LMFDB - Embedded newform 7623.2.a.cs.1.4

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			1.4
Root			$-0.276564$ 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)

$f(q)$	$=$	$q+1.27656 q^{2} -0.370384 q^{4} +4.09144 q^{5} -1.00000 q^{7} -3.02595 q^{8} +O(q^{10})$	$q+1.27656 q^{2} -0.370384 q^{4} +4.09144 q^{5} -1.00000 q^{7} -3.02595 q^{8} +5.22298 q^{10} +4.39091 q^{13} -1.27656 q^{14} -3.12205 q^{16} +4.19146 q^{17} +1.24880 q^{19} -1.51540 q^{20} -4.97180 q^{23} +11.7399 q^{25} +5.60527 q^{26} +0.370384 q^{28} -1.93542 q^{29} -1.56278 q^{31} +2.06640 q^{32} +5.35067 q^{34} -4.09144 q^{35} -0.716296 q^{37} +1.59417 q^{38} -12.3805 q^{40} +4.80626 q^{41} -1.35362 q^{43} -6.34682 q^{46} +10.4662 q^{47} +1.00000 q^{49} +14.9867 q^{50} -1.62632 q^{52} +3.97180 q^{53} +3.02595 q^{56} -2.47069 q^{58} -13.7588 q^{59} +11.7271 q^{61} -1.99499 q^{62} +8.88199 q^{64} +17.9651 q^{65} +7.59274 q^{67} -1.55245 q^{68} -5.22298 q^{70} -0.218316 q^{71} -10.9714 q^{73} -0.914398 q^{74} -0.462535 q^{76} +4.56248 q^{79} -12.7737 q^{80} +6.13550 q^{82} +2.45458 q^{83} +17.1491 q^{85} -1.72798 q^{86} -4.20456 q^{89} -4.39091 q^{91} +1.84148 q^{92} +13.3608 q^{94} +5.10938 q^{95} -10.9249 q^{97} +1.27656 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.27656	0.902667	0.451334	−	0.892355i	$-0.350948\pi$
$2$	1.27656	0.902667	0.451334	+	0.892355i	$0.350948\pi$
$3$	0	0
$3$	0	0
$4$	−0.370384	−0.185192
$5$	4.09144	1.82975	0.914873	−	0.403741i	$-0.132290\pi$
$5$	4.09144	1.82975	0.914873	+	0.403741i	$0.132290\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.02595	−1.06983
$9$	0	0
$10$	5.22298	1.65165
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.39091	1.21782	0.608909	−	0.793240i	$-0.291607\pi$
$13$	4.39091	1.21782	0.608909	+	0.793240i	$0.291607\pi$
$14$	−1.27656	−0.341176
$15$	0	0
$16$	−3.12205	−0.780512
$17$	4.19146	1.01658	0.508289	−	0.861186i	$-0.330278\pi$
$17$	4.19146	1.01658	0.508289	+	0.861186i	$0.330278\pi$
$18$	0	0
$19$	1.24880	0.286494	0.143247	−	0.989687i	$-0.454246\pi$
$19$	1.24880	0.286494	0.143247	+	0.989687i	$0.454246\pi$
$20$	−1.51540	−0.338855
$21$	0	0
$22$	0	0
$23$	−4.97180	−1.03669	−0.518346	−	0.855171i	$-0.673452\pi$
$23$	−4.97180	−1.03669	−0.518346	+	0.855171i	$0.673452\pi$
$24$	0	0
$25$	11.7399	2.34797
$26$	5.60527	1.09928
$27$	0	0
$28$	0.370384	0.0699960
$29$	−1.93542	−0.359399	−0.179699	−	0.983722i	$-0.557512\pi$
$29$	−1.93542	−0.359399	−0.179699	+	0.983722i	$0.557512\pi$
$30$	0	0
$31$	−1.56278	−0.280684	−0.140342	−	0.990103i	$-0.544820\pi$
$31$	−1.56278	−0.280684	−0.140342	+	0.990103i	$0.544820\pi$
$32$	2.06640	0.365292
$33$	0	0
$34$	5.35067	0.917632
$35$	−4.09144	−0.691579
$36$	0	0
$37$	−0.716296	−0.117758	−0.0588792	−	0.998265i	$-0.518753\pi$
$37$	−0.716296	−0.117758	−0.0588792	+	0.998265i	$0.518753\pi$
$38$	1.59417	0.258609
$39$	0	0
$40$	−12.3805	−1.95752
$41$	4.80626	0.750611	0.375306	−	0.926901i	$-0.377538\pi$
$41$	4.80626	0.750611	0.375306	+	0.926901i	$0.377538\pi$
$42$	0	0
$43$	−1.35362	−0.206424	−0.103212	−	0.994659i	$-0.532912\pi$
$43$	−1.35362	−0.206424	−0.103212	+	0.994659i	$0.532912\pi$
$44$	0	0
$45$	0	0
$46$	−6.34682	−0.935788
$47$	10.4662	1.52665	0.763327	−	0.646013i	$-0.223565\pi$
$47$	10.4662	1.52665	0.763327	+	0.646013i	$0.223565\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	14.9867	2.11944
$51$	0	0
$52$	−1.62632	−0.225530
$53$	3.97180	0.545569	0.272784	−	0.962075i	$-0.412055\pi$
$53$	3.97180	0.545569	0.272784	+	0.962075i	$0.412055\pi$
$54$	0	0
$55$	0	0
$56$	3.02595	0.404359
$57$	0	0
$58$	−2.47069	−0.324417
$59$	−13.7588	−1.79125	−0.895624	−	0.444811i	$-0.853271\pi$
$59$	−13.7588	−1.79125	−0.895624	+	0.444811i	$0.853271\pi$
$60$	0	0
$61$	11.7271	1.50151	0.750753	−	0.660583i	$-0.229691\pi$
$61$	11.7271	1.50151	0.750753	+	0.660583i	$0.229691\pi$
$62$	−1.99499	−0.253364
$63$	0	0
$64$	8.88199	1.11025
$65$	17.9651	2.22830
$66$	0	0
$67$	7.59274	0.927600	0.463800	−	0.885940i	$-0.346486\pi$
$67$	7.59274	0.927600	0.463800	+	0.885940i	$0.346486\pi$
$68$	−1.55245	−0.188262
$69$	0	0
$70$	−5.22298	−0.624266
$71$	−0.218316	−0.0259094	−0.0129547	−	0.999916i	$-0.504124\pi$
$71$	−0.218316	−0.0259094	−0.0129547	+	0.999916i	$0.504124\pi$
$72$	0	0
$73$	−10.9714	−1.28411	−0.642054	−	0.766659i	$-0.721918\pi$
$73$	−10.9714	−1.28411	−0.642054	+	0.766659i	$0.721918\pi$
$74$	−0.914398	−0.106297
$75$	0	0
$76$	−0.462535	−0.0530564
$77$	0	0
$78$	0	0
$79$	4.56248	0.513319	0.256659	−	0.966502i	$-0.417378\pi$
$79$	4.56248	0.513319	0.256659	+	0.966502i	$0.417378\pi$
$80$	−12.7737	−1.42814
$81$	0	0
$82$	6.13550	0.677552
$83$	2.45458	0.269425	0.134712	−	0.990885i	$-0.456989\pi$
