LMFDB - Embedded newform 4020.2.f.a.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.a.401.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.370384 - 1.69199i) q^{3} -1.00000 q^{5} +0.0825586i q^{7} +(-2.72563 + 1.25337i) q^{9} +O(q^{10})$	\(q+(-0.370384 - 1.69199i) q^{3} -1.00000 q^{5} +0.0825586i q^{7} +(-2.72563 + 1.25337i) q^{9} +1.67125 q^{11} +4.50431i q^{13} +(0.370384 + 1.69199i) q^{15} -1.74317i q^{17} -1.02008 q^{19} +(0.139688 - 0.0305784i) q^{21} -3.31893i q^{23} +1.00000 q^{25} +(3.13022 + 4.14750i) q^{27} +3.47526i q^{29} -4.73509i q^{31} +(-0.619005 - 2.82773i) q^{33} -0.0825586i q^{35} +1.03140 q^{37} +(7.62123 - 1.66833i) q^{39} -4.96306 q^{41} +6.28802i q^{43} +(2.72563 - 1.25337i) q^{45} -12.7335i q^{47} +6.99318 q^{49} +(-2.94943 + 0.645644i) q^{51} +1.52350 q^{53} -1.67125 q^{55} +(0.377824 + 1.72597i) q^{57} -7.15583i q^{59} +1.24045i q^{61} +(-0.103476 - 0.225024i) q^{63} -4.50431i q^{65} +(-5.31049 - 6.22886i) q^{67} +(-5.61558 + 1.22928i) q^{69} -8.32971i q^{71} +9.54051 q^{73} +(-0.370384 - 1.69199i) q^{75} +0.137976i q^{77} +4.58596i q^{79} +(5.85813 - 6.83245i) q^{81} -2.48005i q^{83} +1.74317i q^{85} +(5.88009 - 1.28718i) q^{87} -2.00636i q^{89} -0.371870 q^{91} +(-8.01170 + 1.75380i) q^{93} +1.02008 q^{95} -8.09526i q^{97} +(-4.55521 + 2.09470i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.370384	−	1.69199i	−0.213842	−	0.976868i
$3$	−0.370384	−	1.69199i	−0.213842	−	0.976868i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			0.0825586i			0.0312042i	0.999878	+	0.0156021i	$0.00496651\pi$
$7$			0.0825586i			0.0312042i	−0.999878	+	0.0156021i	$0.995033\pi$
$8$		0			0
$9$	−2.72563	+	1.25337i	−0.908544	+	0.417790i
$10$		0			0
$11$	1.67125			0.503901			0.251951	−	0.967740i	$-0.418928\pi$
$11$	1.67125			0.503901			0.251951	+	0.967740i	$0.418928\pi$
$12$		0			0
$13$			4.50431i			1.24927i	0.780916	+	0.624635i	$0.214752\pi$
$13$			4.50431i			1.24927i	−0.780916	+	0.624635i	$0.785248\pi$
$14$		0			0
$15$	0.370384	+	1.69199i	0.0956328	+	0.436869i
$16$		0			0
$17$		−	1.74317i		−	0.422782i	−0.977402	−	0.211391i	$-0.932201\pi$
$17$		−	1.74317i		−	0.422782i	0.977402	−	0.211391i	$-0.0677993\pi$
$18$		0			0
$19$	−1.02008			−0.234024			−0.117012	−	0.993131i	$-0.537332\pi$
$19$	−1.02008			−0.234024			−0.117012	+	0.993131i	$0.537332\pi$
$20$		0			0
$21$	0.139688	−	0.0305784i	0.0304824	−	0.00667276i
$22$		0			0
$23$		−	3.31893i		−	0.692045i	−0.938226	−	0.346023i	$-0.887532\pi$
$23$		−	3.31893i		−	0.692045i	0.938226	−	0.346023i	$-0.112468\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	3.13022	+	4.14750i	0.602410	+	0.798187i
$28$		0			0
$29$			3.47526i			0.645339i	0.946512	+	0.322670i	$0.104580\pi$
$29$			3.47526i			0.645339i	−0.946512	+	0.322670i	$0.895420\pi$
$30$		0			0
$31$		−	4.73509i		−	0.850447i	−0.905088	−	0.425223i	$-0.860196\pi$
$31$		−	4.73509i		−	0.850447i	0.905088	−	0.425223i	$-0.139804\pi$
$32$		0			0
$33$	−0.619005	−	2.82773i	−0.107755	−	0.492245i
$34$		0			0
$35$		−	0.0825586i		−	0.0139550i
$36$		0			0
$37$	1.03140			0.169561			0.0847803	−	0.996400i	$-0.472981\pi$
$37$	1.03140			0.169561			0.0847803	+	0.996400i	$0.472981\pi$
$38$		0			0
$39$	7.62123	−	1.66833i	1.22037	−	0.267146i
$40$		0			0
$41$	−4.96306			−0.775100			−0.387550	−	0.921849i	$-0.626679\pi$
$41$	−4.96306			−0.775100			−0.387550	+	0.921849i	$0.626679\pi$
$42$		0			0
$43$			6.28802i			0.958914i	0.877565	+	0.479457i	$0.159166\pi$
$43$			6.28802i			0.958914i	−0.877565	+	0.479457i	$0.840834\pi$
$44$		0			0
$45$	2.72563	−	1.25337i	0.406313	−	0.186841i
$46$		0			0
$47$		−	12.7335i		−	1.85737i	−0.370869	−	0.928685i	$-0.620940\pi$
$47$		−	12.7335i		−	1.85737i	0.370869	−	0.928685i	$-0.379060\pi$
$48$		0			0
$49$	6.99318			0.999026
$50$		0			0
$51$	−2.94943	+	0.645644i	−0.413002	+	0.0904083i
$52$		0			0
$53$	1.52350			0.209269			0.104634	−	0.994511i	$-0.466633\pi$
$53$	1.52350			0.209269			0.104634	+	0.994511i	$0.466633\pi$
$54$		0			0
$55$	−1.67125			−0.225351
$56$		0			0
$57$	0.377824	+	1.72597i	0.0500439	+	0.228610i
$58$		0			0
$59$		−	7.15583i		−	0.931610i	−0.884888	−	0.465805i	$-0.845765\pi$
$59$		−	7.15583i		−	0.931610i	0.884888	−	0.465805i	$-0.154235\pi$
$60$		0			0
$61$			1.24045i			0.158824i	0.996842	+	0.0794118i	$0.0253042\pi$
$61$			1.24045i			0.158824i	−0.996842	+	0.0794118i	$0.974696\pi$
$62$		0			0
$63$	−0.103476	−	0.225024i	−0.0130368	−	0.0283504i
$64$		0			0
$65$		−	4.50431i		−	0.558691i
$66$		0			0
$67$	−5.31049	−	6.22886i	−0.648779	−	0.760976i
$67$	−5.31049	−	6.22886i	−0.648779	−	0.760976i
$68$		0			0
$69$	−5.61558	+	1.22928i	−0.676037	+	0.147988i
$70$		0			0
$71$		−	8.32971i		−	0.988555i	−0.869304	−	0.494277i	$-0.835433\pi$
$71$		−	8.32971i		−	0.988555i	0.869304	−	0.494277i	$-0.164567\pi$
$72$		0			0
$73$	9.54051			1.11663			0.558316	−	0.829628i	$-0.311448\pi$
$73$	9.54051			1.11663			0.558316	+	0.829628i	$0.311448\pi$
$74$		0			0
$75$	−0.370384	−	1.69199i	−0.0427683	−	0.195374i
$76$		0			0
$77$			0.137976i			0.0157238i
$78$		0			0
$79$			4.58596i			0.515960i	0.966150	+	0.257980i	$0.0830569\pi$
$79$			4.58596i			0.515960i	−0.966150	+	0.257980i	$0.916943\pi$
$80$		0			0
$81$	5.85813	−	6.83245i	0.650903	−	0.759161i
$82$		0			0
$83$		−	2.48005i		−	0.272221i	−0.990694	−	0.136110i	$-0.956540\pi$
$83$		−	2.48005i		−	0.272221i	0.990694	−	0.136110i	$-0.0434602\pi$
$84$		0			0
$85$			1.74317i			0.189074i
$86$		0			0
$87$	5.88009	−	1.28718i	0.630412	−	0.138000i
$88$		0			0
$89$		−	2.00636i		−	0.212674i	−0.994330	−	0.106337i	$-0.966088\pi$
