LMFDB - Embedded newform 3432.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3432,2,Mod(1,3432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3432, 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("3432.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.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:	$27.4046579737$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 5x + 16 $ x^4 - x^3 - 9x^2 + 5x + 16
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			1.3
Root			$-2.43292$ of defining polynomial
Character	$\chi$	$=$	3432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.11806 q^{5} -4.35203 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.11806 q^{5} -4.35203 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.11806 q^{15} +2.60425 q^{17} +4.35203 q^{19} +4.35203 q^{21} +0.513812 q^{23} +4.72231 q^{25} -1.00000 q^{27} -3.47010 q^{29} -7.58600 q^{31} +1.00000 q^{33} -13.5699 q^{35} -1.83822 q^{37} -1.00000 q^{39} +7.37966 q^{41} +10.3359 q^{43} +3.11806 q^{45} +2.62972 q^{47} +11.9402 q^{49} -2.60425 q^{51} -8.94019 q^{53} -3.11806 q^{55} -4.35203 q^{57} -3.49556 q^{59} +4.39790 q^{61} -4.35203 q^{63} +3.11806 q^{65} +8.79451 q^{67} -0.513812 q^{69} +3.83822 q^{71} +7.56053 q^{73} -4.72231 q^{75} +4.35203 q^{77} -5.63187 q^{79} +1.00000 q^{81} -3.49556 q^{83} +8.12022 q^{85} +3.47010 q^{87} -2.72447 q^{89} -4.35203 q^{91} +7.58600 q^{93} +13.5699 q^{95} +11.0977 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 2 q^{21} - 4 q^{23} + 8 q^{25} - 4 q^{27} + 10 q^{29} - 8 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} - 4 q^{39} + 2 q^{41} - 4 q^{43} + 4 q^{45} + 6 q^{47} - 8 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 12 q^{59} + 10 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} + 4 q^{69} + 6 q^{71} + 10 q^{73} - 8 q^{75} + 2 q^{77} - 8 q^{79} + 4 q^{81} + 12 q^{83} + 14 q^{85} - 10 q^{87} + 10 q^{89} - 2 q^{91} + 8 q^{93} + 2 q^{95} + 26 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 - 4 * q^15 + 8 * q^17 + 2 * q^19 + 2 * q^21 - 4 * q^23 + 8 * q^25 - 4 * q^27 + 10 * q^29 - 8 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 - 4 * q^39 + 2 * q^41 - 4 * q^43 + 4 * q^45 + 6 * q^47 - 8 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 12 * q^59 + 10 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 + 4 * q^69 + 6 * q^71 + 10 * q^73 - 8 * q^75 + 2 * q^77 - 8 * q^79 + 4 * q^81 + 12 * q^83 + 14 * q^85 - 10 * q^87 + 10 * q^89 - 2 * q^91 + 8 * q^93 + 2 * q^95 + 26 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.11806	1.39444	0.697220	−	0.716857i	$-0.254420\pi$
$5$	3.11806	1.39444	0.697220	+	0.716857i	$0.254420\pi$
$6$	0	0
$7$	−4.35203	−1.64491	−0.822457	−	0.568827i	$-0.807397\pi$
$7$	−4.35203	−1.64491	−0.822457	+	0.568827i	$0.807397\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.11806	−0.805080
$16$	0	0
$17$	2.60425	0.631624	0.315812	−	0.948822i	$-0.397723\pi$
$17$	2.60425	0.631624	0.315812	+	0.948822i	$0.397723\pi$
$18$	0	0
$19$	4.35203	0.998425	0.499212	−	0.866480i	$-0.333623\pi$
$19$	4.35203	0.998425	0.499212	+	0.866480i	$0.333623\pi$
$20$	0	0
$21$	4.35203	0.949691
$22$	0	0
$23$	0.513812	0.107137	0.0535686	−	0.998564i	$-0.482940\pi$
$23$	0.513812	0.107137	0.0535686	+	0.998564i	$0.482940\pi$
$24$	0	0
$25$	4.72231	0.944463
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.47010	−0.644381	−0.322190	−	0.946675i	$-0.604419\pi$
$29$	−3.47010	−0.644381	−0.322190	+	0.946675i	$0.604419\pi$
$30$	0	0
$31$	−7.58600	−1.36249	−0.681243	−	0.732057i	$-0.738560\pi$
$31$	−7.58600	−1.36249	−0.681243	+	0.732057i	$0.738560\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−13.5699	−2.29373
$36$	0	0
$37$	−1.83822	−0.302202	−0.151101	−	0.988518i	$-0.548282\pi$
$37$	−1.83822	−0.302202	−0.151101	+	0.988518i	$0.548282\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.37966	1.15251	0.576254	−	0.817270i	$-0.304514\pi$
$41$	7.37966	1.15251	0.576254	+	0.817270i	$0.304514\pi$
$42$	0	0
$43$	10.3359	1.57622	0.788108	−	0.615537i	$-0.211061\pi$
$43$	10.3359	1.57622	0.788108	+	0.615537i	$0.211061\pi$
$44$	0	0
$45$	3.11806	0.464813
$46$	0	0
$47$	2.62972	0.383584	0.191792	−	0.981436i	$-0.438570\pi$
$47$	2.62972	0.383584	0.191792	+	0.981436i	$0.438570\pi$
$48$	0	0
$49$	11.9402	1.70574
$50$	0	0
$51$	−2.60425	−0.364668
$52$	0	0
$53$	−8.94019	−1.22803	−0.614015	−	0.789294i	$-0.710447\pi$
$53$	−8.94019	−1.22803	−0.614015	+	0.789294i	$0.710447\pi$
$54$	0	0
$55$	−3.11806	−0.420439
$56$	0	0
$57$	−4.35203	−0.576441
$58$	0	0
$59$	−3.49556	−0.455084	−0.227542	−	0.973768i	$-0.573069\pi$
$59$	−3.49556	−0.455084	−0.227542	+	0.973768i	$0.573069\pi$
$60$	0	0
$61$	4.39790	0.563094	0.281547	−	0.959547i	$-0.409152\pi$
