LMFDB - Embedded newform 6864.2.a.by.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.22676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 6x + 2 $ x^4 - x^3 - 6x^2 + 6x + 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			1.1
Root			$-0.266370$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.92905 q^{5} -3.57932 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.92905 q^{5} -3.57932 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.92905 q^{15} +0.349729 q^{17} +0.532739 q^{19} -3.57932 q^{21} +5.67248 q^{23} +10.4374 q^{25} +1.00000 q^{27} +2.55494 q^{29} -1.81699 q^{31} +1.00000 q^{33} +14.0633 q^{35} +3.76494 q^{37} -1.00000 q^{39} +5.43741 q^{41} -0.696849 q^{43} -3.92905 q^{45} +1.06548 q^{47} +5.81152 q^{49} +0.349729 q^{51} -2.00000 q^{53} -3.92905 q^{55} +0.532739 q^{57} +4.11206 q^{59} +3.13974 q^{61} -3.57932 q^{63} +3.92905 q^{65} +8.02220 q^{67} +5.67248 q^{69} -12.9236 q^{71} -10.2788 q^{73} +10.4374 q^{75} -3.57932 q^{77} -1.65027 q^{79} +1.00000 q^{81} -4.22411 q^{83} -1.37410 q^{85} +2.55494 q^{87} -12.7406 q^{89} +3.57932 q^{91} -1.81699 q^{93} -2.09316 q^{95} +3.76494 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 3 q^{15} - 2 q^{19} - 3 q^{21} - 3 q^{23} + 5 q^{25} + 4 q^{27} - 21 q^{29} - 10 q^{31} + 4 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 15 q^{41} + 3 q^{43} - 3 q^{45} - 4 q^{47} + 5 q^{49} - 8 q^{53} - 3 q^{55} - 2 q^{57} + q^{59} - 9 q^{61} - 3 q^{63} + 3 q^{65} + 5 q^{67} - 3 q^{69} - 18 q^{71} - 27 q^{73} + 5 q^{75} - 3 q^{77} - 8 q^{79} + 4 q^{81} + 14 q^{83} - 24 q^{85} - 21 q^{87} - 20 q^{89} + 3 q^{91} - 10 q^{93} + 6 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 3 * q^15 - 2 * q^19 - 3 * q^21 - 3 * q^23 + 5 * q^25 + 4 * q^27 - 21 * q^29 - 10 * q^31 + 4 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 15 * q^41 + 3 * q^43 - 3 * q^45 - 4 * q^47 + 5 * q^49 - 8 * q^53 - 3 * q^55 - 2 * q^57 + q^59 - 9 * q^61 - 3 * q^63 + 3 * q^65 + 5 * q^67 - 3 * q^69 - 18 * q^71 - 27 * q^73 + 5 * q^75 - 3 * q^77 - 8 * q^79 + 4 * q^81 + 14 * q^83 - 24 * q^85 - 21 * q^87 - 20 * q^89 + 3 * q^91 - 10 * q^93 + 6 * q^95 + 4 * 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
$5$	−3.92905	−1.75712	−0.878562	−	0.477629i	$-0.841496\pi$
$5$	−3.92905	−1.75712	−0.878562	+	0.477629i	$0.841496\pi$
$6$	0	0
$7$	−3.57932	−1.35285	−0.676427	−	0.736509i	$-0.736473\pi$
$7$	−3.57932	−1.35285	−0.676427	+	0.736509i	$0.736473\pi$
$8$	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
$15$	−3.92905	−1.01448
$16$	0	0
$17$	0.349729	0.0848219	0.0424109	−	0.999100i	$-0.486496\pi$
$17$	0.349729	0.0848219	0.0424109	+	0.999100i	$0.486496\pi$
$18$	0	0
$19$	0.532739	0.122219	0.0611094	−	0.998131i	$-0.480536\pi$
$19$	0.532739	0.122219	0.0611094	+	0.998131i	$0.480536\pi$
$20$	0	0
$21$	−3.57932	−0.781071
$22$	0	0
$23$	5.67248	1.18279	0.591396	−	0.806381i	$-0.298577\pi$
$23$	5.67248	1.18279	0.591396	+	0.806381i	$0.298577\pi$
$24$	0	0
$25$	10.4374	2.08748
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.55494	0.474441	0.237221	−	0.971456i	$-0.423764\pi$
$29$	2.55494	0.474441	0.237221	+	0.971456i	$0.423764\pi$
$30$	0	0
$31$	−1.81699	−0.326341	−0.163171	−	0.986598i	$-0.552172\pi$
$31$	−1.81699	−0.326341	−0.163171	+	0.986598i	$0.552172\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	14.0633	2.37713
$36$	0	0
$37$	3.76494	0.618952	0.309476	−	0.950907i	$-0.399846\pi$
$37$	3.76494	0.618952	0.309476	+	0.950907i	$0.399846\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.43741	0.849181	0.424591	−	0.905385i	$-0.360418\pi$
$41$	5.43741	0.849181	0.424591	+	0.905385i	$0.360418\pi$
$42$	0	0
$43$	−0.696849	−0.106269	−0.0531343	−	0.998587i	$-0.516921\pi$
$43$	−0.696849	−0.106269	−0.0531343	+	0.998587i	$0.516921\pi$
$44$	0	0
$45$	−3.92905	−0.585708
$46$	0	0
$47$	1.06548	0.155416	0.0777080	−	0.996976i	$-0.475240\pi$
$47$	1.06548	0.155416	0.0777080	+	0.996976i	$0.475240\pi$
$48$	0	0
$49$	5.81152	0.830217
$50$	0	0
$51$	0.349729	0.0489719
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−3.92905	−0.529793
$56$	0	0
$57$	0.532739	0.0705630
$58$	0	0
$59$	4.11206	0.535344	0.267672	−	0.963510i	$-0.413746\pi$
$59$	4.11206	0.535344	0.267672	+	0.963510i	$0.413746\pi$
$60$	0	0
$61$	3.13974	0.402002	0.201001	−	0.979591i	$-0.435581\pi$
$61$	3.13974	0.402002	0.201001	+	0.979591i	$0.435581\pi$
$62$	0	0
$63$	−3.57932	−0.450952
$64$	0	0
$65$	3.92905	0.487338
$66$	0	0
$67$	8.02220	0.980068	0.490034	−	0.871703i	$-0.336984\pi$
$67$	8.02220	0.980068	0.490034	+	0.871703i	$0.336984\pi$
$68$	0	0
$69$	5.67248	0.682886
$70$	0	0
$71$	−12.9236	−1.53375	−0.766873	−	0.641799i	$-0.778188\pi$
