LMFDB - Embedded newform 4020.2.f.b.401.17

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			401.17
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.b.401.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.747634 - 1.56238i) q^{3} +1.00000 q^{5} -2.57301i q^{7} +(-1.88209 + 2.33618i) q^{9} +O(q^{10})$	\(q+(-0.747634 - 1.56238i) q^{3} +1.00000 q^{5} -2.57301i q^{7} +(-1.88209 + 2.33618i) q^{9} -2.96799 q^{11} +6.23256i q^{13} +(-0.747634 - 1.56238i) q^{15} -4.68308i q^{17} +7.77939 q^{19} +(-4.02003 + 1.92367i) q^{21} +0.527989i q^{23} +1.00000 q^{25} +(5.05713 + 1.19393i) q^{27} -5.72560i q^{29} +5.43154i q^{31} +(2.21897 + 4.63714i) q^{33} -2.57301i q^{35} -0.802767 q^{37} +(9.73765 - 4.65967i) q^{39} +8.39649 q^{41} -0.456710i q^{43} +(-1.88209 + 2.33618i) q^{45} -10.9154i q^{47} +0.379602 q^{49} +(-7.31677 + 3.50123i) q^{51} -7.43325 q^{53} -2.96799 q^{55} +(-5.81614 - 12.1544i) q^{57} -9.25119i q^{59} -9.06624i q^{61} +(6.01103 + 4.84264i) q^{63} +6.23256i q^{65} +(3.79930 + 7.25019i) q^{67} +(0.824922 - 0.394743i) q^{69} -16.0505i q^{71} -14.4619 q^{73} +(-0.747634 - 1.56238i) q^{75} +7.63668i q^{77} -10.3834i q^{79} +(-1.91550 - 8.79380i) q^{81} -5.26485i q^{83} -4.68308i q^{85} +(-8.94559 + 4.28066i) q^{87} -1.71863i q^{89} +16.0365 q^{91} +(8.48615 - 4.06080i) q^{93} +7.77939 q^{95} -1.48917i q^{97} +(5.58602 - 6.93377i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.747634	−	1.56238i	−0.431647	−	0.902043i
$3$	−0.747634	−	1.56238i	−0.431647	−	0.902043i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	2.57301i		−	0.972508i	−0.873818	−	0.486254i	$-0.838363\pi$
$7$		−	2.57301i		−	0.972508i	0.873818	−	0.486254i	$-0.161637\pi$
$8$		0			0
$9$	−1.88209	+	2.33618i	−0.627362	+	0.778727i
$10$		0			0
$11$	−2.96799			−0.894883			−0.447442	−	0.894313i	$-0.647665\pi$
$11$	−2.96799			−0.894883			−0.447442	+	0.894313i	$0.647665\pi$
$12$		0			0
$13$			6.23256i			1.72860i	0.502977	+	0.864300i	$0.332238\pi$
$13$			6.23256i			1.72860i	−0.502977	+	0.864300i	$0.667762\pi$
$14$		0			0
$15$	−0.747634	−	1.56238i	−0.193038	−	0.403406i
$16$		0			0
$17$		−	4.68308i		−	1.13581i	−0.823093	−	0.567907i	$-0.807753\pi$
$17$		−	4.68308i		−	1.13581i	0.823093	−	0.567907i	$-0.192247\pi$
$18$		0			0
$19$	7.77939			1.78472			0.892358	−	0.451329i	$-0.149050\pi$
$19$	7.77939			1.78472			0.892358	+	0.451329i	$0.149050\pi$
$20$		0			0
$21$	−4.02003	+	1.92367i	−0.877244	+	0.419780i
$22$		0			0
$23$			0.527989i			0.110093i	0.998484	+	0.0550467i	$0.0175308\pi$
$23$			0.527989i			0.110093i	−0.998484	+	0.0550467i	$0.982469\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	5.05713	+	1.19393i	0.973244	+	0.229773i
$28$		0			0
$29$		−	5.72560i		−	1.06322i	−0.846990	−	0.531609i	$-0.821588\pi$
$29$		−	5.72560i		−	1.06322i	0.846990	−	0.531609i	$-0.178412\pi$
$30$		0			0
$31$			5.43154i			0.975533i	0.872974	+	0.487766i	$0.162188\pi$
$31$			5.43154i			0.975533i	−0.872974	+	0.487766i	$0.837812\pi$
$32$		0			0
$33$	2.21897	+	4.63714i	0.386273	+	0.807223i
$34$		0			0
$35$		−	2.57301i		−	0.434919i
$36$		0			0
$37$	−0.802767			−0.131974			−0.0659870	−	0.997820i	$-0.521020\pi$
$37$	−0.802767			−0.131974			−0.0659870	+	0.997820i	$0.521020\pi$
$38$		0			0
$39$	9.73765	−	4.65967i	1.55927	−	0.746144i
$40$		0			0
$41$	8.39649			1.31131			0.655656	−	0.755060i	$-0.272392\pi$
$41$	8.39649			1.31131			0.655656	+	0.755060i	$0.272392\pi$
$42$		0			0
$43$		−	0.456710i		−	0.0696477i	−0.999393	−	0.0348238i	$-0.988913\pi$
$43$		−	0.456710i		−	0.0696477i	0.999393	−	0.0348238i	$-0.0110870\pi$
$44$		0			0
$45$	−1.88209	+	2.33618i	−0.280565	+	0.348258i
$46$		0			0
$47$		−	10.9154i		−	1.59218i	−0.605181	−	0.796088i	$-0.706899\pi$
$47$		−	10.9154i		−	1.59218i	0.605181	−	0.796088i	$-0.293101\pi$
$48$		0			0
$49$	0.379602			0.0542289
$50$		0			0
$51$	−7.31677	+	3.50123i	−1.02455	+	0.490270i
$52$		0			0
$53$	−7.43325			−1.02103			−0.510517	−	0.859867i	$-0.670546\pi$
$53$	−7.43325			−1.02103			−0.510517	+	0.859867i	$0.670546\pi$
$54$		0			0
$55$	−2.96799			−0.400204
$56$		0			0
$57$	−5.81614	−	12.1544i	−0.770366	−	1.60989i
$58$		0			0
$59$		−	9.25119i		−	1.20440i	−0.798344	−	0.602201i	$-0.794290\pi$
$59$		−	9.25119i		−	1.20440i	0.798344	−	0.602201i	$-0.205710\pi$
$60$		0			0
$61$		−	9.06624i		−	1.16081i	−0.814327	−	0.580407i	$-0.802894\pi$
$61$		−	9.06624i		−	1.16081i	0.814327	−	0.580407i	$-0.197106\pi$
$62$		0			0
$63$	6.01103	+	4.84264i	0.757318	+	0.610115i
$64$		0			0
$65$			6.23256i			0.773053i
$66$		0			0
$67$	3.79930	+	7.25019i	0.464159	+	0.885752i
$67$	3.79930	+	7.25019i	0.464159	+	0.885752i
$68$		0			0
$69$	0.824922	−	0.394743i	0.0993089	−	0.0475214i
$70$		0			0
$71$		−	16.0505i		−	1.90484i	−0.304787	−	0.952420i	$-0.598585\pi$
$71$		−	16.0505i		−	1.90484i	0.304787	−	0.952420i	$-0.401415\pi$
$72$		0			0
$73$	−14.4619			−1.69264			−0.846318	−	0.532679i	$-0.821185\pi$
$73$	−14.4619			−1.69264			−0.846318	+	0.532679i	$0.821185\pi$
$74$		0			0
$75$	−0.747634	−	1.56238i	−0.0863293	−	0.180409i
$76$		0			0
$77$			7.63668i			0.870281i
$78$		0			0
$79$		−	10.3834i		−	1.16822i	−0.811675	−	0.584110i	$-0.801444\pi$
$79$		−	10.3834i		−	1.16822i	0.811675	−	0.584110i	$-0.198556\pi$
$80$		0			0
$81$	−1.91550	−	8.79380i	−0.212833	−	0.977089i
$82$		0			0
$83$		−	5.26485i		−	0.577892i	−0.957345	−	0.288946i	$-0.906695\pi$
$83$		−	5.26485i		−	0.577892i	0.957345	−	0.288946i	$-0.0933049\pi$
$84$		0			0
$85$		−	4.68308i		−	0.507951i
$86$		0			0
$87$	−8.94559	+	4.28066i	−0.959068	+	0.458935i
$88$		0			0