$61$	4.39790	0.563094	0.281547	+	0.959547i	$0.409152\pi$
$62$	0	0
$63$	−4.35203	−0.548305
$64$	0	0
$65$	3.11806	0.386748
$66$	0	0
$67$	8.79451	1.07442	0.537210	−	0.843449i	$-0.319478\pi$
$67$	8.79451	1.07442	0.537210	+	0.843449i	$0.319478\pi$
$68$	0	0
$69$	−0.513812	−0.0618556
$70$	0	0
$71$	3.83822	0.455513	0.227757	−	0.973718i	$-0.426861\pi$
$71$	3.83822	0.455513	0.227757	+	0.973718i	$0.426861\pi$
$72$	0	0
$73$	7.56053	0.884894	0.442447	−	0.896795i	$-0.354111\pi$
$73$	7.56053	0.884894	0.442447	+	0.896795i	$0.354111\pi$
$74$	0	0
$75$	−4.72231	−0.545286
$76$	0	0
$77$	4.35203	0.495960
$78$	0	0
$79$	−5.63187	−0.633635	−0.316818	−	0.948486i	$-0.602614\pi$
$79$	−5.63187	−0.633635	−0.316818	+	0.948486i	$0.602614\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.49556	−0.383688	−0.191844	−	0.981425i	$-0.561447\pi$
$83$	−3.49556	−0.383688	−0.191844	+	0.981425i	$0.561447\pi$
$84$	0	0
$85$	8.12022	0.880761
$86$	0	0
$87$	3.47010	0.372033
$88$	0	0
$89$	−2.72447	−0.288793	−0.144396	−	0.989520i	$-0.546124\pi$
$89$	−2.72447	−0.288793	−0.144396	+	0.989520i	$0.546124\pi$
$90$	0	0
$91$	−4.35203	−0.456217
$92$	0	0
$93$	7.58600	0.786632
$94$	0	0
$95$	13.5699	1.39224
$96$	0	0
$97$	11.0977	1.12680	0.563398	−	0.826185i	$-0.309494\pi$
$97$	11.0977	1.12680	0.563398	+	0.826185i	$0.309494\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	10.0998	1.00497	0.502485	−	0.864586i	$-0.332419\pi$
$101$	10.0998	1.00497	0.502485	+	0.864586i	$0.332419\pi$
$102$	0	0
$103$	9.73169	0.958892	0.479446	−	0.877571i	$-0.340838\pi$
$103$	9.73169	0.958892	0.479446	+	0.877571i	$0.340838\pi$
$104$	0	0
$105$	13.5699	1.32429
$106$	0	0
$107$	7.49556	0.724624	0.362312	−	0.932057i	$-0.381988\pi$
$107$	7.49556	0.724624	0.362312	+	0.932057i	$0.381988\pi$
$108$	0	0
$109$	4.11591	0.394232	0.197116	−	0.980380i	$-0.436842\pi$
$109$	4.11591	0.394232	0.197116	+	0.980380i	$0.436842\pi$
$110$	0	0
$111$	1.83822	0.174476
$112$	0	0
$113$	9.49556	0.893268	0.446634	−	0.894717i	$-0.352623\pi$
$113$	9.49556	0.893268	0.446634	+	0.894717i	$0.352623\pi$
$114$	0	0
$115$	1.60210	0.149396
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−11.3338	−1.03897
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.37966	−0.665401
$124$	0	0
$125$	−0.865845	−0.0774435
$126$	0	0
$127$	5.23828	0.464822	0.232411	−	0.972618i	$-0.425339\pi$
$127$	5.23828	0.464822	0.232411	+	0.972618i	$0.425339\pi$
$128$	0	0
$129$	−10.3359	−0.910029
$130$	0	0
$131$	−11.4446	−0.999922	−0.499961	−	0.866048i	$-0.666652\pi$
$131$	−11.4446	−0.999922	−0.499961	+	0.866048i	$0.666652\pi$
$132$	0	0
$133$	−18.9402	−1.64232
$134$	0	0
$135$	−3.11806	−0.268360
$136$	0	0
$137$	12.1648	1.03931	0.519654	−	0.854377i	$-0.326061\pi$
$137$	12.1648	1.03931	0.519654	+	0.854377i	$0.326061\pi$
$138$	0	0
$139$	10.7295	0.910067	0.455034	−	0.890474i	$-0.349627\pi$
$139$	10.7295	0.910067	0.455034	+	0.890474i	$0.349627\pi$
$140$	0	0
$141$	−2.62972	−0.221462
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−10.8200	−0.898550
$146$	0	0
$147$	−11.9402	−0.984810
$148$	0	0
$149$	9.64797	0.790392	0.395196	−	0.918597i	$-0.370677\pi$
$149$	9.64797	0.790392	0.395196	+	0.918597i	$0.370677\pi$
$150$	0	0
$151$	−0.120217	−0.00978310	−0.00489155	−	0.999988i	$-0.501557\pi$
$151$	−0.120217	−0.00978310	−0.00489155	+	0.999988i	$0.501557\pi$
$152$	0	0
$153$	2.60425	0.210541
$154$	0	0
$155$	−23.6536	−1.89991
$156$	0	0
$157$	−9.45400	−0.754512	−0.377256	−	0.926109i	$-0.623132\pi$
$157$	−9.45400	−0.754512	−0.377256	+	0.926109i	$0.623132\pi$
$158$	0	0
$159$	8.94019	0.709003
$160$	0	0
$161$	−2.23612	−0.176231
$162$	0	0
$163$	19.2433	1.50726	0.753628	−	0.657302i	$-0.228302\pi$
$163$	19.2433	1.50726	0.753628	+	0.657302i	$0.228302\pi$
$164$	0	0
$165$	3.11806	0.242741
$166$	0	0
$167$	10.7593	0.832581	0.416290	−	0.909232i	$-0.363330\pi$
$167$	10.7593	0.832581	0.416290	+	0.909232i	$0.363330\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.35203	0.332808
$172$	0	0
$173$	−17.7441	−1.34906	−0.674528	−	0.738249i	$-0.735653\pi$
$173$	−17.7441	−1.34906	−0.674528	+	0.738249i	$0.735653\pi$
$174$	0	0
$175$	−20.5517	−1.55356
$176$	0	0
$177$	3.49556	0.262743
$178$	0	0
$179$	−11.4540	−0.856112	−0.428056	−	0.903752i	$-0.640801\pi$
$179$	−11.4540	−0.856112	−0.428056	+	0.903752i	$0.640801\pi$
$180$	0	0
$181$	−8.42638	−0.626328	−0.313164	−	0.949699i	$-0.601389\pi$
$181$	−8.42638	−0.626328	−0.313164	+	0.949699i	$0.601389\pi$
$182$	0	0
$183$	−4.39790	−0.325102
$184$	0	0
$185$	−5.73169	−0.421402
$186$	0	0
$187$	−2.60425	−0.190442
$188$	0	0
$189$	4.35203	0.316564
$190$	0	0
$191$	18.7084	1.35369	0.676845	−	0.736125i	$-0.263347\pi$
$191$	18.7084	1.35369	0.676845	+	0.736125i	$0.263347\pi$
$192$	0	0
