LMFDB - Embedded newform 880.2.bd.d.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(417,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("880.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.bd (of order $4$, degree $2$, not 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:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			593.2
Root			$-1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	880.593
Dual form			880.2.bd.d.417.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.15831 + 2.15831i) q^{3} +(1.65831 + 1.50000i) q^{5} +6.31662i q^{9} +O(q^{10})$	$q+(2.15831 + 2.15831i) q^{3} +(1.65831 + 1.50000i) q^{5} +6.31662i q^{9} -3.31662 q^{11} +(0.341688 + 6.81662i) q^{15} +(-2.84169 - 2.84169i) q^{23} +(0.500000 + 4.97494i) q^{25} +(-7.15831 + 7.15831i) q^{27} +9.94987 q^{31} +(-7.15831 - 7.15831i) q^{33} +(1.47494 - 1.47494i) q^{37} +(-9.47494 + 10.4749i) q^{45} +(9.31662 - 9.31662i) q^{47} -7.00000i q^{49} +(3.63325 + 3.63325i) q^{53} +(-5.50000 - 4.97494i) q^{55} -3.31662i q^{59} +(-11.4749 + 11.4749i) q^{67} -12.2665i q^{69} +3.00000 q^{71} +(-9.65831 + 11.8166i) q^{75} -11.9499 q^{81} -9.00000i q^{89} +(21.4749 + 21.4749i) q^{93} +(-3.52506 + 3.52506i) q^{97} -20.9499i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3}+O(q^{10}) $ 4 * q + 2 * q^3	$ 4 q + 2 q^{3} + 8 q^{15} - 18 q^{23} + 2 q^{25} - 22 q^{27} - 22 q^{33} - 14 q^{37} - 18 q^{45} + 24 q^{47} - 12 q^{53} - 22 q^{55} - 26 q^{67} + 12 q^{71} - 32 q^{75} - 8 q^{81} + 66 q^{93} - 34 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 + 8 * q^15 - 18 * q^23 + 2 * q^25 - 22 * q^27 - 22 * q^33 - 14 * q^37 - 18 * q^45 + 24 * q^47 - 12 * q^53 - 22 * q^55 - 26 * q^67 + 12 * q^71 - 32 * q^75 - 8 * q^81 + 66 * q^93 - 34 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$	$1$

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
$3$	2.15831	+	2.15831i	1.24610	+	1.24610i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	2.15831	+	2.15831i	1.24610	+	1.24610i	0.288675	+	0.957427i	$0.406785\pi$
$4$		0			0
$5$	1.65831	+	1.50000i	0.741620	+	0.670820i
$5$	1.65831	+	1.50000i	0.741620	+	0.670820i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$			6.31662i			2.10554i
$10$		0			0
$11$	−3.31662			−1.00000
$11$	−3.31662			−1.00000
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$	0.341688	+	6.81662i	0.0882234	+	1.76004i
$16$		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
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.84169	−	2.84169i	−0.592533	−	0.592533i	0.345782	−	0.938315i	$-0.387614\pi$
$23$	−2.84169	−	2.84169i	−0.592533	−	0.592533i	−0.938315	+	0.345782i	$0.887614\pi$
$24$		0			0
$25$	0.500000	+	4.97494i	0.100000	+	0.994987i
$26$		0			0
$27$	−7.15831	+	7.15831i	−1.37762	+	1.37762i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	9.94987			1.78705			0.893525	−	0.449013i	$-0.148224\pi$
$31$	9.94987			1.78705			0.893525	+	0.449013i	$0.148224\pi$
$32$		0			0
$33$	−7.15831	−	7.15831i	−1.24610	−	1.24610i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.47494	−	1.47494i	0.242478	−	0.242478i	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	1.47494	−	1.47494i	0.242478	−	0.242478i	0.817875	+	0.575396i	$0.195152\pi$
$38$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−9.47494	+	10.4749i	−1.41244	+	1.56151i
$46$		0			0
$47$	9.31662	−	9.31662i	1.35897	−	1.35897i	0.483779	−	0.875190i	$-0.339264\pi$
$47$	9.31662	−	9.31662i	1.35897	−	1.35897i	0.875190	−	0.483779i	$-0.160736\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.63325	+	3.63325i	0.499065	+	0.499065i	0.911147	−	0.412082i	$-0.135198\pi$
$53$	3.63325	+	3.63325i	0.499065	+	0.499065i	−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−5.50000	−	4.97494i	−0.741620	−	0.670820i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	3.31662i		−	0.431788i	−0.976417	−	0.215894i	$-0.930733\pi$
