LMFDB - Embedded newform 4410.2.b.d.881.2

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			881.2
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.d.881.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +1.00000i q^{8} -1.00000i q^{10} -2.76537i q^{11} -6.49661i q^{13} +1.00000 q^{16} -6.95749 q^{17} +5.10973i q^{19} -1.00000 q^{20} -2.76537 q^{22} +9.08881i q^{23} +1.00000 q^{25} -6.49661 q^{26} +1.14545i q^{29} +9.75858i q^{31} -1.00000i q^{32} +6.95749i q^{34} -8.09040 q^{37} +5.10973 q^{38} +1.00000i q^{40} +4.91761 q^{41} -11.3553 q^{43} +2.76537i q^{44} +9.08881 q^{46} -1.65564 q^{47} -1.00000i q^{50} +6.49661i q^{52} +11.1757i q^{53} -2.76537i q^{55} +1.14545 q^{58} +9.50756 q^{59} -8.28130i q^{61} +9.75858 q^{62} -1.00000 q^{64} -6.49661i q^{65} +2.41262 q^{67} +6.95749 q^{68} -5.30411i q^{71} -8.34731i q^{73} +8.09040i q^{74} -5.10973i q^{76} +5.65685 q^{79} +1.00000 q^{80} -4.91761i q^{82} +4.16478 q^{83} -6.95749 q^{85} +11.3553i q^{86} +2.76537 q^{88} -13.1580 q^{89} -9.08881i q^{92} +1.65564i q^{94} +5.10973i q^{95} -2.93853i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	− 2.76537i	− 0.833789i	−0.908955	−	0.416895i	$-0.863118\pi$
$11$	− 2.76537i	− 0.833789i	0.908955	−	0.416895i	$-0.136882\pi$
$12$	0	0
$13$	− 6.49661i	− 1.80183i	−0.433991	−	0.900917i	$-0.642895\pi$
$13$	− 6.49661i	− 1.80183i	0.433991	−	0.900917i	$-0.357105\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.95749	−1.68744	−0.843720	−	0.536784i	$-0.819639\pi$
$17$	−6.95749	−1.68744	−0.843720	+	0.536784i	$0.819639\pi$
$18$	0	0
$19$	5.10973i	1.17225i	0.810220	+	0.586126i	$0.199348\pi$
$19$	5.10973i	1.17225i	−0.810220	+	0.586126i	$0.800652\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.76537	−0.589578
$23$	9.08881i	1.89515i	0.319538	+	0.947574i	$0.396472\pi$
$23$	9.08881i	1.89515i	−0.319538	+	0.947574i	$0.603528\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.49661	−1.27409
$27$	0	0
$28$	0	0
$29$	1.14545i	0.212705i	0.994328	+	0.106353i	$0.0339172\pi$
$29$	1.14545i	0.212705i	−0.994328	+	0.106353i	$0.966083\pi$
$30$	0	0
$31$	9.75858i	1.75269i	0.481682	+	0.876346i	$0.340026\pi$
$31$	9.75858i	1.75269i	−0.481682	+	0.876346i	$0.659974\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	6.95749i	1.19320i
$35$	0	0
$36$	0	0
$37$	−8.09040	−1.33005	−0.665027	−	0.746820i	$-0.731580\pi$
$37$	−8.09040	−1.33005	−0.665027	+	0.746820i	$0.731580\pi$
$38$	5.10973	0.828908
$39$	0	0
$40$	1.00000i	0.158114i
$41$	4.91761	0.768001	0.384001	−	0.923333i	$-0.374546\pi$
$41$	4.91761	0.768001	0.384001	+	0.923333i	$0.374546\pi$
$42$	0	0
$43$	−11.3553	−1.73167	−0.865835	−	0.500330i	$-0.833212\pi$
$43$	−11.3553	−1.73167	−0.865835	+	0.500330i	$0.833212\pi$
$44$	2.76537i	0.416895i
$45$	0	0
$46$	9.08881	1.34007
$47$	−1.65564	−0.241499	−0.120750	−	0.992683i	$-0.538530\pi$
$47$	−1.65564	−0.241499	−0.120750	+	0.992683i	$0.538530\pi$
$48$	0	0
$49$	0	0
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	6.49661i	0.900917i
$53$	11.1757i	1.53511i	0.640986	+	0.767553i	$0.278526\pi$
$53$	11.1757i	1.53511i	−0.640986	+	0.767553i	$0.721474\pi$
$54$	0	0
$55$	− 2.76537i	− 0.372882i
$56$	0	0
$57$	0	0
$58$	1.14545	0.150405
$59$	9.50756	1.23778	0.618889	−	0.785478i	$-0.287583\pi$
$59$	9.50756	1.23778	0.618889	+	0.785478i	$0.287583\pi$
$60$	0	0
$61$	− 8.28130i	− 1.06031i	−0.847900	−	0.530156i	$-0.822133\pi$
$61$	− 8.28130i	− 1.06031i	0.847900	−	0.530156i	$-0.177867\pi$
$62$	9.75858	1.23934
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 6.49661i	− 0.805805i
$66$	0	0
$67$	2.41262	0.294749	0.147374	−	0.989081i	$-0.452918\pi$
$67$	2.41262	0.294749	0.147374	+	0.989081i	$0.452918\pi$
$68$	6.95749	0.843720
$69$	0	0
$70$	0	0
$71$	− 5.30411i	− 0.629482i	−0.949178	−	0.314741i	$-0.898082\pi$
$71$	− 5.30411i	− 0.629482i	0.949178	−	0.314741i	$-0.101918\pi$
$72$	0	0
$73$	− 8.34731i	− 0.976978i	−0.872570	−	0.488489i	$-0.837548\pi$
$73$	− 8.34731i	− 0.976978i	0.872570	−	0.488489i	$-0.162452\pi$
$74$	8.09040i	0.940490i
$75$	0	0
$76$	− 5.10973i	− 0.586126i
$77$	0	0
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	− 4.91761i	− 0.543059i
$83$	4.16478	0.457144	0.228572	−	0.973527i	$-0.426594\pi$
$83$	4.16478	0.457144	0.228572	+	0.973527i	$0.426594\pi$
$84$	0	0
$85$	−6.95749	−0.754646
$86$	11.3553i	1.22448i
$87$	0	0
$88$	2.76537	0.294789
$89$	−13.1580	−1.39474	−0.697372	−	0.716709i	$-0.745648\pi$
$89$	−13.1580	−1.39474	−0.697372	+	0.716709i	$0.745648\pi$
$90$	0	0
$91$	0	0
$92$	− 9.08881i	− 0.947574i
$93$	0	0
$94$	1.65564i	0.170766i
$95$	5.10973i	0.524247i
$96$	0	0
$97$	− 2.93853i	− 0.298363i	−0.988810	−	0.149181i	$-0.952336\pi$
