LMFDB - Embedded newform 3192.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.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:	$25.4882483252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} +3.46410 q^{15} -3.46410 q^{17} -1.00000 q^{19} -1.00000 q^{21} +4.00000 q^{23} +7.00000 q^{25} +1.00000 q^{27} -0.535898 q^{29} +10.9282 q^{31} -3.46410 q^{35} -2.00000 q^{37} -2.00000 q^{39} +6.00000 q^{41} +6.92820 q^{43} +3.46410 q^{45} +9.46410 q^{47} +1.00000 q^{49} -3.46410 q^{51} -3.46410 q^{53} -1.00000 q^{57} +4.00000 q^{59} +6.00000 q^{61} -1.00000 q^{63} -6.92820 q^{65} +14.9282 q^{67} +4.00000 q^{69} +13.4641 q^{71} -0.928203 q^{73} +7.00000 q^{75} +1.00000 q^{81} -5.46410 q^{83} -12.0000 q^{85} -0.535898 q^{87} -15.8564 q^{89} +2.00000 q^{91} +10.9282 q^{93} -3.46410 q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 14 q^{25} + 2 q^{27} - 8 q^{29} + 8 q^{31} - 4 q^{37} - 4 q^{39} + 12 q^{41} + 12 q^{47} + 2 q^{49} - 2 q^{57} + 8 q^{59} + 12 q^{61} - 2 q^{63} + 16 q^{67} + 8 q^{69} + 20 q^{71} + 12 q^{73} + 14 q^{75} + 2 q^{81} - 4 q^{83} - 24 q^{85} - 8 q^{87} - 4 q^{89} + 4 q^{91} + 8 q^{93} + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 14 * q^25 + 2 * q^27 - 8 * q^29 + 8 * q^31 - 4 * q^37 - 4 * q^39 + 12 * q^41 + 12 * q^47 + 2 * q^49 - 2 * q^57 + 8 * q^59 + 12 * q^61 - 2 * q^63 + 16 * q^67 + 8 * q^69 + 20 * q^71 + 12 * q^73 + 14 * q^75 + 2 * q^81 - 4 * q^83 - 24 * q^85 - 8 * q^87 - 4 * q^89 + 4 * q^91 + 8 * q^93 + 20 * q^97

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.535898	−0.0995138	−0.0497569	−	0.998761i	$-0.515845\pi$
$29$	−0.535898	−0.0995138	−0.0497569	+	0.998761i	$0.515845\pi$
$30$	0	0
$31$	10.9282	1.96276	0.981382	−	0.192068i	$-0.0615194\pi$
$31$	10.9282	1.96276	0.981382	+	0.192068i	$0.0615194\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.92820	1.05654	0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	0	0
$45$	3.46410	0.516398
$46$	0	0
$47$	9.46410	1.38048	0.690241	−	0.723580i	$-0.257505\pi$
$47$	9.46410	1.38048	0.690241	+	0.723580i	$0.257505\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.46410	−0.485071
$52$	0	0
$53$	−3.46410	−0.475831	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−3.46410	−0.475831	−0.237915	+	0.971286i	$0.576464\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.92820	−0.859338
$66$	0	0
$67$	14.9282	1.82377	0.911885	−	0.410445i	$-0.134627\pi$
$67$	14.9282	1.82377	0.911885	+	0.410445i	$0.134627\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	13.4641	1.59789	0.798947	−	0.601401i	$-0.205391\pi$
$71$	13.4641	1.59789	0.798947	+	0.601401i	$0.205391\pi$
$72$	0	0
$73$	−0.928203	−0.108638	−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203	−0.108638	−0.0543190	+	0.998524i	$0.517299\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.46410	−0.599763	−0.299882	−	0.953976i	$-0.596947\pi$
$83$	−5.46410	−0.599763	−0.299882	+	0.953976i	$0.596947\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	−0.535898	−0.0574543
$88$	0	0
$89$	−15.8564	−1.68078	−0.840388	−	0.541985i	$-0.817673\pi$
$89$	−15.8564	−1.68078	−0.840388	+	0.541985i	$0.817673\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	10.9282	1.13320
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−3.46410	−0.338062
$106$	0	0
$107$	−12.3923	−1.19801	−0.599005	−	0.800746i	$-0.704437\pi$
$107$	−12.3923	−1.19801	−0.599005	+	0.800746i	$0.704437\pi$
$108$	0	0
$109$	−12.9282	−1.23830	−0.619149	−	0.785274i	$-0.712522\pi$
$109$	−12.9282	−1.23830	−0.619149	+	0.785274i	$0.712522\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	11.4641	1.07845	0.539226	−	0.842161i	$-0.318717\pi$
$113$	11.4641	1.07845	0.539226	+	0.842161i	$0.318717\pi$
$114$	0	0
$115$	13.8564	1.29212
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	3.46410	0.317554
