LMFDB - Embedded newform 7728.2.a.bp.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*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			1.2
Root			$-0.254102$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.25410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.25410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.87086 q^{11} -1.06457 q^{13} +1.25410 q^{15} -7.04399 q^{17} +6.36266 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.42723 q^{25} -1.00000 q^{27} -4.18953 q^{29} +2.69774 q^{31} +4.87086 q^{33} +1.25410 q^{35} -0.826873 q^{37} +1.06457 q^{39} -8.36266 q^{41} +0.556364 q^{43} -1.25410 q^{45} -3.69774 q^{47} +1.00000 q^{49} +7.04399 q^{51} -4.44364 q^{53} +6.10856 q^{55} -6.36266 q^{57} -6.95184 q^{59} -1.87503 q^{61} -1.00000 q^{63} +1.33508 q^{65} -6.48763 q^{67} -1.00000 q^{69} -5.31450 q^{71} -5.17313 q^{73} +3.42723 q^{75} +4.87086 q^{77} +2.69774 q^{79} +1.00000 q^{81} +9.20594 q^{83} +8.83388 q^{85} +4.18953 q^{87} +3.78989 q^{89} +1.06457 q^{91} -2.69774 q^{93} -7.97942 q^{95} -0.351483 q^{97} -4.87086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + q^{11} - 11 q^{13} + 3 q^{15} + 5 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} - 8 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 3 q^{45} - q^{47} + 3 q^{49} - 4 q^{53} + 5 q^{55} - 5 q^{57} - 10 q^{59} - 22 q^{61} - 3 q^{63} + 8 q^{65} + 11 q^{67} - 3 q^{69} + 9 q^{71} - 10 q^{73} + 4 q^{75} - q^{77} - 2 q^{79} + 3 q^{81} + 16 q^{83} - 15 q^{85} + 4 q^{87} - 9 q^{89} + 11 q^{91} + 2 q^{93} + 5 q^{95} - 2 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + q^11 - 11 * q^13 + 3 * q^15 + 5 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 - q^33 + 3 * q^35 - 8 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 3 * q^45 - q^47 + 3 * q^49 - 4 * q^53 + 5 * q^55 - 5 * q^57 - 10 * q^59 - 22 * q^61 - 3 * q^63 + 8 * q^65 + 11 * q^67 - 3 * q^69 + 9 * q^71 - 10 * q^73 + 4 * q^75 - q^77 - 2 * q^79 + 3 * q^81 + 16 * q^83 - 15 * q^85 + 4 * q^87 - 9 * q^89 + 11 * q^91 + 2 * q^93 + 5 * q^95 - 2 * q^97 + 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
$5$	−1.25410	−0.560851	−0.280426	−	0.959876i	$-0.590476\pi$
$5$	−1.25410	−0.560851	−0.280426	+	0.959876i	$0.590476\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.87086	−1.46862	−0.734310	−	0.678814i	$-0.762494\pi$
$11$	−4.87086	−1.46862	−0.734310	+	0.678814i	$0.762494\pi$
$12$	0	0
$13$	−1.06457	−0.295258	−0.147629	−	0.989043i	$-0.547164\pi$
$13$	−1.06457	−0.295258	−0.147629	+	0.989043i	$0.547164\pi$
$14$	0	0
$15$	1.25410	0.323808
$16$	0	0
$17$	−7.04399	−1.70842	−0.854209	−	0.519929i	$-0.825958\pi$
$17$	−7.04399	−1.70842	−0.854209	+	0.519929i	$0.825958\pi$
$18$	0	0
$19$	6.36266	1.45969	0.729847	−	0.683610i	$-0.239591\pi$
$19$	6.36266	1.45969	0.729847	+	0.683610i	$0.239591\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.42723	−0.685446
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.18953	−0.777977	−0.388988	−	0.921243i	$-0.627175\pi$
$29$	−4.18953	−0.777977	−0.388988	+	0.921243i	$0.627175\pi$
$30$	0	0
$31$	2.69774	0.484528	0.242264	−	0.970210i	$-0.422110\pi$
$31$	2.69774	0.484528	0.242264	+	0.970210i	$0.422110\pi$
$32$	0	0
$33$	4.87086	0.847909
$34$	0	0
$35$	1.25410	0.211982
$36$	0	0
$37$	−0.826873	−0.135937	−0.0679685	−	0.997687i	$-0.521652\pi$
$37$	−0.826873	−0.135937	−0.0679685	+	0.997687i	$0.521652\pi$
$38$	0	0
$39$	1.06457	0.170467
$40$	0	0
$41$	−8.36266	−1.30603	−0.653014	−	0.757346i	$-0.726496\pi$
$41$	−8.36266	−1.30603	−0.653014	+	0.757346i	$0.726496\pi$
$42$	0	0
$43$	0.556364	0.0848448	0.0424224	−	0.999100i	$-0.486492\pi$
$43$	0.556364	0.0848448	0.0424224	+	0.999100i	$0.486492\pi$
$44$	0	0
$45$	−1.25410	−0.186950
$46$	0	0
$47$	−3.69774	−0.539370	−0.269685	−	0.962949i	$-0.586920\pi$
$47$	−3.69774	−0.539370	−0.269685	+	0.962949i	$0.586920\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.04399	0.986356
$52$	0	0
$53$	−4.44364	−0.610380	−0.305190	−	0.952291i	$-0.598720\pi$
$53$	−4.44364	−0.610380	−0.305190	+	0.952291i	$0.598720\pi$
$54$	0	0
$55$	6.10856	0.823678
$56$	0	0
$57$	−6.36266	−0.842755
$58$	0	0
$59$	−6.95184	−0.905052	−0.452526	−	0.891751i	$-0.649477\pi$
$59$	−6.95184	−0.905052	−0.452526	+	0.891751i	$0.649477\pi$
$60$	0	0
$61$	−1.87503	−0.240073	−0.120037	−	0.992769i	$-0.538301\pi$
$61$	−1.87503	−0.240073	−0.120037	+	0.992769i	$0.538301\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.33508	0.165596
$66$	0	0
$67$	−6.48763	−0.792590	−0.396295	−	0.918123i	$-0.629704\pi$
$67$	−6.48763	−0.792590	−0.396295	+	0.918123i	$0.629704\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−5.31450	−0.630715	−0.315358	−	0.948973i	$-0.602124\pi$
