LMFDB - Embedded newform 4410.2.b.a.881.5

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			881.5
Root			$0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	4410.881
Dual form			4410.2.b.a.881.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} -1.00000i q^{10} +0.152241i q^{11} -5.19891i q^{13} +1.00000 q^{16} +0.881887 q^{17} +0.116520i q^{19} +1.00000 q^{20} -0.152241 q^{22} +7.88764i q^{23} +1.00000 q^{25} +5.19891 q^{26} +0.367542i q^{29} +4.55007i q^{31} +1.00000i q^{32} +0.881887i q^{34} +1.47727 q^{37} -0.116520 q^{38} +1.00000i q^{40} -3.38687 q^{41} -3.54553 q^{43} -0.152241i q^{44} -7.88764 q^{46} +4.03572 q^{47} +1.00000i q^{50} +5.19891i q^{52} -8.80750i q^{53} -0.152241i q^{55} -0.367542 q^{58} -2.78016 q^{59} -8.94495i q^{61} -4.55007 q^{62} -1.00000 q^{64} +5.19891i q^{65} -7.82458 q^{67} -0.881887 q^{68} -13.6337i q^{71} -11.6359i q^{73} +1.47727i q^{74} -0.116520i q^{76} -5.65685 q^{79} -1.00000 q^{80} -3.38687i q^{82} -7.22625 q^{83} -0.881887 q^{85} -3.54553i q^{86} +0.152241 q^{88} +13.6241 q^{89} -7.88764i q^{92} +4.03572i q^{94} -0.116520i q^{95} -13.3910i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	0.152241i	0.0459024i	0.999737	+	0.0229512i	$0.00730623\pi$
$11$	0.152241i	0.0459024i	−0.999737	+	0.0229512i	$0.992694\pi$
$12$	0	0
$13$	− 5.19891i	− 1.44192i	−0.692977	−	0.720959i	$-0.743701\pi$
$13$	− 5.19891i	− 1.44192i	0.692977	−	0.720959i	$-0.256299\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0.881887	0.213889	0.106945	−	0.994265i	$-0.465893\pi$
$17$	0.881887	0.213889	0.106945	+	0.994265i	$0.465893\pi$
$18$	0	0
$19$	0.116520i	0.0267316i	0.999911	+	0.0133658i	$0.00425459\pi$
$19$	0.116520i	0.0267316i	−0.999911	+	0.0133658i	$0.995745\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−0.152241	−0.0324579
$23$	7.88764i	1.64469i	0.568991	+	0.822344i	$0.307334\pi$
$23$	7.88764i	1.64469i	−0.568991	+	0.822344i	$0.692666\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.19891	1.01959
$27$	0	0
$28$	0	0
$29$	0.367542i	0.0682509i	0.999418	+	0.0341254i	$0.0108646\pi$
$29$	0.367542i	0.0682509i	−0.999418	+	0.0341254i	$0.989135\pi$
$30$	0	0
$31$	4.55007i	0.817216i	0.912710	+	0.408608i	$0.133986\pi$
$31$	4.55007i	0.817216i	−0.912710	+	0.408608i	$0.866014\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0.881887i	0.151242i
$35$	0	0
$36$	0	0
$37$	1.47727	0.242862	0.121431	−	0.992600i	$-0.461252\pi$
$37$	1.47727	0.242862	0.121431	+	0.992600i	$0.461252\pi$
$38$	−0.116520	−0.0189021
$39$	0	0
$40$	1.00000i	0.158114i
$41$	−3.38687	−0.528941	−0.264470	−	0.964394i	$-0.585197\pi$
$41$	−3.38687	−0.528941	−0.264470	+	0.964394i	$0.585197\pi$
$42$	0	0
$43$	−3.54553	−0.540688	−0.270344	−	0.962764i	$-0.587137\pi$
$43$	−3.54553	−0.540688	−0.270344	+	0.962764i	$0.587137\pi$
$44$	− 0.152241i	− 0.0229512i
$45$	0	0
$46$	−7.88764	−1.16297
$47$	4.03572	0.588670	0.294335	−	0.955702i	$-0.404902\pi$
$47$	4.03572	0.588670	0.294335	+	0.955702i	$0.404902\pi$
$48$	0	0
$49$	0	0
$50$	1.00000i	0.141421i
$51$	0	0
$52$	5.19891i	0.720959i
$53$	− 8.80750i	− 1.20980i	−0.796300	−	0.604902i	$-0.793212\pi$
$53$	− 8.80750i	− 1.20980i	0.796300	−	0.604902i	$-0.206788\pi$
$54$	0	0
$55$	− 0.152241i	− 0.0205282i
$56$	0	0
$57$	0	0
$58$	−0.367542	−0.0482606
$59$	−2.78016	−0.361946	−0.180973	−	0.983488i	$-0.557925\pi$
$59$	−2.78016	−0.361946	−0.180973	+	0.983488i	$0.557925\pi$
$60$	0	0
$61$	− 8.94495i	− 1.14528i	−0.819806	−	0.572642i	$-0.805919\pi$
$61$	− 8.94495i	− 1.14528i	0.819806	−	0.572642i	$-0.194081\pi$
$62$	−4.55007	−0.577859
$63$	0	0
$64$	−1.00000	−0.125000
$65$	5.19891i	0.644846i
$66$	0	0
$67$	−7.82458	−0.955925	−0.477962	−	0.878380i	$-0.658624\pi$
$67$	−7.82458	−0.955925	−0.477962	+	0.878380i	$0.658624\pi$
$68$	−0.881887	−0.106945
$69$	0	0
$70$	0	0
$71$	− 13.6337i	− 1.61802i	−0.587795	−	0.809010i	$-0.700004\pi$
$71$	− 13.6337i	− 1.61802i	0.587795	−	0.809010i	$-0.299996\pi$
$72$	0	0
$73$	− 11.6359i	− 1.36188i	−0.732338	−	0.680941i	$-0.761571\pi$
$73$	− 11.6359i	− 1.36188i	0.732338	−	0.680941i	$-0.238429\pi$
$74$	1.47727i	0.171730i
$75$	0	0
$76$	− 0.116520i	− 0.0133658i
$77$	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
$82$	− 3.38687i	− 0.374018i
$83$	−7.22625	−0.793184	−0.396592	−	0.917995i	$-0.629807\pi$
$83$	−7.22625	−0.793184	−0.396592	+	0.917995i	$0.629807\pi$
$84$	0	0
$85$	−0.881887	−0.0956541
$86$	− 3.54553i	− 0.382324i
$87$	0	0
$88$	0.152241	0.0162289
$89$	13.6241	1.44415	0.722075	−	0.691815i	$-0.243189\pi$
$89$	13.6241	1.44415	0.722075	+	0.691815i	$0.243189\pi$
$90$	0	0
$91$	0	0
$92$	− 7.88764i	− 0.822344i
$93$	0	0
$94$	4.03572i	0.416253i
$95$	− 0.116520i	− 0.0119547i
$96$	0	0
$97$	− 13.3910i	− 1.35965i	−0.733373	−	0.679827i	$-0.762055\pi$
