LMFDB - Embedded newform 6384.2.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.864641 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.864641 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.52311 q^{11} +4.00000 q^{13} -0.864641 q^{15} -0.864641 q^{17} +1.00000 q^{19} +1.00000 q^{21} -3.52311 q^{23} -4.25240 q^{25} +1.00000 q^{27} -4.38776 q^{29} -1.52311 q^{31} -3.52311 q^{33} -0.864641 q^{35} +2.00000 q^{37} +4.00000 q^{39} +10.7755 q^{41} +5.52311 q^{43} -0.864641 q^{45} +6.38776 q^{47} +1.00000 q^{49} -0.864641 q^{51} -1.34153 q^{53} +3.04623 q^{55} +1.00000 q^{57} +10.5048 q^{59} +12.5048 q^{61} +1.00000 q^{63} -3.45856 q^{65} +3.52311 q^{67} -3.52311 q^{69} +4.11704 q^{71} +14.7755 q^{73} -4.25240 q^{75} -3.52311 q^{77} +5.25240 q^{79} +1.00000 q^{81} -8.11704 q^{83} +0.747604 q^{85} -4.38776 q^{87} -6.77551 q^{89} +4.00000 q^{91} -1.52311 q^{93} -0.864641 q^{95} +15.2524 q^{97} -3.52311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 12 q^{13} + 3 q^{19} + 3 q^{21} + 2 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 2 q^{33} + 6 q^{37} + 12 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{47} + 3 q^{49} - 14 q^{53} - 16 q^{55} + 3 q^{57} - 4 q^{59} + 2 q^{61} + 3 q^{63} - 2 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 2 q^{77} - 2 q^{79} + 3 q^{81} - 4 q^{83} + 20 q^{85} + 2 q^{87} + 10 q^{89} + 12 q^{91} + 8 q^{93} + 28 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 12 * q^13 + 3 * q^19 + 3 * q^21 + 2 * q^23 + 5 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 2 * q^33 + 6 * q^37 + 12 * q^39 + 2 * q^41 + 4 * q^43 + 4 * q^47 + 3 * q^49 - 14 * q^53 - 16 * q^55 + 3 * q^57 - 4 * q^59 + 2 * q^61 + 3 * q^63 - 2 * q^67 + 2 * q^69 - 8 * q^71 + 14 * q^73 + 5 * q^75 + 2 * q^77 - 2 * q^79 + 3 * q^81 - 4 * q^83 + 20 * q^85 + 2 * q^87 + 10 * q^89 + 12 * q^91 + 8 * q^93 + 28 * q^97 + 2 * 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
$5$	−0.864641	−0.386679	−0.193340	−	0.981132i	$-0.561932\pi$
$5$	−0.864641	−0.386679	−0.193340	+	0.981132i	$0.561932\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.52311	−1.06226	−0.531129	−	0.847291i	$-0.678232\pi$
$11$	−3.52311	−1.06226	−0.531129	+	0.847291i	$0.678232\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−0.864641	−0.223249
$16$	0	0
$17$	−0.864641	−0.209706	−0.104853	−	0.994488i	$-0.533437\pi$
$17$	−0.864641	−0.209706	−0.104853	+	0.994488i	$0.533437\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.52311	−0.734620	−0.367310	−	0.930099i	$-0.619721\pi$
$23$	−3.52311	−0.734620	−0.367310	+	0.930099i	$0.619721\pi$
$24$	0	0
$25$	−4.25240	−0.850479
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.38776	−0.814786	−0.407393	−	0.913253i	$-0.633562\pi$
$29$	−4.38776	−0.814786	−0.407393	+	0.913253i	$0.633562\pi$
$30$	0	0
$31$	−1.52311	−0.273559	−0.136780	−	0.990601i	$-0.543675\pi$
$31$	−1.52311	−0.273559	−0.136780	+	0.990601i	$0.543675\pi$
$32$	0	0
$33$	−3.52311	−0.613295
$34$	0	0
$35$	−0.864641	−0.146151
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	10.7755	1.68285	0.841426	−	0.540372i	$-0.181717\pi$
$41$	10.7755	1.68285	0.841426	+	0.540372i	$0.181717\pi$
$42$	0	0
$43$	5.52311	0.842267	0.421134	−	0.906999i	$-0.361632\pi$
$43$	5.52311	0.842267	0.421134	+	0.906999i	$0.361632\pi$
$44$	0	0
$45$	−0.864641	−0.128893
$46$	0	0
$47$	6.38776	0.931750	0.465875	−	0.884851i	$-0.345740\pi$
$47$	6.38776	0.931750	0.465875	+	0.884851i	$0.345740\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.864641	−0.121074
$52$	0	0
$53$	−1.34153	−0.184273	−0.0921364	−	0.995746i	$-0.529370\pi$
$53$	−1.34153	−0.184273	−0.0921364	+	0.995746i	$0.529370\pi$
$54$	0	0
$55$	3.04623	0.410753
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	10.5048	1.36761	0.683804	−	0.729666i	$-0.260324\pi$
$59$	10.5048	1.36761	0.683804	+	0.729666i	$0.260324\pi$
$60$	0	0
$61$	12.5048	1.60107	0.800537	−	0.599283i	$-0.204548\pi$
$61$	12.5048	1.60107	0.800537	+	0.599283i	$0.204548\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.45856	−0.428982
$66$	0	0
$67$	3.52311	0.430417	0.215208	−	0.976568i	$-0.430957\pi$
$67$	3.52311	0.430417	0.215208	+	0.976568i	$0.430957\pi$
$68$	0	0
$69$	−3.52311	−0.424133
$70$	0	0
$71$	4.11704	0.488602	0.244301	−	0.969699i	$-0.421441\pi$
$71$	4.11704	0.488602	0.244301	+	0.969699i	$0.421441\pi$
$72$	0	0
$73$	14.7755	1.72934	0.864671	−	0.502338i	$-0.167527\pi$
$73$	14.7755	1.72934	0.864671	+	0.502338i	$0.167527\pi$
$74$	0	0
$75$	−4.25240	−0.491024
$76$	0	0
$77$	−3.52311	−0.401496
$78$	0	0
$79$	5.25240	0.590941	0.295470	−	0.955352i	$-0.404524\pi$
