LMFDB - Embedded newform 6384.2.a.ce.1.5

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:	$5$
Coefficient field:	5.5.135076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.15154$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.30308 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.30308 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.56627 q^{11} +2.81429 q^{13} +4.30308 q^{15} -7.96536 q^{17} -1.00000 q^{19} -1.00000 q^{21} +0.814291 q^{23} +13.5165 q^{25} +1.00000 q^{27} +3.48879 q^{29} -3.38056 q^{31} +2.56627 q^{33} -4.30308 q^{35} +3.75198 q^{37} +2.81429 q^{39} +6.60615 q^{41} +4.30308 q^{45} -10.2533 q^{47} +1.00000 q^{49} -7.96536 q^{51} +0.511215 q^{53} +11.0428 q^{55} -1.00000 q^{57} +13.9901 q^{59} +6.00000 q^{61} -1.00000 q^{63} +12.1101 q^{65} -11.9805 q^{67} +0.814291 q^{69} -5.02768 q^{71} +16.0204 q^{73} +13.5165 q^{75} -2.56627 q^{77} +14.8347 q^{79} +1.00000 q^{81} +2.01517 q^{83} -34.2756 q^{85} +3.48879 q^{87} +17.6490 q^{89} -2.81429 q^{91} -3.38056 q^{93} -4.30308 q^{95} -5.66847 q^{97} +2.56627 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} - 5 q^{21} - 2 q^{23} + 19 q^{25} + 5 q^{27} + 4 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{45} + 2 q^{47} + 5 q^{49} - 2 q^{51} + 16 q^{53} + 16 q^{55} - 5 q^{57} + 12 q^{59} + 30 q^{61} - 5 q^{63} + 4 q^{65} - 18 q^{67} - 2 q^{69} + 10 q^{71} + 14 q^{73} + 19 q^{75} + 2 q^{77} + 2 q^{79} + 5 q^{81} + 6 q^{83} + 12 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{93} - 2 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 2 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 5 * q^19 - 5 * q^21 - 2 * q^23 + 19 * q^25 + 5 * q^27 + 4 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 + 2 * q^45 + 2 * q^47 + 5 * q^49 - 2 * q^51 + 16 * q^53 + 16 * q^55 - 5 * q^57 + 12 * q^59 + 30 * q^61 - 5 * q^63 + 4 * q^65 - 18 * q^67 - 2 * q^69 + 10 * q^71 + 14 * q^73 + 19 * q^75 + 2 * q^77 + 2 * q^79 + 5 * q^81 + 6 * q^83 + 12 * q^85 + 4 * q^87 + 10 * q^89 - 8 * q^91 + 4 * q^93 - 2 * q^95 + 8 * 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
$5$	4.30308	1.92439	0.962197	−	0.272354i	$-0.0878023\pi$
$5$	4.30308	1.92439	0.962197	+	0.272354i	$0.0878023\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.56627	0.773759	0.386880	−	0.922130i	$-0.373553\pi$
$11$	2.56627	0.773759	0.386880	+	0.922130i	$0.373553\pi$
$12$	0	0
$13$	2.81429	0.780544	0.390272	−	0.920700i	$-0.372381\pi$
$13$	2.81429	0.780544	0.390272	+	0.920700i	$0.372381\pi$
$14$	0	0
$15$	4.30308	1.11105
$16$	0	0
$17$	−7.96536	−1.93188	−0.965942	−	0.258757i	$-0.916687\pi$
$17$	−7.96536	−1.93188	−0.965942	+	0.258757i	$0.916687\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0.814291	0.169791	0.0848957	−	0.996390i	$-0.472944\pi$
$23$	0.814291	0.169791	0.0848957	+	0.996390i	$0.472944\pi$
$24$	0	0
$25$	13.5165	2.70329
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.48879	0.647851	0.323926	−	0.946083i	$-0.394997\pi$
$29$	3.48879	0.647851	0.323926	+	0.946083i	$0.394997\pi$
$30$	0	0
$31$	−3.38056	−0.607166	−0.303583	−	0.952805i	$-0.598183\pi$
$31$	−3.38056	−0.607166	−0.303583	+	0.952805i	$0.598183\pi$
$32$	0	0
$33$	2.56627	0.446730
$34$	0	0
$35$	−4.30308	−0.727353
$36$	0	0
$37$	3.75198	0.616821	0.308411	−	0.951253i	$-0.400203\pi$
$37$	3.75198	0.616821	0.308411	+	0.951253i	$0.400203\pi$
$38$	0	0
$39$	2.81429	0.450647
$40$	0	0
$41$	6.60615	1.03171	0.515854	−	0.856677i	$-0.327475\pi$
$41$	6.60615	1.03171	0.515854	+	0.856677i	$0.327475\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	4.30308	0.641465
$46$	0	0
$47$	−10.2533	−1.49559	−0.747797	−	0.663928i	$-0.768888\pi$
$47$	−10.2533	−1.49559	−0.747797	+	0.663928i	$0.768888\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.96536	−1.11537
$52$	0	0
$53$	0.511215	0.0702207	0.0351104	−	0.999383i	$-0.488822\pi$
$53$	0.511215	0.0702207	0.0351104	+	0.999383i	$0.488822\pi$
$54$	0	0
$55$	11.0428	1.48902
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	13.9901	1.82135	0.910677	−	0.413120i	$-0.135561\pi$
$59$	13.9901	1.82135	0.910677	+	0.413120i	$0.135561\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	12.1101	1.50207
$66$	0	0
$67$	−11.9805	−1.46366	−0.731828	−	0.681490i	$-0.761332\pi$
$67$	−11.9805	−1.46366	−0.731828	+	0.681490i	$0.761332\pi$
$68$	0	0
$69$	0.814291	0.0980291
$70$	0	0
$71$	−5.02768	−0.596676	−0.298338	−	0.954460i	$-0.596432\pi$
$71$	−5.02768	−0.596676	−0.298338	+	0.954460i	$0.596432\pi$
$72$	0	0
$73$	16.0204	1.87505	0.937524	−	0.347920i	$-0.113112\pi$
$73$	16.0204	1.87505	0.937524	+	0.347920i	$0.113112\pi$
$74$	0	0
$75$	13.5165	1.56075
