LMFDB - Embedded newform 1900.2.l.a.1557.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(493,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1900.493");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.l (of order $4$, degree $2$, 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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.2702336256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			1557.4
Root			$-1.52274 - 1.63746i$ of defining polynomial
Character	$\chi$	$=$	1900.1557
Dual form			1900.2.l.a.493.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.84677 + 2.84677i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(2.84677 + 2.84677i) q^{7} -3.00000i q^{9} -6.50958 q^{11} +(-1.69609 - 1.69609i) q^{17} +4.35890i q^{19} +(-6.35890 + 6.35890i) q^{23} +(-8.74854 + 8.74854i) q^{43} +(-5.35635 - 5.35635i) q^{47} +9.20822i q^{49} +10.8109 q^{61} +(8.54032 - 8.54032i) q^{63} +(-5.11994 + 5.11994i) q^{73} +(-18.5313 - 18.5313i) q^{77} -9.00000 q^{81} +(3.64110 - 3.64110i) q^{83} +19.5287i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{7}+O(q^{10}) $ 8 * q + 6 * q^7	$ 8 q + 6 q^{7} - 14 q^{17} - 16 q^{23} - 2 q^{43} + 26 q^{47} + 18 q^{63} - 22 q^{73} - 26 q^{77} - 72 q^{81} + 64 q^{83}+O(q^{100}) $ 8 * q + 6 * q^7 - 14 * q^17 - 16 * q^23 - 2 * q^43 + 26 * q^47 + 18 * q^63 - 22 * q^73 - 26 * q^77 - 72 * q^81 + 64 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$

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		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.84677	+	2.84677i	1.07598	+	1.07598i	0.996866	+	0.0791130i	$0.0252088\pi$
$7$	2.84677	+	2.84677i	1.07598	+	1.07598i	0.0791130	+	0.996866i	$0.474791\pi$
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−6.50958			−1.96271			−0.981356	−	0.192201i	$-0.938437\pi$
$11$	−6.50958			−1.96271			−0.981356	+	0.192201i	$0.938437\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.69609	−	1.69609i	−0.411363	−	0.411363i	0.470850	−	0.882213i	$-0.343947\pi$
$17$	−1.69609	−	1.69609i	−0.411363	−	0.411363i	−0.882213	+	0.470850i	$0.843947\pi$
$18$		0			0
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.35890	+	6.35890i	−1.32592	+	1.32592i	−0.417029	+	0.908893i	$0.636929\pi$
$23$	−6.35890	+	6.35890i	−1.32592	+	1.32592i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−8.74854	+	8.74854i	−1.33414	+	1.33414i	−0.432511	+	0.901629i	$0.642372\pi$
$43$	−8.74854	+	8.74854i	−1.33414	+	1.33414i	−0.901629	+	0.432511i	$0.857628\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.35635	−	5.35635i	−0.781304	−	0.781304i	0.198747	−	0.980051i	$-0.436313\pi$
$47$	−5.35635	−	5.35635i	−0.781304	−	0.781304i	−0.980051	+	0.198747i	$0.936313\pi$
$48$		0			0
$49$			9.20822i			1.31546i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		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
$61$	10.8109			1.38420			0.692099	−	0.721803i	$-0.256686\pi$
$61$	10.8109			1.38420			0.692099	+	0.721803i	$0.256686\pi$
$62$		0			0
$63$	8.54032	−	8.54032i	1.07598	−	1.07598i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−5.11994	+	5.11994i	−0.599243	+	0.599243i	−0.940111	−	0.340868i	$-0.889279\pi$
$73$	−5.11994	+	5.11994i	−0.599243	+	0.599243i	0.340868	+	0.940111i	$0.389279\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−18.5313	−	18.5313i	−2.11184	−	2.11184i
$78$		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
$83$	3.64110	−	3.64110i	0.399663	−	0.399663i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	3.64110	−	3.64110i	0.399663	−	0.399663i	0.878114	+	0.478451i	$0.158802\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$		0			0
$99$			19.5287i			1.96271i
$100$		0			0
$101$	−17.4356			−1.73491			−0.867453	−	0.497519i	$-0.834245\pi$
$101$	−17.4356			−1.73491			−0.867453	+	0.497519i	$0.834245\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.65679i		−	0.885236i
