LMFDB - Embedded newform 4140.2.i.b.1241.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1241,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4140.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.i (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 $ x^16 + 62x^14 + 1303x^12 + 12842x^10 + 65359x^8 + 170834x^6 + 207293x^4 + 91366*x^2 + 9604
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.13
Root			$-2.59852i$ of defining polynomial
Character	$\chi$	$=$	4140.1241
Dual form			4140.2.i.b.1241.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +3.38743i q^{7} +O(q^{10})$	$q+1.00000 q^{5} +3.38743i q^{7} +1.93787 q^{11} +3.14022 q^{13} +0.203915 q^{17} -0.518638i q^{19} +(0.465354 - 4.77320i) q^{23} +1.00000 q^{25} -4.46938i q^{29} +8.42010 q^{31} +3.38743i q^{35} +0.490199i q^{37} +3.62489i q^{41} -6.61598i q^{43} +0.426051i q^{47} -4.47466 q^{49} +12.5393 q^{53} +1.93787 q^{55} -1.63724i q^{59} +13.6533i q^{61} +3.14022 q^{65} -5.74413i q^{67} -10.7992i q^{71} -8.55201 q^{73} +6.56438i q^{77} +9.91917i q^{79} -4.71928 q^{83} +0.203915 q^{85} +4.48674 q^{89} +10.6373i q^{91} -0.518638i q^{95} +6.61226i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5}+O(q^{10}) $ 16 * q + 16 * q^5	$ 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) $ 16 * q + 16 * q^5 + 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 - 4 * q^53 + 8 * q^73 - 20 * q^83 - 32 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			3.38743i			1.28033i	0.768239	+	0.640163i	$0.221133\pi$
$7$			3.38743i			1.28033i	−0.768239	+	0.640163i	$0.778867\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.93787			0.584288			0.292144	−	0.956374i	$-0.405631\pi$
$11$	1.93787			0.584288			0.292144	+	0.956374i	$0.405631\pi$
$12$		0			0
$13$	3.14022			0.870941			0.435471	−	0.900203i	$-0.356582\pi$
$13$	3.14022			0.870941			0.435471	+	0.900203i	$0.356582\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.203915			0.0494566			0.0247283	−	0.999694i	$-0.492128\pi$
$17$	0.203915			0.0494566			0.0247283	+	0.999694i	$0.492128\pi$
$18$		0			0
$19$		−	0.518638i		−	0.118984i	−0.998229	−	0.0594918i	$-0.981052\pi$
$19$		−	0.518638i		−	0.118984i	0.998229	−	0.0594918i	$-0.0189480\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.465354	−	4.77320i	0.0970331	−	0.995281i
$23$	0.465354	−	4.77320i	0.0970331	−	0.995281i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.46938i		−	0.829944i	−0.909834	−	0.414972i	$-0.863791\pi$
$29$		−	4.46938i		−	0.829944i	0.909834	−	0.414972i	$-0.136209\pi$
$30$		0			0
$31$	8.42010			1.51229			0.756147	−	0.654402i	$-0.227079\pi$
$31$	8.42010			1.51229			0.756147	+	0.654402i	$0.227079\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.38743i			0.572580i
$36$		0			0
$37$			0.490199i			0.0805882i	0.999188	+	0.0402941i	$0.0128295\pi$
$37$			0.490199i			0.0805882i	−0.999188	+	0.0402941i	$0.987171\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.62489i			0.566113i	0.959103	+	0.283056i	$0.0913484\pi$
$41$			3.62489i			0.566113i	−0.959103	+	0.283056i	$0.908652\pi$
$42$		0			0
$43$		−	6.61598i		−	1.00893i	−0.863433	−	0.504464i	$-0.831690\pi$
$43$		−	6.61598i		−	1.00893i	0.863433	−	0.504464i	$-0.168310\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			0.426051i			0.0621459i	0.999517	+	0.0310729i	$0.00989241\pi$
$47$			0.426051i			0.0621459i	−0.999517	+	0.0310729i	$0.990108\pi$
$48$		0			0
$49$	−4.47466			−0.639237
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.5393			1.72241			0.861203	−	0.508260i	$-0.169711\pi$
$53$	12.5393			1.72241			0.861203	+	0.508260i	$0.169711\pi$
$54$		0			0
$55$	1.93787			0.261302
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	1.63724i		−	0.213150i	−0.994305	−	0.106575i	$-0.966012\pi$
$59$		−	1.63724i		−	0.213150i	0.994305	−	0.106575i	$-0.0339884\pi$
$60$		0			0
$61$			13.6533i			1.74812i	0.485814	+	0.874062i	$0.338523\pi$
$61$			13.6533i			1.74812i	−0.485814	+	0.874062i	$0.661477\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.14022			0.389497
$66$		0			0
$67$		−	5.74413i		−	0.701757i	−0.936421	−	0.350878i	$-0.885883\pi$
$67$		−	5.74413i		−	0.701757i	0.936421	−	0.350878i	$-0.114117\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.7992i		−	1.28163i	−0.767696	−	0.640814i	$-0.778597\pi$
$71$		−	10.7992i		−	1.28163i	0.767696	−	0.640814i	$-0.221403\pi$
$72$		0			0
$73$	−8.55201			−1.00094			−0.500468	−	0.865755i	$-0.666839\pi$
$73$	−8.55201			−1.00094			−0.500468	+	0.865755i	$0.666839\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			6.56438i			0.748080i
$78$		0			0
$79$			9.91917i			1.11599i	0.829843	+	0.557997i	$0.188430\pi$
$79$			9.91917i			1.11599i	−0.829843	+	0.557997i	$0.811570\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.71928			−0.518008			−0.259004	−	0.965876i	$-0.583394\pi$
$83$	−4.71928			−0.518008			−0.259004	+	0.965876i	$0.583394\pi$
