LMFDB - Embedded newform 55.5.d.c.54.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,5,Mod(54,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 55 = 5 \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	55.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.68534796961$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			54.2
Root			$0.500000 - 1.65831i$ of defining polynomial
Character	$\chi$	$=$	55.54
Dual form			55.5.d.c.54.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.5831i q^{3} -16.0000 q^{4} +(24.5000 + 4.97494i) q^{5} -194.000 q^{9} +O(q^{10})$	$q+16.5831i q^{3} -16.0000 q^{4} +(24.5000 + 4.97494i) q^{5} -194.000 q^{9} -121.000 q^{11} -265.330i q^{12} +(-82.5000 + 406.287i) q^{15} +256.000 q^{16} +(-392.000 - 79.5990i) q^{20} +1044.74i q^{23} +(575.500 + 243.772i) q^{25} -1873.89i q^{27} +553.000 q^{31} -2006.56i q^{33} +3104.00 q^{36} +1741.23i q^{37} +1936.00 q^{44} +(-4753.00 - 965.138i) q^{45} +3979.95i q^{47} +4245.28i q^{48} -2401.00 q^{49} -5571.93i q^{53} +(-2964.50 - 601.967i) q^{55} +4487.00 q^{59} +(1320.00 - 6500.58i) q^{60} -4096.00 q^{64} +4527.19i q^{67} -17325.0 q^{69} +7607.00 q^{71} +(-4042.50 + 9543.59i) q^{75} +(6272.00 + 1273.58i) q^{80} +15361.0 q^{81} +6433.00 q^{89} -16715.8i q^{92} +9170.47i q^{93} -16069.0i q^{97} +23474.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 49 q^{5} - 388 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 49 * q^5 - 388 * q^9	$ 2 q - 32 q^{4} + 49 q^{5} - 388 q^{9} - 242 q^{11} - 165 q^{15} + 512 q^{16} - 784 q^{20} + 1151 q^{25} + 1106 q^{31} + 6208 q^{36} + 3872 q^{44} - 9506 q^{45} - 4802 q^{49} - 5929 q^{55} + 8974 q^{59} + 2640 q^{60} - 8192 q^{64} - 34650 q^{69} + 15214 q^{71} - 8085 q^{75} + 12544 q^{80} + 30722 q^{81} + 12866 q^{89} + 46948 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 49 * q^5 - 388 * q^9 - 242 * q^11 - 165 * q^15 + 512 * q^16 - 784 * q^20 + 1151 * q^25 + 1106 * q^31 + 6208 * q^36 + 3872 * q^44 - 9506 * q^45 - 4802 * q^49 - 5929 * q^55 + 8974 * q^59 + 2640 * q^60 - 8192 * q^64 - 34650 * q^69 + 15214 * q^71 - 8085 * q^75 + 12544 * q^80 + 30722 * q^81 + 12866 * q^89 + 46948 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$.

$n$	$12$	$46$
$\chi(n)$	$-1$	$-1$

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$			16.5831i			1.84257i	0.388889	+	0.921285i	$0.372859\pi$
$3$			16.5831i			1.84257i	−0.388889	+	0.921285i	$0.627141\pi$
$4$	−16.0000			−1.00000
$5$	24.5000	+	4.97494i	0.980000	+	0.198997i
$5$	24.5000	+	4.97494i	0.980000	+	0.198997i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−194.000			−2.39506
$10$		0			0
$11$	−121.000			−1.00000
$11$	−121.000			−1.00000
$12$		−	265.330i		−	1.84257i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−82.5000	+	406.287i	−0.366667	+	1.80572i
$16$	256.000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−392.000	−	79.5990i	−0.980000	−	0.198997i
$21$		0			0
$22$		0			0
$23$			1044.74i			1.97493i	0.157845	+	0.987464i	$0.449545\pi$
$23$			1044.74i			1.97493i	−0.157845	+	0.987464i	$0.550455\pi$
$24$		0			0
$25$	575.500	+	243.772i	0.920800	+	0.390035i
$26$		0			0
$27$		−	1873.89i		−	2.57050i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	553.000			0.575442			0.287721	−	0.957714i	$-0.407102\pi$
$31$	553.000			0.575442			0.287721	+	0.957714i	$0.407102\pi$
$32$		0			0
$33$		−	2006.56i		−	1.84257i
$34$		0			0
$35$		0			0
$36$	3104.00			2.39506
$37$			1741.23i			1.27190i	0.771731	+	0.635949i	$0.219391\pi$
$37$			1741.23i			1.27190i	−0.771731	+	0.635949i	$0.780609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	1936.00			1.00000
$45$	−4753.00	−	965.138i	−2.34716	−	0.476611i
$46$		0			0
$47$			3979.95i			1.80170i	0.434133	+	0.900849i	$0.357055\pi$
$47$			3979.95i			1.80170i	−0.434133	+	0.900849i	$0.642945\pi$
$48$			4245.28i			1.84257i
