LMFDB - Embedded newform 4400.2.b.p.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (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:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-3.37228i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.p.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.37228i q^{3} -3.37228i q^{7} -8.37228 q^{9} +1.00000 q^{11} -2.00000i q^{13} +1.37228i q^{17} +0.627719 q^{19} -11.3723 q^{21} +2.74456i q^{23} +18.1168i q^{27} -1.37228 q^{29} -3.37228 q^{31} -3.37228i q^{33} +9.37228i q^{37} -6.74456 q^{39} -11.4891 q^{41} -4.00000i q^{43} -2.74456i q^{47} -4.37228 q^{49} +4.62772 q^{51} +4.11684i q^{53} -2.11684i q^{57} -2.74456 q^{59} -5.37228 q^{61} +28.2337i q^{63} -8.00000i q^{67} +9.25544 q^{69} -10.1168 q^{71} +15.4891i q^{73} -3.37228i q^{77} -1.25544 q^{79} +35.9783 q^{81} -2.74456i q^{83} +4.62772i q^{87} +1.37228 q^{89} -6.74456 q^{91} +11.3723i q^{93} -12.7446i q^{97} -8.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{9} + 4 q^{11} + 14 q^{19} - 34 q^{21} + 6 q^{29} - 2 q^{31} - 4 q^{39} - 6 q^{49} + 30 q^{51} + 12 q^{59} - 10 q^{61} + 60 q^{69} - 6 q^{71} - 28 q^{79} + 52 q^{81} - 6 q^{89} - 4 q^{91} - 22 q^{99}+O(q^{100}) $ 4 * q - 22 * q^9 + 4 * q^11 + 14 * q^19 - 34 * q^21 + 6 * q^29 - 2 * q^31 - 4 * q^39 - 6 * q^49 + 30 * q^51 + 12 * q^59 - 10 * q^61 + 60 * q^69 - 6 * q^71 - 28 * q^79 + 52 * q^81 - 6 * q^89 - 4 * q^91 - 22 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 3.37228i	− 1.94699i	−0.228714	−	0.973494i	$-0.573452\pi$
$3$	− 3.37228i	− 1.94699i	0.228714	−	0.973494i	$-0.426548\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.37228i	− 1.27460i	−0.770615	−	0.637301i	$-0.780051\pi$
$7$	− 3.37228i	− 1.27460i	0.770615	−	0.637301i	$-0.219949\pi$
$8$	0	0
$9$	−8.37228	−2.79076
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.37228i	0.332827i	0.986056	+	0.166414i	$0.0532187\pi$
$17$	1.37228i	0.332827i	−0.986056	+	0.166414i	$0.946781\pi$
$18$	0	0
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	0	0
$21$	−11.3723	−2.48164
$22$	0	0
$23$	2.74456i	0.572281i	0.958188	+	0.286140i	$0.0923724\pi$
$23$	2.74456i	0.572281i	−0.958188	+	0.286140i	$0.907628\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	18.1168i	3.48659i
$28$	0	0
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	−3.37228	−0.605680	−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228	−0.605680	−0.302840	+	0.953041i	$0.597935\pi$
$32$	0	0
$33$	− 3.37228i	− 0.587039i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.37228i	1.54079i	0.637565	+	0.770397i	$0.279942\pi$
$37$	9.37228i	1.54079i	−0.637565	+	0.770397i	$0.720058\pi$
$38$	0	0
$39$	−6.74456	−1.07999
$40$	0	0
$41$	−11.4891	−1.79430	−0.897150	−	0.441726i	$-0.854366\pi$
$41$	−11.4891	−1.79430	−0.897150	+	0.441726i	$0.854366\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.74456i	− 0.400336i	−0.979762	−	0.200168i	$-0.935851\pi$
$47$	− 2.74456i	− 0.400336i	0.979762	−	0.200168i	$-0.0641487\pi$
$48$	0	0
$49$	−4.37228	−0.624612
$50$	0	0
$51$	4.62772	0.648010
$52$	0	0
$53$	4.11684i	0.565492i	0.959195	+	0.282746i	$0.0912454\pi$
$53$	4.11684i	0.565492i	−0.959195	+	0.282746i	$0.908755\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.11684i	− 0.280383i
$58$	0	0
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	−5.37228	−0.687850	−0.343925	−	0.938997i	$-0.611757\pi$
$61$	−5.37228	−0.687850	−0.343925	+	0.938997i	$0.611757\pi$
$62$	0	0
$63$	28.2337i	3.55711i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	9.25544	1.11422
$70$	0	0
$71$	−10.1168	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$71$	−10.1168	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$72$	0	0
$73$	15.4891i	1.81286i	0.422351	+	0.906432i	$0.361205\pi$
$73$	15.4891i	1.81286i	−0.422351	+	0.906432i	$0.638795\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.37228i	− 0.384307i
$78$	0	0
$79$	−1.25544	−0.141248	−0.0706239	−	0.997503i	$-0.522499\pi$
$79$	−1.25544	−0.141248	−0.0706239	+	0.997503i	$0.522499\pi$
$80$	0	0
$81$	35.9783	3.99758
$82$	0	0
$83$	− 2.74456i	− 0.301255i	−0.988591	−	0.150627i	$-0.951871\pi$
$83$	− 2.74456i	− 0.301255i	0.988591	−	0.150627i	$-0.0481294\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.62772i	0.496144i
$88$	0	0
$89$	1.37228	0.145462	0.0727308	−	0.997352i	$-0.476829\pi$