$120$		0			0
$121$	31.3746			2.85224
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−22.3746			−1.95488			−0.977438	−	0.211221i	$-0.932256\pi$
$131$	−22.3746			−1.95488			−0.977438	+	0.211221i	$0.932256\pi$
$132$		0			0
$133$	−12.4088	+	12.4088i	−1.07598	+	1.07598i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	16.5070	+	16.5070i	1.41029	+	1.41029i	0.757778	+	0.652512i	$0.226285\pi$
$137$	16.5070	+	16.5070i	1.41029	+	1.41029i	0.652512	+	0.757778i	$0.273715\pi$
$138$		0			0
$139$			14.3746i			1.21924i	0.792695	+	0.609618i	$0.208677\pi$
$139$			14.3746i			1.21924i	−0.792695	+	0.609618i	$0.791323\pi$
$140$		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
$149$			24.3746i			1.99684i	0.0561570	+	0.998422i	$0.482115\pi$
$149$			24.3746i			1.99684i	−0.0561570	+	0.998422i	$0.517885\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−5.08828	+	5.08828i	−0.411363	+	0.411363i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.7178	−	17.7178i	−1.41403	−	1.41403i	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−17.7178	−	17.7178i	−1.41403	−	1.41403i	−0.695756	−	0.718278i	$-0.744931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−36.2047			−2.85333
$162$		0			0
$163$	−16.3589	+	16.3589i	−1.28133	+	1.28133i	−0.341415	+	0.939913i	$0.610906\pi$
$163$	−16.3589	+	16.3589i	−1.28133	+	1.28133i	−0.939913	+	0.341415i	$0.889094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	13.0767			1.00000
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		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
$187$	11.0409	+	11.0409i	0.807387	+	0.807387i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.3746			0.750679			0.375339	−	0.926887i	$-0.377526\pi$
$191$	10.3746			0.750679			0.375339	+	0.926887i	$0.377526\pi$
$192$		0			0
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.28220	+	2.28220i	0.162600	+	0.162600i	0.783718	−	0.621117i	$-0.213321\pi$
$197$	2.28220	+	2.28220i	0.162600	+	0.162600i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$		−	15.1123i		−	1.07128i	−0.844446	−	0.535641i	$-0.820070\pi$
$199$		−	15.1123i		−	1.07128i	0.844446	−	0.535641i	$-0.179930\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	19.0767	+	19.0767i	1.32592	+	1.32592i
$208$		0			0
$209$		−	28.3746i		−	1.96271i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		−	8.37459i		−	0.553408i	−0.960955	−	0.276704i	$-0.910758\pi$
$229$		−	8.37459i		−	0.553408i	0.960955	−	0.276704i	$-0.0892422\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.26715	+	6.26715i	−0.410575	+	0.410575i	−0.881939	−	0.471364i	$-0.843762\pi$
$233$	−6.26715	+	6.26715i	−0.410575	+	0.410575i	0.471364	+	0.881939i	$0.343762\pi$
$234$		0			0
$235$		0			0
$236$		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
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	30.3746			1.91723			0.958613	−	0.284711i	$-0.0918976\pi$
$251$	30.3746			1.91723			0.958613	+	0.284711i	$0.0918976\pi$
$252$		0			0
$253$	41.3937	−	41.3937i	2.60240	−	2.60240i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	22.8534	−	22.8534i	1.40920	−	1.40920i	0.645128	−	0.764075i	$-0.276804\pi$
$263$	22.8534	−	22.8534i	1.40920	−	1.40920i	0.764075	−	0.645128i	$-0.223196\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	26.1534			1.58871			0.794353	−	0.607457i	$-0.207810\pi$
$271$	26.1534			1.58871			0.794353	+	0.607457i	$0.207810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.01660	−	9.01660i	−0.541755	−	0.541755i	0.382288	−	0.924043i	$-0.375136\pi$
$277$	−9.01660	−	9.01660i	−0.541755	−	0.541755i	−0.924043	+	0.382288i	$0.875136\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−18.7802	+	18.7802i	−1.11637	+	1.11637i	−0.124096	+	0.992270i	$0.539603\pi$
