LMFDB - Embedded newform 4410.2.b.e.881.8

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:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.8
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.e.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000i q^{8} +1.00000i q^{10} +5.39595i q^{11} -2.51764i q^{13} +1.00000 q^{16} -4.49938 q^{17} -2.86370i q^{19} -1.00000 q^{20} -5.39595 q^{22} +0.267949i q^{23} +1.00000 q^{25} +2.51764 q^{26} -8.89898i q^{29} -4.82843i q^{31} +1.00000i q^{32} -4.49938i q^{34} -6.51764 q^{37} +2.86370 q^{38} -1.00000i q^{40} +0.760279 q^{41} -5.86370 q^{43} -5.39595i q^{44} -0.267949 q^{46} -7.99536 q^{47} +1.00000i q^{50} +2.51764i q^{52} +8.39836i q^{53} +5.39595i q^{55} +8.89898 q^{58} -12.6715 q^{59} -2.62158i q^{61} +4.82843 q^{62} -1.00000 q^{64} -2.51764i q^{65} +9.82237 q^{67} +4.49938 q^{68} -4.76268i q^{71} -11.6569i q^{73} -6.51764i q^{74} +2.86370i q^{76} +8.59235 q^{79} +1.00000 q^{80} +0.760279i q^{82} +9.45001 q^{83} -4.49938 q^{85} -5.86370i q^{86} +5.39595 q^{88} +7.97005 q^{89} -0.267949i q^{92} -7.99536i q^{94} -2.86370i q^{95} -6.16353i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{25} + 16 q^{26} - 48 q^{37} - 8 q^{38} - 32 q^{41} - 16 q^{43} - 16 q^{46} - 16 q^{47} + 32 q^{58} - 48 q^{59} + 16 q^{62} - 8 q^{64} + 48 q^{67} + 48 q^{79} + 8 q^{80} + 16 q^{83} - 32 q^{89}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^5 + 8 * q^16 - 8 * q^20 + 8 * q^25 + 16 * q^26 - 48 * q^37 - 8 * q^38 - 32 * q^41 - 16 * q^43 - 16 * q^46 - 16 * q^47 + 32 * q^58 - 48 * q^59 + 16 * q^62 - 8 * q^64 + 48 * q^67 + 48 * q^79 + 8 * q^80 + 16 * q^83 - 32 * 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
$11$	5.39595i	1.62694i	0.581606	+	0.813471i	$0.302425\pi$
$11$	5.39595i	1.62694i	−0.581606	+	0.813471i	$0.697575\pi$
$12$	0	0
$13$	− 2.51764i	− 0.698267i	−0.937073	−	0.349134i	$-0.886476\pi$
$13$	− 2.51764i	− 0.698267i	0.937073	−	0.349134i	$-0.113524\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.49938	−1.09126	−0.545630	−	0.838026i	$-0.683709\pi$
$17$	−4.49938	−1.09126	−0.545630	+	0.838026i	$0.683709\pi$
$18$	0	0
$19$	− 2.86370i	− 0.656979i	−0.944508	−	0.328489i	$-0.893461\pi$
$19$	− 2.86370i	− 0.656979i	0.944508	−	0.328489i	$-0.106539\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−5.39595	−1.15042
$23$	0.267949i	0.0558713i	0.999610	+	0.0279356i	$0.00889335\pi$
$23$	0.267949i	0.0558713i	−0.999610	+	0.0279356i	$0.991107\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.51764	0.493749
$27$	0	0
$28$	0	0
$29$	− 8.89898i	− 1.65250i	−0.563304	−	0.826250i	$-0.690470\pi$
$29$	− 8.89898i	− 1.65250i	0.563304	−	0.826250i	$-0.309530\pi$
$30$	0	0
$31$	− 4.82843i	− 0.867211i	−0.901103	−	0.433606i	$-0.857241\pi$
$31$	− 4.82843i	− 0.867211i	0.901103	−	0.433606i	$-0.142759\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 4.49938i	− 0.771637i
$35$	0	0
$36$	0	0
$37$	−6.51764	−1.07149	−0.535747	−	0.844379i	$-0.679970\pi$
$37$	−6.51764	−1.07149	−0.535747	+	0.844379i	$0.679970\pi$
$38$	2.86370	0.464554
$39$	0	0
$40$	− 1.00000i	− 0.158114i
$41$	0.760279	0.118736	0.0593678	−	0.998236i	$-0.481092\pi$
$41$	0.760279	0.118736	0.0593678	+	0.998236i	$0.481092\pi$
$42$	0	0
$43$	−5.86370	−0.894206	−0.447103	−	0.894482i	$-0.647544\pi$
$43$	−5.86370	−0.894206	−0.447103	+	0.894482i	$0.647544\pi$
$44$	− 5.39595i	− 0.813471i
$45$	0	0
$46$	−0.267949	−0.0395070
$47$	−7.99536	−1.16624	−0.583121	−	0.812385i	$-0.698169\pi$
$47$	−7.99536	−1.16624	−0.583121	+	0.812385i	$0.698169\pi$
$48$	0	0
$49$	0	0
$50$	1.00000i	0.141421i
$51$	0	0
$52$	2.51764i	0.349134i
$53$	8.39836i	1.15360i	0.816884	+	0.576802i	$0.195699\pi$
$53$	8.39836i	1.15360i	−0.816884	+	0.576802i	$0.804301\pi$
$54$	0	0
$55$	5.39595i	0.727590i
$56$	0	0
$57$	0	0
$58$	8.89898	1.16849
$59$	−12.6715	−1.64968	−0.824842	−	0.565363i	$-0.808736\pi$
$59$	−12.6715	−1.64968	−0.824842	+	0.565363i	$0.808736\pi$
$60$	0	0
$61$	− 2.62158i	− 0.335659i	−0.985816	−	0.167829i	$-0.946324\pi$
$61$	− 2.62158i	− 0.335659i	0.985816	−	0.167829i	$-0.0536758\pi$
$62$	4.82843	0.613211
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 2.51764i	− 0.312275i
$66$	0	0
$67$	9.82237	1.19999	0.599997	−	0.800002i	$-0.295168\pi$
$67$	9.82237	1.19999	0.599997	+	0.800002i	$0.295168\pi$
$68$	4.49938	0.545630
$69$	0	0
$70$	0	0
$71$	− 4.76268i	− 0.565226i	−0.959234	−	0.282613i	$-0.908799\pi$
$71$	− 4.76268i	− 0.565226i	0.959234	−	0.282613i	$-0.0912013\pi$
$72$	0	0
$73$	− 11.6569i	− 1.36433i	−0.731198	−	0.682166i	$-0.761038\pi$
$73$	− 11.6569i	− 1.36433i	0.731198	−	0.682166i	$-0.238962\pi$
$74$	− 6.51764i	− 0.757660i
$75$	0	0
$76$	2.86370i	0.328489i
$77$	0	0
$78$	0	0
$79$	8.59235	0.966715	0.483358	−	0.875423i	$-0.339417\pi$
$79$	8.59235	0.966715	0.483358	+	0.875423i	$0.339417\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	0.760279i	0.0839587i
$83$	9.45001	1.03727	0.518636	−	0.854995i	$-0.326440\pi$
$83$	9.45001	1.03727	0.518636	+	0.854995i	$0.326440\pi$
$84$	0	0