$193$	−26.2281	−1.88794	−0.943970	−	0.330031i	$-0.892941\pi$
$193$	−26.2281	−1.88794	−0.943970	+	0.330031i	$0.892941\pi$
$194$	0	0
$195$	−3.11806	−0.223289
$196$	0	0
$197$	0.115908	0.00825811	0.00412905	−	0.999991i	$-0.498686\pi$
$197$	0.115908	0.00825811	0.00412905	+	0.999991i	$0.498686\pi$
$198$	0	0
$199$	4.47225	0.317029	0.158515	−	0.987357i	$-0.449329\pi$
$199$	4.47225	0.317029	0.158515	+	0.987357i	$0.449329\pi$
$200$	0	0
$201$	−8.79451	−0.620317
$202$	0	0
$203$	15.1020	1.05995
$204$	0	0
$205$	23.0102	1.60710
$206$	0	0
$207$	0.513812	0.0357124
$208$	0	0
$209$	−4.35203	−0.301036
$210$	0	0
$211$	−18.3359	−1.26230	−0.631149	−	0.775662i	$-0.717416\pi$
$211$	−18.3359	−1.26230	−0.631149	+	0.775662i	$0.717416\pi$
$212$	0	0
$213$	−3.83822	−0.262991
$214$	0	0
$215$	32.2281	2.19794
$216$	0	0
$217$	33.0145	2.24117
$218$	0	0
$219$	−7.56053	−0.510894
$220$	0	0
$221$	2.60425	0.175181
$222$	0	0
$223$	7.58600	0.507996	0.253998	−	0.967205i	$-0.418254\pi$
$223$	7.58600	0.507996	0.253998	+	0.967205i	$0.418254\pi$
$224$	0	0
$225$	4.72231	0.314821
$226$	0	0
$227$	18.3155	1.21564	0.607822	−	0.794073i	$-0.292043\pi$
$227$	18.3155	1.21564	0.607822	+	0.794073i	$0.292043\pi$
$228$	0	0
$229$	−0.629720	−0.0416130	−0.0208065	−	0.999784i	$-0.506623\pi$
$229$	−0.629720	−0.0416130	−0.0208065	+	0.999784i	$0.506623\pi$
$230$	0	0
$231$	−4.35203	−0.286343
$232$	0	0
$233$	6.44247	0.422060	0.211030	−	0.977480i	$-0.432318\pi$
$233$	6.44247	0.422060	0.211030	+	0.977480i	$0.432318\pi$
$234$	0	0
$235$	8.19963	0.534885
$236$	0	0
$237$	5.63187	0.365830
$238$	0	0
$239$	13.1478	0.850463	0.425232	−	0.905085i	$-0.360193\pi$
$239$	13.1478	0.850463	0.425232	+	0.905085i	$0.360193\pi$
$240$	0	0
$241$	18.9954	1.22360	0.611802	−	0.791011i	$-0.290445\pi$
$241$	18.9954	1.22360	0.611802	+	0.791011i	$0.290445\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	37.2303	2.37855
$246$	0	0
$247$	4.35203	0.276913
$248$	0	0
$249$	3.49556	0.221522
$250$	0	0
$251$	18.3848	1.16044	0.580220	−	0.814460i	$-0.302967\pi$
$251$	18.3848	1.16044	0.580220	+	0.814460i	$0.302967\pi$
$252$	0	0
$253$	−0.513812	−0.0323031
$254$	0	0
$255$	−8.12022	−0.508508
$256$	0	0
$257$	6.23182	0.388730	0.194365	−	0.980929i	$-0.437735\pi$
$257$	6.23182	0.388730	0.194365	+	0.980929i	$0.437735\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−3.47010	−0.214794
$262$	0	0
$263$	17.1254	1.05600	0.527998	−	0.849246i	$-0.322943\pi$
$263$	17.1254	1.05600	0.527998	+	0.849246i	$0.322943\pi$
$264$	0	0
$265$	−27.8761	−1.71241
$266$	0	0
$267$	2.72447	0.166735
$268$	0	0
$269$	−26.0248	−1.58676	−0.793379	−	0.608728i	$-0.791680\pi$
$269$	−26.0248	−1.58676	−0.793379	+	0.608728i	$0.791680\pi$
$270$	0	0
$271$	−16.3198	−0.991360	−0.495680	−	0.868505i	$-0.665081\pi$
$271$	−16.3198	−0.991360	−0.495680	+	0.868505i	$0.665081\pi$
$272$	0	0
$273$	4.35203	0.263397
$274$	0	0
$275$	−4.72231	−0.284766
$276$	0	0
$277$	−8.64882	−0.519657	−0.259829	−	0.965655i	$-0.583666\pi$
$277$	−8.64882	−0.519657	−0.259829	+	0.965655i	$0.583666\pi$
$278$	0	0
$279$	−7.58600	−0.454162
$280$	0	0
$281$	22.5517	1.34532	0.672660	−	0.739952i	$-0.265152\pi$
$281$	22.5517	1.34532	0.672660	+	0.739952i	$0.265152\pi$
$282$	0	0
$283$	18.9104	1.12411	0.562054	−	0.827101i	$-0.310011\pi$
$283$	18.9104	1.12411	0.562054	+	0.827101i	$0.310011\pi$
$284$	0	0
$285$	−13.5699	−0.803812
$286$	0	0
$287$	−32.1165	−1.89578
$288$	0	0
$289$	−10.2179	−0.601052
$290$	0	0
$291$	−11.0977	−0.650556
$292$	0	0
$293$	28.5882	1.67014	0.835069	−	0.550145i	$-0.185428\pi$
$293$	28.5882	1.67014	0.835069	+	0.550145i	$0.185428\pi$
$294$	0	0
$295$	−10.8994	−0.634587
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	0.513812	0.0297145
$300$	0	0
$301$	−44.9824	−2.59274
$302$	0	0
$303$	−10.0998	−0.580219
$304$	0	0
$305$	13.7129	0.785201
$306$	0	0
$307$	−1.93503	−0.110438	−0.0552190	−	0.998474i	$-0.517586\pi$
$307$	−1.93503	−0.110438	−0.0552190	+	0.998474i	$0.517586\pi$
$308$	0	0
$309$	−9.73169	−0.553616
$310$	0	0
$311$	−27.6536	−1.56809	−0.784047	−	0.620702i	$-0.786848\pi$
$311$	−27.6536	−1.56809	−0.784047	+	0.620702i	$0.786848\pi$
$312$	0	0
$313$	−0.458565	−0.0259196	−0.0129598	−	0.999916i	$-0.504125\pi$
$313$	−0.458565	−0.0259196	−0.0129598	+	0.999916i	$0.504125\pi$
$314$	0	0
$315$	−13.5699	−0.764578
$316$	0	0
$317$	31.1559	1.74989	0.874945	−	0.484222i	$-0.160897\pi$
$317$	31.1559	1.74989	0.874945	+	0.484222i	$0.160897\pi$
$318$	0	0
$319$	3.47010	0.194288
$320$	0	0
$321$	−7.49556	−0.418362
$322$	0	0
$323$	11.3338	0.630629
$324$	0	0
$325$	4.72231	0.261947
$326$	0	0
$327$	−4.11591	−0.227610
$328$	0	0
$329$	−11.4446	−0.630963
$330$	0	0
$331$	−11.6647	−0.641148	−0.320574	−	0.947224i	$-0.603876\pi$
$331$	−11.6647	−0.641148	−0.320574	+	0.947224i	$0.603876\pi$