$71$	−12.9236	−1.53375	−0.766873	+	0.641799i	$0.778188\pi$
$72$	0	0
$73$	−10.2788	−1.20304	−0.601520	−	0.798858i	$-0.705438\pi$
$73$	−10.2788	−1.20304	−0.601520	+	0.798858i	$0.705438\pi$
$74$	0	0
$75$	10.4374	1.20521
$76$	0	0
$77$	−3.57932	−0.407901
$78$	0	0
$79$	−1.65027	−0.185670	−0.0928350	−	0.995682i	$-0.529593\pi$
$79$	−1.65027	−0.185670	−0.0928350	+	0.995682i	$0.529593\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.22411	−0.463657	−0.231828	−	0.972757i	$-0.574471\pi$
$83$	−4.22411	−0.463657	−0.231828	+	0.972757i	$0.574471\pi$
$84$	0	0
$85$	−1.37410	−0.149042
$86$	0	0
$87$	2.55494	0.273919
$88$	0	0
$89$	−12.7406	−1.35050	−0.675248	−	0.737590i	$-0.735964\pi$
$89$	−12.7406	−1.35050	−0.675248	+	0.737590i	$0.735964\pi$
$90$	0	0
$91$	3.57932	0.375214
$92$	0	0
$93$	−1.81699	−0.188413
$94$	0	0
$95$	−2.09316	−0.214753
$96$	0	0
$97$	3.76494	0.382271	0.191136	−	0.981564i	$-0.438783\pi$
$97$	3.76494	0.382271	0.191136	+	0.981564i	$0.438783\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−12.5738	−1.25114	−0.625572	−	0.780166i	$-0.715134\pi$
$101$	−12.5738	−1.25114	−0.625572	+	0.780166i	$0.715134\pi$
$102$	0	0
$103$	−8.11206	−0.799305	−0.399652	−	0.916667i	$-0.630869\pi$
$103$	−8.11206	−0.799305	−0.399652	+	0.916667i	$0.630869\pi$
$104$	0	0
$105$	14.0633	1.37244
$106$	0	0
$107$	7.04658	0.681218	0.340609	−	0.940205i	$-0.389367\pi$
$107$	7.04658	0.681218	0.340609	+	0.940205i	$0.389367\pi$
$108$	0	0
$109$	−6.71905	−0.643569	−0.321784	−	0.946813i	$-0.604283\pi$
$109$	−6.71905	−0.643569	−0.321784	+	0.946813i	$0.604283\pi$
$110$	0	0
$111$	3.76494	0.357352
$112$	0	0
$113$	−17.4622	−1.64271	−0.821354	−	0.570419i	$-0.806781\pi$
$113$	−17.4622	−1.64271	−0.821354	+	0.570419i	$0.806781\pi$
$114$	0	0
$115$	−22.2874	−2.07831
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.25179	−0.114752
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.43741	0.490275
$124$	0	0
$125$	−21.3638	−1.91084
$126$	0	0
$127$	−2.34973	−0.208505	−0.104252	−	0.994551i	$-0.533245\pi$
$127$	−2.34973	−0.208505	−0.104252	+	0.994551i	$0.533245\pi$
$128$	0	0
$129$	−0.696849	−0.0613541
$130$	0	0
$131$	−8.99783	−0.786144	−0.393072	−	0.919508i	$-0.628588\pi$
$131$	−8.99783	−0.786144	−0.393072	+	0.919508i	$0.628588\pi$
$132$	0	0
$133$	−1.90684	−0.165344
$134$	0	0
$135$	−3.92905	−0.338159
$136$	0	0
$137$	−5.58193	−0.476896	−0.238448	−	0.971155i	$-0.576639\pi$
$137$	−5.58193	−0.476896	−0.238448	+	0.971155i	$0.576639\pi$
$138$	0	0
$139$	6.20782	0.526541	0.263270	−	0.964722i	$-0.415199\pi$
$139$	6.20782	0.526541	0.263270	+	0.964722i	$0.415199\pi$
$140$	0	0
$141$	1.06548	0.0897295
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−10.0385	−0.833652
$146$	0	0
$147$	5.81152	0.479326
$148$	0	0
$149$	−10.0196	−0.820837	−0.410419	−	0.911897i	$-0.634617\pi$
$149$	−10.0196	−0.820837	−0.410419	+	0.911897i	$0.634617\pi$
$150$	0	0
$151$	2.62590	0.213692	0.106846	−	0.994276i	$-0.465925\pi$
$151$	2.62590	0.213692	0.106846	+	0.994276i	$0.465925\pi$
$152$	0	0
$153$	0.349729	0.0282740
$154$	0	0
$155$	7.13904	0.573422
$156$	0	0
$157$	5.69137	0.454221	0.227111	−	0.973869i	$-0.427072\pi$
$157$	5.69137	0.454221	0.227111	+	0.973869i	$0.427072\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−20.3036	−1.60015
$162$	0	0
$163$	14.4814	1.13427	0.567135	−	0.823625i	$-0.308052\pi$
$163$	14.4814	1.13427	0.567135	+	0.823625i	$0.308052\pi$
$164$	0	0
$165$	−3.92905	−0.305876
$166$	0	0
$167$	5.87699	0.454775	0.227388	−	0.973804i	$-0.426982\pi$
$167$	5.87699	0.454775	0.227388	+	0.973804i	$0.426982\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.532739	0.0407396
$172$	0	0
$173$	−21.8555	−1.66164	−0.830821	−	0.556540i	$-0.812129\pi$
$173$	−21.8555	−1.66164	−0.830821	+	0.556540i	$0.812129\pi$
$174$	0	0
$175$	−37.3588	−2.82406
$176$	0	0
$177$	4.11206	0.309081
$178$	0	0
$179$	23.0415	1.72221	0.861103	−	0.508431i	$-0.169774\pi$
$179$	23.0415	1.72221	0.861103	+	0.508431i	$0.169774\pi$
$180$	0	0
$181$	2.49494	0.185447	0.0927237	−	0.995692i	$-0.470443\pi$
$181$	2.49494	0.185447	0.0927237	+	0.995692i	$0.470443\pi$
$182$	0	0
$183$	3.13974	0.232096
$184$	0	0
$185$	−14.7926	−1.08757
$186$	0	0
$187$	0.349729	0.0255748
$188$	0	0
$189$	−3.57932	−0.260357
$190$	0	0
$191$	−5.87699	−0.425244	−0.212622	−	0.977134i	$-0.568200\pi$
$191$	−5.87699	−0.425244	−0.212622	+	0.977134i	$0.568200\pi$
$192$	0	0
$193$	−6.75685	−0.486369	−0.243184	−	0.969980i	$-0.578192\pi$
$193$	−6.75685	−0.486369	−0.243184	+	0.969980i	$0.578192\pi$
$194$	0	0
$195$	3.92905	0.281365
$196$	0	0
$197$	1.54947	0.110395	0.0551976	−	0.998475i	$-0.482421\pi$
$197$	1.54947	0.110395	0.0551976	+	0.998475i	$0.482421\pi$
