LMFDB - Embedded newform 2070.2.d.c.829.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(829,2070)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2070.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.d (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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.4
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	2070.829
Dual form			2070.2.d.c.829.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} -1.85410i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} -1.85410i q^{7} -1.00000i q^{8} +2.23607i q^{10} +5.61803 q^{11} -2.61803i q^{13} +1.85410 q^{14} +1.00000 q^{16} -0.854102i q^{17} +0.145898 q^{19} -2.23607 q^{20} +5.61803i q^{22} +1.00000i q^{23} +5.00000 q^{25} +2.61803 q^{26} +1.85410i q^{28} -9.70820 q^{29} -2.14590 q^{31} +1.00000i q^{32} +0.854102 q^{34} -4.14590i q^{35} -9.70820i q^{37} +0.145898i q^{38} -2.23607i q^{40} +5.61803 q^{41} -11.2361i q^{43} -5.61803 q^{44} -1.00000 q^{46} -1.70820i q^{47} +3.56231 q^{49} +5.00000i q^{50} +2.61803i q^{52} -2.00000i q^{53} +12.5623 q^{55} -1.85410 q^{56} -9.70820i q^{58} -6.00000 q^{59} +2.85410 q^{61} -2.14590i q^{62} -1.00000 q^{64} -5.85410i q^{65} +5.23607i q^{67} +0.854102i q^{68} +4.14590 q^{70} -0.381966 q^{71} +16.4721i q^{73} +9.70820 q^{74} -0.145898 q^{76} -10.4164i q^{77} +7.70820 q^{79} +2.23607 q^{80} +5.61803i q^{82} +7.70820i q^{83} -1.90983i q^{85} +11.2361 q^{86} -5.61803i q^{88} +3.70820 q^{89} -4.85410 q^{91} -1.00000i q^{92} +1.70820 q^{94} +0.326238 q^{95} +13.0344i q^{97} +3.56231i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 18 q^{11} - 6 q^{14} + 4 q^{16} + 14 q^{19} + 20 q^{25} + 6 q^{26} - 12 q^{29} - 22 q^{31} - 10 q^{34} + 18 q^{41} - 18 q^{44} - 4 q^{46} - 26 q^{49} + 10 q^{55} + 6 q^{56} - 24 q^{59} - 2 q^{61} - 4 q^{64} + 30 q^{70} - 6 q^{71} + 12 q^{74} - 14 q^{76} + 4 q^{79} + 36 q^{86} - 12 q^{89} - 6 q^{91} - 20 q^{94} - 30 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 18 * q^11 - 6 * q^14 + 4 * q^16 + 14 * q^19 + 20 * q^25 + 6 * q^26 - 12 * q^29 - 22 * q^31 - 10 * q^34 + 18 * q^41 - 18 * q^44 - 4 * q^46 - 26 * q^49 + 10 * q^55 + 6 * q^56 - 24 * q^59 - 2 * q^61 - 4 * q^64 + 30 * q^70 - 6 * q^71 + 12 * q^74 - 14 * q^76 + 4 * q^79 + 36 * q^86 - 12 * q^89 - 6 * q^91 - 20 * q^94 - 30 * q^95

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.23607	1.00000
$5$	2.23607	1.00000
$6$	0	0
$7$	− 1.85410i	− 0.700785i	−0.936603	−	0.350392i	$-0.886048\pi$
$7$	− 1.85410i	− 0.700785i	0.936603	−	0.350392i	$-0.113952\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	2.23607i	0.707107i
$11$	5.61803	1.69390	0.846950	−	0.531672i	$-0.178436\pi$
$11$	5.61803	1.69390	0.846950	+	0.531672i	$0.178436\pi$
$12$	0	0
$13$	− 2.61803i	− 0.726112i	−0.931767	−	0.363056i	$-0.881733\pi$
$13$	− 2.61803i	− 0.726112i	0.931767	−	0.363056i	$-0.118267\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.00000	0.250000
$17$	− 0.854102i	− 0.207150i	−0.994622	−	0.103575i	$-0.966972\pi$
$17$	− 0.854102i	− 0.207150i	0.994622	−	0.103575i	$-0.0330282\pi$
$18$	0	0
$19$	0.145898	0.0334713	0.0167357	−	0.999860i	$-0.494673\pi$
$19$	0.145898	0.0334713	0.0167357	+	0.999860i	$0.494673\pi$
$20$	−2.23607	−0.500000
$21$	0	0
$22$	5.61803i	1.19777i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	5.00000	1.00000
$26$	2.61803	0.513439
$27$	0	0
$28$	1.85410i	0.350392i
$29$	−9.70820	−1.80277	−0.901384	−	0.433020i	$-0.857448\pi$
$29$	−9.70820	−1.80277	−0.901384	+	0.433020i	$0.857448\pi$
$30$	0	0
$31$	−2.14590	−0.385415	−0.192707	−	0.981256i	$-0.561727\pi$
$31$	−2.14590	−0.385415	−0.192707	+	0.981256i	$0.561727\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0.854102	0.146477
$35$	− 4.14590i	− 0.700785i
$36$	0	0
$37$	− 9.70820i	− 1.59602i	−0.602645	−	0.798009i	$-0.705886\pi$
$37$	− 9.70820i	− 1.59602i	0.602645	−	0.798009i	$-0.294114\pi$
$38$	0.145898i	0.0236678i
$39$	0	0
$40$	− 2.23607i	− 0.353553i
$41$	5.61803	0.877390	0.438695	−	0.898636i	$-0.355441\pi$
$41$	5.61803	0.877390	0.438695	+	0.898636i	$0.355441\pi$
$42$	0	0
$43$	− 11.2361i	− 1.71348i	−0.515745	−	0.856742i	$-0.672485\pi$
$43$	− 11.2361i	− 1.71348i	0.515745	−	0.856742i	$-0.327515\pi$
$44$	−5.61803	−0.846950
$45$	0	0
$46$	−1.00000	−0.147442
$47$	− 1.70820i	− 0.249167i	−0.992209	−	0.124584i	$-0.960241\pi$
$47$	− 1.70820i	− 0.249167i	0.992209	−	0.124584i	$-0.0397595\pi$
$48$	0	0
$49$	3.56231	0.508901
$50$	5.00000i	0.707107i
$51$	0	0
$52$	2.61803i	0.363056i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	12.5623	1.69390
$56$	−1.85410	−0.247765
$57$	0	0
$58$	− 9.70820i	− 1.27475i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.85410	0.365430	0.182715	−	0.983166i	$-0.441511\pi$
$61$	2.85410	0.365430	0.182715	+	0.983166i	$0.441511\pi$
$62$	− 2.14590i	− 0.272529i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 5.85410i	− 0.726112i
$66$	0	0
$67$	5.23607i	0.639688i	0.947470	+	0.319844i	$0.103630\pi$
$67$	5.23607i	0.639688i	−0.947470	+	0.319844i	$0.896370\pi$
$68$	0.854102i	0.103575i
$69$	0	0
$70$	4.14590	0.495530
$71$	−0.381966	−0.0453310	−0.0226655	−	0.999743i	$-0.507215\pi$
