LMFDB - Embedded newform 6384.2.a.cc.1.3

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:	5.5.1240016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 2x^{2} + 16x + 6 $ x^5 - x^4 - 8x^3 + 2x^2 + 16*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.36162$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.06804 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.06804 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.72323 q^{11} +1.56878 q^{13} -1.06804 q^{15} +4.83899 q^{17} -1.00000 q^{19} +1.00000 q^{21} +2.72323 q^{23} -3.85930 q^{25} -1.00000 q^{27} -1.65520 q^{29} -4.72323 q^{31} +2.72323 q^{33} -1.06804 q^{35} +10.0431 q^{37} -1.56878 q^{39} -6.33825 q^{41} +8.76632 q^{43} +1.06804 q^{45} +9.79127 q^{47} +1.00000 q^{49} -4.83899 q^{51} -10.5607 q^{53} -2.90851 q^{55} +1.00000 q^{57} -5.44646 q^{59} -1.77095 q^{61} -1.00000 q^{63} +1.67551 q^{65} -8.92541 q^{67} -2.72323 q^{69} +9.26707 q^{71} +8.76947 q^{73} +3.85930 q^{75} +2.72323 q^{77} -3.01838 q^{79} +1.00000 q^{81} +8.97820 q^{83} +5.16821 q^{85} +1.65520 q^{87} -10.6104 q^{89} -1.56878 q^{91} +4.72323 q^{93} -1.06804 q^{95} -5.83145 q^{97} -2.72323 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 11 q^{25} - 5 q^{27} - 8 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} - 20 q^{43} - 2 q^{45} + 26 q^{47} + 5 q^{49} + 2 q^{51} + 4 q^{53} + 4 q^{55} + 5 q^{57} + 4 q^{59} + 10 q^{61} - 5 q^{63} - 4 q^{65} - 10 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} - 11 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 34 q^{83} - 36 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{93} + 2 q^{95} - 16 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 + 11 * q^25 - 5 * q^27 - 8 * q^31 - 2 * q^33 + 2 * q^35 + 2 * q^37 - 8 * q^39 + 2 * q^41 - 20 * q^43 - 2 * q^45 + 26 * q^47 + 5 * q^49 + 2 * q^51 + 4 * q^53 + 4 * q^55 + 5 * q^57 + 4 * q^59 + 10 * q^61 - 5 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 - 10 * q^71 + 10 * q^73 - 11 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 34 * q^83 - 36 * q^85 + 10 * q^89 - 8 * q^91 + 8 * q^93 + 2 * q^95 - 16 * 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
$5$	1.06804	0.477640	0.238820	−	0.971064i	$-0.423239\pi$
$5$	1.06804	0.477640	0.238820	+	0.971064i	$0.423239\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.72323	−0.821085	−0.410542	−	0.911841i	$-0.634661\pi$
$11$	−2.72323	−0.821085	−0.410542	+	0.911841i	$0.634661\pi$
$12$	0	0
$13$	1.56878	0.435100	0.217550	−	0.976049i	$-0.430193\pi$
$13$	1.56878	0.435100	0.217550	+	0.976049i	$0.430193\pi$
$14$	0	0
$15$	−1.06804	−0.275765
$16$	0	0
$17$	4.83899	1.17363	0.586813	−	0.809722i	$-0.300382\pi$
$17$	4.83899	1.17363	0.586813	+	0.809722i	$0.300382\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	2.72323	0.567833	0.283916	−	0.958849i	$-0.408366\pi$
$23$	2.72323	0.567833	0.283916	+	0.958849i	$0.408366\pi$
$24$	0	0
$25$	−3.85930	−0.771860
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.65520	−0.307362	−0.153681	−	0.988121i	$-0.549113\pi$
$29$	−1.65520	−0.307362	−0.153681	+	0.988121i	$0.549113\pi$
$30$	0	0
$31$	−4.72323	−0.848317	−0.424159	−	0.905588i	$-0.639430\pi$
$31$	−4.72323	−0.848317	−0.424159	+	0.905588i	$0.639430\pi$
$32$	0	0
$33$	2.72323	0.474054
$34$	0	0
$35$	−1.06804	−0.180531
$36$	0	0
$37$	10.0431	1.65107	0.825537	−	0.564348i	$-0.190872\pi$
$37$	10.0431	1.65107	0.825537	+	0.564348i	$0.190872\pi$
$38$	0	0
$39$	−1.56878	−0.251205
$40$	0	0
$41$	−6.33825	−0.989868	−0.494934	−	0.868931i	$-0.664808\pi$
$41$	−6.33825	−0.989868	−0.494934	+	0.868931i	$0.664808\pi$
$42$	0	0
$43$	8.76632	1.33685	0.668426	−	0.743779i	$-0.266968\pi$
$43$	8.76632	1.33685	0.668426	+	0.743779i	$0.266968\pi$
$44$	0	0
$45$	1.06804	0.159213
$46$	0	0
$47$	9.79127	1.42820	0.714101	−	0.700042i	$-0.246836\pi$
$47$	9.79127	1.42820	0.714101	+	0.700042i	$0.246836\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.83899	−0.677594
$52$	0	0
$53$	−10.5607	−1.45063	−0.725314	−	0.688418i	$-0.758306\pi$
$53$	−10.5607	−1.45063	−0.725314	+	0.688418i	$0.758306\pi$
$54$	0	0
$55$	−2.90851	−0.392183
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−5.44646	−0.709069	−0.354534	−	0.935043i	$-0.615361\pi$
$59$	−5.44646	−0.709069	−0.354534	+	0.935043i	$0.615361\pi$
$60$	0	0
$61$	−1.77095	−0.226747	−0.113374	−	0.993552i	$-0.536166\pi$
$61$	−1.77095	−0.226747	−0.113374	+	0.993552i	$0.536166\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.67551	0.207821
$66$	0	0
$67$	−8.92541	−1.09041	−0.545206	−	0.838302i	$-0.683549\pi$
$67$	−8.92541	−1.09041	−0.545206	+	0.838302i	$0.683549\pi$
$68$	0	0
$69$	−2.72323	−0.327838
$70$	0	0
$71$	9.26707	1.09980	0.549899	−	0.835231i	$-0.314666\pi$