$79$	5.25240	0.590941	0.295470	+	0.955352i	$0.404524\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.11704	−0.890961	−0.445480	−	0.895292i	$-0.646967\pi$
$83$	−8.11704	−0.890961	−0.445480	+	0.895292i	$0.646967\pi$
$84$	0	0
$85$	0.747604	0.0810890
$86$	0	0
$87$	−4.38776	−0.470417
$88$	0	0
$89$	−6.77551	−0.718203	−0.359101	−	0.933299i	$-0.616917\pi$
$89$	−6.77551	−0.718203	−0.359101	+	0.933299i	$0.616917\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	−1.52311	−0.157940
$94$	0	0
$95$	−0.864641	−0.0887103
$96$	0	0
$97$	15.2524	1.54865	0.774323	−	0.632790i	$-0.218091\pi$
$97$	15.2524	1.54865	0.774323	+	0.632790i	$0.218091\pi$
$98$	0	0
$99$	−3.52311	−0.354086
$100$	0	0
$101$	0.864641	0.0860350	0.0430175	−	0.999074i	$-0.486303\pi$
$101$	0.864641	0.0860350	0.0430175	+	0.999074i	$0.486303\pi$
$102$	0	0
$103$	−10.5048	−1.03507	−0.517534	−	0.855663i	$-0.673150\pi$
$103$	−10.5048	−1.03507	−0.517534	+	0.855663i	$0.673150\pi$
$104$	0	0
$105$	−0.864641	−0.0843803
$106$	0	0
$107$	−15.1633	−1.46589	−0.732944	−	0.680289i	$-0.761854\pi$
$107$	−15.1633	−1.46589	−0.732944	+	0.680289i	$0.761854\pi$
$108$	0	0
$109$	3.72928	0.357200	0.178600	−	0.983922i	$-0.442843\pi$
$109$	3.72928	0.357200	0.178600	+	0.983922i	$0.442843\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	6.65847	0.626376	0.313188	−	0.949691i	$-0.398603\pi$
$113$	6.65847	0.626376	0.313188	+	0.949691i	$0.398603\pi$
$114$	0	0
$115$	3.04623	0.284062
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−0.864641	−0.0792615
$120$	0	0
$121$	1.41233	0.128394
$122$	0	0
$123$	10.7755	0.971595
$124$	0	0
$125$	8.00000	0.715542
$126$	0	0
$127$	−14.5693	−1.29282	−0.646410	−	0.762990i	$-0.723730\pi$
$127$	−14.5693	−1.29282	−0.646410	+	0.762990i	$0.723730\pi$
$128$	0	0
$129$	5.52311	0.486283
$130$	0	0
$131$	−21.8463	−1.90872	−0.954361	−	0.298656i	$-0.903462\pi$
$131$	−21.8463	−1.90872	−0.954361	+	0.298656i	$0.903462\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−0.864641	−0.0744164
$136$	0	0
$137$	−13.0462	−1.11461	−0.557307	−	0.830306i	$-0.688165\pi$
$137$	−13.0462	−1.11461	−0.557307	+	0.830306i	$0.688165\pi$
$138$	0	0
$139$	8.77551	0.744329	0.372165	−	0.928167i	$-0.378616\pi$
$139$	8.77551	0.744329	0.372165	+	0.928167i	$0.378616\pi$
$140$	0	0
$141$	6.38776	0.537946
$142$	0	0
$143$	−14.0925	−1.17847
$144$	0	0
$145$	3.79383	0.315061
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	11.1878	0.916544	0.458272	−	0.888812i	$-0.348469\pi$
$149$	11.1878	0.916544	0.458272	+	0.888812i	$0.348469\pi$
$150$	0	0
$151$	−7.52311	−0.612222	−0.306111	−	0.951996i	$-0.599028\pi$
$151$	−7.52311	−0.612222	−0.306111	+	0.951996i	$0.599028\pi$
$152$	0	0
$153$	−0.864641	−0.0699021
$154$	0	0
$155$	1.31695	0.105780
$156$	0	0
$157$	13.0462	1.04120	0.520601	−	0.853800i	$-0.325708\pi$
$157$	13.0462	1.04120	0.520601	+	0.853800i	$0.325708\pi$
$158$	0	0
$159$	−1.34153	−0.106390
$160$	0	0
$161$	−3.52311	−0.277660
$162$	0	0
$163$	−2.06455	−0.161708	−0.0808541	−	0.996726i	$-0.525765\pi$
$163$	−2.06455	−0.161708	−0.0808541	+	0.996726i	$0.525765\pi$
$164$	0	0
$165$	3.04623	0.237149
$166$	0	0
$167$	15.8217	1.22432	0.612161	−	0.790733i	$-0.290300\pi$
$167$	15.8217	1.22432	0.612161	+	0.790733i	$0.290300\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	0.270718	0.0205823	0.0102912	−	0.999947i	$-0.496724\pi$
$173$	0.270718	0.0205823	0.0102912	+	0.999947i	$0.496724\pi$
$174$	0	0
$175$	−4.25240	−0.321451
$176$	0	0
$177$	10.5048	0.789589
$178$	0	0
$179$	11.7047	0.874851	0.437425	−	0.899255i	$-0.355890\pi$
$179$	11.7047	0.874851	0.437425	+	0.899255i	$0.355890\pi$
$180$	0	0
$181$	19.7938	1.47126	0.735632	−	0.677381i	$-0.236885\pi$
$181$	19.7938	1.47126	0.735632	+	0.677381i	$0.236885\pi$
$182$	0	0
$183$	12.5048	0.924381
$184$	0	0
$185$	−1.72928	−0.127139
$186$	0	0
$187$	3.04623	0.222762
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	10.9817	0.794606	0.397303	−	0.917687i	$-0.369946\pi$
$191$	10.9817	0.794606	0.397303	+	0.917687i	$0.369946\pi$
$192$	0	0
$193$	−1.45856	−0.104990	−0.0524949	−	0.998621i	$-0.516717\pi$
$193$	−1.45856	−0.104990	−0.0524949	+	0.998621i	$0.516717\pi$
$194$	0	0
$195$	−3.45856	−0.247673
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	−4.77551	−0.338527	−0.169263	−	0.985571i	$-0.554139\pi$
$199$	−4.77551	−0.338527	−0.169263	+	0.985571i	$0.554139\pi$
$200$	0	0
$201$	3.52311	0.248501
$202$	0	0
$203$	−4.38776	−0.307960
$204$	0	0
$205$	−9.31695	−0.650724
$206$	0	0
$207$	−3.52311	−0.244873
$208$	0	0
$209$	−3.52311	−0.243699
$210$	0	0
$211$	−2.33527	−0.160767	−0.0803833	−	0.996764i	$-0.525614\pi$
