pi$
$109$			5.90475i			0.565573i	−0.959183	+	0.282786i	$0.908741\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.33678			−0.313898			−0.156949	−	0.987607i	$-0.550166\pi$
$113$	−3.33678			−0.313898			−0.156949	+	0.987607i	$0.550166\pi$
$114$		0			0
$115$	−2.23033	+	4.24566i	−0.207979	+	0.395910i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			19.9782i			1.83139i
$120$		0			0
$121$	3.81029			0.346390
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	5.39573			0.478794			0.239397	−	0.970922i	$-0.423050\pi$
$127$	5.39573			0.478794			0.239397	+	0.970922i	$0.423050\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			20.2823i			1.77208i	0.463612	+	0.886038i	$0.346553\pi$
$131$			20.2823i			1.77208i	−0.463612	+	0.886038i	$0.653447\pi$
$132$		0			0
$133$	28.8489			2.50152
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−21.4696			−1.83427			−0.917135	−	0.398577i	$-0.869504\pi$
$137$	−21.4696			−1.83427			−0.917135	+	0.398577i	$0.869504\pi$
$138$		0			0
$139$	−7.68441			−0.651783			−0.325892	−	0.945407i	$-0.605664\pi$
$139$	−7.68441			−0.651783			−0.325892	+	0.945407i	$0.605664\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−26.2888			−2.19838
$144$		0			0
$145$		−	1.90324i		−	0.158056i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.834989			−0.0684049			−0.0342025	−	0.999415i	$-0.510889\pi$
$149$	−0.834989			−0.0684049			−0.0342025	+	0.999415i	$0.510889\pi$
$150$		0			0
$151$	8.85860			0.720902			0.360451	−	0.932778i	$-0.382623\pi$
$151$	8.85860			0.720902			0.360451	+	0.932778i	$0.382623\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.07055			−0.407276
$156$		0			0
$157$			22.5413i			1.79899i	0.436932	+	0.899494i	$0.356065\pi$
$157$			22.5413i			1.79899i	−0.436932	+	0.899494i	$0.643935\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−19.7335	−	10.3664i	−1.55522	−	0.816989i
$162$		0			0
$163$	−6.61191			−0.517884			−0.258942	−	0.965893i	$-0.583374\pi$
$163$	−6.61191			−0.517884			−0.258942	+	0.965893i	$0.583374\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.89912i			0.224340i	0.993689	+	0.112170i	$0.0357802\pi$
$167$			2.89912i			0.224340i	−0.993689	+	0.112170i	$0.964220\pi$
$168$		0			0
$169$	33.6636			2.58951
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4.97514i		−	0.378253i	−0.981953	−	0.189127i	$-0.939434\pi$
$173$		−	4.97514i		−	0.378253i	0.981953	−	0.189127i	$-0.0605656\pi$
$174$		0			0
$175$		−	4.64793i		−	0.351351i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			10.9352i			0.817337i	0.912683	+	0.408668i	$0.134007\pi$
$179$			10.9352i			0.817337i	−0.912683	+	0.408668i	$0.865993\pi$
$180$		0			0
$181$			19.3798i			1.44049i	0.693722	+	0.720243i	$0.255970\pi$
$181$			19.3798i			1.44049i	−0.693722	+	0.720243i	$0.744030\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			2.26790i			0.166740i
$186$		0			0
$187$	−16.5416			−1.20964
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.36656			0.533026			0.266513	−	0.963831i	$-0.414129\pi$
$191$	7.36656			0.533026			0.266513	+	0.963831i	$0.414129\pi$
$192$		0			0
$193$	10.5605			0.760164			0.380082	−	0.924953i	$-0.375896\pi$
$193$	10.5605			0.760164			0.380082	+	0.924953i	$0.375896\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.07854i		−	0.646820i	−0.946259	−	0.323410i	$-0.895171\pi$
$197$		−	9.07854i		−	0.646820i	0.946259	−	0.323410i	$-0.104829\pi$
$198$		0			0
$199$			2.82488i			0.200250i	0.994975	+	0.100125i	$0.0319243\pi$
$199$			2.82488i			0.200250i	−0.994975	+	0.100125i	$0.968076\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.84614			0.620877
$204$		0			0
$205$			10.8603i			0.758513i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			23.8864i			1.65226i
$210$		0			0
$211$	21.7417			1.49676			0.748381	−	0.663269i	$-0.230831\pi$
$211$	21.7417			1.49676			0.748381	+	0.663269i	$0.230831\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.10929i		−	0.212052i
$216$		0			0
$217$		−	23.5676i		−	1.59987i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	29.3619			1.97510
$222$		0			0
$223$	24.0275			1.60900			0.804500	−	0.593952i	$-0.202433\pi$
