LMFDB - Embedded newform 6384.2.a.bq.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.82843 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.82843 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} +2.82843 q^{15} -1.00000 q^{19} -1.00000 q^{21} -4.82843 q^{23} +3.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} -1.17157 q^{31} -2.00000 q^{33} -2.82843 q^{35} -6.48528 q^{37} -2.82843 q^{39} -7.65685 q^{41} -8.00000 q^{43} +2.82843 q^{45} -2.82843 q^{47} +1.00000 q^{49} +7.65685 q^{53} -5.65685 q^{55} -1.00000 q^{57} -1.65685 q^{59} +9.31371 q^{61} -1.00000 q^{63} -8.00000 q^{65} -10.0000 q^{67} -4.82843 q^{69} -1.65685 q^{71} -11.6569 q^{73} +3.00000 q^{75} +2.00000 q^{77} +8.82843 q^{79} +1.00000 q^{81} +11.3137 q^{83} -3.65685 q^{87} -10.0000 q^{89} +2.82843 q^{91} -1.17157 q^{93} -2.82843 q^{95} +7.31371 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{19} - 2 q^{21} - 4 q^{23} + 6 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} - 4 q^{33} + 4 q^{37} - 4 q^{41} - 16 q^{43} + 2 q^{49} + 4 q^{53} - 2 q^{57} + 8 q^{59} - 4 q^{61} - 2 q^{63} - 16 q^{65} - 20 q^{67} - 4 q^{69} + 8 q^{71} - 12 q^{73} + 6 q^{75} + 4 q^{77} + 12 q^{79} + 2 q^{81} + 4 q^{87} - 20 q^{89} - 8 q^{93} - 8 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^19 - 2 * q^21 - 4 * q^23 + 6 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 - 4 * q^33 + 4 * q^37 - 4 * q^41 - 16 * q^43 + 2 * q^49 + 4 * q^53 - 2 * q^57 + 8 * q^59 - 4 * q^61 - 2 * q^63 - 16 * q^65 - 20 * q^67 - 4 * q^69 + 8 * q^71 - 12 * q^73 + 6 * q^75 + 4 * q^77 + 12 * q^79 + 2 * q^81 + 4 * q^87 - 20 * q^89 - 8 * q^93 - 8 * 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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−6.48528	−1.06617	−0.533087	−	0.846061i	$-0.678968\pi$
$37$	−6.48528	−1.06617	−0.533087	+	0.846061i	$0.678968\pi$
$38$	0	0
$39$	−2.82843	−0.452911
$40$	0	0
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	9.31371	1.19250	0.596249	−	0.802799i	$-0.296657\pi$
$61$	9.31371	1.19250	0.596249	+	0.802799i	$0.296657\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	−4.82843	−0.581274
$70$	0	0
$71$	−1.65685	−0.196632	−0.0983162	−	0.995155i	$-0.531346\pi$
$71$	−1.65685	−0.196632	−0.0983162	+	0.995155i	$0.531346\pi$
$72$	0	0
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	8.82843	0.993276	0.496638	−	0.867958i	$-0.334568\pi$
$79$	8.82843	0.993276	0.496638	+	0.867958i	$0.334568\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.65685	−0.392056
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	−1.17157	−0.121486
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	7.31371	0.742595	0.371297	−	0.928514i	$-0.378913\pi$
$97$	7.31371	0.742595	0.371297	+	0.928514i	$0.378913\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−16.4853	−1.64035	−0.820173	−	0.572115i	$-0.806123\pi$
$101$	−16.4853	−1.64035	−0.820173	+	0.572115i	$0.806123\pi$
$102$	0	0
$103$	12.4853	1.23021	0.615106	−	0.788445i	$-0.289113\pi$
$103$	12.4853	1.23021	0.615106	+	0.788445i	$0.289113\pi$
$104$	0	0
$105$	−2.82843	−0.276026
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	0	0
$111$	−6.48528	−0.615556
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−13.6569	−1.27351
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−7.65685	−0.690395
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−8.82843	−0.783396	−0.391698	−	0.920094i	$-0.628112\pi$
$127$	−8.82843	−0.783396	−0.391698	+	0.920094i	$0.628112\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	2.82843	0.243432
$136$	0	0
$137$	0.343146	0.0293169	0.0146585	−	0.999893i	$-0.495334\pi$
$137$	0.343146	0.0293169	0.0146585	+	0.999893i	$0.495334\pi$
$138$	0	0
$139$	1.65685	0.140533	0.0702663	−	0.997528i	$-0.477615\pi$
$139$	1.65685	0.140533	0.0702663	+	0.997528i	$0.477615\pi$
$140$	0	0
$141$	−2.82843	−0.238197
