LMFDB - Embedded newform 6384.2.a.ca.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.82405$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.42703 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.42703 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.55164 q^{11} -3.09646 q^{13} +2.42703 q^{15} -5.52349 q^{17} +1.00000 q^{19} -1.00000 q^{21} -3.20597 q^{23} +0.890494 q^{25} +1.00000 q^{27} +8.07514 q^{29} -5.20597 q^{31} -2.55164 q^{33} -2.42703 q^{35} +9.95053 q^{37} -3.09646 q^{39} -2.00000 q^{41} -4.55164 q^{43} +2.42703 q^{45} -6.72946 q^{47} +1.00000 q^{49} -5.52349 q^{51} +8.07514 q^{53} -6.19292 q^{55} +1.00000 q^{57} -13.2962 q^{61} -1.00000 q^{63} -7.51522 q^{65} +9.59863 q^{67} -3.20597 q^{69} +6.07514 q^{71} -14.1503 q^{73} +0.890494 q^{75} +2.55164 q^{77} -11.2060 q^{79} +1.00000 q^{81} -12.2681 q^{83} -13.4057 q^{85} +8.07514 q^{87} +3.10328 q^{89} +3.09646 q^{91} -5.20597 q^{93} +2.42703 q^{95} +11.8478 q^{97} -2.55164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9} - 6 q^{13} - 4 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 8 q^{25} + 4 q^{27} + 2 q^{29} - 10 q^{31} + 4 q^{35} + 6 q^{37} - 6 q^{39} - 8 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 4 q^{57} - 20 q^{61} - 4 q^{63} + 12 q^{65} - 12 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} + 8 q^{75} - 34 q^{79} + 4 q^{81} - 6 q^{83} - 16 q^{85} + 2 q^{87} - 8 q^{89} + 6 q^{91} - 10 q^{93} - 4 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 - 4 * q^7 + 4 * q^9 - 6 * q^13 - 4 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 8 * q^25 + 4 * q^27 + 2 * q^29 - 10 * q^31 + 4 * q^35 + 6 * q^37 - 6 * q^39 - 8 * q^41 - 8 * q^43 - 4 * q^45 + 4 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 4 * q^57 - 20 * q^61 - 4 * q^63 + 12 * q^65 - 12 * q^67 - 2 * q^69 - 6 * q^71 + 4 * q^73 + 8 * q^75 - 34 * q^79 + 4 * q^81 - 6 * q^83 - 16 * q^85 + 2 * q^87 - 8 * q^89 + 6 * q^91 - 10 * q^93 - 4 * q^95 + 4 * 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
$5$	2.42703	1.08540	0.542701	−	0.839926i	$-0.317402\pi$
$5$	2.42703	1.08540	0.542701	+	0.839926i	$0.317402\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.55164	−0.769349	−0.384674	−	0.923052i	$-0.625686\pi$
$11$	−2.55164	−0.769349	−0.384674	+	0.923052i	$0.625686\pi$
$12$	0	0
$13$	−3.09646	−0.858804	−0.429402	−	0.903114i	$-0.641276\pi$
$13$	−3.09646	−0.858804	−0.429402	+	0.903114i	$0.641276\pi$
$14$	0	0
$15$	2.42703	0.626658
$16$	0	0
$17$	−5.52349	−1.33964	−0.669822	−	0.742522i	$-0.733630\pi$
$17$	−5.52349	−1.33964	−0.669822	+	0.742522i	$0.733630\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−3.20597	−0.668490	−0.334245	−	0.942486i	$-0.608481\pi$
$23$	−3.20597	−0.668490	−0.334245	+	0.942486i	$0.608481\pi$
$24$	0	0
$25$	0.890494	0.178099
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.07514	1.49952	0.749758	−	0.661713i	$-0.230170\pi$
$29$	8.07514	1.49952	0.749758	+	0.661713i	$0.230170\pi$
$30$	0	0
$31$	−5.20597	−0.935019	−0.467510	−	0.883988i	$-0.654849\pi$
$31$	−5.20597	−0.935019	−0.467510	+	0.883988i	$0.654849\pi$
$32$	0	0
$33$	−2.55164	−0.444184
$34$	0	0
$35$	−2.42703	−0.410244
$36$	0	0
$37$	9.95053	1.63586	0.817928	−	0.575320i	$-0.195122\pi$
$37$	9.95053	1.63586	0.817928	+	0.575320i	$0.195122\pi$
$38$	0	0
$39$	−3.09646	−0.495831
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−4.55164	−0.694119	−0.347059	−	0.937843i	$-0.612820\pi$
$43$	−4.55164	−0.694119	−0.347059	+	0.937843i	$0.612820\pi$
$44$	0	0
$45$	2.42703	0.361801
$46$	0	0
$47$	−6.72946	−0.981593	−0.490796	−	0.871274i	$-0.663294\pi$
$47$	−6.72946	−0.981593	−0.490796	+	0.871274i	$0.663294\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.52349	−0.773444
$52$	0	0
$53$	8.07514	1.10921	0.554603	−	0.832115i	$-0.312870\pi$
$53$	8.07514	1.10921	0.554603	+	0.832115i	$0.312870\pi$
$54$	0	0
$55$	−6.19292	−0.835053
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−13.2962	−1.70240	−0.851202	−	0.524838i	$-0.824126\pi$
$61$	−13.2962	−1.70240	−0.851202	+	0.524838i	$0.824126\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−7.51522	−0.932148
$66$	0	0
$67$	9.59863	1.17266	0.586330	−	0.810073i	$-0.300572\pi$
$67$	9.59863	1.17266	0.586330	+	0.810073i	$0.300572\pi$
$68$	0	0
$69$	−3.20597	−0.385953
$70$	0	0
$71$	6.07514	0.720986	0.360493	−	0.932762i	$-0.382608\pi$
$71$	6.07514	0.720986	0.360493	+	0.932762i	$0.382608\pi$
$72$	0	0
$73$	−14.1503	−1.65616	−0.828082	−	0.560607i	$-0.810568\pi$
$73$	−14.1503	−1.65616	−0.828082	+	0.560607i	$0.810568\pi$
$74$	0	0
$75$	0.890494	0.102825
$76$	0	0
$77$	2.55164	0.290787
$78$	0	0
