LMFDB - Embedded newform 6864.2.a.cb.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.90996.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} - 3x + 2 $ x^4 - x^3 - 11x^2 - 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.622070$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.29198 q^{5} -3.59300 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.29198 q^{5} -3.59300 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.29198 q^{15} -8.12913 q^{17} -3.59300 q^{19} -3.59300 q^{21} +3.74682 q^{23} +5.83714 q^{25} +1.00000 q^{27} +3.69898 q^{29} -5.29198 q^{31} -1.00000 q^{33} +11.8281 q^{35} -9.82810 q^{37} +1.00000 q^{39} +8.08128 q^{41} -2.94312 q^{43} -3.29198 q^{45} -8.73778 q^{47} +5.90968 q^{49} -8.12913 q^{51} -8.58396 q^{53} +3.29198 q^{55} -3.59300 q^{57} -6.58396 q^{59} +7.24414 q^{61} -3.59300 q^{63} -3.29198 q^{65} +4.38230 q^{67} +3.74682 q^{69} -9.33982 q^{71} -2.99096 q^{73} +5.83714 q^{75} +3.59300 q^{77} -13.5271 q^{79} +1.00000 q^{81} +4.09568 q^{83} +26.7609 q^{85} +3.69898 q^{87} +16.5361 q^{89} -3.59300 q^{91} -5.29198 q^{93} +11.8281 q^{95} +11.0322 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 12 q^{29} - 6 q^{31} - 4 q^{33} + 10 q^{35} - 2 q^{37} + 4 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{45} - 6 q^{47} + 32 q^{49} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 2 q^{57} + 4 q^{59} + 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} + 4 q^{69} - 14 q^{71} + 6 q^{73} + 4 q^{75} + 2 q^{77} - 14 q^{79} + 4 q^{81} + 34 q^{85} + 12 q^{87} + 44 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 12 * q^29 - 6 * q^31 - 4 * q^33 + 10 * q^35 - 2 * q^37 + 4 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^45 - 6 * q^47 + 32 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 2 * q^57 + 4 * q^59 + 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 + 4 * q^69 - 14 * q^71 + 6 * q^73 + 4 * q^75 + 2 * q^77 - 14 * q^79 + 4 * q^81 + 34 * q^85 + 12 * q^87 + 44 * q^89 - 2 * q^91 - 6 * q^93 + 10 * q^95 + 18 * 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
$5$	−3.29198	−1.47222	−0.736109	−	0.676862i	$-0.763339\pi$
$5$	−3.29198	−1.47222	−0.736109	+	0.676862i	$0.763339\pi$
$6$	0	0
$7$	−3.59300	−1.35803	−0.679014	−	0.734125i	$-0.737592\pi$
$7$	−3.59300	−1.35803	−0.679014	+	0.734125i	$0.737592\pi$
$8$	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
$15$	−3.29198	−0.849986
$16$	0	0
$17$	−8.12913	−1.97160	−0.985801	−	0.167917i	$-0.946296\pi$
$17$	−8.12913	−1.97160	−0.985801	+	0.167917i	$0.946296\pi$
$18$	0	0
$19$	−3.59300	−0.824292	−0.412146	−	0.911118i	$-0.635221\pi$
$19$	−3.59300	−0.824292	−0.412146	+	0.911118i	$0.635221\pi$
$20$	0	0
$21$	−3.59300	−0.784058
$22$	0	0
$23$	3.74682	0.781266	0.390633	−	0.920547i	$-0.372256\pi$
$23$	3.74682	0.781266	0.390633	+	0.920547i	$0.372256\pi$
$24$	0	0
$25$	5.83714	1.16743
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.69898	0.686883	0.343441	−	0.939174i	$-0.388407\pi$
$29$	3.69898	0.686883	0.343441	+	0.939174i	$0.388407\pi$
$30$	0	0
$31$	−5.29198	−0.950468	−0.475234	−	0.879859i	$-0.657637\pi$
$31$	−5.29198	−0.950468	−0.475234	+	0.879859i	$0.657637\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	11.8281	1.99931
$36$	0	0
$37$	−9.82810	−1.61573	−0.807865	−	0.589367i	$-0.799377\pi$
$37$	−9.82810	−1.61573	−0.807865	+	0.589367i	$0.799377\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.08128	1.26208	0.631042	−	0.775749i	$-0.282627\pi$
$41$	8.08128	1.26208	0.631042	+	0.775749i	$0.282627\pi$
$42$	0	0
$43$	−2.94312	−0.448821	−0.224411	−	0.974495i	$-0.572046\pi$
$43$	−2.94312	−0.448821	−0.224411	+	0.974495i	$0.572046\pi$
$44$	0	0
$45$	−3.29198	−0.490740
$46$	0	0
$47$	−8.73778	−1.27454	−0.637268	−	0.770642i	$-0.719936\pi$
$47$	−8.73778	−1.27454	−0.637268	+	0.770642i	$0.719936\pi$
$48$	0	0
$49$	5.90968	0.844240
$50$	0	0
$51$	−8.12913	−1.13831
$52$	0	0
$53$	−8.58396	−1.17910	−0.589549	−	0.807733i	$-0.700695\pi$
$53$	−8.58396	−1.17910	−0.589549	+	0.807733i	$0.700695\pi$
$54$	0	0
$55$	3.29198	0.443891
$56$	0	0
$57$	−3.59300	−0.475905
$58$	0	0
$59$	−6.58396	−0.857159	−0.428580	−	0.903504i	$-0.640986\pi$
$59$	−6.58396	−0.857159	−0.428580	+	0.903504i	$0.640986\pi$
$60$	0	0
$61$	7.24414	0.927517	0.463759	−	0.885962i	$-0.346501\pi$
$61$	7.24414	0.927517	0.463759	+	0.885962i	$0.346501\pi$
$62$	0	0
$63$	−3.59300	−0.452676
$64$	0	0
$65$	−3.29198	−0.408320
$66$	0	0
$67$	4.38230	0.535384	0.267692	−	0.963505i	$-0.413739\pi$
$67$	4.38230	0.535384	0.267692	+	0.963505i	$0.413739\pi$
$68$	0	0
$69$	3.74682	0.451064
$70$	0	0
$71$	−9.33982	−1.10843	−0.554217	−	0.832373i	$-0.686982\pi$