$223$	24.0275			1.60900			0.804500	+	0.593952i	$0.202433\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.82958			−0.586040			−0.293020	−	0.956106i	$-0.594660\pi$
$227$	−8.82958			−0.586040			−0.293020	+	0.956106i	$0.594660\pi$
$228$		0			0
$229$			19.9336i			1.31725i	0.752472	+	0.658625i	$0.228861\pi$
$229$			19.9336i			1.31725i	−0.752472	+	0.658625i	$0.771139\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.6603i			1.09145i	0.837963	+	0.545727i	$0.183746\pi$
$233$			16.6603i			1.09145i	−0.837963	+	0.545727i	$0.816254\pi$
$234$		0			0
$235$		−	3.10323i		−	0.202432i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			3.31719i			0.214571i	0.994228	+	0.107286i	$0.0342159\pi$
$239$			3.31719i			0.214571i	−0.994228	+	0.107286i	$0.965784\pi$
$240$		0			0
$241$		−	12.6323i		−	0.813715i	−0.913492	−	0.406857i	$-0.866625\pi$
$241$		−	12.6323i		−	0.813715i	0.913492	−	0.406857i	$-0.133375\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.6033			0.932969
$246$		0			0
$247$		−	42.3993i		−	2.69780i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.681418			−0.0430107			−0.0215054	−	0.999769i	$-0.506846\pi$
$251$	−0.681418			−0.0430107			−0.0215054	+	0.999769i	$0.506846\pi$
$252$		0			0
$253$	8.58323	−	16.3391i	0.539623	−	1.02723i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			20.8424i			1.30012i	0.759885	+	0.650058i	$0.225255\pi$
$257$			20.8424i			1.30012i	−0.759885	+	0.650058i	$0.774745\pi$
$258$		0			0
$259$	−10.5411			−0.654990
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.61790			−0.346415			−0.173207	−	0.984885i	$-0.555413\pi$
$263$	−5.61790			−0.346415			−0.173207	+	0.984885i	$0.555413\pi$
$264$		0			0
$265$	8.56168			0.525940
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	22.4317i		−	1.36768i	−0.729631	−	0.683841i	$-0.760308\pi$
$269$		−	22.4317i		−	1.36768i	0.729631	−	0.683841i	$-0.239692\pi$
$270$		0			0
$271$	28.2038			1.71326			0.856630	−	0.515931i	$-0.172554\pi$
$271$	28.2038			1.71326			0.856630	+	0.515931i	$0.172554\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.84841			0.232068
$276$		0			0
$277$	25.1174			1.50916			0.754580	−	0.656208i	$-0.227841\pi$
$277$	25.1174			1.50916			0.754580	+	0.656208i	$0.227841\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.4945			1.34191			0.670954	−	0.741499i	$-0.265885\pi$
$281$	22.4945			1.34191			0.670954	+	0.741499i	$0.265885\pi$
$282$		0			0
$283$		−	3.78317i		−	0.224886i	−0.993658	−	0.112443i	$-0.964132\pi$
$283$		−	3.78317i		−	0.224886i	0.993658	−	0.112443i	$-0.0358676\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−50.4777			−2.97961
$288$		0			0
$289$	1.47528			0.0867813
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.45783			0.0851673			0.0425836	−	0.999093i	$-0.486441\pi$
$293$	1.45783			0.0851673			0.0425836	+	0.999093i	$0.486441\pi$
$294$		0			0
$295$		−	8.43405i		−	0.491049i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.2356	+	29.0024i	−0.881095	+	1.67725i
$300$		0			0
$301$	14.4517			0.832985
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	8.26761i		−	0.473402i
$306$		0			0
$307$	−34.1122			−1.94688			−0.973442	−	0.228934i	$-0.926476\pi$
$307$	−34.1122			−1.94688			−0.973442	+	0.228934i	$0.926476\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	11.6163i		−	0.658698i	−0.944208	−	0.329349i	$-0.893171\pi$
$311$		−	11.6163i		−	0.658698i	0.944208	−	0.329349i	$-0.106829\pi$
$312$		0			0
$313$			1.12369i			0.0635145i	0.999496	+	0.0317573i	$0.0101103\pi$
$313$			1.12369i			0.0635145i	−0.999496	+	0.0317573i	$0.989890\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.4146i			1.03427i	0.855905	+	0.517134i	$0.173001\pi$
$317$			18.4146i			1.03427i	−0.855905	+	0.517134i	$0.826999\pi$
$318$		0			0
$319$			7.32446i			0.410091i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	26.6787i		−	1.48444i
$324$		0			0
$325$	−6.83108			−0.378920
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.4236			0.795198
$330$		0			0
$331$	−17.2424			−0.947726			−0.473863	−	0.880599i	$-0.657141\pi$
