LMFDB - Embedded newform 7623.2.a.cs.1.3

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:	$0$
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			1.3
Root			$0.879640$ 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)

$f(q)$	$=$	$q+0.120360 q^{2} -1.98551 q^{4} +2.80853 q^{5} -1.00000 q^{7} -0.479696 q^{8} +O(q^{10})$	$q+0.120360 q^{2} -1.98551 q^{4} +2.80853 q^{5} -1.00000 q^{7} -0.479696 q^{8} +0.338034 q^{10} -1.07967 q^{13} -0.120360 q^{14} +3.91329 q^{16} +6.95828 q^{17} -7.54411 q^{19} -5.57636 q^{20} -4.82552 q^{23} +2.88781 q^{25} -0.129949 q^{26} +1.98551 q^{28} +1.22726 q^{29} -8.07409 q^{31} +1.43040 q^{32} +0.837498 q^{34} -2.80853 q^{35} +1.53525 q^{37} -0.908009 q^{38} -1.34724 q^{40} +9.29986 q^{41} +5.23402 q^{43} -0.580799 q^{46} +1.89387 q^{47} +1.00000 q^{49} +0.347577 q^{50} +2.14371 q^{52} +3.82552 q^{53} +0.479696 q^{56} +0.147713 q^{58} +6.66349 q^{59} +9.79952 q^{61} -0.971796 q^{62} -7.65442 q^{64} -3.03229 q^{65} -2.06100 q^{67} -13.8158 q^{68} -0.338034 q^{70} -12.5212 q^{71} -2.56708 q^{73} +0.184783 q^{74} +14.9789 q^{76} +15.3283 q^{79} +10.9906 q^{80} +1.11933 q^{82} +2.04602 q^{83} +19.5425 q^{85} +0.629967 q^{86} -4.76119 q^{89} +1.07967 q^{91} +9.58114 q^{92} +0.227945 q^{94} -21.1878 q^{95} +9.11512 q^{97} +0.120360 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.120360	0.0851073	0.0425536	−	0.999094i	$-0.486451\pi$
$2$	0.120360	0.0851073	0.0425536	+	0.999094i	$0.486451\pi$
$3$	0	0
$3$	0	0
$4$	−1.98551	−0.992757
$5$	2.80853	1.25601	0.628005	−	0.778209i	$-0.283872\pi$
$5$	2.80853	1.25601	0.628005	+	0.778209i	$0.283872\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.479696	−0.169598
$9$	0	0
$10$	0.338034	0.106896
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.07967	−0.299448	−0.149724	−	0.988728i	$-0.547838\pi$
$13$	−1.07967	−0.299448	−0.149724	+	0.988728i	$0.547838\pi$
$14$	−0.120360	−0.0321675
$15$	0	0
$16$	3.91329	0.978323
$17$	6.95828	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$17$	6.95828	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$18$	0	0
$19$	−7.54411	−1.73074	−0.865369	−	0.501135i	$-0.832916\pi$
$19$	−7.54411	−1.73074	−0.865369	+	0.501135i	$0.832916\pi$
$20$	−5.57636	−1.24691
$21$	0	0
$22$	0	0
$23$	−4.82552	−1.00619	−0.503095	−	0.864231i	$-0.667806\pi$
$23$	−4.82552	−1.00619	−0.503095	+	0.864231i	$0.667806\pi$
$24$	0	0
$25$	2.88781	0.577563
$26$	−0.129949	−0.0254852
$27$	0	0
$28$	1.98551	0.375227
$29$	1.22726	0.227897	0.113949	−	0.993487i	$-0.463650\pi$
$29$	1.22726	0.227897	0.113949	+	0.993487i	$0.463650\pi$
$30$	0	0
$31$	−8.07409	−1.45015	−0.725074	−	0.688671i	$-0.758195\pi$
$31$	−8.07409	−1.45015	−0.725074	+	0.688671i	$0.758195\pi$
$32$	1.43040	0.252861
$33$	0	0
$34$	0.837498	0.143630
$35$	−2.80853	−0.474727
$36$	0	0
$37$	1.53525	0.252394	0.126197	−	0.992005i	$-0.459723\pi$
$37$	1.53525	0.252394	0.126197	+	0.992005i	$0.459723\pi$
$38$	−0.908009	−0.147298
$39$	0	0
$40$	−1.34724	−0.213017
$41$	9.29986	1.45239	0.726197	−	0.687487i	$-0.241286\pi$
$41$	9.29986	1.45239	0.726197	+	0.687487i	$0.241286\pi$
$42$	0	0
$43$	5.23402	0.798181	0.399091	−	0.916912i	$-0.369326\pi$
$43$	5.23402	0.798181	0.399091	+	0.916912i	$0.369326\pi$
$44$	0	0
$45$	0	0
$46$	−0.580799	−0.0856341
$47$	1.89387	0.276249	0.138124	−	0.990415i	$-0.455893\pi$
$47$	1.89387	0.276249	0.138124	+	0.990415i	$0.455893\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.347577	0.0491548
$51$	0	0
$52$	2.14371	0.297279
$53$	3.82552	0.525476	0.262738	−	0.964867i	$-0.415375\pi$
$53$	3.82552	0.525476	0.262738	+	0.964867i	$0.415375\pi$
$54$	0	0
$55$	0	0
$56$	0.479696	0.0641021
$57$	0	0
$58$	0.147713	0.0193957
$59$	6.66349	0.867513	0.433756	−	0.901030i	$-0.357188\pi$
$59$	6.66349	0.867513	0.433756	+	0.901030i	$0.357188\pi$
$60$	0	0
$61$	9.79952	1.25470	0.627350	−	0.778737i	$-0.284139\pi$
$61$	9.79952	1.25470	0.627350	+	0.778737i	$0.284139\pi$
$62$	−0.971796	−0.123418
$63$	0	0
$64$	−7.65442	−0.956802
$65$	−3.03229	−0.376110
$66$	0	0
$67$	−2.06100	−0.251792	−0.125896	−	0.992043i	$-0.540181\pi$
$67$	−2.06100	−0.251792	−0.125896	+	0.992043i	$0.540181\pi$
$68$	−13.8158	−1.67541
$69$	0	0
$70$	−0.338034	−0.0404028
$71$	−12.5212	−1.48599	−0.742996	−	0.669296i	$-0.766596\pi$
$71$	−12.5212	−1.48599	−0.742996	+	0.669296i	$0.766596\pi$
$72$	0	0
$73$	−2.56708	−0.300454	−0.150227	−	0.988652i	$-0.548000\pi$
$73$	−2.56708	−0.300454	−0.150227	+	0.988652i	$0.548000\pi$
$74$	0.184783	0.0214806
$75$	0	0
$76$	14.9789	1.71820
$77$	0	0
$78$	0	0
$79$	15.3283	1.72457	0.862287	−	0.506420i	$-0.169031\pi$
$79$	15.3283	1.72457	0.862287	+	0.506420i	$0.169031\pi$
$80$	10.9906	1.22878
$81$	0	0
$82$	1.11933	0.123609
$83$	2.04602	0.224580	0.112290	−	0.993675i	$-0.464181\pi$
$83$	2.04602	0.224580	0.112290	+	0.993675i	$0.464181\pi$
