LMFDB - Embedded newform 3200.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{3} -0.828427 q^{7} +2.82843 q^{9} +O(q^{10})$	$q-2.41421 q^{3} -0.828427 q^{7} +2.82843 q^{9} +5.24264 q^{11} -5.65685 q^{13} +0.171573 q^{17} +1.58579 q^{19} +2.00000 q^{21} +4.82843 q^{23} +0.414214 q^{27} -8.00000 q^{29} -0.828427 q^{31} -12.6569 q^{33} -7.65685 q^{37} +13.6569 q^{39} +10.6569 q^{41} -10.0000 q^{43} +9.65685 q^{47} -6.31371 q^{49} -0.414214 q^{51} +7.65685 q^{53} -3.82843 q^{57} -3.65685 q^{59} -6.00000 q^{61} -2.34315 q^{63} +2.75736 q^{67} -11.6569 q^{69} -9.65685 q^{71} -5.82843 q^{73} -4.34315 q^{77} +12.8284 q^{79} -9.48528 q^{81} -1.24264 q^{83} +19.3137 q^{87} +6.17157 q^{89} +4.68629 q^{91} +2.00000 q^{93} +17.3137 q^{97} +14.8284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^7	$ 2 q - 2 q^{3} + 4 q^{7} + 2 q^{11} + 6 q^{17} + 6 q^{19} + 4 q^{21} + 4 q^{23} - 2 q^{27} - 16 q^{29} + 4 q^{31} - 14 q^{33} - 4 q^{37} + 16 q^{39} + 10 q^{41} - 20 q^{43} + 8 q^{47} + 10 q^{49} + 2 q^{51} + 4 q^{53} - 2 q^{57} + 4 q^{59} - 12 q^{61} - 16 q^{63} + 14 q^{67} - 12 q^{69} - 8 q^{71} - 6 q^{73} - 20 q^{77} + 20 q^{79} - 2 q^{81} + 6 q^{83} + 16 q^{87} + 18 q^{89} + 32 q^{91} + 4 q^{93} + 12 q^{97} + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^11 + 6 * q^17 + 6 * q^19 + 4 * q^21 + 4 * q^23 - 2 * q^27 - 16 * q^29 + 4 * q^31 - 14 * q^33 - 4 * q^37 + 16 * q^39 + 10 * q^41 - 20 * q^43 + 8 * q^47 + 10 * q^49 + 2 * q^51 + 4 * q^53 - 2 * q^57 + 4 * q^59 - 12 * q^61 - 16 * q^63 + 14 * q^67 - 12 * q^69 - 8 * q^71 - 6 * q^73 - 20 * q^77 + 20 * q^79 - 2 * q^81 + 6 * q^83 + 16 * q^87 + 18 * q^89 + 32 * q^91 + 4 * q^93 + 12 * q^97 + 24 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	5.24264	1.58072	0.790358	−	0.612646i	$-0.209895\pi$
$11$	5.24264	1.58072	0.790358	+	0.612646i	$0.209895\pi$
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.171573	0.0416125	0.0208063	−	0.999784i	$-0.493377\pi$
$17$	0.171573	0.0416125	0.0208063	+	0.999784i	$0.493377\pi$
$18$	0	0
$19$	1.58579	0.363804	0.181902	−	0.983317i	$-0.441775\pi$
$19$	1.58579	0.363804	0.181902	+	0.983317i	$0.441775\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−0.828427	−0.148790	−0.0743950	−	0.997229i	$-0.523703\pi$
$31$	−0.828427	−0.148790	−0.0743950	+	0.997229i	$0.523703\pi$
$32$	0	0
$33$	−12.6569	−2.20328
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	0	0
$39$	13.6569	2.18685
$40$	0	0
$41$	10.6569	1.66432	0.832161	−	0.554535i	$-0.187104\pi$
$41$	10.6569	1.66432	0.832161	+	0.554535i	$0.187104\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.65685	1.40860	0.704298	−	0.709904i	$-0.251262\pi$
$47$	9.65685	1.40860	0.704298	+	0.709904i	$0.251262\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	−0.414214	−0.0580015
$52$	0	0
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.82843	−0.507088
$58$	0	0
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−2.34315	−0.295209
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.75736	0.336865	0.168433	−	0.985713i	$-0.446129\pi$
$67$	2.75736	0.336865	0.168433	+	0.985713i	$0.446129\pi$
$68$	0	0
$69$	−11.6569	−1.40332
$70$	0	0
$71$	−9.65685	−1.14606	−0.573029	−	0.819535i	$-0.694232\pi$
$71$	−9.65685	−1.14606	−0.573029	+	0.819535i	$0.694232\pi$
$72$	0	0
$73$	−5.82843	−0.682166	−0.341083	−	0.940033i	$-0.610794\pi$
$73$	−5.82843	−0.682166	−0.341083	+	0.940033i	$0.610794\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.34315	−0.494947
$78$	0	0
$79$	12.8284	1.44331	0.721655	−	0.692252i	$-0.243382\pi$
$79$	12.8284	1.44331	0.721655	+	0.692252i	$0.243382\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−1.24264	−0.136398	−0.0681988	−	0.997672i	$-0.521725\pi$
$83$	−1.24264	−0.136398	−0.0681988	+	0.997672i	$0.521725\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.3137	2.07065
$88$	0	0
$89$	6.17157	0.654185	0.327093	−	0.944992i	$-0.393931\pi$
$89$	6.17157	0.654185	0.327093	+	0.944992i	$0.393931\pi$
$90$	0	0
$91$	4.68629	0.491257
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.3137	1.75794	0.878970	−	0.476876i	$-0.158231\pi$
$97$	17.3137	1.75794	0.878970	+	0.476876i	$0.158231\pi$
$98$	0	0
$99$	14.8284	1.49031
$100$	0	0
$101$	18.9706	1.88764	0.943821	−	0.330458i	$-0.107203\pi$
$101$	18.9706	1.88764	0.943821	+	0.330458i	$0.107203\pi$
$102$	0	0