$332$	0	0
$333$	−1.83822	−0.100734
$334$	0	0
$335$	27.4218	1.49821
$336$	0	0
$337$	−7.44894	−0.405769	−0.202885	−	0.979203i	$-0.565032\pi$
$337$	−7.44894	−0.405769	−0.202885	+	0.979203i	$0.565032\pi$
$338$	0	0
$339$	−9.49556	−0.515728
$340$	0	0
$341$	7.58600	0.410805
$342$	0	0
$343$	−21.4999	−1.16088
$344$	0	0
$345$	−1.60210	−0.0862540
$346$	0	0
$347$	22.5975	1.21310	0.606550	−	0.795046i	$-0.292553\pi$
$347$	22.5975	1.21310	0.606550	+	0.795046i	$0.292553\pi$
$348$	0	0
$349$	14.8849	0.796773	0.398386	−	0.917218i	$-0.369570\pi$
$349$	14.8849	0.796773	0.398386	+	0.917218i	$0.369570\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−28.0626	−1.49362	−0.746810	−	0.665038i	$-0.768415\pi$
$353$	−28.0626	−1.49362	−0.746810	+	0.665038i	$0.768415\pi$
$354$	0	0
$355$	11.9678	0.635186
$356$	0	0
$357$	11.3338	0.599848
$358$	0	0
$359$	14.2137	0.750168	0.375084	−	0.926991i	$-0.377614\pi$
$359$	14.2137	0.750168	0.375084	+	0.926991i	$0.377614\pi$
$360$	0	0
$361$	−0.0598093	−0.00314786
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	23.5742	1.23393
$366$	0	0
$367$	20.3483	1.06217	0.531087	−	0.847317i	$-0.321784\pi$
$367$	20.3483	1.06217	0.531087	+	0.847317i	$0.321784\pi$
$368$	0	0
$369$	7.37966	0.384170
$370$	0	0
$371$	38.9080	2.02000
$372$	0	0
$373$	23.5890	1.22139	0.610696	−	0.791865i	$-0.290890\pi$
$373$	23.5890	1.22139	0.610696	+	0.791865i	$0.290890\pi$
$374$	0	0
$375$	0.865845	0.0447120
$376$	0	0
$377$	−3.47010	−0.178719
$378$	0	0
$379$	−9.18810	−0.471961	−0.235980	−	0.971758i	$-0.575830\pi$
$379$	−9.18810	−0.471961	−0.235980	+	0.971758i	$0.575830\pi$
$380$	0	0
$381$	−5.23828	−0.268365
$382$	0	0
$383$	−10.6166	−0.542485	−0.271242	−	0.962511i	$-0.587434\pi$
$383$	−10.6166	−0.542485	−0.271242	+	0.962511i	$0.587434\pi$
$384$	0	0
$385$	13.5699	0.691587
$386$	0	0
$387$	10.3359	0.525405
$388$	0	0
$389$	9.03193	0.457937	0.228969	−	0.973434i	$-0.426465\pi$
$389$	9.03193	0.457937	0.228969	+	0.973434i	$0.426465\pi$
$390$	0	0
$391$	1.33809	0.0676703
$392$	0	0
$393$	11.4446	0.577305
$394$	0	0
$395$	−17.5605	−0.883566
$396$	0	0
$397$	−26.7827	−1.34419	−0.672093	−	0.740467i	$-0.734605\pi$
$397$	−26.7827	−1.34419	−0.672093	+	0.740467i	$0.734605\pi$
$398$	0	0
$399$	18.9402	0.948196
$400$	0	0
$401$	25.4111	1.26897	0.634486	−	0.772934i	$-0.281212\pi$
$401$	25.4111	1.26897	0.634486	+	0.772934i	$0.281212\pi$
$402$	0	0
$403$	−7.58600	−0.377886
$404$	0	0
$405$	3.11806	0.154938
$406$	0	0
$407$	1.83822	0.0911173
$408$	0	0
$409$	−24.7369	−1.22316	−0.611579	−	0.791183i	$-0.709465\pi$
$409$	−24.7369	−1.22316	−0.611579	+	0.791183i	$0.709465\pi$
$410$	0	0
$411$	−12.1648	−0.600045
$412$	0	0
$413$	15.2128	0.748573
$414$	0	0
$415$	−10.8994	−0.535030
$416$	0	0
$417$	−10.7295	−0.525428
$418$	0	0
$419$	17.7411	0.866708	0.433354	−	0.901224i	$-0.357330\pi$
$419$	17.7411	0.866708	0.433354	+	0.901224i	$0.357330\pi$
$420$	0	0
$421$	29.0567	1.41614	0.708068	−	0.706144i	$-0.249567\pi$
$421$	29.0567	1.41614	0.708068	+	0.706144i	$0.249567\pi$
$422$	0	0
$423$	2.62972	0.127861
$424$	0	0
$425$	12.2981	0.596545
$426$	0	0
$427$	−19.1398	−0.926241
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−7.29593	−0.351433	−0.175716	−	0.984441i	$-0.556224\pi$
$431$	−7.29593	−0.351433	−0.175716	+	0.984441i	$0.556224\pi$
$432$	0	0
$433$	−33.4634	−1.60815	−0.804074	−	0.594530i	$-0.797338\pi$
$433$	−33.4634	−1.60815	−0.804074	+	0.594530i	$0.797338\pi$
$434$	0	0
$435$	10.8200	0.518778
$436$	0	0
$437$	2.23612	0.106968
$438$	0	0
$439$	33.4758	1.59771	0.798855	−	0.601523i	$-0.205439\pi$
$439$	33.4758	1.59771	0.798855	+	0.601523i	$0.205439\pi$
$440$	0	0
$441$	11.9402	0.568581
$442$	0	0
$443$	−22.3526	−1.06201	−0.531003	−	0.847370i	$-0.678185\pi$
$443$	−22.3526	−1.06201	−0.531003	+	0.847370i	$0.678185\pi$
$444$	0	0
$445$	−8.49506	−0.402704
$446$	0	0
$447$	−9.64797	−0.456333
$448$	0	0
$449$	−29.4111	−1.38800	−0.693999	−	0.719976i	$-0.744153\pi$
$449$	−29.4111	−1.38800	−0.693999	+	0.719976i	$0.744153\pi$
$450$	0	0
$451$	−7.37966	−0.347494
$452$	0	0
$453$	0.120217	0.00564828
$454$	0	0
$455$	−13.5699	−0.636167
$456$	0	0
$457$	14.0570	0.657556	0.328778	−	0.944407i	$-0.393363\pi$
$457$	14.0570	0.657556	0.328778	+	0.944407i	$0.393363\pi$
$458$	0	0
$459$	−2.60425	−0.121556
$460$	0	0
$461$	32.3155	1.50508	0.752542	−	0.658544i	$-0.228827\pi$
$461$	32.3155	1.50508	0.752542	+	0.658544i	$0.228827\pi$
$462$	0	0
$463$	−3.40547	−0.158266	−0.0791329	−	0.996864i	$-0.525215\pi$
$463$	−3.40547	−0.158266	−0.0791329	+	0.996864i	$0.525215\pi$
$464$	0	0
$465$	23.6536	1.09691
$466$	0	0
$467$	27.6536	1.27966	0.639829	−	0.768518i	$-0.279005\pi$
$467$	27.6536	1.27966	0.639829	+	0.768518i	$0.279005\pi$
$468$	0	0
$469$	−38.2740	−1.76733