$84$		0			0
$85$	0.203915			0.0221177
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.48674			0.475594			0.237797	−	0.971315i	$-0.423575\pi$
$89$	4.48674			0.475594			0.237797	+	0.971315i	$0.423575\pi$
$90$		0			0
$91$			10.6373i			1.11509i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	0.518638i		−	0.0532111i
$96$		0			0
$97$			6.61226i			0.671373i	0.941974	+	0.335686i	$0.108968\pi$
$97$			6.61226i			0.671373i	−0.941974	+	0.335686i	$0.891032\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			10.2521i			1.02012i	0.860138	+	0.510061i	$0.170377\pi$
$101$			10.2521i			1.02012i	−0.860138	+	0.510061i	$0.829623\pi$
$102$		0			0
$103$			0.565524i			0.0557227i	0.999612	+	0.0278613i	$0.00886969\pi$
$103$			0.565524i			0.0557227i	−0.999612	+	0.0278613i	$0.991130\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.662616			−0.0640575			−0.0320287	−	0.999487i	$-0.510197\pi$
$107$	−0.662616			−0.0640575			−0.0320287	+	0.999487i	$0.510197\pi$
$108$		0			0
$109$			2.97430i			0.284886i	0.989803	+	0.142443i	$0.0454958\pi$
$109$			2.97430i			0.284886i	−0.989803	+	0.142443i	$0.954504\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.3870			1.25935			0.629673	−	0.776860i	$-0.283189\pi$
$113$	13.3870			1.25935			0.629673	+	0.776860i	$0.283189\pi$
$114$		0			0
$115$	0.465354	−	4.77320i	0.0433945	−	0.445103i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.690747i			0.0633206i
$120$		0			0
$121$	−7.24468			−0.658607
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−1.25448			−0.111317			−0.0556586	−	0.998450i	$-0.517726\pi$
$127$	−1.25448			−0.111317			−0.0556586	+	0.998450i	$0.517726\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			11.0521i			0.965625i	0.875724	+	0.482813i	$0.160385\pi$
$131$			11.0521i			0.965625i	−0.875724	+	0.482813i	$0.839615\pi$
$132$		0			0
$133$	1.75685			0.152338
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.40865			0.291221			0.145610	−	0.989342i	$-0.453485\pi$
$137$	3.40865			0.291221			0.145610	+	0.989342i	$0.453485\pi$
$138$		0			0
$139$	−1.87820			−0.159306			−0.0796532	−	0.996823i	$-0.525381\pi$
$139$	−1.87820			−0.159306			−0.0796532	+	0.996823i	$0.525381\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.08533			0.508881
$144$		0			0
$145$		−	4.46938i		−	0.371162i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.0297			0.903589			0.451795	−	0.892122i	$-0.350784\pi$
$149$	11.0297			0.903589			0.451795	+	0.892122i	$0.350784\pi$
$150$		0			0
$151$	−6.64232			−0.540544			−0.270272	−	0.962784i	$-0.587114\pi$
$151$	−6.64232			−0.540544			−0.270272	+	0.962784i	$0.587114\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.42010			0.676318
$156$		0			0
$157$			8.77266i			0.700134i	0.936725	+	0.350067i	$0.113841\pi$
$157$			8.77266i			0.700134i	−0.936725	+	0.350067i	$0.886159\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.1689	+	1.57635i	1.27429	+	0.124234i
$162$		0			0
$163$	−19.2723			−1.50953			−0.754763	−	0.655997i	$-0.772248\pi$
$163$	−19.2723			−1.50953			−0.754763	+	0.655997i	$0.772248\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	13.2454i		−	1.02496i	−0.858699	−	0.512480i	$-0.828727\pi$
$167$		−	13.2454i		−	1.02496i	0.858699	−	0.512480i	$-0.171273\pi$
$168$		0			0
$169$	−3.13900			−0.241462
$170$		0			0
$171$		0			0
$172$		0			0
$173$			12.5523i			0.954331i	0.878814	+	0.477165i	$0.158336\pi$
$173$			12.5523i			0.954331i	−0.878814	+	0.477165i	$0.841664\pi$
$174$		0			0
$175$			3.38743i			0.256065i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1.19174i			0.0890752i	0.999008	+	0.0445376i	$0.0141814\pi$
$179$			1.19174i			0.0890752i	−0.999008	+	0.0445376i	$0.985819\pi$
$180$		0			0
$181$			11.0586i			0.821980i	0.911640	+	0.410990i	$0.134817\pi$
$181$			11.0586i			0.821980i	−0.911640	+	0.410990i	$0.865183\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.490199i			0.0360401i
$186$		0			0
$187$	0.395160			0.0288969
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.9731			0.866344			0.433172	−	0.901311i	$-0.357394\pi$
$191$	11.9731			0.866344			0.433172	+	0.901311i	$0.357394\pi$
$192$		0			0
$193$	22.1938			1.59755			0.798774	−	0.601632i	$-0.205482\pi$
$193$	22.1938			1.59755			0.798774	+	0.601632i	$0.205482\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	13.3812i		−	0.953373i	−0.879073	−	0.476687i	$-0.841838\pi$
$197$		−	13.3812i		−	0.953373i	0.879073	−	0.476687i	$-0.158162\pi$
$198$		0			0
$199$			8.12207i			0.575758i	0.957667	+	0.287879i	$0.0929501\pi$
$199$			8.12207i			0.575758i	−0.957667	+	0.287879i	$0.907050\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.1397			1.06260
$204$		0			0
$205$			3.62489i			0.253173i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.00505i		−	0.0695208i
$210$		0			0
$211$	23.8850			1.64431			0.822155	−	0.569264i	$-0.192772\pi$