$84$	0	0
$85$	19.5425	2.11968
$86$	0.629967	0.0679310
$87$	0	0
$88$	0	0
$89$	−4.76119	−0.504685	−0.252342	−	0.967638i	$-0.581201\pi$
$89$	−4.76119	−0.504685	−0.252342	+	0.967638i	$0.581201\pi$
$90$	0	0
$91$	1.07967	0.113181
$92$	9.58114	0.998902
$93$	0	0
$94$	0.227945	0.0235108
$95$	−21.1878	−2.17383
$96$	0	0
$97$	9.11512	0.925500	0.462750	−	0.886489i	$-0.346863\pi$
$97$	9.11512	0.925500	0.462750	+	0.886489i	$0.346863\pi$
$98$	0.120360	0.0121582
$99$	0	0
$100$	−5.73379	−0.573379
$101$	−4.74385	−0.472031	−0.236015	−	0.971749i	$-0.575842\pi$
$101$	−4.74385	−0.472031	−0.236015	+	0.971749i	$0.575842\pi$
$102$	0	0
$103$	0.350901	0.0345753	0.0172876	−	0.999851i	$-0.494497\pi$
$103$	0.350901	0.0345753	0.0172876	+	0.999851i	$0.494497\pi$
$104$	0.517915	0.0507858
$105$	0	0
$106$	0.460439	0.0447218
$107$	2.54774	0.246299	0.123149	−	0.992388i	$-0.460701\pi$
$107$	2.54774	0.246299	0.123149	+	0.992388i	$0.460701\pi$
$108$	0	0
$109$	3.81522	0.365432	0.182716	−	0.983166i	$-0.441511\pi$
$109$	3.81522	0.365432	0.182716	+	0.983166i	$0.441511\pi$
$110$	0	0
$111$	0	0
$112$	−3.91329	−0.369771
$113$	−2.31468	−0.217747	−0.108873	−	0.994056i	$-0.534724\pi$
$113$	−2.31468	−0.217747	−0.108873	+	0.994056i	$0.534724\pi$
$114$	0	0
$115$	−13.5526	−1.26379
$116$	−2.43675	−0.226246
$117$	0	0
$118$	0.802017	0.0738316
$119$	−6.95828	−0.637865
$120$	0	0
$121$	0	0
$122$	1.17947	0.106784
$123$	0	0
$124$	16.0312	1.43964
$125$	−5.93213	−0.530586
$126$	0	0
$127$	−9.84644	−0.873730	−0.436865	−	0.899527i	$-0.643911\pi$
$127$	−9.84644	−0.873730	−0.436865	+	0.899527i	$0.643911\pi$
$128$	−3.78208	−0.334291
$129$	0	0
$130$	−0.364966	−0.0320097
$131$	16.3782	1.43097	0.715484	−	0.698629i	$-0.246206\pi$
$131$	16.3782	1.43097	0.715484	+	0.698629i	$0.246206\pi$
$132$	0	0
$133$	7.54411	0.654158
$134$	−0.248062	−0.0214293
$135$	0	0
$136$	−3.33786	−0.286219
$137$	14.2142	1.21440	0.607199	−	0.794550i	$-0.292293\pi$
$137$	14.2142	1.21440	0.607199	+	0.794550i	$0.292293\pi$
$138$	0	0
$139$	9.17942	0.778588	0.389294	−	0.921114i	$-0.372719\pi$
$139$	9.17942	0.778588	0.389294	+	0.921114i	$0.372719\pi$
$140$	5.57636	0.471289
$141$	0	0
$142$	−1.50705	−0.126469
$143$	0	0
$144$	0	0
$145$	3.44680	0.286241
$146$	−0.308974	−0.0255708
$147$	0	0
$148$	−3.04827	−0.250566
$149$	11.6200	0.951947	0.475973	−	0.879460i	$-0.342096\pi$
$149$	11.6200	0.951947	0.475973	+	0.879460i	$0.342096\pi$
$150$	0	0
$151$	−17.9150	−1.45790	−0.728952	−	0.684565i	$-0.759992\pi$
$151$	−17.9150	−1.45790	−0.728952	+	0.684565i	$0.759992\pi$
$152$	3.61888	0.293530
$153$	0	0
$154$	0	0
$155$	−22.6763	−1.82140
$156$	0	0
$157$	7.42805	0.592823	0.296412	−	0.955060i	$-0.404210\pi$
$157$	7.42805	0.592823	0.296412	+	0.955060i	$0.404210\pi$
$158$	1.84492	0.146774
$159$	0	0
$160$	4.01730	0.317596
$161$	4.82552	0.380304
$162$	0	0
$163$	7.85296	0.615091	0.307546	−	0.951533i	$-0.400492\pi$
$163$	7.85296	0.615091	0.307546	+	0.951533i	$0.400492\pi$
$164$	−18.4650	−1.44187
$165$	0	0
$166$	0.246259	0.0191134
$167$	−5.40259	−0.418065	−0.209032	−	0.977909i	$-0.567031\pi$
$167$	−5.40259	−0.418065	−0.209032	+	0.977909i	$0.567031\pi$
$168$	0	0
$169$	−11.8343	−0.910331
$170$	2.35214	0.180401
$171$	0	0
$172$	−10.3922	−0.792400
$173$	−19.7211	−1.49937	−0.749685	−	0.661795i	$-0.769795\pi$
$173$	−19.7211	−1.49937	−0.749685	+	0.661795i	$0.769795\pi$
$174$	0	0
$175$	−2.88781	−0.218298
$176$	0	0
$177$	0	0
$178$	−0.573056	−0.0429524
$179$	23.3292	1.74370	0.871851	−	0.489770i	$-0.162919\pi$
$179$	23.3292	1.74370	0.871851	+	0.489770i	$0.162919\pi$
$180$	0	0
$181$	3.08500	0.229306	0.114653	−	0.993406i	$-0.463424\pi$
$181$	3.08500	0.229306	0.114653	+	0.993406i	$0.463424\pi$
$182$	0.129949	0.00963250
$183$	0	0
$184$	2.31478	0.170648
$185$	4.31180	0.317010
$186$	0	0
$187$	0	0
$188$	−3.76030	−0.274248
$189$	0	0
$190$	−2.55017	−0.185008
$191$	−12.5715	−0.909640	−0.454820	−	0.890583i	$-0.650296\pi$
$191$	−12.5715	−0.909640	−0.454820	+	0.890583i	$0.650296\pi$
$192$	0	0
$193$	20.6685	1.48775	0.743877	−	0.668317i	$-0.232985\pi$
$193$	20.6685	1.48775	0.743877	+	0.668317i	$0.232985\pi$
$194$	1.09709	0.0787668
$195$	0	0
$196$	−1.98551	−0.141822
$197$	6.27954	0.447399	0.223699	−	0.974658i	$-0.428187\pi$
$197$	6.27954	0.447399	0.223699	+	0.974658i	$0.428187\pi$
$198$	0	0
$199$	−6.86896	−0.486927	−0.243464	−	0.969910i	$-0.578284\pi$
$199$	−6.86896	−0.486927	−0.243464	+	0.969910i	$0.578284\pi$
$200$	−1.38527	−0.0979536
$201$	0	0
$202$	−0.570969	−0.0401733
$203$	−1.22726	−0.0861370
$204$	0	0
$205$	26.1189	1.82422
$206$	0.0422344	0.00294261
$207$	0	0
$208$	−4.22508	−0.292957
$209$	0	0
$210$	0	0
$211$	−10.2586	−0.706232	−0.353116	−	0.935580i	$-0.614878\pi$
$211$	−10.2586	−0.706232	−0.353116	+	0.935580i	$0.614878\pi$
$212$	−7.59562	−0.521669
$213$	0	0
$214$	0.306645	0.0209618
$215$	14.6999	1.00252
