LMFDB - Embedded newform 4020.2.f.a.401.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(401,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("4020.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.f (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.16
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.963300 + 1.43946i) q^{3} -1.00000 q^{5} -2.54461i q^{7} +(-1.14411 - 2.77327i) q^{9} +O(q^{10})$	\(q+(-0.963300 + 1.43946i) q^{3} -1.00000 q^{5} -2.54461i q^{7} +(-1.14411 - 2.77327i) q^{9} -4.55788 q^{11} -2.49516i q^{13} +(0.963300 - 1.43946i) q^{15} -7.63072i q^{17} +2.99178 q^{19} +(3.66287 + 2.45122i) q^{21} -4.97417i q^{23} +1.00000 q^{25} +(5.09414 + 1.02459i) q^{27} +9.54010i q^{29} -4.41758i q^{31} +(4.39060 - 6.56089i) q^{33} +2.54461i q^{35} -1.29126 q^{37} +(3.59169 + 2.40359i) q^{39} -4.47938 q^{41} +8.78099i q^{43} +(1.14411 + 2.77327i) q^{45} +2.76941i q^{47} +0.524960 q^{49} +(10.9841 + 7.35067i) q^{51} -8.39548 q^{53} +4.55788 q^{55} +(-2.88198 + 4.30655i) q^{57} +0.920152i q^{59} -4.62956i q^{61} +(-7.05689 + 2.91131i) q^{63} +2.49516i q^{65} +(1.56012 + 8.03530i) q^{67} +(7.16013 + 4.79162i) q^{69} -0.0757503i q^{71} -12.9084 q^{73} +(-0.963300 + 1.43946i) q^{75} +11.5980i q^{77} -1.43341i q^{79} +(-6.38204 + 6.34583i) q^{81} -13.5395i q^{83} +7.63072i q^{85} +(-13.7326 - 9.18997i) q^{87} +0.638578i q^{89} -6.34921 q^{91} +(6.35894 + 4.25545i) q^{93} -2.99178 q^{95} -6.99472i q^{97} +(5.21470 + 12.6402i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) $ 46 * q - 46 * q^5 - 4 * q^9	$ 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) $ 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.963300	+	1.43946i	−0.556161	+	0.831074i
$3$	−0.963300	+	1.43946i	−0.556161	+	0.831074i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	2.54461i		−	0.961772i	−0.876783	−	0.480886i	$-0.840315\pi$
$7$		−	2.54461i		−	0.961772i	0.876783	−	0.480886i	$-0.159685\pi$
$8$		0			0
$9$	−1.14411	−	2.77327i	−0.381369	−	0.924423i
$10$		0			0
$11$	−4.55788			−1.37425			−0.687126	−	0.726539i	$-0.741128\pi$
$11$	−4.55788			−1.37425			−0.687126	+	0.726539i	$0.741128\pi$
$12$		0			0
$13$		−	2.49516i		−	0.692033i	−0.938228	−	0.346016i	$-0.887534\pi$
$13$		−	2.49516i		−	0.692033i	0.938228	−	0.346016i	$-0.112466\pi$
$14$		0			0
$15$	0.963300	−	1.43946i	0.248723	−	0.371668i
$16$		0			0
$17$		−	7.63072i		−	1.85072i	−0.379088	−	0.925360i	$-0.623762\pi$
$17$		−	7.63072i		−	1.85072i	0.379088	−	0.925360i	$-0.376238\pi$
$18$		0			0
$19$	2.99178			0.686361			0.343181	−	0.939269i	$-0.388496\pi$
$19$	2.99178			0.686361			0.343181	+	0.939269i	$0.388496\pi$
$20$		0			0
$21$	3.66287	+	2.45122i	0.799304	+	0.534901i
$22$		0			0
$23$		−	4.97417i		−	1.03719i	−0.855021	−	0.518593i	$-0.826456\pi$
$23$		−	4.97417i		−	1.03719i	0.855021	−	0.518593i	$-0.173544\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	5.09414	+	1.02459i	0.980367	+	0.197182i
$28$		0			0
$29$			9.54010i			1.77155i	0.464113	+	0.885776i	$0.346373\pi$
$29$			9.54010i			1.77155i	−0.464113	+	0.885776i	$0.653627\pi$
$30$		0			0
$31$		−	4.41758i		−	0.793421i	−0.917944	−	0.396710i	$-0.870152\pi$
$31$		−	4.41758i		−	0.793421i	0.917944	−	0.396710i	$-0.129848\pi$
$32$		0			0
$33$	4.39060	−	6.56089i	0.764305	−	1.14210i
$34$		0			0
$35$			2.54461i			0.430118i
$36$		0			0
$37$	−1.29126			−0.212281			−0.106141	−	0.994351i	$-0.533849\pi$
$37$	−1.29126			−0.212281			−0.106141	+	0.994351i	$0.533849\pi$
$38$		0			0
$39$	3.59169	+	2.40359i	0.575131	+	0.384882i
$40$		0			0
$41$	−4.47938			−0.699561			−0.349781	−	0.936832i	$-0.613744\pi$
$41$	−4.47938			−0.699561			−0.349781	+	0.936832i	$0.613744\pi$
$42$		0			0
$43$			8.78099i			1.33909i	0.742772	+	0.669544i	$0.233511\pi$
$43$			8.78099i			1.33909i	−0.742772	+	0.669544i	$0.766489\pi$
$44$		0			0
$45$	1.14411	+	2.77327i	0.170553	+	0.413414i
$46$		0			0
$47$			2.76941i			0.403960i	0.979390	+	0.201980i	$0.0647376\pi$
$47$			2.76941i			0.403960i	−0.979390	+	0.201980i	$0.935262\pi$
$48$		0			0
$49$	0.524960			0.0749942
$50$		0			0
$51$	10.9841	+	7.35067i	1.53809	+	1.02930i
$52$		0			0
$53$	−8.39548			−1.15321			−0.576604	−	0.817024i	$-0.695622\pi$
$53$	−8.39548			−1.15321			−0.576604	+	0.817024i	$0.695622\pi$
$54$		0			0
$55$	4.55788			0.614584
$56$		0			0
$57$	−2.88198	+	4.30655i	−0.381727	+	0.570417i
$58$		0			0
$59$			0.920152i			0.119794i	0.998205	+	0.0598968i	$0.0190772\pi$
$59$			0.920152i			0.119794i	−0.998205	+	0.0598968i	$0.980923\pi$
$60$		0			0
$61$		−	4.62956i		−	0.592754i	−0.955071	−	0.296377i	$-0.904221\pi$
$61$		−	4.62956i		−	0.592754i	0.955071	−	0.296377i	$-0.0957785\pi$
$62$		0			0
$63$	−7.05689	+	2.91131i	−0.889084	+	0.366790i
$64$		0			0
$65$			2.49516i			0.309487i
$66$		0			0
$67$	1.56012	+	8.03530i	0.190600	+	0.981668i
$67$	1.56012	+	8.03530i	0.190600	+	0.981668i
$68$		0			0
$69$	7.16013	+	4.79162i	0.861979	+	0.576843i
$70$		0			0
$71$		−	0.0757503i		−	0.00898991i	−0.999990	−	0.00449496i	$-0.998569\pi$
$71$		−	0.0757503i		−	0.00898991i	0.999990	−	0.00449496i	$-0.00143079\pi$
$72$		0			0
$73$	−12.9084			−1.51081			−0.755404	−	0.655259i	$-0.772559\pi$
$73$	−12.9084			−1.51081			−0.755404	+	0.655259i	$0.772559\pi$
$74$		0			0
$75$	−0.963300	+	1.43946i	−0.111232	+	0.166215i
$76$		0			0
$77$			11.5980i			1.32172i
$78$		0			0
$79$		−	1.43341i		−	0.161272i	−0.996744	−	0.0806359i	$-0.974305\pi$
$79$		−	1.43341i		−	0.161272i	0.996744	−	0.0806359i	$-0.0256951\pi$
$80$		0			0
$81$	−6.38204	+	6.34583i	−0.709115	+	0.705093i
$82$		0			0
$83$		−	13.5395i		−	1.48616i	−0.669205	−	0.743078i	$-0.733365\pi$
