LMFDB - Embedded newform 7350.2.a.ds.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.69230$ of defining polynomial
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.979056 q^{11} +1.00000 q^{12} +0.435157 q^{13} +1.00000 q^{16} -2.79881 q^{17} -1.00000 q^{18} -7.34271 q^{19} -0.979056 q^{22} +3.34271 q^{23} -1.00000 q^{24} -0.435157 q^{26} +1.00000 q^{27} +3.74825 q^{29} -4.97906 q^{31} -1.00000 q^{32} +0.979056 q^{33} +2.79881 q^{34} +1.00000 q^{36} -4.63591 q^{37} +7.34271 q^{38} +0.435157 q^{39} -4.94944 q^{41} +9.97862 q^{43} +0.979056 q^{44} -3.34271 q^{46} +4.40554 q^{47} +1.00000 q^{48} -2.79881 q^{51} +0.435157 q^{52} -0.657293 q^{53} -1.00000 q^{54} -7.34271 q^{57} -3.74825 q^{58} -8.27226 q^{59} +11.3336 q^{61} +4.97906 q^{62} +1.00000 q^{64} -0.979056 q^{66} -2.36365 q^{67} -2.79881 q^{68} +3.34271 q^{69} -14.1119 q^{71} -1.00000 q^{72} -15.3928 q^{73} +4.63591 q^{74} -7.34271 q^{76} -0.435157 q^{78} -2.88767 q^{79} +1.00000 q^{81} +4.94944 q^{82} +14.3842 q^{83} -9.97862 q^{86} +3.74825 q^{87} -0.979056 q^{88} -10.1210 q^{89} +3.34271 q^{92} -4.97906 q^{93} -4.40554 q^{94} -1.00000 q^{96} +10.6655 q^{97} +0.979056 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{12} + 4 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} - 4 q^{23} - 4 q^{24} + 4 q^{27} - 8 q^{29} - 16 q^{31} - 4 q^{32} - 4 q^{34} + 4 q^{36} + 8 q^{37} + 12 q^{38} - 12 q^{41} - 4 q^{43} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 4 q^{51} - 20 q^{53} - 4 q^{54} - 12 q^{57} + 8 q^{58} - 20 q^{59} - 12 q^{61} + 16 q^{62} + 4 q^{64} + 4 q^{67} + 4 q^{68} - 4 q^{69} - 20 q^{71} - 4 q^{72} - 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} + 12 q^{82} + 8 q^{83} + 4 q^{86} - 8 q^{87} - 44 q^{89} - 4 q^{92} - 16 q^{93} - 12 q^{94} - 4 q^{96} + 20 q^{97}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^12 + 4 * q^16 + 4 * q^17 - 4 * q^18 - 12 * q^19 - 4 * q^23 - 4 * q^24 + 4 * q^27 - 8 * q^29 - 16 * q^31 - 4 * q^32 - 4 * q^34 + 4 * q^36 + 8 * q^37 + 12 * q^38 - 12 * q^41 - 4 * q^43 + 4 * q^46 + 12 * q^47 + 4 * q^48 + 4 * q^51 - 20 * q^53 - 4 * q^54 - 12 * q^57 + 8 * q^58 - 20 * q^59 - 12 * q^61 + 16 * q^62 + 4 * q^64 + 4 * q^67 + 4 * q^68 - 4 * q^69 - 20 * q^71 - 4 * q^72 - 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 + 12 * q^82 + 8 * q^83 + 4 * q^86 - 8 * q^87 - 44 * q^89 - 4 * q^92 - 16 * q^93 - 12 * q^94 - 4 * q^96 + 20 * q^97

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0.979056	0.295197	0.147598	−	0.989047i	$-0.452846\pi$
$11$	0.979056	0.295197	0.147598	+	0.989047i	$0.452846\pi$
$12$	1.00000	0.288675
$13$	0.435157	0.120691	0.0603455	−	0.998178i	$-0.480780\pi$
$13$	0.435157	0.120691	0.0603455	+	0.998178i	$0.480780\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.79881	−0.678811	−0.339405	−	0.940640i	$-0.610226\pi$
$17$	−2.79881	−0.678811	−0.339405	+	0.940640i	$0.610226\pi$
$18$	−1.00000	−0.235702
$19$	−7.34271	−1.68453	−0.842266	−	0.539062i	$-0.818779\pi$
$19$	−7.34271	−1.68453	−0.842266	+	0.539062i	$0.818779\pi$
$20$	0	0
$21$	0	0
$22$	−0.979056	−0.208735
$23$	3.34271	0.697003	0.348501	−	0.937308i	$-0.386691\pi$
$23$	3.34271	0.697003	0.348501	+	0.937308i	$0.386691\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−0.435157	−0.0853414
$27$	1.00000	0.192450
$28$	0	0
$29$	3.74825	0.696032	0.348016	−	0.937489i	$-0.386856\pi$
$29$	3.74825	0.696032	0.348016	+	0.937489i	$0.386856\pi$
$30$	0	0
$31$	−4.97906	−0.894265	−0.447132	−	0.894468i	$-0.647555\pi$
$31$	−4.97906	−0.894265	−0.447132	+	0.894468i	$0.647555\pi$
$32$	−1.00000	−0.176777
$33$	0.979056	0.170432
$34$	2.79881	0.479992
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.63591	−0.762139	−0.381069	−	0.924546i	$-0.624444\pi$
$37$	−4.63591	−0.762139	−0.381069	+	0.924546i	$0.624444\pi$
$38$	7.34271	1.19114
$39$	0.435157	0.0696809
$40$	0	0
$41$	−4.94944	−0.772972	−0.386486	−	0.922295i	$-0.626311\pi$
$41$	−4.94944	−0.772972	−0.386486	+	0.922295i	$0.626311\pi$
$42$	0	0
$43$	9.97862	1.52172	0.760862	−	0.648913i	$-0.224776\pi$
$43$	9.97862	1.52172	0.760862	+	0.648913i	$0.224776\pi$
$44$	0.979056	0.147598
$45$	0	0
$46$	−3.34271	−0.492855
$47$	4.40554	0.642614	0.321307	−	0.946975i	$-0.395878\pi$
$47$	4.40554	0.642614	0.321307	+	0.946975i	$0.395878\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	−2.79881	−0.391912
$52$	0.435157	0.0603455
$53$	−0.657293	−0.0902861	−0.0451431	−	0.998981i	$-0.514374\pi$
$53$	−0.657293	−0.0902861	−0.0451431	+	0.998981i	$0.514374\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−7.34271	−0.972565
$58$	−3.74825	−0.492169
$59$	−8.27226	−1.07696	−0.538478	−	0.842639i	$-0.681001\pi$
$59$	−8.27226	−1.07696	−0.538478	+	0.842639i	$0.681001\pi$
$60$	0	0
$61$	11.3336	1.45112	0.725559	−	0.688160i	$-0.241581\pi$
$61$	11.3336	1.45112	0.725559	+	0.688160i	$0.241581\pi$
$62$	4.97906	0.632341
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.979056	−0.120513
$67$	−2.36365	−0.288766	−0.144383	−	0.989522i	$-0.546120\pi$
$67$	−2.36365	−0.288766	−0.144383	+	0.989522i	$0.546120\pi$
$68$	−2.79881	−0.339405
$69$	3.34271	0.402415
$70$	0	0
$71$	−14.1119	−1.67477	−0.837387	−	0.546610i	$-0.815918\pi$
$71$	−14.1119	−1.67477	−0.837387	+	0.546610i	$0.815918\pi$
