LMFDB - Embedded newform 4410.2.b.a.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:	$\Q(\zeta_{16})$
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			881.8
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.a.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} -1.00000i q^{10} +3.84776i q^{11} +0.0273395i q^{13} +1.00000 q^{16} -3.71031 q^{17} -2.94495i q^{19} +1.00000 q^{20} -3.84776 q^{22} -9.05921i q^{23} +1.00000 q^{25} -0.0273395 q^{26} +9.28931i q^{29} -2.20692i q^{31} +1.00000i q^{32} -3.71031i q^{34} +3.00801 q^{37} +2.94495 q^{38} +1.00000i q^{40} -8.61313 q^{41} -3.28290 q^{43} -3.84776i q^{44} +9.05921 q^{46} +10.7927 q^{47} +1.00000i q^{50} -0.0273395i q^{52} -4.84935i q^{53} -3.84776i q^{55} -9.28931 q^{58} -4.04826 q^{59} -5.88348i q^{61} +2.20692 q^{62} -1.00000 q^{64} -0.0273395i q^{65} +10.6530 q^{67} +3.71031 q^{68} +1.14840i q^{71} -7.67778i q^{73} +3.00801i q^{74} +2.94495i q^{76} -5.65685 q^{79} -1.00000 q^{80} -8.61313i q^{82} +3.22625 q^{83} +3.71031 q^{85} -3.28290i q^{86} +3.84776 q^{88} -7.28093 q^{89} +9.05921i q^{92} +10.7927i q^{94} +2.94495i q^{95} +1.39104i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	3.84776i	1.16014i	0.814566	+	0.580072i	$0.196975\pi$
$11$	3.84776i	1.16014i	−0.814566	+	0.580072i	$0.803025\pi$
$12$	0	0
$13$	0.0273395i	0.00758261i	0.999993	+	0.00379131i	$0.00120681\pi$
$13$	0.0273395i	0.00758261i	−0.999993	+	0.00379131i	$0.998793\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.71031	−0.899883	−0.449942	−	0.893058i	$-0.648555\pi$
$17$	−3.71031	−0.899883	−0.449942	+	0.893058i	$0.648555\pi$
$18$	0	0
$19$	− 2.94495i	− 0.675617i	−0.941215	−	0.337809i	$-0.890314\pi$
$19$	− 2.94495i	− 0.675617i	0.941215	−	0.337809i	$-0.109686\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.84776	−0.820345
$23$	− 9.05921i	− 1.88898i	−0.328545	−	0.944488i	$-0.606558\pi$
$23$	− 9.05921i	− 1.88898i	0.328545	−	0.944488i	$-0.393442\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.0273395	−0.00536172
$27$	0	0
$28$	0	0
$29$	9.28931i	1.72498i	0.506072	+	0.862491i	$0.331097\pi$
$29$	9.28931i	1.72498i	−0.506072	+	0.862491i	$0.668903\pi$
$30$	0	0
$31$	− 2.20692i	− 0.396375i	−0.980164	−	0.198187i	$-0.936495\pi$
$31$	− 2.20692i	− 0.396375i	0.980164	−	0.198187i	$-0.0635054\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 3.71031i	− 0.636314i
$35$	0	0
$36$	0	0
$37$	3.00801	0.494513	0.247257	−	0.968950i	$-0.420471\pi$
$37$	3.00801	0.494513	0.247257	+	0.968950i	$0.420471\pi$
$38$	2.94495	0.477734
$39$	0	0
$40$	1.00000i	0.158114i
$41$	−8.61313	−1.34514	−0.672572	−	0.740032i	$-0.734811\pi$
$41$	−8.61313	−1.34514	−0.672572	+	0.740032i	$0.734811\pi$
$42$	0	0
$43$	−3.28290	−0.500637	−0.250319	−	0.968164i	$-0.580535\pi$
$43$	−3.28290	−0.500637	−0.250319	+	0.968164i	$0.580535\pi$
$44$	− 3.84776i	− 0.580072i
$45$	0	0
$46$	9.05921	1.33571
$47$	10.7927	1.57428	0.787139	−	0.616776i	$-0.211561\pi$
$47$	10.7927	1.57428	0.787139	+	0.616776i	$0.211561\pi$
$48$	0	0
$49$	0	0
$50$	1.00000i	0.141421i
$51$	0	0
$52$	− 0.0273395i	− 0.00379131i
$53$	− 4.84935i	− 0.666110i	−0.942907	−	0.333055i	$-0.891921\pi$
$53$	− 4.84935i	− 0.666110i	0.942907	−	0.333055i	$-0.108079\pi$
$54$	0	0
$55$	− 3.84776i	− 0.518832i
$56$	0	0
$57$	0	0
$58$	−9.28931	−1.21975
$59$	−4.04826	−0.527039	−0.263520	−	0.964654i	$-0.584883\pi$
$59$	−4.04826	−0.527039	−0.263520	+	0.964654i	$0.584883\pi$
$60$	0	0
$61$	− 5.88348i	− 0.753302i	−0.926355	−	0.376651i	$-0.877076\pi$
$61$	− 5.88348i	− 0.753302i	0.926355	−	0.376651i	$-0.122924\pi$
$62$	2.20692	0.280279
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 0.0273395i	− 0.00339105i
$66$	0	0
$67$	10.6530	1.30147	0.650736	−	0.759304i	$-0.274460\pi$
$67$	10.6530	1.30147	0.650736	+	0.759304i	$0.274460\pi$
$68$	3.71031	0.449942
$69$	0	0
$70$	0	0
$71$	1.14840i	0.136289i	0.997675	+	0.0681447i	$0.0217080\pi$
$71$	1.14840i	0.136289i	−0.997675	+	0.0681447i	$0.978292\pi$
$72$	0	0
$73$	− 7.67778i	− 0.898616i	−0.893377	−	0.449308i	$-0.851671\pi$
$73$	− 7.67778i	− 0.898616i	0.893377	−	0.449308i	$-0.148329\pi$
$74$	3.00801i	0.349674i
$75$	0	0
$76$	2.94495i	0.337809i
$77$	0	0
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	− 8.61313i	− 0.951161i
$83$	3.22625	0.354127	0.177064	−	0.984199i	$-0.443340\pi$
$83$	3.22625	0.354127	0.177064	+	0.984199i	$0.443340\pi$
$84$	0	0
$85$	3.71031	0.402440
$86$	− 3.28290i	− 0.354004i
$87$	0	0
$88$	3.84776	0.410172
$89$	−7.28093	−0.771777	−0.385889	−	0.922545i	$-0.626105\pi$
$89$	−7.28093	−0.771777	−0.385889	+	0.922545i	$0.626105\pi$
$90$	0	0
$91$	0	0
$92$	9.05921i	0.944488i
$93$	0	0
$94$	10.7927i	1.11318i
$95$	2.94495i	0.302145i
$96$	0	0
$97$	1.39104i	0.141238i	0.997503	+	0.0706192i	$0.0224975\pi$