$71$	9.26707	1.09980	0.549899	+	0.835231i	$0.314666\pi$
$72$	0	0
$73$	8.76947	1.02639	0.513194	−	0.858272i	$-0.328462\pi$
$73$	8.76947	1.02639	0.513194	+	0.858272i	$0.328462\pi$
$74$	0	0
$75$	3.85930	0.445634
$76$	0	0
$77$	2.72323	0.310341
$78$	0	0
$79$	−3.01838	−0.339595	−0.169797	−	0.985479i	$-0.554311\pi$
$79$	−3.01838	−0.339595	−0.169797	+	0.985479i	$0.554311\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.97820	0.985486	0.492743	−	0.870175i	$-0.335994\pi$
$83$	8.97820	0.985486	0.492743	+	0.870175i	$0.335994\pi$
$84$	0	0
$85$	5.16821	0.560571
$86$	0	0
$87$	1.65520	0.177456
$88$	0	0
$89$	−10.6104	−1.12470	−0.562349	−	0.826900i	$-0.690102\pi$
$89$	−10.6104	−1.12470	−0.562349	+	0.826900i	$0.690102\pi$
$90$	0	0
$91$	−1.56878	−0.164452
$92$	0	0
$93$	4.72323	0.489776
$94$	0	0
$95$	−1.06804	−0.109578
$96$	0	0
$97$	−5.83145	−0.592094	−0.296047	−	0.955173i	$-0.595668\pi$
$97$	−5.83145	−0.592094	−0.296047	+	0.955173i	$0.595668\pi$
$98$	0	0
$99$	−2.72323	−0.273695
$100$	0	0
$101$	−10.2854	−1.02344	−0.511720	−	0.859152i	$-0.670992\pi$
$101$	−10.2854	−1.02344	−0.511720	+	0.859152i	$0.670992\pi$
$102$	0	0
$103$	14.9516	1.47322	0.736612	−	0.676315i	$-0.236424\pi$
$103$	14.9516	1.47322	0.736612	+	0.676315i	$0.236424\pi$
$104$	0	0
$105$	1.06804	0.104230
$106$	0	0
$107$	8.80650	0.851357	0.425678	−	0.904875i	$-0.360035\pi$
$107$	8.80650	0.851357	0.425678	+	0.904875i	$0.360035\pi$
$108$	0	0
$109$	9.58253	0.917840	0.458920	−	0.888478i	$-0.348236\pi$
$109$	9.58253	0.917840	0.458920	+	0.888478i	$0.348236\pi$
$110$	0	0
$111$	−10.0431	−0.953248
$112$	0	0
$113$	16.4678	1.54916	0.774578	−	0.632478i	$-0.217962\pi$
$113$	16.4678	1.54916	0.774578	+	0.632478i	$0.217962\pi$
$114$	0	0
$115$	2.90851	0.271220
$116$	0	0
$117$	1.56878	0.145033
$118$	0	0
$119$	−4.83899	−0.443589
$120$	0	0
$121$	−3.58401	−0.325820
$122$	0	0
$123$	6.33825	0.571500
$124$	0	0
$125$	−9.46204	−0.846311
$126$	0	0
$127$	−6.88864	−0.611268	−0.305634	−	0.952149i	$-0.598868\pi$
$127$	−6.88864	−0.611268	−0.305634	+	0.952149i	$0.598868\pi$
$128$	0	0
$129$	−8.76632	−0.771832
$130$	0	0
$131$	−1.56222	−0.136492	−0.0682458	−	0.997669i	$-0.521740\pi$
$131$	−1.56222	−0.136492	−0.0682458	+	0.997669i	$0.521740\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−1.06804	−0.0919218
$136$	0	0
$137$	22.3981	1.91360	0.956798	−	0.290754i	$-0.0939062\pi$
$137$	22.3981	1.91360	0.956798	+	0.290754i	$0.0939062\pi$
$138$	0	0
$139$	10.7695	0.913455	0.456727	−	0.889607i	$-0.349022\pi$
$139$	10.7695	0.913455	0.456727	+	0.889607i	$0.349022\pi$
$140$	0	0
$141$	−9.79127	−0.824573
$142$	0	0
$143$	−4.27214	−0.357254
$144$	0	0
$145$	−1.76781	−0.146808
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−1.58253	−0.129646	−0.0648230	−	0.997897i	$-0.520648\pi$
$149$	−1.58253	−0.129646	−0.0648230	+	0.997897i	$0.520648\pi$
$150$	0	0
$151$	−0.570440	−0.0464217	−0.0232109	−	0.999731i	$-0.507389\pi$
$151$	−0.570440	−0.0464217	−0.0232109	+	0.999731i	$0.507389\pi$
$152$	0	0
$153$	4.83899	0.391209
$154$	0	0
$155$	−5.04458	−0.405190
$156$	0	0
$157$	−5.18065	−0.413461	−0.206730	−	0.978398i	$-0.566282\pi$
$157$	−5.18065	−0.413461	−0.206730	+	0.978398i	$0.566282\pi$
$158$	0	0
$159$	10.5607	0.837521
$160$	0	0
$161$	−2.72323	−0.214621
$162$	0	0
$163$	−9.35663	−0.732868	−0.366434	−	0.930444i	$-0.619421\pi$
$163$	−9.35663	−0.732868	−0.366434	+	0.930444i	$0.619421\pi$
$164$	0	0
$165$	2.90851	0.226427
$166$	0	0
$167$	13.2736	1.02714	0.513572	−	0.858047i	$-0.328322\pi$
$167$	13.2736	1.02714	0.513572	+	0.858047i	$0.328322\pi$
$168$	0	0
$169$	−10.5389	−0.810688
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−0.891785	−0.0678012	−0.0339006	−	0.999425i	$-0.510793\pi$
$173$	−0.891785	−0.0678012	−0.0339006	+	0.999425i	$0.510793\pi$
$174$	0	0
$175$	3.85930	0.291736
$176$	0	0
$177$	5.44646	0.409381
$178$	0	0
$179$	−9.72763	−0.727077	−0.363539	−	0.931579i	$-0.618431\pi$
$179$	−9.72763	−0.727077	−0.363539	+	0.931579i	$0.618431\pi$
$180$	0	0
$181$	15.0247	1.11678	0.558389	−	0.829579i	$-0.311420\pi$
$181$	15.0247	1.11678	0.558389	+	0.829579i	$0.311420\pi$
$182$	0	0
$183$	1.77095	0.130913
$184$	0	0
$185$	10.7264	0.788619
$186$	0	0
$187$	−13.1777	−0.963647
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	2.41136	0.174480	0.0872398	−	0.996187i	$-0.472195\pi$
$191$	2.41136	0.174480	0.0872398	+	0.996187i	$0.472195\pi$
$192$	0	0
$193$	4.59663	0.330873	0.165436	−	0.986220i	$-0.447097\pi$
$193$	4.59663	0.330873	0.165436	+	0.986220i	$0.447097\pi$
$194$	0	0
$195$	−1.67551	−0.119986
$196$	0	0
$197$	21.5356	1.53435	0.767174	−	0.641438i	$-0.221662\pi$
$197$	21.5356	1.53435	0.767174	+	0.641438i	$0.221662\pi$
$198$	0	0
$199$	7.31039	0.518220	0.259110	−	0.965848i	$-0.416571\pi$