$72$	−1.00000	−0.117851
$73$	−15.3928	−1.80159	−0.900797	−	0.434240i	$-0.857017\pi$
$73$	−15.3928	−1.80159	−0.900797	+	0.434240i	$0.857017\pi$
$74$	4.63591	0.538914
$75$	0	0
$76$	−7.34271	−0.842266
$77$	0	0
$78$	−0.435157	−0.0492719
$79$	−2.88767	−0.324888	−0.162444	−	0.986718i	$-0.551938\pi$
$79$	−2.88767	−0.324888	−0.162444	+	0.986718i	$0.551938\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.94944	0.546574
$83$	14.3842	1.57887	0.789433	−	0.613837i	$-0.210375\pi$
$83$	14.3842	1.57887	0.789433	+	0.613837i	$0.210375\pi$
$84$	0	0
$85$	0	0
$86$	−9.97862	−1.07602
$87$	3.74825	0.401854
$88$	−0.979056	−0.104368
$89$	−10.1210	−1.07282	−0.536412	−	0.843956i	$-0.680221\pi$
$89$	−10.1210	−1.07282	−0.536412	+	0.843956i	$0.680221\pi$
$90$	0	0
$91$	0	0
$92$	3.34271	0.348501
$93$	−4.97906	−0.516304
$94$	−4.40554	−0.454397
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.6655	1.08292	0.541460	−	0.840726i	$-0.317872\pi$
$97$	10.6655	1.08292	0.541460	+	0.840726i	$0.317872\pi$
$98$	0	0
$99$	0.979056	0.0983989
$100$	0	0
$101$	−14.0328	−1.39631	−0.698157	−	0.715945i	$-0.745996\pi$
$101$	−14.0328	−1.39631	−0.698157	+	0.715945i	$0.745996\pi$
$102$	2.79881	0.277123
$103$	−10.3842	−1.02318	−0.511591	−	0.859229i	$-0.670944\pi$
$103$	−10.3842	−1.02318	−0.511591	+	0.859229i	$0.670944\pi$
$104$	−0.435157	−0.0426707
$105$	0	0
$106$	0.657293	0.0638419
$107$	−20.0410	−1.93744	−0.968719	−	0.248161i	$-0.920174\pi$
$107$	−20.0410	−1.93744	−0.968719	+	0.248161i	$0.920174\pi$
$108$	1.00000	0.0962250
$109$	18.6564	1.78696	0.893480	−	0.449102i	$-0.148256\pi$
$109$	18.6564	1.78696	0.893480	+	0.449102i	$0.148256\pi$
$110$	0	0
$111$	−4.63591	−0.440021
$112$	0	0
$113$	−10.4132	−0.979587	−0.489794	−	0.871838i	$-0.662928\pi$
$113$	−10.4132	−0.979587	−0.489794	+	0.871838i	$0.662928\pi$
$114$	7.34271	0.687708
$115$	0	0
$116$	3.74825	0.348016
$117$	0.435157	0.0402303
$118$	8.27226	0.761523
$119$	0	0
$120$	0	0
$121$	−10.0414	−0.912859
$122$	−11.3336	−1.02610
$123$	−4.94944	−0.446276
$124$	−4.97906	−0.447132
$125$	0	0
$126$	0	0
$127$	11.6987	1.03810	0.519048	−	0.854745i	$-0.326287\pi$
$127$	11.6987	1.03810	0.519048	+	0.854745i	$0.326287\pi$
$128$	−1.00000	−0.0883883
$129$	9.97862	0.878568
$130$	0	0
$131$	−5.38459	−0.470454	−0.235227	−	0.971940i	$-0.575583\pi$
$131$	−5.38459	−0.470454	−0.235227	+	0.971940i	$0.575583\pi$
$132$	0.979056	0.0852159
$133$	0	0
$134$	2.36365	0.204188
$135$	0	0
$136$	2.79881	0.239996
$137$	−3.68585	−0.314904	−0.157452	−	0.987527i	$-0.550328\pi$
$137$	−3.68585	−0.314904	−0.157452	+	0.987527i	$0.550328\pi$
$138$	−3.34271	−0.284550
$139$	−8.15378	−0.691595	−0.345797	−	0.938309i	$-0.612392\pi$
$139$	−8.15378	−0.691595	−0.345797	+	0.938309i	$0.612392\pi$
$140$	0	0
$141$	4.40554	0.371013
$142$	14.1119	1.18424
$143$	0.426043	0.0356275
$144$	1.00000	0.0833333
$145$	0	0
$146$	15.3928	1.27392
$147$	0	0
$148$	−4.63591	−0.381069
$149$	−15.6774	−1.28434	−0.642170	−	0.766563i	$-0.721966\pi$
$149$	−15.6774	−1.28434	−0.642170	+	0.766563i	$0.721966\pi$
$150$	0	0
$151$	4.27226	0.347672	0.173836	−	0.984775i	$-0.444384\pi$
$151$	4.27226	0.347672	0.173836	+	0.984775i	$0.444384\pi$
$152$	7.34271	0.595572
$153$	−2.79881	−0.226270
$154$	0	0
$155$	0	0
$156$	0.435157	0.0348405
$157$	15.2498	1.21707	0.608534	−	0.793528i	$-0.291758\pi$
$157$	15.2498	1.21707	0.608534	+	0.793528i	$0.291758\pi$
$158$	2.88767	0.229730
$159$	−0.657293	−0.0521267
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	7.47599	0.585564	0.292782	−	0.956179i	$-0.405419\pi$
$163$	7.47599	0.585564	0.292782	+	0.956179i	$0.405419\pi$
$164$	−4.94944	−0.386486
$165$	0	0
$166$	−14.3842	−1.11643
$167$	−12.6640	−0.979972	−0.489986	−	0.871730i	$-0.662998\pi$
$167$	−12.6640	−0.979972	−0.489986	+	0.871730i	$0.662998\pi$
$168$	0	0
$169$	−12.8106	−0.985434
$170$	0	0
$171$	−7.34271	−0.561511
$172$	9.97862	0.760862
$173$	7.12057	0.541367	0.270684	−	0.962668i	$-0.412750\pi$
$173$	7.12057	0.541367	0.270684	+	0.962668i	$0.412750\pi$
$174$	−3.74825	−0.284154
$175$	0	0
$176$	0.979056	0.0737991
$177$	−8.27226	−0.621781
$178$	10.1210	0.758602
$179$	3.26420	0.243978	0.121989	−	0.992531i	$-0.461073\pi$
$179$	3.26420	0.243978	0.121989	+	0.992531i	$0.461073\pi$
$180$	0	0
$181$	−1.93823	−0.144067	−0.0720337	−	0.997402i	$-0.522949\pi$
$181$	−1.93823	−0.144067	−0.0720337	+	0.997402i	$0.522949\pi$
$182$	0	0
$183$	11.3336	0.837803
$184$	−3.34271	−0.246428
$185$	0	0
$186$	4.97906	0.365082
$187$	−2.74019	−0.200383
$188$	4.40554	0.321307
$189$	0	0
$190$	0	0
$191$	8.07001	0.583925	0.291963	−	0.956430i	$-0.405692\pi$
$191$	8.07001	0.583925	0.291963	+	0.956430i	$0.405692\pi$
$192$	1.00000	0.0721688
$193$	−1.95811	−0.140948	−0.0704740	−	0.997514i	$-0.522451\pi$
$193$	−1.95811	−0.140948	−0.0704740	+	0.997514i	$0.522451\pi$
$194$	−10.6655	−0.765740
$195$	0	0
$196$	0	0
$197$	17.3418	1.23555	0.617777	−	0.786353i	$-0.288033\pi$
$197$	17.3418	1.23555	0.617777	+	0.786353i	$0.288033\pi$
$198$	−0.979056	−0.0695785
$199$	−2.59490	−0.183948	−0.0919738	−	0.995761i	$-0.529318\pi$
$199$	−2.59490	−0.183948	−0.0919738	+	0.995761i	$0.529318\pi$
$200$	0	0
$201$	−2.36365	−0.166719
$202$	14.0328	0.987343
$203$	0	0
$204$	−2.79881	−0.195956
$205$	0	0
$206$	10.3842	0.723498