$211$	23.8850			1.64431			0.822155	+	0.569264i	$0.192772\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	6.61598i		−	0.451206i
$216$		0			0
$217$			28.5225i			1.93623i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.640338			0.0430738
$222$		0			0
$223$	−8.07043			−0.540436			−0.270218	−	0.962799i	$-0.587096\pi$
$223$	−8.07043			−0.540436			−0.270218	+	0.962799i	$0.587096\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	22.4535			1.49029			0.745146	−	0.666901i	$-0.232380\pi$
$227$	22.4535			1.49029			0.745146	+	0.666901i	$0.232380\pi$
$228$		0			0
$229$		−	2.33954i		−	0.154601i	−0.997008	−	0.0773006i	$-0.975370\pi$
$229$		−	2.33954i		−	0.154601i	0.997008	−	0.0773006i	$-0.0246301\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	10.1072i		−	0.662147i	−0.943605	−	0.331073i	$-0.892589\pi$
$233$		−	10.1072i		−	0.662147i	0.943605	−	0.331073i	$-0.107411\pi$
$234$		0			0
$235$			0.426051i			0.0277925i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	3.76695i		−	0.243664i	−0.992551	−	0.121832i	$-0.961123\pi$
$239$		−	3.76695i		−	0.243664i	0.992551	−	0.121832i	$-0.0388768\pi$
$240$		0			0
$241$			3.54665i			0.228460i	0.993454	+	0.114230i	$0.0364401\pi$
$241$			3.54665i			0.228460i	−0.993454	+	0.114230i	$0.963560\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.47466			−0.285875
$246$		0			0
$247$		−	1.62864i		−	0.103628i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−30.7624			−1.94170			−0.970852	−	0.239680i	$-0.922957\pi$
$251$	−30.7624			−1.94170			−0.970852	+	0.239680i	$0.922957\pi$
$252$		0			0
$253$	0.901794	−	9.24982i	0.0566953	−	0.581531i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.03089i			0.189062i	0.995522	+	0.0945310i	$0.0301351\pi$
$257$			3.03089i			0.189062i	−0.995522	+	0.0945310i	$0.969865\pi$
$258$		0			0
$259$	−1.66051			−0.103179
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.29110			−0.264601			−0.132300	−	0.991210i	$-0.542236\pi$
$263$	−4.29110			−0.264601			−0.132300	+	0.991210i	$0.542236\pi$
$264$		0			0
$265$	12.5393			0.770284
$266$		0			0
$267$		0			0
$268$		0			0
$269$			7.77366i			0.473969i	0.971513	+	0.236984i	$0.0761590\pi$
$269$			7.77366i			0.473969i	−0.971513	+	0.236984i	$0.923841\pi$
$270$		0			0
$271$	18.0971			1.09932			0.549662	−	0.835387i	$-0.314757\pi$
$271$	18.0971			1.09932			0.549662	+	0.835387i	$0.314757\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.93787			0.116858
$276$		0			0
$277$	−24.7445			−1.48675			−0.743377	−	0.668873i	$-0.766777\pi$
$277$	−24.7445			−1.48675			−0.743377	+	0.668873i	$0.766777\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.410293			0.0244760			0.0122380	−	0.999925i	$-0.496104\pi$
$281$	0.410293			0.0244760			0.0122380	+	0.999925i	$0.496104\pi$
$282$		0			0
$283$			7.04547i			0.418810i	0.977829	+	0.209405i	$0.0671527\pi$
$283$			7.04547i			0.418810i	−0.977829	+	0.209405i	$0.932847\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.2791			−0.724810
$288$		0			0
$289$	−16.9584			−0.997554
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.0384			0.644871			0.322435	−	0.946591i	$-0.395498\pi$
$293$	11.0384			0.644871			0.322435	+	0.946591i	$0.395498\pi$
$294$		0			0
$295$		−	1.63724i		−	0.0953236i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.46132	−	14.9889i	0.0845101	−	0.866831i
$300$		0			0
$301$	22.4111			1.29176
$302$		0			0
$303$		0			0
$304$		0			0
$305$			13.6533i			0.781785i
$306$		0			0
$307$	−19.8397			−1.13231			−0.566156	−	0.824298i	$-0.691570\pi$
$307$	−19.8397			−1.13231			−0.566156	+	0.824298i	$0.691570\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.60686i		−	0.374640i	−0.982299	−	0.187320i	$-0.940020\pi$
$311$		−	6.60686i		−	0.374640i	0.982299	−	0.187320i	$-0.0599802\pi$
$312$		0			0
$313$		−	15.8152i		−	0.893931i	−0.894551	−	0.446965i	$-0.852505\pi$
$313$		−	15.8152i		−	0.893931i	0.894551	−	0.446965i	$-0.147495\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	23.7558i		−	1.33426i	−0.744942	−	0.667129i	$-0.767523\pi$
$317$		−	23.7558i		−	1.33426i	0.744942	−	0.667129i	$-0.232477\pi$
$318$		0			0
$319$		−	8.66107i		−	0.484927i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	0.105758i		−	0.00588453i
$324$		0			0
$325$	3.14022			0.174188
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.44321			−0.0795670
$330$		0			0
$331$	12.5625			0.690500			0.345250	−	0.938511i	$-0.387794\pi$
$331$	12.5625			0.690500			0.345250	+	0.938511i	$0.387794\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	5.74413i		−	0.313835i
$336$		0			0
$337$			4.60946i			0.251093i	0.992088	+	0.125547i	$0.0400685\pi$
$337$			4.60946i			0.251093i	−0.992088	+	0.125547i	$0.959932\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.3170			0.883616
$342$		0			0
$343$			8.55442i			0.461895i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.9899i		−	1.01943i	−0.860343	−	0.509716i	$-0.829750\pi$
