LMFDB - Embedded newform 900.3.l.e.757.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(757,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("900.757");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			757.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	900.757
Dual form			900.3.l.e.793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.67423 + 3.67423i) q^{7} +O(q^{10})$	$q+(3.67423 + 3.67423i) q^{7} +(-8.57321 + 8.57321i) q^{13} +11.0000i q^{19} -59.0000 q^{31} +(48.9898 + 48.9898i) q^{37} +(-15.9217 + 15.9217i) q^{43} -22.0000i q^{49} -121.000 q^{61} +(94.3054 + 94.3054i) q^{67} +(-97.9796 + 97.9796i) q^{73} +142.000i q^{79} -63.0000 q^{91} +(69.8105 + 69.8105i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 236 q^{31} - 484 q^{61} - 252 q^{91}+O(q^{100}) $ 4 * q - 236 * q^31 - 484 * q^61 - 252 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.67423	+	3.67423i	0.524891	+	0.524891i	0.919044	−	0.394154i	$-0.128962\pi$
$7$	3.67423	+	3.67423i	0.524891	+	0.524891i	−0.394154	+	0.919044i	$0.628962\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−8.57321	+	8.57321i	−0.659478	+	0.659478i	−0.955257	−	0.295779i	$-0.904421\pi$
$13$	−8.57321	+	8.57321i	−0.659478	+	0.659478i	0.295779	+	0.955257i	$0.404421\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$			11.0000i			0.578947i	0.957186	+	0.289474i	$0.0934803\pi$
$19$			11.0000i			0.578947i	−0.957186	+	0.289474i	$0.906520\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−59.0000			−1.90323			−0.951613	−	0.307299i	$-0.900575\pi$
$31$	−59.0000			−1.90323			−0.951613	+	0.307299i	$0.900575\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	48.9898	+	48.9898i	1.32405	+	1.32405i	0.910467	+	0.413581i	$0.135722\pi$
$37$	48.9898	+	48.9898i	1.32405	+	1.32405i	0.413581	+	0.910467i	$0.364278\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−15.9217	+	15.9217i	−0.370272	+	0.370272i	−0.867576	−	0.497304i	$-0.834323\pi$
$43$	−15.9217	+	15.9217i	−0.370272	+	0.370272i	0.497304	+	0.867576i	$0.334323\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	22.0000i		−	0.448980i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−121.000			−1.98361			−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−121.000			−1.98361			−0.991803	+	0.127774i	$0.959217\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	94.3054	+	94.3054i	1.40754	+	1.40754i	0.772371	+	0.635171i	$0.219071\pi$
$67$	94.3054	+	94.3054i	1.40754	+	1.40754i	0.635171	+	0.772371i	$0.280929\pi$
$68$		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
$73$	−97.9796	+	97.9796i	−1.34219	+	1.34219i	−0.448306	+	0.893880i	$0.647973\pi$
$73$	−97.9796	+	97.9796i	−1.34219	+	1.34219i	−0.893880	+	0.448306i	$0.852027\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			142.000i			1.79747i	0.438494	+	0.898734i	$0.355512\pi$
$79$			142.000i			1.79747i	−0.438494	+	0.898734i	$0.644488\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−63.0000			−0.692308
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	69.8105	+	69.8105i	0.719695	+	0.719695i	0.968543	−	0.248847i	$-0.0800517\pi$
$97$	69.8105	+	69.8105i	0.719695	+	0.719695i	−0.248847	+	0.968543i	$0.580052\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−48.9898	+	48.9898i	−0.475629	+	0.475629i	−0.903731	−	0.428102i	$-0.859183\pi$
$103$	−48.9898	+	48.9898i	−0.475629	+	0.475629i	0.428102	+	0.903731i	$0.359183\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		−	71.0000i		−	0.651376i	−0.945477	−	0.325688i	$-0.894404\pi$
$109$		−	71.0000i		−	0.651376i	0.945477	−	0.325688i	$-0.105596\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−121.000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	146.969	+	146.969i	1.15724	+	1.15724i	0.985067	+	0.172172i	$0.0550786\pi$
$127$	146.969	+	146.969i	1.15724	+	1.15724i	0.172172	+	0.985067i	$0.444921\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−40.4166	+	40.4166i	−0.303884	+	0.303884i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		−	22.0000i		−	0.158273i	−0.996864	−	0.0791367i	$-0.974784\pi$
$139$		−	22.0000i		−	0.158273i	0.996864	−	0.0791367i	$-0.0252164\pi$
$140$		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