$211$	−2.33527	−0.160767	−0.0803833	+	0.996764i	$0.525614\pi$
$212$	0	0
$213$	4.11704	0.282095
$214$	0	0
$215$	−4.77551	−0.325687
$216$	0	0
$217$	−1.52311	−0.103396
$218$	0	0
$219$	14.7755	0.998436
$220$	0	0
$221$	−3.45856	−0.232648
$222$	0	0
$223$	−8.98168	−0.601458	−0.300729	−	0.953710i	$-0.597230\pi$
$223$	−8.98168	−0.601458	−0.300729	+	0.953710i	$0.597230\pi$
$224$	0	0
$225$	−4.25240	−0.283493
$226$	0	0
$227$	8.77551	0.582451	0.291226	−	0.956654i	$-0.405937\pi$
$227$	8.77551	0.582451	0.291226	+	0.956654i	$0.405937\pi$
$228$	0	0
$229$	11.1878	0.739314	0.369657	−	0.929168i	$-0.379475\pi$
$229$	11.1878	0.739314	0.369657	+	0.929168i	$0.379475\pi$
$230$	0	0
$231$	−3.52311	−0.231804
$232$	0	0
$233$	13.8217	0.905492	0.452746	−	0.891639i	$-0.350444\pi$
$233$	13.8217	0.905492	0.452746	+	0.891639i	$0.350444\pi$
$234$	0	0
$235$	−5.52311	−0.360288
$236$	0	0
$237$	5.25240	0.341180
$238$	0	0
$239$	−4.84006	−0.313078	−0.156539	−	0.987672i	$-0.550034\pi$
$239$	−4.84006	−0.313078	−0.156539	+	0.987672i	$0.550034\pi$
$240$	0	0
$241$	7.79383	0.502045	0.251022	−	0.967981i	$-0.419233\pi$
$241$	7.79383	0.502045	0.251022	+	0.967981i	$0.419233\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.864641	−0.0552399
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−8.11704	−0.514396
$250$	0	0
$251$	13.8463	0.873972	0.436986	−	0.899468i	$-0.356046\pi$
$251$	13.8463	0.873972	0.436986	+	0.899468i	$0.356046\pi$
$252$	0	0
$253$	12.4123	0.780357
$254$	0	0
$255$	0.747604	0.0468168
$256$	0	0
$257$	13.0462	0.813801	0.406901	−	0.913472i	$-0.366610\pi$
$257$	13.0462	0.813801	0.406901	+	0.913472i	$0.366610\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−4.38776	−0.271595
$262$	0	0
$263$	23.8863	1.47289	0.736446	−	0.676496i	$-0.236503\pi$
$263$	23.8863	1.47289	0.736446	+	0.676496i	$0.236503\pi$
$264$	0	0
$265$	1.15994	0.0712545
$266$	0	0
$267$	−6.77551	−0.414655
$268$	0	0
$269$	−26.5972	−1.62166	−0.810831	−	0.585280i	$-0.800985\pi$
$269$	−26.5972	−1.62166	−0.810831	+	0.585280i	$0.800985\pi$
$270$	0	0
$271$	17.5510	1.06615	0.533074	−	0.846068i	$-0.321037\pi$
$271$	17.5510	1.06615	0.533074	+	0.846068i	$0.321037\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	14.9817	0.903429
$276$	0	0
$277$	24.7110	1.48474	0.742369	−	0.669991i	$-0.233702\pi$
$277$	24.7110	1.48474	0.742369	+	0.669991i	$0.233702\pi$
$278$	0	0
$279$	−1.52311	−0.0911865
$280$	0	0
$281$	−11.0708	−0.660429	−0.330215	−	0.943906i	$-0.607121\pi$
$281$	−11.0708	−0.660429	−0.330215	+	0.943906i	$0.607121\pi$
$282$	0	0
$283$	−13.7293	−0.816121	−0.408061	−	0.912955i	$-0.633795\pi$
$283$	−13.7293	−0.816121	−0.408061	+	0.912955i	$0.633795\pi$
$284$	0	0
$285$	−0.864641	−0.0512169
$286$	0	0
$287$	10.7755	0.636058
$288$	0	0
$289$	−16.2524	−0.956023
$290$	0	0
$291$	15.2524	0.894111
$292$	0	0
$293$	32.7389	1.91262	0.956312	−	0.292346i	$-0.0944360\pi$
$293$	32.7389	1.91262	0.956312	+	0.292346i	$0.0944360\pi$
$294$	0	0
$295$	−9.08287	−0.528825
$296$	0	0
$297$	−3.52311	−0.204432
$298$	0	0
$299$	−14.0925	−0.814988
$300$	0	0
$301$	5.52311	0.318347
$302$	0	0
$303$	0.864641	0.0496723
$304$	0	0
$305$	−10.8122	−0.619102
$306$	0	0
$307$	−16.5693	−0.945662	−0.472831	−	0.881153i	$-0.656768\pi$
$307$	−16.5693	−0.945662	−0.472831	+	0.881153i	$0.656768\pi$
$308$	0	0
$309$	−10.5048	−0.597597
$310$	0	0
$311$	−7.34153	−0.416300	−0.208150	−	0.978097i	$-0.566744\pi$
$311$	−7.34153	−0.416300	−0.208150	+	0.978097i	$0.566744\pi$
$312$	0	0
$313$	−15.5510	−0.878996	−0.439498	−	0.898244i	$-0.644844\pi$
$313$	−15.5510	−0.878996	−0.439498	+	0.898244i	$0.644844\pi$
$314$	0	0
$315$	−0.864641	−0.0487170
$316$	0	0
$317$	1.57560	0.0884945	0.0442473	−	0.999021i	$-0.485911\pi$
$317$	1.57560	0.0884945	0.0442473	+	0.999021i	$0.485911\pi$
$318$	0	0
$319$	15.4586	0.865513
$320$	0	0
$321$	−15.1633	−0.846331
$322$	0	0
$323$	−0.864641	−0.0481099
$324$	0	0
$325$	−17.0096	−0.943522
$326$	0	0
$327$	3.72928	0.206230
$328$	0	0
$329$	6.38776	0.352168
$330$	0	0
$331$	−2.02791	−0.111464	−0.0557319	−	0.998446i	$-0.517749\pi$
$331$	−2.02791	−0.111464	−0.0557319	+	0.998446i	$0.517749\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	−3.04623	−0.166433
$336$	0	0
$337$	−5.28030	−0.287636	−0.143818	−	0.989604i	$-0.545938\pi$
$337$	−5.28030	−0.287636	−0.143818	+	0.989604i	$0.545938\pi$
$338$	0	0
$339$	6.65847	0.361639
$340$	0	0
$341$	5.36611	0.290591
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.04623	0.164003
$346$	0	0
$347$	−4.29862	−0.230762	−0.115381	−	0.993321i	$-0.536809\pi$
$347$	−4.29862	−0.230762	−0.115381	+	0.993321i	$0.536809\pi$
$348$	0	0
$349$	24.0925	1.28964	0.644820	−	0.764335i	$-0.276932\pi$