$83$	2.45458	0.269425	0.134712	+	0.990885i	$0.456989\pi$
$84$	0	0
$85$	17.1491	1.86008
$86$	−1.72798	−0.186333
$87$	0	0
$88$	0	0
$89$	−4.20456	−0.445683	−0.222841	−	0.974855i	$-0.571533\pi$
$89$	−4.20456	−0.445683	−0.222841	+	0.974855i	$0.571533\pi$
$90$	0	0
$91$	−4.39091	−0.460292
$92$	1.84148	0.191987
$93$	0	0
$94$	13.3608	1.37806
$95$	5.10938	0.524211
$96$	0	0
$97$	−10.9249	−1.10925	−0.554627	−	0.832099i	$-0.687139\pi$
$97$	−10.9249	−1.10925	−0.554627	+	0.832099i	$0.687139\pi$
$98$	1.27656	0.128952
$99$	0	0
$100$	−4.34826	−0.434826
$101$	7.36579	0.732924	0.366462	−	0.930433i	$-0.380569\pi$
$101$	7.36579	0.732924	0.366462	+	0.930433i	$0.380569\pi$
$102$	0	0
$103$	−0.153886	−0.0151629	−0.00758143	−	0.999971i	$-0.502413\pi$
$103$	−0.153886	−0.0151629	−0.00758143	+	0.999971i	$0.502413\pi$
$104$	−13.2867	−1.30286
$105$	0	0
$106$	5.07026	0.492467
$107$	7.94617	0.768185	0.384092	−	0.923295i	$-0.374514\pi$
$107$	7.94617	0.768185	0.384092	+	0.923295i	$0.374514\pi$
$108$	0	0
$109$	4.91440	0.470714	0.235357	−	0.971909i	$-0.424374\pi$
$109$	4.91440	0.470714	0.235357	+	0.971909i	$0.424374\pi$
$110$	0	0
$111$	0	0
$112$	3.12205	0.295006
$113$	−3.04208	−0.286175	−0.143088	−	0.989710i	$-0.545703\pi$
$113$	−3.04208	−0.286175	−0.143088	+	0.989710i	$0.545703\pi$
$114$	0	0
$115$	−20.3418	−1.89688
$116$	0.716849	0.0665578
$117$	0	0
$118$	−17.5640	−1.61690
$119$	−4.19146	−0.384231
$120$	0	0
$121$	0	0
$122$	14.9704	1.35536
$123$	0	0
$124$	0.578830	0.0519804
$125$	27.5757	2.46645
$126$	0	0
$127$	10.7954	0.957932	0.478966	−	0.877833i	$-0.341012\pi$
$127$	10.7954	0.957932	0.478966	+	0.877833i	$0.341012\pi$
$128$	7.20562	0.636893
$129$	0	0
$130$	22.9336	2.01141
$131$	−8.70429	−0.760497	−0.380249	−	0.924884i	$-0.624162\pi$
$131$	−8.70429	−0.760497	−0.380249	+	0.924884i	$0.624162\pi$
$132$	0	0
$133$	−1.24880	−0.108285
$134$	9.69261	0.837314
$135$	0	0
$136$	−12.6831	−1.08757
$137$	−11.9977	−1.02503	−0.512516	−	0.858677i	$-0.671287\pi$
$137$	−11.9977	−1.02503	−0.512516	+	0.858677i	$0.671287\pi$
$138$	0	0
$139$	−8.18967	−0.694639	−0.347319	−	0.937747i	$-0.612908\pi$
$139$	−8.18967	−0.694639	−0.347319	+	0.937747i	$0.612908\pi$
$140$	1.51540	0.128075
$141$	0	0
$142$	−0.278695	−0.0233875
$143$	0	0
$144$	0	0
$145$	−7.91865	−0.657608
$146$	−14.0057	−1.15912
$147$	0	0
$148$	0.265305	0.0218079
$149$	12.0816	0.989766	0.494883	−	0.868960i	$-0.335211\pi$
$149$	12.0816	0.989766	0.494883	+	0.868960i	$0.335211\pi$
$150$	0	0
$151$	6.94850	0.565461	0.282730	−	0.959199i	$-0.408760\pi$
$151$	6.94850	0.565461	0.282730	+	0.959199i	$0.408760\pi$
$152$	−3.77880	−0.306501
$153$	0	0
$154$	0	0
$155$	−6.39402	−0.513580
$156$	0	0
$157$	10.4930	0.837436	0.418718	−	0.908116i	$-0.362480\pi$
$157$	10.4930	0.837436	0.418718	+	0.908116i	$0.362480\pi$
$158$	5.82429	0.463356
$159$	0	0
$160$	8.45455	0.668391
$161$	4.97180	0.391833
$162$	0	0
$163$	−7.81743	−0.612309	−0.306154	−	0.951982i	$-0.599042\pi$
$163$	−7.81743	−0.612309	−0.306154	+	0.951982i	$0.599042\pi$
$164$	−1.78016	−0.139007
$165$	0	0
$166$	3.13342	0.243201
$167$	21.3503	1.65214	0.826068	−	0.563571i	$-0.190573\pi$
$167$	21.3503	1.65214	0.826068	+	0.563571i	$0.190573\pi$
$168$	0	0
$169$	6.28007	0.483082
$170$	21.8919	1.67903
$171$	0	0
$172$	0.501358	0.0382282
$173$	19.4384	1.47787	0.738936	−	0.673776i	$-0.235329\pi$
$173$	19.4384	1.47787	0.738936	+	0.673776i	$0.235329\pi$
$174$	0	0
$175$	−11.7399	−0.887450
$176$	0	0
$177$	0	0
$178$	−5.36740	−0.402303
$179$	−17.9964	−1.34511	−0.672557	−	0.740045i	$-0.734804\pi$
$179$	−17.9964	−1.34511	−0.672557	+	0.740045i	$0.734804\pi$
$180$	0	0
$181$	15.1575	1.12665	0.563326	−	0.826235i	$-0.309522\pi$
$181$	15.1575	1.12665	0.563326	+	0.826235i	$0.309522\pi$
$182$	−5.60527	−0.415491
$183$	0	0
$184$	15.0444	1.10909
$185$	−2.93068	−0.215468
$186$	0	0
$187$	0	0
$188$	−3.87652	−0.282724
$189$	0	0
$190$	6.52245	0.473188
$191$	23.7574	1.71903	0.859513	−	0.511115i	$-0.170767\pi$
$191$	23.7574	1.71903	0.859513	+	0.511115i	$0.170767\pi$
$192$	0	0
$193$	−14.0582	−1.01193	−0.505965	−	0.862554i	$-0.668863\pi$
$193$	−14.0582	−1.01193	−0.505965	+	0.862554i	$0.668863\pi$
$194$	−13.9463	−1.00129
$195$	0	0
$196$	−0.370384	−0.0264560
$197$	18.0665	1.28718	0.643591	−	0.765369i	$-0.277444\pi$
$197$	18.0665	1.28718	0.643591	+	0.765369i	$0.277444\pi$
$198$	0	0
$199$	1.54374	0.109433	0.0547163	−	0.998502i	$-0.482575\pi$
$199$	1.54374	0.109433	0.0547163	+	0.998502i	$0.482575\pi$
$200$	−35.5242	−2.51194
$201$	0	0
$202$	9.40291	0.661586
$203$	1.93542	0.135840
$204$	0	0
$205$	19.6645	1.37343
$206$	−0.196446	−0.0136870
$207$	0	0
$208$	−13.7086	−0.950522
$209$	0	0
$210$	0	0
$211$	22.0836	1.52030	0.760149	−	0.649749i	$-0.225126\pi$