$89$	1.37228	0.145462	0.0727308	+	0.997352i	$0.476829\pi$
$90$	0	0
$91$	−6.74456	−0.707022
$92$	0	0
$93$	11.3723i	1.17925i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12.7446i	− 1.29401i	−0.762484	−	0.647007i	$-0.776020\pi$
$97$	− 12.7446i	− 1.29401i	0.762484	−	0.647007i	$-0.223980\pi$
$98$	0	0
$99$	−8.37228	−0.841446
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 9.48913i	− 0.934991i	−0.883995	−	0.467496i	$-0.845156\pi$
$103$	− 9.48913i	− 0.934991i	0.883995	−	0.467496i	$-0.154844\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	15.4891	1.48359	0.741795	−	0.670627i	$-0.233975\pi$
$109$	15.4891	1.48359	0.741795	+	0.670627i	$0.233975\pi$
$110$	0	0
$111$	31.6060	2.99991
$112$	0	0
$113$	− 3.25544i	− 0.306246i	−0.988207	−	0.153123i	$-0.951067\pi$
$113$	− 3.25544i	− 0.306246i	0.988207	−	0.153123i	$-0.0489330\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.7446i	1.54804i
$118$	0	0
$119$	4.62772	0.424222
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	38.7446i	3.49348i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	−13.4891	−1.18765
$130$	0	0
$131$	−22.1168	−1.93236	−0.966179	−	0.257873i	$-0.916978\pi$
$131$	−22.1168	−1.93236	−0.966179	+	0.257873i	$0.916978\pi$
$132$	0	0
$133$	− 2.11684i	− 0.183554i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.74456i	0.747098i	0.927610	+	0.373549i	$0.121859\pi$
$137$	8.74456i	0.747098i	−0.927610	+	0.373549i	$0.878141\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−9.25544	−0.779448
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.7446i	1.21611i
$148$	0	0
$149$	−21.6060	−1.77003	−0.885015	−	0.465563i	$-0.845852\pi$
$149$	−21.6060	−1.77003	−0.885015	+	0.465563i	$0.845852\pi$
$150$	0	0
$151$	12.2337	0.995563	0.497782	−	0.867302i	$-0.334148\pi$
$151$	12.2337	0.995563	0.497782	+	0.867302i	$0.334148\pi$
$152$	0	0
$153$	− 11.4891i	− 0.928841i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.37228i	0.747989i	0.927431	+	0.373995i	$0.122012\pi$
$157$	9.37228i	0.747989i	−0.927431	+	0.373995i	$0.877988\pi$
$158$	0	0
$159$	13.8832	1.10101
$160$	0	0
$161$	9.25544	0.729431
$162$	0	0
$163$	− 5.88316i	− 0.460804i	−0.973095	−	0.230402i	$-0.925996\pi$
$163$	− 5.88316i	− 0.460804i	0.973095	−	0.230402i	$-0.0740042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.62772i	0.358104i	0.983840	+	0.179052i	$0.0573030\pi$
$167$	4.62772i	0.358104i	−0.983840	+	0.179052i	$0.942697\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−5.25544	−0.401893
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.25544i	0.695681i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	18.1168i	1.33924i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.37228i	0.100351i
$188$	0	0
$189$	61.0951	4.44401
$190$	0	0
$191$	5.48913	0.397179	0.198590	−	0.980083i	$-0.436364\pi$
$191$	5.48913	0.397179	0.198590	+	0.980083i	$0.436364\pi$
$192$	0	0
$193$	− 14.8614i	− 1.06975i	−0.844932	−	0.534874i	$-0.820359\pi$
$193$	− 14.8614i	− 1.06975i	0.844932	−	0.534874i	$-0.179641\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.7446i	− 1.47799i	−0.673712	−	0.738994i	$-0.735301\pi$
$197$	− 20.7446i	− 1.47799i	0.673712	−	0.738994i	$-0.264699\pi$
$198$	0	0
$199$	18.1168	1.28427	0.642135	−	0.766592i	$-0.278049\pi$
$199$	18.1168	1.28427	0.642135	+	0.766592i	$0.278049\pi$
$200$	0	0
$201$	−26.9783	−1.90290
$202$	0	0
$203$	4.62772i	0.324802i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 22.9783i	− 1.59710i
$208$	0	0
$209$	0.627719	0.0434202
$210$	0	0
$211$	−6.11684	−0.421101	−0.210550	−	0.977583i	$-0.567526\pi$
$211$	−6.11684	−0.421101	−0.210550	+	0.977583i	$0.567526\pi$
$212$	0	0
$213$	34.1168i	2.33765i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.3723i	0.772001i
$218$	0	0
$219$	52.2337	3.52963
$220$	0	0
$221$	2.74456	0.184619
$222$	0	0
$223$	− 18.7446i	− 1.25523i	−0.778524	−	0.627614i	$-0.784031\pi$
$223$	− 18.7446i	− 1.25523i	0.778524	−	0.627614i	$-0.215969\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.74456i	0.182163i	0.995843	+	0.0910815i	$0.0290324\pi$
$227$	2.74456i	0.182163i	−0.995843	+	0.0910815i	$0.970968\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−11.3723	−0.748241
$232$	0	0
$233$	− 1.37228i	− 0.0899011i	−0.998989	−	0.0449506i	$-0.985687\pi$
