LMFDB - Embedded newform 6384.2.a.cf.1.1

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:	5.5.368464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.1
Root			$1.17837$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.42801 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.42801 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.50406 q^{11} -6.67091 q^{13} -3.42801 q^{15} +4.28069 q^{17} +1.00000 q^{19} -1.00000 q^{21} -7.57963 q^{23} +6.75126 q^{25} +1.00000 q^{27} -0.151622 q^{29} -10.6073 q^{31} -5.50406 q^{33} +3.42801 q^{35} -5.70870 q^{37} -6.67091 q^{39} +1.20942 q^{41} +6.18511 q^{43} -3.42801 q^{45} +0.0760455 q^{47} +1.00000 q^{49} +4.28069 q^{51} +3.29894 q^{53} +18.8680 q^{55} +1.00000 q^{57} +2.07558 q^{59} +8.00334 q^{61} -1.00000 q^{63} +22.8680 q^{65} +1.51419 q^{67} -7.57963 q^{69} -3.65568 q^{71} +3.07174 q^{73} +6.75126 q^{75} +5.50406 q^{77} -2.99523 q^{79} +1.00000 q^{81} -16.7163 q^{83} -14.6743 q^{85} -0.151622 q^{87} -4.36868 q^{89} +6.67091 q^{91} -10.6073 q^{93} -3.42801 q^{95} +7.34183 q^{97} -5.50406 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} - 5 q^{21} - 12 q^{23} + 15 q^{25} + 5 q^{27} + 4 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{39} + 10 q^{41} + 16 q^{43} + 4 q^{45} + 2 q^{47} + 5 q^{49} + 12 q^{51} + 12 q^{55} + 5 q^{57} + 4 q^{59} - 14 q^{61} - 5 q^{63} + 32 q^{65} + 20 q^{67} - 12 q^{69} + 6 q^{71} + 10 q^{73} + 15 q^{75} + 8 q^{77} + 5 q^{81} - 6 q^{83} + 8 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - 18 q^{97} - 8 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 - 8 * q^11 - 6 * q^13 + 4 * q^15 + 12 * q^17 + 5 * q^19 - 5 * q^21 - 12 * q^23 + 15 * q^25 + 5 * q^27 + 4 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^35 + 2 * q^37 - 6 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 + 2 * q^47 + 5 * q^49 + 12 * q^51 + 12 * q^55 + 5 * q^57 + 4 * q^59 - 14 * q^61 - 5 * q^63 + 32 * q^65 + 20 * q^67 - 12 * q^69 + 6 * q^71 + 10 * q^73 + 15 * q^75 + 8 * q^77 + 5 * q^81 - 6 * q^83 + 8 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - 18 * q^97 - 8 * 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
$5$	−3.42801	−1.53305	−0.766527	−	0.642213i	$-0.778017\pi$
$5$	−3.42801	−1.53305	−0.766527	+	0.642213i	$0.778017\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.50406	−1.65954	−0.829768	−	0.558109i	$-0.811527\pi$
$11$	−5.50406	−1.65954	−0.829768	+	0.558109i	$0.811527\pi$
$12$	0	0
$13$	−6.67091	−1.85018	−0.925089	−	0.379750i	$-0.876010\pi$
$13$	−6.67091	−1.85018	−0.925089	+	0.379750i	$0.876010\pi$
$14$	0	0
$15$	−3.42801	−0.885109
$16$	0	0
$17$	4.28069	1.03822	0.519110	−	0.854707i	$-0.326263\pi$
$17$	4.28069	1.03822	0.519110	+	0.854707i	$0.326263\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−7.57963	−1.58046	−0.790231	−	0.612809i	$-0.790040\pi$
$23$	−7.57963	−1.58046	−0.790231	+	0.612809i	$0.790040\pi$
$24$	0	0
$25$	6.75126	1.35025
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.151622	−0.0281555	−0.0140777	−	0.999901i	$-0.504481\pi$
$29$	−0.151622	−0.0281555	−0.0140777	+	0.999901i	$0.504481\pi$
$30$	0	0
$31$	−10.6073	−1.90512	−0.952562	−	0.304344i	$-0.901563\pi$
$31$	−10.6073	−1.90512	−0.952562	+	0.304344i	$0.901563\pi$
$32$	0	0
$33$	−5.50406	−0.958133
$34$	0	0
$35$	3.42801	0.579440
$36$	0	0
$37$	−5.70870	−0.938505	−0.469252	−	0.883064i	$-0.655477\pi$
$37$	−5.70870	−0.938505	−0.469252	+	0.883064i	$0.655477\pi$
$38$	0	0
$39$	−6.67091	−1.06820
$40$	0	0
$41$	1.20942	0.188879	0.0944395	−	0.995531i	$-0.469894\pi$
$41$	1.20942	0.188879	0.0944395	+	0.995531i	$0.469894\pi$
$42$	0	0
$43$	6.18511	0.943220	0.471610	−	0.881807i	$-0.343673\pi$
$43$	6.18511	0.943220	0.471610	+	0.881807i	$0.343673\pi$
$44$	0	0
$45$	−3.42801	−0.511018
$46$	0	0
$47$	0.0760455	0.0110924	0.00554619	−	0.999985i	$-0.498235\pi$
$47$	0.0760455	0.0110924	0.00554619	+	0.999985i	$0.498235\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.28069	0.599417
$52$	0	0
$53$	3.29894	0.453145	0.226572	−	0.973994i	$-0.427248\pi$
$53$	3.29894	0.453145	0.226572	+	0.973994i	$0.427248\pi$
$54$	0	0
$55$	18.8680	2.54416
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	2.07558	0.270217	0.135109	−	0.990831i	$-0.456862\pi$
$59$	2.07558	0.270217	0.135109	+	0.990831i	$0.456862\pi$
$60$	0	0
$61$	8.00334	1.02472	0.512362	−	0.858770i	$-0.328771\pi$
$61$	8.00334	1.02472	0.512362	+	0.858770i	$0.328771\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	22.8680	2.83642
$66$	0	0
$67$	1.51419	0.184988	0.0924941	−	0.995713i	$-0.470516\pi$
$67$	1.51419	0.184988	0.0924941	+	0.995713i	$0.470516\pi$
$68$	0	0
$69$	−7.57963	−0.912481
$70$	0	0
$71$	−3.65568	−0.433849	−0.216925	−	0.976188i	$-0.569603\pi$
