LMFDB - Embedded newform 7623.2.a.cw.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.55194$ 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)

$f(q)$	$=$	$q+2.55194 q^{2} +4.51241 q^{4} -3.42898 q^{5} -1.00000 q^{7} +6.41153 q^{8} +O(q^{10})$	$q+2.55194 q^{2} +4.51241 q^{4} -3.42898 q^{5} -1.00000 q^{7} +6.41153 q^{8} -8.75055 q^{10} +2.17503 q^{13} -2.55194 q^{14} +7.33702 q^{16} -4.51715 q^{17} +2.42381 q^{19} -15.4729 q^{20} -0.648403 q^{23} +6.75788 q^{25} +5.55056 q^{26} -4.51241 q^{28} -1.25295 q^{29} -8.03946 q^{31} +5.90061 q^{32} -11.5275 q^{34} +3.42898 q^{35} +5.01329 q^{37} +6.18543 q^{38} -21.9850 q^{40} -2.62170 q^{41} +1.46138 q^{43} -1.65469 q^{46} +5.04039 q^{47} +1.00000 q^{49} +17.2457 q^{50} +9.81464 q^{52} -13.3980 q^{53} -6.41153 q^{56} -3.19746 q^{58} -7.66978 q^{59} -14.3348 q^{61} -20.5163 q^{62} +0.383971 q^{64} -7.45814 q^{65} +6.22696 q^{67} -20.3832 q^{68} +8.75055 q^{70} -4.22997 q^{71} +5.92517 q^{73} +12.7936 q^{74} +10.9372 q^{76} -9.76318 q^{79} -25.1585 q^{80} -6.69044 q^{82} -8.37756 q^{83} +15.4892 q^{85} +3.72936 q^{86} -4.76249 q^{89} -2.17503 q^{91} -2.92586 q^{92} +12.8628 q^{94} -8.31119 q^{95} -8.70570 q^{97} +2.55194 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 7 q^{4} - 10 q^{5} - 8 q^{7}+O(q^{10}) $ 8 * q + q^2 + 7 * q^4 - 10 * q^5 - 8 * q^7	$ 8 q + q^{2} + 7 q^{4} - 10 q^{5} - 8 q^{7} - 6 q^{10} + 6 q^{13} - q^{14} + q^{16} - 5 q^{17} + 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} - 7 q^{28} + 9 q^{29} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} + 7 q^{37} + 10 q^{38} - 5 q^{40} - 10 q^{41} + 4 q^{43} - 4 q^{46} - 16 q^{47} + 8 q^{49} + 6 q^{50} + 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} - 19 q^{61} - 18 q^{62} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 6 q^{70} - 13 q^{71} + 25 q^{73} + 33 q^{74} - 26 q^{76} - 4 q^{80} - 13 q^{82} - 25 q^{83} - 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} + 42 q^{94} + 21 q^{95} + 15 q^{97} + q^{98}+O(q^{100}) $ 8 * q + q^2 + 7 * q^4 - 10 * q^5 - 8 * q^7 - 6 * q^10 + 6 * q^13 - q^14 + q^16 - 5 * q^17 + 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 - 7 * q^28 + 9 * q^29 + 9 * q^31 + 16 * q^32 - 12 * q^34 + 10 * q^35 + 7 * q^37 + 10 * q^38 - 5 * q^40 - 10 * q^41 + 4 * q^43 - 4 * q^46 - 16 * q^47 + 8 * q^49 + 6 * q^50 + 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 - 19 * q^61 - 18 * q^62 - 4 * q^64 - 4 * q^65 - 19 * q^67 + 9 * q^68 + 6 * q^70 - 13 * q^71 + 25 * q^73 + 33 * q^74 - 26 * q^76 - 4 * q^80 - 13 * q^82 - 25 * q^83 - 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 + 42 * q^94 + 21 * q^95 + 15 * q^97 + 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 $

$2$	2.55194	1.80450	0.902248	−	0.431218i	$-0.141916\pi$
$2$	2.55194	1.80450	0.902248	+	0.431218i	$0.141916\pi$
$3$	0	0
$3$	0	0
$4$	4.51241	2.25620
$5$	−3.42898	−1.53348	−0.766742	−	0.641955i	$-0.778124\pi$
$5$	−3.42898	−1.53348	−0.766742	+	0.641955i	$0.778124\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	6.41153	2.26682
$9$	0	0
$10$	−8.75055	−2.76717
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.17503	0.603245	0.301623	−	0.953427i	$-0.402472\pi$
$13$	2.17503	0.603245	0.301623	+	0.953427i	$0.402472\pi$
$14$	−2.55194	−0.682035
$15$	0	0
$16$	7.33702	1.83426
$17$	−4.51715	−1.09557	−0.547785	−	0.836619i	$-0.684529\pi$
$17$	−4.51715	−1.09557	−0.547785	+	0.836619i	$0.684529\pi$
$18$	0	0
$19$	2.42381	0.556060	0.278030	−	0.960572i	$-0.410318\pi$
$19$	2.42381	0.556060	0.278030	+	0.960572i	$0.410318\pi$
$20$	−15.4729	−3.45986
$21$	0	0
$22$	0	0
$23$	−0.648403	−0.135201	−0.0676006	−	0.997712i	$-0.521534\pi$
$23$	−0.648403	−0.135201	−0.0676006	+	0.997712i	$0.521534\pi$
$24$	0	0
$25$	6.75788	1.35158
$26$	5.55056	1.08855
$27$	0	0
$28$	−4.51241	−0.852765
$29$	−1.25295	−0.232667	−0.116334	−	0.993210i	$-0.537114\pi$
$29$	−1.25295	−0.232667	−0.116334	+	0.993210i	$0.537114\pi$
$30$	0	0
$31$	−8.03946	−1.44393	−0.721965	−	0.691929i	$-0.756761\pi$
$31$	−8.03946	−1.44393	−0.721965	+	0.691929i	$0.756761\pi$
$32$	5.90061	1.04309
$33$	0	0
$34$	−11.5275	−1.97695
$35$	3.42898	0.579603
$36$	0	0
$37$	5.01329	0.824180	0.412090	−	0.911143i	$-0.364799\pi$
$37$	5.01329	0.824180	0.412090	+	0.911143i	$0.364799\pi$
$38$	6.18543	1.00341
$39$	0	0
$40$	−21.9850	−3.47613
$41$	−2.62170	−0.409441	−0.204721	−	0.978820i	$-0.565629\pi$
$41$	−2.62170	−0.409441	−0.204721	+	0.978820i	$0.565629\pi$
$42$	0	0
$43$	1.46138	0.222858	0.111429	−	0.993772i	$-0.464457\pi$
$43$	1.46138	0.222858	0.111429	+	0.993772i	$0.464457\pi$
$44$	0	0
$45$	0	0
$46$	−1.65469	−0.243970
$47$	5.04039	0.735216	0.367608	−	0.929981i	$-0.380177\pi$
$47$	5.04039	0.735216	0.367608	+	0.929981i	$0.380177\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	17.2457	2.43891
$51$	0	0
$52$	9.81464	1.36105
$53$	−13.3980	−1.84036	−0.920180	−	0.391495i	$-0.871958\pi$
$53$	−13.3980	−1.84036	−0.920180	+	0.391495i	$0.871958\pi$
$54$	0	0
$55$	0	0
$56$	−6.41153	−0.856776
$57$	0	0
$58$	−3.19746	−0.419847
$59$	−7.66978	−0.998520	−0.499260	−	0.866452i	$-0.666395\pi$
$59$	−7.66978	−0.998520	−0.499260	+	0.866452i	$0.666395\pi$
$60$	0	0
$61$	−14.3348	−1.83539	−0.917693	−	0.397290i	$-0.869951\pi$
