LMFDB - Embedded newform 4410.2.b.a.881.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(881,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4410.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.b (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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.3
Root			$0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.a.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} -1.23463i q^{11} +4.33182i q^{13} +1.00000 q^{16} -4.12906 q^{17} +2.28130i q^{19} +1.00000 q^{20} -1.23463 q^{22} -2.26038i q^{23} +1.00000 q^{25} +4.33182 q^{26} +0.511402i q^{29} -3.89828i q^{31} -1.00000i q^{32} +4.12906i q^{34} -4.39488 q^{37} +2.28130 q^{38} -1.00000i q^{40} -7.08239 q^{41} +10.1837 q^{43} +1.23463i q^{44} -2.26038 q^{46} +7.51594 q^{47} -1.00000i q^{50} -4.33182i q^{52} -8.83259i q^{53} +1.23463i q^{55} +0.511402 q^{58} +8.33598 q^{59} +0.890268i q^{61} -3.89828 q^{62} -1.00000 q^{64} -4.33182i q^{65} -5.24105 q^{67} +4.12906 q^{68} +0.818827i q^{71} -11.6610i q^{73} +4.39488i q^{74} -2.28130i q^{76} +5.65685 q^{79} -1.00000 q^{80} +7.08239i q^{82} +0.164784 q^{83} +4.12906 q^{85} -10.1837i q^{86} +1.23463 q^{88} +4.49886 q^{89} +2.26038i q^{92} -7.51594i q^{94} -2.28130i q^{95} +9.06147i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^4 - 8 * q^5	$ 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} - 16 q^{22} + 8 q^{25} + 32 q^{26} - 16 q^{37} - 48 q^{41} - 16 q^{43} + 16 q^{46} + 48 q^{47} - 16 q^{58} - 16 q^{59} - 32 q^{62} - 8 q^{64} - 8 q^{80} - 16 q^{83} + 16 q^{88} + 48 q^{89}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 - 16 * q^22 + 8 * q^25 + 32 * q^26 - 16 * q^37 - 48 * q^41 - 16 * q^43 + 16 * q^46 + 48 * q^47 - 16 * q^58 - 16 * q^59 - 32 * q^62 - 8 * q^64 - 8 * q^80 - 16 * q^83 + 16 * q^88 + 48 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$.

$n$	$1081$	$2647$	$3431$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	0	0
$10$	1.00000i	0.316228i
$11$	− 1.23463i	− 0.372256i	−0.982526	−	0.186128i	$-0.940406\pi$
$11$	− 1.23463i	− 0.372256i	0.982526	−	0.186128i	$-0.0595939\pi$
$12$	0	0
$13$	4.33182i	1.20143i	0.799463	+	0.600716i	$0.205118\pi$
$13$	4.33182i	1.20143i	−0.799463	+	0.600716i	$0.794882\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.12906	−1.00145	−0.500723	−	0.865608i	$-0.666932\pi$
$17$	−4.12906	−1.00145	−0.500723	+	0.865608i	$0.666932\pi$
$18$	0	0
$19$	2.28130i	0.523367i	0.965154	+	0.261684i	$0.0842777\pi$
$19$	2.28130i	0.523367i	−0.965154	+	0.261684i	$0.915722\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.23463	−0.263225
$23$	− 2.26038i	− 0.471322i	−0.971835	−	0.235661i	$-0.924275\pi$
$23$	− 2.26038i	− 0.471322i	0.971835	−	0.235661i	$-0.0757255\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.33182	0.849540
$27$	0	0
$28$	0	0
$29$	0.511402i	0.0949649i	0.998872	+	0.0474825i	$0.0151198\pi$
$29$	0.511402i	0.0949649i	−0.998872	+	0.0474825i	$0.984880\pi$
$30$	0	0
$31$	− 3.89828i	− 0.700151i	−0.936721	−	0.350076i	$-0.886156\pi$
$31$	− 3.89828i	− 0.700151i	0.936721	−	0.350076i	$-0.113844\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	4.12906i	0.708129i
$35$	0	0
$36$	0	0
$37$	−4.39488	−0.722514	−0.361257	−	0.932466i	$-0.617652\pi$
$37$	−4.39488	−0.722514	−0.361257	+	0.932466i	$0.617652\pi$
$38$	2.28130	0.370076
$39$	0	0
$40$	− 1.00000i	− 0.158114i
$41$	−7.08239	−1.10608	−0.553042	−	0.833153i	$-0.686533\pi$
$41$	−7.08239	−1.10608	−0.553042	+	0.833153i	$0.686533\pi$
$42$	0	0
$43$	10.1837	1.55301	0.776503	−	0.630113i	$-0.216992\pi$
$43$	10.1837	1.55301	0.776503	+	0.630113i	$0.216992\pi$
$44$	1.23463i	0.186128i
$45$	0	0
$46$	−2.26038	−0.333275
$47$	7.51594	1.09631	0.548156	−	0.836376i	$-0.315330\pi$
$47$	7.51594	1.09631	0.548156	+	0.836376i	$0.315330\pi$
$48$	0	0
$49$	0	0
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	− 4.33182i	− 0.600716i
$53$	− 8.83259i	− 1.21325i	−0.794988	−	0.606625i	$-0.792523\pi$
$53$	− 8.83259i	− 1.21325i	0.794988	−	0.606625i	$-0.207477\pi$
$54$	0	0
$55$	1.23463i	0.166478i
$56$	0	0
$57$	0	0
$58$	0.511402	0.0671503
$59$	8.33598	1.08525	0.542626	−	0.839974i	$-0.317430\pi$
$59$	8.33598	1.08525	0.542626	+	0.839974i	$0.317430\pi$
$60$	0	0
$61$	0.890268i	0.113987i	0.998375	+	0.0569936i	$0.0181515\pi$
$61$	0.890268i	0.113987i	−0.998375	+	0.0569936i	$0.981849\pi$
$62$	−3.89828	−0.495082
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 4.33182i	− 0.537296i
$66$	0	0
$67$	−5.24105	−0.640296	−0.320148	−	0.947368i	$-0.603733\pi$
$67$	−5.24105	−0.640296	−0.320148	+	0.947368i	$0.603733\pi$
$68$	4.12906	0.500723
$69$	0	0
$70$	0	0
$71$	0.818827i	0.0971769i	0.998819	+	0.0485884i	$0.0154723\pi$
$71$	0.818827i	0.0971769i	−0.998819	+	0.0485884i	$0.984528\pi$
$72$	0	0
$73$	− 11.6610i	− 1.36482i	−0.730970	−	0.682409i	$-0.760932\pi$
$73$	− 11.6610i	− 1.36482i	0.730970	−	0.682409i	$-0.239068\pi$