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	−10.3431	−0.858952
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−18.4853	−1.51437	−0.757187	−	0.653199i	$-0.773427\pi$
$149$	−18.4853	−1.51437	−0.757187	+	0.653199i	$0.773427\pi$
$150$	0	0
$151$	4.14214	0.337082	0.168541	−	0.985695i	$-0.446094\pi$
$151$	4.14214	0.337082	0.168541	+	0.985695i	$0.446094\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.31371	−0.266163
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	7.65685	0.607228
$160$	0	0
$161$	4.82843	0.380533
$162$	0	0
$163$	−3.31371	−0.259550	−0.129775	−	0.991543i	$-0.541425\pi$
$163$	−3.31371	−0.259550	−0.129775	+	0.991543i	$0.541425\pi$
$164$	0	0
$165$	−5.65685	−0.440386
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−19.6569	−1.49448	−0.747241	−	0.664553i	$-0.768622\pi$
$173$	−19.6569	−1.49448	−0.747241	+	0.664553i	$0.768622\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	−1.65685	−0.124537
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	3.51472	0.261247	0.130623	−	0.991432i	$-0.458302\pi$
$181$	3.51472	0.261247	0.130623	+	0.991432i	$0.458302\pi$
$182$	0	0
$183$	9.31371	0.688489
$184$	0	0
$185$	−18.3431	−1.34861
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−16.1421	−1.16800	−0.584002	−	0.811752i	$-0.698514\pi$
$191$	−16.1421	−1.16800	−0.584002	+	0.811752i	$0.698514\pi$
$192$	0	0
$193$	26.9706	1.94138	0.970692	−	0.240328i	$-0.0772549\pi$
$193$	26.9706	1.94138	0.970692	+	0.240328i	$0.0772549\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	0	0
$197$	0.828427	0.0590230	0.0295115	−	0.999564i	$-0.490605\pi$
$197$	0.828427	0.0590230	0.0295115	+	0.999564i	$0.490605\pi$
$198$	0	0
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	3.65685	0.256661
$204$	0	0
$205$	−21.6569	−1.51258
$206$	0	0
$207$	−4.82843	−0.335599
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−4.34315	−0.298994	−0.149497	−	0.988762i	$-0.547766\pi$
$211$	−4.34315	−0.298994	−0.149497	+	0.988762i	$0.547766\pi$
$212$	0	0
$213$	−1.65685	−0.113526
$214$	0	0
$215$	−22.6274	−1.54318
$216$	0	0
$217$	1.17157	0.0795315
$218$	0	0
$219$	−11.6569	−0.787697
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.48528	−0.568216	−0.284108	−	0.958792i	$-0.591698\pi$
$223$	−8.48528	−0.568216	−0.284108	+	0.958792i	$0.591698\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	15.3137	1.01641	0.508203	−	0.861237i	$-0.330310\pi$
$227$	15.3137	1.01641	0.508203	+	0.861237i	$0.330310\pi$
$228$	0	0
$229$	0.343146	0.0226757	0.0113379	−	0.999936i	$-0.496391\pi$
$229$	0.343146	0.0226757	0.0113379	+	0.999936i	$0.496391\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	20.6274	1.35135	0.675674	−	0.737201i	$-0.263853\pi$
$233$	20.6274	1.35135	0.675674	+	0.737201i	$0.263853\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	8.82843	0.573468
$238$	0	0
$239$	8.82843	0.571063	0.285532	−	0.958369i	$-0.407830\pi$
$239$	8.82843	0.571063	0.285532	+	0.958369i	$0.407830\pi$
$240$	0	0
$241$	22.6274	1.45756	0.728780	−	0.684748i	$-0.240088\pi$
$241$	22.6274	1.45756	0.728780	+	0.684748i	$0.240088\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.82843	0.180702
$246$	0	0
$247$	2.82843	0.179969
$248$	0	0
$249$	11.3137	0.716977
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	9.65685	0.607121
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	6.48528	0.402976
$260$	0	0
$261$	−3.65685	−0.226354
$262$	0	0
$263$	16.1421	0.995367	0.497683	−	0.867359i	$-0.334184\pi$
$263$	16.1421	0.995367	0.497683	+	0.867359i	$0.334184\pi$
$264$	0	0
$265$	21.6569	1.33037
$266$	0	0
$267$	−10.0000	−0.611990
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−19.3137	−1.17322	−0.586612	−	0.809868i	$-0.699539\pi$
$271$	−19.3137	−1.17322	−0.586612	+	0.809868i	$0.699539\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	0	0
$279$	−1.17157	−0.0701402