$207$	3.34271	0.232334
$208$	0.435157	0.0301727
$209$	−7.18892	−0.497268
$210$	0	0
$211$	15.4127	1.06106	0.530528	−	0.847668i	$-0.321994\pi$
$211$	15.4127	1.06106	0.530528	+	0.847668i	$0.321994\pi$
$212$	−0.657293	−0.0451431
$213$	−14.1119	−0.966931
$214$	20.0410	1.36998
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−18.6564	−1.26357
$219$	−15.3928	−1.04015
$220$	0	0
$221$	−1.21792	−0.0819263
$222$	4.63591	0.311142
$223$	7.61497	0.509936	0.254968	−	0.966950i	$-0.417935\pi$
$223$	7.61497	0.509936	0.254968	+	0.966950i	$0.417935\pi$
$224$	0	0
$225$	0	0
$226$	10.4132	0.692673
$227$	−8.34227	−0.553696	−0.276848	−	0.960914i	$-0.589290\pi$
$227$	−8.34227	−0.553696	−0.276848	+	0.960914i	$0.589290\pi$
$228$	−7.34271	−0.486283
$229$	−18.6901	−1.23507	−0.617537	−	0.786542i	$-0.711869\pi$
$229$	−18.6901	−1.23507	−0.617537	+	0.786542i	$0.711869\pi$
$230$	0	0
$231$	0	0
$232$	−3.74825	−0.246084
$233$	−20.0829	−1.31567	−0.657837	−	0.753160i	$-0.728529\pi$
$233$	−20.0829	−1.31567	−0.657837	+	0.753160i	$0.728529\pi$
$234$	−0.435157	−0.0284471
$235$	0	0
$236$	−8.27226	−0.538478
$237$	−2.88767	−0.187574
$238$	0	0
$239$	−22.2714	−1.44062	−0.720308	−	0.693654i	$-0.756000\pi$
$239$	−22.2714	−1.44062	−0.720308	+	0.693654i	$0.756000\pi$
$240$	0	0
$241$	6.20075	0.399426	0.199713	−	0.979854i	$-0.435999\pi$
$241$	6.20075	0.399426	0.199713	+	0.979854i	$0.435999\pi$
$242$	10.0414	0.645489
$243$	1.00000	0.0641500
$244$	11.3336	0.725559
$245$	0	0
$246$	4.94944	0.315565
$247$	−3.19523	−0.203308
$248$	4.97906	0.316170
$249$	14.3842	0.911559
$250$	0	0
$251$	27.6560	1.74563	0.872815	−	0.488051i	$-0.162292\pi$
$251$	27.6560	1.74563	0.872815	+	0.488051i	$0.162292\pi$
$252$	0	0
$253$	3.27270	0.205753
$254$	−11.6987	−0.734044
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.7681	−0.734076	−0.367038	−	0.930206i	$-0.619628\pi$
$257$	−11.7681	−0.734076	−0.367038	+	0.930206i	$0.619628\pi$
$258$	−9.97862	−0.621242
$259$	0	0
$260$	0	0
$261$	3.74825	0.232011
$262$	5.38459	0.332661
$263$	0.455042	0.0280591	0.0140295	−	0.999902i	$-0.495534\pi$
$263$	0.455042	0.0280591	0.0140295	+	0.999902i	$0.495534\pi$
$264$	−0.979056	−0.0602567
$265$	0	0
$266$	0	0
$267$	−10.1210	−0.619396
$268$	−2.36365	−0.144383
$269$	−26.1375	−1.59363	−0.796815	−	0.604223i	$-0.793484\pi$
$269$	−26.1375	−1.59363	−0.796815	+	0.604223i	$0.793484\pi$
$270$	0	0
$271$	−16.7768	−1.01912	−0.509559	−	0.860436i	$-0.670192\pi$
$271$	−16.7768	−1.01912	−0.509559	+	0.860436i	$0.670192\pi$
$272$	−2.79881	−0.169703
$273$	0	0
$274$	3.68585	0.222671
$275$	0	0
$276$	3.34271	0.201207
$277$	20.7478	1.24661	0.623307	−	0.781977i	$-0.285789\pi$
$277$	20.7478	1.24661	0.623307	+	0.781977i	$0.285789\pi$
$278$	8.15378	0.489031
$279$	−4.97906	−0.298088
$280$	0	0
$281$	−18.3141	−1.09253	−0.546265	−	0.837612i	$-0.683951\pi$
$281$	−18.3141	−1.09253	−0.546265	+	0.837612i	$0.683951\pi$
$282$	−4.40554	−0.262346
$283$	−11.8525	−0.704560	−0.352280	−	0.935895i	$-0.614593\pi$
$283$	−11.8525	−0.704560	−0.352280	+	0.935895i	$0.614593\pi$
$284$	−14.1119	−0.837387
$285$	0	0
$286$	−0.426043	−0.0251925
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−9.16667	−0.539216
$290$	0	0
$291$	10.6655	0.625224
$292$	−15.3928	−0.900797
$293$	−14.4770	−0.845758	−0.422879	−	0.906186i	$-0.638980\pi$
$293$	−14.4770	−0.845758	−0.422879	+	0.906186i	$0.638980\pi$
$294$	0	0
$295$	0	0
$296$	4.63591	0.269457
$297$	0.979056	0.0568106
$298$	15.6774	0.908165
$299$	1.45460	0.0841219
$300$	0	0
$301$	0	0
$302$	−4.27226	−0.245841
$303$	−14.0328	−0.806162
$304$	−7.34271	−0.421133
$305$	0	0
$306$	2.79881	0.159997
$307$	30.4252	1.73646	0.868228	−	0.496165i	$-0.165259\pi$
$307$	30.4252	1.73646	0.868228	+	0.496165i	$0.165259\pi$
$308$	0	0
$309$	−10.3842	−0.590734
$310$	0	0
$311$	9.02856	0.511963	0.255981	−	0.966682i	$-0.417601\pi$
$311$	9.02856	0.511963	0.255981	+	0.966682i	$0.417601\pi$
$312$	−0.435157	−0.0246359
$313$	17.1732	0.970688	0.485344	−	0.874323i	$-0.338694\pi$
$313$	17.1732	0.970688	0.485344	+	0.874323i	$0.338694\pi$
$314$	−15.2498	−0.860597
$315$	0	0
$316$	−2.88767	−0.162444
$317$	−2.61541	−0.146896	−0.0734479	−	0.997299i	$-0.523400\pi$
$317$	−2.61541	−0.146896	−0.0734479	+	0.997299i	$0.523400\pi$
$318$	0.657293	0.0368592
$319$	3.66974	0.205466
$320$	0	0
$321$	−20.0410	−1.11858
$322$	0	0
$323$	20.5508	1.14348
$324$	1.00000	0.0555556
$325$	0	0
$326$	−7.47599	−0.414057
$327$	18.6564	1.03170
$328$	4.94944	0.273287
$329$	0	0
$330$	0	0
$331$	−17.7559	−0.975950	−0.487975	−	0.872857i	$-0.662264\pi$
$331$	−17.7559	−0.975950	−0.487975	+	0.872857i	$0.662264\pi$
$332$	14.3842	0.789433
$333$	−4.63591	−0.254046
$334$	12.6640	0.692945
$335$	0	0
$336$	0	0
$337$	24.2304	1.31991	0.659956	−	0.751304i	$-0.270575\pi$
$337$	24.2304	1.31991	0.659956	+	0.751304i	$0.270575\pi$
$338$	12.8106	0.696807
$339$	−10.4132	−0.565565
$340$	0	0
$341$	−4.87478	−0.263984
$342$	7.34271	0.397048
$343$	0	0
$344$	−9.97862	−0.538011
$345$	0	0
$346$	−7.12057	−0.382804
$347$	−30.4252	−1.63331	−0.816654	−	0.577127i	$-0.804174\pi$
$347$	−30.4252	−1.63331	−0.816654	+	0.577127i	$0.804174\pi$
$348$	3.74825	0.200927
$349$	−30.2623	−1.61990	−0.809951	−	0.586497i	$-0.800506\pi$
$349$	−30.2623	−1.61990	−0.809951	+	0.586497i	$0.800506\pi$
