LMFDB - Embedded newform 2070.3.c.a.91.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.c (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:	$56.4034147226$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.7
Root			$-1.00527i$ of defining polynomial
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.a.91.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.47532i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.47532i q^{7} -2.82843 q^{8} -3.16228i q^{10} -6.04959i q^{11} -5.21324 q^{13} -2.08642i q^{14} +4.00000 q^{16} +15.7063i q^{17} -4.82663i q^{19} +4.47214i q^{20} +8.55541i q^{22} +(2.58013 - 22.8548i) q^{23} -5.00000 q^{25} +7.37263 q^{26} +2.95065i q^{28} +23.4711 q^{29} -20.4887 q^{31} -5.65685 q^{32} -22.2121i q^{34} -3.29893 q^{35} -15.5248i q^{37} +6.82589i q^{38} -6.32456i q^{40} -20.3093 q^{41} +38.1696i q^{43} -12.0992i q^{44} +(-3.64885 + 32.3216i) q^{46} +13.8273 q^{47} +46.8234 q^{49} +7.07107 q^{50} -10.4265 q^{52} -38.2742i q^{53} +13.5273 q^{55} -4.17285i q^{56} -33.1932 q^{58} +33.5696 q^{59} -100.567i q^{61} +28.9753 q^{62} +8.00000 q^{64} -11.6572i q^{65} +32.4469i q^{67} +31.4126i q^{68} +4.66539 q^{70} +24.1306 q^{71} +15.1818 q^{73} +21.9554i q^{74} -9.65326i q^{76} +8.92511 q^{77} +11.2095i q^{79} +8.94427i q^{80} +28.7216 q^{82} +44.1310i q^{83} -35.1204 q^{85} -53.9800i q^{86} +17.1108i q^{88} -111.039i q^{89} -7.69122i q^{91} +(5.16026 - 45.7096i) q^{92} -19.5547 q^{94} +10.7927 q^{95} -154.126i q^{97} -66.2183 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) $ 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			1.47532i			0.210761i	0.994432	+	0.105380i	$0.0336060\pi$
$7$			1.47532i			0.210761i	−0.994432	+	0.105380i	$0.966394\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$		−	6.04959i		−	0.549963i	−0.961450	−	0.274981i	$-0.911328\pi$
$11$		−	6.04959i		−	0.549963i	0.961450	−	0.274981i	$-0.0886717\pi$
$12$		0			0
$13$	−5.21324			−0.401018			−0.200509	−	0.979692i	$-0.564260\pi$
$13$	−5.21324			−0.401018			−0.200509	+	0.979692i	$0.564260\pi$
$14$		−	2.08642i		−	0.149030i
$15$		0			0
$16$	4.00000			0.250000
$17$			15.7063i			0.923901i	0.886906	+	0.461951i	$0.152850\pi$
$17$			15.7063i			0.923901i	−0.886906	+	0.461951i	$0.847150\pi$
$18$		0			0
$19$		−	4.82663i		−	0.254033i	−0.991901	−	0.127017i	$-0.959460\pi$
$19$		−	4.82663i		−	0.254033i	0.991901	−	0.127017i	$-0.0405402\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$			8.55541i			0.388882i
$23$	2.58013	−	22.8548i	0.112180	−	0.993688i
$23$	2.58013	−	22.8548i	0.112180	−	0.993688i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	7.37263			0.283563
$27$		0			0
$28$			2.95065i			0.105380i
$29$	23.4711			0.809349			0.404674	−	0.914461i	$-0.367385\pi$
$29$	23.4711			0.809349			0.404674	+	0.914461i	$0.367385\pi$
$30$		0			0
$31$	−20.4887			−0.660924			−0.330462	−	0.943819i	$-0.607205\pi$
$31$	−20.4887			−0.660924			−0.330462	+	0.943819i	$0.607205\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$		−	22.2121i		−	0.653297i
$35$	−3.29893			−0.0942550
$36$		0			0
$37$		−	15.5248i		−	0.419589i	−0.977746	−	0.209794i	$-0.932721\pi$
$37$		−	15.5248i		−	0.419589i	0.977746	−	0.209794i	$-0.0672795\pi$
$38$			6.82589i			0.179629i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$	−20.3093			−0.495348			−0.247674	−	0.968843i	$-0.579666\pi$
$41$	−20.3093			−0.495348			−0.247674	+	0.968843i	$0.579666\pi$
$42$		0			0
$43$			38.1696i			0.887666i	0.896109	+	0.443833i	$0.146382\pi$
$43$			38.1696i			0.887666i	−0.896109	+	0.443833i	$0.853618\pi$
$44$		−	12.0992i		−	0.274981i
$45$		0			0
$46$	−3.64885	+	32.3216i	−0.0793229	+	0.702643i
$47$	13.8273			0.294198			0.147099	−	0.989122i	$-0.453006\pi$
$47$	13.8273			0.294198			0.147099	+	0.989122i	$0.453006\pi$
$48$		0			0
$49$	46.8234			0.955580
$50$	7.07107			0.141421
$51$		0			0
$52$	−10.4265			−0.200509
$53$		−	38.2742i		−	0.722154i	−0.932536	−	0.361077i	$-0.882409\pi$
$53$		−	38.2742i		−	0.722154i	0.932536	−	0.361077i	$-0.117591\pi$
$54$		0			0
$55$	13.5273			0.245951
$56$		−	4.17285i		−	0.0745152i
$57$		0			0
$58$	−33.1932			−0.572296
$59$	33.5696			0.568977			0.284488	−	0.958679i	$-0.408176\pi$
$59$	33.5696			0.568977			0.284488	+	0.958679i	$0.408176\pi$
$60$		0			0
$61$		−	100.567i		−	1.64864i	−0.566124	−	0.824320i	$-0.691558\pi$
$61$		−	100.567i		−	1.64864i	0.566124	−	0.824320i	$-0.308442\pi$
$62$	28.9753			0.467344
$63$		0			0
$64$	8.00000			0.125000
$65$		−	11.6572i		−	0.179341i
$66$		0			0
$67$			32.4469i			0.484282i	0.970241	+	0.242141i	$0.0778497\pi$
$67$			32.4469i			0.484282i	−0.970241	+	0.242141i	$0.922150\pi$
$68$			31.4126i			0.461951i
$69$		0			0
$70$	4.66539			0.0666484
$71$	24.1306			0.339868			0.169934	−	0.985455i	$-0.445645\pi$
$71$	24.1306			0.339868			0.169934	+	0.985455i	$0.445645\pi$
$72$		0			0
$73$	15.1818			0.207969			0.103985	−	0.994579i	$-0.466841\pi$
$73$	15.1818			0.207969			0.103985	+	0.994579i	$0.466841\pi$
$74$			21.9554i			0.296694i
$75$		0			0
$76$		−	9.65326i		−	0.127017i
$77$	8.92511			0.115910
$78$		0			0
$79$			11.2095i			0.141893i	0.997480	+	0.0709463i	$0.0226019\pi$
$79$			11.2095i			0.141893i	−0.997480	+	0.0709463i	$0.977398\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$	28.7216			0.350264
$83$			44.1310i			0.531698i	0.964015	+	0.265849i	$0.0856523\pi$
$83$			44.1310i			0.531698i	−0.964015	+	0.265849i	$0.914348\pi$
$84$		0			0
$85$	−35.1204			−0.413181
$86$		−	53.9800i		−	0.627675i
$87$		0			0
$88$			17.1108i			0.194441i
$89$		−	111.039i		−	1.24763i	−0.781572	−	0.623815i	$-0.785582\pi$