$71$	−5.31450	−0.630715	−0.315358	+	0.948973i	$0.602124\pi$
$72$	0	0
$73$	−5.17313	−0.605469	−0.302734	−	0.953075i	$-0.597900\pi$
$73$	−5.17313	−0.605469	−0.302734	+	0.953075i	$0.597900\pi$
$74$	0	0
$75$	3.42723	0.395742
$76$	0	0
$77$	4.87086	0.555087
$78$	0	0
$79$	2.69774	0.303519	0.151760	−	0.988417i	$-0.451506\pi$
$79$	2.69774	0.303519	0.151760	+	0.988417i	$0.451506\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.20594	1.01048	0.505242	−	0.862978i	$-0.331403\pi$
$83$	9.20594	1.01048	0.505242	+	0.862978i	$0.331403\pi$
$84$	0	0
$85$	8.83388	0.958169
$86$	0	0
$87$	4.18953	0.449165
$88$	0	0
$89$	3.78989	0.401728	0.200864	−	0.979619i	$-0.435625\pi$
$89$	3.78989	0.401728	0.200864	+	0.979619i	$0.435625\pi$
$90$	0	0
$91$	1.06457	0.111597
$92$	0	0
$93$	−2.69774	−0.279742
$94$	0	0
$95$	−7.97942	−0.818672
$96$	0	0
$97$	−0.351483	−0.0356877	−0.0178438	−	0.999841i	$-0.505680\pi$
$97$	−0.351483	−0.0356877	−0.0178438	+	0.999841i	$0.505680\pi$
$98$	0	0
$99$	−4.87086	−0.489540
$100$	0	0
$101$	−15.3309	−1.52548	−0.762741	−	0.646704i	$-0.776147\pi$
$101$	−15.3309	−1.52548	−0.762741	+	0.646704i	$0.776147\pi$
$102$	0	0
$103$	6.17313	0.608256	0.304128	−	0.952631i	$-0.401635\pi$
$103$	6.17313	0.608256	0.304128	+	0.952631i	$0.401635\pi$
$104$	0	0
$105$	−1.25410	−0.122388
$106$	0	0
$107$	−9.95184	−0.962081	−0.481040	−	0.876698i	$-0.659741\pi$
$107$	−9.95184	−0.962081	−0.481040	+	0.876698i	$0.659741\pi$
$108$	0	0
$109$	4.14137	0.396672	0.198336	−	0.980134i	$-0.436446\pi$
$109$	4.14137	0.396672	0.198336	+	0.980134i	$0.436446\pi$
$110$	0	0
$111$	0.826873	0.0784833
$112$	0	0
$113$	16.9630	1.59575	0.797873	−	0.602825i	$-0.205958\pi$
$113$	16.9630	1.59575	0.797873	+	0.602825i	$0.205958\pi$
$114$	0	0
$115$	−1.25410	−0.116946
$116$	0	0
$117$	−1.06457	−0.0984193
$118$	0	0
$119$	7.04399	0.645722
$120$	0	0
$121$	12.7253	1.15685
$122$	0	0
$123$	8.36266	0.754036
$124$	0	0
$125$	10.5686	0.945285
$126$	0	0
$127$	8.12497	0.720974	0.360487	−	0.932764i	$-0.382610\pi$
$127$	8.12497	0.720974	0.360487	+	0.932764i	$0.382610\pi$
$128$	0	0
$129$	−0.556364	−0.0489851
$130$	0	0
$131$	−6.74173	−0.589028	−0.294514	−	0.955647i	$-0.595158\pi$
$131$	−6.74173	−0.589028	−0.294514	+	0.955647i	$0.595158\pi$
$132$	0	0
$133$	−6.36266	−0.551713
$134$	0	0
$135$	1.25410	0.107936
$136$	0	0
$137$	−5.59619	−0.478115	−0.239057	−	0.971005i	$-0.576838\pi$
$137$	−5.59619	−0.478115	−0.239057	+	0.971005i	$0.576838\pi$
$138$	0	0
$139$	4.77871	0.405325	0.202663	−	0.979249i	$-0.435041\pi$
$139$	4.77871	0.405325	0.202663	+	0.979249i	$0.435041\pi$
$140$	0	0
$141$	3.69774	0.311406
$142$	0	0
$143$	5.18537	0.433622
$144$	0	0
$145$	5.25410	0.436329
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	1.31867	0.108030	0.0540148	−	0.998540i	$-0.482798\pi$
$149$	1.31867	0.108030	0.0540148	+	0.998540i	$0.482798\pi$
$150$	0	0
$151$	18.0768	1.47107	0.735535	−	0.677487i	$-0.236931\pi$
$151$	18.0768	1.47107	0.735535	+	0.677487i	$0.236931\pi$
$152$	0	0
$153$	−7.04399	−0.569473
$154$	0	0
$155$	−3.38324	−0.271748
$156$	0	0
$157$	−23.4067	−1.86805	−0.934027	−	0.357202i	$-0.883731\pi$
$157$	−23.4067	−1.86805	−0.934027	+	0.357202i	$0.883731\pi$
$158$	0	0
$159$	4.44364	0.352403
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	0.0645677	0.00505733	0.00252867	−	0.999997i	$-0.499195\pi$
$163$	0.0645677	0.00505733	0.00252867	+	0.999997i	$0.499195\pi$
$164$	0	0
$165$	−6.10856	−0.475551
$166$	0	0
$167$	25.1208	1.94391	0.971953	−	0.235176i	$-0.0755666\pi$
$167$	25.1208	1.94391	0.971953	+	0.235176i	$0.0755666\pi$
$168$	0	0
$169$	−11.8667	−0.912823
$170$	0	0
$171$	6.36266	0.486565
$172$	0	0
$173$	18.2775	1.38961	0.694807	−	0.719196i	$-0.255490\pi$
$173$	18.2775	1.38961	0.694807	+	0.719196i	$0.255490\pi$
$174$	0	0
$175$	3.42723	0.259074
$176$	0	0
$177$	6.95184	0.522532
$178$	0	0
$179$	−8.45481	−0.631942	−0.315971	−	0.948769i	$-0.602330\pi$
$179$	−8.45481	−0.631942	−0.315971	+	0.948769i	$0.602330\pi$
$180$	0	0
$181$	−19.0768	−1.41797	−0.708984	−	0.705225i	$-0.750846\pi$
$181$	−19.0768	−1.41797	−0.708984	+	0.705225i	$0.750846\pi$
$182$	0	0
$183$	1.87503	0.138606
$184$	0	0
$185$	1.03698	0.0762405
$186$	0	0
$187$	34.3103	2.50902
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−3.35148	−0.242505	−0.121252	−	0.992622i	$-0.538691\pi$
$191$	−3.35148	−0.242505	−0.121252	+	0.992622i	$0.538691\pi$
$192$	0	0
$193$	12.4231	0.894231	0.447116	−	0.894476i	$-0.352451\pi$
$193$	12.4231	0.894231	0.447116	+	0.894476i	$0.352451\pi$
$194$	0	0
$195$	−1.33508	−0.0956068
$196$	0	0
$197$	−13.0234	−0.927880	−0.463940	−	0.885867i	$-0.653565\pi$
$197$	−13.0234	−0.927880	−0.463940	+	0.885867i	$0.653565\pi$