$79$	−11.2060	−1.26077	−0.630385	−	0.776283i	$-0.717103\pi$
$79$	−11.2060	−1.26077	−0.630385	+	0.776283i	$0.717103\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.2681	−1.34659	−0.673297	−	0.739372i	$-0.735123\pi$
$83$	−12.2681	−1.34659	−0.673297	+	0.739372i	$0.735123\pi$
$84$	0	0
$85$	−13.4057	−1.45405
$86$	0	0
$87$	8.07514	0.865745
$88$	0	0
$89$	3.10328	0.328947	0.164474	−	0.986381i	$-0.447407\pi$
$89$	3.10328	0.328947	0.164474	+	0.986381i	$0.447407\pi$
$90$	0	0
$91$	3.09646	0.324597
$92$	0	0
$93$	−5.20597	−0.539834
$94$	0	0
$95$	2.42703	0.249008
$96$	0	0
$97$	11.8478	1.20297	0.601483	−	0.798885i	$-0.294577\pi$
$97$	11.8478	1.20297	0.601483	+	0.798885i	$0.294577\pi$
$98$	0	0
$99$	−2.55164	−0.256450
$100$	0	0
$101$	−17.7232	−1.76353	−0.881764	−	0.471691i	$-0.843644\pi$
$101$	−17.7232	−1.76353	−0.881764	+	0.471691i	$0.843644\pi$
$102$	0	0
$103$	−18.6419	−1.83684	−0.918419	−	0.395608i	$-0.870534\pi$
$103$	−18.6419	−1.83684	−0.918419	+	0.395608i	$0.870534\pi$
$104$	0	0
$105$	−2.42703	−0.236854
$106$	0	0
$107$	−12.2681	−1.18600	−0.592999	−	0.805203i	$-0.702056\pi$
$107$	−12.2681	−1.18600	−0.592999	+	0.805203i	$0.702056\pi$
$108$	0	0
$109$	9.31547	0.892260	0.446130	−	0.894968i	$-0.352802\pi$
$109$	9.31547	0.892260	0.446130	+	0.894968i	$0.352802\pi$
$110$	0	0
$111$	9.95053	0.944462
$112$	0	0
$113$	−10.5173	−0.989382	−0.494691	−	0.869069i	$-0.664719\pi$
$113$	−10.5173	−0.989382	−0.494691	+	0.869069i	$0.664719\pi$
$114$	0	0
$115$	−7.78099	−0.725581
$116$	0	0
$117$	−3.09646	−0.286268
$118$	0	0
$119$	5.52349	0.506338
$120$	0	0
$121$	−4.48913	−0.408102
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−9.97391	−0.892094
$126$	0	0
$127$	6.25296	0.554860	0.277430	−	0.960746i	$-0.410517\pi$
$127$	6.25296	0.554860	0.277430	+	0.960746i	$0.410517\pi$
$128$	0	0
$129$	−4.55164	−0.400750
$130$	0	0
$131$	−14.9292	−1.30437	−0.652185	−	0.758060i	$-0.726148\pi$
$131$	−14.9292	−1.30437	−0.652185	+	0.758060i	$0.726148\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	2.42703	0.208886
$136$	0	0
$137$	13.2962	1.13597	0.567986	−	0.823038i	$-0.307723\pi$
$137$	13.2962	1.13597	0.567986	+	0.823038i	$0.307723\pi$
$138$	0	0
$139$	12.1503	1.03057	0.515286	−	0.857018i	$-0.327686\pi$
$139$	12.1503	1.03057	0.515286	+	0.857018i	$0.327686\pi$
$140$	0	0
$141$	−6.72946	−0.566723
$142$	0	0
$143$	7.90106	0.660720
$144$	0	0
$145$	19.5986	1.62758
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−17.7150	−1.45127	−0.725633	−	0.688082i	$-0.758453\pi$
$149$	−17.7150	−1.45127	−0.725633	+	0.688082i	$0.758453\pi$
$150$	0	0
$151$	−21.6481	−1.76170	−0.880849	−	0.473398i	$-0.843027\pi$
$151$	−21.6481	−1.76170	−0.880849	+	0.473398i	$0.843027\pi$
$152$	0	0
$153$	−5.52349	−0.446548
$154$	0	0
$155$	−12.6351	−1.01487
$156$	0	0
$157$	17.6820	1.41118	0.705590	−	0.708620i	$-0.250682\pi$
$157$	17.6820	1.41118	0.705590	+	0.708620i	$0.250682\pi$
$158$	0	0
$159$	8.07514	0.640400
$160$	0	0
$161$	3.20597	0.252666
$162$	0	0
$163$	−20.0669	−1.57176	−0.785879	−	0.618381i	$-0.787789\pi$
$163$	−20.0669	−1.57176	−0.785879	+	0.618381i	$0.787789\pi$
$164$	0	0
$165$	−6.19292	−0.482118
$166$	0	0
$167$	−20.7551	−1.60608	−0.803040	−	0.595925i	$-0.796785\pi$
$167$	−20.7551	−1.60608	−0.803040	+	0.595925i	$0.796785\pi$
$168$	0	0
$169$	−3.41193	−0.262456
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	7.48913	0.569388	0.284694	−	0.958618i	$-0.408108\pi$
$173$	7.48913	0.569388	0.284694	+	0.958618i	$0.408108\pi$
$174$	0	0
$175$	−0.890494	−0.0673151
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.5643	1.16333	0.581664	−	0.813429i	$-0.302402\pi$
$179$	15.5643	1.16333	0.581664	+	0.813429i	$0.302402\pi$
$180$	0	0
$181$	22.2103	1.65088	0.825440	−	0.564491i	$-0.190927\pi$
$181$	22.2103	1.65088	0.825440	+	0.564491i	$0.190927\pi$
$182$	0	0
$183$	−13.2962	−0.982884
$184$	0	0
$185$	24.1503	1.77556
$186$	0	0
$187$	14.0940	1.03065
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−2.60111	−0.188210	−0.0941050	−	0.995562i	$-0.529999\pi$
$191$	−2.60111	−0.188210	−0.0941050	+	0.995562i	$0.529999\pi$
$192$	0	0
$193$	7.08964	0.510323	0.255162	−	0.966898i	$-0.417871\pi$
$193$	7.08964	0.510323	0.255162	+	0.966898i	$0.417871\pi$
$194$	0	0
$195$	−7.51522	−0.538176
$196$	0	0
$197$	−1.34568	−0.0958754	−0.0479377	−	0.998850i	$-0.515265\pi$
$197$	−1.34568	−0.0958754	−0.0479377	+	0.998850i	$0.515265\pi$
$198$	0	0
$199$	−16.1366	−1.14390	−0.571948	−	0.820290i	$-0.693812\pi$
$199$	−16.1366	−1.14390	−0.571948	+	0.820290i	$0.693812\pi$
$200$	0	0
$201$	9.59863	0.677035
$202$	0	0
$203$	−8.07514	−0.566763
$204$	0	0
$205$	−4.85407	−0.339023
$206$	0	0