$198$	0	0
$199$	28.0146	1.98590	0.992950	−	0.118536i	$-0.0378201\pi$
$199$	28.0146	1.98590	0.992950	+	0.118536i	$0.0378201\pi$
$200$	0	0
$201$	8.02220	0.565843
$202$	0	0
$203$	−9.14496	−0.641850
$204$	0	0
$205$	−21.3638	−1.49212
$206$	0	0
$207$	5.67248	0.394264
$208$	0	0
$209$	0.532739	0.0368503
$210$	0	0
$211$	−0.808906	−0.0556874	−0.0278437	−	0.999612i	$-0.508864\pi$
$211$	−0.808906	−0.0556874	−0.0278437	+	0.999612i	$0.508864\pi$
$212$	0	0
$213$	−12.9236	−0.885509
$214$	0	0
$215$	2.73795	0.186727
$216$	0	0
$217$	6.50359	0.441492
$218$	0	0
$219$	−10.2788	−0.694575
$220$	0	0
$221$	−0.349729	−0.0235253
$222$	0	0
$223$	25.1997	1.68750	0.843750	−	0.536737i	$-0.180343\pi$
$223$	25.1997	1.68750	0.843750	+	0.536737i	$0.180343\pi$
$224$	0	0
$225$	10.4374	0.695827
$226$	0	0
$227$	−12.3908	−0.822409	−0.411204	−	0.911543i	$-0.634892\pi$
$227$	−12.3908	−0.822409	−0.411204	+	0.911543i	$0.634892\pi$
$228$	0	0
$229$	−29.3529	−1.93969	−0.969847	−	0.243714i	$-0.921634\pi$
$229$	−29.3529	−1.93969	−0.969847	+	0.243714i	$0.921634\pi$
$230$	0	0
$231$	−3.57932	−0.235502
$232$	0	0
$233$	19.2246	1.25944	0.629721	−	0.776821i	$-0.283169\pi$
$233$	19.2246	1.25944	0.629721	+	0.776821i	$0.283169\pi$
$234$	0	0
$235$	−4.18631	−0.273085
$236$	0	0
$237$	−1.65027	−0.107197
$238$	0	0
$239$	−26.1703	−1.69282	−0.846409	−	0.532533i	$-0.821240\pi$
$239$	−26.1703	−1.69282	−0.846409	+	0.532533i	$0.821240\pi$
$240$	0	0
$241$	−8.65071	−0.557241	−0.278621	−	0.960401i	$-0.589877\pi$
$241$	−8.65071	−0.557241	−0.278621	+	0.960401i	$0.589877\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−22.8337	−1.45879
$246$	0	0
$247$	−0.532739	−0.0338974
$248$	0	0
$249$	−4.22411	−0.267692
$250$	0	0
$251$	23.3450	1.47352	0.736760	−	0.676154i	$-0.236355\pi$
$251$	23.3450	1.47352	0.736760	+	0.676154i	$0.236355\pi$
$252$	0	0
$253$	5.67248	0.356625
$254$	0	0
$255$	−1.37410	−0.0860497
$256$	0	0
$257$	−18.4833	−1.15296	−0.576478	−	0.817113i	$-0.695573\pi$
$257$	−18.4833	−1.15296	−0.576478	+	0.817113i	$0.695573\pi$
$258$	0	0
$259$	−13.4759	−0.837352
$260$	0	0
$261$	2.55494	0.158147
$262$	0	0
$263$	11.0167	0.679321	0.339660	−	0.940548i	$-0.389688\pi$
$263$	11.0167	0.679321	0.339660	+	0.940548i	$0.389688\pi$
$264$	0	0
$265$	7.85809	0.482719
$266$	0	0
$267$	−12.7406	−0.779710
$268$	0	0
$269$	−20.5088	−1.25044	−0.625222	−	0.780447i	$-0.714992\pi$
$269$	−20.5088	−1.25044	−0.625222	+	0.780447i	$0.714992\pi$
$270$	0	0
$271$	−10.6259	−0.645477	−0.322739	−	0.946488i	$-0.604603\pi$
$271$	−10.6259	−0.645477	−0.322739	+	0.946488i	$0.604603\pi$
$272$	0	0
$273$	3.57932	0.216630
$274$	0	0
$275$	10.4374	0.629400
$276$	0	0
$277$	16.1011	0.967422	0.483711	−	0.875228i	$-0.339289\pi$
$277$	16.1011	0.967422	0.483711	+	0.875228i	$0.339289\pi$
$278$	0	0
$279$	−1.81699	−0.108780
$280$	0	0
$281$	21.3777	1.27529	0.637644	−	0.770331i	$-0.279909\pi$
$281$	21.3777	1.27529	0.637644	+	0.770331i	$0.279909\pi$
$282$	0	0
$283$	22.3199	1.32678	0.663390	−	0.748274i	$-0.269117\pi$
$283$	22.3199	1.32678	0.663390	+	0.748274i	$0.269117\pi$
$284$	0	0
$285$	−2.09316	−0.123988
$286$	0	0
$287$	−19.4622	−1.14882
$288$	0	0
$289$	−16.8777	−0.992805
$290$	0	0
$291$	3.76494	0.220705
$292$	0	0
$293$	−4.54369	−0.265445	−0.132723	−	0.991153i	$-0.542372\pi$
$293$	−4.54369	−0.265445	−0.132723	+	0.991153i	$0.542372\pi$
$294$	0	0
$295$	−16.1565	−0.940666
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−5.67248	−0.328048
$300$	0	0
$301$	2.49424	0.143766
$302$	0	0
$303$	−12.5738	−0.722348
$304$	0	0
$305$	−12.3362	−0.706367
$306$	0	0
$307$	−30.5662	−1.74451	−0.872253	−	0.489056i	$-0.837341\pi$
$307$	−30.5662	−1.74451	−0.872253	+	0.489056i	$0.837341\pi$
$308$	0	0
$309$	−8.11206	−0.461479
$310$	0	0
$311$	3.32536	0.188564	0.0942818	−	0.995546i	$-0.469945\pi$
$311$	3.32536	0.188564	0.0942818	+	0.995546i	$0.469945\pi$
$312$	0	0
$313$	14.5029	0.819752	0.409876	−	0.912141i	$-0.365572\pi$
$313$	14.5029	0.819752	0.409876	+	0.912141i	$0.365572\pi$
$314$	0	0
$315$	14.0633	0.792378
$316$	0	0
$317$	−5.64958	−0.317312	−0.158656	−	0.987334i	$-0.550716\pi$
$317$	−5.64958	−0.317312	−0.158656	+	0.987334i	$0.550716\pi$
$318$	0	0
$319$	2.55494	0.143049
$320$	0	0
$321$	7.04658	0.393302
$322$	0	0
$323$	0.186315	0.0103668
$324$	0	0
$325$	−10.4374	−0.578963
$326$	0	0
$327$	−6.71905	−0.371565
$328$	0	0
$329$	−3.81369	−0.210255
$330$	0	0
$331$	−5.92905	−0.325890	−0.162945	−	0.986635i	$-0.552099\pi$
$331$	−5.92905	−0.325890	−0.162945	+	0.986635i	$0.552099\pi$
$332$	0	0
$333$	3.76494	0.206317
$334$	0	0
$335$	−31.5196	−1.72210
$336$	0	0
$337$	−24.7329	−1.34729	−0.673644	−	0.739056i	$-0.735272\pi$
$337$	−24.7329	−1.34729	−0.673644	+	0.739056i	$0.735272\pi$
