LMFDB - Embedded newform 8100.2.d.m.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(649,8100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8100.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (of order $2$, degree $1$, not 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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.m.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205i q^{7} +1.73205 q^{11} +5.46410i q^{13} +4.73205i q^{17} +4.46410 q^{19} +3.46410i q^{23} +7.73205 q^{29} +5.92820 q^{31} +6.19615i q^{37} -11.1962 q^{41} +3.26795i q^{43} +1.26795i q^{47} -0.464102 q^{49} -7.26795i q^{53} +7.73205 q^{59} -4.00000 q^{61} -6.39230i q^{67} -11.1962 q^{71} -0.196152i q^{73} +4.73205i q^{77} +14.3923 q^{79} -15.1244i q^{83} -5.19615 q^{89} -14.9282 q^{91} -0.732051i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{19} + 24 q^{29} - 4 q^{31} - 24 q^{41} + 12 q^{49} + 24 q^{59} - 16 q^{61} - 24 q^{71} + 16 q^{79} - 32 q^{91}+O(q^{100}) $ 4 * q + 4 * q^19 + 24 * q^29 - 4 * q^31 - 24 * q^41 + 12 * q^49 + 24 * q^59 - 16 * q^61 - 24 * q^71 + 16 * q^79 - 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.73205i	1.03262i	0.856402	+	0.516309i	$0.172694\pi$
$7$	2.73205i	1.03262i	−0.856402	+	0.516309i	$0.827306\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	5.46410i	1.51547i	0.652563	+	0.757735i	$0.273694\pi$
$13$	5.46410i	1.51547i	−0.652563	+	0.757735i	$0.726306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.73205i	1.14769i	0.818964	+	0.573845i	$0.194549\pi$
$17$	4.73205i	1.14769i	−0.818964	+	0.573845i	$0.805451\pi$
$18$	0	0
$19$	4.46410	1.02414	0.512068	−	0.858945i	$-0.328880\pi$
$19$	4.46410	1.02414	0.512068	+	0.858945i	$0.328880\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.46410i	0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$	3.46410i	0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.73205	1.43581	0.717903	−	0.696143i	$-0.245102\pi$
$29$	7.73205	1.43581	0.717903	+	0.696143i	$0.245102\pi$
$30$	0	0
$31$	5.92820	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$31$	5.92820	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.19615i	1.01864i	0.860577	+	0.509321i	$0.170103\pi$
$37$	6.19615i	1.01864i	−0.860577	+	0.509321i	$0.829897\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.1962	−1.74855	−0.874273	−	0.485435i	$-0.838661\pi$
$41$	−11.1962	−1.74855	−0.874273	+	0.485435i	$0.838661\pi$
$42$	0	0
$43$	3.26795i	0.498358i	0.968458	+	0.249179i	$0.0801607\pi$
$43$	3.26795i	0.498358i	−0.968458	+	0.249179i	$0.919839\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.26795i	0.184949i	0.995715	+	0.0924747i	$0.0294777\pi$
$47$	1.26795i	0.184949i	−0.995715	+	0.0924747i	$0.970522\pi$
$48$	0	0
$49$	−0.464102	−0.0663002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.26795i	− 0.998330i	−0.866507	−	0.499165i	$-0.833640\pi$
$53$	− 7.26795i	− 0.998330i	0.866507	−	0.499165i	$-0.166360\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.73205	1.00663	0.503314	−	0.864104i	$-0.332114\pi$
$59$	7.73205	1.00663	0.503314	+	0.864104i	$0.332114\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.39230i	− 0.780944i	−0.920615	−	0.390472i	$-0.872312\pi$
$67$	− 6.39230i	− 0.780944i	0.920615	−	0.390472i	$-0.127688\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.1962	−1.32874	−0.664369	−	0.747404i	$-0.731300\pi$
$71$	−11.1962	−1.32874	−0.664369	+	0.747404i	$0.731300\pi$
$72$	0	0
$73$	− 0.196152i	− 0.0229579i	−0.999934	−	0.0114790i	$-0.996346\pi$
$73$	− 0.196152i	− 0.0229579i	0.999934	−	0.0114790i	$-0.00365394\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.73205i	0.539267i
$78$	0	0
$79$	14.3923	1.61926	0.809630	−	0.586940i	$-0.199668\pi$
$79$	14.3923	1.61926	0.809630	+	0.586940i	$0.199668\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 15.1244i	− 1.66011i	−0.557679	−	0.830057i	$-0.688308\pi$
$83$	− 15.1244i	− 1.66011i	0.557679	−	0.830057i	$-0.311692\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	−14.9282	−1.56490
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.732051i	− 0.0743285i	−0.999309	−	0.0371642i	$-0.988168\pi$