$61$	−14.3348	−1.83539	−0.917693	+	0.397290i	$0.869951\pi$
$62$	−20.5163	−2.60557
$63$	0	0
$64$	0.383971	0.0479964
$65$	−7.45814	−0.925068
$66$	0	0
$67$	6.22696	0.760744	0.380372	−	0.924834i	$-0.375796\pi$
$67$	6.22696	0.760744	0.380372	+	0.924834i	$0.375796\pi$
$68$	−20.3832	−2.47183
$69$	0	0
$70$	8.75055	1.04589
$71$	−4.22997	−0.502005	−0.251002	−	0.967986i	$-0.580760\pi$
$71$	−4.22997	−0.502005	−0.251002	+	0.967986i	$0.580760\pi$
$72$	0	0
$73$	5.92517	0.693489	0.346744	−	0.937960i	$-0.387287\pi$
$73$	5.92517	0.693489	0.346744	+	0.937960i	$0.387287\pi$
$74$	12.7936	1.48723
$75$	0	0
$76$	10.9372	1.25459
$77$	0	0
$78$	0	0
$79$	−9.76318	−1.09844	−0.549222	−	0.835677i	$-0.685076\pi$
$79$	−9.76318	−1.09844	−0.549222	+	0.835677i	$0.685076\pi$
$80$	−25.1585	−2.81280
$81$	0	0
$82$	−6.69044	−0.738835
$83$	−8.37756	−0.919557	−0.459779	−	0.888034i	$-0.652071\pi$
$83$	−8.37756	−0.919557	−0.459779	+	0.888034i	$0.652071\pi$
$84$	0	0
$85$	15.4892	1.68004
$86$	3.72936	0.402147
$87$	0	0
$88$	0	0
$89$	−4.76249	−0.504823	−0.252412	−	0.967620i	$-0.581224\pi$
$89$	−4.76249	−0.504823	−0.252412	+	0.967620i	$0.581224\pi$
$90$	0	0
$91$	−2.17503	−0.228005
$92$	−2.92586	−0.305042
$93$	0	0
$94$	12.8628	1.32669
$95$	−8.31119	−0.852710
$96$	0	0
$97$	−8.70570	−0.883930	−0.441965	−	0.897032i	$-0.645719\pi$
$97$	−8.70570	−0.883930	−0.441965	+	0.897032i	$0.645719\pi$
$98$	2.55194	0.257785
$99$	0	0
$100$	30.4943	3.04943
$101$	−5.77462	−0.574596	−0.287298	−	0.957841i	$-0.592757\pi$
$101$	−5.77462	−0.574596	−0.287298	+	0.957841i	$0.592757\pi$
$102$	0	0
$103$	−4.64098	−0.457289	−0.228645	−	0.973510i	$-0.573429\pi$
$103$	−4.64098	−0.457289	−0.228645	+	0.973510i	$0.573429\pi$
$104$	13.9453	1.36745
$105$	0	0
$106$	−34.1910	−3.32092
$107$	−4.02202	−0.388824	−0.194412	−	0.980920i	$-0.562280\pi$
$107$	−4.02202	−0.388824	−0.194412	+	0.980920i	$0.562280\pi$
$108$	0	0
$109$	6.14762	0.588835	0.294418	−	0.955677i	$-0.404874\pi$
$109$	6.14762	0.588835	0.294418	+	0.955677i	$0.404874\pi$
$110$	0	0
$111$	0	0
$112$	−7.33702	−0.693284
$113$	2.47837	0.233145	0.116573	−	0.993182i	$-0.462809\pi$
$113$	2.47837	0.233145	0.116573	+	0.993182i	$0.462809\pi$
$114$	0	0
$115$	2.22336	0.207329
$116$	−5.65383	−0.524945
$117$	0	0
$118$	−19.5728	−1.80183
$119$	4.51715	0.414086
$120$	0	0
$121$	0	0
$122$	−36.5817	−3.31195
$123$	0	0
$124$	−36.2774	−3.25780
$125$	−6.02773	−0.539137
$126$	0	0
$127$	6.95397	0.617065	0.308533	−	0.951214i	$-0.400162\pi$
$127$	6.95397	0.617065	0.308533	+	0.951214i	$0.400162\pi$
$128$	−10.8213	−0.956481
$129$	0	0
$130$	−19.0327	−1.66928
$131$	−20.9388	−1.82943	−0.914715	−	0.404101i	$-0.867585\pi$
$131$	−20.9388	−1.82943	−0.914715	+	0.404101i	$0.867585\pi$
$132$	0	0
$133$	−2.42381	−0.210171
$134$	15.8908	1.37276
$135$	0	0
$136$	−28.9618	−2.48346
$137$	−9.13644	−0.780579	−0.390289	−	0.920692i	$-0.627625\pi$
$137$	−9.13644	−0.780579	−0.390289	+	0.920692i	$0.627625\pi$
$138$	0	0
$139$	−10.7519	−0.911961	−0.455981	−	0.889990i	$-0.650711\pi$
$139$	−10.7519	−0.911961	−0.455981	+	0.889990i	$0.650711\pi$
$140$	15.4729	1.30770
$141$	0	0
$142$	−10.7946	−0.905865
$143$	0	0
$144$	0	0
$145$	4.29634	0.356792
$146$	15.1207	1.25140
$147$	0	0
$148$	22.6220	1.85952
$149$	10.7554	0.881118	0.440559	−	0.897724i	$-0.354780\pi$
$149$	10.7554	0.881118	0.440559	+	0.897724i	$0.354780\pi$
$150$	0	0
$151$	7.66422	0.623706	0.311853	−	0.950130i	$-0.399050\pi$
$151$	7.66422	0.623706	0.311853	+	0.950130i	$0.399050\pi$
$152$	15.5403	1.26049
$153$	0	0
$154$	0	0
$155$	27.5671	2.21425
$156$	0	0
$157$	18.3490	1.46441	0.732206	−	0.681083i	$-0.238491\pi$
$157$	18.3490	1.46441	0.732206	+	0.681083i	$0.238491\pi$
$158$	−24.9151	−1.98214
$159$	0	0
$160$	−20.2331	−1.59956
$161$	0.648403	0.0511013
$162$	0	0
$163$	0.377585	0.0295747	0.0147874	−	0.999891i	$-0.495293\pi$
$163$	0.377585	0.0295747	0.0147874	+	0.999891i	$0.495293\pi$
$164$	−11.8302	−0.923784
$165$	0	0
$166$	−21.3791	−1.65934
$167$	−0.114139	−0.00883232	−0.00441616	−	0.999990i	$-0.501406\pi$
$167$	−0.114139	−0.00883232	−0.00441616	+	0.999990i	$0.501406\pi$
$168$	0	0
$169$	−8.26923	−0.636095
$170$	39.5275	3.03162
$171$	0	0
$172$	6.59435	0.502814
$173$	−13.3010	−1.01126	−0.505629	−	0.862751i	$-0.668740\pi$
$173$	−13.3010	−1.01126	−0.505629	+	0.862751i	$0.668740\pi$
$174$	0	0
$175$	−6.75788	−0.510848
$176$	0	0
$177$	0	0
$178$	−12.1536	−0.910952
$179$	−11.3410	−0.847665	−0.423832	−	0.905741i	$-0.639315\pi$
$179$	−11.3410	−0.847665	−0.423832	+	0.905741i	$0.639315\pi$
$180$	0	0
$181$	11.7569	0.873880	0.436940	−	0.899491i	$-0.356062\pi$
$181$	11.7569	0.873880	0.436940	+	0.899491i	$0.356062\pi$
$182$	−5.55056	−0.411435
$183$	0	0
$184$	−4.15725	−0.306477
$185$	−17.1905	−1.26387
$186$	0	0
$187$	0	0
$188$	22.7443	1.65880
$189$	0	0
$190$	−21.2097	−1.53871
$191$	22.5344	1.63053	0.815265	−	0.579087i	$-0.196591\pi$
$191$	22.5344	1.63053	0.815265	+	0.579087i	$0.196591\pi$
$192$	0	0
$193$	−3.44323	−0.247849	−0.123925	−	0.992292i	$-0.539548\pi$
$193$	−3.44323	−0.247849	−0.123925	+	0.992292i	$0.539548\pi$
$194$	−22.2165	−1.59505
$195$	0	0
$196$	4.51241	0.322315
$197$	−2.18213	−0.155470	−0.0777352	−	0.996974i	$-0.524769\pi$
$197$	−2.18213	−0.155470	−0.0777352	+	0.996974i	$0.524769\pi$