$211$	22.0836	1.52030	0.760149	+	0.649749i	$0.225126\pi$
$212$	−1.47109	−0.101035
$213$	0	0
$214$	10.1438	0.693415
$215$	−5.53824	−0.377704
$216$	0	0
$217$	1.56278	0.106089
$218$	6.27354	0.424898
$219$	0	0
$220$	0	0
$221$	18.4043	1.23801
$222$	0	0
$223$	−8.57414	−0.574167	−0.287083	−	0.957906i	$-0.592686\pi$
$223$	−8.57414	−0.574167	−0.287083	+	0.957906i	$0.592686\pi$
$224$	−2.06640	−0.138067
$225$	0	0
$226$	−3.88342	−0.258321
$227$	−26.5709	−1.76357	−0.881786	−	0.471650i	$-0.843659\pi$
$227$	−26.5709	−1.76357	−0.881786	+	0.471650i	$0.843659\pi$
$228$	0	0
$229$	11.4113	0.754081	0.377040	−	0.926197i	$-0.376942\pi$
$229$	11.4113	0.754081	0.377040	+	0.926197i	$0.376942\pi$
$230$	−25.9676	−1.71225
$231$	0	0
$232$	5.85648	0.384497
$233$	26.6130	1.74348	0.871738	−	0.489972i	$-0.162993\pi$
$233$	26.6130	1.74348	0.871738	+	0.489972i	$0.162993\pi$
$234$	0	0
$235$	42.8218	2.79339
$236$	5.09606	0.331725
$237$	0	0
$238$	−5.35067	−0.346832
$239$	−7.51280	−0.485963	−0.242981	−	0.970031i	$-0.578125\pi$
$239$	−7.51280	−0.485963	−0.242981	+	0.970031i	$0.578125\pi$
$240$	0	0
$241$	−27.4388	−1.76749	−0.883744	−	0.467971i	$-0.844985\pi$
$241$	−27.4388	−1.76749	−0.883744	+	0.467971i	$0.844985\pi$
$242$	0	0
$243$	0	0
$244$	−4.34355	−0.278067
$245$	4.09144	0.261392
$246$	0	0
$247$	5.48336	0.348898
$248$	4.72889	0.300285
$249$	0	0
$250$	35.2022	2.22638
$251$	−1.92514	−0.121514	−0.0607568	−	0.998153i	$-0.519351\pi$
$251$	−1.92514	−0.121514	−0.0607568	+	0.998153i	$0.519351\pi$
$252$	0	0
$253$	0	0
$254$	13.7810	0.864694
$255$	0	0
$256$	−8.56553	−0.535346
$257$	−10.1604	−0.633791	−0.316895	−	0.948460i	$-0.602640\pi$
$257$	−10.1604	−0.633791	−0.316895	+	0.948460i	$0.602640\pi$
$258$	0	0
$259$	0.716296	0.0445085
$260$	−6.65400	−0.412663
$261$	0	0
$262$	−11.1116	−0.686476
$263$	−4.38774	−0.270560	−0.135280	−	0.990807i	$-0.543193\pi$
$263$	−4.38774	−0.270560	−0.135280	+	0.990807i	$0.543193\pi$
$264$	0	0
$265$	16.2504	0.998253
$266$	−1.59417	−0.0977449
$267$	0	0
$268$	−2.81223	−0.171784
$269$	−0.625379	−0.0381300	−0.0190650	−	0.999818i	$-0.506069\pi$
$269$	−0.625379	−0.0381300	−0.0190650	+	0.999818i	$0.506069\pi$
$270$	0	0
$271$	11.9375	0.725151	0.362575	−	0.931954i	$-0.381898\pi$
$271$	11.9375	0.725151	0.362575	+	0.931954i	$0.381898\pi$
$272$	−13.0859	−0.793452
$273$	0	0
$274$	−15.3158	−0.925263
$275$	0	0
$276$	0	0
$277$	−28.5571	−1.71583	−0.857915	−	0.513792i	$-0.828240\pi$
$277$	−28.5571	−1.71583	−0.857915	+	0.513792i	$0.828240\pi$
$278$	−10.4546	−0.627028
$279$	0	0
$280$	12.3805	0.739875
$281$	14.6981	0.876814	0.438407	−	0.898777i	$-0.355543\pi$
$281$	14.6981	0.876814	0.438407	+	0.898777i	$0.355543\pi$
$282$	0	0
$283$	−22.5396	−1.33984	−0.669921	−	0.742433i	$-0.733672\pi$
$283$	−22.5396	−1.33984	−0.669921	+	0.742433i	$0.733672\pi$
$284$	0.0808609	0.00479821
$285$	0	0
$286$	0	0
$287$	−4.80626	−0.283704
$288$	0	0
$289$	0.568350	0.0334323
$290$	−10.1087	−0.593601
$291$	0	0
$292$	4.06364	0.237807
$293$	16.8280	0.983105	0.491553	−	0.870848i	$-0.336430\pi$
$293$	16.8280	0.983105	0.491553	+	0.870848i	$0.336430\pi$
$294$	0	0
$295$	−56.2934	−3.27753
$296$	2.16747	0.125982
$297$	0	0
$298$	15.4230	0.893429
$299$	−21.8307	−1.26250
$300$	0	0
$301$	1.35362	0.0780211
$302$	8.87021	0.510423
$303$	0	0
$304$	−3.89881	−0.223612
$305$	47.9808	2.74738
$306$	0	0
$307$	−0.238354	−0.0136036	−0.00680179	−	0.999977i	$-0.502165\pi$
$307$	−0.238354	−0.0136036	−0.00680179	+	0.999977i	$0.502165\pi$
$308$	0	0
$309$	0	0
$310$	−8.16238	−0.463592
$311$	0.656523	0.0372280	0.0186140	−	0.999827i	$-0.494075\pi$
$311$	0.656523	0.0372280	0.0186140	+	0.999827i	$0.494075\pi$
$312$	0	0
$313$	−7.45262	−0.421247	−0.210624	−	0.977567i	$-0.567549\pi$
$313$	−7.45262	−0.421247	−0.210624	+	0.977567i	$0.567549\pi$
$314$	13.3950	0.755926
$315$	0	0
$316$	−1.68987	−0.0950625
$317$	−17.4412	−0.979595	−0.489797	−	0.871836i	$-0.662929\pi$
$317$	−17.4412	−0.979595	−0.489797	+	0.871836i	$0.662929\pi$
$318$	0	0
$319$	0	0
$320$	36.3401	2.03147
$321$	0	0
$322$	6.34682	0.353695
$323$	5.23429	0.291244
$324$	0	0
$325$	51.5486	2.85940
$326$	−9.97946	−0.552711
$327$	0	0
$328$	−14.5435	−0.803030
$329$	−10.4662	−0.577021
$330$	0	0
$331$	−28.6146	−1.57280	−0.786399	−	0.617719i	$-0.788057\pi$
$331$	−28.6146	−1.57280	−0.786399	+	0.617719i	$0.788057\pi$
$332$	−0.909136	−0.0498953
$333$	0	0
$334$	27.2550	1.49133
$335$	31.0652	1.69727
$336$	0	0
$337$	−1.73053	−0.0942679	−0.0471339	−	0.998889i	$-0.515009\pi$
$337$	−1.73053	−0.0942679	−0.0471339	+	0.998889i	$0.515009\pi$
$338$	8.01691	0.436062
$339$	0	0
$340$	−6.35176	−0.344472
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.09597	0.220840
