LMFDB - Embedded newform 4410.2.d.c.4409.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(4409,4410)]

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, 1, 1]))

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

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

chi := DirichletCharacter("4410.4409");

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.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4409.19
Character	$\chi$	$=$	4410.4409
Dual form			4410.2.d.c.4409.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(1.60367 - 1.55828i) q^{5} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +(1.60367 - 1.55828i) q^{5} -1.00000 q^{8} +(-1.60367 + 1.55828i) q^{10} +4.10672i q^{11} -2.67732 q^{13} +1.00000 q^{16} -1.29539i q^{17} +6.96705i q^{19} +(1.60367 - 1.55828i) q^{20} -4.10672i q^{22} +3.53283 q^{23} +(0.143525 - 4.99794i) q^{25} +2.67732 q^{26} -3.22563i q^{29} -8.38019i q^{31} -1.00000 q^{32} +1.29539i q^{34} -11.3287i q^{37} -6.96705i q^{38} +(-1.60367 + 1.55828i) q^{40} -1.99657 q^{41} -0.0984652i q^{43} +4.10672i q^{44} -3.53283 q^{46} +9.68213i q^{47} +(-0.143525 + 4.99794i) q^{50} -2.67732 q^{52} +11.8037 q^{53} +(6.39943 + 6.58584i) q^{55} +3.22563i q^{58} +0.796801 q^{59} +2.69258i q^{61} +8.38019i q^{62} +1.00000 q^{64} +(-4.29354 + 4.17201i) q^{65} +0.696631i q^{67} -1.29539i q^{68} +9.32893i q^{71} +7.26042 q^{73} +11.3287i q^{74} +6.96705i q^{76} +2.73016 q^{79} +(1.60367 - 1.55828i) q^{80} +1.99657 q^{82} +6.79144i q^{83} +(-2.01858 - 2.07738i) q^{85} +0.0984652i q^{86} -4.10672i q^{88} +8.03812 q^{89} +3.53283 q^{92} -9.68213i q^{94} +(10.8566 + 11.1729i) q^{95} +6.29039 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8}+O(q^{10}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{16} + 32 q^{23} - 16 q^{25} - 24 q^{32} - 32 q^{46} + 16 q^{50} - 32 q^{53} + 24 q^{64} - 32 q^{85} + 32 q^{92} + 32 q^{95}+O(q^{100}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^16 + 32 * q^23 - 16 * q^25 - 24 * q^32 - 32 * q^46 + 16 * q^50 - 32 * q^53 + 24 * q^64 - 32 * q^85 + 32 * q^92 + 32 * q^95

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	1.60367	−	1.55828i	0.717184	−	0.696884i
$5$	1.60367	−	1.55828i	0.717184	−	0.696884i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.00000			−0.353553
$9$		0			0
$10$	−1.60367	+	1.55828i	−0.507125	+	0.492772i
$11$			4.10672i			1.23822i	0.785303	+	0.619112i	$0.212507\pi$
$11$			4.10672i			1.23822i	−0.785303	+	0.619112i	$0.787493\pi$
$12$		0			0
$13$	−2.67732			−0.742555			−0.371277	−	0.928522i	$-0.621080\pi$
$13$	−2.67732			−0.742555			−0.371277	+	0.928522i	$0.621080\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		−	1.29539i		−	0.314178i	−0.987584	−	0.157089i	$-0.949789\pi$
$17$		−	1.29539i		−	0.314178i	0.987584	−	0.157089i	$-0.0502110\pi$
$18$		0			0
$19$			6.96705i			1.59835i	0.601097	+	0.799176i	$0.294730\pi$
$19$			6.96705i			1.59835i	−0.601097	+	0.799176i	$0.705270\pi$
$20$	1.60367	−	1.55828i	0.358592	−	0.348442i
$21$		0			0
$22$		−	4.10672i		−	0.875556i
$23$	3.53283			0.736645			0.368323	−	0.929698i	$-0.379932\pi$
$23$	3.53283			0.736645			0.368323	+	0.929698i	$0.379932\pi$
$24$		0			0
$25$	0.143525	−	4.99794i	0.0287050	−	0.999588i
$26$	2.67732			0.525065
$27$		0			0
$28$		0			0
$29$		−	3.22563i		−	0.598984i	−0.954099	−	0.299492i	$-0.903183\pi$
$29$		−	3.22563i		−	0.598984i	0.954099	−	0.299492i	$-0.0968172\pi$
$30$		0			0
$31$		−	8.38019i		−	1.50513i	−0.658520	−	0.752563i	$-0.728817\pi$
$31$		−	8.38019i		−	1.50513i	0.658520	−	0.752563i	$-0.271183\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			1.29539i			0.222158i
$35$		0			0
$36$		0			0
$37$		−	11.3287i		−	1.86243i	−0.364470	−	0.931215i	$-0.618750\pi$
$37$		−	11.3287i		−	1.86243i	0.364470	−	0.931215i	$-0.381250\pi$
$38$		−	6.96705i		−	1.13021i
$39$		0			0
$40$	−1.60367	+	1.55828i	−0.253563	+	0.246386i
$41$	−1.99657			−0.311811			−0.155906	−	0.987772i	$-0.549830\pi$
$41$	−1.99657			−0.311811			−0.155906	+	0.987772i	$0.549830\pi$
$42$		0			0
$43$		−	0.0984652i		−	0.0150158i	−0.999972	−	0.00750790i	$-0.997610\pi$
$43$		−	0.0984652i		−	0.0150158i	0.999972	−	0.00750790i	$-0.00238986\pi$
$44$			4.10672i			0.619112i
$45$		0			0
$46$	−3.53283			−0.520887
$47$			9.68213i			1.41228i	0.708071	+	0.706142i	$0.249566\pi$
$47$			9.68213i			1.41228i	−0.708071	+	0.706142i	$0.750434\pi$
$48$		0			0
$49$		0			0
$50$	−0.143525	+	4.99794i	−0.0202975	+	0.706815i
$51$		0			0
$52$	−2.67732			−0.371277
$53$	11.8037			1.62136			0.810679	−	0.585491i	$-0.199098\pi$
$53$	11.8037			1.62136			0.810679	+	0.585491i	$0.199098\pi$
$54$		0			0
$55$	6.39943	+	6.58584i	0.862899	+	0.888034i
$56$		0			0
$57$		0			0
$58$			3.22563i			0.423546i
$59$	0.796801			0.103735			0.0518673	−	0.998654i	$-0.483483\pi$
$59$	0.796801			0.103735			0.0518673	+	0.998654i	$0.483483\pi$
$60$		0			0
$61$			2.69258i			0.344749i	0.985031	+	0.172375i	$0.0551440\pi$
$61$			2.69258i			0.344749i	−0.985031	+	0.172375i	$0.944856\pi$
$62$			8.38019i			1.06429i
$63$		0			0
$64$	1.00000			0.125000
$65$	−4.29354	+	4.17201i	−0.532548	+	0.517475i
$66$		0			0
$67$			0.696631i			0.0851070i	0.999094	+	0.0425535i	$0.0135493\pi$
$67$			0.696631i			0.0851070i	−0.999094	+	0.0425535i	$0.986451\pi$
$68$		−	1.29539i		−	0.157089i
$69$		0			0
$70$		0			0
$71$			9.32893i			1.10714i	0.832802	+	0.553570i	$0.186735\pi$
$71$			9.32893i			1.10714i	−0.832802	+	0.553570i	$0.813265\pi$
$72$		0			0
$73$	7.26042			0.849768			0.424884	−	0.905248i	$-0.360315\pi$
$73$	7.26042			0.849768			0.424884	+	0.905248i	$0.360315\pi$
$74$			11.3287i			1.31694i
$75$		0			0
$76$			6.96705i			0.799176i
$77$		0			0
$78$		0			0
