LMFDB - Embedded newform 2200.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2200.a (trivial)

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:	yes
Analytic conductor:	$17.5670884447$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.655762$ of defining polynomial
Character	$\chi$	$=$	2200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.655762 q^{3} +0.415806 q^{7} -2.56998 q^{9} +1.00000 q^{11} -4.00000 q^{13} +6.51558 q^{17} +5.20406 q^{19} -0.272670 q^{21} -8.54830 q^{23} +3.65258 q^{27} +0.895717 q^{29} -6.73836 q^{31} -0.655762 q^{33} -8.96410 q^{37} +2.62305 q^{39} +10.0998 q^{41} +4.78825 q^{43} -5.61986 q^{47} -6.82711 q^{49} -4.27267 q^{51} -10.0357 q^{53} -3.41262 q^{57} -1.63408 q^{59} +7.10428 q^{61} -1.06861 q^{63} -10.6914 q^{67} +5.60565 q^{69} -6.19302 q^{71} -3.16839 q^{73} +0.415806 q^{77} -11.2682 q^{79} +5.31471 q^{81} -16.2429 q^{83} -0.587377 q^{87} +9.56998 q^{89} -1.66323 q^{91} +4.41876 q^{93} -0.591657 q^{97} -2.56998 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - q^{7} + 7 q^{9} + 4 q^{11} - 16 q^{13} - 7 q^{17} - 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} - 15 q^{31} - q^{33} - 5 q^{37} + 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49}+ \cdots + 7 q^{99}+O(q^{100}) $ 4 * q - q^3 - q^7 + 7 * q^9 + 4 * q^11 - 16 * q^13 - 7 * q^17 - 9 * q^19 - 7 * q^21 - 6 * q^23 - 13 * q^27 + 3 * q^29 - 15 * q^31 - q^33 - 5 * q^37 + 4 * q^39 + 10 * q^41 - 8 * q^43 + 10 * q^47 + 9 * q^49 - 23 * q^51 - 5 * q^53 + 15 * q^57 + 6 * q^59 + 29 * q^61 - 40 * q^63 - 6 * q^67 - 30 * q^69 - q^71 - 18 * q^73 - q^77 - 20 * q^79 + 44 * q^81 - 26 * q^83 - 31 * q^87 + 21 * q^89 + 4 * q^91 - 25 * q^93 + 4 * q^97 + 7 * q^99

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
$3$	−0.655762	−0.378604	−0.189302	−	0.981919i	$-0.560623\pi$
$3$	−0.655762	−0.378604	−0.189302	+	0.981919i	$0.560623\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.415806	0.157160	0.0785800	−	0.996908i	$-0.474961\pi$
$7$	0.415806	0.157160	0.0785800	+	0.996908i	$0.474961\pi$
$8$	0	0
$9$	−2.56998	−0.856659
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.51558	1.58026	0.790130	−	0.612939i	$-0.210013\pi$
$17$	6.51558	1.58026	0.790130	+	0.612939i	$0.210013\pi$
$18$	0	0
$19$	5.20406	1.19389	0.596946	−	0.802281i	$-0.296381\pi$
$19$	5.20406	1.19389	0.596946	+	0.802281i	$0.296381\pi$
$20$	0	0
$21$	−0.272670	−0.0595015
$22$	0	0
$23$	−8.54830	−1.78244	−0.891221	−	0.453568i	$-0.850151\pi$
$23$	−8.54830	−1.78244	−0.891221	+	0.453568i	$0.850151\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.65258	0.702939
$28$	0	0
$29$	0.895717	0.166331	0.0831653	−	0.996536i	$-0.473497\pi$
$29$	0.895717	0.166331	0.0831653	+	0.996536i	$0.473497\pi$
$30$	0	0
$31$	−6.73836	−1.21025	−0.605123	−	0.796132i	$-0.706876\pi$
$31$	−6.73836	−1.21025	−0.605123	+	0.796132i	$0.706876\pi$
$32$	0	0
$33$	−0.655762	−0.114153
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.96410	−1.47369	−0.736845	−	0.676062i	$-0.763685\pi$
$37$	−8.96410	−1.47369	−0.736845	+	0.676062i	$0.763685\pi$
$38$	0	0
$39$	2.62305	0.420024
$40$	0	0
$41$	10.0998	1.57732	0.788660	−	0.614830i	$-0.210775\pi$
$41$	10.0998	1.57732	0.788660	+	0.614830i	$0.210775\pi$
$42$	0	0
$43$	4.78825	0.730201	0.365101	−	0.930968i	$-0.381035\pi$
$43$	4.78825	0.730201	0.365101	+	0.930968i	$0.381035\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.61986	−0.819741	−0.409871	−	0.912144i	$-0.634426\pi$
$47$	−5.61986	−0.819741	−0.409871	+	0.912144i	$0.634426\pi$
$48$	0	0
$49$	−6.82711	−0.975301
$50$	0	0
$51$	−4.27267	−0.598293
$52$	0	0
$53$	−10.0357	−1.37851	−0.689253	−	0.724521i	$-0.742061\pi$
$53$	−10.0357	−1.37851	−0.689253	+	0.724521i	$0.742061\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.41262	−0.452013
$58$	0	0
$59$	−1.63408	−0.212739	−0.106370	−	0.994327i	$-0.533923\pi$
$59$	−1.63408	−0.212739	−0.106370	+	0.994327i	$0.533923\pi$
$60$	0	0
$61$	7.10428	0.909610	0.454805	−	0.890591i	$-0.349709\pi$
$61$	7.10428	0.909610	0.454805	+	0.890591i	$0.349709\pi$
$62$	0	0
$63$	−1.06861	−0.134633
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.6914	−1.30617	−0.653083	−	0.757286i	$-0.726525\pi$
$67$	−10.6914	−1.30617	−0.653083	+	0.757286i	$0.726525\pi$
$68$	0	0
$69$	5.60565	0.674841
$70$	0	0
$71$	−6.19302	−0.734977	−0.367488	−	0.930028i	$-0.619782\pi$
$71$	−6.19302	−0.734977	−0.367488	+	0.930028i	$0.619782\pi$
$72$	0	0
$73$	−3.16839	−0.370832	−0.185416	−	0.982660i	$-0.559363\pi$
$73$	−3.16839	−0.370832	−0.185416	+	0.982660i	$0.559363\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.415806	0.0473855