$97$	− 13.3910i	− 1.35965i	0.733373	−	0.679827i	$-0.237945\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−1.10973	−0.110422	−0.0552112	−	0.998475i	$-0.517583\pi$
$101$	−1.10973	−0.110422	−0.0552112	+	0.998475i	$0.517583\pi$
$102$	0	0
$103$	− 13.0984i	− 1.29062i	−0.763919	−	0.645312i	$-0.776727\pi$
$103$	− 13.0984i	− 1.29062i	0.763919	−	0.645312i	$-0.223273\pi$
$104$	−5.19891	−0.509795
$105$	0	0
$106$	8.80750	0.855460
$107$	13.9359i	1.34723i	0.739080	+	0.673617i	$0.235260\pi$
$107$	13.9359i	1.34723i	−0.739080	+	0.673617i	$0.764740\pi$
$108$	0	0
$109$	−18.9587	−1.81592	−0.907958	−	0.419062i	$-0.862359\pi$
$109$	−18.9587	−1.81592	−0.907958	+	0.419062i	$0.862359\pi$
$110$	0.152241	0.0145156
$111$	0	0
$112$	0	0
$113$	− 4.59220i	− 0.431998i	−0.976394	−	0.215999i	$-0.930699\pi$
$113$	− 4.59220i	− 0.431998i	0.976394	−	0.215999i	$-0.0693008\pi$
$114$	0	0
$115$	− 7.88764i	− 0.735526i
$116$	− 0.367542i	− 0.0341254i
$117$	0	0
$118$	− 2.78016i	− 0.255935i
$119$	0	0
$120$	0	0
$121$	10.9768	0.997893
$122$	8.94495	0.809837
$123$	0	0
$124$	− 4.55007i	− 0.408608i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.35237	−0.474946	−0.237473	−	0.971394i	$-0.576319\pi$
$127$	−5.35237	−0.474946	−0.237473	+	0.971394i	$0.576319\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	−5.19891	−0.455975
$131$	−0.679129	−0.0593358	−0.0296679	−	0.999560i	$-0.509445\pi$
$131$	−0.679129	−0.0593358	−0.0296679	+	0.999560i	$0.509445\pi$
$132$	0	0
$133$	0	0
$134$	− 7.82458i	− 0.675941i
$135$	0	0
$136$	− 0.881887i	− 0.0756212i
$137$	− 0.619541i	− 0.0529309i	−0.999650	−	0.0264655i	$-0.991575\pi$
$137$	− 0.619541i	− 0.0529309i	0.999650	−	0.0264655i	$-0.00842520\pi$
$138$	0	0
$139$	− 2.29807i	− 0.194920i	−0.995239	−	0.0974598i	$-0.968928\pi$
$139$	− 2.29807i	− 0.194920i	0.995239	−	0.0974598i	$-0.0310717\pi$
$140$	0	0
$141$	0	0
$142$	13.6337	1.14411
$143$	0.791487	0.0661875
$144$	0	0
$145$	− 0.367542i	− 0.0305227i
$146$	11.6359	0.962996
$147$	0	0
$148$	−1.47727	−0.121431
$149$	11.2092i	0.918294i	0.888360	+	0.459147i	$0.151845\pi$
$149$	11.2092i	0.918294i	−0.888360	+	0.459147i	$0.848155\pi$
$150$	0	0
$151$	8.96096	0.729233	0.364616	−	0.931158i	$-0.381200\pi$
$151$	8.96096	0.729233	0.364616	+	0.931158i	$0.381200\pi$
$152$	0.116520	0.00945103
$153$	0	0
$154$	0	0
$155$	− 4.55007i	− 0.365470i
$156$	0	0
$157$	− 7.89308i	− 0.629936i	−0.949102	−	0.314968i	$-0.898006\pi$
$157$	− 7.89308i	− 0.629936i	0.949102	−	0.314968i	$-0.101994\pi$
$158$	− 5.65685i	− 0.450035i
$159$	0	0
$160$	− 1.00000i	− 0.0790569i
$161$	0	0
$162$	0	0
$163$	−9.86325	−0.772549	−0.386275	−	0.922384i	$-0.626238\pi$
$163$	−9.86325	−0.772549	−0.386275	+	0.922384i	$0.626238\pi$
$164$	3.38687	0.264470
$165$	0	0
$166$	− 7.22625i	− 0.560866i
$167$	18.4924	1.43098	0.715492	−	0.698621i	$-0.246203\pi$
$167$	18.4924	1.43098	0.715492	+	0.698621i	$0.246203\pi$
$168$	0	0
$169$	−14.0287	−1.07913
$170$	− 0.881887i	− 0.0676376i
$171$	0	0
$172$	3.54553	0.270344
$173$	9.09425	0.691423	0.345711	−	0.938341i	$-0.387638\pi$
$173$	9.09425	0.691423	0.345711	+	0.938341i	$0.387638\pi$
$174$	0	0
$175$	0	0
$176$	0.152241i	0.0114756i
$177$	0	0
$178$	13.6241i	1.02117i
$179$	− 0.762181i	− 0.0569681i	−0.999594	−	0.0284841i	$-0.990932\pi$
$179$	− 0.762181i	− 0.0569681i	0.999594	−	0.0284841i	$-0.00906798\pi$
$180$	0	0
$181$	− 4.68682i	− 0.348369i	−0.984713	−	0.174184i	$-0.944271\pi$
$181$	− 4.68682i	− 0.348369i	0.984713	−	0.174184i	$-0.0557288\pi$
$182$	0	0
$183$	0	0
$184$	7.88764	0.581485
$185$	−1.47727	−0.108611
$186$	0	0
$187$	0.134259i	0.00981801i
$188$	−4.03572	−0.294335
$189$	0	0
$190$	0.116520	0.00845326
$191$	18.9533i	1.37141i	0.727879	+	0.685705i	$0.240506\pi$
$191$	18.9533i	1.37141i	−0.727879	+	0.685705i	$0.759494\pi$
$192$	0	0
$193$	8.43060	0.606848	0.303424	−	0.952856i	$-0.401870\pi$
$193$	8.43060	0.606848	0.303424	+	0.952856i	$0.401870\pi$
$194$	13.3910	0.961420
$195$	0	0
$196$	0	0
$197$	0.108381i	0.00772185i	0.999993	+	0.00386093i	$0.00122897\pi$
$197$	0.108381i	0.00772185i	−0.999993	+	0.00386093i	$0.998771\pi$
$198$	0	0
$199$	− 23.4524i	− 1.66249i	−0.555903	−	0.831247i	$-0.687627\pi$
$199$	− 23.4524i	− 1.66249i	0.555903	−	0.831247i	$-0.312373\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	− 1.10973i	− 0.0780804i
$203$	0	0
$204$	0	0
$205$	3.38687	0.236550
$206$	13.0984	0.912609
$207$	0	0
$208$	− 5.19891i	− 0.360480i
$209$	−0.0177391	−0.00122704
$210$	0	0
$211$	−2.57128	−0.177014	−0.0885070	−	0.996076i	$-0.528210\pi$
$211$	−2.57128	−0.177014	−0.0885070	+	0.996076i	$0.528210\pi$
$212$	8.80750i	0.604902i
$213$	0	0
$214$	−13.9359	−0.952639
$215$	3.54553	0.241803
$216$	0	0
$217$	0	0
$218$	− 18.9587i	− 1.28405i
$219$	0	0
$220$	0.152241i	0.0102641i
$221$	− 4.58485i	− 0.308411i
$222$	0	0
