LMFDB - Embedded newform 735.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [735,2,Mod(1,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("735.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.334904$ of defining polynomial
Character	$\chi$	$=$	735.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.74912 q^{2} +1.00000 q^{3} +5.55765 q^{4} -1.00000 q^{5} +2.74912 q^{6} +9.78039 q^{8} +1.00000 q^{9} -2.74912 q^{10} -4.44549 q^{11} +5.55765 q^{12} -3.61706 q^{13} -1.00000 q^{15} +15.7721 q^{16} +4.94725 q^{17} +2.74912 q^{18} -2.74441 q^{19} -5.55765 q^{20} -12.2212 q^{22} -4.36147 q^{23} +9.78039 q^{24} +1.00000 q^{25} -9.94372 q^{26} +1.00000 q^{27} +0.660384 q^{29} -2.74912 q^{30} -1.25559 q^{31} +23.7987 q^{32} -4.44549 q^{33} +13.6006 q^{34} +5.55765 q^{36} -2.16804 q^{37} -7.54469 q^{38} -3.61706 q^{39} -9.78039 q^{40} +5.77568 q^{41} -9.11529 q^{43} -24.7064 q^{44} -1.00000 q^{45} -11.9902 q^{46} -6.48528 q^{47} +15.7721 q^{48} +2.74912 q^{50} +4.94725 q^{51} -20.1023 q^{52} +7.03127 q^{53} +2.74912 q^{54} +4.44549 q^{55} -2.74441 q^{57} +1.81547 q^{58} +13.8249 q^{59} -5.55765 q^{60} -9.41421 q^{61} -3.45178 q^{62} +33.8812 q^{64} +3.61706 q^{65} -12.2212 q^{66} +3.06608 q^{67} +27.4951 q^{68} -4.36147 q^{69} +0.277444 q^{71} +9.78039 q^{72} +10.6135 q^{73} -5.96021 q^{74} +1.00000 q^{75} -15.2524 q^{76} -9.94372 q^{78} -5.53803 q^{79} -15.7721 q^{80} +1.00000 q^{81} +15.8780 q^{82} +7.17157 q^{83} -4.94725 q^{85} -25.0590 q^{86} +0.660384 q^{87} -43.4786 q^{88} +0.828427 q^{89} -2.74912 q^{90} -24.2395 q^{92} -1.25559 q^{93} -17.8288 q^{94} +2.74441 q^{95} +23.7987 q^{96} -6.20784 q^{97} -4.44549 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{5} + 4 q^{6} + 12 q^{8} + 4 q^{9} - 4 q^{10} + 8 q^{11} + 8 q^{12} - 4 q^{15} + 12 q^{16} + 8 q^{17} + 4 q^{18} - 8 q^{19} - 8 q^{20} + 12 q^{24} + 4 q^{25} + 4 q^{27}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^5 + 4 * q^6 + 12 * q^8 + 4 * q^9 - 4 * q^10 + 8 * q^11 + 8 * q^12 - 4 * q^15 + 12 * q^16 + 8 * q^17 + 4 * q^18 - 8 * q^19 - 8 * q^20 + 12 * q^24 + 4 * q^25 + 4 * q^27 + 8 * q^29 - 4 * q^30 - 8 * q^31 + 28 * q^32 + 8 * q^33 - 8 * q^34 + 8 * q^36 + 8 * q^37 + 4 * q^38 - 12 * q^40 - 8 * q^43 - 16 * q^44 - 4 * q^45 + 12 * q^46 + 8 * q^47 + 12 * q^48 + 4 * q^50 + 8 * q^51 - 32 * q^52 + 8 * q^53 + 4 * q^54 - 8 * q^55 - 8 * q^57 - 24 * q^58 + 16 * q^59 - 8 * q^60 - 32 * q^61 - 20 * q^62 + 24 * q^64 + 24 * q^68 - 8 * q^71 + 12 * q^72 - 32 * q^74 + 4 * q^75 + 8 * q^76 - 12 * q^80 + 4 * q^81 - 8 * q^82 + 40 * q^83 - 8 * q^85 - 32 * q^86 + 8 * q^87 - 40 * q^88 - 8 * q^89 - 4 * q^90 - 8 * q^92 - 8 * q^93 - 16 * q^94 + 8 * q^95 + 28 * q^96 + 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.74912	1.94392	0.971960	−	0.235147i	$-0.0755571\pi$
$2$	2.74912	1.94392	0.971960	+	0.235147i	$0.0755571\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.55765	2.77882
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.74912	1.12232
$7$	0	0
$7$	0	0
$8$	9.78039	3.45789
$9$	1.00000	0.333333
$10$	−2.74912	−0.869347
$11$	−4.44549	−1.34036	−0.670182	−	0.742197i	$-0.733784\pi$
$11$	−4.44549	−1.34036	−0.670182	+	0.742197i	$0.733784\pi$
$12$	5.55765	1.60435
$13$	−3.61706	−1.00319	−0.501596	−	0.865102i	$-0.667254\pi$
$13$	−3.61706	−1.00319	−0.501596	+	0.865102i	$0.667254\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	15.7721	3.94304
$17$	4.94725	1.19988	0.599942	−	0.800043i	$-0.295190\pi$
