LMFDB - Embedded newform 2898.2.g.d.2575.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(2575,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2898.2575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.g (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:	$23.1406465058$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 966)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2575.2
Root			$-1.32288 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2898.2575
Dual form			2898.2.g.d.2575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +2.64575 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +2.64575 q^{7} +1.00000 q^{8} -2.64575 q^{10} +5.00000i q^{13} +2.64575 q^{14} +1.00000 q^{16} -5.29150 q^{17} -5.29150 q^{19} -2.64575 q^{20} +(-4.00000 - 2.64575i) q^{23} +2.00000 q^{25} +5.00000i q^{26} +2.64575 q^{28} +1.00000 q^{29} -4.00000i q^{31} +1.00000 q^{32} -5.29150 q^{34} -7.00000 q^{35} +7.93725i q^{37} -5.29150 q^{38} -2.64575 q^{40} +9.00000i q^{41} -2.64575i q^{43} +(-4.00000 - 2.64575i) q^{46} +13.0000i q^{47} +7.00000 q^{49} +2.00000 q^{50} +5.00000i q^{52} -5.29150i q^{53} +2.64575 q^{56} +1.00000 q^{58} -14.0000i q^{59} -10.5830 q^{61} -4.00000i q^{62} +1.00000 q^{64} -13.2288i q^{65} -5.29150i q^{67} -5.29150 q^{68} -7.00000 q^{70} -6.00000 q^{71} +4.00000i q^{73} +7.93725i q^{74} -5.29150 q^{76} +5.29150i q^{79} -2.64575 q^{80} +9.00000i q^{82} -15.8745 q^{83} +14.0000 q^{85} -2.64575i q^{86} +13.2288i q^{91} +(-4.00000 - 2.64575i) q^{92} +13.0000i q^{94} +14.0000 q^{95} -2.64575 q^{97} +7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} - 16 q^{23} + 8 q^{25} + 4 q^{29} + 4 q^{32} - 28 q^{35} - 16 q^{46} + 28 q^{49} + 8 q^{50} + 4 q^{58} + 4 q^{64} - 28 q^{70} - 24 q^{71} + 56 q^{85} - 16 q^{92} + 56 q^{95} + 28 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 - 16 * q^23 + 8 * q^25 + 4 * q^29 + 4 * q^32 - 28 * q^35 - 16 * q^46 + 28 * q^49 + 8 * q^50 + 4 * q^58 + 4 * q^64 - 28 * q^70 - 24 * q^71 + 56 * q^85 - 16 * q^92 + 56 * q^95 + 28 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2898\mathbb{Z}\right)^\times$.

$n$	$829$	$1289$	$1891$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−2.64575			−1.18322			−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575			−1.18322			−0.591608	+	0.806226i	$0.701507\pi$
$6$		0			0
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$	1.00000			0.353553
$9$		0			0
$10$	−2.64575			−0.836660
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$			5.00000i			1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$			5.00000i			1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	2.64575			0.707107
$15$		0			0
$16$	1.00000			0.250000
$17$	−5.29150			−1.28338			−0.641689	−	0.766965i	$-0.721766\pi$
$17$	−5.29150			−1.28338			−0.641689	+	0.766965i	$0.721766\pi$
$18$		0			0
$19$	−5.29150			−1.21395			−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150			−1.21395			−0.606977	+	0.794719i	$0.707618\pi$
$20$	−2.64575			−0.591608
$21$		0			0
$22$		0			0
$23$	−4.00000	−	2.64575i	−0.834058	−	0.551677i
$23$	−4.00000	−	2.64575i	−0.834058	−	0.551677i
$24$		0			0
$25$	2.00000			0.400000
$26$			5.00000i			0.980581i
$27$		0			0
$28$	2.64575			0.500000
$29$	1.00000			0.185695			0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	−5.29150			−0.907485
$35$	−7.00000			−1.18322
$36$		0			0
$37$			7.93725i			1.30488i	0.757842	+	0.652438i	$0.226254\pi$
$37$			7.93725i			1.30488i	−0.757842	+	0.652438i	$0.773746\pi$
$38$	−5.29150			−0.858395
$39$		0			0
$40$	−2.64575			−0.418330
$41$			9.00000i			1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$41$			9.00000i			1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$42$		0			0
$43$		−	2.64575i		−	0.403473i	−0.979440	−	0.201737i	$-0.935341\pi$
$43$		−	2.64575i		−	0.403473i	0.979440	−	0.201737i	$-0.0646585\pi$
$44$		0			0
$45$		0			0
$46$	−4.00000	−	2.64575i	−0.589768	−	0.390095i
$47$			13.0000i			1.89624i	0.317905	+	0.948122i	$0.397021\pi$
$47$			13.0000i			1.89624i	−0.317905	+	0.948122i	$0.602979\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	2.00000			0.282843
$51$		0			0
$52$			5.00000i			0.693375i
$53$		−	5.29150i		−	0.726844i	−0.931625	−	0.363422i	$-0.881608\pi$
$53$		−	5.29150i		−	0.726844i	0.931625	−	0.363422i	$-0.118392\pi$
$54$		0			0
$55$		0			0
$56$	2.64575			0.353553
$57$		0			0
$58$	1.00000			0.131306
$59$		−	14.0000i		−	1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$		−	14.0000i		−	1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$		0			0
$61$	−10.5830			−1.35501			−0.677507	−	0.735516i	$-0.736940\pi$
$61$	−10.5830			−1.35501			−0.677507	+	0.735516i	$0.736940\pi$
$62$		−	4.00000i		−	0.508001i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	13.2288i		−	1.64083i
$66$		0			0
$67$		−	5.29150i		−	0.646460i	−0.946320	−	0.323230i	$-0.895231\pi$
$67$		−	5.29150i		−	0.646460i	0.946320	−	0.323230i	$-0.104769\pi$
$68$	−5.29150			−0.641689
$69$		0			0
$70$	−7.00000			−0.836660
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$			7.93725i			0.922687i
$75$		0			0
$76$	−5.29150			−0.606977
$77$		0			0
$78$		0			0