$223$	2.97547i	0.199252i	0.995025	+	0.0996262i	$0.0317647\pi$
$223$	2.97547i	0.199252i	−0.995025	+	0.0996262i	$0.968235\pi$
$224$	0	0
$225$	0	0
$226$	4.59220	0.305469
$227$	13.2809	0.881486	0.440743	−	0.897633i	$-0.354715\pi$
$227$	13.2809	0.881486	0.440743	+	0.897633i	$0.354715\pi$
$228$	0	0
$229$	− 7.17210i	− 0.473946i	−0.971516	−	0.236973i	$-0.923845\pi$
$229$	− 7.17210i	− 0.473946i	0.971516	−	0.236973i	$-0.0761553\pi$
$230$	7.88764	0.520096
$231$	0	0
$232$	0.367542	0.0241303
$233$	− 14.0160i	− 0.918220i	−0.888380	−	0.459110i	$-0.848168\pi$
$233$	− 14.0160i	− 0.918220i	0.888380	−	0.459110i	$-0.151832\pi$
$234$	0	0
$235$	−4.03572	−0.263261
$236$	2.78016	0.180973
$237$	0	0
$238$	0	0
$239$	− 14.8895i	− 0.963123i	−0.876412	−	0.481562i	$-0.840070\pi$
$239$	− 14.8895i	− 0.963123i	0.876412	−	0.481562i	$-0.159930\pi$
$240$	0	0
$241$	− 23.1405i	− 1.49061i	−0.666722	−	0.745307i	$-0.732303\pi$
$241$	− 23.1405i	− 1.49061i	0.666722	−	0.745307i	$-0.267697\pi$
$242$	10.9768i	0.705617i
$243$	0	0
$244$	8.94495i	0.572642i
$245$	0	0
$246$	0	0
$247$	0.605778	0.0385447
$248$	4.55007	0.288929
$249$	0	0
$250$	− 1.00000i	− 0.0632456i
$251$	2.66364	0.168128	0.0840638	−	0.996460i	$-0.473210\pi$
$251$	2.66364	0.168128	0.0840638	+	0.996460i	$0.473210\pi$
$252$	0	0
$253$	−1.20082	−0.0754950
$254$	− 5.35237i	− 0.335838i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.239169	−0.0149190	−0.00745948	−	0.999972i	$-0.502374\pi$
$257$	−0.239169	−0.0149190	−0.00745948	+	0.999972i	$0.502374\pi$
$258$	0	0
$259$	0	0
$260$	− 5.19891i	− 0.322423i
$261$	0	0
$262$	− 0.679129i	− 0.0419567i
$263$	− 21.1139i	− 1.30194i	−0.759104	−	0.650969i	$-0.774363\pi$
$263$	− 21.1139i	− 1.30194i	0.759104	−	0.650969i	$-0.225637\pi$
$264$	0	0
$265$	8.80750i	0.541041i
$266$	0	0
$267$	0	0
$268$	7.82458	0.477962
$269$	−19.6233	−1.19646	−0.598228	−	0.801326i	$-0.704128\pi$
$269$	−19.6233	−1.19646	−0.598228	+	0.801326i	$0.704128\pi$
$270$	0	0
$271$	− 15.8648i	− 0.963715i	−0.876250	−	0.481857i	$-0.839962\pi$
$271$	− 15.8648i	− 0.963715i	0.876250	−	0.481857i	$-0.160038\pi$
$272$	0.881887	0.0534723
$273$	0	0
$274$	0.619541	0.0374278
$275$	0.152241i	0.00918047i
$276$	0	0
$277$	13.0508	0.784149	0.392074	−	0.919934i	$-0.371758\pi$
$277$	13.0508	0.784149	0.392074	+	0.919934i	$0.371758\pi$
$278$	2.29807	0.137829
$279$	0	0
$280$	0	0
$281$	− 20.3133i	− 1.21179i	−0.795544	−	0.605896i	$-0.792815\pi$
$281$	− 20.3133i	− 1.21179i	0.795544	−	0.605896i	$-0.207185\pi$
$282$	0	0
$283$	26.7412i	1.58960i	0.606874	+	0.794798i	$0.292423\pi$
$283$	26.7412i	1.58960i	−0.606874	+	0.794798i	$0.707577\pi$
$284$	13.6337i	0.809010i
$285$	0	0
$286$	0.791487i	0.0468016i
$287$	0	0
$288$	0	0
$289$	−16.2223	−0.954251
$290$	0.367542	0.0215828
$291$	0	0
$292$	11.6359i	0.680941i
$293$	−22.1435	−1.29364	−0.646819	−	0.762644i	$-0.723901\pi$
$293$	−22.1435	−1.29364	−0.646819	+	0.762644i	$0.723901\pi$
$294$	0	0
$295$	2.78016	0.161867
$296$	− 1.47727i	− 0.0858648i
$297$	0	0
$298$	−11.2092	−0.649332
$299$	41.0072	2.37151
$300$	0	0
$301$	0	0
$302$	8.96096i	0.515645i
$303$	0	0
$304$	0.116520i	0.00668289i
$305$	8.94495i	0.512186i
$306$	0	0
$307$	− 26.5458i	− 1.51505i	−0.652805	−	0.757526i	$-0.726408\pi$
$307$	− 26.5458i	− 1.51505i	0.652805	−	0.757526i	$-0.273592\pi$
$308$	0	0
$309$	0	0
$310$	4.55007	0.258426
$311$	−19.9310	−1.13018	−0.565092	−	0.825028i	$-0.691159\pi$
$311$	−19.9310	−1.13018	−0.565092	+	0.825028i	$0.691159\pi$
$312$	0	0
$313$	− 30.3170i	− 1.71362i	−0.515635	−	0.856808i	$-0.672444\pi$
$313$	− 30.3170i	− 1.71362i	0.515635	−	0.856808i	$-0.327556\pi$
$314$	7.89308	0.445432
$315$	0	0
$316$	5.65685	0.318223
$317$	27.6593i	1.55350i	0.629809	+	0.776750i	$0.283133\pi$
$317$	27.6593i	1.55350i	−0.629809	+	0.776750i	$0.716867\pi$
$318$	0	0
$319$	−0.0559550	−0.00313288
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	0.102758i	0.00571759i
$324$	0	0
$325$	− 5.19891i	− 0.288384i
$326$	− 9.86325i	− 0.546275i
$327$	0	0
$328$	3.38687i	0.187009i
$329$	0	0
$330$	0	0
$331$	−20.0428	−1.10165	−0.550827	−	0.834620i	$-0.685687\pi$
$331$	−20.0428	−1.10165	−0.550827	+	0.834620i	$0.685687\pi$
$332$	7.22625	0.396592
$333$	0	0
$334$	18.4924i	1.01186i
$335$	7.82458	0.427503
$336$	0	0
$337$	−24.5368	−1.33660	−0.668302	−	0.743890i	$-0.732978\pi$
$337$	−24.5368	−1.33660	−0.668302	+	0.743890i	$0.732978\pi$
$338$	− 14.0287i	− 0.763060i
$339$	0	0
$340$	0.881887	0.0478270
$341$	−0.692706	−0.0375121
$342$	0	0
$343$	0	0
$344$	3.54553i	0.191162i
$345$	0	0
$346$	9.09425i	0.488910i
$347$	− 30.4845i	− 1.63650i	−0.574866	−	0.818248i	$-0.694946\pi$
$347$	− 30.4845i	− 1.63650i	0.574866	−	0.818248i	$-0.305054\pi$
$348$	0	0
$349$	28.0443i	1.50118i	0.660769	+	0.750589i	$0.270230\pi$
$349$	28.0443i	1.50118i	−0.660769	+	0.750589i	$0.729770\pi$