$97$	− 0.732051i	− 0.0743285i	0.999309	−	0.0371642i	$-0.0118325\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.12436	0.609396	0.304698	−	0.952449i	$-0.401444\pi$
$101$	6.12436	0.609396	0.304698	+	0.952449i	$0.401444\pi$
$102$	0	0
$103$	18.3923i	1.81225i	0.423013	+	0.906124i	$0.360973\pi$
$103$	18.3923i	1.81225i	−0.423013	+	0.906124i	$0.639027\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.46410i	− 0.334887i	−0.985882	−	0.167444i	$-0.946449\pi$
$107$	− 3.46410i	− 0.334887i	0.985882	−	0.167444i	$-0.0535512\pi$
$108$	0	0
$109$	7.92820	0.759384	0.379692	−	0.925113i	$-0.376030\pi$
$109$	7.92820	0.759384	0.379692	+	0.925113i	$0.376030\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 0.339746i	− 0.0319606i	−0.999872	−	0.0159803i	$-0.994913\pi$
$113$	− 0.339746i	− 0.0319606i	0.999872	−	0.0159803i	$-0.00508691\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.9282	−1.18513
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.19615i	− 0.372348i	−0.982517	−	0.186174i	$-0.940391\pi$
$127$	− 4.19615i	− 0.372348i	0.982517	−	0.186174i	$-0.0596089\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.2679	0.897115	0.448557	−	0.893754i	$-0.351938\pi$
$131$	10.2679	0.897115	0.448557	+	0.893754i	$0.351938\pi$
$132$	0	0
$133$	12.1962i	1.05754i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.39230i	− 0.375260i	−0.982240	−	0.187630i	$-0.939919\pi$
$137$	− 4.39230i	− 0.375260i	0.982240	−	0.187630i	$-0.0600806\pi$
$138$	0	0
$139$	−15.3923	−1.30556	−0.652779	−	0.757548i	$-0.726397\pi$
$139$	−15.3923	−1.30556	−0.652779	+	0.757548i	$0.726397\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.46410i	0.791428i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−17.3923	−1.41537	−0.707683	−	0.706530i	$-0.750259\pi$
$151$	−17.3923	−1.41537	−0.707683	+	0.706530i	$0.750259\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.26795i	− 0.260811i	−0.991461	−	0.130405i	$-0.958372\pi$
$157$	− 3.26795i	− 0.260811i	0.991461	−	0.130405i	$-0.0416279\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.46410	−0.745876
$162$	0	0
$163$	18.7321i	1.46721i	0.679577	+	0.733604i	$0.262163\pi$
$163$	18.7321i	1.46721i	−0.679577	+	0.733604i	$0.737837\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.1244i	1.63465i	0.576176	+	0.817326i	$0.304544\pi$
$167$	21.1244i	1.63465i	−0.576176	+	0.817326i	$0.695456\pi$
$168$	0	0
$169$	−16.8564	−1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.2487i	− 1.84360i	−0.387671	−	0.921798i	$-0.626720\pi$
$173$	− 24.2487i	− 1.84360i	0.387671	−	0.921798i	$-0.373280\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.1244	0.906217	0.453108	−	0.891455i	$-0.350315\pi$
$179$	12.1244	0.906217	0.453108	+	0.891455i	$0.350315\pi$
$180$	0	0
$181$	−16.4641	−1.22377	−0.611884	−	0.790948i	$-0.709588\pi$
$181$	−16.4641	−1.22377	−0.611884	+	0.790948i	$0.709588\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.19615i	0.599362i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.0526	1.37859	0.689297	−	0.724479i	$-0.257919\pi$
$191$	19.0526	1.37859	0.689297	+	0.724479i	$0.257919\pi$
$192$	0	0
$193$	− 10.5885i	− 0.762174i	−0.924539	−	0.381087i	$-0.875550\pi$
$193$	− 10.5885i	− 0.762174i	0.924539	−	0.381087i	$-0.124450\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.8564i	0.987228i	0.869681	+	0.493614i	$0.164324\pi$
$197$	13.8564i	0.987228i	−0.869681	+	0.493614i	$0.835676\pi$
$198$	0	0
$199$	−15.8564	−1.12403	−0.562015	−	0.827127i	$-0.689974\pi$
$199$	−15.8564	−1.12403	−0.562015	+	0.827127i	$0.689974\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	21.1244i	1.48264i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.73205	0.534837
$210$	0	0
$211$	−19.9282	−1.37191	−0.685957	−	0.727642i	$-0.740616\pi$
$211$	−19.9282	−1.37191	−0.685957	+	0.727642i	$0.740616\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.1962i	1.09947i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−25.8564	−1.73929
$222$	0	0
$223$	− 5.85641i	− 0.392174i	−0.980587	−	0.196087i	$-0.937177\pi$