$71$	−3.65568	−0.433849	−0.216925	+	0.976188i	$0.569603\pi$
$72$	0	0
$73$	3.07174	0.359520	0.179760	−	0.983710i	$-0.442468\pi$
$73$	3.07174	0.359520	0.179760	+	0.983710i	$0.442468\pi$
$74$	0	0
$75$	6.75126	0.779568
$76$	0	0
$77$	5.50406	0.627245
$78$	0	0
$79$	−2.99523	−0.336990	−0.168495	−	0.985703i	$-0.553891\pi$
$79$	−2.99523	−0.336990	−0.168495	+	0.985703i	$0.553891\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.7163	−1.83486	−0.917429	−	0.397900i	$-0.869739\pi$
$83$	−16.7163	−1.83486	−0.917429	+	0.397900i	$0.869739\pi$
$84$	0	0
$85$	−14.6743	−1.59165
$86$	0	0
$87$	−0.151622	−0.0162556
$88$	0	0
$89$	−4.36868	−0.463079	−0.231540	−	0.972825i	$-0.574376\pi$
$89$	−4.36868	−0.463079	−0.231540	+	0.972825i	$0.574376\pi$
$90$	0	0
$91$	6.67091	0.699302
$92$	0	0
$93$	−10.6073	−1.09992
$94$	0	0
$95$	−3.42801	−0.351707
$96$	0	0
$97$	7.34183	0.745450	0.372725	−	0.927942i	$-0.378424\pi$
$97$	7.34183	0.745450	0.372725	+	0.927942i	$0.378424\pi$
$98$	0	0
$99$	−5.50406	−0.553179
$100$	0	0
$101$	0.711196	0.0707666	0.0353833	−	0.999374i	$-0.488735\pi$
$101$	0.711196	0.0707666	0.0353833	+	0.999374i	$0.488735\pi$
$102$	0	0
$103$	11.0081	1.08466	0.542331	−	0.840165i	$-0.317542\pi$
$103$	11.0081	1.08466	0.542331	+	0.840165i	$0.317542\pi$
$104$	0	0
$105$	3.42801	0.334540
$106$	0	0
$107$	−17.8502	−1.72564	−0.862821	−	0.505509i	$-0.831305\pi$
$107$	−17.8502	−1.72564	−0.862821	+	0.505509i	$0.831305\pi$
$108$	0	0
$109$	7.27485	0.696805	0.348402	−	0.937345i	$-0.386724\pi$
$109$	7.27485	0.696805	0.348402	+	0.937345i	$0.386724\pi$
$110$	0	0
$111$	−5.70870	−0.541846
$112$	0	0
$113$	3.84838	0.362025	0.181012	−	0.983481i	$-0.442063\pi$
$113$	3.84838	0.362025	0.181012	+	0.983481i	$0.442063\pi$
$114$	0	0
$115$	25.9831	2.42293
$116$	0	0
$117$	−6.67091	−0.616726
$118$	0	0
$119$	−4.28069	−0.392410
$120$	0	0
$121$	19.2946	1.75406
$122$	0	0
$123$	1.20942	0.109049
$124$	0	0
$125$	−6.00334	−0.536955
$126$	0	0
$127$	8.78045	0.779139	0.389569	−	0.920997i	$-0.372624\pi$
$127$	8.78045	0.779139	0.389569	+	0.920997i	$0.372624\pi$
$128$	0	0
$129$	6.18511	0.544568
$130$	0	0
$131$	6.27055	0.547861	0.273930	−	0.961750i	$-0.411676\pi$
$131$	6.27055	0.547861	0.273930	+	0.961750i	$0.411676\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−3.42801	−0.295036
$136$	0	0
$137$	−7.42695	−0.634527	−0.317263	−	0.948337i	$-0.602764\pi$
$137$	−7.42695	−0.634527	−0.317263	+	0.948337i	$0.602764\pi$
$138$	0	0
$139$	−16.9445	−1.43721	−0.718606	−	0.695417i	$-0.755220\pi$
$139$	−16.9445	−1.43721	−0.718606	+	0.695417i	$0.755220\pi$
$140$	0	0
$141$	0.0760455	0.00640419
$142$	0	0
$143$	36.7171	3.07044
$144$	0	0
$145$	0.519761	0.0431638
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−2.93160	−0.240166	−0.120083	−	0.992764i	$-0.538316\pi$
$149$	−2.93160	−0.240166	−0.120083	+	0.992764i	$0.538316\pi$
$150$	0	0
$151$	13.8513	1.12720	0.563599	−	0.826048i	$-0.309416\pi$
$151$	13.8513	1.12720	0.563599	+	0.826048i	$0.309416\pi$
$152$	0	0
$153$	4.28069	0.346073
$154$	0	0
$155$	36.3619	2.92066
$156$	0	0
$157$	7.78428	0.621253	0.310627	−	0.950532i	$-0.399461\pi$
$157$	7.78428	0.621253	0.310627	+	0.950532i	$0.399461\pi$
$158$	0	0
$159$	3.29894	0.261623
$160$	0	0
$161$	7.57963	0.597359
$162$	0	0
$163$	−6.44325	−0.504674	−0.252337	−	0.967639i	$-0.581199\pi$
$163$	−6.44325	−0.504674	−0.252337	+	0.967639i	$0.581199\pi$
$164$	0	0
$165$	18.8680	1.46887
$166$	0	0
$167$	6.48581	0.501887	0.250943	−	0.968002i	$-0.419259\pi$
$167$	6.48581	0.501887	0.250943	+	0.968002i	$0.419259\pi$
$168$	0	0
$169$	31.5011	2.42316
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	7.87427	0.598670	0.299335	−	0.954148i	$-0.403235\pi$
$173$	7.87427	0.598670	0.299335	+	0.954148i	$0.403235\pi$
$174$	0	0
$175$	−6.75126	−0.510347
$176$	0	0
$177$	2.07558	0.156010
$178$	0	0
$179$	−20.7062	−1.54765	−0.773827	−	0.633397i	$-0.781660\pi$
$179$	−20.7062	−1.54765	−0.773827	+	0.633397i	$0.781660\pi$
$180$	0	0
$181$	4.70393	0.349640	0.174820	−	0.984600i	$-0.444066\pi$
$181$	4.70393	0.349640	0.174820	+	0.984600i	$0.444066\pi$
$182$	0	0
$183$	8.00334	0.591624
$184$	0	0
$185$	19.5695	1.43878
$186$	0	0
$187$	−23.5612	−1.72296
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	18.3484	1.32764	0.663822	−	0.747891i	$-0.268933\pi$
$191$	18.3484	1.32764	0.663822	+	0.747891i	$0.268933\pi$
$192$	0	0
$193$	5.92777	0.426690	0.213345	−	0.976977i	$-0.431564\pi$
$193$	5.92777	0.426690	0.213345	+	0.976977i	$0.431564\pi$
$194$	0	0
$195$	22.8680	1.63761
$196$	0	0
$197$	−10.9986	−0.783616	−0.391808	−	0.920047i	$-0.628150\pi$
$197$	−10.9986	−0.783616	−0.391808	+	0.920047i	$0.628150\pi$
$198$	0	0
$199$	−0.152091	−0.0107814	−0.00539072	−	0.999985i	$-0.501716\pi$
