LMFDB - Embedded newform 7728.2.a.cb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.820249$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.94523 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.94523 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.72913 q^{11} +2.21610 q^{13} -3.94523 q^{15} -5.14744 q^{17} +1.72913 q^{19} -1.00000 q^{21} +1.00000 q^{23} +10.5648 q^{25} +1.00000 q^{27} +3.01012 q^{29} +8.20126 q^{31} -1.72913 q^{33} +3.94523 q^{35} +7.61958 q^{37} +2.21610 q^{39} +1.14744 q^{41} -7.22998 q^{43} -3.94523 q^{45} -7.34870 q^{47} +1.00000 q^{49} -5.14744 q^{51} -4.21234 q^{53} +6.82179 q^{55} +1.72913 q^{57} +1.33577 q^{59} +14.1743 q^{61} -1.00000 q^{63} -8.74301 q^{65} +0.0825425 q^{67} +1.00000 q^{69} +5.53703 q^{71} -8.33858 q^{73} +10.5648 q^{75} +1.72913 q^{77} -8.20126 q^{79} +1.00000 q^{81} +3.21515 q^{83} +20.3078 q^{85} +3.01012 q^{87} +9.45525 q^{89} -2.21610 q^{91} +8.20126 q^{93} -6.82179 q^{95} -8.33858 q^{97} -1.72913 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 - q^25 + 4 * q^27 - 10 * q^29 + 10 * q^31 - 2 * q^33 + 3 * q^35 + q^39 - 8 * q^41 - 13 * q^43 - 3 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 5 * q^53 - 3 * q^55 + 2 * q^57 + q^59 + 5 * q^61 - 4 * q^63 - 22 * q^65 - 3 * q^67 + 4 * q^69 - 5 * q^71 - 20 * q^73 - q^75 + 2 * q^77 - 10 * q^79 + 4 * q^81 - 18 * q^83 + 25 * q^85 - 10 * q^87 - 31 * q^89 - q^91 + 10 * q^93 + 3 * q^95 - 20 * q^97 - 2 * 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
$5$	−3.94523	−1.76436	−0.882179	−	0.470914i	$-0.843924\pi$
$5$	−3.94523	−1.76436	−0.882179	+	0.470914i	$0.843924\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.72913	−0.521351	−0.260676	−	0.965426i	$-0.583945\pi$
$11$	−1.72913	−0.521351	−0.260676	+	0.965426i	$0.583945\pi$
$12$	0	0
$13$	2.21610	0.614635	0.307318	−	0.951607i	$-0.400569\pi$
$13$	2.21610	0.614635	0.307318	+	0.951607i	$0.400569\pi$
$14$	0	0
$15$	−3.94523	−1.01865
$16$	0	0
$17$	−5.14744	−1.24844	−0.624219	−	0.781250i	$-0.714583\pi$
$17$	−5.14744	−1.24844	−0.624219	+	0.781250i	$0.714583\pi$
$18$	0	0
$19$	1.72913	0.396689	0.198344	−	0.980132i	$-0.436444\pi$
$19$	1.72913	0.396689	0.198344	+	0.980132i	$0.436444\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	10.5648	2.11296
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.01012	0.558966	0.279483	−	0.960151i	$-0.409837\pi$
$29$	3.01012	0.558966	0.279483	+	0.960151i	$0.409837\pi$
$30$	0	0
$31$	8.20126	1.47299	0.736495	−	0.676443i	$-0.236479\pi$
$31$	8.20126	1.47299	0.736495	+	0.676443i	$0.236479\pi$
$32$	0	0
$33$	−1.72913	−0.301002
$34$	0	0
$35$	3.94523	0.666865
$36$	0	0
$37$	7.61958	1.25265	0.626325	−	0.779562i	$-0.284558\pi$
$37$	7.61958	1.25265	0.626325	+	0.779562i	$0.284558\pi$
$38$	0	0
$39$	2.21610	0.354860
$40$	0	0
$41$	1.14744	0.179200	0.0896000	−	0.995978i	$-0.471441\pi$
$41$	1.14744	0.179200	0.0896000	+	0.995978i	$0.471441\pi$
$42$	0	0
$43$	−7.22998	−1.10256	−0.551281	−	0.834320i	$-0.685861\pi$
$43$	−7.22998	−1.10256	−0.551281	+	0.834320i	$0.685861\pi$
$44$	0	0
$45$	−3.94523	−0.588119
$46$	0	0
$47$	−7.34870	−1.07192	−0.535959	−	0.844244i	$-0.680050\pi$
$47$	−7.34870	−1.07192	−0.535959	+	0.844244i	$0.680050\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.14744	−0.720786
$52$	0	0
$53$	−4.21234	−0.578609	−0.289305	−	0.957237i	$-0.593424\pi$
$53$	−4.21234	−0.578609	−0.289305	+	0.957237i	$0.593424\pi$
$54$	0	0
$55$	6.82179	0.919850
$56$	0	0
$57$	1.72913	0.229028
$58$	0	0
$59$	1.33577	0.173903	0.0869513	−	0.996213i	$-0.472288\pi$
$59$	1.33577	0.173903	0.0869513	+	0.996213i	$0.472288\pi$
$60$	0	0
$61$	14.1743	1.81483	0.907414	−	0.420239i	$-0.138054\pi$
$61$	14.1743	1.81483	0.907414	+	0.420239i	$0.138054\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−8.74301	−1.08444
$66$	0	0
$67$	0.0825425	0.0100842	0.00504209	−	0.999987i	$-0.498395\pi$
$67$	0.0825425	0.0100842	0.00504209	+	0.999987i	$0.498395\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	5.53703	0.657125	0.328563	−	0.944482i	$-0.393436\pi$
$71$	5.53703	0.657125	0.328563	+	0.944482i	$0.393436\pi$
$72$	0	0
$73$	−8.33858	−0.975957	−0.487978	−	0.872856i	$-0.662266\pi$
$73$	−8.33858	−0.975957	−0.487978	+	0.872856i	$0.662266\pi$
$74$	0	0
$75$	10.5648	1.21992
$76$	0	0
$77$	1.72913	0.197052
$78$	0	0
$79$	−8.20126	−0.922714	−0.461357	−	0.887215i	$-0.652637\pi$
$79$	−8.20126	−0.922714	−0.461357	+	0.887215i	$0.652637\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.21515	0.352908	0.176454	−	0.984309i	$-0.443537\pi$