$207$	−3.20597	−0.222830
$208$	0	0
$209$	−2.55164	−0.176501
$210$	0	0
$211$	18.2337	1.25526	0.627629	−	0.778512i	$-0.284025\pi$
$211$	18.2337	1.25526	0.627629	+	0.778512i	$0.284025\pi$
$212$	0	0
$213$	6.07514	0.416261
$214$	0	0
$215$	−11.0470	−0.753398
$216$	0	0
$217$	5.20597	0.353404
$218$	0	0
$219$	−14.1503	−0.956187
$220$	0	0
$221$	17.1033	1.15049
$222$	0	0
$223$	0.502170	0.0336278	0.0168139	−	0.999859i	$-0.494648\pi$
$223$	0.502170	0.0336278	0.0168139	+	0.999859i	$0.494648\pi$
$224$	0	0
$225$	0.890494	0.0593663
$226$	0	0
$227$	17.8584	1.18530	0.592652	−	0.805459i	$-0.298081\pi$
$227$	17.8584	1.18530	0.592652	+	0.805459i	$0.298081\pi$
$228$	0	0
$229$	−18.6487	−1.23234	−0.616170	−	0.787613i	$-0.711317\pi$
$229$	−18.6487	−1.23234	−0.616170	+	0.787613i	$0.711317\pi$
$230$	0	0
$231$	2.55164	0.167886
$232$	0	0
$233$	7.33885	0.480784	0.240392	−	0.970676i	$-0.422724\pi$
$233$	7.33885	0.480784	0.240392	+	0.970676i	$0.422724\pi$
$234$	0	0
$235$	−16.3326	−1.06542
$236$	0	0
$237$	−11.2060	−0.727906
$238$	0	0
$239$	18.2530	1.18069	0.590343	−	0.807153i	$-0.298993\pi$
$239$	18.2530	1.18069	0.590343	+	0.807153i	$0.298993\pi$
$240$	0	0
$241$	−14.2461	−0.917674	−0.458837	−	0.888520i	$-0.651734\pi$
$241$	−14.2461	−0.917674	−0.458837	+	0.888520i	$0.651734\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.42703	0.155058
$246$	0	0
$247$	−3.09646	−0.197023
$248$	0	0
$249$	−12.2681	−0.777457
$250$	0	0
$251$	4.61621	0.291373	0.145686	−	0.989331i	$-0.453461\pi$
$251$	4.61621	0.291373	0.145686	+	0.989331i	$0.453461\pi$
$252$	0	0
$253$	8.18048	0.514302
$254$	0	0
$255$	−13.4057	−0.839498
$256$	0	0
$257$	−27.0345	−1.68637	−0.843184	−	0.537625i	$-0.819322\pi$
$257$	−27.0345	−1.68637	−0.843184	+	0.537625i	$0.819322\pi$
$258$	0	0
$259$	−9.95053	−0.618296
$260$	0	0
$261$	8.07514	0.499838
$262$	0	0
$263$	11.1372	0.686751	0.343375	−	0.939198i	$-0.388430\pi$
$263$	11.1372	0.686751	0.343375	+	0.939198i	$0.388430\pi$
$264$	0	0
$265$	19.5986	1.20393
$266$	0	0
$267$	3.10328	0.189918
$268$	0	0
$269$	−18.8541	−1.14955	−0.574776	−	0.818311i	$-0.694911\pi$
$269$	−18.8541	−1.14955	−0.574776	+	0.818311i	$0.694911\pi$
$270$	0	0
$271$	9.10328	0.552985	0.276493	−	0.961016i	$-0.410828\pi$
$271$	9.10328	0.552985	0.276493	+	0.961016i	$0.410828\pi$
$272$	0	0
$273$	3.09646	0.187406
$274$	0	0
$275$	−2.27222	−0.137020
$276$	0	0
$277$	21.1138	1.26861	0.634304	−	0.773084i	$-0.281287\pi$
$277$	21.1138	1.26861	0.634304	+	0.773084i	$0.281287\pi$
$278$	0	0
$279$	−5.20597	−0.311673
$280$	0	0
$281$	14.0188	0.836294	0.418147	−	0.908379i	$-0.362680\pi$
$281$	14.0188	0.836294	0.418147	+	0.908379i	$0.362680\pi$
$282$	0	0
$283$	4.15027	0.246708	0.123354	−	0.992363i	$-0.460635\pi$
$283$	4.15027	0.246708	0.123354	+	0.992363i	$0.460635\pi$
$284$	0	0
$285$	2.42703	0.143765
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	13.5090	0.794647
$290$	0	0
$291$	11.8478	0.694533
$292$	0	0
$293$	17.2962	1.01046	0.505228	−	0.862986i	$-0.331409\pi$
$293$	17.2962	1.01046	0.505228	+	0.862986i	$0.331409\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.55164	−0.148061
$298$	0	0
$299$	9.92715	0.574102
$300$	0	0
$301$	4.55164	0.262352
$302$	0	0
$303$	−17.7232	−1.01817
$304$	0	0
$305$	−32.2703	−1.84779
$306$	0	0
$307$	−21.5423	−1.22949	−0.614743	−	0.788727i	$-0.710740\pi$
$307$	−21.5423	−1.22949	−0.614743	+	0.788727i	$0.710740\pi$
$308$	0	0
$309$	−18.6419	−1.06050
$310$	0	0
$311$	−8.78575	−0.498194	−0.249097	−	0.968478i	$-0.580134\pi$
$311$	−8.78575	−0.498194	−0.249097	+	0.968478i	$0.580134\pi$
$312$	0	0
$313$	25.9147	1.46479	0.732393	−	0.680882i	$-0.238404\pi$
$313$	25.9147	1.46479	0.732393	+	0.680882i	$0.238404\pi$
$314$	0	0
$315$	−2.42703	−0.136748
$316$	0	0
$317$	6.58601	0.369907	0.184954	−	0.982747i	$-0.440787\pi$
$317$	6.58601	0.369907	0.184954	+	0.982747i	$0.440787\pi$
$318$	0	0
$319$	−20.6049	−1.15365
$320$	0	0
$321$	−12.2681	−0.684736
$322$	0	0
$323$	−5.52349	−0.307335
$324$	0	0
$325$	−2.75738	−0.152952
$326$	0	0
$327$	9.31547	0.515147
$328$	0	0
$329$	6.72946	0.371007
$330$	0	0
$331$	−0.388924	−0.0213772	−0.0106886	−	0.999943i	$-0.503402\pi$
$331$	−0.388924	−0.0213772	−0.0106886	+	0.999943i	$0.503402\pi$
$332$	0	0
$333$	9.95053	0.545286
$334$	0	0
$335$	23.2962	1.27281
$336$	0	0
$337$	−25.9313	−1.41257	−0.706283	−	0.707930i	$-0.749629\pi$
$337$	−25.9313	−1.41257	−0.706283	+	0.707930i	$0.749629\pi$
$338$	0	0
$339$	−10.5173	−0.571220
$340$	0	0
$341$	13.2838	0.719356
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.78099	−0.418914
$346$	0	0
$347$	11.3921	0.611558	0.305779	−	0.952103i	$-0.401083\pi$