$345$	0	0
$346$	24.8143	1.33403
$347$	2.73009	0.146559	0.0732796	−	0.997311i	$-0.476653\pi$
$347$	2.73009	0.146559	0.0732796	+	0.997311i	$0.476653\pi$
$348$	0	0
$349$	−30.9063	−1.65437	−0.827187	−	0.561927i	$-0.810060\pi$
$349$	−30.9063	−1.65437	−0.827187	+	0.561927i	$0.810060\pi$
$350$	−14.9867	−0.801072
$351$	0	0
$352$	0	0
$353$	23.9543	1.27496	0.637480	−	0.770467i	$-0.279977\pi$
$353$	23.9543	1.27496	0.637480	+	0.770467i	$0.279977\pi$
$354$	0	0
$355$	−0.893227	−0.0474076
$356$	1.55730	0.0825369
$357$	0	0
$358$	−22.9736	−1.21419
$359$	−1.51611	−0.0800170	−0.0400085	−	0.999199i	$-0.512739\pi$
$359$	−1.51611	−0.0800170	−0.0400085	+	0.999199i	$0.512739\pi$
$360$	0	0
$361$	−17.4405	−0.917921
$362$	19.3496	1.01699
$363$	0	0
$364$	1.62632	0.0852425
$365$	−44.8889	−2.34959
$366$	0	0
$367$	8.72207	0.455288	0.227644	−	0.973744i	$-0.426898\pi$
$367$	8.72207	0.455288	0.227644	+	0.973744i	$0.426898\pi$
$368$	15.5222	0.809151
$369$	0	0
$370$	−3.74120	−0.194496
$371$	−3.97180	−0.206206
$372$	0	0
$373$	36.8111	1.90601	0.953004	−	0.302957i	$-0.0979739\pi$
$373$	36.8111	1.90601	0.953004	+	0.302957i	$0.0979739\pi$
$374$	0	0
$375$	0	0
$376$	−31.6702	−1.63327
$377$	−8.49825	−0.437682
$378$	0	0
$379$	2.59838	0.133470	0.0667349	−	0.997771i	$-0.478742\pi$
$379$	2.59838	0.133470	0.0667349	+	0.997771i	$0.478742\pi$
$380$	−1.89243	−0.0970798
$381$	0	0
$382$	30.3278	1.55171
$383$	−19.2392	−0.983078	−0.491539	−	0.870856i	$-0.663565\pi$
$383$	−19.2392	−0.983078	−0.491539	+	0.870856i	$0.663565\pi$
$384$	0	0
$385$	0	0
$386$	−17.9462	−0.913435
$387$	0	0
$388$	4.04640	0.205425
$389$	28.3331	1.43654	0.718272	−	0.695762i	$-0.244933\pi$
$389$	28.3331	1.43654	0.718272	+	0.695762i	$0.244933\pi$
$390$	0	0
$391$	−20.8391	−1.05388
$392$	−3.02595	−0.152833
$393$	0	0
$394$	23.0630	1.16190
$395$	18.6671	0.939243
$396$	0	0
$397$	−10.3666	−0.520287	−0.260143	−	0.965570i	$-0.583770\pi$
$397$	−10.3666	−0.520287	−0.260143	+	0.965570i	$0.583770\pi$
$398$	1.97068	0.0987812
$399$	0	0
$400$	−36.6524	−1.83262
$401$	12.7878	0.638592	0.319296	−	0.947655i	$-0.396554\pi$
$401$	12.7878	0.638592	0.319296	+	0.947655i	$0.396554\pi$
$402$	0	0
$403$	−6.86203	−0.341822
$404$	−2.72817	−0.135732
$405$	0	0
$406$	2.47069	0.122618
$407$	0	0
$408$	0	0
$409$	32.0217	1.58337	0.791686	−	0.610928i	$-0.209203\pi$
$409$	32.0217	1.58337	0.791686	+	0.610928i	$0.209203\pi$
$410$	25.1030	1.23975
$411$	0	0
$412$	0.0569970	0.00280804
$413$	13.7588	0.677028
$414$	0	0
$415$	10.0427	0.492979
$416$	9.07338	0.444859
$417$	0	0
$418$	0	0
$419$	20.0934	0.981629	0.490815	−	0.871264i	$-0.336699\pi$
$419$	20.0934	0.981629	0.490815	+	0.871264i	$0.336699\pi$
$420$	0	0
$421$	−22.4243	−1.09289	−0.546446	−	0.837494i	$-0.684020\pi$
$421$	−22.4243	−1.09289	−0.546446	+	0.837494i	$0.684020\pi$
$422$	28.1911	1.37232
$423$	0	0
$424$	−12.0185	−0.583668
$425$	49.2072	2.38690
$426$	0	0
$427$	−11.7271	−0.567516
$428$	−2.94313	−0.142262
$429$	0	0
$430$	−7.06991	−0.340941
$431$	12.0856	0.582144	0.291072	−	0.956701i	$-0.405988\pi$
$431$	12.0856	0.582144	0.291072	+	0.956701i	$0.405988\pi$
$432$	0	0
$433$	−0.302453	−0.0145350	−0.00726749	−	0.999974i	$-0.502313\pi$
$433$	−0.302453	−0.0145350	−0.00726749	+	0.999974i	$0.502313\pi$
$434$	1.99499	0.0957626
$435$	0	0
$436$	−1.82022	−0.0871725
$437$	−6.20878	−0.297006
$438$	0	0
$439$	3.52592	0.168283	0.0841416	−	0.996454i	$-0.473185\pi$
$439$	3.52592	0.168283	0.0841416	+	0.996454i	$0.473185\pi$
$440$	0	0
$441$	0	0
$442$	23.4943	1.11751
$443$	21.8443	1.03786	0.518928	−	0.854818i	$-0.326331\pi$
$443$	21.8443	1.03786	0.518928	+	0.854818i	$0.326331\pi$
$444$	0	0
$445$	−17.2027	−0.815487
$446$	−10.9454	−0.518281
$447$	0	0
$448$	−8.88199	−0.419634
$449$	−9.30172	−0.438975	−0.219488	−	0.975615i	$-0.570439\pi$
$449$	−9.30172	−0.438975	−0.219488	+	0.975615i	$0.570439\pi$
$450$	0	0
$451$	0	0
$452$	1.12674	0.0529974
$453$	0	0
$454$	−33.9194	−1.59192
$455$	−17.9651	−0.842218
$456$	0	0
$457$	11.3165	0.529365	0.264683	−	0.964336i	$-0.414733\pi$
$457$	11.3165	0.529365	0.264683	+	0.964336i	$0.414733\pi$
$458$	14.5673	0.680684
$459$	0	0
$460$	7.53429	0.351288
$461$	−0.678821	−0.0316158	−0.0158079	−	0.999875i	$-0.505032\pi$
$461$	−0.678821	−0.0316158	−0.0158079	+	0.999875i	$0.505032\pi$
$462$	0	0
$463$	−13.4936	−0.627103	−0.313551	−	0.949571i	$-0.601519\pi$
$463$	−13.4936	−0.627103	−0.313551	+	0.949571i	$0.601519\pi$
$464$	6.04247	0.280515
$465$	0	0
$466$	33.9732	1.57378
$467$	−29.6568	−1.37235	−0.686176	−	0.727435i	$-0.740712\pi$
$467$	−29.6568	−1.37235	−0.686176	+	0.727435i	$0.740712\pi$
$468$	0	0
$469$	−7.59274	−0.350600
$470$	54.6648	2.52150
$471$	0	0
$472$	41.6335	1.91634
$473$	0	0
$474$	0	0
