LMFDB - Embedded newform 4020.2.f.b.401.19

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			401.19
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.b.401.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.296734 - 1.70644i) q^{3} +1.00000 q^{5} +4.76000i q^{7} +(-2.82390 + 1.01272i) q^{9} +O(q^{10})$	\(q+(-0.296734 - 1.70644i) q^{3} +1.00000 q^{5} +4.76000i q^{7} +(-2.82390 + 1.01272i) q^{9} -4.74465 q^{11} -2.55265i q^{13} +(-0.296734 - 1.70644i) q^{15} -2.48618i q^{17} +1.87453 q^{19} +(8.12267 - 1.41245i) q^{21} -0.122120i q^{23} +1.00000 q^{25} +(2.56610 + 4.51831i) q^{27} -5.57143i q^{29} +1.05987i q^{31} +(1.40790 + 8.09647i) q^{33} +4.76000i q^{35} -6.44944 q^{37} +(-4.35595 + 0.757458i) q^{39} +4.08362 q^{41} -3.58467i q^{43} +(-2.82390 + 1.01272i) q^{45} +4.59365i q^{47} -15.6576 q^{49} +(-4.24253 + 0.737736i) q^{51} +12.4181 q^{53} -4.74465 q^{55} +(-0.556238 - 3.19879i) q^{57} -8.30390i q^{59} +0.253560i q^{61} +(-4.82054 - 13.4417i) q^{63} -2.55265i q^{65} +(8.02933 + 1.59054i) q^{67} +(-0.208392 + 0.0362373i) q^{69} -5.92861i q^{71} +9.97870 q^{73} +(-0.296734 - 1.70644i) q^{75} -22.5845i q^{77} +11.8007i q^{79} +(6.94880 - 5.71963i) q^{81} -6.91111i q^{83} -2.48618i q^{85} +(-9.50733 + 1.65323i) q^{87} -13.5896i q^{89} +12.1506 q^{91} +(1.80861 - 0.314501i) q^{93} +1.87453 q^{95} -15.3389i q^{97} +(13.3984 - 4.80500i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.296734	−	1.70644i	−0.171319	−	0.985216i
$3$	−0.296734	−	1.70644i	−0.171319	−	0.985216i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			4.76000i			1.79911i	0.436808	+	0.899555i	$0.356109\pi$
$7$			4.76000i			1.79911i	−0.436808	+	0.899555i	$0.643891\pi$
$8$		0			0
$9$	−2.82390	+	1.01272i	−0.941299	+	0.337573i
$10$		0			0
$11$	−4.74465			−1.43057			−0.715283	−	0.698835i	$-0.753702\pi$
$11$	−4.74465			−1.43057			−0.715283	+	0.698835i	$0.753702\pi$
$12$		0			0
$13$		−	2.55265i		−	0.707978i	−0.935250	−	0.353989i	$-0.884825\pi$
$13$		−	2.55265i		−	0.707978i	0.935250	−	0.353989i	$-0.115175\pi$
$14$		0			0
$15$	−0.296734	−	1.70644i	−0.0766164	−	0.440602i
$16$		0			0
$17$		−	2.48618i		−	0.602988i	−0.953468	−	0.301494i	$-0.902515\pi$
$17$		−	2.48618i		−	0.602988i	0.953468	−	0.301494i	$-0.0974854\pi$
$18$		0			0
$19$	1.87453			0.430048			0.215024	−	0.976609i	$-0.431017\pi$
$19$	1.87453			0.430048			0.215024	+	0.976609i	$0.431017\pi$
$20$		0			0
$21$	8.12267	−	1.41245i	1.77251	−	0.308222i
$22$		0			0
$23$		−	0.122120i		−	0.0254639i	−0.999919	−	0.0127319i	$-0.995947\pi$
$23$		−	0.122120i		−	0.0254639i	0.999919	−	0.0127319i	$-0.00405281\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	2.56610	+	4.51831i	0.493845	+	0.869550i
$28$		0			0
$29$		−	5.57143i		−	1.03459i	−0.855807	−	0.517295i	$-0.826939\pi$
$29$		−	5.57143i		−	1.03459i	0.855807	−	0.517295i	$-0.173061\pi$
$30$		0			0
$31$			1.05987i			0.190359i	0.995460	+	0.0951795i	$0.0303425\pi$
$31$			1.05987i			0.190359i	−0.995460	+	0.0951795i	$0.969657\pi$
$32$		0			0
$33$	1.40790	+	8.09647i	0.245084	+	1.40942i
$34$		0			0
$35$			4.76000i			0.804586i
$36$		0			0
$37$	−6.44944			−1.06028			−0.530141	−	0.847910i	$-0.677861\pi$
$37$	−6.44944			−1.06028			−0.530141	+	0.847910i	$0.677861\pi$
$38$		0			0
$39$	−4.35595	+	0.757458i	−0.697511	+	0.121290i
$40$		0			0
$41$	4.08362			0.637755			0.318877	−	0.947796i	$-0.396694\pi$
$41$	4.08362			0.637755			0.318877	+	0.947796i	$0.396694\pi$
$42$		0			0
$43$		−	3.58467i		−	0.546658i	−0.961921	−	0.273329i	$-0.911875\pi$
$43$		−	3.58467i		−	0.546658i	0.961921	−	0.273329i	$-0.0881247\pi$
$44$		0			0
$45$	−2.82390	+	1.01272i	−0.420962	+	0.150967i
$46$		0			0
$47$			4.59365i			0.670053i	0.942209	+	0.335027i	$0.108745\pi$
$47$			4.59365i			0.670053i	−0.942209	+	0.335027i	$0.891255\pi$
$48$		0			0
$49$	−15.6576			−2.23680
$50$		0			0
$51$	−4.24253	+	0.737736i	−0.594074	+	0.103304i
$52$		0			0
$53$	12.4181			1.70575			0.852876	−	0.522114i	$-0.174856\pi$
$53$	12.4181			1.70575			0.852876	+	0.522114i	$0.174856\pi$
$54$		0			0
$55$	−4.74465			−0.639768
$56$		0			0
$57$	−0.556238	−	3.19879i	−0.0736755	−	0.423690i
$58$		0			0
$59$		−	8.30390i		−	1.08108i	−0.841320	−	0.540538i	$-0.818221\pi$
$59$		−	8.30390i		−	1.08108i	0.841320	−	0.540538i	$-0.181779\pi$
$60$		0			0
$61$			0.253560i			0.0324650i	0.999868	+	0.0162325i	$0.00516719\pi$
$61$			0.253560i			0.0324650i	−0.999868	+	0.0162325i	$0.994833\pi$
$62$		0			0
$63$	−4.82054	−	13.4417i	−0.607331	−	1.69350i
$64$		0			0
$65$		−	2.55265i		−	0.316617i
$66$		0			0
$67$	8.02933	+	1.59054i	0.980939	+	0.194315i
$67$	8.02933	+	1.59054i	0.980939	+	0.194315i
$68$		0			0
$69$	−0.208392	+	0.0362373i	−0.0250874	+	0.00436246i
$70$		0			0
$71$		−	5.92861i		−	0.703597i	−0.936076	−	0.351798i	$-0.885570\pi$
$71$		−	5.92861i		−	0.703597i	0.936076	−	0.351798i	$-0.114430\pi$
$72$		0			0
$73$	9.97870			1.16792			0.583959	−	0.811783i	$-0.301503\pi$
$73$	9.97870			1.16792			0.583959	+	0.811783i	$0.301503\pi$
$74$		0			0
$75$	−0.296734	−	1.70644i	−0.0342639	−	0.197043i
$76$		0			0
$77$		−	22.5845i		−	2.57374i
$78$		0			0
$79$			11.8007i			1.32768i	0.747875	+	0.663839i	$0.231074\pi$
$79$			11.8007i			1.32768i	−0.747875	+	0.663839i	$0.768926\pi$
$80$		0			0
$81$	6.94880	−	5.71963i	0.772089	−	0.635515i
$82$		0			0
$83$		−	6.91111i		−	0.758593i	−0.925275	−	0.379296i	$-0.876166\pi$
$83$		−	6.91111i		−	0.758593i	0.925275	−	0.379296i	$-0.123834\pi$
$84$		0			0
$85$		−	2.48618i		−	0.269665i
$86$		0			0
$87$	−9.50733	+	1.65323i	−1.01929	+	0.177245i
$88$		0			0