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−59.0000			−0.390728			−0.195364	−	0.980731i	$-0.562589\pi$
$151$	−59.0000			−0.390728			−0.195364	+	0.980731i	$0.562589\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−175.139	−	175.139i	−1.11553	−	1.11553i	−0.992389	−	0.123143i	$-0.960703\pi$
$157$	−175.139	−	175.139i	−1.11553	−	1.11553i	−0.123143	−	0.992389i	$-0.539297\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	229.027	−	229.027i	1.40508	−	1.40508i	0.622283	−	0.782792i	$-0.286205\pi$
$163$	229.027	−	229.027i	1.40508	−	1.40508i	0.782792	−	0.622283i	$-0.213795\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			22.0000i			0.130178i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	1.00000			0.00552486			0.00276243	−	0.999996i	$-0.499121\pi$
$181$	1.00000			0.00552486			0.00276243	+	0.999996i	$0.499121\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−253.522	+	253.522i	−1.31359	+	1.31359i	−0.394834	+	0.918752i	$0.629198\pi$
$193$	−253.522	+	253.522i	−1.31359	+	1.31359i	−0.918752	+	0.394834i	$0.870802\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$			109.000i			0.547739i	0.961767	+	0.273869i	$0.0883036\pi$
$199$			109.000i			0.547739i	−0.961767	+	0.273869i	$0.911696\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	419.000			1.98578			0.992891	−	0.119027i	$-0.0379776\pi$
$211$	419.000			1.98578			0.992891	+	0.119027i	$0.0379776\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−216.780	−	216.780i	−0.998985	−	0.998985i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	309.860	−	309.860i	1.38951	−	1.38951i	0.563163	−	0.826346i	$-0.309584\pi$
$223$	309.860	−	309.860i	1.38951	−	1.38951i	0.826346	−	0.563163i	$-0.190416\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	409.000i		−	1.78603i	−0.450031	−	0.893013i	$-0.648587\pi$
$229$		−	409.000i		−	1.78603i	0.450031	−	0.893013i	$-0.351413\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	479.000			1.98755			0.993776	−	0.111397i	$-0.0355327\pi$
$241$	479.000			1.98755			0.993776	+	0.111397i	$0.0355327\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−94.3054	−	94.3054i	−0.381803	−	0.381803i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			360.000i			1.38996i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		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
$271$	242.000			0.892989			0.446494	−	0.894786i	$-0.352672\pi$
$271$	242.000			0.892989			0.446494	+	0.894786i	$0.352672\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−265.770	−	265.770i	−0.959457	−	0.959457i	0.0397524	−	0.999210i	$-0.487343\pi$
$277$	−265.770	−	265.770i	−0.959457	−	0.959457i	−0.999210	+	0.0397524i	$0.987343\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	162.891	−	162.891i	0.575587	−	0.575587i	−0.358097	−	0.933684i	$-0.616574\pi$
$283$	162.891	−	162.891i	0.575587	−	0.575587i	0.933684	+	0.358097i	$0.116574\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	289.000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−117.000			−0.388704
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−395.593	−	395.593i	−1.28858	−	1.28858i	−0.935652	−	0.352923i	$-0.885188\pi$
$307$	−395.593	−	395.593i	−1.28858	−	1.28858i	−0.352923	−	0.935652i	$-0.614812\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	302.512	−	302.512i	0.966492	−	0.966492i	−0.0329646	−	0.999457i	$-0.510495\pi$
$313$	302.512	−	302.512i	0.966492	−	0.966492i	0.999457	+	0.0329646i	$0.0104948\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	362.000			1.09366			0.546828	−	0.837245i	$-0.315835\pi$
$331$	362.000			1.09366			0.546828	+	0.837245i	$0.315835\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−461.729	−	461.729i	−1.37012	−	1.37012i	−0.860261	−	0.509855i	$-0.829699\pi$
$337$	−461.729	−	461.729i	−1.37012	−	1.37012i	−0.509855	−	0.860261i	$-0.670301\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	260.871	−	260.871i	0.760556	−	0.760556i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	502.000i		−	1.43840i	−0.694805	−	0.719198i	$-0.744510\pi$
$349$		−	502.000i		−	1.43840i	0.694805	−	0.719198i	$-0.255490\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	240.000			0.664820
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	493.572	+	493.572i	1.34488	+	1.34488i	0.891125	+	0.453759i	$0.149917\pi$
