LMFDB - Embedded newform 7728.2.a.v.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:	$\Z[a_1, \ldots, a_{5}]$
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			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.47214 q^{11} -2.38197 q^{13} +1.38197 q^{15} -1.00000 q^{17} +3.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.09017 q^{25} -1.00000 q^{27} -7.47214 q^{29} +3.76393 q^{31} -5.47214 q^{33} +1.38197 q^{35} +1.47214 q^{37} +2.38197 q^{39} -4.70820 q^{41} -8.09017 q^{43} -1.38197 q^{45} -1.70820 q^{47} +1.00000 q^{49} +1.00000 q^{51} -3.38197 q^{53} -7.56231 q^{55} -3.00000 q^{57} +6.14590 q^{59} +13.7984 q^{61} -1.00000 q^{63} +3.29180 q^{65} -4.14590 q^{67} -1.00000 q^{69} +3.90983 q^{71} +2.70820 q^{73} +3.09017 q^{75} -5.47214 q^{77} -0.527864 q^{79} +1.00000 q^{81} +3.00000 q^{83} +1.38197 q^{85} +7.47214 q^{87} +3.14590 q^{89} +2.38197 q^{91} -3.76393 q^{93} -4.14590 q^{95} +5.00000 q^{97} +5.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 7 q^{13} + 5 q^{15} - 2 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} + 5 q^{25} - 2 q^{27} - 6 q^{29} + 12 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + 7 q^{39} + 4 q^{41} - 5 q^{43} - 5 q^{45} + 10 q^{47} + 2 q^{49} + 2 q^{51} - 9 q^{53} + 5 q^{55} - 6 q^{57} + 19 q^{59} + 3 q^{61} - 2 q^{63} + 20 q^{65} - 15 q^{67} - 2 q^{69} + 19 q^{71} - 8 q^{73} - 5 q^{75} - 2 q^{77} - 10 q^{79} + 2 q^{81} + 6 q^{83} + 5 q^{85} + 6 q^{87} + 13 q^{89} + 7 q^{91} - 12 q^{93} - 15 q^{95} + 10 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 7 * q^13 + 5 * q^15 - 2 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^23 + 5 * q^25 - 2 * q^27 - 6 * q^29 + 12 * q^31 - 2 * q^33 + 5 * q^35 - 6 * q^37 + 7 * q^39 + 4 * q^41 - 5 * q^43 - 5 * q^45 + 10 * q^47 + 2 * q^49 + 2 * q^51 - 9 * q^53 + 5 * q^55 - 6 * q^57 + 19 * q^59 + 3 * q^61 - 2 * q^63 + 20 * q^65 - 15 * q^67 - 2 * q^69 + 19 * q^71 - 8 * q^73 - 5 * q^75 - 2 * q^77 - 10 * q^79 + 2 * q^81 + 6 * q^83 + 5 * q^85 + 6 * q^87 + 13 * q^89 + 7 * q^91 - 12 * q^93 - 15 * q^95 + 10 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.38197	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$5$	−1.38197	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.47214	1.64991	0.824956	−	0.565198i	$-0.191200\pi$
$11$	5.47214	1.64991	0.824956	+	0.565198i	$0.191200\pi$
$12$	0	0
$13$	−2.38197	−0.660639	−0.330319	−	0.943869i	$-0.607156\pi$
$13$	−2.38197	−0.660639	−0.330319	+	0.943869i	$0.607156\pi$
$14$	0	0
$15$	1.38197	0.356822
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.09017	−0.618034
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.47214	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$29$	−7.47214	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$30$	0	0
$31$	3.76393	0.676022	0.338011	−	0.941142i	$-0.390246\pi$
$31$	3.76393	0.676022	0.338011	+	0.941142i	$0.390246\pi$
$32$	0	0
$33$	−5.47214	−0.952577
$34$	0	0
$35$	1.38197	0.233595
$36$	0	0
$37$	1.47214	0.242018	0.121009	−	0.992651i	$-0.461387\pi$
$37$	1.47214	0.242018	0.121009	+	0.992651i	$0.461387\pi$
$38$	0	0
$39$	2.38197	0.381420
$40$	0	0
$41$	−4.70820	−0.735298	−0.367649	−	0.929965i	$-0.619837\pi$
$41$	−4.70820	−0.735298	−0.367649	+	0.929965i	$0.619837\pi$
$42$	0	0
$43$	−8.09017	−1.23374	−0.616870	−	0.787065i	$-0.711599\pi$
$43$	−8.09017	−1.23374	−0.616870	+	0.787065i	$0.711599\pi$
$44$	0	0
$45$	−1.38197	−0.206011
$46$	0	0
$47$	−1.70820	−0.249167	−0.124584	−	0.992209i	$-0.539759\pi$
$47$	−1.70820	−0.249167	−0.124584	+	0.992209i	$0.539759\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−3.38197	−0.464549	−0.232274	−	0.972650i	$-0.574617\pi$
$53$	−3.38197	−0.464549	−0.232274	+	0.972650i	$0.574617\pi$
$54$	0	0
$55$	−7.56231	−1.01970
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	6.14590	0.800128	0.400064	−	0.916487i	$-0.368988\pi$
$59$	6.14590	0.800128	0.400064	+	0.916487i	$0.368988\pi$
$60$	0	0
$61$	13.7984	1.76670	0.883350	−	0.468713i	$-0.155282\pi$
$61$	13.7984	1.76670	0.883350	+	0.468713i	$0.155282\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.29180	0.408297
$66$	0	0
$67$	−4.14590	−0.506502	−0.253251	−	0.967401i	$-0.581500\pi$
$67$	−4.14590	−0.506502	−0.253251	+	0.967401i	$0.581500\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.90983	0.464011	0.232006	−	0.972714i	$-0.425471\pi$
$71$	3.90983	0.464011	0.232006	+	0.972714i	$0.425471\pi$
$72$	0	0
$73$	2.70820	0.316971	0.158486	−	0.987361i	$-0.449339\pi$
$73$	2.70820	0.316971	0.158486	+	0.987361i	$0.449339\pi$
$74$	0	0
$75$	3.09017	0.356822
$76$	0	0
$77$	−5.47214	−0.623608
$78$	0	0
