LMFDB - Embedded newform 6384.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$50.9764966504$
Analytic rank:	$0$
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			1.2
Root			$-1.78948$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.430991 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.430991 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.67080 q^{11} -4.61014 q^{13} -0.430991 q^{15} +5.52283 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.98344 q^{23} -4.81425 q^{25} +1.00000 q^{27} +6.41443 q^{29} +6.95226 q^{31} +4.67080 q^{33} +0.430991 q^{35} -3.95382 q^{37} -4.61014 q^{39} -0.487127 q^{41} +2.25184 q^{43} -0.430991 q^{45} -7.10179 q^{47} +1.00000 q^{49} +5.52283 q^{51} -7.50627 q^{53} -2.01307 q^{55} +1.00000 q^{57} +2.68735 q^{59} -2.22985 q^{61} -1.00000 q^{63} +1.98693 q^{65} -0.358299 q^{67} +1.98344 q^{69} +13.0852 q^{71} -1.77919 q^{73} -4.81425 q^{75} -4.67080 q^{77} -13.1117 q^{79} +1.00000 q^{81} -2.40136 q^{83} -2.38029 q^{85} +6.41443 q^{87} +16.4540 q^{89} +4.61014 q^{91} +6.95226 q^{93} -0.430991 q^{95} +3.22028 q^{97} +4.67080 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.430991	−0.192745	−0.0963725	−	0.995345i	$-0.530724\pi$
$5$	−0.430991	−0.192745	−0.0963725	+	0.995345i	$0.530724\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.67080	1.40830	0.704149	−	0.710052i	$-0.251329\pi$
$11$	4.67080	1.40830	0.704149	+	0.710052i	$0.251329\pi$
$12$	0	0
$13$	−4.61014	−1.27862	−0.639311	−	0.768948i	$-0.720781\pi$
$13$	−4.61014	−1.27862	−0.639311	+	0.768948i	$0.720781\pi$
$14$	0	0
$15$	−0.430991	−0.111281
$16$	0	0
$17$	5.52283	1.33948	0.669741	−	0.742595i	$-0.266405\pi$
$17$	5.52283	1.33948	0.669741	+	0.742595i	$0.266405\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.98344	0.413577	0.206788	−	0.978386i	$-0.433699\pi$
$23$	1.98344	0.413577	0.206788	+	0.978386i	$0.433699\pi$
$24$	0	0
$25$	−4.81425	−0.962849
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.41443	1.19113	0.595565	−	0.803307i	$-0.296928\pi$
$29$	6.41443	1.19113	0.595565	+	0.803307i	$0.296928\pi$
$30$	0	0
$31$	6.95226	1.24866	0.624332	−	0.781159i	$-0.285371\pi$
$31$	6.95226	1.24866	0.624332	+	0.781159i	$0.285371\pi$
$32$	0	0
$33$	4.67080	0.813081
$34$	0	0
$35$	0.430991	0.0728508
$36$	0	0
$37$	−3.95382	−0.650003	−0.325002	−	0.945713i	$-0.605365\pi$
$37$	−3.95382	−0.650003	−0.325002	+	0.945713i	$0.605365\pi$
$38$	0	0
$39$	−4.61014	−0.738213
$40$	0	0
$41$	−0.487127	−0.0760765	−0.0380382	−	0.999276i	$-0.512111\pi$
$41$	−0.487127	−0.0760765	−0.0380382	+	0.999276i	$0.512111\pi$
$42$	0	0
$43$	2.25184	0.343403	0.171701	−	0.985149i	$-0.445074\pi$
$43$	2.25184	0.343403	0.171701	+	0.985149i	$0.445074\pi$
$44$	0	0
$45$	−0.430991	−0.0642483
$46$	0	0
$47$	−7.10179	−1.03590	−0.517951	−	0.855410i	$-0.673305\pi$
$47$	−7.10179	−1.03590	−0.517951	+	0.855410i	$0.673305\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.52283	0.773350
$52$	0	0
$53$	−7.50627	−1.03107	−0.515533	−	0.856870i	$-0.672406\pi$
$53$	−7.50627	−1.03107	−0.515533	+	0.856870i	$0.672406\pi$
$54$	0	0
$55$	−2.01307	−0.271442
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	2.68735	0.349863	0.174932	−	0.984581i	$-0.444030\pi$
$59$	2.68735	0.349863	0.174932	+	0.984581i	$0.444030\pi$
$60$	0	0
$61$	−2.22985	−0.285503	−0.142752	−	0.989759i	$-0.545595\pi$
$61$	−2.22985	−0.285503	−0.142752	+	0.989759i	$0.545595\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.98693	0.246448
$66$	0	0
$67$	−0.358299	−0.0437731	−0.0218866	−	0.999760i	$-0.506967\pi$
$67$	−0.358299	−0.0437731	−0.0218866	+	0.999760i	$0.506967\pi$
$68$	0	0
$69$	1.98344	0.238779
$70$	0	0
$71$	13.0852	1.55293	0.776466	−	0.630160i	$-0.217011\pi$
$71$	13.0852	1.55293	0.776466	+	0.630160i	$0.217011\pi$
$72$	0	0
$73$	−1.77919	−0.208238	−0.104119	−	0.994565i	$-0.533202\pi$
$73$	−1.77919	−0.208238	−0.104119	+	0.994565i	$0.533202\pi$
$74$	0	0
$75$	−4.81425	−0.555901
$76$	0	0
$77$	−4.67080	−0.532287
$78$	0	0
$79$	−13.1117	−1.47519	−0.737593	−	0.675246i	$-0.764038\pi$
$79$	−13.1117	−1.47519	−0.737593	+	0.675246i	$0.764038\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.40136	−0.263584	−0.131792	−	0.991277i	$-0.542073\pi$
$83$	−2.40136	−0.263584	−0.131792	+	0.991277i	$0.542073\pi$
$84$	0	0
$85$	−2.38029	−0.258178
$86$	0	0
$87$	6.41443	0.687700
$88$	0	0
$89$	16.4540	1.74412	0.872061	−	0.489397i	$-0.162783\pi$
$89$	16.4540	1.74412	0.872061	+	0.489397i	$0.162783\pi$
$90$	0	0
$91$	4.61014	0.483274