$350$	0	0
$351$	0.435157	0.0232270
$352$	−0.979056	−0.0521839
$353$	4.37189	0.232692	0.116346	−	0.993209i	$-0.462882\pi$
$353$	4.37189	0.232692	0.116346	+	0.993209i	$0.462882\pi$
$354$	8.27226	0.439666
$355$	0	0
$356$	−10.1210	−0.536412
$357$	0	0
$358$	−3.26420	−0.172519
$359$	−15.6850	−0.827821	−0.413911	−	0.910318i	$-0.635837\pi$
$359$	−15.6850	−0.827821	−0.413911	+	0.910318i	$0.635837\pi$
$360$	0	0
$361$	34.9153	1.83765
$362$	1.93823	0.101871
$363$	−10.0414	−0.527039
$364$	0	0
$365$	0	0
$366$	−11.3336	−0.592416
$367$	−4.94478	−0.258116	−0.129058	−	0.991637i	$-0.541195\pi$
$367$	−4.94478	−0.258116	−0.129058	+	0.991637i	$0.541195\pi$
$368$	3.34271	0.174251
$369$	−4.94944	−0.257657
$370$	0	0
$371$	0	0
$372$	−4.97906	−0.258152
$373$	6.02138	0.311775	0.155888	−	0.987775i	$-0.450176\pi$
$373$	6.02138	0.311775	0.155888	+	0.987775i	$0.450176\pi$
$374$	2.74019	0.141692
$375$	0	0
$376$	−4.40554	−0.227198
$377$	1.63108	0.0840047
$378$	0	0
$379$	−36.8030	−1.89044	−0.945222	−	0.326429i	$-0.894155\pi$
$379$	−36.8030	−1.89044	−0.945222	+	0.326429i	$0.894155\pi$
$380$	0	0
$381$	11.6987	0.599345
$382$	−8.07001	−0.412898
$383$	9.03383	0.461607	0.230804	−	0.973000i	$-0.425864\pi$
$383$	9.03383	0.461607	0.230804	+	0.973000i	$0.425864\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	1.95811	0.0996653
$387$	9.97862	0.507242
$388$	10.6655	0.541460
$389$	33.5646	1.70179	0.850896	−	0.525334i	$-0.176060\pi$
$389$	33.5646	1.70179	0.850896	+	0.525334i	$0.176060\pi$
$390$	0	0
$391$	−9.35560	−0.473133
$392$	0	0
$393$	−5.38459	−0.271617
$394$	−17.3418	−0.873669
$395$	0	0
$396$	0.979056	0.0491994
$397$	−33.9792	−1.70537	−0.852685	−	0.522426i	$-0.825027\pi$
$397$	−33.9792	−1.70537	−0.852685	+	0.522426i	$0.825027\pi$
$398$	2.59490	0.130071
$399$	0	0
$400$	0	0
$401$	−4.27226	−0.213346	−0.106673	−	0.994294i	$-0.534020\pi$
$401$	−4.27226	−0.213346	−0.106673	+	0.994294i	$0.534020\pi$
$402$	2.36365	0.117888
$403$	−2.16667	−0.107930
$404$	−14.0328	−0.698157
$405$	0	0
$406$	0	0
$407$	−4.53882	−0.224981
$408$	2.79881	0.138562
$409$	−3.63888	−0.179931	−0.0899656	−	0.995945i	$-0.528676\pi$
$409$	−3.63888	−0.179931	−0.0899656	+	0.995945i	$0.528676\pi$
$410$	0	0
$411$	−3.68585	−0.181810
$412$	−10.3842	−0.511591
$413$	0	0
$414$	−3.34271	−0.164285
$415$	0	0
$416$	−0.435157	−0.0213353
$417$	−8.15378	−0.399293
$418$	7.18892	0.351622
$419$	26.6274	1.30083	0.650417	−	0.759577i	$-0.274594\pi$
$419$	26.6274	1.30083	0.650417	+	0.759577i	$0.274594\pi$
$420$	0	0
$421$	−15.1880	−0.740220	−0.370110	−	0.928988i	$-0.620680\pi$
$421$	−15.1880	−0.740220	−0.370110	+	0.928988i	$0.620680\pi$
$422$	−15.4127	−0.750279
$423$	4.40554	0.214205
$424$	0.657293	0.0319210
$425$	0	0
$426$	14.1119	0.683724
$427$	0	0
$428$	−20.0410	−0.968719
$429$	0.426043	0.0205696
$430$	0	0
$431$	−4.23125	−0.203812	−0.101906	−	0.994794i	$-0.532494\pi$
$431$	−4.23125	−0.203812	−0.101906	+	0.994794i	$0.532494\pi$
$432$	1.00000	0.0481125
$433$	−33.9352	−1.63082	−0.815412	−	0.578882i	$-0.803489\pi$
$433$	−33.9352	−1.63082	−0.815412	+	0.578882i	$0.803489\pi$
$434$	0	0
$435$	0	0
$436$	18.6564	0.893480
$437$	−24.5445	−1.17412
$438$	15.3928	0.735498
$439$	−5.48257	−0.261669	−0.130834	−	0.991404i	$-0.541766\pi$
$439$	−5.48257	−0.261669	−0.130834	+	0.991404i	$0.541766\pi$
$440$	0	0
$441$	0	0
$442$	1.21792	0.0579306
$443$	−1.23081	−0.0584776	−0.0292388	−	0.999572i	$-0.509308\pi$
$443$	−1.23081	−0.0584776	−0.0292388	+	0.999572i	$0.509308\pi$
$444$	−4.63591	−0.220011
$445$	0	0
$446$	−7.61497	−0.360579
$447$	−15.6774	−0.741514
$448$	0	0
$449$	14.6435	0.691071	0.345535	−	0.938406i	$-0.387697\pi$
$449$	14.6435	0.691071	0.345535	+	0.938406i	$0.387697\pi$
$450$	0	0
$451$	−4.84578	−0.228179
$452$	−10.4132	−0.489794
$453$	4.27226	0.200728
$454$	8.34227	0.391522
$455$	0	0
$456$	7.34271	0.343854
$457$	31.3289	1.46551	0.732753	−	0.680495i	$-0.238235\pi$
$457$	31.3289	1.46551	0.732753	+	0.680495i	$0.238235\pi$
$458$	18.6901	0.873329
$459$	−2.79881	−0.130637
$460$	0	0
$461$	12.7667	0.594602	0.297301	−	0.954784i	$-0.403913\pi$
$461$	12.7667	0.594602	0.297301	+	0.954784i	$0.403913\pi$
$462$	0	0
$463$	26.7112	1.24137	0.620687	−	0.784058i	$-0.286854\pi$
$463$	26.7112	1.24137	0.620687	+	0.784058i	$0.286854\pi$
$464$	3.74825	0.174008
$465$	0	0
$466$	20.0829	0.930322
$467$	37.6141	1.74057	0.870286	−	0.492546i	$-0.163934\pi$
$467$	37.6141	1.74057	0.870286	+	0.492546i	$0.163934\pi$
$468$	0.435157	0.0201152
$469$	0	0
$470$	0	0
$471$	15.2498	0.702675
$472$	8.27226	0.380762
$473$	9.76963	0.449208
$474$	2.88767	0.132635
$475$	0	0
$476$	0	0
$477$	−0.657293	−0.0300954
$478$	22.2714	1.01867
$479$	−23.8235	−1.08852	−0.544262	−	0.838915i	$-0.683190\pi$
$479$	−23.8235	−1.08852	−0.544262	+	0.838915i	$0.683190\pi$
$480$	0	0
$481$	−2.01735	−0.0919833
$482$	−6.20075	−0.282437
$483$	0	0
$484$	−10.0414	−0.456429
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	3.75674	0.170234	0.0851170	−	0.996371i	$-0.472874\pi$
$487$	3.75674	0.170234	0.0851170	+	0.996371i	$0.472874\pi$
$488$	−11.3336	−0.513048
$489$	7.47599	0.338076
$490$	0	0
$491$	2.00762	0.0906025	0.0453012	−	0.998973i	$-0.485575\pi$
$491$	2.00762	0.0906025	0.0453012	+	0.998973i	$0.485575\pi$
$492$	−4.94944	−0.223138