$280$	0	0
$281$	−2.68629	−0.160251	−0.0801254	−	0.996785i	$-0.525532\pi$
$281$	−2.68629	−0.160251	−0.0801254	+	0.996785i	$0.525532\pi$
$282$	0	0
$283$	−17.6569	−1.04959	−0.524796	−	0.851228i	$-0.675858\pi$
$283$	−17.6569	−1.04959	−0.524796	+	0.851228i	$0.675858\pi$
$284$	0	0
$285$	−2.82843	−0.167542
$286$	0	0
$287$	7.65685	0.451970
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	7.31371	0.428737
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−4.68629	−0.272846
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	13.6569	0.789796
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	−16.4853	−0.947055
$304$	0	0
$305$	26.3431	1.50840
$306$	0	0
$307$	5.65685	0.322854	0.161427	−	0.986885i	$-0.448390\pi$
$307$	5.65685	0.322854	0.161427	+	0.986885i	$0.448390\pi$
$308$	0	0
$309$	12.4853	0.710263
$310$	0	0
$311$	20.4853	1.16161	0.580807	−	0.814041i	$-0.302737\pi$
$311$	20.4853	1.16161	0.580807	+	0.814041i	$0.302737\pi$
$312$	0	0
$313$	−27.6569	−1.56326	−0.781629	−	0.623744i	$-0.785611\pi$
$313$	−27.6569	−1.56326	−0.781629	+	0.623744i	$0.785611\pi$
$314$	0	0
$315$	−2.82843	−0.159364
$316$	0	0
$317$	30.9706	1.73948	0.869740	−	0.493510i	$-0.164286\pi$
$317$	30.9706	1.73948	0.869740	+	0.493510i	$0.164286\pi$
$318$	0	0
$319$	7.31371	0.409489
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−8.48528	−0.470679
$326$	0	0
$327$	2.48528	0.137436
$328$	0	0
$329$	2.82843	0.155936
$330$	0	0
$331$	−12.6274	−0.694066	−0.347033	−	0.937853i	$-0.612811\pi$
$331$	−12.6274	−0.694066	−0.347033	+	0.937853i	$0.612811\pi$
$332$	0	0
$333$	−6.48528	−0.355391
$334$	0	0
$335$	−28.2843	−1.54533
$336$	0	0
$337$	−13.3137	−0.725244	−0.362622	−	0.931936i	$-0.618118\pi$
$337$	−13.3137	−0.725244	−0.362622	+	0.931936i	$0.618118\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	2.34315	0.126888
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.6569	−0.735260
$346$	0	0
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	−2.82843	−0.150970
$352$	0	0
$353$	−18.3431	−0.976307	−0.488154	−	0.872758i	$-0.662329\pi$
$353$	−18.3431	−0.976307	−0.488154	+	0.872758i	$0.662329\pi$
$354$	0	0
$355$	−4.68629	−0.248723
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.4853	0.553392	0.276696	−	0.960958i	$-0.410761\pi$
$359$	10.4853	0.553392	0.276696	+	0.960958i	$0.410761\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−32.9706	−1.72576
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−7.65685	−0.398600
$370$	0	0
$371$	−7.65685	−0.397524
$372$	0	0
$373$	−34.4853	−1.78558	−0.892790	−	0.450473i	$-0.851255\pi$
$373$	−34.4853	−1.78558	−0.892790	+	0.450473i	$0.851255\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	10.3431	0.532699
$378$	0	0
$379$	−20.6274	−1.05956	−0.529780	−	0.848135i	$-0.677725\pi$
$379$	−20.6274	−1.05956	−0.529780	+	0.848135i	$0.677725\pi$
$380$	0	0
$381$	−8.82843	−0.452294
$382$	0	0
$383$	30.6274	1.56499	0.782494	−	0.622658i	$-0.213947\pi$
$383$	30.6274	1.56499	0.782494	+	0.622658i	$0.213947\pi$
$384$	0	0
$385$	5.65685	0.288300
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	21.7990	1.10525	0.552626	−	0.833429i	$-0.313626\pi$
$389$	21.7990	1.10525	0.552626	+	0.833429i	$0.313626\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	15.3137	0.772474
$394$	0	0
$395$	24.9706	1.25641
$396$	0	0
$397$	38.2843	1.92143	0.960716	−	0.277533i	$-0.0895166\pi$
$397$	38.2843	1.92143	0.960716	+	0.277533i	$0.0895166\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−16.6274	−0.830334	−0.415167	−	0.909745i	$-0.636277\pi$
$401$	−16.6274	−0.830334	−0.415167	+	0.909745i	$0.636277\pi$
$402$	0	0
$403$	3.31371	0.165068
$404$	0	0
$405$	2.82843	0.140546
$406$	0	0
$407$	12.9706	0.642927
$408$	0	0
$409$	−0.686292	−0.0339349	−0.0169675	−	0.999856i	$-0.505401\pi$
$409$	−0.686292	−0.0339349	−0.0169675	+	0.999856i	$0.505401\pi$
$410$	0	0
$411$	0.343146	0.0169261