$233$	− 1.37228i	− 0.0899011i	0.998989	−	0.0449506i	$-0.0143130\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.23369i	0.275008i
$238$	0	0
$239$	−14.7446	−0.953746	−0.476873	−	0.878972i	$-0.658230\pi$
$239$	−14.7446	−0.953746	−0.476873	+	0.878972i	$0.658230\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	− 66.9783i	− 4.29666i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.25544i	− 0.0798816i
$248$	0	0
$249$	−9.25544	−0.586540
$250$	0	0
$251$	−2.74456	−0.173235	−0.0866176	−	0.996242i	$-0.527606\pi$
$251$	−2.74456	−0.173235	−0.0866176	+	0.996242i	$0.527606\pi$
$252$	0	0
$253$	2.74456i	0.172549i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	31.6060	1.96390
$260$	0	0
$261$	11.4891	0.711159
$262$	0	0
$263$	− 24.8614i	− 1.53302i	−0.642232	−	0.766510i	$-0.721992\pi$
$263$	− 24.8614i	− 1.53302i	0.642232	−	0.766510i	$-0.278008\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.62772i	− 0.283212i
$268$	0	0
$269$	8.74456	0.533165	0.266583	−	0.963812i	$-0.414105\pi$
$269$	8.74456	0.533165	0.266583	+	0.963812i	$0.414105\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	22.7446i	1.37656i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.7446i	− 0.765747i	−0.923801	−	0.382873i	$-0.874935\pi$
$277$	− 12.7446i	− 0.765747i	0.923801	−	0.382873i	$-0.125065\pi$
$278$	0	0
$279$	28.2337	1.69031
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	5.25544i	0.312403i	0.987725	+	0.156202i	$0.0499250\pi$
$283$	5.25544i	0.312403i	−0.987725	+	0.156202i	$0.950075\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	38.7446i	2.28702i
$288$	0	0
$289$	15.1168	0.889226
$290$	0	0
$291$	−42.9783	−2.51943
$292$	0	0
$293$	− 23.4891i	− 1.37225i	−0.727484	−	0.686125i	$-0.759310\pi$
$293$	− 23.4891i	− 1.37225i	0.727484	−	0.686125i	$-0.240690\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.1168i	1.05125i
$298$	0	0
$299$	5.48913	0.317444
$300$	0	0
$301$	−13.4891	−0.777500
$302$	0	0
$303$	− 20.2337i	− 1.16240i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.25544i	− 0.299944i	−0.988690	−	0.149972i	$-0.952082\pi$
$307$	− 5.25544i	− 0.299944i	0.988690	−	0.149972i	$-0.0479183\pi$
$308$	0	0
$309$	−32.0000	−1.82042
$310$	0	0
$311$	−19.3723	−1.09850	−0.549251	−	0.835658i	$-0.685087\pi$
$311$	−19.3723	−1.09850	−0.549251	+	0.835658i	$0.685087\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.3505i	− 1.36766i	−0.729640	−	0.683831i	$-0.760313\pi$
$317$	− 24.3505i	− 1.36766i	0.729640	−	0.683831i	$-0.239687\pi$
$318$	0	0
$319$	−1.37228	−0.0768330
$320$	0	0
$321$	40.4674	2.25867
$322$	0	0
$323$	0.861407i	0.0479299i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 52.2337i	− 2.88853i
$328$	0	0
$329$	−9.25544	−0.510269
$330$	0	0
$331$	−30.9783	−1.70272	−0.851359	−	0.524583i	$-0.824221\pi$
$331$	−30.9783	−1.70272	−0.851359	+	0.524583i	$0.824221\pi$
$332$	0	0
$333$	− 78.4674i	− 4.29999i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.1168i	1.31373i	0.754009	+	0.656864i	$0.228117\pi$
$337$	24.1168i	1.31373i	−0.754009	+	0.656864i	$0.771883\pi$
$338$	0	0
$339$	−10.9783	−0.596257
$340$	0	0
$341$	−3.37228	−0.182619
$342$	0	0
$343$	− 8.86141i	− 0.478471i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.2337i	1.73040i	0.501431	+	0.865198i	$0.332807\pi$
$347$	32.2337i	1.73040i	−0.501431	+	0.865198i	$0.667193\pi$
$348$	0	0
$349$	−19.4891	−1.04323	−0.521614	−	0.853181i	$-0.674670\pi$
$349$	−19.4891	−1.04323	−0.521614	+	0.853181i	$0.674670\pi$
$350$	0	0
$351$	36.2337	1.93401
$352$	0	0
$353$	0.510875i	0.0271911i	0.999908	+	0.0135956i	$0.00432773\pi$
$353$	0.510875i	0.0271911i	−0.999908	+	0.0135956i	$0.995672\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 15.6060i	− 0.825955i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	0	0
$363$	− 3.37228i	− 0.176999i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	0	0
$369$	96.1902	5.00746
$370$	0	0
$371$	13.8832	0.720778
$372$	0	0
$373$	− 31.4891i	− 1.63045i	−0.579148	−	0.815223i	$-0.696615\pi$
$373$	− 31.4891i	− 1.63045i	0.579148	−	0.815223i	$-0.303385\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.74456i	0.141352i
$378$	0	0
$379$	−0.233688	−0.0120037	−0.00600187	−	0.999982i	$-0.501910\pi$
$379$	−0.233688	−0.0120037	−0.00600187	+	0.999982i	$0.501910\pi$
$380$	0	0
$381$	−26.9783	−1.38214
$382$	0	0
$383$	32.2337i	1.64706i	0.567269	+	0.823532i	$0.308000\pi$