$76$	0	0
$77$	−2.56627	−0.292453
$78$	0	0
$79$	14.8347	1.66904	0.834518	−	0.550981i	$-0.185746\pi$
$79$	14.8347	1.66904	0.834518	+	0.550981i	$0.185746\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.01517	0.221194	0.110597	−	0.993865i	$-0.464724\pi$
$83$	2.01517	0.221194	0.110597	+	0.993865i	$0.464724\pi$
$84$	0	0
$85$	−34.2756	−3.71771
$86$	0	0
$87$	3.48879	0.374037
$88$	0	0
$89$	17.6490	1.87079	0.935395	−	0.353604i	$-0.115044\pi$
$89$	17.6490	1.87079	0.935395	+	0.353604i	$0.115044\pi$
$90$	0	0
$91$	−2.81429	−0.295018
$92$	0	0
$93$	−3.38056	−0.350548
$94$	0	0
$95$	−4.30308	−0.441486
$96$	0	0
$97$	−5.66847	−0.575545	−0.287773	−	0.957699i	$-0.592915\pi$
$97$	−5.66847	−0.575545	−0.287773	+	0.957699i	$0.592915\pi$
$98$	0	0
$99$	2.56627	0.257920
$100$	0	0
$101$	−4.90923	−0.488486	−0.244243	−	0.969714i	$-0.578540\pi$
$101$	−4.90923	−0.488486	−0.244243	+	0.969714i	$0.578540\pi$
$102$	0	0
$103$	−1.47025	−0.144868	−0.0724339	−	0.997373i	$-0.523077\pi$
$103$	−1.47025	−0.144868	−0.0724339	+	0.997373i	$0.523077\pi$
$104$	0	0
$105$	−4.30308	−0.419937
$106$	0	0
$107$	10.3767	1.00315	0.501575	−	0.865114i	$-0.332754\pi$
$107$	10.3767	1.00315	0.501575	+	0.865114i	$0.332754\pi$
$108$	0	0
$109$	−5.98671	−0.573423	−0.286711	−	0.958017i	$-0.592562\pi$
$109$	−5.98671	−0.573423	−0.286711	+	0.958017i	$0.592562\pi$
$110$	0	0
$111$	3.75198	0.356122
$112$	0	0
$113$	−9.14771	−0.860544	−0.430272	−	0.902699i	$-0.641582\pi$
$113$	−9.14771	−0.860544	−0.430272	+	0.902699i	$0.641582\pi$
$114$	0	0
$115$	3.50396	0.326746
$116$	0	0
$117$	2.81429	0.260181
$118$	0	0
$119$	7.96536	0.730184
$120$	0	0
$121$	−4.41427	−0.401297
$122$	0	0
$123$	6.60615	0.595657
$124$	0	0
$125$	36.6470	3.27781
$126$	0	0
$127$	−0.599976	−0.0532392	−0.0266196	−	0.999646i	$-0.508474\pi$
$127$	−0.599976	−0.0532392	−0.0266196	+	0.999646i	$0.508474\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.14771	−0.624498	−0.312249	−	0.950000i	$-0.601082\pi$
$131$	−7.14771	−0.624498	−0.312249	+	0.950000i	$0.601082\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	4.30308	0.370350
$136$	0	0
$137$	11.6286	0.993497	0.496748	−	0.867895i	$-0.334527\pi$
$137$	11.6286	0.993497	0.496748	+	0.867895i	$0.334527\pi$
$138$	0	0
$139$	3.01251	0.255518	0.127759	−	0.991805i	$-0.459222\pi$
$139$	3.01251	0.255518	0.127759	+	0.991805i	$0.459222\pi$
$140$	0	0
$141$	−10.2533	−0.863481
$142$	0	0
$143$	7.22223	0.603953
$144$	0	0
$145$	15.0125	1.24672
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	12.8542	1.05306	0.526528	−	0.850158i	$-0.323494\pi$
$149$	12.8542	1.05306	0.526528	+	0.850158i	$0.323494\pi$
$150$	0	0
$151$	17.8123	1.44954	0.724771	−	0.688989i	$-0.241945\pi$
$151$	17.8123	1.44954	0.724771	+	0.688989i	$0.241945\pi$
$152$	0	0
$153$	−7.96536	−0.643962
$154$	0	0
$155$	−14.5468	−1.16843
$156$	0	0
$157$	1.92023	0.153251	0.0766256	−	0.997060i	$-0.475585\pi$
$157$	1.92023	0.153251	0.0766256	+	0.997060i	$0.475585\pi$
$158$	0	0
$159$	0.511215	0.0405420
$160$	0	0
$161$	−0.814291	−0.0641751
$162$	0	0
$163$	−22.5468	−1.76600	−0.883001	−	0.469371i	$-0.844481\pi$
$163$	−22.5468	−1.76600	−0.883001	+	0.469371i	$0.844481\pi$
$164$	0	0
$165$	11.0428	0.859685
$166$	0	0
$167$	10.5165	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$167$	10.5165	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$168$	0	0
$169$	−5.07977	−0.390751
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−24.4101	−1.85587	−0.927933	−	0.372746i	$-0.878416\pi$
$173$	−24.4101	−1.85587	−0.927933	+	0.372746i	$0.878416\pi$
$174$	0	0
$175$	−13.5165	−1.02175
$176$	0	0
$177$	13.9901	1.05156
$178$	0	0
$179$	−7.08703	−0.529709	−0.264855	−	0.964288i	$-0.585324\pi$
$179$	−7.08703	−0.529709	−0.264855	+	0.964288i	$0.585324\pi$
$180$	0	0
$181$	2.56964	0.191000	0.0954998	−	0.995429i	$-0.469555\pi$
$181$	2.56964	0.191000	0.0954998	+	0.995429i	$0.469555\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	16.1450	1.18701
$186$	0	0
$187$	−20.4413	−1.49481
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−27.2265	−1.97004	−0.985022	−	0.172429i	$-0.944838\pi$
$191$	−27.2265	−1.97004	−0.985022	+	0.172429i	$0.944838\pi$
$192$	0	0
$193$	−14.5369	−1.04639	−0.523194	−	0.852214i	$-0.675260\pi$
$193$	−14.5369	−1.04639	−0.523194	+	0.852214i	$0.675260\pi$
$194$	0	0
$195$	12.1101	0.867223
$196$	0	0
$197$	−8.32779	−0.593330	−0.296665	−	0.954982i	$-0.595875\pi$
$197$	−8.32779	−0.593330	−0.296665	+	0.954982i	$0.595875\pi$
$198$	0	0
$199$	−14.0204	−0.993881	−0.496941	−	0.867785i	$-0.665543\pi$
$199$	−14.0204	−0.993881	−0.496941	+	0.867785i	$0.665543\pi$
$200$	0	0
$201$	−11.9805	−0.845042
$202$	0	0