$493$	−10.4906	−0.472474
$494$	3.19523	0.143760
$495$	0	0
$496$	−4.97906	−0.223566
$497$	0	0
$498$	−14.3842	−0.644569
$499$	−0.628294	−0.0281263	−0.0140632	−	0.999901i	$-0.504477\pi$
$499$	−0.628294	−0.0281263	−0.0140632	+	0.999901i	$0.504477\pi$
$500$	0	0
$501$	−12.6640	−0.565787
$502$	−27.6560	−1.23435
$503$	−28.9911	−1.29265	−0.646324	−	0.763063i	$-0.723695\pi$
$503$	−28.9911	−1.29265	−0.646324	+	0.763063i	$0.723695\pi$
$504$	0	0
$505$	0	0
$506$	−3.27270	−0.145489
$507$	−12.8106	−0.568940
$508$	11.6987	0.519048
$509$	−15.5056	−0.687274	−0.343637	−	0.939103i	$-0.611659\pi$
$509$	−15.5056	−0.687274	−0.343637	+	0.939103i	$0.611659\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−7.34271	−0.324188
$514$	11.7681	0.519070
$515$	0	0
$516$	9.97862	0.439284
$517$	4.31327	0.189697
$518$	0	0
$519$	7.12057	0.312558
$520$	0	0
$521$	−13.5532	−0.593776	−0.296888	−	0.954912i	$-0.595949\pi$
$521$	−13.5532	−0.593776	−0.296888	+	0.954912i	$0.595949\pi$
$522$	−3.74825	−0.164056
$523$	−22.4542	−0.981852	−0.490926	−	0.871201i	$-0.663341\pi$
$523$	−22.4542	−0.981852	−0.490926	+	0.871201i	$0.663341\pi$
$524$	−5.38459	−0.235227
$525$	0	0
$526$	−0.455042	−0.0198408
$527$	13.9354	0.607037
$528$	0.979056	0.0426080
$529$	−11.8263	−0.514187
$530$	0	0
$531$	−8.27226	−0.358985
$532$	0	0
$533$	−2.15378	−0.0932907
$534$	10.1210	0.437979
$535$	0	0
$536$	2.36365	0.102094
$537$	3.26420	0.140861
$538$	26.1375	1.12687
$539$	0	0
$540$	0	0
$541$	10.8111	0.464804	0.232402	−	0.972620i	$-0.425341\pi$
$541$	10.8111	0.464804	0.232402	+	0.972620i	$0.425341\pi$
$542$	16.7768	0.720625
$543$	−1.93823	−0.0831773
$544$	2.79881	0.119998
$545$	0	0
$546$	0	0
$547$	24.1881	1.03421	0.517103	−	0.855923i	$-0.327010\pi$
$547$	24.1881	1.03421	0.517103	+	0.855923i	$0.327010\pi$
$548$	−3.68585	−0.157452
$549$	11.3336	0.483706
$550$	0	0
$551$	−27.5223	−1.17249
$552$	−3.34271	−0.142275
$553$	0	0
$554$	−20.7478	−0.881490
$555$	0	0
$556$	−8.15378	−0.345797
$557$	26.1529	1.10813	0.554067	−	0.832472i	$-0.313075\pi$
$557$	26.1529	1.10813	0.554067	+	0.832472i	$0.313075\pi$
$558$	4.97906	0.210780
$559$	4.34227	0.183658
$560$	0	0
$561$	−2.74019	−0.115691
$562$	18.3141	0.772536
$563$	−10.5864	−0.446164	−0.223082	−	0.974800i	$-0.571612\pi$
$563$	−10.5864	−0.446164	−0.223082	+	0.974800i	$0.571612\pi$
$564$	4.40554	0.185507
$565$	0	0
$566$	11.8525	0.498199
$567$	0	0
$568$	14.1119	0.592122
$569$	15.6440	0.655829	0.327915	−	0.944707i	$-0.393654\pi$
$569$	15.6440	0.655829	0.327915	+	0.944707i	$0.393654\pi$
$570$	0	0
$571$	8.50982	0.356125	0.178062	−	0.984019i	$-0.443017\pi$
$571$	8.50982	0.356125	0.178062	+	0.984019i	$0.443017\pi$
$572$	0.426043	0.0178138
$573$	8.07001	0.337129
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	9.67798	0.402900	0.201450	−	0.979499i	$-0.435435\pi$
$577$	9.67798	0.402900	0.201450	+	0.979499i	$0.435435\pi$
$578$	9.16667	0.381283
$579$	−1.95811	−0.0813764
$580$	0	0
$581$	0	0
$582$	−10.6655	−0.442100
$583$	−0.643527	−0.0266522
$584$	15.3928	0.636960
$585$	0	0
$586$	14.4770	0.598041
$587$	−16.3013	−0.672825	−0.336412	−	0.941715i	$-0.609214\pi$
$587$	−16.3013	−0.672825	−0.336412	+	0.941715i	$0.609214\pi$
$588$	0	0
$589$	36.5598	1.50642
$590$	0	0
$591$	17.3418	0.713348
$592$	−4.63591	−0.190535
$593$	−12.4802	−0.512500	−0.256250	−	0.966610i	$-0.582487\pi$
$593$	−12.4802	−0.512500	−0.256250	+	0.966610i	$0.582487\pi$
$594$	−0.979056	−0.0401712
$595$	0	0
$596$	−15.6774	−0.642170
$597$	−2.59490	−0.106202
$598$	−1.45460	−0.0594832
$599$	40.7545	1.66519	0.832593	−	0.553886i	$-0.186856\pi$
$599$	40.7545	1.66519	0.832593	+	0.553886i	$0.186856\pi$
$600$	0	0
$601$	4.94541	0.201727	0.100864	−	0.994900i	$-0.467839\pi$
$601$	4.94541	0.201727	0.100864	+	0.994900i	$0.467839\pi$
$602$	0	0
$603$	−2.36365	−0.0962553
$604$	4.27226	0.173836
$605$	0	0
$606$	14.0328	0.570042
$607$	−0.871555	−0.0353753	−0.0176877	−	0.999844i	$-0.505630\pi$
$607$	−0.871555	−0.0353753	−0.0176877	+	0.999844i	$0.505630\pi$
$608$	7.34271	0.297786
$609$	0	0
$610$	0	0
$611$	1.91710	0.0775577
$612$	−2.79881	−0.113135
$613$	−36.9007	−1.49041	−0.745203	−	0.666838i	$-0.767647\pi$
$613$	−36.9007	−1.49041	−0.745203	+	0.666838i	$0.767647\pi$
$614$	−30.4252	−1.22786
$615$	0	0
$616$	0	0
$617$	−28.6911	−1.15506	−0.577530	−	0.816369i	$-0.695984\pi$
$617$	−28.6911	−1.15506	−0.577530	+	0.816369i	$0.695984\pi$
$618$	10.3842	0.417712
$619$	−6.31415	−0.253787	−0.126894	−	0.991916i	$-0.540501\pi$
$619$	−6.31415	−0.253787	−0.126894	+	0.991916i	$0.540501\pi$
$620$	0	0
$621$	3.34271	0.134138
$622$	−9.02856	−0.362012
$623$	0	0
$624$	0.435157	0.0174202
$625$	0	0
$626$	−17.1732	−0.686380
$627$	−7.18892	−0.287098
$628$	15.2498	0.608534
$629$	12.9750	0.517348
$630$	0	0
$631$	−0.237559	−0.00945707	−0.00472853	−	0.999989i	$-0.501505\pi$
$631$	−0.237559	−0.00945707	−0.00472853	+	0.999989i	$0.501505\pi$
$632$	2.88767	0.114865
$633$	15.4127	0.612600
$634$	2.61541	0.103871
$635$	0	0
$636$	−0.657293	−0.0260634
$637$	0	0
$638$	−3.66974	−0.145287
$639$	−14.1119	−0.558258
$640$	0	0
$641$	−15.3490	−0.606250	−0.303125	−	0.952951i	$-0.598030\pi$
$641$	−15.3490	−0.606250	−0.303125	+	0.952951i	$0.598030\pi$
$642$	20.0410	0.790956
$643$	−8.77577	−0.346083	−0.173041	−	0.984915i	$-0.555359\pi$