$347$	11.3921	0.611558	0.305779	+	0.952103i	$0.401083\pi$
$348$	0	0
$349$	−36.7125	−1.96517	−0.982587	−	0.185804i	$-0.940511\pi$
$349$	−36.7125	−1.96517	−0.982587	+	0.185804i	$0.940511\pi$
$350$	0	0
$351$	−3.09646	−0.165277
$352$	0	0
$353$	17.7040	0.942287	0.471144	−	0.882056i	$-0.343841\pi$
$353$	17.7040	0.942287	0.471144	+	0.882056i	$0.343841\pi$
$354$	0	0
$355$	14.7446	0.782560
$356$	0	0
$357$	5.52349	0.292334
$358$	0	0
$359$	8.29560	0.437825	0.218913	−	0.975744i	$-0.429749\pi$
$359$	8.29560	0.437825	0.218913	+	0.975744i	$0.429749\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.48913	−0.235618
$364$	0	0
$365$	−34.3432	−1.79760
$366$	0	0
$367$	−14.6611	−0.765306	−0.382653	−	0.923892i	$-0.624989\pi$
$367$	−14.6611	−0.765306	−0.382653	+	0.923892i	$0.624989\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−8.07514	−0.419240
$372$	0	0
$373$	0.461403	0.0238906	0.0119453	−	0.999929i	$-0.496198\pi$
$373$	0.461403	0.0238906	0.0119453	+	0.999929i	$0.496198\pi$
$374$	0	0
$375$	−9.97391	−0.515050
$376$	0	0
$377$	−25.0043	−1.28779
$378$	0	0
$379$	6.09706	0.313185	0.156592	−	0.987663i	$-0.449949\pi$
$379$	6.09706	0.313185	0.156592	+	0.987663i	$0.449949\pi$
$380$	0	0
$381$	6.25296	0.320349
$382$	0	0
$383$	33.1410	1.69342	0.846712	−	0.532051i	$-0.178579\pi$
$383$	33.1410	1.69342	0.846712	+	0.532051i	$0.178579\pi$
$384$	0	0
$385$	6.19292	0.315620
$386$	0	0
$387$	−4.55164	−0.231373
$388$	0	0
$389$	20.9975	1.06462	0.532308	−	0.846551i	$-0.321325\pi$
$389$	20.9975	1.06462	0.532308	+	0.846551i	$0.321325\pi$
$390$	0	0
$391$	17.7081	0.895539
$392$	0	0
$393$	−14.9292	−0.753079
$394$	0	0
$395$	−27.1973	−1.36844
$396$	0	0
$397$	25.4163	1.27561	0.637803	−	0.770199i	$-0.279843\pi$
$397$	25.4163	1.27561	0.637803	+	0.770199i	$0.279843\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−18.7363	−0.935645	−0.467823	−	0.883822i	$-0.654961\pi$
$401$	−18.7363	−0.935645	−0.467823	+	0.883822i	$0.654961\pi$
$402$	0	0
$403$	16.1201	0.802998
$404$	0	0
$405$	2.42703	0.120600
$406$	0	0
$407$	−25.3902	−1.25854
$408$	0	0
$409$	36.0650	1.78330	0.891649	−	0.452727i	$-0.149549\pi$
$409$	36.0650	1.78330	0.891649	+	0.452727i	$0.149549\pi$
$410$	0	0
$411$	13.2962	0.655853
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−29.7750	−1.46160
$416$	0	0
$417$	12.1503	0.595001
$418$	0	0
$419$	25.5767	1.24950	0.624752	−	0.780823i	$-0.285200\pi$
$419$	25.5767	1.24950	0.624752	+	0.780823i	$0.285200\pi$
$420$	0	0
$421$	−11.9932	−0.584512	−0.292256	−	0.956340i	$-0.594406\pi$
$421$	−11.9932	−0.584512	−0.292256	+	0.956340i	$0.594406\pi$
$422$	0	0
$423$	−6.72946	−0.327198
$424$	0	0
$425$	−4.91864	−0.238589
$426$	0	0
$427$	13.2962	0.643448
$428$	0	0
$429$	7.90106	0.381467
$430$	0	0
$431$	−23.9501	−1.15364	−0.576818	−	0.816873i	$-0.695706\pi$
$431$	−23.9501	−1.15364	−0.576818	+	0.816873i	$0.695706\pi$
$432$	0	0
$433$	23.5684	1.13263	0.566313	−	0.824190i	$-0.308369\pi$
$433$	23.5684	1.13263	0.566313	+	0.824190i	$0.308369\pi$
$434$	0	0
$435$	19.5986	0.939682
$436$	0	0
$437$	−3.20597	−0.153362
$438$	0	0
$439$	−23.5190	−1.12250	−0.561249	−	0.827647i	$-0.689679\pi$
$439$	−23.5190	−1.12250	−0.561249	+	0.827647i	$0.689679\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.7019	0.888555	0.444277	−	0.895889i	$-0.353460\pi$
$443$	18.7019	0.888555	0.444277	+	0.895889i	$0.353460\pi$
$444$	0	0
$445$	7.53177	0.357040
$446$	0	0
$447$	−17.7150	−0.837889
$448$	0	0
$449$	1.27736	0.0602824	0.0301412	−	0.999546i	$-0.490404\pi$
$449$	1.27736	0.0602824	0.0301412	+	0.999546i	$0.490404\pi$
$450$	0	0
$451$	5.10328	0.240304
$452$	0	0
$453$	−21.6481	−1.01712
$454$	0	0
$455$	7.51522	0.352319
$456$	0	0
$457$	7.01987	0.328376	0.164188	−	0.986429i	$-0.447500\pi$
$457$	7.01987	0.328376	0.164188	+	0.986429i	$0.447500\pi$
$458$	0	0
$459$	−5.52349	−0.257815
$460$	0	0
$461$	−1.32375	−0.0616532	−0.0308266	−	0.999525i	$-0.509814\pi$
$461$	−1.32375	−0.0616532	−0.0308266	+	0.999525i	$0.509814\pi$
$462$	0	0
$463$	−35.4163	−1.64593	−0.822967	−	0.568089i	$-0.807683\pi$
$463$	−35.4163	−1.64593	−0.822967	+	0.568089i	$0.807683\pi$
$464$	0	0
$465$	−12.6351	−0.585937
$466$	0	0
$467$	23.8822	1.10514	0.552569	−	0.833467i	$-0.313648\pi$
$467$	23.8822	1.10514	0.552569	+	0.833467i	$0.313648\pi$
$468$	0	0
$469$	−9.59863	−0.443224
$470$	0	0
$471$	17.6820	0.814746
$472$	0	0
$473$	11.6142	0.534020
$474$	0	0
$475$	0.890494	0.0408587
$476$	0	0
$477$	8.07514	0.369735
$478$	0	0
$479$	29.0425	1.32698	0.663492	−	0.748184i	$-0.269074\pi$
$479$	29.0425	1.32698	0.663492	+	0.748184i	$0.269074\pi$
$480$	0	0
$481$	−30.8114	−1.40488
$482$	0	0
$483$	3.20597	0.145877
$484$	0	0
$485$	28.7551	1.30570