$198$	0	0
$199$	−7.80630	−0.553374	−0.276687	−	0.960960i	$-0.589236\pi$
$199$	−7.80630	−0.553374	−0.276687	+	0.960960i	$0.589236\pi$
$200$	0	0
$201$	6.48763	0.457602
$202$	0	0
$203$	4.18953	0.294048
$204$	0	0
$205$	10.4876	0.732488
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−30.9917	−2.14374
$210$	0	0
$211$	21.8545	1.50452	0.752261	−	0.658865i	$-0.228963\pi$
$211$	21.8545	1.50452	0.752261	+	0.658865i	$0.228963\pi$
$212$	0	0
$213$	5.31450	0.364144
$214$	0	0
$215$	−0.697737	−0.0475853
$216$	0	0
$217$	−2.69774	−0.183134
$218$	0	0
$219$	5.17313	0.349568
$220$	0	0
$221$	7.49881	0.504424
$222$	0	0
$223$	−6.35849	−0.425796	−0.212898	−	0.977074i	$-0.568290\pi$
$223$	−6.35849	−0.425796	−0.212898	+	0.977074i	$0.568290\pi$
$224$	0	0
$225$	−3.42723	−0.228482
$226$	0	0
$227$	10.9958	0.729819	0.364909	−	0.931043i	$-0.381100\pi$
$227$	10.9958	0.729819	0.364909	+	0.931043i	$0.381100\pi$
$228$	0	0
$229$	−27.7651	−1.83477	−0.917386	−	0.397998i	$-0.869705\pi$
$229$	−27.7651	−1.83477	−0.917386	+	0.397998i	$0.869705\pi$
$230$	0	0
$231$	−4.87086	−0.320479
$232$	0	0
$233$	19.1250	1.25292	0.626459	−	0.779454i	$-0.284504\pi$
$233$	19.1250	1.25292	0.626459	+	0.779454i	$0.284504\pi$
$234$	0	0
$235$	4.63734	0.302507
$236$	0	0
$237$	−2.69774	−0.175237
$238$	0	0
$239$	7.81748	0.505670	0.252835	−	0.967509i	$-0.418637\pi$
$239$	7.81748	0.505670	0.252835	+	0.967509i	$0.418637\pi$
$240$	0	0
$241$	10.8709	0.700254	0.350127	−	0.936702i	$-0.386138\pi$
$241$	10.8709	0.700254	0.350127	+	0.936702i	$0.386138\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.25410	−0.0801216
$246$	0	0
$247$	−6.77348	−0.430986
$248$	0	0
$249$	−9.20594	−0.583403
$250$	0	0
$251$	−27.3543	−1.72659	−0.863295	−	0.504700i	$-0.831603\pi$
$251$	−27.3543	−1.72659	−0.863295	+	0.504700i	$0.831603\pi$
$252$	0	0
$253$	−4.87086	−0.306229
$254$	0	0
$255$	−8.83388	−0.553199
$256$	0	0
$257$	8.39025	0.523369	0.261685	−	0.965153i	$-0.415722\pi$
$257$	8.39025	0.523369	0.261685	+	0.965153i	$0.415722\pi$
$258$	0	0
$259$	0.826873	0.0513794
$260$	0	0
$261$	−4.18953	−0.259326
$262$	0	0
$263$	21.7581	1.34166	0.670832	−	0.741609i	$-0.265937\pi$
$263$	21.7581	1.34166	0.670832	+	0.741609i	$0.265937\pi$
$264$	0	0
$265$	5.57277	0.342333
$266$	0	0
$267$	−3.78989	−0.231938
$268$	0	0
$269$	20.7651	1.26607	0.633037	−	0.774122i	$-0.281808\pi$
$269$	20.7651	1.26607	0.633037	+	0.774122i	$0.281808\pi$
$270$	0	0
$271$	15.8984	0.965762	0.482881	−	0.875686i	$-0.339590\pi$
$271$	15.8984	0.965762	0.482881	+	0.875686i	$0.339590\pi$
$272$	0	0
$273$	−1.06457	−0.0644306
$274$	0	0
$275$	16.6936	1.00666
$276$	0	0
$277$	−12.7212	−0.764340	−0.382170	−	0.924092i	$-0.624823\pi$
$277$	−12.7212	−0.764340	−0.382170	+	0.924092i	$0.624823\pi$
$278$	0	0
$279$	2.69774	0.161509
$280$	0	0
$281$	−17.3215	−1.03331	−0.516657	−	0.856192i	$-0.672824\pi$
$281$	−17.3215	−1.03331	−0.516657	+	0.856192i	$0.672824\pi$
$282$	0	0
$283$	0.0205757	0.00122310	0.000611550	−	1.00000i	$-0.499805\pi$
$283$	0.0205757	0.00122310	0.000611550		1.00000i	$0.499805\pi$
$284$	0	0
$285$	7.97942	0.472660
$286$	0	0
$287$	8.36266	0.493632
$288$	0	0
$289$	32.6178	1.91870
$290$	0	0
$291$	0.351483	0.0206043
$292$	0	0
$293$	30.2088	1.76482	0.882408	−	0.470485i	$-0.155921\pi$
$293$	30.2088	1.76482	0.882408	+	0.470485i	$0.155921\pi$
$294$	0	0
$295$	8.71831	0.507600
$296$	0	0
$297$	4.87086	0.282636
$298$	0	0
$299$	−1.06457	−0.0615655
$300$	0	0
$301$	−0.556364	−0.0320683
$302$	0	0
$303$	15.3309	0.880738
$304$	0	0
$305$	2.35148	0.134646
$306$	0	0
$307$	−15.0440	−0.858606	−0.429303	−	0.903161i	$-0.641241\pi$
$307$	−15.0440	−0.858606	−0.429303	+	0.903161i	$0.641241\pi$
$308$	0	0
$309$	−6.17313	−0.351177
$310$	0	0
$311$	−18.3749	−1.04194	−0.520972	−	0.853573i	$-0.674431\pi$
$311$	−18.3749	−1.04194	−0.520972	+	0.853573i	$0.674431\pi$
$312$	0	0
$313$	23.5358	1.33032	0.665161	−	0.746700i	$-0.268363\pi$
$313$	23.5358	1.33032	0.665161	+	0.746700i	$0.268363\pi$
$314$	0	0
$315$	1.25410	0.0706606
$316$	0	0
$317$	−11.2130	−0.629782	−0.314891	−	0.949128i	$-0.601968\pi$
$317$	−11.2130	−0.629782	−0.314891	+	0.949128i	$0.601968\pi$
$318$	0	0
$319$	20.4067	1.14255
$320$	0	0
$321$	9.95184	0.555457
$322$	0	0
$323$	−44.8185	−2.49377
$324$	0	0
$325$	3.64852	0.202383
$326$	0	0
$327$	−4.14137	−0.229018
$328$	0	0
$329$	3.69774	0.203863
$330$	0	0
$331$	−8.93126	−0.490907	−0.245453	−	0.969408i	$-0.578937\pi$
$331$	−8.93126	−0.490907	−0.245453	+	0.969408i	$0.578937\pi$
$332$	0	0
$333$	−0.826873	−0.0453123
$334$	0	0
$335$	8.13614	0.444525
$336$	0	0
$337$	7.31450	0.398446	0.199223	−	0.979954i	$-0.436158\pi$
$337$	7.31450	0.398446	0.199223	+	0.979954i	$0.436158\pi$
$338$	0	0
$339$	−16.9630	−0.921305