$71$	−0.381966	−0.0453310	−0.0226655	+	0.999743i	$0.507215\pi$
$72$	0	0
$73$	16.4721i	1.92792i	0.266051	+	0.963959i	$0.414281\pi$
$73$	16.4721i	1.92792i	−0.266051	+	0.963959i	$0.585719\pi$
$74$	9.70820	1.12856
$75$	0	0
$76$	−0.145898	−0.0167357
$77$	− 10.4164i	− 1.18706i
$78$	0	0
$79$	7.70820	0.867241	0.433620	−	0.901096i	$-0.357236\pi$
$79$	7.70820	0.867241	0.433620	+	0.901096i	$0.357236\pi$
$80$	2.23607	0.250000
$81$	0	0
$82$	5.61803i	0.620408i
$83$	7.70820i	0.846085i	0.906110	+	0.423043i	$0.139038\pi$
$83$	7.70820i	0.846085i	−0.906110	+	0.423043i	$0.860962\pi$
$84$	0	0
$85$	− 1.90983i	− 0.207150i
$86$	11.2361	1.21162
$87$	0	0
$88$	− 5.61803i	− 0.598884i
$89$	3.70820	0.393069	0.196534	−	0.980497i	$-0.437031\pi$
$89$	3.70820	0.393069	0.196534	+	0.980497i	$0.437031\pi$
$90$	0	0
$91$	−4.85410	−0.508848
$92$	− 1.00000i	− 0.104257i
$93$	0	0
$94$	1.70820	0.176188
$95$	0.326238	0.0334713
$96$	0	0
$97$	13.0344i	1.32345i	0.749748	+	0.661724i	$0.230175\pi$
$97$	13.0344i	1.32345i	−0.749748	+	0.661724i	$0.769825\pi$
$98$	3.56231i	0.359847i
$99$	0	0
$100$	−5.00000	−0.500000
$101$	1.52786	0.152028	0.0760141	−	0.997107i	$-0.475781\pi$
$101$	1.52786	0.152028	0.0760141	+	0.997107i	$0.475781\pi$
$102$	0	0
$103$	− 10.8541i	− 1.06949i	−0.845015	−	0.534743i	$-0.820408\pi$
$103$	− 10.8541i	− 1.06949i	0.845015	−	0.534743i	$-0.179592\pi$
$104$	−2.61803	−0.256719
$105$	0	0
$106$	2.00000	0.194257
$107$	− 11.7082i	− 1.13187i	−0.824448	−	0.565937i	$-0.808514\pi$
$107$	− 11.7082i	− 1.13187i	0.824448	−	0.565937i	$-0.191486\pi$
$108$	0	0
$109$	−7.56231	−0.724338	−0.362169	−	0.932113i	$-0.617964\pi$
$109$	−7.56231	−0.724338	−0.362169	+	0.932113i	$0.617964\pi$
$110$	12.5623i	1.19777i
$111$	0	0
$112$	− 1.85410i	− 0.175196i
$113$	− 13.4164i	− 1.26211i	−0.775738	−	0.631055i	$-0.782622\pi$
$113$	− 13.4164i	− 1.26211i	0.775738	−	0.631055i	$-0.217378\pi$
$114$	0	0
$115$	2.23607i	0.208514i
$116$	9.70820	0.901384
$117$	0	0
$118$	− 6.00000i	− 0.552345i
$119$	−1.58359	−0.145168
$120$	0	0
$121$	20.5623	1.86930
$122$	2.85410i	0.258398i
$123$	0	0
$124$	2.14590	0.192707
$125$	11.1803	1.00000
$126$	0	0
$127$	− 9.70820i	− 0.861464i	−0.902480	−	0.430732i	$-0.858255\pi$
$127$	− 9.70820i	− 0.861464i	0.902480	−	0.430732i	$-0.141745\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	5.85410	0.513439
$131$	14.1803	1.23894	0.619471	−	0.785020i	$-0.287347\pi$
$131$	14.1803	1.23894	0.619471	+	0.785020i	$0.287347\pi$
$132$	0	0
$133$	− 0.270510i	− 0.0234562i
$134$	−5.23607	−0.452327
$135$	0	0
$136$	−0.854102	−0.0732386
$137$	13.8541i	1.18364i	0.806072	+	0.591818i	$0.201590\pi$
$137$	13.8541i	1.18364i	−0.806072	+	0.591818i	$0.798410\pi$
$138$	0	0
$139$	−4.29180	−0.364025	−0.182013	−	0.983296i	$-0.558261\pi$
$139$	−4.29180	−0.364025	−0.182013	+	0.983296i	$0.558261\pi$
$140$	4.14590i	0.350392i
$141$	0	0
$142$	− 0.381966i	− 0.0320539i
$143$	− 14.7082i	− 1.22996i
$144$	0	0
$145$	−21.7082	−1.80277
$146$	−16.4721	−1.36324
$147$	0	0
$148$	9.70820i	0.798009i
$149$	−2.61803	−0.214478	−0.107239	−	0.994233i	$-0.534201\pi$
$149$	−2.61803	−0.214478	−0.107239	+	0.994233i	$0.534201\pi$
$150$	0	0
$151$	14.2705	1.16132	0.580659	−	0.814147i	$-0.302795\pi$
$151$	14.2705	1.16132	0.580659	+	0.814147i	$0.302795\pi$
$152$	− 0.145898i	− 0.0118339i
$153$	0	0
$154$	10.4164	0.839378
$155$	−4.79837	−0.385415
$156$	0	0
$157$	2.29180i	0.182905i	0.995809	+	0.0914526i	$0.0291510\pi$
$157$	2.29180i	0.182905i	−0.995809	+	0.0914526i	$0.970849\pi$
$158$	7.70820i	0.613232i
$159$	0	0
$160$	2.23607i	0.176777i
$161$	1.85410	0.146124
$162$	0	0
$163$	22.0344i	1.72587i	0.505314	+	0.862935i	$0.331377\pi$
$163$	22.0344i	1.72587i	−0.505314	+	0.862935i	$0.668623\pi$
$164$	−5.61803	−0.438695
$165$	0	0
$166$	−7.70820	−0.598273
$167$	9.70820i	0.751243i	0.926773	+	0.375622i	$0.122571\pi$
$167$	9.70820i	0.751243i	−0.926773	+	0.375622i	$0.877429\pi$
$168$	0	0
$169$	6.14590	0.472761
$170$	1.90983	0.146477
$171$	0	0
$172$	11.2361i	0.856742i
$173$	16.5623i	1.25921i	0.776916	+	0.629604i	$0.216783\pi$
$173$	16.5623i	1.25921i	−0.776916	+	0.629604i	$0.783217\pi$
$174$	0	0
$175$	− 9.27051i	− 0.700785i
$176$	5.61803	0.423475
$177$	0	0
$178$	3.70820i	0.277942i
$179$	7.52786	0.562659	0.281329	−	0.959611i	$-0.409225\pi$
$179$	7.52786	0.562659	0.281329	+	0.959611i	$0.409225\pi$
$180$	0	0
$181$	15.5623	1.15674	0.578369	−	0.815776i	$-0.303690\pi$
$181$	15.5623	1.15674	0.578369	+	0.815776i	$0.303690\pi$
$182$	− 4.85410i	− 0.359810i
$183$	0	0
$184$	1.00000	0.0737210
$185$	− 21.7082i	− 1.59602i
$186$	0	0
$187$	− 4.79837i	− 0.350892i
$188$	1.70820i	0.124584i
$189$	0	0
$190$	0.326238i	0.0236678i
$191$	−25.4164	−1.83907	−0.919533	−	0.393012i	$-0.871433\pi$
$191$	−25.4164	−1.83907	−0.919533	+	0.393012i	$0.871433\pi$
$192$	0	0
$193$	15.7082i	1.13070i	0.824851	+	0.565351i	$0.191259\pi$
$193$	15.7082i	1.13070i	−0.824851	+	0.565351i	$0.808741\pi$
$194$	−13.0344	−0.935818
$195$	0	0
$196$	−3.56231	−0.254450
$197$	20.5623i	1.46500i	0.680765	+	0.732502i	$0.261647\pi$
$197$	20.5623i	1.46500i	−0.680765	+	0.732502i	$0.738353\pi$
$198$	0	0
$199$	−11.4164	−0.809288	−0.404644	−	0.914474i	$-0.632605\pi$