$89$		−	13.5896i		−	1.44049i	−0.693720	−	0.720245i	$-0.744029\pi$
$89$		−	13.5896i		−	1.44049i	0.693720	−	0.720245i	$-0.255971\pi$
$90$		0			0
$91$	12.1506			1.27373
$92$		0			0
$93$	1.80861	−	0.314501i	0.187545	−	0.0326122i
$94$		0			0
$95$	1.87453			0.192323
$96$		0			0
$97$		−	15.3389i		−	1.55743i	−0.627380	−	0.778714i	$-0.715873\pi$
$97$		−	15.3389i		−	1.55743i	0.627380	−	0.778714i	$-0.284127\pi$
$98$		0			0
$99$	13.3984	−	4.80500i	1.34659	−	0.482920i
$100$		0			0
$101$	6.17460			0.614396			0.307198	−	0.951646i	$-0.400609\pi$
$101$	6.17460			0.614396			0.307198	+	0.951646i	$0.400609\pi$
$102$		0			0
$103$	0.909903			0.0896554			0.0448277	−	0.998995i	$-0.485726\pi$
$103$	0.909903			0.0896554			0.0448277	+	0.998995i	$0.485726\pi$
$104$		0			0
$105$	8.12267	−	1.41245i	0.792691	−	0.137841i
$106$		0			0
$107$		−	14.0533i		−	1.35858i	−0.733869	−	0.679291i	$-0.762287\pi$
$107$		−	14.0533i		−	1.35858i	0.733869	−	0.679291i	$-0.237713\pi$
$108$		0			0
$109$		−	4.57405i		−	0.438114i	−0.975712	−	0.219057i	$-0.929702\pi$
$109$		−	4.57405i		−	0.438114i	0.975712	−	0.219057i	$-0.0702981\pi$
$110$		0			0
$111$	1.91377	+	11.0056i	0.181647	+	1.04461i
$112$		0			0
$113$	2.69027			0.253079			0.126539	−	0.991962i	$-0.459613\pi$
$113$	2.69027			0.253079			0.126539	+	0.991962i	$0.459613\pi$
$114$		0			0
$115$		−	0.122120i		−	0.0113878i
$116$		0			0
$117$	2.58512	+	7.20842i	0.238994	+	0.666419i
$118$		0			0
$119$	11.8342			1.08484
$120$		0			0
$121$	11.5117			1.04652
$122$		0			0
$123$	−1.21175	−	6.96847i	−0.109260	−	0.628326i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−0.577470			−0.0512422			−0.0256211	−	0.999672i	$-0.508156\pi$
$127$	−0.577470			−0.0512422			−0.0256211	+	0.999672i	$0.508156\pi$
$128$		0			0
$129$	−6.11704	+	1.06369i	−0.538576	+	0.0936531i
$130$		0			0
$131$		−	17.2744i		−	1.50927i	−0.656144	−	0.754636i	$-0.727814\pi$
$131$		−	17.2744i		−	1.50927i	0.656144	−	0.754636i	$-0.272186\pi$
$132$		0			0
$133$			8.92278i			0.773703i
$134$		0			0
$135$	2.56610	+	4.51831i	0.220854	+	0.388875i
$136$		0			0
$137$	−7.68503			−0.656576			−0.328288	−	0.944578i	$-0.606472\pi$
$137$	−7.68503			−0.656576			−0.328288	+	0.944578i	$0.606472\pi$
$138$		0			0
$139$		−	2.00003i		−	0.169640i	−0.996396	−	0.0848201i	$-0.972968\pi$
$139$		−	2.00003i		−	0.169640i	0.996396	−	0.0848201i	$-0.0270316\pi$
$140$		0			0
$141$	7.83881	−	1.36309i	0.660147	−	0.114793i
$142$		0			0
$143$			12.1114i			1.01281i
$144$		0			0
$145$		−	5.57143i		−	0.462682i
$146$		0			0
$147$	4.64613	+	26.7188i	0.383207	+	2.20373i
$148$		0			0
$149$			3.70488i			0.303516i	0.988418	+	0.151758i	$0.0484934\pi$
$149$			3.70488i			0.303516i	−0.988418	+	0.151758i	$0.951507\pi$
$150$		0			0
$151$	9.94769			0.809532			0.404766	−	0.914420i	$-0.367353\pi$
$151$	9.94769			0.809532			0.404766	+	0.914420i	$0.367353\pi$
$152$		0			0
$153$	2.51781	+	7.02073i	0.203553	+	0.567593i
$154$		0			0
$155$			1.05987i			0.0851311i
$156$		0			0
$157$	−24.4718			−1.95306			−0.976532	−	0.215374i	$-0.930903\pi$
$157$	−24.4718			−1.95306			−0.976532	+	0.215374i	$0.930903\pi$
$158$		0			0
$159$	−3.68486	−	21.1907i	−0.292228	−	1.68053i
$160$		0			0
$161$	0.581293			0.0458123
$162$		0			0
$163$	19.4604			1.52425			0.762127	−	0.647427i	$-0.224155\pi$
$163$	19.4604			1.52425			0.762127	+	0.647427i	$0.224155\pi$
$164$		0			0
$165$	1.40790	+	8.09647i	0.109605	+	0.630310i
$166$		0			0
$167$			5.28582i			0.409029i	0.978864	+	0.204514i	$0.0655615\pi$
$167$			5.28582i			0.409029i	−0.978864	+	0.204514i	$0.934438\pi$
$168$		0			0
$169$	6.48398			0.498767
$170$		0			0
$171$	−5.29349	+	1.89838i	−0.404803	+	0.145173i
$172$		0			0
$173$		−	20.5320i		−	1.56102i	−0.625143	−	0.780510i	$-0.714959\pi$
$173$		−	20.5320i		−	1.56102i	0.625143	−	0.780510i	$-0.285041\pi$
$174$		0			0
$175$			4.76000i			0.359822i
$176$		0			0
$177$	−14.1701	+	2.46405i	−1.06509	+	0.185209i
$178$		0			0
$179$	−10.9483			−0.818312			−0.409156	−	0.912464i	$-0.634177\pi$
$179$	−10.9483			−0.818312			−0.409156	+	0.912464i	$0.634177\pi$
$180$		0			0
$181$	15.8683			1.17948			0.589741	−	0.807593i	$-0.299230\pi$
$181$	15.8683			1.17948			0.589741	+	0.807593i	$0.299230\pi$
$182$		0			0
$183$	0.432685	−	0.0752397i	0.0319850	−	0.00556188i
$184$		0			0
$185$	−6.44944			−0.474172
$186$		0			0
$187$			11.7961i			0.862614i
$188$		0			0
$189$	−21.5072	+	12.2146i	−1.56442	+	0.888482i
$190$		0			0
$191$	15.4570			1.11843			0.559213	−	0.829024i	$-0.311103\pi$
$191$	15.4570			1.11843			0.559213	+	0.829024i	$0.311103\pi$
$192$		0			0
$193$	−15.1188			−1.08828			−0.544138	−	0.838995i	$-0.683143\pi$
$193$	−15.1188			−1.08828			−0.544138	+	0.838995i	$0.683143\pi$
$194$		0			0
$195$	−4.35595	+	0.757458i	−0.311936	+	0.0542427i
$196$		0			0
$197$	5.95793			0.424485			0.212242	−	0.977217i	$-0.431923\pi$
$197$	5.95793			0.424485			0.212242	+	0.977217i	$0.431923\pi$
$198$		0			0
$199$	−17.7334			−1.25709			−0.628544	−	0.777774i	$-0.716349\pi$
$199$	−17.7334			−1.25709			−0.628544	+	0.777774i	$0.716349\pi$
$200$		0			0
$201$	0.331592	−	14.1736i	0.0233887	−	0.999726i
$202$		0			0
$203$	26.5200			1.86134
$204$		0			0
$205$	4.08362			0.285213
$206$		0			0
$207$	0.123674	+	0.344856i	0.00859592	+	0.0239691i
$208$		0			0
$209$	−8.89400			−0.615211
$210$		0			0
$211$	−0.114572			−0.00788748			−0.00394374	−	0.999992i	$-0.501255\pi$
$211$	−0.114572			−0.00788748			−0.00394374	+	0.999992i	$0.501255\pi$
$212$		0			0
$213$	−10.1168	+	1.75922i	−0.693194	+	0.120540i
$214$		0			0
$215$		−	3.58467i		−	0.244473i
$216$		0			0
$217$	−5.04500			−0.342477
$218$		0			0