$203$	−3.48879	−0.244865
$204$	0	0
$205$	28.4268	1.98541
$206$	0	0
$207$	0.814291	0.0565971
$208$	0	0
$209$	−2.56627	−0.177512
$210$	0	0
$211$	−4.44624	−0.306092	−0.153046	−	0.988219i	$-0.548908\pi$
$211$	−4.44624	−0.306092	−0.153046	+	0.988219i	$0.548908\pi$
$212$	0	0
$213$	−5.02768	−0.344491
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.38056	0.229487
$218$	0	0
$219$	16.0204	1.08256
$220$	0	0
$221$	−22.4169	−1.50792
$222$	0	0
$223$	−16.0238	−1.07303	−0.536516	−	0.843890i	$-0.680260\pi$
$223$	−16.0238	−1.07303	−0.536516	+	0.843890i	$0.680260\pi$
$224$	0	0
$225$	13.5165	0.901098
$226$	0	0
$227$	−3.25930	−0.216327	−0.108164	−	0.994133i	$-0.534497\pi$
$227$	−3.25930	−0.216327	−0.108164	+	0.994133i	$0.534497\pi$
$228$	0	0
$229$	23.7387	1.56870	0.784348	−	0.620321i	$-0.212998\pi$
$229$	23.7387	1.56870	0.784348	+	0.620321i	$0.212998\pi$
$230$	0	0
$231$	−2.56627	−0.168848
$232$	0	0
$233$	21.3673	1.39982	0.699908	−	0.714233i	$-0.253224\pi$
$233$	21.3673	1.39982	0.699908	+	0.714233i	$0.253224\pi$
$234$	0	0
$235$	−44.1206	−2.87811
$236$	0	0
$237$	14.8347	0.963618
$238$	0	0
$239$	29.3208	1.89661	0.948303	−	0.317365i	$-0.102798\pi$
$239$	29.3208	1.89661	0.948303	+	0.317365i	$0.102798\pi$
$240$	0	0
$241$	−14.6216	−0.941862	−0.470931	−	0.882170i	$-0.656082\pi$
$241$	−14.6216	−0.941862	−0.470931	+	0.882170i	$0.656082\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.30308	0.274913
$246$	0	0
$247$	−2.81429	−0.179069
$248$	0	0
$249$	2.01517	0.127706
$250$	0	0
$251$	20.8931	1.31876	0.659381	−	0.751809i	$-0.270818\pi$
$251$	20.8931	1.31876	0.659381	+	0.751809i	$0.270818\pi$
$252$	0	0
$253$	2.08969	0.131378
$254$	0	0
$255$	−34.2756	−2.14642
$256$	0	0
$257$	8.54680	0.533135	0.266567	−	0.963816i	$-0.414110\pi$
$257$	8.54680	0.533135	0.266567	+	0.963816i	$0.414110\pi$
$258$	0	0
$259$	−3.75198	−0.233137
$260$	0	0
$261$	3.48879	0.215950
$262$	0	0
$263$	1.24107	0.0765274	0.0382637	−	0.999268i	$-0.487817\pi$
$263$	1.24107	0.0765274	0.0382637	+	0.999268i	$0.487817\pi$
$264$	0	0
$265$	2.19980	0.135132
$266$	0	0
$267$	17.6490	1.08010
$268$	0	0
$269$	−20.6266	−1.25762	−0.628812	−	0.777557i	$-0.716459\pi$
$269$	−20.6266	−1.25762	−0.628812	+	0.777557i	$0.716459\pi$
$270$	0	0
$271$	17.7758	1.07980	0.539900	−	0.841729i	$-0.318462\pi$
$271$	17.7758	1.07980	0.539900	+	0.841729i	$0.318462\pi$
$272$	0	0
$273$	−2.81429	−0.170329
$274$	0	0
$275$	34.6869	2.09170
$276$	0	0
$277$	−22.8613	−1.37360	−0.686801	−	0.726845i	$-0.740986\pi$
$277$	−22.8613	−1.37360	−0.686801	+	0.726845i	$0.740986\pi$
$278$	0	0
$279$	−3.38056	−0.202389
$280$	0	0
$281$	−16.0949	−0.960143	−0.480072	−	0.877229i	$-0.659389\pi$
$281$	−16.0949	−0.960143	−0.480072	+	0.877229i	$0.659389\pi$
$282$	0	0
$283$	−16.4042	−0.975128	−0.487564	−	0.873087i	$-0.662114\pi$
$283$	−16.4042	−0.975128	−0.487564	+	0.873087i	$0.662114\pi$
$284$	0	0
$285$	−4.30308	−0.254892
$286$	0	0
$287$	−6.60615	−0.389949
$288$	0	0
$289$	46.4470	2.73218
$290$	0	0
$291$	−5.66847	−0.332291
$292$	0	0
$293$	−14.3918	−0.840780	−0.420390	−	0.907343i	$-0.638107\pi$
$293$	−14.3918	−0.840780	−0.420390	+	0.907343i	$0.638107\pi$
$294$	0	0
$295$	60.2004	3.50500
$296$	0	0
$297$	2.56627	0.148910
$298$	0	0
$299$	2.29165	0.132530
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.90923	−0.282028
$304$	0	0
$305$	25.8185	1.47836
$306$	0	0
$307$	−17.9901	−1.02675	−0.513374	−	0.858165i	$-0.671605\pi$
$307$	−17.9901	−1.02675	−0.513374	+	0.858165i	$0.671605\pi$
$308$	0	0
$309$	−1.47025	−0.0836395
$310$	0	0
$311$	12.0916	0.685650	0.342825	−	0.939399i	$-0.388616\pi$
$311$	12.0916	0.685650	0.342825	+	0.939399i	$0.388616\pi$
$312$	0	0
$313$	−25.2024	−1.42452	−0.712261	−	0.701914i	$-0.752329\pi$
$313$	−25.2024	−1.42452	−0.712261	+	0.701914i	$0.752329\pi$
$314$	0	0
$315$	−4.30308	−0.242451
$316$	0	0
$317$	−11.8806	−0.667282	−0.333641	−	0.942700i	$-0.608277\pi$
$317$	−11.8806	−0.667282	−0.333641	+	0.942700i	$0.608277\pi$
$318$	0	0
$319$	8.95316	0.501281
$320$	0	0
$321$	10.3767	0.579169
$322$	0	0
$323$	7.96536	0.443205
$324$	0	0
$325$	38.0393	2.11004
$326$	0	0
$327$	−5.98671	−0.331066
$328$	0	0
$329$	10.2533	0.565281
$330$	0	0
$331$	−6.28454	−0.345429	−0.172715	−	0.984972i	$-0.555254\pi$
$331$	−6.28454	−0.345429	−0.172715	+	0.984972i	$0.555254\pi$
$332$	0	0
$333$	3.75198	0.205607
$334$	0	0
$335$	−51.5532	−2.81665
$336$	0	0
$337$	31.1430	1.69647	0.848235	−	0.529621i	$-0.177666\pi$
$337$	31.1430	1.69647	0.848235	+	0.529621i	$0.177666\pi$
$338$	0	0
$339$	−9.14771	−0.496835
$340$	0	0
$341$	−8.67542	−0.469800
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.50396	0.188647
$346$	0	0