$331$	−17.2424			−0.947726			−0.473863	+	0.880599i	$0.657141\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			12.6821i			0.692895i
$336$		0			0
$337$			29.3360i			1.59804i	0.601307	+	0.799018i	$0.294647\pi$
$337$			29.3360i			1.59804i	−0.601307	+	0.799018i	$0.705353\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	19.5136			1.05672
$342$		0			0
$343$			35.3395i			1.90815i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			5.31183i			0.285154i	0.989784	+	0.142577i	$0.0455388\pi$
$347$			5.31183i			0.285154i	−0.989784	+	0.142577i	$0.954461\pi$
$348$		0			0
$349$	−25.2846			−1.35345			−0.676727	−	0.736234i	$-0.736602\pi$
$349$	−25.2846			−1.35345			−0.676727	+	0.736234i	$0.736602\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.0786i			1.06867i	0.845271	+	0.534337i	$0.179439\pi$
$353$			20.0786i			1.06867i	−0.845271	+	0.534337i	$0.820561\pi$
$354$		0			0
$355$			10.5482i			0.559842i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−27.1966			−1.43538			−0.717690	−	0.696363i	$-0.754800\pi$
$359$	−27.1966			−1.43538			−0.717690	+	0.696363i	$0.754800\pi$
$360$		0			0
$361$	−19.5247			−1.02762
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.77129			0.302083
$366$		0			0
$367$		−	21.7769i		−	1.13674i	−0.822772	−	0.568372i	$-0.807574\pi$
$367$		−	21.7769i		−	1.13674i	0.822772	−	0.568372i	$-0.192426\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			39.7941i			2.06601i
$372$		0			0
$373$		−	8.34201i		−	0.431933i	−0.976401	−	0.215967i	$-0.930710\pi$
$373$		−	8.34201i		−	0.431933i	0.976401	−	0.215967i	$-0.0692902\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	13.0012i		−	0.669595i
$378$		0			0
$379$			31.2838i			1.60694i	0.595344	+	0.803471i	$0.297016\pi$
$379$			31.2838i			1.60694i	−0.595344	+	0.803471i	$0.702984\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	27.0099			1.38014			0.690070	−	0.723743i	$-0.257580\pi$
$383$	27.0099			1.38014			0.690070	+	0.723743i	$0.257580\pi$
$384$		0			0
$385$			17.8872i			0.911614i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.2140			0.923485			0.461743	−	0.887014i	$-0.347224\pi$
$389$	18.2140			0.923485			0.461743	+	0.887014i	$0.347224\pi$
$390$		0			0
$391$	−9.58660	+	18.2491i	−0.484815	+	0.922895i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			2.14676i			0.108015i
$396$		0			0
$397$	17.5817			0.882399			0.441199	−	0.897409i	$-0.354553\pi$
$397$	17.5817			0.882399			0.441199	+	0.897409i	$0.354553\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−30.1783			−1.50703			−0.753515	−	0.657431i	$-0.771643\pi$
$401$	−30.1783			−1.50703			−0.753515	+	0.657431i	$0.771643\pi$
$402$		0			0
$403$	−34.6373			−1.72541
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.72783i		−	0.432622i
$408$		0			0
$409$	−3.55891			−0.175977			−0.0879885	−	0.996121i	$-0.528044\pi$
$409$	−3.55891			−0.175977			−0.0879885	+	0.996121i	$0.528044\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	39.2009			1.92895
$414$		0			0
$415$	14.9978			0.736211
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−27.4135			−1.33924			−0.669619	−	0.742705i	$-0.733542\pi$
$419$	−27.4135			−1.33924			−0.669619	+	0.742705i	$0.733542\pi$
$420$		0			0
$421$			29.0950i			1.41800i	0.705206	+	0.709002i	$0.250855\pi$
$421$			29.0950i			1.41800i	−0.705206	+	0.709002i	$0.749145\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.29829			−0.208498
$426$		0			0
$427$	38.4273			1.85963
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.81593			−0.472817			−0.236408	−	0.971654i	$-0.575970\pi$
$431$	−9.81593			−0.472817			−0.236408	+	0.971654i	$0.575970\pi$
$432$		0			0
$433$			28.6704i			1.37781i	0.724851	+	0.688905i	$0.241908\pi$
$433$			28.6704i			1.37781i	−0.724851	+	0.688905i	$0.758092\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	26.3521	+	13.8433i	1.26059	+	0.662214i
$438$		0			0
$439$	32.7417			1.56267			0.781337	−	0.624109i	$-0.214538\pi$
$439$	32.7417			1.56267			0.781337	+	0.624109i	$0.214538\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			35.3787i			1.68089i	0.541894	+	0.840447i	$0.317707\pi$