$219$	−2.96102	−	17.0281i	−0.200087	−	1.15065i
$220$		0			0
$221$	−6.34636			−0.426902
$222$		0			0
$223$	11.0079			0.737142			0.368571	−	0.929600i	$-0.379847\pi$
$223$	11.0079			0.737142			0.368571	+	0.929600i	$0.379847\pi$
$224$		0			0
$225$	−2.82390	+	1.01272i	−0.188260	+	0.0675146i
$226$		0			0
$227$			14.2709i			0.947194i	0.880742	+	0.473597i	$0.157045\pi$
$227$			14.2709i			0.947194i	−0.880742	+	0.473597i	$0.842955\pi$
$228$		0			0
$229$			27.3819i			1.80945i	0.425996	+	0.904725i	$0.359924\pi$
$229$			27.3819i			1.80945i	−0.425996	+	0.904725i	$0.640076\pi$
$230$		0			0
$231$	−38.5392	+	6.70159i	−2.53569	+	0.440932i
$232$		0			0
$233$	−14.4473			−0.946476			−0.473238	−	0.880935i	$-0.656915\pi$
$233$	−14.4473			−0.946476			−0.473238	+	0.880935i	$0.656915\pi$
$234$		0			0
$235$			4.59365i			0.299657i
$236$		0			0
$237$	20.1372	−	3.50166i	1.30805	−	0.227457i
$238$		0			0
$239$	−13.3901			−0.866133			−0.433066	−	0.901362i	$-0.642568\pi$
$239$	−13.3901			−0.866133			−0.433066	+	0.901362i	$0.642568\pi$
$240$		0			0
$241$	4.46088			0.287350			0.143675	−	0.989625i	$-0.454108\pi$
$241$	4.46088			0.287350			0.143675	+	0.989625i	$0.454108\pi$
$242$		0			0
$243$	−11.8222	−	10.1605i	−0.758393	−	0.651798i
$244$		0			0
$245$	−15.6576			−1.00033
$246$		0			0
$247$		−	4.78503i		−	0.304464i
$248$		0			0
$249$	−11.7934	+	2.05076i	−0.747377	+	0.129962i
$250$		0			0
$251$	15.1213			0.954448			0.477224	−	0.878782i	$-0.341643\pi$
$251$	15.1213			0.954448			0.477224	+	0.878782i	$0.341643\pi$
$252$		0			0
$253$			0.579418i			0.0364277i
$254$		0			0
$255$	−4.24253	+	0.737736i	−0.265678	+	0.0461988i
$256$		0			0
$257$		−	27.1601i		−	1.69420i	−0.531433	−	0.847100i	$-0.678346\pi$
$257$		−	27.1601i		−	1.69420i	0.531433	−	0.847100i	$-0.321654\pi$
$258$		0			0
$259$		−	30.6993i		−	1.90756i
$260$		0			0
$261$	5.64230	+	15.7332i	0.349250	+	0.973858i
$262$		0			0
$263$			4.68319i			0.288778i	0.989521	+	0.144389i	$0.0461217\pi$
$263$			4.68319i			0.288778i	−0.989521	+	0.144389i	$0.953878\pi$
$264$		0			0
$265$	12.4181			0.762835
$266$		0			0
$267$	−23.1898	+	4.03248i	−1.41919	+	0.246784i
$268$		0			0
$269$		−	11.0879i		−	0.676043i	−0.941138	−	0.338022i	$-0.890242\pi$
$269$		−	11.0879i		−	0.676043i	0.941138	−	0.338022i	$-0.109758\pi$
$270$		0			0
$271$		−	2.01964i		−	0.122684i	−0.998117	−	0.0613422i	$-0.980462\pi$
$271$		−	2.01964i		−	0.122684i	0.998117	−	0.0613422i	$-0.0195381\pi$
$272$		0			0
$273$	−3.60550	−	20.7343i	−0.218215	−	1.25490i
$274$		0			0
$275$	−4.74465			−0.286113
$276$		0			0
$277$	−4.07418			−0.244794			−0.122397	−	0.992481i	$-0.539058\pi$
$277$	−4.07418			−0.244794			−0.122397	+	0.992481i	$0.539058\pi$
$278$		0			0
$279$	−1.07336	−	2.99298i	−0.0642601	−	0.179185i
$280$		0			0
$281$	−15.5219			−0.925962			−0.462981	−	0.886368i	$-0.653220\pi$
$281$	−15.5219			−0.925962			−0.462981	+	0.886368i	$0.653220\pi$
$282$		0			0
$283$	−29.3652			−1.74558			−0.872791	−	0.488095i	$-0.837692\pi$
$283$	−29.3652			−1.74558			−0.872791	+	0.488095i	$0.837692\pi$
$284$		0			0
$285$	−0.556238	−	3.19879i	−0.0329487	−	0.189480i
$286$		0			0
$287$			19.4380i			1.14739i
$288$		0			0
$289$	10.8189			0.636405
$290$		0			0
$291$	−26.1749	+	4.55157i	−1.53440	+	0.266818i
$292$		0			0
$293$		−	30.4915i		−	1.78133i	−0.454657	−	0.890666i	$-0.650238\pi$
$293$		−	30.4915i		−	1.78133i	0.454657	−	0.890666i	$-0.349762\pi$
$294$		0			0
$295$		−	8.30390i		−	0.483472i
$296$		0			0
$297$	−12.1752	−	21.4378i	−0.706478	−	1.24395i
$298$		0			0
$299$	−0.311731			−0.0180279
$300$		0			0
$301$	17.0630			0.983497
$302$		0			0
$303$	−1.83221	−	10.5366i	−0.105258	−	0.605312i
$304$		0			0
$305$			0.253560i			0.0145188i
$306$		0			0
$307$	12.3322			0.703834			0.351917	−	0.936031i	$-0.385530\pi$
$307$	12.3322			0.703834			0.351917	+	0.936031i	$0.385530\pi$
$308$		0			0
$309$	−0.269999	−	1.55270i	−0.0153597	−	0.0883299i
$310$		0			0
$311$	16.4147			0.930794			0.465397	−	0.885102i	$-0.345911\pi$
$311$	16.4147			0.930794			0.465397	+	0.885102i	$0.345911\pi$
$312$		0			0
$313$		−	9.49913i		−	0.536923i	−0.963290	−	0.268461i	$-0.913485\pi$
$313$		−	9.49913i		−	0.536923i	0.963290	−	0.268461i	$-0.0865152\pi$
$314$		0			0
$315$	−4.82054	−	13.4417i	−0.271607	−	0.757357i
$316$		0			0
$317$			16.9833i			0.953879i	0.878936	+	0.476939i	$0.158254\pi$
$317$			16.9833i			0.953879i	−0.878936	+	0.476939i	$0.841746\pi$
$318$		0			0
$319$			26.4345i			1.48005i
$320$		0			0
$321$	−23.9811	+	4.17009i	−1.33850	+	0.232752i
$322$		0			0
$323$		−	4.66044i		−	0.259314i
$324$		0			0
$325$		−	2.55265i		−	0.141596i
$326$		0			0
$327$	−7.80535	+	1.35727i	−0.431637	+	0.0750575i
$328$		0			0
$329$	−21.8658			−1.20550
$330$		0			0
$331$			9.20324i			0.505856i	0.967485	+	0.252928i	$0.0813936\pi$
$331$			9.20324i			0.505856i	−0.967485	+	0.252928i	$0.918606\pi$
$332$		0			0
$333$	18.2126	−	6.53147i	0.998042	−	0.357923i
$334$		0			0
$335$	8.02933	+	1.59054i	0.438689	+	0.0869005i
$336$		0			0
$337$		−	8.29037i		−	0.451605i	−0.974173	−	0.225803i	$-0.927500\pi$
$337$		−	8.29037i		−	0.451605i	0.974173	−	0.225803i	$-0.0725005\pi$
$338$		0			0
$339$	−0.798293	−	4.59079i	−0.0433573	−	0.249337i
$340$		0			0
$341$		−	5.02873i		−	0.272321i
$342$		0			0
$343$		−	41.2100i		−	2.22513i
$344$		0			0
$345$	−0.208392	+	0.0362373i	−0.0112194	+	0.00195095i
$346$		0			0
$347$	20.0596			1.07686			0.538428	−	0.842671i	$-0.319018\pi$
$347$	20.0596			1.07686			0.538428	+	0.842671i	$0.319018\pi$
$348$		0			0
$349$	4.02583			0.215498			0.107749	−	0.994178i	$-0.465636\pi$
