LMFDB - Embedded newform 4020.2.f.a.401.8

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			401.8
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.45039 + 0.946770i) q^{3} -1.00000 q^{5} +1.74006i q^{7} +(1.20725 - 2.74637i) q^{9} +O(q^{10})$	\(q+(-1.45039 + 0.946770i) q^{3} -1.00000 q^{5} +1.74006i q^{7} +(1.20725 - 2.74637i) q^{9} +0.702140 q^{11} +4.70379i q^{13} +(1.45039 - 0.946770i) q^{15} -1.49494i q^{17} -1.07721 q^{19} +(-1.64743 - 2.52376i) q^{21} -2.17196i q^{23} +1.00000 q^{25} +(0.849189 + 5.12629i) q^{27} +2.17762i q^{29} +3.66633i q^{31} +(-1.01838 + 0.664764i) q^{33} -1.74006i q^{35} -7.97311 q^{37} +(-4.45341 - 6.82233i) q^{39} -1.68328 q^{41} -2.43657i q^{43} +(-1.20725 + 2.74637i) q^{45} +8.94967i q^{47} +3.97220 q^{49} +(1.41537 + 2.16825i) q^{51} +12.5986 q^{53} -0.702140 q^{55} +(1.56238 - 1.01987i) q^{57} +9.27103i q^{59} +2.37446i q^{61} +(4.77884 + 2.10069i) q^{63} -4.70379i q^{65} +(7.89756 + 2.15141i) q^{67} +(2.05634 + 3.15018i) q^{69} -15.2573i q^{71} -3.12540 q^{73} +(-1.45039 + 0.946770i) q^{75} +1.22176i q^{77} +2.10644i q^{79} +(-6.08507 - 6.63113i) q^{81} +12.4148i q^{83} +1.49494i q^{85} +(-2.06170 - 3.15839i) q^{87} -9.88566i q^{89} -8.18488 q^{91} +(-3.47117 - 5.31760i) q^{93} +1.07721 q^{95} +10.0995i q^{97} +(0.847661 - 1.92833i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) $ 46 * q - 46 * q^5 - 4 * q^9	$ 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) $ 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.45039	+	0.946770i	−0.837382	+	0.546618i
$3$	−1.45039	+	0.946770i	−0.837382	+	0.546618i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.74006i			0.657680i	0.944386	+	0.328840i	$0.106658\pi$
$7$			1.74006i			0.657680i	−0.944386	+	0.328840i	$0.893342\pi$
$8$		0			0
$9$	1.20725	−	2.74637i	0.402418	−	0.915456i
$10$		0			0
$11$	0.702140			0.211703			0.105852	−	0.994382i	$-0.466243\pi$
$11$	0.702140			0.211703			0.105852	+	0.994382i	$0.466243\pi$
$12$		0			0
$13$			4.70379i			1.30460i	0.757962	+	0.652299i	$0.226195\pi$
$13$			4.70379i			1.30460i	−0.757962	+	0.652299i	$0.773805\pi$
$14$		0			0
$15$	1.45039	−	0.946770i	0.374489	−	0.244455i
$16$		0			0
$17$		−	1.49494i		−	0.362577i	−0.983430	−	0.181288i	$-0.941973\pi$
$17$		−	1.49494i		−	0.362577i	0.983430	−	0.181288i	$-0.0580268\pi$
$18$		0			0
$19$	−1.07721			−0.247129			−0.123565	−	0.992337i	$-0.539433\pi$
$19$	−1.07721			−0.247129			−0.123565	+	0.992337i	$0.539433\pi$
$20$		0			0
$21$	−1.64743	−	2.52376i	−0.359500	−	0.550730i
$22$		0			0
$23$		−	2.17196i		−	0.452885i	−0.974025	−	0.226442i	$-0.927291\pi$
$23$		−	2.17196i		−	0.452885i	0.974025	−	0.226442i	$-0.0727095\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	0.849189	+	5.12629i	0.163426	+	0.986556i
$28$		0			0
$29$			2.17762i			0.404374i	0.979347	+	0.202187i	$0.0648048\pi$
$29$			2.17762i			0.404374i	−0.979347	+	0.202187i	$0.935195\pi$
$30$		0			0
$31$			3.66633i			0.658492i	0.944244	+	0.329246i	$0.106795\pi$
$31$			3.66633i			0.658492i	−0.944244	+	0.329246i	$0.893205\pi$
$32$		0			0
$33$	−1.01838	+	0.664764i	−0.177276	+	0.115721i
$34$		0			0
$35$		−	1.74006i		−	0.294124i
$36$		0			0
$37$	−7.97311			−1.31077			−0.655386	−	0.755294i	$-0.727494\pi$
$37$	−7.97311			−1.31077			−0.655386	+	0.755294i	$0.727494\pi$
$38$		0			0
$39$	−4.45341	−	6.82233i	−0.713116	−	1.09245i
$40$		0			0
$41$	−1.68328			−0.262884			−0.131442	−	0.991324i	$-0.541961\pi$
$41$	−1.68328			−0.262884			−0.131442	+	0.991324i	$0.541961\pi$
$42$		0			0
$43$		−	2.43657i		−	0.371573i	−0.982590	−	0.185787i	$-0.940517\pi$
$43$		−	2.43657i		−	0.371573i	0.982590	−	0.185787i	$-0.0594833\pi$
$44$		0			0
$45$	−1.20725	+	2.74637i	−0.179967	+	0.409404i
$46$		0			0
$47$			8.94967i			1.30544i	0.757598	+	0.652722i	$0.226373\pi$
$47$			8.94967i			1.30544i	−0.757598	+	0.652722i	$0.773627\pi$
$48$		0			0
$49$	3.97220			0.567457
$50$		0			0
$51$	1.41537	+	2.16825i	0.198191	+	0.303615i
$52$		0			0
$53$	12.5986			1.73055			0.865273	−	0.501302i	$-0.167145\pi$
$53$	12.5986			1.73055			0.865273	+	0.501302i	$0.167145\pi$
$54$		0			0
$55$	−0.702140			−0.0946765
$56$		0			0
$57$	1.56238	−	1.01987i	0.206942	−	0.135085i
$58$		0			0
$59$			9.27103i			1.20698i	0.797369	+	0.603492i	$0.206225\pi$
$59$			9.27103i			1.20698i	−0.797369	+	0.603492i	$0.793775\pi$
$60$		0			0
$61$			2.37446i			0.304018i	0.988379	+	0.152009i	$0.0485743\pi$
$61$			2.37446i			0.304018i	−0.988379	+	0.152009i	$0.951426\pi$
$62$		0			0
$63$	4.77884	+	2.10069i	0.602077	+	0.264663i
$64$		0			0
$65$		−	4.70379i		−	0.583434i
$66$		0			0
$67$	7.89756	+	2.15141i	0.964840	+	0.262837i
$67$	7.89756	+	2.15141i	0.964840	+	0.262837i
$68$		0			0
$69$	2.05634	+	3.15018i	0.247555	+	0.379237i
$70$		0			0
$71$		−	15.2573i		−	1.81071i	−0.424657	−	0.905354i	$-0.639605\pi$
$71$		−	15.2573i		−	1.81071i	0.424657	−	0.905354i	$-0.360395\pi$
$72$		0			0
$73$	−3.12540			−0.365801			−0.182900	−	0.983131i	$-0.558549\pi$
$73$	−3.12540			−0.365801			−0.182900	+	0.983131i	$0.558549\pi$
$74$		0			0
$75$	−1.45039	+	0.946770i	−0.167476	+	0.109324i
$76$		0			0
$77$			1.22176i			0.139233i
$78$		0			0
$79$			2.10644i			0.236994i	0.992954	+	0.118497i	$0.0378075\pi$
$79$			2.10644i			0.236994i	−0.992954	+	0.118497i	$0.962192\pi$
$80$		0			0
$81$	−6.08507	−	6.63113i	−0.676119	−	0.736792i
$82$		0			0
$83$			12.4148i			1.36270i	0.731958	+	0.681349i	$0.238607\pi$
$83$			12.4148i			1.36270i	−0.731958	+	0.681349i	$0.761393\pi$
$84$		0			0
$85$			1.49494i			0.162149i
$86$		0			0
$87$	−2.06170	−	3.15839i	−0.221038	−	0.338615i
