LMFDB - Embedded newform 7728.2.a.bb.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1932)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.85410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.85410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -0.381966 q^{13} -3.85410 q^{15} -1.47214 q^{17} -0.236068 q^{19} +1.00000 q^{21} +1.00000 q^{23} +9.85410 q^{25} -1.00000 q^{27} -5.00000 q^{29} -5.76393 q^{31} +3.00000 q^{33} -3.85410 q^{35} -4.23607 q^{37} +0.381966 q^{39} +3.00000 q^{41} +1.61803 q^{43} +3.85410 q^{45} +3.23607 q^{47} +1.00000 q^{49} +1.47214 q^{51} -2.32624 q^{53} -11.5623 q^{55} +0.236068 q^{57} +13.5623 q^{59} -4.85410 q^{61} -1.00000 q^{63} -1.47214 q^{65} -2.32624 q^{67} -1.00000 q^{69} -7.61803 q^{71} -5.94427 q^{73} -9.85410 q^{75} +3.00000 q^{77} +9.47214 q^{79} +1.00000 q^{81} -12.2361 q^{83} -5.67376 q^{85} +5.00000 q^{87} -6.56231 q^{89} +0.381966 q^{91} +5.76393 q^{93} -0.909830 q^{95} -1.94427 q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} + 6 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 13 q^{25} - 2 q^{27} - 10 q^{29} - 16 q^{31} + 6 q^{33} - q^{35} - 4 q^{37} + 3 q^{39} + 6 q^{41} + q^{43} + q^{45} + 2 q^{47} + 2 q^{49} - 6 q^{51} + 11 q^{53} - 3 q^{55} - 4 q^{57} + 7 q^{59} - 3 q^{61} - 2 q^{63} + 6 q^{65} + 11 q^{67} - 2 q^{69} - 13 q^{71} + 6 q^{73} - 13 q^{75} + 6 q^{77} + 10 q^{79} + 2 q^{81} - 20 q^{83} - 27 q^{85} + 10 q^{87} + 7 q^{89} + 3 q^{91} + 16 q^{93} - 13 q^{95} + 14 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 - 3 * q^13 - q^15 + 6 * q^17 + 4 * q^19 + 2 * q^21 + 2 * q^23 + 13 * q^25 - 2 * q^27 - 10 * q^29 - 16 * q^31 + 6 * q^33 - q^35 - 4 * q^37 + 3 * q^39 + 6 * q^41 + q^43 + q^45 + 2 * q^47 + 2 * q^49 - 6 * q^51 + 11 * q^53 - 3 * q^55 - 4 * q^57 + 7 * q^59 - 3 * q^61 - 2 * q^63 + 6 * q^65 + 11 * q^67 - 2 * q^69 - 13 * q^71 + 6 * q^73 - 13 * q^75 + 6 * q^77 + 10 * q^79 + 2 * q^81 - 20 * q^83 - 27 * q^85 + 10 * q^87 + 7 * q^89 + 3 * q^91 + 16 * q^93 - 13 * q^95 + 14 * 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
$5$	3.85410	1.72361	0.861803	−	0.507242i	$-0.169335\pi$
$5$	3.85410	1.72361	0.861803	+	0.507242i	$0.169335\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−0.381966	−0.105938	−0.0529692	−	0.998596i	$-0.516869\pi$
$13$	−0.381966	−0.105938	−0.0529692	+	0.998596i	$0.516869\pi$
$14$	0	0
$15$	−3.85410	−0.995125
$16$	0	0
$17$	−1.47214	−0.357045	−0.178523	−	0.983936i	$-0.557132\pi$
$17$	−1.47214	−0.357045	−0.178523	+	0.983936i	$0.557132\pi$
$18$	0	0
$19$	−0.236068	−0.0541577	−0.0270789	−	0.999633i	$-0.508621\pi$
$19$	−0.236068	−0.0541577	−0.0270789	+	0.999633i	$0.508621\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	9.85410	1.97082
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−5.76393	−1.03523	−0.517616	−	0.855613i	$-0.673181\pi$
$31$	−5.76393	−1.03523	−0.517616	+	0.855613i	$0.673181\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−3.85410	−0.651462
$36$	0	0
$37$	−4.23607	−0.696405	−0.348203	−	0.937419i	$-0.613208\pi$
$37$	−4.23607	−0.696405	−0.348203	+	0.937419i	$0.613208\pi$
$38$	0	0
$39$	0.381966	0.0611635
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	1.61803	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$43$	1.61803	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$44$	0	0
$45$	3.85410	0.574536
$46$	0	0
$47$	3.23607	0.472029	0.236015	−	0.971750i	$-0.424159\pi$
$47$	3.23607	0.472029	0.236015	+	0.971750i	$0.424159\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.47214	0.206140
$52$	0	0
$53$	−2.32624	−0.319533	−0.159767	−	0.987155i	$-0.551074\pi$
$53$	−2.32624	−0.319533	−0.159767	+	0.987155i	$0.551074\pi$
$54$	0	0
$55$	−11.5623	−1.55906
$56$	0	0
$57$	0.236068	0.0312680
$58$	0	0
$59$	13.5623	1.76566	0.882831	−	0.469691i	$-0.155635\pi$
$59$	13.5623	1.76566	0.882831	+	0.469691i	$0.155635\pi$
$60$	0	0
$61$	−4.85410	−0.621504	−0.310752	−	0.950491i	$-0.600581\pi$
$61$	−4.85410	−0.621504	−0.310752	+	0.950491i	$0.600581\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−1.47214	−0.182596
$66$	0	0
$67$	−2.32624	−0.284195	−0.142098	−	0.989853i	$-0.545385\pi$
$67$	−2.32624	−0.284195	−0.142098	+	0.989853i	$0.545385\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−7.61803	−0.904094	−0.452047	−	0.891994i	$-0.649306\pi$
$71$	−7.61803	−0.904094	−0.452047	+	0.891994i	$0.649306\pi$
$72$	0	0
$73$	−5.94427	−0.695724	−0.347862	−	0.937546i	$-0.613092\pi$
$73$	−5.94427	−0.695724	−0.347862	+	0.937546i	$0.613092\pi$
$74$	0	0
$75$	−9.85410	−1.13785
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	9.47214	1.06570	0.532849	−	0.846210i	$-0.321121\pi$