$49$	−2401.00			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5571.93i		−	1.98360i	−0.127803	−	0.991800i	$-0.540793\pi$
$53$		−	5571.93i		−	1.98360i	0.127803	−	0.991800i	$-0.459207\pi$
$54$		0			0
$55$	−2964.50	−	601.967i	−0.980000	−	0.198997i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4487.00			1.28900			0.644499	−	0.764605i	$-0.277066\pi$
$59$	4487.00			1.28900			0.644499	+	0.764605i	$0.277066\pi$
$60$	1320.00	−	6500.58i	0.366667	−	1.80572i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−4096.00			−1.00000
$65$		0			0
$66$		0			0
$67$			4527.19i			1.00851i	0.863555	+	0.504254i	$0.168232\pi$
$67$			4527.19i			1.00851i	−0.863555	+	0.504254i	$0.831768\pi$
$68$		0			0
$69$	−17325.0			−3.63894
$70$		0			0
$71$	7607.00			1.50903			0.754513	−	0.656285i	$-0.227873\pi$
$71$	7607.00			1.50903			0.754513	+	0.656285i	$0.227873\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−4042.50	+	9543.59i	−0.718667	+	1.69664i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	6272.00	+	1273.58i	0.980000	+	0.198997i
$81$	15361.0			2.34126
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6433.00			0.812145			0.406072	−	0.913841i	$-0.366898\pi$
$89$	6433.00			0.812145			0.406072	+	0.913841i	$0.366898\pi$
$90$		0			0
$91$		0			0
$92$		−	16715.8i		−	1.97493i
$93$			9170.47i			1.06029i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	16069.0i		−	1.70784i	−0.520406	−	0.853919i	$-0.674219\pi$
$97$		−	16069.0i		−	1.70784i	0.520406	−	0.853919i	$-0.325781\pi$
$98$		0			0
$99$	23474.0			2.39506
$100$	−9208.00	−	3900.35i	−0.920800	−	0.390035i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		−	15123.8i		−	1.42556i	−0.701386	−	0.712782i	$-0.747435\pi$
$103$		−	15123.8i		−	1.42556i	0.701386	−	0.712782i	$-0.252565\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$			29982.3i			2.57050i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	−28875.0			−2.34356
$112$		0			0
$113$		−	14974.6i		−	1.17273i	−0.810048	−	0.586364i	$-0.800559\pi$
$113$		−	14974.6i		−	1.17273i	0.810048	−	0.586364i	$-0.199441\pi$
$114$		0			0
$115$	−5197.50	+	25596.1i	−0.393006	+	1.93543i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	14641.0			1.00000
$122$		0			0
$123$		0			0
$124$	−8848.00			−0.575442
$125$	12887.0	+	8835.49i	0.824768	+	0.565471i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$			32104.9i			1.84257i
$133$		0			0
$134$		0			0
$135$	9322.50	−	45910.4i	0.511523	−	2.51909i
$136$		0			0
$137$			18457.0i			0.983378i	0.870771	+	0.491689i	$0.163620\pi$
$137$			18457.0i			0.983378i	−0.870771	+	0.491689i	$0.836380\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−66000.0			−3.31975
$142$		0			0
$143$		0			0
$144$	−49664.0			−2.39506
$145$		0			0
$146$		0			0
$147$		−	39816.1i		−	1.84257i
$148$		−	27859.6i		−	1.27190i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	13548.5	+	2751.14i	0.563933	+	0.114512i
$156$		0			0
$157$			49202.1i			1.99611i	0.0623352	+	0.998055i	$0.480145\pi$
$157$			49202.1i			1.99611i	−0.0623352	+	0.998055i	$0.519855\pi$
$158$		0			0
$159$	92400.0			3.65492
$160$		0			0
$161$		0			0
$162$		0			0
$163$			22287.7i			0.838862i	0.907787	+	0.419431i	$0.137770\pi$
$163$			22287.7i			0.838862i	−0.907787	+	0.419431i	$0.862230\pi$
$164$		0			0
$165$	9982.50	−	49160.7i	0.366667	−	1.80572i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−28561.0			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−30976.0			−1.00000
$177$			74408.5i			2.37507i
$178$		0			0
$179$	57193.0			1.78499			0.892497	−	0.451053i	$-0.148952\pi$
$179$	57193.0			1.78499			0.892497	+	0.451053i	$0.148952\pi$
$180$	76048.0	+	15442.2i	2.34716	+	0.476611i
$181$	−3647.00			−0.111321			−0.0556607	−	0.998450i	$-0.517727\pi$
$181$	−3647.00			−0.111321			−0.0556607	+	0.998450i	$0.517727\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8662.50	+	42660.1i	−0.253104	+	1.24646i