$475$	14.6607	0.672680
$476$	1.55245	0.0711565
$477$	0	0
$478$	−9.59057	−0.438662
$479$	4.70136	0.214811	0.107405	−	0.994215i	$-0.465746\pi$
$479$	4.70136	0.214811	0.107405	+	0.994215i	$0.465746\pi$
$480$	0	0
$481$	−3.14519	−0.143408
$482$	−35.0274	−1.59545
$483$	0	0
$484$	0	0
$485$	−44.6985	−2.02965
$486$	0	0
$487$	32.3738	1.46700	0.733498	−	0.679692i	$-0.237886\pi$
$487$	32.3738	1.46700	0.733498	+	0.679692i	$0.237886\pi$
$488$	−35.4857	−1.60636
$489$	0	0
$490$	5.22298	0.235950
$491$	−6.85594	−0.309404	−0.154702	−	0.987961i	$-0.549442\pi$
$491$	−6.85594	−0.309404	−0.154702	+	0.987961i	$0.549442\pi$
$492$	0	0
$493$	−8.11224	−0.365357
$494$	6.99986	0.314939
$495$	0	0
$496$	4.87908	0.219077
$497$	0.218316	0.00979282
$498$	0	0
$499$	10.9670	0.490952	0.245476	−	0.969403i	$-0.421056\pi$
$499$	10.9670	0.490952	0.245476	+	0.969403i	$0.421056\pi$
$500$	−10.2136	−0.456766
$501$	0	0
$502$	−2.45756	−0.109686
$503$	−26.9334	−1.20090	−0.600451	−	0.799662i	$-0.705012\pi$
$503$	−26.9334	−1.20090	−0.600451	+	0.799662i	$0.705012\pi$
$504$	0	0
$505$	30.1367	1.34106
$506$	0	0
$507$	0	0
$508$	−3.99843	−0.177402
$509$	−30.3958	−1.34727	−0.673635	−	0.739064i	$-0.735268\pi$
$509$	−30.3958	−1.34727	−0.673635	+	0.739064i	$0.735268\pi$
$510$	0	0
$511$	10.9714	0.485347
$512$	−25.3457	−1.12013
$513$	0	0
$514$	−12.9705	−0.572102
$515$	−0.629616	−0.0277442
$516$	0	0
$517$	0	0
$518$	0.914398	0.0401763
$519$	0	0
$520$	−54.3615	−2.38391
$521$	0.935426	0.0409818	0.0204909	−	0.999790i	$-0.493477\pi$
$521$	0.935426	0.0409818	0.0204909	+	0.999790i	$0.493477\pi$
$522$	0	0
$523$	−26.1853	−1.14500	−0.572502	−	0.819903i	$-0.694027\pi$
$523$	−26.1853	−1.14500	−0.572502	+	0.819903i	$0.694027\pi$
$524$	3.22393	0.140838
$525$	0	0
$526$	−5.60124	−0.244226
$527$	−6.55034	−0.285337
$528$	0	0
$529$	1.71881	0.0747307
$530$	20.7446	0.901090
$531$	0	0
$532$	0.462535	0.0200534
$533$	21.1038	0.914109
$534$	0	0
$535$	32.5112	1.40558
$536$	−22.9752	−0.992378
$537$	0	0
$538$	−0.798336	−0.0344187
$539$	0	0
$540$	0	0
$541$	−26.7566	−1.15035	−0.575177	−	0.818029i	$-0.695067\pi$
$541$	−26.7566	−1.15035	−0.575177	+	0.818029i	$0.695067\pi$
$542$	15.2390	0.654570
$543$	0	0
$544$	8.66124	0.371348
$545$	20.1070	0.861287
$546$	0	0
$547$	−46.4581	−1.98641	−0.993203	−	0.116397i	$-0.962865\pi$
$547$	−46.4581	−1.98641	−0.993203	+	0.116397i	$0.962865\pi$
$548$	4.44376	0.189828
$549$	0	0
$550$	0	0
$551$	−2.41695	−0.102966
$552$	0	0
$553$	−4.56248	−0.194016
$554$	−36.4550	−1.54882
$555$	0	0
$556$	3.03332	0.128642
$557$	35.0974	1.48712	0.743562	−	0.668667i	$-0.233135\pi$
$557$	35.0974	1.48712	0.743562	+	0.668667i	$0.233135\pi$
$558$	0	0
$559$	−5.94360	−0.251388
$560$	12.7737	0.539786
$561$	0	0
$562$	18.7630	0.791471
$563$	−8.94791	−0.377109	−0.188555	−	0.982063i	$-0.560380\pi$
$563$	−8.94791	−0.377109	−0.188555	+	0.982063i	$0.560380\pi$
$564$	0	0
$565$	−12.4465	−0.523628
$566$	−28.7733	−1.20943
$567$	0	0
$568$	0.660614	0.0277187
$569$	19.4256	0.814365	0.407183	−	0.913347i	$-0.366511\pi$
$569$	19.4256	0.814365	0.407183	+	0.913347i	$0.366511\pi$
$570$	0	0
$571$	−16.0171	−0.670295	−0.335148	−	0.942166i	$-0.608786\pi$
$571$	−16.0171	−0.670295	−0.335148	+	0.942166i	$0.608786\pi$
$572$	0	0
$573$	0	0
$574$	−6.13550	−0.256091
$575$	−58.3683	−2.43412
$576$	0	0
$577$	−32.9192	−1.37044	−0.685222	−	0.728335i	$-0.740295\pi$
$577$	−32.9192	−1.37044	−0.685222	+	0.728335i	$0.740295\pi$
$578$	0.725535	0.0301783
$579$	0	0
$580$	2.93294	0.121784
$581$	−2.45458	−0.101833
$582$	0	0
$583$	0	0
$584$	33.1990	1.37378
$585$	0	0
$586$	21.4821	0.887417
$587$	−27.4372	−1.13246	−0.566228	−	0.824249i	$-0.691598\pi$
$587$	−27.4372	−1.13246	−0.566228	+	0.824249i	$0.691598\pi$
$588$	0	0
$589$	−1.95160	−0.0804142
$590$	−71.8622	−2.95852
$591$	0	0
$592$	2.23631	0.0919118
$593$	14.6132	0.600092	0.300046	−	0.953925i	$-0.402998\pi$
$593$	14.6132	0.600092	0.300046	+	0.953925i	$0.402998\pi$
$594$	0	0
$595$	−17.1491	−0.703045
$596$	−4.47485	−0.183297
$597$	0	0
$598$	−27.8683	−1.13962
$599$	−25.2765	−1.03277	−0.516384	−	0.856357i	$-0.672722\pi$
$599$	−25.2765	−1.03277	−0.516384	+	0.856357i	$0.672722\pi$
$600$	0	0
$601$	9.30098	0.379395	0.189697	−	0.981843i	$-0.439249\pi$
$601$	9.30098	0.379395	0.189697	+	0.981843i	$0.439249\pi$
$602$	1.72798	0.0704271
$603$	0	0
$604$	−2.57361	−0.104719
$605$	0	0
$606$	0	0
$607$	2.12671	0.0863207	0.0431603	−	0.999068i	$-0.486257\pi$
$607$	2.12671	0.0863207	0.0431603	+	0.999068i	$0.486257\pi$
$608$	2.58052	0.104654
$609$	0	0
$610$	61.2506	2.47997
$611$	45.9561	1.85919
$612$	0	0
$613$	40.9448	1.65374	0.826872	−	0.562390i	$-0.190118\pi$
$613$	40.9448	1.65374	0.826872	+	0.562390i	$0.190118\pi$