$350$	0	0
$351$	0	0
$352$	−0.152241	−0.00811447
$353$	31.7437	1.68955	0.844774	−	0.535123i	$-0.179735\pi$
$353$	31.7437	1.68955	0.844774	+	0.535123i	$0.179735\pi$
$354$	0	0
$355$	13.6337i	0.723600i
$356$	−13.6241	−0.722075
$357$	0	0
$358$	0.762181	0.0402825
$359$	− 24.8245i	− 1.31018i	−0.755549	−	0.655092i	$-0.772630\pi$
$359$	− 24.8245i	− 1.31018i	0.755549	−	0.655092i	$-0.227370\pi$
$360$	0	0
$361$	18.9864	0.999285
$362$	4.68682	0.246334
$363$	0	0
$364$	0	0
$365$	11.6359i	0.609052i
$366$	0	0
$367$	− 7.06147i	− 0.368606i	−0.982870	−	0.184303i	$-0.940997\pi$
$367$	− 7.06147i	− 0.368606i	0.982870	−	0.184303i	$-0.0590027\pi$
$368$	7.88764i	0.411172i
$369$	0	0
$370$	− 1.47727i	− 0.0767998i
$371$	0	0
$372$	0	0
$373$	22.3300	1.15621	0.578103	−	0.815964i	$-0.303793\pi$
$373$	22.3300	1.15621	0.578103	+	0.815964i	$0.303793\pi$
$374$	−0.134259	−0.00694238
$375$	0	0
$376$	− 4.03572i	− 0.208126i
$377$	1.91082	0.0984122
$378$	0	0
$379$	−31.4625	−1.61612	−0.808059	−	0.589101i	$-0.799482\pi$
$379$	−31.4625	−1.61612	−0.808059	+	0.589101i	$0.799482\pi$
$380$	0.116520i	0.00597736i
$381$	0	0
$382$	−18.9533	−0.969734
$383$	5.01389	0.256198	0.128099	−	0.991761i	$-0.459112\pi$
$383$	5.01389	0.256198	0.128099	+	0.991761i	$0.459112\pi$
$384$	0	0
$385$	0	0
$386$	8.43060i	0.429106i
$387$	0	0
$388$	13.3910i	0.679827i
$389$	− 10.4490i	− 0.529787i	−0.964278	−	0.264893i	$-0.914663\pi$
$389$	− 10.4490i	− 0.529787i	0.964278	−	0.264893i	$-0.0853368\pi$
$390$	0	0
$391$	6.95601i	0.351781i
$392$	0	0
$393$	0	0
$394$	−0.108381	−0.00546018
$395$	5.65685	0.284627
$396$	0	0
$397$	− 10.9191i	− 0.548012i	−0.961728	−	0.274006i	$-0.911651\pi$
$397$	− 10.9191i	− 0.548012i	0.961728	−	0.274006i	$-0.0883489\pi$
$398$	23.4524	1.17556
$399$	0	0
$400$	1.00000	0.0500000
$401$	− 8.87351i	− 0.443122i	−0.975147	−	0.221561i	$-0.928885\pi$
$401$	− 8.87351i	− 0.443122i	0.975147	−	0.221561i	$-0.0711151\pi$
$402$	0	0
$403$	23.6554	1.17836
$404$	1.10973	0.0552112
$405$	0	0
$406$	0	0
$407$	0.224902i	0.0111480i
$408$	0	0
$409$	1.54924i	0.0766052i	0.999266	+	0.0383026i	$0.0121951\pi$
$409$	1.54924i	0.0766052i	−0.999266	+	0.0383026i	$0.987805\pi$
$410$	3.38687i	0.167266i
$411$	0	0
$412$	13.0984i	0.645312i
$413$	0	0
$414$	0	0
$415$	7.22625	0.354723
$416$	5.19891	0.254898
$417$	0	0
$418$	− 0.0177391i	0 0.000867650i
$419$	−7.71604	−0.376953	−0.188477	−	0.982078i	$-0.560355\pi$
$419$	−7.71604	−0.376953	−0.188477	+	0.982078i	$0.560355\pi$
$420$	0	0
$421$	−34.1546	−1.66459	−0.832297	−	0.554330i	$-0.812975\pi$
$421$	−34.1546	−1.66459	−0.832297	+	0.554330i	$0.812975\pi$
$422$	− 2.57128i	− 0.125168i
$423$	0	0
$424$	−8.80750	−0.427730
$425$	0.881887	0.0427778
$426$	0	0
$427$	0	0
$428$	− 13.9359i	− 0.673617i
$429$	0	0
$430$	3.54553i	0.170981i
$431$	− 38.3931i	− 1.84933i	−0.380781	−	0.924665i	$-0.624345\pi$
$431$	− 38.3931i	− 1.84933i	0.380781	−	0.924665i	$-0.375655\pi$
$432$	0	0
$433$	− 7.93869i	− 0.381509i	−0.981638	−	0.190754i	$-0.938907\pi$
$433$	− 7.93869i	− 0.381509i	0.981638	−	0.190754i	$-0.0610934\pi$
$434$	0	0
$435$	0	0
$436$	18.9587	0.907958
$437$	−0.919069	−0.0439651
$438$	0	0
$439$	38.4010i	1.83278i	0.400289	+	0.916389i	$0.368910\pi$
$439$	38.4010i	1.83278i	−0.400289	+	0.916389i	$0.631090\pi$
$440$	−0.152241	−0.00725780
$441$	0	0
$442$	4.58485	0.218079
$443$	24.0926i	1.14467i	0.820019	+	0.572337i	$0.193963\pi$
$443$	24.0926i	1.14467i	−0.820019	+	0.572337i	$0.806037\pi$
$444$	0	0
$445$	−13.6241	−0.645843
$446$	−2.97547	−0.140893
$447$	0	0
$448$	0	0
$449$	40.1788i	1.89615i	0.318041	+	0.948077i	$0.396975\pi$
$449$	40.1788i	1.89615i	−0.318041	+	0.948077i	$0.603025\pi$
$450$	0	0
$451$	− 0.515621i	− 0.0242796i
$452$	4.59220i	0.215999i
$453$	0	0
$454$	13.2809i	0.623305i
$455$	0	0
$456$	0	0
$457$	12.8309	0.600203	0.300101	−	0.953907i	$-0.402980\pi$
$457$	12.8309	0.600203	0.300101	+	0.953907i	$0.402980\pi$
$458$	7.17210	0.335130
$459$	0	0
$460$	7.88764i	0.367763i
$461$	7.96722	0.371071	0.185535	−	0.982638i	$-0.440598\pi$
$461$	7.96722	0.371071	0.185535	+	0.982638i	$0.440598\pi$
$462$	0	0
$463$	26.2773	1.22121	0.610605	−	0.791935i	$-0.290926\pi$
$463$	26.2773	1.22121	0.610605	+	0.791935i	$0.290926\pi$
$464$	0.367542i	0.0170627i
$465$	0	0
$466$	14.0160	0.649279
$467$	−22.5709	−1.04446	−0.522229	−	0.852805i	$-0.674899\pi$
$467$	−22.5709	−1.04446	−0.522229	+	0.852805i	$0.674899\pi$
$468$	0	0
$469$	0	0
$470$	− 4.03572i	− 0.186154i
$471$	0	0
$472$	2.78016i	0.127967i
$473$	− 0.539775i	− 0.0248189i
$474$	0	0
$475$	0.116520i	0.00534631i
$476$	0	0
$477$	0	0
$478$	14.8895	0.681031
$479$	−23.9665	−1.09506	−0.547528	−	0.836787i	$-0.684431\pi$
$479$	−23.9665	−1.09506	−0.547528	+	0.836787i	$0.684431\pi$
$480$	0	0
$481$	− 7.68022i	− 0.350188i
