LMFDB - Embedded newform 6864.2.a.cc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23252.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2 $ x^4 - x^3 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.634868$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.962075 q^{5} -1.88053 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.962075 q^{5} -1.88053 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.962075 q^{15} +6.11234 q^{17} -1.88053 q^{19} -1.88053 q^{21} -1.15026 q^{23} -4.07441 q^{25} +1.00000 q^{27} +0.918451 q^{29} +6.72313 q^{31} -1.00000 q^{33} +1.80921 q^{35} +11.5703 q^{37} -1.00000 q^{39} -2.42000 q^{41} -8.11234 q^{43} -0.962075 q^{45} +5.19389 q^{47} -3.46362 q^{49} +6.11234 q^{51} -9.98573 q^{53} +0.962075 q^{55} -1.88053 q^{57} -14.5252 q^{59} +3.19389 q^{61} -1.88053 q^{63} +0.962075 q^{65} +13.0237 q^{67} -1.15026 q^{69} -1.57026 q^{71} +4.49585 q^{73} -4.07441 q^{75} +1.88053 q^{77} -0.727658 q^{79} +1.00000 q^{81} +0.376374 q^{83} -5.88053 q^{85} +0.918451 q^{87} +8.23181 q^{89} +1.88053 q^{91} +6.72313 q^{93} +1.80921 q^{95} -8.87909 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 12 q^{25} + 4 q^{27} + 6 q^{29} + 8 q^{31} - 4 q^{33} - 14 q^{35} + 14 q^{37} - 4 q^{39} + 10 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{47} + 16 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 8 q^{59} - 10 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 26 q^{71} + 14 q^{73} + 12 q^{75} + 2 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{83} - 18 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 8 q^{93} - 14 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 12 * q^25 + 4 * q^27 + 6 * q^29 + 8 * q^31 - 4 * q^33 - 14 * q^35 + 14 * q^37 - 4 * q^39 + 10 * q^41 - 12 * q^43 + 4 * q^45 - 2 * q^47 + 16 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 8 * q^59 - 10 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 26 * q^71 + 14 * q^73 + 12 * q^75 + 2 * q^77 + 16 * q^79 + 4 * q^81 - 8 * q^83 - 18 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 8 * q^93 - 14 * q^95 + 14 * q^97 - 4 * 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
$5$	−0.962075	−0.430253	−0.215126	−	0.976586i	$-0.569016\pi$
$5$	−0.962075	−0.430253	−0.215126	+	0.976586i	$0.569016\pi$
$6$	0	0
$7$	−1.88053	−0.710772	−0.355386	−	0.934720i	$-0.615651\pi$
$7$	−1.88053	−0.710772	−0.355386	+	0.934720i	$0.615651\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.962075	−0.248407
$16$	0	0
$17$	6.11234	1.48246	0.741230	−	0.671251i	$-0.234243\pi$
$17$	6.11234	1.48246	0.741230	+	0.671251i	$0.234243\pi$
$18$	0	0
$19$	−1.88053	−0.431422	−0.215711	−	0.976457i	$-0.569207\pi$
$19$	−1.88053	−0.431422	−0.215711	+	0.976457i	$0.569207\pi$
$20$	0	0
$21$	−1.88053	−0.410364
$22$	0	0
$23$	−1.15026	−0.239846	−0.119923	−	0.992783i	$-0.538265\pi$
$23$	−1.15026	−0.239846	−0.119923	+	0.992783i	$0.538265\pi$
$24$	0	0
$25$	−4.07441	−0.814882
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.918451	0.170552	0.0852761	−	0.996357i	$-0.472823\pi$
$29$	0.918451	0.170552	0.0852761	+	0.996357i	$0.472823\pi$
$30$	0	0
$31$	6.72313	1.20751	0.603755	−	0.797170i	$-0.293671\pi$
$31$	6.72313	1.20751	0.603755	+	0.797170i	$0.293671\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	1.80921	0.305812
$36$	0	0
$37$	11.5703	1.90214	0.951069	−	0.308977i	$-0.0999867\pi$
$37$	11.5703	1.90214	0.951069	+	0.308977i	$0.0999867\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.42000	−0.377940	−0.188970	−	0.981983i	$-0.560515\pi$
$41$	−2.42000	−0.377940	−0.188970	+	0.981983i	$0.560515\pi$
$42$	0	0
$43$	−8.11234	−1.23712	−0.618560	−	0.785738i	$-0.712284\pi$
$43$	−8.11234	−1.23712	−0.618560	+	0.785738i	$0.712284\pi$
$44$	0	0
$45$	−0.962075	−0.143418
$46$	0	0
$47$	5.19389	0.757606	0.378803	−	0.925477i	$-0.376336\pi$
$47$	5.19389	0.757606	0.378803	+	0.925477i	$0.376336\pi$
$48$	0	0
$49$	−3.46362	−0.494803
$50$	0	0
$51$	6.11234	0.855898
$52$	0	0
$53$	−9.98573	−1.37164	−0.685822	−	0.727769i	$-0.740557\pi$
$53$	−9.98573	−1.37164	−0.685822	+	0.727769i	$0.740557\pi$
$54$	0	0
$55$	0.962075	0.129726
$56$	0	0
$57$	−1.88053	−0.249082
$58$	0	0
$59$	−14.5252	−1.89102	−0.945510	−	0.325594i	$-0.894436\pi$
$59$	−14.5252	−1.89102	−0.945510	+	0.325594i	$0.894436\pi$
$60$	0	0
$61$	3.19389	0.408935	0.204468	−	0.978873i	$-0.434454\pi$
$61$	3.19389	0.408935	0.204468	+	0.978873i	$0.434454\pi$
$62$	0	0
$63$	−1.88053	−0.236924
$64$	0	0
$65$	0.962075	0.119331
$66$	0	0
$67$	13.0237	1.59109	0.795546	−	0.605893i	$-0.207184\pi$
$67$	13.0237	1.59109	0.795546	+	0.605893i	$0.207184\pi$
$68$	0	0
$69$	−1.15026	−0.138475
$70$	0	0