$443$			35.3787i			1.68089i	−0.541894	+	0.840447i	$0.682293\pi$
$444$		0			0
$445$	2.95587			0.140122
$446$		0			0
$447$		0			0
$448$		0			0
$449$			12.2277i			0.577061i	0.957471	+	0.288530i	$0.0931666\pi$
$449$			12.2277i			0.577061i	−0.957471	+	0.288530i	$0.906833\pi$
$450$		0			0
$451$		−	41.7947i		−	1.96804i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	31.7504i		−	1.48848i
$456$		0			0
$457$			28.2564i			1.32178i	0.750483	+	0.660890i	$0.229821\pi$
$457$			28.2564i			1.32178i	−0.750483	+	0.660890i	$0.770179\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	41.9856i		−	1.95547i	−0.209854	−	0.977733i	$-0.567299\pi$
$461$		−	41.9856i		−	1.95547i	0.209854	−	0.977733i	$-0.432701\pi$
$462$		0			0
$463$	−24.9376			−1.15895			−0.579474	−	0.814991i	$-0.696742\pi$
$463$	−24.9376			−1.15895			−0.579474	+	0.814991i	$0.696742\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.52227			0.301815			0.150907	−	0.988548i	$-0.451781\pi$
$467$	6.52227			0.301815			0.150907	+	0.988548i	$0.451781\pi$
$468$		0			0
$469$	−58.9454			−2.72184
$470$		0			0
$471$		0			0
$472$		0			0
$473$			11.9658i			0.550189i
$474$		0			0
$475$			6.20683i			0.284789i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.49609			−0.159741			−0.0798703	−	0.996805i	$-0.525451\pi$
$479$	−3.49609			−0.159741			−0.0798703	+	0.996805i	$0.525451\pi$
$480$		0			0
$481$			15.4922i			0.706384i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	3.87584i		−	0.175993i
$486$		0			0
$487$	12.9238			0.585633			0.292816	−	0.956169i	$-0.405408\pi$
$487$	12.9238			0.585633			0.292816	+	0.956169i	$0.405408\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.5515i		−	1.01773i	−0.860845	−	0.508867i	$-0.830064\pi$
$491$		−	22.5515i		−	1.01773i	0.860845	−	0.508867i	$-0.169936\pi$
$492$		0			0
$493$		−	8.18068i		−	0.368439i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−49.0274			−2.19918
$498$		0			0
$499$	35.2380			1.57747			0.788735	−	0.614733i	$-0.210736\pi$
$499$	35.2380			1.57747			0.788735	+	0.614733i	$0.210736\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.67358			0.119209			0.0596046	−	0.998222i	$-0.481016\pi$
$503$	2.67358			0.119209			0.0596046	+	0.998222i	$0.481016\pi$
$504$		0			0
$505$			0.965924i			0.0429830i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	0.925217i		−	0.0410095i	−0.999790	−	0.0205048i	$-0.993473\pi$
$509$		−	0.925217i		−	0.0410095i	0.999790	−	0.0205048i	$-0.00652733\pi$
$510$		0			0
$511$			26.8246i			1.18665i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	4.95069i		−	0.218153i
$516$		0			0
$517$			11.9425i			0.525231i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22.0897			0.967767			0.483884	−	0.875132i	$-0.339226\pi$
$521$	22.0897			0.967767			0.483884	+	0.875132i	$0.339226\pi$
$522$		0			0
$523$			30.9483i			1.35327i	0.736316	+	0.676637i	$0.236564\pi$
$523$			30.9483i			1.35327i	−0.736316	+	0.676637i	$0.763436\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.7947			−0.949392
$528$		0			0
$529$	−13.0513	−	18.9384i	−0.567446	−	0.823411i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			74.1872i			3.21340i
$534$		0			0
$535$	−13.8916			−0.600586
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−56.1994			−2.42068
$540$		0			0
$541$	−9.40237			−0.404240			−0.202120	−	0.979361i	$-0.564783\pi$
$541$	−9.40237			−0.404240			−0.202120	+	0.979361i	$0.564783\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	5.90475i		−	0.252932i
$546$		0			0
$547$	−15.9774			−0.683143			−0.341571	−	0.939856i	$-0.610959\pi$
$547$	−15.9774			−0.683143			−0.341571	+	0.939856i	$0.610959\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11.8131			−0.503255
$552$		0			0
$553$	−9.97801			−0.424308
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−26.3096			−1.11477			−0.557387	−	0.830253i	$-0.688196\pi$
$557$	−26.3096			−1.11477			−0.557387	+	0.830253i	$0.688196\pi$
$558$		0			0
$559$		−	21.2398i		−	0.898346i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.5075			−0.695710			−0.347855	−	0.937548i	$-0.613090\pi$