$85$	−4.49938	−0.488026
$86$	− 5.86370i	− 0.632299i
$87$	0	0
$88$	5.39595	0.575211
$89$	7.97005	0.844823	0.422412	−	0.906404i	$-0.361184\pi$
$89$	7.97005	0.844823	0.422412	+	0.906404i	$0.361184\pi$
$90$	0	0
$91$	0	0
$92$	− 0.267949i	− 0.0279356i
$93$	0	0
$94$	− 7.99536i	− 0.824658i
$95$	− 2.86370i	− 0.293810i
$96$	0	0
$97$	− 6.16353i	− 0.625812i	−0.949784	−	0.312906i	$-0.898698\pi$
$97$	− 6.16353i	− 0.625812i	0.949784	−	0.312906i	$-0.101302\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	14.0492	1.39794	0.698972	−	0.715149i	$-0.253641\pi$
$101$	14.0492	1.39794	0.698972	+	0.715149i	$0.253641\pi$
$102$	0	0
$103$	14.1769i	1.39689i	0.715663	+	0.698446i	$0.246125\pi$
$103$	14.1769i	1.39689i	−0.715663	+	0.698446i	$0.753875\pi$
$104$	−2.51764	−0.246875
$105$	0	0
$106$	−8.39836	−0.815721
$107$	− 1.64173i	− 0.158712i	−0.996846	−	0.0793559i	$-0.974714\pi$
$107$	− 1.64173i	− 0.158712i	0.996846	−	0.0793559i	$-0.0252863\pi$
$108$	0	0
$109$	−19.8977	−1.90586	−0.952929	−	0.303195i	$-0.901947\pi$
$109$	−19.8977	−1.90586	−0.952929	+	0.303195i	$0.901947\pi$
$110$	−5.39595	−0.514484
$111$	0	0
$112$	0	0
$113$	5.95867i	0.560545i	0.959921	+	0.280272i	$0.0904248\pi$
$113$	5.95867i	0.560545i	−0.959921	+	0.280272i	$0.909575\pi$
$114$	0	0
$115$	0.267949i	0.0249864i
$116$	8.89898i	0.826250i
$117$	0	0
$118$	− 12.6715i	− 1.16650i
$119$	0	0
$120$	0	0
$121$	−18.1163	−1.64694
$122$	2.62158	0.237347
$123$	0	0
$124$	4.82843i	0.433606i
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.5103	−1.28758	−0.643792	−	0.765200i	$-0.722640\pi$
$127$	−14.5103	−1.28758	−0.643792	+	0.765200i	$0.722640\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	2.51764	0.220811
$131$	−15.4665	−1.35131	−0.675657	−	0.737216i	$-0.736140\pi$
$131$	−15.4665	−1.35131	−0.675657	+	0.737216i	$0.736140\pi$
$132$	0	0
$133$	0	0
$134$	9.82237i	0.848524i
$135$	0	0
$136$	4.49938i	0.385818i
$137$	− 8.62158i	− 0.736591i	−0.929709	−	0.368296i	$-0.879941\pi$
$137$	− 8.62158i	− 0.736591i	0.929709	−	0.368296i	$-0.120059\pi$
$138$	0	0
$139$	− 10.2512i	− 0.869495i	−0.900552	−	0.434748i	$-0.856838\pi$
$139$	− 10.2512i	− 0.869495i	0.900552	−	0.434748i	$-0.143162\pi$
$140$	0	0
$141$	0	0
$142$	4.76268	0.399675
$143$	13.5851	1.13604
$144$	0	0
$145$	− 8.89898i	− 0.739020i
$146$	11.6569	0.964728
$147$	0	0
$148$	6.51764	0.535747
$149$	− 9.19881i	− 0.753595i	−0.926296	−	0.376798i	$-0.877025\pi$
$149$	− 9.19881i	− 0.753595i	0.926296	−	0.376798i	$-0.122975\pi$
$150$	0	0
$151$	−12.7486	−1.03747	−0.518733	−	0.854937i	$-0.673596\pi$
$151$	−12.7486	−1.03747	−0.518733	+	0.854937i	$0.673596\pi$
$152$	−2.86370	−0.232277
$153$	0	0
$154$	0	0
$155$	− 4.82843i	− 0.387829i
$156$	0	0
$157$	− 13.8447i	− 1.10493i	−0.833537	−	0.552464i	$-0.813688\pi$
$157$	− 13.8447i	− 1.10493i	0.833537	−	0.552464i	$-0.186312\pi$
$158$	8.59235i	0.683571i
$159$	0	0
$160$	1.00000i	0.0790569i
$161$	0	0
$162$	0	0
$163$	−20.6050	−1.61391	−0.806954	−	0.590615i	$-0.798885\pi$
$163$	−20.6050	−1.61391	−0.806954	+	0.590615i	$0.798885\pi$
$164$	−0.760279	−0.0593678
$165$	0	0
$166$	9.45001i	0.733463i
$167$	6.84961	0.530038	0.265019	−	0.964243i	$-0.414622\pi$
$167$	6.84961	0.530038	0.265019	+	0.964243i	$0.414622\pi$
$168$	0	0
$169$	6.66150	0.512423
$170$	− 4.49938i	− 0.345087i
$171$	0	0
$172$	5.86370	0.447103
$173$	12.7580	0.969976	0.484988	−	0.874521i	$-0.338824\pi$
$173$	12.7580	0.969976	0.484988	+	0.874521i	$0.338824\pi$
$174$	0	0
$175$	0	0
$176$	5.39595i	0.406735i
$177$	0	0
$178$	7.97005i	0.597380i
$179$	18.8666i	1.41016i	0.709129	+	0.705079i	$0.249088\pi$
$179$	18.8666i	1.41016i	−0.709129	+	0.705079i	$0.750912\pi$
$180$	0	0
$181$	− 25.5498i	− 1.89910i	−0.313615	−	0.949550i	$-0.601540\pi$
$181$	− 25.5498i	− 1.89910i	0.313615	−	0.949550i	$-0.398460\pi$
$182$	0	0
$183$	0	0
$184$	0.267949	0.0197535
$185$	−6.51764	−0.479186
$186$	0	0
$187$	− 24.2784i	− 1.77541i
$188$	7.99536	0.583121
$189$	0	0
$190$	2.86370	0.207755
$191$	− 8.09049i	− 0.585407i	−0.956203	−	0.292704i	$-0.905445\pi$
$191$	− 8.09049i	− 0.585407i	0.956203	−	0.292704i	$-0.0945549\pi$
$192$	0	0
$193$	−14.1270	−1.01688	−0.508442	−	0.861096i	$-0.669778\pi$
$193$	−14.1270	−1.01688	−0.508442	+	0.861096i	$0.669778\pi$
$194$	6.16353	0.442516
$195$	0	0
$196$	0	0
$197$	− 14.2738i	− 1.01697i	−0.861072	−	0.508483i	$-0.830207\pi$
$197$	− 14.2738i	− 1.01697i	0.861072	−	0.508483i	$-0.169793\pi$
$198$	0	0
$199$	− 3.54195i	− 0.251082i	−0.992088	−	0.125541i	$-0.959933\pi$
$199$	− 3.54195i	− 0.251082i	0.992088	−	0.125541i	$-0.0400667\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	14.0492i	0.988495i
$203$	0	0
$204$	0	0
$205$	0.760279	0.0531002
$206$	−14.1769	−0.987751
$207$	0	0
$208$	− 2.51764i	− 0.174567i
$209$	15.4524	1.06887
$210$	0	0
$211$	−3.92340	−0.270098	−0.135049	−	0.990839i	$-0.543119\pi$
$211$	−3.92340	−0.270098	−0.135049	+	0.990839i	$0.543119\pi$
$212$	− 8.39836i	− 0.576802i
$213$	0	0
$214$	1.64173	0.112226
$215$	−5.86370	−0.399901
$216$	0	0
$217$	0	0
$218$	− 19.8977i	− 1.34764i
$219$	0	0
$220$	− 5.39595i	− 0.363795i