$198$	0	0
$199$	2.33434	0.165477	0.0827386	−	0.996571i	$-0.473633\pi$
$199$	2.33434	0.165477	0.0827386	+	0.996571i	$0.473633\pi$
$200$	43.3283	3.06378
$201$	0	0
$202$	−14.7365	−1.03686
$203$	1.25295	0.0879399
$204$	0	0
$205$	8.98976	0.627872
$206$	−11.8435	−0.825177
$207$	0	0
$208$	15.9583	1.10651
$209$	0	0
$210$	0	0
$211$	13.6679	0.940935	0.470468	−	0.882417i	$-0.344085\pi$
$211$	13.6679	0.940935	0.470468	+	0.882417i	$0.344085\pi$
$212$	−60.4574	−4.15223
$213$	0	0
$214$	−10.2640	−0.701631
$215$	−5.01104	−0.341750
$216$	0	0
$217$	8.03946	0.545754
$218$	15.6884	1.06255
$219$	0	0
$220$	0	0
$221$	−9.82495	−0.660897
$222$	0	0
$223$	−5.36865	−0.359511	−0.179756	−	0.983711i	$-0.557531\pi$
$223$	−5.36865	−0.359511	−0.179756	+	0.983711i	$0.557531\pi$
$224$	−5.90061	−0.394251
$225$	0	0
$226$	6.32465	0.420710
$227$	10.6130	0.704408	0.352204	−	0.935923i	$-0.385432\pi$
$227$	10.6130	0.704408	0.352204	+	0.935923i	$0.385432\pi$
$228$	0	0
$229$	22.6410	1.49616	0.748079	−	0.663610i	$-0.230977\pi$
$229$	22.6410	1.49616	0.748079	+	0.663610i	$0.230977\pi$
$230$	5.67388	0.374125
$231$	0	0
$232$	−8.03333	−0.527414
$233$	14.8236	0.971128	0.485564	−	0.874201i	$-0.338614\pi$
$233$	14.8236	0.971128	0.485564	+	0.874201i	$0.338614\pi$
$234$	0	0
$235$	−17.2834	−1.12744
$236$	−34.6092	−2.25287
$237$	0	0
$238$	11.5275	0.747217
$239$	1.95930	0.126736	0.0633682	−	0.997990i	$-0.479816\pi$
$239$	1.95930	0.126736	0.0633682	+	0.997990i	$0.479816\pi$
$240$	0	0
$241$	−0.790252	−0.0509046	−0.0254523	−	0.999676i	$-0.508103\pi$
$241$	−0.790252	−0.0509046	−0.0254523	+	0.999676i	$0.508103\pi$
$242$	0	0
$243$	0	0
$244$	−64.6846	−4.14101
$245$	−3.42898	−0.219069
$246$	0	0
$247$	5.27187	0.335441
$248$	−51.5452	−3.27313
$249$	0	0
$250$	−15.3824	−0.972870
$251$	20.6154	1.30123	0.650616	−	0.759407i	$-0.274511\pi$
$251$	20.6154	1.30123	0.650616	+	0.759407i	$0.274511\pi$
$252$	0	0
$253$	0	0
$254$	17.7461	1.11349
$255$	0	0
$256$	−28.3834	−1.77396
$257$	−5.17961	−0.323095	−0.161548	−	0.986865i	$-0.551649\pi$
$257$	−5.17961	−0.323095	−0.161548	+	0.986865i	$0.551649\pi$
$258$	0	0
$259$	−5.01329	−0.311511
$260$	−33.6542	−2.08714
$261$	0	0
$262$	−53.4345	−3.30120
$263$	23.1920	1.43008	0.715041	−	0.699083i	$-0.246408\pi$
$263$	23.1920	1.43008	0.715041	+	0.699083i	$0.246408\pi$
$264$	0	0
$265$	45.9415	2.82217
$266$	−6.18543	−0.379253
$267$	0	0
$268$	28.0986	1.71640
$269$	−18.6600	−1.13772	−0.568860	−	0.822435i	$-0.692615\pi$
$269$	−18.6600	−1.13772	−0.568860	+	0.822435i	$0.692615\pi$
$270$	0	0
$271$	−30.2288	−1.83627	−0.918133	−	0.396272i	$-0.870304\pi$
$271$	−30.2288	−1.83627	−0.918133	+	0.396272i	$0.870304\pi$
$272$	−33.1424	−2.00956
$273$	0	0
$274$	−23.3157	−1.40855
$275$	0	0
$276$	0	0
$277$	−13.4921	−0.810660	−0.405330	−	0.914170i	$-0.632843\pi$
$277$	−13.4921	−0.810660	−0.405330	+	0.914170i	$0.632843\pi$
$278$	−27.4381	−1.64563
$279$	0	0
$280$	21.9850	1.31385
$281$	8.10271	0.483367	0.241684	−	0.970355i	$-0.422300\pi$
$281$	8.10271	0.483367	0.241684	+	0.970355i	$0.422300\pi$
$282$	0	0
$283$	−9.99117	−0.593914	−0.296957	−	0.954891i	$-0.595972\pi$
$283$	−9.99117	−0.593914	−0.296957	+	0.954891i	$0.595972\pi$
$284$	−19.0873	−1.13263
$285$	0	0
$286$	0	0
$287$	2.62170	0.154754
$288$	0	0
$289$	3.40464	0.200273
$290$	10.9640	0.643829
$291$	0	0
$292$	26.7368	1.56465
$293$	30.7296	1.79524	0.897622	−	0.440766i	$-0.145293\pi$
$293$	30.7296	1.79524	0.897622	+	0.440766i	$0.145293\pi$
$294$	0	0
$295$	26.2995	1.53122
$296$	32.1429	1.86827
$297$	0	0
$298$	27.4472	1.58997
$299$	−1.41030	−0.0815596
$300$	0	0
$301$	−1.46138	−0.0842326
$302$	19.5587	1.12547
$303$	0	0
$304$	17.7836	1.01996
$305$	49.1538	2.81454
$306$	0	0
$307$	20.0677	1.14532	0.572661	−	0.819792i	$-0.305911\pi$
$307$	20.0677	1.14532	0.572661	+	0.819792i	$0.305911\pi$
$308$	0	0
$309$	0	0
$310$	70.3497	3.99560
$311$	23.4662	1.33065	0.665324	−	0.746555i	$-0.268294\pi$
$311$	23.4662	1.33065	0.665324	+	0.746555i	$0.268294\pi$
$312$	0	0
$313$	−14.6730	−0.829366	−0.414683	−	0.909966i	$-0.636108\pi$
$313$	−14.6730	−0.829366	−0.414683	+	0.909966i	$0.636108\pi$
$314$	46.8257	2.64252
$315$	0	0
$316$	−44.0555	−2.47831
$317$	−18.0509	−1.01384	−0.506921	−	0.861993i	$-0.669216\pi$
$317$	−18.0509	−1.01384	−0.506921	+	0.861993i	$0.669216\pi$
$318$	0	0
$319$	0	0
$320$	−1.31663	−0.0736018
$321$	0	0
$322$	1.65469	0.0922121
$323$	−10.9487	−0.609203
$324$	0	0
$325$	14.6986	0.815332
$326$	0.963575	0.0533675
$327$	0	0
$328$	−16.8091	−0.928129
$329$	−5.04039	−0.277886
$330$	0	0
$331$	−19.8300	−1.08996	−0.544979	−	0.838450i	$-0.683462\pi$
$331$	−19.8300	−1.08996	−0.544979	+	0.838450i	$0.683462\pi$
$332$	−37.8030	−2.07471
$333$	0	0
$334$	−0.291275	−0.0159379
$335$	−21.3521	−1.16659
$336$	0	0
$337$	19.5084	1.06269	0.531345	−	0.847156i	$-0.321687\pi$
$337$	19.5084	1.06269	0.531345	+	0.847156i	$0.321687\pi$
$338$	−21.1026	−1.14783
$339$	0	0
$340$	69.8936	3.79051
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	9.36968	0.505179
$345$	0	0
$346$	−33.9434	−1.82481
$347$	−28.4451	−1.52701	−0.763507	−	0.645799i	$-0.776524\pi$
$347$	−28.4451	−1.52701	−0.763507	+	0.645799i	$0.776524\pi$
$348$	0	0
$349$	−20.0868	−1.07522	−0.537611	−	0.843193i	$-0.680673\pi$