$470$	0	0
$471$	9.45400	0.435617
$472$	0	0
$473$	−10.3359	−0.475247
$474$	0	0
$475$	20.5517	0.942975
$476$	0	0
$477$	−8.94019	−0.409343
$478$	0	0
$479$	−18.0837	−0.826266	−0.413133	−	0.910671i	$-0.635566\pi$
$479$	−18.0837	−0.826266	−0.413133	+	0.910671i	$0.635566\pi$
$480$	0	0
$481$	−1.83822	−0.0838157
$482$	0	0
$483$	2.23612	0.101747
$484$	0	0
$485$	34.6032	1.57125
$486$	0	0
$487$	−42.7535	−1.93735	−0.968674	−	0.248336i	$-0.920116\pi$
$487$	−42.7535	−1.93735	−0.968674	+	0.248336i	$0.920116\pi$
$488$	0	0
$489$	−19.2433	−0.870214
$490$	0	0
$491$	−8.41235	−0.379644	−0.189822	−	0.981819i	$-0.560791\pi$
$491$	−8.41235	−0.379644	−0.189822	+	0.981819i	$0.560791\pi$
$492$	0	0
$493$	−9.03700	−0.407006
$494$	0	0
$495$	−3.11806	−0.140146
$496$	0	0
$497$	−16.7041	−0.749280
$498$	0	0
$499$	−30.4371	−1.36255	−0.681275	−	0.732028i	$-0.738574\pi$
$499$	−30.4371	−1.36255	−0.681275	+	0.732028i	$0.738574\pi$
$500$	0	0
$501$	−10.7593	−0.480691
$502$	0	0
$503$	13.4403	0.599274	0.299637	−	0.954053i	$-0.403134\pi$
$503$	13.4403	0.599274	0.299637	+	0.954053i	$0.403134\pi$
$504$	0	0
$505$	31.4919	1.40137
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−6.54359	−0.290039	−0.145020	−	0.989429i	$-0.546325\pi$
$509$	−6.54359	−0.290039	−0.145020	+	0.989429i	$0.546325\pi$
$510$	0	0
$511$	−32.9037	−1.45557
$512$	0	0
$513$	−4.35203	−0.192147
$514$	0	0
$515$	30.3440	1.33712
$516$	0	0
$517$	−2.62972	−0.115655
$518$	0	0
$519$	17.7441	0.778878
$520$	0	0
$521$	−21.9356	−0.961017	−0.480509	−	0.876990i	$-0.659548\pi$
$521$	−21.9356	−0.961017	−0.480509	+	0.876990i	$0.659548\pi$
$522$	0	0
$523$	−22.4846	−0.983184	−0.491592	−	0.870826i	$-0.663585\pi$
$523$	−22.4846	−0.983184	−0.491592	+	0.870826i	$0.663585\pi$
$524$	0	0
$525$	20.5517	0.896948
$526$	0	0
$527$	−19.7559	−0.860579
$528$	0	0
$529$	−22.7360	−0.988522
$530$	0	0
$531$	−3.49556	−0.151695
$532$	0	0
$533$	7.37966	0.319648
$534$	0	0
$535$	23.3716	1.01044
$536$	0	0
$537$	11.4540	0.494277
$538$	0	0
$539$	−11.9402	−0.514300
$540$	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
$543$	8.42638	0.361611
$544$	0	0
$545$	12.8337	0.549734
$546$	0	0
$547$	19.6145	0.838655	0.419327	−	0.907835i	$-0.362266\pi$
$547$	19.6145	0.838655	0.419327	+	0.907835i	$0.362266\pi$
$548$	0	0
$549$	4.39790	0.187698
$550$	0	0
$551$	−15.1020	−0.643366
$552$	0	0
$553$	24.5101	1.04228
$554$	0	0
$555$	5.73169	0.243297
$556$	0	0
$557$	8.17546	0.346405	0.173203	−	0.984886i	$-0.444588\pi$
$557$	8.17546	0.346405	0.173203	+	0.984886i	$0.444588\pi$
$558$	0	0
$559$	10.3359	0.437164
$560$	0	0
$561$	2.60425	0.109952
$562$	0	0
$563$	−34.1122	−1.43766	−0.718829	−	0.695187i	$-0.755322\pi$
$563$	−34.1122	−1.43766	−0.718829	+	0.695187i	$0.755322\pi$
$564$	0	0
$565$	29.6078	1.24561
$566$	0	0
$567$	−4.35203	−0.182768
$568$	0	0
$569$	6.51511	0.273128	0.136564	−	0.990631i	$-0.456394\pi$
$569$	6.51511	0.273128	0.136564	+	0.990631i	$0.456394\pi$
$570$	0	0
$571$	7.63618	0.319564	0.159782	−	0.987152i	$-0.448921\pi$
$571$	7.63618	0.319564	0.159782	+	0.987152i	$0.448921\pi$
$572$	0	0
$573$	−18.7084	−0.781554
$574$	0	0
$575$	2.42638	0.101187
$576$	0	0
$577$	−35.4999	−1.47788	−0.738940	−	0.673772i	$-0.764673\pi$
$577$	−35.4999	−1.47788	−0.738940	+	0.673772i	$0.764673\pi$
$578$	0	0
$579$	26.2281	1.09000
$580$	0	0
$581$	15.2128	0.631134
$582$	0	0
$583$	8.94019	0.370265
$584$	0	0
$585$	3.11806	0.128916
$586$	0	0
$587$	−39.4226	−1.62714	−0.813572	−	0.581464i	$-0.802480\pi$
$587$	−39.4226	−1.62714	−0.813572	+	0.581464i	$0.802480\pi$
$588$	0	0
$589$	−33.0145	−1.36034
$590$	0	0
$591$	−0.115908	−0.00476782
$592$	0	0
$593$	−6.62465	−0.272042	−0.136021	−	0.990706i	$-0.543431\pi$
$593$	−6.62465	−0.272042	−0.136021	+	0.990706i	$0.543431\pi$
$594$	0	0
$595$	−35.3395	−1.44878
$596$	0	0
$597$	−4.47225	−0.183037
$598$	0	0
$599$	−36.7636	−1.50212	−0.751060	−	0.660233i	$-0.770457\pi$
$599$	−36.7636	−1.50212	−0.751060	+	0.660233i	$0.770457\pi$
$600$	0	0
$601$	−19.8804	−0.810938	−0.405469	−	0.914109i	$-0.632892\pi$
$601$	−19.8804	−0.810938	−0.405469	+	0.914109i	$0.632892\pi$
$602$	0	0
$603$	8.79451	0.358140
$604$	0	0
$605$	3.11806	0.126767
$606$	0	0
$607$	−31.1696	−1.26513	−0.632567	−	0.774505i	$-0.717999\pi$
$607$	−31.1696	−1.26513	−0.632567	+	0.774505i	$0.717999\pi$
$608$	0	0
$609$	−15.1020	−0.611963
$610$	0	0
$611$	2.62972	0.106387
$612$	0	0
$613$	17.3941	0.702541	0.351271	−	0.936274i	$-0.385750\pi$
$613$	17.3941	0.702541	0.351271	+	0.936274i	$0.385750\pi$
$614$	0	0
$615$	−23.0102	−0.927862
$616$	0	0
$617$	42.8601	1.72548	0.862741	−	0.505646i	$-0.168746\pi$
$617$	42.8601	1.72548	0.862741	+	0.505646i	$0.168746\pi$
$618$	0	0
$619$	−36.5684	−1.46981	−0.734903	−	0.678172i	$-0.762772\pi$