$97$	1.39104i	0.141238i	−0.997503	+	0.0706192i	$0.977502\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	6.28130	0.625013	0.312507	−	0.949916i	$-0.398831\pi$
$101$	6.28130	0.625013	0.312507	+	0.949916i	$0.398831\pi$
$102$	0	0
$103$	− 7.87216i	− 0.775667i	−0.921730	−	0.387833i	$-0.873224\pi$
$103$	− 7.87216i	− 0.775667i	0.921730	−	0.387833i	$-0.126776\pi$
$104$	0.0273395	0.00268086
$105$	0	0
$106$	4.84935	0.471011
$107$	− 4.27905i	− 0.413672i	−0.978376	−	0.206836i	$-0.933683\pi$
$107$	− 4.27905i	− 0.413672i	0.978376	−	0.206836i	$-0.0663166\pi$
$108$	0	0
$109$	−2.01185	−0.192701	−0.0963503	−	0.995347i	$-0.530717\pi$
$109$	−2.01185	−0.192701	−0.0963503	+	0.995347i	$0.530717\pi$
$110$	3.84776	0.366869
$111$	0	0
$112$	0	0
$113$	4.59220i	0.431998i	0.976394	+	0.215999i	$0.0693008\pi$
$113$	4.59220i	0.431998i	−0.976394	+	0.215999i	$0.930699\pi$
$114$	0	0
$115$	9.05921i	0.844776i
$116$	− 9.28931i	− 0.862491i
$117$	0	0
$118$	− 4.04826i	− 0.372673i
$119$	0	0
$120$	0	0
$121$	−3.80525	−0.345932
$122$	5.88348	0.532665
$123$	0	0
$124$	2.20692i	0.198187i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.03866	0.180902	0.0904511	−	0.995901i	$-0.471169\pi$
$127$	2.03866	0.180902	0.0904511	+	0.995901i	$0.471169\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0.0273395	0.00239783
$131$	17.1644	1.49966	0.749831	−	0.661630i	$-0.230135\pi$
$131$	17.1644	1.49966	0.749831	+	0.661630i	$0.230135\pi$
$132$	0	0
$133$	0	0
$134$	10.6530i	0.920280i
$135$	0	0
$136$	3.71031i	0.318157i
$137$	13.7911i	1.17825i	0.808040	+	0.589127i	$0.200528\pi$
$137$	13.7911i	1.17825i	−0.808040	+	0.589127i	$0.799472\pi$
$138$	0	0
$139$	− 18.8735i	− 1.60083i	−0.599446	−	0.800415i	$-0.704613\pi$
$139$	− 18.8735i	− 1.60083i	0.599446	−	0.800415i	$-0.295387\pi$
$140$	0	0
$141$	0	0
$142$	−1.14840	−0.0963712
$143$	−0.105196	−0.00879691
$144$	0	0
$145$	− 9.28931i	− 0.771435i
$146$	7.67778	0.635418
$147$	0	0
$148$	−3.00801	−0.247257
$149$	− 12.8661i	− 1.05403i	−0.849856	−	0.527014i	$-0.823311\pi$
$149$	− 12.8661i	− 1.05403i	0.849856	−	0.527014i	$-0.176689\pi$
$150$	0	0
$151$	2.83803	0.230955	0.115478	−	0.993310i	$-0.463160\pi$
$151$	2.83803	0.230955	0.115478	+	0.993310i	$0.463160\pi$
$152$	−2.94495	−0.238867
$153$	0	0
$154$	0	0
$155$	2.20692i	0.177264i
$156$	0	0
$157$	− 17.0775i	− 1.36293i	−0.731850	−	0.681466i	$-0.761343\pi$
$157$	− 17.0775i	− 1.36293i	0.731850	−	0.681466i	$-0.238657\pi$
$158$	− 5.65685i	− 0.450035i
$159$	0	0
$160$	− 1.00000i	− 0.0790569i
$161$	0	0
$162$	0	0
$163$	16.0054	1.25364	0.626819	−	0.779165i	$-0.284357\pi$
$163$	16.0054	1.25364	0.626819	+	0.779165i	$0.284357\pi$
$164$	8.61313	0.672572
$165$	0	0
$166$	3.22625i	0.250406i
$167$	−15.6640	−1.21211	−0.606057	−	0.795421i	$-0.707250\pi$
$167$	−15.6640	−1.21211	−0.606057	+	0.795421i	$0.707250\pi$
$168$	0	0
$169$	12.9993	0.999943
$170$	3.71031i	0.284568i
$171$	0	0
$172$	3.28290	0.250319
$173$	23.8763	1.81528	0.907641	−	0.419746i	$-0.137881\pi$
$173$	23.8763	1.81528	0.907641	+	0.419746i	$0.137881\pi$
$174$	0	0
$175$	0	0
$176$	3.84776i	0.290036i
$177$	0	0
$178$	− 7.28093i	− 0.545729i
$179$	16.0759i	1.20157i	0.799411	+	0.600784i	$0.205145\pi$
$179$	16.0759i	1.20157i	−0.799411	+	0.600784i	$0.794855\pi$
$180$	0	0
$181$	− 23.7985i	− 1.76893i	−0.466611	−	0.884463i	$-0.654525\pi$
$181$	− 23.7985i	− 1.76893i	0.466611	−	0.884463i	$-0.345475\pi$
$182$	0	0
$183$	0	0
$184$	−9.05921	−0.667854
$185$	−3.00801	−0.221153
$186$	0	0
$187$	− 14.2764i	− 1.04399i
$188$	−10.7927	−0.787139
$189$	0	0
$190$	−2.94495	−0.213649
$191$	− 24.1248i	− 1.74561i	−0.488067	−	0.872806i	$-0.662298\pi$
$191$	− 24.1248i	− 1.74561i	0.488067	−	0.872806i	$-0.337702\pi$
$192$	0	0
$193$	18.8831	1.35924	0.679618	−	0.733566i	$-0.262146\pi$
$193$	18.8831	1.35924	0.679618	+	0.733566i	$0.262146\pi$
$194$	−1.39104	−0.0998706
$195$	0	0
$196$	0	0
$197$	14.5190i	1.03444i	0.855853	+	0.517219i	$0.173033\pi$
$197$	14.5190i	1.03444i	−0.855853	+	0.517219i	$0.826967\pi$
$198$	0	0
$199$	− 18.4888i	− 1.31063i	−0.755354	−	0.655316i	$-0.772535\pi$
$199$	− 18.4888i	− 1.31063i	0.755354	−	0.655316i	$-0.227465\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	6.28130i	0.441951i
$203$	0	0
$204$	0	0
$205$	8.61313	0.601567
$206$	7.87216	0.548479
$207$	0	0
$208$	0.0273395i	0.00189565i
$209$	11.3314	0.783813
$210$	0	0
$211$	10.5713	0.727757	0.363878	−	0.931446i	$-0.381452\pi$
$211$	10.5713	0.727757	0.363878	+	0.931446i	$0.381452\pi$
$212$	4.84935i	0.333055i
$213$	0	0
$214$	4.27905	0.292510
$215$	3.28290	0.223892
$216$	0	0
$217$	0	0
$218$	− 2.01185i	− 0.136260i
$219$	0	0
$220$	3.84776i	0.259416i
$221$	− 0.101438i	− 0.00682347i
$222$	0	0
$223$	9.99509i	0.669321i	0.942339	+	0.334660i	$0.108622\pi$