$614$	−0.304274	−0.0122795
$615$	0	0
$616$	0	0
$617$	7.53813	0.303474	0.151737	−	0.988421i	$-0.451513\pi$
$617$	7.53813	0.303474	0.151737	+	0.988421i	$0.451513\pi$
$618$	0	0
$619$	−19.5055	−0.783994	−0.391997	−	0.919967i	$-0.628216\pi$
$619$	−19.5055	−0.783994	−0.391997	+	0.919967i	$0.628216\pi$
$620$	2.36824	0.0951110
$621$	0	0
$622$	0.838094	0.0336045
$623$	4.20456	0.168452
$624$	0	0
$625$	54.1250	2.16500
$626$	−9.51375	−0.380246
$627$	0	0
$628$	−3.88645	−0.155086
$629$	−3.00233	−0.119711
$630$	0	0
$631$	−10.6654	−0.424582	−0.212291	−	0.977207i	$-0.568092\pi$
$631$	−10.6654	−0.424582	−0.212291	+	0.977207i	$0.568092\pi$
$632$	−13.8058	−0.549166
$633$	0	0
$634$	−22.2648	−0.884248
$635$	44.1685	1.75277
$636$	0	0
$637$	4.39091	0.173974
$638$	0	0
$639$	0	0
$640$	29.4814	1.16535
$641$	−0.448349	−0.0177087	−0.00885436	−	0.999961i	$-0.502818\pi$
$641$	−0.448349	−0.0177087	−0.00885436	+	0.999961i	$0.502818\pi$
$642$	0	0
$643$	−31.6440	−1.24792	−0.623958	−	0.781458i	$-0.714476\pi$
$643$	−31.6440	−1.24792	−0.623958	+	0.781458i	$0.714476\pi$
$644$	−1.84148	−0.0725643
$645$	0	0
$646$	6.68191	0.262896
$647$	30.1242	1.18430	0.592152	−	0.805826i	$-0.298279\pi$
$647$	30.1242	1.18430	0.592152	+	0.805826i	$0.298279\pi$
$648$	0	0
$649$	0	0
$650$	65.8051	2.58109
$651$	0	0
$652$	2.89545	0.113395
$653$	−34.6110	−1.35443	−0.677217	−	0.735784i	$-0.736814\pi$
$653$	−34.6110	−1.35443	−0.677217	+	0.735784i	$0.736814\pi$
$654$	0	0
$655$	−35.6131	−1.39152
$656$	−15.0054	−0.585861
$657$	0	0
$658$	−13.3608	−0.520858
$659$	−20.2587	−0.789167	−0.394584	−	0.918860i	$-0.629111\pi$
$659$	−20.2587	−0.789167	−0.394584	+	0.918860i	$0.629111\pi$
$660$	0	0
$661$	40.6737	1.58202	0.791012	−	0.611801i	$-0.209555\pi$
$661$	40.6737	1.58202	0.791012	+	0.611801i	$0.209555\pi$
$662$	−36.5283	−1.41971
$663$	0	0
$664$	−7.42742	−0.288240
$665$	−5.10938	−0.198133
$666$	0	0
$667$	9.62253	0.372586
$668$	−7.90781	−0.305962
$669$	0	0
$670$	39.6567	1.53207
$671$	0	0
$672$	0	0
$673$	−24.7484	−0.953980	−0.476990	−	0.878909i	$-0.658272\pi$
$673$	−24.7484	−0.953980	−0.476990	+	0.878909i	$0.658272\pi$
$674$	−2.20913	−0.0850925
$675$	0	0
$676$	−2.32604	−0.0894630
$677$	−11.3821	−0.437452	−0.218726	−	0.975786i	$-0.570190\pi$
$677$	−11.3821	−0.437452	−0.218726	+	0.975786i	$0.570190\pi$
$678$	0	0
$679$	10.9249	0.419259
$680$	−51.8923	−1.98998
$681$	0	0
$682$	0	0
$683$	0.212700	0.00813875	0.00406938	−	0.999992i	$-0.498705\pi$
$683$	0.212700	0.00813875	0.00406938	+	0.999992i	$0.498705\pi$
$684$	0	0
$685$	−49.0878	−1.87555
$686$	−1.27656	−0.0487394
$687$	0	0
$688$	4.22605	0.161117
$689$	17.4398	0.664404
$690$	0	0
$691$	36.3804	1.38398	0.691988	−	0.721909i	$-0.256735\pi$
$691$	36.3804	1.38398	0.691988	+	0.721909i	$0.256735\pi$
$692$	−7.19966	−0.273690
$693$	0	0
$694$	3.48514	0.132294
$695$	−33.5075	−1.27101
$696$	0	0
$697$	20.1452	0.763056
$698$	−39.4538	−1.49335
$699$	0	0
$700$	4.34826	0.164349
$701$	−18.7394	−0.707779	−0.353889	−	0.935287i	$-0.615141\pi$
$701$	−18.7394	−0.707779	−0.353889	+	0.935287i	$0.615141\pi$
$702$	0	0
$703$	−0.894509	−0.0337371
$704$	0	0
$705$	0	0
$706$	30.5792	1.15086
$707$	−7.36579	−0.277019
$708$	0	0
$709$	−12.7596	−0.479195	−0.239598	−	0.970872i	$-0.577015\pi$
$709$	−12.7596	−0.479195	−0.239598	+	0.970872i	$0.577015\pi$
$710$	−1.14026	−0.0427933
$711$	0	0
$712$	12.7228	0.476807
$713$	7.76984	0.290983
$714$	0	0
$715$	0	0
$716$	6.66558	0.249105
$717$	0	0
$718$	−1.93541	−0.0722287
$719$	17.1228	0.638571	0.319286	−	0.947658i	$-0.396557\pi$
$719$	17.1228	0.638571	0.319286	+	0.947658i	$0.396557\pi$
$720$	0	0
$721$	0.153886	0.00573102
$722$	−22.2639	−0.828577
$723$	0	0
$724$	−5.61411	−0.208647
$725$	−22.7216	−0.843858
$726$	0	0
$727$	−8.65786	−0.321102	−0.160551	−	0.987028i	$-0.551327\pi$
$727$	−8.65786	−0.321102	−0.160551	+	0.987028i	$0.551327\pi$
$728$	13.2867	0.492436
$729$	0	0
$730$	−57.3036	−2.12090
$731$	−5.67363	−0.209847
$732$	0	0
$733$	−28.0866	−1.03740	−0.518702	−	0.854955i	$-0.673584\pi$
$733$	−28.0866	−1.03740	−0.518702	+	0.854955i	$0.673584\pi$
$734$	11.1343	0.410974
$735$	0	0
$736$	−10.2737	−0.378695
$737$	0	0
$738$	0	0
$739$	33.8965	1.24690	0.623451	−	0.781862i	$-0.285730\pi$
$739$	33.8965	1.24690	0.623451	+	0.781862i	$0.285730\pi$
$740$	1.08548	0.0399030
$741$	0	0
$742$	−5.07026	−0.186135
$743$	−5.65321	−0.207396	−0.103698	−	0.994609i	$-0.533068\pi$
$743$	−5.65321	−0.207396	−0.103698	+	0.994609i	$0.533068\pi$
$744$	0	0
$745$	49.4313	1.81102
$746$	46.9918	1.72049
$747$	0	0
$748$	0	0
$749$	−7.94617	−0.290347
$750$	0	0
$751$	44.3409	1.61802	0.809012	−	0.587792i	$-0.200003\pi$
$751$	44.3409	1.61802	0.809012	+	0.587792i	$0.200003\pi$
