LMFDB - Embedded newform 4400.2.b.s.4049.3

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:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$1.79129i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.s.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79129i q^{3} +4.79129i q^{7} -0.208712 q^{9} +1.00000 q^{11} +1.00000i q^{13} -3.79129i q^{17} -2.58258 q^{19} -8.58258 q^{21} +0.791288i q^{23} +5.00000i q^{27} -2.20871 q^{29} -0.582576 q^{31} +1.79129i q^{33} +6.58258i q^{37} -1.79129 q^{39} -10.5826 q^{41} -10.0000i q^{43} +10.5826i q^{47} -15.9564 q^{49} +6.79129 q^{51} +2.37386i q^{53} -4.62614i q^{57} -1.41742 q^{59} +8.79129 q^{61} -1.00000i q^{63} +4.00000i q^{67} -1.41742 q^{69} -16.7477 q^{71} +3.20871i q^{73} +4.79129i q^{77} +16.5390 q^{79} -9.58258 q^{81} +12.9564i q^{83} -3.95644i q^{87} -3.79129 q^{89} -4.79129 q^{91} -1.04356i q^{93} -10.7913i q^{97} -0.208712 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{9} + 4 q^{11} + 8 q^{19} - 16 q^{21} - 18 q^{29} + 16 q^{31} + 2 q^{39} - 24 q^{41} - 18 q^{49} + 18 q^{51} - 24 q^{59} + 26 q^{61} - 24 q^{69} - 12 q^{71} + 2 q^{79} - 20 q^{81} - 6 q^{89}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q - 10 * q^9 + 4 * q^11 + 8 * q^19 - 16 * q^21 - 18 * q^29 + 16 * q^31 + 2 * q^39 - 24 * q^41 - 18 * q^49 + 18 * q^51 - 24 * q^59 + 26 * q^61 - 24 * q^69 - 12 * q^71 + 2 * q^79 - 20 * q^81 - 6 * q^89 - 10 * q^91 - 10 * 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
$3$	1.79129i	1.03420i	0.855925	+	0.517100i	$0.172989\pi$
$3$	1.79129i	1.03420i	−0.855925	+	0.517100i	$0.827011\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.79129i	1.81094i	0.424414	+	0.905468i	$0.360480\pi$
$7$	4.79129i	1.81094i	−0.424414	+	0.905468i	$0.639520\pi$
$8$	0	0
$9$	−0.208712	−0.0695707
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.79129i	− 0.919522i	−0.888043	−	0.459761i	$-0.847935\pi$
$17$	− 3.79129i	− 0.919522i	0.888043	−	0.459761i	$-0.152065\pi$
$18$	0	0
$19$	−2.58258	−0.592483	−0.296242	−	0.955113i	$-0.595733\pi$
$19$	−2.58258	−0.592483	−0.296242	+	0.955113i	$0.595733\pi$
$20$	0	0
$21$	−8.58258	−1.87287
$22$	0	0
$23$	0.791288i	0.164995i	0.996591	+	0.0824975i	$0.0262896\pi$
$23$	0.791288i	0.164995i	−0.996591	+	0.0824975i	$0.973710\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	−2.20871	−0.410148	−0.205074	−	0.978747i	$-0.565743\pi$
$29$	−2.20871	−0.410148	−0.205074	+	0.978747i	$0.565743\pi$
$30$	0	0
$31$	−0.582576	−0.104634	−0.0523168	−	0.998631i	$-0.516661\pi$
$31$	−0.582576	−0.104634	−0.0523168	+	0.998631i	$0.516661\pi$
$32$	0	0
$33$	1.79129i	0.311823i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.58258i	1.08217i	0.840968	+	0.541084i	$0.181986\pi$
$37$	6.58258i	1.08217i	−0.840968	+	0.541084i	$0.818014\pi$
$38$	0	0
$39$	−1.79129	−0.286836
$40$	0	0
$41$	−10.5826	−1.65272	−0.826360	−	0.563142i	$-0.809593\pi$
$41$	−10.5826	−1.65272	−0.826360	+	0.563142i	$0.809593\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.5826i	1.54363i	0.635849	+	0.771814i	$0.280650\pi$
$47$	10.5826i	1.54363i	−0.635849	+	0.771814i	$0.719350\pi$
$48$	0	0
$49$	−15.9564	−2.27949
$50$	0	0
$51$	6.79129	0.950971
$52$	0	0
$53$	2.37386i	0.326075i	0.986620	+	0.163038i	$0.0521292\pi$
$53$	2.37386i	0.326075i	−0.986620	+	0.163038i	$0.947871\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.62614i	− 0.612747i
$58$	0	0
$59$	−1.41742	−0.184533	−0.0922665	−	0.995734i	$-0.529411\pi$
$59$	−1.41742	−0.184533	−0.0922665	+	0.995734i	$0.529411\pi$
$60$	0	0
$61$	8.79129	1.12561	0.562805	−	0.826590i	$-0.309722\pi$
$61$	8.79129	1.12561	0.562805	+	0.826590i	$0.309722\pi$
$62$	0	0
$63$	− 1.00000i	− 0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	−1.41742	−0.170638
$70$	0	0
$71$	−16.7477	−1.98759	−0.993795	−	0.111229i	$-0.964521\pi$
$71$	−16.7477	−1.98759	−0.993795	+	0.111229i	$0.964521\pi$
$72$	0	0
$73$	3.20871i	0.375551i	0.982212	+	0.187776i	$0.0601278\pi$
$73$	3.20871i	0.375551i	−0.982212	+	0.187776i	$0.939872\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.79129i	0.546018i
$78$	0	0
$79$	16.5390	1.86078	0.930392	−	0.366565i	$-0.119466\pi$
$79$	16.5390	1.86078	0.930392	+	0.366565i	$0.119466\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	0	0
$83$	12.9564i	1.42215i	0.703114	+	0.711077i	$0.251792\pi$
$83$	12.9564i	1.42215i	−0.703114	+	0.711077i	$0.748208\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.95644i	− 0.424175i
$88$	0	0
$89$	−3.79129	−0.401876	−0.200938	−	0.979604i	$-0.564399\pi$