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−18.9282	−1.67961	−0.839803	−	0.542891i	$-0.817330\pi$
$127$	−18.9282	−1.67961	−0.839803	+	0.542891i	$0.817330\pi$
$128$	0	0
$129$	6.92820	0.609994
$130$	0	0
$131$	−8.39230	−0.733239	−0.366620	−	0.930371i	$-0.619485\pi$
$131$	−8.39230	−0.733239	−0.366620	+	0.930371i	$0.619485\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	9.46410	0.797021
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.85641	−0.154166
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−4.92820	−0.403734	−0.201867	−	0.979413i	$-0.564701\pi$
$149$	−4.92820	−0.403734	−0.201867	+	0.979413i	$0.564701\pi$
$150$	0	0
$151$	−2.92820	−0.238294	−0.119147	−	0.992877i	$-0.538016\pi$
$151$	−2.92820	−0.238294	−0.119147	+	0.992877i	$0.538016\pi$
$152$	0	0
$153$	−3.46410	−0.280056
$154$	0	0
$155$	37.8564	3.04070
$156$	0	0
$157$	11.8564	0.946244	0.473122	−	0.880997i	$-0.343127\pi$
$157$	11.8564	0.946244	0.473122	+	0.880997i	$0.343127\pi$
$158$	0	0
$159$	−3.46410	−0.274721
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−14.9282	−1.16927	−0.584634	−	0.811297i	$-0.698762\pi$
$163$	−14.9282	−1.16927	−0.584634	+	0.811297i	$0.698762\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.92820	−0.226591	−0.113296	−	0.993561i	$-0.536141\pi$
$167$	−2.92820	−0.226591	−0.113296	+	0.993561i	$0.536141\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−7.00000	−0.529150
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	6.53590	0.488516	0.244258	−	0.969710i	$-0.421456\pi$
$179$	6.53590	0.488516	0.244258	+	0.969710i	$0.421456\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−6.92820	−0.509372
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−17.8564	−1.29204	−0.646022	−	0.763319i	$-0.723569\pi$
$191$	−17.8564	−1.29204	−0.646022	+	0.763319i	$0.723569\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−6.92820	−0.496139
$196$	0	0
$197$	19.8564	1.41471	0.707355	−	0.706858i	$-0.249888\pi$
$197$	19.8564	1.41471	0.707355	+	0.706858i	$0.249888\pi$
$198$	0	0
$199$	−5.07180	−0.359530	−0.179765	−	0.983710i	$-0.557534\pi$
$199$	−5.07180	−0.359530	−0.179765	+	0.983710i	$0.557534\pi$
$200$	0	0
$201$	14.9282	1.05295
$202$	0	0
$203$	0.535898	0.0376127
$204$	0	0
$205$	20.7846	1.45166
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	25.8564	1.78003	0.890014	−	0.455933i	$-0.150694\pi$
$211$	25.8564	1.78003	0.890014	+	0.455933i	$0.150694\pi$
$212$	0	0
$213$	13.4641	0.922545
$214$	0	0
$215$	24.0000	1.63679
$216$	0	0
$217$	−10.9282	−0.741855
$218$	0	0
$219$	−0.928203	−0.0627222
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	−24.7846	−1.65970	−0.829850	−	0.557986i	$-0.811574\pi$
$223$	−24.7846	−1.65970	−0.829850	+	0.557986i	$0.811574\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	−18.7846	−1.24132	−0.620661	−	0.784079i	$-0.713136\pi$
$229$	−18.7846	−1.24132	−0.620661	+	0.784079i	$0.713136\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.92820	0.322857	0.161429	−	0.986884i	$-0.448390\pi$
$233$	4.92820	0.322857	0.161429	+	0.986884i	$0.448390\pi$
$234$	0	0
$235$	32.7846	2.13863
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	−27.8564	−1.79439	−0.897194	−	0.441636i	$-0.854398\pi$
$241$	−27.8564	−1.79439	−0.897194	+	0.441636i	$0.854398\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.46410	0.221313
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−5.46410	−0.346273
$250$	0	0
$251$	−8.39230	−0.529718	−0.264859	−	0.964287i	$-0.585325\pi$
$251$	−8.39230	−0.529718	−0.264859	+	0.964287i	$0.585325\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	22.7846	1.42126	0.710632	−	0.703563i	$-0.248409\pi$
$257$	22.7846	1.42126	0.710632	+	0.703563i	$0.248409\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−0.535898	−0.0331713