$383$	32.2337i	1.64706i	−0.567269	+	0.823532i	$0.692000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	33.4891i	1.70235i
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−3.76631	−0.190471
$392$	0	0
$393$	74.5842i	3.76228i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.9783i	1.25362i	0.779171	+	0.626811i	$0.215640\pi$
$397$	24.9783i	1.25362i	−0.779171	+	0.626811i	$0.784360\pi$
$398$	0	0
$399$	−7.13859	−0.357377
$400$	0	0
$401$	13.3723	0.667780	0.333890	−	0.942612i	$-0.391639\pi$
$401$	13.3723	0.667780	0.333890	+	0.942612i	$0.391639\pi$
$402$	0	0
$403$	6.74456i	0.335971i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.37228i	0.464567i
$408$	0	0
$409$	1.76631	0.0873385	0.0436693	−	0.999046i	$-0.486095\pi$
$409$	1.76631	0.0873385	0.0436693	+	0.999046i	$0.486095\pi$
$410$	0	0
$411$	29.4891	1.45459
$412$	0	0
$413$	9.25544i	0.455430i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.4891i	0.660565i
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	10.2337	0.498759	0.249380	−	0.968406i	$-0.419773\pi$
$421$	10.2337	0.498759	0.249380	+	0.968406i	$0.419773\pi$
$422$	0	0
$423$	22.9783i	1.11724i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.1168i	0.876736i
$428$	0	0
$429$	−6.74456	−0.325631
$430$	0	0
$431$	34.9783	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$431$	34.9783	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$432$	0	0
$433$	− 27.7228i	− 1.33227i	−0.745830	−	0.666137i	$-0.767947\pi$
$433$	− 27.7228i	− 1.33227i	0.745830	−	0.666137i	$-0.232053\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.72281i	0.0824133i
$438$	0	0
$439$	18.9783	0.905782	0.452891	−	0.891566i	$-0.350393\pi$
$439$	18.9783	0.905782	0.452891	+	0.891566i	$0.350393\pi$
$440$	0	0
$441$	36.6060	1.74314
$442$	0	0
$443$	29.4891i	1.40107i	0.713618	+	0.700535i	$0.247055\pi$
$443$	29.4891i	1.40107i	−0.713618	+	0.700535i	$0.752945\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	72.8614i	3.44623i
$448$	0	0
$449$	−28.9783	−1.36757	−0.683784	−	0.729684i	$-0.739667\pi$
$449$	−28.9783	−1.36757	−0.683784	+	0.729684i	$0.739667\pi$
$450$	0	0
$451$	−11.4891	−0.541002
$452$	0	0
$453$	− 41.2554i	− 1.93835i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16.3505i	− 0.764846i	−0.923987	−	0.382423i	$-0.875090\pi$
$457$	− 16.3505i	− 0.764846i	0.923987	−	0.382423i	$-0.124910\pi$
$458$	0	0
$459$	−24.8614	−1.16043
$460$	0	0
$461$	16.1168	0.750636	0.375318	−	0.926896i	$-0.377533\pi$
$461$	16.1168	0.750636	0.375318	+	0.926896i	$0.377533\pi$
$462$	0	0
$463$	− 0.233688i	− 0.0108604i	−0.999985	−	0.00543020i	$-0.998272\pi$
$463$	− 0.233688i	− 0.0108604i	0.999985	−	0.00543020i	$-0.00172849\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.3723i	0.896442i	0.893923	+	0.448221i	$0.147942\pi$
$467$	19.3723i	0.896442i	−0.893923	+	0.448221i	$0.852058\pi$
$468$	0	0
$469$	−26.9783	−1.24574
$470$	0	0
$471$	31.6060	1.45633
$472$	0	0
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 34.4674i	− 1.57815i
$478$	0	0
$479$	5.48913	0.250805	0.125402	−	0.992106i	$-0.459978\pi$
$479$	5.48913	0.250805	0.125402	+	0.992106i	$0.459978\pi$
$480$	0	0
$481$	18.7446	0.854678
$482$	0	0
$483$	− 31.2119i	− 1.42019i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	0	0
$489$	−19.8397	−0.897180
$490$	0	0
$491$	7.37228	0.332706	0.166353	−	0.986066i	$-0.446801\pi$
$491$	7.37228	0.332706	0.166353	+	0.986066i	$0.446801\pi$
$492$	0	0
$493$	− 1.88316i	− 0.0848131i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.1168i	1.53035i
$498$	0	0
$499$	−33.4891	−1.49918	−0.749590	−	0.661903i	$-0.769749\pi$
$499$	−33.4891	−1.49918	−0.749590	+	0.661903i	$0.769749\pi$
$500$	0	0
$501$	15.6060	0.697223
$502$	0	0
$503$	− 34.9783i	− 1.55960i	−0.626027	−	0.779802i	$-0.715320\pi$
$503$	− 34.9783i	− 1.55960i	0.626027	−	0.779802i	$-0.284680\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 30.3505i	− 1.34791i
$508$	0	0
$509$	−9.76631	−0.432884	−0.216442	−	0.976295i	$-0.569445\pi$
$509$	−9.76631	−0.432884	−0.216442	+	0.976295i	$0.569445\pi$
$510$	0	0
$511$	52.2337	2.31068
$512$	0	0
$513$	11.3723i	0.502098i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.74456i	− 0.120706i
$518$	0	0
$519$	20.2337	0.888160
$520$	0	0
$521$	12.5109	0.548111	0.274056	−	0.961714i	$-0.411635\pi$
$521$	12.5109	0.548111	0.274056	+	0.961714i	$0.411635\pi$
