LMFDB - Embedded newform 7728.2.a.bx.1.3

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:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.52892$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.52892 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.52892 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.86651 q^{11} -2.66241 q^{13} +2.52892 q^{15} -4.92434 q^{17} -2.13349 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.39543 q^{25} +1.00000 q^{27} +5.00000 q^{29} -3.86651 q^{31} -1.86651 q^{33} -2.52892 q^{35} +2.05784 q^{37} -2.66241 q^{39} +7.79085 q^{41} -5.72025 q^{43} +2.52892 q^{45} -5.92434 q^{47} +1.00000 q^{49} -4.92434 q^{51} -8.45326 q^{53} -4.72025 q^{55} -2.13349 q^{57} -14.1863 q^{59} -1.26193 q^{61} -1.00000 q^{63} -6.73302 q^{65} +11.2441 q^{67} +1.00000 q^{69} +1.26193 q^{71} +14.7730 q^{73} +1.39543 q^{75} +1.86651 q^{77} -15.9822 q^{79} +1.00000 q^{81} +17.8309 q^{83} -12.4533 q^{85} +5.00000 q^{87} -1.26193 q^{89} +2.66241 q^{91} -3.86651 q^{93} -5.39543 q^{95} -12.3248 q^{97} -1.86651 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 - 6 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 + 3 * q^27 + 15 * q^29 - 12 * q^31 - 6 * q^33 - 9 * q^37 + 9 * q^41 + 6 * q^43 - 3 * q^47 + 3 * q^49 - 3 * q^53 + 9 * q^55 - 6 * q^57 - 21 * q^59 + 3 * q^61 - 3 * q^63 - 21 * q^65 - 3 * q^67 + 3 * q^69 - 3 * q^71 - 3 * q^75 + 6 * q^77 - 18 * q^79 + 3 * q^81 - 6 * q^83 - 15 * q^85 + 15 * q^87 + 3 * q^89 - 12 * q^93 - 9 * q^95 - 21 * q^97 - 6 * 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
$5$	2.52892	1.13097	0.565483	−	0.824760i	$-0.308690\pi$
$5$	2.52892	1.13097	0.565483	+	0.824760i	$0.308690\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.86651	−0.562773	−0.281387	−	0.959594i	$-0.590794\pi$
$11$	−1.86651	−0.562773	−0.281387	+	0.959594i	$0.590794\pi$
$12$	0	0
$13$	−2.66241	−0.738420	−0.369210	−	0.929346i	$-0.620372\pi$
$13$	−2.66241	−0.738420	−0.369210	+	0.929346i	$0.620372\pi$
$14$	0	0
$15$	2.52892	0.652964
$16$	0	0
$17$	−4.92434	−1.19433	−0.597164	−	0.802119i	$-0.703706\pi$
$17$	−4.92434	−1.19433	−0.597164	+	0.802119i	$0.703706\pi$
$18$	0	0
$19$	−2.13349	−0.489457	−0.244728	−	0.969592i	$-0.578699\pi$
$19$	−2.13349	−0.489457	−0.244728	+	0.969592i	$0.578699\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.39543	0.279085
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−3.86651	−0.694445	−0.347223	−	0.937783i	$-0.612875\pi$
$31$	−3.86651	−0.694445	−0.347223	+	0.937783i	$0.612875\pi$
$32$	0	0
$33$	−1.86651	−0.324917
$34$	0	0
$35$	−2.52892	−0.427465
$36$	0	0
$37$	2.05784	0.338306	0.169153	−	0.985590i	$-0.445897\pi$
$37$	2.05784	0.338306	0.169153	+	0.985590i	$0.445897\pi$
$38$	0	0
$39$	−2.66241	−0.426327
$40$	0	0
$41$	7.79085	1.21673	0.608363	−	0.793659i	$-0.291826\pi$
$41$	7.79085	1.21673	0.608363	+	0.793659i	$0.291826\pi$
$42$	0	0
$43$	−5.72025	−0.872329	−0.436165	−	0.899867i	$-0.643663\pi$
$43$	−5.72025	−0.872329	−0.436165	+	0.899867i	$0.643663\pi$
$44$	0	0
$45$	2.52892	0.376989
$46$	0	0
$47$	−5.92434	−0.864154	−0.432077	−	0.901837i	$-0.642219\pi$
$47$	−5.92434	−0.864154	−0.432077	+	0.901837i	$0.642219\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.92434	−0.689546
$52$	0	0
$53$	−8.45326	−1.16114	−0.580572	−	0.814209i	$-0.697171\pi$
$53$	−8.45326	−1.16114	−0.580572	+	0.814209i	$0.697171\pi$
$54$	0	0
$55$	−4.72025	−0.636478
$56$	0	0
$57$	−2.13349	−0.282588
$58$	0	0
$59$	−14.1863	−1.84690	−0.923448	−	0.383723i	$-0.874642\pi$
$59$	−14.1863	−1.84690	−0.923448	+	0.383723i	$0.874642\pi$
$60$	0	0
$61$	−1.26193	−0.161574	−0.0807871	−	0.996731i	$-0.525743\pi$
$61$	−1.26193	−0.161574	−0.0807871	+	0.996731i	$0.525743\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.73302	−0.835128
$66$	0	0
$67$	11.2441	1.37369	0.686844	−	0.726805i	$-0.258996\pi$
$67$	11.2441	1.37369	0.686844	+	0.726805i	$0.258996\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	1.26193	0.149764	0.0748820	−	0.997192i	$-0.476142\pi$
$71$	1.26193	0.149764	0.0748820	+	0.997192i	$0.476142\pi$
$72$	0	0
$73$	14.7730	1.72905	0.864526	−	0.502588i	$-0.167619\pi$
$73$	14.7730	1.72905	0.864526	+	0.502588i	$0.167619\pi$
$74$	0	0
$75$	1.39543	0.161130
$76$	0	0
$77$	1.86651	0.212708
$78$	0	0
$79$	−15.9822	−1.79814	−0.899068	−	0.437809i	$-0.855755\pi$
$79$	−15.9822	−1.79814	−0.899068	+	0.437809i	$0.855755\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.8309	1.95719	0.978596	−	0.205791i	$-0.0659766\pi$