$74$	4.39488i	0.510895i
$75$	0	0
$76$	− 2.28130i	− 0.261684i
$77$	0	0
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	7.08239i	0.782119i
$83$	0.164784	0.0180874	0.00904372	−	0.999959i	$-0.497121\pi$
$83$	0.164784	0.0180874	0.00904372	+	0.999959i	$0.497121\pi$
$84$	0	0
$85$	4.12906	0.447860
$86$	− 10.1837i	− 1.09814i
$87$	0	0
$88$	1.23463	0.131612
$89$	4.49886	0.476878	0.238439	−	0.971157i	$-0.423364\pi$
$89$	4.49886	0.476878	0.238439	+	0.971157i	$0.423364\pi$
$90$	0	0
$91$	0	0
$92$	2.26038i	0.235661i
$93$	0	0
$94$	− 7.51594i	− 0.775210i
$95$	− 2.28130i	− 0.234057i
$96$	0	0
$97$	9.06147i	0.920053i	0.887905	+	0.460026i	$0.152160\pi$
$97$	9.06147i	0.920053i	−0.887905	+	0.460026i	$0.847840\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	3.88348	0.386421	0.193210	−	0.981157i	$-0.438110\pi$
$101$	3.88348	0.386421	0.193210	+	0.981157i	$0.438110\pi$
$102$	0	0
$103$	− 7.56767i	− 0.745665i	−0.927899	−	0.372833i	$-0.878387\pi$
$103$	− 7.56767i	− 0.745665i	0.927899	−	0.372833i	$-0.121613\pi$
$104$	−4.33182	−0.424770
$105$	0	0
$106$	−8.83259	−0.857897
$107$	− 13.7679i	− 1.33100i	−0.746399	−	0.665498i	$-0.768219\pi$
$107$	− 13.7679i	− 1.33100i	0.746399	−	0.665498i	$-0.231781\pi$
$108$	0	0
$109$	0.810688	0.0776498	0.0388249	−	0.999246i	$-0.487639\pi$
$109$	0.810688	0.0776498	0.0388249	+	0.999246i	$0.487639\pi$
$110$	1.23463	0.117718
$111$	0	0
$112$	0	0
$113$	− 11.0866i	− 1.04294i	−0.853271	−	0.521468i	$-0.825385\pi$
$113$	− 11.0866i	− 1.04294i	0.853271	−	0.521468i	$-0.174615\pi$
$114$	0	0
$115$	2.26038i	0.210782i
$116$	− 0.511402i	− 0.0474825i
$117$	0	0
$118$	− 8.33598i	− 0.767389i
$119$	0	0
$120$	0	0
$121$	9.47568	0.861426
$122$	0.890268	0.0806011
$123$	0	0
$124$	3.89828i	0.350076i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.12612	0.721077	0.360538	−	0.932744i	$-0.382593\pi$
$127$	8.12612	0.721077	0.360538	+	0.932744i	$0.382593\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−4.33182	−0.379926
$131$	0.391410	0.0341976	0.0170988	−	0.999854i	$-0.494557\pi$
$131$	0.391410	0.0341976	0.0170988	+	0.999854i	$0.494557\pi$
$132$	0	0
$133$	0	0
$134$	5.24105i	0.452758i
$135$	0	0
$136$	− 4.12906i	− 0.354064i
$137$	− 21.5832i	− 1.84397i	−0.387221	−	0.921987i	$-0.626565\pi$
$137$	− 21.5832i	− 1.84397i	0.387221	−	0.921987i	$-0.373435\pi$
$138$	0	0
$139$	22.9700i	1.94829i	0.225918	+	0.974146i	$0.427462\pi$
$139$	22.9700i	1.94829i	−0.225918	+	0.974146i	$0.572538\pi$
$140$	0	0
$141$	0	0
$142$	0.818827	0.0687144
$143$	5.34821	0.447240
$144$	0	0
$145$	− 0.511402i	− 0.0424696i
$146$	−11.6610	−0.965073
$147$	0	0
$148$	4.39488	0.361257
$149$	− 22.8033i	− 1.86812i	−0.357119	−	0.934059i	$-0.616241\pi$
$149$	− 22.8033i	− 1.86812i	0.357119	−	0.934059i	$-0.383759\pi$
$150$	0	0
$151$	−21.2905	−1.73260	−0.866299	−	0.499525i	$-0.833508\pi$
$151$	−21.2905	−1.73260	−0.866299	+	0.499525i	$0.833508\pi$
$152$	−2.28130	−0.185038
$153$	0	0
$154$	0	0
$155$	3.89828i	0.313117i
$156$	0	0
$157$	6.60127i	0.526839i	0.964681	+	0.263419i	$0.0848503\pi$
$157$	6.60127i	0.526839i	−0.964681	+	0.263419i	$0.915150\pi$
$158$	− 5.65685i	− 0.450035i
$159$	0	0
$160$	1.00000i	0.0790569i
$161$	0	0
$162$	0	0
$163$	−16.4286	−1.28679	−0.643395	−	0.765534i	$-0.722475\pi$
$163$	−16.4286	−1.28679	−0.643395	+	0.765534i	$0.722475\pi$
$164$	7.08239	0.553042
$165$	0	0
$166$	− 0.164784i	− 0.0127897i
$167$	−0.834556	−0.0645799	−0.0322899	−	0.999479i	$-0.510280\pi$
$167$	−0.834556	−0.0645799	−0.0322899	+	0.999479i	$0.510280\pi$
$168$	0	0
$169$	−5.76468	−0.443437
$170$	− 4.12906i	− 0.316685i
$171$	0	0
$172$	−10.1837	−0.776503
$173$	−3.54675	−0.269654	−0.134827	−	0.990869i	$-0.543048\pi$
$173$	−3.54675	−0.269654	−0.134827	+	0.990869i	$0.543048\pi$
$174$	0	0
$175$	0	0
$176$	− 1.23463i	− 0.0930640i
$177$	0	0
$178$	− 4.49886i	− 0.337204i
$179$	− 16.6685i	− 1.24586i	−0.782277	−	0.622931i	$-0.785942\pi$
$179$	− 16.6685i	− 1.24586i	0.782277	−	0.622931i	$-0.214058\pi$
$180$	0	0
$181$	− 2.53036i	− 0.188080i	−0.995568	−	0.0940401i	$-0.970022\pi$
$181$	− 2.53036i	− 0.188080i	0.995568	−	0.0940401i	$-0.0299782\pi$
$182$	0	0
$183$	0	0
$184$	2.26038	0.166637
$185$	4.39488	0.323118
$186$	0	0
$187$	5.09788i	0.372794i
$188$	−7.51594	−0.548156
$189$	0	0
$190$	−2.28130	−0.165503
$191$	5.15158i	0.372755i	0.982478	+	0.186378i	$0.0596748\pi$
$191$	5.15158i	0.372755i	−0.982478	+	0.186378i	$0.940325\pi$
$192$	0	0
$193$	4.50793	0.324488	0.162244	−	0.986751i	$-0.448127\pi$
$193$	4.50793	0.324488	0.162244	+	0.986751i	$0.448127\pi$
$194$	9.06147	0.650575
$195$	0	0
$196$	0	0
$197$	3.14476i	0.224055i	0.993705	+	0.112028i	$0.0357345\pi$
$197$	3.14476i	0.224055i	−0.993705	+	0.112028i	$0.964266\pi$
$198$	0	0
$199$	− 3.28343i	− 0.232756i	−0.993205	−	0.116378i	$-0.962872\pi$
$199$	− 3.28343i	− 0.232756i	0.993205	−	0.116378i	$-0.0371284\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	− 3.88348i	− 0.273241i
$203$	0	0
