LMFDB - Embedded newform 7623.2.a.cp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.10939$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10939 q^{2} +2.44952 q^{4} -0.492391 q^{5} +1.00000 q^{7} -0.948212 q^{8} +O(q^{10})$	$q-2.10939 q^{2} +2.44952 q^{4} -0.492391 q^{5} +1.00000 q^{7} -0.948212 q^{8} +1.03864 q^{10} +5.30029 q^{13} -2.10939 q^{14} -2.89889 q^{16} -3.03721 q^{17} -4.66622 q^{19} -1.20612 q^{20} +5.63835 q^{23} -4.75755 q^{25} -11.1804 q^{26} +2.44952 q^{28} -6.92295 q^{29} -1.26565 q^{31} +8.01131 q^{32} +6.40665 q^{34} -0.492391 q^{35} +10.8759 q^{37} +9.84288 q^{38} +0.466891 q^{40} +1.44322 q^{41} +2.88224 q^{43} -11.8935 q^{46} +8.75522 q^{47} +1.00000 q^{49} +10.0355 q^{50} +12.9832 q^{52} -6.63835 q^{53} -0.948212 q^{56} +14.6032 q^{58} +8.35733 q^{59} -13.8953 q^{61} +2.66975 q^{62} -11.1012 q^{64} -2.60982 q^{65} -9.70431 q^{67} -7.43970 q^{68} +1.03864 q^{70} -5.94751 q^{71} -3.77421 q^{73} -22.9414 q^{74} -11.4300 q^{76} -8.80383 q^{79} +1.42739 q^{80} -3.04431 q^{82} -11.0898 q^{83} +1.49549 q^{85} -6.07976 q^{86} -3.10324 q^{89} +5.30029 q^{91} +13.8113 q^{92} -18.4682 q^{94} +2.29761 q^{95} -6.31676 q^{97} -2.10939 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 + 6 * q^7 - 12 * q^8	$ 6 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8} - 8 q^{10} + 4 q^{13} - 4 q^{14} + 8 q^{16} - 22 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{23} + 4 q^{25} - 6 q^{26} + 4 q^{28} - 12 q^{29} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 14 q^{37} + 22 q^{38} + 18 q^{40} - 26 q^{41} - 4 q^{43} + 12 q^{46} + 16 q^{47} + 6 q^{49} + 4 q^{50} + 12 q^{52} - 4 q^{53} - 12 q^{56} - 2 q^{58} + 4 q^{59} - 8 q^{61} - 20 q^{62} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 8 q^{70} - 22 q^{71} + 14 q^{73} - 44 q^{74} - 30 q^{76} - 28 q^{79} + 4 q^{80} - 4 q^{82} - 22 q^{83} - 24 q^{85} + 30 q^{86} + 4 q^{91} - 10 q^{92} - 38 q^{94} + 24 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 + 6 * q^7 - 12 * q^8 - 8 * q^10 + 4 * q^13 - 4 * q^14 + 8 * q^16 - 22 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^23 + 4 * q^25 - 6 * q^26 + 4 * q^28 - 12 * q^29 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 14 * q^37 + 22 * q^38 + 18 * q^40 - 26 * q^41 - 4 * q^43 + 12 * q^46 + 16 * q^47 + 6 * q^49 + 4 * q^50 + 12 * q^52 - 4 * q^53 - 12 * q^56 - 2 * q^58 + 4 * q^59 - 8 * q^61 - 20 * q^62 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 8 * q^70 - 22 * q^71 + 14 * q^73 - 44 * q^74 - 30 * q^76 - 28 * q^79 + 4 * q^80 - 4 * q^82 - 22 * q^83 - 24 * q^85 + 30 * q^86 + 4 * q^91 - 10 * q^92 - 38 * q^94 + 24 * q^95 - 4 * q^97 - 4 * q^98

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 $

$2$	−2.10939	−1.49156	−0.745781	−	0.666191i	$-0.767924\pi$
$2$	−2.10939	−1.49156	−0.745781	+	0.666191i	$0.767924\pi$
$3$	0	0
$3$	0	0
$4$	2.44952	1.22476
$5$	−0.492391	−0.220204	−0.110102	−	0.993920i	$-0.535118\pi$
$5$	−0.492391	−0.220204	−0.110102	+	0.993920i	$0.535118\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.948212	−0.335243
$9$	0	0
$10$	1.03864	0.328448
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.30029	1.47004	0.735018	−	0.678047i	$-0.237174\pi$
$13$	5.30029	1.47004	0.735018	+	0.678047i	$0.237174\pi$
$14$	−2.10939	−0.563758
$15$	0	0
$16$	−2.89889	−0.724723
$17$	−3.03721	−0.736631	−0.368315	−	0.929701i	$-0.620065\pi$
$17$	−3.03721	−0.736631	−0.368315	+	0.929701i	$0.620065\pi$
$18$	0	0
$19$	−4.66622	−1.07050	−0.535252	−	0.844692i	$-0.679784\pi$
$19$	−4.66622	−1.07050	−0.535252	+	0.844692i	$0.679784\pi$
$20$	−1.20612	−0.269697
$21$	0	0
$22$	0	0
$23$	5.63835	1.17568	0.587839	−	0.808978i	$-0.299979\pi$
$23$	5.63835	1.17568	0.587839	+	0.808978i	$0.299979\pi$
$24$	0	0
$25$	−4.75755	−0.951510
$26$	−11.1804	−2.19265
$27$	0	0
$28$	2.44952	0.462916
$29$	−6.92295	−1.28556	−0.642780	−	0.766051i	$-0.722219\pi$
$29$	−6.92295	−1.28556	−0.642780	+	0.766051i	$0.722219\pi$
$30$	0	0
$31$	−1.26565	−0.227317	−0.113659	−	0.993520i	$-0.536257\pi$
$31$	−1.26565	−0.227317	−0.113659	+	0.993520i	$0.536257\pi$
$32$	8.01131	1.41621
$33$	0	0
$34$	6.40665	1.09873
$35$	−0.492391	−0.0832293
$36$	0	0
$37$	10.8759	1.78798	0.893990	−	0.448087i	$-0.147895\pi$
$37$	10.8759	1.78798	0.893990	+	0.448087i	$0.147895\pi$
$38$	9.84288	1.59673
$39$	0	0
$40$	0.466891	0.0738220
$41$	1.44322	0.225393	0.112696	−	0.993629i	$-0.464051\pi$
$41$	1.44322	0.225393	0.112696	+	0.993629i	$0.464051\pi$
$42$	0	0
$43$	2.88224	0.439537	0.219769	−	0.975552i	$-0.429470\pi$
$43$	2.88224	0.439537	0.219769	+	0.975552i	$0.429470\pi$
$44$	0	0
$45$	0	0
$46$	−11.8935	−1.75360
$47$	8.75522	1.27708	0.638540	−	0.769589i	$-0.279539\pi$
$47$	8.75522	1.27708	0.638540	+	0.769589i	$0.279539\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.0355	1.41924
$51$	0	0
$52$	12.9832	1.80044
$53$	−6.63835	−0.911848	−0.455924	−	0.890019i	$-0.650691\pi$
$53$	−6.63835	−0.911848	−0.455924	+	0.890019i	$0.650691\pi$
$54$	0	0
$55$	0	0
$56$	−0.948212	−0.126710
$57$	0	0
$58$	14.6032	1.91749
$59$	8.35733	1.08803	0.544016	−	0.839075i	$-0.316903\pi$
$59$	8.35733	1.08803	0.544016	+	0.839075i	$0.316903\pi$