$349$	−20.0868	−1.07522	−0.537611	+	0.843193i	$0.680673\pi$
$350$	−17.2457	−0.921823
$351$	0	0
$352$	0	0
$353$	−8.99700	−0.478862	−0.239431	−	0.970913i	$-0.576961\pi$
$353$	−8.99700	−0.478862	−0.239431	+	0.970913i	$0.576961\pi$
$354$	0	0
$355$	14.5045	0.769817
$356$	−21.4903	−1.13898
$357$	0	0
$358$	−28.9415	−1.52961
$359$	9.46601	0.499597	0.249798	−	0.968298i	$-0.419636\pi$
$359$	9.46601	0.499597	0.249798	+	0.968298i	$0.419636\pi$
$360$	0	0
$361$	−13.1251	−0.690797
$362$	30.0028	1.57691
$363$	0	0
$364$	−9.81464	−0.514427
$365$	−20.3173	−1.06345
$366$	0	0
$367$	1.37513	0.0717813	0.0358907	−	0.999356i	$-0.488573\pi$
$367$	1.37513	0.0717813	0.0358907	+	0.999356i	$0.488573\pi$
$368$	−4.75735	−0.247994
$369$	0	0
$370$	−43.8691	−2.28064
$371$	13.3980	0.695591
$372$	0	0
$373$	−5.07180	−0.262608	−0.131304	−	0.991342i	$-0.541916\pi$
$373$	−5.07180	−0.262608	−0.131304	+	0.991342i	$0.541916\pi$
$374$	0	0
$375$	0	0
$376$	32.3166	1.66660
$377$	−2.72521	−0.140355
$378$	0	0
$379$	−27.9822	−1.43735	−0.718675	−	0.695346i	$-0.755251\pi$
$379$	−27.9822	−1.43735	−0.718675	+	0.695346i	$0.755251\pi$
$380$	−37.5035	−1.92389
$381$	0	0
$382$	57.5065	2.94229
$383$	31.2805	1.59836	0.799181	−	0.601091i	$-0.205267\pi$
$383$	31.2805	1.59836	0.799181	+	0.601091i	$0.205267\pi$
$384$	0	0
$385$	0	0
$386$	−8.78692	−0.447243
$387$	0	0
$388$	−39.2837	−1.99433
$389$	−10.6536	−0.540158	−0.270079	−	0.962838i	$-0.587050\pi$
$389$	−10.6536	−0.540158	−0.270079	+	0.962838i	$0.587050\pi$
$390$	0	0
$391$	2.92893	0.148122
$392$	6.41153	0.323831
$393$	0	0
$394$	−5.56868	−0.280546
$395$	33.4777	1.68445
$396$	0	0
$397$	21.1755	1.06277	0.531385	−	0.847131i	$-0.321672\pi$
$397$	21.1755	1.06277	0.531385	+	0.847131i	$0.321672\pi$
$398$	5.95711	0.298603
$399$	0	0
$400$	49.5827	2.47914
$401$	33.7109	1.68344	0.841722	−	0.539911i	$-0.181542\pi$
$401$	33.7109	1.68344	0.841722	+	0.539911i	$0.181542\pi$
$402$	0	0
$403$	−17.4861	−0.871044
$404$	−26.0575	−1.29641
$405$	0	0
$406$	3.19746	0.158687
$407$	0	0
$408$	0	0
$409$	2.64404	0.130739	0.0653696	−	0.997861i	$-0.479177\pi$
$409$	2.64404	0.130739	0.0653696	+	0.997861i	$0.479177\pi$
$410$	22.9414	1.13299
$411$	0	0
$412$	−20.9420	−1.03174
$413$	7.66978	0.377405
$414$	0	0
$415$	28.7265	1.41013
$416$	12.8340	0.629240
$417$	0	0
$418$	0	0
$419$	−33.0757	−1.61585	−0.807926	−	0.589284i	$-0.799410\pi$
$419$	−33.0757	−1.61585	−0.807926	+	0.589284i	$0.799410\pi$
$420$	0	0
$421$	33.6819	1.64155	0.820777	−	0.571249i	$-0.193541\pi$
$421$	33.6819	1.64155	0.820777	+	0.571249i	$0.193541\pi$
$422$	34.8796	1.69791
$423$	0	0
$424$	−85.9018	−4.17176
$425$	−30.5264	−1.48075
$426$	0	0
$427$	14.3348	0.693711
$428$	−18.1490	−0.877266
$429$	0	0
$430$	−12.7879	−0.616686
$431$	−29.7526	−1.43313	−0.716566	−	0.697520i	$-0.754287\pi$
$431$	−29.7526	−1.43313	−0.716566	+	0.697520i	$0.754287\pi$
$432$	0	0
$433$	−23.1510	−1.11256	−0.556282	−	0.830994i	$-0.687773\pi$
$433$	−23.1510	−1.11256	−0.556282	+	0.830994i	$0.687773\pi$
$434$	20.5163	0.984812
$435$	0	0
$436$	27.7406	1.32853
$437$	−1.57161	−0.0751801
$438$	0	0
$439$	−15.8166	−0.754884	−0.377442	−	0.926033i	$-0.623196\pi$
$439$	−15.8166	−0.754884	−0.377442	+	0.926033i	$0.623196\pi$
$440$	0	0
$441$	0	0
$442$	−25.0727	−1.19259
$443$	10.0267	0.476381	0.238191	−	0.971218i	$-0.423446\pi$
$443$	10.0267	0.476381	0.238191	+	0.971218i	$0.423446\pi$
$444$	0	0
$445$	16.3305	0.774139
$446$	−13.7005	−0.648737
$447$	0	0
$448$	−0.383971	−0.0181409
$449$	−1.60627	−0.0758044	−0.0379022	−	0.999281i	$-0.512068\pi$
$449$	−1.60627	−0.0758044	−0.0379022	+	0.999281i	$0.512068\pi$
$450$	0	0
$451$	0	0
$452$	11.1834	0.526023
$453$	0	0
$454$	27.0837	1.27110
$455$	7.45814	0.349643
$456$	0	0
$457$	−3.75695	−0.175743	−0.0878714	−	0.996132i	$-0.528006\pi$
$457$	−3.75695	−0.175743	−0.0878714	+	0.996132i	$0.528006\pi$
$458$	57.7784	2.69981
$459$	0	0
$460$	10.0327	0.467777
$461$	−14.1849	−0.660657	−0.330329	−	0.943866i	$-0.607160\pi$
$461$	−14.1849	−0.660657	−0.330329	+	0.943866i	$0.607160\pi$
$462$	0	0
$463$	−5.34265	−0.248294	−0.124147	−	0.992264i	$-0.539619\pi$
$463$	−5.34265	−0.248294	−0.124147	+	0.992264i	$0.539619\pi$
$464$	−9.19293	−0.426771
$465$	0	0
$466$	37.8290	1.75240
$467$	23.8696	1.10455	0.552276	−	0.833662i	$-0.313760\pi$
$467$	23.8696	1.10455	0.552276	+	0.833662i	$0.313760\pi$
$468$	0	0
$469$	−6.22696	−0.287534
$470$	−44.1062	−2.03447
$471$	0	0
$472$	−49.1750	−2.26346
$473$	0	0
$474$	0	0
$475$	16.3798	0.751558
$476$	20.3832	0.934264
$477$	0	0
$478$	5.00001	0.228695
$479$	−0.436516	−0.0199449	−0.00997247	−	0.999950i	$-0.503174\pi$
$479$	−0.436516	−0.0199449	−0.00997247	+	0.999950i	$0.503174\pi$
$480$	0	0
$481$	10.9041	0.497183
$482$	−2.01668	−0.0918572
$483$	0	0
$484$	0	0
$485$	29.8517	1.35549
$486$	0	0
$487$	0.733680	0.0332462	0.0166231	−	0.999862i	$-0.494708\pi$
$487$	0.733680	0.0332462	0.0166231	+	0.999862i	$0.494708\pi$
$488$	−91.9081	−4.16048
$489$	0	0
$490$	−8.75055	−0.395310
$491$	2.60751	0.117675	0.0588377	−	0.998268i	$-0.481261\pi$
$491$	2.60751	0.117675	0.0588377	+	0.998268i	$0.481261\pi$
$492$	0	0
$493$	5.65977	0.254903
$494$	13.4535	0.605302
$495$	0	0
$496$	−58.9857	−2.64854
$497$	4.22997	0.189740
$498$	0	0
$499$	−8.25801	−0.369679	−0.184840	−	0.982769i	$-0.559177\pi$