$83$	17.8309	1.95719	0.978596	+	0.205791i	$0.0659766\pi$
$84$	0	0
$85$	−12.4533	−1.35075
$86$	0	0
$87$	5.00000	0.536056
$88$	0	0
$89$	−1.26193	−0.133765	−0.0668824	−	0.997761i	$-0.521305\pi$
$89$	−1.26193	−0.133765	−0.0668824	+	0.997761i	$0.521305\pi$
$90$	0	0
$91$	2.66241	0.279096
$92$	0	0
$93$	−3.86651	−0.400938
$94$	0	0
$95$	−5.39543	−0.553559
$96$	0	0
$97$	−12.3248	−1.25140	−0.625698	−	0.780065i	$-0.715186\pi$
$97$	−12.3248	−1.25140	−0.625698	+	0.780065i	$0.715186\pi$
$98$	0	0
$99$	−1.86651	−0.187591
$100$	0	0
$101$	7.66241	0.762438	0.381219	−	0.924485i	$-0.375504\pi$
$101$	7.66241	0.762438	0.381219	+	0.924485i	$0.375504\pi$
$102$	0	0
$103$	−15.2492	−1.50254	−0.751272	−	0.659992i	$-0.770560\pi$
$103$	−15.2492	−1.50254	−0.751272	+	0.659992i	$0.770560\pi$
$104$	0	0
$105$	−2.52892	−0.246797
$106$	0	0
$107$	−4.47108	−0.432236	−0.216118	−	0.976367i	$-0.569340\pi$
$107$	−4.47108	−0.432236	−0.216118	+	0.976367i	$0.569340\pi$
$108$	0	0
$109$	−18.3776	−1.76026	−0.880128	−	0.474737i	$-0.842543\pi$
$109$	−18.3776	−1.76026	−0.880128	+	0.474737i	$0.842543\pi$
$110$	0	0
$111$	2.05784	0.195321
$112$	0	0
$113$	5.85374	0.550673	0.275337	−	0.961348i	$-0.411211\pi$
$113$	5.85374	0.550673	0.275337	+	0.961348i	$0.411211\pi$
$114$	0	0
$115$	2.52892	0.235823
$116$	0	0
$117$	−2.66241	−0.246140
$118$	0	0
$119$	4.92434	0.451414
$120$	0	0
$121$	−7.51615	−0.683286
$122$	0	0
$123$	7.79085	0.702477
$124$	0	0
$125$	−9.11567	−0.815330
$126$	0	0
$127$	−13.5689	−1.20405	−0.602024	−	0.798478i	$-0.705639\pi$
$127$	−13.5689	−1.20405	−0.602024	+	0.798478i	$0.705639\pi$
$128$	0	0
$129$	−5.72025	−0.503640
$130$	0	0
$131$	−18.9243	−1.65343	−0.826714	−	0.562623i	$-0.809792\pi$
$131$	−18.9243	−1.65343	−0.826714	+	0.562623i	$0.809792\pi$
$132$	0	0
$133$	2.13349	0.184997
$134$	0	0
$135$	2.52892	0.217655
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	−19.1685	−1.62585	−0.812924	−	0.582370i	$-0.802125\pi$
$139$	−19.1685	−1.62585	−0.812924	+	0.582370i	$0.802125\pi$
$140$	0	0
$141$	−5.92434	−0.498920
$142$	0	0
$143$	4.96941	0.415563
$144$	0	0
$145$	12.6446	1.05008
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	3.05784	0.250508	0.125254	−	0.992125i	$-0.460025\pi$
$149$	3.05784	0.250508	0.125254	+	0.992125i	$0.460025\pi$
$150$	0	0
$151$	8.04002	0.654287	0.327144	−	0.944975i	$-0.393914\pi$
$151$	8.04002	0.654287	0.327144	+	0.944975i	$0.393914\pi$
$152$	0	0
$153$	−4.92434	−0.398110
$154$	0	0
$155$	−9.77808	−0.785394
$156$	0	0
$157$	4.67518	0.373120	0.186560	−	0.982444i	$-0.440266\pi$
$157$	4.67518	0.373120	0.186560	+	0.982444i	$0.440266\pi$
$158$	0	0
$159$	−8.45326	−0.670387
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−13.6624	−1.07012	−0.535061	−	0.844813i	$-0.679711\pi$
$163$	−13.6624	−1.07012	−0.535061	+	0.844813i	$0.679711\pi$
$164$	0	0
$165$	−4.72025	−0.367471
$166$	0	0
$167$	11.3070	0.874962	0.437481	−	0.899228i	$-0.355871\pi$
$167$	11.3070	0.874962	0.437481	+	0.899228i	$0.355871\pi$
$168$	0	0
$169$	−5.91157	−0.454736
$170$	0	0
$171$	−2.13349	−0.163152
$172$	0	0
$173$	−18.8887	−1.43608	−0.718041	−	0.696001i	$-0.754961\pi$
$173$	−18.8887	−1.43608	−0.718041	+	0.696001i	$0.754961\pi$
$174$	0	0
$175$	−1.39543	−0.105484
$176$	0	0
$177$	−14.1863	−1.06631
$178$	0	0
$179$	−15.5868	−1.16501	−0.582504	−	0.812828i	$-0.697927\pi$
$179$	−15.5868	−1.16501	−0.582504	+	0.812828i	$0.697927\pi$
$180$	0	0
$181$	21.5740	1.60358	0.801791	−	0.597605i	$-0.203881\pi$
$181$	21.5740	1.60358	0.801791	+	0.597605i	$0.203881\pi$
$182$	0	0
$183$	−1.26193	−0.0932849
$184$	0	0
$185$	5.20410	0.382613
$186$	0	0
$187$	9.19133	0.672136
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−18.5239	−1.34034	−0.670170	−	0.742208i	$-0.733779\pi$
$191$	−18.5239	−1.34034	−0.670170	+	0.742208i	$0.733779\pi$
$192$	0	0
$193$	−4.07566	−0.293372	−0.146686	−	0.989183i	$-0.546861\pi$
$193$	−4.07566	−0.293372	−0.146686	+	0.989183i	$0.546861\pi$
$194$	0	0
$195$	−6.73302	−0.482161
$196$	0	0
$197$	17.9771	1.28082	0.640409	−	0.768034i	$-0.278765\pi$
$197$	17.9771	1.28082	0.640409	+	0.768034i	$0.278765\pi$
$198$	0	0
$199$	25.2841	1.79234	0.896172	−	0.443706i	$-0.146337\pi$
$199$	25.2841	1.79234	0.896172	+	0.443706i	$0.146337\pi$
$200$	0	0
$201$	11.2441	0.793099
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	19.7024	1.37608
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	3.98218	0.275453
$210$	0	0
$211$	9.19133	0.632757	0.316379	−	0.948633i	$-0.397533\pi$
$211$	9.19133	0.632757	0.316379	+	0.948633i	$0.397533\pi$
$212$	0	0
$213$	1.26193	0.0864663
