LMFDB - Embedded newform 4140.2.s.a.737.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 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("4140.737");

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.s (of order $4$, degree $2$, 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:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.14
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.340177 + 2.21004i) q^{5} +(-3.49741 - 3.49741i) q^{7} +O(q^{10})$	$q+(0.340177 + 2.21004i) q^{5} +(-3.49741 - 3.49741i) q^{7} +1.20436i q^{11} +(0.0289513 - 0.0289513i) q^{13} +(4.88095 - 4.88095i) q^{17} +6.86518i q^{19} +(0.707107 + 0.707107i) q^{23} +(-4.76856 + 1.50361i) q^{25} -2.41815 q^{29} +4.34787 q^{31} +(6.53968 - 8.91916i) q^{35} +(-7.27883 - 7.27883i) q^{37} -1.65317i q^{41} +(-2.38031 + 2.38031i) q^{43} +(-1.54998 + 1.54998i) q^{47} +17.4638i q^{49} +(0.188256 + 0.188256i) q^{53} +(-2.66168 + 0.409695i) q^{55} +0.182425 q^{59} +10.5168 q^{61} +(0.0738321 + 0.0541350i) q^{65} +(-4.11911 - 4.11911i) q^{67} -8.99344i q^{71} +(5.12495 - 5.12495i) q^{73} +(4.21214 - 4.21214i) q^{77} -10.4176i q^{79} +(-7.95167 - 7.95167i) q^{83} +(12.4475 + 9.12672i) q^{85} +13.8105 q^{89} -0.202509 q^{91} +(-15.1723 + 2.33537i) q^{95} +(3.45647 + 3.45647i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 12 q^{7}+O(q^{10}) $ 44 * q - 12 * q^7	$ 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) $ 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$5$	0.340177	+	2.21004i	0.152132	+	0.988360i
$5$	0.340177	+	2.21004i	0.152132	+	0.988360i
$6$		0			0
$7$	−3.49741	−	3.49741i	−1.32190	−	1.32190i	−0.912239	−	0.409657i	$-0.865648\pi$
$7$	−3.49741	−	3.49741i	−1.32190	−	1.32190i	−0.409657	−	0.912239i	$-0.634352\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.20436i			0.363128i	0.983379	+	0.181564i	$0.0581160\pi$
$11$			1.20436i			0.363128i	−0.983379	+	0.181564i	$0.941884\pi$
$12$		0			0
$13$	0.0289513	−	0.0289513i	0.00802965	−	0.00802965i	−0.703081	−	0.711110i	$-0.748193\pi$
$13$	0.0289513	−	0.0289513i	0.00802965	−	0.00802965i	0.711110	+	0.703081i	$0.248193\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.88095	−	4.88095i	1.18380	−	1.18380i	0.205054	−	0.978751i	$-0.434263\pi$
$17$	4.88095	−	4.88095i	1.18380	−	1.18380i	0.978751	−	0.205054i	$-0.0657370\pi$
$18$		0			0
$19$			6.86518i			1.57498i	0.616327	+	0.787490i	$0.288620\pi$
$19$			6.86518i			1.57498i	−0.616327	+	0.787490i	$0.711380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$24$		0			0
$25$	−4.76856	+	1.50361i	−0.953712	+	0.300722i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.41815			−0.449039			−0.224520	−	0.974470i	$-0.572081\pi$
$29$	−2.41815			−0.449039			−0.224520	+	0.974470i	$0.572081\pi$
$30$		0			0
$31$	4.34787			0.780901			0.390451	−	0.920624i	$-0.372319\pi$
$31$	4.34787			0.780901			0.390451	+	0.920624i	$0.372319\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.53968	−	8.91916i	1.10541	−	1.50761i
$36$		0			0
$37$	−7.27883	−	7.27883i	−1.19663	−	1.19663i	−0.975168	−	0.221465i	$-0.928916\pi$
$37$	−7.27883	−	7.27883i	−1.19663	−	1.19663i	−0.221465	−	0.975168i	$-0.571084\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.65317i		−	0.258181i	−0.991633	−	0.129091i	$-0.958794\pi$
$41$		−	1.65317i		−	0.258181i	0.991633	−	0.129091i	$-0.0412058\pi$
$42$		0			0
$43$	−2.38031	+	2.38031i	−0.362994	+	0.362994i	−0.864914	−	0.501920i	$-0.832627\pi$
$43$	−2.38031	+	2.38031i	−0.362994	+	0.362994i	0.501920	+	0.864914i	$0.332627\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.54998	+	1.54998i	−0.226088	+	0.226088i	−0.811056	−	0.584968i	$-0.801107\pi$
$47$	−1.54998	+	1.54998i	−0.226088	+	0.226088i	0.584968	+	0.811056i	$0.301107\pi$
$48$		0			0
$49$			17.4638i			2.49482i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.188256	+	0.188256i	0.0258589	+	0.0258589i	0.719918	−	0.694059i	$-0.244179\pi$
$53$	0.188256	+	0.188256i	0.0258589	+	0.0258589i	−0.694059	+	0.719918i	$0.744179\pi$
$54$		0			0
$55$	−2.66168	+	0.409695i	−0.358901	+	0.0552433i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.182425			0.0237497			0.0118748	−	0.999929i	$-0.496220\pi$
$59$	0.182425			0.0237497			0.0118748	+	0.999929i	$0.496220\pi$
$60$		0			0
$61$	10.5168			1.34654			0.673268	−	0.739398i	$-0.264890\pi$
$61$	10.5168			1.34654			0.673268	+	0.739398i	$0.264890\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.0738321	+	0.0541350i	0.00915775	+	0.00671462i
$66$		0			0
$67$	−4.11911	−	4.11911i	−0.503230	−	0.503230i	0.409210	−	0.912440i	$-0.365804\pi$
$67$	−4.11911	−	4.11911i	−0.503230	−	0.503230i	−0.912440	+	0.409210i	$0.865804\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.99344i		−	1.06732i	−0.845698	−	0.533662i	$-0.820815\pi$
$71$		−	8.99344i		−	1.06732i	0.845698	−	0.533662i	$-0.179185\pi$
$72$		0			0
$73$	5.12495	−	5.12495i	0.599830	−	0.599830i	−0.340438	−	0.940267i	$-0.610575\pi$
$73$	5.12495	−	5.12495i	0.599830	−	0.599830i	0.940267	+	0.340438i	$0.110575\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.21214	−	4.21214i	0.480018	−	0.480018i
$78$		0			0
$79$		−	10.4176i		−	1.17207i	−0.810284	−	0.586037i	$-0.800687\pi$
$79$		−	10.4176i		−	1.17207i	0.810284	−	0.586037i	$-0.199313\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.95167	−	7.95167i	−0.872809	−	0.872809i	0.119969	−	0.992778i	$-0.461721\pi$
$83$	−7.95167	−	7.95167i	−0.872809	−	0.872809i	−0.992778	+	0.119969i	$0.961721\pi$
$84$		0			0
$85$	12.4475	+	9.12672i	1.35012	+	0.989931i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.8105			1.46391			0.731953	−	0.681355i	$-0.238609\pi$