$349$	24.0925	1.28964	0.644820	+	0.764335i	$0.276932\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−22.0525	−1.17374	−0.586868	−	0.809683i	$-0.699639\pi$
$353$	−22.0525	−1.17374	−0.586868	+	0.809683i	$0.699639\pi$
$354$	0	0
$355$	−3.55976	−0.188932
$356$	0	0
$357$	−0.864641	−0.0457616
$358$	0	0
$359$	−33.0741	−1.74559	−0.872793	−	0.488090i	$-0.837694\pi$
$359$	−33.0741	−1.74559	−0.872793	+	0.488090i	$0.837694\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.41233	0.0741284
$364$	0	0
$365$	−12.7755	−0.668701
$366$	0	0
$367$	7.82174	0.408291	0.204146	−	0.978941i	$-0.434558\pi$
$367$	7.82174	0.408291	0.204146	+	0.978941i	$0.434558\pi$
$368$	0	0
$369$	10.7755	0.560951
$370$	0	0
$371$	−1.34153	−0.0696486
$372$	0	0
$373$	9.45856	0.489746	0.244873	−	0.969555i	$-0.421254\pi$
$373$	9.45856	0.489746	0.244873	+	0.969555i	$0.421254\pi$
$374$	0	0
$375$	8.00000	0.413118
$376$	0	0
$377$	−17.5510	−0.903924
$378$	0	0
$379$	−8.06455	−0.414248	−0.207124	−	0.978315i	$-0.566410\pi$
$379$	−8.06455	−0.414248	−0.207124	+	0.978315i	$0.566410\pi$
$380$	0	0
$381$	−14.5693	−0.746410
$382$	0	0
$383$	12.3632	0.631729	0.315864	−	0.948804i	$-0.397706\pi$
$383$	12.3632	0.631729	0.315864	+	0.948804i	$0.397706\pi$
$384$	0	0
$385$	3.04623	0.155250
$386$	0	0
$387$	5.52311	0.280756
$388$	0	0
$389$	20.8680	1.05805	0.529024	−	0.848607i	$-0.322558\pi$
$389$	20.8680	1.05805	0.529024	+	0.848607i	$0.322558\pi$
$390$	0	0
$391$	3.04623	0.154054
$392$	0	0
$393$	−21.8463	−1.10200
$394$	0	0
$395$	−4.54144	−0.228504
$396$	0	0
$397$	24.5048	1.22986	0.614930	−	0.788582i	$-0.289184\pi$
$397$	24.5048	1.22986	0.614930	+	0.788582i	$0.289184\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−13.3415	−0.666244	−0.333122	−	0.942884i	$-0.608102\pi$
$401$	−13.3415	−0.666244	−0.333122	+	0.942884i	$0.608102\pi$
$402$	0	0
$403$	−6.09246	−0.303487
$404$	0	0
$405$	−0.864641	−0.0429644
$406$	0	0
$407$	−7.04623	−0.349269
$408$	0	0
$409$	11.4586	0.566590	0.283295	−	0.959033i	$-0.408573\pi$
$409$	11.4586	0.566590	0.283295	+	0.959033i	$0.408573\pi$
$410$	0	0
$411$	−13.0462	−0.643523
$412$	0	0
$413$	10.5048	0.516907
$414$	0	0
$415$	7.01832	0.344516
$416$	0	0
$417$	8.77551	0.429739
$418$	0	0
$419$	5.61224	0.274176	0.137088	−	0.990559i	$-0.456226\pi$
$419$	5.61224	0.274176	0.137088	+	0.990559i	$0.456226\pi$
$420$	0	0
$421$	36.3265	1.77045	0.885223	−	0.465166i	$-0.154005\pi$
$421$	36.3265	1.77045	0.885223	+	0.465166i	$0.154005\pi$
$422$	0	0
$423$	6.38776	0.310583
$424$	0	0
$425$	3.67680	0.178351
$426$	0	0
$427$	12.5048	0.605149
$428$	0	0
$429$	−14.0925	−0.680390
$430$	0	0
$431$	−13.6681	−0.658367	−0.329184	−	0.944266i	$-0.606774\pi$
$431$	−13.6681	−0.658367	−0.329184	+	0.944266i	$0.606774\pi$
$432$	0	0
$433$	3.38150	0.162504	0.0812522	−	0.996694i	$-0.474108\pi$
$433$	3.38150	0.162504	0.0812522	+	0.996694i	$0.474108\pi$
$434$	0	0
$435$	3.79383	0.181900
$436$	0	0
$437$	−3.52311	−0.168533
$438$	0	0
$439$	−37.0375	−1.76770	−0.883852	−	0.467768i	$-0.845058\pi$
$439$	−37.0375	−1.76770	−0.883852	+	0.467768i	$0.845058\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	35.2158	1.67315	0.836575	−	0.547852i	$-0.184554\pi$
$443$	35.2158	1.67315	0.836575	+	0.547852i	$0.184554\pi$
$444$	0	0
$445$	5.85838	0.277714
$446$	0	0
$447$	11.1878	0.529167
$448$	0	0
$449$	20.0804	0.947652	0.473826	−	0.880618i	$-0.342873\pi$
$449$	20.0804	0.947652	0.473826	+	0.880618i	$0.342873\pi$
$450$	0	0
$451$	−37.9634	−1.78762
$452$	0	0
$453$	−7.52311	−0.353467
$454$	0	0
$455$	−3.45856	−0.162140
$456$	0	0
$457$	−13.3815	−0.625960	−0.312980	−	0.949760i	$-0.601327\pi$
$457$	−13.3815	−0.625960	−0.312980	+	0.949760i	$0.601327\pi$
$458$	0	0
$459$	−0.864641	−0.0403580
$460$	0	0
$461$	4.32320	0.201352	0.100676	−	0.994919i	$-0.467899\pi$
$461$	4.32320	0.201352	0.100676	+	0.994919i	$0.467899\pi$
$462$	0	0
$463$	27.1020	1.25954	0.629769	−	0.776782i	$-0.283149\pi$
$463$	27.1020	1.25954	0.629769	+	0.776782i	$0.283149\pi$
$464$	0	0
$465$	1.31695	0.0610720
$466$	0	0
$467$	17.6681	0.817580	0.408790	−	0.912628i	$-0.365951\pi$
$467$	17.6681	0.817580	0.408790	+	0.912628i	$0.365951\pi$
$468$	0	0
$469$	3.52311	0.162682
$470$	0	0
$471$	13.0462	0.601138
$472$	0	0
$473$	−19.4586	−0.894706
$474$	0	0
$475$	−4.25240	−0.195113
$476$	0	0
$477$	−1.34153	−0.0614243
$478$	0	0
$479$	34.2095	1.56307	0.781536	−	0.623860i	$-0.214437\pi$
$479$	34.2095	1.56307	0.781536	+	0.623860i	$0.214437\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	−3.52311	−0.160307
$484$	0	0
$485$	−13.1878	−0.598829
$486$	0	0
$487$	−0.298625	−0.0135320	−0.00676599	−	0.999977i	$-0.502154\pi$