$79$	9.47214	1.06570	0.532849	+	0.846210i	$0.321121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.2361	−1.34308	−0.671541	−	0.740967i	$-0.734367\pi$
$83$	−12.2361	−1.34308	−0.671541	+	0.740967i	$0.734367\pi$
$84$	0	0
$85$	−5.67376	−0.615406
$86$	0	0
$87$	5.00000	0.536056
$88$	0	0
$89$	−6.56231	−0.695603	−0.347802	−	0.937568i	$-0.613072\pi$
$89$	−6.56231	−0.695603	−0.347802	+	0.937568i	$0.613072\pi$
$90$	0	0
$91$	0.381966	0.0400409
$92$	0	0
$93$	5.76393	0.597692
$94$	0	0
$95$	−0.909830	−0.0933466
$96$	0	0
$97$	−1.94427	−0.197411	−0.0987055	−	0.995117i	$-0.531470\pi$
$97$	−1.94427	−0.197411	−0.0987055	+	0.995117i	$0.531470\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−7.56231	−0.752478	−0.376239	−	0.926523i	$-0.622783\pi$
$101$	−7.56231	−0.752478	−0.376239	+	0.926523i	$0.622783\pi$
$102$	0	0
$103$	−12.9443	−1.27544	−0.637719	−	0.770270i	$-0.720122\pi$
$103$	−12.9443	−1.27544	−0.637719	+	0.770270i	$0.720122\pi$
$104$	0	0
$105$	3.85410	0.376122
$106$	0	0
$107$	−12.5623	−1.21444	−0.607222	−	0.794532i	$-0.707716\pi$
$107$	−12.5623	−1.21444	−0.607222	+	0.794532i	$0.707716\pi$
$108$	0	0
$109$	−4.90983	−0.470276	−0.235138	−	0.971962i	$-0.575554\pi$
$109$	−4.90983	−0.470276	−0.235138	+	0.971962i	$0.575554\pi$
$110$	0	0
$111$	4.23607	0.402070
$112$	0	0
$113$	−8.03444	−0.755817	−0.377908	−	0.925843i	$-0.623357\pi$
$113$	−8.03444	−0.755817	−0.377908	+	0.925843i	$0.623357\pi$
$114$	0	0
$115$	3.85410	0.359397
$116$	0	0
$117$	−0.381966	−0.0353128
$118$	0	0
$119$	1.47214	0.134950
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	18.7082	1.67331
$126$	0	0
$127$	−3.32624	−0.295156	−0.147578	−	0.989050i	$-0.547148\pi$
$127$	−3.32624	−0.295156	−0.147578	+	0.989050i	$0.547148\pi$
$128$	0	0
$129$	−1.61803	−0.142460
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	0.236068	0.0204697
$134$	0	0
$135$	−3.85410	−0.331708
$136$	0	0
$137$	8.41641	0.719062	0.359531	−	0.933133i	$-0.382937\pi$
$137$	8.41641	0.719062	0.359531	+	0.933133i	$0.382937\pi$
$138$	0	0
$139$	10.7984	0.915906	0.457953	−	0.888976i	$-0.348583\pi$
$139$	10.7984	0.915906	0.457953	+	0.888976i	$0.348583\pi$
$140$	0	0
$141$	−3.23607	−0.272526
$142$	0	0
$143$	1.14590	0.0958248
$144$	0	0
$145$	−19.2705	−1.60033
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	16.1803	1.32555	0.662773	−	0.748821i	$-0.269380\pi$
$149$	16.1803	1.32555	0.662773	+	0.748821i	$0.269380\pi$
$150$	0	0
$151$	22.6525	1.84343	0.921716	−	0.387865i	$-0.126787\pi$
$151$	22.6525	1.84343	0.921716	+	0.387865i	$0.126787\pi$
$152$	0	0
$153$	−1.47214	−0.119015
$154$	0	0
$155$	−22.2148	−1.78433
$156$	0	0
$157$	−6.76393	−0.539821	−0.269910	−	0.962885i	$-0.586994\pi$
$157$	−6.76393	−0.539821	−0.269910	+	0.962885i	$0.586994\pi$
$158$	0	0
$159$	2.32624	0.184483
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−5.67376	−0.444403	−0.222202	−	0.975001i	$-0.571324\pi$
$163$	−5.67376	−0.444403	−0.222202	+	0.975001i	$0.571324\pi$
$164$	0	0
$165$	11.5623	0.900124
$166$	0	0
$167$	−8.41641	−0.651281	−0.325641	−	0.945494i	$-0.605580\pi$
$167$	−8.41641	−0.651281	−0.325641	+	0.945494i	$0.605580\pi$
$168$	0	0
$169$	−12.8541	−0.988777
$170$	0	0
$171$	−0.236068	−0.0180526
$172$	0	0
$173$	−17.4721	−1.32838	−0.664191	−	0.747563i	$-0.731224\pi$
$173$	−17.4721	−1.32838	−0.664191	+	0.747563i	$0.731224\pi$
$174$	0	0
$175$	−9.85410	−0.744900
$176$	0	0
$177$	−13.5623	−1.01941
$178$	0	0
$179$	−21.5623	−1.61164	−0.805821	−	0.592159i	$-0.798276\pi$
$179$	−21.5623	−1.61164	−0.805821	+	0.592159i	$0.798276\pi$
$180$	0	0
$181$	6.41641	0.476928	0.238464	−	0.971151i	$-0.423356\pi$
$181$	6.41641	0.476928	0.238464	+	0.971151i	$0.423356\pi$
$182$	0	0
$183$	4.85410	0.358826
$184$	0	0
$185$	−16.3262	−1.20033
$186$	0	0
$187$	4.41641	0.322960
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−12.7639	−0.923566	−0.461783	−	0.886993i	$-0.652790\pi$
$191$	−12.7639	−0.923566	−0.461783	+	0.886993i	$0.652790\pi$
$192$	0	0
$193$	24.6525	1.77452	0.887262	−	0.461266i	$-0.152605\pi$
$193$	24.6525	1.77452	0.887262	+	0.461266i	$0.152605\pi$
$194$	0	0
$195$	1.47214	0.105422
$196$	0	0
$197$	4.09017	0.291413	0.145706	−	0.989328i	$-0.453455\pi$
$197$	4.09017	0.291413	0.145706	+	0.989328i	$0.453455\pi$
$198$	0	0
$199$	−10.7984	−0.765476	−0.382738	−	0.923857i	$-0.625019\pi$
$199$	−10.7984	−0.765476	−0.382738	+	0.923857i	$0.625019\pi$
$200$	0	0
$201$	2.32624	0.164080
$202$	0	0
$203$	5.00000	0.350931
$204$	0	0
$205$	11.5623	0.807546
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0.708204	0.0489875
$210$	0	0