$79$	−0.527864	−0.0593893	−0.0296947	−	0.999559i	$-0.509453\pi$
$79$	−0.527864	−0.0593893	−0.0296947	+	0.999559i	$0.509453\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	1.38197	0.149895
$86$	0	0
$87$	7.47214	0.801097
$88$	0	0
$89$	3.14590	0.333465	0.166732	−	0.986002i	$-0.446678\pi$
$89$	3.14590	0.333465	0.166732	+	0.986002i	$0.446678\pi$
$90$	0	0
$91$	2.38197	0.249698
$92$	0	0
$93$	−3.76393	−0.390302
$94$	0	0
$95$	−4.14590	−0.425360
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	5.47214	0.549970
$100$	0	0
$101$	−1.85410	−0.184490	−0.0922450	−	0.995736i	$-0.529404\pi$
$101$	−1.85410	−0.184490	−0.0922450	+	0.995736i	$0.529404\pi$
$102$	0	0
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	0	0
$105$	−1.38197	−0.134866
$106$	0	0
$107$	−11.7984	−1.14059	−0.570296	−	0.821439i	$-0.693171\pi$
$107$	−11.7984	−1.14059	−0.570296	+	0.821439i	$0.693171\pi$
$108$	0	0
$109$	−6.32624	−0.605944	−0.302972	−	0.953000i	$-0.597979\pi$
$109$	−6.32624	−0.605944	−0.302972	+	0.953000i	$0.597979\pi$
$110$	0	0
$111$	−1.47214	−0.139729
$112$	0	0
$113$	6.32624	0.595122	0.297561	−	0.954703i	$-0.403827\pi$
$113$	6.32624	0.595122	0.297561	+	0.954703i	$0.403827\pi$
$114$	0	0
$115$	−1.38197	−0.128869
$116$	0	0
$117$	−2.38197	−0.220213
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	18.9443	1.72221
$122$	0	0
$123$	4.70820	0.424524
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	−18.8541	−1.67303	−0.836516	−	0.547943i	$-0.815411\pi$
$127$	−18.8541	−1.67303	−0.836516	+	0.547943i	$0.815411\pi$
$128$	0	0
$129$	8.09017	0.712300
$130$	0	0
$131$	17.9443	1.56780	0.783899	−	0.620888i	$-0.213228\pi$
$131$	17.9443	1.56780	0.783899	+	0.620888i	$0.213228\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	1.38197	0.118941
$136$	0	0
$137$	9.18034	0.784329	0.392165	−	0.919895i	$-0.371726\pi$
$137$	9.18034	0.784329	0.392165	+	0.919895i	$0.371726\pi$
$138$	0	0
$139$	16.3262	1.38477	0.692387	−	0.721527i	$-0.256559\pi$
$139$	16.3262	1.38477	0.692387	+	0.721527i	$0.256559\pi$
$140$	0	0
$141$	1.70820	0.143857
$142$	0	0
$143$	−13.0344	−1.08999
$144$	0	0
$145$	10.3262	0.857547
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−16.1803	−1.32555	−0.662773	−	0.748821i	$-0.730620\pi$
$149$	−16.1803	−1.32555	−0.662773	+	0.748821i	$0.730620\pi$
$150$	0	0
$151$	−0.763932	−0.0621679	−0.0310840	−	0.999517i	$-0.509896\pi$
$151$	−0.763932	−0.0621679	−0.0310840	+	0.999517i	$0.509896\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−5.20163	−0.417805
$156$	0	0
$157$	−20.6525	−1.64825	−0.824124	−	0.566410i	$-0.808332\pi$
$157$	−20.6525	−1.64825	−0.824124	+	0.566410i	$0.808332\pi$
$158$	0	0
$159$	3.38197	0.268207
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−2.79837	−0.219186	−0.109593	−	0.993977i	$-0.534955\pi$
$163$	−2.79837	−0.219186	−0.109593	+	0.993977i	$0.534955\pi$
$164$	0	0
$165$	7.56231	0.588725
$166$	0	0
$167$	−14.5279	−1.12420	−0.562100	−	0.827069i	$-0.690006\pi$
$167$	−14.5279	−1.12420	−0.562100	+	0.827069i	$0.690006\pi$
$168$	0	0
$169$	−7.32624	−0.563557
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	−9.18034	−0.697968	−0.348984	−	0.937129i	$-0.613473\pi$
$173$	−9.18034	−0.697968	−0.348984	+	0.937129i	$0.613473\pi$
$174$	0	0
$175$	3.09017	0.233595
$176$	0	0
$177$	−6.14590	−0.461954
$178$	0	0
$179$	−7.85410	−0.587043	−0.293522	−	0.955952i	$-0.594827\pi$
$179$	−7.85410	−0.587043	−0.293522	+	0.955952i	$0.594827\pi$
$180$	0	0
$181$	7.47214	0.555399	0.277700	−	0.960668i	$-0.410428\pi$
$181$	7.47214	0.555399	0.277700	+	0.960668i	$0.410428\pi$
$182$	0	0
$183$	−13.7984	−1.02001
$184$	0	0
$185$	−2.03444	−0.149575
$186$	0	0
$187$	−5.47214	−0.400162
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−6.29180	−0.455258	−0.227629	−	0.973748i	$-0.573097\pi$
$191$	−6.29180	−0.455258	−0.227629	+	0.973748i	$0.573097\pi$
$192$	0	0
$193$	−14.1803	−1.02072	−0.510362	−	0.859960i	$-0.670488\pi$
$193$	−14.1803	−1.02072	−0.510362	+	0.859960i	$0.670488\pi$
$194$	0	0
$195$	−3.29180	−0.235730
$196$	0	0
$197$	−12.8541	−0.915817	−0.457908	−	0.888999i	$-0.651401\pi$
$197$	−12.8541	−0.915817	−0.457908	+	0.888999i	$0.651401\pi$
$198$	0	0
$199$	18.0344	1.27843	0.639214	−	0.769029i	$-0.279260\pi$
$199$	18.0344	1.27843	0.639214	+	0.769029i	$0.279260\pi$
$200$	0	0
$201$	4.14590	0.292429
$202$	0	0
$203$	7.47214	0.524441
$204$	0	0
$205$	6.50658	0.454439
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	16.4164	1.13555
$210$	0	0
$211$	−26.4164	−1.81858	−0.909290	−	0.416163i	$-0.863375\pi$
$211$	−26.4164	−1.81858	−0.909290	+	0.416163i	$0.863375\pi$
$212$	0	0
$213$	−3.90983	−0.267897
$214$	0	0