$89$	13.8105			1.46391			0.731953	+	0.681355i	$0.238609\pi$
$90$		0			0
$91$	−0.202509			−0.0212287
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.1723	+	2.33537i	−1.55665	+	0.239604i
$96$		0			0
$97$	3.45647	+	3.45647i	0.350952	+	0.350952i	0.860464	−	0.509512i	$-0.170174\pi$
$97$	3.45647	+	3.45647i	0.350952	+	0.350952i	−0.509512	+	0.860464i	$0.670174\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	8.65198i		−	0.860904i	−0.902613	−	0.430452i	$-0.858354\pi$
$101$		−	8.65198i		−	0.860904i	0.902613	−	0.430452i	$-0.141646\pi$
$102$		0			0
$103$	10.5735	−	10.5735i	1.04184	−	1.04184i	0.0427511	−	0.999086i	$-0.486388\pi$
$103$	10.5735	−	10.5735i	1.04184	−	1.04184i	0.999086	−	0.0427511i	$-0.0136123\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.13635	+	3.13635i	−0.303202	+	0.303202i	−0.842265	−	0.539063i	$-0.818778\pi$
$107$	−3.13635	+	3.13635i	−0.303202	+	0.303202i	0.539063	+	0.842265i	$0.318778\pi$
$108$		0			0
$109$		−	10.8112i		−	1.03552i	−0.855525	−	0.517762i	$-0.826765\pi$
$109$		−	10.8112i		−	1.03552i	0.855525	−	0.517762i	$-0.173235\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.6808	−	12.6808i	−1.19291	−	1.19291i	−0.976246	−	0.216667i	$-0.930481\pi$
$113$	−12.6808	−	12.6808i	−1.19291	−	1.19291i	−0.216667	−	0.976246i	$-0.569519\pi$
$114$		0			0
$115$	−1.32219	+	1.80328i	−0.123295	+	0.168156i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−34.1414			−3.12974
$120$		0			0
$121$	9.54952			0.868138
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.94519	−	10.0272i	−0.442311	−	0.896862i
$126$		0			0
$127$	−10.4012	−	10.4012i	−0.922959	−	0.922959i	0.0742784	−	0.997238i	$-0.476335\pi$
$127$	−10.4012	−	10.4012i	−0.922959	−	0.922959i	−0.997238	+	0.0742784i	$0.976335\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			2.91986i			0.255109i	0.991832	+	0.127555i	$0.0407128\pi$
$131$			2.91986i			0.255109i	−0.991832	+	0.127555i	$0.959287\pi$
$132$		0			0
$133$	24.0104	−	24.0104i	2.08196	−	2.08196i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.7694	−	11.7694i	1.00553	−	1.00553i	0.00554568	−	0.999985i	$-0.498235\pi$
$137$	11.7694	−	11.7694i	1.00553	−	1.00553i	0.999985	−	0.00554568i	$-0.00176525\pi$
$138$		0			0
$139$		−	3.02188i		−	0.256312i	−0.991754	−	0.128156i	$-0.959094\pi$
$139$		−	3.02188i		−	0.256312i	0.991754	−	0.128156i	$-0.0409059\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.0348678	+	0.0348678i	0.00291579	+	0.00291579i
$144$		0			0
$145$	−0.822598	−	5.34421i	−0.0683130	−	0.443812i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	16.3017			1.33549			0.667744	−	0.744391i	$-0.267260\pi$
$149$	16.3017			1.33549			0.667744	+	0.744391i	$0.267260\pi$
$150$		0			0
$151$	12.0084			0.977230			0.488615	−	0.872500i	$-0.337502\pi$
$151$	12.0084			0.977230			0.488615	+	0.872500i	$0.337502\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.47904	+	9.60898i	0.118800	+	0.771812i
$156$		0			0
$157$	15.2593	+	15.2593i	1.21782	+	1.21782i	0.968393	+	0.249430i	$0.0802433\pi$
$157$	15.2593	+	15.2593i	1.21782	+	1.21782i	0.249430	+	0.968393i	$0.419757\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.94609i		−	0.389806i
$162$		0			0
$163$	5.85902	−	5.85902i	0.458914	−	0.458914i	−0.439385	−	0.898299i	$-0.644804\pi$
$163$	5.85902	−	5.85902i	0.458914	−	0.458914i	0.898299	+	0.439385i	$0.144804\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.81380	−	1.81380i	0.140356	−	0.140356i	−0.633438	−	0.773794i	$-0.718357\pi$
$167$	1.81380	−	1.81380i	0.140356	−	0.140356i	0.773794	+	0.633438i	$0.218357\pi$
$168$		0			0
$169$			12.9983i			0.999871i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.61875	+	9.61875i	0.731300	+	0.731300i	0.970877	−	0.239577i	$-0.0770089\pi$
$173$	9.61875	+	9.61875i	0.731300	+	0.731300i	−0.239577	+	0.970877i	$0.577009\pi$
$174$		0			0
$175$	21.9363	+	11.4189i	1.65823	+	0.863186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.7802			−1.32896			−0.664478	−	0.747308i	$-0.731346\pi$
$179$	−17.7802			−1.32896			−0.664478	+	0.747308i	$0.731346\pi$
$180$		0			0
$181$	2.72744			0.202729			0.101364	−	0.994849i	$-0.467679\pi$
$181$	2.72744			0.202729			0.101364	+	0.994849i	$0.467679\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.6104	−	18.5626i	1.00066	−	1.36475i
$186$		0			0
$187$	5.87842	+	5.87842i	0.429873	+	0.429873i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.63136i		−	0.624543i	−0.949993	−	0.312272i	$-0.898910\pi$
$191$		−	8.63136i		−	0.624543i	0.949993	−	0.312272i	$-0.101090\pi$
$192$		0			0
$193$	−8.94883	+	8.94883i	−0.644151	+	0.644151i	−0.951573	−	0.307422i	$-0.900534\pi$
$193$	−8.94883	+	8.94883i	−0.644151	+	0.644151i	0.307422	+	0.951573i	$0.400534\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.3316	−	16.3316i	1.16358	−	1.16358i	0.179893	−	0.983686i	$-0.442425\pi$
$197$	16.3316	−	16.3316i	1.16358	−	1.16358i	0.983686	−	0.179893i	$-0.0575752\pi$
$198$		0			0
$199$			7.72478i			0.547595i	0.961787	+	0.273797i	$0.0882798\pi$
$199$			7.72478i			0.547595i	−0.961787	+	0.273797i	$0.911720\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.45726	+	8.45726i	0.593583	+	0.593583i
$204$		0			0
$205$	3.65357	−	0.562369i	0.255176	−	0.0392775i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.26815			−0.571920
$210$		0			0
$211$	4.63475			0.319070			0.159535	−	0.987192i	$-0.449001\pi$
$211$	4.63475			0.319070			0.159535	+	0.987192i	$0.449001\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.07032	−	4.45086i	−0.413992	−	0.303546i