$60$	0	0
$61$	−13.8953	−1.77911	−0.889554	−	0.456829i	$-0.848985\pi$
$61$	−13.8953	−1.77911	−0.889554	+	0.456829i	$0.848985\pi$
$62$	2.66975	0.339058
$63$	0	0
$64$	−11.1012	−1.38765
$65$	−2.60982	−0.323708
$66$	0	0
$67$	−9.70431	−1.18557	−0.592785	−	0.805361i	$-0.701972\pi$
$67$	−9.70431	−1.18557	−0.592785	+	0.805361i	$0.701972\pi$
$68$	−7.43970	−0.902196
$69$	0	0
$70$	1.03864	0.124142
$71$	−5.94751	−0.705839	−0.352920	−	0.935654i	$-0.614811\pi$
$71$	−5.94751	−0.705839	−0.352920	+	0.935654i	$0.614811\pi$
$72$	0	0
$73$	−3.77421	−0.441737	−0.220869	−	0.975304i	$-0.570889\pi$
$73$	−3.77421	−0.441737	−0.220869	+	0.975304i	$0.570889\pi$
$74$	−22.9414	−2.66688
$75$	0	0
$76$	−11.4300	−1.31111
$77$	0	0
$78$	0	0
$79$	−8.80383	−0.990508	−0.495254	−	0.868748i	$-0.664925\pi$
$79$	−8.80383	−0.990508	−0.495254	+	0.868748i	$0.664925\pi$
$80$	1.42739	0.159587
$81$	0	0
$82$	−3.04431	−0.336188
$83$	−11.0898	−1.21726	−0.608632	−	0.793453i	$-0.708281\pi$
$83$	−11.0898	−1.21726	−0.608632	+	0.793453i	$0.708281\pi$
$84$	0	0
$85$	1.49549	0.162209
$86$	−6.07976	−0.655597
$87$	0	0
$88$	0	0
$89$	−3.10324	−0.328943	−0.164472	−	0.986382i	$-0.552592\pi$
$89$	−3.10324	−0.328943	−0.164472	+	0.986382i	$0.552592\pi$
$90$	0	0
$91$	5.30029	0.555622
$92$	13.8113	1.43992
$93$	0	0
$94$	−18.4682	−1.90484
$95$	2.29761	0.235730
$96$	0	0
$97$	−6.31676	−0.641370	−0.320685	−	0.947186i	$-0.603913\pi$
$97$	−6.31676	−0.641370	−0.320685	+	0.947186i	$0.603913\pi$
$98$	−2.10939	−0.213080
$99$	0	0
$100$	−11.6537	−1.16537
$101$	−11.7984	−1.17399	−0.586993	−	0.809592i	$-0.699688\pi$
$101$	−11.7984	−1.17399	−0.586993	+	0.809592i	$0.699688\pi$
$102$	0	0
$103$	7.00565	0.690287	0.345144	−	0.938550i	$-0.387830\pi$
$103$	7.00565	0.690287	0.345144	+	0.938550i	$0.387830\pi$
$104$	−5.02580	−0.492820
$105$	0	0
$106$	14.0029	1.36008
$107$	−11.3547	−1.09770	−0.548850	−	0.835921i	$-0.684934\pi$
$107$	−11.3547	−1.09770	−0.548850	+	0.835921i	$0.684934\pi$
$108$	0	0
$109$	18.9414	1.81426	0.907129	−	0.420853i	$-0.138269\pi$
$109$	18.9414	1.81426	0.907129	+	0.420853i	$0.138269\pi$
$110$	0	0
$111$	0	0
$112$	−2.89889	−0.273920
$113$	13.4961	1.26961	0.634804	−	0.772673i	$-0.281081\pi$
$113$	13.4961	1.26961	0.634804	+	0.772673i	$0.281081\pi$
$114$	0	0
$115$	−2.77627	−0.258889
$116$	−16.9579	−1.57450
$117$	0	0
$118$	−17.6289	−1.62287
$119$	−3.03721	−0.278420
$120$	0	0
$121$	0	0
$122$	29.3106	2.65365
$123$	0	0
$124$	−3.10023	−0.278409
$125$	4.80453	0.429730
$126$	0	0
$127$	−4.66064	−0.413565	−0.206782	−	0.978387i	$-0.566299\pi$
$127$	−4.66064	−0.413565	−0.206782	+	0.978387i	$0.566299\pi$
$128$	7.39409	0.653551
$129$	0	0
$130$	5.50512	0.482831
$131$	−9.03676	−0.789545	−0.394773	−	0.918779i	$-0.629177\pi$
$131$	−9.03676	−0.789545	−0.394773	+	0.918779i	$0.629177\pi$
$132$	0	0
$133$	−4.66622	−0.404613
$134$	20.4702	1.76835
$135$	0	0
$136$	2.87991	0.246951
$137$	1.63772	0.139920	0.0699600	−	0.997550i	$-0.477713\pi$
$137$	1.63772	0.139920	0.0699600	+	0.997550i	$0.477713\pi$
$138$	0	0
$139$	−1.53472	−0.130173	−0.0650866	−	0.997880i	$-0.520732\pi$
$139$	−1.53472	−0.130173	−0.0650866	+	0.997880i	$0.520732\pi$
$140$	−1.20612	−0.101936
$141$	0	0
$142$	12.5456	1.05280
$143$	0	0
$144$	0	0
$145$	3.40880	0.283086
$146$	7.96127	0.658879
$147$	0	0
$148$	26.6406	2.18985
$149$	−13.4909	−1.10522	−0.552610	−	0.833440i	$-0.686368\pi$
$149$	−13.4909	−1.10522	−0.552610	+	0.833440i	$0.686368\pi$
$150$	0	0
$151$	−12.2370	−0.995835	−0.497917	−	0.867225i	$-0.665902\pi$
$151$	−12.2370	−0.995835	−0.497917	+	0.867225i	$0.665902\pi$
$152$	4.42457	0.358880
$153$	0	0
$154$	0	0
$155$	0.623194	0.0500562
$156$	0	0
$157$	2.52042	0.201152	0.100576	−	0.994929i	$-0.467932\pi$
$157$	2.52042	0.201152	0.100576	+	0.994929i	$0.467932\pi$
$158$	18.5707	1.47741
$159$	0	0
$160$	−3.94470	−0.311856
$161$	5.63835	0.444364
$162$	0	0
$163$	7.87905	0.617135	0.308567	−	0.951203i	$-0.400151\pi$
$163$	7.87905	0.617135	0.308567	+	0.951203i	$0.400151\pi$
$164$	3.53519	0.276052
$165$	0	0
$166$	23.3927	1.81562
$167$	2.05485	0.159009	0.0795047	−	0.996834i	$-0.474666\pi$
$167$	2.05485	0.159009	0.0795047	+	0.996834i	$0.474666\pi$
$168$	0	0
$169$	15.0931	1.16101
$170$	−3.15458	−0.241945
$171$	0	0
$172$	7.06010	0.538327
$173$	23.2707	1.76923	0.884617	−	0.466318i	$-0.154420\pi$
$173$	23.2707	1.76923	0.884617	+	0.466318i	$0.154420\pi$
$174$	0	0
$175$	−4.75755	−0.359637
$176$	0	0
$177$	0	0
$178$	6.54595	0.490640
$179$	−17.6596	−1.31994	−0.659969	−	0.751293i	$-0.729431\pi$
$179$	−17.6596	−1.31994	−0.659969	+	0.751293i	$0.729431\pi$
$180$	0	0
$181$	15.4701	1.14988	0.574941	−	0.818195i	$-0.305025\pi$
$181$	15.4701	1.14988	0.574941	+	0.818195i	$0.305025\pi$
$182$	−11.1804	−0.828745
$183$	0	0
$184$	−5.34635	−0.394138
$185$	−5.35518	−0.393720
$186$	0	0
$187$	0	0
$188$	21.4461	1.56412
$189$	0	0
$190$	−4.84655	−0.351605
$191$	−15.9385	−1.15327	−0.576635	−	0.817002i	$-0.695634\pi$
$191$	−15.9385	−1.15327	−0.576635	+	0.817002i	$0.695634\pi$
$192$	0	0
$193$	8.45386	0.608522	0.304261	−	0.952589i	$-0.401590\pi$
$193$	8.45386	0.608522	0.304261	+	0.952589i	$0.401590\pi$
$194$	13.3245	0.956643
$195$	0	0