$216$	0	0
$217$	8.07409	0.548105
$218$	0.459199	0.0311009
$219$	0	0
$220$	0	0
$221$	−7.51268	−0.505358
$222$	0	0
$223$	−13.5221	−0.905506	−0.452753	−	0.891636i	$-0.649558\pi$
$223$	−13.5221	−0.905506	−0.452753	+	0.891636i	$0.649558\pi$
$224$	−1.43040	−0.0955723
$225$	0	0
$226$	−0.278595	−0.0185319
$227$	13.5764	0.901100	0.450550	−	0.892751i	$-0.351228\pi$
$227$	13.5764	0.901100	0.450550	+	0.892751i	$0.351228\pi$
$228$	0	0
$229$	−1.45296	−0.0960141	−0.0480070	−	0.998847i	$-0.515287\pi$
$229$	−1.45296	−0.0960141	−0.0480070	+	0.998847i	$0.515287\pi$
$230$	−1.63119	−0.107557
$231$	0	0
$232$	−0.588713	−0.0386509
$233$	8.20387	0.537453	0.268727	−	0.963216i	$-0.413397\pi$
$233$	8.20387	0.537453	0.268727	+	0.963216i	$0.413397\pi$
$234$	0	0
$235$	5.31897	0.346971
$236$	−13.2305	−0.861229
$237$	0	0
$238$	−0.837498	−0.0542870
$239$	10.3835	0.671655	0.335827	−	0.941924i	$-0.390984\pi$
$239$	10.3835	0.671655	0.335827	+	0.941924i	$0.390984\pi$
$240$	0	0
$241$	8.20445	0.528495	0.264248	−	0.964455i	$-0.414876\pi$
$241$	8.20445	0.528495	0.264248	+	0.964455i	$0.414876\pi$
$242$	0	0
$243$	0	0
$244$	−19.4571	−1.24561
$245$	2.80853	0.179430
$246$	0	0
$247$	8.14519	0.518266
$248$	3.87311	0.245943
$249$	0	0
$250$	−0.713990	−0.0451567
$251$	16.4452	1.03801	0.519005	−	0.854771i	$-0.326302\pi$
$251$	16.4452	1.03801	0.519005	+	0.854771i	$0.326302\pi$
$252$	0	0
$253$	0	0
$254$	−1.18512	−0.0743608
$255$	0	0
$256$	14.8536	0.928352
$257$	12.0440	0.751282	0.375641	−	0.926765i	$-0.377423\pi$
$257$	12.0440	0.751282	0.375641	+	0.926765i	$0.377423\pi$
$258$	0	0
$259$	−1.53525	−0.0953960
$260$	6.02066	0.373385
$261$	0	0
$262$	1.97127	0.121786
$263$	1.87602	0.115681	0.0578403	−	0.998326i	$-0.481579\pi$
$263$	1.87602	0.115681	0.0578403	+	0.998326i	$0.481579\pi$
$264$	0	0
$265$	10.7441	0.660003
$266$	0.908009	0.0556736
$267$	0	0
$268$	4.09215	0.249968
$269$	14.0290	0.855362	0.427681	−	0.903930i	$-0.359331\pi$
$269$	14.0290	0.855362	0.427681	+	0.903930i	$0.359331\pi$
$270$	0	0
$271$	−1.35328	−0.0822056	−0.0411028	−	0.999155i	$-0.513087\pi$
$271$	−1.35328	−0.0822056	−0.0411028	+	0.999155i	$0.513087\pi$
$272$	27.2298	1.65105
$273$	0	0
$274$	1.71081	0.103354
$275$	0	0
$276$	0	0
$277$	13.3835	0.804138	0.402069	−	0.915609i	$-0.368291\pi$
$277$	13.3835	0.804138	0.402069	+	0.915609i	$0.368291\pi$
$278$	1.10483	0.0662635
$279$	0	0
$280$	1.34724	0.0805129
$281$	2.31887	0.138332	0.0691661	−	0.997605i	$-0.477966\pi$
$281$	2.31887	0.138332	0.0691661	+	0.997605i	$0.477966\pi$
$282$	0	0
$283$	22.2679	1.32369	0.661845	−	0.749641i	$-0.269774\pi$
$283$	22.2679	1.32369	0.661845	+	0.749641i	$0.269774\pi$
$284$	24.8610	1.47523
$285$	0	0
$286$	0	0
$287$	−9.29986	−0.548953
$288$	0	0
$289$	31.4177	1.84810
$290$	0.414857	0.0243612
$291$	0	0
$292$	5.09698	0.298278
$293$	−2.43981	−0.142535	−0.0712675	−	0.997457i	$-0.522704\pi$
$293$	−2.43981	−0.142535	−0.0712675	+	0.997457i	$0.522704\pi$
$294$	0	0
$295$	18.7146	1.08961
$296$	−0.736455	−0.0428056
$297$	0	0
$298$	1.39858	0.0810176
$299$	5.20999	0.301301
$300$	0	0
$301$	−5.23402	−0.301684
$302$	−2.15625	−0.124078
$303$	0	0
$304$	−29.5223	−1.69322
$305$	27.5222	1.57592
$306$	0	0
$307$	8.89055	0.507410	0.253705	−	0.967282i	$-0.418351\pi$
$307$	8.89055	0.507410	0.253705	+	0.967282i	$0.418351\pi$
$308$	0	0
$309$	0	0
$310$	−2.72931	−0.155015
$311$	11.8347	0.671085	0.335543	−	0.942025i	$-0.391080\pi$
$311$	11.8347	0.671085	0.335543	+	0.942025i	$0.391080\pi$
$312$	0	0
$313$	−29.5406	−1.66973	−0.834865	−	0.550454i	$-0.814455\pi$
$313$	−29.5406	−1.66973	−0.834865	+	0.550454i	$0.814455\pi$
$314$	0.894039	0.0504536
$315$	0	0
$316$	−30.4346	−1.71208
$317$	25.1545	1.41282	0.706409	−	0.707804i	$-0.250314\pi$
$317$	25.1545	1.41282	0.706409	+	0.707804i	$0.250314\pi$
$318$	0	0
$319$	0	0
$320$	−21.4976	−1.20175
$321$	0	0
$322$	0.580799	0.0323667
$323$	−52.4941	−2.92085
$324$	0	0
$325$	−3.11790	−0.172950
$326$	0.945181	0.0523488
$327$	0	0
$328$	−4.46110	−0.246323
$329$	−1.89387	−0.104412
$330$	0	0
$331$	−9.46333	−0.520152	−0.260076	−	0.965588i	$-0.583748\pi$
$331$	−9.46333	−0.520152	−0.260076	+	0.965588i	$0.583748\pi$
$332$	−4.06241	−0.222954
$333$	0	0
$334$	−0.650255	−0.0355804
$335$	−5.78838	−0.316253
$336$	0	0
$337$	17.2248	0.938297	0.469148	−	0.883119i	$-0.344561\pi$
$337$	17.2248	0.938297	0.469148	+	0.883119i	$0.344561\pi$
$338$	−1.42438	−0.0774758
$339$	0	0
$340$	−38.8019	−2.10433
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−2.51074	−0.135370
$345$	0	0
$346$	−2.37363	−0.127607
$347$	−8.73485	−0.468911	−0.234456	−	0.972127i	$-0.575331\pi$
$347$	−8.73485	−0.468911	−0.234456	+	0.972127i	$0.575331\pi$
$348$	0	0
$349$	29.9851	1.60506	0.802532	−	0.596609i	$-0.203486\pi$
$349$	29.9851	1.60506	0.802532	+	0.596609i	$0.203486\pi$
$350$	−0.347577	−0.0185788
$351$	0	0