$71$	−9.33982	−1.10843	−0.554217	+	0.832373i	$0.686982\pi$
$72$	0	0
$73$	−2.99096	−0.350065	−0.175033	−	0.984563i	$-0.556003\pi$
$73$	−2.99096	−0.350065	−0.175033	+	0.984563i	$0.556003\pi$
$74$	0	0
$75$	5.83714	0.674015
$76$	0	0
$77$	3.59300	0.409461
$78$	0	0
$79$	−13.5271	−1.52192	−0.760958	−	0.648801i	$-0.775271\pi$
$79$	−13.5271	−1.52192	−0.760958	+	0.648801i	$0.775271\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.09568	0.449560	0.224780	−	0.974410i	$-0.427834\pi$
$83$	4.09568	0.449560	0.224780	+	0.974410i	$0.427834\pi$
$84$	0	0
$85$	26.7609	2.90263
$86$	0	0
$87$	3.69898	0.396572
$88$	0	0
$89$	16.5361	1.75283	0.876413	−	0.481561i	$-0.159930\pi$
$89$	16.5361	1.75283	0.876413	+	0.481561i	$0.159930\pi$
$90$	0	0
$91$	−3.59300	−0.376649
$92$	0	0
$93$	−5.29198	−0.548753
$94$	0	0
$95$	11.8281	1.21354
$96$	0	0
$97$	11.0322	1.12015	0.560075	−	0.828442i	$-0.310772\pi$
$97$	11.0322	1.12015	0.560075	+	0.828442i	$0.310772\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	8.94312	0.889873	0.444937	−	0.895562i	$-0.353226\pi$
$101$	8.94312	0.889873	0.444937	+	0.895562i	$0.353226\pi$
$102$	0	0
$103$	−9.98192	−0.983548	−0.491774	−	0.870723i	$-0.663651\pi$
$103$	−9.98192	−0.983548	−0.491774	+	0.870723i	$0.663651\pi$
$104$	0	0
$105$	11.8281	1.15430
$106$	0	0
$107$	−10.5840	−1.02319	−0.511595	−	0.859227i	$-0.670945\pi$
$107$	−10.5840	−1.02319	−0.511595	+	0.859227i	$0.670945\pi$
$108$	0	0
$109$	−6.17697	−0.591646	−0.295823	−	0.955243i	$-0.595594\pi$
$109$	−6.17697	−0.591646	−0.295823	+	0.955243i	$0.595594\pi$
$110$	0	0
$111$	−9.82810	−0.932842
$112$	0	0
$113$	15.7700	1.48351	0.741757	−	0.670669i	$-0.233993\pi$
$113$	15.7700	1.48351	0.741757	+	0.670669i	$0.233993\pi$
$114$	0	0
$115$	−12.3345	−1.15019
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	29.2080	2.67749
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.08128	0.728665
$124$	0	0
$125$	−2.75586	−0.246492
$126$	0	0
$127$	7.46895	0.662762	0.331381	−	0.943497i	$-0.392485\pi$
$127$	7.46895	0.662762	0.331381	+	0.943497i	$0.392485\pi$
$128$	0	0
$129$	−2.94312	−0.259127
$130$	0	0
$131$	17.6743	1.54421	0.772105	−	0.635495i	$-0.219204\pi$
$131$	17.6743	1.54421	0.772105	+	0.635495i	$0.219204\pi$
$132$	0	0
$133$	12.9097	1.11941
$134$	0	0
$135$	−3.29198	−0.283329
$136$	0	0
$137$	18.9081	1.61543	0.807716	−	0.589572i	$-0.200704\pi$
$137$	18.9081	1.61543	0.807716	+	0.589572i	$0.200704\pi$
$138$	0	0
$139$	8.28294	0.702550	0.351275	−	0.936272i	$-0.385748\pi$
$139$	8.28294	0.702550	0.351275	+	0.936272i	$0.385748\pi$
$140$	0	0
$141$	−8.73778	−0.735854
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−12.1770	−1.01124
$146$	0	0
$147$	5.90968	0.487422
$148$	0	0
$149$	−0.0812828	−0.00665895	−0.00332947	−	0.999994i	$-0.501060\pi$
$149$	−0.0812828	−0.00665895	−0.00332947	+	0.999994i	$0.501060\pi$
$150$	0	0
$151$	16.0632	1.30720	0.653602	−	0.756838i	$-0.273257\pi$
$151$	16.0632	1.30720	0.653602	+	0.756838i	$0.273257\pi$
$152$	0	0
$153$	−8.12913	−0.657201
$154$	0	0
$155$	17.4211	1.39930
$156$	0	0
$157$	8.02315	0.640317	0.320159	−	0.947364i	$-0.396264\pi$
$157$	8.02315	0.640317	0.320159	+	0.947364i	$0.396264\pi$
$158$	0	0
$159$	−8.58396	−0.680753
$160$	0	0
$161$	−13.4623	−1.06098
$162$	0	0
$163$	−14.0298	−1.09890	−0.549448	−	0.835528i	$-0.685162\pi$
$163$	−14.0298	−1.09890	−0.549448	+	0.835528i	$0.685162\pi$
$164$	0	0
$165$	3.29198	0.256280
$166$	0	0
$167$	−15.1860	−1.17513	−0.587564	−	0.809177i	$-0.699913\pi$
$167$	−15.1860	−1.17513	−0.587564	+	0.809177i	$0.699913\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.59300	−0.274764
$172$	0	0
$173$	8.43676	0.641435	0.320717	−	0.947175i	$-0.396076\pi$
$173$	8.43676	0.641435	0.320717	+	0.947175i	$0.396076\pi$
$174$	0	0
$175$	−20.9729	−1.58540
$176$	0	0
$177$	−6.58396	−0.494881
$178$	0	0
$179$	11.8425	0.885150	0.442575	−	0.896731i	$-0.354065\pi$
$179$	11.8425	0.885150	0.442575	+	0.896731i	$0.354065\pi$
$180$	0	0
$181$	−4.02315	−0.299038	−0.149519	−	0.988759i	$-0.547773\pi$
$181$	−4.02315	−0.299038	−0.149519	+	0.988759i	$0.547773\pi$
$182$	0	0
$183$	7.24414	0.535502
$184$	0	0
$185$	32.3539	2.37871
$186$	0	0
$187$	8.12913	0.594460
$188$	0	0
$189$	−3.59300	−0.261353
$190$	0	0
$191$	9.41604	0.681321	0.340660	−	0.940186i	$-0.389349\pi$
$191$	9.41604	0.681321	0.340660	+	0.940186i	$0.389349\pi$
$192$	0	0
$193$	−25.8332	−1.85951	−0.929756	−	0.368176i	$-0.879982\pi$
$193$	−25.8332	−1.85951	−0.929756	+	0.368176i	$0.879982\pi$
$194$	0	0
$195$	−3.29198	−0.235744
$196$	0	0
$197$	−3.00904	−0.214385	−0.107193	−	0.994238i	$-0.534186\pi$
$197$	−3.00904	−0.214385	−0.107193	+	0.994238i	$0.534186\pi$
$198$	0	0