$643$	−8.77577	−0.346083	−0.173041	+	0.984915i	$0.555359\pi$
$644$	0	0
$645$	0	0
$646$	−20.5508	−0.808562
$647$	19.2504	0.756813	0.378406	−	0.925640i	$-0.376472\pi$
$647$	19.2504	0.756813	0.378406	+	0.925640i	$0.376472\pi$
$648$	−1.00000	−0.0392837
$649$	−8.09901	−0.317914
$650$	0	0
$651$	0	0
$652$	7.47599	0.292782
$653$	−7.45460	−0.291721	−0.145861	−	0.989305i	$-0.546595\pi$
$653$	−7.45460	−0.291721	−0.145861	+	0.989305i	$0.546595\pi$
$654$	−18.6564	−0.729524
$655$	0	0
$656$	−4.94944	−0.193243
$657$	−15.3928	−0.600532
$658$	0	0
$659$	2.93717	0.114416	0.0572079	−	0.998362i	$-0.481780\pi$
$659$	2.93717	0.114416	0.0572079	+	0.998362i	$0.481780\pi$
$660$	0	0
$661$	23.6361	0.919337	0.459669	−	0.888090i	$-0.347968\pi$
$661$	23.6361	0.919337	0.459669	+	0.888090i	$0.347968\pi$
$662$	17.7559	0.690101
$663$	−1.21792	−0.0473002
$664$	−14.3842	−0.558214
$665$	0	0
$666$	4.63591	0.179638
$667$	12.5293	0.485136
$668$	−12.6640	−0.489986
$669$	7.61497	0.294412
$670$	0	0
$671$	11.0962	0.428365
$672$	0	0
$673$	50.9568	1.96424	0.982120	−	0.188256i	$-0.0602833\pi$
$673$	50.9568	1.96424	0.982120	+	0.188256i	$0.0602833\pi$
$674$	−24.2304	−0.933319
$675$	0	0
$676$	−12.8106	−0.492717
$677$	−5.23992	−0.201387	−0.100693	−	0.994918i	$-0.532106\pi$
$677$	−5.23992	−0.201387	−0.100693	+	0.994918i	$0.532106\pi$
$678$	10.4132	0.399915
$679$	0	0
$680$	0	0
$681$	−8.34227	−0.319676
$682$	4.87478	0.186665
$683$	−32.2648	−1.23458	−0.617289	−	0.786736i	$-0.711769\pi$
$683$	−32.2648	−1.23458	−0.617289	+	0.786736i	$0.711769\pi$
$684$	−7.34271	−0.280755
$685$	0	0
$686$	0	0
$687$	−18.6901	−0.713071
$688$	9.97862	0.380431
$689$	−0.286026	−0.0108967
$690$	0	0
$691$	20.4470	0.777840	0.388920	−	0.921272i	$-0.372848\pi$
$691$	20.4470	0.777840	0.388920	+	0.921272i	$0.372848\pi$
$692$	7.12057	0.270684
$693$	0	0
$694$	30.4252	1.15492
$695$	0	0
$696$	−3.74825	−0.142077
$697$	13.8525	0.524702
$698$	30.2623	1.14544
$699$	−20.0829	−0.759605
$700$	0	0
$701$	−15.4969	−0.585311	−0.292655	−	0.956218i	$-0.594539\pi$
$701$	−15.4969	−0.585311	−0.292655	+	0.956218i	$0.594539\pi$
$702$	−0.435157	−0.0164240
$703$	34.0401	1.28385
$704$	0.979056	0.0368996
$705$	0	0
$706$	−4.37189	−0.164538
$707$	0	0
$708$	−8.27226	−0.310891
$709$	38.9212	1.46172	0.730859	−	0.682529i	$-0.239120\pi$
$709$	38.9212	1.46172	0.730859	+	0.682529i	$0.239120\pi$
$710$	0	0
$711$	−2.88767	−0.108296
$712$	10.1210	0.379301
$713$	−16.6435	−0.623305
$714$	0	0
$715$	0	0
$716$	3.26420	0.121989
$717$	−22.2714	−0.831740
$718$	15.6850	0.585358
$719$	−3.53119	−0.131691	−0.0658456	−	0.997830i	$-0.520974\pi$
$719$	−3.53119	−0.131691	−0.0658456	+	0.997830i	$0.520974\pi$
$720$	0	0
$721$	0	0
$722$	−34.9153	−1.29941
$723$	6.20075	0.230608
$724$	−1.93823	−0.0720337
$725$	0	0
$726$	10.0414	0.372673
$727$	−45.2934	−1.67984	−0.839919	−	0.542712i	$-0.817398\pi$
$727$	−45.2934	−1.67984	−0.839919	+	0.542712i	$0.817398\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.9282	−1.03296
$732$	11.3336	0.418902
$733$	−48.5462	−1.79309	−0.896547	−	0.442949i	$-0.853932\pi$
$733$	−48.5462	−1.79309	−0.896547	+	0.442949i	$0.853932\pi$
$734$	4.94478	0.182515
$735$	0	0
$736$	−3.34271	−0.123214
$737$	−2.31415	−0.0852427
$738$	4.94944	0.182191
$739$	25.7969	0.948953	0.474477	−	0.880268i	$-0.342637\pi$
$739$	25.7969	0.948953	0.474477	+	0.880268i	$0.342637\pi$
$740$	0	0
$741$	−3.19523	−0.117380
$742$	0	0
$743$	28.7192	1.05361	0.526803	−	0.849987i	$-0.323390\pi$
$743$	28.7192	1.05361	0.526803	+	0.849987i	$0.323390\pi$
$744$	4.97906	0.182541
$745$	0	0
$746$	−6.02138	−0.220458
$747$	14.3842	0.526289
$748$	−2.74019	−0.100191
$749$	0	0
$750$	0	0
$751$	−34.1876	−1.24752	−0.623762	−	0.781615i	$-0.714396\pi$
$751$	−34.1876	−1.24752	−0.623762	+	0.781615i	$0.714396\pi$
$752$	4.40554	0.160653
$753$	27.6560	1.00784
$754$	−1.63108	−0.0594003
$755$	0	0
$756$	0	0
$757$	41.4622	1.50697	0.753485	−	0.657465i	$-0.228371\pi$
$757$	41.4622	1.50697	0.753485	+	0.657465i	$0.228371\pi$
$758$	36.8030	1.33675
$759$	3.27270	0.118791
$760$	0	0
$761$	27.6941	1.00391	0.501955	−	0.864894i	$-0.332614\pi$
$761$	27.6941	1.00391	0.501955	+	0.864894i	$0.332614\pi$
$762$	−11.6987	−0.423801
$763$	0	0
$764$	8.07001	0.291963
$765$	0	0
$766$	−9.03383	−0.326406
$767$	−3.59973	−0.129979
$768$	1.00000	0.0360844
$769$	9.74001	0.351234	0.175617	−	0.984459i	$-0.443808\pi$
$769$	9.74001	0.351234	0.175617	+	0.984459i	$0.443808\pi$
$770$	0	0
$771$	−11.7681	−0.423819
$772$	−1.95811	−0.0704740
$773$	−30.6171	−1.10122	−0.550610	−	0.834763i	$-0.685605\pi$
$773$	−30.6171	−1.10122	−0.550610	+	0.834763i	$0.685605\pi$
$774$	−9.97862	−0.358674
$775$	0	0
$776$	−10.6655	−0.382870
$777$	0	0
$778$	−33.5646	−1.20335
$779$	36.3423	1.30210
$780$	0	0
$781$	−13.8163	−0.494388
$782$	9.35560	0.334555
$783$	3.74825	0.133951
$784$	0	0
$785$	0	0
$786$	5.38459	0.192062
$787$	−15.7977	−0.563129	−0.281564	−	0.959542i	$-0.590853\pi$
$787$	−15.7977	−0.563129	−0.281564	+	0.959542i	$0.590853\pi$
$788$	17.3418	0.617777
$789$	0.455042	0.0161999
$790$	0	0
$791$	0	0
$792$	−0.979056	−0.0347892
$793$	4.93190	0.175137
$794$	33.9792	1.20588
$795$	0	0
$796$	−2.59490	−0.0919738
$797$	−47.6651	−1.68838	−0.844192	−	0.536041i	$-0.819919\pi$
$797$	−47.6651	−1.68838	−0.844192	+	0.536041i	$0.819919\pi$