$215$	11.1803	0.762493
$216$	0	0
$217$	−3.76393	−0.255512
$218$	0	0
$219$	−2.70820	−0.183003
$220$	0	0
$221$	2.38197	0.160228
$222$	0	0
$223$	20.3262	1.36115	0.680573	−	0.732680i	$-0.261731\pi$
$223$	20.3262	1.36115	0.680573	+	0.732680i	$0.261731\pi$
$224$	0	0
$225$	−3.09017	−0.206011
$226$	0	0
$227$	8.79837	0.583969	0.291984	−	0.956423i	$-0.405684\pi$
$227$	8.79837	0.583969	0.291984	+	0.956423i	$0.405684\pi$
$228$	0	0
$229$	−21.5623	−1.42488	−0.712439	−	0.701734i	$-0.752409\pi$
$229$	−21.5623	−1.42488	−0.712439	+	0.701734i	$0.752409\pi$
$230$	0	0
$231$	5.47214	0.360040
$232$	0	0
$233$	19.2705	1.26245	0.631227	−	0.775599i	$-0.282552\pi$
$233$	19.2705	1.26245	0.631227	+	0.775599i	$0.282552\pi$
$234$	0	0
$235$	2.36068	0.153994
$236$	0	0
$237$	0.527864	0.0342885
$238$	0	0
$239$	17.8541	1.15489	0.577443	−	0.816431i	$-0.304051\pi$
$239$	17.8541	1.15489	0.577443	+	0.816431i	$0.304051\pi$
$240$	0	0
$241$	−20.4164	−1.31514	−0.657568	−	0.753395i	$-0.728415\pi$
$241$	−20.4164	−1.31514	−0.657568	+	0.753395i	$0.728415\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.38197	−0.0882906
$246$	0	0
$247$	−7.14590	−0.454683
$248$	0	0
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−1.23607	−0.0780199	−0.0390100	−	0.999239i	$-0.512420\pi$
$251$	−1.23607	−0.0780199	−0.0390100	+	0.999239i	$0.512420\pi$
$252$	0	0
$253$	5.47214	0.344030
$254$	0	0
$255$	−1.38197	−0.0865421
$256$	0	0
$257$	−31.5967	−1.97095	−0.985475	−	0.169818i	$-0.945682\pi$
$257$	−31.5967	−1.97095	−0.985475	+	0.169818i	$0.945682\pi$
$258$	0	0
$259$	−1.47214	−0.0914741
$260$	0	0
$261$	−7.47214	−0.462514
$262$	0	0
$263$	−23.6525	−1.45847	−0.729237	−	0.684261i	$-0.760125\pi$
$263$	−23.6525	−1.45847	−0.729237	+	0.684261i	$0.760125\pi$
$264$	0	0
$265$	4.67376	0.287107
$266$	0	0
$267$	−3.14590	−0.192526
$268$	0	0
$269$	−14.2705	−0.870088	−0.435044	−	0.900409i	$-0.643267\pi$
$269$	−14.2705	−0.870088	−0.435044	+	0.900409i	$0.643267\pi$
$270$	0	0
$271$	−31.8328	−1.93371	−0.966853	−	0.255334i	$-0.917815\pi$
$271$	−31.8328	−1.93371	−0.966853	+	0.255334i	$0.917815\pi$
$272$	0	0
$273$	−2.38197	−0.144163
$274$	0	0
$275$	−16.9098	−1.01970
$276$	0	0
$277$	−19.1459	−1.15037	−0.575183	−	0.818025i	$-0.695069\pi$
$277$	−19.1459	−1.15037	−0.575183	+	0.818025i	$0.695069\pi$
$278$	0	0
$279$	3.76393	0.225341
$280$	0	0
$281$	27.7082	1.65293	0.826466	−	0.562986i	$-0.190348\pi$
$281$	27.7082	1.65293	0.826466	+	0.562986i	$0.190348\pi$
$282$	0	0
$283$	27.0902	1.61034	0.805172	−	0.593042i	$-0.202073\pi$
$283$	27.0902	1.61034	0.805172	+	0.593042i	$0.202073\pi$
$284$	0	0
$285$	4.14590	0.245582
$286$	0	0
$287$	4.70820	0.277916
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−5.00000	−0.293105
$292$	0	0
$293$	3.41641	0.199589	0.0997943	−	0.995008i	$-0.468182\pi$
$293$	3.41641	0.199589	0.0997943	+	0.995008i	$0.468182\pi$
$294$	0	0
$295$	−8.49342	−0.494506
$296$	0	0
$297$	−5.47214	−0.317526
$298$	0	0
$299$	−2.38197	−0.137753
$300$	0	0
$301$	8.09017	0.466310
$302$	0	0
$303$	1.85410	0.106515
$304$	0	0
$305$	−19.0689	−1.09188
$306$	0	0
$307$	22.7082	1.29603	0.648013	−	0.761629i	$-0.275600\pi$
$307$	22.7082	1.29603	0.648013	+	0.761629i	$0.275600\pi$
$308$	0	0
$309$	−4.94427	−0.281270
$310$	0	0
$311$	−21.5066	−1.21953	−0.609763	−	0.792584i	$-0.708735\pi$
$311$	−21.5066	−1.21953	−0.609763	+	0.792584i	$0.708735\pi$
$312$	0	0
$313$	−2.47214	−0.139733	−0.0698667	−	0.997556i	$-0.522257\pi$
$313$	−2.47214	−0.139733	−0.0698667	+	0.997556i	$0.522257\pi$
$314$	0	0
$315$	1.38197	0.0778650
$316$	0	0
$317$	−24.7984	−1.39282	−0.696408	−	0.717646i	$-0.745219\pi$
$317$	−24.7984	−1.39282	−0.696408	+	0.717646i	$0.745219\pi$
$318$	0	0
$319$	−40.8885	−2.28932
$320$	0	0
$321$	11.7984	0.658521
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	7.36068	0.408297
$326$	0	0
$327$	6.32624	0.349842
$328$	0	0
$329$	1.70820	0.0941763
$330$	0	0
$331$	−18.9443	−1.04127	−0.520636	−	0.853779i	$-0.674305\pi$
$331$	−18.9443	−1.04127	−0.520636	+	0.853779i	$0.674305\pi$
$332$	0	0
$333$	1.47214	0.0806726
$334$	0	0
$335$	5.72949	0.313035
$336$	0	0
$337$	12.6738	0.690384	0.345192	−	0.938532i	$-0.387814\pi$
$337$	12.6738	0.690384	0.345192	+	0.938532i	$0.387814\pi$
$338$	0	0
$339$	−6.32624	−0.343594
$340$	0	0
$341$	20.5967	1.11538
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.38197	0.0744025
$346$	0	0
$347$	6.41641	0.344451	0.172225	−	0.985058i	$-0.444904\pi$
$347$	6.41641	0.344451	0.172225	+	0.985058i	$0.444904\pi$
$348$	0	0
$349$	−36.2148	−1.93853	−0.969266	−	0.246013i	$-0.920879\pi$
$349$	−36.2148	−1.93853	−0.969266	+	0.246013i	$0.920879\pi$
$350$	0	0