$199$	−0.152091	−0.0107814	−0.00539072	+	0.999985i	$0.501716\pi$
$200$	0	0
$201$	1.51419	0.106803
$202$	0	0
$203$	0.151622	0.0106418
$204$	0	0
$205$	−4.14589	−0.289562
$206$	0	0
$207$	−7.57963	−0.526821
$208$	0	0
$209$	−5.50406	−0.380724
$210$	0	0
$211$	3.62978	0.249885	0.124942	−	0.992164i	$-0.460125\pi$
$211$	3.62978	0.249885	0.124942	+	0.992164i	$0.460125\pi$
$212$	0	0
$213$	−3.65568	−0.250483
$214$	0	0
$215$	−21.2026	−1.44601
$216$	0	0
$217$	10.6073	0.720069
$218$	0	0
$219$	3.07174	0.207569
$220$	0	0
$221$	−28.5561	−1.92089
$222$	0	0
$223$	18.8654	1.26332	0.631661	−	0.775245i	$-0.282373\pi$
$223$	18.8654	1.26332	0.631661	+	0.775245i	$0.282373\pi$
$224$	0	0
$225$	6.75126	0.450084
$226$	0	0
$227$	−11.5695	−0.767894	−0.383947	−	0.923355i	$-0.625436\pi$
$227$	−11.5695	−0.767894	−0.383947	+	0.923355i	$0.625436\pi$
$228$	0	0
$229$	−20.3585	−1.34533	−0.672665	−	0.739947i	$-0.734851\pi$
$229$	−20.3585	−1.34533	−0.672665	+	0.739947i	$0.734851\pi$
$230$	0	0
$231$	5.50406	0.362140
$232$	0	0
$233$	−16.2734	−1.06611	−0.533054	−	0.846081i	$-0.678956\pi$
$233$	−16.2734	−1.06611	−0.533054	+	0.846081i	$0.678956\pi$
$234$	0	0
$235$	−0.260685	−0.0170052
$236$	0	0
$237$	−2.99523	−0.194561
$238$	0	0
$239$	−1.69522	−0.109655	−0.0548274	−	0.998496i	$-0.517461\pi$
$239$	−1.69522	−0.109655	−0.0548274	+	0.998496i	$0.517461\pi$
$240$	0	0
$241$	1.28653	0.0828725	0.0414362	−	0.999141i	$-0.486807\pi$
$241$	1.28653	0.0828725	0.0414362	+	0.999141i	$0.486807\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.42801	−0.219008
$246$	0	0
$247$	−6.67091	−0.424460
$248$	0	0
$249$	−16.7163	−1.05936
$250$	0	0
$251$	−22.0913	−1.39439	−0.697196	−	0.716880i	$-0.745569\pi$
$251$	−22.0913	−1.39439	−0.697196	+	0.716880i	$0.745569\pi$
$252$	0	0
$253$	41.7187	2.62283
$254$	0	0
$255$	−14.6743	−0.918938
$256$	0	0
$257$	−19.0736	−1.18978	−0.594888	−	0.803809i	$-0.702804\pi$
$257$	−19.0736	−1.18978	−0.594888	+	0.803809i	$0.702804\pi$
$258$	0	0
$259$	5.70870	0.354721
$260$	0	0
$261$	−0.151622	−0.00938516
$262$	0	0
$263$	15.6466	0.964811	0.482406	−	0.875948i	$-0.339763\pi$
$263$	15.6466	0.964811	0.482406	+	0.875948i	$0.339763\pi$
$264$	0	0
$265$	−11.3088	−0.694695
$266$	0	0
$267$	−4.36868	−0.267359
$268$	0	0
$269$	26.1477	1.59425	0.797127	−	0.603812i	$-0.206352\pi$
$269$	26.1477	1.59425	0.797127	+	0.603812i	$0.206352\pi$
$270$	0	0
$271$	10.9531	0.665353	0.332676	−	0.943041i	$-0.392048\pi$
$271$	10.9531	0.665353	0.332676	+	0.943041i	$0.392048\pi$
$272$	0	0
$273$	6.67091	0.403742
$274$	0	0
$275$	−37.1593	−2.24079
$276$	0	0
$277$	19.4633	1.16944	0.584718	−	0.811236i	$-0.301205\pi$
$277$	19.4633	1.16944	0.584718	+	0.811236i	$0.301205\pi$
$278$	0	0
$279$	−10.6073	−0.635041
$280$	0	0
$281$	18.4131	1.09843	0.549217	−	0.835680i	$-0.314926\pi$
$281$	18.4131	1.09843	0.549217	+	0.835680i	$0.314926\pi$
$282$	0	0
$283$	7.47769	0.444503	0.222251	−	0.974989i	$-0.428659\pi$
$283$	7.47769	0.444503	0.222251	+	0.974989i	$0.428659\pi$
$284$	0	0
$285$	−3.42801	−0.203058
$286$	0	0
$287$	−1.20942	−0.0713896
$288$	0	0
$289$	1.32432	0.0779009
$290$	0	0
$291$	7.34183	0.430386
$292$	0	0
$293$	−9.12430	−0.533047	−0.266524	−	0.963828i	$-0.585875\pi$
$293$	−9.12430	−0.533047	−0.266524	+	0.963828i	$0.585875\pi$
$294$	0	0
$295$	−7.11510	−0.414257
$296$	0	0
$297$	−5.50406	−0.319378
$298$	0	0
$299$	50.5631	2.92414
$300$	0	0
$301$	−6.18511	−0.356504
$302$	0	0
$303$	0.711196	0.0408571
$304$	0	0
$305$	−27.4355	−1.57095
$306$	0	0
$307$	19.5992	1.11858	0.559292	−	0.828971i	$-0.311073\pi$
$307$	19.5992	1.11858	0.559292	+	0.828971i	$0.311073\pi$
$308$	0	0
$309$	11.0081	0.626230
$310$	0	0
$311$	5.86126	0.332362	0.166181	−	0.986095i	$-0.446856\pi$
$311$	5.86126	0.332362	0.166181	+	0.986095i	$0.446856\pi$
$312$	0	0
$313$	29.9517	1.69297	0.846484	−	0.532414i	$-0.178715\pi$
$313$	29.9517	1.69297	0.846484	+	0.532414i	$0.178715\pi$
$314$	0	0
$315$	3.42801	0.193147
$316$	0	0
$317$	−11.1597	−0.626793	−0.313397	−	0.949622i	$-0.601467\pi$
$317$	−11.1597	−0.626793	−0.313397	+	0.949622i	$0.601467\pi$
$318$	0	0
$319$	0.834535	0.0467250
$320$	0	0
$321$	−17.8502	−0.996300
$322$	0	0
$323$	4.28069	0.238184
$324$	0	0
$325$	−45.0371	−2.49821
$326$	0	0
$327$	7.27485	0.402300
$328$	0	0
$329$	−0.0760455	−0.00419252
$330$	0	0
$331$	3.03524	0.166832	0.0834160	−	0.996515i	$-0.473417\pi$
$331$	3.03524	0.166832	0.0834160	+	0.996515i	$0.473417\pi$
$332$	0	0
$333$	−5.70870	−0.312835
$334$	0	0
$335$	−5.19068	−0.283597
$336$	0	0
$337$	−7.99140	−0.435319	−0.217660	−	0.976025i	$-0.569842\pi$
$337$	−7.99140	−0.435319	−0.217660	+	0.976025i	$0.569842\pi$
$338$	0	0
$339$	3.84838	0.209015
$340$	0	0
$341$	58.3831	3.16162
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	25.9831	1.39888
$346$	0	0