$199$	−11.4164	−0.809288	−0.404644	+	0.914474i	$0.632605\pi$
$200$	− 5.00000i	− 0.353553i
$201$	0	0
$202$	1.52786i	0.107500i
$203$	18.0000i	1.26335i
$204$	0	0
$205$	12.5623	0.877390
$206$	10.8541	0.756241
$207$	0	0
$208$	− 2.61803i	− 0.181528i
$209$	0.819660	0.0566971
$210$	0	0
$211$	−7.70820	−0.530655	−0.265327	−	0.964158i	$-0.585480\pi$
$211$	−7.70820	−0.530655	−0.265327	+	0.964158i	$0.585480\pi$
$212$	2.00000i	0.137361i
$213$	0	0
$214$	11.7082	0.800356
$215$	− 25.1246i	− 1.71348i
$216$	0	0
$217$	3.97871i	0.270093i
$218$	− 7.56231i	− 0.512184i
$219$	0	0
$220$	−12.5623	−0.846950
$221$	−2.23607	−0.150414
$222$	0	0
$223$	− 22.3607i	− 1.49738i	−0.662919	−	0.748691i	$-0.730683\pi$
$223$	− 22.3607i	− 1.49738i	0.662919	−	0.748691i	$-0.269317\pi$
$224$	1.85410	0.123882
$225$	0	0
$226$	13.4164	0.892446
$227$	− 10.2918i	− 0.683090i	−0.939865	−	0.341545i	$-0.889050\pi$
$227$	− 10.2918i	− 0.683090i	0.939865	−	0.341545i	$-0.110950\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−2.23607	−0.147442
$231$	0	0
$232$	9.70820i	0.637375i
$233$	21.1246i	1.38392i	0.721936	+	0.691960i	$0.243252\pi$
$233$	21.1246i	1.38392i	−0.721936	+	0.691960i	$0.756748\pi$
$234$	0	0
$235$	− 3.81966i	− 0.249167i
$236$	6.00000	0.390567
$237$	0	0
$238$	− 1.58359i	− 0.102649i
$239$	−10.4721	−0.677386	−0.338693	−	0.940897i	$-0.609985\pi$
$239$	−10.4721	−0.677386	−0.338693	+	0.940897i	$0.609985\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	20.5623i	1.32180i
$243$	0	0
$244$	−2.85410	−0.182715
$245$	7.96556	0.508901
$246$	0	0
$247$	− 0.381966i	− 0.0243039i
$248$	2.14590i	0.136265i
$249$	0	0
$250$	11.1803i	0.707107i
$251$	16.7984	1.06030	0.530152	−	0.847903i	$-0.322135\pi$
$251$	16.7984	1.06030	0.530152	+	0.847903i	$0.322135\pi$
$252$	0	0
$253$	5.61803i	0.353203i
$254$	9.70820	0.609147
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.4164i	0.712136i	0.934460	+	0.356068i	$0.115883\pi$
$257$	11.4164i	0.712136i	−0.934460	+	0.356068i	$0.884117\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	5.85410i	0.363056i
$261$	0	0
$262$	14.1803i	0.876064i
$263$	− 18.2705i	− 1.12661i	−0.826250	−	0.563304i	$-0.809530\pi$
$263$	− 18.2705i	− 1.12661i	0.826250	−	0.563304i	$-0.190470\pi$
$264$	0	0
$265$	− 4.47214i	− 0.274721i
$266$	0.270510	0.0165860
$267$	0	0
$268$	− 5.23607i	− 0.319844i
$269$	−1.52786	−0.0931555	−0.0465778	−	0.998915i	$-0.514832\pi$
$269$	−1.52786	−0.0931555	−0.0465778	+	0.998915i	$0.514832\pi$
$270$	0	0
$271$	25.8541	1.57052	0.785262	−	0.619163i	$-0.212528\pi$
$271$	25.8541	1.57052	0.785262	+	0.619163i	$0.212528\pi$
$272$	− 0.854102i	− 0.0517875i
$273$	0	0
$274$	−13.8541	−0.836957
$275$	28.0902	1.69390
$276$	0	0
$277$	7.52786i	0.452306i	0.974092	+	0.226153i	$0.0726149\pi$
$277$	7.52786i	0.452306i	−0.974092	+	0.226153i	$0.927385\pi$
$278$	− 4.29180i	− 0.257405i
$279$	0	0
$280$	−4.14590	−0.247765
$281$	−30.6525	−1.82857	−0.914287	−	0.405068i	$-0.867248\pi$
$281$	−30.6525	−1.82857	−0.914287	+	0.405068i	$0.867248\pi$
$282$	0	0
$283$	− 20.9443i	− 1.24501i	−0.782617	−	0.622504i	$-0.786115\pi$
$283$	− 20.9443i	− 1.24501i	0.782617	−	0.622504i	$-0.213885\pi$
$284$	0.381966	0.0226655
$285$	0	0
$286$	14.7082	0.869714
$287$	− 10.4164i	− 0.614861i
$288$	0	0
$289$	16.2705	0.957089
$290$	− 21.7082i	− 1.27475i
$291$	0	0
$292$	− 16.4721i	− 0.963959i
$293$	− 14.2918i	− 0.834936i	−0.908692	−	0.417468i	$-0.862918\pi$
$293$	− 14.2918i	− 0.834936i	0.908692	−	0.417468i	$-0.137082\pi$
$294$	0	0
$295$	−13.4164	−0.781133
$296$	−9.70820	−0.564278
$297$	0	0
$298$	− 2.61803i	− 0.151659i
$299$	2.61803	0.151405
$300$	0	0
$301$	−20.8328	−1.20078
$302$	14.2705i	0.821176i
$303$	0	0
$304$	0.145898	0.00836783
$305$	6.38197	0.365430
$306$	0	0
$307$	16.8541i	0.961914i	0.876744	+	0.480957i	$0.159711\pi$
$307$	16.8541i	0.961914i	−0.876744	+	0.480957i	$0.840289\pi$
$308$	10.4164i	0.593530i
$309$	0	0
$310$	− 4.79837i	− 0.272529i
$311$	22.4721	1.27428	0.637139	−	0.770749i	$-0.280118\pi$
$311$	22.4721	1.27428	0.637139	+	0.770749i	$0.280118\pi$
$312$	0	0
$313$	− 25.8541i	− 1.46136i	−0.682720	−	0.730680i	$-0.739203\pi$
$313$	− 25.8541i	− 1.46136i	0.682720	−	0.730680i	$-0.260797\pi$
$314$	−2.29180	−0.129334
$315$	0	0
$316$	−7.70820	−0.433620
$317$	7.27051i	0.408353i	0.978934	+	0.204176i	$0.0654516\pi$
$317$	7.27051i	0.408353i	−0.978934	+	0.204176i	$0.934548\pi$
$318$	0	0
$319$	−54.5410	−3.05371
$320$	−2.23607	−0.125000
$321$	0	0
$322$	1.85410i	0.103325i
$323$	− 0.124612i	− 0.00693359i
$324$	0	0
$325$	− 13.0902i	− 0.726112i
$326$	−22.0344	−1.22037
$327$	0	0
$328$	− 5.61803i	− 0.310204i
$329$	−3.16718	−0.174613
$330$	0	0
$331$	−25.1246	−1.38097	−0.690487	−	0.723345i	$-0.742604\pi$
$331$	−25.1246	−1.38097	−0.690487	+	0.723345i	$0.742604\pi$
$332$	− 7.70820i	− 0.423043i
$333$	0	0
$334$	−9.70820	−0.531209
$335$	11.7082i	0.639688i
$336$	0	0
$337$	− 3.38197i	− 0.184227i	−0.995748	−	0.0921137i	$-0.970638\pi$
$337$	− 3.38197i	− 0.184227i	0.995748	−	0.0921137i	$-0.0293623\pi$
$338$	6.14590i	0.334293i
$339$	0	0
$340$	1.90983i	0.103575i
$341$	−12.0557	−0.652854
$342$	0	0
$343$	− 19.5836i	− 1.05741i
$344$	−11.2361	−0.605808