$223$	9.99509i	0.669321i	−0.942339	+	0.334660i	$0.891378\pi$
$224$	0	0
$225$	0	0
$226$	−4.59220	−0.305469
$227$	−7.62408	−0.506028	−0.253014	−	0.967463i	$-0.581422\pi$
$227$	−7.62408	−0.506028	−0.253014	+	0.967463i	$0.581422\pi$
$228$	0	0
$229$	− 26.2837i	− 1.73688i	−0.495795	−	0.868439i	$-0.665123\pi$
$229$	− 26.2837i	− 1.73688i	0.495795	−	0.868439i	$-0.334877\pi$
$230$	−9.05921	−0.597347
$231$	0	0
$232$	9.28931	0.609873
$233$	− 10.9545i	− 0.717656i	−0.933404	−	0.358828i	$-0.883176\pi$
$233$	− 10.9545i	− 0.717656i	0.933404	−	0.358828i	$-0.116824\pi$
$234$	0	0
$235$	−10.7927	−0.704039
$236$	4.04826	0.263520
$237$	0	0
$238$	0	0
$239$	4.74739i	0.307083i	0.988142	+	0.153541i	$0.0490678\pi$
$239$	4.74739i	0.307083i	−0.988142	+	0.153541i	$0.950932\pi$
$240$	0	0
$241$	− 15.4869i	− 0.997597i	−0.866718	−	0.498799i	$-0.833775\pi$
$241$	− 15.4869i	− 0.997597i	0.866718	−	0.498799i	$-0.166225\pi$
$242$	− 3.80525i	− 0.244611i
$243$	0	0
$244$	5.88348i	0.376651i
$245$	0	0
$246$	0	0
$247$	0.0805134	0.00512294
$248$	−2.20692	−0.140140
$249$	0	0
$250$	− 1.00000i	− 0.0632456i
$251$	6.99321	0.441408	0.220704	−	0.975341i	$-0.429165\pi$
$251$	6.99321	0.441408	0.220704	+	0.975341i	$0.429165\pi$
$252$	0	0
$253$	34.8577	2.19148
$254$	2.03866i	0.127917i
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.72445	0.294703	0.147352	−	0.989084i	$-0.452925\pi$
$257$	4.72445	0.294703	0.147352	+	0.989084i	$0.452925\pi$
$258$	0	0
$259$	0	0
$260$	0.0273395i	0.00169552i
$261$	0	0
$262$	17.1644i	1.06042i
$263$	6.28547i	0.387578i	0.981043	+	0.193789i	$0.0620778\pi$
$263$	6.28547i	0.387578i	−0.981043	+	0.193789i	$0.937922\pi$
$264$	0	0
$265$	4.84935i	0.297893i
$266$	0	0
$267$	0	0
$268$	−10.6530	−0.650736
$269$	28.3096	1.72607	0.863034	−	0.505146i	$-0.168561\pi$
$269$	28.3096	1.72607	0.863034	+	0.505146i	$0.168561\pi$
$270$	0	0
$271$	26.2079i	1.59202i	0.605286	+	0.796008i	$0.293059\pi$
$271$	26.2079i	1.59202i	−0.605286	+	0.796008i	$0.706941\pi$
$272$	−3.71031	−0.224971
$273$	0	0
$274$	−13.7911	−0.833152
$275$	3.84776i	0.232029i
$276$	0	0
$277$	−15.8793	−0.954092	−0.477046	−	0.878878i	$-0.658293\pi$
$277$	−15.8793	−0.954092	−0.477046	+	0.878878i	$0.658293\pi$
$278$	18.8735	1.13196
$279$	0	0
$280$	0	0
$281$	− 6.79936i	− 0.405616i	−0.979219	−	0.202808i	$-0.934993\pi$
$281$	− 6.79936i	− 0.405616i	0.979219	−	0.202808i	$-0.0650067\pi$
$282$	0	0
$283$	− 6.25587i	− 0.371873i	−0.982562	−	0.185937i	$-0.940468\pi$
$283$	− 6.25587i	− 0.371873i	0.982562	−	0.185937i	$-0.0595319\pi$
$284$	− 1.14840i	− 0.0681447i
$285$	0	0
$286$	− 0.105196i	− 0.00622036i
$287$	0	0
$288$	0	0
$289$	−3.23357	−0.190210
$290$	9.28931	0.545487
$291$	0	0
$292$	7.67778i	0.449308i
$293$	27.8004	1.62412	0.812058	−	0.583577i	$-0.198348\pi$
$293$	27.8004	1.62412	0.812058	+	0.583577i	$0.198348\pi$
$294$	0	0
$295$	4.04826	0.235699
$296$	− 3.00801i	− 0.174837i
$297$	0	0
$298$	12.8661	0.745311
$299$	0.247674	0.0143234
$300$	0	0
$301$	0	0
$302$	2.83803i	0.163310i
$303$	0	0
$304$	− 2.94495i	− 0.168904i
$305$	5.88348i	0.336887i
$306$	0	0
$307$	12.2027i	0.696445i	0.937412	+	0.348222i	$0.113215\pi$
$307$	12.2027i	0.696445i	−0.937412	+	0.348222i	$0.886785\pi$
$308$	0	0
$309$	0	0
$310$	−2.20692	−0.125345
$311$	5.30358	0.300738	0.150369	−	0.988630i	$-0.451954\pi$
$311$	5.30358	0.300738	0.150369	+	0.988630i	$0.451954\pi$
$312$	0	0
$313$	22.3170i	1.26143i	0.776015	+	0.630715i	$0.217238\pi$
$313$	22.3170i	1.26143i	−0.776015	+	0.630715i	$0.782762\pi$
$314$	17.0775	0.963738
$315$	0	0
$316$	5.65685	0.318223
$317$	15.9387i	0.895206i	0.894232	+	0.447603i	$0.147722\pi$
$317$	15.9387i	0.895206i	−0.894232	+	0.447603i	$0.852278\pi$
$318$	0	0
$319$	−35.7430	−2.00123
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	10.9267i	0.607977i
$324$	0	0
$325$	0.0273395i	0.00151652i
$326$	16.0054i	0.886456i
$327$	0	0
$328$	8.61313i	0.475580i
$329$	0	0
$330$	0	0
$331$	7.35653	0.404352	0.202176	−	0.979349i	$-0.435199\pi$
$331$	7.35653	0.404352	0.202176	+	0.979349i	$0.435199\pi$
$332$	−3.22625	−0.177064
$333$	0	0
$334$	− 15.6640i	− 0.857094i
$335$	−10.6530	−0.582036
$336$	0	0
$337$	1.22307	0.0666247	0.0333123	−	0.999445i	$-0.489394\pi$
$337$	1.22307	0.0666247	0.0333123	+	0.999445i	$0.489394\pi$
$338$	12.9993i	0.707066i
$339$	0	0
$340$	−3.71031	−0.201220
$341$	8.49170	0.459851
$342$	0	0
$343$	0	0
$344$	3.28290i	0.177002i
$345$	0	0
$346$	23.8763i	1.28360i
$347$	− 3.45659i	− 0.185560i	−0.995687	−	0.0927798i	$-0.970425\pi$
$347$	− 3.45659i	− 0.185560i	0.995687	−	0.0927798i	$-0.0295753\pi$
$348$	0	0
$349$	− 15.5590i	− 0.832857i	−0.909169	−	0.416428i	$-0.863282\pi$
$349$	− 15.5590i	− 0.832857i	0.909169	−	0.416428i	$-0.136718\pi$
$350$	0	0
$351$	0	0
$352$	−3.84776	−0.205086