$352$	0	0
$353$	−7.31999	−0.389604	−0.194802	−	0.980843i	$-0.562406\pi$
$353$	−7.31999	−0.389604	−0.194802	+	0.980843i	$0.562406\pi$
$354$	0	0
$355$	−35.1661	−1.86642
$356$	9.45340	0.501029
$357$	0	0
$358$	2.80789	0.148402
$359$	4.96996	0.262304	0.131152	−	0.991362i	$-0.458132\pi$
$359$	4.96996	0.262304	0.131152	+	0.991362i	$0.458132\pi$
$360$	0	0
$361$	37.9137	1.99546
$362$	0.371311	0.0195156
$363$	0	0
$364$	−2.14371	−0.112361
$365$	−7.20971	−0.377374
$366$	0	0
$367$	−14.2042	−0.741455	−0.370727	−	0.928742i	$-0.620892\pi$
$367$	−14.2042	−0.741455	−0.370727	+	0.928742i	$0.620892\pi$
$368$	−18.8837	−0.984379
$369$	0	0
$370$	0.518967	0.0269798
$371$	−3.82552	−0.198611
$372$	0	0
$373$	−8.97781	−0.464853	−0.232427	−	0.972614i	$-0.574667\pi$
$373$	−8.97781	−0.464853	−0.232427	+	0.972614i	$0.574667\pi$
$374$	0	0
$375$	0	0
$376$	−0.908480	−0.0468513
$377$	−1.32504	−0.0682433
$378$	0	0
$379$	10.7896	0.554223	0.277112	−	0.960838i	$-0.410623\pi$
$379$	10.7896	0.554223	0.277112	+	0.960838i	$0.410623\pi$
$380$	42.0687	2.15808
$381$	0	0
$382$	−1.51310	−0.0774170
$383$	23.5100	1.20130	0.600652	−	0.799511i	$-0.294908\pi$
$383$	23.5100	1.20130	0.600652	+	0.799511i	$0.294908\pi$
$384$	0	0
$385$	0	0
$386$	2.48766	0.126619
$387$	0	0
$388$	−18.0982	−0.918797
$389$	11.8745	0.602063	0.301032	−	0.953614i	$-0.402669\pi$
$389$	11.8745	0.602063	0.301032	+	0.953614i	$0.402669\pi$
$390$	0	0
$391$	−33.5773	−1.69808
$392$	−0.479696	−0.0242283
$393$	0	0
$394$	0.755804	0.0380769
$395$	43.0501	2.16608
$396$	0	0
$397$	−23.7264	−1.19079	−0.595397	−	0.803431i	$-0.703005\pi$
$397$	−23.7264	−1.19079	−0.595397	+	0.803431i	$0.703005\pi$
$398$	−0.826747	−0.0414411
$399$	0	0
$400$	11.3009	0.565043
$401$	3.80121	0.189823	0.0949117	−	0.995486i	$-0.469743\pi$
$401$	3.80121	0.189823	0.0949117	+	0.995486i	$0.469743\pi$
$402$	0	0
$403$	8.71738	0.434244
$404$	9.41898	0.468612
$405$	0	0
$406$	−0.147713	−0.00733089
$407$	0	0
$408$	0	0
$409$	5.98291	0.295836	0.147918	−	0.989000i	$-0.452743\pi$
$409$	5.98291	0.295836	0.147918	+	0.989000i	$0.452743\pi$
$410$	3.14367	0.155255
$411$	0	0
$412$	−0.696718	−0.0343248
$413$	−6.66349	−0.327889
$414$	0	0
$415$	5.74631	0.282075
$416$	−1.54436	−0.0757185
$417$	0	0
$418$	0	0
$419$	4.16889	0.203664	0.101832	−	0.994802i	$-0.467530\pi$
$419$	4.16889	0.203664	0.101832	+	0.994802i	$0.467530\pi$
$420$	0	0
$421$	23.4555	1.14315	0.571575	−	0.820550i	$-0.306333\pi$
$421$	23.4555	1.14315	0.571575	+	0.820550i	$0.306333\pi$
$422$	−1.23473	−0.0601055
$423$	0	0
$424$	−1.83509	−0.0891197
$425$	20.0942	0.974713
$426$	0	0
$427$	−9.79952	−0.474232
$428$	−5.05856	−0.244515
$429$	0	0
$430$	1.76928	0.0853221
$431$	37.2730	1.79538	0.897689	−	0.440630i	$-0.145245\pi$
$431$	37.2730	1.79538	0.897689	+	0.440630i	$0.145245\pi$
$432$	0	0
$433$	−22.7863	−1.09504	−0.547520	−	0.836793i	$-0.684428\pi$
$433$	−22.7863	−1.09504	−0.547520	+	0.836793i	$0.684428\pi$
$434$	0.971796	0.0466477
$435$	0	0
$436$	−7.57517	−0.362785
$437$	36.4043	1.74145
$438$	0	0
$439$	1.42974	0.0682379	0.0341189	−	0.999418i	$-0.489137\pi$
$439$	1.42974	0.0682379	0.0341189	+	0.999418i	$0.489137\pi$
$440$	0	0
$441$	0	0
$442$	−0.904225	−0.0430096
$443$	26.4301	1.25573	0.627866	−	0.778321i	$-0.283928\pi$
$443$	26.4301	1.25573	0.627866	+	0.778321i	$0.283928\pi$
$444$	0	0
$445$	−13.3719	−0.633890
$446$	−1.62752	−0.0770651
$447$	0	0
$448$	7.65442	0.361637
$449$	−14.0870	−0.664806	−0.332403	−	0.943137i	$-0.607859\pi$
$449$	−14.0870	−0.664806	−0.332403	+	0.943137i	$0.607859\pi$
$450$	0	0
$451$	0	0
$452$	4.59583	0.216170
$453$	0	0
$454$	1.63406	0.0766902
$455$	3.03229	0.142156
$456$	0	0
$457$	−29.4829	−1.37915	−0.689575	−	0.724214i	$-0.742203\pi$
$457$	−29.4829	−1.37915	−0.689575	+	0.724214i	$0.742203\pi$
$458$	−0.174878	−0.00817150
$459$	0	0
$460$	26.9089	1.25463
$461$	31.7282	1.47773	0.738866	−	0.673853i	$-0.235362\pi$
$461$	31.7282	1.47773	0.738866	+	0.673853i	$0.235362\pi$
$462$	0	0
$463$	−12.7839	−0.594117	−0.297059	−	0.954859i	$-0.596006\pi$
$463$	−12.7839	−0.594117	−0.297059	+	0.954859i	$0.596006\pi$
$464$	4.80264	0.222957
$465$	0	0
$466$	0.987417	0.0457412
$467$	−29.5768	−1.36865	−0.684325	−	0.729177i	$-0.739903\pi$
$467$	−29.5768	−1.36865	−0.684325	+	0.729177i	$0.739903\pi$
$468$	0	0
$469$	2.06100	0.0951683
$470$	0.640191	0.0295298
$471$	0	0
$472$	−3.19645	−0.147129
$473$	0	0
$474$	0	0
$475$	−21.7860	−0.999610
$476$	13.8158	0.633245
$477$	0	0
$478$	1.24976	0.0571627
$479$	−28.5574	−1.30482	−0.652411	−	0.757866i	$-0.726242\pi$
$479$	−28.5574	−1.30482	−0.652411	+	0.757866i	$0.726242\pi$
$480$	0	0
$481$	−1.65757	−0.0755788
$482$	0.987487	0.0449788
$483$	0	0
$484$	0	0
$485$	25.6000	1.16244
$486$	0	0
$487$	1.05828	0.0479552	0.0239776	−	0.999712i	$-0.492367\pi$
$487$	1.05828	0.0479552	0.0239776	+	0.999712i	$0.492367\pi$