$199$	7.31039	0.518220	0.259110	+	0.965848i	$0.416571\pi$
$200$	0	0
$201$	8.92541	0.629550
$202$	0	0
$203$	1.65520	0.116172
$204$	0	0
$205$	−6.76947	−0.472800
$206$	0	0
$207$	2.72323	0.189278
$208$	0	0
$209$	2.72323	0.188370
$210$	0	0
$211$	−8.10359	−0.557874	−0.278937	−	0.960309i	$-0.589982\pi$
$211$	−8.10359	−0.557874	−0.278937	+	0.960309i	$0.589982\pi$
$212$	0	0
$213$	−9.26707	−0.634969
$214$	0	0
$215$	9.36274	0.638534
$216$	0	0
$217$	4.72323	0.320634
$218$	0	0
$219$	−8.76947	−0.592586
$220$	0	0
$221$	7.59129	0.510645
$222$	0	0
$223$	−0.265137	−0.0177549	−0.00887743	−	0.999961i	$-0.502826\pi$
$223$	−0.265137	−0.0177549	−0.00887743	+	0.999961i	$0.502826\pi$
$224$	0	0
$225$	−3.85930	−0.257287
$226$	0	0
$227$	−4.23151	−0.280855	−0.140428	−	0.990091i	$-0.544848\pi$
$227$	−4.23151	−0.280855	−0.140428	+	0.990091i	$0.544848\pi$
$228$	0	0
$229$	12.5404	0.828694	0.414347	−	0.910119i	$-0.364010\pi$
$229$	12.5404	0.828694	0.414347	+	0.910119i	$0.364010\pi$
$230$	0	0
$231$	−2.72323	−0.179175
$232$	0	0
$233$	21.6750	1.41998	0.709989	−	0.704213i	$-0.248700\pi$
$233$	21.6750	1.41998	0.709989	+	0.704213i	$0.248700\pi$
$234$	0	0
$235$	10.4574	0.682167
$236$	0	0
$237$	3.01838	0.196065
$238$	0	0
$239$	21.2574	1.37502	0.687512	−	0.726173i	$-0.258703\pi$
$239$	21.2574	1.37502	0.687512	+	0.726173i	$0.258703\pi$
$240$	0	0
$241$	27.9544	1.80070	0.900351	−	0.435165i	$-0.143310\pi$
$241$	27.9544	1.80070	0.900351	+	0.435165i	$0.143310\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.06804	0.0682343
$246$	0	0
$247$	−1.56878	−0.0998189
$248$	0	0
$249$	−8.97820	−0.568971
$250$	0	0
$251$	15.2964	0.965500	0.482750	−	0.875758i	$-0.339638\pi$
$251$	15.2964	0.965500	0.482750	+	0.875758i	$0.339638\pi$
$252$	0	0
$253$	−7.41599	−0.466239
$254$	0	0
$255$	−5.16821	−0.323646
$256$	0	0
$257$	3.52568	0.219926	0.109963	−	0.993936i	$-0.464927\pi$
$257$	3.52568	0.219926	0.109963	+	0.993936i	$0.464927\pi$
$258$	0	0
$259$	−10.0431	−0.624047
$260$	0	0
$261$	−1.65520	−0.102454
$262$	0	0
$263$	6.95475	0.428848	0.214424	−	0.976741i	$-0.431213\pi$
$263$	6.95475	0.428848	0.214424	+	0.976741i	$0.431213\pi$
$264$	0	0
$265$	−11.2792	−0.692878
$266$	0	0
$267$	10.6104	0.649345
$268$	0	0
$269$	20.2884	1.23700	0.618502	−	0.785783i	$-0.287740\pi$
$269$	20.2884	1.23700	0.618502	+	0.785783i	$0.287740\pi$
$270$	0	0
$271$	−17.8939	−1.08698	−0.543489	−	0.839416i	$-0.682897\pi$
$271$	−17.8939	−1.08698	−0.543489	+	0.839416i	$0.682897\pi$
$272$	0	0
$273$	1.56878	0.0949467
$274$	0	0
$275$	10.5098	0.633763
$276$	0	0
$277$	−20.9852	−1.26088	−0.630440	−	0.776238i	$-0.717125\pi$
$277$	−20.9852	−1.26088	−0.630440	+	0.776238i	$0.717125\pi$
$278$	0	0
$279$	−4.72323	−0.282772
$280$	0	0
$281$	0.165641	0.00988130	0.00494065	−	0.999988i	$-0.498427\pi$
$281$	0.165641	0.00988130	0.00494065	+	0.999988i	$0.498427\pi$
$282$	0	0
$283$	28.8430	1.71454	0.857270	−	0.514867i	$-0.172159\pi$
$283$	28.8430	1.71454	0.857270	+	0.514867i	$0.172159\pi$
$284$	0	0
$285$	1.06804	0.0632649
$286$	0	0
$287$	6.33825	0.374135
$288$	0	0
$289$	6.41581	0.377400
$290$	0	0
$291$	5.83145	0.341845
$292$	0	0
$293$	−33.8152	−1.97550	−0.987752	−	0.156032i	$-0.950130\pi$
$293$	−33.8152	−1.97550	−0.987752	+	0.156032i	$0.950130\pi$
$294$	0	0
$295$	−5.81701	−0.338680
$296$	0	0
$297$	2.72323	0.158018
$298$	0	0
$299$	4.27214	0.247064
$300$	0	0
$301$	−8.76632	−0.505283
$302$	0	0
$303$	10.2854	0.590884
$304$	0	0
$305$	−1.89144	−0.108304
$306$	0	0
$307$	−6.44812	−0.368014	−0.184007	−	0.982925i	$-0.558907\pi$
$307$	−6.44812	−0.368014	−0.184007	+	0.982925i	$0.558907\pi$
$308$	0	0
$309$	−14.9516	−0.850567
$310$	0	0
$311$	−6.69681	−0.379741	−0.189871	−	0.981809i	$-0.560807\pi$
$311$	−6.69681	−0.379741	−0.189871	+	0.981809i	$0.560807\pi$
$312$	0	0
$313$	0.324492	0.0183413	0.00917067	−	0.999958i	$-0.497081\pi$
$313$	0.324492	0.0183413	0.00917067	+	0.999958i	$0.497081\pi$
$314$	0	0
$315$	−1.06804	−0.0601770
$316$	0	0
$317$	2.06637	0.116059	0.0580295	−	0.998315i	$-0.481518\pi$
$317$	2.06637	0.116059	0.0580295	+	0.998315i	$0.481518\pi$
$318$	0	0
$319$	4.50748	0.252370
$320$	0	0
$321$	−8.80650	−0.491531
$322$	0	0
$323$	−4.83899	−0.269248
$324$	0	0
$325$	−6.05438	−0.335837
$326$	0	0
$327$	−9.58253	−0.529915
$328$	0	0
$329$	−9.79127	−0.539810
$330$	0	0
$331$	21.5588	1.18498	0.592490	−	0.805578i	$-0.298145\pi$
$331$	21.5588	1.18498	0.592490	+	0.805578i	$0.298145\pi$
$332$	0	0
$333$	10.0431	0.550358
$334$	0	0
$335$	−9.53265	−0.520824
$336$	0	0
$337$	−35.5195	−1.93487	−0.967436	−	0.253115i	$-0.918545\pi$
$337$	−35.5195	−1.93487	−0.967436	+	0.253115i	$0.918545\pi$
$338$	0	0
$339$	−16.4678	−0.894406
$340$	0	0
$341$	12.8624	0.696541
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.90851	−0.156589