$199$	−3.30227	−0.234092	−0.117046	−	0.993127i	$-0.537342\pi$
$199$	−3.30227	−0.234092	−0.117046	+	0.993127i	$0.537342\pi$
$200$	0	0
$201$	4.38230	0.309104
$202$	0	0
$203$	−13.2904	−0.932806
$204$	0	0
$205$	−26.6034	−1.85806
$206$	0	0
$207$	3.74682	0.260422
$208$	0	0
$209$	3.59300	0.248533
$210$	0	0
$211$	−10.0334	−0.690731	−0.345365	−	0.938468i	$-0.612245\pi$
$211$	−10.0334	−0.690731	−0.345365	+	0.938468i	$0.612245\pi$
$212$	0	0
$213$	−9.33982	−0.639954
$214$	0	0
$215$	9.68869	0.660763
$216$	0	0
$217$	19.0141	1.29076
$218$	0	0
$219$	−2.99096	−0.202110
$220$	0	0
$221$	−8.12913	−0.546824
$222$	0	0
$223$	23.6640	1.58466	0.792329	−	0.610094i	$-0.208868\pi$
$223$	23.6640	1.58466	0.792329	+	0.610094i	$0.208868\pi$
$224$	0	0
$225$	5.83714	0.389143
$226$	0	0
$227$	11.4124	0.757465	0.378732	−	0.925506i	$-0.376360\pi$
$227$	11.4124	0.757465	0.378732	+	0.925506i	$0.376360\pi$
$228$	0	0
$229$	8.91843	0.589346	0.294673	−	0.955598i	$-0.404789\pi$
$229$	8.91843	0.589346	0.294673	+	0.955598i	$0.404789\pi$
$230$	0	0
$231$	3.59300	0.236402
$232$	0	0
$233$	11.7947	0.772694	0.386347	−	0.922354i	$-0.373737\pi$
$233$	11.7947	0.772694	0.386347	+	0.922354i	$0.373737\pi$
$234$	0	0
$235$	28.7646	1.87640
$236$	0	0
$237$	−13.5271	−0.878678
$238$	0	0
$239$	13.7556	0.889774	0.444887	−	0.895587i	$-0.353244\pi$
$239$	13.7556	0.889774	0.444887	+	0.895587i	$0.353244\pi$
$240$	0	0
$241$	−29.0542	−1.87154	−0.935772	−	0.352607i	$-0.885295\pi$
$241$	−29.0542	−1.87154	−0.935772	+	0.352607i	$0.885295\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−19.4545	−1.24291
$246$	0	0
$247$	−3.59300	−0.228617
$248$	0	0
$249$	4.09568	0.259553
$250$	0	0
$251$	11.7881	0.744055	0.372028	−	0.928222i	$-0.378663\pi$
$251$	11.7881	0.744055	0.372028	+	0.928222i	$0.378663\pi$
$252$	0	0
$253$	−3.74682	−0.235561
$254$	0	0
$255$	26.7609	1.67583
$256$	0	0
$257$	−12.4883	−0.778997	−0.389499	−	0.921027i	$-0.627352\pi$
$257$	−12.4883	−0.778997	−0.389499	+	0.921027i	$0.627352\pi$
$258$	0	0
$259$	35.3124	2.19421
$260$	0	0
$261$	3.69898	0.228961
$262$	0	0
$263$	−22.2402	−1.37139	−0.685694	−	0.727890i	$-0.740501\pi$
$263$	−22.2402	−1.37139	−0.685694	+	0.727890i	$0.740501\pi$
$264$	0	0
$265$	28.2583	1.73589
$266$	0	0
$267$	16.5361	1.01199
$268$	0	0
$269$	24.9379	1.52049	0.760245	−	0.649636i	$-0.225079\pi$
$269$	24.9379	1.52049	0.760245	+	0.649636i	$0.225079\pi$
$270$	0	0
$271$	−22.4846	−1.36584	−0.682921	−	0.730492i	$-0.739291\pi$
$271$	−22.4846	−1.36584	−0.682921	+	0.730492i	$0.739291\pi$
$272$	0	0
$273$	−3.59300	−0.217458
$274$	0	0
$275$	−5.83714	−0.351993
$276$	0	0
$277$	7.98192	0.479587	0.239794	−	0.970824i	$-0.422920\pi$
$277$	7.98192	0.479587	0.239794	+	0.970824i	$0.422920\pi$
$278$	0	0
$279$	−5.29198	−0.316823
$280$	0	0
$281$	0.990959	0.0591157	0.0295578	−	0.999563i	$-0.490590\pi$
$281$	0.990959	0.0591157	0.0295578	+	0.999563i	$0.490590\pi$
$282$	0	0
$283$	−6.30102	−0.374557	−0.187278	−	0.982307i	$-0.559967\pi$
$283$	−6.30102	−0.374557	−0.187278	+	0.982307i	$0.559967\pi$
$284$	0	0
$285$	11.8281	0.700636
$286$	0	0
$287$	−29.0361	−1.71395
$288$	0	0
$289$	49.0827	2.88722
$290$	0	0
$291$	11.0322	0.646719
$292$	0	0
$293$	−15.0090	−0.876838	−0.438419	−	0.898771i	$-0.644461\pi$
$293$	−15.0090	−0.876838	−0.438419	+	0.898771i	$0.644461\pi$
$294$	0	0
$295$	21.6743	1.26193
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	3.74682	0.216684
$300$	0	0
$301$	10.5746	0.609512
$302$	0	0
$303$	8.94312	0.513769
$304$	0	0
$305$	−23.8476	−1.36551
$306$	0	0
$307$	−6.77901	−0.386899	−0.193449	−	0.981110i	$-0.561968\pi$
$307$	−6.77901	−0.386899	−0.193449	+	0.981110i	$0.561968\pi$
$308$	0	0
$309$	−9.98192	−0.567852
$310$	0	0
$311$	−6.23510	−0.353560	−0.176780	−	0.984250i	$-0.556568\pi$
$311$	−6.23510	−0.353560	−0.176780	+	0.984250i	$0.556568\pi$
$312$	0	0
$313$	8.93283	0.504913	0.252457	−	0.967608i	$-0.418761\pi$
$313$	8.93283	0.504913	0.252457	+	0.967608i	$0.418761\pi$
$314$	0	0
$315$	11.8281	0.666438
$316$	0	0
$317$	−31.1201	−1.74788	−0.873939	−	0.486036i	$-0.838443\pi$
$317$	−31.1201	−1.74788	−0.873939	+	0.486036i	$0.838443\pi$
$318$	0	0
$319$	−3.69898	−0.207103
$320$	0	0
$321$	−10.5840	−0.590739
$322$	0	0
$323$	29.2080	1.62518
$324$	0	0
$325$	5.83714	0.323786
$326$	0	0
$327$	−6.17697	−0.341587
$328$	0	0
$329$	31.3949	1.73086
$330$	0	0
$331$	0.259790	0.0142794	0.00713968	−	0.999975i	$-0.497727\pi$
$331$	0.259790	0.0142794	0.00713968	+	0.999975i	$0.497727\pi$
$332$	0	0
$333$	−9.82810	−0.538577
$334$	0	0
$335$	−14.4265	−0.788202
$336$	0	0
$337$	0.583963	0.0318105	0.0159053	−	0.999874i	$-0.494937\pi$
$337$	0.583963	0.0318105	0.0159053	+	0.999874i	$0.494937\pi$
$338$	0	0
$339$	15.7700	0.856507