$71$	−1.57026	−0.186356	−0.0931778	−	0.995649i	$-0.529702\pi$
$71$	−1.57026	−0.186356	−0.0931778	+	0.995649i	$0.529702\pi$
$72$	0	0
$73$	4.49585	0.526199	0.263100	−	0.964769i	$-0.415255\pi$
$73$	4.49585	0.526199	0.263100	+	0.964769i	$0.415255\pi$
$74$	0	0
$75$	−4.07441	−0.470473
$76$	0	0
$77$	1.88053	0.214306
$78$	0	0
$79$	−0.727658	−0.0818680	−0.0409340	−	0.999162i	$-0.513033\pi$
$79$	−0.727658	−0.0818680	−0.0409340	+	0.999162i	$0.513033\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.376374	0.0413124	0.0206562	−	0.999787i	$-0.493424\pi$
$83$	0.376374	0.0413124	0.0206562	+	0.999787i	$0.493424\pi$
$84$	0	0
$85$	−5.88053	−0.637833
$86$	0	0
$87$	0.918451	0.0984683
$88$	0	0
$89$	8.23181	0.872570	0.436285	−	0.899808i	$-0.356294\pi$
$89$	8.23181	0.872570	0.436285	+	0.899808i	$0.356294\pi$
$90$	0	0
$91$	1.88053	0.197133
$92$	0	0
$93$	6.72313	0.697156
$94$	0	0
$95$	1.80921	0.185621
$96$	0	0
$97$	−8.87909	−0.901535	−0.450767	−	0.892641i	$-0.648850\pi$
$97$	−8.87909	−0.901535	−0.450767	+	0.892641i	$0.648850\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	16.0365	1.59569	0.797845	−	0.602863i	$-0.205973\pi$
$101$	16.0365	1.59569	0.797845	+	0.602863i	$0.205973\pi$
$102$	0	0
$103$	16.3621	1.61221	0.806103	−	0.591776i	$-0.201573\pi$
$103$	16.3621	1.61221	0.806103	+	0.591776i	$0.201573\pi$
$104$	0	0
$105$	1.80921	0.176561
$106$	0	0
$107$	−3.38468	−0.327209	−0.163605	−	0.986526i	$-0.552312\pi$
$107$	−3.38468	−0.327209	−0.163605	+	0.986526i	$0.552312\pi$
$108$	0	0
$109$	13.5657	1.29936	0.649681	−	0.760207i	$-0.274903\pi$
$109$	13.5657	1.29936	0.649681	+	0.760207i	$0.274903\pi$
$110$	0	0
$111$	11.5703	1.09820
$112$	0	0
$113$	16.7641	1.57704	0.788519	−	0.615010i	$-0.210848\pi$
$113$	16.7641	1.57704	0.788519	+	0.615010i	$0.210848\pi$
$114$	0	0
$115$	1.10664	0.103195
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−11.4944	−1.05369
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.42000	−0.218204
$124$	0	0
$125$	8.73026	0.780859
$126$	0	0
$127$	1.69687	0.150573	0.0752864	−	0.997162i	$-0.476013\pi$
$127$	1.69687	0.150573	0.0752864	+	0.997162i	$0.476013\pi$
$128$	0	0
$129$	−8.11234	−0.714251
$130$	0	0
$131$	−12.8400	−1.12184	−0.560918	−	0.827872i	$-0.689552\pi$
$131$	−12.8400	−1.12184	−0.560918	+	0.827872i	$0.689552\pi$
$132$	0	0
$133$	3.53638	0.306643
$134$	0	0
$135$	−0.962075	−0.0828022
$136$	0	0
$137$	13.5406	1.15685	0.578427	−	0.815734i	$-0.303667\pi$
$137$	13.5406	1.15685	0.578427	+	0.815734i	$0.303667\pi$
$138$	0	0
$139$	−13.6826	−1.16054	−0.580271	−	0.814423i	$-0.697053\pi$
$139$	−13.6826	−1.16054	−0.580271	+	0.814423i	$0.697053\pi$
$140$	0	0
$141$	5.19389	0.437404
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.883619	−0.0733806
$146$	0	0
$147$	−3.46362	−0.285675
$148$	0	0
$149$	−8.72052	−0.714413	−0.357206	−	0.934025i	$-0.616271\pi$
$149$	−8.72052	−0.714413	−0.357206	+	0.934025i	$0.616271\pi$
$150$	0	0
$151$	13.6416	1.11014	0.555068	−	0.831805i	$-0.312692\pi$
$151$	13.6416	1.11014	0.555068	+	0.831805i	$0.312692\pi$
$152$	0	0
$153$	6.11234	0.494153
$154$	0	0
$155$	−6.46815	−0.519535
$156$	0	0
$157$	2.53494	0.202310	0.101155	−	0.994871i	$-0.467746\pi$
$157$	2.53494	0.202310	0.101155	+	0.994871i	$0.467746\pi$
$158$	0	0
$159$	−9.98573	−0.791920
$160$	0	0
$161$	2.16310	0.170476
$162$	0	0
$163$	−1.61649	−0.126613	−0.0633066	−	0.997994i	$-0.520165\pi$
$163$	−1.61649	−0.126613	−0.0633066	+	0.997994i	$0.520165\pi$
$164$	0	0
$165$	0.962075	0.0748974
$166$	0	0
$167$	7.60935	0.588829	0.294415	−	0.955678i	$-0.404875\pi$
$167$	7.60935	0.588829	0.294415	+	0.955678i	$0.404875\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.88053	−0.143807
$172$	0	0
$173$	24.8765	1.89132	0.945662	−	0.325151i	$-0.105415\pi$
$173$	24.8765	1.89132	0.945662	+	0.325151i	$0.105415\pi$
$174$	0	0
$175$	7.66204	0.579196
$176$	0	0
$177$	−14.5252	−1.09178
$178$	0	0
$179$	5.61388	0.419601	0.209801	−	0.977744i	$-0.432718\pi$
$179$	5.61388	0.419601	0.209801	+	0.977744i	$0.432718\pi$
$180$	0	0
$181$	1.46506	0.108897	0.0544485	−	0.998517i	$-0.482660\pi$
$181$	1.46506	0.108897	0.0544485	+	0.998517i	$0.482660\pi$
$182$	0	0
$183$	3.19389	0.236099
$184$	0	0
$185$	−11.1315	−0.818401
$186$	0	0
$187$	−6.11234	−0.446978
$188$	0	0
$189$	−1.88053	−0.136788
$190$	0	0
$191$	24.8257	1.79633	0.898163	−	0.439662i	$-0.144902\pi$
$191$	24.8257	1.79633	0.898163	+	0.439662i	$0.144902\pi$
$192$	0	0
$193$	−18.4057	−1.32487	−0.662436	−	0.749118i	$-0.730478\pi$
$193$	−18.4057	−1.32487	−0.662436	+	0.749118i	$0.730478\pi$
$194$	0	0
$195$	0.962075	0.0688956
$196$	0	0
$197$	26.5574	1.89214	0.946069	−	0.323965i	$-0.105016\pi$
$197$	26.5574	1.89214	0.946069	+	0.323965i	$0.105016\pi$
$198$	0	0