$262$	0	0
$263$	−9.07180	−0.559391	−0.279695	−	0.960089i	$-0.590234\pi$
$263$	−9.07180	−0.559391	−0.279695	+	0.960089i	$0.590234\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−15.8564	−0.970396
$268$	0	0
$269$	−3.07180	−0.187291	−0.0936454	−	0.995606i	$-0.529852\pi$
$269$	−3.07180	−0.187291	−0.0936454	+	0.995606i	$0.529852\pi$
$270$	0	0
$271$	−29.8564	−1.81365	−0.906824	−	0.421510i	$-0.861500\pi$
$271$	−29.8564	−1.81365	−0.906824	+	0.421510i	$0.861500\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.85641	0.231709	0.115855	−	0.993266i	$-0.463039\pi$
$277$	3.85641	0.231709	0.115855	+	0.993266i	$0.463039\pi$
$278$	0	0
$279$	10.9282	0.654254
$280$	0	0
$281$	17.3205	1.03325	0.516627	−	0.856210i	$-0.327187\pi$
$281$	17.3205	1.03325	0.516627	+	0.856210i	$0.327187\pi$
$282$	0	0
$283$	−9.07180	−0.539262	−0.269631	−	0.962964i	$-0.586902\pi$
$283$	−9.07180	−0.539262	−0.269631	+	0.962964i	$0.586902\pi$
$284$	0	0
$285$	−3.46410	−0.205196
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	13.8564	0.806751
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.00000	−0.462652
$300$	0	0
$301$	−6.92820	−0.399335
$302$	0	0
$303$	−10.3923	−0.597022
$304$	0	0
$305$	20.7846	1.19012
$306$	0	0
$307$	1.07180	0.0611707	0.0305853	−	0.999532i	$-0.490263\pi$
$307$	1.07180	0.0611707	0.0305853	+	0.999532i	$0.490263\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	17.4641	0.990298	0.495149	−	0.868808i	$-0.335113\pi$
$311$	17.4641	0.990298	0.495149	+	0.868808i	$0.335113\pi$
$312$	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
$315$	−3.46410	−0.195180
$316$	0	0
$317$	7.46410	0.419226	0.209613	−	0.977784i	$-0.432780\pi$
$317$	7.46410	0.419226	0.209613	+	0.977784i	$0.432780\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.3923	−0.691671
$322$	0	0
$323$	3.46410	0.192748
$324$	0	0
$325$	−14.0000	−0.776580
$326$	0	0
$327$	−12.9282	−0.714931
$328$	0	0
$329$	−9.46410	−0.521773
$330$	0	0
$331$	14.9282	0.820528	0.410264	−	0.911967i	$-0.365437\pi$
$331$	14.9282	0.820528	0.410264	+	0.911967i	$0.365437\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	51.7128	2.82537
$336$	0	0
$337$	20.9282	1.14003	0.570016	−	0.821634i	$-0.306937\pi$
$337$	20.9282	1.14003	0.570016	+	0.821634i	$0.306937\pi$
$338$	0	0
$339$	11.4641	0.622645
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	13.8564	0.746004
$346$	0	0
$347$	32.7846	1.75997	0.879985	−	0.475001i	$-0.157552\pi$
$347$	32.7846	1.75997	0.879985	+	0.475001i	$0.157552\pi$
$348$	0	0
$349$	11.8564	0.634659	0.317329	−	0.948315i	$-0.397214\pi$
$349$	11.8564	0.634659	0.317329	+	0.948315i	$0.397214\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−19.4641	−1.03597	−0.517985	−	0.855390i	$-0.673318\pi$
$353$	−19.4641	−1.03597	−0.517985	+	0.855390i	$0.673318\pi$
$354$	0	0
$355$	46.6410	2.47545
$356$	0	0
$357$	3.46410	0.183340
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	−3.21539	−0.168301
$366$	0	0
$367$	2.92820	0.152851	0.0764255	−	0.997075i	$-0.475649\pi$
$367$	2.92820	0.152851	0.0764255	+	0.997075i	$0.475649\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	3.46410	0.179847
$372$	0	0
$373$	−7.85641	−0.406789	−0.203395	−	0.979097i	$-0.565197\pi$
$373$	−7.85641	−0.406789	−0.203395	+	0.979097i	$0.565197\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	1.07180	0.0552003
$378$	0	0
$379$	−28.7846	−1.47857	−0.739283	−	0.673395i	$-0.764835\pi$
$379$	−28.7846	−1.47857	−0.739283	+	0.673395i	$0.764835\pi$
$380$	0	0
$381$	−18.9282	−0.969721
$382$	0	0
$383$	18.9282	0.967186	0.483593	−	0.875293i	$-0.339331\pi$
$383$	18.9282	0.967186	0.483593	+	0.875293i	$0.339331\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.92820	0.352180
$388$	0	0
$389$	−7.07180	−0.358554	−0.179277	−	0.983799i	$-0.557376\pi$