$92$	0	0
$93$	6.95226	0.720916
$94$	0	0
$95$	−0.430991	−0.0442187
$96$	0	0
$97$	3.22028	0.326970	0.163485	−	0.986546i	$-0.447726\pi$
$97$	3.22028	0.326970	0.163485	+	0.986546i	$0.447726\pi$
$98$	0	0
$99$	4.67080	0.469433
$100$	0	0
$101$	19.8188	1.97204	0.986020	−	0.166624i	$-0.0532867\pi$
$101$	19.8188	1.97204	0.986020	+	0.166624i	$0.0532867\pi$
$102$	0	0
$103$	−9.34159	−0.920454	−0.460227	−	0.887801i	$-0.652232\pi$
$103$	−9.34159	−0.920454	−0.460227	+	0.887801i	$0.652232\pi$
$104$	0	0
$105$	0.430991	0.0420604
$106$	0	0
$107$	−1.22688	−0.118607	−0.0593037	−	0.998240i	$-0.518888\pi$
$107$	−1.22688	−0.118607	−0.0593037	+	0.998240i	$0.518888\pi$
$108$	0	0
$109$	−2.11227	−0.202319	−0.101159	−	0.994870i	$-0.532255\pi$
$109$	−2.11227	−0.202319	−0.101159	+	0.994870i	$0.532255\pi$
$110$	0	0
$111$	−3.95382	−0.375280
$112$	0	0
$113$	10.4144	0.979708	0.489854	−	0.871805i	$-0.337050\pi$
$113$	10.4144	0.979708	0.489854	+	0.871805i	$0.337050\pi$
$114$	0	0
$115$	−0.854846	−0.0797148
$116$	0	0
$117$	−4.61014	−0.426208
$118$	0	0
$119$	−5.52283	−0.506277
$120$	0	0
$121$	10.8163	0.983303
$122$	0	0
$123$	−0.487127	−0.0439228
$124$	0	0
$125$	4.22985	0.378329
$126$	0	0
$127$	2.17463	0.192967	0.0964836	−	0.995335i	$-0.469240\pi$
$127$	2.17463	0.192967	0.0964836	+	0.995335i	$0.469240\pi$
$128$	0	0
$129$	2.25184	0.198264
$130$	0	0
$131$	−0.789672	−0.0689939	−0.0344970	−	0.999405i	$-0.510983\pi$
$131$	−0.789672	−0.0689939	−0.0344970	+	0.999405i	$0.510983\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−0.430991	−0.0370938
$136$	0	0
$137$	16.3158	1.39396	0.696978	−	0.717092i	$-0.254527\pi$
$137$	16.3158	1.39396	0.696978	+	0.717092i	$0.254527\pi$
$138$	0	0
$139$	18.9040	1.60342	0.801708	−	0.597716i	$-0.203925\pi$
$139$	18.9040	1.60342	0.801708	+	0.597716i	$0.203925\pi$
$140$	0	0
$141$	−7.10179	−0.598078
$142$	0	0
$143$	−21.5330	−1.80068
$144$	0	0
$145$	−2.76456	−0.229585
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	2.45067	0.200766	0.100383	−	0.994949i	$-0.467993\pi$
$149$	2.45067	0.200766	0.100383	+	0.994949i	$0.467993\pi$
$150$	0	0
$151$	17.9737	1.46268	0.731340	−	0.682013i	$-0.238895\pi$
$151$	17.9737	1.46268	0.731340	+	0.682013i	$0.238895\pi$
$152$	0	0
$153$	5.52283	0.446494
$154$	0	0
$155$	−2.99636	−0.240674
$156$	0	0
$157$	6.64117	0.530023	0.265011	−	0.964245i	$-0.414624\pi$
$157$	6.64117	0.530023	0.265011	+	0.964245i	$0.414624\pi$
$158$	0	0
$159$	−7.50627	−0.595286
$160$	0	0
$161$	−1.98344	−0.156317
$162$	0	0
$163$	−18.1264	−1.41977	−0.709883	−	0.704320i	$-0.751252\pi$
$163$	−18.1264	−1.41977	−0.709883	+	0.704320i	$0.751252\pi$
$164$	0	0
$165$	−2.01307	−0.156717
$166$	0	0
$167$	8.35830	0.646785	0.323392	−	0.946265i	$-0.395177\pi$
$167$	8.35830	0.646785	0.323392	+	0.946265i	$0.395177\pi$
$168$	0	0
$169$	8.25339	0.634876
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−10.1671	−0.772991	−0.386496	−	0.922291i	$-0.626315\pi$
$173$	−10.1671	−0.772991	−0.386496	+	0.922291i	$0.626315\pi$
$174$	0	0
$175$	4.81425	0.363923
$176$	0	0
$177$	2.68735	0.201994
$178$	0	0
$179$	1.91113	0.142845	0.0714224	−	0.997446i	$-0.477246\pi$
$179$	1.91113	0.142845	0.0714224	+	0.997446i	$0.477246\pi$
$180$	0	0
$181$	13.0656	0.971155	0.485578	−	0.874194i	$-0.338609\pi$
$181$	13.0656	0.971155	0.485578	+	0.874194i	$0.338609\pi$
$182$	0	0
$183$	−2.22985	−0.164835
$184$	0	0
$185$	1.70406	0.125285
$186$	0	0
$187$	25.7960	1.88639
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−19.0790	−1.38051	−0.690254	−	0.723567i	$-0.742501\pi$
$191$	−19.0790	−1.38051	−0.690254	+	0.723567i	$0.742501\pi$
$192$	0	0
$193$	−4.91720	−0.353948	−0.176974	−	0.984216i	$-0.556631\pi$
$193$	−4.91720	−0.353948	−0.176974	+	0.984216i	$0.556631\pi$
$194$	0	0
$195$	1.98693	0.142287
$196$	0	0
$197$	−10.8819	−0.775302	−0.387651	−	0.921806i	$-0.626713\pi$
$197$	−10.8819	−0.775302	−0.387651	+	0.921806i	$0.626713\pi$
$198$	0	0
$199$	14.2036	1.00686	0.503432	−	0.864035i	$-0.332070\pi$
$199$	14.2036	1.00686	0.503432	+	0.864035i	$0.332070\pi$
$200$	0	0
$201$	−0.358299	−0.0252724
$202$	0	0
$203$	−6.41443	−0.450205
$204$	0	0
$205$	0.209947	0.0146634
$206$	0	0
$207$	1.98344	0.137859
$208$	0	0
$209$	4.67080	0.323086
$210$	0	0
$211$	11.4963	0.791439	0.395720	−	0.918371i	$-0.370495\pi$
$211$	11.4963	0.791439	0.395720	+	0.918371i	$0.370495\pi$
$212$	0	0
$213$	13.0852	0.896585
$214$	0	0
$215$	−0.970524	−0.0661892
$216$	0	0
$217$	−6.95226	−0.471950
$218$	0	0
$219$	−1.77919	−0.120226