$89$		−	111.039i		−	1.24763i	0.781572	−	0.623815i	$-0.214418\pi$
$90$		0			0
$91$		−	7.69122i		−	0.0845189i
$92$	5.16026	−	45.7096i	0.0560898	−	0.496844i
$93$		0			0
$94$	−19.5547			−0.208029
$95$	10.7927			0.113607
$96$		0			0
$97$		−	154.126i		−	1.58893i	−0.607312	−	0.794463i	$-0.707752\pi$
$97$		−	154.126i		−	1.58893i	0.607312	−	0.794463i	$-0.292248\pi$
$98$	−66.2183			−0.675697
$99$		0			0
$100$	−10.0000			−0.100000
$101$	58.8607			0.582779			0.291390	−	0.956604i	$-0.405882\pi$
$101$	58.8607			0.582779			0.291390	+	0.956604i	$0.405882\pi$
$102$		0			0
$103$			54.2662i			0.526856i	0.964679	+	0.263428i	$0.0848531\pi$
$103$			54.2662i			0.526856i	−0.964679	+	0.263428i	$0.915147\pi$
$104$	14.7453			0.141781
$105$		0			0
$106$			54.1279i			0.510640i
$107$			119.124i			1.11331i	0.830745	+	0.556653i	$0.187915\pi$
$107$			119.124i			1.11331i	−0.830745	+	0.556653i	$0.812085\pi$
$108$		0			0
$109$			149.223i			1.36902i	0.729003	+	0.684510i	$0.239984\pi$
$109$			149.223i			1.36902i	−0.729003	+	0.684510i	$0.760016\pi$
$110$	−19.1305			−0.173913
$111$		0			0
$112$			5.90130i			0.0526902i
$113$			35.3339i			0.312690i	0.987703	+	0.156345i	$0.0499711\pi$
$113$			35.3339i			0.312690i	−0.987703	+	0.156345i	$0.950029\pi$
$114$		0			0
$115$	51.1049	+	5.76935i	0.444391	+	0.0501682i
$116$	46.9422			0.404674
$117$		0			0
$118$	−47.4746			−0.402327
$119$	−23.1719			−0.194722
$120$		0			0
$121$	84.4025			0.697541
$122$			142.223i			1.16576i
$123$		0			0
$124$	−40.9773			−0.330462
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	92.8398			0.731022			0.365511	−	0.930807i	$-0.380894\pi$
$127$	92.8398			0.731022			0.365511	+	0.930807i	$0.380894\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$			16.4857i			0.126813i
$131$	−151.709			−1.15808			−0.579042	−	0.815297i	$-0.696573\pi$
$131$	−151.709			−1.15808			−0.579042	+	0.815297i	$0.696573\pi$
$132$		0			0
$133$	7.12085			0.0535402
$134$		−	45.8869i		−	0.342439i
$135$		0			0
$136$		−	44.4242i		−	0.326648i
$137$			267.464i			1.95229i	0.217121	+	0.976145i	$0.430333\pi$
$137$			267.464i			1.95229i	−0.217121	+	0.976145i	$0.569667\pi$
$138$		0			0
$139$	239.779			1.72503			0.862516	−	0.506030i	$-0.168888\pi$
$139$	239.779			1.72503			0.862516	+	0.506030i	$0.168888\pi$
$140$	−6.59785			−0.0471275
$141$		0			0
$142$	−34.1258			−0.240323
$143$			31.5380i			0.220545i
$144$		0			0
$145$			52.4830i			0.361952i
$146$	−21.4703			−0.147057
$147$		0			0
$148$		−	31.0496i		−	0.209794i
$149$		−	201.647i		−	1.35333i	−0.736290	−	0.676667i	$-0.763424\pi$
$149$		−	201.647i		−	1.35333i	0.736290	−	0.676667i	$-0.236576\pi$
$150$		0			0
$151$	120.787			0.799912			0.399956	−	0.916534i	$-0.369025\pi$
$151$	120.787			0.799912			0.399956	+	0.916534i	$0.369025\pi$
$152$			13.6518i			0.0898143i
$153$		0			0
$154$	−12.6220			−0.0819611
$155$		−	45.8140i		−	0.295574i
$156$		0			0
$157$		−	194.897i		−	1.24138i	−0.784055	−	0.620692i	$-0.786852\pi$
$157$		−	194.897i		−	1.24138i	0.784055	−	0.620692i	$-0.213148\pi$
$158$		−	15.8526i		−	0.100333i
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	33.7183	+	3.80653i	0.209430	+	0.0236430i
$162$		0			0
$163$	153.813			0.943635			0.471817	−	0.881696i	$-0.343598\pi$
$163$	153.813			0.943635			0.471817	+	0.881696i	$0.343598\pi$
$164$	−40.6185			−0.247674
$165$		0			0
$166$		−	62.4106i		−	0.375967i
$167$	−48.4949			−0.290389			−0.145194	−	0.989403i	$-0.546381\pi$
$167$	−48.4949			−0.290389			−0.145194	+	0.989403i	$0.546381\pi$
$168$		0			0
$169$	−141.822			−0.839184
$170$	49.6678			0.292163
$171$		0			0
$172$			76.3393i			0.443833i
$173$	111.269			0.643174			0.321587	−	0.946880i	$-0.395784\pi$
$173$	111.269			0.643174			0.321587	+	0.946880i	$0.395784\pi$
$174$		0			0
$175$		−	7.37662i		−	0.0421521i
$176$		−	24.1984i		−	0.137491i
$177$		0			0
$178$			157.033i			0.882207i
$179$	−262.590			−1.46698			−0.733492	−	0.679698i	$-0.762111\pi$
$179$	−262.590			−1.46698			−0.733492	+	0.679698i	$0.762111\pi$
$180$		0			0
$181$			211.167i			1.16667i	0.812231	+	0.583335i	$0.198253\pi$
$181$			211.167i			1.16667i	−0.812231	+	0.583335i	$0.801747\pi$
$182$			10.8770i			0.0597639i
$183$		0			0
$184$	−7.29771	+	64.6432i	−0.0396615	+	0.351322i
$185$	34.7145			0.187646
$186$		0			0
$187$	95.0168			0.508111
$188$	27.6546			0.147099
$189$		0			0
$190$	−15.2631			−0.0803324
$191$		−	72.8209i		−	0.381261i	−0.981662	−	0.190631i	$-0.938947\pi$
$191$		−	72.8209i		−	0.381261i	0.981662	−	0.190631i	$-0.0610533\pi$
$192$		0			0
$193$	198.550			1.02875			0.514377	−	0.857564i	$-0.328023\pi$
$193$	198.550			1.02875			0.514377	+	0.857564i	$0.328023\pi$
$194$			217.967i			1.12354i
$195$		0			0
$196$	93.6468			0.477790
$197$	−28.2055			−0.143175			−0.0715876	−	0.997434i	$-0.522807\pi$
$197$	−28.2055			−0.143175			−0.0715876	+	0.997434i	$0.522807\pi$
$198$		0			0
$199$		−	214.499i		−	1.07788i	−0.842343	−	0.538941i	$-0.818825\pi$
$199$		−	214.499i		−	1.07788i	0.842343	−	0.538941i	$-0.181175\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	−83.2416			−0.412087
$203$			34.6275i			0.170579i
$204$		0			0
$205$		−	45.4129i		−	0.221526i
$206$		−	76.7439i		−	0.372543i
$207$		0			0
$208$	−20.8530			−0.100255
$209$	−29.1991			−0.139709
$210$		0			0
$211$	240.262			1.13868			0.569340	−	0.822102i	$-0.307199\pi$
$211$	240.262			1.13868			0.569340	+	0.822102i	$0.307199\pi$
$212$		−	76.5483i		−	0.361077i
$213$		0			0
$214$		−	168.466i		−	0.787226i
$215$	−85.3499			−0.396976
$216$		0			0
$217$		−	30.2274i		−	0.139297i
$218$		−	211.034i		−	0.968044i
$219$		0			0
$220$	27.0546			0.122975
$221$		−	81.8808i		−	0.370501i