$347$	−31.3196	−1.68132	−0.840662	−	0.541560i	$-0.817834\pi$
$347$	−31.3196	−1.68132	−0.840662	+	0.541560i	$0.817834\pi$
$348$	0	0
$349$	−14.6432	−0.783834	−0.391917	−	0.920001i	$-0.628188\pi$
$349$	−14.6432	−0.783834	−0.391917	+	0.920001i	$0.628188\pi$
$350$	0	0
$351$	2.81429	0.150216
$352$	0	0
$353$	−9.85309	−0.524427	−0.262214	−	0.965010i	$-0.584452\pi$
$353$	−9.85309	−0.524427	−0.262214	+	0.965010i	$0.584452\pi$
$354$	0	0
$355$	−21.6345	−1.14824
$356$	0	0
$357$	7.96536	0.421572
$358$	0	0
$359$	−3.26548	−0.172345	−0.0861726	−	0.996280i	$-0.527464\pi$
$359$	−3.26548	−0.172345	−0.0861726	+	0.996280i	$0.527464\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.41427	−0.231689
$364$	0	0
$365$	68.9371	3.60833
$366$	0	0
$367$	4.03034	0.210382	0.105191	−	0.994452i	$-0.466455\pi$
$367$	4.03034	0.210382	0.105191	+	0.994452i	$0.466455\pi$
$368$	0	0
$369$	6.60615	0.343903
$370$	0	0
$371$	−0.511215	−0.0265409
$372$	0	0
$373$	14.6194	0.756966	0.378483	−	0.925608i	$-0.376446\pi$
$373$	14.6194	0.756966	0.378483	+	0.925608i	$0.376446\pi$
$374$	0	0
$375$	36.6470	1.89244
$376$	0	0
$377$	9.81846	0.505676
$378$	0	0
$379$	−19.3175	−0.992272	−0.496136	−	0.868245i	$-0.665248\pi$
$379$	−19.3175	−0.992272	−0.496136	+	0.868245i	$0.665248\pi$
$380$	0	0
$381$	−0.599976	−0.0307377
$382$	0	0
$383$	14.2954	0.730462	0.365231	−	0.930917i	$-0.380990\pi$
$383$	14.2954	0.730462	0.365231	+	0.930917i	$0.380990\pi$
$384$	0	0
$385$	−11.0428	−0.562796
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.9946	−1.06447	−0.532235	−	0.846597i	$-0.678648\pi$
$389$	−20.9946	−1.06447	−0.532235	+	0.846597i	$0.678648\pi$
$390$	0	0
$391$	−6.48612	−0.328017
$392$	0	0
$393$	−7.14771	−0.360554
$394$	0	0
$395$	63.8349	3.21188
$396$	0	0
$397$	−10.6365	−0.533830	−0.266915	−	0.963720i	$-0.586004\pi$
$397$	−10.6365	−0.533830	−0.266915	+	0.963720i	$0.586004\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−35.0011	−1.74787	−0.873936	−	0.486041i	$-0.838440\pi$
$401$	−35.0011	−1.74787	−0.873936	+	0.486041i	$0.838440\pi$
$402$	0	0
$403$	−9.51388	−0.473920
$404$	0	0
$405$	4.30308	0.213822
$406$	0	0
$407$	9.62858	0.477271
$408$	0	0
$409$	4.98845	0.246663	0.123331	−	0.992366i	$-0.460642\pi$
$409$	4.98845	0.246663	0.123331	+	0.992366i	$0.460642\pi$
$410$	0	0
$411$	11.6286	0.573596
$412$	0	0
$413$	−13.9901	−0.688407
$414$	0	0
$415$	8.67143	0.425664
$416$	0	0
$417$	3.01251	0.147523
$418$	0	0
$419$	−13.6991	−0.669245	−0.334623	−	0.942352i	$-0.608609\pi$
$419$	−13.6991	−0.669245	−0.334623	+	0.942352i	$0.608609\pi$
$420$	0	0
$421$	18.9399	0.923073	0.461536	−	0.887121i	$-0.347298\pi$
$421$	18.9399	0.923073	0.461536	+	0.887121i	$0.347298\pi$
$422$	0	0
$423$	−10.2533	−0.498531
$424$	0	0
$425$	−107.664	−5.22245
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	7.22223	0.348692
$430$	0	0
$431$	−9.20706	−0.443488	−0.221744	−	0.975105i	$-0.571175\pi$
$431$	−9.20706	−0.443488	−0.221744	+	0.975105i	$0.571175\pi$
$432$	0	0
$433$	−11.6236	−0.558595	−0.279297	−	0.960205i	$-0.590102\pi$
$433$	−11.6236	−0.558595	−0.279297	+	0.960205i	$0.590102\pi$
$434$	0	0
$435$	15.0125	0.719795
$436$	0	0
$437$	−0.814291	−0.0389528
$438$	0	0
$439$	−4.43333	−0.211591	−0.105796	−	0.994388i	$-0.533739\pi$
$439$	−4.43333	−0.211591	−0.105796	+	0.994388i	$0.533739\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−13.2847	−0.631175	−0.315587	−	0.948897i	$-0.602202\pi$
$443$	−13.2847	−0.631175	−0.315587	+	0.948897i	$0.602202\pi$
$444$	0	0
$445$	75.9450	3.60014
$446$	0	0
$447$	12.8542	0.607982
$448$	0	0
$449$	−38.9213	−1.83681	−0.918406	−	0.395640i	$-0.870523\pi$
$449$	−38.9213	−1.83681	−0.918406	+	0.395640i	$0.870523\pi$
$450$	0	0
$451$	16.9532	0.798293
$452$	0	0
$453$	17.8123	0.836894
$454$	0	0
$455$	−12.1101	−0.567731
$456$	0	0
$457$	19.1820	0.897294	0.448647	−	0.893709i	$-0.351906\pi$
$457$	19.1820	0.897294	0.448647	+	0.893709i	$0.351906\pi$
$458$	0	0
$459$	−7.96536	−0.371791
$460$	0	0
$461$	10.6356	0.495348	0.247674	−	0.968843i	$-0.420334\pi$
$461$	10.6356	0.495348	0.247674	+	0.968843i	$0.420334\pi$
$462$	0	0
$463$	12.1794	0.566024	0.283012	−	0.959116i	$-0.408666\pi$
$463$	12.1794	0.566024	0.283012	+	0.959116i	$0.408666\pi$
$464$	0	0
$465$	−14.5468	−0.674592
$466$	0	0
$467$	19.3765	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$467$	19.3765	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$468$	0	0
$469$	11.9805	0.553210
$470$	0	0
$471$	1.92023	0.0884797
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−13.5165	−0.620178
$476$	0	0
$477$	0.511215	0.0234069
$478$	0	0
$479$	−4.65503	−0.212694	−0.106347	−	0.994329i	$-0.533915\pi$