$563$	−16.5075			−0.695710			−0.347855	+	0.937548i	$0.613090\pi$
$564$		0			0
$565$	3.33678			0.140379
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.10502			−0.297858			−0.148929	−	0.988848i	$-0.547583\pi$
$569$	−7.10502			−0.297858			−0.148929	+	0.988848i	$0.547583\pi$
$570$		0			0
$571$		−	40.2844i		−	1.68585i	−0.538031	−	0.842925i	$-0.680832\pi$
$571$		−	40.2844i		−	1.68585i	0.538031	−	0.842925i	$-0.319168\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.23033	−	4.24566i	0.0930112	−	0.177056i
$576$		0			0
$577$	−32.6969			−1.36119			−0.680594	−	0.732661i	$-0.738278\pi$
$577$	−32.6969			−1.36119			−0.680594	+	0.732661i	$0.738278\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			69.7086i			2.89200i
$582$		0			0
$583$	−32.9489			−1.36460
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	32.3737i		−	1.33620i	−0.744070	−	0.668102i	$-0.767107\pi$
$587$		−	32.3737i		−	1.33620i	0.744070	−	0.668102i	$-0.232893\pi$
$588$		0			0
$589$			31.4720i			1.29678i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	12.5763i		−	0.516446i	−0.966085	−	0.258223i	$-0.916863\pi$
$593$		−	12.5763i		−	0.516446i	0.966085	−	0.258223i	$-0.0831368\pi$
$594$		0			0
$595$		−	19.9782i		−	0.819024i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	31.4007i		−	1.28300i	−0.767124	−	0.641499i	$-0.778313\pi$
$599$		−	31.4007i		−	1.28300i	0.767124	−	0.641499i	$-0.221687\pi$
$600$		0			0
$601$	−36.0180			−1.46920			−0.734602	−	0.678499i	$-0.762631\pi$
$601$	−36.0180			−1.46920			−0.734602	+	0.678499i	$0.762631\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.81029			−0.154910
$606$		0			0
$607$	27.4609			1.11460			0.557302	−	0.830310i	$-0.311837\pi$
$607$	27.4609			1.11460			0.557302	+	0.830310i	$0.311837\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	21.1984i		−	0.857595i
$612$		0			0
$613$			34.4453i			1.39123i	0.718413	+	0.695617i	$0.244869\pi$
$613$			34.4453i			1.39123i	−0.718413	+	0.695617i	$0.755131\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.6321			−0.589068			−0.294534	−	0.955641i	$-0.595164\pi$
$617$	−14.6321			−0.589068			−0.294534	+	0.955641i	$0.595164\pi$
$618$		0			0
$619$		−	4.45456i		−	0.179044i	−0.995985	−	0.0895220i	$-0.971466\pi$
$619$		−	4.45456i		−	0.179044i	0.995985	−	0.0895220i	$-0.0285339\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			13.7387i			0.550429i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.74810i			0.388682i
$630$		0			0
$631$			18.8420i			0.750088i	0.927007	+	0.375044i	$0.122372\pi$
$631$			18.8420i			0.750088i	−0.927007	+	0.375044i	$0.877628\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.39573			−0.214123
$636$		0			0
$637$	99.7561			3.95248
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.7464			−0.700942			−0.350471	−	0.936574i	$-0.613979\pi$
$641$	−17.7464			−0.700942			−0.350471	+	0.936574i	$0.613979\pi$
$642$		0			0
$643$			44.9439i			1.77241i	0.463289	+	0.886207i	$0.346669\pi$
$643$			44.9439i			1.77241i	−0.463289	+	0.886207i	$0.653331\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			25.2710i			0.993505i	0.867892	+	0.496752i	$0.165474\pi$
$647$			25.2710i			0.993505i	−0.867892	+	0.496752i	$0.834526\pi$
$648$		0			0
$649$			32.4577i			1.27408i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	43.1865i		−	1.69002i	−0.534751	−	0.845010i	$-0.679595\pi$
$653$		−	43.1865i		−	1.69002i	0.534751	−	0.845010i	$-0.320405\pi$
$654$		0			0
$655$		−	20.2823i		−	0.792497i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.79943			0.186959			0.0934796	−	0.995621i	$-0.470201\pi$
$659$	4.79943			0.186959			0.0934796	+	0.995621i	$0.470201\pi$
$660$		0			0
$661$			4.06371i			0.158060i	0.996872	+	0.0790300i	$0.0251823\pi$
$661$			4.06371i			0.158060i	−0.996872	+	0.0790300i	$0.974818\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−28.8489			−1.11871
$666$		0			0
$667$	8.08052	+	4.24486i	0.312879	+	0.164362i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			31.8172i			1.22829i
$672$		0			0
$673$	45.4339			1.75135			0.875674	−	0.482903i	$-0.160418\pi$