$186$		0			0
$187$		0			0
$188$		−	63679.2i		−	1.80170i
$189$		0			0
$190$		0			0
$191$	48313.0			1.32433			0.662167	−	0.749357i	$-0.269637\pi$
$191$	48313.0			1.32433			0.662167	+	0.749357i	$0.269637\pi$
$192$		−	67924.5i		−	1.84257i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	38416.0			1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−79198.0			−1.99990			−0.999949	−	0.0100501i	$-0.996801\pi$
$199$	−79198.0			−1.99990			−0.999949	+	0.0100501i	$0.996801\pi$
$200$		0			0
$201$	−75075.0			−1.85825
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	202679.i		−	4.73007i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			89150.9i			1.98360i
$213$			126148.i			2.78048i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	47432.0	+	9631.48i	0.980000	+	0.198997i
$221$		0			0
$222$		0			0
$223$		−	13283.1i		−	0.267109i	−0.991041	−	0.133555i	$-0.957361\pi$
$223$		−	13283.1i		−	0.267109i	0.991041	−	0.133555i	$-0.0426392\pi$
$224$		0			0
$225$	−111647.	−	47291.8i	−2.20537	−	0.934158i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	82607.0			1.57524			0.787618	−	0.616163i	$-0.211314\pi$
$229$	82607.0			1.57524			0.787618	+	0.616163i	$0.211314\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−19800.0	+	97508.8i	−0.358533	+	1.76566i
$236$	−71792.0			−1.28900
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−21120.0	+	104009.i	−0.366667	+	1.80572i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$			102948.i			1.74343i
$244$		0			0
$245$	−58824.5	−	11944.8i	−0.980000	−	0.198997i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−74473.0			−1.18209			−0.591046	−	0.806638i	$-0.701285\pi$
$251$	−74473.0			−1.18209			−0.591046	+	0.806638i	$0.701285\pi$
$252$		0			0
$253$		−	126413.i		−	1.97493i
$254$		0			0
$255$		0			0
$256$	65536.0			1.00000
$257$		−	90742.9i		−	1.37387i	−0.726718	−	0.686936i	$-0.758955\pi$
$257$		−	90742.9i		−	1.37387i	0.726718	−	0.686936i	$-0.241045\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	27720.0	−	136512.i	0.394731	−	1.94393i
$266$		0			0
$267$			106679.i			1.49643i
$268$		−	72435.1i		−	1.00851i
$269$	−13678.0			−0.189024			−0.0945122	−	0.995524i	$-0.530129\pi$
$269$	−13678.0			−0.189024			−0.0945122	+	0.995524i	$0.530129\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−69635.5	−	29496.4i	−0.920800	−	0.390035i
$276$	277200.			3.63894
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$	−107282.			−1.37822
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	−121712.			−1.50903
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−83521.0			−1.00000
$290$		0			0
$291$	266475.			3.14681
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	109931.	+	22322.5i	1.26322	+	0.256507i
$296$		0			0
$297$			226741.i			2.57050i
$298$		0			0
$299$		0			0
$300$	64680.0	−	152697.i	0.718667	−	1.69664i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	250800.			2.62670
$310$		0			0
$311$	35042.0			0.362300			0.181150	−	0.983455i	$-0.442018\pi$
$311$	35042.0			0.362300			0.181150	+	0.983455i	$0.442018\pi$
$312$		0			0
$313$			150293.i			1.53409i	0.641596	+	0.767043i	$0.278273\pi$
$313$			150293.i			1.53409i	−0.641596	+	0.767043i	$0.721727\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			59550.0i			0.592602i	0.955095	+	0.296301i	$0.0957532\pi$
$317$			59550.0i			0.592602i	−0.955095	+	0.296301i	$0.904247\pi$
$318$		0			0
$319$		0			0
$320$	−100352.	−	20377.3i	−0.980000	−	0.198997i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−245776.			−2.34126
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−97847.0			−0.893082			−0.446541	−	0.894763i	$-0.647344\pi$
$331$	−97847.0			−0.893082			−0.446541	+	0.894763i	$0.647344\pi$
$332$		0			0
$333$		−	337798.i		−	3.04627i
$334$		0			0