$97$	− 2.93853i	− 0.298363i	0.988810	−	0.149181i	$-0.0476638\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−6.94495	−0.691048	−0.345524	−	0.938410i	$-0.612299\pi$
$101$	−6.94495	−0.691048	−0.345524	+	0.938410i	$0.612299\pi$
$102$	0	0
$103$	5.40289i	0.532362i	0.963923	+	0.266181i	$0.0857620\pi$
$103$	5.40289i	0.532362i	−0.963923	+	0.266181i	$0.914238\pi$
$104$	6.49661	0.637045
$105$	0	0
$106$	11.1757	1.08548
$107$	15.4248i	1.49117i	0.666410	+	0.745585i	$0.267830\pi$
$107$	15.4248i	1.49117i	−0.666410	+	0.745585i	$0.732170\pi$
$108$	0	0
$109$	12.1599	1.16470	0.582352	−	0.812936i	$-0.302132\pi$
$109$	12.1599	1.16470	0.582352	+	0.812936i	$0.302132\pi$
$110$	−2.76537	−0.263667
$111$	0	0
$112$	0	0
$113$	11.0866i	1.04294i	0.853271	+	0.521468i	$0.174615\pi$
$113$	11.0866i	1.04294i	−0.853271	+	0.521468i	$0.825385\pi$
$114$	0	0
$115$	9.08881i	0.847536i
$116$	− 1.14545i	− 0.106353i
$117$	0	0
$118$	− 9.50756i	− 0.875242i
$119$	0	0
$120$	0	0
$121$	3.35275	0.304795
$122$	−8.28130	−0.749754
$123$	0	0
$124$	− 9.75858i	− 0.876346i
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.1876	0.992738	0.496369	−	0.868112i	$-0.334666\pi$
$127$	11.1876	0.992738	0.496369	+	0.868112i	$0.334666\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−6.49661	−0.569790
$131$	0.876691	0.0765969	0.0382984	−	0.999266i	$-0.487806\pi$
$131$	0.876691	0.0765969	0.0382984	+	0.999266i	$0.487806\pi$
$132$	0	0
$133$	0	0
$134$	− 2.41262i	− 0.208419i
$135$	0	0
$136$	− 6.95749i	− 0.596600i
$137$	2.75473i	0.235353i	0.993052	+	0.117676i	$0.0375445\pi$
$137$	2.75473i	0.235353i	−0.993052	+	0.117676i	$0.962455\pi$
$138$	0	0
$139$	− 3.85839i	− 0.327265i	−0.986521	−	0.163632i	$-0.947679\pi$
$139$	− 3.85839i	− 0.327265i	0.986521	−	0.163632i	$-0.0523211\pi$
$140$	0	0
$141$	0	0
$142$	−5.30411	−0.445111
$143$	−17.9655	−1.50235
$144$	0	0
$145$	1.14545i	0.0951246i
$146$	−8.34731	−0.690828
$147$	0	0
$148$	8.09040	0.665027
$149$	13.1464i	1.07700i	0.842626	+	0.538499i	$0.181008\pi$
$149$	13.1464i	1.07700i	−0.842626	+	0.538499i	$0.818992\pi$
$150$	0	0
$151$	−6.50846	−0.529651	−0.264825	−	0.964296i	$-0.585314\pi$
$151$	−6.50846	−0.529651	−0.264825	+	0.964296i	$0.585314\pi$
$152$	−5.10973	−0.414454
$153$	0	0
$154$	0	0
$155$	9.75858i	0.783828i
$156$	0	0
$157$	15.5718i	1.24277i	0.783506	+	0.621384i	$0.213429\pi$
$157$	15.5718i	1.24277i	−0.783506	+	0.621384i	$0.786571\pi$
$158$	− 5.65685i	− 0.450035i
$159$	0	0
$160$	− 1.00000i	− 0.0790569i
$161$	0	0
$162$	0	0
$163$	−5.71350	−0.447516	−0.223758	−	0.974645i	$-0.571833\pi$
$163$	−5.71350	−0.447516	−0.223758	+	0.974645i	$0.571833\pi$
$164$	−4.91761	−0.384001
$165$	0	0
$166$	− 4.16478i	− 0.323250i
$167$	1.99387	0.154290	0.0771452	−	0.997020i	$-0.475419\pi$
$167$	1.99387	0.154290	0.0771452	+	0.997020i	$0.475419\pi$
$168$	0	0
$169$	−29.2059	−2.24661
$170$	6.95749i	0.533615i
$171$	0	0
$172$	11.3553	0.865835
$173$	−2.57619	−0.195864	−0.0979319	−	0.995193i	$-0.531223\pi$
$173$	−2.57619	−0.195864	−0.0979319	+	0.995193i	$0.531223\pi$
$174$	0	0
$175$	0	0
$176$	− 2.76537i	− 0.208447i
$177$	0	0
$178$	13.1580i	0.986233i
$179$	23.9822i	1.79251i	0.443535	+	0.896257i	$0.353724\pi$
$179$	23.9822i	1.79251i	−0.443535	+	0.896257i	$0.646276\pi$
$180$	0	0
$181$	− 14.0451i	− 1.04396i	−0.852957	−	0.521981i	$-0.825193\pi$
$181$	− 14.0451i	− 1.04396i	0.852957	−	0.521981i	$-0.174807\pi$
$182$	0	0
$183$	0	0
$184$	−9.08881	−0.670036
$185$	−8.09040	−0.594818
$186$	0	0
$187$	19.2400i	1.40697i
$188$	1.65564	0.120750
$189$	0	0
$190$	5.10973	0.370699
$191$	5.67685i	0.410762i	0.978682	+	0.205381i	$0.0658434\pi$
$191$	5.67685i	0.410762i	−0.978682	+	0.205381i	$0.934157\pi$
$192$	0	0
$193$	0.178361	0.0128387	0.00641937	−	0.999979i	$-0.497957\pi$
$193$	0.178361	0.0128387	0.00641937	+	0.999979i	$0.497957\pi$
$194$	−2.93853	−0.210974
$195$	0	0
$196$	0	0
$197$	27.4827i	1.95806i	0.203721	+	0.979029i	$0.434697\pi$
$197$	27.4827i	1.95806i	−0.203721	+	0.979029i	$0.565303\pi$
$198$	0	0
$199$	22.6577i	1.60616i	0.595870	+	0.803081i	$0.296807\pi$
$199$	22.6577i	1.60616i	−0.595870	+	0.803081i	$0.703193\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	6.94495i	0.488645i
$203$	0	0
$204$	0	0
$205$	4.91761	0.343461
$206$	5.40289	0.376437
$207$	0	0
$208$	− 6.49661i	− 0.450459i
$209$	14.1303	0.977412
$210$	0	0
$211$	−17.0907	−1.17657	−0.588287	−	0.808652i	$-0.700197\pi$
$211$	−17.0907	−1.17657	−0.588287	+	0.808652i	$0.700197\pi$
$212$	− 11.1757i	− 0.767553i
$213$	0	0
$214$	15.4248	1.05442
$215$	−11.3553	−0.774426
$216$	0	0
$217$	0	0
$218$	− 12.1599i	− 0.823571i
$219$	0	0
$220$	2.76537i	0.186441i
$221$	45.2001i	3.04049i
$222$	0	0
$223$	− 24.1850i	− 1.61954i	−0.586744	−	0.809772i	$-0.699591\pi$
$223$	− 24.1850i	− 1.61954i	0.586744	−	0.809772i	$-0.300409\pi$
$224$	0	0
$225$	0	0