$389$	−7.07180	−0.358554	−0.179277	+	0.983799i	$0.557376\pi$
$390$	0	0
$391$	−13.8564	−0.700749
$392$	0	0
$393$	−8.39230	−0.423336
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.143594	0.00720675	0.00360338	−	0.999994i	$-0.498853\pi$
$397$	0.143594	0.00720675	0.00360338	+	0.999994i	$0.498853\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	6.39230	0.319216	0.159608	−	0.987180i	$-0.448977\pi$
$401$	6.39230	0.319216	0.159608	+	0.987180i	$0.448977\pi$
$402$	0	0
$403$	−21.8564	−1.08875
$404$	0	0
$405$	3.46410	0.172133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−18.9282	−0.929149
$416$	0	0
$417$	−6.92820	−0.339276
$418$	0	0
$419$	2.53590	0.123887	0.0619434	−	0.998080i	$-0.480270\pi$
$419$	2.53590	0.123887	0.0619434	+	0.998080i	$0.480270\pi$
$420$	0	0
$421$	−15.0718	−0.734554	−0.367277	−	0.930112i	$-0.619710\pi$
$421$	−15.0718	−0.734554	−0.367277	+	0.930112i	$0.619710\pi$
$422$	0	0
$423$	9.46410	0.460160
$424$	0	0
$425$	−24.2487	−1.17624
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	37.4641	1.80458	0.902291	−	0.431127i	$-0.141884\pi$
$431$	37.4641	1.80458	0.902291	+	0.431127i	$0.141884\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−1.85641	−0.0890079
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	0.784610	0.0374474	0.0187237	−	0.999825i	$-0.494040\pi$
$439$	0.784610	0.0374474	0.0187237	+	0.999825i	$0.494040\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−16.7846	−0.797461	−0.398730	−	0.917068i	$-0.630549\pi$
$443$	−16.7846	−0.797461	−0.398730	+	0.917068i	$0.630549\pi$
$444$	0	0
$445$	−54.9282	−2.60385
$446$	0	0
$447$	−4.92820	−0.233096
$448$	0	0
$449$	−7.46410	−0.352253	−0.176126	−	0.984368i	$-0.556357\pi$
$449$	−7.46410	−0.352253	−0.176126	+	0.984368i	$0.556357\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.92820	−0.137579
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	−3.85641	−0.180395	−0.0901975	−	0.995924i	$-0.528750\pi$
$457$	−3.85641	−0.180395	−0.0901975	+	0.995924i	$0.528750\pi$
$458$	0	0
$459$	−3.46410	−0.161690
$460$	0	0
$461$	−12.5359	−0.583855	−0.291927	−	0.956440i	$-0.594297\pi$
$461$	−12.5359	−0.583855	−0.291927	+	0.956440i	$0.594297\pi$
$462$	0	0
$463$	−19.7128	−0.916132	−0.458066	−	0.888918i	$-0.651458\pi$
$463$	−19.7128	−0.916132	−0.458066	+	0.888918i	$0.651458\pi$
$464$	0	0
$465$	37.8564	1.75555
$466$	0	0
$467$	37.4641	1.73363	0.866816	−	0.498628i	$-0.166163\pi$
$467$	37.4641	1.73363	0.866816	+	0.498628i	$0.166163\pi$
$468$	0	0
$469$	−14.9282	−0.689320
$470$	0	0
$471$	11.8564	0.546314
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	−3.46410	−0.158610
$478$	0	0
$479$	22.5359	1.02969	0.514846	−	0.857283i	$-0.327849\pi$
$479$	22.5359	1.02969	0.514846	+	0.857283i	$0.327849\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	34.6410	1.57297
$486$	0	0
$487$	−27.7128	−1.25579	−0.627894	−	0.778299i	$-0.716083\pi$
$487$	−27.7128	−1.25579	−0.627894	+	0.778299i	$0.716083\pi$
$488$	0	0
$489$	−14.9282	−0.675077
$490$	0	0
$491$	5.07180	0.228887	0.114443	−	0.993430i	$-0.463492\pi$
$491$	5.07180	0.228887	0.114443	+	0.993430i	$0.463492\pi$
$492$	0	0
$493$	1.85641	0.0836083
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.4641	−0.603947
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−2.92820	−0.130822
$502$	0	0
$503$	37.1769	1.65764	0.828818	−	0.559518i	$-0.189014\pi$
$503$	37.1769	1.65764	0.828818	+	0.559518i	$0.189014\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	1.21539	0.0538712	0.0269356	−	0.999637i	$-0.491425\pi$
$509$	1.21539	0.0538712	0.0269356	+	0.999637i	$0.491425\pi$
$510$	0	0
$511$	0.928203	0.0410613
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	27.7128	1.22117
$516$	0	0
$517$	0	0
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−26.7846	−1.17346	−0.586728	−	0.809784i	$-0.699584\pi$