$223$	− 5.85641i	− 0.392174i	0.980587	−	0.196087i	$-0.0628235\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 26.1962i	− 1.73870i	−0.494197	−	0.869350i	$-0.664538\pi$
$227$	− 26.1962i	− 1.73870i	0.494197	−	0.869350i	$-0.335462\pi$
$228$	0	0
$229$	17.8564	1.17998	0.589992	−	0.807409i	$-0.299131\pi$
$229$	17.8564	1.17998	0.589992	+	0.807409i	$0.299131\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.5885i	1.61084i	0.592702	+	0.805422i	$0.298061\pi$
$233$	24.5885i	1.61084i	−0.592702	+	0.805422i	$0.701939\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.53590	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$239$	−8.53590	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$240$	0	0
$241$	10.3205	0.664802	0.332401	−	0.943138i	$-0.392141\pi$
$241$	10.3205	0.664802	0.332401	+	0.943138i	$0.392141\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	24.3923i	1.55205i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.5359	1.29621	0.648107	−	0.761549i	$-0.275561\pi$
$251$	20.5359	1.29621	0.648107	+	0.761549i	$0.275561\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 15.4641i	− 0.964624i	−0.875999	−	0.482312i	$-0.839797\pi$
$257$	− 15.4641i	− 0.964624i	0.875999	−	0.482312i	$-0.160203\pi$
$258$	0	0
$259$	−16.9282	−1.05187
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 24.2487i	− 1.49524i	−0.664127	−	0.747620i	$-0.731197\pi$
$263$	− 24.2487i	− 1.49524i	0.664127	−	0.747620i	$-0.268803\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.2679	0.991874	0.495937	−	0.868358i	$-0.334825\pi$
$269$	16.2679	0.991874	0.495937	+	0.868358i	$0.334825\pi$
$270$	0	0
$271$	16.7846	1.01959	0.509796	−	0.860295i	$-0.329721\pi$
$271$	16.7846	1.01959	0.509796	+	0.860295i	$0.329721\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.12436i	− 0.307893i	−0.988079	−	0.153946i	$-0.950802\pi$
$277$	− 5.12436i	− 0.307893i	0.988079	−	0.153946i	$-0.0491983\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.3205	1.74911	0.874557	−	0.484922i	$-0.161152\pi$
$281$	29.3205	1.74911	0.874557	+	0.484922i	$0.161152\pi$
$282$	0	0
$283$	16.5359i	0.982957i	0.870890	+	0.491479i	$0.163543\pi$
$283$	16.5359i	0.982957i	−0.870890	+	0.491479i	$0.836457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 30.5885i	− 1.80558i
$288$	0	0
$289$	−5.39230	−0.317194
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.5885i	1.43647i	0.695799	+	0.718237i	$0.255050\pi$
$293$	24.5885i	1.43647i	−0.695799	+	0.718237i	$0.744950\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.9282	−1.09465
$300$	0	0
$301$	−8.92820	−0.514613
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.80385i	0.102951i	0.998674	+	0.0514755i	$0.0163924\pi$
$307$	1.80385i	0.102951i	−0.998674	+	0.0514755i	$0.983608\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.5167	−0.936574	−0.468287	−	0.883576i	$-0.655129\pi$
$311$	−16.5167	−0.936574	−0.468287	+	0.883576i	$0.655129\pi$
$312$	0	0
$313$	14.9282i	0.843792i	0.906644	+	0.421896i	$0.138635\pi$
$313$	14.9282i	0.843792i	−0.906644	+	0.421896i	$0.861365\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.12436i	− 0.175481i	−0.996143	−	0.0877406i	$-0.972035\pi$
$317$	− 3.12436i	− 0.175481i	0.996143	−	0.0877406i	$-0.0279647\pi$
$318$	0	0
$319$	13.3923	0.749825
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.1244i	1.17539i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.46410	−0.190982
$330$	0	0
$331$	−29.3923	−1.61555	−0.807774	−	0.589493i	$-0.799328\pi$
$331$	−29.3923	−1.61555	−0.807774	+	0.589493i	$0.799328\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	34.2487i	1.86565i	0.360335	+	0.932823i	$0.382662\pi$
$337$	34.2487i	1.86565i	−0.360335	+	0.932823i	$0.617338\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.2679	0.556041
$342$	0	0
$343$	17.8564i	0.964155i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 21.8038i	− 1.17049i	−0.810856	−	0.585246i	$-0.800998\pi$
$347$	− 21.8038i	− 1.17049i	0.810856	−	0.585246i	$-0.199002\pi$
$348$	0	0
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 28.9808i	− 1.54249i	−0.636538	−	0.771245i	$-0.719634\pi$