$347$	−5.69522	−0.305736	−0.152868	−	0.988247i	$-0.548851\pi$
$347$	−5.69522	−0.305736	−0.152868	+	0.988247i	$0.548851\pi$
$348$	0	0
$349$	−9.50252	−0.508658	−0.254329	−	0.967118i	$-0.581855\pi$
$349$	−9.50252	−0.508658	−0.254329	+	0.967118i	$0.581855\pi$
$350$	0	0
$351$	−6.67091	−0.356067
$352$	0	0
$353$	1.72265	0.0916875	0.0458438	−	0.998949i	$-0.485402\pi$
$353$	1.72265	0.0916875	0.0458438	+	0.998949i	$0.485402\pi$
$354$	0	0
$355$	12.5317	0.665114
$356$	0	0
$357$	−4.28069	−0.226558
$358$	0	0
$359$	−24.0807	−1.27093	−0.635466	−	0.772129i	$-0.719192\pi$
$359$	−24.0807	−1.27093	−0.635466	+	0.772129i	$0.719192\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	19.2946	1.01271
$364$	0	0
$365$	−10.5300	−0.551164
$366$	0	0
$367$	11.9397	0.623248	0.311624	−	0.950206i	$-0.399127\pi$
$367$	11.9397	0.623248	0.311624	+	0.950206i	$0.399127\pi$
$368$	0	0
$369$	1.20942	0.0629597
$370$	0	0
$371$	−3.29894	−0.171273
$372$	0	0
$373$	−34.9783	−1.81111	−0.905554	−	0.424231i	$-0.860544\pi$
$373$	−34.9783	−1.81111	−0.905554	+	0.424231i	$0.860544\pi$
$374$	0	0
$375$	−6.00334	−0.310011
$376$	0	0
$377$	1.01146	0.0520926
$378$	0	0
$379$	3.72685	0.191435	0.0957176	−	0.995409i	$-0.469485\pi$
$379$	3.72685	0.191435	0.0957176	+	0.995409i	$0.469485\pi$
$380$	0	0
$381$	8.78045	0.449836
$382$	0	0
$383$	13.0589	0.667277	0.333638	−	0.942701i	$-0.391724\pi$
$383$	13.0589	0.667277	0.333638	+	0.942701i	$0.391724\pi$
$384$	0	0
$385$	−18.8680	−0.961601
$386$	0	0
$387$	6.18511	0.314407
$388$	0	0
$389$	−20.3585	−1.03222	−0.516110	−	0.856523i	$-0.672620\pi$
$389$	−20.3585	−1.03222	−0.516110	+	0.856523i	$0.672620\pi$
$390$	0	0
$391$	−32.4461	−1.64087
$392$	0	0
$393$	6.27055	0.316308
$394$	0	0
$395$	10.2677	0.516623
$396$	0	0
$397$	−20.5797	−1.03287	−0.516434	−	0.856327i	$-0.672741\pi$
$397$	−20.5797	−1.03287	−0.516434	+	0.856327i	$0.672741\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−12.9862	−0.648498	−0.324249	−	0.945972i	$-0.605112\pi$
$401$	−12.9862	−0.648498	−0.324249	+	0.945972i	$0.605112\pi$
$402$	0	0
$403$	70.7603	3.52482
$404$	0	0
$405$	−3.42801	−0.170339
$406$	0	0
$407$	31.4210	1.55748
$408$	0	0
$409$	37.2085	1.83984	0.919921	−	0.392103i	$-0.128252\pi$
$409$	37.2085	1.83984	0.919921	+	0.392103i	$0.128252\pi$
$410$	0	0
$411$	−7.42695	−0.366344
$412$	0	0
$413$	−2.07558	−0.102132
$414$	0	0
$415$	57.3038	2.81293
$416$	0	0
$417$	−16.9445	−0.829775
$418$	0	0
$419$	27.8034	1.35828	0.679142	−	0.734007i	$-0.262352\pi$
$419$	27.8034	1.35828	0.679142	+	0.734007i	$0.262352\pi$
$420$	0	0
$421$	17.6417	0.859805	0.429903	−	0.902875i	$-0.358548\pi$
$421$	17.6417	0.859805	0.429903	+	0.902875i	$0.358548\pi$
$422$	0	0
$423$	0.0760455	0.00369746
$424$	0	0
$425$	28.9001	1.40186
$426$	0	0
$427$	−8.00334	−0.387309
$428$	0	0
$429$	36.7171	1.77272
$430$	0	0
$431$	−31.9937	−1.54108	−0.770541	−	0.637391i	$-0.780014\pi$
$431$	−31.9937	−1.54108	−0.770541	+	0.637391i	$0.780014\pi$
$432$	0	0
$433$	−35.0685	−1.68528	−0.842642	−	0.538473i	$-0.819001\pi$
$433$	−35.0685	−1.68528	−0.842642	+	0.538473i	$0.819001\pi$
$434$	0	0
$435$	0.519761	0.0249206
$436$	0	0
$437$	−7.57963	−0.362583
$438$	0	0
$439$	−19.6910	−0.939799	−0.469899	−	0.882720i	$-0.655710\pi$
$439$	−19.6910	−0.939799	−0.469899	+	0.882720i	$0.655710\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	15.8626	0.753655	0.376827	−	0.926283i	$-0.377015\pi$
$443$	15.8626	0.753655	0.376827	+	0.926283i	$0.377015\pi$
$444$	0	0
$445$	14.9759	0.709925
$446$	0	0
$447$	−2.93160	−0.138660
$448$	0	0
$449$	−25.6996	−1.21284	−0.606420	−	0.795144i	$-0.707395\pi$
$449$	−25.6996	−1.21284	−0.606420	+	0.795144i	$0.707395\pi$
$450$	0	0
$451$	−6.65670	−0.313452
$452$	0	0
$453$	13.8513	0.650789
$454$	0	0
$455$	−22.8680	−1.07207
$456$	0	0
$457$	6.09759	0.285233	0.142617	−	0.989778i	$-0.454448\pi$
$457$	6.09759	0.285233	0.142617	+	0.989778i	$0.454448\pi$
$458$	0	0
$459$	4.28069	0.199806
$460$	0	0
$461$	9.17322	0.427239	0.213620	−	0.976917i	$-0.431475\pi$
$461$	9.17322	0.427239	0.213620	+	0.976917i	$0.431475\pi$
$462$	0	0
$463$	−19.8727	−0.923565	−0.461782	−	0.886993i	$-0.652790\pi$
$463$	−19.8727	−0.923565	−0.461782	+	0.886993i	$0.652790\pi$
$464$	0	0
$465$	36.3619	1.68624
$466$	0	0
$467$	−0.452461	−0.0209374	−0.0104687	−	0.999945i	$-0.503332\pi$
$467$	−0.452461	−0.0209374	−0.0104687	+	0.999945i	$0.503332\pi$
$468$	0	0
$469$	−1.51419	−0.0699190
$470$	0	0
$471$	7.78428	0.358681
$472$	0	0
$473$	−34.0432	−1.56531
$474$	0	0
$475$	6.75126	0.309769
$476$	0	0
$477$	3.29894	0.151048
$478$	0	0
$479$	−13.6541	−0.623874	−0.311937	−	0.950103i	$-0.600978\pi$
$479$	−13.6541	−0.623874	−0.311937	+	0.950103i	$0.600978\pi$
$480$	0	0
$481$	38.0823	1.73640
$482$	0	0
$483$	7.57963	0.344885
$484$	0	0
