LMFDB - Embedded newform 4014.2.d.a.4013.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(4013,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4014.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.d (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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.12
Character	$\chi$	$=$	4014.4013
Dual form			4014.2.d.a.4013.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -2.58876 q^{5} -1.62417 q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -2.58876 q^{5} -1.62417 q^{7} -1.00000i q^{8} -2.58876i q^{10} -3.35493 q^{11} +1.39220i q^{13} -1.62417i q^{14} +1.00000 q^{16} -2.51094i q^{17} -7.76552 q^{19} +2.58876 q^{20} -3.35493i q^{22} +6.33943 q^{23} +1.70167 q^{25} -1.39220 q^{26} +1.62417 q^{28} -3.18597i q^{29} -6.49737 q^{31} +1.00000i q^{32} +2.51094 q^{34} +4.20460 q^{35} -4.70491 q^{37} -7.76552i q^{38} +2.58876i q^{40} -5.10053i q^{41} +1.63574 q^{43} +3.35493 q^{44} +6.33943i q^{46} +0.189135i q^{47} -4.36206 q^{49} +1.70167i q^{50} -1.39220i q^{52} +7.39901i q^{53} +8.68509 q^{55} +1.62417i q^{56} +3.18597 q^{58} +6.15231 q^{59} -12.1736i q^{61} -6.49737i q^{62} -1.00000 q^{64} -3.60406i q^{65} +1.08548i q^{67} +2.51094i q^{68} +4.20460i q^{70} +7.66501 q^{71} -6.27565 q^{73} -4.70491i q^{74} +7.76552 q^{76} +5.44898 q^{77} -0.667910i q^{79} -2.58876 q^{80} +5.10053 q^{82} +14.8497i q^{83} +6.50022i q^{85} +1.63574i q^{86} +3.35493i q^{88} +12.1942i q^{89} -2.26117i q^{91} -6.33943 q^{92} -0.189135 q^{94} +20.1031 q^{95} +18.4113i q^{97} -4.36206i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) $ 72 * q - 72 * q^4 + 16 * q^7	$ 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) $ 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−2.58876			−1.15773			−0.578864	−	0.815424i	$-0.696504\pi$
$5$	−2.58876			−1.15773			−0.578864	+	0.815424i	$0.696504\pi$
$6$		0			0
$7$	−1.62417			−0.613880			−0.306940	−	0.951729i	$-0.599305\pi$
$7$	−1.62417			−0.613880			−0.306940	+	0.951729i	$0.599305\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	2.58876i		−	0.818637i
$11$	−3.35493			−1.01155			−0.505774	−	0.862666i	$-0.668793\pi$
$11$	−3.35493			−1.01155			−0.505774	+	0.862666i	$0.668793\pi$
$12$		0			0
$13$			1.39220i			0.386126i	0.981186	+	0.193063i	$0.0618421\pi$
$13$			1.39220i			0.386126i	−0.981186	+	0.193063i	$0.938158\pi$
$14$		−	1.62417i		−	0.434079i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	2.51094i		−	0.608992i	−0.952514	−	0.304496i	$-0.901512\pi$
$17$		−	2.51094i		−	0.608992i	0.952514	−	0.304496i	$-0.0984881\pi$
$18$		0			0
$19$	−7.76552			−1.78153			−0.890766	−	0.454462i	$-0.849831\pi$
$19$	−7.76552			−1.78153			−0.890766	+	0.454462i	$0.849831\pi$
$20$	2.58876			0.578864
$21$		0			0
$22$		−	3.35493i		−	0.715272i
$23$	6.33943			1.32186			0.660931	−	0.750447i	$-0.270162\pi$
$23$	6.33943			1.32186			0.660931	+	0.750447i	$0.270162\pi$
$24$		0			0
$25$	1.70167			0.340334
$26$	−1.39220			−0.273032
$27$		0			0
$28$	1.62417			0.306940
$29$		−	3.18597i		−	0.591619i	−0.955247	−	0.295810i	$-0.904411\pi$
$29$		−	3.18597i		−	0.591619i	0.955247	−	0.295810i	$-0.0955894\pi$
$30$		0			0
$31$	−6.49737			−1.16696			−0.583481	−	0.812127i	$-0.698310\pi$
$31$	−6.49737			−1.16696			−0.583481	+	0.812127i	$0.698310\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	2.51094			0.430623
$35$	4.20460			0.710706
$36$		0			0
$37$	−4.70491			−0.773482			−0.386741	−	0.922188i	$-0.626399\pi$
$37$	−4.70491			−0.773482			−0.386741	+	0.922188i	$0.626399\pi$
$38$		−	7.76552i		−	1.25973i
$39$		0			0
$40$			2.58876i			0.409319i
$41$		−	5.10053i		−	0.796568i	−0.917262	−	0.398284i	$-0.869606\pi$
$41$		−	5.10053i		−	0.796568i	0.917262	−	0.398284i	$-0.130394\pi$
$42$		0			0
$43$	1.63574			0.249447			0.124724	−	0.992192i	$-0.460196\pi$
$43$	1.63574			0.249447			0.124724	+	0.992192i	$0.460196\pi$
$44$	3.35493			0.505774
$45$		0			0
$46$			6.33943i			0.934697i
$47$			0.189135i			0.0275882i	0.999905	+	0.0137941i	$0.00439094\pi$
$47$			0.189135i			0.0275882i	−0.999905	+	0.0137941i	$0.995609\pi$
$48$		0			0
$49$	−4.36206			−0.623151
$50$			1.70167i			0.240653i
$51$		0			0
$52$		−	1.39220i		−	0.193063i
$53$			7.39901i			1.01633i	0.861259	+	0.508166i	$0.169676\pi$
$53$			7.39901i			1.01633i	−0.861259	+	0.508166i	$0.830324\pi$
$54$		0			0
$55$	8.68509			1.17110
$56$			1.62417i			0.217039i
$57$		0			0
$58$	3.18597			0.418338
$59$	6.15231			0.800962			0.400481	−	0.916305i	$-0.368843\pi$
$59$	6.15231			0.800962			0.400481	+	0.916305i	$0.368843\pi$
$60$		0			0
$61$		−	12.1736i		−	1.55867i	−0.626610	−	0.779333i	$-0.715558\pi$
$61$		−	12.1736i		−	1.55867i	0.626610	−	0.779333i	$-0.284442\pi$
$62$		−	6.49737i		−	0.825167i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	3.60406i		−	0.447028i
$66$		0			0
$67$			1.08548i			0.132612i	0.997799	+	0.0663062i	$0.0211214\pi$
$67$			1.08548i			0.132612i	−0.997799	+	0.0663062i	$0.978879\pi$
$68$			2.51094i			0.304496i
$69$		0			0
$70$			4.20460i			0.502545i
$71$	7.66501			0.909670			0.454835	−	0.890576i	$-0.349698\pi$
$71$	7.66501			0.909670			0.454835	+	0.890576i	$0.349698\pi$
$72$		0			0
$73$	−6.27565			−0.734509			−0.367255	−	0.930120i	$-0.619702\pi$
$73$	−6.27565			−0.734509			−0.367255	+	0.930120i	$0.619702\pi$
$74$		−	4.70491i		−	0.546935i
$75$		0			0
$76$	7.76552			0.890766
$77$	5.44898			0.620969
$78$		0			0
$79$		−	0.667910i		−	0.0751457i	−0.999294	−	0.0375729i	$-0.988037\pi$