$335$	−22522.5	+	110916.i	−0.200691	+	0.988338i
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	248325.			2.16083
$340$		0			0
$341$	−66913.0			−0.575442
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−424462.	−	86190.8i	−3.56616	−	0.724140i
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	205216.i		−	1.64688i	−0.567403	−	0.823440i	$-0.692052\pi$
$353$		−	205216.i		−	1.64688i	0.567403	−	0.823440i	$-0.307948\pi$
$354$		0			0
$355$	186372.	+	37844.3i	1.47885	+	0.300292i
$356$	−102928.			−0.812145
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	130321.			1.00000
$362$		0			0
$363$			242794.i			1.84257i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	233772.i		−	1.73565i	−0.496874	−	0.867823i	$-0.665519\pi$
$367$		−	233772.i		−	1.73565i	0.496874	−	0.867823i	$-0.334481\pi$
$368$			267453.i			1.97493i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		−	146727.i		−	1.06029i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−146520.	+	213707.i	−1.04192	+	1.51969i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−269593.			−1.87685			−0.938426	−	0.345479i	$-0.887716\pi$
$379$	−269593.			−1.87685			−0.938426	+	0.345479i	$0.887716\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			97658.0i			0.665749i	0.942971	+	0.332874i	$0.108019\pi$
$383$			97658.0i			0.665749i	−0.942971	+	0.332874i	$0.891981\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$			257105.i			1.70784i
$389$	3167.00			0.0209290			0.0104645	−	0.999945i	$-0.496669\pi$
$389$	3167.00			0.0209290			0.0104645	+	0.999945i	$0.496669\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−375584.			−2.39506
$397$		−	62883.2i		−	0.398982i	−0.979900	−	0.199491i	$-0.936071\pi$
$397$		−	62883.2i		−	0.398982i	0.979900	−	0.199491i	$-0.0639289\pi$
$398$		0			0
$399$		0			0
$400$	147328.	+	62405.6i	0.920800	+	0.390035i
$401$	311998.			1.94027			0.970137	−	0.242558i	$-0.0779863\pi$
$401$	311998.			1.94027			0.970137	+	0.242558i	$0.0779863\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	376344.	+	76420.0i	2.29443	+	0.465905i
$406$		0			0
$407$		−	210689.i		−	1.27190i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−306075.			−1.81194
$412$			241981.i			1.42556i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	282478.			1.60900			0.804501	−	0.593951i	$-0.202433\pi$
$419$	282478.			1.60900			0.804501	+	0.593951i	$0.202433\pi$
$420$		0			0
$421$	−196082.			−1.10630			−0.553151	−	0.833081i	$-0.686575\pi$
$421$	−196082.			−1.10630			−0.553151	+	0.833081i	$0.686575\pi$
$422$		0			0
$423$		−	772110.i		−	4.31518i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	479717.i		−	2.57050i
$433$			21641.0i			0.115425i	0.998333	+	0.0577127i	$0.0183807\pi$
$433$			21641.0i			0.115425i	−0.998333	+	0.0577127i	$0.981619\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	465794.			2.39506
$442$		0			0
$443$			340932.i			1.73724i	0.495475	+	0.868622i	$0.334994\pi$
$443$			340932.i			1.73724i	−0.495475	+	0.868622i	$0.665006\pi$
$444$	462000.			2.34356
$445$	157608.	+	32003.8i	0.795902	+	0.161615i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15073.0			0.0747665			0.0373832	−	0.999301i	$-0.488098\pi$
$449$	15073.0			0.0747665			0.0373832	+	0.999301i	$0.488098\pi$
$450$		0			0
$451$		0			0
$452$			239593.i			1.17273i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$	83160.0	−	409537.i	0.393006	−	1.93543i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$			32386.8i			0.151080i	0.997143	+	0.0755399i	$0.0240680\pi$
$463$			32386.8i			0.151080i	−0.997143	+	0.0755399i	$0.975932\pi$
$464$		0			0
$465$	−45622.5	+	224676.i	−0.210995	+	1.03909i
$466$		0			0
$467$		−	119647.i		−	0.548617i	−0.961642	−	0.274308i	$-0.911551\pi$
$467$		−	119647.i		−	0.548617i	0.961642	−	0.274308i	$-0.0884489\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−815925.			−3.67797