$353$	−28.9153	−1.53901	−0.769503	−	0.638643i	$-0.779496\pi$
$353$	−28.9153	−1.53901	−0.769503	+	0.638643i	$0.779496\pi$
$354$	0	0
$355$	− 1.14840i	− 0.0609505i
$356$	7.28093	0.385889
$357$	0	0
$358$	−16.0759	−0.849637
$359$	− 22.2882i	− 1.17633i	−0.808741	−	0.588164i	$-0.799851\pi$
$359$	− 22.2882i	− 1.17633i	0.808741	−	0.588164i	$-0.200149\pi$
$360$	0	0
$361$	10.3273	0.543541
$362$	23.7985	1.25082
$363$	0	0
$364$	0	0
$365$	7.67778i	0.401873i
$366$	0	0
$367$	− 0.938533i	− 0.0489910i	−0.999700	−	0.0244955i	$-0.992202\pi$
$367$	− 0.938533i	− 0.0489910i	0.999700	−	0.0244955i	$-0.00779794\pi$
$368$	− 9.05921i	− 0.472244i
$369$	0	0
$370$	− 3.00801i	− 0.156379i
$371$	0	0
$372$	0	0
$373$	−19.2173	−0.995036	−0.497518	−	0.867454i	$-0.665755\pi$
$373$	−19.2173	−0.995036	−0.497518	+	0.867454i	$0.665755\pi$
$374$	14.2764	0.738215
$375$	0	0
$376$	− 10.7927i	− 0.556591i
$377$	−0.253965	−0.0130799
$378$	0	0
$379$	−30.1944	−1.55098	−0.775490	−	0.631359i	$-0.782497\pi$
$379$	−30.1944	−1.55098	−0.775490	+	0.631359i	$0.782497\pi$
$380$	− 2.94495i	− 0.151073i
$381$	0	0
$382$	24.1248	1.23433
$383$	−24.8129	−1.26788	−0.633940	−	0.773383i	$-0.718563\pi$
$383$	−24.8129	−1.26788	−0.633940	+	0.773383i	$0.718563\pi$
$384$	0	0
$385$	0	0
$386$	18.8831i	0.961125i
$387$	0	0
$388$	− 1.39104i	− 0.0706192i
$389$	15.4196i	0.781804i	0.920432	+	0.390902i	$0.127837\pi$
$389$	15.4196i	0.781804i	−0.920432	+	0.390902i	$0.872163\pi$
$390$	0	0
$391$	33.6125i	1.69986i
$392$	0	0
$393$	0	0
$394$	−14.5190	−0.731458
$395$	5.65685	0.284627
$396$	0	0
$397$	− 36.6789i	− 1.84086i	−0.390905	−	0.920431i	$-0.627838\pi$
$397$	− 36.6789i	− 1.84086i	0.390905	−	0.920431i	$-0.372162\pi$
$398$	18.4888	0.926757
$399$	0	0
$400$	1.00000	0.0500000
$401$	7.70193i	0.384616i	0.981335	+	0.192308i	$0.0615973\pi$
$401$	7.70193i	0.384616i	−0.981335	+	0.192308i	$0.938403\pi$
$402$	0	0
$403$	0.0603361	0.00300555
$404$	−6.28130	−0.312507
$405$	0	0
$406$	0	0
$407$	11.5741i	0.573706i
$408$	0	0
$409$	35.7056i	1.76553i	0.469817	+	0.882764i	$0.344320\pi$
$409$	35.7056i	1.76553i	−0.469817	+	0.882764i	$0.655680\pi$
$410$	8.61313i	0.425372i
$411$	0	0
$412$	7.87216i	0.387833i
$413$	0	0
$414$	0	0
$415$	−3.22625	−0.158370
$416$	−0.0273395	−0.00134043
$417$	0	0
$418$	11.3314i	0.554239i
$419$	−18.9114	−0.923881	−0.461941	−	0.886911i	$-0.652847\pi$
$419$	−18.9114	−0.923881	−0.461941	+	0.886911i	$0.652847\pi$
$420$	0	0
$421$	25.8704	1.26084	0.630422	−	0.776253i	$-0.282882\pi$
$421$	25.8704	1.26084	0.630422	+	0.776253i	$0.282882\pi$
$422$	10.5713i	0.514602i
$423$	0	0
$424$	−4.84935	−0.235505
$425$	−3.71031	−0.179977
$426$	0	0
$427$	0	0
$428$	4.27905i	0.206836i
$429$	0	0
$430$	3.28290i	0.158315i
$431$	1.62355i	0.0782036i	0.999235	+	0.0391018i	$0.0124497\pi$
$431$	1.62355i	0.0782036i	−0.999235	+	0.0391018i	$0.987550\pi$
$432$	0	0
$433$	− 19.6593i	− 0.944765i	−0.881394	−	0.472383i	$-0.843394\pi$
$433$	− 19.6593i	− 0.944765i	0.881394	−	0.472383i	$-0.156606\pi$
$434$	0	0
$435$	0	0
$436$	2.01185	0.0963503
$437$	−26.6789	−1.27623
$438$	0	0
$439$	− 8.74410i	− 0.417333i	−0.977987	−	0.208667i	$-0.933088\pi$
$439$	− 8.74410i	− 0.417333i	0.977987	−	0.208667i	$-0.0669123\pi$
$440$	−3.84776	−0.183435
$441$	0	0
$442$	0.101438	0.00482492
$443$	− 20.7789i	− 0.987235i	−0.869679	−	0.493617i	$-0.835674\pi$
$443$	− 20.7789i	− 0.987235i	0.869679	−	0.493617i	$-0.164326\pi$
$444$	0	0
$445$	7.28093	0.345149
$446$	−9.99509	−0.473281
$447$	0	0
$448$	0	0
$449$	− 40.3798i	− 1.90564i	−0.303536	−	0.952820i	$-0.598167\pi$
$449$	− 40.3798i	− 1.90564i	0.303536	−	0.952820i	$-0.401833\pi$
$450$	0	0
$451$	− 33.1412i	− 1.56056i
$452$	− 4.59220i	− 0.215999i
$453$	0	0
$454$	− 7.62408i	− 0.357816i
$455$	0	0
$456$	0	0
$457$	1.11026	0.0519358	0.0259679	−	0.999663i	$-0.491733\pi$
$457$	1.11026	0.0519358	0.0259679	+	0.999663i	$0.491733\pi$
$458$	26.2837	1.22816
$459$	0	0
$460$	− 9.05921i	− 0.422388i
$461$	−12.9378	−0.602573	−0.301286	−	0.953534i	$-0.597416\pi$
$461$	−12.9378	−0.602573	−0.301286	+	0.953534i	$0.597416\pi$
$462$	0	0
$463$	16.3501	0.759853	0.379927	−	0.925017i	$-0.375949\pi$
$463$	16.3501	0.759853	0.379927	+	0.925017i	$0.375949\pi$
$464$	9.28931i	0.431245i
$465$	0	0
$466$	10.9545	0.507460
$467$	32.2278	1.49132	0.745662	−	0.666324i	$-0.232133\pi$
$467$	32.2278	1.49132	0.745662	+	0.666324i	$0.232133\pi$
$468$	0	0
$469$	0	0
$470$	− 10.7927i	− 0.497830i
$471$	0	0
$472$	4.04826i	0.186336i
$473$	− 12.6318i	− 0.580811i
$474$	0	0
$475$	− 2.94495i	− 0.135123i
$476$	0	0
$477$	0	0
$478$	−4.74739	−0.217140
$479$	23.9665	1.09506	0.547528	−	0.836787i	$-0.315569\pi$
$479$	23.9665	1.09506	0.547528	+	0.836787i	$0.315569\pi$
$480$	0	0
$481$	0.0822374i	0.00374970i
$482$	15.4869	0.705408
$483$	0	0
$484$	3.80525	0.172966