$619$	−36.5684	−1.46981	−0.734903	+	0.678172i	$0.762772\pi$
$620$	0	0
$621$	−0.513812	−0.0206185
$622$	0	0
$623$	11.8570	0.475040
$624$	0	0
$625$	−26.3113	−1.05245
$626$	0	0
$627$	4.35203	0.173803
$628$	0	0
$629$	−4.78719	−0.190878
$630$	0	0
$631$	−22.4052	−0.891938	−0.445969	−	0.895048i	$-0.647141\pi$
$631$	−22.4052	−0.891938	−0.445969	+	0.895048i	$0.647141\pi$
$632$	0	0
$633$	18.3359	0.728788
$634$	0	0
$635$	16.3333	0.648167
$636$	0	0
$637$	11.9402	0.473088
$638$	0	0
$639$	3.83822	0.151838
$640$	0	0
$641$	5.58470	0.220582	0.110291	−	0.993899i	$-0.464822\pi$
$641$	5.58470	0.220582	0.110291	+	0.993899i	$0.464822\pi$
$642$	0	0
$643$	17.5581	0.692425	0.346212	−	0.938156i	$-0.387468\pi$
$643$	17.5581	0.692425	0.346212	+	0.938156i	$0.387468\pi$
$644$	0	0
$645$	−32.2281	−1.26898
$646$	0	0
$647$	−33.5976	−1.32086	−0.660430	−	0.750888i	$-0.729626\pi$
$647$	−33.5976	−1.32086	−0.660430	+	0.750888i	$0.729626\pi$
$648$	0	0
$649$	3.49556	0.137213
$650$	0	0
$651$	−33.0145	−1.29394
$652$	0	0
$653$	8.50874	0.332973	0.166486	−	0.986044i	$-0.446758\pi$
$653$	8.50874	0.332973	0.166486	+	0.986044i	$0.446758\pi$
$654$	0	0
$655$	−35.6851	−1.39433
$656$	0	0
$657$	7.56053	0.294965
$658$	0	0
$659$	−43.5930	−1.69814	−0.849071	−	0.528280i	$-0.822837\pi$
$659$	−43.5930	−1.69814	−0.849071	+	0.528280i	$0.822837\pi$
$660$	0	0
$661$	−37.7360	−1.46776	−0.733880	−	0.679279i	$-0.762293\pi$
$661$	−37.7360	−1.46776	−0.733880	+	0.679279i	$0.762293\pi$
$662$	0	0
$663$	−2.60425	−0.101141
$664$	0	0
$665$	−59.0567	−2.29012
$666$	0	0
$667$	−1.78298	−0.0690371
$668$	0	0
$669$	−7.58600	−0.293292
$670$	0	0
$671$	−4.39790	−0.169779
$672$	0	0
$673$	10.6910	0.412107	0.206053	−	0.978541i	$-0.433938\pi$
$673$	10.6910	0.412107	0.206053	+	0.978541i	$0.433938\pi$
$674$	0	0
$675$	−4.72231	−0.181762
$676$	0	0
$677$	−1.25272	−0.0481460	−0.0240730	−	0.999710i	$-0.507663\pi$
$677$	−1.25272	−0.0481460	−0.0240730	+	0.999710i	$0.507663\pi$
$678$	0	0
$679$	−48.2974	−1.85348
$680$	0	0
$681$	−18.3155	−0.701853
$682$	0	0
$683$	18.5788	0.710897	0.355449	−	0.934696i	$-0.384328\pi$
$683$	18.5788	0.710897	0.355449	+	0.934696i	$0.384328\pi$
$684$	0	0
$685$	37.9306	1.44925
$686$	0	0
$687$	0.629720	0.0240253
$688$	0	0
$689$	−8.94019	−0.340594
$690$	0	0
$691$	33.5304	1.27556	0.637778	−	0.770220i	$-0.279854\pi$
$691$	33.5304	1.27556	0.637778	+	0.770220i	$0.279854\pi$
$692$	0	0
$693$	4.35203	0.165320
$694$	0	0
$695$	33.4554	1.26903
$696$	0	0
$697$	19.2185	0.727952
$698$	0	0
$699$	−6.44247	−0.243677
$700$	0	0
$701$	−9.26169	−0.349809	−0.174905	−	0.984585i	$-0.555962\pi$
$701$	−9.26169	−0.349809	−0.174905	+	0.984585i	$0.555962\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−8.19963	−0.308816
$706$	0	0
$707$	−43.9547	−1.65309
$708$	0	0
$709$	−37.5844	−1.41151	−0.705755	−	0.708456i	$-0.749392\pi$
$709$	−37.5844	−1.41151	−0.705755	+	0.708456i	$0.749392\pi$
$710$	0	0
$711$	−5.63187	−0.211212
$712$	0	0
$713$	−3.89778	−0.145973
$714$	0	0
$715$	−3.11806	−0.116609
$716$	0	0
$717$	−13.1478	−0.491015
$718$	0	0
$719$	−2.91543	−0.108727	−0.0543635	−	0.998521i	$-0.517313\pi$
$719$	−2.91543	−0.108727	−0.0543635	+	0.998521i	$0.517313\pi$
$720$	0	0
$721$	−42.3526	−1.57729
$722$	0	0
$723$	−18.9954	−0.706448
$724$	0	0
$725$	−16.3869	−0.608593
$726$	0	0
$727$	31.4124	1.16502	0.582511	−	0.812822i	$-0.302070\pi$
$727$	31.4124	1.16502	0.582511	+	0.812822i	$0.302070\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	26.9174	0.995575
$732$	0	0
$733$	−2.85647	−0.105506	−0.0527530	−	0.998608i	$-0.516800\pi$
$733$	−2.85647	−0.105506	−0.0527530	+	0.998608i	$0.516800\pi$
$734$	0	0
$735$	−37.2303	−1.37326
$736$	0	0
$737$	−8.79451	−0.323950
$738$	0	0
$739$	38.7047	1.42377	0.711887	−	0.702294i	$-0.247841\pi$
$739$	38.7047	1.42377	0.711887	+	0.702294i	$0.247841\pi$
$740$	0	0
$741$	−4.35203	−0.159876
$742$	0	0
$743$	−35.7012	−1.30975	−0.654875	−	0.755737i	$-0.727279\pi$
$743$	−35.7012	−1.30975	−0.654875	+	0.755737i	$0.727279\pi$
$744$	0	0
$745$	30.0830	1.10215
$746$	0	0
$747$	−3.49556	−0.127896
$748$	0	0
$749$	−32.6209	−1.19194
$750$	0	0
$751$	−28.0248	−1.02264	−0.511319	−	0.859391i	$-0.670843\pi$
$751$	−28.0248	−1.02264	−0.511319	+	0.859391i	$0.670843\pi$
$752$	0	0
$753$	−18.3848	−0.669980
$754$	0	0
$755$	−0.374843	−0.0136419
$756$	0	0
$757$	−10.7864	−0.392040	−0.196020	−	0.980600i	$-0.562802\pi$
$757$	−10.7864	−0.392040	−0.196020	+	0.980600i	$0.562802\pi$
$758$	0	0
$759$	0.513812	0.0186502
$760$	0	0
$761$	−35.7194	−1.29483	−0.647413	−	0.762139i	$-0.724149\pi$
$761$	−35.7194	−1.29483	−0.647413	+	0.762139i	$0.724149\pi$
$762$	0	0
$763$	−17.9126	−0.648478
$764$	0	0
$765$	8.12022	0.293587
$766$	0	0
$767$	−3.49556	−0.126218
$768$	0	0
$769$	44.9650	1.62148	0.810739	−	0.585408i	$-0.199066\pi$