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			1.08095e6i			4.75084i
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−234256.			−1.00000
$485$	79942.5	−	393692.i	0.339855	−	1.67368i
$486$		0			0
$487$			374364.i			1.57847i	0.614092	+	0.789235i	$0.289523\pi$
$487$			374364.i			1.57847i	−0.614092	+	0.789235i	$0.710477\pi$
$488$		0			0
$489$	−369600.			−1.54566
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	575113.	+	116782.i	2.34716	+	0.476611i
$496$	141568.			0.575442
$497$		0			0
$498$		0			0
$499$	−135598.			−0.544568			−0.272284	−	0.962217i	$-0.587779\pi$
$499$	−135598.			−0.544568			−0.272284	+	0.962217i	$0.587779\pi$
$500$	−206192.	−	141368.i	−0.824768	−	0.565471i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	473631.i		−	1.84257i
$508$		0			0
$509$	495887.			1.91402			0.957012	−	0.290050i	$-0.0936719\pi$
$509$	495887.			1.91402			0.957012	+	0.290050i	$0.0936719\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	75240.0	−	370533.i	0.283684	−	1.39705i
$516$		0			0
$517$		−	481574.i		−	1.80170i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−124607.			−0.459057			−0.229529	−	0.973302i	$-0.573718\pi$
$521$	−124607.			−0.459057			−0.229529	+	0.973302i	$0.573718\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		−	513679.i		−	1.84257i
$529$	−811634.			−2.90034
$530$		0			0
$531$	−870478.			−3.08723
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$			948439.i			3.28898i
$538$		0			0
$539$	290521.			1.00000
$540$	−149160.	+	734566.i	−0.511523	+	2.51909i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	60478.7i		−	0.205117i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		−	295312.i		−	0.983378i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−707438.	−	143651.i	−2.29669	−	0.466363i
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	1.05600e6			3.31975
$565$	74497.5	−	366877.i	0.233370	−	1.14927i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$			801180.i			2.44018i
$574$		0			0
$575$	−254678.	+	601246.i	−0.770291	+	1.81851i
$576$	794624.			2.39506
$577$		−	493365.i		−	1.48189i	−0.671565	−	0.740946i	$-0.734378\pi$
$577$		−	493365.i		−	1.48189i	0.671565	−	0.740946i	$-0.265622\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			674203.i			1.98360i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	359787.i		−	1.04417i	−0.852894	−	0.522083i	$-0.825155\pi$
$587$		−	359787.i		−	1.04417i	0.852894	−	0.522083i	$-0.174845\pi$
$588$			637057.i			1.84257i
$589$		0			0
$590$		0			0
$591$		0			0
$592$			445754.i			1.27190i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1.31335e6i		−	3.68495i
$598$		0			0
$599$	707998.			1.97323			0.986617	−	0.163058i	$-0.0521357\pi$
$599$	707998.			1.97323			0.986617	+	0.163058i	$0.0521357\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		−	878275.i		−	2.41544i
$604$		0			0
$605$	358704.	+	72838.1i	0.980000	+	0.198997i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	590625.i		−	1.55146i	−0.631064	−	0.775731i	$-0.717381\pi$
$617$		−	590625.i		−	1.55146i	0.631064	−	0.775731i	$-0.282619\pi$
$618$		0			0
$619$	−763847.			−1.99354			−0.996770	−	0.0803056i	$-0.974410\pi$
$619$	−763847.			−1.99354			−0.996770	+	0.0803056i	$0.974410\pi$
$620$	−216776.	−	44018.2i	−0.563933	−	0.114512i
$621$	1.95772e6			5.07655
$622$		0			0
$623$		0			0
$624$		0			0
$625$	271776.	+	280581.i	0.695745	+	0.718289i
$626$		0			0
$627$		0			0
$628$		−	787234.i		−	1.99611i
$629$		0			0
$630$		0			0
$631$	675047.			1.69541			0.847706	−	0.530466i	$-0.177983\pi$
$631$	675047.			1.69541			0.847706	+	0.530466i	$0.177983\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.47840e6			−3.65492
$637$		0			0
$638$		0			0
$639$	−1.47576e6			−3.61421
$640$		0			0
$641$	269713.			0.656426			0.328213	−	0.944604i	$-0.393554\pi$