$196$	2.44952	0.174966
$197$	14.3384	1.02157	0.510785	−	0.859708i	$-0.329355\pi$
$197$	14.3384	1.02157	0.510785	+	0.859708i	$0.329355\pi$
$198$	0	0
$199$	−22.1343	−1.56906	−0.784528	−	0.620094i	$-0.787095\pi$
$199$	−22.1343	−1.56906	−0.784528	+	0.620094i	$0.787095\pi$
$200$	4.51117	0.318988
$201$	0	0
$202$	24.8874	1.75107
$203$	−6.92295	−0.485896
$204$	0	0
$205$	−0.710628	−0.0496324
$206$	−14.7776	−1.02961
$207$	0	0
$208$	−15.3650	−1.06537
$209$	0	0
$210$	0	0
$211$	2.18302	0.150286	0.0751428	−	0.997173i	$-0.476059\pi$
$211$	2.18302	0.150286	0.0751428	+	0.997173i	$0.476059\pi$
$212$	−16.2608	−1.11679
$213$	0	0
$214$	23.9515	1.63729
$215$	−1.41919	−0.0967878
$216$	0	0
$217$	−1.26565	−0.0859178
$218$	−39.9548	−2.70608
$219$	0	0
$220$	0	0
$221$	−16.0981	−1.08287
$222$	0	0
$223$	27.2603	1.82549	0.912744	−	0.408533i	$-0.133960\pi$
$223$	27.2603	1.82549	0.912744	+	0.408533i	$0.133960\pi$
$224$	8.01131	0.535278
$225$	0	0
$226$	−28.4685	−1.89370
$227$	13.5892	0.901945	0.450973	−	0.892538i	$-0.351077\pi$
$227$	13.5892	0.901945	0.450973	+	0.892538i	$0.351077\pi$
$228$	0	0
$229$	−8.30141	−0.548573	−0.274286	−	0.961648i	$-0.588442\pi$
$229$	−8.30141	−0.548573	−0.274286	+	0.961648i	$0.588442\pi$
$230$	5.85624	0.386149
$231$	0	0
$232$	6.56443	0.430976
$233$	−12.9476	−0.848229	−0.424114	−	0.905609i	$-0.639415\pi$
$233$	−12.9476	−0.848229	−0.424114	+	0.905609i	$0.639415\pi$
$234$	0	0
$235$	−4.31099	−0.281218
$236$	20.4715	1.33258
$237$	0	0
$238$	6.40665	0.415281
$239$	−1.89342	−0.122475	−0.0612375	−	0.998123i	$-0.519505\pi$
$239$	−1.89342	−0.122475	−0.0612375	+	0.998123i	$0.519505\pi$
$240$	0	0
$241$	−11.6983	−0.753557	−0.376778	−	0.926303i	$-0.622968\pi$
$241$	−11.6983	−0.753557	−0.376778	+	0.926303i	$0.622968\pi$
$242$	0	0
$243$	0	0
$244$	−34.0368	−2.17898
$245$	−0.492391	−0.0314577
$246$	0	0
$247$	−24.7323	−1.57368
$248$	1.20010	0.0762066
$249$	0	0
$250$	−10.1346	−0.640970
$251$	13.1860	0.832291	0.416146	−	0.909298i	$-0.363381\pi$
$251$	13.1860	0.832291	0.416146	+	0.909298i	$0.363381\pi$
$252$	0	0
$253$	0	0
$254$	9.83110	0.616858
$255$	0	0
$256$	6.60537	0.412836
$257$	15.3445	0.957165	0.478582	−	0.878043i	$-0.341151\pi$
$257$	15.3445	0.957165	0.478582	+	0.878043i	$0.341151\pi$
$258$	0	0
$259$	10.8759	0.675793
$260$	−6.39280	−0.396465
$261$	0	0
$262$	19.0620	1.17766
$263$	−10.4197	−0.642507	−0.321254	−	0.946993i	$-0.604104\pi$
$263$	−10.4197	−0.642507	−0.321254	+	0.946993i	$0.604104\pi$
$264$	0	0
$265$	3.26867	0.200792
$266$	9.84288	0.603506
$267$	0	0
$268$	−23.7709	−1.45204
$269$	−16.2712	−0.992074	−0.496037	−	0.868301i	$-0.665212\pi$
$269$	−16.2712	−0.992074	−0.496037	+	0.868301i	$0.665212\pi$
$270$	0	0
$271$	−5.83922	−0.354707	−0.177354	−	0.984147i	$-0.556754\pi$
$271$	−5.83922	−0.354707	−0.177354	+	0.984147i	$0.556754\pi$
$272$	8.80454	0.533853
$273$	0	0
$274$	−3.45459	−0.208700
$275$	0	0
$276$	0	0
$277$	15.9255	0.956868	0.478434	−	0.878123i	$-0.341205\pi$
$277$	15.9255	0.956868	0.478434	+	0.878123i	$0.341205\pi$
$278$	3.23732	0.194162
$279$	0	0
$280$	0.466891	0.0279021
$281$	−10.2004	−0.608504	−0.304252	−	0.952592i	$-0.598406\pi$
$281$	−10.2004	−0.608504	−0.304252	+	0.952592i	$0.598406\pi$
$282$	0	0
$283$	−16.1634	−0.960812	−0.480406	−	0.877046i	$-0.659511\pi$
$283$	−16.1634	−0.960812	−0.480406	+	0.877046i	$0.659511\pi$
$284$	−14.5685	−0.864483
$285$	0	0
$286$	0	0
$287$	1.44322	0.0851905
$288$	0	0
$289$	−7.77538	−0.457375
$290$	−7.19049	−0.422240
$291$	0	0
$292$	−9.24499	−0.541022
$293$	−27.0517	−1.58038	−0.790188	−	0.612864i	$-0.790017\pi$
$293$	−27.0517	−1.58038	−0.790188	+	0.612864i	$0.790017\pi$
$294$	0	0
$295$	−4.11508	−0.239589
$296$	−10.3126	−0.599408
$297$	0	0
$298$	28.4576	1.64851
$299$	29.8849	1.72829
$300$	0	0
$301$	2.88224	0.166129
$302$	25.8126	1.48535
$303$	0	0
$304$	13.5269	0.775820
$305$	6.84192	0.391767
$306$	0	0
$307$	29.7251	1.69650	0.848250	−	0.529596i	$-0.177657\pi$
$307$	29.7251	1.69650	0.848250	+	0.529596i	$0.177657\pi$
$308$	0	0
$309$	0	0
$310$	−1.31456	−0.0746619
$311$	−22.4029	−1.27035	−0.635176	−	0.772367i	$-0.719072\pi$
$311$	−22.4029	−1.27035	−0.635176	+	0.772367i	$0.719072\pi$
$312$	0	0
$313$	9.94833	0.562313	0.281157	−	0.959662i	$-0.409282\pi$
$313$	9.94833	0.562313	0.281157	+	0.959662i	$0.409282\pi$
$314$	−5.31655	−0.300030
$315$	0	0
$316$	−21.5652	−1.21313
$317$	11.1420	0.625796	0.312898	−	0.949787i	$-0.398700\pi$
$317$	11.1420	0.625796	0.312898	+	0.949787i	$0.398700\pi$
$318$	0	0
$319$	0	0
$320$	5.46613	0.305566
$321$	0	0
$322$	−11.8935	−0.662797
$323$	14.1723	0.788567
$324$	0	0
$325$	−25.2164	−1.39875
$326$	−16.6200	−0.920495
$327$	0	0
$328$	−1.36848	−0.0755615
$329$	8.75522	0.482691
$330$	0	0
$331$	14.5950	0.802214	0.401107	−	0.916031i	$-0.368626\pi$
$331$	14.5950	0.802214	0.401107	+	0.916031i	$0.368626\pi$
$332$	−27.1647	−1.49086
$333$	0	0
$334$	−4.33449	−0.237172
$335$	4.77832	0.261067
$336$	0	0
$337$	12.6059	0.686688	0.343344	−	0.939210i	$-0.388440\pi$
$337$	12.6059	0.686688	0.343344	+	0.939210i	$0.388440\pi$
$338$	−31.8372	−1.73172
$339$	0	0
$340$	3.66324	0.198667
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−2.73297	−0.147352