$479$	−4.65503	−0.212694	−0.106347	+	0.994329i	$0.533915\pi$
$480$	0	0
$481$	10.5592	0.481456
$482$	0	0
$483$	−0.814291	−0.0370515
$484$	0	0
$485$	−24.3918	−1.10758
$486$	0	0
$487$	−28.5431	−1.29341	−0.646705	−	0.762740i	$-0.723853\pi$
$487$	−28.5431	−1.29341	−0.646705	+	0.762740i	$0.723853\pi$
$488$	0	0
$489$	−22.5468	−1.01960
$490$	0	0
$491$	−20.2443	−0.913611	−0.456806	−	0.889566i	$-0.651007\pi$
$491$	−20.2443	−0.913611	−0.456806	+	0.889566i	$0.651007\pi$
$492$	0	0
$493$	−27.7894	−1.25157
$494$	0	0
$495$	11.0428	0.496339
$496$	0	0
$497$	5.02768	0.225522
$498$	0	0
$499$	29.6345	1.32662	0.663311	−	0.748344i	$-0.269151\pi$
$499$	29.6345	1.32662	0.663311	+	0.748344i	$0.269151\pi$
$500$	0	0
$501$	10.5165	0.469841
$502$	0	0
$503$	4.03089	0.179729	0.0898643	−	0.995954i	$-0.471357\pi$
$503$	4.03089	0.179729	0.0898643	+	0.995954i	$0.471357\pi$
$504$	0	0
$505$	−21.1248	−0.940040
$506$	0	0
$507$	−5.07977	−0.225600
$508$	0	0
$509$	−14.5409	−0.644513	−0.322256	−	0.946652i	$-0.604441\pi$
$509$	−14.5409	−0.644513	−0.322256	+	0.946652i	$0.604441\pi$
$510$	0	0
$511$	−16.0204	−0.708702
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−6.32659	−0.278783
$516$	0	0
$517$	−26.3126	−1.15723
$518$	0	0
$519$	−24.4101	−1.07149
$520$	0	0
$521$	−32.0631	−1.40471	−0.702355	−	0.711827i	$-0.747868\pi$
$521$	−32.0631	−1.40471	−0.702355	+	0.711827i	$0.747868\pi$
$522$	0	0
$523$	39.0837	1.70901	0.854505	−	0.519443i	$-0.173861\pi$
$523$	39.0837	1.70901	0.854505	+	0.519443i	$0.173861\pi$
$524$	0	0
$525$	−13.5165	−0.589907
$526$	0	0
$527$	26.9274	1.17298
$528$	0	0
$529$	−22.3369	−0.971171
$530$	0	0
$531$	13.9901	0.607118
$532$	0	0
$533$	18.5916	0.805293
$534$	0	0
$535$	44.6516	1.93046
$536$	0	0
$537$	−7.08703	−0.305828
$538$	0	0
$539$	2.56627	0.110537
$540$	0	0
$541$	28.3166	1.21743	0.608714	−	0.793390i	$-0.291686\pi$
$541$	28.3166	1.21743	0.608714	+	0.793390i	$0.291686\pi$
$542$	0	0
$543$	2.56964	0.110274
$544$	0	0
$545$	−25.7613	−1.10349
$546$	0	0
$547$	25.2621	1.08013	0.540065	−	0.841623i	$-0.318400\pi$
$547$	25.2621	1.08013	0.540065	+	0.841623i	$0.318400\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−3.48879	−0.148627
$552$	0	0
$553$	−14.8347	−0.630836
$554$	0	0
$555$	16.1450	0.685319
$556$	0	0
$557$	8.67480	0.367563	0.183781	−	0.982967i	$-0.441166\pi$
$557$	8.67480	0.367563	0.183781	+	0.982967i	$0.441166\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−20.4413	−0.863031
$562$	0	0
$563$	5.97958	0.252009	0.126005	−	0.992030i	$-0.459785\pi$
$563$	5.97958	0.252009	0.126005	+	0.992030i	$0.459785\pi$
$564$	0	0
$565$	−39.3633	−1.65603
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−30.7830	−1.29049	−0.645246	−	0.763975i	$-0.723245\pi$
$569$	−30.7830	−1.29049	−0.645246	+	0.763975i	$0.723245\pi$
$570$	0	0
$571$	−11.5389	−0.482888	−0.241444	−	0.970415i	$-0.577621\pi$
$571$	−11.5389	−0.482888	−0.241444	+	0.970415i	$0.577621\pi$
$572$	0	0
$573$	−27.2265	−1.13741
$574$	0	0
$575$	11.0063	0.458996
$576$	0	0
$577$	24.7345	1.02971	0.514856	−	0.857277i	$-0.327845\pi$
$577$	24.7345	1.02971	0.514856	+	0.857277i	$0.327845\pi$
$578$	0	0
$579$	−14.5369	−0.604132
$580$	0	0
$581$	−2.01517	−0.0836033
$582$	0	0
$583$	1.31191	0.0543339
$584$	0	0
$585$	12.1101	0.500691
$586$	0	0
$587$	23.5745	0.973023	0.486511	−	0.873674i	$-0.338269\pi$
$587$	23.5745	0.973023	0.486511	+	0.873674i	$0.338269\pi$
$588$	0	0
$589$	3.38056	0.139294
$590$	0	0
$591$	−8.32779	−0.342559
$592$	0	0
$593$	−13.0608	−0.536344	−0.268172	−	0.963371i	$-0.586420\pi$
$593$	−13.0608	−0.536344	−0.268172	+	0.963371i	$0.586420\pi$
$594$	0	0
$595$	34.2756	1.40516
$596$	0	0
$597$	−14.0204	−0.573818
$598$	0	0
$599$	5.38643	0.220084	0.110042	−	0.993927i	$-0.464902\pi$
$599$	5.38643	0.220084	0.110042	+	0.993927i	$0.464902\pi$
$600$	0	0
$601$	−25.8213	−1.05327	−0.526636	−	0.850091i	$-0.676547\pi$
$601$	−25.8213	−1.05327	−0.526636	+	0.850091i	$0.676547\pi$
$602$	0	0
$603$	−11.9805	−0.487885
$604$	0	0
$605$	−18.9949	−0.772253
$606$	0	0
$607$	26.6095	1.08005	0.540024	−	0.841650i	$-0.318415\pi$
$607$	26.6095	1.08005	0.540024	+	0.841650i	$0.318415\pi$
$608$	0	0
$609$	−3.48879	−0.141373
$610$	0	0
$611$	−28.8557	−1.16738
$612$	0	0
$613$	−12.4101	−0.501240	−0.250620	−	0.968086i	$-0.580634\pi$
$613$	−12.4101	−0.501240	−0.250620	+	0.968086i	$0.580634\pi$
$614$	0	0
$615$	28.4268	1.14628
$616$	0	0
$617$	−36.3448	−1.46319	−0.731594	−	0.681740i	$-0.761223\pi$
$617$	−36.3448	−1.46319	−0.731594	+	0.681740i	$0.761223\pi$
$618$	0	0
$619$	26.3371	1.05858	0.529288	−	0.848442i	$-0.322459\pi$
$619$	26.3371	1.05858	0.529288	+	0.848442i	$0.322459\pi$
$620$	0	0
$621$	0.814291	0.0326764
$622$	0	0