$488$	−4.70079	−0.212795
$489$	0	0
$490$	0.338034	0.0152708
$491$	18.4489	0.832586	0.416293	−	0.909230i	$-0.363329\pi$
$491$	18.4489	0.832586	0.416293	+	0.909230i	$0.363329\pi$
$492$	0	0
$493$	8.53965	0.384606
$494$	0.980354	0.0441082
$495$	0	0
$496$	−31.5962	−1.41871
$497$	12.5212	0.561652
$498$	0	0
$499$	31.7293	1.42040	0.710199	−	0.704001i	$-0.248605\pi$
$499$	31.7293	1.42040	0.710199	+	0.704001i	$0.248605\pi$
$500$	11.7783	0.526742
$501$	0	0
$502$	1.97934	0.0883423
$503$	−29.0283	−1.29431	−0.647154	−	0.762359i	$-0.724041\pi$
$503$	−29.0283	−1.29431	−0.647154	+	0.762359i	$0.724041\pi$
$504$	0	0
$505$	−13.3232	−0.592876
$506$	0	0
$507$	0	0
$508$	19.5502	0.867401
$509$	−2.88831	−0.128022	−0.0640110	−	0.997949i	$-0.520389\pi$
$509$	−2.88831	−0.128022	−0.0640110	+	0.997949i	$0.520389\pi$
$510$	0	0
$511$	2.56708	0.113561
$512$	9.35193	0.413301
$513$	0	0
$514$	1.44961	0.0639396
$515$	0.985513	0.0434269
$516$	0	0
$517$	0	0
$518$	−0.184783	−0.00811890
$519$	0	0
$520$	1.45458	0.0637875
$521$	−31.6708	−1.38752	−0.693762	−	0.720204i	$-0.744048\pi$
$521$	−31.6708	−1.38752	−0.693762	+	0.720204i	$0.744048\pi$
$522$	0	0
$523$	16.0380	0.701295	0.350647	−	0.936508i	$-0.385962\pi$
$523$	16.0380	0.701295	0.350647	+	0.936508i	$0.385962\pi$
$524$	−32.5191	−1.42060
$525$	0	0
$526$	0.225798	0.00984526
$527$	−56.1818	−2.44732
$528$	0	0
$529$	0.285644	0.0124193
$530$	1.29316	0.0561711
$531$	0	0
$532$	−14.9789	−0.649419
$533$	−10.0408	−0.434916
$534$	0	0
$535$	7.15538	0.309354
$536$	0.988655	0.0427034
$537$	0	0
$538$	1.68853	0.0727976
$539$	0	0
$540$	0	0
$541$	−13.5946	−0.584480	−0.292240	−	0.956345i	$-0.594401\pi$
$541$	−13.5946	−0.584480	−0.292240	+	0.956345i	$0.594401\pi$
$542$	−0.162880	−0.00699630
$543$	0	0
$544$	9.95310	0.426735
$545$	10.7151	0.458986
$546$	0	0
$547$	8.82486	0.377324	0.188662	−	0.982042i	$-0.439585\pi$
$547$	8.82486	0.377324	0.188662	+	0.982042i	$0.439585\pi$
$548$	−28.2224	−1.20560
$549$	0	0
$550$	0	0
$551$	−9.25862	−0.394430
$552$	0	0
$553$	−15.3283	−0.651828
$554$	1.61084	0.0684380
$555$	0	0
$556$	−18.2259	−0.772949
$557$	33.9920	1.44029	0.720145	−	0.693824i	$-0.244075\pi$
$557$	33.9920	1.44029	0.720145	+	0.693824i	$0.244075\pi$
$558$	0	0
$559$	−5.65104	−0.239014
$560$	−10.9906	−0.464437
$561$	0	0
$562$	0.279099	0.0117731
$563$	−20.2256	−0.852406	−0.426203	−	0.904628i	$-0.640149\pi$
$563$	−20.2256	−0.852406	−0.426203	+	0.904628i	$0.640149\pi$
$564$	0	0
$565$	−6.50084	−0.273492
$566$	2.68016	0.112656
$567$	0	0
$568$	6.00637	0.252022
$569$	28.9330	1.21293	0.606467	−	0.795109i	$-0.292586\pi$
$569$	28.9330	1.21293	0.606467	+	0.795109i	$0.292586\pi$
$570$	0	0
$571$	7.51312	0.314414	0.157207	−	0.987566i	$-0.449751\pi$
$571$	7.51312	0.314414	0.157207	+	0.987566i	$0.449751\pi$
$572$	0	0
$573$	0	0
$574$	−1.11933	−0.0467199
$575$	−13.9352	−0.581138
$576$	0	0
$577$	34.3748	1.43104	0.715521	−	0.698592i	$-0.246190\pi$
$577$	34.3748	1.43104	0.715521	+	0.698592i	$0.246190\pi$
$578$	3.78143	0.157287
$579$	0	0
$580$	−6.84367	−0.284168
$581$	−2.04602	−0.0848834
$582$	0	0
$583$	0	0
$584$	1.23142	0.0509565
$585$	0	0
$586$	−0.293655	−0.0121308
$587$	−42.0392	−1.73514	−0.867571	−	0.497313i	$-0.834320\pi$
$587$	−42.0392	−1.73514	−0.867571	+	0.497313i	$0.834320\pi$
$588$	0	0
$589$	60.9118	2.50983
$590$	2.25249	0.0927333
$591$	0	0
$592$	6.00789	0.246923
$593$	29.2867	1.20266	0.601331	−	0.799000i	$-0.294637\pi$
$593$	29.2867	1.20266	0.601331	+	0.799000i	$0.294637\pi$
$594$	0	0
$595$	−19.5425	−0.801165
$596$	−23.0716	−0.945051
$597$	0	0
$598$	0.627074	0.0256430
$599$	15.1577	0.619327	0.309663	−	0.950846i	$-0.399784\pi$
$599$	15.1577	0.619327	0.309663	+	0.950846i	$0.399784\pi$
$600$	0	0
$601$	−31.8735	−1.30015	−0.650073	−	0.759872i	$-0.725262\pi$
$601$	−31.8735	−1.30015	−0.650073	+	0.759872i	$0.725262\pi$
$602$	−0.629967	−0.0256755
$603$	0	0
$604$	35.5705	1.44734
$605$	0	0
$606$	0	0
$607$	20.1463	0.817714	0.408857	−	0.912598i	$-0.365927\pi$
$607$	20.1463	0.817714	0.408857	+	0.912598i	$0.365927\pi$
$608$	−10.7911	−0.437635
$609$	0	0
$610$	3.31257	0.134122
$611$	−2.04476	−0.0827220
$612$	0	0
$613$	3.89132	0.157169	0.0785844	−	0.996907i	$-0.474960\pi$
$613$	3.89132	0.157169	0.0785844	+	0.996907i	$0.474960\pi$
$614$	1.07007	0.0431843
$615$	0	0
$616$	0	0
$617$	−28.6122	−1.15189	−0.575943	−	0.817490i	$-0.695365\pi$
$617$	−28.6122	−1.15189	−0.575943	+	0.817490i	$0.695365\pi$
$618$	0	0
$619$	−26.1546	−1.05124	−0.525621	−	0.850719i	$-0.676167\pi$
$619$	−26.1546	−1.05124	−0.525621	+	0.850719i	$0.676167\pi$
$620$	45.0240	1.80821
$621$	0	0
$622$	1.42443	0.0571143
$623$	4.76119	0.190753
$624$	0	0
$625$	−31.0996	−1.24398
$626$	−3.55550	−0.142106
$627$	0	0
$628$	−14.7485	−0.588529
$629$	10.6827	0.425948
$630$	0	0
$631$	−5.68272	−0.226225	−0.113113	−	0.993582i	$-0.536082\pi$