$79$		−	0.667910i		−	0.0751457i	0.999294	−	0.0375729i	$-0.0119626\pi$
$80$	−2.58876			−0.289432
$81$		0			0
$82$	5.10053			0.563259
$83$			14.8497i			1.62997i	0.579483	+	0.814984i	$0.303255\pi$
$83$			14.8497i			1.62997i	−0.579483	+	0.814984i	$0.696745\pi$
$84$		0			0
$85$			6.50022i			0.705047i
$86$			1.63574i			0.176386i
$87$		0			0
$88$			3.35493i			0.357636i
$89$			12.1942i			1.29258i	0.763091	+	0.646291i	$0.223681\pi$
$89$			12.1942i			1.29258i	−0.763091	+	0.646291i	$0.776319\pi$
$90$		0			0
$91$		−	2.26117i		−	0.237035i
$92$	−6.33943			−0.660931
$93$		0			0
$94$	−0.189135			−0.0195078
$95$	20.1031			2.06253
$96$		0			0
$97$			18.4113i			1.86939i	0.355453	+	0.934694i	$0.384327\pi$
$97$			18.4113i			1.86939i	−0.355453	+	0.934694i	$0.615673\pi$
$98$		−	4.36206i		−	0.440634i
$99$		0			0
$100$	−1.70167			−0.170167
$101$			2.27696i			0.226566i	0.993563	+	0.113283i	$0.0361367\pi$
$101$			2.27696i			0.226566i	−0.993563	+	0.113283i	$0.963863\pi$
$102$		0			0
$103$		−	16.7571i		−	1.65112i	−0.564311	−	0.825562i	$-0.690858\pi$
$103$		−	16.7571i		−	1.65112i	0.564311	−	0.825562i	$-0.309142\pi$
$104$	1.39220			0.136516
$105$		0			0
$106$	−7.39901			−0.718655
$107$	15.0160			1.45165			0.725827	−	0.687878i	$-0.241457\pi$
$107$	15.0160			1.45165			0.725827	+	0.687878i	$0.241457\pi$
$108$		0			0
$109$	18.1096			1.73458			0.867290	−	0.497803i	$-0.165860\pi$
$109$	18.1096			1.73458			0.867290	+	0.497803i	$0.165860\pi$
$110$			8.68509i			0.828091i
$111$		0			0
$112$	−1.62417			−0.153470
$113$	−13.9862			−1.31571			−0.657855	−	0.753144i	$-0.728536\pi$
$113$	−13.9862			−1.31571			−0.657855	+	0.753144i	$0.728536\pi$
$114$		0			0
$115$	−16.4112			−1.53036
$116$			3.18597i			0.295810i
$117$		0			0
$118$			6.15231i			0.566366i
$119$			4.07820i			0.373848i
$120$		0			0
$121$	0.255522			0.0232293
$122$	12.1736			1.10214
$123$		0			0
$124$	6.49737			0.583481
$125$	8.53858			0.763713
$126$		0			0
$127$	5.50304			0.488316			0.244158	−	0.969735i	$-0.421488\pi$
$127$	5.50304			0.488316			0.244158	+	0.969735i	$0.421488\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	3.60406			0.316097
$131$		−	16.4298i		−	1.43548i	−0.696312	−	0.717740i	$-0.745177\pi$
$131$		−	16.4298i		−	1.43548i	0.696312	−	0.717740i	$-0.254823\pi$
$132$		0			0
$133$	12.6126			1.09365
$134$	−1.08548			−0.0937711
$135$		0			0
$136$	−2.51094			−0.215311
$137$	17.2073			1.47011			0.735057	−	0.678005i	$-0.237155\pi$
$137$	17.2073			1.47011			0.735057	+	0.678005i	$0.237155\pi$
$138$		0			0
$139$	6.34416			0.538105			0.269052	−	0.963126i	$-0.413290\pi$
$139$	6.34416			0.538105			0.269052	+	0.963126i	$0.413290\pi$
$140$	−4.20460			−0.355353
$141$		0			0
$142$			7.66501i			0.643234i
$143$		−	4.67071i		−	0.390584i
$144$		0			0
$145$			8.24770i			0.684934i
$146$		−	6.27565i		−	0.519376i
$147$		0			0
$148$	4.70491			0.386741
$149$	−12.4200			−1.01748			−0.508741	−	0.860920i	$-0.669889\pi$
$149$	−12.4200			−1.01748			−0.508741	+	0.860920i	$0.669889\pi$
$150$		0			0
$151$			16.1204i			1.31186i	0.754821	+	0.655931i	$0.227724\pi$
$151$			16.1204i			1.31186i	−0.754821	+	0.655931i	$0.772276\pi$
$152$			7.76552i			0.629867i
$153$		0			0
$154$			5.44898i			0.439092i
$155$	16.8201			1.35102
$156$		0			0
$157$			18.0451i			1.44016i	0.693893	+	0.720078i	$0.255894\pi$
$157$			18.0451i			1.44016i	−0.693893	+	0.720078i	$0.744106\pi$
$158$	0.667910			0.0531361
$159$		0			0
$160$		−	2.58876i		−	0.204659i
$161$	−10.2963			−0.811465
$162$		0			0
$163$		−	9.50513i		−	0.744499i	−0.928133	−	0.372250i	$-0.878587\pi$
$163$		−	9.50513i		−	0.744499i	0.928133	−	0.372250i	$-0.121413\pi$
$164$			5.10053i			0.398284i
$165$		0			0
$166$	−14.8497			−1.15256
$167$	5.48032			0.424080			0.212040	−	0.977261i	$-0.431989\pi$
$167$	5.48032			0.424080			0.212040	+	0.977261i	$0.431989\pi$
$168$		0			0
$169$	11.0618			0.850907
$170$	−6.50022			−0.498544
$171$		0			0
$172$	−1.63574			−0.124724
$173$	−10.5770			−0.804155			−0.402077	−	0.915606i	$-0.631712\pi$
$173$	−10.5770			−0.804155			−0.402077	+	0.915606i	$0.631712\pi$
$174$		0			0
$175$	−2.76381			−0.208925
$176$	−3.35493			−0.252887
$177$		0			0
$178$	−12.1942			−0.913993
$179$		−	1.57594i		−	0.117792i	−0.998264	−	0.0588958i	$-0.981242\pi$
$179$		−	1.57594i		−	0.117792i	0.998264	−	0.0588958i	$-0.0187580\pi$
$180$		0			0
$181$	−16.8077			−1.24931			−0.624653	−	0.780902i	$-0.714760\pi$
$181$	−16.8077			−1.24931			−0.624653	+	0.780902i	$0.714760\pi$
$182$	2.26117			0.167609
$183$		0			0
$184$		−	6.33943i		−	0.467349i
$185$	12.1799			0.895482
$186$		0			0
$187$			8.42401i			0.616025i
$188$		−	0.189135i		−	0.0137941i
$189$		0			0
$190$			20.1031i			1.45843i
$191$	21.6968			1.56993			0.784963	−	0.619542i	$-0.212682\pi$
$191$	21.6968			1.56993			0.784963	+	0.619542i	$0.212682\pi$
$192$		0			0
$193$			2.30781i			0.166120i	0.996545	+	0.0830600i	$0.0264693\pi$
$193$			2.30781i			0.166120i	−0.996545	+	0.0830600i	$0.973531\pi$
$194$	−18.4113			−1.32186
$195$		0			0
$196$	4.36206			0.311576
$197$			5.18835i			0.369655i	0.982771	+	0.184827i	$0.0591726\pi$
$197$			5.18835i			0.369655i	−0.982771	+	0.184827i	$0.940827\pi$
$198$		0			0
$199$	14.8520			1.05283			0.526416	−	0.850227i	$-0.323536\pi$
$199$	14.8520			1.05283			0.526416	+	0.850227i	$0.323536\pi$
$200$		−	1.70167i		−	0.120326i
$201$		0			0
$202$	−2.27696			−0.160206
$203$			5.17457i			0.363183i
$204$		0			0
$205$			13.2040i			0.922210i
$206$	16.7571			1.16752
$207$		0			0
$208$			1.39220i			0.0965314i
$209$	26.0527			1.80210
$210$		0			0