$214$	0	0
$215$	−14.4660	−0.986575
$216$	0	0
$217$	3.86651	0.262476
$218$	0	0
$219$	14.7730	0.998269
$220$	0	0
$221$	13.1106	0.881916
$222$	0	0
$223$	9.51110	0.636910	0.318455	−	0.947938i	$-0.396836\pi$
$223$	9.51110	0.636910	0.318455	+	0.947938i	$0.396836\pi$
$224$	0	0
$225$	1.39543	0.0930284
$226$	0	0
$227$	−4.68023	−0.310638	−0.155319	−	0.987864i	$-0.549641\pi$
$227$	−4.68023	−0.310638	−0.155319	+	0.987864i	$0.549641\pi$
$228$	0	0
$229$	−4.45326	−0.294280	−0.147140	−	0.989116i	$-0.547007\pi$
$229$	−4.45326	−0.294280	−0.147140	+	0.989116i	$0.547007\pi$
$230$	0	0
$231$	1.86651	0.122807
$232$	0	0
$233$	6.72025	0.440258	0.220129	−	0.975471i	$-0.429352\pi$
$233$	6.72025	0.440258	0.220129	+	0.975471i	$0.429352\pi$
$234$	0	0
$235$	−14.9822	−0.977330
$236$	0	0
$237$	−15.9822	−1.03815
$238$	0	0
$239$	10.5689	0.683647	0.341824	−	0.939764i	$-0.388955\pi$
$239$	10.5689	0.683647	0.341824	+	0.939764i	$0.388955\pi$
$240$	0	0
$241$	2.74312	0.176700	0.0883498	−	0.996090i	$-0.471841\pi$
$241$	2.74312	0.176700	0.0883498	+	0.996090i	$0.471841\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.52892	0.161567
$246$	0	0
$247$	5.68023	0.361424
$248$	0	0
$249$	17.8309	1.12999
$250$	0	0
$251$	−27.9243	−1.76257	−0.881284	−	0.472586i	$-0.843321\pi$
$251$	−27.9243	−1.76257	−0.881284	+	0.472586i	$0.843321\pi$
$252$	0	0
$253$	−1.86651	−0.117346
$254$	0	0
$255$	−12.4533	−0.779854
$256$	0	0
$257$	22.9065	1.42887	0.714435	−	0.699702i	$-0.246684\pi$
$257$	22.9065	1.42887	0.714435	+	0.699702i	$0.246684\pi$
$258$	0	0
$259$	−2.05784	−0.127868
$260$	0	0
$261$	5.00000	0.309492
$262$	0	0
$263$	−10.5918	−0.653119	−0.326559	−	0.945177i	$-0.605889\pi$
$263$	−10.5918	−0.653119	−0.326559	+	0.945177i	$0.605889\pi$
$264$	0	0
$265$	−21.3776	−1.31322
$266$	0	0
$267$	−1.26193	−0.0772291
$268$	0	0
$269$	30.0272	1.83079	0.915397	−	0.402553i	$-0.131877\pi$
$269$	30.0272	1.83079	0.915397	+	0.402553i	$0.131877\pi$
$270$	0	0
$271$	2.80867	0.170615	0.0853073	−	0.996355i	$-0.472813\pi$
$271$	2.80867	0.170615	0.0853073	+	0.996355i	$0.472813\pi$
$272$	0	0
$273$	2.66241	0.161136
$274$	0	0
$275$	−2.60457	−0.157062
$276$	0	0
$277$	11.2619	0.676664	0.338332	−	0.941027i	$-0.390137\pi$
$277$	11.2619	0.676664	0.338332	+	0.941027i	$0.390137\pi$
$278$	0	0
$279$	−3.86651	−0.231482
$280$	0	0
$281$	−7.39038	−0.440873	−0.220436	−	0.975401i	$-0.570748\pi$
$281$	−7.39038	−0.440873	−0.220436	+	0.975401i	$0.570748\pi$
$282$	0	0
$283$	0.337590	0.0200676	0.0100338	−	0.999950i	$-0.496806\pi$
$283$	0.337590	0.0200676	0.0100338	+	0.999950i	$0.496806\pi$
$284$	0	0
$285$	−5.39543	−0.319597
$286$	0	0
$287$	−7.79085	−0.459879
$288$	0	0
$289$	7.24916	0.426421
$290$	0	0
$291$	−12.3248	−0.722494
$292$	0	0
$293$	−19.3147	−1.12838	−0.564189	−	0.825646i	$-0.690811\pi$
$293$	−19.3147	−1.12838	−0.564189	+	0.825646i	$0.690811\pi$
$294$	0	0
$295$	−35.8759	−2.08878
$296$	0	0
$297$	−1.86651	−0.108306
$298$	0	0
$299$	−2.66241	−0.153971
$300$	0	0
$301$	5.72025	0.329709
$302$	0	0
$303$	7.66241	0.440194
$304$	0	0
$305$	−3.19133	−0.182735
$306$	0	0
$307$	21.0800	1.20310	0.601550	−	0.798835i	$-0.294550\pi$
$307$	21.0800	1.20310	0.601550	+	0.798835i	$0.294550\pi$
$308$	0	0
$309$	−15.2492	−0.867495
$310$	0	0
$311$	6.72797	0.381508	0.190754	−	0.981638i	$-0.438907\pi$
$311$	6.72797	0.381508	0.190754	+	0.981638i	$0.438907\pi$
$312$	0	0
$313$	−24.8964	−1.40723	−0.703615	−	0.710582i	$-0.748432\pi$
$313$	−24.8964	−1.40723	−0.703615	+	0.710582i	$0.748432\pi$
$314$	0	0
$315$	−2.52892	−0.142488
$316$	0	0
$317$	6.26193	0.351705	0.175853	−	0.984417i	$-0.443732\pi$
$317$	6.26193	0.351705	0.175853	+	0.984417i	$0.443732\pi$
$318$	0	0
$319$	−9.33254	−0.522522
$320$	0	0
$321$	−4.47108	−0.249551
$322$	0	0
$323$	10.5060	0.584572
$324$	0	0
$325$	−3.71520	−0.206082
$326$	0	0
$327$	−18.3776	−1.01628
$328$	0	0
$329$	5.92434	0.326620
$330$	0	0
$331$	0.266984	0.0146748	0.00733738	−	0.999973i	$-0.497664\pi$
$331$	0.266984	0.0146748	0.00733738	+	0.999973i	$0.497664\pi$
$332$	0	0
$333$	2.05784	0.112769
$334$	0	0
$335$	28.4354	1.55359
$336$	0	0
$337$	−5.00505	−0.272642	−0.136321	−	0.990665i	$-0.543528\pi$
$337$	−5.00505	−0.272642	−0.136321	+	0.990665i	$0.543528\pi$
$338$	0	0
$339$	5.85374	0.317931
$340$	0	0
$341$	7.21687	0.390815
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.52892	0.136152
$346$	0	0
$347$	−8.85879	−0.475565	−0.237782	−	0.971318i	$-0.576420\pi$
$347$	−8.85879	−0.475565	−0.237782	+	0.971318i	$0.576420\pi$
$348$	0	0