$641$	269713.			0.656426			0.328213	+	0.944604i	$0.393554\pi$
$642$		0			0
$643$			483415.i			1.16922i	0.811313	+	0.584612i	$0.198753\pi$
$643$			483415.i			1.16922i	−0.811313	+	0.584612i	$0.801247\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			657736.i			1.57124i	0.618707	+	0.785621i	$0.287657\pi$
$647$			657736.i			1.57124i	−0.618707	+	0.785621i	$0.712343\pi$
$648$		0			0
$649$	−542927.			−1.28900
$650$		0			0
$651$		0			0
$652$		−	356603.i		−	0.838862i
$653$		−	219046.i		−	0.513700i	−0.966451	−	0.256850i	$-0.917315\pi$
$653$		−	219046.i		−	0.513700i	0.966451	−	0.256850i	$-0.0826847\pi$
$654$		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			$\pi$
$660$	−159720.	+	786571.i	−0.366667	+	1.80572i
$661$	455567.			1.04268			0.521338	−	0.853350i	$-0.325433\pi$
$661$	455567.			1.04268			0.521338	+	0.853350i	$0.325433\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	220275.			0.492168
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$	456802.	−	1.07843e6i	1.00258	−	2.36691i
$676$	456976.			1.00000
$677$		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
$683$			880365.i			1.88721i	0.331067	+	0.943607i	$0.392591\pi$
$683$			880365.i			1.88721i	−0.331067	+	0.943607i	$0.607409\pi$
$684$		0			0
$685$	−91822.5	+	452197.i	−0.195690	+	0.963710i
$686$		0			0
$687$			1.36988e6i			2.90248i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	239687.			0.501982			0.250991	−	0.967989i	$-0.419243\pi$
$691$	239687.			0.501982			0.250991	+	0.967989i	$0.419243\pi$
$692$		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		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	495616.			1.00000
$705$	−1.61700e6	−	328346.i	−3.25336	−	0.660622i
$706$		0			0
$707$		0			0
$708$		−	1.19054e6i		−	2.37507i
$709$	−111887.			−0.222581			−0.111290	−	0.993788i	$-0.535498\pi$
$709$	−111887.			−0.222581			−0.111290	+	0.993788i	$0.535498\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			577739.i			1.13646i
$714$		0			0
$715$		0			0
$716$	−915088.			−1.78499
$717$		0			0
$718$		0			0
$719$	−318647.			−0.616385			−0.308192	−	0.951324i	$-0.599724\pi$
$719$	−318647.			−0.616385			−0.308192	+	0.951324i	$0.599724\pi$
$720$	−1.21677e6	−	247075.i	−2.34716	−	0.476611i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	58352.0			0.111321
$725$		0			0
$726$		0			0
$727$			714550.i			1.35196i	0.736920	+	0.675980i	$0.236280\pi$
$727$			714550.i			1.35196i	−0.736920	+	0.675980i	$0.763720\pi$
$728$		0			0
$729$	−462959.			−0.871139
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	198082.	−	975494.i	0.366667	−	1.80572i
$736$		0			0
$737$		−	547790.i		−	1.00851i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	138600.	−	682561.i	0.253104	−	1.24646i
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−181273.			−0.321405			−0.160703	−	0.987003i	$-0.551376\pi$
$751$	−181273.			−0.321405			−0.160703	+	0.987003i	$0.551376\pi$
$752$			1.01887e6i			1.80170i
$753$		−	1.23499e6i		−	2.17809i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			105867.i			0.184743i	0.995725	+	0.0923714i	$0.0294447\pi$
$757$			105867.i			0.184743i	−0.995725	+	0.0923714i	$0.970555\pi$
$758$		0			0
$759$	2.09632e6			3.63894
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	−773008.			−1.32433
$765$		0			0
$766$		0			0
$767$		0			0
$768$			1.08679e6i			1.84257i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	1.50480e6			2.53145
$772$		0			0
$773$		−	981456.i		−	1.64252i	−0.570551	−	0.821262i	$-0.693270\pi$
$773$		−	981456.i		−	1.64252i	0.570551	−	0.821262i	$-0.306730\pi$
$774$		0			0
$775$	318252.	+	134806.i	0.529867	+	0.224443i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−920447.			−1.50903
$782$		0			0
$783$		0			0
$784$	−614656.			−1.00000
$785$	−244778.	+	1.20545e6i	−0.397221	+	1.95619i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	2.26380e6	+	459684.i	3.58182	+	0.727320i