$89$		−	1.71863i		−	0.182175i	−0.995843	−	0.0910874i	$-0.970966\pi$
$89$		−	1.71863i		−	0.182175i	0.995843	−	0.0910874i	$-0.0290343\pi$
$90$		0			0
$91$	16.0365			1.68108
$92$		0			0
$93$	8.48615	−	4.06080i	0.879972	−	0.421085i
$94$		0			0
$95$	7.77939			0.798149
$96$		0			0
$97$		−	1.48917i		−	0.151203i	−0.997138	−	0.0756014i	$-0.975912\pi$
$97$		−	1.48917i		−	0.151203i	0.997138	−	0.0756014i	$-0.0240877\pi$
$98$		0			0
$99$	5.58602	−	6.93377i	0.561416	−	0.696870i
$100$		0			0
$101$	−12.4507			−1.23889			−0.619444	−	0.785041i	$-0.712642\pi$
$101$	−12.4507			−1.23889			−0.619444	+	0.785041i	$0.712642\pi$
$102$		0			0
$103$	4.31540			0.425209			0.212605	−	0.977138i	$-0.431805\pi$
$103$	4.31540			0.425209			0.212605	+	0.977138i	$0.431805\pi$
$104$		0			0
$105$	−4.02003	+	1.92367i	−0.392315	+	0.187731i
$106$		0			0
$107$			18.3915i			1.77798i	0.457929	+	0.888989i	$0.348592\pi$
$107$			18.3915i			1.77798i	−0.457929	+	0.888989i	$0.651408\pi$
$108$		0			0
$109$			9.87730i			0.946074i	0.881043	+	0.473037i	$0.156842\pi$
$109$			9.87730i			0.946074i	−0.881043	+	0.473037i	$0.843158\pi$
$110$		0			0
$111$	0.600176	+	1.25423i	0.0569662	+	0.119046i
$112$		0			0
$113$	5.79898			0.545522			0.272761	−	0.962082i	$-0.412063\pi$
$113$	5.79898			0.545522			0.272761	+	0.962082i	$0.412063\pi$
$114$		0			0
$115$			0.527989i			0.0492352i
$116$		0			0
$117$	−14.5604	−	11.7302i	−1.34611	−	1.08446i
$118$		0			0
$119$	−12.0496			−1.10459
$120$		0			0
$121$	−2.19103			−0.199184
$122$		0			0
$123$	−6.27750	−	13.1185i	−0.566023	−	1.18286i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	13.9522			1.23806			0.619029	−	0.785368i	$-0.287526\pi$
$127$	13.9522			1.23806			0.619029	+	0.785368i	$0.287526\pi$
$128$		0			0
$129$	−0.713557	+	0.341452i	−0.0628252	+	0.0300632i
$130$		0			0
$131$			16.8927i			1.47592i	0.674842	+	0.737962i	$0.264212\pi$
$131$			16.8927i			1.47592i	−0.674842	+	0.737962i	$0.735788\pi$
$132$		0			0
$133$		−	20.0165i		−	1.73565i
$134$		0			0
$135$	5.05713	+	1.19393i	0.435248	+	0.102757i
$136$		0			0
$137$	−14.6923			−1.25525			−0.627625	−	0.778516i	$-0.715973\pi$
$137$	−14.6923			−1.25525			−0.627625	+	0.778516i	$0.715973\pi$
$138$		0			0
$139$		−	21.2620i		−	1.80342i	−0.432345	−	0.901708i	$-0.642314\pi$
$139$		−	21.2620i		−	1.80342i	0.432345	−	0.901708i	$-0.357686\pi$
$140$		0			0
$141$	−17.0541	+	8.16073i	−1.43621	+	0.687258i
$142$		0			0
$143$		−	18.4982i		−	1.54689i
$144$		0			0
$145$		−	5.72560i		−	0.475486i
$146$		0			0
$147$	−0.283803	−	0.593084i	−0.0234077	−	0.0489168i
$148$		0			0
$149$		−	0.997313i		−	0.0817030i	−0.999165	−	0.0408515i	$-0.986993\pi$
$149$		−	0.997313i		−	0.0817030i	0.999165	−	0.0408515i	$-0.0130071\pi$
$150$		0			0
$151$	23.5722			1.91828			0.959139	−	0.282935i	$-0.0913080\pi$
$151$	23.5722			1.91828			0.959139	+	0.282935i	$0.0913080\pi$
$152$		0			0
$153$	10.9405	+	8.81396i	0.884489	+	0.712567i
$154$		0			0
$155$			5.43154i			0.436272i
$156$		0			0
$157$	−3.48301			−0.277975			−0.138987	−	0.990294i	$-0.544385\pi$
$157$	−3.48301			−0.277975			−0.138987	+	0.990294i	$0.544385\pi$
$158$		0			0
$159$	5.55735	+	11.6136i	0.440726	+	0.921017i
$160$		0			0
$161$	1.35852			0.107067
$162$		0			0
$163$	−13.8645			−1.08595			−0.542976	−	0.839748i	$-0.682703\pi$
$163$	−13.8645			−1.08595			−0.542976	+	0.839748i	$0.682703\pi$
$164$		0			0
$165$	2.21897	+	4.63714i	0.172747	+	0.361001i
$166$		0			0
$167$			7.74374i			0.599229i	0.954060	+	0.299614i	$0.0968580\pi$
$167$			7.74374i			0.599229i	−0.954060	+	0.299614i	$0.903142\pi$
$168$		0			0
$169$	−25.8448			−1.98806
$170$		0			0
$171$	−14.6415	+	18.1741i	−1.11966	+	1.38981i
$172$		0			0
$173$		−	0.928844i		−	0.0706187i	−0.999376	−	0.0353094i	$-0.988758\pi$
$173$		−	0.928844i		−	0.0706187i	0.999376	−	0.0353094i	$-0.0112417\pi$
$174$		0			0
$175$		−	2.57301i		−	0.194502i
$176$		0			0
$177$	−14.4539	+	6.91651i	−1.08642	+	0.519876i
$178$		0			0
$179$	12.2793			0.917796			0.458898	−	0.888489i	$-0.348244\pi$
$179$	12.2793			0.917796			0.458898	+	0.888489i	$0.348244\pi$
$180$		0			0
$181$	−14.9753			−1.11310			−0.556551	−	0.830813i	$-0.687876\pi$
$181$	−14.9753			−1.11310			−0.556551	+	0.830813i	$0.687876\pi$
$182$		0			0
$183$	−14.1650	+	6.77823i	−1.04710	+	0.501061i
$184$		0			0
$185$	−0.802767			−0.0590206
$186$		0			0
$187$			13.8993i			1.01642i
$188$		0			0
$189$	3.07201	−	13.0121i	0.223456	−	0.946488i
$190$		0			0
$191$	−14.6745			−1.06181			−0.530903	−	0.847433i	$-0.678147\pi$
$191$	−14.6745			−1.06181			−0.530903	+	0.847433i	$0.678147\pi$
$192$		0			0
$193$	−0.563709			−0.0405766			−0.0202883	−	0.999794i	$-0.506458\pi$
$193$	−0.563709			−0.0405766			−0.0202883	+	0.999794i	$0.506458\pi$
$194$		0			0
$195$	9.73765	−	4.65967i	0.697327	−	0.333686i
$196$		0			0
$197$	−12.8963			−0.918823			−0.459412	−	0.888224i	$-0.651940\pi$
$197$	−12.8963			−0.918823			−0.459412	+	0.888224i	$0.651940\pi$
$198$		0			0
$199$	9.00936			0.638657			0.319328	−	0.947644i	$-0.396543\pi$
$199$	9.00936			0.638657			0.319328	+	0.947644i	$0.396543\pi$
$200$		0			0
$201$	8.48710	−	11.3565i	0.598634	−	0.801023i
$202$		0			0
$203$	−14.7321			−1.03399
$204$		0			0
$205$	8.39649			0.586437
$206$		0			0
$207$	−1.23348	−	0.993722i	−0.0857327	−	0.0690684i
$208$		0			0
$209$	−23.0892			−1.59711
$210$		0			0
$211$	27.9936			1.92716			0.963579	−	0.267425i	$-0.0861727\pi$
$211$	27.9936			1.92716			0.963579	+	0.267425i	$0.0861727\pi$
$212$		0			0
$213$	−25.0770	+	11.9999i	−1.71825	+	0.822218i
$214$		0			0
$215$		−	0.456710i		−	0.0311474i
$216$		0			0