$485$	− 1.39104i	− 0.0631637i
$486$	0	0
$487$	−16.4288	−0.744460	−0.372230	−	0.928141i	$-0.621407\pi$
$487$	−16.4288	−0.744460	−0.372230	+	0.928141i	$0.621407\pi$
$488$	−5.88348	−0.266333
$489$	0	0
$490$	0	0
$491$	− 24.5246i	− 1.10678i	−0.832922	−	0.553391i	$-0.813334\pi$
$491$	− 24.5246i	− 1.10678i	0.832922	−	0.553391i	$-0.186666\pi$
$492$	0	0
$493$	− 34.4663i	− 1.55228i
$494$	0.0805134i	0.00362247i
$495$	0	0
$496$	− 2.20692i	− 0.0990936i
$497$	0	0
$498$	0	0
$499$	−31.5492	−1.41234	−0.706168	−	0.708045i	$-0.749578\pi$
$499$	−31.5492	−1.41234	−0.706168	+	0.708045i	$0.749578\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	6.99321i	0.312122i
$503$	40.0883	1.78745	0.893725	−	0.448615i	$-0.148082\pi$
$503$	40.0883	1.78745	0.893725	+	0.448615i	$0.148082\pi$
$504$	0	0
$505$	−6.28130	−0.279514
$506$	34.8577i	1.54961i
$507$	0	0
$508$	−2.03866	−0.0904511
$509$	−32.1350	−1.42436	−0.712179	−	0.701998i	$-0.752291\pi$
$509$	−32.1350	−1.42436	−0.712179	+	0.701998i	$0.752291\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	4.72445i	0.208387i
$515$	7.87216i	0.346889i
$516$	0	0
$517$	41.5277i	1.82639i
$518$	0	0
$519$	0	0
$520$	−0.0273395	−0.00119892
$521$	−37.7822	−1.65527	−0.827635	−	0.561267i	$-0.810314\pi$
$521$	−37.7822	−1.65527	−0.827635	+	0.561267i	$0.810314\pi$
$522$	0	0
$523$	35.9674i	1.57274i	0.617753	+	0.786372i	$0.288043\pi$
$523$	35.9674i	1.57274i	−0.617753	+	0.786372i	$0.711957\pi$
$524$	−17.1644	−0.749831
$525$	0	0
$526$	−6.28547	−0.274059
$527$	8.18837i	0.356691i
$528$	0	0
$529$	−59.0694	−2.56823
$530$	−4.84935	−0.210642
$531$	0	0
$532$	0	0
$533$	− 0.235478i	− 0.0101997i
$534$	0	0
$535$	4.27905i	0.185000i
$536$	− 10.6530i	− 0.460140i
$537$	0	0
$538$	28.3096i	1.22051i
$539$	0	0
$540$	0	0
$541$	−1.86308	−0.0801002	−0.0400501	−	0.999198i	$-0.512752\pi$
$541$	−1.86308	−0.0801002	−0.0400501	+	0.999198i	$0.512752\pi$
$542$	−26.2079	−1.12573
$543$	0	0
$544$	− 3.71031i	− 0.159078i
$545$	2.01185	0.0861783
$546$	0	0
$547$	6.75577	0.288856	0.144428	−	0.989515i	$-0.453866\pi$
$547$	6.75577	0.288856	0.144428	+	0.989515i	$0.453866\pi$
$548$	− 13.7911i	− 0.589127i
$549$	0	0
$550$	−3.84776	−0.164069
$551$	27.3565	1.16543
$552$	0	0
$553$	0	0
$554$	− 15.8793i	− 0.674645i
$555$	0	0
$556$	18.8735i	0.800415i
$557$	22.3592i	0.947388i	0.880689	+	0.473694i	$0.157080\pi$
$557$	22.3592i	0.947388i	−0.880689	+	0.473694i	$0.842920\pi$
$558$	0	0
$559$	− 0.0897527i	− 0.00379614i
$560$	0	0
$561$	0	0
$562$	6.79936	0.286814
$563$	−21.8185	−0.919538	−0.459769	−	0.888039i	$-0.652068\pi$
$563$	−21.8185	−0.919538	−0.459769	+	0.888039i	$0.652068\pi$
$564$	0	0
$565$	− 4.59220i	− 0.193195i
$566$	6.25587	0.262954
$567$	0	0
$568$	1.14840	0.0481856
$569$	− 23.7566i	− 0.995929i	−0.867197	−	0.497964i	$-0.834081\pi$
$569$	− 23.7566i	− 0.995929i	0.867197	−	0.497964i	$-0.165919\pi$
$570$	0	0
$571$	1.94214	0.0812758	0.0406379	−	0.999174i	$-0.487061\pi$
$571$	1.94214	0.0812758	0.0406379	+	0.999174i	$0.487061\pi$
$572$	0.105196	0.00439846
$573$	0	0
$574$	0	0
$575$	− 9.05921i	− 0.377795i
$576$	0	0
$577$	− 12.6395i	− 0.526191i	−0.964770	−	0.263095i	$-0.915257\pi$
$577$	− 12.6395i	− 0.526191i	0.964770	−	0.263095i	$-0.0847434\pi$
$578$	− 3.23357i	− 0.134499i
$579$	0	0
$580$	9.28931i	0.385718i
$581$	0	0
$582$	0	0
$583$	18.6591	0.772783
$584$	−7.67778	−0.317709
$585$	0	0
$586$	27.8004i	1.14842i
$587$	−7.52034	−0.310398	−0.155199	−	0.987883i	$-0.549602\pi$
$587$	−7.52034	−0.310398	−0.155199	+	0.987883i	$0.549602\pi$
$588$	0	0
$589$	−6.49926	−0.267797
$590$	4.04826i	0.166664i
$591$	0	0
$592$	3.00801	0.123628
$593$	−25.1359	−1.03221	−0.516103	−	0.856527i	$-0.672618\pi$
$593$	−25.1359	−1.03221	−0.516103	+	0.856527i	$0.672618\pi$
$594$	0	0
$595$	0	0
$596$	12.8661i	0.527014i
$597$	0	0
$598$	0.247674i	0.0101282i
$599$	45.8246i	1.87234i	0.351544	+	0.936171i	$0.385657\pi$
$599$	45.8246i	1.87234i	−0.351544	+	0.936171i	$0.614343\pi$
$600$	0	0
$601$	18.9060i	0.771193i	0.922667	+	0.385597i	$0.126004\pi$
$601$	18.9060i	0.771193i	−0.922667	+	0.385597i	$0.873996\pi$
$602$	0	0
$603$	0	0
$604$	−2.83803	−0.115478
$605$	3.80525	0.154705
$606$	0	0
$607$	36.2609i	1.47178i	0.677098	+	0.735892i	$0.263237\pi$
$607$	36.2609i	1.47178i	−0.677098	+	0.735892i	$0.736763\pi$
$608$	2.94495	0.119433
$609$	0	0
$610$	−5.88348	−0.238215
$611$	0.295067i	0.0119371i
$612$	0	0
$613$	21.2278	0.857383	0.428692	−	0.903451i	$-0.358975\pi$
$613$	21.2278	0.857383	0.428692	+	0.903451i	$0.358975\pi$
$614$	−12.2027	−0.492461
$615$	0	0
$616$	0	0
$617$	− 42.7282i	− 1.72017i	−0.510150	−	0.860085i	$-0.670410\pi$
$617$	− 42.7282i	− 1.72017i	0.510150	−	0.860085i	$-0.329590\pi$
$618$	0	0
$619$	− 5.80615i	− 0.233369i	−0.993169	−	0.116684i	$-0.962773\pi$