$103$	−1.65685	−0.163255	−0.0816274	−	0.996663i	$-0.526012\pi$
$103$	−1.65685	−0.163255	−0.0816274	+	0.996663i	$0.526012\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.92893	0.186477	0.0932385	−	0.995644i	$-0.470278\pi$
$107$	1.92893	0.186477	0.0932385	+	0.995644i	$0.470278\pi$
$108$	0	0
$109$	−11.6569	−1.11652	−0.558262	−	0.829665i	$-0.688532\pi$
$109$	−11.6569	−1.11652	−0.558262	+	0.829665i	$0.688532\pi$
$110$	0	0
$111$	18.4853	1.75455
$112$	0	0
$113$	−3.34315	−0.314497	−0.157248	−	0.987559i	$-0.550262\pi$
$113$	−3.34315	−0.314497	−0.157248	+	0.987559i	$0.550262\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−16.0000	−1.47920
$118$	0	0
$119$	−0.142136	−0.0130296
$120$	0	0
$121$	16.4853	1.49866
$122$	0	0
$123$	−25.7279	−2.31981
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−20.1421	−1.78733	−0.893663	−	0.448739i	$-0.851873\pi$
$127$	−20.1421	−1.78733	−0.893663	+	0.448739i	$0.851873\pi$
$128$	0	0
$129$	24.1421	2.12560
$130$	0	0
$131$	5.31371	0.464261	0.232130	−	0.972685i	$-0.425430\pi$
$131$	5.31371	0.464261	0.232130	+	0.972685i	$0.425430\pi$
$132$	0	0
$133$	−1.31371	−0.113913
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	10.8995	0.924483	0.462242	−	0.886754i	$-0.347045\pi$
$139$	10.8995	0.924483	0.462242	+	0.886754i	$0.347045\pi$
$140$	0	0
$141$	−23.3137	−1.96337
$142$	0	0
$143$	−29.6569	−2.48003
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.2426	1.25719
$148$	0	0
$149$	3.31371	0.271470	0.135735	−	0.990745i	$-0.456660\pi$
$149$	3.31371	0.271470	0.135735	+	0.990745i	$0.456660\pi$
$150$	0	0
$151$	6.48528	0.527765	0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528	0.527765	0.263882	+	0.964555i	$0.414997\pi$
$152$	0	0
$153$	0.485281	0.0392327
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.6569	1.56879	0.784394	−	0.620263i	$-0.212974\pi$
$157$	19.6569	1.56879	0.784394	+	0.620263i	$0.212974\pi$
$158$	0	0
$159$	−18.4853	−1.46598
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	19.7279	1.54521	0.772605	−	0.634887i	$-0.218953\pi$
$163$	19.7279	1.54521	0.772605	+	0.634887i	$0.218953\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.34315	0.181318	0.0906590	−	0.995882i	$-0.471103\pi$
$167$	2.34315	0.181318	0.0906590	+	0.995882i	$0.471103\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	4.48528	0.342998
$172$	0	0
$173$	9.65685	0.734197	0.367099	−	0.930182i	$-0.380351\pi$
$173$	9.65685	0.734197	0.367099	+	0.930182i	$0.380351\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.82843	0.663585
$178$	0	0
$179$	21.5858	1.61340	0.806699	−	0.590963i	$-0.201252\pi$
$179$	21.5858	1.61340	0.806699	+	0.590963i	$0.201252\pi$
$180$	0	0
$181$	−15.6569	−1.16376	−0.581882	−	0.813273i	$-0.697684\pi$
$181$	−15.6569	−1.16376	−0.581882	+	0.813273i	$0.697684\pi$
$182$	0	0
$183$	14.4853	1.07078
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.899495	0.0657776
$188$	0	0
$189$	−0.343146	−0.0249602
$190$	0	0
$191$	1.51472	0.109601	0.0548006	−	0.998497i	$-0.482548\pi$
$191$	1.51472	0.109601	0.0548006	+	0.998497i	$0.482548\pi$
$192$	0	0
$193$	8.17157	0.588203	0.294101	−	0.955774i	$-0.404980\pi$
$193$	8.17157	0.588203	0.294101	+	0.955774i	$0.404980\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.65685	0.545528	0.272764	−	0.962081i	$-0.412062\pi$
$197$	7.65685	0.545528	0.272764	+	0.962081i	$0.412062\pi$
$198$	0	0
$199$	25.6569	1.81877	0.909383	−	0.415960i	$-0.136554\pi$
$199$	25.6569	1.81877	0.909383	+	0.415960i	$0.136554\pi$
$200$	0	0
$201$	−6.65685	−0.469538
$202$	0	0
$203$	6.62742	0.465153
$204$	0	0
$205$	0	0
$206$	0	0
$207$	13.6569	0.949217
$208$	0	0
$209$	8.31371	0.575071
$210$	0	0
$211$	−10.8995	−0.750352	−0.375176	−	0.926954i	$-0.622418\pi$
$211$	−10.8995	−0.750352	−0.375176	+	0.926954i	$0.622418\pi$
$212$	0	0
$213$	23.3137	1.59743
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.686292	0.0465885
$218$	0	0
$219$	14.0711	0.950835
$220$	0	0
$221$	−0.970563	−0.0652871
$222$	0	0
$223$	−13.6569	−0.914531	−0.457265	−	0.889330i	$-0.651171\pi$
$223$	−13.6569	−0.914531	−0.457265	+	0.889330i	$0.651171\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.3137	1.41464	0.707320	−	0.706893i	$-0.249904\pi$
$227$	21.3137	1.41464	0.707320	+	0.706893i	$0.249904\pi$
$228$	0	0
$229$	−0.686292	−0.0453514	−0.0226757	−	0.999743i	$-0.507219\pi$
$229$	−0.686292	−0.0453514	−0.0226757	+	0.999743i	$0.507219\pi$