$204$	0	0
$205$	7.08239	0.494656
$206$	−7.56767	−0.527265
$207$	0	0
$208$	4.33182i	0.300358i
$209$	2.81657	0.194827
$210$	0	0
$211$	25.0907	1.72732	0.863658	−	0.504078i	$-0.168168\pi$
$211$	25.0907	1.72732	0.863658	+	0.504078i	$0.168168\pi$
$212$	8.83259i	0.606625i
$213$	0	0
$214$	−13.7679	−0.941157
$215$	−10.1837	−0.694525
$216$	0	0
$217$	0	0
$218$	− 0.810688i	− 0.0549067i
$219$	0	0
$220$	− 1.23463i	− 0.0832389i
$221$	− 17.8864i	− 1.20317i
$222$	0	0
$223$	− 3.21440i	− 0.215252i	−0.994191	−	0.107626i	$-0.965675\pi$
$223$	− 3.21440i	− 0.215252i	0.994191	−	0.107626i	$-0.0343249\pi$
$224$	0	0
$225$	0	0
$226$	−11.0866	−0.737467
$227$	−7.15800	−0.475093	−0.237546	−	0.971376i	$-0.576343\pi$
$227$	−7.15800	−0.475093	−0.237546	+	0.971376i	$0.576343\pi$
$228$	0	0
$229$	− 17.0156i	− 1.12443i	−0.826993	−	0.562213i	$-0.809950\pi$
$229$	− 17.0156i	− 1.12443i	0.826993	−	0.562213i	$-0.190050\pi$
$230$	2.26038	0.149045
$231$	0	0
$232$	−0.511402	−0.0335752
$233$	− 8.18080i	− 0.535942i	−0.963427	−	0.267971i	$-0.913647\pi$
$233$	− 8.18080i	− 0.535942i	0.963427	−	0.267971i	$-0.0863531\pi$
$234$	0	0
$235$	−7.51594	−0.490286
$236$	−8.33598	−0.542626
$237$	0	0
$238$	0	0
$239$	− 22.3224i	− 1.44392i	−0.691937	−	0.721958i	$-0.743242\pi$
$239$	− 22.3224i	− 1.44392i	0.691937	−	0.721958i	$-0.256758\pi$
$240$	0	0
$241$	− 12.5525i	− 0.808578i	−0.914631	−	0.404289i	$-0.867519\pi$
$241$	− 12.5525i	− 0.808578i	0.914631	−	0.404289i	$-0.132481\pi$
$242$	− 9.47568i	− 0.609120i
$243$	0	0
$244$	− 0.890268i	− 0.0569936i
$245$	0	0
$246$	0	0
$247$	−9.88220	−0.628790
$248$	3.89828	0.247541
$249$	0	0
$250$	1.00000i	0.0632456i
$251$	−6.05468	−0.382168	−0.191084	−	0.981574i	$-0.561200\pi$
$251$	−6.05468	−0.382168	−0.191084	+	0.981574i	$0.561200\pi$
$252$	0	0
$253$	−2.79074	−0.175452
$254$	− 8.12612i	− 0.509878i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.9298	−0.993672	−0.496836	−	0.867844i	$-0.665505\pi$
$257$	−15.9298	−0.993672	−0.496836	+	0.867844i	$0.665505\pi$
$258$	0	0
$259$	0	0
$260$	4.33182i	0.268648i
$261$	0	0
$262$	− 0.391410i	− 0.0241814i
$263$	8.09560i	0.499196i	0.968350	+	0.249598i	$0.0802984\pi$
$263$	8.09560i	0.499196i	−0.968350	+	0.249598i	$0.919702\pi$
$264$	0	0
$265$	8.83259i	0.542582i
$266$	0	0
$267$	0	0
$268$	5.24105	0.320148
$269$	31.7070	1.93321	0.966606	−	0.256268i	$-0.0824929\pi$
$269$	31.7070	1.93321	0.966606	+	0.256268i	$0.0824929\pi$
$270$	0	0
$271$	− 30.2571i	− 1.83799i	−0.394272	−	0.918994i	$-0.629003\pi$
$271$	− 30.2571i	− 1.83799i	0.394272	−	0.918994i	$-0.370997\pi$
$272$	−4.12906	−0.250361
$273$	0	0
$274$	−21.5832	−1.30389
$275$	− 1.23463i	− 0.0744512i
$276$	0	0
$277$	3.07626	0.184835	0.0924174	−	0.995720i	$-0.470541\pi$
$277$	3.07626	0.184835	0.0924174	+	0.995720i	$0.470541\pi$
$278$	22.9700	1.37765
$279$	0	0
$280$	0	0
$281$	− 23.4167i	− 1.39692i	−0.715649	−	0.698460i	$-0.753869\pi$
$281$	− 23.4167i	− 1.39692i	0.715649	−	0.698460i	$-0.246131\pi$
$282$	0	0
$283$	− 19.4152i	− 1.15411i	−0.816704	−	0.577057i	$-0.804201\pi$
$283$	− 19.4152i	− 1.15411i	0.816704	−	0.577057i	$-0.195799\pi$
$284$	− 0.818827i	− 0.0485884i
$285$	0	0
$286$	− 5.34821i	− 0.316246i
$287$	0	0
$288$	0	0
$289$	0.0491665	0.00289215
$290$	−0.511402	−0.0300305
$291$	0	0
$292$	11.6610i	0.682409i
$293$	−26.1609	−1.52833	−0.764167	−	0.645019i	$-0.776849\pi$
$293$	−26.1609	−1.52833	−0.764167	+	0.645019i	$0.776849\pi$
$294$	0	0
$295$	−8.33598	−0.485340
$296$	− 4.39488i	− 0.255447i
$297$	0	0
$298$	−22.8033	−1.32096
$299$	9.79156	0.566261
$300$	0	0
$301$	0	0
$302$	21.2905i	1.22513i
$303$	0	0
$304$	2.28130i	0.130842i
$305$	− 0.890268i	− 0.0509766i
$306$	0	0
$307$	7.86481i	0.448868i	0.974489	+	0.224434i	$0.0720534\pi$
$307$	7.86481i	0.448868i	−0.974489	+	0.224434i	$0.927947\pi$
$308$	0	0
$309$	0	0
$310$	3.89828	0.221407
$311$	14.4170	0.817515	0.408757	−	0.912643i	$-0.365962\pi$
$311$	14.4170	0.817515	0.408757	+	0.912643i	$0.365962\pi$
$312$	0	0
$313$	8.40649i	0.475163i	0.971368	+	0.237582i	$0.0763547\pi$
$313$	8.40649i	0.475163i	−0.971368	+	0.237582i	$0.923645\pi$
$314$	6.60127	0.372531
$315$	0	0
$316$	−5.65685	−0.318223
$317$	11.0420i	0.620181i	0.950707	+	0.310090i	$0.100359\pi$
$317$	11.0420i	0.620181i	−0.950707	+	0.310090i	$0.899641\pi$
$318$	0	0
$319$	0.631394	0.0353513
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	− 9.41965i	− 0.524123i
$324$	0	0
$325$	4.33182i	0.240286i
$326$	16.4286i	0.909898i
$327$	0	0
$328$	− 7.08239i	− 0.391060i
$329$	0	0
$330$	0	0
$331$	−21.1667	−1.16343	−0.581713	−	0.813394i	$-0.697617\pi$
$331$	−21.1667	−1.16343	−0.581713	+	0.813394i	$0.697617\pi$
$332$	−0.164784	−0.00904372
$333$	0	0
$334$	0.834556i	0.0456649i
$335$	5.24105	0.286349
$336$	0	0
$337$	20.2992	1.10577	0.552885	−	0.833258i	$-0.313527\pi$
$337$	20.2992	1.10577	0.552885	+	0.833258i	$0.313527\pi$
$338$	5.76468i	0.313557i
$339$	0	0
$340$	−4.12906	−0.223930
$341$	−4.81294	−0.260635