$353$	− 28.9808i	− 1.54249i	0.636538	−	0.771245i	$-0.280366\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.33975	−0.492933	−0.246466	−	0.969151i	$-0.579270\pi$
$359$	−9.33975	−0.492933	−0.246466	+	0.969151i	$0.579270\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.3923i	1.37767i	0.724920	+	0.688834i	$0.241877\pi$
$367$	26.3923i	1.37767i	−0.724920	+	0.688834i	$0.758123\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	19.8564	1.03089
$372$	0	0
$373$	− 14.0526i	− 0.727614i	−0.931474	−	0.363807i	$-0.881477\pi$
$373$	− 14.0526i	− 0.727614i	0.931474	−	0.363807i	$-0.118523\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	42.2487i	2.17592i
$378$	0	0
$379$	−4.53590	−0.232993	−0.116497	−	0.993191i	$-0.537166\pi$
$379$	−4.53590	−0.232993	−0.116497	+	0.993191i	$0.537166\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.2487i	0.932466i	0.884662	+	0.466233i	$0.154389\pi$
$383$	18.2487i	0.932466i	−0.884662	+	0.466233i	$0.845611\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−27.4641	−1.39249	−0.696243	−	0.717807i	$-0.745146\pi$
$389$	−27.4641	−1.39249	−0.696243	+	0.717807i	$0.745146\pi$
$390$	0	0
$391$	−16.3923	−0.828994
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 37.3205i	− 1.87306i	−0.350584	−	0.936531i	$-0.614017\pi$
$397$	− 37.3205i	− 1.87306i	0.350584	−	0.936531i	$-0.385983\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.7846	−0.738308	−0.369154	−	0.929368i	$-0.620353\pi$
$401$	−14.7846	−0.738308	−0.369154	+	0.929368i	$0.620353\pi$
$402$	0	0
$403$	32.3923i	1.61358i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.7321i	0.531968i
$408$	0	0
$409$	17.8564	0.882942	0.441471	−	0.897275i	$-0.354457\pi$
$409$	17.8564	0.882942	0.441471	+	0.897275i	$0.354457\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21.1244i	1.03946i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.4641	−1.04859	−0.524295	−	0.851537i	$-0.675671\pi$
$419$	−21.4641	−1.04859	−0.524295	+	0.851537i	$0.675671\pi$
$420$	0	0
$421$	13.7846	0.671821	0.335910	−	0.941894i	$-0.390956\pi$
$421$	13.7846	0.671821	0.335910	+	0.941894i	$0.390956\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 10.9282i	− 0.528853i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.5885	−1.61790	−0.808950	−	0.587878i	$-0.799963\pi$
$431$	−33.5885	−1.61790	−0.808950	+	0.587878i	$0.799963\pi$
$432$	0	0
$433$	11.4641i	0.550930i	0.961311	+	0.275465i	$0.0888317\pi$
$433$	11.4641i	0.550930i	−0.961311	+	0.275465i	$0.911168\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.4641i	0.739748i
$438$	0	0
$439$	−6.60770	−0.315368	−0.157684	−	0.987490i	$-0.550403\pi$
$439$	−6.60770	−0.315368	−0.157684	+	0.987490i	$0.550403\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.2679i	1.77065i	0.464969	+	0.885327i	$0.346065\pi$
$443$	37.2679i	1.77065i	−0.464969	+	0.885327i	$0.653935\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.1244	−1.13850	−0.569249	−	0.822165i	$-0.692766\pi$
$449$	−24.1244	−1.13850	−0.569249	+	0.822165i	$0.692766\pi$
$450$	0	0
$451$	−19.3923	−0.913148
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.1962i	− 1.03829i	−0.854686	−	0.519146i	$-0.826250\pi$
$457$	− 22.1962i	− 1.03829i	0.854686	−	0.519146i	$-0.173750\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.58846	0.446579	0.223289	−	0.974752i	$-0.428320\pi$
$461$	9.58846	0.446579	0.223289	+	0.974752i	$0.428320\pi$
$462$	0	0
$463$	− 2.39230i	− 0.111180i	−0.998454	−	0.0555899i	$-0.982296\pi$
$463$	− 2.39230i	− 0.111180i	0.998454	−	0.0555899i	$-0.0177039\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 26.1962i	− 1.21221i	−0.795383	−	0.606107i	$-0.792730\pi$
$467$	− 26.1962i	− 1.21221i	0.795383	−	0.606107i	$-0.207270\pi$
$468$	0	0
$469$	17.4641	0.806417
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.66025i	0.260259i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.33975	0.152597	0.0762984	−	0.997085i	$-0.475690\pi$
$479$	3.33975	0.152597	0.0762984	+	0.997085i	$0.475690\pi$
$480$	0	0
$481$	−33.8564	−1.54372
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.5359i	1.38371i	0.722035	+	0.691857i	$0.243207\pi$