$89$		−	2.00636i		−	0.212674i	0.994330	−	0.106337i	$-0.0339123\pi$
$90$		0			0
$91$	−0.371870			−0.0389825
$92$		0			0
$93$	−8.01170	+	1.75380i	−0.830775	+	0.181861i
$94$		0			0
$95$	1.02008			0.104659
$96$		0			0
$97$		−	8.09526i		−	0.821950i	−0.911647	−	0.410975i	$-0.865188\pi$
$97$		−	8.09526i		−	0.821950i	0.911647	−	0.410975i	$-0.134812\pi$
$98$		0			0
$99$	−4.55521	+	2.09470i	−0.457816	+	0.210525i
$100$		0			0
$101$	−2.97812			−0.296334			−0.148167	−	0.988962i	$-0.547337\pi$
$101$	−2.97812			−0.296334			−0.148167	+	0.988962i	$0.547337\pi$
$102$		0			0
$103$	17.9755			1.77118			0.885591	−	0.464467i	$-0.153754\pi$
$103$	17.9755			1.77118			0.885591	+	0.464467i	$0.153754\pi$
$104$		0			0
$105$	−0.139688	+	0.0305784i	−0.0136322	+	0.00298415i
$106$		0			0
$107$		−	10.0685i		−	0.973362i	−0.873580	−	0.486681i	$-0.838207\pi$
$107$		−	10.0685i		−	0.973362i	0.873580	−	0.486681i	$-0.161793\pi$
$108$		0			0
$109$		−	1.31942i		−	0.126378i	−0.998002	−	0.0631888i	$-0.979873\pi$
$109$		−	1.31942i		−	0.126378i	0.998002	−	0.0631888i	$-0.0201270\pi$
$110$		0			0
$111$	−0.382013	−	1.74511i	−0.0362591	−	0.165638i
$112$		0			0
$113$	−9.45036			−0.889015			−0.444508	−	0.895775i	$-0.646621\pi$
$113$	−9.45036			−0.889015			−0.444508	+	0.895775i	$0.646621\pi$
$114$		0			0
$115$			3.31893i			0.309492i
$116$		0			0
$117$	−5.64557	−	12.2771i	−0.521933	−	1.13502i
$118$		0			0
$119$	0.143914			0.0131926
$120$		0			0
$121$	−8.20692			−0.746084
$122$		0			0
$123$	1.83824	+	8.39743i	0.165749	+	0.757171i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.54221			−0.580527			−0.290264	−	0.956947i	$-0.593743\pi$
$127$	−6.54221			−0.580527			−0.290264	+	0.956947i	$0.593743\pi$
$128$		0			0
$129$	10.6392	−	2.32898i	0.936733	−	0.205056i
$130$		0			0
$131$		−	15.8097i		−	1.38130i	−0.723188	−	0.690651i	$-0.757324\pi$
$131$		−	15.8097i		−	1.38130i	0.723188	−	0.690651i	$-0.242676\pi$
$132$		0			0
$133$		−	0.0842168i		−	0.00730252i
$134$		0			0
$135$	−3.13022	−	4.14750i	−0.269406	−	0.356960i
$136$		0			0
$137$	−9.06224			−0.774240			−0.387120	−	0.922029i	$-0.626530\pi$
$137$	−9.06224			−0.774240			−0.387120	+	0.922029i	$0.626530\pi$
$138$		0			0
$139$			0.631689i			0.0535792i	0.999641	+	0.0267896i	$0.00852841\pi$
$139$			0.631689i			0.0535792i	−0.999641	+	0.0267896i	$0.991472\pi$
$140$		0			0
$141$	−21.5449	+	4.71629i	−1.81441	+	0.397183i
$142$		0			0
$143$			7.52783i			0.629509i
$144$		0			0
$145$		−	3.47526i		−	0.288605i
$146$		0			0
$147$	−2.59017	−	11.8324i	−0.213633	−	0.975917i
$148$		0			0
$149$		−	2.34874i		−	0.192416i	−0.995361	−	0.0962079i	$-0.969329\pi$
$149$		−	2.34874i		−	0.192416i	0.995361	−	0.0962079i	$-0.0306714\pi$
$150$		0			0
$151$	−6.81193			−0.554347			−0.277173	−	0.960820i	$-0.589398\pi$
$151$	−6.81193			−0.554347			−0.277173	+	0.960820i	$0.589398\pi$
$152$		0			0
$153$	2.18484	+	4.75125i	0.176634	+	0.384116i
$154$		0			0
$155$			4.73509i			0.380331i
$156$		0			0
$157$	−3.28192			−0.261925			−0.130963	−	0.991387i	$-0.541807\pi$
$157$	−3.28192			−0.261925			−0.130963	+	0.991387i	$0.541807\pi$
$158$		0			0
$159$	−0.564280	−	2.57774i	−0.0447503	−	0.204428i
$160$		0			0
$161$	0.274006			0.0215947
$162$		0			0
$163$	10.4909			0.821710			0.410855	−	0.911701i	$-0.365230\pi$
$163$	10.4909			0.821710			0.410855	+	0.911701i	$0.365230\pi$
$164$		0			0
$165$	0.619005	+	2.82773i	0.0481895	+	0.220139i
$166$		0			0
$167$			1.89936i			0.146977i	0.997296	+	0.0734885i	$0.0234132\pi$
$167$			1.89936i			0.146977i	−0.997296	+	0.0734885i	$0.976587\pi$
$168$		0			0
$169$	−7.28881			−0.560678
$170$		0			0
$171$	2.78037	−	1.27854i	0.212621	−	0.0977727i
$172$		0			0
$173$		−	15.8055i		−	1.20167i	−0.799374	−	0.600833i	$-0.794836\pi$
$173$		−	15.8055i		−	1.20167i	0.799374	−	0.600833i	$-0.205164\pi$
$174$		0			0
$175$			0.0825586i			0.00624084i
$176$		0			0
$177$	−12.1076	+	2.65041i	−0.910060	+	0.199217i
$178$		0			0
$179$	−11.1981			−0.836985			−0.418493	−	0.908220i	$-0.637441\pi$
$179$	−11.1981			−0.836985			−0.418493	+	0.908220i	$0.637441\pi$
$180$		0			0
$181$	−12.2874			−0.913313			−0.456656	−	0.889643i	$-0.650953\pi$
$181$	−12.2874			−0.913313			−0.456656	+	0.889643i	$0.650953\pi$
$182$		0			0
$183$	2.09883	−	0.459444i	0.155150	−	0.0339631i
$184$		0			0
$185$	−1.03140			−0.0758298
$186$		0			0
$187$		−	2.91328i		−	0.213040i
$188$		0			0
$189$	−0.342412	+	0.258426i	−0.0249068	+	0.0187977i
$190$		0			0
$191$	13.9708			1.01089			0.505445	−	0.862859i	$-0.331328\pi$
$191$	13.9708			1.01089			0.505445	+	0.862859i	$0.331328\pi$
$192$		0			0
$193$	−5.64847			−0.406586			−0.203293	−	0.979118i	$-0.565164\pi$
$193$	−5.64847			−0.406586			−0.203293	+	0.979118i	$0.565164\pi$
$194$		0			0
$195$	−7.62123	+	1.66833i	−0.545768	+	0.119471i
$196$		0			0
$197$	9.28217			0.661327			0.330664	−	0.943749i	$-0.392727\pi$
$197$	9.28217			0.661327			0.330664	+	0.943749i	$0.392727\pi$
$198$		0			0
$199$	−15.9556			−1.13106			−0.565531	−	0.824727i	$-0.691329\pi$
$199$	−15.9556			−1.13106			−0.565531	+	0.824727i	$0.691329\pi$
$200$		0			0
$201$	−8.57222	+	11.2923i	−0.604638	+	0.796501i
$202$		0			0
$203$	−0.286913			−0.0201373
$204$		0			0
$205$	4.96306			0.346635
$206$		0			0
$207$	4.15985	+	9.04618i	0.289130	+	0.628753i
$208$		0			0
$209$	−1.70482			−0.117925
$210$		0			0
$211$	−8.81324			−0.606728			−0.303364	−	0.952875i	$-0.598110\pi$
$211$	−8.81324			−0.606728			−0.303364	+	0.952875i	$0.598110\pi$
$212$		0			0
$213$	−14.0938	+	3.08519i	−0.965688	+	0.211394i
$214$		0			0