$221$	11.3278i	0.761991i
$222$	0	0
$223$	− 14.6904i	− 0.983740i	−0.870669	−	0.491870i	$-0.836314\pi$
$223$	− 14.6904i	− 0.983740i	0.870669	−	0.491870i	$-0.163686\pi$
$224$	0	0
$225$	0	0
$226$	−5.95867	−0.396365
$227$	22.7826	1.51214	0.756068	−	0.654493i	$-0.227118\pi$
$227$	22.7826	1.51214	0.756068	+	0.654493i	$0.227118\pi$
$228$	0	0
$229$	− 20.2175i	− 1.33601i	−0.744157	−	0.668005i	$-0.767149\pi$
$229$	− 20.2175i	− 1.33601i	0.744157	−	0.668005i	$-0.232851\pi$
$230$	−0.267949	−0.0176680
$231$	0	0
$232$	−8.89898	−0.584247
$233$	2.63087i	0.172354i	0.996280	+	0.0861769i	$0.0274650\pi$
$233$	2.63087i	0.172354i	−0.996280	+	0.0861769i	$0.972535\pi$
$234$	0	0
$235$	−7.99536	−0.521560
$236$	12.6715	0.824842
$237$	0	0
$238$	0	0
$239$	− 16.8766i	− 1.09165i	−0.837898	−	0.545827i	$-0.816216\pi$
$239$	− 16.8766i	− 1.09165i	0.837898	−	0.545827i	$-0.183784\pi$
$240$	0	0
$241$	14.5254i	0.935661i	0.883818	+	0.467831i	$0.154964\pi$
$241$	14.5254i	0.935661i	−0.883818	+	0.467831i	$0.845036\pi$
$242$	− 18.1163i	− 1.16456i
$243$	0	0
$244$	2.62158i	0.167829i
$245$	0	0
$246$	0	0
$247$	−7.20977	−0.458747
$248$	−4.82843	−0.306605
$249$	0	0
$250$	1.00000i	0.0632456i
$251$	15.7243	0.992507	0.496254	−	0.868178i	$-0.334709\pi$
$251$	15.7243	0.992507	0.496254	+	0.868178i	$0.334709\pi$
$252$	0	0
$253$	−1.44584	−0.0908993
$254$	− 14.5103i	− 0.910460i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.6617	−1.22646	−0.613230	−	0.789904i	$-0.710130\pi$
$257$	−19.6617	−1.22646	−0.613230	+	0.789904i	$0.710130\pi$
$258$	0	0
$259$	0	0
$260$	2.51764i	0.156137i
$261$	0	0
$262$	− 15.4665i	− 0.955524i
$263$	4.32175i	0.266491i	0.991083	+	0.133245i	$0.0425398\pi$
$263$	4.32175i	0.266491i	−0.991083	+	0.133245i	$0.957460\pi$
$264$	0	0
$265$	8.39836i	0.515907i
$266$	0	0
$267$	0	0
$268$	−9.82237	−0.599997
$269$	−17.5979	−1.07296	−0.536482	−	0.843912i	$-0.680247\pi$
$269$	−17.5979	−1.07296	−0.536482	+	0.843912i	$0.680247\pi$
$270$	0	0
$271$	− 10.5359i	− 0.640010i	−0.947416	−	0.320005i	$-0.896315\pi$
$271$	− 10.5359i	− 0.640010i	0.947416	−	0.320005i	$-0.103685\pi$
$272$	−4.49938	−0.272815
$273$	0	0
$274$	8.62158	0.520849
$275$	5.39595i	0.325388i
$276$	0	0
$277$	3.01942	0.181419	0.0907097	−	0.995877i	$-0.471086\pi$
$277$	3.01942	0.181419	0.0907097	+	0.995877i	$0.471086\pi$
$278$	10.2512	0.614826
$279$	0	0
$280$	0	0
$281$	− 10.6880i	− 0.637591i	−0.947824	−	0.318795i	$-0.896722\pi$
$281$	− 10.6880i	− 0.637591i	0.947824	−	0.318795i	$-0.103278\pi$
$282$	0	0
$283$	17.5687i	1.04435i	0.852838	+	0.522175i	$0.174879\pi$
$283$	17.5687i	1.04435i	−0.852838	+	0.522175i	$0.825121\pi$
$284$	4.76268i	0.282613i
$285$	0	0
$286$	13.5851i	0.803301i
$287$	0	0
$288$	0	0
$289$	3.24440	0.190847
$290$	8.89898	0.522566
$291$	0	0
$292$	11.6569i	0.682166i
$293$	−14.6710	−0.857086	−0.428543	−	0.903521i	$-0.640973\pi$
$293$	−14.6710	−0.857086	−0.428543	+	0.903521i	$0.640973\pi$
$294$	0	0
$295$	−12.6715	−0.737761
$296$	6.51764i	0.378830i
$297$	0	0
$298$	9.19881	0.532872
$299$	0.674599	0.0390131
$300$	0	0
$301$	0	0
$302$	− 12.7486i	− 0.733599i
$303$	0	0
$304$	− 2.86370i	− 0.164245i
$305$	− 2.62158i	− 0.150111i
$306$	0	0
$307$	21.2772i	1.21435i	0.794567	+	0.607177i	$0.207698\pi$
$307$	21.2772i	1.21435i	−0.794567	+	0.607177i	$0.792302\pi$
$308$	0	0
$309$	0	0
$310$	4.82843	0.274236
$311$	11.8345	0.671072	0.335536	−	0.942027i	$-0.391083\pi$
$311$	11.8345	0.671072	0.335536	+	0.942027i	$0.391083\pi$
$312$	0	0
$313$	− 4.50970i	− 0.254903i	−0.991845	−	0.127452i	$-0.959320\pi$
$313$	− 4.50970i	− 0.254903i	0.991845	−	0.127452i	$-0.0406797\pi$
$314$	13.8447	0.781302
$315$	0	0
$316$	−8.59235	−0.483358
$317$	− 18.3830i	− 1.03249i	−0.856440	−	0.516247i	$-0.827329\pi$
$317$	− 18.3830i	− 1.03249i	0.856440	−	0.516247i	$-0.172671\pi$
$318$	0	0
$319$	48.0185	2.68852
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	12.8849i	0.716934i
$324$	0	0
$325$	− 2.51764i	− 0.139653i
$326$	− 20.6050i	− 1.14121i
$327$	0	0
$328$	− 0.760279i	− 0.0419794i
$329$	0	0
$330$	0	0
$331$	7.96132	0.437594	0.218797	−	0.975770i	$-0.429787\pi$
$331$	7.96132	0.437594	0.218797	+	0.975770i	$0.429787\pi$
$332$	−9.45001	−0.518636
$333$	0	0
$334$	6.84961i	0.374794i
$335$	9.82237	0.536654
$336$	0	0
$337$	−6.89417	−0.375549	−0.187775	−	0.982212i	$-0.560127\pi$
$337$	−6.89417	−0.375549	−0.187775	+	0.982212i	$0.560127\pi$
$338$	6.66150i	0.362338i
$339$	0	0
$340$	4.49938	0.244013
$341$	26.0540	1.41090
$342$	0	0
$343$	0	0
$344$	5.86370i	0.316150i
$345$	0	0
$346$	12.7580i	0.685876i
$347$	16.3558i	0.878025i	0.898481	+	0.439012i	$0.144672\pi$
$347$	16.3558i	0.878025i	−0.898481	+	0.439012i	$0.855328\pi$
$348$	0	0
$349$	− 24.5851i	− 1.31601i	−0.753014	−	0.658004i	$-0.771401\pi$
$349$	− 24.5851i	− 1.31601i	0.753014	−	0.658004i	$-0.228599\pi$
$350$	0	0
$351$	0	0
$352$	−5.39595	−0.287605
$353$	−13.7181	−0.730142	−0.365071	−	0.930980i	$-0.618955\pi$
$353$	−13.7181	−0.730142	−0.365071	+	0.930980i	$0.618955\pi$
$354$	0	0
$355$	− 4.76268i	− 0.252777i
$356$	−7.97005	−0.422412