$487$	30.5359i	1.38371i	−0.722035	+	0.691857i	$0.756793\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.26795	−0.192610	−0.0963049	−	0.995352i	$-0.530702\pi$
$491$	−4.26795	−0.192610	−0.0963049	+	0.995352i	$0.530702\pi$
$492$	0	0
$493$	36.5885i	1.64786i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 30.5885i	− 1.37208i
$498$	0	0
$499$	−27.3923	−1.22625	−0.613124	−	0.789987i	$-0.710087\pi$
$499$	−27.3923	−1.22625	−0.613124	+	0.789987i	$0.710087\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 35.3205i	− 1.57486i	−0.616402	−	0.787432i	$-0.711410\pi$
$503$	− 35.3205i	− 1.57486i	0.616402	−	0.787432i	$-0.288590\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.7846	1.18721	0.593603	−	0.804758i	$-0.297705\pi$
$509$	26.7846	1.18721	0.593603	+	0.804758i	$0.297705\pi$
$510$	0	0
$511$	0.535898	0.0237067
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.19615i	0.0965867i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	0	0
$523$	3.60770i	0.157753i	0.996884	+	0.0788767i	$0.0251334\pi$
$523$	3.60770i	0.157753i	−0.996884	+	0.0788767i	$0.974867\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.0526i	1.22199i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 61.1769i	− 2.64987i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.803848	−0.0346242
$540$	0	0
$541$	2.46410	0.105940	0.0529700	−	0.998596i	$-0.483131\pi$
$541$	2.46410	0.105940	0.0529700	+	0.998596i	$0.483131\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.78461i	− 0.204575i	−0.994755	−	0.102288i	$-0.967384\pi$
$547$	− 4.78461i	− 0.204575i	0.994755	−	0.102288i	$-0.0326162\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	34.5167	1.47046
$552$	0	0
$553$	39.3205i	1.67208i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.39230i	0.186108i	0.995661	+	0.0930540i	$0.0296629\pi$
$557$	4.39230i	0.186108i	−0.995661	+	0.0930540i	$0.970337\pi$
$558$	0	0
$559$	−17.8564	−0.755246
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.7321i	0.958042i	0.877804	+	0.479021i	$0.159008\pi$
$563$	22.7321i	0.958042i	−0.877804	+	0.479021i	$0.840992\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	43.0526	1.80486	0.902429	−	0.430839i	$-0.141782\pi$
$569$	43.0526	1.80486	0.902429	+	0.430839i	$0.141782\pi$
$570$	0	0
$571$	0.856406	0.0358395	0.0179197	−	0.999839i	$-0.494296\pi$
$571$	0.856406	0.0358395	0.0179197	+	0.999839i	$0.494296\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 11.8038i	− 0.491401i	−0.969346	−	0.245700i	$-0.920982\pi$
$577$	− 11.8038i	− 0.491401i	0.969346	−	0.245700i	$-0.0790179\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	41.3205	1.71426
$582$	0	0
$583$	− 12.5885i	− 0.521361i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.80385i	− 0.157002i	−0.996914	−	0.0785008i	$-0.974987\pi$
$587$	− 3.80385i	− 0.157002i	0.996914	−	0.0785008i	$-0.0250133\pi$
$588$	0	0
$589$	26.4641	1.09043
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 23.0718i	− 0.947445i	−0.880674	−	0.473723i	$-0.842910\pi$
$593$	− 23.0718i	− 0.947445i	0.880674	−	0.473723i	$-0.157090\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.4115	0.588840	0.294420	−	0.955676i	$-0.404874\pi$
$599$	14.4115	0.588840	0.294420	+	0.955676i	$0.404874\pi$
$600$	0	0
$601$	−24.3205	−0.992054	−0.496027	−	0.868307i	$-0.665208\pi$
$601$	−24.3205	−0.992054	−0.496027	+	0.868307i	$0.665208\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.58846i	− 0.105062i	−0.998619	−	0.0525311i	$-0.983271\pi$
$607$	− 2.58846i	− 0.105062i	0.998619	−	0.0525311i	$-0.0167289\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.92820	−0.280285
$612$	0	0
$613$	24.3923i	0.985196i	0.870257	+	0.492598i	$0.163953\pi$
$613$	24.3923i	0.985196i	−0.870257	+	0.492598i	$0.836047\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 14.1962i	− 0.568757i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.3205	−1.16909
$630$	0	0
$631$	−0.0717968	−0.00285818	−0.00142909	−	0.999999i	$-0.500455\pi$
$631$	−0.0717968	−0.00285818	−0.00142909	+	0.999999i	$0.500455\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.53590i	− 0.100476i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.8038	−0.505722	−0.252861	−	0.967503i	$-0.581371\pi$