$78$	0	0
$79$	−11.2682	−1.26777	−0.633884	−	0.773428i	$-0.718540\pi$
$79$	−11.2682	−1.26777	−0.633884	+	0.773428i	$0.718540\pi$
$80$	0	0
$81$	5.31471	0.590523
$82$	0	0
$83$	−16.2429	−1.78289	−0.891446	−	0.453128i	$-0.850308\pi$
$83$	−16.2429	−1.78289	−0.891446	+	0.453128i	$0.850308\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.587377	−0.0629735
$88$	0	0
$89$	9.56998	1.01442	0.507208	−	0.861824i	$-0.330678\pi$
$89$	9.56998	1.01442	0.507208	+	0.861824i	$0.330678\pi$
$90$	0	0
$91$	−1.66323	−0.174353
$92$	0	0
$93$	4.41876	0.458204
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.591657	−0.0600737	−0.0300368	−	0.999549i	$-0.509562\pi$
$97$	−0.591657	−0.0600737	−0.0300368	+	0.999549i	$0.509562\pi$
$98$	0	0
$99$	−2.56998	−0.258292
$100$	0	0
$101$	−4.62305	−0.460010	−0.230005	−	0.973189i	$-0.573874\pi$
$101$	−4.62305	−0.460010	−0.230005	+	0.973189i	$0.573874\pi$
$102$	0	0
$103$	8.24291	0.812198	0.406099	−	0.913829i	$-0.366889\pi$
$103$	8.24291	0.812198	0.406099	+	0.913829i	$0.366889\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.6199	−1.70338	−0.851688	−	0.524049i	$-0.824421\pi$
$107$	−17.6199	−1.70338	−0.851688	+	0.524049i	$0.824421\pi$
$108$	0	0
$109$	3.66323	0.350873	0.175437	−	0.984491i	$-0.443866\pi$
$109$	3.66323	0.350873	0.175437	+	0.984491i	$0.443866\pi$
$110$	0	0
$111$	5.87832	0.557945
$112$	0	0
$113$	11.4797	1.07992	0.539959	−	0.841691i	$-0.318440\pi$
$113$	11.4797	1.07992	0.539959	+	0.841691i	$0.318440\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.2799	0.950378
$118$	0	0
$119$	2.70922	0.248354
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.62305	−0.597180
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.13995	0.811040	0.405520	−	0.914086i	$-0.367091\pi$
$127$	9.13995	0.811040	0.405520	+	0.914086i	$0.367091\pi$
$128$	0	0
$129$	−3.13995	−0.276457
$130$	0	0
$131$	−2.48123	−0.216787	−0.108393	−	0.994108i	$-0.534571\pi$
$131$	−2.48123	−0.216787	−0.108393	+	0.994108i	$0.534571\pi$
$132$	0	0
$133$	2.16388	0.187632
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.40834	−0.632937	−0.316469	−	0.948603i	$-0.602497\pi$
$137$	−7.40834	−0.632937	−0.316469	+	0.948603i	$0.602497\pi$
$138$	0	0
$139$	−1.69166	−0.143485	−0.0717424	−	0.997423i	$-0.522856\pi$
$139$	−1.69166	−0.143485	−0.0717424	+	0.997423i	$0.522856\pi$
$140$	0	0
$141$	3.68529	0.310358
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.47696	0.369253
$148$	0	0
$149$	3.74940	0.307163	0.153581	−	0.988136i	$-0.450919\pi$
$149$	3.74940	0.307163	0.153581	+	0.988136i	$0.450919\pi$
$150$	0	0
$151$	−15.1400	−1.23207	−0.616036	−	0.787718i	$-0.711262\pi$
$151$	−15.1400	−1.23207	−0.616036	+	0.787718i	$0.711262\pi$
$152$	0	0
$153$	−16.7449	−1.35374
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.5871	−0.924755	−0.462378	−	0.886683i	$-0.653004\pi$
$157$	−11.5871	−0.924755	−0.462378	+	0.886683i	$0.653004\pi$
$158$	0	0
$159$	6.58101	0.521908
$160$	0	0
$161$	−3.55444	−0.280129
$162$	0	0
$163$	−6.82392	−0.534491	−0.267245	−	0.963629i	$-0.586113\pi$
$163$	−6.82392	−0.534491	−0.267245	+	0.963629i	$0.586113\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.10428	−0.549746	−0.274873	−	0.961481i	$-0.588636\pi$
$167$	−7.10428	−0.549746	−0.274873	+	0.961481i	$0.588636\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−13.3743	−1.02276
$172$	0	0
$173$	−9.31152	−0.707942	−0.353971	−	0.935256i	$-0.615169\pi$
$173$	−9.31152	−0.707942	−0.353971	+	0.935256i	$0.615169\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.07157	0.0805440
$178$	0	0
$179$	−26.6368	−1.99093	−0.995464	−	0.0951359i	$-0.969671\pi$
$179$	−26.6368	−1.99093	−0.995464	+	0.0951359i	$0.969671\pi$
$180$	0	0
$181$	1.01103	0.0751495	0.0375748	−	0.999294i	$-0.488037\pi$
$181$	1.01103	0.0751495	0.0375748	+	0.999294i	$0.488037\pi$
$182$	0	0
$183$	−4.65872	−0.344382
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.51558	0.476466
$188$	0	0
$189$	1.51877	0.110474
$190$	0	0
$191$	10.5655	0.764490	0.382245	−	0.924061i	$-0.375151\pi$
$191$	10.5655	0.764490	0.382245	+	0.924061i	$0.375151\pi$
$192$	0	0
$193$	−11.8271	−0.851334	−0.425667	−	0.904880i	$-0.639960\pi$
$193$	−11.8271	−0.851334	−0.425667	+	0.904880i	$0.639960\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.31152	0.0934422	0.0467211	−	0.998908i	$-0.485123\pi$
$197$	1.31152	0.0934422	0.0467211	+	0.998908i	$0.485123\pi$
$198$	0	0
$199$	−11.8271	−0.838401	−0.419201	−	0.907894i	$-0.637690\pi$
$199$	−11.8271	−0.838401	−0.419201	+	0.907894i	$0.637690\pi$
$200$	0	0
$201$	7.01103	0.494520
$202$	0	0
$203$	0.372445	0.0261405
$204$	0	0
$205$	0	0
$206$	0	0