$79$			5.29150i			0.595341i	0.954669	+	0.297670i	$0.0962096\pi$
$79$			5.29150i			0.595341i	−0.954669	+	0.297670i	$0.903790\pi$
$80$	−2.64575			−0.295804
$81$		0			0
$82$			9.00000i			0.993884i
$83$	−15.8745			−1.74245			−0.871227	−	0.490881i	$-0.836675\pi$
$83$	−15.8745			−1.74245			−0.871227	+	0.490881i	$0.836675\pi$
$84$		0			0
$85$	14.0000			1.51851
$86$		−	2.64575i		−	0.285299i
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$			13.2288i			1.38675i
$92$	−4.00000	−	2.64575i	−0.417029	−	0.275839i
$93$		0			0
$94$			13.0000i			1.34085i
$95$	14.0000			1.43637
$96$		0			0
$97$	−2.64575			−0.268635			−0.134318	−	0.990938i	$-0.542884\pi$
$97$	−2.64575			−0.268635			−0.134318	+	0.990938i	$0.542884\pi$
$98$	7.00000			0.707107
$99$		0			0
$100$	2.00000			0.200000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−2.64575			−0.260694			−0.130347	−	0.991468i	$-0.541609\pi$
$103$	−2.64575			−0.260694			−0.130347	+	0.991468i	$0.541609\pi$
$104$			5.00000i			0.490290i
$105$		0			0
$106$		−	5.29150i		−	0.513956i
$107$			5.29150i			0.511549i	0.966736	+	0.255774i	$0.0823304\pi$
$107$			5.29150i			0.511549i	−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$			18.5203i			1.77392i	0.461847	+	0.886960i	$0.347187\pi$
$109$			18.5203i			1.77392i	−0.461847	+	0.886960i	$0.652813\pi$
$110$		0			0
$111$		0			0
$112$	2.64575			0.250000
$113$		−	2.64575i		−	0.248891i	−0.992226	−	0.124446i	$-0.960285\pi$
$113$		−	2.64575i		−	0.248891i	0.992226	−	0.124446i	$-0.0397153\pi$
$114$		0			0
$115$	10.5830	+	7.00000i	0.986870	+	0.652753i
$116$	1.00000			0.0928477
$117$		0			0
$118$		−	14.0000i		−	1.28880i
$119$	−14.0000			−1.28338
$120$		0			0
$121$	11.0000			1.00000
$122$	−10.5830			−0.958140
$123$		0			0
$124$		−	4.00000i		−	0.359211i
$125$	7.93725			0.709930
$126$		0			0
$127$	−13.0000			−1.15356			−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000			−1.15356			−0.576782	+	0.816898i	$0.695692\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	13.2288i		−	1.16024i
$131$			22.0000i			1.92215i	0.276289	+	0.961074i	$0.410895\pi$
$131$			22.0000i			1.92215i	−0.276289	+	0.961074i	$0.589105\pi$
$132$		0			0
$133$	−14.0000			−1.21395
$134$		−	5.29150i		−	0.457116i
$135$		0			0
$136$	−5.29150			−0.453743
$137$		−	7.93725i		−	0.678125i	−0.940764	−	0.339063i	$-0.889890\pi$
$137$		−	7.93725i		−	0.678125i	0.940764	−	0.339063i	$-0.110110\pi$
$138$		0			0
$139$		−	5.00000i		−	0.424094i	−0.977259	−	0.212047i	$-0.931987\pi$
$139$		−	5.00000i		−	0.424094i	0.977259	−	0.212047i	$-0.0680131\pi$
$140$	−7.00000			−0.591608
$141$		0			0
$142$	−6.00000			−0.503509
$143$		0			0
$144$		0			0
$145$	−2.64575			−0.219718
$146$			4.00000i			0.331042i
$147$		0			0
$148$			7.93725i			0.652438i
$149$			15.8745i			1.30049i	0.759724	+	0.650245i	$0.225334\pi$
$149$			15.8745i			1.30049i	−0.759724	+	0.650245i	$0.774666\pi$
$150$		0			0
$151$	−5.00000			−0.406894			−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000			−0.406894			−0.203447	+	0.979086i	$0.565214\pi$
$152$	−5.29150			−0.429198
$153$		0			0
$154$		0			0
$155$			10.5830i			0.850047i
$156$		0			0
$157$	−10.5830			−0.844616			−0.422308	−	0.906452i	$-0.638780\pi$
$157$	−10.5830			−0.844616			−0.422308	+	0.906452i	$0.638780\pi$
$158$			5.29150i			0.420969i
$159$		0			0
$160$	−2.64575			−0.209165
$161$	−10.5830	−	7.00000i	−0.834058	−	0.551677i
$162$		0			0
$163$	−18.0000			−1.40987			−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000			−1.40987			−0.704934	+	0.709273i	$0.749024\pi$
$164$			9.00000i			0.702782i
$165$		0			0
$166$	−15.8745			−1.23210
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$	14.0000			1.07375
$171$		0			0
$172$		−	2.64575i		−	0.201737i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	5.29150			0.400000
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.00000			−0.672692			−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000			−0.672692			−0.336346	+	0.941739i	$0.609191\pi$
$180$		0			0
$181$	15.8745			1.17994			0.589971	−	0.807424i	$-0.299139\pi$
$181$	15.8745			1.17994			0.589971	+	0.807424i	$0.299139\pi$
$182$			13.2288i			0.980581i
$183$		0			0
$184$	−4.00000	−	2.64575i	−0.294884	−	0.195047i
$185$		−	21.0000i		−	1.54395i
$186$		0			0
$187$		0			0
$188$			13.0000i			0.948122i
$189$		0			0
$190$	14.0000			1.01567
$191$			15.8745i			1.14864i	0.818631	+	0.574320i	$0.194733\pi$
$191$			15.8745i			1.14864i	−0.818631	+	0.574320i	$0.805267\pi$
$192$		0			0
$193$	19.0000			1.36765			0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000			1.36765			0.683825	+	0.729646i	$0.260315\pi$
$194$	−2.64575			−0.189954
$195$		0			0
$196$	7.00000			0.500000
$197$	1.00000			0.0712470			0.0356235	−	0.999365i	$-0.488658\pi$
$197$	1.00000			0.0712470			0.0356235	+	0.999365i	$0.488658\pi$
$198$		0			0
$199$	13.2288			0.937762			0.468881	−	0.883261i	$-0.344657\pi$
$199$	13.2288			0.937762			0.468881	+	0.883261i	$0.344657\pi$
$200$	2.00000			0.141421
$201$		0			0
$202$		0			0
$203$	2.64575			0.185695
$204$		0			0
$205$		−	23.8118i		−	1.66309i
$206$	−2.64575			−0.184338
$207$		0			0
$208$			5.00000i			0.346688i
$209$		0			0