$349$	−10.8615	−0.581401	−0.290700	−	0.956814i	$-0.593888\pi$
$349$	−10.8615	−0.581401	−0.290700	+	0.956814i	$0.593888\pi$
$350$	0	0
$351$	−2.66241	−0.142109
$352$	0	0
$353$	20.7075	1.10215	0.551074	−	0.834456i	$-0.314218\pi$
$353$	20.7075	1.10215	0.551074	+	0.834456i	$0.314218\pi$
$354$	0	0
$355$	3.19133	0.169378
$356$	0	0
$357$	4.92434	0.260624
$358$	0	0
$359$	30.4933	1.60937	0.804687	−	0.593700i	$-0.202333\pi$
$359$	30.4933	1.60937	0.804687	+	0.593700i	$0.202333\pi$
$360$	0	0
$361$	−14.4482	−0.760432
$362$	0	0
$363$	−7.51615	−0.394495
$364$	0	0
$365$	37.3598	1.95550
$366$	0	0
$367$	−17.4533	−0.911053	−0.455526	−	0.890222i	$-0.650549\pi$
$367$	−17.4533	−0.911053	−0.455526	+	0.890222i	$0.650549\pi$
$368$	0	0
$369$	7.79085	0.405576
$370$	0	0
$371$	8.45326	0.438871
$372$	0	0
$373$	−7.99228	−0.413825	−0.206912	−	0.978360i	$-0.566341\pi$
$373$	−7.99228	−0.413825	−0.206912	+	0.978360i	$0.566341\pi$
$374$	0	0
$375$	−9.11567	−0.470731
$376$	0	0
$377$	−13.3120	−0.685605
$378$	0	0
$379$	26.5639	1.36450	0.682248	−	0.731121i	$-0.261003\pi$
$379$	26.5639	1.36450	0.682248	+	0.731121i	$0.261003\pi$
$380$	0	0
$381$	−13.5689	−0.695158
$382$	0	0
$383$	−29.8309	−1.52429	−0.762143	−	0.647409i	$-0.775853\pi$
$383$	−29.8309	−1.52429	−0.762143	+	0.647409i	$0.775853\pi$
$384$	0	0
$385$	4.72025	0.240566
$386$	0	0
$387$	−5.72025	−0.290776
$388$	0	0
$389$	12.5417	0.635889	0.317944	−	0.948109i	$-0.397007\pi$
$389$	12.5417	0.635889	0.317944	+	0.948109i	$0.397007\pi$
$390$	0	0
$391$	−4.92434	−0.249035
$392$	0	0
$393$	−18.9243	−0.954607
$394$	0	0
$395$	−40.4176	−2.03363
$396$	0	0
$397$	−11.4405	−0.574182	−0.287091	−	0.957903i	$-0.592688\pi$
$397$	−11.4405	−0.574182	−0.287091	+	0.957903i	$0.592688\pi$
$398$	0	0
$399$	2.13349	0.106808
$400$	0	0
$401$	6.92434	0.345785	0.172893	−	0.984941i	$-0.444689\pi$
$401$	6.92434	0.345785	0.172893	+	0.984941i	$0.444689\pi$
$402$	0	0
$403$	10.2942	0.512792
$404$	0	0
$405$	2.52892	0.125663
$406$	0	0
$407$	−3.84097	−0.190390
$408$	0	0
$409$	31.1456	1.54005	0.770025	−	0.638014i	$-0.220244\pi$
$409$	31.1456	1.54005	0.770025	+	0.638014i	$0.220244\pi$
$410$	0	0
$411$	−13.0000	−0.641243
$412$	0	0
$413$	14.1863	0.698061
$414$	0	0
$415$	45.0928	2.21352
$416$	0	0
$417$	−19.1685	−0.938683
$418$	0	0
$419$	−0.987230	−0.0482293	−0.0241147	−	0.999709i	$-0.507677\pi$
$419$	−0.987230	−0.0482293	−0.0241147	+	0.999709i	$0.507677\pi$
$420$	0	0
$421$	−5.08071	−0.247618	−0.123809	−	0.992306i	$-0.539511\pi$
$421$	−5.08071	−0.247618	−0.123809	+	0.992306i	$0.539511\pi$
$422$	0	0
$423$	−5.92434	−0.288051
$424$	0	0
$425$	−6.87156	−0.333320
$426$	0	0
$427$	1.26193	0.0610693
$428$	0	0
$429$	4.96941	0.239925
$430$	0	0
$431$	38.2186	1.84092	0.920462	−	0.390832i	$-0.127813\pi$
$431$	38.2186	1.84092	0.920462	+	0.390832i	$0.127813\pi$
$432$	0	0
$433$	21.8887	1.05190	0.525952	−	0.850514i	$-0.323709\pi$
$433$	21.8887	1.05190	0.525952	+	0.850514i	$0.323709\pi$
$434$	0	0
$435$	12.6446	0.606262
$436$	0	0
$437$	−2.13349	−0.102059
$438$	0	0
$439$	1.71520	0.0818618	0.0409309	−	0.999162i	$-0.486968\pi$
$439$	1.71520	0.0818618	0.0409309	+	0.999162i	$0.486968\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−10.7152	−0.509094	−0.254547	−	0.967060i	$-0.581926\pi$
$443$	−10.7152	−0.509094	−0.254547	+	0.967060i	$0.581926\pi$
$444$	0	0
$445$	−3.19133	−0.151283
$446$	0	0
$447$	3.05784	0.144631
$448$	0	0
$449$	0.977130	0.0461136	0.0230568	−	0.999734i	$-0.492660\pi$
$449$	0.977130	0.0461136	0.0230568	+	0.999734i	$0.492660\pi$
$450$	0	0
$451$	−14.5417	−0.684741
$452$	0	0
$453$	8.04002	0.377753
$454$	0	0
$455$	6.73302	0.315649
$456$	0	0
$457$	−12.1863	−0.570050	−0.285025	−	0.958520i	$-0.592002\pi$
$457$	−12.1863	−0.570050	−0.285025	+	0.958520i	$0.592002\pi$
$458$	0	0
$459$	−4.92434	−0.229849
$460$	0	0
$461$	8.71015	0.405672	0.202836	−	0.979213i	$-0.434984\pi$
$461$	8.71015	0.405672	0.202836	+	0.979213i	$0.434984\pi$
$462$	0	0
$463$	33.3827	1.55142	0.775712	−	0.631087i	$-0.217391\pi$
$463$	33.3827	1.55142	0.775712	+	0.631087i	$0.217391\pi$
$464$	0	0
$465$	−9.77808	−0.453448
$466$	0	0
$467$	−8.46603	−0.391761	−0.195881	−	0.980628i	$-0.562757\pi$
$467$	−8.46603	−0.391761	−0.195881	+	0.980628i	$0.562757\pi$
$468$	0	0
$469$	−11.2441	−0.519205
$470$	0	0
$471$	4.67518	0.215421
$472$	0	0
$473$	10.6769	0.490924
$474$	0	0
$475$	−2.97713	−0.136600
$476$	0	0
$477$	−8.45326	−0.387048
$478$	0	0
$479$	30.5861	1.39751	0.698757	−	0.715359i	$-0.253737\pi$
$479$	30.5861	1.39751	0.698757	+	0.715359i	$0.253737\pi$
$480$	0	0
$481$	−5.47880	−0.249812