$631$	−5.68272	−0.226225	−0.113113	+	0.993582i	$0.536082\pi$
$632$	−7.35295	−0.292485
$633$	0	0
$634$	3.02759	0.120241
$635$	−27.6540	−1.09741
$636$	0	0
$637$	−1.07967	−0.0427782
$638$	0	0
$639$	0	0
$640$	−10.6221	−0.419874
$641$	20.7292	0.818756	0.409378	−	0.912365i	$-0.365746\pi$
$641$	20.7292	0.818756	0.409378	+	0.912365i	$0.365746\pi$
$642$	0	0
$643$	21.2756	0.839029	0.419515	−	0.907749i	$-0.362200\pi$
$643$	21.2756	0.839029	0.419515	+	0.907749i	$0.362200\pi$
$644$	−9.58114	−0.377550
$645$	0	0
$646$	−6.31818	−0.248586
$647$	17.2718	0.679023	0.339512	−	0.940602i	$-0.389738\pi$
$647$	17.2718	0.679023	0.339512	+	0.940602i	$0.389738\pi$
$648$	0	0
$649$	0	0
$650$	−0.375270	−0.0147193
$651$	0	0
$652$	−15.5922	−0.610636
$653$	−41.0788	−1.60754	−0.803768	−	0.594942i	$-0.797175\pi$
$653$	−41.0788	−1.60754	−0.803768	+	0.594942i	$0.797175\pi$
$654$	0	0
$655$	45.9985	1.79731
$656$	36.3930	1.42091
$657$	0	0
$658$	−0.227945	−0.00888624
$659$	6.00410	0.233887	0.116943	−	0.993139i	$-0.462690\pi$
$659$	6.00410	0.233887	0.116943	+	0.993139i	$0.462690\pi$
$660$	0	0
$661$	−1.83502	−0.0713739	−0.0356870	−	0.999363i	$-0.511362\pi$
$661$	−1.83502	−0.0713739	−0.0356870	+	0.999363i	$0.511362\pi$
$662$	−1.13901	−0.0442687
$663$	0	0
$664$	−0.981469	−0.0380884
$665$	21.1878	0.821629
$666$	0	0
$667$	−5.92218	−0.229308
$668$	10.7269	0.415037
$669$	0	0
$670$	−0.696689	−0.0269154
$671$	0	0
$672$	0	0
$673$	−44.4403	−1.71305	−0.856524	−	0.516107i	$-0.827381\pi$
$673$	−44.4403	−1.71305	−0.856524	+	0.516107i	$0.827381\pi$
$674$	2.07318	0.0798559
$675$	0	0
$676$	23.4972	0.903737
$677$	15.5471	0.597525	0.298762	−	0.954327i	$-0.403426\pi$
$677$	15.5471	0.597525	0.298762	+	0.954327i	$0.403426\pi$
$678$	0	0
$679$	−9.11512	−0.349806
$680$	−9.37447	−0.359494
$681$	0	0
$682$	0	0
$683$	36.8979	1.41186	0.705930	−	0.708282i	$-0.250529\pi$
$683$	36.8979	1.41186	0.705930	+	0.708282i	$0.250529\pi$
$684$	0	0
$685$	39.9208	1.52530
$686$	−0.120360	−0.00459536
$687$	0	0
$688$	20.4823	0.780879
$689$	−4.13032	−0.157352
$690$	0	0
$691$	−18.6726	−0.710339	−0.355169	−	0.934802i	$-0.615577\pi$
$691$	−18.6726	−0.710339	−0.355169	+	0.934802i	$0.615577\pi$
$692$	39.1566	1.48851
$693$	0	0
$694$	−1.05133	−0.0399078
$695$	25.7806	0.977915
$696$	0	0
$697$	64.7110	2.45111
$698$	3.60900	0.136603
$699$	0	0
$700$	5.73379	0.216717
$701$	−26.1328	−0.987022	−0.493511	−	0.869739i	$-0.664287\pi$
$701$	−26.1328	−0.987022	−0.493511	+	0.869739i	$0.664287\pi$
$702$	0	0
$703$	−11.5821	−0.436828
$704$	0	0
$705$	0	0
$706$	−0.881033	−0.0331581
$707$	4.74385	0.178411
$708$	0	0
$709$	−16.4449	−0.617602	−0.308801	−	0.951127i	$-0.599928\pi$
$709$	−16.4449	−0.617602	−0.308801	+	0.951127i	$0.599928\pi$
$710$	−4.23259	−0.158846
$711$	0	0
$712$	2.28392	0.0855936
$713$	38.9617	1.45913
$714$	0	0
$715$	0	0
$716$	−46.3204	−1.73107
$717$	0	0
$718$	0.598184	0.0223240
$719$	9.34913	0.348664	0.174332	−	0.984687i	$-0.444223\pi$
$719$	9.34913	0.348664	0.174332	+	0.984687i	$0.444223\pi$
$720$	0	0
$721$	−0.350901	−0.0130682
$722$	4.56328	0.169828
$723$	0	0
$724$	−6.12531	−0.227646
$725$	3.54411	0.131625
$726$	0	0
$727$	27.7523	1.02928	0.514638	−	0.857408i	$-0.327927\pi$
$727$	27.7523	1.02928	0.514638	+	0.857408i	$0.327927\pi$
$728$	−0.517915	−0.0191952
$729$	0	0
$730$	−0.867760	−0.0321173
$731$	36.4198	1.34704
$732$	0	0
$733$	−1.45630	−0.0537896	−0.0268948	−	0.999638i	$-0.508562\pi$
$733$	−1.45630	−0.0537896	−0.0268948	+	0.999638i	$0.508562\pi$
$734$	−1.70962	−0.0631032
$735$	0	0
$736$	−6.90240	−0.254426
$737$	0	0
$738$	0	0
$739$	27.5966	1.01516	0.507579	−	0.861605i	$-0.330541\pi$
$739$	27.5966	1.01516	0.507579	+	0.861605i	$0.330541\pi$
$740$	−8.56113	−0.314713
$741$	0	0
$742$	−0.460439	−0.0169033
$743$	27.9773	1.02639	0.513194	−	0.858273i	$-0.328462\pi$
$743$	27.9773	1.02639	0.513194	+	0.858273i	$0.328462\pi$
$744$	0	0
$745$	32.6350	1.19566
$746$	−1.08057	−0.0395624
$747$	0	0
$748$	0	0
$749$	−2.54774	−0.0930922
$750$	0	0
$751$	34.0957	1.24417	0.622085	−	0.782950i	$-0.286286\pi$
$751$	34.0957	1.24417	0.622085	+	0.782950i	$0.286286\pi$
$752$	7.41125	0.270260
$753$	0	0
$754$	−0.159482	−0.00580800
$755$	−50.3148	−1.83114
$756$	0	0
$757$	−39.7629	−1.44521	−0.722604	−	0.691262i	$-0.757055\pi$
$757$	−39.7629	−1.44521	−0.722604	+	0.691262i	$0.757055\pi$
$758$	1.29863	0.0471684
$759$	0	0
$760$	10.1637	0.368677
$761$	−3.82415	−0.138625	−0.0693127	−	0.997595i	$-0.522081\pi$
$761$	−3.82415	−0.138625	−0.0693127	+	0.997595i	$0.522081\pi$
$762$	0	0
$763$	−3.81522	−0.138120
$764$	24.9608	0.903051
$765$	0	0
$766$	2.82966	0.102240
$767$	−7.19440	−0.259775
$768$	0	0
$769$	19.6583	0.708897	0.354449	−	0.935075i	$-0.384669\pi$
$769$	19.6583	0.708897	0.354449	+	0.935075i	$0.384669\pi$
$770$	0	0
$771$	0	0
$772$	−41.0376	−1.47698
$773$	17.5966	0.632905	0.316452	−	0.948608i	$-0.397508\pi$