$89$	−3.79129	−0.401876	−0.200938	+	0.979604i	$0.564399\pi$
$90$	0	0
$91$	−4.79129	−0.502263
$92$	0	0
$93$	− 1.04356i	− 0.108212i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.7913i	− 1.09569i	−0.836580	−	0.547845i	$-0.815448\pi$
$97$	− 10.7913i	− 1.09569i	0.836580	−	0.547845i	$-0.184552\pi$
$98$	0	0
$99$	−0.208712	−0.0209764
$100$	0	0
$101$	−3.62614	−0.360814	−0.180407	−	0.983592i	$-0.557741\pi$
$101$	−3.62614	−0.360814	−0.180407	+	0.983592i	$0.557741\pi$
$102$	0	0
$103$	− 16.9564i	− 1.67077i	−0.549667	−	0.835384i	$-0.685245\pi$
$103$	− 16.9564i	− 1.67077i	0.549667	−	0.835384i	$-0.314755\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.74773i	0.749001i	0.927227	+	0.374501i	$0.122186\pi$
$107$	7.74773i	0.749001i	−0.927227	+	0.374501i	$0.877814\pi$
$108$	0	0
$109$	0.208712	0.0199910	0.00999550	−	0.999950i	$-0.496818\pi$
$109$	0.208712	0.0199910	0.00999550	+	0.999950i	$0.496818\pi$
$110$	0	0
$111$	−11.7913	−1.11918
$112$	0	0
$113$	− 10.5826i	− 0.995525i	−0.867313	−	0.497762i	$-0.834155\pi$
$113$	− 10.5826i	− 0.995525i	0.867313	−	0.497762i	$-0.165845\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.208712i	− 0.0192954i
$118$	0	0
$119$	18.1652	1.66520
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 18.9564i	− 1.70924i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.62614i	− 0.233032i	−0.993189	−	0.116516i	$-0.962827\pi$
$127$	− 2.62614i	− 0.233032i	0.993189	−	0.116516i	$-0.0371726\pi$
$128$	0	0
$129$	17.9129	1.57714
$130$	0	0
$131$	14.3739	1.25585	0.627925	−	0.778274i	$-0.283904\pi$
$131$	14.3739	1.25585	0.627925	+	0.778274i	$0.283904\pi$
$132$	0	0
$133$	− 12.3739i	− 1.07295i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.5390i	1.75477i	0.479790	+	0.877383i	$0.340713\pi$
$137$	20.5390i	1.75477i	−0.479790	+	0.877383i	$0.659287\pi$
$138$	0	0
$139$	−17.7477	−1.50534	−0.752671	−	0.658396i	$-0.771235\pi$
$139$	−17.7477	−1.50534	−0.752671	+	0.658396i	$0.771235\pi$
$140$	0	0
$141$	−18.9564	−1.59642
$142$	0	0
$143$	1.00000i	0.0836242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 28.5826i	− 2.35745i
$148$	0	0
$149$	15.1652	1.24238	0.621189	−	0.783661i	$-0.286650\pi$
$149$	15.1652	1.24238	0.621189	+	0.783661i	$0.286650\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	0.791288i	0.0639718i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1.16515i	− 0.0929892i	−0.998919	−	0.0464946i	$-0.985195\pi$
$157$	− 1.16515i	− 0.0929892i	0.998919	−	0.0464946i	$-0.0148050\pi$
$158$	0	0
$159$	−4.25227	−0.337227
$160$	0	0
$161$	−3.79129	−0.298795
$162$	0	0
$163$	− 7.62614i	− 0.597325i	−0.954359	−	0.298663i	$-0.903459\pi$
$163$	− 7.62614i	− 0.597325i	0.954359	−	0.298663i	$-0.0965405\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.41742i	0.573978i	0.957934	+	0.286989i	$0.0926542\pi$
$167$	7.41742i	0.573978i	−0.957934	+	0.286989i	$0.907346\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0.539015	0.0412195
$172$	0	0
$173$	− 13.7477i	− 1.04522i	−0.852572	−	0.522610i	$-0.824958\pi$
$173$	− 13.7477i	− 1.04522i	0.852572	−	0.522610i	$-0.175042\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2.53901i	− 0.190844i
$178$	0	0
$179$	−19.1216	−1.42921	−0.714607	−	0.699526i	$-0.753395\pi$
$179$	−19.1216	−1.42921	−0.714607	+	0.699526i	$0.753395\pi$
$180$	0	0
$181$	10.3739	0.771083	0.385542	−	0.922690i	$-0.374015\pi$
$181$	10.3739	0.771083	0.385542	+	0.922690i	$0.374015\pi$
$182$	0	0
$183$	15.7477i	1.16411i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.79129i	− 0.277246i
$188$	0	0
$189$	−23.9564	−1.74257
$190$	0	0
$191$	−21.7913	−1.57676	−0.788381	−	0.615187i	$-0.789080\pi$
$191$	−21.7913	−1.57676	−0.788381	+	0.615187i	$0.789080\pi$
$192$	0	0
$193$	− 11.1652i	− 0.803685i	−0.915709	−	0.401843i	$-0.868370\pi$
$193$	− 11.1652i	− 0.803685i	0.915709	−	0.401843i	$-0.131630\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.53901i	− 0.394638i	−0.980339	−	0.197319i	$-0.936776\pi$
$197$	− 5.53901i	− 0.394638i	0.980339	−	0.197319i	$-0.0632235\pi$
$198$	0	0
$199$	−15.3739	−1.08982	−0.544912	−	0.838493i	$-0.683437\pi$
$199$	−15.3739	−1.08982	−0.544912	+	0.838493i	$0.683437\pi$
$200$	0	0
$201$	−7.16515	−0.505391
$202$	0	0
$203$	− 10.5826i	− 0.742751i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.165151i	− 0.0114788i
$208$	0	0
$209$	−2.58258	−0.178640
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	− 30.0000i	− 2.05557i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.79129i	− 0.189485i
$218$	0	0
$219$	−5.74773	−0.388395
$220$	0	0
$221$	3.79129	0.255030
$222$	0	0