$619$	− 5.80615i	− 0.233369i	0.993169	−	0.116684i	$-0.0372266\pi$
$620$	− 2.20692i	− 0.0886320i
$621$	0	0
$622$	5.30358i	0.212654i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.3170	−0.891965
$627$	0	0
$628$	17.0775i	0.681466i
$629$	−11.1607	−0.445004
$630$	0	0
$631$	−1.55980	−0.0620945	−0.0310473	−	0.999518i	$-0.509884\pi$
$631$	−1.55980	−0.0620945	−0.0310473	+	0.999518i	$0.509884\pi$
$632$	5.65685i	0.225018i
$633$	0	0
$634$	−15.9387	−0.633006
$635$	−2.03866	−0.0809019
$636$	0	0
$637$	0	0
$638$	− 35.7430i	− 1.41508i
$639$	0	0
$640$	1.00000i	0.0395285i
$641$	− 33.4044i	− 1.31939i	−0.751532	−	0.659697i	$-0.770685\pi$
$641$	− 33.4044i	− 1.31939i	0.751532	−	0.659697i	$-0.229315\pi$
$642$	0	0
$643$	7.66154i	0.302142i	0.988523	+	0.151071i	$0.0482722\pi$
$643$	7.66154i	0.302142i	−0.988523	+	0.151071i	$0.951728\pi$
$644$	0	0
$645$	0	0
$646$	−10.9267	−0.429904
$647$	7.21880	0.283800	0.141900	−	0.989881i	$-0.454679\pi$
$647$	7.21880	0.283800	0.141900	+	0.989881i	$0.454679\pi$
$648$	0	0
$649$	− 15.5767i	− 0.611441i
$650$	−0.0273395	−0.00107234
$651$	0	0
$652$	−16.0054	−0.626819
$653$	10.5458i	0.412691i	0.978479	+	0.206345i	$0.0661571\pi$
$653$	10.5458i	0.412691i	−0.978479	+	0.206345i	$0.933843\pi$
$654$	0	0
$655$	−17.1644	−0.670669
$656$	−8.61313	−0.336286
$657$	0	0
$658$	0	0
$659$	43.9724i	1.71292i	0.516213	+	0.856460i	$0.327341\pi$
$659$	43.9724i	1.71292i	−0.516213	+	0.856460i	$0.672659\pi$
$660$	0	0
$661$	16.8450i	0.655193i	0.944818	+	0.327597i	$0.106239\pi$
$661$	16.8450i	0.655193i	−0.944818	+	0.327597i	$0.893761\pi$
$662$	7.35653i	0.285920i
$663$	0	0
$664$	− 3.22625i	− 0.125203i
$665$	0	0
$666$	0	0
$667$	84.1539	3.25845
$668$	15.6640	0.606057
$669$	0	0
$670$	− 10.6530i	− 0.411562i
$671$	22.6382	0.873939
$672$	0	0
$673$	25.4844	0.982351	0.491175	−	0.871061i	$-0.336567\pi$
$673$	25.4844	0.982351	0.491175	+	0.871061i	$0.336567\pi$
$674$	1.22307i	0.0471108i
$675$	0	0
$676$	−12.9993	−0.499971
$677$	−21.8931	−0.841419	−0.420710	−	0.907195i	$-0.638219\pi$
$677$	−21.8931	−0.841419	−0.420710	+	0.907195i	$0.638219\pi$
$678$	0	0
$679$	0	0
$680$	− 3.71031i	− 0.142284i
$681$	0	0
$682$	8.49170i	0.325164i
$683$	− 13.9153i	− 0.532455i	−0.963910	−	0.266228i	$-0.914223\pi$
$683$	− 13.9153i	− 0.532455i	0.963910	−	0.266228i	$-0.0857773\pi$
$684$	0	0
$685$	− 13.7911i	− 0.526931i
$686$	0	0
$687$	0	0
$688$	−3.28290	−0.125159
$689$	0.132579	0.00505085
$690$	0	0
$691$	− 26.6579i	− 1.01411i	−0.861913	−	0.507056i	$-0.830734\pi$
$691$	− 26.6579i	− 1.01411i	0.861913	−	0.507056i	$-0.169266\pi$
$692$	−23.8763	−0.907641
$693$	0	0
$694$	3.45659	0.131210
$695$	18.8735i	0.715913i
$696$	0	0
$697$	31.9574	1.21047
$698$	15.5590	0.588919
$699$	0	0
$700$	0	0
$701$	− 43.6610i	− 1.64905i	−0.565824	−	0.824526i	$-0.691442\pi$
$701$	− 43.6610i	− 1.64905i	0.565824	−	0.824526i	$-0.308558\pi$
$702$	0	0
$703$	− 8.85842i	− 0.334102i
$704$	− 3.84776i	− 0.145018i
$705$	0	0
$706$	− 28.9153i	− 1.08824i
$707$	0	0
$708$	0	0
$709$	−1.12252	−0.0421570	−0.0210785	−	0.999778i	$-0.506710\pi$
$709$	−1.12252	−0.0421570	−0.0210785	+	0.999778i	$0.506710\pi$
$710$	1.14840	0.0430985
$711$	0	0
$712$	7.28093i	0.272864i
$713$	−19.9930	−0.748742
$714$	0	0
$715$	0.105196	0.00393410
$716$	− 16.0759i	− 0.600784i
$717$	0	0
$718$	22.2882	0.831790
$719$	−46.9151	−1.74964	−0.874818	−	0.484451i	$-0.839019\pi$
$719$	−46.9151	−1.74964	−0.874818	+	0.484451i	$0.839019\pi$
$720$	0	0
$721$	0	0
$722$	10.3273i	0.384342i
$723$	0	0
$724$	23.7985i	0.884463i
$725$	9.28931i	0.344996i
$726$	0	0
$727$	− 37.6397i	− 1.39598i	−0.716107	−	0.697991i	$-0.754078\pi$
$727$	− 37.6397i	− 1.39598i	0.716107	−	0.697991i	$-0.245922\pi$
$728$	0	0
$729$	0	0
$730$	−7.67778	−0.284167
$731$	12.1806	0.450515
$732$	0	0
$733$	33.4767i	1.23649i	0.785986	+	0.618245i	$0.212156\pi$
$733$	33.4767i	1.23649i	−0.785986	+	0.618245i	$0.787844\pi$
$734$	0.938533	0.0346419
$735$	0	0
$736$	9.05921	0.333927
$737$	40.9902i	1.50989i
$738$	0	0
$739$	−40.9642	−1.50689	−0.753446	−	0.657510i	$-0.771610\pi$
$739$	−40.9642	−1.50689	−0.753446	+	0.657510i	$0.771610\pi$
$740$	3.00801	0.110577
$741$	0	0
$742$	0	0
$743$	− 2.11843i	− 0.0777177i	−0.999245	−	0.0388588i	$-0.987628\pi$
$743$	− 2.11843i	− 0.0777177i	0.999245	−	0.0388588i	$-0.0123723\pi$
$744$	0	0
$745$	12.8661i	0.471376i
$746$	− 19.2173i	− 0.703597i
$747$	0	0
$748$	14.2764i	0.521997i
$749$	0	0
$750$	0	0
$751$	5.11292	0.186573	0.0932865	−	0.995639i	$-0.470263\pi$
$751$	5.11292	0.186573	0.0932865	+	0.995639i	$0.470263\pi$
$752$	10.7927	0.393570
$753$	0	0
$754$	− 0.253965i	− 0.00924886i
$755$	−2.83803	−0.103286
$756$	0	0
$757$	1.87828	0.0682674	0.0341337	−	0.999417i	$-0.489133\pi$
$757$	1.87828	0.0682674	0.0341337	+	0.999417i	$0.489133\pi$
$758$	− 30.1944i	− 1.09671i
