LMFDB - Embedded newform 1520.2.g.d.1519.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1519,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1520.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1519.1
Root			$2.13746 + 0.656712i$ of defining polynomial
Character	$\chi$	$=$	1520.1519
Dual form			1520.2.g.d.1519.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.63746 - 1.52274i) q^{5} +5.27492 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+(-1.63746 - 1.52274i) q^{5} +5.27492 q^{7} +3.00000 q^{9} -6.50958i q^{11} -3.88273i q^{17} +4.35890i q^{19} -4.00000 q^{23} +(0.362541 + 4.98684i) q^{25} +(-8.63746 - 8.03231i) q^{35} -11.8248 q^{43} +(-4.91238 - 4.56821i) q^{45} +2.72508 q^{47} +20.8248 q^{49} +(-9.91238 + 10.6592i) q^{55} +11.2749 q^{61} +15.8248 q^{63} -16.0646i q^{73} -34.3375i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(-5.91238 + 6.35781i) q^{85} +(6.63746 - 7.13752i) q^{95} -19.5287i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5} + 6 q^{7} + 12 q^{9}+O(q^{10}) $ 4 * q + q^5 + 6 * q^7 + 12 * q^9	$ 4 q + q^{5} + 6 q^{7} + 12 q^{9} - 16 q^{23} + 9 q^{25} - 27 q^{35} - 2 q^{43} + 3 q^{45} + 26 q^{47} + 38 q^{49} - 17 q^{55} + 30 q^{61} + 18 q^{63} + 36 q^{81} - 64 q^{83} - q^{85} + 19 q^{95}+O(q^{100}) $ 4 * q + q^5 + 6 * q^7 + 12 * q^9 - 16 * q^23 + 9 * q^25 - 27 * q^35 - 2 * q^43 + 3 * q^45 + 26 * q^47 + 38 * q^49 - 17 * q^55 + 30 * q^61 + 18 * q^63 + 36 * q^81 - 64 * q^83 - q^85 + 19 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$		0			0
$5$	−1.63746	−	1.52274i	−0.732294	−	0.680989i
$5$	−1.63746	−	1.52274i	−0.732294	−	0.680989i
$6$		0			0
$7$	5.27492			1.99373			0.996866	−	0.0791130i	$-0.0252088\pi$
$7$	5.27492			1.99373			0.996866	+	0.0791130i	$0.0252088\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		−	6.50958i		−	1.96271i	−0.192201	−	0.981356i	$-0.561563\pi$
$11$		−	6.50958i		−	1.96271i	0.192201	−	0.981356i	$-0.438437\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.88273i		−	0.941700i	−0.882213	−	0.470850i	$-0.843947\pi$
$17$		−	3.88273i		−	0.941700i	0.882213	−	0.470850i	$-0.156053\pi$
$18$		0			0
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	0.362541	+	4.98684i	0.0725083	+	0.997368i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.63746	−	8.03231i	−1.46000	−	1.35771i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−11.8248			−1.80326			−0.901629	−	0.432511i	$-0.857628\pi$
$43$	−11.8248			−1.80326			−0.901629	+	0.432511i	$0.857628\pi$
$44$		0			0
$45$	−4.91238	−	4.56821i	−0.732294	−	0.680989i
$46$		0			0
$47$	2.72508			0.397494			0.198747	−	0.980051i	$-0.436313\pi$
$47$	2.72508			0.397494			0.198747	+	0.980051i	$0.436313\pi$
$48$		0			0
$49$	20.8248			2.97496
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−9.91238	+	10.6592i	−1.33658	+	1.43728i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	11.2749			1.44361			0.721803	−	0.692099i	$-0.243314\pi$
$61$	11.2749			1.44361			0.721803	+	0.692099i	$0.243314\pi$
$62$		0			0
$63$	15.8248			1.99373
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	16.0646i		−	1.88022i	−0.340868	−	0.940111i	$-0.610721\pi$
$73$		−	16.0646i		−	1.88022i	0.340868	−	0.940111i	$-0.389279\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	34.3375i		−	3.91312i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	−5.91238	+	6.35781i	−0.641287	+	0.689601i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.63746	−	7.13752i	0.680989	−	0.732294i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		−	19.5287i		−	1.96271i
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	6.54983	+	6.09095i	0.610775	+	0.567984i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	20.4811i		−	1.87750i
$120$		0			0
$121$	−31.3746			−2.85224