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		−	5.29150i		−	0.363422i
$213$		0			0
$214$			5.29150i			0.361720i
$215$			7.00000i			0.477396i
$216$		0			0
$217$		−	10.5830i		−	0.718421i
$218$			18.5203i			1.25435i
$219$		0			0
$220$		0			0
$221$		−	26.4575i		−	1.77972i
$222$		0			0
$223$		−	28.0000i		−	1.87502i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$		−	28.0000i		−	1.87502i	0.347960	−	0.937509i	$-0.386874\pi$
$224$	2.64575			0.176777
$225$		0			0
$226$		−	2.64575i		−	0.175993i
$227$	−7.93725			−0.526814			−0.263407	−	0.964685i	$-0.584846\pi$
$227$	−7.93725			−0.526814			−0.263407	+	0.964685i	$0.584846\pi$
$228$		0			0
$229$	−10.5830			−0.699345			−0.349672	−	0.936872i	$-0.613707\pi$
$229$	−10.5830			−0.699345			−0.349672	+	0.936872i	$0.613707\pi$
$230$	10.5830	+	7.00000i	0.697823	+	0.461566i
$231$		0			0
$232$	1.00000			0.0656532
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		−	34.3948i		−	2.24367i
$236$		−	14.0000i		−	0.911322i
$237$		0			0
$238$	−14.0000			−0.907485
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	23.8118			1.53385			0.766925	−	0.641736i	$-0.221786\pi$
$241$	23.8118			1.53385			0.766925	+	0.641736i	$0.221786\pi$
$242$	11.0000			0.707107
$243$		0			0
$244$	−10.5830			−0.677507
$245$	−18.5203			−1.18322
$246$		0			0
$247$		−	26.4575i		−	1.68345i
$248$		−	4.00000i		−	0.254000i
$249$		0			0
$250$	7.93725			0.501996
$251$	18.5203			1.16899			0.584494	−	0.811398i	$-0.301293\pi$
$251$	18.5203			1.16899			0.584494	+	0.811398i	$0.301293\pi$
$252$		0			0
$253$		0			0
$254$	−13.0000			−0.815693
$255$		0			0
$256$	1.00000			0.0625000
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$			21.0000i			1.30488i
$260$		−	13.2288i		−	0.820413i
$261$		0			0
$262$			22.0000i			1.35916i
$263$			23.8118i			1.46830i	0.678989	+	0.734148i	$0.262418\pi$
$263$			23.8118i			1.46830i	−0.678989	+	0.734148i	$0.737582\pi$
$264$		0			0
$265$			14.0000i			0.860013i
$266$	−14.0000			−0.858395
$267$		0			0
$268$		−	5.29150i		−	0.323230i
$269$			10.0000i			0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$			10.0000i			0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$		0			0
$271$		−	14.0000i		−	0.850439i	−0.905090	−	0.425220i	$-0.860197\pi$
$271$		−	14.0000i		−	0.850439i	0.905090	−	0.425220i	$-0.139803\pi$
$272$	−5.29150			−0.320844
$273$		0			0
$274$		−	7.93725i		−	0.479507i
$275$		0			0
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		−	5.00000i		−	0.299880i
$279$		0			0
$280$	−7.00000			−0.418330
$281$		−	7.93725i		−	0.473497i	−0.971571	−	0.236748i	$-0.923918\pi$
$281$		−	7.93725i		−	0.473497i	0.971571	−	0.236748i	$-0.0760817\pi$
$282$		0			0
$283$	−10.5830			−0.629094			−0.314547	−	0.949242i	$-0.601853\pi$
$283$	−10.5830			−0.629094			−0.314547	+	0.949242i	$0.601853\pi$
$284$	−6.00000			−0.356034
$285$		0			0
$286$		0			0
$287$			23.8118i			1.40556i
$288$		0			0
$289$	11.0000			0.647059
$290$	−2.64575			−0.155364
$291$		0			0
$292$			4.00000i			0.234082i
$293$	21.1660			1.23653			0.618266	−	0.785969i	$-0.287836\pi$
$293$	21.1660			1.23653			0.618266	+	0.785969i	$0.287836\pi$
$294$		0			0
$295$			37.0405i			2.15658i
$296$			7.93725i			0.461344i
$297$		0			0
$298$			15.8745i			0.919586i
$299$	13.2288	−	20.0000i	0.765039	−	1.15663i
$300$		0			0
$301$		−	7.00000i		−	0.403473i
$302$	−5.00000			−0.287718
$303$		0			0
$304$	−5.29150			−0.303488
$305$	28.0000			1.60328
$306$		0			0
$307$		−	21.0000i		−	1.19853i	−0.800549	−	0.599267i	$-0.795459\pi$
$307$		−	21.0000i		−	1.19853i	0.800549	−	0.599267i	$-0.204541\pi$
$308$		0			0
$309$		0			0
$310$			10.5830i			0.601074i
$311$		−	24.0000i		−	1.36092i	−0.732787	−	0.680458i	$-0.761781\pi$
$311$		−	24.0000i		−	1.36092i	0.732787	−	0.680458i	$-0.238219\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	−10.5830			−0.597234
$315$		0			0
$316$			5.29150i			0.297670i
$317$	−27.0000			−1.51647			−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000			−1.51647			−0.758236	+	0.651981i	$0.773938\pi$
$318$		0			0
$319$		0			0
$320$	−2.64575			−0.147902
$321$		0			0
$322$	−10.5830	−	7.00000i	−0.589768	−	0.390095i
$323$	28.0000			1.55796
$324$		0			0
$325$			10.0000i			0.554700i
$326$	−18.0000			−0.996928
$327$		0			0
$328$			9.00000i			0.496942i
$329$			34.3948i			1.89624i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$	−15.8745			−0.871227
$333$		0			0
$334$		0			0
$335$			14.0000i			0.764902i
$336$		0			0
$337$		−	26.4575i		−	1.44123i	−0.693334	−	0.720616i	$-0.743859\pi$
$337$		−	26.4575i		−	1.44123i	0.693334	−	0.720616i	$-0.256141\pi$
$338$	−12.0000			−0.652714
$339$		0			0
$340$	14.0000			0.759257
$341$		0			0
$342$		0			0
$343$	18.5203			1.00000
$344$		−	2.64575i		−	0.142649i
$345$		0			0
$346$		0			0
$347$	11.0000			0.590511			0.295255	−	0.955418i	$-0.404595\pi$
$347$	11.0000			0.590511			0.295255	+	0.955418i	$0.404595\pi$
$348$		0			0
$349$		−	30.0000i		−	1.60586i	−0.596071	−	0.802932i	$-0.703272\pi$
$349$		−	30.0000i		−	1.60586i	0.596071	−	0.802932i	$-0.296728\pi$
$350$	5.29150			0.282843
$351$		0			0
$352$		0			0
$353$			25.0000i			1.33062i	0.746569	+	0.665308i	$0.231700\pi$
$353$			25.0000i			1.33062i	−0.746569	+	0.665308i	$0.768300\pi$