$223$	− 11.4174i	− 0.764567i	−0.924045	−	0.382284i	$-0.875138\pi$
$223$	− 11.4174i	− 0.764567i	0.924045	−	0.382284i	$-0.124862\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.5390i	1.36322i	0.731715	+	0.681611i	$0.238720\pi$
$227$	20.5390i	1.36322i	−0.731715	+	0.681611i	$0.761280\pi$
$228$	0	0
$229$	21.3739	1.41242	0.706212	−	0.708000i	$-0.250402\pi$
$229$	21.3739	1.41242	0.706212	+	0.708000i	$0.250402\pi$
$230$	0	0
$231$	−8.58258	−0.564692
$232$	0	0
$233$	− 11.2087i	− 0.734307i	−0.930160	−	0.367154i	$-0.880332\pi$
$233$	− 11.2087i	− 0.734307i	0.930160	−	0.367154i	$-0.119668\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	29.6261i	1.92442i
$238$	0	0
$239$	−6.79129	−0.439292	−0.219646	−	0.975580i	$-0.570490\pi$
$239$	−6.79129	−0.439292	−0.219646	+	0.975580i	$0.570490\pi$
$240$	0	0
$241$	18.1216	1.16731	0.583657	−	0.812000i	$-0.301621\pi$
$241$	18.1216	1.16731	0.583657	+	0.812000i	$0.301621\pi$
$242$	0	0
$243$	− 2.16515i	− 0.138895i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.58258i	− 0.164325i
$248$	0	0
$249$	−23.2087	−1.47079
$250$	0	0
$251$	−20.5390	−1.29641	−0.648206	−	0.761465i	$-0.724480\pi$
$251$	−20.5390	−1.29641	−0.648206	+	0.761465i	$0.724480\pi$
$252$	0	0
$253$	0.791288i	0.0497478i
$254$	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
$259$	−31.5390	−1.95974
$260$	0	0
$261$	0.460985	0.0285343
$262$	0	0
$263$	5.83485i	0.359792i	0.983686	+	0.179896i	$0.0575762\pi$
$263$	5.83485i	0.359792i	−0.983686	+	0.179896i	$0.942424\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.79129i	− 0.415620i
$268$	0	0
$269$	−29.7042	−1.81109	−0.905547	−	0.424245i	$-0.860540\pi$
$269$	−29.7042	−1.81109	−0.905547	+	0.424245i	$0.860540\pi$
$270$	0	0
$271$	11.7477	0.713624	0.356812	−	0.934176i	$-0.383864\pi$
$271$	11.7477	0.713624	0.356812	+	0.934176i	$0.383864\pi$
$272$	0	0
$273$	− 8.58258i	− 0.519441i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.582576i	0.0350036i	0.999847	+	0.0175018i	$0.00557128\pi$
$277$	0.582576i	0.0350036i	−0.999847	+	0.0175018i	$0.994429\pi$
$278$	0	0
$279$	0.121591	0.00727944
$280$	0	0
$281$	13.7477	0.820121	0.410060	−	0.912058i	$-0.365508\pi$
$281$	13.7477	0.820121	0.410060	+	0.912058i	$0.365508\pi$
$282$	0	0
$283$	19.7042i	1.17129i	0.810567	+	0.585646i	$0.199159\pi$
$283$	19.7042i	1.17129i	−0.810567	+	0.585646i	$0.800841\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 50.7042i	− 2.99297i
$288$	0	0
$289$	2.62614	0.154479
$290$	0	0
$291$	19.3303	1.13316
$292$	0	0
$293$	21.1652i	1.23648i	0.785989	+	0.618241i	$0.212154\pi$
$293$	21.1652i	1.23648i	−0.785989	+	0.618241i	$0.787846\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000i	0.290129i
$298$	0	0
$299$	−0.791288	−0.0457614
$300$	0	0
$301$	47.9129	2.76165
$302$	0	0
$303$	− 6.49545i	− 0.373154i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.7913i	1.81442i	0.420673	+	0.907212i	$0.361794\pi$
$307$	31.7913i	1.81442i	−0.420673	+	0.907212i	$0.638206\pi$
$308$	0	0
$309$	30.3739	1.72791
$310$	0	0
$311$	24.1652	1.37028	0.685140	−	0.728411i	$-0.259741\pi$
$311$	24.1652	1.37028	0.685140	+	0.728411i	$0.259741\pi$
$312$	0	0
$313$	20.7477i	1.17273i	0.810047	+	0.586365i	$0.199442\pi$
$313$	20.7477i	1.17273i	−0.810047	+	0.586365i	$0.800558\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.37386i	0.301826i	0.988547	+	0.150913i	$0.0482214\pi$
$317$	5.37386i	0.301826i	−0.988547	+	0.150913i	$0.951779\pi$
$318$	0	0
$319$	−2.20871	−0.123664
$320$	0	0
$321$	−13.8784	−0.774617
$322$	0	0
$323$	9.79129i	0.544802i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.373864i	0.0206747i
$328$	0	0
$329$	−50.7042	−2.79541
$330$	0	0
$331$	19.3303	1.06249	0.531245	−	0.847218i	$-0.321724\pi$
$331$	19.3303	1.06249	0.531245	+	0.847218i	$0.321724\pi$
$332$	0	0
$333$	− 1.37386i	− 0.0752873i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	0	0
$339$	18.9564	1.02957
$340$	0	0
$341$	−0.582576	−0.0315482
$342$	0	0
$343$	− 42.9129i	− 2.31708i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.3739i	− 1.41582i	−0.706301	−	0.707912i	$-0.749638\pi$
$347$	− 26.3739i	− 1.41582i	0.706301	−	0.707912i	$-0.250362\pi$
$348$	0	0
$349$	5.41742	0.289988	0.144994	−	0.989433i	$-0.453684\pi$
$349$	5.41742	0.289988	0.144994	+	0.989433i	$0.453684\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	0.165151i	0.00879012i	0.999990	+	0.00439506i	$0.00139900\pi$
$353$	0.165151i	0.00879012i	−0.999990	+	0.00439506i	$0.998601\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	32.5390i	1.72215i