$487$	−0.298625	−0.0135320	−0.00676599	+	0.999977i	$0.502154\pi$
$488$	0	0
$489$	−2.06455	−0.0933622
$490$	0	0
$491$	−19.2158	−0.867195	−0.433597	−	0.901107i	$-0.642756\pi$
$491$	−19.2158	−0.867195	−0.433597	+	0.901107i	$0.642756\pi$
$492$	0	0
$493$	3.79383	0.170866
$494$	0	0
$495$	3.04623	0.136918
$496$	0	0
$497$	4.11704	0.184674
$498$	0	0
$499$	41.9634	1.87854	0.939269	−	0.343182i	$-0.111505\pi$
$499$	41.9634	1.87854	0.939269	+	0.343182i	$0.111505\pi$
$500$	0	0
$501$	15.8217	0.706863
$502$	0	0
$503$	20.1170	0.896974	0.448487	−	0.893789i	$-0.351963\pi$
$503$	20.1170	0.896974	0.448487	+	0.893789i	$0.351963\pi$
$504$	0	0
$505$	−0.747604	−0.0332679
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−15.9634	−0.707563	−0.353782	−	0.935328i	$-0.615104\pi$
$509$	−15.9634	−0.707563	−0.353782	+	0.935328i	$0.615104\pi$
$510$	0	0
$511$	14.7755	0.653630
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	9.08287	0.400239
$516$	0	0
$517$	−22.5048	−0.989760
$518$	0	0
$519$	0.270718	0.0118832
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	0	0
$523$	−3.55976	−0.155657	−0.0778287	−	0.996967i	$-0.524799\pi$
$523$	−3.55976	−0.155657	−0.0778287	+	0.996967i	$0.524799\pi$
$524$	0	0
$525$	−4.25240	−0.185590
$526$	0	0
$527$	1.31695	0.0573671
$528$	0	0
$529$	−10.5877	−0.460333
$530$	0	0
$531$	10.5048	0.455869
$532$	0	0
$533$	43.1020	1.86696
$534$	0	0
$535$	13.1108	0.566828
$536$	0	0
$537$	11.7047	0.505095
$538$	0	0
$539$	−3.52311	−0.151751
$540$	0	0
$541$	−27.0096	−1.16123	−0.580616	−	0.814177i	$-0.697188\pi$
$541$	−27.0096	−1.16123	−0.580616	+	0.814177i	$0.697188\pi$
$542$	0	0
$543$	19.7938	0.849435
$544$	0	0
$545$	−3.22449	−0.138122
$546$	0	0
$547$	−44.3544	−1.89646	−0.948229	−	0.317586i	$-0.897128\pi$
$547$	−44.3544	−1.89646	−0.948229	+	0.317586i	$0.897128\pi$
$548$	0	0
$549$	12.5048	0.533692
$550$	0	0
$551$	−4.38776	−0.186925
$552$	0	0
$553$	5.25240	0.223355
$554$	0	0
$555$	−1.72928	−0.0734039
$556$	0	0
$557$	−28.8680	−1.22318	−0.611588	−	0.791177i	$-0.709469\pi$
$557$	−28.8680	−1.22318	−0.611588	+	0.791177i	$0.709469\pi$
$558$	0	0
$559$	22.0925	0.934411
$560$	0	0
$561$	3.04623	0.128612
$562$	0	0
$563$	−25.1878	−1.06154	−0.530771	−	0.847516i	$-0.678097\pi$
$563$	−25.1878	−1.06154	−0.530771	+	0.847516i	$0.678097\pi$
$564$	0	0
$565$	−5.75719	−0.242207
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−33.2924	−1.39569	−0.697844	−	0.716249i	$-0.745857\pi$
$569$	−33.2924	−1.39569	−0.697844	+	0.716249i	$0.745857\pi$
$570$	0	0
$571$	15.4586	0.646921	0.323460	−	0.946242i	$-0.395154\pi$
$571$	15.4586	0.646921	0.323460	+	0.946242i	$0.395154\pi$
$572$	0	0
$573$	10.9817	0.458766
$574$	0	0
$575$	14.9817	0.624779
$576$	0	0
$577$	−17.4586	−0.726810	−0.363405	−	0.931631i	$-0.618386\pi$
$577$	−17.4586	−0.726810	−0.363405	+	0.931631i	$0.618386\pi$
$578$	0	0
$579$	−1.45856	−0.0606158
$580$	0	0
$581$	−8.11704	−0.336751
$582$	0	0
$583$	4.72635	0.195745
$584$	0	0
$585$	−3.45856	−0.142994
$586$	0	0
$587$	−39.7605	−1.64109	−0.820546	−	0.571580i	$-0.806331\pi$
$587$	−39.7605	−1.64109	−0.820546	+	0.571580i	$0.806331\pi$
$588$	0	0
$589$	−1.52311	−0.0627588
$590$	0	0
$591$	14.0000	0.575883
$592$	0	0
$593$	19.6035	0.805020	0.402510	−	0.915416i	$-0.368138\pi$
$593$	19.6035	0.805020	0.402510	+	0.915416i	$0.368138\pi$
$594$	0	0
$595$	0.747604	0.0306488
$596$	0	0
$597$	−4.77551	−0.195449
$598$	0	0
$599$	14.7509	0.602707	0.301353	−	0.953513i	$-0.402562\pi$
$599$	14.7509	0.602707	0.301353	+	0.953513i	$0.402562\pi$
$600$	0	0
$601$	3.79383	0.154754	0.0773768	−	0.997002i	$-0.475346\pi$
$601$	3.79383	0.154754	0.0773768	+	0.997002i	$0.475346\pi$
$602$	0	0
$603$	3.52311	0.143472
$604$	0	0
$605$	−1.22116	−0.0496473
$606$	0	0
$607$	9.08287	0.368662	0.184331	−	0.982864i	$-0.440988\pi$
$607$	9.08287	0.368662	0.184331	+	0.982864i	$0.440988\pi$
$608$	0	0
$609$	−4.38776	−0.177801
$610$	0	0
$611$	25.5510	1.03368
$612$	0	0
$613$	−44.7668	−1.80811	−0.904056	−	0.427413i	$-0.859425\pi$
$613$	−44.7668	−1.80811	−0.904056	+	0.427413i	$0.859425\pi$
$614$	0	0
$615$	−9.31695	−0.375696
$616$	0	0
$617$	−8.09246	−0.325790	−0.162895	−	0.986643i	$-0.552083\pi$
$617$	−8.09246	−0.325790	−0.162895	+	0.986643i	$0.552083\pi$
$618$	0	0
$619$	−12.2341	−0.491729	−0.245864	−	0.969304i	$-0.579072\pi$
$619$	−12.2341	−0.491729	−0.245864	+	0.969304i	$0.579072\pi$
$620$	0	0
$621$	−3.52311	−0.141378
$622$	0	0
$623$	−6.77551	−0.271455
$624$	0	0
$625$	14.3449	0.573794
$626$	0	0
$627$	−3.52311	−0.140700
$628$	0	0
$629$	−1.72928	−0.0689510
$630$	0	0
$631$	20.1570	0.802438	0.401219	−	0.915982i	$-0.368587\pi$