$351$	2.38197	0.127140
$352$	0	0
$353$	11.6525	0.620199	0.310099	−	0.950704i	$-0.399638\pi$
$353$	11.6525	0.620199	0.310099	+	0.950704i	$0.399638\pi$
$354$	0	0
$355$	−5.40325	−0.286775
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	16.5066	0.871184	0.435592	−	0.900144i	$-0.356539\pi$
$359$	16.5066	0.871184	0.435592	+	0.900144i	$0.356539\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−18.9443	−0.994316
$364$	0	0
$365$	−3.74265	−0.195899
$366$	0	0
$367$	−24.5623	−1.28214	−0.641071	−	0.767482i	$-0.721510\pi$
$367$	−24.5623	−1.28214	−0.641071	+	0.767482i	$0.721510\pi$
$368$	0	0
$369$	−4.70820	−0.245099
$370$	0	0
$371$	3.38197	0.175583
$372$	0	0
$373$	−12.0557	−0.624222	−0.312111	−	0.950046i	$-0.601036\pi$
$373$	−12.0557	−0.624222	−0.312111	+	0.950046i	$0.601036\pi$
$374$	0	0
$375$	−11.1803	−0.577350
$376$	0	0
$377$	17.7984	0.916663
$378$	0	0
$379$	−32.3607	−1.66226	−0.831128	−	0.556081i	$-0.812304\pi$
$379$	−32.3607	−1.66226	−0.831128	+	0.556081i	$0.812304\pi$
$380$	0	0
$381$	18.8541	0.965925
$382$	0	0
$383$	27.4721	1.40376	0.701880	−	0.712295i	$-0.252344\pi$
$383$	27.4721	1.40376	0.701880	+	0.712295i	$0.252344\pi$
$384$	0	0
$385$	7.56231	0.385411
$386$	0	0
$387$	−8.09017	−0.411246
$388$	0	0
$389$	8.41641	0.426729	0.213364	−	0.976973i	$-0.431558\pi$
$389$	8.41641	0.426729	0.213364	+	0.976973i	$0.431558\pi$
$390$	0	0
$391$	−1.00000	−0.0505722
$392$	0	0
$393$	−17.9443	−0.905169
$394$	0	0
$395$	0.729490	0.0367046
$396$	0	0
$397$	−29.7082	−1.49101	−0.745506	−	0.666499i	$-0.767792\pi$
$397$	−29.7082	−1.49101	−0.745506	+	0.666499i	$0.767792\pi$
$398$	0	0
$399$	3.00000	0.150188
$400$	0	0
$401$	−5.58359	−0.278831	−0.139416	−	0.990234i	$-0.544522\pi$
$401$	−5.58359	−0.278831	−0.139416	+	0.990234i	$0.544522\pi$
$402$	0	0
$403$	−8.96556	−0.446606
$404$	0	0
$405$	−1.38197	−0.0686704
$406$	0	0
$407$	8.05573	0.399308
$408$	0	0
$409$	22.4164	1.10842	0.554210	−	0.832377i	$-0.313020\pi$
$409$	22.4164	1.10842	0.554210	+	0.832377i	$0.313020\pi$
$410$	0	0
$411$	−9.18034	−0.452833
$412$	0	0
$413$	−6.14590	−0.302420
$414$	0	0
$415$	−4.14590	−0.203514
$416$	0	0
$417$	−16.3262	−0.799499
$418$	0	0
$419$	−15.2705	−0.746013	−0.373007	−	0.927829i	$-0.621673\pi$
$419$	−15.2705	−0.746013	−0.373007	+	0.927829i	$0.621673\pi$
$420$	0	0
$421$	−15.6180	−0.761176	−0.380588	−	0.924745i	$-0.624278\pi$
$421$	−15.6180	−0.761176	−0.380588	+	0.924745i	$0.624278\pi$
$422$	0	0
$423$	−1.70820	−0.0830557
$424$	0	0
$425$	3.09017	0.149895
$426$	0	0
$427$	−13.7984	−0.667750
$428$	0	0
$429$	13.0344	0.629309
$430$	0	0
$431$	32.1591	1.54905	0.774524	−	0.632545i	$-0.217990\pi$
$431$	32.1591	1.54905	0.774524	+	0.632545i	$0.217990\pi$
$432$	0	0
$433$	−14.1803	−0.681464	−0.340732	−	0.940161i	$-0.610675\pi$
$433$	−14.1803	−0.681464	−0.340732	+	0.940161i	$0.610675\pi$
$434$	0	0
$435$	−10.3262	−0.495105
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	3.18034	0.151789	0.0758947	−	0.997116i	$-0.475819\pi$
$439$	3.18034	0.151789	0.0758947	+	0.997116i	$0.475819\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.58359	−0.217773	−0.108887	−	0.994054i	$-0.534729\pi$
$443$	−4.58359	−0.217773	−0.108887	+	0.994054i	$0.534729\pi$
$444$	0	0
$445$	−4.34752	−0.206092
$446$	0	0
$447$	16.1803	0.765304
$448$	0	0
$449$	36.5066	1.72285	0.861426	−	0.507883i	$-0.169572\pi$
$449$	36.5066	1.72285	0.861426	+	0.507883i	$0.169572\pi$
$450$	0	0
$451$	−25.7639	−1.21318
$452$	0	0
$453$	0.763932	0.0358927
$454$	0	0
$455$	−3.29180	−0.154322
$456$	0	0
$457$	18.9098	0.884565	0.442282	−	0.896876i	$-0.354169\pi$
$457$	18.9098	0.884565	0.442282	+	0.896876i	$0.354169\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	0.618034	0.0287847	0.0143924	−	0.999896i	$-0.495419\pi$
$461$	0.618034	0.0287847	0.0143924	+	0.999896i	$0.495419\pi$
$462$	0	0
$463$	−37.6525	−1.74986	−0.874929	−	0.484250i	$-0.839092\pi$
$463$	−37.6525	−1.74986	−0.874929	+	0.484250i	$0.839092\pi$
$464$	0	0
$465$	5.20163	0.241220
$466$	0	0
$467$	−38.7771	−1.79439	−0.897195	−	0.441635i	$-0.854399\pi$
$467$	−38.7771	−1.79439	−0.897195	+	0.441635i	$0.854399\pi$
$468$	0	0
$469$	4.14590	0.191440
$470$	0	0
$471$	20.6525	0.951616
$472$	0	0
$473$	−44.2705	−2.03556
$474$	0	0
$475$	−9.27051	−0.425360
$476$	0	0
$477$	−3.38197	−0.154850
$478$	0	0
$479$	−18.2361	−0.833227	−0.416614	−	0.909084i	$-0.636783\pi$
$479$	−18.2361	−0.833227	−0.416614	+	0.909084i	$0.636783\pi$
$480$	0	0
$481$	−3.50658	−0.159886
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−6.90983	−0.313759
$486$	0	0
$487$	12.7082	0.575864	0.287932	−	0.957651i	$-0.407032\pi$
$487$	12.7082	0.575864	0.287932	+	0.957651i	$0.407032\pi$