$357$	0	0
$358$	−18.8666	−0.997132
$359$	− 11.9002i	− 0.628070i	−0.949411	−	0.314035i	$-0.898319\pi$
$359$	− 11.9002i	− 0.628070i	0.949411	−	0.314035i	$-0.101681\pi$
$360$	0	0
$361$	10.7992	0.568379
$362$	25.5498	1.34287
$363$	0	0
$364$	0	0
$365$	− 11.6569i	− 0.610148i
$366$	0	0
$367$	12.5892i	0.657152i	0.944478	+	0.328576i	$0.106569\pi$
$367$	12.5892i	0.657152i	−0.944478	+	0.328576i	$0.893431\pi$
$368$	0.267949i	0.0139678i
$369$	0	0
$370$	− 6.51764i	− 0.338836i
$371$	0	0
$372$	0	0
$373$	−27.1737	−1.40700	−0.703499	−	0.710696i	$-0.748380\pi$
$373$	−27.1737	−1.40700	−0.703499	+	0.710696i	$0.748380\pi$
$374$	24.2784	1.25541
$375$	0	0
$376$	7.99536i	0.412329i
$377$	−22.4044	−1.15389
$378$	0	0
$379$	15.7335	0.808174	0.404087	−	0.914721i	$-0.367589\pi$
$379$	15.7335	0.808174	0.404087	+	0.914721i	$0.367589\pi$
$380$	2.86370i	0.146905i
$381$	0	0
$382$	8.09049	0.413945
$383$	15.7812	0.806382	0.403191	−	0.915116i	$-0.367901\pi$
$383$	15.7812	0.806382	0.403191	+	0.915116i	$0.367901\pi$
$384$	0	0
$385$	0	0
$386$	− 14.1270i	− 0.719046i
$387$	0	0
$388$	6.16353i	0.312906i
$389$	15.8836i	0.805332i	0.915347	+	0.402666i	$0.131916\pi$
$389$	15.8836i	0.805332i	−0.915347	+	0.402666i	$0.868084\pi$
$390$	0	0
$391$	− 1.20560i	− 0.0609700i
$392$	0	0
$393$	0	0
$394$	14.2738	0.719103
$395$	8.59235	0.432328
$396$	0	0
$397$	37.3611i	1.87510i	0.347850	+	0.937550i	$0.386912\pi$
$397$	37.3611i	1.87510i	−0.347850	+	0.937550i	$0.613088\pi$
$398$	3.54195	0.177542
$399$	0	0
$400$	1.00000	0.0500000
$401$	28.2735i	1.41191i	0.708255	+	0.705957i	$0.249483\pi$
$401$	28.2735i	1.41191i	−0.708255	+	0.705957i	$0.750517\pi$
$402$	0	0
$403$	−12.1562	−0.605545
$404$	−14.0492	−0.698972
$405$	0	0
$406$	0	0
$407$	− 35.1689i	− 1.74326i
$408$	0	0
$409$	− 16.0096i	− 0.791625i	−0.918331	−	0.395812i	$-0.870463\pi$
$409$	− 16.0096i	− 0.791625i	0.918331	−	0.395812i	$-0.129537\pi$
$410$	0.760279i	0.0375475i
$411$	0	0
$412$	− 14.1769i	− 0.698446i
$413$	0	0
$414$	0	0
$415$	9.45001	0.463883
$416$	2.51764	0.123437
$417$	0	0
$418$	15.4524i	0.755802i
$419$	29.5137	1.44184	0.720919	−	0.693020i	$-0.243720\pi$
$419$	29.5137	1.44184	0.720919	+	0.693020i	$0.243720\pi$
$420$	0	0
$421$	0.309114	0.0150653	0.00753265	−	0.999972i	$-0.497602\pi$
$421$	0.309114	0.0150653	0.00753265	+	0.999972i	$0.497602\pi$
$422$	− 3.92340i	− 0.190988i
$423$	0	0
$424$	8.39836	0.407860
$425$	−4.49938	−0.218252
$426$	0	0
$427$	0	0
$428$	1.64173i	0.0793559i
$429$	0	0
$430$	− 5.86370i	− 0.282773i
$431$	− 8.82010i	− 0.424849i	−0.977177	−	0.212425i	$-0.931864\pi$
$431$	− 8.82010i	− 0.424849i	0.977177	−	0.212425i	$-0.0681360\pi$
$432$	0	0
$433$	9.56388i	0.459611i	0.973237	+	0.229805i	$0.0738089\pi$
$433$	9.56388i	0.459611i	−0.973237	+	0.229805i	$0.926191\pi$
$434$	0	0
$435$	0	0
$436$	19.8977	0.952929
$437$	0.767327	0.0367062
$438$	0	0
$439$	− 36.1810i	− 1.72682i	−0.504500	−	0.863412i	$-0.668323\pi$
$439$	− 36.1810i	− 1.72682i	0.504500	−	0.863412i	$-0.331677\pi$
$440$	5.39595	0.257242
$441$	0	0
$442$	−11.3278	−0.538809
$443$	− 4.08568i	− 0.194116i	−0.995279	−	0.0970582i	$-0.969057\pi$
$443$	− 4.08568i	− 0.194116i	0.995279	−	0.0970582i	$-0.0309433\pi$
$444$	0	0
$445$	7.97005	0.377816
$446$	14.6904	0.695609
$447$	0	0
$448$	0	0
$449$	19.9377i	0.940918i	0.882422	+	0.470459i	$0.155912\pi$
$449$	19.9377i	0.940918i	−0.882422	+	0.470459i	$0.844088\pi$
$450$	0	0
$451$	4.10243i	0.193176i
$452$	− 5.95867i	− 0.280272i
$453$	0	0
$454$	22.7826i	1.06924i
$455$	0	0
$456$	0	0
$457$	−20.1245	−0.941385	−0.470693	−	0.882297i	$-0.655996\pi$
$457$	−20.1245	−0.941385	−0.470693	+	0.882297i	$0.655996\pi$
$458$	20.2175	0.944702
$459$	0	0
$460$	− 0.267949i	− 0.0124932i
$461$	−0.909299	−0.0423503	−0.0211751	−	0.999776i	$-0.506741\pi$
$461$	−0.909299	−0.0423503	−0.0211751	+	0.999776i	$0.506741\pi$
$462$	0	0
$463$	21.4280	0.995843	0.497922	−	0.867222i	$-0.334097\pi$
$463$	21.4280	0.995843	0.497922	+	0.867222i	$0.334097\pi$
$464$	− 8.89898i	− 0.413125i
$465$	0	0
$466$	−2.63087	−0.121873
$467$	12.6921	0.587322	0.293661	−	0.955910i	$-0.405126\pi$
$467$	12.6921	0.587322	0.293661	+	0.955910i	$0.405126\pi$
$468$	0	0
$469$	0	0
$470$	− 7.99536i	− 0.368798i
$471$	0	0
$472$	12.6715i	0.583252i
$473$	− 31.6403i	− 1.45482i
$474$	0	0
$475$	− 2.86370i	− 0.131396i
$476$	0	0
$477$	0	0
$478$	16.8766	0.771916
$479$	−12.8766	−0.588345	−0.294172	−	0.955752i	$-0.595044\pi$
$479$	−12.8766	−0.588345	−0.294172	+	0.955752i	$0.595044\pi$
$480$	0	0
$481$	16.4091i	0.748188i
$482$	−14.5254	−0.661612
$483$	0	0
$484$	18.1163	0.823469
$485$	− 6.16353i	− 0.279871i
$486$	0	0
$487$	−20.8194	−0.943416	−0.471708	−	0.881755i	$-0.656362\pi$
$487$	−20.8194	−0.943416	−0.471708	+	0.881755i	$0.656362\pi$
$488$	−2.62158	−0.118673
$489$	0	0
$490$	0	0
$491$	27.3271i	1.23325i	0.787256	+	0.616627i	$0.211501\pi$
$491$	27.3271i	1.23325i	−0.787256	+	0.616627i	$0.788499\pi$
$492$	0	0
$493$	40.0399i	1.80331i
$494$	− 7.20977i	− 0.324383i
$495$	0	0
$496$	− 4.82843i	− 0.216803i
$497$	0	0
$498$	0	0