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−31.1685	−1.41529
$486$	0	0
$487$	−14.0477	−0.636564	−0.318282	−	0.947996i	$-0.603106\pi$
$487$	−14.0477	−0.636564	−0.318282	+	0.947996i	$0.603106\pi$
$488$	0	0
$489$	−13.6624	−0.617836
$490$	0	0
$491$	−35.9015	−1.62021	−0.810105	−	0.586284i	$-0.800590\pi$
$491$	−35.9015	−1.62021	−0.810105	+	0.586284i	$0.800590\pi$
$492$	0	0
$493$	−24.6217	−1.10891
$494$	0	0
$495$	−4.72025	−0.212159
$496$	0	0
$497$	−1.26193	−0.0566055
$498$	0	0
$499$	2.75589	0.123370	0.0616852	−	0.998096i	$-0.480353\pi$
$499$	2.75589	0.123370	0.0616852	+	0.998096i	$0.480353\pi$
$500$	0	0
$501$	11.3070	0.505159
$502$	0	0
$503$	−28.6745	−1.27853	−0.639267	−	0.768985i	$-0.720762\pi$
$503$	−28.6745	−1.27853	−0.639267	+	0.768985i	$0.720762\pi$
$504$	0	0
$505$	19.3776	0.862292
$506$	0	0
$507$	−5.91157	−0.262542
$508$	0	0
$509$	−2.01010	−0.0890961	−0.0445480	−	0.999007i	$-0.514185\pi$
$509$	−2.01010	−0.0890961	−0.0445480	+	0.999007i	$0.514185\pi$
$510$	0	0
$511$	−14.7730	−0.653520
$512$	0	0
$513$	−2.13349	−0.0941960
$514$	0	0
$515$	−38.5639	−1.69933
$516$	0	0
$517$	11.0578	0.486323
$518$	0	0
$519$	−18.8887	−0.829122
$520$	0	0
$521$	−3.58170	−0.156917	−0.0784587	−	0.996917i	$-0.525000\pi$
$521$	−3.58170	−0.156917	−0.0784587	+	0.996917i	$0.525000\pi$
$522$	0	0
$523$	2.53397	0.110803	0.0554013	−	0.998464i	$-0.482356\pi$
$523$	2.53397	0.110803	0.0554013	+	0.998464i	$0.482356\pi$
$524$	0	0
$525$	−1.39543	−0.0609014
$526$	0	0
$527$	19.0400	0.829396
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.1863	−0.615632
$532$	0	0
$533$	−20.7424	−0.898455
$534$	0	0
$535$	−11.3070	−0.488844
$536$	0	0
$537$	−15.5868	−0.672618
$538$	0	0
$539$	−1.86651	−0.0803962
$540$	0	0
$541$	3.28918	0.141413	0.0707064	−	0.997497i	$-0.477475\pi$
$541$	3.28918	0.141413	0.0707064	+	0.997497i	$0.477475\pi$
$542$	0	0
$543$	21.5740	0.925828
$544$	0	0
$545$	−46.4755	−1.99079
$546$	0	0
$547$	4.64021	0.198401	0.0992006	−	0.995067i	$-0.468371\pi$
$547$	4.64021	0.198401	0.0992006	+	0.995067i	$0.468371\pi$
$548$	0	0
$549$	−1.26193	−0.0538580
$550$	0	0
$551$	−10.6675	−0.454449
$552$	0	0
$553$	15.9822	0.679631
$554$	0	0
$555$	5.20410	0.220902
$556$	0	0
$557$	−11.0222	−0.467025	−0.233513	−	0.972354i	$-0.575022\pi$
$557$	−11.0222	−0.467025	−0.233513	+	0.972354i	$0.575022\pi$
$558$	0	0
$559$	15.2296	0.644145
$560$	0	0
$561$	9.19133	0.388058
$562$	0	0
$563$	−22.8837	−0.964431	−0.482216	−	0.876053i	$-0.660168\pi$
$563$	−22.8837	−0.964431	−0.482216	+	0.876053i	$0.660168\pi$
$564$	0	0
$565$	14.8036	0.622793
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	11.0979	0.465246	0.232623	−	0.972567i	$-0.425269\pi$
$569$	11.0979	0.465246	0.232623	+	0.972567i	$0.425269\pi$
$570$	0	0
$571$	39.9109	1.67022	0.835110	−	0.550084i	$-0.185404\pi$
$571$	39.9109	1.67022	0.835110	+	0.550084i	$0.185404\pi$
$572$	0	0
$573$	−18.5239	−0.773846
$574$	0	0
$575$	1.39543	0.0581933
$576$	0	0
$577$	37.1634	1.54713	0.773566	−	0.633715i	$-0.218471\pi$
$577$	37.1634	1.54713	0.773566	+	0.633715i	$0.218471\pi$
$578$	0	0
$579$	−4.07566	−0.169378
$580$	0	0
$581$	−17.8309	−0.739749
$582$	0	0
$583$	15.7781	0.653461
$584$	0	0
$585$	−6.73302	−0.278376
$586$	0	0
$587$	3.29757	0.136105	0.0680527	−	0.997682i	$-0.478321\pi$
$587$	3.29757	0.136105	0.0680527	+	0.997682i	$0.478321\pi$
$588$	0	0
$589$	8.24916	0.339901
$590$	0	0
$591$	17.9771	0.739480
$592$	0	0
$593$	−29.7374	−1.22117	−0.610584	−	0.791951i	$-0.709065\pi$
$593$	−29.7374	−1.22117	−0.610584	+	0.791951i	$0.709065\pi$
$594$	0	0
$595$	12.4533	0.510534
$596$	0	0
$597$	25.2841	1.03481
$598$	0	0
$599$	1.14626	0.0468350	0.0234175	−	0.999726i	$-0.492545\pi$
$599$	1.14626	0.0468350	0.0234175	+	0.999726i	$0.492545\pi$
$600$	0	0
$601$	−28.9516	−1.18096	−0.590480	−	0.807052i	$-0.701062\pi$
$601$	−28.9516	−1.18096	−0.590480	+	0.807052i	$0.701062\pi$
$602$	0	0
$603$	11.2441	0.457896
$604$	0	0
$605$	−19.0077	−0.772774
$606$	0	0
$607$	31.7858	1.29015	0.645073	−	0.764121i	$-0.276827\pi$
$607$	31.7858	1.29015	0.645073	+	0.764121i	$0.276827\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	15.7730	0.638109
$612$	0	0
$613$	26.0477	1.05206	0.526029	−	0.850467i	$-0.323680\pi$
$613$	26.0477	1.05206	0.526029	+	0.850467i	$0.323680\pi$
$614$	0	0
$615$	19.7024	0.794478
$616$	0	0
$617$	−24.5511	−0.988391	−0.494195	−	0.869351i	$-0.664537\pi$
$617$	−24.5511	−0.988391	−0.494195	+	0.869351i	$0.664537\pi$
$618$	0	0
$619$	8.72797	0.350807	0.175403	−	0.984497i	$-0.443877\pi$