$482$	23.1405	1.05402
$483$	0	0
$484$	−10.9768	−0.498946
$485$	13.3910i	0.608056i
$486$	0	0
$487$	38.3699	1.73871	0.869354	−	0.494191i	$-0.164535\pi$
$487$	38.3699	1.73871	0.869354	+	0.494191i	$0.164535\pi$
$488$	−8.94495	−0.404919
$489$	0	0
$490$	0	0
$491$	7.83834i	0.353740i	0.984234	+	0.176870i	$0.0565971\pi$
$491$	7.83834i	0.353740i	−0.984234	+	0.176870i	$0.943403\pi$
$492$	0	0
$493$	0.324131i	0.0145981i
$494$	0.605778i	0.0272552i
$495$	0	0
$496$	4.55007i	0.204304i
$497$	0	0
$498$	0	0
$499$	−13.7056	−0.613549	−0.306775	−	0.951782i	$-0.599250\pi$
$499$	−13.7056	−0.613549	−0.306775	+	0.951782i	$0.599250\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	2.66364i	0.118884i
$503$	3.02437	0.134850	0.0674249	−	0.997724i	$-0.478522\pi$
$503$	3.02437	0.134850	0.0674249	+	0.997724i	$0.478522\pi$
$504$	0	0
$505$	1.10973	0.0493824
$506$	− 1.20082i	− 0.0533830i
$507$	0	0
$508$	5.35237	0.237473
$509$	−14.2914	−0.633457	−0.316728	−	0.948516i	$-0.602584\pi$
$509$	−14.2914	−0.633457	−0.316728	+	0.948516i	$0.602584\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 0.239169i	− 0.0105493i
$515$	13.0984i	0.577185i
$516$	0	0
$517$	0.614402i	0.0270214i
$518$	0	0
$519$	0	0
$520$	5.19891	0.227987
$521$	−13.8158	−0.605279	−0.302640	−	0.953105i	$-0.597868\pi$
$521$	−13.8158	−0.605279	−0.302640	+	0.953105i	$0.597868\pi$
$522$	0	0
$523$	− 7.48213i	− 0.327171i	−0.986529	−	0.163585i	$-0.947694\pi$
$523$	− 7.48213i	− 0.327171i	0.986529	−	0.163585i	$-0.0523059\pi$
$524$	0.679129	0.0296679
$525$	0	0
$526$	21.1139	0.920610
$527$	4.01264i	0.174794i
$528$	0	0
$529$	−39.2149	−1.70500
$530$	−8.80750	−0.382573
$531$	0	0
$532$	0	0
$533$	17.6081i	0.762690i
$534$	0	0
$535$	− 13.9359i	− 0.602502i
$536$	7.82458i	0.337970i
$537$	0	0
$538$	− 19.6233i	− 0.846022i
$539$	0	0
$540$	0	0
$541$	−20.0780	−0.863222	−0.431611	−	0.902060i	$-0.642055\pi$
$541$	−20.0780	−0.863222	−0.431611	+	0.902060i	$0.642055\pi$
$542$	15.8648	0.681449
$543$	0	0
$544$	0.881887i	0.0378106i
$545$	18.9587	0.812102
$546$	0	0
$547$	−0.897902	−0.0383915	−0.0191958	−	0.999816i	$-0.506111\pi$
$547$	−0.897902	−0.0383915	−0.0191958	+	0.999816i	$0.506111\pi$
$548$	0.619541i	0.0264655i
$549$	0	0
$550$	−0.152241	−0.00649158
$551$	−0.0428261	−0.00182445
$552$	0	0
$553$	0	0
$554$	13.0508i	0.554477i
$555$	0	0
$556$	2.29807i	0.0974598i
$557$	19.2977i	0.817670i	0.912608	+	0.408835i	$0.134065\pi$
$557$	19.2977i	0.817670i	−0.912608	+	0.408835i	$0.865935\pi$
$558$	0	0
$559$	18.4329i	0.779629i
$560$	0	0
$561$	0	0
$562$	20.3133	0.856866
$563$	−2.18155	−0.0919412	−0.0459706	−	0.998943i	$-0.514638\pi$
$563$	−2.18155	−0.0919412	−0.0459706	+	0.998943i	$0.514638\pi$
$564$	0	0
$565$	4.59220i	0.193195i
$566$	−26.7412	−1.12401
$567$	0	0
$568$	−13.6337	−0.572056
$569$	3.27133i	0.137141i	0.997646	+	0.0685707i	$0.0218439\pi$
$569$	3.27133i	0.137141i	−0.997646	+	0.0685707i	$0.978156\pi$
$570$	0	0
$571$	−2.91270	−0.121893	−0.0609463	−	0.998141i	$-0.519412\pi$
$571$	−2.91270	−0.121893	−0.0609463	+	0.998141i	$0.519412\pi$
$572$	−0.791487	−0.0330937
$573$	0	0
$574$	0	0
$575$	7.88764i	0.328937i
$576$	0	0
$577$	2.29639i	0.0955998i	0.998857	+	0.0477999i	$0.0152210\pi$
$577$	2.29639i	0.0955998i	−0.998857	+	0.0477999i	$0.984779\pi$
$578$	− 16.2223i	− 0.674758i
$579$	0	0
$580$	0.367542i	0.0152614i
$581$	0	0
$582$	0	0
$583$	1.34086	0.0555328
$584$	−11.6359	−0.481498
$585$	0	0
$586$	− 22.1435i	− 0.914740i
$587$	−11.1071	−0.458438	−0.229219	−	0.973375i	$-0.573617\pi$
$587$	−11.1071	−0.458438	−0.229219	+	0.973375i	$0.573617\pi$
$588$	0	0
$589$	−0.530174	−0.0218455
$590$	2.78016i	0.114457i
$591$	0	0
$592$	1.47727	0.0607156
$593$	−18.3789	−0.754730	−0.377365	−	0.926065i	$-0.623170\pi$
$593$	−18.3789	−0.754730	−0.377365	+	0.926065i	$0.623170\pi$
$594$	0	0
$595$	0	0
$596$	− 11.2092i	− 0.459147i
$597$	0	0
$598$	41.0072i	1.67691i
$599$	48.8861i	1.99743i	0.0506712	+	0.998715i	$0.483864\pi$
$599$	48.8861i	1.99743i	−0.0506712	+	0.998715i	$0.516136\pi$
$600$	0	0
$601$	− 6.21975i	− 0.253709i	−0.991921	−	0.126854i	$-0.959512\pi$
$601$	− 6.21975i	− 0.253709i	0.991921	−	0.126854i	$-0.0404881\pi$
$602$	0	0
$603$	0	0
$604$	−8.96096	−0.364616
$605$	−10.9768	−0.446271
$606$	0	0
$607$	44.0234i	1.78685i	0.449209	+	0.893427i	$0.351706\pi$
$607$	44.0234i	1.78685i	−0.449209	+	0.893427i	$0.648294\pi$
$608$	−0.116520	−0.00472552
$609$	0	0
$610$	−8.94495	−0.362170
$611$	− 20.9814i	− 0.848815i
$612$	0	0
$613$	46.5712	1.88099	0.940496	−	0.339805i	$-0.110361\pi$
$613$	46.5712	1.88099	0.940496	+	0.339805i	$0.110361\pi$
$614$	26.5458	1.07130
$615$	0	0
$616$	0	0
$617$	19.6155i	0.789688i	0.918748	+	0.394844i	$0.129201\pi$
$617$	19.6155i	0.789688i	−0.918748	+	0.394844i	$0.870799\pi$
$618$	0	0
$619$	− 23.6497i	− 0.950561i	−0.879834	−	0.475281i	$-0.842347\pi$