$345$	0	0
$346$	−49.0868	−2.63893
$347$	−0.410734	−0.0220494	−0.0110247	−	0.999939i	$-0.503509\pi$
$347$	−0.410734	−0.0220494	−0.0110247	+	0.999939i	$0.503509\pi$
$348$	0	0
$349$	13.4025	0.717422	0.358711	−	0.933449i	$-0.383216\pi$
$349$	13.4025	0.717422	0.358711	+	0.933449i	$0.383216\pi$
$350$	10.0355	0.536421
$351$	0	0
$352$	0	0
$353$	−12.3419	−0.656892	−0.328446	−	0.944523i	$-0.606525\pi$
$353$	−12.3419	−0.656892	−0.328446	+	0.944523i	$0.606525\pi$
$354$	0	0
$355$	2.92850	0.155429
$356$	−7.60146	−0.402877
$357$	0	0
$358$	37.2509	1.96877
$359$	25.0097	1.31996	0.659981	−	0.751283i	$-0.270565\pi$
$359$	25.0097	1.31996	0.659981	+	0.751283i	$0.270565\pi$
$360$	0	0
$361$	2.77364	0.145981
$362$	−32.6324	−1.71512
$363$	0	0
$364$	12.9832	0.680503
$365$	1.85839	0.0972724
$366$	0	0
$367$	−2.06915	−0.108009	−0.0540043	−	0.998541i	$-0.517198\pi$
$367$	−2.06915	−0.108009	−0.0540043	+	0.998541i	$0.517198\pi$
$368$	−16.3450	−0.852041
$369$	0	0
$370$	11.2961	0.587259
$371$	−6.63835	−0.344646
$372$	0	0
$373$	−14.7623	−0.764365	−0.382183	−	0.924087i	$-0.624828\pi$
$373$	−14.7623	−0.764365	−0.382183	+	0.924087i	$0.624828\pi$
$374$	0	0
$375$	0	0
$376$	−8.30180	−0.428133
$377$	−36.6937	−1.88982
$378$	0	0
$379$	27.7508	1.42546	0.712730	−	0.701438i	$-0.247458\pi$
$379$	27.7508	1.42546	0.712730	+	0.701438i	$0.247458\pi$
$380$	5.62803	0.288712
$381$	0	0
$382$	33.6205	1.72017
$383$	18.0334	0.921464	0.460732	−	0.887539i	$-0.347587\pi$
$383$	18.0334	0.921464	0.460732	+	0.887539i	$0.347587\pi$
$384$	0	0
$385$	0	0
$386$	−17.8325	−0.907649
$387$	0	0
$388$	−15.4730	−0.785524
$389$	−13.5412	−0.686564	−0.343282	−	0.939232i	$-0.611539\pi$
$389$	−13.5412	−0.686564	−0.343282	+	0.939232i	$0.611539\pi$
$390$	0	0
$391$	−17.1248	−0.866040
$392$	−0.948212	−0.0478919
$393$	0	0
$394$	−30.2453	−1.52374
$395$	4.33493	0.218114
$396$	0	0
$397$	−24.6525	−1.23727	−0.618637	−	0.785677i	$-0.712315\pi$
$397$	−24.6525	−1.23727	−0.618637	+	0.785677i	$0.712315\pi$
$398$	46.6897	2.34035
$399$	0	0
$400$	13.7916	0.689581
$401$	−18.7389	−0.935778	−0.467889	−	0.883787i	$-0.654985\pi$
$401$	−18.7389	−0.935778	−0.467889	+	0.883787i	$0.654985\pi$
$402$	0	0
$403$	−6.70831	−0.334165
$404$	−28.9004	−1.43785
$405$	0	0
$406$	14.6032	0.724745
$407$	0	0
$408$	0	0
$409$	15.7098	0.776801	0.388401	−	0.921491i	$-0.373028\pi$
$409$	15.7098	0.776801	0.388401	+	0.921491i	$0.373028\pi$
$410$	1.49899	0.0740299
$411$	0	0
$412$	17.1605	0.845436
$413$	8.35733	0.411238
$414$	0	0
$415$	5.46052	0.268046
$416$	42.4623	2.08189
$417$	0	0
$418$	0	0
$419$	−22.6034	−1.10425	−0.552125	−	0.833761i	$-0.686183\pi$
$419$	−22.6034	−1.10425	−0.552125	+	0.833761i	$0.686183\pi$
$420$	0	0
$421$	−23.3311	−1.13709	−0.568544	−	0.822653i	$-0.692493\pi$
$421$	−23.3311	−1.13709	−0.568544	+	0.822653i	$0.692493\pi$
$422$	−4.60485	−0.224160
$423$	0	0
$424$	6.29456	0.305691
$425$	14.4497	0.700912
$426$	0	0
$427$	−13.8953	−0.672440
$428$	−27.8136	−1.34442
$429$	0	0
$430$	2.99362	0.144365
$431$	9.53898	0.459476	0.229738	−	0.973252i	$-0.426213\pi$
$431$	9.53898	0.459476	0.229738	+	0.973252i	$0.426213\pi$
$432$	0	0
$433$	−23.0105	−1.10581	−0.552907	−	0.833243i	$-0.686482\pi$
$433$	−23.0105	−1.10581	−0.552907	+	0.833243i	$0.686482\pi$
$434$	2.66975	0.128152
$435$	0	0
$436$	46.3973	2.22203
$437$	−26.3098	−1.25857
$438$	0	0
$439$	−27.6434	−1.31935	−0.659673	−	0.751553i	$-0.729305\pi$
$439$	−27.6434	−1.31935	−0.659673	+	0.751553i	$0.729305\pi$
$440$	0	0
$441$	0	0
$442$	33.9571	1.61518
$443$	−14.7713	−0.701807	−0.350904	−	0.936412i	$-0.614125\pi$
$443$	−14.7713	−0.701807	−0.350904	+	0.936412i	$0.614125\pi$
$444$	0	0
$445$	1.52801	0.0724346
$446$	−57.5026	−2.72283
$447$	0	0
$448$	−11.1012	−0.524482
$449$	−30.2669	−1.42838	−0.714192	−	0.699950i	$-0.753205\pi$
$449$	−30.2669	−1.42838	−0.714192	+	0.699950i	$0.753205\pi$
$450$	0	0
$451$	0	0
$452$	33.0590	1.55496
$453$	0	0
$454$	−28.6648	−1.34531
$455$	−2.60982	−0.122350
$456$	0	0
$457$	−20.2565	−0.947558	−0.473779	−	0.880644i	$-0.657111\pi$
$457$	−20.2565	−0.947558	−0.473779	+	0.880644i	$0.657111\pi$
$458$	17.5109	0.818231
$459$	0	0
$460$	−6.80054	−0.317077
$461$	−8.51184	−0.396436	−0.198218	−	0.980158i	$-0.563515\pi$
$461$	−8.51184	−0.396436	−0.198218	+	0.980158i	$0.563515\pi$
$462$	0	0
$463$	−0.591469	−0.0274879	−0.0137440	−	0.999906i	$-0.504375\pi$
$463$	−0.591469	−0.0274879	−0.0137440	+	0.999906i	$0.504375\pi$
$464$	20.0689	0.931675
$465$	0	0
$466$	27.3116	1.26519
$467$	41.0347	1.89886	0.949430	−	0.313979i	$-0.101662\pi$
$467$	41.0347	1.89886	0.949430	+	0.313979i	$0.101662\pi$
$468$	0	0
$469$	−9.70431	−0.448103
$470$	9.09356	0.419454
$471$	0	0
$472$	−7.92452	−0.364756
$473$	0	0
$474$	0	0
$475$	22.1998	1.01860
$476$	−7.43970	−0.340998
$477$	0	0
$478$	3.99395	0.182679
$479$	−20.3437	−0.929527	−0.464763	−	0.885435i	$-0.653861\pi$
$479$	−20.3437	−0.929527	−0.464763	+	0.885435i	$0.653861\pi$
$480$	0	0
$481$	57.6452	2.62840
$482$	24.6764	1.12398
$483$	0	0
$484$	0	0
$485$	3.11032	0.141232
$486$	0	0
$487$	−28.8165	−1.30580	−0.652900	−	0.757444i	$-0.726448\pi$
$487$	−28.8165	−1.30580	−0.652900	+	0.757444i	$0.726448\pi$
$488$	13.1757	0.596435