$619$	8.72797	0.350807	0.175403	+	0.984497i	$0.443877\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	1.26193	0.0505583
$624$	0	0
$625$	−30.0299	−1.20120
$626$	0	0
$627$	3.98218	0.159033
$628$	0	0
$629$	−10.1335	−0.404049
$630$	0	0
$631$	0.621720	0.0247503	0.0123751	−	0.999923i	$-0.496061\pi$
$631$	0.621720	0.0247503	0.0123751	+	0.999923i	$0.496061\pi$
$632$	0	0
$633$	9.19133	0.365322
$634$	0	0
$635$	−34.3147	−1.36174
$636$	0	0
$637$	−2.66241	−0.105489
$638$	0	0
$639$	1.26193	0.0499213
$640$	0	0
$641$	25.0094	0.987813	0.493906	−	0.869515i	$-0.335569\pi$
$641$	25.0094	0.987813	0.493906	+	0.869515i	$0.335569\pi$
$642$	0	0
$643$	29.4933	1.16310	0.581551	−	0.813510i	$-0.302446\pi$
$643$	29.4933	1.16310	0.581551	+	0.813510i	$0.302446\pi$
$644$	0	0
$645$	−14.4660	−0.569599
$646$	0	0
$647$	31.7424	1.24792	0.623962	−	0.781455i	$-0.285522\pi$
$647$	31.7424	1.24792	0.623962	+	0.781455i	$0.285522\pi$
$648$	0	0
$649$	26.4788	1.03938
$650$	0	0
$651$	3.86651	0.151540
$652$	0	0
$653$	−13.4253	−0.525374	−0.262687	−	0.964881i	$-0.584609\pi$
$653$	−13.4253	−0.525374	−0.262687	+	0.964881i	$0.584609\pi$
$654$	0	0
$655$	−47.8581	−1.86997
$656$	0	0
$657$	14.7730	0.576351
$658$	0	0
$659$	17.3171	0.674578	0.337289	−	0.941401i	$-0.390490\pi$
$659$	17.3171	0.674578	0.337289	+	0.941401i	$0.390490\pi$
$660$	0	0
$661$	37.5740	1.46146	0.730729	−	0.682667i	$-0.239180\pi$
$661$	37.5740	1.46146	0.730729	+	0.682667i	$0.239180\pi$
$662$	0	0
$663$	13.1106	0.509174
$664$	0	0
$665$	5.39543	0.209226
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	9.51110	0.367720
$670$	0	0
$671$	2.35541	0.0909296
$672$	0	0
$673$	−36.9644	−1.42487	−0.712436	−	0.701737i	$-0.752408\pi$
$673$	−36.9644	−1.42487	−0.712436	+	0.701737i	$0.752408\pi$
$674$	0	0
$675$	1.39543	0.0537100
$676$	0	0
$677$	45.4576	1.74708	0.873539	−	0.486753i	$-0.161819\pi$
$677$	45.4576	1.74708	0.873539	+	0.486753i	$0.161819\pi$
$678$	0	0
$679$	12.3248	0.472983
$680$	0	0
$681$	−4.68023	−0.179347
$682$	0	0
$683$	−41.9566	−1.60543	−0.802713	−	0.596365i	$-0.796611\pi$
$683$	−41.9566	−1.60543	−0.802713	+	0.596365i	$0.796611\pi$
$684$	0	0
$685$	−32.8759	−1.25612
$686$	0	0
$687$	−4.45326	−0.169903
$688$	0	0
$689$	22.5060	0.857412
$690$	0	0
$691$	19.3120	0.734665	0.367332	−	0.930090i	$-0.380271\pi$
$691$	19.3120	0.734665	0.367332	+	0.930090i	$0.380271\pi$
$692$	0	0
$693$	1.86651	0.0709028
$694$	0	0
$695$	−48.4755	−1.83878
$696$	0	0
$697$	−38.3648	−1.45317
$698$	0	0
$699$	6.72025	0.254183
$700$	0	0
$701$	20.4533	0.772509	0.386255	−	0.922392i	$-0.373769\pi$
$701$	20.4533	0.772509	0.386255	+	0.922392i	$0.373769\pi$
$702$	0	0
$703$	−4.39038	−0.165586
$704$	0	0
$705$	−14.9822	−0.564262
$706$	0	0
$707$	−7.66241	−0.288175
$708$	0	0
$709$	−47.2008	−1.77266	−0.886331	−	0.463053i	$-0.846754\pi$
$709$	−47.2008	−1.77266	−0.886331	+	0.463053i	$0.846754\pi$
$710$	0	0
$711$	−15.9822	−0.599379
$712$	0	0
$713$	−3.86651	−0.144802
$714$	0	0
$715$	12.5672	0.469988
$716$	0	0
$717$	10.5689	0.394704
$718$	0	0
$719$	−3.38266	−0.126152	−0.0630759	−	0.998009i	$-0.520091\pi$
$719$	−3.38266	−0.126152	−0.0630759	+	0.998009i	$0.520091\pi$
$720$	0	0
$721$	15.2492	0.567909
$722$	0	0
$723$	2.74312	0.102018
$724$	0	0
$725$	6.97713	0.259124
$726$	0	0
$727$	−0.516148	−0.0191429	−0.00957143	−	0.999954i	$-0.503047\pi$
$727$	−0.516148	−0.0191429	−0.00957143	+	0.999954i	$0.503047\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.1685	1.04185
$732$	0	0
$733$	−45.4405	−1.67838	−0.839191	−	0.543836i	$-0.816971\pi$
$733$	−45.4405	−1.67838	−0.839191	+	0.543836i	$0.816971\pi$
$734$	0	0
$735$	2.52892	0.0932805
$736$	0	0
$737$	−20.9872	−0.773075
$738$	0	0
$739$	22.9243	0.843286	0.421643	−	0.906762i	$-0.361454\pi$
$739$	22.9243	0.843286	0.421643	+	0.906762i	$0.361454\pi$
$740$	0	0
$741$	5.68023	0.208668
$742$	0	0
$743$	15.7781	0.578842	0.289421	−	0.957202i	$-0.406537\pi$
$743$	15.7781	0.578842	0.289421	+	0.957202i	$0.406537\pi$
$744$	0	0
$745$	7.73302	0.283316
$746$	0	0
$747$	17.8309	0.652397
$748$	0	0
$749$	4.47108	0.163370
$750$	0	0
$751$	49.7680	1.81606	0.908030	−	0.418906i	$-0.137586\pi$
$751$	49.7680	1.81606	0.908030	+	0.418906i	$0.137586\pi$
$752$	0	0
$753$	−27.9243	−1.01762
$754$	0	0
$755$	20.3325	0.739977
$756$	0	0
$757$	21.5740	0.784120	0.392060	−	0.919940i	$-0.371763\pi$
$757$	21.5740	0.784120	0.392060	+	0.919940i	$0.371763\pi$
$758$	0	0
$759$	−1.86651	−0.0677500
$760$	0	0
$761$	−30.1957	−1.09459	−0.547297	−	0.836939i	$-0.684343\pi$
$761$	−30.1957	−1.09459	−0.547297	+	0.836939i	$0.684343\pi$