$226$	11.0866	0.737467
$227$	−1.50114	−0.0996343	−0.0498171	−	0.998758i	$-0.515864\pi$
$227$	−1.50114	−0.0996343	−0.0498171	+	0.998758i	$0.515864\pi$
$228$	0	0
$229$	0.440203i	0.0290894i	0.999894	+	0.0145447i	$0.00462989\pi$
$229$	0.440203i	0.0290894i	−0.999894	+	0.0145447i	$0.995370\pi$
$230$	9.08881	0.599298
$231$	0	0
$232$	−1.14545	−0.0752026
$233$	− 0.789763i	− 0.0517391i	−0.999665	−	0.0258696i	$-0.991765\pi$
$233$	− 0.789763i	− 0.0517391i	0.999665	−	0.0258696i	$-0.00823545\pi$
$234$	0	0
$235$	−1.65564	−0.108002
$236$	−9.50756	−0.618889
$237$	0	0
$238$	0	0
$239$	4.18027i	0.270399i	0.990818	+	0.135200i	$0.0431676\pi$
$239$	4.18027i	0.270399i	−0.990818	+	0.135200i	$0.956832\pi$
$240$	0	0
$241$	− 5.92509i	− 0.381668i	−0.981622	−	0.190834i	$-0.938881\pi$
$241$	− 5.92509i	− 0.381668i	0.981622	−	0.190834i	$-0.0611193\pi$
$242$	− 3.35275i	− 0.215523i
$243$	0	0
$244$	8.28130i	0.530156i
$245$	0	0
$246$	0	0
$247$	33.1959	2.11221
$248$	−9.75858	−0.619670
$249$	0	0
$250$	− 1.00000i	− 0.0632456i
$251$	−4.39782	−0.277588	−0.138794	−	0.990321i	$-0.544323\pi$
$251$	−4.39782	−0.277588	−0.138794	+	0.990321i	$0.544323\pi$
$252$	0	0
$253$	25.1339	1.58015
$254$	− 11.1876i	− 0.701972i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.44450	−0.214862	−0.107431	−	0.994213i	$-0.534262\pi$
$257$	−3.44450	−0.214862	−0.107431	+	0.994213i	$0.534262\pi$
$258$	0	0
$259$	0	0
$260$	6.49661i	0.402902i
$261$	0	0
$262$	− 0.876691i	− 0.0541622i
$263$	1.07598i	0.0663476i	0.999450	+	0.0331738i	$0.0105615\pi$
$263$	1.07598i	0.0663476i	−0.999450	+	0.0331738i	$0.989439\pi$
$264$	0	0
$265$	11.1757i	0.686520i
$266$	0	0
$267$	0	0
$268$	−2.41262	−0.147374
$269$	0.393320	0.0239811	0.0119906	−	0.999928i	$-0.496183\pi$
$269$	0.393320	0.0239811	0.0119906	+	0.999928i	$0.496183\pi$
$270$	0	0
$271$	− 8.60024i	− 0.522427i	−0.965281	−	0.261214i	$-0.915877\pi$
$271$	− 8.60024i	− 0.522427i	0.965281	−	0.261214i	$-0.0841227\pi$
$272$	−6.95749	−0.421860
$273$	0	0
$274$	2.75473	0.166420
$275$	− 2.76537i	− 0.166758i
$276$	0	0
$277$	−0.247836	−0.0148910	−0.00744552	−	0.999972i	$-0.502370\pi$
$277$	−0.247836	−0.0148910	−0.00744552	+	0.999972i	$0.502370\pi$
$278$	−3.85839	−0.231411
$279$	0	0
$280$	0	0
$281$	− 11.6960i	− 0.697728i	−0.937173	−	0.348864i	$-0.886568\pi$
$281$	− 11.6960i	− 0.697728i	0.937173	−	0.348864i	$-0.113432\pi$
$282$	0	0
$283$	− 15.9005i	− 0.945185i	−0.881281	−	0.472592i	$-0.843318\pi$
$283$	− 15.9005i	− 0.945185i	0.881281	−	0.472592i	$-0.156682\pi$
$284$	5.30411i	0.314741i
$285$	0	0
$286$	17.9655i	1.06232i
$287$	0	0
$288$	0	0
$289$	31.4067	1.84745
$290$	1.14545	0.0672633
$291$	0	0
$292$	8.34731i	0.488489i
$293$	−20.5040	−1.19786	−0.598928	−	0.800803i	$-0.704407\pi$
$293$	−20.5040	−1.19786	−0.598928	+	0.800803i	$0.704407\pi$
$294$	0	0
$295$	9.50756	0.553551
$296$	− 8.09040i	− 0.470245i
$297$	0	0
$298$	13.1464	0.761552
$299$	59.0464	3.41474
$300$	0	0
$301$	0	0
$302$	6.50846i	0.374520i
$303$	0	0
$304$	5.10973i	0.293063i
$305$	− 8.28130i	− 0.474186i
$306$	0	0
$307$	− 17.7920i	− 1.01545i	−0.861520	−	0.507723i	$-0.830487\pi$
$307$	− 17.7920i	− 1.01545i	0.861520	−	0.507723i	$-0.169513\pi$
$308$	0	0
$309$	0	0
$310$	9.75858	0.554250
$311$	−16.2104	−0.919207	−0.459603	−	0.888124i	$-0.652008\pi$
$311$	−16.2104	−0.919207	−0.459603	+	0.888124i	$0.652008\pi$
$312$	0	0
$313$	0.406492i	0.0229763i	0.999934	+	0.0114881i	$0.00365687\pi$
$313$	0.406492i	0.0229763i	−0.999934	+	0.0114881i	$0.996343\pi$
$314$	15.5718	0.878770
$315$	0	0
$316$	−5.65685	−0.318223
$317$	24.5560i	1.37920i	0.724190	+	0.689600i	$0.242214\pi$
$317$	24.5560i	1.37920i	−0.724190	+	0.689600i	$0.757786\pi$
$318$	0	0
$319$	3.16760	0.177351
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	− 35.5509i	− 1.97811i
$324$	0	0
$325$	− 6.49661i	− 0.360367i
$326$	5.71350i	0.316441i
$327$	0	0
$328$	4.91761i	0.271529i
$329$	0	0
$330$	0	0
$331$	−14.1470	−0.777592	−0.388796	−	0.921324i	$-0.627109\pi$
$331$	−14.1470	−0.777592	−0.388796	+	0.921324i	$0.627109\pi$
$332$	−4.16478	−0.228572
$333$	0	0
$334$	− 1.99387i	− 0.109100i
$335$	2.41262	0.131816
$336$	0	0
$337$	−20.9855	−1.14315	−0.571577	−	0.820548i	$-0.693668\pi$
$337$	−20.9855	−1.14315	−0.571577	+	0.820548i	$0.693668\pi$
$338$	29.2059i	1.58859i
$339$	0	0
$340$	6.95749	0.377323
$341$	26.9860	1.46138
$342$	0	0
$343$	0	0
$344$	− 11.3553i	− 0.612238i
$345$	0	0
$346$	2.57619i	0.138497i
$347$	− 5.24996i	− 0.281832i	−0.990022	−	0.140916i	$-0.954995\pi$
$347$	− 5.24996i	− 0.281832i	0.990022	−	0.140916i	$-0.0450048\pi$
$348$	0	0
$349$	19.0338i	1.01886i	0.860514	+	0.509428i	$0.170143\pi$
$349$	19.0338i	1.01886i	−0.860514	+	0.509428i	$0.829857\pi$
$350$	0	0
$351$	0	0
$352$	−2.76537	−0.147395