$521$	−26.7846	−1.17346	−0.586728	+	0.809784i	$0.699584\pi$
$522$	0	0
$523$	22.9282	1.00258	0.501290	−	0.865279i	$-0.332859\pi$
$523$	22.9282	1.00258	0.501290	+	0.865279i	$0.332859\pi$
$524$	0	0
$525$	−7.00000	−0.305505
$526$	0	0
$527$	−37.8564	−1.64905
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−42.9282	−1.85595
$536$	0	0
$537$	6.53590	0.282045
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	−44.7846	−1.91836
$546$	0	0
$547$	39.7128	1.69800	0.848999	−	0.528395i	$-0.177206\pi$
$547$	39.7128	1.69800	0.848999	+	0.528395i	$0.177206\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	0.535898	0.0228300
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.92820	−0.294086
$556$	0	0
$557$	−18.7846	−0.795929	−0.397965	−	0.917401i	$-0.630283\pi$
$557$	−18.7846	−0.795929	−0.397965	+	0.917401i	$0.630283\pi$
$558$	0	0
$559$	−13.8564	−0.586064
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.07180	−0.0451708	−0.0225854	−	0.999745i	$-0.507190\pi$
$563$	−1.07180	−0.0451708	−0.0225854	+	0.999745i	$0.507190\pi$
$564$	0	0
$565$	39.7128	1.67073
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	27.4641	1.15136	0.575678	−	0.817677i	$-0.304738\pi$
$569$	27.4641	1.15136	0.575678	+	0.817677i	$0.304738\pi$
$570$	0	0
$571$	−39.7128	−1.66193	−0.830965	−	0.556325i	$-0.812211\pi$
$571$	−39.7128	−1.66193	−0.830965	+	0.556325i	$0.812211\pi$
$572$	0	0
$573$	−17.8564	−0.745962
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	5.46410	0.226689
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.92820	−0.286446
$586$	0	0
$587$	−8.39230	−0.346387	−0.173194	−	0.984888i	$-0.555409\pi$
$587$	−8.39230	−0.346387	−0.173194	+	0.984888i	$0.555409\pi$
$588$	0	0
$589$	−10.9282	−0.450289
$590$	0	0
$591$	19.8564	0.816783
$592$	0	0
$593$	15.4641	0.635035	0.317517	−	0.948252i	$-0.397151\pi$
$593$	15.4641	0.635035	0.317517	+	0.948252i	$0.397151\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	−5.07180	−0.207575
$598$	0	0
$599$	−15.6077	−0.637713	−0.318857	−	0.947803i	$-0.603299\pi$
$599$	−15.6077	−0.637713	−0.318857	+	0.947803i	$0.603299\pi$
$600$	0	0
$601$	37.7128	1.53834	0.769169	−	0.639046i	$-0.220670\pi$
$601$	37.7128	1.53834	0.769169	+	0.639046i	$0.220670\pi$
$602$	0	0
$603$	14.9282	0.607923
$604$	0	0
$605$	−38.1051	−1.54919
$606$	0	0
$607$	−19.7128	−0.800118	−0.400059	−	0.916489i	$-0.631010\pi$
$607$	−19.7128	−0.800118	−0.400059	+	0.916489i	$0.631010\pi$
$608$	0	0
$609$	0.535898	0.0217157
$610$	0	0
$611$	−18.9282	−0.765753
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	20.7846	0.838116
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−38.9282	−1.56466	−0.782328	−	0.622866i	$-0.785968\pi$
$619$	−38.9282	−1.56466	−0.782328	+	0.622866i	$0.785968\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	15.8564	0.635274
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.92820	0.276246
$630$	0	0
$631$	−0.784610	−0.0312348	−0.0156174	−	0.999878i	$-0.504971\pi$
$631$	−0.784610	−0.0312348	−0.0156174	+	0.999878i	$0.504971\pi$
$632$	0	0
$633$	25.8564	1.02770
$634$	0	0
$635$	−65.5692	−2.60204
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	13.4641	0.532632
$640$	0	0
$641$	−43.1769	−1.70539	−0.852693	−	0.522413i	$-0.825032\pi$
$641$	−43.1769	−1.70539	−0.852693	+	0.522413i	$0.825032\pi$
$642$	0	0
$643$	3.21539	0.126803	0.0634013	−	0.997988i	$-0.479805\pi$
$643$	3.21539	0.126803	0.0634013	+	0.997988i	$0.479805\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	−42.2487	−1.66097	−0.830484	−	0.557042i	$-0.811936\pi$
$647$	−42.2487	−1.66097	−0.830484	+	0.557042i	$0.811936\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−10.9282	−0.428310
$652$	0	0
$653$	−45.7128	−1.78888	−0.894440	−	0.447187i	$-0.852426\pi$
$653$	−45.7128	−1.78888	−0.894440	+	0.447187i	$0.852426\pi$