$88$		0			0
$89$		−	9.88566i		−	1.04788i	−0.851756	−	0.523939i	$-0.824462\pi$
$89$		−	9.88566i		−	1.04788i	0.851756	−	0.523939i	$-0.175538\pi$
$90$		0			0
$91$	−8.18488			−0.858008
$92$		0			0
$93$	−3.47117	−	5.31760i	−0.359943	−	0.551409i
$94$		0			0
$95$	1.07721			0.110520
$96$		0			0
$97$			10.0995i			1.02545i	0.858552	+	0.512726i	$0.171364\pi$
$97$			10.0995i			1.02545i	−0.858552	+	0.512726i	$0.828636\pi$
$98$		0			0
$99$	0.847661	−	1.92833i	0.0851932	−	0.193805i
$100$		0			0
$101$	−18.2502			−1.81596			−0.907979	−	0.419016i	$-0.862375\pi$
$101$	−18.2502			−1.81596			−0.907979	+	0.419016i	$0.862375\pi$
$102$		0			0
$103$	−12.1257			−1.19478			−0.597390	−	0.801951i	$-0.703796\pi$
$103$	−12.1257			−1.19478			−0.597390	+	0.801951i	$0.703796\pi$
$104$		0			0
$105$	1.64743	+	2.52376i	0.160773	+	0.246294i
$106$		0			0
$107$			12.2667i			1.18587i	0.805251	+	0.592934i	$0.202030\pi$
$107$			12.2667i			1.18587i	−0.805251	+	0.592934i	$0.797970\pi$
$108$		0			0
$109$		−	15.6304i		−	1.49712i	−0.663069	−	0.748559i	$-0.730746\pi$
$109$		−	15.6304i		−	1.49712i	0.663069	−	0.748559i	$-0.269254\pi$
$110$		0			0
$111$	11.5641	−	7.54870i	1.09762	−	0.716491i
$112$		0			0
$113$	−8.66363			−0.815006			−0.407503	−	0.913204i	$-0.633600\pi$
$113$	−8.66363			−0.815006			−0.407503	+	0.913204i	$0.633600\pi$
$114$		0			0
$115$			2.17196i			0.202536i
$116$		0			0
$117$	12.9183	+	5.67868i	1.19430	+	0.524994i
$118$		0			0
$119$	2.60129			0.238460
$120$		0			0
$121$	−10.5070			−0.955182
$122$		0			0
$123$	2.44141	−	1.59368i	0.220134	−	0.143697i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−16.1229			−1.43067			−0.715336	−	0.698781i	$-0.753726\pi$
$127$	−16.1229			−1.43067			−0.715336	+	0.698781i	$0.753726\pi$
$128$		0			0
$129$	2.30687	+	3.53397i	0.203108	+	0.311149i
$130$		0			0
$131$		−	13.9305i		−	1.21711i	−0.793511	−	0.608556i	$-0.791749\pi$
$131$		−	13.9305i		−	1.21711i	0.793511	−	0.608556i	$-0.208251\pi$
$132$		0			0
$133$		−	1.87441i		−	0.162532i
$134$		0			0
$135$	−0.849189	−	5.12629i	−0.0730865	−	0.441201i
$136$		0			0
$137$	1.17971			0.100790			0.0503948	−	0.998729i	$-0.483952\pi$
$137$	1.17971			0.100790			0.0503948	+	0.998729i	$0.483952\pi$
$138$		0			0
$139$			18.7735i			1.59235i	0.605069	+	0.796173i	$0.293145\pi$
$139$			18.7735i			1.59235i	−0.605069	+	0.796173i	$0.706855\pi$
$140$		0			0
$141$	−8.47328	−	12.9805i	−0.713579	−	1.09316i
$142$		0			0
$143$			3.30272i			0.276187i
$144$		0			0
$145$		−	2.17762i		−	0.180841i
$146$		0			0
$147$	−5.76123	+	3.76075i	−0.475178	+	0.310182i
$148$		0			0
$149$		−	12.0053i		−	0.983511i	−0.870733	−	0.491756i	$-0.836355\pi$
$149$		−	12.0053i		−	0.983511i	0.870733	−	0.491756i	$-0.163645\pi$
$150$		0			0
$151$	22.7737			1.85330			0.926650	−	0.375925i	$-0.122675\pi$
$151$	22.7737			1.85330			0.926650	+	0.375925i	$0.122675\pi$
$152$		0			0
$153$	−4.10566	−	1.80478i	−0.331923	−	0.145908i
$154$		0			0
$155$		−	3.66633i		−	0.294486i
$156$		0			0
$157$	8.41112			0.671281			0.335640	−	0.941990i	$-0.391047\pi$
$157$	8.41112			0.671281			0.335640	+	0.941990i	$0.391047\pi$
$158$		0			0
$159$	−18.2728	+	11.9279i	−1.44913	+	0.945947i
$160$		0			0
$161$	3.77933			0.297853
$162$		0			0
$163$	19.4332			1.52212			0.761061	−	0.648680i	$-0.224679\pi$
$163$	19.4332			1.52212			0.761061	+	0.648680i	$0.224679\pi$
$164$		0			0
$165$	1.01838	−	0.664764i	0.0792804	−	0.0517518i
$166$		0			0
$167$		−	10.8194i		−	0.837231i	−0.908163	−	0.418616i	$-0.862515\pi$
$167$		−	10.8194i		−	0.837231i	0.908163	−	0.418616i	$-0.137485\pi$
$168$		0			0
$169$	−9.12567			−0.701975
$170$		0			0
$171$	−1.30047	+	2.95842i	−0.0994494	+	0.226236i
$172$		0			0
$173$		−	6.11300i		−	0.464763i	−0.972625	−	0.232382i	$-0.925348\pi$
$173$		−	6.11300i		−	0.464763i	0.972625	−	0.232382i	$-0.0746518\pi$
$174$		0			0
$175$			1.74006i			0.131536i
$176$		0			0
$177$	−8.77752	−	13.4466i	−0.659759	−	1.01071i
$178$		0			0
$179$	−3.89351			−0.291014			−0.145507	−	0.989357i	$-0.546481\pi$
$179$	−3.89351			−0.291014			−0.145507	+	0.989357i	$0.546481\pi$
$180$		0			0
$181$	−24.0521			−1.78778			−0.893889	−	0.448289i	$-0.852034\pi$
$181$	−24.0521			−1.78778			−0.893889	+	0.448289i	$0.852034\pi$
$182$		0			0
$183$	−2.24806	−	3.44389i	−0.166182	−	0.254579i
$184$		0			0
$185$	7.97311			0.586195
$186$		0			0
$187$		−	1.04966i		−	0.0767586i
$188$		0			0
$189$	−8.92005	+	1.47764i	−0.648838	+	0.107482i
$190$		0			0
$191$	−21.1865			−1.53300			−0.766499	−	0.642245i	$-0.778003\pi$
$191$	−21.1865			−1.53300			−0.766499	+	0.642245i	$0.778003\pi$
$192$		0			0
$193$	−15.7490			−1.13364			−0.566819	−	0.823842i	$-0.691826\pi$
$193$	−15.7490			−1.13364			−0.566819	+	0.823842i	$0.691826\pi$
$194$		0			0
$195$	4.45341	+	6.82233i	0.318915	+	0.488557i
$196$		0			0
$197$	−26.1065			−1.86001			−0.930006	−	0.367544i	$-0.880199\pi$
$197$	−26.1065			−1.86001			−0.930006	+	0.367544i	$0.880199\pi$
$198$		0			0
$199$	10.3438			0.733253			0.366626	−	0.930368i	$-0.380513\pi$
$199$	10.3438			0.733253			0.366626	+	0.930368i	$0.380513\pi$
$200$		0			0
$201$	−13.4914	+	4.35678i	−0.951611	+	0.307304i
$202$		0			0
$203$	−3.78918			−0.265949
$204$		0			0
$205$	1.68328			0.117565
$206$		0			0
$207$	−5.96499	−	2.62211i	−0.414596	−	0.182249i
$208$		0			0
$209$	−0.756353			−0.0523181
$210$		0			0
$211$	−3.64072			−0.250637			−0.125319	−	0.992117i	$-0.539995\pi$
$211$	−3.64072			−0.250637			−0.125319	+	0.992117i	$0.539995\pi$
$212$		0			0
$213$	14.4451	+	22.1290i	0.989765	+	1.51625i
$214$		0			0
$215$			2.43657i			0.166173i
$216$		0			0