$673$	45.4339			1.75135			0.875674	+	0.482903i	$0.160418\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−38.6560			−1.48567			−0.742835	−	0.669474i	$-0.766519\pi$
$677$	−38.6560			−1.48567			−0.742835	+	0.669474i	$0.766519\pi$
$678$		0			0
$679$	18.0146			0.691339
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	35.3394i		−	1.35223i	−0.736798	−	0.676113i	$-0.763663\pi$
$683$		−	35.3394i		−	1.35223i	0.736798	−	0.676113i	$-0.236337\pi$
$684$		0			0
$685$	21.4696			0.820310
$686$		0			0
$687$		0			0
$688$		0			0
$689$	58.4854			2.22812
$690$		0			0
$691$	−19.2810			−0.733483			−0.366742	−	0.930323i	$-0.619527\pi$
$691$	−19.2810			−0.733483			−0.366742	+	0.930323i	$0.619527\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.68441			0.291486
$696$		0			0
$697$			46.6805i			1.76815i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8.59946			−0.324797			−0.162399	−	0.986725i	$-0.551923\pi$
$701$	−8.59946			−0.324797			−0.162399	+	0.986725i	$0.551923\pi$
$702$		0			0
$703$	14.0765			0.530905
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.48955			−0.168847
$708$		0			0
$709$			1.77246i			0.0665660i	0.999446	+	0.0332830i	$0.0105963\pi$
$709$			1.77246i			0.0665660i	−0.999446	+	0.0332830i	$0.989404\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.3090	−	21.5278i	0.423525	−	0.806224i
$714$		0			0
$715$	26.2888			0.983145
$716$		0			0
$717$		0			0
$718$		0			0
$719$			18.4697i			0.688804i	0.938822	+	0.344402i	$0.111918\pi$
$719$			18.4697i			0.688804i	−0.938822	+	0.344402i	$0.888082\pi$
$720$		0			0
$721$	23.0105			0.856954
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1.90324i			0.0706846i
$726$		0			0
$727$			14.1803i			0.525917i	0.964807	+	0.262959i	$0.0846983\pi$
$727$			14.1803i			0.525917i	−0.964807	+	0.262959i	$0.915302\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	13.3646i		−	0.494308i
$732$		0			0
$733$		−	21.4452i		−	0.792096i	−0.918230	−	0.396048i	$-0.870381\pi$
$733$		−	21.4452i		−	0.792096i	0.918230	−	0.396048i	$-0.129619\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	48.8058i		−	1.79779i
$738$		0			0
$739$	−31.2095			−1.14806			−0.574031	−	0.818834i	$-0.694621\pi$
$739$	−31.2095			−1.14806			−0.574031	+	0.818834i	$0.694621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.65948			−0.280999			−0.140499	−	0.990081i	$-0.544871\pi$
$743$	−7.65948			−0.280999			−0.140499	+	0.990081i	$0.544871\pi$
$744$		0			0
$745$	0.834989			0.0305916
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	64.5672i		−	2.35923i
$750$		0			0
$751$		−	0.296355i		−	0.0108141i	−0.999985	−	0.00540706i	$-0.998279\pi$
$751$		−	0.296355i		−	0.0108141i	0.999985	−	0.00540706i	$-0.00172113\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.85860			−0.322397
$756$		0			0
$757$		−	5.41555i		−	0.196831i	−0.995145	−	0.0984157i	$-0.968623\pi$
$757$		−	5.41555i		−	0.196831i	0.995145	−	0.0984157i	$-0.0313775\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	28.0323i		−	1.01617i	−0.861307	−	0.508085i	$-0.830354\pi$
$761$		−	28.0323i		−	1.01617i	0.861307	−	0.508085i	$-0.169646\pi$
$762$		0			0
$763$	27.4449			0.993572
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	57.6136i		−	2.08031i
$768$		0			0
$769$		−	10.2138i		−	0.368321i	−0.982896	−	0.184160i	$-0.941043\pi$
$769$		−	10.2138i		−	0.368321i	0.982896	−	0.184160i	$-0.0589566\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−10.4077			−0.374338			−0.187169	−	0.982328i	$-0.559931\pi$
$773$	−10.4077			−0.374338			−0.187169	+	0.982328i	$0.559931\pi$
$774$		0			0
$775$	5.07055			0.182140
$776$		0			0
$777$		0			0
$778$		0			0
$779$	67.4077			2.41513
$780$		0			0
$781$		−	40.5939i		−	1.45256i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	22.5413i		−	0.804532i
$786$		0			0
$787$			1.93489i			0.0689712i	0.999405	+	0.0344856i	$0.0109793\pi$
$787$			1.93489i			0.0689712i	−0.999405	+	0.0344856i	$0.989021\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			15.5091i			0.551441i
$792$		0			0
$793$		−	56.4767i		−	2.00555i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.0936			−0.853439			−0.426720	−	0.904384i	$-0.640331\pi$
