LMFDB - Embedded newform 6384.2.a.cd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$50.9764966504$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.401584.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 12x - 4 $ x^5 - 2x^4 - 7x^3 + 8x^2 + 12x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.82751$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.60789 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.60789 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.65501 q^{11} +4.54633 q^{13} -1.60789 q^{15} -1.17878 q^{17} -1.00000 q^{19} +1.00000 q^{21} -8.44169 q^{23} -2.41467 q^{25} -1.00000 q^{27} +3.61800 q^{29} -1.84379 q^{31} -1.65501 q^{33} -1.60789 q^{35} -4.30599 q^{37} -4.54633 q^{39} -10.8834 q^{41} +7.77223 q^{43} +1.60789 q^{45} -11.1398 q^{47} +1.00000 q^{49} +1.17878 q^{51} +3.61800 q^{53} +2.66109 q^{55} +1.00000 q^{57} -10.4518 q^{59} -2.00000 q^{61} -1.00000 q^{63} +7.31003 q^{65} -8.01258 q^{67} +8.44169 q^{69} +3.56890 q^{71} -1.21579 q^{73} +2.41467 q^{75} -1.65501 q^{77} -8.44169 q^{79} +1.00000 q^{81} +7.19136 q^{83} -1.89536 q^{85} -3.61800 q^{87} +5.79071 q^{89} -4.54633 q^{91} +1.84379 q^{93} -1.60789 q^{95} +3.17433 q^{97} +1.65501 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - 2 q^{15} + 6 q^{17} - 5 q^{19} + 5 q^{21} - 10 q^{23} + 7 q^{25} - 5 q^{27} + 4 q^{29} + 4 q^{31} + 6 q^{33} - 2 q^{35} + 6 q^{37} + 10 q^{41} - 4 q^{43} + 2 q^{45} - 2 q^{47} + 5 q^{49} - 6 q^{51} + 4 q^{53} - 8 q^{55} + 5 q^{57} - 12 q^{59} - 10 q^{61} - 5 q^{63} + 8 q^{65} - 2 q^{67} + 10 q^{69} - 30 q^{71} + 6 q^{73} - 7 q^{75} + 6 q^{77} - 10 q^{79} + 5 q^{81} - 14 q^{83} - 4 q^{87} + 10 q^{89} - 4 q^{93} - 2 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 6 * q^11 - 2 * q^15 + 6 * q^17 - 5 * q^19 + 5 * q^21 - 10 * q^23 + 7 * q^25 - 5 * q^27 + 4 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 6 * q^37 + 10 * q^41 - 4 * q^43 + 2 * q^45 - 2 * q^47 + 5 * q^49 - 6 * q^51 + 4 * q^53 - 8 * q^55 + 5 * q^57 - 12 * q^59 - 10 * q^61 - 5 * q^63 + 8 * q^65 - 2 * q^67 + 10 * q^69 - 30 * q^71 + 6 * q^73 - 7 * q^75 + 6 * q^77 - 10 * q^79 + 5 * q^81 - 14 * q^83 - 4 * q^87 + 10 * q^89 - 4 * q^93 - 2 * q^95 - 8 * q^97 - 6 * 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
$5$	1.60789	0.719073	0.359536	−	0.933131i	$-0.382935\pi$
$5$	1.60789	0.719073	0.359536	+	0.933131i	$0.382935\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.65501	0.499006	0.249503	−	0.968374i	$-0.419733\pi$
$11$	1.65501	0.499006	0.249503	+	0.968374i	$0.419733\pi$
$12$	0	0
$13$	4.54633	1.26093	0.630463	−	0.776219i	$-0.282865\pi$
$13$	4.54633	1.26093	0.630463	+	0.776219i	$0.282865\pi$
$14$	0	0
$15$	−1.60789	−0.415157
$16$	0	0
$17$	−1.17878	−0.285896	−0.142948	−	0.989730i	$-0.545658\pi$
$17$	−1.17878	−0.285896	−0.142948	+	0.989730i	$0.545658\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−8.44169	−1.76021	−0.880107	−	0.474775i	$-0.842529\pi$
$23$	−8.44169	−1.76021	−0.880107	+	0.474775i	$0.842529\pi$
$24$	0	0
$25$	−2.41467	−0.482935
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.61800	0.671847	0.335923	−	0.941889i	$-0.390952\pi$
$29$	3.61800	0.671847	0.335923	+	0.941889i	$0.390952\pi$
$30$	0	0
$31$	−1.84379	−0.331154	−0.165577	−	0.986197i	$-0.552949\pi$
$31$	−1.84379	−0.331154	−0.165577	+	0.986197i	$0.552949\pi$
$32$	0	0
$33$	−1.65501	−0.288101
$34$	0	0
$35$	−1.60789	−0.271784
$36$	0	0
$37$	−4.30599	−0.707901	−0.353951	−	0.935264i	$-0.615162\pi$
$37$	−4.30599	−0.707901	−0.353951	+	0.935264i	$0.615162\pi$
$38$	0	0
$39$	−4.54633	−0.727996
$40$	0	0
$41$	−10.8834	−1.69970	−0.849849	−	0.527026i	$-0.823307\pi$
$41$	−10.8834	−1.69970	−0.849849	+	0.527026i	$0.823307\pi$
$42$	0	0
$43$	7.77223	1.18525	0.592627	−	0.805477i	$-0.298091\pi$
$43$	7.77223	1.18525	0.592627	+	0.805477i	$0.298091\pi$
$44$	0	0
$45$	1.60789	0.239691
$46$	0	0
$47$	−11.1398	−1.62491	−0.812453	−	0.583027i	$-0.801868\pi$
$47$	−11.1398	−1.62491	−0.812453	+	0.583027i	$0.801868\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.17878	0.165062
$52$	0	0
$53$	3.61800	0.496971	0.248486	−	0.968636i	$-0.420067\pi$
$53$	3.61800	0.496971	0.248486	+	0.968636i	$0.420067\pi$
$54$	0	0
$55$	2.66109	0.358821
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−10.4518	−1.36071	−0.680354	−	0.732883i	$-0.738174\pi$
$59$	−10.4518	−1.36071	−0.680354	+	0.732883i	$0.738174\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	7.31003	0.906697
$66$	0	0
$67$	−8.01258	−0.978892	−0.489446	−	0.872034i	$-0.662801\pi$
$67$	−8.01258	−0.978892	−0.489446	+	0.872034i	$0.662801\pi$
$68$	0	0
$69$	8.44169	1.01626
$70$	0	0
$71$	3.56890	0.423551	0.211775	−	0.977318i	$-0.432075\pi$