$211$	−4.23607	−0.291623	−0.145811	−	0.989312i	$-0.546579\pi$
$211$	−4.23607	−0.291623	−0.145811	+	0.989312i	$0.546579\pi$
$212$	0	0
$213$	7.61803	0.521979
$214$	0	0
$215$	6.23607	0.425296
$216$	0	0
$217$	5.76393	0.391281
$218$	0	0
$219$	5.94427	0.401677
$220$	0	0
$221$	0.562306	0.0378248
$222$	0	0
$223$	4.32624	0.289706	0.144853	−	0.989453i	$-0.453729\pi$
$223$	4.32624	0.289706	0.144853	+	0.989453i	$0.453729\pi$
$224$	0	0
$225$	9.85410	0.656940
$226$	0	0
$227$	−0.618034	−0.0410204	−0.0205102	−	0.999790i	$-0.506529\pi$
$227$	−0.618034	−0.0410204	−0.0205102	+	0.999790i	$0.506529\pi$
$228$	0	0
$229$	−2.43769	−0.161087	−0.0805437	−	0.996751i	$-0.525666\pi$
$229$	−2.43769	−0.161087	−0.0805437	+	0.996751i	$0.525666\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	−16.1459	−1.05775	−0.528876	−	0.848699i	$-0.677387\pi$
$233$	−16.1459	−1.05775	−0.528876	+	0.848699i	$0.677387\pi$
$234$	0	0
$235$	12.4721	0.813592
$236$	0	0
$237$	−9.47214	−0.615281
$238$	0	0
$239$	5.67376	0.367005	0.183503	−	0.983019i	$-0.441256\pi$
$239$	5.67376	0.367005	0.183503	+	0.983019i	$0.441256\pi$
$240$	0	0
$241$	−2.05573	−0.132421	−0.0662105	−	0.997806i	$-0.521091\pi$
$241$	−2.05573	−0.132421	−0.0662105	+	0.997806i	$0.521091\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.85410	0.246230
$246$	0	0
$247$	0.0901699	0.00573738
$248$	0	0
$249$	12.2361	0.775429
$250$	0	0
$251$	−13.2361	−0.835453	−0.417727	−	0.908573i	$-0.637173\pi$
$251$	−13.2361	−0.835453	−0.417727	+	0.908573i	$0.637173\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	5.67376	0.355305
$256$	0	0
$257$	−0.763932	−0.0476528	−0.0238264	−	0.999716i	$-0.507585\pi$
$257$	−0.763932	−0.0476528	−0.0238264	+	0.999716i	$0.507585\pi$
$258$	0	0
$259$	4.23607	0.263216
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0	0
$263$	−17.2918	−1.06626	−0.533129	−	0.846034i	$-0.678984\pi$
$263$	−17.2918	−1.06626	−0.533129	+	0.846034i	$0.678984\pi$
$264$	0	0
$265$	−8.96556	−0.550750
$266$	0	0
$267$	6.56231	0.401607
$268$	0	0
$269$	8.67376	0.528849	0.264424	−	0.964406i	$-0.414818\pi$
$269$	8.67376	0.528849	0.264424	+	0.964406i	$0.414818\pi$
$270$	0	0
$271$	−24.4164	−1.48319	−0.741596	−	0.670847i	$-0.765931\pi$
$271$	−24.4164	−1.48319	−0.741596	+	0.670847i	$0.765931\pi$
$272$	0	0
$273$	−0.381966	−0.0231176
$274$	0	0
$275$	−29.5623	−1.78267
$276$	0	0
$277$	3.32624	0.199854	0.0999271	−	0.994995i	$-0.468139\pi$
$277$	3.32624	0.199854	0.0999271	+	0.994995i	$0.468139\pi$
$278$	0	0
$279$	−5.76393	−0.345078
$280$	0	0
$281$	−19.1246	−1.14088	−0.570439	−	0.821340i	$-0.693227\pi$
$281$	−19.1246	−1.14088	−0.570439	+	0.821340i	$0.693227\pi$
$282$	0	0
$283$	26.0344	1.54759	0.773793	−	0.633438i	$-0.218357\pi$
$283$	26.0344	1.54759	0.773793	+	0.633438i	$0.218357\pi$
$284$	0	0
$285$	0.909830	0.0538937
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−14.8328	−0.872519
$290$	0	0
$291$	1.94427	0.113975
$292$	0	0
$293$	12.9443	0.756212	0.378106	−	0.925762i	$-0.376575\pi$
$293$	12.9443	0.756212	0.378106	+	0.925762i	$0.376575\pi$
$294$	0	0
$295$	52.2705	3.04331
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	−0.381966	−0.0220897
$300$	0	0
$301$	−1.61803	−0.0932619
$302$	0	0
$303$	7.56231	0.434443
$304$	0	0
$305$	−18.7082	−1.07123
$306$	0	0
$307$	23.6525	1.34992	0.674959	−	0.737855i	$-0.264161\pi$
$307$	23.6525	1.34992	0.674959	+	0.737855i	$0.264161\pi$
$308$	0	0
$309$	12.9443	0.736374
$310$	0	0
$311$	16.3820	0.928936	0.464468	−	0.885590i	$-0.346246\pi$
$311$	16.3820	0.928936	0.464468	+	0.885590i	$0.346246\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	0	0
$315$	−3.85410	−0.217154
$316$	0	0
$317$	30.0344	1.68690	0.843451	−	0.537206i	$-0.180520\pi$
$317$	30.0344	1.68690	0.843451	+	0.537206i	$0.180520\pi$
$318$	0	0
$319$	15.0000	0.839839
$320$	0	0
$321$	12.5623	0.701160
$322$	0	0
$323$	0.347524	0.0193368
$324$	0	0
$325$	−3.76393	−0.208785
$326$	0	0
$327$	4.90983	0.271514
$328$	0	0
$329$	−3.23607	−0.178410
$330$	0	0
$331$	18.3607	1.00919	0.504597	−	0.863355i	$-0.331641\pi$
$331$	18.3607	1.00919	0.504597	+	0.863355i	$0.331641\pi$
$332$	0	0
$333$	−4.23607	−0.232135
$334$	0	0
$335$	−8.96556	−0.489841
$336$	0	0
$337$	−4.20163	−0.228877	−0.114439	−	0.993430i	$-0.536507\pi$
$337$	−4.20163	−0.228877	−0.114439	+	0.993430i	$0.536507\pi$
$338$	0	0
$339$	8.03444	0.436371
$340$	0	0
$341$	17.2918	0.936403
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.85410	−0.207498
$346$	0	0
$347$	28.5967	1.53515	0.767577	−	0.640957i	$-0.221462\pi$
$347$	28.5967	1.53515	0.767577	+	0.640957i	$0.221462\pi$
$348$	0	0
$349$	−27.4508	−1.46941	−0.734705	−	0.678387i	$-0.762679\pi$