$349$	4.02583			0.215498			0.107749	+	0.994178i	$0.465636\pi$
$350$		0			0
$351$	11.5337	−	6.55034i	0.615622	−	0.349631i
$352$		0			0
$353$	−27.3740			−1.45697			−0.728485	−	0.685061i	$-0.759775\pi$
$353$	−27.3740			−1.45697			−0.728485	+	0.685061i	$0.759775\pi$
$354$		0			0
$355$		−	5.92861i		−	0.314658i
$356$		0			0
$357$	−3.51162	−	20.1944i	−0.185855	−	1.06880i
$358$		0			0
$359$			33.1291i			1.74849i	0.485485	+	0.874245i	$0.338643\pi$
$359$			33.1291i			1.74849i	−0.485485	+	0.874245i	$0.661357\pi$
$360$		0			0
$361$	−15.4861			−0.815059
$362$		0			0
$363$	−3.41591	−	19.6440i	−0.179289	−	1.03104i
$364$		0			0
$365$	9.97870			0.522309
$366$		0			0
$367$		−	34.0474i		−	1.77726i	−0.458624	−	0.888630i	$-0.651657\pi$
$367$		−	34.0474i		−	1.77726i	0.458624	−	0.888630i	$-0.348343\pi$
$368$		0			0
$369$	−11.5317	+	4.13556i	−0.600318	+	0.215289i
$370$		0			0
$371$			59.1099i			3.06883i
$372$		0			0
$373$			0.619600i			0.0320817i	0.999871	+	0.0160408i	$0.00510617\pi$
$373$			0.619600i			0.0320817i	−0.999871	+	0.0160408i	$0.994894\pi$
$374$		0			0
$375$	−0.296734	−	1.70644i	−0.0153233	−	0.0881204i
$376$		0			0
$377$	−14.2219			−0.732466
$378$		0			0
$379$			16.3250i			0.838558i	0.907857	+	0.419279i	$0.137717\pi$
$379$			16.3250i			0.838558i	−0.907857	+	0.419279i	$0.862283\pi$
$380$		0			0
$381$	0.171355	+	0.985420i	0.00877878	+	0.0504846i
$382$		0			0
$383$	−11.5187			−0.588576			−0.294288	−	0.955717i	$-0.595082\pi$
$383$	−11.5187			−0.588576			−0.294288	+	0.955717i	$0.595082\pi$
$384$		0			0
$385$		−	22.5845i		−	1.15101i
$386$		0			0
$387$	3.63027	+	10.1228i	0.184537	+	0.514568i
$388$		0			0
$389$		−	23.0933i		−	1.17088i	−0.810717	−	0.585438i	$-0.800923\pi$
$389$		−	23.0933i		−	1.17088i	0.810717	−	0.585438i	$-0.199077\pi$
$390$		0			0
$391$	−0.303614			−0.0153544
$392$		0			0
$393$	−29.4778	+	5.12590i	−1.48696	+	0.258568i
$394$		0			0
$395$			11.8007i			0.593756i
$396$		0			0
$397$	−9.42066			−0.472809			−0.236405	−	0.971655i	$-0.575969\pi$
$397$	−9.42066			−0.472809			−0.236405	+	0.971655i	$0.575969\pi$
$398$		0			0
$399$	15.2262	−	2.64769i	0.762264	−	0.132550i
$400$		0			0
$401$	−33.0276			−1.64932			−0.824659	−	0.565630i	$-0.808633\pi$
$401$	−33.0276			−1.64932			−0.824659	+	0.565630i	$0.808633\pi$
$402$		0			0
$403$	2.70549			0.134770
$404$		0			0
$405$	6.94880	−	5.71963i	0.345289	−	0.284211i
$406$		0			0
$407$	30.6003			1.51680
$408$		0			0
$409$			8.17826i			0.404389i	0.979345	+	0.202194i	$0.0648073\pi$
$409$			8.17826i			0.404389i	−0.979345	+	0.202194i	$0.935193\pi$
$410$		0			0
$411$	2.28041	+	13.1141i	0.112484	+	0.646869i
$412$		0			0
$413$	39.5265			1.94497
$414$		0			0
$415$		−	6.91111i		−	0.339253i
$416$		0			0
$417$	−3.41294	+	0.593477i	−0.167132	+	0.0290627i
$418$		0			0
$419$		−	22.7695i		−	1.11236i	−0.831061	−	0.556181i	$-0.812266\pi$
$419$		−	22.7695i		−	1.11236i	0.831061	−	0.556181i	$-0.187734\pi$
$420$		0			0
$421$	4.69557			0.228848			0.114424	−	0.993432i	$-0.463498\pi$
$421$	4.69557			0.228848			0.114424	+	0.993432i	$0.463498\pi$
$422$		0			0
$423$	−4.65208	−	12.9720i	−0.226192	−	0.630721i
$424$		0			0
$425$		−	2.48618i		−	0.120598i
$426$		0			0
$427$	−1.20694			−0.0584080
$428$		0			0
$429$	20.6675	−	3.59387i	0.997835	−	0.173514i
$430$		0			0
$431$			18.2505i			0.879098i	0.898219	+	0.439549i	$0.144862\pi$
$431$			18.2505i			0.879098i	−0.898219	+	0.439549i	$0.855138\pi$
$432$		0			0
$433$		−	2.45444i		−	0.117953i	−0.998259	−	0.0589765i	$-0.981216\pi$
$433$		−	2.45444i		−	0.117953i	0.998259	−	0.0589765i	$-0.0187837\pi$
$434$		0			0
$435$	−9.50733	+	1.65323i	−0.455842	+	0.0792665i
$436$		0			0
$437$		−	0.228919i		−	0.0109507i
$438$		0			0
$439$	24.4383			1.16638			0.583189	−	0.812336i	$-0.301805\pi$
$439$	24.4383			1.16638			0.583189	+	0.812336i	$0.301805\pi$
$440$		0			0
$441$	44.2154	−	15.8567i	2.10549	−	0.755082i
$442$		0			0
$443$	36.2643			1.72297			0.861484	−	0.507784i	$-0.169535\pi$
$443$	36.2643			1.72297			0.861484	+	0.507784i	$0.169535\pi$
$444$		0			0
$445$		−	13.5896i		−	0.644207i
$446$		0			0
$447$	6.32217	−	1.09936i	0.299028	−	0.0519981i
$448$		0			0
$449$			2.51957i			0.118906i	0.998231	+	0.0594530i	$0.0189357\pi$
$449$			2.51957i			0.118906i	−0.998231	+	0.0594530i	$0.981064\pi$
$450$		0			0
$451$	−19.3753			−0.912350
$452$		0			0
$453$	−2.95182	−	16.9752i	−0.138689	−	0.797563i
$454$		0			0
$455$	12.1506			0.569629
$456$		0			0
$457$	−12.1798			−0.569746			−0.284873	−	0.958565i	$-0.591951\pi$
$457$	−12.1798			−0.569746			−0.284873	+	0.958565i	$0.591951\pi$
$458$		0			0
$459$	11.2334	−	6.37979i	0.524328	−	0.297783i
$460$		0			0
$461$		−	0.264273i		−	0.0123084i	−0.999981	−	0.00615421i	$-0.998041\pi$
$461$		−	0.264273i		−	0.0123084i	0.999981	−	0.00615421i	$-0.00195896\pi$
$462$		0			0
$463$			14.6295i			0.679892i	0.940445	+	0.339946i	$0.110409\pi$
$463$			14.6295i			0.679892i	−0.940445	+	0.339946i	$0.889591\pi$
$464$		0			0
$465$	1.80861	−	0.314501i	0.0838725	−	0.0145846i
$466$		0			0
$467$			7.68221i			0.355490i	0.984077	+	0.177745i	$0.0568803\pi$
$467$			7.68221i			0.355490i	−0.984077	+	0.177745i	$0.943120\pi$
$468$		0			0
$469$	−7.57097	+	38.2196i	−0.349595	+	1.76482i
$470$		0			0
$471$	7.26162	+	41.7598i	0.334598	+	1.92419i
$472$		0			0
$473$			17.0080i			0.782029i
$474$		0			0
$475$	1.87453			0.0860095
$476$		0			0
$477$	−35.0673	+	12.5760i	−1.60562	+	0.575816i
$478$		0			0
$479$			5.96989i			0.272771i	0.990656	+	0.136386i	$0.0435486\pi$
$479$			5.96989i			0.272771i	−0.990656	+	0.136386i	$0.956451\pi$
$480$		0			0
$481$			16.4632i			0.750656i