$71$	3.56890	0.423551	0.211775	+	0.977318i	$0.432075\pi$
$72$	0	0
$73$	−1.21579	−0.142297	−0.0711487	−	0.997466i	$-0.522666\pi$
$73$	−1.21579	−0.142297	−0.0711487	+	0.997466i	$0.522666\pi$
$74$	0	0
$75$	2.41467	0.278822
$76$	0	0
$77$	−1.65501	−0.188606
$78$	0	0
$79$	−8.44169	−0.949764	−0.474882	−	0.880049i	$-0.657509\pi$
$79$	−8.44169	−0.949764	−0.474882	+	0.880049i	$0.657509\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.19136	0.789354	0.394677	−	0.918820i	$-0.370857\pi$
$83$	7.19136	0.789354	0.394677	+	0.918820i	$0.370857\pi$
$84$	0	0
$85$	−1.89536	−0.205580
$86$	0	0
$87$	−3.61800	−0.387891
$88$	0	0
$89$	5.79071	0.613814	0.306907	−	0.951739i	$-0.400706\pi$
$89$	5.79071	0.613814	0.306907	+	0.951739i	$0.400706\pi$
$90$	0	0
$91$	−4.54633	−0.476585
$92$	0	0
$93$	1.84379	0.191192
$94$	0	0
$95$	−1.60789	−0.164967
$96$	0	0
$97$	3.17433	0.322305	0.161152	−	0.986930i	$-0.448479\pi$
$97$	3.17433	0.322305	0.161152	+	0.986930i	$0.448479\pi$
$98$	0	0
$99$	1.65501	0.166335
$100$	0	0
$101$	16.7457	1.66626	0.833130	−	0.553078i	$-0.186547\pi$
$101$	16.7457	1.66626	0.833130	+	0.553078i	$0.186547\pi$
$102$	0	0
$103$	6.78668	0.668711	0.334355	−	0.942447i	$-0.391481\pi$
$103$	6.78668	0.668711	0.334355	+	0.942447i	$0.391481\pi$
$104$	0	0
$105$	1.60789	0.156914
$106$	0	0
$107$	4.14382	0.400599	0.200299	−	0.979735i	$-0.435809\pi$
$107$	4.14382	0.400599	0.200299	+	0.979735i	$0.435809\pi$
$108$	0	0
$109$	−9.36160	−0.896678	−0.448339	−	0.893864i	$-0.647984\pi$
$109$	−9.36160	−0.896678	−0.448339	+	0.893864i	$0.647984\pi$
$110$	0	0
$111$	4.30599	0.408707
$112$	0	0
$113$	−9.26537	−0.871613	−0.435807	−	0.900040i	$-0.643537\pi$
$113$	−9.26537	−0.871613	−0.435807	+	0.900040i	$0.643537\pi$
$114$	0	0
$115$	−13.5734	−1.26572
$116$	0	0
$117$	4.54633	0.420309
$118$	0	0
$119$	1.17878	0.108059
$120$	0	0
$121$	−8.26093	−0.750993
$122$	0	0
$123$	10.8834	0.981321
$124$	0	0
$125$	−11.9220	−1.06634
$126$	0	0
$127$	2.56481	0.227590	0.113795	−	0.993504i	$-0.463699\pi$
$127$	2.56481	0.227590	0.113795	+	0.993504i	$0.463699\pi$
$128$	0	0
$129$	−7.77223	−0.684307
$130$	0	0
$131$	0.833795	0.0728490	0.0364245	−	0.999336i	$-0.488403\pi$
$131$	0.833795	0.0728490	0.0364245	+	0.999336i	$0.488403\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−1.60789	−0.138386
$136$	0	0
$137$	12.4518	1.06383	0.531915	−	0.846798i	$-0.321473\pi$
$137$	12.4518	1.06383	0.531915	+	0.846798i	$0.321473\pi$
$138$	0	0
$139$	−11.9309	−1.01197	−0.505983	−	0.862543i	$-0.668870\pi$
$139$	−11.9309	−1.01197	−0.505983	+	0.862543i	$0.668870\pi$
$140$	0	0
$141$	11.1398	0.938139
$142$	0	0
$143$	7.52425	0.629209
$144$	0	0
$145$	5.81737	0.483106
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−8.88091	−0.727553	−0.363776	−	0.931486i	$-0.618513\pi$
$149$	−8.88091	−0.727553	−0.363776	+	0.931486i	$0.618513\pi$
$150$	0	0
$151$	2.79432	0.227399	0.113699	−	0.993515i	$-0.463730\pi$
$151$	2.79432	0.227399	0.113699	+	0.993515i	$0.463730\pi$
$152$	0	0
$153$	−1.17878	−0.0952987
$154$	0	0
$155$	−2.96462	−0.238124
$156$	0	0
$157$	10.2344	0.816797	0.408399	−	0.912804i	$-0.366087\pi$
$157$	10.2344	0.816797	0.408399	+	0.912804i	$0.366087\pi$
$158$	0	0
$159$	−3.61800	−0.286926
$160$	0	0
$161$	8.44169	0.665298
$162$	0	0
$163$	0.462206	0.0362027	0.0181014	−	0.999836i	$-0.494238\pi$
$163$	0.462206	0.0362027	0.0181014	+	0.999836i	$0.494238\pi$
$164$	0	0
$165$	−2.66109	−0.207166
$166$	0	0
$167$	−3.92598	−0.303802	−0.151901	−	0.988396i	$-0.548539\pi$
$167$	−3.92598	−0.303802	−0.151901	+	0.988396i	$0.548539\pi$
$168$	0	0
$169$	7.66916	0.589935
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−9.14177	−0.695036	−0.347518	−	0.937673i	$-0.612975\pi$
$173$	−9.14177	−0.695036	−0.347518	+	0.937673i	$0.612975\pi$
$174$	0	0
$175$	2.41467	0.182532
$176$	0	0
$177$	10.4518	0.785605
$178$	0	0
$179$	−15.6631	−1.17072	−0.585359	−	0.810774i	$-0.699046\pi$
$179$	−15.6631	−1.17072	−0.585359	+	0.810774i	$0.699046\pi$
$180$	0	0
$181$	2.27343	0.168983	0.0844914	−	0.996424i	$-0.473073\pi$
$181$	2.27343	0.168983	0.0844914	+	0.996424i	$0.473073\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−6.92359	−0.509032
$186$	0	0
$187$	−1.95090	−0.142664
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−19.0826	−1.38077	−0.690383	−	0.723444i	$-0.742558\pi$
$191$	−19.0826	−1.38077	−0.690383	+	0.723444i	$0.742558\pi$
$192$	0	0
$193$	−12.8343	−0.923832	−0.461916	−	0.886924i	$-0.652838\pi$
$193$	−12.8343	−0.923832	−0.461916	+	0.886924i	$0.652838\pi$
$194$	0	0
$195$	−7.31003	−0.523482
$196$	0	0
$197$	−8.47425	−0.603765	−0.301883	−	0.953345i	$-0.597615\pi$
$197$	−8.47425	−0.603765	−0.301883	+	0.953345i	$0.597615\pi$
$198$	0	0