$211$	−15.5611			−1.07127			−0.535636	−	0.844449i	$-0.679928\pi$
$211$	−15.5611			−1.07127			−0.535636	+	0.844449i	$0.679928\pi$
$212$		−	7.39901i		−	0.508166i
$213$		0			0
$214$			15.0160i			1.02647i
$215$	−4.23452			−0.288792
$216$		0			0
$217$	10.5529			0.716375
$218$			18.1096i			1.22653i
$219$		0			0
$220$	−8.68509			−0.585549
$221$	3.49572			0.235147
$222$		0			0
$223$	0.496576	+	14.9249i	0.0332532	+	0.999447i
$223$	0.496576	+	14.9249i	0.0332532	+	0.999447i
$224$		−	1.62417i		−	0.108520i
$225$		0			0
$226$		−	13.9862i		−	0.930348i
$227$			16.1086i			1.06917i	0.845116	+	0.534584i	$0.179532\pi$
$227$			16.1086i			1.06917i	−0.845116	+	0.534584i	$0.820468\pi$
$228$		0			0
$229$		−	12.5095i		−	0.826652i	−0.910583	−	0.413326i	$-0.864367\pi$
$229$		−	12.5095i		−	0.826652i	0.910583	−	0.413326i	$-0.135633\pi$
$230$		−	16.4112i		−	1.08213i
$231$		0			0
$232$	−3.18597			−0.209169
$233$	29.2714			1.91763			0.958815	−	0.284031i	$-0.0916717\pi$
$233$	29.2714			1.91763			0.958815	+	0.284031i	$0.0916717\pi$
$234$		0			0
$235$		−	0.489625i		−	0.0319396i
$236$	−6.15231			−0.400481
$237$		0			0
$238$	−4.07820			−0.264351
$239$		−	9.64506i		−	0.623887i	−0.950101	−	0.311943i	$-0.899020\pi$
$239$		−	9.64506i		−	0.623887i	0.950101	−	0.311943i	$-0.100980\pi$
$240$		0			0
$241$	2.11008			0.135922			0.0679611	−	0.997688i	$-0.478351\pi$
$241$	2.11008			0.135922			0.0679611	+	0.997688i	$0.478351\pi$
$242$			0.255522i			0.0164256i
$243$		0			0
$244$			12.1736i			0.779333i
$245$	11.2923			0.721440
$246$		0			0
$247$		−	10.8111i		−	0.687895i
$248$			6.49737i			0.412583i
$249$		0			0
$250$			8.53858i			0.540027i
$251$		−	3.51692i		−	0.221986i	−0.993821	−	0.110993i	$-0.964597\pi$
$251$		−	3.51692i		−	0.221986i	0.993821	−	0.110993i	$-0.0354031\pi$
$252$		0			0
$253$	−21.2683			−1.33713
$254$			5.50304i			0.345292i
$255$		0			0
$256$	1.00000			0.0625000
$257$			2.73540i			0.170629i	0.996354	+	0.0853147i	$0.0271896\pi$
$257$			2.73540i			0.170629i	−0.996354	+	0.0853147i	$0.972810\pi$
$258$		0			0
$259$	7.64159			0.474826
$260$			3.60406i			0.223514i
$261$		0			0
$262$	16.4298			1.01504
$263$	−3.27374			−0.201868			−0.100934	−	0.994893i	$-0.532183\pi$
$263$	−3.27374			−0.201868			−0.100934	+	0.994893i	$0.532183\pi$
$264$		0			0
$265$		−	19.1543i		−	1.17664i
$266$			12.6126i			0.773325i
$267$		0			0
$268$		−	1.08548i		−	0.0663062i
$269$	−12.4319			−0.757988			−0.378994	−	0.925399i	$-0.623730\pi$
$269$	−12.4319			−0.757988			−0.378994	+	0.925399i	$0.623730\pi$
$270$		0			0
$271$			3.76369i			0.228628i	0.993445	+	0.114314i	$0.0364670\pi$
$271$			3.76369i			0.228628i	−0.993445	+	0.114314i	$0.963533\pi$
$272$		−	2.51094i		−	0.152248i
$273$		0			0
$274$			17.2073i			1.03953i
$275$	−5.70898			−0.344265
$276$		0			0
$277$			8.61449i			0.517594i	0.965932	+	0.258797i	$0.0833261\pi$
$277$			8.61449i			0.517594i	−0.965932	+	0.258797i	$0.916674\pi$
$278$			6.34416i			0.380497i
$279$		0			0
$280$		−	4.20460i		−	0.251273i
$281$		−	25.8228i		−	1.54046i	−0.637765	−	0.770231i	$-0.720141\pi$
$281$		−	25.8228i		−	1.54046i	0.637765	−	0.770231i	$-0.279859\pi$
$282$		0			0
$283$	19.0721			1.13372			0.566860	−	0.823814i	$-0.308158\pi$
$283$	19.0721			1.13372			0.566860	+	0.823814i	$0.308158\pi$
$284$	−7.66501			−0.454835
$285$		0			0
$286$	4.67071			0.276185
$287$			8.28414i			0.488998i
$288$		0			0
$289$	10.6952			0.629128
$290$	−8.24770			−0.484322
$291$		0			0
$292$	6.27565			0.367255
$293$	−15.4907			−0.904975			−0.452487	−	0.891771i	$-0.649463\pi$
$293$	−15.4907			−0.904975			−0.452487	+	0.891771i	$0.649463\pi$
$294$		0			0
$295$	−15.9268			−0.927296
$296$			4.70491i			0.273467i
$297$		0			0
$298$		−	12.4200i		−	0.719468i
$299$			8.82572i			0.510405i
$300$		0			0
$301$	−2.65672			−0.153131
$302$	−16.1204			−0.927627
$303$		0			0
$304$	−7.76552			−0.445383
$305$			31.5144i			1.80451i
$306$		0			0
$307$			19.8512i			1.13297i	0.824073	+	0.566484i	$0.191697\pi$
$307$			19.8512i			1.13297i	−0.824073	+	0.566484i	$0.808303\pi$
$308$	−5.44898			−0.310485
$309$		0			0
$310$			16.8201i			0.955318i
$311$	−33.8063			−1.91698			−0.958490	−	0.285127i	$-0.907964\pi$
$311$	−33.8063			−1.91698			−0.958490	+	0.285127i	$0.907964\pi$
$312$		0			0
$313$			6.12183i			0.346027i	0.984919	+	0.173013i	$0.0553504\pi$
$313$			6.12183i			0.346027i	−0.984919	+	0.173013i	$0.944650\pi$
$314$	−18.0451			−1.01834
$315$		0			0
$316$			0.667910i			0.0375729i
$317$		−	22.4354i		−	1.26010i	−0.776556	−	0.630048i	$-0.783035\pi$
$317$		−	22.4354i		−	1.26010i	0.776556	−	0.630048i	$-0.216965\pi$
$318$		0			0
$319$			10.6887i			0.598451i
$320$	2.58876			0.144716
$321$		0			0
$322$		−	10.2963i		−	0.573792i
$323$			19.4987i			1.08494i
$324$		0			0
$325$			2.36906i			0.131412i
$326$	9.50513			0.526440
$327$		0			0
$328$	−5.10053			−0.281629
$329$		−	0.307189i		−	0.0169359i
$330$		0			0
$331$		−	24.4268i		−	1.34262i	−0.741177	−	0.671309i	$-0.765732\pi$
$331$		−	24.4268i		−	1.34262i	0.741177	−	0.671309i	$-0.234268\pi$
$332$		−	14.8497i		−	0.814984i
$333$		0			0
$334$			5.48032i			0.299870i
$335$		−	2.81004i		−	0.153529i
$336$		0			0
$337$		−	4.65829i		−	0.253753i	−0.991919	−	0.126877i	$-0.959505\pi$
$337$		−	4.65829i		−	0.253753i	0.991919	−	0.126877i	$-0.0404952\pi$
$338$			11.0618i			0.601682i
$339$		0			0
$340$		−	6.50022i		−	0.352524i
$341$	21.7982			1.18044
$342$		0			0
$343$	18.4540			0.996420
$344$		−	1.63574i		−	0.0881929i
$345$		0			0
$346$		−	10.5770i		−	0.568623i