$340$	0	0
$341$	5.29198	0.286577
$342$	0	0
$343$	3.91754	0.211527
$344$	0	0
$345$	−12.3345	−0.664065
$346$	0	0
$347$	21.8462	1.17276	0.586382	−	0.810034i	$-0.300552\pi$
$347$	21.8462	1.17276	0.586382	+	0.810034i	$0.300552\pi$
$348$	0	0
$349$	34.5659	1.85027	0.925135	−	0.379639i	$-0.123952\pi$
$349$	34.5659	1.85027	0.925135	+	0.379639i	$0.123952\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	4.80370	0.255675	0.127838	−	0.991795i	$-0.459196\pi$
$353$	4.80370	0.255675	0.127838	+	0.991795i	$0.459196\pi$
$354$	0	0
$355$	30.7465	1.63186
$356$	0	0
$357$	29.2080	1.54585
$358$	0	0
$359$	−24.7428	−1.30588	−0.652939	−	0.757411i	$-0.726464\pi$
$359$	−24.7428	−1.30588	−0.652939	+	0.757411i	$0.726464\pi$
$360$	0	0
$361$	−6.09032	−0.320543
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	9.84618	0.515373
$366$	0	0
$367$	−6.27633	−0.327622	−0.163811	−	0.986492i	$-0.552379\pi$
$367$	−6.27633	−0.327622	−0.163811	+	0.986492i	$0.552379\pi$
$368$	0	0
$369$	8.08128	0.420695
$370$	0	0
$371$	30.8422	1.60125
$372$	0	0
$373$	−27.0722	−1.40175	−0.700874	−	0.713285i	$-0.747206\pi$
$373$	−27.0722	−1.40175	−0.700874	+	0.713285i	$0.747206\pi$
$374$	0	0
$375$	−2.75586	−0.142312
$376$	0	0
$377$	3.69898	0.190507
$378$	0	0
$379$	28.4018	1.45890	0.729451	−	0.684033i	$-0.239776\pi$
$379$	28.4018	1.45890	0.729451	+	0.684033i	$0.239776\pi$
$380$	0	0
$381$	7.46895	0.382646
$382$	0	0
$383$	−16.2583	−0.830758	−0.415379	−	0.909648i	$-0.636351\pi$
$383$	−16.2583	−0.830758	−0.415379	+	0.909648i	$0.636351\pi$
$384$	0	0
$385$	−11.8281	−0.602816
$386$	0	0
$387$	−2.94312	−0.149607
$388$	0	0
$389$	15.9325	0.807812	0.403906	−	0.914801i	$-0.367652\pi$
$389$	15.9325	0.807812	0.403906	+	0.914801i	$0.367652\pi$
$390$	0	0
$391$	−30.4584	−1.54035
$392$	0	0
$393$	17.6743	0.891550
$394$	0	0
$395$	44.5309	2.24059
$396$	0	0
$397$	17.7118	0.888932	0.444466	−	0.895796i	$-0.353393\pi$
$397$	17.7118	0.888932	0.444466	+	0.895796i	$0.353393\pi$
$398$	0	0
$399$	12.9097	0.646292
$400$	0	0
$401$	−10.5737	−0.528024	−0.264012	−	0.964519i	$-0.585046\pi$
$401$	−10.5737	−0.528024	−0.264012	+	0.964519i	$0.585046\pi$
$402$	0	0
$403$	−5.29198	−0.263612
$404$	0	0
$405$	−3.29198	−0.163580
$406$	0	0
$407$	9.82810	0.487161
$408$	0	0
$409$	33.1642	1.63987	0.819933	−	0.572459i	$-0.194010\pi$
$409$	33.1642	1.63987	0.819933	+	0.572459i	$0.194010\pi$
$410$	0	0
$411$	18.9081	0.932670
$412$	0	0
$413$	23.6562	1.16405
$414$	0	0
$415$	−13.4829	−0.661850
$416$	0	0
$417$	8.28294	0.405617
$418$	0	0
$419$	11.3073	0.552400	0.276200	−	0.961100i	$-0.410925\pi$
$419$	11.3073	0.552400	0.276200	+	0.961100i	$0.410925\pi$
$420$	0	0
$421$	−1.30227	−0.0634688	−0.0317344	−	0.999496i	$-0.510103\pi$
$421$	−1.30227	−0.0634688	−0.0317344	+	0.999496i	$0.510103\pi$
$422$	0	0
$423$	−8.73778	−0.424845
$424$	0	0
$425$	−47.4509	−2.30171
$426$	0	0
$427$	−26.0282	−1.25959
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	19.1498	0.922416	0.461208	−	0.887292i	$-0.347416\pi$
$431$	19.1498	0.922416	0.461208	+	0.887292i	$0.347416\pi$
$432$	0	0
$433$	17.9638	0.863287	0.431643	−	0.902044i	$-0.357934\pi$
$433$	17.9638	0.863287	0.431643	+	0.902044i	$0.357934\pi$
$434$	0	0
$435$	−12.1770	−0.583841
$436$	0	0
$437$	−13.4623	−0.643991
$438$	0	0
$439$	−37.3151	−1.78095	−0.890477	−	0.455028i	$-0.849629\pi$
$439$	−37.3151	−1.78095	−0.890477	+	0.455028i	$0.849629\pi$
$440$	0	0
$441$	5.90968	0.281413
$442$	0	0
$443$	26.3720	1.25297	0.626486	−	0.779433i	$-0.284493\pi$
$443$	26.3720	1.25297	0.626486	+	0.779433i	$0.284493\pi$
$444$	0	0
$445$	−54.4366	−2.58054
$446$	0	0
$447$	−0.0812828	−0.00384455
$448$	0	0
$449$	−17.1576	−0.809719	−0.404859	−	0.914379i	$-0.632680\pi$
$449$	−17.1576	−0.809719	−0.404859	+	0.914379i	$0.632680\pi$
$450$	0	0
$451$	−8.08128	−0.380533
$452$	0	0
$453$	16.0632	0.754715
$454$	0	0
$455$	11.8281	0.554510
$456$	0	0
$457$	−10.5064	−0.491467	−0.245733	−	0.969337i	$-0.579029\pi$
$457$	−10.5064	−0.491467	−0.245733	+	0.969337i	$0.579029\pi$
$458$	0	0
$459$	−8.12913	−0.379435
$460$	0	0
$461$	−20.2438	−0.942850	−0.471425	−	0.881906i	$-0.656260\pi$
$461$	−20.2438	−0.942850	−0.471425	+	0.881906i	$0.656260\pi$
$462$	0	0
$463$	−16.0478	−0.745806	−0.372903	−	0.927870i	$-0.621638\pi$
$463$	−16.0478	−0.745806	−0.372903	+	0.927870i	$0.621638\pi$
$464$	0	0
$465$	17.4211	0.807884
$466$	0	0
$467$	32.0720	1.48411	0.742056	−	0.670337i	$-0.233851\pi$
$467$	32.0720	1.48411	0.742056	+	0.670337i	$0.233851\pi$
$468$	0	0
$469$	−15.7456	−0.727066
$470$	0	0
$471$	8.02315	0.369687
$472$	0	0
$473$	2.94312	0.135325
$474$	0	0
$475$	−20.9729	−0.962302
$476$	0	0
$477$	−8.58396	−0.393033
$478$	0	0