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.00000	−	8.71780i	0.626099	−	0.779744i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			4.83507i			0.422442i	0.977438	+	0.211221i	$0.0677440\pi$
$131$			4.83507i			0.422442i	−0.977438	+	0.211221i	$0.932256\pi$
$132$		0			0
$133$			22.9928i			1.99373i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			17.7391i			1.51556i	0.652512	+	0.757778i	$0.273715\pi$
$137$			17.7391i			1.51556i	−0.652512	+	0.757778i	$0.726285\pi$
$138$		0			0
$139$		−	18.6915i		−	1.58539i	−0.609618	−	0.792695i	$-0.708677\pi$
$139$		−	18.6915i		−	1.58539i	0.609618	−	0.792695i	$-0.291323\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	24.3746			1.99684			0.998422	−	0.0561570i	$-0.0178847\pi$
$149$	24.3746			1.99684			0.998422	+	0.0561570i	$0.0178847\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	11.6482i		−	0.941700i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			17.4356i			1.39151i	0.718278	+	0.695756i	$0.244931\pi$
$157$			17.4356i			1.39151i	−0.718278	+	0.695756i	$0.755069\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−21.0997			−1.66289
$162$		0			0
$163$	24.0000			1.87983			0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$			13.0767i			1.00000i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	1.91238	+	26.3052i	0.144562	+	1.98848i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−25.2749			−1.84828
$188$		0			0
$189$		0			0
$190$		0			0
$191$			25.6197i			1.85377i	0.375339	+	0.926887i	$0.377526\pi$
$191$			25.6197i			1.85377i	−0.375339	+	0.926887i	$0.622474\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.4356i			1.24223i	0.783718	+	0.621117i	$0.213321\pi$
$197$			17.4356i			1.24223i	−0.783718	+	0.621117i	$0.786679\pi$
$198$		0			0
$199$			15.1123i			1.07128i	0.844446	+	0.535641i	$0.179930\pi$
$199$			15.1123i			1.07128i	−0.844446	+	0.535641i	$0.820070\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−12.0000			−0.834058
$208$		0			0
$209$	28.3746			1.96271
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.3625	+	18.0060i	1.32051	+	1.22800i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	1.08762	+	14.9605i	0.0725083	+	0.997368i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	8.37459			0.553408			0.276704	−	0.960955i	$-0.410758\pi$
$229$	8.37459			0.553408			0.276704	+	0.960955i	$0.410758\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	14.3901i		−	0.942728i	−0.881939	−	0.471364i	$-0.843762\pi$
$233$		−	14.3901i		−	0.942728i	0.881939	−	0.471364i	$-0.156238\pi$
$234$		0			0
$235$	−4.46221	−	4.14959i	−0.291083	−	0.270689i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.9260i		−	0.706745i	−0.935483	−	0.353373i	$-0.885035\pi$
$239$		−	10.9260i		−	0.706745i	0.935483	−	0.353373i	$-0.114965\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−34.0997	−	31.7106i	−2.17855	−	2.02592i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	9.02134i		−	0.569422i	−0.958613	−	0.284711i	$-0.908102\pi$
$251$		−	9.02134i		−	0.569422i	0.958613	−	0.284711i	$-0.0918976\pi$
$252$		0			0
$253$			26.0383i			1.63701i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−20.9244			−1.29026			−0.645128	−	0.764075i	$-0.723196\pi$
$263$	−20.9244			−1.29026			−0.645128	+	0.764075i	$0.723196\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		−	26.1534i		−	1.58871i	−0.607457	−	0.794353i	$-0.707810\pi$
$271$		−	26.1534i		−	1.58871i	0.607457	−	0.794353i	$-0.292190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	32.4622	−	2.35999i	1.95754	−	0.142313i
$276$		0			0
$277$			30.7583i			1.84809i	0.382288	+	0.924043i	$0.375136\pi$
$277$			30.7583i			1.84809i	−0.382288	+	0.924043i	$0.624864\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−4.17525			−0.248193			−0.124096	−	0.992270i	$-0.539603\pi$
$283$	−4.17525			−0.248193			−0.124096	+	0.992270i	$0.539603\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.92442			0.113201