$353$	11.8123	0.628707	0.314353	−	0.949306i	$-0.398212\pi$
$353$	11.8123	0.628707	0.314353	+	0.949306i	$0.398212\pi$
$354$	0	0
$355$	− 5.30411i	− 0.281513i
$356$	13.1580	0.697372
$357$	0	0
$358$	23.9822	1.26750
$359$	− 25.3999i	− 1.34055i	−0.742111	−	0.670277i	$-0.766175\pi$
$359$	− 25.3999i	− 1.34055i	0.742111	−	0.670277i	$-0.233825\pi$
$360$	0	0
$361$	−7.10936	−0.374177
$362$	−14.0451	−0.738193
$363$	0	0
$364$	0	0
$365$	− 8.34731i	− 0.436918i
$366$	0	0
$367$	− 11.3910i	− 0.594607i	−0.954783	−	0.297304i	$-0.903913\pi$
$367$	− 11.3910i	− 0.594607i	0.954783	−	0.297304i	$-0.0960874\pi$
$368$	9.08881i	0.473787i
$369$	0	0
$370$	8.09040i	0.420600i
$371$	0	0
$372$	0	0
$373$	−31.6667	−1.63964	−0.819821	−	0.572620i	$-0.805927\pi$
$373$	−31.6667	−1.63964	−0.819821	+	0.572620i	$0.805927\pi$
$374$	19.2400	0.994877
$375$	0	0
$376$	− 1.65564i	− 0.0853829i
$377$	7.44155	0.383259
$378$	0	0
$379$	−16.2498	−0.834696	−0.417348	−	0.908747i	$-0.637040\pi$
$379$	−16.2498	−0.834696	−0.417348	+	0.908747i	$0.637040\pi$
$380$	− 5.10973i	− 0.262124i
$381$	0	0
$382$	5.67685	0.290453
$383$	−14.5461	−0.743270	−0.371635	−	0.928379i	$-0.621203\pi$
$383$	−14.5461	−0.743270	−0.371635	+	0.928379i	$0.621203\pi$
$384$	0	0
$385$	0	0
$386$	− 0.178361i	− 0.00907835i
$387$	0	0
$388$	2.93853i	0.149181i
$389$	9.12771i	0.462793i	0.972859	+	0.231397i	$0.0743295\pi$
$389$	9.12771i	0.462793i	−0.972859	+	0.231397i	$0.925670\pi$
$390$	0	0
$391$	− 63.2353i	− 3.19795i
$392$	0	0
$393$	0	0
$394$	27.4827	1.38456
$395$	5.65685	0.284627
$396$	0	0
$397$	36.4414i	1.82894i	0.404654	+	0.914470i	$0.367392\pi$
$397$	36.4414i	1.82894i	−0.404654	+	0.914470i	$0.632608\pi$
$398$	22.6577	1.13573
$399$	0	0
$400$	1.00000	0.0500000
$401$	12.9700i	0.647693i	0.946110	+	0.323846i	$0.104976\pi$
$401$	12.9700i	0.647693i	−0.946110	+	0.323846i	$0.895024\pi$
$402$	0	0
$403$	63.3976	3.15806
$404$	6.94495	0.345524
$405$	0	0
$406$	0	0
$407$	22.3729i	1.10898i
$408$	0	0
$409$	− 26.0478i	− 1.28798i	−0.765034	−	0.643989i	$-0.777278\pi$
$409$	− 26.0478i	− 1.28798i	0.765034	−	0.643989i	$-0.222722\pi$
$410$	− 4.91761i	− 0.242863i
$411$	0	0
$412$	− 5.40289i	− 0.266181i
$413$	0	0
$414$	0	0
$415$	4.16478	0.204441
$416$	−6.49661	−0.318522
$417$	0	0
$418$	− 14.1303i	− 0.691135i
$419$	18.9823	0.927348	0.463674	−	0.886006i	$-0.346531\pi$
$419$	18.9823	0.927348	0.463674	+	0.886006i	$0.346531\pi$
$420$	0	0
$421$	30.0794	1.46598	0.732989	−	0.680240i	$-0.238125\pi$
$421$	30.0794	1.46598	0.732989	+	0.680240i	$0.238125\pi$
$422$	17.0907i	0.831963i
$423$	0	0
$424$	−11.1757	−0.542742
$425$	−6.95749	−0.337488
$426$	0	0
$427$	0	0
$428$	− 15.4248i	− 0.745585i
$429$	0	0
$430$	11.3553i	0.547602i
$431$	− 22.3429i	− 1.07622i	−0.842874	−	0.538110i	$-0.819138\pi$
$431$	− 22.3429i	− 1.07622i	0.842874	−	0.538110i	$-0.180862\pi$
$432$	0	0
$433$	19.0420i	0.915100i	0.889184	+	0.457550i	$0.151273\pi$
$433$	19.0420i	0.915100i	−0.889184	+	0.457550i	$0.848727\pi$
$434$	0	0
$435$	0	0
$436$	−12.1599	−0.582352
$437$	−46.4414	−2.22159
$438$	0	0
$439$	− 7.08684i	− 0.338236i	−0.985596	−	0.169118i	$-0.945908\pi$
$439$	− 7.08684i	− 0.338236i	0.985596	−	0.169118i	$-0.0540920\pi$
$440$	2.76537	0.131834
$441$	0	0
$442$	45.2001	2.14995
$443$	− 2.69780i	− 0.128176i	−0.997944	−	0.0640882i	$-0.979586\pi$
$443$	− 2.69780i	− 0.128176i	0.997944	−	0.0640882i	$-0.0204139\pi$
$444$	0	0
$445$	−13.1580	−0.623749
$446$	−24.1850	−1.14519
$447$	0	0
$448$	0	0
$449$	6.27674i	0.296218i	0.988971	+	0.148109i	$0.0473186\pi$
$449$	6.27674i	0.296218i	−0.988971	+	0.148109i	$0.952681\pi$
$450$	0	0
$451$	− 13.5990i	− 0.640351i
$452$	− 11.0866i	− 0.521468i
$453$	0	0
$454$	1.50114i	0.0704521i
$455$	0	0
$456$	0	0
$457$	−33.7275	−1.57771	−0.788854	−	0.614581i	$-0.789325\pi$
$457$	−33.7275	−1.57771	−0.788854	+	0.614581i	$0.789325\pi$
$458$	0.440203	0.0205693
$459$	0	0
$460$	− 9.08881i	− 0.423768i
$461$	−18.8149	−0.876295	−0.438147	−	0.898903i	$-0.644365\pi$
$461$	−18.8149	−0.876295	−0.438147	+	0.898903i	$0.644365\pi$
$462$	0	0
$463$	−20.6880	−0.961452	−0.480726	−	0.876871i	$-0.659627\pi$
$463$	−20.6880	−0.961452	−0.480726	+	0.876871i	$0.659627\pi$
$464$	1.14545i	0.0531763i
$465$	0	0
$466$	−0.789763	−0.0365851
$467$	−6.19119	−0.286494	−0.143247	−	0.989687i	$-0.545754\pi$
$467$	−6.19119	−0.286494	−0.143247	+	0.989687i	$0.545754\pi$
$468$	0	0
$469$	0	0
$470$	1.65564i	0.0763688i
$471$	0	0
$472$	9.50756i	0.437621i
$473$	31.4016i	1.44385i
$474$	0	0
$475$	5.10973i	0.234451i
$476$	0	0
$477$	0	0
$478$	4.18027	0.191201
$479$	16.0502	0.733351	0.366676	−	0.930349i	$-0.380496\pi$
$479$	16.0502	0.733351	0.366676	+	0.930349i	$0.380496\pi$
$480$	0	0
$481$	52.5601i	2.39654i
$482$	−5.92509	−0.269880
$483$	0	0
$484$	−3.35275	−0.152398
