LMFDB - Embedded newform 4020.2.f.a.401.6

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			401.6
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.65289 + 0.517632i) q^{3} -1.00000 q^{5} -0.387683i q^{7} +(2.46411 - 1.71118i) q^{9} +O(q^{10})$	\(q+(-1.65289 + 0.517632i) q^{3} -1.00000 q^{5} -0.387683i q^{7} +(2.46411 - 1.71118i) q^{9} -4.30349 q^{11} -3.73654i q^{13} +(1.65289 - 0.517632i) q^{15} +3.85421i q^{17} -5.95792 q^{19} +(0.200677 + 0.640798i) q^{21} +4.48285i q^{23} +1.00000 q^{25} +(-3.18716 + 4.10391i) q^{27} +7.61818i q^{29} -3.91096i q^{31} +(7.11322 - 2.22763i) q^{33} +0.387683i q^{35} -1.55292 q^{37} +(1.93415 + 6.17611i) q^{39} -0.142673 q^{41} -8.29503i q^{43} +(-2.46411 + 1.71118i) q^{45} +0.257298i q^{47} +6.84970 q^{49} +(-1.99506 - 6.37059i) q^{51} -5.19804 q^{53} +4.30349 q^{55} +(9.84781 - 3.08401i) q^{57} -3.12064i q^{59} +8.47069i q^{61} +(-0.663395 - 0.955294i) q^{63} +3.73654i q^{65} +(-7.70770 - 2.75526i) q^{67} +(-2.32047 - 7.40967i) q^{69} +0.503657i q^{71} +5.39528 q^{73} +(-1.65289 + 0.517632i) q^{75} +1.66839i q^{77} -2.98877i q^{79} +(3.14371 - 8.43309i) q^{81} -5.05155i q^{83} -3.85421i q^{85} +(-3.94342 - 12.5920i) q^{87} -12.6057i q^{89} -1.44859 q^{91} +(2.02444 + 6.46440i) q^{93} +5.95792 q^{95} +3.95193i q^{97} +(-10.6043 + 7.36406i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.65289	+	0.517632i	−0.954299	+	0.298855i
$3$	−1.65289	+	0.517632i	−0.954299	+	0.298855i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	0.387683i		−	0.146530i	−0.997313	−	0.0732651i	$-0.976658\pi$
$7$		−	0.387683i		−	0.146530i	0.997313	−	0.0732651i	$-0.0233419\pi$
$8$		0			0
$9$	2.46411	−	1.71118i	0.821371	−	0.570394i
$10$		0			0
$11$	−4.30349			−1.29755			−0.648776	−	0.760979i	$-0.724719\pi$
$11$	−4.30349			−1.29755			−0.648776	+	0.760979i	$0.724719\pi$
$12$		0			0
$13$		−	3.73654i		−	1.03633i	−0.855281	−	0.518165i	$-0.826615\pi$
$13$		−	3.73654i		−	1.03633i	0.855281	−	0.518165i	$-0.173385\pi$
$14$		0			0
$15$	1.65289	−	0.517632i	0.426775	−	0.133652i
$16$		0			0
$17$			3.85421i			0.934783i	0.884051	+	0.467391i	$0.154806\pi$
$17$			3.85421i			0.934783i	−0.884051	+	0.467391i	$0.845194\pi$
$18$		0			0
$19$	−5.95792			−1.36684			−0.683421	−	0.730025i	$-0.739509\pi$
$19$	−5.95792			−1.36684			−0.683421	+	0.730025i	$0.739509\pi$
$20$		0			0
$21$	0.200677	+	0.640798i	0.0437913	+	0.139834i
$22$		0			0
$23$			4.48285i			0.934738i	0.884062	+	0.467369i	$0.154798\pi$
$23$			4.48285i			0.934738i	−0.884062	+	0.467369i	$0.845202\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−3.18716	+	4.10391i	−0.613368	+	0.789797i
$28$		0			0
$29$			7.61818i			1.41466i	0.706883	+	0.707330i	$0.250101\pi$
$29$			7.61818i			1.41466i	−0.706883	+	0.707330i	$0.749899\pi$
$30$		0			0
$31$		−	3.91096i		−	0.702429i	−0.936295	−	0.351214i	$-0.885769\pi$
$31$		−	3.91096i		−	0.702429i	0.936295	−	0.351214i	$-0.114231\pi$
$32$		0			0
$33$	7.11322	−	2.22763i	1.23825	−	0.387780i
$34$		0			0
$35$			0.387683i			0.0655303i
$36$		0			0
$37$	−1.55292			−0.255298			−0.127649	−	0.991819i	$-0.540743\pi$
$37$	−1.55292			−0.255298			−0.127649	+	0.991819i	$0.540743\pi$
$38$		0			0
$39$	1.93415	+	6.17611i	0.309713	+	0.988969i
$40$		0			0
$41$	−0.142673			−0.0222818			−0.0111409	−	0.999938i	$-0.503546\pi$
$41$	−0.142673			−0.0222818			−0.0111409	+	0.999938i	$0.503546\pi$
$42$		0			0
$43$		−	8.29503i		−	1.26498i	−0.774568	−	0.632490i	$-0.782033\pi$
$43$		−	8.29503i		−	1.26498i	0.774568	−	0.632490i	$-0.217967\pi$
$44$		0			0
$45$	−2.46411	+	1.71118i	−0.367328	+	0.255088i
$46$		0			0
$47$			0.257298i			0.0375307i	0.999824	+	0.0187654i	$0.00597355\pi$
$47$			0.257298i			0.0375307i	−0.999824	+	0.0187654i	$0.994026\pi$
$48$		0			0
$49$	6.84970			0.978529
$50$		0			0
$51$	−1.99506	−	6.37059i	−0.279364	−	0.892062i
$52$		0			0
$53$	−5.19804			−0.714006			−0.357003	−	0.934103i	$-0.616201\pi$
$53$	−5.19804			−0.714006			−0.357003	+	0.934103i	$0.616201\pi$
$54$		0			0
$55$	4.30349			0.580283
$56$		0			0
$57$	9.84781	−	3.08401i	1.30437	−	0.408488i
$58$		0			0
$59$		−	3.12064i		−	0.406272i	−0.979151	−	0.203136i	$-0.934887\pi$
$59$		−	3.12064i		−	0.406272i	0.979151	−	0.203136i	$-0.0651134\pi$
$60$		0			0
$61$			8.47069i			1.08456i	0.840198	+	0.542280i	$0.182439\pi$
$61$			8.47069i			1.08456i	−0.840198	+	0.542280i	$0.817561\pi$
$62$		0			0
$63$	−0.663395	−	0.955294i	−0.0835799	−	0.120356i
$64$		0			0
$65$			3.73654i			0.463461i
$66$		0			0
$67$	−7.70770	−	2.75526i	−0.941645	−	0.336608i
$67$	−7.70770	−	2.75526i	−0.941645	−	0.336608i
$68$		0			0
$69$	−2.32047	−	7.40967i	−0.279351	−	0.892019i
$70$		0			0
$71$			0.503657i			0.0597731i	0.999553	+	0.0298866i	$0.00951460\pi$
$71$			0.503657i			0.0597731i	−0.999553	+	0.0298866i	$0.990485\pi$
$72$		0			0
$73$	5.39528			0.631470			0.315735	−	0.948847i	$-0.397749\pi$
$73$	5.39528			0.631470			0.315735	+	0.948847i	$0.397749\pi$
$74$		0			0
$75$	−1.65289	+	0.517632i	−0.190860	+	0.0597710i
$76$		0			0
$77$			1.66839i			0.190131i
$78$		0			0
$79$		−	2.98877i		−	0.336262i	−0.985765	−	0.168131i	$-0.946227\pi$
$79$		−	2.98877i		−	0.336262i	0.985765	−	0.168131i	$-0.0537732\pi$
$80$		0			0
$81$	3.14371	−	8.43309i	0.349302	−	0.937010i
$82$		0			0
$83$		−	5.05155i		−	0.554479i	−0.960801	−	0.277240i	$-0.910580\pi$
$83$		−	5.05155i		−	0.554479i	0.960801	−	0.277240i	$-0.0894196\pi$
$84$		0			0
$85$		−	3.85421i		−	0.418047i
$86$		0			0
$87$	−3.94342	−	12.5920i	−0.422779	−	1.35001i
$88$		0			0
$89$		−	12.6057i		−	1.33620i	−0.744069	−	0.668102i	$-0.767107\pi$