$346$	0	0
$347$	−8.58087	−0.460645	−0.230323	−	0.973114i	$-0.573978\pi$
$347$	−8.58087	−0.460645	−0.230323	+	0.973114i	$0.573978\pi$
$348$	0	0
$349$	−18.4044	−0.985162	−0.492581	−	0.870266i	$-0.663947\pi$
$349$	−18.4044	−0.985162	−0.492581	+	0.870266i	$0.663947\pi$
$350$	0	0
$351$	−1.56878	−0.0837351
$352$	0	0
$353$	−30.3872	−1.61735	−0.808674	−	0.588257i	$-0.799814\pi$
$353$	−30.3872	−1.61735	−0.808674	+	0.588257i	$0.799814\pi$
$354$	0	0
$355$	9.89755	0.525307
$356$	0	0
$357$	4.83899	0.256106
$358$	0	0
$359$	−7.98374	−0.421366	−0.210683	−	0.977554i	$-0.567569\pi$
$359$	−7.98374	−0.421366	−0.210683	+	0.977554i	$0.567569\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.58401	0.188112
$364$	0	0
$365$	9.36610	0.490244
$366$	0	0
$367$	18.1361	0.946695	0.473348	−	0.880876i	$-0.343045\pi$
$367$	18.1361	0.946695	0.473348	+	0.880876i	$0.343045\pi$
$368$	0	0
$369$	−6.33825	−0.329956
$370$	0	0
$371$	10.5607	0.548286
$372$	0	0
$373$	19.2018	0.994233	0.497117	−	0.867684i	$-0.334392\pi$
$373$	19.2018	0.994233	0.497117	+	0.867684i	$0.334392\pi$
$374$	0	0
$375$	9.46204	0.488618
$376$	0	0
$377$	−2.59663	−0.133733
$378$	0	0
$379$	7.46336	0.383367	0.191684	−	0.981457i	$-0.438605\pi$
$379$	7.46336	0.383367	0.191684	+	0.981457i	$0.438605\pi$
$380$	0	0
$381$	6.88864	0.352916
$382$	0	0
$383$	13.2736	0.678250	0.339125	−	0.940741i	$-0.389869\pi$
$383$	13.2736	0.678250	0.339125	+	0.940741i	$0.389869\pi$
$384$	0	0
$385$	2.90851	0.148231
$386$	0	0
$387$	8.76632	0.445617
$388$	0	0
$389$	−5.64750	−0.286340	−0.143170	−	0.989698i	$-0.545730\pi$
$389$	−5.64750	−0.286340	−0.143170	+	0.989698i	$0.545730\pi$
$390$	0	0
$391$	13.1777	0.666424
$392$	0	0
$393$	1.56222	0.0788035
$394$	0	0
$395$	−3.22374	−0.162204
$396$	0	0
$397$	10.8562	0.544855	0.272427	−	0.962176i	$-0.412174\pi$
$397$	10.8562	0.544855	0.272427	+	0.962176i	$0.412174\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	7.43630	0.371351	0.185676	−	0.982611i	$-0.440553\pi$
$401$	7.43630	0.371351	0.185676	+	0.982611i	$0.440553\pi$
$402$	0	0
$403$	−7.40969	−0.369103
$404$	0	0
$405$	1.06804	0.0530711
$406$	0	0
$407$	−27.3497	−1.35567
$408$	0	0
$409$	13.6947	0.677159	0.338580	−	0.940938i	$-0.390054\pi$
$409$	13.6947	0.677159	0.338580	+	0.940938i	$0.390054\pi$
$410$	0	0
$411$	−22.3981	−1.10481
$412$	0	0
$413$	5.44646	0.268003
$414$	0	0
$415$	9.58904	0.470707
$416$	0	0
$417$	−10.7695	−0.527383
$418$	0	0
$419$	12.9757	0.633906	0.316953	−	0.948441i	$-0.397340\pi$
$419$	12.9757	0.633906	0.316953	+	0.948441i	$0.397340\pi$
$420$	0	0
$421$	14.0300	0.683779	0.341890	−	0.939740i	$-0.388933\pi$
$421$	14.0300	0.683779	0.341890	+	0.939740i	$0.388933\pi$
$422$	0	0
$423$	9.79127	0.476068
$424$	0	0
$425$	−18.6751	−0.905876
$426$	0	0
$427$	1.77095	0.0857024
$428$	0	0
$429$	4.27214	0.206261
$430$	0	0
$431$	19.3063	0.929952	0.464976	−	0.885323i	$-0.346063\pi$
$431$	19.3063	0.929952	0.464976	+	0.885323i	$0.346063\pi$
$432$	0	0
$433$	25.0552	1.20408	0.602038	−	0.798468i	$-0.294356\pi$
$433$	25.0552	1.20408	0.602038	+	0.798468i	$0.294356\pi$
$434$	0	0
$435$	1.76781	0.0847599
$436$	0	0
$437$	−2.72323	−0.130270
$438$	0	0
$439$	9.12758	0.435636	0.217818	−	0.975989i	$-0.430106\pi$
$439$	9.12758	0.435636	0.217818	+	0.975989i	$0.430106\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	38.5779	1.83289	0.916446	−	0.400159i	$-0.131045\pi$
$443$	38.5779	1.83289	0.916446	+	0.400159i	$0.131045\pi$
$444$	0	0
$445$	−11.3323	−0.537201
$446$	0	0
$447$	1.58253	0.0748512
$448$	0	0
$449$	22.9719	1.08411	0.542056	−	0.840343i	$-0.317646\pi$
$449$	22.9719	1.08411	0.542056	+	0.840343i	$0.317646\pi$
$450$	0	0
$451$	17.2605	0.812766
$452$	0	0
$453$	0.570440	0.0268016
$454$	0	0
$455$	−1.67551	−0.0785490
$456$	0	0
$457$	11.1344	0.520846	0.260423	−	0.965495i	$-0.416138\pi$
$457$	11.1344	0.520846	0.260423	+	0.965495i	$0.416138\pi$
$458$	0	0
$459$	−4.83899	−0.225865
$460$	0	0
$461$	25.1567	1.17166	0.585832	−	0.810433i	$-0.300768\pi$
$461$	25.1567	1.17166	0.585832	+	0.810433i	$0.300768\pi$
$462$	0	0
$463$	3.81405	0.177254	0.0886269	−	0.996065i	$-0.471752\pi$
$463$	3.81405	0.177254	0.0886269	+	0.996065i	$0.471752\pi$
$464$	0	0
$465$	5.04458	0.233937
$466$	0	0
$467$	23.2402	1.07543	0.537714	−	0.843127i	$-0.319288\pi$
$467$	23.2402	1.07543	0.537714	+	0.843127i	$0.319288\pi$
$468$	0	0
$469$	8.92541	0.412137
$470$	0	0
$471$	5.18065	0.238712
$472$	0	0
$473$	−23.8727	−1.09767
$474$	0	0
$475$	3.85930	0.177077
$476$	0	0
$477$	−10.5607	−0.483543
$478$	0	0
$479$	−7.36994	−0.336741	−0.168371	−	0.985724i	$-0.553851\pi$
$479$	−7.36994	−0.336741	−0.168371	+	0.985724i	$0.553851\pi$
$480$	0	0
$481$	15.7554	0.718383
$482$	0	0
$483$	2.72323	0.123911
$484$	0	0
$485$	−6.22819	−0.282807
$486$	0	0