$488$	0	0
$489$	2.79837	0.126547
$490$	0	0
$491$	−15.9787	−0.721109	−0.360555	−	0.932738i	$-0.617413\pi$
$491$	−15.9787	−0.721109	−0.360555	+	0.932738i	$0.617413\pi$
$492$	0	0
$493$	7.47214	0.336528
$494$	0	0
$495$	−7.56231	−0.339900
$496$	0	0
$497$	−3.90983	−0.175380
$498$	0	0
$499$	10.0344	0.449203	0.224602	−	0.974451i	$-0.427892\pi$
$499$	10.0344	0.449203	0.224602	+	0.974451i	$0.427892\pi$
$500$	0	0
$501$	14.5279	0.649057
$502$	0	0
$503$	37.7426	1.68286	0.841431	−	0.540365i	$-0.181714\pi$
$503$	37.7426	1.68286	0.841431	+	0.540365i	$0.181714\pi$
$504$	0	0
$505$	2.56231	0.114021
$506$	0	0
$507$	7.32624	0.325370
$508$	0	0
$509$	8.18034	0.362587	0.181294	−	0.983429i	$-0.441972\pi$
$509$	8.18034	0.362587	0.181294	+	0.983429i	$0.441972\pi$
$510$	0	0
$511$	−2.70820	−0.119804
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	−6.83282	−0.301090
$516$	0	0
$517$	−9.34752	−0.411104
$518$	0	0
$519$	9.18034	0.402972
$520$	0	0
$521$	33.4164	1.46400	0.732000	−	0.681305i	$-0.238587\pi$
$521$	33.4164	1.46400	0.732000	+	0.681305i	$0.238587\pi$
$522$	0	0
$523$	14.3607	0.627949	0.313974	−	0.949431i	$-0.398339\pi$
$523$	14.3607	0.627949	0.313974	+	0.949431i	$0.398339\pi$
$524$	0	0
$525$	−3.09017	−0.134866
$526$	0	0
$527$	−3.76393	−0.163959
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.14590	0.266709
$532$	0	0
$533$	11.2148	0.485766
$534$	0	0
$535$	16.3050	0.704925
$536$	0	0
$537$	7.85410	0.338930
$538$	0	0
$539$	5.47214	0.235702
$540$	0	0
$541$	−38.6525	−1.66180	−0.830900	−	0.556422i	$-0.812174\pi$
$541$	−38.6525	−1.66180	−0.830900	+	0.556422i	$0.812174\pi$
$542$	0	0
$543$	−7.47214	−0.320660
$544$	0	0
$545$	8.74265	0.374494
$546$	0	0
$547$	−7.27051	−0.310865	−0.155432	−	0.987847i	$-0.549677\pi$
$547$	−7.27051	−0.310865	−0.155432	+	0.987847i	$0.549677\pi$
$548$	0	0
$549$	13.7984	0.588900
$550$	0	0
$551$	−22.4164	−0.954971
$552$	0	0
$553$	0.527864	0.0224471
$554$	0	0
$555$	2.03444	0.0863572
$556$	0	0
$557$	27.7082	1.17403	0.587017	−	0.809575i	$-0.300302\pi$
$557$	27.7082	1.17403	0.587017	+	0.809575i	$0.300302\pi$
$558$	0	0
$559$	19.2705	0.815056
$560$	0	0
$561$	5.47214	0.231034
$562$	0	0
$563$	−32.7426	−1.37994	−0.689969	−	0.723839i	$-0.742376\pi$
$563$	−32.7426	−1.37994	−0.689969	+	0.723839i	$0.742376\pi$
$564$	0	0
$565$	−8.74265	−0.367806
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−18.8328	−0.789513	−0.394756	−	0.918786i	$-0.629171\pi$
$569$	−18.8328	−0.789513	−0.394756	+	0.918786i	$0.629171\pi$
$570$	0	0
$571$	−38.8885	−1.62743	−0.813717	−	0.581261i	$-0.802560\pi$
$571$	−38.8885	−1.62743	−0.813717	+	0.581261i	$0.802560\pi$
$572$	0	0
$573$	6.29180	0.262844
$574$	0	0
$575$	−3.09017	−0.128869
$576$	0	0
$577$	−16.4721	−0.685744	−0.342872	−	0.939382i	$-0.611400\pi$
$577$	−16.4721	−0.685744	−0.342872	+	0.939382i	$0.611400\pi$
$578$	0	0
$579$	14.1803	0.589315
$580$	0	0
$581$	−3.00000	−0.124461
$582$	0	0
$583$	−18.5066	−0.766464
$584$	0	0
$585$	3.29180	0.136099
$586$	0	0
$587$	−3.72949	−0.153933	−0.0769663	−	0.997034i	$-0.524523\pi$
$587$	−3.72949	−0.153933	−0.0769663	+	0.997034i	$0.524523\pi$
$588$	0	0
$589$	11.2918	0.465270
$590$	0	0
$591$	12.8541	0.528747
$592$	0	0
$593$	19.1246	0.785354	0.392677	−	0.919677i	$-0.371549\pi$
$593$	19.1246	0.785354	0.392677	+	0.919677i	$0.371549\pi$
$594$	0	0
$595$	−1.38197	−0.0566551
$596$	0	0
$597$	−18.0344	−0.738101
$598$	0	0
$599$	−14.0902	−0.575709	−0.287854	−	0.957674i	$-0.592942\pi$
$599$	−14.0902	−0.575709	−0.287854	+	0.957674i	$0.592942\pi$
$600$	0	0
$601$	−15.6738	−0.639346	−0.319673	−	0.947528i	$-0.603573\pi$
$601$	−15.6738	−0.639346	−0.319673	+	0.947528i	$0.603573\pi$
$602$	0	0
$603$	−4.14590	−0.168834
$604$	0	0
$605$	−26.1803	−1.06438
$606$	0	0
$607$	−20.0902	−0.815435	−0.407717	−	0.913108i	$-0.633675\pi$
$607$	−20.0902	−0.815435	−0.407717	+	0.913108i	$0.633675\pi$
$608$	0	0
$609$	−7.47214	−0.302786
$610$	0	0
$611$	4.06888	0.164609
$612$	0	0
$613$	−1.65248	−0.0667429	−0.0333714	−	0.999443i	$-0.510624\pi$
$613$	−1.65248	−0.0667429	−0.0333714	+	0.999443i	$0.510624\pi$
$614$	0	0
$615$	−6.50658	−0.262371
$616$	0	0
$617$	−33.3951	−1.34444	−0.672218	−	0.740353i	$-0.734658\pi$
$617$	−33.3951	−1.34444	−0.672218	+	0.740353i	$0.734658\pi$
$618$	0	0
$619$	−43.6180	−1.75316	−0.876578	−	0.481259i	$-0.840180\pi$
$619$	−43.6180	−1.75316	−0.876578	+	0.481259i	$0.840180\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−3.14590	−0.126038
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−16.4164	−0.655608
$628$	0	0
$629$	−1.47214	−0.0586979
$630$	0	0
$631$	20.7082	0.824381	0.412190	−	0.911098i	$-0.364764\pi$