$59$		−	3.31662i		−	0.431788i	0.976417	−	0.215894i	$-0.0692665\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−11.4749	+	11.4749i	−1.40189	+	1.40189i	−0.607785	+	0.794101i	$0.707942\pi$
$67$	−11.4749	+	11.4749i	−1.40189	+	1.40189i	−0.794101	+	0.607785i	$0.792058\pi$
$68$		0			0
$69$		−	12.2665i		−	1.47671i
$70$		0			0
$71$	3.00000			0.356034			0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000			0.356034			0.178017	+	0.984027i	$0.443032\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$	−9.65831	+	11.8166i	−1.11525	+	1.36447i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−11.9499			−1.32776
$82$		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
$89$		−	9.00000i		−	0.953998i	−0.878904	−	0.476999i	$-0.841725\pi$
$89$		−	9.00000i		−	0.953998i	0.878904	−	0.476999i	$-0.158275\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	21.4749	+	21.4749i	2.22685	+	2.22685i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.52506	+	3.52506i	−0.357916	+	0.357916i	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−3.52506	+	3.52506i	−0.357916	+	0.357916i	0.505128	+	0.863044i	$0.331445\pi$
$98$		0			0
$99$		−	20.9499i		−	2.10554i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	7.94987	+	7.94987i	0.783324	+	0.783324i	0.980390	−	0.197066i	$-0.0631413\pi$
$103$	7.94987	+	7.94987i	0.783324	+	0.783324i	−0.197066	+	0.980390i	$0.563141\pi$
$104$		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
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	6.36675			0.604305
$112$		0			0
$113$	−12.1583	−	12.1583i	−1.14376	−	1.14376i	−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−12.1583	−	12.1583i	−1.14376	−	1.14376i	−0.156001	−	0.987757i	$-0.549860\pi$
$114$		0			0
$115$	−0.449874	−	8.97494i	−0.0419510	−	0.836917i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.63325	+	9.00000i	−0.593296	+	0.804984i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−22.6082	+	1.13325i	−1.94580	+	0.0975346i
$136$		0			0
$137$	−10.1082	+	10.1082i	−0.863601	+	0.863601i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	−10.1082	+	10.1082i	−0.863601	+	0.863601i	0.128154	+	0.991754i	$0.459095\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	40.2164			3.38683
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	15.1082	−	15.1082i	1.24610	−	1.24610i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.5000	+	14.9248i	1.32531	+	1.19879i
$156$		0			0
$157$	16.4749	−	16.4749i	1.31484	−	1.31484i	0.397043	−	0.917800i	$-0.370036\pi$
$157$	16.4749	−	16.4749i	1.31484	−	1.31484i	0.917800	−	0.397043i	$-0.129964\pi$
$158$		0			0
$159$			15.6834i			1.24377i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	17.9499	+	17.9499i	1.40594	+	1.40594i	0.779334	+	0.626608i	$0.215557\pi$
$163$	17.9499	+	17.9499i	1.40594	+	1.40594i	0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$	−1.13325	−	22.6082i	−0.0882234	−	1.76004i
$166$		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
$169$			13.0000i			1.00000i
$170$		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
$177$	7.15831	−	7.15831i	0.538052	−	0.538052i
$178$		0			0
$179$		−	21.0000i		−	1.56961i	−0.619740	−	0.784807i	$-0.712762\pi$
$179$		−	21.0000i		−	1.56961i	0.619740	−	0.784807i	$-0.287238\pi$
$180$		0			0
$181$	−9.94987			−0.739568			−0.369784	−	0.929118i	$-0.620568\pi$
$181$	−9.94987			−0.739568			−0.369784	+	0.929118i	$0.620568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.65831	−	0.233501i	0.342486	−	0.0171673i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−23.2164			−1.67988			−0.839939	−	0.542681i	$-0.817409\pi$
$191$	−23.2164			−1.67988			−0.839939	+	0.542681i	$0.817409\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		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