$412$	0	0
$413$	1.65685	0.0815285
$414$	0	0
$415$	32.0000	1.57082
$416$	0	0
$417$	1.65685	0.0811365
$418$	0	0
$419$	−20.9706	−1.02448	−0.512240	−	0.858843i	$-0.671184\pi$
$419$	−20.9706	−1.02448	−0.512240	+	0.858843i	$0.671184\pi$
$420$	0	0
$421$	8.82843	0.430271	0.215136	−	0.976584i	$-0.430981\pi$
$421$	8.82843	0.430271	0.215136	+	0.976584i	$0.430981\pi$
$422$	0	0
$423$	−2.82843	−0.137523
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9.31371	−0.450722
$428$	0	0
$429$	5.65685	0.273115
$430$	0	0
$431$	−6.34315	−0.305539	−0.152769	−	0.988262i	$-0.548819\pi$
$431$	−6.34315	−0.305539	−0.152769	+	0.988262i	$0.548819\pi$
$432$	0	0
$433$	7.31371	0.351474	0.175737	−	0.984437i	$-0.443769\pi$
$433$	7.31371	0.351474	0.175737	+	0.984437i	$0.443769\pi$
$434$	0	0
$435$	−10.3431	−0.495916
$436$	0	0
$437$	4.82843	0.230975
$438$	0	0
$439$	3.51472	0.167748	0.0838742	−	0.996476i	$-0.473271\pi$
$439$	3.51472	0.167748	0.0838742	+	0.996476i	$0.473271\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	17.3137	0.822599	0.411300	−	0.911500i	$-0.365075\pi$
$443$	17.3137	0.822599	0.411300	+	0.911500i	$0.365075\pi$
$444$	0	0
$445$	−28.2843	−1.34080
$446$	0	0
$447$	−18.4853	−0.874324
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	15.3137	0.721094
$452$	0	0
$453$	4.14214	0.194615
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−32.6274	−1.52625	−0.763123	−	0.646253i	$-0.776335\pi$
$457$	−32.6274	−1.52625	−0.763123	+	0.646253i	$0.776335\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.85786	0.459127	0.229563	−	0.973294i	$-0.426270\pi$
$461$	9.85786	0.459127	0.229563	+	0.973294i	$0.426270\pi$
$462$	0	0
$463$	4.68629	0.217790	0.108895	−	0.994053i	$-0.465269\pi$
$463$	4.68629	0.217790	0.108895	+	0.994053i	$0.465269\pi$
$464$	0	0
$465$	−3.31371	−0.153670
$466$	0	0
$467$	−12.6863	−0.587052	−0.293526	−	0.955951i	$-0.594829\pi$
$467$	−12.6863	−0.587052	−0.293526	+	0.955951i	$0.594829\pi$
$468$	0	0
$469$	10.0000	0.461757
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	−3.00000	−0.137649
$476$	0	0
$477$	7.65685	0.350583
$478$	0	0
$479$	−27.1127	−1.23881	−0.619405	−	0.785071i	$-0.712626\pi$
$479$	−27.1127	−1.23881	−0.619405	+	0.785071i	$0.712626\pi$
$480$	0	0
$481$	18.3431	0.836375
$482$	0	0
$483$	4.82843	0.219701
$484$	0	0
$485$	20.6863	0.939316
$486$	0	0
$487$	24.1421	1.09398	0.546992	−	0.837138i	$-0.315773\pi$
$487$	24.1421	1.09398	0.546992	+	0.837138i	$0.315773\pi$
$488$	0	0
$489$	−3.31371	−0.149851
$490$	0	0
$491$	4.34315	0.196003	0.0980017	−	0.995186i	$-0.468755\pi$
$491$	4.34315	0.196003	0.0980017	+	0.995186i	$0.468755\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−5.65685	−0.254257
$496$	0	0
$497$	1.65685	0.0743201
$498$	0	0
$499$	−4.68629	−0.209787	−0.104894	−	0.994483i	$-0.533450\pi$
$499$	−4.68629	−0.209787	−0.104894	+	0.994483i	$0.533450\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11.5147	−0.513416	−0.256708	−	0.966489i	$-0.582638\pi$
$503$	−11.5147	−0.513416	−0.256708	+	0.966489i	$0.582638\pi$
$504$	0	0
$505$	−46.6274	−2.07489
$506$	0	0
$507$	−5.00000	−0.222058
$508$	0	0
$509$	29.3137	1.29931	0.649654	−	0.760230i	$-0.274914\pi$
$509$	29.3137	1.29931	0.649654	+	0.760230i	$0.274914\pi$
$510$	0	0
$511$	11.6569	0.515669
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	35.3137	1.55611
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	−19.6569	−0.862840
$520$	0	0
$521$	23.9411	1.04888	0.524440	−	0.851447i	$-0.324275\pi$
$521$	23.9411	1.04888	0.524440	+	0.851447i	$0.324275\pi$
$522$	0	0
$523$	−13.6569	−0.597173	−0.298586	−	0.954383i	$-0.596515\pi$
$523$	−13.6569	−0.597173	−0.298586	+	0.954383i	$0.596515\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	0	0
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	−1.65685	−0.0719014
$532$	0	0
$533$	21.6569	0.938062