$485$	− 2.93853i	− 0.133432i
$486$	0	0
$487$	−29.9902	−1.35898	−0.679492	−	0.733683i	$-0.737800\pi$
$487$	−29.9902	−1.35898	−0.679492	+	0.733683i	$0.737800\pi$
$488$	8.28130	0.374877
$489$	0	0
$490$	0	0
$491$	32.8538i	1.48267i	0.671134	+	0.741336i	$0.265807\pi$
$491$	32.8538i	1.48267i	−0.671134	+	0.741336i	$0.734193\pi$
$492$	0	0
$493$	− 7.96947i	− 0.358927i
$494$	− 33.1959i	− 1.49355i
$495$	0	0
$496$	9.75858i	0.438173i
$497$	0	0
$498$	0	0
$499$	23.2615	1.04133	0.520663	−	0.853762i	$-0.325685\pi$
$499$	23.2615	1.04133	0.520663	+	0.853762i	$0.325685\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	4.39782i	0.196285i
$503$	−24.0973	−1.07444	−0.537222	−	0.843441i	$-0.680526\pi$
$503$	−24.0973	−1.07444	−0.537222	+	0.843441i	$0.680526\pi$
$504$	0	0
$505$	−6.94495	−0.309046
$506$	− 25.1339i	− 1.11734i
$507$	0	0
$508$	−11.1876	−0.496369
$509$	−19.8473	−0.879714	−0.439857	−	0.898068i	$-0.644971\pi$
$509$	−19.8473	−0.879714	−0.439857	+	0.898068i	$0.644971\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	3.44450i	0.151930i
$515$	5.40289i	0.238080i
$516$	0	0
$517$	4.57844i	0.201360i
$518$	0	0
$519$	0	0
$520$	6.49661	0.284895
$521$	−21.8241	−0.956130	−0.478065	−	0.878324i	$-0.658662\pi$
$521$	−21.8241	−0.956130	−0.478065	+	0.878324i	$0.658662\pi$
$522$	0	0
$523$	21.2504i	0.929215i	0.885517	+	0.464608i	$0.153805\pi$
$523$	21.2504i	0.929215i	−0.885517	+	0.464608i	$0.846195\pi$
$524$	−0.876691	−0.0382984
$525$	0	0
$526$	1.07598	0.0469148
$527$	− 67.8952i	− 2.95756i
$528$	0	0
$529$	−59.6064	−2.59158
$530$	11.1757	0.485443
$531$	0	0
$532$	0	0
$533$	− 31.9478i	− 1.38381i
$534$	0	0
$535$	15.4248i	0.666872i
$536$	2.41262i	0.104209i
$537$	0	0
$538$	− 0.393320i	− 0.0169572i
$539$	0	0
$540$	0	0
$541$	37.5669	1.61513	0.807564	−	0.589780i	$-0.200785\pi$
$541$	37.5669	1.61513	0.807564	+	0.589780i	$0.200785\pi$
$542$	−8.60024	−0.369412
$543$	0	0
$544$	6.95749i	0.298300i
$545$	12.1599	0.520872
$546$	0	0
$547$	7.83227	0.334884	0.167442	−	0.985882i	$-0.446449\pi$
$547$	7.83227	0.334884	0.167442	+	0.985882i	$0.446449\pi$
$548$	− 2.75473i	− 0.117676i
$549$	0	0
$550$	−2.76537	−0.117916
$551$	−5.85295	−0.249344
$552$	0	0
$553$	0	0
$554$	0.247836i	0.0105296i
$555$	0	0
$556$	3.85839i	0.163632i
$557$	− 11.4761i	− 0.486256i	−0.969994	−	0.243128i	$-0.921826\pi$
$557$	− 11.4761i	− 0.486256i	0.969994	−	0.243128i	$-0.0781735\pi$
$558$	0	0
$559$	73.7710i	3.12018i
$560$	0	0
$561$	0	0
$562$	−11.6960	−0.493368
$563$	−1.25134	−0.0527376	−0.0263688	−	0.999652i	$-0.508394\pi$
$563$	−1.25134	−0.0527376	−0.0263688	+	0.999652i	$0.508394\pi$
$564$	0	0
$565$	11.0866i	0.466415i
$566$	−15.9005	−0.668347
$567$	0	0
$568$	5.30411	0.222555
$569$	− 9.96325i	− 0.417681i	−0.977950	−	0.208840i	$-0.933031\pi$
$569$	− 9.96325i	− 0.417681i	0.977950	−	0.208840i	$-0.0669689\pi$
$570$	0	0
$571$	0.172475	0.00721786	0.00360893	−	0.999993i	$-0.498851\pi$
$571$	0.172475	0.00721786	0.00360893	+	0.999993i	$0.498851\pi$
$572$	17.9655	0.751175
$573$	0	0
$574$	0	0
$575$	9.08881i	0.379029i
$576$	0	0
$577$	22.8796i	0.952489i	0.879313	+	0.476245i	$0.158002\pi$
$577$	22.8796i	0.952489i	−0.879313	+	0.476245i	$0.841998\pi$
$578$	− 31.4067i	− 1.30635i
$579$	0	0
$580$	− 1.14545i	− 0.0475623i
$581$	0	0
$582$	0	0
$583$	30.9050	1.27995
$584$	8.34731	0.345414
$585$	0	0
$586$	20.5040i	0.847012i
$587$	11.9209	0.492027	0.246013	−	0.969266i	$-0.420879\pi$
$587$	11.9209	0.492027	0.246013	+	0.969266i	$0.420879\pi$
$588$	0	0
$589$	−49.8637	−2.05460
$590$	− 9.50756i	− 0.391420i
$591$	0	0
$592$	−8.09040	−0.332513
$593$	27.3125	1.12159	0.560795	−	0.827955i	$-0.310496\pi$
$593$	27.3125	1.12159	0.560795	+	0.827955i	$0.310496\pi$
$594$	0	0
$595$	0	0
$596$	− 13.1464i	− 0.538499i
$597$	0	0
$598$	− 59.0464i	− 2.41459i
$599$	19.6598i	0.803279i	0.915798	+	0.401639i	$0.131559\pi$
$599$	19.6598i	0.803279i	−0.915798	+	0.401639i	$0.868441\pi$
$600$	0	0
$601$	− 7.44641i	− 0.303745i	−0.988400	−	0.151873i	$-0.951470\pi$
$601$	− 7.44641i	− 0.303745i	0.988400	−	0.151873i	$-0.0485303\pi$
$602$	0	0
$603$	0	0
$604$	6.50846	0.264825
$605$	3.35275	0.136309
$606$	0	0
$607$	− 4.90328i	− 0.199018i	−0.995037	−	0.0995091i	$-0.968273\pi$
$607$	− 4.90328i	− 0.199018i	0.995037	−	0.0995091i	$-0.0317272\pi$
$608$	5.10973	0.207227
$609$	0	0
$610$	−8.28130	−0.335300
$611$	10.7560i	0.435142i
$612$	0	0
$613$	−12.7961	−0.516830	−0.258415	−	0.966034i	$-0.583200\pi$
$613$	−12.7961	−0.516830	−0.258415	+	0.966034i	$0.583200\pi$
$614$	−17.7920	−0.718029
$615$	0	0
$616$	0	0
$617$	− 47.7755i	− 1.92337i	−0.274160	−	0.961684i	$-0.588400\pi$
$617$	− 47.7755i	− 1.92337i	0.274160	−	0.961684i	$-0.411600\pi$
$618$	0	0
$619$	10.0939i	0.405707i	0.979209	+	0.202853i	$0.0650215\pi$