$340$	0	0
$341$	−13.1403	−0.711588
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.25410	0.0675186
$346$	0	0
$347$	13.9396	0.748317	0.374159	−	0.927365i	$-0.377932\pi$
$347$	13.9396	0.748317	0.374159	+	0.927365i	$0.377932\pi$
$348$	0	0
$349$	−13.8175	−0.739633	−0.369816	−	0.929105i	$-0.620579\pi$
$349$	−13.8175	−0.739633	−0.369816	+	0.929105i	$0.620579\pi$
$350$	0	0
$351$	1.06457	0.0568224
$352$	0	0
$353$	11.8656	0.631544	0.315772	−	0.948835i	$-0.397737\pi$
$353$	11.8656	0.631544	0.315772	+	0.948835i	$0.397737\pi$
$354$	0	0
$355$	6.66492	0.353737
$356$	0	0
$357$	−7.04399	−0.372808
$358$	0	0
$359$	−14.9630	−0.789718	−0.394859	−	0.918742i	$-0.629207\pi$
$359$	−14.9630	−0.789718	−0.394859	+	0.918742i	$0.629207\pi$
$360$	0	0
$361$	21.4835	1.13071
$362$	0	0
$363$	−12.7253	−0.667906
$364$	0	0
$365$	6.48763	0.339578
$366$	0	0
$367$	−10.2130	−0.533112	−0.266556	−	0.963819i	$-0.585886\pi$
$367$	−10.2130	−0.533112	−0.266556	+	0.963819i	$0.585886\pi$
$368$	0	0
$369$	−8.36266	−0.435343
$370$	0	0
$371$	4.44364	0.230702
$372$	0	0
$373$	24.9477	1.29174	0.645871	−	0.763447i	$-0.276495\pi$
$373$	24.9477	1.29174	0.645871	+	0.763447i	$0.276495\pi$
$374$	0	0
$375$	−10.5686	−0.545760
$376$	0	0
$377$	4.46004	0.229704
$378$	0	0
$379$	13.1044	0.673127	0.336564	−	0.941661i	$-0.390735\pi$
$379$	13.1044	0.673127	0.336564	+	0.941661i	$0.390735\pi$
$380$	0	0
$381$	−8.12497	−0.416255
$382$	0	0
$383$	−16.8820	−0.862632	−0.431316	−	0.902201i	$-0.641951\pi$
$383$	−16.8820	−0.862632	−0.431316	+	0.902201i	$0.641951\pi$
$384$	0	0
$385$	−6.10856	−0.311321
$386$	0	0
$387$	0.556364	0.0282816
$388$	0	0
$389$	2.85969	0.144992	0.0724959	−	0.997369i	$-0.476904\pi$
$389$	2.85969	0.144992	0.0724959	+	0.997369i	$0.476904\pi$
$390$	0	0
$391$	−7.04399	−0.356230
$392$	0	0
$393$	6.74173	0.340075
$394$	0	0
$395$	−3.38324	−0.170229
$396$	0	0
$397$	−19.7334	−0.990391	−0.495195	−	0.868782i	$-0.664903\pi$
$397$	−19.7334	−0.990391	−0.495195	+	0.868782i	$0.664903\pi$
$398$	0	0
$399$	6.36266	0.318531
$400$	0	0
$401$	28.4178	1.41912	0.709559	−	0.704646i	$-0.248894\pi$
$401$	28.4178	1.41912	0.709559	+	0.704646i	$0.248894\pi$
$402$	0	0
$403$	−2.87192	−0.143061
$404$	0	0
$405$	−1.25410	−0.0623168
$406$	0	0
$407$	4.02759	0.199640
$408$	0	0
$409$	−14.0604	−0.695242	−0.347621	−	0.937635i	$-0.613010\pi$
$409$	−14.0604	−0.695242	−0.347621	+	0.937635i	$0.613010\pi$
$410$	0	0
$411$	5.59619	0.276040
$412$	0	0
$413$	6.95184	0.342078
$414$	0	0
$415$	−11.5452	−0.566731
$416$	0	0
$417$	−4.77871	−0.234015
$418$	0	0
$419$	1.50403	0.0734769	0.0367384	−	0.999325i	$-0.488303\pi$
$419$	1.50403	0.0734769	0.0367384	+	0.999325i	$0.488303\pi$
$420$	0	0
$421$	−26.5069	−1.29187	−0.645933	−	0.763394i	$-0.723531\pi$
$421$	−26.5069	−1.29187	−0.645933	+	0.763394i	$0.723531\pi$
$422$	0	0
$423$	−3.69774	−0.179790
$424$	0	0
$425$	24.1414	1.17103
$426$	0	0
$427$	1.87503	0.0907392
$428$	0	0
$429$	−5.18537	−0.250352
$430$	0	0
$431$	21.8995	1.05486	0.527431	−	0.849598i	$-0.323155\pi$
$431$	21.8995	1.05486	0.527431	+	0.849598i	$0.323155\pi$
$432$	0	0
$433$	6.64852	0.319507	0.159754	−	0.987157i	$-0.448930\pi$
$433$	6.64852	0.319507	0.159754	+	0.987157i	$0.448930\pi$
$434$	0	0
$435$	−5.25410	−0.251915
$436$	0	0
$437$	6.36266	0.304367
$438$	0	0
$439$	1.33508	0.0637197	0.0318599	−	0.999492i	$-0.489857\pi$
$439$	1.33508	0.0637197	0.0318599	+	0.999492i	$0.489857\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	16.7581	0.796203	0.398102	−	0.917341i	$-0.369669\pi$
$443$	16.7581	0.796203	0.398102	+	0.917341i	$0.369669\pi$
$444$	0	0
$445$	−4.75291	−0.225309
$446$	0	0
$447$	−1.31867	−0.0623709
$448$	0	0
$449$	24.7047	1.16589	0.582945	−	0.812512i	$-0.301900\pi$
$449$	24.7047	1.16589	0.582945	+	0.812512i	$0.301900\pi$
$450$	0	0
$451$	40.7334	1.91806
$452$	0	0
$453$	−18.0768	−0.849322
$454$	0	0
$455$	−1.33508	−0.0625893
$456$	0	0
$457$	27.0615	1.26588	0.632941	−	0.774200i	$-0.281848\pi$
$457$	27.0615	1.26588	0.632941	+	0.774200i	$0.281848\pi$
$458$	0	0
$459$	7.04399	0.328785
$460$	0	0
$461$	15.9466	0.742708	0.371354	−	0.928491i	$-0.378894\pi$
$461$	15.9466	0.742708	0.371354	+	0.928491i	$0.378894\pi$
$462$	0	0
$463$	9.25827	0.430268	0.215134	−	0.976585i	$-0.430981\pi$
$463$	9.25827	0.430268	0.215134	+	0.976585i	$0.430981\pi$
$464$	0	0
$465$	3.38324	0.156894
$466$	0	0
$467$	28.0521	1.29809	0.649047	−	0.760748i	$-0.275168\pi$
$467$	28.0521	1.29809	0.649047	+	0.760748i	$0.275168\pi$
$468$	0	0
$469$	6.48763	0.299571
$470$	0	0
$471$	23.4067	1.07852
$472$	0	0
$473$	−2.70998	−0.124605
$474$	0	0
$475$	−21.8063	−1.00054
$476$	0	0
$477$	−4.44364	−0.203460
$478$	0	0
$479$	1.26945	0.0580026	0.0290013	−	0.999579i	$-0.490767\pi$