$347$			26.5708i			1.42639i	0.700964	+	0.713197i	$0.252753\pi$
$347$			26.5708i			1.42639i	−0.700964	+	0.713197i	$0.747247\pi$
$348$		0			0
$349$	−34.9363			−1.87010			−0.935048	−	0.354520i	$-0.884644\pi$
$349$	−34.9363			−1.87010			−0.935048	+	0.354520i	$0.884644\pi$
$350$		−	2.76381i		−	0.147732i
$351$		0			0
$352$		−	3.35493i		−	0.178818i
$353$		−	26.3264i		−	1.40121i	−0.713548	−	0.700606i	$-0.752913\pi$
$353$		−	26.3264i		−	1.40121i	0.713548	−	0.700606i	$-0.247087\pi$
$354$		0			0
$355$	−19.8429			−1.05315
$356$		−	12.1942i		−	0.646291i
$357$		0			0
$358$	1.57594			0.0832912
$359$		−	26.2832i		−	1.38717i	−0.720373	−	0.693587i	$-0.756029\pi$
$359$		−	26.2832i		−	1.38717i	0.720373	−	0.693587i	$-0.243971\pi$
$360$		0			0
$361$	41.3033			2.17386
$362$		−	16.8077i		−	0.883393i
$363$		0			0
$364$			2.26117i			0.118517i
$365$	16.2461			0.850362
$366$		0			0
$367$	10.2698			0.536080			0.268040	−	0.963408i	$-0.413624\pi$
$367$	10.2698			0.536080			0.268040	+	0.963408i	$0.413624\pi$
$368$	6.33943			0.330465
$369$		0			0
$370$			12.1799i			0.633202i
$371$		−	12.0173i		−	0.623906i
$372$		0			0
$373$		−	7.20389i		−	0.373003i	−0.982455	−	0.186502i	$-0.940285\pi$
$373$		−	7.20389i		−	0.373003i	0.982455	−	0.186502i	$-0.0597149\pi$
$374$	−8.42401			−0.435595
$375$		0			0
$376$	0.189135			0.00975390
$377$	4.43549			0.228439
$378$		0			0
$379$	−8.37620			−0.430256			−0.215128	−	0.976586i	$-0.569017\pi$
$379$	−8.37620			−0.430256			−0.215128	+	0.976586i	$0.569017\pi$
$380$	−20.1031			−1.03126
$381$		0			0
$382$			21.6968i			1.11011i
$383$	0.636946			0.0325464			0.0162732	−	0.999868i	$-0.494820\pi$
$383$	0.636946			0.0325464			0.0162732	+	0.999868i	$0.494820\pi$
$384$		0			0
$385$	−14.1061			−0.718914
$386$	−2.30781			−0.117465
$387$		0			0
$388$		−	18.4113i		−	0.934694i
$389$		−	1.84031i		−	0.0933072i	−0.998911	−	0.0466536i	$-0.985144\pi$
$389$		−	1.84031i		−	0.0933072i	0.998911	−	0.0466536i	$-0.0148557\pi$
$390$		0			0
$391$		−	15.9179i		−	0.805003i
$392$			4.36206i			0.220317i
$393$		0			0
$394$	−5.18835			−0.261385
$395$			1.72906i			0.0869983i
$396$		0			0
$397$			8.99607i			0.451500i	0.974185	+	0.225750i	$0.0724832\pi$
$397$			8.99607i			0.451500i	−0.974185	+	0.225750i	$0.927517\pi$
$398$			14.8520i			0.744464i
$399$		0			0
$400$	1.70167			0.0850836
$401$		−	15.4885i		−	0.773458i	−0.922193	−	0.386729i	$-0.873605\pi$
$401$		−	15.4885i		−	0.773458i	0.922193	−	0.386729i	$-0.126395\pi$
$402$		0			0
$403$		−	9.04561i		−	0.450594i
$404$		−	2.27696i		−	0.113283i
$405$		0			0
$406$	−5.17457			−0.256809
$407$	15.7846			0.782415
$408$		0			0
$409$			6.83590i			0.338013i	0.985615	+	0.169007i	$0.0540559\pi$
$409$			6.83590i			0.338013i	−0.985615	+	0.169007i	$0.945944\pi$
$410$	−13.2040			−0.652101
$411$		0			0
$412$			16.7571i			0.825562i
$413$	−9.99242			−0.491695
$414$		0			0
$415$		−	38.4423i		−	1.88706i
$416$	−1.39220			−0.0682580
$417$		0			0
$418$			26.0527i			1.27428i
$419$		−	39.0439i		−	1.90742i	−0.300728	−	0.953710i	$-0.597230\pi$
$419$		−	39.0439i		−	1.90742i	0.300728	−	0.953710i	$-0.402770\pi$
$420$		0			0
$421$			32.4049i			1.57932i	0.613545	+	0.789660i	$0.289743\pi$
$421$			32.4049i			1.57932i	−0.613545	+	0.789660i	$0.710257\pi$
$422$		−	15.5611i		−	0.757503i
$423$		0			0
$424$	7.39901			0.359328
$425$		−	4.27280i		−	0.207261i
$426$		0			0
$427$			19.7720i			0.956834i
$428$	−15.0160			−0.725827
$429$		0			0
$430$		−	4.23452i		−	0.204207i
$431$	20.3792			0.981629			0.490815	−	0.871264i	$-0.336699\pi$
$431$	20.3792			0.981629			0.490815	+	0.871264i	$0.336699\pi$
$432$		0			0
$433$	32.3710			1.55565			0.777827	−	0.628479i	$-0.216322\pi$
$433$	32.3710			1.55565			0.777827	+	0.628479i	$0.216322\pi$
$434$			10.5529i			0.506553i
$435$		0			0
$436$	−18.1096			−0.867290
$437$	−49.2289			−2.35494
$438$		0			0
$439$		−	3.51894i		−	0.167950i	−0.996468	−	0.0839749i	$-0.973238\pi$
$439$		−	3.51894i		−	0.167950i	0.996468	−	0.0839749i	$-0.0267616\pi$
$440$		−	8.68509i		−	0.414045i
$441$		0			0
$442$			3.49572i			0.166274i
$443$		−	12.3553i		−	0.587017i	−0.955956	−	0.293508i	$-0.905177\pi$
$443$		−	12.3553i		−	0.587017i	0.955956	−	0.293508i	$-0.0948229\pi$
$444$		0			0
$445$		−	31.5678i		−	1.49646i
$446$	−14.9249	+	0.496576i	−0.706716	+	0.0235136i
$447$		0			0
$448$	1.62417			0.0767350
$449$	6.27640			0.296202			0.148101	−	0.988972i	$-0.452684\pi$
$449$	6.27640			0.296202			0.148101	+	0.988972i	$0.452684\pi$
$450$		0			0
$451$			17.1119i			0.805767i
$452$	13.9862			0.657855
$453$		0			0
$454$	−16.1086			−0.756016
$455$			5.85362i			0.274422i
$456$		0			0
$457$			4.60785i			0.215546i	0.994176	+	0.107773i	$0.0343720\pi$
$457$			4.60785i			0.215546i	−0.994176	+	0.107773i	$0.965628\pi$
$458$	12.5095			0.584532
$459$		0			0
$460$	16.4112			0.765178
$461$			23.1819i			1.07969i	0.841765	+	0.539844i	$0.181517\pi$
$461$			23.1819i			1.07969i	−0.841765	+	0.539844i	$0.818483\pi$
$462$		0			0
$463$	13.2234			0.614543			0.307272	−	0.951622i	$-0.400584\pi$
$463$	13.2234			0.614543			0.307272	+	0.951622i	$0.400584\pi$
$464$		−	3.18597i		−	0.147905i
$465$		0			0
$466$			29.2714i			1.35597i
$467$	23.9309			1.10739			0.553694	−	0.832720i	$-0.313218\pi$
$467$	23.9309			1.10739			0.553694	+	0.832720i	$0.313218\pi$
$468$		0			0
$469$		−	1.76301i		−	0.0814081i
$470$	0.489625			0.0225847
$471$		0			0
$472$		−	6.15231i		−	0.283183i
$473$	−5.48777			−0.252328
$474$		0			0
$475$	−13.2144			−0.606317
$476$		−	4.07820i		−	0.186924i
$477$		0			0
$478$	9.64506			0.441155