$216$		0			0
$217$	−15.2063	−	15.2063i	−1.03227	−	1.03227i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.282620i		−	0.0190111i
$222$		0			0
$223$	0.883812	−	0.883812i	0.0591844	−	0.0591844i	−0.676895	−	0.736080i	$-0.736675\pi$
$223$	0.883812	−	0.883812i	0.0591844	−	0.0591844i	0.736080	+	0.676895i	$0.236675\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.3080	−	14.3080i	0.949656	−	0.949656i	−0.0491357	−	0.998792i	$-0.515647\pi$
$227$	14.3080	−	14.3080i	0.949656	−	0.949656i	0.998792	+	0.0491357i	$0.0156467\pi$
$228$		0			0
$229$			8.30395i			0.548741i	0.961624	+	0.274370i	$0.0884694\pi$
$229$			8.30395i			0.548741i	−0.961624	+	0.274370i	$0.911531\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.5686	−	10.5686i	−0.692370	−	0.692370i	0.270383	−	0.962753i	$-0.412850\pi$
$233$	−10.5686	−	10.5686i	−0.692370	−	0.692370i	−0.962753	+	0.270383i	$0.912850\pi$
$234$		0			0
$235$	−3.95279	−	2.89825i	−0.257852	−	0.189061i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.0914			−0.976182			−0.488091	−	0.872793i	$-0.662307\pi$
$239$	−15.0914			−0.976182			−0.488091	+	0.872793i	$0.662307\pi$
$240$		0			0
$241$	1.38199			0.0890220			0.0445110	−	0.999009i	$-0.485827\pi$
$241$	1.38199			0.0890220			0.0445110	+	0.999009i	$0.485827\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−38.5956	+	5.94076i	−2.46578	+	0.379541i
$246$		0			0
$247$	0.198756	+	0.198756i	0.0126465	+	0.0126465i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			13.0606i			0.824380i	0.911098	+	0.412190i	$0.135236\pi$
$251$			13.0606i			0.824380i	−0.911098	+	0.412190i	$0.864764\pi$
$252$		0			0
$253$	−0.851611	+	0.851611i	−0.0535403	+	0.0535403i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.19330	−	5.19330i	0.323949	−	0.323949i	−0.526331	−	0.850280i	$-0.676433\pi$
$257$	5.19330	−	5.19330i	0.323949	−	0.323949i	0.850280	+	0.526331i	$0.176433\pi$
$258$		0			0
$259$			50.9141i			3.16365i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.31761	+	4.31761i	0.266236	+	0.266236i	0.827581	−	0.561346i	$-0.189716\pi$
$263$	4.31761	+	4.31761i	0.266236	+	0.266236i	−0.561346	+	0.827581i	$0.689716\pi$
$264$		0			0
$265$	−0.352012	+	0.480093i	−0.0216239	+	0.0294919i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.5832			0.950125			0.475063	−	0.879952i	$-0.342425\pi$
$269$	15.5832			0.950125			0.475063	+	0.879952i	$0.342425\pi$
$270$		0			0
$271$	−25.6217			−1.55641			−0.778203	−	0.628012i	$-0.783869\pi$
$271$	−25.6217			−1.55641			−0.778203	+	0.628012i	$0.783869\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.81089	−	5.74306i	−0.109201	−	0.346320i
$276$		0			0
$277$	−14.2721	−	14.2721i	−0.857525	−	0.857525i	0.133521	−	0.991046i	$-0.457372\pi$
$277$	−14.2721	−	14.2721i	−0.857525	−	0.857525i	−0.991046	+	0.133521i	$0.957372\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6.95179i			0.414709i	0.978266	+	0.207354i	$0.0664853\pi$
$281$			6.95179i			0.414709i	−0.978266	+	0.207354i	$0.933515\pi$
$282$		0			0
$283$	7.53076	−	7.53076i	0.447657	−	0.447657i	−0.446918	−	0.894575i	$-0.647478\pi$
$283$	7.53076	−	7.53076i	0.447657	−	0.447657i	0.894575	+	0.446918i	$0.147478\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.78180	+	5.78180i	−0.341289	+	0.341289i
$288$		0			0
$289$		−	30.6474i		−	1.80279i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	19.9953	+	19.9953i	1.16814	+	1.16814i	0.982646	+	0.185491i	$0.0593877\pi$
$293$	19.9953	+	19.9953i	1.16814	+	1.16814i	0.185491	+	0.982646i	$0.440612\pi$
$294$		0			0
$295$	0.0620566	+	0.403166i	0.00361307	+	0.0234732i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.0409433			0.00236781
$300$		0			0
$301$	16.6499			0.959682
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.57756	+	23.2425i	0.204851	+	1.33086i
$306$		0			0
$307$	17.8844	+	17.8844i	1.02072	+	1.02072i	0.999781	+	0.0209372i	$0.00666500\pi$
$307$	17.8844	+	17.8844i	1.02072	+	1.02072i	0.0209372	+	0.999781i	$0.493335\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	31.0578i		−	1.76112i	−0.473931	−	0.880562i	$-0.657165\pi$
$311$		−	31.0578i		−	1.76112i	0.473931	−	0.880562i	$-0.342835\pi$
$312$		0			0
$313$	−18.1884	+	18.1884i	−1.02807	+	1.02807i	−0.0284719	+	0.999595i	$0.509064\pi$
$313$	−18.1884	+	18.1884i	−1.02807	+	1.02807i	−0.999595	+	0.0284719i	$0.990936\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.1173	−	22.1173i	1.24223	−	1.24223i	0.283158	−	0.959073i	$-0.408618\pi$
$317$	22.1173	−	22.1173i	1.24223	−	1.24223i	0.959073	−	0.283158i	$-0.0913821\pi$
$318$		0			0
$319$		−	2.91232i		−	0.163059i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	33.5086	+	33.5086i	1.86447	+	1.86447i
$324$		0			0
$325$	−0.0945246	+	0.181587i	−0.00524328	+	0.0100727i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	10.8418			0.597730
$330$		0			0
$331$	−26.9109			−1.47916			−0.739578	−	0.673071i	$-0.764975\pi$
$331$	−26.9109			−1.47916			−0.739578	+	0.673071i	$0.764975\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.70218	−	10.5046i	0.420815	−	0.573929i
$336$		0			0
$337$	9.03268	+	9.03268i	0.492041	+	0.492041i	0.908949	−	0.416908i	$-0.136886\pi$
$337$	9.03268	+	9.03268i	0.492041	+	0.492041i	−0.416908	+	0.908949i	$0.636886\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			5.23640i			0.283567i
$342$		0			0
$343$	36.5961	−	36.5961i	1.97600	−	1.97600i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.11067	+	5.11067i	−0.274355	+	0.274355i	−0.830851	−	0.556495i	$-0.812146\pi$
$347$	−5.11067	+	5.11067i	−0.274355	+	0.274355i	0.556495	+	0.830851i	$0.312146\pi$