$89$		−	12.6057i		−	1.33620i	0.744069	−	0.668102i	$-0.232893\pi$
$90$		0			0
$91$	−1.44859			−0.151854
$92$		0			0
$93$	2.02444	+	6.46440i	0.209924	+	0.670327i
$94$		0			0
$95$	5.95792			0.611270
$96$		0			0
$97$			3.95193i			0.401258i	0.979667	+	0.200629i	$0.0642985\pi$
$97$			3.95193i			0.401258i	−0.979667	+	0.200629i	$0.935701\pi$
$98$		0			0
$99$	−10.6043	+	7.36406i	−1.06577	+	0.740116i
$100$		0			0
$101$	15.1867			1.51113			0.755567	−	0.655072i	$-0.227362\pi$
$101$	15.1867			1.51113			0.755567	+	0.655072i	$0.227362\pi$
$102$		0			0
$103$	−7.13098			−0.702637			−0.351318	−	0.936256i	$-0.614266\pi$
$103$	−7.13098			−0.702637			−0.351318	+	0.936256i	$0.614266\pi$
$104$		0			0
$105$	−0.200677	−	0.640798i	−0.0195841	−	0.0625355i
$106$		0			0
$107$		−	9.47371i		−	0.915858i	−0.888989	−	0.457929i	$-0.848591\pi$
$107$		−	9.47371i		−	0.915858i	0.888989	−	0.457929i	$-0.151409\pi$
$108$		0			0
$109$			6.98600i			0.669137i	0.942371	+	0.334569i	$0.108591\pi$
$109$			6.98600i			0.669137i	−0.942371	+	0.334569i	$0.891409\pi$
$110$		0			0
$111$	2.56680	−	0.803839i	0.243630	−	0.0762970i
$112$		0			0
$113$	−1.40484			−0.132157			−0.0660784	−	0.997814i	$-0.521049\pi$
$113$	−1.40484			−0.132157			−0.0660784	+	0.997814i	$0.521049\pi$
$114$		0			0
$115$		−	4.48285i		−	0.418028i
$116$		0			0
$117$	−6.39390	−	9.20727i	−0.591117	−	0.851212i
$118$		0			0
$119$	1.49421			0.136974
$120$		0			0
$121$	7.52006			0.683642
$122$		0			0
$123$	0.235824	−	0.0738523i	0.0212635	−	0.00665904i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	3.29690			0.292552			0.146276	−	0.989244i	$-0.453271\pi$
$127$	3.29690			0.292552			0.146276	+	0.989244i	$0.453271\pi$
$128$		0			0
$129$	4.29378	+	13.7108i	0.378046	+	1.20717i
$130$		0			0
$131$			4.44230i			0.388125i	0.980989	+	0.194063i	$0.0621665\pi$
$131$			4.44230i			0.388125i	−0.980989	+	0.194063i	$0.937834\pi$
$132$		0			0
$133$			2.30978i			0.200284i
$134$		0			0
$135$	3.18716	−	4.10391i	0.274307	−	0.353208i
$136$		0			0
$137$	9.42862			0.805542			0.402771	−	0.915301i	$-0.368047\pi$
$137$	9.42862			0.805542			0.402771	+	0.915301i	$0.368047\pi$
$138$		0			0
$139$			21.3537i			1.81120i	0.424135	+	0.905599i	$0.360578\pi$
$139$			21.3537i			1.81120i	−0.424135	+	0.905599i	$0.639422\pi$
$140$		0			0
$141$	−0.133186	−	0.425286i	−0.0112162	−	0.0358155i
$142$		0			0
$143$			16.0802i			1.34469i
$144$		0			0
$145$		−	7.61818i		−	0.632656i
$146$		0			0
$147$	−11.3218	+	3.54563i	−0.933809	+	0.292438i
$148$		0			0
$149$		−	17.9261i		−	1.46856i	−0.678846	−	0.734281i	$-0.737520\pi$
$149$		−	17.9261i		−	1.46856i	0.678846	−	0.734281i	$-0.262480\pi$
$150$		0			0
$151$	20.2577			1.64855			0.824274	−	0.566191i	$-0.191583\pi$
$151$	20.2577			1.64855			0.824274	+	0.566191i	$0.191583\pi$
$152$		0			0
$153$	6.59525	+	9.49721i	0.533194	+	0.767804i
$154$		0			0
$155$			3.91096i			0.314136i
$156$		0			0
$157$	−9.57333			−0.764035			−0.382017	−	0.924155i	$-0.624771\pi$
$157$	−9.57333			−0.764035			−0.382017	+	0.924155i	$0.624771\pi$
$158$		0			0
$159$	8.59181	−	2.69067i	0.681375	−	0.213384i
$160$		0			0
$161$	1.73792			0.136967
$162$		0			0
$163$	20.9735			1.64277			0.821384	−	0.570376i	$-0.193202\pi$
$163$	20.9735			1.64277			0.821384	+	0.570376i	$0.193202\pi$
$164$		0			0
$165$	−7.11322	+	2.22763i	−0.553763	+	0.173421i
$166$		0			0
$167$		−	8.69008i		−	0.672459i	−0.941780	−	0.336229i	$-0.890848\pi$
$167$		−	8.69008i		−	0.672459i	0.941780	−	0.336229i	$-0.109152\pi$
$168$		0			0
$169$	−0.961752			−0.0739809
$170$		0			0
$171$	−14.6810	+	10.1951i	−1.12268	+	0.779638i
$172$		0			0
$173$			23.2776i			1.76976i	0.465816	+	0.884882i	$0.345761\pi$
$173$			23.2776i			1.76976i	−0.465816	+	0.884882i	$0.654239\pi$
$174$		0			0
$175$		−	0.387683i		−	0.0293060i
$176$		0			0
$177$	1.61534	+	5.15808i	0.121417	+	0.387705i
$178$		0			0
$179$	16.0590			1.20031			0.600154	−	0.799885i	$-0.295106\pi$
$179$	16.0590			1.20031			0.600154	+	0.799885i	$0.295106\pi$
$180$		0			0
$181$	20.5687			1.52886			0.764428	−	0.644709i	$-0.223021\pi$
$181$	20.5687			1.52886			0.764428	+	0.644709i	$0.223021\pi$
$182$		0			0
$183$	−4.38470	−	14.0011i	−0.324126	−	1.03499i
$184$		0			0
$185$	1.55292			0.114173
$186$		0			0
$187$		−	16.5866i		−	1.21293i
$188$		0			0
$189$	1.59101	+	1.23560i	0.115729	+	0.0898770i
$190$		0			0
$191$	0.229298			0.0165914			0.00829572	−	0.999966i	$-0.497359\pi$
$191$	0.229298			0.0165914			0.00829572	+	0.999966i	$0.497359\pi$
$192$		0			0
$193$	13.5060			0.972184			0.486092	−	0.873908i	$-0.338422\pi$
$193$	13.5060			0.972184			0.486092	+	0.873908i	$0.338422\pi$
$194$		0			0
$195$	−1.93415	−	6.17611i	−0.138508	−	0.442280i
$196$		0			0
$197$	−15.2548			−1.08686			−0.543429	−	0.839455i	$-0.682874\pi$
$197$	−15.2548			−1.08686			−0.543429	+	0.839455i	$0.682874\pi$
$198$		0			0
$199$	−6.16700			−0.437167			−0.218583	−	0.975818i	$-0.570144\pi$
$199$	−6.16700			−0.437167			−0.218583	+	0.975818i	$0.570144\pi$
$200$		0			0
$201$	14.1662	+	0.564395i	0.999207	+	0.0398093i
$202$		0			0
$203$	2.95344			0.207291
$204$		0			0
$205$	0.142673			0.00996474
$206$		0			0
$207$	7.67097	+	11.0462i	0.533169	+	0.767767i
$208$		0			0
$209$	25.6399			1.77355
$210$		0			0
$211$	15.1919			1.04586			0.522928	−	0.852377i	$-0.324840\pi$
$211$	15.1919			1.04586			0.522928	+	0.852377i	$0.324840\pi$
$212$		0			0
$213$	−0.260709	−	0.832492i	−0.0178635	−	0.0570414i
$214$		0			0
$215$			8.29503i			0.565717i
$216$		0			0
$217$	−1.51621			−0.102927
$218$		0			0
$219$	−8.91783	+	2.79277i	−0.602611	+	0.188718i
$220$		0			0