$499$	33.3604	1.49342	0.746708	−	0.665152i	$-0.231633\pi$
$499$	33.3604	1.49342	0.746708	+	0.665152i	$0.231633\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	15.7243i	0.701809i
$503$	−16.2936	−0.726494	−0.363247	−	0.931693i	$-0.618332\pi$
$503$	−16.2936	−0.726494	−0.363247	+	0.931693i	$0.618332\pi$
$504$	0	0
$505$	14.0492	0.625179
$506$	− 1.44584i	− 0.0642755i
$507$	0	0
$508$	14.5103	0.643792
$509$	−24.6800	−1.09392	−0.546961	−	0.837158i	$-0.684216\pi$
$509$	−24.6800	−1.09392	−0.546961	+	0.837158i	$0.684216\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 19.6617i	− 0.867239i
$515$	14.1769i	0.624709i
$516$	0	0
$517$	− 43.1426i	− 1.89741i
$518$	0	0
$519$	0	0
$520$	−2.51764	−0.110406
$521$	−0.283325	−0.0124127	−0.00620635	−	0.999981i	$-0.501976\pi$
$521$	−0.283325	−0.0124127	−0.00620635	+	0.999981i	$0.501976\pi$
$522$	0	0
$523$	− 11.2817i	− 0.493313i	−0.969103	−	0.246656i	$-0.920668\pi$
$523$	− 11.2817i	− 0.493313i	0.969103	−	0.246656i	$-0.0793319\pi$
$524$	15.4665	0.675657
$525$	0	0
$526$	−4.32175	−0.188437
$527$	21.7249i	0.946352i
$528$	0	0
$529$	22.9282	0.996878
$530$	−8.39836	−0.364801
$531$	0	0
$532$	0	0
$533$	− 1.91411i	− 0.0829092i
$534$	0	0
$535$	− 1.64173i	− 0.0709780i
$536$	− 9.82237i	− 0.424262i
$537$	0	0
$538$	− 17.5979i	− 0.758700i
$539$	0	0
$540$	0	0
$541$	−34.8250	−1.49724	−0.748621	−	0.662998i	$-0.769284\pi$
$541$	−34.8250	−1.49724	−0.748621	+	0.662998i	$0.769284\pi$
$542$	10.5359	0.452556
$543$	0	0
$544$	− 4.49938i	− 0.192909i
$545$	−19.8977	−0.852325
$546$	0	0
$547$	35.4261	1.51471	0.757356	−	0.653002i	$-0.226491\pi$
$547$	35.4261	1.51471	0.757356	+	0.653002i	$0.226491\pi$
$548$	8.62158i	0.368296i
$549$	0	0
$550$	−5.39595	−0.230084
$551$	−25.4840	−1.08566
$552$	0	0
$553$	0	0
$554$	3.01942i	0.128283i
$555$	0	0
$556$	10.2512i	0.434748i
$557$	− 8.14185i	− 0.344981i	−0.985011	−	0.172491i	$-0.944819\pi$
$557$	− 8.14185i	− 0.344981i	0.985011	−	0.172491i	$-0.0551815\pi$
$558$	0	0
$559$	14.7627i	0.624395i
$560$	0	0
$561$	0	0
$562$	10.6880	0.450845
$563$	−20.4175	−0.860497	−0.430248	−	0.902711i	$-0.641574\pi$
$563$	−20.4175	−0.860497	−0.430248	+	0.902711i	$0.641574\pi$
$564$	0	0
$565$	5.95867i	0.250683i
$566$	−17.5687	−0.738467
$567$	0	0
$568$	−4.76268	−0.199838
$569$	26.0433i	1.09179i	0.837853	+	0.545896i	$0.183811\pi$
$569$	26.0433i	1.09179i	−0.837853	+	0.545896i	$0.816189\pi$
$570$	0	0
$571$	−12.3762	−0.517929	−0.258964	−	0.965887i	$-0.583381\pi$
$571$	−12.3762	−0.517929	−0.258964	+	0.965887i	$0.583381\pi$
$572$	−13.5851	−0.568020
$573$	0	0
$574$	0	0
$575$	0.267949i	0.0111743i
$576$	0	0
$577$	− 26.9506i	− 1.12197i	−0.827826	−	0.560985i	$-0.810423\pi$
$577$	− 26.9506i	− 1.12197i	0.827826	−	0.560985i	$-0.189577\pi$
$578$	3.24440i	0.134949i
$579$	0	0
$580$	8.89898i	0.369510i
$581$	0	0
$582$	0	0
$583$	−45.3171	−1.87684
$584$	−11.6569	−0.482364
$585$	0	0
$586$	− 14.6710i	− 0.606051i
$587$	−35.3511	−1.45910	−0.729548	−	0.683930i	$-0.760269\pi$
$587$	−35.3511	−1.45910	−0.729548	+	0.683930i	$0.760269\pi$
$588$	0	0
$589$	−13.8272	−0.569739
$590$	− 12.6715i	− 0.521676i
$591$	0	0
$592$	−6.51764	−0.267873
$593$	−29.4115	−1.20779	−0.603893	−	0.797066i	$-0.706384\pi$
$593$	−29.4115	−1.20779	−0.603893	+	0.797066i	$0.706384\pi$
$594$	0	0
$595$	0	0
$596$	9.19881i	0.376798i
$597$	0	0
$598$	0.674599i	0.0275864i
$599$	− 30.6566i	− 1.25260i	−0.779584	−	0.626298i	$-0.784569\pi$
$599$	− 30.6566i	− 1.25260i	0.779584	−	0.626298i	$-0.215431\pi$
$600$	0	0
$601$	34.3407i	1.40078i	0.713758	+	0.700392i	$0.246992\pi$
$601$	34.3407i	1.40078i	−0.713758	+	0.700392i	$0.753008\pi$
$602$	0	0
$603$	0	0
$604$	12.7486	0.518733
$605$	−18.1163	−0.736533
$606$	0	0
$607$	32.6174i	1.32390i	0.749548	+	0.661950i	$0.230271\pi$
$607$	32.6174i	1.32390i	−0.749548	+	0.661950i	$0.769729\pi$
$608$	2.86370	0.116139
$609$	0	0
$610$	2.62158	0.106145
$611$	20.1294i	0.814349i
$612$	0	0
$613$	−15.9969	−0.646107	−0.323054	−	0.946381i	$-0.604709\pi$
$613$	−15.9969	−0.646107	−0.323054	+	0.946381i	$0.604709\pi$
$614$	−21.2772	−0.858677
$615$	0	0
$616$	0	0
$617$	25.1429i	1.01221i	0.862471	+	0.506107i	$0.168916\pi$
$617$	25.1429i	1.01221i	−0.862471	+	0.506107i	$0.831084\pi$
$618$	0	0
$619$	− 37.3406i	− 1.50085i	−0.660958	−	0.750423i	$-0.729850\pi$
$619$	− 37.3406i	− 1.50085i	0.660958	−	0.750423i	$-0.270150\pi$
$620$	4.82843i	0.193914i
$621$	0	0
$622$	11.8345i	0.474519i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.50970	0.180244
$627$	0	0
$628$	13.8447i	0.552464i
$629$	29.3253	1.16928
$630$	0	0
$631$	49.5015	1.97062	0.985311	−	0.170767i	$-0.0546245\pi$
$631$	49.5015	1.97062	0.985311	+	0.170767i	$0.0546245\pi$
$632$	− 8.59235i	− 0.341786i
$633$	0	0
$634$	18.3830	0.730083
$635$	−14.5103	−0.575825
$636$	0	0
$637$	0	0
$638$	48.0185i	1.90107i
$639$	0	0
$640$	− 1.00000i	− 0.0395285i
$641$	36.3083i	1.43409i	0.697026	+	0.717046i	$0.254506\pi$
$641$	36.3083i	1.43409i	−0.697026	+	0.717046i	$0.745494\pi$
$642$	0	0
$643$	10.2653i	0.404824i	0.979300	+	0.202412i	$0.0648780\pi$