$348$		0			0
$349$		−	7.49177i		−	0.401025i	−0.979691	−	0.200513i	$-0.935739\pi$
$349$		−	7.49177i		−	0.401025i	0.979691	−	0.200513i	$-0.0642608\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.2326	−	10.2326i	−0.544624	−	0.544624i	0.380257	−	0.924881i	$-0.375836\pi$
$353$	−10.2326	−	10.2326i	−0.544624	−	0.544624i	−0.924881	+	0.380257i	$0.875836\pi$
$354$		0			0
$355$	19.8759	−	3.05936i	1.05490	−	0.162374i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−35.9037			−1.89493			−0.947463	−	0.319865i	$-0.896362\pi$
$359$	−35.9037			−1.89493			−0.947463	+	0.319865i	$0.896362\pi$
$360$		0			0
$361$	−28.1307			−1.48056
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.0697	+	9.58295i	0.684101	+	0.501595i
$366$		0			0
$367$	−6.31994	−	6.31994i	−0.329898	−	0.329898i	0.522650	−	0.852548i	$-0.324944\pi$
$367$	−6.31994	−	6.31994i	−0.329898	−	0.329898i	−0.852548	+	0.522650i	$0.824944\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.31681i		−	0.0683656i
$372$		0			0
$373$	−3.83433	+	3.83433i	−0.198534	+	0.198534i	−0.799371	−	0.600837i	$-0.794834\pi$
$373$	−3.83433	+	3.83433i	−0.198534	+	0.198534i	0.600837	+	0.799371i	$0.294834\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.0700086	+	0.0700086i	−0.00360562	+	0.00360562i
$378$		0			0
$379$		−	2.30860i		−	0.118585i	−0.998241	−	0.0592924i	$-0.981116\pi$
$379$		−	2.30860i		−	0.118585i	0.998241	−	0.0592924i	$-0.0188844\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.2647	−	16.2647i	−0.831090	−	0.831090i	0.156576	−	0.987666i	$-0.449954\pi$
$383$	−16.2647	−	16.2647i	−0.831090	−	0.831090i	−0.987666	+	0.156576i	$0.949954\pi$
$384$		0			0
$385$	10.7419	+	7.87613i	0.547457	+	0.401405i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.1567			0.819175			0.409588	−	0.912271i	$-0.365673\pi$
$389$	16.1567			0.819175			0.409588	+	0.912271i	$0.365673\pi$
$390$		0			0
$391$	6.90271			0.349085
$392$		0			0
$393$		0			0
$394$		0			0
$395$	23.0234	−	3.54383i	1.15843	−	0.178310i
$396$		0			0
$397$	24.3552	+	24.3552i	1.22235	+	1.22235i	0.966794	+	0.255557i	$0.0822588\pi$
$397$	24.3552	+	24.3552i	1.22235	+	1.22235i	0.255557	+	0.966794i	$0.417741\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.39119i			0.169348i	0.996409	+	0.0846741i	$0.0269849\pi$
$401$			3.39119i			0.169348i	−0.996409	+	0.0846741i	$0.973015\pi$
$402$		0			0
$403$	0.125877	−	0.125877i	0.00627036	−	0.00627036i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.76634	−	8.76634i	0.434531	−	0.434531i
$408$		0			0
$409$		−	20.2815i		−	1.00286i	−0.865199	−	0.501429i	$-0.832808\pi$
$409$		−	20.2815i		−	1.00286i	0.865199	−	0.501429i	$-0.167192\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.638014	−	0.638014i	−0.0313946	−	0.0313946i
$414$		0			0
$415$	14.8685	−	20.2785i	0.729868	−	0.995431i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.9646			0.535658			0.267829	−	0.963466i	$-0.413694\pi$
$419$	10.9646			0.535658			0.267829	+	0.963466i	$0.413694\pi$
$420$		0			0
$421$	−23.0871			−1.12520			−0.562598	−	0.826730i	$-0.690198\pi$
$421$	−23.0871			−1.12520			−0.562598	+	0.826730i	$0.690198\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.9361	+	30.6141i	−0.773013	+	1.48500i
$426$		0			0
$427$	−36.7815	−	36.7815i	−1.77998	−	1.77998i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	31.9164i		−	1.53736i	−0.639633	−	0.768681i	$-0.720914\pi$
$431$		−	31.9164i		−	1.53736i	0.639633	−	0.768681i	$-0.279086\pi$
$432$		0			0
$433$	−10.1290	+	10.1290i	−0.486768	+	0.486768i	−0.907285	−	0.420517i	$-0.861849\pi$
$433$	−10.1290	+	10.1290i	−0.486768	+	0.486768i	0.420517	+	0.907285i	$0.361849\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.85442	+	4.85442i	−0.232218	+	0.232218i
$438$		0			0
$439$			28.4992i			1.36019i	0.733124	+	0.680096i	$0.238062\pi$
$439$			28.4992i			1.36019i	−0.733124	+	0.680096i	$0.761938\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.9728	−	26.9728i	−1.28151	−	1.28151i	−0.939806	−	0.341708i	$-0.888995\pi$
$443$	−26.9728	−	26.9728i	−1.28151	−	1.28151i	−0.341708	−	0.939806i	$-0.611005\pi$
$444$		0			0
$445$	4.69800	+	30.5217i	0.222706	+	1.44687i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.8676			−0.890416			−0.445208	−	0.895427i	$-0.646870\pi$
$449$	−18.8676			−0.890416			−0.445208	+	0.895427i	$0.646870\pi$
$450$		0			0
$451$	1.99101			0.0937529
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.0688889	−	0.447554i	−0.00322956	−	0.0209816i
$456$		0			0
$457$	−8.59139	−	8.59139i	−0.401888	−	0.401888i	0.477010	−	0.878898i	$-0.341721\pi$
$457$	−8.59139	−	8.59139i	−0.401888	−	0.401888i	−0.878898	+	0.477010i	$0.841721\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	33.1535i		−	1.54411i	−0.635553	−	0.772057i	$-0.719228\pi$
$461$		−	33.1535i		−	1.54411i	0.635553	−	0.772057i	$-0.280772\pi$
$462$		0			0
$463$	−14.6583	+	14.6583i	−0.681230	+	0.681230i	−0.960277	−	0.279047i	$-0.909981\pi$
$463$	−14.6583	+	14.6583i	−0.681230	+	0.681230i	0.279047	+	0.960277i	$0.409981\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.0539110	−	0.0539110i	0.00249470	−	0.00249470i	−0.705858	−	0.708353i	$-0.749438\pi$
$467$	0.0539110	−	0.0539110i	0.00249470	−	0.00249470i	0.708353	+	0.705858i	$0.249438\pi$
$468$		0			0
$469$			28.8124i			1.33044i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.86675	−	2.86675i	−0.131813	−	0.131813i
$474$		0			0
$475$	−10.3225	−	32.7370i	−0.473631	−	1.50208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.6440			−0.486337			−0.243168	−	0.969984i	$-0.578187\pi$