$342$	0	0
$343$	0	0
$344$	10.1837i	0.549071i
$345$	0	0
$346$	3.54675i	0.190674i
$347$	− 28.6912i	− 1.54022i	−0.637910	−	0.770111i	$-0.720201\pi$
$347$	− 28.6912i	− 1.54022i	0.637910	−	0.770111i	$-0.279799\pi$
$348$	0	0
$349$	23.5191i	1.25895i	0.777022	+	0.629473i	$0.216729\pi$
$349$	23.5191i	1.25895i	−0.777022	+	0.629473i	$0.783271\pi$
$350$	0	0
$351$	0	0
$352$	−1.23463	−0.0658062
$353$	8.98390	0.478165	0.239082	−	0.970999i	$-0.423153\pi$
$353$	8.98390	0.478165	0.239082	+	0.970999i	$0.423153\pi$
$354$	0	0
$355$	− 0.818827i	− 0.0434588i
$356$	−4.49886	−0.238439
$357$	0	0
$358$	−16.6685	−0.880957
$359$	10.2872i	0.542937i	0.962447	+	0.271469i	$0.0875093\pi$
$359$	10.2872i	0.542937i	−0.962447	+	0.271469i	$0.912491\pi$
$360$	0	0
$361$	13.7956	0.726087
$362$	−2.53036	−0.132993
$363$	0	0
$364$	0	0
$365$	11.6610i	0.610365i
$366$	0	0
$367$	− 3.39104i	− 0.177011i	−0.996076	−	0.0885053i	$-0.971791\pi$
$367$	− 3.39104i	− 0.177011i	0.996076	−	0.0885053i	$-0.0282090\pi$
$368$	− 2.26038i	− 0.117830i
$369$	0	0
$370$	− 4.39488i	− 0.228479i
$371$	0	0
$372$	0	0
$373$	−27.4460	−1.42110	−0.710549	−	0.703648i	$-0.751553\pi$
$373$	−27.4460	−1.42110	−0.710549	+	0.703648i	$0.751553\pi$
$374$	5.09788	0.263605
$375$	0	0
$376$	7.51594i	0.387605i
$377$	−2.21530	−0.114094
$378$	0	0
$379$	−34.0933	−1.75126	−0.875629	−	0.482985i	$-0.839553\pi$
$379$	−34.0933	−1.75126	−0.875629	+	0.482985i	$0.839553\pi$
$380$	2.28130i	0.117028i
$381$	0	0
$382$	5.15158	0.263578
$383$	5.25290	0.268411	0.134205	−	0.990954i	$-0.457152\pi$
$383$	5.25290	0.268411	0.134205	+	0.990954i	$0.457152\pi$
$384$	0	0
$385$	0	0
$386$	− 4.50793i	− 0.229448i
$387$	0	0
$388$	− 9.06147i	− 0.460026i
$389$	19.8428i	1.00607i	0.864265	+	0.503036i	$0.167784\pi$
$389$	19.8428i	1.00607i	−0.864265	+	0.503036i	$0.832216\pi$
$390$	0	0
$391$	9.33325i	0.472003i
$392$	0	0
$393$	0	0
$394$	3.14476	0.158431
$395$	−5.65685	−0.284627
$396$	0	0
$397$	4.84339i	0.243083i	0.992586	+	0.121541i	$0.0387837\pi$
$397$	4.84339i	0.243083i	−0.992586	+	0.121541i	$0.961216\pi$
$398$	−3.28343	−0.164583
$399$	0	0
$400$	1.00000	0.0500000
$401$	− 6.14161i	− 0.306697i	−0.988172	−	0.153349i	$-0.950994\pi$
$401$	− 6.14161i	− 0.306697i	0.988172	−	0.153349i	$-0.0490057\pi$
$402$	0	0
$403$	16.8866	0.841183
$404$	−3.88348	−0.193210
$405$	0	0
$406$	0	0
$407$	5.42607i	0.268960i
$408$	0	0
$409$	27.2071i	1.34530i	0.739959	+	0.672652i	$0.234845\pi$
$409$	27.2071i	1.34530i	−0.739959	+	0.672652i	$0.765155\pi$
$410$	− 7.08239i	− 0.349774i
$411$	0	0
$412$	7.56767i	0.372833i
$413$	0	0
$414$	0	0
$415$	−0.164784	−0.00808895
$416$	4.33182	0.212385
$417$	0	0
$418$	− 2.81657i	− 0.137763i
$419$	37.6098	1.83736	0.918678	−	0.395007i	$-0.129258\pi$
$419$	37.6098	1.83736	0.918678	+	0.395007i	$0.129258\pi$
$420$	0	0
$421$	18.2049	0.887253	0.443627	−	0.896212i	$-0.353692\pi$
$421$	18.2049	0.887253	0.443627	+	0.896212i	$0.353692\pi$
$422$	− 25.0907i	− 1.22140i
$423$	0	0
$424$	8.83259	0.428948
$425$	−4.12906	−0.200289
$426$	0	0
$427$	0	0
$428$	13.7679i	0.665498i
$429$	0	0
$430$	10.1837i	0.491104i
$431$	− 14.4266i	− 0.694906i	−0.937697	−	0.347453i	$-0.887047\pi$
$431$	− 14.4266i	− 0.694906i	0.937697	−	0.347453i	$-0.112953\pi$
$432$	0	0
$433$	− 32.5560i	− 1.56454i	−0.622939	−	0.782270i	$-0.714062\pi$
$433$	− 32.5560i	− 1.56454i	0.622939	−	0.782270i	$-0.285938\pi$
$434$	0	0
$435$	0	0
$436$	−0.810688	−0.0388249
$437$	5.15661	0.246674
$438$	0	0
$439$	− 25.4300i	− 1.21371i	−0.794814	−	0.606854i	$-0.792431\pi$
$439$	− 25.4300i	− 1.21371i	0.794814	−	0.606854i	$-0.207569\pi$
$440$	−1.23463	−0.0588588
$441$	0	0
$442$	−17.8864	−0.850768
$443$	22.0115i	1.04580i	0.852395	+	0.522899i	$0.175150\pi$
$443$	22.0115i	1.04580i	−0.852395	+	0.522899i	$0.824850\pi$
$444$	0	0
$445$	−4.49886	−0.213266
$446$	−3.21440	−0.152206
$447$	0	0
$448$	0	0
$449$	33.5223i	1.58201i	0.611808	+	0.791006i	$0.290443\pi$
$449$	33.5223i	1.58201i	−0.611808	+	0.791006i	$0.709557\pi$
$450$	0	0
$451$	8.74416i	0.411746i
$452$	11.0866i	0.521468i
$453$	0	0
$454$	7.15800i	0.335941i
$455$	0	0
$456$	0	0
$457$	−20.2136	−0.945551	−0.472776	−	0.881183i	$-0.656748\pi$
$457$	−20.2136	−0.945551	−0.472776	+	0.881183i	$0.656748\pi$
$458$	−17.0156	−0.795089
$459$	0	0
$460$	− 2.26038i	− 0.105391i
$461$	10.1557	0.472999	0.236499	−	0.971632i	$-0.424000\pi$
$461$	10.1557	0.472999	0.236499	+	0.971632i	$0.424000\pi$
$462$	0	0
$463$	18.0606	0.839346	0.419673	−	0.907675i	$-0.362145\pi$
$463$	18.0606	0.839346	0.419673	+	0.907675i	$0.362145\pi$
$464$	0.511402i	0.0237412i
$465$	0	0
$466$	−8.18080	−0.378968
$467$	−7.84805	−0.363164	−0.181582	−	0.983376i	$-0.558122\pi$
$467$	−7.84805	−0.363164	−0.181582	+	0.983376i	$0.558122\pi$
$468$	0	0
$469$	0	0
$470$	7.51594i	0.346684i
$471$	0	0
$472$	8.33598i	0.383695i
$473$	− 12.5732i	− 0.578116i
$474$	0	0
$475$	2.28130i	0.104673i
$476$	0	0
$477$	0	0
$478$	−22.3224	−1.02100