$349$	−27.4508	−1.46941	−0.734705	+	0.678387i	$0.762679\pi$
$350$	0	0
$351$	0.381966	0.0203878
$352$	0	0
$353$	30.4164	1.61890	0.809451	−	0.587187i	$-0.199765\pi$
$353$	30.4164	1.61890	0.809451	+	0.587187i	$0.199765\pi$
$354$	0	0
$355$	−29.3607	−1.55830
$356$	0	0
$357$	−1.47214	−0.0779137
$358$	0	0
$359$	23.8541	1.25897	0.629486	−	0.777012i	$-0.283266\pi$
$359$	23.8541	1.25897	0.629486	+	0.777012i	$0.283266\pi$
$360$	0	0
$361$	−18.9443	−0.997067
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−22.9098	−1.19916
$366$	0	0
$367$	−1.43769	−0.0750470	−0.0375235	−	0.999296i	$-0.511947\pi$
$367$	−1.43769	−0.0750470	−0.0375235	+	0.999296i	$0.511947\pi$
$368$	0	0
$369$	3.00000	0.156174
$370$	0	0
$371$	2.32624	0.120772
$372$	0	0
$373$	−16.2361	−0.840672	−0.420336	−	0.907369i	$-0.638088\pi$
$373$	−16.2361	−0.840672	−0.420336	+	0.907369i	$0.638088\pi$
$374$	0	0
$375$	−18.7082	−0.966087
$376$	0	0
$377$	1.90983	0.0983613
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	3.32624	0.170408
$382$	0	0
$383$	2.70820	0.138383	0.0691914	−	0.997603i	$-0.477958\pi$
$383$	2.70820	0.138383	0.0691914	+	0.997603i	$0.477958\pi$
$384$	0	0
$385$	11.5623	0.589270
$386$	0	0
$387$	1.61803	0.0822493
$388$	0	0
$389$	−15.1803	−0.769674	−0.384837	−	0.922985i	$-0.625742\pi$
$389$	−15.1803	−0.769674	−0.384837	+	0.922985i	$0.625742\pi$
$390$	0	0
$391$	−1.47214	−0.0744491
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	36.5066	1.83685
$396$	0	0
$397$	4.76393	0.239095	0.119547	−	0.992828i	$-0.461856\pi$
$397$	4.76393	0.239095	0.119547	+	0.992828i	$0.461856\pi$
$398$	0	0
$399$	−0.236068	−0.0118182
$400$	0	0
$401$	35.5410	1.77483	0.887417	−	0.460968i	$-0.152498\pi$
$401$	35.5410	1.77483	0.887417	+	0.460968i	$0.152498\pi$
$402$	0	0
$403$	2.20163	0.109671
$404$	0	0
$405$	3.85410	0.191512
$406$	0	0
$407$	12.7082	0.629922
$408$	0	0
$409$	1.18034	0.0583641	0.0291820	−	0.999574i	$-0.490710\pi$
$409$	1.18034	0.0583641	0.0291820	+	0.999574i	$0.490710\pi$
$410$	0	0
$411$	−8.41641	−0.415151
$412$	0	0
$413$	−13.5623	−0.667357
$414$	0	0
$415$	−47.1591	−2.31495
$416$	0	0
$417$	−10.7984	−0.528799
$418$	0	0
$419$	−28.6869	−1.40145	−0.700724	−	0.713433i	$-0.747139\pi$
$419$	−28.6869	−1.40145	−0.700724	+	0.713433i	$0.747139\pi$
$420$	0	0
$421$	−36.2705	−1.76772	−0.883858	−	0.467755i	$-0.845063\pi$
$421$	−36.2705	−1.76772	−0.883858	+	0.467755i	$0.845063\pi$
$422$	0	0
$423$	3.23607	0.157343
$424$	0	0
$425$	−14.5066	−0.703672
$426$	0	0
$427$	4.85410	0.234906
$428$	0	0
$429$	−1.14590	−0.0553245
$430$	0	0
$431$	31.0344	1.49488	0.747438	−	0.664331i	$-0.231284\pi$
$431$	31.0344	1.49488	0.747438	+	0.664331i	$0.231284\pi$
$432$	0	0
$433$	−15.7082	−0.754888	−0.377444	−	0.926032i	$-0.623197\pi$
$433$	−15.7082	−0.754888	−0.377444	+	0.926032i	$0.623197\pi$
$434$	0	0
$435$	19.2705	0.923950
$436$	0	0
$437$	−0.236068	−0.0112927
$438$	0	0
$439$	−17.1803	−0.819973	−0.409986	−	0.912092i	$-0.634467\pi$
$439$	−17.1803	−0.819973	−0.409986	+	0.912092i	$0.634467\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−27.4164	−1.30259	−0.651296	−	0.758823i	$-0.725775\pi$
$443$	−27.4164	−1.30259	−0.651296	+	0.758823i	$0.725775\pi$
$444$	0	0
$445$	−25.2918	−1.19895
$446$	0	0
$447$	−16.1803	−0.765304
$448$	0	0
$449$	−27.2705	−1.28697	−0.643487	−	0.765457i	$-0.722513\pi$
$449$	−27.2705	−1.28697	−0.643487	+	0.765457i	$0.722513\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	0	0
$453$	−22.6525	−1.06431
$454$	0	0
$455$	1.47214	0.0690148
$456$	0	0
$457$	−4.97871	−0.232894	−0.116447	−	0.993197i	$-0.537151\pi$
$457$	−4.97871	−0.232894	−0.116447	+	0.993197i	$0.537151\pi$
$458$	0	0
$459$	1.47214	0.0687134
$460$	0	0
$461$	−3.56231	−0.165913	−0.0829566	−	0.996553i	$-0.526436\pi$
$461$	−3.56231	−0.165913	−0.0829566	+	0.996553i	$0.526436\pi$
$462$	0	0
$463$	19.3607	0.899767	0.449884	−	0.893087i	$-0.351465\pi$
$463$	19.3607	0.899767	0.449884	+	0.893087i	$0.351465\pi$
$464$	0	0
$465$	22.2148	1.03019
$466$	0	0
$467$	−17.6525	−0.816859	−0.408430	−	0.912790i	$-0.633923\pi$
$467$	−17.6525	−0.816859	−0.408430	+	0.912790i	$0.633923\pi$
$468$	0	0
$469$	2.32624	0.107416
$470$	0	0
$471$	6.76393	0.311666
$472$	0	0
$473$	−4.85410	−0.223192
$474$	0	0
$475$	−2.32624	−0.106735
$476$	0	0
$477$	−2.32624	−0.106511
$478$	0	0
$479$	20.0557	0.916370	0.458185	−	0.888857i	$-0.348500\pi$
$479$	20.0557	0.916370	0.458185	+	0.888857i	$0.348500\pi$
$480$	0	0
$481$	1.61803	0.0737760
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−7.49342	−0.340259
$486$	0	0
$487$	23.8328	1.07997	0.539984	−	0.841675i	$-0.318430\pi$
$487$	23.8328	1.07997	0.539984	+	0.841675i	$0.318430\pi$