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−12.3303	−0.648963
$362$	0	0
$363$	1.79129i	0.0940182i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 22.3739i	− 1.16791i	−0.811787	−	0.583953i	$-0.801505\pi$
$367$	− 22.3739i	− 1.16791i	0.811787	−	0.583953i	$-0.198495\pi$
$368$	0	0
$369$	2.20871	0.114981
$370$	0	0
$371$	−11.3739	−0.590502
$372$	0	0
$373$	− 17.3303i	− 0.897329i	−0.893700	−	0.448665i	$-0.851900\pi$
$373$	− 17.3303i	− 0.897329i	0.893700	−	0.448665i	$-0.148100\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.20871i	− 0.113754i
$378$	0	0
$379$	−31.0000	−1.59236	−0.796182	−	0.605058i	$-0.793150\pi$
$379$	−31.0000	−1.59236	−0.796182	+	0.605058i	$0.793150\pi$
$380$	0	0
$381$	4.70417	0.241002
$382$	0	0
$383$	− 18.0000i	− 0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$	− 18.0000i	− 0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.08712i	0.106094i
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	25.7477i	1.29880i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.7913i	0.591788i	0.955221	+	0.295894i	$0.0956175\pi$
$397$	11.7913i	0.591788i	−0.955221	+	0.295894i	$0.904383\pi$
$398$	0	0
$399$	22.1652	1.10965
$400$	0	0
$401$	−33.1652	−1.65619	−0.828094	−	0.560589i	$-0.810575\pi$
$401$	−33.1652	−1.65619	−0.828094	+	0.560589i	$0.810575\pi$
$402$	0	0
$403$	− 0.582576i	− 0.0290202i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.58258i	0.326286i
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	−36.7913	−1.81478
$412$	0	0
$413$	− 6.79129i	− 0.334177i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 31.7913i	− 1.55683i
$418$	0	0
$419$	27.0000	1.31904	0.659518	−	0.751689i	$-0.270760\pi$
$419$	27.0000	1.31904	0.659518	+	0.751689i	$0.270760\pi$
$420$	0	0
$421$	11.6261	0.566623	0.283312	−	0.959028i	$-0.408567\pi$
$421$	11.6261	0.566623	0.283312	+	0.959028i	$0.408567\pi$
$422$	0	0
$423$	− 2.20871i	− 0.107391i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.1216i	2.03841i
$428$	0	0
$429$	−1.79129	−0.0864842
$430$	0	0
$431$	34.9129	1.68169	0.840847	−	0.541273i	$-0.182057\pi$
$431$	34.9129	1.68169	0.840847	+	0.541273i	$0.182057\pi$
$432$	0	0
$433$	− 23.3303i	− 1.12118i	−0.828093	−	0.560591i	$-0.810574\pi$
$433$	− 23.3303i	− 1.12118i	0.828093	−	0.560591i	$-0.189426\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.04356i	− 0.0977568i
$438$	0	0
$439$	−10.9564	−0.522922	−0.261461	−	0.965214i	$-0.584204\pi$
$439$	−10.9564	−0.522922	−0.261461	+	0.965214i	$0.584204\pi$
$440$	0	0
$441$	3.33030	0.158586
$442$	0	0
$443$	31.5826i	1.50053i	0.661135	+	0.750267i	$0.270075\pi$
$443$	31.5826i	1.50053i	−0.661135	+	0.750267i	$0.729925\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	27.1652i	1.28487i
$448$	0	0
$449$	−30.7913	−1.45313	−0.726565	−	0.687097i	$-0.758885\pi$
$449$	−30.7913	−1.45313	−0.726565	+	0.687097i	$0.758885\pi$
$450$	0	0
$451$	−10.5826	−0.498314
$452$	0	0
$453$	− 8.95644i	− 0.420810i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.373864i	− 0.0174886i	−0.999962	−	0.00874430i	$-0.997217\pi$
$457$	− 0.373864i	− 0.0174886i	0.999962	−	0.00874430i	$-0.00278343\pi$
$458$	0	0
$459$	18.9564	0.884811
$460$	0	0
$461$	30.4955	1.42031	0.710157	−	0.704043i	$-0.248624\pi$
$461$	30.4955	1.42031	0.710157	+	0.704043i	$0.248624\pi$
$462$	0	0
$463$	− 29.7477i	− 1.38249i	−0.722619	−	0.691247i	$-0.757062\pi$
$463$	− 29.7477i	− 1.38249i	0.722619	−	0.691247i	$-0.242938\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.41742i	− 0.0655906i	−0.999462	−	0.0327953i	$-0.989559\pi$
$467$	− 1.41742i	− 0.0655906i	0.999462	−	0.0327953i	$-0.0104409\pi$
$468$	0	0
$469$	−19.1652	−0.884964
$470$	0	0
$471$	2.08712	0.0961695
$472$	0	0
$473$	− 10.0000i	− 0.459800i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.495454i	− 0.0226853i
$478$	0	0
$479$	−18.1652	−0.829987	−0.414993	−	0.909824i	$-0.636216\pi$
$479$	−18.1652	−0.829987	−0.414993	+	0.909824i	$0.636216\pi$
$480$	0	0
$481$	−6.58258	−0.300140
$482$	0	0
$483$	− 6.79129i	− 0.309014i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.25227i	0.102060i	0.998697	+	0.0510301i	$0.0162504\pi$
$487$	2.25227i	0.102060i	−0.998697	+	0.0510301i	$0.983750\pi$
$488$	0	0
$489$	13.6606	0.617754
$490$	0	0
$491$	16.5826	0.748361	0.374181	−	0.927356i	$-0.377924\pi$
$491$	16.5826	0.748361	0.374181	+	0.927356i	$0.377924\pi$
$492$	0	0
$493$	8.37386i	0.377140i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 80.2432i	− 3.59940i
$498$	0	0
$499$	−7.95644	−0.356179	−0.178090	−	0.984014i	$-0.556992\pi$
$499$	−7.95644	−0.356179	−0.178090	+	0.984014i	$0.556992\pi$