$773$	17.5966	0.632905	0.316452	+	0.948608i	$0.397508\pi$
$774$	0	0
$775$	−23.3165	−0.837552
$776$	−4.37249	−0.156963
$777$	0	0
$778$	1.42922	0.0512400
$779$	−70.1592	−2.51371
$780$	0	0
$781$	0	0
$782$	−4.04136	−0.144519
$783$	0	0
$784$	3.91329	0.139760
$785$	20.8619	0.744592
$786$	0	0
$787$	54.0967	1.92834	0.964169	−	0.265287i	$-0.0854667\pi$
$787$	54.0967	1.92834	0.964169	+	0.265287i	$0.0854667\pi$
$788$	−12.4681	−0.444158
$789$	0	0
$790$	5.18150	0.184349
$791$	2.31468	0.0823006
$792$	0	0
$793$	−10.5803	−0.375717
$794$	−2.85571	−0.101345
$795$	0	0
$796$	13.6384	0.483401
$797$	−36.5244	−1.29376	−0.646881	−	0.762591i	$-0.723927\pi$
$797$	−36.5244	−1.29376	−0.646881	+	0.762591i	$0.723927\pi$
$798$	0	0
$799$	13.1781	0.466206
$800$	4.13072	0.146043
$801$	0	0
$802$	0.457513	0.0161554
$803$	0	0
$804$	0	0
$805$	13.5526	0.477666
$806$	1.04922	0.0369573
$807$	0	0
$808$	2.27561	0.0800555
$809$	46.9354	1.65016	0.825081	−	0.565015i	$-0.191130\pi$
$809$	46.9354	1.65016	0.825081	+	0.565015i	$0.191130\pi$
$810$	0	0
$811$	12.6615	0.444605	0.222303	−	0.974978i	$-0.428643\pi$
$811$	12.6615	0.444605	0.222303	+	0.974978i	$0.428643\pi$
$812$	2.43675	0.0855131
$813$	0	0
$814$	0	0
$815$	22.0552	0.772561
$816$	0	0
$817$	−39.4861	−1.38144
$818$	0.720103	0.0251778
$819$	0	0
$820$	−51.8594	−1.81101
$821$	−49.2938	−1.72037	−0.860183	−	0.509986i	$-0.829651\pi$
$821$	−49.2938	−1.72037	−0.860183	+	0.509986i	$0.829651\pi$
$822$	0	0
$823$	19.2259	0.670173	0.335087	−	0.942187i	$-0.391234\pi$
$823$	19.2259	0.670173	0.335087	+	0.942187i	$0.391234\pi$
$824$	−0.168326	−0.00586390
$825$	0	0
$826$	−0.802017	−0.0279057
$827$	43.6419	1.51758	0.758789	−	0.651337i	$-0.225792\pi$
$827$	43.6419	1.51758	0.758789	+	0.651337i	$0.225792\pi$
$828$	0	0
$829$	−36.7072	−1.27489	−0.637447	−	0.770494i	$-0.720010\pi$
$829$	−36.7072	−1.27489	−0.637447	+	0.770494i	$0.720010\pi$
$830$	0.691625	0.0240067
$831$	0	0
$832$	8.26428	0.286512
$833$	6.95828	0.241090
$834$	0	0
$835$	−15.1733	−0.525094
$836$	0	0
$837$	0	0
$838$	0.501768	0.0173333
$839$	−40.4545	−1.39665	−0.698323	−	0.715783i	$-0.746070\pi$
$839$	−40.4545	−1.39665	−0.698323	+	0.715783i	$0.746070\pi$
$840$	0	0
$841$	−27.4938	−0.948063
$842$	2.82310	0.0972904
$843$	0	0
$844$	20.3686	0.701117
$845$	−33.2369	−1.14339
$846$	0	0
$847$	0	0
$848$	14.9704	0.514085
$849$	0	0
$850$	2.41854	0.0829552
$851$	−7.40840	−0.253957
$852$	0	0
$853$	−25.4948	−0.872927	−0.436463	−	0.899722i	$-0.643769\pi$
$853$	−25.4948	−0.872927	−0.436463	+	0.899722i	$0.643769\pi$
$854$	−1.17947	−0.0403606
$855$	0	0
$856$	−1.22214	−0.0417718
$857$	49.4756	1.69005	0.845027	−	0.534723i	$-0.179584\pi$
$857$	49.4756	1.69005	0.845027	+	0.534723i	$0.179584\pi$
$858$	0	0
$859$	21.6779	0.739641	0.369821	−	0.929103i	$-0.379419\pi$
$859$	21.6779	0.739641	0.369821	+	0.929103i	$0.379419\pi$
$860$	−29.1868	−0.995262
$861$	0	0
$862$	4.48618	0.152800
$863$	−13.6398	−0.464306	−0.232153	−	0.972679i	$-0.574577\pi$
$863$	−13.6398	−0.464306	−0.232153	+	0.972679i	$0.574577\pi$
$864$	0	0
$865$	−55.3873	−1.88322
$866$	−2.74256	−0.0931959
$867$	0	0
$868$	−16.0312	−0.544135
$869$	0	0
$870$	0	0
$871$	2.22521	0.0753985
$872$	−1.83014	−0.0619765
$873$	0	0
$874$	4.38161	0.148210
$875$	5.93213	0.200542
$876$	0	0
$877$	−29.1290	−0.983618	−0.491809	−	0.870703i	$-0.663664\pi$
$877$	−29.1290	−0.983618	−0.491809	+	0.870703i	$0.663664\pi$
$878$	0.172084	0.00580754
$879$	0	0
$880$	0	0
$881$	48.8256	1.64498	0.822488	−	0.568783i	$-0.192586\pi$
$881$	48.8256	1.64498	0.822488	+	0.568783i	$0.192586\pi$
$882$	0	0
$883$	−24.2131	−0.814837	−0.407419	−	0.913242i	$-0.633571\pi$
$883$	−24.2131	−0.814837	−0.407419	+	0.913242i	$0.633571\pi$
$884$	14.9165	0.501697
$885$	0	0
$886$	3.18113	0.106872
$887$	6.64749	0.223201	0.111600	−	0.993753i	$-0.464402\pi$
$887$	6.64749	0.223201	0.111600	+	0.993753i	$0.464402\pi$
$888$	0	0
$889$	9.84644	0.330239
$890$	−1.60944	−0.0539486
$891$	0	0
$892$	26.8483	0.898947
$893$	−14.2875	−0.478114
$894$	0	0
$895$	65.5205	2.19011
$896$	3.78208	0.126350
$897$	0	0
$898$	−1.69551	−0.0565799
$899$	−9.90903	−0.330485
$900$	0	0
$901$	26.6191	0.886809
$902$	0	0
$903$	0	0
$904$	1.11034	0.0369295
$905$	8.66431	0.288011
$906$	0	0
$907$	−8.37806	−0.278189	−0.139094	−	0.990279i	$-0.544419\pi$
$907$	−8.37806	−0.278189	−0.139094	+	0.990279i	$0.544419\pi$
$908$	−26.9562	−0.894573
$909$	0	0
$910$	0.364966	0.0120985
$911$	18.7868	0.622434	0.311217	−	0.950339i	$-0.399263\pi$
$911$	18.7868	0.622434	0.311217	+	0.950339i	$0.399263\pi$
$912$	0	0
$913$	0	0
$914$	−3.54856	−0.117376
$915$	0	0
$916$	2.88486	0.0953186
$917$	−16.3782	−0.540855
$918$	0	0
$919$	−18.0039	−0.593894	−0.296947	−	0.954894i	$-0.595968\pi$
$919$	−18.0039	−0.593894	−0.296947	+	0.954894i	$0.595968\pi$
$920$	6.50113	0.214336
$921$	0	0
$922$	3.81881	0.125766
