LMFDB - Embedded newform 7728.2.a.cd.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:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.46506$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.653724 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.653724 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.42281 q^{11} +4.26520 q^{13} -0.653724 q^{15} +2.38853 q^{17} +5.35294 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.57264 q^{25} +1.00000 q^{27} +3.23415 q^{29} -0.388529 q^{31} +2.42281 q^{33} +0.653724 q^{35} +9.31865 q^{37} +4.26520 q^{39} +5.73026 q^{41} -10.8180 q^{43} -0.653724 q^{45} -1.18866 q^{47} +1.00000 q^{49} +2.38853 q^{51} -1.38056 q^{53} -1.58385 q^{55} +5.35294 q^{57} -12.9480 q^{59} -2.00666 q^{61} -1.00000 q^{63} -2.78826 q^{65} -10.8913 q^{67} +1.00000 q^{69} +1.73480 q^{71} +12.2264 q^{73} -4.57264 q^{75} -2.42281 q^{77} +5.69598 q^{79} +1.00000 q^{81} -6.07977 q^{83} -1.56144 q^{85} +3.23415 q^{87} +5.57264 q^{89} -4.26520 q^{91} -0.388529 q^{93} -3.49934 q^{95} +0.0112053 q^{97} +2.42281 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39} + 3 q^{41} - 9 q^{43} + 5 q^{45} - 7 q^{47} + 4 q^{49} + 12 q^{51} - 6 q^{53} + 21 q^{55} - 3 q^{57} + 2 q^{59} + 24 q^{61} - 4 q^{63} - 14 q^{65} - q^{67} + 4 q^{69} + 17 q^{71} + 16 q^{73} + 7 q^{75} - 5 q^{77} + 10 q^{79} + 4 q^{81} - 8 q^{83} + 17 q^{85} + 6 q^{87} - 3 q^{89} - 7 q^{91} - 4 q^{93} + 3 q^{95} - 2 q^{97} + 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9 + 5 * q^11 + 7 * q^13 + 5 * q^15 + 12 * q^17 - 3 * q^19 - 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 + 5 * q^33 - 5 * q^35 + 20 * q^37 + 7 * q^39 + 3 * q^41 - 9 * q^43 + 5 * q^45 - 7 * q^47 + 4 * q^49 + 12 * q^51 - 6 * q^53 + 21 * q^55 - 3 * q^57 + 2 * q^59 + 24 * q^61 - 4 * q^63 - 14 * q^65 - q^67 + 4 * q^69 + 17 * q^71 + 16 * q^73 + 7 * q^75 - 5 * q^77 + 10 * q^79 + 4 * q^81 - 8 * q^83 + 17 * q^85 + 6 * q^87 - 3 * q^89 - 7 * q^91 - 4 * q^93 + 3 * q^95 - 2 * q^97 + 5 * 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
$5$	−0.653724	−0.292354	−0.146177	−	0.989258i	$-0.546697\pi$
$5$	−0.653724	−0.292354	−0.146177	+	0.989258i	$0.546697\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.42281	0.730505	0.365252	−	0.930909i	$-0.380983\pi$
$11$	2.42281	0.730505	0.365252	+	0.930909i	$0.380983\pi$
$12$	0	0
$13$	4.26520	1.18295	0.591476	−	0.806322i	$-0.298545\pi$
$13$	4.26520	1.18295	0.591476	+	0.806322i	$0.298545\pi$
$14$	0	0
$15$	−0.653724	−0.168791
$16$	0	0
$17$	2.38853	0.579303	0.289652	−	0.957132i	$-0.406461\pi$
$17$	2.38853	0.579303	0.289652	+	0.957132i	$0.406461\pi$
$18$	0	0
$19$	5.35294	1.22805	0.614024	−	0.789288i	$-0.289550\pi$
$19$	5.35294	1.22805	0.614024	+	0.789288i	$0.289550\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.57264	−0.914529
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.23415	0.600566	0.300283	−	0.953850i	$-0.402919\pi$
$29$	3.23415	0.600566	0.300283	+	0.953850i	$0.402919\pi$
$30$	0	0
$31$	−0.388529	−0.0697818	−0.0348909	−	0.999391i	$-0.511108\pi$
$31$	−0.388529	−0.0697818	−0.0348909	+	0.999391i	$0.511108\pi$
$32$	0	0
$33$	2.42281	0.421757
$34$	0	0
$35$	0.653724	0.110500
$36$	0	0
$37$	9.31865	1.53198	0.765989	−	0.642854i	$-0.222250\pi$
$37$	9.31865	1.53198	0.765989	+	0.642854i	$0.222250\pi$
$38$	0	0
$39$	4.26520	0.682978
$40$	0	0
$41$	5.73026	0.894916	0.447458	−	0.894305i	$-0.352329\pi$
$41$	5.73026	0.894916	0.447458	+	0.894305i	$0.352329\pi$
$42$	0	0
$43$	−10.8180	−1.64973	−0.824865	−	0.565330i	$-0.808749\pi$
$43$	−10.8180	−1.64973	−0.824865	+	0.565330i	$0.808749\pi$
$44$	0	0
$45$	−0.653724	−0.0974515
$46$	0	0
$47$	−1.18866	−0.173384	−0.0866921	−	0.996235i	$-0.527630\pi$
$47$	−1.18866	−0.173384	−0.0866921	+	0.996235i	$0.527630\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.38853	0.334461
$52$	0	0
$53$	−1.38056	−0.189634	−0.0948170	−	0.995495i	$-0.530227\pi$
$53$	−1.38056	−0.189634	−0.0948170	+	0.995495i	$0.530227\pi$
$54$	0	0
$55$	−1.58385	−0.213566
$56$	0	0
$57$	5.35294	0.709014
$58$	0	0
$59$	−12.9480	−1.68568	−0.842842	−	0.538160i	$-0.819119\pi$
$59$	−12.9480	−1.68568	−0.842842	+	0.538160i	$0.819119\pi$
$60$	0	0
$61$	−2.00666	−0.256926	−0.128463	−	0.991714i	$-0.541004\pi$
$61$	−2.00666	−0.256926	−0.128463	+	0.991714i	$0.541004\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.78826	−0.345841
$66$	0	0
$67$	−10.8913	−1.33058	−0.665292	−	0.746583i	$-0.731693\pi$
$67$	−10.8913	−1.33058	−0.665292	+	0.746583i	$0.731693\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	1.73480	0.205883	0.102942	−	0.994687i	$-0.467174\pi$
$71$	1.73480	0.205883	0.102942	+	0.994687i	$0.467174\pi$
$72$	0	0