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−62.3746			−3.59521
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−18.4622	−	17.1687i	−1.05714	−	0.983079i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			28.1314i			1.59519i	0.603195	+	0.797594i	$0.293894\pi$
$311$			28.1314i			1.59519i	−0.603195	+	0.797594i	$0.706106\pi$
$312$		0			0
$313$		−	34.8712i		−	1.97104i	−0.169570	−	0.985518i	$-0.554238\pi$
$313$		−	34.8712i		−	1.97104i	0.169570	−	0.985518i	$-0.445762\pi$
$314$		0			0
$315$	−25.9124	−	24.0969i	−1.46000	−	1.35771i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.9244			0.941700
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.3746			0.792497
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	72.9244			3.93755
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.7251			−0.790484			−0.395242	−	0.918577i	$-0.629339\pi$
$347$	−14.7251			−0.790484			−0.395242	+	0.918577i	$0.629339\pi$
$348$		0			0
$349$	−6.17525			−0.330553			−0.165277	−	0.986247i	$-0.552852\pi$
$349$	−6.17525			−0.330553			−0.165277	+	0.986247i	$0.552852\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			34.8712i			1.85601i	0.372572	+	0.928003i	$0.378476\pi$
$353$			34.8712i			1.85601i	−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			15.9495i			0.841785i	0.907111	+	0.420892i	$0.138283\pi$
$359$			15.9495i			0.841785i	−0.907111	+	0.420892i	$0.861717\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−24.4622	+	26.3052i	−1.28041	+	1.37688i
$366$		0			0
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	−52.2870	+	56.2262i	−2.66479	+	2.86555i
$386$		0			0
$387$	−35.4743			−1.80326
$388$		0			0
$389$	38.9244			1.97355			0.986773	−	0.162107i	$-0.0518289\pi$
$389$	38.9244			1.97355			0.986773	+	0.162107i	$0.0518289\pi$
$390$		0			0
$391$			15.5309i			0.785432i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	13.5529i		−	0.680199i	−0.940389	−	0.340099i	$-0.889539\pi$
$397$		−	13.5529i		−	0.680199i	0.940389	−	0.340099i	$-0.110461\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−14.7371	−	13.7046i	−0.732294	−	0.680989i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	26.1993	+	24.3638i	1.28607	+	1.19597i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.71780i			0.425892i	0.977064	+	0.212946i	$0.0683059\pi$
$419$			8.71780i			0.425892i	−0.977064	+	0.212946i	$0.931694\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	8.17525			0.397494
$424$		0			0
$425$	19.3625	−	1.40765i	0.939221	−	0.0682810i
$426$		0			0
$427$	59.4743			2.87816
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	17.4356i		−	0.834058i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	62.4743			2.97496
$442$		0			0
$443$	−40.9244			−1.94438			−0.972189	−	0.234198i	$-0.924754\pi$
$443$	−40.9244			−1.94438			−0.972189	+	0.234198i	$0.924754\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			34.3375i			1.60624i	0.595818	+	0.803120i	$0.296828\pi$
$457$			34.3375i			1.60624i	−0.595818	+	0.803120i	$0.703172\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.374586			−0.0174462			−0.00872311	−	0.999962i	$-0.502777\pi$
$461$	−0.374586			−0.0174462			−0.00872311	+	0.999962i	$0.502777\pi$
$462$		0			0
$463$	−31.8248			−1.47902			−0.739511	−	0.673145i	$-0.764943\pi$
$463$	−31.8248			−1.47902			−0.739511	+	0.673145i	$0.764943\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.2749			1.16958			0.584792	−	0.811183i	$-0.301176\pi$
$467$	25.2749			1.16958			0.584792	+	0.811183i	$0.301176\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			76.9741i			3.53927i
$474$		0			0
$475$	−21.7371	+	1.58028i	−0.997368	+	0.0725083i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			43.5890i			1.99163i	0.0913823	+	0.995816i	$0.470871\pi$
$479$			43.5890i			1.99163i	−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			43.5890i			1.96714i	0.180517	+	0.983572i	$0.442223\pi$