$199$	17.1918	1.21870	0.609349	−	0.792902i	$-0.291431\pi$
$199$	17.1918	1.21870	0.609349	+	0.792902i	$0.291431\pi$
$200$	0	0
$201$	8.01258	0.565163
$202$	0	0
$203$	−3.61800	−0.253934
$204$	0	0
$205$	−17.4993	−1.22221
$206$	0	0
$207$	−8.44169	−0.586738
$208$	0	0
$209$	−1.65501	−0.114480
$210$	0	0
$211$	12.6986	0.874206	0.437103	−	0.899411i	$-0.356004\pi$
$211$	12.6986	0.874206	0.437103	+	0.899411i	$0.356004\pi$
$212$	0	0
$213$	−3.56890	−0.244537
$214$	0	0
$215$	12.4969	0.852284
$216$	0	0
$217$	1.84379	0.125165
$218$	0	0
$219$	1.21579	0.0821555
$220$	0	0
$221$	−5.35913	−0.360494
$222$	0	0
$223$	−20.5098	−1.37344	−0.686719	−	0.726923i	$-0.740950\pi$
$223$	−20.5098	−1.37344	−0.686719	+	0.726923i	$0.740950\pi$
$224$	0	0
$225$	−2.41467	−0.160978
$226$	0	0
$227$	9.18691	0.609757	0.304878	−	0.952391i	$-0.401384\pi$
$227$	9.18691	0.609757	0.304878	+	0.952391i	$0.401384\pi$
$228$	0	0
$229$	−10.3035	−0.680876	−0.340438	−	0.940267i	$-0.610575\pi$
$229$	−10.3035	−0.680876	−0.340438	+	0.940267i	$0.610575\pi$
$230$	0	0
$231$	1.65501	0.108892
$232$	0	0
$233$	7.39620	0.484541	0.242271	−	0.970209i	$-0.422108\pi$
$233$	7.39620	0.484541	0.242271	+	0.970209i	$0.422108\pi$
$234$	0	0
$235$	−17.9116	−1.16842
$236$	0	0
$237$	8.44169	0.548347
$238$	0	0
$239$	0.179193	0.0115910	0.00579552	−	0.999983i	$-0.498155\pi$
$239$	0.179193	0.0115910	0.00579552	+	0.999983i	$0.498155\pi$
$240$	0	0
$241$	29.4218	1.89522	0.947612	−	0.319425i	$-0.103490\pi$
$241$	29.4218	1.89522	0.947612	+	0.319425i	$0.103490\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.60789	0.102725
$246$	0	0
$247$	−4.54633	−0.289276
$248$	0	0
$249$	−7.19136	−0.455734
$250$	0	0
$251$	−26.0947	−1.64708	−0.823542	−	0.567255i	$-0.808005\pi$
$251$	−26.0947	−1.64708	−0.823542	+	0.567255i	$0.808005\pi$
$252$	0	0
$253$	−13.9711	−0.878357
$254$	0	0
$255$	1.89536	0.118692
$256$	0	0
$257$	−27.9020	−1.74048	−0.870240	−	0.492627i	$-0.836037\pi$
$257$	−27.9020	−1.74048	−0.870240	+	0.492627i	$0.836037\pi$
$258$	0	0
$259$	4.30599	0.267561
$260$	0	0
$261$	3.61800	0.223949
$262$	0	0
$263$	19.7888	1.22023	0.610114	−	0.792313i	$-0.291123\pi$
$263$	19.7888	1.22023	0.610114	+	0.792313i	$0.291123\pi$
$264$	0	0
$265$	5.81737	0.357358
$266$	0	0
$267$	−5.79071	−0.354386
$268$	0	0
$269$	−4.08617	−0.249138	−0.124569	−	0.992211i	$-0.539755\pi$
$269$	−4.08617	−0.249138	−0.124569	+	0.992211i	$0.539755\pi$
$270$	0	0
$271$	9.07245	0.551112	0.275556	−	0.961285i	$-0.411138\pi$
$271$	9.07245	0.551112	0.275556	+	0.961285i	$0.411138\pi$
$272$	0	0
$273$	4.54633	0.275157
$274$	0	0
$275$	−3.99632	−0.240987
$276$	0	0
$277$	−4.81977	−0.289592	−0.144796	−	0.989462i	$-0.546253\pi$
$277$	−4.81977	−0.289592	−0.144796	+	0.989462i	$0.546253\pi$
$278$	0	0
$279$	−1.84379	−0.110385
$280$	0	0
$281$	−8.74763	−0.521840	−0.260920	−	0.965360i	$-0.584026\pi$
$281$	−8.74763	−0.521840	−0.260920	+	0.965360i	$0.584026\pi$
$282$	0	0
$283$	18.5510	1.10274	0.551370	−	0.834261i	$-0.314105\pi$
$283$	18.5510	1.10274	0.551370	+	0.834261i	$0.314105\pi$
$284$	0	0
$285$	1.60789	0.0952435
$286$	0	0
$287$	10.8834	0.642426
$288$	0	0
$289$	−15.6105	−0.918263
$290$	0	0
$291$	−3.17433	−0.186083
$292$	0	0
$293$	30.1983	1.76421	0.882103	−	0.471057i	$-0.156127\pi$
$293$	30.1983	1.76421	0.882103	+	0.471057i	$0.156127\pi$
$294$	0	0
$295$	−16.8054	−0.978448
$296$	0	0
$297$	−1.65501	−0.0960337
$298$	0	0
$299$	−38.3787	−2.21950
$300$	0	0
$301$	−7.77223	−0.447984
$302$	0	0
$303$	−16.7457	−0.962015
$304$	0	0
$305$	−3.21579	−0.184136
$306$	0	0
$307$	−10.5814	−0.603910	−0.301955	−	0.953322i	$-0.597639\pi$
$307$	−10.5814	−0.603910	−0.301955	+	0.953322i	$0.597639\pi$
$308$	0	0
$309$	−6.78668	−0.386080
$310$	0	0
$311$	−23.0045	−1.30447	−0.652233	−	0.758018i	$-0.726168\pi$
$311$	−23.0045	−1.30447	−0.652233	+	0.758018i	$0.726168\pi$
$312$	0	0
$313$	−32.5421	−1.83939	−0.919693	−	0.392637i	$-0.871563\pi$
$313$	−32.5421	−1.83939	−0.919693	+	0.392637i	$0.871563\pi$
$314$	0	0
$315$	−1.60789	−0.0905946
$316$	0	0
$317$	−4.77976	−0.268458	−0.134229	−	0.990950i	$-0.542856\pi$
$317$	−4.77976	−0.268458	−0.134229	+	0.990950i	$0.542856\pi$
$318$	0	0
$319$	5.98785	0.335255
$320$	0	0
$321$	−4.14382	−0.231286
$322$	0	0
$323$	1.17878	0.0655891
$324$	0	0
$325$	−10.9779	−0.608945
$326$	0	0
$327$	9.36160	0.517697
$328$	0	0
$329$	11.1398	0.614156
$330$	0	0
$331$	−12.2840	−0.675188	−0.337594	−	0.941292i	$-0.609613\pi$
$331$	−12.2840	−0.675188	−0.337594	+	0.941292i	$0.609613\pi$
$332$	0	0
$333$	−4.30599	−0.235967
$334$	0	0
$335$	−12.8834	−0.703894
$336$	0	0
$337$	21.4092	1.16623	0.583117	−	0.812388i	$-0.301833\pi$
$337$	21.4092	1.16623	0.583117	+	0.812388i	$0.301833\pi$
$338$	0	0
$339$	9.26537	0.503226
$340$	0	0