$207$	21.9689	1.52695
$208$	0	0
$209$	5.20406	0.359972
$210$	0	0
$211$	−8.37244	−0.576383	−0.288191	−	0.957573i	$-0.593054\pi$
$211$	−8.37244	−0.576383	−0.288191	+	0.957573i	$0.593054\pi$
$212$	0	0
$213$	4.06115	0.278265
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.80185	−0.190202
$218$	0	0
$219$	2.07771	0.140398
$220$	0	0
$221$	−26.0623	−1.75314
$222$	0	0
$223$	22.1461	1.48301	0.741506	−	0.670946i	$-0.234112\pi$
$223$	22.1461	1.48301	0.741506	+	0.670946i	$0.234112\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.8194	1.44821	0.724103	−	0.689692i	$-0.242254\pi$
$227$	21.8194	1.44821	0.724103	+	0.689692i	$0.242254\pi$
$228$	0	0
$229$	2.80247	0.185192	0.0925962	−	0.995704i	$-0.470483\pi$
$229$	2.80247	0.185192	0.0925962	+	0.995704i	$0.470483\pi$
$230$	0	0
$231$	−0.272670	−0.0179404
$232$	0	0
$233$	−8.24424	−0.540098	−0.270049	−	0.962847i	$-0.587040\pi$
$233$	−8.24424	−0.540098	−0.270049	+	0.962847i	$0.587040\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.38923	0.479982
$238$	0	0
$239$	−11.2682	−0.728877	−0.364438	−	0.931227i	$-0.618739\pi$
$239$	−11.2682	−0.728877	−0.364438	+	0.931227i	$0.618739\pi$
$240$	0	0
$241$	22.3860	1.44201	0.721006	−	0.692929i	$-0.243680\pi$
$241$	22.3860	1.44201	0.721006	+	0.692929i	$0.243680\pi$
$242$	0	0
$243$	−14.4429	−0.926514
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−20.8162	−1.32451
$248$	0	0
$249$	10.6515	0.675010
$250$	0	0
$251$	3.84265	0.242546	0.121273	−	0.992619i	$-0.461302\pi$
$251$	3.84265	0.242546	0.121273	+	0.992619i	$0.461302\pi$
$252$	0	0
$253$	−8.54830	−0.537427
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.6230	−0.662648	−0.331324	−	0.943517i	$-0.607495\pi$
$257$	−10.6230	−0.662648	−0.331324	+	0.943517i	$0.607495\pi$
$258$	0	0
$259$	−3.72733	−0.231605
$260$	0	0
$261$	−2.30197	−0.142489
$262$	0	0
$263$	−5.79276	−0.357197	−0.178598	−	0.983922i	$-0.557156\pi$
$263$	−5.79276	−0.357197	−0.178598	+	0.983922i	$0.557156\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.27563	−0.384062
$268$	0	0
$269$	−8.20857	−0.500485	−0.250243	−	0.968183i	$-0.580510\pi$
$269$	−8.20857	−0.500485	−0.250243	+	0.968183i	$0.580510\pi$
$270$	0	0
$271$	27.3111	1.65903	0.829515	−	0.558485i	$-0.188617\pi$
$271$	27.3111	1.65903	0.829515	+	0.558485i	$0.188617\pi$
$272$	0	0
$273$	1.09068	0.0660109
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.4735	0.869631	0.434815	−	0.900520i	$-0.356814\pi$
$277$	14.4735	0.869631	0.434815	+	0.900520i	$0.356814\pi$
$278$	0	0
$279$	17.3174	1.03677
$280$	0	0
$281$	16.2799	0.971178	0.485589	−	0.874187i	$-0.338605\pi$
$281$	16.2799	0.971178	0.485589	+	0.874187i	$0.338605\pi$
$282$	0	0
$283$	−10.5169	−0.625165	−0.312583	−	0.949891i	$-0.601194\pi$
$283$	−10.5169	−0.625165	−0.312583	+	0.949891i	$0.601194\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.19955	0.247892
$288$	0	0
$289$	25.4528	1.49722
$290$	0	0
$291$	0.387986	0.0227441
$292$	0	0
$293$	9.37695	0.547807	0.273904	−	0.961757i	$-0.411685\pi$
$293$	9.37695	0.547807	0.273904	+	0.961757i	$0.411685\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.65258	0.211944
$298$	0	0
$299$	34.1932	1.97744
$300$	0	0
$301$	1.99099	0.114758
$302$	0	0
$303$	3.03162	0.174162
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.09978	−0.119840	−0.0599202	−	0.998203i	$-0.519085\pi$
$307$	−2.09978	−0.119840	−0.0599202	+	0.998203i	$0.519085\pi$
$308$	0	0
$309$	−5.40539	−0.307502
$310$	0	0
$311$	−20.3724	−1.15522	−0.577608	−	0.816314i	$-0.696014\pi$
$311$	−20.3724	−1.15522	−0.577608	+	0.816314i	$0.696014\pi$
$312$	0	0
$313$	23.8968	1.35073	0.675364	−	0.737485i	$-0.263987\pi$
$313$	23.8968	1.35073	0.675364	+	0.737485i	$0.263987\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.86114	0.329195	0.164597	−	0.986361i	$-0.447368\pi$
$317$	5.86114	0.329195	0.164597	+	0.986361i	$0.447368\pi$
$318$	0	0
$319$	0.895717	0.0501506
$320$	0	0
$321$	11.5544	0.644906
$322$	0	0
$323$	33.9075	1.88666
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.40220	−0.132842
$328$	0	0
$329$	−2.33677	−0.128831
$330$	0	0
$331$	−0.574484	−0.0315765	−0.0157882	−	0.999875i	$-0.505026\pi$
$331$	−0.574484	−0.0315765	−0.0157882	+	0.999875i	$0.505026\pi$
$332$	0	0
$333$	23.0375	1.26245
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.10747	0.223748	0.111874	−	0.993722i	$-0.464315\pi$
$337$	4.10747	0.223748	0.111874	+	0.993722i	$0.464315\pi$
$338$	0	0
$339$	−7.52794	−0.408862
$340$	0	0
$341$	−6.73836	−0.364903
$342$	0	0
$343$	−5.74940	−0.310438
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.30834	0.446015	0.223008	−	0.974817i	$-0.428413\pi$