$619$	10.0939i	0.405707i	−0.979209	+	0.202853i	$0.934978\pi$
$620$	− 9.75858i	− 0.391914i
$621$	0	0
$622$	16.2104i	0.649977i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.406492	0.0162467
$627$	0	0
$628$	− 15.5718i	− 0.621384i
$629$	56.2889	2.24438
$630$	0	0
$631$	−9.17210	−0.365136	−0.182568	−	0.983193i	$-0.558441\pi$
$631$	−9.17210	−0.365136	−0.182568	+	0.983193i	$0.558441\pi$
$632$	5.65685i	0.225018i
$633$	0	0
$634$	24.5560	0.975242
$635$	11.1876	0.443966
$636$	0	0
$637$	0	0
$638$	− 3.16760i	− 0.125406i
$639$	0	0
$640$	1.00000i	0.0395285i
$641$	− 37.7841i	− 1.49238i	−0.665731	−	0.746192i	$-0.731880\pi$
$641$	− 37.7841i	− 1.49238i	0.665731	−	0.746192i	$-0.268120\pi$
$642$	0	0
$643$	− 41.5218i	− 1.63746i	−0.574177	−	0.818731i	$-0.694678\pi$
$643$	− 41.5218i	− 1.63746i	0.574177	−	0.818731i	$-0.305322\pi$
$644$	0	0
$645$	0	0
$646$	−35.5509	−1.39873
$647$	−15.1233	−0.594556	−0.297278	−	0.954791i	$-0.596079\pi$
$647$	−15.1233	−0.594556	−0.297278	+	0.954791i	$0.596079\pi$
$648$	0	0
$649$	− 26.2919i	− 1.03205i
$650$	−6.49661	−0.254818
$651$	0	0
$652$	5.71350	0.223758
$653$	8.13519i	0.318355i	0.987250	+	0.159177i	$0.0508842\pi$
$653$	8.13519i	0.318355i	−0.987250	+	0.159177i	$0.949116\pi$
$654$	0	0
$655$	0.876691	0.0342552
$656$	4.91761	0.192000
$657$	0	0
$658$	0	0
$659$	47.6077i	1.85453i	0.374400	+	0.927267i	$0.377849\pi$
$659$	47.6077i	1.85453i	−0.374400	+	0.927267i	$0.622151\pi$
$660$	0	0
$661$	− 40.8533i	− 1.58901i	−0.607258	−	0.794505i	$-0.707731\pi$
$661$	− 40.8533i	− 1.58901i	0.607258	−	0.794505i	$-0.292269\pi$
$662$	14.1470i	0.549841i
$663$	0	0
$664$	4.16478i	0.161625i
$665$	0	0
$666$	0	0
$667$	−10.4108	−0.403108
$668$	−1.99387	−0.0771452
$669$	0	0
$670$	− 2.41262i	− 0.0932077i
$671$	−22.9008	−0.884077
$672$	0	0
$673$	17.2558	0.665164	0.332582	−	0.943074i	$-0.392080\pi$
$673$	17.2558	0.665164	0.332582	+	0.943074i	$0.392080\pi$
$674$	20.9855i	0.808332i
$675$	0	0
$676$	29.2059	1.12330
$677$	20.6013	0.791771	0.395886	−	0.918300i	$-0.370438\pi$
$677$	20.6013	0.791771	0.395886	+	0.918300i	$0.370438\pi$
$678$	0	0
$679$	0	0
$680$	− 6.95749i	− 0.266808i
$681$	0	0
$682$	− 26.9860i	− 1.03335i
$683$	− 11.1750i	− 0.427599i	−0.976878	−	0.213800i	$-0.931416\pi$
$683$	− 11.1750i	− 0.427599i	0.976878	−	0.213800i	$-0.0685839\pi$
$684$	0	0
$685$	2.75473i	0.105253i
$686$	0	0
$687$	0	0
$688$	−11.3553	−0.432917
$689$	72.6043	2.76601
$690$	0	0
$691$	− 26.5432i	− 1.00975i	−0.863192	−	0.504875i	$-0.831538\pi$
$691$	− 26.5432i	− 1.00975i	0.863192	−	0.504875i	$-0.168462\pi$
$692$	2.57619	0.0979319
$693$	0	0
$694$	−5.24996	−0.199286
$695$	− 3.85839i	− 0.146357i
$696$	0	0
$697$	−34.2142	−1.29596
$698$	19.0338	0.720439
$699$	0	0
$700$	0	0
$701$	− 1.73217i	− 0.0654232i	−0.999465	−	0.0327116i	$-0.989586\pi$
$701$	− 1.73217i	− 0.0654232i	0.999465	−	0.0327116i	$-0.0104143\pi$
$702$	0	0
$703$	− 41.3398i	− 1.55916i
$704$	2.76537i	0.104224i
$705$	0	0
$706$	− 11.8123i	− 0.444563i
$707$	0	0
$708$	0	0
$709$	−7.25723	−0.272551	−0.136275	−	0.990671i	$-0.543513\pi$
$709$	−7.25723	−0.272551	−0.136275	+	0.990671i	$0.543513\pi$
$710$	−5.30411	−0.199060
$711$	0	0
$712$	− 13.1580i	− 0.493117i
$713$	−88.6938	−3.32161
$714$	0	0
$715$	−17.9655	−0.671872
$716$	− 23.9822i	− 0.896257i
$717$	0	0
$718$	−25.3999	−0.947915
$719$	20.6632	0.770609	0.385304	−	0.922790i	$-0.374096\pi$
$719$	20.6632	0.770609	0.385304	+	0.922790i	$0.374096\pi$
$720$	0	0
$721$	0	0
$722$	7.10936i	0.264583i
$723$	0	0
$724$	14.0451i	0.521981i
$725$	1.14545i	0.0425410i
$726$	0	0
$727$	− 19.2568i	− 0.714196i	−0.934067	−	0.357098i	$-0.883766\pi$
$727$	− 19.2568i	− 0.714196i	0.934067	−	0.357098i	$-0.116234\pi$
$728$	0	0
$729$	0	0
$730$	−8.34731	−0.308948
$731$	79.0045	2.92209
$732$	0	0
$733$	31.3948i	1.15959i	0.814761	+	0.579797i	$0.196868\pi$
$733$	31.3948i	1.15959i	−0.814761	+	0.579797i	$0.803132\pi$
$734$	−11.3910	−0.420451
$735$	0	0
$736$	9.08881	0.335018
$737$	− 6.67178i	− 0.245758i
$738$	0	0
$739$	22.6120	0.831797	0.415899	−	0.909411i	$-0.363467\pi$
$739$	22.6120	0.831797	0.415899	+	0.909411i	$0.363467\pi$
$740$	8.09040	0.297409
$741$	0	0
$742$	0	0
$743$	2.17761i	0.0798889i	0.999202	+	0.0399445i	$0.0127181\pi$
$743$	2.17761i	0.0798889i	−0.999202	+	0.0399445i	$0.987282\pi$
$744$	0	0
$745$	13.1464i	0.481648i
$746$	31.6667i	1.15940i
$747$	0	0
$748$	− 19.2400i	− 0.703485i
$749$	0	0
$750$	0	0
$751$	14.9373	0.545068	0.272534	−	0.962146i	$-0.412138\pi$
$751$	14.9373	0.545068	0.272534	+	0.962146i	$0.412138\pi$
$752$	−1.65564	−0.0603748
$753$	0	0
$754$	− 7.44155i	− 0.271005i
$755$	−6.50846	−0.236867
$756$	0	0
$757$	9.08950	0.330363	0.165182	−	0.986263i	$-0.447179\pi$
$757$	9.08950	0.330363	0.165182	+	0.986263i	$0.447179\pi$
$758$	16.2498i	0.590219i