$479$		−	29.3689i		−	1.34190i	−0.741503	−	0.670950i	$-0.765886\pi$
$479$		−	29.3689i		−	1.34190i	0.741503	−	0.670950i	$-0.234114\pi$
$480$		0			0
$481$		−	6.55015i		−	0.298661i
$482$			2.11008i			0.0961115i
$483$		0			0
$484$	−0.255522			−0.0116146
$485$		−	47.6625i		−	2.16424i
$486$		0			0
$487$	−23.5847			−1.06873			−0.534363	−	0.845255i	$-0.679448\pi$
$487$	−23.5847			−1.06873			−0.534363	+	0.845255i	$0.679448\pi$
$488$	−12.1736			−0.551072
$489$		0			0
$490$			11.2923i			0.510135i
$491$	−5.32884			−0.240487			−0.120243	−	0.992744i	$-0.538368\pi$
$491$	−5.32884			−0.240487			−0.120243	+	0.992744i	$0.538368\pi$
$492$		0			0
$493$	−7.99977			−0.360292
$494$	10.8111			0.486415
$495$		0			0
$496$	−6.49737			−0.291740
$497$	−12.4493			−0.558428
$498$		0			0
$499$	−30.8614			−1.38155			−0.690773	−	0.723071i	$-0.742730\pi$
$499$	−30.8614			−1.38155			−0.690773	+	0.723071i	$0.742730\pi$
$500$	−8.53858			−0.381857
$501$		0			0
$502$	3.51692			0.156968
$503$	−5.98426			−0.266825			−0.133413	−	0.991061i	$-0.542594\pi$
$503$	−5.98426			−0.266825			−0.133413	+	0.991061i	$0.542594\pi$
$504$		0			0
$505$		−	5.89451i		−	0.262302i
$506$		−	21.2683i		−	0.945491i
$507$		0			0
$508$	−5.50304			−0.244158
$509$			20.6128i			0.913647i	0.889557	+	0.456824i	$0.151013\pi$
$509$			20.6128i			0.913647i	−0.889557	+	0.456824i	$0.848987\pi$
$510$		0			0
$511$	10.1927			0.450901
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−2.73540			−0.120653
$515$			43.3800i			1.91155i
$516$		0			0
$517$		−	0.634535i		−	0.0279068i
$518$			7.64159i			0.335752i
$519$		0			0
$520$	−3.60406			−0.158048
$521$	33.0121			1.44628			0.723142	−	0.690699i	$-0.242697\pi$
$521$	33.0121			1.44628			0.723142	+	0.690699i	$0.242697\pi$
$522$		0			0
$523$		−	10.5069i		−	0.459435i	−0.973257	−	0.229718i	$-0.926220\pi$
$523$		−	10.5069i		−	0.459435i	0.973257	−	0.229718i	$-0.0737803\pi$
$524$			16.4298i			0.717740i
$525$		0			0
$526$		−	3.27374i		−	0.142742i
$527$			16.3145i			0.710671i
$528$		0			0
$529$	17.1883			0.747318
$530$	19.1543			0.832008
$531$		0			0
$532$	−12.6126			−0.546824
$533$	7.10093			0.307575
$534$		0			0
$535$	−38.8728			−1.68062
$536$	1.08548			0.0468855
$537$		0			0
$538$		−	12.4319i		−	0.535979i
$539$	14.6344			0.630347
$540$		0			0
$541$			33.6498i			1.44672i	0.690472	+	0.723359i	$0.257403\pi$
$541$			33.6498i			1.44672i	−0.690472	+	0.723359i	$0.742597\pi$
$542$	−3.76369			−0.161664
$543$		0			0
$544$	2.51094			0.107656
$545$	−46.8813			−2.00817
$546$		0			0
$547$	−28.9529			−1.23794			−0.618968	−	0.785416i	$-0.712449\pi$
$547$	−28.9529			−1.23794			−0.618968	+	0.785416i	$0.712449\pi$
$548$	−17.2073			−0.735057
$549$		0			0
$550$		−	5.70898i		−	0.243432i
$551$			24.7407i			1.05399i
$552$		0			0
$553$			1.08480i			0.0461305i
$554$	−8.61449			−0.365994
$555$		0			0
$556$	−6.34416			−0.269052
$557$	−0.545586			−0.0231172			−0.0115586	−	0.999933i	$-0.503679\pi$
$557$	−0.545586			−0.0231172			−0.0115586	+	0.999933i	$0.503679\pi$
$558$		0			0
$559$			2.27726i			0.0963179i
$560$	4.20460			0.177677
$561$		0			0
$562$	25.8228			1.08927
$563$	−6.34062			−0.267225			−0.133613	−	0.991034i	$-0.542658\pi$
$563$	−6.34062			−0.267225			−0.133613	+	0.991034i	$0.542658\pi$
$564$		0			0
$565$	36.2069			1.52324
$566$			19.0721i			0.801660i
$567$		0			0
$568$		−	7.66501i		−	0.321617i
$569$	−20.2226			−0.847777			−0.423888	−	0.905714i	$-0.639335\pi$
$569$	−20.2226			−0.847777			−0.423888	+	0.905714i	$0.639335\pi$
$570$		0			0
$571$			2.23042i			0.0933401i	0.998910	+	0.0466701i	$0.0148609\pi$
$571$			2.23042i			0.0933401i	−0.998910	+	0.0466701i	$0.985139\pi$
$572$			4.67071i			0.195292i
$573$		0			0
$574$	−8.28414			−0.345773
$575$	10.7876			0.449875
$576$		0			0
$577$	18.7340			0.779907			0.389953	−	0.920835i	$-0.372491\pi$
$577$	18.7340			0.779907			0.389953	+	0.920835i	$0.372491\pi$
$578$			10.6952i			0.444861i
$579$		0			0
$580$		−	8.24770i		−	0.342467i
$581$		−	24.1185i		−	1.00061i
$582$		0			0
$583$		−	24.8231i		−	1.02807i
$584$			6.27565i			0.259688i
$585$		0			0
$586$		−	15.4907i		−	0.639914i
$587$	2.76772			0.114236			0.0571181	−	0.998367i	$-0.481809\pi$
$587$	2.76772			0.114236			0.0571181	+	0.998367i	$0.481809\pi$
$588$		0			0
$589$	50.4554			2.07898
$590$		−	15.9268i		−	0.655697i
$591$		0			0
$592$	−4.70491			−0.193371
$593$	−15.5199			−0.637326			−0.318663	−	0.947868i	$-0.603234\pi$
$593$	−15.5199			−0.637326			−0.318663	+	0.947868i	$0.603234\pi$
$594$		0			0
$595$		−	10.5575i		−	0.432815i
$596$	12.4200			0.508741
$597$		0			0
$598$	−8.82572			−0.360910
$599$		−	3.66674i		−	0.149819i	−0.997190	−	0.0749094i	$-0.976133\pi$
$599$		−	3.66674i		−	0.149819i	0.997190	−	0.0749094i	$-0.0238668\pi$
$600$		0			0
$601$		−	31.8727i		−	1.30011i	−0.759886	−	0.650057i	$-0.774745\pi$
$601$		−	31.8727i		−	1.30011i	0.759886	−	0.650057i	$-0.225255\pi$
$602$		−	2.65672i		−	0.108280i
$603$		0			0
$604$		−	16.1204i		−	0.655931i
$605$	−0.661485			−0.0268932
$606$		0			0
$607$		−	38.8110i		−	1.57529i	−0.616129	−	0.787645i	$-0.711300\pi$
$607$		−	38.8110i		−	1.57529i	0.616129	−	0.787645i	$-0.288700\pi$
$608$		−	7.76552i		−	0.314933i
$609$		0			0
$610$	−31.5144			−1.27598
$611$	−0.263313			−0.0106525
$612$		0			0
$613$		−	33.6606i		−	1.35954i	−0.733427	−	0.679769i	$-0.762080\pi$
$613$		−	33.6606i		−	1.35954i	0.733427	−	0.679769i	$-0.237920\pi$
$614$	−19.8512			−0.801129
$615$		0			0
$616$		−	5.44898i		−	0.219546i
$617$			12.7096i			0.511669i	0.966721	+	0.255835i	$0.0823502\pi$