$491$			43.5890i			1.96714i	−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−29.7371	+	31.9775i	−1.33658	+	1.43728i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			22.8777i			1.02415i	0.858941	+	0.512074i	$0.171123\pi$
$499$			22.8777i			1.02415i	−0.858941	+	0.512074i	$0.828877\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−44.0000			−1.96186			−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000			−1.96186			−0.980932	+	0.194354i	$0.937739\pi$
$504$		0			0
$505$	−16.3746	−	15.2274i	−0.728660	−	0.677609i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		−	84.7396i		−	3.74866i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	17.7391i		−	0.780166i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	135.560i		−	5.83900i
$540$		0			0
$541$	21.4743			0.923250			0.461625	−	0.887075i	$-0.347267\pi$
$541$	21.4743			0.923250			0.461625	+	0.887075i	$0.347267\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	33.8248			1.44361
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	38.5237i		−	1.63230i	−0.577838	−	0.816152i	$-0.696103\pi$
$557$		−	38.5237i		−	1.63230i	0.577838	−	0.816152i	$-0.303897\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	47.4743			1.99373
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$			26.1534i			1.09449i	0.836974	+	0.547243i	$0.184323\pi$
$571$			26.1534i			1.09449i	−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.45017	−	19.9474i	−0.0604761	−	0.831862i
$576$		0			0
$577$		−	26.5720i		−	1.10621i	−0.833112	−	0.553104i	$-0.813443\pi$
$577$		−	26.5720i		−	1.10621i	0.833112	−	0.553104i	$-0.186557\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−84.3987			−3.50145
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.0241			−1.56942			−0.784711	−	0.619862i	$-0.787189\pi$
$587$	−38.0241			−1.56942			−0.784711	+	0.619862i	$0.787189\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	34.8712i		−	1.43199i	−0.698106	−	0.715994i	$-0.745974\pi$
$593$		−	34.8712i		−	1.43199i	0.698106	−	0.715994i	$-0.254026\pi$
$594$		0			0
$595$	−31.1873	+	33.5369i	−1.27855	+	1.37488i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	51.3746	+	47.7753i	2.08867	+	1.94234i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	49.0311i		−	1.98035i	−0.139837	−	0.990174i	$-0.544658\pi$
$613$		−	49.0311i		−	1.98035i	0.139837	−	0.990174i	$-0.455342\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.7156i			0.511911i	0.966689	+	0.255956i	$0.0823901\pi$
$617$			12.7156i			0.511911i	−0.966689	+	0.255956i	$0.917610\pi$
$618$		0			0
$619$		−	43.5890i		−	1.75199i	−0.482321	−	0.875995i	$-0.660206\pi$
$619$		−	43.5890i		−	1.75199i	0.482321	−	0.875995i	$-0.339794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.7371	+	3.61587i	−0.989485	+	0.144635i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			36.9643i			1.47153i	0.677239	+	0.735763i	$0.263176\pi$
$631$			36.9643i			1.47153i	−0.677239	+	0.735763i	$0.736824\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	35.8248			1.41279			0.706395	−	0.707818i	$-0.250320\pi$
$643$	35.8248			1.41279			0.706395	+	0.707818i	$0.250320\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	50.0241			1.96665			0.983325	−	0.181857i	$-0.0582109\pi$
$647$	50.0241			1.96665			0.983325	+	0.181857i	$0.0582109\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			20.2509i			0.792479i	0.918147	+	0.396239i	$0.129685\pi$
$653$			20.2509i			0.792479i	−0.918147	+	0.396239i	$0.870315\pi$
$654$		0			0
$655$	7.36254	−	7.91723i	0.287678	−	0.309352i
$656$		0			0
$657$		−	48.1939i		−	1.88022i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	35.0120	−	37.6498i	1.35771	−	1.46000i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	73.3949i		−	2.83338i