$221$	14.4014			0.968744
$222$		0			0
$223$	10.0142			0.670600			0.335300	−	0.942111i	$-0.391162\pi$
$223$	10.0142			0.670600			0.335300	+	0.942111i	$0.391162\pi$
$224$		0			0
$225$	2.46411	−	1.71118i	0.164274	−	0.114079i
$226$		0			0
$227$			4.27404i			0.283678i	0.989890	+	0.141839i	$0.0453015\pi$
$227$			4.27404i			0.283678i	−0.989890	+	0.141839i	$0.954698\pi$
$228$		0			0
$229$		−	22.8795i		−	1.51192i	−0.654618	−	0.755960i	$-0.727171\pi$
$229$		−	22.8795i		−	1.51192i	0.654618	−	0.755960i	$-0.272829\pi$
$230$		0			0
$231$	−0.863612	−	2.75767i	−0.0568215	−	0.181441i
$232$		0			0
$233$	−14.2579			−0.934068			−0.467034	−	0.884239i	$-0.654677\pi$
$233$	−14.2579			−0.934068			−0.467034	+	0.884239i	$0.654677\pi$
$234$		0			0
$235$		−	0.257298i		−	0.0167843i
$236$		0			0
$237$	1.54708	+	4.94011i	0.100494	+	0.320895i
$238$		0			0
$239$	1.78395			0.115394			0.0576971	−	0.998334i	$-0.481624\pi$
$239$	1.78395			0.115394			0.0576971	+	0.998334i	$0.481624\pi$
$240$		0			0
$241$	12.2311			0.787876			0.393938	−	0.919137i	$-0.371112\pi$
$241$	12.2311			0.787876			0.393938	+	0.919137i	$0.371112\pi$
$242$		0			0
$243$	−0.830986	+	15.5663i	−0.0533078	+	0.998578i
$244$		0			0
$245$	−6.84970			−0.437611
$246$		0			0
$247$			22.2620i			1.41650i
$248$		0			0
$249$	2.61484	+	8.34967i	0.165709	+	0.529139i
$250$		0			0
$251$	7.41958			0.468320			0.234160	−	0.972198i	$-0.424766\pi$
$251$	7.41958			0.468320			0.234160	+	0.972198i	$0.424766\pi$
$252$		0			0
$253$		−	19.2919i		−	1.21287i
$254$		0			0
$255$	1.99506	+	6.37059i	0.124936	+	0.398942i
$256$		0			0
$257$			11.5162i			0.718363i	0.933268	+	0.359182i	$0.116944\pi$
$257$			11.5162i			0.718363i	−0.933268	+	0.359182i	$0.883056\pi$
$258$		0			0
$259$			0.602038i			0.0374088i
$260$		0			0
$261$	13.0361	+	18.7721i	0.806914	+	1.16196i
$262$		0			0
$263$		−	25.0272i		−	1.54324i	−0.636083	−	0.771621i	$-0.719446\pi$
$263$		−	25.0272i		−	1.54324i	0.636083	−	0.771621i	$-0.280554\pi$
$264$		0			0
$265$	5.19804			0.319313
$266$		0			0
$267$	6.52513	+	20.8359i	0.399332	+	1.27514i
$268$		0			0
$269$		−	17.6467i		−	1.07594i	−0.842965	−	0.537968i	$-0.819192\pi$
$269$		−	17.6467i		−	1.07594i	0.842965	−	0.537968i	$-0.180808\pi$
$270$		0			0
$271$		−	17.8696i		−	1.08550i	−0.839893	−	0.542752i	$-0.817382\pi$
$271$		−	17.8696i		−	1.08550i	0.839893	−	0.542752i	$-0.182618\pi$
$272$		0			0
$273$	2.39437	−	0.749838i	0.144914	−	0.0453823i
$274$		0			0
$275$	−4.30349			−0.259510
$276$		0			0
$277$	−4.13513			−0.248456			−0.124228	−	0.992254i	$-0.539645\pi$
$277$	−4.13513			−0.248456			−0.124228	+	0.992254i	$0.539645\pi$
$278$		0			0
$279$	−6.69236	−	9.63705i	−0.400661	−	0.576955i
$280$		0			0
$281$	21.1023			1.25886			0.629429	−	0.777058i	$-0.283289\pi$
$281$	21.1023			1.25886			0.629429	+	0.777058i	$0.283289\pi$
$282$		0			0
$283$	−22.8482			−1.35818			−0.679092	−	0.734053i	$-0.737626\pi$
$283$	−22.8482			−1.35818			−0.679092	+	0.734053i	$0.737626\pi$
$284$		0			0
$285$	−9.84781	+	3.08401i	−0.583334	+	0.182681i
$286$		0			0
$287$			0.0553120i			0.00326496i
$288$		0			0
$289$	2.14509			0.126182
$290$		0			0
$291$	−2.04565	−	6.53212i	−0.119918	−	0.382920i
$292$		0			0
$293$		−	3.68886i		−	0.215505i	−0.994178	−	0.107753i	$-0.965635\pi$
$293$		−	3.68886i		−	0.215505i	0.994178	−	0.107753i	$-0.0343655\pi$
$294$		0			0
$295$			3.12064i			0.181690i
$296$		0			0
$297$	13.7159	−	17.6611i	0.795877	−	1.02480i
$298$		0			0
$299$	16.7504			0.968698
$300$		0			0
$301$	−3.21584			−0.185358
$302$		0			0
$303$	−25.1020	+	7.86113i	−1.44207	+	0.451610i
$304$		0			0
$305$		−	8.47069i		−	0.485030i
$306$		0			0
$307$	7.96221			0.454427			0.227214	−	0.973845i	$-0.427038\pi$
$307$	7.96221			0.454427			0.227214	+	0.973845i	$0.427038\pi$
$308$		0			0
$309$	11.7868	−	3.69123i	0.670525	−	0.209986i
$310$		0			0
$311$	10.5121			0.596089			0.298045	−	0.954552i	$-0.403666\pi$
$311$	10.5121			0.596089			0.298045	+	0.954552i	$0.403666\pi$
$312$		0			0
$313$			26.7401i			1.51144i	0.654895	+	0.755720i	$0.272713\pi$
$313$			26.7401i			1.51144i	−0.654895	+	0.755720i	$0.727287\pi$
$314$		0			0
$315$	0.663395	+	0.955294i	0.0373781	+	0.0538247i
$316$		0			0
$317$		−	21.5689i		−	1.21143i	−0.795681	−	0.605715i	$-0.792887\pi$
$317$		−	21.5689i		−	1.21143i	0.795681	−	0.605715i	$-0.207113\pi$
$318$		0			0
$319$		−	32.7848i		−	1.83560i
$320$		0			0
$321$	4.90390	+	15.6590i	0.273709	+	0.874002i
$322$		0			0
$323$		−	22.9631i		−	1.27770i
$324$		0			0
$325$		−	3.73654i		−	0.207266i
$326$		0			0
$327$	−3.61618	−	11.5471i	−0.199975	−	0.638557i
$328$		0			0
$329$	0.0997498			0.00549939
$330$		0			0
$331$			9.13197i			0.501938i	0.967995	+	0.250969i	$0.0807493\pi$
$331$			9.13197i			0.501938i	−0.967995	+	0.250969i	$0.919251\pi$
$332$		0			0
$333$	−3.82656	+	2.65732i	−0.209694	+	0.145620i
$334$		0			0
$335$	7.70770	+	2.75526i	0.421116	+	0.150536i
$336$		0			0
$337$			23.4951i			1.27986i	0.768434	+	0.639929i	$0.221036\pi$
$337$			23.4951i			1.27986i	−0.768434	+	0.639929i	$0.778964\pi$
$338$		0			0
$339$	2.32206	−	0.727193i	0.126117	−	0.0394957i
$340$		0			0
$341$			16.8308i			0.911438i
$342$		0			0
$343$		−	5.36929i		−	0.289914i
$344$		0			0
$345$	2.32047	+	7.40967i	0.124930	+	0.398923i
$346$		0			0
$347$	34.6502			1.86012			0.930059	−	0.367409i	$-0.119755\pi$
$347$	34.6502			1.86012			0.930059	+	0.367409i	$0.119755\pi$
$348$		0			0
$349$	11.4239			0.611510			0.305755	−	0.952110i	$-0.401091\pi$
$349$	11.4239			0.611510			0.305755	+	0.952110i	$0.401091\pi$
$350$		0			0
$351$	15.3344	+	11.9089i	0.818491	+	0.635652i