$230$	0	0
$231$	10.4853	0.689881
$232$	0	0
$233$	13.3137	0.872210	0.436105	−	0.899896i	$-0.356358\pi$
$233$	13.3137	0.872210	0.436105	+	0.899896i	$0.356358\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−30.9706	−2.01175
$238$	0	0
$239$	18.6274	1.20491	0.602454	−	0.798154i	$-0.294190\pi$
$239$	18.6274	1.20491	0.602454	+	0.798154i	$0.294190\pi$
$240$	0	0
$241$	9.82843	0.633105	0.316552	−	0.948575i	$-0.397475\pi$
$241$	9.82843	0.633105	0.316552	+	0.948575i	$0.397475\pi$
$242$	0	0
$243$	21.6569	1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.97056	−0.570783
$248$	0	0
$249$	3.00000	0.190117
$250$	0	0
$251$	8.75736	0.552760	0.276380	−	0.961048i	$-0.410865\pi$
$251$	8.75736	0.552760	0.276380	+	0.961048i	$0.410865\pi$
$252$	0	0
$253$	25.3137	1.59146
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.31371	−0.331460	−0.165730	−	0.986171i	$-0.552998\pi$
$257$	−5.31371	−0.331460	−0.165730	+	0.986171i	$0.552998\pi$
$258$	0	0
$259$	6.34315	0.394144
$260$	0	0
$261$	−22.6274	−1.40060
$262$	0	0
$263$	−10.4853	−0.646550	−0.323275	−	0.946305i	$-0.604784\pi$
$263$	−10.4853	−0.646550	−0.323275	+	0.946305i	$0.604784\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.8995	−0.911834
$268$	0	0
$269$	6.34315	0.386748	0.193374	−	0.981125i	$-0.438057\pi$
$269$	6.34315	0.386748	0.193374	+	0.981125i	$0.438057\pi$
$270$	0	0
$271$	19.1716	1.16459	0.582295	−	0.812978i	$-0.302155\pi$
$271$	19.1716	1.16459	0.582295	+	0.812978i	$0.302155\pi$
$272$	0	0
$273$	−11.3137	−0.684737
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.65685	0.580224	0.290112	−	0.956993i	$-0.406307\pi$
$277$	9.65685	0.580224	0.290112	+	0.956993i	$0.406307\pi$
$278$	0	0
$279$	−2.34315	−0.140280
$280$	0	0
$281$	−20.6274	−1.23053	−0.615264	−	0.788321i	$-0.710951\pi$
$281$	−20.6274	−1.23053	−0.615264	+	0.788321i	$0.710951\pi$
$282$	0	0
$283$	9.72792	0.578265	0.289132	−	0.957289i	$-0.406633\pi$
$283$	9.72792	0.578265	0.289132	+	0.957289i	$0.406633\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.82843	−0.521126
$288$	0	0
$289$	−16.9706	−0.998268
$290$	0	0
$291$	−41.7990	−2.45030
$292$	0	0
$293$	17.3137	1.01148	0.505739	−	0.862687i	$-0.331220\pi$
$293$	17.3137	1.01148	0.505739	+	0.862687i	$0.331220\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.17157	0.126007
$298$	0	0
$299$	−27.3137	−1.57959
$300$	0	0
$301$	8.28427	0.477497
$302$	0	0
$303$	−45.7990	−2.63108
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.24264	0.185067	0.0925336	−	0.995710i	$-0.470503\pi$
$307$	3.24264	0.185067	0.0925336	+	0.995710i	$0.470503\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−22.4853	−1.27502	−0.637512	−	0.770441i	$-0.720036\pi$
$311$	−22.4853	−1.27502	−0.637512	+	0.770441i	$0.720036\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.34315	0.131604	0.0658021	−	0.997833i	$-0.479039\pi$
$317$	2.34315	0.131604	0.0658021	+	0.997833i	$0.479039\pi$
$318$	0	0
$319$	−41.9411	−2.34825
$320$	0	0
$321$	−4.65685	−0.259920
$322$	0	0
$323$	0.272078	0.0151388
$324$	0	0
$325$	0	0
$326$	0	0
$327$	28.1421	1.55626
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	22.8995	1.25867	0.629335	−	0.777134i	$-0.283327\pi$
$331$	22.8995	1.25867	0.629335	+	0.777134i	$0.283327\pi$
$332$	0	0
$333$	−21.6569	−1.18679
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.34315	−0.400007	−0.200003	−	0.979795i	$-0.564095\pi$
$337$	−7.34315	−0.400007	−0.200003	+	0.979795i	$0.564095\pi$
$338$	0	0
$339$	8.07107	0.438360
$340$	0	0
$341$	−4.34315	−0.235195
$342$	0	0
$343$	11.0294	0.595534
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.8995	1.01458	0.507289	−	0.861776i	$-0.330648\pi$
$347$	18.8995	1.01458	0.507289	+	0.861776i	$0.330648\pi$
$348$	0	0
$349$	−29.3137	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$349$	−29.3137	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$350$	0	0
$351$	−2.34315	−0.125068
$352$	0	0
$353$	12.6274	0.672090	0.336045	−	0.941846i	$-0.390911\pi$
$353$	12.6274	0.672090	0.336045	+	0.941846i	$0.390911\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.343146	0.0181612
$358$	0	0
$359$	8.82843	0.465947	0.232973	−	0.972483i	$-0.425155\pi$
$359$	8.82843	0.465947	0.232973	+	0.972483i	$0.425155\pi$
$360$	0	0
$361$	−16.4853	−0.867646
$362$	0	0
$363$	−39.7990	−2.08891