$619$	− 23.6497i	− 0.950561i	0.879834	−	0.475281i	$-0.157653\pi$
$620$	4.55007i	0.182735i
$621$	0	0
$622$	− 19.9310i	− 0.799160i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	30.3170	1.21171
$627$	0	0
$628$	7.89308i	0.314968i
$629$	1.30279	0.0519456
$630$	0	0
$631$	15.0156	0.597763	0.298882	−	0.954290i	$-0.403386\pi$
$631$	15.0156	0.597763	0.298882	+	0.954290i	$0.403386\pi$
$632$	5.65685i	0.225018i
$633$	0	0
$634$	−27.6593	−1.09849
$635$	5.35237	0.212402
$636$	0	0
$637$	0	0
$638$	− 0.0559550i	− 0.00221528i
$639$	0	0
$640$	1.00000i	0.0395285i
$641$	− 19.3652i	− 0.764878i	−0.923981	−	0.382439i	$-0.875084\pi$
$641$	− 19.3652i	− 0.764878i	0.923981	−	0.382439i	$-0.124916\pi$
$642$	0	0
$643$	6.76486i	0.266780i	0.991064	+	0.133390i	$0.0425863\pi$
$643$	6.76486i	0.266780i	−0.991064	+	0.133390i	$0.957414\pi$
$644$	0	0
$645$	0	0
$646$	−0.102758	−0.00404294
$647$	−10.7335	−0.421978	−0.210989	−	0.977488i	$-0.567668\pi$
$647$	−10.7335	−0.421978	−0.210989	+	0.977488i	$0.567668\pi$
$648$	0	0
$649$	− 0.423255i	− 0.0166142i
$650$	5.19891	0.203918
$651$	0	0
$652$	9.86325	0.386275
$653$	− 28.2027i	− 1.10366i	−0.833958	−	0.551828i	$-0.813930\pi$
$653$	− 28.2027i	− 1.10366i	0.833958	−	0.551828i	$-0.186070\pi$
$654$	0	0
$655$	0.679129	0.0265358
$656$	−3.38687	−0.132235
$657$	0	0
$658$	0	0
$659$	− 10.7176i	− 0.417497i	−0.977969	−	0.208748i	$-0.933061\pi$
$659$	− 10.7176i	− 0.417497i	0.977969	−	0.208748i	$-0.0669389\pi$
$660$	0	0
$661$	32.8951i	1.27947i	0.768594	+	0.639736i	$0.220956\pi$
$661$	32.8951i	1.27947i	−0.768594	+	0.639736i	$0.779044\pi$
$662$	− 20.0428i	− 0.778987i
$663$	0	0
$664$	7.22625i	0.280433i
$665$	0	0
$666$	0	0
$667$	−2.89904	−0.112251
$668$	−18.4924	−0.715492
$669$	0	0
$670$	7.82458i	0.302290i
$671$	1.36179	0.0525712
$672$	0	0
$673$	−7.14123	−0.275274	−0.137637	−	0.990483i	$-0.543951\pi$
$673$	−7.14123	−0.275274	−0.137637	+	0.990483i	$0.543951\pi$
$674$	− 24.5368i	− 0.945121i
$675$	0	0
$676$	14.0287	0.539565
$677$	−31.0775	−1.19440	−0.597202	−	0.802091i	$-0.703721\pi$
$677$	−31.0775	−1.19440	−0.597202	+	0.802091i	$0.703721\pi$
$678$	0	0
$679$	0	0
$680$	0.881887i	0.0338188i
$681$	0	0
$682$	− 0.692706i	− 0.0265251i
$683$	− 33.3984i	− 1.27795i	−0.769226	−	0.638977i	$-0.779358\pi$
$683$	− 33.3984i	− 1.27795i	0.769226	−	0.638977i	$-0.220642\pi$
$684$	0	0
$685$	0.619541i	0.0236714i
$686$	0	0
$687$	0	0
$688$	−3.54553	−0.135172
$689$	−45.7894	−1.74444
$690$	0	0
$691$	36.8000i	1.39994i	0.714173	+	0.699969i	$0.246803\pi$
$691$	36.8000i	1.39994i	−0.714173	+	0.699969i	$0.753197\pi$
$692$	−9.09425	−0.345711
$693$	0	0
$694$	30.4845	1.15718
$695$	2.29807i	0.0871707i
$696$	0	0
$697$	−2.98684	−0.113135
$698$	−28.0443	−1.06149
$699$	0	0
$700$	0	0
$701$	28.3473i	1.07066i	0.844643	+	0.535330i	$0.179813\pi$
$701$	28.3473i	1.07066i	−0.844643	+	0.535330i	$0.820187\pi$
$702$	0	0
$703$	0.172132i	0.00649209i
$704$	− 0.152241i	− 0.00573780i
$705$	0	0
$706$	31.7437i	1.19469i
$707$	0	0
$708$	0	0
$709$	−15.1618	−0.569412	−0.284706	−	0.958615i	$-0.591896\pi$
$709$	−15.1618	−0.569412	−0.284706	+	0.958615i	$0.591896\pi$
$710$	−13.6337	−0.511663
$711$	0	0
$712$	− 13.6241i	− 0.510584i
$713$	−35.8893	−1.34406
$714$	0	0
$715$	−0.791487	−0.0295999
$716$	0.762181i	0.0284841i
$717$	0	0
$718$	24.8245	0.926440
$719$	43.1993	1.61106	0.805532	−	0.592553i	$-0.201880\pi$
$719$	43.1993	1.61106	0.805532	+	0.592553i	$0.201880\pi$
$720$	0	0
$721$	0	0
$722$	18.9864i	0.706601i
$723$	0	0
$724$	4.68682i	0.174184i
$725$	0.367542i	0.0136502i
$726$	0	0
$727$	27.9829i	1.03783i	0.854826	+	0.518914i	$0.173664\pi$
$727$	27.9829i	1.03783i	−0.854826	+	0.518914i	$0.826336\pi$
$728$	0	0
$729$	0	0
$730$	−11.6359	−0.430665
$731$	−3.12676	−0.115647
$732$	0	0
$733$	− 36.1041i	− 1.33354i	−0.745266	−	0.666768i	$-0.767677\pi$
$733$	− 36.1041i	− 1.33354i	0.745266	−	0.666768i	$-0.232323\pi$
$734$	7.06147	0.260644
$735$	0	0
$736$	−7.88764	−0.290742
$737$	− 1.19122i	− 0.0438792i
$738$	0	0
$739$	−10.3495	−0.380713	−0.190357	−	0.981715i	$-0.560964\pi$
$739$	−10.3495	−0.380713	−0.190357	+	0.981715i	$0.560964\pi$
$740$	1.47727	0.0543057
$741$	0	0
$742$	0	0
$743$	31.7753i	1.16572i	0.812572	+	0.582861i	$0.198067\pi$
$743$	31.7753i	1.16572i	−0.812572	+	0.582861i	$0.801933\pi$
$744$	0	0
$745$	− 11.2092i	− 0.410673i
$746$	22.3300i	0.817561i
$747$	0	0
$748$	− 0.134259i	− 0.00490901i
$749$	0	0
$750$	0	0
$751$	13.0292	0.475443	0.237721	−	0.971333i	$-0.423599\pi$
$751$	13.0292	0.475443	0.237721	+	0.971333i	$0.423599\pi$
$752$	4.03572	0.147168
$753$	0	0
$754$	1.91082i	0.0695879i
$755$	−8.96096	−0.326123
$756$	0	0
$757$	8.26385	0.300355	0.150177	−	0.988659i	$-0.452016\pi$
$757$	8.26385	0.300355	0.150177	+	0.988659i	$0.452016\pi$
