LMFDB - Embedded newform 4761.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1587)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} -3.34607 q^{5} -3.86370 q^{7} -1.73205 q^{8} -5.79555 q^{10} -4.89898 q^{11} -3.00000 q^{13} -6.69213 q^{14} -5.00000 q^{16} +4.24264 q^{17} -1.03528 q^{19} -3.34607 q^{20} -8.48528 q^{22} +6.19615 q^{25} -5.19615 q^{26} -3.86370 q^{28} +4.26795 q^{29} +3.46410 q^{31} -5.19615 q^{32} +7.34847 q^{34} +12.9282 q^{35} -8.38375 q^{37} -1.79315 q^{38} +5.79555 q^{40} +0.464102 q^{41} -5.93426 q^{43} -4.89898 q^{44} -6.92820 q^{47} +7.92820 q^{49} +10.7321 q^{50} -3.00000 q^{52} +0.896575 q^{53} +16.3923 q^{55} +6.69213 q^{56} +7.39230 q^{58} -0.928203 q^{59} +0.138701 q^{61} +6.00000 q^{62} +1.00000 q^{64} +10.0382 q^{65} +7.72741 q^{67} +4.24264 q^{68} +22.3923 q^{70} +3.46410 q^{71} -12.1244 q^{73} -14.5211 q^{74} -1.03528 q^{76} +18.9282 q^{77} +4.14110 q^{79} +16.7303 q^{80} +0.803848 q^{82} -6.69213 q^{83} -14.1962 q^{85} -10.2784 q^{86} +8.48528 q^{88} -0.896575 q^{89} +11.5911 q^{91} -12.0000 q^{94} +3.46410 q^{95} -18.9024 q^{97} +13.7321 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 12 q^{13} - 20 q^{16} + 4 q^{25} + 24 q^{29} + 24 q^{35} - 12 q^{41} + 4 q^{49} + 36 q^{50} - 12 q^{52} + 24 q^{55} - 12 q^{58} + 24 q^{59} + 24 q^{62} + 4 q^{64} + 48 q^{70} + 48 q^{77}+ \cdots + 48 q^{98}+O(q^{100}) $ 4 * q + 4 * q^4 - 12 * q^13 - 20 * q^16 + 4 * q^25 + 24 * q^29 + 24 * q^35 - 12 * q^41 + 4 * q^49 + 36 * q^50 - 12 * q^52 + 24 * q^55 - 12 * q^58 + 24 * q^59 + 24 * q^62 + 4 * q^64 + 48 * q^70 + 48 * q^77 + 24 * q^82 - 36 * q^85 - 48 * q^94 + 48 * q^98

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.34607	−1.49641	−0.748203	−	0.663470i	$-0.769083\pi$
$5$	−3.34607	−1.49641	−0.748203	+	0.663470i	$0.769083\pi$
$6$	0	0
$7$	−3.86370	−1.46034	−0.730171	−	0.683264i	$-0.760560\pi$
$7$	−3.86370	−1.46034	−0.730171	+	0.683264i	$0.760560\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	−5.79555	−1.83272
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−6.69213	−1.78855
$15$	0	0
$16$	−5.00000	−1.25000
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	−1.03528	−0.237509	−0.118754	−	0.992924i	$-0.537890\pi$
$19$	−1.03528	−0.237509	−0.118754	+	0.992924i	$0.537890\pi$
$20$	−3.34607	−0.748203
$21$	0	0
$22$	−8.48528	−1.80907
$23$	0	0
$23$	0	0
$24$	0	0
$25$	6.19615	1.23923
$26$	−5.19615	−1.01905
$27$	0	0
$28$	−3.86370	−0.730171
$29$	4.26795	0.792538	0.396269	−	0.918134i	$-0.370305\pi$
$29$	4.26795	0.792538	0.396269	+	0.918134i	$0.370305\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	7.34847	1.26025
$35$	12.9282	2.18527
$36$	0	0
$37$	−8.38375	−1.37828	−0.689140	−	0.724629i	$-0.742011\pi$
$37$	−8.38375	−1.37828	−0.689140	+	0.724629i	$0.742011\pi$
$38$	−1.79315	−0.290887
$39$	0	0
$40$	5.79555	0.916358
$41$	0.464102	0.0724805	0.0362402	−	0.999343i	$-0.488462\pi$
$41$	0.464102	0.0724805	0.0362402	+	0.999343i	$0.488462\pi$
$42$	0	0
$43$	−5.93426	−0.904966	−0.452483	−	0.891773i	$-0.649462\pi$
$43$	−5.93426	−0.904966	−0.452483	+	0.891773i	$0.649462\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	7.92820	1.13260
$50$	10.7321	1.51774
$51$	0	0
$52$	−3.00000	−0.416025
$53$	0.896575	0.123154	0.0615771	−	0.998102i	$-0.480387\pi$
$53$	0.896575	0.123154	0.0615771	+	0.998102i	$0.480387\pi$
$54$	0	0
$55$	16.3923	2.21034
$56$	6.69213	0.894274
$57$	0	0
$58$	7.39230	0.970657
$59$	−0.928203	−0.120842	−0.0604209	−	0.998173i	$-0.519244\pi$
$59$	−0.928203	−0.120842	−0.0604209	+	0.998173i	$0.519244\pi$
$60$	0	0
$61$	0.138701	0.0177588	0.00887940	−	0.999961i	$-0.497174\pi$
$61$	0.138701	0.0177588	0.00887940	+	0.999961i	$0.497174\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	10.0382	1.24508
$66$	0	0
$67$	7.72741	0.944053	0.472026	−	0.881584i	$-0.343523\pi$
$67$	7.72741	0.944053	0.472026	+	0.881584i	$0.343523\pi$
$68$	4.24264	0.514496
$69$	0	0
$70$	22.3923	2.67639
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	−12.1244	−1.41905	−0.709524	−	0.704681i	$-0.751090\pi$
$73$	−12.1244	−1.41905	−0.709524	+	0.704681i	$0.751090\pi$
$74$	−14.5211	−1.68804
$75$	0	0
$76$	−1.03528	−0.118754
$77$	18.9282	2.15707
$78$	0	0
$79$	4.14110	0.465911	0.232955	−	0.972487i	$-0.425160\pi$
$79$	4.14110	0.465911	0.232955	+	0.972487i	$0.425160\pi$
$80$	16.7303	1.87051
$81$	0	0
$82$	0.803848	0.0887701
$83$	−6.69213	−0.734557	−0.367278	−	0.930111i	$-0.619710\pi$
$83$	−6.69213	−0.734557	−0.367278	+	0.930111i	$0.619710\pi$
$84$	0	0
$85$	−14.1962	−1.53979
$86$	−10.2784	−1.10835
$87$	0	0
$88$	8.48528	0.904534
$89$	−0.896575	−0.0950368	−0.0475184	−	0.998870i	$-0.515131\pi$