$367$	493.572	+	493.572i	1.34488	+	1.34488i	0.453759	+	0.891125i	$0.350083\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−334.355	+	334.355i	−0.896395	+	0.896395i	−0.995115	−	0.0987202i	$-0.968525\pi$
$373$	−334.355	+	334.355i	−0.896395	+	0.896395i	0.0987202	+	0.995115i	$0.468525\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			611.000i			1.61214i	0.591822	+	0.806069i	$0.298409\pi$
$379$			611.000i			1.61214i	−0.591822	+	0.806069i	$0.701591\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	28.1691	+	28.1691i	0.0709550	+	0.0709550i	0.741694	−	0.670739i	$-0.234023\pi$
$397$	28.1691	+	28.1691i	0.0709550	+	0.0709550i	−0.670739	+	0.741694i	$0.734023\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	505.820	−	505.820i	1.25514	−	1.25514i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	769.000i		−	1.88020i	−0.340905	−	0.940098i	$-0.610733\pi$
$409$		−	769.000i		−	1.88020i	0.340905	−	0.940098i	$-0.389267\pi$
$410$		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
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	358.000			0.850356			0.425178	−	0.905110i	$-0.360211\pi$
$421$	358.000			0.850356			0.425178	+	0.905110i	$0.360211\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−444.582	−	444.582i	−1.04118	−	1.04118i
$428$		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
$433$	−498.471	+	498.471i	−1.15120	+	1.15120i	−0.164892	+	0.986312i	$0.552728\pi$
$433$	−498.471	+	498.471i	−1.15120	+	1.15120i	−0.986312	+	0.164892i	$0.947272\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			709.000i			1.61503i	0.589844	+	0.807517i	$0.299189\pi$
$439$			709.000i			1.61503i	−0.589844	+	0.807517i	$0.700811\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		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
$457$	293.939	+	293.939i	0.643192	+	0.643192i	0.951339	−	0.308147i	$-0.0997088\pi$
$457$	293.939	+	293.939i	0.643192	+	0.643192i	−0.308147	+	0.951339i	$0.599709\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−538.888	+	538.888i	−1.16390	+	1.16390i	−0.180291	+	0.983613i	$0.557704\pi$
$463$	−538.888	+	538.888i	−1.16390	+	1.16390i	−0.983613	+	0.180291i	$0.942296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$			693.000i			1.47761i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	−840.000			−1.74636
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	535.214	+	535.214i	1.09900	+	1.09900i	0.994528	+	0.104473i	$0.0333157\pi$
$487$	535.214	+	535.214i	1.09900	+	1.09900i	0.104473	+	0.994528i	$0.466684\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	851.000i		−	1.70541i	−0.522392	−	0.852705i	$-0.674960\pi$
$499$		−	851.000i		−	1.70541i	0.522392	−	0.852705i	$-0.325040\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	−720.000			−1.40900
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	473.976	−	473.976i	0.906264	−	0.906264i	−0.0897041	−	0.995968i	$-0.528592\pi$
$523$	473.976	−	473.976i	0.906264	−	0.906264i	0.995968	+	0.0897041i	$0.0285921\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			529.000i			1.00000i
$530$		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
$539$		0			0
$540$		0			0
$541$	−241.000			−0.445471			−0.222736	−	0.974879i	$-0.571499\pi$
$541$	−241.000			−0.445471			−0.222736	+	0.974879i	$0.571499\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	685.857	+	685.857i	1.25385	+	1.25385i	0.953979	+	0.299873i	$0.0969443\pi$
$547$	685.857	+	685.857i	1.25385	+	1.25385i	0.299873	+	0.953979i	$0.403056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−521.741	+	521.741i	−0.943474	+	0.943474i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		−	273.000i		−	0.488372i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−181.000			−0.316988			−0.158494	−	0.987360i	$-0.550664\pi$
$571$	−181.000			−0.316988			−0.158494	+	0.987360i	$0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	363.749	+	363.749i	0.630415	+	0.630415i	0.948172	−	0.317757i	$-0.102930\pi$
$577$	363.749	+	363.749i	0.630415	+	0.630415i	−0.317757	+	0.948172i	$0.602930\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		−	649.000i		−	1.10187i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−1199.00			−1.99501			−0.997504	−	0.0706077i	$-0.977506\pi$