$217$	13.9754			0.948713
$218$		0			0
$219$	10.8122	+	22.5950i	0.730620	+	1.52683i
$220$		0			0
$221$	29.1876			1.96337
$222$		0			0
$223$	−21.2414			−1.42243			−0.711215	−	0.702975i	$-0.751855\pi$
$223$	−21.2414			−1.42243			−0.711215	+	0.702975i	$0.751855\pi$
$224$		0			0
$225$	−1.88209	+	2.33618i	−0.125472	+	0.155745i
$226$		0			0
$227$		−	3.59524i		−	0.238625i	−0.992857	−	0.119312i	$-0.961931\pi$
$227$		−	3.59524i		−	0.238625i	0.992857	−	0.119312i	$-0.0380690\pi$
$228$		0			0
$229$		−	18.5376i		−	1.22500i	−0.790470	−	0.612500i	$-0.790164\pi$
$229$		−	18.5376i		−	1.22500i	0.790470	−	0.612500i	$-0.209836\pi$
$230$		0			0
$231$	11.9314	−	5.70944i	0.785030	−	0.375654i
$232$		0			0
$233$	21.8466			1.43122			0.715609	−	0.698501i	$-0.246149\pi$
$233$	21.8466			1.43122			0.715609	+	0.698501i	$0.246149\pi$
$234$		0			0
$235$		−	10.9154i		−	0.712043i
$236$		0			0
$237$	−16.2228	+	7.76295i	−1.05378	+	0.504258i
$238$		0			0
$239$	5.21754			0.337495			0.168747	−	0.985659i	$-0.446028\pi$
$239$	5.21754			0.337495			0.168747	+	0.985659i	$0.446028\pi$
$240$		0			0
$241$	−4.39320			−0.282991			−0.141496	−	0.989939i	$-0.545191\pi$
$241$	−4.39320			−0.282991			−0.141496	+	0.989939i	$0.545191\pi$
$242$		0			0
$243$	−12.3072	+	9.56728i	−0.789507	+	0.613742i
$244$		0			0
$245$	0.379602			0.0242519
$246$		0			0
$247$			48.4855i			3.08506i
$248$		0			0
$249$	−8.22572	+	3.93618i	−0.521284	+	0.249445i
$250$		0			0
$251$	11.8167			0.745861			0.372930	−	0.927859i	$-0.378353\pi$
$251$	11.8167			0.745861			0.372930	+	0.927859i	$0.378353\pi$
$252$		0			0
$253$		−	1.56707i		−	0.0985207i
$254$		0			0
$255$	−7.31677	+	3.50123i	−0.458194	+	0.219255i
$256$		0			0
$257$		−	12.6199i		−	0.787206i	−0.919280	−	0.393603i	$-0.871228\pi$
$257$		−	12.6199i		−	0.787206i	0.919280	−	0.393603i	$-0.128772\pi$
$258$		0			0
$259$			2.06553i			0.128346i
$260$		0			0
$261$	13.3761	+	10.7761i	0.827957	+	0.667023i
$262$		0			0
$263$		−	26.3538i		−	1.62504i	−0.582930	−	0.812522i	$-0.698094\pi$
$263$		−	26.3538i		−	1.62504i	0.582930	−	0.812522i	$-0.301906\pi$
$264$		0			0
$265$	−7.43325			−0.456621
$266$		0			0
$267$	−2.68517	+	1.28491i	−0.164329	+	0.0786352i
$268$		0			0
$269$			10.9011i			0.664655i	0.943164	+	0.332327i	$0.107834\pi$
$269$			10.9011i			0.664655i	−0.943164	+	0.332327i	$0.892166\pi$
$270$		0			0
$271$		−	10.8314i		−	0.657963i	−0.944336	−	0.328982i	$-0.893295\pi$
$271$		−	10.8314i		−	0.657963i	0.944336	−	0.328982i	$-0.106705\pi$
$272$		0			0
$273$	−11.9894	−	25.0551i	−0.725631	−	1.51640i
$274$		0			0
$275$	−2.96799			−0.178977
$276$		0			0
$277$	22.7549			1.36721			0.683606	−	0.729851i	$-0.260411\pi$
$277$	22.7549			1.36721			0.683606	+	0.729851i	$0.260411\pi$
$278$		0			0
$279$	−12.6891	−	10.2226i	−0.759674	−	0.612013i
$280$		0			0
$281$	−7.25922			−0.433049			−0.216524	−	0.976277i	$-0.569472\pi$
$281$	−7.25922			−0.433049			−0.216524	+	0.976277i	$0.569472\pi$
$282$		0			0
$283$	−30.6155			−1.81990			−0.909952	−	0.414713i	$-0.863882\pi$
$283$	−30.6155			−1.81990			−0.909952	+	0.414713i	$0.863882\pi$
$284$		0			0
$285$	−5.81614	−	12.1544i	−0.344518	−	0.719964i
$286$		0			0
$287$		−	21.6043i		−	1.27526i
$288$		0			0
$289$	−4.93123			−0.290072
$290$		0			0
$291$	−2.32666	+	1.11336i	−0.136391	+	0.0652662i
$292$		0			0
$293$		−	19.4496i		−	1.13626i	−0.822939	−	0.568130i	$-0.807667\pi$
$293$		−	19.4496i		−	1.13626i	0.822939	−	0.568130i	$-0.192333\pi$
$294$		0			0
$295$		−	9.25119i		−	0.538625i
$296$		0			0
$297$	−15.0095	−	3.54358i	−0.870940	−	0.205620i
$298$		0			0
$299$	−3.29072			−0.190307
$300$		0			0
$301$	−1.17512			−0.0677329
$302$		0			0
$303$	9.30855	+	19.4527i	0.534762	+	1.11753i
$304$		0			0
$305$		−	9.06624i		−	0.519132i
$306$		0			0
$307$	−14.2572			−0.813701			−0.406851	−	0.913495i	$-0.633373\pi$
$307$	−14.2572			−0.813701			−0.406851	+	0.913495i	$0.633373\pi$
$308$		0			0
$309$	−3.22634	−	6.74232i	−0.183540	−	0.383557i
$310$		0			0
$311$	7.41173			0.420281			0.210140	−	0.977671i	$-0.432608\pi$
$311$	7.41173			0.420281			0.210140	+	0.977671i	$0.432608\pi$
$312$		0			0
$313$			4.00371i			0.226303i	0.993578	+	0.113151i	$0.0360945\pi$
$313$			4.00371i			0.226303i	−0.993578	+	0.113151i	$0.963905\pi$
$314$		0			0
$315$	6.01103	+	4.84264i	0.338683	+	0.272852i
$316$		0			0
$317$		−	28.9195i		−	1.62428i	−0.583460	−	0.812142i	$-0.698302\pi$
$317$		−	28.9195i		−	1.62428i	0.583460	−	0.812142i	$-0.301698\pi$
$318$		0			0
$319$			16.9935i			0.951456i
$320$		0			0
$321$	28.7347	−	13.7501i	1.60381	−	0.767458i
$322$		0			0
$323$		−	36.4315i		−	2.02710i
$324$		0			0
$325$			6.23256i			0.345720i
$326$		0			0
$327$	15.4321	−	7.38461i	0.853399	−	0.408370i
$328$		0			0
$329$	−28.0855			−1.54840
$330$		0			0
$331$		−	30.8077i		−	1.69335i	−0.532114	−	0.846673i	$-0.678602\pi$
$331$		−	30.8077i		−	1.69335i	0.532114	−	0.846673i	$-0.321398\pi$
$332$		0			0
$333$	1.51088	−	1.87541i	0.0827955	−	0.102772i
$334$		0			0
$335$	3.79930	+	7.25019i	0.207578	+	0.396120i
$336$		0			0
$337$			5.50639i			0.299952i	0.988690	+	0.149976i	$0.0479196\pi$
$337$			5.50639i			0.299952i	−0.988690	+	0.149976i	$0.952080\pi$
$338$		0			0
$339$	−4.33552	−	9.06024i	−0.235473	−	0.492085i
$340$		0			0
$341$		−	16.1208i		−	0.872988i
$342$		0			0
$343$		−	18.9878i		−	1.02525i
$344$		0			0
$345$	0.824922	−	0.394743i	0.0444123	−	0.0212522i
$346$		0			0
$347$	−30.3683			−1.63026			−0.815128	−	0.579281i	$-0.803334\pi$
$347$	−30.3683			−1.63026			−0.815128	+	0.579281i	$0.803334\pi$
$348$		0			0
$349$	−1.36866			−0.0732625			−0.0366313	−	0.999329i	$-0.511663\pi$