$217$	−6.37962			−0.433077
$218$		0			0
$219$	4.53305	−	2.95904i	0.306315	−	0.199953i
$220$		0			0
$221$	7.03190			0.473017
$222$		0			0
$223$	−6.10424			−0.408770			−0.204385	−	0.978891i	$-0.565519\pi$
$223$	−6.10424			−0.408770			−0.204385	+	0.978891i	$0.565519\pi$
$224$		0			0
$225$	1.20725	−	2.74637i	0.0804837	−	0.183091i
$226$		0			0
$227$			6.15026i			0.408207i	0.978949	+	0.204104i	$0.0654280\pi$
$227$			6.15026i			0.408207i	−0.978949	+	0.204104i	$0.934572\pi$
$228$		0			0
$229$		−	18.4489i		−	1.21914i	−0.792732	−	0.609570i	$-0.791342\pi$
$229$		−	18.4489i		−	1.21914i	0.792732	−	0.609570i	$-0.208658\pi$
$230$		0			0
$231$	−1.15673	−	1.77203i	−0.0761072	−	0.116591i
$232$		0			0
$233$	1.64569			0.107813			0.0539063	−	0.998546i	$-0.482833\pi$
$233$	1.64569			0.107813			0.0539063	+	0.998546i	$0.482833\pi$
$234$		0			0
$235$		−	8.94967i		−	0.583812i
$236$		0			0
$237$	−1.99432	−	3.05516i	−0.129545	−	0.198454i
$238$		0			0
$239$	−28.1564			−1.82129			−0.910644	−	0.413192i	$-0.864414\pi$
$239$	−28.1564			−1.82129			−0.910644	+	0.413192i	$0.864414\pi$
$240$		0			0
$241$	−22.0413			−1.41981			−0.709903	−	0.704299i	$-0.751261\pi$
$241$	−22.0413			−1.41981			−0.709903	+	0.704299i	$0.751261\pi$
$242$		0			0
$243$	15.1039	+	3.85656i	0.968914	+	0.247398i
$244$		0			0
$245$	−3.97220			−0.253774
$246$		0			0
$247$		−	5.06698i		−	0.322404i
$248$		0			0
$249$	−11.7539	−	18.0063i	−0.744875	−	1.14110i
$250$		0			0
$251$	−24.8254			−1.56696			−0.783482	−	0.621414i	$-0.786558\pi$
$251$	−24.8254			−1.56696			−0.783482	+	0.621414i	$0.786558\pi$
$252$		0			0
$253$		−	1.52502i		−	0.0958770i
$254$		0			0
$255$	−1.41537	−	2.16825i	−0.0886337	−	0.135781i
$256$		0			0
$257$			4.24506i			0.264800i	0.991196	+	0.132400i	$0.0422683\pi$
$257$			4.24506i			0.264800i	−0.991196	+	0.132400i	$0.957732\pi$
$258$		0			0
$259$		−	13.8737i		−	0.862069i
$260$		0			0
$261$	5.98054	+	2.62894i	0.370186	+	0.162727i
$262$		0			0
$263$		−	12.9465i		−	0.798314i	−0.916883	−	0.399157i	$-0.869303\pi$
$263$		−	12.9465i		−	0.798314i	0.916883	−	0.399157i	$-0.130697\pi$
$264$		0			0
$265$	−12.5986			−0.773923
$266$		0			0
$267$	9.35944	+	14.3380i	0.572789	+	0.877474i
$268$		0			0
$269$		−	18.1180i		−	1.10467i	−0.833621	−	0.552337i	$-0.813736\pi$
$269$		−	18.1180i		−	1.10467i	0.833621	−	0.552337i	$-0.186264\pi$
$270$		0			0
$271$			5.91279i			0.359176i	0.983742	+	0.179588i	$0.0574765\pi$
$271$			5.91279i			0.359176i	−0.983742	+	0.179588i	$0.942523\pi$
$272$		0			0
$273$	11.8713	−	7.74919i	0.718481	−	0.469002i
$274$		0			0
$275$	0.702140			0.0423406
$276$		0			0
$277$	−1.40079			−0.0841652			−0.0420826	−	0.999114i	$-0.513399\pi$
$277$	−1.40079			−0.0841652			−0.0420826	+	0.999114i	$0.513399\pi$
$278$		0			0
$279$	10.0691	+	4.42619i	0.602820	+	0.264989i
$280$		0			0
$281$	12.1700			0.726000			0.363000	−	0.931789i	$-0.381753\pi$
$281$	12.1700			0.726000			0.363000	+	0.931789i	$0.381753\pi$
$282$		0			0
$283$	−12.9449			−0.769493			−0.384747	−	0.923022i	$-0.625711\pi$
$283$	−12.9449			−0.769493			−0.384747	+	0.923022i	$0.625711\pi$
$284$		0			0
$285$	−1.56238	+	1.01987i	−0.0925472	+	0.0604120i
$286$		0			0
$287$		−	2.92900i		−	0.172893i
$288$		0			0
$289$	14.7651			0.868538
$290$		0			0
$291$	−9.56193	−	14.6483i	−0.560530	−	0.858696i
$292$		0			0
$293$		−	14.1164i		−	0.824689i	−0.911028	−	0.412345i	$-0.864710\pi$
$293$		−	14.1164i		−	0.824689i	0.911028	−	0.412345i	$-0.135290\pi$
$294$		0			0
$295$		−	9.27103i		−	0.539780i
$296$		0			0
$297$	0.596249	+	3.59937i	0.0345979	+	0.208857i
$298$		0			0
$299$	10.2164			0.590832
$300$		0			0
$301$	4.23977			0.244376
$302$		0			0
$303$	26.4698	−	17.2787i	1.52065	−	0.992635i
$304$		0			0
$305$		−	2.37446i		−	0.135961i
$306$		0			0
$307$	30.8686			1.76177			0.880883	−	0.473335i	$-0.156950\pi$
$307$	30.8686			1.76177			0.880883	+	0.473335i	$0.156950\pi$
$308$		0			0
$309$	17.5870	−	11.4802i	1.00049	−	0.653088i
$310$		0			0
$311$	−6.01054			−0.340827			−0.170413	−	0.985373i	$-0.554510\pi$
$311$	−6.01054			−0.340827			−0.170413	+	0.985373i	$0.554510\pi$
$312$		0			0
$313$		−	10.2122i		−	0.577227i	−0.957446	−	0.288613i	$-0.906806\pi$
$313$		−	10.2122i		−	0.577227i	0.957446	−	0.288613i	$-0.0931942\pi$
$314$		0			0
$315$	−4.77884	−	2.10069i	−0.269257	−	0.118361i
$316$		0			0
$317$			1.10744i			0.0621998i	0.999516	+	0.0310999i	$0.00990100\pi$
$317$			1.10744i			0.0621998i	−0.999516	+	0.0310999i	$0.990099\pi$
$318$		0			0
$319$			1.52899i			0.0856071i
$320$		0			0
$321$	−11.6137	−	17.7915i	−0.648216	−	0.993024i
$322$		0			0
$323$			1.61037i			0.0896034i
$324$		0			0
$325$			4.70379i			0.260920i
$326$		0			0
$327$	14.7984	+	22.6701i	0.818351	+	1.25366i
$328$		0			0
$329$	−15.5730			−0.858565
$330$		0			0
$331$			3.89325i			0.213992i	0.994259	+	0.106996i	$0.0341233\pi$
$331$			3.89325i			0.213992i	−0.994259	+	0.106996i	$0.965877\pi$
$332$		0			0
$333$	−9.62558	+	21.8971i	−0.527478	+	1.19995i
$334$		0			0
$335$	−7.89756	−	2.15141i	−0.431490	−	0.117544i
$336$		0			0
$337$		−	26.9475i		−	1.46792i	−0.679191	−	0.733961i	$-0.737669\pi$
$337$		−	26.9475i		−	1.46792i	0.679191	−	0.733961i	$-0.262331\pi$
$338$		0			0
$339$	12.5656	−	8.20247i	0.682472	−	0.445497i
$340$		0			0
$341$			2.57427i			0.139405i
$342$		0			0
$343$			19.0923i			1.03089i
$344$		0			0
$345$	−2.05634	−	3.15018i	−0.110710	−	0.169600i
$346$		0			0
$347$	−7.19725			−0.386369			−0.193184	−	0.981162i	$-0.561882\pi$
$347$	−7.19725			−0.386369			−0.193184	+	0.981162i	$0.561882\pi$
$348$		0			0