$199$	−10.3005	−0.730185	−0.365092	−	0.930971i	$-0.618963\pi$
$199$	−10.3005	−0.730185	−0.365092	+	0.930971i	$0.618963\pi$
$200$	0	0
$201$	13.0237	0.918618
$202$	0	0
$203$	−1.72717	−0.121224
$204$	0	0
$205$	2.32822	0.162610
$206$	0	0
$207$	−1.15026	−0.0799487
$208$	0	0
$209$	1.88053	0.130079
$210$	0	0
$211$	−9.56765	−0.658664	−0.329332	−	0.944214i	$-0.606824\pi$
$211$	−9.56765	−0.658664	−0.329332	+	0.944214i	$0.606824\pi$
$212$	0	0
$213$	−1.57026	−0.107592
$214$	0	0
$215$	7.80468	0.532274
$216$	0	0
$217$	−12.6430	−0.858264
$218$	0	0
$219$	4.49585	0.303801
$220$	0	0
$221$	−6.11234	−0.411160
$222$	0	0
$223$	−6.49324	−0.434820	−0.217410	−	0.976080i	$-0.569761\pi$
$223$	−6.49324	−0.434820	−0.217410	+	0.976080i	$0.569761\pi$
$224$	0	0
$225$	−4.07441	−0.271627
$226$	0	0
$227$	12.0293	0.798416	0.399208	−	0.916860i	$-0.369285\pi$
$227$	12.0293	0.798416	0.399208	+	0.916860i	$0.369285\pi$
$228$	0	0
$229$	20.7776	1.37302	0.686510	−	0.727120i	$-0.259142\pi$
$229$	20.7776	1.37302	0.686510	+	0.727120i	$0.259142\pi$
$230$	0	0
$231$	1.88053	0.123730
$232$	0	0
$233$	29.2304	1.91495	0.957473	−	0.288524i	$-0.0931645\pi$
$233$	29.2304	1.91495	0.957473	+	0.288524i	$0.0931645\pi$
$234$	0	0
$235$	−4.99691	−0.325962
$236$	0	0
$237$	−0.727658	−0.0472665
$238$	0	0
$239$	−9.25999	−0.598979	−0.299490	−	0.954100i	$-0.596816\pi$
$239$	−9.25999	−0.598979	−0.299490	+	0.954100i	$0.596816\pi$
$240$	0	0
$241$	24.3735	1.57003	0.785017	−	0.619474i	$-0.212654\pi$
$241$	24.3735	1.57003	0.785017	+	0.619474i	$0.212654\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.33226	0.212890
$246$	0	0
$247$	1.88053	0.119655
$248$	0	0
$249$	0.376374	0.0238517
$250$	0	0
$251$	−1.08415	−0.0684312	−0.0342156	−	0.999414i	$-0.510893\pi$
$251$	−1.08415	−0.0684312	−0.0342156	+	0.999414i	$0.510893\pi$
$252$	0	0
$253$	1.15026	0.0723163
$254$	0	0
$255$	−5.88053	−0.368253
$256$	0	0
$257$	−12.0616	−0.752380	−0.376190	−	0.926542i	$-0.622766\pi$
$257$	−12.0616	−0.752380	−0.376190	+	0.926542i	$0.622766\pi$
$258$	0	0
$259$	−21.7582	−1.35199
$260$	0	0
$261$	0.918451	0.0568507
$262$	0	0
$263$	−7.86257	−0.484827	−0.242414	−	0.970173i	$-0.577939\pi$
$263$	−7.86257	−0.484827	−0.242414	+	0.970173i	$0.577939\pi$
$264$	0	0
$265$	9.60702	0.590154
$266$	0	0
$267$	8.23181	0.503779
$268$	0	0
$269$	16.9940	1.03614	0.518072	−	0.855337i	$-0.326650\pi$
$269$	16.9940	1.03614	0.518072	+	0.855337i	$0.326650\pi$
$270$	0	0
$271$	−10.0180	−0.608547	−0.304274	−	0.952585i	$-0.598414\pi$
$271$	−10.0180	−0.608547	−0.304274	+	0.952585i	$0.598414\pi$
$272$	0	0
$273$	1.88053	0.113815
$274$	0	0
$275$	4.07441	0.245696
$276$	0	0
$277$	−9.28316	−0.557771	−0.278885	−	0.960324i	$-0.589965\pi$
$277$	−9.28316	−0.557771	−0.278885	+	0.960324i	$0.589965\pi$
$278$	0	0
$279$	6.72313	0.402503
$280$	0	0
$281$	31.2367	1.86342	0.931711	−	0.363200i	$-0.118316\pi$
$281$	31.2367	1.86342	0.931711	+	0.363200i	$0.118316\pi$
$282$	0	0
$283$	−13.0673	−0.776769	−0.388384	−	0.921497i	$-0.626967\pi$
$283$	−13.0673	−0.776769	−0.388384	+	0.921497i	$0.626967\pi$
$284$	0	0
$285$	1.80921	0.107168
$286$	0	0
$287$	4.55087	0.268629
$288$	0	0
$289$	20.3607	1.19769
$290$	0	0
$291$	−8.87909	−0.520501
$292$	0	0
$293$	5.95638	0.347975	0.173988	−	0.984748i	$-0.444335\pi$
$293$	5.95638	0.347975	0.173988	+	0.984748i	$0.444335\pi$
$294$	0	0
$295$	13.9743	0.813617
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	1.15026	0.0665214
$300$	0	0
$301$	15.2555	0.879310
$302$	0	0
$303$	16.0365	0.921272
$304$	0	0
$305$	−3.07276	−0.175946
$306$	0	0
$307$	18.3299	1.04614	0.523071	−	0.852289i	$-0.324786\pi$
$307$	18.3299	1.04614	0.523071	+	0.852289i	$0.324786\pi$
$308$	0	0
$309$	16.3621	0.930807
$310$	0	0
$311$	19.0601	1.08080	0.540401	−	0.841408i	$-0.318273\pi$
$311$	19.0601	1.08080	0.540401	+	0.841408i	$0.318273\pi$
$312$	0	0
$313$	11.4765	0.648688	0.324344	−	0.945939i	$-0.394857\pi$
$313$	11.4765	0.648688	0.324344	+	0.945939i	$0.394857\pi$
$314$	0	0
$315$	1.80921	0.101937
$316$	0	0
$317$	3.15287	0.177083	0.0885413	−	0.996073i	$-0.471779\pi$
$317$	3.15287	0.177083	0.0885413	+	0.996073i	$0.471779\pi$
$318$	0	0
$319$	−0.918451	−0.0514234
$320$	0	0
$321$	−3.38468	−0.188914
$322$	0	0
$323$	−11.4944	−0.639566
$324$	0	0
$325$	4.07441	0.226008
$326$	0	0
$327$	13.5657	0.750186
$328$	0	0
$329$	−9.76724	−0.538485
$330$	0	0
$331$	17.8412	0.980639	0.490319	−	0.871543i	$-0.336880\pi$
$331$	17.8412	0.980639	0.490319	+	0.871543i	$0.336880\pi$
$332$	0	0
$333$	11.5703	0.634046
$334$	0	0
$335$	−12.5297	−0.684572
$336$	0	0
$337$	−32.5252	−1.77176	−0.885880	−	0.463914i	$-0.846445\pi$
$337$	−32.5252	−1.77176	−0.885880	+	0.463914i	$0.846445\pi$
$338$	0	0
$339$	16.7641	0.910503
$340$	0	0