$797$	−24.0936			−0.853439			−0.426720	+	0.904384i	$0.640331\pi$
$798$		0			0
$799$		−	13.3386i		−	0.471885i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22.2103			−0.783786
$804$		0			0
$805$	19.7335	+	10.3664i	0.695516	+	0.365368i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			2.81387i			0.0989303i	0.998776	+	0.0494652i	$0.0157517\pi$
$809$			2.81387i			0.0989303i	−0.998776	+	0.0494652i	$0.984248\pi$
$810$		0			0
$811$	−50.7198			−1.78101			−0.890506	−	0.454971i	$-0.849650\pi$
$811$	−50.7198			−1.78101			−0.890506	+	0.454971i	$0.849650\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.61191			0.231605
$816$		0			0
$817$	−19.2988			−0.675180
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19.7842i			0.690472i	0.938516	+	0.345236i	$0.112201\pi$
$821$			19.7842i			0.690472i	−0.938516	+	0.345236i	$0.887799\pi$
$822$		0			0
$823$	−9.43987			−0.329053			−0.164527	−	0.986373i	$-0.552610\pi$
$823$	−9.43987			−0.329053			−0.164527	+	0.986373i	$0.552610\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.22858			−0.320909			−0.160455	−	0.987043i	$-0.551296\pi$
$827$	−9.22858			−0.320909			−0.160455	+	0.987043i	$0.551296\pi$
$828$		0			0
$829$	−13.6277			−0.473310			−0.236655	−	0.971594i	$-0.576051\pi$
$829$	−13.6277			−0.473310			−0.236655	+	0.971594i	$0.576051\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	62.7691			2.17482
$834$		0			0
$835$		−	2.89912i		−	0.100328i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−31.6300			−1.09199			−0.545995	−	0.837789i	$-0.683848\pi$
$839$	−31.6300			−1.09199			−0.545995	+	0.837789i	$0.683848\pi$
$840$		0			0
$841$	25.3777			0.875092
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−33.6636			−1.15806
$846$		0			0
$847$		−	17.7100i		−	0.608521i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.62874	−	5.05817i	−0.330069	−	0.173392i
$852$		0			0
$853$	−41.5196			−1.42160			−0.710802	−	0.703393i	$-0.751668\pi$
$853$	−41.5196			−1.42160			−0.710802	+	0.703393i	$0.751668\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.81325i			0.0619395i	0.999520	+	0.0309697i	$0.00985955\pi$
$857$			1.81325i			0.0619395i	−0.999520	+	0.0309697i	$0.990140\pi$
$858$		0			0
$859$	16.0249			0.546763			0.273382	−	0.961906i	$-0.411858\pi$
$859$	16.0249			0.546763			0.273382	+	0.961906i	$0.411858\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			23.6549i			0.805222i	0.915371	+	0.402611i	$0.131897\pi$
$863$			23.6549i			0.805222i	−0.915371	+	0.402611i	$0.868103\pi$
$864$		0			0
$865$			4.97514i			0.169160i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	8.26164i		−	0.280257i
$870$		0			0
$871$			86.6321i			2.93542i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			4.64793i			0.157129i
$876$		0			0
$877$	−26.9760			−0.910914			−0.455457	−	0.890258i	$-0.650524\pi$
$877$	−26.9760			−0.910914			−0.455457	+	0.890258i	$0.650524\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	21.6174			0.728310			0.364155	−	0.931338i	$-0.381358\pi$
$881$	21.6174			0.728310			0.364155	+	0.931338i	$0.381358\pi$
$882$		0			0
$883$	−1.93559			−0.0651377			−0.0325689	−	0.999469i	$-0.510369\pi$
$883$	−1.93559			−0.0651377			−0.0325689	+	0.999469i	$0.510369\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	49.0804i		−	1.64796i	−0.566620	−	0.823979i	$-0.691749\pi$
$887$		−	49.0804i		−	1.64796i	0.566620	−	0.823979i	$-0.308251\pi$
$888$		0			0
$889$		−	25.0790i		−	0.841123i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−19.2612			−0.644552
$894$		0			0
$895$		−	10.9352i		−	0.365524i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			9.65048i			0.321862i
$900$		0			0
$901$	36.8006			1.22600
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	19.3798i		−	0.644205i
$906$		0			0
$907$			11.6243i			0.385980i	0.981201	+	0.192990i	$0.0618185\pi$
$907$			11.6243i			0.385980i	−0.981201	+	0.192990i	$0.938182\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−33.6776			−1.11579			−0.557895	−	0.829912i	$-0.688391\pi$
$911$	−33.6776			−1.11579			−0.557895	+	0.829912i	$0.688391\pi$
$912$		0			0