$83$		−	13.5395i		−	1.48616i	0.669205	−	0.743078i	$-0.266635\pi$
$84$		0			0
$85$			7.63072i			0.827668i
$86$		0			0
$87$	−13.7326	−	9.18997i	−1.47229	−	0.985269i
$88$		0			0
$89$			0.638578i			0.0676891i	0.999427	+	0.0338446i	$0.0107751\pi$
$89$			0.638578i			0.0676891i	−0.999427	+	0.0338446i	$0.989225\pi$
$90$		0			0
$91$	−6.34921			−0.665578
$92$		0			0
$93$	6.35894	+	4.25545i	0.659392	+	0.441270i
$94$		0			0
$95$	−2.99178			−0.306950
$96$		0			0
$97$		−	6.99472i		−	0.710206i	−0.934827	−	0.355103i	$-0.884446\pi$
$97$		−	6.99472i		−	0.710206i	0.934827	−	0.355103i	$-0.115554\pi$
$98$		0			0
$99$	5.21470	+	12.6402i	0.524097	+	1.27039i
$100$		0			0
$101$	−12.2900			−1.22290			−0.611448	−	0.791284i	$-0.709413\pi$
$101$	−12.2900			−1.22290			−0.611448	+	0.791284i	$0.709413\pi$
$102$		0			0
$103$	14.0466			1.38405			0.692024	−	0.721875i	$-0.256719\pi$
$103$	14.0466			1.38405			0.692024	+	0.721875i	$0.256719\pi$
$104$		0			0
$105$	−3.66287	−	2.45122i	−0.357460	−	0.239215i
$106$		0			0
$107$			19.5473i			1.88971i	0.327490	+	0.944855i	$0.393797\pi$
$107$			19.5473i			1.88971i	−0.327490	+	0.944855i	$0.606203\pi$
$108$		0			0
$109$			15.0935i			1.44569i	0.691009	+	0.722846i	$0.257167\pi$
$109$			15.0935i			1.44569i	−0.691009	+	0.722846i	$0.742833\pi$
$110$		0			0
$111$	1.24387	−	1.85872i	0.118063	−	0.176421i
$112$		0			0
$113$	−18.0102			−1.69426			−0.847128	−	0.531390i	$-0.821670\pi$
$113$	−18.0102			−1.69426			−0.847128	+	0.531390i	$0.821670\pi$
$114$		0			0
$115$			4.97417i			0.463844i
$116$		0			0
$117$	−6.91975	+	2.85473i	−0.639731	+	0.263920i
$118$		0			0
$119$	−19.4172			−1.77997
$120$		0			0
$121$	9.77423			0.888566
$122$		0			0
$123$	4.31498	−	6.44790i	0.389069	−	0.581387i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	7.46508			0.662418			0.331209	−	0.943557i	$-0.392543\pi$
$127$	7.46508			0.662418			0.331209	+	0.943557i	$0.392543\pi$
$128$		0			0
$129$	−12.6399	−	8.45873i	−1.11288	−	0.744749i
$130$		0			0
$131$		−	4.56466i		−	0.398817i	−0.979916	−	0.199408i	$-0.936098\pi$
$131$		−	4.56466i		−	0.398817i	0.979916	−	0.199408i	$-0.0639020\pi$
$132$		0			0
$133$		−	7.61291i		−	0.660123i
$134$		0			0
$135$	−5.09414	−	1.02459i	−0.438433	−	0.0881826i
$136$		0			0
$137$	17.4073			1.48720			0.743601	−	0.668624i	$-0.233116\pi$
$137$	17.4073			1.48720			0.743601	+	0.668624i	$0.233116\pi$
$138$		0			0
$139$		−	0.341211i		−	0.0289411i	−0.999895	−	0.0144706i	$-0.995394\pi$
$139$		−	0.341211i		−	0.0289411i	0.999895	−	0.0144706i	$-0.00460628\pi$
$140$		0			0
$141$	−3.98646	−	2.66777i	−0.335721	−	0.224667i
$142$		0			0
$143$			11.3726i			0.951027i
$144$		0			0
$145$		−	9.54010i		−	0.792262i
$146$		0			0
$147$	−0.505694	+	0.755660i	−0.0417089	+	0.0623258i
$148$		0			0
$149$		−	6.62577i		−	0.542804i	−0.962466	−	0.271402i	$-0.912513\pi$
$149$		−	6.62577i		−	0.542804i	0.962466	−	0.271402i	$-0.0874873\pi$
$150$		0			0
$151$	−1.61556			−0.131473			−0.0657363	−	0.997837i	$-0.520940\pi$
$151$	−1.61556			−0.131473			−0.0657363	+	0.997837i	$0.520940\pi$
$152$		0			0
$153$	−21.1620	+	8.73036i	−1.71085	+	0.705808i
$154$		0			0
$155$			4.41758i			0.354829i
$156$		0			0
$157$	17.6280			1.40687			0.703434	−	0.710761i	$-0.251649\pi$
$157$	17.6280			1.40687			0.703434	+	0.710761i	$0.251649\pi$
$158$		0			0
$159$	8.08736	−	12.0850i	0.641370	−	0.958401i
$160$		0			0
$161$	−12.6573			−0.997537
$162$		0			0
$163$	1.66222			0.130195			0.0650974	−	0.997879i	$-0.479264\pi$
$163$	1.66222			0.130195			0.0650974	+	0.997879i	$0.479264\pi$
$164$		0			0
$165$	−4.39060	+	6.56089i	−0.341808	+	0.510765i
$166$		0			0
$167$			10.8058i			0.836178i	0.908406	+	0.418089i	$0.137300\pi$
$167$			10.8058i			0.836178i	−0.908406	+	0.418089i	$0.862700\pi$
$168$		0			0
$169$	6.77418			0.521090
$170$		0			0
$171$	−3.42292	−	8.29701i	−0.261757	−	0.634488i
$172$		0			0
$173$		−	22.4010i		−	1.70312i	−0.524260	−	0.851558i	$-0.675658\pi$
$173$		−	22.4010i		−	1.70312i	0.524260	−	0.851558i	$-0.324342\pi$
$174$		0			0
$175$		−	2.54461i		−	0.192354i
$176$		0			0
$177$	−1.32452	−	0.886382i	−0.0995574	−	0.0666246i
$178$		0			0
$179$	3.80314			0.284260			0.142130	−	0.989848i	$-0.454605\pi$
$179$	3.80314			0.284260			0.142130	+	0.989848i	$0.454605\pi$
$180$		0			0
$181$	−13.5774			−1.00920			−0.504602	−	0.863352i	$-0.668361\pi$
$181$	−13.5774			−1.00920			−0.504602	+	0.863352i	$0.668361\pi$
$182$		0			0
$183$	6.66408	+	4.45965i	0.492623	+	0.329667i
$184$		0			0
$185$	1.29126			0.0949350
$186$		0			0
$187$			34.7799i			2.54336i
$188$		0			0
$189$	2.60718	−	12.9626i	0.189644	−	0.942890i
$190$		0			0
$191$	−6.07318			−0.439440			−0.219720	−	0.975563i	$-0.570514\pi$
$191$	−6.07318			−0.439440			−0.219720	+	0.975563i	$0.570514\pi$
$192$		0			0
$193$	21.5678			1.55249			0.776244	−	0.630433i	$-0.217123\pi$
$193$	21.5678			1.55249			0.776244	+	0.630433i	$0.217123\pi$
$194$		0			0
$195$	−3.59169	−	2.40359i	−0.257206	−	0.172124i
$196$		0			0
$197$	13.9321			0.992620			0.496310	−	0.868145i	$-0.334688\pi$
$197$	13.9321			0.992620			0.496310	+	0.868145i	$0.334688\pi$
$198$		0			0
$199$	−24.6912			−1.75031			−0.875157	−	0.483839i	$-0.839242\pi$
$199$	−24.6912			−1.75031			−0.875157	+	0.483839i	$0.839242\pi$
$200$		0			0
$201$	−13.0694	−	5.49466i	−0.921843	−	0.387563i
$202$		0			0
$203$	24.2758			1.70383
$204$		0			0
$205$	4.47938			0.312853
$206$		0			0
$207$	−13.7947	+	5.69098i	−0.958799	+	0.395551i
$208$		0			0
$209$	−13.6362			−0.943232
$210$		0			0
$211$	−20.5722			−1.41625			−0.708125	−	0.706088i	$-0.750458\pi$
$211$	−20.5722			−1.41625			−0.708125	+	0.706088i	$0.750458\pi$