$341$	−6.72313	−0.364078
$342$	0	0
$343$	19.6771	1.06246
$344$	0	0
$345$	1.10664	0.0595794
$346$	0	0
$347$	−2.80090	−0.150360	−0.0751802	−	0.997170i	$-0.523953\pi$
$347$	−2.80090	−0.150360	−0.0751802	+	0.997170i	$0.523953\pi$
$348$	0	0
$349$	−29.6041	−1.58467	−0.792337	−	0.610084i	$-0.791136\pi$
$349$	−29.6041	−1.58467	−0.792337	+	0.610084i	$0.791136\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	9.11378	0.485077	0.242539	−	0.970142i	$-0.422020\pi$
$353$	9.11378	0.485077	0.242539	+	0.970142i	$0.422020\pi$
$354$	0	0
$355$	1.51071	0.0801800
$356$	0	0
$357$	−11.4944	−0.608349
$358$	0	0
$359$	−20.5431	−1.08423	−0.542113	−	0.840306i	$-0.682376\pi$
$359$	−20.5431	−1.08423	−0.542113	+	0.840306i	$0.682376\pi$
$360$	0	0
$361$	−15.4636	−0.813875
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−4.32534	−0.226399
$366$	0	0
$367$	−32.5868	−1.70102	−0.850508	−	0.525962i	$-0.823705\pi$
$367$	−32.5868	−1.70102	−0.850508	+	0.525962i	$0.823705\pi$
$368$	0	0
$369$	−2.42000	−0.125980
$370$	0	0
$371$	18.7784	0.974927
$372$	0	0
$373$	−14.6153	−0.756753	−0.378376	−	0.925652i	$-0.623517\pi$
$373$	−14.6153	−0.756753	−0.378376	+	0.925652i	$0.623517\pi$
$374$	0	0
$375$	8.73026	0.450829
$376$	0	0
$377$	−0.918451	−0.0473026
$378$	0	0
$379$	19.2965	0.991194	0.495597	−	0.868553i	$-0.334949\pi$
$379$	19.2965	0.991194	0.495597	+	0.868553i	$0.334949\pi$
$380$	0	0
$381$	1.69687	0.0869333
$382$	0	0
$383$	−7.89848	−0.403593	−0.201797	−	0.979427i	$-0.564678\pi$
$383$	−7.89848	−0.403593	−0.201797	+	0.979427i	$0.564678\pi$
$384$	0	0
$385$	−1.80921	−0.0922057
$386$	0	0
$387$	−8.11234	−0.412373
$388$	0	0
$389$	−16.7551	−0.849516	−0.424758	−	0.905307i	$-0.639641\pi$
$389$	−16.7551	−0.849516	−0.424758	+	0.905307i	$0.639641\pi$
$390$	0	0
$391$	−7.03079	−0.355562
$392$	0	0
$393$	−12.8400	−0.647692
$394$	0	0
$395$	0.700062	0.0352239
$396$	0	0
$397$	6.96921	0.349775	0.174887	−	0.984588i	$-0.444044\pi$
$397$	6.96921	0.349775	0.174887	+	0.984588i	$0.444044\pi$
$398$	0	0
$399$	3.53638	0.177040
$400$	0	0
$401$	26.4031	1.31851	0.659254	−	0.751920i	$-0.270872\pi$
$401$	26.4031	1.31851	0.659254	+	0.751920i	$0.270872\pi$
$402$	0	0
$403$	−6.72313	−0.334903
$404$	0	0
$405$	−0.962075	−0.0478059
$406$	0	0
$407$	−11.5703	−0.573516
$408$	0	0
$409$	−12.2569	−0.606065	−0.303032	−	0.952980i	$-0.597999\pi$
$409$	−12.2569	−0.606065	−0.303032	+	0.952980i	$0.597999\pi$
$410$	0	0
$411$	13.5406	0.667910
$412$	0	0
$413$	27.3150	1.34408
$414$	0	0
$415$	−0.362100	−0.0177748
$416$	0	0
$417$	−13.6826	−0.670040
$418$	0	0
$419$	7.82716	0.382382	0.191191	−	0.981553i	$-0.438765\pi$
$419$	7.82716	0.382382	0.191191	+	0.981553i	$0.438765\pi$
$420$	0	0
$421$	−38.9631	−1.89895	−0.949474	−	0.313846i	$-0.898382\pi$
$421$	−38.9631	−1.89895	−0.949474	+	0.313846i	$0.898382\pi$
$422$	0	0
$423$	5.19389	0.252535
$424$	0	0
$425$	−24.9042	−1.20803
$426$	0	0
$427$	−6.00619	−0.290660
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−3.00830	−0.144905	−0.0724525	−	0.997372i	$-0.523083\pi$
$431$	−3.00830	−0.144905	−0.0724525	+	0.997372i	$0.523083\pi$
$432$	0	0
$433$	4.92106	0.236491	0.118245	−	0.992984i	$-0.462273\pi$
$433$	4.92106	0.236491	0.118245	+	0.992984i	$0.462273\pi$
$434$	0	0
$435$	−0.883619	−0.0423663
$436$	0	0
$437$	2.16310	0.103475
$438$	0	0
$439$	−28.4100	−1.35593	−0.677967	−	0.735092i	$-0.737139\pi$
$439$	−28.4100	−1.35593	−0.677967	+	0.735092i	$0.737139\pi$
$440$	0	0
$441$	−3.46362	−0.164934
$442$	0	0
$443$	15.5449	0.738560	0.369280	−	0.929318i	$-0.379604\pi$
$443$	15.5449	0.738560	0.369280	+	0.929318i	$0.379604\pi$
$444$	0	0
$445$	−7.91962	−0.375426
$446$	0	0
$447$	−8.72052	−0.412467
$448$	0	0
$449$	−7.41430	−0.349902	−0.174951	−	0.984577i	$-0.555977\pi$
$449$	−7.41430	−0.349902	−0.174951	+	0.984577i	$0.555977\pi$
$450$	0	0
$451$	2.42000	0.113953
$452$	0	0
$453$	13.6416	0.640937
$454$	0	0
$455$	−1.80921	−0.0848169
$456$	0	0
$457$	31.4410	1.47075	0.735375	−	0.677660i	$-0.237006\pi$
$457$	31.4410	1.47075	0.735375	+	0.677660i	$0.237006\pi$
$458$	0	0
$459$	6.11234	0.285299
$460$	0	0
$461$	2.33275	0.108647	0.0543235	−	0.998523i	$-0.482700\pi$
$461$	2.33275	0.108647	0.0543235	+	0.998523i	$0.482700\pi$
$462$	0	0
$463$	−21.9786	−1.02143	−0.510716	−	0.859750i	$-0.670620\pi$
$463$	−21.9786	−1.02143	−0.510716	+	0.859750i	$0.670620\pi$
$464$	0	0
$465$	−6.46815	−0.299953
$466$	0	0
$467$	−18.3074	−0.847165	−0.423582	−	0.905858i	$-0.639228\pi$
$467$	−18.3074	−0.847165	−0.423582	+	0.905858i	$0.639228\pi$
$468$	0	0
$469$	−24.4913	−1.13090
$470$	0	0
$471$	2.53494	0.116804
$472$	0	0
$473$	8.11234	0.373006
$474$	0	0
$475$	7.66204	0.351558
$476$	0	0
$477$	−9.98573	−0.457215
$478$	0	0
$479$	39.5515	1.80715	0.903576	−	0.428428i	$-0.140932\pi$