$220$	0	0
$221$	−25.4610	−1.71269
$222$	0	0
$223$	16.9223	1.13320	0.566599	−	0.823994i	$-0.308259\pi$
$223$	16.9223	1.13320	0.566599	+	0.823994i	$0.308259\pi$
$224$	0	0
$225$	−4.81425	−0.320950
$226$	0	0
$227$	6.29594	0.417876	0.208938	−	0.977929i	$-0.432999\pi$
$227$	6.29594	0.417876	0.208938	+	0.977929i	$0.432999\pi$
$228$	0	0
$229$	8.76651	0.579307	0.289654	−	0.957132i	$-0.406460\pi$
$229$	8.76651	0.579307	0.289654	+	0.957132i	$0.406460\pi$
$230$	0	0
$231$	−4.67080	−0.307316
$232$	0	0
$233$	−6.76961	−0.443492	−0.221746	−	0.975104i	$-0.571176\pi$
$233$	−6.76961	−0.443492	−0.221746	+	0.975104i	$0.571176\pi$
$234$	0	0
$235$	3.06081	0.199665
$236$	0	0
$237$	−13.1117	−0.851699
$238$	0	0
$239$	−1.87117	−0.121036	−0.0605180	−	0.998167i	$-0.519275\pi$
$239$	−1.87117	−0.121036	−0.0605180	+	0.998167i	$0.519275\pi$
$240$	0	0
$241$	13.1579	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$241$	13.1579	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.430991	−0.0275350
$246$	0	0
$247$	−4.61014	−0.293336
$248$	0	0
$249$	−2.40136	−0.152180
$250$	0	0
$251$	10.2067	0.644241	0.322120	−	0.946699i	$-0.395604\pi$
$251$	10.2067	0.644241	0.322120	+	0.946699i	$0.395604\pi$
$252$	0	0
$253$	9.26426	0.582439
$254$	0	0
$255$	−2.38029	−0.149059
$256$	0	0
$257$	8.96674	0.559330	0.279665	−	0.960098i	$-0.409777\pi$
$257$	8.96674	0.559330	0.279665	+	0.960098i	$0.409777\pi$
$258$	0	0
$259$	3.95382	0.245678
$260$	0	0
$261$	6.41443	0.397044
$262$	0	0
$263$	11.3491	0.699816	0.349908	−	0.936784i	$-0.386213\pi$
$263$	11.3491	0.699816	0.349908	+	0.936784i	$0.386213\pi$
$264$	0	0
$265$	3.23513	0.198733
$266$	0	0
$267$	16.4540	1.00697
$268$	0	0
$269$	−1.39750	−0.0852069	−0.0426035	−	0.999092i	$-0.513565\pi$
$269$	−1.39750	−0.0852069	−0.0426035	+	0.999092i	$0.513565\pi$
$270$	0	0
$271$	−29.5492	−1.79499	−0.897493	−	0.441029i	$-0.854614\pi$
$271$	−29.5492	−1.79499	−0.897493	+	0.441029i	$0.854614\pi$
$272$	0	0
$273$	4.61014	0.279018
$274$	0	0
$275$	−22.4864	−1.35598
$276$	0	0
$277$	−4.09028	−0.245761	−0.122881	−	0.992421i	$-0.539213\pi$
$277$	−4.09028	−0.245761	−0.122881	+	0.992421i	$0.539213\pi$
$278$	0	0
$279$	6.95226	0.416221
$280$	0	0
$281$	17.2302	1.02787	0.513935	−	0.857829i	$-0.328187\pi$
$281$	17.2302	1.02787	0.513935	+	0.857829i	$0.328187\pi$
$282$	0	0
$283$	29.6999	1.76548	0.882738	−	0.469866i	$-0.155698\pi$
$283$	29.6999	1.76548	0.882738	+	0.469866i	$0.155698\pi$
$284$	0	0
$285$	−0.430991	−0.0255297
$286$	0	0
$287$	0.487127	0.0287542
$288$	0	0
$289$	13.5016	0.794212
$290$	0	0
$291$	3.22028	0.188776
$292$	0	0
$293$	−27.0490	−1.58022	−0.790110	−	0.612965i	$-0.789976\pi$
$293$	−27.0490	−1.58022	−0.790110	+	0.612965i	$0.789976\pi$
$294$	0	0
$295$	−1.15822	−0.0674344
$296$	0	0
$297$	4.67080	0.271027
$298$	0	0
$299$	−9.14395	−0.528808
$300$	0	0
$301$	−2.25184	−0.129794
$302$	0	0
$303$	19.8188	1.13856
$304$	0	0
$305$	0.961046	0.0550293
$306$	0	0
$307$	22.3893	1.27783	0.638913	−	0.769279i	$-0.279384\pi$
$307$	22.3893	1.27783	0.638913	+	0.769279i	$0.279384\pi$
$308$	0	0
$309$	−9.34159	−0.531425
$310$	0	0
$311$	−18.0389	−1.02289	−0.511446	−	0.859315i	$-0.670890\pi$
$311$	−18.0389	−1.02289	−0.511446	+	0.859315i	$0.670890\pi$
$312$	0	0
$313$	−10.6673	−0.602952	−0.301476	−	0.953474i	$-0.597479\pi$
$313$	−10.6673	−0.602952	−0.301476	+	0.953474i	$0.597479\pi$
$314$	0	0
$315$	0.430991	0.0242836
$316$	0	0
$317$	15.7560	0.884947	0.442473	−	0.896782i	$-0.354101\pi$
$317$	15.7560	0.884947	0.442473	+	0.896782i	$0.354101\pi$
$318$	0	0
$319$	29.9605	1.67747
$320$	0	0
$321$	−1.22688	−0.0684780
$322$	0	0
$323$	5.52283	0.307298
$324$	0	0
$325$	22.1944	1.23112
$326$	0	0
$327$	−2.11227	−0.116809
$328$	0	0
$329$	7.10179	0.391534
$330$	0	0
$331$	22.2790	1.22456	0.612282	−	0.790639i	$-0.290252\pi$
$331$	22.2790	1.22456	0.612282	+	0.790639i	$0.290252\pi$
$332$	0	0
$333$	−3.95382	−0.216668
$334$	0	0
$335$	0.154423	0.00843705
$336$	0	0
$337$	−12.6452	−0.688828	−0.344414	−	0.938818i	$-0.611922\pi$
$337$	−12.6452	−0.688828	−0.344414	+	0.938818i	$0.611922\pi$
$338$	0	0
$339$	10.4144	0.565634
$340$	0	0
$341$	32.4726	1.75849
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.854846	−0.0460234
$346$	0	0
$347$	−5.87117	−0.315181	−0.157590	−	0.987505i	$-0.550373\pi$
$347$	−5.87117	−0.315181	−0.157590	+	0.987505i	$0.550373\pi$
$348$	0	0
$349$	13.6285	0.729517	0.364758	−	0.931102i	$-0.381152\pi$
$349$	13.6285	0.729517	0.364758	+	0.931102i	$0.381152\pi$
$350$	0	0
$351$	−4.61014	−0.246071