$349$	−12.3932			−0.663391			−0.331696	−	0.943386i	$-0.607621\pi$
$349$	−12.3932			−0.663391			−0.331696	+	0.943386i	$0.607621\pi$
$350$		0			0
$351$	−24.1130	+	3.99441i	−1.28706	+	0.213206i
$352$		0			0
$353$	0.905014			0.0481690			0.0240845	−	0.999710i	$-0.492333\pi$
$353$	0.905014			0.0481690			0.0240845	+	0.999710i	$0.492333\pi$
$354$		0			0
$355$			15.2573i			0.809773i
$356$		0			0
$357$	−3.77288	+	2.46282i	−0.199682	+	0.130346i
$358$		0			0
$359$		−	7.76504i		−	0.409823i	−0.978780	−	0.204912i	$-0.934309\pi$
$359$		−	7.76504i		−	0.409823i	0.978780	−	0.204912i	$-0.0656907\pi$
$360$		0			0
$361$	−17.8396			−0.938927
$362$		0			0
$363$	15.2392	−	9.94771i	0.799852	−	0.522119i
$364$		0			0
$365$	3.12540			0.163591
$366$		0			0
$367$		−	28.2125i		−	1.47268i	−0.676611	−	0.736340i	$-0.736552\pi$
$367$		−	28.2125i		−	1.47268i	0.676611	−	0.736340i	$-0.263448\pi$
$368$		0			0
$369$	−2.03214	+	4.62290i	−0.105789	+	0.240659i
$370$		0			0
$371$			21.9222i			1.13815i
$372$		0			0
$373$			0.360074i			0.0186439i	0.999957	+	0.00932196i	$0.00296732\pi$
$373$			0.360074i			0.0186439i	−0.999957	+	0.00932196i	$0.997033\pi$
$374$		0			0
$375$	1.45039	−	0.946770i	0.0748978	−	0.0488910i
$376$		0			0
$377$	−10.2431			−0.527545
$378$		0			0
$379$			29.9071i			1.53622i	0.640315	+	0.768112i	$0.278804\pi$
$379$			29.9071i			1.53622i	−0.640315	+	0.768112i	$0.721196\pi$
$380$		0			0
$381$	23.3844	−	15.2646i	1.19802	−	0.782030i
$382$		0			0
$383$	6.74193			0.344497			0.172248	−	0.985054i	$-0.444897\pi$
$383$	6.74193			0.344497			0.172248	+	0.985054i	$0.444897\pi$
$384$		0			0
$385$		−	1.22176i		−	0.0622669i
$386$		0			0
$387$	−6.69171	−	2.94156i	−0.340159	−	0.149528i
$388$		0			0
$389$			38.4310i			1.94853i	0.225408	+	0.974264i	$0.427628\pi$
$389$			38.4310i			1.94853i	−0.225408	+	0.974264i	$0.572372\pi$
$390$		0			0
$391$	−3.24695			−0.164205
$392$		0			0
$393$	13.1890	+	20.2046i	0.665295	+	1.01919i
$394$		0			0
$395$		−	2.10644i		−	0.105987i
$396$		0			0
$397$	−11.4882			−0.576578			−0.288289	−	0.957543i	$-0.593086\pi$
$397$	−11.4882			−0.576578			−0.288289	+	0.957543i	$0.593086\pi$
$398$		0			0
$399$	1.77464	+	2.71863i	0.0888430	+	0.136102i
$400$		0			0
$401$	−27.5106			−1.37381			−0.686906	−	0.726746i	$-0.741032\pi$
$401$	−27.5106			−1.37381			−0.686906	+	0.726746i	$0.741032\pi$
$402$		0			0
$403$	−17.2456			−0.859067
$404$		0			0
$405$	6.08507	+	6.63113i	0.302370	+	0.329504i
$406$		0			0
$407$	−5.59824			−0.277494
$408$		0			0
$409$			34.1771i			1.68995i	0.534805	+	0.844976i	$0.320385\pi$
$409$			34.1771i			1.68995i	−0.534805	+	0.844976i	$0.679615\pi$
$410$		0			0
$411$	−1.71104	+	1.11692i	−0.0843994	+	0.0550934i
$412$		0			0
$413$	−16.1321			−0.793810
$414$		0			0
$415$		−	12.4148i		−	0.609417i
$416$		0			0
$417$	−17.7742	−	27.2288i	−0.870404	−	1.33340i
$418$		0			0
$419$			8.34103i			0.407486i	0.979024	+	0.203743i	$0.0653107\pi$
$419$			8.34103i			0.407486i	−0.979024	+	0.203743i	$0.934689\pi$
$420$		0			0
$421$	−7.19229			−0.350531			−0.175265	−	0.984521i	$-0.556078\pi$
$421$	−7.19229			−0.350531			−0.175265	+	0.984521i	$0.556078\pi$
$422$		0			0
$423$	24.5791	+	10.8045i	1.19508	+	0.525334i
$424$		0			0
$425$		−	1.49494i		−	0.0725154i
$426$		0			0
$427$	−4.13169			−0.199947
$428$		0			0
$429$	−3.12691	−	4.79023i	−0.150969	−	0.231274i
$430$		0			0
$431$		−	18.0703i		−	0.870415i	−0.900330	−	0.435208i	$-0.856675\pi$
$431$		−	18.0703i		−	0.870415i	0.900330	−	0.435208i	$-0.143325\pi$
$432$		0			0
$433$			19.2523i			0.925205i	0.886566	+	0.462602i	$0.153084\pi$
$433$			19.2523i			0.925205i	−0.886566	+	0.462602i	$0.846916\pi$
$434$		0			0
$435$	2.06170	+	3.15839i	0.0988511	+	0.151433i
$436$		0			0
$437$			2.33966i			0.111921i
$438$		0			0
$439$	5.64185			0.269271			0.134635	−	0.990895i	$-0.457014\pi$
$439$	5.64185			0.269271			0.134635	+	0.990895i	$0.457014\pi$
$440$		0			0
$441$	4.79545	−	10.9091i	0.228355	−	0.519482i
$442$		0			0
$443$	25.2564			1.19997			0.599983	−	0.800012i	$-0.295174\pi$
$443$	25.2564			1.19997			0.599983	+	0.800012i	$0.295174\pi$
$444$		0			0
$445$			9.88566i			0.468625i
$446$		0			0
$447$	11.3662	+	17.4123i	0.537605	+	0.823575i
$448$		0			0
$449$			0.566830i			0.0267504i	0.999911	+	0.0133752i	$0.00425758\pi$
$449$			0.566830i			0.0267504i	−0.999911	+	0.0133752i	$0.995742\pi$
$450$		0			0
$451$	−1.18190			−0.0556533
$452$		0			0
$453$	−33.0308	+	21.5615i	−1.55192	+	1.01305i
$454$		0			0
$455$	8.18488			0.383713
$456$		0			0
$457$	12.6220			0.590434			0.295217	−	0.955430i	$-0.404608\pi$
$457$	12.6220			0.590434			0.295217	+	0.955430i	$0.404608\pi$
$458$		0			0
$459$	7.66352	−	1.26949i	0.357702	−	0.0592547i
$460$		0			0
$461$		−	12.5921i		−	0.586471i	−0.956040	−	0.293235i	$-0.905268\pi$
$461$		−	12.5921i		−	0.586471i	0.956040	−	0.293235i	$-0.0947320\pi$
$462$		0			0
$463$			22.4105i			1.04151i	0.853707	+	0.520753i	$0.174349\pi$
$463$			22.4105i			1.04151i	−0.853707	+	0.520753i	$0.825651\pi$
$464$		0			0
$465$	3.47117	+	5.31760i	0.160972	+	0.246598i
$466$		0			0
$467$			29.6412i			1.37163i	0.727775	+	0.685816i	$0.240555\pi$
$467$			29.6412i			1.37163i	−0.727775	+	0.685816i	$0.759445\pi$
$468$		0			0
$469$	−3.74358	+	13.7422i	−0.172863	+	0.634556i
$470$		0			0
$471$	−12.1994	+	7.96339i	−0.562118	+	0.366934i
$472$		0			0
$473$		−	1.71081i		−	0.0786632i
$474$		0			0
$475$	−1.07721			−0.0494259
$476$		0			0
$477$	15.2097	−	34.6003i	0.696403	−	1.58424i
$478$		0			0
$479$			5.34964i			0.244431i	0.992504	+	0.122216i	$0.0389999\pi$
$479$			5.34964i			0.244431i	−0.992504	+	0.122216i	$0.961000\pi$