$798$	0	0
$799$	−12.3303	−0.436213
$800$	0	0
$801$	−10.1210	−0.357608
$802$	4.27226	0.150859
$803$	−15.0704	−0.531825
$804$	−2.36365	−0.0833595
$805$	0	0
$806$	2.16667	0.0763178
$807$	−26.1375	−0.920083
$808$	14.0328	0.493671
$809$	−17.2709	−0.607214	−0.303607	−	0.952797i	$-0.598191\pi$
$809$	−17.2709	−0.607214	−0.303607	+	0.952797i	$0.598191\pi$
$810$	0	0
$811$	−14.7901	−0.519351	−0.259676	−	0.965696i	$-0.583616\pi$
$811$	−14.7901	−0.519351	−0.259676	+	0.965696i	$0.583616\pi$
$812$	0	0
$813$	−16.7768	−0.588388
$814$	4.53882	0.159085
$815$	0	0
$816$	−2.79881	−0.0979779
$817$	−73.2701	−2.56340
$818$	3.63888	0.127231
$819$	0	0
$820$	0	0
$821$	−45.6917	−1.59465	−0.797326	−	0.603549i	$-0.793753\pi$
$821$	−45.6917	−1.59465	−0.797326	+	0.603549i	$0.793753\pi$
$822$	3.68585	0.128559
$823$	4.99386	0.174075	0.0870375	−	0.996205i	$-0.472260\pi$
$823$	4.99386	0.174075	0.0870375	+	0.996205i	$0.472260\pi$
$824$	10.3842	0.361749
$825$	0	0
$826$	0	0
$827$	−53.5946	−1.86367	−0.931834	−	0.362885i	$-0.881792\pi$
$827$	−53.5946	−1.86367	−0.931834	+	0.362885i	$0.881792\pi$
$828$	3.34271	0.116167
$829$	−1.16692	−0.0405288	−0.0202644	−	0.999795i	$-0.506451\pi$
$829$	−1.16692	−0.0405288	−0.0202644	+	0.999795i	$0.506451\pi$
$830$	0	0
$831$	20.7478	0.719733
$832$	0.435157	0.0150864
$833$	0	0
$834$	8.15378	0.282342
$835$	0	0
$836$	−7.18892	−0.248634
$837$	−4.97906	−0.172101
$838$	−26.6274	−0.919829
$839$	31.2709	1.07959	0.539796	−	0.841796i	$-0.318501\pi$
$839$	31.2709	1.07959	0.539796	+	0.841796i	$0.318501\pi$
$840$	0	0
$841$	−14.9507	−0.515540
$842$	15.1880	0.523415
$843$	−18.3141	−0.630773
$844$	15.4127	0.530528
$845$	0	0
$846$	−4.40554	−0.151466
$847$	0	0
$848$	−0.657293	−0.0225715
$849$	−11.8525	−0.406778
$850$	0	0
$851$	−15.4965	−0.531213
$852$	−14.1119	−0.483466
$853$	11.3154	0.387431	0.193715	−	0.981058i	$-0.437946\pi$
$853$	11.3154	0.387431	0.193715	+	0.981058i	$0.437946\pi$
$854$	0	0
$855$	0	0
$856$	20.0410	0.684988
$857$	−36.8572	−1.25902	−0.629508	−	0.776994i	$-0.716744\pi$
$857$	−36.8572	−1.25902	−0.629508	+	0.776994i	$0.716744\pi$
$858$	−0.426043	−0.0145449
$859$	−18.6974	−0.637948	−0.318974	−	0.947763i	$-0.603338\pi$
$859$	−18.6974	−0.637948	−0.318974	+	0.947763i	$0.603338\pi$
$860$	0	0
$861$	0	0
$862$	4.23125	0.144117
$863$	−29.5384	−1.00550	−0.502749	−	0.864432i	$-0.667678\pi$
$863$	−29.5384	−1.00550	−0.502749	+	0.864432i	$0.667678\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	33.9352	1.15317
$867$	−9.16667	−0.311317
$868$	0	0
$869$	−2.82719	−0.0959057
$870$	0	0
$871$	−1.02856	−0.0348514
$872$	−18.6564	−0.631786
$873$	10.6655	0.360973
$874$	24.5445	0.830231
$875$	0	0
$876$	−15.3928	−0.520076
$877$	14.2581	0.481461	0.240730	−	0.970592i	$-0.422613\pi$
$877$	14.2581	0.481461	0.240730	+	0.970592i	$0.422613\pi$
$878$	5.48257	0.185028
$879$	−14.4770	−0.488299
$880$	0	0
$881$	−31.2744	−1.05366	−0.526830	−	0.849971i	$-0.676620\pi$
$881$	−31.2744	−1.05366	−0.526830	+	0.849971i	$0.676620\pi$
$882$	0	0
$883$	−33.3566	−1.12254	−0.561270	−	0.827633i	$-0.689687\pi$
$883$	−33.3566	−1.12254	−0.561270	+	0.827633i	$0.689687\pi$
$884$	−1.21792	−0.0409631
$885$	0	0
$886$	1.23081	0.0413499
$887$	42.1444	1.41507	0.707535	−	0.706678i	$-0.249807\pi$
$887$	42.1444	1.41507	0.707535	+	0.706678i	$0.249807\pi$
$888$	4.63591	0.155571
$889$	0	0
$890$	0	0
$891$	0.979056	0.0327996
$892$	7.61497	0.254968
$893$	−32.3486	−1.08250
$894$	15.6774	0.524329
$895$	0	0
$896$	0	0
$897$	1.45460	0.0485678
$898$	−14.6435	−0.488661
$899$	−18.6627	−0.622437
$900$	0	0
$901$	1.83964	0.0612872
$902$	4.84578	0.161347
$903$	0	0
$904$	10.4132	0.346336
$905$	0	0
$906$	−4.27226	−0.141936
$907$	−55.9840	−1.85892	−0.929460	−	0.368923i	$-0.879727\pi$
$907$	−55.9840	−1.85892	−0.929460	+	0.368923i	$0.879727\pi$
$908$	−8.34227	−0.276848
$909$	−14.0328	−0.465438
$910$	0	0
$911$	30.4389	1.00849	0.504243	−	0.863562i	$-0.331771\pi$
$911$	30.4389	1.00849	0.504243	+	0.863562i	$0.331771\pi$
$912$	−7.34271	−0.243141
$913$	14.0829	0.466076
$914$	−31.3289	−1.03627
$915$	0	0
$916$	−18.6901	−0.617537
$917$	0	0
$918$	2.79881	0.0923744
$919$	−28.1538	−0.928708	−0.464354	−	0.885650i	$-0.653713\pi$
$919$	−28.1538	−0.928708	−0.464354	+	0.885650i	$0.653713\pi$
$920$	0	0
$921$	30.4252	1.00254
$922$	−12.7667	−0.420447
$923$	−6.14089	−0.202130
$924$	0	0
$925$	0	0
$926$	−26.7112	−0.877784
$927$	−10.3842	−0.341060
$928$	−3.74825	−0.123042
$929$	25.8955	0.849603	0.424801	−	0.905287i	$-0.360344\pi$
$929$	25.8955	0.849603	0.424801	+	0.905287i	$0.360344\pi$
$930$	0	0
$931$	0	0
$932$	−20.0829	−0.657837
$933$	9.02856	0.295582
$934$	−37.6141	−1.23077
$935$	0	0
$936$	−0.435157	−0.0142236
$937$	48.1184	1.57196	0.785979	−	0.618253i	$-0.212159\pi$
$937$	48.1184	1.57196	0.785979	+	0.618253i	$0.212159\pi$
$938$	0	0
$939$	17.1732	0.560427
$940$	0	0
$941$	−52.2066	−1.70189	−0.850944	−	0.525257i	$-0.823969\pi$
$941$	−52.2066	−1.70189	−0.850944	+	0.525257i	$0.823969\pi$
$942$	−15.2498	−0.496866
$943$	−16.5445	−0.538764
$944$	−8.27226	−0.269239
$945$	0	0
$946$	−9.76963	−0.317638
$947$	20.2995	0.659645	0.329823	−	0.944043i	$-0.393011\pi$
$947$	20.2995	0.659645	0.329823	+	0.944043i	$0.393011\pi$
$948$	−2.88767	−0.0937870