$488$	0	0
$489$	5.67376	0.256576
$490$	0	0
$491$	−9.03444	−0.407719	−0.203859	−	0.979000i	$-0.565349\pi$
$491$	−9.03444	−0.407719	−0.203859	+	0.979000i	$0.565349\pi$
$492$	0	0
$493$	7.36068	0.331508
$494$	0	0
$495$	−11.5623	−0.519687
$496$	0	0
$497$	7.61803	0.341716
$498$	0	0
$499$	−35.4508	−1.58700	−0.793499	−	0.608572i	$-0.791743\pi$
$499$	−35.4508	−1.58700	−0.793499	+	0.608572i	$0.791743\pi$
$500$	0	0
$501$	8.41641	0.376017
$502$	0	0
$503$	−0.0344419	−0.00153569	−0.000767843	−	1.00000i	$-0.500244\pi$
$503$	−0.0344419	−0.00153569	−0.000767843		1.00000i	$0.500244\pi$
$504$	0	0
$505$	−29.1459	−1.29698
$506$	0	0
$507$	12.8541	0.570871
$508$	0	0
$509$	−25.7082	−1.13950	−0.569748	−	0.821819i	$-0.692959\pi$
$509$	−25.7082	−1.13950	−0.569748	+	0.821819i	$0.692959\pi$
$510$	0	0
$511$	5.94427	0.262959
$512$	0	0
$513$	0.236068	0.0104227
$514$	0	0
$515$	−49.8885	−2.19835
$516$	0	0
$517$	−9.70820	−0.426966
$518$	0	0
$519$	17.4721	0.766942
$520$	0	0
$521$	−41.4164	−1.81449	−0.907243	−	0.420607i	$-0.861817\pi$
$521$	−41.4164	−1.81449	−0.907243	+	0.420607i	$0.861817\pi$
$522$	0	0
$523$	−28.4721	−1.24500	−0.622500	−	0.782620i	$-0.713883\pi$
$523$	−28.4721	−1.24500	−0.622500	+	0.782620i	$0.713883\pi$
$524$	0	0
$525$	9.85410	0.430068
$526$	0	0
$527$	8.48529	0.369625
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	13.5623	0.588554
$532$	0	0
$533$	−1.14590	−0.0496344
$534$	0	0
$535$	−48.4164	−2.09322
$536$	0	0
$537$	21.5623	0.930482
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−6.29180	−0.270505	−0.135253	−	0.990811i	$-0.543185\pi$
$541$	−6.29180	−0.270505	−0.135253	+	0.990811i	$0.543185\pi$
$542$	0	0
$543$	−6.41641	−0.275354
$544$	0	0
$545$	−18.9230	−0.810572
$546$	0	0
$547$	17.8541	0.763386	0.381693	−	0.924289i	$-0.375341\pi$
$547$	17.8541	0.763386	0.381693	+	0.924289i	$0.375341\pi$
$548$	0	0
$549$	−4.85410	−0.207168
$550$	0	0
$551$	1.18034	0.0502842
$552$	0	0
$553$	−9.47214	−0.402796
$554$	0	0
$555$	16.3262	0.693010
$556$	0	0
$557$	−25.2361	−1.06929	−0.534643	−	0.845078i	$-0.679554\pi$
$557$	−25.2361	−1.06929	−0.534643	+	0.845078i	$0.679554\pi$
$558$	0	0
$559$	−0.618034	−0.0261401
$560$	0	0
$561$	−4.41641	−0.186461
$562$	0	0
$563$	−3.50658	−0.147785	−0.0738923	−	0.997266i	$-0.523542\pi$
$563$	−3.50658	−0.147785	−0.0738923	+	0.997266i	$0.523542\pi$
$564$	0	0
$565$	−30.9656	−1.30273
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	21.8885	0.917615	0.458808	−	0.888536i	$-0.348277\pi$
$569$	21.8885	0.917615	0.458808	+	0.888536i	$0.348277\pi$
$570$	0	0
$571$	−24.5279	−1.02646	−0.513230	−	0.858251i	$-0.671551\pi$
$571$	−24.5279	−1.02646	−0.513230	+	0.858251i	$0.671551\pi$
$572$	0	0
$573$	12.7639	0.533221
$574$	0	0
$575$	9.85410	0.410944
$576$	0	0
$577$	10.3607	0.431321	0.215660	−	0.976468i	$-0.430810\pi$
$577$	10.3607	0.431321	0.215660	+	0.976468i	$0.430810\pi$
$578$	0	0
$579$	−24.6525	−1.02452
$580$	0	0
$581$	12.2361	0.507638
$582$	0	0
$583$	6.97871	0.289029
$584$	0	0
$585$	−1.47214	−0.0608653
$586$	0	0
$587$	−16.6738	−0.688200	−0.344100	−	0.938933i	$-0.611816\pi$
$587$	−16.6738	−0.688200	−0.344100	+	0.938933i	$0.611816\pi$
$588$	0	0
$589$	1.36068	0.0560658
$590$	0	0
$591$	−4.09017	−0.168247
$592$	0	0
$593$	39.1246	1.60666	0.803328	−	0.595537i	$-0.203061\pi$
$593$	39.1246	1.60666	0.803328	+	0.595537i	$0.203061\pi$
$594$	0	0
$595$	5.67376	0.232602
$596$	0	0
$597$	10.7984	0.441948
$598$	0	0
$599$	18.7426	0.765804	0.382902	−	0.923789i	$-0.374925\pi$
$599$	18.7426	0.765804	0.382902	+	0.923789i	$0.374925\pi$
$600$	0	0
$601$	31.5623	1.28745	0.643727	−	0.765256i	$-0.277387\pi$
$601$	31.5623	1.28745	0.643727	+	0.765256i	$0.277387\pi$
$602$	0	0
$603$	−2.32624	−0.0947317
$604$	0	0
$605$	−7.70820	−0.313383
$606$	0	0
$607$	−35.5066	−1.44117	−0.720584	−	0.693368i	$-0.756126\pi$
$607$	−35.5066	−1.44117	−0.720584	+	0.693368i	$0.756126\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	−1.23607	−0.0500060
$612$	0	0
$613$	−38.4164	−1.55162	−0.775812	−	0.630964i	$-0.782660\pi$
$613$	−38.4164	−1.55162	−0.775812	+	0.630964i	$0.782660\pi$
$614$	0	0
$615$	−11.5623	−0.466237
$616$	0	0
$617$	4.56231	0.183672	0.0918358	−	0.995774i	$-0.470727\pi$
$617$	4.56231	0.183672	0.0918358	+	0.995774i	$0.470727\pi$
$618$	0	0
$619$	−3.32624	−0.133693	−0.0668464	−	0.997763i	$-0.521294\pi$
$619$	−3.32624	−0.133693	−0.0668464	+	0.997763i	$0.521294\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	6.56231	0.262913
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	−0.708204	−0.0282829
$628$	0	0
$629$	6.23607	0.248648
$630$	0	0
