LMFDB - Embedded newform 4140.2.s.a.737.6

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			737.6
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.77166 + 1.36427i) q^{5} +(3.33197 + 3.33197i) q^{7} +O(q^{10})$	$q+(-1.77166 + 1.36427i) q^{5} +(3.33197 + 3.33197i) q^{7} -1.66971i q^{11} +(-2.55795 + 2.55795i) q^{13} +(-1.21550 + 1.21550i) q^{17} +2.64188i q^{19} +(-0.707107 - 0.707107i) q^{23} +(1.27752 - 4.83404i) q^{25} -1.63157 q^{29} +1.29243 q^{31} +(-10.4488 - 1.35739i) q^{35} +(5.51115 + 5.51115i) q^{37} +10.2362i q^{41} +(-3.39690 + 3.39690i) q^{43} +(4.00206 - 4.00206i) q^{47} +15.2041i q^{49} +(-2.13690 - 2.13690i) q^{53} +(2.27794 + 2.95815i) q^{55} -4.14478 q^{59} -13.4611 q^{61} +(1.04207 - 8.02153i) q^{65} +(3.61788 + 3.61788i) q^{67} -12.0837i q^{71} +(3.28632 - 3.28632i) q^{73} +(5.56343 - 5.56343i) q^{77} +1.68432i q^{79} +(-7.92698 - 7.92698i) q^{83} +(0.495176 - 3.81173i) q^{85} -2.20916 q^{89} -17.0460 q^{91} +(-3.60425 - 4.68051i) q^{95} +(-7.00880 - 7.00880i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.77166	+	1.36427i	−0.792308	+	0.610121i
$5$	−1.77166	+	1.36427i	−0.792308	+	0.610121i
$6$		0			0
$7$	3.33197	+	3.33197i	1.25937	+	1.25937i	0.951395	+	0.307973i	$0.0996505\pi$
$7$	3.33197	+	3.33197i	1.25937	+	1.25937i	0.307973	+	0.951395i	$0.400349\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.66971i		−	0.503437i	−0.967801	−	0.251718i	$-0.919004\pi$
$11$		−	1.66971i		−	0.503437i	0.967801	−	0.251718i	$-0.0809956\pi$
$12$		0			0
$13$	−2.55795	+	2.55795i	−0.709447	+	0.709447i	−0.966419	−	0.256972i	$-0.917275\pi$
$13$	−2.55795	+	2.55795i	−0.709447	+	0.709447i	0.256972	+	0.966419i	$0.417275\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.21550	+	1.21550i	−0.294803	+	0.294803i	−0.838974	−	0.544171i	$-0.816844\pi$
$17$	−1.21550	+	1.21550i	−0.294803	+	0.294803i	0.544171	+	0.838974i	$0.316844\pi$
$18$		0			0
$19$			2.64188i			0.606090i	0.952976	+	0.303045i	$0.0980032\pi$
$19$			2.64188i			0.606090i	−0.952976	+	0.303045i	$0.901997\pi$
$20$		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
$25$	1.27752	−	4.83404i	0.255505	−	0.966808i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.63157			−0.302975			−0.151488	−	0.988459i	$-0.548406\pi$
$29$	−1.63157			−0.302975			−0.151488	+	0.988459i	$0.548406\pi$
$30$		0			0
$31$	1.29243			0.232127			0.116063	−	0.993242i	$-0.462972\pi$
$31$	1.29243			0.232127			0.116063	+	0.993242i	$0.462972\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.4488	−	1.35739i	−1.76617	−	0.229441i
$36$		0			0
$37$	5.51115	+	5.51115i	0.906027	+	0.906027i	0.995949	−	0.0899214i	$-0.0286616\pi$
$37$	5.51115	+	5.51115i	0.906027	+	0.906027i	−0.0899214	+	0.995949i	$0.528662\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.2362i			1.59863i	0.600915	+	0.799313i	$0.294803\pi$
$41$			10.2362i			1.59863i	−0.600915	+	0.799313i	$0.705197\pi$
$42$		0			0
$43$	−3.39690	+	3.39690i	−0.518023	+	0.518023i	−0.916973	−	0.398950i	$-0.869375\pi$
$43$	−3.39690	+	3.39690i	−0.518023	+	0.518023i	0.398950	+	0.916973i	$0.369375\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00206	−	4.00206i	0.583760	−	0.583760i	−0.352175	−	0.935934i	$-0.614558\pi$
$47$	4.00206	−	4.00206i	0.583760	−	0.583760i	0.935934	+	0.352175i	$0.114558\pi$
$48$		0			0
$49$			15.2041i			2.17201i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.13690	−	2.13690i	−0.293526	−	0.293526i	0.544945	−	0.838472i	$-0.316550\pi$
$53$	−2.13690	−	2.13690i	−0.293526	−	0.293526i	−0.838472	+	0.544945i	$0.816550\pi$
$54$		0			0
$55$	2.27794	+	2.95815i	0.307157	+	0.398877i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.14478			−0.539604			−0.269802	−	0.962916i	$-0.586958\pi$
$59$	−4.14478			−0.539604			−0.269802	+	0.962916i	$0.586958\pi$
$60$		0			0
$61$	−13.4611			−1.72351			−0.861756	−	0.507322i	$-0.830635\pi$
$61$	−13.4611			−1.72351			−0.861756	+	0.507322i	$0.830635\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.04207	−	8.02153i	0.129252	−	0.994949i
$66$		0			0
$67$	3.61788	+	3.61788i	0.441995	+	0.441995i	0.892682	−	0.450687i	$-0.148821\pi$
$67$	3.61788	+	3.61788i	0.441995	+	0.441995i	−0.450687	+	0.892682i	$0.648821\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.0837i		−	1.43407i	−0.697038	−	0.717034i	$-0.745499\pi$
$71$		−	12.0837i		−	1.43407i	0.697038	−	0.717034i	$-0.254501\pi$
$72$		0			0
$73$	3.28632	−	3.28632i	0.384634	−	0.384634i	−0.488134	−	0.872769i	$-0.662322\pi$
$73$	3.28632	−	3.28632i	0.384634	−	0.384634i	0.872769	+	0.488134i	$0.162322\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.56343	−	5.56343i	0.634012	−	0.634012i
$78$		0			0
$79$			1.68432i			0.189501i	0.995501	+	0.0947503i	$0.0302053\pi$
$79$			1.68432i			0.189501i	−0.995501	+	0.0947503i	$0.969795\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.92698	−	7.92698i	−0.870099	−	0.870099i	0.122383	−	0.992483i	$-0.460946\pi$
$83$	−7.92698	−	7.92698i	−0.870099	−	0.870099i	−0.992483	+	0.122383i	$0.960946\pi$
$84$		0			0
$85$	0.495176	−	3.81173i	0.0537094	−	0.413440i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.20916			−0.234171			−0.117085	−	0.993122i	$-0.537355\pi$
$89$	−2.20916			−0.234171			−0.117085	+	0.993122i	$0.537355\pi$
$90$		0			0
$91$	−17.0460			−1.78691
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.60425	−	4.68051i	−0.369788	−	0.480210i
$96$		0			0
$97$	−7.00880	−	7.00880i	−0.711636	−	0.711636i	0.255241	−	0.966877i	$-0.417845\pi$