$489$	0	0
$490$	1.03864	0.0469212
$491$	−2.68045	−0.120967	−0.0604834	−	0.998169i	$-0.519264\pi$
$491$	−2.68045	−0.120967	−0.0604834	+	0.998169i	$0.519264\pi$
$492$	0	0
$493$	21.0264	0.946983
$494$	52.1701	2.34725
$495$	0	0
$496$	3.66898	0.164742
$497$	−5.94751	−0.266782
$498$	0	0
$499$	22.3425	1.00019	0.500095	−	0.865971i	$-0.333298\pi$
$499$	22.3425	1.00019	0.500095	+	0.865971i	$0.333298\pi$
$500$	11.7688	0.526317
$501$	0	0
$502$	−27.8143	−1.24141
$503$	4.47599	0.199575	0.0997873	−	0.995009i	$-0.468184\pi$
$503$	4.47599	0.199575	0.0997873	+	0.995009i	$0.468184\pi$
$504$	0	0
$505$	5.80943	0.258516
$506$	0	0
$507$	0	0
$508$	−11.4163	−0.506518
$509$	17.1547	0.760368	0.380184	−	0.924911i	$-0.375861\pi$
$509$	17.1547	0.760368	0.380184	+	0.924911i	$0.375861\pi$
$510$	0	0
$511$	−3.77421	−0.166961
$512$	−28.7215	−1.26932
$513$	0	0
$514$	−32.3676	−1.42767
$515$	−3.44952	−0.152004
$516$	0	0
$517$	0	0
$518$	−22.9414	−1.00799
$519$	0	0
$520$	2.47466	0.108521
$521$	−1.00957	−0.0442300	−0.0221150	−	0.999755i	$-0.507040\pi$
$521$	−1.00957	−0.0442300	−0.0221150	+	0.999755i	$0.507040\pi$
$522$	0	0
$523$	−13.6433	−0.596578	−0.298289	−	0.954476i	$-0.596416\pi$
$523$	−13.6433	−0.596578	−0.298289	+	0.954476i	$0.596416\pi$
$524$	−22.1357	−0.967004
$525$	0	0
$526$	21.9792	0.958340
$527$	3.84404	0.167449
$528$	0	0
$529$	8.79099	0.382217
$530$	−6.89488	−0.299495
$531$	0	0
$532$	−11.4300	−0.495554
$533$	7.64948	0.331336
$534$	0	0
$535$	5.59095	0.241718
$536$	9.20174	0.397455
$537$	0	0
$538$	34.3223	1.47974
$539$	0	0
$540$	0	0
$541$	−2.76335	−0.118806	−0.0594028	−	0.998234i	$-0.518920\pi$
$541$	−2.76335	−0.118806	−0.0594028	+	0.998234i	$0.518920\pi$
$542$	12.3172	0.529068
$543$	0	0
$544$	−24.3320	−1.04323
$545$	−9.32658	−0.399507
$546$	0	0
$547$	15.2417	0.651687	0.325843	−	0.945424i	$-0.394352\pi$
$547$	15.2417	0.651687	0.325843	+	0.945424i	$0.394352\pi$
$548$	4.01163	0.171368
$549$	0	0
$550$	0	0
$551$	32.3041	1.37620
$552$	0	0
$553$	−8.80383	−0.374377
$554$	−33.5930	−1.42723
$555$	0	0
$556$	−3.75933	−0.159431
$557$	−3.56730	−0.151151	−0.0755757	−	0.997140i	$-0.524079\pi$
$557$	−3.56730	−0.151151	−0.0755757	+	0.997140i	$0.524079\pi$
$558$	0	0
$559$	15.2767	0.646136
$560$	1.42739	0.0603182
$561$	0	0
$562$	21.5166	0.907621
$563$	−36.7500	−1.54883	−0.774414	−	0.632679i	$-0.781955\pi$
$563$	−36.7500	−1.54883	−0.774414	+	0.632679i	$0.781955\pi$
$564$	0	0
$565$	−6.64537	−0.279573
$566$	34.0948	1.43311
$567$	0	0
$568$	5.63949	0.236628
$569$	34.0802	1.42872	0.714359	−	0.699779i	$-0.246718\pi$
$569$	34.0802	1.42872	0.714359	+	0.699779i	$0.246718\pi$
$570$	0	0
$571$	−5.79312	−0.242434	−0.121217	−	0.992626i	$-0.538680\pi$
$571$	−5.79312	−0.242434	−0.121217	+	0.992626i	$0.538680\pi$
$572$	0	0
$573$	0	0
$574$	−3.04431	−0.127067
$575$	−26.8247	−1.11867
$576$	0	0
$577$	−31.1231	−1.29567	−0.647837	−	0.761779i	$-0.724326\pi$
$577$	−31.1231	−1.29567	−0.647837	+	0.761779i	$0.724326\pi$
$578$	16.4013	0.682204
$579$	0	0
$580$	8.34993	0.346712
$581$	−11.0898	−0.460082
$582$	0	0
$583$	0	0
$584$	3.57875	0.148090
$585$	0	0
$586$	57.0625	2.35723
$587$	47.6946	1.96857	0.984283	−	0.176601i	$-0.0565102\pi$
$587$	47.6946	1.96857	0.984283	+	0.176601i	$0.0565102\pi$
$588$	0	0
$589$	5.90580	0.243344
$590$	8.68030	0.357362
$591$	0	0
$592$	−31.5279	−1.29579
$593$	−44.5859	−1.83092	−0.915462	−	0.402404i	$-0.868175\pi$
$593$	−44.5859	−1.83092	−0.915462	+	0.402404i	$0.868175\pi$
$594$	0	0
$595$	1.49549	0.0613093
$596$	−33.0463	−1.35363
$597$	0	0
$598$	−63.0389	−2.57785
$599$	8.06937	0.329706	0.164853	−	0.986318i	$-0.447285\pi$
$599$	8.06937	0.329706	0.164853	+	0.986318i	$0.447285\pi$
$600$	0	0
$601$	−28.9086	−1.17921	−0.589603	−	0.807693i	$-0.700716\pi$
$601$	−28.9086	−1.17921	−0.589603	+	0.807693i	$0.700716\pi$
$602$	−6.07976	−0.247792
$603$	0	0
$604$	−29.9748	−1.21966
$605$	0	0
$606$	0	0
$607$	−9.85310	−0.399925	−0.199962	−	0.979804i	$-0.564082\pi$
$607$	−9.85310	−0.399925	−0.199962	+	0.979804i	$0.564082\pi$
$608$	−37.3826	−1.51606
$609$	0	0
$610$	−14.4323	−0.584345
$611$	46.4052	1.87735
$612$	0	0
$613$	−35.8329	−1.44728	−0.723638	−	0.690180i	$-0.757531\pi$
$613$	−35.8329	−1.44728	−0.723638	+	0.690180i	$0.757531\pi$
$614$	−62.7017	−2.53044
$615$	0	0
$616$	0	0
$617$	38.4398	1.54753	0.773764	−	0.633474i	$-0.218372\pi$
$617$	38.4398	1.54753	0.773764	+	0.633474i	$0.218372\pi$
$618$	0	0
$619$	4.19552	0.168632	0.0843162	−	0.996439i	$-0.473129\pi$
$619$	4.19552	0.168632	0.0843162	+	0.996439i	$0.473129\pi$
$620$	1.52653	0.0613068
$621$	0	0
$622$	47.2564	1.89481
$623$	−3.10324	−0.124329
$624$	0	0
$625$	21.4220	0.856882
$626$	−20.9849	−0.838725
$627$	0	0
$628$	6.17382	0.246362
$629$	−33.0322	−1.31708
$630$	0	0
$631$	−3.51798	−0.140049	−0.0700243	−	0.997545i	$-0.522308\pi$
$631$	−3.51798	−0.140049	−0.0700243	+	0.997545i	$0.522308\pi$
$632$	8.34790	0.332061
$633$	0	0
$634$	−23.5028	−0.933414
$635$	2.29486	0.0910686
$636$	0	0
$637$	5.30029	0.210005
$638$	0	0
$639$	0	0
$640$	−3.64078	−0.143915
$641$	−16.0952	−0.635723	−0.317862	−	0.948137i	$-0.602965\pi$
$641$	−16.0952	−0.635723	−0.317862	+	0.948137i	$0.602965\pi$
$642$	0	0