$345$	0	0
$346$	−16.5623	−0.890395
$347$	− 14.5623i	− 0.781746i	−0.920445	−	0.390873i	$-0.872173\pi$
$347$	− 14.5623i	− 0.781746i	0.920445	−	0.390873i	$-0.127827\pi$
$348$	0	0
$349$	−27.7082	−1.48319	−0.741593	−	0.670850i	$-0.765929\pi$
$349$	−27.7082	−1.48319	−0.741593	+	0.670850i	$0.765929\pi$
$350$	9.27051	0.495530
$351$	0	0
$352$	5.61803i	0.299442i
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	0	0
$355$	−0.854102	−0.0453310
$356$	−3.70820	−0.196534
$357$	0	0
$358$	7.52786i	0.397860i
$359$	−13.4164	−0.708091	−0.354045	−	0.935228i	$-0.615194\pi$
$359$	−13.4164	−0.708091	−0.354045	+	0.935228i	$0.615194\pi$
$360$	0	0
$361$	−18.9787	−0.998880
$362$	15.5623i	0.817937i
$363$	0	0
$364$	4.85410	0.254424
$365$	36.8328i	1.92792i
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	1.00000i	0.0521286i
$369$	0	0
$370$	21.7082	1.12856
$371$	−3.70820	−0.192520
$372$	0	0
$373$	20.9443i	1.08445i	0.840232	+	0.542227i	$0.182419\pi$
$373$	20.9443i	1.08445i	−0.840232	+	0.542227i	$0.817581\pi$
$374$	4.79837	0.248118
$375$	0	0
$376$	−1.70820	−0.0880939
$377$	25.4164i	1.30901i
$378$	0	0
$379$	24.2705	1.24669	0.623346	−	0.781946i	$-0.285773\pi$
$379$	24.2705	1.24669	0.623346	+	0.781946i	$0.285773\pi$
$380$	−0.326238	−0.0167357
$381$	0	0
$382$	− 25.4164i	− 1.30042i
$383$	0.583592i	0.0298202i	0.999889	+	0.0149101i	$0.00474620\pi$
$383$	0.583592i	0.0298202i	−0.999889	+	0.0149101i	$0.995254\pi$
$384$	0	0
$385$	− 23.2918i	− 1.18706i
$386$	−15.7082	−0.799527
$387$	0	0
$388$	− 13.0344i	− 0.661724i
$389$	−21.3262	−1.08128	−0.540642	−	0.841253i	$-0.681818\pi$
$389$	−21.3262	−1.08128	−0.540642	+	0.841253i	$0.681818\pi$
$390$	0	0
$391$	0.854102	0.0431938
$392$	− 3.56231i	− 0.179924i
$393$	0	0
$394$	−20.5623	−1.03591
$395$	17.2361	0.867241
$396$	0	0
$397$	15.4377i	0.774796i	0.921913	+	0.387398i	$0.126626\pi$
$397$	15.4377i	0.774796i	−0.921913	+	0.387398i	$0.873374\pi$
$398$	− 11.4164i	− 0.572253i
$399$	0	0
$400$	5.00000	0.250000
$401$	25.5279	1.27480	0.637400	−	0.770533i	$-0.280010\pi$
$401$	25.5279	1.27480	0.637400	+	0.770533i	$0.280010\pi$
$402$	0	0
$403$	5.61803i	0.279854i
$404$	−1.52786	−0.0760141
$405$	0	0
$406$	−18.0000	−0.893325
$407$	− 54.5410i	− 2.70350i
$408$	0	0
$409$	−13.5623	−0.670613	−0.335306	−	0.942109i	$-0.608840\pi$
$409$	−13.5623	−0.670613	−0.335306	+	0.942109i	$0.608840\pi$
$410$	12.5623i	0.620408i
$411$	0	0
$412$	10.8541i	0.534743i
$413$	11.1246i	0.547406i
$414$	0	0
$415$	17.2361i	0.846085i
$416$	2.61803	0.128360
$417$	0	0
$418$	0.819660i	0.0400909i
$419$	−29.8885	−1.46015	−0.730075	−	0.683367i	$-0.760515\pi$
$419$	−29.8885	−1.46015	−0.730075	+	0.683367i	$0.760515\pi$
$420$	0	0
$421$	0.145898	0.00711064	0.00355532	−	0.999994i	$-0.498868\pi$
$421$	0.145898	0.00711064	0.00355532	+	0.999994i	$0.498868\pi$
$422$	− 7.70820i	− 0.375229i
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	− 4.27051i	− 0.207150i
$426$	0	0
$427$	− 5.29180i	− 0.256088i
$428$	11.7082i	0.565937i
$429$	0	0
$430$	25.1246	1.21162
$431$	11.2361	0.541222	0.270611	−	0.962689i	$-0.412774\pi$
$431$	11.2361	0.541222	0.270611	+	0.962689i	$0.412774\pi$
$432$	0	0
$433$	27.3820i	1.31589i	0.753065	+	0.657947i	$0.228575\pi$
$433$	27.3820i	1.31589i	−0.753065	+	0.657947i	$0.771425\pi$
$434$	−3.97871	−0.190984
$435$	0	0
$436$	7.56231	0.362169
$437$	0.145898i	0.00697925i
$438$	0	0
$439$	−33.2705	−1.58791	−0.793957	−	0.607973i	$-0.791983\pi$
$439$	−33.2705	−1.58791	−0.793957	+	0.607973i	$0.791983\pi$
$440$	− 12.5623i	− 0.598884i
$441$	0	0
$442$	− 2.23607i	− 0.106359i
$443$	0.145898i	0.00693182i	0.999994	+	0.00346591i	$0.00110324\pi$
$443$	0.145898i	0.00693182i	−0.999994	+	0.00346591i	$0.998897\pi$
$444$	0	0
$445$	8.29180	0.393069
$446$	22.3607	1.05881
$447$	0	0
$448$	1.85410i	0.0875981i
$449$	−17.5623	−0.828816	−0.414408	−	0.910091i	$-0.636011\pi$
$449$	−17.5623	−0.828816	−0.414408	+	0.910091i	$0.636011\pi$
$450$	0	0
$451$	31.5623	1.48621
$452$	13.4164i	0.631055i
$453$	0	0
$454$	10.2918	0.483018
$455$	−10.8541	−0.508848
$456$	0	0
$457$	1.41641i	0.0662568i	0.999451	+	0.0331284i	$0.0105470\pi$
$457$	1.41641i	0.0662568i	−0.999451	+	0.0331284i	$0.989453\pi$
$458$	10.0000i	0.467269i
$459$	0	0
$460$	− 2.23607i	− 0.104257i
$461$	−37.3050	−1.73746	−0.868732	−	0.495282i	$-0.835065\pi$
$461$	−37.3050	−1.73746	−0.868732	+	0.495282i	$0.835065\pi$
$462$	0	0
$463$	15.7082i	0.730022i	0.931003	+	0.365011i	$0.118935\pi$
$463$	15.7082i	0.730022i	−0.931003	+	0.365011i	$0.881065\pi$
$464$	−9.70820	−0.450692
$465$	0	0
$466$	−21.1246	−0.978579
$467$	− 23.1246i	− 1.07008i	−0.844827	−	0.535040i	$-0.820297\pi$
$467$	− 23.1246i	− 1.07008i	0.844827	−	0.535040i	$-0.179703\pi$
$468$	0	0
$469$	9.70820	0.448283
$470$	3.81966	0.176188
$471$	0	0
$472$	6.00000i	0.276172i
$473$	− 63.1246i	− 2.90247i
$474$	0	0
$475$	0.729490	0.0334713
$476$	1.58359	0.0725838
$477$	0	0
$478$	− 10.4721i	− 0.478984i
$479$	−28.4721	−1.30093	−0.650463	−	0.759538i	$-0.725425\pi$
$479$	−28.4721	−1.30093	−0.650463	+	0.759538i	$0.725425\pi$
$480$	0	0
$481$	−25.4164	−1.15889
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	−20.5623	−0.934650