$215$		−	6.28802i		−	0.428839i
$216$		0			0
$217$	0.390922			0.0265375
$218$		0			0
$219$	−3.53366	−	16.1424i	−0.238782	−	1.09080i
$220$		0			0
$221$	7.85180			0.528169
$222$		0			0
$223$	−6.01845			−0.403025			−0.201513	−	0.979486i	$-0.564586\pi$
$223$	−6.01845			−0.403025			−0.201513	+	0.979486i	$0.564586\pi$
$224$		0			0
$225$	−2.72563	+	1.25337i	−0.181709	+	0.0835580i
$226$		0			0
$227$			20.2941i			1.34697i	0.739202	+	0.673484i	$0.235203\pi$
$227$			20.2941i			1.34697i	−0.739202	+	0.673484i	$0.764797\pi$
$228$		0			0
$229$		−	20.4680i		−	1.35256i	−0.736644	−	0.676280i	$-0.763591\pi$
$229$		−	20.4680i		−	1.35256i	0.736644	−	0.676280i	$-0.236409\pi$
$230$		0			0
$231$	0.233454	−	0.0511042i	0.0153601	−	0.00336241i
$232$		0			0
$233$	12.5372			0.821339			0.410670	−	0.911784i	$-0.365295\pi$
$233$	12.5372			0.821339			0.410670	+	0.911784i	$0.365295\pi$
$234$		0			0
$235$			12.7335i			0.830641i
$236$		0			0
$237$	7.75937	−	1.69857i	0.504025	−	0.110334i
$238$		0			0
$239$	−6.51061			−0.421136			−0.210568	−	0.977579i	$-0.567531\pi$
$239$	−6.51061			−0.421136			−0.210568	+	0.977579i	$0.567531\pi$
$240$		0			0
$241$	−9.20512			−0.592954			−0.296477	−	0.955040i	$-0.595812\pi$
$241$	−9.20512			−0.592954			−0.296477	+	0.955040i	$0.595812\pi$
$242$		0			0
$243$	−13.7302	−	7.38123i	−0.880790	−	0.473506i
$244$		0			0
$245$	−6.99318			−0.446778
$246$		0			0
$247$		−	4.59478i		−	0.292359i
$248$		0			0
$249$	−4.19621	+	0.918571i	−0.265924	+	0.0582121i
$250$		0			0
$251$	−5.90434			−0.372679			−0.186339	−	0.982485i	$-0.559662\pi$
$251$	−5.90434			−0.372679			−0.186339	+	0.982485i	$0.559662\pi$
$252$		0			0
$253$		−	5.54677i		−	0.348722i
$254$		0			0
$255$	2.94943	−	0.645644i	0.184700	−	0.0404318i
$256$		0			0
$257$			1.52830i			0.0953330i	0.998863	+	0.0476665i	$0.0151785\pi$
$257$			1.52830i			0.0953330i	−0.998863	+	0.0476665i	$0.984822\pi$
$258$		0			0
$259$			0.0851507i			0.00529101i
$260$		0			0
$261$	−4.35579	−	9.47227i	−0.269616	−	0.586319i
$262$		0			0
$263$		−	14.7297i		−	0.908273i	−0.890932	−	0.454136i	$-0.849948\pi$
$263$		−	14.7297i		−	0.908273i	0.890932	−	0.454136i	$-0.150052\pi$
$264$		0			0
$265$	−1.52350			−0.0935878
$266$		0			0
$267$	−3.39474	+	0.743126i	−0.207755	+	0.0454786i
$268$		0			0
$269$			11.6644i			0.711189i	0.934640	+	0.355595i	$0.115722\pi$
$269$			11.6644i			0.711189i	−0.934640	+	0.355595i	$0.884278\pi$
$270$		0			0
$271$		−	12.5149i		−	0.760227i	−0.924940	−	0.380113i	$-0.875885\pi$
$271$		−	12.5149i		−	0.760227i	0.924940	−	0.380113i	$-0.124115\pi$
$272$		0			0
$273$	0.137735	+	0.629198i	0.00833608	+	0.0380808i
$274$		0			0
$275$	1.67125			0.100780
$276$		0			0
$277$	17.9499			1.07850			0.539252	−	0.842144i	$-0.318707\pi$
$277$	17.9499			1.07850			0.539252	+	0.842144i	$0.318707\pi$
$278$		0			0
$279$	5.93482	+	12.9061i	0.355308	+	0.772668i
$280$		0			0
$281$	−28.2406			−1.68469			−0.842346	−	0.538937i	$-0.818826\pi$
$281$	−28.2406			−1.68469			−0.842346	+	0.538937i	$0.818826\pi$
$282$		0			0
$283$	19.4016			1.15331			0.576653	−	0.816989i	$-0.304359\pi$
$283$	19.4016			1.15331			0.576653	+	0.816989i	$0.304359\pi$
$284$		0			0
$285$	−0.377824	−	1.72597i	−0.0223803	−	0.102238i
$286$		0			0
$287$		−	0.409743i		−	0.0241864i
$288$		0			0
$289$	13.9613			0.821256
$290$		0			0
$291$	−13.6971	+	2.99836i	−0.802937	+	0.175767i
$292$		0			0
$293$		−	1.09562i		−	0.0640070i	−0.999488	−	0.0320035i	$-0.989811\pi$
$293$		−	1.09562i		−	0.0640070i	0.999488	−	0.0320035i	$-0.0101888\pi$
$294$		0			0
$295$			7.15583i			0.416628i
$296$		0			0
$297$	5.23138	+	6.93151i	0.303555	+	0.402207i
$298$		0			0
$299$	14.9495			0.864552
$300$		0			0
$301$	−0.519130			−0.0299222
$302$		0			0
$303$	1.10305	+	5.03893i	0.0633684	+	0.289479i
$304$		0			0
$305$		−	1.24045i		−	0.0710281i
$306$		0			0
$307$	7.55832			0.431376			0.215688	−	0.976462i	$-0.430801\pi$
$307$	7.55832			0.431376			0.215688	+	0.976462i	$0.430801\pi$
$308$		0			0
$309$	−6.65785	−	30.4143i	−0.378752	−	1.73021i
$310$		0			0
$311$	8.75315			0.496346			0.248173	−	0.968716i	$-0.420170\pi$
$311$	8.75315			0.496346			0.248173	+	0.968716i	$0.420170\pi$
$312$		0			0
$313$			26.2470i			1.48357i	0.670639	+	0.741784i	$0.266020\pi$
$313$			26.2470i			1.48357i	−0.670639	+	0.741784i	$0.733980\pi$
$314$		0			0
$315$	0.103476	+	0.225024i	0.00583024	+	0.0126787i
$316$		0			0
$317$		−	23.5840i		−	1.32461i	−0.749235	−	0.662304i	$-0.769579\pi$
$317$		−	23.5840i		−	1.32461i	0.749235	−	0.662304i	$-0.230421\pi$
$318$		0			0
$319$			5.80803i			0.325187i
$320$		0			0
$321$	−17.0358	+	3.72923i	−0.950847	+	0.208145i
$322$		0			0
$323$			1.77819i			0.0989409i
$324$		0			0
$325$			4.50431i			0.249854i
$326$		0			0
$327$	−2.23244	+	0.488693i	−0.123454	+	0.0270248i
$328$		0			0
$329$	1.05126			0.0579578
$330$		0			0
$331$			7.67554i			0.421886i	0.977498	+	0.210943i	$0.0676534\pi$
$331$			7.67554i			0.421886i	−0.977498	+	0.210943i	$0.932347\pi$
$332$		0			0
$333$	−2.81121	+	1.29272i	−0.154053	+	0.0708408i
$334$		0			0
$335$	5.31049	+	6.22886i	0.290143	+	0.340319i
$336$		0			0
$337$		−	23.0371i		−	1.25491i	−0.778653	−	0.627455i	$-0.784097\pi$
$337$		−	23.0371i		−	1.25491i	0.778653	−	0.627455i	$-0.215903\pi$
$338$		0			0
$339$	3.50027	+	15.9899i	0.190108	+	0.868451i
$340$		0			0
$341$		−	7.91352i		−	0.428541i
$342$		0			0
$343$			1.15526i			0.0623781i
$344$		0			0
$345$	5.61558	−	1.22928i	0.302333	−	0.0661822i
$346$		0			0
$347$	35.7380			1.91852			0.959258	−	0.282532i	$-0.0911744\pi$