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.970563	0.0506630	0.0253315	−	0.999679i	$-0.491936\pi$
$367$	0.970563	0.0506630	0.0253315	+	0.999679i	$0.491936\pi$
$368$	0	0
$369$	30.1421	1.56914
$370$	0	0
$371$	−6.34315	−0.329320
$372$	0	0
$373$	−33.3137	−1.72492	−0.862459	−	0.506127i	$-0.831077\pi$
$373$	−33.3137	−1.72492	−0.862459	+	0.506127i	$0.831077\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.2548	2.33074
$378$	0	0
$379$	−16.8995	−0.868069	−0.434034	−	0.900896i	$-0.642910\pi$
$379$	−16.8995	−0.868069	−0.434034	+	0.900896i	$0.642910\pi$
$380$	0	0
$381$	48.6274	2.49126
$382$	0	0
$383$	27.4558	1.40293	0.701464	−	0.712705i	$-0.252530\pi$
$383$	27.4558	1.40293	0.701464	+	0.712705i	$0.252530\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.2843	−1.43777
$388$	0	0
$389$	−5.31371	−0.269416	−0.134708	−	0.990885i	$-0.543010\pi$
$389$	−5.31371	−0.269416	−0.134708	+	0.990885i	$0.543010\pi$
$390$	0	0
$391$	0.828427	0.0418954
$392$	0	0
$393$	−12.8284	−0.647109
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.68629	−0.435952	−0.217976	−	0.975954i	$-0.569946\pi$
$397$	−8.68629	−0.435952	−0.217976	+	0.975954i	$0.569946\pi$
$398$	0	0
$399$	3.17157	0.158777
$400$	0	0
$401$	20.4558	1.02152	0.510758	−	0.859724i	$-0.329365\pi$
$401$	20.4558	1.02152	0.510758	+	0.859724i	$0.329365\pi$
$402$	0	0
$403$	4.68629	0.233441
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−40.1421	−1.98977
$408$	0	0
$409$	5.97056	0.295225	0.147613	−	0.989045i	$-0.452841\pi$
$409$	5.97056	0.295225	0.147613	+	0.989045i	$0.452841\pi$
$410$	0	0
$411$	−12.0711	−0.595422
$412$	0	0
$413$	3.02944	0.149069
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−26.3137	−1.28859
$418$	0	0
$419$	6.55635	0.320299	0.160149	−	0.987093i	$-0.448802\pi$
$419$	6.55635	0.320299	0.160149	+	0.987093i	$0.448802\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	27.3137	1.32804
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.97056	0.240542
$428$	0	0
$429$	71.5980	3.45678
$430$	0	0
$431$	12.1421	0.584866	0.292433	−	0.956286i	$-0.405535\pi$
$431$	12.1421	0.584866	0.292433	+	0.956286i	$0.405535\pi$
$432$	0	0
$433$	21.8284	1.04901	0.524504	−	0.851408i	$-0.324251\pi$
$433$	21.8284	1.04901	0.524504	+	0.851408i	$0.324251\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.65685	0.366277
$438$	0	0
$439$	0.686292	0.0327549	0.0163775	−	0.999866i	$-0.494787\pi$
$439$	0.686292	0.0327549	0.0163775	+	0.999866i	$0.494787\pi$
$440$	0	0
$441$	−17.8579	−0.850374
$442$	0	0
$443$	−22.5563	−1.07168	−0.535842	−	0.844318i	$-0.680006\pi$
$443$	−22.5563	−1.07168	−0.535842	+	0.844318i	$0.680006\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.00000	−0.378387
$448$	0	0
$449$	−34.4558	−1.62607	−0.813036	−	0.582214i	$-0.802187\pi$
$449$	−34.4558	−1.62607	−0.813036	+	0.582214i	$0.802187\pi$
$450$	0	0
$451$	55.8701	2.63082
$452$	0	0
$453$	−15.6569	−0.735623
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.3431	−0.811278	−0.405639	−	0.914033i	$-0.632951\pi$
$457$	−17.3431	−0.811278	−0.405639	+	0.914033i	$0.632951\pi$
$458$	0	0
$459$	0.0710678	0.00331716
$460$	0	0
$461$	−5.65685	−0.263466	−0.131733	−	0.991285i	$-0.542054\pi$
$461$	−5.65685	−0.263466	−0.131733	+	0.991285i	$0.542054\pi$
$462$	0	0
$463$	12.9706	0.602793	0.301397	−	0.953499i	$-0.402547\pi$
$463$	12.9706	0.602793	0.301397	+	0.953499i	$0.402547\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.31371	−0.430987	−0.215494	−	0.976505i	$-0.569136\pi$
$467$	−9.31371	−0.430987	−0.215494	+	0.976505i	$0.569136\pi$
$468$	0	0
$469$	−2.28427	−0.105478
$470$	0	0
$471$	−47.4558	−2.18665
$472$	0	0
$473$	−52.4264	−2.41057
$474$	0	0
$475$	0	0
$476$	0	0
$477$	21.6569	0.991599
$478$	0	0
$479$	−35.1716	−1.60703	−0.803515	−	0.595284i	$-0.797039\pi$
$479$	−35.1716	−1.60703	−0.803515	+	0.595284i	$0.797039\pi$
$480$	0	0
$481$	43.3137	1.97494
$482$	0	0
$483$	9.65685	0.439402
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.1127	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$487$	25.1127	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$488$	0	0
$489$	−47.6274	−2.15379
$490$	0	0
$491$	−16.6274	−0.750385	−0.375192	−	0.926947i	$-0.622423\pi$
$491$	−16.6274	−0.750385	−0.375192	+	0.926947i	$0.622423\pi$
$492$	0	0
$493$	−1.37258	−0.0618180
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	−5.65685	−0.252730