$479$	2.76093	0.126150	0.0630751	−	0.998009i	$-0.479909\pi$
$479$	2.76093	0.126150	0.0630751	+	0.998009i	$0.479909\pi$
$480$	0	0
$481$	−9.82810	−0.448123
$482$	0	0
$483$	−13.4623	−0.612558
$484$	0	0
$485$	−36.3178	−1.64911
$486$	0	0
$487$	38.8901	1.76228	0.881138	−	0.472859i	$-0.156778\pi$
$487$	38.8901	1.76228	0.881138	+	0.472859i	$0.156778\pi$
$488$	0	0
$489$	−14.0298	−0.634448
$490$	0	0
$491$	−0.363266	−0.0163940	−0.00819698	−	0.999966i	$-0.502609\pi$
$491$	−0.363266	−0.0163940	−0.00819698	+	0.999966i	$0.502609\pi$
$492$	0	0
$493$	−30.0695	−1.35426
$494$	0	0
$495$	3.29198	0.147964
$496$	0	0
$497$	33.5580	1.50528
$498$	0	0
$499$	−18.3436	−0.821174	−0.410587	−	0.911821i	$-0.634676\pi$
$499$	−18.3436	−0.821174	−0.410587	+	0.911821i	$0.634676\pi$
$500$	0	0
$501$	−15.1860	−0.678461
$502$	0	0
$503$	−32.7284	−1.45929	−0.729645	−	0.683826i	$-0.760315\pi$
$503$	−32.7284	−1.45929	−0.729645	+	0.683826i	$0.760315\pi$
$504$	0	0
$505$	−29.4406	−1.31009
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−1.74021	−0.0771334	−0.0385667	−	0.999256i	$-0.512279\pi$
$509$	−1.74021	−0.0771334	−0.0385667	+	0.999256i	$0.512279\pi$
$510$	0	0
$511$	10.7465	0.475398
$512$	0	0
$513$	−3.59300	−0.158635
$514$	0	0
$515$	32.8603	1.44800
$516$	0	0
$517$	8.73778	0.384287
$518$	0	0
$519$	8.43676	0.370333
$520$	0	0
$521$	−21.5399	−0.943682	−0.471841	−	0.881684i	$-0.656410\pi$
$521$	−21.5399	−0.943682	−0.471841	+	0.881684i	$0.656410\pi$
$522$	0	0
$523$	−31.5090	−1.37779	−0.688896	−	0.724860i	$-0.741904\pi$
$523$	−31.5090	−1.37779	−0.688896	+	0.724860i	$0.741904\pi$
$524$	0	0
$525$	−20.9729	−0.915331
$526$	0	0
$527$	43.0192	1.87394
$528$	0	0
$529$	−8.96134	−0.389623
$530$	0	0
$531$	−6.58396	−0.285720
$532$	0	0
$533$	8.08128	0.350039
$534$	0	0
$535$	34.8422	1.50636
$536$	0	0
$537$	11.8425	0.511042
$538$	0	0
$539$	−5.90968	−0.254548
$540$	0	0
$541$	−30.5527	−1.31356	−0.656781	−	0.754082i	$-0.728082\pi$
$541$	−30.5527	−1.31356	−0.656781	+	0.754082i	$0.728082\pi$
$542$	0	0
$543$	−4.02315	−0.172650
$544$	0	0
$545$	20.3345	0.871033
$546$	0	0
$547$	−22.4636	−0.960474	−0.480237	−	0.877139i	$-0.659449\pi$
$547$	−22.4636	−0.960474	−0.480237	+	0.877139i	$0.659449\pi$
$548$	0	0
$549$	7.24414	0.309172
$550$	0	0
$551$	−13.2904	−0.566192
$552$	0	0
$553$	48.6029	2.06680
$554$	0	0
$555$	32.3539	1.37335
$556$	0	0
$557$	16.8591	0.714344	0.357172	−	0.934039i	$-0.383741\pi$
$557$	16.8591	0.714344	0.357172	+	0.934039i	$0.383741\pi$
$558$	0	0
$559$	−2.94312	−0.124481
$560$	0	0
$561$	8.12913	0.343212
$562$	0	0
$563$	0.470198	0.0198165	0.00990824	−	0.999951i	$-0.496846\pi$
$563$	0.470198	0.0198165	0.00990824	+	0.999951i	$0.496846\pi$
$564$	0	0
$565$	−51.9145	−2.18406
$566$	0	0
$567$	−3.59300	−0.150892
$568$	0	0
$569$	−3.41082	−0.142989	−0.0714944	−	0.997441i	$-0.522777\pi$
$569$	−3.41082	−0.142989	−0.0714944	+	0.997441i	$0.522777\pi$
$570$	0	0
$571$	−6.53980	−0.273682	−0.136841	−	0.990593i	$-0.543695\pi$
$571$	−6.53980	−0.273682	−0.136841	+	0.990593i	$0.543695\pi$
$572$	0	0
$573$	9.41604	0.393361
$574$	0	0
$575$	21.8707	0.912072
$576$	0	0
$577$	−17.3774	−0.723430	−0.361715	−	0.932289i	$-0.617809\pi$
$577$	−17.3774	−0.723430	−0.361715	+	0.932289i	$0.617809\pi$
$578$	0	0
$579$	−25.8332	−1.07359
$580$	0	0
$581$	−14.7158	−0.610515
$582$	0	0
$583$	8.58396	0.355511
$584$	0	0
$585$	−3.29198	−0.136107
$586$	0	0
$587$	−14.2402	−0.587755	−0.293877	−	0.955843i	$-0.594946\pi$
$587$	−14.2402	−0.587755	−0.293877	+	0.955843i	$0.594946\pi$
$588$	0	0
$589$	19.0141	0.783463
$590$	0	0
$591$	−3.00904	−0.123775
$592$	0	0
$593$	16.8747	0.692961	0.346480	−	0.938057i	$-0.387377\pi$
$593$	16.8747	0.692961	0.346480	+	0.938057i	$0.387377\pi$
$594$	0	0
$595$	−96.1521	−3.94185
$596$	0	0
$597$	−3.30227	−0.135153
$598$	0	0
$599$	−27.1366	−1.10877	−0.554386	−	0.832260i	$-0.687047\pi$
$599$	−27.1366	−1.10877	−0.554386	+	0.832260i	$0.687047\pi$
$600$	0	0
$601$	−42.4002	−1.72954	−0.864771	−	0.502167i	$-0.832536\pi$
$601$	−42.4002	−1.72954	−0.864771	+	0.502167i	$0.832536\pi$
$602$	0	0
$603$	4.38230	0.178461
$604$	0	0
$605$	−3.29198	−0.133838
$606$	0	0
$607$	8.48167	0.344260	0.172130	−	0.985074i	$-0.444935\pi$
$607$	8.48167	0.344260	0.172130	+	0.985074i	$0.444935\pi$
$608$	0	0
$609$	−13.2904	−0.538556
$610$	0	0
$611$	−8.73778	−0.353493
$612$	0	0
$613$	−43.5436	−1.75871	−0.879355	−	0.476166i	$-0.842026\pi$
$613$	−43.5436	−1.75871	−0.879355	+	0.476166i	$0.842026\pi$
$614$	0	0
$615$	−26.6034	−1.07275
$616$	0	0
$617$	−20.8138	−0.837934	−0.418967	−	0.908002i	$-0.637608\pi$
$617$	−20.8138	−0.837934	−0.418967	+	0.908002i	$0.637608\pi$
$618$	0	0
$619$	1.55956	0.0626841	0.0313421	−	0.999509i	$-0.490022\pi$