$341$	−3.05150	−0.165248
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	13.5734	0.730765
$346$	0	0
$347$	−3.63010	−0.194874	−0.0974369	−	0.995242i	$-0.531064\pi$
$347$	−3.63010	−0.194874	−0.0974369	+	0.995242i	$0.531064\pi$
$348$	0	0
$349$	−23.3843	−1.25173	−0.625866	−	0.779931i	$-0.715254\pi$
$349$	−23.3843	−1.25173	−0.625866	+	0.779931i	$0.715254\pi$
$350$	0	0
$351$	−4.54633	−0.242665
$352$	0	0
$353$	21.1008	1.12308	0.561541	−	0.827449i	$-0.310209\pi$
$353$	21.1008	1.12308	0.561541	+	0.827449i	$0.310209\pi$
$354$	0	0
$355$	5.73842	0.304564
$356$	0	0
$357$	−1.17878	−0.0623877
$358$	0	0
$359$	−35.1077	−1.85291	−0.926457	−	0.376401i	$-0.877162\pi$
$359$	−35.1077	−1.85291	−0.926457	+	0.376401i	$0.877162\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.26093	0.433586
$364$	0	0
$365$	−1.95486	−0.102322
$366$	0	0
$367$	0.554701	0.0289551	0.0144776	−	0.999895i	$-0.495391\pi$
$367$	0.554701	0.0289551	0.0144776	+	0.999895i	$0.495391\pi$
$368$	0	0
$369$	−10.8834	−0.566566
$370$	0	0
$371$	−3.61800	−0.187837
$372$	0	0
$373$	2.75779	0.142793	0.0713966	−	0.997448i	$-0.477254\pi$
$373$	2.75779	0.142793	0.0713966	+	0.997448i	$0.477254\pi$
$374$	0	0
$375$	11.9220	0.615650
$376$	0	0
$377$	16.4487	0.847149
$378$	0	0
$379$	33.4620	1.71883	0.859413	−	0.511282i	$-0.170829\pi$
$379$	33.4620	1.71883	0.859413	+	0.511282i	$0.170829\pi$
$380$	0	0
$381$	−2.56481	−0.131399
$382$	0	0
$383$	−0.406662	−0.0207795	−0.0103897	−	0.999946i	$-0.503307\pi$
$383$	−0.406662	−0.0207795	−0.0103897	+	0.999946i	$0.503307\pi$
$384$	0	0
$385$	−2.66109	−0.135622
$386$	0	0
$387$	7.77223	0.395085
$388$	0	0
$389$	14.6225	0.741391	0.370696	−	0.928754i	$-0.379119\pi$
$389$	14.6225	0.741391	0.370696	+	0.928754i	$0.379119\pi$
$390$	0	0
$391$	9.95090	0.503239
$392$	0	0
$393$	−0.833795	−0.0420594
$394$	0	0
$395$	−13.5734	−0.682949
$396$	0	0
$397$	−33.9220	−1.70250	−0.851249	−	0.524763i	$-0.824154\pi$
$397$	−33.9220	−1.70250	−0.851249	+	0.524763i	$0.824154\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	3.05198	0.152409	0.0762043	−	0.997092i	$-0.475720\pi$
$401$	3.05198	0.152409	0.0762043	+	0.997092i	$0.475720\pi$
$402$	0	0
$403$	−8.38248	−0.417561
$404$	0	0
$405$	1.60789	0.0798969
$406$	0	0
$407$	−7.12648	−0.353247
$408$	0	0
$409$	−35.3171	−1.74632	−0.873160	−	0.487434i	$-0.837933\pi$
$409$	−35.3171	−1.74632	−0.873160	+	0.487434i	$0.837933\pi$
$410$	0	0
$411$	−12.4518	−0.614202
$412$	0	0
$413$	10.4518	0.514299
$414$	0	0
$415$	11.5629	0.567603
$416$	0	0
$417$	11.9309	0.584259
$418$	0	0
$419$	8.89998	0.434792	0.217396	−	0.976083i	$-0.430244\pi$
$419$	8.89998	0.434792	0.217396	+	0.976083i	$0.430244\pi$
$420$	0	0
$421$	19.3125	0.941233	0.470617	−	0.882338i	$-0.344031\pi$
$421$	19.3125	0.941233	0.470617	+	0.882338i	$0.344031\pi$
$422$	0	0
$423$	−11.1398	−0.541635
$424$	0	0
$425$	2.84637	0.138069
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−7.52425	−0.363274
$430$	0	0
$431$	8.83873	0.425746	0.212873	−	0.977080i	$-0.431718\pi$
$431$	8.83873	0.425746	0.212873	+	0.977080i	$0.431718\pi$
$432$	0	0
$433$	−31.3106	−1.50469	−0.752345	−	0.658769i	$-0.771077\pi$
$433$	−31.3106	−1.50469	−0.752345	+	0.658769i	$0.771077\pi$
$434$	0	0
$435$	−5.81737	−0.278922
$436$	0	0
$437$	8.44169	0.403821
$438$	0	0
$439$	23.4171	1.11764	0.558820	−	0.829289i	$-0.311254\pi$
$439$	23.4171	1.11764	0.558820	+	0.829289i	$0.311254\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	20.4355	0.970920	0.485460	−	0.874259i	$-0.338652\pi$
$443$	20.4355	0.970920	0.485460	+	0.874259i	$0.338652\pi$
$444$	0	0
$445$	9.31085	0.441377
$446$	0	0
$447$	8.88091	0.420053
$448$	0	0
$449$	0.266944	0.0125979	0.00629894	−	0.999980i	$-0.497995\pi$
$449$	0.266944	0.0125979	0.00629894	+	0.999980i	$0.497995\pi$
$450$	0	0
$451$	−18.0121	−0.848159
$452$	0	0
$453$	−2.79432	−0.131289
$454$	0	0
$455$	−7.31003	−0.342699
$456$	0	0
$457$	30.9843	1.44938	0.724692	−	0.689073i	$-0.241982\pi$
$457$	30.9843	1.44938	0.724692	+	0.689073i	$0.241982\pi$
$458$	0	0
$459$	1.17878	0.0550208
$460$	0	0
$461$	−42.6847	−1.98802	−0.994012	−	0.109271i	$-0.965148\pi$
$461$	−42.6847	−1.98802	−0.994012	+	0.109271i	$0.965148\pi$
$462$	0	0
$463$	19.9018	0.924915	0.462457	−	0.886641i	$-0.346968\pi$
$463$	19.9018	0.924915	0.462457	+	0.886641i	$0.346968\pi$
$464$	0	0
$465$	2.96462	0.137481
$466$	0	0
$467$	3.99579	0.184903	0.0924514	−	0.995717i	$-0.470530\pi$
$467$	3.99579	0.184903	0.0924514	+	0.995717i	$0.470530\pi$
$468$	0	0
$469$	8.01258	0.369986
$470$	0	0
$471$	−10.2344	−0.471578
$472$	0	0
$473$	12.8632	0.591449
$474$	0	0
$475$	2.41467	0.110793
$476$	0	0
$477$	3.61800	0.165657
$478$	0	0
$479$	−1.88670	−0.0862054	−0.0431027	−	0.999071i	$-0.513724\pi$