$796$	1.26717e6			1.99990
$797$			268099.i			0.422065i	0.977479	+	0.211032i	$0.0676826\pi$
$797$			268099.i			0.422065i	−0.977479	+	0.211032i	$0.932317\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.24800e6			−1.94514
$802$		0			0
$803$		0			0
$804$	1.20120e6			1.85825
$805$		0			0
$806$		0			0
$807$		−	226824.i		−	0.348291i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−110880.	+	546049.i	−0.166931	+	0.822085i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		−	1.10429e6i		−	1.63036i	−0.579211	−	0.815178i	$-0.696639\pi$
$823$		−	1.10429e6i		−	1.63036i	0.579211	−	0.815178i	$-0.303361\pi$
$824$		0			0
$825$	489142.	−	1.15477e6i	0.718667	−	1.69664i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$			3.24286e6i			4.73007i
$829$	706993.			1.02874			0.514371	−	0.857568i	$-0.328026\pi$
$829$	706993.			1.02874			0.514371	+	0.857568i	$0.328026\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	1.03626e6i		−	1.47917i
$838$		0			0
$839$	−1.28743e6			−1.82895			−0.914473	−	0.404648i	$-0.867394\pi$
$839$	−1.28743e6			−1.82895			−0.914473	+	0.404648i	$0.867394\pi$
$840$		0			0
$841$	707281.			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−699744.	−	142089.i	−0.980000	−	0.198997i
$846$		0			0
$847$		0			0
$848$		−	1.42641e6i		−	1.98360i
$849$		0			0
$850$		0			0
$851$	−1.81912e6			−2.51191
$852$		−	2.01837e6i		−	2.78048i
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	902713.			1.22339			0.611693	−	0.791095i	$-0.290489\pi$
$859$	902713.			1.22339			0.611693	+	0.791095i	$0.290489\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	718779.i		−	0.965103i	−0.875868	−	0.482552i	$-0.839710\pi$
$863$		−	718779.i		−	0.965103i	0.875868	−	0.482552i	$-0.160290\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1.38504e6i		−	1.84257i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$			3.11740e6i			4.09038i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$	−758912.	−	154104.i	−0.980000	−	0.198997i
$881$	658847.			0.848854			0.424427	−	0.905462i	$-0.360476\pi$
$881$	658847.			0.848854			0.424427	+	0.905462i	$0.360476\pi$
$882$		0			0
$883$		−	1.52671e6i		−	1.95810i	−0.203621	−	0.979050i	$-0.565271\pi$
$883$		−	1.52671e6i		−	1.95810i	0.203621	−	0.979050i	$-0.434729\pi$
$884$		0			0
$885$	−370178.	+	1.82301e6i	−0.472632	+	2.32757i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.85868e6			−2.34126
$892$			212529.i			0.267109i
$893$		0			0
$894$		0			0
$895$	1.40123e6	+	284532.i	1.74929	+	0.355209i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.78635e6	+	756668.i	2.20537	+	0.934158i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−89351.5	−	18143.6i	−0.109095	−	0.0221527i
$906$		0			0
$907$		−	835789.i		−	1.01597i	−0.861365	−	0.507987i	$-0.830390\pi$
$907$		−	835789.i		−	1.01597i	0.861365	−	0.507987i	$-0.169610\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.50144e6			−1.80914			−0.904569	−	0.426327i	$-0.859807\pi$
$911$	−1.50144e6			−1.80914			−0.904569	+	0.426327i	$0.859807\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.32171e6			−1.57524
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−424462.	+	1.00208e6i	−0.496085	+	1.17116i
$926$		0			0
$927$			2.93402e6i			3.41431i
$928$		0			0
$929$	−808318.			−0.936593			−0.468296	−	0.883571i	$-0.655132\pi$
$929$	−808318.			−0.936593			−0.468296	+	0.883571i	$0.655132\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			581106.i			0.667563i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$	−2.49232e6			−2.82666
$940$	316800.	−	1.56014e6i	0.358533	−	1.76566i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	1.14867e6			1.28900
$945$		0			0
$946$		0			0
$947$			1.79312e6i			1.99944i	0.0236433	+	0.999720i	$0.492473\pi$