$283$	−18.7802	+	18.7802i	−1.11637	+	1.11637i	−0.992270	+	0.124096i	$0.960397\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	11.2465i		−	0.661560i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−49.8102			−2.87101
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	28.1314			1.59519			0.797594	−	0.603195i	$-0.206106\pi$
$311$	28.1314			1.59519			0.797594	+	0.603195i	$0.206106\pi$
$312$		0			0
$313$	20.4356	−	20.4356i	1.15509	−	1.15509i	0.169570	−	0.985518i	$-0.445762\pi$
$313$	20.4356	−	20.4356i	1.15509	−	1.15509i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.39310	−	7.39310i	0.411363	−	0.411363i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	30.4966i		−	1.68133i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.28630	+	6.28630i	−0.339428	+	0.339428i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−24.4737	−	24.4737i	−1.31382	−	1.31382i	−0.918577	−	0.395242i	$-0.870661\pi$
$347$	−24.4737	−	24.4737i	−1.31382	−	1.31382i	−0.395242	−	0.918577i	$-0.629339\pi$
$348$		0			0
$349$			36.8492i			1.97249i	0.165277	+	0.986247i	$0.447148\pi$
$349$			36.8492i			1.97249i	−0.165277	+	0.986247i	$0.552852\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.4356	−	10.4356i	0.555431	−	0.555431i	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	10.4356	−	10.4356i	0.555431	−	0.555431i	0.928003	+	0.372572i	$0.121524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			34.3746i			1.81422i	0.420892	+	0.907111i	$0.361717\pi$
$359$			34.3746i			1.81422i	−0.420892	+	0.907111i	$0.638283\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.923303	−	0.923303i	−0.0481960	−	0.0481960i	0.682598	−	0.730794i	$-0.260850\pi$
$367$	−0.923303	−	0.923303i	−0.0481960	−	0.0481960i	−0.730794	+	0.682598i	$0.760850\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		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		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	26.2456	+	26.2456i	1.33414	+	1.33414i
$388$		0			0
$389$			6.39449i			0.324213i	0.986773	+	0.162107i	$0.0518289\pi$
$389$			6.39449i			0.324213i	−0.986773	+	0.162107i	$0.948171\pi$
$390$		0			0
$391$	21.5706			1.09087
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−11.9607	−	11.9607i	−0.600290	−	0.600290i	0.340099	−	0.940389i	$-0.389539\pi$
$397$	−11.9607	−	11.9607i	−0.600290	−	0.600290i	−0.940389	+	0.340099i	$0.889539\pi$
$398$		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
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		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
$423$	−16.0690	+	16.0690i	−0.781304	+	0.781304i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	30.7763	+	30.7763i	1.48937	+	1.48937i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−27.7178	−	27.7178i	−1.32592	−	1.32592i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	27.6247			1.31546
$442$		0			0
$443$	15.5329	−	15.5329i	0.737991	−	0.737991i	−0.234198	−	0.972189i	$-0.575246\pi$
$443$	15.5329	−	15.5329i	0.737991	−	0.737991i	0.972189	+	0.234198i	$0.0752464\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	29.9059	+	29.9059i	1.39894	+	1.39894i	0.803120	+	0.595818i	$0.203172\pi$
$457$	29.9059	+	29.9059i	1.39894	+	1.39894i	0.595818	+	0.803120i	$0.296828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.374586			0.0174462			0.00872311	−	0.999962i	$-0.497223\pi$
$461$	0.374586			0.0174462			0.00872311	+	0.999962i	$0.497223\pi$
$462$		0			0
$463$	−1.42803	+	1.42803i	−0.0663661	+	0.0663661i	−0.739511	−	0.673145i	$-0.764943\pi$
$463$	−1.42803	+	1.42803i	−0.0663661	+	0.0663661i	0.673145	+	0.739511i	$0.264943\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.1673	+	30.1673i	1.39598	+	1.39598i	0.811183	+	0.584792i	$0.198824\pi$
$467$	30.1673	+	30.1673i	1.39598	+	1.39598i	0.584792	+	0.811183i	$0.301176\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	56.9493	−	56.9493i	2.61853	−	2.61853i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			4.00000i			0.182765i	0.995816	+	0.0913823i	$0.0291285\pi$