$89$	−0.896575	−0.0950368	−0.0475184	+	0.998870i	$0.515131\pi$
$90$	0	0
$91$	11.5911	1.21508
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	3.46410	0.355409
$96$	0	0
$97$	−18.9024	−1.91925	−0.959625	−	0.281284i	$-0.909240\pi$
$97$	−18.9024	−1.91925	−0.959625	+	0.281284i	$0.909240\pi$
$98$	13.7321	1.38715
$99$	0	0
$100$	6.19615	0.619615
$101$	18.4641	1.83725	0.918623	−	0.395134i	$-0.129302\pi$
$101$	18.4641	1.83725	0.918623	+	0.395134i	$0.129302\pi$
$102$	0	0
$103$	17.2480	1.69949	0.849746	−	0.527192i	$-0.176755\pi$
$103$	17.2480	1.69949	0.849746	+	0.527192i	$0.176755\pi$
$104$	5.19615	0.509525
$105$	0	0
$106$	1.55291	0.150832
$107$	16.4901	1.59416	0.797079	−	0.603876i	$-0.206378\pi$
$107$	16.4901	1.59416	0.797079	+	0.603876i	$0.206378\pi$
$108$	0	0
$109$	10.7961	1.03408	0.517038	−	0.855962i	$-0.327035\pi$
$109$	10.7961	1.03408	0.517038	+	0.855962i	$0.327035\pi$
$110$	28.3923	2.70710
$111$	0	0
$112$	19.3185	1.82543
$113$	−3.34607	−0.314771	−0.157386	−	0.987537i	$-0.550307\pi$
$113$	−3.34607	−0.314771	−0.157386	+	0.987537i	$0.550307\pi$
$114$	0	0
$115$	0	0
$116$	4.26795	0.396269
$117$	0	0
$118$	−1.60770	−0.148000
$119$	−16.3923	−1.50268
$120$	0	0
$121$	13.0000	1.18182
$122$	0.240237	0.0217500
$123$	0	0
$124$	3.46410	0.311086
$125$	−4.00240	−0.357986
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	17.3867	1.52491
$131$	7.85641	0.686417	0.343209	−	0.939259i	$-0.388486\pi$
$131$	7.85641	0.686417	0.343209	+	0.939259i	$0.388486\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	13.3843	1.15622
$135$	0	0
$136$	−7.34847	−0.630126
$137$	−2.68973	−0.229799	−0.114899	−	0.993377i	$-0.536655\pi$
$137$	−2.68973	−0.229799	−0.114899	+	0.993377i	$0.536655\pi$
$138$	0	0
$139$	−18.3923	−1.56001	−0.780007	−	0.625770i	$-0.784785\pi$
$139$	−18.3923	−1.56001	−0.780007	+	0.625770i	$0.784785\pi$
$140$	12.9282	1.09263
$141$	0	0
$142$	6.00000	0.503509
$143$	14.6969	1.22902
$144$	0	0
$145$	−14.2808	−1.18596
$146$	−21.0000	−1.73797
$147$	0	0
$148$	−8.38375	−0.689140
$149$	2.68973	0.220351	0.110175	−	0.993912i	$-0.464859\pi$
$149$	2.68973	0.220351	0.110175	+	0.993912i	$0.464859\pi$
$150$	0	0
$151$	−12.3923	−1.00847	−0.504236	−	0.863566i	$-0.668226\pi$
$151$	−12.3923	−1.00847	−0.504236	+	0.863566i	$0.668226\pi$
$152$	1.79315	0.145444
$153$	0	0
$154$	32.7846	2.64186
$155$	−11.5911	−0.931020
$156$	0	0
$157$	−13.9019	−1.10949	−0.554746	−	0.832020i	$-0.687185\pi$
$157$	−13.9019	−1.10949	−0.554746	+	0.832020i	$0.687185\pi$
$158$	7.17260	0.570622
$159$	0	0
$160$	17.3867	1.37454
$161$	0	0
$162$	0	0
$163$	−15.4641	−1.21124	−0.605621	−	0.795753i	$-0.707075\pi$
$163$	−15.4641	−1.21124	−0.605621	+	0.795753i	$0.707075\pi$
$164$	0.464102	0.0362402
$165$	0	0
$166$	−11.5911	−0.899645
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	−24.5885	−1.88585
$171$	0	0
$172$	−5.93426	−0.452483
$173$	−11.1962	−0.851228	−0.425614	−	0.904905i	$-0.639942\pi$
$173$	−11.1962	−0.851228	−0.425614	+	0.904905i	$0.639942\pi$
$174$	0	0
$175$	−23.9401	−1.80970
$176$	24.4949	1.84637
$177$	0	0
$178$	−1.55291	−0.116396
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−1.41421	−0.105118	−0.0525588	−	0.998618i	$-0.516738\pi$
$181$	−1.41421	−0.105118	−0.0525588	+	0.998618i	$0.516738\pi$
$182$	20.0764	1.48816
$183$	0	0
$184$	0	0
$185$	28.0526	2.06247
$186$	0	0
$187$	−20.7846	−1.51992
$188$	−6.92820	−0.505291
$189$	0	0
$190$	6.00000	0.435286
$191$	16.4901	1.19318	0.596590	−	0.802546i	$-0.296522\pi$
$191$	16.4901	1.19318	0.596590	+	0.802546i	$0.296522\pi$
$192$	0	0
$193$	−25.1962	−1.81366	−0.906829	−	0.421498i	$-0.861504\pi$
$193$	−25.1962	−1.81366	−0.906829	+	0.421498i	$0.861504\pi$
$194$	−32.7399	−2.35059
$195$	0	0
$196$	7.92820	0.566300
$197$	−6.12436	−0.436342	−0.218171	−	0.975911i	$-0.570009\pi$
$197$	−6.12436	−0.436342	−0.218171	+	0.975911i	$0.570009\pi$
$198$	0	0
$199$	5.65685	0.401004	0.200502	−	0.979693i	$-0.435743\pi$
$199$	5.65685	0.401004	0.200502	+	0.979693i	$0.435743\pi$
$200$	−10.7321	−0.758871
$201$	0	0
$202$	31.9808	2.25016
$203$	−16.4901	−1.15738
$204$	0	0
$205$	−1.55291	−0.108460
$206$	29.8744	2.08144
$207$	0	0
$208$	15.0000	1.04006
$209$	5.07180	0.350824
$210$	0	0
$211$	14.5359	1.00069	0.500346	−	0.865825i	$-0.333206\pi$
$211$	14.5359	1.00069	0.500346	+	0.865825i	$0.333206\pi$
$212$	0.896575	0.0615771
$213$	0	0
$214$	28.5617	1.95244
$215$	19.8564	1.35420
$216$	0	0
$217$	−13.3843	−0.908583
$218$	18.6993	1.26648
$219$	0	0
$220$	16.3923	1.10517
$221$	−12.7279	−0.856173
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	20.0764	1.34141
$225$	0	0
$226$	−5.79555	−0.385515
$227$	−18.2832	−1.21350	−0.606751	−	0.794892i	$-0.707527\pi$