$352$		0			0
$353$	−12.2562			−0.652331			−0.326165	−	0.945313i	$-0.605757\pi$
$353$	−12.2562			−0.652331			−0.326165	+	0.945313i	$0.605757\pi$
$354$		0			0
$355$		−	0.503657i		−	0.0267314i
$356$		0			0
$357$	−2.46977	+	0.773451i	−0.130714	+	0.0409353i
$358$		0			0
$359$			14.7170i			0.776732i	0.921505	+	0.388366i	$0.126960\pi$
$359$			14.7170i			0.776732i	−0.921505	+	0.388366i	$0.873040\pi$
$360$		0			0
$361$	16.4969			0.868256
$362$		0			0
$363$	−12.4299	+	3.89263i	−0.652398	+	0.204310i
$364$		0			0
$365$	−5.39528			−0.282402
$366$		0			0
$367$			7.91646i			0.413236i	0.978422	+	0.206618i	$0.0662457\pi$
$367$			7.91646i			0.413236i	−0.978422	+	0.206618i	$0.933754\pi$
$368$		0			0
$369$	−0.351563	+	0.244140i	−0.0183017	+	0.0127094i
$370$		0			0
$371$			2.01519i			0.104623i
$372$		0			0
$373$		−	11.1052i		−	0.575008i	−0.957779	−	0.287504i	$-0.907175\pi$
$373$		−	11.1052i		−	0.575008i	0.957779	−	0.287504i	$-0.0928254\pi$
$374$		0			0
$375$	1.65289	−	0.517632i	0.0853551	−	0.0267304i
$376$		0			0
$377$	28.4657			1.46606
$378$		0			0
$379$			30.8624i			1.58529i	0.609681	+	0.792647i	$0.291297\pi$
$379$			30.8624i			1.58529i	−0.609681	+	0.792647i	$0.708703\pi$
$380$		0			0
$381$	−5.44942	+	1.70658i	−0.279182	+	0.0874307i
$382$		0			0
$383$	−11.9853			−0.612420			−0.306210	−	0.951964i	$-0.599061\pi$
$383$	−11.9853			−0.612420			−0.306210	+	0.951964i	$0.599061\pi$
$384$		0			0
$385$		−	1.66839i		−	0.0850290i
$386$		0			0
$387$	−14.1943	−	20.4399i	−0.721537	−	1.03902i
$388$		0			0
$389$		−	0.322831i		−	0.0163682i	−0.999967	−	0.00818409i	$-0.997395\pi$
$389$		−	0.322831i		−	0.0163682i	0.999967	−	0.00818409i	$-0.00260511\pi$
$390$		0			0
$391$	−17.2778			−0.873777
$392$		0			0
$393$	−2.29948	−	7.34264i	−0.115993	−	0.370387i
$394$		0			0
$395$			2.98877i			0.150381i
$396$		0			0
$397$	28.1682			1.41372			0.706860	−	0.707354i	$-0.250111\pi$
$397$	28.1682			1.41372			0.706860	+	0.707354i	$0.250111\pi$
$398$		0			0
$399$	−1.19562	−	3.81783i	−0.0598558	−	0.191130i
$400$		0			0
$401$	4.17860			0.208669			0.104335	−	0.994542i	$-0.466729\pi$
$401$	4.17860			0.208669			0.104335	+	0.994542i	$0.466729\pi$
$402$		0			0
$403$	−14.6135			−0.727949
$404$		0			0
$405$	−3.14371	+	8.43309i	−0.156212	+	0.419044i
$406$		0			0
$407$	6.68296			0.331262
$408$		0			0
$409$			10.0429i			0.496590i	0.968684	+	0.248295i	$0.0798702\pi$
$409$			10.0429i			0.496590i	−0.968684	+	0.248295i	$0.920130\pi$
$410$		0			0
$411$	−15.5845	+	4.88056i	−0.768727	+	0.240740i
$412$		0			0
$413$	−1.20982			−0.0595312
$414$		0			0
$415$			5.05155i			0.247971i
$416$		0			0
$417$	−11.0534	−	35.2954i	−0.541286	−	1.72842i
$418$		0			0
$419$		−	1.95532i		−	0.0955236i	−0.998859	−	0.0477618i	$-0.984791\pi$
$419$		−	1.95532i		−	0.0955236i	0.998859	−	0.0477618i	$-0.0152088\pi$
$420$		0			0
$421$	30.9583			1.50882			0.754408	−	0.656406i	$-0.227924\pi$
$421$	30.9583			1.50882			0.754408	+	0.656406i	$0.227924\pi$
$422$		0			0
$423$	0.440283	+	0.634011i	0.0214073	+	0.0308267i
$424$		0			0
$425$			3.85421i			0.186957i
$426$		0			0
$427$	3.28394			0.158921
$428$		0			0
$429$	−8.32362	−	26.5788i	−0.401868	−	1.28324i
$430$		0			0
$431$		−	9.46825i		−	0.456069i	−0.973653	−	0.228035i	$-0.926770\pi$
$431$		−	9.46825i		−	0.456069i	0.973653	−	0.228035i	$-0.0732300\pi$
$432$		0			0
$433$		−	36.8532i		−	1.77105i	−0.464591	−	0.885525i	$-0.653799\pi$
$433$		−	36.8532i		−	1.77105i	0.464591	−	0.885525i	$-0.346201\pi$
$434$		0			0
$435$	3.94342	+	12.5920i	0.189072	+	0.603742i
$436$		0			0
$437$		−	26.7085i		−	1.27764i
$438$		0			0
$439$	−22.7100			−1.08389			−0.541945	−	0.840414i	$-0.682312\pi$
$439$	−22.7100			−1.08389			−0.541945	+	0.840414i	$0.682312\pi$
$440$		0			0
$441$	16.8784	−	11.7211i	0.803736	−	0.558147i
$442$		0			0
$443$	−33.6752			−1.59996			−0.799979	−	0.600028i	$-0.795156\pi$
$443$	−33.6752			−1.59996			−0.799979	+	0.600028i	$0.795156\pi$
$444$		0			0
$445$			12.6057i			0.597569i
$446$		0			0
$447$	9.27911	+	29.6299i	0.438887	+	1.40145i
$448$		0			0
$449$		−	7.15314i		−	0.337577i	−0.985652	−	0.168789i	$-0.946014\pi$
$449$		−	7.15314i		−	0.337577i	0.985652	−	0.168789i	$-0.0539855\pi$
$450$		0			0
$451$	0.613994			0.0289118
$452$		0			0
$453$	−33.4838	+	10.4860i	−1.57321	+	0.492677i
$454$		0			0
$455$	1.44859			0.0679111
$456$		0			0
$457$	−11.5352			−0.539592			−0.269796	−	0.962917i	$-0.586956\pi$
$457$	−11.5352			−0.539592			−0.269796	+	0.962917i	$0.586956\pi$
$458$		0			0
$459$	−15.8173	−	12.2840i	−0.738288	−	0.573366i
$460$		0			0
$461$			21.6422i			1.00798i	0.863710	+	0.503989i	$0.168135\pi$
$461$			21.6422i			1.00798i	−0.863710	+	0.503989i	$0.831865\pi$
$462$		0			0
$463$			12.8842i			0.598782i	0.954131	+	0.299391i	$0.0967835\pi$
$463$			12.8842i			0.598782i	−0.954131	+	0.299391i	$0.903217\pi$
$464$		0			0
$465$	−2.02444	−	6.46440i	−0.0938811	−	0.299779i
$466$		0			0
$467$		−	16.8683i		−	0.780572i	−0.920694	−	0.390286i	$-0.872376\pi$
$467$		−	16.8683i		−	0.780572i	0.920694	−	0.390286i	$-0.127624\pi$
$468$		0			0
$469$	−1.06816	+	2.98814i	−0.0493233	+	0.137979i
$470$		0			0
$471$	15.8237	−	4.95546i	0.729117	−	0.228336i
$472$		0			0
$473$			35.6976i			1.64138i
$474$		0			0
$475$	−5.95792			−0.273368
$476$		0			0
$477$	−12.8086	+	8.89479i	−0.586464	+	0.407265i
$478$		0			0
$479$			40.6384i			1.85682i	0.371561	+	0.928408i	$0.378823\pi$
$479$			40.6384i			1.85682i	−0.371561	+	0.928408i	$0.621177\pi$
$480$		0			0
$481$			5.80253i			0.264573i
$482$		0			0
$483$	−2.87260	+	0.899604i	−0.130708	+	0.0409334i
$484$		0			0