$502$	0	0
$503$	16.9706	0.756680	0.378340	−	0.925667i	$-0.376495\pi$
$503$	16.9706	0.756680	0.378340	+	0.925667i	$0.376495\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−45.8701	−2.03716
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	4.82843	0.213597
$512$	0	0
$513$	0.656854	0.0290008
$514$	0	0
$515$	0	0
$516$	0	0
$517$	50.6274	2.22659
$518$	0	0
$519$	−23.3137	−1.02336
$520$	0	0
$521$	16.3137	0.714717	0.357358	−	0.933967i	$-0.383678\pi$
$521$	16.3137	0.714717	0.357358	+	0.933967i	$0.383678\pi$
$522$	0	0
$523$	19.5858	0.856427	0.428213	−	0.903678i	$-0.359143\pi$
$523$	19.5858	0.856427	0.428213	+	0.903678i	$0.359143\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.142136	−0.00619153
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	−10.3431	−0.448854
$532$	0	0
$533$	−60.2843	−2.61120
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−52.1127	−2.24883
$538$	0	0
$539$	−33.1005	−1.42574
$540$	0	0
$541$	17.6569	0.759127	0.379564	−	0.925166i	$-0.376074\pi$
$541$	17.6569	0.759127	0.379564	+	0.925166i	$0.376074\pi$
$542$	0	0
$543$	37.7990	1.62211
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.5563	0.622385	0.311192	−	0.950347i	$-0.399272\pi$
$547$	14.5563	0.622385	0.311192	+	0.950347i	$0.399272\pi$
$548$	0	0
$549$	−16.9706	−0.724286
$550$	0	0
$551$	−12.6863	−0.540454
$552$	0	0
$553$	−10.6274	−0.451924
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.3431	−1.11619	−0.558097	−	0.829775i	$-0.688468\pi$
$557$	−26.3431	−1.11619	−0.558097	+	0.829775i	$0.688468\pi$
$558$	0	0
$559$	56.5685	2.39259
$560$	0	0
$561$	−2.17157	−0.0916839
$562$	0	0
$563$	6.97056	0.293774	0.146887	−	0.989153i	$-0.453075\pi$
$563$	6.97056	0.293774	0.146887	+	0.989153i	$0.453075\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.85786	0.329999
$568$	0	0
$569$	−26.3137	−1.10313	−0.551564	−	0.834133i	$-0.685969\pi$
$569$	−26.3137	−1.10313	−0.551564	+	0.834133i	$0.685969\pi$
$570$	0	0
$571$	31.9411	1.33669	0.668347	−	0.743849i	$-0.267002\pi$
$571$	31.9411	1.33669	0.668347	+	0.743849i	$0.267002\pi$
$572$	0	0
$573$	−3.65685	−0.152767
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−41.2843	−1.71869	−0.859343	−	0.511399i	$-0.829127\pi$
$577$	−41.2843	−1.71869	−0.859343	+	0.511399i	$0.829127\pi$
$578$	0	0
$579$	−19.7279	−0.819864
$580$	0	0
$581$	1.02944	0.0427083
$582$	0	0
$583$	40.1421	1.66252
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.0710678	0.00293328	0.00146664	−	0.999999i	$-0.499533\pi$
$587$	0.0710678	0.00293328	0.00146664	+	0.999999i	$0.499533\pi$
$588$	0	0
$589$	−1.31371	−0.0541304
$590$	0	0
$591$	−18.4853	−0.760383
$592$	0	0
$593$	7.00000	0.287456	0.143728	−	0.989617i	$-0.454091\pi$
$593$	7.00000	0.287456	0.143728	+	0.989617i	$0.454091\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−61.9411	−2.53508
$598$	0	0
$599$	26.7696	1.09377	0.546887	−	0.837206i	$-0.315813\pi$
$599$	26.7696	1.09377	0.546887	+	0.837206i	$0.315813\pi$
$600$	0	0
$601$	35.1421	1.43348	0.716739	−	0.697342i	$-0.245634\pi$
$601$	35.1421	1.43348	0.716739	+	0.697342i	$0.245634\pi$
$602$	0	0
$603$	7.79899	0.317599
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	0	0
$609$	−16.0000	−0.648353
$610$	0	0
$611$	−54.6274	−2.20999
$612$	0	0
$613$	−6.68629	−0.270057	−0.135028	−	0.990842i	$-0.543113\pi$
$613$	−6.68629	−0.270057	−0.135028	+	0.990842i	$0.543113\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−13.3137	−0.535123	−0.267562	−	0.963541i	$-0.586218\pi$
$619$	−13.3137	−0.535123	−0.267562	+	0.963541i	$0.586218\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	−5.11270	−0.204836
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−20.0711	−0.801561
$628$	0	0
$629$	−1.31371	−0.0523810
$630$	0	0
$631$	15.8579	0.631292	0.315646	−	0.948877i	$-0.397779\pi$
$631$	15.8579	0.631292	0.315646	+	0.948877i	$0.397779\pi$
$632$	0	0
$633$	26.3137	1.04588
$634$	0	0
$635$	0	0
$636$	0	0
$637$	35.7157	1.41511
$638$	0	0
$639$	−27.3137	−1.08051
$640$	0	0
$641$	−45.3137	−1.78978	−0.894892	−	0.446283i	$-0.852748\pi$
$641$	−45.3137	−1.78978	−0.894892	+	0.446283i	$0.852748\pi$
$642$	0	0
$643$	14.6863	0.579171	0.289585	−	0.957152i	$-0.406483\pi$
$643$	14.6863	0.579171	0.289585	+	0.957152i	$0.406483\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.65685	0.222394	0.111197	−	0.993798i	$-0.464532\pi$