$227$	−18.2832	−1.21350	−0.606751	+	0.794892i	$0.707527\pi$
$228$	0	0
$229$	−16.1112	−1.06465	−0.532327	−	0.846539i	$-0.678682\pi$
$229$	−16.1112	−1.06465	−0.532327	+	0.846539i	$0.678682\pi$
$230$	0	0
$231$	0	0
$232$	−7.39230	−0.485329
$233$	−6.46410	−0.423477	−0.211739	−	0.977326i	$-0.567913\pi$
$233$	−6.46410	−0.423477	−0.211739	+	0.977326i	$0.567913\pi$
$234$	0	0
$235$	23.1822	1.51224
$236$	−0.928203	−0.0604209
$237$	0	0
$238$	−28.3923	−1.84040
$239$	9.46410	0.612182	0.306091	−	0.952002i	$-0.400979\pi$
$239$	9.46410	0.612182	0.306091	+	0.952002i	$0.400979\pi$
$240$	0	0
$241$	−0.0371647	−0.00239399	−0.00119700	−	0.999999i	$-0.500381\pi$
$241$	−0.0371647	−0.00239399	−0.00119700	+	0.999999i	$0.500381\pi$
$242$	22.5167	1.44743
$243$	0	0
$244$	0.138701	0.00887940
$245$	−26.5283	−1.69483
$246$	0	0
$247$	3.10583	0.197619
$248$	−6.00000	−0.381000
$249$	0	0
$250$	−6.93237	−0.438441
$251$	5.37945	0.339548	0.169774	−	0.985483i	$-0.445696\pi$
$251$	5.37945	0.339548	0.169774	+	0.985483i	$0.445696\pi$
$252$	0	0
$253$	0	0
$254$	−10.3923	−0.652071
$255$	0	0
$256$	19.0000	1.18750
$257$	11.5359	0.719590	0.359795	−	0.933031i	$-0.382847\pi$
$257$	11.5359	0.719590	0.359795	+	0.933031i	$0.382847\pi$
$258$	0	0
$259$	32.3923	2.01276
$260$	10.0382	0.622542
$261$	0	0
$262$	13.6077	0.840686
$263$	−23.6627	−1.45910	−0.729552	−	0.683925i	$-0.760271\pi$
$263$	−23.6627	−1.45910	−0.729552	+	0.683925i	$0.760271\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	6.92820	0.424795
$267$	0	0
$268$	7.72741	0.472026
$269$	−18.9282	−1.15407	−0.577036	−	0.816718i	$-0.695791\pi$
$269$	−18.9282	−1.15407	−0.577036	+	0.816718i	$0.695791\pi$
$270$	0	0
$271$	−4.39230	−0.266814	−0.133407	−	0.991061i	$-0.542592\pi$
$271$	−4.39230	−0.266814	−0.133407	+	0.991061i	$0.542592\pi$
$272$	−21.2132	−1.28624
$273$	0	0
$274$	−4.65874	−0.281445
$275$	−30.3548	−1.83046
$276$	0	0
$277$	−30.9282	−1.85830	−0.929148	−	0.369708i	$-0.879458\pi$
$277$	−30.9282	−1.85830	−0.929148	+	0.369708i	$0.879458\pi$
$278$	−31.8564	−1.91062
$279$	0	0
$280$	−22.3923	−1.33820
$281$	−14.0406	−0.837592	−0.418796	−	0.908080i	$-0.637548\pi$
$281$	−14.0406	−0.837592	−0.418796	+	0.908080i	$0.637548\pi$
$282$	0	0
$283$	8.76268	0.520887	0.260444	−	0.965489i	$-0.416131\pi$
$283$	8.76268	0.520887	0.260444	+	0.965489i	$0.416131\pi$
$284$	3.46410	0.205557
$285$	0	0
$286$	25.4558	1.50524
$287$	−1.79315	−0.105846
$288$	0	0
$289$	1.00000	0.0588235
$290$	−24.7351	−1.45250
$291$	0	0
$292$	−12.1244	−0.709524
$293$	−16.9062	−0.987670	−0.493835	−	0.869556i	$-0.664405\pi$
$293$	−16.9062	−0.987670	−0.493835	+	0.869556i	$0.664405\pi$
$294$	0	0
$295$	3.10583	0.180828
$296$	14.5211	0.844020
$297$	0	0
$298$	4.65874	0.269874
$299$	0	0
$300$	0	0
$301$	22.9282	1.32156
$302$	−21.4641	−1.23512
$303$	0	0
$304$	5.17638	0.296886
$305$	−0.464102	−0.0265744
$306$	0	0
$307$	30.9282	1.76517	0.882583	−	0.470157i	$-0.155803\pi$
$307$	30.9282	1.76517	0.882583	+	0.470157i	$0.155803\pi$
$308$	18.9282	1.07853
$309$	0	0
$310$	−20.0764	−1.14026
$311$	3.46410	0.196431	0.0982156	−	0.995165i	$-0.468687\pi$
$311$	3.46410	0.196431	0.0982156	+	0.995165i	$0.468687\pi$
$312$	0	0
$313$	9.65926	0.545974	0.272987	−	0.962018i	$-0.411988\pi$
$313$	9.65926	0.545974	0.272987	+	0.962018i	$0.411988\pi$
$314$	−24.0788	−1.35885
$315$	0	0
$316$	4.14110	0.232955
$317$	35.1962	1.97681	0.988406	−	0.151831i	$-0.0485170\pi$
$317$	35.1962	1.97681	0.988406	+	0.151831i	$0.0485170\pi$
$318$	0	0
$319$	−20.9086	−1.17066
$320$	−3.34607	−0.187051
$321$	0	0
$322$	0	0
$323$	−4.39230	−0.244394
$324$	0	0
$325$	−18.5885	−1.03110
$326$	−26.7846	−1.48346
$327$	0	0
$328$	−0.803848	−0.0443851
$329$	26.7685	1.47580
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	−6.69213	−0.367278
$333$	0	0
$334$	30.0000	1.64153
$335$	−25.8564	−1.41269
$336$	0	0
$337$	−3.06866	−0.167161	−0.0835804	−	0.996501i	$-0.526636\pi$
$337$	−3.06866	−0.167161	−0.0835804	+	0.996501i	$0.526636\pi$
$338$	−6.92820	−0.376845
$339$	0	0
$340$	−14.1962	−0.769894
$341$	−16.9706	−0.919007
$342$	0	0
$343$	−3.58630	−0.193642
$344$	10.2784	0.554176
$345$	0	0
$346$	−19.3923	−1.04254
$347$	4.39230	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$347$	4.39230	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$348$	0	0
$349$	24.7846	1.32669	0.663345	−	0.748314i	$-0.269136\pi$
$349$	24.7846	1.32669	0.663345	+	0.748314i	$0.269136\pi$
$350$	−41.4655	−2.21642
$351$	0	0
$352$	25.4558	1.35680
$353$	−7.73205	−0.411536	−0.205768	−	0.978601i	$-0.565969\pi$
$353$	−7.73205	−0.411536	−0.205768	+	0.978601i	$0.565969\pi$
$354$	0	0
$355$	−11.5911	−0.615192
$356$	−0.896575	−0.0475184
$357$	0	0