$487$	32.5603	1.47545	0.737725	−	0.675102i	$-0.235900\pi$
$487$	32.5603	1.47545	0.737725	+	0.675102i	$0.235900\pi$
$488$	0	0
$489$	9.35663	0.423121
$490$	0	0
$491$	−21.1987	−0.956683	−0.478342	−	0.878174i	$-0.658762\pi$
$491$	−21.1987	−0.956683	−0.478342	+	0.878174i	$0.658762\pi$
$492$	0	0
$493$	−8.00947	−0.360728
$494$	0	0
$495$	−2.90851	−0.130728
$496$	0	0
$497$	−9.26707	−0.415685
$498$	0	0
$499$	−0.642691	−0.0287708	−0.0143854	−	0.999897i	$-0.504579\pi$
$499$	−0.642691	−0.0287708	−0.0143854	+	0.999897i	$0.504579\pi$
$500$	0	0
$501$	−13.2736	−0.593022
$502$	0	0
$503$	27.1477	1.21046	0.605228	−	0.796052i	$-0.293082\pi$
$503$	27.1477	1.21046	0.605228	+	0.796052i	$0.293082\pi$
$504$	0	0
$505$	−10.9852	−0.488836
$506$	0	0
$507$	10.5389	0.468051
$508$	0	0
$509$	8.47432	0.375617	0.187809	−	0.982206i	$-0.439861\pi$
$509$	8.47432	0.375617	0.187809	+	0.982206i	$0.439861\pi$
$510$	0	0
$511$	−8.76947	−0.387939
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	15.9688	0.703671
$516$	0	0
$517$	−26.6639	−1.17268
$518$	0	0
$519$	0.891785	0.0391450
$520$	0	0
$521$	2.10673	0.0922976	0.0461488	−	0.998935i	$-0.485305\pi$
$521$	2.10673	0.0922976	0.0461488	+	0.998935i	$0.485305\pi$
$522$	0	0
$523$	−25.9499	−1.13471	−0.567356	−	0.823473i	$-0.692034\pi$
$523$	−25.9499	−1.13471	−0.567356	+	0.823473i	$0.692034\pi$
$524$	0	0
$525$	−3.85930	−0.168434
$526$	0	0
$527$	−22.8557	−0.995608
$528$	0	0
$529$	−15.5840	−0.677566
$530$	0	0
$531$	−5.44646	−0.236356
$532$	0	0
$533$	−9.94329	−0.430692
$534$	0	0
$535$	9.40566	0.406642
$536$	0	0
$537$	9.72763	0.419778
$538$	0	0
$539$	−2.72323	−0.117298
$540$	0	0
$541$	16.4601	0.707673	0.353837	−	0.935307i	$-0.384877\pi$
$541$	16.4601	0.707673	0.353837	+	0.935307i	$0.384877\pi$
$542$	0	0
$543$	−15.0247	−0.644772
$544$	0	0
$545$	10.2345	0.438397
$546$	0	0
$547$	−32.8295	−1.40369	−0.701843	−	0.712331i	$-0.747639\pi$
$547$	−32.8295	−1.40369	−0.701843	+	0.712331i	$0.747639\pi$
$548$	0	0
$549$	−1.77095	−0.0755824
$550$	0	0
$551$	1.65520	0.0705137
$552$	0	0
$553$	3.01838	0.128355
$554$	0	0
$555$	−10.7264	−0.455309
$556$	0	0
$557$	−20.1947	−0.855679	−0.427839	−	0.903855i	$-0.640725\pi$
$557$	−20.1947	−0.855679	−0.427839	+	0.903855i	$0.640725\pi$
$558$	0	0
$559$	13.7524	0.581665
$560$	0	0
$561$	13.1777	0.556362
$562$	0	0
$563$	32.4436	1.36734	0.683668	−	0.729793i	$-0.260384\pi$
$563$	32.4436	1.36734	0.683668	+	0.729793i	$0.260384\pi$
$564$	0	0
$565$	17.5881	0.739939
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	41.3801	1.73475	0.867373	−	0.497659i	$-0.165807\pi$
$569$	41.3801	1.73475	0.867373	+	0.497659i	$0.165807\pi$
$570$	0	0
$571$	−36.7627	−1.53847	−0.769235	−	0.638966i	$-0.779363\pi$
$571$	−36.7627	−1.53847	−0.769235	+	0.638966i	$0.779363\pi$
$572$	0	0
$573$	−2.41136	−0.100736
$574$	0	0
$575$	−10.5098	−0.438288
$576$	0	0
$577$	21.4866	0.894498	0.447249	−	0.894409i	$-0.352404\pi$
$577$	21.4866	0.894498	0.447249	+	0.894409i	$0.352404\pi$
$578$	0	0
$579$	−4.59663	−0.191030
$580$	0	0
$581$	−8.97820	−0.372479
$582$	0	0
$583$	28.7593	1.19109
$584$	0	0
$585$	1.67551	0.0692738
$586$	0	0
$587$	45.4194	1.87466	0.937329	−	0.348446i	$-0.113291\pi$
$587$	45.4194	1.87466	0.937329	+	0.348446i	$0.113291\pi$
$588$	0	0
$589$	4.72323	0.194617
$590$	0	0
$591$	−21.5356	−0.885857
$592$	0	0
$593$	40.5977	1.66715	0.833574	−	0.552407i	$-0.186291\pi$
$593$	40.5977	1.66715	0.833574	+	0.552407i	$0.186291\pi$
$594$	0	0
$595$	−5.16821	−0.211876
$596$	0	0
$597$	−7.31039	−0.299194
$598$	0	0
$599$	−31.4462	−1.28486	−0.642429	−	0.766345i	$-0.722073\pi$
$599$	−31.4462	−1.28486	−0.642429	+	0.766345i	$0.722073\pi$
$600$	0	0
$601$	37.2799	1.52068	0.760339	−	0.649527i	$-0.225033\pi$
$601$	37.2799	1.52068	0.760339	+	0.649527i	$0.225033\pi$
$602$	0	0
$603$	−8.92541	−0.363471
$604$	0	0
$605$	−3.82785	−0.155624
$606$	0	0
$607$	−13.4465	−0.545775	−0.272888	−	0.962046i	$-0.587979\pi$
$607$	−13.4465	−0.545775	−0.272888	+	0.962046i	$0.587979\pi$
$608$	0	0
$609$	−1.65520	−0.0670719
$610$	0	0
$611$	15.3603	0.621412
$612$	0	0
$613$	−32.9793	−1.33202	−0.666010	−	0.745942i	$-0.731999\pi$
$613$	−32.9793	−1.33202	−0.666010	+	0.745942i	$0.731999\pi$
$614$	0	0
$615$	6.76947	0.272971
$616$	0	0
$617$	14.2184	0.572411	0.286206	−	0.958168i	$-0.407606\pi$
$617$	14.2184	0.572411	0.286206	+	0.958168i	$0.407606\pi$
$618$	0	0
$619$	−16.9322	−0.680561	−0.340280	−	0.940324i	$-0.610522\pi$
$619$	−16.9322	−0.680561	−0.340280	+	0.940324i	$0.610522\pi$
$620$	0	0
$621$	−2.72323	−0.109279
$622$	0	0
$623$	10.6104	0.425096
$624$	0	0
$625$	9.19071	0.367628
$626$	0	0
$627$	−2.72323	−0.108755
$628$	0	0
$629$	48.5984	1.93775
$630$	0	0
$631$	−37.8074	−1.50509	−0.752545	−	0.658541i	$-0.771174\pi$
$631$	−37.8074	−1.50509	−0.752545	+	0.658541i	$0.771174\pi$