$79$	2.73016			0.307168			0.153584	−	0.988136i	$-0.450919\pi$
$79$	2.73016			0.307168			0.153584	+	0.988136i	$0.450919\pi$
$80$	1.60367	−	1.55828i	0.179296	−	0.174221i
$81$		0			0
$82$	1.99657			0.220484
$83$			6.79144i			0.745457i	0.927940	+	0.372728i	$0.121578\pi$
$83$			6.79144i			0.745457i	−0.927940	+	0.372728i	$0.878422\pi$
$84$		0			0
$85$	−2.01858	−	2.07738i	−0.218946	−	0.225324i
$86$			0.0984652i			0.0106178i
$87$		0			0
$88$		−	4.10672i		−	0.437778i
$89$	8.03812			0.852039			0.426019	−	0.904714i	$-0.359916\pi$
$89$	8.03812			0.852039			0.426019	+	0.904714i	$0.359916\pi$
$90$		0			0
$91$		0			0
$92$	3.53283			0.368323
$93$		0			0
$94$		−	9.68213i		−	0.998635i
$95$	10.8566	+	11.1729i	1.11387	+	1.14631i
$96$		0			0
$97$	6.29039			0.638692			0.319346	−	0.947638i	$-0.396537\pi$
$97$	6.29039			0.638692			0.319346	+	0.947638i	$0.396537\pi$
$98$		0			0
$99$		0			0
$100$	0.143525	−	4.99794i	0.0143525	−	0.499794i
$101$	12.0292			1.19695			0.598477	−	0.801140i	$-0.295773\pi$
$101$	12.0292			1.19695			0.598477	+	0.801140i	$0.295773\pi$
$102$		0			0
$103$	−15.1612			−1.49388			−0.746939	−	0.664892i	$-0.768477\pi$
$103$	−15.1612			−1.49388			−0.746939	+	0.664892i	$0.768477\pi$
$104$	2.67732			0.262533
$105$		0			0
$106$	−11.8037			−1.14647
$107$	5.95843			0.576023			0.288011	−	0.957627i	$-0.407006\pi$
$107$	5.95843			0.576023			0.288011	+	0.957627i	$0.407006\pi$
$108$		0			0
$109$	15.1027			1.44657			0.723287	−	0.690547i	$-0.242630\pi$
$109$	15.1027			1.44657			0.723287	+	0.690547i	$0.242630\pi$
$110$	−6.39943	−	6.58584i	−0.610161	−	0.627935i
$111$		0			0
$112$		0			0
$113$	0.471895			0.0443922			0.0221961	−	0.999754i	$-0.492934\pi$
$113$	0.471895			0.0443922			0.0221961	+	0.999754i	$0.492934\pi$
$114$		0			0
$115$	5.66549	−	5.50513i	0.528310	−	0.513356i
$116$		−	3.22563i		−	0.299492i
$117$		0			0
$118$	−0.796801			−0.0733515
$119$		0			0
$120$		0			0
$121$	−5.86518			−0.533198
$122$		−	2.69258i		−	0.243775i
$123$		0			0
$124$		−	8.38019i		−	0.752563i
$125$	−7.55802	−	8.23871i	−0.676010	−	0.736892i
$126$		0			0
$127$			10.2712i			0.911426i	0.890127	+	0.455713i	$0.150616\pi$
$127$			10.2712i			0.911426i	−0.890127	+	0.455713i	$0.849384\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	4.29354	−	4.17201i	0.376568	−	0.365910i
$131$	7.93947			0.693675			0.346838	−	0.937925i	$-0.387256\pi$
$131$	7.93947			0.693675			0.346838	+	0.937925i	$0.387256\pi$
$132$		0			0
$133$		0			0
$134$		−	0.696631i		−	0.0601797i
$135$		0			0
$136$			1.29539i			0.111079i
$137$	14.2024			1.21339			0.606695	−	0.794935i	$-0.292495\pi$
$137$	14.2024			1.21339			0.606695	+	0.794935i	$0.292495\pi$
$138$		0			0
$139$		−	2.26467i		−	0.192087i	−0.995377	−	0.0960433i	$-0.969381\pi$
$139$		−	2.26467i		−	0.192087i	0.995377	−	0.0960433i	$-0.0306187\pi$
$140$		0			0
$141$		0			0
$142$		−	9.32893i		−	0.782867i
$143$		−	10.9950i		−	0.919449i
$144$		0			0
$145$	−5.02643	−	5.17285i	−0.417423	−	0.429582i
$146$	−7.26042			−0.600877
$147$		0			0
$148$		−	11.3287i		−	0.931215i
$149$			23.6322i			1.93602i	0.250906	+	0.968012i	$0.419272\pi$
$149$			23.6322i			1.93602i	−0.250906	+	0.968012i	$0.580728\pi$
$150$		0			0
$151$	17.5657			1.42948			0.714740	−	0.699391i	$-0.246545\pi$
$151$	17.5657			1.42948			0.714740	+	0.699391i	$0.246545\pi$
$152$		−	6.96705i		−	0.565103i
$153$		0			0
$154$		0			0
$155$	−13.0587	−	13.4391i	−1.04890	−	1.07945i
$156$		0			0
$157$	−15.7736			−1.25887			−0.629436	−	0.777052i	$-0.716714\pi$
$157$	−15.7736			−1.25887			−0.629436	+	0.777052i	$0.716714\pi$
$158$	−2.73016			−0.217200
$159$		0			0
$160$	−1.60367	+	1.55828i	−0.126781	+	0.123193i
$161$		0			0
$162$		0			0
$163$		−	8.18877i		−	0.641394i	−0.947182	−	0.320697i	$-0.896083\pi$
$163$		−	8.18877i		−	0.641394i	0.947182	−	0.320697i	$-0.103917\pi$
$164$	−1.99657			−0.155906
$165$		0			0
$166$		−	6.79144i		−	0.527118i
$167$			25.5577i			1.97771i	0.148866	+	0.988857i	$0.452438\pi$
$167$			25.5577i			1.97771i	−0.148866	+	0.988857i	$0.547562\pi$
$168$		0			0
$169$	−5.83196			−0.448613
$170$	2.01858	+	2.07738i	0.154818	+	0.159328i
$171$		0			0
$172$		−	0.0984652i		−	0.00750790i
$173$			4.41919i			0.335985i	0.985788	+	0.167992i	$0.0537284\pi$
$173$			4.41919i			0.335985i	−0.985788	+	0.167992i	$0.946272\pi$
$174$		0			0
$175$		0			0
$176$			4.10672i			0.309556i
$177$		0			0
$178$	−8.03812			−0.602482
$179$		−	5.56112i		−	0.415657i	−0.978165	−	0.207829i	$-0.933360\pi$
$179$		−	5.56112i		−	0.415657i	0.978165	−	0.207829i	$-0.0666397\pi$
$180$		0			0
$181$			2.88829i			0.214685i	0.994222	+	0.107343i	$0.0342342\pi$
$181$			2.88829i			0.214685i	−0.994222	+	0.107343i	$0.965766\pi$
$182$		0			0
$183$		0			0
$184$	−3.53283			−0.260443
$185$	−17.6533	−	18.1675i	−1.29790	−	1.33570i
$186$		0			0
$187$	5.31981			0.389023
$188$			9.68213i			0.706142i
$189$		0			0
$190$	−10.8566	−	11.1729i	−0.787622	−	0.810565i
$191$			16.5552i			1.19789i	0.800788	+	0.598947i	$0.204414\pi$
$191$			16.5552i			1.19789i	−0.800788	+	0.598947i	$0.795586\pi$
$192$		0			0
$193$		−	24.8202i		−	1.78660i	−0.449465	−	0.893298i	$-0.648385\pi$
$193$		−	24.8202i		−	1.78660i	0.449465	−	0.893298i	$-0.351615\pi$
$194$	−6.29039			−0.451624
$195$		0			0
$196$		0			0
$197$	5.94419			0.423506			0.211753	−	0.977323i	$-0.432083\pi$
$197$	5.94419			0.423506			0.211753	+	0.977323i	$0.432083\pi$
$198$		0			0
$199$		−	4.95485i		−	0.351240i	−0.984458	−	0.175620i	$-0.943807\pi$
$199$		−	4.95485i		−	0.351240i	0.984458	−	0.175620i	$-0.0561929\pi$
$200$	−0.143525	+	4.99794i	−0.0101487	+	0.353408i
$201$		0			0
$202$	−12.0292			−0.846374
$203$		0			0
$204$		0			0
$205$	−3.20183	+	3.11121i	−0.223626	+	0.217296i
$206$	15.1612			1.05633
$207$		0			0
$208$	−2.67732			−0.185639