$623$	−17.6490	−0.707092
$624$	0	0
$625$	90.1125	3.60450
$626$	0	0
$627$	−2.56627	−0.102487
$628$	0	0
$629$	−29.8859	−1.19163
$630$	0	0
$631$	23.2024	0.923672	0.461836	−	0.886965i	$-0.347191\pi$
$631$	23.2024	0.923672	0.461836	+	0.886965i	$0.347191\pi$
$632$	0	0
$633$	−4.44624	−0.176722
$634$	0	0
$635$	−2.58174	−0.102453
$636$	0	0
$637$	2.81429	0.111506
$638$	0	0
$639$	−5.02768	−0.198892
$640$	0	0
$641$	27.8540	1.10017	0.550084	−	0.835109i	$-0.314596\pi$
$641$	27.8540	1.10017	0.550084	+	0.835109i	$0.314596\pi$
$642$	0	0
$643$	−33.5593	−1.32345	−0.661725	−	0.749747i	$-0.730175\pi$
$643$	−33.5593	−1.32345	−0.661725	+	0.749747i	$0.730175\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.0534	0.867010	0.433505	−	0.901151i	$-0.357277\pi$
$647$	22.0534	0.867010	0.433505	+	0.901151i	$0.357277\pi$
$648$	0	0
$649$	35.9023	1.40929
$650$	0	0
$651$	3.38056	0.132495
$652$	0	0
$653$	−1.41949	−0.0555489	−0.0277745	−	0.999614i	$-0.508842\pi$
$653$	−1.41949	−0.0555489	−0.0277745	+	0.999614i	$0.508842\pi$
$654$	0	0
$655$	−30.7571	−1.20178
$656$	0	0
$657$	16.0204	0.625016
$658$	0	0
$659$	2.46852	0.0961600	0.0480800	−	0.998843i	$-0.484690\pi$
$659$	2.46852	0.0961600	0.0480800	+	0.998843i	$0.484690\pi$
$660$	0	0
$661$	−6.78771	−0.264011	−0.132006	−	0.991249i	$-0.542142\pi$
$661$	−6.78771	−0.264011	−0.132006	+	0.991249i	$0.542142\pi$
$662$	0	0
$663$	−22.4169	−0.870598
$664$	0	0
$665$	4.30308	0.166866
$666$	0	0
$667$	2.84089	0.110000
$668$	0	0
$669$	−16.0238	−0.619515
$670$	0	0
$671$	15.3976	0.594418
$672$	0	0
$673$	23.6087	0.910050	0.455025	−	0.890479i	$-0.349630\pi$
$673$	23.6087	0.910050	0.455025	+	0.890479i	$0.349630\pi$
$674$	0	0
$675$	13.5165	0.520249
$676$	0	0
$677$	−14.2817	−0.548891	−0.274446	−	0.961603i	$-0.588494\pi$
$677$	−14.2817	−0.548891	−0.274446	+	0.961603i	$0.588494\pi$
$678$	0	0
$679$	5.66847	0.217536
$680$	0	0
$681$	−3.25930	−0.124897
$682$	0	0
$683$	−38.0560	−1.45617	−0.728086	−	0.685485i	$-0.759590\pi$
$683$	−38.0560	−1.45617	−0.728086	+	0.685485i	$0.759590\pi$
$684$	0	0
$685$	50.0387	1.91188
$686$	0	0
$687$	23.7387	0.905687
$688$	0	0
$689$	1.43871	0.0548104
$690$	0	0
$691$	18.3838	0.699352	0.349676	−	0.936871i	$-0.386292\pi$
$691$	18.3838	0.699352	0.349676	+	0.936871i	$0.386292\pi$
$692$	0	0
$693$	−2.56627	−0.0974845
$694$	0	0
$695$	12.9631	0.491717
$696$	0	0
$697$	−52.6204	−1.99314
$698$	0	0
$699$	21.3673	0.808184
$700$	0	0
$701$	32.7583	1.23726	0.618632	−	0.785681i	$-0.287687\pi$
$701$	32.7583	1.23726	0.618632	+	0.785681i	$0.287687\pi$
$702$	0	0
$703$	−3.75198	−0.141509
$704$	0	0
$705$	−44.1206	−1.66168
$706$	0	0
$707$	4.90923	0.184631
$708$	0	0
$709$	2.39583	0.0899773	0.0449886	−	0.998987i	$-0.485675\pi$
$709$	2.39583	0.0899773	0.0449886	+	0.998987i	$0.485675\pi$
$710$	0	0
$711$	14.8347	0.556345
$712$	0	0
$713$	−2.75276	−0.103092
$714$	0	0
$715$	31.0778	1.16224
$716$	0	0
$717$	29.3208	1.09501
$718$	0	0
$719$	−17.7999	−0.663825	−0.331912	−	0.943310i	$-0.607694\pi$
$719$	−17.7999	−0.663825	−0.331912	+	0.943310i	$0.607694\pi$
$720$	0	0
$721$	1.47025	0.0547549
$722$	0	0
$723$	−14.6216	−0.543784
$724$	0	0
$725$	47.1560	1.75133
$726$	0	0
$727$	20.0921	0.745175	0.372588	−	0.927997i	$-0.378471\pi$
$727$	20.0921	0.745175	0.372588	+	0.927997i	$0.378471\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	45.3474	1.67495	0.837473	−	0.546479i	$-0.184032\pi$
$733$	45.3474	1.67495	0.837473	+	0.546479i	$0.184032\pi$
$734$	0	0
$735$	4.30308	0.158721
$736$	0	0
$737$	−30.7453	−1.13252
$738$	0	0
$739$	6.15710	0.226493	0.113246	−	0.993567i	$-0.463875\pi$
$739$	6.15710	0.226493	0.113246	+	0.993567i	$0.463875\pi$
$740$	0	0
$741$	−2.81429	−0.103386
$742$	0	0
$743$	−31.5785	−1.15850	−0.579251	−	0.815149i	$-0.696655\pi$
$743$	−31.5785	−1.15850	−0.579251	+	0.815149i	$0.696655\pi$
$744$	0	0
$745$	55.3125	2.02649
$746$	0	0
$747$	2.01517	0.0737312
$748$	0	0
$749$	−10.3767	−0.379155
$750$	0	0
$751$	38.3207	1.39834	0.699171	−	0.714955i	$-0.253553\pi$
$751$	38.3207	1.39834	0.699171	+	0.714955i	$0.253553\pi$
$752$	0	0
$753$	20.8931	0.761388
$754$	0	0
$755$	76.6476	2.78949
$756$	0	0
$757$	−28.0924	−1.02104	−0.510518	−	0.859867i	$-0.670546\pi$
$757$	−28.0924	−1.02104	−0.510518	+	0.859867i	$0.670546\pi$
$758$	0	0
$759$	2.08969	0.0758509
$760$	0	0
$761$	16.4348	0.595762	0.297881	−	0.954603i	$-0.403720\pi$
$761$	16.4348	0.595762	0.297881	+	0.954603i	$0.403720\pi$
$762$	0	0
$763$	5.98671	0.216733
$764$	0	0
$765$	−34.2756	−1.23924
$766$	0	0
$767$	39.3721	1.42165
$768$	0	0
$769$	11.0679	0.399117	0.199559	−	0.979886i	$-0.436049\pi$
$769$	11.0679	0.399117	0.199559	+	0.979886i	$0.436049\pi$
$770$	0	0