$759$	0	0
$760$	−5.10973	−0.185349
$761$	16.8106	0.609384	0.304692	−	0.952451i	$-0.401446\pi$
$761$	16.8106	0.609384	0.304692	+	0.952451i	$0.401446\pi$
$762$	0	0
$763$	0	0
$764$	− 5.67685i	− 0.205381i
$765$	0	0
$766$	14.5461i	0.525572i
$767$	− 61.7668i	− 2.23027i
$768$	0	0
$769$	− 2.22328i	− 0.0801735i	−0.999196	−	0.0400867i	$-0.987237\pi$
$769$	− 2.22328i	− 0.0801735i	0.999196	−	0.0400867i	$-0.0127634\pi$
$770$	0	0
$771$	0	0
$772$	−0.178361	−0.00641937
$773$	−16.0056	−0.575682	−0.287841	−	0.957678i	$-0.592938\pi$
$773$	−16.0056	−0.575682	−0.287841	+	0.957678i	$0.592938\pi$
$774$	0	0
$775$	9.75858i	0.350538i
$776$	2.93853	0.105487
$777$	0	0
$778$	9.12771	0.327244
$779$	25.1277i	0.900292i
$780$	0	0
$781$	−14.6678	−0.524855
$782$	−63.2353	−2.26129
$783$	0	0
$784$	0	0
$785$	15.5718i	0.555783i
$786$	0	0
$787$	1.48963i	0.0530996i	0.999647	+	0.0265498i	$0.00845206\pi$
$787$	1.48963i	0.0530996i	−0.999647	+	0.0265498i	$0.991548\pi$
$788$	− 27.4827i	− 0.979029i
$789$	0	0
$790$	− 5.65685i	− 0.201262i
$791$	0	0
$792$	0	0
$793$	−53.8004	−1.91051
$794$	36.4414	1.29326
$795$	0	0
$796$	− 22.6577i	− 0.803081i
$797$	1.19209	0.0422261	0.0211131	−	0.999777i	$-0.493279\pi$
$797$	1.19209	0.0422261	0.0211131	+	0.999777i	$0.493279\pi$
$798$	0	0
$799$	11.5191	0.407515
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	12.9700	0.457988
$803$	−23.0834	−0.814594
$804$	0	0
$805$	0	0
$806$	− 63.3976i	− 2.23309i
$807$	0	0
$808$	− 6.94495i	− 0.244322i
$809$	29.3201i	1.03084i	0.856938	+	0.515420i	$0.172364\pi$
$809$	29.3201i	1.03084i	−0.856938	+	0.515420i	$0.827636\pi$
$810$	0	0
$811$	44.3203i	1.55630i	0.628081	+	0.778148i	$0.283841\pi$
$811$	44.3203i	1.55630i	−0.628081	+	0.778148i	$0.716159\pi$
$812$	0	0
$813$	0	0
$814$	22.3729	0.784171
$815$	−5.71350	−0.200135
$816$	0	0
$817$	− 58.0226i	− 2.02995i
$818$	−26.0478	−0.910739
$819$	0	0
$820$	−4.91761	−0.171730
$821$	− 5.17748i	− 0.180695i	−0.995910	−	0.0903477i	$-0.971202\pi$
$821$	− 5.17748i	− 0.180695i	0.995910	−	0.0903477i	$-0.0287978\pi$
$822$	0	0
$823$	−43.9714	−1.53274	−0.766372	−	0.642396i	$-0.777940\pi$
$823$	−43.9714	−1.53274	−0.766372	+	0.642396i	$0.777940\pi$
$824$	−5.40289	−0.188219
$825$	0	0
$826$	0	0
$827$	7.99756i	0.278102i	0.990285	+	0.139051i	$0.0444053\pi$
$827$	7.99756i	0.278102i	−0.990285	+	0.139051i	$0.955595\pi$
$828$	0	0
$829$	13.4103i	0.465758i	0.972506	+	0.232879i	$0.0748147\pi$
$829$	13.4103i	0.465758i	−0.972506	+	0.232879i	$0.925185\pi$
$830$	− 4.16478i	− 0.144562i
$831$	0	0
$832$	6.49661i	0.225229i
$833$	0	0
$834$	0	0
$835$	1.99387	0.0690008
$836$	−14.1303	−0.488706
$837$	0	0
$838$	− 18.9823i	− 0.655734i
$839$	−49.6938	−1.71562	−0.857810	−	0.513967i	$-0.828175\pi$
$839$	−49.6938	−1.71562	−0.857810	+	0.513967i	$0.828175\pi$
$840$	0	0
$841$	27.6879	0.954757
$842$	− 30.0794i	− 1.03660i
$843$	0	0
$844$	17.0907	0.588287
$845$	−29.2059	−1.00471
$846$	0	0
$847$	0	0
$848$	11.1757i	0.383776i
$849$	0	0
$850$	6.95749i	0.238640i
$851$	− 73.5321i	− 2.52065i
$852$	0	0
$853$	− 39.5178i	− 1.35306i	−0.736414	−	0.676532i	$-0.763482\pi$
$853$	− 39.5178i	− 1.35306i	0.736414	−	0.676532i	$-0.236518\pi$
$854$	0	0
$855$	0	0
$856$	−15.4248	−0.527208
$857$	11.9407	0.407885	0.203942	−	0.978983i	$-0.434624\pi$
$857$	11.9407	0.407885	0.203942	+	0.978983i	$0.434624\pi$
$858$	0	0
$859$	19.3905i	0.661596i	0.943702	+	0.330798i	$0.107318\pi$
$859$	19.3905i	0.661596i	−0.943702	+	0.330798i	$0.892682\pi$
$860$	11.3553	0.387213
$861$	0	0
$862$	−22.3429	−0.761003
$863$	− 6.23574i	− 0.212267i	−0.994352	−	0.106134i	$-0.966153\pi$
$863$	− 6.23574i	− 0.212267i	0.994352	−	0.106134i	$-0.0338471\pi$
$864$	0	0
$865$	−2.57619	−0.0875930
$866$	19.0420	0.647074
$867$	0	0
$868$	0	0
$869$	− 15.6433i	− 0.530662i
$870$	0	0
$871$	− 15.6738i	− 0.531088i
$872$	12.1599i	0.411785i
$873$	0	0
$874$	46.4414i	1.57090i
$875$	0	0
$876$	0	0
$877$	−18.4355	−0.622523	−0.311261	−	0.950324i	$-0.600751\pi$
$877$	−18.4355	−0.622523	−0.311261	+	0.950324i	$0.600751\pi$
$878$	−7.08684	−0.239169
$879$	0	0
$880$	− 2.76537i	− 0.0932205i
$881$	34.6921	1.16881	0.584404	−	0.811463i	$-0.301328\pi$
$881$	34.6921	1.16881	0.584404	+	0.811463i	$0.301328\pi$
$882$	0	0
$883$	31.9748	1.07604	0.538019	−	0.842933i	$-0.319173\pi$
$883$	31.9748	1.07604	0.538019	+	0.842933i	$0.319173\pi$
$884$	− 45.2001i	− 1.52024i
$885$	0	0
$886$	−2.69780	−0.0906344
$887$	17.7376	0.595571	0.297785	−	0.954633i	$-0.403752\pi$
$887$	17.7376	0.595571	0.297785	+	0.954633i	$0.403752\pi$
$888$	0	0
$889$	0	0
$890$	13.1580i	0.441057i
$891$	0	0
$892$	24.1850i	0.809772i
$893$	− 8.45985i	− 0.283098i
$894$	0	0
$895$	23.9822i	0.801637i
$896$	0	0
$897$	0	0
$898$	6.27674	0.209457
$899$	−11.1780	−0.372807
$900$	0	0
$901$	− 77.7551i	− 2.59040i