$349$	−1.36866			−0.0732625			−0.0366313	+	0.999329i	$0.511663\pi$
$350$		0			0
$351$	−7.44126	+	31.5188i	−0.397185	+	1.68235i
$352$		0			0
$353$	0.155180			0.00825942			0.00412971	−	0.999991i	$-0.498685\pi$
$353$	0.155180			0.00825942			0.00412971	+	0.999991i	$0.498685\pi$
$354$		0			0
$355$		−	16.0505i		−	0.851871i
$356$		0			0
$357$	9.00871	+	18.8261i	0.476791	+	0.996385i
$358$		0			0
$359$		−	3.54249i		−	0.186966i	−0.995621	−	0.0934828i	$-0.970200\pi$
$359$		−	3.54249i		−	0.186966i	0.995621	−	0.0934828i	$-0.0298000\pi$
$360$		0			0
$361$	41.5190			2.18521
$362$		0			0
$363$	1.63809	+	3.42323i	0.0859773	+	0.179673i
$364$		0			0
$365$	−14.4619			−0.756969
$366$		0			0
$367$		−	14.5403i		−	0.759000i	−0.925192	−	0.379500i	$-0.876096\pi$
$367$		−	14.5403i		−	0.759000i	0.925192	−	0.379500i	$-0.123904\pi$
$368$		0			0
$369$	−15.8029	+	19.6157i	−0.822668	+	1.02115i
$370$		0			0
$371$			19.1258i			0.992964i
$372$		0			0
$373$		−	2.71140i		−	0.140391i	−0.997533	−	0.0701955i	$-0.977638\pi$
$373$		−	2.71140i		−	0.140391i	0.997533	−	0.0701955i	$-0.0223623\pi$
$374$		0			0
$375$	−0.747634	−	1.56238i	−0.0386077	−	0.0806812i
$376$		0			0
$377$	35.6852			1.83788
$378$		0			0
$379$			20.7910i			1.06796i	0.845497	+	0.533980i	$0.179304\pi$
$379$			20.7910i			1.06796i	−0.845497	+	0.533980i	$0.820696\pi$
$380$		0			0
$381$	−10.4312	−	21.7987i	−0.534404	−	1.11678i
$382$		0			0
$383$	26.4817			1.35315			0.676576	−	0.736373i	$-0.263463\pi$
$383$	26.4817			1.35315			0.676576	+	0.736373i	$0.263463\pi$
$384$		0			0
$385$			7.63668i			0.389201i
$386$		0			0
$387$	1.06696	+	0.859569i	0.0542366	+	0.0436943i
$388$		0			0
$389$			10.7522i			0.545156i	0.962134	+	0.272578i	$0.0878763\pi$
$389$			10.7522i			0.545156i	−0.962134	+	0.272578i	$0.912124\pi$
$390$		0			0
$391$	2.47261			0.125046
$392$		0			0
$393$	26.3929	−	12.6296i	1.33135	−	0.637078i
$394$		0			0
$395$		−	10.3834i		−	0.522444i
$396$		0			0
$397$	17.9792			0.902352			0.451176	−	0.892435i	$-0.351005\pi$
$397$	17.9792			0.902352			0.451176	+	0.892435i	$0.351005\pi$
$398$		0			0
$399$	−31.2734	+	14.9650i	−1.56563	+	0.749187i
$400$		0			0
$401$	24.9376			1.24532			0.622662	−	0.782491i	$-0.286051\pi$
$401$	24.9376			1.24532			0.622662	+	0.782491i	$0.286051\pi$
$402$		0			0
$403$	−33.8524			−1.68631
$404$		0			0
$405$	−1.91550	−	8.79380i	−0.0951818	−	0.436967i
$406$		0			0
$407$	2.38260			0.118101
$408$		0			0
$409$		−	21.9809i		−	1.08688i	−0.839447	−	0.543442i	$-0.817121\pi$
$409$		−	21.9809i		−	1.08688i	0.839447	−	0.543442i	$-0.182879\pi$
$410$		0			0
$411$	10.9845	+	22.9551i	0.541824	+	1.13229i
$412$		0			0
$413$	−23.8034			−1.17129
$414$		0			0
$415$		−	5.26485i		−	0.258441i
$416$		0			0
$417$	−33.2194	+	15.8962i	−1.62676	+	0.778439i
$418$		0			0
$419$			16.8656i			0.823938i	0.911198	+	0.411969i	$0.135159\pi$
$419$			16.8656i			0.823938i	−0.911198	+	0.411969i	$0.864841\pi$
$420$		0			0
$421$	23.8377			1.16178			0.580889	−	0.813983i	$-0.302705\pi$
$421$	23.8377			1.16178			0.580889	+	0.813983i	$0.302705\pi$
$422$		0			0
$423$	25.5004	+	20.5438i	1.23987	+	0.998871i
$424$		0			0
$425$		−	4.68308i		−	0.227163i
$426$		0			0
$427$	−23.3276			−1.12890
$428$		0			0
$429$	−28.9012	+	13.8299i	−1.39537	+	0.667712i
$430$		0			0
$431$		−	3.69569i		−	0.178015i	−0.996031	−	0.0890075i	$-0.971630\pi$
$431$		−	3.69569i		−	0.178015i	0.996031	−	0.0890075i	$-0.0283695\pi$
$432$		0			0
$433$		−	13.6559i		−	0.656260i	−0.944633	−	0.328130i	$-0.893582\pi$
$433$		−	13.6559i		−	0.656260i	0.944633	−	0.328130i	$-0.106418\pi$
$434$		0			0
$435$	−8.94559	+	4.28066i	−0.428908	+	0.205242i
$436$		0			0
$437$			4.10744i			0.196485i
$438$		0			0
$439$	−20.3025			−0.968984			−0.484492	−	0.874796i	$-0.660996\pi$
$439$	−20.3025			−0.968984			−0.484492	+	0.874796i	$0.660996\pi$
$440$		0			0
$441$	−0.714444	+	0.886820i	−0.0340211	+	0.0422295i
$442$		0			0
$443$	−10.1349			−0.481521			−0.240761	−	0.970585i	$-0.577397\pi$
$443$	−10.1349			−0.481521			−0.240761	+	0.970585i	$0.577397\pi$
$444$		0			0
$445$		−	1.71863i		−	0.0814711i
$446$		0			0
$447$	−1.55819	+	0.745625i	−0.0736996	+	0.0352668i
$448$		0			0
$449$		−	11.0290i		−	0.520490i	−0.965543	−	0.260245i	$-0.916197\pi$
$449$		−	11.0290i		−	0.520490i	0.965543	−	0.260245i	$-0.0838034\pi$
$450$		0			0
$451$	−24.9207			−1.17347
$452$		0			0
$453$	−17.6234	−	36.8288i	−0.828018	−	1.73037i
$454$		0			0
$455$	16.0365			0.751800
$456$		0			0
$457$	−16.2306			−0.759234			−0.379617	−	0.925144i	$-0.623944\pi$
$457$	−16.2306			−0.759234			−0.379617	+	0.925144i	$0.623944\pi$
$458$		0			0
$459$	5.59128	−	23.6829i	0.260979	−	1.10542i
$460$		0			0
$461$			26.5307i			1.23566i	0.786313	+	0.617829i	$0.211988\pi$
$461$			26.5307i			1.23566i	−0.786313	+	0.617829i	$0.788012\pi$
$462$		0			0
$463$			15.9109i			0.739441i	0.929143	+	0.369721i	$0.120547\pi$
$463$			15.9109i			0.739441i	−0.929143	+	0.369721i	$0.879453\pi$
$464$		0			0
$465$	8.48615	−	4.06080i	0.393536	−	0.188315i
$466$		0			0
$467$		−	28.6699i		−	1.32669i	−0.748315	−	0.663343i	$-0.769137\pi$
$467$		−	28.6699i		−	1.32669i	0.748315	−	0.663343i	$-0.230863\pi$
$468$		0			0
$469$	18.6548	−	9.77566i	0.861401	−	0.451398i
$470$		0			0
$471$	2.60402	+	5.44180i	0.119987	+	0.250745i
$472$		0			0
$473$			1.35551i			0.0623265i
$474$		0			0
$475$	7.77939			0.356943
$476$		0			0
$477$	13.9900	−	17.3654i	0.640559	−	0.795108i
$478$		0			0
$479$			13.3502i			0.609986i	0.952355	+	0.304993i	$0.0986541\pi$
$479$			13.3502i			0.609986i	−0.952355	+	0.304993i	$0.901346\pi$