$347$		−	18.9899i		−	1.01943i	0.860343	−	0.509716i	$-0.170250\pi$
$348$		0			0
$349$	9.27385			0.496418			0.248209	−	0.968707i	$-0.420158\pi$
$349$	9.27385			0.496418			0.248209	+	0.968707i	$0.420158\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	22.5537i		−	1.20041i	−0.799845	−	0.600207i	$-0.795085\pi$
$353$		−	22.5537i		−	1.20041i	0.799845	−	0.600207i	$-0.204915\pi$
$354$		0			0
$355$		−	10.7992i		−	0.573161i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.2850			1.01782			0.508910	−	0.860819i	$-0.330048\pi$
$359$	19.2850			1.01782			0.508910	+	0.860819i	$0.330048\pi$
$360$		0			0
$361$	18.7310			0.985843
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.55201			−0.447633
$366$		0			0
$367$		−	25.4970i		−	1.33093i	−0.746427	−	0.665467i	$-0.768232\pi$
$367$		−	25.4970i		−	1.33093i	0.746427	−	0.665467i	$-0.231768\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			42.4760i			2.20524i
$372$		0			0
$373$			14.5326i			0.752471i	0.926524	+	0.376235i	$0.122782\pi$
$373$			14.5326i			0.752471i	−0.926524	+	0.376235i	$0.877218\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.0349i		−	0.722832i
$378$		0			0
$379$			25.5829i			1.31411i	0.753844	+	0.657054i	$0.228198\pi$
$379$			25.5829i			1.31411i	−0.753844	+	0.657054i	$0.771802\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.5546			−0.794804			−0.397402	−	0.917645i	$-0.630088\pi$
$383$	−15.5546			−0.794804			−0.397402	+	0.917645i	$0.630088\pi$
$384$		0			0
$385$			6.56438i			0.334552i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.4426			1.39140			0.695698	−	0.718334i	$-0.255095\pi$
$389$	27.4426			1.39140			0.695698	+	0.718334i	$0.255095\pi$
$390$		0			0
$391$	0.0948927	−	0.973327i	0.00479893	−	0.0492232i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			9.91917i			0.499087i
$396$		0			0
$397$	−21.1092			−1.05944			−0.529720	−	0.848173i	$-0.677703\pi$
$397$	−21.1092			−1.05944			−0.529720	+	0.848173i	$0.677703\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.09928			0.254646			0.127323	−	0.991861i	$-0.459362\pi$
$401$	5.09928			0.254646			0.127323	+	0.991861i	$0.459362\pi$
$402$		0			0
$403$	26.4410			1.31712
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.949939i			0.0470867i
$408$		0			0
$409$	29.3044			1.44901			0.724505	−	0.689270i	$-0.242069\pi$
$409$	29.3044			1.44901			0.724505	+	0.689270i	$0.242069\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.54602			0.272902
$414$		0			0
$415$	−4.71928			−0.231660
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.94001			−0.0947756			−0.0473878	−	0.998877i	$-0.515090\pi$
$419$	−1.94001			−0.0947756			−0.0473878	+	0.998877i	$0.515090\pi$
$420$		0			0
$421$		−	35.3431i		−	1.72252i	−0.508165	−	0.861260i	$-0.669676\pi$
$421$		−	35.3431i		−	1.72252i	0.508165	−	0.861260i	$-0.330324\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.203915			0.00989132
$426$		0			0
$427$	−46.2495			−2.23817
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.14170			−0.392172			−0.196086	−	0.980587i	$-0.562823\pi$
$431$	−8.14170			−0.392172			−0.196086	+	0.980587i	$0.562823\pi$
$432$		0			0
$433$			34.9375i			1.67899i	0.543367	+	0.839495i	$0.317149\pi$
$433$			34.9375i			1.67899i	−0.543367	+	0.839495i	$0.682851\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.47556	−	0.241350i	−0.118422	−	0.0115454i
$438$		0			0
$439$	−19.8273			−0.946307			−0.473154	−	0.880980i	$-0.656884\pi$
$439$	−19.8273			−0.946307			−0.473154	+	0.880980i	$0.656884\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			10.8206i			0.514103i	0.966398	+	0.257052i	$0.0827510\pi$
$443$			10.8206i			0.514103i	−0.966398	+	0.257052i	$0.917249\pi$
$444$		0			0
$445$	4.48674			0.212692
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3.45386i			0.162998i	0.996673	+	0.0814990i	$0.0259707\pi$
$449$			3.45386i			0.162998i	−0.996673	+	0.0814990i	$0.974029\pi$
$450$		0			0
$451$			7.02455i			0.330773i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			10.6373i			0.498683i
$456$		0			0
$457$			10.0252i			0.468959i	0.972121	+	0.234479i	$0.0753386\pi$
$457$			10.0252i			0.468959i	−0.972121	+	0.234479i	$0.924661\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			14.7319i			0.686133i	0.939311	+	0.343067i	$0.111466\pi$
$461$			14.7319i			0.686133i	−0.939311	+	0.343067i	$0.888534\pi$
$462$		0			0
$463$	−5.55722			−0.258266			−0.129133	−	0.991627i	$-0.541219\pi$
$463$	−5.55722			−0.258266			−0.129133	+	0.991627i	$0.541219\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.35331			0.155173			0.0775864	−	0.996986i	$-0.475279\pi$
$467$	3.35331			0.155173			0.0775864	+	0.996986i	$0.475279\pi$
$468$		0			0
$469$	19.4578			0.898478
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	12.8209i		−	0.589504i
$474$		0			0
$475$		−	0.518638i		−	0.0237967i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−40.7506			−1.86194			−0.930972	−	0.365090i	$-0.881038\pi$