$480$		0			0
$481$		−	37.5039i		−	1.71003i
$482$		0			0
$483$	−5.48150	+	3.57816i	−0.249417	+	0.162812i
$484$		0			0
$485$		−	10.0995i		−	0.458596i
$486$		0			0
$487$		−	42.5706i		−	1.92906i	−0.263980	−	0.964528i	$-0.585035\pi$
$487$		−	42.5706i		−	1.92906i	0.263980	−	0.964528i	$-0.414965\pi$
$488$		0			0
$489$	−28.1856	+	18.3987i	−1.27460	+	0.832019i
$490$		0			0
$491$			25.8771i			1.16782i	0.811820	+	0.583908i	$0.198477\pi$
$491$			25.8771i			1.16782i	−0.811820	+	0.583908i	$0.801523\pi$
$492$		0			0
$493$	3.25542			0.146617
$494$		0			0
$495$	−0.847661	+	1.92833i	−0.0380995	+	0.0866721i
$496$		0			0
$497$	26.5486			1.19087
$498$		0			0
$499$		−	9.76701i		−	0.437232i	−0.975811	−	0.218616i	$-0.929846\pi$
$499$		−	9.76701i		−	0.437232i	0.975811	−	0.218616i	$-0.0701541\pi$
$500$		0			0
$501$	10.2435	+	15.6924i	0.457645	+	0.701083i
$502$		0			0
$503$	4.42529			0.197314			0.0986569	−	0.995122i	$-0.468545\pi$
$503$	4.42529			0.197314			0.0986569	+	0.995122i	$0.468545\pi$
$504$		0			0
$505$	18.2502			0.812121
$506$		0			0
$507$	13.2358	−	8.63991i	0.587821	−	0.383712i
$508$		0			0
$509$		−	0.0706133i		−	0.00312988i	−0.999999	−	0.00156494i	$-0.999502\pi$
$509$		−	0.0706133i		−	0.00312988i	0.999999	−	0.00156494i	$-0.000498136\pi$
$510$		0			0
$511$		−	5.43838i		−	0.240580i
$512$		0			0
$513$	−0.914757	−	5.52211i	−0.0403875	−	0.243807i
$514$		0			0
$515$	12.1257			0.534322
$516$		0			0
$517$			6.28392i			0.276366i
$518$		0			0
$519$	5.78761	+	8.86623i	0.254048	+	0.389184i
$520$		0			0
$521$	−30.9621			−1.35647			−0.678236	−	0.734844i	$-0.737255\pi$
$521$	−30.9621			−1.35647			−0.678236	+	0.734844i	$0.737255\pi$
$522$		0			0
$523$	19.8295			0.867085			0.433542	−	0.901133i	$-0.357263\pi$
$523$	19.8295			0.867085			0.433542	+	0.901133i	$0.357263\pi$
$524$		0			0
$525$	−1.64743	−	2.52376i	−0.0718999	−	0.110146i
$526$		0			0
$527$	5.48095			0.238754
$528$		0			0
$529$	18.2826			0.794896
$530$		0			0
$531$	25.4616	+	11.1925i	1.10494	+	0.485713i
$532$		0			0
$533$		−	7.91779i		−	0.342958i
$534$		0			0
$535$		−	12.2667i		−	0.530336i
$536$		0			0
$537$	5.64710	−	3.68626i	0.243690	−	0.159074i
$538$		0			0
$539$	2.78904			0.120132
$540$		0			0
$541$			24.4505i			1.05121i	0.850730	+	0.525603i	$0.176160\pi$
$541$			24.4505i			1.05121i	−0.850730	+	0.525603i	$0.823840\pi$
$542$		0			0
$543$	34.8849	−	22.7718i	1.49705	−	0.977231i
$544$		0			0
$545$			15.6304i			0.669531i
$546$		0			0
$547$		−	1.85263i		−	0.0792126i	−0.999215	−	0.0396063i	$-0.987390\pi$
$547$		−	1.85263i		−	0.0792126i	0.999215	−	0.0396063i	$-0.0126104\pi$
$548$		0			0
$549$	6.52113	+	2.86657i	0.278315	+	0.122342i
$550$		0			0
$551$		−	2.34576i		−	0.0999326i
$552$		0			0
$553$	−3.66534			−0.155866
$554$		0			0
$555$	−11.5641	+	7.54870i	−0.490869	+	0.320424i
$556$		0			0
$557$		−	2.06857i		−	0.0876483i	−0.999039	−	0.0438241i	$-0.986046\pi$
$557$		−	2.06857i		−	0.0876483i	0.999039	−	0.0438241i	$-0.0139541\pi$
$558$		0			0
$559$	11.4611			0.484753
$560$		0			0
$561$	0.993785	+	1.52241i	0.0419576	+	0.0642763i
$562$		0			0
$563$	−8.71387			−0.367246			−0.183623	−	0.982997i	$-0.558783\pi$
$563$	−8.71387			−0.367246			−0.183623	+	0.982997i	$0.558783\pi$
$564$		0			0
$565$	8.66363			0.364482
$566$		0			0
$567$	11.5386	−	10.5884i	0.484574	−	0.444670i
$568$		0			0
$569$			12.0152i			0.503705i	0.967766	+	0.251853i	$0.0810398\pi$
$569$			12.0152i			0.503705i	−0.967766	+	0.251853i	$0.918960\pi$
$570$		0			0
$571$	23.4721			0.982276			0.491138	−	0.871082i	$-0.336581\pi$
$571$	23.4721			0.982276			0.491138	+	0.871082i	$0.336581\pi$
$572$		0			0
$573$	30.7286	−	20.0587i	1.28371	−	0.837964i
$574$		0			0
$575$		−	2.17196i		−	0.0905769i
$576$		0			0
$577$		−	4.39504i		−	0.182968i	−0.995807	−	0.0914839i	$-0.970839\pi$
$577$		−	4.39504i		−	0.182968i	0.995807	−	0.0914839i	$-0.0291610\pi$
$578$		0			0
$579$	22.8422	−	14.9107i	0.949289	−	0.619667i
$580$		0			0
$581$	−21.6024			−0.896220
$582$		0			0
$583$	8.84595			0.366362
$584$		0			0
$585$	−12.9183	−	5.67868i	−0.534108	−	0.234784i
$586$		0			0
$587$	−32.2450			−1.33089			−0.665447	−	0.746445i	$-0.731759\pi$
$587$	−32.2450			−1.33089			−0.665447	+	0.746445i	$0.731759\pi$
$588$		0			0
$589$		−	3.94941i		−	0.162733i
$590$		0			0
$591$	37.8646	−	24.7169i	1.55754	−	1.01672i
$592$		0			0
$593$	−11.2368			−0.461440			−0.230720	−	0.973020i	$-0.574108\pi$
$593$	−11.2368			−0.461440			−0.230720	+	0.973020i	$0.574108\pi$
$594$		0			0
$595$	−2.60129			−0.106642
$596$		0			0
$597$	−15.0025	+	9.79319i	−0.614013	+	0.400809i
$598$		0			0
$599$	14.7407			0.602289			0.301145	−	0.953578i	$-0.402631\pi$
$599$	14.7407			0.602289			0.301145	+	0.953578i	$0.402631\pi$
$600$		0			0
$601$	−11.8462			−0.483217			−0.241608	−	0.970374i	$-0.577675\pi$
$601$	−11.8462			−0.483217			−0.241608	+	0.970374i	$0.577675\pi$
$602$		0			0
$603$	15.4429	−	19.0923i	0.628885	−	0.777498i
$604$		0			0
$605$	10.5070			0.427170
$606$		0			0
$607$	−31.6110			−1.28305			−0.641525	−	0.767102i	$-0.721698\pi$
$607$	−31.6110			−1.28305			−0.641525	+	0.767102i	$0.721698\pi$
$608$		0			0
$609$	5.49579	−	3.58748i	0.222701	−	0.145372i
$610$		0			0
$611$	−42.0974			−1.70308
$612$		0			0
$613$	−22.8703			−0.923724			−0.461862	−	0.886952i	$-0.652818\pi$
$613$	−22.8703			−0.923724			−0.461862	+	0.886952i	$0.652818\pi$
$614$		0			0
$615$	−2.44141	+	1.59368i	−0.0984470	+	0.0642632i
$616$		0			0
$617$			28.5570i			1.14966i	0.818273	+	0.574830i	$0.194932\pi$
$617$			28.5570i			1.14966i	−0.818273	+	0.574830i	$0.805068\pi$
$618$		0			0