$485$		−	3.95193i		−	0.179448i
$486$		0			0
$487$			1.38118i			0.0625873i	0.999510	+	0.0312936i	$0.00996270\pi$
$487$			1.38118i			0.0625873i	−0.999510	+	0.0312936i	$0.990037\pi$
$488$		0			0
$489$	−34.6669	+	10.8565i	−1.56769	+	0.490949i
$490$		0			0
$491$			11.6741i			0.526847i	0.964680	+	0.263423i	$0.0848516\pi$
$491$			11.6741i			0.526847i	−0.964680	+	0.263423i	$0.915148\pi$
$492$		0			0
$493$	−29.3621			−1.32240
$494$		0			0
$495$	10.6043	−	7.36406i	0.476628	−	0.330990i
$496$		0			0
$497$	0.195259			0.00875857
$498$		0			0
$499$			1.39871i			0.0626150i	0.999510	+	0.0313075i	$0.00996712\pi$
$499$			1.39871i			0.0626150i	−0.999510	+	0.0313075i	$0.990033\pi$
$500$		0			0
$501$	4.49827	+	14.3638i	0.200968	+	0.641726i
$502$		0			0
$503$	3.22638			0.143857			0.0719287	−	0.997410i	$-0.477085\pi$
$503$	3.22638			0.143857			0.0719287	+	0.997410i	$0.477085\pi$
$504$		0			0
$505$	−15.1867			−0.675799
$506$		0			0
$507$	1.58967	−	0.497834i	0.0705999	−	0.0221096i
$508$		0			0
$509$		−	33.4352i		−	1.48199i	−0.671511	−	0.740995i	$-0.734354\pi$
$509$		−	33.4352i		−	1.48199i	0.671511	−	0.740995i	$-0.265646\pi$
$510$		0			0
$511$		−	2.09166i		−	0.0925295i
$512$		0			0
$513$	18.9888	−	24.4508i	0.838377	−	1.07953i
$514$		0			0
$515$	7.13098			0.314229
$516$		0			0
$517$		−	1.10728i		−	0.0486981i
$518$		0			0
$519$	−12.0492	−	38.4754i	−0.528903	−	1.68888i
$520$		0			0
$521$	36.5272			1.60028			0.800142	−	0.599811i	$-0.204758\pi$
$521$	36.5272			1.60028			0.800142	+	0.599811i	$0.204758\pi$
$522$		0			0
$523$	−8.45066			−0.369521			−0.184761	−	0.982784i	$-0.559151\pi$
$523$	−8.45066			−0.369521			−0.184761	+	0.982784i	$0.559151\pi$
$524$		0			0
$525$	0.200677	+	0.640798i	0.00875826	+	0.0279667i
$526$		0			0
$527$	15.0736			0.656618
$528$		0			0
$529$	2.90408			0.126264
$530$		0			0
$531$	−5.33998	−	7.68960i	−0.231735	−	0.333700i
$532$		0			0
$533$			0.533105i			0.0230913i
$534$		0			0
$535$			9.47371i			0.409584i
$536$		0			0
$537$	−26.5438	+	8.31266i	−1.14545	+	0.358718i
$538$		0			0
$539$	−29.4777			−1.26969
$540$		0			0
$541$		−	22.5036i		−	0.967507i	−0.875204	−	0.483754i	$-0.839273\pi$
$541$		−	22.5036i		−	0.967507i	0.875204	−	0.483754i	$-0.160727\pi$
$542$		0			0
$543$	−33.9978	+	10.6470i	−1.45899	+	0.456906i
$544$		0			0
$545$		−	6.98600i		−	0.299247i
$546$		0			0
$547$		−	28.2814i		−	1.20922i	−0.796520	−	0.604612i	$-0.793328\pi$
$547$		−	28.2814i		−	1.20922i	0.796520	−	0.604612i	$-0.206672\pi$
$548$		0			0
$549$	14.4949	+	20.8727i	0.618627	+	0.890827i
$550$		0			0
$551$		−	45.3886i		−	1.93362i
$552$		0			0
$553$	−1.15869			−0.0492726
$554$		0			0
$555$	−2.56680	+	0.803839i	−0.108955	+	0.0341211i
$556$		0			0
$557$			44.8981i			1.90239i	0.308588	+	0.951196i	$0.400144\pi$
$557$			44.8981i			1.90239i	−0.308588	+	0.951196i	$0.599856\pi$
$558$		0			0
$559$	−30.9948			−1.31094
$560$		0			0
$561$	8.58574	+	27.4158i	0.362490	+	1.15750i
$562$		0			0
$563$	−19.2966			−0.813256			−0.406628	−	0.913594i	$-0.633296\pi$
$563$	−19.2966			−0.813256			−0.406628	+	0.913594i	$0.633296\pi$
$564$		0			0
$565$	1.40484			0.0591023
$566$		0			0
$567$	−3.26936	−	1.21876i	−0.137300	−	0.0511832i
$568$		0			0
$569$			14.8494i			0.622519i	0.950325	+	0.311259i	$0.100751\pi$
$569$			14.8494i			0.622519i	−0.950325	+	0.311259i	$0.899249\pi$
$570$		0			0
$571$	8.93175			0.373782			0.186891	−	0.982381i	$-0.440159\pi$
$571$	8.93175			0.373782			0.186891	+	0.982381i	$0.440159\pi$
$572$		0			0
$573$	−0.379006	+	0.118692i	−0.0158332	+	0.00495844i
$574$		0			0
$575$			4.48285i			0.186948i
$576$		0			0
$577$		−	3.32982i		−	0.138622i	−0.997595	−	0.0693111i	$-0.977920\pi$
$577$		−	3.32982i		−	0.138622i	0.997595	−	0.0693111i	$-0.0220801\pi$
$578$		0			0
$579$	−22.3240	+	6.99115i	−0.927754	+	0.290542i
$580$		0			0
$581$	−1.95840			−0.0812480
$582$		0			0
$583$	22.3697			0.926460
$584$		0			0
$585$	6.39390	+	9.20727i	0.264355	+	0.380674i
$586$		0			0
$587$	4.09780			0.169134			0.0845672	−	0.996418i	$-0.473049\pi$
$587$	4.09780			0.169134			0.0845672	+	0.996418i	$0.473049\pi$
$588$		0			0
$589$			23.3012i			0.960109i
$590$		0			0
$591$	25.2145	−	7.89636i	1.03719	−	0.324813i
$592$		0			0
$593$	−25.1296			−1.03195			−0.515975	−	0.856604i	$-0.672570\pi$
$593$	−25.1296			−1.03195			−0.515975	+	0.856604i	$0.672570\pi$
$594$		0			0
$595$	−1.49421			−0.0612566
$596$		0			0
$597$	10.1934	−	3.19224i	0.417188	−	0.130650i
$598$		0			0
$599$	9.61008			0.392657			0.196329	−	0.980538i	$-0.437098\pi$
$599$	9.61008			0.392657			0.196329	+	0.980538i	$0.437098\pi$
$600$		0			0
$601$	−19.3558			−0.789538			−0.394769	−	0.918780i	$-0.629175\pi$
$601$	−19.3558			−0.789538			−0.394769	+	0.918780i	$0.629175\pi$
$602$		0			0
$603$	−23.7074	+	6.40000i	−0.965439	+	0.260628i
$604$		0			0
$605$	−7.52006			−0.305734
$606$		0			0
$607$	2.59232			0.105219			0.0526095	−	0.998615i	$-0.483246\pi$
$607$	2.59232			0.105219			0.0526095	+	0.998615i	$0.483246\pi$
$608$		0			0
$609$	−4.88172	+	1.52879i	−0.197817	+	0.0619498i
$610$		0			0
$611$	0.961404			0.0388942
$612$		0			0
$613$	23.1392			0.934583			0.467292	−	0.884103i	$-0.345230\pi$
$613$	23.1392			0.934583			0.467292	+	0.884103i	$0.345230\pi$
$614$		0			0
$615$	−0.235824	+	0.0738523i	−0.00950933	+	0.00297801i
$616$		0			0
$617$		−	30.1969i		−	1.21568i	−0.794059	−	0.607841i	$-0.792036\pi$
$617$		−	30.1969i		−	1.21568i	0.794059	−	0.607841i	$-0.207964\pi$
$618$		0			0
$619$	−32.1393			−1.29179			−0.645893	−	0.763428i	$-0.723515\pi$
$619$	−32.1393			−1.29179			−0.645893	+	0.763428i	$0.723515\pi$
$620$		0			0