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	27.0120	−	29.0471i	1.03208	−	1.10983i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			10.6958i			0.406889i	0.979086	+	0.203445i	$0.0652137\pi$
$691$			10.6958i			0.406889i	−0.979086	+	0.203445i	$0.934786\pi$
$692$		0			0
$693$		−	103.012i		−	3.91312i
$694$		0			0
$695$	−28.4622	+	30.6065i	−1.07963	+	1.16097i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	52.7492			1.98384
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	53.3325i		−	1.98897i	−0.104896	−	0.994483i	$-0.533451\pi$
$719$		−	53.3325i		−	1.98897i	0.104896	−	0.994483i	$-0.466549\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.4743			−0.573908			−0.286954	−	0.957944i	$-0.592643\pi$
$727$	−15.4743			−0.573908			−0.286954	+	0.957944i	$0.592643\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$			45.9123i			1.69813i
$732$		0			0
$733$			52.3068i			1.93200i	0.258551	+	0.965998i	$0.416755\pi$
$733$			52.3068i			1.93200i	−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			23.7150i			0.872370i	0.899857	+	0.436185i	$0.143671\pi$
$739$			23.7150i			0.872370i	−0.899857	+	0.436185i	$0.856329\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−39.9124	−	37.1161i	−1.46228	−	1.35983i
$746$		0			0
$747$	−48.0000			−1.75623
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			47.3566i			1.72121i	0.509276	+	0.860603i	$0.329913\pi$
$757$			47.3566i			1.72121i	−0.509276	+	0.860603i	$0.670087\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31.2749			1.13371			0.566857	−	0.823816i	$-0.308159\pi$
$761$	31.2749			1.13371			0.566857	+	0.823816i	$0.308159\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−17.7371	+	19.0734i	−0.641287	+	0.689601i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	44.3746			1.60019			0.800094	−	0.599874i	$-0.204783\pi$
$769$	44.3746			1.60019			0.800094	+	0.599874i	$0.204783\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	26.5498	−	28.5501i	0.947604	−	1.01900i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		−	10.5808i		−	0.374320i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−104.574			−3.69033
$804$		0			0
$805$	34.5498	+	32.1293i	1.21772	+	1.13241i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−46.5739			−1.63745			−0.818726	−	0.574184i	$-0.805319\pi$
$809$	−46.5739			−1.63745			−0.818726	+	0.574184i	$0.805319\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−39.2990	−	36.5457i	−1.37658	−	1.28014i
$816$		0			0
$817$		−	51.5429i		−	1.80326i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.62541			0.266129			0.133064	−	0.991107i	$-0.457518\pi$
$821$	7.62541			0.266129			0.133064	+	0.991107i	$0.457518\pi$
$822$		0			0
$823$	44.5739			1.55375			0.776875	−	0.629655i	$-0.216804\pi$
$823$	44.5739			1.55375			0.776875	+	0.629655i	$0.216804\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	80.8569i		−	2.80152i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	21.2870	+	19.7956i	0.732294	+	0.680989i
$846$		0			0
$847$	−165.498			−5.68659
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			52.3068i			1.79095i	0.445112	+	0.895475i	$0.353164\pi$
$853$			52.3068i			1.79095i	−0.445112	+	0.895475i	$0.646836\pi$
$854$		0			0
$855$	19.9124	−	21.4125i	0.680989	−	0.732294i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			15.3425i			0.523478i	0.965139	+	0.261739i	$0.0842960\pi$
$859$			15.3425i			0.523478i	−0.965139	+	0.261739i	$0.915704\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	36.9244	−	45.9857i	1.24827	−	1.55460i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.0241			−0.809392			−0.404696	−	0.914451i	$-0.632623\pi$
$881$	−24.0241			−0.809392			−0.404696	+	0.914451i	$0.632623\pi$
$882$		0			0
$883$	−0.924421			−0.0311092			−0.0155546	−	0.999879i	$-0.504951\pi$
$883$	−0.924421			−0.0311092			−0.0155546	+	0.999879i	$0.504951\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	58.5862i		−	1.96271i