$83$	3.21515	0.352908	0.176454	+	0.984309i	$0.443537\pi$
$84$	0	0
$85$	20.3078	2.20269
$86$	0	0
$87$	3.01012	0.322719
$88$	0	0
$89$	9.45525	1.00225	0.501127	−	0.865374i	$-0.332919\pi$
$89$	9.45525	1.00225	0.501127	+	0.865374i	$0.332919\pi$
$90$	0	0
$91$	−2.21610	−0.232310
$92$	0	0
$93$	8.20126	0.850431
$94$	0	0
$95$	−6.82179	−0.699901
$96$	0	0
$97$	−8.33858	−0.846654	−0.423327	−	0.905977i	$-0.639138\pi$
$97$	−8.33858	−0.846654	−0.423327	+	0.905977i	$0.639138\pi$
$98$	0	0
$99$	−1.72913	−0.173784
$100$	0	0
$101$	−17.2540	−1.71684	−0.858418	−	0.512951i	$-0.828552\pi$
$101$	−17.2540	−1.71684	−0.858418	+	0.512951i	$0.828552\pi$
$102$	0	0
$103$	6.02777	0.593934	0.296967	−	0.954888i	$-0.404025\pi$
$103$	6.02777	0.593934	0.296967	+	0.954888i	$0.404025\pi$
$104$	0	0
$105$	3.94523	0.385015
$106$	0	0
$107$	−11.2225	−1.08492	−0.542458	−	0.840083i	$-0.682506\pi$
$107$	−11.2225	−1.08492	−0.542458	+	0.840083i	$0.682506\pi$
$108$	0	0
$109$	−11.0268	−1.05618	−0.528089	−	0.849189i	$-0.677091\pi$
$109$	−11.0268	−1.05618	−0.528089	+	0.849189i	$0.677091\pi$
$110$	0	0
$111$	7.61958	0.723218
$112$	0	0
$113$	−16.2800	−1.53150	−0.765749	−	0.643140i	$-0.777631\pi$
$113$	−16.2800	−1.53150	−0.765749	+	0.643140i	$0.777631\pi$
$114$	0	0
$115$	−3.94523	−0.367894
$116$	0	0
$117$	2.21610	0.204878
$118$	0	0
$119$	5.14744	0.471865
$120$	0	0
$121$	−8.01012	−0.728193
$122$	0	0
$123$	1.14744	0.103461
$124$	0	0
$125$	−21.9544	−1.96366
$126$	0	0
$127$	−1.64658	−0.146111	−0.0730553	−	0.997328i	$-0.523275\pi$
$127$	−1.64658	−0.146111	−0.0730553	+	0.997328i	$0.523275\pi$
$128$	0	0
$129$	−7.22998	−0.636565
$130$	0	0
$131$	6.60569	0.577142	0.288571	−	0.957458i	$-0.406820\pi$
$131$	6.60569	0.577142	0.288571	+	0.957458i	$0.406820\pi$
$132$	0	0
$133$	−1.72913	−0.149934
$134$	0	0
$135$	−3.94523	−0.339551
$136$	0	0
$137$	−20.2290	−1.72828	−0.864141	−	0.503249i	$-0.832138\pi$
$137$	−20.2290	−1.72828	−0.864141	+	0.503249i	$0.832138\pi$
$138$	0	0
$139$	−5.58073	−0.473352	−0.236676	−	0.971589i	$-0.576058\pi$
$139$	−5.58073	−0.473352	−0.236676	+	0.971589i	$0.576058\pi$
$140$	0	0
$141$	−7.34870	−0.618872
$142$	0	0
$143$	−3.83191	−0.320441
$144$	0	0
$145$	−11.8756	−0.986216
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−8.22718	−0.673996	−0.336998	−	0.941505i	$-0.609412\pi$
$149$	−8.22718	−0.673996	−0.336998	+	0.941505i	$0.609412\pi$
$150$	0	0
$151$	−1.32093	−0.107496	−0.0537481	−	0.998555i	$-0.517117\pi$
$151$	−1.32093	−0.107496	−0.0537481	+	0.998555i	$0.517117\pi$
$152$	0	0
$153$	−5.14744	−0.416146
$154$	0	0
$155$	−32.3558	−2.59888
$156$	0	0
$157$	−1.93229	−0.154214	−0.0771069	−	0.997023i	$-0.524568\pi$
$157$	−1.93229	−0.154214	−0.0771069	+	0.997023i	$0.524568\pi$
$158$	0	0
$159$	−4.21234	−0.334060
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−0.801546	−0.0627819	−0.0313909	−	0.999507i	$-0.509994\pi$
$163$	−0.801546	−0.0627819	−0.0313909	+	0.999507i	$0.509994\pi$
$164$	0	0
$165$	6.82179	0.531076
$166$	0	0
$167$	19.7372	1.52731	0.763655	−	0.645624i	$-0.223403\pi$
$167$	19.7372	1.52731	0.763655	+	0.645624i	$0.223403\pi$
$168$	0	0
$169$	−8.08890	−0.622223
$170$	0	0
$171$	1.72913	0.132230
$172$	0	0
$173$	18.1397	1.37914	0.689569	−	0.724220i	$-0.257800\pi$
$173$	18.1397	1.37914	0.689569	+	0.724220i	$0.257800\pi$
$174$	0	0
$175$	−10.5648	−0.798624
$176$	0	0
$177$	1.33577	0.100403
$178$	0	0
$179$	22.3281	1.66888	0.834439	−	0.551101i	$-0.185792\pi$
$179$	22.3281	1.66888	0.834439	+	0.551101i	$0.185792\pi$
$180$	0	0
$181$	−8.78295	−0.652831	−0.326416	−	0.945226i	$-0.605841\pi$
$181$	−8.78295	−0.652831	−0.326416	+	0.945226i	$0.605841\pi$
$182$	0	0
$183$	14.1743	1.04779
$184$	0	0
$185$	−30.0609	−2.21012
$186$	0	0
$187$	8.90057	0.650874
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	1.32093	0.0955795	0.0477897	−	0.998857i	$-0.484782\pi$
$191$	1.32093	0.0955795	0.0477897	+	0.998857i	$0.484782\pi$
$192$	0	0
$193$	−11.2411	−0.809149	−0.404575	−	0.914505i	$-0.632580\pi$
$193$	−11.2411	−0.809149	−0.404575	+	0.914505i	$0.632580\pi$
$194$	0	0
$195$	−8.74301	−0.626100
$196$	0	0
$197$	−5.13260	−0.365683	−0.182841	−	0.983142i	$-0.558529\pi$
$197$	−5.13260	−0.365683	−0.182841	+	0.983142i	$0.558529\pi$
$198$	0	0
$199$	17.2262	1.22113	0.610567	−	0.791964i	$-0.290941\pi$
$199$	17.2262	1.22113	0.610567	+	0.791964i	$0.290941\pi$
$200$	0	0
$201$	0.0825425	0.00582210
$202$	0	0
$203$	−3.01012	−0.211269
$204$	0	0
$205$	−4.52691	−0.316173
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−2.98988	−0.206814
$210$	0	0