$485$	29.1459i	1.32345i
$486$	0	0
$487$	− 8.29180i	− 0.375737i	−0.982194	−	0.187869i	$-0.939842\pi$
$487$	− 8.29180i	− 0.375737i	0.982194	−	0.187869i	$-0.0601579\pi$
$488$	− 2.85410i	− 0.129199i
$489$	0	0
$490$	7.96556i	0.359847i
$491$	−26.8328	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$491$	−26.8328	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$492$	0	0
$493$	8.29180i	0.373444i
$494$	0.381966	0.0171855
$495$	0	0
$496$	−2.14590	−0.0963537
$497$	0.708204i	0.0317673i
$498$	0	0
$499$	15.4164	0.690133	0.345067	−	0.938578i	$-0.387856\pi$
$499$	15.4164	0.690133	0.345067	+	0.938578i	$0.387856\pi$
$500$	−11.1803	−0.500000
$501$	0	0
$502$	16.7984i	0.749748i
$503$	10.1459i	0.452383i	0.974083	+	0.226192i	$0.0726276\pi$
$503$	10.1459i	0.452383i	−0.974083	+	0.226192i	$0.927372\pi$
$504$	0	0
$505$	3.41641	0.152028
$506$	−5.61803	−0.249752
$507$	0	0
$508$	9.70820i	0.430732i
$509$	6.65248	0.294866	0.147433	−	0.989072i	$-0.452899\pi$
$509$	6.65248	0.294866	0.147433	+	0.989072i	$0.452899\pi$
$510$	0	0
$511$	30.5410	1.35106
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−11.4164	−0.503556
$515$	− 24.2705i	− 1.06949i
$516$	0	0
$517$	− 9.59675i	− 0.422064i
$518$	− 18.0000i	− 0.790875i
$519$	0	0
$520$	−5.85410	−0.256719
$521$	38.0689	1.66783	0.833914	−	0.551894i	$-0.186095\pi$
$521$	38.0689	1.66783	0.833914	+	0.551894i	$0.186095\pi$
$522$	0	0
$523$	− 4.36068i	− 0.190679i	−0.995445	−	0.0953396i	$-0.969606\pi$
$523$	− 4.36068i	− 0.190679i	0.995445	−	0.0953396i	$-0.0303937\pi$
$524$	−14.1803	−0.619471
$525$	0	0
$526$	18.2705	0.796632
$527$	1.83282i	0.0798387i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	4.47214	0.194257
$531$	0	0
$532$	0.270510i	0.0117281i
$533$	− 14.7082i	− 0.637083i
$534$	0	0
$535$	− 26.1803i	− 1.13187i
$536$	5.23607	0.226164
$537$	0	0
$538$	− 1.52786i	− 0.0658709i
$539$	20.0132	0.862028
$540$	0	0
$541$	−0.583592	−0.0250906	−0.0125453	−	0.999921i	$-0.503993\pi$
$541$	−0.583592	−0.0250906	−0.0125453	+	0.999921i	$0.503993\pi$
$542$	25.8541i	1.11053i
$543$	0	0
$544$	0.854102	0.0366193
$545$	−16.9098	−0.724338
$546$	0	0
$547$	− 7.85410i	− 0.335817i	−0.985803	−	0.167909i	$-0.946299\pi$
$547$	− 7.85410i	− 0.335817i	0.985803	−	0.167909i	$-0.0537013\pi$
$548$	− 13.8541i	− 0.591818i
$549$	0	0
$550$	28.0902i	1.19777i
$551$	−1.41641	−0.0603410
$552$	0	0
$553$	− 14.2918i	− 0.607749i
$554$	−7.52786	−0.319828
$555$	0	0
$556$	4.29180	0.182013
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	−29.4164	−1.24418
$560$	− 4.14590i	− 0.175196i
$561$	0	0
$562$	− 30.6525i	− 1.29300i
$563$	− 15.7082i	− 0.662022i	−0.943627	−	0.331011i	$-0.892610\pi$
$563$	− 15.7082i	− 0.662022i	0.943627	−	0.331011i	$-0.107390\pi$
$564$	0	0
$565$	− 30.0000i	− 1.26211i
$566$	20.9443	0.880353
$567$	0	0
$568$	0.381966i	0.0160269i
$569$	−16.3607	−0.685875	−0.342938	−	0.939358i	$-0.611422\pi$
$569$	−16.3607	−0.685875	−0.342938	+	0.939358i	$0.611422\pi$
$570$	0	0
$571$	8.27051	0.346110	0.173055	−	0.984912i	$-0.444636\pi$
$571$	8.27051	0.346110	0.173055	+	0.984912i	$0.444636\pi$
$572$	14.7082i	0.614981i
$573$	0	0
$574$	10.4164	0.434772
$575$	5.00000i	0.208514i
$576$	0	0
$577$	− 39.7082i	− 1.65307i	−0.562882	−	0.826537i	$-0.690308\pi$
$577$	− 39.7082i	− 1.65307i	0.562882	−	0.826537i	$-0.309692\pi$
$578$	16.2705i	0.676764i
$579$	0	0
$580$	21.7082	0.901384
$581$	14.2918	0.592924
$582$	0	0
$583$	− 11.2361i	− 0.465350i
$584$	16.4721	0.681622
$585$	0	0
$586$	14.2918	0.590389
$587$	8.85410i	0.365448i	0.983164	+	0.182724i	$0.0584915\pi$
$587$	8.85410i	0.365448i	−0.983164	+	0.182724i	$0.941509\pi$
$588$	0	0
$589$	−0.313082	−0.0129003
$590$	− 13.4164i	− 0.552345i
$591$	0	0
$592$	− 9.70820i	− 0.399005i
$593$	6.58359i	0.270356i	0.990821	+	0.135178i	$0.0431606\pi$
$593$	6.58359i	0.270356i	−0.990821	+	0.135178i	$0.956839\pi$
$594$	0	0
$595$	−3.54102	−0.145168
$596$	2.61803	0.107239
$597$	0	0
$598$	2.61803i	0.107059i
$599$	23.6180	0.965007	0.482503	−	0.875894i	$-0.339728\pi$
$599$	23.6180	0.965007	0.482503	+	0.875894i	$0.339728\pi$
$600$	0	0
$601$	−9.85410	−0.401957	−0.200979	−	0.979596i	$-0.564412\pi$
$601$	−9.85410	−0.401957	−0.200979	+	0.979596i	$0.564412\pi$
$602$	− 20.8328i	− 0.849082i
$603$	0	0
$604$	−14.2705	−0.580659
$605$	45.9787	1.86930
$606$	0	0
$607$	21.0557i	0.854626i	0.904104	+	0.427313i	$0.140540\pi$
$607$	21.0557i	0.854626i	−0.904104	+	0.427313i	$0.859460\pi$
$608$	0.145898i	0.00591695i
$609$	0	0
$610$	6.38197i	0.258398i
$611$	−4.47214	−0.180923
$612$	0	0
$613$	0.763932i	0.0308549i	0.999881	+	0.0154275i	$0.00491091\pi$
$613$	0.763932i	0.0308549i	−0.999881	+	0.0154275i	$0.995089\pi$
$614$	−16.8541	−0.680176
$615$	0	0
$616$	−10.4164	−0.419689
$617$	2.56231i	0.103155i	0.998669	+	0.0515773i	$0.0164248\pi$
$617$	2.56231i	0.103155i	−0.998669	+	0.0515773i	$0.983575\pi$
$618$	0	0
$619$	27.8541	1.11955	0.559775	−	0.828644i	$-0.310887\pi$
$619$	27.8541	1.11955	0.559775	+	0.828644i	$0.310887\pi$
$620$	4.79837	0.192707
$621$	0	0
$622$	22.4721i	0.901051i
$623$	− 6.87539i	− 0.275457i
$624$	0	0
$625$	25.0000	1.00000
$626$	25.8541	1.03334
$627$	0	0
$628$	− 2.29180i	− 0.0914526i
$629$	−8.29180	−0.330616