$347$	35.7380			1.91852			0.959258	+	0.282532i	$0.0911744\pi$
$348$		0			0
$349$	9.01469			0.482545			0.241273	−	0.970457i	$-0.422435\pi$
$349$	9.01469			0.482545			0.241273	+	0.970457i	$0.422435\pi$
$350$		0			0
$351$	−18.6816	+	14.0995i	−0.997151	+	0.752574i
$352$		0			0
$353$	−35.0606			−1.86609			−0.933044	−	0.359763i	$-0.882857\pi$
$353$	−35.0606			−1.86609			−0.933044	+	0.359763i	$0.882857\pi$
$354$		0			0
$355$			8.32971i			0.442095i
$356$		0			0
$357$	−0.0533035	−	0.243500i	−0.00282112	−	0.0128874i
$358$		0			0
$359$		−	13.8485i		−	0.730898i	−0.930831	−	0.365449i	$-0.880915\pi$
$359$		−	13.8485i		−	0.730898i	0.930831	−	0.365449i	$-0.119085\pi$
$360$		0			0
$361$	−17.9594			−0.945233
$362$		0			0
$363$	3.03972	+	13.8860i	0.159544	+	0.728825i
$364$		0			0
$365$	−9.54051			−0.499373
$366$		0			0
$367$		−	18.5326i		−	0.967393i	−0.875236	−	0.483696i	$-0.839294\pi$
$367$		−	18.5326i		−	0.967393i	0.875236	−	0.483696i	$-0.160706\pi$
$368$		0			0
$369$	13.5275	−	6.22055i	0.704212	−	0.323829i
$370$		0			0
$371$			0.125778i			0.00653006i
$372$		0			0
$373$			4.09260i			0.211907i	0.994371	+	0.105953i	$0.0337895\pi$
$373$			4.09260i			0.211907i	−0.994371	+	0.105953i	$0.966211\pi$
$374$		0			0
$375$	0.370384	+	1.69199i	0.0191266	+	0.0873738i
$376$		0			0
$377$	−15.6536			−0.806204
$378$		0			0
$379$			25.8469i			1.32767i	0.747880	+	0.663833i	$0.231072\pi$
$379$			25.8469i			1.32767i	−0.747880	+	0.663833i	$0.768928\pi$
$380$		0			0
$381$	2.42313	+	11.0693i	0.124141	+	0.567099i
$382$		0			0
$383$	6.83819			0.349415			0.174708	−	0.984620i	$-0.444102\pi$
$383$	6.83819			0.349415			0.174708	+	0.984620i	$0.444102\pi$
$384$		0			0
$385$		−	0.137976i		−	0.00703192i
$386$		0			0
$387$	−7.88121	−	17.1388i	−0.400625	−	0.871215i
$388$		0			0
$389$		−	28.0052i		−	1.41992i	−0.704242	−	0.709960i	$-0.748713\pi$
$389$		−	28.0052i		−	1.41992i	0.704242	−	0.709960i	$-0.251287\pi$
$390$		0			0
$391$	−5.78548			−0.292584
$392$		0			0
$393$	−26.7498	+	5.85568i	−1.34935	+	0.295380i
$394$		0			0
$395$		−	4.58596i		−	0.230745i
$396$		0			0
$397$	27.8697			1.39874			0.699371	−	0.714759i	$-0.253463\pi$
$397$	27.8697			1.39874			0.699371	+	0.714759i	$0.253463\pi$
$398$		0			0
$399$	−0.142494	+	0.0311926i	−0.00713360	+	0.00156158i
$400$		0			0
$401$	−17.2623			−0.862036			−0.431018	−	0.902343i	$-0.641845\pi$
$401$	−17.2623			−0.862036			−0.431018	+	0.902343i	$0.641845\pi$
$402$		0			0
$403$	21.3283			1.06244
$404$		0			0
$405$	−5.85813	+	6.83245i	−0.291093	+	0.339507i
$406$		0			0
$407$	1.72372			0.0854418
$408$		0			0
$409$		−	29.9625i		−	1.48155i	−0.671754	−	0.740774i	$-0.734459\pi$
$409$		−	29.9625i		−	1.48155i	0.671754	−	0.740774i	$-0.265541\pi$
$410$		0			0
$411$	3.35651	+	15.3332i	0.165565	+	0.756330i
$412$		0			0
$413$	0.590775			0.0290702
$414$		0			0
$415$			2.48005i			0.121741i
$416$		0			0
$417$	1.06881	−	0.233968i	0.0523398	−	0.0114575i
$418$		0			0
$419$			8.94013i			0.436754i	0.975864	+	0.218377i	$0.0700763\pi$
$419$			8.94013i			0.436754i	−0.975864	+	0.218377i	$0.929924\pi$
$420$		0			0
$421$	−10.7017			−0.521567			−0.260783	−	0.965397i	$-0.583981\pi$
$421$	−10.7017			−0.521567			−0.260783	+	0.965397i	$0.583981\pi$
$422$		0			0
$423$	15.9598	+	34.7068i	0.775991	+	1.68750i
$424$		0			0
$425$		−	1.74317i		−	0.0845564i
$426$		0			0
$427$	−0.102410			−0.00495597
$428$		0			0
$429$	12.7370	−	2.78819i	0.614947	−	0.134615i
$430$		0			0
$431$		−	35.4760i		−	1.70882i	−0.519600	−	0.854410i	$-0.673919\pi$
$431$		−	35.4760i		−	1.70882i	0.519600	−	0.854410i	$-0.326081\pi$
$432$		0			0
$433$		−	6.50984i		−	0.312843i	−0.987690	−	0.156421i	$-0.950004\pi$
$433$		−	6.50984i		−	0.312843i	0.987690	−	0.156421i	$-0.0499958\pi$
$434$		0			0
$435$	−5.88009	+	1.28718i	−0.281929	+	0.0617156i
$436$		0			0
$437$			3.38559i			0.161955i
$438$		0			0
$439$	−2.13152			−0.101732			−0.0508660	−	0.998705i	$-0.516198\pi$
$439$	−2.13152			−0.101732			−0.0508660	+	0.998705i	$0.516198\pi$
$440$		0			0
$441$	−19.0608	+	8.76505i	−0.907659	+	0.417383i
$442$		0			0
$443$	−22.8818			−1.08715			−0.543574	−	0.839361i	$-0.682929\pi$
$443$	−22.8818			−1.08715			−0.543574	+	0.839361i	$0.682929\pi$
$444$		0			0
$445$			2.00636i			0.0951108i
$446$		0			0
$447$	−3.97403	+	0.869935i	−0.187965	+	0.0411465i
$448$		0			0
$449$			5.39630i			0.254667i	0.991860	+	0.127333i	$0.0406418\pi$
$449$			5.39630i			0.254667i	−0.991860	+	0.127333i	$0.959358\pi$
$450$		0			0
$451$	−8.29452			−0.390574
$452$		0			0
$453$	2.52303	+	11.5257i	0.118542	+	0.541524i
$454$		0			0
$455$	0.371870			0.0174335
$456$		0			0
$457$	−22.4137			−1.04847			−0.524235	−	0.851574i	$-0.675649\pi$
$457$	−22.4137			−1.04847			−0.524235	+	0.851574i	$0.675649\pi$
$458$		0			0
$459$	7.22981	−	5.45651i	0.337459	−	0.254688i
$460$		0			0
$461$		−	7.77307i		−	0.362028i	−0.983481	−	0.181014i	$-0.942062\pi$
$461$		−	7.77307i		−	0.362028i	0.983481	−	0.181014i	$-0.0579379\pi$
$462$		0			0
$463$		−	1.63614i		−	0.0760380i	−0.999277	−	0.0380190i	$-0.987895\pi$
$463$		−	1.63614i		−	0.0760380i	0.999277	−	0.0380190i	$-0.0121047\pi$
$464$		0			0
$465$	8.01170	−	1.75380i	0.371534	−	0.0813306i
$466$		0			0
$467$			21.2579i			0.983699i	0.870680	+	0.491850i	$0.163679\pi$
$467$			21.2579i			0.983699i	−0.870680	+	0.491850i	$0.836321\pi$
$468$		0			0
$469$	0.514246	−	0.438427i	0.0237457	−	0.0202447i
$470$		0			0
$471$	1.21557	+	5.55296i	0.0560105	+	0.255867i
$472$		0			0
$473$			10.5089i			0.483198i
$474$		0			0
$475$	−1.02008			−0.0468047
$476$		0			0
$477$	−4.15249	+	1.90951i	−0.190130	+	0.0874303i