$485$	−25.1679	−1.14281
$486$	0	0
$487$	15.5686	0.705479	0.352739	−	0.935722i	$-0.385250\pi$
$487$	15.5686	0.705479	0.352739	+	0.935722i	$0.385250\pi$
$488$	0	0
$489$	−6.44325	−0.291374
$490$	0	0
$491$	−14.9519	−0.674771	−0.337386	−	0.941367i	$-0.609543\pi$
$491$	−14.9519	−0.674771	−0.337386	+	0.941367i	$0.609543\pi$
$492$	0	0
$493$	−0.649046	−0.0292316
$494$	0	0
$495$	18.8680	0.848052
$496$	0	0
$497$	3.65568	0.163980
$498$	0	0
$499$	−33.9702	−1.52071	−0.760357	−	0.649505i	$-0.774976\pi$
$499$	−33.9702	−1.52071	−0.760357	+	0.649505i	$0.774976\pi$
$500$	0	0
$501$	6.48581	0.289764
$502$	0	0
$503$	−24.0007	−1.07014	−0.535070	−	0.844808i	$-0.679715\pi$
$503$	−24.0007	−1.07014	−0.535070	+	0.844808i	$0.679715\pi$
$504$	0	0
$505$	−2.43799	−0.108489
$506$	0	0
$507$	31.5011	1.39901
$508$	0	0
$509$	34.7303	1.53939	0.769697	−	0.638410i	$-0.220407\pi$
$509$	34.7303	1.53939	0.769697	+	0.638410i	$0.220407\pi$
$510$	0	0
$511$	−3.07174	−0.135886
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−37.7359	−1.66284
$516$	0	0
$517$	−0.418559	−0.0184082
$518$	0	0
$519$	7.87427	0.345642
$520$	0	0
$521$	16.9331	0.741854	0.370927	−	0.928662i	$-0.379040\pi$
$521$	16.9331	0.741854	0.370927	+	0.928662i	$0.379040\pi$
$522$	0	0
$523$	10.4561	0.457215	0.228607	−	0.973519i	$-0.426583\pi$
$523$	10.4561	0.457215	0.228607	+	0.973519i	$0.426583\pi$
$524$	0	0
$525$	−6.75126	−0.294649
$526$	0	0
$527$	−45.4065	−1.97794
$528$	0	0
$529$	34.4508	1.49786
$530$	0	0
$531$	2.07558	0.0900723
$532$	0	0
$533$	−8.06791	−0.349460
$534$	0	0
$535$	61.1906	2.64550
$536$	0	0
$537$	−20.7062	−0.893539
$538$	0	0
$539$	−5.50406	−0.237077
$540$	0	0
$541$	−3.38787	−0.145656	−0.0728280	−	0.997345i	$-0.523202\pi$
$541$	−3.38787	−0.145656	−0.0728280	+	0.997345i	$0.523202\pi$
$542$	0	0
$543$	4.70393	0.201865
$544$	0	0
$545$	−24.9383	−1.06824
$546$	0	0
$547$	−42.0070	−1.79609	−0.898044	−	0.439906i	$-0.855012\pi$
$547$	−42.0070	−1.79609	−0.898044	+	0.439906i	$0.855012\pi$
$548$	0	0
$549$	8.00334	0.341574
$550$	0	0
$551$	−0.151622	−0.00645931
$552$	0	0
$553$	2.99523	0.127370
$554$	0	0
$555$	19.5695	0.830679
$556$	0	0
$557$	−28.5316	−1.20892	−0.604461	−	0.796635i	$-0.706611\pi$
$557$	−28.5316	−1.20892	−0.604461	+	0.796635i	$0.706611\pi$
$558$	0	0
$559$	−41.2603	−1.74513
$560$	0	0
$561$	−23.5612	−0.994753
$562$	0	0
$563$	36.3748	1.53301	0.766507	−	0.642236i	$-0.221993\pi$
$563$	36.3748	1.53301	0.766507	+	0.642236i	$0.221993\pi$
$564$	0	0
$565$	−13.1923	−0.555004
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	12.1848	0.510813	0.255406	−	0.966834i	$-0.417791\pi$
$569$	12.1848	0.510813	0.255406	+	0.966834i	$0.417791\pi$
$570$	0	0
$571$	31.8525	1.33298	0.666492	−	0.745512i	$-0.267795\pi$
$571$	31.8525	1.33298	0.666492	+	0.745512i	$0.267795\pi$
$572$	0	0
$573$	18.3484	0.766516
$574$	0	0
$575$	−51.1721	−2.13402
$576$	0	0
$577$	38.3898	1.59819	0.799094	−	0.601206i	$-0.205313\pi$
$577$	38.3898	1.59819	0.799094	+	0.601206i	$0.205313\pi$
$578$	0	0
$579$	5.92777	0.246350
$580$	0	0
$581$	16.7163	0.693511
$582$	0	0
$583$	−18.1576	−0.752009
$584$	0	0
$585$	22.8680	0.945474
$586$	0	0
$587$	9.65414	0.398469	0.199235	−	0.979952i	$-0.436154\pi$
$587$	9.65414	0.398469	0.199235	+	0.979952i	$0.436154\pi$
$588$	0	0
$589$	−10.6073	−0.437066
$590$	0	0
$591$	−10.9986	−0.452421
$592$	0	0
$593$	42.4388	1.74275	0.871376	−	0.490615i	$-0.163228\pi$
$593$	42.4388	1.74275	0.871376	+	0.490615i	$0.163228\pi$
$594$	0	0
$595$	14.6743	0.601586
$596$	0	0
$597$	−0.152091	−0.00622467
$598$	0	0
$599$	44.1246	1.80288	0.901440	−	0.432904i	$-0.142511\pi$
$599$	44.1246	1.80288	0.901440	+	0.432904i	$0.142511\pi$
$600$	0	0
$601$	−37.7673	−1.54056	−0.770281	−	0.637705i	$-0.779884\pi$
$601$	−37.7673	−1.54056	−0.770281	+	0.637705i	$0.779884\pi$
$602$	0	0
$603$	1.51419	0.0616628
$604$	0	0
$605$	−66.1422	−2.68906
$606$	0	0
$607$	−20.4351	−0.829433	−0.414717	−	0.909951i	$-0.636119\pi$
$607$	−20.4351	−0.829433	−0.414717	+	0.909951i	$0.636119\pi$
$608$	0	0
$609$	0.151622	0.00614403
$610$	0	0
$611$	−0.507293	−0.0205229
$612$	0	0
$613$	−11.3715	−0.459291	−0.229645	−	0.973274i	$-0.573757\pi$
$613$	−11.3715	−0.459291	−0.229645	+	0.973274i	$0.573757\pi$
$614$	0	0
$615$	−4.14589	−0.167179
$616$	0	0
$617$	10.2191	0.411404	0.205702	−	0.978615i	$-0.434052\pi$
$617$	10.2191	0.411404	0.205702	+	0.978615i	$0.434052\pi$
$618$	0	0
$619$	−27.1119	−1.08972	−0.544859	−	0.838528i	$-0.683417\pi$
$619$	−27.1119	−1.08972	−0.544859	+	0.838528i	$0.683417\pi$
$620$	0	0
$621$	−7.57963	−0.304160
$622$	0	0
$623$	4.36868	0.175028
$624$	0	0
$625$	−13.1768	−0.527071
$626$	0	0
$627$	−5.50406	−0.219811
$628$	0	0
$629$	−24.4372	−0.974375
$630$	0	0
$631$	45.2847	1.80276	0.901379	−	0.433032i	$-0.142556\pi$