$222$		0			0
$223$	257.402			1.15427			0.577136	−	0.816648i	$-0.304170\pi$
$223$	257.402			1.15427			0.577136	+	0.816648i	$0.304170\pi$
$224$		−	8.34570i		−	0.0372576i
$225$		0			0
$226$		−	49.9697i		−	0.221105i
$227$		−	106.401i		−	0.468727i	−0.972149	−	0.234363i	$-0.924699\pi$
$227$		−	106.401i		−	0.468727i	0.972149	−	0.234363i	$-0.0753005\pi$
$228$		0			0
$229$		−	328.507i		−	1.43453i	−0.696801	−	0.717265i	$-0.745394\pi$
$229$		−	328.507i		−	1.43453i	0.696801	−	0.717265i	$-0.254606\pi$
$230$	−72.2733	−	8.15909i	−0.314232	−	0.0354743i
$231$		0			0
$232$	−66.3863			−0.286148
$233$	149.213			0.640400			0.320200	−	0.947350i	$-0.396250\pi$
$233$	149.213			0.640400			0.320200	+	0.947350i	$0.396250\pi$
$234$		0			0
$235$			30.9187i			0.131569i
$236$	67.1393			0.284488
$237$		0			0
$238$	32.7701			0.137689
$239$	343.168			1.43585			0.717925	−	0.696121i	$-0.245092\pi$
$239$	343.168			1.43585			0.717925	+	0.696121i	$0.245092\pi$
$240$		0			0
$241$			348.748i			1.44709i	0.690278	+	0.723544i	$0.257488\pi$
$241$			348.748i			1.44709i	−0.690278	+	0.723544i	$0.742512\pi$
$242$	−119.363			−0.493236
$243$		0			0
$244$		−	201.134i		−	0.824320i
$245$			104.700i			0.427348i
$246$		0			0
$247$			25.1624i			0.101872i
$248$	57.9507			0.233672
$249$		0			0
$250$			15.8114i			0.0632456i
$251$		−	225.099i		−	0.896809i	−0.893831	−	0.448405i	$-0.851992\pi$
$251$		−	225.099i		−	0.896809i	0.893831	−	0.448405i	$-0.148008\pi$
$252$		0			0
$253$	−138.262	−	15.6087i	−0.546491	−	0.0616946i
$254$	−131.295			−0.516911
$255$		0			0
$256$	16.0000			0.0625000
$257$	393.632			1.53164			0.765821	−	0.643054i	$-0.222333\pi$
$257$	393.632			1.53164			0.765821	+	0.643054i	$0.222333\pi$
$258$		0			0
$259$	22.9041			0.0884328
$260$		−	23.3143i		−	0.0896704i
$261$		0			0
$262$	214.549			0.818890
$263$			132.720i			0.504640i	0.967644	+	0.252320i	$0.0811935\pi$
$263$			132.720i			0.504640i	−0.967644	+	0.252320i	$0.918806\pi$
$264$		0			0
$265$	85.5836			0.322957
$266$	−10.0704			−0.0378586
$267$		0			0
$268$			64.8938i			0.242141i
$269$	438.116			1.62868			0.814341	−	0.580386i	$-0.197098\pi$
$269$	438.116			1.62868			0.814341	+	0.580386i	$0.197098\pi$
$270$		0			0
$271$	−204.698			−0.755342			−0.377671	−	0.925940i	$-0.623275\pi$
$271$	−204.698			−0.755342			−0.377671	+	0.925940i	$0.623275\pi$
$272$			62.8253i			0.230975i
$273$		0			0
$274$		−	378.251i		−	1.38048i
$275$			30.2479i			0.109993i
$276$		0			0
$277$	173.804			0.627452			0.313726	−	0.949514i	$-0.398423\pi$
$277$	173.804			0.627452			0.313726	+	0.949514i	$0.398423\pi$
$278$	−339.099			−1.21978
$279$		0			0
$280$	9.33077			0.0333242
$281$		−	143.605i		−	0.511051i	−0.966802	−	0.255526i	$-0.917752\pi$
$281$		−	143.605i		−	0.511051i	0.966802	−	0.255526i	$-0.0822485\pi$
$282$		0			0
$283$		−	72.1714i		−	0.255023i	−0.991837	−	0.127511i	$-0.959301\pi$
$283$		−	72.1714i		−	0.255023i	0.991837	−	0.127511i	$-0.0406989\pi$
$284$	48.2612			0.169934
$285$		0			0
$286$		−	44.6014i		−	0.155949i
$287$		−	29.9627i		−	0.104400i
$288$		0			0
$289$	42.3114			0.146406
$290$		−	74.2222i		−	0.255939i
$291$		0			0
$292$	30.3635			0.103985
$293$			124.199i			0.423889i	0.977282	+	0.211944i	$0.0679796\pi$
$293$			124.199i			0.423889i	−0.977282	+	0.211944i	$0.932020\pi$
$294$		0			0
$295$			75.0640i			0.254454i
$296$			43.9107i			0.148347i
$297$		0			0
$298$			285.171i			0.956951i
$299$	−13.4508	+	119.148i	−0.0449861	+	0.398487i
$300$		0			0
$301$	−56.3126			−0.187085
$302$	−170.818			−0.565623
$303$		0			0
$304$		−	19.3065i		−	0.0635083i
$305$	224.875			0.737294
$306$		0			0
$307$	219.717			0.715690			0.357845	−	0.933781i	$-0.383512\pi$
$307$	219.717			0.715690			0.357845	+	0.933781i	$0.383512\pi$
$308$	17.8502			0.0579552
$309$		0			0
$310$			64.7908i			0.209003i
$311$	317.069			1.01951			0.509757	−	0.860319i	$-0.329735\pi$
$311$	317.069			1.01951			0.509757	+	0.860319i	$0.329735\pi$
$312$		0			0
$313$			484.654i			1.54841i	0.632932	+	0.774207i	$0.281851\pi$
$313$			484.654i			1.54841i	−0.632932	+	0.774207i	$0.718149\pi$
$314$			275.626i			0.877791i
$315$		0			0
$316$			22.4190i			0.0709463i
$317$	521.910			1.64640			0.823202	−	0.567748i	$-0.192185\pi$
$317$	521.910			1.64640			0.823202	+	0.567748i	$0.192185\pi$
$318$		0			0
$319$		−	141.991i		−	0.445111i
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	−47.6849	−	5.38325i	−0.148090	−	0.0167182i
$323$	75.8086			0.234702
$324$		0			0
$325$	26.0662			0.0802037
$326$	−217.524			−0.667251
$327$		0			0
$328$	57.4432			0.175132
$329$			20.3997i			0.0620053i
$330$		0			0
$331$	−318.270			−0.961540			−0.480770	−	0.876847i	$-0.659643\pi$
$331$	−318.270			−0.961540			−0.480770	+	0.876847i	$0.659643\pi$
$332$			88.2619i			0.265849i
$333$		0			0
$334$	68.5822			0.205336
$335$	−72.5535			−0.216578
$336$		0			0
$337$		−	116.539i		−	0.345812i	−0.984938	−	0.172906i	$-0.944684\pi$
$337$		−	116.539i		−	0.345812i	0.984938	−	0.172906i	$-0.0553156\pi$
$338$	200.567			0.593393
$339$		0			0
$340$	−70.2408			−0.206591
$341$			123.948i			0.363484i
$342$		0			0
$343$			141.371i			0.412159i
$344$		−	107.960i		−	0.313837i
$345$		0			0
$346$	−157.358			−0.454793
$347$	100.715			0.290245			0.145122	−	0.989414i	$-0.453642\pi$
$347$	100.715			0.290245			0.145122	+	0.989414i	$0.453642\pi$
$348$		0			0
$349$	143.476			0.411106			0.205553	−	0.978646i	$-0.434101\pi$
$349$	143.476			0.411106			0.205553	+	0.978646i	$0.434101\pi$
$350$			10.4321i			0.0298061i
$351$		0			0
$352$			34.2216i			0.0972206i
$353$	274.394			0.777320			0.388660	−	0.921381i	$-0.372938\pi$