$73$	12.2264	1.43099	0.715494	−	0.698619i	$-0.246202\pi$
$73$	12.2264	1.43099	0.715494	+	0.698619i	$0.246202\pi$
$74$	0	0
$75$	−4.57264	−0.528004
$76$	0	0
$77$	−2.42281	−0.276105
$78$	0	0
$79$	5.69598	0.640848	0.320424	−	0.947274i	$-0.396175\pi$
$79$	5.69598	0.640848	0.320424	+	0.947274i	$0.396175\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.07977	−0.667341	−0.333671	−	0.942690i	$-0.608287\pi$
$83$	−6.07977	−0.667341	−0.333671	+	0.942690i	$0.608287\pi$
$84$	0	0
$85$	−1.56144	−0.169362
$86$	0	0
$87$	3.23415	0.346737
$88$	0	0
$89$	5.57264	0.590699	0.295350	−	0.955389i	$-0.404564\pi$
$89$	5.57264	0.590699	0.295350	+	0.955389i	$0.404564\pi$
$90$	0	0
$91$	−4.26520	−0.447114
$92$	0	0
$93$	−0.388529	−0.0402885
$94$	0	0
$95$	−3.49934	−0.359025
$96$	0	0
$97$	0.0112053	0.00113772	0.000568862	−	1.00000i	$-0.499819\pi$
$97$	0.0112053	0.00113772	0.000568862		1.00000i	$0.499819\pi$
$98$	0	0
$99$	2.42281	0.243502
$100$	0	0
$101$	13.8557	1.37869	0.689347	−	0.724431i	$-0.257898\pi$
$101$	13.8557	1.37869	0.689347	+	0.724431i	$0.257898\pi$
$102$	0	0
$103$	7.41029	0.730158	0.365079	−	0.930977i	$-0.381042\pi$
$103$	7.41029	0.730158	0.365079	+	0.930977i	$0.381042\pi$
$104$	0	0
$105$	0.653724	0.0637970
$106$	0	0
$107$	10.2652	0.992374	0.496187	−	0.868216i	$-0.334733\pi$
$107$	10.2652	0.992374	0.496187	+	0.868216i	$0.334733\pi$
$108$	0	0
$109$	−14.2897	−1.36871	−0.684353	−	0.729150i	$-0.739915\pi$
$109$	−14.2897	−1.36871	−0.684353	+	0.729150i	$0.739915\pi$
$110$	0	0
$111$	9.31865	0.884487
$112$	0	0
$113$	−2.50066	−0.235242	−0.117621	−	0.993059i	$-0.537527\pi$
$113$	−2.50066	−0.235242	−0.117621	+	0.993059i	$0.537527\pi$
$114$	0	0
$115$	−0.653724	−0.0609601
$116$	0	0
$117$	4.26520	0.394317
$118$	0	0
$119$	−2.38853	−0.218956
$120$	0	0
$121$	−5.12999	−0.466363
$122$	0	0
$123$	5.73026	0.516680
$124$	0	0
$125$	6.25787	0.559721
$126$	0	0
$127$	14.9592	1.32741	0.663707	−	0.747993i	$-0.268982\pi$
$127$	14.9592	1.32741	0.663707	+	0.747993i	$0.268982\pi$
$128$	0	0
$129$	−10.8180	−0.952472
$130$	0	0
$131$	−9.64575	−0.842753	−0.421377	−	0.906886i	$-0.638453\pi$
$131$	−9.64575	−0.842753	−0.421377	+	0.906886i	$0.638453\pi$
$132$	0	0
$133$	−5.35294	−0.464158
$134$	0	0
$135$	−0.653724	−0.0562636
$136$	0	0
$137$	7.96098	0.680153	0.340076	−	0.940398i	$-0.389547\pi$
$137$	7.96098	0.680153	0.340076	+	0.940398i	$0.389547\pi$
$138$	0	0
$139$	−6.94654	−0.589198	−0.294599	−	0.955621i	$-0.595186\pi$
$139$	−6.94654	−0.589198	−0.294599	+	0.955621i	$0.595186\pi$
$140$	0	0
$141$	−1.18866	−0.100103
$142$	0	0
$143$	10.3338	0.864152
$144$	0	0
$145$	−2.11424	−0.175578
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	5.82597	0.477282	0.238641	−	0.971108i	$-0.423298\pi$
$149$	5.82597	0.477282	0.238641	+	0.971108i	$0.423298\pi$
$150$	0	0
$151$	10.9803	0.893568	0.446784	−	0.894642i	$-0.352569\pi$
$151$	10.9803	0.893568	0.446784	+	0.894642i	$0.352569\pi$
$152$	0	0
$153$	2.38853	0.193101
$154$	0	0
$155$	0.253991	0.0204010
$156$	0	0
$157$	1.56598	0.124979	0.0624896	−	0.998046i	$-0.480096\pi$
$157$	1.56598	0.124979	0.0624896	+	0.998046i	$0.480096\pi$
$158$	0	0
$159$	−1.38056	−0.109485
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−15.6867	−1.22868	−0.614338	−	0.789043i	$-0.710577\pi$
$163$	−15.6867	−1.22868	−0.614338	+	0.789043i	$0.710577\pi$
$164$	0	0
$165$	−1.58385	−0.123303
$166$	0	0
$167$	−9.03771	−0.699359	−0.349679	−	0.936869i	$-0.613709\pi$
$167$	−9.03771	−0.699359	−0.349679	+	0.936869i	$0.613709\pi$
$168$	0	0
$169$	5.19190	0.399377
$170$	0	0
$171$	5.35294	0.409349
$172$	0	0
$173$	7.98879	0.607377	0.303688	−	0.952771i	$-0.401782\pi$
$173$	7.98879	0.607377	0.303688	+	0.952771i	$0.401782\pi$
$174$	0	0
$175$	4.57264	0.345659
$176$	0	0
$177$	−12.9480	−0.973231
$178$	0	0
$179$	14.9137	1.11470	0.557351	−	0.830277i	$-0.311818\pi$
$179$	14.9137	1.11470	0.557351	+	0.830277i	$0.311818\pi$
$180$	0	0
$181$	2.45709	0.182634	0.0913171	−	0.995822i	$-0.470892\pi$
$181$	2.45709	0.182634	0.0913171	+	0.995822i	$0.470892\pi$
$182$	0	0
$183$	−2.00666	−0.148337
$184$	0	0
$185$	−6.09183	−0.447880
$186$	0	0
$187$	5.78695	0.423184
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−9.66381	−0.699249	−0.349624	−	0.936890i	$-0.613691\pi$
$191$	−9.66381	−0.699249	−0.349624	+	0.936890i	$0.613691\pi$
$192$	0	0
$193$	−3.35767	−0.241691	−0.120845	−	0.992671i	$-0.538560\pi$
$193$	−3.35767	−0.241691	−0.120845	+	0.992671i	$0.538560\pi$
$194$	0	0
$195$	−2.78826	−0.199672
$196$	0	0
$197$	1.79690	0.128024	0.0640119	−	0.997949i	$-0.479610\pi$
$197$	1.79690	0.128024	0.0640119	+	0.997949i	$0.479610\pi$
$198$	0	0
$199$	8.39519	0.595119	0.297560	−	0.954703i	$-0.403827\pi$