$97$	−7.00880	−	7.00880i	−0.711636	−	0.711636i	−0.966877	+	0.255241i	$0.917845\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			7.65636i			0.761836i	0.924609	+	0.380918i	$0.124392\pi$
$101$			7.65636i			0.761836i	−0.924609	+	0.380918i	$0.875608\pi$
$102$		0			0
$103$	1.97583	−	1.97583i	0.194685	−	0.194685i	−0.603032	−	0.797717i	$-0.706041\pi$
$103$	1.97583	−	1.97583i	0.194685	−	0.194685i	0.797717	+	0.603032i	$0.206041\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.54379	−	4.54379i	0.439265	−	0.439265i	−0.452500	−	0.891765i	$-0.649468\pi$
$107$	4.54379	−	4.54379i	0.439265	−	0.439265i	0.891765	+	0.452500i	$0.149468\pi$
$108$		0			0
$109$			3.09955i			0.296883i	0.988921	+	0.148441i	$0.0474256\pi$
$109$			3.09955i			0.296883i	−0.988921	+	0.148441i	$0.952574\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.18841	+	6.18841i	0.582157	+	0.582157i	0.935496	−	0.353339i	$-0.114954\pi$
$113$	6.18841	+	6.18841i	0.582157	+	0.582157i	−0.353339	+	0.935496i	$0.614954\pi$
$114$		0			0
$115$	2.21744	+	0.288064i	0.206777	+	0.0268621i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.10005			−0.742530
$120$		0			0
$121$	8.21207			0.746552
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.33161	+	10.3071i	0.387431	+	0.921899i
$126$		0			0
$127$	−5.09047	−	5.09047i	−0.451706	−	0.451706i	0.444215	−	0.895920i	$-0.353483\pi$
$127$	−5.09047	−	5.09047i	−0.451706	−	0.451706i	−0.895920	+	0.444215i	$0.853483\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			0.704703i			0.0615702i	0.999526	+	0.0307851i	$0.00980075\pi$
$131$			0.704703i			0.0615702i	−0.999526	+	0.0307851i	$0.990199\pi$
$132$		0			0
$133$	−8.80269	+	8.80269i	−0.763290	+	0.763290i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.94793	+	1.94793i	−0.166423	+	0.166423i	−0.785405	−	0.618982i	$-0.787545\pi$
$137$	−1.94793	+	1.94793i	−0.166423	+	0.166423i	0.618982	+	0.785405i	$0.287545\pi$
$138$		0			0
$139$		−	10.1806i		−	0.863510i	−0.901991	−	0.431755i	$-0.857895\pi$
$139$		−	10.1806i		−	0.863510i	0.901991	−	0.431755i	$-0.142105\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.27103	+	4.27103i	0.357161	+	0.357161i
$144$		0			0
$145$	2.89058	−	2.22591i	0.240050	−	0.184852i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.85216			−0.725197			−0.362599	−	0.931945i	$-0.618110\pi$
$149$	−8.85216			−0.725197			−0.362599	+	0.931945i	$0.618110\pi$
$150$		0			0
$151$	−8.86089			−0.721089			−0.360545	−	0.932742i	$-0.617409\pi$
$151$	−8.86089			−0.721089			−0.360545	+	0.932742i	$0.617409\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.28974	+	1.76322i	−0.183916	+	0.141625i
$156$		0			0
$157$	9.11450	+	9.11450i	0.727416	+	0.727416i	0.970104	−	0.242688i	$-0.0780291\pi$
$157$	9.11450	+	9.11450i	0.727416	+	0.727416i	−0.242688	+	0.970104i	$0.578029\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.71212i		−	0.371367i
$162$		0			0
$163$	−2.40151	+	2.40151i	−0.188101	+	0.188101i	−0.794875	−	0.606774i	$-0.792463\pi$
$163$	−2.40151	+	2.40151i	−0.188101	+	0.188101i	0.606774	+	0.794875i	$0.292463\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.373594	−	0.373594i	0.0289095	−	0.0289095i	−0.692504	−	0.721414i	$-0.743493\pi$
$167$	0.373594	−	0.373594i	0.0289095	−	0.0289095i	0.721414	+	0.692504i	$0.243493\pi$
$168$		0			0
$169$		−	0.0861786i		−	0.00662912i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.2935	−	12.2935i	−0.934659	−	0.934659i	0.0633337	−	0.997992i	$-0.479827\pi$
$173$	−12.2935	−	12.2935i	−0.934659	−	0.934659i	−0.997992	+	0.0633337i	$0.979827\pi$
$174$		0			0
$175$	20.3636	−	11.8502i	1.53934	−	0.895792i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−11.0255			−0.824088			−0.412044	−	0.911164i	$-0.635185\pi$
$179$	−11.0255			−0.824088			−0.412044	+	0.911164i	$0.635185\pi$
$180$		0			0
$181$	−25.2745			−1.87864			−0.939319	−	0.343044i	$-0.888542\pi$
$181$	−25.2745			−1.87864			−0.939319	+	0.343044i	$0.888542\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−17.2826	−	2.24515i	−1.27064	−	0.165067i
$186$		0			0
$187$	2.02954	+	2.02954i	0.148415	+	0.148415i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.87463i			0.208001i	0.994577	+	0.104000i	$0.0331643\pi$
$191$			2.87463i			0.208001i	−0.994577	+	0.104000i	$0.966836\pi$
$192$		0			0
$193$	−5.70822	+	5.70822i	−0.410887	+	0.410887i	−0.882047	−	0.471161i	$-0.843835\pi$
$193$	−5.70822	+	5.70822i	−0.410887	+	0.410887i	0.471161	+	0.882047i	$0.343835\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.52123	−	8.52123i	0.607112	−	0.607112i	−0.335078	−	0.942190i	$-0.608763\pi$
$197$	8.52123	−	8.52123i	0.607112	−	0.607112i	0.942190	+	0.335078i	$0.108763\pi$
$198$		0			0
$199$			3.97080i			0.281482i	0.990046	+	0.140741i	$0.0449485\pi$
$199$			3.97080i			0.281482i	−0.990046	+	0.140741i	$0.955051\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.43636	−	5.43636i	−0.381557	−	0.381557i
$204$		0			0
$205$	−13.9650	−	18.1350i	−0.975355	−	1.26660i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.41118			0.305128
$210$		0			0
$211$	17.3500			1.19442			0.597211	−	0.802084i	$-0.296275\pi$
$211$	17.3500			1.19442			0.597211	+	0.802084i	$0.296275\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.38384	−	10.6524i	0.0943772	−	0.726490i
$216$		0			0
$217$	4.30633	+	4.30633i	0.292333	+	0.292333i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6.21838i		−	0.418294i
$222$		0			0
$223$	2.97365	−	2.97365i	0.199130	−	0.199130i	−0.600497	−	0.799627i	$-0.705030\pi$