$619$	30.0360			1.20725			0.603624	−	0.797269i	$-0.293723\pi$
$619$	30.0360			1.20725			0.603624	+	0.797269i	$0.293723\pi$
$620$		0			0
$621$	11.1341	−	1.84440i	0.446796	−	0.0740133i
$622$		0			0
$623$	17.2016			0.689169
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	1.09701	−	0.716092i	0.0438102	−	0.0285980i
$628$		0			0
$629$			11.9193i			0.475256i
$630$		0			0
$631$		−	20.4655i		−	0.814718i	−0.913268	−	0.407359i	$-0.866450\pi$
$631$		−	20.4655i		−	0.814718i	0.913268	−	0.407359i	$-0.133550\pi$
$632$		0			0
$633$	5.28045	−	3.44692i	0.209879	−	0.137003i
$634$		0			0
$635$	16.1229			0.639816
$636$		0			0
$637$			18.6844i			0.740303i
$638$		0			0
$639$	−41.9022	−	18.4194i	−1.65762	−	0.728662i
$640$		0			0
$641$	18.4518			0.728803			0.364401	−	0.931242i	$-0.381274\pi$
$641$	18.4518			0.728803			0.364401	+	0.931242i	$0.381274\pi$
$642$		0			0
$643$	14.1845			0.559382			0.279691	−	0.960090i	$-0.409768\pi$
$643$	14.1845			0.559382			0.279691	+	0.960090i	$0.409768\pi$
$644$		0			0
$645$	−2.30687	−	3.53397i	−0.0908329	−	0.139150i
$646$		0			0
$647$	−25.8869			−1.01772			−0.508860	−	0.860849i	$-0.669933\pi$
$647$	−25.8869			−1.01772			−0.508860	+	0.860849i	$0.669933\pi$
$648$		0			0
$649$			6.50955i			0.255522i
$650$		0			0
$651$	9.25293	−	6.04003i	0.362651	−	0.236728i
$652$		0			0
$653$	43.6003			1.70621			0.853106	−	0.521737i	$-0.174716\pi$
$653$	43.6003			1.70621			0.853106	+	0.521737i	$0.174716\pi$
$654$		0			0
$655$			13.9305i			0.544309i
$656$		0			0
$657$	−3.77316	+	8.58350i	−0.147205	+	0.334874i
$658$		0			0
$659$		−	7.99828i		−	0.311569i	−0.987791	−	0.155784i	$-0.950209\pi$
$659$		−	7.99828i		−	0.311569i	0.987791	−	0.155784i	$-0.0497905\pi$
$660$		0			0
$661$		−	3.98369i		−	0.154947i	−0.996994	−	0.0774737i	$-0.975315\pi$
$661$		−	3.98369i		−	0.154947i	0.996994	−	0.0774737i	$-0.0246854\pi$
$662$		0			0
$663$	−10.1990	+	6.65759i	−0.396096	+	0.258559i
$664$		0			0
$665$			1.87441i			0.0726866i
$666$		0			0
$667$	4.72970			0.183135
$668$		0			0
$669$	8.85352	−	5.77931i	0.342297	−	0.223441i
$670$		0			0
$671$			1.66720i			0.0643615i
$672$		0			0
$673$			44.1445i			1.70164i	0.525454	+	0.850822i	$0.323896\pi$
$673$			44.1445i			1.70164i	−0.525454	+	0.850822i	$0.676104\pi$
$674$		0			0
$675$	0.849189	+	5.12629i	0.0326853	+	0.197311i
$676$		0			0
$677$	−0.784682			−0.0301578			−0.0150789	−	0.999886i	$-0.504800\pi$
$677$	−0.784682			−0.0301578			−0.0150789	+	0.999886i	$0.504800\pi$
$678$		0			0
$679$	−17.5738			−0.674420
$680$		0			0
$681$	−5.82288	−	8.92027i	−0.223133	−	0.341826i
$682$		0			0
$683$	14.9024			0.570224			0.285112	−	0.958494i	$-0.407969\pi$
$683$	14.9024			0.570224			0.285112	+	0.958494i	$0.407969\pi$
$684$		0			0
$685$	−1.17971			−0.0450745
$686$		0			0
$687$	17.4669	+	26.7581i	0.666403	+	1.02089i
$688$		0			0
$689$			59.2610i			2.25766i
$690$		0			0
$691$	6.69509			0.254693			0.127347	−	0.991858i	$-0.459354\pi$
$691$	6.69509			0.254693			0.127347	+	0.991858i	$0.459354\pi$
$692$		0			0
$693$	3.35541	+	1.47498i	0.127462	+	0.0560299i
$694$		0			0
$695$		−	18.7735i		−	0.712119i
$696$		0			0
$697$			2.51640i			0.0953156i
$698$		0			0
$699$	−2.38689	+	1.55809i	−0.0902804	+	0.0589323i
$700$		0			0
$701$	−11.1459			−0.420975			−0.210488	−	0.977597i	$-0.567505\pi$
$701$	−11.1459			−0.420975			−0.210488	+	0.977597i	$0.567505\pi$
$702$		0			0
$703$	8.58873			0.323930
$704$		0			0
$705$	8.47328	+	12.9805i	0.319122	+	0.488874i
$706$		0			0
$707$		−	31.7563i		−	1.19432i
$708$		0			0
$709$	−9.83155			−0.369232			−0.184616	−	0.982811i	$-0.559104\pi$
$709$	−9.83155			−0.369232			−0.184616	+	0.982811i	$0.559104\pi$
$710$		0			0
$711$	5.78507	+	2.54302i	0.216957	+	0.0953705i
$712$		0			0
$713$	7.96311			0.298221
$714$		0			0
$715$		−	3.30272i		−	0.123515i
$716$		0			0
$717$	40.8378	−	26.6576i	1.52511	−	0.995548i
$718$		0			0
$719$			6.64137i			0.247681i	0.992302	+	0.123841i	$0.0395212\pi$
$719$			6.64137i			0.247681i	−0.992302	+	0.123841i	$0.960479\pi$
$720$		0			0
$721$		−	21.0994i		−	0.785783i
$722$		0			0
$723$	31.9685	−	20.8680i	1.18892	−	0.776091i
$724$		0			0
$725$			2.17762i			0.0808747i
$726$		0			0
$727$			29.8459i			1.10692i	0.832875	+	0.553461i	$0.186693\pi$
$727$			29.8459i			1.10692i	−0.832875	+	0.553461i	$0.813307\pi$
$728$		0			0
$729$	−25.5578	+	8.70638i	−0.946584	+	0.322459i
$730$		0			0
$731$	−3.64253			−0.134724
$732$		0			0
$733$			42.1397i			1.55646i	0.627977	+	0.778232i	$0.283883\pi$
$733$			42.1397i			1.55646i	−0.627977	+	0.778232i	$0.716117\pi$
$734$		0			0
$735$	5.76123	−	3.76075i	0.212506	−	0.138718i
$736$		0			0
$737$	5.54519	+	1.51059i	0.204260	+	0.0556434i
$738$		0			0
$739$		−	16.1943i		−	0.595717i	−0.954610	−	0.297858i	$-0.903728\pi$
$739$		−	16.1943i		−	0.595717i	0.954610	−	0.297858i	$-0.0962724\pi$
$740$		0			0
$741$	4.79727	+	7.34910i	0.176232	+	0.269976i
$742$		0			0
$743$			42.0160i			1.54142i	0.637188	+	0.770708i	$0.280097\pi$
$743$			42.0160i			1.54142i	−0.637188	+	0.770708i	$0.719903\pi$
$744$		0			0
$745$			12.0053i			0.439840i
$746$		0			0
$747$	34.0955	+	14.9878i	1.24749	+	0.548375i
$748$		0			0
$749$	−21.3448			−0.779921
$750$		0			0
$751$	28.5335			1.04120			0.520600	−	0.853801i	$-0.325708\pi$
$751$	28.5335			1.04120			0.520600	+	0.853801i	$0.325708\pi$
$752$		0			0
$753$	36.0065	−	23.5039i	1.31215	−	0.856531i
$754$		0			0
$755$	−22.7737			−0.828821
$756$		0			0
$757$			14.3021i			0.519820i	0.965633	+	0.259910i	$0.0836929\pi$
$757$			14.3021i			0.519820i	−0.965633	+	0.259910i	$0.916307\pi$
$758$		0			0
$759$	1.44384	+	2.21187i	0.0524081	+	0.0802857i