$762$	0	0
$763$	18.3776	0.665314
$764$	0	0
$765$	−12.4533	−0.450249
$766$	0	0
$767$	37.7697	1.36378
$768$	0	0
$769$	−15.5417	−0.560448	−0.280224	−	0.959935i	$-0.590409\pi$
$769$	−15.5417	−0.560448	−0.280224	+	0.959935i	$0.590409\pi$
$770$	0	0
$771$	22.9065	0.824958
$772$	0	0
$773$	−30.5663	−1.09939	−0.549696	−	0.835365i	$-0.685256\pi$
$773$	−30.5663	−1.09939	−0.549696	+	0.835365i	$0.685256\pi$
$774$	0	0
$775$	−5.39543	−0.193809
$776$	0	0
$777$	−2.05784	−0.0738245
$778$	0	0
$779$	−16.6217	−0.595535
$780$	0	0
$781$	−2.35541	−0.0842832
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	11.8231	0.421986
$786$	0	0
$787$	−22.9415	−0.817776	−0.408888	−	0.912585i	$-0.634083\pi$
$787$	−22.9415	−0.817776	−0.408888	+	0.912585i	$0.634083\pi$
$788$	0	0
$789$	−10.5918	−0.377078
$790$	0	0
$791$	−5.85374	−0.208135
$792$	0	0
$793$	3.35979	0.119309
$794$	0	0
$795$	−21.3776	−0.758186
$796$	0	0
$797$	−49.0878	−1.73878	−0.869389	−	0.494129i	$-0.835487\pi$
$797$	−49.0878	−1.73878	−0.869389	+	0.494129i	$0.835487\pi$
$798$	0	0
$799$	29.1735	1.03208
$800$	0	0
$801$	−1.26193	−0.0445882
$802$	0	0
$803$	−27.5740	−0.973065
$804$	0	0
$805$	−2.52892	−0.0891326
$806$	0	0
$807$	30.0272	1.05701
$808$	0	0
$809$	−28.5767	−1.00470	−0.502351	−	0.864664i	$-0.667531\pi$
$809$	−28.5767	−1.00470	−0.502351	+	0.864664i	$0.667531\pi$
$810$	0	0
$811$	−23.3369	−0.819470	−0.409735	−	0.912205i	$-0.634379\pi$
$811$	−23.3369	−0.819470	−0.409735	+	0.912205i	$0.634379\pi$
$812$	0	0
$813$	2.80867	0.0985044
$814$	0	0
$815$	−34.5511	−1.21027
$816$	0	0
$817$	12.2041	0.426967
$818$	0	0
$819$	2.66241	0.0930321
$820$	0	0
$821$	3.74312	0.130636	0.0653178	−	0.997865i	$-0.479194\pi$
$821$	3.74312	0.130636	0.0653178	+	0.997865i	$0.479194\pi$
$822$	0	0
$823$	26.6089	0.927530	0.463765	−	0.885958i	$-0.346498\pi$
$823$	26.6089	0.927530	0.463765	+	0.885958i	$0.346498\pi$
$824$	0	0
$825$	−2.60457	−0.0906796
$826$	0	0
$827$	56.6389	1.96953	0.984763	−	0.173901i	$-0.0556372\pi$
$827$	56.6389	1.96953	0.984763	+	0.173901i	$0.0556372\pi$
$828$	0	0
$829$	−33.9465	−1.17901	−0.589506	−	0.807764i	$-0.700678\pi$
$829$	−33.9465	−1.17901	−0.589506	+	0.807764i	$0.700678\pi$
$830$	0	0
$831$	11.2619	0.390672
$832$	0	0
$833$	−4.92434	−0.170618
$834$	0	0
$835$	28.5945	0.989553
$836$	0	0
$837$	−3.86651	−0.133646
$838$	0	0
$839$	7.64459	0.263921	0.131960	−	0.991255i	$-0.457873\pi$
$839$	7.64459	0.263921	0.131960	+	0.991255i	$0.457873\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−7.39038	−0.254538
$844$	0	0
$845$	−14.9499	−0.514292
$846$	0	0
$847$	7.51615	0.258258
$848$	0	0
$849$	0.337590	0.0115861
$850$	0	0
$851$	2.05784	0.0705417
$852$	0	0
$853$	49.8887	1.70816	0.854078	−	0.520144i	$-0.174122\pi$
$853$	49.8887	1.70816	0.854078	+	0.520144i	$0.174122\pi$
$854$	0	0
$855$	−5.39543	−0.184520
$856$	0	0
$857$	−27.6217	−0.943540	−0.471770	−	0.881722i	$-0.656385\pi$
$857$	−27.6217	−0.943540	−0.471770	+	0.881722i	$0.656385\pi$
$858$	0	0
$859$	−6.44821	−0.220010	−0.110005	−	0.993931i	$-0.535087\pi$
$859$	−6.44821	−0.220010	−0.110005	+	0.993931i	$0.535087\pi$
$860$	0	0
$861$	−7.79085	−0.265512
$862$	0	0
$863$	−50.7196	−1.72651	−0.863257	−	0.504764i	$-0.831579\pi$
$863$	−50.7196	−1.72651	−0.863257	+	0.504764i	$0.831579\pi$
$864$	0	0
$865$	−47.7680	−1.62416
$866$	0	0
$867$	7.24916	0.246195
$868$	0	0
$869$	29.8309	1.01194
$870$	0	0
$871$	−29.9364	−1.01436
$872$	0	0
$873$	−12.3248	−0.417132
$874$	0	0
$875$	9.11567	0.308166
$876$	0	0
$877$	−25.2993	−0.854296	−0.427148	−	0.904182i	$-0.640482\pi$
$877$	−25.2993	−0.854296	−0.427148	+	0.904182i	$0.640482\pi$
$878$	0	0
$879$	−19.3147	−0.651469
$880$	0	0
$881$	−47.7075	−1.60731	−0.803653	−	0.595098i	$-0.797113\pi$
$881$	−47.7075	−1.60731	−0.803653	+	0.595098i	$0.797113\pi$
$882$	0	0
$883$	33.0851	1.11340	0.556701	−	0.830713i	$-0.312067\pi$
$883$	33.0851	1.11340	0.556701	+	0.830713i	$0.312067\pi$
$884$	0	0
$885$	−35.8759	−1.20596
$886$	0	0
$887$	−53.7347	−1.80424	−0.902118	−	0.431490i	$-0.857988\pi$
$887$	−53.7347	−1.80424	−0.902118	+	0.431490i	$0.857988\pi$
$888$	0	0
$889$	13.5689	0.455087
$890$	0	0
$891$	−1.86651	−0.0625304
$892$	0	0
$893$	12.6395	0.422966
$894$	0	0
$895$	−39.4176	−1.31759
$896$	0	0
$897$	−2.66241	−0.0888953
$898$	0	0
$899$	−19.3325	−0.644776
$900$	0	0
$901$	41.6268	1.38679
$902$	0	0
$903$	5.72025	0.190358
$904$	0	0
$905$	54.5588	1.81360
$906$	0	0
$907$	20.1207	0.668098	0.334049	−	0.942556i	$-0.391585\pi$
$907$	20.1207	0.668098	0.334049	+	0.942556i	$0.391585\pi$
$908$	0	0