$631$	20.1570	0.802438	0.401219	+	0.915982i	$0.368587\pi$
$632$	0	0
$633$	−2.33527	−0.0928186
$634$	0	0
$635$	12.5972	0.499907
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	4.11704	0.162867
$640$	0	0
$641$	−23.1999	−0.916341	−0.458171	−	0.888864i	$-0.651495\pi$
$641$	−23.1999	−0.916341	−0.458171	+	0.888864i	$0.651495\pi$
$642$	0	0
$643$	4.90461	0.193419	0.0967095	−	0.995313i	$-0.469168\pi$
$643$	4.90461	0.193419	0.0967095	+	0.995313i	$0.469168\pi$
$644$	0	0
$645$	−4.77551	−0.188036
$646$	0	0
$647$	−34.2095	−1.34491	−0.672457	−	0.740136i	$-0.734761\pi$
$647$	−34.2095	−1.34491	−0.672457	+	0.740136i	$0.734761\pi$
$648$	0	0
$649$	−37.0096	−1.45275
$650$	0	0
$651$	−1.52311	−0.0596956
$652$	0	0
$653$	−11.4219	−0.446974	−0.223487	−	0.974707i	$-0.571744\pi$
$653$	−11.4219	−0.446974	−0.223487	+	0.974707i	$0.571744\pi$
$654$	0	0
$655$	18.8892	0.738063
$656$	0	0
$657$	14.7755	0.576448
$658$	0	0
$659$	0.0120646	0.000469970 0	0.000234985	−	1.00000i	$-0.499925\pi$
$659$	0.0120646	0.000469970 0	0.000234985		1.00000i	$0.499925\pi$
$660$	0	0
$661$	−39.6435	−1.54195	−0.770976	−	0.636864i	$-0.780231\pi$
$661$	−39.6435	−1.54195	−0.770976	+	0.636864i	$0.780231\pi$
$662$	0	0
$663$	−3.45856	−0.134319
$664$	0	0
$665$	−0.864641	−0.0335293
$666$	0	0
$667$	15.4586	0.598558
$668$	0	0
$669$	−8.98168	−0.347252
$670$	0	0
$671$	−44.0558	−1.70076
$672$	0	0
$673$	28.5048	1.09878	0.549389	−	0.835566i	$-0.314860\pi$
$673$	28.5048	1.09878	0.549389	+	0.835566i	$0.314860\pi$
$674$	0	0
$675$	−4.25240	−0.163675
$676$	0	0
$677$	10.9538	0.420988	0.210494	−	0.977595i	$-0.432493\pi$
$677$	10.9538	0.420988	0.210494	+	0.977595i	$0.432493\pi$
$678$	0	0
$679$	15.2524	0.585333
$680$	0	0
$681$	8.77551	0.336278
$682$	0	0
$683$	−9.43398	−0.360981	−0.180491	−	0.983577i	$-0.557769\pi$
$683$	−9.43398	−0.360981	−0.180491	+	0.983577i	$0.557769\pi$
$684$	0	0
$685$	11.2803	0.430998
$686$	0	0
$687$	11.1878	0.426843
$688$	0	0
$689$	−5.36611	−0.204432
$690$	0	0
$691$	−50.5606	−1.92342	−0.961708	−	0.274076i	$-0.911628\pi$
$691$	−50.5606	−1.92342	−0.961708	+	0.274076i	$0.911628\pi$
$692$	0	0
$693$	−3.52311	−0.133832
$694$	0	0
$695$	−7.58767	−0.287817
$696$	0	0
$697$	−9.31695	−0.352905
$698$	0	0
$699$	13.8217	0.522786
$700$	0	0
$701$	−26.5414	−1.00246	−0.501228	−	0.865315i	$-0.667118\pi$
$701$	−26.5414	−1.00246	−0.501228	+	0.865315i	$0.667118\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	−5.52311	−0.208013
$706$	0	0
$707$	0.864641	0.0325182
$708$	0	0
$709$	44.3544	1.66577	0.832883	−	0.553449i	$-0.186689\pi$
$709$	44.3544	1.66577	0.832883	+	0.553449i	$0.186689\pi$
$710$	0	0
$711$	5.25240	0.196980
$712$	0	0
$713$	5.36611	0.200962
$714$	0	0
$715$	12.1849	0.455690
$716$	0	0
$717$	−4.84006	−0.180755
$718$	0	0
$719$	3.88296	0.144810	0.0724050	−	0.997375i	$-0.476933\pi$
$719$	3.88296	0.144810	0.0724050	+	0.997375i	$0.476933\pi$
$720$	0	0
$721$	−10.5048	−0.391219
$722$	0	0
$723$	7.79383	0.289856
$724$	0	0
$725$	18.6585	0.692958
$726$	0	0
$727$	17.4952	0.648861	0.324431	−	0.945909i	$-0.394827\pi$
$727$	17.4952	0.648861	0.324431	+	0.945909i	$0.394827\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.77551	−0.176629
$732$	0	0
$733$	16.2707	0.600973	0.300486	−	0.953786i	$-0.402851\pi$
$733$	16.2707	0.600973	0.300486	+	0.953786i	$0.402851\pi$
$734$	0	0
$735$	−0.864641	−0.0318928
$736$	0	0
$737$	−12.4123	−0.457214
$738$	0	0
$739$	28.1570	1.03577	0.517886	−	0.855450i	$-0.326719\pi$
$739$	28.1570	1.03577	0.517886	+	0.855450i	$0.326719\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	6.38776	0.234344	0.117172	−	0.993112i	$-0.462617\pi$
$743$	6.38776	0.234344	0.117172	+	0.993112i	$0.462617\pi$
$744$	0	0
$745$	−9.67347	−0.354408
$746$	0	0
$747$	−8.11704	−0.296987
$748$	0	0
$749$	−15.1633	−0.554054
$750$	0	0
$751$	−28.6618	−1.04588	−0.522942	−	0.852368i	$-0.675166\pi$
$751$	−28.6618	−1.04588	−0.522942	+	0.852368i	$0.675166\pi$
$752$	0	0
$753$	13.8463	0.504588
$754$	0	0
$755$	6.50479	0.236734
$756$	0	0
$757$	47.1387	1.71328	0.856642	−	0.515911i	$-0.172546\pi$
$757$	47.1387	1.71328	0.856642	+	0.515911i	$0.172546\pi$
$758$	0	0
$759$	12.4123	0.450539
$760$	0	0
$761$	−3.78177	−0.137089	−0.0685445	−	0.997648i	$-0.521836\pi$
$761$	−3.78177	−0.137089	−0.0685445	+	0.997648i	$0.521836\pi$
$762$	0	0
$763$	3.72928	0.135009
$764$	0	0
$765$	0.747604	0.0270297
$766$	0	0
$767$	42.0192	1.51722
$768$	0	0
$769$	−37.6435	−1.35746	−0.678729	−	0.734389i	$-0.737469\pi$
$769$	−37.6435	−1.35746	−0.678729	+	0.734389i	$0.737469\pi$
$770$	0	0
$771$	13.0462	0.469848
$772$	0	0
$773$	−41.2803	−1.48475	−0.742375	−	0.669985i	$-0.766301\pi$