$358$	10.3923	0.549250
$359$	9.31749	0.491758	0.245879	−	0.969301i	$-0.420923\pi$
$359$	9.31749	0.491758	0.245879	+	0.969301i	$0.420923\pi$
$360$	0	0
$361$	−17.9282	−0.943590
$362$	−2.44949	−0.128742
$363$	0	0
$364$	11.5911	0.607539
$365$	40.5689	2.12347
$366$	0	0
$367$	12.8295	0.669692	0.334846	−	0.942273i	$-0.391316\pi$
$367$	12.8295	0.669692	0.334846	+	0.942273i	$0.391316\pi$
$368$	0	0
$369$	0	0
$370$	48.5885	2.52599
$371$	−3.46410	−0.179847
$372$	0	0
$373$	8.38375	0.434094	0.217047	−	0.976161i	$-0.430358\pi$
$373$	8.38375	0.434094	0.217047	+	0.976161i	$0.430358\pi$
$374$	−36.0000	−1.86152
$375$	0	0
$376$	12.0000	0.618853
$377$	−12.8038	−0.659432
$378$	0	0
$379$	−1.23835	−0.0636097	−0.0318048	−	0.999494i	$-0.510126\pi$
$379$	−1.23835	−0.0636097	−0.0318048	+	0.999494i	$0.510126\pi$
$380$	3.46410	0.177705
$381$	0	0
$382$	28.5617	1.46134
$383$	24.9754	1.27618	0.638091	−	0.769961i	$-0.279724\pi$
$383$	24.9754	1.27618	0.638091	+	0.769961i	$0.279724\pi$
$384$	0	0
$385$	−63.3350	−3.22785
$386$	−43.6410	−2.22127
$387$	0	0
$388$	−18.9024	−0.959625
$389$	−9.14162	−0.463499	−0.231749	−	0.972776i	$-0.574445\pi$
$389$	−9.14162	−0.463499	−0.231749	+	0.972776i	$0.574445\pi$
$390$	0	0
$391$	0	0
$392$	−13.7321	−0.693573
$393$	0	0
$394$	−10.6077	−0.534408
$395$	−13.8564	−0.697191
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	9.79796	0.491127
$399$	0	0
$400$	−30.9808	−1.54904
$401$	19.6603	0.981788	0.490894	−	0.871219i	$-0.336670\pi$
$401$	19.6603	0.981788	0.490894	+	0.871219i	$0.336670\pi$
$402$	0	0
$403$	−10.3923	−0.517678
$404$	18.4641	0.918623
$405$	0	0
$406$	−28.5617	−1.41749
$407$	41.0718	2.03585
$408$	0	0
$409$	−12.7846	−0.632158	−0.316079	−	0.948733i	$-0.602367\pi$
$409$	−12.7846	−0.632158	−0.316079	+	0.948733i	$0.602367\pi$
$410$	−2.68973	−0.132836
$411$	0	0
$412$	17.2480	0.849746
$413$	3.58630	0.176470
$414$	0	0
$415$	22.3923	1.09920
$416$	15.5885	0.764287
$417$	0	0
$418$	8.78461	0.429669
$419$	3.58630	0.175202	0.0876012	−	0.996156i	$-0.472080\pi$
$419$	3.58630	0.175202	0.0876012	+	0.996156i	$0.472080\pi$
$420$	0	0
$421$	8.38375	0.408599	0.204299	−	0.978908i	$-0.434508\pi$
$421$	8.38375	0.408599	0.204299	+	0.978908i	$0.434508\pi$
$422$	25.1769	1.22559
$423$	0	0
$424$	−1.55291	−0.0754162
$425$	26.2880	1.27516
$426$	0	0
$427$	−0.535898	−0.0259339
$428$	16.4901	0.797079
$429$	0	0
$430$	34.3923	1.65854
$431$	35.2538	1.69812	0.849058	−	0.528300i	$-0.177170\pi$
$431$	35.2538	1.69812	0.849058	+	0.528300i	$0.177170\pi$
$432$	0	0
$433$	39.6351	1.90474	0.952372	−	0.304939i	$-0.0986362\pi$
$433$	39.6351	1.90474	0.952372	+	0.304939i	$0.0986362\pi$
$434$	−23.1822	−1.11278
$435$	0	0
$436$	10.7961	0.517038
$437$	0	0
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	−28.3923	−1.35355
$441$	0	0
$442$	−22.0454	−1.04859
$443$	−40.6410	−1.93091	−0.965456	−	0.260564i	$-0.916091\pi$
$443$	−40.6410	−1.93091	−0.965456	+	0.260564i	$0.916091\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	6.92820	0.328060
$447$	0	0
$448$	−3.86370	−0.182543
$449$	−33.2487	−1.56910	−0.784552	−	0.620063i	$-0.787107\pi$
$449$	−33.2487	−1.56910	−0.784552	+	0.620063i	$0.787107\pi$
$450$	0	0
$451$	−2.27362	−0.107061
$452$	−3.34607	−0.157386
$453$	0	0
$454$	−31.6675	−1.48623
$455$	−38.7846	−1.81825
$456$	0	0
$457$	30.3920	1.42168	0.710839	−	0.703355i	$-0.248316\pi$
$457$	30.3920	1.42168	0.710839	+	0.703355i	$0.248316\pi$
$458$	−27.9053	−1.30393
$459$	0	0
$460$	0	0
$461$	−6.12436	−0.285240	−0.142620	−	0.989778i	$-0.545553\pi$
$461$	−6.12436	−0.285240	−0.142620	+	0.989778i	$0.545553\pi$
$462$	0	0
$463$	−0.392305	−0.0182320	−0.00911598	−	0.999958i	$-0.502902\pi$
$463$	−0.392305	−0.0182320	−0.00911598	+	0.999958i	$0.502902\pi$
$464$	−21.3397	−0.990673
$465$	0	0
$466$	−11.1962	−0.518652
$467$	−12.0716	−0.558606	−0.279303	−	0.960203i	$-0.590103\pi$
$467$	−12.0716	−0.558606	−0.279303	+	0.960203i	$0.590103\pi$
$468$	0	0
$469$	−29.8564	−1.37864
$470$	40.1528	1.85211
$471$	0	0
$472$	1.60770	0.0740002
$473$	29.0718	1.33672
$474$	0	0
$475$	−6.41473	−0.294328
$476$	−16.3923	−0.751340
$477$	0	0
$478$	16.3923	0.749767
$479$	8.96575	0.409656	0.204828	−	0.978798i	$-0.434337\pi$
$479$	8.96575	0.409656	0.204828	+	0.978798i	$0.434337\pi$
$480$	0	0
$481$	25.1512	1.14680
$482$	−0.0643712	−0.00293203
$483$	0	0
$484$	13.0000	0.590909
$485$	63.2487	2.87198
$486$	0	0
$487$	−17.6077	−0.797881	−0.398940	−	0.916977i	$-0.630622\pi$
$487$	−17.6077	−0.797881	−0.398940	+	0.916977i	$0.630622\pi$
$488$	−0.240237	−0.0108750
$489$	0	0
$490$	−45.9483	−2.07573
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	18.1074	0.815515
$494$	5.37945	0.242033
$495$	0	0