$923$	13.5188	0.444977
$924$	0	0
$925$	4.43353	0.145773
$926$	−1.53867	−0.0505637
$927$	0	0
$928$	1.75547	0.0576262
$929$	10.1225	0.332109	0.166054	−	0.986117i	$-0.446897\pi$
$929$	10.1225	0.332109	0.166054	+	0.986117i	$0.446897\pi$
$930$	0	0
$931$	−7.54411	−0.247248
$932$	−16.2889	−0.533560
$933$	0	0
$934$	−3.55986	−0.116482
$935$	0	0
$936$	0	0
$937$	28.2113	0.921622	0.460811	−	0.887498i	$-0.347559\pi$
$937$	28.2113	0.921622	0.460811	+	0.887498i	$0.347559\pi$
$938$	0.248062	0.00809952
$939$	0	0
$940$	−10.5609	−0.344458
$941$	−20.1501	−0.656875	−0.328437	−	0.944526i	$-0.606522\pi$
$941$	−20.1501	−0.656875	−0.328437	+	0.944526i	$0.606522\pi$
$942$	0	0
$943$	−44.8766	−1.46138
$944$	26.0762	0.848707
$945$	0	0
$946$	0	0
$947$	−0.125141	−0.00406653	−0.00203326	−	0.999998i	$-0.500647\pi$
$947$	−0.125141	−0.00406653	−0.00203326	+	0.999998i	$0.500647\pi$
$948$	0	0
$949$	2.77161	0.0899703
$950$	−2.62216	−0.0850741
$951$	0	0
$952$	3.33786	0.108181
$953$	−29.3495	−0.950723	−0.475362	−	0.879790i	$-0.657683\pi$
$953$	−29.3495	−0.950723	−0.475362	+	0.879790i	$0.657683\pi$
$954$	0	0
$955$	−35.3073	−1.14252
$956$	−20.6166	−0.666790
$957$	0	0
$958$	−3.43717	−0.111050
$959$	−14.2142	−0.458999
$960$	0	0
$961$	34.1909	1.10293
$962$	−0.199505	−0.00643231
$963$	0	0
$964$	−16.2900	−0.524667
$965$	58.0481	1.86863
$966$	0	0
$967$	51.9463	1.67048	0.835240	−	0.549885i	$-0.185328\pi$
$967$	51.9463	1.67048	0.835240	+	0.549885i	$0.185328\pi$
$968$	0	0
$969$	0	0
$970$	3.08122	0.0989320
$971$	25.0066	0.802500	0.401250	−	0.915969i	$-0.368576\pi$
$971$	25.0066	0.802500	0.401250	+	0.915969i	$0.368576\pi$
$972$	0	0
$973$	−9.17942	−0.294279
$974$	0.127374	0.00408134
$975$	0	0
$976$	38.3484	1.22750
$977$	17.4156	0.557176	0.278588	−	0.960411i	$-0.410134\pi$
$977$	17.4156	0.557176	0.278588	+	0.960411i	$0.410134\pi$
$978$	0	0
$979$	0	0
$980$	−5.57636	−0.178130
$981$	0	0
$982$	2.22051	0.0708592
$983$	42.3349	1.35028	0.675138	−	0.737692i	$-0.264084\pi$
$983$	42.3349	1.35028	0.675138	+	0.737692i	$0.264084\pi$
$984$	0	0
$985$	17.6362	0.561937
$986$	1.02783	0.0327328
$987$	0	0
$988$	−16.1724	−0.514512
$989$	−25.2569	−0.803122
$990$	0	0
$991$	55.6263	1.76703	0.883513	−	0.468406i	$-0.155172\pi$
$991$	55.6263	1.76703	0.883513	+	0.468406i	$0.155172\pi$
$992$	−11.5491	−0.366685
$993$	0	0
$994$	1.50705	0.0478007
$995$	−19.2916	−0.611586
$996$	0	0
$997$	16.4267	0.520240	0.260120	−	0.965576i	$-0.416238\pi$
$997$	16.4267	0.520240	0.260120	+	0.965576i	$0.416238\pi$
$998$	3.81894	0.120886
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cs.1.3		6
3.2	odd	2		847.2.a.m.1.4	✓	6
11.10	odd	2		7623.2.a.cp.1.4		6
21.20	even	2		5929.2.a.bj.1.4		6
33.2	even	10		847.2.f.y.323.4		24
33.5	odd	10		847.2.f.z.729.3		24
33.8	even	10		847.2.f.y.372.3		24
33.14	odd	10		847.2.f.z.372.4		24
33.17	even	10		847.2.f.y.729.4		24
33.20	odd	10		847.2.f.z.323.3		24
33.26	odd	10		847.2.f.z.148.4		24
33.29	even	10		847.2.f.y.148.3		24
33.32	even	2		847.2.a.n.1.3	yes	6
231.230	odd	2		5929.2.a.bm.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.4	✓	6	3.2	odd	2
847.2.a.n.1.3	yes	6	33.32	even	2
847.2.f.y.148.3		24	33.29	even	10
847.2.f.y.323.4		24	33.2	even	10
847.2.f.y.372.3		24	33.8	even	10
847.2.f.y.729.4		24	33.17	even	10
847.2.f.z.148.4		24	33.26	odd	10
847.2.f.z.323.3		24	33.20	odd	10
847.2.f.z.372.4		24	33.14	odd	10
847.2.f.z.729.3		24	33.5	odd	10
5929.2.a.bj.1.4		6	21.20	even	2
5929.2.a.bm.1.3		6	231.230	odd	2
7623.2.a.cp.1.4		6	11.10	odd	2
7623.2.a.cs.1.3		6	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)

\(f(q)\)	\(=\)	\(q+0.120360 q^{2} -1.98551 q^{4} +2.80853 q^{5} -1.00000 q^{7} -0.479696 q^{8} +O(q^{10})\)	\(q+0.120360 q^{2} -1.98551 q^{4} +2.80853 q^{5} -1.00000 q^{7} -0.479696 q^{8} +0.338034 q^{10} -1.07967 q^{13} -0.120360 q^{14} +3.91329 q^{16} +6.95828 q^{17} -7.54411 q^{19} -5.57636 q^{20} -4.82552 q^{23} +2.88781 q^{25} -0.129949 q^{26} +1.98551 q^{28} +1.22726 q^{29} -8.07409 q^{31} +1.43040 q^{32} +0.837498 q^{34} -2.80853 q^{35} +1.53525 q^{37} -0.908009 q^{38} -1.34724 q^{40} +9.29986 q^{41} +5.23402 q^{43} -0.580799 q^{46} +1.89387 q^{47} +1.00000 q^{49} +0.347577 q^{50} +2.14371 q^{52} +3.82552 q^{53} +0.479696 q^{56} +0.147713 q^{58} +6.66349 q^{59} +9.79952 q^{61} -0.971796 q^{62} -7.65442 q^{64} -3.03229 q^{65} -2.06100 q^{67} -13.8158 q^{68} -0.338034 q^{70} -12.5212 q^{71} -2.56708 q^{73} +0.184783 q^{74} +14.9789 q^{76} +15.3283 q^{79} +10.9906 q^{80} +1.11933 q^{82} +2.04602 q^{83} +19.5425 q^{85} +0.629967 q^{86} -4.76119 q^{89} +1.07967 q^{91} +9.58114 q^{92} +0.227945 q^{94} -21.1878 q^{95} +9.11512 q^{97} +0.120360 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 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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