$212$		0			0
$213$	0.109040	+	0.0729703i	0.00747128	+	0.00499984i
$214$		0			0
$215$		−	8.78099i		−	0.598859i
$216$		0			0
$217$	−11.2410			−0.763090
$218$		0			0
$219$	12.4346	−	18.5811i	0.840253	−	1.25559i
$220$		0			0
$221$	−19.0399			−1.28076
$222$		0			0
$223$	8.53648			0.571645			0.285823	−	0.958283i	$-0.407733\pi$
$223$	8.53648			0.571645			0.285823	+	0.958283i	$0.407733\pi$
$224$		0			0
$225$	−1.14411	−	2.77327i	−0.0762738	−	0.184885i
$226$		0			0
$227$			1.19706i			0.0794516i	0.999211	+	0.0397258i	$0.0126484\pi$
$227$			1.19706i			0.0794516i	−0.999211	+	0.0397258i	$0.987352\pi$
$228$		0			0
$229$			18.1101i			1.19675i	0.801216	+	0.598375i	$0.204187\pi$
$229$			18.1101i			1.19675i	−0.801216	+	0.598375i	$0.795813\pi$
$230$		0			0
$231$	−16.6949	−	11.1724i	−1.09844	−	0.735088i
$232$		0			0
$233$	−20.6090			−1.35014			−0.675070	−	0.737754i	$-0.735887\pi$
$233$	−20.6090			−1.35014			−0.675070	+	0.737754i	$0.735887\pi$
$234$		0			0
$235$		−	2.76941i		−	0.180656i
$236$		0			0
$237$	2.06335	+	1.38081i	0.134029	+	0.0896931i
$238$		0			0
$239$	22.6811			1.46712			0.733559	−	0.679626i	$-0.237858\pi$
$239$	22.6811			1.46712			0.733559	+	0.679626i	$0.237858\pi$
$240$		0			0
$241$	−11.6842			−0.752643			−0.376321	−	0.926489i	$-0.622811\pi$
$241$	−11.6842			−0.752643			−0.376321	+	0.926489i	$0.622811\pi$
$242$		0			0
$243$	−2.98678	−	15.2996i	−0.191602	−	0.981473i
$244$		0			0
$245$	−0.524960			−0.0335384
$246$		0			0
$247$		−	7.46497i		−	0.474984i
$248$		0			0
$249$	19.4896	+	13.0426i	1.23511	+	0.826542i
$250$		0			0
$251$	−3.85422			−0.243276			−0.121638	−	0.992575i	$-0.538815\pi$
$251$	−3.85422			−0.243276			−0.121638	+	0.992575i	$0.538815\pi$
$252$		0			0
$253$			22.6716i			1.42535i
$254$		0			0
$255$	−10.9841	−	7.35067i	−0.687853	−	0.460317i
$256$		0			0
$257$		−	2.28883i		−	0.142773i	−0.997449	−	0.0713867i	$-0.977258\pi$
$257$		−	2.28883i		−	0.142773i	0.997449	−	0.0713867i	$-0.0227424\pi$
$258$		0			0
$259$			3.28574i			0.204166i
$260$		0			0
$261$	26.4573	−	10.9149i	1.63766	−	0.675615i
$262$		0			0
$263$			15.6993i			0.968058i	0.875052	+	0.484029i	$0.160827\pi$
$263$			15.6993i			0.968058i	−0.875052	+	0.484029i	$0.839173\pi$
$264$		0			0
$265$	8.39548			0.515730
$266$		0			0
$267$	−0.919209	−	0.615142i	−0.0562547	−	0.0376461i
$268$		0			0
$269$			11.4475i			0.697965i	0.937129	+	0.348982i	$0.113473\pi$
$269$			11.4475i			0.697965i	−0.937129	+	0.348982i	$0.886527\pi$
$270$		0			0
$271$			0.0972657i			0.00590847i	0.999996	+	0.00295423i	$0.000940363\pi$
$271$			0.0972657i			0.00590847i	−0.999996	+	0.00295423i	$0.999060\pi$
$272$		0			0
$273$	6.11619	−	9.13945i	0.370169	−	0.553145i
$274$		0			0
$275$	−4.55788			−0.274850
$276$		0			0
$277$	−30.6859			−1.84373			−0.921867	−	0.387506i	$-0.873337\pi$
$277$	−30.6859			−1.84373			−0.921867	+	0.387506i	$0.873337\pi$
$278$		0			0
$279$	−12.2511	+	5.05418i	−0.733456	+	0.302586i
$280$		0			0
$281$	−6.98085			−0.416443			−0.208221	−	0.978082i	$-0.566767\pi$
$281$	−6.98085			−0.416443			−0.208221	+	0.978082i	$0.566767\pi$
$282$		0			0
$283$	2.28993			0.136122			0.0680611	−	0.997681i	$-0.478319\pi$
$283$	2.28993			0.136122			0.0680611	+	0.997681i	$0.478319\pi$
$284$		0			0
$285$	2.88198	−	4.30655i	0.170714	−	0.255098i
$286$		0			0
$287$			11.3983i			0.672819i
$288$		0			0
$289$	−41.2279			−2.42517
$290$		0			0
$291$	10.0686	+	6.73801i	0.590234	+	0.394989i
$292$		0			0
$293$			23.5617i			1.37649i	0.725478	+	0.688245i	$0.241619\pi$
$293$			23.5617i			1.37649i	−0.725478	+	0.688245i	$0.758381\pi$
$294$		0			0
$295$		−	0.920152i		−	0.0535733i
$296$		0			0
$297$	−23.2184	−	4.66995i	−1.34727	−	0.270978i
$298$		0			0
$299$	−12.4114			−0.717767
$300$		0			0
$301$	22.3442			1.28790
$302$		0			0
$303$	11.8389	−	17.6909i	0.680128	−	1.01632i
$304$		0			0
$305$			4.62956i			0.265088i
$306$		0			0
$307$	−9.10787			−0.519814			−0.259907	−	0.965634i	$-0.583692\pi$
$307$	−9.10787			−0.519814			−0.259907	+	0.965634i	$0.583692\pi$
$308$		0			0
$309$	−13.5310	+	20.2195i	−0.769754	+	1.15025i
$310$		0			0
$311$	0.866773			0.0491502			0.0245751	−	0.999698i	$-0.492177\pi$
$311$	0.866773			0.0491502			0.0245751	+	0.999698i	$0.492177\pi$
$312$		0			0
$313$			19.0229i			1.07524i	0.843188	+	0.537619i	$0.180676\pi$
$313$			19.0229i			1.07524i	−0.843188	+	0.537619i	$0.819324\pi$
$314$		0			0
$315$	7.05689	−	2.91131i	0.397611	−	0.164034i
$316$		0			0
$317$		−	30.0111i		−	1.68559i	−0.538235	−	0.842795i	$-0.680909\pi$
$317$		−	30.0111i		−	1.68559i	0.538235	−	0.842795i	$-0.319091\pi$
$318$		0			0
$319$		−	43.4826i		−	2.43456i
$320$		0			0
$321$	−28.1376	−	18.8299i	−1.57049	−	1.05098i
$322$		0			0
$323$		−	22.8294i		−	1.27026i
$324$		0			0
$325$		−	2.49516i		−	0.138407i
$326$		0			0
$327$	−21.7265	−	14.5395i	−1.20148	−	0.804038i
$328$		0			0
$329$	7.04706			0.388517
$330$		0			0
$331$		−	20.3551i		−	1.11882i	−0.828892	−	0.559408i	$-0.811028\pi$
$331$		−	20.3551i		−	1.11882i	0.828892	−	0.559408i	$-0.188972\pi$
$332$		0			0
$333$	1.47734	+	3.58100i	0.0809575	+	0.196238i
$334$		0			0
$335$	−1.56012	−	8.03530i	−0.0852387	−	0.439015i
$336$		0			0
$337$			29.6044i			1.61265i	0.591470	+	0.806327i	$0.298548\pi$
$337$			29.6044i			1.61265i	−0.591470	+	0.806327i	$0.701452\pi$
$338$		0			0
$339$	17.3492	−	25.9250i	0.942279	−	1.40805i
$340$		0			0
$341$			20.1348i			1.09036i
$342$		0			0
$343$		−	19.1481i		−	1.03390i
$344$		0			0
$345$	−7.16013	−	4.79162i	−0.385489	−	0.257972i
$346$		0			0
$347$	−29.8418			−1.60199			−0.800995	−	0.598671i	$-0.795696\pi$