$631$	20.7082	0.824381	0.412190	+	0.911098i	$0.364764\pi$
$632$	0	0
$633$	26.4164	1.04996
$634$	0	0
$635$	26.0557	1.03399
$636$	0	0
$637$	−2.38197	−0.0943769
$638$	0	0
$639$	3.90983	0.154670
$640$	0	0
$641$	27.2148	1.07492	0.537460	−	0.843289i	$-0.319384\pi$
$641$	27.2148	1.07492	0.537460	+	0.843289i	$0.319384\pi$
$642$	0	0
$643$	−5.14590	−0.202934	−0.101467	−	0.994839i	$-0.532354\pi$
$643$	−5.14590	−0.202934	−0.101467	+	0.994839i	$0.532354\pi$
$644$	0	0
$645$	−11.1803	−0.440225
$646$	0	0
$647$	−18.3262	−0.720479	−0.360239	−	0.932860i	$-0.617305\pi$
$647$	−18.3262	−0.720479	−0.360239	+	0.932860i	$0.617305\pi$
$648$	0	0
$649$	33.6312	1.32014
$650$	0	0
$651$	3.76393	0.147520
$652$	0	0
$653$	−39.5623	−1.54819	−0.774096	−	0.633068i	$-0.781795\pi$
$653$	−39.5623	−1.54819	−0.774096	+	0.633068i	$0.781795\pi$
$654$	0	0
$655$	−24.7984	−0.968953
$656$	0	0
$657$	2.70820	0.105657
$658$	0	0
$659$	29.4721	1.14807	0.574036	−	0.818830i	$-0.305377\pi$
$659$	29.4721	1.14807	0.574036	+	0.818830i	$0.305377\pi$
$660$	0	0
$661$	32.7771	1.27488	0.637440	−	0.770500i	$-0.279993\pi$
$661$	32.7771	1.27488	0.637440	+	0.770500i	$0.279993\pi$
$662$	0	0
$663$	−2.38197	−0.0925079
$664$	0	0
$665$	4.14590	0.160771
$666$	0	0
$667$	−7.47214	−0.289322
$668$	0	0
$669$	−20.3262	−0.785858
$670$	0	0
$671$	75.5066	2.91490
$672$	0	0
$673$	−32.8885	−1.26776	−0.633880	−	0.773431i	$-0.718539\pi$
$673$	−32.8885	−1.26776	−0.633880	+	0.773431i	$0.718539\pi$
$674$	0	0
$675$	3.09017	0.118941
$676$	0	0
$677$	−3.61803	−0.139052	−0.0695262	−	0.997580i	$-0.522149\pi$
$677$	−3.61803	−0.139052	−0.0695262	+	0.997580i	$0.522149\pi$
$678$	0	0
$679$	−5.00000	−0.191882
$680$	0	0
$681$	−8.79837	−0.337154
$682$	0	0
$683$	34.8885	1.33497	0.667487	−	0.744622i	$-0.267370\pi$
$683$	34.8885	1.33497	0.667487	+	0.744622i	$0.267370\pi$
$684$	0	0
$685$	−12.6869	−0.484742
$686$	0	0
$687$	21.5623	0.822653
$688$	0	0
$689$	8.05573	0.306899
$690$	0	0
$691$	1.03444	0.0393520	0.0196760	−	0.999806i	$-0.493737\pi$
$691$	1.03444	0.0393520	0.0196760	+	0.999806i	$0.493737\pi$
$692$	0	0
$693$	−5.47214	−0.207869
$694$	0	0
$695$	−22.5623	−0.855837
$696$	0	0
$697$	4.70820	0.178336
$698$	0	0
$699$	−19.2705	−0.728878
$700$	0	0
$701$	−20.0344	−0.756690	−0.378345	−	0.925665i	$-0.623507\pi$
$701$	−20.0344	−0.756690	−0.378345	+	0.925665i	$0.623507\pi$
$702$	0	0
$703$	4.41641	0.166568
$704$	0	0
$705$	−2.36068	−0.0889083
$706$	0	0
$707$	1.85410	0.0697307
$708$	0	0
$709$	30.9230	1.16134	0.580669	−	0.814140i	$-0.302791\pi$
$709$	30.9230	1.16134	0.580669	+	0.814140i	$0.302791\pi$
$710$	0	0
$711$	−0.527864	−0.0197964
$712$	0	0
$713$	3.76393	0.140960
$714$	0	0
$715$	18.0132	0.673654
$716$	0	0
$717$	−17.8541	−0.666774
$718$	0	0
$719$	28.7082	1.07064	0.535318	−	0.844651i	$-0.320192\pi$
$719$	28.7082	1.07064	0.535318	+	0.844651i	$0.320192\pi$
$720$	0	0
$721$	−4.94427	−0.184134
$722$	0	0
$723$	20.4164	0.759294
$724$	0	0
$725$	23.0902	0.857547
$726$	0	0
$727$	−3.36068	−0.124641	−0.0623204	−	0.998056i	$-0.519850\pi$
$727$	−3.36068	−0.124641	−0.0623204	+	0.998056i	$0.519850\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.09017	0.299226
$732$	0	0
$733$	2.18034	0.0805327	0.0402663	−	0.999189i	$-0.487179\pi$
$733$	2.18034	0.0805327	0.0402663	+	0.999189i	$0.487179\pi$
$734$	0	0
$735$	1.38197	0.0509746
$736$	0	0
$737$	−22.6869	−0.835683
$738$	0	0
$739$	−4.88854	−0.179828	−0.0899140	−	0.995950i	$-0.528659\pi$
$739$	−4.88854	−0.179828	−0.0899140	+	0.995950i	$0.528659\pi$
$740$	0	0
$741$	7.14590	0.262511
$742$	0	0
$743$	22.0344	0.808365	0.404183	−	0.914678i	$-0.367556\pi$
$743$	22.0344	0.808365	0.404183	+	0.914678i	$0.367556\pi$
$744$	0	0
$745$	22.3607	0.819232
$746$	0	0
$747$	3.00000	0.109764
$748$	0	0
$749$	11.7984	0.431103
$750$	0	0
$751$	44.3951	1.62000	0.810000	−	0.586429i	$-0.199467\pi$
$751$	44.3951	1.62000	0.810000	+	0.586429i	$0.199467\pi$
$752$	0	0
$753$	1.23607	0.0450448
$754$	0	0
$755$	1.05573	0.0384219
$756$	0	0
$757$	3.11146	0.113088	0.0565439	−	0.998400i	$-0.481992\pi$
$757$	3.11146	0.113088	0.0565439	+	0.998400i	$0.481992\pi$
$758$	0	0
$759$	−5.47214	−0.198626
$760$	0	0
$761$	13.3050	0.482304	0.241152	−	0.970487i	$-0.422475\pi$
$761$	13.3050	0.482304	0.241152	+	0.970487i	$0.422475\pi$
$762$	0	0
$763$	6.32624	0.229025
$764$	0	0
$765$	1.38197	0.0499651
$766$	0	0
$767$	−14.6393	−0.528595
$768$	0	0
$769$	−17.5967	−0.634555	−0.317277	−	0.948333i	$-0.602769\pi$
$769$	−17.5967	−0.634555	−0.317277	+	0.948333i	$0.602769\pi$
$770$	0	0
$771$	31.5967	1.13793
$772$	0	0
$773$	−40.0132	−1.43917	−0.719587	−	0.694403i	$-0.755669\pi$