$654$	0	0
$655$	−29.0718	−1.13593
$656$	0	0
$657$	−0.928203	−0.0362127
$658$	0	0
$659$	32.1051	1.25064	0.625319	−	0.780369i	$-0.284969\pi$
$659$	32.1051	1.25064	0.625319	+	0.780369i	$0.284969\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	6.92820	0.269069
$664$	0	0
$665$	3.46410	0.134332
$666$	0	0
$667$	−2.14359	−0.0830003
$668$	0	0
$669$	−24.7846	−0.958228
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	7.00000	0.269430
$676$	0	0
$677$	1.21539	0.0467112	0.0233556	−	0.999727i	$-0.492565\pi$
$677$	1.21539	0.0467112	0.0233556	+	0.999727i	$0.492565\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−6.92820	−0.265489
$682$	0	0
$683$	36.3923	1.39251	0.696256	−	0.717793i	$-0.254848\pi$
$683$	36.3923	1.39251	0.696256	+	0.717793i	$0.254848\pi$
$684$	0	0
$685$	6.92820	0.264713
$686$	0	0
$687$	−18.7846	−0.716678
$688$	0	0
$689$	6.92820	0.263944
$690$	0	0
$691$	9.85641	0.374955	0.187478	−	0.982269i	$-0.439969\pi$
$691$	9.85641	0.374955	0.187478	+	0.982269i	$0.439969\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	−20.7846	−0.787273
$698$	0	0
$699$	4.92820	0.186402
$700$	0	0
$701$	−7.85641	−0.296732	−0.148366	−	0.988932i	$-0.547401\pi$
$701$	−7.85641	−0.296732	−0.148366	+	0.988932i	$0.547401\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	32.7846	1.23474
$706$	0	0
$707$	10.3923	0.390843
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	43.7128	1.63706
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.92820	−0.258738
$718$	0	0
$719$	26.2487	0.978912	0.489456	−	0.872028i	$-0.337195\pi$
$719$	26.2487	0.978912	0.489456	+	0.872028i	$0.337195\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−27.8564	−1.03599
$724$	0	0
$725$	−3.75129	−0.139319
$726$	0	0
$727$	5.85641	0.217202	0.108601	−	0.994085i	$-0.465363\pi$
$727$	5.85641	0.217202	0.108601	+	0.994085i	$0.465363\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	22.7846	0.841569	0.420784	−	0.907161i	$-0.361755\pi$
$733$	22.7846	0.841569	0.420784	+	0.907161i	$0.361755\pi$
$734$	0	0
$735$	3.46410	0.127775
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−52.7846	−1.94171	−0.970857	−	0.239661i	$-0.922964\pi$
$739$	−52.7846	−1.94171	−0.970857	+	0.239661i	$0.922964\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	7.60770	0.279099	0.139550	−	0.990215i	$-0.455435\pi$
$743$	7.60770	0.279099	0.139550	+	0.990215i	$0.455435\pi$
$744$	0	0
$745$	−17.0718	−0.625462
$746$	0	0
$747$	−5.46410	−0.199921
$748$	0	0
$749$	12.3923	0.452805
$750$	0	0
$751$	5.07180	0.185072	0.0925362	−	0.995709i	$-0.470503\pi$
$751$	5.07180	0.185072	0.0925362	+	0.995709i	$0.470503\pi$
$752$	0	0
$753$	−8.39230	−0.305833
$754$	0	0
$755$	−10.1436	−0.369163
$756$	0	0
$757$	−43.5692	−1.58355	−0.791775	−	0.610813i	$-0.790843\pi$
$757$	−43.5692	−1.58355	−0.791775	+	0.610813i	$0.790843\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.1769	0.405163	0.202581	−	0.979265i	$-0.435067\pi$
$761$	11.1769	0.405163	0.202581	+	0.979265i	$0.435067\pi$
$762$	0	0
$763$	12.9282	0.468032
$764$	0	0
$765$	−12.0000	−0.433861
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	51.5692	1.85963	0.929817	−	0.368023i	$-0.119965\pi$
$769$	51.5692	1.85963	0.929817	+	0.368023i	$0.119965\pi$
$770$	0	0
$771$	22.7846	0.820568
$772$	0	0
$773$	−11.8564	−0.426445	−0.213223	−	0.977004i	$-0.568396\pi$
$773$	−11.8564	−0.426445	−0.213223	+	0.977004i	$0.568396\pi$
$774$	0	0
$775$	76.4974	2.74787
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.535898	−0.0191514
$784$	0	0
$785$	41.0718	1.46592
$786$	0	0
$787$	36.7846	1.31123	0.655615	−	0.755095i	$-0.272410\pi$
$787$	36.7846	1.31123	0.655615	+	0.755095i	$0.272410\pi$
$788$	0	0
$789$	−9.07180	−0.322965
$790$	0	0
$791$	−11.4641	−0.407617