$347$	−29.8418			−1.60199			−0.800995	+	0.598671i	$0.795696\pi$
$348$		0			0
$349$	−26.4316			−1.41485			−0.707424	−	0.706789i	$-0.750143\pi$
$349$	−26.4316			−1.41485			−0.707424	+	0.706789i	$0.750143\pi$
$350$		0			0
$351$	2.55651	−	12.7107i	0.136457	−	0.678446i
$352$		0			0
$353$	−8.72715			−0.464499			−0.232250	−	0.972656i	$-0.574609\pi$
$353$	−8.72715			−0.464499			−0.232250	+	0.972656i	$0.574609\pi$
$354$		0			0
$355$			0.0757503i			0.00402041i
$356$		0			0
$357$	18.7046	−	27.9503i	0.989952	−	1.47929i
$358$		0			0
$359$			26.3051i			1.38833i	0.719816	+	0.694165i	$0.244226\pi$
$359$			26.3051i			1.38833i	−0.719816	+	0.694165i	$0.755774\pi$
$360$		0			0
$361$	−10.0493			−0.528909
$362$		0			0
$363$	−9.41551	+	14.0696i	−0.494186	+	0.738464i
$364$		0			0
$365$	12.9084			0.675654
$366$		0			0
$367$		−	15.8937i		−	0.829647i	−0.909902	−	0.414823i	$-0.863843\pi$
$367$		−	15.8937i		−	0.829647i	0.909902	−	0.414823i	$-0.136157\pi$
$368$		0			0
$369$	5.12489	+	12.4225i	0.266791	+	0.646690i
$370$		0			0
$371$			21.3632i			1.10912i
$372$		0			0
$373$			0.566452i			0.0293298i	0.999892	+	0.0146649i	$0.00466815\pi$
$373$			0.566452i			0.0293298i	−0.999892	+	0.0146649i	$0.995332\pi$
$374$		0			0
$375$	0.963300	−	1.43946i	0.0497446	−	0.0743335i
$376$		0			0
$377$	23.8041			1.22597
$378$		0			0
$379$			28.4093i			1.45929i	0.683828	+	0.729643i	$0.260314\pi$
$379$			28.4093i			1.45929i	−0.683828	+	0.729643i	$0.739686\pi$
$380$		0			0
$381$	−7.19110	+	10.7457i	−0.368411	+	0.550519i
$382$		0			0
$383$	16.2283			0.829229			0.414614	−	0.909997i	$-0.363916\pi$
$383$	16.2283			0.829229			0.414614	+	0.909997i	$0.363916\pi$
$384$		0			0
$385$		−	11.5980i		−	0.591090i
$386$		0			0
$387$	24.3521	−	10.0464i	1.23788	−	0.510687i
$388$		0			0
$389$		−	14.1138i		−	0.715597i	−0.933799	−	0.357798i	$-0.883528\pi$
$389$		−	14.1138i		−	0.715597i	0.933799	−	0.357798i	$-0.116472\pi$
$390$		0			0
$391$	−37.9565			−1.91954
$392$		0			0
$393$	6.57066	+	4.39714i	0.331446	+	0.221806i
$394$		0			0
$395$			1.43341i			0.0721229i
$396$		0			0
$397$	−2.95108			−0.148111			−0.0740553	−	0.997254i	$-0.523594\pi$
$397$	−2.95108			−0.148111			−0.0740553	+	0.997254i	$0.523594\pi$
$398$		0			0
$399$	10.9585	+	7.33351i	0.548611	+	0.367135i
$400$		0			0
$401$	−22.6363			−1.13040			−0.565201	−	0.824953i	$-0.691201\pi$
$401$	−22.6363			−1.13040			−0.565201	+	0.824953i	$0.691201\pi$
$402$		0			0
$403$	−11.0226			−0.549073
$404$		0			0
$405$	6.38204	−	6.34583i	0.317126	−	0.315327i
$406$		0			0
$407$	5.88538			0.291728
$408$		0			0
$409$		−	13.4339i		−	0.664266i	−0.943233	−	0.332133i	$-0.892232\pi$
$409$		−	13.4339i		−	0.664266i	0.943233	−	0.332133i	$-0.107768\pi$
$410$		0			0
$411$	−16.7684	+	25.0571i	−0.827124	+	1.23598i
$412$		0			0
$413$	2.34143			0.115214
$414$		0			0
$415$			13.5395i			0.664629i
$416$		0			0
$417$	0.491160	+	0.328688i	0.0240522	+	0.0160959i
$418$		0			0
$419$			23.9401i			1.16955i	0.811196	+	0.584774i	$0.198817\pi$
$419$			23.9401i			1.16955i	−0.811196	+	0.584774i	$0.801183\pi$
$420$		0			0
$421$	−34.7579			−1.69400			−0.846998	−	0.531596i	$-0.821592\pi$
$421$	−34.7579			−1.69400			−0.846998	+	0.531596i	$0.821592\pi$
$422$		0			0
$423$	7.68031	−	3.16850i	0.373430	−	0.154058i
$424$		0			0
$425$		−	7.63072i		−	0.370144i
$426$		0			0
$427$	−11.7804			−0.570095
$428$		0			0
$429$	−16.3705	−	10.9552i	−0.790374	−	0.528924i
$430$		0			0
$431$			24.5796i			1.18396i	0.805953	+	0.591979i	$0.201653\pi$
$431$			24.5796i			1.18396i	−0.805953	+	0.591979i	$0.798347\pi$
$432$		0			0
$433$			18.7997i			0.903456i	0.892156	+	0.451728i	$0.149192\pi$
$433$			18.7997i			0.903456i	−0.892156	+	0.451728i	$0.850808\pi$
$434$		0			0
$435$	13.7326	+	9.18997i	0.658429	+	0.440626i
$436$		0			0
$437$		−	14.8816i		−	0.711884i
$438$		0			0
$439$	31.2138			1.48975			0.744877	−	0.667202i	$-0.232508\pi$
$439$	31.2138			1.48975			0.744877	+	0.667202i	$0.232508\pi$
$440$		0			0
$441$	−0.600610	−	1.45585i	−0.0286005	−	0.0693264i
$442$		0			0
$443$	38.7155			1.83943			0.919715	−	0.392586i	$-0.128419\pi$
$443$	38.7155			1.83943			0.919715	+	0.392586i	$0.128419\pi$
$444$		0			0
$445$		−	0.638578i		−	0.0302715i
$446$		0			0
$447$	9.53755	+	6.38260i	0.451110	+	0.301887i
$448$		0			0
$449$		−	4.52283i		−	0.213446i	−0.994289	−	0.106723i	$-0.965964\pi$
$449$		−	4.52283i		−	0.213446i	0.994289	−	0.106723i	$-0.0340358\pi$
$450$		0			0
$451$	20.4164			0.961373
$452$		0			0
$453$	1.55627	−	2.32554i	0.0731199	−	0.109263i
$454$		0			0
$455$	6.34921			0.297656
$456$		0			0
$457$	−0.249672			−0.0116792			−0.00583958	−	0.999983i	$-0.501859\pi$
$457$	−0.249672			−0.0116792			−0.00583958	+	0.999983i	$0.501859\pi$
$458$		0			0
$459$	7.81835	−	38.8719i	0.364929	−	1.81439i
$460$		0			0
$461$			4.92680i			0.229464i	0.993396	+	0.114732i	$0.0366010\pi$
$461$			4.92680i			0.229464i	−0.993396	+	0.114732i	$0.963399\pi$
$462$		0			0
$463$			24.6709i			1.14655i	0.819362	+	0.573276i	$0.194328\pi$
$463$			24.6709i			1.14655i	−0.819362	+	0.573276i	$0.805672\pi$
$464$		0			0
$465$	−6.35894	−	4.25545i	−0.294889	−	0.197342i
$466$		0			0
$467$			17.2143i			0.796581i	0.917259	+	0.398291i	$0.130396\pi$
$467$			17.2143i			0.796581i	−0.917259	+	0.398291i	$0.869604\pi$
$468$		0			0
$469$	20.4467	−	3.96991i	0.944141	−	0.183313i
$470$		0			0
$471$	−16.9811	+	25.3749i	−0.782446	+	1.16921i
$472$		0			0
$473$		−	40.0227i		−	1.84024i
$474$		0			0
$475$	2.99178			0.137272
$476$		0			0
$477$	9.60533	+	23.2829i	0.439798	+	1.06605i
$478$		0			0
$479$			27.3534i			1.24981i	0.780701	+	0.624905i	$0.214862\pi$