$352$	0	0
$353$	−9.75268	−0.519083	−0.259541	−	0.965732i	$-0.583571\pi$
$353$	−9.75268	−0.519083	−0.259541	+	0.965732i	$0.583571\pi$
$354$	0	0
$355$	−5.63962	−0.299320
$356$	0	0
$357$	−5.52283	−0.292299
$358$	0	0
$359$	8.73005	0.460754	0.230377	−	0.973101i	$-0.426004\pi$
$359$	8.73005	0.460754	0.230377	+	0.973101i	$0.426004\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.8163	0.567710
$364$	0	0
$365$	0.766813	0.0401368
$366$	0	0
$367$	−13.7923	−0.719950	−0.359975	−	0.932962i	$-0.617215\pi$
$367$	−13.7923	−0.719950	−0.359975	+	0.932962i	$0.617215\pi$
$368$	0	0
$369$	−0.487127	−0.0253588
$370$	0	0
$371$	7.50627	0.389706
$372$	0	0
$373$	−18.2569	−0.945306	−0.472653	−	0.881249i	$-0.656704\pi$
$373$	−18.2569	−0.945306	−0.472653	+	0.881249i	$0.656704\pi$
$374$	0	0
$375$	4.22985	0.218429
$376$	0	0
$377$	−29.5714	−1.52301
$378$	0	0
$379$	−22.9149	−1.17706	−0.588529	−	0.808476i	$-0.700293\pi$
$379$	−22.9149	−1.17706	−0.588529	+	0.808476i	$0.700293\pi$
$380$	0	0
$381$	2.17463	0.111410
$382$	0	0
$383$	38.6741	1.97616	0.988078	−	0.153953i	$-0.0492003\pi$
$383$	38.6741	1.97616	0.988078	+	0.153953i	$0.0492003\pi$
$384$	0	0
$385$	2.01307	0.102596
$386$	0	0
$387$	2.25184	0.114468
$388$	0	0
$389$	8.76651	0.444480	0.222240	−	0.974992i	$-0.428663\pi$
$389$	8.76651	0.444480	0.222240	+	0.974992i	$0.428663\pi$
$390$	0	0
$391$	10.9542	0.553978
$392$	0	0
$393$	−0.789672	−0.0398337
$394$	0	0
$395$	5.65104	0.284335
$396$	0	0
$397$	−23.8561	−1.19731	−0.598653	−	0.801009i	$-0.704297\pi$
$397$	−23.8561	−1.19731	−0.598653	+	0.801009i	$0.704297\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−35.5461	−1.77509	−0.887543	−	0.460724i	$-0.847590\pi$
$401$	−35.5461	−1.77509	−0.887543	+	0.460724i	$0.847590\pi$
$402$	0	0
$403$	−32.0509	−1.59657
$404$	0	0
$405$	−0.430991	−0.0214161
$406$	0	0
$407$	−18.4675	−0.915398
$408$	0	0
$409$	−12.1947	−0.602987	−0.301493	−	0.953468i	$-0.597485\pi$
$409$	−12.1947	−0.602987	−0.301493	+	0.953468i	$0.597485\pi$
$410$	0	0
$411$	16.3158	0.804801
$412$	0	0
$413$	−2.68735	−0.132236
$414$	0	0
$415$	1.03497	0.0508045
$416$	0	0
$417$	18.9040	0.925733
$418$	0	0
$419$	−16.4827	−0.805234	−0.402617	−	0.915369i	$-0.631899\pi$
$419$	−16.4827	−0.805234	−0.402617	+	0.915369i	$0.631899\pi$
$420$	0	0
$421$	10.6213	0.517649	0.258824	−	0.965924i	$-0.416665\pi$
$421$	10.6213	0.517649	0.258824	+	0.965924i	$0.416665\pi$
$422$	0	0
$423$	−7.10179	−0.345301
$424$	0	0
$425$	−26.5882	−1.28972
$426$	0	0
$427$	2.22985	0.107910
$428$	0	0
$429$	−21.5330	−1.03962
$430$	0	0
$431$	−5.66851	−0.273043	−0.136521	−	0.990637i	$-0.543592\pi$
$431$	−5.66851	−0.273043	−0.136521	+	0.990637i	$0.543592\pi$
$432$	0	0
$433$	39.1499	1.88142	0.940712	−	0.339207i	$-0.110159\pi$
$433$	39.1499	1.88142	0.940712	+	0.339207i	$0.110159\pi$
$434$	0	0
$435$	−2.76456	−0.132551
$436$	0	0
$437$	1.98344	0.0948810
$438$	0	0
$439$	17.6065	0.840313	0.420156	−	0.907452i	$-0.361975\pi$
$439$	17.6065	0.840313	0.420156	+	0.907452i	$0.361975\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−23.4373	−1.11354	−0.556770	−	0.830667i	$-0.687960\pi$
$443$	−23.4373	−1.11354	−0.556770	+	0.830667i	$0.687960\pi$
$444$	0	0
$445$	−7.09153	−0.336171
$446$	0	0
$447$	2.45067	0.115913
$448$	0	0
$449$	−36.3882	−1.71726	−0.858632	−	0.512593i	$-0.828685\pi$
$449$	−36.3882	−1.71726	−0.858632	+	0.512593i	$0.828685\pi$
$450$	0	0
$451$	−2.27527	−0.107138
$452$	0	0
$453$	17.9737	0.844479
$454$	0	0
$455$	−1.98693	−0.0931487
$456$	0	0
$457$	16.4395	0.769009	0.384505	−	0.923123i	$-0.374372\pi$
$457$	16.4395	0.769009	0.384505	+	0.923123i	$0.374372\pi$
$458$	0	0
$459$	5.52283	0.257783
$460$	0	0
$461$	−19.6734	−0.916281	−0.458140	−	0.888880i	$-0.651484\pi$
$461$	−19.6734	−0.916281	−0.458140	+	0.888880i	$0.651484\pi$
$462$	0	0
$463$	11.1248	0.517014	0.258507	−	0.966009i	$-0.416770\pi$
$463$	11.1248	0.517014	0.258507	+	0.966009i	$0.416770\pi$
$464$	0	0
$465$	−2.99636	−0.138953
$466$	0	0
$467$	24.5917	1.13797	0.568985	−	0.822348i	$-0.307336\pi$
$467$	24.5917	1.13797	0.568985	+	0.822348i	$0.307336\pi$
$468$	0	0
$469$	0.358299	0.0165447
$470$	0	0
$471$	6.64117	0.306009
$472$	0	0
$473$	10.5179	0.483613
$474$	0	0
$475$	−4.81425	−0.220893
$476$	0	0
$477$	−7.50627	−0.343688
$478$	0	0
$479$	16.0429	0.733020	0.366510	−	0.930414i	$-0.380553\pi$
$479$	16.0429	0.733020	0.366510	+	0.930414i	$0.380553\pi$
$480$	0	0
$481$	18.2276	0.831109
$482$	0	0
$483$	−1.98344	−0.0902498
$484$	0	0
$485$	−1.38791	−0.0630218
$486$	0	0