$631$	11.6525	0.463878	0.231939	−	0.972730i	$-0.425493\pi$
$631$	11.6525	0.463878	0.231939	+	0.972730i	$0.425493\pi$
$632$	0	0
$633$	4.23607	0.168369
$634$	0	0
$635$	−12.8197	−0.508733
$636$	0	0
$637$	−0.381966	−0.0151340
$638$	0	0
$639$	−7.61803	−0.301365
$640$	0	0
$641$	13.9787	0.552126	0.276063	−	0.961140i	$-0.410970\pi$
$641$	13.9787	0.552126	0.276063	+	0.961140i	$0.410970\pi$
$642$	0	0
$643$	−10.7426	−0.423649	−0.211824	−	0.977308i	$-0.567940\pi$
$643$	−10.7426	−0.423649	−0.211824	+	0.977308i	$0.567940\pi$
$644$	0	0
$645$	−6.23607	−0.245545
$646$	0	0
$647$	−2.32624	−0.0914538	−0.0457269	−	0.998954i	$-0.514560\pi$
$647$	−2.32624	−0.0914538	−0.0457269	+	0.998954i	$0.514560\pi$
$648$	0	0
$649$	−40.6869	−1.59710
$650$	0	0
$651$	−5.76393	−0.225906
$652$	0	0
$653$	−11.5623	−0.452468	−0.226234	−	0.974073i	$-0.572641\pi$
$653$	−11.5623	−0.452468	−0.226234	+	0.974073i	$0.572641\pi$
$654$	0	0
$655$	−11.5623	−0.451777
$656$	0	0
$657$	−5.94427	−0.231908
$658$	0	0
$659$	13.0000	0.506408	0.253204	−	0.967413i	$-0.418516\pi$
$659$	13.0000	0.506408	0.253204	+	0.967413i	$0.418516\pi$
$660$	0	0
$661$	34.4164	1.33864	0.669322	−	0.742973i	$-0.266585\pi$
$661$	34.4164	1.33864	0.669322	+	0.742973i	$0.266585\pi$
$662$	0	0
$663$	−0.562306	−0.0218382
$664$	0	0
$665$	0.909830	0.0352817
$666$	0	0
$667$	−5.00000	−0.193601
$668$	0	0
$669$	−4.32624	−0.167262
$670$	0	0
$671$	14.5623	0.562172
$672$	0	0
$673$	−22.4164	−0.864089	−0.432045	−	0.901852i	$-0.642208\pi$
$673$	−22.4164	−0.864089	−0.432045	+	0.901852i	$0.642208\pi$
$674$	0	0
$675$	−9.85410	−0.379285
$676$	0	0
$677$	−29.9098	−1.14953	−0.574764	−	0.818319i	$-0.694906\pi$
$677$	−29.9098	−1.14953	−0.574764	+	0.818319i	$0.694906\pi$
$678$	0	0
$679$	1.94427	0.0746143
$680$	0	0
$681$	0.618034	0.0236831
$682$	0	0
$683$	33.0689	1.26535	0.632673	−	0.774419i	$-0.281958\pi$
$683$	33.0689	1.26535	0.632673	+	0.774419i	$0.281958\pi$
$684$	0	0
$685$	32.4377	1.23938
$686$	0	0
$687$	2.43769	0.0930038
$688$	0	0
$689$	0.888544	0.0338508
$690$	0	0
$691$	−15.3262	−0.583038	−0.291519	−	0.956565i	$-0.594161\pi$
$691$	−15.3262	−0.583038	−0.291519	+	0.956565i	$0.594161\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	41.6180	1.57866
$696$	0	0
$697$	−4.41641	−0.167283
$698$	0	0
$699$	16.1459	0.610694
$700$	0	0
$701$	0.437694	0.0165315	0.00826574	−	0.999966i	$-0.497369\pi$
$701$	0.437694	0.0165315	0.00826574	+	0.999966i	$0.497369\pi$
$702$	0	0
$703$	1.00000	0.0377157
$704$	0	0
$705$	−12.4721	−0.469728
$706$	0	0
$707$	7.56231	0.284410
$708$	0	0
$709$	−34.6738	−1.30220	−0.651100	−	0.758992i	$-0.725692\pi$
$709$	−34.6738	−1.30220	−0.651100	+	0.758992i	$0.725692\pi$
$710$	0	0
$711$	9.47214	0.355233
$712$	0	0
$713$	−5.76393	−0.215861
$714$	0	0
$715$	4.41641	0.165164
$716$	0	0
$717$	−5.67376	−0.211891
$718$	0	0
$719$	−28.2361	−1.05303	−0.526514	−	0.850167i	$-0.676501\pi$
$719$	−28.2361	−1.05303	−0.526514	+	0.850167i	$0.676501\pi$
$720$	0	0
$721$	12.9443	0.482070
$722$	0	0
$723$	2.05573	0.0764534
$724$	0	0
$725$	−49.2705	−1.82986
$726$	0	0
$727$	−17.0689	−0.633050	−0.316525	−	0.948584i	$-0.602516\pi$
$727$	−17.0689	−0.633050	−0.316525	+	0.948584i	$0.602516\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.38197	−0.0881002
$732$	0	0
$733$	−9.59675	−0.354464	−0.177232	−	0.984169i	$-0.556714\pi$
$733$	−9.59675	−0.354464	−0.177232	+	0.984169i	$0.556714\pi$
$734$	0	0
$735$	−3.85410	−0.142161
$736$	0	0
$737$	6.97871	0.257064
$738$	0	0
$739$	35.1803	1.29413	0.647065	−	0.762435i	$-0.275996\pi$
$739$	35.1803	1.29413	0.647065	+	0.762435i	$0.275996\pi$
$740$	0	0
$741$	−0.0901699	−0.00331248
$742$	0	0
$743$	−8.14590	−0.298844	−0.149422	−	0.988774i	$-0.547741\pi$
$743$	−8.14590	−0.298844	−0.149422	+	0.988774i	$0.547741\pi$
$744$	0	0
$745$	62.3607	2.28472
$746$	0	0
$747$	−12.2361	−0.447694
$748$	0	0
$749$	12.5623	0.459017
$750$	0	0
$751$	−2.50658	−0.0914663	−0.0457332	−	0.998954i	$-0.514562\pi$
$751$	−2.50658	−0.0914663	−0.0457332	+	0.998954i	$0.514562\pi$
$752$	0	0
$753$	13.2361	0.482349
$754$	0	0
$755$	87.3050	3.17735
$756$	0	0
$757$	−3.18034	−0.115591	−0.0577957	−	0.998328i	$-0.518407\pi$
$757$	−3.18034	−0.115591	−0.0577957	+	0.998328i	$0.518407\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	37.5279	1.36038	0.680192	−	0.733034i	$-0.261896\pi$
$761$	37.5279	1.36038	0.680192	+	0.733034i	$0.261896\pi$
$762$	0	0
$763$	4.90983	0.177748
$764$	0	0
$765$	−5.67376	−0.205135
$766$	0	0
$767$	−5.18034	−0.187051
$768$	0	0
$769$	−36.0689	−1.30068	−0.650339	−	0.759644i	$-0.725373\pi$
$769$	−36.0689	−1.30068	−0.650339	+	0.759644i	$0.725373\pi$
$770$	0	0
$771$	0.763932	0.0275123