$478$		0			0
$479$			7.63268i			0.348746i	0.984680	+	0.174373i	$0.0557898\pi$
$479$			7.63268i			0.348746i	−0.984680	+	0.174373i	$0.944210\pi$
$480$		0			0
$481$			4.64573i			0.211827i
$482$		0			0
$483$	−0.101488	−	0.463615i	−0.00461785	−	0.0210952i
$484$		0			0
$485$			8.09526i			0.367587i
$486$		0			0
$487$			5.06161i			0.229364i	0.993402	+	0.114682i	$0.0365848\pi$
$487$			5.06161i			0.229364i	−0.993402	+	0.114682i	$0.963415\pi$
$488$		0			0
$489$	−3.88566	−	17.7504i	−0.175716	−	0.802703i
$490$		0			0
$491$		−	3.57933i		−	0.161533i	−0.996733	−	0.0807664i	$-0.974263\pi$
$491$		−	3.57933i		−	0.161533i	0.996733	−	0.0807664i	$-0.0257368\pi$
$492$		0			0
$493$	6.05798			0.272838
$494$		0			0
$495$	4.55521	−	2.09470i	0.204742	−	0.0941496i
$496$		0			0
$497$	0.687689			0.0308471
$498$		0			0
$499$			7.79623i			0.349007i	0.984657	+	0.174504i	$0.0558321\pi$
$499$			7.79623i			0.349007i	−0.984657	+	0.174504i	$0.944168\pi$
$500$		0			0
$501$	3.21369	−	0.703494i	0.143577	−	0.0314298i
$502$		0			0
$503$	29.3416			1.30828			0.654138	−	0.756376i	$-0.273032\pi$
$503$	29.3416			1.30828			0.654138	+	0.756376i	$0.273032\pi$
$504$		0			0
$505$	2.97812			0.132524
$506$		0			0
$507$	2.69966	+	12.3326i	0.119896	+	0.547708i
$508$		0			0
$509$		−	6.89521i		−	0.305625i	−0.988255	−	0.152812i	$-0.951167\pi$
$509$		−	6.89521i		−	0.305625i	0.988255	−	0.152812i	$-0.0488330\pi$
$510$		0			0
$511$			0.787652i			0.0348437i
$512$		0			0
$513$	−3.19309	−	4.23080i	−0.140978	−	0.186794i
$514$		0			0
$515$	−17.9755			−0.792096
$516$		0			0
$517$		−	21.2809i		−	0.935931i
$518$		0			0
$519$	−26.7426	+	5.85409i	−1.17387	+	0.256966i
$520$		0			0
$521$	−6.30793			−0.276355			−0.138178	−	0.990407i	$-0.544125\pi$
$521$	−6.30793			−0.276355			−0.138178	+	0.990407i	$0.544125\pi$
$522$		0			0
$523$	24.2258			1.05932			0.529661	−	0.848210i	$-0.322319\pi$
$523$	24.2258			1.05932			0.529661	+	0.848210i	$0.322319\pi$
$524$		0			0
$525$	0.139688	−	0.0305784i	0.00609648	−	0.00133455i
$526$		0			0
$527$	−8.25408			−0.359553
$528$		0			0
$529$	11.9847			0.521074
$530$		0			0
$531$	8.96890	+	19.5041i	0.389217	+	0.846408i
$532$		0			0
$533$		−	22.3552i		−	0.968310i
$534$		0			0
$535$			10.0685i			0.435301i
$536$		0			0
$537$	4.14760	+	18.9470i	0.178982	+	0.817624i
$538$		0			0
$539$	11.6874			0.503410
$540$		0			0
$541$		−	12.9778i		−	0.557957i	−0.960297	−	0.278979i	$-0.910004\pi$
$541$		−	12.9778i		−	0.557957i	0.960297	−	0.278979i	$-0.0899959\pi$
$542$		0			0
$543$	4.55105	+	20.7900i	0.195304	+	0.892186i
$544$		0			0
$545$			1.31942i			0.0565177i
$546$		0			0
$547$		−	33.6389i		−	1.43830i	−0.694856	−	0.719149i	$-0.744532\pi$
$547$		−	33.6389i		−	1.43830i	0.694856	−	0.719149i	$-0.255468\pi$
$548$		0			0
$549$	−1.55475	−	3.38101i	−0.0663549	−	0.144298i
$550$		0			0
$551$		−	3.54506i		−	0.151025i
$552$		0			0
$553$	−0.378610			−0.0161001
$554$		0			0
$555$	0.382013	+	1.74511i	0.0162156	+	0.0740758i
$556$		0			0
$557$		−	21.2382i		−	0.899892i	−0.893056	−	0.449946i	$-0.851443\pi$
$557$		−	21.2382i		−	0.899892i	0.893056	−	0.449946i	$-0.148557\pi$
$558$		0			0
$559$	−28.3232			−1.19794
$560$		0			0
$561$	−4.92923	+	1.07903i	−0.208112	+	0.0455568i
$562$		0			0
$563$	40.5625			1.70951			0.854754	−	0.519034i	$-0.173708\pi$
$563$	40.5625			1.70951			0.854754	+	0.519034i	$0.173708\pi$
$564$		0			0
$565$	9.45036			0.397580
$566$		0			0
$567$	0.564077	+	0.483639i	0.0236890	+	0.0203109i
$568$		0			0
$569$			0.181936i			0.00762714i	0.999993	+	0.00381357i	$0.00121390\pi$
$569$			0.181936i			0.00762714i	−0.999993	+	0.00381357i	$0.998786\pi$
$570$		0			0
$571$	−28.7075			−1.20137			−0.600687	−	0.799485i	$-0.705106\pi$
$571$	−28.7075			−1.20137			−0.600687	+	0.799485i	$0.705106\pi$
$572$		0			0
$573$	−5.17456	−	23.6383i	−0.216170	−	0.987506i
$574$		0			0
$575$		−	3.31893i		−	0.138409i
$576$		0			0
$577$			9.33608i			0.388666i	0.980936	+	0.194333i	$0.0622543\pi$
$577$			9.33608i			0.388666i	−0.980936	+	0.194333i	$0.937746\pi$
$578$		0			0
$579$	2.09210	+	9.55712i	0.0869449	+	0.397181i
$580$		0			0
$581$	0.204749			0.00849443
$582$		0			0
$583$	2.54615			0.105451
$584$		0			0
$585$	5.64557	+	12.2771i	0.233416	+	0.507595i
$586$		0			0
$587$	40.6231			1.67670			0.838348	−	0.545135i	$-0.183522\pi$
$587$	40.6231			1.67670			0.838348	+	0.545135i	$0.183522\pi$
$588$		0			0
$589$			4.83019i			0.199025i
$590$		0			0
$591$	−3.43797	−	15.7053i	−0.141419	−	0.646030i
$592$		0			0
$593$	20.1595			0.827853			0.413926	−	0.910310i	$-0.364157\pi$
$593$	20.1595			0.827853			0.413926	+	0.910310i	$0.364157\pi$
$594$		0			0
$595$	−0.143914			−0.00589990
$596$		0			0
$597$	5.90970	+	26.9966i	0.241868	+	1.10490i
$598$		0			0
$599$	41.5770			1.69879			0.849395	−	0.527758i	$-0.176967\pi$
$599$	41.5770			1.69879			0.849395	+	0.527758i	$0.176967\pi$
$600$		0			0
$601$	8.97900			0.366261			0.183130	−	0.983089i	$-0.441377\pi$
$601$	8.97900			0.366261			0.183130	+	0.983089i	$0.441377\pi$
$602$		0			0
$603$	22.2815	+	10.3216i	0.907373	+	0.420327i
$604$		0			0
$605$	8.20692			0.333659
$606$		0			0
$607$	−5.30965			−0.215512			−0.107756	−	0.994177i	$-0.534367\pi$
$607$	−5.30965			−0.215512			−0.107756	+	0.994177i	$0.534367\pi$
$608$		0			0
$609$	0.106268	+	0.485452i	0.00430619	+	0.0196715i
$610$		0			0
$611$	57.3556			2.32036
$612$		0			0
$613$	23.7879			0.960785			0.480393	−	0.877053i	$-0.340494\pi$
$613$	23.7879			0.960785			0.480393	+	0.877053i	$0.340494\pi$
$614$		0			0
$615$	−1.83824	−	8.39743i	−0.0741250	−	0.338617i