$479$	1.26945	0.0580026	0.0290013	+	0.999579i	$0.490767\pi$
$480$	0	0
$481$	0.880262	0.0401365
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	0.440795	0.0200155
$486$	0	0
$487$	3.14031	0.142301	0.0711506	−	0.997466i	$-0.477333\pi$
$487$	3.14031	0.142301	0.0711506	+	0.997466i	$0.477333\pi$
$488$	0	0
$489$	−0.0645677	−0.00291985
$490$	0	0
$491$	17.9435	0.809779	0.404889	−	0.914366i	$-0.367310\pi$
$491$	17.9435	0.809779	0.404889	+	0.914366i	$0.367310\pi$
$492$	0	0
$493$	29.5110	1.32911
$494$	0	0
$495$	6.10856	0.274559
$496$	0	0
$497$	5.31450	0.238388
$498$	0	0
$499$	0.303322	0.0135786	0.00678928	−	0.999977i	$-0.497839\pi$
$499$	0.303322	0.0135786	0.00678928	+	0.999977i	$0.497839\pi$
$500$	0	0
$501$	−25.1208	−1.12231
$502$	0	0
$503$	5.30927	0.236729	0.118364	−	0.992970i	$-0.462235\pi$
$503$	5.30927	0.236729	0.118364	+	0.992970i	$0.462235\pi$
$504$	0	0
$505$	19.2265	0.855569
$506$	0	0
$507$	11.8667	0.527018
$508$	0	0
$509$	−12.1096	−0.536749	−0.268375	−	0.963315i	$-0.586487\pi$
$509$	−12.1096	−0.536749	−0.268375	+	0.963315i	$0.586487\pi$
$510$	0	0
$511$	5.17313	0.228846
$512$	0	0
$513$	−6.36266	−0.280918
$514$	0	0
$515$	−7.74173	−0.341141
$516$	0	0
$517$	18.0112	0.792131
$518$	0	0
$519$	−18.2775	−0.802294
$520$	0	0
$521$	−17.7417	−0.777279	−0.388640	−	0.921390i	$-0.627055\pi$
$521$	−17.7417	−0.777279	−0.388640	+	0.921390i	$0.627055\pi$
$522$	0	0
$523$	−6.87609	−0.300671	−0.150335	−	0.988635i	$-0.548035\pi$
$523$	−6.87609	−0.300671	−0.150335	+	0.988635i	$0.548035\pi$
$524$	0	0
$525$	−3.42723	−0.149577
$526$	0	0
$527$	−19.0028	−0.827777
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.95184	−0.301684
$532$	0	0
$533$	8.90262	0.385615
$534$	0	0
$535$	12.4806	0.539584
$536$	0	0
$537$	8.45481	0.364852
$538$	0	0
$539$	−4.87086	−0.209803
$540$	0	0
$541$	−1.06040	−0.0455901	−0.0227951	−	0.999740i	$-0.507257\pi$
$541$	−1.06040	−0.0455901	−0.0227951	+	0.999740i	$0.507257\pi$
$542$	0	0
$543$	19.0768	0.818664
$544$	0	0
$545$	−5.19370	−0.222474
$546$	0	0
$547$	−18.9875	−0.811847	−0.405923	−	0.913907i	$-0.633050\pi$
$547$	−18.9875	−0.811847	−0.405923	+	0.913907i	$0.633050\pi$
$548$	0	0
$549$	−1.87503	−0.0800245
$550$	0	0
$551$	−26.6566	−1.13561
$552$	0	0
$553$	−2.69774	−0.114719
$554$	0	0
$555$	−1.03698	−0.0440175
$556$	0	0
$557$	−24.3051	−1.02984	−0.514920	−	0.857238i	$-0.672178\pi$
$557$	−24.3051	−1.02984	−0.514920	+	0.857238i	$0.672178\pi$
$558$	0	0
$559$	−0.592287	−0.0250511
$560$	0	0
$561$	−34.3103	−1.44858
$562$	0	0
$563$	−17.9107	−0.754845	−0.377423	−	0.926041i	$-0.623190\pi$
$563$	−17.9107	−0.754845	−0.377423	+	0.926041i	$0.623190\pi$
$564$	0	0
$565$	−21.2733	−0.894977
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	28.4642	1.19328	0.596641	−	0.802508i	$-0.296502\pi$
$569$	28.4642	1.19328	0.596641	+	0.802508i	$0.296502\pi$
$570$	0	0
$571$	33.7141	1.41089	0.705447	−	0.708763i	$-0.250747\pi$
$571$	33.7141	1.41089	0.705447	+	0.708763i	$0.250747\pi$
$572$	0	0
$573$	3.35148	0.140010
$574$	0	0
$575$	−3.42723	−0.142925
$576$	0	0
$577$	24.7253	1.02933	0.514664	−	0.857392i	$-0.327917\pi$
$577$	24.7253	1.02933	0.514664	+	0.857392i	$0.327917\pi$
$578$	0	0
$579$	−12.4231	−0.516285
$580$	0	0
$581$	−9.20594	−0.381927
$582$	0	0
$583$	21.6443	0.896417
$584$	0	0
$585$	1.33508	0.0551986
$586$	0	0
$587$	4.05934	0.167547	0.0837734	−	0.996485i	$-0.473303\pi$
$587$	4.05934	0.167547	0.0837734	+	0.996485i	$0.473303\pi$
$588$	0	0
$589$	17.1648	0.707263
$590$	0	0
$591$	13.0234	0.535712
$592$	0	0
$593$	−22.2723	−0.914613	−0.457307	−	0.889309i	$-0.651186\pi$
$593$	−22.2723	−0.914613	−0.457307	+	0.889309i	$0.651186\pi$
$594$	0	0
$595$	−8.83388	−0.362154
$596$	0	0
$597$	7.80630	0.319490
$598$	0	0
$599$	35.9107	1.46727	0.733635	−	0.679543i	$-0.237822\pi$
$599$	35.9107	1.46727	0.733635	+	0.679543i	$0.237822\pi$
$600$	0	0
$601$	−10.5839	−0.431728	−0.215864	−	0.976423i	$-0.569257\pi$
$601$	−10.5839	−0.431728	−0.215864	+	0.976423i	$0.569257\pi$
$602$	0	0
$603$	−6.48763	−0.264197
$604$	0	0
$605$	−15.9588	−0.648819
$606$	0	0
$607$	4.50119	0.182698	0.0913489	−	0.995819i	$-0.470882\pi$
$607$	4.50119	0.182698	0.0913489	+	0.995819i	$0.470882\pi$
$608$	0	0
$609$	−4.18953	−0.169768
$610$	0	0
$611$	3.93649	0.159253
$612$	0	0
$613$	13.3574	0.539502	0.269751	−	0.962930i	$-0.413059\pi$
$613$	13.3574	0.539502	0.269751	+	0.962930i	$0.413059\pi$
$614$	0	0
$615$	−10.4876	−0.422902
$616$	0	0
$617$	−1.34731	−0.0542408	−0.0271204	−	0.999632i	$-0.508634\pi$
$617$	−1.34731	−0.0542408	−0.0271204	+	0.999632i	$0.508634\pi$
$618$	0	0
$619$	29.8451	1.19957	0.599787	−	0.800160i	$-0.295252\pi$
$619$	29.8451	1.19957	0.599787	+	0.800160i	$0.295252\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−3.78989	−0.151839