$758$	− 31.4625i	− 1.14277i
$759$	0	0
$760$	−0.116520	−0.00422663
$761$	−50.8346	−1.84275	−0.921377	−	0.388670i	$-0.872934\pi$
$761$	−50.8346	−1.84275	−0.921377	+	0.388670i	$0.872934\pi$
$762$	0	0
$763$	0	0
$764$	− 18.9533i	− 0.685705i
$765$	0	0
$766$	5.01389i	0.181159i
$767$	14.4538i	0.521897i
$768$	0	0
$769$	44.8781i	1.61835i	0.587570	+	0.809174i	$0.300085\pi$
$769$	44.8781i	1.61835i	−0.587570	+	0.809174i	$0.699915\pi$
$770$	0	0
$771$	0	0
$772$	−8.43060	−0.303424
$773$	3.43559	0.123569	0.0617847	−	0.998089i	$-0.480321\pi$
$773$	3.43559	0.123569	0.0617847	+	0.998089i	$0.480321\pi$
$774$	0	0
$775$	4.55007i	0.163443i
$776$	−13.3910	−0.480710
$777$	0	0
$778$	10.4490	0.374616
$779$	− 0.394639i	− 0.0141394i
$780$	0	0
$781$	2.07560	0.0742709
$782$	−6.95601	−0.248746
$783$	0	0
$784$	0	0
$785$	7.89308i	0.281716i
$786$	0	0
$787$	14.5020i	0.516942i	0.966019	+	0.258471i	$0.0832187\pi$
$787$	14.5020i	0.516942i	−0.966019	+	0.258471i	$0.916781\pi$
$788$	− 0.108381i	− 0.00386093i
$789$	0	0
$790$	5.65685i	0.201262i
$791$	0	0
$792$	0	0
$793$	−46.5040	−1.65141
$794$	10.9191	0.387503
$795$	0	0
$796$	23.4524i	0.831247i
$797$	36.3121	1.28624	0.643120	−	0.765766i	$-0.277640\pi$
$797$	36.3121	1.28624	0.643120	+	0.765766i	$0.277640\pi$
$798$	0	0
$799$	3.55905	0.125910
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	8.87351	0.313334
$803$	1.77146	0.0625136
$804$	0	0
$805$	0	0
$806$	23.6554i	0.833226i
$807$	0	0
$808$	1.10973i	0.0390402i
$809$	11.8145i	0.415375i	0.978195	+	0.207687i	$0.0665937\pi$
$809$	11.8145i	0.415375i	−0.978195	+	0.207687i	$0.933406\pi$
$810$	0	0
$811$	− 47.7615i	− 1.67713i	−0.544799	−	0.838566i	$-0.683394\pi$
$811$	− 47.7615i	− 1.67713i	0.544799	−	0.838566i	$-0.316606\pi$
$812$	0	0
$813$	0	0
$814$	−0.224902	−0.00788280
$815$	9.86325	0.345494
$816$	0	0
$817$	− 0.413126i	− 0.0144534i
$818$	−1.54924	−0.0541680
$819$	0	0
$820$	−3.38687	−0.118275
$821$	− 40.7291i	− 1.42146i	−0.703467	−	0.710728i	$-0.748365\pi$
$821$	− 40.7291i	− 1.42146i	0.703467	−	0.710728i	$-0.251635\pi$
$822$	0	0
$823$	50.8205	1.77149	0.885745	−	0.464172i	$-0.153648\pi$
$823$	50.8205	1.77149	0.885745	+	0.464172i	$0.153648\pi$
$824$	−13.0984	−0.456305
$825$	0	0
$826$	0	0
$827$	− 35.3852i	− 1.23046i	−0.788347	−	0.615231i	$-0.789063\pi$
$827$	− 35.3852i	− 1.23046i	0.788347	−	0.615231i	$-0.210937\pi$
$828$	0	0
$829$	− 46.4408i	− 1.61296i	−0.591263	−	0.806478i	$-0.701371\pi$
$829$	− 46.4408i	− 1.61296i	0.591263	−	0.806478i	$-0.298629\pi$
$830$	7.22625i	0.250827i
$831$	0	0
$832$	5.19891i	0.180240i
$833$	0	0
$834$	0	0
$835$	−18.4924	−0.639955
$836$	0.0177391	0.000613521 0
$837$	0	0
$838$	− 7.71604i	− 0.266546i
$839$	28.1665	0.972416	0.486208	−	0.873843i	$-0.338380\pi$
$839$	28.1665	0.972416	0.486208	+	0.873843i	$0.338380\pi$
$840$	0	0
$841$	28.8649	0.995342
$842$	− 34.1546i	− 1.17705i
$843$	0	0
$844$	2.57128	0.0885070
$845$	14.0287	0.482602
$846$	0	0
$847$	0	0
$848$	− 8.80750i	− 0.302451i
$849$	0	0
$850$	0.881887i	0.0302485i
$851$	11.6522i	0.399433i
$852$	0	0
$853$	− 19.1972i	− 0.657301i	−0.944452	−	0.328650i	$-0.893406\pi$
$853$	− 19.1972i	− 0.657301i	0.944452	−	0.328650i	$-0.106594\pi$
$854$	0	0
$855$	0	0
$856$	13.9359	0.476319
$857$	0.0426640	0.00145738	0.000728688	−	1.00000i	$-0.499768\pi$
$857$	0.0426640	0.00145738	0.000728688		1.00000i	$0.499768\pi$
$858$	0	0
$859$	44.9055i	1.53216i	0.642747	+	0.766079i	$0.277795\pi$
$859$	44.9055i	1.53216i	−0.642747	+	0.766079i	$0.722205\pi$
$860$	−3.54553	−0.120902
$861$	0	0
$862$	38.3931	1.30767
$863$	− 2.04004i	− 0.0694439i	−0.999397	−	0.0347220i	$-0.988945\pi$
$863$	− 2.04004i	− 0.0694439i	0.999397	−	0.0347220i	$-0.0110546\pi$
$864$	0	0
$865$	−9.09425	−0.309214
$866$	7.93869	0.269768
$867$	0	0
$868$	0	0
$869$	− 0.861205i	− 0.0292144i
$870$	0	0
$871$	40.6793i	1.37837i
$872$	18.9587i	0.642023i
$873$	0	0
$874$	− 0.919069i	− 0.0310880i
$875$	0	0
$876$	0	0
$877$	25.5037	0.861200	0.430600	−	0.902543i	$-0.358302\pi$
$877$	25.5037	0.861200	0.430600	+	0.902543i	$0.358302\pi$
$878$	−38.4010	−1.29597
$879$	0	0
$880$	− 0.152241i	− 0.00513204i
$881$	20.2983	0.683866	0.341933	−	0.939724i	$-0.388918\pi$
$881$	20.2983	0.683866	0.341933	+	0.939724i	$0.388918\pi$
$882$	0	0
$883$	50.1046	1.68615	0.843077	−	0.537793i	$-0.180742\pi$
$883$	50.1046	1.68615	0.843077	+	0.537793i	$0.180742\pi$
$884$	4.58485i	0.154205i
$885$	0	0
$886$	−24.0926	−0.809407
$887$	−48.6881	−1.63479	−0.817394	−	0.576079i	$-0.804582\pi$
$887$	−48.6881	−1.63479	−0.817394	+	0.576079i	$0.804582\pi$
$888$	0	0
$889$	0	0
$890$	− 13.6241i	− 0.456680i
$891$	0	0
$892$	− 2.97547i	− 0.0996262i
$893$	0.470243i	0.0157361i
$894$	0	0
$895$	0.762181i	0.0254769i
$896$	0	0
$897$	0	0
$898$	−40.1788	−1.34078
$899$	−1.67234	−0.0557757
$900$	0	0
$901$	− 7.76722i	− 0.258764i