$199$		−	19.8997i		−	1.41066i	−0.708881	−	0.705328i	$-0.750800\pi$
$199$		−	19.8997i		−	1.41066i	0.708881	−	0.705328i	$-0.249200\pi$
$200$		0			0
$201$	−49.5330			−3.49379
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	17.9499	−	17.9499i	1.24760	−	1.24760i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	6.47494	+	6.47494i	0.443655	+	0.443655i
$214$		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
$223$	−14.4248	−	14.4248i	−0.965957	−	0.965957i	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	−14.4248	−	14.4248i	−0.965957	−	0.965957i	−0.999439	+	0.0334825i	$0.989340\pi$
$224$		0			0
$225$	−31.4248	+	3.15831i	−2.09499	+	0.210554i
$226$		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
$229$		−	29.8496i		−	1.97252i	−0.165205	−	0.986259i	$-0.552828\pi$
$229$		−	29.8496i		−	1.97252i	0.165205	−	0.986259i	$-0.447172\pi$
$230$		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
$235$	29.4248	−	1.47494i	1.91946	−	0.0962143i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−4.31662	−	4.31662i	−0.276912	−	0.276912i
$244$		0			0
$245$	10.5000	−	11.6082i	0.670820	−	0.741620i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.0000			−1.70422			−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000			−1.70422			−0.852112	+	0.523359i	$0.824679\pi$
$252$		0			0
$253$	9.42481	+	9.42481i	0.592533	+	0.592533i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22.2665	−	22.2665i	1.38895	−	1.38895i	0.561405	−	0.827541i	$-0.310261\pi$
$257$	22.2665	−	22.2665i	1.38895	−	1.38895i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$		0			0
$260$		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
$265$	0.575188	+	11.4749i	0.0353335	+	0.704900i
$266$		0			0
$267$	19.4248	−	19.4248i	1.18878	−	1.18878i
$268$		0			0
$269$		−	13.2665i		−	0.808873i	−0.914566	−	0.404436i	$-0.867468\pi$
$269$		−	13.2665i		−	0.808873i	0.914566	−	0.404436i	$-0.132532\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.65831	−	16.5000i	−0.100000	−	0.994987i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$			62.8496i			3.76271i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	−15.2164			−0.892000
$292$		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
$295$	4.97494	−	5.50000i	0.289652	−	0.320222i
$296$		0			0
$297$	23.7414	−	23.7414i	1.37762	−	1.37762i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$			34.3166i			1.95220i
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	24.4248	+	24.4248i	1.38057	+	1.38057i	0.843600	+	0.536972i	$0.180432\pi$
$313$	24.4248	+	24.4248i	1.38057	+	1.38057i	0.536972	+	0.843600i	$0.319568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−25.1082	+	25.1082i	−1.41022	+	1.41022i	−0.651981	+	0.758236i	$0.726062\pi$
$317$	−25.1082	+	25.1082i	−1.41022	+	1.41022i	−0.758236	+	0.651981i	$0.773938\pi$
$318$		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
$331$	9.94987			0.546895			0.273447	−	0.961887i	$-0.411836\pi$
$331$	9.94987			0.546895			0.273447	+	0.961887i	$0.411836\pi$
$332$		0			0
$333$	9.31662	+	9.31662i	0.510548	+	0.510548i
$334$		0			0
$335$	−36.2414	+	1.81662i	−1.98008	+	0.0992528i
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		−	52.4829i		−	2.85048i
$340$		0			0
$341$	−33.0000			−1.78705
$342$		0			0
$343$		0			0
$344$		0			0
$345$	18.3997	−	20.3417i	0.990609	−	1.09516i
$346$		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
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.7414	−	13.7414i	−0.731383	−	0.731383i	0.239511	−	0.970894i	$-0.423013\pi$
$353$	−13.7414	−	13.7414i	−0.731383	−	0.731383i	−0.970894	+	0.239511i	$0.923013\pi$
$354$		0			0
$355$	4.97494	+	4.50000i	0.264042	+	0.238835i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	23.7414	+	23.7414i	1.24610	+	1.24610i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	13.5251	−	13.5251i	0.706003	−	0.706003i	−0.259690	−	0.965692i	$-0.583620\pi$