$479$	−1.88670	−0.0862054	−0.0431027	+	0.999071i	$0.513724\pi$
$480$	0	0
$481$	−19.5765	−0.892611
$482$	0	0
$483$	−8.44169	−0.384110
$484$	0	0
$485$	5.10399	0.231760
$486$	0	0
$487$	14.4336	0.654050	0.327025	−	0.945016i	$-0.393954\pi$
$487$	14.4336	0.654050	0.327025	+	0.945016i	$0.393954\pi$
$488$	0	0
$489$	−0.462206	−0.0209017
$490$	0	0
$491$	−19.7253	−0.890190	−0.445095	−	0.895483i	$-0.646830\pi$
$491$	−19.7253	−0.890190	−0.445095	+	0.895483i	$0.646830\pi$
$492$	0	0
$493$	−4.26483	−0.192078
$494$	0	0
$495$	2.66109	0.119607
$496$	0	0
$497$	−3.56890	−0.160087
$498$	0	0
$499$	1.88577	0.0844189	0.0422094	−	0.999109i	$-0.486560\pi$
$499$	1.88577	0.0844189	0.0422094	+	0.999109i	$0.486560\pi$
$500$	0	0
$501$	3.92598	0.175400
$502$	0	0
$503$	−10.5599	−0.470844	−0.235422	−	0.971893i	$-0.575647\pi$
$503$	−10.5599	−0.470844	−0.235422	+	0.971893i	$0.575647\pi$
$504$	0	0
$505$	26.9253	1.19816
$506$	0	0
$507$	−7.66916	−0.340599
$508$	0	0
$509$	−16.3324	−0.723921	−0.361961	−	0.932193i	$-0.617893\pi$
$509$	−16.3324	−0.723921	−0.361961	+	0.932193i	$0.617893\pi$
$510$	0	0
$511$	1.21579	0.0537834
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	10.9123	0.480852
$516$	0	0
$517$	−18.4365	−0.810837
$518$	0	0
$519$	9.14177	0.401279
$520$	0	0
$521$	−16.7805	−0.735166	−0.367583	−	0.929991i	$-0.619815\pi$
$521$	−16.7805	−0.735166	−0.367583	+	0.929991i	$0.619815\pi$
$522$	0	0
$523$	15.7520	0.688788	0.344394	−	0.938825i	$-0.388084\pi$
$523$	15.7520	0.688788	0.344394	+	0.938825i	$0.388084\pi$
$524$	0	0
$525$	−2.41467	−0.105385
$526$	0	0
$527$	2.17342	0.0946757
$528$	0	0
$529$	48.2621	2.09835
$530$	0	0
$531$	−10.4518	−0.453569
$532$	0	0
$533$	−49.4795	−2.14319
$534$	0	0
$535$	6.66283	0.288059
$536$	0	0
$537$	15.6631	0.675914
$538$	0	0
$539$	1.65501	0.0712865
$540$	0	0
$541$	−0.677821	−0.0291418	−0.0145709	−	0.999894i	$-0.504638\pi$
$541$	−0.677821	−0.0291418	−0.0145709	+	0.999894i	$0.504638\pi$
$542$	0	0
$543$	−2.27343	−0.0975623
$544$	0	0
$545$	−15.0525	−0.644777
$546$	0	0
$547$	25.0110	1.06939	0.534697	−	0.845044i	$-0.320426\pi$
$547$	25.0110	1.06939	0.534697	+	0.845044i	$0.320426\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−3.61800	−0.154132
$552$	0	0
$553$	8.44169	0.358977
$554$	0	0
$555$	6.92359	0.293890
$556$	0	0
$557$	−24.2369	−1.02695	−0.513475	−	0.858104i	$-0.671642\pi$
$557$	−24.2369	−1.02695	−0.513475	+	0.858104i	$0.671642\pi$
$558$	0	0
$559$	35.3352	1.49452
$560$	0	0
$561$	1.95090	0.0823670
$562$	0	0
$563$	−23.6847	−0.998190	−0.499095	−	0.866547i	$-0.666334\pi$
$563$	−23.6847	−0.998190	−0.499095	+	0.866547i	$0.666334\pi$
$564$	0	0
$565$	−14.8977	−0.626753
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−17.1914	−0.720699	−0.360350	−	0.932817i	$-0.617343\pi$
$569$	−17.1914	−0.720699	−0.360350	+	0.932817i	$0.617343\pi$
$570$	0	0
$571$	38.0012	1.59030	0.795150	−	0.606412i	$-0.207392\pi$
$571$	38.0012	1.59030	0.795150	+	0.606412i	$0.207392\pi$
$572$	0	0
$573$	19.0826	0.797185
$574$	0	0
$575$	20.3839	0.850068
$576$	0	0
$577$	−13.5445	−0.563864	−0.281932	−	0.959434i	$-0.590975\pi$
$577$	−13.5445	−0.563864	−0.281932	+	0.959434i	$0.590975\pi$
$578$	0	0
$579$	12.8343	0.533374
$580$	0	0
$581$	−7.19136	−0.298348
$582$	0	0
$583$	5.98785	0.247991
$584$	0	0
$585$	7.31003	0.302232
$586$	0	0
$587$	−14.7398	−0.608376	−0.304188	−	0.952612i	$-0.598385\pi$
$587$	−14.7398	−0.608376	−0.304188	+	0.952612i	$0.598385\pi$
$588$	0	0
$589$	1.84379	0.0759720
$590$	0	0
$591$	8.47425	0.348584
$592$	0	0
$593$	3.80280	0.156162	0.0780812	−	0.996947i	$-0.475121\pi$
$593$	3.80280	0.156162	0.0780812	+	0.996947i	$0.475121\pi$
$594$	0	0
$595$	1.89536	0.0777020
$596$	0	0
$597$	−17.1918	−0.703615
$598$	0	0
$599$	8.73956	0.357089	0.178544	−	0.983932i	$-0.442861\pi$
$599$	8.73956	0.357089	0.178544	+	0.983932i	$0.442861\pi$
$600$	0	0
$601$	−25.4377	−1.03762	−0.518812	−	0.854888i	$-0.673625\pi$
$601$	−25.4377	−1.03762	−0.518812	+	0.854888i	$0.673625\pi$
$602$	0	0
$603$	−8.01258	−0.326297
$604$	0	0
$605$	−13.2827	−0.540019
$606$	0	0
$607$	13.1931	0.535492	0.267746	−	0.963490i	$-0.413721\pi$
$607$	13.1931	0.535492	0.267746	+	0.963490i	$0.413721\pi$
$608$	0	0
$609$	3.61800	0.146609
$610$	0	0
$611$	−50.6452	−2.04889
$612$	0	0
$613$	8.18697	0.330668	0.165334	−	0.986238i	$-0.447130\pi$
$613$	8.18697	0.330668	0.165334	+	0.986238i	$0.447130\pi$
$614$	0	0
$615$	17.4993	0.705641
$616$	0	0
$617$	−15.7047	−0.632246	−0.316123	−	0.948718i	$-0.602381\pi$
$617$	−15.7047	−0.632246	−0.316123	+	0.948718i	$0.602381\pi$
$618$	0	0
$619$	−37.4465	−1.50510	−0.752551	−	0.658534i	$-0.771177\pi$
$619$	−37.4465	−1.50510	−0.752551	+	0.658534i	$0.771177\pi$
$620$	0	0
$621$	8.44169	0.338753
$622$	0	0