$209$	−28.6118			−1.97912
$210$		0			0
$211$	23.8746			1.64360			0.821798	−	0.569779i	$-0.192971\pi$
$211$	23.8746			1.64360			0.821798	+	0.569779i	$0.192971\pi$
$212$	11.8037			0.810679
$213$		0			0
$214$	−5.95843			−0.407310
$215$	−0.153436	−	0.157906i	−0.0104643	−	0.0107691i
$216$		0			0
$217$		0			0
$218$	−15.1027			−1.02288
$219$		0			0
$220$	6.39943	+	6.58584i	0.431449	+	0.444017i
$221$			3.46818i			0.233295i
$222$		0			0
$223$	22.4667			1.50448			0.752241	−	0.658888i	$-0.228973\pi$
$223$	22.4667			1.50448			0.752241	+	0.658888i	$0.228973\pi$
$224$		0			0
$225$		0			0
$226$	−0.471895			−0.0313900
$227$		−	7.98520i		−	0.529996i	−0.964249	−	0.264998i	$-0.914629\pi$
$227$		−	7.98520i		−	0.529996i	0.964249	−	0.264998i	$-0.0853714\pi$
$228$		0			0
$229$			8.04775i			0.531810i	0.963999	+	0.265905i	$0.0856708\pi$
$229$			8.04775i			0.531810i	−0.963999	+	0.265905i	$0.914329\pi$
$230$	−5.66549	+	5.50513i	−0.373572	+	0.362998i
$231$		0			0
$232$			3.22563i			0.211773i
$233$	15.4065			1.00932			0.504658	−	0.863319i	$-0.331618\pi$
$233$	15.4065			1.00932			0.504658	+	0.863319i	$0.331618\pi$
$234$		0			0
$235$	15.0875	+	15.5270i	0.984198	+	1.01287i
$236$	0.796801			0.0518673
$237$		0			0
$238$		0			0
$239$		−	4.54137i		−	0.293757i	−0.989155	−	0.146878i	$-0.953077\pi$
$239$		−	4.54137i		−	0.293757i	0.989155	−	0.146878i	$-0.0469226\pi$
$240$		0			0
$241$		−	20.0641i		−	1.29244i	−0.763150	−	0.646222i	$-0.776348\pi$
$241$		−	20.0641i		−	1.29244i	0.763150	−	0.646222i	$-0.223652\pi$
$242$	5.86518			0.377028
$243$		0			0
$244$			2.69258i			0.172375i
$245$		0			0
$246$		0			0
$247$		−	18.6530i		−	1.18686i
$248$			8.38019i			0.532143i
$249$		0			0
$250$	7.55802	+	8.23871i	0.478011	+	0.521062i
$251$	−16.3967			−1.03495			−0.517476	−	0.855698i	$-0.673128\pi$
$251$	−16.3967			−1.03495			−0.517476	+	0.855698i	$0.673128\pi$
$252$		0			0
$253$			14.5083i			0.912132i
$254$		−	10.2712i		−	0.644476i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	18.5675i		−	1.15821i	−0.815254	−	0.579103i	$-0.803403\pi$
$257$		−	18.5675i		−	1.15821i	0.815254	−	0.579103i	$-0.196597\pi$
$258$		0			0
$259$		0			0
$260$	−4.29354	+	4.17201i	−0.266274	+	0.258737i
$261$		0			0
$262$	−7.93947			−0.490502
$263$	−9.17316			−0.565642			−0.282821	−	0.959173i	$-0.591270\pi$
$263$	−9.17316			−0.565642			−0.282821	+	0.959173i	$0.591270\pi$
$264$		0			0
$265$	18.9292	−	18.3934i	1.16281	−	1.12990i
$266$		0			0
$267$		0			0
$268$			0.696631i			0.0425535i
$269$	23.5057			1.43317			0.716583	−	0.697502i	$-0.245705\pi$
$269$	23.5057			1.43317			0.716583	+	0.697502i	$0.245705\pi$
$270$		0			0
$271$		−	24.3044i		−	1.47639i	−0.674590	−	0.738193i	$-0.735680\pi$
$271$		−	24.3044i		−	1.47639i	0.674590	−	0.738193i	$-0.264320\pi$
$272$		−	1.29539i		−	0.0785446i
$273$		0			0
$274$	−14.2024			−0.857997
$275$	20.5252	+	0.589417i	1.23771	+	0.0355432i
$276$		0			0
$277$			3.89198i			0.233847i	0.993141	+	0.116923i	$0.0373032\pi$
$277$			3.89198i			0.233847i	−0.993141	+	0.116923i	$0.962697\pi$
$278$			2.26467i			0.135826i
$279$		0			0
$280$		0			0
$281$		−	12.5692i		−	0.749813i	−0.927063	−	0.374907i	$-0.877675\pi$
$281$		−	12.5692i		−	0.749813i	0.927063	−	0.374907i	$-0.122325\pi$
$282$		0			0
$283$	−10.7033			−0.636248			−0.318124	−	0.948049i	$-0.603053\pi$
$283$	−10.7033			−0.636248			−0.318124	+	0.948049i	$0.603053\pi$
$284$			9.32893i			0.553570i
$285$		0			0
$286$			10.9950i			0.650148i
$287$		0			0
$288$		0			0
$289$	15.3220			0.901292
$290$	5.02643	+	5.17285i	0.295162	+	0.303760i
$291$		0			0
$292$	7.26042			0.424884
$293$		−	13.3118i		−	0.777681i	−0.921305	−	0.388841i	$-0.872876\pi$
$293$		−	13.3118i		−	0.777681i	0.921305	−	0.388841i	$-0.127124\pi$
$294$		0			0
$295$	1.27781	−	1.24164i	0.0743968	−	0.0722910i
$296$			11.3287i			0.658469i
$297$		0			0
$298$		−	23.6322i		−	1.36897i
$299$	−9.45850			−0.546999
$300$		0			0
$301$		0			0
$302$	−17.5657			−1.01079
$303$		0			0
$304$			6.96705i			0.399588i
$305$	4.19579	+	4.31801i	0.240250	+	0.247249i
$306$		0			0
$307$	−17.7191			−1.01128			−0.505641	−	0.862744i	$-0.668744\pi$
$307$	−17.7191			−1.01128			−0.505641	+	0.862744i	$0.668744\pi$
$308$		0			0
$309$		0			0
$310$	13.0587	+	13.4391i	0.741684	+	0.763288i
$311$	11.6990			0.663391			0.331695	−	0.943387i	$-0.392379\pi$
$311$	11.6990			0.663391			0.331695	+	0.943387i	$0.392379\pi$
$312$		0			0
$313$	−23.0112			−1.30067			−0.650336	−	0.759647i	$-0.725372\pi$
$313$	−23.0112			−1.30067			−0.650336	+	0.759647i	$0.725372\pi$
$314$	15.7736			0.890157
$315$		0			0
$316$	2.73016			0.153584
$317$	−31.0601			−1.74451			−0.872254	−	0.489052i	$-0.837343\pi$
$317$	−31.0601			−1.74451			−0.872254	+	0.489052i	$0.837343\pi$
$318$		0			0
$319$	13.2468			0.741677
$320$	1.60367	−	1.55828i	0.0896480	−	0.0871105i
$321$		0			0
$322$		0			0
$323$	9.02506			0.502168
$324$		0			0
$325$	−0.384262	+	13.3811i	−0.0213150	+	0.742249i
$326$			8.18877i			0.453534i
$327$		0			0
$328$	1.99657			0.110242
$329$		0			0
$330$		0			0
$331$	5.57625			0.306498			0.153249	−	0.988188i	$-0.451026\pi$
$331$	5.57625			0.306498			0.153249	+	0.988188i	$0.451026\pi$
$332$			6.79144i			0.372728i
$333$		0			0
$334$		−	25.5577i		−	1.39846i
$335$	1.08555	+	1.11717i	0.0593097	+	0.0610374i
$336$		0			0
$337$			29.5071i			1.60735i	0.595066	+	0.803677i	$0.297126\pi$
$337$			29.5071i			1.60735i	−0.595066	+	0.803677i	$0.702874\pi$
$338$	5.83196			0.317217
$339$		0			0
$340$	−2.01858	−	2.07738i	−0.109473	−	0.112662i
$341$	34.4151			1.86368
$342$		0			0
$343$		0			0
$344$			0.0984652i			0.00530889i
$345$		0			0
$346$		−	4.41919i		−	0.237577i
$347$	31.2172			1.67583			0.837914	−	0.545803i	$-0.183775\pi$