$479$	−10.6440			−0.486337			−0.243168	+	0.969984i	$0.578187\pi$
$480$		0			0
$481$	−0.421463			−0.0192171
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.46314	+	8.81476i	−0.293476	+	0.400258i
$486$		0			0
$487$	−16.0624	−	16.0624i	−0.727859	−	0.727859i	0.242334	−	0.970193i	$-0.422087\pi$
$487$	−16.0624	−	16.0624i	−0.727859	−	0.727859i	−0.970193	+	0.242334i	$0.922087\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9.45718i		−	0.426796i	−0.976965	−	0.213398i	$-0.931547\pi$
$491$		−	9.45718i		−	0.426796i	0.976965	−	0.213398i	$-0.0684532\pi$
$492$		0			0
$493$	−11.8029	+	11.8029i	−0.531574	+	0.531574i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.4537	+	31.4537i	−1.41089	+	1.41089i
$498$		0			0
$499$			15.8797i			0.710872i	0.934701	+	0.355436i	$0.115668\pi$
$499$			15.8797i			0.710872i	−0.934701	+	0.355436i	$0.884332\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.0950	+	15.0950i	0.673051	+	0.673051i	0.958418	−	0.285367i	$-0.0921156\pi$
$503$	15.0950	+	15.0950i	0.673051	+	0.673051i	−0.285367	+	0.958418i	$0.592116\pi$
$504$		0			0
$505$	19.1212	−	2.94320i	0.850883	−	0.130971i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.21209			−0.275346			−0.137673	−	0.990478i	$-0.543962\pi$
$509$	−6.21209			−0.275346			−0.137673	+	0.990478i	$0.543962\pi$
$510$		0			0
$511$	−35.8481			−1.58583
$512$		0			0
$513$		0			0
$514$		0			0
$515$	26.9647	+	19.7710i	1.18821	+	0.871214i
$516$		0			0
$517$	−1.86674	−	1.86674i	−0.0820989	−	0.0820989i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	41.2899i		−	1.80895i	−0.426532	−	0.904473i	$-0.640265\pi$
$521$		−	41.2899i		−	1.80895i	0.426532	−	0.904473i	$-0.359735\pi$
$522$		0			0
$523$	6.89452	−	6.89452i	0.301476	−	0.301476i	−0.540115	−	0.841591i	$-0.681619\pi$
$523$	6.89452	−	6.89452i	0.301476	−	0.301476i	0.841591	+	0.540115i	$0.181619\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.2218	−	21.2218i	0.924434	−	0.924434i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.0478613	−	0.0478613i	−0.00207310	−	0.00207310i
$534$		0			0
$535$	−7.99837	−	5.86454i	−0.345799	−	0.253546i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−21.0326			−0.905940
$540$		0			0
$541$	14.7467			0.634008			0.317004	−	0.948424i	$-0.397323\pi$
$541$	14.7467			0.634008			0.317004	+	0.948424i	$0.397323\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	23.8932	−	3.67772i	1.02347	−	0.157536i
$546$		0			0
$547$	13.8146	+	13.8146i	0.590670	+	0.590670i	0.937812	−	0.347143i	$-0.112848\pi$
$547$	13.8146	+	13.8146i	0.590670	+	0.590670i	−0.347143	+	0.937812i	$0.612848\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	16.6010i		−	0.707228i
$552$		0			0
$553$	−36.4347	+	36.4347i	−1.54936	+	1.54936i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.06189	+	3.06189i	−0.129736	+	0.129736i	−0.768993	−	0.639257i	$-0.779242\pi$
$557$	−3.06189	+	3.06189i	−0.129736	+	0.129736i	0.639257	+	0.768993i	$0.279242\pi$
$558$		0			0
$559$			0.137826i			0.00582943i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.19362	−	2.19362i	−0.0924500	−	0.0924500i	0.659369	−	0.751819i	$-0.270823\pi$
$563$	−2.19362	−	2.19362i	−0.0924500	−	0.0924500i	−0.751819	+	0.659369i	$0.770823\pi$
$564$		0			0
$565$	23.7114	−	32.3389i	0.997548	−	1.36051i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	42.4872			1.78116			0.890578	−	0.454830i	$-0.150300\pi$
$569$	42.4872			1.78116			0.890578	+	0.454830i	$0.150300\pi$
$570$		0			0
$571$	6.26816			0.262314			0.131157	−	0.991362i	$-0.458131\pi$
$571$	6.26816			0.262314			0.131157	+	0.991362i	$0.458131\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.43509	−	2.30867i	−0.184956	−	0.0962782i
$576$		0			0
$577$	−15.3804	−	15.3804i	−0.640293	−	0.640293i	0.310335	−	0.950627i	$-0.399559\pi$
$577$	−15.3804	−	15.3804i	−0.640293	−	0.640293i	−0.950627	+	0.310335i	$0.899559\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			55.6205i			2.30753i
$582$		0			0
$583$	−0.226727	+	0.226727i	−0.00939009	+	0.00939009i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−28.0541	+	28.0541i	−1.15792	+	1.15792i	−0.172996	+	0.984923i	$0.555345\pi$
$587$	−28.0541	+	28.0541i	−1.15792	+	1.15792i	−0.984923	+	0.172996i	$0.944655\pi$
$588$		0			0
$589$			29.8489i			1.22990i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.28356	−	8.28356i	−0.340165	−	0.340165i	0.516264	−	0.856429i	$-0.327322\pi$
$593$	−8.28356	−	8.28356i	−0.340165	−	0.340165i	−0.856429	+	0.516264i	$0.827322\pi$
$594$		0			0
$595$	−11.6141	−	75.4538i	−0.476132	−	3.09331i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.9978			−0.776227			−0.388114	−	0.921612i	$-0.626873\pi$
$599$	−18.9978			−0.776227			−0.388114	+	0.921612i	$0.626873\pi$
$600$		0			0
$601$	−2.20388			−0.0898983			−0.0449491	−	0.998989i	$-0.514313\pi$
$601$	−2.20388			−0.0898983			−0.0449491	+	0.998989i	$0.514313\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.24852	+	21.1048i	0.132071	+	0.858033i
$606$		0			0
$607$	20.2527	+	20.2527i	0.822032	+	0.822032i	0.986399	−	0.164367i	$-0.0525581\pi$
$607$	20.2527	+	20.2527i	0.822032	+	0.822032i	−0.164367	+	0.986399i	$0.552558\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			0.0897480i			0.00363081i
$612$		0			0
$613$	7.65158	−	7.65158i	0.309044	−	0.309044i	−0.535494	−	0.844539i	$-0.679875\pi$
$613$	7.65158	−	7.65158i	0.309044	−	0.309044i	0.844539	+	0.535494i	$0.179875\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−12.9174	+	12.9174i	−0.520034	+	0.520034i	−0.917582	−	0.397547i	$-0.869861\pi$