$769$	44.9650	1.62148	0.810739	+	0.585408i	$0.199066\pi$
$770$	0	0
$771$	−6.23182	−0.224433
$772$	0	0
$773$	−17.5407	−0.630896	−0.315448	−	0.948943i	$-0.602155\pi$
$773$	−17.5407	−0.630896	−0.315448	+	0.948943i	$0.602155\pi$
$774$	0	0
$775$	−35.8235	−1.28682
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	32.1165	1.15069
$780$	0	0
$781$	−3.83822	−0.137342
$782$	0	0
$783$	3.47010	0.124011
$784$	0	0
$785$	−29.4782	−1.05212
$786$	0	0
$787$	47.0664	1.67774	0.838868	−	0.544334i	$-0.183218\pi$
$787$	47.0664	1.67774	0.838868	+	0.544334i	$0.183218\pi$
$788$	0	0
$789$	−17.1254	−0.609680
$790$	0	0
$791$	−41.3250	−1.46935
$792$	0	0
$793$	4.39790	0.156174
$794$	0	0
$795$	27.8761	0.988662
$796$	0	0
$797$	18.3993	0.651735	0.325868	−	0.945415i	$-0.394344\pi$
$797$	18.3993	0.651735	0.325868	+	0.945415i	$0.394344\pi$
$798$	0	0
$799$	6.84845	0.242281
$800$	0	0
$801$	−2.72447	−0.0962643
$802$	0	0
$803$	−7.56053	−0.266805
$804$	0	0
$805$	−6.97238	−0.245744
$806$	0	0
$807$	26.0248	0.916115
$808$	0	0
$809$	20.0719	0.705692	0.352846	−	0.935681i	$-0.385214\pi$
$809$	20.0719	0.705692	0.352846	+	0.935681i	$0.385214\pi$
$810$	0	0
$811$	−25.1860	−0.884401	−0.442201	−	0.896916i	$-0.645802\pi$
$811$	−25.1860	−0.884401	−0.442201	+	0.896916i	$0.645802\pi$
$812$	0	0
$813$	16.3198	0.572362
$814$	0	0
$815$	60.0020	2.10178
$816$	0	0
$817$	44.9824	1.57373
$818$	0	0
$819$	−4.35203	−0.152072
$820$	0	0
$821$	−15.2557	−0.532429	−0.266214	−	0.963914i	$-0.585773\pi$
$821$	−15.2557	−0.532429	−0.266214	+	0.963914i	$0.585773\pi$
$822$	0	0
$823$	−15.8379	−0.552074	−0.276037	−	0.961147i	$-0.589021\pi$
$823$	−15.8379	−0.552074	−0.276037	+	0.961147i	$0.589021\pi$
$824$	0	0
$825$	4.72231	0.164410
$826$	0	0
$827$	−46.2410	−1.60796	−0.803979	−	0.594658i	$-0.797288\pi$
$827$	−46.2410	−1.60796	−0.803979	+	0.594658i	$0.797288\pi$
$828$	0	0
$829$	−36.4170	−1.26481	−0.632407	−	0.774636i	$-0.717933\pi$
$829$	−36.4170	−1.26481	−0.632407	+	0.774636i	$0.717933\pi$
$830$	0	0
$831$	8.64882	0.300024
$832$	0	0
$833$	31.0953	1.07739
$834$	0	0
$835$	33.5482	1.16098
$836$	0	0
$837$	7.58600	0.262211
$838$	0	0
$839$	14.4271	0.498080	0.249040	−	0.968493i	$-0.419885\pi$
$839$	14.4271	0.498080	0.249040	+	0.968493i	$0.419885\pi$
$840$	0	0
$841$	−16.9584	−0.584774
$842$	0	0
$843$	−22.5517	−0.776721
$844$	0	0
$845$	3.11806	0.107265
$846$	0	0
$847$	−4.35203	−0.149538
$848$	0	0
$849$	−18.9104	−0.649004
$850$	0	0
$851$	−0.944499	−0.0323770
$852$	0	0
$853$	18.1674	0.622041	0.311021	−	0.950403i	$-0.399329\pi$
$853$	18.1674	0.622041	0.311021	+	0.950403i	$0.399329\pi$
$854$	0	0
$855$	13.5699	0.464081
$856$	0	0
$857$	−40.7034	−1.39040	−0.695200	−	0.718816i	$-0.744684\pi$
$857$	−40.7034	−1.39040	−0.695200	+	0.718816i	$0.744684\pi$
$858$	0	0
$859$	−4.52319	−0.154329	−0.0771646	−	0.997018i	$-0.524587\pi$
$859$	−4.52319	−0.154329	−0.0771646	+	0.997018i	$0.524587\pi$
$860$	0	0
$861$	32.1165	1.09453
$862$	0	0
$863$	−39.8761	−1.35740	−0.678699	−	0.734417i	$-0.737456\pi$
$863$	−39.8761	−1.35740	−0.678699	+	0.734417i	$0.737456\pi$
$864$	0	0
$865$	−55.3271	−1.88118
$866$	0	0
$867$	10.2179	0.347017
$868$	0	0
$869$	5.63187	0.191048
$870$	0	0
$871$	8.79451	0.297990
$872$	0	0
$873$	11.0977	0.375599
$874$	0	0
$875$	3.76818	0.127388
$876$	0	0
$877$	39.9695	1.34967	0.674837	−	0.737966i	$-0.264214\pi$
$877$	39.9695	1.34967	0.674837	+	0.737966i	$0.264214\pi$
$878$	0	0
$879$	−28.5882	−0.964255
$880$	0	0
$881$	−20.8893	−0.703777	−0.351888	−	0.936042i	$-0.614460\pi$
$881$	−20.8893	−0.703777	−0.351888	+	0.936042i	$0.614460\pi$
$882$	0	0
$883$	34.5275	1.16194	0.580971	−	0.813924i	$-0.302673\pi$
$883$	34.5275	1.16194	0.580971	+	0.813924i	$0.302673\pi$
$884$	0	0
$885$	10.8994	0.366379
$886$	0	0
$887$	−21.8527	−0.733741	−0.366870	−	0.930272i	$-0.619571\pi$
$887$	−21.8527	−0.733741	−0.366870	+	0.930272i	$0.619571\pi$
$888$	0	0
$889$	−22.7972	−0.764592
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	11.4446	0.382980
$894$	0	0
$895$	−35.7143	−1.19380
$896$	0	0
$897$	−0.513812	−0.0171557
$898$	0	0
$899$	26.3242	0.877960
$900$	0	0
$901$	−23.2825	−0.775653
$902$	0	0
$903$	44.9824	1.49692
$904$	0	0
$905$	−26.2740	−0.873377
$906$	0	0
$907$	1.42157	0.0472023	0.0236012	−	0.999721i	$-0.492487\pi$
$907$	1.42157	0.0472023	0.0236012	+	0.999721i	$0.492487\pi$
$908$	0	0
$909$	10.0998	0.334990
$910$	0	0
$911$	39.7200	1.31598	0.657990	−	0.753026i	$-0.271407\pi$
$911$	39.7200	1.31598	0.657990	+	0.753026i	$0.271407\pi$
$912$	0	0
$913$	3.49556	0.115686
$914$	0	0
$915$	−13.7129	−0.453336
$916$	0	0
$917$	49.8074	1.64478
$918$	0	0
$919$	0.949512	0.0313215	0.0156607	−	0.999877i	$-0.495015\pi$
$919$	0.949512	0.0313215	0.0156607	+	0.999877i	$0.495015\pi$