$601$	−1199.00			−1.99501			−0.997504	+	0.0706077i	$0.977506\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	636.867	+	636.867i	1.04920	+	1.04920i	0.998725	+	0.0504797i	$0.0160750\pi$
$607$	636.867	+	636.867i	1.04920	+	1.04920i	0.0504797	+	0.998725i	$0.483925\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−342.929	+	342.929i	−0.559427	+	0.559427i	−0.929144	−	0.369718i	$-0.879454\pi$
$613$	−342.929	+	342.929i	−0.559427	+	0.559427i	0.369718	+	0.929144i	$0.379454\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$			949.000i			1.53312i	0.642174	+	0.766559i	$0.278033\pi$
$619$			949.000i			1.53312i	−0.642174	+	0.766559i	$0.721967\pi$
$620$		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
$631$	−1261.00			−1.99842			−0.999208	−	0.0398015i	$-0.987327\pi$
$631$	−1261.00			−1.99842			−0.999208	+	0.0398015i	$0.987327\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	188.611	+	188.611i	0.296092	+	0.296092i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−881.816	+	881.816i	−1.37141	+	1.37141i	−0.513052	+	0.858358i	$0.671485\pi$
$643$	−881.816	+	881.816i	−1.37141	+	1.37141i	−0.858358	+	0.513052i	$0.828515\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−122.000			−0.184569			−0.0922844	−	0.995733i	$-0.529417\pi$
$661$	−122.000			−0.184569			−0.0922844	+	0.995733i	$0.529417\pi$
$662$		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
$671$		0			0
$672$		0			0
$673$	489.898	−	489.898i	0.727932	−	0.727932i	−0.242276	−	0.970207i	$-0.577894\pi$
$673$	489.898	−	489.898i	0.727932	−	0.727932i	0.970207	+	0.242276i	$0.0778939\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$			513.000i			0.755523i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1318.00			1.90738			0.953690	−	0.300790i	$-0.0972504\pi$
$691$	1318.00			1.90738			0.953690	+	0.300790i	$0.0972504\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−538.888	+	538.888i	−0.766554	+	0.766554i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			1391.00i			1.96192i	0.194214	+	0.980959i	$0.437784\pi$
$709$			1391.00i			1.96192i	−0.194214	+	0.980959i	$0.562216\pi$
$710$		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
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−360.000			−0.499307
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	780.162	+	780.162i	1.07313	+	1.07313i	0.997106	+	0.0760195i	$0.0242211\pi$
$727$	780.162	+	780.162i	1.07313	+	1.07313i	0.0760195	+	0.997106i	$0.475779\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−734.847	+	734.847i	−1.00252	+	1.00252i	−0.00252286	+	0.999997i	$0.500803\pi$
$733$	−734.847	+	734.847i	−1.00252	+	1.00252i	−0.999997	+	0.00252286i	$0.999197\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	1222.00i		−	1.65359i	−0.562506	−	0.826793i	$-0.690163\pi$
$739$		−	1222.00i		−	1.65359i	0.562506	−	0.826793i	$-0.309837\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1202.00			−1.60053			−0.800266	−	0.599645i	$-0.795309\pi$
$751$	−1202.00			−1.60053			−0.800266	+	0.599645i	$0.795309\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	958.975	+	958.975i	1.26681	+	1.26681i	0.947727	+	0.319083i	$0.103375\pi$
$757$	958.975	+	958.975i	1.26681	+	1.26681i	0.319083	+	0.947727i	$0.396625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	260.871	−	260.871i	0.341901	−	0.341901i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	671.000i		−	0.872562i	−0.899811	−	0.436281i	$-0.856295\pi$
$769$		−	671.000i		−	0.872562i	0.899811	−	0.436281i	$-0.143705\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		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
$787$	−1025.11	−	1025.11i	−1.30256	−	1.30256i	−0.926663	−	0.375893i	$-0.877336\pi$
$787$	−1025.11	−	1025.11i	−1.30256	−	1.30256i	−0.375893	−	0.926663i	$-0.622664\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1037.36	−	1037.36i	1.30814	−	1.30814i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	1261.00			1.55487			0.777435	−	0.628963i	$-0.216520\pi$
$811$	1261.00			1.55487			0.777435	+	0.628963i	$0.216520\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−175.139	−	175.139i	−0.214368	−	0.214368i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	−1093.70	+	1093.70i	−1.32892	+	1.32892i	−0.422598	+	0.906317i	$0.638882\pi$