$617$	−12.9174	+	12.9174i	−0.520034	+	0.520034i	0.397547	+	0.917582i	$0.369861\pi$
$618$		0			0
$619$		−	32.9234i		−	1.32330i	−0.749812	−	0.661651i	$-0.769856\pi$
$619$		−	32.9234i		−	1.32330i	0.749812	−	0.661651i	$-0.230144\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−48.3008	−	48.3008i	−1.93513	−	1.93513i
$624$		0			0
$625$	20.4783	−	14.3401i	0.819133	−	0.573604i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−71.0553			−2.83316
$630$		0			0
$631$	9.97857			0.397241			0.198620	−	0.980076i	$-0.436354\pi$
$631$	9.97857			0.397241			0.198620	+	0.980076i	$0.436354\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.4489	−	26.5254i	0.771805	−	1.05263i
$636$		0			0
$637$	0.505599	+	0.505599i	0.0200325	+	0.0200325i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			19.5614i			0.772630i	0.922367	+	0.386315i	$0.126252\pi$
$641$			19.5614i			0.772630i	−0.922367	+	0.386315i	$0.873748\pi$
$642$		0			0
$643$	−14.3645	+	14.3645i	−0.566481	+	0.566481i	−0.931141	−	0.364660i	$-0.881185\pi$
$643$	−14.3645	+	14.3645i	−0.566481	+	0.566481i	0.364660	+	0.931141i	$0.381185\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.06978	+	4.06978i	−0.160000	+	0.160000i	−0.782567	−	0.622567i	$-0.786090\pi$
$647$	−4.06978	+	4.06978i	−0.160000	+	0.160000i	0.622567	+	0.782567i	$0.286090\pi$
$648$		0			0
$649$			0.219705i			0.00862417i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.6491	−	24.6491i	−0.964593	−	0.964593i	0.0348013	−	0.999394i	$-0.488920\pi$
$653$	−24.6491	−	24.6491i	−0.964593	−	0.964593i	−0.999394	+	0.0348013i	$0.988920\pi$
$654$		0			0
$655$	−6.45301	+	0.993268i	−0.252140	+	0.0388102i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14.0327			−0.546636			−0.273318	−	0.961924i	$-0.588121\pi$
$659$	−14.0327			−0.546636			−0.273318	+	0.961924i	$0.588121\pi$
$660$		0			0
$661$	−37.4773			−1.45770			−0.728850	−	0.684674i	$-0.759945\pi$
$661$	−37.4773			−1.45770			−0.728850	+	0.684674i	$0.759945\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	61.2316	+	44.8961i	2.37446	+	1.74100i
$666$		0			0
$667$	−1.70989	−	1.70989i	−0.0662072	−	0.0662072i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			12.6660i			0.488965i
$672$		0			0
$673$	11.5404	−	11.5404i	0.444852	−	0.444852i	−0.448787	−	0.893639i	$-0.648144\pi$
$673$	11.5404	−	11.5404i	0.444852	−	0.444852i	0.893639	+	0.448787i	$0.148144\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.49698	+	2.49698i	−0.0959669	+	0.0959669i	−0.753460	−	0.657493i	$-0.771617\pi$
$677$	−2.49698	+	2.49698i	−0.0959669	+	0.0959669i	0.657493	+	0.753460i	$0.271617\pi$
$678$		0			0
$679$		−	24.1774i		−	0.927844i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.09940	+	7.09940i	0.271651	+	0.271651i	0.829765	−	0.558114i	$-0.188475\pi$
$683$	7.09940	+	7.09940i	0.271651	+	0.271651i	−0.558114	+	0.829765i	$0.688475\pi$
$684$		0			0
$685$	30.0146	+	22.0072i	1.14680	+	0.840853i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.0109005			0.000415275
$690$		0			0
$691$	44.3667			1.68779			0.843895	−	0.536509i	$-0.180257\pi$
$691$	44.3667			1.68779			0.843895	+	0.536509i	$0.180257\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.67848	−	1.02797i	0.253329	−	0.0389932i
$696$		0			0
$697$	−8.06903	−	8.06903i	−0.305636	−	0.305636i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	32.8801i		−	1.24186i	−0.783865	−	0.620932i	$-0.786754\pi$
$701$		−	32.8801i		−	1.24186i	0.783865	−	0.620932i	$-0.213246\pi$
$702$		0			0
$703$	49.9705	−	49.9705i	1.88467	−	1.88467i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−30.2595	+	30.2595i	−1.13803	+	1.13803i
$708$		0			0
$709$		−	1.14419i		−	0.0429711i	−0.999769	−	0.0214856i	$-0.993160\pi$
$709$		−	1.14419i		−	0.0429711i	0.999769	−	0.0214856i	$-0.00683960\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.07441	+	3.07441i	0.115138	+	0.115138i
$714$		0			0
$715$	−0.0651980	+	0.0889204i	−0.00243827	+	0.00332544i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.4006			0.686227			0.343113	−	0.939294i	$-0.388518\pi$
$719$	18.4006			0.686227			0.343113	+	0.939294i	$0.388518\pi$
$720$		0			0
$721$	−73.9597			−2.75440
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.5311	−	3.63595i	0.428254	−	0.135036i
$726$		0			0
$727$	34.2722	+	34.2722i	1.27109	+	1.27109i	0.945519	+	0.325566i	$0.105555\pi$
$727$	34.2722	+	34.2722i	1.27109	+	1.27109i	0.325566	+	0.945519i	$0.394445\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			23.2364i			0.859429i
$732$		0			0
$733$	3.15029	−	3.15029i	0.116359	−	0.116359i	−0.646530	−	0.762889i	$-0.723780\pi$
$733$	3.15029	−	3.15029i	0.116359	−	0.116359i	0.762889	+	0.646530i	$0.223780\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.96089	−	4.96089i	0.182737	−	0.182737i
$738$		0			0
$739$			40.9531i			1.50648i	0.657744	+	0.753241i	$0.271511\pi$
$739$			40.9531i			1.50648i	−0.657744	+	0.753241i	$0.728489\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	32.2691	+	32.2691i	1.18384	+	1.18384i	0.978741	+	0.205098i	$0.0657514\pi$
$743$	32.2691	+	32.2691i	1.18384	+	1.18384i	0.205098	+	0.978741i	$0.434249\pi$
$744$		0			0
$745$	5.54546	+	36.0274i	0.203170	+	1.31994i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21.9382			0.801604
$750$		0			0
$751$	4.71689			0.172122			0.0860608	−	0.996290i	$-0.472572\pi$
$751$	4.71689			0.172122			0.0860608	+	0.996290i	$0.472572\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.08498	+	26.5391i	0.148668	+	0.965855i
$756$		0			0
$757$	−20.1048	−	20.1048i	−0.730721	−	0.730721i	0.240042	−	0.970763i	$-0.422839\pi$