$347$	8.30834	0.446015	0.223008	+	0.974817i	$0.428413\pi$
$348$	0	0
$349$	−18.4858	−0.989523	−0.494762	−	0.869029i	$-0.664745\pi$
$349$	−18.4858	−0.989523	−0.494762	+	0.869029i	$0.664745\pi$
$350$	0	0
$351$	−14.6103	−0.779841
$352$	0	0
$353$	−26.5199	−1.41151	−0.705755	−	0.708456i	$-0.749392\pi$
$353$	−26.5199	−1.41151	−0.705755	+	0.708456i	$0.749392\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.77660	−0.0940278
$358$	0	0
$359$	−18.8226	−0.993419	−0.496709	−	0.867917i	$-0.665459\pi$
$359$	−18.8226	−0.993419	−0.496709	+	0.867917i	$0.665459\pi$
$360$	0	0
$361$	8.08222	0.425380
$362$	0	0
$363$	−0.655762	−0.0344186
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.2115	0.533037	0.266519	−	0.963830i	$-0.414127\pi$
$367$	10.2115	0.533037	0.266519	+	0.963830i	$0.414127\pi$
$368$	0	0
$369$	−25.9562	−1.35122
$370$	0	0
$371$	−4.17289	−0.216646
$372$	0	0
$373$	32.3363	1.67431	0.837156	−	0.546965i	$-0.184217\pi$
$373$	32.3363	1.67431	0.837156	+	0.546965i	$0.184217\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.58287	−0.184527
$378$	0	0
$379$	15.3686	0.789434	0.394717	−	0.918803i	$-0.370843\pi$
$379$	15.3686	0.789434	0.394717	+	0.918803i	$0.370843\pi$
$380$	0	0
$381$	−5.99363	−0.307063
$382$	0	0
$383$	13.6662	0.698309	0.349155	−	0.937065i	$-0.386469\pi$
$383$	13.6662	0.698309	0.349155	+	0.937065i	$0.386469\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.3057	−0.625534
$388$	0	0
$389$	−9.01103	−0.456878	−0.228439	−	0.973558i	$-0.573362\pi$
$389$	−9.01103	−0.456878	−0.228439	+	0.973558i	$0.573362\pi$
$390$	0	0
$391$	−55.6971	−2.81672
$392$	0	0
$393$	1.62710	0.0820763
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.45466	0.173384	0.0866922	−	0.996235i	$-0.472370\pi$
$397$	3.45466	0.173384	0.0866922	+	0.996235i	$0.472370\pi$
$398$	0	0
$399$	−1.41899	−0.0710384
$400$	0	0
$401$	−1.62756	−0.0812762	−0.0406381	−	0.999174i	$-0.512939\pi$
$401$	−1.62756	−0.0812762	−0.0406381	+	0.999174i	$0.512939\pi$
$402$	0	0
$403$	26.9535	1.34265
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.96410	−0.444334
$408$	0	0
$409$	6.54534	0.323646	0.161823	−	0.986820i	$-0.448263\pi$
$409$	6.54534	0.323646	0.161823	+	0.986820i	$0.448263\pi$
$410$	0	0
$411$	4.85811	0.239633
$412$	0	0
$413$	−0.679461	−0.0334341
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.10933	0.0543239
$418$	0	0
$419$	−34.4795	−1.68443	−0.842216	−	0.539141i	$-0.818749\pi$
$419$	−34.4795	−1.68443	−0.842216	+	0.539141i	$0.818749\pi$
$420$	0	0
$421$	6.54534	0.319000	0.159500	−	0.987198i	$-0.449012\pi$
$421$	6.54534	0.319000	0.159500	+	0.987198i	$0.449012\pi$
$422$	0	0
$423$	14.4429	0.702239
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.95401	0.142954
$428$	0	0
$429$	2.62305	0.126642
$430$	0	0
$431$	36.1711	1.74230	0.871151	−	0.491016i	$-0.163374\pi$
$431$	36.1711	1.74230	0.871151	+	0.491016i	$0.163374\pi$
$432$	0	0
$433$	38.1027	1.83110	0.915550	−	0.402204i	$-0.131756\pi$
$433$	38.1027	1.83110	0.915550	+	0.402204i	$0.131756\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−44.4858	−2.12805
$438$	0	0
$439$	12.1282	0.578848	0.289424	−	0.957201i	$-0.406536\pi$
$439$	12.1282	0.578848	0.289424	+	0.957201i	$0.406536\pi$
$440$	0	0
$441$	17.5455	0.835500
$442$	0	0
$443$	16.5483	0.786233	0.393117	−	0.919489i	$-0.371397\pi$
$443$	16.5483	0.786233	0.393117	+	0.919489i	$0.371397\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.45871	−0.116293
$448$	0	0
$449$	15.6056	0.736476	0.368238	−	0.929732i	$-0.379961\pi$
$449$	15.6056	0.736476	0.368238	+	0.929732i	$0.379961\pi$
$450$	0	0
$451$	10.0998	0.475580
$452$	0	0
$453$	9.92820	0.466468
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.87761	0.274943	0.137471	−	0.990506i	$-0.456102\pi$
$457$	5.87761	0.274943	0.137471	+	0.990506i	$0.456102\pi$
$458$	0	0
$459$	23.7987	1.11083
$460$	0	0
$461$	36.7495	1.71159	0.855797	−	0.517312i	$-0.173067\pi$
$461$	36.7495	1.71159	0.855797	+	0.517312i	$0.173067\pi$
$462$	0	0
$463$	19.3081	0.897324	0.448662	−	0.893702i	$-0.351901\pi$
$463$	19.3081	0.897324	0.448662	+	0.893702i	$0.351901\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.4129	0.528127	0.264064	−	0.964505i	$-0.414937\pi$
$467$	11.4129	0.528127	0.264064	+	0.964505i	$0.414937\pi$
$468$	0	0
$469$	−4.44556	−0.205277
$470$	0	0
$471$	7.59841	0.350116
$472$	0	0
$473$	4.78825	0.220164
$474$	0	0
$475$	0	0
$476$	0	0
$477$	25.7914	1.18091
$478$	0	0
$479$	29.2307	1.33559	0.667793	−	0.744347i	$-0.267239\pi$
$479$	29.2307	1.33559	0.667793	+	0.744347i	$0.267239\pi$
$480$	0	0
$481$	35.8564	1.63491
$482$	0	0
$483$	2.33086	0.106058
$484$	0	0
$485$	0	0