$487$	13.2823	0.601880	0.300940	−	0.953643i	$-0.402700\pi$
$487$	13.2823	0.601880	0.300940	+	0.953643i	$0.402700\pi$
$488$	0	0
$489$	−18.1264	−0.819702
$490$	0	0
$491$	−30.6276	−1.38220	−0.691102	−	0.722758i	$-0.742874\pi$
$491$	−30.6276	−1.38220	−0.691102	+	0.722758i	$0.742874\pi$
$492$	0	0
$493$	35.4258	1.59550
$494$	0	0
$495$	−2.01307	−0.0904808
$496$	0	0
$497$	−13.0852	−0.586953
$498$	0	0
$499$	−37.5985	−1.68314	−0.841570	−	0.540149i	$-0.818368\pi$
$499$	−37.5985	−1.68314	−0.841570	+	0.540149i	$0.818368\pi$
$500$	0	0
$501$	8.35830	0.373421
$502$	0	0
$503$	19.9344	0.888830	0.444415	−	0.895821i	$-0.353412\pi$
$503$	19.9344	0.888830	0.444415	+	0.895821i	$0.353412\pi$
$504$	0	0
$505$	−8.54171	−0.380101
$506$	0	0
$507$	8.25339	0.366546
$508$	0	0
$509$	10.6949	0.474042	0.237021	−	0.971505i	$-0.423829\pi$
$509$	10.6949	0.474042	0.237021	+	0.971505i	$0.423829\pi$
$510$	0	0
$511$	1.77919	0.0787066
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	4.02614	0.177413
$516$	0	0
$517$	−33.1710	−1.45886
$518$	0	0
$519$	−10.1671	−0.446287
$520$	0	0
$521$	24.5070	1.07367	0.536836	−	0.843686i	$-0.319619\pi$
$521$	24.5070	1.07367	0.536836	+	0.843686i	$0.319619\pi$
$522$	0	0
$523$	−8.32697	−0.364113	−0.182056	−	0.983288i	$-0.558275\pi$
$523$	−8.32697	−0.364113	−0.182056	+	0.983288i	$0.558275\pi$
$524$	0	0
$525$	4.81425	0.210111
$526$	0	0
$527$	38.3961	1.67256
$528$	0	0
$529$	−19.0660	−0.828954
$530$	0	0
$531$	2.68735	0.116621
$532$	0	0
$533$	2.24572	0.0972731
$534$	0	0
$535$	0.528776	0.0228610
$536$	0	0
$537$	1.91113	0.0824715
$538$	0	0
$539$	4.67080	0.201185
$540$	0	0
$541$	45.0614	1.93734	0.968670	−	0.248351i	$-0.0798886\pi$
$541$	45.0614	1.93734	0.968670	+	0.248351i	$0.0798886\pi$
$542$	0	0
$543$	13.0656	0.560697
$544$	0	0
$545$	0.910370	0.0389960
$546$	0	0
$547$	14.6050	0.624463	0.312231	−	0.950006i	$-0.398924\pi$
$547$	14.6050	0.624463	0.312231	+	0.950006i	$0.398924\pi$
$548$	0	0
$549$	−2.22985	−0.0951678
$550$	0	0
$551$	6.41443	0.273264
$552$	0	0
$553$	13.1117	0.557568
$554$	0	0
$555$	1.70406	0.0723333
$556$	0	0
$557$	−34.6441	−1.46792	−0.733960	−	0.679193i	$-0.762330\pi$
$557$	−34.6441	−1.46792	−0.733960	+	0.679193i	$0.762330\pi$
$558$	0	0
$559$	−10.3813	−0.439082
$560$	0	0
$561$	25.7960	1.08911
$562$	0	0
$563$	−33.4497	−1.40974	−0.704868	−	0.709338i	$-0.748994\pi$
$563$	−33.4497	−1.40974	−0.704868	+	0.709338i	$0.748994\pi$
$564$	0	0
$565$	−4.48853	−0.188834
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−8.24277	−0.345555	−0.172777	−	0.984961i	$-0.555274\pi$
$569$	−8.24277	−0.345555	−0.172777	+	0.984961i	$0.555274\pi$
$570$	0	0
$571$	−15.7498	−0.659109	−0.329554	−	0.944137i	$-0.606899\pi$
$571$	−15.7498	−0.659109	−0.329554	+	0.944137i	$0.606899\pi$
$572$	0	0
$573$	−19.0790	−0.797037
$574$	0	0
$575$	−9.54879	−0.398212
$576$	0	0
$577$	−20.4094	−0.849652	−0.424826	−	0.905275i	$-0.639665\pi$
$577$	−20.4094	−0.849652	−0.424826	+	0.905275i	$0.639665\pi$
$578$	0	0
$579$	−4.91720	−0.204352
$580$	0	0
$581$	2.40136	0.0996254
$582$	0	0
$583$	−35.0602	−1.45205
$584$	0	0
$585$	1.98693	0.0821494
$586$	0	0
$587$	−20.0429	−0.827260	−0.413630	−	0.910445i	$-0.635739\pi$
$587$	−20.0429	−0.827260	−0.413630	+	0.910445i	$0.635739\pi$
$588$	0	0
$589$	6.95226	0.286463
$590$	0	0
$591$	−10.8819	−0.447621
$592$	0	0
$593$	−11.7074	−0.480766	−0.240383	−	0.970678i	$-0.577273\pi$
$593$	−11.7074	−0.480766	−0.240383	+	0.970678i	$0.577273\pi$
$594$	0	0
$595$	2.38029	0.0975823
$596$	0	0
$597$	14.2036	0.581314
$598$	0	0
$599$	2.41822	0.0988059	0.0494029	−	0.998779i	$-0.484268\pi$
$599$	2.41822	0.0988059	0.0494029	+	0.998779i	$0.484268\pi$
$600$	0	0
$601$	−9.78632	−0.399192	−0.199596	−	0.979878i	$-0.563963\pi$
$601$	−9.78632	−0.399192	−0.199596	+	0.979878i	$0.563963\pi$
$602$	0	0
$603$	−0.358299	−0.0145910
$604$	0	0
$605$	−4.66174	−0.189527
$606$	0	0
$607$	23.6574	0.960226	0.480113	−	0.877207i	$-0.340596\pi$
$607$	23.6574	0.960226	0.480113	+	0.877207i	$0.340596\pi$
$608$	0	0
$609$	−6.41443	−0.259926
$610$	0	0
$611$	32.7402	1.32453
$612$	0	0
$613$	−27.9055	−1.12709	−0.563547	−	0.826084i	$-0.690564\pi$
$613$	−27.9055	−1.12709	−0.563547	+	0.826084i	$0.690564\pi$
$614$	0	0
$615$	0.209947	0.00846590
$616$	0	0
$617$	1.12898	0.0454510	0.0227255	−	0.999742i	$-0.492766\pi$
$617$	1.12898	0.0454510	0.0227255	+	0.999742i	$0.492766\pi$
$618$	0	0
$619$	48.2125	1.93782	0.968911	−	0.247408i	$-0.0795787\pi$
$619$	48.2125	1.93782	0.968911	+	0.247408i	$0.0795787\pi$