$499$	−8.25801	−0.369679	−0.184840	+	0.982769i	$0.559177\pi$
$500$	−27.1996	−1.21640
$501$	0	0
$502$	52.6093	2.34807
$503$	33.9117	1.51205	0.756024	−	0.654544i	$-0.227139\pi$
$503$	33.9117	1.51205	0.756024	+	0.654544i	$0.227139\pi$
$504$	0	0
$505$	19.8010	0.881135
$506$	0	0
$507$	0	0
$508$	31.3792	1.39223
$509$	29.2437	1.29620	0.648102	−	0.761554i	$-0.275563\pi$
$509$	29.2437	1.29620	0.648102	+	0.761554i	$0.275563\pi$
$510$	0	0
$511$	−5.92517	−0.262114
$512$	−50.7901	−2.24463
$513$	0	0
$514$	−13.2181	−0.583024
$515$	15.9138	0.701246
$516$	0	0
$517$	0	0
$518$	−12.7936	−0.562120
$519$	0	0
$520$	−47.8180	−2.09696
$521$	9.71550	0.425644	0.212822	−	0.977091i	$-0.431735\pi$
$521$	9.71550	0.425644	0.212822	+	0.977091i	$0.431735\pi$
$522$	0	0
$523$	12.8257	0.560831	0.280415	−	0.959879i	$-0.409528\pi$
$523$	12.8257	0.560831	0.280415	+	0.959879i	$0.409528\pi$
$524$	−94.4843	−4.12757
$525$	0	0
$526$	59.1847	2.58058
$527$	36.3155	1.58193
$528$	0	0
$529$	−22.5796	−0.981721
$530$	117.240	5.09259
$531$	0	0
$532$	−10.9372	−0.474189
$533$	−5.70229	−0.246994
$534$	0	0
$535$	13.7914	0.596255
$536$	39.9243	1.72447
$537$	0	0
$538$	−47.6192	−2.05301
$539$	0	0
$540$	0	0
$541$	6.90126	0.296708	0.148354	−	0.988934i	$-0.452602\pi$
$541$	6.90126	0.296708	0.148354	+	0.988934i	$0.452602\pi$
$542$	−77.1420	−3.31353
$543$	0	0
$544$	−26.6539	−1.14278
$545$	−21.0801	−0.902970
$546$	0	0
$547$	17.7218	0.757729	0.378865	−	0.925452i	$-0.376315\pi$
$547$	17.7218	0.757729	0.378865	+	0.925452i	$0.376315\pi$
$548$	−41.2274	−1.76115
$549$	0	0
$550$	0	0
$551$	−3.03692	−0.129377
$552$	0	0
$553$	9.76318	0.415173
$554$	−34.4310	−1.46283
$555$	0	0
$556$	−48.5168	−2.05757
$557$	43.4273	1.84007	0.920037	−	0.391832i	$-0.128159\pi$
$557$	43.4273	1.84007	0.920037	+	0.391832i	$0.128159\pi$
$558$	0	0
$559$	3.17855	0.134438
$560$	25.1585	1.06314
$561$	0	0
$562$	20.6777	0.872234
$563$	22.3137	0.940412	0.470206	−	0.882557i	$-0.344180\pi$
$563$	22.3137	0.940412	0.470206	+	0.882557i	$0.344180\pi$
$564$	0	0
$565$	−8.49827	−0.357525
$566$	−25.4969	−1.07171
$567$	0	0
$568$	−27.1205	−1.13795
$569$	22.6691	0.950337	0.475169	−	0.879895i	$-0.342387\pi$
$569$	22.6691	0.950337	0.475169	+	0.879895i	$0.342387\pi$
$570$	0	0
$571$	−35.0994	−1.46886	−0.734432	−	0.678683i	$-0.762551\pi$
$571$	−35.0994	−1.46886	−0.734432	+	0.678683i	$0.762551\pi$
$572$	0	0
$573$	0	0
$574$	6.69044	0.279254
$575$	−4.38183	−0.182735
$576$	0	0
$577$	−32.6358	−1.35864	−0.679322	−	0.733840i	$-0.737726\pi$
$577$	−32.6358	−1.35864	−0.679322	+	0.733840i	$0.737726\pi$
$578$	8.68844	0.361392
$579$	0	0
$580$	19.3869	0.804995
$581$	8.37756	0.347560
$582$	0	0
$583$	0	0
$584$	37.9894	1.57201
$585$	0	0
$586$	78.4203	3.23951
$587$	−37.0174	−1.52787	−0.763936	−	0.645292i	$-0.776736\pi$
$587$	−37.0174	−1.52787	−0.763936	+	0.645292i	$0.776736\pi$
$588$	0	0
$589$	−19.4861	−0.802912
$590$	67.1148	2.76307
$591$	0	0
$592$	36.7826	1.51176
$593$	−16.2161	−0.665915	−0.332958	−	0.942942i	$-0.608047\pi$
$593$	−16.2161	−0.665915	−0.332958	+	0.942942i	$0.608047\pi$
$594$	0	0
$595$	−15.4892	−0.634995
$596$	48.5329	1.98798
$597$	0	0
$598$	−3.59900	−0.147174
$599$	−10.5475	−0.430960	−0.215480	−	0.976508i	$-0.569132\pi$
$599$	−10.5475	−0.430960	−0.215480	+	0.976508i	$0.569132\pi$
$600$	0	0
$601$	36.4911	1.48851	0.744253	−	0.667898i	$-0.232806\pi$
$601$	36.4911	1.48851	0.744253	+	0.667898i	$0.232806\pi$
$602$	−3.72936	−0.151997
$603$	0	0
$604$	34.5841	1.40721
$605$	0	0
$606$	0	0
$607$	33.7114	1.36830	0.684152	−	0.729340i	$-0.260173\pi$
$607$	33.7114	1.36830	0.684152	+	0.729340i	$0.260173\pi$
$608$	14.3020	0.580021
$609$	0	0
$610$	125.438	5.07882
$611$	10.9630	0.443516
$612$	0	0
$613$	−24.2067	−0.977697	−0.488849	−	0.872369i	$-0.662583\pi$
$613$	−24.2067	−0.977697	−0.488849	+	0.872369i	$0.662583\pi$
$614$	51.2115	2.06673
$615$	0	0
$616$	0	0
$617$	−17.6040	−0.708710	−0.354355	−	0.935111i	$-0.615300\pi$
$617$	−17.6040	−0.708710	−0.354355	+	0.935111i	$0.615300\pi$
$618$	0	0
$619$	−24.3176	−0.977407	−0.488703	−	0.872450i	$-0.662530\pi$
$619$	−24.3176	−0.977407	−0.488703	+	0.872450i	$0.662530\pi$
$620$	124.394	4.99579
$621$	0	0
$622$	59.8844	2.40115
$623$	4.76249	0.190805
$624$	0	0
$625$	−13.1204	−0.524818
$626$	−37.4446	−1.49659
$627$	0	0
$628$	82.7983	3.30401
$629$	−22.6458	−0.902947
$630$	0	0
$631$	4.04047	0.160848	0.0804242	−	0.996761i	$-0.474372\pi$
$631$	4.04047	0.160848	0.0804242	+	0.996761i	$0.474372\pi$
$632$	−62.5969	−2.48997
$633$	0	0
$634$	−46.0649	−1.82947
$635$	−23.8450	−0.946260
$636$	0	0
$637$	2.17503	0.0861779
$638$	0	0
$639$	0	0
$640$	37.1062	1.46675
$641$	21.3682	0.843994	0.421997	−	0.906597i	$-0.361329\pi$
$641$	21.3682	0.843994	0.421997	+	0.906597i	$0.361329\pi$
$642$	0	0
$643$	22.5750	0.890273	0.445136	−	0.895463i	$-0.353155\pi$
$643$	22.5750	0.890273	0.445136	+	0.895463i	$0.353155\pi$
$644$	2.92586	0.115295
$645$	0	0
$646$	−27.9405	−1.09930
$647$	−17.1171	−0.672942	−0.336471	−	0.941694i	$-0.609233\pi$
$647$	−17.1171	−0.672942	−0.336471	+	0.941694i	$0.609233\pi$
$648$	0	0
$649$	0	0
$650$	37.5100	1.47126
$651$	0	0
$652$	1.70382	0.0667267
$653$	13.2411	0.518165	0.259082	−	0.965855i	$-0.416580\pi$
$653$	13.2411	0.518165	0.259082	+	0.965855i	$0.416580\pi$