$479$			4.00000i			0.182765i	−0.995816	+	0.0913823i	$0.970871\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	38.3746i		−	1.71788i	−0.512074	−	0.858941i	$-0.671123\pi$
$499$		−	38.3746i		−	1.71788i	0.512074	−	0.858941i	$-0.328877\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.3589	+	26.3589i	−1.17529	+	1.17529i	−0.194354	+	0.980932i	$0.562261\pi$
$503$	−26.3589	+	26.3589i	−1.17529	+	1.17529i	−0.980932	+	0.194354i	$0.937739\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−29.1506			−1.28955
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	34.8676	+	34.8676i	1.53347	+	1.53347i
$518$		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		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	57.8712i		−	2.51614i
$530$		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
$539$		−	59.9416i		−	2.58187i
$540$		0			0
$541$	41.2657			1.77415			0.887075	−	0.461625i	$-0.152733\pi$
$541$	41.2657			1.77415			0.887075	+	0.461625i	$0.152733\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		−	32.4328i		−	1.38420i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.62441	+	5.62441i	0.238314	+	0.238314i	0.816152	−	0.577838i	$-0.196103\pi$
$557$	5.62441	+	5.62441i	0.238314	+	0.238314i	−0.577838	+	0.816152i	$0.696103\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−25.6209	−	25.6209i	−1.07598	−	1.07598i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	26.1534			1.09449			0.547243	−	0.836974i	$-0.315677\pi$
$571$	26.1534			1.09449			0.547243	+	0.836974i	$0.315677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−33.2981	−	33.2981i	−1.38622	−	1.38622i	−0.833112	−	0.553104i	$-0.813443\pi$
$577$	−33.2981	−	33.2981i	−1.38622	−	1.38622i	−0.553104	−	0.833112i	$-0.686557\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	20.7308			0.860057
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.99398	+	3.99398i	0.164849	+	0.164849i	0.784711	−	0.619862i	$-0.212811\pi$
$587$	3.99398	+	3.99398i	0.164849	+	0.164849i	−0.619862	+	0.784711i	$0.712811\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.435596	−	0.435596i	0.0178878	−	0.0178878i	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	0.435596	−	0.435596i	0.0178878	−	0.0178878i	0.715994	+	0.698106i	$0.245974\pi$
$594$		0			0
$595$		0			0
$596$		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
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	21.0534	−	21.0534i	0.850337	−	0.850337i	−0.139837	−	0.990174i	$-0.544658\pi$
$613$	21.0534	−	21.0534i	0.850337	−	0.850337i	0.990174	+	0.139837i	$0.0446580\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.6542	+	17.6542i	0.710733	+	0.710733i	0.966689	−	0.255956i	$-0.0823901\pi$
$617$	17.6542	+	17.6542i	0.710733	+	0.710733i	−0.255956	+	0.966689i	$0.582390\pi$
$618$		0			0
$619$			24.0000i			0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$			24.0000i			0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−36.9643			−1.47153			−0.735763	−	0.677239i	$-0.763176\pi$
$631$	−36.9643			−1.47153			−0.735763	+	0.677239i	$0.763176\pi$
$632$		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
$643$	35.8608	−	35.8608i	1.41421	−	1.41421i	0.706395	−	0.707818i	$-0.250320\pi$
$643$	35.8608	−	35.8608i	1.41421	−	1.41421i	0.707818	−	0.706395i	$-0.249680\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−29.6378	−	29.6378i	−1.16518	−	1.16518i	−0.983325	−	0.181857i	$-0.941789\pi$
$647$	−29.6378	−	29.6378i	−1.16518	−	1.16518i	−0.181857	−	0.983325i	$-0.558211\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−33.5877	+	33.5877i	−1.31439	+	1.31439i	−0.396239	+	0.918147i	$0.629685\pi$
$653$	−33.5877	+	33.5877i	−1.31439	+	1.31439i	−0.918147	+	0.396239i	$0.870315\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	15.3598	+	15.3598i	0.599243	+	0.599243i