$620$	0	0
$621$	1.98344	0.0795929
$622$	0	0
$623$	−16.4540	−0.659216
$624$	0	0
$625$	22.2482	0.889928
$626$	0	0
$627$	4.67080	0.186534
$628$	0	0
$629$	−21.8362	−0.870668
$630$	0	0
$631$	29.4819	1.17366	0.586828	−	0.809711i	$-0.300376\pi$
$631$	29.4819	1.17366	0.586828	+	0.809711i	$0.300376\pi$
$632$	0	0
$633$	11.4963	0.456938
$634$	0	0
$635$	−0.937246	−0.0371935
$636$	0	0
$637$	−4.61014	−0.182660
$638$	0	0
$639$	13.0852	0.517644
$640$	0	0
$641$	−4.03313	−0.159299	−0.0796495	−	0.996823i	$-0.525380\pi$
$641$	−4.03313	−0.159299	−0.0796495	+	0.996823i	$0.525380\pi$
$642$	0	0
$643$	−29.0555	−1.14584	−0.572918	−	0.819613i	$-0.694188\pi$
$643$	−29.0555	−1.14584	−0.572918	+	0.819613i	$0.694188\pi$
$644$	0	0
$645$	−0.970524	−0.0382143
$646$	0	0
$647$	−9.75943	−0.383683	−0.191841	−	0.981426i	$-0.561446\pi$
$647$	−9.75943	−0.383683	−0.191841	+	0.981426i	$0.561446\pi$
$648$	0	0
$649$	12.5521	0.492712
$650$	0	0
$651$	−6.95226	−0.272481
$652$	0	0
$653$	−20.1174	−0.787256	−0.393628	−	0.919270i	$-0.628780\pi$
$653$	−20.1174	−0.787256	−0.393628	+	0.919270i	$0.628780\pi$
$654$	0	0
$655$	0.340341	0.0132982
$656$	0	0
$657$	−1.77919	−0.0694127
$658$	0	0
$659$	−17.4676	−0.680441	−0.340221	−	0.940346i	$-0.610502\pi$
$659$	−17.4676	−0.680441	−0.340221	+	0.940346i	$0.610502\pi$
$660$	0	0
$661$	−16.2717	−0.632898	−0.316449	−	0.948610i	$-0.602491\pi$
$661$	−16.2717	−0.632898	−0.316449	+	0.948610i	$0.602491\pi$
$662$	0	0
$663$	−25.4610	−0.988823
$664$	0	0
$665$	0.430991	0.0167131
$666$	0	0
$667$	12.7227	0.492624
$668$	0	0
$669$	16.9223	0.654252
$670$	0	0
$671$	−10.4152	−0.402074
$672$	0	0
$673$	41.0755	1.58335	0.791673	−	0.610946i	$-0.209211\pi$
$673$	41.0755	1.58335	0.791673	+	0.610946i	$0.209211\pi$
$674$	0	0
$675$	−4.81425	−0.185300
$676$	0	0
$677$	2.81542	0.108205	0.0541026	−	0.998535i	$-0.482770\pi$
$677$	2.81542	0.108205	0.0541026	+	0.998535i	$0.482770\pi$
$678$	0	0
$679$	−3.22028	−0.123583
$680$	0	0
$681$	6.29594	0.241261
$682$	0	0
$683$	28.9174	1.10649	0.553247	−	0.833017i	$-0.313388\pi$
$683$	28.9174	1.10649	0.553247	+	0.833017i	$0.313388\pi$
$684$	0	0
$685$	−7.03198	−0.268678
$686$	0	0
$687$	8.76651	0.334463
$688$	0	0
$689$	34.6050	1.31834
$690$	0	0
$691$	−35.4760	−1.34957	−0.674785	−	0.738015i	$-0.735764\pi$
$691$	−35.4760	−1.34957	−0.674785	+	0.738015i	$0.735764\pi$
$692$	0	0
$693$	−4.67080	−0.177429
$694$	0	0
$695$	−8.14745	−0.309050
$696$	0	0
$697$	−2.69032	−0.101903
$698$	0	0
$699$	−6.76961	−0.256050
$700$	0	0
$701$	30.5104	1.15236	0.576181	−	0.817322i	$-0.304542\pi$
$701$	30.5104	1.15236	0.576181	+	0.817322i	$0.304542\pi$
$702$	0	0
$703$	−3.95382	−0.149121
$704$	0	0
$705$	3.06081	0.115277
$706$	0	0
$707$	−19.8188	−0.745361
$708$	0	0
$709$	43.7808	1.64422	0.822112	−	0.569326i	$-0.192796\pi$
$709$	43.7808	1.64422	0.822112	+	0.569326i	$0.192796\pi$
$710$	0	0
$711$	−13.1117	−0.491729
$712$	0	0
$713$	13.7894	0.516418
$714$	0	0
$715$	9.28054	0.347072
$716$	0	0
$717$	−1.87117	−0.0698802
$718$	0	0
$719$	−14.5597	−0.542985	−0.271492	−	0.962441i	$-0.587517\pi$
$719$	−14.5597	−0.542985	−0.271492	+	0.962441i	$0.587517\pi$
$720$	0	0
$721$	9.34159	0.347899
$722$	0	0
$723$	13.1579	0.489348
$724$	0	0
$725$	−30.8807	−1.14688
$726$	0	0
$727$	−35.4537	−1.31491	−0.657453	−	0.753496i	$-0.728366\pi$
$727$	−35.4537	−1.31491	−0.657453	+	0.753496i	$0.728366\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.4365	0.459982
$732$	0	0
$733$	2.00373	0.0740095	0.0370047	−	0.999315i	$-0.488218\pi$
$733$	2.00373	0.0740095	0.0370047	+	0.999315i	$0.488218\pi$
$734$	0	0
$735$	−0.430991	−0.0158973
$736$	0	0
$737$	−1.67354	−0.0616456
$738$	0	0
$739$	−33.6847	−1.23911	−0.619556	−	0.784952i	$-0.712688\pi$
$739$	−33.6847	−1.23911	−0.619556	+	0.784952i	$0.712688\pi$
$740$	0	0
$741$	−4.61014	−0.169358
$742$	0	0
$743$	−18.0757	−0.663132	−0.331566	−	0.943432i	$-0.607577\pi$
$743$	−18.0757	−0.663132	−0.331566	+	0.943432i	$0.607577\pi$
$744$	0	0
$745$	−1.05622	−0.0386967
$746$	0	0
$747$	−2.40136	−0.0878613
$748$	0	0
$749$	1.22688	0.0448294
$750$	0	0
$751$	−6.55814	−0.239310	−0.119655	−	0.992816i	$-0.538179\pi$
$751$	−6.55814	−0.239310	−0.119655	+	0.992816i	$0.538179\pi$
$752$	0	0
$753$	10.2067	0.371953
$754$	0	0
$755$	−7.74651	−0.281924
$756$	0	0
$757$	−44.0279	−1.60022	−0.800110	−	0.599853i	$-0.795226\pi$
$757$	−44.0279	−1.60022	−0.800110	+	0.599853i	$0.795226\pi$
$758$	0	0
$759$	9.26426	0.336271
$760$	0	0
$761$	−8.08173	−0.292963	−0.146481	−	0.989213i	$-0.546795\pi$