$617$			12.7096i			0.511669i	−0.966721	+	0.255835i	$0.917650\pi$
$618$		0			0
$619$		−	35.1471i		−	1.41268i	−0.707873	−	0.706340i	$-0.750345\pi$
$619$		−	35.1471i		−	1.41268i	0.707873	−	0.706340i	$-0.249655\pi$
$620$	−16.8201			−0.675512
$621$		0			0
$622$		−	33.8063i		−	1.35551i
$623$		−	19.8055i		−	0.793490i
$624$		0			0
$625$	−30.6127			−1.22451
$626$	−6.12183			−0.244678
$627$		0			0
$628$		−	18.0451i		−	0.720078i
$629$			11.8137i			0.471045i
$630$		0			0
$631$			44.7000i			1.77948i	0.456468	+	0.889740i	$0.349114\pi$
$631$			44.7000i			1.77948i	−0.456468	+	0.889740i	$0.650886\pi$
$632$	−0.667910			−0.0265680
$633$		0			0
$634$	22.4354			0.891023
$635$	−14.2460			−0.565337
$636$		0			0
$637$		−	6.07284i		−	0.240615i
$638$	−10.6887			−0.423169
$639$		0			0
$640$			2.58876i			0.102330i
$641$	−17.9127			−0.707507			−0.353754	−	0.935339i	$-0.615095\pi$
$641$	−17.9127			−0.707507			−0.353754	+	0.935339i	$0.615095\pi$
$642$		0			0
$643$	32.7838			1.29287			0.646434	−	0.762970i	$-0.276260\pi$
$643$	32.7838			1.29287			0.646434	+	0.762970i	$0.276260\pi$
$644$	10.2963			0.405732
$645$		0			0
$646$	−19.4987			−0.767168
$647$		−	25.1654i		−	0.989353i	−0.869077	−	0.494676i	$-0.835287\pi$
$647$		−	25.1654i		−	0.989353i	0.869077	−	0.494676i	$-0.164713\pi$
$648$		0			0
$649$	−20.6405			−0.810212
$650$	−2.36906			−0.0929222
$651$		0			0
$652$			9.50513i			0.372250i
$653$	5.11558			0.200188			0.100094	−	0.994978i	$-0.468086\pi$
$653$	5.11558			0.200188			0.100094	+	0.994978i	$0.468086\pi$
$654$		0			0
$655$			42.5328i			1.66189i
$656$		−	5.10053i		−	0.199142i
$657$		0			0
$658$	0.307189			0.0119755
$659$			25.2944i			0.985330i	0.870219	+	0.492665i	$0.163977\pi$
$659$			25.2944i			0.985330i	−0.870219	+	0.492665i	$0.836023\pi$
$660$		0			0
$661$			7.55626i			0.293905i	0.989144	+	0.146952i	$0.0469464\pi$
$661$			7.55626i			0.293905i	−0.989144	+	0.146952i	$0.953054\pi$
$662$	24.4268			0.949375
$663$		0			0
$664$	14.8497			0.576281
$665$	−32.6509			−1.26615
$666$		0			0
$667$		−	20.1972i		−	0.782039i
$668$	−5.48032			−0.212040
$669$		0			0
$670$	2.81004			0.108561
$671$			40.8414i			1.57667i
$672$		0			0
$673$	−17.0575			−0.657520			−0.328760	−	0.944413i	$-0.606631\pi$
$673$	−17.0575			−0.657520			−0.328760	+	0.944413i	$0.606631\pi$
$674$	4.65829			0.179431
$675$		0			0
$676$	−11.0618			−0.425454
$677$		−	31.1163i		−	1.19590i	−0.801535	−	0.597948i	$-0.795983\pi$
$677$		−	31.1163i		−	1.19590i	0.801535	−	0.597948i	$-0.204017\pi$
$678$		0			0
$679$		−	29.9032i		−	1.14758i
$680$	6.50022			0.249272
$681$		0			0
$682$			21.7982i			0.834696i
$683$			23.1499i			0.885806i	0.896569	+	0.442903i	$0.146051\pi$
$683$			23.1499i			0.885806i	−0.896569	+	0.442903i	$0.853949\pi$
$684$		0			0
$685$	−44.5454			−1.70199
$686$			18.4540i			0.704576i
$687$		0			0
$688$	1.63574			0.0623618
$689$	−10.3009			−0.392432
$690$		0			0
$691$		−	18.1431i		−	0.690195i	−0.938567	−	0.345097i	$-0.887846\pi$
$691$		−	18.1431i		−	0.690195i	0.938567	−	0.345097i	$-0.112154\pi$
$692$	10.5770			0.402077
$693$		0			0
$694$	−26.5708			−1.00861
$695$	−16.4235			−0.622979
$696$		0			0
$697$	−12.8071			−0.485104
$698$		−	34.9363i		−	1.32236i
$699$		0			0
$700$	2.76381			0.104462
$701$			31.6109i			1.19393i	0.802268	+	0.596964i	$0.203627\pi$
$701$			31.6109i			1.19393i	−0.802268	+	0.596964i	$0.796373\pi$
$702$		0			0
$703$	36.5361			1.37798
$704$	3.35493			0.126443
$705$		0			0
$706$	26.3264			0.990807
$707$		−	3.69818i		−	0.139084i
$708$		0			0
$709$			10.3749i			0.389638i	0.980839	+	0.194819i	$0.0624119\pi$
$709$			10.3749i			0.389638i	−0.980839	+	0.194819i	$0.937588\pi$
$710$		−	19.8429i		−	0.744690i
$711$		0			0
$712$	12.1942			0.456996
$713$	−41.1896			−1.54256
$714$		0			0
$715$			12.0913i			0.452191i
$716$			1.57594i			0.0588958i
$717$		0			0
$718$	26.2832			0.980880
$719$			30.1095i			1.12290i	0.827512	+	0.561448i	$0.189755\pi$
$719$			30.1095i			1.12290i	−0.827512	+	0.561448i	$0.810245\pi$
$720$		0			0
$721$			27.2164i			1.01359i
$722$			41.3033i			1.53715i
$723$		0			0
$724$	16.8077			0.624653
$725$		−	5.42147i		−	0.201348i
$726$		0			0
$727$	25.2530			0.936582			0.468291	−	0.883574i	$-0.344870\pi$
$727$	25.2530			0.936582			0.468291	+	0.883574i	$0.344870\pi$
$728$	−2.26117			−0.0838045
$729$		0			0
$730$			16.2461i			0.601297i
$731$		−	4.10723i		−	0.151911i
$732$		0			0
$733$	−33.0287			−1.21994			−0.609972	−	0.792423i	$-0.708819\pi$
$733$	−33.0287			−1.21994			−0.609972	+	0.792423i	$0.708819\pi$
$734$			10.2698i			0.379066i
$735$		0			0
$736$			6.33943i			0.233674i
$737$		−	3.64170i		−	0.134144i
$738$		0			0
$739$		−	8.02661i		−	0.295264i	−0.989042	−	0.147632i	$-0.952835\pi$
$739$		−	8.02661i		−	0.295264i	0.989042	−	0.147632i	$-0.0471651\pi$
$740$	−12.1799			−0.447741
$741$		0			0
$742$	12.0173			0.441168
$743$			6.70267i			0.245897i	0.992413	+	0.122949i	$0.0392350\pi$
$743$			6.70267i			0.245897i	−0.992413	+	0.122949i	$0.960765\pi$
$744$		0			0
$745$	32.1523			1.17797
$746$	7.20389			0.263753
$747$		0			0
$748$		−	8.42401i		−	0.308012i
$749$	−24.3886			−0.891141
$750$		0			0
$751$	39.2685			1.43293			0.716464	−	0.697624i	$-0.245759\pi$
$751$	39.2685			1.43293			0.716464	+	0.697624i	$0.245759\pi$
$752$			0.189135i			0.00689705i
$753$		0			0
$754$			4.43549i			0.161531i
$755$		−	41.7319i		−	1.51878i
$756$		0			0
$757$			24.4680i			0.889306i	0.895703	+	0.444653i	$0.146673\pi$
$757$			24.4680i			0.889306i	−0.895703	+	0.444653i	$0.853327\pi$