$643$	−34.8261	−1.37341	−0.686704	−	0.726938i	$-0.740943\pi$
$643$	−34.8261	−1.37341	−0.686704	+	0.726938i	$0.740943\pi$
$644$	13.8113	0.544239
$645$	0	0
$646$	−29.8948	−1.17620
$647$	−16.9051	−0.664607	−0.332304	−	0.943172i	$-0.607826\pi$
$647$	−16.9051	−0.664607	−0.332304	+	0.943172i	$0.607826\pi$
$648$	0	0
$649$	0	0
$650$	53.1912	2.08633
$651$	0	0
$652$	19.2999	0.755842
$653$	−3.52799	−0.138061	−0.0690304	−	0.997615i	$-0.521991\pi$
$653$	−3.52799	−0.138061	−0.0690304	+	0.997615i	$0.521991\pi$
$654$	0	0
$655$	4.44962	0.173861
$656$	−4.18374	−0.163347
$657$	0	0
$658$	−18.4682	−0.719964
$659$	−29.4409	−1.14686	−0.573428	−	0.819256i	$-0.694387\pi$
$659$	−29.4409	−1.14686	−0.573428	+	0.819256i	$0.694387\pi$
$660$	0	0
$661$	15.2989	0.595059	0.297529	−	0.954713i	$-0.403837\pi$
$661$	15.2989	0.595059	0.297529	+	0.954713i	$0.403837\pi$
$662$	−30.7865	−1.19655
$663$	0	0
$664$	10.5155	0.408080
$665$	2.29761	0.0890974
$666$	0	0
$667$	−39.0340	−1.51140
$668$	5.03341	0.194748
$669$	0	0
$670$	−10.0793	−0.389398
$671$	0	0
$672$	0	0
$673$	12.1652	0.468936	0.234468	−	0.972124i	$-0.424665\pi$
$673$	12.1652	0.468936	0.234468	+	0.972124i	$0.424665\pi$
$674$	−26.5908	−1.02424
$675$	0	0
$676$	36.9709	1.42196
$677$	0.313619	0.0120533	0.00602667	−	0.999982i	$-0.498082\pi$
$677$	0.313619	0.0120533	0.00602667	+	0.999982i	$0.498082\pi$
$678$	0	0
$679$	−6.31676	−0.242415
$680$	−1.41804	−0.0543795
$681$	0	0
$682$	0	0
$683$	−38.2419	−1.46328	−0.731642	−	0.681689i	$-0.761246\pi$
$683$	−38.2419	−1.46328	−0.731642	+	0.681689i	$0.761246\pi$
$684$	0	0
$685$	−0.806400	−0.0308110
$686$	−2.10939	−0.0805368
$687$	0	0
$688$	−8.35530	−0.318543
$689$	−35.1852	−1.34045
$690$	0	0
$691$	−10.2754	−0.390895	−0.195448	−	0.980714i	$-0.562616\pi$
$691$	−10.2754	−0.390895	−0.195448	+	0.980714i	$0.562616\pi$
$692$	57.0019	2.16689
$693$	0	0
$694$	0.866398	0.0328880
$695$	0.755683	0.0286647
$696$	0	0
$697$	−4.38335	−0.166031
$698$	−28.2712	−1.07008
$699$	0	0
$700$	−11.6537	−0.440469
$701$	−18.9188	−0.714552	−0.357276	−	0.933999i	$-0.616294\pi$
$701$	−18.9188	−0.714552	−0.357276	+	0.933999i	$0.616294\pi$
$702$	0	0
$703$	−50.7492	−1.91404
$704$	0	0
$705$	0	0
$706$	26.0338	0.979795
$707$	−11.7984	−0.443725
$708$	0	0
$709$	−14.1498	−0.531409	−0.265704	−	0.964055i	$-0.585605\pi$
$709$	−14.1498	−0.531409	−0.265704	+	0.964055i	$0.585605\pi$
$710$	−6.17734	−0.231832
$711$	0	0
$712$	2.94253	0.110276
$713$	−7.13617	−0.267252
$714$	0	0
$715$	0	0
$716$	−43.2575	−1.61661
$717$	0	0
$718$	−52.7552	−1.96880
$719$	15.9330	0.594201	0.297101	−	0.954846i	$-0.403980\pi$
$719$	15.9330	0.594201	0.297101	+	0.954846i	$0.403980\pi$
$720$	0	0
$721$	7.00565	0.260904
$722$	−5.85068	−0.217740
$723$	0	0
$724$	37.8943	1.40833
$725$	32.9363	1.22322
$726$	0	0
$727$	−4.20455	−0.155938	−0.0779691	−	0.996956i	$-0.524844\pi$
$727$	−4.20455	−0.155938	−0.0779691	+	0.996956i	$0.524844\pi$
$728$	−5.02580	−0.186269
$729$	0	0
$730$	−3.92006	−0.145088
$731$	−8.75395	−0.323777
$732$	0	0
$733$	34.0777	1.25869	0.629345	−	0.777126i	$-0.283323\pi$
$733$	34.0777	1.25869	0.629345	+	0.777126i	$0.283323\pi$
$734$	4.36464	0.161102
$735$	0	0
$736$	45.1706	1.66501
$737$	0	0
$738$	0	0
$739$	26.1306	0.961230	0.480615	−	0.876932i	$-0.340413\pi$
$739$	26.1306	0.961230	0.480615	+	0.876932i	$0.340413\pi$
$740$	−13.1176	−0.482213
$741$	0	0
$742$	14.0029	0.514061
$743$	23.6263	0.866765	0.433383	−	0.901210i	$-0.357320\pi$
$743$	23.6263	0.866765	0.433383	+	0.901210i	$0.357320\pi$
$744$	0	0
$745$	6.64282	0.243374
$746$	31.1395	1.14010
$747$	0	0
$748$	0	0
$749$	−11.3547	−0.414892
$750$	0	0
$751$	−2.32359	−0.0847889	−0.0423945	−	0.999101i	$-0.513499\pi$
$751$	−2.32359	−0.0847889	−0.0423945	+	0.999101i	$0.513499\pi$
$752$	−25.3804	−0.925529
$753$	0	0
$754$	77.4012	2.81879
$755$	6.02540	0.219287
$756$	0	0
$757$	−14.7651	−0.536647	−0.268324	−	0.963329i	$-0.586470\pi$
$757$	−14.7651	−0.536647	−0.268324	+	0.963329i	$0.586470\pi$
$758$	−58.5371	−2.12616
$759$	0	0
$760$	−2.17862	−0.0790268
$761$	47.1876	1.71055	0.855274	−	0.518175i	$-0.173389\pi$
$761$	47.1876	1.71055	0.855274	+	0.518175i	$0.173389\pi$
$762$	0	0
$763$	18.9414	0.685725
$764$	−39.0417	−1.41248
$765$	0	0
$766$	−38.0395	−1.37442
$767$	44.2963	1.59945
$768$	0	0
$769$	−12.4418	−0.448662	−0.224331	−	0.974513i	$-0.572020\pi$
$769$	−12.4418	−0.448662	−0.224331	+	0.974513i	$0.572020\pi$
$770$	0	0
$771$	0	0
$772$	20.7079	0.745294
$773$	35.2698	1.26857	0.634284	−	0.773101i	$-0.281295\pi$
$773$	35.2698	1.26857	0.634284	+	0.773101i	$0.281295\pi$
$774$	0	0
$775$	6.02139	0.216295
$776$	5.98962	0.215015
$777$	0	0
$778$	28.5636	1.02405
$779$	−6.73438	−0.241284
$780$	0	0
$781$	0	0
$782$	36.1229	1.29175
$783$	0	0
$784$	−2.89889	−0.103532
$785$	−1.24103	−0.0442944
$786$	0	0
$787$	−16.3383	−0.582397	−0.291198	−	0.956663i	$-0.594054\pi$
$787$	−16.3383	−0.582397	−0.291198	+	0.956663i	$0.594054\pi$
$788$	35.1223	1.25118
$789$	0	0
$790$	−9.14405	−0.325331
$791$	13.4961	0.479867
$792$	0	0
$793$	−73.6491	−2.61536
$794$	52.0017	1.84547
$795$	0	0
$796$	−54.2183	−1.92172
$797$	0.292902	0.0103751	0.00518755	−	0.999987i	$-0.498349\pi$