$500$	0	0
$501$	−13.2867	−0.593608
$502$	0	0
$503$	25.5826i	1.14067i	0.821412	+	0.570335i	$0.193187\pi$
$503$	25.5826i	1.14067i	−0.821412	+	0.570335i	$0.806813\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	21.4955i	0.954647i
$508$	0	0
$509$	31.1216	1.37944	0.689720	−	0.724076i	$-0.257734\pi$
$509$	31.1216	1.37944	0.689720	+	0.724076i	$0.257734\pi$
$510$	0	0
$511$	−15.3739	−0.680100
$512$	0	0
$513$	− 12.9129i	− 0.570118i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.5826i	0.465421i
$518$	0	0
$519$	24.6261	1.08097
$520$	0	0
$521$	−7.58258	−0.332199	−0.166099	−	0.986109i	$-0.553117\pi$
$521$	−7.58258	−0.332199	−0.166099	+	0.986109i	$0.553117\pi$
$522$	0	0
$523$	− 6.83485i	− 0.298867i	−0.988772	−	0.149434i	$-0.952255\pi$
$523$	− 6.83485i	− 0.298867i	0.988772	−	0.149434i	$-0.0477450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.20871i	0.0962130i
$528$	0	0
$529$	22.3739	0.972777
$530$	0	0
$531$	0.295834	0.0128381
$532$	0	0
$533$	− 10.5826i	− 0.458382i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 34.2523i	− 1.47809i
$538$	0	0
$539$	−15.9564	−0.687292
$540$	0	0
$541$	−24.3739	−1.04791	−0.523957	−	0.851745i	$-0.675545\pi$
$541$	−24.3739	−1.04791	−0.523957	+	0.851745i	$0.675545\pi$
$542$	0	0
$543$	18.5826i	0.797455i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.37386i	0.272527i	0.990673	+	0.136263i	$0.0435093\pi$
$547$	6.37386i	0.272527i	−0.990673	+	0.136263i	$0.956491\pi$
$548$	0	0
$549$	−1.83485	−0.0783094
$550$	0	0
$551$	5.70417	0.243006
$552$	0	0
$553$	79.2432i	3.36976i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.6606i	1.17202i	0.810305	+	0.586009i	$0.199302\pi$
$557$	27.6606i	1.17202i	−0.810305	+	0.586009i	$0.800698\pi$
$558$	0	0
$559$	10.0000	0.422955
$560$	0	0
$561$	6.79129	0.286728
$562$	0	0
$563$	1.12159i	0.0472694i	0.999721	+	0.0236347i	$0.00752386\pi$
$563$	1.12159i	0.0472694i	−0.999721	+	0.0236347i	$0.992476\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 45.9129i	− 1.92816i
$568$	0	0
$569$	3.95644	0.165863	0.0829313	−	0.996555i	$-0.473572\pi$
$569$	3.95644	0.165863	0.0829313	+	0.996555i	$0.473572\pi$
$570$	0	0
$571$	32.2867	1.35116	0.675579	−	0.737288i	$-0.263894\pi$
$571$	32.2867	1.35116	0.675579	+	0.737288i	$0.263894\pi$
$572$	0	0
$573$	− 39.0345i	− 1.63069i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10.9564i	− 0.456123i	−0.973647	−	0.228061i	$-0.926761\pi$
$577$	− 10.9564i	− 0.456123i	0.973647	−	0.228061i	$-0.0732387\pi$
$578$	0	0
$579$	20.0000	0.831172
$580$	0	0
$581$	−62.0780	−2.57543
$582$	0	0
$583$	2.37386i	0.0983154i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.95644i	0.287123i	0.989641	+	0.143561i	$0.0458555\pi$
$587$	6.95644i	0.287123i	−0.989641	+	0.143561i	$0.954145\pi$
$588$	0	0
$589$	1.50455	0.0619937
$590$	0	0
$591$	9.92197	0.408135
$592$	0	0
$593$	13.4174i	0.550988i	0.961303	+	0.275494i	$0.0888413\pi$
$593$	13.4174i	0.550988i	−0.961303	+	0.275494i	$0.911159\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 27.5390i	− 1.12710i
$598$	0	0
$599$	−5.70417	−0.233066	−0.116533	−	0.993187i	$-0.537178\pi$
$599$	−5.70417	−0.233066	−0.116533	+	0.993187i	$0.537178\pi$
$600$	0	0
$601$	−3.53901	−0.144359	−0.0721797	−	0.997392i	$-0.522996\pi$
$601$	−3.53901	−0.144359	−0.0721797	+	0.997392i	$0.522996\pi$
$602$	0	0
$603$	− 0.834849i	− 0.0339977i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.0000i	1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$	34.0000i	1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$	0	0
$609$	18.9564	0.768154
$610$	0	0
$611$	−10.5826	−0.428125
$612$	0	0
$613$	44.4519i	1.79540i	0.440612	+	0.897698i	$0.354761\pi$
$613$	44.4519i	1.79540i	−0.440612	+	0.897698i	$0.645239\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.25227i	− 0.0504146i	−0.999682	−	0.0252073i	$-0.991975\pi$
$617$	− 1.25227i	− 0.0504146i	0.999682	−	0.0252073i	$-0.00802458\pi$
$618$	0	0
$619$	33.7477	1.35644	0.678218	−	0.734861i	$-0.262753\pi$
$619$	33.7477	1.35644	0.678218	+	0.734861i	$0.262753\pi$
$620$	0	0
$621$	−3.95644	−0.158766
$622$	0	0
$623$	− 18.1652i	− 0.727771i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.62614i	− 0.184750i
$628$	0	0
$629$	24.9564	0.995078
$630$	0	0
$631$	23.1216	0.920456	0.460228	−	0.887801i	$-0.347768\pi$
$631$	23.1216	0.920456	0.460228	+	0.887801i	$0.347768\pi$
$632$	0	0
$633$	− 8.95644i	− 0.355987i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 15.9564i	− 0.632217i
$638$	0	0
$639$	3.49545	0.138278
$640$	0	0
$641$	−42.1652	−1.66542	−0.832712	−	0.553707i	$-0.813213\pi$