$486$	0	0
$487$	9.39889	0.425904	0.212952	−	0.977063i	$-0.431692\pi$
$487$	9.39889	0.425904	0.212952	+	0.977063i	$0.431692\pi$
$488$	0	0
$489$	−20.0669	−0.907455
$490$	0	0
$491$	6.13560	0.276896	0.138448	−	0.990370i	$-0.455789\pi$
$491$	6.13560	0.276896	0.138448	+	0.990370i	$0.455789\pi$
$492$	0	0
$493$	−44.6030	−2.00882
$494$	0	0
$495$	−6.19292	−0.278351
$496$	0	0
$497$	−6.07514	−0.272507
$498$	0	0
$499$	−24.2869	−1.08723	−0.543615	−	0.839334i	$-0.682945\pi$
$499$	−24.2869	−1.08723	−0.543615	+	0.839334i	$0.682945\pi$
$500$	0	0
$501$	−20.7551	−0.927271
$502$	0	0
$503$	23.2142	1.03507	0.517536	−	0.855661i	$-0.326849\pi$
$503$	23.2142	1.03507	0.517536	+	0.855661i	$0.326849\pi$
$504$	0	0
$505$	−43.0149	−1.91414
$506$	0	0
$507$	−3.41193	−0.151529
$508$	0	0
$509$	−31.2399	−1.38468	−0.692342	−	0.721569i	$-0.743421\pi$
$509$	−31.2399	−1.38468	−0.692342	+	0.721569i	$0.743421\pi$
$510$	0	0
$511$	14.1503	0.625971
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−45.2445	−1.99371
$516$	0	0
$517$	17.1712	0.755187
$518$	0	0
$519$	7.48913	0.328736
$520$	0	0
$521$	3.77688	0.165468	0.0827339	−	0.996572i	$-0.473635\pi$
$521$	3.77688	0.165468	0.0827339	+	0.996572i	$0.473635\pi$
$522$	0	0
$523$	−9.65492	−0.422180	−0.211090	−	0.977467i	$-0.567701\pi$
$523$	−9.65492	−0.422180	−0.211090	+	0.977467i	$0.567701\pi$
$524$	0	0
$525$	−0.890494	−0.0388644
$526$	0	0
$527$	28.7551	1.25259
$528$	0	0
$529$	−12.7218	−0.553121
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.19292	0.268245
$534$	0	0
$535$	−29.7750	−1.28729
$536$	0	0
$537$	15.5643	0.671647
$538$	0	0
$539$	−2.55164	−0.109907
$540$	0	0
$541$	−18.8239	−0.809301	−0.404651	−	0.914471i	$-0.632607\pi$
$541$	−18.8239	−0.809301	−0.404651	+	0.914471i	$0.632607\pi$
$542$	0	0
$543$	22.2103	0.953135
$544$	0	0
$545$	22.6090	0.968462
$546$	0	0
$547$	37.5821	1.60689	0.803447	−	0.595377i	$-0.202997\pi$
$547$	37.5821	1.60689	0.803447	+	0.595377i	$0.202997\pi$
$548$	0	0
$549$	−13.2962	−0.567468
$550$	0	0
$551$	8.07514	0.344012
$552$	0	0
$553$	11.2060	0.476526
$554$	0	0
$555$	24.1503	1.02512
$556$	0	0
$557$	26.8046	1.13575	0.567874	−	0.823116i	$-0.307766\pi$
$557$	26.8046	1.13575	0.567874	+	0.823116i	$0.307766\pi$
$558$	0	0
$559$	14.0940	0.596112
$560$	0	0
$561$	14.0940	0.595048
$562$	0	0
$563$	−10.9406	−0.461090	−0.230545	−	0.973062i	$-0.574051\pi$
$563$	−10.9406	−0.461090	−0.230545	+	0.973062i	$0.574051\pi$
$564$	0	0
$565$	−25.5258	−1.07388
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	7.05424	0.295729	0.147864	−	0.989008i	$-0.452760\pi$
$569$	7.05424	0.295729	0.147864	+	0.989008i	$0.452760\pi$
$570$	0	0
$571$	29.3902	1.22994	0.614970	−	0.788550i	$-0.289168\pi$
$571$	29.3902	1.22994	0.614970	+	0.788550i	$0.289168\pi$
$572$	0	0
$573$	−2.60111	−0.108663
$574$	0	0
$575$	−2.85490	−0.119057
$576$	0	0
$577$	20.3734	0.848156	0.424078	−	0.905626i	$-0.360598\pi$
$577$	20.3734	0.848156	0.424078	+	0.905626i	$0.360598\pi$
$578$	0	0
$579$	7.08964	0.294635
$580$	0	0
$581$	12.2681	0.508965
$582$	0	0
$583$	−20.6049	−0.853366
$584$	0	0
$585$	−7.51522	−0.310716
$586$	0	0
$587$	30.0064	1.23850	0.619248	−	0.785195i	$-0.287437\pi$
$587$	30.0064	1.23850	0.619248	+	0.785195i	$0.287437\pi$
$588$	0	0
$589$	−5.20597	−0.214508
$590$	0	0
$591$	−1.34568	−0.0553537
$592$	0	0
$593$	−35.9354	−1.47569	−0.737846	−	0.674970i	$-0.764157\pi$
$593$	−35.9354	−1.47569	−0.737846	+	0.674970i	$0.764157\pi$
$594$	0	0
$595$	13.4057	0.549581
$596$	0	0
$597$	−16.1366	−0.660428
$598$	0	0
$599$	−29.5080	−1.20566	−0.602831	−	0.797869i	$-0.705961\pi$
$599$	−29.5080	−1.20566	−0.602831	+	0.797869i	$0.705961\pi$
$600$	0	0
$601$	−23.4695	−0.957340	−0.478670	−	0.877995i	$-0.658881\pi$
$601$	−23.4695	−0.957340	−0.478670	+	0.877995i	$0.658881\pi$
$602$	0	0
$603$	9.59863	0.390886
$604$	0	0
$605$	−10.8953	−0.442955
$606$	0	0
$607$	−5.53860	−0.224805	−0.112402	−	0.993663i	$-0.535855\pi$
$607$	−5.53860	−0.224805	−0.112402	+	0.993663i	$0.535855\pi$
$608$	0	0
$609$	−8.07514	−0.327221
$610$	0	0
$611$	20.8375	0.842995
$612$	0	0
$613$	−33.6802	−1.36033	−0.680165	−	0.733059i	$-0.738092\pi$
$613$	−33.6802	−1.36033	−0.680165	+	0.733059i	$0.738092\pi$
$614$	0	0
$615$	−4.85407	−0.195735
$616$	0	0
$617$	−17.5289	−0.705685	−0.352843	−	0.935683i	$-0.614785\pi$
$617$	−17.5289	−0.705685	−0.352843	+	0.935683i	$0.614785\pi$
$618$	0	0
$619$	−17.7247	−0.712416	−0.356208	−	0.934407i	$-0.615930\pi$
$619$	−17.7247	−0.712416	−0.356208	+	0.934407i	$0.615930\pi$
$620$	0	0
$621$	−3.20597	−0.128651
$622$	0	0
$623$	−3.10328	−0.124330
$624$	0	0
$625$	−28.6595	−1.14638
$626$	0	0
$627$	−2.55164	−0.101903
$628$	0	0