$658$		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
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−70.3746			−2.71678
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−10.6958			−0.406889			−0.203445	−	0.979086i	$-0.565214\pi$
$691$	−10.6958			−0.406889			−0.203445	+	0.979086i	$0.565214\pi$
$692$		0			0
$693$	−55.5938	+	55.5938i	−2.11184	+	2.11184i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.4356			−0.658533			−0.329267	−	0.944237i	$-0.606802\pi$
$701$	−17.4356			−0.658533			−0.329267	+	0.944237i	$0.606802\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−49.6352	−	49.6352i	−1.86672	−	1.86672i
$708$		0			0
$709$			52.3068i			1.96442i	0.187779	+	0.982211i	$0.439871\pi$
$709$			52.3068i			1.96442i	−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
$719$		−	5.62541i		−	0.209793i	−0.994483	−	0.104896i	$-0.966549\pi$
$719$		−	5.62541i		−	0.209793i	0.994483	−	0.104896i	$-0.0334511\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	33.5661	+	33.5661i	1.24490	+	1.24490i	0.957944	+	0.286954i	$0.0926427\pi$
$727$	33.5661	+	33.5661i	1.24490	+	1.24490i	0.286954	+	0.957944i	$0.407357\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$	29.6767			1.09763
$732$		0			0
$733$	−33.1534	+	33.1534i	−1.22455	+	1.22455i	−0.258551	+	0.965998i	$0.583245\pi$
$733$	−33.1534	+	33.1534i	−1.22455	+	1.22455i	−0.965998	+	0.258551i	$0.916755\pi$
$734$		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		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−10.9233	−	10.9233i	−0.399663	−	0.399663i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−9.66627	−	9.66627i	−0.351327	−	0.351327i	0.509276	−	0.860603i	$-0.329913\pi$
$757$	−9.66627	−	9.66627i	−0.351327	−	0.351327i	−0.860603	+	0.509276i	$0.829913\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	45.4519			1.64763			0.823816	−	0.566857i	$-0.191841\pi$
$761$	45.4519			1.64763			0.823816	+	0.566857i	$0.191841\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			44.3746i			1.60019i	0.599874	+	0.800094i	$0.295217\pi$
$769$			44.3746i			1.60019i	−0.599874	+	0.800094i	$0.704783\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		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
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$			18.1697i			0.642799i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	33.3286	−	33.3286i	1.17614	−	1.17614i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			32.6630i			1.14837i	0.818726	+	0.574184i	$0.194681\pi$
$809$			32.6630i			1.14837i	−0.818726	+	0.574184i	$0.805319\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−38.1340	−	38.1340i	−1.33414	−	1.33414i
$818$		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
$823$	−40.3505	+	40.3505i	−1.40653	+	1.40653i	−0.629655	+	0.776875i	$0.716804\pi$
$823$	−40.3505	+	40.3505i	−1.40653	+	1.40653i	−0.776875	+	0.629655i	$0.783196\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.6180	−	15.6180i	0.541132	−	0.541132i
$834$		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
$845$		0			0
$846$		0			0
$847$	89.3163	+	89.3163i	3.06894	+	3.06894i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−13.1534	+	13.1534i	−0.450364	+	0.450364i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	−13.1534	+	13.1534i	−0.450364	+	0.450364i	0.445112	+	0.895475i	$0.353164\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		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		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		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
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−54.2848			−1.82890			−0.914451	−	0.404696i	$-0.867377\pi$
$881$	−54.2848			−1.82890			−0.914451	+	0.404696i	$0.867377\pi$
$882$		0			0
$883$	30.1739	−	30.1739i	1.01543	−	1.01543i	0.0155546	−	0.999879i	$-0.495049\pi$