$630$	0	0
$631$	−23.4164	−0.932192	−0.466096	−	0.884734i	$-0.654340\pi$
$631$	−23.4164	−0.932192	−0.466096	+	0.884734i	$0.654340\pi$
$632$	− 7.70820i	− 0.306616i
$633$	0	0
$634$	−7.27051	−0.288749
$635$	− 21.7082i	− 0.861464i
$636$	0	0
$637$	− 9.32624i	− 0.369519i
$638$	− 54.5410i	− 2.15930i
$639$	0	0
$640$	− 2.23607i	− 0.0883883i
$641$	0.652476	0.0257712	0.0128856	−	0.999917i	$-0.495898\pi$
$641$	0.652476	0.0257712	0.0128856	+	0.999917i	$0.495898\pi$
$642$	0	0
$643$	41.0132i	1.61740i	0.588221	+	0.808700i	$0.299829\pi$
$643$	41.0132i	1.61740i	−0.588221	+	0.808700i	$0.700171\pi$
$644$	−1.85410	−0.0730619
$645$	0	0
$646$	0.124612	0.00490279
$647$	13.7082i	0.538925i	0.963011	+	0.269463i	$0.0868460\pi$
$647$	13.7082i	0.538925i	−0.963011	+	0.269463i	$0.913154\pi$
$648$	0	0
$649$	−33.7082	−1.32316
$650$	13.0902	0.513439
$651$	0	0
$652$	− 22.0344i	− 0.862935i
$653$	34.9787i	1.36882i	0.729096	+	0.684411i	$0.239941\pi$
$653$	34.9787i	1.36882i	−0.729096	+	0.684411i	$0.760059\pi$
$654$	0	0
$655$	31.7082	1.23894
$656$	5.61803	0.219347
$657$	0	0
$658$	− 3.16718i	− 0.123470i
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−39.3951	−1.53229	−0.766146	−	0.642666i	$-0.777828\pi$
$661$	−39.3951	−1.53229	−0.766146	+	0.642666i	$0.777828\pi$
$662$	− 25.1246i	− 0.976496i
$663$	0	0
$664$	7.70820	0.299136
$665$	− 0.604878i	− 0.0234562i
$666$	0	0
$667$	− 9.70820i	− 0.375903i
$668$	− 9.70820i	− 0.375622i
$669$	0	0
$670$	−11.7082	−0.452327
$671$	16.0344	0.619003
$672$	0	0
$673$	27.7082i	1.06807i	0.845461	+	0.534036i	$0.179325\pi$
$673$	27.7082i	1.06807i	−0.845461	+	0.534036i	$0.820675\pi$
$674$	3.38197	0.130268
$675$	0	0
$676$	−6.14590	−0.236381
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	0	0
$679$	24.1672	0.927451
$680$	−1.90983	−0.0732386
$681$	0	0
$682$	− 12.0557i	− 0.461638i
$683$	− 10.9787i	− 0.420089i	−0.977692	−	0.210044i	$-0.932639\pi$
$683$	− 10.9787i	− 0.420089i	0.977692	−	0.210044i	$-0.0673609\pi$
$684$	0	0
$685$	30.9787i	1.18364i
$686$	19.5836	0.747705
$687$	0	0
$688$	− 11.2361i	− 0.428371i
$689$	−5.23607	−0.199478
$690$	0	0
$691$	−21.1246	−0.803618	−0.401809	−	0.915723i	$-0.631618\pi$
$691$	−21.1246	−0.803618	−0.401809	+	0.915723i	$0.631618\pi$
$692$	− 16.5623i	− 0.629604i
$693$	0	0
$694$	14.5623	0.552778
$695$	−9.59675	−0.364025
$696$	0	0
$697$	− 4.79837i	− 0.181751i
$698$	− 27.7082i	− 1.04877i
$699$	0	0
$700$	9.27051i	0.350392i
$701$	27.9230	1.05464	0.527318	−	0.849668i	$-0.323198\pi$
$701$	27.9230	1.05464	0.527318	+	0.849668i	$0.323198\pi$
$702$	0	0
$703$	− 1.41641i	− 0.0534208i
$704$	−5.61803	−0.211738
$705$	0	0
$706$	−8.00000	−0.301084
$707$	− 2.83282i	− 0.106539i
$708$	0	0
$709$	−8.56231	−0.321564	−0.160782	−	0.986990i	$-0.551402\pi$
$709$	−8.56231	−0.321564	−0.160782	+	0.986990i	$0.551402\pi$
$710$	− 0.854102i	− 0.0320539i
$711$	0	0
$712$	− 3.70820i	− 0.138971i
$713$	− 2.14590i	− 0.0803645i
$714$	0	0
$715$	− 32.8885i	− 1.22996i
$716$	−7.52786	−0.281329
$717$	0	0
$718$	− 13.4164i	− 0.500696i
$719$	23.4508	0.874569	0.437285	−	0.899323i	$-0.355940\pi$
$719$	23.4508	0.874569	0.437285	+	0.899323i	$0.355940\pi$
$720$	0	0
$721$	−20.1246	−0.749480
$722$	− 18.9787i	− 0.706315i
$723$	0	0
$724$	−15.5623	−0.578369
$725$	−48.5410	−1.80277
$726$	0	0
$727$	40.7426i	1.51106i	0.655113	+	0.755531i	$0.272621\pi$
$727$	40.7426i	1.51106i	−0.655113	+	0.755531i	$0.727379\pi$
$728$	4.85410i	0.179905i
$729$	0	0
$730$	−36.8328	−1.36324
$731$	−9.59675	−0.354949
$732$	0	0
$733$	− 0.111456i	− 0.00411673i	−0.999998	−	0.00205836i	$-0.999345\pi$
$733$	− 0.111456i	− 0.00411673i	0.999998	−	0.00205836i	$-0.000655198\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	29.4164i	1.08357i
$738$	0	0
$739$	−34.5410	−1.27061	−0.635306	−	0.772261i	$-0.719126\pi$
$739$	−34.5410	−1.27061	−0.635306	+	0.772261i	$0.719126\pi$
$740$	21.7082i	0.798009i
$741$	0	0
$742$	− 3.70820i	− 0.136132i
$743$	36.9787i	1.35662i	0.734777	+	0.678309i	$0.237287\pi$
$743$	36.9787i	1.35662i	−0.734777	+	0.678309i	$0.762713\pi$
$744$	0	0
$745$	−5.85410	−0.214478
$746$	−20.9443	−0.766824
$747$	0	0
$748$	4.79837i	0.175446i
$749$	−21.7082	−0.793201
$750$	0	0
$751$	−0.875388	−0.0319434	−0.0159717	−	0.999872i	$-0.505084\pi$
$751$	−0.875388	−0.0319434	−0.0159717	+	0.999872i	$0.505084\pi$
$752$	− 1.70820i	− 0.0622918i
$753$	0	0
$754$	−25.4164	−0.925611
$755$	31.9098	1.16132
$756$	0	0
$757$	29.0132i	1.05450i	0.849710	+	0.527251i	$0.176777\pi$
$757$	29.0132i	1.05450i	−0.849710	+	0.527251i	$0.823223\pi$
$758$	24.2705i	0.881545i
$759$	0	0
$760$	− 0.326238i	− 0.0118339i
$761$	−26.5066	−0.960863	−0.480431	−	0.877032i	$-0.659520\pi$
$761$	−26.5066	−0.960863	−0.480431	+	0.877032i	$0.659520\pi$
$762$	0	0
$763$	14.0213i	0.507605i
$764$	25.4164	0.919533
$765$	0	0
$766$	−0.583592	−0.0210860
$767$	15.7082i	0.567190i
$768$	0	0
$769$	47.7082	1.72040	0.860201	−	0.509955i	$-0.170338\pi$
$769$	47.7082	1.72040	0.860201	+	0.509955i	$0.170338\pi$
$770$	23.2918	0.839378
$771$	0	0
$772$	− 15.7082i	− 0.565351i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	−10.7295	−0.385415
$776$	13.0344	0.467909
$777$	0	0
$778$	− 21.3262i	− 0.764583i