$211$	6.33858	0.436366	0.218183	−	0.975908i	$-0.429987\pi$
$211$	6.33858	0.436366	0.218183	+	0.975908i	$0.429987\pi$
$212$	0	0
$213$	5.53703	0.379391
$214$	0	0
$215$	28.5239	1.94531
$216$	0	0
$217$	−8.20126	−0.556738
$218$	0	0
$219$	−8.33858	−0.563469
$220$	0	0
$221$	−11.4072	−0.767334
$222$	0	0
$223$	−16.1427	−1.08100	−0.540498	−	0.841345i	$-0.681764\pi$
$223$	−16.1427	−1.08100	−0.540498	+	0.841345i	$0.681764\pi$
$224$	0	0
$225$	10.5648	0.704320
$226$	0	0
$227$	−11.5888	−0.769176	−0.384588	−	0.923088i	$-0.625657\pi$
$227$	−11.5888	−0.769176	−0.384588	+	0.923088i	$0.625657\pi$
$228$	0	0
$229$	−6.51474	−0.430506	−0.215253	−	0.976558i	$-0.569058\pi$
$229$	−6.51474	−0.430506	−0.215253	+	0.976558i	$0.569058\pi$
$230$	0	0
$231$	1.72913	0.113768
$232$	0	0
$233$	−25.7661	−1.68799	−0.843995	−	0.536350i	$-0.819803\pi$
$233$	−25.7661	−1.68799	−0.843995	+	0.536350i	$0.819803\pi$
$234$	0	0
$235$	28.9923	1.89125
$236$	0	0
$237$	−8.20126	−0.532729
$238$	0	0
$239$	16.1427	1.04419	0.522093	−	0.852889i	$-0.325151\pi$
$239$	16.1427	1.04419	0.522093	+	0.852889i	$0.325151\pi$
$240$	0	0
$241$	23.6257	1.52187	0.760934	−	0.648829i	$-0.224741\pi$
$241$	23.6257	1.52187	0.760934	+	0.648829i	$0.224741\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.94523	−0.252051
$246$	0	0
$247$	3.83191	0.243819
$248$	0	0
$249$	3.21515	0.203752
$250$	0	0
$251$	11.9803	0.756190	0.378095	−	0.925767i	$-0.376579\pi$
$251$	11.9803	0.756190	0.378095	+	0.925767i	$0.376579\pi$
$252$	0	0
$253$	−1.72913	−0.108709
$254$	0	0
$255$	20.3078	1.27172
$256$	0	0
$257$	−15.5880	−0.972356	−0.486178	−	0.873860i	$-0.661609\pi$
$257$	−15.5880	−0.972356	−0.486178	+	0.873860i	$0.661609\pi$
$258$	0	0
$259$	−7.61958	−0.473457
$260$	0	0
$261$	3.01012	0.186322
$262$	0	0
$263$	−13.4423	−0.828889	−0.414445	−	0.910075i	$-0.636024\pi$
$263$	−13.4423	−0.828889	−0.414445	+	0.910075i	$0.636024\pi$
$264$	0	0
$265$	16.6186	1.02087
$266$	0	0
$267$	9.45525	0.578652
$268$	0	0
$269$	−6.83772	−0.416903	−0.208452	−	0.978033i	$-0.566842\pi$
$269$	−6.83772	−0.416903	−0.208452	+	0.978033i	$0.566842\pi$
$270$	0	0
$271$	−19.8726	−1.20718	−0.603588	−	0.797297i	$-0.706263\pi$
$271$	−19.8726	−1.20718	−0.603588	+	0.797297i	$0.706263\pi$
$272$	0	0
$273$	−2.21610	−0.134124
$274$	0	0
$275$	−18.2679	−1.10159
$276$	0	0
$277$	−4.24387	−0.254989	−0.127495	−	0.991839i	$-0.540694\pi$
$277$	−4.24387	−0.254989	−0.127495	+	0.991839i	$0.540694\pi$
$278$	0	0
$279$	8.20126	0.490997
$280$	0	0
$281$	28.1016	1.67640	0.838202	−	0.545360i	$-0.183607\pi$
$281$	28.1016	1.67640	0.838202	+	0.545360i	$0.183607\pi$
$282$	0	0
$283$	−11.1863	−0.664956	−0.332478	−	0.943111i	$-0.607885\pi$
$283$	−11.1863	−0.664956	−0.332478	+	0.943111i	$0.607885\pi$
$284$	0	0
$285$	−6.82179	−0.404088
$286$	0	0
$287$	−1.14744	−0.0677313
$288$	0	0
$289$	9.49614	0.558597
$290$	0	0
$291$	−8.33858	−0.488816
$292$	0	0
$293$	13.9026	0.812200	0.406100	−	0.913829i	$-0.366889\pi$
$293$	13.9026	0.812200	0.406100	+	0.913829i	$0.366889\pi$
$294$	0	0
$295$	−5.26992	−0.306827
$296$	0	0
$297$	−1.72913	−0.100334
$298$	0	0
$299$	2.21610	0.128160
$300$	0	0
$301$	7.22998	0.416729
$302$	0	0
$303$	−17.2540	−0.991216
$304$	0	0
$305$	−55.9206	−3.20201
$306$	0	0
$307$	2.18102	0.124477	0.0622386	−	0.998061i	$-0.480176\pi$
$307$	2.18102	0.124477	0.0622386	+	0.998061i	$0.480176\pi$
$308$	0	0
$309$	6.02777	0.342908
$310$	0	0
$311$	−11.1806	−0.633995	−0.316997	−	0.948426i	$-0.602675\pi$
$311$	−11.1806	−0.633995	−0.316997	+	0.948426i	$0.602675\pi$
$312$	0	0
$313$	−19.5416	−1.10455	−0.552277	−	0.833661i	$-0.686241\pi$
$313$	−19.5416	−1.10455	−0.552277	+	0.833661i	$0.686241\pi$
$314$	0	0
$315$	3.94523	0.222288
$316$	0	0
$317$	6.79402	0.381590	0.190795	−	0.981630i	$-0.438893\pi$
$317$	6.79402	0.381590	0.190795	+	0.981630i	$0.438893\pi$
$318$	0	0
$319$	−5.20488	−0.291417
$320$	0	0
$321$	−11.2225	−0.626377
$322$	0	0
$323$	−8.90057	−0.495241
$324$	0	0
$325$	23.4126	1.29870
$326$	0	0
$327$	−11.0268	−0.609784
$328$	0	0
$329$	7.34870	0.405147
$330$	0	0
$331$	11.8683	0.652341	0.326170	−	0.945311i	$-0.394242\pi$
$331$	11.8683	0.652341	0.326170	+	0.945311i	$0.394242\pi$
$332$	0	0
$333$	7.61958	0.417550
$334$	0	0
$335$	−0.325649	−0.0177921
$336$	0	0
$337$	−15.7577	−0.858374	−0.429187	−	0.903216i	$-0.641200\pi$
$337$	−15.7577	−0.858374	−0.429187	+	0.903216i	$0.641200\pi$
$338$	0	0
$339$	−16.2800	−0.884211
$340$	0	0
$341$	−14.1810	−0.767945
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.94523	−0.212404
$346$	0	0
$347$	−26.1120	−1.40176	−0.700882	−	0.713278i	$-0.747210\pi$