$522$	0	0
$523$	30.9783i	1.35458i	0.735714	+	0.677292i	$0.236847\pi$
$523$	30.9783i	1.35458i	−0.735714	+	0.677292i	$0.763153\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.62772i	− 0.201587i
$528$	0	0
$529$	15.4674	0.672495
$530$	0	0
$531$	22.9783	0.997171
$532$	0	0
$533$	22.9783i	0.995299i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	40.4674i	1.74630i
$538$	0	0
$539$	−4.37228	−0.188327
$540$	0	0
$541$	−20.1168	−0.864891	−0.432445	−	0.901660i	$-0.642349\pi$
$541$	−20.1168	−0.864891	−0.432445	+	0.901660i	$0.642349\pi$
$542$	0	0
$543$	33.7228i	1.44718i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	0	0
$549$	44.9783	1.91962
$550$	0	0
$551$	−0.861407	−0.0366972
$552$	0	0
$553$	4.23369i	0.180035i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.97825i	0.210935i	0.994423	+	0.105468i	$0.0336339\pi$
$557$	4.97825i	0.210935i	−0.994423	+	0.105468i	$0.966366\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	4.62772	0.195382
$562$	0	0
$563$	− 8.23369i	− 0.347009i	−0.984833	−	0.173504i	$-0.944491\pi$
$563$	− 8.23369i	− 0.347009i	0.984833	−	0.173504i	$-0.0555090\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 121.329i	− 5.09533i
$568$	0	0
$569$	15.2554	0.639541	0.319771	−	0.947495i	$-0.396394\pi$
$569$	15.2554	0.639541	0.319771	+	0.947495i	$0.396394\pi$
$570$	0	0
$571$	−15.3723	−0.643310	−0.321655	−	0.946857i	$-0.604239\pi$
$571$	−15.3723	−0.643310	−0.321655	+	0.946857i	$0.604239\pi$
$572$	0	0
$573$	− 18.5109i	− 0.773303i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	36.9783i	1.53942i	0.638391	+	0.769712i	$0.279600\pi$
$577$	36.9783i	1.53942i	−0.638391	+	0.769712i	$0.720400\pi$
$578$	0	0
$579$	−50.1168	−2.08278
$580$	0	0
$581$	−9.25544	−0.383980
$582$	0	0
$583$	4.11684i	0.170502i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.8614i	− 1.02614i	−0.858347	−	0.513070i	$-0.828508\pi$
$587$	− 24.8614i	− 1.02614i	0.858347	−	0.513070i	$-0.171492\pi$
$588$	0	0
$589$	−2.11684	−0.0872230
$590$	0	0
$591$	−69.9565	−2.87763
$592$	0	0
$593$	12.5109i	0.513760i	0.966443	+	0.256880i	$0.0826945\pi$
$593$	12.5109i	0.513760i	−0.966443	+	0.256880i	$0.917305\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 61.0951i	− 2.50046i
$598$	0	0
$599$	−39.6060	−1.61826	−0.809128	−	0.587632i	$-0.800060\pi$
$599$	−39.6060	−1.61826	−0.809128	+	0.587632i	$0.800060\pi$
$600$	0	0
$601$	−16.5109	−0.673493	−0.336746	−	0.941595i	$-0.609326\pi$
$601$	−16.5109	−0.673493	−0.336746	+	0.941595i	$0.609326\pi$
$602$	0	0
$603$	66.9783i	2.72757i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.88316i	0.238790i	0.992847	+	0.119395i	$0.0380955\pi$
$607$	5.88316i	0.238790i	−0.992847	+	0.119395i	$0.961905\pi$
$608$	0	0
$609$	15.6060	0.632386
$610$	0	0
$611$	−5.48913	−0.222066
$612$	0	0
$613$	− 20.5109i	− 0.828426i	−0.910180	−	0.414213i	$-0.864057\pi$
$613$	− 20.5109i	− 0.828426i	0.910180	−	0.414213i	$-0.135943\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.23369i	− 0.0899249i	−0.998989	−	0.0449624i	$-0.985683\pi$
$617$	− 2.23369i	− 0.0899249i	0.998989	−	0.0449624i	$-0.0143168\pi$
$618$	0	0
$619$	−44.4674	−1.78729	−0.893647	−	0.448770i	$-0.851862\pi$
$619$	−44.4674	−1.78729	−0.893647	+	0.448770i	$0.851862\pi$
$620$	0	0
$621$	−49.7228	−1.99531
$622$	0	0
$623$	− 4.62772i	− 0.185406i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.11684i	− 0.0845386i
$628$	0	0
$629$	−12.8614	−0.512818
$630$	0	0
$631$	−42.1168	−1.67665	−0.838323	−	0.545175i	$-0.816463\pi$
$631$	−42.1168	−1.67665	−0.838323	+	0.545175i	$0.816463\pi$
$632$	0	0
$633$	20.6277i	0.819878i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.74456i	0.346472i
$638$	0	0
$639$	84.7011	3.35072
$640$	0	0
$641$	−27.0951	−1.07019	−0.535096	−	0.844791i	$-0.679725\pi$
$641$	−27.0951	−1.07019	−0.535096	+	0.844791i	$0.679725\pi$
$642$	0	0
$643$	− 5.88316i	− 0.232009i	−0.993249	−	0.116005i	$-0.962991\pi$
$643$	− 5.88316i	− 0.232009i	0.993249	−	0.116005i	$-0.0370087\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.7228i	1.48304i	0.670933	+	0.741518i	$0.265894\pi$
$647$	37.7228i	1.48304i	−0.670933	+	0.741518i	$0.734106\pi$
$648$	0	0
$649$	−2.74456	−0.107734
$650$	0	0
$651$	38.3505	1.50308
$652$	0	0
$653$	− 10.6277i	− 0.415895i	−0.978140	−	0.207947i	$-0.933322\pi$