$616$		0			0
$617$			3.77319i			0.151903i	0.997112	+	0.0759515i	$0.0241994\pi$
$617$			3.77319i			0.151903i	−0.997112	+	0.0759515i	$0.975801\pi$
$618$		0			0
$619$	−18.8844			−0.759029			−0.379515	−	0.925186i	$-0.623909\pi$
$619$	−18.8844			−0.759029			−0.379515	+	0.925186i	$0.623909\pi$
$620$		0			0
$621$	13.7653	−	10.3890i	0.552381	−	0.416895i
$622$		0			0
$623$	0.165643			0.00663633
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	0.631438	+	2.88453i	0.0252172	+	0.115197i
$628$		0			0
$629$		−	1.79790i		−	0.0716872i
$630$		0			0
$631$			2.32608i			0.0925996i	0.998928	+	0.0462998i	$0.0147430\pi$
$631$			2.32608i			0.0925996i	−0.998928	+	0.0462998i	$0.985257\pi$
$632$		0			0
$633$	3.26429	+	14.9119i	0.129744	+	0.592694i
$634$		0			0
$635$	6.54221			0.259620
$636$		0			0
$637$			31.4995i			1.24805i
$638$		0			0
$639$	10.4402	+	22.7037i	0.413008	+	0.898145i
$640$		0			0
$641$	8.42327			0.332699			0.166350	−	0.986067i	$-0.446802\pi$
$641$	8.42327			0.332699			0.166350	+	0.986067i	$0.446802\pi$
$642$		0			0
$643$	24.8929			0.981680			0.490840	−	0.871250i	$-0.336690\pi$
$643$	24.8929			0.981680			0.490840	+	0.871250i	$0.336690\pi$
$644$		0			0
$645$	−10.6392	+	2.32898i	−0.418920	+	0.0917036i
$646$		0			0
$647$	2.53991			0.0998541			0.0499271	−	0.998753i	$-0.484101\pi$
$647$	2.53991			0.0998541			0.0499271	+	0.998753i	$0.484101\pi$
$648$		0			0
$649$		−	11.9592i		−	0.469439i
$650$		0			0
$651$	−0.144792	−	0.661435i	−0.00567483	−	0.0259237i
$652$		0			0
$653$	30.2026			1.18192			0.590959	−	0.806702i	$-0.298750\pi$
$653$	30.2026			1.18192			0.590959	+	0.806702i	$0.298750\pi$
$654$		0			0
$655$			15.8097i			0.617737i
$656$		0			0
$657$	−26.0039	+	11.9578i	−1.01451	+	0.466518i
$658$		0			0
$659$			6.55024i			0.255161i	0.991828	+	0.127581i	$0.0407211\pi$
$659$			6.55024i			0.255161i	−0.991828	+	0.127581i	$0.959279\pi$
$660$		0			0
$661$		−	42.0516i		−	1.63562i	−0.575489	−	0.817810i	$-0.695188\pi$
$661$		−	42.0516i		−	1.63562i	0.575489	−	0.817810i	$-0.304812\pi$
$662$		0			0
$663$	−2.90818	−	13.2851i	−0.112944	−	0.515952i
$664$		0			0
$665$			0.0842168i			0.00326579i
$666$		0			0
$667$	11.5341			0.446604
$668$		0			0
$669$	2.22914	+	10.1831i	0.0861835	+	0.393703i
$670$		0			0
$671$			2.07311i			0.0800314i
$672$		0			0
$673$		−	29.3791i		−	1.13248i	−0.824240	−	0.566240i	$-0.808397\pi$
$673$		−	29.3791i		−	1.13248i	0.824240	−	0.566240i	$-0.191603\pi$
$674$		0			0
$675$	3.13022	+	4.14750i	0.120482	+	0.159637i
$676$		0			0
$677$	−0.806869			−0.0310105			−0.0155052	−	0.999880i	$-0.504936\pi$
$677$	−0.806869			−0.0310105			−0.0155052	+	0.999880i	$0.504936\pi$
$678$		0			0
$679$	0.668334			0.0256483
$680$		0			0
$681$	34.3373	−	7.51662i	1.31581	−	0.288038i
$682$		0			0
$683$	−5.99923			−0.229554			−0.114777	−	0.993391i	$-0.536615\pi$
$683$	−5.99923			−0.229554			−0.114777	+	0.993391i	$0.536615\pi$
$684$		0			0
$685$	9.06224			0.346250
$686$		0			0
$687$	−34.6315	+	7.58101i	−1.32127	+	0.289234i
$688$		0			0
$689$			6.86231i			0.261433i
$690$		0			0
$691$	−39.0972			−1.48733			−0.743663	−	0.668554i	$-0.766913\pi$
$691$	−39.0972			−1.48733			−0.743663	+	0.668554i	$0.766913\pi$
$692$		0			0
$693$	−0.172935	−	0.376072i	−0.00656927	−	0.0142858i
$694$		0			0
$695$		−	0.631689i		−	0.0239613i
$696$		0			0
$697$			8.65148i			0.327698i
$698$		0			0
$699$	−4.64358	−	21.2128i	−0.175636	−	0.802341i
$700$		0			0
$701$	29.7656			1.12423			0.562115	−	0.827059i	$-0.309988\pi$
$701$	29.7656			1.12423			0.562115	+	0.827059i	$0.309988\pi$
$702$		0			0
$703$	−1.05211			−0.0396812
$704$		0			0
$705$	21.5449	−	4.71629i	0.811427	−	0.177626i
$706$		0			0
$707$		−	0.245869i		−	0.00924686i
$708$		0			0
$709$	−26.3122			−0.988174			−0.494087	−	0.869412i	$-0.664498\pi$
$709$	−26.3122			−0.988174			−0.494087	+	0.869412i	$0.664498\pi$
$710$		0			0
$711$	−5.74790	−	12.4996i	−0.215563	−	0.468773i
$712$		0			0
$713$	−15.7154			−0.588548
$714$		0			0
$715$		−	7.52783i		−	0.281525i
$716$		0			0
$717$	2.41143	+	11.0159i	0.0900564	+	0.411395i
$718$		0			0
$719$		−	40.6459i		−	1.51584i	−0.652349	−	0.757919i	$-0.726216\pi$
$719$		−	40.6459i		−	1.51584i	0.652349	−	0.757919i	$-0.273784\pi$
$720$		0			0
$721$			1.48403i			0.0552683i
$722$		0			0
$723$	3.40943	+	15.5749i	0.126798	+	0.579238i
$724$		0			0
$725$			3.47526i			0.129068i
$726$		0			0
$727$			30.8644i			1.14470i	0.820011	+	0.572348i	$0.193967\pi$
$727$			30.8644i			1.14470i	−0.820011	+	0.572348i	$0.806033\pi$
$728$		0			0
$729$	−7.40350	+	25.9651i	−0.274204	+	0.961672i
$730$		0			0
$731$	10.9611			0.405411
$732$		0			0
$733$			41.2464i			1.52347i	0.647889	+	0.761735i	$0.275652\pi$
$733$			41.2464i			1.52347i	−0.647889	+	0.761735i	$0.724348\pi$
$734$		0			0
$735$	2.59017	+	11.8324i	0.0955397	+	0.436443i
$736$		0			0
$737$	−8.87516	−	10.4100i	−0.326921	−	0.383457i
$738$		0			0
$739$			25.6924i			0.945111i	0.881301	+	0.472556i	$0.156668\pi$
$739$			25.6924i			0.945111i	−0.881301	+	0.472556i	$0.843332\pi$
$740$		0			0
$741$	−7.77430	+	1.70183i	−0.285596	+	0.0625184i
$742$		0			0
$743$			32.2510i			1.18318i	0.806240	+	0.591588i	$0.201499\pi$
$743$			32.2510i			1.18318i	−0.806240	+	0.591588i	$0.798501\pi$
$744$		0			0
$745$			2.34874i			0.0860510i
$746$		0			0
$747$	3.10842	+	6.75970i	0.113731	+	0.247324i
$748$		0			0
$749$	0.831245			0.0303730
$750$		0			0
$751$	25.2785			0.922424			0.461212	−	0.887290i	$-0.347415\pi$
$751$	25.2785			0.922424			0.461212	+	0.887290i	$0.347415\pi$
$752$		0			0
$753$	2.18688	+	9.99006i	0.0796942	+	0.364058i