$641$	−12.8038	−0.505722	−0.252861	+	0.967503i	$0.581371\pi$
$642$	0	0
$643$	14.5885i	0.575313i	0.957734	+	0.287656i	$0.0928761\pi$
$643$	14.5885i	0.575313i	−0.957734	+	0.287656i	$0.907124\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 0.248711i	− 0.00977785i	−0.999988	−	0.00488893i	$-0.998444\pi$
$647$	− 0.248711i	− 0.00977785i	0.999988	−	0.00488893i	$-0.00155620\pi$
$648$	0	0
$649$	13.3923	0.525694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.53590i	− 0.0992374i	−0.998768	−	0.0496187i	$-0.984199\pi$
$653$	− 2.53590i	− 0.0992374i	0.998768	−	0.0496187i	$-0.0158006\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.53590	0.0987846	0.0493923	−	0.998779i	$-0.484272\pi$
$659$	2.53590	0.0987846	0.0493923	+	0.998779i	$0.484272\pi$
$660$	0	0
$661$	15.3923	0.598691	0.299346	−	0.954145i	$-0.403232\pi$
$661$	15.3923	0.598691	0.299346	+	0.954145i	$0.403232\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	26.7846i	1.03710i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.92820	−0.267460
$672$	0	0
$673$	− 38.3923i	− 1.47991i	−0.672654	−	0.739957i	$-0.734846\pi$
$673$	− 38.3923i	− 1.47991i	0.672654	−	0.739957i	$-0.265154\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 40.6410i	− 1.56196i	−0.624555	−	0.780981i	$-0.714720\pi$
$677$	− 40.6410i	− 1.56196i	0.624555	−	0.780981i	$-0.285280\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.4641i	0.821301i	0.911793	+	0.410651i	$0.134698\pi$
$683$	21.4641i	0.821301i	−0.911793	+	0.410651i	$0.865302\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	39.7128	1.51294
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 52.9808i	− 2.00679i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17.8756	−0.675154	−0.337577	−	0.941298i	$-0.609607\pi$
$701$	−17.8756	−0.675154	−0.337577	+	0.941298i	$0.609607\pi$
$702$	0	0
$703$	27.6603i	1.04323i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.7321i	0.629274i
$708$	0	0
$709$	−10.5359	−0.395684	−0.197842	−	0.980234i	$-0.563393\pi$
$709$	−10.5359	−0.395684	−0.197842	+	0.980234i	$0.563393\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.5359i	0.769075i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.1962	0.641308	0.320654	−	0.947196i	$-0.396097\pi$
$719$	17.1962	0.641308	0.320654	+	0.947196i	$0.396097\pi$
$720$	0	0
$721$	−50.2487	−1.87136
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.1769i	1.08211i	0.840987	+	0.541056i	$0.181975\pi$
$727$	29.1769i	1.08211i	−0.840987	+	0.541056i	$0.818025\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.4641	−0.571960
$732$	0	0
$733$	43.5692i	1.60927i	0.593773	+	0.804633i	$0.297638\pi$
$733$	43.5692i	1.60927i	−0.593773	+	0.804633i	$0.702362\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.0718i	− 0.407835i
$738$	0	0
$739$	26.1769	0.962933	0.481467	−	0.876464i	$-0.340104\pi$
$739$	26.1769	0.962933	0.481467	+	0.876464i	$0.340104\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 5.41154i	− 0.198530i	−0.995061	−	0.0992651i	$-0.968351\pi$
$743$	− 5.41154i	− 0.198530i	0.995061	−	0.0992651i	$-0.0316492\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.46410	0.345811
$750$	0	0
$751$	40.7846	1.48825	0.744126	−	0.668040i	$-0.232866\pi$
$751$	40.7846	1.48825	0.744126	+	0.668040i	$0.232866\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.3923i	0.741171i	0.928798	+	0.370585i	$0.120843\pi$
$757$	20.3923i	0.741171i	−0.928798	+	0.370585i	$0.879157\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.1244	−1.09201	−0.546004	−	0.837783i	$-0.683851\pi$
$761$	−30.1244	−1.09201	−0.546004	+	0.837783i	$0.683851\pi$
$762$	0	0
$763$	21.6603i	0.784154i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.2487i	1.52551i
$768$	0	0
$769$	−35.2487	−1.27110	−0.635551	−	0.772059i	$-0.719227\pi$
$769$	−35.2487	−1.27110	−0.635551	+	0.772059i	$0.719227\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17.6603i	0.635195i	0.948226	+	0.317598i	$0.102876\pi$
$773$	17.6603i	0.635195i	−0.948226	+	0.317598i	$0.897124\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−49.9808	−1.79075