$632$	0	0
$633$	8.10359	0.322089
$634$	0	0
$635$	−7.35731	−0.291966
$636$	0	0
$637$	1.56878	0.0621572
$638$	0	0
$639$	9.26707	0.366599
$640$	0	0
$641$	38.0198	1.50169	0.750846	−	0.660477i	$-0.229646\pi$
$641$	38.0198	1.50169	0.750846	+	0.660477i	$0.229646\pi$
$642$	0	0
$643$	−24.6833	−0.973415	−0.486708	−	0.873565i	$-0.661802\pi$
$643$	−24.6833	−0.973415	−0.486708	+	0.873565i	$0.661802\pi$
$644$	0	0
$645$	−9.36274	−0.368658
$646$	0	0
$647$	45.1075	1.77336	0.886679	−	0.462385i	$-0.153006\pi$
$647$	45.1075	1.77336	0.886679	+	0.462385i	$0.153006\pi$
$648$	0	0
$649$	14.8320	0.582206
$650$	0	0
$651$	−4.72323	−0.185118
$652$	0	0
$653$	−10.0862	−0.394703	−0.197351	−	0.980333i	$-0.563234\pi$
$653$	−10.0862	−0.394703	−0.197351	+	0.980333i	$0.563234\pi$
$654$	0	0
$655$	−1.66850	−0.0651939
$656$	0	0
$657$	8.76947	0.342130
$658$	0	0
$659$	−14.9983	−0.584249	−0.292124	−	0.956380i	$-0.594362\pi$
$659$	−14.9983	−0.584249	−0.292124	+	0.956380i	$0.594362\pi$
$660$	0	0
$661$	−26.9493	−1.04821	−0.524103	−	0.851655i	$-0.675599\pi$
$661$	−26.9493	−1.04821	−0.524103	+	0.851655i	$0.675599\pi$
$662$	0	0
$663$	−7.59129	−0.294821
$664$	0	0
$665$	1.06804	0.0414166
$666$	0	0
$667$	−4.50748	−0.174530
$668$	0	0
$669$	0.265137	0.0102508
$670$	0	0
$671$	4.82271	0.186179
$672$	0	0
$673$	12.2645	0.472760	0.236380	−	0.971661i	$-0.424039\pi$
$673$	12.2645	0.472760	0.236380	+	0.971661i	$0.424039\pi$
$674$	0	0
$675$	3.85930	0.148545
$676$	0	0
$677$	−5.92707	−0.227796	−0.113898	−	0.993492i	$-0.536334\pi$
$677$	−5.92707	−0.227796	−0.113898	+	0.993492i	$0.536334\pi$
$678$	0	0
$679$	5.83145	0.223790
$680$	0	0
$681$	4.23151	0.162152
$682$	0	0
$683$	−19.4923	−0.745850	−0.372925	−	0.927861i	$-0.621645\pi$
$683$	−19.4923	−0.745850	−0.372925	+	0.927861i	$0.621645\pi$
$684$	0	0
$685$	23.9219	0.914009
$686$	0	0
$687$	−12.5404	−0.478447
$688$	0	0
$689$	−16.5674	−0.631169
$690$	0	0
$691$	−23.9781	−0.912171	−0.456085	−	0.889936i	$-0.650749\pi$
$691$	−23.9781	−0.912171	−0.456085	+	0.889936i	$0.650749\pi$
$692$	0	0
$693$	2.72323	0.103447
$694$	0	0
$695$	11.5022	0.436302
$696$	0	0
$697$	−30.6707	−1.16174
$698$	0	0
$699$	−21.6750	−0.819824
$700$	0	0
$701$	−26.4842	−1.00030	−0.500148	−	0.865940i	$-0.666721\pi$
$701$	−26.4842	−1.00030	−0.500148	+	0.865940i	$0.666721\pi$
$702$	0	0
$703$	−10.0431	−0.378782
$704$	0	0
$705$	−10.4574	−0.393849
$706$	0	0
$707$	10.2854	0.386824
$708$	0	0
$709$	−39.6587	−1.48942	−0.744708	−	0.667391i	$-0.767411\pi$
$709$	−39.6587	−1.48942	−0.744708	+	0.667391i	$0.767411\pi$
$710$	0	0
$711$	−3.01838	−0.113198
$712$	0	0
$713$	−12.8624	−0.481702
$714$	0	0
$715$	−4.56280	−0.170639
$716$	0	0
$717$	−21.2574	−0.793871
$718$	0	0
$719$	37.0915	1.38328	0.691641	−	0.722242i	$-0.256888\pi$
$719$	37.0915	1.38328	0.691641	+	0.722242i	$0.256888\pi$
$720$	0	0
$721$	−14.9516	−0.556827
$722$	0	0
$723$	−27.9544	−1.03964
$724$	0	0
$725$	6.38790	0.237241
$726$	0	0
$727$	24.9055	0.923695	0.461848	−	0.886959i	$-0.347187\pi$
$727$	24.9055	0.923695	0.461848	+	0.886959i	$0.347187\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	42.4201	1.56897
$732$	0	0
$733$	30.3608	1.12140	0.560701	−	0.828019i	$-0.310532\pi$
$733$	30.3608	1.12140	0.560701	+	0.828019i	$0.310532\pi$
$734$	0	0
$735$	−1.06804	−0.0393951
$736$	0	0
$737$	24.3059	0.895321
$738$	0	0
$739$	14.8681	0.546930	0.273465	−	0.961882i	$-0.411830\pi$
$739$	14.8681	0.546930	0.273465	+	0.961882i	$0.411830\pi$
$740$	0	0
$741$	1.56878	0.0576304
$742$	0	0
$743$	−30.1726	−1.10693	−0.553463	−	0.832874i	$-0.686694\pi$
$743$	−30.1726	−1.10693	−0.553463	+	0.832874i	$0.686694\pi$
$744$	0	0
$745$	−1.69020	−0.0619241
$746$	0	0
$747$	8.97820	0.328495
$748$	0	0
$749$	−8.80650	−0.321783
$750$	0	0
$751$	−18.7806	−0.685313	−0.342657	−	0.939461i	$-0.611327\pi$
$751$	−18.7806	−0.685313	−0.342657	+	0.939461i	$0.611327\pi$
$752$	0	0
$753$	−15.2964	−0.557432
$754$	0	0
$755$	−0.609250	−0.0221729
$756$	0	0
$757$	14.3119	0.520174	0.260087	−	0.965585i	$-0.416249\pi$
$757$	14.3119	0.520174	0.260087	+	0.965585i	$0.416249\pi$
$758$	0	0
$759$	7.41599	0.269183
$760$	0	0
$761$	45.7382	1.65801	0.829005	−	0.559241i	$-0.188907\pi$
$761$	45.7382	1.65801	0.829005	+	0.559241i	$0.188907\pi$
$762$	0	0
$763$	−9.58253	−0.346911
$764$	0	0
$765$	5.16821	0.186857
$766$	0	0
$767$	−8.54428	−0.308516
$768$	0	0
$769$	−41.1147	−1.48263	−0.741317	−	0.671155i	$-0.765798\pi$
$769$	−41.1147	−1.48263	−0.741317	+	0.671155i	$0.765798\pi$
$770$	0	0
$771$	−3.52568	−0.126974
$772$	0	0
$773$	20.1891	0.726150	0.363075	−	0.931760i	$-0.381727\pi$
$773$	20.1891	0.726150	0.363075	+	0.931760i	$0.381727\pi$
$774$	0	0
$775$	18.2284	0.654782
$776$	0	0
$777$	10.0431	0.360294
$778$	0	0
$779$	6.33825	0.227091
$780$	0	0