$883$	30.1739	−	30.1739i	1.01543	−	1.01543i	0.999879	−	0.0155546i	$-0.00495139\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	58.5862			1.96271
$892$		0			0
$893$	23.3478	−	23.3478i	0.781304	−	0.781304i
$894$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$			52.3068i			1.73491i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−23.7020	+	23.7020i	−0.784423	+	0.784423i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−63.6953	−	63.6953i	−2.10341	−	2.10341i
$918$		0			0
$919$			8.71780i			0.287574i	0.989609	+	0.143787i	$0.0459280\pi$
$919$			8.71780i			0.287574i	−0.989609	+	0.143787i	$0.954072\pi$
$920$		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
$929$		−	34.8712i		−	1.14409i	−0.820223	−	0.572043i	$-0.806151\pi$
$929$		−	34.8712i		−	1.14409i	0.820223	−	0.572043i	$-0.193849\pi$
$930$		0			0
$931$	−40.1377			−1.31546
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−39.2812	−	39.2812i	−1.28326	−	1.28326i	−0.938801	−	0.344460i	$-0.888062\pi$
$937$	−39.2812	−	39.2812i	−1.28326	−	1.28326i	−0.344460	−	0.938801i	$-0.611938\pi$
$938$		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
$947$	−34.5123	−	34.5123i	−1.12150	−	1.12150i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	−34.5123	−	34.5123i	−1.12150	−	1.12150i	−0.129983	−	0.991516i	$-0.541492\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			93.9835i			3.03488i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−24.5123	−	24.5123i	−0.788262	−	0.788262i	0.192947	−	0.981209i	$-0.438195\pi$
$967$	−24.5123	−	24.5123i	−0.788262	−	0.788262i	−0.981209	+	0.192947i	$0.938195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−40.9212	+	40.9212i	−1.31187	+	1.31187i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	111.262i		−	3.53793i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	43.8275	+	43.8275i	1.38803	+	1.38803i	0.829448	+	0.558584i	$0.188655\pi$
$997$	43.8275	+	43.8275i	1.38803	+	1.38803i	0.558584	+	0.829448i	$0.311345\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.l.a.1557.4		8
5.2	odd	4		380.2.l.a.113.4	yes	8
5.3	odd	4	inner	1900.2.l.a.493.4		8
5.4	even	2		380.2.l.a.37.3	✓	8
15.2	even	4		3420.2.bb.c.2773.1		8
15.14	odd	2		3420.2.bb.c.37.2		8
19.18	odd	2	CM	1900.2.l.a.1557.4		8
95.18	even	4	inner	1900.2.l.a.493.4		8
95.37	even	4		380.2.l.a.113.4	yes	8
95.94	odd	2		380.2.l.a.37.3	✓	8
285.227	odd	4		3420.2.bb.c.2773.1		8
285.284	even	2		3420.2.bb.c.37.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.l.a.37.3	✓	8	5.4	even	2
380.2.l.a.37.3	✓	8	95.94	odd	2
380.2.l.a.113.4	yes	8	5.2	odd	4
380.2.l.a.113.4	yes	8	95.37	even	4
1900.2.l.a.493.4		8	5.3	odd	4	inner
1900.2.l.a.493.4		8	95.18	even	4	inner
1900.2.l.a.1557.4		8	1.1	even	1	trivial
1900.2.l.a.1557.4		8	19.18	odd	2	CM
3420.2.bb.c.37.2		8	15.14	odd	2
3420.2.bb.c.37.2		8	285.284	even	2
3420.2.bb.c.2773.1		8	15.2	even	4
3420.2.bb.c.2773.1		8	285.227	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.84677 + 2.84677i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(2.84677 + 2.84677i) q^{7} -3.00000i q^{9} -6.50958 q^{11} +(-1.69609 - 1.69609i) q^{17} +4.35890i q^{19} +(-6.35890 + 6.35890i) q^{23} +(-8.74854 + 8.74854i) q^{43} +(-5.35635 - 5.35635i) q^{47} +9.20822i q^{49} +10.8109 q^{61} +(8.54032 - 8.54032i) q^{63} +(-5.11994 + 5.11994i) q^{73} +(-18.5313 - 18.5313i) q^{77} -9.00000 q^{81} +(3.64110 - 3.64110i) q^{83} +19.5287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{7}+O(q^{10}) \) 8 * q + 6 * q^7	\( 8 q + 6 q^{7} - 14 q^{17} - 16 q^{23} - 2 q^{43} + 26 q^{47} + 18 q^{63} - 22 q^{73} - 26 q^{77} - 72 q^{81} + 64 q^{83}+O(q^{100}) \) 8 * q + 6 * q^7 - 14 * q^17 - 16 * q^23 - 2 * q^43 + 26 * q^47 + 18 * q^63 - 22 * q^73 - 26 * q^77 - 72 * q^81 + 64 * q^83

Introduction

L-functions

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