$479$	−40.7506			−1.86194			−0.930972	+	0.365090i	$0.881038\pi$
$480$		0			0
$481$			1.53933i			0.0701875i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			6.61226i			0.300247i
$486$		0			0
$487$	16.0182			0.725852			0.362926	−	0.931818i	$-0.381778\pi$
$487$	16.0182			0.725852			0.362926	+	0.931818i	$0.381778\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			32.5280i			1.46797i	0.679168	+	0.733983i	$0.262341\pi$
$491$			32.5280i			1.46797i	−0.679168	+	0.733983i	$0.737659\pi$
$492$		0			0
$493$		−	0.911374i		−	0.0410462i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	36.5814			1.64090
$498$		0			0
$499$	−15.4415			−0.691258			−0.345629	−	0.938371i	$-0.612334\pi$
$499$	−15.4415			−0.691258			−0.345629	+	0.938371i	$0.612334\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−25.3118			−1.12860			−0.564298	−	0.825571i	$-0.690853\pi$
$503$	−25.3118			−1.12860			−0.564298	+	0.825571i	$0.690853\pi$
$504$		0			0
$505$			10.2521i			0.456213i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	12.3573i		−	0.547727i	−0.961769	−	0.273863i	$-0.911698\pi$
$509$		−	12.3573i		−	0.547727i	0.961769	−	0.273863i	$-0.0883016\pi$
$510$		0			0
$511$		−	28.9693i		−	1.28153i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			0.565524i			0.0249199i
$516$		0			0
$517$			0.825629i			0.0363111i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.8059			1.17439			0.587193	−	0.809447i	$-0.300233\pi$
$521$	26.8059			1.17439			0.587193	+	0.809447i	$0.300233\pi$
$522$		0			0
$523$			6.50211i			0.284317i	0.989844	+	0.142159i	$0.0454043\pi$
$523$			6.50211i			0.284317i	−0.989844	+	0.142159i	$0.954596\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.71698			0.0747930
$528$		0			0
$529$	−22.5669	−	4.44246i	−0.981169	−	0.193150i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11.3830i			0.493051i
$534$		0			0
$535$	−0.662616			−0.0286474
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.67128			−0.373499
$540$		0			0
$541$	37.1033			1.59520			0.797598	−	0.603189i	$-0.206104\pi$
$541$	37.1033			1.59520			0.797598	+	0.603189i	$0.206104\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.97430i			0.127405i
$546$		0			0
$547$	0.757562			0.0323910			0.0161955	−	0.999869i	$-0.494845\pi$
$547$	0.757562			0.0323910			0.0161955	+	0.999869i	$0.494845\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.31799			−0.0987497
$552$		0			0
$553$	−33.6004			−1.42884
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.62765			0.153708			0.0768542	−	0.997042i	$-0.475512\pi$
$557$	3.62765			0.153708			0.0768542	+	0.997042i	$0.475512\pi$
$558$		0			0
$559$		−	20.7756i		−	0.878716i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−32.8756			−1.38554			−0.692771	−	0.721157i	$-0.743611\pi$
$563$	−32.8756			−1.38554			−0.692771	+	0.721157i	$0.743611\pi$
$564$		0			0
$565$	13.3870			0.563197
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.2153			−0.721702			−0.360851	−	0.932624i	$-0.617514\pi$
$569$	−17.2153			−0.721702			−0.360851	+	0.932624i	$0.617514\pi$
$570$		0			0
$571$		−	36.2532i		−	1.51715i	−0.651586	−	0.758575i	$-0.725896\pi$
$571$		−	36.2532i		−	1.51715i	0.651586	−	0.758575i	$-0.274104\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.465354	−	4.77320i	0.0194066	−	0.199056i
$576$		0			0
$577$	−9.37820			−0.390419			−0.195210	−	0.980762i	$-0.562539\pi$
$577$	−9.37820			−0.390419			−0.195210	+	0.980762i	$0.562539\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	15.9862i		−	0.663219i
$582$		0			0
$583$	24.2995			1.00638
$584$		0			0
$585$		0			0
$586$		0			0
$587$			41.1707i			1.69930i	0.527351	+	0.849648i	$0.323185\pi$
$587$			41.1707i			1.69930i	−0.527351	+	0.849648i	$0.676815\pi$
$588$		0			0
$589$		−	4.36698i		−	0.179938i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			36.9287i			1.51648i	0.651977	+	0.758239i	$0.273940\pi$
$593$			36.9287i			1.51648i	−0.651977	+	0.758239i	$0.726060\pi$
$594$		0			0
$595$			0.690747i			0.0283178i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			4.80124i			0.196173i	0.995178	+	0.0980867i	$0.0312723\pi$
$599$			4.80124i			0.196173i	−0.995178	+	0.0980867i	$0.968728\pi$
$600$		0			0
$601$	−16.9946			−0.693223			−0.346611	−	0.938009i	$-0.612668\pi$
$601$	−16.9946			−0.693223			−0.346611	+	0.938009i	$0.612668\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.24468			−0.294538
$606$		0			0
$607$	5.84848			0.237383			0.118691	−	0.992931i	$-0.462130\pi$
$607$	5.84848			0.237383			0.118691	+	0.992931i	$0.462130\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.33789i			0.0541254i
$612$		0			0
$613$			18.4821i			0.746484i	0.927734	+	0.373242i	$0.121754\pi$
$613$			18.4821i			0.746484i	−0.927734	+	0.373242i	$0.878246\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.2292			−0.814396			−0.407198	−	0.913340i	$-0.633494\pi$