$353$	274.394			0.777320			0.388660	+	0.921381i	$0.372938\pi$
$354$		0			0
$355$			53.9577i			0.151994i
$356$		−	222.078i		−	0.623815i
$357$		0			0
$358$	371.358			1.03731
$359$		−	476.871i		−	1.32833i	−0.747586	−	0.664166i	$-0.768787\pi$
$359$		−	476.871i		−	1.32833i	0.747586	−	0.664166i	$-0.231213\pi$
$360$		0			0
$361$	337.704			0.935467
$362$		−	298.636i		−	0.824961i
$363$		0			0
$364$		−	15.3824i		−	0.0422595i
$365$			33.9475i			0.0930067i
$366$		0			0
$367$			14.0319i			0.0382340i	0.999817	+	0.0191170i	$0.00608551\pi$
$367$			14.0319i			0.0382340i	−0.999817	+	0.0191170i	$0.993914\pi$
$368$	10.3205	−	91.4193i	0.0280449	−	0.248422i
$369$		0			0
$370$	−49.0937			−0.132686
$371$	56.4668			0.152202
$372$		0			0
$373$			105.662i			0.283277i	0.989918	+	0.141639i	$0.0452371\pi$
$373$			105.662i			0.283277i	−0.989918	+	0.141639i	$0.954763\pi$
$374$	−134.374			−0.359289
$375$		0			0
$376$	−39.1095			−0.104015
$377$	−122.361			−0.324564
$378$		0			0
$379$		−	339.983i		−	0.897053i	−0.893770	−	0.448527i	$-0.851949\pi$
$379$		−	339.983i		−	0.897053i	0.893770	−	0.448527i	$-0.148051\pi$
$380$	21.5854			0.0568036
$381$		0			0
$382$			102.984i			0.269592i
$383$			699.796i			1.82714i	0.406676	+	0.913572i	$0.366688\pi$
$383$			699.796i			1.82714i	−0.406676	+	0.913572i	$0.633312\pi$
$384$		0			0
$385$			19.9571i			0.0518367i
$386$	−280.792			−0.727439
$387$		0			0
$388$		−	308.252i		−	0.794463i
$389$		−	124.817i		−	0.320867i	−0.987047	−	0.160433i	$-0.948711\pi$
$389$		−	124.817i		−	0.320867i	0.987047	−	0.160433i	$-0.0512892\pi$
$390$		0			0
$391$	358.965	+	40.5244i	0.918070	+	0.103643i
$392$	−132.437			−0.337849
$393$		0			0
$394$	39.8886			0.101240
$395$	−25.0652			−0.0634563
$396$		0			0
$397$	−149.557			−0.376719			−0.188360	−	0.982100i	$-0.560317\pi$
$397$	−149.557			−0.376719			−0.188360	+	0.982100i	$0.560317\pi$
$398$			303.347i			0.762178i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$		−	426.696i		−	1.06408i	−0.846719	−	0.532040i	$-0.821426\pi$
$401$		−	426.696i		−	1.06408i	0.846719	−	0.532040i	$-0.178574\pi$
$402$		0			0
$403$	106.812			0.265043
$404$	117.721			0.291390
$405$		0			0
$406$		−	48.9707i		−	0.120617i
$407$	−93.9186			−0.230758
$408$		0			0
$409$	346.263			0.846610			0.423305	−	0.905987i	$-0.360870\pi$
$409$	346.263			0.846610			0.423305	+	0.905987i	$0.360870\pi$
$410$			64.2235i			0.156643i
$411$		0			0
$412$			108.532i			0.263428i
$413$			49.5261i			0.119918i
$414$		0			0
$415$	−98.6798			−0.237783
$416$	29.4905			0.0708907
$417$		0			0
$418$	41.2938			0.0987890
$419$		−	158.433i		−	0.378121i	−0.981965	−	0.189060i	$-0.939456\pi$
$419$		−	158.433i		−	0.378121i	0.981965	−	0.189060i	$-0.0605442\pi$
$420$		0			0
$421$			18.0568i			0.0428902i	0.999770	+	0.0214451i	$0.00682671\pi$
$421$			18.0568i			0.0428902i	−0.999770	+	0.0214451i	$0.993173\pi$
$422$	−339.781			−0.805169
$423$		0			0
$424$			108.256i			0.255320i
$425$		−	78.5316i		−	0.184780i
$426$		0			0
$427$	148.369			0.347469
$428$			238.247i			0.556653i
$429$		0			0
$430$	120.703			0.280705
$431$		−	151.730i		−	0.352041i	−0.984387	−	0.176020i	$-0.943678\pi$
$431$		−	151.730i		−	0.352041i	0.984387	−	0.176020i	$-0.0563225\pi$
$432$		0			0
$433$		−	625.471i		−	1.44451i	−0.691629	−	0.722253i	$-0.743106\pi$
$433$		−	625.471i		−	1.44451i	0.691629	−	0.722253i	$-0.256894\pi$
$434$			42.7480i			0.0984978i
$435$		0			0
$436$			298.447i			0.684510i
$437$	−110.312	−	12.4533i	−0.252430	−	0.0284973i
$438$		0			0
$439$	272.842			0.621509			0.310755	−	0.950490i	$-0.399418\pi$
$439$	272.842			0.621509			0.310755	+	0.950490i	$0.399418\pi$
$440$	−38.2610			−0.0869567
$441$		0			0
$442$			115.797i			0.261984i
$443$	−5.57528			−0.0125853			−0.00629264	−	0.999980i	$-0.502003\pi$
$443$	−5.57528			−0.0125853			−0.00629264	+	0.999980i	$0.502003\pi$
$444$		0			0
$445$	248.291			0.557957
$446$	−364.022			−0.816193
$447$		0			0
$448$			11.8026i			0.0263451i
$449$	−455.331			−1.01410			−0.507050	−	0.861916i	$-0.669264\pi$
$449$	−455.331			−1.01410			−0.507050	+	0.861916i	$0.669264\pi$
$450$		0			0
$451$			122.863i			0.272423i
$452$			70.6679i			0.156345i
$453$		0			0
$454$			150.474i			0.331440i
$455$	17.1981			0.0377980
$456$		0			0
$457$		−	597.785i		−	1.30806i	−0.756467	−	0.654032i	$-0.773076\pi$
$457$		−	597.785i		−	1.30806i	0.756467	−	0.654032i	$-0.226924\pi$
$458$			464.579i			1.01437i
$459$		0			0
$460$	102.210	+	11.5387i	0.222195	+	0.0250841i
$461$	340.556			0.738734			0.369367	−	0.929284i	$-0.379575\pi$
$461$	340.556			0.738734			0.369367	+	0.929284i	$0.379575\pi$
$462$		0			0
$463$	−471.395			−1.01813			−0.509066	−	0.860727i	$-0.670009\pi$
$463$	−471.395			−1.01813			−0.509066	+	0.860727i	$0.670009\pi$
$464$	93.8844			0.202337
$465$		0			0
$466$	−211.019			−0.452831
$467$		−	245.681i		−	0.526084i	−0.964784	−	0.263042i	$-0.915274\pi$
$467$		−	245.681i		−	0.526084i	0.964784	−	0.263042i	$-0.0847257\pi$
$468$		0			0
$469$	−47.8697			−0.102068
$470$		−	43.7257i		−	0.0930334i
$471$		0			0
$472$	−94.9493			−0.201164
$473$	230.911			0.488183
$474$		0			0
$475$			24.1332i			0.0508066i
$476$	−46.3439			−0.0973610
$477$		0			0
$478$	−485.313			−1.01530
$479$			203.940i			0.425763i	0.977078	+	0.212881i	$0.0682848\pi$
$479$			203.940i			0.425763i	−0.977078	+	0.212881i	$0.931715\pi$
$480$		0			0
$481$			80.9345i			0.168263i
$482$		−	493.205i		−	1.02325i
$483$		0			0
$484$	168.805			0.348771
$485$	344.636			0.710589
$486$		0			0
$487$	−681.456			−1.39929			−0.699647	−	0.714489i	$-0.746659\pi$
$487$	−681.456			−1.39929			−0.699647	+	0.714489i	$0.746659\pi$
$488$			284.447i			0.582882i