$780$	0	0
$781$	−19.3923	−0.693911
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.1962i	1.29025i	0.764076	+	0.645127i	$0.223195\pi$
$787$	36.1962i	1.29025i	−0.764076	+	0.645127i	$0.776805\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.928203	0.0330031
$792$	0	0
$793$	− 21.8564i	− 0.776144i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23.3205i	− 0.826055i	−0.910719	−	0.413027i	$-0.864471\pi$
$797$	− 23.3205i	− 0.826055i	0.910719	−	0.413027i	$-0.135529\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 0.339746i	− 0.0119894i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.41154	0.295734	0.147867	−	0.989007i	$-0.452759\pi$
$809$	8.41154	0.295734	0.147867	+	0.989007i	$0.452759\pi$
$810$	0	0
$811$	−25.2487	−0.886602	−0.443301	−	0.896373i	$-0.646193\pi$
$811$	−25.2487	−0.886602	−0.443301	+	0.896373i	$0.646193\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.5885i	0.510386i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.33975	0.116558	0.0582790	−	0.998300i	$-0.481439\pi$
$821$	3.33975	0.116558	0.0582790	+	0.998300i	$0.481439\pi$
$822$	0	0
$823$	− 16.9282i	− 0.590080i	−0.955485	−	0.295040i	$-0.904667\pi$
$823$	− 16.9282i	− 0.590080i	0.955485	−	0.295040i	$-0.0953330\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.4641i	1.58094i	0.612500	+	0.790471i	$0.290164\pi$
$827$	45.4641i	1.58094i	−0.612500	+	0.790471i	$0.709836\pi$
$828$	0	0
$829$	51.7846	1.79855	0.899277	−	0.437380i	$-0.144093\pi$
$829$	51.7846	1.79855	0.899277	+	0.437380i	$0.144093\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.19615i	− 0.0760922i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.41154	0.290399	0.145199	−	0.989402i	$-0.453618\pi$
$839$	8.41154	0.290399	0.145199	+	0.989402i	$0.453618\pi$
$840$	0	0
$841$	30.7846	1.06154
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 21.8564i	− 0.750995i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21.4641	−0.735780
$852$	0	0
$853$	− 0.196152i	− 0.00671613i	−0.999994	−	0.00335807i	$-0.998931\pi$
$853$	− 0.196152i	− 0.00671613i	0.999994	−	0.00335807i	$-0.00106891\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.0718i	0.993074i	0.868016	+	0.496537i	$0.165395\pi$
$857$	29.0718i	0.993074i	−0.868016	+	0.496537i	$0.834605\pi$
$858$	0	0
$859$	−7.78461	−0.265607	−0.132804	−	0.991142i	$-0.542398\pi$
$859$	−7.78461	−0.265607	−0.132804	+	0.991142i	$0.542398\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 49.5167i	− 1.68557i	−0.538253	−	0.842783i	$-0.680915\pi$
$863$	− 49.5167i	− 1.68557i	0.538253	−	0.842783i	$-0.319085\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.9282	0.845631
$870$	0	0
$871$	34.9282	1.18350
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.2487i	− 0.481145i	−0.970631	−	0.240572i	$-0.922665\pi$
$877$	− 14.2487i	− 0.481145i	0.970631	−	0.240572i	$-0.0773351\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.80385	0.229227	0.114614	−	0.993410i	$-0.463437\pi$
$881$	6.80385	0.229227	0.114614	+	0.993410i	$0.463437\pi$
$882$	0	0
$883$	− 46.8372i	− 1.57620i	−0.615550	−	0.788098i	$-0.711066\pi$
$883$	− 46.8372i	− 1.57620i	0.615550	−	0.788098i	$-0.288934\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 13.2679i	− 0.445494i	−0.974876	−	0.222747i	$-0.928498\pi$
$887$	− 13.2679i	− 0.445494i	0.974876	−	0.222747i	$-0.0715024\pi$
$888$	0	0
$889$	11.4641	0.384494
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.66025i	0.189413i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45.8372	1.52876
$900$	0	0
$901$	34.3923	1.14577
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.21539i	0.0403564i	0.999796	+	0.0201782i	$0.00642335\pi$
$907$	1.21539i	0.0403564i	−0.999796	+	0.0201782i	$0.993577\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.80385	−0.225422	−0.112711	−	0.993628i	$-0.535953\pi$
$911$	−6.80385	−0.225422	−0.112711	+	0.993628i	$0.535953\pi$
$912$	0	0
$913$	− 26.1962i	− 0.866966i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.0526i	0.926377i
$918$	0	0
$919$	−51.3923	−1.69528	−0.847638	−	0.530575i	$-0.821976\pi$