$913$	−57.7176			−1.91017
$914$		0			0
$915$		0			0
$916$		0			0
$917$	94.2709			3.11310
$918$		0			0
$919$		−	40.0554i		−	1.32131i	−0.750691	−	0.660653i	$-0.770279\pi$
$919$		−	40.0554i		−	1.32131i	0.750691	−	0.660653i	$-0.229721\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			72.0557i			2.37174i
$924$		0			0
$925$		−	2.26790i		−	0.0745682i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	34.0020i		−	1.11557i	−0.829985	−	0.557786i	$-0.811651\pi$
$929$		−	34.0020i		−	1.11557i	0.829985	−	0.557786i	$-0.188349\pi$
$930$		0			0
$931$		−	90.6400i		−	2.97061i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	16.5416			0.540968
$936$		0			0
$937$		−	2.21207i		−	0.0722653i	−0.999347	−	0.0361326i	$-0.988496\pi$
$937$		−	2.21207i		−	0.0722653i	0.999347	−	0.0361326i	$-0.0115039\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	37.2347			1.21382			0.606909	−	0.794772i	$-0.292409\pi$
$941$	37.2347			1.21382			0.606909	+	0.794772i	$0.292409\pi$
$942$		0			0
$943$	−46.1089	−	24.2220i	−1.50151	−	0.788775i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			10.4633i			0.340012i	0.985443	+	0.170006i	$0.0543787\pi$
$947$			10.4633i			0.340012i	−0.985443	+	0.170006i	$0.945621\pi$
$948$		0			0
$949$	39.4241			1.27976
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.71629			0.0555960			0.0277980	−	0.999614i	$-0.491150\pi$
$953$	1.71629			0.0555960			0.0277980	+	0.999614i	$0.491150\pi$
$954$		0			0
$955$	−7.36656			−0.238376
$956$		0			0
$957$		0			0
$958$		0			0
$959$			99.7892i			3.22236i
$960$		0			0
$961$	−5.28952			−0.170630
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.5605			−0.339955
$966$		0			0
$967$	38.1878			1.22804			0.614019	−	0.789291i	$-0.289552\pi$
$967$	38.1878			1.22804			0.614019	+	0.789291i	$0.289552\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.6808			0.663679			0.331840	−	0.943336i	$-0.392331\pi$
$971$	20.6808			0.663679			0.331840	+	0.943336i	$0.392331\pi$
$972$		0			0
$973$			35.7166i			1.14502i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.600512			0.0192121			0.00960604	−	0.999954i	$-0.496942\pi$
$977$	0.600512			0.0192121			0.00960604	+	0.999954i	$0.496942\pi$
$978$		0			0
$979$	−11.3754			−0.363560
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.66432			−0.212559			−0.106279	−	0.994336i	$-0.533894\pi$
$983$	−6.66432			−0.212559			−0.106279	+	0.994336i	$0.533894\pi$
$984$		0			0
$985$			9.07854i			0.289266i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.2010	+	6.93473i	0.419766	+	0.220512i
$990$		0			0
$991$	5.29945			0.168343			0.0841713	−	0.996451i	$-0.473176\pi$
$991$	5.29945			0.168343			0.0841713	+	0.996451i	$0.473176\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	2.82488i		−	0.0895547i
$996$		0			0
$997$	31.1223			0.985654			0.492827	−	0.870127i	$-0.335964\pi$
$997$	31.1223			0.985654			0.492827	+	0.870127i	$0.335964\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.2	✓	48
3.2	odd	2		8280.2.p.b.1241.2	yes	48
23.22	odd	2		8280.2.p.b.1241.47	yes	48
69.68	even	2	inner	8280.2.p.a.1241.47	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.2	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.47	yes	48	69.68	even	2	inner
8280.2.p.b.1241.2	yes	48	3.2	odd	2
8280.2.p.b.1241.47	yes	48	23.22	odd	2

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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.64793i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -4.64793i q^{7} +3.84841 q^{11} -6.83108 q^{13} -4.29829 q^{17} +6.20683i q^{19} +(2.23033 - 4.24566i) q^{23} +1.00000 q^{25} +1.90324i q^{29} +5.07055 q^{31} +4.64793i q^{35} -2.26790i q^{37} -10.8603i q^{41} +3.10929i q^{43} +3.10323i q^{47} -14.6033 q^{49} -8.56168 q^{53} -3.84841 q^{55} +8.43405i q^{59} +8.26761i q^{61} +6.83108 q^{65} -12.6821i q^{67} -10.5482i q^{71} -5.77129 q^{73} -17.8872i q^{77} -2.14676i q^{79} -14.9978 q^{83} +4.29829 q^{85} -2.95587 q^{89} +31.7504i q^{91} -6.20683i q^{95} +3.87584i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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