$773$	−40.0132	−1.43917	−0.719587	+	0.694403i	$0.755669\pi$
$774$	0	0
$775$	−11.6312	−0.417805
$776$	0	0
$777$	1.47214	0.0528126
$778$	0	0
$779$	−14.1246	−0.506067
$780$	0	0
$781$	21.3951	0.765578
$782$	0	0
$783$	7.47214	0.267032
$784$	0	0
$785$	28.5410	1.01867
$786$	0	0
$787$	30.0344	1.07061	0.535306	−	0.844658i	$-0.320196\pi$
$787$	30.0344	1.07061	0.535306	+	0.844658i	$0.320196\pi$
$788$	0	0
$789$	23.6525	0.842050
$790$	0	0
$791$	−6.32624	−0.224935
$792$	0	0
$793$	−32.8673	−1.16715
$794$	0	0
$795$	−4.67376	−0.165761
$796$	0	0
$797$	−35.5967	−1.26090	−0.630451	−	0.776229i	$-0.717130\pi$
$797$	−35.5967	−1.26090	−0.630451	+	0.776229i	$0.717130\pi$
$798$	0	0
$799$	1.70820	0.0604319
$800$	0	0
$801$	3.14590	0.111155
$802$	0	0
$803$	14.8197	0.522974
$804$	0	0
$805$	1.38197	0.0487079
$806$	0	0
$807$	14.2705	0.502346
$808$	0	0
$809$	0.965558	0.0339472	0.0169736	−	0.999856i	$-0.494597\pi$
$809$	0.965558	0.0339472	0.0169736	+	0.999856i	$0.494597\pi$
$810$	0	0
$811$	10.5279	0.369683	0.184842	−	0.982768i	$-0.440823\pi$
$811$	10.5279	0.369683	0.184842	+	0.982768i	$0.440823\pi$
$812$	0	0
$813$	31.8328	1.11643
$814$	0	0
$815$	3.86726	0.135464
$816$	0	0
$817$	−24.2705	−0.849118
$818$	0	0
$819$	2.38197	0.0832326
$820$	0	0
$821$	20.1803	0.704299	0.352149	−	0.935944i	$-0.385451\pi$
$821$	20.1803	0.704299	0.352149	+	0.935944i	$0.385451\pi$
$822$	0	0
$823$	20.4377	0.712413	0.356207	−	0.934407i	$-0.384070\pi$
$823$	20.4377	0.712413	0.356207	+	0.934407i	$0.384070\pi$
$824$	0	0
$825$	16.9098	0.588725
$826$	0	0
$827$	33.5623	1.16708	0.583538	−	0.812086i	$-0.301668\pi$
$827$	33.5623	1.16708	0.583538	+	0.812086i	$0.301668\pi$
$828$	0	0
$829$	−30.0557	−1.04388	−0.521939	−	0.852983i	$-0.674791\pi$
$829$	−30.0557	−1.04388	−0.521939	+	0.852983i	$0.674791\pi$
$830$	0	0
$831$	19.1459	0.664164
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	20.0770	0.694794
$836$	0	0
$837$	−3.76393	−0.130101
$838$	0	0
$839$	15.1459	0.522894	0.261447	−	0.965218i	$-0.415800\pi$
$839$	15.1459	0.522894	0.261447	+	0.965218i	$0.415800\pi$
$840$	0	0
$841$	26.8328	0.925270
$842$	0	0
$843$	−27.7082	−0.954321
$844$	0	0
$845$	10.1246	0.348297
$846$	0	0
$847$	−18.9443	−0.650933
$848$	0	0
$849$	−27.0902	−0.929732
$850$	0	0
$851$	1.47214	0.0504642
$852$	0	0
$853$	11.3475	0.388532	0.194266	−	0.980949i	$-0.437768\pi$
$853$	11.3475	0.388532	0.194266	+	0.980949i	$0.437768\pi$
$854$	0	0
$855$	−4.14590	−0.141787
$856$	0	0
$857$	27.1246	0.926559	0.463280	−	0.886212i	$-0.346673\pi$
$857$	27.1246	0.926559	0.463280	+	0.886212i	$0.346673\pi$
$858$	0	0
$859$	−16.8328	−0.574328	−0.287164	−	0.957881i	$-0.592713\pi$
$859$	−16.8328	−0.574328	−0.287164	+	0.957881i	$0.592713\pi$
$860$	0	0
$861$	−4.70820	−0.160455
$862$	0	0
$863$	−14.1803	−0.482704	−0.241352	−	0.970438i	$-0.577591\pi$
$863$	−14.1803	−0.482704	−0.241352	+	0.970438i	$0.577591\pi$
$864$	0	0
$865$	12.6869	0.431368
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	−2.88854	−0.0979871
$870$	0	0
$871$	9.87539	0.334615
$872$	0	0
$873$	5.00000	0.169224
$874$	0	0
$875$	−11.1803	−0.377964
$876$	0	0
$877$	−15.4164	−0.520575	−0.260288	−	0.965531i	$-0.583817\pi$
$877$	−15.4164	−0.520575	−0.260288	+	0.965531i	$0.583817\pi$
$878$	0	0
$879$	−3.41641	−0.115233
$880$	0	0
$881$	−47.4853	−1.59982	−0.799910	−	0.600120i	$-0.795120\pi$
$881$	−47.4853	−1.59982	−0.799910	+	0.600120i	$0.795120\pi$
$882$	0	0
$883$	−7.50658	−0.252616	−0.126308	−	0.991991i	$-0.540313\pi$
$883$	−7.50658	−0.252616	−0.126308	+	0.991991i	$0.540313\pi$
$884$	0	0
$885$	8.49342	0.285503
$886$	0	0
$887$	−37.1459	−1.24724	−0.623619	−	0.781729i	$-0.714338\pi$
$887$	−37.1459	−1.24724	−0.623619	+	0.781729i	$0.714338\pi$
$888$	0	0
$889$	18.8541	0.632346
$890$	0	0
$891$	5.47214	0.183323
$892$	0	0
$893$	−5.12461	−0.171489
$894$	0	0
$895$	10.8541	0.362813
$896$	0	0
$897$	2.38197	0.0795315
$898$	0	0
$899$	−28.1246	−0.938008
$900$	0	0
$901$	3.38197	0.112670
$902$	0	0
$903$	−8.09017	−0.269224
$904$	0	0
$905$	−10.3262	−0.343256
$906$	0	0
$907$	−23.9656	−0.795763	−0.397882	−	0.917437i	$-0.630255\pi$
$907$	−23.9656	−0.795763	−0.397882	+	0.917437i	$0.630255\pi$
$908$	0	0
$909$	−1.85410	−0.0614967
$910$	0	0
$911$	36.1803	1.19871	0.599354	−	0.800484i	$-0.295424\pi$
$911$	36.1803	1.19871	0.599354	+	0.800484i	$0.295424\pi$
$912$	0	0
$913$	16.4164	0.543304
$914$	0	0
$915$	19.0689	0.630398
$916$	0	0
$917$	−17.9443	−0.592572
$918$	0	0
$919$	−15.0689	−0.497077	−0.248538	−	0.968622i	$-0.579950\pi$
$919$	−15.0689	−0.497077	−0.248538	+	0.968622i	$0.579950\pi$
$920$	0	0
$921$	−22.7082	−0.748261
$922$	0	0
$923$	−9.31308	−0.306544
$924$	0	0
$925$	−4.54915	−0.149575