$892$		0			0
$893$			11.8784i			0.397494i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	30.0000			0.995037
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$			104.153i			3.44697i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			25.5046i			0.842236i
$918$		0			0
$919$		−	8.71780i		−	0.287574i	−0.989609	−	0.143787i	$-0.954072\pi$
$919$		−	8.71780i		−	0.287574i	0.989609	−	0.143787i	$-0.0459280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	50.0000			1.64045			0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000			1.64045			0.820223	+	0.572043i	$0.193849\pi$
$930$		0			0
$931$			90.7730i			2.97496i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	41.3866	+	38.4871i	1.35349	+	1.25866i
$936$		0			0
$937$			21.0881i			0.688920i	0.938801	+	0.344460i	$0.111938\pi$
$937$			21.0881i			0.688920i	−0.938801	+	0.344460i	$0.888062\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.00000			0.259965			0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000			0.259965			0.129983	+	0.991516i	$0.458508\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	39.0120	−	41.9512i	1.26240	−	1.35751i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			93.5725i			3.02161i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	12.0000			0.385894			0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000			0.385894			0.192947	+	0.981209i	$0.438195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		−	98.5960i		−	3.16084i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	26.5498	−	28.5501i	0.845948	−	0.909681i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	47.2990			1.50402
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	23.0120	−	24.7457i	0.729531	−	0.784493i
$996$		0			0
$997$			52.3802i			1.65890i	0.558584	+	0.829448i	$0.311345\pi$
$997$			52.3802i			1.65890i	−0.558584	+	0.829448i	$0.688655\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.g.d.1519.1	yes	4
4.3	odd	2		1520.2.g.c.1519.1	✓	4
5.4	even	2		1520.2.g.c.1519.2	yes	4
19.18	odd	2	CM	1520.2.g.d.1519.1	yes	4
20.19	odd	2	inner	1520.2.g.d.1519.2	yes	4
76.75	even	2		1520.2.g.c.1519.1	✓	4
95.94	odd	2		1520.2.g.c.1519.2	yes	4
380.379	even	2	inner	1520.2.g.d.1519.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.g.c.1519.1	✓	4	4.3	odd	2
1520.2.g.c.1519.1	✓	4	76.75	even	2
1520.2.g.c.1519.2	yes	4	5.4	even	2
1520.2.g.c.1519.2	yes	4	95.94	odd	2
1520.2.g.d.1519.1	yes	4	1.1	even	1	trivial
1520.2.g.d.1519.1	yes	4	19.18	odd	2	CM
1520.2.g.d.1519.2	yes	4	20.19	odd	2	inner
1520.2.g.d.1519.2	yes	4	380.379	even	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.63746 - 1.52274i) q^{5} +5.27492 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+(-1.63746 - 1.52274i) q^{5} +5.27492 q^{7} +3.00000 q^{9} -6.50958i q^{11} -3.88273i q^{17} +4.35890i q^{19} -4.00000 q^{23} +(0.362541 + 4.98684i) q^{25} +(-8.63746 - 8.03231i) q^{35} -11.8248 q^{43} +(-4.91238 - 4.56821i) q^{45} +2.72508 q^{47} +20.8248 q^{49} +(-9.91238 + 10.6592i) q^{55} +11.2749 q^{61} +15.8248 q^{63} -16.0646i q^{73} -34.3375i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(-5.91238 + 6.35781i) q^{85} +(6.63746 - 7.13752i) q^{95} -19.5287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} + 6 q^{7} + 12 q^{9}+O(q^{10}) \) 4 * q + q^5 + 6 * q^7 + 12 * q^9	\( 4 q + q^{5} + 6 q^{7} + 12 q^{9} - 16 q^{23} + 9 q^{25} - 27 q^{35} - 2 q^{43} + 3 q^{45} + 26 q^{47} + 38 q^{49} - 17 q^{55} + 30 q^{61} + 18 q^{63} + 36 q^{81} - 64 q^{83} - q^{85} + 19 q^{95}+O(q^{100}) \) 4 * q + q^5 + 6 * q^7 + 12 * q^9 - 16 * q^23 + 9 * q^25 - 27 * q^35 - 2 * q^43 + 3 * q^45 + 26 * q^47 + 38 * q^49 - 17 * q^55 + 30 * q^61 + 18 * q^63 + 36 * q^81 - 64 * q^83 - q^85 + 19 * q^95

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