$759$	0	0
$760$	2.94495	0.106824
$761$	−33.7339	−1.22285	−0.611427	−	0.791301i	$-0.709404\pi$
$761$	−33.7339	−1.22285	−0.611427	+	0.791301i	$0.709404\pi$
$762$	0	0
$763$	0	0
$764$	24.1248i	0.872806i
$765$	0	0
$766$	− 24.8129i	− 0.896526i
$767$	− 0.110677i	− 0.00399633i
$768$	0	0
$769$	− 24.5939i	− 0.886878i	−0.896305	−	0.443439i	$-0.853758\pi$
$769$	− 24.5939i	− 0.886878i	0.896305	−	0.443439i	$-0.146242\pi$
$770$	0	0
$771$	0	0
$772$	−18.8831	−0.679618
$773$	−47.0336	−1.69168	−0.845840	−	0.533437i	$-0.820900\pi$
$773$	−47.0336	−1.69168	−0.845840	+	0.533437i	$0.820900\pi$
$774$	0	0
$775$	− 2.20692i	− 0.0792749i
$776$	1.39104	0.0499353
$777$	0	0
$778$	−15.4196	−0.552819
$779$	25.3652i	0.908803i
$780$	0	0
$781$	−4.41875	−0.158115
$782$	−33.6125	−1.20198
$783$	0	0
$784$	0	0
$785$	17.0775i	0.609521i
$786$	0	0
$787$	38.4685i	1.37125i	0.727953	+	0.685627i	$0.240472\pi$
$787$	38.4685i	1.37125i	−0.727953	+	0.685627i	$0.759528\pi$
$788$	− 14.5190i	− 0.517219i
$789$	0	0
$790$	5.65685i	0.201262i
$791$	0	0
$792$	0	0
$793$	0.160851	0.00571200
$794$	36.6789	1.30169
$795$	0	0
$796$	18.4888i	0.655316i
$797$	−23.3415	−0.826798	−0.413399	−	0.910550i	$-0.635659\pi$
$797$	−23.3415	−0.826798	−0.413399	+	0.910550i	$0.635659\pi$
$798$	0	0
$799$	−40.0443	−1.41667
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	−7.70193	−0.271965
$803$	29.5422	1.04252
$804$	0	0
$805$	0	0
$806$	0.0603361i	0.00212525i
$807$	0	0
$808$	− 6.28130i	− 0.220976i
$809$	− 10.3586i	− 0.364190i	−0.983281	−	0.182095i	$-0.941712\pi$
$809$	− 10.3586i	− 0.364190i	0.983281	−	0.182095i	$-0.0582879\pi$
$810$	0	0
$811$	2.70766i	0.0950790i	0.998869	+	0.0475395i	$0.0151380\pi$
$811$	2.70766i	0.0950790i	−0.998869	+	0.0475395i	$0.984862\pi$
$812$	0	0
$813$	0	0
$814$	−11.5741	−0.405672
$815$	−16.0054	−0.560644
$816$	0	0
$817$	9.66796i	0.338239i
$818$	−35.7056	−1.24842
$819$	0	0
$820$	−8.61313	−0.300783
$821$	− 34.8688i	− 1.21693i	−0.793580	−	0.608465i	$-0.791785\pi$
$821$	− 34.8688i	− 1.21693i	0.793580	−	0.608465i	$-0.208215\pi$
$822$	0	0
$823$	45.7481	1.59468	0.797339	−	0.603532i	$-0.206240\pi$
$823$	45.7481	1.59468	0.797339	+	0.603532i	$0.206240\pi$
$824$	−7.87216	−0.274240
$825$	0	0
$826$	0	0
$827$	− 48.8991i	− 1.70039i	−0.526469	−	0.850195i	$-0.676484\pi$
$827$	− 48.8991i	− 1.70039i	0.526469	−	0.850195i	$-0.323516\pi$
$828$	0	0
$829$	13.9556i	0.484696i	0.970189	+	0.242348i	$0.0779176\pi$
$829$	13.9556i	0.484696i	−0.970189	+	0.242348i	$0.922082\pi$
$830$	− 3.22625i	− 0.111985i
$831$	0	0
$832$	− 0.0273395i	0 0.000947826i
$833$	0	0
$834$	0	0
$835$	15.6640	0.542074
$836$	−11.3314	−0.391906
$837$	0	0
$838$	− 18.9114i	− 0.653283i
$839$	15.5492	0.536819	0.268409	−	0.963305i	$-0.413502\pi$
$839$	15.5492	0.536819	0.268409	+	0.963305i	$0.413502\pi$
$840$	0	0
$841$	−57.2913	−1.97556
$842$	25.8704i	0.891551i
$843$	0	0
$844$	−10.5713	−0.363878
$845$	−12.9993	−0.447188
$846$	0	0
$847$	0	0
$848$	− 4.84935i	− 0.166527i
$849$	0	0
$850$	− 3.71031i	− 0.127263i
$851$	− 27.2502i	− 0.934124i
$852$	0	0
$853$	17.5404i	0.600571i	0.953849	+	0.300286i	$0.0970820\pi$
$853$	17.5404i	0.600571i	−0.953849	+	0.300286i	$0.902918\pi$
$854$	0	0
$855$	0	0
$856$	−4.27905	−0.146255
$857$	16.7269	0.571380	0.285690	−	0.958322i	$-0.407777\pi$
$857$	16.7269	0.571380	0.285690	+	0.958322i	$0.407777\pi$
$858$	0	0
$859$	22.2072i	0.757699i	0.925458	+	0.378849i	$0.123680\pi$
$859$	22.2072i	0.757699i	−0.925458	+	0.378849i	$0.876320\pi$
$860$	−3.28290	−0.111946
$861$	0	0
$862$	−1.62355	−0.0552983
$863$	55.2949i	1.88226i	0.338045	+	0.941130i	$0.390234\pi$
$863$	55.2949i	1.88226i	−0.338045	+	0.941130i	$0.609766\pi$
$864$	0	0
$865$	−23.8763	−0.811819
$866$	19.6593	0.668050
$867$	0	0
$868$	0	0
$869$	− 21.7662i	− 0.738368i
$870$	0	0
$871$	0.291248i	0.00986856i
$872$	2.01185i	0.0681299i
$873$	0	0
$874$	− 26.6789i	− 0.902428i
$875$	0	0
$876$	0	0
$877$	−38.9596	−1.31557	−0.657786	−	0.753205i	$-0.728507\pi$
$877$	−38.9596	−1.31557	−0.657786	+	0.753205i	$0.728507\pi$
$878$	8.74410	0.295099
$879$	0	0
$880$	− 3.84776i	− 0.129708i
$881$	14.3292	0.482762	0.241381	−	0.970430i	$-0.422400\pi$
$881$	14.3292	0.482762	0.241381	+	0.970430i	$0.422400\pi$
$882$	0	0
$883$	−21.9036	−0.737115	−0.368557	−	0.929605i	$-0.620148\pi$
$883$	−21.9036	−0.737115	−0.368557	+	0.929605i	$0.620148\pi$
$884$	0.101438i	0.00341173i
$885$	0	0
$886$	20.7789	0.698080
$887$	34.1440	1.14644	0.573221	−	0.819400i	$-0.305693\pi$
$887$	34.1440	1.14644	0.573221	+	0.819400i	$0.305693\pi$
$888$	0	0
$889$	0	0
$890$	7.28093i	0.244057i
$891$	0	0
$892$	− 9.99509i	− 0.334660i
$893$	− 31.7840i	− 1.06361i
$894$	0	0
$895$	− 16.0759i	− 0.537358i
$896$	0	0
$897$	0	0
$898$	40.3798	1.34749
$899$	20.5008	0.683739
$900$	0	0