$643$	10.2653i	0.404824i	−0.979300	+	0.202412i	$0.935122\pi$
$644$	0	0
$645$	0	0
$646$	−12.8849	−0.506949
$647$	21.8215	0.857892	0.428946	−	0.903330i	$-0.358885\pi$
$647$	21.8215	0.857892	0.428946	+	0.903330i	$0.358885\pi$
$648$	0	0
$649$	− 68.3746i	− 2.68394i
$650$	2.51764	0.0987499
$651$	0	0
$652$	20.6050	0.806954
$653$	47.2914i	1.85066i	0.379166	+	0.925329i	$0.376211\pi$
$653$	47.2914i	1.85066i	−0.379166	+	0.925329i	$0.623789\pi$
$654$	0	0
$655$	−15.4665	−0.604326
$656$	0.760279	0.0296839
$657$	0	0
$658$	0	0
$659$	3.58255i	0.139556i	0.997563	+	0.0697782i	$0.0222291\pi$
$659$	3.58255i	0.139556i	−0.997563	+	0.0697782i	$0.977771\pi$
$660$	0	0
$661$	1.63692i	0.0636688i	0.999493	+	0.0318344i	$0.0101349\pi$
$661$	1.63692i	0.0636688i	−0.999493	+	0.0318344i	$0.989865\pi$
$662$	7.96132i	0.309426i
$663$	0	0
$664$	− 9.45001i	− 0.366731i
$665$	0	0
$666$	0	0
$667$	2.38447	0.0923272
$668$	−6.84961	−0.265019
$669$	0	0
$670$	9.82237i	0.379471i
$671$	14.1459	0.546097
$672$	0	0
$673$	−2.02242	−0.0779587	−0.0389794	−	0.999240i	$-0.512411\pi$
$673$	−2.02242	−0.0779587	−0.0389794	+	0.999240i	$0.512411\pi$
$674$	− 6.89417i	− 0.265554i
$675$	0	0
$676$	−6.66150	−0.256211
$677$	−22.8826	−0.879450	−0.439725	−	0.898132i	$-0.644924\pi$
$677$	−22.8826	−0.879450	−0.439725	+	0.898132i	$0.644924\pi$
$678$	0	0
$679$	0	0
$680$	4.49938i	0.172543i
$681$	0	0
$682$	26.0540i	0.997658i
$683$	− 31.4052i	− 1.20169i	−0.799366	−	0.600844i	$-0.794831\pi$
$683$	− 31.4052i	− 1.20169i	0.799366	−	0.600844i	$-0.205169\pi$
$684$	0	0
$685$	− 8.62158i	− 0.329414i
$686$	0	0
$687$	0	0
$688$	−5.86370	−0.223552
$689$	21.1440	0.805523
$690$	0	0
$691$	33.9109i	1.29003i	0.764170	+	0.645015i	$0.223149\pi$
$691$	33.9109i	1.29003i	−0.764170	+	0.645015i	$0.776851\pi$
$692$	−12.7580	−0.484988
$693$	0	0
$694$	−16.3558	−0.620857
$695$	− 10.2512i	− 0.388850i
$696$	0	0
$697$	−3.42078	−0.129571
$698$	24.5851	0.930558
$699$	0	0
$700$	0	0
$701$	− 10.5296i	− 0.397699i	−0.980030	−	0.198849i	$-0.936280\pi$
$701$	− 10.5296i	− 0.397699i	0.980030	−	0.198849i	$-0.0637205\pi$
$702$	0	0
$703$	18.6646i	0.703948i
$704$	− 5.39595i	− 0.203368i
$705$	0	0
$706$	− 13.7181i	− 0.516288i
$707$	0	0
$708$	0	0
$709$	15.0514	0.565268	0.282634	−	0.959228i	$-0.408792\pi$
$709$	15.0514	0.565268	0.282634	+	0.959228i	$0.408792\pi$
$710$	4.76268	0.178740
$711$	0	0
$712$	− 7.97005i	− 0.298690i
$713$	1.29377	0.0484522
$714$	0	0
$715$	13.5851	0.508052
$716$	− 18.8666i	− 0.705079i
$717$	0	0
$718$	11.9002	0.444112
$719$	−24.4274	−0.910987	−0.455494	−	0.890239i	$-0.650537\pi$
$719$	−24.4274	−0.910987	−0.455494	+	0.890239i	$0.650537\pi$
$720$	0	0
$721$	0	0
$722$	10.7992i	0.401905i
$723$	0	0
$724$	25.5498i	0.949550i
$725$	− 8.89898i	− 0.330500i
$726$	0	0
$727$	43.7349i	1.62204i	0.585020	+	0.811019i	$0.301087\pi$
$727$	43.7349i	1.62204i	−0.585020	+	0.811019i	$0.698913\pi$
$728$	0	0
$729$	0	0
$730$	11.6569	0.431440
$731$	26.3830	0.975811
$732$	0	0
$733$	44.7193i	1.65174i	0.563857	+	0.825872i	$0.309317\pi$
$733$	44.7193i	1.65174i	−0.563857	+	0.825872i	$0.690683\pi$
$734$	−12.5892	−0.464677
$735$	0	0
$736$	−0.267949	−0.00987674
$737$	53.0011i	1.95232i
$738$	0	0
$739$	−21.4721	−0.789864	−0.394932	−	0.918710i	$-0.629232\pi$
$739$	−21.4721	−0.789864	−0.394932	+	0.918710i	$0.629232\pi$
$740$	6.51764	0.239593
$741$	0	0
$742$	0	0
$743$	− 29.5637i	− 1.08459i	−0.840190	−	0.542293i	$-0.817556\pi$
$743$	− 29.5637i	− 1.08459i	0.840190	−	0.542293i	$-0.182444\pi$
$744$	0	0
$745$	− 9.19881i	− 0.337018i
$746$	− 27.1737i	− 0.994898i
$747$	0	0
$748$	24.2784i	0.887707i
$749$	0	0
$750$	0	0
$751$	1.19350	0.0435514	0.0217757	−	0.999763i	$-0.493068\pi$
$751$	1.19350	0.0435514	0.0217757	+	0.999763i	$0.493068\pi$
$752$	−7.99536	−0.291561
$753$	0	0
$754$	− 22.4044i	− 0.815920i
$755$	−12.7486	−0.463969
$756$	0	0
$757$	−26.8915	−0.977386	−0.488693	−	0.872456i	$-0.662526\pi$
$757$	−26.8915	−0.977386	−0.488693	+	0.872456i	$0.662526\pi$
$758$	15.7335i	0.571465i
$759$	0	0
$760$	−2.86370	−0.103877
$761$	1.87915	0.0681190	0.0340595	−	0.999420i	$-0.489156\pi$
$761$	1.87915	0.0681190	0.0340595	+	0.999420i	$0.489156\pi$
$762$	0	0
$763$	0	0
$764$	8.09049i	0.292704i
$765$	0	0
$766$	15.7812i	0.570198i
$767$	31.9022i	1.15192i
$768$	0	0
$769$	50.6544i	1.82664i	0.407239	+	0.913322i	$0.366492\pi$
$769$	50.6544i	1.82664i	−0.407239	+	0.913322i	$0.633508\pi$
$770$	0	0
$771$	0	0
$772$	14.1270	0.508442
$773$	48.6705	1.75056	0.875279	−	0.483619i	$-0.160678\pi$
$773$	48.6705	1.75056	0.875279	+	0.483619i	$0.160678\pi$
$774$	0	0
$775$	− 4.82843i	− 0.173442i
$776$	−6.16353	−0.221258
$777$	0	0
$778$	−15.8836	−0.569456
$779$	− 2.17721i	− 0.0780067i
$780$	0	0
$781$	25.6992	0.919590
$782$	1.20560	0.0431123
$783$	0	0
$784$	0	0
$785$	− 13.8447i	− 0.494139i
$786$	0	0
$787$	14.9461i	0.532773i	0.963866	+	0.266386i	$0.0858297\pi$
$787$	14.9461i	0.532773i	−0.963866	+	0.266386i	$0.914170\pi$
$788$	14.2738i	0.508483i
$789$	0	0
$790$	8.59235i	0.305702i
$791$	0	0
$792$	0	0
$793$	−6.60018	−0.234379
$794$	−37.3611	−1.32590
$795$	0	0