$919$	−51.3923	−1.69528	−0.847638	+	0.530575i	$0.821976\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 61.1769i	− 2.01366i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.48334	−0.0486668	−0.0243334	−	0.999704i	$-0.507746\pi$
$929$	−1.48334	−0.0486668	−0.0243334	+	0.999704i	$0.507746\pi$
$930$	0	0
$931$	−2.07180	−0.0679004
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 19.0718i	− 0.623048i	−0.950238	−	0.311524i	$-0.899160\pi$
$937$	− 19.0718i	− 0.623048i	0.950238	−	0.311524i	$-0.100840\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.53590	−0.278262	−0.139131	−	0.990274i	$-0.544431\pi$
$941$	−8.53590	−0.278262	−0.139131	+	0.990274i	$0.544431\pi$
$942$	0	0
$943$	− 38.7846i	− 1.26300i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.6410i	1.71060i	0.518130	+	0.855302i	$0.326628\pi$
$947$	52.6410i	1.71060i	−0.518130	+	0.855302i	$0.673372\pi$
$948$	0	0
$949$	1.07180	0.0347920
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.53590i	− 0.0821458i	−0.999156	−	0.0410729i	$-0.986922\pi$
$953$	− 2.53590i	− 0.0821458i	0.999156	−	0.0410729i	$-0.0130776\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	4.14359	0.133664
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 7.41154i	− 0.238339i	−0.992874	−	0.119170i	$-0.961977\pi$
$967$	− 7.41154i	− 0.238339i	0.992874	−	0.119170i	$-0.0380232\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.2679	0.714612	0.357306	−	0.933987i	$-0.383695\pi$
$971$	22.2679	0.714612	0.357306	+	0.933987i	$0.383695\pi$
$972$	0	0
$973$	− 42.0526i	− 1.34814i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.39230i	− 0.140522i	−0.997529	−	0.0702611i	$-0.977617\pi$
$977$	− 4.39230i	− 0.140522i	0.997529	−	0.0702611i	$-0.0223833\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 52.7321i	− 1.68189i	−0.541120	−	0.840946i	$-0.681999\pi$
$983$	− 52.7321i	− 1.68189i	0.541120	−	0.840946i	$-0.318001\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.3205	−0.359971
$990$	0	0
$991$	55.7846	1.77206	0.886028	−	0.463631i	$-0.153454\pi$
$991$	55.7846	1.77206	0.886028	+	0.463631i	$0.153454\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 52.1962i	− 1.65307i	−0.562886	−	0.826534i	$-0.690309\pi$
$997$	− 52.1962i	− 1.65307i	0.562886	−	0.826534i	$-0.309691\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.m.649.4		4
3.2	odd	2		8100.2.d.l.649.4		4
5.2	odd	4		1620.2.a.g.1.1	✓	2
5.3	odd	4		8100.2.a.t.1.2		2
5.4	even	2	inner	8100.2.d.m.649.1		4
15.2	even	4		1620.2.a.h.1.1	yes	2
15.8	even	4		8100.2.a.s.1.2		2
15.14	odd	2		8100.2.d.l.649.1		4
20.7	even	4		6480.2.a.bh.1.2		2
45.2	even	12		1620.2.i.m.1081.2		4
45.7	odd	12		1620.2.i.n.1081.2		4
45.22	odd	12		1620.2.i.n.541.2		4
45.32	even	12		1620.2.i.m.541.2		4
60.47	odd	4		6480.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.a.g.1.1	✓	2	5.2	odd	4
1620.2.a.h.1.1	yes	2	15.2	even	4
1620.2.i.m.541.2		4	45.32	even	12
1620.2.i.m.1081.2		4	45.2	even	12
1620.2.i.n.541.2		4	45.22	odd	12
1620.2.i.n.1081.2		4	45.7	odd	12
6480.2.a.bh.1.2		2	20.7	even	4
6480.2.a.bp.1.2		2	60.47	odd	4
8100.2.a.s.1.2		2	15.8	even	4
8100.2.a.t.1.2		2	5.3	odd	4
8100.2.d.l.649.1		4	15.14	odd	2
8100.2.d.l.649.4		4	3.2	odd	2
8100.2.d.m.649.1		4	5.4	even	2	inner
8100.2.d.m.649.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.73205i q^{7} +1.73205 q^{11} +5.46410i q^{13} +4.73205i q^{17} +4.46410 q^{19} +3.46410i q^{23} +7.73205 q^{29} +5.92820 q^{31} +6.19615i q^{37} -11.1962 q^{41} +3.26795i q^{43} +1.26795i q^{47} -0.464102 q^{49} -7.26795i q^{53} +7.73205 q^{59} -4.00000 q^{61} -6.39230i q^{67} -11.1962 q^{71} -0.196152i q^{73} +4.73205i q^{77} +14.3923 q^{79} -15.1244i q^{83} -5.19615 q^{89} -14.9282 q^{91} -0.732051i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{19} + 24 q^{29} - 4 q^{31} - 24 q^{41} + 12 q^{49} + 24 q^{59} - 16 q^{61} - 24 q^{71} + 16 q^{79} - 32 q^{91}+O(q^{100}) \) 4 * q + 4 * q^19 + 24 * q^29 - 4 * q^31 - 24 * q^41 + 12 * q^49 + 24 * q^59 - 16 * q^61 - 24 * q^71 + 16 * q^79 - 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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