$947$			1.79312e6i			1.99944i	−0.0236433	+	0.999720i	$0.507527\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−987525.			−1.09191
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	1.18367e6	+	240354.i	1.29785	+	0.263539i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	337920.	−	1.66415e6i	0.366667	−	1.80572i
$961$	−617712.			−0.668866
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.87879e6			1.99269			0.996347	−	0.0854008i	$-0.0272171\pi$
$971$	1.87879e6			1.99269			0.996347	+	0.0854008i	$0.0272171\pi$
$972$		−	1.64717e6i		−	1.74343i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.86486e6i		−	1.95369i	−0.213945	−	0.976846i	$-0.568631\pi$
$977$		−	1.86486e6i		−	1.95369i	0.213945	−	0.976846i	$-0.431369\pi$
$978$		0			0
$979$	−778393.			−0.812145
$980$	941192.	+	191117.i	0.980000	+	0.198997i
$981$		0			0
$982$		0			0
$983$		−	1.18150e6i		−	1.22272i	−0.791354	−	0.611358i	$-0.790623\pi$
$983$		−	1.18150e6i		−	1.22272i	0.791354	−	0.611358i	$-0.209377\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−538562.			−0.548389			−0.274194	−	0.961674i	$-0.588411\pi$
$991$	−538562.			−0.548389			−0.274194	+	0.961674i	$0.588411\pi$
$992$		0			0
$993$		−	1.62261e6i		−	1.64557i
$994$		0			0
$995$	−1.94035e6	−	394005.i	−1.95990	−	0.397975i
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$	3.26288e6			3.26941

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.5.d.c.54.2	yes	2
5.2	odd	4		275.5.c.d.76.2		2
5.3	odd	4		275.5.c.d.76.1		2
5.4	even	2	inner	55.5.d.c.54.1	✓	2
11.10	odd	2	CM	55.5.d.c.54.2	yes	2
55.32	even	4		275.5.c.d.76.2		2
55.43	even	4		275.5.c.d.76.1		2
55.54	odd	2	inner	55.5.d.c.54.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.5.d.c.54.1	✓	2	5.4	even	2	inner
55.5.d.c.54.1	✓	2	55.54	odd	2	inner
55.5.d.c.54.2	yes	2	1.1	even	1	trivial
55.5.d.c.54.2	yes	2	11.10	odd	2	CM
275.5.c.d.76.1		2	5.3	odd	4
275.5.c.d.76.1		2	55.43	even	4
275.5.c.d.76.2		2	5.2	odd	4
275.5.c.d.76.2		2	55.32	even	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:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	55.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+16.5831i q^{3} -16.0000 q^{4} +(24.5000 + 4.97494i) q^{5} -194.000 q^{9} +O(q^{10})\)	\(q+16.5831i q^{3} -16.0000 q^{4} +(24.5000 + 4.97494i) q^{5} -194.000 q^{9} -121.000 q^{11} -265.330i q^{12} +(-82.5000 + 406.287i) q^{15} +256.000 q^{16} +(-392.000 - 79.5990i) q^{20} +1044.74i q^{23} +(575.500 + 243.772i) q^{25} -1873.89i q^{27} +553.000 q^{31} -2006.56i q^{33} +3104.00 q^{36} +1741.23i q^{37} +1936.00 q^{44} +(-4753.00 - 965.138i) q^{45} +3979.95i q^{47} +4245.28i q^{48} -2401.00 q^{49} -5571.93i q^{53} +(-2964.50 - 601.967i) q^{55} +4487.00 q^{59} +(1320.00 - 6500.58i) q^{60} -4096.00 q^{64} +4527.19i q^{67} -17325.0 q^{69} +7607.00 q^{71} +(-4042.50 + 9543.59i) q^{75} +(6272.00 + 1273.58i) q^{80} +15361.0 q^{81} +6433.00 q^{89} -16715.8i q^{92} +9170.47i q^{93} -16069.0i q^{97} +23474.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 49 q^{5} - 388 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 49 * q^5 - 388 * q^9	\( 2 q - 32 q^{4} + 49 q^{5} - 388 q^{9} - 242 q^{11} - 165 q^{15} + 512 q^{16} - 784 q^{20} + 1151 q^{25} + 1106 q^{31} + 6208 q^{36} + 3872 q^{44} - 9506 q^{45} - 4802 q^{49} - 5929 q^{55} + 8974 q^{59} + 2640 q^{60} - 8192 q^{64} - 34650 q^{69} + 15214 q^{71} - 8085 q^{75} + 12544 q^{80} + 30722 q^{81} + 12866 q^{89} + 46948 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 49 * q^5 - 388 * q^9 - 242 * q^11 - 165 * q^15 + 512 * q^16 - 784 * q^20 + 1151 * q^25 + 1106 * q^31 + 6208 * q^36 + 3872 * q^44 - 9506 * q^45 - 4802 * q^49 - 5929 * q^55 + 8974 * q^59 + 2640 * q^60 - 8192 * q^64 - 34650 * q^69 + 15214 * q^71 - 8085 * q^75 + 12544 * q^80 + 30722 * q^81 + 12866 * q^89 + 46948 * q^99

Introduction

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