$654$	0	0
$655$	71.7986	2.80540
$656$	−19.2355	−0.751020
$657$	0	0
$658$	−12.8628	−0.501443
$659$	28.3747	1.10532	0.552660	−	0.833407i	$-0.313613\pi$
$659$	28.3747	1.10532	0.552660	+	0.833407i	$0.313613\pi$
$660$	0	0
$661$	18.9657	0.737679	0.368840	−	0.929493i	$-0.379755\pi$
$661$	18.9657	0.737679	0.368840	+	0.929493i	$0.379755\pi$
$662$	−50.6051	−1.96682
$663$	0	0
$664$	−53.7130	−2.08447
$665$	8.31119	0.322294
$666$	0	0
$667$	0.812417	0.0314569
$668$	−0.515040	−0.0199275
$669$	0	0
$670$	−54.4893	−2.10511
$671$	0	0
$672$	0	0
$673$	−46.5443	−1.79415	−0.897075	−	0.441878i	$-0.854312\pi$
$673$	−46.5443	−1.79415	−0.897075	+	0.441878i	$0.854312\pi$
$674$	49.7843	1.91762
$675$	0	0
$676$	−37.3142	−1.43516
$677$	28.5695	1.09801	0.549007	−	0.835818i	$-0.315006\pi$
$677$	28.5695	1.09801	0.549007	+	0.835818i	$0.315006\pi$
$678$	0	0
$679$	8.70570	0.334094
$680$	99.3094	3.80834
$681$	0	0
$682$	0	0
$683$	21.9450	0.839704	0.419852	−	0.907593i	$-0.362082\pi$
$683$	21.9450	0.839704	0.419852	+	0.907593i	$0.362082\pi$
$684$	0	0
$685$	31.3286	1.19701
$686$	−2.55194	−0.0974336
$687$	0	0
$688$	10.7222	0.408779
$689$	−29.1411	−1.11019
$690$	0	0
$691$	11.1296	0.423390	0.211695	−	0.977336i	$-0.432102\pi$
$691$	11.1296	0.423390	0.211695	+	0.977336i	$0.432102\pi$
$692$	−60.0197	−2.28161
$693$	0	0
$694$	−72.5903	−2.75549
$695$	36.8679	1.39848
$696$	0	0
$697$	11.8426	0.448572
$698$	−51.2604	−1.94023
$699$	0	0
$700$	−30.4943	−1.15258
$701$	−38.3963	−1.45021	−0.725104	−	0.688639i	$-0.758208\pi$
$701$	−38.3963	−1.45021	−0.725104	+	0.688639i	$0.758208\pi$
$702$	0	0
$703$	12.1513	0.458294
$704$	0	0
$705$	0	0
$706$	−22.9598	−0.864104
$707$	5.77462	0.217177
$708$	0	0
$709$	−0.480502	−0.0180456	−0.00902282	−	0.999959i	$-0.502872\pi$
$709$	−0.480502	−0.0180456	−0.00902282	+	0.999959i	$0.502872\pi$
$710$	37.0145	1.38913
$711$	0	0
$712$	−30.5348	−1.14434
$713$	5.21281	0.195221
$714$	0	0
$715$	0	0
$716$	−51.1752	−1.91251
$717$	0	0
$718$	24.1567	0.901521
$719$	−36.4697	−1.36009	−0.680046	−	0.733169i	$-0.738040\pi$
$719$	−36.4697	−1.36009	−0.680046	+	0.733169i	$0.738040\pi$
$720$	0	0
$721$	4.64098	0.172839
$722$	−33.4946	−1.24654
$723$	0	0
$724$	53.0517	1.97165
$725$	−8.46730	−0.314467
$726$	0	0
$727$	−33.6867	−1.24937	−0.624686	−	0.780876i	$-0.714773\pi$
$727$	−33.6867	−1.24937	−0.624686	+	0.780876i	$0.714773\pi$
$728$	−13.9453	−0.516846
$729$	0	0
$730$	−51.8485	−1.91900
$731$	−6.60127	−0.244157
$732$	0	0
$733$	27.0637	0.999619	0.499810	−	0.866135i	$-0.333403\pi$
$733$	27.0637	0.999619	0.499810	+	0.866135i	$0.333403\pi$
$734$	3.50926	0.129529
$735$	0	0
$736$	−3.82597	−0.141027
$737$	0	0
$738$	0	0
$739$	10.4126	0.383035	0.191517	−	0.981489i	$-0.438659\pi$
$739$	10.4126	0.383035	0.191517	+	0.981489i	$0.438659\pi$
$740$	−77.5704	−2.85155
$741$	0	0
$742$	34.1910	1.25519
$743$	47.5521	1.74452	0.872260	−	0.489043i	$-0.162654\pi$
$743$	47.5521	1.74452	0.872260	+	0.489043i	$0.162654\pi$
$744$	0	0
$745$	−36.8801	−1.35118
$746$	−12.9429	−0.473874
$747$	0	0
$748$	0	0
$749$	4.02202	0.146961
$750$	0	0
$751$	−29.6748	−1.08285	−0.541424	−	0.840749i	$-0.682115\pi$
$751$	−29.6748	−1.08285	−0.541424	+	0.840749i	$0.682115\pi$
$752$	36.9815	1.34857
$753$	0	0
$754$	−6.95458	−0.253271
$755$	−26.2804	−0.956443
$756$	0	0
$757$	−30.9790	−1.12595	−0.562976	−	0.826473i	$-0.690344\pi$
$757$	−30.9790	−1.12595	−0.562976	+	0.826473i	$0.690344\pi$
$758$	−71.4090	−2.59369
$759$	0	0
$760$	−53.2874	−1.93294
$761$	11.7673	0.426564	0.213282	−	0.976991i	$-0.431585\pi$
$761$	11.7673	0.426564	0.213282	+	0.976991i	$0.431585\pi$
$762$	0	0
$763$	−6.14762	−0.222559
$764$	101.684	3.67881
$765$	0	0
$766$	79.8261	2.88424
$767$	−16.6820	−0.602353
$768$	0	0
$769$	−4.17897	−0.150697	−0.0753487	−	0.997157i	$-0.524007\pi$
$769$	−4.17897	−0.150697	−0.0753487	+	0.997157i	$0.524007\pi$
$770$	0	0
$771$	0	0
$772$	−15.5373	−0.559198
$773$	−0.0837940	−0.00301386	−0.00150693	−	0.999999i	$-0.500480\pi$
$773$	−0.0837940	−0.00301386	−0.00150693	+	0.999999i	$0.500480\pi$
$774$	0	0
$775$	−54.3297	−1.95158
$776$	−55.8168	−2.00371
$777$	0	0
$778$	−27.1873	−0.974713
$779$	−6.35452	−0.227674
$780$	0	0
$781$	0	0
$782$	7.47446	0.267286
$783$	0	0
$784$	7.33702	0.262037
$785$	−62.9184	−2.24565
$786$	0	0
$787$	−13.4395	−0.479065	−0.239533	−	0.970888i	$-0.576994\pi$
$787$	−13.4395	−0.479065	−0.239533	+	0.970888i	$0.576994\pi$
$788$	−9.84668	−0.350773
$789$	0	0
$790$	85.4332	3.03958
$791$	−2.47837	−0.0881206
$792$	0	0
$793$	−31.1787	−1.10719
$794$	54.0387	1.91776
$795$	0	0
$796$	10.5335	0.373350
$797$	13.5979	0.481661	0.240831	−	0.970567i	$-0.422580\pi$
$797$	13.5979	0.481661	0.240831	+	0.970567i	$0.422580\pi$
$798$	0	0
$799$	−22.7682	−0.805481
$800$	39.8756	1.40982
$801$	0	0
$802$	86.0284	3.03777
$803$	0	0
$804$	0	0
$805$	−2.22336	−0.0783631
$806$	−44.6235	−1.57180
$807$	0	0
$808$	−37.0241	−1.30250
$809$	44.5000	1.56454	0.782269	−	0.622941i	$-0.214062\pi$
$809$	44.5000	1.56454	0.782269	+	0.622941i	$0.214062\pi$
$810$	0	0
$811$	24.2045	0.849935	0.424968	−	0.905209i	$-0.360285\pi$
$811$	24.2045	0.849935	0.424968	+	0.905209i	$0.360285\pi$
$812$	5.65383	0.198411
$813$	0	0
$814$	0	0
$815$	−1.29473	−0.0453524
$816$	0	0
$817$	3.54211	0.123923
$818$	6.74743	0.235918