$619$	1.55956	0.0626841	0.0313421	+	0.999509i	$0.490022\pi$
$620$	0	0
$621$	3.74682	0.150355
$622$	0	0
$623$	−59.4143	−2.38039
$624$	0	0
$625$	−20.1135	−0.804539
$626$	0	0
$627$	3.59300	0.143491
$628$	0	0
$629$	79.8939	3.18558
$630$	0	0
$631$	−17.0244	−0.677731	−0.338865	−	0.940835i	$-0.610043\pi$
$631$	−17.0244	−0.677731	−0.338865	+	0.940835i	$0.610043\pi$
$632$	0	0
$633$	−10.0334	−0.398793
$634$	0	0
$635$	−24.5876	−0.975731
$636$	0	0
$637$	5.90968	0.234150
$638$	0	0
$639$	−9.33982	−0.369478
$640$	0	0
$641$	−9.80127	−0.387127	−0.193563	−	0.981088i	$-0.562005\pi$
$641$	−9.80127	−0.387127	−0.193563	+	0.981088i	$0.562005\pi$
$642$	0	0
$643$	33.3383	1.31473	0.657367	−	0.753571i	$-0.271670\pi$
$643$	33.3383	1.31473	0.657367	+	0.753571i	$0.271670\pi$
$644$	0	0
$645$	9.68869	0.381492
$646$	0	0
$647$	29.3799	1.15504	0.577521	−	0.816376i	$-0.304020\pi$
$647$	29.3799	1.15504	0.577521	+	0.816376i	$0.304020\pi$
$648$	0	0
$649$	6.58396	0.258443
$650$	0	0
$651$	19.0141	0.745222
$652$	0	0
$653$	27.0517	1.05861	0.529307	−	0.848431i	$-0.322452\pi$
$653$	27.0517	1.05861	0.529307	+	0.848431i	$0.322452\pi$
$654$	0	0
$655$	−58.1834	−2.27341
$656$	0	0
$657$	−2.99096	−0.116688
$658$	0	0
$659$	−9.51561	−0.370676	−0.185338	−	0.982675i	$-0.559338\pi$
$659$	−9.51561	−0.370676	−0.185338	+	0.982675i	$0.559338\pi$
$660$	0	0
$661$	48.9379	1.90346	0.951732	−	0.306931i	$-0.0993020\pi$
$661$	48.9379	1.90346	0.951732	+	0.306931i	$0.0993020\pi$
$662$	0	0
$663$	−8.12913	−0.315709
$664$	0	0
$665$	−42.4984	−1.64802
$666$	0	0
$667$	13.8594	0.536638
$668$	0	0
$669$	23.6640	0.914903
$670$	0	0
$671$	−7.24414	−0.279657
$672$	0	0
$673$	−34.4897	−1.32948	−0.664740	−	0.747075i	$-0.731457\pi$
$673$	−34.4897	−1.32948	−0.664740	+	0.747075i	$0.731457\pi$
$674$	0	0
$675$	5.83714	0.224672
$676$	0	0
$677$	8.18476	0.314566	0.157283	−	0.987554i	$-0.449727\pi$
$677$	8.18476	0.314566	0.157283	+	0.987554i	$0.449727\pi$
$678$	0	0
$679$	−39.6387	−1.52119
$680$	0	0
$681$	11.4124	0.437322
$682$	0	0
$683$	−10.5078	−0.402068	−0.201034	−	0.979584i	$-0.564430\pi$
$683$	−10.5078	−0.402068	−0.201034	+	0.979584i	$0.564430\pi$
$684$	0	0
$685$	−62.2452	−2.37827
$686$	0	0
$687$	8.91843	0.340259
$688$	0	0
$689$	−8.58396	−0.327023
$690$	0	0
$691$	−19.3603	−0.736499	−0.368249	−	0.929727i	$-0.620043\pi$
$691$	−19.3603	−0.736499	−0.368249	+	0.929727i	$0.620043\pi$
$692$	0	0
$693$	3.59300	0.136487
$694$	0	0
$695$	−27.2673	−1.03431
$696$	0	0
$697$	−65.6938	−2.48833
$698$	0	0
$699$	11.7947	0.446115
$700$	0	0
$701$	22.1472	0.836488	0.418244	−	0.908335i	$-0.362646\pi$
$701$	22.1472	0.836488	0.418244	+	0.908335i	$0.362646\pi$
$702$	0	0
$703$	35.3124	1.33183
$704$	0	0
$705$	28.7646	1.08334
$706$	0	0
$707$	−32.1327	−1.20847
$708$	0	0
$709$	−5.02733	−0.188805	−0.0944027	−	0.995534i	$-0.530094\pi$
$709$	−5.02733	−0.188805	−0.0944027	+	0.995534i	$0.530094\pi$
$710$	0	0
$711$	−13.5271	−0.507305
$712$	0	0
$713$	−19.8281	−0.742568
$714$	0	0
$715$	3.29198	0.123113
$716$	0	0
$717$	13.7556	0.513711
$718$	0	0
$719$	46.2997	1.72669	0.863344	−	0.504617i	$-0.168366\pi$
$719$	46.2997	1.72669	0.863344	+	0.504617i	$0.168366\pi$
$720$	0	0
$721$	35.8651	1.33569
$722$	0	0
$723$	−29.0542	−1.08054
$724$	0	0
$725$	21.5915	0.801887
$726$	0	0
$727$	−35.3618	−1.31150	−0.655748	−	0.754980i	$-0.727647\pi$
$727$	−35.3618	−1.31150	−0.655748	+	0.754980i	$0.727647\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.9250	0.884897
$732$	0	0
$733$	−17.7068	−0.654014	−0.327007	−	0.945022i	$-0.606040\pi$
$733$	−17.7068	−0.654014	−0.327007	+	0.945022i	$0.606040\pi$
$734$	0	0
$735$	−19.4545	−0.717592
$736$	0	0
$737$	−4.38230	−0.161424
$738$	0	0
$739$	−14.1101	−0.519048	−0.259524	−	0.965737i	$-0.583566\pi$
$739$	−14.1101	−0.519048	−0.259524	+	0.965737i	$0.583566\pi$
$740$	0	0
$741$	−3.59300	−0.131992
$742$	0	0
$743$	8.37451	0.307231	0.153616	−	0.988131i	$-0.450908\pi$
$743$	8.37451	0.307231	0.153616	+	0.988131i	$0.450908\pi$
$744$	0	0
$745$	0.267582	0.00980343
$746$	0	0
$747$	4.09568	0.149853
$748$	0	0
$749$	38.0282	1.38952
$750$	0	0
$751$	11.5605	0.421849	0.210925	−	0.977502i	$-0.432353\pi$
$751$	11.5605	0.421849	0.210925	+	0.977502i	$0.432353\pi$
$752$	0	0
$753$	11.7881	0.429580
$754$	0	0
$755$	−52.8798	−1.92449
$756$	0	0
$757$	27.7155	1.00734	0.503669	−	0.863897i	$-0.331983\pi$
$757$	27.7155	1.00734	0.503669	+	0.863897i	$0.331983\pi$
$758$	0	0
$759$	−3.74682	−0.136001
$760$	0	0
$761$	−31.0192	−1.12444	−0.562222	−	0.826986i	$-0.690053\pi$
$761$	−31.0192	−1.12444	−0.562222	+	0.826986i	$0.690053\pi$
$762$	0	0
$763$	22.1939	0.803472
$764$	0	0
$765$	26.7609	0.967543
$766$	0	0
$767$	−6.58396	−0.237733
$768$	0	0