$338$	0	0
$339$	−17.4622	−0.948418
$340$	0	0
$341$	−1.81699	−0.0983955
$342$	0	0
$343$	4.25396	0.229692
$344$	0	0
$345$	−22.2874	−1.19991
$346$	0	0
$347$	−20.7817	−1.11562	−0.557809	−	0.829969i	$-0.688358\pi$
$347$	−20.7817	−1.11562	−0.557809	+	0.829969i	$0.688358\pi$
$348$	0	0
$349$	−21.5313	−1.15254	−0.576271	−	0.817259i	$-0.695493\pi$
$349$	−21.5313	−1.15254	−0.576271	+	0.817259i	$0.695493\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−23.3861	−1.24471	−0.622357	−	0.782733i	$-0.713825\pi$
$353$	−23.3861	−1.24471	−0.622357	+	0.782733i	$0.713825\pi$
$354$	0	0
$355$	50.7773	2.69498
$356$	0	0
$357$	−1.25179	−0.0662519
$358$	0	0
$359$	−25.8966	−1.36677	−0.683385	−	0.730058i	$-0.739493\pi$
$359$	−25.8966	−1.36677	−0.683385	+	0.730058i	$0.739493\pi$
$360$	0	0
$361$	−18.7162	−0.985063
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	40.3858	2.11389
$366$	0	0
$367$	3.52987	0.184258	0.0921290	−	0.995747i	$-0.470633\pi$
$367$	3.52987	0.184258	0.0921290	+	0.995747i	$0.470633\pi$
$368$	0	0
$369$	5.43741	0.283060
$370$	0	0
$371$	7.15864	0.371658
$372$	0	0
$373$	26.3362	1.36364	0.681818	−	0.731522i	$-0.261190\pi$
$373$	26.3362	1.36364	0.681818	+	0.731522i	$0.261190\pi$
$374$	0	0
$375$	−21.3638	−1.10322
$376$	0	0
$377$	−2.55494	−0.131586
$378$	0	0
$379$	10.9647	0.563218	0.281609	−	0.959529i	$-0.409132\pi$
$379$	10.9647	0.563218	0.281609	+	0.959529i	$0.409132\pi$
$380$	0	0
$381$	−2.34973	−0.120380
$382$	0	0
$383$	−36.8261	−1.88172	−0.940862	−	0.338789i	$-0.889983\pi$
$383$	−36.8261	−1.88172	−0.940862	+	0.338789i	$0.889983\pi$
$384$	0	0
$385$	14.0633	0.716733
$386$	0	0
$387$	−0.696849	−0.0354228
$388$	0	0
$389$	11.4811	0.582116	0.291058	−	0.956705i	$-0.405993\pi$
$389$	11.4811	0.582116	0.291058	+	0.956705i	$0.405993\pi$
$390$	0	0
$391$	1.98383	0.100327
$392$	0	0
$393$	−8.99783	−0.453881
$394$	0	0
$395$	6.48399	0.326245
$396$	0	0
$397$	−34.1020	−1.71153	−0.855765	−	0.517365i	$-0.826913\pi$
$397$	−34.1020	−1.71153	−0.855765	+	0.517365i	$0.826913\pi$
$398$	0	0
$399$	−1.90684	−0.0954615
$400$	0	0
$401$	15.8614	0.792080	0.396040	−	0.918233i	$-0.370384\pi$
$401$	15.8614	0.792080	0.396040	+	0.918233i	$0.370384\pi$
$402$	0	0
$403$	1.81699	0.0905107
$404$	0	0
$405$	−3.92905	−0.195236
$406$	0	0
$407$	3.76494	0.186621
$408$	0	0
$409$	21.2133	1.04893	0.524465	−	0.851432i	$-0.324265\pi$
$409$	21.2133	1.04893	0.524465	+	0.851432i	$0.324265\pi$
$410$	0	0
$411$	−5.58193	−0.275336
$412$	0	0
$413$	−14.7184	−0.724243
$414$	0	0
$415$	16.5967	0.814702
$416$	0	0
$417$	6.20782	0.303998
$418$	0	0
$419$	20.2437	0.988970	0.494485	−	0.869186i	$-0.335357\pi$
$419$	20.2437	0.988970	0.494485	+	0.869186i	$0.335357\pi$
$420$	0	0
$421$	31.9658	1.55792	0.778959	−	0.627075i	$-0.215748\pi$
$421$	31.9658	1.55792	0.778959	+	0.627075i	$0.215748\pi$
$422$	0	0
$423$	1.06548	0.0518053
$424$	0	0
$425$	3.65027	0.177064
$426$	0	0
$427$	−11.2381	−0.543850
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−23.3827	−1.12631	−0.563154	−	0.826352i	$-0.690412\pi$
$431$	−23.3827	−1.12631	−0.563154	+	0.826352i	$0.690412\pi$
$432$	0	0
$433$	−0.0241191	−0.00115909	−0.000579545	−	1.00000i	$-0.500184\pi$
$433$	−0.0241191	−0.00115909	−0.000579545		1.00000i	$0.500184\pi$
$434$	0	0
$435$	−10.0385	−0.481309
$436$	0	0
$437$	3.02195	0.144559
$438$	0	0
$439$	−20.4429	−0.975686	−0.487843	−	0.872931i	$-0.662216\pi$
$439$	−20.4429	−0.975686	−0.487843	+	0.872931i	$0.662216\pi$
$440$	0	0
$441$	5.81152	0.276739
$442$	0	0
$443$	−10.0335	−0.476704	−0.238352	−	0.971179i	$-0.576607\pi$
$443$	−10.0335	−0.476704	−0.238352	+	0.971179i	$0.576607\pi$
$444$	0	0
$445$	50.0583	2.37299
$446$	0	0
$447$	−10.0196	−0.473911
$448$	0	0
$449$	−35.9866	−1.69831	−0.849157	−	0.528141i	$-0.822889\pi$
$449$	−35.9866	−1.69831	−0.849157	+	0.528141i	$0.822889\pi$
$450$	0	0
$451$	5.43741	0.256038
$452$	0	0
$453$	2.62590	0.123375
$454$	0	0
$455$	−14.0633	−0.659298
$456$	0	0
$457$	35.8173	1.67546	0.837731	−	0.546083i	$-0.183882\pi$
$457$	35.8173	1.67546	0.837731	+	0.546083i	$0.183882\pi$
$458$	0	0
$459$	0.349729	0.0163240
$460$	0	0
$461$	21.5065	1.00166	0.500828	−	0.865547i	$-0.333029\pi$
$461$	21.5065	1.00166	0.500828	+	0.865547i	$0.333029\pi$
$462$	0	0
$463$	3.25371	0.151213	0.0756063	−	0.997138i	$-0.475911\pi$
$463$	3.25371	0.151213	0.0756063	+	0.997138i	$0.475911\pi$
$464$	0	0
$465$	7.13904	0.331065
$466$	0	0
$467$	−33.8289	−1.56542	−0.782708	−	0.622389i	$-0.786162\pi$
$467$	−33.8289	−1.56542	−0.782708	+	0.622389i	$0.786162\pi$
$468$	0	0
$469$	−28.7140	−1.32589
$470$	0	0
$471$	5.69137	0.262245
$472$	0	0
$473$	−0.696849	−0.0320412
$474$	0	0
$475$	5.56042	0.255129