$347$	31.2172			1.67583			0.837914	+	0.545803i	$0.183775\pi$
$348$		0			0
$349$			14.9313i			0.799252i	0.916678	+	0.399626i	$0.130860\pi$
$349$			14.9313i			0.799252i	−0.916678	+	0.399626i	$0.869140\pi$
$350$		0			0
$351$		0			0
$352$		−	4.10672i		−	0.218889i
$353$		−	4.79695i		−	0.255316i	−0.991818	−	0.127658i	$-0.959254\pi$
$353$		−	4.79695i		−	0.255316i	0.991818	−	0.127658i	$-0.0407460\pi$
$354$		0			0
$355$	14.5371	+	14.9605i	0.771549	+	0.794023i
$356$	8.03812			0.426019
$357$		0			0
$358$			5.56112i			0.293914i
$359$		−	11.5464i		−	0.609396i	−0.952449	−	0.304698i	$-0.901445\pi$
$359$		−	11.5464i		−	0.609396i	0.952449	−	0.304698i	$-0.0985555\pi$
$360$		0			0
$361$	−29.5398			−1.55473
$362$		−	2.88829i		−	0.151805i
$363$		0			0
$364$		0			0
$365$	11.6433	−	11.3138i	0.609440	−	0.592190i
$366$		0			0
$367$	−6.57159			−0.343034			−0.171517	−	0.985181i	$-0.554867\pi$
$367$	−6.57159			−0.343034			−0.171517	+	0.985181i	$0.554867\pi$
$368$	3.53283			0.184161
$369$		0			0
$370$	17.6533	+	18.1675i	0.917753	+	0.944486i
$371$		0			0
$372$		0			0
$373$			17.2318i			0.892227i	0.894976	+	0.446113i	$0.147192\pi$
$373$			17.2318i			0.892227i	−0.894976	+	0.446113i	$0.852808\pi$
$374$	−5.31981			−0.275081
$375$		0			0
$376$		−	9.68213i		−	0.499318i
$377$			8.63604i			0.444779i
$378$		0			0
$379$	−1.90232			−0.0977157			−0.0488579	−	0.998806i	$-0.515558\pi$
$379$	−1.90232			−0.0977157			−0.0488579	+	0.998806i	$0.515558\pi$
$380$	10.8566	+	11.1729i	0.556933	+	0.573156i
$381$		0			0
$382$		−	16.5552i		−	0.847039i
$383$			8.85433i			0.452435i	0.974077	+	0.226218i	$0.0726360\pi$
$383$			8.85433i			0.452435i	−0.974077	+	0.226218i	$0.927364\pi$
$384$		0			0
$385$		0			0
$386$			24.8202i			1.26331i
$387$		0			0
$388$	6.29039			0.319346
$389$			33.9348i			1.72056i	0.509821	+	0.860280i	$0.329712\pi$
$389$			33.9348i			1.72056i	−0.509821	+	0.860280i	$0.670288\pi$
$390$		0			0
$391$		−	4.57639i		−	0.231438i
$392$		0			0
$393$		0			0
$394$	−5.94419			−0.299464
$395$	4.37829	−	4.25436i	0.220296	−	0.214060i
$396$		0			0
$397$	−36.6819			−1.84101			−0.920507	−	0.390726i	$-0.872224\pi$
$397$	−36.6819			−1.84101			−0.920507	+	0.390726i	$0.872224\pi$
$398$			4.95485i			0.248364i
$399$		0			0
$400$	0.143525	−	4.99794i	0.00717625	−	0.249897i
$401$		−	32.2327i		−	1.60962i	−0.593530	−	0.804812i	$-0.702266\pi$
$401$		−	32.2327i		−	1.60962i	0.593530	−	0.804812i	$-0.297734\pi$
$402$		0			0
$403$			22.4364i			1.11764i
$404$	12.0292			0.598477
$405$		0			0
$406$		0			0
$407$	46.5239			2.30611
$408$		0			0
$409$			14.3827i			0.711177i	0.934643	+	0.355589i	$0.115720\pi$
$409$			14.3827i			0.711177i	−0.934643	+	0.355589i	$0.884280\pi$
$410$	3.20183	−	3.11121i	0.158127	−	0.153652i
$411$		0			0
$412$	−15.1612			−0.746939
$413$		0			0
$414$		0			0
$415$	10.5830	+	10.8912i	0.519497	+	0.534630i
$416$	2.67732			0.131266
$417$		0			0
$418$	28.6118			1.39945
$419$	5.32282			0.260037			0.130018	−	0.991512i	$-0.458496\pi$
$419$	5.32282			0.260037			0.130018	+	0.991512i	$0.458496\pi$
$420$		0			0
$421$	15.3672			0.748953			0.374476	−	0.927236i	$-0.377822\pi$
$421$	15.3672			0.748953			0.374476	+	0.927236i	$0.377822\pi$
$422$	−23.8746			−1.16220
$423$		0			0
$424$	−11.8037			−0.573237
$425$	−6.47429	−	0.185921i	−0.314049	−	0.00901849i
$426$		0			0
$427$		0			0
$428$	5.95843			0.288011
$429$		0			0
$430$	0.153436	+	0.157906i	0.00739936	+	0.00761489i
$431$		−	32.3577i		−	1.55862i	−0.626641	−	0.779308i	$-0.715571\pi$
$431$		−	32.3577i		−	1.55862i	0.626641	−	0.779308i	$-0.284429\pi$
$432$		0			0
$433$	26.1611			1.25722			0.628610	−	0.777720i	$-0.283624\pi$
$433$	26.1611			1.25722			0.628610	+	0.777720i	$0.283624\pi$
$434$		0			0
$435$		0			0
$436$	15.1027			0.723287
$437$			24.6134i			1.17742i
$438$		0			0
$439$			22.2993i			1.06429i	0.846655	+	0.532143i	$0.178613\pi$
$439$			22.2993i			1.06429i	−0.846655	+	0.532143i	$0.821387\pi$
$440$	−6.39943	−	6.58584i	−0.305081	−	0.313967i
$441$		0			0
$442$		−	3.46818i		−	0.164964i
$443$	31.7237			1.50724			0.753619	−	0.657312i	$-0.228307\pi$
$443$	31.7237			1.50724			0.753619	+	0.657312i	$0.228307\pi$
$444$		0			0
$445$	12.8905	−	12.5256i	0.611068	−	0.593772i
$446$	−22.4667			−1.06383
$447$		0			0
$448$		0			0
$449$		−	4.85438i		−	0.229092i	−0.993418	−	0.114546i	$-0.963459\pi$
$449$		−	4.85438i		−	0.229092i	0.993418	−	0.114546i	$-0.0365414\pi$
$450$		0			0
$451$		−	8.19934i		−	0.386092i
$452$	0.471895			0.0221961
$453$		0			0
$454$			7.98520i			0.374764i
$455$		0			0
$456$		0			0
$457$		−	6.94478i		−	0.324863i	−0.986720	−	0.162432i	$-0.948066\pi$
$457$		−	6.94478i		−	0.324863i	0.986720	−	0.162432i	$-0.0519337\pi$
$458$		−	8.04775i		−	0.376046i
$459$		0			0
$460$	5.66549	−	5.50513i	0.264155	−	0.256678i
$461$	0.734361			0.0342026			0.0171013	−	0.999854i	$-0.494556\pi$
$461$	0.734361			0.0342026			0.0171013	+	0.999854i	$0.494556\pi$
$462$		0			0
$463$		−	17.7420i		−	0.824542i	−0.911061	−	0.412271i	$-0.864736\pi$
$463$		−	17.7420i		−	0.824542i	0.911061	−	0.412271i	$-0.135264\pi$
$464$		−	3.22563i		−	0.149746i
$465$		0			0
$466$	−15.4065			−0.713695
$467$		−	17.9353i		−	0.829947i	−0.909834	−	0.414973i	$-0.863791\pi$
$467$		−	17.9353i		−	0.829947i	0.909834	−	0.414973i	$-0.136209\pi$
$468$		0			0
$469$		0			0
$470$	−15.0875	−	15.5270i	−0.695933	−	0.716205i
$471$		0			0
$472$	−0.796801			−0.0366757
$473$	0.404369			0.0185929
$474$		0			0
$475$	34.8209	+	0.999946i	1.59769	+	0.0458807i
$476$		0			0
$477$		0			0
$478$			4.54137i			0.207717i
$479$	−29.8425			−1.36354			−0.681769	−	0.731567i	$-0.738789\pi$
$479$	−29.8425			−1.36354			−0.681769	+	0.731567i	$0.738789\pi$
$480$		0			0
$481$			30.3306i			1.38296i
$482$			20.0641i			0.913896i