$920$	0	0
$921$	1.93503	0.0637614
$922$	0	0
$923$	3.83822	0.126337
$924$	0	0
$925$	−8.68066	−0.285418
$926$	0	0
$927$	9.73169	0.319631
$928$	0	0
$929$	−33.1516	−1.08767	−0.543834	−	0.839193i	$-0.683028\pi$
$929$	−33.1516	−1.08767	−0.543834	+	0.839193i	$0.683028\pi$
$930$	0	0
$931$	51.9641	1.70305
$932$	0	0
$933$	27.6536	0.905339
$934$	0	0
$935$	−8.12022	−0.265559
$936$	0	0
$937$	47.4037	1.54861	0.774305	−	0.632812i	$-0.218100\pi$
$937$	47.4037	1.54861	0.774305	+	0.632812i	$0.218100\pi$
$938$	0	0
$939$	0.458565	0.0149647
$940$	0	0
$941$	46.1959	1.50594	0.752972	−	0.658053i	$-0.228620\pi$
$941$	46.1959	1.50594	0.752972	+	0.658053i	$0.228620\pi$
$942$	0	0
$943$	3.79175	0.123476
$944$	0	0
$945$	13.5699	0.441429
$946$	0	0
$947$	24.9254	0.809967	0.404983	−	0.914324i	$-0.367277\pi$
$947$	24.9254	0.809967	0.404983	+	0.914324i	$0.367277\pi$
$948$	0	0
$949$	7.56053	0.245425
$950$	0	0
$951$	−31.1559	−1.01030
$952$	0	0
$953$	12.5869	0.407728	0.203864	−	0.978999i	$-0.434650\pi$
$953$	12.5869	0.407728	0.203864	+	0.978999i	$0.434650\pi$
$954$	0	0
$955$	58.3339	1.88764
$956$	0	0
$957$	−3.47010	−0.112172
$958$	0	0
$959$	−52.9415	−1.70957
$960$	0	0
$961$	26.5474	0.856369
$962$	0	0
$963$	7.49556	0.241541
$964$	0	0
$965$	−81.7809	−2.63262
$966$	0	0
$967$	27.1361	0.872638	0.436319	−	0.899792i	$-0.356282\pi$
$967$	27.1361	0.872638	0.436319	+	0.899792i	$0.356282\pi$
$968$	0	0
$969$	−11.3338	−0.364094
$970$	0	0
$971$	18.9496	0.608121	0.304060	−	0.952653i	$-0.401658\pi$
$971$	18.9496	0.608121	0.304060	+	0.952653i	$0.401658\pi$
$972$	0	0
$973$	−46.6953	−1.49698
$974$	0	0
$975$	−4.72231	−0.151235
$976$	0	0
$977$	−9.49721	−0.303843	−0.151921	−	0.988393i	$-0.548546\pi$
$977$	−9.49721	−0.303843	−0.151921	+	0.988393i	$0.548546\pi$
$978$	0	0
$979$	2.72447	0.0870744
$980$	0	0
$981$	4.11591	0.131411
$982$	0	0
$983$	0.665866	0.0212378	0.0106189	−	0.999944i	$-0.496620\pi$
$983$	0.665866	0.0212378	0.0106189	+	0.999944i	$0.496620\pi$
$984$	0	0
$985$	0.361409	0.0115154
$986$	0	0
$987$	11.4446	0.364286
$988$	0	0
$989$	5.31073	0.168871
$990$	0	0
$991$	31.4694	0.999658	0.499829	−	0.866124i	$-0.333396\pi$
$991$	31.4694	0.999658	0.499829	+	0.866124i	$0.333396\pi$
$992$	0	0
$993$	11.6647	0.370167
$994$	0	0
$995$	13.9448	0.442078
$996$	0	0
$997$	34.7550	1.10070	0.550351	−	0.834933i	$-0.314494\pi$
$997$	34.7550	1.10070	0.550351	+	0.834933i	$0.314494\pi$
$998$	0	0
$999$	1.83822	0.0581588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.a.s.1.3	✓	4
4.3	odd	2		6864.2.a.cd.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.s.1.3	✓	4	1.1	even	1	trivial
6864.2.a.cd.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:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.11806 q^{5} -4.35203 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.11806 q^{5} -4.35203 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.11806 q^{15} +2.60425 q^{17} +4.35203 q^{19} +4.35203 q^{21} +0.513812 q^{23} +4.72231 q^{25} -1.00000 q^{27} -3.47010 q^{29} -7.58600 q^{31} +1.00000 q^{33} -13.5699 q^{35} -1.83822 q^{37} -1.00000 q^{39} +7.37966 q^{41} +10.3359 q^{43} +3.11806 q^{45} +2.62972 q^{47} +11.9402 q^{49} -2.60425 q^{51} -8.94019 q^{53} -3.11806 q^{55} -4.35203 q^{57} -3.49556 q^{59} +4.39790 q^{61} -4.35203 q^{63} +3.11806 q^{65} +8.79451 q^{67} -0.513812 q^{69} +3.83822 q^{71} +7.56053 q^{73} -4.72231 q^{75} +4.35203 q^{77} -5.63187 q^{79} +1.00000 q^{81} -3.49556 q^{83} +8.12022 q^{85} +3.47010 q^{87} -2.72447 q^{89} -4.35203 q^{91} +7.58600 q^{93} +13.5699 q^{95} +11.0977 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 2 q^{21} - 4 q^{23} + 8 q^{25} - 4 q^{27} + 10 q^{29} - 8 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} - 4 q^{39} + 2 q^{41} - 4 q^{43} + 4 q^{45} + 6 q^{47} - 8 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 12 q^{59} + 10 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} + 4 q^{69} + 6 q^{71} + 10 q^{73} - 8 q^{75} + 2 q^{77} - 8 q^{79} + 4 q^{81} + 12 q^{83} + 14 q^{85} - 10 q^{87} + 10 q^{89} - 2 q^{91} + 8 q^{93} + 2 q^{95} + 26 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 - 4 * q^15 + 8 * q^17 + 2 * q^19 + 2 * q^21 - 4 * q^23 + 8 * q^25 - 4 * q^27 + 10 * q^29 - 8 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 - 4 * q^39 + 2 * q^41 - 4 * q^43 + 4 * q^45 + 6 * q^47 - 8 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 12 * q^59 + 10 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 + 4 * q^69 + 6 * q^71 + 10 * q^73 - 8 * q^75 + 2 * q^77 - 8 * q^79 + 4 * q^81 + 12 * q^83 + 14 * q^85 - 10 * q^87 + 10 * q^89 - 2 * q^91 + 8 * q^93 + 2 * q^95 + 26 * q^97 - 4 * q^99

Introduction

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