$771$	8.54680	0.307806
$772$	0	0
$773$	−0.596231	−0.0214449	−0.0107225	−	0.999943i	$-0.503413\pi$
$773$	−0.596231	−0.0214449	−0.0107225	+	0.999943i	$0.503413\pi$
$774$	0	0
$775$	−45.6932	−1.64135
$776$	0	0
$777$	−3.75198	−0.134601
$778$	0	0
$779$	−6.60615	−0.236690
$780$	0	0
$781$	−12.9024	−0.461683
$782$	0	0
$783$	3.48879	0.124679
$784$	0	0
$785$	8.26291	0.294916
$786$	0	0
$787$	26.8567	0.957339	0.478670	−	0.877995i	$-0.341119\pi$
$787$	26.8567	0.957339	0.478670	+	0.877995i	$0.341119\pi$
$788$	0	0
$789$	1.24107	0.0441831
$790$	0	0
$791$	9.14771	0.325255
$792$	0	0
$793$	16.8857	0.599630
$794$	0	0
$795$	2.19980	0.0780187
$796$	0	0
$797$	7.04742	0.249632	0.124816	−	0.992180i	$-0.460166\pi$
$797$	7.04742	0.249632	0.124816	+	0.992180i	$0.460166\pi$
$798$	0	0
$799$	81.6710	2.88931
$800$	0	0
$801$	17.6490	0.623597
$802$	0	0
$803$	41.1127	1.45084
$804$	0	0
$805$	−3.50396	−0.123498
$806$	0	0
$807$	−20.6266	−0.726090
$808$	0	0
$809$	28.1531	0.989811	0.494905	−	0.868947i	$-0.335203\pi$
$809$	28.1531	0.989811	0.494905	+	0.868947i	$0.335203\pi$
$810$	0	0
$811$	6.75550	0.237218	0.118609	−	0.992941i	$-0.462157\pi$
$811$	6.75550	0.237218	0.118609	+	0.992941i	$0.462157\pi$
$812$	0	0
$813$	17.7758	0.623423
$814$	0	0
$815$	−97.0206	−3.39848
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.81429	−0.0983393
$820$	0	0
$821$	35.1260	1.22591	0.612953	−	0.790120i	$-0.289982\pi$
$821$	35.1260	1.22591	0.612953	+	0.790120i	$0.289982\pi$
$822$	0	0
$823$	11.4393	0.398748	0.199374	−	0.979923i	$-0.436109\pi$
$823$	11.4393	0.398748	0.199374	+	0.979923i	$0.436109\pi$
$824$	0	0
$825$	34.6869	1.20764
$826$	0	0
$827$	−18.7134	−0.650730	−0.325365	−	0.945588i	$-0.605487\pi$
$827$	−18.7134	−0.650730	−0.325365	+	0.945588i	$0.605487\pi$
$828$	0	0
$829$	37.2244	1.29286	0.646429	−	0.762974i	$-0.276262\pi$
$829$	37.2244	1.29286	0.646429	+	0.762974i	$0.276262\pi$
$830$	0	0
$831$	−22.8613	−0.793050
$832$	0	0
$833$	−7.96536	−0.275984
$834$	0	0
$835$	45.2531	1.56605
$836$	0	0
$837$	−3.38056	−0.116849
$838$	0	0
$839$	32.6127	1.12591	0.562957	−	0.826486i	$-0.309664\pi$
$839$	32.6127	1.12591	0.562957	+	0.826486i	$0.309664\pi$
$840$	0	0
$841$	−16.8284	−0.580289
$842$	0	0
$843$	−16.0949	−0.554339
$844$	0	0
$845$	−21.8586	−0.751960
$846$	0	0
$847$	4.41427	0.151676
$848$	0	0
$849$	−16.4042	−0.562990
$850$	0	0
$851$	3.05520	0.104731
$852$	0	0
$853$	6.56346	0.224729	0.112364	−	0.993667i	$-0.464158\pi$
$853$	6.56346	0.224729	0.112364	+	0.993667i	$0.464158\pi$
$854$	0	0
$855$	−4.30308	−0.147162
$856$	0	0
$857$	19.6939	0.672729	0.336365	−	0.941732i	$-0.390803\pi$
$857$	19.6939	0.672729	0.336365	+	0.941732i	$0.390803\pi$
$858$	0	0
$859$	1.41988	0.0484458	0.0242229	−	0.999707i	$-0.492289\pi$
$859$	1.41988	0.0484458	0.0242229	+	0.999707i	$0.492289\pi$
$860$	0	0
$861$	−6.60615	−0.225137
$862$	0	0
$863$	31.9991	1.08926	0.544631	−	0.838676i	$-0.316670\pi$
$863$	31.9991	1.08926	0.544631	+	0.838676i	$0.316670\pi$
$864$	0	0
$865$	−105.039	−3.57142
$866$	0	0
$867$	46.4470	1.57742
$868$	0	0
$869$	38.0698	1.29143
$870$	0	0
$871$	−33.7167	−1.14245
$872$	0	0
$873$	−5.66847	−0.191848
$874$	0	0
$875$	−36.6470	−1.23889
$876$	0	0
$877$	−26.4933	−0.894614	−0.447307	−	0.894381i	$-0.647617\pi$
$877$	−26.4933	−0.894614	−0.447307	+	0.894381i	$0.647617\pi$
$878$	0	0
$879$	−14.3918	−0.485425
$880$	0	0
$881$	−17.5630	−0.591713	−0.295856	−	0.955232i	$-0.595605\pi$
$881$	−17.5630	−0.591713	−0.295856	+	0.955232i	$0.595605\pi$
$882$	0	0
$883$	5.05551	0.170132	0.0850658	−	0.996375i	$-0.472890\pi$
$883$	5.05551	0.170132	0.0850658	+	0.996375i	$0.472890\pi$
$884$	0	0
$885$	60.2004	2.02361
$886$	0	0
$887$	15.7390	0.528464	0.264232	−	0.964459i	$-0.414882\pi$
$887$	15.7390	0.528464	0.264232	+	0.964459i	$0.414882\pi$
$888$	0	0
$889$	0.599976	0.0201225
$890$	0	0
$891$	2.56627	0.0859732
$892$	0	0
$893$	10.2533	0.343113
$894$	0	0
$895$	−30.4960	−1.01937
$896$	0	0
$897$	2.29165	0.0765160
$898$	0	0
$899$	−11.7940	−0.393353
$900$	0	0
$901$	−4.07201	−0.135658
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.0573	0.367558
$906$	0	0
$907$	−19.0379	−0.632142	−0.316071	−	0.948736i	$-0.602364\pi$
$907$	−19.0379	−0.632142	−0.316071	+	0.948736i	$0.602364\pi$
$908$	0	0
$909$	−4.90923	−0.162829
$910$	0	0
$911$	−52.9775	−1.75522	−0.877611	−	0.479373i	$-0.840864\pi$
$911$	−52.9775	−1.75522	−0.877611	+	0.479373i	$0.840864\pi$
$912$	0	0
$913$	5.17147	0.171151
$914$	0	0
$915$	25.8185	0.853532
$916$	0	0
$917$	7.14771	0.236038
$918$	0	0
$919$	37.5222	1.23774	0.618872	−	0.785492i	$-0.287590\pi$
$919$	37.5222	1.23774	0.618872	+	0.785492i	$0.287590\pi$
$920$	0	0
$921$	−17.9901	−0.592793