$486$	0	0
$487$	33.2932	1.50866	0.754329	−	0.656496i	$-0.227962\pi$
$487$	33.2932	1.50866	0.754329	+	0.656496i	$0.227962\pi$
$488$	0	0
$489$	4.47487	0.202361
$490$	0	0
$491$	−24.5151	−1.10635	−0.553176	−	0.833064i	$-0.686584\pi$
$491$	−24.5151	−1.10635	−0.553176	+	0.833064i	$0.686584\pi$
$492$	0	0
$493$	5.83612	0.262846
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.57510	−0.115509
$498$	0	0
$499$	−18.9093	−0.846497	−0.423249	−	0.906014i	$-0.639110\pi$
$499$	−18.9093	−0.846497	−0.423249	+	0.906014i	$0.639110\pi$
$500$	0	0
$501$	4.65872	0.208136
$502$	0	0
$503$	−9.42031	−0.420031	−0.210016	−	0.977698i	$-0.567351\pi$
$503$	−9.42031	−0.420031	−0.210016	+	0.977698i	$0.567351\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.96729	−0.0873702
$508$	0	0
$509$	−16.7804	−0.743778	−0.371889	−	0.928277i	$-0.621290\pi$
$509$	−16.7804	−0.743778	−0.371889	+	0.928277i	$0.621290\pi$
$510$	0	0
$511$	−1.31744	−0.0582799
$512$	0	0
$513$	19.0082	0.839234
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.61986	−0.247161
$518$	0	0
$519$	6.10614	0.268030
$520$	0	0
$521$	−37.8273	−1.65724	−0.828621	−	0.559810i	$-0.810874\pi$
$521$	−37.8273	−1.65724	−0.828621	+	0.559810i	$0.810874\pi$
$522$	0	0
$523$	10.0998	0.441632	0.220816	−	0.975315i	$-0.429128\pi$
$523$	10.0998	0.441632	0.220816	+	0.975315i	$0.429128\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−43.9044	−1.91250
$528$	0	0
$529$	50.0734	2.17710
$530$	0	0
$531$	4.19955	0.182245
$532$	0	0
$533$	−40.3991	−1.74988
$534$	0	0
$535$	0	0
$536$	0	0
$537$	17.4674	0.753774
$538$	0	0
$539$	−6.82711	−0.294064
$540$	0	0
$541$	−29.9269	−1.28666	−0.643329	−	0.765590i	$-0.722447\pi$
$541$	−29.9269	−1.28666	−0.643329	+	0.765590i	$0.722447\pi$
$542$	0	0
$543$	−0.662997	−0.0284519
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23.7630	1.01603	0.508016	−	0.861347i	$-0.330379\pi$
$547$	23.7630	1.01603	0.508016	+	0.861347i	$0.330379\pi$
$548$	0	0
$549$	−18.2578	−0.779226
$550$	0	0
$551$	4.66137	0.198581
$552$	0	0
$553$	−4.68537	−0.199242
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.3363	−1.03116	−0.515581	−	0.856841i	$-0.672424\pi$
$557$	−24.3363	−1.03116	−0.515581	+	0.856841i	$0.672424\pi$
$558$	0	0
$559$	−19.1530	−0.810086
$560$	0	0
$561$	−4.27267	−0.180392
$562$	0	0
$563$	−8.85368	−0.373138	−0.186569	−	0.982442i	$-0.559737\pi$
$563$	−8.85368	−0.373138	−0.186569	+	0.982442i	$0.559737\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.20989	0.0928066
$568$	0	0
$569$	26.3860	1.10616	0.553080	−	0.833128i	$-0.313452\pi$
$569$	26.3860	1.10616	0.553080	+	0.833128i	$0.313452\pi$
$570$	0	0
$571$	6.45280	0.270041	0.135021	−	0.990843i	$-0.456890\pi$
$571$	6.45280	0.270041	0.135021	+	0.990843i	$0.456890\pi$
$572$	0	0
$573$	−6.92843	−0.289439
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.1745	0.506832	0.253416	−	0.967357i	$-0.418446\pi$
$577$	12.1745	0.506832	0.253416	+	0.967357i	$0.418446\pi$
$578$	0	0
$579$	7.75576	0.322319
$580$	0	0
$581$	−6.75390	−0.280199
$582$	0	0
$583$	−10.0357	−0.415635
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.9521	−0.947336	−0.473668	−	0.880704i	$-0.657070\pi$
$587$	−22.9521	−0.947336	−0.473668	+	0.880704i	$0.657070\pi$
$588$	0	0
$589$	−35.0668	−1.44490
$590$	0	0
$591$	−0.860047	−0.0353776
$592$	0	0
$593$	10.4081	0.427410	0.213705	−	0.976898i	$-0.431447\pi$
$593$	10.4081	0.427410	0.213705	+	0.976898i	$0.431447\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.75576	0.317422
$598$	0	0
$599$	−8.98913	−0.367286	−0.183643	−	0.982993i	$-0.558789\pi$
$599$	−8.98913	−0.367286	−0.183643	+	0.982993i	$0.558789\pi$
$600$	0	0
$601$	−3.57650	−0.145889	−0.0729443	−	0.997336i	$-0.523240\pi$
$601$	−3.57650	−0.145889	−0.0729443	+	0.997336i	$0.523240\pi$
$602$	0	0
$603$	27.4767	1.11894
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.0235	1.09685	0.548424	−	0.836200i	$-0.315228\pi$
$607$	27.0235	1.09685	0.548424	+	0.836200i	$0.315228\pi$
$608$	0	0
$609$	−0.244235	−0.00989691
$610$	0	0
$611$	22.4795	0.909421
$612$	0	0
$613$	−44.5572	−1.79965	−0.899823	−	0.436254i	$-0.856305\pi$
$613$	−44.5572	−1.79965	−0.899823	+	0.436254i	$0.856305\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.2966	−0.615818	−0.307909	−	0.951416i	$-0.599629\pi$
$617$	−15.2966	−0.615818	−0.307909	+	0.951416i	$0.599629\pi$
$618$	0	0
$619$	14.5655	0.585436	0.292718	−	0.956199i	$-0.405440\pi$
$619$	14.5655	0.585436	0.292718	+	0.956199i	$0.405440\pi$
$620$	0	0
$621$	−31.2233	−1.25295
$622$	0	0
$623$	3.97926	0.159426
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.41262	−0.136287
$628$	0	0
$629$	−58.4063	−2.32881