$623$	−5.79071	−0.232000
$624$	0	0
$625$	−7.09598	−0.283839
$626$	0	0
$627$	1.65501	0.0660949
$628$	0	0
$629$	5.07582	0.202386
$630$	0	0
$631$	−27.8023	−1.10679	−0.553396	−	0.832918i	$-0.686669\pi$
$631$	−27.8023	−1.10679	−0.553396	+	0.832918i	$0.686669\pi$
$632$	0	0
$633$	−12.6986	−0.504723
$634$	0	0
$635$	4.12395	0.163654
$636$	0	0
$637$	4.54633	0.180132
$638$	0	0
$639$	3.56890	0.141184
$640$	0	0
$641$	−13.4909	−0.532859	−0.266430	−	0.963854i	$-0.585844\pi$
$641$	−13.4909	−0.532859	−0.266430	+	0.963854i	$0.585844\pi$
$642$	0	0
$643$	−8.55963	−0.337559	−0.168779	−	0.985654i	$-0.553983\pi$
$643$	−8.55963	−0.337559	−0.168779	+	0.985654i	$0.553983\pi$
$644$	0	0
$645$	−12.4969	−0.492066
$646$	0	0
$647$	−41.3049	−1.62386	−0.811932	−	0.583752i	$-0.801584\pi$
$647$	−41.3049	−1.62386	−0.811932	+	0.583752i	$0.801584\pi$
$648$	0	0
$649$	−17.2979	−0.679001
$650$	0	0
$651$	−1.84379	−0.0722638
$652$	0	0
$653$	−15.6940	−0.614154	−0.307077	−	0.951685i	$-0.599351\pi$
$653$	−15.6940	−0.614154	−0.307077	+	0.951685i	$0.599351\pi$
$654$	0	0
$655$	1.34065	0.0523837
$656$	0	0
$657$	−1.21579	−0.0474325
$658$	0	0
$659$	−24.5387	−0.955891	−0.477946	−	0.878389i	$-0.658618\pi$
$659$	−24.5387	−0.955891	−0.477946	+	0.878389i	$0.658618\pi$
$660$	0	0
$661$	−3.93435	−0.153028	−0.0765142	−	0.997068i	$-0.524379\pi$
$661$	−3.93435	−0.153028	−0.0765142	+	0.997068i	$0.524379\pi$
$662$	0	0
$663$	5.35913	0.208131
$664$	0	0
$665$	1.60789	0.0623515
$666$	0	0
$667$	−30.5421	−1.18259
$668$	0	0
$669$	20.5098	0.792955
$670$	0	0
$671$	−3.31003	−0.127782
$672$	0	0
$673$	10.1891	0.392760	0.196380	−	0.980528i	$-0.437081\pi$
$673$	10.1891	0.392760	0.196380	+	0.980528i	$0.437081\pi$
$674$	0	0
$675$	2.41467	0.0929408
$676$	0	0
$677$	−33.9550	−1.30500	−0.652498	−	0.757790i	$-0.726279\pi$
$677$	−33.9550	−1.30500	−0.652498	+	0.757790i	$0.726279\pi$
$678$	0	0
$679$	−3.17433	−0.121820
$680$	0	0
$681$	−9.18691	−0.352043
$682$	0	0
$683$	42.9243	1.64245	0.821226	−	0.570603i	$-0.193290\pi$
$683$	42.9243	1.64245	0.821226	+	0.570603i	$0.193290\pi$
$684$	0	0
$685$	20.0212	0.764970
$686$	0	0
$687$	10.3035	0.393104
$688$	0	0
$689$	16.4487	0.626644
$690$	0	0
$691$	25.9760	0.988175	0.494088	−	0.869412i	$-0.335502\pi$
$691$	25.9760	0.988175	0.494088	+	0.869412i	$0.335502\pi$
$692$	0	0
$693$	−1.65501	−0.0628688
$694$	0	0
$695$	−19.1837	−0.727677
$696$	0	0
$697$	12.8291	0.485937
$698$	0	0
$699$	−7.39620	−0.279750
$700$	0	0
$701$	−11.1931	−0.422758	−0.211379	−	0.977404i	$-0.567795\pi$
$701$	−11.1931	−0.422758	−0.211379	+	0.977404i	$0.567795\pi$
$702$	0	0
$703$	4.30599	0.162404
$704$	0	0
$705$	17.9116	0.674590
$706$	0	0
$707$	−16.7457	−0.629787
$708$	0	0
$709$	−6.34282	−0.238209	−0.119105	−	0.992882i	$-0.538002\pi$
$709$	−6.34282	−0.238209	−0.119105	+	0.992882i	$0.538002\pi$
$710$	0	0
$711$	−8.44169	−0.316588
$712$	0	0
$713$	15.5647	0.582902
$714$	0	0
$715$	12.0982	0.452447
$716$	0	0
$717$	−0.179193	−0.00669209
$718$	0	0
$719$	4.19105	0.156300	0.0781499	−	0.996942i	$-0.475099\pi$
$719$	4.19105	0.156300	0.0781499	+	0.996942i	$0.475099\pi$
$720$	0	0
$721$	−6.78668	−0.252749
$722$	0	0
$723$	−29.4218	−1.09421
$724$	0	0
$725$	−8.73630	−0.324458
$726$	0	0
$727$	−2.28847	−0.0848747	−0.0424374	−	0.999099i	$-0.513512\pi$
$727$	−2.28847	−0.0848747	−0.0424374	+	0.999099i	$0.513512\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.16176	−0.338860
$732$	0	0
$733$	30.0874	1.11130	0.555651	−	0.831416i	$-0.312469\pi$
$733$	30.0874	1.11130	0.555651	+	0.831416i	$0.312469\pi$
$734$	0	0
$735$	−1.60789	−0.0593081
$736$	0	0
$737$	−13.2609	−0.488473
$738$	0	0
$739$	−34.7096	−1.27681	−0.638407	−	0.769699i	$-0.720407\pi$
$739$	−34.7096	−1.27681	−0.638407	+	0.769699i	$0.720407\pi$
$740$	0	0
$741$	4.54633	0.167014
$742$	0	0
$743$	24.1438	0.885751	0.442875	−	0.896583i	$-0.353958\pi$
$743$	24.1438	0.885751	0.442875	+	0.896583i	$0.353958\pi$
$744$	0	0
$745$	−14.2796	−0.523163
$746$	0	0
$747$	7.19136	0.263118
$748$	0	0
$749$	−4.14382	−0.151412
$750$	0	0
$751$	38.3027	1.39768	0.698842	−	0.715276i	$-0.253699\pi$
$751$	38.3027	1.39768	0.698842	+	0.715276i	$0.253699\pi$
$752$	0	0
$753$	26.0947	0.950945
$754$	0	0
$755$	4.49297	0.163516
$756$	0	0
$757$	−11.4045	−0.414504	−0.207252	−	0.978288i	$-0.566452\pi$
$757$	−11.4045	−0.414504	−0.207252	+	0.978288i	$0.566452\pi$
$758$	0	0
$759$	13.9711	0.507119
$760$	0	0
$761$	−24.3906	−0.884159	−0.442079	−	0.896976i	$-0.645759\pi$
$761$	−24.3906	−0.884159	−0.442079	+	0.896976i	$0.645759\pi$
$762$	0	0
$763$	9.36160	0.338913
$764$	0	0
$765$	−1.89536	−0.0685267
$766$	0	0
$767$	−47.5174	−1.71575
$768$	0	0
$769$	42.4070	1.52924	0.764618	−	0.644483i	$-0.222927\pi$
$769$	42.4070	1.52924	0.764618	+	0.644483i	$0.222927\pi$