$797$	0.292902	0.0103751	0.00518755	+	0.999987i	$0.498349\pi$
$798$	0	0
$799$	−26.5914	−0.940736
$800$	−38.1142	−1.34754
$801$	0	0
$802$	39.5277	1.39577
$803$	0	0
$804$	0	0
$805$	−2.77627	−0.0978508
$806$	14.1504	0.498428
$807$	0	0
$808$	11.1874	0.393571
$809$	11.0373	0.388050	0.194025	−	0.980997i	$-0.437846\pi$
$809$	11.0373	0.388050	0.194025	+	0.980997i	$0.437846\pi$
$810$	0	0
$811$	−5.21763	−0.183216	−0.0916078	−	0.995795i	$-0.529201\pi$
$811$	−5.21763	−0.183216	−0.0916078	+	0.995795i	$0.529201\pi$
$812$	−16.9579	−0.595106
$813$	0	0
$814$	0	0
$815$	−3.87957	−0.135896
$816$	0	0
$817$	−13.4492	−0.470527
$818$	−33.1381	−1.15865
$819$	0	0
$820$	−1.74070	−0.0607878
$821$	−26.4468	−0.923000	−0.461500	−	0.887140i	$-0.652689\pi$
$821$	−26.4468	−0.923000	−0.461500	+	0.887140i	$0.652689\pi$
$822$	0	0
$823$	−49.1895	−1.71464	−0.857319	−	0.514786i	$-0.827871\pi$
$823$	−49.1895	−1.71464	−0.857319	+	0.514786i	$0.827871\pi$
$824$	−6.64284	−0.231414
$825$	0	0
$826$	−17.6289	−0.613387
$827$	41.8006	1.45355	0.726774	−	0.686876i	$-0.241019\pi$
$827$	41.8006	1.45355	0.726774	+	0.686876i	$0.241019\pi$
$828$	0	0
$829$	−24.8988	−0.864771	−0.432385	−	0.901689i	$-0.642328\pi$
$829$	−24.8988	−0.864771	−0.432385	+	0.901689i	$0.642328\pi$
$830$	−11.5184	−0.399808
$831$	0	0
$832$	−58.8395	−2.03989
$833$	−3.03721	−0.105233
$834$	0	0
$835$	−1.01179	−0.0350145
$836$	0	0
$837$	0	0
$838$	47.6794	1.64706
$839$	53.2912	1.83982	0.919908	−	0.392135i	$-0.128263\pi$
$839$	53.2912	1.83982	0.919908	+	0.392135i	$0.128263\pi$
$840$	0	0
$841$	18.9273	0.652666
$842$	49.2143	1.69604
$843$	0	0
$844$	5.34736	0.184064
$845$	−7.43171	−0.255659
$846$	0	0
$847$	0	0
$848$	19.2439	0.660837
$849$	0	0
$850$	−30.4800	−1.04545
$851$	61.3219	2.10209
$852$	0	0
$853$	13.3706	0.457800	0.228900	−	0.973450i	$-0.426487\pi$
$853$	13.3706	0.457800	0.228900	+	0.973450i	$0.426487\pi$
$854$	29.3106	1.00299
$855$	0	0
$856$	10.7667	0.367997
$857$	−43.4419	−1.48395	−0.741974	−	0.670429i	$-0.766110\pi$
$857$	−43.4419	−1.48395	−0.741974	+	0.670429i	$0.766110\pi$
$858$	0	0
$859$	−7.96898	−0.271898	−0.135949	−	0.990716i	$-0.543408\pi$
$859$	−7.96898	−0.271898	−0.135949	+	0.990716i	$0.543408\pi$
$860$	−3.47633	−0.118542
$861$	0	0
$862$	−20.1214	−0.685338
$863$	−50.3471	−1.71384	−0.856918	−	0.515452i	$-0.827624\pi$
$863$	−50.3471	−1.71384	−0.856918	+	0.515452i	$0.827624\pi$
$864$	0	0
$865$	−11.4583	−0.389593
$866$	48.5381	1.64939
$867$	0	0
$868$	−3.10023	−0.105229
$869$	0	0
$870$	0	0
$871$	−51.4357	−1.74283
$872$	−17.9605	−0.608218
$873$	0	0
$874$	55.4976	1.87723
$875$	4.80453	0.162423
$876$	0	0
$877$	−55.1801	−1.86330	−0.931649	−	0.363359i	$-0.881630\pi$
$877$	−55.1801	−1.86330	−0.931649	+	0.363359i	$0.881630\pi$
$878$	58.3106	1.96789
$879$	0	0
$880$	0	0
$881$	22.6475	0.763014	0.381507	−	0.924366i	$-0.375405\pi$
$881$	22.6475	0.763014	0.381507	+	0.924366i	$0.375405\pi$
$882$	0	0
$883$	−12.7050	−0.427557	−0.213779	−	0.976882i	$-0.568577\pi$
$883$	−12.7050	−0.427557	−0.213779	+	0.976882i	$0.568577\pi$
$884$	−39.4326	−1.32626
$885$	0	0
$886$	31.1585	1.04679
$887$	−6.70004	−0.224965	−0.112483	−	0.993654i	$-0.535880\pi$
$887$	−6.70004	−0.224965	−0.112483	+	0.993654i	$0.535880\pi$
$888$	0	0
$889$	−4.66064	−0.156313
$890$	−3.22317	−0.108041
$891$	0	0
$892$	66.7747	2.23578
$893$	−40.8538	−1.36712
$894$	0	0
$895$	8.69542	0.290656
$896$	7.39409	0.247019
$897$	0	0
$898$	63.8446	2.13052
$899$	8.76203	0.292230
$900$	0	0
$901$	20.1620	0.671695
$902$	0	0
$903$	0	0
$904$	−12.7972	−0.425628
$905$	−7.61733	−0.253209
$906$	0	0
$907$	−57.7184	−1.91651	−0.958254	−	0.285918i	$-0.907701\pi$
$907$	−57.7184	−1.91651	−0.958254	+	0.285918i	$0.907701\pi$
$908$	33.2869	1.10467
$909$	0	0
$910$	5.50512	0.182493
$911$	2.84362	0.0942134	0.0471067	−	0.998890i	$-0.485000\pi$
$911$	2.84362	0.0942134	0.0471067	+	0.998890i	$0.485000\pi$
$912$	0	0
$913$	0	0
$914$	42.7288	1.41334
$915$	0	0
$916$	−20.3345	−0.671870
$917$	−9.03676	−0.298420
$918$	0	0
$919$	−9.36824	−0.309030	−0.154515	−	0.987990i	$-0.549381\pi$
$919$	−9.36824	−0.309030	−0.154515	+	0.987990i	$0.549381\pi$
$920$	2.63250	0.0867908
$921$	0	0
$922$	17.9548	0.591309
$923$	−31.5235	−1.03761
$924$	0	0
$925$	−51.7424	−1.70128
$926$	1.24764	0.0410000
$927$	0	0
$928$	−55.4620	−1.82063
$929$	43.8437	1.43847	0.719233	−	0.694769i	$-0.244494\pi$
$929$	43.8437	1.43847	0.719233	+	0.694769i	$0.244494\pi$
$930$	0	0
$931$	−4.66622	−0.152929
$932$	−31.7155	−1.03888
$933$	0	0
$934$	−86.5581	−2.83227
$935$	0	0
$936$	0	0
$937$	3.20925	0.104842	0.0524208	−	0.998625i	$-0.483306\pi$
$937$	3.20925	0.104842	0.0524208	+	0.998625i	$0.483306\pi$
$938$	20.4702	0.668374
$939$	0	0
$940$	−10.5599	−0.344425
$941$	−11.1320	−0.362893	−0.181446	−	0.983401i	$-0.558078\pi$
$941$	−11.1320	−0.362893	−0.181446	+	0.983401i	$0.558078\pi$
$942$	0	0
$943$	8.13737	0.264989
$944$	−24.2270	−0.788522
$945$	0	0
$946$	0	0
$947$	−51.6934	−1.67981	−0.839905	−	0.542733i	$-0.817390\pi$
$947$	−51.6934	−1.67981	−0.839905	+	0.542733i	$0.817390\pi$
$948$	0	0
$949$	−20.0044	−0.649370
$950$	−46.8280	−1.51930
$951$	0	0
$952$	2.87991	0.0933386
$953$	−28.0305	−0.907996	−0.453998	−	0.891003i	$-0.650003\pi$