$754$		0			0
$755$	6.81193			0.247911
$756$		0			0
$757$			21.5605i			0.783630i	0.920044	+	0.391815i	$0.128153\pi$
$757$			21.5605i			0.783630i	−0.920044	+	0.391815i	$0.871847\pi$
$758$		0			0
$759$	−9.38505	+	2.05444i	−0.340656	+	0.0745713i
$760$		0			0
$761$		−	29.1557i		−	1.05689i	−0.848966	−	0.528447i	$-0.822774\pi$
$761$		−	29.1557i		−	1.05689i	0.848966	−	0.528447i	$-0.177226\pi$
$762$		0			0
$763$	0.108929			0.00394351
$764$		0			0
$765$	−2.18484	−	4.75125i	−0.0789931	−	0.171782i
$766$		0			0
$767$	32.2321			1.16383
$768$		0			0
$769$			25.0579i			0.903612i	0.892116	+	0.451806i	$0.149220\pi$
$769$			25.0579i			0.903612i	−0.892116	+	0.451806i	$0.850780\pi$
$770$		0			0
$771$	2.58587	−	0.566060i	0.0931278	−	0.0203861i
$772$		0			0
$773$			7.92683i			0.285108i	0.989787	+	0.142554i	$0.0455315\pi$
$773$			7.92683i			0.285108i	−0.989787	+	0.142554i	$0.954469\pi$
$774$		0			0
$775$		−	4.73509i		−	0.170089i
$776$		0			0
$777$	0.144074	−	0.0315385i	0.00516862	−	0.00113144i
$778$		0			0
$779$	5.06274			0.181392
$780$		0			0
$781$		−	13.9210i		−	0.498134i
$782$		0			0
$783$	−14.4136	+	10.8783i	−0.515101	+	0.388759i
$784$		0			0
$785$	3.28192			0.117137
$786$		0			0
$787$		−	18.7578i		−	0.668642i	−0.942459	−	0.334321i	$-0.891493\pi$
$787$		−	18.7578i		−	0.668642i	0.942459	−	0.334321i	$-0.108507\pi$
$788$		0			0
$789$	−24.9224	+	5.45565i	−0.887263	+	0.194226i
$790$		0			0
$791$		−	0.780209i		−	0.0277410i
$792$		0			0
$793$	−5.58738			−0.198414
$794$		0			0
$795$	0.564280	+	2.57774i	0.0200130	+	0.0914229i
$796$		0			0
$797$			11.1759i			0.395871i	0.980215	+	0.197935i	$0.0634236\pi$
$797$			11.1759i			0.395871i	−0.980215	+	0.197935i	$0.936576\pi$
$798$		0			0
$799$	−22.1967			−0.785262
$800$		0			0
$801$	2.51472	+	5.46861i	0.0888532	+	0.193224i
$802$		0			0
$803$	15.9446			0.562673
$804$		0			0
$805$	−0.274006			−0.00965746
$806$		0			0
$807$	19.7360	−	4.32030i	0.694738	−	0.152082i
$808$		0			0
$809$	34.1568			1.20089			0.600445	−	0.799666i	$-0.294990\pi$
$809$	34.1568			1.20089			0.600445	+	0.799666i	$0.294990\pi$
$810$		0			0
$811$		−	5.12940i		−	0.180118i	−0.995936	−	0.0900588i	$-0.971295\pi$
$811$		−	5.12940i		−	0.180118i	0.995936	−	0.0900588i	$-0.0287055\pi$
$812$		0			0
$813$	−21.1750	+	4.63533i	−0.742641	+	0.162568i
$814$		0			0
$815$	−10.4909			−0.367480
$816$		0			0
$817$		−	6.41431i		−	0.224408i
$818$		0			0
$819$	1.01358	−	0.466090i	0.0354173	−	0.0162865i
$820$		0			0
$821$			17.4855i			0.610249i	0.952313	+	0.305124i	$0.0986980\pi$
$821$			17.4855i			0.610249i	−0.952313	+	0.305124i	$0.901302\pi$
$822$		0			0
$823$	20.4402			0.712499			0.356250	−	0.934391i	$-0.384055\pi$
$823$	20.4402			0.712499			0.356250	+	0.934391i	$0.384055\pi$
$824$		0			0
$825$	−0.619005	−	2.82773i	−0.0215510	−	0.0984490i
$826$		0			0
$827$		−	40.8488i		−	1.42045i	−0.703975	−	0.710225i	$-0.748593\pi$
$827$		−	40.8488i		−	1.42045i	0.703975	−	0.710225i	$-0.251407\pi$
$828$		0			0
$829$	−14.1121			−0.490135			−0.245067	−	0.969506i	$-0.578810\pi$
$829$	−14.1121			−0.490135			−0.245067	+	0.969506i	$0.578810\pi$
$830$		0			0
$831$	−6.64836	−	30.3710i	−0.230629	−	1.05356i
$832$		0			0
$833$		−	12.1903i		−	0.422370i
$834$		0			0
$835$		−	1.89936i		−	0.0657301i
$836$		0			0
$837$	19.6388	−	14.8218i	0.678815	−	0.512318i
$838$		0			0
$839$			1.64281i			0.0567161i	0.999598	+	0.0283581i	$0.00902786\pi$
$839$			1.64281i			0.0567161i	−0.999598	+	0.0283581i	$0.990972\pi$
$840$		0			0
$841$	16.9226			0.583537
$842$		0			0
$843$	10.4599	+	47.7827i	0.360257	+	1.64572i
$844$		0			0
$845$	7.28881			0.250743
$846$		0			0
$847$		−	0.677552i		−	0.0232810i
$848$		0			0
$849$	−7.18605	−	32.8272i	−0.246625	−	1.12663i
$850$		0			0
$851$		−	3.42314i		−	0.117344i
$852$		0			0
$853$	35.6193			1.21958			0.609791	−	0.792562i	$-0.291253\pi$
$853$	35.6193			1.21958			0.609791	+	0.792562i	$0.291253\pi$
$854$		0			0
$855$	−2.78037	+	1.27854i	−0.0950868	+	0.0437253i
$856$		0			0
$857$	−19.8486			−0.678016			−0.339008	−	0.940783i	$-0.610091\pi$
$857$	−19.8486			−0.678016			−0.339008	+	0.940783i	$0.610091\pi$
$858$		0			0
$859$	21.9240			0.748036			0.374018	−	0.927421i	$-0.377980\pi$
$859$	21.9240			0.748036			0.374018	+	0.927421i	$0.377980\pi$
$860$		0			0
$861$	−0.693280	+	0.151763i	−0.0236269	+	0.00517205i
$862$		0			0
$863$			27.7881i			0.945918i	0.881084	+	0.472959i	$0.156814\pi$
$863$			27.7881i			0.945918i	−0.881084	+	0.472959i	$0.843186\pi$
$864$		0			0
$865$			15.8055i			0.537401i
$866$		0			0
$867$	−5.17106	−	23.6224i	−0.175619	−	0.802259i
$868$		0			0
$869$			7.66429i			0.259993i
$870$		0			0
$871$	28.0567	−	23.9201i	0.950666	−	0.810501i
$872$		0			0
$873$	10.1464	+	22.0647i	0.343402	+	0.746777i
$874$		0			0
$875$		−	0.0825586i		−	0.00279099i
$876$		0			0
$877$	−46.5189			−1.57083			−0.785415	−	0.618969i	$-0.787551\pi$
$877$	−46.5189			−1.57083			−0.785415	+	0.618969i	$0.787551\pi$
$878$		0			0
$879$	−1.85378	+	0.405802i	−0.0625264	+	0.0136874i
$880$		0			0
$881$		−	16.5396i		−	0.557235i	−0.960402	−	0.278617i	$-0.910124\pi$
$881$		−	16.5396i		−	0.557235i	0.960402	−	0.278617i	$-0.0898761\pi$
$882$		0			0
$883$			19.6612i			0.661654i	0.943691	+	0.330827i	$0.107328\pi$
$883$			19.6612i			0.661654i	−0.943691	+	0.330827i	$0.892672\pi$
$884$		0			0
$885$	12.1076	−	2.65041i	0.406991	−	0.0890925i
$886$		0			0
$887$			6.60795i			0.221873i	0.993827	+	0.110937i	$0.0353851\pi$
$887$			6.60795i			0.221873i	−0.993827	+	0.110937i	$0.964615\pi$
$888$		0			0
$889$		−	0.540116i		−	0.0181149i