$483$		0			0
$484$	−5.86518			−0.266599
$485$	10.0877	−	9.80219i	0.458060	−	0.445095i
$486$		0			0
$487$			23.3967i			1.06021i	0.847934	+	0.530103i	$0.177847\pi$
$487$			23.3967i			1.06021i	−0.847934	+	0.530103i	$0.822153\pi$
$488$		−	2.69258i		−	0.121887i
$489$		0			0
$490$		0			0
$491$		−	0.854553i		−	0.0385654i	−0.999814	−	0.0192827i	$-0.993862\pi$
$491$		−	0.854553i		−	0.0385654i	0.999814	−	0.0192827i	$-0.00613826\pi$
$492$		0			0
$493$	−4.17845			−0.188188
$494$			18.6530i			0.839239i
$495$		0			0
$496$		−	8.38019i		−	0.376282i
$497$		0			0
$498$		0			0
$499$	25.5425			1.14344			0.571719	−	0.820450i	$-0.306277\pi$
$499$	25.5425			1.14344			0.571719	+	0.820450i	$0.306277\pi$
$500$	−7.55802	−	8.23871i	−0.338005	−	0.368446i
$501$		0			0
$502$	16.3967			0.731822
$503$			3.40477i			0.151811i	0.997115	+	0.0759055i	$0.0241847\pi$
$503$			3.40477i			0.151811i	−0.997115	+	0.0759055i	$0.975815\pi$
$504$		0			0
$505$	19.2909	−	18.7449i	0.858435	−	0.834138i
$506$		−	14.5083i		−	0.644975i
$507$		0			0
$508$			10.2712i			0.455713i
$509$	−33.4991			−1.48482			−0.742410	−	0.669945i	$-0.766318\pi$
$509$	−33.4991			−1.48482			−0.742410	+	0.669945i	$0.766318\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			18.5675i			0.818975i
$515$	−24.3136	+	23.6254i	−1.07139	+	1.04106i
$516$		0			0
$517$	−39.7618			−1.74872
$518$		0			0
$519$		0			0
$520$	4.29354	−	4.17201i	0.188284	−	0.182955i
$521$	29.6835			1.30046			0.650230	−	0.759738i	$-0.274673\pi$
$521$	29.6835			1.30046			0.650230	+	0.759738i	$0.274673\pi$
$522$		0			0
$523$	19.2177			0.840333			0.420167	−	0.907447i	$-0.361972\pi$
$523$	19.2177			0.840333			0.420167	+	0.907447i	$0.361972\pi$
$524$	7.93947			0.346838
$525$		0			0
$526$	9.17316			0.399969
$527$	−10.8556			−0.472879
$528$		0			0
$529$	−10.5191			−0.457354
$530$	−18.9292	+	18.3934i	−0.822232	+	0.798959i
$531$		0			0
$532$		0			0
$533$	5.34544			0.231537
$534$		0			0
$535$	9.55536	−	9.28490i	0.413114	−	0.401421i
$536$		−	0.696631i		−	0.0300899i
$537$		0			0
$538$	−23.5057			−1.01340
$539$		0			0
$540$		0			0
$541$	17.0941			0.734934			0.367467	−	0.930036i	$-0.380225\pi$
$541$	17.0941			0.734934			0.367467	+	0.930036i	$0.380225\pi$
$542$			24.3044i			1.04396i
$543$		0			0
$544$			1.29539i			0.0555394i
$545$	24.2197	−	23.5342i	1.03746	−	1.00810i
$546$		0			0
$547$			42.5352i			1.81867i	0.416061	+	0.909337i	$0.363410\pi$
$547$			42.5352i			1.81867i	−0.416061	+	0.909337i	$0.636590\pi$
$548$	14.2024			0.606695
$549$		0			0
$550$	−20.5252	−	0.589417i	−0.875196	−	0.0251328i
$551$	22.4731			0.957388
$552$		0			0
$553$		0			0
$554$		−	3.89198i		−	0.165354i
$555$		0			0
$556$		−	2.26467i		−	0.0960433i
$557$	−10.8670			−0.460450			−0.230225	−	0.973137i	$-0.573946\pi$
$557$	−10.8670			−0.460450			−0.230225	+	0.973137i	$0.573946\pi$
$558$		0			0
$559$			0.263623i			0.0111500i
$560$		0			0
$561$		0			0
$562$			12.5692i			0.530198i
$563$		−	28.4240i		−	1.19793i	−0.800776	−	0.598964i	$-0.795579\pi$
$563$		−	28.4240i		−	1.19793i	0.800776	−	0.598964i	$-0.204421\pi$
$564$		0			0
$565$	0.756765	−	0.735345i	0.0318373	−	0.0309362i
$566$	10.7033			0.449895
$567$		0			0
$568$		−	9.32893i		−	0.391433i
$569$			19.6809i			0.825068i	0.910942	+	0.412534i	$0.135356\pi$
$569$			19.6809i			0.825068i	−0.910942	+	0.412534i	$0.864644\pi$
$570$		0			0
$571$	26.7887			1.12107			0.560537	−	0.828129i	$-0.310595\pi$
$571$	26.7887			1.12107			0.560537	+	0.828129i	$0.310595\pi$
$572$		−	10.9950i		−	0.459724i
$573$		0			0
$574$		0			0
$575$	0.507049	−	17.6569i	0.0211454	−	0.736342i
$576$		0			0
$577$	−47.9657			−1.99684			−0.998419	−	0.0562078i	$-0.982099\pi$
$577$	−47.9657			−1.99684			−0.998419	+	0.0562078i	$0.982099\pi$
$578$	−15.3220			−0.637310
$579$		0			0
$580$	−5.02643	−	5.17285i	−0.208711	−	0.214791i
$581$		0			0
$582$		0			0
$583$			48.4744i			2.00760i
$584$	−7.26042			−0.300438
$585$		0			0
$586$			13.3118i			0.549904i
$587$		−	29.2950i		−	1.20913i	−0.796554	−	0.604567i	$-0.793346\pi$
$587$		−	29.2950i		−	1.20913i	0.796554	−	0.604567i	$-0.206654\pi$
$588$		0			0
$589$	58.3853			2.40572
$590$	−1.27781	+	1.24164i	−0.0526065	+	0.0511175i
$591$		0			0
$592$		−	11.3287i		−	0.465608i
$593$			0.573952i			0.0235694i	0.999931	+	0.0117847i	$0.00375127\pi$
$593$			0.573952i			0.0235694i	−0.999931	+	0.0117847i	$0.996249\pi$
$594$		0			0
$595$		0			0
$596$			23.6322i			0.968012i
$597$		0			0
$598$	9.45850			0.386787
$599$		−	46.2445i		−	1.88950i	−0.327790	−	0.944750i	$-0.606304\pi$
$599$		−	46.2445i		−	1.88950i	0.327790	−	0.944750i	$-0.393696\pi$
$600$		0			0
$601$		−	28.5615i		−	1.16505i	−0.812813	−	0.582525i	$-0.802065\pi$
$601$		−	28.5615i		−	1.16505i	0.812813	−	0.582525i	$-0.197935\pi$
$602$		0			0
$603$		0			0
$604$	17.5657			0.714740
$605$	−9.40582	+	9.13959i	−0.382401	+	0.371577i
$606$		0			0
$607$	35.8214			1.45394			0.726972	−	0.686667i	$-0.240927\pi$
$607$	35.8214			1.45394			0.726972	+	0.686667i	$0.240927\pi$
$608$		−	6.96705i		−	0.282551i
$609$		0			0
$610$	−4.19579	−	4.31801i	−0.169883	−	0.174831i
$611$		−	25.9221i		−	1.04870i
$612$		0			0
$613$			4.39427i			0.177483i	0.996055	+	0.0887414i	$0.0282845\pi$
$613$			4.39427i			0.177483i	−0.996055	+	0.0887414i	$0.971716\pi$
$614$	17.7191			0.715084
$615$		0			0
$616$		0			0
$617$	−2.19978			−0.0885599			−0.0442799	−	0.999019i	$-0.514099\pi$
$617$	−2.19978			−0.0885599			−0.0442799	+	0.999019i	$0.514099\pi$
$618$		0			0
$619$			10.6408i			0.427688i	0.976868	+	0.213844i	$0.0685985\pi$
$619$			10.6408i			0.427688i	−0.976868	+	0.213844i	$0.931402\pi$
$620$	−13.0587	−	13.4391i	−0.524450	−	0.539726i
$621$		0			0
$622$	−11.6990			−0.469088
$623$		0			0
$624$		0			0