$199$	8.39519	0.595119	0.297560	+	0.954703i	$0.403827\pi$
$200$	0	0
$201$	−10.8913	−0.768213
$202$	0	0
$203$	−3.23415	−0.226993
$204$	0	0
$205$	−3.74601	−0.261633
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	12.9691	0.897094
$210$	0	0
$211$	−21.7371	−1.49644	−0.748222	−	0.663448i	$-0.769092\pi$
$211$	−21.7371	−1.49644	−0.748222	+	0.663448i	$0.769092\pi$
$212$	0	0
$213$	1.73480	0.118867
$214$	0	0
$215$	7.07199	0.482306
$216$	0	0
$217$	0.388529	0.0263750
$218$	0	0
$219$	12.2264	0.826181
$220$	0	0
$221$	10.1875	0.685288
$222$	0	0
$223$	−26.1500	−1.75113	−0.875566	−	0.483099i	$-0.839511\pi$
$223$	−26.1500	−1.75113	−0.875566	+	0.483099i	$0.839511\pi$
$224$	0	0
$225$	−4.57264	−0.304843
$226$	0	0
$227$	16.6606	1.10580	0.552901	−	0.833247i	$-0.313521\pi$
$227$	16.6606	1.10580	0.552901	+	0.833247i	$0.313521\pi$
$228$	0	0
$229$	28.8082	1.90370	0.951851	−	0.306561i	$-0.0991783\pi$
$229$	28.8082	1.90370	0.951851	+	0.306561i	$0.0991783\pi$
$230$	0	0
$231$	−2.42281	−0.159409
$232$	0	0
$233$	11.8214	0.774447	0.387224	−	0.921986i	$-0.373434\pi$
$233$	11.8214	0.774447	0.387224	+	0.921986i	$0.373434\pi$
$234$	0	0
$235$	0.777057	0.0506896
$236$	0	0
$237$	5.69598	0.369993
$238$	0	0
$239$	19.7236	1.27581	0.637907	−	0.770114i	$-0.279801\pi$
$239$	19.7236	1.27581	0.637907	+	0.770114i	$0.279801\pi$
$240$	0	0
$241$	4.95320	0.319064	0.159532	−	0.987193i	$-0.449002\pi$
$241$	4.95320	0.319064	0.159532	+	0.987193i	$0.449002\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.653724	−0.0417649
$246$	0	0
$247$	22.8313	1.45272
$248$	0	0
$249$	−6.07977	−0.385290
$250$	0	0
$251$	31.4053	1.98228	0.991142	−	0.132809i	$-0.0423997\pi$
$251$	31.4053	1.98228	0.991142	+	0.132809i	$0.0423997\pi$
$252$	0	0
$253$	2.42281	0.152321
$254$	0	0
$255$	−1.56144	−0.0977811
$256$	0	0
$257$	−26.3697	−1.64490	−0.822448	−	0.568841i	$-0.807392\pi$
$257$	−26.3697	−1.64490	−0.822448	+	0.568841i	$0.807392\pi$
$258$	0	0
$259$	−9.31865	−0.579033
$260$	0	0
$261$	3.23415	0.200189
$262$	0	0
$263$	29.1196	1.79559	0.897795	−	0.440413i	$-0.145168\pi$
$263$	29.1196	1.79559	0.897795	+	0.440413i	$0.145168\pi$
$264$	0	0
$265$	0.902504	0.0554404
$266$	0	0
$267$	5.57264	0.341040
$268$	0	0
$269$	3.12676	0.190642	0.0953209	−	0.995447i	$-0.469612\pi$
$269$	3.12676	0.190642	0.0953209	+	0.995447i	$0.469612\pi$
$270$	0	0
$271$	16.5484	1.00525	0.502623	−	0.864506i	$-0.332368\pi$
$271$	16.5484	1.00525	0.502623	+	0.864506i	$0.332368\pi$
$272$	0	0
$273$	−4.26520	−0.258141
$274$	0	0
$275$	−11.0786	−0.668068
$276$	0	0
$277$	9.70052	0.582848	0.291424	−	0.956594i	$-0.405871\pi$
$277$	9.70052	0.582848	0.291424	+	0.956594i	$0.405871\pi$
$278$	0	0
$279$	−0.388529	−0.0232606
$280$	0	0
$281$	−23.4670	−1.39992	−0.699961	−	0.714181i	$-0.746800\pi$
$281$	−23.4670	−1.39992	−0.699961	+	0.714181i	$0.746800\pi$
$282$	0	0
$283$	25.3372	1.50614	0.753070	−	0.657941i	$-0.228572\pi$
$283$	25.3372	1.50614	0.753070	+	0.657941i	$0.228572\pi$
$284$	0	0
$285$	−3.49934	−0.207283
$286$	0	0
$287$	−5.73026	−0.338246
$288$	0	0
$289$	−11.2949	−0.664408
$290$	0	0
$291$	0.0112053	0.000656865 0
$292$	0	0
$293$	−6.25998	−0.365712	−0.182856	−	0.983140i	$-0.558534\pi$
$293$	−6.25998	−0.365712	−0.182856	+	0.983140i	$0.558534\pi$
$294$	0	0
$295$	8.46442	0.492817
$296$	0	0
$297$	2.42281	0.140586
$298$	0	0
$299$	4.26520	0.246663
$300$	0	0
$301$	10.8180	0.623539
$302$	0	0
$303$	13.8557	0.795989
$304$	0	0
$305$	1.31180	0.0751136
$306$	0	0
$307$	20.1776	1.15160	0.575798	−	0.817592i	$-0.304691\pi$
$307$	20.1776	1.15160	0.575798	+	0.817592i	$0.304691\pi$
$308$	0	0
$309$	7.41029	0.421557
$310$	0	0
$311$	−21.6888	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$311$	−21.6888	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$312$	0	0
$313$	27.9790	1.58147	0.790734	−	0.612159i	$-0.209699\pi$
$313$	27.9790	1.58147	0.790734	+	0.612159i	$0.209699\pi$
$314$	0	0
$315$	0.653724	0.0368332
$316$	0	0
$317$	19.3372	1.08608	0.543042	−	0.839705i	$-0.317272\pi$
$317$	19.3372	1.08608	0.543042	+	0.839705i	$0.317272\pi$
$318$	0	0
$319$	7.83573	0.438716
$320$	0	0
$321$	10.2652	0.572947
$322$	0	0
$323$	12.7856	0.711412
$324$	0	0
$325$	−19.5032	−1.08184
$326$	0	0
$327$	−14.2897	−0.790223
$328$	0	0
$329$	1.18866	0.0655330
$330$	0	0
$331$	17.1796	0.944275	0.472137	−	0.881525i	$-0.343483\pi$
$331$	17.1796	0.944275	0.472137	+	0.881525i	$0.343483\pi$
$332$	0	0
$333$	9.31865	0.510659
$334$	0	0
$335$	7.11991	0.389002
$336$	0	0
$337$	−20.7403	−1.12980	−0.564899	−	0.825160i	$-0.691085\pi$
$337$	−20.7403	−1.12980	−0.564899	+	0.825160i	$0.691085\pi$
$338$	0	0
$339$	−2.50066	−0.135817
$340$	0	0
$341$	−0.941331	−0.0509759
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.653724	−0.0351953