$479$	39.5515	1.80715	0.903576	+	0.428428i	$0.140932\pi$
$480$	0	0
$481$	−11.5703	−0.527558
$482$	0	0
$483$	2.16310	0.0984243
$484$	0	0
$485$	8.54235	0.387888
$486$	0	0
$487$	23.0770	1.04572	0.522860	−	0.852419i	$-0.324865\pi$
$487$	23.0770	1.04572	0.522860	+	0.852419i	$0.324865\pi$
$488$	0	0
$489$	−1.61649	−0.0731002
$490$	0	0
$491$	28.3201	1.27807	0.639035	−	0.769178i	$-0.279334\pi$
$491$	28.3201	1.27807	0.639035	+	0.769178i	$0.279334\pi$
$492$	0	0
$493$	5.61388	0.252837
$494$	0	0
$495$	0.962075	0.0432421
$496$	0	0
$497$	2.95291	0.132456
$498$	0	0
$499$	23.0208	1.03055	0.515276	−	0.857024i	$-0.327690\pi$
$499$	23.0208	1.03055	0.515276	+	0.857024i	$0.327690\pi$
$500$	0	0
$501$	7.60935	0.339961
$502$	0	0
$503$	6.22755	0.277673	0.138836	−	0.990315i	$-0.455664\pi$
$503$	6.22755	0.277673	0.138836	+	0.990315i	$0.455664\pi$
$504$	0	0
$505$	−15.4283	−0.686550
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−34.9726	−1.55013	−0.775067	−	0.631879i	$-0.782284\pi$
$509$	−34.9726	−1.55013	−0.775067	+	0.631879i	$0.782284\pi$
$510$	0	0
$511$	−8.45456	−0.374008
$512$	0	0
$513$	−1.88053	−0.0830273
$514$	0	0
$515$	−15.7416	−0.693656
$516$	0	0
$517$	−5.19389	−0.228427
$518$	0	0
$519$	24.8765	1.09196
$520$	0	0
$521$	−29.9715	−1.31307	−0.656537	−	0.754294i	$-0.727979\pi$
$521$	−29.9715	−1.31307	−0.656537	+	0.754294i	$0.727979\pi$
$522$	0	0
$523$	−6.65702	−0.291091	−0.145546	−	0.989352i	$-0.546494\pi$
$523$	−6.65702	−0.291091	−0.145546	+	0.989352i	$0.546494\pi$
$524$	0	0
$525$	7.66204	0.334399
$526$	0	0
$527$	41.0940	1.79008
$528$	0	0
$529$	−21.6769	−0.942474
$530$	0	0
$531$	−14.5252	−0.630340
$532$	0	0
$533$	2.42000	0.104822
$534$	0	0
$535$	3.25631	0.140783
$536$	0	0
$537$	5.61388	0.242257
$538$	0	0
$539$	3.46362	0.149189
$540$	0	0
$541$	29.0670	1.24969	0.624844	−	0.780750i	$-0.285163\pi$
$541$	29.0670	1.24969	0.624844	+	0.780750i	$0.285163\pi$
$542$	0	0
$543$	1.46506	0.0628717
$544$	0	0
$545$	−13.0512	−0.559054
$546$	0	0
$547$	12.4520	0.532407	0.266204	−	0.963917i	$-0.414231\pi$
$547$	12.4520	0.532407	0.266204	+	0.963917i	$0.414231\pi$
$548$	0	0
$549$	3.19389	0.136312
$550$	0	0
$551$	−1.72717	−0.0735800
$552$	0	0
$553$	1.36838	0.0581895
$554$	0	0
$555$	−11.1315	−0.472504
$556$	0	0
$557$	19.0992	0.809260	0.404630	−	0.914480i	$-0.367400\pi$
$557$	19.0992	0.809260	0.404630	+	0.914480i	$0.367400\pi$
$558$	0	0
$559$	8.11234	0.343115
$560$	0	0
$561$	−6.11234	−0.258063
$562$	0	0
$563$	−40.2104	−1.69467	−0.847333	−	0.531062i	$-0.821793\pi$
$563$	−40.2104	−1.69467	−0.847333	+	0.531062i	$0.821793\pi$
$564$	0	0
$565$	−16.1284	−0.678525
$566$	0	0
$567$	−1.88053	−0.0789747
$568$	0	0
$569$	7.79754	0.326890	0.163445	−	0.986552i	$-0.447739\pi$
$569$	7.79754	0.326890	0.163445	+	0.986552i	$0.447739\pi$
$570$	0	0
$571$	−30.0079	−1.25579	−0.627897	−	0.778297i	$-0.716084\pi$
$571$	−30.0079	−1.25579	−0.627897	+	0.778297i	$0.716084\pi$
$572$	0	0
$573$	24.8257	1.03711
$574$	0	0
$575$	4.68664	0.195446
$576$	0	0
$577$	47.1376	1.96237	0.981183	−	0.193081i	$-0.0618479\pi$
$577$	47.1376	1.96237	0.981183	+	0.193081i	$0.0618479\pi$
$578$	0	0
$579$	−18.4057	−0.764916
$580$	0	0
$581$	−0.707781	−0.0293637
$582$	0	0
$583$	9.98573	0.413567
$584$	0	0
$585$	0.962075	0.0397769
$586$	0	0
$587$	−35.8340	−1.47903	−0.739514	−	0.673141i	$-0.764945\pi$
$587$	−35.8340	−1.47903	−0.739514	+	0.673141i	$0.764945\pi$
$588$	0	0
$589$	−12.6430	−0.520946
$590$	0	0
$591$	26.5574	1.09243
$592$	0	0
$593$	−36.5545	−1.50112	−0.750558	−	0.660805i	$-0.770215\pi$
$593$	−36.5545	−1.50112	−0.750558	+	0.660805i	$0.770215\pi$
$594$	0	0
$595$	11.0585	0.453354
$596$	0	0
$597$	−10.3005	−0.421572
$598$	0	0
$599$	1.60702	0.0656609	0.0328305	−	0.999461i	$-0.489548\pi$
$599$	1.60702	0.0656609	0.0328305	+	0.999461i	$0.489548\pi$
$600$	0	0
$601$	−44.2075	−1.80326	−0.901631	−	0.432506i	$-0.857630\pi$
$601$	−44.2075	−1.80326	−0.901631	+	0.432506i	$0.857630\pi$
$602$	0	0
$603$	13.0237	0.530364
$604$	0	0
$605$	−0.962075	−0.0391139
$606$	0	0
$607$	4.61793	0.187436	0.0937179	−	0.995599i	$-0.470125\pi$
$607$	4.61793	0.187436	0.0937179	+	0.995599i	$0.470125\pi$
$608$	0	0
$609$	−1.72717	−0.0699885
$610$	0	0
$611$	−5.19389	−0.210122
$612$	0	0
$613$	−5.44258	−0.219824	−0.109912	−	0.993941i	$-0.535057\pi$
$613$	−5.44258	−0.219824	−0.109912	+	0.993941i	$0.535057\pi$
$614$	0	0
$615$	2.32822	0.0938829
$616$	0	0
$617$	8.38266	0.337473	0.168737	−	0.985661i	$-0.446031\pi$
$617$	8.38266	0.337473	0.168737	+	0.985661i	$0.446031\pi$
$618$	0	0
$619$	29.6524	1.19183	0.595915	−	0.803047i	$-0.296789\pi$
$619$	29.6524	1.19183	0.595915	+	0.803047i	$0.296789\pi$
$620$	0	0
$621$	−1.15026	−0.0461584
$622$	0	0