$367$	13.5251	−	13.5251i	0.706003	−	0.706003i	0.965692	+	0.259690i	$0.0836203\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	−33.7414	+	5.10819i	−1.74240	+	0.263786i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			29.8496i			1.53327i	0.642082	+	0.766636i	$0.278071\pi$
$379$			29.8496i			1.53327i	−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.8417	−	17.8417i	−0.911668	−	0.911668i	0.0847358	−	0.996403i	$-0.472995\pi$
$383$	−17.8417	−	17.8417i	−0.911668	−	0.911668i	−0.996403	+	0.0847358i	$0.972995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			36.4829i			1.84976i	0.380265	+	0.924878i	$0.375833\pi$
$389$			36.4829i			1.84976i	−0.380265	+	0.924878i	$0.624167\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−20.8997	+	20.8997i	−1.04893	+	1.04893i	−0.0501886	+	0.998740i	$0.515982\pi$
$397$	−20.8997	+	20.8997i	−1.04893	+	1.04893i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−26.5330			−1.32499			−0.662497	−	0.749064i	$-0.730503\pi$
$401$	−26.5330			−1.32499			−0.662497	+	0.749064i	$0.730503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−19.8166	−	17.9248i	−0.984696	−	0.890691i
$406$		0			0
$407$	−4.89181	+	4.89181i	−0.242478	+	0.242478i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−43.6332			−2.15227
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			24.0000i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$	39.7995			1.93971			0.969854	−	0.243685i	$-0.0783563\pi$
$421$	39.7995			1.93971			0.969854	+	0.243685i	$0.0783563\pi$
$422$		0			0
$423$	58.8496	+	58.8496i	2.86137	+	2.86137i
$424$		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			$\pi$
$432$		0			0
$433$	29.4248	+	29.4248i	1.41407	+	1.41407i	0.717241	+	0.696826i	$0.245405\pi$
$433$	29.4248	+	29.4248i	1.41407	+	1.41407i	0.696826	+	0.717241i	$0.254595\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	44.2164			2.10554
$442$		0			0
$443$	28.7414	+	28.7414i	1.36555	+	1.36555i	0.866677	+	0.498870i	$0.166252\pi$
$443$	28.7414	+	28.7414i	1.36555	+	1.36555i	0.498870	+	0.866677i	$0.333748\pi$
$444$		0			0
$445$	13.5000	−	14.9248i	0.639961	−	0.707504i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	39.0000i		−	1.84052i	−0.391303	−	0.920262i	$-0.627976\pi$
$449$		−	39.0000i		−	1.84052i	0.391303	−	0.920262i	$-0.372024\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	0.575188	+	0.575188i	0.0267313	+	0.0267313i	0.720346	−	0.693615i	$-0.243983\pi$
$463$	0.575188	+	0.575188i	0.0267313	+	0.0267313i	−0.693615	+	0.720346i	$0.743983\pi$
$464$		0			0
$465$	3.39975	+	67.8246i	0.157660	+	3.14529i
$466$		0			0
$467$	−23.0581	+	23.0581i	−1.06700	+	1.06700i	−0.0694117	+	0.997588i	$0.522112\pi$
$467$	−23.0581	+	23.0581i	−1.06700	+	1.06700i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	71.1161			3.27686
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−22.9499	+	22.9499i	−1.05080	+	1.05080i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.1332	+	0.558061i	−0.505535	+	0.0253403i
$486$		0			0
$487$	−26.4749	+	26.4749i	−1.19969	+	1.19969i	−0.225436	+	0.974258i	$0.572381\pi$
$487$	−26.4749	+	26.4749i	−1.19969	+	1.19969i	−0.974258	+	0.225436i	$0.927619\pi$
$488$		0			0
$489$			77.4829i			3.50390i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	31.4248	−	34.7414i	1.41244	−	1.56151i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	19.8997i		−	0.890835i	−0.895323	−	0.445418i	$-0.853055\pi$
$499$		−	19.8997i		−	0.890835i	0.895323	−	0.445418i	$-0.146945\pi$
$500$		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
$507$	−28.0581	+	28.0581i	−1.24610	+	1.24610i
$508$		0			0
$509$			3.31662i			0.147007i	0.997295	+	0.0735034i	$0.0234180\pi$