$346$	0	0
$347$	21.0008	1.12738	0.563691	−	0.825986i	$-0.309381\pi$
$347$	21.0008	1.12738	0.563691	+	0.825986i	$0.309381\pi$
$348$	0	0
$349$	23.2819	1.24625	0.623127	−	0.782121i	$-0.285862\pi$
$349$	23.2819	1.24625	0.623127	+	0.782121i	$0.285862\pi$
$350$	0	0
$351$	4.26520	0.227659
$352$	0	0
$353$	33.5471	1.78553	0.892767	−	0.450519i	$-0.148761\pi$
$353$	33.5471	1.78553	0.892767	+	0.450519i	$0.148761\pi$
$354$	0	0
$355$	−1.13408	−0.0601909
$356$	0	0
$357$	−2.38853	−0.126414
$358$	0	0
$359$	−33.1025	−1.74708	−0.873542	−	0.486749i	$-0.838183\pi$
$359$	−33.1025	−1.74708	−0.873542	+	0.486749i	$0.838183\pi$
$360$	0	0
$361$	9.65392	0.508101
$362$	0	0
$363$	−5.12999	−0.269255
$364$	0	0
$365$	−7.99267	−0.418356
$366$	0	0
$367$	−20.3576	−1.06266	−0.531329	−	0.847165i	$-0.678307\pi$
$367$	−20.3576	−1.06266	−0.531329	+	0.847165i	$0.678307\pi$
$368$	0	0
$369$	5.73026	0.298305
$370$	0	0
$371$	1.38056	0.0716749
$372$	0	0
$373$	6.21043	0.321564	0.160782	−	0.986990i	$-0.448598\pi$
$373$	6.21043	0.321564	0.160782	+	0.986990i	$0.448598\pi$
$374$	0	0
$375$	6.25787	0.323155
$376$	0	0
$377$	13.7943	0.710441
$378$	0	0
$379$	−13.9288	−0.715475	−0.357738	−	0.933822i	$-0.616452\pi$
$379$	−13.9288	−0.715475	−0.357738	+	0.933822i	$0.616452\pi$
$380$	0	0
$381$	14.9592	0.766383
$382$	0	0
$383$	18.0798	0.923833	0.461916	−	0.886923i	$-0.347162\pi$
$383$	18.0798	0.923833	0.461916	+	0.886923i	$0.347162\pi$
$384$	0	0
$385$	1.58385	0.0807205
$386$	0	0
$387$	−10.8180	−0.549910
$388$	0	0
$389$	−3.03230	−0.153744	−0.0768720	−	0.997041i	$-0.524493\pi$
$389$	−3.03230	−0.153744	−0.0768720	+	0.997041i	$0.524493\pi$
$390$	0	0
$391$	2.38853	0.120793
$392$	0	0
$393$	−9.64575	−0.486564
$394$	0	0
$395$	−3.72360	−0.187355
$396$	0	0
$397$	−37.6428	−1.88924	−0.944620	−	0.328166i	$-0.893570\pi$
$397$	−37.6428	−1.88924	−0.944620	+	0.328166i	$0.893570\pi$
$398$	0	0
$399$	−5.35294	−0.267982
$400$	0	0
$401$	−20.6604	−1.03173	−0.515865	−	0.856670i	$-0.672529\pi$
$401$	−20.6604	−1.03173	−0.515865	+	0.856670i	$0.672529\pi$
$402$	0	0
$403$	−1.65715	−0.0825485
$404$	0	0
$405$	−0.653724	−0.0324838
$406$	0	0
$407$	22.5773	1.11912
$408$	0	0
$409$	−10.8728	−0.537624	−0.268812	−	0.963193i	$-0.586631\pi$
$409$	−10.8728	−0.537624	−0.268812	+	0.963193i	$0.586631\pi$
$410$	0	0
$411$	7.96098	0.392686
$412$	0	0
$413$	12.9480	0.637129
$414$	0	0
$415$	3.97449	0.195100
$416$	0	0
$417$	−6.94654	−0.340174
$418$	0	0
$419$	−35.4574	−1.73221	−0.866104	−	0.499864i	$-0.833383\pi$
$419$	−35.4574	−1.73221	−0.866104	+	0.499864i	$0.833383\pi$
$420$	0	0
$421$	5.91028	0.288050	0.144025	−	0.989574i	$-0.453995\pi$
$421$	5.91028	0.288050	0.144025	+	0.989574i	$0.453995\pi$
$422$	0	0
$423$	−1.18866	−0.0577947
$424$	0	0
$425$	−10.9219	−0.529790
$426$	0	0
$427$	2.00666	0.0971091
$428$	0	0
$429$	10.3338	0.498919
$430$	0	0
$431$	30.2826	1.45866	0.729330	−	0.684162i	$-0.239832\pi$
$431$	30.2826	1.45866	0.729330	+	0.684162i	$0.239832\pi$
$432$	0	0
$433$	−2.48964	−0.119645	−0.0598223	−	0.998209i	$-0.519053\pi$
$433$	−2.48964	−0.119645	−0.0598223	+	0.998209i	$0.519053\pi$
$434$	0	0
$435$	−2.11424	−0.101370
$436$	0	0
$437$	5.35294	0.256066
$438$	0	0
$439$	−32.6527	−1.55843	−0.779215	−	0.626757i	$-0.784382\pi$
$439$	−32.6527	−1.55843	−0.779215	+	0.626757i	$0.784382\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	26.3354	1.25123	0.625616	−	0.780131i	$-0.284848\pi$
$443$	26.3354	1.25123	0.625616	+	0.780131i	$0.284848\pi$
$444$	0	0
$445$	−3.64297	−0.172693
$446$	0	0
$447$	5.82597	0.275559
$448$	0	0
$449$	13.8279	0.652579	0.326289	−	0.945270i	$-0.394202\pi$
$449$	13.8279	0.652579	0.326289	+	0.945270i	$0.394202\pi$
$450$	0	0
$451$	13.8833	0.653740
$452$	0	0
$453$	10.9803	0.515902
$454$	0	0
$455$	2.78826	0.130716
$456$	0	0
$457$	−21.8055	−1.02002	−0.510009	−	0.860169i	$-0.670358\pi$
$457$	−21.8055	−1.02002	−0.510009	+	0.860169i	$0.670358\pi$
$458$	0	0
$459$	2.38853	0.111487
$460$	0	0
$461$	−31.4415	−1.46438	−0.732188	−	0.681103i	$-0.761501\pi$
$461$	−31.4415	−1.46438	−0.732188	+	0.681103i	$0.761501\pi$
$462$	0	0
$463$	−39.5102	−1.83620	−0.918098	−	0.396353i	$-0.870276\pi$
$463$	−39.5102	−1.83620	−0.918098	+	0.396353i	$0.870276\pi$
$464$	0	0
$465$	0.253991	0.0117785
$466$	0	0
$467$	−21.7779	−1.00776	−0.503880	−	0.863774i	$-0.668094\pi$
$467$	−21.7779	−1.00776	−0.503880	+	0.863774i	$0.668094\pi$
$468$	0	0
$469$	10.8913	0.502913
$470$	0	0
$471$	1.56598	0.0721568
$472$	0	0
$473$	−26.2100	−1.20513
$474$	0	0
$475$	−24.4771	−1.12309
$476$	0	0
$477$	−1.38056	−0.0632114
$478$	0	0
$479$	2.54160	0.116129	0.0580643	−	0.998313i	$-0.481507\pi$
$479$	2.54160	0.116129	0.0580643	+	0.998313i	$0.481507\pi$