$354$		0			0
$355$	15.8745			0.842531
$356$		0			0
$357$		0			0
$358$	−9.00000			−0.475665
$359$		−	23.8118i		−	1.25674i	−0.777916	−	0.628368i	$-0.783723\pi$
$359$		−	23.8118i		−	1.25674i	0.777916	−	0.628368i	$-0.216277\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$	15.8745			0.834346
$363$		0			0
$364$			13.2288i			0.693375i
$365$		−	10.5830i		−	0.553940i
$366$		0			0
$367$	18.5203			0.966750			0.483375	−	0.875413i	$-0.339411\pi$
$367$	18.5203			0.966750			0.483375	+	0.875413i	$0.339411\pi$
$368$	−4.00000	−	2.64575i	−0.208514	−	0.137919i
$369$		0			0
$370$		−	21.0000i		−	1.09174i
$371$		−	14.0000i		−	0.726844i
$372$		0			0
$373$			21.1660i			1.09593i	0.836500	+	0.547967i	$0.184598\pi$
$373$			21.1660i			1.09593i	−0.836500	+	0.547967i	$0.815402\pi$
$374$		0			0
$375$		0			0
$376$			13.0000i			0.670424i
$377$			5.00000i			0.257513i
$378$		0			0
$379$		−	23.8118i		−	1.22313i	−0.791195	−	0.611564i	$-0.790541\pi$
$379$		−	23.8118i		−	1.22313i	0.791195	−	0.611564i	$-0.209459\pi$
$380$	14.0000			0.718185
$381$		0			0
$382$			15.8745i			0.812210i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	19.0000			0.967075
$387$		0			0
$388$	−2.64575			−0.134318
$389$		−	21.1660i		−	1.07316i	−0.843850	−	0.536580i	$-0.819716\pi$
$389$		−	21.1660i		−	1.07316i	0.843850	−	0.536580i	$-0.180284\pi$
$390$		0			0
$391$	21.1660	+	14.0000i	1.07041	+	0.708010i
$392$	7.00000			0.353553
$393$		0			0
$394$	1.00000			0.0503793
$395$		−	14.0000i		−	0.704416i
$396$		0			0
$397$		−	34.0000i		−	1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$		−	34.0000i		−	1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	13.2288			0.663098
$399$		0			0
$400$	2.00000			0.100000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	20.0000			0.996271
$404$		0			0
$405$		0			0
$406$	2.64575			0.131306
$407$		0			0
$408$		0			0
$409$			18.0000i			0.890043i	0.895520	+	0.445021i	$0.146804\pi$
$409$			18.0000i			0.890043i	−0.895520	+	0.445021i	$0.853196\pi$
$410$		−	23.8118i		−	1.17598i
$411$		0			0
$412$	−2.64575			−0.130347
$413$		−	37.0405i		−	1.82264i
$414$		0			0
$415$	42.0000			2.06170
$416$			5.00000i			0.245145i
$417$		0			0
$418$		0			0
$419$	15.8745			0.775520			0.387760	−	0.921760i	$-0.373249\pi$
$419$	15.8745			0.775520			0.387760	+	0.921760i	$0.373249\pi$
$420$		0			0
$421$			29.1033i			1.41841i	0.705004	+	0.709203i	$0.250945\pi$
$421$			29.1033i			1.41841i	−0.705004	+	0.709203i	$0.749055\pi$
$422$	−16.0000			−0.778868
$423$		0			0
$424$		−	5.29150i		−	0.256978i
$425$	−10.5830			−0.513351
$426$		0			0
$427$	−28.0000			−1.35501
$428$			5.29150i			0.255774i
$429$		0			0
$430$			7.00000i			0.337570i
$431$		−	13.2288i		−	0.637207i	−0.947888	−	0.318603i	$-0.896786\pi$
$431$		−	13.2288i		−	0.637207i	0.947888	−	0.318603i	$-0.103214\pi$
$432$		0			0
$433$	13.2288			0.635733			0.317867	−	0.948135i	$-0.397034\pi$
$433$	13.2288			0.635733			0.317867	+	0.948135i	$0.397034\pi$
$434$		−	10.5830i		−	0.508001i
$435$		0			0
$436$			18.5203i			0.886960i
$437$	21.1660	+	14.0000i	1.01251	+	0.669711i
$438$		0			0
$439$		−	6.00000i		−	0.286364i	−0.989696	−	0.143182i	$-0.954267\pi$
$439$		−	6.00000i		−	0.286364i	0.989696	−	0.143182i	$-0.0457335\pi$
$440$		0			0
$441$		0			0
$442$		−	26.4575i		−	1.25846i
$443$	−3.00000			−0.142534			−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000			−0.142534			−0.0712672	+	0.997457i	$0.522704\pi$
$444$		0			0
$445$		0			0
$446$		−	28.0000i		−	1.32584i
$447$		0			0
$448$	2.64575			0.125000
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$		0			0
$452$		−	2.64575i		−	0.124446i
$453$		0			0
$454$	−7.93725			−0.372514
$455$		−	35.0000i		−	1.64083i
$456$		0			0
$457$			5.29150i			0.247526i	0.992312	+	0.123763i	$0.0394963\pi$
$457$			5.29150i			0.247526i	−0.992312	+	0.123763i	$0.960504\pi$
$458$	−10.5830			−0.494511
$459$		0			0
$460$	10.5830	+	7.00000i	0.493435	+	0.326377i
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$		0			0
$463$	19.0000			0.883005			0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000			0.883005			0.441502	+	0.897260i	$0.354446\pi$
$464$	1.00000			0.0464238
$465$		0			0
$466$	18.0000			0.833834
$467$	−13.2288			−0.612154			−0.306077	−	0.952007i	$-0.599016\pi$
$467$	−13.2288			−0.612154			−0.306077	+	0.952007i	$0.599016\pi$
$468$		0			0
$469$		−	14.0000i		−	0.646460i
$470$		−	34.3948i		−	1.58651i
$471$		0			0
$472$		−	14.0000i		−	0.644402i
$473$		0			0
$474$		0			0
$475$	−10.5830			−0.485582
$476$	−14.0000			−0.641689
$477$		0			0
$478$	20.0000			0.914779
$479$	5.29150			0.241775			0.120887	−	0.992666i	$-0.461426\pi$
$479$	5.29150			0.241775			0.120887	+	0.992666i	$0.461426\pi$
$480$		0			0
$481$	−39.6863			−1.80954
$482$	23.8118			1.08460
$483$		0			0
$484$	11.0000			0.500000
$485$	7.00000			0.317854
$486$		0			0
$487$	−23.0000			−1.04223			−0.521115	−	0.853487i	$-0.674484\pi$
$487$	−23.0000			−1.04223			−0.521115	+	0.853487i	$0.674484\pi$
$488$	−10.5830			−0.479070
$489$		0			0
$490$	−18.5203			−0.836660
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$	−5.29150			−0.238317
$494$		−	26.4575i		−	1.19038i
$495$		0			0
$496$		−	4.00000i		−	0.179605i