$496$	−17.3205	−0.777714
$497$	−13.3843	−0.600366
$498$	0	0
$499$	−6.92820	−0.310149	−0.155074	−	0.987903i	$-0.549562\pi$
$499$	−6.92820	−0.310149	−0.155074	+	0.987903i	$0.549562\pi$
$500$	−4.00240	−0.178993
$501$	0	0
$502$	9.31749	0.415860
$503$	23.1822	1.03364	0.516822	−	0.856093i	$-0.327115\pi$
$503$	23.1822	1.03364	0.516822	+	0.856093i	$0.327115\pi$
$504$	0	0
$505$	−61.7821	−2.74927
$506$	0	0
$507$	0	0
$508$	−6.00000	−0.266207
$509$	31.7321	1.40650	0.703249	−	0.710943i	$-0.251732\pi$
$509$	31.7321	1.40650	0.703249	+	0.710943i	$0.251732\pi$
$510$	0	0
$511$	46.8449	2.07230
$512$	8.66025	0.382733
$513$	0	0
$514$	19.9808	0.881314
$515$	−57.7128	−2.54313
$516$	0	0
$517$	33.9411	1.49273
$518$	56.1051	2.46512
$519$	0	0
$520$	−17.3867	−0.762456
$521$	−14.0406	−0.615130	−0.307565	−	0.951527i	$-0.599514\pi$
$521$	−14.0406	−0.615130	−0.307565	+	0.951527i	$0.599514\pi$
$522$	0	0
$523$	−16.7675	−0.733191	−0.366596	−	0.930380i	$-0.619477\pi$
$523$	−16.7675	−0.733191	−0.366596	+	0.930380i	$0.619477\pi$
$524$	7.85641	0.343209
$525$	0	0
$526$	−40.9850	−1.78703
$527$	14.6969	0.640209
$528$	0	0
$529$	0	0
$530$	−5.19615	−0.225706
$531$	0	0
$532$	4.00000	0.173422
$533$	−1.39230	−0.0603074
$534$	0	0
$535$	−55.1769	−2.38551
$536$	−13.3843	−0.578112
$537$	0	0
$538$	−32.7846	−1.41344
$539$	−38.8401	−1.67296
$540$	0	0
$541$	−35.5885	−1.53007	−0.765034	−	0.643990i	$-0.777278\pi$
$541$	−35.5885	−1.53007	−0.765034	+	0.643990i	$0.777278\pi$
$542$	−7.60770	−0.326778
$543$	0	0
$544$	−22.0454	−0.945189
$545$	−36.1244	−1.54740
$546$	0	0
$547$	5.32051	0.227488	0.113744	−	0.993510i	$-0.463716\pi$
$547$	5.32051	0.227488	0.113744	+	0.993510i	$0.463716\pi$
$548$	−2.68973	−0.114899
$549$	0	0
$550$	−52.5761	−2.24185
$551$	−4.41851	−0.188235
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−53.5692	−2.27594
$555$	0	0
$556$	−18.3923	−0.780007
$557$	23.2466	0.984990	0.492495	−	0.870315i	$-0.336085\pi$
$557$	23.2466	0.984990	0.492495	+	0.870315i	$0.336085\pi$
$558$	0	0
$559$	17.8028	0.752977
$560$	−64.6410	−2.73158
$561$	0	0
$562$	−24.3190	−1.02584
$563$	−4.41851	−0.186218	−0.0931089	−	0.995656i	$-0.529680\pi$
$563$	−4.41851	−0.186218	−0.0931089	+	0.995656i	$0.529680\pi$
$564$	0	0
$565$	11.1962	0.471026
$566$	15.1774	0.637954
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−0.240237	−0.0100712	−0.00503562	−	0.999987i	$-0.501603\pi$
$569$	−0.240237	−0.0100712	−0.00503562	+	0.999987i	$0.501603\pi$
$570$	0	0
$571$	−26.9716	−1.12873	−0.564363	−	0.825527i	$-0.690878\pi$
$571$	−26.9716	−1.12873	−0.564363	+	0.825527i	$0.690878\pi$
$572$	14.6969	0.614510
$573$	0	0
$574$	−3.10583	−0.129635
$575$	0	0
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	1.73205	0.0720438
$579$	0	0
$580$	−14.2808	−0.592979
$581$	25.8564	1.07270
$582$	0	0
$583$	−4.39230	−0.181911
$584$	21.0000	0.868986
$585$	0	0
$586$	−29.2824	−1.20964
$587$	−34.6410	−1.42979	−0.714894	−	0.699233i	$-0.753525\pi$
$587$	−34.6410	−1.42979	−0.714894	+	0.699233i	$0.753525\pi$
$588$	0	0
$589$	−3.58630	−0.147771
$590$	5.37945	0.221469
$591$	0	0
$592$	41.9187	1.72285
$593$	38.7846	1.59269	0.796347	−	0.604841i	$-0.206763\pi$
$593$	38.7846	1.59269	0.796347	+	0.604841i	$0.206763\pi$
$594$	0	0
$595$	54.8497	2.24862
$596$	2.68973	0.110175
$597$	0	0
$598$	0	0
$599$	29.3205	1.19800	0.599002	−	0.800748i	$-0.295564\pi$
$599$	29.3205	1.19800	0.599002	+	0.800748i	$0.295564\pi$
$600$	0	0
$601$	21.1962	0.864609	0.432305	−	0.901728i	$-0.357701\pi$
$601$	21.1962	0.864609	0.432305	+	0.901728i	$0.357701\pi$
$602$	39.7128	1.61857
$603$	0	0
$604$	−12.3923	−0.504236
$605$	−43.4988	−1.76848
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	5.37945	0.218166
$609$	0	0
$610$	−0.803848	−0.0325468
$611$	20.7846	0.840855
$612$	0	0
$613$	−17.6641	−0.713445	−0.356722	−	0.934210i	$-0.616106\pi$
$613$	−17.6641	−0.713445	−0.356722	+	0.934210i	$0.616106\pi$
$614$	53.5692	2.16188
$615$	0	0
$616$	−32.7846	−1.32093
$617$	43.4345	1.74861	0.874303	−	0.485380i	$-0.161319\pi$
$617$	43.4345	1.74861	0.874303	+	0.485380i	$0.161319\pi$
$618$	0	0
$619$	−24.6980	−0.992695	−0.496348	−	0.868124i	$-0.665326\pi$
$619$	−24.6980	−0.992695	−0.496348	+	0.868124i	$0.665326\pi$
$620$	−11.5911	−0.465510
$621$	0	0
$622$	6.00000	0.240578
$623$	3.46410	0.138786
$624$	0	0
$625$	−17.5885	−0.703538
$626$	16.7303	0.668678
$627$	0	0
$628$	−13.9019	−0.554746
$629$	−35.5692	−1.41824
$630$	0	0
$631$	27.0459	1.07668	0.538341	−	0.842727i	$-0.319051\pi$
$631$	27.0459	1.07668	0.538341	+	0.842727i	$0.319051\pi$
$632$	−7.17260	−0.285311
$633$	0	0
$634$	60.9615	2.42109
$635$	20.0764	0.796707
$636$	0	0
$637$	−23.7846	−0.942381
$638$	−36.2147	−1.43376