$641$	−42.1652	−1.66542	−0.832712	+	0.553707i	$0.813213\pi$
$642$	0	0
$643$	11.4955i	0.453336i	0.973972	+	0.226668i	$0.0727833\pi$
$643$	11.4955i	0.453336i	−0.973972	+	0.226668i	$0.927217\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.3303i	0.602696i	0.953514	+	0.301348i	$0.0974366\pi$
$647$	15.3303i	0.602696i	−0.953514	+	0.301348i	$0.902563\pi$
$648$	0	0
$649$	−1.41742	−0.0556388
$650$	0	0
$651$	5.00000	0.195965
$652$	0	0
$653$	23.7042i	0.927616i	0.885936	+	0.463808i	$0.153517\pi$
$653$	23.7042i	0.927616i	−0.885936	+	0.463808i	$0.846483\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 0.669697i	− 0.0261274i
$658$	0	0
$659$	32.5390	1.26754	0.633770	−	0.773522i	$-0.281507\pi$
$659$	32.5390	1.26754	0.633770	+	0.773522i	$0.281507\pi$
$660$	0	0
$661$	21.4174	0.833041	0.416521	−	0.909126i	$-0.363249\pi$
$661$	21.4174	0.833041	0.416521	+	0.909126i	$0.363249\pi$
$662$	0	0
$663$	6.79129i	0.263752i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.74773i	− 0.0676723i
$668$	0	0
$669$	20.4519	0.790716
$670$	0	0
$671$	8.79129	0.339384
$672$	0	0
$673$	− 6.91288i	− 0.266472i	−0.991084	−	0.133236i	$-0.957463\pi$
$673$	− 6.91288i	− 0.266472i	0.991084	−	0.133236i	$-0.0425368\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.58258i	0.291422i	0.989327	+	0.145711i	$0.0465470\pi$
$677$	7.58258i	0.291422i	−0.989327	+	0.145711i	$0.953453\pi$
$678$	0	0
$679$	51.7042	1.98422
$680$	0	0
$681$	−36.7913	−1.40985
$682$	0	0
$683$	− 10.9129i	− 0.417570i	−0.977962	−	0.208785i	$-0.933049\pi$
$683$	− 10.9129i	− 0.417570i	0.977962	−	0.208785i	$-0.0669508\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	38.2867i	1.46073i
$688$	0	0
$689$	−2.37386	−0.0904370
$690$	0	0
$691$	−9.12159	−0.347002	−0.173501	−	0.984834i	$-0.555508\pi$
$691$	−9.12159	−0.347002	−0.173501	+	0.984834i	$0.555508\pi$
$692$	0	0
$693$	− 1.00000i	− 0.0379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	40.1216i	1.51971i
$698$	0	0
$699$	20.0780	0.759421
$700$	0	0
$701$	39.1652	1.47925	0.739624	−	0.673021i	$-0.235004\pi$
$701$	39.1652	1.47925	0.739624	+	0.673021i	$0.235004\pi$
$702$	0	0
$703$	− 17.0000i	− 0.641167i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.3739i	− 0.653411i
$708$	0	0
$709$	25.4955	0.957502	0.478751	−	0.877951i	$-0.341090\pi$
$709$	25.4955	0.957502	0.478751	+	0.877951i	$0.341090\pi$
$710$	0	0
$711$	−3.45189	−0.129456
$712$	0	0
$713$	− 0.460985i	− 0.0172640i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 12.1652i	− 0.454316i
$718$	0	0
$719$	−48.4955	−1.80858	−0.904288	−	0.426924i	$-0.859597\pi$
$719$	−48.4955	−1.80858	−0.904288	+	0.426924i	$0.859597\pi$
$720$	0	0
$721$	81.2432	3.02565
$722$	0	0
$723$	32.4610i	1.20724i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 18.1216i	− 0.672093i	−0.941845	−	0.336046i	$-0.890910\pi$
$727$	− 18.1216i	− 0.672093i	0.941845	−	0.336046i	$-0.109090\pi$
$728$	0	0
$729$	−24.8693	−0.921086
$730$	0	0
$731$	−37.9129	−1.40226
$732$	0	0
$733$	51.2432i	1.89271i	0.323129	+	0.946355i	$0.395265\pi$
$733$	51.2432i	1.89271i	−0.323129	+	0.946355i	$0.604735\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000i	0.147342i
$738$	0	0
$739$	0.878409	0.0323128	0.0161564	−	0.999869i	$-0.494857\pi$
$739$	0.878409	0.0323128	0.0161564	+	0.999869i	$0.494857\pi$
$740$	0	0
$741$	4.62614	0.169945
$742$	0	0
$743$	35.2087i	1.29168i	0.763472	+	0.645841i	$0.223493\pi$
$743$	35.2087i	1.29168i	−0.763472	+	0.645841i	$0.776507\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.70417i	− 0.0989403i
$748$	0	0
$749$	−37.1216	−1.35639
$750$	0	0
$751$	−17.7913	−0.649213	−0.324607	−	0.945849i	$-0.605232\pi$
$751$	−17.7913	−0.649213	−0.324607	+	0.945849i	$0.605232\pi$
$752$	0	0
$753$	− 36.7913i	− 1.34075i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.3303i	1.06603i	0.846106	+	0.533014i	$0.178941\pi$
$757$	29.3303i	1.06603i	−0.846106	+	0.533014i	$0.821059\pi$
$758$	0	0
$759$	−1.41742	−0.0514492
$760$	0	0
$761$	−6.33030	−0.229473	−0.114737	−	0.993396i	$-0.536602\pi$
$761$	−6.33030	−0.229473	−0.114737	+	0.993396i	$0.536602\pi$
$762$	0	0
$763$	1.00000i	0.0362024i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.41742i	− 0.0511802i
$768$	0	0
$769$	14.2523	0.513950	0.256975	−	0.966418i	$-0.417274\pi$
$769$	14.2523	0.513950	0.256975	+	0.966418i	$0.417274\pi$
$770$	0	0
$771$	−32.2432	−1.16121
$772$	0	0
$773$	− 30.7913i	− 1.10749i	−0.832688	−	0.553743i	$-0.813199\pi$
$773$	− 30.7913i	− 1.10749i	0.832688	−	0.553743i	$-0.186801\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 56.4955i	− 2.02676i