$482$		0			0
$483$	−0.172489	−	0.991943i	−0.00784854	−	0.0451350i
$484$		0			0
$485$		−	15.3389i		−	0.696503i
$486$		0			0
$487$			12.2597i			0.555540i	0.960648	+	0.277770i	$0.0895952\pi$
$487$			12.2597i			0.555540i	−0.960648	+	0.277770i	$0.910405\pi$
$488$		0			0
$489$	−5.77456	−	33.2080i	−0.261135	−	1.50172i
$490$		0			0
$491$		−	28.3098i		−	1.27761i	−0.769371	−	0.638803i	$-0.779430\pi$
$491$		−	28.3098i		−	1.27761i	0.769371	−	0.638803i	$-0.220570\pi$
$492$		0			0
$493$	−13.8516			−0.623845
$494$		0			0
$495$	13.3984	−	4.80500i	0.602213	−	0.215969i
$496$		0			0
$497$	28.2202			1.26585
$498$		0			0
$499$		−	14.6901i		−	0.657619i	−0.944396	−	0.328809i	$-0.893353\pi$
$499$		−	14.6901i		−	0.657619i	0.944396	−	0.328809i	$-0.106647\pi$
$500$		0			0
$501$	9.01995	−	1.56848i	0.402982	−	0.0700746i
$502$		0			0
$503$	26.5675			1.18459			0.592294	−	0.805722i	$-0.298223\pi$
$503$	26.5675			1.18459			0.592294	+	0.805722i	$0.298223\pi$
$504$		0			0
$505$	6.17460			0.274766
$506$		0			0
$507$	−1.92402	−	11.0645i	−0.0854486	−	0.491393i
$508$		0			0
$509$			10.4050i			0.461193i	0.973049	+	0.230597i	$0.0740678\pi$
$509$			10.4050i			0.461193i	−0.973049	+	0.230597i	$0.925932\pi$
$510$		0			0
$511$			47.4986i			2.10121i
$512$		0			0
$513$	4.81023	+	8.46973i	0.212377	+	0.373948i
$514$		0			0
$515$	0.909903			0.0400951
$516$		0			0
$517$		−	21.7953i		−	0.958555i
$518$		0			0
$519$	−35.0367	+	6.09255i	−1.53794	+	0.267433i
$520$		0			0
$521$	41.8429			1.83317			0.916586	−	0.399837i	$-0.130933\pi$
$521$	41.8429			1.83317			0.916586	+	0.399837i	$0.130933\pi$
$522$		0			0
$523$	−1.07036			−0.0468035			−0.0234017	−	0.999726i	$-0.507450\pi$
$523$	−1.07036			−0.0468035			−0.0234017	+	0.999726i	$0.507450\pi$
$524$		0			0
$525$	8.12267	−	1.41245i	0.354502	−	0.0616445i
$526$		0			0
$527$	2.63504			0.114784
$528$		0			0
$529$	22.9851			0.999352
$530$		0			0
$531$	8.40952	+	23.4494i	0.364942	+	1.01762i
$532$		0			0
$533$		−	10.4241i		−	0.451516i
$534$		0			0
$535$		−	14.0533i		−	0.607576i
$536$		0			0
$537$	3.24873	+	18.6826i	0.140193	+	0.806214i
$538$		0			0
$539$	74.2897			3.19988
$540$		0			0
$541$			29.0294i			1.24807i	0.781395	+	0.624036i	$0.214508\pi$
$541$			29.0294i			1.24807i	−0.781395	+	0.624036i	$0.785492\pi$
$542$		0			0
$543$	−4.70867	−	27.0784i	−0.202068	−	1.16204i
$544$		0			0
$545$		−	4.57405i		−	0.195931i
$546$		0			0
$547$			1.88401i			0.0805547i	0.999189	+	0.0402773i	$0.0128241\pi$
$547$			1.88401i			0.0805547i	−0.999189	+	0.0402773i	$0.987176\pi$
$548$		0			0
$549$	−0.256785	−	0.716026i	−0.0109593	−	0.0305593i
$550$		0			0
$551$		−	10.4438i		−	0.444923i
$552$		0			0
$553$	−56.1711			−2.38864
$554$		0			0
$555$	1.91377	+	11.0056i	0.0812349	+	0.467162i
$556$		0			0
$557$			0.378305i			0.0160293i	0.999968	+	0.00801465i	$0.00255117\pi$
$557$			0.378305i			0.0160293i	−0.999968	+	0.00801465i	$0.997449\pi$
$558$		0			0
$559$	−9.15042			−0.387021
$560$		0			0
$561$	20.1293	−	3.50030i	0.849861	−	0.147783i
$562$		0			0
$563$	−22.8843			−0.964459			−0.482230	−	0.876045i	$-0.660173\pi$
$563$	−22.8843			−0.964459			−0.482230	+	0.876045i	$0.660173\pi$
$564$		0			0
$565$	2.69027			0.113180
$566$		0			0
$567$	27.2254	+	33.0763i	1.14336	+	1.38907i
$568$		0			0
$569$			46.4061i			1.94544i	0.231974	+	0.972722i	$0.425482\pi$
$569$			46.4061i			1.94544i	−0.231974	+	0.972722i	$0.574518\pi$
$570$		0			0
$571$	−24.4875			−1.02477			−0.512386	−	0.858755i	$-0.671238\pi$
$571$	−24.4875			−1.02477			−0.512386	+	0.858755i	$0.671238\pi$
$572$		0			0
$573$	−4.58660	−	26.3764i	−0.191608	−	1.10189i
$574$		0			0
$575$		−	0.122120i		−	0.00509277i
$576$		0			0
$577$		−	13.1365i		−	0.546880i	−0.961889	−	0.273440i	$-0.911839\pi$
$577$		−	13.1365i		−	0.546880i	0.961889	−	0.273440i	$-0.0881614\pi$
$578$		0			0
$579$	4.48627	+	25.7994i	0.186443	+	1.07219i
$580$		0			0
$581$	32.8969			1.36479
$582$		0			0
$583$	−58.9193			−2.44019
$584$		0			0
$585$	2.58512	+	7.20842i	0.106882	+	0.298032i
$586$		0			0
$587$	−12.1564			−0.501747			−0.250874	−	0.968020i	$-0.580718\pi$
$587$	−12.1564			−0.501747			−0.250874	+	0.968020i	$0.580718\pi$
$588$		0			0
$589$			1.98677i			0.0818634i
$590$		0			0
$591$	−1.76792	−	10.1669i	−0.0727225	−	0.418209i
$592$		0			0
$593$	−9.81932			−0.403231			−0.201615	−	0.979465i	$-0.564619\pi$
$593$	−9.81932			−0.403231			−0.201615	+	0.979465i	$0.564619\pi$
$594$		0			0
$595$	11.8342			0.485156
$596$		0			0
$597$	5.26211	+	30.2611i	0.215364	+	1.23850i
$598$		0			0
$599$	−21.3196			−0.871094			−0.435547	−	0.900166i	$-0.643445\pi$
$599$	−21.3196			−0.871094			−0.435547	+	0.900166i	$0.643445\pi$
$600$		0			0
$601$	−12.3647			−0.504366			−0.252183	−	0.967680i	$-0.581149\pi$
$601$	−12.3647			−0.504366			−0.252183	+	0.967680i	$0.581149\pi$
$602$		0			0
$603$	−24.2848	+	3.63994i	−0.988953	+	0.148230i
$604$		0			0
$605$	11.5117			0.468017
$606$		0			0
$607$	−28.9923			−1.17676			−0.588381	−	0.808584i	$-0.700234\pi$
$607$	−28.9923			−1.17676			−0.588381	+	0.808584i	$0.700234\pi$
$608$		0			0
$609$	−7.86939	−	45.2549i	−0.318884	−	1.83382i
$610$		0			0
$611$	11.7260			0.474383
$612$		0			0
$613$	37.7742			1.52569			0.762843	−	0.646583i	$-0.223803\pi$
$613$	37.7742			1.52569			0.762843	+	0.646583i	$0.223803\pi$
$614$		0			0
$615$	−1.21175	−	6.96847i	−0.0488625	−	0.280996i
$616$		0			0
$617$		−	24.9081i		−	1.00276i	−0.865227	−	0.501381i	$-0.832826\pi$
$617$		−	24.9081i		−	1.00276i	0.865227	−	0.501381i	$-0.167174\pi$
$618$		0			0
$619$	38.9985			1.56748			0.783742	−	0.621087i	$-0.213309\pi$