$653$	− 10.6277i	− 0.415895i	0.978140	−	0.207947i	$-0.0666783\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 129.679i	− 5.05927i
$658$	0	0
$659$	12.8614	0.501009	0.250505	−	0.968115i	$-0.419403\pi$
$659$	12.8614	0.501009	0.250505	+	0.968115i	$0.419403\pi$
$660$	0	0
$661$	8.51087	0.331035	0.165517	−	0.986207i	$-0.447071\pi$
$661$	8.51087	0.331035	0.165517	+	0.986207i	$0.447071\pi$
$662$	0	0
$663$	− 9.25544i	− 0.359451i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.76631i	− 0.145832i
$668$	0	0
$669$	−63.2119	−2.44391
$670$	0	0
$671$	−5.37228	−0.207395
$672$	0	0
$673$	− 14.8614i	− 0.572865i	−0.958100	−	0.286433i	$-0.907531\pi$
$673$	− 14.8614i	− 0.572865i	0.958100	−	0.286433i	$-0.0924694\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.25544i	0.125117i	0.998041	+	0.0625583i	$0.0199259\pi$
$677$	3.25544i	0.125117i	−0.998041	+	0.0625583i	$0.980074\pi$
$678$	0	0
$679$	−42.9783	−1.64935
$680$	0	0
$681$	9.25544	0.354669
$682$	0	0
$683$	− 28.6277i	− 1.09541i	−0.836672	−	0.547705i	$-0.815502\pi$
$683$	− 28.6277i	− 1.09541i	0.836672	−	0.547705i	$-0.184498\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 33.7228i	− 1.28661i
$688$	0	0
$689$	8.23369	0.313679
$690$	0	0
$691$	−40.2337	−1.53056	−0.765281	−	0.643697i	$-0.777400\pi$
$691$	−40.2337	−1.53056	−0.765281	+	0.643697i	$0.777400\pi$
$692$	0	0
$693$	28.2337i	1.07251i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 15.7663i	− 0.597192i
$698$	0	0
$699$	−4.62772	−0.175036
$700$	0	0
$701$	−37.3723	−1.41153	−0.705766	−	0.708445i	$-0.749397\pi$
$701$	−37.3723	−1.41153	−0.705766	+	0.708445i	$0.749397\pi$
$702$	0	0
$703$	5.88316i	0.221887i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 20.2337i	− 0.760966i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	10.5109	0.394189
$712$	0	0
$713$	− 9.25544i	− 0.346619i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.7228i	1.85693i
$718$	0	0
$719$	13.8832	0.517754	0.258877	−	0.965910i	$-0.416648\pi$
$719$	13.8832	0.517754	0.258877	+	0.965910i	$0.416648\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	74.1902i	2.75916i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.2337i	0.898778i	0.893336	+	0.449389i	$0.148358\pi$
$727$	24.2337i	0.898778i	−0.893336	+	0.449389i	$0.851642\pi$
$728$	0	0
$729$	−117.935	−4.36795
$730$	0	0
$731$	5.48913	0.203023
$732$	0	0
$733$	− 46.2337i	− 1.70768i	−0.520535	−	0.853840i	$-0.674268\pi$
$733$	− 46.2337i	− 1.70768i	0.520535	−	0.853840i	$-0.325732\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	−20.4674	−0.752905	−0.376452	−	0.926436i	$-0.622856\pi$
$739$	−20.4674	−0.752905	−0.376452	+	0.926436i	$0.622856\pi$
$740$	0	0
$741$	−4.23369	−0.155528
$742$	0	0
$743$	− 4.62772i	− 0.169775i	−0.996391	−	0.0848873i	$-0.972947\pi$
$743$	− 4.62772i	− 0.169775i	0.996391	−	0.0848873i	$-0.0270530\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22.9783i	0.840730i
$748$	0	0
$749$	40.4674	1.47865
$750$	0	0
$751$	−8.86141	−0.323357	−0.161679	−	0.986843i	$-0.551691\pi$
$751$	−8.86141	−0.323357	−0.161679	+	0.986843i	$0.551691\pi$
$752$	0	0
$753$	9.25544i	0.337287i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 20.9783i	− 0.762467i	−0.924479	−	0.381234i	$-0.875499\pi$
$757$	− 20.9783i	− 0.762467i	0.924479	−	0.381234i	$-0.124501\pi$
$758$	0	0
$759$	9.25544	0.335951
$760$	0	0
$761$	4.97825	0.180461	0.0902307	−	0.995921i	$-0.471240\pi$
$761$	4.97825	0.180461	0.0902307	+	0.995921i	$0.471240\pi$
$762$	0	0
$763$	− 52.2337i	− 1.89099i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.48913i	0.198201i
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	60.7011	2.18610
$772$	0	0
$773$	33.6060i	1.20872i	0.796710	+	0.604361i	$0.206572\pi$
$773$	33.6060i	1.20872i	−0.796710	+	0.604361i	$0.793428\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 106.584i	− 3.82369i
$778$	0	0
$779$	−7.21194	−0.258395
$780$	0	0
$781$	−10.1168	−0.362009
$782$	0	0
$783$	− 24.8614i	− 0.888474i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.4674i	1.58509i	0.609813	+	0.792545i	$0.291245\pi$
$787$	44.4674i	1.58509i	−0.609813	+	0.792545i	$0.708755\pi$
$788$	0	0
$789$	−83.8397	−2.98477
$790$	0	0
$791$	−10.9783	−0.390342
$792$	0	0
$793$	10.7446i	0.381551i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.4891i	0.406966i	0.979079	+	0.203483i	$0.0652261\pi$