$769$	47.6844	1.71954	0.859772	−	0.510678i	$-0.170605\pi$
$769$	47.6844	1.71954	0.859772	+	0.510678i	$0.170605\pi$
$770$	0	0
$771$	−12.4883	−0.449754
$772$	0	0
$773$	24.6524	0.886685	0.443342	−	0.896352i	$-0.353793\pi$
$773$	24.6524	0.886685	0.443342	+	0.896352i	$0.353793\pi$
$774$	0	0
$775$	−30.8901	−1.10960
$776$	0	0
$777$	35.3124	1.26683
$778$	0	0
$779$	−29.0361	−1.04033
$780$	0	0
$781$	9.33982	0.334205
$782$	0	0
$783$	3.69898	0.132191
$784$	0	0
$785$	−26.4121	−0.942687
$786$	0	0
$787$	31.2492	1.11391	0.556957	−	0.830541i	$-0.311969\pi$
$787$	31.2492	1.11391	0.556957	+	0.830541i	$0.311969\pi$
$788$	0	0
$789$	−22.2402	−0.791771
$790$	0	0
$791$	−56.6616	−2.01465
$792$	0	0
$793$	7.24414	0.257247
$794$	0	0
$795$	28.2583	1.00222
$796$	0	0
$797$	25.3774	0.898913	0.449456	−	0.893302i	$-0.351618\pi$
$797$	25.3774	0.898913	0.449456	+	0.893302i	$0.351618\pi$
$798$	0	0
$799$	71.0305	2.51288
$800$	0	0
$801$	16.5361	0.584275
$802$	0	0
$803$	2.99096	0.105549
$804$	0	0
$805$	44.3178	1.56200
$806$	0	0
$807$	24.9379	0.877856
$808$	0	0
$809$	49.7961	1.75074	0.875368	−	0.483457i	$-0.160619\pi$
$809$	49.7961	1.75074	0.875368	+	0.483457i	$0.160619\pi$
$810$	0	0
$811$	20.5696	0.722295	0.361148	−	0.932509i	$-0.382385\pi$
$811$	20.5696	0.722295	0.361148	+	0.932509i	$0.382385\pi$
$812$	0	0
$813$	−22.4846	−0.788569
$814$	0	0
$815$	46.1857	1.61782
$816$	0	0
$817$	10.5746	0.369960
$818$	0	0
$819$	−3.59300	−0.125550
$820$	0	0
$821$	9.50746	0.331813	0.165906	−	0.986142i	$-0.446945\pi$
$821$	9.50746	0.331813	0.165906	+	0.986142i	$0.446945\pi$
$822$	0	0
$823$	−4.68215	−0.163209	−0.0816047	−	0.996665i	$-0.526004\pi$
$823$	−4.68215	−0.163209	−0.0816047	+	0.996665i	$0.526004\pi$
$824$	0	0
$825$	−5.83714	−0.203223
$826$	0	0
$827$	−19.0398	−0.662077	−0.331039	−	0.943617i	$-0.607399\pi$
$827$	−19.0398	−0.662077	−0.331039	+	0.943617i	$0.607399\pi$
$828$	0	0
$829$	3.81935	0.132652	0.0663258	−	0.997798i	$-0.478872\pi$
$829$	3.81935	0.132652	0.0663258	+	0.997798i	$0.478872\pi$
$830$	0	0
$831$	7.98192	0.276890
$832$	0	0
$833$	−48.0405	−1.66450
$834$	0	0
$835$	49.9921	1.73005
$836$	0	0
$837$	−5.29198	−0.182918
$838$	0	0
$839$	40.6791	1.40440	0.702199	−	0.711981i	$-0.252202\pi$
$839$	40.6791	1.40440	0.702199	+	0.711981i	$0.252202\pi$
$840$	0	0
$841$	−15.3176	−0.528192
$842$	0	0
$843$	0.990959	0.0341305
$844$	0	0
$845$	−3.29198	−0.113248
$846$	0	0
$847$	−3.59300	−0.123457
$848$	0	0
$849$	−6.30102	−0.216251
$850$	0	0
$851$	−36.8241	−1.26231
$852$	0	0
$853$	11.3667	0.389187	0.194593	−	0.980884i	$-0.437661\pi$
$853$	11.3667	0.389187	0.194593	+	0.980884i	$0.437661\pi$
$854$	0	0
$855$	11.8281	0.404513
$856$	0	0
$857$	7.27508	0.248512	0.124256	−	0.992250i	$-0.460346\pi$
$857$	7.27508	0.248512	0.124256	+	0.992250i	$0.460346\pi$
$858$	0	0
$859$	28.5659	0.974655	0.487328	−	0.873219i	$-0.337972\pi$
$859$	28.5659	0.974655	0.487328	+	0.873219i	$0.337972\pi$
$860$	0	0
$861$	−29.0361	−0.989547
$862$	0	0
$863$	−45.0824	−1.53462	−0.767311	−	0.641275i	$-0.778406\pi$
$863$	−45.0824	−1.53462	−0.767311	+	0.641275i	$0.778406\pi$
$864$	0	0
$865$	−27.7737	−0.944332
$866$	0	0
$867$	49.0827	1.66693
$868$	0	0
$869$	13.5271	0.458875
$870$	0	0
$871$	4.38230	0.148489
$872$	0	0
$873$	11.0322	0.373383
$874$	0	0
$875$	9.90182	0.334743
$876$	0	0
$877$	45.5707	1.53881	0.769406	−	0.638760i	$-0.220552\pi$
$877$	45.5707	1.53881	0.769406	+	0.638760i	$0.220552\pi$
$878$	0	0
$879$	−15.0090	−0.506242
$880$	0	0
$881$	2.65143	0.0893288	0.0446644	−	0.999002i	$-0.485778\pi$
$881$	2.65143	0.0893288	0.0446644	+	0.999002i	$0.485778\pi$
$882$	0	0
$883$	45.1781	1.52036	0.760181	−	0.649711i	$-0.225110\pi$
$883$	45.1781	1.52036	0.760181	+	0.649711i	$0.225110\pi$
$884$	0	0
$885$	21.6743	0.728573
$886$	0	0
$887$	26.1795	0.879023	0.439511	−	0.898237i	$-0.355152\pi$
$887$	26.1795	0.879023	0.439511	+	0.898237i	$0.355152\pi$
$888$	0	0
$889$	−26.8360	−0.900049
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	31.3949	1.05059
$894$	0	0
$895$	−38.9853	−1.30314
$896$	0	0
$897$	3.74682	0.125103
$898$	0	0
$899$	−19.5749	−0.652860
$900$	0	0
$901$	69.7801	2.32471
$902$	0	0
$903$	10.5746	0.351902
$904$	0	0
$905$	13.2441	0.440250
$906$	0	0
$907$	−50.7928	−1.68655	−0.843274	−	0.537484i	$-0.819375\pi$
$907$	−50.7928	−1.68655	−0.843274	+	0.537484i	$0.819375\pi$
$908$	0	0
$909$	8.94312	0.296624
$910$	0	0
$911$	−23.8682	−0.790787	−0.395394	−	0.918512i	$-0.629392\pi$
$911$	−23.8682	−0.790787	−0.395394	+	0.918512i	$0.629392\pi$
$912$	0	0
$913$	−4.09568	−0.135547
$914$	0	0
$915$	−23.8476	−0.788376
$916$	0	0
$917$	−63.5038	−2.09708
$918$	0	0
$919$	9.32446	0.307586	0.153793	−	0.988103i	$-0.450851\pi$