$509$			3.31662i			0.147007i	−0.997295	+	0.0735034i	$0.976582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.25856	+	25.1082i	0.0554589	+	1.10640i
$516$		0			0
$517$	−30.8997	+	30.8997i	−1.35897	+	1.35897i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−43.1161			−1.88895			−0.944476	−	0.328581i	$-0.893430\pi$
$521$	−43.1161			−1.88895			−0.944476	+	0.328581i	$0.893430\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	6.84962i		−	0.297810i
$530$		0			0
$531$	20.9499			0.909147
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	45.3246	−	45.3246i	1.95590	−	1.95590i
$538$		0			0
$539$			23.2164i			1.00000i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−21.4749	−	21.4749i	−0.921578	−	0.921578i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	10.5581	+	9.55013i	0.448165	+	0.405380i
$556$		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
$559$		0			0
$560$		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
$565$	−1.92481	−	38.3997i	−0.0809774	−	1.61549i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−50.1082	−	50.1082i	−2.09330	−	2.09330i
$574$		0			0
$575$	12.7164	−	15.5581i	0.530309	−	0.648816i
$576$		0			0
$577$	−18.5251	+	18.5251i	−0.771208	+	0.771208i	−0.978318	−	0.207109i	$-0.933594\pi$
$577$	−18.5251	+	18.5251i	−0.771208	+	0.771208i	0.207109	+	0.978318i	$0.433594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−12.0501	−	12.0501i	−0.499065	−	0.499065i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.6834	+	20.6834i	−0.853694	+	0.853694i	−0.990586	−	0.136892i	$-0.956289\pi$
$587$	−20.6834	+	20.6834i	−0.853694	+	0.853694i	0.136892	+	0.990586i	$0.456289\pi$
$588$		0			0
$589$		0			0
$590$		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
$597$	42.9499	−	42.9499i	1.75782	−	1.75782i
$598$		0			0
$599$		−	36.0000i		−	1.47092i	−0.677568	−	0.735460i	$-0.736966\pi$
$599$		−	36.0000i		−	1.47092i	0.677568	−	0.735460i	$-0.263034\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−72.4829	−	72.4829i	−2.95173	−	2.95173i
$604$		0			0
$605$	18.2414	+	16.5000i	0.741620	+	0.670820i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.73350	+	7.73350i	−0.311339	+	0.311339i	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−7.73350	+	7.73350i	−0.311339	+	0.311339i	0.534089	+	0.845428i	$0.320655\pi$
$618$		0			0
$619$		−	1.00000i		−	0.0401934i	−0.999798	−	0.0200967i	$-0.993603\pi$
$619$		−	1.00000i		−	0.0401934i	0.999798	−	0.0200967i	$-0.00639741\pi$
$620$		0			0
$621$	40.6834			1.63257
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.5000	+	4.97494i	−0.980000	+	0.198997i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−7.00000			−0.278666			−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000			−0.278666			−0.139333	+	0.990246i	$0.544496\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$			18.9499i			0.749645i
$640$		0			0
$641$	23.2164			0.916992			0.458496	−	0.888697i	$-0.348388\pi$
$641$	23.2164			0.916992			0.458496	+	0.888697i	$0.348388\pi$
$642$		0			0
$643$	5.57519	+	5.57519i	0.219864	+	0.219864i	0.808441	−	0.588577i	$-0.200312\pi$
$643$	5.57519	+	5.57519i	0.219864	+	0.219864i	−0.588577	+	0.808441i	$0.700312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.05806	+	8.05806i	−0.316795	+	0.316795i	−0.847535	−	0.530740i	$-0.821914\pi$
$647$	−8.05806	+	8.05806i	−0.316795	+	0.316795i	0.530740	+	0.847535i	$0.321914\pi$
$648$		0			0
$649$			11.0000i			0.431788i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−27.1583	−	27.1583i	−1.06279	−	1.06279i	−0.997892	−	0.0648948i	$-0.979329\pi$
$653$	−27.1583	−	27.1583i	−1.06279	−	1.06279i	−0.0648948	−	0.997892i	$-0.520671\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	62.2665i		−	2.40736i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$	−39.1913	−	32.0330i	−1.50847	−	1.23295i