$901$	17.9926i	0.599421i
$902$	33.1412	1.10348
$903$	0	0
$904$	4.59220	0.152734
$905$	23.7985i	0.791088i
$906$	0	0
$907$	−9.05534	−0.300678	−0.150339	−	0.988635i	$-0.548036\pi$
$907$	−9.05534	−0.300678	−0.150339	+	0.988635i	$0.548036\pi$
$908$	7.62408	0.253014
$909$	0	0
$910$	0	0
$911$	− 16.1266i	− 0.534300i	−0.963655	−	0.267150i	$-0.913918\pi$
$911$	− 16.1266i	− 0.534300i	0.963655	−	0.267150i	$-0.0860819\pi$
$912$	0	0
$913$	12.4138i	0.410838i
$914$	1.11026i	0.0367241i
$915$	0	0
$916$	26.2837i	0.868439i
$917$	0	0
$918$	0	0
$919$	52.6798	1.73775	0.868873	−	0.495036i	$-0.164845\pi$
$919$	52.6798	1.73775	0.868873	+	0.495036i	$0.164845\pi$
$920$	9.05921	0.298673
$921$	0	0
$922$	− 12.9378i	− 0.426083i
$923$	−0.0313966	−0.00103343
$924$	0	0
$925$	3.00801	0.0989027
$926$	16.3501i	0.537297i
$927$	0	0
$928$	−9.28931	−0.304937
$929$	−28.7343	−0.942743	−0.471372	−	0.881935i	$-0.656241\pi$
$929$	−28.7343	−0.942743	−0.471372	+	0.881935i	$0.656241\pi$
$930$	0	0
$931$	0	0
$932$	10.9545i	0.358828i
$933$	0	0
$934$	32.2278i	1.05453i
$935$	14.2764i	0.466888i
$936$	0	0
$937$	19.1607i	0.625953i	0.949761	+	0.312976i	$0.101326\pi$
$937$	19.1607i	0.625953i	−0.949761	+	0.312976i	$0.898674\pi$
$938$	0	0
$939$	0	0
$940$	10.7927	0.352019
$941$	33.2028	1.08238	0.541190	−	0.840900i	$-0.317974\pi$
$941$	33.2028	1.08238	0.541190	+	0.840900i	$0.317974\pi$
$942$	0	0
$943$	78.0282i	2.54095i
$944$	−4.04826	−0.131760
$945$	0	0
$946$	12.6318	0.410695
$947$	− 33.3945i	− 1.08517i	−0.840000	−	0.542587i	$-0.817445\pi$
$947$	− 33.3945i	− 1.08517i	0.840000	−	0.542587i	$-0.182555\pi$
$948$	0	0
$949$	0.209907	0.00681386
$950$	2.94495	0.0955467
$951$	0	0
$952$	0	0
$953$	− 25.2742i	− 0.818711i	−0.912375	−	0.409355i	$-0.865754\pi$
$953$	− 25.2742i	− 0.818711i	0.912375	−	0.409355i	$-0.134246\pi$
$954$	0	0
$955$	24.1248i	0.780662i
$956$	− 4.74739i	− 0.153541i
$957$	0	0
$958$	23.9665i	0.774322i
$959$	0	0
$960$	0	0
$961$	26.1295	0.842887
$962$	−0.0822374	−0.00265144
$963$	0	0
$964$	15.4869i	0.498799i
$965$	−18.8831	−0.607869
$966$	0	0
$967$	−21.3858	−0.687720	−0.343860	−	0.939021i	$-0.611735\pi$
$967$	−21.3858	−0.687720	−0.343860	+	0.939021i	$0.611735\pi$
$968$	3.80525i	0.122305i
$969$	0	0
$970$	1.39104	0.0446635
$971$	−25.0452	−0.803738	−0.401869	−	0.915697i	$-0.631639\pi$
$971$	−25.0452	−0.803738	−0.401869	+	0.915697i	$0.631639\pi$
$972$	0	0
$973$	0	0
$974$	− 16.4288i	− 0.526413i
$975$	0	0
$976$	− 5.88348i	− 0.188326i
$977$	− 51.9612i	− 1.66239i	−0.555983	−	0.831193i	$-0.687658\pi$
$977$	− 51.9612i	− 1.66239i	0.555983	−	0.831193i	$-0.312342\pi$
$978$	0	0
$979$	− 28.0153i	− 0.895372i
$980$	0	0
$981$	0	0
$982$	24.5246	0.782613
$983$	−22.4481	−0.715983	−0.357992	−	0.933725i	$-0.616538\pi$
$983$	−22.4481	−0.715983	−0.357992	+	0.933725i	$0.616538\pi$
$984$	0	0
$985$	− 14.5190i	− 0.462615i
$986$	34.4663	1.09763
$987$	0	0
$988$	−0.0805134	−0.00256147
$989$	29.7405i	0.945692i
$990$	0	0
$991$	−18.9082	−0.600640	−0.300320	−	0.953839i	$-0.597093\pi$
$991$	−18.9082	−0.600640	−0.300320	+	0.953839i	$0.597093\pi$
$992$	2.20692	0.0700698
$993$	0	0
$994$	0	0
$995$	18.4888i	0.586133i
$996$	0	0
$997$	− 14.7055i	− 0.465728i	−0.972509	−	0.232864i	$-0.925190\pi$
$997$	− 14.7055i	− 0.465728i	0.972509	−	0.232864i	$-0.0748096\pi$
$998$	− 31.5492i	− 0.998672i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.a.881.8	yes	8
3.2	odd	2		4410.2.b.d.881.1	yes	8
7.6	odd	2		4410.2.b.d.881.8	yes	8
21.20	even	2	inner	4410.2.b.a.881.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.a.881.1	✓	8	21.20	even	2	inner
4410.2.b.a.881.8	yes	8	1.1	even	1	trivial
4410.2.b.d.881.1	yes	8	3.2	odd	2
4410.2.b.d.881.8	yes	8	7.6	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} -1.00000i q^{10} +3.84776i q^{11} +0.0273395i q^{13} +1.00000 q^{16} -3.71031 q^{17} -2.94495i q^{19} +1.00000 q^{20} -3.84776 q^{22} -9.05921i q^{23} +1.00000 q^{25} -0.0273395 q^{26} +9.28931i q^{29} -2.20692i q^{31} +1.00000i q^{32} -3.71031i q^{34} +3.00801 q^{37} +2.94495 q^{38} +1.00000i q^{40} -8.61313 q^{41} -3.28290 q^{43} -3.84776i q^{44} +9.05921 q^{46} +10.7927 q^{47} +1.00000i q^{50} -0.0273395i q^{52} -4.84935i q^{53} -3.84776i q^{55} -9.28931 q^{58} -4.04826 q^{59} -5.88348i q^{61} +2.20692 q^{62} -1.00000 q^{64} -0.0273395i q^{65} +10.6530 q^{67} +3.71031 q^{68} +1.14840i q^{71} -7.67778i q^{73} +3.00801i q^{74} +2.94495i q^{76} -5.65685 q^{79} -1.00000 q^{80} -8.61313i q^{82} +3.22625 q^{83} +3.71031 q^{85} -3.28290i q^{86} +3.84776 q^{88} -7.28093 q^{89} +9.05921i q^{92} +10.7927i q^{94} +2.94495i q^{95} +1.39104i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 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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