$770$	0	0
$771$	27.9020	1.00487
$772$	0	0
$773$	−7.86496	−0.282883	−0.141442	−	0.989947i	$-0.545174\pi$
$773$	−7.86496	−0.282883	−0.141442	+	0.989947i	$0.545174\pi$
$774$	0	0
$775$	4.45215	0.159926
$776$	0	0
$777$	−4.30599	−0.154477
$778$	0	0
$779$	10.8834	0.389938
$780$	0	0
$781$	5.90659	0.211354
$782$	0	0
$783$	−3.61800	−0.129297
$784$	0	0
$785$	16.4559	0.587336
$786$	0	0
$787$	35.4899	1.26508	0.632539	−	0.774528i	$-0.282013\pi$
$787$	35.4899	1.26508	0.632539	+	0.774528i	$0.282013\pi$
$788$	0	0
$789$	−19.7888	−0.704499
$790$	0	0
$791$	9.26537	0.329439
$792$	0	0
$793$	−9.09267	−0.322890
$794$	0	0
$795$	−5.81737	−0.206321
$796$	0	0
$797$	−19.9307	−0.705981	−0.352990	−	0.935627i	$-0.614835\pi$
$797$	−19.9307	−0.705981	−0.352990	+	0.935627i	$0.614835\pi$
$798$	0	0
$799$	13.1314	0.464554
$800$	0	0
$801$	5.79071	0.204605
$802$	0	0
$803$	−2.01215	−0.0710072
$804$	0	0
$805$	13.5734	0.478398
$806$	0	0
$807$	4.08617	0.143840
$808$	0	0
$809$	39.8729	1.40186	0.700929	−	0.713232i	$-0.252769\pi$
$809$	39.8729	1.40186	0.700929	+	0.713232i	$0.252769\pi$
$810$	0	0
$811$	46.3136	1.62629	0.813146	−	0.582060i	$-0.197753\pi$
$811$	46.3136	1.62629	0.813146	+	0.582060i	$0.197753\pi$
$812$	0	0
$813$	−9.07245	−0.318185
$814$	0	0
$815$	0.743178	0.0260324
$816$	0	0
$817$	−7.77223	−0.271916
$818$	0	0
$819$	−4.54633	−0.158862
$820$	0	0
$821$	−7.20936	−0.251608	−0.125804	−	0.992055i	$-0.540151\pi$
$821$	−7.20936	−0.251608	−0.125804	+	0.992055i	$0.540151\pi$
$822$	0	0
$823$	8.22777	0.286802	0.143401	−	0.989665i	$-0.454196\pi$
$823$	8.22777	0.286802	0.143401	+	0.989665i	$0.454196\pi$
$824$	0	0
$825$	3.99632	0.139134
$826$	0	0
$827$	56.4982	1.96464	0.982318	−	0.187219i	$-0.0599473\pi$
$827$	56.4982	1.96464	0.982318	+	0.187219i	$0.0599473\pi$
$828$	0	0
$829$	−35.5294	−1.23399	−0.616994	−	0.786968i	$-0.711650\pi$
$829$	−35.5294	−1.23399	−0.616994	+	0.786968i	$0.711650\pi$
$830$	0	0
$831$	4.81977	0.167196
$832$	0	0
$833$	−1.17878	−0.0408423
$834$	0	0
$835$	−6.31256	−0.218455
$836$	0	0
$837$	1.84379	0.0637306
$838$	0	0
$839$	26.6452	0.919895	0.459947	−	0.887946i	$-0.347868\pi$
$839$	26.6452	0.919895	0.459947	+	0.887946i	$0.347868\pi$
$840$	0	0
$841$	−15.9100	−0.548622
$842$	0	0
$843$	8.74763	0.301284
$844$	0	0
$845$	12.3312	0.424206
$846$	0	0
$847$	8.26093	0.283849
$848$	0	0
$849$	−18.5510	−0.636668
$850$	0	0
$851$	36.3499	1.24606
$852$	0	0
$853$	−28.6912	−0.982367	−0.491183	−	0.871056i	$-0.663436\pi$
$853$	−28.6912	−0.982367	−0.491183	+	0.871056i	$0.663436\pi$
$854$	0	0
$855$	−1.60789	−0.0549888
$856$	0	0
$857$	−25.3352	−0.865433	−0.432717	−	0.901530i	$-0.642445\pi$
$857$	−25.3352	−0.865433	−0.432717	+	0.901530i	$0.642445\pi$
$858$	0	0
$859$	−6.45673	−0.220301	−0.110150	−	0.993915i	$-0.535133\pi$
$859$	−6.45673	−0.220301	−0.110150	+	0.993915i	$0.535133\pi$
$860$	0	0
$861$	−10.8834	−0.370905
$862$	0	0
$863$	−51.6270	−1.75740	−0.878702	−	0.477370i	$-0.841590\pi$
$863$	−51.6270	−1.75740	−0.878702	+	0.477370i	$0.841590\pi$
$864$	0	0
$865$	−14.6990	−0.499781
$866$	0	0
$867$	15.6105	0.530160
$868$	0	0
$869$	−13.9711	−0.473938
$870$	0	0
$871$	−36.4278	−1.23431
$872$	0	0
$873$	3.17433	0.107435
$874$	0	0
$875$	11.9220	0.403038
$876$	0	0
$877$	−3.68115	−0.124304	−0.0621518	−	0.998067i	$-0.519796\pi$
$877$	−3.68115	−0.124304	−0.0621518	+	0.998067i	$0.519796\pi$
$878$	0	0
$879$	−30.1983	−1.01856
$880$	0	0
$881$	3.06350	0.103212	0.0516059	−	0.998668i	$-0.483566\pi$
$881$	3.06350	0.103212	0.0516059	+	0.998668i	$0.483566\pi$
$882$	0	0
$883$	−23.3964	−0.787353	−0.393676	−	0.919249i	$-0.628797\pi$
$883$	−23.3964	−0.787353	−0.393676	+	0.919249i	$0.628797\pi$
$884$	0	0
$885$	16.8054	0.564907
$886$	0	0
$887$	18.7143	0.628365	0.314182	−	0.949363i	$-0.398270\pi$
$887$	18.7143	0.628365	0.314182	+	0.949363i	$0.398270\pi$
$888$	0	0
$889$	−2.56481	−0.0860210
$890$	0	0
$891$	1.65501	0.0554451
$892$	0	0
$893$	11.1398	0.372779
$894$	0	0
$895$	−25.1847	−0.841831
$896$	0	0
$897$	38.3787	1.28143
$898$	0	0
$899$	−6.67083	−0.222485
$900$	0	0
$901$	−4.26483	−0.142082
$902$	0	0
$903$	7.77223	0.258644
$904$	0	0
$905$	3.65544	0.121511
$906$	0	0
$907$	−1.69197	−0.0561808	−0.0280904	−	0.999605i	$-0.508943\pi$
$907$	−1.69197	−0.0561808	−0.0280904	+	0.999605i	$0.508943\pi$
$908$	0	0
$909$	16.7457	0.555420
$910$	0	0
$911$	−37.8045	−1.25252	−0.626260	−	0.779614i	$-0.715415\pi$
$911$	−37.8045	−1.25252	−0.626260	+	0.779614i	$0.715415\pi$
$912$	0	0
$913$	11.9018	0.393892
$914$	0	0
$915$	3.21579	0.106311
$916$	0	0
$917$	−0.833795	−0.0275343
$918$	0	0
$919$	27.9018	0.920395	0.460198	−	0.887816i	$-0.347779\pi$
$919$	27.9018	0.920395	0.460198	+	0.887816i	$0.347779\pi$
$920$	0	0