$760$		0			0
$761$		−	34.1115i		−	1.23654i	−0.785965	−	0.618271i	$-0.787834\pi$
$761$		−	34.1115i		−	1.23654i	0.785965	−	0.618271i	$-0.212166\pi$
$762$		0			0
$763$	27.1977			0.984624
$764$		0			0
$765$	4.10566	+	1.80478i	0.148441	+	0.0652518i
$766$		0			0
$767$	−43.6090			−1.57463
$768$		0			0
$769$		−	26.7724i		−	0.965437i	−0.875776	−	0.482719i	$-0.839649\pi$
$769$		−	26.7724i		−	0.965437i	0.875776	−	0.482719i	$-0.160351\pi$
$770$		0			0
$771$	−4.01910	−	6.15699i	−0.144744	−	0.221739i
$772$		0			0
$773$		−	48.3814i		−	1.74016i	−0.492914	−	0.870078i	$-0.664068\pi$
$773$		−	48.3814i		−	1.74016i	0.492914	−	0.870078i	$-0.335932\pi$
$774$		0			0
$775$			3.66633i			0.131698i
$776$		0			0
$777$	13.1352	+	20.1222i	0.471222	+	0.721881i
$778$		0			0
$779$	1.81325			0.0649663
$780$		0			0
$781$		−	10.7128i		−	0.383332i
$782$		0			0
$783$	−11.1631	+	1.84921i	−0.398937	+	0.0660854i
$784$		0			0
$785$	−8.41112			−0.300206
$786$		0			0
$787$		−	18.2987i		−	0.652279i	−0.945322	−	0.326139i	$-0.894252\pi$
$787$		−	18.2987i		−	0.652279i	0.945322	−	0.326139i	$-0.105748\pi$
$788$		0			0
$789$	12.2573	+	18.7774i	0.436373	+	0.668494i
$790$		0			0
$791$		−	15.0752i		−	0.536013i
$792$		0			0
$793$	−11.1690			−0.396621
$794$		0			0
$795$	18.2728	−	11.9279i	0.648070	−	0.423040i
$796$		0			0
$797$		−	26.2965i		−	0.931470i	−0.884924	−	0.465735i	$-0.845790\pi$
$797$		−	26.2965i		−	0.931470i	0.884924	−	0.465735i	$-0.154210\pi$
$798$		0			0
$799$	13.3792			0.473324
$800$		0			0
$801$	−27.1497	−	11.9345i	−0.959286	−	0.421685i
$802$		0			0
$803$	−2.19447			−0.0774411
$804$		0			0
$805$	−3.77933			−0.133204
$806$		0			0
$807$	17.1536	+	26.2781i	0.603834	+	0.925034i
$808$		0			0
$809$	−34.1582			−1.20094			−0.600469	−	0.799648i	$-0.705019\pi$
$809$	−34.1582			−1.20094			−0.600469	+	0.799648i	$0.705019\pi$
$810$		0			0
$811$		−	24.8096i		−	0.871183i	−0.900144	−	0.435592i	$-0.856539\pi$
$811$		−	24.8096i		−	0.871183i	0.900144	−	0.435592i	$-0.143461\pi$
$812$		0			0
$813$	−5.59805	−	8.57584i	−0.196332	−	0.300768i
$814$		0			0
$815$	−19.4332			−0.680714
$816$		0			0
$817$			2.62470i			0.0918267i
$818$		0			0
$819$	−9.88123	+	22.4787i	−0.345278	+	0.785469i
$820$		0			0
$821$			6.08910i			0.212511i	0.994339	+	0.106256i	$0.0338861\pi$
$821$			6.08910i			0.212511i	−0.994339	+	0.106256i	$0.966114\pi$
$822$		0			0
$823$	32.8212			1.14408			0.572038	−	0.820227i	$-0.306153\pi$
$823$	32.8212			1.14408			0.572038	+	0.820227i	$0.306153\pi$
$824$		0			0
$825$	−1.01838	+	0.664764i	−0.0354553	+	0.0231441i
$826$		0			0
$827$			42.2107i			1.46781i	0.679252	+	0.733905i	$0.262304\pi$
$827$			42.2107i			1.46781i	−0.679252	+	0.733905i	$0.737696\pi$
$828$		0			0
$829$	37.9196			1.31700			0.658502	−	0.752579i	$-0.271190\pi$
$829$	37.9196			1.31700			0.658502	+	0.752579i	$0.271190\pi$
$830$		0			0
$831$	2.03169	−	1.32622i	0.0704785	−	0.0460062i
$832$		0			0
$833$		−	5.93821i		−	0.205747i
$834$		0			0
$835$			10.8194i			0.374421i
$836$		0			0
$837$	−18.7947	+	3.11340i	−0.649639	+	0.107615i
$838$		0			0
$839$			4.12659i			0.142466i	0.997460	+	0.0712328i	$0.0226933\pi$
$839$			4.12659i			0.142466i	−0.997460	+	0.0712328i	$0.977307\pi$
$840$		0			0
$841$	24.2580			0.836482
$842$		0			0
$843$	−17.6512	+	11.5222i	−0.607939	+	0.396844i
$844$		0			0
$845$	9.12567			0.313933
$846$		0			0
$847$		−	18.2828i		−	0.628204i
$848$		0			0
$849$	18.7751	−	12.2558i	0.644360	−	0.420619i
$850$		0			0
$851$			17.3173i			0.593628i
$852$		0			0
$853$	17.0664			0.584343			0.292171	−	0.956366i	$-0.405622\pi$
$853$	17.0664			0.584343			0.292171	+	0.956366i	$0.405622\pi$
$854$		0			0
$855$	1.30047	−	2.95842i	0.0444751	−	0.101176i
$856$		0			0
$857$	−21.1415			−0.722180			−0.361090	−	0.932531i	$-0.617595\pi$
$857$	−21.1415			−0.722180			−0.361090	+	0.932531i	$0.617595\pi$
$858$		0			0
$859$	32.9373			1.12381			0.561904	−	0.827203i	$-0.310069\pi$
$859$	32.9373			1.12381			0.561904	+	0.827203i	$0.310069\pi$
$860$		0			0
$861$	2.77309	+	4.24819i	0.0945066	+	0.144778i
$862$		0			0
$863$			26.3474i			0.896876i	0.893814	+	0.448438i	$0.148020\pi$
$863$			26.3474i			0.896876i	−0.893814	+	0.448438i	$0.851980\pi$
$864$		0			0
$865$			6.11300i			0.207848i
$866$		0			0
$867$	−21.4152	+	13.9792i	−0.727298	+	0.474758i
$868$		0			0
$869$			1.47902i			0.0501723i
$870$		0			0
$871$	−10.1198	+	37.1485i	−0.342896	+	1.25873i
$872$		0			0
$873$	27.7370	+	12.1927i	0.938757	+	0.412661i
$874$		0			0
$875$		−	1.74006i		−	0.0588247i
$876$		0			0
$877$	−34.4639			−1.16376			−0.581881	−	0.813274i	$-0.697683\pi$
$877$	−34.4639			−1.16376			−0.581881	+	0.813274i	$0.697683\pi$
$878$		0			0
$879$	13.3650	+	20.4743i	0.450790	+	0.690580i
$880$		0			0
$881$			15.0431i			0.506813i	0.967360	+	0.253407i	$0.0815511\pi$
$881$			15.0431i			0.506813i	−0.967360	+	0.253407i	$0.918449\pi$
$882$		0			0
$883$			15.3001i			0.514890i	0.966293	+	0.257445i	$0.0828806\pi$
$883$			15.3001i			0.514890i	−0.966293	+	0.257445i	$0.917119\pi$
$884$		0			0
$885$	8.77752	+	13.4466i	0.295053	+	0.452002i
$886$		0			0
$887$			18.8837i			0.634053i	0.948417	+	0.317027i	$0.102684\pi$
$887$			18.8837i			0.634053i	−0.948417	+	0.317027i	$0.897316\pi$
$888$		0			0
$889$		−	28.0547i		−	0.940925i
$890$		0			0
$891$	−4.27257	−	4.65598i	−0.143136	−	0.155981i
$892$		0			0
$893$		−	9.64070i		−	0.322614i
$894$		0			0
$895$	3.89351			0.130146
$896$		0			0
$897$	−14.8178	+	9.67262i	−0.494752	+	0.322959i
$898$		0			0
$899$	−7.98386			−0.266277
$900$		0			0
$901$		−	18.8341i		−	0.627456i
$902$		0			0