$757$	−20.1048	−	20.1048i	−0.730721	−	0.730721i	−0.970763	+	0.240042i	$0.922839\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			11.6751i			0.423221i	0.977354	+	0.211611i	$0.0678708\pi$
$761$			11.6751i			0.423221i	−0.977354	+	0.211611i	$0.932129\pi$
$762$		0			0
$763$	−37.8112	+	37.8112i	−1.36886	+	1.36886i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.00528143	−	0.00528143i	0.000190701	−	0.000190701i
$768$		0			0
$769$			29.1746i			1.05206i	0.850465	+	0.526031i	$0.176320\pi$
$769$			29.1746i			1.05206i	−0.850465	+	0.526031i	$0.823680\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15.8607	+	15.8607i	0.570470	+	0.570470i	0.932260	−	0.361789i	$-0.117834\pi$
$773$	15.8607	+	15.8607i	0.570470	+	0.570470i	−0.361789	+	0.932260i	$0.617834\pi$
$774$		0			0
$775$	−20.7331	+	6.53750i	−0.744755	+	0.234834i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11.3493			0.406631
$780$		0			0
$781$	10.8313			0.387576
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−28.5328	+	38.9145i	−1.01838	+	1.38892i
$786$		0			0
$787$	−14.6732	−	14.6732i	−0.523042	−	0.523042i	0.395447	−	0.918489i	$-0.370590\pi$
$787$	−14.6732	−	14.6732i	−0.523042	−	0.523042i	−0.918489	+	0.395447i	$0.870590\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			88.7002i			3.15382i
$792$		0			0
$793$	0.304475	−	0.304475i	0.0108122	−	0.0108122i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.57529	+	8.57529i	−0.303752	+	0.303752i	−0.842480	−	0.538728i	$-0.818905\pi$
$797$	−8.57529	+	8.57529i	−0.303752	+	0.303752i	0.538728	+	0.842480i	$0.318905\pi$
$798$		0			0
$799$			15.1308i			0.535288i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.17228	+	6.17228i	0.217815	+	0.217815i
$804$		0			0
$805$	10.9310	−	1.68254i	0.385269	−	0.0593018i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.8898			0.945394			0.472697	−	0.881225i	$-0.343281\pi$
$809$	26.8898			0.945394			0.472697	+	0.881225i	$0.343281\pi$
$810$		0			0
$811$	−2.76045			−0.0969326			−0.0484663	−	0.998825i	$-0.515433\pi$
$811$	−2.76045			−0.0969326			−0.0484663	+	0.998825i	$0.515433\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.9418	+	10.9556i	0.523387	+	0.383757i
$816$		0			0
$817$	−16.3413	−	16.3413i	−0.571709	−	0.571709i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			7.09138i			0.247491i	0.992314	+	0.123745i	$0.0394906\pi$
$821$			7.09138i			0.247491i	−0.992314	+	0.123745i	$0.960509\pi$
$822$		0			0
$823$	17.4737	−	17.4737i	0.609096	−	0.609096i	−0.333614	−	0.942710i	$-0.608268\pi$
$823$	17.4737	−	17.4737i	0.609096	−	0.609096i	0.942710	+	0.333614i	$0.108268\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.9446	+	13.9446i	−0.484901	+	0.484901i	−0.906693	−	0.421792i	$-0.861401\pi$
$827$	−13.9446	+	13.9446i	−0.484901	+	0.484901i	0.421792	+	0.906693i	$0.361401\pi$
$828$		0			0
$829$		−	1.80730i		−	0.0627701i	−0.999507	−	0.0313850i	$-0.990008\pi$
$829$		−	1.80730i		−	0.0627701i	0.999507	−	0.0313850i	$-0.00999181\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	85.2398	+	85.2398i	2.95338	+	2.95338i
$834$		0			0
$835$	4.62557	+	3.39155i	0.160075	+	0.117369i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	51.1808			1.76696			0.883478	−	0.468472i	$-0.155195\pi$
$839$	51.1808			1.76696			0.883478	+	0.468472i	$0.155195\pi$
$840$		0			0
$841$	−23.1526			−0.798364
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−28.7268	+	4.42173i	−0.988233	+	0.152112i
$846$		0			0
$847$	−33.3986	−	33.3986i	−1.14759	−	1.14759i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	10.2938i		−	0.352868i
$852$		0			0
$853$	−6.30899	+	6.30899i	−0.216016	+	0.216016i	−0.806817	−	0.590801i	$-0.798812\pi$
$853$	−6.30899	+	6.30899i	−0.216016	+	0.216016i	0.590801	+	0.806817i	$0.298812\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.5190	−	16.5190i	0.564278	−	0.564278i	−0.366242	−	0.930520i	$-0.619356\pi$
$857$	16.5190	−	16.5190i	0.564278	−	0.564278i	0.930520	+	0.366242i	$0.119356\pi$
$858$		0			0
$859$			4.51418i			0.154022i	0.997030	+	0.0770109i	$0.0245376\pi$
$859$			4.51418i			0.154022i	−0.997030	+	0.0770109i	$0.975462\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33.6202	+	33.6202i	1.14445	+	1.14445i	0.987627	+	0.156819i	$0.0501238\pi$
$863$	33.6202	+	33.6202i	1.14445	+	1.14445i	0.156819	+	0.987627i	$0.449876\pi$
$864$		0			0
$865$	−17.9857	+	24.5299i	−0.611534	+	0.834041i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.5466			0.425613
$870$		0			0
$871$	−0.238507			−0.00808151
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17.7739	+	52.3647i	−0.600869	+	1.77025i
$876$		0			0
$877$	10.4832	+	10.4832i	0.353993	+	0.353993i	0.861593	−	0.507600i	$-0.169467\pi$
$877$	10.4832	+	10.4832i	0.353993	+	0.353993i	−0.507600	+	0.861593i	$0.669467\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	51.6565i		−	1.74035i	−0.492741	−	0.870176i	$-0.664005\pi$
$881$		−	51.6565i		−	1.74035i	0.492741	−	0.870176i	$-0.335995\pi$
$882$		0			0
$883$	−17.0352	+	17.0352i	−0.573281	+	0.573281i	−0.933044	−	0.359763i	$-0.882858\pi$
$883$	−17.0352	+	17.0352i	−0.573281	+	0.573281i	0.359763	+	0.933044i	$0.382858\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.9465	+	15.9465i	−0.535431	+	0.535431i	−0.922183	−	0.386753i	$-0.873597\pi$
$887$	−15.9465	+	15.9465i	−0.535431	+	0.535431i	0.386753	+	0.922183i	$0.373597\pi$
$888$		0			0
$889$			72.7547i			2.44011i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.6409	−	10.6409i	−0.356084	−	0.356084i
$894$		0			0