$623$	−15.4801	−0.620199
$624$	0	0
$625$	11.9729	0.478916
$626$	0	0
$627$	1.88053	0.0751010
$628$	0	0
$629$	70.7213	2.81984
$630$	0	0
$631$	−23.7140	−0.944038	−0.472019	−	0.881588i	$-0.656475\pi$
$631$	−23.7140	−0.944038	−0.472019	+	0.881588i	$0.656475\pi$
$632$	0	0
$633$	−9.56765	−0.380280
$634$	0	0
$635$	−1.63252	−0.0647844
$636$	0	0
$637$	3.46362	0.137234
$638$	0	0
$639$	−1.57026	−0.0621185
$640$	0	0
$641$	22.9631	0.906990	0.453495	−	0.891259i	$-0.350177\pi$
$641$	22.9631	0.906990	0.453495	+	0.891259i	$0.350177\pi$
$642$	0	0
$643$	−4.79377	−0.189048	−0.0945238	−	0.995523i	$-0.530133\pi$
$643$	−4.79377	−0.189048	−0.0945238	+	0.995523i	$0.530133\pi$
$644$	0	0
$645$	7.80468	0.307309
$646$	0	0
$647$	41.5785	1.63462	0.817309	−	0.576199i	$-0.195465\pi$
$647$	41.5785	1.63462	0.817309	+	0.576199i	$0.195465\pi$
$648$	0	0
$649$	14.5252	0.570164
$650$	0	0
$651$	−12.6430	−0.495519
$652$	0	0
$653$	5.39895	0.211277	0.105639	−	0.994405i	$-0.466311\pi$
$653$	5.39895	0.211277	0.105639	+	0.994405i	$0.466311\pi$
$654$	0	0
$655$	12.3530	0.482673
$656$	0	0
$657$	4.49585	0.175400
$658$	0	0
$659$	−14.6544	−0.570855	−0.285427	−	0.958400i	$-0.592136\pi$
$659$	−14.6544	−0.570855	−0.285427	+	0.958400i	$0.592136\pi$
$660$	0	0
$661$	−22.8257	−0.887818	−0.443909	−	0.896072i	$-0.646409\pi$
$661$	−22.8257	−0.887818	−0.443909	+	0.896072i	$0.646409\pi$
$662$	0	0
$663$	−6.11234	−0.237383
$664$	0	0
$665$	−3.40226	−0.131934
$666$	0	0
$667$	−1.05646	−0.0409063
$668$	0	0
$669$	−6.49324	−0.251043
$670$	0	0
$671$	−3.19389	−0.123299
$672$	0	0
$673$	22.5591	0.869589	0.434794	−	0.900530i	$-0.356821\pi$
$673$	22.5591	0.869589	0.434794	+	0.900530i	$0.356821\pi$
$674$	0	0
$675$	−4.07441	−0.156824
$676$	0	0
$677$	−9.24465	−0.355301	−0.177650	−	0.984094i	$-0.556850\pi$
$677$	−9.24465	−0.355301	−0.177650	+	0.984094i	$0.556850\pi$
$678$	0	0
$679$	16.6974	0.640786
$680$	0	0
$681$	12.0293	0.460966
$682$	0	0
$683$	−2.01961	−0.0772781	−0.0386391	−	0.999253i	$-0.512302\pi$
$683$	−2.01961	−0.0772781	−0.0386391	+	0.999253i	$0.512302\pi$
$684$	0	0
$685$	−13.0271	−0.497740
$686$	0	0
$687$	20.7776	0.792714
$688$	0	0
$689$	9.98573	0.380426
$690$	0	0
$691$	41.8861	1.59342	0.796712	−	0.604360i	$-0.206571\pi$
$691$	41.8861	1.59342	0.796712	+	0.604360i	$0.206571\pi$
$692$	0	0
$693$	1.88053	0.0714353
$694$	0	0
$695$	13.1637	0.499327
$696$	0	0
$697$	−14.7918	−0.560281
$698$	0	0
$699$	29.2304	1.10559
$700$	0	0
$701$	−47.6765	−1.80072	−0.900359	−	0.435148i	$-0.856696\pi$
$701$	−47.6765	−1.80072	−0.900359	+	0.435148i	$0.856696\pi$
$702$	0	0
$703$	−21.7582	−0.820625
$704$	0	0
$705$	−4.99691	−0.188194
$706$	0	0
$707$	−30.1570	−1.13417
$708$	0	0
$709$	−19.0781	−0.716493	−0.358246	−	0.933627i	$-0.616625\pi$
$709$	−19.0781	−0.716493	−0.358246	+	0.933627i	$0.616625\pi$
$710$	0	0
$711$	−0.727658	−0.0272893
$712$	0	0
$713$	−7.73336	−0.289616
$714$	0	0
$715$	−0.962075	−0.0359796
$716$	0	0
$717$	−9.25999	−0.345821
$718$	0	0
$719$	6.69117	0.249539	0.124769	−	0.992186i	$-0.460181\pi$
$719$	6.69117	0.249539	0.124769	+	0.992186i	$0.460181\pi$
$720$	0	0
$721$	−30.7694	−1.14591
$722$	0	0
$723$	24.3735	0.906460
$724$	0	0
$725$	−3.74215	−0.138980
$726$	0	0
$727$	19.6686	0.729468	0.364734	−	0.931112i	$-0.381160\pi$
$727$	19.6686	0.729468	0.364734	+	0.931112i	$0.381160\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−49.5853	−1.83398
$732$	0	0
$733$	−5.21269	−0.192535	−0.0962676	−	0.995355i	$-0.530690\pi$
$733$	−5.21269	−0.192535	−0.0962676	+	0.995355i	$0.530690\pi$
$734$	0	0
$735$	3.33226	0.122912
$736$	0	0
$737$	−13.0237	−0.479732
$738$	0	0
$739$	49.5486	1.82267	0.911337	−	0.411661i	$-0.135051\pi$
$739$	49.5486	1.82267	0.911337	+	0.411661i	$0.135051\pi$
$740$	0	0
$741$	1.88053	0.0690829
$742$	0	0
$743$	−1.32310	−0.0485399	−0.0242700	−	0.999705i	$-0.507726\pi$
$743$	−1.32310	−0.0485399	−0.0242700	+	0.999705i	$0.507726\pi$
$744$	0	0
$745$	8.38980	0.307378
$746$	0	0
$747$	0.376374	0.0137708
$748$	0	0
$749$	6.36498	0.232571
$750$	0	0
$751$	35.0031	1.27728	0.638640	−	0.769505i	$-0.279497\pi$
$751$	35.0031	1.27728	0.638640	+	0.769505i	$0.279497\pi$
$752$	0	0
$753$	−1.08415	−0.0395088
$754$	0	0
$755$	−13.1242	−0.477639
$756$	0	0
$757$	−24.6056	−0.894305	−0.447152	−	0.894458i	$-0.647562\pi$
$757$	−24.6056	−0.894305	−0.447152	+	0.894458i	$0.647562\pi$
$758$	0	0
$759$	1.15026	0.0417519
$760$	0	0
$761$	−1.57479	−0.0570861	−0.0285431	−	0.999593i	$-0.509087\pi$
$761$	−1.57479	−0.0570861	−0.0285431	+	0.999593i	$0.509087\pi$
$762$	0	0
$763$	−25.5107	−0.923550
$764$	0	0
$765$	−5.88053	−0.212611
$766$	0	0
$767$	14.5252	0.524474
$768$	0	0
$769$	−18.0963	−0.652570	−0.326285	−	0.945272i	$-0.605797\pi$