$797$	11.4891i	0.406966i	−0.979079	+	0.203483i	$0.934774\pi$
$798$	0	0
$799$	3.76631	0.133243
$800$	0	0
$801$	−11.4891	−0.405948
$802$	0	0
$803$	15.4891i	0.546599i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 29.4891i	− 1.03807i
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−44.8614	−1.57530	−0.787649	−	0.616125i	$-0.788702\pi$
$811$	−44.8614	−1.57530	−0.787649	+	0.616125i	$0.788702\pi$
$812$	0	0
$813$	− 53.9565i	− 1.89234i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.51087i	− 0.0878444i
$818$	0	0
$819$	56.4674	1.97313
$820$	0	0
$821$	11.4891	0.400973	0.200487	−	0.979696i	$-0.435748\pi$
$821$	11.4891	0.400973	0.200487	+	0.979696i	$0.435748\pi$
$822$	0	0
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46.9783i	− 1.63359i	−0.576925	−	0.816797i	$-0.695748\pi$
$827$	− 46.9783i	− 1.63359i	0.576925	−	0.816797i	$-0.304252\pi$
$828$	0	0
$829$	24.7446	0.859414	0.429707	−	0.902968i	$-0.358617\pi$
$829$	24.7446	0.859414	0.429707	+	0.902968i	$0.358617\pi$
$830$	0	0
$831$	−42.9783	−1.49090
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 61.0951i	− 2.11176i
$838$	0	0
$839$	−10.9783	−0.379011	−0.189506	−	0.981880i	$-0.560689\pi$
$839$	−10.9783	−0.379011	−0.189506	+	0.981880i	$0.560689\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	− 60.7011i	− 2.09066i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.37228i	− 0.115873i
$848$	0	0
$849$	17.7228	0.608245
$850$	0	0
$851$	−25.7228	−0.881767
$852$	0	0
$853$	38.4674i	1.31710i	0.752538	+	0.658549i	$0.228829\pi$
$853$	38.4674i	1.31710i	−0.752538	+	0.658549i	$0.771171\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.3505i	1.24171i	0.783925	+	0.620855i	$0.213215\pi$
$857$	36.3505i	1.24171i	−0.783925	+	0.620855i	$0.786785\pi$
$858$	0	0
$859$	−42.7446	−1.45843	−0.729213	−	0.684287i	$-0.760114\pi$
$859$	−42.7446	−1.45843	−0.729213	+	0.684287i	$0.760114\pi$
$860$	0	0
$861$	130.658	4.45280
$862$	0	0
$863$	21.2554i	0.723544i	0.932267	+	0.361772i	$0.117828\pi$
$863$	21.2554i	0.723544i	−0.932267	+	0.361772i	$0.882172\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 50.9783i	− 1.73131i
$868$	0	0
$869$	−1.25544	−0.0425878
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	106.701i	3.61128i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.9783i	1.24867i	0.781158	+	0.624333i	$0.214629\pi$
$877$	36.9783i	1.24867i	−0.781158	+	0.624333i	$0.785371\pi$
$878$	0	0
$879$	−79.2119	−2.67175
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	3.37228i	0.113486i	0.998389	+	0.0567432i	$0.0180716\pi$
$883$	3.37228i	0.113486i	−0.998389	+	0.0567432i	$0.981928\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.9783i	0.368614i	0.982869	+	0.184307i	$0.0590040\pi$
$887$	10.9783i	0.368614i	−0.982869	+	0.184307i	$0.940996\pi$
$888$	0	0
$889$	−26.9783	−0.904821
$890$	0	0
$891$	35.9783	1.20532
$892$	0	0
$893$	− 1.72281i	− 0.0576517i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 18.5109i	− 0.618060i
$898$	0	0
$899$	4.62772	0.154343
$900$	0	0
$901$	−5.64947	−0.188211
$902$	0	0
$903$	45.4891i	1.51378i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0.394031i	0.0130836i	0.999979	+	0.00654179i	$0.00208233\pi$
$907$	0.394031i	0.0130836i	−0.999979	+	0.00654179i	$0.997918\pi$
$908$	0	0
$909$	−50.2337	−1.66615
$910$	0	0
$911$	8.39403	0.278107	0.139053	−	0.990285i	$-0.455594\pi$
$911$	8.39403	0.278107	0.139053	+	0.990285i	$0.455594\pi$
$912$	0	0
$913$	− 2.74456i	− 0.0908318i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	74.5842i	2.46299i
$918$	0	0
$919$	18.9783	0.626035	0.313017	−	0.949747i	$-0.398660\pi$
$919$	18.9783	0.626035	0.313017	+	0.949747i	$0.398660\pi$
$920$	0	0
$921$	−17.7228	−0.583987
$922$	0	0
$923$	20.2337i	0.666000i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	79.4456i	2.60934i
$928$	0	0
$929$	−24.3505	−0.798915	−0.399458	−	0.916752i	$-0.630801\pi$
$929$	−24.3505	−0.798915	−0.399458	+	0.916752i	$0.630801\pi$
$930$	0	0
$931$	−2.74456	−0.0899494
$932$	0	0
$933$	65.3288i	2.13877i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28.5109i	− 0.931410i	−0.884940	−	0.465705i	$-0.845801\pi$
$937$	− 28.5109i	− 0.931410i	0.884940	−	0.465705i	$-0.154199\pi$
$938$	0	0
$939$	74.1902	2.42111
$940$	0	0
$941$	−15.0951	−0.492086	−0.246043	−	0.969259i	$-0.579130\pi$
$941$	−15.0951	−0.492086	−0.246043	+	0.969259i	$0.579130\pi$