$629$	−54.9617	−2.19147
$630$	0	0
$631$	−13.4570	−0.535716	−0.267858	−	0.963458i	$-0.586316\pi$
$631$	−13.4570	−0.535716	−0.267858	+	0.963458i	$0.586316\pi$
$632$	0	0
$633$	18.2337	0.724724
$634$	0	0
$635$	15.1761	0.602247
$636$	0	0
$637$	−3.09646	−0.122686
$638$	0	0
$639$	6.07514	0.240329
$640$	0	0
$641$	33.7531	1.33317	0.666583	−	0.745431i	$-0.267756\pi$
$641$	33.7531	1.33317	0.666583	+	0.745431i	$0.267756\pi$
$642$	0	0
$643$	−10.9406	−0.431454	−0.215727	−	0.976454i	$-0.569212\pi$
$643$	−10.9406	−0.431454	−0.215727	+	0.976454i	$0.569212\pi$
$644$	0	0
$645$	−11.0470	−0.434975
$646$	0	0
$647$	41.6912	1.63905	0.819524	−	0.573045i	$-0.194238\pi$
$647$	41.6912	1.63905	0.819524	+	0.573045i	$0.194238\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	5.20597	0.204038
$652$	0	0
$653$	27.2592	1.06673	0.533367	−	0.845884i	$-0.320926\pi$
$653$	27.2592	1.06673	0.533367	+	0.845884i	$0.320926\pi$
$654$	0	0
$655$	−36.2337	−1.41577
$656$	0	0
$657$	−14.1503	−0.552055
$658$	0	0
$659$	17.4015	0.677868	0.338934	−	0.940810i	$-0.389934\pi$
$659$	17.4015	0.677868	0.338934	+	0.940810i	$0.389934\pi$
$660$	0	0
$661$	−5.38832	−0.209582	−0.104791	−	0.994494i	$-0.533417\pi$
$661$	−5.38832	−0.209582	−0.104791	+	0.994494i	$0.533417\pi$
$662$	0	0
$663$	17.1033	0.664236
$664$	0	0
$665$	−2.42703	−0.0941163
$666$	0	0
$667$	−25.8886	−1.00241
$668$	0	0
$669$	0.502170	0.0194150
$670$	0	0
$671$	33.9271	1.30974
$672$	0	0
$673$	7.30865	0.281728	0.140864	−	0.990029i	$-0.455012\pi$
$673$	7.30865	0.281728	0.140864	+	0.990029i	$0.455012\pi$
$674$	0	0
$675$	0.890494	0.0342751
$676$	0	0
$677$	−17.0606	−0.655693	−0.327847	−	0.944731i	$-0.606323\pi$
$677$	−17.0606	−0.655693	−0.327847	+	0.944731i	$0.606323\pi$
$678$	0	0
$679$	−11.8478	−0.454679
$680$	0	0
$681$	17.8584	0.684336
$682$	0	0
$683$	17.9898	0.688362	0.344181	−	0.938903i	$-0.388157\pi$
$683$	17.9898	0.688362	0.344181	+	0.938903i	$0.388157\pi$
$684$	0	0
$685$	32.2703	1.23299
$686$	0	0
$687$	−18.6487	−0.711492
$688$	0	0
$689$	−25.0043	−0.952590
$690$	0	0
$691$	−42.6988	−1.62434	−0.812170	−	0.583421i	$-0.801714\pi$
$691$	−42.6988	−1.62434	−0.812170	+	0.583421i	$0.801714\pi$
$692$	0	0
$693$	2.55164	0.0969288
$694$	0	0
$695$	29.4891	1.11859
$696$	0	0
$697$	11.0470	0.418435
$698$	0	0
$699$	7.33885	0.277581
$700$	0	0
$701$	−31.4260	−1.18694	−0.593472	−	0.804855i	$-0.702243\pi$
$701$	−31.4260	−1.18694	−0.593472	+	0.804855i	$0.702243\pi$
$702$	0	0
$703$	9.95053	0.375291
$704$	0	0
$705$	−16.3326	−0.615122
$706$	0	0
$707$	17.7232	0.666551
$708$	0	0
$709$	−36.4101	−1.36741	−0.683704	−	0.729759i	$-0.739632\pi$
$709$	−36.4101	−1.36741	−0.683704	+	0.729759i	$0.739632\pi$
$710$	0	0
$711$	−11.2060	−0.420257
$712$	0	0
$713$	16.6902	0.625051
$714$	0	0
$715$	19.1761	0.717147
$716$	0	0
$717$	18.2530	0.681669
$718$	0	0
$719$	28.3010	1.05545	0.527724	−	0.849416i	$-0.323046\pi$
$719$	28.3010	1.05545	0.527724	+	0.849416i	$0.323046\pi$
$720$	0	0
$721$	18.6419	0.694260
$722$	0	0
$723$	−14.2461	−0.529819
$724$	0	0
$725$	7.19086	0.267062
$726$	0	0
$727$	36.2869	1.34581	0.672903	−	0.739730i	$-0.265047\pi$
$727$	36.2869	1.34581	0.672903	+	0.739730i	$0.265047\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	25.1410	0.929872
$732$	0	0
$733$	10.8541	0.400904	0.200452	−	0.979704i	$-0.435759\pi$
$733$	10.8541	0.400904	0.200452	+	0.979704i	$0.435759\pi$
$734$	0	0
$735$	2.42703	0.0895225
$736$	0	0
$737$	−24.4923	−0.902184
$738$	0	0
$739$	−11.8478	−0.435830	−0.217915	−	0.975968i	$-0.569926\pi$
$739$	−11.8478	−0.435830	−0.217915	+	0.975968i	$0.569926\pi$
$740$	0	0
$741$	−3.09646	−0.113751
$742$	0	0
$743$	−11.6843	−0.428656	−0.214328	−	0.976762i	$-0.568756\pi$
$743$	−11.6843	−0.428656	−0.214328	+	0.976762i	$0.568756\pi$
$744$	0	0
$745$	−42.9948	−1.57521
$746$	0	0
$747$	−12.2681	−0.448865
$748$	0	0
$749$	12.2681	0.448265
$750$	0	0
$751$	−39.0768	−1.42593	−0.712967	−	0.701198i	$-0.752649\pi$
$751$	−39.0768	−1.42593	−0.712967	+	0.701198i	$0.752649\pi$
$752$	0	0
$753$	4.61621	0.168224
$754$	0	0
$755$	−52.5407	−1.91215
$756$	0	0
$757$	−40.3266	−1.46570	−0.732848	−	0.680392i	$-0.761809\pi$
$757$	−40.3266	−1.46570	−0.732848	+	0.680392i	$0.761809\pi$
$758$	0	0
$759$	8.18048	0.296932
$760$	0	0
$761$	45.2026	1.63859	0.819297	−	0.573369i	$-0.194364\pi$
$761$	45.2026	1.63859	0.819297	+	0.573369i	$0.194364\pi$
$762$	0	0
$763$	−9.31547	−0.337243
$764$	0	0
$765$	−13.4057	−0.484684
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.06355	−0.0383525	−0.0191763	−	0.999816i	$-0.506104\pi$
$769$	−1.06355	−0.0383525	−0.0191763	+	0.999816i	$0.506104\pi$
$770$	0	0
$771$	−27.0345	−0.973625
$772$	0	0