$621$	−18.3972	−	14.2875i	−0.738253	−	0.573339i
$622$		0			0
$623$	−4.88702			−0.195794
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−42.3800	+	13.2720i	−1.69249	+	0.530034i
$628$		0			0
$629$		−	5.98526i		−	0.238648i
$630$		0			0
$631$			33.3936i			1.32938i	0.747120	+	0.664689i	$0.231436\pi$
$631$			33.3936i			1.32938i	−0.747120	+	0.664689i	$0.768564\pi$
$632$		0			0
$633$	−25.1106	+	7.86383i	−0.998058	+	0.312559i
$634$		0			0
$635$	−3.29690			−0.130833
$636$		0			0
$637$		−	25.5942i		−	1.01408i
$638$		0			0
$639$	0.861849	+	1.24107i	0.0340942	+	0.0490959i
$640$		0			0
$641$	12.7627			0.504096			0.252048	−	0.967715i	$-0.418896\pi$
$641$	12.7627			0.504096			0.252048	+	0.967715i	$0.418896\pi$
$642$		0			0
$643$	−37.0566			−1.46137			−0.730685	−	0.682714i	$-0.760799\pi$
$643$	−37.0566			−1.46137			−0.730685	+	0.682714i	$0.760799\pi$
$644$		0			0
$645$	−4.29378	−	13.7108i	−0.169067	−	0.539863i
$646$		0			0
$647$	31.6257			1.24333			0.621667	−	0.783282i	$-0.286456\pi$
$647$	31.6257			1.24333			0.621667	+	0.783282i	$0.286456\pi$
$648$		0			0
$649$			13.4296i			0.527160i
$650$		0			0
$651$	2.50613	−	0.784839i	0.0982232	−	0.0307603i
$652$		0			0
$653$	−3.06196			−0.119824			−0.0599119	−	0.998204i	$-0.519082\pi$
$653$	−3.06196			−0.119824			−0.0599119	+	0.998204i	$0.519082\pi$
$654$		0			0
$655$		−	4.44230i		−	0.173575i
$656$		0			0
$657$	13.2946	−	9.23231i	0.518672	−	0.360187i
$658$		0			0
$659$			21.4267i			0.834667i	0.908753	+	0.417333i	$0.137035\pi$
$659$			21.4267i			0.834667i	−0.908753	+	0.417333i	$0.862965\pi$
$660$		0			0
$661$			23.8692i			0.928404i	0.885729	+	0.464202i	$0.153659\pi$
$661$			23.8692i			0.928404i	−0.885729	+	0.464202i	$0.846341\pi$
$662$		0			0
$663$	−23.8040	+	7.45463i	−0.924471	+	0.289514i
$664$		0			0
$665$		−	2.30978i		−	0.0895696i
$666$		0			0
$667$	−34.1511			−1.32234
$668$		0			0
$669$	−16.5524	+	5.18367i	−0.639953	+	0.200412i
$670$		0			0
$671$		−	36.4536i		−	1.40727i
$672$		0			0
$673$		−	43.2063i		−	1.66548i	−0.553664	−	0.832740i	$-0.686771\pi$
$673$		−	43.2063i		−	1.66548i	0.553664	−	0.832740i	$-0.313229\pi$
$674$		0			0
$675$	−3.18716	+	4.10391i	−0.122674	+	0.157959i
$676$		0			0
$677$	−15.1221			−0.581189			−0.290595	−	0.956846i	$-0.593853\pi$
$677$	−15.1221			−0.581189			−0.290595	+	0.956846i	$0.593853\pi$
$678$		0			0
$679$	1.53209			0.0587964
$680$		0			0
$681$	−2.21238	−	7.06454i	−0.0847786	−	0.270714i
$682$		0			0
$683$	27.6919			1.05960			0.529800	−	0.848123i	$-0.322267\pi$
$683$	27.6919			1.05960			0.529800	+	0.848123i	$0.322267\pi$
$684$		0			0
$685$	−9.42862			−0.360249
$686$		0			0
$687$	11.8432	+	37.8174i	0.451845	+	1.44282i
$688$		0			0
$689$			19.4227i			0.739946i
$690$		0			0
$691$	−37.7044			−1.43434			−0.717172	−	0.696896i	$-0.754564\pi$
$691$	−37.7044			−1.43434			−0.717172	+	0.696896i	$0.754564\pi$
$692$		0			0
$693$	2.85492	+	4.11110i	0.108449	+	0.156168i
$694$		0			0
$695$		−	21.3537i		−	0.809992i
$696$		0			0
$697$		−	0.549893i		−	0.0208287i
$698$		0			0
$699$	23.5668	−	7.38036i	0.891379	−	0.279151i
$700$		0			0
$701$	24.4857			0.924813			0.462406	−	0.886668i	$-0.346986\pi$
$701$	24.4857			0.924813			0.462406	+	0.886668i	$0.346986\pi$
$702$		0			0
$703$	9.25215			0.348952
$704$		0			0
$705$	0.133186	+	0.425286i	0.00501606	+	0.0160172i
$706$		0			0
$707$		−	5.88762i		−	0.221427i
$708$		0			0
$709$	8.55752			0.321385			0.160692	−	0.987005i	$-0.448627\pi$
$709$	8.55752			0.321385			0.160692	+	0.987005i	$0.448627\pi$
$710$		0			0
$711$	−5.11432	−	7.36466i	−0.191802	−	0.276196i
$712$		0			0
$713$	17.5322			0.656587
$714$		0			0
$715$		−	16.0802i		−	0.601365i
$716$		0			0
$717$	−2.94868	+	0.923431i	−0.110121	+	0.0344862i
$718$		0			0
$719$		−	42.9823i		−	1.60297i	−0.598014	−	0.801485i	$-0.704043\pi$
$719$		−	42.9823i		−	1.60297i	0.598014	−	0.801485i	$-0.295957\pi$
$720$		0			0
$721$			2.76456i			0.102957i
$722$		0			0
$723$	−20.2168	+	6.33123i	−0.751869	+	0.235461i
$724$		0			0
$725$			7.61818i			0.282932i
$726$		0			0
$727$		−	2.34039i		−	0.0868001i	−0.999058	−	0.0434000i	$-0.986181\pi$
$727$		−	2.34039i		−	0.0868001i	0.999058	−	0.0434000i	$-0.0138190\pi$
$728$		0			0
$729$	−6.68408	−	26.1596i	−0.247559	−	0.968873i
$730$		0			0
$731$	31.9708			1.18248
$732$		0			0
$733$		−	35.4849i		−	1.31067i	−0.755340	−	0.655333i	$-0.772528\pi$
$733$		−	35.4849i		−	1.31067i	0.755340	−	0.655333i	$-0.227472\pi$
$734$		0			0
$735$	11.3218	−	3.54563i	0.417612	−	0.130782i
$736$		0			0
$737$	33.1700	+	11.8572i	1.22183	+	0.436767i
$738$		0			0
$739$		−	22.2601i		−	0.818849i	−0.912344	−	0.409425i	$-0.865729\pi$
$739$		−	22.2601i		−	0.818849i	0.912344	−	0.409425i	$-0.134271\pi$
$740$		0			0
$741$	−11.5235	−	36.7968i	−0.423328	−	1.35176i
$742$		0			0
$743$			12.8795i			0.472504i	0.971692	+	0.236252i	$0.0759192\pi$
$743$			12.8795i			0.472504i	−0.971692	+	0.236252i	$0.924081\pi$
$744$		0			0
$745$			17.9261i			0.656761i
$746$		0			0
$747$	−8.64411	−	12.4476i	−0.316272	−	0.455433i
$748$		0			0
$749$	−3.67279			−0.134201
$750$		0			0
$751$	−24.8781			−0.907813			−0.453907	−	0.891049i	$-0.649970\pi$
$751$	−24.8781			−0.907813			−0.453907	+	0.891049i	$0.649970\pi$
$752$		0			0
$753$	−12.2638	+	3.84061i	−0.446917	+	0.139960i
$754$		0			0
$755$	−20.2577			−0.737253
$756$		0			0
$757$			27.8603i			1.01260i	0.862357	+	0.506301i	$0.168987\pi$
$757$			27.8603i			1.01260i	−0.862357	+	0.506301i	$0.831013\pi$
$758$		0			0
$759$	9.98611	+	31.8875i	0.362473	+	1.15744i
$760$		0			0
$761$		−	19.7701i		−	0.716665i	−0.933594	−	0.358332i	$-0.883345\pi$