$347$	−26.1120	−1.40176	−0.700882	+	0.713278i	$0.747210\pi$
$348$	0	0
$349$	−8.82179	−0.472220	−0.236110	−	0.971726i	$-0.575873\pi$
$349$	−8.82179	−0.472220	−0.236110	+	0.971726i	$0.575873\pi$
$350$	0	0
$351$	2.21610	0.118287
$352$	0	0
$353$	−11.6614	−0.620675	−0.310337	−	0.950626i	$-0.600442\pi$
$353$	−11.6614	−0.620675	−0.310337	+	0.950626i	$0.600442\pi$
$354$	0	0
$355$	−21.8448	−1.15940
$356$	0	0
$357$	5.14744	0.272431
$358$	0	0
$359$	−21.6363	−1.14192	−0.570959	−	0.820978i	$-0.693429\pi$
$359$	−21.6363	−1.14192	−0.570959	+	0.820978i	$0.693429\pi$
$360$	0	0
$361$	−16.0101	−0.842638
$362$	0	0
$363$	−8.01012	−0.420422
$364$	0	0
$365$	32.8976	1.72194
$366$	0	0
$367$	−8.27163	−0.431776	−0.215888	−	0.976418i	$-0.569265\pi$
$367$	−8.27163	−0.431776	−0.215888	+	0.976418i	$0.569265\pi$
$368$	0	0
$369$	1.14744	0.0597334
$370$	0	0
$371$	4.21234	0.218694
$372$	0	0
$373$	4.11331	0.212979	0.106490	−	0.994314i	$-0.466039\pi$
$373$	4.11331	0.212979	0.106490	+	0.994314i	$0.466039\pi$
$374$	0	0
$375$	−21.9544	−1.13372
$376$	0	0
$377$	6.67073	0.343560
$378$	0	0
$379$	−18.5277	−0.951703	−0.475851	−	0.879526i	$-0.657860\pi$
$379$	−18.5277	−0.951703	−0.475851	+	0.879526i	$0.657860\pi$
$380$	0	0
$381$	−1.64658	−0.0843570
$382$	0	0
$383$	−5.96192	−0.304640	−0.152320	−	0.988331i	$-0.548674\pi$
$383$	−5.96192	−0.304640	−0.152320	+	0.988331i	$0.548674\pi$
$384$	0	0
$385$	−6.82179	−0.347671
$386$	0	0
$387$	−7.22998	−0.367521
$388$	0	0
$389$	17.5081	0.887697	0.443848	−	0.896102i	$-0.353613\pi$
$389$	17.5081	0.887697	0.443848	+	0.896102i	$0.353613\pi$
$390$	0	0
$391$	−5.14744	−0.260317
$392$	0	0
$393$	6.60569	0.333213
$394$	0	0
$395$	32.3558	1.62800
$396$	0	0
$397$	12.1929	0.611942	0.305971	−	0.952041i	$-0.401019\pi$
$397$	12.1929	0.611942	0.305971	+	0.952041i	$0.401019\pi$
$398$	0	0
$399$	−1.72913	−0.0865646
$400$	0	0
$401$	−10.3108	−0.514897	−0.257449	−	0.966292i	$-0.582882\pi$
$401$	−10.3108	−0.514897	−0.257449	+	0.966292i	$0.582882\pi$
$402$	0	0
$403$	18.1748	0.905352
$404$	0	0
$405$	−3.94523	−0.196040
$406$	0	0
$407$	−13.1752	−0.653071
$408$	0	0
$409$	−18.0377	−0.891907	−0.445953	−	0.895056i	$-0.647135\pi$
$409$	−18.0377	−0.891907	−0.445953	+	0.895056i	$0.647135\pi$
$410$	0	0
$411$	−20.2290	−0.997824
$412$	0	0
$413$	−1.33577	−0.0657290
$414$	0	0
$415$	−12.6845	−0.622656
$416$	0	0
$417$	−5.58073	−0.273290
$418$	0	0
$419$	14.2457	0.695949	0.347975	−	0.937504i	$-0.386870\pi$
$419$	14.2457	0.695949	0.347975	+	0.937504i	$0.386870\pi$
$420$	0	0
$421$	24.4136	1.18985	0.594923	−	0.803783i	$-0.297183\pi$
$421$	24.4136	1.18985	0.594923	+	0.803783i	$0.297183\pi$
$422$	0	0
$423$	−7.34870	−0.357306
$424$	0	0
$425$	−54.3817	−2.63790
$426$	0	0
$427$	−14.1743	−0.685940
$428$	0	0
$429$	−3.83191	−0.185007
$430$	0	0
$431$	−38.1823	−1.83918	−0.919589	−	0.392882i	$-0.871478\pi$
$431$	−38.1823	−1.83918	−0.919589	+	0.392882i	$0.871478\pi$
$432$	0	0
$433$	−22.7473	−1.09317	−0.546583	−	0.837405i	$-0.684072\pi$
$433$	−22.7473	−1.09317	−0.546583	+	0.837405i	$0.684072\pi$
$434$	0	0
$435$	−11.8756	−0.569392
$436$	0	0
$437$	1.72913	0.0827153
$438$	0	0
$439$	37.4757	1.78862	0.894309	−	0.447450i	$-0.147668\pi$
$439$	37.4757	1.78862	0.894309	+	0.447450i	$0.147668\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.61753	0.314408	0.157204	−	0.987566i	$-0.449752\pi$
$443$	6.61753	0.314408	0.157204	+	0.987566i	$0.449752\pi$
$444$	0	0
$445$	−37.3031	−1.76834
$446$	0	0
$447$	−8.22718	−0.389132
$448$	0	0
$449$	−28.7678	−1.35764	−0.678818	−	0.734307i	$-0.737507\pi$
$449$	−28.7678	−1.35764	−0.678818	+	0.734307i	$0.737507\pi$
$450$	0	0
$451$	−1.98407	−0.0934262
$452$	0	0
$453$	−1.32093	−0.0620629
$454$	0	0
$455$	8.74301	0.409879
$456$	0	0
$457$	3.61041	0.168888	0.0844438	−	0.996428i	$-0.473089\pi$
$457$	3.61041	0.168888	0.0844438	+	0.996428i	$0.473089\pi$
$458$	0	0
$459$	−5.14744	−0.240262
$460$	0	0
$461$	−9.92363	−0.462189	−0.231095	−	0.972931i	$-0.574231\pi$
$461$	−9.92363	−0.462189	−0.231095	+	0.972931i	$0.574231\pi$
$462$	0	0
$463$	−11.0622	−0.514105	−0.257053	−	0.966397i	$-0.582751\pi$
$463$	−11.0622	−0.514105	−0.257053	+	0.966397i	$0.582751\pi$
$464$	0	0
$465$	−32.3558	−1.50047
$466$	0	0
$467$	8.42653	0.389933	0.194967	−	0.980810i	$-0.437540\pi$
$467$	8.42653	0.389933	0.194967	+	0.980810i	$0.437540\pi$
$468$	0	0
$469$	−0.0825425	−0.00381146
$470$	0	0
$471$	−1.93229	−0.0890354
$472$	0	0
$473$	12.5016	0.574822
$474$	0	0
$475$	18.2679	0.838188
$476$	0	0
$477$	−4.21234	−0.192870
$478$	0	0
$479$	24.0842	1.10043	0.550217	−	0.835021i	$-0.314545\pi$