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	18.4541	0.843191	0.421596	−	0.906784i	$-0.361470\pi$
$479$	18.4541	0.843191	0.421596	+	0.906784i	$0.361470\pi$
$480$	0	0
$481$	−3.76494	−0.171666
$482$	0	0
$483$	−20.3036	−0.923845
$484$	0	0
$485$	−14.7926	−0.671698
$486$	0	0
$487$	−8.50550	−0.385421	−0.192711	−	0.981256i	$-0.561728\pi$
$487$	−8.50550	−0.385421	−0.192711	+	0.981256i	$0.561728\pi$
$488$	0	0
$489$	14.4814	0.654871
$490$	0	0
$491$	−33.9658	−1.53286	−0.766428	−	0.642330i	$-0.777968\pi$
$491$	−33.9658	−1.53286	−0.766428	+	0.642330i	$0.777968\pi$
$492$	0	0
$493$	0.893539	0.0402430
$494$	0	0
$495$	−3.92905	−0.176598
$496$	0	0
$497$	46.2576	2.07494
$498$	0	0
$499$	−26.7055	−1.19550	−0.597751	−	0.801682i	$-0.703939\pi$
$499$	−26.7055	−1.19550	−0.597751	+	0.801682i	$0.703939\pi$
$500$	0	0
$501$	5.87699	0.262565
$502$	0	0
$503$	−19.3894	−0.864529	−0.432264	−	0.901747i	$-0.642285\pi$
$503$	−19.3894	−0.864529	−0.432264	+	0.901747i	$0.642285\pi$
$504$	0	0
$505$	49.4032	2.19841
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	26.1190	1.15770	0.578852	−	0.815433i	$-0.303501\pi$
$509$	26.1190	1.15770	0.578852	+	0.815433i	$0.303501\pi$
$510$	0	0
$511$	36.7910	1.62754
$512$	0	0
$513$	0.532739	0.0235210
$514$	0	0
$515$	31.8727	1.40448
$516$	0	0
$517$	1.06548	0.0468597
$518$	0	0
$519$	−21.8555	−0.959349
$520$	0	0
$521$	−20.8072	−0.911579	−0.455789	−	0.890088i	$-0.650643\pi$
$521$	−20.8072	−0.911579	−0.455789	+	0.890088i	$0.650643\pi$
$522$	0	0
$523$	−2.98949	−0.130721	−0.0653607	−	0.997862i	$-0.520820\pi$
$523$	−2.98949	−0.130721	−0.0653607	+	0.997862i	$0.520820\pi$
$524$	0	0
$525$	−37.3588	−1.63047
$526$	0	0
$527$	−0.635455	−0.0276809
$528$	0	0
$529$	9.17697	0.398999
$530$	0	0
$531$	4.11206	0.178448
$532$	0	0
$533$	−5.43741	−0.235520
$534$	0	0
$535$	−27.6863	−1.19698
$536$	0	0
$537$	23.0415	0.994316
$538$	0	0
$539$	5.81152	0.250320
$540$	0	0
$541$	−23.5633	−1.01307	−0.506533	−	0.862220i	$-0.669073\pi$
$541$	−23.5633	−1.01307	−0.506533	+	0.862220i	$0.669073\pi$
$542$	0	0
$543$	2.49494	0.107068
$544$	0	0
$545$	26.3995	1.13083
$546$	0	0
$547$	25.7742	1.10202	0.551012	−	0.834498i	$-0.314242\pi$
$547$	25.7742	1.10202	0.551012	+	0.834498i	$0.314242\pi$
$548$	0	0
$549$	3.13974	0.134001
$550$	0	0
$551$	1.36112	0.0579856
$552$	0	0
$553$	5.90684	0.251185
$554$	0	0
$555$	−14.7926	−0.627912
$556$	0	0
$557$	−15.7401	−0.666930	−0.333465	−	0.942762i	$-0.608218\pi$
$557$	−15.7401	−0.666930	−0.333465	+	0.942762i	$0.608218\pi$
$558$	0	0
$559$	0.696849	0.0294736
$560$	0	0
$561$	0.349729	0.0147656
$562$	0	0
$563$	32.7817	1.38158	0.690791	−	0.723054i	$-0.257262\pi$
$563$	32.7817	1.38158	0.690791	+	0.723054i	$0.257262\pi$
$564$	0	0
$565$	68.6099	2.88644
$566$	0	0
$567$	−3.57932	−0.150317
$568$	0	0
$569$	−26.4821	−1.11019	−0.555093	−	0.831788i	$-0.687317\pi$
$569$	−26.4821	−1.11019	−0.555093	+	0.831788i	$0.687317\pi$
$570$	0	0
$571$	8.97059	0.375408	0.187704	−	0.982226i	$-0.439895\pi$
$571$	8.97059	0.375408	0.187704	+	0.982226i	$0.439895\pi$
$572$	0	0
$573$	−5.87699	−0.245515
$574$	0	0
$575$	59.2060	2.46906
$576$	0	0
$577$	−8.27947	−0.344679	−0.172339	−	0.985038i	$-0.555133\pi$
$577$	−8.27947	−0.344679	−0.172339	+	0.985038i	$0.555133\pi$
$578$	0	0
$579$	−6.75685	−0.280805
$580$	0	0
$581$	15.1194	0.627260
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	3.92905	0.162446
$586$	0	0
$587$	21.0422	0.868506	0.434253	−	0.900791i	$-0.357012\pi$
$587$	21.0422	0.868506	0.434253	+	0.900791i	$0.357012\pi$
$588$	0	0
$589$	−0.967982	−0.0398850
$590$	0	0
$591$	1.54947	0.0637366
$592$	0	0
$593$	−46.2933	−1.90104	−0.950520	−	0.310664i	$-0.899449\pi$
$593$	−46.2933	−1.90104	−0.950520	+	0.310664i	$0.899449\pi$
$594$	0	0
$595$	4.91835	0.201633
$596$	0	0
$597$	28.0146	1.14656
$598$	0	0
$599$	−14.1011	−0.576156	−0.288078	−	0.957607i	$-0.593016\pi$
$599$	−14.1011	−0.576156	−0.288078	+	0.957607i	$0.593016\pi$
$600$	0	0
$601$	−20.8748	−0.851502	−0.425751	−	0.904840i	$-0.639990\pi$
$601$	−20.8748	−0.851502	−0.425751	+	0.904840i	$0.639990\pi$
$602$	0	0
$603$	8.02220	0.326689
$604$	0	0
$605$	−3.92905	−0.159738
$606$	0	0
$607$	28.7988	1.16891	0.584454	−	0.811427i	$-0.301309\pi$
$607$	28.7988	1.16891	0.584454	+	0.811427i	$0.301309\pi$
$608$	0	0
$609$	−9.14496	−0.370572
$610$	0	0
$611$	−1.06548	−0.0431046
$612$	0	0
$613$	25.0363	1.01121	0.505604	−	0.862766i	$-0.331270\pi$
$613$	25.0363	1.01121	0.505604	+	0.862766i	$0.331270\pi$
$614$	0	0
$615$	−21.3638	−0.861474
$616$	0	0
$617$	−23.3316	−0.939294	−0.469647	−	0.882854i	$-0.655619\pi$
$617$	−23.3316	−0.939294	−0.469647	+	0.882854i	$0.655619\pi$
$618$	0	0
$619$	15.8359	0.636498	0.318249	−	0.948007i	$-0.396905\pi$