$761$	−8.08173	−0.292963	−0.146481	+	0.989213i	$0.546795\pi$
$762$	0	0
$763$	2.11227	0.0764694
$764$	0	0
$765$	−2.38029	−0.0860595
$766$	0	0
$767$	−12.3891	−0.447343
$768$	0	0
$769$	−12.9570	−0.467242	−0.233621	−	0.972328i	$-0.575057\pi$
$769$	−12.9570	−0.467242	−0.233621	+	0.972328i	$0.575057\pi$
$770$	0	0
$771$	8.96674	0.322929
$772$	0	0
$773$	−6.70111	−0.241022	−0.120511	−	0.992712i	$-0.538453\pi$
$773$	−6.70111	−0.241022	−0.120511	+	0.992712i	$0.538453\pi$
$774$	0	0
$775$	−33.4699	−1.20227
$776$	0	0
$777$	3.95382	0.141842
$778$	0	0
$779$	−0.487127	−0.0174531
$780$	0	0
$781$	61.1184	2.18699
$782$	0	0
$783$	6.41443	0.229233
$784$	0	0
$785$	−2.86228	−0.102159
$786$	0	0
$787$	5.64379	0.201179	0.100590	−	0.994928i	$-0.467927\pi$
$787$	5.64379	0.201179	0.100590	+	0.994928i	$0.467927\pi$
$788$	0	0
$789$	11.3491	0.404039
$790$	0	0
$791$	−10.4144	−0.370295
$792$	0	0
$793$	10.2799	0.365051
$794$	0	0
$795$	3.23513	0.114738
$796$	0	0
$797$	−34.0525	−1.20620	−0.603101	−	0.797665i	$-0.706068\pi$
$797$	−34.0525	−1.20620	−0.603101	+	0.797665i	$0.706068\pi$
$798$	0	0
$799$	−39.2219	−1.38757
$800$	0	0
$801$	16.4540	0.581374
$802$	0	0
$803$	−8.31022	−0.293261
$804$	0	0
$805$	0.854846	0.0301294
$806$	0	0
$807$	−1.39750	−0.0491942
$808$	0	0
$809$	3.90376	0.137249	0.0686245	−	0.997643i	$-0.478139\pi$
$809$	3.90376	0.137249	0.0686245	+	0.997643i	$0.478139\pi$
$810$	0	0
$811$	−24.6511	−0.865618	−0.432809	−	0.901486i	$-0.642478\pi$
$811$	−24.6511	−0.865618	−0.432809	+	0.901486i	$0.642478\pi$
$812$	0	0
$813$	−29.5492	−1.03634
$814$	0	0
$815$	7.81230	0.273653
$816$	0	0
$817$	2.25184	0.0787820
$818$	0	0
$819$	4.61014	0.161091
$820$	0	0
$821$	17.9234	0.625530	0.312765	−	0.949831i	$-0.398745\pi$
$821$	17.9234	0.625530	0.312765	+	0.949831i	$0.398745\pi$
$822$	0	0
$823$	−41.8653	−1.45933	−0.729666	−	0.683803i	$-0.760325\pi$
$823$	−41.8653	−1.45933	−0.729666	+	0.683803i	$0.760325\pi$
$824$	0	0
$825$	−22.4864	−0.782875
$826$	0	0
$827$	−40.8151	−1.41928	−0.709641	−	0.704564i	$-0.751143\pi$
$827$	−40.8151	−1.41928	−0.709641	+	0.704564i	$0.751143\pi$
$828$	0	0
$829$	−50.9370	−1.76911	−0.884557	−	0.466432i	$-0.845539\pi$
$829$	−50.9370	−1.76911	−0.884557	+	0.466432i	$0.845539\pi$
$830$	0	0
$831$	−4.09028	−0.141890
$832$	0	0
$833$	5.52283	0.191355
$834$	0	0
$835$	−3.60235	−0.124665
$836$	0	0
$837$	6.95226	0.240305
$838$	0	0
$839$	−12.2527	−0.423011	−0.211505	−	0.977377i	$-0.567837\pi$
$839$	−12.2527	−0.423011	−0.211505	+	0.977377i	$0.567837\pi$
$840$	0	0
$841$	12.1450	0.418792
$842$	0	0
$843$	17.2302	0.593441
$844$	0	0
$845$	−3.55714	−0.122369
$846$	0	0
$847$	−10.8163	−0.371654
$848$	0	0
$849$	29.6999	1.01930
$850$	0	0
$851$	−7.84217	−0.268826
$852$	0	0
$853$	−31.7198	−1.08607	−0.543033	−	0.839712i	$-0.682724\pi$
$853$	−31.7198	−1.08607	−0.543033	+	0.839712i	$0.682724\pi$
$854$	0	0
$855$	−0.430991	−0.0147396
$856$	0	0
$857$	35.4906	1.21234	0.606168	−	0.795336i	$-0.292706\pi$
$857$	35.4906	1.21234	0.606168	+	0.795336i	$0.292706\pi$
$858$	0	0
$859$	−24.2527	−0.827492	−0.413746	−	0.910392i	$-0.635780\pi$
$859$	−24.2527	−0.827492	−0.413746	+	0.910392i	$0.635780\pi$
$860$	0	0
$861$	0.487127	0.0166012
$862$	0	0
$863$	22.2259	0.756579	0.378289	−	0.925687i	$-0.376512\pi$
$863$	22.2259	0.756579	0.378289	+	0.925687i	$0.376512\pi$
$864$	0	0
$865$	4.38193	0.148990
$866$	0	0
$867$	13.5016	0.458538
$868$	0	0
$869$	−61.2423	−2.07750
$870$	0	0
$871$	1.65181	0.0559693
$872$	0	0
$873$	3.22028	0.108990
$874$	0	0
$875$	−4.22985	−0.142995
$876$	0	0
$877$	25.3600	0.856345	0.428173	−	0.903697i	$-0.359158\pi$
$877$	25.3600	0.856345	0.428173	+	0.903697i	$0.359158\pi$
$878$	0	0
$879$	−27.0490	−0.912340
$880$	0	0
$881$	7.42362	0.250108	0.125054	−	0.992150i	$-0.460090\pi$
$881$	7.42362	0.250108	0.125054	+	0.992150i	$0.460090\pi$
$882$	0	0
$883$	14.7424	0.496123	0.248061	−	0.968744i	$-0.420207\pi$
$883$	14.7424	0.496123	0.248061	+	0.968744i	$0.420207\pi$
$884$	0	0
$885$	−1.15822	−0.0389333
$886$	0	0
$887$	40.5456	1.36139	0.680693	−	0.732568i	$-0.261679\pi$
$887$	40.5456	1.36139	0.680693	+	0.732568i	$0.261679\pi$
$888$	0	0
$889$	−2.17463	−0.0729348
$890$	0	0
$891$	4.67080	0.156478
$892$	0	0
$893$	−7.10179	−0.237652
$894$	0	0
$895$	−0.823681	−0.0275326
$896$	0	0
$897$	−9.14395	−0.305308
$898$	0	0
$899$	44.5948	1.48732
$900$	0	0
$901$	−41.4558	−1.38109
$902$	0	0
$903$	−2.25184	−0.0749366
$904$	0	0
$905$	−5.63114	−0.187185
$906$	0	0
$907$	45.5330	1.51190	0.755950	−	0.654630i	$-0.227175\pi$