$480$	0	0
$481$	39.7459	1.81226
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−0.00732516	−0.000332619 0
$486$	0	0
$487$	42.7148	1.93559	0.967797	−	0.251732i	$-0.0810001\pi$
$487$	42.7148	1.93559	0.967797	+	0.251732i	$0.0810001\pi$
$488$	0	0
$489$	−15.6867	−0.709377
$490$	0	0
$491$	18.9184	0.853778	0.426889	−	0.904304i	$-0.359610\pi$
$491$	18.9184	0.853778	0.426889	+	0.904304i	$0.359610\pi$
$492$	0	0
$493$	7.72486	0.347910
$494$	0	0
$495$	−1.58385	−0.0711888
$496$	0	0
$497$	−1.73480	−0.0778166
$498$	0	0
$499$	−17.9771	−0.804766	−0.402383	−	0.915471i	$-0.631818\pi$
$499$	−17.9771	−0.804766	−0.402383	+	0.915471i	$0.631818\pi$
$500$	0	0
$501$	−9.03771	−0.403775
$502$	0	0
$503$	14.7447	0.657434	0.328717	−	0.944429i	$-0.393384\pi$
$503$	14.7447	0.657434	0.328717	+	0.944429i	$0.393384\pi$
$504$	0	0
$505$	−9.05781	−0.403067
$506$	0	0
$507$	5.19190	0.230580
$508$	0	0
$509$	−18.6492	−0.826610	−0.413305	−	0.910593i	$-0.635626\pi$
$509$	−18.6492	−0.826610	−0.413305	+	0.910593i	$0.635626\pi$
$510$	0	0
$511$	−12.2264	−0.540863
$512$	0	0
$513$	5.35294	0.236338
$514$	0	0
$515$	−4.84429	−0.213465
$516$	0	0
$517$	−2.87990	−0.126658
$518$	0	0
$519$	7.98879	0.350669
$520$	0	0
$521$	−23.0608	−1.01031	−0.505156	−	0.863028i	$-0.668565\pi$
$521$	−23.0608	−1.01031	−0.505156	+	0.863028i	$0.668565\pi$
$522$	0	0
$523$	−34.6781	−1.51637	−0.758183	−	0.652042i	$-0.773913\pi$
$523$	−34.6781	−1.51637	−0.758183	+	0.652042i	$0.773913\pi$
$524$	0	0
$525$	4.57264	0.199567
$526$	0	0
$527$	−0.928011	−0.0404248
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−12.9480	−0.561895
$532$	0	0
$533$	24.4407	1.05864
$534$	0	0
$535$	−6.71061	−0.290125
$536$	0	0
$537$	14.9137	0.643574
$538$	0	0
$539$	2.42281	0.104358
$540$	0	0
$541$	−1.72552	−0.0741859	−0.0370930	−	0.999312i	$-0.511810\pi$
$541$	−1.72552	−0.0741859	−0.0370930	+	0.999312i	$0.511810\pi$
$542$	0	0
$543$	2.45709	0.105444
$544$	0	0
$545$	9.34154	0.400148
$546$	0	0
$547$	10.6670	0.456090	0.228045	−	0.973651i	$-0.426767\pi$
$547$	10.6670	0.456090	0.228045	+	0.973651i	$0.426767\pi$
$548$	0	0
$549$	−2.00666	−0.0856421
$550$	0	0
$551$	17.3122	0.737524
$552$	0	0
$553$	−5.69598	−0.242218
$554$	0	0
$555$	−6.09183	−0.258584
$556$	0	0
$557$	10.0819	0.427183	0.213592	−	0.976923i	$-0.431484\pi$
$557$	10.0819	0.427183	0.213592	+	0.976923i	$0.431484\pi$
$558$	0	0
$559$	−46.1409	−1.95155
$560$	0	0
$561$	5.78695	0.244325
$562$	0	0
$563$	4.50924	0.190042	0.0950209	−	0.995475i	$-0.469708\pi$
$563$	4.50924	0.190042	0.0950209	+	0.995475i	$0.469708\pi$
$564$	0	0
$565$	1.63474	0.0687740
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	43.1376	1.80842	0.904212	−	0.427084i	$-0.140459\pi$
$569$	43.1376	1.80842	0.904212	+	0.427084i	$0.140459\pi$
$570$	0	0
$571$	33.1565	1.38756	0.693778	−	0.720189i	$-0.255945\pi$
$571$	33.1565	1.38756	0.693778	+	0.720189i	$0.255945\pi$
$572$	0	0
$573$	−9.66381	−0.403711
$574$	0	0
$575$	−4.57264	−0.190692
$576$	0	0
$577$	4.65972	0.193987	0.0969933	−	0.995285i	$-0.469077\pi$
$577$	4.65972	0.193987	0.0969933	+	0.995285i	$0.469077\pi$
$578$	0	0
$579$	−3.35767	−0.139540
$580$	0	0
$581$	6.07977	0.252231
$582$	0	0
$583$	−3.34483	−0.138529
$584$	0	0
$585$	−2.78826	−0.115280
$586$	0	0
$587$	18.8272	0.777084	0.388542	−	0.921431i	$-0.372979\pi$
$587$	18.8272	0.777084	0.388542	+	0.921431i	$0.372979\pi$
$588$	0	0
$589$	−2.07977	−0.0856953
$590$	0	0
$591$	1.79690	0.0739146
$592$	0	0
$593$	−16.8658	−0.692595	−0.346298	−	0.938125i	$-0.612561\pi$
$593$	−16.8658	−0.692595	−0.346298	+	0.938125i	$0.612561\pi$
$594$	0	0
$595$	1.56144	0.0640128
$596$	0	0
$597$	8.39519	0.343592
$598$	0	0
$599$	−15.3246	−0.626148	−0.313074	−	0.949729i	$-0.601359\pi$
$599$	−15.3246	−0.626148	−0.313074	+	0.949729i	$0.601359\pi$
$600$	0	0
$601$	−16.8556	−0.687553	−0.343776	−	0.939052i	$-0.611706\pi$
$601$	−16.8556	−0.687553	−0.343776	+	0.939052i	$0.611706\pi$
$602$	0	0
$603$	−10.8913	−0.443528
$604$	0	0
$605$	3.35360	0.136343
$606$	0	0
$607$	−25.0267	−1.01580	−0.507901	−	0.861415i	$-0.669578\pi$
$607$	−25.0267	−1.01580	−0.507901	+	0.861415i	$0.669578\pi$
$608$	0	0
$609$	−3.23415	−0.131054
$610$	0	0
$611$	−5.06987	−0.205105
$612$	0	0
$613$	29.4100	1.18786	0.593930	−	0.804517i	$-0.297576\pi$
$613$	29.4100	1.18786	0.593930	+	0.804517i	$0.297576\pi$
$614$	0	0
$615$	−3.74601	−0.151054
$616$	0	0
$617$	−27.8167	−1.11986	−0.559929	−	0.828541i	$-0.689172\pi$
$617$	−27.8167	−1.11986	−0.559929	+	0.828541i	$0.689172\pi$
$618$	0	0
$619$	10.0504	0.403960	0.201980	−	0.979390i	$-0.435262\pi$
$619$	10.0504	0.403960	0.201980	+	0.979390i	$0.435262\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−5.57264	−0.223263
$624$	0	0