$619$	15.8359	0.636498	0.318249	+	0.948007i	$0.396905\pi$
$620$	0	0
$621$	5.67248	0.227629
$622$	0	0
$623$	45.6025	1.82703
$624$	0	0
$625$	31.7525	1.27010
$626$	0	0
$627$	0.532739	0.0212756
$628$	0	0
$629$	1.31671	0.0525006
$630$	0	0
$631$	15.1954	0.604919	0.302460	−	0.953162i	$-0.402192\pi$
$631$	15.1954	0.604919	0.302460	+	0.953162i	$0.402192\pi$
$632$	0	0
$633$	−0.808906	−0.0321511
$634$	0	0
$635$	9.23220	0.366369
$636$	0	0
$637$	−5.81152	−0.230261
$638$	0	0
$639$	−12.9236	−0.511249
$640$	0	0
$641$	−24.2874	−0.959296	−0.479648	−	0.877461i	$-0.659236\pi$
$641$	−24.2874	−0.959296	−0.479648	+	0.877461i	$0.659236\pi$
$642$	0	0
$643$	16.5825	0.653950	0.326975	−	0.945033i	$-0.393971\pi$
$643$	16.5825	0.653950	0.326975	+	0.945033i	$0.393971\pi$
$644$	0	0
$645$	2.73795	0.107807
$646$	0	0
$647$	10.1806	0.400240	0.200120	−	0.979771i	$-0.435867\pi$
$647$	10.1806	0.400240	0.200120	+	0.979771i	$0.435867\pi$
$648$	0	0
$649$	4.11206	0.161412
$650$	0	0
$651$	6.50359	0.254896
$652$	0	0
$653$	34.1710	1.33722	0.668608	−	0.743615i	$-0.266891\pi$
$653$	34.1710	1.33722	0.668608	+	0.743615i	$0.266891\pi$
$654$	0	0
$655$	35.3529	1.38135
$656$	0	0
$657$	−10.2788	−0.401013
$658$	0	0
$659$	15.4650	0.602429	0.301215	−	0.953556i	$-0.402608\pi$
$659$	15.4650	0.602429	0.301215	+	0.953556i	$0.402608\pi$
$660$	0	0
$661$	1.77067	0.0688710	0.0344355	−	0.999407i	$-0.489037\pi$
$661$	1.77067	0.0688710	0.0344355	+	0.999407i	$0.489037\pi$
$662$	0	0
$663$	−0.349729	−0.0135824
$664$	0	0
$665$	7.49208	0.290530
$666$	0	0
$667$	14.4929	0.561166
$668$	0	0
$669$	25.1997	0.974278
$670$	0	0
$671$	3.13974	0.121208
$672$	0	0
$673$	−17.6069	−0.678695	−0.339347	−	0.940661i	$-0.610206\pi$
$673$	−17.6069	−0.678695	−0.339347	+	0.940661i	$0.610206\pi$
$674$	0	0
$675$	10.4374	0.401736
$676$	0	0
$677$	−4.93986	−0.189854	−0.0949272	−	0.995484i	$-0.530262\pi$
$677$	−4.93986	−0.189854	−0.0949272	+	0.995484i	$0.530262\pi$
$678$	0	0
$679$	−13.4759	−0.517158
$680$	0	0
$681$	−12.3908	−0.474818
$682$	0	0
$683$	42.7409	1.63543	0.817717	−	0.575621i	$-0.195239\pi$
$683$	42.7409	1.63543	0.817717	+	0.575621i	$0.195239\pi$
$684$	0	0
$685$	21.9317	0.837966
$686$	0	0
$687$	−29.3529	−1.11988
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	−8.83372	−0.336051	−0.168025	−	0.985783i	$-0.553739\pi$
$691$	−8.83372	−0.336051	−0.168025	+	0.985783i	$0.553739\pi$
$692$	0	0
$693$	−3.57932	−0.135967
$694$	0	0
$695$	−24.3908	−0.925197
$696$	0	0
$697$	1.90162	0.0720291
$698$	0	0
$699$	19.2246	0.727139
$700$	0	0
$701$	1.67578	0.0632933	0.0316467	−	0.999499i	$-0.489925\pi$
$701$	1.67578	0.0632933	0.0316467	+	0.999499i	$0.489925\pi$
$702$	0	0
$703$	2.00573	0.0756475
$704$	0	0
$705$	−4.18631	−0.157666
$706$	0	0
$707$	45.0058	1.69262
$708$	0	0
$709$	14.5496	0.546422	0.273211	−	0.961954i	$-0.411914\pi$
$709$	14.5496	0.546422	0.273211	+	0.961954i	$0.411914\pi$
$710$	0	0
$711$	−1.65027	−0.0618900
$712$	0	0
$713$	−10.3068	−0.385994
$714$	0	0
$715$	3.92905	0.146938
$716$	0	0
$717$	−26.1703	−0.977349
$718$	0	0
$719$	−28.9003	−1.07780	−0.538900	−	0.842370i	$-0.681160\pi$
$719$	−28.9003	−1.07780	−0.538900	+	0.842370i	$0.681160\pi$
$720$	0	0
$721$	29.0356	1.08134
$722$	0	0
$723$	−8.65071	−0.321723
$724$	0	0
$725$	26.6670	0.990388
$726$	0	0
$727$	2.36029	0.0875383	0.0437692	−	0.999042i	$-0.486063\pi$
$727$	2.36029	0.0875383	0.0437692	+	0.999042i	$0.486063\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.243709	−0.00901389
$732$	0	0
$733$	−35.0903	−1.29609	−0.648045	−	0.761602i	$-0.724413\pi$
$733$	−35.0903	−1.29609	−0.648045	+	0.761602i	$0.724413\pi$
$734$	0	0
$735$	−22.8337	−0.842234
$736$	0	0
$737$	8.02220	0.295502
$738$	0	0
$739$	−8.52752	−0.313690	−0.156845	−	0.987623i	$-0.550132\pi$
$739$	−8.52752	−0.313690	−0.156845	+	0.987623i	$0.550132\pi$
$740$	0	0
$741$	−0.532739	−0.0195707
$742$	0	0
$743$	3.37341	0.123758	0.0618792	−	0.998084i	$-0.480291\pi$
$743$	3.37341	0.123758	0.0618792	+	0.998084i	$0.480291\pi$
$744$	0	0
$745$	39.3675	1.44231
$746$	0	0
$747$	−4.22411	−0.154552
$748$	0	0
$749$	−25.2219	−0.921590
$750$	0	0
$751$	24.0576	0.877874	0.438937	−	0.898518i	$-0.355355\pi$
$751$	24.0576	0.877874	0.438937	+	0.898518i	$0.355355\pi$
$752$	0	0
$753$	23.3450	0.850737
$754$	0	0
$755$	−10.3173	−0.375484
$756$	0	0
$757$	9.50506	0.345467	0.172734	−	0.984969i	$-0.444740\pi$
$757$	9.50506	0.345467	0.172734	+	0.984969i	$0.444740\pi$
$758$	0	0
$759$	5.67248	0.205898
$760$	0	0
$761$	23.2401	0.842455	0.421227	−	0.906955i	$-0.361599\pi$
$761$	23.2401	0.842455	0.421227	+	0.906955i	$0.361599\pi$
$762$	0	0
$763$	24.0496	0.870655
$764$	0	0
$765$	−1.37410	−0.0496808
$766$	0	0
$767$	−4.11206	−0.148478
$768$	0	0