$619$	38.9985			1.56748			0.783742	+	0.621087i	$0.213309\pi$
$620$		0			0
$621$	0.551778	−	0.313373i	0.0221421	−	0.0125752i
$622$		0			0
$623$	64.6863			2.59160
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	2.63915	+	15.1771i	0.105398	+	0.606116i
$628$		0			0
$629$			16.0345i			0.639337i
$630$		0			0
$631$			40.2213i			1.60119i	0.599208	+	0.800593i	$0.295482\pi$
$631$			40.2213i			1.60119i	−0.599208	+	0.800593i	$0.704518\pi$
$632$		0			0
$633$	0.0339975	+	0.195511i	0.00135128	+	0.00777087i
$634$		0			0
$635$	−0.577470			−0.0229162
$636$		0			0
$637$			39.9683i			1.58360i
$638$		0			0
$639$	6.00402	+	16.7418i	0.237515	+	0.662295i
$640$		0			0
$641$	−38.6270			−1.52567			−0.762837	−	0.646591i	$-0.776194\pi$
$641$	−38.6270			−1.52567			−0.762837	+	0.646591i	$0.776194\pi$
$642$		0			0
$643$	−30.6784			−1.20984			−0.604919	−	0.796287i	$-0.706795\pi$
$643$	−30.6784			−1.20984			−0.604919	+	0.796287i	$0.706795\pi$
$644$		0			0
$645$	−6.11704	+	1.06369i	−0.240858	+	0.0418829i
$646$		0			0
$647$	−12.5718			−0.494249			−0.247124	−	0.968984i	$-0.579486\pi$
$647$	−12.5718			−0.494249			−0.247124	+	0.968984i	$0.579486\pi$
$648$		0			0
$649$			39.3991i			1.54655i
$650$		0			0
$651$	1.49702	+	8.60900i	0.0586729	+	0.337413i
$652$		0			0
$653$	−24.6023			−0.962762			−0.481381	−	0.876511i	$-0.659865\pi$
$653$	−24.6023			−0.962762			−0.481381	+	0.876511i	$0.659865\pi$
$654$		0			0
$655$		−	17.2744i		−	0.674967i
$656$		0			0
$657$	−28.1788	+	10.1056i	−1.09936	+	0.394258i
$658$		0			0
$659$			41.7811i			1.62756i	0.581173	+	0.813780i	$0.302594\pi$
$659$			41.7811i			1.62756i	−0.581173	+	0.813780i	$0.697406\pi$
$660$		0			0
$661$			15.5622i			0.605298i	0.953102	+	0.302649i	$0.0978711\pi$
$661$			15.5622i			0.605298i	−0.953102	+	0.302649i	$0.902129\pi$
$662$		0			0
$663$	1.88318	+	10.8297i	0.0731367	+	0.420591i
$664$		0			0
$665$			8.92278i			0.346010i
$666$		0			0
$667$	−0.680386			−0.0263446
$668$		0			0
$669$	−3.26641	−	18.7843i	−0.126287	−	0.726244i
$670$		0			0
$671$		−	1.20305i		−	0.0464433i
$672$		0			0
$673$		−	41.1730i		−	1.58710i	−0.608504	−	0.793551i	$-0.708230\pi$
$673$		−	41.1730i		−	1.58710i	0.608504	−	0.793551i	$-0.291770\pi$
$674$		0			0
$675$	2.56610	+	4.51831i	0.0987690	+	0.173910i
$676$		0			0
$677$	27.5867			1.06024			0.530122	−	0.847921i	$-0.322146\pi$
$677$	27.5867			1.06024			0.530122	+	0.847921i	$0.322146\pi$
$678$		0			0
$679$	73.0130			2.80198
$680$		0			0
$681$	24.3525	−	4.23467i	0.933191	−	0.162273i
$682$		0			0
$683$	33.5640			1.28429			0.642145	−	0.766583i	$-0.278045\pi$
$683$	33.5640			1.28429			0.642145	+	0.766583i	$0.278045\pi$
$684$		0			0
$685$	−7.68503			−0.293630
$686$		0			0
$687$	46.7257	−	8.12515i	1.78270	−	0.309994i
$688$		0			0
$689$		−	31.6990i		−	1.20763i
$690$		0			0
$691$	−1.58145			−0.0601613			−0.0300807	−	0.999547i	$-0.509576\pi$
$691$	−1.58145			−0.0601613			−0.0300807	+	0.999547i	$0.509576\pi$
$692$		0			0
$693$	22.8718	+	63.7764i	0.868827	+	2.42266i
$694$		0			0
$695$		−	2.00003i		−	0.0758654i
$696$		0			0
$697$		−	10.1526i		−	0.384559i
$698$		0			0
$699$	4.28701	+	24.6535i	0.162150	+	0.932482i
$700$		0			0
$701$	10.2256			0.386215			0.193107	−	0.981178i	$-0.438143\pi$
$701$	10.2256			0.386215			0.193107	+	0.981178i	$0.438143\pi$
$702$		0			0
$703$	−12.0897			−0.455971
$704$		0			0
$705$	7.83881	−	1.36309i	0.295227	−	0.0513371i
$706$		0			0
$707$			29.3911i			1.10537i
$708$		0			0
$709$	−1.78898			−0.0671865			−0.0335933	−	0.999436i	$-0.510695\pi$
$709$	−1.78898			−0.0671865			−0.0335933	+	0.999436i	$0.510695\pi$
$710$		0			0
$711$	−11.9508	−	33.3239i	−0.448189	−	1.24974i
$712$		0			0
$713$	0.129432			0.00484728
$714$		0			0
$715$			12.1114i			0.452942i
$716$		0			0
$717$	3.97329	+	22.8494i	0.148385	+	0.853328i
$718$		0			0
$719$			15.4921i			0.577756i	0.957366	+	0.288878i	$0.0932822\pi$
$719$			15.4921i			0.577756i	−0.957366	+	0.288878i	$0.906718\pi$
$720$		0			0
$721$			4.33114i			0.161300i
$722$		0			0
$723$	−1.32369	−	7.61223i	−0.0492287	−	0.283102i
$724$		0			0
$725$		−	5.57143i		−	0.206918i
$726$		0			0
$727$			2.80557i			0.104053i	0.998646	+	0.0520263i	$0.0165680\pi$
$727$			2.80557i			0.104053i	−0.998646	+	0.0520263i	$0.983432\pi$
$728$		0			0
$729$	−13.8303	+	23.1888i	−0.512234	+	0.858846i
$730$		0			0
$731$	−8.91216			−0.329628
$732$		0			0
$733$		−	40.7412i		−	1.50481i	−0.658700	−	0.752406i	$-0.728893\pi$
$733$		−	40.7412i		−	1.50481i	0.658700	−	0.752406i	$-0.271107\pi$
$734$		0			0
$735$	4.64613	+	26.7188i	0.171375	+	0.985536i
$736$		0			0
$737$	−38.0964	−	7.54656i	−1.40330	−	0.277981i
$738$		0			0
$739$		−	25.8680i		−	0.951570i	−0.879562	−	0.475785i	$-0.842164\pi$
$739$		−	25.8680i		−	0.951570i	0.879562	−	0.475785i	$-0.157836\pi$
$740$		0			0
$741$	−8.16538	+	1.41988i	−0.299963	+	0.0521606i
$742$		0			0
$743$		−	21.0197i		−	0.771136i	−0.922679	−	0.385568i	$-0.874005\pi$
$743$		−	21.0197i		−	0.771136i	0.922679	−	0.385568i	$-0.125995\pi$
$744$		0			0
$745$			3.70488i			0.135736i
$746$		0			0
$747$	6.99902	+	19.5163i	0.256081	+	0.714063i
$748$		0			0
$749$	66.8936			2.44424
$750$		0			0
$751$	−17.8558			−0.651567			−0.325783	−	0.945444i	$-0.605628\pi$
$751$	−17.8558			−0.651567			−0.325783	+	0.945444i	$0.605628\pi$
$752$		0			0
$753$	−4.48700	−	25.8036i	−0.163515	−	0.940337i
$754$		0			0
$755$	9.94769			0.362034
$756$		0			0
$757$		−	20.4740i		−	0.744142i	−0.928204	−	0.372071i	$-0.878648\pi$
$757$		−	20.4740i		−	0.744142i	0.928204	−	0.372071i	$-0.121352\pi$
$758$		0			0
$759$	0.988745	−	0.171933i	0.0358892	−	0.00624078i
$760$		0			0