$479$			27.3534i			1.24981i	−0.780701	+	0.624905i	$0.785138\pi$
$480$		0			0
$481$			3.22189i			0.146906i
$482$		0			0
$483$	12.1928	−	18.2197i	0.554791	−	0.829027i
$484$		0			0
$485$			6.99472i			0.317614i
$486$		0			0
$487$			16.2098i			0.734535i	0.930115	+	0.367267i	$0.119707\pi$
$487$			16.2098i			0.734535i	−0.930115	+	0.367267i	$0.880293\pi$
$488$		0			0
$489$	−1.60121	+	2.39270i	−0.0724093	+	0.108202i
$490$		0			0
$491$		−	25.0659i		−	1.13121i	−0.824677	−	0.565604i	$-0.808643\pi$
$491$		−	25.0659i		−	1.13121i	0.824677	−	0.565604i	$-0.191357\pi$
$492$		0			0
$493$	72.7978			3.27865
$494$		0			0
$495$	−5.21470	−	12.6402i	−0.234383	−	0.568135i
$496$		0			0
$497$	−0.192755			−0.00864625
$498$		0			0
$499$		−	36.7145i		−	1.64357i	−0.569800	−	0.821784i	$-0.692979\pi$
$499$		−	36.7145i		−	1.64357i	0.569800	−	0.821784i	$-0.307021\pi$
$500$		0			0
$501$	−15.5545	−	10.4092i	−0.694926	−	0.465050i
$502$		0			0
$503$	42.6829			1.90314			0.951568	−	0.307439i	$-0.0994721\pi$
$503$	42.6829			1.90314			0.951568	+	0.307439i	$0.0994721\pi$
$504$		0			0
$505$	12.2900			0.546896
$506$		0			0
$507$	−6.52556	+	9.75118i	−0.289810	+	0.433065i
$508$		0			0
$509$			14.0369i			0.622173i	0.950382	+	0.311087i	$0.100693\pi$
$509$			14.0369i			0.622173i	−0.950382	+	0.311087i	$0.899307\pi$
$510$		0			0
$511$			32.8467i			1.45305i
$512$		0			0
$513$	15.2405	+	3.06534i	0.672886	+	0.135338i
$514$		0			0
$515$	−14.0466			−0.618965
$516$		0			0
$517$		−	12.6226i		−	0.555142i
$518$		0			0
$519$	32.2454	+	21.5789i	1.41542	+	0.947207i
$520$		0			0
$521$	25.5126			1.11773			0.558864	−	0.829260i	$-0.311238\pi$
$521$	25.5126			1.11773			0.558864	+	0.829260i	$0.311238\pi$
$522$		0			0
$523$	−5.77150			−0.252370			−0.126185	−	0.992007i	$-0.540273\pi$
$523$	−5.77150			−0.252370			−0.126185	+	0.992007i	$0.540273\pi$
$524$		0			0
$525$	3.66287	+	2.45122i	0.159861	+	0.106980i
$526$		0			0
$527$	−33.7093			−1.46840
$528$		0			0
$529$	−1.74237			−0.0757552
$530$		0			0
$531$	2.55183	−	1.05275i	0.110740	−	0.0456856i
$532$		0			0
$533$			11.1768i			0.484119i
$534$		0			0
$535$		−	19.5473i		−	0.845104i
$536$		0			0
$537$	−3.66356	+	5.47447i	−0.158094	+	0.236241i
$538$		0			0
$539$	−2.39270			−0.103061
$540$		0			0
$541$			18.3167i			0.787496i	0.919218	+	0.393748i	$0.128822\pi$
$541$			18.3167i			0.787496i	−0.919218	+	0.393748i	$0.871178\pi$
$542$		0			0
$543$	13.0791	−	19.5442i	0.561280	−	0.838723i
$544$		0			0
$545$		−	15.0935i		−	0.646533i
$546$		0			0
$547$		−	42.1775i		−	1.80338i	−0.432385	−	0.901689i	$-0.642328\pi$
$547$		−	42.1775i		−	1.80338i	0.432385	−	0.901689i	$-0.357672\pi$
$548$		0			0
$549$	−12.8390	+	5.29671i	−0.547956	+	0.226058i
$550$		0			0
$551$			28.5419i			1.21592i
$552$		0			0
$553$	−3.64748			−0.155107
$554$		0			0
$555$	−1.24387	+	1.85872i	−0.0527992	+	0.0788981i
$556$		0			0
$557$			18.4264i			0.780751i	0.920656	+	0.390376i	$0.127655\pi$
$557$			18.4264i			0.780751i	−0.920656	+	0.390376i	$0.872345\pi$
$558$		0			0
$559$	21.9100			0.926694
$560$		0			0
$561$	−50.0643	−	33.5034i	−2.11372	−	1.41452i
$562$		0			0
$563$	−1.51492			−0.0638461			−0.0319230	−	0.999490i	$-0.510163\pi$
$563$	−1.51492			−0.0638461			−0.0319230	+	0.999490i	$0.510163\pi$
$564$		0			0
$565$	18.0102			0.757694
$566$		0			0
$567$	16.1477	+	16.2398i	0.678138	+	0.682007i
$568$		0			0
$569$			3.22658i			0.135265i	0.997710	+	0.0676327i	$0.0215446\pi$
$569$			3.22658i			0.135265i	−0.997710	+	0.0676327i	$0.978455\pi$
$570$		0			0
$571$	44.8382			1.87642			0.938210	−	0.346065i	$-0.112482\pi$
$571$	44.8382			1.87642			0.938210	+	0.346065i	$0.112482\pi$
$572$		0			0
$573$	5.85029	−	8.74212i	0.244399	−	0.365207i
$574$		0			0
$575$		−	4.97417i		−	0.207437i
$576$		0			0
$577$			13.8424i			0.576268i	0.957590	+	0.288134i	$0.0930348\pi$
$577$			13.8424i			0.576268i	−0.957590	+	0.288134i	$0.906965\pi$
$578$		0			0
$579$	−20.7763	+	31.0461i	−0.863433	+	1.29023i
$580$		0			0
$581$	−34.4528			−1.42934
$582$		0			0
$583$	38.2655			1.58480
$584$		0			0
$585$	6.91975	−	2.85473i	0.286096	−	0.118029i
$586$		0			0
$587$	15.6778			0.647091			0.323545	−	0.946213i	$-0.395125\pi$
$587$	15.6778			0.647091			0.323545	+	0.946213i	$0.395125\pi$
$588$		0			0
$589$		−	13.2164i		−	0.544573i
$590$		0			0
$591$	−13.4208	+	20.0547i	−0.552057	+	0.824941i
$592$		0			0
$593$	−32.0831			−1.31750			−0.658748	−	0.752364i	$-0.728913\pi$
$593$	−32.0831			−1.31750			−0.658748	+	0.752364i	$0.728913\pi$
$594$		0			0
$595$	19.4172			0.796028
$596$		0			0
$597$	23.7850	−	35.5421i	0.973457	−	1.45464i
$598$		0			0
$599$	−11.6346			−0.475376			−0.237688	−	0.971342i	$-0.576390\pi$
$599$	−11.6346			−0.475376			−0.237688	+	0.971342i	$0.576390\pi$
$600$		0			0
$601$	−23.2537			−0.948536			−0.474268	−	0.880380i	$-0.657287\pi$
$601$	−23.2537			−0.948536			−0.474268	+	0.880380i	$0.657287\pi$
$602$		0			0
$603$	20.4991	−	13.5199i	0.834787	−	0.550572i
$604$		0			0
$605$	−9.77423			−0.397379
$606$		0			0
$607$	46.4202			1.88414			0.942069	−	0.335418i	$-0.108878\pi$
$607$	46.4202			1.88414			0.942069	+	0.335418i	$0.108878\pi$
$608$		0			0
$609$	−23.3849	+	34.9442i	−0.947604	+	1.41601i
$610$		0			0
$611$	6.91012			0.279553
$612$		0			0
$613$	−9.07011			−0.366338			−0.183169	−	0.983081i	$-0.558636\pi$
$613$	−9.07011			−0.366338			−0.183169	+	0.983081i	$0.558636\pi$
$614$		0			0
$615$	−4.31498	+	6.44790i	−0.173997	+	0.260004i
$616$		0			0
$617$			19.0302i			0.766126i	0.923722	+	0.383063i	$0.125131\pi$
$617$			19.0302i			0.766126i	−0.923722	+	0.383063i	$0.874869\pi$
$618$		0			0
$619$	15.1588			0.609285			0.304642	−	0.952467i	$-0.401463\pi$