$479$	16.0502	0.733351	0.366676	−	0.930349i	$-0.380496\pi$
$479$	16.0502	0.733351	0.366676	+	0.930349i	$0.380496\pi$
$480$	0	0
$481$	− 19.0378i	− 0.868051i
$482$	−12.5525	−0.571751
$483$	0	0
$484$	−9.47568	−0.430713
$485$	− 9.06147i	− 0.411460i
$486$	0	0
$487$	−15.9509	−0.722806	−0.361403	−	0.932410i	$-0.617702\pi$
$487$	−15.9509	−0.722806	−0.361403	+	0.932410i	$0.617702\pi$
$488$	−0.890268	−0.0403006
$489$	0	0
$490$	0	0
$491$	6.45991i	0.291532i	0.989319	+	0.145766i	$0.0465646\pi$
$491$	6.45991i	0.291532i	−0.989319	+	0.145766i	$0.953435\pi$
$492$	0	0
$493$	− 2.11161i	− 0.0951022i
$494$	9.88220i	0.444621i
$495$	0	0
$496$	− 3.89828i	− 0.175038i
$497$	0	0
$498$	0	0
$499$	21.9934	0.984558	0.492279	−	0.870437i	$-0.336164\pi$
$499$	21.9934	0.984558	0.492279	+	0.870437i	$0.336164\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	6.05468i	0.270234i
$503$	−43.2100	−1.92664	−0.963318	−	0.268361i	$-0.913518\pi$
$503$	−43.2100	−1.92664	−0.963318	+	0.268361i	$0.913518\pi$
$504$	0	0
$505$	−3.88348	−0.172813
$506$	2.79074i	0.124064i
$507$	0	0
$508$	−8.12612	−0.360538
$509$	18.5792	0.823506	0.411753	−	0.911295i	$-0.364917\pi$
$509$	18.5792	0.823506	0.411753	+	0.911295i	$0.364917\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	15.9298i	0.702632i
$515$	7.56767i	0.333472i
$516$	0	0
$517$	− 9.27943i	− 0.408109i
$518$	0	0
$519$	0	0
$520$	4.33182	0.189963
$521$	5.77390	0.252959	0.126480	−	0.991969i	$-0.459632\pi$
$521$	5.77390	0.252959	0.126480	+	0.991969i	$0.459632\pi$
$522$	0	0
$523$	9.73569i	0.425712i	0.977084	+	0.212856i	$0.0682765\pi$
$523$	9.73569i	0.425712i	−0.977084	+	0.212856i	$0.931723\pi$
$524$	−0.391410	−0.0170988
$525$	0	0
$526$	8.09560	0.352985
$527$	16.0962i	0.701163i
$528$	0	0
$529$	17.8907	0.777856
$530$	8.83259	0.383663
$531$	0	0
$532$	0	0
$533$	− 30.6797i	− 1.32888i
$534$	0	0
$535$	13.7679i	0.595240i
$536$	− 5.24105i	− 0.226379i
$537$	0	0
$538$	− 31.7070i	− 1.36699i
$539$	0	0
$540$	0	0
$541$	8.37420	0.360035	0.180017	−	0.983663i	$-0.442385\pi$
$541$	8.37420	0.360035	0.180017	+	0.983663i	$0.442385\pi$
$542$	−30.2571	−1.29965
$543$	0	0
$544$	4.12906i	0.177032i
$545$	−0.810688	−0.0347261
$546$	0	0
$547$	26.3099	1.12493	0.562464	−	0.826822i	$-0.309853\pi$
$547$	26.3099	1.12493	0.562464	+	0.826822i	$0.309853\pi$
$548$	21.5832i	0.921987i
$549$	0	0
$550$	−1.23463	−0.0526449
$551$	−1.16666	−0.0497015
$552$	0	0
$553$	0	0
$554$	− 3.07626i	− 0.130698i
$555$	0	0
$556$	− 22.9700i	− 0.974146i
$557$	− 18.8671i	− 0.799424i	−0.916641	−	0.399712i	$-0.869110\pi$
$557$	− 18.8671i	− 0.799424i	0.916641	−	0.399712i	$-0.130890\pi$
$558$	0	0
$559$	44.1142i	1.86583i
$560$	0	0
$561$	0	0
$562$	−23.4167	−0.987772
$563$	−25.2513	−1.06422	−0.532109	−	0.846676i	$-0.678600\pi$
$563$	−25.2513	−1.06422	−0.532109	+	0.846676i	$0.678600\pi$
$564$	0	0
$565$	11.0866i	0.466415i
$566$	−19.4152	−0.816081
$567$	0	0
$568$	−0.818827	−0.0343572
$569$	13.4780i	0.565026i	0.959264	+	0.282513i	$0.0911680\pi$
$569$	13.4780i	0.565026i	−0.959264	+	0.282513i	$0.908832\pi$
$570$	0	0
$571$	32.7981	1.37256	0.686279	−	0.727339i	$-0.259243\pi$
$571$	32.7981	1.37256	0.686279	+	0.727339i	$0.259243\pi$
$572$	−5.34821	−0.223620
$573$	0	0
$574$	0	0
$575$	− 2.26038i	− 0.0942644i
$576$	0	0
$577$	44.5364i	1.85408i	0.374967	+	0.927038i	$0.377654\pi$
$577$	44.5364i	1.85408i	−0.374967	+	0.927038i	$0.622346\pi$
$578$	− 0.0491665i	− 0.00204506i
$579$	0	0
$580$	0.511402i	0.0212348i
$581$	0	0
$582$	0	0
$583$	−10.9050	−0.451639
$584$	11.6610	0.482536
$585$	0	0
$586$	26.1609i	1.08069i
$587$	38.5483	1.59106	0.795529	−	0.605916i	$-0.207193\pi$
$587$	38.5483	1.59106	0.795529	+	0.605916i	$0.207193\pi$
$588$	0	0
$589$	8.89315	0.366436
$590$	8.33598i	0.343187i
$591$	0	0
$592$	−4.39488	−0.180629
$593$	−33.1728	−1.36224	−0.681122	−	0.732170i	$-0.738508\pi$
$593$	−33.1728	−1.36224	−0.681122	+	0.732170i	$0.738508\pi$
$594$	0	0
$595$	0	0
$596$	22.8033i	0.934059i
$597$	0	0
$598$	− 9.79156i	− 0.400407i
$599$	27.0509i	1.10527i	0.833424	+	0.552634i	$0.186377\pi$
$599$	27.0509i	1.10527i	−0.833424	+	0.552634i	$0.813623\pi$
$600$	0	0
$601$	− 42.7601i	− 1.74422i	−0.489309	−	0.872111i	$-0.662751\pi$
$601$	− 42.7601i	− 1.74422i	0.489309	−	0.872111i	$-0.337249\pi$
$602$	0	0
$603$	0	0
$604$	21.2905	0.866299
$605$	−9.47568	−0.385241
$606$	0	0
$607$	− 28.6190i	− 1.16161i	−0.814043	−	0.580805i	$-0.802738\pi$
$607$	− 28.6190i	− 1.16161i	0.814043	−	0.580805i	$-0.197262\pi$
$608$	2.28130	0.0925191
$609$	0	0
$610$	−0.890268	−0.0360459
$611$	32.5577i	1.31714i
$612$	0	0
$613$	40.9971	1.65586	0.827929	−	0.560833i	$-0.189519\pi$
$613$	40.9971	1.65586	0.827929	+	0.560833i	$0.189519\pi$
$614$	7.86481	0.317398
$615$	0	0
$616$	0	0
$617$	8.66277i	0.348750i	0.984679	+	0.174375i	$0.0557905\pi$
$617$	8.66277i	0.348750i	−0.984679	+	0.174375i	$0.944209\pi$
$618$	0	0
$619$	− 11.3620i	− 0.456676i	−0.973582	−	0.228338i	$-0.926671\pi$