$779$	0.819660	0.0293674
$780$	0	0
$781$	−2.14590	−0.0767863
$782$	0.854102i	0.0305426i
$783$	0	0
$784$	3.56231	0.127225
$785$	5.12461i	0.182905i
$786$	0	0
$787$	4.58359i	0.163387i	0.996657	+	0.0816937i	$0.0260329\pi$
$787$	4.58359i	0.163387i	−0.996657	+	0.0816937i	$0.973967\pi$
$788$	− 20.5623i	− 0.732502i
$789$	0	0
$790$	17.2361i	0.613232i
$791$	−24.8754	−0.884467
$792$	0	0
$793$	− 7.47214i	− 0.265343i
$794$	−15.4377	−0.547863
$795$	0	0
$796$	11.4164	0.404644
$797$	− 45.4164i	− 1.60873i	−0.594134	−	0.804366i	$-0.702505\pi$
$797$	− 45.4164i	− 1.60873i	0.594134	−	0.804366i	$-0.297495\pi$
$798$	0	0
$799$	−1.45898	−0.0516150
$800$	5.00000i	0.176777i
$801$	0	0
$802$	25.5279i	0.901420i
$803$	92.5410i	3.26570i
$804$	0	0
$805$	4.14590	0.146124
$806$	−5.61803	−0.197887
$807$	0	0
$808$	− 1.52786i	− 0.0537501i
$809$	−42.1591	−1.48223	−0.741117	−	0.671376i	$-0.765703\pi$
$809$	−42.1591	−1.48223	−0.741117	+	0.671376i	$0.765703\pi$
$810$	0	0
$811$	5.70820	0.200442	0.100221	−	0.994965i	$-0.468045\pi$
$811$	5.70820	0.200442	0.100221	+	0.994965i	$0.468045\pi$
$812$	− 18.0000i	− 0.631676i
$813$	0	0
$814$	54.5410	1.91166
$815$	49.2705i	1.72587i
$816$	0	0
$817$	− 1.63932i	− 0.0573526i
$818$	− 13.5623i	− 0.474195i
$819$	0	0
$820$	−12.5623	−0.438695
$821$	−14.9443	−0.521559	−0.260779	−	0.965398i	$-0.583980\pi$
$821$	−14.9443	−0.521559	−0.260779	+	0.965398i	$0.583980\pi$
$822$	0	0
$823$	32.8328i	1.14448i	0.820086	+	0.572240i	$0.193925\pi$
$823$	32.8328i	1.14448i	−0.820086	+	0.572240i	$0.806075\pi$
$824$	−10.8541	−0.378121
$825$	0	0
$826$	−11.1246	−0.387075
$827$	32.2492i	1.12142i	0.828014	+	0.560708i	$0.189471\pi$
$827$	32.2492i	1.12142i	−0.828014	+	0.560708i	$0.810529\pi$
$828$	0	0
$829$	34.5410	1.19966	0.599830	−	0.800128i	$-0.295235\pi$
$829$	34.5410	1.19966	0.599830	+	0.800128i	$0.295235\pi$
$830$	−17.2361	−0.598273
$831$	0	0
$832$	2.61803i	0.0907640i
$833$	− 3.04257i	− 0.105419i
$834$	0	0
$835$	21.7082i	0.751243i
$836$	−0.819660	−0.0283485
$837$	0	0
$838$	− 29.8885i	− 1.03248i
$839$	−11.2361	−0.387912	−0.193956	−	0.981010i	$-0.562132\pi$
$839$	−11.2361	−0.387912	−0.193956	+	0.981010i	$0.562132\pi$
$840$	0	0
$841$	65.2492	2.24997
$842$	0.145898i	0.00502798i
$843$	0	0
$844$	7.70820	0.265327
$845$	13.7426	0.472761
$846$	0	0
$847$	− 38.1246i	− 1.30998i
$848$	− 2.00000i	− 0.0686803i
$849$	0	0
$850$	4.27051	0.146477
$851$	9.70820	0.332793
$852$	0	0
$853$	− 0.214782i	− 0.00735399i	−0.999993	−	0.00367699i	$-0.998830\pi$
$853$	− 0.214782i	− 0.00735399i	0.999993	−	0.00367699i	$-0.00117043\pi$
$854$	5.29180	0.181082
$855$	0	0
$856$	−11.7082	−0.400178
$857$	36.0000i	1.22974i	0.788630	+	0.614868i	$0.210791\pi$
$857$	36.0000i	1.22974i	−0.788630	+	0.614868i	$0.789209\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	25.1246i	0.856742i
$861$	0	0
$862$	11.2361i	0.382702i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	37.0344i	1.25921i
$866$	−27.3820	−0.930477
$867$	0	0
$868$	− 3.97871i	− 0.135046i
$869$	43.3050	1.46902
$870$	0	0
$871$	13.7082	0.464485
$872$	7.56231i	0.256092i
$873$	0	0
$874$	−0.145898	−0.00493507
$875$	− 20.7295i	− 0.700785i
$876$	0	0
$877$	25.6869i	0.867386i	0.901061	+	0.433693i	$0.142790\pi$
$877$	25.6869i	0.867386i	−0.901061	+	0.433693i	$0.857210\pi$
$878$	− 33.2705i	− 1.12283i
$879$	0	0
$880$	12.5623	0.423475
$881$	51.7082	1.74209	0.871047	−	0.491200i	$-0.163442\pi$
$881$	51.7082	1.74209	0.871047	+	0.491200i	$0.163442\pi$
$882$	0	0
$883$	51.2705i	1.72539i	0.505725	+	0.862695i	$0.331225\pi$
$883$	51.2705i	1.72539i	−0.505725	+	0.862695i	$0.668775\pi$
$884$	2.23607	0.0752071
$885$	0	0
$886$	−0.145898	−0.00490154
$887$	− 30.5410i	− 1.02547i	−0.858548	−	0.512734i	$-0.828633\pi$
$887$	− 30.5410i	− 1.02547i	0.858548	−	0.512734i	$-0.171367\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	8.29180i	0.277942i
$891$	0	0
$892$	22.3607i	0.748691i
$893$	− 0.249224i	− 0.00833995i
$894$	0	0
$895$	16.8328	0.562659
$896$	−1.85410	−0.0619412
$897$	0	0
$898$	− 17.5623i	− 0.586062i
$899$	20.8328	0.694813
$900$	0	0
$901$	−1.70820	−0.0569085
$902$	31.5623i	1.05091i
$903$	0	0
$904$	−13.4164	−0.446223
$905$	34.7984	1.15674
$906$	0	0
$907$	− 36.5410i	− 1.21332i	−0.794960	−	0.606662i	$-0.792508\pi$
$907$	− 36.5410i	− 1.21332i	0.794960	−	0.606662i	$-0.207492\pi$
$908$	10.2918i	0.341545i
$909$	0	0
$910$	− 10.8541i	− 0.359810i
$911$	22.3607	0.740842	0.370421	−	0.928864i	$-0.379213\pi$
$911$	22.3607	0.740842	0.370421	+	0.928864i	$0.379213\pi$
$912$	0	0
$913$	43.3050i	1.43318i
$914$	−1.41641	−0.0468506
$915$	0	0
$916$	−10.0000	−0.330409
$917$	− 26.2918i	− 0.868232i
$918$	0	0
$919$	41.4164	1.36620	0.683101	−	0.730324i	$-0.260631\pi$
$919$	41.4164	1.36620	0.683101	+	0.730324i	$0.260631\pi$
$920$	2.23607	0.0737210
$921$	0	0
$922$	− 37.3050i	− 1.22857i
$923$	1.00000i	0.0329154i
$924$	0	0
$925$	− 48.5410i	− 1.59602i
$926$	−15.7082	−0.516204
$927$	0	0
$928$	− 9.70820i	− 0.318687i
$929$	−43.3050	−1.42079	−0.710395	−	0.703804i	$-0.751484\pi$
$929$	−43.3050	−1.42079	−0.710395	+	0.703804i	$0.751484\pi$
$930$	0	0
$931$	0.519733	0.0170336
$932$	− 21.1246i	− 0.691960i
$933$	0	0
$934$	23.1246	0.756660