$796$	3.54195i	0.125541i
$797$	−15.0557	−0.533301	−0.266650	−	0.963793i	$-0.585917\pi$
$797$	−15.0557	−0.533301	−0.266650	+	0.963793i	$0.585917\pi$
$798$	0	0
$799$	35.9741	1.27267
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	−28.2735	−0.998373
$803$	62.8998	2.21969
$804$	0	0
$805$	0	0
$806$	− 12.1562i	− 0.428185i
$807$	0	0
$808$	− 14.0492i	− 0.494248i
$809$	− 35.4985i	− 1.24806i	−0.781399	−	0.624031i	$-0.785494\pi$
$809$	− 35.4985i	− 1.24806i	0.781399	−	0.624031i	$-0.214506\pi$
$810$	0	0
$811$	− 24.5935i	− 0.863594i	−0.901971	−	0.431797i	$-0.857880\pi$
$811$	− 24.5935i	− 0.863594i	0.901971	−	0.431797i	$-0.142120\pi$
$812$	0	0
$813$	0	0
$814$	35.1689	1.23267
$815$	−20.6050	−0.721761
$816$	0	0
$817$	16.7919i	0.587474i
$818$	16.0096	0.559763
$819$	0	0
$820$	−0.760279	−0.0265501
$821$	− 14.2068i	− 0.495822i	−0.968783	−	0.247911i	$-0.920256\pi$
$821$	− 14.2068i	− 0.495822i	0.968783	−	0.247911i	$-0.0797441\pi$
$822$	0	0
$823$	1.89117	0.0659219	0.0329610	−	0.999457i	$-0.489506\pi$
$823$	1.89117	0.0659219	0.0329610	+	0.999457i	$0.489506\pi$
$824$	14.1769	0.493876
$825$	0	0
$826$	0	0
$827$	− 31.9280i	− 1.11024i	−0.831769	−	0.555122i	$-0.812671\pi$
$827$	− 31.9280i	− 1.11024i	0.831769	−	0.555122i	$-0.187329\pi$
$828$	0	0
$829$	− 3.91286i	− 0.135899i	−0.997689	−	0.0679497i	$-0.978354\pi$
$829$	− 3.91286i	− 0.135899i	0.997689	−	0.0679497i	$-0.0216457\pi$
$830$	9.45001i	0.328014i
$831$	0	0
$832$	2.51764i	0.0872834i
$833$	0	0
$834$	0	0
$835$	6.84961	0.237040
$836$	−15.4524	−0.534433
$837$	0	0
$838$	29.5137i	1.01953i
$839$	−15.3513	−0.529984	−0.264992	−	0.964251i	$-0.585369\pi$
$839$	−15.3513	−0.529984	−0.264992	+	0.964251i	$0.585369\pi$
$840$	0	0
$841$	−50.1918	−1.73075
$842$	0.309114i	0.0106528i
$843$	0	0
$844$	3.92340	0.135049
$845$	6.66150	0.229163
$846$	0	0
$847$	0	0
$848$	8.39836i	0.288401i
$849$	0	0
$850$	− 4.49938i	− 0.154327i
$851$	− 1.74640i	− 0.0598657i
$852$	0	0
$853$	− 22.2302i	− 0.761148i	−0.924750	−	0.380574i	$-0.875726\pi$
$853$	− 22.2302i	− 0.761148i	0.924750	−	0.380574i	$-0.124274\pi$
$854$	0	0
$855$	0	0
$856$	−1.64173	−0.0561131
$857$	−52.8197	−1.80429	−0.902143	−	0.431438i	$-0.858007\pi$
$857$	−52.8197	−1.80429	−0.902143	+	0.431438i	$0.858007\pi$
$858$	0	0
$859$	− 17.6572i	− 0.602455i	−0.953552	−	0.301228i	$-0.902604\pi$
$859$	− 17.6572i	− 0.602455i	0.953552	−	0.301228i	$-0.0973964\pi$
$860$	5.86370	0.199951
$861$	0	0
$862$	8.82010	0.300414
$863$	− 16.4489i	− 0.559928i	−0.960011	−	0.279964i	$-0.909677\pi$
$863$	− 16.4489i	− 0.559928i	0.960011	−	0.279964i	$-0.0903225\pi$
$864$	0	0
$865$	12.7580	0.433786
$866$	−9.56388	−0.324994
$867$	0	0
$868$	0	0
$869$	46.3639i	1.57279i
$870$	0	0
$871$	− 24.7292i	− 0.837916i
$872$	19.8977i	0.673822i
$873$	0	0
$874$	0.767327i	0.0259552i
$875$	0	0
$876$	0	0
$877$	54.2156	1.83073	0.915366	−	0.402623i	$-0.131902\pi$
$877$	54.2156	1.83073	0.915366	+	0.402623i	$0.131902\pi$
$878$	36.1810	1.22105
$879$	0	0
$880$	5.39595i	0.181898i
$881$	−50.1647	−1.69009	−0.845046	−	0.534694i	$-0.820427\pi$
$881$	−50.1647	−1.69009	−0.845046	+	0.534694i	$0.820427\pi$
$882$	0	0
$883$	−0.841563	−0.0283208	−0.0141604	−	0.999900i	$-0.504508\pi$
$883$	−0.841563	−0.0283208	−0.0141604	+	0.999900i	$0.504508\pi$
$884$	− 11.3278i	− 0.380995i
$885$	0	0
$886$	4.08568	0.137261
$887$	40.0983	1.34637	0.673185	−	0.739474i	$-0.264926\pi$
$887$	40.0983	1.34637	0.673185	+	0.739474i	$0.264926\pi$
$888$	0	0
$889$	0	0
$890$	7.97005i	0.267157i
$891$	0	0
$892$	14.6904i	0.491870i
$893$	22.8963i	0.766196i
$894$	0	0
$895$	18.8666i	0.630642i
$896$	0	0
$897$	0	0
$898$	−19.9377	−0.665329
$899$	−42.9681	−1.43307
$900$	0	0
$901$	− 37.7874i	− 1.25888i
$902$	−4.10243	−0.136596
$903$	0	0
$904$	5.95867	0.198182
$905$	− 25.5498i	− 0.849303i
$906$	0	0
$907$	−45.1890	−1.50048	−0.750238	−	0.661167i	$-0.770061\pi$
$907$	−45.1890	−1.50048	−0.750238	+	0.661167i	$0.770061\pi$
$908$	−22.7826	−0.756068
$909$	0	0
$910$	0	0
$911$	58.2281i	1.92918i	0.263746	+	0.964592i	$0.415042\pi$
$911$	58.2281i	1.92918i	−0.263746	+	0.964592i	$0.584958\pi$
$912$	0	0
$913$	50.9918i	1.68758i
$914$	− 20.1245i	− 0.665660i
$915$	0	0
$916$	20.2175i	0.668005i
$917$	0	0
$918$	0	0
$919$	13.2349	0.436579	0.218290	−	0.975884i	$-0.429952\pi$
$919$	13.2349	0.436579	0.218290	+	0.975884i	$0.429952\pi$
$920$	0.267949	0.00883402
$921$	0	0
$922$	− 0.909299i	− 0.0299462i
$923$	−11.9907	−0.394679
$924$	0	0
$925$	−6.51764	−0.214299
$926$	21.4280i	0.704168i
$927$	0	0
$928$	8.89898	0.292123
$929$	−20.8868	−0.685273	−0.342637	−	0.939468i	$-0.611320\pi$
$929$	−20.8868	−0.685273	−0.342637	+	0.939468i	$0.611320\pi$
$930$	0	0
$931$	0	0
$932$	− 2.63087i	− 0.0861769i
$933$	0	0
$934$	12.6921i	0.415299i
$935$	− 24.2784i	− 0.793990i
$936$	0	0
$937$	− 23.2465i	− 0.759430i	−0.925103	−	0.379715i	$-0.876022\pi$
$937$	− 23.2465i	− 0.759430i	0.925103	−	0.379715i	$-0.123978\pi$
$938$	0	0
$939$	0	0
$940$	7.99536	0.260780
$941$	−1.50510	−0.0490649	−0.0245325	−	0.999699i	$-0.507810\pi$