$489$		0			0
$490$		−	148.069i		−	0.302181i
$491$	329.666			0.671417			0.335709	−	0.941966i	$-0.391024\pi$
$491$	329.666			0.671417			0.335709	+	0.941966i	$0.391024\pi$
$492$		0			0
$493$			368.645i			0.747758i
$494$		−	35.5850i		−	0.0720344i
$495$		0			0
$496$	−81.9546			−0.165231
$497$			35.6005i			0.0716308i
$498$		0			0
$499$	−993.708			−1.99140			−0.995700	−	0.0926377i	$-0.970470\pi$
$499$	−993.708			−1.99140			−0.995700	+	0.0926377i	$0.970470\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$			318.338i			0.634140i
$503$			880.515i			1.75053i	0.483646	+	0.875264i	$0.339312\pi$
$503$			880.515i			1.75053i	−0.483646	+	0.875264i	$0.660688\pi$
$504$		0			0
$505$			131.617i			0.260627i
$506$	195.532	+	22.0741i	0.386428	+	0.0436246i
$507$		0			0
$508$	185.680			0.365511
$509$	689.907			1.35542			0.677708	−	0.735331i	$-0.262973\pi$
$509$	689.907			1.35542			0.677708	+	0.735331i	$0.262973\pi$
$510$		0			0
$511$			22.3980i			0.0438318i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−556.680			−1.08303
$515$	−121.343			−0.235617
$516$		0			0
$517$		−	83.6494i		−	0.161798i
$518$	−32.3913			−0.0625315
$519$		0			0
$520$			32.9714i			0.0634066i
$521$		−	408.559i		−	0.784183i	−0.919926	−	0.392092i	$-0.871752\pi$
$521$		−	408.559i		−	0.784183i	0.919926	−	0.392092i	$-0.128248\pi$
$522$		0			0
$523$			516.837i			0.988216i	0.869400	+	0.494108i	$0.164505\pi$
$523$			516.837i			0.988216i	−0.869400	+	0.494108i	$0.835495\pi$
$524$	−303.418			−0.579042
$525$		0			0
$526$		−	187.695i		−	0.356834i
$527$		−	321.801i		−	0.610629i
$528$		0			0
$529$	−515.686	−	117.937i	−0.974831	−	0.222943i
$530$	−121.034			−0.228365
$531$		0			0
$532$	14.2417			0.0267701
$533$	105.877			0.198644
$534$		0			0
$535$	−266.369			−0.497886
$536$		−	91.7737i		−	0.171220i
$537$		0			0
$538$	−619.589			−1.15165
$539$		−	283.262i		−	0.525533i
$540$		0			0
$541$	−825.026			−1.52500			−0.762501	−	0.646987i	$-0.776029\pi$
$541$	−825.026			−1.52500			−0.762501	+	0.646987i	$0.776029\pi$
$542$	289.486			0.534107
$543$		0			0
$544$		−	88.8484i		−	0.163324i
$545$	−333.673			−0.612245
$546$		0			0
$547$	−749.566			−1.37032			−0.685161	−	0.728392i	$-0.740268\pi$
$547$	−749.566			−1.37032			−0.685161	+	0.728392i	$0.740268\pi$
$548$			534.927i			0.976145i
$549$		0			0
$550$		−	42.7770i		−	0.0777765i
$551$		−	113.286i		−	0.205601i
$552$		0			0
$553$	−16.5377			−0.0299054
$554$	−245.796			−0.443675
$555$		0			0
$556$	479.559			0.862516
$557$			406.149i			0.729173i	0.931170	+	0.364586i	$0.118790\pi$
$557$			406.149i			0.729173i	−0.931170	+	0.364586i	$0.881210\pi$
$558$		0			0
$559$		−	198.988i		−	0.355971i
$560$	−13.1957			−0.0235638
$561$		0			0
$562$			203.089i			0.361368i
$563$		−	285.771i		−	0.507587i	−0.967258	−	0.253793i	$-0.918322\pi$
$563$		−	285.771i		−	0.507587i	0.967258	−	0.253793i	$-0.0816784\pi$
$564$		0			0
$565$	−79.0091			−0.139839
$566$			102.066i			0.180328i
$567$		0			0
$568$	−68.2517			−0.120161
$569$		−	315.624i		−	0.554699i	−0.960769	−	0.277349i	$-0.910544\pi$
$569$		−	315.624i		−	0.554699i	0.960769	−	0.277349i	$-0.0894560\pi$
$570$		0			0
$571$		−	383.907i		−	0.672342i	−0.941801	−	0.336171i	$-0.890868\pi$
$571$		−	383.907i		−	0.672342i	0.941801	−	0.336171i	$-0.109132\pi$
$572$			63.0759i			0.110273i
$573$		0			0
$574$			42.3737i			0.0738218i
$575$	−12.9007	+	114.274i	−0.0224359	+	0.198738i
$576$		0			0
$577$	−717.832			−1.24408			−0.622038	−	0.782987i	$-0.713695\pi$
$577$	−717.832			−1.24408			−0.622038	+	0.782987i	$0.713695\pi$
$578$	−59.8374			−0.103525
$579$		0			0
$580$			104.966i			0.180976i
$581$	−65.1075			−0.112061
$582$		0			0
$583$	−231.543			−0.397158
$584$	−42.9405			−0.0735283
$585$		0			0
$586$		−	175.644i		−	0.299735i
$587$	296.393			0.504928			0.252464	−	0.967606i	$-0.418759\pi$
$587$	296.393			0.504928			0.252464	+	0.967606i	$0.418759\pi$
$588$		0			0
$589$			98.8912i			0.167897i
$590$		−	106.157i		−	0.179926i
$591$		0			0
$592$		−	62.0992i		−	0.104897i
$593$	−654.259			−1.10330			−0.551652	−	0.834075i	$-0.686002\pi$
$593$	−654.259			−1.10330			−0.551652	+	0.834075i	$0.686002\pi$
$594$		0			0
$595$		−	51.8140i		−	0.0870824i
$596$		−	403.293i		−	0.676667i
$597$		0			0
$598$	19.0224	−	168.500i	0.0318100	−	0.281773i
$599$	−368.673			−0.615480			−0.307740	−	0.951470i	$-0.599573\pi$
$599$	−368.673			−0.615480			−0.307740	+	0.951470i	$0.599573\pi$
$600$		0			0
$601$	148.316			0.246782			0.123391	−	0.992358i	$-0.460623\pi$
$601$	148.316			0.246782			0.123391	+	0.992358i	$0.460623\pi$
$602$	79.6381			0.132289
$603$		0			0
$604$	241.573			0.399956
$605$			188.730i			0.311950i
$606$		0			0
$607$	939.932			1.54849			0.774244	−	0.632887i	$-0.218130\pi$
$607$	939.932			1.54849			0.774244	+	0.632887i	$0.218130\pi$
$608$			27.3036i			0.0449072i
$609$		0			0
$610$	−318.021			−0.521346
$611$	−72.0850			−0.117979
$612$		0			0
$613$			602.963i			0.983626i	0.870701	+	0.491813i	$0.163666\pi$
$613$			602.963i			0.983626i	−0.870701	+	0.491813i	$0.836334\pi$
$614$	−310.727			−0.506069
$615$		0			0
$616$	−25.2440			−0.0409805
$617$			885.697i			1.43549i	0.696306	+	0.717745i	$0.254826\pi$
$617$			885.697i			1.43549i	−0.696306	+	0.717745i	$0.745174\pi$
$618$		0			0
$619$		−	85.9838i		−	0.138908i	−0.997585	−	0.0694538i	$-0.977874\pi$
$619$		−	85.9838i		−	0.138908i	0.997585	−	0.0694538i	$-0.0221257\pi$
$620$		−	91.6280i		−	0.147787i
$621$		0			0
$622$	−448.403			−0.720905
$623$	163.819			0.262951
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	685.404i		−	1.09489i
$627$		0			0
$628$		−	389.794i		−	0.620692i
$629$	243.837			0.387659