$773$	−41.2803	−1.48475	−0.742375	+	0.669985i	$0.766301\pi$
$774$	0	0
$775$	6.47689	0.232657
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	10.7755	0.386073
$780$	0	0
$781$	−14.5048	−0.519022
$782$	0	0
$783$	−4.38776	−0.156806
$784$	0	0
$785$	−11.2803	−0.402611
$786$	0	0
$787$	3.55976	0.126892	0.0634458	−	0.997985i	$-0.479791\pi$
$787$	3.55976	0.126892	0.0634458	+	0.997985i	$0.479791\pi$
$788$	0	0
$789$	23.8863	0.850374
$790$	0	0
$791$	6.65847	0.236748
$792$	0	0
$793$	50.0192	1.77623
$794$	0	0
$795$	1.15994	0.0411388
$796$	0	0
$797$	8.68305	0.307570	0.153785	−	0.988104i	$-0.450854\pi$
$797$	8.68305	0.307570	0.153785	+	0.988104i	$0.450854\pi$
$798$	0	0
$799$	−5.52311	−0.195394
$800$	0	0
$801$	−6.77551	−0.239401
$802$	0	0
$803$	−52.0558	−1.83701
$804$	0	0
$805$	3.04623	0.107365
$806$	0	0
$807$	−26.5972	−0.936268
$808$	0	0
$809$	−31.4952	−1.10731	−0.553656	−	0.832745i	$-0.686768\pi$
$809$	−31.4952	−1.10731	−0.553656	+	0.832745i	$0.686768\pi$
$810$	0	0
$811$	1.96336	0.0689427	0.0344714	−	0.999406i	$-0.489025\pi$
$811$	1.96336	0.0689427	0.0344714	+	0.999406i	$0.489025\pi$
$812$	0	0
$813$	17.5510	0.615541
$814$	0	0
$815$	1.78510	0.0625292
$816$	0	0
$817$	5.52311	0.193229
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	12.7389	0.444590	0.222295	−	0.974980i	$-0.428645\pi$
$821$	12.7389	0.444590	0.222295	+	0.974980i	$0.428645\pi$
$822$	0	0
$823$	−47.5423	−1.65722	−0.828610	−	0.559826i	$-0.810868\pi$
$823$	−47.5423	−1.65722	−0.828610	+	0.559826i	$0.810868\pi$
$824$	0	0
$825$	14.9817	0.521595
$826$	0	0
$827$	3.70470	0.128825	0.0644126	−	0.997923i	$-0.479483\pi$
$827$	3.70470	0.128825	0.0644126	+	0.997923i	$0.479483\pi$
$828$	0	0
$829$	−17.7572	−0.616733	−0.308366	−	0.951268i	$-0.599782\pi$
$829$	−17.7572	−0.616733	−0.308366	+	0.951268i	$0.599782\pi$
$830$	0	0
$831$	24.7110	0.857214
$832$	0	0
$833$	−0.864641	−0.0299580
$834$	0	0
$835$	−13.6801	−0.473420
$836$	0	0
$837$	−1.52311	−0.0526465
$838$	0	0
$839$	−0.670538	−0.0231495	−0.0115748	−	0.999933i	$-0.503684\pi$
$839$	−0.670538	−0.0231495	−0.0115748	+	0.999933i	$0.503684\pi$
$840$	0	0
$841$	−9.74760	−0.336124
$842$	0	0
$843$	−11.0708	−0.381299
$844$	0	0
$845$	−2.59392	−0.0892337
$846$	0	0
$847$	1.41233	0.0485284
$848$	0	0
$849$	−13.7293	−0.471188
$850$	0	0
$851$	−7.04623	−0.241542
$852$	0	0
$853$	−42.0558	−1.43996	−0.719982	−	0.693993i	$-0.755850\pi$
$853$	−42.0558	−1.43996	−0.719982	+	0.693993i	$0.755850\pi$
$854$	0	0
$855$	−0.864641	−0.0295701
$856$	0	0
$857$	−35.7293	−1.22049	−0.610245	−	0.792213i	$-0.708929\pi$
$857$	−35.7293	−1.22049	−0.610245	+	0.792213i	$0.708929\pi$
$858$	0	0
$859$	36.8805	1.25835	0.629173	−	0.777265i	$-0.283394\pi$
$859$	36.8805	1.25835	0.629173	+	0.777265i	$0.283394\pi$
$860$	0	0
$861$	10.7755	0.367228
$862$	0	0
$863$	−6.38776	−0.217442	−0.108721	−	0.994072i	$-0.534675\pi$
$863$	−6.38776	−0.217442	−0.108721	+	0.994072i	$0.534675\pi$
$864$	0	0
$865$	−0.234074	−0.00795876
$866$	0	0
$867$	−16.2524	−0.551960
$868$	0	0
$869$	−18.5048	−0.627732
$870$	0	0
$871$	14.0925	0.477505
$872$	0	0
$873$	15.2524	0.516215
$874$	0	0
$875$	8.00000	0.270449
$876$	0	0
$877$	15.6801	0.529480	0.264740	−	0.964320i	$-0.414714\pi$
$877$	15.6801	0.529480	0.264740	+	0.964320i	$0.414714\pi$
$878$	0	0
$879$	32.7389	1.10425
$880$	0	0
$881$	−16.6864	−0.562178	−0.281089	−	0.959682i	$-0.590696\pi$
$881$	−16.6864	−0.562178	−0.281089	+	0.959682i	$0.590696\pi$
$882$	0	0
$883$	−52.5972	−1.77004	−0.885019	−	0.465555i	$-0.845855\pi$
$883$	−52.5972	−1.77004	−0.885019	+	0.465555i	$0.845855\pi$
$884$	0	0
$885$	−9.08287	−0.305317
$886$	0	0
$887$	1.31695	0.0442188	0.0221094	−	0.999756i	$-0.492962\pi$
$887$	1.31695	0.0442188	0.0221094	+	0.999756i	$0.492962\pi$
$888$	0	0
$889$	−14.5693	−0.488640
$890$	0	0
$891$	−3.52311	−0.118029
$892$	0	0
$893$	6.38776	0.213758
$894$	0	0
$895$	−10.1204	−0.338286
$896$	0	0
$897$	−14.0925	−0.470533
$898$	0	0
$899$	6.68305	0.222892
$900$	0	0
$901$	1.15994	0.0386432
$902$	0	0
$903$	5.52311	0.183798
$904$	0	0
$905$	−17.1146	−0.568907
$906$	0	0
$907$	51.9421	1.72471	0.862355	−	0.506305i	$-0.168989\pi$
$907$	51.9421	1.72471	0.862355	+	0.506305i	$0.168989\pi$
$908$	0	0
$909$	0.864641	0.0286783
$910$	0	0
$911$	−30.7509	−1.01882	−0.509412	−	0.860523i	$-0.670137\pi$
$911$	−30.7509	−1.01882	−0.509412	+	0.860523i	$0.670137\pi$
$912$	0	0
$913$	28.5972	0.946431
$914$	0	0
$915$	−10.8122	−0.357439
$916$	0	0
$917$	−21.8463	−0.721429
$918$	0	0
$919$	6.09246	0.200972	0.100486	−	0.994938i	$-0.467960\pi$
$919$	6.09246	0.200972	0.100486	+	0.994938i	$0.467960\pi$
$920$	0	0
$921$	−16.5693	−0.545978
$922$	0	0