$479$	24.0842	1.10043	0.550217	+	0.835021i	$0.314545\pi$
$480$	0	0
$481$	16.8857	0.769923
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	32.8976	1.49380
$486$	0	0
$487$	8.73549	0.395843	0.197921	−	0.980218i	$-0.436581\pi$
$487$	8.73549	0.395843	0.197921	+	0.980218i	$0.436581\pi$
$488$	0	0
$489$	−0.801546	−0.0362471
$490$	0	0
$491$	28.6408	1.29254	0.646270	−	0.763109i	$-0.276328\pi$
$491$	28.6408	1.29254	0.646270	+	0.763109i	$0.276328\pi$
$492$	0	0
$493$	−15.4944	−0.697834
$494$	0	0
$495$	6.82179	0.306617
$496$	0	0
$497$	−5.53703	−0.248370
$498$	0	0
$499$	−15.6307	−0.699724	−0.349862	−	0.936801i	$-0.613772\pi$
$499$	−15.6307	−0.699724	−0.349862	+	0.936801i	$0.613772\pi$
$500$	0	0
$501$	19.7372	0.881793
$502$	0	0
$503$	−21.1167	−0.941546	−0.470773	−	0.882254i	$-0.656025\pi$
$503$	−21.1167	−0.941546	−0.470773	+	0.882254i	$0.656025\pi$
$504$	0	0
$505$	68.0709	3.02911
$506$	0	0
$507$	−8.08890	−0.359241
$508$	0	0
$509$	−0.946178	−0.0419386	−0.0209693	−	0.999780i	$-0.506675\pi$
$509$	−0.946178	−0.0419386	−0.0209693	+	0.999780i	$0.506675\pi$
$510$	0	0
$511$	8.33858	0.368877
$512$	0	0
$513$	1.72913	0.0763428
$514$	0	0
$515$	−23.7809	−1.04791
$516$	0	0
$517$	12.7068	0.558846
$518$	0	0
$519$	18.1397	0.796246
$520$	0	0
$521$	−36.1632	−1.58434	−0.792169	−	0.610302i	$-0.791048\pi$
$521$	−36.1632	−1.58434	−0.792169	+	0.610302i	$0.791048\pi$
$522$	0	0
$523$	−12.7670	−0.558263	−0.279131	−	0.960253i	$-0.590046\pi$
$523$	−12.7670	−0.558263	−0.279131	+	0.960253i	$0.590046\pi$
$524$	0	0
$525$	−10.5648	−0.461086
$526$	0	0
$527$	−42.2155	−1.83894
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.33577	0.0579676
$532$	0	0
$533$	2.54284	0.110143
$534$	0	0
$535$	44.2751	1.91418
$536$	0	0
$537$	22.3281	0.963527
$538$	0	0
$539$	−1.72913	−0.0744787
$540$	0	0
$541$	−10.1057	−0.434480	−0.217240	−	0.976118i	$-0.569705\pi$
$541$	−10.1057	−0.434480	−0.217240	+	0.976118i	$0.569705\pi$
$542$	0	0
$543$	−8.78295	−0.376912
$544$	0	0
$545$	43.5033	1.86348
$546$	0	0
$547$	35.3020	1.50940	0.754702	−	0.656067i	$-0.227781\pi$
$547$	35.3020	1.50940	0.754702	+	0.656067i	$0.227781\pi$
$548$	0	0
$549$	14.1743	0.604942
$550$	0	0
$551$	5.20488	0.221735
$552$	0	0
$553$	8.20126	0.348753
$554$	0	0
$555$	−30.0609	−1.27602
$556$	0	0
$557$	36.7408	1.55676	0.778378	−	0.627796i	$-0.216043\pi$
$557$	36.7408	1.55676	0.778378	+	0.627796i	$0.216043\pi$
$558$	0	0
$559$	−16.0224	−0.677674
$560$	0	0
$561$	8.90057	0.375783
$562$	0	0
$563$	−32.4969	−1.36958	−0.684791	−	0.728740i	$-0.740106\pi$
$563$	−32.4969	−1.36958	−0.684791	+	0.728740i	$0.740106\pi$
$564$	0	0
$565$	64.2284	2.70211
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	5.67735	0.238007	0.119003	−	0.992894i	$-0.462030\pi$
$569$	5.67735	0.238007	0.119003	+	0.992894i	$0.462030\pi$
$570$	0	0
$571$	31.8923	1.33465	0.667326	−	0.744766i	$-0.267439\pi$
$571$	31.8923	1.33465	0.667326	+	0.744766i	$0.267439\pi$
$572$	0	0
$573$	1.32093	0.0551828
$574$	0	0
$575$	10.5648	0.440583
$576$	0	0
$577$	−7.61772	−0.317130	−0.158565	−	0.987349i	$-0.550687\pi$
$577$	−7.61772	−0.317130	−0.158565	+	0.987349i	$0.550687\pi$
$578$	0	0
$579$	−11.2411	−0.467162
$580$	0	0
$581$	−3.21515	−0.133387
$582$	0	0
$583$	7.28366	0.301659
$584$	0	0
$585$	−8.74301	−0.361479
$586$	0	0
$587$	6.45697	0.266507	0.133254	−	0.991082i	$-0.457457\pi$
$587$	6.45697	0.266507	0.133254	+	0.991082i	$0.457457\pi$
$588$	0	0
$589$	14.1810	0.584319
$590$	0	0
$591$	−5.13260	−0.211127
$592$	0	0
$593$	32.3763	1.32953	0.664767	−	0.747051i	$-0.268531\pi$
$593$	32.3763	1.32953	0.664767	+	0.747051i	$0.268531\pi$
$594$	0	0
$595$	−20.3078	−0.832539
$596$	0	0
$597$	17.2262	0.705022
$598$	0	0
$599$	−19.3754	−0.791656	−0.395828	−	0.918325i	$-0.629542\pi$
$599$	−19.3754	−0.791656	−0.395828	+	0.918325i	$0.629542\pi$
$600$	0	0
$601$	−29.4854	−1.20274	−0.601368	−	0.798972i	$-0.705377\pi$
$601$	−29.4854	−1.20274	−0.601368	+	0.798972i	$0.705377\pi$
$602$	0	0
$603$	0.0825425	0.00336139
$604$	0	0
$605$	31.6017	1.28479
$606$	0	0
$607$	39.5033	1.60339	0.801694	−	0.597735i	$-0.203932\pi$
$607$	39.5033	1.60339	0.801694	+	0.597735i	$0.203932\pi$
$608$	0	0
$609$	−3.01012	−0.121976
$610$	0	0
$611$	−16.2855	−0.658839
$612$	0	0
$613$	−37.7212	−1.52355	−0.761773	−	0.647844i	$-0.775671\pi$
$613$	−37.7212	−1.52355	−0.761773	+	0.647844i	$0.775671\pi$
$614$	0	0
$615$	−4.52691	−0.182543
$616$	0	0
$617$	−34.3995	−1.38487	−0.692436	−	0.721479i	$-0.743463\pi$
$617$	−34.3995	−1.38487	−0.692436	+	0.721479i	$0.743463\pi$
$618$	0	0