$639$	0	0
$640$	−40.5689	−1.60363
$641$	−39.4321	−1.55747	−0.778737	−	0.627351i	$-0.784139\pi$
$641$	−39.4321	−1.55747	−0.778737	+	0.627351i	$0.784139\pi$
$642$	0	0
$643$	24.2175	0.955045	0.477522	−	0.878620i	$-0.341535\pi$
$643$	24.2175	0.955045	0.477522	+	0.878620i	$0.341535\pi$
$644$	0	0
$645$	0	0
$646$	−7.60770	−0.299321
$647$	6.92820	0.272376	0.136188	−	0.990683i	$-0.456515\pi$
$647$	6.92820	0.272376	0.136188	+	0.990683i	$0.456515\pi$
$648$	0	0
$649$	4.54725	0.178495
$650$	−32.1962	−1.26284
$651$	0	0
$652$	−15.4641	−0.605621
$653$	17.5359	0.686233	0.343116	−	0.939293i	$-0.388517\pi$
$653$	17.5359	0.686233	0.343116	+	0.939293i	$0.388517\pi$
$654$	0	0
$655$	−26.2880	−1.02716
$656$	−2.32051	−0.0906006
$657$	0	0
$658$	46.3644	1.80747
$659$	−45.0518	−1.75497	−0.877484	−	0.479606i	$-0.840779\pi$
$659$	−45.0518	−1.75497	−0.877484	+	0.479606i	$0.840779\pi$
$660$	0	0
$661$	2.89280	0.112517	0.0562584	−	0.998416i	$-0.482083\pi$
$661$	2.89280	0.112517	0.0562584	+	0.998416i	$0.482083\pi$
$662$	31.1769	1.21173
$663$	0	0
$664$	11.5911	0.449822
$665$	−13.3843	−0.519019
$666$	0	0
$667$	0	0
$668$	17.3205	0.670151
$669$	0	0
$670$	−44.7846	−1.73018
$671$	−0.679492	−0.0262315
$672$	0	0
$673$	33.7128	1.29953	0.649767	−	0.760134i	$-0.274867\pi$
$673$	33.7128	1.29953	0.649767	+	0.760134i	$0.274867\pi$
$674$	−5.31508	−0.204729
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−9.86233	−0.379040	−0.189520	−	0.981877i	$-0.560693\pi$
$677$	−9.86233	−0.379040	−0.189520	+	0.981877i	$0.560693\pi$
$678$	0	0
$679$	73.0333	2.80276
$680$	24.5885	0.942924
$681$	0	0
$682$	−29.3939	−1.12555
$683$	−16.3923	−0.627234	−0.313617	−	0.949550i	$-0.601541\pi$
$683$	−16.3923	−0.627234	−0.313617	+	0.949550i	$0.601541\pi$
$684$	0	0
$685$	9.00000	0.343872
$686$	−6.21166	−0.237162
$687$	0	0
$688$	29.6713	1.13121
$689$	−2.68973	−0.102470
$690$	0	0
$691$	−24.2487	−0.922464	−0.461232	−	0.887279i	$-0.652592\pi$
$691$	−24.2487	−0.922464	−0.461232	+	0.887279i	$0.652592\pi$
$692$	−11.1962	−0.425614
$693$	0	0
$694$	7.60770	0.288784
$695$	61.5419	2.33442
$696$	0	0
$697$	1.96902	0.0745818
$698$	42.9282	1.62486
$699$	0	0
$700$	−23.9401	−0.904851
$701$	−31.0112	−1.17128	−0.585638	−	0.810573i	$-0.699156\pi$
$701$	−31.0112	−1.17128	−0.585638	+	0.810573i	$0.699156\pi$
$702$	0	0
$703$	8.67949	0.327353
$704$	−4.89898	−0.184637
$705$	0	0
$706$	−13.3923	−0.504026
$707$	−71.3398	−2.68301
$708$	0	0
$709$	16.8319	0.632134	0.316067	−	0.948737i	$-0.397638\pi$
$709$	16.8319	0.632134	0.316067	+	0.948737i	$0.397638\pi$
$710$	−20.0764	−0.753454
$711$	0	0
$712$	1.55291	0.0581979
$713$	0	0
$714$	0	0
$715$	−49.1769	−1.83911
$716$	6.00000	0.224231
$717$	0	0
$718$	16.1384	0.602278
$719$	43.8564	1.63557	0.817784	−	0.575525i	$-0.195202\pi$
$719$	43.8564	1.63557	0.817784	+	0.575525i	$0.195202\pi$
$720$	0	0
$721$	−66.6410	−2.48184
$722$	−31.0526	−1.15566
$723$	0	0
$724$	−1.41421	−0.0525588
$725$	26.4449	0.982138
$726$	0	0
$727$	−17.2480	−0.639692	−0.319846	−	0.947470i	$-0.603631\pi$
$727$	−17.2480	−0.639692	−0.319846	+	0.947470i	$0.603631\pi$
$728$	−20.0764	−0.744081
$729$	0	0
$730$	70.2674	2.60071
$731$	−25.1769	−0.931202
$732$	0	0
$733$	12.8666	0.475240	0.237620	−	0.971358i	$-0.423633\pi$
$733$	12.8666	0.475240	0.237620	+	0.971358i	$0.423633\pi$
$734$	22.2213	0.820202
$735$	0	0
$736$	0	0
$737$	−37.8564	−1.39446
$738$	0	0
$739$	39.1769	1.44115	0.720573	−	0.693379i	$-0.243879\pi$
$739$	39.1769	1.44115	0.720573	+	0.693379i	$0.243879\pi$
$740$	28.0526	1.03123
$741$	0	0
$742$	−6.00000	−0.220267
$743$	28.0812	1.03020	0.515100	−	0.857130i	$-0.327755\pi$
$743$	28.0812	1.03020	0.515100	+	0.857130i	$0.327755\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	14.5211	0.531654
$747$	0	0
$748$	−20.7846	−0.759961
$749$	−63.7128	−2.32802
$750$	0	0
$751$	31.1127	1.13532	0.567659	−	0.823264i	$-0.307849\pi$
$751$	31.1127	1.13532	0.567659	+	0.823264i	$0.307849\pi$
$752$	34.6410	1.26323
$753$	0	0
$754$	−22.1769	−0.807636
$755$	41.4655	1.50908
$756$	0	0
$757$	32.0093	1.16340	0.581698	−	0.813405i	$-0.302388\pi$
$757$	32.0093	1.16340	0.581698	+	0.813405i	$0.302388\pi$
$758$	−2.14488	−0.0779056
$759$	0	0
$760$	−6.00000	−0.217643
$761$	−31.9808	−1.15930	−0.579651	−	0.814865i	$-0.696811\pi$
$761$	−31.9808	−1.15930	−0.579651	+	0.814865i	$0.696811\pi$
$762$	0	0
$763$	−41.7128	−1.51011
$764$	16.4901	0.596590
$765$	0	0
$766$	43.2586	1.56300
$767$	2.78461	0.100546
$768$	0	0
$769$	16.1112	0.580983	0.290492	−	0.956878i	$-0.406181\pi$
$769$	16.1112	0.580983	0.290492	+	0.956878i	$0.406181\pi$
$770$	−109.699	−3.95329
$771$	0	0
$772$	−25.1962	−0.906829
$773$	33.2848	1.19717	0.598585	−	0.801059i	$-0.295730\pi$