$823$	−1093.70	+	1093.70i	−1.32892	+	1.32892i	−0.906317	+	0.422598i	$0.861118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			458.000i			0.552473i	0.961090	+	0.276236i	$0.0890873\pi$
$829$			458.000i			0.552473i	−0.961090	+	0.276236i	$0.910913\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−444.582	−	444.582i	−0.524891	−	0.524891i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	873.243	−	873.243i	1.02373	−	1.02373i	0.0240202	−	0.999711i	$-0.492353\pi$
$853$	873.243	−	873.243i	1.02373	−	1.02373i	0.999711	−	0.0240202i	$-0.00764659\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		−	1418.00i		−	1.65076i	−0.564580	−	0.825378i	$-0.690962\pi$
$859$		−	1418.00i		−	1.65076i	0.564580	−	0.825378i	$-0.309038\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1617.00			−1.85649
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−216.780	−	216.780i	−0.247183	−	0.247183i	0.572630	−	0.819814i	$-0.305923\pi$
$877$	−216.780	−	216.780i	−0.247183	−	0.247183i	−0.819814	+	0.572630i	$0.805923\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	1208.82	−	1208.82i	1.36900	−	1.36900i	0.507121	−	0.861875i	$-0.330710\pi$
$883$	1208.82	−	1208.82i	1.36900	−	1.36900i	0.861875	−	0.507121i	$-0.169290\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$			1080.00i			1.21485i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		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
$907$	−1273.73	−	1273.73i	−1.40434	−	1.40434i	−0.785587	−	0.618751i	$-0.787639\pi$
$907$	−1273.73	−	1273.73i	−1.40434	−	1.40434i	−0.618751	−	0.785587i	$-0.712361\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			971.000i			1.05658i	0.849063	+	0.528292i	$0.177167\pi$
$919$			971.000i			1.05658i	−0.849063	+	0.528292i	$0.822833\pi$
$920$		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
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	242.000			0.259936
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	224.128	+	224.128i	0.239198	+	0.239198i	0.816518	−	0.577320i	$-0.195902\pi$
$937$	224.128	+	224.128i	0.239198	+	0.239198i	−0.577320	+	0.816518i	$0.695902\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		−	1680.00i		−	1.77028i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2520.00			2.62227
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−832.827	−	832.827i	−0.861248	−	0.861248i	0.130235	−	0.991483i	$-0.458427\pi$
$967$	−832.827	−	832.827i	−0.861248	−	0.861248i	−0.991483	+	0.130235i	$0.958427\pi$
$968$		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
$973$	80.8332	−	80.8332i	0.0830762	−	0.0830762i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1739.00			1.75479			0.877397	−	0.479766i	$-0.159278\pi$
$991$	1739.00			1.75479			0.877397	+	0.479766i	$0.159278\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−440.908	−	440.908i	−0.442235	−	0.442235i	0.450528	−	0.892762i	$-0.351236\pi$
$997$	−440.908	−	440.908i	−0.442235	−	0.442235i	−0.892762	+	0.450528i	$0.851236\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.l.e.757.2	yes	4
3.2	odd	2	CM	900.3.l.e.757.2	yes	4
5.2	odd	4	inner	900.3.l.e.793.1	yes	4
5.3	odd	4	inner	900.3.l.e.793.2	yes	4
5.4	even	2	inner	900.3.l.e.757.1	✓	4
15.2	even	4	inner	900.3.l.e.793.1	yes	4
15.8	even	4	inner	900.3.l.e.793.2	yes	4
15.14	odd	2	inner	900.3.l.e.757.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.3.l.e.757.1	✓	4	5.4	even	2	inner
900.3.l.e.757.1	✓	4	15.14	odd	2	inner
900.3.l.e.757.2	yes	4	1.1	even	1	trivial
900.3.l.e.757.2	yes	4	3.2	odd	2	CM
900.3.l.e.793.1	yes	4	5.2	odd	4	inner
900.3.l.e.793.1	yes	4	15.2	even	4	inner
900.3.l.e.793.2	yes	4	5.3	odd	4	inner
900.3.l.e.793.2	yes	4	15.8	even	4	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.67423 + 3.67423i) q^{7} +O(q^{10})\)	\(q+(3.67423 + 3.67423i) q^{7} +(-8.57321 + 8.57321i) q^{13} +11.0000i q^{19} -59.0000 q^{31} +(48.9898 + 48.9898i) q^{37} +(-15.9217 + 15.9217i) q^{43} -22.0000i q^{49} -121.000 q^{61} +(94.3054 + 94.3054i) q^{67} +(-97.9796 + 97.9796i) q^{73} +142.000i q^{79} -63.0000 q^{91} +(69.8105 + 69.8105i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 236 q^{31} - 484 q^{61} - 252 q^{91}+O(q^{100}) \) 4 * q - 236 * q^31 - 484 * q^61 - 252 * q^91

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