$619$	−30.3299	−1.21906	−0.609531	−	0.792762i	$-0.708642\pi$
$619$	−30.3299	−1.21906	−0.609531	+	0.792762i	$0.708642\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−9.45525	−0.378817
$624$	0	0
$625$	33.7910	1.35164
$626$	0	0
$627$	−2.98988	−0.119404
$628$	0	0
$629$	−39.2213	−1.56386
$630$	0	0
$631$	15.7447	0.626788	0.313394	−	0.949623i	$-0.398534\pi$
$631$	15.7447	0.626788	0.313394	+	0.949623i	$0.398534\pi$
$632$	0	0
$633$	6.33858	0.251936
$634$	0	0
$635$	6.49614	0.257792
$636$	0	0
$637$	2.21610	0.0878050
$638$	0	0
$639$	5.53703	0.219042
$640$	0	0
$641$	−30.6111	−1.20907	−0.604533	−	0.796580i	$-0.706640\pi$
$641$	−30.6111	−1.20907	−0.604533	+	0.796580i	$0.706640\pi$
$642$	0	0
$643$	−27.9071	−1.10055	−0.550275	−	0.834983i	$-0.685477\pi$
$643$	−27.9071	−1.10055	−0.550275	+	0.834983i	$0.685477\pi$
$644$	0	0
$645$	28.5239	1.12313
$646$	0	0
$647$	19.9608	0.784741	0.392370	−	0.919807i	$-0.371655\pi$
$647$	19.9608	0.784741	0.392370	+	0.919807i	$0.371655\pi$
$648$	0	0
$649$	−2.30972	−0.0906644
$650$	0	0
$651$	−8.20126	−0.321433
$652$	0	0
$653$	−1.90529	−0.0745597	−0.0372798	−	0.999305i	$-0.511869\pi$
$653$	−1.90529	−0.0745597	−0.0372798	+	0.999305i	$0.511869\pi$
$654$	0	0
$655$	−26.0609	−1.01829
$656$	0	0
$657$	−8.33858	−0.325319
$658$	0	0
$659$	32.8291	1.27884	0.639419	−	0.768858i	$-0.279175\pi$
$659$	32.8291	1.27884	0.639419	+	0.768858i	$0.279175\pi$
$660$	0	0
$661$	−24.3447	−0.946901	−0.473450	−	0.880820i	$-0.656992\pi$
$661$	−24.3447	−0.946901	−0.473450	+	0.880820i	$0.656992\pi$
$662$	0	0
$663$	−11.4072	−0.443020
$664$	0	0
$665$	6.82179	0.264538
$666$	0	0
$667$	3.01012	0.116552
$668$	0	0
$669$	−16.1427	−0.624114
$670$	0	0
$671$	−24.5091	−0.946162
$672$	0	0
$673$	21.2176	0.817879	0.408939	−	0.912562i	$-0.365899\pi$
$673$	21.2176	0.817879	0.408939	+	0.912562i	$0.365899\pi$
$674$	0	0
$675$	10.5648	0.406639
$676$	0	0
$677$	9.92929	0.381614	0.190807	−	0.981628i	$-0.438890\pi$
$677$	9.92929	0.381614	0.190807	+	0.981628i	$0.438890\pi$
$678$	0	0
$679$	8.33858	0.320005
$680$	0	0
$681$	−11.5888	−0.444084
$682$	0	0
$683$	−22.3386	−0.854762	−0.427381	−	0.904072i	$-0.640564\pi$
$683$	−22.3386	−0.854762	−0.427381	+	0.904072i	$0.640564\pi$
$684$	0	0
$685$	79.8081	3.04931
$686$	0	0
$687$	−6.51474	−0.248553
$688$	0	0
$689$	−9.33496	−0.355634
$690$	0	0
$691$	32.7687	1.24658	0.623289	−	0.781991i	$-0.285796\pi$
$691$	32.7687	1.24658	0.623289	+	0.781991i	$0.285796\pi$
$692$	0	0
$693$	1.72913	0.0656841
$694$	0	0
$695$	22.0172	0.835162
$696$	0	0
$697$	−5.90638	−0.223720
$698$	0	0
$699$	−25.7661	−0.974562
$700$	0	0
$701$	−19.8432	−0.749467	−0.374734	−	0.927133i	$-0.622266\pi$
$701$	−19.8432	−0.749467	−0.374734	+	0.927133i	$0.622266\pi$
$702$	0	0
$703$	13.1752	0.496912
$704$	0	0
$705$	28.9923	1.09191
$706$	0	0
$707$	17.2540	0.648903
$708$	0	0
$709$	−28.5727	−1.07307	−0.536535	−	0.843878i	$-0.680267\pi$
$709$	−28.5727	−1.07307	−0.536535	+	0.843878i	$0.680267\pi$
$710$	0	0
$711$	−8.20126	−0.307571
$712$	0	0
$713$	8.20126	0.307140
$714$	0	0
$715$	15.1178	0.565372
$716$	0	0
$717$	16.1427	0.602861
$718$	0	0
$719$	16.2643	0.606557	0.303278	−	0.952902i	$-0.401919\pi$
$719$	16.2643	0.606557	0.303278	+	0.952902i	$0.401919\pi$
$720$	0	0
$721$	−6.02777	−0.224486
$722$	0	0
$723$	23.6257	0.878651
$724$	0	0
$725$	31.8014	1.18107
$726$	0	0
$727$	3.30500	0.122576	0.0612879	−	0.998120i	$-0.480479\pi$
$727$	3.30500	0.122576	0.0612879	+	0.998120i	$0.480479\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	37.2159	1.37648
$732$	0	0
$733$	−25.2750	−0.933553	−0.466777	−	0.884375i	$-0.654585\pi$
$733$	−25.2750	−0.933553	−0.466777	+	0.884375i	$0.654585\pi$
$734$	0	0
$735$	−3.94523	−0.145522
$736$	0	0
$737$	−0.142726	−0.00525739
$738$	0	0
$739$	−47.9450	−1.76368	−0.881842	−	0.471545i	$-0.843697\pi$
$739$	−47.9450	−1.76368	−0.881842	+	0.471545i	$0.843697\pi$
$740$	0	0
$741$	3.83191	0.140769
$742$	0	0
$743$	−17.7604	−0.651568	−0.325784	−	0.945444i	$-0.605628\pi$
$743$	−17.7604	−0.651568	−0.325784	+	0.945444i	$0.605628\pi$
$744$	0	0
$745$	32.4581	1.18917
$746$	0	0
$747$	3.21515	0.117636
$748$	0	0
$749$	11.2225	0.410060
$750$	0	0
$751$	31.9716	1.16666	0.583331	−	0.812235i	$-0.301749\pi$
$751$	31.9716	1.16666	0.583331	+	0.812235i	$0.301749\pi$
$752$	0	0
$753$	11.9803	0.436587
$754$	0	0
$755$	5.21139	0.189662
$756$	0	0
$757$	9.18928	0.333990	0.166995	−	0.985958i	$-0.446594\pi$
$757$	9.18928	0.333990	0.166995	+	0.985958i	$0.446594\pi$
$758$	0	0
$759$	−1.72913	−0.0627633
$760$	0	0
$761$	−13.0482	−0.472997	−0.236499	−	0.971632i	$-0.576000\pi$