$773$	33.2848	1.19717	0.598585	+	0.801059i	$0.295730\pi$
$774$	0	0
$775$	21.4641	0.771013
$776$	32.7399	1.17530
$777$	0	0
$778$	−15.8338	−0.567667
$779$	−0.480473	−0.0172147
$780$	0	0
$781$	−16.9706	−0.607254
$782$	0	0
$783$	0	0
$784$	−39.6410	−1.41575
$785$	46.5167	1.66025
$786$	0	0
$787$	39.3949	1.40428	0.702138	−	0.712040i	$-0.252229\pi$
$787$	39.3949	1.40428	0.702138	+	0.712040i	$0.252229\pi$
$788$	−6.12436	−0.218171
$789$	0	0
$790$	−24.0000	−0.853882
$791$	12.9282	0.459674
$792$	0	0
$793$	−0.416102	−0.0147762
$794$	−43.3013	−1.53670
$795$	0	0
$796$	5.65685	0.200502
$797$	−15.3533	−0.543841	−0.271920	−	0.962320i	$-0.587659\pi$
$797$	−15.3533	−0.543841	−0.271920	+	0.962320i	$0.587659\pi$
$798$	0	0
$799$	−29.3939	−1.03988
$800$	−32.1962	−1.13831
$801$	0	0
$802$	34.0526	1.20244
$803$	59.3970	2.09607
$804$	0	0
$805$	0	0
$806$	−18.0000	−0.634023
$807$	0	0
$808$	−31.9808	−1.12508
$809$	1.85641	0.0652678	0.0326339	−	0.999467i	$-0.489610\pi$
$809$	1.85641	0.0652678	0.0326339	+	0.999467i	$0.489610\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	−16.4901	−0.578689
$813$	0	0
$814$	71.1384	2.49340
$815$	51.7439	1.81251
$816$	0	0
$817$	6.14359	0.214937
$818$	−22.1436	−0.774233
$819$	0	0
$820$	−1.55291	−0.0542301
$821$	−36.0333	−1.25757	−0.628786	−	0.777579i	$-0.716448\pi$
$821$	−36.0333	−1.25757	−0.628786	+	0.777579i	$0.716448\pi$
$822$	0	0
$823$	−9.46410	−0.329898	−0.164949	−	0.986302i	$-0.552746\pi$
$823$	−9.46410	−0.329898	−0.164949	+	0.986302i	$0.552746\pi$
$824$	−29.8744	−1.04072
$825$	0	0
$826$	6.21166	0.216131
$827$	−16.0096	−0.556709	−0.278354	−	0.960478i	$-0.589789\pi$
$827$	−16.0096	−0.556709	−0.278354	+	0.960478i	$0.589789\pi$
$828$	0	0
$829$	31.5885	1.09711	0.548556	−	0.836114i	$-0.315178\pi$
$829$	31.5885	1.09711	0.548556	+	0.836114i	$0.315178\pi$
$830$	38.7846	1.34623
$831$	0	0
$832$	−3.00000	−0.104006
$833$	33.6365	1.16544
$834$	0	0
$835$	−57.9555	−2.00563
$836$	5.07180	0.175412
$837$	0	0
$838$	6.21166	0.214578
$839$	32.1480	1.10987	0.554936	−	0.831893i	$-0.312743\pi$
$839$	32.1480	1.10987	0.554936	+	0.831893i	$0.312743\pi$
$840$	0	0
$841$	−10.7846	−0.371883
$842$	14.5211	0.500429
$843$	0	0
$844$	14.5359	0.500346
$845$	13.3843	0.460433
$846$	0	0
$847$	−50.2281	−1.72586
$848$	−4.48288	−0.153943
$849$	0	0
$850$	45.5322	1.56174
$851$	0	0
$852$	0	0
$853$	−5.21539	−0.178572	−0.0892858	−	0.996006i	$-0.528458\pi$
$853$	−5.21539	−0.178572	−0.0892858	+	0.996006i	$0.528458\pi$
$854$	−0.928203	−0.0317625
$855$	0	0
$856$	−28.5617	−0.976218
$857$	51.2487	1.75062	0.875311	−	0.483560i	$-0.160656\pi$
$857$	51.2487	1.75062	0.875311	+	0.483560i	$0.160656\pi$
$858$	0	0
$859$	36.3923	1.24169	0.620845	−	0.783934i	$-0.286790\pi$
$859$	36.3923	1.24169	0.620845	+	0.783934i	$0.286790\pi$
$860$	19.8564	0.677098
$861$	0	0
$862$	61.0614	2.07976
$863$	−8.78461	−0.299032	−0.149516	−	0.988759i	$-0.547771\pi$
$863$	−8.78461	−0.299032	−0.149516	+	0.988759i	$0.547771\pi$
$864$	0	0
$865$	37.4631	1.27378
$866$	68.6501	2.33282
$867$	0	0
$868$	−13.3843	−0.454291
$869$	−20.2872	−0.688196
$870$	0	0
$871$	−23.1822	−0.785500
$872$	−18.6993	−0.633240
$873$	0	0
$874$	0	0
$875$	15.4641	0.522782
$876$	0	0
$877$	−32.7846	−1.10706	−0.553529	−	0.832830i	$-0.686719\pi$
$877$	−32.7846	−1.10706	−0.553529	+	0.832830i	$0.686719\pi$
$878$	45.0333	1.51980
$879$	0	0
$880$	−81.9615	−2.76292
$881$	0.896575	0.0302064	0.0151032	−	0.999886i	$-0.495192\pi$
$881$	0.896575	0.0302064	0.0151032	+	0.999886i	$0.495192\pi$
$882$	0	0
$883$	21.7128	0.730694	0.365347	−	0.930871i	$-0.380950\pi$
$883$	21.7128	0.730694	0.365347	+	0.930871i	$0.380950\pi$
$884$	−12.7279	−0.428086
$885$	0	0
$886$	−70.3923	−2.36488
$887$	−17.0718	−0.573215	−0.286607	−	0.958048i	$-0.592528\pi$
$887$	−17.0718	−0.573215	−0.286607	+	0.958048i	$0.592528\pi$
$888$	0	0
$889$	23.1822	0.777507
$890$	5.19615	0.174175
$891$	0	0
$892$	4.00000	0.133930
$893$	7.17260	0.240022
$894$	0	0
$895$	−20.0764	−0.671080
$896$	−46.8449	−1.56498
$897$	0	0
$898$	−57.5885	−1.92175
$899$	14.7846	0.493094
$900$	0	0
$901$	3.80385	0.126725
$902$	−3.93803	−0.131122
$903$	0	0
$904$	5.79555	0.192757
$905$	4.73205	0.157299
$906$	0	0
$907$	−44.4970	−1.47750	−0.738749	−	0.673981i	$-0.764583\pi$
$907$	−44.4970	−1.47750	−0.738749	+	0.673981i	$0.764583\pi$
$908$	−18.2832	−0.606751
$909$	0	0
$910$	−67.1769	−2.22689
$911$	−11.5911	−0.384031	−0.192015	−	0.981392i	$-0.561502\pi$
$911$	−11.5911	−0.384031	−0.192015	+	0.981392i	$0.561502\pi$
$912$	0	0
$913$	32.7846	1.08501
$914$	52.6405	1.74119
$915$	0	0
$916$	−16.1112	−0.532327
$917$	−30.3548	−1.00240
$918$	0	0
$919$	−24.4206	−0.805560	−0.402780	−	0.915297i	$-0.631956\pi$