$761$			27.9719i			1.01398i	0.861951	+	0.506991i	$0.169242\pi$
$761$			27.9719i			1.01398i	−0.861951	+	0.506991i	$0.830758\pi$
$762$		0			0
$763$	21.7724			0.788215
$764$		0			0
$765$	2.51781	+	7.02073i	0.0910315	+	0.253835i
$766$		0			0
$767$	−21.1969			−0.765377
$768$		0			0
$769$		−	24.1505i		−	0.870888i	−0.900216	−	0.435444i	$-0.856591\pi$
$769$		−	24.1505i		−	0.870888i	0.900216	−	0.435444i	$-0.143409\pi$
$770$		0			0
$771$	−46.3472	+	8.05933i	−1.66915	+	0.290250i
$772$		0			0
$773$			19.4800i			0.700646i	0.936629	+	0.350323i	$0.113928\pi$
$773$			19.4800i			0.700646i	−0.936629	+	0.350323i	$0.886072\pi$
$774$		0			0
$775$			1.05987i			0.0380718i
$776$		0			0
$777$	−52.3866	+	9.10953i	−1.87936	+	0.326803i
$778$		0			0
$779$	7.65489			0.274265
$780$		0			0
$781$			28.1292i			1.00654i
$782$		0			0
$783$	25.1735	−	14.2968i	0.899627	−	0.510927i
$784$		0			0
$785$	−24.4718			−0.873436
$786$		0			0
$787$			4.24168i			0.151199i	0.997138	+	0.0755997i	$0.0240871\pi$
$787$			4.24168i			0.151199i	−0.997138	+	0.0755997i	$0.975913\pi$
$788$		0			0
$789$	7.99160	−	1.38966i	0.284509	−	0.0494733i
$790$		0			0
$791$			12.8057i			0.455317i
$792$		0			0
$793$	0.647249			0.0229845
$794$		0			0
$795$	−3.68486	−	21.1907i	−0.130689	−	0.751557i
$796$		0			0
$797$		−	9.47397i		−	0.335585i	−0.985822	−	0.167793i	$-0.946336\pi$
$797$		−	9.47397i		−	0.335585i	0.985822	−	0.167793i	$-0.0536639\pi$
$798$		0			0
$799$	11.4207			0.404034
$800$		0			0
$801$	13.7624	+	38.3755i	0.486271	+	1.35593i
$802$		0			0
$803$	−47.3454			−1.67078
$804$		0			0
$805$	0.581293			0.0204879
$806$		0			0
$807$	−18.9209	+	3.29017i	−0.666048	+	0.115819i
$808$		0			0
$809$	19.5485			0.687288			0.343644	−	0.939100i	$-0.388339\pi$
$809$	19.5485			0.687288			0.343644	+	0.939100i	$0.388339\pi$
$810$		0			0
$811$		−	25.0212i		−	0.878614i	−0.898337	−	0.439307i	$-0.855224\pi$
$811$		−	25.0212i		−	0.878614i	0.898337	−	0.439307i	$-0.144776\pi$
$812$		0			0
$813$	−3.44640	+	0.599296i	−0.120871	+	0.0210182i
$814$		0			0
$815$	19.4604			0.681667
$816$		0			0
$817$		−	6.71959i		−	0.235089i
$818$		0			0
$819$	−34.3121	+	12.3052i	−1.19896	+	0.429977i
$820$		0			0
$821$			15.3895i			0.537097i	0.963266	+	0.268548i	$0.0865439\pi$
$821$			15.3895i			0.537097i	−0.963266	+	0.268548i	$0.913456\pi$
$822$		0			0
$823$	−40.7469			−1.42035			−0.710174	−	0.704026i	$-0.751384\pi$
$823$	−40.7469			−1.42035			−0.710174	+	0.704026i	$0.751384\pi$
$824$		0			0
$825$	1.40790	+	8.09647i	0.0490167	+	0.281883i
$826$		0			0
$827$			18.4811i			0.642652i	0.946969	+	0.321326i	$0.104129\pi$
$827$			18.4811i			0.642652i	−0.946969	+	0.321326i	$0.895871\pi$
$828$		0			0
$829$	−10.2439			−0.355785			−0.177893	−	0.984050i	$-0.556928\pi$
$829$	−10.2439			−0.355785			−0.177893	+	0.984050i	$0.556928\pi$
$830$		0			0
$831$	1.20895	+	6.95235i	0.0419379	+	0.241174i
$832$		0			0
$833$			38.9276i			1.34876i
$834$		0			0
$835$			5.28582i			0.182923i
$836$		0			0
$837$	−4.78884	+	2.71974i	−0.165527	+	0.0940079i
$838$		0			0
$839$			19.1595i			0.661459i	0.943726	+	0.330730i	$0.107295\pi$
$839$			19.1595i			0.661459i	−0.943726	+	0.330730i	$0.892705\pi$
$840$		0			0
$841$	−2.04086			−0.0703745
$842$		0			0
$843$	4.60589	+	26.4873i	0.158635	+	0.912272i
$844$		0			0
$845$	6.48398			0.223056
$846$		0			0
$847$			54.7956i			1.88280i
$848$		0			0
$849$	8.71366	+	50.1101i	0.299052	+	1.71977i
$850$		0			0
$851$			0.787608i			0.0269989i
$852$		0			0
$853$	−39.4708			−1.35146			−0.675728	−	0.737151i	$-0.736171\pi$
$853$	−39.4708			−1.35146			−0.675728	+	0.737151i	$0.736171\pi$
$854$		0			0
$855$	−5.29349	+	1.89838i	−0.181034	+	0.0649231i
$856$		0			0
$857$	−15.9517			−0.544898			−0.272449	−	0.962170i	$-0.587834\pi$
$857$	−15.9517			−0.544898			−0.272449	+	0.962170i	$0.587834\pi$
$858$		0			0
$859$	39.5677			1.35003			0.675017	−	0.737802i	$-0.264136\pi$
$859$	39.5677			1.35003			0.675017	+	0.737802i	$0.264136\pi$
$860$		0			0
$861$	33.1699	−	5.76792i	1.13043	−	0.196570i
$862$		0			0
$863$			7.44762i			0.253520i	0.991933	+	0.126760i	$0.0404578\pi$
$863$			7.44762i			0.253520i	−0.991933	+	0.126760i	$0.959542\pi$
$864$		0			0
$865$		−	20.5320i		−	0.698110i
$866$		0			0
$867$	−3.21033	−	18.4618i	−0.109029	−	0.626996i
$868$		0			0
$869$		−	55.9900i		−	1.89933i
$870$		0			0
$871$	4.06009	−	20.4961i	0.137571	−	0.694483i
$872$		0			0
$873$	15.5340	+	43.3154i	0.525746	+	1.46601i
$874$		0			0
$875$			4.76000i			0.160917i
$876$		0			0
$877$	8.44240			0.285080			0.142540	−	0.989789i	$-0.454473\pi$
$877$	8.44240			0.285080			0.142540	+	0.989789i	$0.454473\pi$
$878$		0			0
$879$	−52.0320	+	9.04787i	−1.75500	+	0.305177i
$880$		0			0
$881$		−	47.7752i		−	1.60959i	−0.593555	−	0.804793i	$-0.702276\pi$
$881$		−	47.7752i		−	1.60959i	0.593555	−	0.804793i	$-0.297724\pi$
$882$		0			0
$883$			9.91564i			0.333688i	0.985983	+	0.166844i	$0.0533576\pi$
$883$			9.91564i			0.333688i	−0.985983	+	0.166844i	$0.946642\pi$
$884$		0			0
$885$	−14.1701	+	2.46405i	−0.476324	+	0.0828281i
$886$		0			0
$887$		−	35.8825i		−	1.20482i	−0.798188	−	0.602408i	$-0.794208\pi$
$887$		−	35.8825i		−	1.20482i	0.798188	−	0.602408i	$-0.205792\pi$
$888$		0			0
$889$		−	2.74875i		−	0.0921903i
$890$		0			0
$891$	−32.9696	+	27.1376i	−1.10452	+	0.909145i
$892$		0			0
$893$			8.61096i			0.288155i
$894$		0			0
$895$	−10.9483			−0.365960
$896$		0			0
$897$	0.0925011	+	0.531951i	0.00308852	+	0.0177613i
$898$		0			0
$899$	5.90502			0.196943
$900$		0			0
$901$		−	30.8736i		−	1.02855i
$902$		0			0
$903$	−5.06318	−	29.1171i	−0.168492	−	0.968957i
$904$		0			0