$819$	0	0
$820$	40.5655	1.41661
$821$	−9.28037	−0.323887	−0.161943	−	0.986800i	$-0.551776\pi$
$821$	−9.28037	−0.323887	−0.161943	+	0.986800i	$0.551776\pi$
$822$	0	0
$823$	−23.4833	−0.818576	−0.409288	−	0.912405i	$-0.634223\pi$
$823$	−23.4833	−0.818576	−0.409288	+	0.912405i	$0.634223\pi$
$824$	−29.7558	−1.03659
$825$	0	0
$826$	19.5728	0.681026
$827$	−19.7388	−0.686385	−0.343193	−	0.939265i	$-0.611508\pi$
$827$	−19.7388	−0.686385	−0.343193	+	0.939265i	$0.611508\pi$
$828$	0	0
$829$	−39.9938	−1.38904	−0.694521	−	0.719472i	$-0.744384\pi$
$829$	−39.9938	−1.38904	−0.694521	+	0.719472i	$0.744384\pi$
$830$	73.3083	2.54457
$831$	0	0
$832$	0.835150	0.0289536
$833$	−4.51715	−0.156510
$834$	0	0
$835$	0.391379	0.0135442
$836$	0	0
$837$	0	0
$838$	−84.4072	−2.91580
$839$	7.84815	0.270948	0.135474	−	0.990781i	$-0.456744\pi$
$839$	7.84815	0.270948	0.135474	+	0.990781i	$0.456744\pi$
$840$	0	0
$841$	−27.4301	−0.945866
$842$	85.9542	2.96218
$843$	0	0
$844$	61.6750	2.12294
$845$	28.3550	0.975442
$846$	0	0
$847$	0	0
$848$	−98.3017	−3.37569
$849$	0	0
$850$	−77.9015	−2.67200
$851$	−3.25063	−0.111430
$852$	0	0
$853$	40.9990	1.40378	0.701890	−	0.712286i	$-0.252340\pi$
$853$	40.9990	1.40378	0.701890	+	0.712286i	$0.252340\pi$
$854$	36.5817	1.25180
$855$	0	0
$856$	−25.7873	−0.881392
$857$	31.1892	1.06540	0.532702	−	0.846303i	$-0.321177\pi$
$857$	31.1892	1.06540	0.532702	+	0.846303i	$0.321177\pi$
$858$	0	0
$859$	7.95825	0.271532	0.135766	−	0.990741i	$-0.456650\pi$
$859$	7.95825	0.271532	0.135766	+	0.990741i	$0.456650\pi$
$860$	−22.6119	−0.771058
$861$	0	0
$862$	−75.9269	−2.58608
$863$	−31.1042	−1.05880	−0.529400	−	0.848372i	$-0.677583\pi$
$863$	−31.1042	−1.05880	−0.529400	+	0.848372i	$0.677583\pi$
$864$	0	0
$865$	45.6089	1.55075
$866$	−59.0799	−2.00762
$867$	0	0
$868$	36.2774	1.23133
$869$	0	0
$870$	0	0
$871$	13.5438	0.458916
$872$	39.4156	1.33478
$873$	0	0
$874$	−4.01065	−0.135662
$875$	6.02773	0.203775
$876$	0	0
$877$	51.8971	1.75244	0.876221	−	0.481910i	$-0.160057\pi$
$877$	51.8971	1.75244	0.876221	+	0.481910i	$0.160057\pi$
$878$	−40.3630	−1.36219
$879$	0	0
$880$	0	0
$881$	−42.1448	−1.41989	−0.709947	−	0.704256i	$-0.751281\pi$
$881$	−42.1448	−1.41989	−0.709947	+	0.704256i	$0.751281\pi$
$882$	0	0
$883$	−17.6578	−0.594233	−0.297116	−	0.954841i	$-0.596025\pi$
$883$	−17.6578	−0.594233	−0.297116	+	0.954841i	$0.596025\pi$
$884$	−44.3342	−1.49112
$885$	0	0
$886$	25.5875	0.859628
$887$	−10.6530	−0.357692	−0.178846	−	0.983877i	$-0.557236\pi$
$887$	−10.6530	−0.357692	−0.178846	+	0.983877i	$0.557236\pi$
$888$	0	0
$889$	−6.95397	−0.233229
$890$	41.6744	1.39693
$891$	0	0
$892$	−24.2255	−0.811131
$893$	12.2169	0.408825
$894$	0	0
$895$	38.8880	1.29988
$896$	10.8213	0.361516
$897$	0	0
$898$	−4.09910	−0.136789
$899$	10.0731	0.335955
$900$	0	0
$901$	60.5209	2.01624
$902$	0	0
$903$	0	0
$904$	15.8901	0.528497
$905$	−40.3140	−1.34008
$906$	0	0
$907$	50.3005	1.67020	0.835100	−	0.550098i	$-0.185410\pi$
$907$	50.3005	1.67020	0.835100	+	0.550098i	$0.185410\pi$
$908$	47.8901	1.58929
$909$	0	0
$910$	19.0327	0.630929
$911$	23.0373	0.763259	0.381629	−	0.924315i	$-0.375363\pi$
$911$	23.0373	0.763259	0.381629	+	0.924315i	$0.375363\pi$
$912$	0	0
$913$	0	0
$914$	−9.58753	−0.317127
$915$	0	0
$916$	102.165	3.37564
$917$	20.9388	0.691459
$918$	0	0
$919$	38.0655	1.25566	0.627832	−	0.778349i	$-0.283942\pi$
$919$	38.0655	1.25566	0.627832	+	0.778349i	$0.283942\pi$
$920$	14.2551	0.469977
$921$	0	0
$922$	−36.1991	−1.19215
$923$	−9.20031	−0.302832
$924$	0	0
$925$	33.8792	1.11394
$926$	−13.6341	−0.448045
$927$	0	0
$928$	−7.39318	−0.242693
$929$	17.5382	0.575409	0.287705	−	0.957719i	$-0.407108\pi$
$929$	17.5382	0.575409	0.287705	+	0.957719i	$0.407108\pi$
$930$	0	0
$931$	2.42381	0.0794372
$932$	66.8903	2.19106
$933$	0	0
$934$	60.9137	1.99316
$935$	0	0
$936$	0	0
$937$	56.7590	1.85423	0.927117	−	0.374772i	$-0.122279\pi$
$937$	56.7590	1.85423	0.927117	+	0.374772i	$0.122279\pi$
$938$	−15.8908	−0.518854
$939$	0	0
$940$	−77.9897	−2.54374
$941$	−22.2986	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$941$	−22.2986	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$942$	0	0
$943$	1.69992	0.0553570
$944$	−56.2734	−1.83154
$945$	0	0
$946$	0	0
$947$	−44.7782	−1.45510	−0.727548	−	0.686057i	$-0.759340\pi$
$947$	−44.7782	−1.45510	−0.727548	+	0.686057i	$0.759340\pi$
$948$	0	0
$949$	12.8874	0.418344
$950$	41.8004	1.35618
$951$	0	0
$952$	28.9618	0.938658
$953$	1.85181	0.0599861	0.0299931	−	0.999550i	$-0.490451\pi$
$953$	1.85181	0.0599861	0.0299931	+	0.999550i	$0.490451\pi$
$954$	0	0
$955$	−77.2699	−2.50039
$956$	8.84115	0.285943
$957$	0	0
$958$	−1.11396	−0.0359906
$959$	9.13644	0.295031
$960$	0	0
$961$	33.6330	1.08494
$962$	27.8266	0.897165
$963$	0	0
$964$	−3.56594	−0.114851
$965$	11.8068	0.380073
$966$	0	0
$967$	33.5800	1.07986	0.539930	−	0.841710i	$-0.318451\pi$
$967$	33.5800	1.07986	0.539930	+	0.841710i	$0.318451\pi$
$968$	0	0
$969$	0	0
$970$	76.1797	2.44598
$971$	−42.4743	−1.36307	−0.681533	−	0.731788i	$-0.738686\pi$
$971$	−42.4743	−1.36307	−0.681533	+	0.731788i	$0.738686\pi$
$972$	0	0
$973$	10.7519	0.344689
$974$	1.87231	0.0599926
$975$	0	0
$976$	−105.175	−3.36657
$977$	−46.3327	−1.48231	−0.741157	−	0.671332i	$-0.765722\pi$