$497$	−15.8745			−0.712069
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$	7.93725			0.354965
$501$		0			0
$502$	18.5203			0.826600
$503$	26.4575			1.17968			0.589841	−	0.807519i	$-0.299190\pi$
$503$	26.4575			1.17968			0.589841	+	0.807519i	$0.299190\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−13.0000			−0.576782
$509$			20.0000i			0.886484i	0.896402	+	0.443242i	$0.146172\pi$
$509$			20.0000i			0.886484i	−0.896402	+	0.443242i	$0.853828\pi$
$510$		0			0
$511$			10.5830i			0.468165i
$512$	1.00000			0.0441942
$513$		0			0
$514$			6.00000i			0.264649i
$515$	7.00000			0.308457
$516$		0			0
$517$		0			0
$518$			21.0000i			0.922687i
$519$		0			0
$520$		−	13.2288i		−	0.580119i
$521$	15.8745			0.695475			0.347737	−	0.937592i	$-0.386950\pi$
$521$	15.8745			0.695475			0.347737	+	0.937592i	$0.386950\pi$
$522$		0			0
$523$	21.1660			0.925525			0.462763	−	0.886482i	$-0.346858\pi$
$523$	21.1660			0.925525			0.462763	+	0.886482i	$0.346858\pi$
$524$			22.0000i			0.961074i
$525$		0			0
$526$			23.8118i			1.03824i
$527$			21.1660i			0.922006i
$528$		0			0
$529$	9.00000	+	21.1660i	0.391304	+	0.920261i
$530$			14.0000i			0.608121i
$531$		0			0
$532$	−14.0000			−0.606977
$533$	−45.0000			−1.94917
$534$		0			0
$535$		−	14.0000i		−	0.605273i
$536$		−	5.29150i		−	0.228558i
$537$		0			0
$538$			10.0000i			0.431131i
$539$		0			0
$540$		0			0
$541$	−24.0000			−1.03184			−0.515920	−	0.856637i	$-0.672550\pi$
$541$	−24.0000			−1.03184			−0.515920	+	0.856637i	$0.672550\pi$
$542$		−	14.0000i		−	0.601351i
$543$		0			0
$544$	−5.29150			−0.226871
$545$		−	49.0000i		−	2.09893i
$546$		0			0
$547$	−34.0000			−1.45374			−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000			−1.45374			−0.726868	+	0.686778i	$0.759025\pi$
$548$		−	7.93725i		−	0.339063i
$549$		0			0
$550$		0			0
$551$	−5.29150			−0.225426
$552$		0			0
$553$			14.0000i			0.595341i
$554$	2.00000			0.0849719
$555$		0			0
$556$		−	5.00000i		−	0.212047i
$557$			31.7490i			1.34525i	0.739984	+	0.672624i	$0.234833\pi$
$557$			31.7490i			1.34525i	−0.739984	+	0.672624i	$0.765167\pi$
$558$		0			0
$559$	13.2288			0.559517
$560$	−7.00000			−0.295804
$561$		0			0
$562$		−	7.93725i		−	0.334813i
$563$	23.8118			1.00355			0.501773	−	0.864999i	$-0.332681\pi$
$563$	23.8118			1.00355			0.501773	+	0.864999i	$0.332681\pi$
$564$		0			0
$565$			7.00000i			0.294492i
$566$	−10.5830			−0.444837
$567$		0			0
$568$	−6.00000			−0.251754
$569$			23.8118i			0.998241i	0.866533	+	0.499120i	$0.166343\pi$
$569$			23.8118i			0.998241i	−0.866533	+	0.499120i	$0.833657\pi$
$570$		0			0
$571$			26.4575i			1.10721i	0.832779	+	0.553606i	$0.186749\pi$
$571$			26.4575i			1.10721i	−0.832779	+	0.553606i	$0.813251\pi$
$572$		0			0
$573$		0			0
$574$			23.8118i			0.993884i
$575$	−8.00000	−	5.29150i	−0.333623	−	0.220671i
$576$		0			0
$577$			10.0000i			0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$			10.0000i			0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	11.0000			0.457540
$579$		0			0
$580$	−2.64575			−0.109859
$581$	−42.0000			−1.74245
$582$		0			0
$583$		0			0
$584$			4.00000i			0.165521i
$585$		0			0
$586$	21.1660			0.874360
$587$			2.00000i			0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$		0			0
$589$			21.1660i			0.872130i
$590$			37.0405i			1.52493i
$591$		0			0
$592$			7.93725i			0.326219i
$593$		−	21.0000i		−	0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$		−	21.0000i		−	0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$		0			0
$595$	37.0405			1.51851
$596$			15.8745i			0.650245i
$597$		0			0
$598$	13.2288	−	20.0000i	0.540964	−	0.817861i
$599$	46.0000			1.87951			0.939755	−	0.341850i	$-0.111053\pi$
$599$	46.0000			1.87951			0.939755	+	0.341850i	$0.111053\pi$
$600$		0			0
$601$			2.00000i			0.0815817i	0.999168	+	0.0407909i	$0.0129877\pi$
$601$			2.00000i			0.0815817i	−0.999168	+	0.0407909i	$0.987012\pi$
$602$		−	7.00000i		−	0.285299i
$603$		0			0
$604$	−5.00000			−0.203447
$605$	−29.1033			−1.18322
$606$		0			0
$607$			14.0000i			0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$			14.0000i			0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	−5.29150			−0.214599
$609$		0			0
$610$	28.0000			1.13369
$611$	−65.0000			−2.62962
$612$		0			0
$613$		−	13.2288i		−	0.534304i	−0.963654	−	0.267152i	$-0.913917\pi$
$613$		−	13.2288i		−	0.534304i	0.963654	−	0.267152i	$-0.0860827\pi$
$614$		−	21.0000i		−	0.847491i
$615$		0			0
$616$		0			0
$617$		−	21.1660i		−	0.852111i	−0.904697	−	0.426056i	$-0.859903\pi$
$617$		−	21.1660i		−	0.852111i	0.904697	−	0.426056i	$-0.140097\pi$
$618$		0			0
$619$	21.1660			0.850734			0.425367	−	0.905021i	$-0.360145\pi$
$619$	21.1660			0.850734			0.425367	+	0.905021i	$0.360145\pi$
$620$			10.5830i			0.425024i
$621$		0			0
$622$		−	24.0000i		−	0.962312i
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$		0			0
$628$	−10.5830			−0.422308
$629$		−	42.0000i		−	1.67465i
$630$		0			0
$631$			31.7490i			1.26391i	0.775006	+	0.631954i	$0.217747\pi$
$631$			31.7490i			1.26391i	−0.775006	+	0.631954i	$0.782253\pi$
$632$			5.29150i			0.210485i
$633$		0			0
$634$	−27.0000			−1.07231
$635$	34.3948			1.36491
$636$		0			0
$637$			35.0000i			1.38675i