$625$	−24.9588	−	1.43466i	−0.998352	−	0.0573863i
$626$	23.0112			0.919714
$627$		0			0
$628$	−15.7736			−0.629436
$629$	−14.6751			−0.585136
$630$		0			0
$631$	−10.0363			−0.399540			−0.199770	−	0.979843i	$-0.564020\pi$
$631$	−10.0363			−0.399540			−0.199770	+	0.979843i	$0.564020\pi$
$632$	−2.73016			−0.108600
$633$		0			0
$634$	31.0601			1.23355
$635$	16.0055	+	16.4717i	0.635158	+	0.653660i
$636$		0			0
$637$		0			0
$638$	−13.2468			−0.524444
$639$		0			0
$640$	−1.60367	+	1.55828i	−0.0633907	+	0.0615964i
$641$		−	8.44410i		−	0.333522i	−0.985997	−	0.166761i	$-0.946669\pi$
$641$		−	8.44410i		−	0.333522i	0.985997	−	0.166761i	$-0.0533308\pi$
$642$		0			0
$643$	−20.4190			−0.805248			−0.402624	−	0.915366i	$-0.631902\pi$
$643$	−20.4190			−0.805248			−0.402624	+	0.915366i	$0.631902\pi$
$644$		0			0
$645$		0			0
$646$	−9.02506			−0.355086
$647$			15.9942i			0.628796i	0.949291	+	0.314398i	$0.101803\pi$
$647$			15.9942i			0.628796i	−0.949291	+	0.314398i	$0.898197\pi$
$648$		0			0
$649$			3.27224i			0.128447i
$650$	0.384262	−	13.3811i	0.0150720	−	0.524849i
$651$		0			0
$652$		−	8.18877i		−	0.320697i
$653$	−29.6701			−1.16108			−0.580541	−	0.814231i	$-0.697159\pi$
$653$	−29.6701			−1.16108			−0.580541	+	0.814231i	$0.697159\pi$
$654$		0			0
$655$	12.7323	−	12.3719i	0.497492	−	0.483411i
$656$	−1.99657			−0.0779528
$657$		0			0
$658$		0			0
$659$		−	6.28916i		−	0.244991i	−0.992469	−	0.122495i	$-0.960910\pi$
$659$		−	6.28916i		−	0.244991i	0.992469	−	0.122495i	$-0.0390897\pi$
$660$		0			0
$661$			5.26725i			0.204872i	0.994740	+	0.102436i	$0.0326638\pi$
$661$			5.26725i			0.204872i	−0.994740	+	0.102436i	$0.967336\pi$
$662$	−5.57625			−0.216727
$663$		0			0
$664$		−	6.79144i		−	0.263559i
$665$		0			0
$666$		0			0
$667$		−	11.3956i		−	0.441239i
$668$			25.5577i			0.988857i
$669$		0			0
$670$	−1.08555	−	1.11717i	−0.0419383	−	0.0431599i
$671$	−11.0577			−0.426877
$672$		0			0
$673$		−	6.10722i		−	0.235416i	−0.993048	−	0.117708i	$-0.962445\pi$
$673$		−	6.10722i		−	0.235416i	0.993048	−	0.117708i	$-0.0375547\pi$
$674$		−	29.5071i		−	1.13657i
$675$		0			0
$676$	−5.83196			−0.224306
$677$			10.3238i			0.396777i	0.980123	+	0.198389i	$0.0635708\pi$
$677$			10.3238i			0.396777i	−0.980123	+	0.198389i	$0.936429\pi$
$678$		0			0
$679$		0			0
$680$	2.01858	+	2.07738i	0.0774091	+	0.0796640i
$681$		0			0
$682$	−34.4151			−1.31782
$683$	19.3208			0.739291			0.369646	−	0.929173i	$-0.379479\pi$
$683$	19.3208			0.739291			0.369646	+	0.929173i	$0.379479\pi$
$684$		0			0
$685$	22.7759	−	22.1313i	0.870224	−	0.845592i
$686$		0			0
$687$		0			0
$688$		−	0.0984652i		−	0.00375395i
$689$	−31.6022			−1.20395
$690$		0			0
$691$		−	50.5898i		−	1.92453i	−0.272122	−	0.962263i	$-0.587726\pi$
$691$		−	50.5898i		−	1.92453i	0.272122	−	0.962263i	$-0.412274\pi$
$692$			4.41919i			0.167992i
$693$		0			0
$694$	−31.2172			−1.18499
$695$	−3.52899	−	3.63178i	−0.133862	−	0.137761i
$696$		0			0
$697$			2.58633i			0.0979643i
$698$		−	14.9313i		−	0.565157i
$699$		0			0
$700$		0			0
$701$			4.85900i			0.183522i	0.995781	+	0.0917610i	$0.0292496\pi$
$701$			4.85900i			0.183522i	−0.995781	+	0.0917610i	$0.970750\pi$
$702$		0			0
$703$	78.9278			2.97682
$704$			4.10672i			0.154778i
$705$		0			0
$706$			4.79695i			0.180536i
$707$		0			0
$708$		0			0
$709$	−39.3199			−1.47669			−0.738344	−	0.674424i	$-0.764392\pi$
$709$	−39.3199			−1.47669			−0.738344	+	0.674424i	$0.764392\pi$
$710$	−14.5371	−	14.9605i	−0.545567	−	0.561459i
$711$		0			0
$712$	−8.03812			−0.301241
$713$		−	29.6058i		−	1.10874i
$714$		0			0
$715$	−17.1333	−	17.6324i	−0.640749	−	0.659414i
$716$		−	5.56112i		−	0.207829i
$717$		0			0
$718$			11.5464i			0.430908i
$719$	−31.9349			−1.19097			−0.595486	−	0.803366i	$-0.703040\pi$
$719$	−31.9349			−1.19097			−0.595486	+	0.803366i	$0.703040\pi$
$720$		0			0
$721$		0			0
$722$	29.5398			1.09936
$723$		0			0
$724$			2.88829i			0.107343i
$725$	−16.1215	−	0.462958i	−0.598737	−	0.0171938i
$726$		0			0
$727$	−49.5695			−1.83843			−0.919216	−	0.393754i	$-0.871176\pi$
$727$	−49.5695			−1.83843			−0.919216	+	0.393754i	$0.871176\pi$
$728$		0			0
$729$		0			0
$730$	−11.6433	+	11.3138i	−0.430939	+	0.418741i
$731$	−0.127551			−0.00471764
$732$		0			0
$733$	22.7160			0.839036			0.419518	−	0.907747i	$-0.362199\pi$
$733$	22.7160			0.839036			0.419518	+	0.907747i	$0.362199\pi$
$734$	6.57159			0.242562
$735$		0			0
$736$	−3.53283			−0.130222
$737$	−2.86087			−0.105382
$738$		0			0
$739$	−34.2098			−1.25843			−0.629213	−	0.777233i	$-0.716623\pi$
$739$	−34.2098			−1.25843			−0.629213	+	0.777233i	$0.716623\pi$
$740$	−17.6533	−	18.1675i	−0.648949	−	0.667852i
$741$		0			0
$742$		0			0
$743$	−19.7338			−0.723961			−0.361981	−	0.932186i	$-0.617899\pi$
$743$	−19.7338			−0.723961			−0.361981	+	0.932186i	$0.617899\pi$
$744$		0			0
$745$	36.8255	+	37.8982i	1.34918	+	1.38848i
$746$		−	17.2318i		−	0.630900i
$747$		0			0
$748$	5.31981			0.194512
$749$		0			0
$750$		0			0
$751$	−40.1064			−1.46350			−0.731752	−	0.681571i	$-0.761297\pi$
$751$	−40.1064			−1.46350			−0.731752	+	0.681571i	$0.761297\pi$
$752$			9.68213i			0.353071i
$753$		0			0
$754$		−	8.63604i		−	0.314506i
$755$	28.1697	−	27.3723i	1.02520	−	0.996182i
$756$		0			0
$757$			41.6404i			1.51345i	0.653735	+	0.756724i	$0.273201\pi$
$757$			41.6404i			1.51345i	−0.653735	+	0.756724i	$0.726799\pi$
$758$	1.90232			0.0690954
$759$		0			0
$760$	−10.8566	−	11.1729i	−0.393811	−	0.405282i
$761$	1.12253			0.0406916			0.0203458	−	0.999793i	$-0.493523\pi$
$761$	1.12253			0.0406916			0.0203458	+	0.999793i	$0.493523\pi$
$762$		0			0
$763$		0			0
$764$			16.5552i			0.598947i
$765$		0			0
$766$		−	8.85433i		−	0.319920i
$767$	−2.13329			−0.0770286
$768$		0			0