$624$	0	0
$625$	3.88204	0.155282
$626$	0	0
$627$	30.9917	1.23769
$628$	0	0
$629$	5.82448	0.232237
$630$	0	0
$631$	−6.60142	−0.262798	−0.131399	−	0.991330i	$-0.541947\pi$
$631$	−6.60142	−0.262798	−0.131399	+	0.991330i	$0.541947\pi$
$632$	0	0
$633$	−21.8545	−0.868637
$634$	0	0
$635$	−10.1895	−0.404359
$636$	0	0
$637$	−1.06457	−0.0421797
$638$	0	0
$639$	−5.31450	−0.210238
$640$	0	0
$641$	7.06741	0.279146	0.139573	−	0.990212i	$-0.455427\pi$
$641$	7.06741	0.279146	0.139573	+	0.990212i	$0.455427\pi$
$642$	0	0
$643$	4.60036	0.181420	0.0907102	−	0.995877i	$-0.471086\pi$
$643$	4.60036	0.181420	0.0907102	+	0.995877i	$0.471086\pi$
$644$	0	0
$645$	0.697737	0.0274734
$646$	0	0
$647$	−39.3009	−1.54508	−0.772540	−	0.634967i	$-0.781014\pi$
$647$	−39.3009	−1.54508	−0.772540	+	0.634967i	$0.781014\pi$
$648$	0	0
$649$	33.8615	1.32918
$650$	0	0
$651$	2.69774	0.105733
$652$	0	0
$653$	44.8339	1.75449	0.877243	−	0.480047i	$-0.159380\pi$
$653$	44.8339	1.75449	0.877243	+	0.480047i	$0.159380\pi$
$654$	0	0
$655$	8.45481	0.330357
$656$	0	0
$657$	−5.17313	−0.201823
$658$	0	0
$659$	−13.2694	−0.516904	−0.258452	−	0.966024i	$-0.583212\pi$
$659$	−13.2694	−0.516904	−0.258452	+	0.966024i	$0.583212\pi$
$660$	0	0
$661$	43.1453	1.67816	0.839078	−	0.544011i	$-0.183095\pi$
$661$	43.1453	1.67816	0.839078	+	0.544011i	$0.183095\pi$
$662$	0	0
$663$	−7.49881	−0.291229
$664$	0	0
$665$	7.97942	0.309429
$666$	0	0
$667$	−4.18953	−0.162219
$668$	0	0
$669$	6.35849	0.245834
$670$	0	0
$671$	9.13304	0.352577
$672$	0	0
$673$	34.1924	1.31802	0.659010	−	0.752135i	$-0.270976\pi$
$673$	34.1924	1.31802	0.659010	+	0.752135i	$0.270976\pi$
$674$	0	0
$675$	3.42723	0.131914
$676$	0	0
$677$	−12.2569	−0.471073	−0.235536	−	0.971866i	$-0.575685\pi$
$677$	−12.2569	−0.471073	−0.235536	+	0.971866i	$0.575685\pi$
$678$	0	0
$679$	0.351483	0.0134887
$680$	0	0
$681$	−10.9958	−0.421361
$682$	0	0
$683$	20.8737	0.798710	0.399355	−	0.916796i	$-0.369234\pi$
$683$	20.8737	0.798710	0.399355	+	0.916796i	$0.369234\pi$
$684$	0	0
$685$	7.01819	0.268151
$686$	0	0
$687$	27.7651	1.05931
$688$	0	0
$689$	4.73055	0.180220
$690$	0	0
$691$	−25.3697	−0.965108	−0.482554	−	0.875866i	$-0.660291\pi$
$691$	−25.3697	−0.965108	−0.482554	+	0.875866i	$0.660291\pi$
$692$	0	0
$693$	4.87086	0.185029
$694$	0	0
$695$	−5.99299	−0.227327
$696$	0	0
$697$	58.9065	2.23124
$698$	0	0
$699$	−19.1250	−0.723373
$700$	0	0
$701$	19.3226	0.729803	0.364902	−	0.931046i	$-0.381103\pi$
$701$	19.3226	0.729803	0.364902	+	0.931046i	$0.381103\pi$
$702$	0	0
$703$	−5.26111	−0.198427
$704$	0	0
$705$	−4.63734	−0.174652
$706$	0	0
$707$	15.3309	0.576578
$708$	0	0
$709$	−22.4600	−0.843505	−0.421752	−	0.906711i	$-0.638585\pi$
$709$	−22.4600	−0.843505	−0.421752	+	0.906711i	$0.638585\pi$
$710$	0	0
$711$	2.69774	0.101173
$712$	0	0
$713$	2.69774	0.101031
$714$	0	0
$715$	−6.50298	−0.243197
$716$	0	0
$717$	−7.81748	−0.291949
$718$	0	0
$719$	−29.9557	−1.11716	−0.558580	−	0.829450i	$-0.688654\pi$
$719$	−29.9557	−1.11716	−0.558580	+	0.829450i	$0.688654\pi$
$720$	0	0
$721$	−6.17313	−0.229899
$722$	0	0
$723$	−10.8709	−0.404292
$724$	0	0
$725$	14.3585	0.533261
$726$	0	0
$727$	11.6702	0.432822	0.216411	−	0.976302i	$-0.430565\pi$
$727$	11.6702	0.432822	0.216411	+	0.976302i	$0.430565\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.91903	−0.144950
$732$	0	0
$733$	15.0961	0.557585	0.278793	−	0.960351i	$-0.410066\pi$
$733$	15.0961	0.557585	0.278793	+	0.960351i	$0.410066\pi$
$734$	0	0
$735$	1.25410	0.0462582
$736$	0	0
$737$	31.6004	1.16401
$738$	0	0
$739$	53.1320	1.95449	0.977246	−	0.212111i	$-0.0680337\pi$
$739$	53.1320	1.95449	0.977246	+	0.212111i	$0.0680337\pi$
$740$	0	0
$741$	6.77348	0.248830
$742$	0	0
$743$	−25.1138	−0.921336	−0.460668	−	0.887573i	$-0.652390\pi$
$743$	−25.1138	−0.921336	−0.460668	+	0.887573i	$0.652390\pi$
$744$	0	0
$745$	−1.65375	−0.0605885
$746$	0	0
$747$	9.20594	0.336828
$748$	0	0
$749$	9.95184	0.363632
$750$	0	0
$751$	−9.89144	−0.360944	−0.180472	−	0.983580i	$-0.557762\pi$
$751$	−9.89144	−0.360944	−0.180472	+	0.983580i	$0.557762\pi$
$752$	0	0
$753$	27.3543	0.996847
$754$	0	0
$755$	−22.6702	−0.825051
$756$	0	0
$757$	−12.6014	−0.458006	−0.229003	−	0.973426i	$-0.573547\pi$
$757$	−12.6014	−0.458006	−0.229003	+	0.973426i	$0.573547\pi$
$758$	0	0
$759$	4.87086	0.176801
$760$	0	0
$761$	−41.4863	−1.50388	−0.751939	−	0.659233i	$-0.770881\pi$
$761$	−41.4863	−1.50388	−0.751939	+	0.659233i	$0.770881\pi$
$762$	0	0
$763$	−4.14137	−0.149928
$764$	0	0
$765$	8.83388	0.319390
$766$	0	0
$767$	7.40070	0.267224
$768$	0	0
$769$	−48.8050	−1.75995	−0.879976	−	0.475018i	$-0.842442\pi$