$942$	0	0
$943$	− 31.5326i	− 1.02684i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 48.8614i	− 1.58778i	−0.608060	−	0.793891i	$-0.708052\pi$
$947$	− 48.8614i	− 1.58778i	0.608060	−	0.793891i	$-0.291948\pi$
$948$	0	0
$949$	30.9783	1.00560
$950$	0	0
$951$	−82.1168	−2.66282
$952$	0	0
$953$	− 40.1168i	− 1.29951i	−0.760143	−	0.649756i	$-0.774871\pi$
$953$	− 40.1168i	− 1.29951i	0.760143	−	0.649756i	$-0.225129\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.62772i	0.149593i
$958$	0	0
$959$	29.4891	0.952254
$960$	0	0
$961$	−19.6277	−0.633152
$962$	0	0
$963$	− 100.467i	− 3.23752i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 47.6060i	− 1.53090i	−0.643493	−	0.765452i	$-0.722515\pi$
$967$	− 47.6060i	− 1.53090i	0.643493	−	0.765452i	$-0.277485\pi$
$968$	0	0
$969$	2.90491	0.0933190
$970$	0	0
$971$	1.02175	0.0327895	0.0163947	−	0.999866i	$-0.494781\pi$
$971$	1.02175	0.0327895	0.0163947	+	0.999866i	$0.494781\pi$
$972$	0	0
$973$	13.4891i	0.432442i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.2337i	0.455376i	0.973734	+	0.227688i	$0.0731166\pi$
$977$	14.2337i	0.455376i	−0.973734	+	0.227688i	$0.926883\pi$
$978$	0	0
$979$	1.37228	0.0438583
$980$	0	0
$981$	−129.679	−4.14034
$982$	0	0
$983$	− 13.7228i	− 0.437690i	−0.975760	−	0.218845i	$-0.929771\pi$
$983$	− 13.7228i	− 0.437690i	0.975760	−	0.218845i	$-0.0702289\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	31.2119i	0.993487i
$988$	0	0
$989$	10.9783	0.349088
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	104.467i	3.31517i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	22.2337i	0.704148i	0.935972	+	0.352074i	$0.114523\pi$
$997$	22.2337i	0.704148i	−0.935972	+	0.352074i	$0.885477\pi$
$998$	0	0
$999$	−169.796	−5.37211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.p.4049.1		4
4.3	odd	2		550.2.b.f.199.2		4
5.2	odd	4		4400.2.a.bl.1.1		2
5.3	odd	4		880.2.a.n.1.2		2
5.4	even	2	inner	4400.2.b.p.4049.4		4
12.11	even	2		4950.2.c.bc.199.4		4
15.8	even	4		7920.2.a.bq.1.1		2
20.3	even	4		110.2.a.d.1.1	✓	2
20.7	even	4		550.2.a.n.1.2		2
20.19	odd	2		550.2.b.f.199.3		4
40.3	even	4		3520.2.a.bq.1.2		2
40.13	odd	4		3520.2.a.bj.1.1		2
55.43	even	4		9680.2.a.bt.1.2		2
60.23	odd	4		990.2.a.m.1.2		2
60.47	odd	4		4950.2.a.bw.1.1		2
60.59	even	2		4950.2.c.bc.199.1		4
140.83	odd	4		5390.2.a.bp.1.2		2
220.43	odd	4		1210.2.a.r.1.1		2
220.87	odd	4		6050.2.a.cb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.1	✓	2	20.3	even	4
550.2.a.n.1.2		2	20.7	even	4
550.2.b.f.199.2		4	4.3	odd	2
550.2.b.f.199.3		4	20.19	odd	2
880.2.a.n.1.2		2	5.3	odd	4
990.2.a.m.1.2		2	60.23	odd	4
1210.2.a.r.1.1		2	220.43	odd	4
3520.2.a.bj.1.1		2	40.13	odd	4
3520.2.a.bq.1.2		2	40.3	even	4
4400.2.a.bl.1.1		2	5.2	odd	4
4400.2.b.p.4049.1		4	1.1	even	1	trivial
4400.2.b.p.4049.4		4	5.4	even	2	inner
4950.2.a.bw.1.1		2	60.47	odd	4
4950.2.c.bc.199.1		4	60.59	even	2
4950.2.c.bc.199.4		4	12.11	even	2
5390.2.a.bp.1.2		2	140.83	odd	4
6050.2.a.cb.1.2		2	220.87	odd	4
7920.2.a.bq.1.1		2	15.8	even	4
9680.2.a.bt.1.2		2	55.43	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.37228i q^{3} -3.37228i q^{7} -8.37228 q^{9} +1.00000 q^{11} -2.00000i q^{13} +1.37228i q^{17} +0.627719 q^{19} -11.3723 q^{21} +2.74456i q^{23} +18.1168i q^{27} -1.37228 q^{29} -3.37228 q^{31} -3.37228i q^{33} +9.37228i q^{37} -6.74456 q^{39} -11.4891 q^{41} -4.00000i q^{43} -2.74456i q^{47} -4.37228 q^{49} +4.62772 q^{51} +4.11684i q^{53} -2.11684i q^{57} -2.74456 q^{59} -5.37228 q^{61} +28.2337i q^{63} -8.00000i q^{67} +9.25544 q^{69} -10.1168 q^{71} +15.4891i q^{73} -3.37228i q^{77} -1.25544 q^{79} +35.9783 q^{81} -2.74456i q^{83} +4.62772i q^{87} +1.37228 q^{89} -6.74456 q^{91} +11.3723i q^{93} -12.7446i q^{97} -8.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{9} + 4 q^{11} + 14 q^{19} - 34 q^{21} + 6 q^{29} - 2 q^{31} - 4 q^{39} - 6 q^{49} + 30 q^{51} + 12 q^{59} - 10 q^{61} + 60 q^{69} - 6 q^{71} - 28 q^{79} + 52 q^{81} - 6 q^{89} - 4 q^{91} - 22 q^{99}+O(q^{100}) \) 4 * q - 22 * q^9 + 4 * q^11 + 14 * q^19 - 34 * q^21 + 6 * q^29 - 2 * q^31 - 4 * q^39 - 6 * q^49 + 30 * q^51 + 12 * q^59 - 10 * q^61 + 60 * q^69 - 6 * q^71 - 28 * q^79 + 52 * q^81 - 6 * q^89 - 4 * q^91 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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