$630$	0	0
$631$	−22.0779	−0.878906	−0.439453	−	0.898266i	$-0.644828\pi$
$631$	−22.0779	−0.878906	−0.439453	+	0.898266i	$0.644828\pi$
$632$	0	0
$633$	5.49033	0.218221
$634$	0	0
$635$	0	0
$636$	0	0
$637$	27.3084	1.08200
$638$	0	0
$639$	15.9159	0.629624
$640$	0	0
$641$	−31.2152	−1.23293	−0.616463	−	0.787384i	$-0.711435\pi$
$641$	−31.2152	−1.23293	−0.616463	+	0.787384i	$0.711435\pi$
$642$	0	0
$643$	−10.4498	−0.412102	−0.206051	−	0.978541i	$-0.566061\pi$
$643$	−10.4498	−0.412102	−0.206051	+	0.978541i	$0.566061\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.1307	1.02730	0.513652	−	0.857999i	$-0.328292\pi$
$647$	26.1307	1.02730	0.513652	+	0.857999i	$0.328292\pi$
$648$	0	0
$649$	−1.63408	−0.0641433
$650$	0	0
$651$	1.83735	0.0720114
$652$	0	0
$653$	44.2097	1.73006	0.865030	−	0.501719i	$-0.167299\pi$
$653$	44.2097	1.73006	0.865030	+	0.501719i	$0.167299\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.14268	0.317676
$658$	0	0
$659$	6.96706	0.271398	0.135699	−	0.990750i	$-0.456672\pi$
$659$	6.96706	0.271398	0.135699	+	0.990750i	$0.456672\pi$
$660$	0	0
$661$	−9.19753	−0.357743	−0.178871	−	0.983872i	$-0.557245\pi$
$661$	−9.19753	−0.357743	−0.178871	+	0.983872i	$0.557245\pi$
$662$	0	0
$663$	17.0907	0.663747
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.65686	−0.296475
$668$	0	0
$669$	−14.5226	−0.561475
$670$	0	0
$671$	7.10428	0.274258
$672$	0	0
$673$	−8.72415	−0.336291	−0.168146	−	0.985762i	$-0.553778\pi$
$673$	−8.72415	−0.336291	−0.168146	+	0.985762i	$0.553778\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.76618	0.336912	0.168456	−	0.985709i	$-0.446122\pi$
$677$	8.76618	0.336912	0.168456	+	0.985709i	$0.446122\pi$
$678$	0	0
$679$	−0.246015	−0.00944118
$680$	0	0
$681$	−14.3083	−0.548297
$682$	0	0
$683$	−49.2320	−1.88381	−0.941906	−	0.335877i	$-0.890967\pi$
$683$	−49.2320	−1.88381	−0.941906	+	0.335877i	$0.890967\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.83775	−0.0701146
$688$	0	0
$689$	40.1427	1.52931
$690$	0	0
$691$	−18.1483	−0.690395	−0.345198	−	0.938530i	$-0.612188\pi$
$691$	−18.1483	−0.690395	−0.345198	+	0.938530i	$0.612188\pi$
$692$	0	0
$693$	−1.06861	−0.0405932
$694$	0	0
$695$	0	0
$696$	0	0
$697$	65.8059	2.49258
$698$	0	0
$699$	5.40626	0.204483
$700$	0	0
$701$	22.6587	0.855808	0.427904	−	0.903824i	$-0.359252\pi$
$701$	22.6587	0.855808	0.427904	+	0.903824i	$0.359252\pi$
$702$	0	0
$703$	−46.6497	−1.75943
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.92229	−0.0722952
$708$	0	0
$709$	−36.7366	−1.37967	−0.689836	−	0.723966i	$-0.742317\pi$
$709$	−36.7366	−1.37967	−0.689836	+	0.723966i	$0.742317\pi$
$710$	0	0
$711$	28.9589	1.08604
$712$	0	0
$713$	57.6015	2.15719
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.38923	0.275956
$718$	0	0
$719$	8.74738	0.326222	0.163111	−	0.986608i	$-0.447847\pi$
$719$	8.74738	0.326222	0.163111	+	0.986608i	$0.447847\pi$
$720$	0	0
$721$	3.42745	0.127645
$722$	0	0
$723$	−14.6799	−0.545952
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.20887	−0.0819226	−0.0409613	−	0.999161i	$-0.513042\pi$
$727$	−2.20887	−0.0819226	−0.0409613	+	0.999161i	$0.513042\pi$
$728$	0	0
$729$	−6.47301	−0.239741
$730$	0	0
$731$	31.1982	1.15391
$732$	0	0
$733$	3.58552	0.132434	0.0662171	−	0.997805i	$-0.478907\pi$
$733$	3.58552	0.132434	0.0662171	+	0.997805i	$0.478907\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.6914	−0.393824
$738$	0	0
$739$	−24.9314	−0.917116	−0.458558	−	0.888665i	$-0.651634\pi$
$739$	−24.9314	−0.917116	−0.458558	+	0.888665i	$0.651634\pi$
$740$	0	0
$741$	13.6505	0.501463
$742$	0	0
$743$	21.4257	0.786032	0.393016	−	0.919532i	$-0.371432\pi$
$743$	21.4257	0.786032	0.393016	+	0.919532i	$0.371432\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	41.7439	1.52733
$748$	0	0
$749$	−7.32645	−0.267703
$750$	0	0
$751$	28.1582	1.02751	0.513754	−	0.857938i	$-0.328254\pi$
$751$	28.1582	1.02751	0.513754	+	0.857938i	$0.328254\pi$
$752$	0	0
$753$	−2.51986	−0.0918288
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.7513	−0.390761	−0.195381	−	0.980728i	$-0.562594\pi$
$757$	−10.7513	−0.390761	−0.195381	+	0.980728i	$0.562594\pi$
$758$	0	0
$759$	5.60565	0.203472
$760$	0	0
$761$	−38.0429	−1.37905	−0.689527	−	0.724260i	$-0.742182\pi$
$761$	−38.0429	−1.37905	−0.689527	+	0.724260i	$0.742182\pi$
$762$	0	0
$763$	1.52319	0.0551433
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.53632	0.236013
$768$	0	0
$769$	26.2280	0.945805	0.472903	−	0.881115i	$-0.343206\pi$
$769$	26.2280	0.945805	0.472903	+	0.881115i	$0.343206\pi$
$770$	0	0
$771$	6.96619	0.250881
$772$	0	0
$773$	−8.94499	−0.321729	−0.160864	−	0.986977i	$-0.551428\pi$