$758$		−	8.37620i		−	0.304237i
$759$		0			0
$760$		−	20.1031i		−	0.729214i
$761$	−1.46621			−0.0531502			−0.0265751	−	0.999647i	$-0.508460\pi$
$761$	−1.46621			−0.0531502			−0.0265751	+	0.999647i	$0.508460\pi$
$762$		0			0
$763$	−29.4131			−1.06482
$764$	−21.6968			−0.784963
$765$		0			0
$766$			0.636946i			0.0230138i
$767$			8.56521i			0.309272i
$768$		0			0
$769$	21.1851			0.763954			0.381977	−	0.924172i	$-0.375243\pi$
$769$	21.1851			0.763954			0.381977	+	0.924172i	$0.375243\pi$
$770$		−	14.1061i		−	0.508349i
$771$		0			0
$772$		−	2.30781i		−	0.0830600i
$773$	−25.0632			−0.901461			−0.450731	−	0.892660i	$-0.648837\pi$
$773$	−25.0632			−0.901461			−0.450731	+	0.892660i	$0.648837\pi$
$774$		0			0
$775$	−11.0564			−0.397157
$776$	18.4113			0.660929
$777$		0			0
$778$	1.84031			0.0659781
$779$			39.6082i			1.41911i
$780$		0			0
$781$	−25.7155			−0.920174
$782$	15.9179			0.569223
$783$		0			0
$784$	−4.36206			−0.155788
$785$		−	46.7144i		−	1.66731i
$786$		0			0
$787$		−	46.0427i		−	1.64124i	−0.571471	−	0.820622i	$-0.693627\pi$
$787$		−	46.0427i		−	1.64124i	0.571471	−	0.820622i	$-0.306373\pi$
$788$		−	5.18835i		−	0.184827i
$789$		0			0
$790$	−1.72906			−0.0615171
$791$	22.7160			0.807689
$792$		0			0
$793$	16.9480			0.601841
$794$	−8.99607			−0.319259
$795$		0			0
$796$	−14.8520			−0.526416
$797$			30.7422i			1.08894i	0.838779	+	0.544472i	$0.183270\pi$
$797$			30.7422i			1.08894i	−0.838779	+	0.544472i	$0.816730\pi$
$798$		0			0
$799$	0.474907			0.0168010
$800$			1.70167i			0.0601632i
$801$		0			0
$802$	15.4885			0.546917
$803$	21.0543			0.742991
$804$		0			0
$805$	26.6547			0.939455
$806$	9.04561			0.318618
$807$		0			0
$808$	2.27696			0.0801032
$809$	45.5332			1.60086			0.800431	−	0.599424i	$-0.204604\pi$
$809$	45.5332			1.60086			0.800431	+	0.599424i	$0.204604\pi$
$810$		0			0
$811$			14.6287i			0.513683i	0.966453	+	0.256842i	$0.0826819\pi$
$811$			14.6287i			0.513683i	−0.966453	+	0.256842i	$0.917318\pi$
$812$		−	5.17457i		−	0.181592i
$813$		0			0
$814$			15.7846i			0.553251i
$815$			24.6065i			0.861928i
$816$		0			0
$817$	−12.7023			−0.444398
$818$	−6.83590			−0.239011
$819$		0			0
$820$		−	13.2040i		−	0.461105i
$821$			20.0802i			0.700804i	0.936599	+	0.350402i	$0.113955\pi$
$821$			20.0802i			0.700804i	−0.936599	+	0.350402i	$0.886045\pi$
$822$		0			0
$823$		−	41.4392i		−	1.44448i	−0.691643	−	0.722239i	$-0.743113\pi$
$823$		−	41.4392i		−	1.44448i	0.691643	−	0.722239i	$-0.256887\pi$
$824$	−16.7571			−0.583761
$825$		0			0
$826$		−	9.99242i		−	0.347681i
$827$	44.9920			1.56453			0.782263	−	0.622948i	$-0.214065\pi$
$827$	44.9920			1.56453			0.782263	+	0.622948i	$0.214065\pi$
$828$		0			0
$829$		−	3.05090i		−	0.105962i	−0.998596	−	0.0529811i	$-0.983128\pi$
$829$		−	3.05090i		−	0.105962i	0.998596	−	0.0529811i	$-0.0168723\pi$
$830$	38.4423			1.33435
$831$		0			0
$832$		−	1.39220i		−	0.0482657i
$833$			10.9529i			0.379494i
$834$		0			0
$835$	−14.1872			−0.490969
$836$	−26.0527			−0.901052
$837$		0			0
$838$	39.0439			1.34875
$839$	29.0284			1.00217			0.501086	−	0.865398i	$-0.332934\pi$
$839$	29.0284			1.00217			0.501086	+	0.865398i	$0.332934\pi$
$840$		0			0
$841$	18.8496			0.649987
$842$	−32.4049			−1.11675
$843$		0			0
$844$	15.5611			0.535636
$845$	−28.6363			−0.985119
$846$		0			0
$847$	−0.415012			−0.0142600
$848$			7.39901i			0.254083i
$849$		0			0
$850$	4.27280			0.146556
$851$	−29.8264			−1.02244
$852$		0			0
$853$		−	4.32413i		−	0.148055i	−0.997256	−	0.0740277i	$-0.976415\pi$
$853$		−	4.32413i		−	0.148055i	0.997256	−	0.0740277i	$-0.0235853\pi$
$854$	−19.7720			−0.676584
$855$		0			0
$856$		−	15.0160i		−	0.513237i
$857$		−	15.0291i		−	0.513383i	−0.966493	−	0.256692i	$-0.917367\pi$
$857$		−	15.0291i		−	0.513383i	0.966493	−	0.256692i	$-0.0826325\pi$
$858$		0			0
$859$			27.1313i			0.925710i	0.886434	+	0.462855i	$0.153175\pi$
$859$			27.1313i			0.925710i	−0.886434	+	0.462855i	$0.846825\pi$
$860$	4.23452			0.144396
$861$		0			0
$862$			20.3792i			0.694117i
$863$	9.11095			0.310140			0.155070	−	0.987903i	$-0.450440\pi$
$863$	9.11095			0.310140			0.155070	+	0.987903i	$0.450440\pi$
$864$		0			0
$865$	27.3813			0.930993
$866$			32.3710i			1.10001i
$867$		0			0
$868$	−10.5529			−0.358187
$869$			2.24079i			0.0760135i
$870$		0			0
$871$	−1.51120			−0.0512050
$872$		−	18.1096i		−	0.613267i
$873$		0			0
$874$		−	49.2289i		−	1.66519i
$875$	−13.8681			−0.468829
$876$		0			0
$877$		−	2.55628i		−	0.0863196i	−0.999068	−	0.0431598i	$-0.986258\pi$
$877$		−	2.55628i		−	0.0863196i	0.999068	−	0.0431598i	$-0.0137425\pi$
$878$	3.51894			0.118758
$879$		0			0
$880$	8.68509			0.292774
$881$			14.1308i			0.476078i	0.971256	+	0.238039i	$0.0765047\pi$
$881$			14.1308i			0.476078i	−0.971256	+	0.238039i	$0.923495\pi$
$882$		0			0
$883$		−	47.4003i		−	1.59515i	−0.603221	−	0.797574i	$-0.706116\pi$
$883$		−	47.4003i		−	1.59515i	0.603221	−	0.797574i	$-0.293884\pi$
$884$	−3.49572			−0.117574
$885$		0			0
$886$	12.3553			0.415084
$887$			21.2118i			0.712221i	0.934444	+	0.356111i	$0.115897\pi$
$887$			21.2118i			0.712221i	−0.934444	+	0.356111i	$0.884103\pi$
$888$		0			0
$889$	−8.93790			−0.299767
$890$	31.5678			1.05816
$891$		0			0
$892$	−0.496576	−	14.9249i	−0.0166266	−	0.499723i
$893$		−	1.46873i		−	0.0491493i
$894$		0			0
$895$			4.07974i			0.136371i
$896$			1.62417i			0.0542599i
$897$		0			0
$898$			6.27640i			0.209446i
$899$			20.7004i			0.690397i
$900$		0			0
$901$	18.5785			0.618938
$902$	−17.1119			−0.569763
$903$		0			0
$904$			13.9862i			0.465174i