$619$	15.1588			0.609285			0.304642	+	0.952467i	$0.401463\pi$
$620$		0			0
$621$	5.09648	−	25.3391i	0.204515	−	1.01682i
$622$		0			0
$623$	1.62493			0.0651015
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	13.1357	−	19.6287i	0.524589	−	0.783896i
$628$		0			0
$629$			9.85321i			0.392873i
$630$		0			0
$631$			6.69510i			0.266528i	0.991081	+	0.133264i	$0.0425457\pi$
$631$			6.69510i			0.266528i	−0.991081	+	0.133264i	$0.957454\pi$
$632$		0			0
$633$	19.8172	−	29.6129i	0.787663	−	1.17701i
$634$		0			0
$635$	−7.46508			−0.296242
$636$		0			0
$637$		−	1.30986i		−	0.0518985i
$638$		0			0
$639$	−0.210076	+	0.0866665i	−0.00831048	+	0.00342847i
$640$		0			0
$641$	24.2520			0.957898			0.478949	−	0.877843i	$-0.341018\pi$
$641$	24.2520			0.957898			0.478949	+	0.877843i	$0.341018\pi$
$642$		0			0
$643$	−39.3336			−1.55116			−0.775582	−	0.631246i	$-0.782544\pi$
$643$	−39.3336			−1.55116			−0.775582	+	0.631246i	$0.782544\pi$
$644$		0			0
$645$	12.6399	+	8.45873i	0.497696	+	0.333062i
$646$		0			0
$647$	38.8358			1.52679			0.763397	−	0.645929i	$-0.223530\pi$
$647$	38.8358			1.52679			0.763397	+	0.645929i	$0.223530\pi$
$648$		0			0
$649$		−	4.19394i		−	0.164626i
$650$		0			0
$651$	10.8285	−	16.1810i	0.424401	−	0.634184i
$652$		0			0
$653$	−18.2375			−0.713688			−0.356844	−	0.934164i	$-0.616147\pi$
$653$	−18.2375			−0.713688			−0.356844	+	0.934164i	$0.616147\pi$
$654$		0			0
$655$			4.56466i			0.178356i
$656$		0			0
$657$	14.7685	+	35.7983i	0.576176	+	1.39663i
$658$		0			0
$659$			6.69911i			0.260960i	0.991451	+	0.130480i	$0.0416519\pi$
$659$			6.69911i			0.260960i	−0.991451	+	0.130480i	$0.958348\pi$
$660$		0			0
$661$		−	27.0948i		−	1.05387i	−0.849907	−	0.526933i	$-0.823342\pi$
$661$		−	27.0948i		−	1.05387i	0.849907	−	0.526933i	$-0.176658\pi$
$662$		0			0
$663$	18.3411	−	27.4072i	0.712309	−	1.06441i
$664$		0			0
$665$			7.61291i			0.295216i
$666$		0			0
$667$	47.4541			1.83743
$668$		0			0
$669$	−8.22319	+	12.2880i	−0.317927	+	0.475080i
$670$		0			0
$671$			21.1010i			0.814593i
$672$		0			0
$673$		−	7.21091i		−	0.277960i	−0.990295	−	0.138980i	$-0.955618\pi$
$673$		−	7.21091i		−	0.277960i	0.990295	−	0.138980i	$-0.0443824\pi$
$674$		0			0
$675$	5.09414	+	1.02459i	0.196073	+	0.0394364i
$676$		0			0
$677$	−45.4629			−1.74728			−0.873640	−	0.486573i	$-0.838247\pi$
$677$	−45.4629			−1.74728			−0.873640	+	0.486573i	$0.838247\pi$
$678$		0			0
$679$	−17.7988			−0.683056
$680$		0			0
$681$	−1.72312	−	1.15313i	−0.0660302	−	0.0441879i
$682$		0			0
$683$	−14.6828			−0.561821			−0.280911	−	0.959734i	$-0.590636\pi$
$683$	−14.6828			−0.561821			−0.280911	+	0.959734i	$0.590636\pi$
$684$		0			0
$685$	−17.4073			−0.665097
$686$		0			0
$687$	−26.0688	−	17.4455i	−0.994588	−	0.665586i
$688$		0			0
$689$			20.9481i			0.798058i
$690$		0			0
$691$	−0.921162			−0.0350426			−0.0175213	−	0.999846i	$-0.505577\pi$
$691$	−0.921162			−0.0350426			−0.0175213	+	0.999846i	$0.505577\pi$
$692$		0			0
$693$	32.1644	−	13.2694i	1.22182	−	0.504062i
$694$		0			0
$695$			0.341211i			0.0129429i
$696$		0			0
$697$			34.1809i			1.29469i
$698$		0			0
$699$	19.8526	−	29.6659i	0.750896	−	1.12207i
$700$		0			0
$701$	34.8904			1.31779			0.658896	−	0.752234i	$-0.271024\pi$
$701$	34.8904			1.31779			0.658896	+	0.752234i	$0.271024\pi$
$702$		0			0
$703$	−3.86315			−0.145702
$704$		0			0
$705$	3.98646	+	2.66777i	0.150139	+	0.100474i
$706$		0			0
$707$			31.2732i			1.17615i
$708$		0			0
$709$	10.1784			0.382259			0.191129	−	0.981565i	$-0.438785\pi$
$709$	10.1784			0.382259			0.191129	+	0.981565i	$0.438785\pi$
$710$		0			0
$711$	−3.97524	+	1.63998i	−0.149083	+	0.0615040i
$712$		0			0
$713$	−21.9738			−0.822925
$714$		0			0
$715$		−	11.3726i		−	0.425312i
$716$		0			0
$717$	−21.8487	+	32.6486i	−0.815954	+	1.21928i
$718$		0			0
$719$			29.6813i			1.10693i	0.832874	+	0.553463i	$0.186694\pi$
$719$			29.6813i			1.10693i	−0.832874	+	0.553463i	$0.813306\pi$
$720$		0			0
$721$		−	35.7430i		−	1.33114i
$722$		0			0
$723$	11.2553	−	16.8189i	0.418591	−	0.625502i
$724$		0			0
$725$			9.54010i			0.354310i
$726$		0			0
$727$			1.55839i			0.0577974i	0.999582	+	0.0288987i	$0.00920002\pi$
$727$			1.55839i			0.0577974i	−0.999582	+	0.0288987i	$0.990800\pi$
$728$		0			0
$729$	24.9004	+	10.4388i	0.922238	+	0.386622i
$730$		0			0
$731$	67.0053			2.47828
$732$		0			0
$733$		−	15.1378i		−	0.559129i	−0.960127	−	0.279565i	$-0.909810\pi$
$733$		−	15.1378i		−	0.559129i	0.960127	−	0.279565i	$-0.0901901\pi$
$734$		0			0
$735$	0.505694	−	0.755660i	0.0186528	−	0.0278729i
$736$		0			0
$737$	−7.11085	−	36.6239i	−0.261932	−	1.34906i
$738$		0			0
$739$		−	38.1619i		−	1.40381i	−0.712271	−	0.701905i	$-0.752333\pi$
$739$		−	38.1619i		−	1.40381i	0.712271	−	0.701905i	$-0.247667\pi$
$740$		0			0
$741$	10.7455	+	7.19100i	0.394747	+	0.264168i
$742$		0			0
$743$			2.31373i			0.0848825i	0.999099	+	0.0424413i	$0.0135135\pi$
$743$			2.31373i			0.0848825i	−0.999099	+	0.0424413i	$0.986486\pi$
$744$		0			0
$745$			6.62577i			0.242749i
$746$		0			0
$747$	−37.5487	+	15.4907i	−1.37384	+	0.566774i
$748$		0			0
$749$	49.7403			1.81747
$750$		0			0
$751$	−35.3392			−1.28955			−0.644773	−	0.764374i	$-0.723048\pi$
$751$	−35.3392			−1.28955			−0.644773	+	0.764374i	$0.723048\pi$
$752$		0			0
$753$	3.71277	−	5.54801i	0.135301	−	0.202181i
$754$		0			0
$755$	1.61556			0.0587963
$756$		0			0
$757$		−	1.52824i		−	0.0555448i	−0.999614	−	0.0277724i	$-0.991159\pi$
$757$		−	1.52824i		−	0.0555448i	0.999614	−	0.0277724i	$-0.00884136\pi$
$758$		0			0
$759$	−32.6350	−	21.8396i	−1.18458	−	0.792727i
$760$		0			0