$638$		0			0
$639$		0			0
$640$	−2.64575			−0.104583
$641$		−	2.64575i		−	0.104501i	−0.998634	−	0.0522504i	$-0.983361\pi$
$641$		−	2.64575i		−	0.104501i	0.998634	−	0.0522504i	$-0.0166394\pi$
$642$		0			0
$643$	−31.7490			−1.25206			−0.626029	−	0.779799i	$-0.715321\pi$
$643$	−31.7490			−1.25206			−0.626029	+	0.779799i	$0.715321\pi$
$644$	−10.5830	−	7.00000i	−0.417029	−	0.275839i
$645$		0			0
$646$	28.0000			1.10165
$647$		−	24.0000i		−	0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$		−	24.0000i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$		0			0
$649$		0			0
$650$			10.0000i			0.392232i
$651$		0			0
$652$	−18.0000			−0.704934
$653$	3.00000			0.117399			0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000			0.117399			0.0586995	+	0.998276i	$0.481305\pi$
$654$		0			0
$655$		−	58.2065i		−	2.27432i
$656$			9.00000i			0.351391i
$657$		0			0
$658$			34.3948i			1.34085i
$659$			42.3320i			1.64902i	0.565846	+	0.824511i	$0.308550\pi$
$659$			42.3320i			1.64902i	−0.565846	+	0.824511i	$0.691450\pi$
$660$		0			0
$661$	−26.4575			−1.02908			−0.514539	−	0.857467i	$-0.672037\pi$
$661$	−26.4575			−1.02908			−0.514539	+	0.857467i	$0.672037\pi$
$662$	4.00000			0.155464
$663$		0			0
$664$	−15.8745			−0.616050
$665$	37.0405			1.43637
$666$		0			0
$667$	−4.00000	−	2.64575i	−0.154881	−	0.102444i
$668$		0			0
$669$		0			0
$670$			14.0000i			0.540867i
$671$		0			0
$672$		0			0
$673$	−29.0000			−1.11787			−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000			−1.11787			−0.558934	+	0.829212i	$0.688789\pi$
$674$		−	26.4575i		−	1.01911i
$675$		0			0
$676$	−12.0000			−0.461538
$677$	31.7490			1.22021			0.610107	−	0.792319i	$-0.291126\pi$
$677$	31.7490			1.22021			0.610107	+	0.792319i	$0.291126\pi$
$678$		0			0
$679$	−7.00000			−0.268635
$680$	14.0000			0.536875
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$			21.0000i			0.802369i
$686$	18.5203			0.707107
$687$		0			0
$688$		−	2.64575i		−	0.100868i
$689$	26.4575			1.00795
$690$		0			0
$691$		−	35.0000i		−	1.33146i	−0.746191	−	0.665731i	$-0.768120\pi$
$691$		−	35.0000i		−	1.33146i	0.746191	−	0.665731i	$-0.231880\pi$
$692$		0			0
$693$		0			0
$694$	11.0000			0.417554
$695$			13.2288i			0.501795i
$696$		0			0
$697$		−	47.6235i		−	1.80387i
$698$		−	30.0000i		−	1.13552i
$699$		0			0
$700$	5.29150			0.200000
$701$			10.5830i			0.399715i	0.979825	+	0.199857i	$0.0640479\pi$
$701$			10.5830i			0.399715i	−0.979825	+	0.199857i	$0.935952\pi$
$702$		0			0
$703$		−	42.0000i		−	1.58406i
$704$		0			0
$705$		0			0
$706$			25.0000i			0.940887i
$707$		0			0
$708$		0			0
$709$			42.3320i			1.58981i	0.606732	+	0.794906i	$0.292480\pi$
$709$			42.3320i			1.58981i	−0.606732	+	0.794906i	$0.707520\pi$
$710$	15.8745			0.595760
$711$		0			0
$712$		0			0
$713$	−10.5830	+	16.0000i	−0.396337	+	0.599205i
$714$		0			0
$715$		0			0
$716$	−9.00000			−0.336346
$717$		0			0
$718$		−	23.8118i		−	0.888647i
$719$			21.0000i			0.783168i	0.920142	+	0.391584i	$0.128073\pi$
$719$			21.0000i			0.783168i	−0.920142	+	0.391584i	$0.871927\pi$
$720$		0			0
$721$	−7.00000			−0.260694
$722$	9.00000			0.334945
$723$		0			0
$724$	15.8745			0.589971
$725$	2.00000			0.0742781
$726$		0			0
$727$	−47.6235			−1.76626			−0.883129	−	0.469130i	$-0.844568\pi$
$727$	−47.6235			−1.76626			−0.883129	+	0.469130i	$0.844568\pi$
$728$			13.2288i			0.490290i
$729$		0			0
$730$		−	10.5830i		−	0.391695i
$731$			14.0000i			0.517809i
$732$		0			0
$733$	37.0405			1.36812			0.684061	−	0.729424i	$-0.260212\pi$
$733$	37.0405			1.36812			0.684061	+	0.729424i	$0.260212\pi$
$734$	18.5203			0.683595
$735$		0			0
$736$	−4.00000	−	2.64575i	−0.147442	−	0.0975237i
$737$		0			0
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$		−	21.0000i		−	0.771975i
$741$		0			0
$742$		−	14.0000i		−	0.513956i
$743$		−	15.8745i		−	0.582379i	−0.956665	−	0.291190i	$-0.905949\pi$
$743$		−	15.8745i		−	0.582379i	0.956665	−	0.291190i	$-0.0940511\pi$
$744$		0			0
$745$		−	42.0000i		−	1.53876i
$746$			21.1660i			0.774943i
$747$		0			0
$748$		0			0
$749$			14.0000i			0.511549i
$750$		0			0
$751$		−	15.8745i		−	0.579269i	−0.957137	−	0.289635i	$-0.906466\pi$
$751$		−	15.8745i		−	0.579269i	0.957137	−	0.289635i	$-0.0935338\pi$
$752$			13.0000i			0.474061i
$753$		0			0
$754$			5.00000i			0.182089i
$755$	13.2288			0.481444
$756$		0			0
$757$			52.9150i			1.92323i	0.274403	+	0.961615i	$0.411520\pi$
$757$			52.9150i			1.92323i	−0.274403	+	0.961615i	$0.588480\pi$
$758$		−	23.8118i		−	0.864882i
$759$		0			0
$760$	14.0000			0.507833
$761$			6.00000i			0.217500i	0.994069	+	0.108750i	$0.0346848\pi$
$761$			6.00000i			0.217500i	−0.994069	+	0.108750i	$0.965315\pi$
$762$		0			0
$763$			49.0000i			1.77392i
$764$			15.8745i			0.574320i
$765$		0			0
$766$		0			0
$767$	70.0000			2.52755
$768$		0			0
$769$	−29.1033			−1.04949			−0.524745	−	0.851259i	$-0.675839\pi$
$769$	−29.1033			−1.04949			−0.524745	+	0.851259i	$0.675839\pi$
$770$		0			0
$771$		0			0
$772$	19.0000			0.683825
$773$	−34.3948			−1.23709			−0.618547	−	0.785748i	$-0.712278\pi$
$773$	−34.3948			−1.23709			−0.618547	+	0.785748i	$0.712278\pi$
$774$		0			0
$775$		−	8.00000i		−	0.287368i