$480$		0			0
$481$		−	5.00329i		−	0.228130i
$482$		0			0
$483$	−1.01568	−	2.12253i	−0.0462150	−	0.0965787i
$484$		0			0
$485$		−	1.48917i		−	0.0676199i
$486$		0			0
$487$		−	21.2839i		−	0.964467i	−0.876043	−	0.482233i	$-0.839826\pi$
$487$		−	21.2839i		−	0.964467i	0.876043	−	0.482233i	$-0.160174\pi$
$488$		0			0
$489$	10.3656	+	21.6617i	0.468748	+	0.979575i
$490$		0			0
$491$		−	6.41799i		−	0.289640i	−0.989458	−	0.144820i	$-0.953740\pi$
$491$		−	6.41799i		−	0.289640i	0.989458	−	0.144820i	$-0.0462603\pi$
$492$		0			0
$493$	−26.8135			−1.20762
$494$		0			0
$495$	5.58602	−	6.93377i	0.251073	−	0.311650i
$496$		0			0
$497$	−41.2981			−1.85247
$498$		0			0
$499$			0.114328i			0.00511803i	0.999997	+	0.00255902i	$0.000814561\pi$
$499$			0.114328i			0.00511803i	−0.999997	+	0.00255902i	$0.999185\pi$
$500$		0			0
$501$	12.0987	−	5.78949i	0.540530	−	0.258655i
$502$		0			0
$503$	−28.1593			−1.25556			−0.627780	−	0.778391i	$-0.716036\pi$
$503$	−28.1593			−1.25556			−0.627780	+	0.778391i	$0.716036\pi$
$504$		0			0
$505$	−12.4507			−0.554048
$506$		0			0
$507$	19.3224	+	40.3794i	0.858139	+	1.79331i
$508$		0			0
$509$		−	8.09759i		−	0.358919i	−0.983765	−	0.179460i	$-0.942565\pi$
$509$		−	8.09759i		−	0.358919i	0.983765	−	0.179460i	$-0.0574350\pi$
$510$		0			0
$511$			37.2106i			1.64610i
$512$		0			0
$513$	39.3414	+	9.28808i	1.73696	+	0.410079i
$514$		0			0
$515$	4.31540			0.190159
$516$		0			0
$517$			32.3968i			1.42481i
$518$		0			0
$519$	−1.45121	+	0.694436i	−0.0637011	+	0.0304823i
$520$		0			0
$521$	−36.8142			−1.61286			−0.806429	−	0.591330i	$-0.798603\pi$
$521$	−36.8142			−1.61286			−0.806429	+	0.591330i	$0.798603\pi$
$522$		0			0
$523$	27.9097			1.22041			0.610204	−	0.792244i	$-0.291088\pi$
$523$	27.9097			1.22041			0.610204	+	0.792244i	$0.291088\pi$
$524$		0			0
$525$	−4.02003	+	1.92367i	−0.175449	+	0.0839559i
$526$		0			0
$527$	25.4363			1.10802
$528$		0			0
$529$	22.7212			0.987879
$530$		0			0
$531$	21.6125	+	17.4115i	0.937901	+	0.755597i
$532$		0			0
$533$			52.3316i			2.26673i
$534$		0			0
$535$			18.3915i			0.795136i
$536$		0			0
$537$	−9.18040	−	19.1849i	−0.396163	−	0.827891i
$538$		0			0
$539$	−1.12666			−0.0485285
$540$		0			0
$541$			20.7049i			0.890175i	0.895487	+	0.445087i	$0.146827\pi$
$541$			20.7049i			0.890175i	−0.895487	+	0.445087i	$0.853173\pi$
$542$		0			0
$543$	11.1960	+	23.3971i	0.480467	+	1.00407i
$544$		0			0
$545$			9.87730i			0.423097i
$546$		0			0
$547$			12.5445i			0.536366i	0.963368	+	0.268183i	$0.0864232\pi$
$547$			12.5445i			0.536366i	−0.963368	+	0.268183i	$0.913577\pi$
$548$		0			0
$549$	21.1804	+	17.0635i	0.903957	+	0.728251i
$550$		0			0
$551$		−	44.5417i		−	1.89754i
$552$		0			0
$553$	−26.7165			−1.13610
$554$		0			0
$555$	0.600176	+	1.25423i	0.0254760	+	0.0532391i
$556$		0			0
$557$		−	9.89310i		−	0.419184i	−0.977789	−	0.209592i	$-0.932786\pi$
$557$		−	9.89310i		−	0.419184i	0.977789	−	0.209592i	$-0.0672136\pi$
$558$		0			0
$559$	2.84647			0.120393
$560$		0			0
$561$	21.7161	−	10.3916i	0.916855	−	0.438734i
$562$		0			0
$563$	16.7720			0.706854			0.353427	−	0.935462i	$-0.385016\pi$
$563$	16.7720			0.706854			0.353427	+	0.935462i	$0.385016\pi$
$564$		0			0
$565$	5.79898			0.243965
$566$		0			0
$567$	−22.6266	+	4.92860i	−0.950226	+	0.206982i
$568$		0			0
$569$		−	11.4398i		−	0.479581i	−0.970825	−	0.239791i	$-0.922921\pi$
$569$		−	11.4398i		−	0.479581i	0.970825	−	0.239791i	$-0.0770788\pi$
$570$		0			0
$571$	2.81133			0.117651			0.0588253	−	0.998268i	$-0.481265\pi$
$571$	2.81133			0.117651			0.0588253	+	0.998268i	$0.481265\pi$
$572$		0			0
$573$	10.9711	+	22.9271i	0.458325	+	0.957795i
$574$		0			0
$575$			0.527989i			0.0220187i
$576$		0			0
$577$		−	5.58386i		−	0.232459i	−0.993222	−	0.116230i	$-0.962919\pi$
$577$		−	5.58386i		−	0.232459i	0.993222	−	0.116230i	$-0.0370809\pi$
$578$		0			0
$579$	0.421448	+	0.880729i	0.0175148	+	0.0366019i
$580$		0			0
$581$	−13.5465			−0.562005
$582$		0			0
$583$	22.0618			0.913707
$584$		0			0
$585$	−14.5604	−	11.7302i	−0.601998	−	0.484985i
$586$		0			0
$587$	−17.4620			−0.720735			−0.360368	−	0.932810i	$-0.617349\pi$
$587$	−17.4620			−0.720735			−0.360368	+	0.932810i	$0.617349\pi$
$588$		0			0
$589$			42.2541i			1.74105i
$590$		0			0
$591$	9.64171	+	20.1490i	0.396607	+	0.828818i
$592$		0			0
$593$	−13.9840			−0.574252			−0.287126	−	0.957893i	$-0.592700\pi$
$593$	−13.9840			−0.574252			−0.287126	+	0.957893i	$0.592700\pi$
$594$		0			0
$595$	−12.0496			−0.493986
$596$		0			0
$597$	−6.73571	−	14.0761i	−0.275674	−	0.576096i
$598$		0			0
$599$	13.8172			0.564555			0.282278	−	0.959333i	$-0.408910\pi$
$599$	13.8172			0.564555			0.282278	+	0.959333i	$0.408910\pi$
$600$		0			0
$601$	−36.9144			−1.50577			−0.752886	−	0.658151i	$-0.771339\pi$
$601$	−36.9144			−1.50577			−0.752886	+	0.658151i	$0.771339\pi$
$602$		0			0
$603$	−24.0884	−	4.76963i	−0.980955	−	0.194234i
$604$		0			0
$605$	−2.19103			−0.0890780
$606$		0			0
$607$	23.6932			0.961679			0.480839	−	0.876809i	$-0.340332\pi$
$607$	23.6932			0.961679			0.480839	+	0.876809i	$0.340332\pi$
$608$		0			0
$609$	11.0142	+	23.0171i	0.446317	+	0.932701i
$610$		0			0
$611$	68.0309			2.75224
$612$		0			0
$613$	−9.46592			−0.382325			−0.191162	−	0.981558i	$-0.561226\pi$
$613$	−9.46592			−0.382325			−0.191162	+	0.981558i	$0.561226\pi$
$614$		0			0
$615$	−6.27750	−	13.1185i	−0.253133	−	0.528991i
$616$		0			0
$617$			20.5476i			0.827215i	0.910455	+	0.413608i	$0.135731\pi$
$617$			20.5476i			0.827215i	−0.910455	+	0.413608i	$0.864269\pi$
$618$		0			0