$630$		0			0
$631$			645.385i			1.02280i	0.859344	+	0.511398i	$0.170872\pi$
$631$			645.385i			1.02280i	−0.859344	+	0.511398i	$0.829128\pi$
$632$		−	31.7053i		−	0.0501666i
$633$		0			0
$634$	−738.092			−1.16418
$635$			207.596i			0.326923i
$636$		0			0
$637$	−244.102			−0.383205
$638$			200.805i			0.314741i
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$		−	140.119i		−	0.218594i	−0.994009	−	0.109297i	$-0.965140\pi$
$641$		−	140.119i		−	0.218594i	0.994009	−	0.109297i	$-0.0348599\pi$
$642$		0			0
$643$			1028.11i			1.59892i	0.600718	+	0.799461i	$0.294882\pi$
$643$			1028.11i			1.59892i	−0.600718	+	0.799461i	$0.705118\pi$
$644$	67.4366	+	7.61306i	0.104715	+	0.0118215i
$645$		0			0
$646$	−107.210			−0.165959
$647$	−1248.29			−1.92934			−0.964672	−	0.263455i	$-0.915138\pi$
$647$	−1248.29			−1.92934			−0.964672	+	0.263455i	$0.915138\pi$
$648$		0			0
$649$		−	203.082i		−	0.312916i
$650$	−36.8632			−0.0567126
$651$		0			0
$652$	307.625			0.471817
$653$	457.504			0.700618			0.350309	−	0.936634i	$-0.386077\pi$
$653$	457.504			0.700618			0.350309	+	0.936634i	$0.386077\pi$
$654$		0			0
$655$		−	339.232i		−	0.517911i
$656$	−81.2370			−0.123837
$657$		0			0
$658$		−	28.8496i		−	0.0438444i
$659$			173.014i			0.262540i	0.991347	+	0.131270i	$0.0419055\pi$
$659$			173.014i			0.262540i	−0.991347	+	0.131270i	$0.958095\pi$
$660$		0			0
$661$			582.673i			0.881502i	0.897629	+	0.440751i	$0.145288\pi$
$661$			582.673i			0.881502i	−0.897629	+	0.440751i	$0.854712\pi$
$662$	450.101			0.679911
$663$		0			0
$664$		−	124.821i		−	0.187984i
$665$			15.9227i			0.0239439i
$666$		0			0
$667$	60.5585	−	536.428i	0.0907924	−	0.804240i
$668$	−96.9899			−0.145194
$669$		0			0
$670$	102.606			0.153144
$671$	−608.389			−0.906690
$672$		0			0
$673$	−390.678			−0.580502			−0.290251	−	0.956951i	$-0.593739\pi$
$673$	−390.678			−0.580502			−0.290251	+	0.956951i	$0.593739\pi$
$674$			164.810i			0.244526i
$675$		0			0
$676$	−283.644			−0.419592
$677$		−	952.947i		−	1.40760i	−0.710397	−	0.703801i	$-0.751485\pi$
$677$		−	952.947i		−	1.40760i	0.710397	−	0.703801i	$-0.248515\pi$
$678$		0			0
$679$	227.386			0.334883
$680$	99.3355			0.146082
$681$		0			0
$682$		−	175.289i		−	0.257022i
$683$	−299.732			−0.438847			−0.219423	−	0.975630i	$-0.570418\pi$
$683$	−299.732			−0.438847			−0.219423	+	0.975630i	$0.570418\pi$
$684$		0			0
$685$	−598.067			−0.873090
$686$		−	199.928i		−	0.291441i
$687$		0			0
$688$			152.679i			0.221917i
$689$			199.532i			0.289597i
$690$		0			0
$691$	165.585			0.239631			0.119815	−	0.992796i	$-0.461770\pi$
$691$	165.585			0.239631			0.119815	+	0.992796i	$0.461770\pi$
$692$	222.538			0.321587
$693$		0			0
$694$	−142.432			−0.205234
$695$			536.163i			0.771458i
$696$		0			0
$697$		−	318.984i		−	0.457652i
$698$	−202.906			−0.290696
$699$		0			0
$700$		−	14.7532i		−	0.0210761i
$701$		−	831.750i		−	1.18652i	−0.805011	−	0.593260i	$-0.797841\pi$
$701$		−	831.750i		−	1.18652i	0.805011	−	0.593260i	$-0.202159\pi$
$702$		0			0
$703$	−74.9324			−0.106590
$704$		−	48.3967i		−	0.0687453i
$705$		0			0
$706$	−388.052			−0.549649
$707$			86.8387i			0.122827i
$708$		0			0
$709$		−	835.784i		−	1.17882i	−0.807834	−	0.589410i	$-0.799360\pi$
$709$		−	835.784i		−	1.17882i	0.807834	−	0.589410i	$-0.200640\pi$
$710$		−	76.3077i		−	0.107476i
$711$		0			0
$712$			314.066i			0.441104i
$713$	−52.8634	+	468.265i	−0.0741422	+	0.656753i
$714$		0			0
$715$	−70.5210			−0.0986308
$716$	−525.180			−0.733492
$717$		0			0
$718$			674.397i			0.939272i
$719$	−881.656			−1.22623			−0.613113	−	0.789996i	$-0.710083\pi$
$719$	−881.656			−1.22623			−0.613113	+	0.789996i	$0.710083\pi$
$720$		0			0
$721$	−80.0602			−0.111041
$722$	−477.585			−0.661475
$723$		0			0
$724$			422.335i			0.583335i
$725$	−117.356			−0.161870
$726$		0			0
$727$			531.577i			0.731193i	0.930774	+	0.365596i	$0.119135\pi$
$727$			531.577i			0.731193i	−0.930774	+	0.365596i	$0.880865\pi$
$728$			21.7541i			0.0298820i
$729$		0			0
$730$		−	48.0090i		−	0.0657657i
$731$	−599.505			−0.820116
$732$		0			0
$733$			18.4847i			0.0252178i	0.999921	+	0.0126089i	$0.00401364\pi$
$733$			18.4847i			0.0252178i	−0.999921	+	0.0126089i	$0.995986\pi$
$734$		−	19.8441i		−	0.0270356i
$735$		0			0
$736$	−14.5954	+	129.286i	−0.0198307	+	0.175661i
$737$	196.290			0.266337
$738$		0			0
$739$	−960.371			−1.29955			−0.649777	−	0.760125i	$-0.725138\pi$
$739$	−960.371			−1.29955			−0.649777	+	0.760125i	$0.725138\pi$
$740$	69.4290			0.0938229
$741$		0			0
$742$	−79.8562			−0.107623
$743$			248.966i			0.335082i	0.985865	+	0.167541i	$0.0535826\pi$
$743$			248.966i			0.335082i	−0.985865	+	0.167541i	$0.946417\pi$
$744$		0			0
$745$	450.896			0.605229
$746$		−	149.429i		−	0.200307i
$747$		0			0
$748$	190.034			0.254056
$749$	−175.746			−0.234641
$750$		0			0
$751$		−	24.0757i		−	0.0320582i	−0.999872	−	0.0160291i	$-0.994898\pi$
$751$		−	24.0757i		−	0.0320582i	0.999872	−	0.0160291i	$-0.00510244\pi$
$752$	55.3091			0.0735494
$753$		0			0
$754$	173.044			0.229501
$755$			270.087i			0.357731i
$756$		0			0
$757$		−	914.334i		−	1.20784i	−0.797045	−	0.603920i	$-0.793605\pi$
$757$		−	914.334i		−	1.20784i	0.797045	−	0.603920i	$-0.206395\pi$
$758$			480.809i			0.634312i
$759$		0			0
$760$	−30.5263			−0.0401662
$761$	−313.770			−0.412312			−0.206156	−	0.978519i	$-0.566095\pi$
$761$	−313.770			−0.412312			−0.206156	+	0.978519i	$0.566095\pi$
$762$		0			0
$763$	−220.153			−0.288536
$764$		−	145.642i		−	0.190631i
$765$		0			0
$766$		−	989.662i		−	1.29199i
$767$	−175.007			−0.228170
$768$		0			0
$769$			309.335i			0.402257i	0.979565	+	0.201128i	$0.0644608\pi$