$773$	−28.3266	−1.01884	−0.509419	−	0.860519i	$-0.670140\pi$
$773$	−28.3266	−1.01884	−0.509419	+	0.860519i	$0.670140\pi$
$774$	0	0
$775$	−4.63588	−0.166526
$776$	0	0
$777$	−9.95053	−0.356973
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−15.5016	−0.554690
$782$	0	0
$783$	8.07514	0.288582
$784$	0	0
$785$	42.9149	1.53170
$786$	0	0
$787$	42.3277	1.50882	0.754409	−	0.656404i	$-0.227923\pi$
$787$	42.3277	1.50882	0.754409	+	0.656404i	$0.227923\pi$
$788$	0	0
$789$	11.1372	0.396496
$790$	0	0
$791$	10.5173	0.373951
$792$	0	0
$793$	41.1712	1.46203
$794$	0	0
$795$	19.5986	0.695092
$796$	0	0
$797$	−33.4163	−1.18367	−0.591833	−	0.806061i	$-0.701595\pi$
$797$	−33.4163	−1.18367	−0.591833	+	0.806061i	$0.701595\pi$
$798$	0	0
$799$	37.1701	1.31499
$800$	0	0
$801$	3.10328	0.109649
$802$	0	0
$803$	36.1064	1.27417
$804$	0	0
$805$	7.78099	0.274244
$806$	0	0
$807$	−18.8541	−0.663695
$808$	0	0
$809$	20.7125	0.728212	0.364106	−	0.931358i	$-0.381375\pi$
$809$	20.7125	0.728212	0.364106	+	0.931358i	$0.381375\pi$
$810$	0	0
$811$	−13.3902	−0.470193	−0.235096	−	0.971972i	$-0.575541\pi$
$811$	−13.3902	−0.470193	−0.235096	+	0.971972i	$0.575541\pi$
$812$	0	0
$813$	9.10328	0.319266
$814$	0	0
$815$	−48.7029	−1.70599
$816$	0	0
$817$	−4.55164	−0.159242
$818$	0	0
$819$	3.09646	0.108199
$820$	0	0
$821$	−39.9079	−1.39279	−0.696397	−	0.717656i	$-0.745215\pi$
$821$	−39.9079	−1.39279	−0.696397	+	0.717656i	$0.745215\pi$
$822$	0	0
$823$	−12.0669	−0.420624	−0.210312	−	0.977634i	$-0.567448\pi$
$823$	−12.0669	−0.420624	−0.210312	+	0.977634i	$0.567448\pi$
$824$	0	0
$825$	−2.27222	−0.0791086
$826$	0	0
$827$	19.7833	0.687932	0.343966	−	0.938982i	$-0.388230\pi$
$827$	19.7833	0.687932	0.343966	+	0.938982i	$0.388230\pi$
$828$	0	0
$829$	10.6814	0.370982	0.185491	−	0.982646i	$-0.440612\pi$
$829$	10.6814	0.370982	0.185491	+	0.982646i	$0.440612\pi$
$830$	0	0
$831$	21.1138	0.732431
$832$	0	0
$833$	−5.52349	−0.191378
$834$	0	0
$835$	−50.3734	−1.74324
$836$	0	0
$837$	−5.20597	−0.179945
$838$	0	0
$839$	10.0124	0.345668	0.172834	−	0.984951i	$-0.444708\pi$
$839$	10.0124	0.345668	0.172834	+	0.984951i	$0.444708\pi$
$840$	0	0
$841$	36.2078	1.24855
$842$	0	0
$843$	14.0188	0.482835
$844$	0	0
$845$	−8.28088	−0.284871
$846$	0	0
$847$	4.48913	0.154248
$848$	0	0
$849$	4.15027	0.142437
$850$	0	0
$851$	−31.9011	−1.09355
$852$	0	0
$853$	44.0803	1.50928	0.754641	−	0.656138i	$-0.227811\pi$
$853$	44.0803	1.50928	0.754641	+	0.656138i	$0.227811\pi$
$854$	0	0
$855$	2.42703	0.0830028
$856$	0	0
$857$	4.20657	0.143694	0.0718468	−	0.997416i	$-0.477111\pi$
$857$	4.20657	0.143694	0.0718468	+	0.997416i	$0.477111\pi$
$858$	0	0
$859$	13.1033	0.447078	0.223539	−	0.974695i	$-0.428239\pi$
$859$	13.1033	0.447078	0.223539	+	0.974695i	$0.428239\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	26.4746	0.901207	0.450603	−	0.892724i	$-0.351209\pi$
$863$	26.4746	0.901207	0.450603	+	0.892724i	$0.351209\pi$
$864$	0	0
$865$	18.1764	0.618015
$866$	0	0
$867$	13.5090	0.458789
$868$	0	0
$869$	28.5936	0.969972
$870$	0	0
$871$	−29.7218	−1.00708
$872$	0	0
$873$	11.8478	0.400989
$874$	0	0
$875$	9.97391	0.337180
$876$	0	0
$877$	−24.9673	−0.843086	−0.421543	−	0.906808i	$-0.638511\pi$
$877$	−24.9673	−0.843086	−0.421543	+	0.906808i	$0.638511\pi$
$878$	0	0
$879$	17.2962	0.583386
$880$	0	0
$881$	35.4121	1.19306	0.596532	−	0.802589i	$-0.296545\pi$
$881$	35.4121	1.19306	0.596532	+	0.802589i	$0.296545\pi$
$882$	0	0
$883$	−36.2016	−1.21828	−0.609140	−	0.793062i	$-0.708485\pi$
$883$	−36.2016	−1.21828	−0.609140	+	0.793062i	$0.708485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.7812	1.16784	0.583919	−	0.811812i	$-0.301518\pi$
$887$	34.7812	1.16784	0.583919	+	0.811812i	$0.301518\pi$
$888$	0	0
$889$	−6.25296	−0.209717
$890$	0	0
$891$	−2.55164	−0.0854832
$892$	0	0
$893$	−6.72946	−0.225193
$894$	0	0
$895$	37.7750	1.26268
$896$	0	0
$897$	9.92715	0.331458
$898$	0	0
$899$	−42.0389	−1.40208
$900$	0	0
$901$	−44.6030	−1.48594
$902$	0	0
$903$	4.55164	0.151469
$904$	0	0
$905$	53.9052	1.79187
$906$	0	0
$907$	13.8342	0.459357	0.229679	−	0.973267i	$-0.426233\pi$
$907$	13.8342	0.459357	0.229679	+	0.973267i	$0.426233\pi$
$908$	0	0
$909$	−17.7232	−0.587843
$910$	0	0
$911$	−1.66200	−0.0550646	−0.0275323	−	0.999621i	$-0.508765\pi$
$911$	−1.66200	−0.0550646	−0.0275323	+	0.999621i	$0.508765\pi$
$912$	0	0
$913$	31.3037	1.03600
$914$	0	0
$915$	−32.2703	−1.06682
$916$	0	0
$917$	14.9292	0.493006
$918$	0	0
$919$	−17.4076	−0.574223	−0.287112	−	0.957897i	$-0.592695\pi$
$919$	−17.4076	−0.574223	−0.287112	+	0.957897i	$0.592695\pi$
$920$	0	0
$921$	−21.5423	−0.709844
$922$	0	0
$923$	−18.8114	−0.619185