$619$	38.8800			1.56272			0.781360	−	0.624081i	$-0.214526\pi$
$619$	38.8800			1.56272			0.781360	+	0.624081i	$0.214526\pi$
$620$		0			0
$621$	−0.630384	+	2.67011i	−0.0252964	+	0.107148i
$622$		0			0
$623$	−4.42207			−0.177166
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	17.2622	+	36.0742i	0.689388	+	1.44066i
$628$		0			0
$629$			3.75942i			0.149898i
$630$		0			0
$631$			45.4430i			1.80906i	0.426415	+	0.904528i	$0.359777\pi$
$631$			45.4430i			1.80906i	−0.426415	+	0.904528i	$0.640223\pi$
$632$		0			0
$633$	−20.9289	−	43.7367i	−0.831851	−	1.73838i
$634$		0			0
$635$	13.9522			0.553677
$636$		0			0
$637$			2.36589i			0.0937400i
$638$		0			0
$639$	37.4968	+	30.2084i	1.48335	+	1.19503i
$640$		0			0
$641$	−39.0345			−1.54177			−0.770886	−	0.636973i	$-0.780186\pi$
$641$	−39.0345			−1.54177			−0.770886	+	0.636973i	$0.780186\pi$
$642$		0			0
$643$	−34.9625			−1.37879			−0.689393	−	0.724388i	$-0.742123\pi$
$643$	−34.9625			−1.37879			−0.689393	+	0.724388i	$0.742123\pi$
$644$		0			0
$645$	−0.713557	+	0.341452i	−0.0280963	+	0.0134447i
$646$		0			0
$647$	11.8961			0.467685			0.233843	−	0.972274i	$-0.424870\pi$
$647$	11.8961			0.467685			0.233843	+	0.972274i	$0.424870\pi$
$648$		0			0
$649$			27.4575i			1.07780i
$650$		0			0
$651$	−10.4485	−	21.8350i	−0.409509	−	0.855780i
$652$		0			0
$653$	32.6451			1.27750			0.638751	−	0.769413i	$-0.279451\pi$
$653$	32.6451			1.27750			0.638751	+	0.769413i	$0.279451\pi$
$654$		0			0
$655$			16.8927i			0.660053i
$656$		0			0
$657$	27.2185	−	33.7856i	1.06190	−	1.31810i
$658$		0			0
$659$		−	42.6342i		−	1.66079i	−0.557173	−	0.830397i	$-0.688114\pi$
$659$		−	42.6342i		−	1.66079i	0.557173	−	0.830397i	$-0.311886\pi$
$660$		0			0
$661$			26.6356i			1.03600i	0.855380	+	0.518002i	$0.173324\pi$
$661$			26.6356i			1.03600i	−0.855380	+	0.518002i	$0.826676\pi$
$662$		0			0
$663$	−21.8216	−	45.6022i	−0.847481	−	1.77104i
$664$		0			0
$665$		−	20.0165i		−	0.776206i
$666$		0			0
$667$	3.02306			0.117053
$668$		0			0
$669$	15.8808	+	33.1872i	0.613987	+	1.28309i
$670$		0			0
$671$			26.9085i			1.03879i
$672$		0			0
$673$		−	13.2731i		−	0.511639i	−0.966725	−	0.255820i	$-0.917655\pi$
$673$		−	13.2731i		−	0.511639i	0.966725	−	0.255820i	$-0.0823453\pi$
$674$		0			0
$675$	5.05713	+	1.19393i	0.194649	+	0.0459545i
$676$		0			0
$677$	41.3714			1.59003			0.795017	−	0.606588i	$-0.207462\pi$
$677$	41.3714			1.59003			0.795017	+	0.606588i	$0.207462\pi$
$678$		0			0
$679$	−3.83167			−0.147046
$680$		0			0
$681$	−5.61715	+	2.68793i	−0.215250	+	0.103002i
$682$		0			0
$683$	41.3034			1.58043			0.790215	−	0.612829i	$-0.209969\pi$
$683$	41.3034			1.58043			0.790215	+	0.612829i	$0.209969\pi$
$684$		0			0
$685$	−14.6923			−0.561365
$686$		0			0
$687$	−28.9629	+	13.8594i	−1.10500	+	0.528768i
$688$		0			0
$689$		−	46.3281i		−	1.76496i
$690$		0			0
$691$	40.2877			1.53262			0.766308	−	0.642473i	$-0.222092\pi$
$691$	40.2877			1.53262			0.766308	+	0.642473i	$0.222092\pi$
$692$		0			0
$693$	−17.8407	−	14.3729i	−0.677711	−	0.545981i
$694$		0			0
$695$		−	21.2620i		−	0.806513i
$696$		0			0
$697$		−	39.3214i		−	1.48941i
$698$		0			0
$699$	−16.3333	−	34.1328i	−0.617781	−	1.29102i
$700$		0			0
$701$	37.2476			1.40682			0.703411	−	0.710783i	$-0.251659\pi$
$701$	37.2476			1.40682			0.703411	+	0.710783i	$0.251659\pi$
$702$		0			0
$703$	−6.24504			−0.235536
$704$		0			0
$705$	−17.0541	+	8.16073i	−0.642293	+	0.307351i
$706$		0			0
$707$			32.0358i			1.20483i
$708$		0			0
$709$	14.5470			0.546324			0.273162	−	0.961968i	$-0.411931\pi$
$709$	14.5470			0.546324			0.273162	+	0.961968i	$0.411931\pi$
$710$		0			0
$711$	24.2574	+	19.5424i	0.909725	+	0.732897i
$712$		0			0
$713$	−2.86779			−0.107400
$714$		0			0
$715$		−	18.4982i		−	0.691792i
$716$		0			0
$717$	−3.90081	−	8.15180i	−0.145678	−	0.304435i
$718$		0			0
$719$			10.0647i			0.375351i	0.982231	+	0.187676i	$0.0600954\pi$
$719$			10.0647i			0.375351i	−0.982231	+	0.187676i	$0.939905\pi$
$720$		0			0
$721$		−	11.1036i		−	0.413519i
$722$		0			0
$723$	3.28451	+	6.86387i	0.122152	+	0.255270i
$724$		0			0
$725$		−	5.72560i		−	0.212644i
$726$		0			0
$727$		−	34.1482i		−	1.26649i	−0.773952	−	0.633244i	$-0.781723\pi$
$727$		−	34.1482i		−	1.26649i	0.773952	−	0.633244i	$-0.218277\pi$
$728$		0			0
$729$	24.1490	+	12.0757i	0.894409	+	0.447250i
$730$		0			0
$731$	−2.13881			−0.0791068
$732$		0			0
$733$			6.43186i			0.237566i	0.992920	+	0.118783i	$0.0378993\pi$
$733$			6.43186i			0.237566i	−0.992920	+	0.118783i	$0.962101\pi$
$734$		0			0
$735$	−0.283803	−	0.593084i	−0.0104682	−	0.0218762i
$736$		0			0
$737$	−11.2763	−	21.5185i	−0.415368	−	0.792644i
$738$		0			0
$739$			12.5699i			0.462390i	0.972907	+	0.231195i	$0.0742635\pi$
$739$			12.5699i			0.462390i	−0.972907	+	0.231195i	$0.925736\pi$
$740$		0			0
$741$	75.7530	−	36.2494i	2.78286	−	1.33166i
$742$		0			0
$743$			5.27574i			0.193548i	0.995306	+	0.0967740i	$0.0308524\pi$
$743$			5.27574i			0.193548i	−0.995306	+	0.0967740i	$0.969148\pi$
$744$		0			0
$745$		−	0.997313i		−	0.0365387i
$746$		0			0
$747$	12.2996	+	9.90890i	0.450021	+	0.362548i
$748$		0			0
$749$	47.3217			1.72910
$750$		0			0
$751$	−10.4971			−0.383045			−0.191523	−	0.981488i	$-0.561343\pi$
$751$	−10.4971			−0.383045			−0.191523	+	0.981488i	$0.561343\pi$
$752$		0			0
$753$	−8.83453	−	18.4622i	−0.321948	−	0.672798i
$754$		0			0
$755$	23.5722			0.857880
$756$		0			0
$757$		−	18.4980i		−	0.672322i	−0.941805	−	0.336161i	$-0.890871\pi$
$757$		−	18.4980i		−	0.672322i	0.941805	−	0.336161i	$-0.109129\pi$
$758$		0			0