$761$		−	24.6700i		−	0.894287i	−0.894462	−	0.447144i	$-0.852441\pi$
$761$		−	24.6700i		−	0.894287i	0.894462	−	0.447144i	$-0.147559\pi$
$762$		0			0
$763$	38.4070			1.39043
$764$		0			0
$765$	21.1620	−	8.73036i	0.765115	−	0.315647i
$766$		0			0
$767$	2.29593			0.0829011
$768$		0			0
$769$		−	44.1558i		−	1.59230i	−0.605099	−	0.796150i	$-0.706867\pi$
$769$		−	44.1558i		−	1.59230i	0.605099	−	0.796150i	$-0.293133\pi$
$770$		0			0
$771$	3.29469	+	2.20483i	0.118655	+	0.0794050i
$772$		0			0
$773$			7.13798i			0.256735i	0.991727	+	0.128368i	$0.0409737\pi$
$773$			7.13798i			0.256735i	−0.991727	+	0.128368i	$0.959026\pi$
$774$		0			0
$775$		−	4.41758i		−	0.158684i
$776$		0			0
$777$	−4.72971	−	3.16516i	−0.169677	−	0.113549i
$778$		0			0
$779$	−13.4013			−0.480152
$780$		0			0
$781$			0.345261i			0.0123544i
$782$		0			0
$783$	−9.77468	+	48.5986i	−0.349318	+	1.73677i
$784$		0			0
$785$	−17.6280			−0.629171
$786$		0			0
$787$		−	49.4237i		−	1.76177i	−0.473334	−	0.880883i	$-0.656950\pi$
$787$		−	49.4237i		−	1.76177i	0.473334	−	0.880883i	$-0.343050\pi$
$788$		0			0
$789$	−22.5985	−	15.1231i	−0.804528	−	0.538397i
$790$		0			0
$791$			45.8289i			1.62949i
$792$		0			0
$793$	−11.5515			−0.410205
$794$		0			0
$795$	−8.08736	+	12.0850i	−0.286829	+	0.428610i
$796$		0			0
$797$			19.1157i			0.677111i	0.940946	+	0.338556i	$0.109938\pi$
$797$			19.1157i			0.677111i	−0.940946	+	0.338556i	$0.890062\pi$
$798$		0			0
$799$	21.1326			0.747617
$800$		0			0
$801$	1.77095	−	0.730601i	0.0625734	−	0.0258145i
$802$		0			0
$803$	58.8347			2.07623
$804$		0			0
$805$	12.6573			0.446112
$806$		0			0
$807$	−16.4782	−	11.0273i	−0.580061	−	0.388181i
$808$		0			0
$809$	−22.5176			−0.791677			−0.395839	−	0.918320i	$-0.629546\pi$
$809$	−22.5176			−0.791677			−0.395839	+	0.918320i	$0.629546\pi$
$810$		0			0
$811$			8.39730i			0.294869i	0.989072	+	0.147435i	$0.0471016\pi$
$811$			8.39730i			0.294869i	−0.989072	+	0.147435i	$0.952898\pi$
$812$		0			0
$813$	−0.140010	−	0.0936960i	−0.00491038	−	0.00328606i
$814$		0			0
$815$	−1.66222			−0.0582249
$816$		0			0
$817$			26.2708i			0.919098i
$818$		0			0
$819$	7.26418	+	17.6081i	0.253831	+	0.615276i
$820$		0			0
$821$			48.1301i			1.67975i	0.542779	+	0.839875i	$0.317372\pi$
$821$			48.1301i			1.67975i	−0.542779	+	0.839875i	$0.682628\pi$
$822$		0			0
$823$	−19.4609			−0.678366			−0.339183	−	0.940720i	$-0.610151\pi$
$823$	−19.4609			−0.678366			−0.339183	+	0.940720i	$0.610151\pi$
$824$		0			0
$825$	4.39060	−	6.56089i	0.152861	−	0.228421i
$826$		0			0
$827$			13.1534i			0.457387i	0.973498	+	0.228694i	$0.0734454\pi$
$827$			13.1534i			0.457387i	−0.973498	+	0.228694i	$0.926555\pi$
$828$		0			0
$829$	−29.8044			−1.03515			−0.517574	−	0.855638i	$-0.673165\pi$
$829$	−29.8044			−1.03515			−0.517574	+	0.855638i	$0.673165\pi$
$830$		0			0
$831$	29.5597	−	44.1711i	1.02541	−	1.53228i
$832$		0			0
$833$		−	4.00582i		−	0.138793i
$834$		0			0
$835$		−	10.8058i		−	0.373950i
$836$		0			0
$837$	4.52620	−	22.5037i	0.156448	−	0.777843i
$838$		0			0
$839$		−	54.5708i		−	1.88399i	−0.335624	−	0.941996i	$-0.608947\pi$
$839$		−	54.5708i		−	1.88399i	0.335624	−	0.941996i	$-0.391053\pi$
$840$		0			0
$841$	−62.0135			−2.13840
$842$		0			0
$843$	6.72465	−	10.0487i	0.231609	−	0.346095i
$844$		0			0
$845$	−6.77418			−0.233039
$846$		0			0
$847$		−	24.8716i		−	0.854598i
$848$		0			0
$849$	−2.20589	+	3.29627i	−0.0757059	+	0.113128i
$850$		0			0
$851$			6.42293i			0.220175i
$852$		0			0
$853$	4.56295			0.156233			0.0781163	−	0.996944i	$-0.475109\pi$
$853$	4.56295			0.156233			0.0781163	+	0.996944i	$0.475109\pi$
$854$		0			0
$855$	3.42292	+	8.29701i	0.117061	+	0.283752i
$856$		0			0
$857$	25.7659			0.880147			0.440073	−	0.897962i	$-0.354952\pi$
$857$	25.7659			0.880147			0.440073	+	0.897962i	$0.354952\pi$
$858$		0			0
$859$	−13.9590			−0.476274			−0.238137	−	0.971232i	$-0.576537\pi$
$859$	−13.9590			−0.476274			−0.238137	+	0.971232i	$0.576537\pi$
$860$		0			0
$861$	−16.4074	−	10.9799i	−0.559162	−	0.374196i
$862$		0			0
$863$			11.0156i			0.374977i	0.982267	+	0.187488i	$0.0600347\pi$
$863$			11.0156i			0.374977i	−0.982267	+	0.187488i	$0.939965\pi$
$864$		0			0
$865$			22.4010i			0.761657i
$866$		0			0
$867$	39.7148	−	59.3460i	1.34878	−	2.01549i
$868$		0			0
$869$			6.53332i			0.221628i
$870$		0			0
$871$	20.0494	−	3.89276i	0.679346	−	0.131901i
$872$		0			0
$873$	−19.3982	+	8.00271i	−0.656531	+	0.270851i
$874$		0			0
$875$			2.54461i			0.0860235i
$876$		0			0
$877$	−5.58218			−0.188497			−0.0942483	−	0.995549i	$-0.530045\pi$
$877$	−5.58218			−0.188497			−0.0942483	+	0.995549i	$0.530045\pi$
$878$		0			0
$879$	−33.9162	−	22.6970i	−1.14397	−	0.765551i
$880$		0			0
$881$		−	27.9580i		−	0.941929i	−0.882152	−	0.470965i	$-0.843906\pi$
$881$		−	27.9580i		−	0.941929i	0.882152	−	0.470965i	$-0.156094\pi$
$882$		0			0
$883$			18.3974i			0.619120i	0.950880	+	0.309560i	$0.100182\pi$
$883$			18.3974i			0.619120i	−0.950880	+	0.309560i	$0.899818\pi$
$884$		0			0
$885$	1.32452	+	0.886382i	0.0445234	+	0.0297954i
$886$		0			0
$887$		−	13.5380i		−	0.454562i	−0.973829	−	0.227281i	$-0.927016\pi$
$887$		−	13.5380i		−	0.454562i	0.973829	−	0.227281i	$-0.0729837\pi$
$888$		0			0
$889$		−	18.9957i		−	0.637095i
$890$		0			0
$891$	29.0885	−	28.9235i	0.974502	−	0.968974i
$892$		0			0
$893$			8.28546i			0.277262i
$894$		0			0
$895$	−3.80314			−0.127125
$896$		0			0
$897$	11.9559	−	17.8657i	0.399194	−	0.596518i
$898$		0			0
$899$	42.1441			1.40559
$900$		0			0
$901$			64.0635i			2.13427i
$902$		0			0
$903$	−21.5242	+	32.1637i	−0.716279	+	1.07034i
$904$		0			0
$905$	13.5774			0.451330