$953$	−28.0305	−0.907996	−0.453998	+	0.891003i	$0.650003\pi$
$954$	0	0
$955$	7.84798	0.253955
$956$	−4.63796	−0.150002
$957$	0	0
$958$	42.9127	1.38645
$959$	1.63772	0.0528848
$960$	0	0
$961$	−29.3981	−0.948327
$962$	−121.596	−3.92042
$963$	0	0
$964$	−28.6553	−0.922926
$965$	−4.16261	−0.133999
$966$	0	0
$967$	−25.5912	−0.822957	−0.411479	−	0.911419i	$-0.634988\pi$
$967$	−25.5912	−0.822957	−0.411479	+	0.911419i	$0.634988\pi$
$968$	0	0
$969$	0	0
$970$	−6.56086	−0.210657
$971$	−12.5737	−0.403508	−0.201754	−	0.979436i	$-0.564664\pi$
$971$	−12.5737	−0.403508	−0.201754	+	0.979436i	$0.564664\pi$
$972$	0	0
$973$	−1.53472	−0.0492009
$974$	60.7852	1.94768
$975$	0	0
$976$	40.2809	1.28936
$977$	14.5164	0.464420	0.232210	−	0.972666i	$-0.425404\pi$
$977$	14.5164	0.464420	0.232210	+	0.972666i	$0.425404\pi$
$978$	0	0
$979$	0	0
$980$	−1.20612	−0.0385281
$981$	0	0
$982$	5.65410	0.180430
$983$	12.9883	0.414264	0.207132	−	0.978313i	$-0.433587\pi$
$983$	12.9883	0.414264	0.207132	+	0.978313i	$0.433587\pi$
$984$	0	0
$985$	−7.06012	−0.224954
$986$	−44.3529	−1.41249
$987$	0	0
$988$	−60.5824	−1.92738
$989$	16.2511	0.516754
$990$	0	0
$991$	7.01006	0.222682	0.111341	−	0.993782i	$-0.464485\pi$
$991$	7.01006	0.222682	0.111341	+	0.993782i	$0.464485\pi$
$992$	−10.1395	−0.321930
$993$	0	0
$994$	12.5456	0.397922
$995$	10.8987	0.345512
$996$	0	0
$997$	−31.8789	−1.00961	−0.504807	−	0.863232i	$-0.668436\pi$
$997$	−31.8789	−1.00961	−0.504807	+	0.863232i	$0.668436\pi$
$998$	−47.1291	−1.49185
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cp.1.2		6
3.2	odd	2		847.2.a.n.1.5	yes	6
11.10	odd	2		7623.2.a.cs.1.5		6
21.20	even	2		5929.2.a.bm.1.5		6
33.2	even	10		847.2.f.z.323.5		24
33.5	odd	10		847.2.f.y.729.2		24
33.8	even	10		847.2.f.z.372.2		24
33.14	odd	10		847.2.f.y.372.5		24
33.17	even	10		847.2.f.z.729.5		24
33.20	odd	10		847.2.f.y.323.2		24
33.26	odd	10		847.2.f.y.148.5		24
33.29	even	10		847.2.f.z.148.2		24
33.32	even	2		847.2.a.m.1.2	✓	6
231.230	odd	2		5929.2.a.bj.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.2	✓	6	33.32	even	2
847.2.a.n.1.5	yes	6	3.2	odd	2
847.2.f.y.148.5		24	33.26	odd	10
847.2.f.y.323.2		24	33.20	odd	10
847.2.f.y.372.5		24	33.14	odd	10
847.2.f.y.729.2		24	33.5	odd	10
847.2.f.z.148.2		24	33.29	even	10
847.2.f.z.323.5		24	33.2	even	10
847.2.f.z.372.2		24	33.8	even	10
847.2.f.z.729.5		24	33.17	even	10
5929.2.a.bj.1.2		6	231.230	odd	2
5929.2.a.bm.1.5		6	21.20	even	2
7623.2.a.cp.1.2		6	1.1	even	1	trivial
7623.2.a.cs.1.5		6	11.10	odd	2

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10939 q^{2} +2.44952 q^{4} -0.492391 q^{5} +1.00000 q^{7} -0.948212 q^{8} +O(q^{10})\)	\(q-2.10939 q^{2} +2.44952 q^{4} -0.492391 q^{5} +1.00000 q^{7} -0.948212 q^{8} +1.03864 q^{10} +5.30029 q^{13} -2.10939 q^{14} -2.89889 q^{16} -3.03721 q^{17} -4.66622 q^{19} -1.20612 q^{20} +5.63835 q^{23} -4.75755 q^{25} -11.1804 q^{26} +2.44952 q^{28} -6.92295 q^{29} -1.26565 q^{31} +8.01131 q^{32} +6.40665 q^{34} -0.492391 q^{35} +10.8759 q^{37} +9.84288 q^{38} +0.466891 q^{40} +1.44322 q^{41} +2.88224 q^{43} -11.8935 q^{46} +8.75522 q^{47} +1.00000 q^{49} +10.0355 q^{50} +12.9832 q^{52} -6.63835 q^{53} -0.948212 q^{56} +14.6032 q^{58} +8.35733 q^{59} -13.8953 q^{61} +2.66975 q^{62} -11.1012 q^{64} -2.60982 q^{65} -9.70431 q^{67} -7.43970 q^{68} +1.03864 q^{70} -5.94751 q^{71} -3.77421 q^{73} -22.9414 q^{74} -11.4300 q^{76} -8.80383 q^{79} +1.42739 q^{80} -3.04431 q^{82} -11.0898 q^{83} +1.49549 q^{85} -6.07976 q^{86} -3.10324 q^{89} +5.30029 q^{91} +13.8113 q^{92} -18.4682 q^{94} +2.29761 q^{95} -6.31676 q^{97} -2.10939 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 + 6 * q^7 - 12 * q^8	\( 6 q - 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8} - 8 q^{10} + 4 q^{13} - 4 q^{14} + 8 q^{16} - 22 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{23} + 4 q^{25} - 6 q^{26} + 4 q^{28} - 12 q^{29} - 2 q^{31} - 8 q^{32} + 24 q^{34} + 4 q^{35} + 14 q^{37} + 22 q^{38} + 18 q^{40} - 26 q^{41} - 4 q^{43} + 12 q^{46} + 16 q^{47} + 6 q^{49} + 4 q^{50} + 12 q^{52} - 4 q^{53} - 12 q^{56} - 2 q^{58} + 4 q^{59} - 8 q^{61} - 20 q^{62} + 26 q^{64} - 24 q^{65} + 6 q^{67} - 12 q^{68} - 8 q^{70} - 22 q^{71} + 14 q^{73} - 44 q^{74} - 30 q^{76} - 28 q^{79} + 4 q^{80} - 4 q^{82} - 22 q^{83} - 24 q^{85} + 30 q^{86} + 4 q^{91} - 10 q^{92} - 38 q^{94} + 24 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 + 6 * q^7 - 12 * q^8 - 8 * q^10 + 4 * q^13 - 4 * q^14 + 8 * q^16 - 22 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^23 + 4 * q^25 - 6 * q^26 + 4 * q^28 - 12 * q^29 - 2 * q^31 - 8 * q^32 + 24 * q^34 + 4 * q^35 + 14 * q^37 + 22 * q^38 + 18 * q^40 - 26 * q^41 - 4 * q^43 + 12 * q^46 + 16 * q^47 + 6 * q^49 + 4 * q^50 + 12 * q^52 - 4 * q^53 - 12 * q^56 - 2 * q^58 + 4 * q^59 - 8 * q^61 - 20 * q^62 + 26 * q^64 - 24 * q^65 + 6 * q^67 - 12 * q^68 - 8 * q^70 - 22 * q^71 + 14 * q^73 - 44 * q^74 - 30 * q^76 - 28 * q^79 + 4 * q^80 - 4 * q^82 - 22 * q^83 - 24 * q^85 + 30 * q^86 + 4 * q^91 - 10 * q^92 - 38 * q^94 + 24 * q^95 - 4 * q^97 - 4 * q^98

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