$631$	45.2847	1.80276	0.901379	+	0.433032i	$0.142556\pi$
$632$	0	0
$633$	3.62978	0.144271
$634$	0	0
$635$	−30.0995	−1.19446
$636$	0	0
$637$	−6.67091	−0.264311
$638$	0	0
$639$	−3.65568	−0.144616
$640$	0	0
$641$	33.4119	1.31969	0.659846	−	0.751401i	$-0.270621\pi$
$641$	33.4119	1.31969	0.659846	+	0.751401i	$0.270621\pi$
$642$	0	0
$643$	−34.3192	−1.35342	−0.676708	−	0.736251i	$-0.736594\pi$
$643$	−34.3192	−1.35342	−0.676708	+	0.736251i	$0.736594\pi$
$644$	0	0
$645$	−21.2026	−0.834852
$646$	0	0
$647$	−39.3008	−1.54507	−0.772537	−	0.634970i	$-0.781013\pi$
$647$	−39.3008	−1.54507	−0.772537	+	0.634970i	$0.781013\pi$
$648$	0	0
$649$	−11.4241	−0.448435
$650$	0	0
$651$	10.6073	0.415732
$652$	0	0
$653$	26.6132	1.04145	0.520727	−	0.853723i	$-0.325661\pi$
$653$	26.6132	1.04145	0.520727	+	0.853723i	$0.325661\pi$
$654$	0	0
$655$	−21.4955	−0.839900
$656$	0	0
$657$	3.07174	0.119840
$658$	0	0
$659$	−33.0645	−1.28801	−0.644005	−	0.765022i	$-0.722728\pi$
$659$	−33.0645	−1.28801	−0.644005	+	0.765022i	$0.722728\pi$
$660$	0	0
$661$	−14.3277	−0.557281	−0.278641	−	0.960395i	$-0.589884\pi$
$661$	−14.3277	−0.557281	−0.278641	+	0.960395i	$0.589884\pi$
$662$	0	0
$663$	−28.5561	−1.10903
$664$	0	0
$665$	3.42801	0.132933
$666$	0	0
$667$	1.14924	0.0444987
$668$	0	0
$669$	18.8654	0.729379
$670$	0	0
$671$	−44.0509	−1.70056
$672$	0	0
$673$	−26.7960	−1.03291	−0.516455	−	0.856314i	$-0.672749\pi$
$673$	−26.7960	−1.03291	−0.516455	+	0.856314i	$0.672749\pi$
$674$	0	0
$675$	6.75126	0.259856
$676$	0	0
$677$	18.7491	0.720586	0.360293	−	0.932839i	$-0.382677\pi$
$677$	18.7491	0.720586	0.360293	+	0.932839i	$0.382677\pi$
$678$	0	0
$679$	−7.34183	−0.281754
$680$	0	0
$681$	−11.5695	−0.443344
$682$	0	0
$683$	20.5744	0.787257	0.393629	−	0.919270i	$-0.371220\pi$
$683$	20.5744	0.787257	0.393629	+	0.919270i	$0.371220\pi$
$684$	0	0
$685$	25.4597	0.972763
$686$	0	0
$687$	−20.3585	−0.776727
$688$	0	0
$689$	−22.0070	−0.838398
$690$	0	0
$691$	30.2260	1.14985	0.574926	−	0.818205i	$-0.305031\pi$
$691$	30.2260	1.14985	0.574926	+	0.818205i	$0.305031\pi$
$692$	0	0
$693$	5.50406	0.209082
$694$	0	0
$695$	58.0859	2.20332
$696$	0	0
$697$	5.17714	0.196098
$698$	0	0
$699$	−16.2734	−0.615518
$700$	0	0
$701$	7.49605	0.283122	0.141561	−	0.989930i	$-0.454788\pi$
$701$	7.49605	0.283122	0.141561	+	0.989930i	$0.454788\pi$
$702$	0	0
$703$	−5.70870	−0.215308
$704$	0	0
$705$	−0.260685	−0.00981796
$706$	0	0
$707$	−0.711196	−0.0267473
$708$	0	0
$709$	−4.73910	−0.177981	−0.0889903	−	0.996032i	$-0.528364\pi$
$709$	−4.73910	−0.177981	−0.0889903	+	0.996032i	$0.528364\pi$
$710$	0	0
$711$	−2.99523	−0.112330
$712$	0	0
$713$	80.3993	3.01098
$714$	0	0
$715$	−125.867	−4.70714
$716$	0	0
$717$	−1.69522	−0.0633092
$718$	0	0
$719$	28.1519	1.04989	0.524944	−	0.851137i	$-0.324086\pi$
$719$	28.1519	1.04989	0.524944	+	0.851137i	$0.324086\pi$
$720$	0	0
$721$	−11.0081	−0.409964
$722$	0	0
$723$	1.28653	0.0478465
$724$	0	0
$725$	−1.02364	−0.0380170
$726$	0	0
$727$	40.1677	1.48974	0.744868	−	0.667212i	$-0.232512\pi$
$727$	40.1677	1.48974	0.744868	+	0.667212i	$0.232512\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	26.4765	0.979270
$732$	0	0
$733$	−21.6215	−0.798607	−0.399303	−	0.916819i	$-0.630748\pi$
$733$	−21.6215	−0.798607	−0.399303	+	0.916819i	$0.630748\pi$
$734$	0	0
$735$	−3.42801	−0.126444
$736$	0	0
$737$	−8.33421	−0.306995
$738$	0	0
$739$	−12.2998	−0.452454	−0.226227	−	0.974075i	$-0.572639\pi$
$739$	−12.2998	−0.452454	−0.226227	+	0.974075i	$0.572639\pi$
$740$	0	0
$741$	−6.67091	−0.245062
$742$	0	0
$743$	−15.6895	−0.575592	−0.287796	−	0.957692i	$-0.592922\pi$
$743$	−15.6895	−0.575592	−0.287796	+	0.957692i	$0.592922\pi$
$744$	0	0
$745$	10.0496	0.368187
$746$	0	0
$747$	−16.7163	−0.611619
$748$	0	0
$749$	17.8502	0.652232
$750$	0	0
$751$	−13.9552	−0.509231	−0.254616	−	0.967042i	$-0.581949\pi$
$751$	−13.9552	−0.509231	−0.254616	+	0.967042i	$0.581949\pi$
$752$	0	0
$753$	−22.0913	−0.805053
$754$	0	0
$755$	−47.4822	−1.72806
$756$	0	0
$757$	−21.6076	−0.785343	−0.392671	−	0.919679i	$-0.628449\pi$
$757$	−21.6076	−0.785343	−0.392671	+	0.919679i	$0.628449\pi$
$758$	0	0
$759$	41.7187	1.51429
$760$	0	0
$761$	2.13288	0.0773169	0.0386584	−	0.999252i	$-0.487692\pi$
$761$	2.13288	0.0773169	0.0386584	+	0.999252i	$0.487692\pi$
$762$	0	0
$763$	−7.27485	−0.263367
$764$	0	0
$765$	−14.6743	−0.530549
$766$	0	0
$767$	−13.8460	−0.499950
$768$	0	0
$769$	9.64267	0.347723	0.173862	−	0.984770i	$-0.444375\pi$
$769$	9.64267	0.347723	0.173862	+	0.984770i	$0.444375\pi$
$770$	0	0
$771$	−19.0736	−0.686917
$772$	0	0
$773$	5.14819	0.185168	0.0925838	−	0.995705i	$-0.470487\pi$
$773$	5.14819	0.185168	0.0925838	+	0.995705i	$0.470487\pi$
$774$	0	0
$775$	−71.6125	−2.57240
$776$	0	0