$223$	2.97365	−	2.97365i	0.199130	−	0.199130i	0.799627	+	0.600497i	$0.205030\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.6569	+	19.6569i	−1.30468	+	1.30468i	−0.379472	+	0.925203i	$0.623894\pi$
$227$	−19.6569	+	19.6569i	−1.30468	+	1.30468i	−0.925203	+	0.379472i	$0.876106\pi$
$228$		0			0
$229$		−	15.4350i		−	1.01997i	−0.860182	−	0.509987i	$-0.829650\pi$
$229$		−	15.4350i		−	1.01997i	0.860182	−	0.509987i	$-0.170350\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.32226	+	4.32226i	0.283161	+	0.283161i	0.834368	−	0.551208i	$-0.185833\pi$
$233$	4.32226	+	4.32226i	0.283161	+	0.283161i	−0.551208	+	0.834368i	$0.685833\pi$
$234$		0			0
$235$	−1.63037	+	12.5502i	−0.106354	+	0.818682i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.83325			0.247952			0.123976	−	0.992285i	$-0.460435\pi$
$239$	3.83325			0.247952			0.123976	+	0.992285i	$0.460435\pi$
$240$		0			0
$241$	−17.6148			−1.13467			−0.567336	−	0.823487i	$-0.692026\pi$
$241$	−17.6148			−1.13467			−0.567336	+	0.823487i	$0.692026\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−20.7425	−	26.9364i	−1.32519	−	1.72091i
$246$		0			0
$247$	−6.75779	−	6.75779i	−0.429988	−	0.429988i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.4377i			0.785062i	0.919739	+	0.392531i	$0.128400\pi$
$251$			12.4377i			0.785062i	−0.919739	+	0.392531i	$0.871600\pi$
$252$		0			0
$253$	−1.18066	+	1.18066i	−0.0742277	+	0.0742277i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.8535	+	13.8535i	−0.864160	+	0.864160i	−0.991818	−	0.127658i	$-0.959254\pi$
$257$	−13.8535	+	13.8535i	−0.864160	+	0.864160i	0.127658	+	0.991818i	$0.459254\pi$
$258$		0			0
$259$			36.7260i			2.28204i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.7116	−	18.7116i	−1.15381	−	1.15381i	−0.985783	−	0.168022i	$-0.946262\pi$
$263$	−18.7116	−	18.7116i	−1.15381	−	1.15381i	−0.168022	−	0.985783i	$-0.553738\pi$
$264$		0			0
$265$	6.70117	+	0.870539i	0.411650	+	0.0534768i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.1915			−0.926241			−0.463121	−	0.886295i	$-0.653270\pi$
$269$	−15.1915			−0.926241			−0.463121	+	0.886295i	$0.653270\pi$
$270$		0			0
$271$	−11.5498			−0.701600			−0.350800	−	0.936450i	$-0.614090\pi$
$271$	−11.5498			−0.701600			−0.350800	+	0.936450i	$0.614090\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.07144	−	2.13310i	−0.486726	−	0.128631i
$276$		0			0
$277$	−5.45625	−	5.45625i	−0.327834	−	0.327834i	0.523928	−	0.851762i	$-0.324466\pi$
$277$	−5.45625	−	5.45625i	−0.327834	−	0.327834i	−0.851762	+	0.523928i	$0.824466\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8.81882i			0.526087i	0.964784	+	0.263043i	$0.0847263\pi$
$281$			8.81882i			0.526087i	−0.964784	+	0.263043i	$0.915274\pi$
$282$		0			0
$283$	−10.8374	+	10.8374i	−0.644217	+	0.644217i	−0.951589	−	0.307372i	$-0.900550\pi$
$283$	−10.8374	+	10.8374i	−0.644217	+	0.644217i	0.307372	+	0.951589i	$0.400550\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−34.1068	+	34.1068i	−2.01326	+	2.01326i
$288$		0			0
$289$			14.0451i			0.826183i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.73800	−	7.73800i	−0.452059	−	0.452059i	0.443978	−	0.896037i	$-0.353567\pi$
$293$	−7.73800	−	7.73800i	−0.452059	−	0.452059i	−0.896037	+	0.443978i	$0.853567\pi$
$294$		0			0
$295$	7.34312	−	5.65460i	0.427533	−	0.329224i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.61748			0.209204
$300$		0			0
$301$	−22.6368			−1.30476
$302$		0			0
$303$		0			0
$304$		0			0
$305$	23.8484	−	18.3646i	1.36555	−	1.05155i
$306$		0			0
$307$	16.8643	+	16.8643i	0.962498	+	0.962498i	0.999322	−	0.0368236i	$-0.0117240\pi$
$307$	16.8643	+	16.8643i	0.962498	+	0.962498i	−0.0368236	+	0.999322i	$0.511724\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.5002i			0.765526i	0.923847	+	0.382763i	$0.125027\pi$
$311$			13.5002i			0.765526i	−0.923847	+	0.382763i	$0.874973\pi$
$312$		0			0
$313$	−2.77208	+	2.77208i	−0.156687	+	0.156687i	−0.781097	−	0.624410i	$-0.785340\pi$
$313$	−2.77208	+	2.77208i	−0.156687	+	0.156687i	0.624410	+	0.781097i	$0.285340\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.9778	−	18.9778i	1.06590	−	1.06590i	0.0682290	−	0.997670i	$-0.478265\pi$
$317$	18.9778	−	18.9778i	1.06590	−	1.06590i	0.997670	−	0.0682290i	$-0.0217348\pi$
$318$		0			0
$319$			2.72425i			0.152529i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.21122	−	3.21122i	−0.178677	−	0.178677i
$324$		0			0
$325$	9.09737	+	15.6331i	0.504631	+	0.867166i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	26.6695			1.47034
$330$		0			0
$331$	34.2495			1.88252			0.941260	−	0.337683i	$-0.109643\pi$
$331$	34.2495			1.88252			0.941260	+	0.337683i	$0.109643\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.3454	−	1.47387i	−0.619866	−	0.0805259i
$336$		0			0
$337$	13.9698	+	13.9698i	0.760985	+	0.760985i	0.976500	−	0.215516i	$-0.0691433\pi$
$337$	13.9698	+	13.9698i	0.760985	+	0.760985i	−0.215516	+	0.976500i	$0.569143\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	2.15798i		−	0.116861i
$342$		0			0
$343$	−27.3359	+	27.3359i	−1.47600	+	1.47600i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.7354	+	13.7354i	−0.737354	+	0.737354i	−0.972065	−	0.234711i	$-0.924586\pi$
$347$	−13.7354	+	13.7354i	−0.737354	+	0.737354i	0.234711	+	0.972065i	$0.424586\pi$
$348$		0			0
$349$			1.94394i			0.104057i	0.998646	+	0.0520285i	$0.0165687\pi$