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−41.3137	−1.77622	−0.888108	−	0.459636i	$-0.847980\pi$
$541$	−41.3137	−1.77622	−0.888108	+	0.459636i	$0.847980\pi$
$542$	0	0
$543$	3.51472	0.150831
$544$	0	0
$545$	7.02944	0.301108
$546$	0	0
$547$	−43.9411	−1.87879	−0.939393	−	0.342841i	$-0.888611\pi$
$547$	−43.9411	−1.87879	−0.939393	+	0.342841i	$0.888611\pi$
$548$	0	0
$549$	9.31371	0.397499
$550$	0	0
$551$	3.65685	0.155787
$552$	0	0
$553$	−8.82843	−0.375423
$554$	0	0
$555$	−18.3431	−0.778623
$556$	0	0
$557$	16.8284	0.713043	0.356522	−	0.934287i	$-0.383963\pi$
$557$	16.8284	0.713043	0.356522	+	0.934287i	$0.383963\pi$
$558$	0	0
$559$	22.6274	0.957038
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.5980	−1.50028	−0.750138	−	0.661281i	$-0.770013\pi$
$563$	−35.5980	−1.50028	−0.750138	+	0.661281i	$0.770013\pi$
$564$	0	0
$565$	−16.9706	−0.713957
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−5.31371	−0.222762	−0.111381	−	0.993778i	$-0.535527\pi$
$569$	−5.31371	−0.222762	−0.111381	+	0.993778i	$0.535527\pi$
$570$	0	0
$571$	−42.6274	−1.78390	−0.891951	−	0.452132i	$-0.850664\pi$
$571$	−42.6274	−1.78390	−0.891951	+	0.452132i	$0.850664\pi$
$572$	0	0
$573$	−16.1421	−0.674347
$574$	0	0
$575$	−14.4853	−0.604078
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	26.9706	1.12086
$580$	0	0
$581$	−11.3137	−0.469372
$582$	0	0
$583$	−15.3137	−0.634229
$584$	0	0
$585$	−8.00000	−0.330759
$586$	0	0
$587$	−10.3431	−0.426907	−0.213454	−	0.976953i	$-0.568471\pi$
$587$	−10.3431	−0.426907	−0.213454	+	0.976953i	$0.568471\pi$
$588$	0	0
$589$	1.17157	0.0482738
$590$	0	0
$591$	0.828427	0.0340769
$592$	0	0
$593$	33.6569	1.38212	0.691061	−	0.722797i	$-0.257144\pi$
$593$	33.6569	1.38212	0.691061	+	0.722797i	$0.257144\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.9706	0.694559
$598$	0	0
$599$	−20.9706	−0.856834	−0.428417	−	0.903581i	$-0.640929\pi$
$599$	−20.9706	−0.856834	−0.428417	+	0.903581i	$0.640929\pi$
$600$	0	0
$601$	−33.6569	−1.37289	−0.686446	−	0.727181i	$-0.740830\pi$
$601$	−33.6569	−1.37289	−0.686446	+	0.727181i	$0.740830\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	0	0
$605$	−19.7990	−0.804943
$606$	0	0
$607$	−26.8284	−1.08893	−0.544466	−	0.838783i	$-0.683268\pi$
$607$	−26.8284	−1.08893	−0.544466	+	0.838783i	$0.683268\pi$
$608$	0	0
$609$	3.65685	0.148183
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−30.2843	−1.22317	−0.611585	−	0.791179i	$-0.709468\pi$
$613$	−30.2843	−1.22317	−0.611585	+	0.791179i	$0.709468\pi$
$614$	0	0
$615$	−21.6569	−0.873289
$616$	0	0
$617$	10.6863	0.430214	0.215107	−	0.976590i	$-0.430990\pi$
$617$	10.6863	0.430214	0.215107	+	0.976590i	$0.430990\pi$
$618$	0	0
$619$	−7.31371	−0.293963	−0.146981	−	0.989139i	$-0.546956\pi$
$619$	−7.31371	−0.293963	−0.146981	+	0.989139i	$0.546956\pi$
$620$	0	0
$621$	−4.82843	−0.193758
$622$	0	0
$623$	10.0000	0.400642
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	0	0
$630$	0	0
$631$	41.6569	1.65833	0.829167	−	0.559002i	$-0.188815\pi$
$631$	41.6569	1.65833	0.829167	+	0.559002i	$0.188815\pi$
$632$	0	0
$633$	−4.34315	−0.172625
$634$	0	0
$635$	−24.9706	−0.990927
$636$	0	0
$637$	−2.82843	−0.112066
$638$	0	0
$639$	−1.65685	−0.0655441
$640$	0	0
$641$	−26.6863	−1.05405	−0.527023	−	0.849851i	$-0.676692\pi$
$641$	−26.6863	−1.05405	−0.527023	+	0.849851i	$0.676692\pi$
$642$	0	0
$643$	44.9706	1.77347	0.886733	−	0.462282i	$-0.152969\pi$
$643$	44.9706	1.77347	0.886733	+	0.462282i	$0.152969\pi$
$644$	0	0
$645$	−22.6274	−0.890954
$646$	0	0
$647$	−5.85786	−0.230296	−0.115148	−	0.993348i	$-0.536734\pi$
$647$	−5.85786	−0.230296	−0.115148	+	0.993348i	$0.536734\pi$
$648$	0	0
$649$	3.31371	0.130074
$650$	0	0
$651$	1.17157	0.0459176
$652$	0	0
$653$	25.5147	0.998468	0.499234	−	0.866467i	$-0.333615\pi$
$653$	25.5147	0.998468	0.499234	+	0.866467i	$0.333615\pi$
$654$	0	0
$655$	43.3137	1.69241
$656$	0	0
$657$	−11.6569	−0.454777