$769$	−48.8050	−1.75995	−0.879976	+	0.475018i	$0.842442\pi$
$770$	0	0
$771$	−8.39025	−0.302167
$772$	0	0
$773$	−40.5910	−1.45996	−0.729978	−	0.683471i	$-0.760470\pi$
$773$	−40.5910	−1.45996	−0.729978	+	0.683471i	$0.760470\pi$
$774$	0	0
$775$	−9.24576	−0.332118
$776$	0	0
$777$	−0.826873	−0.0296639
$778$	0	0
$779$	−53.2088	−1.90640
$780$	0	0
$781$	25.8862	0.926281
$782$	0	0
$783$	4.18953	0.149722
$784$	0	0
$785$	29.3543	1.04770
$786$	0	0
$787$	20.9742	0.747649	0.373825	−	0.927499i	$-0.378046\pi$
$787$	20.9742	0.747649	0.373825	+	0.927499i	$0.378046\pi$
$788$	0	0
$789$	−21.7581	−0.774610
$790$	0	0
$791$	−16.9630	−0.603135
$792$	0	0
$793$	1.99610	0.0708836
$794$	0	0
$795$	−5.57277	−0.197646
$796$	0	0
$797$	0.832101	0.0294745	0.0147373	−	0.999891i	$-0.495309\pi$
$797$	0.832101	0.0294745	0.0147373	+	0.999891i	$0.495309\pi$
$798$	0	0
$799$	26.0468	0.921471
$800$	0	0
$801$	3.78989	0.133909
$802$	0	0
$803$	25.1976	0.889204
$804$	0	0
$805$	1.25410	0.0442013
$806$	0	0
$807$	−20.7651	−0.730968
$808$	0	0
$809$	−49.2405	−1.73121	−0.865603	−	0.500732i	$-0.833064\pi$
$809$	−49.2405	−1.73121	−0.865603	+	0.500732i	$0.833064\pi$
$810$	0	0
$811$	−3.37623	−0.118555	−0.0592777	−	0.998242i	$-0.518880\pi$
$811$	−3.37623	−0.118555	−0.0592777	+	0.998242i	$0.518880\pi$
$812$	0	0
$813$	−15.8984	−0.557583
$814$	0	0
$815$	−0.0809744	−0.00283641
$816$	0	0
$817$	3.53996	0.123847
$818$	0	0
$819$	1.06457	0.0371990
$820$	0	0
$821$	−34.1125	−1.19053	−0.595267	−	0.803528i	$-0.702954\pi$
$821$	−34.1125	−1.19053	−0.595267	+	0.803528i	$0.702954\pi$
$822$	0	0
$823$	−20.9435	−0.730045	−0.365022	−	0.930999i	$-0.618939\pi$
$823$	−20.9435	−0.730045	−0.365022	+	0.930999i	$0.618939\pi$
$824$	0	0
$825$	−16.6936	−0.581195
$826$	0	0
$827$	−46.8779	−1.63010	−0.815052	−	0.579388i	$-0.803292\pi$
$827$	−46.8779	−1.63010	−0.815052	+	0.579388i	$0.803292\pi$
$828$	0	0
$829$	−17.1648	−0.596158	−0.298079	−	0.954541i	$-0.596346\pi$
$829$	−17.1648	−0.596158	−0.298079	+	0.954541i	$0.596346\pi$
$830$	0	0
$831$	12.7212	0.441292
$832$	0	0
$833$	−7.04399	−0.244060
$834$	0	0
$835$	−31.5040	−1.09024
$836$	0	0
$837$	−2.69774	−0.0932474
$838$	0	0
$839$	−46.1195	−1.59222	−0.796110	−	0.605151i	$-0.793113\pi$
$839$	−46.1195	−1.59222	−0.796110	+	0.605151i	$0.793113\pi$
$840$	0	0
$841$	−11.4478	−0.394752
$842$	0	0
$843$	17.3215	0.596584
$844$	0	0
$845$	14.8820	0.511958
$846$	0	0
$847$	−12.7253	−0.437247
$848$	0	0
$849$	−0.0205757	−0.000706157 0
$850$	0	0
$851$	−0.826873	−0.0283448
$852$	0	0
$853$	26.0573	0.892185	0.446092	−	0.894987i	$-0.352815\pi$
$853$	26.0573	0.892185	0.446092	+	0.894987i	$0.352815\pi$
$854$	0	0
$855$	−7.97942	−0.272891
$856$	0	0
$857$	−41.6811	−1.42380	−0.711899	−	0.702282i	$-0.752165\pi$
$857$	−41.6811	−1.42380	−0.711899	+	0.702282i	$0.752165\pi$
$858$	0	0
$859$	45.0716	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$859$	45.0716	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$860$	0	0
$861$	−8.36266	−0.284999
$862$	0	0
$863$	−9.14554	−0.311318	−0.155659	−	0.987811i	$-0.549750\pi$
$863$	−9.14554	−0.311318	−0.155659	+	0.987811i	$0.549750\pi$
$864$	0	0
$865$	−22.9219	−0.779367
$866$	0	0
$867$	−32.6178	−1.10776
$868$	0	0
$869$	−13.1403	−0.445755
$870$	0	0
$871$	6.90652	0.234018
$872$	0	0
$873$	−0.351483	−0.0118959
$874$	0	0
$875$	−10.5686	−0.357284
$876$	0	0
$877$	−40.6402	−1.37232	−0.686161	−	0.727450i	$-0.740705\pi$
$877$	−40.6402	−1.37232	−0.686161	+	0.727450i	$0.740705\pi$
$878$	0	0
$879$	−30.2088	−1.01892
$880$	0	0
$881$	54.3379	1.83069	0.915345	−	0.402669i	$-0.131918\pi$
$881$	54.3379	1.83069	0.915345	+	0.402669i	$0.131918\pi$
$882$	0	0
$883$	52.4434	1.76486	0.882430	−	0.470444i	$-0.155906\pi$
$883$	52.4434	1.76486	0.882430	+	0.470444i	$0.155906\pi$
$884$	0	0
$885$	−8.71831	−0.293063
$886$	0	0
$887$	48.2981	1.62169	0.810846	−	0.585260i	$-0.199008\pi$
$887$	48.2981	1.62169	0.810846	+	0.585260i	$0.199008\pi$
$888$	0	0
$889$	−8.12497	−0.272503
$890$	0	0
$891$	−4.87086	−0.163180
$892$	0	0
$893$	−23.5275	−0.787316
$894$	0	0
$895$	10.6032	0.354426
$896$	0	0
$897$	1.06457	0.0355449
$898$	0	0
$899$	−11.3023	−0.376952
$900$	0	0
$901$	31.3009	1.04279
$902$	0	0
$903$	0.556364	0.0185146
$904$	0	0
$905$	23.9243	0.795269
$906$	0	0
$907$	−29.5675	−0.981774	−0.490887	−	0.871223i	$-0.663327\pi$
$907$	−29.5675	−0.981774	−0.490887	+	0.871223i	$0.663327\pi$
$908$	0	0
$909$	−15.3309	−0.508494
$910$	0	0
$911$	29.9365	0.991840	0.495920	−	0.868368i	$-0.334831\pi$
$911$	29.9365	0.991840	0.495920	+	0.868368i	$0.334831\pi$
$912$	0	0
$913$	−44.8409	−1.48402
$914$	0	0
$915$	−2.35148	−0.0777376
$916$	0	0
$917$	6.74173	0.222632
$918$	0	0
$919$	−3.87609	−0.127861	−0.0639303	−	0.997954i	$-0.520364\pi$