$922$	0	0
$923$	−14.1493	−0.465731
$924$	0	0
$925$	50.7135	1.66745
$926$	0	0
$927$	−1.47025	−0.0482893
$928$	0	0
$929$	−57.3372	−1.88117	−0.940586	−	0.339555i	$-0.889724\pi$
$929$	−57.3372	−1.88117	−0.940586	+	0.339555i	$0.889724\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	12.0916	0.395860
$934$	0	0
$935$	−87.9603	−2.87661
$936$	0	0
$937$	−11.3469	−0.370685	−0.185343	−	0.982674i	$-0.559340\pi$
$937$	−11.3469	−0.370685	−0.185343	+	0.982674i	$0.559340\pi$
$938$	0	0
$939$	−25.2024	−0.822449
$940$	0	0
$941$	−34.7123	−1.13159	−0.565794	−	0.824547i	$-0.691430\pi$
$941$	−34.7123	−1.13159	−0.565794	+	0.824547i	$0.691430\pi$
$942$	0	0
$943$	5.37933	0.175175
$944$	0	0
$945$	−4.30308	−0.139979
$946$	0	0
$947$	16.2357	0.527589	0.263794	−	0.964579i	$-0.415026\pi$
$947$	16.2357	0.527589	0.263794	+	0.964579i	$0.415026\pi$
$948$	0	0
$949$	45.0861	1.46356
$950$	0	0
$951$	−11.8806	−0.385256
$952$	0	0
$953$	−19.4158	−0.628938	−0.314469	−	0.949268i	$-0.601826\pi$
$953$	−19.4158	−0.628938	−0.314469	+	0.949268i	$0.601826\pi$
$954$	0	0
$955$	−117.158	−3.79114
$956$	0	0
$957$	8.95316	0.289415
$958$	0	0
$959$	−11.6286	−0.375506
$960$	0	0
$961$	−19.5718	−0.631349
$962$	0	0
$963$	10.3767	0.334383
$964$	0	0
$965$	−62.5533	−2.01366
$966$	0	0
$967$	1.38578	0.0445638	0.0222819	−	0.999752i	$-0.492907\pi$
$967$	1.38578	0.0445638	0.0222819	+	0.999752i	$0.492907\pi$
$968$	0	0
$969$	7.96536	0.255884
$970$	0	0
$971$	1.33450	0.0428261	0.0214131	−	0.999771i	$-0.493183\pi$
$971$	1.33450	0.0428261	0.0214131	+	0.999771i	$0.493183\pi$
$972$	0	0
$973$	−3.01251	−0.0965766
$974$	0	0
$975$	38.0393	1.21823
$976$	0	0
$977$	−13.0460	−0.417377	−0.208689	−	0.977982i	$-0.566920\pi$
$977$	−13.0460	−0.417377	−0.208689	+	0.977982i	$0.566920\pi$
$978$	0	0
$979$	45.2921	1.44754
$980$	0	0
$981$	−5.98671	−0.191141
$982$	0	0
$983$	−13.0877	−0.417433	−0.208717	−	0.977976i	$-0.566929\pi$
$983$	−13.0877	−0.417433	−0.208717	+	0.977976i	$0.566929\pi$
$984$	0	0
$985$	−35.8351	−1.14180
$986$	0	0
$987$	10.2533	0.326365
$988$	0	0
$989$	0	0
$990$	0	0
$991$	4.81643	0.152999	0.0764994	−	0.997070i	$-0.475626\pi$
$991$	4.81643	0.152999	0.0764994	+	0.997070i	$0.475626\pi$
$992$	0	0
$993$	−6.28454	−0.199434
$994$	0	0
$995$	−60.3309	−1.91262
$996$	0	0
$997$	−11.1127	−0.351943	−0.175971	−	0.984395i	$-0.556307\pi$
$997$	−11.1127	−0.351943	−0.175971	+	0.984395i	$0.556307\pi$
$998$	0	0
$999$	3.75198	0.118707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.ce.1.5		5
4.3	odd	2		3192.2.a.ba.1.5	✓	5
12.11	even	2		9576.2.a.co.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.ba.1.5	✓	5	4.3	odd	2
6384.2.a.ce.1.5		5	1.1	even	1	trivial
9576.2.a.co.1.1		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.30308 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.30308 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.56627 q^{11} +2.81429 q^{13} +4.30308 q^{15} -7.96536 q^{17} -1.00000 q^{19} -1.00000 q^{21} +0.814291 q^{23} +13.5165 q^{25} +1.00000 q^{27} +3.48879 q^{29} -3.38056 q^{31} +2.56627 q^{33} -4.30308 q^{35} +3.75198 q^{37} +2.81429 q^{39} +6.60615 q^{41} +4.30308 q^{45} -10.2533 q^{47} +1.00000 q^{49} -7.96536 q^{51} +0.511215 q^{53} +11.0428 q^{55} -1.00000 q^{57} +13.9901 q^{59} +6.00000 q^{61} -1.00000 q^{63} +12.1101 q^{65} -11.9805 q^{67} +0.814291 q^{69} -5.02768 q^{71} +16.0204 q^{73} +13.5165 q^{75} -2.56627 q^{77} +14.8347 q^{79} +1.00000 q^{81} +2.01517 q^{83} -34.2756 q^{85} +3.48879 q^{87} +17.6490 q^{89} -2.81429 q^{91} -3.38056 q^{93} -4.30308 q^{95} -5.66847 q^{97} +2.56627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} - 5 q^{21} - 2 q^{23} + 19 q^{25} + 5 q^{27} + 4 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{45} + 2 q^{47} + 5 q^{49} - 2 q^{51} + 16 q^{53} + 16 q^{55} - 5 q^{57} + 12 q^{59} + 30 q^{61} - 5 q^{63} + 4 q^{65} - 18 q^{67} - 2 q^{69} + 10 q^{71} + 14 q^{73} + 19 q^{75} + 2 q^{77} + 2 q^{79} + 5 q^{81} + 6 q^{83} + 12 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{93} - 2 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 2 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 5 * q^19 - 5 * q^21 - 2 * q^23 + 19 * q^25 + 5 * q^27 + 4 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 + 2 * q^45 + 2 * q^47 + 5 * q^49 - 2 * q^51 + 16 * q^53 + 16 * q^55 - 5 * q^57 + 12 * q^59 + 30 * q^61 - 5 * q^63 + 4 * q^65 - 18 * q^67 - 2 * q^69 + 10 * q^71 + 14 * q^73 + 19 * q^75 + 2 * q^77 + 2 * q^79 + 5 * q^81 + 6 * q^83 + 12 * q^85 + 4 * q^87 + 10 * q^89 - 8 * q^91 + 4 * q^93 - 2 * q^95 + 8 * q^97 - 2 * q^99

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