$619$	− 11.3620i	− 0.456676i	0.973582	−	0.228338i	$-0.0733292\pi$
$620$	− 3.89828i	− 0.156559i
$621$	0	0
$622$	− 14.4170i	− 0.578070i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.40649	0.335991
$627$	0	0
$628$	− 6.60127i	− 0.263419i
$629$	18.1467	0.723558
$630$	0	0
$631$	−28.2837	−1.12596	−0.562979	−	0.826471i	$-0.690345\pi$
$631$	−28.2837	−1.12596	−0.562979	+	0.826471i	$0.690345\pi$
$632$	5.65685i	0.225018i
$633$	0	0
$634$	11.0420	0.438534
$635$	−8.12612	−0.322475
$636$	0	0
$637$	0	0
$638$	− 0.631394i	− 0.0249971i
$639$	0	0
$640$	− 1.00000i	− 0.0395285i
$641$	17.0146i	0.672036i	0.941856	+	0.336018i	$0.109080\pi$
$641$	17.0146i	0.672036i	−0.941856	+	0.336018i	$0.890920\pi$
$642$	0	0
$643$	28.9046i	1.13988i	0.821685	+	0.569942i	$0.193034\pi$
$643$	28.9046i	1.13988i	−0.821685	+	0.569942i	$0.806966\pi$
$644$	0	0
$645$	0	0
$646$	−9.41965	−0.370611
$647$	−35.6085	−1.39992	−0.699958	−	0.714184i	$-0.746798\pi$
$647$	−35.6085	−1.39992	−0.699958	+	0.714184i	$0.746798\pi$
$648$	0	0
$649$	− 10.2919i	− 0.403992i
$650$	4.33182	0.169908
$651$	0	0
$652$	16.4286	0.643395
$653$	− 1.79205i	− 0.0701282i	−0.999385	−	0.0350641i	$-0.988836\pi$
$653$	− 1.79205i	− 0.0701282i	0.999385	−	0.0350641i	$-0.0111635\pi$
$654$	0	0
$655$	−0.391410	−0.0152937
$656$	−7.08239	−0.276521
$657$	0	0
$658$	0	0
$659$	9.64709i	0.375797i	0.982188	+	0.187899i	$0.0601677\pi$
$659$	9.64709i	0.375797i	−0.982188	+	0.187899i	$0.939832\pi$
$660$	0	0
$661$	16.8868i	0.656821i	0.944535	+	0.328411i	$0.106513\pi$
$661$	16.8868i	0.656821i	−0.944535	+	0.328411i	$0.893487\pi$
$662$	21.1667i	0.822666i
$663$	0	0
$664$	0.164784i	0.00639487i
$665$	0	0
$666$	0	0
$667$	1.15596	0.0447590
$668$	0.834556	0.0322899
$669$	0	0
$670$	− 5.24105i	− 0.202479i
$671$	1.09915	0.0424324
$672$	0	0
$673$	12.4010	0.478024	0.239012	−	0.971017i	$-0.423177\pi$
$673$	12.4010	0.478024	0.239012	+	0.971017i	$0.423177\pi$
$674$	− 20.2992i	− 0.781897i
$675$	0	0
$676$	5.76468	0.221718
$677$	1.57184	0.0604106	0.0302053	−	0.999544i	$-0.490384\pi$
$677$	1.57184	0.0604106	0.0302053	+	0.999544i	$0.490384\pi$
$678$	0	0
$679$	0	0
$680$	4.12906i	0.158342i
$681$	0	0
$682$	4.81294i	0.184297i
$683$	35.8613i	1.37219i	0.727510	+	0.686097i	$0.240677\pi$
$683$	35.8613i	1.37219i	−0.727510	+	0.686097i	$0.759323\pi$
$684$	0	0
$685$	21.5832i	0.824650i
$686$	0	0
$687$	0	0
$688$	10.1837	0.388252
$689$	38.2612	1.45764
$690$	0	0
$691$	− 8.40105i	− 0.319591i	−0.987150	−	0.159796i	$-0.948916\pi$
$691$	− 8.40105i	− 0.319591i	0.987150	−	0.159796i	$-0.0510835\pi$
$692$	3.54675	0.134827
$693$	0	0
$694$	−28.6912	−1.08910
$695$	− 22.9700i	− 0.871303i
$696$	0	0
$697$	29.2436	1.10768
$698$	23.5191	0.890210
$699$	0	0
$700$	0	0
$701$	− 5.58154i	− 0.210812i	−0.994429	−	0.105406i	$-0.966386\pi$
$701$	− 5.58154i	− 0.210812i	0.994429	−	0.105406i	$-0.0336142\pi$
$702$	0	0
$703$	− 10.0261i	− 0.378140i
$704$	1.23463i	0.0465320i
$705$	0	0
$706$	− 8.98390i	− 0.338113i
$707$	0	0
$708$	0	0
$709$	47.5415	1.78546	0.892729	−	0.450594i	$-0.148788\pi$
$709$	47.5415	1.78546	0.892729	+	0.450594i	$0.148788\pi$
$710$	−0.818827	−0.0307300
$711$	0	0
$712$	4.49886i	0.168602i
$713$	−8.81158	−0.329996
$714$	0	0
$715$	−5.34821	−0.200012
$716$	16.6685i	0.622931i
$717$	0	0
$718$	10.2872	0.383915
$719$	−39.6210	−1.47762	−0.738808	−	0.673916i	$-0.764611\pi$
$719$	−39.6210	−1.47762	−0.738808	+	0.673916i	$0.764611\pi$
$720$	0	0
$721$	0	0
$722$	− 13.7956i	− 0.513421i
$723$	0	0
$724$	2.53036i	0.0940401i
$725$	0.511402i	0.0189930i
$726$	0	0
$727$	− 20.9137i	− 0.775645i	−0.921734	−	0.387823i	$-0.873227\pi$
$727$	− 20.9137i	− 0.775645i	0.921734	−	0.387823i	$-0.126773\pi$
$728$	0	0
$729$	0	0
$730$	11.6610	0.431594
$731$	−42.0493	−1.55525
$732$	0	0
$733$	− 11.2326i	− 0.414886i	−0.978247	−	0.207443i	$-0.933486\pi$
$733$	− 11.2326i	− 0.414886i	0.978247	−	0.207443i	$-0.0665142\pi$
$734$	−3.39104	−0.125165
$735$	0	0
$736$	−2.26038	−0.0833187
$737$	6.47077i	0.238354i
$738$	0	0
$739$	−51.2983	−1.88704	−0.943520	−	0.331316i	$-0.892507\pi$
$739$	−51.2983	−1.88704	−0.943520	+	0.331316i	$0.892507\pi$
$740$	−4.39488	−0.161559
$741$	0	0
$742$	0	0
$743$	− 20.5208i	− 0.752834i	−0.926450	−	0.376417i	$-0.877156\pi$
$743$	− 20.5208i	− 0.752834i	0.926450	−	0.376417i	$-0.122844\pi$
$744$	0	0
$745$	22.8033i	0.835448i
$746$	27.4460i	1.00487i
$747$	0	0
$748$	− 5.09788i	− 0.186397i
$749$	0	0
$750$	0	0
$751$	−25.0794	−0.915160	−0.457580	−	0.889168i	$-0.651284\pi$
$751$	−25.0794	−0.915160	−0.457580	+	0.889168i	$0.651284\pi$
$752$	7.51594	0.274078
$753$	0	0
$754$	2.21530i	0.0806765i
$755$	21.2905	0.774842
$756$	0	0
$757$	−27.2316	−0.989751	−0.494875	−	0.868964i	$-0.664786\pi$
$757$	−27.2316	−0.989751	−0.494875	+	0.868964i	$0.664786\pi$
$758$	34.0933i	1.23833i
$759$	0	0
$760$	2.28130	0.0827516
$761$	45.3792	1.64499	0.822496	−	0.568770i	$-0.192581\pi$
$761$	45.3792	1.64499	0.822496	+	0.568770i	$0.192581\pi$
$762$	0	0
$763$	0	0