$761$		−	19.7701i		−	0.716665i	0.933594	−	0.358332i	$-0.116655\pi$
$762$		0			0
$763$	2.70835			0.0980488
$764$		0			0
$765$	−6.59525	−	9.49721i	−0.238452	−	0.343372i
$766$		0			0
$767$	−11.6604			−0.421032
$768$		0			0
$769$			1.95860i			0.0706291i	0.999376	+	0.0353145i	$0.0112433\pi$
$769$			1.95860i			0.0706291i	−0.999376	+	0.0353145i	$0.988757\pi$
$770$		0			0
$771$	−5.96118	−	19.0351i	−0.214687	−	0.685533i
$772$		0			0
$773$		−	33.3558i		−	1.19972i	−0.800104	−	0.599862i	$-0.795222\pi$
$773$		−	33.3558i		−	1.19972i	0.800104	−	0.599862i	$-0.204778\pi$
$774$		0			0
$775$		−	3.91096i		−	0.140486i
$776$		0			0
$777$	−0.311634	−	0.995105i	−0.0111798	−	0.0356992i
$778$		0			0
$779$	0.850037			0.0304557
$780$		0			0
$781$		−	2.16749i		−	0.0775587i
$782$		0			0
$783$	−31.2643	−	24.2803i	−1.11729	−	0.867708i
$784$		0			0
$785$	9.57333			0.341687
$786$		0			0
$787$			19.7057i			0.702432i	0.936294	+	0.351216i	$0.114232\pi$
$787$			19.7057i			0.702432i	−0.936294	+	0.351216i	$0.885768\pi$
$788$		0			0
$789$	12.9549	+	41.3673i	0.461206	+	1.47271i
$790$		0			0
$791$			0.544634i			0.0193650i
$792$		0			0
$793$	31.6511			1.12396
$794$		0			0
$795$	−8.59181	+	2.69067i	−0.304720	+	0.0954284i
$796$		0			0
$797$			28.6041i			1.01321i	0.862179	+	0.506604i	$0.169099\pi$
$797$			28.6041i			1.01321i	−0.862179	+	0.506604i	$0.830901\pi$
$798$		0			0
$799$	−0.991679			−0.0350831
$800$		0			0
$801$	−21.5707	−	31.0620i	−0.762163	−	1.09752i
$802$		0			0
$803$	−23.2186			−0.819366
$804$		0			0
$805$	−1.73792			−0.0612537
$806$		0			0
$807$	9.13448	+	29.1681i	0.321549	+	1.02676i
$808$		0			0
$809$	33.3884			1.17387			0.586937	−	0.809633i	$-0.300334\pi$
$809$	33.3884			1.17387			0.586937	+	0.809633i	$0.300334\pi$
$810$		0			0
$811$		−	28.7011i		−	1.00783i	−0.863752	−	0.503917i	$-0.831892\pi$
$811$		−	28.7011i		−	1.00783i	0.863752	−	0.503917i	$-0.168108\pi$
$812$		0			0
$813$	9.24990	+	29.5366i	0.324408	+	1.03589i
$814$		0			0
$815$	−20.9735			−0.734668
$816$		0			0
$817$			49.4212i			1.72903i
$818$		0			0
$819$	−3.56950	+	2.47880i	−0.124728	+	0.0866164i
$820$		0			0
$821$			15.5795i			0.543730i	0.962335	+	0.271865i	$0.0876404\pi$
$821$			15.5795i			0.543730i	−0.962335	+	0.271865i	$0.912360\pi$
$822$		0			0
$823$	0.593809			0.0206989			0.0103494	−	0.999946i	$-0.496706\pi$
$823$	0.593809			0.0206989			0.0103494	+	0.999946i	$0.496706\pi$
$824$		0			0
$825$	7.11322	−	2.22763i	0.247650	−	0.0775560i
$826$		0			0
$827$		−	33.4798i		−	1.16421i	−0.813115	−	0.582103i	$-0.802230\pi$
$827$		−	33.4798i		−	1.16421i	0.813115	−	0.582103i	$-0.197770\pi$
$828$		0			0
$829$	−19.2438			−0.668363			−0.334182	−	0.942509i	$-0.608460\pi$
$829$	−19.2438			−0.668363			−0.334182	+	0.942509i	$0.608460\pi$
$830$		0			0
$831$	6.83492	−	2.14047i	0.237101	−	0.0742522i
$832$		0			0
$833$			26.4002i			0.914712i
$834$		0			0
$835$			8.69008i			0.300733i
$836$		0			0
$837$	16.0502	+	12.4648i	0.554776	+	0.430848i
$838$		0			0
$839$			15.1880i			0.524349i	0.965020	+	0.262175i	$0.0844396\pi$
$839$			15.1880i			0.524349i	−0.965020	+	0.262175i	$0.915560\pi$
$840$		0			0
$841$	−29.0367			−1.00127
$842$		0			0
$843$	−34.8799	+	10.9232i	−1.20133	+	0.376216i
$844$		0			0
$845$	0.961752			0.0330853
$846$		0			0
$847$		−	2.91540i		−	0.100174i
$848$		0			0
$849$	37.7656	−	11.8270i	1.29611	−	0.405900i
$850$		0			0
$851$		−	6.96148i		−	0.238637i
$852$		0			0
$853$	30.9929			1.06118			0.530589	−	0.847629i	$-0.321971\pi$
$853$	30.9929			1.06118			0.530589	+	0.847629i	$0.321971\pi$
$854$		0			0
$855$	14.6810	−	10.1951i	0.502080	−	0.348665i
$856$		0			0
$857$	−25.1966			−0.860700			−0.430350	−	0.902662i	$-0.641610\pi$
$857$	−25.1966			−0.860700			−0.430350	+	0.902662i	$0.641610\pi$
$858$		0			0
$859$	−7.18577			−0.245175			−0.122588	−	0.992458i	$-0.539119\pi$
$859$	−7.18577			−0.245175			−0.122588	+	0.992458i	$0.539119\pi$
$860$		0			0
$861$	−0.0286312	−	0.0914248i	−0.000975750	−	0.00311575i
$862$		0			0
$863$			21.0274i			0.715782i	0.933763	+	0.357891i	$0.116504\pi$
$863$			21.0274i			0.715782i	−0.933763	+	0.357891i	$0.883496\pi$
$864$		0			0
$865$		−	23.2776i		−	0.791462i
$866$		0			0
$867$	−3.54560	+	1.11037i	−0.120415	+	0.0377100i
$868$		0			0
$869$			12.8621i			0.436318i
$870$		0			0
$871$	−10.2951	+	28.8001i	−0.348837	+	0.975855i
$872$		0			0
$873$	6.76247	+	9.73800i	0.228875	+	0.329582i
$874$		0			0
$875$			0.387683i			0.0131061i
$876$		0			0
$877$	36.1524			1.22078			0.610390	−	0.792101i	$-0.291013\pi$
$877$	36.1524			1.22078			0.610390	+	0.792101i	$0.291013\pi$
$878$		0			0
$879$	1.90947	+	6.09729i	0.0644049	+	0.205656i
$880$		0			0
$881$		−	39.0019i		−	1.31401i	−0.753888	−	0.657003i	$-0.771824\pi$
$881$		−	39.0019i		−	1.31401i	0.753888	−	0.657003i	$-0.228176\pi$
$882$		0			0
$883$			30.8877i			1.03945i	0.854333	+	0.519726i	$0.173966\pi$
$883$			30.8877i			1.03945i	−0.854333	+	0.519726i	$0.826034\pi$
$884$		0			0
$885$	−1.61534	−	5.15808i	−0.0542991	−	0.173387i
$886$		0			0
$887$		−	43.9428i		−	1.47545i	−0.675099	−	0.737727i	$-0.735899\pi$
$887$		−	43.9428i		−	1.47545i	0.675099	−	0.737727i	$-0.264101\pi$
$888$		0			0
$889$		−	1.27815i		−	0.0428677i
$890$		0			0
$891$	−13.5290	+	36.2918i	−0.453237	+	1.21582i
$892$		0			0
$893$		−	1.53296i		−	0.0512986i
$894$		0			0
$895$	−16.0590			−0.536794
$896$		0			0
$897$	−27.6865	+	8.67052i	−0.924427	+	0.289500i
$898$		0			0
$899$	29.7944			0.993699
$900$		0			0
$901$		−	20.0343i		−	0.667440i
$902$		0			0
$903$	5.31544	−	1.66462i	0.176887	−	0.0553952i
$904$		0			0