$769$		−	43.4547i		−	1.56702i	−0.621382	−	0.783508i	$-0.713429\pi$
$769$		−	43.4547i		−	1.56702i	0.621382	−	0.783508i	$-0.286571\pi$
$770$		0			0
$771$		0			0
$772$		−	24.8202i		−	0.893298i
$773$			23.7066i			0.852668i	0.904566	+	0.426334i	$0.140195\pi$
$773$			23.7066i			0.852668i	−0.904566	+	0.426334i	$0.859805\pi$
$774$		0			0
$775$	−41.8837	−	1.20277i	−1.50451	−	0.0432047i
$776$	−6.29039			−0.225812
$777$		0			0
$778$		−	33.9348i		−	1.21662i
$779$		−	13.9102i		−	0.498384i
$780$		0			0
$781$	−38.3113			−1.37089
$782$			4.57639i			0.163651i
$783$		0			0
$784$		0			0
$785$	−25.2957	+	24.5797i	−0.902843	+	0.877288i
$786$		0			0
$787$	−15.8300			−0.564279			−0.282140	−	0.959373i	$-0.591044\pi$
$787$	−15.8300			−0.564279			−0.282140	+	0.959373i	$0.591044\pi$
$788$	5.94419			0.211753
$789$		0			0
$790$	−4.37829	+	4.25436i	−0.155772	+	0.151363i
$791$		0			0
$792$		0			0
$793$		−	7.20889i		−	0.255995i
$794$	36.6819			1.30179
$795$		0			0
$796$		−	4.95485i		−	0.175620i
$797$			34.8696i			1.23514i	0.786515	+	0.617571i	$0.211883\pi$
$797$			34.8696i			1.23514i	−0.786515	+	0.617571i	$0.788117\pi$
$798$		0			0
$799$	12.5421			0.443709
$800$	−0.143525	+	4.99794i	−0.00507437	+	0.176704i
$801$		0			0
$802$			32.2327i			1.13818i
$803$			29.8165i			1.05220i
$804$		0			0
$805$		0			0
$806$		−	22.4364i		−	0.790290i
$807$		0			0
$808$	−12.0292			−0.423187
$809$		−	35.1888i		−	1.23717i	−0.785716	−	0.618587i	$-0.787705\pi$
$809$		−	35.1888i		−	1.23717i	0.785716	−	0.618587i	$-0.212295\pi$
$810$		0			0
$811$			17.5831i			0.617428i	0.951155	+	0.308714i	$0.0998985\pi$
$811$			17.5831i			0.617428i	−0.951155	+	0.308714i	$0.900101\pi$
$812$		0			0
$813$		0			0
$814$	−46.5239			−1.63066
$815$	−12.7604	−	13.1321i	−0.446977	−	0.459997i
$816$		0			0
$817$	0.686012			0.0240005
$818$		−	14.3827i		−	0.502878i
$819$		0			0
$820$	−3.20183	+	3.11121i	−0.111813	+	0.108648i
$821$			38.9656i			1.35991i	0.733255	+	0.679954i	$0.238000\pi$
$821$			38.9656i			1.35991i	−0.733255	+	0.679954i	$0.762000\pi$
$822$		0			0
$823$			18.8786i			0.658065i	0.944319	+	0.329033i	$0.106723\pi$
$823$			18.8786i			0.658065i	−0.944319	+	0.329033i	$0.893277\pi$
$824$	15.1612			0.528166
$825$		0			0
$826$		0			0
$827$	25.7654			0.895950			0.447975	−	0.894046i	$-0.352145\pi$
$827$	25.7654			0.895950			0.447975	+	0.894046i	$0.352145\pi$
$828$		0			0
$829$		−	9.10894i		−	0.316366i	−0.987410	−	0.158183i	$-0.949436\pi$
$829$		−	9.10894i		−	0.316366i	0.987410	−	0.158183i	$-0.0505637\pi$
$830$	−10.5830	−	10.8912i	−0.367340	−	0.378040i
$831$		0			0
$832$	−2.67732			−0.0928193
$833$		0			0
$834$		0			0
$835$	39.8261	+	40.9862i	1.37824	+	1.41838i
$836$	−28.6118			−0.989559
$837$		0			0
$838$	−5.32282			−0.183874
$839$	11.5225			0.397800			0.198900	−	0.980020i	$-0.436263\pi$
$839$	11.5225			0.397800			0.198900	+	0.980020i	$0.436263\pi$
$840$		0			0
$841$	18.5953			0.641218
$842$	−15.3672			−0.529590
$843$		0			0
$844$	23.8746			0.821798
$845$	−9.35255	+	9.08783i	−0.321738	+	0.312631i
$846$		0			0
$847$		0			0
$848$	11.8037			0.405339
$849$		0			0
$850$	6.47429	+	0.185921i	0.222066	+	0.00637704i
$851$		−	40.0224i		−	1.37195i
$852$		0			0
$853$	45.2713			1.55006			0.775030	−	0.631925i	$-0.217735\pi$
$853$	45.2713			1.55006			0.775030	+	0.631925i	$0.217735\pi$
$854$		0			0
$855$		0			0
$856$	−5.95843			−0.203655
$857$			53.1721i			1.81632i	0.418620	+	0.908161i	$0.362514\pi$
$857$			53.1721i			1.81632i	−0.418620	+	0.908161i	$0.637486\pi$
$858$		0			0
$859$			17.0126i			0.580462i	0.956957	+	0.290231i	$0.0937322\pi$
$859$			17.0126i			0.580462i	−0.956957	+	0.290231i	$0.906268\pi$
$860$	−0.153436	−	0.157906i	−0.00523214	−	0.00538454i
$861$		0			0
$862$			32.3577i			1.10211i
$863$	−32.7306			−1.11416			−0.557080	−	0.830459i	$-0.688079\pi$
$863$	−32.7306			−1.11416			−0.557080	+	0.830459i	$0.688079\pi$
$864$		0			0
$865$	6.88633	+	7.08692i	0.234142	+	0.240963i
$866$	−26.1611			−0.888989
$867$		0			0
$868$		0			0
$869$			11.2120i			0.380342i
$870$		0			0
$871$		−	1.86510i		−	0.0631966i
$872$	−15.1027			−0.511441
$873$		0			0
$874$		−	24.6134i		−	0.832560i
$875$		0			0
$876$		0			0
$877$		−	46.3164i		−	1.56399i	−0.623283	−	0.781996i	$-0.714202\pi$
$877$		−	46.3164i		−	1.56399i	0.623283	−	0.781996i	$-0.285798\pi$
$878$		−	22.2993i		−	0.752564i
$879$		0			0
$880$	6.39943	+	6.58584i	0.215725	+	0.222008i
$881$	48.6850			1.64024			0.820119	−	0.572193i	$-0.193907\pi$
$881$	48.6850			1.64024			0.820119	+	0.572193i	$0.193907\pi$
$882$		0			0
$883$			40.9532i			1.37818i	0.724674	+	0.689092i	$0.241990\pi$
$883$			40.9532i			1.37818i	−0.724674	+	0.689092i	$0.758010\pi$
$884$			3.46818i			0.116647i
$885$		0			0
$886$	−31.7237			−1.06578
$887$			47.8647i			1.60714i	0.595210	+	0.803570i	$0.297069\pi$
$887$			47.8647i			1.60714i	−0.595210	+	0.803570i	$0.702931\pi$
$888$		0			0
$889$		0			0
$890$	−12.8905	+	12.5256i	−0.432091	+	0.419860i
$891$		0			0
$892$	22.4667			0.752241
$893$	−67.4559			−2.25733
$894$		0			0
$895$	−8.66578	−	8.91821i	−0.289665	−	0.298103i
$896$		0			0
$897$		0			0
$898$			4.85438i			0.161993i
$899$	−27.0314			−0.901547
$900$		0			0
$901$		−	15.2904i		−	0.509396i
$902$			8.19934i			0.273008i
$903$		0			0
$904$	−0.471895			−0.0156950
$905$	4.50077	+	4.63187i	0.149611	+	0.153969i
$906$		0			0
$907$			44.4729i			1.47670i	0.674418	+	0.738350i	$0.264395\pi$
$907$			44.4729i			1.47670i	−0.674418	+	0.738350i	$0.735605\pi$
$908$		−	7.98520i		−	0.264998i
$909$		0			0
$910$		0			0
$911$			27.7147i			0.918230i	0.888377	+	0.459115i	$0.151833\pi$
$911$			27.7147i			0.918230i	−0.888377	+	0.459115i	$0.848167\pi$
$912$		0			0
$913$	−27.8905			−0.923042
$914$			6.94478i			0.229713i
$915$		0			0