$769$			309.335i			0.402257i	−0.979565	+	0.201128i	$0.935539\pi$
$770$		−	28.2237i		−	0.0366541i
$771$		0			0
$772$	397.099			0.514377
$773$			233.356i			0.301884i	0.988543	+	0.150942i	$0.0482306\pi$
$773$			233.356i			0.301884i	−0.988543	+	0.150942i	$0.951769\pi$
$774$		0			0
$775$	102.443			0.132185
$776$			435.934i			0.561770i
$777$		0			0
$778$			176.518i			0.226887i
$779$			98.0253i			0.125835i
$780$		0			0
$781$		−	145.980i		−	0.186915i
$782$	−507.653	−	57.3101i	−0.649173	−	0.0732866i
$783$		0			0
$784$	187.294			0.238895
$785$	435.803			0.555164
$786$		0			0
$787$		−	690.751i		−	0.877702i	−0.898560	−	0.438851i	$-0.855386\pi$
$787$		−	690.751i		−	0.877702i	0.898560	−	0.438851i	$-0.144614\pi$
$788$	−56.4111			−0.0715876
$789$		0			0
$790$	35.4476			0.0448704
$791$	−52.1290			−0.0659027
$792$		0			0
$793$			524.280i			0.661135i
$794$	211.506			0.266381
$795$		0			0
$796$		−	428.997i		−	0.538941i
$797$		−	91.6732i		−	0.115023i	−0.998345	−	0.0575114i	$-0.981683\pi$
$797$		−	91.6732i		−	0.115023i	0.998345	−	0.0575114i	$-0.0183166\pi$
$798$		0			0
$799$			217.176i			0.271810i
$800$	28.2843			0.0353553
$801$		0			0
$802$			603.439i			0.752418i
$803$		−	91.8434i		−	0.114375i
$804$		0			0
$805$	−8.51166	+	75.3964i	−0.0105735	+	0.0936601i
$806$	−151.055			−0.187414
$807$		0			0
$808$	−166.483			−0.206044
$809$	−1071.73			−1.32476			−0.662382	−	0.749166i	$-0.730454\pi$
$809$	−1071.73			−1.32476			−0.662382	+	0.749166i	$0.730454\pi$
$810$		0			0
$811$	−257.388			−0.317371			−0.158685	−	0.987329i	$-0.550726\pi$
$811$	−257.388			−0.317371			−0.158685	+	0.987329i	$0.550726\pi$
$812$			69.2550i			0.0852894i
$813$		0			0
$814$	132.821			0.163171
$815$			343.935i			0.422006i
$816$		0			0
$817$	184.231			0.225497
$818$	−489.690			−0.598643
$819$		0			0
$820$		−	90.8257i		−	0.110763i
$821$	349.772			0.426032			0.213016	−	0.977049i	$-0.431671\pi$
$821$	349.772			0.426032			0.213016	+	0.977049i	$0.431671\pi$
$822$		0			0
$823$	1438.73			1.74815			0.874076	−	0.485789i	$-0.161468\pi$
$823$	1438.73			1.74815			0.874076	+	0.485789i	$0.161468\pi$
$824$		−	153.488i		−	0.186272i
$825$		0			0
$826$		−	70.0405i		−	0.0847948i
$827$		−	996.093i		−	1.20447i	−0.798320	−	0.602233i	$-0.794278\pi$
$827$		−	996.093i		−	1.20447i	0.798320	−	0.602233i	$-0.205722\pi$
$828$		0			0
$829$	−234.343			−0.282681			−0.141341	−	0.989961i	$-0.545141\pi$
$829$	−234.343			−0.282681			−0.141341	+	0.989961i	$0.545141\pi$
$830$	139.554			0.168138
$831$		0			0
$832$	−41.7059			−0.0501273
$833$			735.424i			0.882862i
$834$		0			0
$835$		−	108.438i		−	0.129866i
$836$	−58.3983			−0.0698544
$837$		0			0
$838$			224.058i			0.267372i
$839$		−	742.848i		−	0.885397i	−0.896670	−	0.442699i	$-0.854021\pi$
$839$		−	742.848i		−	0.885397i	0.896670	−	0.442699i	$-0.145979\pi$
$840$		0			0
$841$	−290.107			−0.344955
$842$		−	25.5361i		−	0.0303280i
$843$		0			0
$844$	480.523			0.569340
$845$		−	317.124i		−	0.375295i
$846$		0			0
$847$			124.521i			0.147014i
$848$		−	153.097i		−	0.180539i
$849$		0			0
$850$			111.060i			0.130659i
$851$	−354.816	−	40.0560i	−0.416940	−	0.0470693i
$852$		0			0
$853$	482.186			0.565283			0.282642	−	0.959226i	$-0.408789\pi$
$853$	482.186			0.565283			0.282642	+	0.959226i	$0.408789\pi$
$854$	−209.826			−0.245697
$855$		0			0
$856$		−	336.933i		−	0.393613i
$857$	−685.909			−0.800361			−0.400180	−	0.916436i	$-0.631053\pi$
$857$	−685.909			−0.800361			−0.400180	+	0.916436i	$0.631053\pi$
$858$		0			0
$859$	670.668			0.780754			0.390377	−	0.920655i	$-0.372345\pi$
$859$	670.668			0.780754			0.390377	+	0.920655i	$0.372345\pi$
$860$	−170.700			−0.198488
$861$		0			0
$862$			214.578i			0.248931i
$863$	−891.818			−1.03339			−0.516696	−	0.856169i	$-0.672838\pi$
$863$	−891.818			−1.03339			−0.516696	+	0.856169i	$0.672838\pi$
$864$		0			0
$865$			248.805i			0.287636i
$866$			884.550i			1.02142i
$867$		0			0
$868$		−	60.4548i		−	0.0696484i
$869$	67.8129			0.0780356
$870$		0			0
$871$		−	169.154i		−	0.194206i
$872$		−	422.067i		−	0.484022i
$873$		0			0
$874$	156.004	+	17.6117i	0.178495	+	0.0201507i
$875$	16.4946			0.0188510
$876$		0			0
$877$	181.676			0.207156			0.103578	−	0.994621i	$-0.466971\pi$
$877$	181.676			0.207156			0.103578	+	0.994621i	$0.466971\pi$
$878$	−385.858			−0.439473
$879$		0			0
$880$	54.1092			0.0614877
$881$			316.887i			0.359691i	0.983695	+	0.179845i	$0.0575597\pi$
$881$			316.887i			0.359691i	−0.983695	+	0.179845i	$0.942440\pi$
$882$		0			0
$883$	−1361.85			−1.54230			−0.771148	−	0.636656i	$-0.780317\pi$
$883$	−1361.85			−1.54230			−0.771148	+	0.636656i	$0.780317\pi$
$884$		−	163.762i		−	0.185251i
$885$		0			0
$886$	7.88463			0.00889914
$887$	−1628.63			−1.83611			−0.918057	−	0.396449i	$-0.870242\pi$
$887$	−1628.63			−1.83611			−0.918057	+	0.396449i	$0.870242\pi$
$888$		0			0
$889$			136.969i			0.154071i
$890$	−351.136			−0.394535
$891$		0			0
$892$	514.805			0.577136
$893$		−	66.7392i		−	0.0747360i
$894$		0			0
$895$		−	587.169i		−	0.656055i
$896$		−	16.6914i		−	0.0186288i
$897$		0			0
$898$	643.936			0.717078
$899$	−480.891			−0.534918
$900$		0			0
$901$	601.146			0.667199
$902$		−	173.754i		−	0.192632i
$903$		0			0
$904$		−	99.9395i		−	0.110552i
$905$	−472.185			−0.521751
$906$		0			0
$907$		−	1551.21i		−	1.71026i	−0.518413	−	0.855130i	$-0.673477\pi$
$907$		−	1551.21i		−	1.71026i	0.518413	−	0.855130i	$-0.326523\pi$
$908$		−	212.802i		−	0.234363i
$909$		0			0
$910$	−24.3218			−0.0267272
$911$		−	1672.31i		−	1.83569i	−0.396937	−	0.917846i	$-0.629927\pi$
$911$		−	1672.31i		−	1.83569i	0.396937	−	0.917846i	$-0.370073\pi$