$769$	−18.0963	−0.652570	−0.326285	+	0.945272i	$0.605797\pi$
$770$	0	0
$771$	−12.0616	−0.434387
$772$	0	0
$773$	0.736549	0.0264918	0.0132459	−	0.999912i	$-0.495784\pi$
$773$	0.736549	0.0264918	0.0132459	+	0.999912i	$0.495784\pi$
$774$	0	0
$775$	−27.3928	−0.983978
$776$	0	0
$777$	−21.7582	−0.780570
$778$	0	0
$779$	4.55087	0.163052
$780$	0	0
$781$	1.57026	0.0561883
$782$	0	0
$783$	0.918451	0.0328228
$784$	0	0
$785$	−2.43880	−0.0870446
$786$	0	0
$787$	−10.4171	−0.371330	−0.185665	−	0.982613i	$-0.559444\pi$
$787$	−10.4171	−0.371330	−0.185665	+	0.982613i	$0.559444\pi$
$788$	0	0
$789$	−7.86257	−0.279915
$790$	0	0
$791$	−31.5254	−1.12091
$792$	0	0
$793$	−3.19389	−0.113418
$794$	0	0
$795$	9.60702	0.340726
$796$	0	0
$797$	24.4522	0.866142	0.433071	−	0.901360i	$-0.357430\pi$
$797$	24.4522	0.866142	0.433071	+	0.901360i	$0.357430\pi$
$798$	0	0
$799$	31.7468	1.12312
$800$	0	0
$801$	8.23181	0.290857
$802$	0	0
$803$	−4.49585	−0.158655
$804$	0	0
$805$	−2.08106	−0.0733478
$806$	0	0
$807$	16.9940	0.598218
$808$	0	0
$809$	7.08976	0.249263	0.124631	−	0.992203i	$-0.460225\pi$
$809$	7.08976	0.249263	0.124631	+	0.992203i	$0.460225\pi$
$810$	0	0
$811$	−43.4733	−1.52655	−0.763276	−	0.646072i	$-0.776411\pi$
$811$	−43.4733	−1.52655	−0.763276	+	0.646072i	$0.776411\pi$
$812$	0	0
$813$	−10.0180	−0.351345
$814$	0	0
$815$	1.55518	0.0544757
$816$	0	0
$817$	15.2555	0.533721
$818$	0	0
$819$	1.88053	0.0657109
$820$	0	0
$821$	−42.5198	−1.48395	−0.741976	−	0.670427i	$-0.766111\pi$
$821$	−42.5198	−1.48395	−0.741976	+	0.670427i	$0.766111\pi$
$822$	0	0
$823$	−36.0444	−1.25643	−0.628215	−	0.778040i	$-0.716214\pi$
$823$	−36.0444	−1.25643	−0.628215	+	0.778040i	$0.716214\pi$
$824$	0	0
$825$	4.07441	0.141853
$826$	0	0
$827$	−23.0826	−0.802661	−0.401331	−	0.915933i	$-0.631452\pi$
$827$	−23.0826	−0.802661	−0.401331	+	0.915933i	$0.631452\pi$
$828$	0	0
$829$	20.4660	0.710812	0.355406	−	0.934712i	$-0.384343\pi$
$829$	20.4660	0.710812	0.355406	+	0.934712i	$0.384343\pi$
$830$	0	0
$831$	−9.28316	−0.322029
$832$	0	0
$833$	−21.1708	−0.733525
$834$	0	0
$835$	−7.32077	−0.253346
$836$	0	0
$837$	6.72313	0.232385
$838$	0	0
$839$	35.6094	1.22937	0.614686	−	0.788772i	$-0.289283\pi$
$839$	35.6094	1.22937	0.614686	+	0.788772i	$0.289283\pi$
$840$	0	0
$841$	−28.1564	−0.970912
$842$	0	0
$843$	31.2367	1.07585
$844$	0	0
$845$	−0.962075	−0.0330964
$846$	0	0
$847$	−1.88053	−0.0646156
$848$	0	0
$849$	−13.0673	−0.448468
$850$	0	0
$851$	−13.3088	−0.456221
$852$	0	0
$853$	−30.8114	−1.05496	−0.527482	−	0.849566i	$-0.676864\pi$
$853$	−30.8114	−1.05496	−0.527482	+	0.849566i	$0.676864\pi$
$854$	0	0
$855$	1.80921	0.0618736
$856$	0	0
$857$	12.6600	0.432458	0.216229	−	0.976343i	$-0.430624\pi$
$857$	12.6600	0.432458	0.216229	+	0.976343i	$0.430624\pi$
$858$	0	0
$859$	−23.0903	−0.787832	−0.393916	−	0.919146i	$-0.628880\pi$
$859$	−23.0903	−0.787832	−0.393916	+	0.919146i	$0.628880\pi$
$860$	0	0
$861$	4.55087	0.155093
$862$	0	0
$863$	39.8145	1.35530	0.677651	−	0.735383i	$-0.262998\pi$
$863$	39.8145	1.35530	0.677651	+	0.735383i	$0.262998\pi$
$864$	0	0
$865$	−23.9330	−0.813748
$866$	0	0
$867$	20.3607	0.691484
$868$	0	0
$869$	0.727658	0.0246841
$870$	0	0
$871$	−13.0237	−0.441290
$872$	0	0
$873$	−8.87909	−0.300512
$874$	0	0
$875$	−16.4175	−0.555012
$876$	0	0
$877$	−49.6823	−1.67765	−0.838826	−	0.544399i	$-0.816758\pi$
$877$	−49.6823	−1.67765	−0.838826	+	0.544399i	$0.816758\pi$
$878$	0	0
$879$	5.95638	0.200904
$880$	0	0
$881$	54.7470	1.84447	0.922237	−	0.386626i	$-0.126360\pi$
$881$	54.7470	1.84447	0.922237	+	0.386626i	$0.126360\pi$
$882$	0	0
$883$	−33.6709	−1.13312	−0.566558	−	0.824022i	$-0.691725\pi$
$883$	−33.6709	−1.13312	−0.566558	+	0.824022i	$0.691725\pi$
$884$	0	0
$885$	13.9743	0.469742
$886$	0	0
$887$	−37.0136	−1.24280	−0.621398	−	0.783495i	$-0.713435\pi$
$887$	−37.0136	−1.24280	−0.621398	+	0.783495i	$0.713435\pi$
$888$	0	0
$889$	−3.19101	−0.107023
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−9.76724	−0.326848
$894$	0	0
$895$	−5.40098	−0.180535
$896$	0	0
$897$	1.15026	0.0384061
$898$	0	0
$899$	6.17486	0.205943
$900$	0	0
$901$	−61.0361	−2.03341
$902$	0	0
$903$	15.2555	0.507670
$904$	0	0
$905$	−1.40950	−0.0468533
$906$	0	0
$907$	−27.8626	−0.925162	−0.462581	−	0.886577i	$-0.653076\pi$
$907$	−27.8626	−0.925162	−0.462581	+	0.886577i	$0.653076\pi$
$908$	0	0
$909$	16.0365	0.531897
$910$	0	0
$911$	49.0271	1.62434	0.812169	−	0.583422i	$-0.198286\pi$
$911$	49.0271	1.62434	0.812169	+	0.583422i	$0.198286\pi$
$912$	0	0
$913$	−0.376374	−0.0124562
$914$	0	0
$915$	−3.07276	−0.101582
$916$	0	0
$917$	24.1459	0.797369
$918$	0	0
$919$	30.0499	0.991255	0.495628	−	0.868535i	$-0.334938\pi$
$919$	30.0499	0.991255	0.495628	+	0.868535i	$0.334938\pi$