$902$	0.515621	0.0171683
$903$	0	0
$904$	−4.59220	−0.152734
$905$	4.68682i	0.155795i
$906$	0	0
$907$	−1.77309	−0.0588744	−0.0294372	−	0.999567i	$-0.509372\pi$
$907$	−1.77309	−0.0588744	−0.0294372	+	0.999567i	$0.509372\pi$
$908$	−13.2809	−0.440743
$909$	0	0
$910$	0	0
$911$	− 38.2998i	− 1.26893i	−0.772953	−	0.634464i	$-0.781221\pi$
$911$	− 38.2998i	− 1.26893i	0.772953	−	0.634464i	$-0.218779\pi$
$912$	0	0
$913$	− 1.10013i	− 0.0364090i
$914$	12.8309i	0.424407i
$915$	0	0
$916$	7.17210i	0.236973i
$917$	0	0
$918$	0	0
$919$	−42.1356	−1.38993	−0.694963	−	0.719046i	$-0.744579\pi$
$919$	−42.1356	−1.38993	−0.694963	+	0.719046i	$0.744579\pi$
$920$	−7.88764	−0.260048
$921$	0	0
$922$	7.96722i	0.262386i
$923$	−70.8803	−2.33305
$924$	0	0
$925$	1.47727	0.0485725
$926$	26.2773i	0.863527i
$927$	0	0
$928$	−0.367542	−0.0120652
$929$	−19.5499	−0.641413	−0.320706	−	0.947179i	$-0.603920\pi$
$929$	−19.5499	−0.641413	−0.320706	+	0.947179i	$0.603920\pi$
$930$	0	0
$931$	0	0
$932$	14.0160i	0.459110i
$933$	0	0
$934$	− 22.5709i	− 0.738544i
$935$	− 0.134259i	− 0.00439075i
$936$	0	0
$937$	− 33.1018i	− 1.08139i	−0.841219	−	0.540695i	$-0.818161\pi$
$937$	− 33.1018i	− 1.08139i	0.841219	−	0.540695i	$-0.181839\pi$
$938$	0	0
$939$	0	0
$940$	4.03572	0.131631
$941$	39.8510	1.29911	0.649553	−	0.760316i	$-0.274956\pi$
$941$	39.8510	1.29911	0.649553	+	0.760316i	$0.274956\pi$
$942$	0	0
$943$	− 26.7145i	− 0.869942i
$944$	−2.78016	−0.0904866
$945$	0	0
$946$	0.539775	0.0175496
$947$	28.4239i	0.923652i	0.886970	+	0.461826i	$0.152806\pi$
$947$	28.4239i	0.923652i	−0.886970	+	0.461826i	$0.847194\pi$
$948$	0	0
$949$	−60.4942	−1.96372
$950$	−0.116520	−0.00378041
$951$	0	0
$952$	0	0
$953$	28.1026i	0.910333i	0.890406	+	0.455166i	$0.150420\pi$
$953$	28.1026i	0.910333i	−0.890406	+	0.455166i	$0.849580\pi$
$954$	0	0
$955$	− 18.9533i	− 0.613313i
$956$	14.8895i	0.481562i
$957$	0	0
$958$	− 23.9665i	− 0.774322i
$959$	0	0
$960$	0	0
$961$	10.2969	0.332158
$962$	7.68022	0.247620
$963$	0	0
$964$	23.1405i	0.745307i
$965$	−8.43060	−0.271391
$966$	0	0
$967$	42.0721	1.35295	0.676473	−	0.736467i	$-0.263507\pi$
$967$	42.0721	1.35295	0.676473	+	0.736467i	$0.263507\pi$
$968$	− 10.9768i	− 0.352808i
$969$	0	0
$970$	−13.3910	−0.429960
$971$	−53.8665	−1.72866	−0.864329	−	0.502927i	$-0.832257\pi$
$971$	−53.8665	−1.72866	−0.864329	+	0.502927i	$0.832257\pi$
$972$	0	0
$973$	0	0
$974$	38.3699i	1.22945i
$975$	0	0
$976$	− 8.94495i	− 0.286321i
$977$	44.6475i	1.42840i	0.699941	+	0.714200i	$0.253209\pi$
$977$	44.6475i	1.42840i	−0.699941	+	0.714200i	$0.746791\pi$
$978$	0	0
$979$	2.07414i	0.0662899i
$980$	0	0
$981$	0	0
$982$	−7.83834	−0.250132
$983$	7.90394	0.252097	0.126048	−	0.992024i	$-0.459771\pi$
$983$	7.90394	0.252097	0.126048	+	0.992024i	$0.459771\pi$
$984$	0	0
$985$	− 0.108381i	− 0.00345332i
$986$	−0.324131	−0.0103224
$987$	0	0
$988$	−0.605778	−0.0192724
$989$	− 27.9659i	− 0.889263i
$990$	0	0
$991$	47.3935	1.50550	0.752752	−	0.658304i	$-0.228726\pi$
$991$	47.3935	1.50550	0.752752	+	0.658304i	$0.228726\pi$
$992$	−4.55007	−0.144465
$993$	0	0
$994$	0	0
$995$	23.4524i	0.743490i
$996$	0	0
$997$	− 26.9514i	− 0.853558i	−0.904356	−	0.426779i	$-0.859648\pi$
$997$	− 26.9514i	− 0.853558i	0.904356	−	0.426779i	$-0.140352\pi$
$998$	− 13.7056i	− 0.433845i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.b.a.881.5	yes	8
3.2	odd	2		4410.2.b.d.881.4	yes	8
7.6	odd	2		4410.2.b.d.881.5	yes	8
21.20	even	2	inner	4410.2.b.a.881.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.b.a.881.4	✓	8	21.20	even	2	inner
4410.2.b.a.881.5	yes	8	1.1	even	1	trivial
4410.2.b.d.881.4	yes	8	3.2	odd	2
4410.2.b.d.881.5	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000i q^{8} -1.00000i q^{10} +0.152241i q^{11} -5.19891i q^{13} +1.00000 q^{16} +0.881887 q^{17} +0.116520i q^{19} +1.00000 q^{20} -0.152241 q^{22} +7.88764i q^{23} +1.00000 q^{25} +5.19891 q^{26} +0.367542i q^{29} +4.55007i q^{31} +1.00000i q^{32} +0.881887i q^{34} +1.47727 q^{37} -0.116520 q^{38} +1.00000i q^{40} -3.38687 q^{41} -3.54553 q^{43} -0.152241i q^{44} -7.88764 q^{46} +4.03572 q^{47} +1.00000i q^{50} +5.19891i q^{52} -8.80750i q^{53} -0.152241i q^{55} -0.367542 q^{58} -2.78016 q^{59} -8.94495i q^{61} -4.55007 q^{62} -1.00000 q^{64} +5.19891i q^{65} -7.82458 q^{67} -0.881887 q^{68} -13.6337i q^{71} -11.6359i q^{73} +1.47727i q^{74} -0.116520i q^{76} -5.65685 q^{79} -1.00000 q^{80} -3.38687i q^{82} -7.22625 q^{83} -0.881887 q^{85} -3.54553i q^{86} +0.152241 q^{88} +13.6241 q^{89} -7.88764i q^{92} +4.03572i q^{94} -0.116520i q^{95} -13.3910i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{5}+O(q^{10}) \) 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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