$349$			1.94394i			0.104057i	−0.998646	+	0.0520285i	$0.983431\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−17.1505	−	17.1505i	−0.912830	−	0.912830i	0.0836636	−	0.996494i	$-0.473338\pi$
$353$	−17.1505	−	17.1505i	−0.912830	−	0.912830i	−0.996494	+	0.0836636i	$0.973338\pi$
$354$		0			0
$355$	16.4854	+	21.4081i	0.874955	+	1.13622i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.46015			−0.288175			−0.144088	−	0.989565i	$-0.546025\pi$
$359$	−5.46015			−0.288175			−0.144088	+	0.989565i	$0.546025\pi$
$360$		0			0
$361$	12.0205			0.632655
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.33879	+	10.3057i	−0.0700755	+	0.539422i
$366$		0			0
$367$	−14.9840	−	14.9840i	−0.782160	−	0.782160i	0.198035	−	0.980195i	$-0.436544\pi$
$367$	−14.9840	−	14.9840i	−0.782160	−	0.782160i	−0.980195	+	0.198035i	$0.936544\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	14.2402i		−	0.739315i
$372$		0			0
$373$	20.8247	−	20.8247i	1.07826	−	1.07826i	0.0815961	−	0.996665i	$-0.473998\pi$
$373$	20.8247	−	20.8247i	1.07826	−	1.07826i	0.996665	−	0.0815961i	$-0.0260018\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.17347	−	4.17347i	0.214945	−	0.214945i
$378$		0			0
$379$			16.8739i			0.866756i	0.901212	+	0.433378i	$0.142678\pi$
$379$			16.8739i			0.866756i	−0.901212	+	0.433378i	$0.857322\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	25.7551	+	25.7551i	1.31602	+	1.31602i	0.916896	+	0.399127i	$0.130687\pi$
$383$	25.7551	+	25.7551i	1.31602	+	1.31602i	0.399127	+	0.916896i	$0.369313\pi$
$384$		0			0
$385$	−2.26645	+	17.4465i	−0.115509	+	0.889157i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.0154			−0.710609			−0.355304	−	0.934751i	$-0.615623\pi$
$389$	−14.0154			−0.710609			−0.355304	+	0.934751i	$0.615623\pi$
$390$		0			0
$391$	1.71898			0.0869326
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.29787	−	2.98403i	−0.115618	−	0.150143i
$396$		0			0
$397$	20.7845	+	20.7845i	1.04314	+	1.04314i	0.999026	+	0.0441171i	$0.0140475\pi$
$397$	20.7845	+	20.7845i	1.04314	+	1.04314i	0.0441171	+	0.999026i	$0.485953\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.74101i			0.136879i	0.997655	+	0.0684397i	$0.0218021\pi$
$401$			2.74101i			0.136879i	−0.997655	+	0.0684397i	$0.978198\pi$
$402$		0			0
$403$	−3.30596	+	3.30596i	−0.164682	+	0.164682i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.20202	−	9.20202i	0.456127	−	0.456127i
$408$		0			0
$409$			0.888866i			0.0439516i	0.999759	+	0.0219758i	$0.00699567\pi$
$409$			0.888866i			0.0439516i	−0.999759	+	0.0219758i	$0.993004\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−13.8103	−	13.8103i	−0.679560	−	0.679560i
$414$		0			0
$415$	24.8584	+	3.22932i	1.22025	+	0.158521i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−21.5588			−1.05322			−0.526608	−	0.850108i	$-0.676537\pi$
$419$	−21.5588			−1.05322			−0.526608	+	0.850108i	$0.676537\pi$
$420$		0			0
$421$	−33.1939			−1.61777			−0.808887	−	0.587964i	$-0.799930\pi$
$421$	−33.1939			−1.61777			−0.808887	+	0.587964i	$0.799930\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.32295	+	7.42862i	0.209694	+	0.360341i
$426$		0			0
$427$	−44.8519	−	44.8519i	−2.17054	−	2.17054i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			9.56008i			0.460493i	0.973132	+	0.230247i	$0.0739533\pi$
$431$			9.56008i			0.460493i	−0.973132	+	0.230247i	$0.926047\pi$
$432$		0			0
$433$	9.65173	−	9.65173i	0.463833	−	0.463833i	−0.436077	−	0.899909i	$-0.643632\pi$
$433$	9.65173	−	9.65173i	0.463833	−	0.463833i	0.899909	+	0.436077i	$0.143632\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.86809	−	1.86809i	0.0893630	−	0.0893630i
$438$		0			0
$439$			2.00856i			0.0958634i	0.998851	+	0.0479317i	$0.0152630\pi$
$439$			2.00856i			0.0958634i	−0.998851	+	0.0479317i	$0.984737\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.4437	+	13.4437i	0.638730	+	0.638730i	0.950242	−	0.311512i	$-0.100835\pi$
$443$	13.4437	+	13.4437i	0.638730	+	0.638730i	−0.311512	+	0.950242i	$0.600835\pi$
$444$		0			0
$445$	3.91388	−	3.01390i	0.185536	−	0.142873i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	36.0990			1.70362			0.851809	−	0.523853i	$-0.175506\pi$
$449$	36.0990			1.70362			0.851809	+	0.523853i	$0.175506\pi$
$450$		0			0
$451$	17.0915			0.804807
$452$		0			0
$453$		0			0
$454$		0			0
$455$	30.1997	−	23.2554i	1.41578	−	1.09023i
$456$		0			0
$457$	−15.5874	−	15.5874i	−0.729146	−	0.729146i	0.241304	−	0.970450i	$-0.422425\pi$
$457$	−15.5874	−	15.5874i	−0.729146	−	0.729146i	−0.970450	+	0.241304i	$0.922425\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.04378i			0.188337i	0.995556	+	0.0941687i	$0.0300193\pi$
$461$			4.04378i			0.188337i	−0.995556	+	0.0941687i	$0.969981\pi$
$462$		0			0
$463$	25.8303	−	25.8303i	1.20043	−	1.20043i	0.226399	−	0.974035i	$-0.427305\pi$
$463$	25.8303	−	25.8303i	1.20043	−	1.20043i	0.974035	−	0.226399i	$-0.0726953\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.0742	+	25.0742i	−1.16030	+	1.16030i	−0.175885	+	0.984411i	$0.556279\pi$
$467$	−25.0742	+	25.0742i	−1.16030	+	1.16030i	−0.984411	+	0.175885i	$0.943721\pi$
$468$		0			0
$469$			24.1094i			1.11327i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.67184	+	5.67184i	0.260791	+	0.260791i
$474$		0			0
$475$	12.7710	+	3.37507i	0.585972	+	0.154859i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	33.2857			1.52086			0.760432	−	0.649418i	$-0.224987\pi$
$479$	33.2857			1.52086			0.760432	+	0.649418i	$0.224987\pi$
$480$		0			0
$481$	−28.1945			−1.28556
$482$		0			0
$483$		0			0
$484$		0			0