$902$	−13.5990	−0.452797
$903$	0	0
$904$	−11.0866	−0.368733
$905$	− 14.0451i	− 0.466874i
$906$	0	0
$907$	21.8834	0.726627	0.363314	−	0.931667i	$-0.381645\pi$
$907$	21.8834	0.726627	0.363314	+	0.931667i	$0.381645\pi$
$908$	1.50114	0.0498171
$909$	0	0
$910$	0	0
$911$	− 19.8054i	− 0.656182i	−0.944646	−	0.328091i	$-0.893595\pi$
$911$	− 19.8054i	− 0.656182i	0.944646	−	0.328091i	$-0.106405\pi$
$912$	0	0
$913$	− 11.5172i	− 0.381162i
$914$	33.7275i	1.11561i
$915$	0	0
$916$	− 0.440203i	− 0.0145447i
$917$	0	0
$918$	0	0
$919$	41.7057	1.37574	0.687872	−	0.725832i	$-0.258545\pi$
$919$	41.7057	1.37574	0.687872	+	0.725832i	$0.258545\pi$
$920$	−9.08881	−0.299649
$921$	0	0
$922$	18.8149i	0.619634i
$923$	−34.4587	−1.13422
$924$	0	0
$925$	−8.09040	−0.266011
$926$	20.6880i	0.679849i
$927$	0	0
$928$	1.14545	0.0376013
$929$	−15.2287	−0.499637	−0.249819	−	0.968293i	$-0.580371\pi$
$929$	−15.2287	−0.499637	−0.249819	+	0.968293i	$0.580371\pi$
$930$	0	0
$931$	0	0
$932$	0.789763i	0.0258696i
$933$	0	0
$934$	6.19119i	0.202582i
$935$	19.2400i	0.629216i
$936$	0	0
$937$	16.1466i	0.527488i	0.964593	+	0.263744i	$0.0849574\pi$
$937$	16.1466i	0.527488i	−0.964593	+	0.263744i	$0.915043\pi$
$938$	0	0
$939$	0	0
$940$	1.65564	0.0540009
$941$	−0.403182	−0.0131434	−0.00657169	−	0.999978i	$-0.502092\pi$
$941$	−0.403182	−0.0131434	−0.00657169	+	0.999978i	$0.502092\pi$
$942$	0	0
$943$	44.6952i	1.45548i
$944$	9.50756	0.309445
$945$	0	0
$946$	31.4016	1.02095
$947$	− 21.1653i	− 0.687781i	−0.939010	−	0.343891i	$-0.888255\pi$
$947$	− 21.1653i	− 0.687781i	0.939010	−	0.343891i	$-0.111745\pi$
$948$	0	0
$949$	−54.2292	−1.76035
$950$	5.10973	0.165782
$951$	0	0
$952$	0	0
$953$	36.2815i	1.17527i	0.809125	+	0.587637i	$0.199942\pi$
$953$	36.2815i	1.17527i	−0.809125	+	0.587637i	$0.800058\pi$
$954$	0	0
$955$	5.67685i	0.183698i
$956$	− 4.18027i	− 0.135200i
$957$	0	0
$958$	− 16.0502i	− 0.518558i
$959$	0	0
$960$	0	0
$961$	−64.2299	−2.07193
$962$	52.5601	1.69461
$963$	0	0
$964$	5.92509i	0.190834i
$965$	0.178361	0.00574166
$966$	0	0
$967$	4.18473	0.134572	0.0672860	−	0.997734i	$-0.478566\pi$
$967$	4.18473	0.134572	0.0672860	+	0.997734i	$0.478566\pi$
$968$	3.35275i	0.107761i
$969$	0	0
$970$	−2.93853	−0.0943506
$971$	12.8820	0.413405	0.206702	−	0.978404i	$-0.433727\pi$
$971$	12.8820	0.413405	0.206702	+	0.978404i	$0.433727\pi$
$972$	0	0
$973$	0	0
$974$	29.9902i	0.960947i
$975$	0	0
$976$	− 8.28130i	− 0.265078i
$977$	− 9.29637i	− 0.297417i	−0.988881	−	0.148709i	$-0.952488\pi$
$977$	− 9.29637i	− 0.297417i	0.988881	−	0.148709i	$-0.0475117\pi$
$978$	0	0
$979$	36.3867i	1.16292i
$980$	0	0
$981$	0	0
$982$	32.8538	1.04841
$983$	6.54227	0.208666	0.104333	−	0.994542i	$-0.466729\pi$
$983$	6.54227	0.208666	0.104333	+	0.994542i	$0.466729\pi$
$984$	0	0
$985$	27.4827i	0.875670i
$986$	−7.96947	−0.253800
$987$	0	0
$988$	−33.1959	−1.05610
$989$	− 103.206i	− 3.28177i
$990$	0	0
$991$	43.9806	1.39709	0.698546	−	0.715566i	$-0.253831\pi$
$991$	43.9806	1.39709	0.698546	+	0.715566i	$0.253831\pi$
$992$	9.75858	0.309835
$993$	0	0
$994$	0	0
$995$	22.6577i	0.718297i
$996$	0	0
$997$	− 29.9536i	− 0.948641i	−0.880352	−	0.474321i	$-0.842694\pi$
$997$	− 29.9536i	− 0.948641i	0.880352	−	0.474321i	$-0.157306\pi$
$998$	− 23.2615i	− 0.736329i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.d.881.2	yes	8
3.2	odd	2		4410.2.b.a.881.7	yes	8
7.6	odd	2		4410.2.b.a.881.2	✓	8
21.20	even	2	inner	4410.2.b.d.881.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.a.881.2	✓	8	7.6	odd	2
4410.2.b.a.881.7	yes	8	3.2	odd	2
4410.2.b.d.881.2	yes	8	1.1	even	1	trivial
4410.2.b.d.881.7	yes	8	21.20	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +1.00000i q^{8} -1.00000i q^{10} -2.76537i q^{11} -6.49661i q^{13} +1.00000 q^{16} -6.95749 q^{17} +5.10973i q^{19} -1.00000 q^{20} -2.76537 q^{22} +9.08881i q^{23} +1.00000 q^{25} -6.49661 q^{26} +1.14545i q^{29} +9.75858i q^{31} -1.00000i q^{32} +6.95749i q^{34} -8.09040 q^{37} +5.10973 q^{38} +1.00000i q^{40} +4.91761 q^{41} -11.3553 q^{43} +2.76537i q^{44} +9.08881 q^{46} -1.65564 q^{47} -1.00000i q^{50} +6.49661i q^{52} +11.1757i q^{53} -2.76537i q^{55} +1.14545 q^{58} +9.50756 q^{59} -8.28130i q^{61} +9.75858 q^{62} -1.00000 q^{64} -6.49661i q^{65} +2.41262 q^{67} +6.95749 q^{68} -5.30411i q^{71} -8.34731i q^{73} +8.09040i q^{74} -5.10973i q^{76} +5.65685 q^{79} +1.00000 q^{80} -4.91761i q^{82} +4.16478 q^{83} -6.95749 q^{85} +11.3553i q^{86} +2.76537 q^{88} -13.1580 q^{89} -9.08881i q^{92} +1.65564i q^{94} +5.10973i q^{95} -2.93853i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 8 q^{5}+O(q^{10}) \) 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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