$903$	−6.14932	+	4.01409i	−0.204636	+	0.133580i
$904$		0			0
$905$	24.0521			0.799518
$906$		0			0
$907$	28.4931			0.946099			0.473050	−	0.881036i	$-0.343153\pi$
$907$	28.4931			0.946099			0.473050	+	0.881036i	$0.343153\pi$
$908$		0			0
$909$	−22.0326	+	50.1216i	−0.730775	+	1.66243i
$910$		0			0
$911$			0.139047i			0.00460683i	0.999997	+	0.00230342i	$0.000733200\pi$
$911$			0.139047i			0.00460683i	−0.999997	+	0.00230342i	$0.999267\pi$
$912$		0			0
$913$			8.71691i			0.288488i
$914$		0			0
$915$	2.24806	+	3.44389i	0.0743187	+	0.113851i
$916$		0			0
$917$	24.2399			0.800471
$918$		0			0
$919$			48.0721i			1.58575i	0.609383	+	0.792876i	$0.291417\pi$
$919$			48.0721i			1.58575i	−0.609383	+	0.792876i	$0.708583\pi$
$920$		0			0
$921$	−44.7715	+	29.2255i	−1.47527	+	0.963012i
$922$		0			0
$923$	71.7672			2.36225
$924$		0			0
$925$	−7.97311			−0.262154
$926$		0			0
$927$	−14.6388	+	33.3016i	−0.480801	+	1.09377i
$928$		0			0
$929$	16.5765			0.543856			0.271928	−	0.962318i	$-0.412339\pi$
$929$	16.5765			0.543856			0.271928	+	0.962318i	$0.412339\pi$
$930$		0			0
$931$	−4.27890			−0.140235
$932$		0			0
$933$	8.71762	−	5.69060i	0.285402	−	0.186302i
$934$		0			0
$935$			1.04966i			0.0343275i
$936$		0			0
$937$		−	37.4112i		−	1.22217i	−0.791565	−	0.611085i	$-0.790733\pi$
$937$		−	37.4112i		−	1.22217i	0.791565	−	0.611085i	$-0.209267\pi$
$938$		0			0
$939$	9.66858	+	14.8116i	0.315522	+	0.483359i
$940$		0			0
$941$	−34.5496			−1.12629			−0.563143	−	0.826360i	$-0.690408\pi$
$941$	−34.5496			−1.12629			−0.563143	+	0.826360i	$0.690408\pi$
$942$		0			0
$943$			3.65601i			0.119056i
$944$		0			0
$945$	8.92005	−	1.47764i	0.290169	−	0.0480676i
$946$		0			0
$947$			21.3155i			0.692661i	0.938113	+	0.346330i	$0.112572\pi$
$947$			21.3155i			0.692661i	−0.938113	+	0.346330i	$0.887428\pi$
$948$		0			0
$949$		−	14.7012i		−	0.477223i
$950$		0			0
$951$	−1.04849	−	1.60621i	−0.0339995	−	0.0520850i
$952$		0			0
$953$			27.9054i			0.903944i	0.892032	+	0.451972i	$0.149279\pi$
$953$			27.9054i			0.903944i	−0.892032	+	0.451972i	$0.850721\pi$
$954$		0			0
$955$	21.1865			0.685578
$956$		0			0
$957$	−1.44760	−	2.21763i	−0.0467944	−	0.0716859i
$958$		0			0
$959$			2.05277i			0.0662873i
$960$		0			0
$961$	17.5580			0.566389
$962$		0			0
$963$	33.6889	+	14.8090i	1.08561	+	0.477215i
$964$		0			0
$965$	15.7490			0.506979
$966$		0			0
$967$	−20.7521			−0.667343			−0.333672	−	0.942689i	$-0.608288\pi$
$967$	−20.7521			−0.667343			−0.333672	+	0.942689i	$0.608288\pi$
$968$		0			0
$969$	−1.52465	−	2.33566i	−0.0489788	−	0.0750323i
$970$		0			0
$971$			53.6095i			1.72041i	0.509947	+	0.860206i	$0.329665\pi$
$971$			53.6095i			1.72041i	−0.509947	+	0.860206i	$0.670335\pi$
$972$		0			0
$973$	−32.6669			−1.04725
$974$		0			0
$975$	−4.45341	−	6.82233i	−0.142623	−	0.218489i
$976$		0			0
$977$		−	8.37430i		−	0.267917i	−0.990987	−	0.133959i	$-0.957231\pi$
$977$		−	8.37430i		−	0.267917i	0.990987	−	0.133959i	$-0.0427690\pi$
$978$		0			0
$979$		−	6.94111i		−	0.221839i
$980$		0			0
$981$	−42.9267	−	18.8698i	−1.37054	−	0.602467i
$982$		0			0
$983$	−2.98941			−0.0953475			−0.0476737	−	0.998863i	$-0.515181\pi$
$983$	−2.98941			−0.0953475			−0.0476737	+	0.998863i	$0.515181\pi$
$984$		0			0
$985$	26.1065			0.831823
$986$		0			0
$987$	22.5868	−	14.7440i	0.718947	−	0.469307i
$988$		0			0
$989$	−5.29212			−0.168280
$990$		0			0
$991$		−	26.5765i		−	0.844231i	−0.906542	−	0.422115i	$-0.861288\pi$
$991$		−	26.5765i		−	0.844231i	0.906542	−	0.422115i	$-0.138712\pi$
$992$		0			0
$993$	−3.68601	−	5.64673i	−0.116972	−	0.179194i
$994$		0			0
$995$	−10.3438			−0.327920
$996$		0			0
$997$	6.68104			0.211591			0.105795	−	0.994388i	$-0.466261\pi$
$997$	6.68104			0.211591			0.105795	+	0.994388i	$0.466261\pi$
$998$		0			0
$999$	−6.77068	−	40.8725i	−0.214215	−	1.29315i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.8	yes	46
3.2	odd	2		4020.2.f.b.401.40	yes	46
67.66	odd	2		4020.2.f.b.401.39	yes	46
201.200	even	2	inner	4020.2.f.a.401.7	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.7	✓	46	201.200	even	2	inner
4020.2.f.a.401.8	yes	46	1.1	even	1	trivial
4020.2.f.b.401.39	yes	46	67.66	odd	2
4020.2.f.b.401.40	yes	46	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.45039 + 0.946770i) q^{3} -1.00000 q^{5} +1.74006i q^{7} +(1.20725 - 2.74637i) q^{9} +O(q^{10})\)	\(q+(-1.45039 + 0.946770i) q^{3} -1.00000 q^{5} +1.74006i q^{7} +(1.20725 - 2.74637i) q^{9} +0.702140 q^{11} +4.70379i q^{13} +(1.45039 - 0.946770i) q^{15} -1.49494i q^{17} -1.07721 q^{19} +(-1.64743 - 2.52376i) q^{21} -2.17196i q^{23} +1.00000 q^{25} +(0.849189 + 5.12629i) q^{27} +2.17762i q^{29} +3.66633i q^{31} +(-1.01838 + 0.664764i) q^{33} -1.74006i q^{35} -7.97311 q^{37} +(-4.45341 - 6.82233i) q^{39} -1.68328 q^{41} -2.43657i q^{43} +(-1.20725 + 2.74637i) q^{45} +8.94967i q^{47} +3.97220 q^{49} +(1.41537 + 2.16825i) q^{51} +12.5986 q^{53} -0.702140 q^{55} +(1.56238 - 1.01987i) q^{57} +9.27103i q^{59} +2.37446i q^{61} +(4.77884 + 2.10069i) q^{63} -4.70379i q^{65} +(7.89756 + 2.15141i) q^{67} +(2.05634 + 3.15018i) q^{69} -15.2573i q^{71} -3.12540 q^{73} +(-1.45039 + 0.946770i) q^{75} +1.22176i q^{77} +2.10644i q^{79} +(-6.08507 - 6.63113i) q^{81} +12.4148i q^{83} +1.49494i q^{85} +(-2.06170 - 3.15839i) q^{87} -9.88566i q^{89} -8.18488 q^{91} +(-3.47117 - 5.31760i) q^{93} +1.07721 q^{95} +10.0995i q^{97} +(0.847661 - 1.92833i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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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