$485$	21.9791	+	2.85527i	0.998019	+	0.129651i
$486$		0			0
$487$	−20.8782	−	20.8782i	−0.946080	−	0.946080i	0.0525391	−	0.998619i	$-0.483269\pi$
$487$	−20.8782	−	20.8782i	−0.946080	−	0.946080i	−0.998619	+	0.0525391i	$0.983269\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			1.11642i			0.0503834i	0.999683	+	0.0251917i	$0.00801962\pi$
$491$			1.11642i			0.0503834i	−0.999683	+	0.0251917i	$0.991980\pi$
$492$		0			0
$493$	1.98318	−	1.98318i	0.0893180	−	0.0893180i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	40.2625	−	40.2625i	1.80602	−	1.80602i
$498$		0			0
$499$			16.6576i			0.745698i	0.927892	+	0.372849i	$0.121619\pi$
$499$			16.6576i			0.745698i	−0.927892	+	0.372849i	$0.878381\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.5445	+	19.5445i	0.871445	+	0.871445i	0.992630	−	0.121185i	$-0.0386693\pi$
$503$	19.5445	+	19.5445i	0.871445	+	0.871445i	−0.121185	+	0.992630i	$0.538669\pi$
$504$		0			0
$505$	−10.4454	−	13.5644i	−0.464812	−	0.603609i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.62913			0.382479			0.191240	−	0.981543i	$-0.438749\pi$
$509$	8.62913			0.382479			0.191240	+	0.981543i	$0.438749\pi$
$510$		0			0
$511$	21.8998			0.968792
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.804922	+	6.19607i	−0.0354691	+	0.273032i
$516$		0			0
$517$	−6.68227	−	6.68227i	−0.293886	−	0.293886i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	34.0156i		−	1.49025i	−0.666926	−	0.745124i	$-0.732390\pi$
$521$		−	34.0156i		−	1.49025i	0.666926	−	0.745124i	$-0.267610\pi$
$522$		0			0
$523$	20.7355	−	20.7355i	0.906699	−	0.906699i	−0.0893052	−	0.996004i	$-0.528465\pi$
$523$	20.7355	−	20.7355i	0.906699	−	0.906699i	0.996004	+	0.0893052i	$0.0284647\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.57095	+	1.57095i	−0.0684316	+	0.0684316i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−26.1837	−	26.1837i	−1.13414	−	1.13414i
$534$		0			0
$535$	−1.85107	+	14.2490i	−0.0800285	+	0.616038i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.3865			1.09347
$540$		0			0
$541$	3.56762			0.153384			0.0766920	−	0.997055i	$-0.475564\pi$
$541$	3.56762			0.153384			0.0766920	+	0.997055i	$0.475564\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.22862	−	5.49133i	−0.181134	−	0.235223i
$546$		0			0
$547$	−10.2573	−	10.2573i	−0.438570	−	0.438570i	0.452960	−	0.891531i	$-0.350368\pi$
$547$	−10.2573	−	10.2573i	−0.438570	−	0.438570i	−0.891531	+	0.452960i	$0.850368\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	4.31042i		−	0.183630i
$552$		0			0
$553$	−5.61210	+	5.61210i	−0.238651	+	0.238651i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−28.7262	+	28.7262i	−1.21717	+	1.21717i	−0.248549	+	0.968619i	$0.579954\pi$
$557$	−28.7262	+	28.7262i	−1.21717	+	1.21717i	−0.968619	+	0.248549i	$0.920046\pi$
$558$		0			0
$559$		−	17.3782i		−	0.735019i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.3549	+	29.3549i	1.23716	+	1.23716i	0.961156	+	0.276006i	$0.0890110\pi$
$563$	29.3549	+	29.3549i	1.23716	+	1.23716i	0.276006	+	0.961156i	$0.410989\pi$
$564$		0			0
$565$	−19.4064	−	2.52106i	−0.816434	−	0.106062i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	44.2825			1.85642			0.928210	−	0.372057i	$-0.121347\pi$
$569$	44.2825			1.85642			0.928210	+	0.372057i	$0.121347\pi$
$570$		0			0
$571$	41.7528			1.74730			0.873650	−	0.486555i	$-0.161747\pi$
$571$	41.7528			1.74730			0.873650	+	0.486555i	$0.161747\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.32153	+	2.51483i	−0.180220	+	0.104876i
$576$		0			0
$577$	−28.8305	−	28.8305i	−1.20023	−	1.20023i	−0.974096	−	0.226135i	$-0.927391\pi$
$577$	−28.8305	−	28.8305i	−1.20023	−	1.20023i	−0.226135	−	0.974096i	$-0.572609\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	52.8250i		−	2.19155i
$582$		0			0
$583$	−3.56801	+	3.56801i	−0.147772	+	0.147772i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23.4923	+	23.4923i	−0.969629	+	0.969629i	−0.999552	−	0.0299228i	$-0.990474\pi$
$587$	−23.4923	+	23.4923i	−0.969629	+	0.969629i	0.0299228	+	0.999552i	$0.490474\pi$
$588$		0			0
$589$			3.41444i			0.140690i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.9801	−	18.9801i	−0.779418	−	0.779418i	0.200314	−	0.979732i	$-0.435804\pi$
$593$	−18.9801	−	18.9801i	−0.779418	−	0.779418i	−0.979732	+	0.200314i	$0.935804\pi$
$594$		0			0
$595$	14.3505	−	11.0507i	0.588313	−	0.453033i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.2599			0.623501			0.311750	−	0.950164i	$-0.399085\pi$
$599$	15.2599			0.623501			0.311750	+	0.950164i	$0.399085\pi$
$600$		0			0
$601$	36.2699			1.47948			0.739739	−	0.672894i	$-0.234949\pi$
$601$	36.2699			1.47948			0.739739	+	0.672894i	$0.234949\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.5490	+	11.2035i	−0.591499	+	0.455487i
$606$		0			0
$607$	6.18693	+	6.18693i	0.251120	+	0.251120i	0.821430	−	0.570310i	$-0.193177\pi$
$607$	6.18693	+	6.18693i	0.251120	+	0.251120i	−0.570310	+	0.821430i	$0.693177\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			20.4741i			0.828293i
$612$		0			0
$613$	26.4724	−	26.4724i	1.06921	−	1.06921i	0.0717920	−	0.997420i	$-0.477128\pi$
$613$	26.4724	−	26.4724i	1.06921	−	1.06921i	0.997420	−	0.0717920i	$-0.0228718\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.29015	−	9.29015i	0.374007	−	0.374007i	−0.494927	−	0.868935i	$-0.664805\pi$
$617$	9.29015	−	9.29015i	0.374007	−	0.374007i	0.868935	+	0.494927i	$0.164805\pi$
$618$		0			0
$619$			26.8954i			1.08102i	0.841339	+	0.540508i	$0.181768\pi$