$759$	−2.44836	+	1.17159i	−0.0888699	+	0.0425261i
$760$		0			0
$761$			49.6489i			1.79977i	0.436127	+	0.899885i	$0.356350\pi$
$761$			49.6489i			1.79977i	−0.436127	+	0.899885i	$0.643650\pi$
$762$		0			0
$763$	25.4144			0.920064
$764$		0			0
$765$	10.9405	+	8.81396i	0.395556	+	0.318669i
$766$		0			0
$767$	57.6586			2.08193
$768$		0			0
$769$		−	21.5298i		−	0.776384i	−0.921578	−	0.388192i	$-0.873100\pi$
$769$		−	21.5298i		−	0.776384i	0.921578	−	0.388192i	$-0.126900\pi$
$770$		0			0
$771$	−19.7171	+	9.43505i	−0.710094	+	0.339795i
$772$		0			0
$773$		−	16.2746i		−	0.585356i	−0.956211	−	0.292678i	$-0.905454\pi$
$773$		−	16.2746i		−	0.585356i	0.956211	−	0.292678i	$-0.0945465\pi$
$774$		0			0
$775$			5.43154i			0.195107i
$776$		0			0
$777$	3.22715	−	1.54426i	0.115773	−	0.0554000i
$778$		0			0
$779$	65.3196			2.34032
$780$		0			0
$781$			47.6377i			1.70461i
$782$		0			0
$783$	6.83599	−	28.9551i	0.244298	−	1.03477i
$784$		0			0
$785$	−3.48301			−0.124314
$786$		0			0
$787$			49.8683i			1.77761i	0.458281	+	0.888807i	$0.348465\pi$
$787$			49.8683i			1.77761i	−0.458281	+	0.888807i	$0.651535\pi$
$788$		0			0
$789$	−41.1748	+	19.7030i	−1.46586	+	0.701445i
$790$		0			0
$791$		−	14.9209i		−	0.530525i
$792$		0			0
$793$	56.5059			2.00658
$794$		0			0
$795$	5.55735	+	11.6136i	0.197099	+	0.411891i
$796$		0			0
$797$		−	32.9385i		−	1.16674i	−0.812207	−	0.583370i	$-0.801734\pi$
$797$		−	32.9385i		−	1.16674i	0.812207	−	0.583370i	$-0.198266\pi$
$798$		0			0
$799$	−51.1177			−1.80842
$800$		0			0
$801$	4.01504	+	3.23462i	0.141865	+	0.114290i
$802$		0			0
$803$	42.9227			1.51471
$804$		0			0
$805$	1.35852			0.0478817
$806$		0			0
$807$	17.0318	−	8.15007i	0.599547	−	0.286896i
$808$		0			0
$809$	−46.0841			−1.62023			−0.810116	−	0.586270i	$-0.800596\pi$
$809$	−46.0841			−1.62023			−0.810116	+	0.586270i	$0.800596\pi$
$810$		0			0
$811$		−	10.7234i		−	0.376548i	−0.982116	−	0.188274i	$-0.939711\pi$
$811$		−	10.7234i		−	0.376548i	0.982116	−	0.188274i	$-0.0602894\pi$
$812$		0			0
$813$	−16.9229	+	8.09796i	−0.593511	+	0.284008i
$814$		0			0
$815$	−13.8645			−0.485653
$816$		0			0
$817$		−	3.55293i		−	0.124301i
$818$		0			0
$819$	−30.1820	+	37.4641i	−1.05464	+	1.30910i
$820$		0			0
$821$			14.1461i			0.493703i	0.969053	+	0.246852i	$0.0793960\pi$
$821$			14.1461i			0.493703i	−0.969053	+	0.246852i	$0.920604\pi$
$822$		0			0
$823$	−44.4535			−1.54955			−0.774776	−	0.632235i	$-0.782138\pi$
$823$	−44.4535			−1.54955			−0.774776	+	0.632235i	$0.782138\pi$
$824$		0			0
$825$	2.21897	+	4.63714i	0.0772547	+	0.161445i
$826$		0			0
$827$			50.1891i			1.74524i	0.488395	+	0.872622i	$0.337582\pi$
$827$			50.1891i			1.74524i	−0.488395	+	0.872622i	$0.662418\pi$
$828$		0			0
$829$	10.3026			0.357823			0.178911	−	0.983865i	$-0.442742\pi$
$829$	10.3026			0.357823			0.178911	+	0.983865i	$0.442742\pi$
$830$		0			0
$831$	−17.0124	−	35.5520i	−0.590152	−	1.23328i
$832$		0			0
$833$		−	1.77771i		−	0.0615939i
$834$		0			0
$835$			7.74374i			0.267983i
$836$		0			0
$837$	−6.48489	+	27.4680i	−0.224151	+	0.949432i
$838$		0			0
$839$		−	8.60336i		−	0.297021i	−0.988911	−	0.148511i	$-0.952552\pi$
$839$		−	8.60336i		−	0.297021i	0.988911	−	0.148511i	$-0.0474479\pi$
$840$		0			0
$841$	−3.78255			−0.130433
$842$		0			0
$843$	5.42724	+	11.3417i	0.186924	+	0.390628i
$844$		0			0
$845$	−25.8448			−0.889087
$846$		0			0
$847$			5.63754i			0.193708i
$848$		0			0
$849$	22.8892	+	47.8332i	0.785556	+	1.64163i
$850$		0			0
$851$		−	0.423852i		−	0.0145295i
$852$		0			0
$853$	26.0381			0.891526			0.445763	−	0.895151i	$-0.352932\pi$
$853$	26.0381			0.891526			0.445763	+	0.895151i	$0.352932\pi$
$854$		0			0
$855$	−14.6415	+	18.1741i	−0.500729	+	0.621541i
$856$		0			0
$857$	24.8664			0.849422			0.424711	−	0.905329i	$-0.360376\pi$
$857$	24.8664			0.849422			0.424711	+	0.905329i	$0.360376\pi$
$858$		0			0
$859$	−15.9274			−0.543437			−0.271718	−	0.962377i	$-0.587592\pi$
$859$	−15.9274			−0.543437			−0.271718	+	0.962377i	$0.587592\pi$
$860$		0			0
$861$	−33.7542	+	16.1521i	−1.15034	+	0.550462i
$862$		0			0
$863$			5.76860i			0.196366i	0.995168	+	0.0981828i	$0.0313030\pi$
$863$			5.76860i			0.196366i	−0.995168	+	0.0981828i	$0.968697\pi$
$864$		0			0
$865$		−	0.928844i		−	0.0315817i
$866$		0			0
$867$	3.68675	+	7.70447i	0.125209	+	0.261658i
$868$		0			0
$869$			30.8177i			1.04542i
$870$		0			0
$871$	−45.1872	+	23.6794i	−1.53111	+	0.802345i
$872$		0			0
$873$	3.47898	+	2.80276i	0.117746	+	0.0948589i
$874$		0			0
$875$		−	2.57301i		−	0.0869837i
$876$		0			0
$877$	36.6213			1.23661			0.618307	−	0.785937i	$-0.287819\pi$
$877$	36.6213			1.23661			0.618307	+	0.785937i	$0.287819\pi$
$878$		0			0
$879$	−30.3878	+	14.5412i	−1.02495	+	0.490462i
$880$		0			0
$881$		−	1.08958i		−	0.0367090i	−0.999832	−	0.0183545i	$-0.994157\pi$
$881$		−	1.08958i		−	0.0367090i	0.999832	−	0.0183545i	$-0.00584275\pi$
$882$		0			0
$883$		−	20.1862i		−	0.679320i	−0.940548	−	0.339660i	$-0.889688\pi$
$883$		−	20.1862i		−	0.679320i	0.940548	−	0.339660i	$-0.110312\pi$
$884$		0			0
$885$	−14.4539	+	6.91651i	−0.485863	+	0.232496i
$886$		0			0
$887$			2.89356i			0.0971562i	0.998819	+	0.0485781i	$0.0154690\pi$
$887$			2.89356i			0.0971562i	−0.998819	+	0.0485781i	$0.984531\pi$
$888$		0			0
$889$		−	35.8992i		−	1.20402i
$890$		0			0
$891$	5.68518	+	26.0999i	0.190461	+	0.874380i
$892$		0			0
$893$		−	84.9153i		−	2.84158i
$894$		0			0
$895$	12.2793			0.410451
$896$		0			0
$897$	2.46026	+	5.14137i	0.0821455	+	0.171665i
$898$		0			0
$899$	31.0988			1.03720
$900$		0			0
$901$			34.8105i			1.15971i