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	13.7128	0.485733	0.242866	−	0.970060i	$-0.421912\pi$
$797$	13.7128	0.485733	0.242866	+	0.970060i	$0.421912\pi$
$798$	0	0
$799$	−32.7846	−1.15984
$800$	0	0
$801$	−15.8564	−0.560259
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−13.8564	−0.488374
$806$	0	0
$807$	−3.07180	−0.108132
$808$	0	0
$809$	−11.8564	−0.416849	−0.208425	−	0.978038i	$-0.566834\pi$
$809$	−11.8564	−0.416849	−0.208425	+	0.978038i	$0.566834\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	−29.8564	−1.04711
$814$	0	0
$815$	−51.7128	−1.81142
$816$	0	0
$817$	−6.92820	−0.242387
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−31.0718	−1.08441	−0.542207	−	0.840245i	$-0.682411\pi$
$821$	−31.0718	−1.08441	−0.542207	+	0.840245i	$0.682411\pi$
$822$	0	0
$823$	10.9282	0.380933	0.190467	−	0.981694i	$-0.439000\pi$
$823$	10.9282	0.380933	0.190467	+	0.981694i	$0.439000\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.3205	0.532746	0.266373	−	0.963870i	$-0.414175\pi$
$827$	15.3205	0.532746	0.266373	+	0.963870i	$0.414175\pi$
$828$	0	0
$829$	−20.1436	−0.699616	−0.349808	−	0.936821i	$-0.613753\pi$
$829$	−20.1436	−0.699616	−0.349808	+	0.936821i	$0.613753\pi$
$830$	0	0
$831$	3.85641	0.133777
$832$	0	0
$833$	−3.46410	−0.120024
$834$	0	0
$835$	−10.1436	−0.351034
$836$	0	0
$837$	10.9282	0.377734
$838$	0	0
$839$	−10.1436	−0.350196	−0.175098	−	0.984551i	$-0.556024\pi$
$839$	−10.1436	−0.350196	−0.175098	+	0.984551i	$0.556024\pi$
$840$	0	0
$841$	−28.7128	−0.990097
$842$	0	0
$843$	17.3205	0.596550
$844$	0	0
$845$	−31.1769	−1.07252
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	−9.07180	−0.311343
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−35.5692	−1.21787	−0.608933	−	0.793221i	$-0.708402\pi$
$853$	−35.5692	−1.21787	−0.608933	+	0.793221i	$0.708402\pi$
$854$	0	0
$855$	−3.46410	−0.118470
$856$	0	0
$857$	−31.8564	−1.08819	−0.544097	−	0.839022i	$-0.683128\pi$
$857$	−31.8564	−1.08819	−0.544097	+	0.839022i	$0.683128\pi$
$858$	0	0
$859$	−23.7128	−0.809071	−0.404535	−	0.914522i	$-0.632567\pi$
$859$	−23.7128	−0.809071	−0.404535	+	0.914522i	$0.632567\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	−27.3205	−0.930001	−0.465000	−	0.885310i	$-0.653946\pi$
$863$	−27.3205	−0.930001	−0.465000	+	0.885310i	$0.653946\pi$
$864$	0	0
$865$	6.92820	0.235566
$866$	0	0
$867$	−5.00000	−0.169809
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−29.8564	−1.01165
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	−22.0000	−0.742042
$880$	0	0
$881$	6.67949	0.225038	0.112519	−	0.993650i	$-0.464108\pi$
$881$	6.67949	0.225038	0.112519	+	0.993650i	$0.464108\pi$
$882$	0	0
$883$	−6.14359	−0.206748	−0.103374	−	0.994643i	$-0.532964\pi$
$883$	−6.14359	−0.206748	−0.103374	+	0.994643i	$0.532964\pi$
$884$	0	0
$885$	13.8564	0.465778
$886$	0	0
$887$	46.6410	1.56605	0.783026	−	0.621989i	$-0.213675\pi$
$887$	46.6410	1.56605	0.783026	+	0.621989i	$0.213675\pi$
$888$	0	0
$889$	18.9282	0.634832
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.46410	−0.316704
$894$	0	0
$895$	22.6410	0.756806
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	−5.85641	−0.195322
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−6.92820	−0.230556
$904$	0	0
$905$	−6.92820	−0.230301
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	0	0
$909$	−10.3923	−0.344691
$910$	0	0
$911$	50.5359	1.67433	0.837165	−	0.546951i	$-0.184212\pi$
$911$	50.5359	1.67433	0.837165	+	0.546951i	$0.184212\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	20.7846	0.687118
$916$	0	0
$917$	8.39230	0.277138
$918$	0	0
$919$	3.71281	0.122474	0.0612372	−	0.998123i	$-0.480495\pi$
$919$	3.71281	0.122474	0.0612372	+	0.998123i	$0.480495\pi$
$920$	0	0
$921$	1.07180	0.0353169
$922$	0	0
$923$	−26.9282	−0.886353
$924$	0	0
$925$	−14.0000	−0.460317