$764$	− 5.15158i	− 0.186378i
$765$	0	0
$766$	− 5.25290i	− 0.189795i
$767$	36.1100i	1.30386i
$768$	0	0
$769$	34.0610i	1.22827i	0.789201	+	0.614135i	$0.210495\pi$
$769$	34.0610i	1.22827i	−0.789201	+	0.614135i	$0.789505\pi$
$770$	0	0
$771$	0	0
$772$	−4.50793	−0.162244
$773$	19.5924	0.704688	0.352344	−	0.935871i	$-0.385385\pi$
$773$	19.5924	0.704688	0.352344	+	0.935871i	$0.385385\pi$
$774$	0	0
$775$	− 3.89828i	− 0.140030i
$776$	−9.06147	−0.325288
$777$	0	0
$778$	19.8428	0.711401
$779$	− 16.1571i	− 0.578888i
$780$	0	0
$781$	1.01095	0.0361747
$782$	9.33325	0.333756
$783$	0	0
$784$	0	0
$785$	− 6.60127i	− 0.235610i
$786$	0	0
$787$	− 17.5398i	− 0.625227i	−0.949880	−	0.312613i	$-0.898796\pi$
$787$	− 17.5398i	− 0.625227i	0.949880	−	0.312613i	$-0.101204\pi$
$788$	− 3.14476i	− 0.112028i
$789$	0	0
$790$	5.65685i	0.201262i
$791$	0	0
$792$	0	0
$793$	−3.85648	−0.136948
$794$	4.84339	0.171885
$795$	0	0
$796$	3.28343i	0.116378i
$797$	−19.7785	−0.700589	−0.350295	−	0.936640i	$-0.613919\pi$
$797$	−19.7785	−0.700589	−0.350295	+	0.936640i	$0.613919\pi$
$798$	0	0
$799$	−31.0338	−1.09790
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	−6.14161	−0.216868
$803$	−14.3971	−0.508062
$804$	0	0
$805$	0	0
$806$	− 16.8866i	− 0.594806i
$807$	0	0
$808$	3.88348i	0.136620i
$809$	20.1357i	0.707934i	0.935258	+	0.353967i	$0.115168\pi$
$809$	20.1357i	0.707934i	−0.935258	+	0.353967i	$0.884832\pi$
$810$	0	0
$811$	− 40.7335i	− 1.43035i	−0.698946	−	0.715174i	$-0.746347\pi$
$811$	− 40.7335i	− 1.43035i	0.698946	−	0.715174i	$-0.253653\pi$
$812$	0	0
$813$	0	0
$814$	5.42607	0.190184
$815$	16.4286	0.575470
$816$	0	0
$817$	23.2322i	0.812792i
$818$	27.2071	0.951273
$819$	0	0
$820$	−7.08239	−0.247328
$821$	1.57950i	0.0551250i	0.999620	+	0.0275625i	$0.00877453\pi$
$821$	1.57950i	0.0551250i	−0.999620	+	0.0275625i	$0.991225\pi$
$822$	0	0
$823$	27.4028	0.955202	0.477601	−	0.878577i	$-0.341506\pi$
$823$	27.4028	0.955202	0.477601	+	0.878577i	$0.341506\pi$
$824$	7.56767	0.263632
$825$	0	0
$826$	0	0
$827$	19.7182i	0.685668i	0.939396	+	0.342834i	$0.111387\pi$
$827$	19.7182i	0.685668i	−0.939396	+	0.342834i	$0.888613\pi$
$828$	0	0
$829$	28.9250i	1.00461i	0.864691	+	0.502303i	$0.167514\pi$
$829$	28.9250i	1.00461i	−0.864691	+	0.502303i	$0.832486\pi$
$830$	0.164784i	0.00571975i
$831$	0	0
$832$	− 4.33182i	− 0.150179i
$833$	0	0
$834$	0	0
$835$	0.834556	0.0288810
$836$	−2.81657	−0.0974133
$837$	0	0
$838$	− 37.6098i	− 1.29921i
$839$	50.5905	1.74658	0.873289	−	0.487203i	$-0.161983\pi$
$839$	50.5905	1.74658	0.873289	+	0.487203i	$0.161983\pi$
$840$	0	0
$841$	28.7385	0.990982
$842$	− 18.2049i	− 0.627383i
$843$	0	0
$844$	−25.0907	−0.863658
$845$	5.76468	0.198311
$846$	0	0
$847$	0	0
$848$	− 8.83259i	− 0.303312i
$849$	0	0
$850$	4.12906i	0.141626i
$851$	9.93410i	0.340537i
$852$	0	0
$853$	− 49.1746i	− 1.68371i	−0.539706	−	0.841854i	$-0.681464\pi$
$853$	− 49.1746i	− 1.68371i	0.539706	−	0.841854i	$-0.318536\pi$
$854$	0	0
$855$	0	0
$856$	13.7679	0.470578
$857$	−44.8289	−1.53133	−0.765663	−	0.643242i	$-0.777589\pi$
$857$	−44.8289	−1.53133	−0.765663	+	0.643242i	$0.777589\pi$
$858$	0	0
$859$	14.5032i	0.494843i	0.968908	+	0.247422i	$0.0795832\pi$
$859$	14.5032i	0.494843i	−0.968908	+	0.247422i	$0.920417\pi$
$860$	10.1837	0.347263
$861$	0	0
$862$	−14.4266	−0.491373
$863$	43.4906i	1.48044i	0.672367	+	0.740218i	$0.265278\pi$
$863$	43.4906i	1.48044i	−0.672367	+	0.740218i	$0.734722\pi$
$864$	0	0
$865$	3.54675	0.120593
$866$	−32.5560	−1.10630
$867$	0	0
$868$	0	0
$869$	− 6.98414i	− 0.236921i
$870$	0	0
$871$	− 22.7033i	− 0.769271i
$872$	0.810688i	0.0274534i
$873$	0	0
$874$	− 5.15661i	− 0.174425i
$875$	0	0
$876$	0	0
$877$	55.8913	1.88732	0.943658	−	0.330922i	$-0.107360\pi$
$877$	55.8913	1.88732	0.943658	+	0.330922i	$0.107360\pi$
$878$	−25.4300	−0.858220
$879$	0	0
$880$	1.23463i	0.0416195i
$881$	24.0647	0.810761	0.405381	−	0.914148i	$-0.367139\pi$
$881$	24.0647	0.810761	0.405381	+	0.914148i	$0.367139\pi$
$882$	0	0
$883$	35.8242	1.20558	0.602790	−	0.797900i	$-0.294056\pi$
$883$	35.8242	1.20558	0.602790	+	0.797900i	$0.294056\pi$
$884$	17.8864i	0.601584i
$885$	0	0
$886$	22.0115	0.739491
$887$	−47.7182	−1.60222	−0.801111	−	0.598516i	$-0.795757\pi$
$887$	−47.7182	−1.60222	−0.801111	+	0.598516i	$0.795757\pi$
$888$	0	0
$889$	0	0
$890$	4.49886i	0.150802i
$891$	0	0
$892$	3.21440i	0.107626i
$893$	17.1461i	0.573774i
$894$	0	0
$895$	16.6685i	0.557166i
$896$	0	0
$897$	0	0
$898$	33.5223	1.11865
$899$	1.99359	0.0664898
$900$	0	0
$901$	36.4703i	1.21500i
$902$	8.74416	0.291149
$903$	0	0
$904$	11.0866	0.368733
$905$	2.53036i	0.0841120i
$906$	0	0
$907$	−27.0550	−0.898346	−0.449173	−	0.893445i	$-0.648281\pi$
$907$	−27.0550	−0.898346	−0.449173	+	0.893445i	$0.648281\pi$
$908$	7.15800	0.237546
$909$	0	0
$910$	0	0
$911$	− 10.6210i	− 0.351890i	−0.984400	−	0.175945i	$-0.943702\pi$
$911$	− 10.6210i	− 0.351890i	0.984400	−	0.175945i	$-0.0562980\pi$
$912$	0	0