$941$	−1.50510	−0.0490649	−0.0245325	+	0.999699i	$0.507810\pi$
$942$	0	0
$943$	0.203716i	0.00663391i
$944$	−12.6715	−0.412421
$945$	0	0
$946$	31.6403	1.02871
$947$	− 24.2628i	− 0.788435i	−0.919017	−	0.394218i	$-0.871016\pi$
$947$	− 24.2628i	− 0.788435i	0.919017	−	0.394218i	$-0.128984\pi$
$948$	0	0
$949$	−29.3477	−0.952668
$950$	2.86370	0.0929108
$951$	0	0
$952$	0	0
$953$	56.7061i	1.83689i	0.395547	+	0.918446i	$0.370555\pi$
$953$	56.7061i	1.83689i	−0.395547	+	0.918446i	$0.629445\pi$
$954$	0	0
$955$	− 8.09049i	− 0.261802i
$956$	16.8766i	0.545827i
$957$	0	0
$958$	− 12.8766i	− 0.416023i
$959$	0	0
$960$	0	0
$961$	7.68629	0.247945
$962$	−16.4091	−0.529049
$963$	0	0
$964$	− 14.5254i	− 0.467831i
$965$	−14.1270	−0.454764
$966$	0	0
$967$	7.23556	0.232680	0.116340	−	0.993209i	$-0.462884\pi$
$967$	7.23556	0.232680	0.116340	+	0.993209i	$0.462884\pi$
$968$	18.1163i	0.582280i
$969$	0	0
$970$	6.16353	0.197899
$971$	38.7120	1.24233	0.621163	−	0.783681i	$-0.286660\pi$
$971$	38.7120	1.24233	0.621163	+	0.783681i	$0.286660\pi$
$972$	0	0
$973$	0	0
$974$	− 20.8194i	− 0.667096i
$975$	0	0
$976$	− 2.62158i	− 0.0839147i
$977$	− 18.1610i	− 0.581023i	−0.956871	−	0.290512i	$-0.906175\pi$
$977$	− 18.1610i	− 0.581023i	0.956871	−	0.290512i	$-0.0938255\pi$
$978$	0	0
$979$	43.0060i	1.37448i
$980$	0	0
$981$	0	0
$982$	−27.3271	−0.872042
$983$	−16.3998	−0.523071	−0.261536	−	0.965194i	$-0.584229\pi$
$983$	−16.3998	−0.523071	−0.261536	+	0.965194i	$0.584229\pi$
$984$	0	0
$985$	− 14.2738i	− 0.454801i
$986$	−40.0399	−1.27513
$987$	0	0
$988$	7.20977	0.229373
$989$	− 1.57117i	− 0.0499604i
$990$	0	0
$991$	−10.8917	−0.345986	−0.172993	−	0.984923i	$-0.555344\pi$
$991$	−10.8917	−0.345986	−0.172993	+	0.984923i	$0.555344\pi$
$992$	4.82843	0.153303
$993$	0	0
$994$	0	0
$995$	− 3.54195i	− 0.112287i
$996$	0	0
$997$	− 35.5481i	− 1.12582i	−0.826518	−	0.562910i	$-0.809682\pi$
$997$	− 35.5481i	− 1.12582i	0.826518	−	0.562910i	$-0.190318\pi$
$998$	33.3604i	1.05600i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.e.881.8		8
3.2	odd	2		4410.2.b.b.881.1		8
7.2	even	3		630.2.be.a.521.3	yes	8
7.3	odd	6		630.2.be.b.341.1	yes	8
7.6	odd	2		4410.2.b.b.881.8		8
21.2	odd	6		630.2.be.b.521.1	yes	8
21.17	even	6		630.2.be.a.341.3	✓	8
21.20	even	2	inner	4410.2.b.e.881.1		8
35.2	odd	12		3150.2.bp.d.899.2		8
35.3	even	12		3150.2.bp.c.1349.2		8
35.9	even	6		3150.2.bf.b.1151.2		8
35.17	even	12		3150.2.bp.f.1349.3		8
35.23	odd	12		3150.2.bp.a.899.3		8
35.24	odd	6		3150.2.bf.c.1601.4		8
105.2	even	12		3150.2.bp.c.899.2		8
105.17	odd	12		3150.2.bp.a.1349.3		8
105.23	even	12		3150.2.bp.f.899.3		8
105.38	odd	12		3150.2.bp.d.1349.2		8
105.44	odd	6		3150.2.bf.c.1151.4		8
105.59	even	6		3150.2.bf.b.1601.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.be.a.341.3	✓	8	21.17	even	6
630.2.be.a.521.3	yes	8	7.2	even	3
630.2.be.b.341.1	yes	8	7.3	odd	6
630.2.be.b.521.1	yes	8	21.2	odd	6
3150.2.bf.b.1151.2		8	35.9	even	6
3150.2.bf.b.1601.2		8	105.59	even	6
3150.2.bf.c.1151.4		8	105.44	odd	6
3150.2.bf.c.1601.4		8	35.24	odd	6
3150.2.bp.a.899.3		8	35.23	odd	12
3150.2.bp.a.1349.3		8	105.17	odd	12
3150.2.bp.c.899.2		8	105.2	even	12
3150.2.bp.c.1349.2		8	35.3	even	12
3150.2.bp.d.899.2		8	35.2	odd	12
3150.2.bp.d.1349.2		8	105.38	odd	12
3150.2.bp.f.899.3		8	105.23	even	12
3150.2.bp.f.1349.3		8	35.17	even	12
4410.2.b.b.881.1		8	3.2	odd	2
4410.2.b.b.881.8		8	7.6	odd	2
4410.2.b.e.881.1		8	21.20	even	2	inner
4410.2.b.e.881.8		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000i q^{8} +1.00000i q^{10} +5.39595i q^{11} -2.51764i q^{13} +1.00000 q^{16} -4.49938 q^{17} -2.86370i q^{19} -1.00000 q^{20} -5.39595 q^{22} +0.267949i q^{23} +1.00000 q^{25} +2.51764 q^{26} -8.89898i q^{29} -4.82843i q^{31} +1.00000i q^{32} -4.49938i q^{34} -6.51764 q^{37} +2.86370 q^{38} -1.00000i q^{40} +0.760279 q^{41} -5.86370 q^{43} -5.39595i q^{44} -0.267949 q^{46} -7.99536 q^{47} +1.00000i q^{50} +2.51764i q^{52} +8.39836i q^{53} +5.39595i q^{55} +8.89898 q^{58} -12.6715 q^{59} -2.62158i q^{61} +4.82843 q^{62} -1.00000 q^{64} -2.51764i q^{65} +9.82237 q^{67} +4.49938 q^{68} -4.76268i q^{71} -11.6569i q^{73} -6.51764i q^{74} +2.86370i q^{76} +8.59235 q^{79} +1.00000 q^{80} +0.760279i q^{82} +9.45001 q^{83} -4.49938 q^{85} -5.86370i q^{86} +5.39595 q^{88} +7.97005 q^{89} -0.267949i q^{92} -7.99536i q^{94} -2.86370i q^{95} -6.16353i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^4 + 8 * q^5	\( 8 q - 8 q^{4} + 8 q^{5} + 8 q^{16} - 8 q^{20} + 8 q^{25} + 16 q^{26} - 48 q^{37} - 8 q^{38} - 32 q^{41} - 16 q^{43} - 16 q^{46} - 16 q^{47} + 32 q^{58} - 48 q^{59} + 16 q^{62} - 8 q^{64} + 48 q^{67} + 48 q^{79} + 8 q^{80} + 16 q^{83} - 32 q^{89}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^5 + 8 * q^16 - 8 * q^20 + 8 * q^25 + 16 * q^26 - 48 * q^37 - 8 * q^38 - 32 * q^41 - 16 * q^43 - 16 * q^46 - 16 * q^47 + 32 * q^58 - 48 * q^59 + 16 * q^62 - 8 * q^64 + 48 * q^67 + 48 * q^79 + 8 * q^80 + 16 * q^83 - 32 * q^89

Introduction

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