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	20.5359	0.673761	0.336880	−	0.941547i	$-0.390628\pi$
$929$	20.5359	0.673761	0.336880	+	0.941547i	$0.390628\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	17.4641	0.571749
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.7128	1.75472	0.877361	−	0.479832i	$-0.159302\pi$
$937$	53.7128	1.75472	0.877361	+	0.479832i	$0.159302\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	50.7846	1.65553	0.827765	−	0.561074i	$-0.189612\pi$
$941$	50.7846	1.65553	0.827765	+	0.561074i	$0.189612\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	−3.46410	−0.112687
$946$	0	0
$947$	2.14359	0.0696574	0.0348287	−	0.999393i	$-0.488911\pi$
$947$	2.14359	0.0696574	0.0348287	+	0.999393i	$0.488911\pi$
$948$	0	0
$949$	1.85641	0.0602615
$950$	0	0
$951$	7.46410	0.242040
$952$	0	0
$953$	22.3923	0.725358	0.362679	−	0.931914i	$-0.381862\pi$
$953$	22.3923	0.725358	0.362679	+	0.931914i	$0.381862\pi$
$954$	0	0
$955$	−61.8564	−2.00163
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	88.4256	2.85244
$962$	0	0
$963$	−12.3923	−0.399336
$964$	0	0
$965$	−48.4974	−1.56119
$966$	0	0
$967$	−10.9282	−0.351427	−0.175714	−	0.984441i	$-0.556223\pi$
$967$	−10.9282	−0.351427	−0.175714	+	0.984441i	$0.556223\pi$
$968$	0	0
$969$	3.46410	0.111283
$970$	0	0
$971$	6.14359	0.197157	0.0985786	−	0.995129i	$-0.468570\pi$
$971$	6.14359	0.197157	0.0985786	+	0.995129i	$0.468570\pi$
$972$	0	0
$973$	6.92820	0.222108
$974$	0	0
$975$	−14.0000	−0.448359
$976$	0	0
$977$	−43.1769	−1.38135	−0.690676	−	0.723164i	$-0.742687\pi$
$977$	−43.1769	−1.38135	−0.690676	+	0.723164i	$0.742687\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−12.9282	−0.412766
$982$	0	0
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	0	0
$985$	68.7846	2.19166
$986$	0	0
$987$	−9.46410	−0.301246
$988$	0	0
$989$	27.7128	0.881216
$990$	0	0
$991$	−57.5692	−1.82875	−0.914373	−	0.404872i	$-0.867316\pi$
$991$	−57.5692	−1.82875	−0.914373	+	0.404872i	$0.867316\pi$
$992$	0	0
$993$	14.9282	0.473732
$994$	0	0
$995$	−17.5692	−0.556982
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.t.1.2	✓	2
3.2	odd	2		9576.2.a.bk.1.1		2
4.3	odd	2		6384.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.t.1.2	✓	2	1.1	even	1	trivial
6384.2.a.bk.1.2		2	4.3	odd	2
9576.2.a.bk.1.1		2	3.2	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:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} +3.46410 q^{15} -3.46410 q^{17} -1.00000 q^{19} -1.00000 q^{21} +4.00000 q^{23} +7.00000 q^{25} +1.00000 q^{27} -0.535898 q^{29} +10.9282 q^{31} -3.46410 q^{35} -2.00000 q^{37} -2.00000 q^{39} +6.00000 q^{41} +6.92820 q^{43} +3.46410 q^{45} +9.46410 q^{47} +1.00000 q^{49} -3.46410 q^{51} -3.46410 q^{53} -1.00000 q^{57} +4.00000 q^{59} +6.00000 q^{61} -1.00000 q^{63} -6.92820 q^{65} +14.9282 q^{67} +4.00000 q^{69} +13.4641 q^{71} -0.928203 q^{73} +7.00000 q^{75} +1.00000 q^{81} -5.46410 q^{83} -12.0000 q^{85} -0.535898 q^{87} -15.8564 q^{89} +2.00000 q^{91} +10.9282 q^{93} -3.46410 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 14 q^{25} + 2 q^{27} - 8 q^{29} + 8 q^{31} - 4 q^{37} - 4 q^{39} + 12 q^{41} + 12 q^{47} + 2 q^{49} - 2 q^{57} + 8 q^{59} + 12 q^{61} - 2 q^{63} + 16 q^{67} + 8 q^{69} + 20 q^{71} + 12 q^{73} + 14 q^{75} + 2 q^{81} - 4 q^{83} - 24 q^{85} - 8 q^{87} - 4 q^{89} + 4 q^{91} + 8 q^{93} + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 14 * q^25 + 2 * q^27 - 8 * q^29 + 8 * q^31 - 4 * q^37 - 4 * q^39 + 12 * q^41 + 12 * q^47 + 2 * q^49 - 2 * q^57 + 8 * q^59 + 12 * q^61 - 2 * q^63 + 16 * q^67 + 8 * q^69 + 20 * q^71 + 12 * q^73 + 14 * q^75 + 2 * q^81 - 4 * q^83 - 24 * q^85 - 8 * q^87 - 4 * q^89 + 4 * q^91 + 8 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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