$913$	− 0.203448i	− 0.00673316i
$914$	20.2136i	0.668606i
$915$	0	0
$916$	17.0156i	0.562213i
$917$	0	0
$918$	0	0
$919$	19.7502	0.651497	0.325749	−	0.945456i	$-0.394384\pi$
$919$	19.7502	0.651497	0.325749	+	0.945456i	$0.394384\pi$
$920$	−2.26038	−0.0745225
$921$	0	0
$922$	− 10.1557i	− 0.334461i
$923$	−3.54701	−0.116751
$924$	0	0
$925$	−4.39488	−0.144503
$926$	− 18.0606i	− 0.593507i
$927$	0	0
$928$	0.511402	0.0167876
$929$	−6.94442	−0.227839	−0.113919	−	0.993490i	$-0.536341\pi$
$929$	−6.94442	−0.227839	−0.113919	+	0.993490i	$0.536341\pi$
$930$	0	0
$931$	0	0
$932$	8.18080i	0.267971i
$933$	0	0
$934$	7.84805i	0.256796i
$935$	− 5.09788i	− 0.166718i
$936$	0	0
$937$	− 37.7945i	− 1.23469i	−0.786692	−	0.617346i	$-0.788208\pi$
$937$	− 37.7945i	− 1.23469i	0.786692	−	0.617346i	$-0.211792\pi$
$938$	0	0
$939$	0	0
$940$	7.51594	0.245143
$941$	−57.4570	−1.87304	−0.936522	−	0.350608i	$-0.885975\pi$
$941$	−57.4570	−1.87304	−0.936522	+	0.350608i	$0.885975\pi$
$942$	0	0
$943$	16.0089i	0.521321i
$944$	8.33598	0.271313
$945$	0	0
$946$	−12.5732	−0.408790
$947$	− 7.80522i	− 0.253636i	−0.991926	−	0.126818i	$-0.959524\pi$
$947$	− 7.80522i	− 0.253636i	0.991926	−	0.126818i	$-0.0404763\pi$
$948$	0	0
$949$	50.5134	1.63974
$950$	2.28130	0.0740153
$951$	0	0
$952$	0	0
$953$	− 33.4531i	− 1.08365i	−0.840491	−	0.541826i	$-0.817733\pi$
$953$	− 33.4531i	− 1.08365i	0.840491	−	0.541826i	$-0.182267\pi$
$954$	0	0
$955$	− 5.15158i	− 0.166701i
$956$	22.3224i	0.721958i
$957$	0	0
$958$	− 16.0502i	− 0.518558i
$959$	0	0
$960$	0	0
$961$	15.8034	0.509789
$962$	−19.0378	−0.613805
$963$	0	0
$964$	12.5525i	0.404289i
$965$	−4.50793	−0.145115
$966$	0	0
$967$	39.1290	1.25830	0.629152	−	0.777283i	$-0.283402\pi$
$967$	39.1290	1.25830	0.629152	+	0.777283i	$0.283402\pi$
$968$	9.47568i	0.304560i
$969$	0	0
$970$	−9.06147	−0.290946
$971$	35.7937	1.14868	0.574338	−	0.818618i	$-0.305260\pi$
$971$	35.7937	1.14868	0.574338	+	0.818618i	$0.305260\pi$
$972$	0	0
$973$	0	0
$974$	15.9509i	0.511101i
$975$	0	0
$976$	0.890268i	0.0284968i
$977$	− 6.01733i	− 0.192512i	−0.995357	−	0.0962558i	$-0.969313\pi$
$977$	− 6.01733i	− 0.192512i	0.995357	−	0.0962558i	$-0.0306867\pi$
$978$	0	0
$979$	− 5.55444i	− 0.177521i
$980$	0	0
$981$	0	0
$982$	6.45991	0.206144
$983$	−58.9136	−1.87905	−0.939526	−	0.342479i	$-0.888734\pi$
$983$	−58.9136	−1.87905	−0.939526	+	0.342479i	$0.888734\pi$
$984$	0	0
$985$	− 3.14476i	− 0.100200i
$986$	−2.11161	−0.0672474
$987$	0	0
$988$	9.88220	0.314395
$989$	− 23.0191i	− 0.731966i
$990$	0	0
$991$	−32.4659	−1.03131	−0.515657	−	0.856795i	$-0.672452\pi$
$991$	−32.4659	−1.03131	−0.515657	+	0.856795i	$0.672452\pi$
$992$	−3.89828	−0.123770
$993$	0	0
$994$	0	0
$995$	3.28343i	0.104092i
$996$	0	0
$997$	0.389500i	0.0123356i	0.999981	+	0.00616780i	$0.00196328\pi$
$997$	0.389500i	0.0123356i	−0.999981	+	0.00616780i	$0.998037\pi$
$998$	− 21.9934i	− 0.696188i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.a.881.3	✓	8
3.2	odd	2		4410.2.b.d.881.6	yes	8
7.6	odd	2		4410.2.b.d.881.3	yes	8
21.20	even	2	inner	4410.2.b.a.881.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.a.881.3	✓	8	1.1	even	1	trivial
4410.2.b.a.881.6	yes	8	21.20	even	2	inner
4410.2.b.d.881.3	yes	8	7.6	odd	2
4410.2.b.d.881.6	yes	8	3.2	odd	2

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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000i q^{8} +1.00000i q^{10} -1.23463i q^{11} +4.33182i q^{13} +1.00000 q^{16} -4.12906 q^{17} +2.28130i q^{19} +1.00000 q^{20} -1.23463 q^{22} -2.26038i q^{23} +1.00000 q^{25} +4.33182 q^{26} +0.511402i q^{29} -3.89828i q^{31} -1.00000i q^{32} +4.12906i q^{34} -4.39488 q^{37} +2.28130 q^{38} -1.00000i q^{40} -7.08239 q^{41} +10.1837 q^{43} +1.23463i q^{44} -2.26038 q^{46} +7.51594 q^{47} -1.00000i q^{50} -4.33182i q^{52} -8.83259i q^{53} +1.23463i q^{55} +0.511402 q^{58} +8.33598 q^{59} +0.890268i q^{61} -3.89828 q^{62} -1.00000 q^{64} -4.33182i q^{65} -5.24105 q^{67} +4.12906 q^{68} +0.818827i q^{71} -11.6610i q^{73} +4.39488i q^{74} -2.28130i q^{76} +5.65685 q^{79} -1.00000 q^{80} +7.08239i q^{82} +0.164784 q^{83} +4.12906 q^{85} -10.1837i q^{86} +1.23463 q^{88} +4.49886 q^{89} +2.26038i q^{92} -7.51594i q^{94} -2.28130i q^{95} +9.06147i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 - 8 * q^5	\( 8 q - 8 q^{4} - 8 q^{5} + 8 q^{16} + 8 q^{20} - 16 q^{22} + 8 q^{25} + 32 q^{26} - 16 q^{37} - 48 q^{41} - 16 q^{43} + 16 q^{46} + 48 q^{47} - 16 q^{58} - 16 q^{59} - 32 q^{62} - 8 q^{64} - 8 q^{80} - 16 q^{83} + 16 q^{88} + 48 q^{89}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^5 + 8 * q^16 + 8 * q^20 - 16 * q^22 + 8 * q^25 + 32 * q^26 - 16 * q^37 - 48 * q^41 - 16 * q^43 + 16 * q^46 + 48 * q^47 - 16 * q^58 - 16 * q^59 - 32 * q^62 - 8 * q^64 - 8 * q^80 - 16 * q^83 + 16 * q^88 + 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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