$658$	0	0
$659$	29.9411	1.16634	0.583170	−	0.812350i	$-0.301812\pi$
$659$	29.9411	1.16634	0.583170	+	0.812350i	$0.301812\pi$
$660$	0	0
$661$	15.7990	0.614509	0.307255	−	0.951627i	$-0.400590\pi$
$661$	15.7990	0.614509	0.307255	+	0.951627i	$0.400590\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	17.6569	0.683676
$668$	0	0
$669$	−8.48528	−0.328060
$670$	0	0
$671$	−18.6274	−0.719103
$672$	0	0
$673$	−34.9706	−1.34802	−0.674008	−	0.738724i	$-0.735429\pi$
$673$	−34.9706	−1.34802	−0.674008	+	0.738724i	$0.735429\pi$
$674$	0	0
$675$	3.00000	0.115470
$676$	0	0
$677$	−14.6863	−0.564440	−0.282220	−	0.959350i	$-0.591071\pi$
$677$	−14.6863	−0.564440	−0.282220	+	0.959350i	$0.591071\pi$
$678$	0	0
$679$	−7.31371	−0.280674
$680$	0	0
$681$	15.3137	0.586823
$682$	0	0
$683$	−33.9411	−1.29872	−0.649361	−	0.760481i	$-0.724963\pi$
$683$	−33.9411	−1.29872	−0.649361	+	0.760481i	$0.724963\pi$
$684$	0	0
$685$	0.970563	0.0370833
$686$	0	0
$687$	0.343146	0.0130918
$688$	0	0
$689$	−21.6569	−0.825060
$690$	0	0
$691$	0.686292	0.0261078	0.0130539	−	0.999915i	$-0.495845\pi$
$691$	0.686292	0.0261078	0.0130539	+	0.999915i	$0.495845\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	4.68629	0.177761
$696$	0	0
$697$	0	0
$698$	0	0
$699$	20.6274	0.780201
$700$	0	0
$701$	23.1716	0.875178	0.437589	−	0.899175i	$-0.355832\pi$
$701$	23.1716	0.875178	0.437589	+	0.899175i	$0.355832\pi$
$702$	0	0
$703$	6.48528	0.244597
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	16.4853	0.619993
$708$	0	0
$709$	−18.2843	−0.686680	−0.343340	−	0.939211i	$-0.611558\pi$
$709$	−18.2843	−0.686680	−0.343340	+	0.939211i	$0.611558\pi$
$710$	0	0
$711$	8.82843	0.331092
$712$	0	0
$713$	5.65685	0.211851
$714$	0	0
$715$	16.0000	0.598366
$716$	0	0
$717$	8.82843	0.329704
$718$	0	0
$719$	−2.14214	−0.0798882	−0.0399441	−	0.999202i	$-0.512718\pi$
$719$	−2.14214	−0.0798882	−0.0399441	+	0.999202i	$0.512718\pi$
$720$	0	0
$721$	−12.4853	−0.464976
$722$	0	0
$723$	22.6274	0.841523
$724$	0	0
$725$	−10.9706	−0.407436
$726$	0	0
$727$	0.970563	0.0359962	0.0179981	−	0.999838i	$-0.494271\pi$
$727$	0.970563	0.0359962	0.0179981	+	0.999838i	$0.494271\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	2.82843	0.104328
$736$	0	0
$737$	20.0000	0.736709
$738$	0	0
$739$	−10.6274	−0.390936	−0.195468	−	0.980710i	$-0.562623\pi$
$739$	−10.6274	−0.390936	−0.195468	+	0.980710i	$0.562623\pi$
$740$	0	0
$741$	2.82843	0.103905
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−52.2843	−1.91555
$746$	0	0
$747$	11.3137	0.413947
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−13.1127	−0.478489	−0.239245	−	0.970959i	$-0.576900\pi$
$751$	−13.1127	−0.478489	−0.239245	+	0.970959i	$0.576900\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	11.7157	0.426379
$756$	0	0
$757$	48.9117	1.77773	0.888863	−	0.458174i	$-0.151496\pi$
$757$	48.9117	1.77773	0.888863	+	0.458174i	$0.151496\pi$
$758$	0	0
$759$	9.65685	0.350522
$760$	0	0
$761$	−42.6274	−1.54524	−0.772621	−	0.634867i	$-0.781055\pi$
$761$	−42.6274	−1.54524	−0.772621	+	0.634867i	$0.781055\pi$
$762$	0	0
$763$	−2.48528	−0.0899732
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.68629	0.169212
$768$	0	0
$769$	33.5980	1.21157	0.605787	−	0.795627i	$-0.292858\pi$
$769$	33.5980	1.21157	0.605787	+	0.795627i	$0.292858\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	40.6274	1.46127	0.730633	−	0.682770i	$-0.239225\pi$
$773$	40.6274	1.46127	0.730633	+	0.682770i	$0.239225\pi$
$774$	0	0
$775$	−3.51472	−0.126252
$776$	0	0
$777$	6.48528	0.232658
$778$	0	0
$779$	7.65685	0.274335
$780$	0	0
$781$	3.31371	0.118574
$782$	0	0
$783$	−3.65685	−0.130685
$784$	0	0
$785$	28.2843	1.00951
$786$	0	0
$787$	−35.3137	−1.25880	−0.629399	−	0.777082i	$-0.716699\pi$
$787$	−35.3137	−1.25880	−0.629399	+	0.777082i	$0.716699\pi$
$788$	0	0
$789$	16.1421	0.574675