$647$	5.65685	0.222394	0.111197	+	0.993798i	$0.464532\pi$
$648$	0	0
$649$	−19.1716	−0.752550
$650$	0	0
$651$	−1.65685	−0.0649372
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−16.4853	−0.643152
$658$	0	0
$659$	−51.1838	−1.99384	−0.996918	−	0.0784478i	$-0.975004\pi$
$659$	−51.1838	−1.99384	−0.996918	+	0.0784478i	$0.975004\pi$
$660$	0	0
$661$	−6.34315	−0.246720	−0.123360	−	0.992362i	$-0.539367\pi$
$661$	−6.34315	−0.246720	−0.123360	+	0.992362i	$0.539367\pi$
$662$	0	0
$663$	2.34315	0.0910002
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−38.6274	−1.49566
$668$	0	0
$669$	32.9706	1.27472
$670$	0	0
$671$	−31.4558	−1.21434
$672$	0	0
$673$	25.3137	0.975772	0.487886	−	0.872907i	$-0.337768\pi$
$673$	25.3137	0.975772	0.487886	+	0.872907i	$0.337768\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.6863	1.10250	0.551252	−	0.834339i	$-0.314150\pi$
$677$	28.6863	1.10250	0.551252	+	0.834339i	$0.314150\pi$
$678$	0	0
$679$	−14.3431	−0.550439
$680$	0	0
$681$	−51.4558	−1.97179
$682$	0	0
$683$	−16.6985	−0.638950	−0.319475	−	0.947595i	$-0.603507\pi$
$683$	−16.6985	−0.638950	−0.319475	+	0.947595i	$0.603507\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.65685	0.0632129
$688$	0	0
$689$	−43.3137	−1.65012
$690$	0	0
$691$	36.2132	1.37762	0.688808	−	0.724944i	$-0.258134\pi$
$691$	36.2132	1.37762	0.688808	+	0.724944i	$0.258134\pi$
$692$	0	0
$693$	−12.2843	−0.466641
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.82843	0.0692566
$698$	0	0
$699$	−32.1421	−1.21573
$700$	0	0
$701$	47.3137	1.78701	0.893507	−	0.449049i	$-0.148237\pi$
$701$	47.3137	1.78701	0.893507	+	0.449049i	$0.148237\pi$
$702$	0	0
$703$	−12.1421	−0.457949
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.7157	−0.591051
$708$	0	0
$709$	21.9411	0.824016	0.412008	−	0.911180i	$-0.364828\pi$
$709$	21.9411	0.824016	0.412008	+	0.911180i	$0.364828\pi$
$710$	0	0
$711$	36.2843	1.36077
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−44.9706	−1.67946
$718$	0	0
$719$	−23.4558	−0.874755	−0.437378	−	0.899278i	$-0.644093\pi$
$719$	−23.4558	−0.874755	−0.437378	+	0.899278i	$0.644093\pi$
$720$	0	0
$721$	1.37258	0.0511177
$722$	0	0
$723$	−23.7279	−0.882451
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−1.71573	−0.0634585
$732$	0	0
$733$	−26.3431	−0.973006	−0.486503	−	0.873679i	$-0.661728\pi$
$733$	−26.3431	−0.973006	−0.486503	+	0.873679i	$0.661728\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.4558	0.532488
$738$	0	0
$739$	−28.6274	−1.05308	−0.526538	−	0.850151i	$-0.676510\pi$
$739$	−28.6274	−1.05308	−0.526538	+	0.850151i	$0.676510\pi$
$740$	0	0
$741$	21.6569	0.795584
$742$	0	0
$743$	−25.7990	−0.946473	−0.473237	−	0.880935i	$-0.656914\pi$
$743$	−25.7990	−0.946473	−0.473237	+	0.880935i	$0.656914\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.51472	−0.128597
$748$	0	0
$749$	−1.59798	−0.0583889
$750$	0	0
$751$	−34.6274	−1.26357	−0.631786	−	0.775143i	$-0.717678\pi$
$751$	−34.6274	−1.26357	−0.631786	+	0.775143i	$0.717678\pi$
$752$	0	0
$753$	−21.1421	−0.770462
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.6569	0.932514	0.466257	−	0.884649i	$-0.345602\pi$
$757$	25.6569	0.932514	0.466257	+	0.884649i	$0.345602\pi$
$758$	0	0
$759$	−61.1127	−2.21825
$760$	0	0
$761$	−39.2843	−1.42405	−0.712027	−	0.702152i	$-0.752223\pi$
$761$	−39.2843	−1.42405	−0.712027	+	0.702152i	$0.752223\pi$
$762$	0	0
$763$	9.65685	0.349602
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.6863	0.746939
$768$	0	0
$769$	34.1127	1.23014	0.615068	−	0.788474i	$-0.289129\pi$
$769$	34.1127	1.23014	0.615068	+	0.788474i	$0.289129\pi$
$770$	0	0
$771$	12.8284	0.462005
$772$	0	0
$773$	19.3137	0.694666	0.347333	−	0.937742i	$-0.387087\pi$
$773$	19.3137	0.694666	0.347333	+	0.937742i	$0.387087\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.3137	−0.549376
$778$	0	0
$779$	16.8995	0.605487
$780$	0	0
$781$	−50.6274	−1.81159
$782$	0	0
$783$	−3.31371	−0.118422
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16.6274	0.592703	0.296352	−	0.955079i	$-0.404230\pi$
$787$	16.6274	0.592703	0.296352	+	0.955079i	$0.404230\pi$
$788$	0	0
$789$	25.3137	0.901192
$790$	0	0
$791$	2.76955	0.0984740
$792$	0	0
$793$	33.9411	1.20528
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	1.65685	0.0586153
$800$	0	0
$801$	17.4558	0.616772
$802$	0	0
$803$	−30.5563	−1.07831