$619$			26.8954i			1.08102i	−0.841339	+	0.540508i	$0.818232\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.36088	−	7.36088i	−0.294907	−	0.294907i
$624$		0			0
$625$	−21.7359	−	12.3512i	−0.869434	−	0.494048i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13.3976			−0.534199
$630$		0			0
$631$	30.9341			1.23147			0.615734	−	0.787954i	$-0.288860\pi$
$631$	30.9341			1.23147			0.615734	+	0.787954i	$0.288860\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.9633	+	2.07377i	0.633486	+	0.0822951i
$636$		0			0
$637$	−38.8913	−	38.8913i	−1.54093	−	1.54093i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			34.4713i			1.36154i	0.732499	+	0.680768i	$0.238354\pi$
$641$			34.4713i			1.36154i	−0.732499	+	0.680768i	$0.761646\pi$
$642$		0			0
$643$	15.6519	−	15.6519i	0.617252	−	0.617252i	−0.327573	−	0.944826i	$-0.606231\pi$
$643$	15.6519	−	15.6519i	0.617252	−	0.617252i	0.944826	+	0.327573i	$0.106231\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.782495	−	0.782495i	0.0307631	−	0.0307631i	−0.691558	−	0.722321i	$-0.743075\pi$
$647$	0.782495	−	0.782495i	0.0307631	−	0.0307631i	0.722321	+	0.691558i	$0.243075\pi$
$648$		0			0
$649$			6.92058i			0.271656i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.9394	−	11.9394i	−0.467226	−	0.467226i	0.433788	−	0.901015i	$-0.357177\pi$
$653$	−11.9394	−	11.9394i	−0.467226	−	0.467226i	−0.901015	+	0.433788i	$0.857177\pi$
$654$		0			0
$655$	−0.961407	−	1.24849i	−0.0375653	−	0.0487826i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	32.7117			1.27427			0.637133	−	0.770754i	$-0.280120\pi$
$659$	32.7117			1.27427			0.637133	+	0.770754i	$0.280120\pi$
$660$		0			0
$661$	−17.1057			−0.665334			−0.332667	−	0.943044i	$-0.607948\pi$
$661$	−17.1057			−0.665334			−0.332667	+	0.943044i	$0.607948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.58607	−	27.6046i	0.139062	−	1.07046i
$666$		0			0
$667$	1.15370	+	1.15370i	0.0446713	+	0.0446713i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			22.4761i			0.867679i
$672$		0			0
$673$	20.4124	−	20.4124i	0.786841	−	0.786841i	−0.194134	−	0.980975i	$-0.562190\pi$
$673$	20.4124	−	20.4124i	0.786841	−	0.786841i	0.980975	+	0.194134i	$0.0621897\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.88719	+	4.88719i	−0.187830	+	0.187830i	−0.794757	−	0.606927i	$-0.792402\pi$
$677$	−4.88719	+	4.88719i	−0.187830	+	0.187830i	0.606927	+	0.794757i	$0.292402\pi$
$678$		0			0
$679$		−	46.7063i		−	1.79242i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.09853	+	6.09853i	0.233354	+	0.233354i	0.814091	−	0.580737i	$-0.197236\pi$
$683$	6.09853	+	6.09853i	0.233354	+	0.233354i	−0.580737	+	0.814091i	$0.697236\pi$
$684$		0			0
$685$	0.793556	−	6.10858i	0.0303202	−	0.233397i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.9322			0.416482
$690$		0			0
$691$	−37.2305			−1.41631			−0.708157	−	0.706055i	$-0.750473\pi$
$691$	−37.2305			−1.41631			−0.708157	+	0.706055i	$0.750473\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.8891	+	18.0366i	0.526845	+	0.684166i
$696$		0			0
$697$	−12.4421	−	12.4421i	−0.471279	−	0.471279i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			32.7684i			1.23764i	0.785531	+	0.618822i	$0.212390\pi$
$701$			32.7684i			1.23764i	−0.785531	+	0.618822i	$0.787610\pi$
$702$		0			0
$703$	−14.5598	+	14.5598i	−0.549134	+	0.549134i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25.5108	+	25.5108i	−0.959432	+	0.959432i
$708$		0			0
$709$			30.4400i			1.14320i	0.820533	+	0.571600i	$0.193677\pi$
$709$			30.4400i			1.14320i	−0.820533	+	0.571600i	$0.806323\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.913884	−	0.913884i	−0.0342252	−	0.0342252i
$714$		0			0
$715$	−13.3936	−	1.73995i	−0.500894	−	0.0650703i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	50.5967			1.88694			0.943470	−	0.331459i	$-0.107541\pi$
$719$	50.5967			1.88694			0.943470	+	0.331459i	$0.107541\pi$
$720$		0			0
$721$	13.1669			0.490359
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.08437	+	7.88708i	−0.0774117	+	0.292919i
$726$		0			0
$727$	13.4649	+	13.4649i	0.499386	+	0.499386i	0.911247	−	0.411861i	$-0.135121\pi$
$727$	13.4649	+	13.4649i	0.499386	+	0.499386i	−0.411861	+	0.911247i	$0.635121\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	8.25789i		−	0.305429i
$732$		0			0
$733$	12.6959	−	12.6959i	0.468935	−	0.468935i	−0.432634	−	0.901569i	$-0.642416\pi$
$733$	12.6959	−	12.6959i	0.468935	−	0.468935i	0.901569	+	0.432634i	$0.142416\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.04081	−	6.04081i	0.222516	−	0.222516i
$738$		0			0
$739$			19.2229i			0.707127i	0.935411	+	0.353564i	$0.115030\pi$
$739$			19.2229i			0.707127i	−0.935411	+	0.353564i	$0.884970\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.72197	+	7.72197i	0.283292	+	0.283292i	0.834420	−	0.551129i	$-0.185803\pi$
$743$	7.72197	+	7.72197i	0.283292	+	0.283292i	−0.551129	+	0.834420i	$0.685803\pi$
$744$		0			0
$745$	15.6830	−	12.0768i	0.574580	−	0.442458i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	30.2796			1.10639
$750$		0			0
$751$	−52.4760			−1.91488			−0.957439	−	0.288637i	$-0.906798\pi$
$751$	−52.4760			−1.91488			−0.957439	+	0.288637i	$0.906798\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.6985	−	12.0887i	0.571325	−	0.439952i
$756$		0			0
$757$	−18.9098	−	18.9098i	−0.687287	−	0.687287i	0.274344	−	0.961631i	$-0.411539\pi$