$769$	−3.13313	−0.112983	−0.0564917	−	0.998403i	$-0.517991\pi$
$769$	−3.13313	−0.112983	−0.0564917	+	0.998403i	$0.517991\pi$
$770$	0	0
$771$	−18.4833	−0.665660
$772$	0	0
$773$	34.2987	1.23364	0.616819	−	0.787105i	$-0.288421\pi$
$773$	34.2987	1.23364	0.616819	+	0.787105i	$0.288421\pi$
$774$	0	0
$775$	−18.9647	−0.681231
$776$	0	0
$777$	−13.4759	−0.483445
$778$	0	0
$779$	2.89672	0.103786
$780$	0	0
$781$	−12.9236	−0.462442
$782$	0	0
$783$	2.55494	0.0913062
$784$	0	0
$785$	−22.3617	−0.798123
$786$	0	0
$787$	−21.3807	−0.762140	−0.381070	−	0.924546i	$-0.624444\pi$
$787$	−21.3807	−0.762140	−0.381070	+	0.924546i	$0.624444\pi$
$788$	0	0
$789$	11.0167	0.392206
$790$	0	0
$791$	62.5029	2.22235
$792$	0	0
$793$	−3.13974	−0.111495
$794$	0	0
$795$	7.85809	0.278698
$796$	0	0
$797$	24.5844	0.870824	0.435412	−	0.900231i	$-0.356603\pi$
$797$	24.5844	0.870824	0.435412	+	0.900231i	$0.356603\pi$
$798$	0	0
$799$	0.372629	0.0131827
$800$	0	0
$801$	−12.7406	−0.450166
$802$	0	0
$803$	−10.2788	−0.362730
$804$	0	0
$805$	79.7738	2.81166
$806$	0	0
$807$	−20.5088	−0.721944
$808$	0	0
$809$	−31.4865	−1.10701	−0.553503	−	0.832847i	$-0.686709\pi$
$809$	−31.4865	−1.10701	−0.553503	+	0.832847i	$0.686709\pi$
$810$	0	0
$811$	−41.3043	−1.45039	−0.725196	−	0.688543i	$-0.758251\pi$
$811$	−41.3043	−1.45039	−0.725196	+	0.688543i	$0.758251\pi$
$812$	0	0
$813$	−10.6259	−0.372666
$814$	0	0
$815$	−56.8980	−1.99305
$816$	0	0
$817$	−0.371239	−0.0129880
$818$	0	0
$819$	3.57932	0.125071
$820$	0	0
$821$	45.1734	1.57656	0.788281	−	0.615315i	$-0.210971\pi$
$821$	45.1734	1.57656	0.788281	+	0.615315i	$0.210971\pi$
$822$	0	0
$823$	−35.6711	−1.24341	−0.621707	−	0.783250i	$-0.713561\pi$
$823$	−35.6711	−1.24341	−0.621707	+	0.783250i	$0.713561\pi$
$824$	0	0
$825$	10.4374	0.363384
$826$	0	0
$827$	−55.6369	−1.93468	−0.967342	−	0.253474i	$-0.918427\pi$
$827$	−55.6369	−1.93468	−0.967342	+	0.253474i	$0.918427\pi$
$828$	0	0
$829$	−43.9360	−1.52596	−0.762980	−	0.646423i	$-0.776264\pi$
$829$	−43.9360	−1.52596	−0.762980	+	0.646423i	$0.776264\pi$
$830$	0	0
$831$	16.1011	0.558541
$832$	0	0
$833$	2.03246	0.0704205
$834$	0	0
$835$	−23.0910	−0.799097
$836$	0	0
$837$	−1.81699	−0.0628044
$838$	0	0
$839$	−50.2576	−1.73508	−0.867542	−	0.497364i	$-0.834301\pi$
$839$	−50.2576	−1.73508	−0.867542	+	0.497364i	$0.834301\pi$
$840$	0	0
$841$	−22.4723	−0.774906
$842$	0	0
$843$	21.3777	0.736288
$844$	0	0
$845$	−3.92905	−0.135163
$846$	0	0
$847$	−3.57932	−0.122987
$848$	0	0
$849$	22.3199	0.766016
$850$	0	0
$851$	21.3565	0.732092
$852$	0	0
$853$	−22.9302	−0.785114	−0.392557	−	0.919728i	$-0.628410\pi$
$853$	−22.9302	−0.785114	−0.392557	+	0.919728i	$0.628410\pi$
$854$	0	0
$855$	−2.09316	−0.0715845
$856$	0	0
$857$	−47.8213	−1.63354	−0.816772	−	0.576960i	$-0.804239\pi$
$857$	−47.8213	−1.63354	−0.816772	+	0.576960i	$0.804239\pi$
$858$	0	0
$859$	−15.7592	−0.537697	−0.268849	−	0.963182i	$-0.586643\pi$
$859$	−15.7592	−0.537697	−0.268849	+	0.963182i	$0.586643\pi$
$860$	0	0
$861$	−19.4622	−0.663271
$862$	0	0
$863$	55.1811	1.87839	0.939194	−	0.343387i	$-0.111574\pi$
$863$	55.1811	1.87839	0.939194	+	0.343387i	$0.111574\pi$
$864$	0	0
$865$	85.8712	2.91971
$866$	0	0
$867$	−16.8777	−0.573196
$868$	0	0
$869$	−1.65027	−0.0559816
$870$	0	0
$871$	−8.02220	−0.271822
$872$	0	0
$873$	3.76494	0.127424
$874$	0	0
$875$	76.4680	2.58509
$876$	0	0
$877$	−0.688510	−0.0232493	−0.0116247	−	0.999932i	$-0.503700\pi$
$877$	−0.688510	−0.0232493	−0.0116247	+	0.999932i	$0.503700\pi$
$878$	0	0
$879$	−4.54369	−0.153255
$880$	0	0
$881$	7.76767	0.261699	0.130850	−	0.991402i	$-0.458229\pi$
$881$	7.76767	0.261699	0.130850	+	0.991402i	$0.458229\pi$
$882$	0	0
$883$	−16.1915	−0.544889	−0.272444	−	0.962172i	$-0.587832\pi$
$883$	−16.1915	−0.544889	−0.272444	+	0.962172i	$0.587832\pi$
$884$	0	0
$885$	−16.1565	−0.543094
$886$	0	0
$887$	54.8217	1.84073	0.920367	−	0.391056i	$-0.127890\pi$
$887$	54.8217	1.84073	0.920367	+	0.391056i	$0.127890\pi$
$888$	0	0
$889$	8.41043	0.282077
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0.567622	0.0189947
$894$	0	0
$895$	−90.5313	−3.02613
$896$	0	0
$897$	−5.67248	−0.189398
$898$	0	0
$899$	−4.64231	−0.154830
$900$	0	0
$901$	−0.699459	−0.0233024
$902$	0	0
$903$	2.49424	0.0830033
$904$	0	0
$905$	−9.80274	−0.325854
$906$	0	0
$907$	−20.4482	−0.678972	−0.339486	−	0.940611i	$-0.610253\pi$
$907$	−20.4482	−0.678972	−0.339486	+	0.940611i	$0.610253\pi$
$908$	0	0
$909$	−12.5738	−0.417048
$910$	0	0
$911$	−49.1543	−1.62855	−0.814277	−	0.580476i	$-0.802866\pi$
$911$	−49.1543	−1.62855	−0.814277	+	0.580476i	$0.802866\pi$
$912$	0	0
$913$	−4.22411	−0.139798
$914$	0	0
$915$	−12.3362	−0.407821
$916$	0	0
$917$	32.2061	1.06354