$919$	−3.87609	−0.127861	−0.0639303	+	0.997954i	$0.520364\pi$
$920$	0	0
$921$	15.0440	0.495716
$922$	0	0
$923$	5.65765	0.186224
$924$	0	0
$925$	2.83388	0.0931775
$926$	0	0
$927$	6.17313	0.202752
$928$	0	0
$929$	−44.8255	−1.47068	−0.735339	−	0.677699i	$-0.762977\pi$
$929$	−44.8255	−1.47068	−0.735339	+	0.677699i	$0.762977\pi$
$930$	0	0
$931$	6.36266	0.208528
$932$	0	0
$933$	18.3749	0.601567
$934$	0	0
$935$	−43.0286	−1.40719
$936$	0	0
$937$	29.7581	0.972156	0.486078	−	0.873915i	$-0.338427\pi$
$937$	29.7581	0.972156	0.486078	+	0.873915i	$0.338427\pi$
$938$	0	0
$939$	−23.5358	−0.768061
$940$	0	0
$941$	21.5163	0.701410	0.350705	−	0.936486i	$-0.385942\pi$
$941$	21.5163	0.701410	0.350705	+	0.936486i	$0.385942\pi$
$942$	0	0
$943$	−8.36266	−0.272326
$944$	0	0
$945$	−1.25410	−0.0407959
$946$	0	0
$947$	−49.3132	−1.60246	−0.801231	−	0.598355i	$-0.795821\pi$
$947$	−49.3132	−1.60246	−0.801231	+	0.598355i	$0.795821\pi$
$948$	0	0
$949$	5.50714	0.178769
$950$	0	0
$951$	11.2130	0.363605
$952$	0	0
$953$	40.0510	1.29738	0.648690	−	0.761053i	$-0.275317\pi$
$953$	40.0510	1.29738	0.648690	+	0.761053i	$0.275317\pi$
$954$	0	0
$955$	4.20310	0.136009
$956$	0	0
$957$	−20.4067	−0.659653
$958$	0	0
$959$	5.59619	0.180710
$960$	0	0
$961$	−23.7222	−0.765233
$962$	0	0
$963$	−9.95184	−0.320694
$964$	0	0
$965$	−15.5798	−0.501531
$966$	0	0
$967$	−30.7805	−0.989834	−0.494917	−	0.868940i	$-0.664802\pi$
$967$	−30.7805	−0.989834	−0.494917	+	0.868940i	$0.664802\pi$
$968$	0	0
$969$	44.8185	1.43978
$970$	0	0
$971$	−18.5012	−0.593732	−0.296866	−	0.954919i	$-0.595941\pi$
$971$	−18.5012	−0.593732	−0.296866	+	0.954919i	$0.595941\pi$
$972$	0	0
$973$	−4.77871	−0.153198
$974$	0	0
$975$	−3.64852	−0.116846
$976$	0	0
$977$	43.4137	1.38893	0.694463	−	0.719528i	$-0.255642\pi$
$977$	43.4137	1.38893	0.694463	+	0.719528i	$0.255642\pi$
$978$	0	0
$979$	−18.4600	−0.589986
$980$	0	0
$981$	4.14137	0.132224
$982$	0	0
$983$	40.5519	1.29341	0.646703	−	0.762742i	$-0.276147\pi$
$983$	40.5519	1.29341	0.646703	+	0.762742i	$0.276147\pi$
$984$	0	0
$985$	16.3327	0.520403
$986$	0	0
$987$	−3.69774	−0.117700
$988$	0	0
$989$	0.556364	0.0176914
$990$	0	0
$991$	−10.0042	−0.317793	−0.158896	−	0.987295i	$-0.550794\pi$
$991$	−10.0042	−0.317793	−0.158896	+	0.987295i	$0.550794\pi$
$992$	0	0
$993$	8.93126	0.283425
$994$	0	0
$995$	9.78989	0.310360
$996$	0	0
$997$	−32.2364	−1.02094	−0.510468	−	0.859897i	$-0.670528\pi$
$997$	−32.2364	−1.02094	−0.510468	+	0.859897i	$0.670528\pi$
$998$	0	0
$999$	0.826873	0.0261611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bp.1.2		3
4.3	odd	2		3864.2.a.p.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.p.1.2	✓	3	4.3	odd	2
7728.2.a.bp.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.25410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.25410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.87086 q^{11} -1.06457 q^{13} +1.25410 q^{15} -7.04399 q^{17} +6.36266 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.42723 q^{25} -1.00000 q^{27} -4.18953 q^{29} +2.69774 q^{31} +4.87086 q^{33} +1.25410 q^{35} -0.826873 q^{37} +1.06457 q^{39} -8.36266 q^{41} +0.556364 q^{43} -1.25410 q^{45} -3.69774 q^{47} +1.00000 q^{49} +7.04399 q^{51} -4.44364 q^{53} +6.10856 q^{55} -6.36266 q^{57} -6.95184 q^{59} -1.87503 q^{61} -1.00000 q^{63} +1.33508 q^{65} -6.48763 q^{67} -1.00000 q^{69} -5.31450 q^{71} -5.17313 q^{73} +3.42723 q^{75} +4.87086 q^{77} +2.69774 q^{79} +1.00000 q^{81} +9.20594 q^{83} +8.83388 q^{85} +4.18953 q^{87} +3.78989 q^{89} +1.06457 q^{91} -2.69774 q^{93} -7.97942 q^{95} -0.351483 q^{97} -4.87086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + q^{11} - 11 q^{13} + 3 q^{15} + 5 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} - 8 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 3 q^{45} - q^{47} + 3 q^{49} - 4 q^{53} + 5 q^{55} - 5 q^{57} - 10 q^{59} - 22 q^{61} - 3 q^{63} + 8 q^{65} + 11 q^{67} - 3 q^{69} + 9 q^{71} - 10 q^{73} + 4 q^{75} - q^{77} - 2 q^{79} + 3 q^{81} + 16 q^{83} - 15 q^{85} + 4 q^{87} - 9 q^{89} + 11 q^{91} + 2 q^{93} + 5 q^{95} - 2 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + q^11 - 11 * q^13 + 3 * q^15 + 5 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 - q^33 + 3 * q^35 - 8 * q^37 + 11 * q^39 - 11 * q^41 + 11 * q^43 - 3 * q^45 - q^47 + 3 * q^49 - 4 * q^53 + 5 * q^55 - 5 * q^57 - 10 * q^59 - 22 * q^61 - 3 * q^63 + 8 * q^65 + 11 * q^67 - 3 * q^69 + 9 * q^71 - 10 * q^73 + 4 * q^75 - q^77 - 2 * q^79 + 3 * q^81 + 16 * q^83 - 15 * q^85 + 4 * q^87 - 9 * q^89 + 11 * q^91 + 2 * q^93 + 5 * q^95 - 2 * q^97 + q^99

Introduction

L-functions

Modular forms

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