$757$	−18.9098	−	18.9098i	−0.687287	−	0.687287i	−0.961631	+	0.274344i	$0.911539\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	34.0225i		−	1.23331i	−0.787232	−	0.616657i	$-0.788486\pi$
$761$		−	34.0225i		−	1.23331i	0.787232	−	0.616657i	$-0.211514\pi$
$762$		0			0
$763$	−10.3276	+	10.3276i	−0.373885	+	0.373885i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.6021	−	10.6021i	0.382820	−	0.382820i
$768$		0			0
$769$			47.9443i			1.72892i	0.502705	+	0.864458i	$0.332338\pi$
$769$			47.9443i			1.72892i	−0.502705	+	0.864458i	$0.667662\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15.5125	−	15.5125i	−0.557945	−	0.557945i	0.370777	−	0.928722i	$-0.379091\pi$
$773$	−15.5125	−	15.5125i	−0.557945	−	0.557945i	−0.928722	+	0.370777i	$0.879091\pi$
$774$		0			0
$775$	1.65111	−	6.24764i	0.0593095	−	0.224422i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.0428			−0.968911
$780$		0			0
$781$	−20.1762			−0.721962
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−28.5824	−	3.71310i	−1.02015	−	0.132526i
$786$		0			0
$787$	−30.2028	−	30.2028i	−1.07661	−	1.07661i	−0.996811	−	0.0798033i	$-0.974571\pi$
$787$	−30.2028	−	30.2028i	−1.07661	−	1.07661i	−0.0798033	−	0.996811i	$-0.525429\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			41.2393i			1.46630i
$792$		0			0
$793$	34.4327	−	34.4327i	1.22274	−	1.22274i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.420765	−	0.420765i	0.0149043	−	0.0149043i	−0.699615	−	0.714520i	$-0.746645\pi$
$797$	0.420765	−	0.420765i	0.0149043	−	0.0149043i	0.714520	+	0.699615i	$0.246645\pi$
$798$		0			0
$799$			9.72902i			0.344188i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.48720	−	5.48720i	−0.193639	−	0.193639i
$804$		0			0
$805$	6.42862	+	8.34826i	0.226579	+	0.294237i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	28.5320			1.00313			0.501565	−	0.865120i	$-0.332758\pi$
$809$	28.5320			1.00313			0.501565	+	0.865120i	$0.332758\pi$
$810$		0			0
$811$	39.1272			1.37394			0.686972	−	0.726684i	$-0.258940\pi$
$811$	39.1272			1.37394			0.686972	+	0.726684i	$0.258940\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.978335	−	7.53096i	0.0342696	−	0.263798i
$816$		0			0
$817$	−8.97421	−	8.97421i	−0.313968	−	0.313968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			49.3575i			1.72259i	0.508105	+	0.861295i	$0.330346\pi$
$821$			49.3575i			1.72259i	−0.508105	+	0.861295i	$0.669654\pi$
$822$		0			0
$823$	−0.0150100	+	0.0150100i	−0.000523216	+	0.000523216i	−0.707368	−	0.706845i	$-0.750118\pi$
$823$	−0.0150100	+	0.0150100i	−0.000523216	+	0.000523216i	0.706845	+	0.707368i	$0.250118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.8744	+	18.8744i	−0.656327	+	0.656327i	−0.954509	−	0.298182i	$-0.903620\pi$
$827$	−18.8744	+	18.8744i	−0.656327	+	0.656327i	0.298182	+	0.954509i	$0.403620\pi$
$828$		0			0
$829$			22.9885i			0.798425i	0.916858	+	0.399212i	$0.130716\pi$
$829$			22.9885i			0.798425i	−0.916858	+	0.399212i	$0.869284\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.4806	−	18.4806i	−0.640316	−	0.640316i
$834$		0			0
$835$	−0.152196	+	1.17156i	−0.00526695	+	0.0405436i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.9493			−0.895871			−0.447935	−	0.894066i	$-0.647841\pi$
$839$	−25.9493			−0.895871			−0.447935	+	0.894066i	$0.647841\pi$
$840$		0			0
$841$	−26.3380			−0.908206
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.117571	+	0.152679i	0.00404457	+	0.00525231i
$846$		0			0
$847$	27.3624	+	27.3624i	0.940183	+	0.940183i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	7.79394i		−	0.267173i
$852$		0			0
$853$	29.0714	−	29.0714i	0.995386	−	0.995386i	−0.00460387	−	0.999989i	$-0.501465\pi$
$853$	29.0714	−	29.0714i	0.995386	−	0.995386i	0.999989	+	0.00460387i	$0.00146546\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.6447	+	19.6447i	−0.671049	+	0.671049i	−0.957958	−	0.286909i	$-0.907372\pi$
$857$	−19.6447	+	19.6447i	−0.671049	+	0.671049i	0.286909	+	0.957958i	$0.407372\pi$
$858$		0			0
$859$			6.05457i			0.206579i	0.994651	+	0.103290i	$0.0329369\pi$
$859$			6.05457i			0.206579i	−0.994651	+	0.103290i	$0.967063\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.7777	−	14.7777i	−0.503037	−	0.503037i	0.409343	−	0.912381i	$-0.365758\pi$
$863$	−14.7777	−	14.7777i	−0.503037	−	0.503037i	−0.912381	+	0.409343i	$0.865758\pi$
$864$		0			0
$865$	38.5516	+	5.00818i	1.31079	+	0.170283i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.81232			0.0954015
$870$		0			0
$871$	−18.5087			−0.627143
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.9103	+	48.7759i	−0.673092	+	1.64893i
$876$		0			0
$877$	6.34190	+	6.34190i	0.214151	+	0.214151i	0.806028	−	0.591877i	$-0.201613\pi$
$877$	6.34190	+	6.34190i	0.214151	+	0.214151i	−0.591877	+	0.806028i	$0.701613\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			18.0148i			0.606934i	0.952842	+	0.303467i	$0.0981443\pi$
$881$			18.0148i			0.606934i	−0.952842	+	0.303467i	$0.901856\pi$
$882$		0			0
$883$	29.0038	−	29.0038i	0.976056	−	0.976056i	−0.0236641	−	0.999720i	$-0.507533\pi$
$883$	29.0038	−	29.0038i	0.976056	−	0.976056i	0.999720	+	0.0236641i	$0.00753322\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.28070	+	6.28070i	−0.210885	+	0.210885i	−0.804643	−	0.593758i	$-0.797644\pi$
$887$	−6.28070	+	6.28070i	−0.210885	+	0.210885i	0.593758	+	0.804643i	$0.297644\pi$
$888$		0			0
$889$		−	33.9226i		−	1.13773i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.5730	+	10.5730i	0.353811	+	0.353811i
$894$		0			0