$676$		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
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.2164	−	35.2164i	−1.34752	−	1.34752i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	−35.2164	−	35.2164i	−1.34752	−	1.34752i	−0.459167	−	0.888350i	$-0.651852\pi$
$684$		0			0
$685$	−31.9248	+	1.60025i	−1.21978	+	0.0611425i
$686$		0			0
$687$	64.4248	−	64.4248i	2.45796	−	2.45796i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−17.0000			−0.646710			−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000			−0.646710			−0.323355	+	0.946278i	$0.604811\pi$
$692$		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
$704$		0			0
$705$	66.6913	+	60.3246i	2.51174	+	2.27195i
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	19.0000i		−	0.713560i	−0.934188	−	0.356780i	$-0.883875\pi$
$709$		−	19.0000i		−	0.713560i	0.934188	−	0.356780i	$-0.116125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−28.2744	−	28.2744i	−1.05889	−	1.05889i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	51.0000i		−	1.90198i	−0.309223	−	0.950990i	$-0.600069\pi$
$719$		−	51.0000i		−	1.90198i	0.309223	−	0.950990i	$-0.399931\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−31.4749	+	31.4749i	−1.16734	+	1.16734i	−0.184510	+	0.982831i	$0.559070\pi$
$727$	−31.4749	+	31.4749i	−1.16734	+	1.16734i	−0.982831	+	0.184510i	$0.940930\pi$
$728$		0			0
$729$			17.2164i			0.637643i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$	47.7164	−	2.39181i	1.76004	−	0.0882234i
$736$		0			0
$737$	38.0581	−	38.0581i	1.40189	−	1.40189i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		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
$751$	23.0000			0.839282			0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000			0.839282			0.419641	+	0.907690i	$0.362156\pi$
$752$		0			0
$753$	−58.2744	−	58.2744i	−2.12364	−	2.12364i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.899749	+	0.899749i	−0.0327019	+	0.0327019i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	−0.899749	+	0.899749i	−0.0327019	+	0.0327019i	0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$			40.6834i			1.47671i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	96.1161			3.46154
$772$		0			0
$773$	33.6332	+	33.6332i	1.20970	+	1.20970i	0.971123	+	0.238581i	$0.0766824\pi$
$773$	33.6332	+	33.6332i	1.20970	+	1.20970i	0.238581	+	0.971123i	$0.423318\pi$
$774$		0			0
$775$	4.97494	+	49.5000i	0.178705	+	1.77809i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−9.94987			−0.356034
$782$		0			0
$783$		0			0
$784$		0			0
$785$	52.0330	−	2.60819i	1.85714	−	0.0930902i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−23.5251	+	26.0079i	−0.834348	+	0.922406i
$796$		0			0
$797$	−26.6913	+	26.6913i	−0.945455	+	0.945455i	−0.998587	−	0.0531327i	$-0.983079\pi$
$797$	−26.6913	+	26.6913i	−0.945455	+	0.945455i	0.0531327	+	0.998587i	$0.483079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	56.8496			2.00868
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	28.6332	−	28.6332i	1.00794	−	1.00794i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.84169	+	56.6913i	0.0995400	+	1.98581i
$816$		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
$823$	−39.4248	−	39.4248i	−1.37426	−	1.37426i	−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−39.4248	−	39.4248i	−1.37426	−	1.37426i	−0.520246	−	0.854016i	$-0.674160\pi$
$824$		0			0
$825$	32.0330	−	39.1913i	1.11525	−	1.36447i
$826$		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
$829$		−	29.0000i		−	1.00721i	−0.863934	−	0.503606i	$-0.832006\pi$
$829$		−	29.0000i		−	1.00721i	0.863934	−	0.503606i	$-0.167994\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−71.2243	+	71.2243i	−2.46187	+	2.46187i
$838$		0			0
$839$		−	36.4829i		−	1.25953i	−0.776786	−	0.629764i	$-0.783151\pi$