$909$	7.66241	0.254146
$910$	0	0
$911$	−34.9721	−1.15868	−0.579338	−	0.815087i	$-0.696689\pi$
$911$	−34.9721	−1.15868	−0.579338	+	0.815087i	$0.696689\pi$
$912$	0	0
$913$	−33.2815	−1.10146
$914$	0	0
$915$	−3.19133	−0.105502
$916$	0	0
$917$	18.9243	0.624937
$918$	0	0
$919$	−3.30700	−0.109088	−0.0545439	−	0.998511i	$-0.517370\pi$
$919$	−3.30700	−0.109088	−0.0545439	+	0.998511i	$0.517370\pi$
$920$	0	0
$921$	21.0800	0.694611
$922$	0	0
$923$	−3.35979	−0.110589
$924$	0	0
$925$	2.87156	0.0944162
$926$	0	0
$927$	−15.2492	−0.500848
$928$	0	0
$929$	−22.7959	−0.747909	−0.373955	−	0.927447i	$-0.621998\pi$
$929$	−22.7959	−0.747909	−0.373955	+	0.927447i	$0.621998\pi$
$930$	0	0
$931$	−2.13349	−0.0699224
$932$	0	0
$933$	6.72797	0.220264
$934$	0	0
$935$	23.2441	0.760164
$936$	0	0
$937$	27.7653	0.907053	0.453527	−	0.891243i	$-0.350166\pi$
$937$	27.7653	0.907053	0.453527	+	0.891243i	$0.350166\pi$
$938$	0	0
$939$	−24.8964	−0.812464
$940$	0	0
$941$	56.8309	1.85263	0.926317	−	0.376746i	$-0.122957\pi$
$941$	56.8309	1.85263	0.926317	+	0.376746i	$0.122957\pi$
$942$	0	0
$943$	7.79085	0.253705
$944$	0	0
$945$	−2.52892	−0.0822657
$946$	0	0
$947$	6.59952	0.214456	0.107228	−	0.994234i	$-0.465803\pi$
$947$	6.59952	0.214456	0.107228	+	0.994234i	$0.465803\pi$
$948$	0	0
$949$	−39.3319	−1.27677
$950$	0	0
$951$	6.26193	0.203057
$952$	0	0
$953$	−34.4354	−1.11547	−0.557737	−	0.830018i	$-0.688330\pi$
$953$	−34.4354	−1.11547	−0.557737	+	0.830018i	$0.688330\pi$
$954$	0	0
$955$	−46.8453	−1.51588
$956$	0	0
$957$	−9.33254	−0.301678
$958$	0	0
$959$	13.0000	0.419792
$960$	0	0
$961$	−16.0501	−0.517746
$962$	0	0
$963$	−4.47108	−0.144079
$964$	0	0
$965$	−10.3070	−0.331794
$966$	0	0
$967$	−5.03230	−0.161828	−0.0809139	−	0.996721i	$-0.525784\pi$
$967$	−5.03230	−0.161828	−0.0809139	+	0.996721i	$0.525784\pi$
$968$	0	0
$969$	10.5060	0.337503
$970$	0	0
$971$	−24.5969	−0.789351	−0.394675	−	0.918821i	$-0.629143\pi$
$971$	−24.5969	−0.789351	−0.394675	+	0.918821i	$0.629143\pi$
$972$	0	0
$973$	19.1685	0.614513
$974$	0	0
$975$	−3.71520	−0.118981
$976$	0	0
$977$	−37.4176	−1.19710	−0.598548	−	0.801087i	$-0.704255\pi$
$977$	−37.4176	−1.19710	−0.598548	+	0.801087i	$0.704255\pi$
$978$	0	0
$979$	2.35541	0.0752792
$980$	0	0
$981$	−18.3776	−0.586752
$982$	0	0
$983$	−41.6796	−1.32937	−0.664686	−	0.747123i	$-0.731435\pi$
$983$	−41.6796	−1.32937	−0.664686	+	0.747123i	$0.731435\pi$
$984$	0	0
$985$	45.4627	1.44856
$986$	0	0
$987$	5.92434	0.188574
$988$	0	0
$989$	−5.72025	−0.181893
$990$	0	0
$991$	−5.44889	−0.173090	−0.0865448	−	0.996248i	$-0.527583\pi$
$991$	−5.44889	−0.173090	−0.0865448	+	0.996248i	$0.527583\pi$
$992$	0	0
$993$	0.266984	0.00847248
$994$	0	0
$995$	63.9415	2.02708
$996$	0	0
$997$	47.2815	1.49742	0.748709	−	0.662898i	$-0.230674\pi$
$997$	47.2815	1.49742	0.748709	+	0.662898i	$0.230674\pi$
$998$	0	0
$999$	2.05784	0.0651070

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bx.1.3		3
4.3	odd	2		3864.2.a.m.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.m.1.3	✓	3	4.3	odd	2
7728.2.a.bx.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.52892 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.52892 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.86651 q^{11} -2.66241 q^{13} +2.52892 q^{15} -4.92434 q^{17} -2.13349 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.39543 q^{25} +1.00000 q^{27} +5.00000 q^{29} -3.86651 q^{31} -1.86651 q^{33} -2.52892 q^{35} +2.05784 q^{37} -2.66241 q^{39} +7.79085 q^{41} -5.72025 q^{43} +2.52892 q^{45} -5.92434 q^{47} +1.00000 q^{49} -4.92434 q^{51} -8.45326 q^{53} -4.72025 q^{55} -2.13349 q^{57} -14.1863 q^{59} -1.26193 q^{61} -1.00000 q^{63} -6.73302 q^{65} +11.2441 q^{67} +1.00000 q^{69} +1.26193 q^{71} +14.7730 q^{73} +1.39543 q^{75} +1.86651 q^{77} -15.9822 q^{79} +1.00000 q^{81} +17.8309 q^{83} -12.4533 q^{85} +5.00000 q^{87} -1.26193 q^{89} +2.66241 q^{91} -3.86651 q^{93} -5.39543 q^{95} -12.3248 q^{97} -1.86651 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 - 6 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 + 3 * q^27 + 15 * q^29 - 12 * q^31 - 6 * q^33 - 9 * q^37 + 9 * q^41 + 6 * q^43 - 3 * q^47 + 3 * q^49 - 3 * q^53 + 9 * q^55 - 6 * q^57 - 21 * q^59 + 3 * q^61 - 3 * q^63 - 21 * q^65 - 3 * q^67 + 3 * q^69 - 3 * q^71 - 3 * q^75 + 6 * q^77 - 18 * q^79 + 3 * q^81 - 6 * q^83 - 15 * q^85 + 15 * q^87 + 3 * q^89 - 12 * q^93 - 9 * q^95 - 21 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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