$777$	5.70870	0.204799
$778$	0	0
$779$	1.20942	0.0433318
$780$	0	0
$781$	20.1211	0.719988
$782$	0	0
$783$	−0.151622	−0.00541852
$784$	0	0
$785$	−26.6846	−0.952414
$786$	0	0
$787$	27.5601	0.982411	0.491206	−	0.871044i	$-0.336556\pi$
$787$	27.5601	0.982411	0.491206	+	0.871044i	$0.336556\pi$
$788$	0	0
$789$	15.6466	0.557034
$790$	0	0
$791$	−3.84838	−0.136833
$792$	0	0
$793$	−53.3896	−1.89592
$794$	0	0
$795$	−11.3088	−0.401082
$796$	0	0
$797$	−9.60150	−0.340103	−0.170051	−	0.985435i	$-0.554393\pi$
$797$	−9.60150	−0.340103	−0.170051	+	0.985435i	$0.554393\pi$
$798$	0	0
$799$	0.325527	0.0115163
$800$	0	0
$801$	−4.36868	−0.154360
$802$	0	0
$803$	−16.9071	−0.596637
$804$	0	0
$805$	−25.9831	−0.915783
$806$	0	0
$807$	26.1477	0.920443
$808$	0	0
$809$	−14.8458	−0.521951	−0.260976	−	0.965345i	$-0.584044\pi$
$809$	−14.8458	−0.521951	−0.260976	+	0.965345i	$0.584044\pi$
$810$	0	0
$811$	33.7157	1.18392	0.591959	−	0.805968i	$-0.298355\pi$
$811$	33.7157	1.18392	0.591959	+	0.805968i	$0.298355\pi$
$812$	0	0
$813$	10.9531	0.384142
$814$	0	0
$815$	22.0875	0.773692
$816$	0	0
$817$	6.18511	0.216390
$818$	0	0
$819$	6.67091	0.233101
$820$	0	0
$821$	−24.5318	−0.856167	−0.428084	−	0.903739i	$-0.640811\pi$
$821$	−24.5318	−0.856167	−0.428084	+	0.903739i	$0.640811\pi$
$822$	0	0
$823$	26.6269	0.928154	0.464077	−	0.885795i	$-0.346386\pi$
$823$	26.6269	0.928154	0.464077	+	0.885795i	$0.346386\pi$
$824$	0	0
$825$	−37.1593	−1.29372
$826$	0	0
$827$	17.6507	0.613775	0.306887	−	0.951746i	$-0.400713\pi$
$827$	17.6507	0.613775	0.306887	+	0.951746i	$0.400713\pi$
$828$	0	0
$829$	11.6699	0.405312	0.202656	−	0.979250i	$-0.435043\pi$
$829$	11.6699	0.405312	0.202656	+	0.979250i	$0.435043\pi$
$830$	0	0
$831$	19.4633	0.675175
$832$	0	0
$833$	4.28069	0.148317
$834$	0	0
$835$	−22.2334	−0.769419
$836$	0	0
$837$	−10.6073	−0.366641
$838$	0	0
$839$	11.1135	0.383681	0.191840	−	0.981426i	$-0.438554\pi$
$839$	11.1135	0.383681	0.191840	+	0.981426i	$0.438554\pi$
$840$	0	0
$841$	−28.9770	−0.999207
$842$	0	0
$843$	18.4131	0.634181
$844$	0	0
$845$	−107.986	−3.71483
$846$	0	0
$847$	−19.2946	−0.662972
$848$	0	0
$849$	7.47769	0.256634
$850$	0	0
$851$	43.2699	1.48327
$852$	0	0
$853$	−3.62024	−0.123955	−0.0619774	−	0.998078i	$-0.519741\pi$
$853$	−3.62024	−0.123955	−0.0619774	+	0.998078i	$0.519741\pi$
$854$	0	0
$855$	−3.42801	−0.117236
$856$	0	0
$857$	27.2678	0.931450	0.465725	−	0.884930i	$-0.345794\pi$
$857$	27.2678	0.931450	0.465725	+	0.884930i	$0.345794\pi$
$858$	0	0
$859$	−0.886493	−0.0302468	−0.0151234	−	0.999886i	$-0.504814\pi$
$859$	−0.886493	−0.0302468	−0.0151234	+	0.999886i	$0.504814\pi$
$860$	0	0
$861$	−1.20942	−0.0412168
$862$	0	0
$863$	−25.5930	−0.871195	−0.435598	−	0.900141i	$-0.643463\pi$
$863$	−25.5930	−0.871195	−0.435598	+	0.900141i	$0.643463\pi$
$864$	0	0
$865$	−26.9931	−0.917793
$866$	0	0
$867$	1.32432	0.0449761
$868$	0	0
$869$	16.4859	0.559246
$870$	0	0
$871$	−10.1011	−0.342261
$872$	0	0
$873$	7.34183	0.248483
$874$	0	0
$875$	6.00334	0.202950
$876$	0	0
$877$	−7.32971	−0.247507	−0.123753	−	0.992313i	$-0.539493\pi$
$877$	−7.32971	−0.247507	−0.123753	+	0.992313i	$0.539493\pi$
$878$	0	0
$879$	−9.12430	−0.307755
$880$	0	0
$881$	−5.31242	−0.178980	−0.0894900	−	0.995988i	$-0.528524\pi$
$881$	−5.31242	−0.178980	−0.0894900	+	0.995988i	$0.528524\pi$
$882$	0	0
$883$	−48.5929	−1.63528	−0.817641	−	0.575729i	$-0.804718\pi$
$883$	−48.5929	−1.63528	−0.817641	+	0.575729i	$0.804718\pi$
$884$	0	0
$885$	−7.11510	−0.239171
$886$	0	0
$887$	−39.3150	−1.32007	−0.660034	−	0.751236i	$-0.729458\pi$
$887$	−39.3150	−1.32007	−0.660034	+	0.751236i	$0.729458\pi$
$888$	0	0
$889$	−8.78045	−0.294487
$890$	0	0
$891$	−5.50406	−0.184393
$892$	0	0
$893$	0.0760455	0.00254477
$894$	0	0
$895$	70.9811	2.37264
$896$	0	0
$897$	50.5631	1.68825
$898$	0	0
$899$	1.60830	0.0536397
$900$	0	0
$901$	14.1218	0.470464
$902$	0	0
$903$	−6.18511	−0.205828
$904$	0	0
$905$	−16.1251	−0.536017
$906$	0	0
$907$	−12.7171	−0.422264	−0.211132	−	0.977458i	$-0.567715\pi$
$907$	−12.7171	−0.422264	−0.211132	+	0.977458i	$0.567715\pi$
$908$	0	0
$909$	0.711196	0.0235889
$910$	0	0
$911$	−4.25642	−0.141021	−0.0705107	−	0.997511i	$-0.522463\pi$
$911$	−4.25642	−0.141021	−0.0705107	+	0.997511i	$0.522463\pi$
$912$	0	0
$913$	92.0077	3.04501
$914$	0	0
$915$	−27.4355	−0.906991
$916$	0	0
$917$	−6.27055	−0.207072
$918$	0	0
$919$	23.9594	0.790349	0.395175	−	0.918606i	$-0.370684\pi$
$919$	23.9594	0.790349	0.395175	+	0.918606i	$0.370684\pi$
$920$	0	0
$921$	19.5992	0.645815
$922$	0	0
$923$	24.3867	0.802699
$924$	0	0
$925$	−38.5409	−1.26722
$926$	0	0
$927$	11.0081	0.361554
$928$	0	0
$929$	−17.8836	−0.586743	−0.293371	−	0.955999i	$-0.594777\pi$