$919$	−24.4206	−0.805560	−0.402780	+	0.915297i	$0.631956\pi$
$920$	0	0
$921$	0	0
$922$	−10.6077	−0.349346
$923$	−10.3923	−0.342067
$924$	0	0
$925$	−51.9470	−1.70801
$926$	−0.679492	−0.0223295
$927$	0	0
$928$	−22.1769	−0.727993
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−8.20788	−0.269002
$932$	−6.46410	−0.211739
$933$	0	0
$934$	−20.9086	−0.684150
$935$	69.5467	2.27442
$936$	0	0
$937$	48.5365	1.58562	0.792810	−	0.609469i	$-0.208617\pi$
$937$	48.5365	1.58562	0.792810	+	0.609469i	$0.208617\pi$
$938$	−51.7128	−1.68848
$939$	0	0
$940$	23.1822	0.756121
$941$	3.28169	0.106980	0.0534901	−	0.998568i	$-0.482965\pi$
$941$	3.28169	0.106980	0.0534901	+	0.998568i	$0.482965\pi$
$942$	0	0
$943$	0	0
$944$	4.64102	0.151052
$945$	0	0
$946$	50.3538	1.63714
$947$	−6.92820	−0.225136	−0.112568	−	0.993644i	$-0.535908\pi$
$947$	−6.92820	−0.225136	−0.112568	+	0.993644i	$0.535908\pi$
$948$	0	0
$949$	36.3731	1.18072
$950$	−11.1106	−0.360477
$951$	0	0
$952$	28.3923	0.920200
$953$	9.07725	0.294041	0.147020	−	0.989133i	$-0.453032\pi$
$953$	9.07725	0.294041	0.147020	+	0.989133i	$0.453032\pi$
$954$	0	0
$955$	−55.1769	−1.78548
$956$	9.46410	0.306091
$957$	0	0
$958$	15.5291	0.501724
$959$	10.3923	0.335585
$960$	0	0
$961$	−19.0000	−0.612903
$962$	43.5632	1.40453
$963$	0	0
$964$	−0.0371647	−0.00119700
$965$	84.3080	2.71397
$966$	0	0
$967$	47.5692	1.52972	0.764861	−	0.644195i	$-0.222807\pi$
$967$	47.5692	1.52972	0.764861	+	0.644195i	$0.222807\pi$
$968$	−22.5167	−0.723713
$969$	0	0
$970$	109.550	3.51744
$971$	−14.3452	−0.460360	−0.230180	−	0.973148i	$-0.573931\pi$
$971$	−14.3452	−0.460360	−0.230180	+	0.973148i	$0.573931\pi$
$972$	0	0
$973$	71.0624	2.27816
$974$	−30.4974	−0.977200
$975$	0	0
$976$	−0.693504	−0.0221985
$977$	17.2108	0.550622	0.275311	−	0.961355i	$-0.411219\pi$
$977$	17.2108	0.550622	0.275311	+	0.961355i	$0.411219\pi$
$978$	0	0
$979$	4.39230	0.140379
$980$	−26.5283	−0.847415
$981$	0	0
$982$	31.1769	0.994895
$983$	−4.06678	−0.129710	−0.0648550	−	0.997895i	$-0.520658\pi$
$983$	−4.06678	−0.129710	−0.0648550	+	0.997895i	$0.520658\pi$
$984$	0	0
$985$	20.4925	0.652945
$986$	31.3629	0.998798
$987$	0	0
$988$	3.10583	0.0988096
$989$	0	0
$990$	0	0
$991$	−22.6410	−0.719216	−0.359608	−	0.933104i	$-0.617090\pi$
$991$	−22.6410	−0.719216	−0.359608	+	0.933104i	$0.617090\pi$
$992$	−18.0000	−0.571501
$993$	0	0
$994$	−23.1822	−0.735295
$995$	−18.9282	−0.600064
$996$	0	0
$997$	13.1962	0.417926	0.208963	−	0.977924i	$-0.432991\pi$
$997$	13.1962	0.417926	0.208963	+	0.977924i	$0.432991\pi$
$998$	−12.0000	−0.379853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bd.1.3		4
3.2	odd	2		1587.2.a.n.1.2	yes	4
23.22	odd	2	inner	4761.2.a.bd.1.4		4
69.68	even	2		1587.2.a.n.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1587.2.a.n.1.1	✓	4	69.68	even	2
1587.2.a.n.1.2	yes	4	3.2	odd	2
4761.2.a.bd.1.3		4	1.1	even	1	trivial
4761.2.a.bd.1.4		4	23.22	odd	2	inner

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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -3.34607 q^{5} -3.86370 q^{7} -1.73205 q^{8} -5.79555 q^{10} -4.89898 q^{11} -3.00000 q^{13} -6.69213 q^{14} -5.00000 q^{16} +4.24264 q^{17} -1.03528 q^{19} -3.34607 q^{20} -8.48528 q^{22} +6.19615 q^{25} -5.19615 q^{26} -3.86370 q^{28} +4.26795 q^{29} +3.46410 q^{31} -5.19615 q^{32} +7.34847 q^{34} +12.9282 q^{35} -8.38375 q^{37} -1.79315 q^{38} +5.79555 q^{40} +0.464102 q^{41} -5.93426 q^{43} -4.89898 q^{44} -6.92820 q^{47} +7.92820 q^{49} +10.7321 q^{50} -3.00000 q^{52} +0.896575 q^{53} +16.3923 q^{55} +6.69213 q^{56} +7.39230 q^{58} -0.928203 q^{59} +0.138701 q^{61} +6.00000 q^{62} +1.00000 q^{64} +10.0382 q^{65} +7.72741 q^{67} +4.24264 q^{68} +22.3923 q^{70} +3.46410 q^{71} -12.1244 q^{73} -14.5211 q^{74} -1.03528 q^{76} +18.9282 q^{77} +4.14110 q^{79} +16.7303 q^{80} +0.803848 q^{82} -6.69213 q^{83} -14.1962 q^{85} -10.2784 q^{86} +8.48528 q^{88} -0.896575 q^{89} +11.5911 q^{91} -12.0000 q^{94} +3.46410 q^{95} -18.9024 q^{97} +13.7321 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 12 q^{13} - 20 q^{16} + 4 q^{25} + 24 q^{29} + 24 q^{35} - 12 q^{41} + 4 q^{49} + 36 q^{50} - 12 q^{52} + 24 q^{55} - 12 q^{58} + 24 q^{59} + 24 q^{62} + 4 q^{64} + 48 q^{70} + 48 q^{77}+ \cdots + 48 q^{98}+O(q^{100}) \) 4 * q + 4 * q^4 - 12 * q^13 - 20 * q^16 + 4 * q^25 + 24 * q^29 + 24 * q^35 - 12 * q^41 + 4 * q^49 + 36 * q^50 - 12 * q^52 + 24 * q^55 - 12 * q^58 + 24 * q^59 + 24 * q^62 + 4 * q^64 + 48 * q^70 + 48 * q^77 + 24 * q^82 - 36 * q^85 - 48 * q^94 + 48 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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