$773$	−8.94499	−0.321729	−0.160864	+	0.986977i	$0.551428\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.44424	0.0876867
$778$	0	0
$779$	52.5598	1.88315
$780$	0	0
$781$	−6.19302	−0.221604
$782$	0	0
$783$	3.27168	0.116920
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−29.9716	−1.06837	−0.534185	−	0.845367i	$-0.679382\pi$
$787$	−29.9716	−1.06837	−0.534185	+	0.845367i	$0.679382\pi$
$788$	0	0
$789$	3.79867	0.135236
$790$	0	0
$791$	4.77332	0.169720
$792$	0	0
$793$	−28.4171	−1.00912
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.80022	−0.240876	−0.120438	−	0.992721i	$-0.538430\pi$
$797$	−6.80022	−0.240876	−0.120438	+	0.992721i	$0.538430\pi$
$798$	0	0
$799$	−36.6167	−1.29541
$800$	0	0
$801$	−24.5946	−0.869008
$802$	0	0
$803$	−3.16839	−0.111810
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.38286	0.189486
$808$	0	0
$809$	53.1246	1.86776	0.933880	−	0.357586i	$-0.116400\pi$
$809$	53.1246	1.86776	0.933880	+	0.357586i	$0.116400\pi$
$810$	0	0
$811$	−30.0487	−1.05515	−0.527577	−	0.849507i	$-0.676899\pi$
$811$	−30.0487	−1.05515	−0.527577	+	0.849507i	$0.676899\pi$
$812$	0	0
$813$	−17.9096	−0.628116
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.9183	0.871782
$818$	0	0
$819$	4.27445	0.149361
$820$	0	0
$821$	19.9069	0.694756	0.347378	−	0.937725i	$-0.387072\pi$
$821$	19.9069	0.694756	0.347378	+	0.937725i	$0.387072\pi$
$822$	0	0
$823$	33.2491	1.15899	0.579495	−	0.814976i	$-0.303250\pi$
$823$	33.2491	1.15899	0.579495	+	0.814976i	$0.303250\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.3770	−0.986766	−0.493383	−	0.869812i	$-0.664240\pi$
$827$	−28.3770	−0.986766	−0.493383	+	0.869812i	$0.664240\pi$
$828$	0	0
$829$	3.63143	0.126125	0.0630625	−	0.998010i	$-0.479913\pi$
$829$	3.63143	0.126125	0.0630625	+	0.998010i	$0.479913\pi$
$830$	0	0
$831$	−9.49120	−0.329246
$832$	0	0
$833$	−44.4826	−1.54123
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−24.6124	−0.850729
$838$	0	0
$839$	−3.45660	−0.119335	−0.0596675	−	0.998218i	$-0.519004\pi$
$839$	−3.45660	−0.119335	−0.0596675	+	0.998218i	$0.519004\pi$
$840$	0	0
$841$	−28.1977	−0.972334
$842$	0	0
$843$	−10.6757	−0.367692
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.415806	0.0142873
$848$	0	0
$849$	6.89659	0.236690
$850$	0	0
$851$	76.6278	2.62677
$852$	0	0
$853$	47.4452	1.62449	0.812246	−	0.583315i	$-0.198245\pi$
$853$	47.4452	1.62449	0.812246	+	0.583315i	$0.198245\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.8271	−1.22383	−0.611915	−	0.790923i	$-0.709601\pi$
$857$	−35.8271	−1.22383	−0.611915	+	0.790923i	$0.709601\pi$
$858$	0	0
$859$	−39.0539	−1.33250	−0.666252	−	0.745727i	$-0.732102\pi$
$859$	−39.0539	−1.33250	−0.666252	+	0.745727i	$0.732102\pi$
$860$	0	0
$861$	−2.75390	−0.0938528
$862$	0	0
$863$	5.75072	0.195757	0.0978784	−	0.995198i	$-0.468794\pi$
$863$	5.75072	0.195757	0.0978784	+	0.995198i	$0.468794\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.6910	−0.566855
$868$	0	0
$869$	−11.2682	−0.382246
$870$	0	0
$871$	42.7657	1.44906
$872$	0	0
$873$	1.52054	0.0514626
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.32776	0.281208	0.140604	−	0.990066i	$-0.455095\pi$
$877$	8.32776	0.281208	0.140604	+	0.990066i	$0.455095\pi$
$878$	0	0
$879$	−6.14905	−0.207402
$880$	0	0
$881$	−49.0954	−1.65407	−0.827033	−	0.562153i	$-0.809973\pi$
$881$	−49.0954	−1.65407	−0.827033	+	0.562153i	$0.809973\pi$
$882$	0	0
$883$	−10.7526	−0.361853	−0.180927	−	0.983497i	$-0.557910\pi$
$883$	−10.7526	−0.361853	−0.180927	+	0.983497i	$0.557910\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−53.0591	−1.78155	−0.890776	−	0.454443i	$-0.849838\pi$
$887$	−53.0591	−1.78155	−0.890776	+	0.454443i	$0.849838\pi$
$888$	0	0
$889$	3.80045	0.127463
$890$	0	0
$891$	5.31471	0.178049
$892$	0	0
$893$	−29.2461	−0.978683
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−22.4226	−0.748668
$898$	0	0
$899$	−6.03567	−0.201301
$900$	0	0
$901$	−65.3882	−2.17840
$902$	0	0
$903$	−1.30561	−0.0434481
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.03885	0.233721	0.116861	−	0.993148i	$-0.462717\pi$
$907$	7.03885	0.233721	0.116861	+	0.993148i	$0.462717\pi$
$908$	0	0
$909$	11.8811	0.394072
$910$	0	0
$911$	32.0980	1.06345	0.531727	−	0.846916i	$-0.321543\pi$
$911$	32.0980	1.06345	0.531727	+	0.846916i	$0.321543\pi$
$912$	0	0
$913$	−16.2429	−0.537562
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.03171	−0.0340702
$918$	0	0
$919$	−1.05960	−0.0349529	−0.0174764	−	0.999847i	$-0.505563\pi$
$919$	−1.05960	−0.0349529	−0.0174764	+	0.999847i	$0.505563\pi$
$920$	0	0
$921$	1.37695	0.0453721