$905$	43.5111			1.44636
$906$		0			0
$907$	−24.7655			−0.822325			−0.411163	−	0.911562i	$-0.634877\pi$
$907$	−24.7655			−0.822325			−0.411163	+	0.911562i	$0.634877\pi$
$908$		−	16.1086i		−	0.534584i
$909$		0			0
$910$	−5.85362			−0.194046
$911$		−	27.2715i		−	0.903545i	−0.892133	−	0.451773i	$-0.850792\pi$
$911$		−	27.2715i		−	0.903545i	0.892133	−	0.451773i	$-0.149208\pi$
$912$		0			0
$913$		−	49.8197i		−	1.64879i
$914$	−4.60785			−0.152414
$915$		0			0
$916$			12.5095i			0.413326i
$917$			26.6849i			0.881212i
$918$		0			0
$919$		−	17.0461i		−	0.562298i	−0.959664	−	0.281149i	$-0.909284\pi$
$919$		−	17.0461i		−	0.562298i	0.959664	−	0.281149i	$-0.0907155\pi$
$920$			16.4112i			0.541063i
$921$		0			0
$922$	−23.1819			−0.763454
$923$			10.6712i			0.351247i
$924$		0			0
$925$	−8.00621			−0.263243
$926$			13.2234i			0.434548i
$927$		0			0
$928$	3.18597			0.104585
$929$		−	22.4865i		−	0.737757i	−0.929478	−	0.368878i	$-0.879742\pi$
$929$		−	22.4865i		−	0.737757i	0.929478	−	0.368878i	$-0.120258\pi$
$930$		0			0
$931$	33.8736			1.11016
$932$	−29.2714			−0.958815
$933$		0			0
$934$			23.9309i			0.783042i
$935$		−	21.8077i		−	0.713189i
$936$		0			0
$937$			19.2231i			0.627990i	0.949425	+	0.313995i	$0.101668\pi$
$937$			19.2231i			0.627990i	−0.949425	+	0.313995i	$0.898332\pi$
$938$	1.76301			0.0575642
$939$		0			0
$940$			0.489625i			0.0159698i
$941$			1.08869i			0.0354902i	0.999843	+	0.0177451i	$0.00564873\pi$
$941$			1.08869i			0.0354902i	−0.999843	+	0.0177451i	$0.994351\pi$
$942$		0			0
$943$		−	32.3344i		−	1.05295i
$944$	6.15231			0.200241
$945$		0			0
$946$		−	5.48777i		−	0.178423i
$947$			16.8048i			0.546084i	0.962002	+	0.273042i	$0.0880298\pi$
$947$			16.8048i			0.546084i	−0.962002	+	0.273042i	$0.911970\pi$
$948$		0			0
$949$		−	8.73693i		−	0.283613i
$950$		−	13.2144i		−	0.428731i
$951$		0			0
$952$	4.07820			0.132175
$953$	−36.6351			−1.18673			−0.593363	−	0.804935i	$-0.702200\pi$
$953$	−36.6351			−1.18673			−0.593363	+	0.804935i	$0.702200\pi$
$954$		0			0
$955$	−56.1678			−1.81755
$956$			9.64506i			0.311943i
$957$		0			0
$958$	29.3689			0.948867
$959$	−27.9476			−0.902474
$960$		0			0
$961$	11.2158			0.361800
$962$	6.55015			0.211185
$963$		0			0
$964$	−2.11008			−0.0679611
$965$		−	5.97437i		−	0.192322i
$966$		0			0
$967$		−	17.8688i		−	0.574622i	−0.957837	−	0.287311i	$-0.907239\pi$
$967$		−	17.8688i		−	0.574622i	0.957837	−	0.287311i	$-0.0927613\pi$
$968$		−	0.255522i		−	0.00821279i
$969$		0			0
$970$	47.6625			1.53035
$971$	20.9192			0.671328			0.335664	−	0.941982i	$-0.391039\pi$
$971$	20.9192			0.671328			0.335664	+	0.941982i	$0.391039\pi$
$972$		0			0
$973$	−10.3040			−0.330332
$974$		−	23.5847i		−	0.755703i
$975$		0			0
$976$		−	12.1736i		−	0.389667i
$977$	16.8810			0.540070			0.270035	−	0.962851i	$-0.412965\pi$
$977$	16.8810			0.540070			0.270035	+	0.962851i	$0.412965\pi$
$978$		0			0
$979$		−	40.9106i		−	1.30751i
$980$	−11.2923			−0.360720
$981$		0			0
$982$		−	5.32884i		−	0.170050i
$983$	46.8233			1.49343			0.746716	−	0.665142i	$-0.231629\pi$
$983$	46.8233			1.49343			0.746716	+	0.665142i	$0.231629\pi$
$984$		0			0
$985$		−	13.4314i		−	0.427960i
$986$		−	7.99977i		−	0.254765i
$987$		0			0
$988$			10.8111i			0.343947i
$989$	10.3696			0.329735
$990$		0			0
$991$			4.49403i			0.142758i	0.997449	+	0.0713788i	$0.0227399\pi$
$991$			4.49403i			0.142758i	−0.997449	+	0.0713788i	$0.977260\pi$
$992$		−	6.49737i		−	0.206292i
$993$		0			0
$994$		−	12.4493i		−	0.394868i
$995$	−38.4483			−1.21889
$996$		0			0
$997$	21.0379			0.666276			0.333138	−	0.942878i	$-0.391893\pi$
$997$	21.0379			0.666276			0.333138	+	0.942878i	$0.391893\pi$
$998$		−	30.8614i		−	0.976901i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.d.a.4013.12	yes	72
3.2	odd	2	inner	4014.2.d.a.4013.33	yes	72
223.222	odd	2	inner	4014.2.d.a.4013.34	yes	72
669.668	even	2	inner	4014.2.d.a.4013.11	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.d.a.4013.11	✓	72	669.668	even	2	inner
4014.2.d.a.4013.12	yes	72	1.1	even	1	trivial
4014.2.d.a.4013.33	yes	72	3.2	odd	2	inner
4014.2.d.a.4013.34	yes	72	223.222	odd	2	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.58876 q^{5} -1.62417 q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.58876 q^{5} -1.62417 q^{7} -1.00000i q^{8} -2.58876i q^{10} -3.35493 q^{11} +1.39220i q^{13} -1.62417i q^{14} +1.00000 q^{16} -2.51094i q^{17} -7.76552 q^{19} +2.58876 q^{20} -3.35493i q^{22} +6.33943 q^{23} +1.70167 q^{25} -1.39220 q^{26} +1.62417 q^{28} -3.18597i q^{29} -6.49737 q^{31} +1.00000i q^{32} +2.51094 q^{34} +4.20460 q^{35} -4.70491 q^{37} -7.76552i q^{38} +2.58876i q^{40} -5.10053i q^{41} +1.63574 q^{43} +3.35493 q^{44} +6.33943i q^{46} +0.189135i q^{47} -4.36206 q^{49} +1.70167i q^{50} -1.39220i q^{52} +7.39901i q^{53} +8.68509 q^{55} +1.62417i q^{56} +3.18597 q^{58} +6.15231 q^{59} -12.1736i q^{61} -6.49737i q^{62} -1.00000 q^{64} -3.60406i q^{65} +1.08548i q^{67} +2.51094i q^{68} +4.20460i q^{70} +7.66501 q^{71} -6.27565 q^{73} -4.70491i q^{74} +7.76552 q^{76} +5.44898 q^{77} -0.667910i q^{79} -2.58876 q^{80} +5.10053 q^{82} +14.8497i q^{83} +6.50022i q^{85} +1.63574i q^{86} +3.35493i q^{88} +12.1942i q^{89} -2.26117i q^{91} -6.33943 q^{92} -0.189135 q^{94} +20.1031 q^{95} +18.4113i q^{97} -4.36206i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) 72 * q - 72 * q^4 + 16 * q^7	\( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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