$921$	10.5814	0.348668
$922$	0	0
$923$	16.2254	0.534067
$924$	0	0
$925$	10.3976	0.341870
$926$	0	0
$927$	6.78668	0.222904
$928$	0	0
$929$	−18.4767	−0.606200	−0.303100	−	0.952959i	$-0.598022\pi$
$929$	−18.4767	−0.606200	−0.303100	+	0.952959i	$0.598022\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	23.0045	0.753134
$934$	0	0
$935$	−3.13684	−0.102586
$936$	0	0
$937$	−41.0228	−1.34016	−0.670078	−	0.742291i	$-0.733739\pi$
$937$	−41.0228	−1.34016	−0.670078	+	0.742291i	$0.733739\pi$
$938$	0	0
$939$	32.5421	1.06197
$940$	0	0
$941$	6.17656	0.201350	0.100675	−	0.994919i	$-0.467900\pi$
$941$	6.17656	0.201350	0.100675	+	0.994919i	$0.467900\pi$
$942$	0	0
$943$	91.8741	2.99183
$944$	0	0
$945$	1.60789	0.0523048
$946$	0	0
$947$	20.8427	0.677298	0.338649	−	0.940913i	$-0.390030\pi$
$947$	20.8427	0.677298	0.338649	+	0.940913i	$0.390030\pi$
$948$	0	0
$949$	−5.52739	−0.179427
$950$	0	0
$951$	4.77976	0.154994
$952$	0	0
$953$	22.9330	0.742872	0.371436	−	0.928459i	$-0.378866\pi$
$953$	22.9330	0.742872	0.371436	+	0.928459i	$0.378866\pi$
$954$	0	0
$955$	−30.6828	−0.992871
$956$	0	0
$957$	−5.98785	−0.193560
$958$	0	0
$959$	−12.4518	−0.402090
$960$	0	0
$961$	−27.6004	−0.890337
$962$	0	0
$963$	4.14382	0.133533
$964$	0	0
$965$	−20.6362	−0.664302
$966$	0	0
$967$	24.3191	0.782049	0.391025	−	0.920380i	$-0.372121\pi$
$967$	24.3191	0.782049	0.391025	+	0.920380i	$0.372121\pi$
$968$	0	0
$969$	−1.17878	−0.0378679
$970$	0	0
$971$	22.6452	0.726719	0.363360	−	0.931649i	$-0.381630\pi$
$971$	22.6452	0.726719	0.363360	+	0.931649i	$0.381630\pi$
$972$	0	0
$973$	11.9309	0.382487
$974$	0	0
$975$	10.9779	0.351575
$976$	0	0
$977$	21.3755	0.683863	0.341931	−	0.939725i	$-0.388919\pi$
$977$	21.3755	0.683863	0.341931	+	0.939725i	$0.388919\pi$
$978$	0	0
$979$	9.58371	0.306297
$980$	0	0
$981$	−9.36160	−0.298893
$982$	0	0
$983$	−33.4292	−1.06623	−0.533113	−	0.846044i	$-0.678978\pi$
$983$	−33.4292	−1.06623	−0.533113	+	0.846044i	$0.678978\pi$
$984$	0	0
$985$	−13.6257	−0.434151
$986$	0	0
$987$	−11.1398	−0.354583
$988$	0	0
$989$	−65.6108	−2.08630
$990$	0	0
$991$	−23.1649	−0.735857	−0.367928	−	0.929854i	$-0.619933\pi$
$991$	−23.1649	−0.735857	−0.367928	+	0.929854i	$0.619933\pi$
$992$	0	0
$993$	12.2840	0.389820
$994$	0	0
$995$	27.6427	0.876332
$996$	0	0
$997$	1.76556	0.0559158	0.0279579	−	0.999609i	$-0.491100\pi$
$997$	1.76556	0.0559158	0.0279579	+	0.999609i	$0.491100\pi$
$998$	0	0
$999$	4.30599	0.136236

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cd.1.4		5
4.3	odd	2		3192.2.a.bb.1.4	✓	5
12.11	even	2		9576.2.a.cn.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.bb.1.4	✓	5	4.3	odd	2
6384.2.a.cd.1.4		5	1.1	even	1	trivial
9576.2.a.cn.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.60789 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.60789 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.65501 q^{11} +4.54633 q^{13} -1.60789 q^{15} -1.17878 q^{17} -1.00000 q^{19} +1.00000 q^{21} -8.44169 q^{23} -2.41467 q^{25} -1.00000 q^{27} +3.61800 q^{29} -1.84379 q^{31} -1.65501 q^{33} -1.60789 q^{35} -4.30599 q^{37} -4.54633 q^{39} -10.8834 q^{41} +7.77223 q^{43} +1.60789 q^{45} -11.1398 q^{47} +1.00000 q^{49} +1.17878 q^{51} +3.61800 q^{53} +2.66109 q^{55} +1.00000 q^{57} -10.4518 q^{59} -2.00000 q^{61} -1.00000 q^{63} +7.31003 q^{65} -8.01258 q^{67} +8.44169 q^{69} +3.56890 q^{71} -1.21579 q^{73} +2.41467 q^{75} -1.65501 q^{77} -8.44169 q^{79} +1.00000 q^{81} +7.19136 q^{83} -1.89536 q^{85} -3.61800 q^{87} +5.79071 q^{89} -4.54633 q^{91} +1.84379 q^{93} -1.60789 q^{95} +3.17433 q^{97} +1.65501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 6 q^{11} - 2 q^{15} + 6 q^{17} - 5 q^{19} + 5 q^{21} - 10 q^{23} + 7 q^{25} - 5 q^{27} + 4 q^{29} + 4 q^{31} + 6 q^{33} - 2 q^{35} + 6 q^{37} + 10 q^{41} - 4 q^{43} + 2 q^{45} - 2 q^{47} + 5 q^{49} - 6 q^{51} + 4 q^{53} - 8 q^{55} + 5 q^{57} - 12 q^{59} - 10 q^{61} - 5 q^{63} + 8 q^{65} - 2 q^{67} + 10 q^{69} - 30 q^{71} + 6 q^{73} - 7 q^{75} + 6 q^{77} - 10 q^{79} + 5 q^{81} - 14 q^{83} - 4 q^{87} + 10 q^{89} - 4 q^{93} - 2 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 6 * q^11 - 2 * q^15 + 6 * q^17 - 5 * q^19 + 5 * q^21 - 10 * q^23 + 7 * q^25 - 5 * q^27 + 4 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 6 * q^37 + 10 * q^41 - 4 * q^43 + 2 * q^45 - 2 * q^47 + 5 * q^49 - 6 * q^51 + 4 * q^53 - 8 * q^55 + 5 * q^57 - 12 * q^59 - 10 * q^61 - 5 * q^63 + 8 * q^65 - 2 * q^67 + 10 * q^69 - 30 * q^71 + 6 * q^73 - 7 * q^75 + 6 * q^77 - 10 * q^79 + 5 * q^81 - 14 * q^83 - 4 * q^87 + 10 * q^89 - 4 * q^93 - 2 * q^95 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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