$625$	18.7723	0.750892
$626$	0	0
$627$	12.9691	0.517938
$628$	0	0
$629$	22.2579	0.887479
$630$	0	0
$631$	−40.8542	−1.62638	−0.813190	−	0.581998i	$-0.802271\pi$
$631$	−40.8542	−1.62638	−0.813190	+	0.581998i	$0.802271\pi$
$632$	0	0
$633$	−21.7371	−0.863973
$634$	0	0
$635$	−9.77919	−0.388075
$636$	0	0
$637$	4.26520	0.168993
$638$	0	0
$639$	1.73480	0.0686278
$640$	0	0
$641$	1.15195	0.0454992	0.0227496	−	0.999741i	$-0.492758\pi$
$641$	1.15195	0.0454992	0.0227496	+	0.999741i	$0.492758\pi$
$642$	0	0
$643$	−9.15419	−0.361006	−0.180503	−	0.983574i	$-0.557773\pi$
$643$	−9.15419	−0.361006	−0.180503	+	0.983574i	$0.557773\pi$
$644$	0	0
$645$	7.07199	0.278459
$646$	0	0
$647$	−2.79994	−0.110077	−0.0550385	−	0.998484i	$-0.517528\pi$
$647$	−2.79994	−0.110077	−0.0550385	+	0.998484i	$0.517528\pi$
$648$	0	0
$649$	−31.3705	−1.23140
$650$	0	0
$651$	0.388529	0.0152276
$652$	0	0
$653$	36.1276	1.41378	0.706890	−	0.707323i	$-0.250097\pi$
$653$	36.1276	1.41378	0.706890	+	0.707323i	$0.250097\pi$
$654$	0	0
$655$	6.30566	0.246383
$656$	0	0
$657$	12.2264	0.476996
$658$	0	0
$659$	40.3794	1.57296	0.786480	−	0.617616i	$-0.211901\pi$
$659$	40.3794	1.57296	0.786480	+	0.617616i	$0.211901\pi$
$660$	0	0
$661$	−37.4732	−1.45754	−0.728769	−	0.684760i	$-0.759907\pi$
$661$	−37.4732	−1.45754	−0.728769	+	0.684760i	$0.759907\pi$
$662$	0	0
$663$	10.1875	0.395651
$664$	0	0
$665$	3.49934	0.135699
$666$	0	0
$667$	3.23415	0.125227
$668$	0	0
$669$	−26.1500	−1.01102
$670$	0	0
$671$	−4.86175	−0.187686
$672$	0	0
$673$	−25.9122	−0.998842	−0.499421	−	0.866359i	$-0.666454\pi$
$673$	−25.9122	−0.998842	−0.499421	+	0.866359i	$0.666454\pi$
$674$	0	0
$675$	−4.57264	−0.176001
$676$	0	0
$677$	−31.2444	−1.20082	−0.600409	−	0.799693i	$-0.704996\pi$
$677$	−31.2444	−1.20082	−0.600409	+	0.799693i	$0.704996\pi$
$678$	0	0
$679$	−0.0112053	−0.000430019 0
$680$	0	0
$681$	16.6606	0.638435
$682$	0	0
$683$	−33.8559	−1.29546	−0.647730	−	0.761870i	$-0.724281\pi$
$683$	−33.8559	−1.29546	−0.647730	+	0.761870i	$0.724281\pi$
$684$	0	0
$685$	−5.20429	−0.198846
$686$	0	0
$687$	28.8082	1.09910
$688$	0	0
$689$	−5.88835	−0.224328
$690$	0	0
$691$	−20.3273	−0.773287	−0.386643	−	0.922229i	$-0.626366\pi$
$691$	−20.3273	−0.773287	−0.386643	+	0.922229i	$0.626366\pi$
$692$	0	0
$693$	−2.42281	−0.0920349
$694$	0	0
$695$	4.54112	0.172255
$696$	0	0
$697$	13.6869	0.518428
$698$	0	0
$699$	11.8214	0.447127
$700$	0	0
$701$	44.7211	1.68909	0.844547	−	0.535482i	$-0.179870\pi$
$701$	44.7211	1.68909	0.844547	+	0.535482i	$0.179870\pi$
$702$	0	0
$703$	49.8822	1.88134
$704$	0	0
$705$	0.777057	0.0292657
$706$	0	0
$707$	−13.8557	−0.521097
$708$	0	0
$709$	−0.740344	−0.0278042	−0.0139021	−	0.999903i	$-0.504425\pi$
$709$	−0.740344	−0.0278042	−0.0139021	+	0.999903i	$0.504425\pi$
$710$	0	0
$711$	5.69598	0.213616
$712$	0	0
$713$	−0.388529	−0.0145505
$714$	0	0
$715$	−6.75543	−0.252639
$716$	0	0
$717$	19.7236	0.736591
$718$	0	0
$719$	45.1142	1.68248	0.841238	−	0.540666i	$-0.181828\pi$
$719$	45.1142	1.68248	0.841238	+	0.540666i	$0.181828\pi$
$720$	0	0
$721$	−7.41029	−0.275974
$722$	0	0
$723$	4.95320	0.184212
$724$	0	0
$725$	−14.7886	−0.549235
$726$	0	0
$727$	4.53395	0.168155	0.0840775	−	0.996459i	$-0.473206\pi$
$727$	4.53395	0.168155	0.0840775	+	0.996459i	$0.473206\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−25.8391	−0.955694
$732$	0	0
$733$	48.4621	1.78999	0.894994	−	0.446078i	$-0.147179\pi$
$733$	48.4621	1.78999	0.894994	+	0.446078i	$0.147179\pi$
$734$	0	0
$735$	−0.653724	−0.0241130
$736$	0	0
$737$	−26.3875	−0.971998
$738$	0	0
$739$	14.1802	0.521628	0.260814	−	0.965389i	$-0.416009\pi$
$739$	14.1802	0.521628	0.260814	+	0.965389i	$0.416009\pi$
$740$	0	0
$741$	22.8313	0.838729
$742$	0	0
$743$	−5.97171	−0.219081	−0.109540	−	0.993982i	$-0.534938\pi$
$743$	−5.97171	−0.219081	−0.109540	+	0.993982i	$0.534938\pi$
$744$	0	0
$745$	−3.80858	−0.139536
$746$	0	0
$747$	−6.07977	−0.222447
$748$	0	0
$749$	−10.2652	−0.375082
$750$	0	0
$751$	−28.3902	−1.03597	−0.517986	−	0.855389i	$-0.673318\pi$
$751$	−28.3902	−1.03597	−0.517986	+	0.855389i	$0.673318\pi$
$752$	0	0
$753$	31.4053	1.14447
$754$	0	0
$755$	−7.17812	−0.261239
$756$	0	0
$757$	−33.9995	−1.23573	−0.617866	−	0.786283i	$-0.712003\pi$
$757$	−33.9995	−1.23573	−0.617866	+	0.786283i	$0.712003\pi$
$758$	0	0
$759$	2.42281	0.0879424
$760$	0	0
$761$	0.844171	0.0306012	0.0153006	−	0.999883i	$-0.495129\pi$
$761$	0.844171	0.0306012	0.0153006	+	0.999883i	$0.495129\pi$
$762$	0	0
$763$	14.2897	0.517323
$764$	0	0
$765$	−1.56144	−0.0564540
$766$	0	0
$767$	−55.2257	−1.99408
$768$	0	0
$769$	−9.02110	−0.325309	−0.162655	−	0.986683i	$-0.552006\pi$
$769$	−9.02110	−0.325309	−0.162655	+	0.986683i	$0.552006\pi$