$918$	0	0
$919$	−7.00478	−0.231066	−0.115533	−	0.993304i	$-0.536858\pi$
$919$	−7.00478	−0.231066	−0.115533	+	0.993304i	$0.536858\pi$
$920$	0	0
$921$	−30.5662	−1.00719
$922$	0	0
$923$	12.9236	0.425385
$924$	0	0
$925$	39.2962	1.29205
$926$	0	0
$927$	−8.11206	−0.266435
$928$	0	0
$929$	−4.77975	−0.156819	−0.0784093	−	0.996921i	$-0.524984\pi$
$929$	−4.77975	−0.156819	−0.0784093	+	0.996921i	$0.524984\pi$
$930$	0	0
$931$	3.09602	0.101468
$932$	0	0
$933$	3.32536	0.108867
$934$	0	0
$935$	−1.37410	−0.0449380
$936$	0	0
$937$	−59.8371	−1.95479	−0.977396	−	0.211417i	$-0.932192\pi$
$937$	−59.8371	−1.95479	−0.977396	+	0.211417i	$0.932192\pi$
$938$	0	0
$939$	14.5029	0.473284
$940$	0	0
$941$	18.8835	0.615584	0.307792	−	0.951454i	$-0.400410\pi$
$941$	18.8835	0.615584	0.307792	+	0.951454i	$0.400410\pi$
$942$	0	0
$943$	30.8436	1.00441
$944$	0	0
$945$	14.0633	0.457479
$946$	0	0
$947$	−49.6522	−1.61348	−0.806739	−	0.590908i	$-0.798770\pi$
$947$	−49.6522	−1.61348	−0.806739	+	0.590908i	$0.798770\pi$
$948$	0	0
$949$	10.2788	0.333663
$950$	0	0
$951$	−5.64958	−0.183200
$952$	0	0
$953$	−33.1210	−1.07289	−0.536447	−	0.843934i	$-0.680234\pi$
$953$	−33.1210	−1.07289	−0.536447	+	0.843934i	$0.680234\pi$
$954$	0	0
$955$	23.0910	0.747207
$956$	0	0
$957$	2.55494	0.0825896
$958$	0	0
$959$	19.9795	0.645171
$960$	0	0
$961$	−27.6985	−0.893501
$962$	0	0
$963$	7.04658	0.227073
$964$	0	0
$965$	26.5480	0.854610
$966$	0	0
$967$	−31.7556	−1.02119	−0.510595	−	0.859821i	$-0.670575\pi$
$967$	−31.7556	−1.02119	−0.510595	+	0.859821i	$0.670575\pi$
$968$	0	0
$969$	0.186315	0.00598529
$970$	0	0
$971$	30.8174	0.988978	0.494489	−	0.869184i	$-0.335355\pi$
$971$	30.8174	0.988978	0.494489	+	0.869184i	$0.335355\pi$
$972$	0	0
$973$	−22.2198	−0.712333
$974$	0	0
$975$	−10.4374	−0.334265
$976$	0	0
$977$	2.08985	0.0668603	0.0334302	−	0.999441i	$-0.489357\pi$
$977$	2.08985	0.0668603	0.0334302	+	0.999441i	$0.489357\pi$
$978$	0	0
$979$	−12.7406	−0.407190
$980$	0	0
$981$	−6.71905	−0.214523
$982$	0	0
$983$	−1.35068	−0.0430800	−0.0215400	−	0.999768i	$-0.506857\pi$
$983$	−1.35068	−0.0430800	−0.0215400	+	0.999768i	$0.506857\pi$
$984$	0	0
$985$	−6.08794	−0.193978
$986$	0	0
$987$	−3.81369	−0.121391
$988$	0	0
$989$	−3.95286	−0.125694
$990$	0	0
$991$	−34.0088	−1.08033	−0.540163	−	0.841560i	$-0.681637\pi$
$991$	−34.0088	−1.08033	−0.540163	+	0.841560i	$0.681637\pi$
$992$	0	0
$993$	−5.92905	−0.188153
$994$	0	0
$995$	−110.071	−3.48947
$996$	0	0
$997$	0.970980	0.0307512	0.0153756	−	0.999882i	$-0.495106\pi$
$997$	0.970980	0.0307512	0.0153756	+	0.999882i	$0.495106\pi$
$998$	0	0
$999$	3.76494	0.119117

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.q.1.1	✓	4	4.3	odd	2
6864.2.a.by.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.92905 q^{5} -3.57932 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.92905 q^{5} -3.57932 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.92905 q^{15} +0.349729 q^{17} +0.532739 q^{19} -3.57932 q^{21} +5.67248 q^{23} +10.4374 q^{25} +1.00000 q^{27} +2.55494 q^{29} -1.81699 q^{31} +1.00000 q^{33} +14.0633 q^{35} +3.76494 q^{37} -1.00000 q^{39} +5.43741 q^{41} -0.696849 q^{43} -3.92905 q^{45} +1.06548 q^{47} +5.81152 q^{49} +0.349729 q^{51} -2.00000 q^{53} -3.92905 q^{55} +0.532739 q^{57} +4.11206 q^{59} +3.13974 q^{61} -3.57932 q^{63} +3.92905 q^{65} +8.02220 q^{67} +5.67248 q^{69} -12.9236 q^{71} -10.2788 q^{73} +10.4374 q^{75} -3.57932 q^{77} -1.65027 q^{79} +1.00000 q^{81} -4.22411 q^{83} -1.37410 q^{85} +2.55494 q^{87} -12.7406 q^{89} +3.57932 q^{91} -1.81699 q^{93} -2.09316 q^{95} +3.76494 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 3 q^{15} - 2 q^{19} - 3 q^{21} - 3 q^{23} + 5 q^{25} + 4 q^{27} - 21 q^{29} - 10 q^{31} + 4 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 15 q^{41} + 3 q^{43} - 3 q^{45} - 4 q^{47} + 5 q^{49} - 8 q^{53} - 3 q^{55} - 2 q^{57} + q^{59} - 9 q^{61} - 3 q^{63} + 3 q^{65} + 5 q^{67} - 3 q^{69} - 18 q^{71} - 27 q^{73} + 5 q^{75} - 3 q^{77} - 8 q^{79} + 4 q^{81} + 14 q^{83} - 24 q^{85} - 21 q^{87} - 20 q^{89} + 3 q^{91} - 10 q^{93} + 6 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 3 * q^15 - 2 * q^19 - 3 * q^21 - 3 * q^23 + 5 * q^25 + 4 * q^27 - 21 * q^29 - 10 * q^31 + 4 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 15 * q^41 + 3 * q^43 - 3 * q^45 - 4 * q^47 + 5 * q^49 - 8 * q^53 - 3 * q^55 - 2 * q^57 + q^59 - 9 * q^61 - 3 * q^63 + 3 * q^65 + 5 * q^67 - 3 * q^69 - 18 * q^71 - 27 * q^73 + 5 * q^75 - 3 * q^77 - 8 * q^79 + 4 * q^81 + 14 * q^83 - 24 * q^85 - 21 * q^87 - 20 * q^89 + 3 * q^91 - 10 * q^93 + 6 * q^95 + 4 * q^97 + 4 * q^99

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