$617$	−20.2292			−0.814396			−0.407198	+	0.913340i	$0.633494\pi$
$618$		0			0
$619$			13.5937i			0.546375i	0.961961	+	0.273188i	$0.0880780\pi$
$619$			13.5937i			0.546375i	−0.961961	+	0.273188i	$0.911922\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			15.1985i			0.608915i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			0.0999588i			0.00398562i
$630$		0			0
$631$		−	19.0015i		−	0.756438i	−0.925716	−	0.378219i	$-0.876537\pi$
$631$		−	19.0015i		−	0.756438i	0.925716	−	0.378219i	$-0.123463\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.25448			−0.0497826
$636$		0			0
$637$	−14.0514			−0.556737
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.4695			−0.611009			−0.305504	−	0.952191i	$-0.598825\pi$
$641$	−15.4695			−0.611009			−0.305504	+	0.952191i	$0.598825\pi$
$642$		0			0
$643$		−	36.0492i		−	1.42164i	−0.703373	−	0.710821i	$-0.748324\pi$
$643$		−	36.0492i		−	1.42164i	0.703373	−	0.710821i	$-0.251676\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	20.8734i		−	0.820617i	−0.911947	−	0.410309i	$-0.865421\pi$
$647$		−	20.8734i		−	0.820617i	0.911947	−	0.410309i	$-0.134579\pi$
$648$		0			0
$649$		−	3.17274i		−	0.124541i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	19.4196i		−	0.759946i	−0.924997	−	0.379973i	$-0.875933\pi$
$653$		−	19.4196i		−	0.759946i	0.924997	−	0.379973i	$-0.124067\pi$
$654$		0			0
$655$			11.0521i			0.431841i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.5315			−1.26725			−0.633624	−	0.773641i	$-0.718433\pi$
$659$	−32.5315			−1.26725			−0.633624	+	0.773641i	$0.718433\pi$
$660$		0			0
$661$			15.4785i			0.602043i	0.953617	+	0.301022i	$0.0973276\pi$
$661$			15.4785i			0.602043i	−0.953617	+	0.301022i	$0.902672\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.75685			0.0681276
$666$		0			0
$667$	−21.3333	−	2.07985i	−0.826027	−	0.0805320i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			26.4582i			1.02141i
$672$		0			0
$673$	−31.4955			−1.21406			−0.607031	−	0.794678i	$-0.707640\pi$
$673$	−31.4955			−1.21406			−0.607031	+	0.794678i	$0.707640\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.49751			−0.0959870			−0.0479935	−	0.998848i	$-0.515283\pi$
$677$	−2.49751			−0.0959870			−0.0479935	+	0.998848i	$0.515283\pi$
$678$		0			0
$679$	−22.3985			−0.859577
$680$		0			0
$681$		0			0
$682$		0			0
$683$			19.7486i			0.755659i	0.925875	+	0.377829i	$0.123329\pi$
$683$			19.7486i			0.755659i	−0.925875	+	0.377829i	$0.876671\pi$
$684$		0			0
$685$	3.40865			0.130238
$686$		0			0
$687$		0			0
$688$		0			0
$689$	39.3762			1.50011
$690$		0			0
$691$	35.4403			1.34821			0.674107	−	0.738634i	$-0.264529\pi$
$691$	35.4403			1.34821			0.674107	+	0.738634i	$0.264529\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.87820			−0.0712440
$696$		0			0
$697$			0.739169i			0.0279980i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16.9669			−0.640832			−0.320416	−	0.947277i	$-0.603823\pi$
$701$	−16.9669			−0.640832			−0.320416	+	0.947277i	$0.603823\pi$
$702$		0			0
$703$	0.254235			0.00958867
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−34.7282			−1.30609
$708$		0			0
$709$		−	27.7334i		−	1.04155i	−0.853694	−	0.520774i	$-0.825643\pi$
$709$		−	27.7334i		−	1.04155i	0.853694	−	0.520774i	$-0.174357\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.91833	−	40.1908i	0.146743	−	1.50516i
$714$		0			0
$715$	6.08533			0.227578
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	42.1793i		−	1.57302i	−0.617577	−	0.786510i	$-0.711886\pi$
$719$		−	42.1793i		−	1.57302i	0.617577	−	0.786510i	$-0.288114\pi$
$720$		0			0
$721$	−1.91567			−0.0713432
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	4.46938i		−	0.165989i
$726$		0			0
$727$		−	45.8099i		−	1.69899i	−0.527594	−	0.849497i	$-0.676906\pi$
$727$		−	45.8099i		−	1.69899i	0.527594	−	0.849497i	$-0.323094\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.34910i		−	0.0498981i
$732$		0			0
$733$		−	3.07043i		−	0.113409i	−0.998391	−	0.0567044i	$-0.981941\pi$
$733$		−	3.07043i		−	0.113409i	0.998391	−	0.0567044i	$-0.0180592\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	11.1313i		−	0.410028i
$738$		0			0
$739$	−45.1930			−1.66245			−0.831226	−	0.555935i	$-0.812360\pi$
$739$	−45.1930			−1.66245			−0.831226	+	0.555935i	$0.812360\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.6625			−0.537913			−0.268957	−	0.963152i	$-0.586679\pi$
$743$	−14.6625			−0.537913			−0.268957	+	0.963152i	$0.586679\pi$
$744$		0			0
$745$	11.0297			0.404097
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	2.24456i		−	0.0820145i
$750$		0			0
$751$		−	2.34759i		−	0.0856648i	−0.999082	−	0.0428324i	$-0.986362\pi$
$751$		−	2.34759i		−	0.0856648i	0.999082	−	0.0428324i	$-0.0136381\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.64232			−0.241739
$756$		0			0