$895$	19.5335	−	15.0418i	0.652932	−	0.502793i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.10869			−0.0703287
$900$		0			0
$901$	5.19482			0.173065
$902$		0			0
$903$		0			0
$904$		0			0
$905$	44.7777	−	34.4813i	1.48846	−	1.14620i
$906$		0			0
$907$	−1.77375	−	1.77375i	−0.0588965	−	0.0588965i	0.677045	−	0.735942i	$-0.263260\pi$
$907$	−1.77375	−	1.77375i	−0.0588965	−	0.0588965i	−0.735942	+	0.677045i	$0.763260\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	19.6741i		−	0.651833i	−0.945399	−	0.325917i	$-0.894327\pi$
$911$		−	19.6741i		−	0.651833i	0.945399	−	0.325917i	$-0.105673\pi$
$912$		0			0
$913$	−13.2358	+	13.2358i	−0.438040	+	0.438040i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.34805	+	2.34805i	−0.0775395	+	0.0775395i
$918$		0			0
$919$		−	22.4294i		−	0.739879i	−0.929056	−	0.369940i	$-0.879378\pi$
$919$		−	22.4294i		−	0.739879i	0.929056	−	0.369940i	$-0.120622\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	30.9094	+	30.9094i	1.01739	+	1.01739i
$924$		0			0
$925$	33.6817	−	19.6005i	1.10745	−	0.644460i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.6551			−1.10419			−0.552095	−	0.833781i	$-0.686171\pi$
$929$	−33.6551			−1.10419			−0.552095	+	0.833781i	$0.686171\pi$
$930$		0			0
$931$	−40.1675			−1.31644
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.36448	−	0.826800i	−0.208141	−	0.0270393i
$936$		0			0
$937$	28.7308	+	28.7308i	0.938596	+	0.938596i	0.998221	−	0.0596253i	$-0.0189906\pi$
$937$	28.7308	+	28.7308i	0.938596	+	0.938596i	−0.0596253	+	0.998221i	$0.518991\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			25.5486i			0.832862i	0.909167	+	0.416431i	$0.136719\pi$
$941$			25.5486i			0.832862i	−0.909167	+	0.416431i	$0.863281\pi$
$942$		0			0
$943$	7.23809	−	7.23809i	0.235705	−	0.235705i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.8817	−	12.8817i	0.418601	−	0.418601i	−0.466121	−	0.884721i	$-0.654349\pi$
$947$	12.8817	−	12.8817i	0.418601	−	0.418601i	0.884721	+	0.466121i	$0.154349\pi$
$948$		0			0
$949$			16.8124i			0.545755i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23.1487	+	23.1487i	0.749859	+	0.749859i	0.974453	−	0.224594i	$-0.0721055\pi$
$953$	23.1487	+	23.1487i	0.749859	+	0.749859i	−0.224594	+	0.974453i	$0.572105\pi$
$954$		0			0
$955$	−3.92177	−	5.09285i	−0.126906	−	0.164801i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.9809			−0.419176
$960$		0			0
$961$	−29.3296			−0.946117
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.32543	−	17.9006i	0.0748584	−	0.576239i
$966$		0			0
$967$	−19.0588	−	19.0588i	−0.612889	−	0.612889i	0.330809	−	0.943698i	$-0.392679\pi$
$967$	−19.0588	−	19.0588i	−0.612889	−	0.612889i	−0.943698	+	0.330809i	$0.892679\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			34.0087i			1.09139i	0.837984	+	0.545695i	$0.183734\pi$
$971$			34.0087i			1.09139i	−0.837984	+	0.545695i	$0.816266\pi$
$972$		0			0
$973$	33.9216	−	33.9216i	1.08748	−	1.08748i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.44696	−	7.44696i	0.238249	−	0.238249i	−0.577876	−	0.816125i	$-0.696118\pi$
$977$	7.44696	−	7.44696i	0.238249	−	0.238249i	0.816125	+	0.577876i	$0.196118\pi$
$978$		0			0
$979$			3.68866i			0.117890i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	29.9988	+	29.9988i	0.956812	+	0.956812i	0.999105	−	0.0422936i	$-0.0134665\pi$
$983$	29.9988	+	29.9988i	0.956812	+	0.956812i	−0.0422936	+	0.999105i	$0.513466\pi$
$984$		0			0
$985$	−3.47141	+	26.7220i	−0.110608	+	0.851432i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.80394			0.152757
$990$		0			0
$991$	34.8709			1.10771			0.553855	−	0.832613i	$-0.313156\pi$
$991$	34.8709			1.10771			0.553855	+	0.832613i	$0.313156\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.41725	−	7.03488i	−0.171738	−	0.223021i
$996$		0			0
$997$	−20.1705	−	20.1705i	−0.638807	−	0.638807i	0.311454	−	0.950261i	$-0.399184\pi$
$997$	−20.1705	−	20.1705i	−0.638807	−	0.638807i	−0.950261	+	0.311454i	$0.899184\pi$
$998$		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.737.6	✓	44
3.2	odd	2	inner	4140.2.s.a.737.17	yes	44
5.3	odd	4	inner	4140.2.s.a.2393.17	yes	44
15.8	even	4	inner	4140.2.s.a.2393.6	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.6	✓	44	1.1	even	1	trivial
4140.2.s.a.737.17	yes	44	3.2	odd	2	inner
4140.2.s.a.2393.6	yes	44	15.8	even	4	inner
4140.2.s.a.2393.17	yes	44	5.3	odd	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)

\(f(q)\)	\(=\)	\(q+(-1.77166 + 1.36427i) q^{5} +(3.33197 + 3.33197i) q^{7} +O(q^{10})\)	\(q+(-1.77166 + 1.36427i) q^{5} +(3.33197 + 3.33197i) q^{7} -1.66971i q^{11} +(-2.55795 + 2.55795i) q^{13} +(-1.21550 + 1.21550i) q^{17} +2.64188i q^{19} +(-0.707107 - 0.707107i) q^{23} +(1.27752 - 4.83404i) q^{25} -1.63157 q^{29} +1.29243 q^{31} +(-10.4488 - 1.35739i) q^{35} +(5.51115 + 5.51115i) q^{37} +10.2362i q^{41} +(-3.39690 + 3.39690i) q^{43} +(4.00206 - 4.00206i) q^{47} +15.2041i q^{49} +(-2.13690 - 2.13690i) q^{53} +(2.27794 + 2.95815i) q^{55} -4.14478 q^{59} -13.4611 q^{61} +(1.04207 - 8.02153i) q^{65} +(3.61788 + 3.61788i) q^{67} -12.0837i q^{71} +(3.28632 - 3.28632i) q^{73} +(5.56343 - 5.56343i) q^{77} +1.68432i q^{79} +(-7.92698 - 7.92698i) q^{83} +(0.495176 - 3.81173i) q^{85} -2.20916 q^{89} -17.0460 q^{91} +(-3.60425 - 4.68051i) q^{95} +(-7.00880 - 7.00880i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 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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