$905$	15.8683			0.527480
$906$		0			0
$907$	0.835003			0.0277258			0.0138629	−	0.999904i	$-0.495587\pi$
$907$	0.835003			0.0277258			0.0138629	+	0.999904i	$0.495587\pi$
$908$		0			0
$909$	−17.4364	+	6.25314i	−0.578330	+	0.207404i
$910$		0			0
$911$		−	1.54756i		−	0.0512728i	−0.999671	−	0.0256364i	$-0.991839\pi$
$911$		−	1.54756i		−	0.0512728i	0.999671	−	0.0256364i	$-0.00816122\pi$
$912$		0			0
$913$			32.7908i			1.08522i
$914$		0			0
$915$	0.432685	−	0.0752397i	0.0143041	−	0.00248735i
$916$		0			0
$917$	82.2261			2.71535
$918$		0			0
$919$		−	7.30964i		−	0.241123i	−0.992706	−	0.120561i	$-0.961531\pi$
$919$		−	7.30964i		−	0.241123i	0.992706	−	0.120561i	$-0.0384695\pi$
$920$		0			0
$921$	−3.65938	−	21.0442i	−0.120581	−	0.693429i
$922$		0			0
$923$	−15.1337			−0.498131
$924$		0			0
$925$	−6.44944			−0.212056
$926$		0			0
$927$	−2.56947	+	0.921477i	−0.0843926	+	0.0302653i
$928$		0			0
$929$	−5.92164			−0.194283			−0.0971413	−	0.995271i	$-0.530970\pi$
$929$	−5.92164			−0.194283			−0.0971413	+	0.995271i	$0.530970\pi$
$930$		0			0
$931$	−29.3506			−0.961929
$932$		0			0
$933$	−4.87081	−	28.0108i	−0.159463	−	0.917033i
$934$		0			0
$935$			11.7961i			0.385773i
$936$		0			0
$937$		−	9.19519i		−	0.300394i	−0.988656	−	0.150197i	$-0.952009\pi$
$937$		−	9.19519i		−	0.300394i	0.988656	−	0.150197i	$-0.0479907\pi$
$938$		0			0
$939$	−16.2097	+	2.81872i	−0.528985	+	0.0919853i
$940$		0			0
$941$	−10.6270			−0.346429			−0.173214	−	0.984884i	$-0.555415\pi$
$941$	−10.6270			−0.346429			−0.173214	+	0.984884i	$0.555415\pi$
$942$		0			0
$943$		−	0.498694i		−	0.0162397i
$944$		0			0
$945$	−21.5072	+	12.2146i	−0.699628	+	0.397341i
$946$		0			0
$947$		−	50.6584i		−	1.64618i	−0.567914	−	0.823088i	$-0.692249\pi$
$947$		−	50.6584i		−	1.64618i	0.567914	−	0.823088i	$-0.307751\pi$
$948$		0			0
$949$		−	25.4721i		−	0.826861i
$950$		0			0
$951$	28.9811	−	5.03953i	0.939776	−	0.163418i
$952$		0			0
$953$			26.5100i			0.858744i	0.903128	+	0.429372i	$0.141265\pi$
$953$			26.5100i			0.858744i	−0.903128	+	0.429372i	$0.858735\pi$
$954$		0			0
$955$	15.4570			0.500175
$956$		0			0
$957$	45.1090	−	7.84401i	1.45817	−	0.253561i
$958$		0			0
$959$		−	36.5807i		−	1.18125i
$960$		0			0
$961$	29.8767			0.963763
$962$		0			0
$963$	14.2320	+	39.6850i	0.458621	+	1.27883i
$964$		0			0
$965$	−15.1188			−0.486692
$966$		0			0
$967$	−45.5264			−1.46403			−0.732015	−	0.681288i	$-0.761420\pi$
$967$	−45.5264			−1.46403			−0.732015	+	0.681288i	$0.761420\pi$
$968$		0			0
$969$	−7.95277	+	1.38291i	−0.255480	+	0.0444255i
$970$		0			0
$971$		−	3.44662i		−	0.110607i	−0.998470	−	0.0553037i	$-0.982387\pi$
$971$		−	3.44662i		−	0.110607i	0.998470	−	0.0553037i	$-0.0176127\pi$
$972$		0			0
$973$	9.52013			0.305201
$974$		0			0
$975$	−4.35595	+	0.757458i	−0.139502	+	0.0242581i
$976$		0			0
$977$		−	53.7583i		−	1.71988i	−0.510395	−	0.859940i	$-0.670501\pi$
$977$		−	53.7583i		−	1.71988i	0.510395	−	0.859940i	$-0.329499\pi$
$978$		0			0
$979$			64.4777i			2.06072i
$980$		0			0
$981$	4.63222	+	12.9166i	0.147896	+	0.412396i
$982$		0			0
$983$	−25.2963			−0.806826			−0.403413	−	0.915018i	$-0.632176\pi$
$983$	−25.2963			−0.806826			−0.403413	+	0.915018i	$0.632176\pi$
$984$		0			0
$985$	5.95793			0.189835
$986$		0			0
$987$	6.48832	+	37.3127i	0.206525	+	1.18768i
$988$		0			0
$989$	−0.437762			−0.0139200
$990$		0			0
$991$			16.3009i			0.517816i	0.965902	+	0.258908i	$0.0833626\pi$
$991$			16.3009i			0.517816i	−0.965902	+	0.258908i	$0.916637\pi$
$992$		0			0
$993$	15.7048	−	2.73091i	0.498377	−	0.0866630i
$994$		0			0
$995$	−17.7334			−0.562187
$996$		0			0
$997$	59.6448			1.88897			0.944485	−	0.328556i	$-0.106562\pi$
$997$	59.6448			1.88897			0.944485	+	0.328556i	$0.106562\pi$
$998$		0			0
$999$	−16.5499	−	29.1406i	−0.523615	−	0.921968i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.b.401.19	yes	46
3.2	odd	2		4020.2.f.a.401.27	✓	46
67.66	odd	2		4020.2.f.a.401.28	yes	46
201.200	even	2	inner	4020.2.f.b.401.20	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.27	✓	46	3.2	odd	2
4020.2.f.a.401.28	yes	46	67.66	odd	2
4020.2.f.b.401.19	yes	46	1.1	even	1	trivial
4020.2.f.b.401.20	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)

\(f(q)\)	\(=\)	\(q+(-0.296734 - 1.70644i) q^{3} +1.00000 q^{5} +4.76000i q^{7} +(-2.82390 + 1.01272i) q^{9} +O(q^{10})\)	\(q+(-0.296734 - 1.70644i) q^{3} +1.00000 q^{5} +4.76000i q^{7} +(-2.82390 + 1.01272i) q^{9} -4.74465 q^{11} -2.55265i q^{13} +(-0.296734 - 1.70644i) q^{15} -2.48618i q^{17} +1.87453 q^{19} +(8.12267 - 1.41245i) q^{21} -0.122120i q^{23} +1.00000 q^{25} +(2.56610 + 4.51831i) q^{27} -5.57143i q^{29} +1.05987i q^{31} +(1.40790 + 8.09647i) q^{33} +4.76000i q^{35} -6.44944 q^{37} +(-4.35595 + 0.757458i) q^{39} +4.08362 q^{41} -3.58467i q^{43} +(-2.82390 + 1.01272i) q^{45} +4.59365i q^{47} -15.6576 q^{49} +(-4.24253 + 0.737736i) q^{51} +12.4181 q^{53} -4.74465 q^{55} +(-0.556238 - 3.19879i) q^{57} -8.30390i q^{59} +0.253560i q^{61} +(-4.82054 - 13.4417i) q^{63} -2.55265i q^{65} +(8.02933 + 1.59054i) q^{67} +(-0.208392 + 0.0362373i) q^{69} -5.92861i q^{71} +9.97870 q^{73} +(-0.296734 - 1.70644i) q^{75} -22.5845i q^{77} +11.8007i q^{79} +(6.94880 - 5.71963i) q^{81} -6.91111i q^{83} -2.48618i q^{85} +(-9.50733 + 1.65323i) q^{87} -13.5896i q^{89} +12.1506 q^{91} +(1.80861 - 0.314501i) q^{93} +1.87453 q^{95} -15.3389i q^{97} +(13.3984 - 4.80500i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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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