$772$	0	0
$773$	4.23607	0.152361	0.0761804	−	0.997094i	$-0.475728\pi$
$773$	4.23607	0.152361	0.0761804	+	0.997094i	$0.475728\pi$
$774$	0	0
$775$	−56.7984	−2.04026
$776$	0	0
$777$	−4.23607	−0.151968
$778$	0	0
$779$	−0.708204	−0.0253740
$780$	0	0
$781$	22.8541	0.817784
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	−26.0689	−0.930438
$786$	0	0
$787$	−22.4377	−0.799817	−0.399909	−	0.916555i	$-0.630958\pi$
$787$	−22.4377	−0.799817	−0.399909	+	0.916555i	$0.630958\pi$
$788$	0	0
$789$	17.2918	0.615604
$790$	0	0
$791$	8.03444	0.285672
$792$	0	0
$793$	1.85410	0.0658411
$794$	0	0
$795$	8.96556	0.317976
$796$	0	0
$797$	18.2918	0.647929	0.323964	−	0.946069i	$-0.394984\pi$
$797$	18.2918	0.647929	0.323964	+	0.946069i	$0.394984\pi$
$798$	0	0
$799$	−4.76393	−0.168536
$800$	0	0
$801$	−6.56231	−0.231868
$802$	0	0
$803$	17.8328	0.629306
$804$	0	0
$805$	−3.85410	−0.135839
$806$	0	0
$807$	−8.67376	−0.305331
$808$	0	0
$809$	43.7984	1.53987	0.769934	−	0.638123i	$-0.220289\pi$
$809$	43.7984	1.53987	0.769934	+	0.638123i	$0.220289\pi$
$810$	0	0
$811$	28.7771	1.01050	0.505250	−	0.862973i	$-0.331400\pi$
$811$	28.7771	1.01050	0.505250	+	0.862973i	$0.331400\pi$
$812$	0	0
$813$	24.4164	0.856321
$814$	0	0
$815$	−21.8673	−0.765977
$816$	0	0
$817$	−0.381966	−0.0133633
$818$	0	0
$819$	0.381966	0.0133470
$820$	0	0
$821$	6.29180	0.219585	0.109793	−	0.993955i	$-0.464981\pi$
$821$	6.29180	0.219585	0.109793	+	0.993955i	$0.464981\pi$
$822$	0	0
$823$	−55.7426	−1.94307	−0.971533	−	0.236903i	$-0.923868\pi$
$823$	−55.7426	−1.94307	−0.971533	+	0.236903i	$0.923868\pi$
$824$	0	0
$825$	29.5623	1.02923
$826$	0	0
$827$	−35.0902	−1.22020	−0.610102	−	0.792323i	$-0.708872\pi$
$827$	−35.0902	−1.22020	−0.610102	+	0.792323i	$0.708872\pi$
$828$	0	0
$829$	17.2918	0.600569	0.300284	−	0.953850i	$-0.402918\pi$
$829$	17.2918	0.600569	0.300284	+	0.953850i	$0.402918\pi$
$830$	0	0
$831$	−3.32624	−0.115386
$832$	0	0
$833$	−1.47214	−0.0510065
$834$	0	0
$835$	−32.4377	−1.12255
$836$	0	0
$837$	5.76393	0.199231
$838$	0	0
$839$	37.6869	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$839$	37.6869	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	19.1246	0.658687
$844$	0	0
$845$	−49.5410	−1.70426
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−26.0344	−0.893500
$850$	0	0
$851$	−4.23607	−0.145211
$852$	0	0
$853$	−17.5967	−0.602501	−0.301251	−	0.953545i	$-0.597404\pi$
$853$	−17.5967	−0.602501	−0.301251	+	0.953545i	$0.597404\pi$
$854$	0	0
$855$	−0.909830	−0.0311155
$856$	0	0
$857$	−20.0689	−0.685540	−0.342770	−	0.939419i	$-0.611365\pi$
$857$	−20.0689	−0.685540	−0.342770	+	0.939419i	$0.611365\pi$
$858$	0	0
$859$	−52.4721	−1.79033	−0.895163	−	0.445739i	$-0.852941\pi$
$859$	−52.4721	−1.79033	−0.895163	+	0.445739i	$0.852941\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	0	0
$863$	−6.76393	−0.230247	−0.115123	−	0.993351i	$-0.536726\pi$
$863$	−6.76393	−0.230247	−0.115123	+	0.993351i	$0.536726\pi$
$864$	0	0
$865$	−67.3394	−2.28961
$866$	0	0
$867$	14.8328	0.503749
$868$	0	0
$869$	−28.4164	−0.963961
$870$	0	0
$871$	0.888544	0.0301072
$872$	0	0
$873$	−1.94427	−0.0658036
$874$	0	0
$875$	−18.7082	−0.632453
$876$	0	0
$877$	−4.94427	−0.166956	−0.0834781	−	0.996510i	$-0.526603\pi$
$877$	−4.94427	−0.166956	−0.0834781	+	0.996510i	$0.526603\pi$
$878$	0	0
$879$	−12.9443	−0.436599
$880$	0	0
$881$	33.5967	1.13190	0.565952	−	0.824438i	$-0.308509\pi$
$881$	33.5967	1.13190	0.565952	+	0.824438i	$0.308509\pi$
$882$	0	0
$883$	−3.97871	−0.133894	−0.0669472	−	0.997757i	$-0.521326\pi$
$883$	−3.97871	−0.133894	−0.0669472	+	0.997757i	$0.521326\pi$
$884$	0	0
$885$	−52.2705	−1.75705
$886$	0	0
$887$	20.7426	0.696470	0.348235	−	0.937407i	$-0.386781\pi$
$887$	20.7426	0.696470	0.348235	+	0.937407i	$0.386781\pi$
$888$	0	0
$889$	3.32624	0.111558
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−0.763932	−0.0255640
$894$	0	0
$895$	−83.1033	−2.77784
$896$	0	0
$897$	0.381966	0.0127535
$898$	0	0
$899$	28.8197	0.961189
$900$	0	0
$901$	3.42454	0.114088
$902$	0	0
$903$	1.61803	0.0538448
$904$	0	0
$905$	24.7295	0.822036
$906$	0	0
$907$	3.38197	0.112296	0.0561482	−	0.998422i	$-0.482118\pi$
$907$	3.38197	0.112296	0.0561482	+	0.998422i	$0.482118\pi$
$908$	0	0
$909$	−7.56231	−0.250826
$910$	0	0
$911$	−35.5967	−1.17937	−0.589686	−	0.807632i	$-0.700749\pi$
$911$	−35.5967	−1.17937	−0.589686	+	0.807632i	$0.700749\pi$
$912$	0	0
$913$	36.7082	1.21486
$914$	0	0
$915$	18.7082	0.618474
$916$	0	0
$917$	3.00000	0.0990687
$918$	0	0
$919$	52.5967	1.73501	0.867503	−	0.497431i	$-0.165723\pi$
$919$	52.5967	1.73501	0.867503	+	0.497431i	$0.165723\pi$
$920$	0	0