$778$	0	0
$779$	27.3303	0.979210
$780$	0	0
$781$	−16.7477	−0.599281
$782$	0	0
$783$	− 11.0436i	− 0.394665i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 9.74773i	− 0.347469i	−0.984793	−	0.173734i	$-0.944417\pi$
$787$	− 9.74773i	− 0.347469i	0.984793	−	0.173734i	$-0.0555834\pi$
$788$	0	0
$789$	−10.4519	−0.372097
$790$	0	0
$791$	50.7042	1.80283
$792$	0	0
$793$	8.79129i	0.312188i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 25.2867i	− 0.895702i	−0.894108	−	0.447851i	$-0.852189\pi$
$797$	− 25.2867i	− 0.895702i	0.894108	−	0.447851i	$-0.147811\pi$
$798$	0	0
$799$	40.1216	1.41940
$800$	0	0
$801$	0.791288	0.0279588
$802$	0	0
$803$	3.20871i	0.113233i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 53.2087i	− 1.87304i
$808$	0	0
$809$	−9.49545	−0.333842	−0.166921	−	0.985970i	$-0.553383\pi$
$809$	−9.49545	−0.333842	−0.166921	+	0.985970i	$0.553383\pi$
$810$	0	0
$811$	−17.6606	−0.620148	−0.310074	−	0.950712i	$-0.600354\pi$
$811$	−17.6606	−0.620148	−0.310074	+	0.950712i	$0.600354\pi$
$812$	0	0
$813$	21.0436i	0.738030i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25.8258i	0.903529i
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−24.6606	−0.860661	−0.430331	−	0.902671i	$-0.641603\pi$
$821$	−24.6606	−0.860661	−0.430331	+	0.902671i	$0.641603\pi$
$822$	0	0
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.5826i	0.680953i	0.940253	+	0.340476i	$0.110588\pi$
$827$	19.5826i	0.680953i	−0.940253	+	0.340476i	$0.889412\pi$
$828$	0	0
$829$	15.0436	0.522484	0.261242	−	0.965273i	$-0.415868\pi$
$829$	15.0436	0.522484	0.261242	+	0.965273i	$0.415868\pi$
$830$	0	0
$831$	−1.04356	−0.0362007
$832$	0	0
$833$	60.4955i	2.09604i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.91288i	− 0.100684i
$838$	0	0
$839$	26.7042	0.921930	0.460965	−	0.887418i	$-0.347503\pi$
$839$	26.7042	0.921930	0.460965	+	0.887418i	$0.347503\pi$
$840$	0	0
$841$	−24.1216	−0.831779
$842$	0	0
$843$	24.6261i	0.848169i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.79129i	0.164631i
$848$	0	0
$849$	−35.2958	−1.21135
$850$	0	0
$851$	−5.20871	−0.178552
$852$	0	0
$853$	38.1216i	1.30526i	0.757677	+	0.652629i	$0.226334\pi$
$853$	38.1216i	1.30526i	−0.757677	+	0.652629i	$0.773666\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.4955i	0.836749i	0.908275	+	0.418374i	$0.137400\pi$
$857$	24.4955i	0.836749i	−0.908275	+	0.418374i	$0.862600\pi$
$858$	0	0
$859$	−47.9129	−1.63477	−0.817383	−	0.576094i	$-0.804576\pi$
$859$	−47.9129	−1.63477	−0.817383	+	0.576094i	$0.804576\pi$
$860$	0	0
$861$	90.8258	3.09533
$862$	0	0
$863$	9.33030i	0.317607i	0.987310	+	0.158804i	$0.0507637\pi$
$863$	9.33030i	0.317607i	−0.987310	+	0.158804i	$0.949236\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.70417i	0.159762i
$868$	0	0
$869$	16.5390	0.561048
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	2.25227i	0.0762279i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.6606i	1.91329i	0.291252	+	0.956646i	$0.405928\pi$
$877$	56.6606i	1.91329i	−0.291252	+	0.956646i	$0.594072\pi$
$878$	0	0
$879$	−37.9129	−1.27877
$880$	0	0
$881$	43.1216	1.45280	0.726402	−	0.687270i	$-0.241191\pi$
$881$	43.1216	1.45280	0.726402	+	0.687270i	$0.241191\pi$
$882$	0	0
$883$	7.83485i	0.263664i	0.991272	+	0.131832i	$0.0420859\pi$
$883$	7.83485i	0.263664i	−0.991272	+	0.131832i	$0.957914\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.66970i	0.0896397i	0.998995	+	0.0448198i	$0.0142714\pi$
$887$	2.66970i	0.0896397i	−0.998995	+	0.0448198i	$0.985729\pi$
$888$	0	0
$889$	12.5826	0.422006
$890$	0	0
$891$	−9.58258	−0.321028
$892$	0	0
$893$	− 27.3303i	− 0.914574i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.41742i	− 0.0473264i
$898$	0	0
$899$	1.28674	0.0429152
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	85.8258i	2.85610i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.6606i	1.35011i	0.737766	+	0.675057i	$0.235881\pi$
$907$	40.6606i	1.35011i	−0.737766	+	0.675057i	$0.764119\pi$
$908$	0	0
$909$	0.756819	0.0251021
$910$	0	0
$911$	−24.3303	−0.806099	−0.403049	−	0.915178i	$-0.632050\pi$
$911$	−24.3303	−0.806099	−0.403049	+	0.915178i	$0.632050\pi$
$912$	0	0
$913$	12.9564i	0.428796i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	68.8693i	2.27427i
$918$	0	0
$919$	47.1652	1.55583	0.777917	−	0.628367i	$-0.216276\pi$
$919$	47.1652	1.55583	0.777917	+	0.628367i	$0.216276\pi$
$920$	0	0
$921$	−56.9473	−1.87648
$922$	0	0
$923$	− 16.7477i	− 0.551258i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.53901i	0.116237i