$781$	−25.2364	−0.903028
$782$	0	0
$783$	1.65520	0.0591519
$784$	0	0
$785$	−5.53311	−0.197485
$786$	0	0
$787$	2.50680	0.0893578	0.0446789	−	0.999001i	$-0.485774\pi$
$787$	2.50680	0.0893578	0.0446789	+	0.999001i	$0.485774\pi$
$788$	0	0
$789$	−6.95475	−0.247595
$790$	0	0
$791$	−16.4678	−0.585526
$792$	0	0
$793$	−2.77823	−0.0986578
$794$	0	0
$795$	11.2792	0.400033
$796$	0	0
$797$	−51.3144	−1.81765	−0.908824	−	0.417179i	$-0.863019\pi$
$797$	−51.3144	−1.81765	−0.908824	+	0.417179i	$0.863019\pi$
$798$	0	0
$799$	47.3798	1.67618
$800$	0	0
$801$	−10.6104	−0.374900
$802$	0	0
$803$	−23.8813	−0.842752
$804$	0	0
$805$	−2.90851	−0.102511
$806$	0	0
$807$	−20.2884	−0.714184
$808$	0	0
$809$	43.8635	1.54216	0.771079	−	0.636740i	$-0.219717\pi$
$809$	43.8635	1.54216	0.771079	+	0.636740i	$0.219717\pi$
$810$	0	0
$811$	−51.7206	−1.81615	−0.908077	−	0.418802i	$-0.862450\pi$
$811$	−51.7206	−1.81615	−0.908077	+	0.418802i	$0.862450\pi$
$812$	0	0
$813$	17.8939	0.627567
$814$	0	0
$815$	−9.99321	−0.350047
$816$	0	0
$817$	−8.76632	−0.306695
$818$	0	0
$819$	−1.56878	−0.0548175
$820$	0	0
$821$	12.9060	0.450424	0.225212	−	0.974310i	$-0.427693\pi$
$821$	12.9060	0.450424	0.225212	+	0.974310i	$0.427693\pi$
$822$	0	0
$823$	−21.3165	−0.743048	−0.371524	−	0.928423i	$-0.621165\pi$
$823$	−21.3165	−0.743048	−0.371524	+	0.928423i	$0.621165\pi$
$824$	0	0
$825$	−10.5098	−0.365903
$826$	0	0
$827$	−15.4767	−0.538177	−0.269088	−	0.963116i	$-0.586722\pi$
$827$	−15.4767	−0.538177	−0.269088	+	0.963116i	$0.586722\pi$
$828$	0	0
$829$	−7.98557	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$829$	−7.98557	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$830$	0	0
$831$	20.9852	0.727969
$832$	0	0
$833$	4.83899	0.167661
$834$	0	0
$835$	14.1767	0.490605
$836$	0	0
$837$	4.72323	0.163259
$838$	0	0
$839$	12.3706	0.427079	0.213539	−	0.976934i	$-0.431501\pi$
$839$	12.3706	0.427079	0.213539	+	0.976934i	$0.431501\pi$
$840$	0	0
$841$	−26.2603	−0.905529
$842$	0	0
$843$	−0.165641	−0.00570497
$844$	0	0
$845$	−11.2560	−0.387217
$846$	0	0
$847$	3.58401	0.123148
$848$	0	0
$849$	−28.8430	−0.989891
$850$	0	0
$851$	27.3497	0.937534
$852$	0	0
$853$	7.07455	0.242228	0.121114	−	0.992639i	$-0.461353\pi$
$853$	7.07455	0.242228	0.121114	+	0.992639i	$0.461353\pi$
$854$	0	0
$855$	−1.06804	−0.0365260
$856$	0	0
$857$	27.0144	0.922794	0.461397	−	0.887194i	$-0.347348\pi$
$857$	27.0144	0.922794	0.461397	+	0.887194i	$0.347348\pi$
$858$	0	0
$859$	2.63973	0.0900663	0.0450331	−	0.998985i	$-0.485661\pi$
$859$	2.63973	0.0900663	0.0450331	+	0.998985i	$0.485661\pi$
$860$	0	0
$861$	−6.33825	−0.216007
$862$	0	0
$863$	22.2588	0.757698	0.378849	−	0.925458i	$-0.376320\pi$
$863$	22.2588	0.757698	0.378849	+	0.925458i	$0.376320\pi$
$864$	0	0
$865$	−0.952458	−0.0323845
$866$	0	0
$867$	−6.41581	−0.217892
$868$	0	0
$869$	8.21976	0.278836
$870$	0	0
$871$	−14.0020	−0.474439
$872$	0	0
$873$	−5.83145	−0.197365
$874$	0	0
$875$	9.46204	0.319875
$876$	0	0
$877$	−10.6237	−0.358738	−0.179369	−	0.983782i	$-0.557406\pi$
$877$	−10.6237	−0.358738	−0.179369	+	0.983782i	$0.557406\pi$
$878$	0	0
$879$	33.8152	1.14056
$880$	0	0
$881$	26.7063	0.899757	0.449879	−	0.893090i	$-0.351467\pi$
$881$	26.7063	0.899757	0.449879	+	0.893090i	$0.351467\pi$
$882$	0	0
$883$	−5.28674	−0.177913	−0.0889565	−	0.996036i	$-0.528353\pi$
$883$	−5.28674	−0.177913	−0.0889565	+	0.996036i	$0.528353\pi$
$884$	0	0
$885$	5.81701	0.195537
$886$	0	0
$887$	−16.5646	−0.556185	−0.278093	−	0.960554i	$-0.589702\pi$
$887$	−16.5646	−0.556185	−0.278093	+	0.960554i	$0.589702\pi$
$888$	0	0
$889$	6.88864	0.231038
$890$	0	0
$891$	−2.72323	−0.0912317
$892$	0	0
$893$	−9.79127	−0.327652
$894$	0	0
$895$	−10.3894	−0.347281
$896$	0	0
$897$	−4.27214	−0.142643
$898$	0	0
$899$	7.81787	0.260741
$900$	0	0
$901$	−51.1033	−1.70250
$902$	0	0
$903$	8.76632	0.291725
$904$	0	0
$905$	16.0469	0.533418
$906$	0	0
$907$	1.30314	0.0432701	0.0216351	−	0.999766i	$-0.493113\pi$
$907$	1.30314	0.0432701	0.0216351	+	0.999766i	$0.493113\pi$
$908$	0	0
$909$	−10.2854	−0.341147
$910$	0	0
$911$	−39.0195	−1.29277	−0.646386	−	0.763010i	$-0.723721\pi$
$911$	−39.0195	−1.29277	−0.646386	+	0.763010i	$0.723721\pi$
$912$	0	0
$913$	−24.4497	−0.809168
$914$	0	0
$915$	1.89144	0.0625291
$916$	0	0
$917$	1.56222	0.0515890
$918$	0	0
$919$	−51.0653	−1.68449	−0.842244	−	0.539096i	$-0.818766\pi$
$919$	−51.0653	−1.68449	−0.842244	+	0.539096i	$0.818766\pi$
$920$	0	0
$921$	6.44812	0.212473
$922$	0	0
$923$	14.5380	0.478523
$924$	0	0
$925$	−38.7593	−1.27440
$926$	0	0
$927$	14.9516	0.491075
$928$	0	0
$929$	50.6776	1.66268	0.831339	−	0.555766i	$-0.187575\pi$
$929$	50.6776	1.66268	0.831339	+	0.555766i	$0.187575\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	6.69681	0.219244