$921$	−23.6525	−0.779376
$922$	0	0
$923$	2.90983	0.0957782
$924$	0	0
$925$	−41.7426	−1.37249
$926$	0	0
$927$	−12.9443	−0.425146
$928$	0	0
$929$	41.5623	1.36362	0.681808	−	0.731532i	$-0.261194\pi$
$929$	41.5623	1.36362	0.681808	+	0.731532i	$0.261194\pi$
$930$	0	0
$931$	−0.236068	−0.00773682
$932$	0	0
$933$	−16.3820	−0.536321
$934$	0	0
$935$	17.0213	0.556656
$936$	0	0
$937$	45.5410	1.48776	0.743880	−	0.668313i	$-0.232983\pi$
$937$	45.5410	1.48776	0.743880	+	0.668313i	$0.232983\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	−16.5836	−0.540610	−0.270305	−	0.962775i	$-0.587124\pi$
$941$	−16.5836	−0.540610	−0.270305	+	0.962775i	$0.587124\pi$
$942$	0	0
$943$	3.00000	0.0976934
$944$	0	0
$945$	3.85410	0.125374
$946$	0	0
$947$	−44.4721	−1.44515	−0.722575	−	0.691292i	$-0.757042\pi$
$947$	−44.4721	−1.44515	−0.722575	+	0.691292i	$0.757042\pi$
$948$	0	0
$949$	2.27051	0.0737039
$950$	0	0
$951$	−30.0344	−0.973934
$952$	0	0
$953$	−4.96556	−0.160850	−0.0804251	−	0.996761i	$-0.525628\pi$
$953$	−4.96556	−0.160850	−0.0804251	+	0.996761i	$0.525628\pi$
$954$	0	0
$955$	−49.1935	−1.59186
$956$	0	0
$957$	−15.0000	−0.484881
$958$	0	0
$959$	−8.41641	−0.271780
$960$	0	0
$961$	2.22291	0.0717069
$962$	0	0
$963$	−12.5623	−0.404815
$964$	0	0
$965$	95.0132	3.05858
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	−0.347524	−0.0111641
$970$	0	0
$971$	−32.2705	−1.03561	−0.517805	−	0.855499i	$-0.673251\pi$
$971$	−32.2705	−1.03561	−0.517805	+	0.855499i	$0.673251\pi$
$972$	0	0
$973$	−10.7984	−0.346180
$974$	0	0
$975$	3.76393	0.120542
$976$	0	0
$977$	−9.43769	−0.301939	−0.150969	−	0.988538i	$-0.548239\pi$
$977$	−9.43769	−0.301939	−0.150969	+	0.988538i	$0.548239\pi$
$978$	0	0
$979$	19.6869	0.629197
$980$	0	0
$981$	−4.90983	−0.156759
$982$	0	0
$983$	40.5967	1.29484	0.647418	−	0.762135i	$-0.275849\pi$
$983$	40.5967	1.29484	0.647418	+	0.762135i	$0.275849\pi$
$984$	0	0
$985$	15.7639	0.502281
$986$	0	0
$987$	3.23607	0.103005
$988$	0	0
$989$	1.61803	0.0514505
$990$	0	0
$991$	−55.8673	−1.77468	−0.887341	−	0.461114i	$-0.847450\pi$
$991$	−55.8673	−1.77468	−0.887341	+	0.461114i	$0.847450\pi$
$992$	0	0
$993$	−18.3607	−0.582659
$994$	0	0
$995$	−41.6180	−1.31938
$996$	0	0
$997$	−24.5967	−0.778987	−0.389493	−	0.921029i	$-0.627350\pi$
$997$	−24.5967	−0.778987	−0.389493	+	0.921029i	$0.627350\pi$
$998$	0	0
$999$	4.23607	0.134023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bb.1.2		2
4.3	odd	2		1932.2.a.h.1.2	✓	2
12.11	even	2		5796.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.h.1.2	✓	2	4.3	odd	2
5796.2.a.j.1.1		2	12.11	even	2
7728.2.a.bb.1.2		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.85410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.85410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -0.381966 q^{13} -3.85410 q^{15} -1.47214 q^{17} -0.236068 q^{19} +1.00000 q^{21} +1.00000 q^{23} +9.85410 q^{25} -1.00000 q^{27} -5.00000 q^{29} -5.76393 q^{31} +3.00000 q^{33} -3.85410 q^{35} -4.23607 q^{37} +0.381966 q^{39} +3.00000 q^{41} +1.61803 q^{43} +3.85410 q^{45} +3.23607 q^{47} +1.00000 q^{49} +1.47214 q^{51} -2.32624 q^{53} -11.5623 q^{55} +0.236068 q^{57} +13.5623 q^{59} -4.85410 q^{61} -1.00000 q^{63} -1.47214 q^{65} -2.32624 q^{67} -1.00000 q^{69} -7.61803 q^{71} -5.94427 q^{73} -9.85410 q^{75} +3.00000 q^{77} +9.47214 q^{79} +1.00000 q^{81} -12.2361 q^{83} -5.67376 q^{85} +5.00000 q^{87} -6.56231 q^{89} +0.381966 q^{91} +5.76393 q^{93} -0.909830 q^{95} -1.94427 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} + 6 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 13 q^{25} - 2 q^{27} - 10 q^{29} - 16 q^{31} + 6 q^{33} - q^{35} - 4 q^{37} + 3 q^{39} + 6 q^{41} + q^{43} + q^{45} + 2 q^{47} + 2 q^{49} - 6 q^{51} + 11 q^{53} - 3 q^{55} - 4 q^{57} + 7 q^{59} - 3 q^{61} - 2 q^{63} + 6 q^{65} + 11 q^{67} - 2 q^{69} - 13 q^{71} + 6 q^{73} - 13 q^{75} + 6 q^{77} + 10 q^{79} + 2 q^{81} - 20 q^{83} - 27 q^{85} + 10 q^{87} + 7 q^{89} + 3 q^{91} + 16 q^{93} - 13 q^{95} + 14 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 - 3 * q^13 - q^15 + 6 * q^17 + 4 * q^19 + 2 * q^21 + 2 * q^23 + 13 * q^25 - 2 * q^27 - 10 * q^29 - 16 * q^31 + 6 * q^33 - q^35 - 4 * q^37 + 3 * q^39 + 6 * q^41 + q^43 + q^45 + 2 * q^47 + 2 * q^49 - 6 * q^51 + 11 * q^53 - 3 * q^55 - 4 * q^57 + 7 * q^59 - 3 * q^61 - 2 * q^63 + 6 * q^65 + 11 * q^67 - 2 * q^69 - 13 * q^71 + 6 * q^73 - 13 * q^75 + 6 * q^77 + 10 * q^79 + 2 * q^81 - 20 * q^83 - 27 * q^85 + 10 * q^87 + 7 * q^89 + 3 * q^91 + 16 * q^93 - 13 * q^95 + 14 * q^97 - 6 * q^99

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