$907$	45.5330	1.51190	0.755950	+	0.654630i	$0.227175\pi$
$908$	0	0
$909$	19.8188	0.657347
$910$	0	0
$911$	33.8615	1.12188	0.560941	−	0.827855i	$-0.310439\pi$
$911$	33.8615	1.12188	0.560941	+	0.827855i	$0.310439\pi$
$912$	0	0
$913$	−11.2163	−0.371205
$914$	0	0
$915$	0.961046	0.0317712
$916$	0	0
$917$	0.789672	0.0260773
$918$	0	0
$919$	−9.24999	−0.305129	−0.152564	−	0.988294i	$-0.548753\pi$
$919$	−9.24999	−0.305129	−0.152564	+	0.988294i	$0.548753\pi$
$920$	0	0
$921$	22.3893	0.737753
$922$	0	0
$923$	−60.3247	−1.98561
$924$	0	0
$925$	19.0346	0.625855
$926$	0	0
$927$	−9.34159	−0.306818
$928$	0	0
$929$	48.7467	1.59933	0.799664	−	0.600448i	$-0.205011\pi$
$929$	48.7467	1.59933	0.799664	+	0.600448i	$0.205011\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−18.0389	−0.590567
$934$	0	0
$935$	−11.1178	−0.363592
$936$	0	0
$937$	33.4921	1.09414	0.547070	−	0.837087i	$-0.315743\pi$
$937$	33.4921	1.09414	0.547070	+	0.837087i	$0.315743\pi$
$938$	0	0
$939$	−10.6673	−0.348115
$940$	0	0
$941$	−23.3742	−0.761977	−0.380988	−	0.924580i	$-0.624416\pi$
$941$	−23.3742	−0.761977	−0.380988	+	0.924580i	$0.624416\pi$
$942$	0	0
$943$	−0.966189	−0.0314634
$944$	0	0
$945$	0.430991	0.0140201
$946$	0	0
$947$	23.9643	0.778735	0.389368	−	0.921082i	$-0.372694\pi$
$947$	23.9643	0.778735	0.389368	+	0.921082i	$0.372694\pi$
$948$	0	0
$949$	8.20230	0.266258
$950$	0	0
$951$	15.7560	0.510924
$952$	0	0
$953$	36.5266	1.18321	0.591606	−	0.806227i	$-0.298494\pi$
$953$	36.5266	1.18321	0.591606	+	0.806227i	$0.298494\pi$
$954$	0	0
$955$	8.22288	0.266086
$956$	0	0
$957$	29.9605	0.968486
$958$	0	0
$959$	−16.3158	−0.526866
$960$	0	0
$961$	17.3340	0.559161
$962$	0	0
$963$	−1.22688	−0.0395358
$964$	0	0
$965$	2.11927	0.0682217
$966$	0	0
$967$	15.6636	0.503706	0.251853	−	0.967766i	$-0.418960\pi$
$967$	15.6636	0.503706	0.251853	+	0.967766i	$0.418960\pi$
$968$	0	0
$969$	5.52283	0.177419
$970$	0	0
$971$	5.25339	0.168589	0.0842947	−	0.996441i	$-0.473136\pi$
$971$	5.25339	0.168589	0.0842947	+	0.996441i	$0.473136\pi$
$972$	0	0
$973$	−18.9040	−0.606034
$974$	0	0
$975$	22.1944	0.710788
$976$	0	0
$977$	21.9525	0.702322	0.351161	−	0.936315i	$-0.385787\pi$
$977$	21.9525	0.702322	0.351161	+	0.936315i	$0.385787\pi$
$978$	0	0
$979$	76.8533	2.45624
$980$	0	0
$981$	−2.11227	−0.0674397
$982$	0	0
$983$	18.7244	0.597214	0.298607	−	0.954376i	$-0.403478\pi$
$983$	18.7244	0.597214	0.298607	+	0.954376i	$0.403478\pi$
$984$	0	0
$985$	4.69000	0.149436
$986$	0	0
$987$	7.10179	0.226052
$988$	0	0
$989$	4.46640	0.142023
$990$	0	0
$991$	−20.1060	−0.638689	−0.319345	−	0.947639i	$-0.603463\pi$
$991$	−20.1060	−0.638689	−0.319345	+	0.947639i	$0.603463\pi$
$992$	0	0
$993$	22.2790	0.707003
$994$	0	0
$995$	−6.12161	−0.194068
$996$	0	0
$997$	29.5676	0.936415	0.468207	−	0.883619i	$-0.344900\pi$
$997$	29.5676	0.936415	0.468207	+	0.883619i	$0.344900\pi$
$998$	0	0
$999$	−3.95382	−0.125093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cf.1.2		5
4.3	odd	2		399.2.a.g.1.5	✓	5
12.11	even	2		1197.2.a.o.1.1		5
20.19	odd	2		9975.2.a.bp.1.1		5
28.27	even	2		2793.2.a.bg.1.5		5
76.75	even	2		7581.2.a.w.1.1		5
84.83	odd	2		8379.2.a.cb.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.g.1.5	✓	5	4.3	odd	2
1197.2.a.o.1.1		5	12.11	even	2
2793.2.a.bg.1.5		5	28.27	even	2
6384.2.a.cf.1.2		5	1.1	even	1	trivial
7581.2.a.w.1.1		5	76.75	even	2
8379.2.a.cb.1.1		5	84.83	odd	2
9975.2.a.bp.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.430991 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.430991 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.67080 q^{11} -4.61014 q^{13} -0.430991 q^{15} +5.52283 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.98344 q^{23} -4.81425 q^{25} +1.00000 q^{27} +6.41443 q^{29} +6.95226 q^{31} +4.67080 q^{33} +0.430991 q^{35} -3.95382 q^{37} -4.61014 q^{39} -0.487127 q^{41} +2.25184 q^{43} -0.430991 q^{45} -7.10179 q^{47} +1.00000 q^{49} +5.52283 q^{51} -7.50627 q^{53} -2.01307 q^{55} +1.00000 q^{57} +2.68735 q^{59} -2.22985 q^{61} -1.00000 q^{63} +1.98693 q^{65} -0.358299 q^{67} +1.98344 q^{69} +13.0852 q^{71} -1.77919 q^{73} -4.81425 q^{75} -4.67080 q^{77} -13.1117 q^{79} +1.00000 q^{81} -2.40136 q^{83} -2.38029 q^{85} +6.41443 q^{87} +16.4540 q^{89} +4.61014 q^{91} +6.95226 q^{93} -0.430991 q^{95} +3.22028 q^{97} +4.67080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 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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