$895$	−6.04841	−	39.2950i	−0.202176	−	1.31349i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−10.5138			−0.350655
$900$		0			0
$901$	1.83773			0.0612237
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.927810	+	6.02775i	0.0308415	+	0.200369i
$906$		0			0
$907$	31.0400	+	31.0400i	1.03067	+	1.03067i	0.999515	+	0.0311515i	$0.00991743\pi$
$907$	31.0400	+	31.0400i	1.03067	+	1.03067i	0.0311515	+	0.999515i	$0.490083\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	6.28369i		−	0.208188i	−0.994567	−	0.104094i	$-0.966806\pi$
$911$		−	6.28369i		−	0.208188i	0.994567	−	0.104094i	$-0.0331943\pi$
$912$		0			0
$913$	9.57667	−	9.57667i	0.316941	−	0.316941i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.2119	−	10.2119i	0.337228	−	0.337228i
$918$		0			0
$919$		−	22.2922i		−	0.735352i	−0.929954	−	0.367676i	$-0.880154\pi$
$919$		−	22.2922i		−	0.735352i	0.929954	−	0.367676i	$-0.119846\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.260372	−	0.260372i	−0.00857024	−	0.00857024i
$924$		0			0
$925$	45.6541	+	23.7650i	1.50110	+	0.781390i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.603857			−0.0198119			−0.00990595	−	0.999951i	$-0.503153\pi$
$929$	−0.603857			−0.0198119			−0.00990595	+	0.999951i	$0.503153\pi$
$930$		0			0
$931$	−119.892			−3.92930
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.9918	+	14.9913i	−0.359472	+	0.490266i
$936$		0			0
$937$	−13.5741	−	13.5741i	−0.443446	−	0.443446i	0.449722	−	0.893168i	$-0.351523\pi$
$937$	−13.5741	−	13.5741i	−0.443446	−	0.443446i	−0.893168	+	0.449722i	$0.851523\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	17.8250i		−	0.581078i	−0.956863	−	0.290539i	$-0.906165\pi$
$941$		−	17.8250i		−	0.581078i	0.956863	−	0.290539i	$-0.0938346\pi$
$942$		0			0
$943$	1.16897	−	1.16897i	0.0380668	−	0.0380668i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.47697	+	7.47697i	−0.242969	+	0.242969i	−0.818077	−	0.575108i	$-0.804960\pi$
$947$	−7.47697	+	7.47697i	−0.242969	+	0.242969i	0.575108	+	0.818077i	$0.304960\pi$
$948$		0			0
$949$		−	0.296748i		−	0.00963284i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	38.6008	+	38.6008i	1.25040	+	1.25040i	0.955539	+	0.294863i	$0.0952742\pi$
$953$	38.6008	+	38.6008i	1.25040	+	1.25040i	0.294863	+	0.955539i	$0.404726\pi$
$954$		0			0
$955$	19.0757	−	2.93619i	0.617274	−	0.0950128i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−82.3251			−2.65841
$960$		0			0
$961$	−12.0960			−0.390193
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.8215	−	16.7331i	−0.734649	−	0.538657i
$966$		0			0
$967$	−14.5713	−	14.5713i	−0.468581	−	0.468581i	0.432873	−	0.901455i	$-0.357500\pi$
$967$	−14.5713	−	14.5713i	−0.468581	−	0.468581i	−0.901455	+	0.432873i	$0.857500\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	9.34795i		−	0.299990i	−0.988687	−	0.149995i	$-0.952074\pi$
$971$		−	9.34795i		−	0.299990i	0.988687	−	0.149995i	$-0.0479258\pi$
$972$		0			0
$973$	−10.5688	+	10.5688i	−0.338819	+	0.338819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	29.8649	−	29.8649i	0.955464	−	0.955464i	−0.0435856	−	0.999050i	$-0.513878\pi$
$977$	29.8649	−	29.8649i	0.955464	−	0.955464i	0.999050	+	0.0435856i	$0.0138781\pi$
$978$		0			0
$979$			16.6328i			0.531585i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25.3895	+	25.3895i	0.809799	+	0.809799i	0.984603	−	0.174804i	$-0.0559291\pi$
$983$	25.3895	+	25.3895i	0.809799	+	0.809799i	−0.174804	+	0.984603i	$0.555929\pi$
$984$		0			0
$985$	41.6492	+	30.5379i	1.32705	+	0.973018i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.36627			−0.107041
$990$		0			0
$991$	−47.1920			−1.49910			−0.749552	−	0.661945i	$-0.769731\pi$
$991$	−47.1920			−1.49910			−0.749552	+	0.661945i	$0.769731\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17.0721	+	2.62779i	−0.541221	+	0.0833065i
$996$		0			0
$997$	25.9953	+	25.9953i	0.823280	+	0.823280i	0.986577	−	0.163297i	$-0.0522128\pi$
$997$	25.9953	+	25.9953i	0.823280	+	0.823280i	−0.163297	+	0.986577i	$0.552213\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.a.737.14	yes	44
3.2	odd	2	inner	4140.2.s.a.737.9	✓	44
5.3	odd	4	inner	4140.2.s.a.2393.9	yes	44
15.8	even	4	inner	4140.2.s.a.2393.14	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.9	✓	44	3.2	odd	2	inner
4140.2.s.a.737.14	yes	44	1.1	even	1	trivial
4140.2.s.a.2393.9	yes	44	5.3	odd	4	inner
4140.2.s.a.2393.14	yes	44	15.8	even	4	inner

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.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.340177 + 2.21004i) q^{5} +(-3.49741 - 3.49741i) q^{7} +O(q^{10})\)	\(q+(0.340177 + 2.21004i) q^{5} +(-3.49741 - 3.49741i) q^{7} +1.20436i q^{11} +(0.0289513 - 0.0289513i) q^{13} +(4.88095 - 4.88095i) q^{17} +6.86518i q^{19} +(0.707107 + 0.707107i) q^{23} +(-4.76856 + 1.50361i) q^{25} -2.41815 q^{29} +4.34787 q^{31} +(6.53968 - 8.91916i) q^{35} +(-7.27883 - 7.27883i) q^{37} -1.65317i q^{41} +(-2.38031 + 2.38031i) q^{43} +(-1.54998 + 1.54998i) q^{47} +17.4638i q^{49} +(0.188256 + 0.188256i) q^{53} +(-2.66168 + 0.409695i) q^{55} +0.182425 q^{59} +10.5168 q^{61} +(0.0738321 + 0.0541350i) q^{65} +(-4.11911 - 4.11911i) q^{67} -8.99344i q^{71} +(5.12495 - 5.12495i) q^{73} +(4.21214 - 4.21214i) q^{77} -10.4176i q^{79} +(-7.95167 - 7.95167i) q^{83} +(12.4475 + 9.12672i) q^{85} +13.8105 q^{89} -0.202509 q^{91} +(-15.1723 + 2.33537i) q^{95} +(3.45647 + 3.45647i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 44 * q - 12 * q^7	\( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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