$906$		0			0
$907$	−34.9982			−1.16210			−0.581048	−	0.813870i	$-0.697357\pi$
$907$	−34.9982			−1.16210			−0.581048	+	0.813870i	$0.697357\pi$
$908$		0			0
$909$	14.0610	+	34.0834i	0.466375	+	1.13047i
$910$		0			0
$911$		−	6.27974i		−	0.208057i	−0.994574	−	0.104029i	$-0.966827\pi$
$911$		−	6.27974i		−	0.208057i	0.994574	−	0.104029i	$-0.0331733\pi$
$912$		0			0
$913$			61.7114i			2.04235i
$914$		0			0
$915$	−6.66408	−	4.45965i	−0.220308	−	0.147432i
$916$		0			0
$917$	−11.6153			−0.383571
$918$		0			0
$919$		−	33.5833i		−	1.10781i	−0.832580	−	0.553906i	$-0.813137\pi$
$919$		−	33.5833i		−	1.10781i	0.832580	−	0.553906i	$-0.186863\pi$
$920$		0			0
$921$	8.77361	−	13.1104i	0.289100	−	0.432004i
$922$		0			0
$923$	−0.189009			−0.00622131
$924$		0			0
$925$	−1.29126			−0.0424562
$926$		0			0
$927$	−16.0708	−	38.9549i	−0.527833	−	1.27945i
$928$		0			0
$929$	24.6563			0.808946			0.404473	−	0.914550i	$-0.367455\pi$
$929$	24.6563			0.808946			0.404473	+	0.914550i	$0.367455\pi$
$930$		0			0
$931$	1.57056			0.0514731
$932$		0			0
$933$	−0.834962	+	1.24769i	−0.0273354	+	0.0408475i
$934$		0			0
$935$		−	34.7799i		−	1.13742i
$936$		0			0
$937$		−	42.8569i		−	1.40007i	−0.714107	−	0.700036i	$-0.753167\pi$
$937$		−	42.8569i		−	1.40007i	0.714107	−	0.700036i	$-0.246833\pi$
$938$		0			0
$939$	−27.3828	−	18.3248i	−0.893603	−	0.598006i
$940$		0			0
$941$	−20.9013			−0.681363			−0.340682	−	0.940179i	$-0.610658\pi$
$941$	−20.9013			−0.681363			−0.340682	+	0.940179i	$0.610658\pi$
$942$		0			0
$943$			22.2812i			0.725575i
$944$		0			0
$945$	−2.60718	+	12.9626i	−0.0848115	+	0.421673i
$946$		0			0
$947$			18.5716i			0.603495i	0.953388	+	0.301748i	$0.0975700\pi$
$947$			18.5716i			0.603495i	−0.953388	+	0.301748i	$0.902430\pi$
$948$		0			0
$949$			32.2084i			1.04553i
$950$		0			0
$951$	43.1998	+	28.9097i	1.40085	+	0.937460i
$952$		0			0
$953$		−	18.2434i		−	0.590961i	−0.955349	−	0.295480i	$-0.904520\pi$
$953$		−	18.2434i		−	0.590961i	0.955349	−	0.295480i	$-0.0954797\pi$
$954$		0			0
$955$	6.07318			0.196523
$956$		0			0
$957$	62.5916	+	41.8868i	2.02330	+	1.35401i
$958$		0			0
$959$		−	44.2947i		−	1.43035i
$960$		0			0
$961$	11.4850			0.370484
$962$		0			0
$963$	54.2099	−	22.3642i	1.74689	−	0.720677i
$964$		0			0
$965$	−21.5678			−0.694293
$966$		0			0
$967$	−49.2898			−1.58505			−0.792527	−	0.609837i	$-0.791235\pi$
$967$	−49.2898			−1.58505			−0.792527	+	0.609837i	$0.791235\pi$
$968$		0			0
$969$	32.8621	+	21.9916i	1.05568	+	0.706471i
$970$		0			0
$971$			12.3496i			0.396318i	0.980170	+	0.198159i	$0.0634962\pi$
$971$			12.3496i			0.396318i	−0.980170	+	0.198159i	$0.936504\pi$
$972$		0			0
$973$	−0.868248			−0.0278348
$974$		0			0
$975$	3.59169	+	2.40359i	0.115026	+	0.0769764i
$976$		0			0
$977$			8.79549i			0.281393i	0.990053	+	0.140696i	$0.0449341\pi$
$977$			8.79549i			0.281393i	−0.990053	+	0.140696i	$0.955066\pi$
$978$		0			0
$979$		−	2.91056i		−	0.0930218i
$980$		0			0
$981$	41.8583	−	17.2686i	1.33643	−	0.551342i
$982$		0			0
$983$	−14.8763			−0.474482			−0.237241	−	0.971451i	$-0.576243\pi$
$983$	−14.8763			−0.474482			−0.237241	+	0.971451i	$0.576243\pi$
$984$		0			0
$985$	−13.9321			−0.443913
$986$		0			0
$987$	−6.78844	+	10.1440i	−0.216078	+	0.322887i
$988$		0			0
$989$	43.6782			1.38888
$990$		0			0
$991$			17.1324i			0.544229i	0.962265	+	0.272114i	$0.0877229\pi$
$991$			17.1324i			0.544229i	−0.962265	+	0.272114i	$0.912277\pi$
$992$		0			0
$993$	29.3004	+	19.6081i	0.929820	+	0.622243i
$994$		0			0
$995$	24.6912			0.782764
$996$		0			0
$997$	−10.0126			−0.317102			−0.158551	−	0.987351i	$-0.550682\pi$
$997$	−10.0126			−0.317102			−0.158551	+	0.987351i	$0.550682\pi$
$998$		0			0
$999$	−6.57783	−	1.32301i	−0.208113	−	0.0418581i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.16	yes	46
3.2	odd	2		4020.2.f.b.401.32	yes	46
67.66	odd	2		4020.2.f.b.401.31	yes	46
201.200	even	2	inner	4020.2.f.a.401.15	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.15	✓	46	201.200	even	2	inner
4020.2.f.a.401.16	yes	46	1.1	even	1	trivial
4020.2.f.b.401.31	yes	46	67.66	odd	2
4020.2.f.b.401.32	yes	46	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.963300 + 1.43946i) q^{3} -1.00000 q^{5} -2.54461i q^{7} +(-1.14411 - 2.77327i) q^{9} +O(q^{10})\)	\(q+(-0.963300 + 1.43946i) q^{3} -1.00000 q^{5} -2.54461i q^{7} +(-1.14411 - 2.77327i) q^{9} -4.55788 q^{11} -2.49516i q^{13} +(0.963300 - 1.43946i) q^{15} -7.63072i q^{17} +2.99178 q^{19} +(3.66287 + 2.45122i) q^{21} -4.97417i q^{23} +1.00000 q^{25} +(5.09414 + 1.02459i) q^{27} +9.54010i q^{29} -4.41758i q^{31} +(4.39060 - 6.56089i) q^{33} +2.54461i q^{35} -1.29126 q^{37} +(3.59169 + 2.40359i) q^{39} -4.47938 q^{41} +8.78099i q^{43} +(1.14411 + 2.77327i) q^{45} +2.76941i q^{47} +0.524960 q^{49} +(10.9841 + 7.35067i) q^{51} -8.39548 q^{53} +4.55788 q^{55} +(-2.88198 + 4.30655i) q^{57} +0.920152i q^{59} -4.62956i q^{61} +(-7.05689 + 2.91131i) q^{63} +2.49516i q^{65} +(1.56012 + 8.03530i) q^{67} +(7.16013 + 4.79162i) q^{69} -0.0757503i q^{71} -12.9084 q^{73} +(-0.963300 + 1.43946i) q^{75} +11.5980i q^{77} -1.43341i q^{79} +(-6.38204 + 6.34583i) q^{81} -13.5395i q^{83} +7.63072i q^{85} +(-13.7326 - 9.18997i) q^{87} +0.638578i q^{89} -6.34921 q^{91} +(6.35894 + 4.25545i) q^{93} -2.99178 q^{95} -6.99472i q^{97} +(5.21470 + 12.6402i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 46 * q - 46 * q^5 - 4 * q^9	\( 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) \) 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Introduction

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