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−26.3431	−0.935473
$794$	0	0
$795$	21.6569	0.768089
$796$	0	0
$797$	10.9706	0.388597	0.194299	−	0.980942i	$-0.437757\pi$
$797$	10.9706	0.388597	0.194299	+	0.980942i	$0.437757\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	23.3137	0.822723
$804$	0	0
$805$	13.6569	0.481341
$806$	0	0
$807$	−14.0000	−0.492823
$808$	0	0
$809$	−34.2843	−1.20537	−0.602685	−	0.797979i	$-0.705903\pi$
$809$	−34.2843	−1.20537	−0.602685	+	0.797979i	$0.705903\pi$
$810$	0	0
$811$	−5.65685	−0.198639	−0.0993195	−	0.995056i	$-0.531667\pi$
$811$	−5.65685	−0.198639	−0.0993195	+	0.995056i	$0.531667\pi$
$812$	0	0
$813$	−19.3137	−0.677361
$814$	0	0
$815$	−9.37258	−0.328307
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	−35.4558	−1.23742	−0.618709	−	0.785620i	$-0.712344\pi$
$821$	−35.4558	−1.23742	−0.618709	+	0.785620i	$0.712344\pi$
$822$	0	0
$823$	−30.6274	−1.06760	−0.533802	−	0.845609i	$-0.679237\pi$
$823$	−30.6274	−1.06760	−0.533802	+	0.845609i	$0.679237\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	17.4558	0.606267	0.303133	−	0.952948i	$-0.401967\pi$
$829$	17.4558	0.606267	0.303133	+	0.952948i	$0.401967\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.17157	−0.0404955
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	−2.68629	−0.0925208
$844$	0	0
$845$	−14.1421	−0.486504
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−17.6569	−0.605982
$850$	0	0
$851$	31.3137	1.07342
$852$	0	0
$853$	−21.3137	−0.729767	−0.364884	−	0.931053i	$-0.618891\pi$
$853$	−21.3137	−0.729767	−0.364884	+	0.931053i	$0.618891\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	4.62742	0.158070	0.0790348	−	0.996872i	$-0.474816\pi$
$857$	4.62742	0.158070	0.0790348	+	0.996872i	$0.474816\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	7.65685	0.260945
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	−55.5980	−1.89039
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−17.6569	−0.598968
$870$	0	0
$871$	28.2843	0.958376
$872$	0	0
$873$	7.31371	0.247532
$874$	0	0
$875$	5.65685	0.191237
$876$	0	0
$877$	35.4558	1.19726	0.598629	−	0.801026i	$-0.295712\pi$
$877$	35.4558	1.19726	0.598629	+	0.801026i	$0.295712\pi$
$878$	0	0
$879$	10.0000	0.337292
$880$	0	0
$881$	32.9706	1.11081	0.555403	−	0.831581i	$-0.312564\pi$
$881$	32.9706	1.11081	0.555403	+	0.831581i	$0.312564\pi$
$882$	0	0
$883$	28.6863	0.965371	0.482685	−	0.875794i	$-0.339662\pi$
$883$	28.6863	0.965371	0.482685	+	0.875794i	$0.339662\pi$
$884$	0	0
$885$	−4.68629	−0.157528
$886$	0	0
$887$	−34.3431	−1.15313	−0.576565	−	0.817051i	$-0.695607\pi$
$887$	−34.3431	−1.15313	−0.576565	+	0.817051i	$0.695607\pi$
$888$	0	0
$889$	8.82843	0.296096
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	2.82843	0.0946497
$894$	0	0
$895$	11.3137	0.378176
$896$	0	0
$897$	13.6569	0.455989
$898$	0	0
$899$	4.28427	0.142888
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	9.94113	0.330454
$906$	0	0
$907$	−12.6274	−0.419286	−0.209643	−	0.977778i	$-0.567230\pi$
$907$	−12.6274	−0.419286	−0.209643	+	0.977778i	$0.567230\pi$
$908$	0	0
$909$	−16.4853	−0.546782
$910$	0	0
$911$	49.6569	1.64520	0.822602	−	0.568617i	$-0.192521\pi$
$911$	49.6569	1.64520	0.822602	+	0.568617i	$0.192521\pi$
$912$	0	0
$913$	−22.6274	−0.748858
$914$	0	0
$915$	26.3431	0.870878
$916$	0	0
$917$	−15.3137	−0.505703
$918$	0	0
$919$	−43.3137	−1.42879	−0.714394	−	0.699744i	$-0.753297\pi$
$919$	−43.3137	−1.42879	−0.714394	+	0.699744i	$0.753297\pi$
$920$	0	0
$921$	5.65685	0.186400
$922$	0	0
$923$	4.68629	0.154251
$924$	0	0
$925$	−19.4558	−0.639704
$926$	0	0
$927$	12.4853	0.410070
$928$	0	0
$929$	−27.3137	−0.896134	−0.448067	−	0.894000i	$-0.647887\pi$
$929$	−27.3137	−0.896134	−0.448067	+	0.894000i	$0.647887\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	20.4853	0.670658
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.2843	−0.727995	−0.363998	−	0.931400i	$-0.618588\pi$