$916$			8.04775i			0.265905i
$917$		0			0
$918$		0			0
$919$	−1.16663			−0.0384837			−0.0192418	−	0.999815i	$-0.506125\pi$
$919$	−1.16663			−0.0384837			−0.0192418	+	0.999815i	$0.506125\pi$
$920$	−5.66549	+	5.50513i	−0.186786	+	0.181499i
$921$		0			0
$922$	−0.734361			−0.0241849
$923$		−	24.9765i		−	0.822112i
$924$		0			0
$925$	−56.6203	−	1.62595i	−1.86166	−	0.0534611i
$926$			17.7420i			0.583040i
$927$		0			0
$928$			3.22563i			0.105886i
$929$	−8.12440			−0.266553			−0.133276	−	0.991079i	$-0.542550\pi$
$929$	−8.12440			−0.266553			−0.133276	+	0.991079i	$0.542550\pi$
$930$		0			0
$931$		0			0
$932$	15.4065			0.504658
$933$		0			0
$934$			17.9353i			0.586861i
$935$	8.53123	−	8.28976i	0.279001	−	0.271104i
$936$		0			0
$937$	−11.2806			−0.368521			−0.184260	−	0.982877i	$-0.558989\pi$
$937$	−11.2806			−0.368521			−0.184260	+	0.982877i	$0.558989\pi$
$938$		0			0
$939$		0			0
$940$	15.0875	+	15.5270i	0.492099	+	0.506433i
$941$	−18.0948			−0.589872			−0.294936	−	0.955517i	$-0.595298\pi$
$941$	−18.0948			−0.589872			−0.294936	+	0.955517i	$0.595298\pi$
$942$		0			0
$943$	−7.05352			−0.229694
$944$	0.796801			0.0259337
$945$		0			0
$946$	−0.404369			−0.0131472
$947$	−43.2000			−1.40381			−0.701906	−	0.712270i	$-0.747667\pi$
$947$	−43.2000			−1.40381			−0.701906	+	0.712270i	$0.747667\pi$
$948$		0			0
$949$	−19.4385			−0.630999
$950$	−34.8209	−	0.999946i	−1.12974	−	0.0324425i
$951$		0			0
$952$		0			0
$953$	−7.86393			−0.254738			−0.127369	−	0.991855i	$-0.540653\pi$
$953$	−7.86393			−0.254738			−0.127369	+	0.991855i	$0.540653\pi$
$954$		0			0
$955$	25.7977	+	26.5492i	0.834794	+	0.859110i
$956$		−	4.54137i		−	0.146878i
$957$		0			0
$958$	29.8425			0.964167
$959$		0			0
$960$		0			0
$961$	−39.2276			−1.26541
$962$		−	30.3306i		−	0.977898i
$963$		0			0
$964$		−	20.0641i		−	0.646222i
$965$	−38.6768	−	39.8034i	−1.24505	−	1.28132i
$966$		0			0
$967$		−	45.4858i		−	1.46273i	−0.681989	−	0.731363i	$-0.738885\pi$
$967$		−	45.4858i		−	1.46273i	0.681989	−	0.731363i	$-0.261115\pi$
$968$	5.86518			0.188514
$969$		0			0
$970$	−10.0877	+	9.80219i	−0.323897	+	0.314729i
$971$	−35.3116			−1.13320			−0.566601	−	0.823992i	$-0.691742\pi$
$971$	−35.3116			−1.13320			−0.566601	+	0.823992i	$0.691742\pi$
$972$		0			0
$973$		0			0
$974$		−	23.3967i		−	0.749678i
$975$		0			0
$976$			2.69258i			0.0861873i
$977$	−36.9189			−1.18114			−0.590570	−	0.806987i	$-0.701097\pi$
$977$	−36.9189			−1.18114			−0.590570	+	0.806987i	$0.701097\pi$
$978$		0			0
$979$			33.0103i			1.05501i
$980$		0			0
$981$		0			0
$982$			0.854553i			0.0272699i
$983$			36.6036i			1.16747i	0.811943	+	0.583737i	$0.198410\pi$
$983$			36.6036i			1.16747i	−0.811943	+	0.583737i	$0.801590\pi$
$984$		0			0
$985$	9.53253	−	9.26272i	0.303732	−	0.295135i
$986$	4.17845			0.133069
$987$		0			0
$988$		−	18.6530i		−	0.593432i
$989$		−	0.347860i		−	0.0110613i
$990$		0			0
$991$	16.5134			0.524565			0.262283	−	0.964991i	$-0.415525\pi$
$991$	16.5134			0.524565			0.262283	+	0.964991i	$0.415525\pi$
$992$			8.38019i			0.266071i
$993$		0			0
$994$		0			0
$995$	−7.72104	−	7.94594i	−0.244773	−	0.251903i
$996$		0			0
$997$	38.6792			1.22498			0.612492	−	0.790477i	$-0.290167\pi$
$997$	38.6792			1.22498			0.612492	+	0.790477i	$0.290167\pi$
$998$	−25.5425			−0.808532
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.d.c.4409.19	yes	24
3.2	odd	2		4410.2.d.d.4409.6	yes	24
5.4	even	2		4410.2.d.d.4409.20	yes	24
7.6	odd	2	inner	4410.2.d.c.4409.6	yes	24
15.14	odd	2	inner	4410.2.d.c.4409.5	✓	24
21.20	even	2		4410.2.d.d.4409.19	yes	24
35.34	odd	2		4410.2.d.d.4409.5	yes	24
105.104	even	2	inner	4410.2.d.c.4409.20	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.d.c.4409.5	✓	24	15.14	odd	2	inner
4410.2.d.c.4409.6	yes	24	7.6	odd	2	inner
4410.2.d.c.4409.19	yes	24	1.1	even	1	trivial
4410.2.d.c.4409.20	yes	24	105.104	even	2	inner
4410.2.d.d.4409.5	yes	24	35.34	odd	2
4410.2.d.d.4409.6	yes	24	3.2	odd	2
4410.2.d.d.4409.19	yes	24	21.20	even	2
4410.2.d.d.4409.20	yes	24	5.4	even	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(1.60367 - 1.55828i) q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(1.60367 - 1.55828i) q^{5} -1.00000 q^{8} +(-1.60367 + 1.55828i) q^{10} +4.10672i q^{11} -2.67732 q^{13} +1.00000 q^{16} -1.29539i q^{17} +6.96705i q^{19} +(1.60367 - 1.55828i) q^{20} -4.10672i q^{22} +3.53283 q^{23} +(0.143525 - 4.99794i) q^{25} +2.67732 q^{26} -3.22563i q^{29} -8.38019i q^{31} -1.00000 q^{32} +1.29539i q^{34} -11.3287i q^{37} -6.96705i q^{38} +(-1.60367 + 1.55828i) q^{40} -1.99657 q^{41} -0.0984652i q^{43} +4.10672i q^{44} -3.53283 q^{46} +9.68213i q^{47} +(-0.143525 + 4.99794i) q^{50} -2.67732 q^{52} +11.8037 q^{53} +(6.39943 + 6.58584i) q^{55} +3.22563i q^{58} +0.796801 q^{59} +2.69258i q^{61} +8.38019i q^{62} +1.00000 q^{64} +(-4.29354 + 4.17201i) q^{65} +0.696631i q^{67} -1.29539i q^{68} +9.32893i q^{71} +7.26042 q^{73} +11.3287i q^{74} +6.96705i q^{76} +2.73016 q^{79} +(1.60367 - 1.55828i) q^{80} +1.99657 q^{82} +6.79144i q^{83} +(-2.01858 - 2.07738i) q^{85} +0.0984652i q^{86} -4.10672i q^{88} +8.03812 q^{89} +3.53283 q^{92} -9.68213i q^{94} +(10.8566 + 11.1729i) q^{95} +6.29039 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8}+O(q^{10}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{16} + 32 q^{23} - 16 q^{25} - 24 q^{32} - 32 q^{46} + 16 q^{50} - 32 q^{53} + 24 q^{64} - 32 q^{85} + 32 q^{92} + 32 q^{95}+O(q^{100}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^16 + 32 * q^23 - 16 * q^25 - 24 * q^32 - 32 * q^46 + 16 * q^50 - 32 * q^53 + 24 * q^64 - 32 * q^85 + 32 * q^92 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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