$757$			2.18679i			0.0794803i	0.999210	+	0.0397401i	$0.0126530\pi$
$757$			2.18679i			0.0794803i	−0.999210	+	0.0397401i	$0.987347\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	27.9399i		−	1.01282i	−0.862293	−	0.506410i	$-0.830972\pi$
$761$		−	27.9399i		−	1.01282i	0.862293	−	0.506410i	$-0.169028\pi$
$762$		0			0
$763$	−10.0752			−0.364748
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	5.14129i		−	0.185641i
$768$		0			0
$769$		−	21.9653i		−	0.792089i	−0.918231	−	0.396045i	$-0.870383\pi$
$769$		−	21.9653i		−	0.792089i	0.918231	−	0.396045i	$-0.129617\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−42.5662			−1.53100			−0.765500	−	0.643436i	$-0.777508\pi$
$773$	−42.5662			−1.53100			−0.765500	+	0.643436i	$0.777508\pi$
$774$		0			0
$775$	8.42010			0.302459
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.88001			0.0673582
$780$		0			0
$781$		−	20.9274i		−	0.748840i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			8.77266i			0.313110i
$786$		0			0
$787$		−	18.6705i		−	0.665531i	−0.943010	−	0.332765i	$-0.892018\pi$
$787$		−	18.6705i		−	0.665531i	0.943010	−	0.332765i	$-0.107982\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			45.3476i			1.61237i
$792$		0			0
$793$			42.8744i			1.52251i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−34.5752			−1.22472			−0.612359	−	0.790580i	$-0.709779\pi$
$797$	−34.5752			−1.22472			−0.612359	+	0.790580i	$0.709779\pi$
$798$		0			0
$799$			0.0868780i			0.00307352i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16.5726			−0.584836
$804$		0			0
$805$	16.1689	+	1.57635i	0.569878	+	0.0555592i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	19.3970i		−	0.681963i	−0.940070	−	0.340981i	$-0.889241\pi$
$809$		−	19.3970i		−	0.681963i	0.940070	−	0.340981i	$-0.110759\pi$
$810$		0			0
$811$	23.3284			0.819172			0.409586	−	0.912271i	$-0.365673\pi$
$811$	23.3284			0.819172			0.409586	+	0.912271i	$0.365673\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.2723			−0.675081
$816$		0			0
$817$	−3.43130			−0.120046
$818$		0			0
$819$		0			0
$820$		0			0
$821$			16.8435i			0.587844i	0.955829	+	0.293922i	$0.0949605\pi$
$821$			16.8435i			0.587844i	−0.955829	+	0.293922i	$0.905039\pi$
$822$		0			0
$823$	18.9216			0.659566			0.329783	−	0.944057i	$-0.393024\pi$
$823$	18.9216			0.659566			0.329783	+	0.944057i	$0.393024\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.66771			−0.197086			−0.0985428	−	0.995133i	$-0.531418\pi$
$827$	−5.66771			−0.197086			−0.0985428	+	0.995133i	$0.531418\pi$
$828$		0			0
$829$	1.20697			0.0419198			0.0209599	−	0.999780i	$-0.493328\pi$
$829$	1.20697			0.0419198			0.0209599	+	0.999780i	$0.493328\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.912449			−0.0316145
$834$		0			0
$835$		−	13.2454i		−	0.458376i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.1404			−0.833419			−0.416709	−	0.909040i	$-0.636817\pi$
$839$	−24.1404			−0.833419			−0.416709	+	0.909040i	$0.636817\pi$
$840$		0			0
$841$	9.02460			0.311193
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.13900			−0.107985
$846$		0			0
$847$		−	24.5408i		−	0.843232i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.33982	+	0.228116i	0.0802079	+	0.00781972i
$852$		0			0
$853$	−33.7803			−1.15661			−0.578307	−	0.815819i	$-0.696286\pi$
$853$	−33.7803			−1.15661			−0.578307	+	0.815819i	$0.696286\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	46.3661i		−	1.58384i	−0.610628	−	0.791918i	$-0.709083\pi$
$857$		−	46.3661i		−	1.58384i	0.610628	−	0.791918i	$-0.290917\pi$
$858$		0			0
$859$	−3.60210			−0.122902			−0.0614510	−	0.998110i	$-0.519573\pi$
$859$	−3.60210			−0.122902			−0.0614510	+	0.998110i	$0.519573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	10.9853i		−	0.373945i	−0.982365	−	0.186972i	$-0.940132\pi$
$863$		−	10.9853i		−	0.373945i	0.982365	−	0.186972i	$-0.0598675\pi$
$864$		0			0
$865$			12.5523i			0.426790i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			19.2220i			0.652062i
$870$		0			0
$871$		−	18.0378i		−	0.611189i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			3.38743i			0.114516i
$876$		0			0
$877$	−0.104707			−0.00353571			−0.00176785	−	0.999998i	$-0.500563\pi$
$877$	−0.104707			−0.00353571			−0.00176785	+	0.999998i	$0.500563\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	11.5483			0.389072			0.194536	−	0.980895i	$-0.437680\pi$
$881$	11.5483			0.389072			0.194536	+	0.980895i	$0.437680\pi$
$882$		0			0
$883$	8.32093			0.280022			0.140011	−	0.990150i	$-0.455286\pi$
$883$	8.32093			0.280022			0.140011	+	0.990150i	$0.455286\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	49.2806i		−	1.65468i	−0.561700	−	0.827341i	$-0.689853\pi$
$887$		−	49.2806i		−	1.65468i	0.561700	−	0.827341i	$-0.310147\pi$
$888$		0			0
$889$		−	4.24946i		−	0.142522i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.220966			0.00739434