$776$	−2.64575			−0.0949769
$777$		0			0
$778$		−	21.1660i		−	0.758838i
$779$		−	47.6235i		−	1.70629i
$780$		0			0
$781$		0			0
$782$	21.1660	+	14.0000i	0.756895	+	0.500639i
$783$		0			0
$784$	7.00000			0.250000
$785$	28.0000			0.999363
$786$		0			0
$787$	−42.3320			−1.50897			−0.754487	−	0.656315i	$-0.772114\pi$
$787$	−42.3320			−1.50897			−0.754487	+	0.656315i	$0.772114\pi$
$788$	1.00000			0.0356235
$789$		0			0
$790$		−	14.0000i		−	0.498098i
$791$		−	7.00000i		−	0.248891i
$792$		0			0
$793$		−	52.9150i		−	1.87907i
$794$		−	34.0000i		−	1.20661i
$795$		0			0
$796$	13.2288			0.468881
$797$	−50.2693			−1.78063			−0.890315	−	0.455346i	$-0.849516\pi$
$797$	−50.2693			−1.78063			−0.890315	+	0.455346i	$0.849516\pi$
$798$		0			0
$799$		−	68.7895i		−	2.43360i
$800$	2.00000			0.0707107
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	28.0000	+	18.5203i	0.986870	+	0.652753i
$806$	20.0000			0.704470
$807$		0			0
$808$		0			0
$809$	−4.00000			−0.140633			−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000			−0.140633			−0.0703163	+	0.997525i	$0.522401\pi$
$810$		0			0
$811$			5.00000i			0.175574i	0.996139	+	0.0877869i	$0.0279794\pi$
$811$			5.00000i			0.175574i	−0.996139	+	0.0877869i	$0.972021\pi$
$812$	2.64575			0.0928477
$813$		0			0
$814$		0			0
$815$	47.6235			1.66818
$816$		0			0
$817$			14.0000i			0.489798i
$818$			18.0000i			0.629355i
$819$		0			0
$820$		−	23.8118i		−	0.831543i
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$		0			0
$823$	47.0000			1.63832			0.819159	−	0.573567i	$-0.194441\pi$
$823$	47.0000			1.63832			0.819159	+	0.573567i	$0.194441\pi$
$824$	−2.64575			−0.0921691
$825$		0			0
$826$		−	37.0405i		−	1.28880i
$827$			47.6235i			1.65603i	0.560704	+	0.828016i	$0.310530\pi$
$827$			47.6235i			1.65603i	−0.560704	+	0.828016i	$0.689470\pi$
$828$		0			0
$829$			14.0000i			0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$			14.0000i			0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$	42.0000			1.45784
$831$		0			0
$832$			5.00000i			0.173344i
$833$	−37.0405			−1.28338
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	15.8745			0.548376
$839$	5.29150			0.182683			0.0913415	−	0.995820i	$-0.470885\pi$
$839$	5.29150			0.182683			0.0913415	+	0.995820i	$0.470885\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$			29.1033i			1.00296i
$843$		0			0
$844$	−16.0000			−0.550743
$845$	31.7490			1.09220
$846$		0			0
$847$	29.1033			1.00000
$848$		−	5.29150i		−	0.181711i
$849$		0			0
$850$	−10.5830			−0.362994
$851$	21.0000	−	31.7490i	0.719871	−	1.08834i
$852$		0			0
$853$		−	7.00000i		−	0.239675i	−0.992793	−	0.119838i	$-0.961763\pi$
$853$		−	7.00000i		−	0.239675i	0.992793	−	0.119838i	$-0.0382374\pi$
$854$	−28.0000			−0.958140
$855$		0			0
$856$			5.29150i			0.180860i
$857$			3.00000i			0.102478i	0.998686	+	0.0512390i	$0.0163170\pi$
$857$			3.00000i			0.102478i	−0.998686	+	0.0512390i	$0.983683\pi$
$858$		0			0
$859$			13.0000i			0.443554i	0.975097	+	0.221777i	$0.0711857\pi$
$859$			13.0000i			0.443554i	−0.975097	+	0.221777i	$0.928814\pi$
$860$			7.00000i			0.238698i
$861$		0			0
$862$		−	13.2288i		−	0.450573i
$863$	18.0000			0.612727			0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000			0.612727			0.306364	+	0.951915i	$0.400888\pi$
$864$		0			0
$865$		0			0
$866$	13.2288			0.449531
$867$		0			0
$868$		−	10.5830i		−	0.359211i
$869$		0			0
$870$		0			0
$871$	26.4575			0.896479
$872$			18.5203i			0.627175i
$873$		0			0
$874$	21.1660	+	14.0000i	0.715951	+	0.473557i
$875$	21.0000			0.709930
$876$		0			0
$877$	22.0000			0.742887			0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000			0.742887			0.371444	+	0.928456i	$0.378863\pi$
$878$		−	6.00000i		−	0.202490i
$879$		0			0
$880$		0			0
$881$	−10.5830			−0.356551			−0.178275	−	0.983981i	$-0.557052\pi$
$881$	−10.5830			−0.356551			−0.178275	+	0.983981i	$0.557052\pi$
$882$		0			0
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$		−	26.4575i		−	0.889862i
$885$		0			0
$886$	−3.00000			−0.100787
$887$			8.00000i			0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$			8.00000i			0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$		0			0
$889$	−34.3948			−1.15356
$890$		0			0
$891$		0			0
$892$		−	28.0000i		−	0.937509i
$893$		−	68.7895i		−	2.30195i
$894$		0			0
$895$	23.8118			0.795939
$896$	2.64575			0.0883883
$897$		0			0
$898$	−12.0000			−0.400445
$899$		−	4.00000i		−	0.133407i
$900$		0			0
$901$			28.0000i			0.932815i
$902$		0			0
$903$		0			0
$904$		−	2.64575i		−	0.0879964i
$905$	−42.0000			−1.39613
$906$		0			0
$907$		−	50.2693i		−	1.66916i	−0.550884	−	0.834582i	$-0.685709\pi$
$907$		−	50.2693i		−	1.66916i	0.550884	−	0.834582i	$-0.314291\pi$
$908$	−7.93725			−0.263407
$909$		0			0
$910$		−	35.0000i		−	1.16024i
$911$		−	39.6863i		−	1.31486i	−0.753514	−	0.657432i	$-0.771643\pi$
$911$		−	39.6863i		−	1.31486i	0.753514	−	0.657432i	$-0.228357\pi$
$912$		0			0
$913$		0			0
$914$			5.29150i			0.175027i
$915$		0			0
$916$	−10.5830			−0.349672
$917$			58.2065i			1.92215i
$918$		0			0
$919$			10.5830i			0.349101i	0.984648	+	0.174551i	$0.0558472\pi$
$919$			10.5830i			0.349101i	−0.984648	+	0.174551i	$0.944153\pi$