$770$	0	0
$771$	−26.3697	−0.949681
$772$	0	0
$773$	−16.3083	−0.586567	−0.293284	−	0.956026i	$-0.594748\pi$
$773$	−16.3083	−0.586567	−0.293284	+	0.956026i	$0.594748\pi$
$774$	0	0
$775$	1.77660	0.0638175
$776$	0	0
$777$	−9.31865	−0.334305
$778$	0	0
$779$	30.6737	1.09900
$780$	0	0
$781$	4.20310	0.150399
$782$	0	0
$783$	3.23415	0.115579
$784$	0	0
$785$	−1.02372	−0.0365382
$786$	0	0
$787$	−3.79756	−0.135369	−0.0676843	−	0.997707i	$-0.521561\pi$
$787$	−3.79756	−0.135369	−0.0676843	+	0.997707i	$0.521561\pi$
$788$	0	0
$789$	29.1196	1.03668
$790$	0	0
$791$	2.50066	0.0889131
$792$	0	0
$793$	−8.55880	−0.303932
$794$	0	0
$795$	0.902504	0.0320085
$796$	0	0
$797$	12.5502	0.444550	0.222275	−	0.974984i	$-0.428652\pi$
$797$	12.5502	0.444550	0.222275	+	0.974984i	$0.428652\pi$
$798$	0	0
$799$	−2.83915	−0.100442
$800$	0	0
$801$	5.57264	0.196900
$802$	0	0
$803$	29.6222	1.04534
$804$	0	0
$805$	0.653724	0.0230408
$806$	0	0
$807$	3.12676	0.110067
$808$	0	0
$809$	48.2047	1.69479	0.847394	−	0.530965i	$-0.178171\pi$
$809$	48.2047	1.69479	0.847394	+	0.530965i	$0.178171\pi$
$810$	0	0
$811$	20.3014	0.712879	0.356439	−	0.934318i	$-0.383991\pi$
$811$	20.3014	0.712879	0.356439	+	0.934318i	$0.383991\pi$
$812$	0	0
$813$	16.5484	0.580379
$814$	0	0
$815$	10.2548	0.359209
$816$	0	0
$817$	−57.9080	−2.02595
$818$	0	0
$819$	−4.26520	−0.149038
$820$	0	0
$821$	−35.8778	−1.25214	−0.626072	−	0.779765i	$-0.715338\pi$
$821$	−35.8778	−1.25214	−0.626072	+	0.779765i	$0.715338\pi$
$822$	0	0
$823$	−26.9324	−0.938806	−0.469403	−	0.882984i	$-0.655531\pi$
$823$	−26.9324	−0.938806	−0.469403	+	0.882984i	$0.655531\pi$
$824$	0	0
$825$	−11.0786	−0.385709
$826$	0	0
$827$	−23.0101	−0.800139	−0.400070	−	0.916485i	$-0.631014\pi$
$827$	−23.0101	−0.800139	−0.400070	+	0.916485i	$0.631014\pi$
$828$	0	0
$829$	−32.7106	−1.13609	−0.568043	−	0.822999i	$-0.692299\pi$
$829$	−32.7106	−1.13609	−0.568043	+	0.822999i	$0.692299\pi$
$830$	0	0
$831$	9.70052	0.336507
$832$	0	0
$833$	2.38853	0.0827576
$834$	0	0
$835$	5.90817	0.204461
$836$	0	0
$837$	−0.388529	−0.0134295
$838$	0	0
$839$	45.8300	1.58223	0.791114	−	0.611669i	$-0.209502\pi$
$839$	45.8300	1.58223	0.791114	+	0.611669i	$0.209502\pi$
$840$	0	0
$841$	−18.5403	−0.639320
$842$	0	0
$843$	−23.4670	−0.808246
$844$	0	0
$845$	−3.39407	−0.116760
$846$	0	0
$847$	5.12999	0.176269
$848$	0	0
$849$	25.3372	0.869570
$850$	0	0
$851$	9.31865	0.319439
$852$	0	0
$853$	3.35360	0.114825	0.0574126	−	0.998351i	$-0.481715\pi$
$853$	3.35360	0.114825	0.0574126	+	0.998351i	$0.481715\pi$
$854$	0	0
$855$	−3.49934	−0.119675
$856$	0	0
$857$	−27.7324	−0.947320	−0.473660	−	0.880708i	$-0.657067\pi$
$857$	−27.7324	−0.947320	−0.473660	+	0.880708i	$0.657067\pi$
$858$	0	0
$859$	−29.2970	−0.999602	−0.499801	−	0.866140i	$-0.666593\pi$
$859$	−29.2970	−0.999602	−0.499801	+	0.866140i	$0.666593\pi$
$860$	0	0
$861$	−5.73026	−0.195287
$862$	0	0
$863$	−19.0673	−0.649057	−0.324528	−	0.945876i	$-0.605206\pi$
$863$	−19.0673	−0.649057	−0.324528	+	0.945876i	$0.605206\pi$
$864$	0	0
$865$	−5.22247	−0.177569
$866$	0	0
$867$	−11.2949	−0.383596
$868$	0	0
$869$	13.8003	0.468142
$870$	0	0
$871$	−46.4535	−1.57402
$872$	0	0
$873$	0.0112053	0.000379241 0
$874$	0	0
$875$	−6.25787	−0.211555
$876$	0	0
$877$	−19.4591	−0.657086	−0.328543	−	0.944489i	$-0.606558\pi$
$877$	−19.4591	−0.657086	−0.328543	+	0.944489i	$0.606558\pi$
$878$	0	0
$879$	−6.25998	−0.211144
$880$	0	0
$881$	−36.1111	−1.21662	−0.608308	−	0.793701i	$-0.708151\pi$
$881$	−36.1111	−1.21662	−0.608308	+	0.793701i	$0.708151\pi$
$882$	0	0
$883$	−45.6061	−1.53477	−0.767384	−	0.641187i	$-0.778442\pi$
$883$	−45.6061	−1.53477	−0.767384	+	0.641187i	$0.778442\pi$
$884$	0	0
$885$	8.46442	0.284528
$886$	0	0
$887$	6.71371	0.225424	0.112712	−	0.993628i	$-0.464046\pi$
$887$	6.71371	0.225424	0.112712	+	0.993628i	$0.464046\pi$
$888$	0	0
$889$	−14.9592	−0.501715
$890$	0	0
$891$	2.42281	0.0811672
$892$	0	0
$893$	−6.36283	−0.212924
$894$	0	0
$895$	−9.74945	−0.325888
$896$	0	0
$897$	4.26520	0.142411
$898$	0	0
$899$	−1.25656	−0.0419086
$900$	0	0
$901$	−3.29750	−0.109856
$902$	0	0
$903$	10.8180	0.360000
$904$	0	0
$905$	−1.60626	−0.0533939
$906$	0	0
$907$	−9.74339	−0.323524	−0.161762	−	0.986830i	$-0.551718\pi$
$907$	−9.74339	−0.323524	−0.161762	+	0.986830i	$0.551718\pi$
$908$	0	0
$909$	13.8557	0.459565
$910$	0	0
$911$	59.9266	1.98546	0.992728	−	0.120379i	$-0.0384110\pi$
$911$	59.9266	1.98546	0.992728	+	0.120379i	$0.0384110\pi$
$912$	0	0
$913$	−14.7301	−0.487496
$914$	0	0
$915$	1.31180	0.0433668
$916$	0	0
$917$	9.64575	0.318531
$918$	0	0
$919$	42.5816	1.40464	0.702318	−	0.711863i	$-0.252149\pi$
$919$	42.5816	1.40464	0.702318	+	0.711863i	$0.252149\pi$
$920$	0	0