$929$	−17.8836	−0.586743	−0.293371	+	0.955999i	$0.594777\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	5.86126	0.191889
$934$	0	0
$935$	80.7679	2.64139
$936$	0	0
$937$	−14.4620	−0.472454	−0.236227	−	0.971698i	$-0.575911\pi$
$937$	−14.4620	−0.472454	−0.236227	+	0.971698i	$0.575911\pi$
$938$	0	0
$939$	29.9517	0.977435
$940$	0	0
$941$	15.5492	0.506888	0.253444	−	0.967350i	$-0.418437\pi$
$941$	15.5492	0.506888	0.253444	+	0.967350i	$0.418437\pi$
$942$	0	0
$943$	−9.16693	−0.298516
$944$	0	0
$945$	3.42801	0.111513
$946$	0	0
$947$	43.1107	1.40091	0.700455	−	0.713697i	$-0.252980\pi$
$947$	43.1107	1.40091	0.700455	+	0.713697i	$0.252980\pi$
$948$	0	0
$949$	−20.4913	−0.665177
$950$	0	0
$951$	−11.1597	−0.361879
$952$	0	0
$953$	−25.3112	−0.819909	−0.409954	−	0.912106i	$-0.634455\pi$
$953$	−25.3112	−0.819909	−0.409954	+	0.912106i	$0.634455\pi$
$954$	0	0
$955$	−62.8985	−2.03535
$956$	0	0
$957$	0.834535	0.0269767
$958$	0	0
$959$	7.42695	0.239829
$960$	0	0
$961$	81.5145	2.62950
$962$	0	0
$963$	−17.8502	−0.575214
$964$	0	0
$965$	−20.3204	−0.654138
$966$	0	0
$967$	−14.1820	−0.456064	−0.228032	−	0.973654i	$-0.573229\pi$
$967$	−14.1820	−0.456064	−0.228032	+	0.973654i	$0.573229\pi$
$968$	0	0
$969$	4.28069	0.137516
$970$	0	0
$971$	28.5011	0.914644	0.457322	−	0.889301i	$-0.348809\pi$
$971$	28.5011	0.914644	0.457322	+	0.889301i	$0.348809\pi$
$972$	0	0
$973$	16.9445	0.543215
$974$	0	0
$975$	−45.0371	−1.44234
$976$	0	0
$977$	−52.1509	−1.66846	−0.834228	−	0.551420i	$-0.814086\pi$
$977$	−52.1509	−1.66846	−0.834228	+	0.551420i	$0.814086\pi$
$978$	0	0
$979$	24.0455	0.768497
$980$	0	0
$981$	7.27485	0.232268
$982$	0	0
$983$	−14.4427	−0.460651	−0.230326	−	0.973114i	$-0.573979\pi$
$983$	−14.4427	−0.460651	−0.230326	+	0.973114i	$0.573979\pi$
$984$	0	0
$985$	37.7032	1.20132
$986$	0	0
$987$	−0.0760455	−0.00242055
$988$	0	0
$989$	−46.8809	−1.49072
$990$	0	0
$991$	−46.6036	−1.48041	−0.740207	−	0.672380i	$-0.765272\pi$
$991$	−46.6036	−1.48041	−0.740207	+	0.672380i	$0.765272\pi$
$992$	0	0
$993$	3.03524	0.0963205
$994$	0	0
$995$	0.521370	0.0165285
$996$	0	0
$997$	−23.2747	−0.737117	−0.368558	−	0.929605i	$-0.620149\pi$
$997$	−23.2747	−0.737117	−0.368558	+	0.929605i	$0.620149\pi$
$998$	0	0
$999$	−5.70870	−0.180615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cf.1.1		5
4.3	odd	2		399.2.a.g.1.2	✓	5
12.11	even	2		1197.2.a.o.1.4		5
20.19	odd	2		9975.2.a.bp.1.4		5
28.27	even	2		2793.2.a.bg.1.2		5
76.75	even	2		7581.2.a.w.1.4		5
84.83	odd	2		8379.2.a.cb.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.g.1.2	✓	5	4.3	odd	2
1197.2.a.o.1.4		5	12.11	even	2
2793.2.a.bg.1.2		5	28.27	even	2
6384.2.a.cf.1.1		5	1.1	even	1	trivial
7581.2.a.w.1.4		5	76.75	even	2
8379.2.a.cb.1.4		5	84.83	odd	2
9975.2.a.bp.1.4		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.42801 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.42801 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.50406 q^{11} -6.67091 q^{13} -3.42801 q^{15} +4.28069 q^{17} +1.00000 q^{19} -1.00000 q^{21} -7.57963 q^{23} +6.75126 q^{25} +1.00000 q^{27} -0.151622 q^{29} -10.6073 q^{31} -5.50406 q^{33} +3.42801 q^{35} -5.70870 q^{37} -6.67091 q^{39} +1.20942 q^{41} +6.18511 q^{43} -3.42801 q^{45} +0.0760455 q^{47} +1.00000 q^{49} +4.28069 q^{51} +3.29894 q^{53} +18.8680 q^{55} +1.00000 q^{57} +2.07558 q^{59} +8.00334 q^{61} -1.00000 q^{63} +22.8680 q^{65} +1.51419 q^{67} -7.57963 q^{69} -3.65568 q^{71} +3.07174 q^{73} +6.75126 q^{75} +5.50406 q^{77} -2.99523 q^{79} +1.00000 q^{81} -16.7163 q^{83} -14.6743 q^{85} -0.151622 q^{87} -4.36868 q^{89} +6.67091 q^{91} -10.6073 q^{93} -3.42801 q^{95} +7.34183 q^{97} -5.50406 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} - 5 q^{21} - 12 q^{23} + 15 q^{25} + 5 q^{27} + 4 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{39} + 10 q^{41} + 16 q^{43} + 4 q^{45} + 2 q^{47} + 5 q^{49} + 12 q^{51} + 12 q^{55} + 5 q^{57} + 4 q^{59} - 14 q^{61} - 5 q^{63} + 32 q^{65} + 20 q^{67} - 12 q^{69} + 6 q^{71} + 10 q^{73} + 15 q^{75} + 8 q^{77} + 5 q^{81} - 6 q^{83} + 8 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - 18 q^{97} - 8 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 - 8 * q^11 - 6 * q^13 + 4 * q^15 + 12 * q^17 + 5 * q^19 - 5 * q^21 - 12 * q^23 + 15 * q^25 + 5 * q^27 + 4 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^35 + 2 * q^37 - 6 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 + 2 * q^47 + 5 * q^49 + 12 * q^51 + 12 * q^55 + 5 * q^57 + 4 * q^59 - 14 * q^61 - 5 * q^63 + 32 * q^65 + 20 * q^67 - 12 * q^69 + 6 * q^71 + 10 * q^73 + 15 * q^75 + 8 * q^77 + 5 * q^81 - 6 * q^83 + 8 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - 18 * q^97 - 8 * q^99

Introduction

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