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15.3137	−0.539068
$808$	0	0
$809$	40.6274	1.42838	0.714192	−	0.699950i	$-0.246794\pi$
$809$	40.6274	1.42838	0.714192	+	0.699950i	$0.246794\pi$
$810$	0	0
$811$	−35.9411	−1.26206	−0.631032	−	0.775757i	$-0.717368\pi$
$811$	−35.9411	−1.26206	−0.631032	+	0.775757i	$0.717368\pi$
$812$	0	0
$813$	−46.2843	−1.62326
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−15.8579	−0.554796
$818$	0	0
$819$	13.2548	0.463161
$820$	0	0
$821$	40.6274	1.41791	0.708953	−	0.705255i	$-0.249168\pi$
$821$	40.6274	1.41791	0.708953	+	0.705255i	$0.249168\pi$
$822$	0	0
$823$	12.9706	0.452125	0.226063	−	0.974113i	$-0.427415\pi$
$823$	12.9706	0.452125	0.226063	+	0.974113i	$0.427415\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.0711	1.67159	0.835797	−	0.549038i	$-0.185006\pi$
$827$	48.0711	1.67159	0.835797	+	0.549038i	$0.185006\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	−23.3137	−0.808744
$832$	0	0
$833$	−1.08326	−0.0375328
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.343146	−0.0118609
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	49.7990	1.71517
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.6569	−0.469255
$848$	0	0
$849$	−23.4853	−0.806013
$850$	0	0
$851$	−36.9706	−1.26733
$852$	0	0
$853$	16.6274	0.569312	0.284656	−	0.958630i	$-0.408121\pi$
$853$	16.6274	0.569312	0.284656	+	0.958630i	$0.408121\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.0000	−0.375753	−0.187876	−	0.982193i	$-0.560160\pi$
$857$	−11.0000	−0.375753	−0.187876	+	0.982193i	$0.560160\pi$
$858$	0	0
$859$	54.2132	1.84973	0.924865	−	0.380295i	$-0.124177\pi$
$859$	54.2132	1.84973	0.924865	+	0.380295i	$0.124177\pi$
$860$	0	0
$861$	21.3137	0.726369
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	40.9706	1.39143
$868$	0	0
$869$	67.2548	2.28146
$870$	0	0
$871$	−15.5980	−0.528517
$872$	0	0
$873$	48.9706	1.65740
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.3431	−0.889545	−0.444772	−	0.895644i	$-0.646715\pi$
$877$	−26.3431	−0.889545	−0.444772	+	0.895644i	$0.646715\pi$
$878$	0	0
$879$	−41.7990	−1.40984
$880$	0	0
$881$	−14.6863	−0.494794	−0.247397	−	0.968914i	$-0.579575\pi$
$881$	−14.6863	−0.494794	−0.247397	+	0.968914i	$0.579575\pi$
$882$	0	0
$883$	43.2426	1.45523	0.727615	−	0.685985i	$-0.240629\pi$
$883$	43.2426	1.45523	0.727615	+	0.685985i	$0.240629\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.3431	−0.750209	−0.375105	−	0.926982i	$-0.622393\pi$
$887$	−22.3431	−0.750209	−0.375105	+	0.926982i	$0.622393\pi$
$888$	0	0
$889$	16.6863	0.559640
$890$	0	0
$891$	−49.7279	−1.66595
$892$	0	0
$893$	15.3137	0.512454
$894$	0	0
$895$	0	0
$896$	0	0
$897$	65.9411	2.20171
$898$	0	0
$899$	6.62742	0.221037
$900$	0	0
$901$	1.31371	0.0437660
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−45.3137	−1.50462	−0.752308	−	0.658811i	$-0.771060\pi$
$907$	−45.3137	−1.50462	−0.752308	+	0.658811i	$0.771060\pi$
$908$	0	0
$909$	53.6569	1.77969
$910$	0	0
$911$	36.2843	1.20215	0.601076	−	0.799192i	$-0.294739\pi$
$911$	36.2843	1.20215	0.601076	+	0.799192i	$0.294739\pi$
$912$	0	0
$913$	−6.51472	−0.215606
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.40202	−0.145368
$918$	0	0
$919$	−39.1716	−1.29215	−0.646075	−	0.763274i	$-0.723591\pi$
$919$	−39.1716	−1.29215	−0.646075	+	0.763274i	$0.723591\pi$
$920$	0	0
$921$	−7.82843	−0.257955
$922$	0	0
$923$	54.6274	1.79808
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.68629	−0.153918
$928$	0	0
$929$	0.627417	0.0205849	0.0102924	−	0.999947i	$-0.496724\pi$
$929$	0.627417	0.0205849	0.0102924	+	0.999947i	$0.496724\pi$
$930$	0	0
$931$	−10.0122	−0.328136
$932$	0	0
$933$	54.2843	1.77719
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.1127	0.657053	0.328527	−	0.944495i	$-0.393448\pi$
$937$	20.1127	0.657053	0.328527	+	0.944495i	$0.393448\pi$
$938$	0	0
$939$	−24.1421	−0.787849
$940$	0	0
$941$	−40.9706	−1.33560	−0.667801	−	0.744340i	$-0.732764\pi$
$941$	−40.9706	−1.33560	−0.667801	+	0.744340i	$0.732764\pi$
$942$	0	0
$943$	51.4558	1.67563
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.5980	−1.09179	−0.545894	−	0.837854i	$-0.683810\pi$
$947$	−33.5980	−1.09179	−0.545894	+	0.837854i	$0.683810\pi$
$948$	0	0
$949$	32.9706	1.07027
$950$	0	0
$951$	−5.65685	−0.183436
$952$	0	0
$953$	12.3137	0.398880	0.199440	−	0.979910i	$-0.436088\pi$