$839$		−	36.4829i		−	1.25953i	0.776786	−	0.629764i	$-0.216849\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.5000	+	21.5581i	−0.670820	+	0.741620i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.38262			−0.287353
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		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
$859$		−	31.0000i		−	1.05771i	−0.848713	−	0.528853i	$-0.822622\pi$
$859$		−	31.0000i		−	1.05771i	0.848713	−	0.528853i	$-0.177378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.21637	−	5.21637i	−0.177567	−	0.177567i	0.612727	−	0.790295i	$-0.290072\pi$
$863$	−5.21637	−	5.21637i	−0.177567	−	0.177567i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	36.6913	−	36.6913i	1.24610	−	1.24610i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−22.2665	−	22.2665i	−0.753607	−	0.753607i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	57.0000			1.92038			0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000			1.92038			0.960189	+	0.279350i	$0.0901189\pi$
$882$		0			0
$883$	37.9499	+	37.9499i	1.27711	+	1.27711i	0.942275	+	0.334840i	$0.108682\pi$
$883$	37.9499	+	37.9499i	1.27711	+	1.27711i	0.334840	+	0.942275i	$0.391318\pi$
$884$		0			0
$885$	22.6082	−	1.13325i	0.759966	−	0.0380938i
$886$		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
$889$		0			0
$890$		0			0
$891$	39.6332			1.32776
$892$		0			0
$893$		0			0
$894$		0			0
$895$	31.5000	−	34.8246i	1.05293	−	1.16406i
$896$		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
$905$	−16.5000	−	14.9248i	−0.548479	−	0.496118i
$906$		0			0
$907$	−33.8496	+	33.8496i	−1.12396	+	1.12396i	−0.132818	+	0.991140i	$0.542403\pi$
$907$	−33.8496	+	33.8496i	−1.12396	+	1.12396i	−0.991140	+	0.132818i	$0.957597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.63325			−0.219769			−0.109885	−	0.993944i	$-0.535048\pi$
$911$	−6.63325			−0.219769			−0.109885	+	0.993944i	$0.535048\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	8.07519	+	6.60025i	0.265511	+	0.217015i
$926$		0			0
$927$	−50.2164	+	50.2164i	−1.64932	+	1.64932i
$928$		0			0
$929$			53.0660i			1.74104i	0.492134	+	0.870519i	$0.336217\pi$
$929$			53.0660i			1.74104i	−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−25.8997	−	25.8997i	−0.847920	−	0.847920i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$			105.433i			3.44067i
$940$		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
$947$	40.1082	−	40.1082i	1.30334	−	1.30334i	0.377215	−	0.926126i	$-0.376882\pi$
$947$	40.1082	−	40.1082i	1.30334	−	1.30334i	0.926126	−	0.377215i	$-0.123118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−108.383			−3.51455
$952$		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
$955$	−38.5000	−	34.8246i	−1.24583	−	1.12690i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	68.0000			2.19355
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	43.1161			1.38366			0.691831	−	0.722059i	$-0.256804\pi$
$971$	43.1161			1.38366			0.691831	+	0.722059i	$0.256804\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−41.6913	+	41.6913i	−1.33382	+	1.33382i	−0.431903	+	0.901920i	$0.642158\pi$
$977$	−41.6913	+	41.6913i	−1.33382	+	1.33382i	−0.901920	+	0.431903i	$0.857842\pi$
$978$		0			0
$979$			29.8496i			0.953998i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	43.7414	+	43.7414i	1.39514	+	1.39514i	0.813324	+	0.581811i	$0.197656\pi$
$983$	43.7414	+	43.7414i	1.39514	+	1.39514i	0.581811	+	0.813324i	$0.302344\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	59.6992			1.89641			0.948205	−	0.317660i	$-0.102897\pi$
$991$	59.6992			1.89641			0.948205	+	0.317660i	$0.102897\pi$
$992$		0			0
$993$	21.4749	+	21.4749i	0.681487	+	0.681487i
$994$		0			0
$995$	29.8496	−	33.0000i	0.946297	−	1.04617i