$924$	0	0
$925$	8.86089	0.291344
$926$	0	0
$927$	−18.6419	−0.612280
$928$	0	0
$929$	−2.10722	−0.0691357	−0.0345679	−	0.999402i	$-0.511005\pi$
$929$	−2.10722	−0.0691357	−0.0345679	+	0.999402i	$0.511005\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−8.78575	−0.287633
$934$	0	0
$935$	34.2066	1.11867
$936$	0	0
$937$	−37.8625	−1.23691	−0.618457	−	0.785818i	$-0.712242\pi$
$937$	−37.8625	−1.23691	−0.618457	+	0.785818i	$0.712242\pi$
$938$	0	0
$939$	25.9147	0.845694
$940$	0	0
$941$	−36.2579	−1.18197	−0.590987	−	0.806681i	$-0.701261\pi$
$941$	−36.2579	−1.18197	−0.590987	+	0.806681i	$0.701261\pi$
$942$	0	0
$943$	6.41193	0.208801
$944$	0	0
$945$	−2.42703	−0.0789514
$946$	0	0
$947$	−25.1304	−0.816628	−0.408314	−	0.912841i	$-0.633883\pi$
$947$	−25.1304	−0.816628	−0.408314	+	0.912841i	$0.633883\pi$
$948$	0	0
$949$	43.8158	1.42232
$950$	0	0
$951$	6.58601	0.213566
$952$	0	0
$953$	29.7833	0.964775	0.482387	−	0.875958i	$-0.339770\pi$
$953$	29.7833	0.964775	0.482387	+	0.875958i	$0.339770\pi$
$954$	0	0
$955$	−6.31299	−0.204284
$956$	0	0
$957$	−20.6049	−0.666060
$958$	0	0
$959$	−13.2962	−0.429357
$960$	0	0
$961$	−3.89792	−0.125739
$962$	0	0
$963$	−12.2681	−0.395333
$964$	0	0
$965$	17.2068	0.553906
$966$	0	0
$967$	−6.75821	−0.217329	−0.108665	−	0.994078i	$-0.534657\pi$
$967$	−6.75821	−0.217329	−0.108665	+	0.994078i	$0.534657\pi$
$968$	0	0
$969$	−5.52349	−0.177440
$970$	0	0
$971$	45.9023	1.47307	0.736537	−	0.676397i	$-0.236460\pi$
$971$	45.9023	1.47307	0.736537	+	0.676397i	$0.236460\pi$
$972$	0	0
$973$	−12.1503	−0.389520
$974$	0	0
$975$	−2.75738	−0.0883069
$976$	0	0
$977$	−25.8271	−0.826283	−0.413141	−	0.910667i	$-0.635568\pi$
$977$	−25.8271	−0.826283	−0.413141	+	0.910667i	$0.635568\pi$
$978$	0	0
$979$	−7.91847	−0.253075
$980$	0	0
$981$	9.31547	0.297420
$982$	0	0
$983$	−36.7249	−1.17134	−0.585672	−	0.810548i	$-0.699169\pi$
$983$	−36.7249	−1.17134	−0.585672	+	0.810548i	$0.699169\pi$
$984$	0	0
$985$	−3.26600	−0.104063
$986$	0	0
$987$	6.72946	0.214201
$988$	0	0
$989$	14.5924	0.464012
$990$	0	0
$991$	42.2906	1.34341	0.671703	−	0.740820i	$-0.265563\pi$
$991$	42.2906	1.34341	0.671703	+	0.740820i	$0.265563\pi$
$992$	0	0
$993$	−0.388924	−0.0123421
$994$	0	0
$995$	−39.1641	−1.24159
$996$	0	0
$997$	−26.7038	−0.845718	−0.422859	−	0.906196i	$-0.638973\pi$
$997$	−26.7038	−0.845718	−0.422859	+	0.906196i	$0.638973\pi$
$998$	0	0
$999$	9.95053	0.314821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.ca.1.4		4
4.3	odd	2		3192.2.a.w.1.4	✓	4
12.11	even	2		9576.2.a.cl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.w.1.4	✓	4	4.3	odd	2
6384.2.a.ca.1.4		4	1.1	even	1	trivial
9576.2.a.cl.1.1		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.42703 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.42703 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.55164 q^{11} -3.09646 q^{13} +2.42703 q^{15} -5.52349 q^{17} +1.00000 q^{19} -1.00000 q^{21} -3.20597 q^{23} +0.890494 q^{25} +1.00000 q^{27} +8.07514 q^{29} -5.20597 q^{31} -2.55164 q^{33} -2.42703 q^{35} +9.95053 q^{37} -3.09646 q^{39} -2.00000 q^{41} -4.55164 q^{43} +2.42703 q^{45} -6.72946 q^{47} +1.00000 q^{49} -5.52349 q^{51} +8.07514 q^{53} -6.19292 q^{55} +1.00000 q^{57} -13.2962 q^{61} -1.00000 q^{63} -7.51522 q^{65} +9.59863 q^{67} -3.20597 q^{69} +6.07514 q^{71} -14.1503 q^{73} +0.890494 q^{75} +2.55164 q^{77} -11.2060 q^{79} +1.00000 q^{81} -12.2681 q^{83} -13.4057 q^{85} +8.07514 q^{87} +3.10328 q^{89} +3.09646 q^{91} -5.20597 q^{93} +2.42703 q^{95} +11.8478 q^{97} -2.55164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9} - 6 q^{13} - 4 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 8 q^{25} + 4 q^{27} + 2 q^{29} - 10 q^{31} + 4 q^{35} + 6 q^{37} - 6 q^{39} - 8 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 4 q^{57} - 20 q^{61} - 4 q^{63} + 12 q^{65} - 12 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} + 8 q^{75} - 34 q^{79} + 4 q^{81} - 6 q^{83} - 16 q^{85} + 2 q^{87} - 8 q^{89} + 6 q^{91} - 10 q^{93} - 4 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 - 4 * q^7 + 4 * q^9 - 6 * q^13 - 4 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 8 * q^25 + 4 * q^27 + 2 * q^29 - 10 * q^31 + 4 * q^35 + 6 * q^37 - 6 * q^39 - 8 * q^41 - 8 * q^43 - 4 * q^45 + 4 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 4 * q^57 - 20 * q^61 - 4 * q^63 + 12 * q^65 - 12 * q^67 - 2 * q^69 - 6 * q^71 + 4 * q^73 + 8 * q^75 - 34 * q^79 + 4 * q^81 - 6 * q^83 - 16 * q^85 + 2 * q^87 - 8 * q^89 + 6 * q^91 - 10 * q^93 - 4 * q^95 + 4 * q^97

Introduction

L-functions

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