$905$	−20.5687			−0.683725
$906$		0			0
$907$	6.90485			0.229272			0.114636	−	0.993408i	$-0.463430\pi$
$907$	6.90485			0.229272			0.114636	+	0.993408i	$0.463430\pi$
$908$		0			0
$909$	37.4218	−	25.9872i	1.24120	−	0.861941i
$910$		0			0
$911$			24.7571i			0.820239i	0.912032	+	0.410119i	$0.134513\pi$
$911$			24.7571i			0.820239i	−0.912032	+	0.410119i	$0.865487\pi$
$912$		0			0
$913$			21.7393i			0.719466i
$914$		0			0
$915$	4.38470	+	14.0011i	0.144954	+	0.462864i
$916$		0			0
$917$	1.72220			0.0568721
$918$		0			0
$919$		−	3.20908i		−	0.105858i	−0.998598	−	0.0529289i	$-0.983144\pi$
$919$		−	3.20908i		−	0.105858i	0.998598	−	0.0529289i	$-0.0168557\pi$
$920$		0			0
$921$	−13.1607	+	4.12149i	−0.433659	+	0.135808i
$922$		0			0
$923$	1.88194			0.0619447
$924$		0			0
$925$	−1.55292			−0.0510595
$926$		0			0
$927$	−17.5716	+	12.2024i	−0.577125	+	0.400780i
$928$		0			0
$929$	−51.2250			−1.68064			−0.840318	−	0.542093i	$-0.817632\pi$
$929$	−51.2250			−1.68064			−0.840318	+	0.542093i	$0.817632\pi$
$930$		0			0
$931$	−40.8100			−1.33749
$932$		0			0
$933$	−17.3755	+	5.44142i	−0.568847	+	0.178144i
$934$		0			0
$935$			16.5866i			0.542438i
$936$		0			0
$937$		−	17.1276i		−	0.559533i	−0.960068	−	0.279766i	$-0.909743\pi$
$937$		−	17.1276i		−	0.559533i	0.960068	−	0.279766i	$-0.0902570\pi$
$938$		0			0
$939$	−13.8415	−	44.1985i	−0.451701	−	1.44236i
$940$		0			0
$941$	−7.00088			−0.228222			−0.114111	−	0.993468i	$-0.536402\pi$
$941$	−7.00088			−0.228222			−0.114111	+	0.993468i	$0.536402\pi$
$942$		0			0
$943$		−	0.639583i		−	0.0208277i
$944$		0			0
$945$	−1.59101	−	1.23560i	−0.0517556	−	0.0401942i
$946$		0			0
$947$		−	30.4416i		−	0.989219i	−0.869115	−	0.494609i	$-0.835311\pi$
$947$		−	30.4416i		−	0.989219i	0.869115	−	0.494609i	$-0.164689\pi$
$948$		0			0
$949$		−	20.1597i		−	0.654412i
$950$		0			0
$951$	11.1648	+	35.6511i	0.362042	+	1.15607i
$952$		0			0
$953$		−	17.6975i		−	0.573277i	−0.958039	−	0.286638i	$-0.907462\pi$
$953$		−	17.6975i		−	0.573277i	0.958039	−	0.286638i	$-0.0925378\pi$
$954$		0			0
$955$	−0.229298			−0.00741992
$956$		0			0
$957$	16.9705	+	54.1898i	0.548577	+	1.75171i
$958$		0			0
$959$		−	3.65531i		−	0.118036i
$960$		0			0
$961$	15.7044			0.506594
$962$		0			0
$963$	−16.2112	−	23.3443i	−0.522400	−	0.752260i
$964$		0			0
$965$	−13.5060			−0.434774
$966$		0			0
$967$	−15.0479			−0.483907			−0.241954	−	0.970288i	$-0.577788\pi$
$967$	−15.0479			−0.483907			−0.241954	+	0.970288i	$0.577788\pi$
$968$		0			0
$969$	11.8864	+	37.9555i	0.381847	+	1.21931i
$970$		0			0
$971$		−	12.3461i		−	0.396204i	−0.980181	−	0.198102i	$-0.936522\pi$
$971$		−	12.3461i		−	0.396204i	0.980181	−	0.198102i	$-0.0634777\pi$
$972$		0			0
$973$	8.27846			0.265395
$974$		0			0
$975$	1.93415	+	6.17611i	0.0619425	+	0.197794i
$976$		0			0
$977$			2.05648i			0.0657927i	0.999459	+	0.0328963i	$0.0104731\pi$
$977$			2.05648i			0.0657927i	−0.999459	+	0.0328963i	$0.989527\pi$
$978$		0			0
$979$			54.2487i			1.73380i
$980$		0			0
$981$	11.9543	+	17.2143i	0.381672	+	0.549610i
$982$		0			0
$983$	−38.0100			−1.21233			−0.606166	−	0.795338i	$-0.707293\pi$
$983$	−38.0100			−1.21233			−0.606166	+	0.795338i	$0.707293\pi$
$984$		0			0
$985$	15.2548			0.486057
$986$		0			0
$987$	−0.164876	+	0.0516337i	−0.00524806	+	0.00164352i
$988$		0			0
$989$	37.1854			1.18243
$990$		0			0
$991$			10.4204i			0.331015i	0.986209	+	0.165507i	$0.0529262\pi$
$991$			10.4204i			0.331015i	−0.986209	+	0.165507i	$0.947074\pi$
$992$		0			0
$993$	−4.72700	−	15.0942i	−0.150007	−	0.478999i
$994$		0			0
$995$	6.16700			0.195507
$996$		0			0
$997$	−19.1275			−0.605773			−0.302886	−	0.953027i	$-0.597950\pi$
$997$	−19.1275			−0.605773			−0.302886	+	0.953027i	$0.597950\pi$
$998$		0			0
$999$	4.94938	−	6.37302i	0.156592	−	0.201633i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.6	yes	46
3.2	odd	2		4020.2.f.b.401.42	yes	46
67.66	odd	2		4020.2.f.b.401.41	yes	46
201.200	even	2	inner	4020.2.f.a.401.5	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.5	✓	46	201.200	even	2	inner
4020.2.f.a.401.6	yes	46	1.1	even	1	trivial
4020.2.f.b.401.41	yes	46	67.66	odd	2
4020.2.f.b.401.42	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)

\(f(q)\)	\(=\)	\(q+(-1.65289 + 0.517632i) q^{3} -1.00000 q^{5} -0.387683i q^{7} +(2.46411 - 1.71118i) q^{9} +O(q^{10})\)	\(q+(-1.65289 + 0.517632i) q^{3} -1.00000 q^{5} -0.387683i q^{7} +(2.46411 - 1.71118i) q^{9} -4.30349 q^{11} -3.73654i q^{13} +(1.65289 - 0.517632i) q^{15} +3.85421i q^{17} -5.95792 q^{19} +(0.200677 + 0.640798i) q^{21} +4.48285i q^{23} +1.00000 q^{25} +(-3.18716 + 4.10391i) q^{27} +7.61818i q^{29} -3.91096i q^{31} +(7.11322 - 2.22763i) q^{33} +0.387683i q^{35} -1.55292 q^{37} +(1.93415 + 6.17611i) q^{39} -0.142673 q^{41} -8.29503i q^{43} +(-2.46411 + 1.71118i) q^{45} +0.257298i q^{47} +6.84970 q^{49} +(-1.99506 - 6.37059i) q^{51} -5.19804 q^{53} +4.30349 q^{55} +(9.84781 - 3.08401i) q^{57} -3.12064i q^{59} +8.47069i q^{61} +(-0.663395 - 0.955294i) q^{63} +3.73654i q^{65} +(-7.70770 - 2.75526i) q^{67} +(-2.32047 - 7.40967i) q^{69} +0.503657i q^{71} +5.39528 q^{73} +(-1.65289 + 0.517632i) q^{75} +1.66839i q^{77} -2.98877i q^{79} +(3.14371 - 8.43309i) q^{81} -5.05155i q^{83} -3.85421i q^{85} +(-3.94342 - 12.5920i) q^{87} -12.6057i q^{89} -1.44859 q^{91} +(2.02444 + 6.46440i) q^{93} +5.95792 q^{95} +3.95193i q^{97} +(-10.6043 + 7.36406i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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

L-functions

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