$934$	0	0
$935$	−14.0742	−0.460276
$936$	0	0
$937$	−5.41599	−0.176933	−0.0884663	−	0.996079i	$-0.528197\pi$
$937$	−5.41599	−0.176933	−0.0884663	+	0.996079i	$0.528197\pi$
$938$	0	0
$939$	−0.324492	−0.0105894
$940$	0	0
$941$	−16.6941	−0.544211	−0.272106	−	0.962267i	$-0.587720\pi$
$941$	−16.6941	−0.544211	−0.272106	+	0.962267i	$0.587720\pi$
$942$	0	0
$943$	−17.2605	−0.562079
$944$	0	0
$945$	1.06804	0.0347432
$946$	0	0
$947$	−15.3010	−0.497214	−0.248607	−	0.968604i	$-0.579973\pi$
$947$	−15.3010	−0.497214	−0.248607	+	0.968604i	$0.579973\pi$
$948$	0	0
$949$	13.7573	0.446582
$950$	0	0
$951$	−2.06637	−0.0670067
$952$	0	0
$953$	−61.1007	−1.97925	−0.989623	−	0.143690i	$-0.954103\pi$
$953$	−61.1007	−1.97925	−0.989623	+	0.143690i	$0.954103\pi$
$954$	0	0
$955$	2.57541	0.0833384
$956$	0	0
$957$	−4.50748	−0.145706
$958$	0	0
$959$	−22.3981	−0.723271
$960$	0	0
$961$	−8.69109	−0.280358
$962$	0	0
$963$	8.80650	0.283786
$964$	0	0
$965$	4.90936	0.158038
$966$	0	0
$967$	−42.5373	−1.36791	−0.683954	−	0.729525i	$-0.739741\pi$
$967$	−42.5373	−1.36791	−0.683954	+	0.729525i	$0.739741\pi$
$968$	0	0
$969$	4.83899	0.155451
$970$	0	0
$971$	−24.4233	−0.783781	−0.391890	−	0.920012i	$-0.628179\pi$
$971$	−24.4233	−0.783781	−0.391890	+	0.920012i	$0.628179\pi$
$972$	0	0
$973$	−10.7695	−0.345253
$974$	0	0
$975$	6.05438	0.193895
$976$	0	0
$977$	−23.8150	−0.761908	−0.380954	−	0.924594i	$-0.624404\pi$
$977$	−23.8150	−0.761908	−0.380954	+	0.924594i	$0.624404\pi$
$978$	0	0
$979$	28.8945	0.923473
$980$	0	0
$981$	9.58253	0.305947
$982$	0	0
$983$	−49.3070	−1.57265	−0.786324	−	0.617814i	$-0.788018\pi$
$983$	−49.3070	−1.57265	−0.786324	+	0.617814i	$0.788018\pi$
$984$	0	0
$985$	23.0008	0.732866
$986$	0	0
$987$	9.79127	0.311659
$988$	0	0
$989$	23.8727	0.759108
$990$	0	0
$991$	7.43203	0.236086	0.118043	−	0.993008i	$-0.462338\pi$
$991$	7.43203	0.236086	0.118043	+	0.993008i	$0.462338\pi$
$992$	0	0
$993$	−21.5588	−0.684148
$994$	0	0
$995$	7.80775	0.247522
$996$	0	0
$997$	25.2668	0.800209	0.400104	−	0.916470i	$-0.368974\pi$
$997$	25.2668	0.800209	0.400104	+	0.916470i	$0.368974\pi$
$998$	0	0
$999$	−10.0431	−0.317749

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cc.1.3		5
4.3	odd	2		399.2.a.f.1.3	✓	5
12.11	even	2		1197.2.a.p.1.3		5
20.19	odd	2		9975.2.a.bq.1.3		5
28.27	even	2		2793.2.a.be.1.3		5
76.75	even	2		7581.2.a.x.1.3		5
84.83	odd	2		8379.2.a.ce.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.f.1.3	✓	5	4.3	odd	2
1197.2.a.p.1.3		5	12.11	even	2
2793.2.a.be.1.3		5	28.27	even	2
6384.2.a.cc.1.3		5	1.1	even	1	trivial
7581.2.a.x.1.3		5	76.75	even	2
8379.2.a.ce.1.3		5	84.83	odd	2
9975.2.a.bq.1.3		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.06804 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.06804 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.72323 q^{11} +1.56878 q^{13} -1.06804 q^{15} +4.83899 q^{17} -1.00000 q^{19} +1.00000 q^{21} +2.72323 q^{23} -3.85930 q^{25} -1.00000 q^{27} -1.65520 q^{29} -4.72323 q^{31} +2.72323 q^{33} -1.06804 q^{35} +10.0431 q^{37} -1.56878 q^{39} -6.33825 q^{41} +8.76632 q^{43} +1.06804 q^{45} +9.79127 q^{47} +1.00000 q^{49} -4.83899 q^{51} -10.5607 q^{53} -2.90851 q^{55} +1.00000 q^{57} -5.44646 q^{59} -1.77095 q^{61} -1.00000 q^{63} +1.67551 q^{65} -8.92541 q^{67} -2.72323 q^{69} +9.26707 q^{71} +8.76947 q^{73} +3.85930 q^{75} +2.72323 q^{77} -3.01838 q^{79} +1.00000 q^{81} +8.97820 q^{83} +5.16821 q^{85} +1.65520 q^{87} -10.6104 q^{89} -1.56878 q^{91} +4.72323 q^{93} -1.06804 q^{95} -5.83145 q^{97} -2.72323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 2 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} + 5 q^{21} - 2 q^{23} + 11 q^{25} - 5 q^{27} - 8 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} - 20 q^{43} - 2 q^{45} + 26 q^{47} + 5 q^{49} + 2 q^{51} + 4 q^{53} + 4 q^{55} + 5 q^{57} + 4 q^{59} + 10 q^{61} - 5 q^{63} - 4 q^{65} - 10 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} - 11 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 34 q^{83} - 36 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{93} + 2 q^{95} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 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 + 11 * q^25 - 5 * q^27 - 8 * q^31 - 2 * q^33 + 2 * q^35 + 2 * q^37 - 8 * q^39 + 2 * q^41 - 20 * q^43 - 2 * q^45 + 26 * q^47 + 5 * q^49 + 2 * q^51 + 4 * q^53 + 4 * q^55 + 5 * q^57 + 4 * q^59 + 10 * q^61 - 5 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 - 10 * q^71 + 10 * q^73 - 11 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 34 * q^83 - 36 * q^85 + 10 * q^89 - 8 * q^91 + 8 * q^93 + 2 * q^95 - 16 * q^97 + 2 * q^99

Introduction

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