$912$		0			0
$913$	266.974			0.292414
$914$			845.396i			0.924941i
$915$		0			0
$916$		−	657.014i		−	0.717265i
$917$		−	223.820i		−	0.244079i
$918$		0			0
$919$			1670.00i			1.81719i	0.417673	+	0.908597i	$0.362846\pi$
$919$			1670.00i			1.81719i	−0.417673	+	0.908597i	$0.637154\pi$
$920$	−144.547	−	16.3182i	−0.157116	−	0.0177371i
$921$		0			0
$922$	−481.619			−0.522364
$923$	−125.799			−0.136293
$924$		0			0
$925$			77.6240i			0.0839178i
$926$	666.653			0.719928
$927$		0			0
$928$	−132.773			−0.143074
$929$	877.260			0.944306			0.472153	−	0.881517i	$-0.343477\pi$
$929$	877.260			0.944306			0.472153	+	0.881517i	$0.343477\pi$
$930$		0			0
$931$		−	225.999i		−	0.242749i
$932$	298.426			0.320200
$933$		0			0
$934$			347.445i			0.371997i
$935$			212.464i			0.227234i
$936$		0			0
$937$			726.689i			0.775549i	0.921754	+	0.387774i	$0.126756\pi$
$937$			726.689i			0.775549i	−0.921754	+	0.387774i	$0.873244\pi$
$938$	67.6980			0.0721727
$939$		0			0
$940$			61.8375i			0.0657846i
$941$		−	289.839i		−	0.308011i	−0.988070	−	0.154006i	$-0.950783\pi$
$941$		−	289.839i		−	0.308011i	0.988070	−	0.154006i	$-0.0492174\pi$
$942$		0			0
$943$	−52.4005	+	464.164i	−0.0555679	+	0.492221i
$944$	134.279			0.142244
$945$		0			0
$946$	−326.557			−0.345198
$947$	1420.59			1.50010			0.750050	−	0.661381i	$-0.230030\pi$
$947$	1420.59			1.50010			0.750050	+	0.661381i	$0.230030\pi$
$948$		0			0
$949$	−79.1462			−0.0833996
$950$		−	34.1294i		−	0.0359257i
$951$		0			0
$952$	65.5401			0.0688446
$953$			1465.45i			1.53772i	0.639418	+	0.768860i	$0.279176\pi$
$953$			1465.45i			1.53772i	−0.639418	+	0.768860i	$0.720824\pi$
$954$		0			0
$955$	162.832			0.170505
$956$	686.336			0.717925
$957$		0			0
$958$		−	288.415i		−	0.301060i
$959$	−394.596			−0.411466
$960$		0			0
$961$	−541.215			−0.563179
$962$		−	114.459i		−	0.118980i
$963$		0			0
$964$			697.497i			0.723544i
$965$			443.970i			0.460073i
$966$		0			0
$967$	−1536.47			−1.58890			−0.794451	−	0.607328i	$-0.792241\pi$
$967$	−1536.47			−1.58890			−0.794451	+	0.607328i	$0.792241\pi$
$968$	−238.726			−0.246618
$969$		0			0
$970$	−487.389			−0.502463
$971$			1693.09i			1.74366i	0.489812	+	0.871828i	$0.337065\pi$
$971$			1693.09i			1.74366i	−0.489812	+	0.871828i	$0.662935\pi$
$972$		0			0
$973$			353.752i			0.363569i
$974$	963.724			0.989450
$975$		0			0
$976$		−	402.268i		−	0.412160i
$977$		−	1295.96i		−	1.32647i	−0.748411	−	0.663235i	$-0.769183\pi$
$977$		−	1295.96i		−	1.32647i	0.748411	−	0.663235i	$-0.230817\pi$
$978$		0			0
$979$	−671.740			−0.686150
$980$			209.401i			0.213674i
$981$		0			0
$982$	−466.218			−0.474764
$983$			1692.07i			1.72133i	0.509168	+	0.860667i	$0.329953\pi$
$983$			1692.07i			1.72133i	−0.509168	+	0.860667i	$0.670047\pi$
$984$		0			0
$985$		−	63.0695i		−	0.0640299i
$986$		−	521.343i		−	0.528745i
$987$		0			0
$988$			50.3248i			0.0509360i
$989$	872.361	+	98.4827i	0.882063	+	0.0995780i
$990$		0			0
$991$	45.3192			0.0457308			0.0228654	−	0.999739i	$-0.492721\pi$
$991$	45.3192			0.0457308			0.0228654	+	0.999739i	$0.492721\pi$
$992$	115.901			0.116836
$993$		0			0
$994$		−	50.3467i		−	0.0506506i
$995$	479.633			0.482044
$996$		0			0
$997$	1358.89			1.36298			0.681488	−	0.731830i	$-0.261333\pi$
$997$	1358.89			1.36298			0.681488	+	0.731830i	$0.261333\pi$
$998$	1405.32			1.40813
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.a.91.7		16
3.2	odd	2		230.3.d.a.91.15	✓	16
12.11	even	2		1840.3.k.d.321.3		16
15.2	even	4		1150.3.c.c.1149.14		32
15.8	even	4		1150.3.c.c.1149.19		32
15.14	odd	2		1150.3.d.b.551.1		16
23.22	odd	2	inner	2070.3.c.a.91.2		16
69.68	even	2		230.3.d.a.91.16	yes	16
276.275	odd	2		1840.3.k.d.321.4		16
345.68	odd	4		1150.3.c.c.1149.13		32
345.137	odd	4		1150.3.c.c.1149.20		32
345.344	even	2		1150.3.d.b.551.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.15	✓	16	3.2	odd	2
230.3.d.a.91.16	yes	16	69.68	even	2
1150.3.c.c.1149.13		32	345.68	odd	4
1150.3.c.c.1149.14		32	15.2	even	4
1150.3.c.c.1149.19		32	15.8	even	4
1150.3.c.c.1149.20		32	345.137	odd	4
1150.3.d.b.551.1		16	15.14	odd	2
1150.3.d.b.551.2		16	345.344	even	2
1840.3.k.d.321.3		16	12.11	even	2
1840.3.k.d.321.4		16	276.275	odd	2
2070.3.c.a.91.2		16	23.22	odd	2	inner
2070.3.c.a.91.7		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.47532i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +1.47532i q^{7} -2.82843 q^{8} -3.16228i q^{10} -6.04959i q^{11} -5.21324 q^{13} -2.08642i q^{14} +4.00000 q^{16} +15.7063i q^{17} -4.82663i q^{19} +4.47214i q^{20} +8.55541i q^{22} +(2.58013 - 22.8548i) q^{23} -5.00000 q^{25} +7.37263 q^{26} +2.95065i q^{28} +23.4711 q^{29} -20.4887 q^{31} -5.65685 q^{32} -22.2121i q^{34} -3.29893 q^{35} -15.5248i q^{37} +6.82589i q^{38} -6.32456i q^{40} -20.3093 q^{41} +38.1696i q^{43} -12.0992i q^{44} +(-3.64885 + 32.3216i) q^{46} +13.8273 q^{47} +46.8234 q^{49} +7.07107 q^{50} -10.4265 q^{52} -38.2742i q^{53} +13.5273 q^{55} -4.17285i q^{56} -33.1932 q^{58} +33.5696 q^{59} -100.567i q^{61} +28.9753 q^{62} +8.00000 q^{64} -11.6572i q^{65} +32.4469i q^{67} +31.4126i q^{68} +4.66539 q^{70} +24.1306 q^{71} +15.1818 q^{73} +21.9554i q^{74} -9.65326i q^{76} +8.92511 q^{77} +11.2095i q^{79} +8.94427i q^{80} +28.7216 q^{82} +44.1310i q^{83} -35.1204 q^{85} -53.9800i q^{86} +17.1108i q^{88} -111.039i q^{89} -7.69122i q^{91} +(5.16026 - 45.7096i) q^{92} -19.5547 q^{94} +10.7927 q^{95} -154.126i q^{97} -66.2183 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) \) 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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