$890$		0			0
$891$	9.79040	−	11.4187i	0.327991	−	0.382542i
$892$		0			0
$893$			12.9892i			0.434668i
$894$		0			0
$895$	11.1981			0.374311
$896$		0			0
$897$	−5.53706	−	25.2943i	−0.184877	−	0.844553i
$898$		0			0
$899$	16.4557			0.548827
$900$		0			0
$901$		−	2.65572i		−	0.0884750i
$902$		0			0
$903$	0.192278	+	0.878361i	0.00639860	+	0.0292300i
$904$		0			0
$905$	12.2874			0.408446
$906$		0			0
$907$	−42.1478			−1.39949			−0.699747	−	0.714391i	$-0.746704\pi$
$907$	−42.1478			−1.39949			−0.699747	+	0.714391i	$0.746704\pi$
$908$		0			0
$909$	8.11725	−	3.73268i	0.269232	−	0.123805i
$910$		0			0
$911$			37.9245i			1.25650i	0.778013	+	0.628248i	$0.216228\pi$
$911$			37.9245i			1.25650i	−0.778013	+	0.628248i	$0.783772\pi$
$912$		0			0
$913$		−	4.14478i		−	0.137172i
$914$		0			0
$915$	−2.09883	+	0.459444i	−0.0693851	+	0.0151888i
$916$		0			0
$917$	1.30523			0.0431025
$918$		0			0
$919$		−	7.67863i		−	0.253295i	−0.991948	−	0.126647i	$-0.959578\pi$
$919$		−	7.67863i		−	0.253295i	0.991948	−	0.126647i	$-0.0404216\pi$
$920$		0			0
$921$	−2.79948	−	12.7886i	−0.0922461	−	0.421398i
$922$		0			0
$923$	37.5196			1.23497
$924$		0			0
$925$	1.03140			0.0339121
$926$		0			0
$927$	−48.9947	+	22.5300i	−1.60920	+	0.739982i
$928$		0			0
$929$	−5.33354			−0.174988			−0.0874939	−	0.996165i	$-0.527886\pi$
$929$	−5.33354			−0.174988			−0.0874939	+	0.996165i	$0.527886\pi$
$930$		0			0
$931$	−7.13364			−0.233796
$932$		0			0
$933$	−3.24203	−	14.8102i	−0.106139	−	0.484864i
$934$		0			0
$935$			2.91328i			0.0952745i
$936$		0			0
$937$		−	8.70022i		−	0.284224i	−0.989851	−	0.142112i	$-0.954611\pi$
$937$		−	8.70022i		−	0.284224i	0.989851	−	0.142112i	$-0.0453893\pi$
$938$		0			0
$939$	44.4096	−	9.72148i	1.44925	−	0.317249i
$940$		0			0
$941$	15.3896			0.501688			0.250844	−	0.968028i	$-0.419292\pi$
$941$	15.3896			0.501688			0.250844	+	0.968028i	$0.419292\pi$
$942$		0			0
$943$			16.4721i			0.536404i
$944$		0			0
$945$	0.342412	−	0.258426i	0.0111387	−	0.00840661i
$946$		0			0
$947$			35.2347i			1.14497i	0.819914	+	0.572487i	$0.194021\pi$
$947$			35.2347i			1.14497i	−0.819914	+	0.572487i	$0.805979\pi$
$948$		0			0
$949$			42.9734i			1.39498i
$950$		0			0
$951$	−39.9037	+	8.73513i	−1.29397	+	0.283256i
$952$		0			0
$953$			5.67383i			0.183793i	0.995769	+	0.0918966i	$0.0292929\pi$
$953$			5.67383i			0.183793i	−0.995769	+	0.0918966i	$0.970707\pi$
$954$		0			0
$955$	−13.9708			−0.452084
$956$		0			0
$957$	9.82710	−	2.15120i	0.317665	−	0.0695385i
$958$		0			0
$959$		−	0.748166i		−	0.0241595i
$960$		0			0
$961$	8.57894			0.276740
$962$		0			0
$963$	12.6196	+	27.4431i	0.406661	+	0.884342i
$964$		0			0
$965$	5.64847			0.181831
$966$		0			0
$967$	47.3754			1.52349			0.761745	−	0.647877i	$-0.224343\pi$
$967$	47.3754			1.52349			0.761745	+	0.647877i	$0.224343\pi$
$968$		0			0
$969$	3.00866	−	0.658612i	0.0966522	−	0.0211577i
$970$		0			0
$971$			30.4366i			0.976757i	0.872632	+	0.488378i	$0.162411\pi$
$971$			30.4366i			0.976757i	−0.872632	+	0.488378i	$0.837589\pi$
$972$		0			0
$973$	−0.0521514			−0.00167190
$974$		0			0
$975$	7.62123	−	1.66833i	0.244075	−	0.0534292i
$976$		0			0
$977$		−	10.3655i		−	0.331623i	−0.986157	−	0.165812i	$-0.946976\pi$
$977$		−	10.3655i		−	0.331623i	0.986157	−	0.165812i	$-0.0530244\pi$
$978$		0			0
$979$		−	3.35314i		−	0.107167i
$980$		0			0
$981$	1.65372	+	3.59625i	0.0527993	+	0.114819i
$982$		0			0
$983$	−25.9630			−0.828090			−0.414045	−	0.910256i	$-0.635884\pi$
$983$	−25.9630			−0.828090			−0.414045	+	0.910256i	$0.635884\pi$
$984$		0			0
$985$	−9.28217			−0.295754
$986$		0			0
$987$	−0.389370	−	1.77872i	−0.0123938	−	0.0566171i
$988$		0			0
$989$	20.8695			0.663612
$990$		0			0
$991$		−	34.0668i		−	1.08217i	−0.840969	−	0.541084i	$-0.818014\pi$
$991$		−	34.0668i		−	1.08217i	0.840969	−	0.541084i	$-0.181986\pi$
$992$		0			0
$993$	12.9869	−	2.84290i	0.412127	−	0.0902167i
$994$		0			0
$995$	15.9556			0.505826
$996$		0			0
$997$	−32.4827			−1.02874			−0.514369	−	0.857569i	$-0.671974\pi$
$997$	−32.4827			−1.02874			−0.514369	+	0.857569i	$0.671974\pi$
$998$		0			0
$999$	3.22850	+	4.27772i	0.102145	+	0.135341i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.370384 - 1.69199i) q^{3} -1.00000 q^{5} +0.0825586i q^{7} +(-2.72563 + 1.25337i) q^{9} +O(q^{10})\)	\(q+(-0.370384 - 1.69199i) q^{3} -1.00000 q^{5} +0.0825586i q^{7} +(-2.72563 + 1.25337i) q^{9} +1.67125 q^{11} +4.50431i q^{13} +(0.370384 + 1.69199i) q^{15} -1.74317i q^{17} -1.02008 q^{19} +(0.139688 - 0.0305784i) q^{21} -3.31893i q^{23} +1.00000 q^{25} +(3.13022 + 4.14750i) q^{27} +3.47526i q^{29} -4.73509i q^{31} +(-0.619005 - 2.82773i) q^{33} -0.0825586i q^{35} +1.03140 q^{37} +(7.62123 - 1.66833i) q^{39} -4.96306 q^{41} +6.28802i q^{43} +(2.72563 - 1.25337i) q^{45} -12.7335i q^{47} +6.99318 q^{49} +(-2.94943 + 0.645644i) q^{51} +1.52350 q^{53} -1.67125 q^{55} +(0.377824 + 1.72597i) q^{57} -7.15583i q^{59} +1.24045i q^{61} +(-0.103476 - 0.225024i) q^{63} -4.50431i q^{65} +(-5.31049 - 6.22886i) q^{67} +(-5.61558 + 1.22928i) q^{69} -8.32971i q^{71} +9.54051 q^{73} +(-0.370384 - 1.69199i) q^{75} +0.137976i q^{77} +4.58596i q^{79} +(5.85813 - 6.83245i) q^{81} -2.48005i q^{83} +1.74317i q^{85} +(5.88009 - 1.28718i) q^{87} -2.00636i q^{89} -0.371870 q^{91} +(-8.01170 + 1.75380i) q^{93} +1.02008 q^{95} -8.09526i q^{97} +(-4.55521 + 2.09470i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 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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