$977$	−46.3327	−1.48231	−0.741157	+	0.671332i	$0.765722\pi$
$978$	0	0
$979$	0	0
$980$	−15.4729	−0.494265
$981$	0	0
$982$	6.65422	0.212345
$983$	−25.0907	−0.800268	−0.400134	−	0.916457i	$-0.631036\pi$
$983$	−25.0907	−0.800268	−0.400134	+	0.916457i	$0.631036\pi$
$984$	0	0
$985$	7.48248	0.238412
$986$	14.4434	0.459972
$987$	0	0
$988$	23.7888	0.756823
$989$	−0.947563	−0.0301307
$990$	0	0
$991$	37.5645	1.19328	0.596638	−	0.802510i	$-0.296503\pi$
$991$	37.5645	1.19328	0.596638	+	0.802510i	$0.296503\pi$
$992$	−47.4378	−1.50615
$993$	0	0
$994$	10.7946	0.342385
$995$	−8.00440	−0.253757
$996$	0	0
$997$	−35.8390	−1.13503	−0.567517	−	0.823362i	$-0.692096\pi$
$997$	−35.8390	−1.13503	−0.567517	+	0.823362i	$0.692096\pi$
$998$	−21.0740	−0.667084
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cw.1.8		8
3.2	odd	2		847.2.a.o.1.1		8
11.7	odd	10		693.2.m.i.379.1		16
11.8	odd	10		693.2.m.i.64.1		16
11.10	odd	2		7623.2.a.ct.1.1		8
21.20	even	2		5929.2.a.bs.1.1		8
33.2	even	10		847.2.f.w.323.1		16
33.5	odd	10		847.2.f.v.729.4		16
33.8	even	10		77.2.f.b.64.4	✓	16
33.14	odd	10		847.2.f.x.372.1		16
33.17	even	10		847.2.f.w.729.1		16
33.20	odd	10		847.2.f.v.323.4		16
33.26	odd	10		847.2.f.x.148.1		16
33.29	even	10		77.2.f.b.71.4	yes	16
33.32	even	2		847.2.a.p.1.8		8
231.41	odd	10		539.2.f.e.295.4		16
231.62	odd	10		539.2.f.e.148.4		16
231.74	even	30		539.2.q.g.471.4		32
231.95	even	30		539.2.q.g.324.1		32
231.107	even	30		539.2.q.g.361.1		32
231.128	even	30		539.2.q.g.214.4		32
231.173	odd	30		539.2.q.f.361.1		32
231.194	odd	30		539.2.q.f.214.4		32
231.206	odd	30		539.2.q.f.471.4		32
231.227	odd	30		539.2.q.f.324.1		32
231.230	odd	2		5929.2.a.bt.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.64.4	✓	16	33.8	even	10
77.2.f.b.71.4	yes	16	33.29	even	10
539.2.f.e.148.4		16	231.62	odd	10
539.2.f.e.295.4		16	231.41	odd	10
539.2.q.f.214.4		32	231.194	odd	30
539.2.q.f.324.1		32	231.227	odd	30
539.2.q.f.361.1		32	231.173	odd	30
539.2.q.f.471.4		32	231.206	odd	30
539.2.q.g.214.4		32	231.128	even	30
539.2.q.g.324.1		32	231.95	even	30
539.2.q.g.361.1		32	231.107	even	30
539.2.q.g.471.4		32	231.74	even	30
693.2.m.i.64.1		16	11.8	odd	10
693.2.m.i.379.1		16	11.7	odd	10
847.2.a.o.1.1		8	3.2	odd	2
847.2.a.p.1.8		8	33.32	even	2
847.2.f.v.323.4		16	33.20	odd	10
847.2.f.v.729.4		16	33.5	odd	10
847.2.f.w.323.1		16	33.2	even	10
847.2.f.w.729.1		16	33.17	even	10
847.2.f.x.148.1		16	33.26	odd	10
847.2.f.x.372.1		16	33.14	odd	10
5929.2.a.bs.1.1		8	21.20	even	2
5929.2.a.bt.1.8		8	231.230	odd	2
7623.2.a.ct.1.1		8	11.10	odd	2
7623.2.a.cw.1.8		8	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+2.55194 q^{2} +4.51241 q^{4} -3.42898 q^{5} -1.00000 q^{7} +6.41153 q^{8} +O(q^{10})\)	\(q+2.55194 q^{2} +4.51241 q^{4} -3.42898 q^{5} -1.00000 q^{7} +6.41153 q^{8} -8.75055 q^{10} +2.17503 q^{13} -2.55194 q^{14} +7.33702 q^{16} -4.51715 q^{17} +2.42381 q^{19} -15.4729 q^{20} -0.648403 q^{23} +6.75788 q^{25} +5.55056 q^{26} -4.51241 q^{28} -1.25295 q^{29} -8.03946 q^{31} +5.90061 q^{32} -11.5275 q^{34} +3.42898 q^{35} +5.01329 q^{37} +6.18543 q^{38} -21.9850 q^{40} -2.62170 q^{41} +1.46138 q^{43} -1.65469 q^{46} +5.04039 q^{47} +1.00000 q^{49} +17.2457 q^{50} +9.81464 q^{52} -13.3980 q^{53} -6.41153 q^{56} -3.19746 q^{58} -7.66978 q^{59} -14.3348 q^{61} -20.5163 q^{62} +0.383971 q^{64} -7.45814 q^{65} +6.22696 q^{67} -20.3832 q^{68} +8.75055 q^{70} -4.22997 q^{71} +5.92517 q^{73} +12.7936 q^{74} +10.9372 q^{76} -9.76318 q^{79} -25.1585 q^{80} -6.69044 q^{82} -8.37756 q^{83} +15.4892 q^{85} +3.72936 q^{86} -4.76249 q^{89} -2.17503 q^{91} -2.92586 q^{92} +12.8628 q^{94} -8.31119 q^{95} -8.70570 q^{97} +2.55194 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 7 q^{4} - 10 q^{5} - 8 q^{7}+O(q^{10}) \) 8 * q + q^2 + 7 * q^4 - 10 * q^5 - 8 * q^7	\( 8 q + q^{2} + 7 q^{4} - 10 q^{5} - 8 q^{7} - 6 q^{10} + 6 q^{13} - q^{14} + q^{16} - 5 q^{17} + 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} - 7 q^{28} + 9 q^{29} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} + 7 q^{37} + 10 q^{38} - 5 q^{40} - 10 q^{41} + 4 q^{43} - 4 q^{46} - 16 q^{47} + 8 q^{49} + 6 q^{50} + 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} - 19 q^{61} - 18 q^{62} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 6 q^{70} - 13 q^{71} + 25 q^{73} + 33 q^{74} - 26 q^{76} - 4 q^{80} - 13 q^{82} - 25 q^{83} - 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} + 42 q^{94} + 21 q^{95} + 15 q^{97} + q^{98}+O(q^{100}) \) 8 * q + q^2 + 7 * q^4 - 10 * q^5 - 8 * q^7 - 6 * q^10 + 6 * q^13 - q^14 + q^16 - 5 * q^17 + 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 - 7 * q^28 + 9 * q^29 + 9 * q^31 + 16 * q^32 - 12 * q^34 + 10 * q^35 + 7 * q^37 + 10 * q^38 - 5 * q^40 - 10 * q^41 + 4 * q^43 - 4 * q^46 - 16 * q^47 + 8 * q^49 + 6 * q^50 + 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 - 19 * q^61 - 18 * q^62 - 4 * q^64 - 4 * q^65 - 19 * q^67 + 9 * q^68 + 6 * q^70 - 13 * q^71 + 25 * q^73 + 33 * q^74 - 26 * q^76 - 4 * q^80 - 13 * q^82 - 25 * q^83 - 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 + 42 * q^94 + 21 * q^95 + 15 * q^97 + q^98

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