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$			21.1161i			0.668085i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.bd.d.593.2		4
4.3	odd	2		55.2.e.b.43.1	yes	4
5.2	odd	4	inner	880.2.bd.d.417.2		4
11.10	odd	2	CM	880.2.bd.d.593.2		4
12.11	even	2		495.2.k.a.208.1		4
20.3	even	4		275.2.e.a.32.2		4
20.7	even	4		55.2.e.b.32.1	✓	4
20.19	odd	2		275.2.e.a.43.2		4
44.3	odd	10		605.2.m.a.233.2		16
44.7	even	10		605.2.m.a.578.2		16
44.15	odd	10		605.2.m.a.578.2		16
44.19	even	10		605.2.m.a.233.2		16
44.27	odd	10		605.2.m.a.118.2		16
44.31	odd	10		605.2.m.a.403.1		16
44.35	even	10		605.2.m.a.403.1		16
44.39	even	10		605.2.m.a.118.2		16
44.43	even	2		55.2.e.b.43.1	yes	4
55.32	even	4	inner	880.2.bd.d.417.2		4
60.47	odd	4		495.2.k.a.307.1		4
132.131	odd	2		495.2.k.a.208.1		4
220.7	odd	20		605.2.m.a.457.2		16
220.27	even	20		605.2.m.a.602.1		16
220.43	odd	4		275.2.e.a.32.2		4
220.47	even	20		605.2.m.a.112.2		16
220.87	odd	4		55.2.e.b.32.1	✓	4
220.107	odd	20		605.2.m.a.112.2		16
220.127	odd	20		605.2.m.a.602.1		16
220.147	even	20		605.2.m.a.457.2		16
220.167	odd	20		605.2.m.a.282.2		16
220.207	even	20		605.2.m.a.282.2		16
220.219	even	2		275.2.e.a.43.2		4
660.527	even	4		495.2.k.a.307.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.1	✓	4	20.7	even	4
55.2.e.b.32.1	✓	4	220.87	odd	4
55.2.e.b.43.1	yes	4	4.3	odd	2
55.2.e.b.43.1	yes	4	44.43	even	2
275.2.e.a.32.2		4	20.3	even	4
275.2.e.a.32.2		4	220.43	odd	4
275.2.e.a.43.2		4	20.19	odd	2
275.2.e.a.43.2		4	220.219	even	2
495.2.k.a.208.1		4	12.11	even	2
495.2.k.a.208.1		4	132.131	odd	2
495.2.k.a.307.1		4	60.47	odd	4
495.2.k.a.307.1		4	660.527	even	4
605.2.m.a.112.2		16	220.47	even	20
605.2.m.a.112.2		16	220.107	odd	20
605.2.m.a.118.2		16	44.27	odd	10
605.2.m.a.118.2		16	44.39	even	10
605.2.m.a.233.2		16	44.3	odd	10
605.2.m.a.233.2		16	44.19	even	10
605.2.m.a.282.2		16	220.167	odd	20
605.2.m.a.282.2		16	220.207	even	20
605.2.m.a.403.1		16	44.31	odd	10
605.2.m.a.403.1		16	44.35	even	10
605.2.m.a.457.2		16	220.7	odd	20
605.2.m.a.457.2		16	220.147	even	20
605.2.m.a.578.2		16	44.7	even	10
605.2.m.a.578.2		16	44.15	odd	10
605.2.m.a.602.1		16	220.27	even	20
605.2.m.a.602.1		16	220.127	odd	20
880.2.bd.d.417.2		4	5.2	odd	4	inner
880.2.bd.d.417.2		4	55.32	even	4	inner
880.2.bd.d.593.2		4	1.1	even	1	trivial
880.2.bd.d.593.2		4	11.10	odd	2	CM

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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.bd (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.15831 + 2.15831i) q^{3} +(1.65831 + 1.50000i) q^{5} +6.31662i q^{9} +O(q^{10})\)	\(q+(2.15831 + 2.15831i) q^{3} +(1.65831 + 1.50000i) q^{5} +6.31662i q^{9} -3.31662 q^{11} +(0.341688 + 6.81662i) q^{15} +(-2.84169 - 2.84169i) q^{23} +(0.500000 + 4.97494i) q^{25} +(-7.15831 + 7.15831i) q^{27} +9.94987 q^{31} +(-7.15831 - 7.15831i) q^{33} +(1.47494 - 1.47494i) q^{37} +(-9.47494 + 10.4749i) q^{45} +(9.31662 - 9.31662i) q^{47} -7.00000i q^{49} +(3.63325 + 3.63325i) q^{53} +(-5.50000 - 4.97494i) q^{55} -3.31662i q^{59} +(-11.4749 + 11.4749i) q^{67} -12.2665i q^{69} +3.00000 q^{71} +(-9.65831 + 11.8166i) q^{75} -11.9499 q^{81} -9.00000i q^{89} +(21.4749 + 21.4749i) q^{93} +(-3.52506 + 3.52506i) q^{97} -20.9499i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3}+O(q^{10}) \) 4 * q + 2 * q^3	\( 4 q + 2 q^{3} + 8 q^{15} - 18 q^{23} + 2 q^{25} - 22 q^{27} - 22 q^{33} - 14 q^{37} - 18 q^{45} + 24 q^{47} - 12 q^{53} - 22 q^{55} - 26 q^{67} + 12 q^{71} - 32 q^{75} - 8 q^{81} + 66 q^{93} - 34 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 + 8 * q^15 - 18 * q^23 + 2 * q^25 - 22 * q^27 - 22 * q^33 - 14 * q^37 - 18 * q^45 + 24 * q^47 - 12 * q^53 - 22 * q^55 - 26 * q^67 + 12 * q^71 - 32 * q^75 - 8 * q^81 + 66 * q^93 - 34 * q^97

Introduction

L-functions

Modular forms

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