$949$	−6.69830	−0.217436
$950$	0	0
$951$	−2.61541	−0.0848103
$952$	0	0
$953$	16.9511	0.549100	0.274550	−	0.961573i	$-0.411471\pi$
$953$	16.9511	0.549100	0.274550	+	0.961573i	$0.411471\pi$
$954$	0.657293	0.0212806
$955$	0	0
$956$	−22.2714	−0.720308
$957$	3.66974	0.118626
$958$	23.8235	0.769703
$959$	0	0
$960$	0	0
$961$	−6.20900	−0.200290
$962$	2.01735	0.0650420
$963$	−20.0410	−0.645813
$964$	6.20075	0.199713
$965$	0	0
$966$	0	0
$967$	23.0294	0.740577	0.370288	−	0.928917i	$-0.379259\pi$
$967$	23.0294	0.740577	0.370288	+	0.928917i	$0.379259\pi$
$968$	10.0414	0.322744
$969$	20.5508	0.660188
$970$	0	0
$971$	−12.3770	−0.397196	−0.198598	−	0.980081i	$-0.563639\pi$
$971$	−12.3770	−0.397196	−0.198598	+	0.980081i	$0.563639\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−3.75674	−0.120374
$975$	0	0
$976$	11.3336	0.362779
$977$	1.00614	0.0321893	0.0160946	−	0.999870i	$-0.494877\pi$
$977$	1.00614	0.0321893	0.0160946	+	0.999870i	$0.494877\pi$
$978$	−7.47599	−0.239056
$979$	−9.90904	−0.316694
$980$	0	0
$981$	18.6564	0.595654
$982$	−2.00762	−0.0640656
$983$	−42.1444	−1.34420	−0.672099	−	0.740461i	$-0.734607\pi$
$983$	−42.1444	−1.34420	−0.672099	+	0.740461i	$0.734607\pi$
$984$	4.94944	0.157782
$985$	0	0
$986$	10.4906	0.334089
$987$	0	0
$988$	−3.19523	−0.101654
$989$	33.3556	1.06065
$990$	0	0
$991$	−34.0691	−1.08224	−0.541121	−	0.840945i	$-0.682000\pi$
$991$	−34.0691	−1.08224	−0.541121	+	0.840945i	$0.682000\pi$
$992$	4.97906	0.158085
$993$	−17.7559	−0.563465
$994$	0	0
$995$	0	0
$996$	14.3842	0.455779
$997$	−6.02076	−0.190679	−0.0953397	−	0.995445i	$-0.530394\pi$
$997$	−6.02076	−0.190679	−0.0953397	+	0.995445i	$0.530394\pi$
$998$	0.628294	0.0198883
$999$	−4.63591	−0.146674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.ds.1.3		4
5.2	odd	4		1470.2.g.j.589.4	✓	8
5.3	odd	4		1470.2.g.j.589.8	yes	8
5.4	even	2		7350.2.a.dt.1.3		4
7.6	odd	2		7350.2.a.dr.1.3		4
35.2	odd	12		1470.2.n.l.949.1		16
35.3	even	12		1470.2.n.k.79.4		16
35.12	even	12		1470.2.n.k.949.4		16
35.13	even	4		1470.2.g.k.589.5	yes	8
35.17	even	12		1470.2.n.k.79.7		16
35.18	odd	12		1470.2.n.l.79.1		16
35.23	odd	12		1470.2.n.l.949.6		16
35.27	even	4		1470.2.g.k.589.1	yes	8
35.32	odd	12		1470.2.n.l.79.6		16
35.33	even	12		1470.2.n.k.949.7		16
35.34	odd	2		7350.2.a.du.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.g.j.589.4	✓	8	5.2	odd	4
1470.2.g.j.589.8	yes	8	5.3	odd	4
1470.2.g.k.589.1	yes	8	35.27	even	4
1470.2.g.k.589.5	yes	8	35.13	even	4
1470.2.n.k.79.4		16	35.3	even	12
1470.2.n.k.79.7		16	35.17	even	12
1470.2.n.k.949.4		16	35.12	even	12
1470.2.n.k.949.7		16	35.33	even	12
1470.2.n.l.79.1		16	35.18	odd	12
1470.2.n.l.79.6		16	35.32	odd	12
1470.2.n.l.949.1		16	35.2	odd	12
1470.2.n.l.949.6		16	35.23	odd	12
7350.2.a.dr.1.3		4	7.6	odd	2
7350.2.a.ds.1.3		4	1.1	even	1	trivial
7350.2.a.dt.1.3		4	5.4	even	2
7350.2.a.du.1.3		4	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +0.979056 q^{11} +1.00000 q^{12} +0.435157 q^{13} +1.00000 q^{16} -2.79881 q^{17} -1.00000 q^{18} -7.34271 q^{19} -0.979056 q^{22} +3.34271 q^{23} -1.00000 q^{24} -0.435157 q^{26} +1.00000 q^{27} +3.74825 q^{29} -4.97906 q^{31} -1.00000 q^{32} +0.979056 q^{33} +2.79881 q^{34} +1.00000 q^{36} -4.63591 q^{37} +7.34271 q^{38} +0.435157 q^{39} -4.94944 q^{41} +9.97862 q^{43} +0.979056 q^{44} -3.34271 q^{46} +4.40554 q^{47} +1.00000 q^{48} -2.79881 q^{51} +0.435157 q^{52} -0.657293 q^{53} -1.00000 q^{54} -7.34271 q^{57} -3.74825 q^{58} -8.27226 q^{59} +11.3336 q^{61} +4.97906 q^{62} +1.00000 q^{64} -0.979056 q^{66} -2.36365 q^{67} -2.79881 q^{68} +3.34271 q^{69} -14.1119 q^{71} -1.00000 q^{72} -15.3928 q^{73} +4.63591 q^{74} -7.34271 q^{76} -0.435157 q^{78} -2.88767 q^{79} +1.00000 q^{81} +4.94944 q^{82} +14.3842 q^{83} -9.97862 q^{86} +3.74825 q^{87} -0.979056 q^{88} -10.1210 q^{89} +3.34271 q^{92} -4.97906 q^{93} -4.40554 q^{94} -1.00000 q^{96} +10.6655 q^{97} +0.979056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{12} + 4 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} - 4 q^{23} - 4 q^{24} + 4 q^{27} - 8 q^{29} - 16 q^{31} - 4 q^{32} - 4 q^{34} + 4 q^{36} + 8 q^{37} + 12 q^{38} - 12 q^{41} - 4 q^{43} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 4 q^{51} - 20 q^{53} - 4 q^{54} - 12 q^{57} + 8 q^{58} - 20 q^{59} - 12 q^{61} + 16 q^{62} + 4 q^{64} + 4 q^{67} + 4 q^{68} - 4 q^{69} - 20 q^{71} - 4 q^{72} - 12 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{81} + 12 q^{82} + 8 q^{83} + 4 q^{86} - 8 q^{87} - 44 q^{89} - 4 q^{92} - 16 q^{93} - 12 q^{94} - 4 q^{96} + 20 q^{97}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^12 + 4 * q^16 + 4 * q^17 - 4 * q^18 - 12 * q^19 - 4 * q^23 - 4 * q^24 + 4 * q^27 - 8 * q^29 - 16 * q^31 - 4 * q^32 - 4 * q^34 + 4 * q^36 + 8 * q^37 + 12 * q^38 - 12 * q^41 - 4 * q^43 + 4 * q^46 + 12 * q^47 + 4 * q^48 + 4 * q^51 - 20 * q^53 - 4 * q^54 - 12 * q^57 + 8 * q^58 - 20 * q^59 - 12 * q^61 + 16 * q^62 + 4 * q^64 + 4 * q^67 + 4 * q^68 - 4 * q^69 - 20 * q^71 - 4 * q^72 - 12 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^81 + 12 * q^82 + 8 * q^83 + 4 * q^86 - 8 * q^87 - 44 * q^89 - 4 * q^92 - 16 * q^93 - 12 * q^94 - 4 * q^96 + 20 * q^97

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