$926$	0	0
$927$	4.94427	0.162391
$928$	0	0
$929$	−54.6180	−1.79196	−0.895980	−	0.444095i	$-0.853525\pi$
$929$	−54.6180	−1.79196	−0.895980	+	0.444095i	$0.853525\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	21.5066	0.704094
$934$	0	0
$935$	7.56231	0.247314
$936$	0	0
$937$	44.1246	1.44149	0.720744	−	0.693201i	$-0.243800\pi$
$937$	44.1246	1.44149	0.720744	+	0.693201i	$0.243800\pi$
$938$	0	0
$939$	2.47214	0.0806751
$940$	0	0
$941$	26.8328	0.874725	0.437362	−	0.899285i	$-0.355913\pi$
$941$	26.8328	0.874725	0.437362	+	0.899285i	$0.355913\pi$
$942$	0	0
$943$	−4.70820	−0.153320
$944$	0	0
$945$	−1.38197	−0.0449554
$946$	0	0
$947$	45.4164	1.47583	0.737917	−	0.674891i	$-0.235809\pi$
$947$	45.4164	1.47583	0.737917	+	0.674891i	$0.235809\pi$
$948$	0	0
$949$	−6.45085	−0.209403
$950$	0	0
$951$	24.7984	0.804142
$952$	0	0
$953$	59.6869	1.93345	0.966724	−	0.255820i	$-0.0823454\pi$
$953$	59.6869	1.93345	0.966724	+	0.255820i	$0.0823454\pi$
$954$	0	0
$955$	8.69505	0.281365
$956$	0	0
$957$	40.8885	1.32174
$958$	0	0
$959$	−9.18034	−0.296449
$960$	0	0
$961$	−16.8328	−0.542994
$962$	0	0
$963$	−11.7984	−0.380197
$964$	0	0
$965$	19.5967	0.630842
$966$	0	0
$967$	21.7771	0.700304	0.350152	−	0.936693i	$-0.386130\pi$
$967$	21.7771	0.700304	0.350152	+	0.936693i	$0.386130\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	−35.9787	−1.15461	−0.577306	−	0.816528i	$-0.695896\pi$
$971$	−35.9787	−1.15461	−0.577306	+	0.816528i	$0.695896\pi$
$972$	0	0
$973$	−16.3262	−0.523395
$974$	0	0
$975$	−7.36068	−0.235730
$976$	0	0
$977$	−24.2016	−0.774279	−0.387139	−	0.922021i	$-0.626537\pi$
$977$	−24.2016	−0.774279	−0.387139	+	0.922021i	$0.626537\pi$
$978$	0	0
$979$	17.2148	0.550187
$980$	0	0
$981$	−6.32624	−0.201981
$982$	0	0
$983$	33.3607	1.06404	0.532020	−	0.846732i	$-0.321433\pi$
$983$	33.3607	1.06404	0.532020	+	0.846732i	$0.321433\pi$
$984$	0	0
$985$	17.7639	0.566006
$986$	0	0
$987$	−1.70820	−0.0543727
$988$	0	0
$989$	−8.09017	−0.257252
$990$	0	0
$991$	−14.0902	−0.447589	−0.223794	−	0.974636i	$-0.571844\pi$
$991$	−14.0902	−0.447589	−0.223794	+	0.974636i	$0.571844\pi$
$992$	0	0
$993$	18.9443	0.601178
$994$	0	0
$995$	−24.9230	−0.790112
$996$	0	0
$997$	−40.8885	−1.29495	−0.647477	−	0.762085i	$-0.724176\pi$
$997$	−40.8885	−1.29495	−0.647477	+	0.762085i	$0.724176\pi$
$998$	0	0
$999$	−1.47214	−0.0465763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.v.1.2		2
4.3	odd	2		483.2.a.c.1.2	✓	2
12.11	even	2		1449.2.a.k.1.1		2
28.27	even	2		3381.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.c.1.2	✓	2	4.3	odd	2
1449.2.a.k.1.1		2	12.11	even	2
3381.2.a.n.1.2		2	28.27	even	2
7728.2.a.v.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.47214 q^{11} -2.38197 q^{13} +1.38197 q^{15} -1.00000 q^{17} +3.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.09017 q^{25} -1.00000 q^{27} -7.47214 q^{29} +3.76393 q^{31} -5.47214 q^{33} +1.38197 q^{35} +1.47214 q^{37} +2.38197 q^{39} -4.70820 q^{41} -8.09017 q^{43} -1.38197 q^{45} -1.70820 q^{47} +1.00000 q^{49} +1.00000 q^{51} -3.38197 q^{53} -7.56231 q^{55} -3.00000 q^{57} +6.14590 q^{59} +13.7984 q^{61} -1.00000 q^{63} +3.29180 q^{65} -4.14590 q^{67} -1.00000 q^{69} +3.90983 q^{71} +2.70820 q^{73} +3.09017 q^{75} -5.47214 q^{77} -0.527864 q^{79} +1.00000 q^{81} +3.00000 q^{83} +1.38197 q^{85} +7.47214 q^{87} +3.14590 q^{89} +2.38197 q^{91} -3.76393 q^{93} -4.14590 q^{95} +5.00000 q^{97} +5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 7 q^{13} + 5 q^{15} - 2 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} + 5 q^{25} - 2 q^{27} - 6 q^{29} + 12 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + 7 q^{39} + 4 q^{41} - 5 q^{43} - 5 q^{45} + 10 q^{47} + 2 q^{49} + 2 q^{51} - 9 q^{53} + 5 q^{55} - 6 q^{57} + 19 q^{59} + 3 q^{61} - 2 q^{63} + 20 q^{65} - 15 q^{67} - 2 q^{69} + 19 q^{71} - 8 q^{73} - 5 q^{75} - 2 q^{77} - 10 q^{79} + 2 q^{81} + 6 q^{83} + 5 q^{85} + 6 q^{87} + 13 q^{89} + 7 q^{91} - 12 q^{93} - 15 q^{95} + 10 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 7 * q^13 + 5 * q^15 - 2 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^23 + 5 * q^25 - 2 * q^27 - 6 * q^29 + 12 * q^31 - 2 * q^33 + 5 * q^35 - 6 * q^37 + 7 * q^39 + 4 * q^41 - 5 * q^43 - 5 * q^45 + 10 * q^47 + 2 * q^49 + 2 * q^51 - 9 * q^53 + 5 * q^55 - 6 * q^57 + 19 * q^59 + 3 * q^61 - 2 * q^63 + 20 * q^65 - 15 * q^67 - 2 * q^69 + 19 * q^71 - 8 * q^73 - 5 * q^75 - 2 * q^77 - 10 * q^79 + 2 * q^81 + 6 * q^83 + 5 * q^85 + 6 * q^87 + 13 * q^89 + 7 * q^91 - 12 * q^93 - 15 * q^95 + 10 * q^97 + 2 * q^99

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