$928$	0	0
$929$	13.7477	0.451048	0.225524	−	0.974238i	$-0.427591\pi$
$929$	13.7477	0.451048	0.225524	+	0.974238i	$0.427591\pi$
$930$	0	0
$931$	41.2087	1.35056
$932$	0	0
$933$	43.2867i	1.41714i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.8348i	0.549971i	0.961448	+	0.274985i	$0.0886730\pi$
$937$	16.8348i	0.549971i	−0.961448	+	0.274985i	$0.911327\pi$
$938$	0	0
$939$	−37.1652	−1.21284
$940$	0	0
$941$	14.0780	0.458931	0.229465	−	0.973317i	$-0.426302\pi$
$941$	14.0780	0.458931	0.229465	+	0.973317i	$0.426302\pi$
$942$	0	0
$943$	− 8.37386i	− 0.272691i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 40.9129i	− 1.32949i	−0.747070	−	0.664745i	$-0.768540\pi$
$947$	− 40.9129i	− 1.32949i	0.747070	−	0.664745i	$-0.231460\pi$
$948$	0	0
$949$	−3.20871	−0.104159
$950$	0	0
$951$	−9.62614	−0.312149
$952$	0	0
$953$	42.0000i	1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$	42.0000i	1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.95644i	− 0.127894i
$958$	0	0
$959$	−98.4083	−3.17777
$960$	0	0
$961$	−30.6606	−0.989052
$962$	0	0
$963$	− 1.61704i	− 0.0521085i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 11.4610i	− 0.368560i	−0.982874	−	0.184280i	$-0.941005\pi$
$967$	− 11.4610i	− 0.368560i	0.982874	−	0.184280i	$-0.0589954\pi$
$968$	0	0
$969$	−17.5390	−0.563434
$970$	0	0
$971$	3.95644	0.126968	0.0634841	−	0.997983i	$-0.479779\pi$
$971$	3.95644	0.126968	0.0634841	+	0.997983i	$0.479779\pi$
$972$	0	0
$973$	− 85.0345i	− 2.72608i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.74773i	0.0559147i	0.999609	+	0.0279574i	$0.00890027\pi$
$977$	1.74773i	0.0559147i	−0.999609	+	0.0279574i	$0.991100\pi$
$978$	0	0
$979$	−3.79129	−0.121170
$980$	0	0
$981$	−0.0435608	−0.00139079
$982$	0	0
$983$	24.6606i	0.786551i	0.919421	+	0.393276i	$0.128658\pi$
$983$	24.6606i	0.786551i	−0.919421	+	0.393276i	$0.871342\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 90.8258i	− 2.89102i
$988$	0	0
$989$	7.91288	0.251615
$990$	0	0
$991$	12.8693	0.408807	0.204404	−	0.978887i	$-0.434474\pi$
$991$	12.8693	0.408807	0.204404	+	0.978887i	$0.434474\pi$
$992$	0	0
$993$	34.6261i	1.09883i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.6261i	0.843258i	0.906768	+	0.421629i	$0.138542\pi$
$997$	26.6261i	0.843258i	−0.906768	+	0.421629i	$0.861458\pi$
$998$	0	0
$999$	−32.9129	−1.04132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.s.4049.3		4
4.3	odd	2		1100.2.b.d.749.2		4
5.2	odd	4		4400.2.a.bi.1.2		2
5.3	odd	4		4400.2.a.bu.1.1		2
5.4	even	2	inner	4400.2.b.s.4049.2		4
12.11	even	2		9900.2.c.x.5149.1		4
20.3	even	4		1100.2.a.g.1.2	✓	2
20.7	even	4		1100.2.a.h.1.1	yes	2
20.19	odd	2		1100.2.b.d.749.3		4
60.23	odd	4		9900.2.a.bh.1.1		2
60.47	odd	4		9900.2.a.bz.1.2		2
60.59	even	2		9900.2.c.x.5149.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.g.1.2	✓	2	20.3	even	4
1100.2.a.h.1.1	yes	2	20.7	even	4
1100.2.b.d.749.2		4	4.3	odd	2
1100.2.b.d.749.3		4	20.19	odd	2
4400.2.a.bi.1.2		2	5.2	odd	4
4400.2.a.bu.1.1		2	5.3	odd	4
4400.2.b.s.4049.2		4	5.4	even	2	inner
4400.2.b.s.4049.3		4	1.1	even	1	trivial
9900.2.a.bh.1.1		2	60.23	odd	4
9900.2.a.bz.1.2		2	60.47	odd	4
9900.2.c.x.5149.1		4	12.11	even	2
9900.2.c.x.5149.4		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.79129i q^{3} +4.79129i q^{7} -0.208712 q^{9} +1.00000 q^{11} +1.00000i q^{13} -3.79129i q^{17} -2.58258 q^{19} -8.58258 q^{21} +0.791288i q^{23} +5.00000i q^{27} -2.20871 q^{29} -0.582576 q^{31} +1.79129i q^{33} +6.58258i q^{37} -1.79129 q^{39} -10.5826 q^{41} -10.0000i q^{43} +10.5826i q^{47} -15.9564 q^{49} +6.79129 q^{51} +2.37386i q^{53} -4.62614i q^{57} -1.41742 q^{59} +8.79129 q^{61} -1.00000i q^{63} +4.00000i q^{67} -1.41742 q^{69} -16.7477 q^{71} +3.20871i q^{73} +4.79129i q^{77} +16.5390 q^{79} -9.58258 q^{81} +12.9564i q^{83} -3.95644i q^{87} -3.79129 q^{89} -4.79129 q^{91} -1.04356i q^{93} -10.7913i q^{97} -0.208712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{9} + 4 q^{11} + 8 q^{19} - 16 q^{21} - 18 q^{29} + 16 q^{31} + 2 q^{39} - 24 q^{41} - 18 q^{49} + 18 q^{51} - 24 q^{59} + 26 q^{61} - 24 q^{69} - 12 q^{71} + 2 q^{79} - 20 q^{81} - 6 q^{89}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q - 10 * q^9 + 4 * q^11 + 8 * q^19 - 16 * q^21 - 18 * q^29 + 16 * q^31 + 2 * q^39 - 24 * q^41 - 18 * q^49 + 18 * q^51 - 24 * q^59 + 26 * q^61 - 24 * q^69 - 12 * q^71 + 2 * q^79 - 20 * q^81 - 6 * q^89 - 10 * q^91 - 10 * q^99

Introduction

L-functions

Modular forms

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