$902$		0			0
$903$	0.878561	+	1.83599i	0.0292367	+	0.0610980i
$904$		0			0
$905$	−14.9753			−0.497795
$906$		0			0
$907$	15.6945			0.521128			0.260564	−	0.965457i	$-0.416091\pi$
$907$	15.6945			0.521128			0.260564	+	0.965457i	$0.416091\pi$
$908$		0			0
$909$	23.4333	−	29.0871i	0.777232	−	0.964757i
$910$		0			0
$911$			29.1406i			0.965470i	0.875767	+	0.482735i	$0.160357\pi$
$911$			29.1406i			0.965470i	−0.875767	+	0.482735i	$0.839643\pi$
$912$		0			0
$913$			15.6260i			0.517146i
$914$		0			0
$915$	−14.1650	+	6.77823i	−0.468279	+	0.224081i
$916$		0			0
$917$	43.4652			1.43535
$918$		0			0
$919$		−	25.0761i		−	0.827184i	−0.910462	−	0.413592i	$-0.864274\pi$
$919$		−	25.0761i		−	0.827184i	0.910462	−	0.413592i	$-0.135726\pi$
$920$		0			0
$921$	10.6592	+	22.2752i	0.351231	+	0.733993i
$922$		0			0
$923$	100.035			3.29271
$924$		0			0
$925$	−0.802767			−0.0263948
$926$		0			0
$927$	−8.12196	+	10.0816i	−0.266760	+	0.331122i
$928$		0			0
$929$	52.8462			1.73383			0.866913	−	0.498459i	$-0.166100\pi$
$929$	52.8462			1.73383			0.866913	+	0.498459i	$0.166100\pi$
$930$		0			0
$931$	2.95307			0.0967831
$932$		0			0
$933$	−5.54126	−	11.5800i	−0.181413	−	0.379111i
$934$		0			0
$935$			13.8993i			0.454557i
$936$		0			0
$937$			47.9529i			1.56655i	0.621674	+	0.783276i	$0.286453\pi$
$937$			47.9529i			1.56655i	−0.621674	+	0.783276i	$0.713547\pi$
$938$		0			0
$939$	6.25533	−	2.99331i	0.204135	−	0.0976829i
$940$		0			0
$941$	5.70563			0.185998			0.0929990	−	0.995666i	$-0.470355\pi$
$941$	5.70563			0.185998			0.0929990	+	0.995666i	$0.470355\pi$
$942$		0			0
$943$			4.43326i			0.144367i
$944$		0			0
$945$	3.07201	−	13.0121i	0.0999324	−	0.423282i
$946$		0			0
$947$			16.5790i			0.538745i	0.963036	+	0.269373i	$0.0868163\pi$
$947$			16.5790i			0.538745i	−0.963036	+	0.269373i	$0.913184\pi$
$948$		0			0
$949$		−	90.1345i		−	2.92589i
$950$		0			0
$951$	−45.1834	+	21.6212i	−1.46517	+	0.701116i
$952$		0			0
$953$			31.4070i			1.01737i	0.860952	+	0.508686i	$0.169868\pi$
$953$			31.4070i			1.01737i	−0.860952	+	0.508686i	$0.830132\pi$
$954$		0			0
$955$	−14.6745			−0.474854
$956$		0			0
$957$	26.5504	−	12.7050i	0.858254	−	0.410693i
$958$		0			0
$959$			37.8036i			1.22074i
$960$		0			0
$961$	1.49841			0.0483357
$962$		0			0
$963$	−42.9660	−	34.6145i	−1.38456	−	1.11544i
$964$		0			0
$965$	−0.563709			−0.0181464
$966$		0			0
$967$	37.5508			1.20755			0.603777	−	0.797153i	$-0.293662\pi$
$967$	37.5508			1.20755			0.603777	+	0.797153i	$0.293662\pi$
$968$		0			0
$969$	−56.9200	+	27.2374i	−1.82853	+	0.874993i
$970$		0			0
$971$		−	7.15505i		−	0.229617i	−0.993388	−	0.114808i	$-0.963375\pi$
$971$		−	7.15505i		−	0.229617i	0.993388	−	0.114808i	$-0.0366254\pi$
$972$		0			0
$973$	−54.7073			−1.75384
$974$		0			0
$975$	9.73765	−	4.65967i	0.311854	−	0.149229i
$976$		0			0
$977$		−	19.4565i		−	0.622469i	−0.950333	−	0.311235i	$-0.899258\pi$
$977$		−	19.4565i		−	0.622469i	0.950333	−	0.311235i	$-0.100742\pi$
$978$		0			0
$979$			5.10089i			0.163025i
$980$		0			0
$981$	−23.0752	−	18.5899i	−0.736734	−	0.593531i
$982$		0			0
$983$	13.9099			0.443657			0.221828	−	0.975086i	$-0.428798\pi$
$983$	13.9099			0.443657			0.221828	+	0.975086i	$0.428798\pi$
$984$		0			0
$985$	−12.8963			−0.410910
$986$		0			0
$987$	20.9977	+	43.8803i	0.668363	+	1.39673i
$988$		0			0
$989$	0.241138			0.00766775
$990$		0			0
$991$			13.1194i			0.416752i	0.978049	+	0.208376i	$0.0668177\pi$
$991$			13.1194i			0.416752i	−0.978049	+	0.208376i	$0.933182\pi$
$992$		0			0
$993$	−48.1335	+	23.0329i	−1.52747	+	0.730927i
$994$		0			0
$995$	9.00936			0.285616
$996$		0			0
$997$	31.2750			0.990489			0.495245	−	0.868754i	$-0.335078\pi$
$997$	31.2750			0.990489			0.495245	+	0.868754i	$0.335078\pi$
$998$		0			0
$999$	−4.05969	−	0.958450i	−0.128443	−	0.0303240i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.b.401.17	yes	46
3.2	odd	2		4020.2.f.a.401.29	✓	46
67.66	odd	2		4020.2.f.a.401.30	yes	46
201.200	even	2	inner	4020.2.f.b.401.18	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.29	✓	46	3.2	odd	2
4020.2.f.a.401.30	yes	46	67.66	odd	2
4020.2.f.b.401.17	yes	46	1.1	even	1	trivial
4020.2.f.b.401.18	yes	46	201.200	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.747634 - 1.56238i) q^{3} +1.00000 q^{5} -2.57301i q^{7} +(-1.88209 + 2.33618i) q^{9} +O(q^{10})\)	\(q+(-0.747634 - 1.56238i) q^{3} +1.00000 q^{5} -2.57301i q^{7} +(-1.88209 + 2.33618i) q^{9} -2.96799 q^{11} +6.23256i q^{13} +(-0.747634 - 1.56238i) q^{15} -4.68308i q^{17} +7.77939 q^{19} +(-4.02003 + 1.92367i) q^{21} +0.527989i q^{23} +1.00000 q^{25} +(5.05713 + 1.19393i) q^{27} -5.72560i q^{29} +5.43154i q^{31} +(2.21897 + 4.63714i) q^{33} -2.57301i q^{35} -0.802767 q^{37} +(9.73765 - 4.65967i) q^{39} +8.39649 q^{41} -0.456710i q^{43} +(-1.88209 + 2.33618i) q^{45} -10.9154i q^{47} +0.379602 q^{49} +(-7.31677 + 3.50123i) q^{51} -7.43325 q^{53} -2.96799 q^{55} +(-5.81614 - 12.1544i) q^{57} -9.25119i q^{59} -9.06624i q^{61} +(6.01103 + 4.84264i) q^{63} +6.23256i q^{65} +(3.79930 + 7.25019i) q^{67} +(0.824922 - 0.394743i) q^{69} -16.0505i q^{71} -14.4619 q^{73} +(-0.747634 - 1.56238i) q^{75} +7.63668i q^{77} -10.3834i q^{79} +(-1.91550 - 8.79380i) q^{81} -5.26485i q^{83} -4.68308i q^{85} +(-8.94559 + 4.28066i) q^{87} -1.71863i q^{89} +16.0365 q^{91} +(8.48615 - 4.06080i) q^{93} +7.77939 q^{95} -1.48917i q^{97} +(5.58602 - 6.93377i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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

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