$920$	0	0
$921$	18.3299	0.603990
$922$	0	0
$923$	1.57026	0.0516857
$924$	0	0
$925$	−47.1420	−1.55002
$926$	0	0
$927$	16.3621	0.537402
$928$	0	0
$929$	−39.5970	−1.29914	−0.649568	−	0.760304i	$-0.725050\pi$
$929$	−39.5970	−1.29914	−0.649568	+	0.760304i	$0.725050\pi$
$930$	0	0
$931$	6.51343	0.213469
$932$	0	0
$933$	19.0601	0.624001
$934$	0	0
$935$	5.88053	0.192314
$936$	0	0
$937$	−10.4718	−0.342100	−0.171050	−	0.985262i	$-0.554716\pi$
$937$	−10.4718	−0.342100	−0.171050	+	0.985262i	$0.554716\pi$
$938$	0	0
$939$	11.4765	0.374520
$940$	0	0
$941$	2.50725	0.0817339	0.0408669	−	0.999165i	$-0.486988\pi$
$941$	2.50725	0.0817339	0.0408669	+	0.999165i	$0.486988\pi$
$942$	0	0
$943$	2.78363	0.0906475
$944$	0	0
$945$	1.80921	0.0588535
$946$	0	0
$947$	51.3877	1.66988	0.834938	−	0.550345i	$-0.185504\pi$
$947$	51.3877	1.66988	0.834938	+	0.550345i	$0.185504\pi$
$948$	0	0
$949$	−4.49585	−0.145941
$950$	0	0
$951$	3.15287	0.102239
$952$	0	0
$953$	15.4489	0.500438	0.250219	−	0.968189i	$-0.419497\pi$
$953$	15.4489	0.500438	0.250219	+	0.968189i	$0.419497\pi$
$954$	0	0
$955$	−23.8842	−0.772875
$956$	0	0
$957$	−0.918451	−0.0296893
$958$	0	0
$959$	−25.4635	−0.822260
$960$	0	0
$961$	14.2004	0.458079
$962$	0	0
$963$	−3.38468	−0.109070
$964$	0	0
$965$	17.7077	0.570031
$966$	0	0
$967$	−44.1758	−1.42060	−0.710300	−	0.703899i	$-0.751441\pi$
$967$	−44.1758	−1.42060	−0.710300	+	0.703899i	$0.751441\pi$
$968$	0	0
$969$	−11.4944	−0.369254
$970$	0	0
$971$	16.8469	0.540642	0.270321	−	0.962770i	$-0.412870\pi$
$971$	16.8469	0.540642	0.270321	+	0.962770i	$0.412870\pi$
$972$	0	0
$973$	25.7305	0.824881
$974$	0	0
$975$	4.07441	0.130486
$976$	0	0
$977$	44.9703	1.43873	0.719363	−	0.694634i	$-0.244434\pi$
$977$	44.9703	1.43873	0.719363	+	0.694634i	$0.244434\pi$
$978$	0	0
$979$	−8.23181	−0.263090
$980$	0	0
$981$	13.5657	0.433120
$982$	0	0
$983$	−1.92936	−0.0615371	−0.0307685	−	0.999527i	$-0.509795\pi$
$983$	−1.92936	−0.0615371	−0.0307685	+	0.999527i	$0.509795\pi$
$984$	0	0
$985$	−25.5502	−0.814098
$986$	0	0
$987$	−9.76724	−0.310895
$988$	0	0
$989$	9.33131	0.296718
$990$	0	0
$991$	2.55375	0.0811224	0.0405612	−	0.999177i	$-0.487085\pi$
$991$	2.55375	0.0811224	0.0405612	+	0.999177i	$0.487085\pi$
$992$	0	0
$993$	17.8412	0.566172
$994$	0	0
$995$	9.90988	0.314164
$996$	0	0
$997$	−14.0925	−0.446313	−0.223156	−	0.974783i	$-0.571636\pi$
$997$	−14.0925	−0.446313	−0.223156	+	0.974783i	$0.571636\pi$
$998$	0	0
$999$	11.5703	0.366067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cc.1.2		4
4.3	odd	2		3432.2.a.t.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.t.1.2	✓	4	4.3	odd	2
6864.2.a.cc.1.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.962075 q^{5} -1.88053 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.962075 q^{5} -1.88053 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.962075 q^{15} +6.11234 q^{17} -1.88053 q^{19} -1.88053 q^{21} -1.15026 q^{23} -4.07441 q^{25} +1.00000 q^{27} +0.918451 q^{29} +6.72313 q^{31} -1.00000 q^{33} +1.80921 q^{35} +11.5703 q^{37} -1.00000 q^{39} -2.42000 q^{41} -8.11234 q^{43} -0.962075 q^{45} +5.19389 q^{47} -3.46362 q^{49} +6.11234 q^{51} -9.98573 q^{53} +0.962075 q^{55} -1.88053 q^{57} -14.5252 q^{59} +3.19389 q^{61} -1.88053 q^{63} +0.962075 q^{65} +13.0237 q^{67} -1.15026 q^{69} -1.57026 q^{71} +4.49585 q^{73} -4.07441 q^{75} +1.88053 q^{77} -0.727658 q^{79} +1.00000 q^{81} +0.376374 q^{83} -5.88053 q^{85} +0.918451 q^{87} +8.23181 q^{89} +1.88053 q^{91} +6.72313 q^{93} +1.80921 q^{95} -8.87909 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 8 q^{23} + 12 q^{25} + 4 q^{27} + 6 q^{29} + 8 q^{31} - 4 q^{33} - 14 q^{35} + 14 q^{37} - 4 q^{39} + 10 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{47} + 16 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{55} - 2 q^{57} + 8 q^{59} - 10 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} + 26 q^{71} + 14 q^{73} + 12 q^{75} + 2 q^{77} + 16 q^{79} + 4 q^{81} - 8 q^{83} - 18 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 8 q^{93} - 14 q^{95} + 14 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^17 - 2 * q^19 - 2 * q^21 + 8 * q^23 + 12 * q^25 + 4 * q^27 + 6 * q^29 + 8 * q^31 - 4 * q^33 - 14 * q^35 + 14 * q^37 - 4 * q^39 + 10 * q^41 - 12 * q^43 + 4 * q^45 - 2 * q^47 + 16 * q^49 + 4 * q^51 + 12 * q^53 - 4 * q^55 - 2 * q^57 + 8 * q^59 - 10 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 26 * q^71 + 14 * q^73 + 12 * q^75 + 2 * q^77 + 16 * q^79 + 4 * q^81 - 8 * q^83 - 18 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 8 * q^93 - 14 * q^95 + 14 * q^97 - 4 * q^99

Introduction

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