$919$	9.32446	0.307586	0.153793	+	0.988103i	$0.450851\pi$
$920$	0	0
$921$	−6.77901	−0.223376
$922$	0	0
$923$	−9.33982	−0.307424
$924$	0	0
$925$	−57.3680	−1.88625
$926$	0	0
$927$	−9.98192	−0.327849
$928$	0	0
$929$	−7.51018	−0.246401	−0.123201	−	0.992382i	$-0.539316\pi$
$929$	−7.51018	−0.246401	−0.123201	+	0.992382i	$0.539316\pi$
$930$	0	0
$931$	−21.2335	−0.695900
$932$	0	0
$933$	−6.23510	−0.204128
$934$	0	0
$935$	−26.7609	−0.875176
$936$	0	0
$937$	−3.68362	−0.120339	−0.0601693	−	0.998188i	$-0.519164\pi$
$937$	−3.68362	−0.120339	−0.0601693	+	0.998188i	$0.519164\pi$
$938$	0	0
$939$	8.93283	0.291512
$940$	0	0
$941$	−5.51054	−0.179638	−0.0898192	−	0.995958i	$-0.528629\pi$
$941$	−5.51054	−0.179638	−0.0898192	+	0.995958i	$0.528629\pi$
$942$	0	0
$943$	30.2791	0.986024
$944$	0	0
$945$	11.8281	0.384768
$946$	0	0
$947$	0.0762128	0.00247658	0.00123829	−	0.999999i	$-0.499606\pi$
$947$	0.0762128	0.00247658	0.00123829	+	0.999999i	$0.499606\pi$
$948$	0	0
$949$	−2.99096	−0.0970907
$950$	0	0
$951$	−31.1201	−1.00914
$952$	0	0
$953$	10.8850	0.352599	0.176300	−	0.984337i	$-0.443587\pi$
$953$	10.8850	0.352599	0.176300	+	0.984337i	$0.443587\pi$
$954$	0	0
$955$	−30.9974	−1.00305
$956$	0	0
$957$	−3.69898	−0.119571
$958$	0	0
$959$	−67.9370	−2.19380
$960$	0	0
$961$	−2.99493	−0.0966106
$962$	0	0
$963$	−10.5840	−0.341063
$964$	0	0
$965$	85.0423	2.73761
$966$	0	0
$967$	0.158886	0.00510944	0.00255472	−	0.999997i	$-0.499187\pi$
$967$	0.158886	0.00510944	0.00255472	+	0.999997i	$0.499187\pi$
$968$	0	0
$969$	29.2080	0.938295
$970$	0	0
$971$	9.53737	0.306069	0.153034	−	0.988221i	$-0.451095\pi$
$971$	9.53737	0.306069	0.153034	+	0.988221i	$0.451095\pi$
$972$	0	0
$973$	−29.7606	−0.954082
$974$	0	0
$975$	5.83714	0.186938
$976$	0	0
$977$	−7.06056	−0.225887	−0.112944	−	0.993601i	$-0.536028\pi$
$977$	−7.06056	−0.225887	−0.112944	+	0.993601i	$0.536028\pi$
$978$	0	0
$979$	−16.5361	−0.528497
$980$	0	0
$981$	−6.17697	−0.197215
$982$	0	0
$983$	61.8470	1.97261	0.986306	−	0.164924i	$-0.0527377\pi$
$983$	61.8470	1.97261	0.986306	+	0.164924i	$0.0527377\pi$
$984$	0	0
$985$	9.90571	0.315622
$986$	0	0
$987$	31.3949	0.999310
$988$	0	0
$989$	−11.0273	−0.350649
$990$	0	0
$991$	35.7157	1.13455	0.567274	−	0.823529i	$-0.307998\pi$
$991$	35.7157	1.13455	0.567274	+	0.823529i	$0.307998\pi$
$992$	0	0
$993$	0.259790	0.00824419
$994$	0	0
$995$	10.8710	0.344634
$996$	0	0
$997$	−13.1366	−0.416041	−0.208021	−	0.978124i	$-0.566702\pi$
$997$	−13.1366	−0.416041	−0.208021	+	0.978124i	$0.566702\pi$
$998$	0	0
$999$	−9.82810	−0.310947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cb.1.1		4
4.3	odd	2		1716.2.a.i.1.1	✓	4
12.11	even	2		5148.2.a.q.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.i.1.1	✓	4	4.3	odd	2
5148.2.a.q.1.4		4	12.11	even	2
6864.2.a.cb.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.29198 q^{5} -3.59300 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.29198 q^{5} -3.59300 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.29198 q^{15} -8.12913 q^{17} -3.59300 q^{19} -3.59300 q^{21} +3.74682 q^{23} +5.83714 q^{25} +1.00000 q^{27} +3.69898 q^{29} -5.29198 q^{31} -1.00000 q^{33} +11.8281 q^{35} -9.82810 q^{37} +1.00000 q^{39} +8.08128 q^{41} -2.94312 q^{43} -3.29198 q^{45} -8.73778 q^{47} +5.90968 q^{49} -8.12913 q^{51} -8.58396 q^{53} +3.29198 q^{55} -3.59300 q^{57} -6.58396 q^{59} +7.24414 q^{61} -3.59300 q^{63} -3.29198 q^{65} +4.38230 q^{67} +3.74682 q^{69} -9.33982 q^{71} -2.99096 q^{73} +5.83714 q^{75} +3.59300 q^{77} -13.5271 q^{79} +1.00000 q^{81} +4.09568 q^{83} +26.7609 q^{85} +3.69898 q^{87} +16.5361 q^{89} -3.59300 q^{91} -5.29198 q^{93} +11.8281 q^{95} +11.0322 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 12 q^{29} - 6 q^{31} - 4 q^{33} + 10 q^{35} - 2 q^{37} + 4 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{45} - 6 q^{47} + 32 q^{49} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 2 q^{57} + 4 q^{59} + 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} + 4 q^{69} - 14 q^{71} + 6 q^{73} + 4 q^{75} + 2 q^{77} - 14 q^{79} + 4 q^{81} + 34 q^{85} + 12 q^{87} + 44 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 12 * q^29 - 6 * q^31 - 4 * q^33 + 10 * q^35 - 2 * q^37 + 4 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^45 - 6 * q^47 + 32 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 2 * q^57 + 4 * q^59 + 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 + 4 * q^69 - 14 * q^71 + 6 * q^73 + 4 * q^75 + 2 * q^77 - 14 * q^79 + 4 * q^81 + 34 * q^85 + 12 * q^87 + 44 * q^89 - 2 * q^91 - 6 * q^93 + 10 * q^95 + 18 * q^97 - 4 * q^99

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