$752$	−32.6760	−1.19157
$753$	0	0
$754$	−10.8486	−0.395081
$755$	28.4294	1.03465
$756$	0	0
$757$	−21.3781	−0.777001	−0.388500	−	0.921449i	$-0.627007\pi$
$757$	−21.3781	−0.777001	−0.388500	+	0.921449i	$0.627007\pi$
$758$	3.31700	0.120479
$759$	0	0
$760$	−15.4607	−0.560819
$761$	−4.58574	−0.166233	−0.0831164	−	0.996540i	$-0.526487\pi$
$761$	−4.58574	−0.166233	−0.0831164	+	0.996540i	$0.526487\pi$
$762$	0	0
$763$	−4.91440	−0.177913
$764$	−8.79936	−0.318350
$765$	0	0
$766$	−24.5601	−0.887392
$767$	−60.4138	−2.18142
$768$	0	0
$769$	−12.8223	−0.462385	−0.231193	−	0.972908i	$-0.574263\pi$
$769$	−12.8223	−0.462385	−0.231193	+	0.972908i	$0.574263\pi$
$770$	0	0
$771$	0	0
$772$	5.20692	0.187401
$773$	−54.0639	−1.94454	−0.972272	−	0.233855i	$-0.924866\pi$
$773$	−54.0639	−1.94454	−0.972272	+	0.233855i	$0.924866\pi$
$774$	0	0
$775$	−18.3468	−0.659038
$776$	33.0581	1.18672
$777$	0	0
$778$	36.1690	1.29672
$779$	6.00205	0.215046
$780$	0	0
$781$	0	0
$782$	−26.6025	−0.951302
$783$	0	0
$784$	−3.12205	−0.111502
$785$	42.9316	1.53229
$786$	0	0
$787$	43.8514	1.56314	0.781568	−	0.623821i	$-0.214420\pi$
$787$	43.8514	1.56314	0.781568	+	0.623821i	$0.214420\pi$
$788$	−6.69153	−0.238376
$789$	0	0
$790$	23.8297	0.847824
$791$	3.04208	0.108164
$792$	0	0
$793$	51.4928	1.82856
$794$	−13.2337	−0.469646
$795$	0	0
$796$	−0.571776	−0.0202660
$797$	35.7697	1.26703	0.633514	−	0.773731i	$-0.281612\pi$
$797$	35.7697	1.26703	0.633514	+	0.773731i	$0.281612\pi$
$798$	0	0
$799$	43.8687	1.55196
$800$	24.2593	0.857694
$801$	0	0
$802$	16.3244	0.576436
$803$	0	0
$804$	0	0
$805$	20.3418	0.716955
$806$	−8.75982	−0.308551
$807$	0	0
$808$	−22.2885	−0.784107
$809$	−30.7068	−1.07959	−0.539797	−	0.841795i	$-0.681499\pi$
$809$	−30.7068	−1.07959	−0.539797	+	0.841795i	$0.681499\pi$
$810$	0	0
$811$	−17.0984	−0.600405	−0.300202	−	0.953876i	$-0.597054\pi$
$811$	−17.0984	−0.600405	−0.300202	+	0.953876i	$0.597054\pi$
$812$	−0.716849	−0.0251565
$813$	0	0
$814$	0	0
$815$	−31.9845	−1.12037
$816$	0	0
$817$	−1.69039	−0.0591394
$818$	40.8778	1.42926
$819$	0	0
$820$	−7.28342	−0.254348
$821$	−12.3822	−0.432142	−0.216071	−	0.976378i	$-0.569324\pi$
$821$	−12.3822	−0.432142	−0.216071	+	0.976378i	$0.569324\pi$
$822$	0	0
$823$	46.1384	1.60829	0.804143	−	0.594436i	$-0.202625\pi$
$823$	46.1384	1.60829	0.804143	+	0.594436i	$0.202625\pi$
$824$	0.465652	0.0162217
$825$	0	0
$826$	17.5640	0.611131
$827$	46.7103	1.62428	0.812138	−	0.583466i	$-0.198304\pi$
$827$	46.7103	1.62428	0.812138	+	0.583466i	$0.198304\pi$
$828$	0	0
$829$	4.02555	0.139813	0.0699065	−	0.997554i	$-0.477730\pi$
$829$	4.02555	0.139813	0.0699065	+	0.997554i	$0.477730\pi$
$830$	12.8202	0.444996
$831$	0	0
$832$	39.0000	1.35208
$833$	4.19146	0.145226
$834$	0	0
$835$	87.3534	3.02299
$836$	0	0
$837$	0	0
$838$	25.6506	0.886084
$839$	14.3021	0.493765	0.246882	−	0.969045i	$-0.420594\pi$
$839$	14.3021	0.493765	0.246882	+	0.969045i	$0.420594\pi$
$840$	0	0
$841$	−25.2541	−0.870833
$842$	−28.6260	−0.986518
$843$	0	0
$844$	−8.17941	−0.281547
$845$	25.6945	0.883918
$846$	0	0
$847$	0	0
$848$	−12.4002	−0.425823
$849$	0	0
$850$	62.8161	2.15457
$851$	3.56128	0.122079
$852$	0	0
$853$	32.3007	1.10595	0.552977	−	0.833197i	$-0.313492\pi$
$853$	32.3007	1.10595	0.552977	+	0.833197i	$0.313492\pi$
$854$	−14.9704	−0.512278
$855$	0	0
$856$	−24.0447	−0.821830
$857$	16.2318	0.554468	0.277234	−	0.960802i	$-0.410582\pi$
$857$	16.2318	0.554468	0.277234	+	0.960802i	$0.410582\pi$
$858$	0	0
$859$	−28.2893	−0.965220	−0.482610	−	0.875835i	$-0.660311\pi$
$859$	−28.2893	−0.965220	−0.482610	+	0.875835i	$0.660311\pi$
$860$	2.05127	0.0699479
$861$	0	0
$862$	15.4281	0.525482
$863$	−31.9865	−1.08883	−0.544417	−	0.838815i	$-0.683249\pi$
$863$	−31.9865	−1.08883	−0.544417	+	0.838815i	$0.683249\pi$
$864$	0	0
$865$	79.5309	2.70413
$866$	−0.386101	−0.0131203
$867$	0	0
$868$	−0.578830	−0.0196468
$869$	0	0
$870$	0	0
$871$	33.3390	1.12965
$872$	−14.8707	−0.503586
$873$	0	0
$874$	−7.92590	−0.268098
$875$	−27.5757	−0.932229
$876$	0	0
$877$	−45.7968	−1.54645	−0.773224	−	0.634133i	$-0.781357\pi$
$877$	−45.7968	−1.54645	−0.773224	+	0.634133i	$0.781357\pi$
$878$	4.50107	0.151904
$879$	0	0
$880$	0	0
$881$	−26.7142	−0.900025	−0.450012	−	0.893022i	$-0.648580\pi$
$881$	−26.7142	−0.900025	−0.450012	+	0.893022i	$0.648580\pi$
$882$	0	0
$883$	−28.4869	−0.958662	−0.479331	−	0.877634i	$-0.659121\pi$
$883$	−28.4869	−0.958662	−0.479331	+	0.877634i	$0.659121\pi$
$884$	−6.81667	−0.229269
$885$	0	0
$886$	27.8857	0.936838
$887$	−23.6139	−0.792879	−0.396439	−	0.918061i	$-0.629754\pi$
$887$	−23.6139	−0.792879	−0.396439	+	0.918061i	$0.629754\pi$
$888$	0	0
$889$	−10.7954	−0.362064
$890$	−21.9604	−0.736113