$953$	12.3137	0.398880	0.199440	+	0.979910i	$0.436088\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	101.255	3.27310
$958$	0	0
$959$	−4.14214	−0.133757
$960$	0	0
$961$	−30.3137	−0.977862
$962$	0	0
$963$	5.45584	0.175812
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−42.6274	−1.37081	−0.685403	−	0.728164i	$-0.740374\pi$
$967$	−42.6274	−1.37081	−0.685403	+	0.728164i	$0.740374\pi$
$968$	0	0
$969$	−0.656854	−0.0211012
$970$	0	0
$971$	42.8995	1.37671	0.688355	−	0.725374i	$-0.258333\pi$
$971$	42.8995	1.37671	0.688355	+	0.725374i	$0.258333\pi$
$972$	0	0
$973$	−9.02944	−0.289470
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.1127	−0.771434	−0.385717	−	0.922617i	$-0.626046\pi$
$977$	−24.1127	−0.771434	−0.385717	+	0.922617i	$0.626046\pi$
$978$	0	0
$979$	32.3553	1.03408
$980$	0	0
$981$	−32.9706	−1.05267
$982$	0	0
$983$	−33.7990	−1.07802	−0.539010	−	0.842299i	$-0.681202\pi$
$983$	−33.7990	−1.07802	−0.539010	+	0.842299i	$0.681202\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	19.3137	0.614762
$988$	0	0
$989$	−48.2843	−1.53535
$990$	0	0
$991$	62.0833	1.97214	0.986070	−	0.166331i	$-0.0531922\pi$
$991$	62.0833	1.97214	0.986070	+	0.166331i	$0.0531922\pi$
$992$	0	0
$993$	−55.2843	−1.75439
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.9706	0.537463	0.268732	−	0.963215i	$-0.413396\pi$
$997$	16.9706	0.537463	0.268732	+	0.963215i	$0.413396\pi$
$998$	0	0
$999$	−3.17157	−0.100344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.a.bh.1.1	yes	2
4.3	odd	2		3200.2.a.bi.1.2	yes	2
5.2	odd	4		3200.2.c.bb.2049.4		4
5.3	odd	4		3200.2.c.bb.2049.1		4
5.4	even	2		3200.2.a.bj.1.2	yes	2
8.3	odd	2		3200.2.a.bd.1.1	yes	2
8.5	even	2		3200.2.a.bm.1.2	yes	2
20.3	even	4		3200.2.c.z.2049.4		4
20.7	even	4		3200.2.c.z.2049.1		4
20.19	odd	2		3200.2.a.bg.1.1	yes	2
40.3	even	4		3200.2.c.ba.2049.1		4
40.13	odd	4		3200.2.c.y.2049.4		4
40.19	odd	2		3200.2.a.bn.1.2	yes	2
40.27	even	4		3200.2.c.ba.2049.4		4
40.29	even	2		3200.2.a.bc.1.1	✓	2
40.37	odd	4		3200.2.c.y.2049.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.a.bc.1.1	✓	2	40.29	even	2
3200.2.a.bd.1.1	yes	2	8.3	odd	2
3200.2.a.bg.1.1	yes	2	20.19	odd	2
3200.2.a.bh.1.1	yes	2	1.1	even	1	trivial
3200.2.a.bi.1.2	yes	2	4.3	odd	2
3200.2.a.bj.1.2	yes	2	5.4	even	2
3200.2.a.bm.1.2	yes	2	8.5	even	2
3200.2.a.bn.1.2	yes	2	40.19	odd	2
3200.2.c.y.2049.1		4	40.37	odd	4
3200.2.c.y.2049.4		4	40.13	odd	4
3200.2.c.z.2049.1		4	20.7	even	4
3200.2.c.z.2049.4		4	20.3	even	4
3200.2.c.ba.2049.1		4	40.3	even	4
3200.2.c.ba.2049.4		4	40.27	even	4
3200.2.c.bb.2049.1		4	5.3	odd	4
3200.2.c.bb.2049.4		4	5.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} -0.828427 q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q-2.41421 q^{3} -0.828427 q^{7} +2.82843 q^{9} +5.24264 q^{11} -5.65685 q^{13} +0.171573 q^{17} +1.58579 q^{19} +2.00000 q^{21} +4.82843 q^{23} +0.414214 q^{27} -8.00000 q^{29} -0.828427 q^{31} -12.6569 q^{33} -7.65685 q^{37} +13.6569 q^{39} +10.6569 q^{41} -10.0000 q^{43} +9.65685 q^{47} -6.31371 q^{49} -0.414214 q^{51} +7.65685 q^{53} -3.82843 q^{57} -3.65685 q^{59} -6.00000 q^{61} -2.34315 q^{63} +2.75736 q^{67} -11.6569 q^{69} -9.65685 q^{71} -5.82843 q^{73} -4.34315 q^{77} +12.8284 q^{79} -9.48528 q^{81} -1.24264 q^{83} +19.3137 q^{87} +6.17157 q^{89} +4.68629 q^{91} +2.00000 q^{93} +17.3137 q^{97} +14.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^7	\( 2 q - 2 q^{3} + 4 q^{7} + 2 q^{11} + 6 q^{17} + 6 q^{19} + 4 q^{21} + 4 q^{23} - 2 q^{27} - 16 q^{29} + 4 q^{31} - 14 q^{33} - 4 q^{37} + 16 q^{39} + 10 q^{41} - 20 q^{43} + 8 q^{47} + 10 q^{49} + 2 q^{51} + 4 q^{53} - 2 q^{57} + 4 q^{59} - 12 q^{61} - 16 q^{63} + 14 q^{67} - 12 q^{69} - 8 q^{71} - 6 q^{73} - 20 q^{77} + 20 q^{79} - 2 q^{81} + 6 q^{83} + 16 q^{87} + 18 q^{89} + 32 q^{91} + 4 q^{93} + 12 q^{97} + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^7 + 2 * q^11 + 6 * q^17 + 6 * q^19 + 4 * q^21 + 4 * q^23 - 2 * q^27 - 16 * q^29 + 4 * q^31 - 14 * q^33 - 4 * q^37 + 16 * q^39 + 10 * q^41 - 20 * q^43 + 8 * q^47 + 10 * q^49 + 2 * q^51 + 4 * q^53 - 2 * q^57 + 4 * q^59 - 12 * q^61 - 16 * q^63 + 14 * q^67 - 12 * q^69 - 8 * q^71 - 6 * q^73 - 20 * q^77 + 20 * q^79 - 2 * q^81 + 6 * q^83 + 16 * q^87 + 18 * q^89 + 32 * q^91 + 4 * q^93 + 12 * q^97 + 24 * q^99

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