$922$	0	0
$923$	24.7721	0.815383
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−21.1841	−0.695777
$928$	0	0
$929$	−12.9981	−0.426455	−0.213228	−	0.977003i	$-0.568398\pi$
$929$	−12.9981	−0.426455	−0.213228	+	0.977003i	$0.568398\pi$
$930$	0	0
$931$	−35.5286	−1.16440
$932$	0	0
$933$	13.3595	0.437370
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39.9846	−1.30624	−0.653120	−	0.757254i	$-0.726540\pi$
$937$	−39.9846	−1.30624	−0.653120	+	0.757254i	$0.726540\pi$
$938$	0	0
$939$	−15.6706	−0.511391
$940$	0	0
$941$	34.0267	1.10924	0.554619	−	0.832105i	$-0.312864\pi$
$941$	34.0267	1.10924	0.554619	+	0.832105i	$0.312864\pi$
$942$	0	0
$943$	−86.3359	−2.81148
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.1506	−0.882276	−0.441138	−	0.897439i	$-0.645425\pi$
$947$	−27.1506	−0.882276	−0.441138	+	0.897439i	$0.645425\pi$
$948$	0	0
$949$	12.6735	0.411401
$950$	0	0
$951$	−3.84351	−0.124634
$952$	0	0
$953$	43.1169	1.39669	0.698346	−	0.715760i	$-0.253920\pi$
$953$	43.1169	1.39669	0.698346	+	0.715760i	$0.253920\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.587377	−0.0189872
$958$	0	0
$959$	−3.08044	−0.0994725
$960$	0	0
$961$	14.4055	0.464695
$962$	0	0
$963$	45.2826	1.45921
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.0641	0.387955	0.193978	−	0.981006i	$-0.437861\pi$
$967$	12.0641	0.387955	0.193978	+	0.981006i	$0.437861\pi$
$968$	0	0
$969$	−22.2352	−0.714298
$970$	0	0
$971$	−3.81156	−0.122319	−0.0611595	−	0.998128i	$-0.519480\pi$
$971$	−3.81156	−0.122319	−0.0611595	+	0.998128i	$0.519480\pi$
$972$	0	0
$973$	−0.703403	−0.0225501
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.4202	−1.26116	−0.630581	−	0.776123i	$-0.717184\pi$
$977$	−39.4202	−1.26116	−0.630581	+	0.776123i	$0.717184\pi$
$978$	0	0
$979$	9.56998	0.305858
$980$	0	0
$981$	−9.41440	−0.300579
$982$	0	0
$983$	8.67651	0.276738	0.138369	−	0.990381i	$-0.455814\pi$
$983$	8.67651	0.276738	0.138369	+	0.990381i	$0.455814\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.53237	0.0487758
$988$	0	0
$989$	−40.9314	−1.30154
$990$	0	0
$991$	35.7256	1.13486	0.567430	−	0.823422i	$-0.307938\pi$
$991$	35.7256	1.13486	0.567430	+	0.823422i	$0.307938\pi$
$992$	0	0
$993$	0.376725	0.0119550
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.3453	0.834365	0.417183	−	0.908823i	$-0.363018\pi$
$997$	26.3453	0.834365	0.417183	+	0.908823i	$0.363018\pi$
$998$	0	0
$999$	−32.7421	−1.03591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.2.a.x.1.2		4
4.3	odd	2		4400.2.a.ce.1.3		4
5.2	odd	4		440.2.b.d.89.6	yes	8
5.3	odd	4		440.2.b.d.89.3	✓	8
5.4	even	2		2200.2.a.y.1.3		4
15.2	even	4		3960.2.d.f.3169.7		8
15.8	even	4		3960.2.d.f.3169.8		8
20.3	even	4		880.2.b.j.529.6		8
20.7	even	4		880.2.b.j.529.3		8
20.19	odd	2		4400.2.a.cb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.3	✓	8	5.3	odd	4
440.2.b.d.89.6	yes	8	5.2	odd	4
880.2.b.j.529.3		8	20.7	even	4
880.2.b.j.529.6		8	20.3	even	4
2200.2.a.x.1.2		4	1.1	even	1	trivial
2200.2.a.y.1.3		4	5.4	even	2
3960.2.d.f.3169.7		8	15.2	even	4
3960.2.d.f.3169.8		8	15.8	even	4
4400.2.a.cb.1.2		4	20.19	odd	2
4400.2.a.ce.1.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.a (trivial)

\(f(q)\)	\(=\)	\(q-0.655762 q^{3} +0.415806 q^{7} -2.56998 q^{9} +1.00000 q^{11} -4.00000 q^{13} +6.51558 q^{17} +5.20406 q^{19} -0.272670 q^{21} -8.54830 q^{23} +3.65258 q^{27} +0.895717 q^{29} -6.73836 q^{31} -0.655762 q^{33} -8.96410 q^{37} +2.62305 q^{39} +10.0998 q^{41} +4.78825 q^{43} -5.61986 q^{47} -6.82711 q^{49} -4.27267 q^{51} -10.0357 q^{53} -3.41262 q^{57} -1.63408 q^{59} +7.10428 q^{61} -1.06861 q^{63} -10.6914 q^{67} +5.60565 q^{69} -6.19302 q^{71} -3.16839 q^{73} +0.415806 q^{77} -11.2682 q^{79} +5.31471 q^{81} -16.2429 q^{83} -0.587377 q^{87} +9.56998 q^{89} -1.66323 q^{91} +4.41876 q^{93} -0.591657 q^{97} -2.56998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - q^{7} + 7 q^{9} + 4 q^{11} - 16 q^{13} - 7 q^{17} - 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} - 15 q^{31} - q^{33} - 5 q^{37} + 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49}+ \cdots + 7 q^{99}+O(q^{100}) \) 4 * q - q^3 - q^7 + 7 * q^9 + 4 * q^11 - 16 * q^13 - 7 * q^17 - 9 * q^19 - 7 * q^21 - 6 * q^23 - 13 * q^27 + 3 * q^29 - 15 * q^31 - q^33 - 5 * q^37 + 4 * q^39 + 10 * q^41 - 8 * q^43 + 10 * q^47 + 9 * q^49 - 23 * q^51 - 5 * q^53 + 15 * q^57 + 6 * q^59 + 29 * q^61 - 40 * q^63 - 6 * q^67 - 30 * q^69 - q^71 - 18 * q^73 - q^77 - 20 * q^79 + 44 * q^81 - 26 * q^83 - 31 * q^87 + 21 * q^89 + 4 * q^91 - 25 * q^93 + 4 * q^97 + 7 * q^99

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