$935$	− 10.7295i	− 0.350892i
$936$	0	0
$937$	− 24.3262i	− 0.794704i	−0.917666	−	0.397352i	$-0.869929\pi$
$937$	− 24.3262i	− 0.794704i	0.917666	−	0.397352i	$-0.130071\pi$
$938$	9.70820i	0.316984i
$939$	0	0
$940$	3.81966i	0.124584i
$941$	−27.2148	−0.887177	−0.443588	−	0.896231i	$-0.646295\pi$
$941$	−27.2148	−0.887177	−0.443588	+	0.896231i	$0.646295\pi$
$942$	0	0
$943$	5.61803i	0.182948i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	63.1246	2.05236
$947$	− 26.3951i	− 0.857726i	−0.903369	−	0.428863i	$-0.858914\pi$
$947$	− 26.3951i	− 0.857726i	0.903369	−	0.428863i	$-0.141086\pi$
$948$	0	0
$949$	43.1246	1.39988
$950$	0.729490i	0.0236678i
$951$	0	0
$952$	1.58359i	0.0513245i
$953$	37.3951i	1.21135i	0.795713	+	0.605673i	$0.207096\pi$
$953$	37.3951i	1.21135i	−0.795713	+	0.605673i	$0.792904\pi$
$954$	0	0
$955$	−56.8328	−1.83907
$956$	10.4721	0.338693
$957$	0	0
$958$	− 28.4721i	− 0.919893i
$959$	25.6869	0.829474
$960$	0	0
$961$	−26.3951	−0.851456
$962$	− 25.4164i	− 0.819458i
$963$	0	0
$964$	2.00000	0.0644157
$965$	35.1246i	1.13070i
$966$	0	0
$967$	29.2361i	0.940169i	0.882622	+	0.470084i	$0.155776\pi$
$967$	29.2361i	0.940169i	−0.882622	+	0.470084i	$0.844224\pi$
$968$	− 20.5623i	− 0.660898i
$969$	0	0
$970$	−29.1459	−0.935818
$971$	−19.1459	−0.614421	−0.307211	−	0.951642i	$-0.599396\pi$
$971$	−19.1459	−0.614421	−0.307211	+	0.951642i	$0.599396\pi$
$972$	0	0
$973$	7.95743i	0.255103i
$974$	8.29180	0.265686
$975$	0	0
$976$	2.85410	0.0913576
$977$	− 1.27051i	− 0.0406472i	−0.999793	−	0.0203236i	$-0.993530\pi$
$977$	− 1.27051i	− 0.0406472i	0.999793	−	0.0203236i	$-0.00646965\pi$
$978$	0	0
$979$	20.8328	0.665820
$980$	−7.96556	−0.254450
$981$	0	0
$982$	− 26.8328i	− 0.856270i
$983$	− 47.3951i	− 1.51167i	−0.654762	−	0.755835i	$-0.727231\pi$
$983$	− 47.3951i	− 1.51167i	0.654762	−	0.755835i	$-0.272769\pi$
$984$	0	0
$985$	45.9787i	1.46500i
$986$	−8.29180	−0.264065
$987$	0	0
$988$	0.381966i	0.0121520i
$989$	11.2361	0.357286
$990$	0	0
$991$	17.7295	0.563196	0.281598	−	0.959532i	$-0.409136\pi$
$991$	17.7295	0.563196	0.281598	+	0.959532i	$0.409136\pi$
$992$	− 2.14590i	− 0.0681323i
$993$	0	0
$994$	−0.708204	−0.0224629
$995$	−25.5279	−0.809288
$996$	0	0
$997$	− 62.7214i	− 1.98641i	−0.116398	−	0.993203i	$-0.537135\pi$
$997$	− 62.7214i	− 1.98641i	0.116398	−	0.993203i	$-0.462865\pi$
$998$	15.4164i	0.487998i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.d.c.829.4		4
3.2	odd	2		230.2.b.a.139.1	✓	4
5.4	even	2	inner	2070.2.d.c.829.2		4
12.11	even	2		1840.2.e.c.369.4		4
15.2	even	4		1150.2.a.n.1.1		2
15.8	even	4		1150.2.a.l.1.2		2
15.14	odd	2		230.2.b.a.139.4	yes	4
60.23	odd	4		9200.2.a.bo.1.1		2
60.47	odd	4		9200.2.a.by.1.2		2
60.59	even	2		1840.2.e.c.369.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.a.139.1	✓	4	3.2	odd	2
230.2.b.a.139.4	yes	4	15.14	odd	2
1150.2.a.l.1.2		2	15.8	even	4
1150.2.a.n.1.1		2	15.2	even	4
1840.2.e.c.369.1		4	60.59	even	2
1840.2.e.c.369.4		4	12.11	even	2
2070.2.d.c.829.2		4	5.4	even	2	inner
2070.2.d.c.829.4		4	1.1	even	1	trivial
9200.2.a.bo.1.1		2	60.23	odd	4
9200.2.a.by.1.2		2	60.47	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} -1.85410i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.23607 q^{5} -1.85410i q^{7} -1.00000i q^{8} +2.23607i q^{10} +5.61803 q^{11} -2.61803i q^{13} +1.85410 q^{14} +1.00000 q^{16} -0.854102i q^{17} +0.145898 q^{19} -2.23607 q^{20} +5.61803i q^{22} +1.00000i q^{23} +5.00000 q^{25} +2.61803 q^{26} +1.85410i q^{28} -9.70820 q^{29} -2.14590 q^{31} +1.00000i q^{32} +0.854102 q^{34} -4.14590i q^{35} -9.70820i q^{37} +0.145898i q^{38} -2.23607i q^{40} +5.61803 q^{41} -11.2361i q^{43} -5.61803 q^{44} -1.00000 q^{46} -1.70820i q^{47} +3.56231 q^{49} +5.00000i q^{50} +2.61803i q^{52} -2.00000i q^{53} +12.5623 q^{55} -1.85410 q^{56} -9.70820i q^{58} -6.00000 q^{59} +2.85410 q^{61} -2.14590i q^{62} -1.00000 q^{64} -5.85410i q^{65} +5.23607i q^{67} +0.854102i q^{68} +4.14590 q^{70} -0.381966 q^{71} +16.4721i q^{73} +9.70820 q^{74} -0.145898 q^{76} -10.4164i q^{77} +7.70820 q^{79} +2.23607 q^{80} +5.61803i q^{82} +7.70820i q^{83} -1.90983i q^{85} +11.2361 q^{86} -5.61803i q^{88} +3.70820 q^{89} -4.85410 q^{91} -1.00000i q^{92} +1.70820 q^{94} +0.326238 q^{95} +13.0344i q^{97} +3.56231i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 18 q^{11} - 6 q^{14} + 4 q^{16} + 14 q^{19} + 20 q^{25} + 6 q^{26} - 12 q^{29} - 22 q^{31} - 10 q^{34} + 18 q^{41} - 18 q^{44} - 4 q^{46} - 26 q^{49} + 10 q^{55} + 6 q^{56} - 24 q^{59} - 2 q^{61} - 4 q^{64} + 30 q^{70} - 6 q^{71} + 12 q^{74} - 14 q^{76} + 4 q^{79} + 36 q^{86} - 12 q^{89} - 6 q^{91} - 20 q^{94} - 30 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 18 * q^11 - 6 * q^14 + 4 * q^16 + 14 * q^19 + 20 * q^25 + 6 * q^26 - 12 * q^29 - 22 * q^31 - 10 * q^34 + 18 * q^41 - 18 * q^44 - 4 * q^46 - 26 * q^49 + 10 * q^55 + 6 * q^56 - 24 * q^59 - 2 * q^61 - 4 * q^64 + 30 * q^70 - 6 * q^71 + 12 * q^74 - 14 * q^76 + 4 * q^79 + 36 * q^86 - 12 * q^89 - 6 * q^91 - 20 * q^94 - 30 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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