$894$		0			0
$895$			1.19174i			0.0398356i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	37.6327i		−	1.25512i
$900$		0			0
$901$	2.55695			0.0851844
$902$		0			0
$903$		0			0
$904$		0			0
$905$			11.0586i			0.367601i
$906$		0			0
$907$		−	26.3925i		−	0.876347i	−0.898890	−	0.438173i	$-0.855626\pi$
$907$		−	26.3925i		−	0.876347i	0.898890	−	0.438173i	$-0.144374\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.43475			0.0806668			0.0403334	−	0.999186i	$-0.487158\pi$
$911$	2.43475			0.0806668			0.0403334	+	0.999186i	$0.487158\pi$
$912$		0			0
$913$	−9.14532			−0.302666
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−37.4381			−1.23632
$918$		0			0
$919$		−	10.4481i		−	0.344651i	−0.985040	−	0.172325i	$-0.944872\pi$
$919$		−	10.4481i		−	0.344651i	0.985040	−	0.172325i	$-0.0551280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	33.9118i		−	1.11622i
$924$		0			0
$925$			0.490199i			0.0161176i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			17.9068i			0.587502i	0.955882	+	0.293751i	$0.0949036\pi$
$929$			17.9068i			0.587502i	−0.955882	+	0.293751i	$0.905096\pi$
$930$		0			0
$931$			2.32073i			0.0760587i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.395160			0.0129231
$936$		0			0
$937$			21.6115i			0.706017i	0.935620	+	0.353008i	$0.114841\pi$
$937$			21.6115i			0.706017i	−0.935620	+	0.353008i	$0.885159\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.635634			0.0207211			0.0103605	−	0.999946i	$-0.496702\pi$
$941$	0.635634			0.0207211			0.0103605	+	0.999946i	$0.496702\pi$
$942$		0			0
$943$	17.3023	+	1.68686i	0.563442	+	0.0549317i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	35.0912i		−	1.14031i	−0.821537	−	0.570155i	$-0.806883\pi$
$947$		−	35.0912i		−	1.14031i	0.821537	−	0.570155i	$-0.193117\pi$
$948$		0			0
$949$	−26.8552			−0.871757
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29.9970			0.971698			0.485849	−	0.874043i	$-0.338510\pi$
$953$	29.9970			0.971698			0.485849	+	0.874043i	$0.338510\pi$
$954$		0			0
$955$	11.9731			0.387441
$956$		0			0
$957$		0			0
$958$		0			0
$959$			11.5466i			0.372858i
$960$		0			0
$961$	39.8980			1.28703
$962$		0			0
$963$		0			0
$964$		0			0
$965$	22.1938			0.714445
$966$		0			0
$967$	−29.4860			−0.948207			−0.474103	−	0.880469i	$-0.657228\pi$
$967$	−29.4860			−0.948207			−0.474103	+	0.880469i	$0.657228\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.77001			0.0888940			0.0444470	−	0.999012i	$-0.485847\pi$
$971$	2.77001			0.0888940			0.0444470	+	0.999012i	$0.485847\pi$
$972$		0			0
$973$		−	6.36225i		−	0.203964i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.02326			0.160708			0.0803542	−	0.996766i	$-0.474395\pi$
$977$	5.02326			0.160708			0.0803542	+	0.996766i	$0.474395\pi$
$978$		0			0
$979$	8.69470			0.277884
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−43.8932			−1.39998			−0.699988	−	0.714155i	$-0.746811\pi$
$983$	−43.8932			−1.39998			−0.699988	+	0.714155i	$0.746811\pi$
$984$		0			0
$985$		−	13.3812i		−	0.426361i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−31.5794	−	3.07877i	−1.00417	−	0.0978993i
$990$		0			0
$991$	32.8042			1.04206			0.521031	−	0.853538i	$-0.325548\pi$
$991$	32.8042			1.04206			0.521031	+	0.853538i	$0.325548\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			8.12207i			0.257487i
$996$		0			0
$997$	−48.2716			−1.52878			−0.764389	−	0.644756i	$-0.776959\pi$
$997$	−48.2716			−1.52878			−0.764389	+	0.644756i	$0.776959\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.i.b.1241.13	yes	16
3.2	odd	2		4140.2.i.a.1241.13	yes	16
23.22	odd	2		4140.2.i.a.1241.4	✓	16
69.68	even	2	inner	4140.2.i.b.1241.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.i.a.1241.4	✓	16	23.22	odd	2
4140.2.i.a.1241.13	yes	16	3.2	odd	2
4140.2.i.b.1241.4	yes	16	69.68	even	2	inner
4140.2.i.b.1241.13	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +3.38743i q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +3.38743i q^{7} +1.93787 q^{11} +3.14022 q^{13} +0.203915 q^{17} -0.518638i q^{19} +(0.465354 - 4.77320i) q^{23} +1.00000 q^{25} -4.46938i q^{29} +8.42010 q^{31} +3.38743i q^{35} +0.490199i q^{37} +3.62489i q^{41} -6.61598i q^{43} +0.426051i q^{47} -4.47466 q^{49} +12.5393 q^{53} +1.93787 q^{55} -1.63724i q^{59} +13.6533i q^{61} +3.14022 q^{65} -5.74413i q^{67} -10.7992i q^{71} -8.55201 q^{73} +6.56438i q^{77} +9.91917i q^{79} -4.71928 q^{83} +0.203915 q^{85} +4.48674 q^{89} +10.6373i q^{91} -0.518638i q^{95} +6.61226i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5}+O(q^{10}) \) 16 * q + 16 * q^5	\( 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) \) 16 * q + 16 * q^5 + 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 - 4 * q^53 + 8 * q^73 - 20 * q^83 - 32 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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