$761$	−13.0482	−0.472997	−0.236499	+	0.971632i	$0.576000\pi$
$762$	0	0
$763$	11.0268	0.399197
$764$	0	0
$765$	20.3078	0.734230
$766$	0	0
$767$	2.96020	0.106887
$768$	0	0
$769$	21.9385	0.791121	0.395561	−	0.918440i	$-0.370550\pi$
$769$	21.9385	0.791121	0.395561	+	0.918440i	$0.370550\pi$
$770$	0	0
$771$	−15.5880	−0.561390
$772$	0	0
$773$	−46.4745	−1.67157	−0.835787	−	0.549054i	$-0.814988\pi$
$773$	−46.4745	−1.67157	−0.835787	+	0.549054i	$0.814988\pi$
$774$	0	0
$775$	86.6447	3.11237
$776$	0	0
$777$	−7.61958	−0.273351
$778$	0	0
$779$	1.98407	0.0710867
$780$	0	0
$781$	−9.57423	−0.342593
$782$	0	0
$783$	3.01012	0.107573
$784$	0	0
$785$	7.62334	0.272089
$786$	0	0
$787$	26.3996	0.941043	0.470522	−	0.882388i	$-0.344066\pi$
$787$	26.3996	0.941043	0.470522	+	0.882388i	$0.344066\pi$
$788$	0	0
$789$	−13.4423	−0.478559
$790$	0	0
$791$	16.2800	0.578852
$792$	0	0
$793$	31.4116	1.11546
$794$	0	0
$795$	16.6186	0.589402
$796$	0	0
$797$	47.3346	1.67668	0.838339	−	0.545149i	$-0.183527\pi$
$797$	47.3346	1.67668	0.838339	+	0.545149i	$0.183527\pi$
$798$	0	0
$799$	37.8270	1.33822
$800$	0	0
$801$	9.45525	0.334085
$802$	0	0
$803$	14.4185	0.508816
$804$	0	0
$805$	3.94523	0.139051
$806$	0	0
$807$	−6.83772	−0.240699
$808$	0	0
$809$	−23.5446	−0.827782	−0.413891	−	0.910326i	$-0.635831\pi$
$809$	−23.5446	−0.827782	−0.413891	+	0.910326i	$0.635831\pi$
$810$	0	0
$811$	1.07918	0.0378952	0.0189476	−	0.999820i	$-0.493968\pi$
$811$	1.07918	0.0378952	0.0189476	+	0.999820i	$0.493968\pi$
$812$	0	0
$813$	−19.8726	−0.696963
$814$	0	0
$815$	3.16228	0.110770
$816$	0	0
$817$	−12.5016	−0.437374
$818$	0	0
$819$	−2.21610	−0.0774368
$820$	0	0
$821$	−11.2335	−0.392053	−0.196027	−	0.980599i	$-0.562804\pi$
$821$	−11.2335	−0.392053	−0.196027	+	0.980599i	$0.562804\pi$
$822$	0	0
$823$	−42.3577	−1.47650	−0.738249	−	0.674528i	$-0.764347\pi$
$823$	−42.3577	−1.47650	−0.738249	+	0.674528i	$0.764347\pi$
$824$	0	0
$825$	−18.2679	−0.636006
$826$	0	0
$827$	57.2264	1.98996	0.994978	−	0.100091i	$-0.0319134\pi$
$827$	57.2264	1.98996	0.994978	+	0.100091i	$0.0319134\pi$
$828$	0	0
$829$	37.7874	1.31241	0.656205	−	0.754582i	$-0.272160\pi$
$829$	37.7874	1.31241	0.656205	+	0.754582i	$0.272160\pi$
$830$	0	0
$831$	−4.24387	−0.147218
$832$	0	0
$833$	−5.14744	−0.178348
$834$	0	0
$835$	−77.8677	−2.69472
$836$	0	0
$837$	8.20126	0.283477
$838$	0	0
$839$	−19.0452	−0.657514	−0.328757	−	0.944415i	$-0.606630\pi$
$839$	−19.0452	−0.657514	−0.328757	+	0.944415i	$0.606630\pi$
$840$	0	0
$841$	−19.9392	−0.687557
$842$	0	0
$843$	28.1016	0.967872
$844$	0	0
$845$	31.9126	1.09783
$846$	0	0
$847$	8.01012	0.275231
$848$	0	0
$849$	−11.1863	−0.383912
$850$	0	0
$851$	7.61958	0.261196
$852$	0	0
$853$	25.9385	0.888116	0.444058	−	0.895998i	$-0.353538\pi$
$853$	25.9385	0.888116	0.444058	+	0.895998i	$0.353538\pi$
$854$	0	0
$855$	−6.82179	−0.233300
$856$	0	0
$857$	49.9347	1.70574	0.852868	−	0.522127i	$-0.174861\pi$
$857$	49.9347	1.70574	0.852868	+	0.522127i	$0.174861\pi$
$858$	0	0
$859$	1.97975	0.0675483	0.0337742	−	0.999429i	$-0.489247\pi$
$859$	1.97975	0.0675483	0.0337742	+	0.999429i	$0.489247\pi$
$860$	0	0
$861$	−1.14744	−0.0391047
$862$	0	0
$863$	16.2855	0.554363	0.277182	−	0.960818i	$-0.410600\pi$
$863$	16.2855	0.554363	0.277182	+	0.960818i	$0.410600\pi$
$864$	0	0
$865$	−71.5653	−2.43329
$866$	0	0
$867$	9.49614	0.322506
$868$	0	0
$869$	14.1810	0.481058
$870$	0	0
$871$	0.182922	0.00619809
$872$	0	0
$873$	−8.33858	−0.282218
$874$	0	0
$875$	21.9544	0.742194
$876$	0	0
$877$	9.89645	0.334179	0.167090	−	0.985942i	$-0.446563\pi$
$877$	9.89645	0.334179	0.167090	+	0.985942i	$0.446563\pi$
$878$	0	0
$879$	13.9026	0.468924
$880$	0	0
$881$	3.91451	0.131883	0.0659416	−	0.997823i	$-0.478995\pi$
$881$	3.91451	0.131883	0.0659416	+	0.997823i	$0.478995\pi$
$882$	0	0
$883$	−57.3303	−1.92932	−0.964659	−	0.263503i	$-0.915122\pi$
$883$	−57.3303	−1.92932	−0.964659	+	0.263503i	$0.915122\pi$
$884$	0	0
$885$	−5.26992	−0.177146
$886$	0	0
$887$	39.6927	1.33275	0.666375	−	0.745617i	$-0.267845\pi$
$887$	39.6927	1.33275	0.666375	+	0.745617i	$0.267845\pi$
$888$	0	0
$889$	1.64658	0.0552246
$890$	0	0
$891$	−1.72913	−0.0579279
$892$	0	0
$893$	−12.7068	−0.425218
$894$	0	0
$895$	−88.0892	−2.94450
$896$	0	0
$897$	2.21610	0.0739934
$898$	0	0
$899$	24.6868	0.823351
$900$	0	0
$901$	21.6828	0.722357
$902$	0	0
$903$	7.22998	0.240599
$904$	0	0
$905$	34.6507	1.15183
$906$	0	0
$907$	54.4194	1.80697	0.903484	−	0.428622i	$-0.141001\pi$
$907$	54.4194	1.80697	0.903484	+	0.428622i	$0.141001\pi$
$908$	0	0