$891$	0	0
$892$	3.17572	0.106331
$893$	13.0702	0.437377
$894$	0	0
$895$	−73.6312	−2.46122
$896$	−7.20562	−0.240723
$897$	0	0
$898$	−11.8742	−0.396249
$899$	3.02464	0.100877
$900$	0	0
$901$	16.6477	0.554614
$902$	0	0
$903$	0	0
$904$	9.20519	0.306160
$905$	62.0161	2.06149
$906$	0	0
$907$	23.9434	0.795029	0.397514	−	0.917596i	$-0.369873\pi$
$907$	23.9434	0.795029	0.397514	+	0.917596i	$0.369873\pi$
$908$	9.84143	0.326599
$909$	0	0
$910$	−22.9336	−0.760242
$911$	−18.2388	−0.604280	−0.302140	−	0.953264i	$-0.597701\pi$
$911$	−18.2388	−0.604280	−0.302140	+	0.953264i	$0.597701\pi$
$912$	0	0
$913$	0	0
$914$	14.4463	0.477841
$915$	0	0
$916$	−4.22657	−0.139650
$917$	8.70429	0.287441
$918$	0	0
$919$	−28.4024	−0.936909	−0.468455	−	0.883488i	$-0.655189\pi$
$919$	−28.4024	−0.936909	−0.468455	+	0.883488i	$0.655189\pi$
$920$	61.5533	2.02935
$921$	0	0
$922$	−0.866558	−0.0285386
$923$	−0.958607	−0.0315529
$924$	0	0
$925$	−8.40922	−0.276493
$926$	−17.2255	−0.566065
$927$	0	0
$928$	−3.99936	−0.131285
$929$	−3.38963	−0.111210	−0.0556051	−	0.998453i	$-0.517709\pi$
$929$	−3.38963	−0.111210	−0.0556051	+	0.998453i	$0.517709\pi$
$930$	0	0
$931$	1.24880	0.0409277
$932$	−9.85704	−0.322878
$933$	0	0
$934$	−37.8588	−1.23878
$935$	0	0
$936$	0	0
$937$	−45.3871	−1.48273	−0.741367	−	0.671100i	$-0.765822\pi$
$937$	−45.3871	−1.48273	−0.741367	+	0.671100i	$0.765822\pi$
$938$	−9.69261	−0.316475
$939$	0	0
$940$	−15.8605	−0.517313
$941$	21.5279	0.701789	0.350894	−	0.936415i	$-0.385878\pi$
$941$	21.5279	0.701789	0.350894	+	0.936415i	$0.385878\pi$
$942$	0	0
$943$	−23.8958	−0.778153
$944$	42.9558	1.39809
$945$	0	0
$946$	0	0
$947$	11.2673	0.366140	0.183070	−	0.983100i	$-0.441397\pi$
$947$	11.2673	0.366140	0.183070	+	0.983100i	$0.441397\pi$
$948$	0	0
$949$	−48.1745	−1.56381
$950$	18.7153	0.607206
$951$	0	0
$952$	12.6831	0.411063
$953$	−23.8696	−0.773213	−0.386607	−	0.922245i	$-0.626353\pi$
$953$	−23.8696	−0.773213	−0.386607	+	0.922245i	$0.626353\pi$
$954$	0	0
$955$	97.2019	3.14538
$956$	2.78262	0.0899964
$957$	0	0
$958$	6.00159	0.193902
$959$	11.9977	0.387426
$960$	0	0
$961$	−28.5577	−0.921217
$962$	−4.01504	−0.129450
$963$	0	0
$964$	10.1629	0.327325
$965$	−57.5181	−1.85157
$966$	0	0
$967$	−20.5165	−0.659765	−0.329882	−	0.944022i	$-0.607009\pi$
$967$	−20.5165	−0.659765	−0.329882	+	0.944022i	$0.607009\pi$
$968$	0	0
$969$	0	0
$970$	−57.0605	−1.83210
$971$	34.7645	1.11565	0.557824	−	0.829960i	$-0.311637\pi$
$971$	34.7645	1.11565	0.557824	+	0.829960i	$0.311637\pi$
$972$	0	0
$973$	8.18967	0.262549
$974$	41.3272	1.32421
$975$	0	0
$976$	−36.6127	−1.17194
$977$	−5.78294	−0.185013	−0.0925064	−	0.995712i	$-0.529488\pi$
$977$	−5.78294	−0.185013	−0.0925064	+	0.995712i	$0.529488\pi$
$978$	0	0
$979$	0	0
$980$	−1.51540	−0.0484078
$981$	0	0
$982$	−8.75204	−0.279289
$983$	17.2321	0.549618	0.274809	−	0.961499i	$-0.411385\pi$
$983$	17.2321	0.549618	0.274809	+	0.961499i	$0.411385\pi$
$984$	0	0
$985$	73.9178	2.35522
$986$	−10.3558	−0.329796
$987$	0	0
$988$	−2.03095	−0.0646131
$989$	6.72991	0.213999
$990$	0	0
$991$	−24.0077	−0.762630	−0.381315	−	0.924445i	$-0.624529\pi$
$991$	−24.0077	−0.762630	−0.381315	+	0.924445i	$0.624529\pi$
$992$	−3.22933	−0.102531
$993$	0	0
$994$	0.278695	0.00883966
$995$	6.31610	0.200234
$996$	0	0
$997$	−36.0335	−1.14119	−0.570596	−	0.821231i	$-0.693288\pi$
$997$	−36.0335	−1.14119	−0.570596	+	0.821231i	$0.693288\pi$
$998$	14.0001	0.443166
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.27656 q^{2} -0.370384 q^{4} +4.09144 q^{5} -1.00000 q^{7} -3.02595 q^{8} +O(q^{10})\)	\(q+1.27656 q^{2} -0.370384 q^{4} +4.09144 q^{5} -1.00000 q^{7} -3.02595 q^{8} +5.22298 q^{10} +4.39091 q^{13} -1.27656 q^{14} -3.12205 q^{16} +4.19146 q^{17} +1.24880 q^{19} -1.51540 q^{20} -4.97180 q^{23} +11.7399 q^{25} +5.60527 q^{26} +0.370384 q^{28} -1.93542 q^{29} -1.56278 q^{31} +2.06640 q^{32} +5.35067 q^{34} -4.09144 q^{35} -0.716296 q^{37} +1.59417 q^{38} -12.3805 q^{40} +4.80626 q^{41} -1.35362 q^{43} -6.34682 q^{46} +10.4662 q^{47} +1.00000 q^{49} +14.9867 q^{50} -1.62632 q^{52} +3.97180 q^{53} +3.02595 q^{56} -2.47069 q^{58} -13.7588 q^{59} +11.7271 q^{61} -1.99499 q^{62} +8.88199 q^{64} +17.9651 q^{65} +7.59274 q^{67} -1.55245 q^{68} -5.22298 q^{70} -0.218316 q^{71} -10.9714 q^{73} -0.914398 q^{74} -0.462535 q^{76} +4.56248 q^{79} -12.7737 q^{80} +6.13550 q^{82} +2.45458 q^{83} +17.1491 q^{85} -1.72798 q^{86} -4.20456 q^{89} -4.39091 q^{91} +1.84148 q^{92} +13.3608 q^{94} +5.10938 q^{95} -10.9249 q^{97} +1.27656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 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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