$921$	20.1776	0.664874
$922$	0	0
$923$	7.39928	0.243550
$924$	0	0
$925$	−42.6109	−1.40104
$926$	0	0
$927$	7.41029	0.243386
$928$	0	0
$929$	1.83831	0.0603131	0.0301566	−	0.999545i	$-0.490399\pi$
$929$	1.83831	0.0603131	0.0301566	+	0.999545i	$0.490399\pi$
$930$	0	0
$931$	5.35294	0.175435
$932$	0	0
$933$	−21.6888	−0.710060
$934$	0	0
$935$	−3.78307	−0.123720
$936$	0	0
$937$	−4.03863	−0.131936	−0.0659682	−	0.997822i	$-0.521014\pi$
$937$	−4.03863	−0.131936	−0.0659682	+	0.997822i	$0.521014\pi$
$938$	0	0
$939$	27.9790	0.913061
$940$	0	0
$941$	8.07541	0.263251	0.131625	−	0.991300i	$-0.457980\pi$
$941$	8.07541	0.263251	0.131625	+	0.991300i	$0.457980\pi$
$942$	0	0
$943$	5.73026	0.186603
$944$	0	0
$945$	0.653724	0.0212657
$946$	0	0
$947$	−36.1732	−1.17547	−0.587736	−	0.809053i	$-0.699981\pi$
$947$	−36.1732	−1.17547	−0.587736	+	0.809053i	$0.699981\pi$
$948$	0	0
$949$	52.1479	1.69279
$950$	0	0
$951$	19.3372	0.627051
$952$	0	0
$953$	−29.8867	−0.968125	−0.484063	−	0.875033i	$-0.660839\pi$
$953$	−29.8867	−0.968125	−0.484063	+	0.875033i	$0.660839\pi$
$954$	0	0
$955$	6.31747	0.204428
$956$	0	0
$957$	7.83573	0.253293
$958$	0	0
$959$	−7.96098	−0.257073
$960$	0	0
$961$	−30.8490	−0.995131
$962$	0	0
$963$	10.2652	0.330791
$964$	0	0
$965$	2.19499	0.0706593
$966$	0	0
$967$	0.283726	0.00912402	0.00456201	−	0.999990i	$-0.498548\pi$
$967$	0.283726	0.00912402	0.00456201	+	0.999990i	$0.498548\pi$
$968$	0	0
$969$	12.7856	0.410734
$970$	0	0
$971$	−39.8102	−1.27757	−0.638785	−	0.769385i	$-0.720563\pi$
$971$	−39.8102	−1.27757	−0.638785	+	0.769385i	$0.720563\pi$
$972$	0	0
$973$	6.94654	0.222696
$974$	0	0
$975$	−19.5032	−0.624603
$976$	0	0
$977$	50.4503	1.61405	0.807024	−	0.590518i	$-0.201077\pi$
$977$	50.4503	1.61405	0.807024	+	0.590518i	$0.201077\pi$
$978$	0	0
$979$	13.5015	0.431508
$980$	0	0
$981$	−14.2897	−0.456236
$982$	0	0
$983$	−25.3027	−0.807031	−0.403516	−	0.914973i	$-0.632212\pi$
$983$	−25.3027	−0.807031	−0.403516	+	0.914973i	$0.632212\pi$
$984$	0	0
$985$	−1.17468	−0.0374283
$986$	0	0
$987$	1.18866	0.0378355
$988$	0	0
$989$	−10.8180	−0.343992
$990$	0	0
$991$	−12.8751	−0.408990	−0.204495	−	0.978868i	$-0.565555\pi$
$991$	−12.8751	−0.408990	−0.204495	+	0.978868i	$0.565555\pi$
$992$	0	0
$993$	17.1796	0.545177
$994$	0	0
$995$	−5.48814	−0.173986
$996$	0	0
$997$	20.8464	0.660212	0.330106	−	0.943944i	$-0.392915\pi$
$997$	20.8464	0.660212	0.330106	+	0.943944i	$0.392915\pi$
$998$	0	0
$999$	9.31865	0.294829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cd.1.2		4
4.3	odd	2		483.2.a.i.1.1	✓	4
12.11	even	2		1449.2.a.p.1.4		4
28.27	even	2		3381.2.a.w.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.1	✓	4	4.3	odd	2
1449.2.a.p.1.4		4	12.11	even	2
3381.2.a.w.1.1		4	28.27	even	2
7728.2.a.cd.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.653724 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.653724 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.42281 q^{11} +4.26520 q^{13} -0.653724 q^{15} +2.38853 q^{17} +5.35294 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.57264 q^{25} +1.00000 q^{27} +3.23415 q^{29} -0.388529 q^{31} +2.42281 q^{33} +0.653724 q^{35} +9.31865 q^{37} +4.26520 q^{39} +5.73026 q^{41} -10.8180 q^{43} -0.653724 q^{45} -1.18866 q^{47} +1.00000 q^{49} +2.38853 q^{51} -1.38056 q^{53} -1.58385 q^{55} +5.35294 q^{57} -12.9480 q^{59} -2.00666 q^{61} -1.00000 q^{63} -2.78826 q^{65} -10.8913 q^{67} +1.00000 q^{69} +1.73480 q^{71} +12.2264 q^{73} -4.57264 q^{75} -2.42281 q^{77} +5.69598 q^{79} +1.00000 q^{81} -6.07977 q^{83} -1.56144 q^{85} +3.23415 q^{87} +5.57264 q^{89} -4.26520 q^{91} -0.388529 q^{93} -3.49934 q^{95} +0.0112053 q^{97} +2.42281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39} + 3 q^{41} - 9 q^{43} + 5 q^{45} - 7 q^{47} + 4 q^{49} + 12 q^{51} - 6 q^{53} + 21 q^{55} - 3 q^{57} + 2 q^{59} + 24 q^{61} - 4 q^{63} - 14 q^{65} - q^{67} + 4 q^{69} + 17 q^{71} + 16 q^{73} + 7 q^{75} - 5 q^{77} + 10 q^{79} + 4 q^{81} - 8 q^{83} + 17 q^{85} + 6 q^{87} - 3 q^{89} - 7 q^{91} - 4 q^{93} + 3 q^{95} - 2 q^{97} + 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9 + 5 * q^11 + 7 * q^13 + 5 * q^15 + 12 * q^17 - 3 * q^19 - 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 + 5 * q^33 - 5 * q^35 + 20 * q^37 + 7 * q^39 + 3 * q^41 - 9 * q^43 + 5 * q^45 - 7 * q^47 + 4 * q^49 + 12 * q^51 - 6 * q^53 + 21 * q^55 - 3 * q^57 + 2 * q^59 + 24 * q^61 - 4 * q^63 - 14 * q^65 - q^67 + 4 * q^69 + 17 * q^71 + 16 * q^73 + 7 * q^75 - 5 * q^77 + 10 * q^79 + 4 * q^81 - 8 * q^83 + 17 * q^85 + 6 * q^87 - 3 * q^89 - 7 * q^91 - 4 * q^93 + 3 * q^95 - 2 * q^97 + 5 * q^99

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