$909$	−17.2540	−0.572279
$910$	0	0
$911$	−9.10184	−0.301557	−0.150779	−	0.988568i	$-0.548178\pi$
$911$	−9.10184	−0.301557	−0.150779	+	0.988568i	$0.548178\pi$
$912$	0	0
$913$	−5.55939	−0.183989
$914$	0	0
$915$	−55.9206	−1.84868
$916$	0	0
$917$	−6.60569	−0.218139
$918$	0	0
$919$	−48.5461	−1.60139	−0.800693	−	0.599074i	$-0.795535\pi$
$919$	−48.5461	−1.60139	−0.800693	+	0.599074i	$0.795535\pi$
$920$	0	0
$921$	2.18102	0.0718669
$922$	0	0
$923$	12.2706	0.403892
$924$	0	0
$925$	80.4993	2.64680
$926$	0	0
$927$	6.02777	0.197978
$928$	0	0
$929$	−56.4785	−1.85300	−0.926500	−	0.376295i	$-0.877198\pi$
$929$	−56.4785	−1.85300	−0.926500	+	0.376295i	$0.877198\pi$
$930$	0	0
$931$	1.72913	0.0566698
$932$	0	0
$933$	−11.1806	−0.366037
$934$	0	0
$935$	−35.1148	−1.14838
$936$	0	0
$937$	40.6316	1.32738	0.663688	−	0.748010i	$-0.268990\pi$
$937$	40.6316	1.32738	0.663688	+	0.748010i	$0.268990\pi$
$938$	0	0
$939$	−19.5416	−0.637715
$940$	0	0
$941$	−37.7634	−1.23105	−0.615526	−	0.788117i	$-0.711056\pi$
$941$	−37.7634	−1.23105	−0.615526	+	0.788117i	$0.711056\pi$
$942$	0	0
$943$	1.14744	0.0373658
$944$	0	0
$945$	3.94523	0.128338
$946$	0	0
$947$	−60.0997	−1.95298	−0.976490	−	0.215565i	$-0.930841\pi$
$947$	−60.0997	−1.95298	−0.976490	+	0.215565i	$0.930841\pi$
$948$	0	0
$949$	−18.4791	−0.599858
$950$	0	0
$951$	6.79402	0.220311
$952$	0	0
$953$	17.5130	0.567301	0.283650	−	0.958928i	$-0.408454\pi$
$953$	17.5130	0.567301	0.283650	+	0.958928i	$0.408454\pi$
$954$	0	0
$955$	−5.21139	−0.168636
$956$	0	0
$957$	−5.20488	−0.168250
$958$	0	0
$959$	20.2290	0.653229
$960$	0	0
$961$	36.2607	1.16970
$962$	0	0
$963$	−11.2225	−0.361639
$964$	0	0
$965$	44.3485	1.42763
$966$	0	0
$967$	−41.7768	−1.34345	−0.671726	−	0.740800i	$-0.734447\pi$
$967$	−41.7768	−1.34345	−0.671726	+	0.740800i	$0.734447\pi$
$968$	0	0
$969$	−8.90057	−0.285928
$970$	0	0
$971$	47.0589	1.51019	0.755096	−	0.655614i	$-0.227590\pi$
$971$	47.0589	1.51019	0.755096	+	0.655614i	$0.227590\pi$
$972$	0	0
$973$	5.58073	0.178910
$974$	0	0
$975$	23.4126	0.749805
$976$	0	0
$977$	−21.9171	−0.701191	−0.350595	−	0.936527i	$-0.614021\pi$
$977$	−21.9171	−0.701191	−0.350595	+	0.936527i	$0.614021\pi$
$978$	0	0
$979$	−16.3493	−0.522527
$980$	0	0
$981$	−11.0268	−0.352059
$982$	0	0
$983$	−35.7706	−1.14090	−0.570452	−	0.821331i	$-0.693232\pi$
$983$	−35.7706	−1.14090	−0.570452	+	0.821331i	$0.693232\pi$
$984$	0	0
$985$	20.2493	0.645196
$986$	0	0
$987$	7.34870	0.233912
$988$	0	0
$989$	−7.22998	−0.229900
$990$	0	0
$991$	−2.76346	−0.0877843	−0.0438921	−	0.999036i	$-0.513976\pi$
$991$	−2.76346	−0.0877843	−0.0438921	+	0.999036i	$0.513976\pi$
$992$	0	0
$993$	11.8683	0.376629
$994$	0	0
$995$	−67.9613	−2.15452
$996$	0	0
$997$	−55.3290	−1.75229	−0.876143	−	0.482052i	$-0.839892\pi$
$997$	−55.3290	−1.75229	−0.876143	+	0.482052i	$0.839892\pi$
$998$	0	0
$999$	7.61958	0.241073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cb.1.1		4
4.3	odd	2		3864.2.a.r.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.r.1.1	✓	4	4.3	odd	2
7728.2.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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.94523 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.94523 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.72913 q^{11} +2.21610 q^{13} -3.94523 q^{15} -5.14744 q^{17} +1.72913 q^{19} -1.00000 q^{21} +1.00000 q^{23} +10.5648 q^{25} +1.00000 q^{27} +3.01012 q^{29} +8.20126 q^{31} -1.72913 q^{33} +3.94523 q^{35} +7.61958 q^{37} +2.21610 q^{39} +1.14744 q^{41} -7.22998 q^{43} -3.94523 q^{45} -7.34870 q^{47} +1.00000 q^{49} -5.14744 q^{51} -4.21234 q^{53} +6.82179 q^{55} +1.72913 q^{57} +1.33577 q^{59} +14.1743 q^{61} -1.00000 q^{63} -8.74301 q^{65} +0.0825425 q^{67} +1.00000 q^{69} +5.53703 q^{71} -8.33858 q^{73} +10.5648 q^{75} +1.72913 q^{77} -8.20126 q^{79} +1.00000 q^{81} +3.21515 q^{83} +20.3078 q^{85} +3.01012 q^{87} +9.45525 q^{89} -2.21610 q^{91} +8.20126 q^{93} -6.82179 q^{95} -8.33858 q^{97} -1.72913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 - q^25 + 4 * q^27 - 10 * q^29 + 10 * q^31 - 2 * q^33 + 3 * q^35 + q^39 - 8 * q^41 - 13 * q^43 - 3 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 5 * q^53 - 3 * q^55 + 2 * q^57 + q^59 + 5 * q^61 - 4 * q^63 - 22 * q^65 - 3 * q^67 + 4 * q^69 - 5 * q^71 - 20 * q^73 - q^75 + 2 * q^77 - 10 * q^79 + 4 * q^81 - 18 * q^83 + 25 * q^85 - 10 * q^87 - 31 * q^89 - q^91 + 10 * q^93 + 3 * q^95 - 20 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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