$937$	−22.2843	−0.727995	−0.363998	+	0.931400i	$0.618588\pi$
$938$	0	0
$939$	−27.6569	−0.902547
$940$	0	0
$941$	5.31371	0.173222	0.0866110	−	0.996242i	$-0.472396\pi$
$941$	5.31371	0.173222	0.0866110	+	0.996242i	$0.472396\pi$
$942$	0	0
$943$	36.9706	1.20393
$944$	0	0
$945$	−2.82843	−0.0920087
$946$	0	0
$947$	54.2843	1.76400	0.882001	−	0.471248i	$-0.156196\pi$
$947$	54.2843	1.76400	0.882001	+	0.471248i	$0.156196\pi$
$948$	0	0
$949$	32.9706	1.07027
$950$	0	0
$951$	30.9706	1.00429
$952$	0	0
$953$	−4.62742	−0.149897	−0.0749484	−	0.997187i	$-0.523879\pi$
$953$	−4.62742	−0.149897	−0.0749484	+	0.997187i	$0.523879\pi$
$954$	0	0
$955$	−45.6569	−1.47742
$956$	0	0
$957$	7.31371	0.236419
$958$	0	0
$959$	−0.343146	−0.0110808
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	76.2843	2.45568
$966$	0	0
$967$	−30.3431	−0.975770	−0.487885	−	0.872908i	$-0.662231\pi$
$967$	−30.3431	−0.975770	−0.487885	+	0.872908i	$0.662231\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	55.3137	1.77510	0.887551	−	0.460710i	$-0.152405\pi$
$971$	55.3137	1.77510	0.887551	+	0.460710i	$0.152405\pi$
$972$	0	0
$973$	−1.65685	−0.0531163
$974$	0	0
$975$	−8.48528	−0.271746
$976$	0	0
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	2.48528	0.0793489
$982$	0	0
$983$	42.9117	1.36867	0.684335	−	0.729168i	$-0.260093\pi$
$983$	42.9117	1.36867	0.684335	+	0.729168i	$0.260093\pi$
$984$	0	0
$985$	2.34315	0.0746588
$986$	0	0
$987$	2.82843	0.0900298
$988$	0	0
$989$	38.6274	1.22828
$990$	0	0
$991$	−6.20101	−0.196982	−0.0984908	−	0.995138i	$-0.531402\pi$
$991$	−6.20101	−0.196982	−0.0984908	+	0.995138i	$0.531402\pi$
$992$	0	0
$993$	−12.6274	−0.400719
$994$	0	0
$995$	48.0000	1.52170
$996$	0	0
$997$	−4.62742	−0.146552	−0.0732759	−	0.997312i	$-0.523345\pi$
$997$	−4.62742	−0.146552	−0.0732759	+	0.997312i	$0.523345\pi$
$998$	0	0
$999$	−6.48528	−0.205185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bq.1.2		2
4.3	odd	2		798.2.a.l.1.2	✓	2
12.11	even	2		2394.2.a.r.1.1		2
28.27	even	2		5586.2.a.bp.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.l.1.2	✓	2	4.3	odd	2
2394.2.a.r.1.1		2	12.11	even	2
5586.2.a.bp.1.1		2	28.27	even	2
6384.2.a.bq.1.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.82843 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.82843 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} +2.82843 q^{15} -1.00000 q^{19} -1.00000 q^{21} -4.82843 q^{23} +3.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} -1.17157 q^{31} -2.00000 q^{33} -2.82843 q^{35} -6.48528 q^{37} -2.82843 q^{39} -7.65685 q^{41} -8.00000 q^{43} +2.82843 q^{45} -2.82843 q^{47} +1.00000 q^{49} +7.65685 q^{53} -5.65685 q^{55} -1.00000 q^{57} -1.65685 q^{59} +9.31371 q^{61} -1.00000 q^{63} -8.00000 q^{65} -10.0000 q^{67} -4.82843 q^{69} -1.65685 q^{71} -11.6569 q^{73} +3.00000 q^{75} +2.00000 q^{77} +8.82843 q^{79} +1.00000 q^{81} +11.3137 q^{83} -3.65685 q^{87} -10.0000 q^{89} +2.82843 q^{91} -1.17157 q^{93} -2.82843 q^{95} +7.31371 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{19} - 2 q^{21} - 4 q^{23} + 6 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} - 4 q^{33} + 4 q^{37} - 4 q^{41} - 16 q^{43} + 2 q^{49} + 4 q^{53} - 2 q^{57} + 8 q^{59} - 4 q^{61} - 2 q^{63} - 16 q^{65} - 20 q^{67} - 4 q^{69} + 8 q^{71} - 12 q^{73} + 6 q^{75} + 4 q^{77} + 12 q^{79} + 2 q^{81} + 4 q^{87} - 20 q^{89} - 8 q^{93} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^19 - 2 * q^21 - 4 * q^23 + 6 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 - 4 * q^33 + 4 * q^37 - 4 * q^41 - 16 * q^43 + 2 * q^49 + 4 * q^53 - 2 * q^57 + 8 * q^59 - 4 * q^61 - 2 * q^63 - 16 * q^65 - 20 * q^67 - 4 * q^69 + 8 * q^71 - 12 * q^73 + 6 * q^75 + 4 * q^77 + 12 * q^79 + 2 * q^81 + 4 * q^87 - 20 * q^89 - 8 * q^93 - 8 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more