$923$	16.4681	0.542056
$924$	0	0
$925$	−8.50479	−0.279636
$926$	0	0
$927$	−10.5048	−0.345023
$928$	0	0
$929$	15.6035	0.511934	0.255967	−	0.966685i	$-0.417606\pi$
$929$	15.6035	0.511934	0.255967	+	0.966685i	$0.417606\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−7.34153	−0.240351
$934$	0	0
$935$	−2.63389	−0.0861375
$936$	0	0
$937$	23.1387	0.755908	0.377954	−	0.925824i	$-0.376628\pi$
$937$	23.1387	0.755908	0.377954	+	0.925824i	$0.376628\pi$
$938$	0	0
$939$	−15.5510	−0.507488
$940$	0	0
$941$	−40.6339	−1.32463	−0.662314	−	0.749227i	$-0.730425\pi$
$941$	−40.6339	−1.32463	−0.662314	+	0.749227i	$0.730425\pi$
$942$	0	0
$943$	−37.9634	−1.23626
$944$	0	0
$945$	−0.864641	−0.0281268
$946$	0	0
$947$	42.0279	1.36572	0.682862	−	0.730548i	$-0.260735\pi$
$947$	42.0279	1.36572	0.682862	+	0.730548i	$0.260735\pi$
$948$	0	0
$949$	59.1020	1.91853
$950$	0	0
$951$	1.57560	0.0510923
$952$	0	0
$953$	−3.56309	−0.115420	−0.0577098	−	0.998333i	$-0.518380\pi$
$953$	−3.56309	−0.115420	−0.0577098	+	0.998333i	$0.518380\pi$
$954$	0	0
$955$	−9.49521	−0.307258
$956$	0	0
$957$	15.4586	0.499704
$958$	0	0
$959$	−13.0462	−0.421285
$960$	0	0
$961$	−28.6801	−0.925165
$962$	0	0
$963$	−15.1633	−0.488629
$964$	0	0
$965$	1.26113	0.0405973
$966$	0	0
$967$	−14.8892	−0.478805	−0.239403	−	0.970920i	$-0.576952\pi$
$967$	−14.8892	−0.478805	−0.239403	+	0.970920i	$0.576952\pi$
$968$	0	0
$969$	−0.864641	−0.0277763
$970$	0	0
$971$	21.9634	0.704838	0.352419	−	0.935842i	$-0.385359\pi$
$971$	21.9634	0.704838	0.352419	+	0.935842i	$0.385359\pi$
$972$	0	0
$973$	8.77551	0.281330
$974$	0	0
$975$	−17.0096	−0.544743
$976$	0	0
$977$	0.0245796	0.000786370 0	0.000393185	−	1.00000i	$-0.499875\pi$
$977$	0.0245796	0.000786370 0	0.000393185		1.00000i	$0.499875\pi$
$978$	0	0
$979$	23.8709	0.762917
$980$	0	0
$981$	3.72928	0.119067
$982$	0	0
$983$	−31.1753	−0.994339	−0.497169	−	0.867654i	$-0.665627\pi$
$983$	−31.1753	−0.994339	−0.497169	+	0.867654i	$0.665627\pi$
$984$	0	0
$985$	−12.1050	−0.385696
$986$	0	0
$987$	6.38776	0.203324
$988$	0	0
$989$	−19.4586	−0.618746
$990$	0	0
$991$	22.7476	0.722601	0.361301	−	0.932449i	$-0.382333\pi$
$991$	22.7476	0.722601	0.361301	+	0.932449i	$0.382333\pi$
$992$	0	0
$993$	−2.02791	−0.0643537
$994$	0	0
$995$	4.12910	0.130901
$996$	0	0
$997$	40.1483	1.27151	0.635754	−	0.771892i	$-0.280689\pi$
$997$	40.1483	1.27151	0.635754	+	0.771892i	$0.280689\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bw.1.2		3
4.3	odd	2		3192.2.a.u.1.2	✓	3
12.11	even	2		9576.2.a.cb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.u.1.2	✓	3	4.3	odd	2
6384.2.a.bw.1.2		3	1.1	even	1	trivial
9576.2.a.cb.1.2		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.864641 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.864641 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.52311 q^{11} +4.00000 q^{13} -0.864641 q^{15} -0.864641 q^{17} +1.00000 q^{19} +1.00000 q^{21} -3.52311 q^{23} -4.25240 q^{25} +1.00000 q^{27} -4.38776 q^{29} -1.52311 q^{31} -3.52311 q^{33} -0.864641 q^{35} +2.00000 q^{37} +4.00000 q^{39} +10.7755 q^{41} +5.52311 q^{43} -0.864641 q^{45} +6.38776 q^{47} +1.00000 q^{49} -0.864641 q^{51} -1.34153 q^{53} +3.04623 q^{55} +1.00000 q^{57} +10.5048 q^{59} +12.5048 q^{61} +1.00000 q^{63} -3.45856 q^{65} +3.52311 q^{67} -3.52311 q^{69} +4.11704 q^{71} +14.7755 q^{73} -4.25240 q^{75} -3.52311 q^{77} +5.25240 q^{79} +1.00000 q^{81} -8.11704 q^{83} +0.747604 q^{85} -4.38776 q^{87} -6.77551 q^{89} +4.00000 q^{91} -1.52311 q^{93} -0.864641 q^{95} +15.2524 q^{97} -3.52311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 12 q^{13} + 3 q^{19} + 3 q^{21} + 2 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 2 q^{33} + 6 q^{37} + 12 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{47} + 3 q^{49} - 14 q^{53} - 16 q^{55} + 3 q^{57} - 4 q^{59} + 2 q^{61} + 3 q^{63} - 2 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 2 q^{77} - 2 q^{79} + 3 q^{81} - 4 q^{83} + 20 q^{85} + 2 q^{87} + 10 q^{89} + 12 q^{91} + 8 q^{93} + 28 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 12 * q^13 + 3 * q^19 + 3 * q^21 + 2 * q^23 + 5 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 2 * q^33 + 6 * q^37 + 12 * q^39 + 2 * q^41 + 4 * q^43 + 4 * q^47 + 3 * q^49 - 14 * q^53 - 16 * q^55 + 3 * q^57 - 4 * q^59 + 2 * q^61 + 3 * q^63 - 2 * q^67 + 2 * q^69 - 8 * q^71 + 14 * q^73 + 5 * q^75 + 2 * q^77 - 2 * q^79 + 3 * q^81 - 4 * q^83 + 20 * q^85 + 2 * q^87 + 10 * q^89 + 12 * q^91 + 8 * q^93 + 28 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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