$920$	10.5830	+	7.00000i	0.348911	+	0.230783i
$921$		0			0
$922$			12.0000i			0.395199i
$923$		−	30.0000i		−	0.987462i
$924$		0			0
$925$			15.8745i			0.521951i
$926$	19.0000			0.624379
$927$		0			0
$928$	1.00000			0.0328266
$929$			29.0000i			0.951459i	0.879592	+	0.475730i	$0.157816\pi$
$929$			29.0000i			0.951459i	−0.879592	+	0.475730i	$0.842184\pi$
$930$		0			0
$931$	−37.0405			−1.21395
$932$	18.0000			0.589610
$933$		0			0
$934$	−13.2288			−0.432858
$935$		0			0
$936$		0			0
$937$	−7.93725			−0.259299			−0.129649	−	0.991560i	$-0.541385\pi$
$937$	−7.93725			−0.259299			−0.129649	+	0.991560i	$0.541385\pi$
$938$		−	14.0000i		−	0.457116i
$939$		0			0
$940$		−	34.3948i		−	1.12183i
$941$	50.2693			1.63873			0.819366	−	0.573271i	$-0.194326\pi$
$941$	50.2693			1.63873			0.819366	+	0.573271i	$0.194326\pi$
$942$		0			0
$943$	23.8118	−	36.0000i	0.775418	−	1.17232i
$944$		−	14.0000i		−	0.455661i
$945$		0			0
$946$		0			0
$947$	−11.0000			−0.357452			−0.178726	−	0.983899i	$-0.557198\pi$
$947$	−11.0000			−0.357452			−0.178726	+	0.983899i	$0.557198\pi$
$948$		0			0
$949$	−20.0000			−0.649227
$950$	−10.5830			−0.343358
$951$		0			0
$952$	−14.0000			−0.453743
$953$			10.5830i			0.342817i	0.985200	+	0.171409i	$0.0548318\pi$
$953$			10.5830i			0.342817i	−0.985200	+	0.171409i	$0.945168\pi$
$954$		0			0
$955$		−	42.0000i		−	1.35909i
$956$	20.0000			0.646846
$957$		0			0
$958$	5.29150			0.170961
$959$		−	21.0000i		−	0.678125i
$960$		0			0
$961$	15.0000			0.483871
$962$	−39.6863			−1.27954
$963$		0			0
$964$	23.8118			0.766925
$965$	−50.2693			−1.61823
$966$		0			0
$967$	−8.00000			−0.257263			−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000			−0.257263			−0.128631	+	0.991692i	$0.541058\pi$
$968$	11.0000			0.353553
$969$		0			0
$970$	7.00000			0.224756
$971$	−47.6235			−1.52831			−0.764156	−	0.645032i	$-0.776844\pi$
$971$	−47.6235			−1.52831			−0.764156	+	0.645032i	$0.776844\pi$
$972$		0			0
$973$		−	13.2288i		−	0.424094i
$974$	−23.0000			−0.736968
$975$		0			0
$976$	−10.5830			−0.338754
$977$			39.6863i			1.26968i	0.772645	+	0.634838i	$0.218933\pi$
$977$			39.6863i			1.26968i	−0.772645	+	0.634838i	$0.781067\pi$
$978$		0			0
$979$		0			0
$980$	−18.5203			−0.591608
$981$		0			0
$982$	−20.0000			−0.638226
$983$	−52.9150			−1.68773			−0.843864	−	0.536558i	$-0.819724\pi$
$983$	−52.9150			−1.68773			−0.843864	+	0.536558i	$0.819724\pi$
$984$		0			0
$985$	−2.64575			−0.0843006
$986$	−5.29150			−0.168516
$987$		0			0
$988$		−	26.4575i		−	0.841726i
$989$	−7.00000	+	10.5830i	−0.222587	+	0.336520i
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$		−	4.00000i		−	0.127000i
$993$		0			0
$994$	−15.8745			−0.503509
$995$	−35.0000			−1.10957
$996$		0			0
$997$			42.0000i			1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$			42.0000i			1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	4.00000			0.126618
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.g.d.2575.2		4
3.2	odd	2		966.2.g.d.643.4	yes	4
7.6	odd	2	inner	2898.2.g.d.2575.3		4
21.20	even	2		966.2.g.d.643.1	✓	4
23.22	odd	2	inner	2898.2.g.d.2575.4		4
69.68	even	2		966.2.g.d.643.3	yes	4
161.160	even	2	inner	2898.2.g.d.2575.1		4
483.482	odd	2		966.2.g.d.643.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.g.d.643.1	✓	4	21.20	even	2
966.2.g.d.643.2	yes	4	483.482	odd	2
966.2.g.d.643.3	yes	4	69.68	even	2
966.2.g.d.643.4	yes	4	3.2	odd	2
2898.2.g.d.2575.1		4	161.160	even	2	inner
2898.2.g.d.2575.2		4	1.1	even	1	trivial
2898.2.g.d.2575.3		4	7.6	odd	2	inner
2898.2.g.d.2575.4		4	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +2.64575 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +2.64575 q^{7} +1.00000 q^{8} -2.64575 q^{10} +5.00000i q^{13} +2.64575 q^{14} +1.00000 q^{16} -5.29150 q^{17} -5.29150 q^{19} -2.64575 q^{20} +(-4.00000 - 2.64575i) q^{23} +2.00000 q^{25} +5.00000i q^{26} +2.64575 q^{28} +1.00000 q^{29} -4.00000i q^{31} +1.00000 q^{32} -5.29150 q^{34} -7.00000 q^{35} +7.93725i q^{37} -5.29150 q^{38} -2.64575 q^{40} +9.00000i q^{41} -2.64575i q^{43} +(-4.00000 - 2.64575i) q^{46} +13.0000i q^{47} +7.00000 q^{49} +2.00000 q^{50} +5.00000i q^{52} -5.29150i q^{53} +2.64575 q^{56} +1.00000 q^{58} -14.0000i q^{59} -10.5830 q^{61} -4.00000i q^{62} +1.00000 q^{64} -13.2288i q^{65} -5.29150i q^{67} -5.29150 q^{68} -7.00000 q^{70} -6.00000 q^{71} +4.00000i q^{73} +7.93725i q^{74} -5.29150 q^{76} +5.29150i q^{79} -2.64575 q^{80} +9.00000i q^{82} -15.8745 q^{83} +14.0000 q^{85} -2.64575i q^{86} +13.2288i q^{91} +(-4.00000 - 2.64575i) q^{92} +13.0000i q^{94} +14.0000 q^{95} -2.64575 q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} - 16 q^{23} + 8 q^{25} + 4 q^{29} + 4 q^{32} - 28 q^{35} - 16 q^{46} + 28 q^{49} + 8 q^{50} + 4 q^{58} + 4 q^{64} - 28 q^{70} - 24 q^{71} + 56 q^{85} - 16 q^{92} + 56 q^{95} + 28 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 - 16 * q^23 + 8 * q^25 + 4 * q^29 + 4 * q^32 - 28 * q^35 - 16 * q^46 + 28 * q^49 + 8 * q^50 + 4 * q^58 + 4 * q^64 - 28 * q^70 - 24 * q^71 + 56 * q^85 - 16 * q^92 + 56 * q^95 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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