$17$	4.94725	1.19988	0.599942	+	0.800043i	$0.295190\pi$
$18$	2.74912	0.647973
$19$	−2.74441	−0.629610	−0.314805	−	0.949156i	$-0.601939\pi$
$19$	−2.74441	−0.629610	−0.314805	+	0.949156i	$0.601939\pi$
$20$	−5.55765	−1.24273
$21$	0	0
$22$	−12.2212	−2.60556
$23$	−4.36147	−0.909428	−0.454714	−	0.890637i	$-0.650259\pi$
$23$	−4.36147	−0.909428	−0.454714	+	0.890637i	$0.650259\pi$
$24$	9.78039	1.99641
$25$	1.00000	0.200000
$26$	−9.94372	−1.95012
$27$	1.00000	0.192450
$28$	0	0
$29$	0.660384	0.122630	0.0613151	−	0.998118i	$-0.480471\pi$
$29$	0.660384	0.122630	0.0613151	+	0.998118i	$0.480471\pi$
$30$	−2.74912	−0.501918
$31$	−1.25559	−0.225511	−0.112756	−	0.993623i	$-0.535968\pi$
$31$	−1.25559	−0.225511	−0.112756	+	0.993623i	$0.535968\pi$
$32$	23.7987	4.20706
$33$	−4.44549	−0.773860
$34$	13.6006	2.33248
$35$	0	0
$36$	5.55765	0.926275
$37$	−2.16804	−0.356424	−0.178212	−	0.983992i	$-0.557031\pi$
$37$	−2.16804	−0.356424	−0.178212	+	0.983992i	$0.557031\pi$
$38$	−7.54469	−1.22391
$39$	−3.61706	−0.579193
$40$	−9.78039	−1.54642
$41$	5.77568	0.902009	0.451005	−	0.892522i	$-0.351066\pi$
$41$	5.77568	0.902009	0.451005	+	0.892522i	$0.351066\pi$
$42$	0	0
$43$	−9.11529	−1.39007	−0.695035	−	0.718976i	$-0.744611\pi$
$43$	−9.11529	−1.39007	−0.695035	+	0.718976i	$0.744611\pi$
$44$	−24.7064	−3.72464
$45$	−1.00000	−0.149071
$46$	−11.9902	−1.76786
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	15.7721	2.27651
$49$	0	0
$50$	2.74912	0.388784
$51$	4.94725	0.692754
$52$	−20.1023	−2.78769
$53$	7.03127	0.965820	0.482910	−	0.875670i	$-0.339580\pi$
$53$	7.03127	0.965820	0.482910	+	0.875670i	$0.339580\pi$
$54$	2.74912	0.374108
$55$	4.44549	0.599429
$56$	0	0
$57$	−2.74441	−0.363505
$58$	1.81547	0.238383
$59$	13.8249	1.79985	0.899924	−	0.436046i	$-0.143622\pi$
$59$	13.8249	1.79985	0.899924	+	0.436046i	$0.143622\pi$
$60$	−5.55765	−0.717489
$61$	−9.41421	−1.20537	−0.602683	−	0.797981i	$-0.705902\pi$
$61$	−9.41421	−1.20537	−0.602683	+	0.797981i	$0.705902\pi$
$62$	−3.45178	−0.438376
$63$	0	0
$64$	33.8812	4.23514
$65$	3.61706	0.448641
$66$	−12.2212	−1.50432
$67$	3.06608	0.374581	0.187290	−	0.982305i	$-0.440029\pi$
$67$	3.06608	0.374581	0.187290	+	0.982305i	$0.440029\pi$
$68$	27.4951	3.33427
$69$	−4.36147	−0.525059
$70$	0	0
$71$	0.277444	0.0329265	0.0164632	−	0.999864i	$-0.494759\pi$
$71$	0.277444	0.0329265	0.0164632	+	0.999864i	$0.494759\pi$
$72$	9.78039	1.15263
$73$	10.6135	1.24222	0.621110	−	0.783724i	$-0.286682\pi$
$73$	10.6135	1.24222	0.621110	+	0.783724i	$0.286682\pi$
$74$	−5.96021	−0.692860
$75$	1.00000	0.115470
$76$	−15.2524	−1.74957
$77$	0	0
$78$	−9.94372	−1.12590
$79$	−5.53803	−0.623077	−0.311539	−	0.950234i	$-0.600844\pi$
$79$	−5.53803	−0.623077	−0.311539	+	0.950234i	$0.600844\pi$
$80$	−15.7721	−1.76338
$81$	1.00000	0.111111
$82$	15.8780	1.75343
$83$	7.17157	0.787182	0.393591	−	0.919286i	$-0.371233\pi$
$83$	7.17157	0.787182	0.393591	+	0.919286i	$0.371233\pi$
$84$	0	0
$85$	−4.94725	−0.536605
$86$	−25.0590	−2.70218
$87$	0.660384	0.0708006
$88$	−43.4786	−4.63483
$89$	0.828427	0.0878131	0.0439065	−	0.999036i	$-0.486020\pi$
$89$	0.828427	0.0878131	0.0439065	+	0.999036i	$0.486020\pi$
$90$	−2.74912	−0.289782
$91$	0	0
$92$	−24.2395	−2.52714
$93$	−1.25559	−0.130199
$94$	−17.8288	−1.83890
$95$	2.74441	0.281570
$96$	23.7987	2.42895
$97$	−6.20784	−0.630310	−0.315155	−	0.949040i	$-0.602057\pi$
$97$	−6.20784	−0.630310	−0.315155	+	0.949040i	$0.602057\pi$
$98$	0	0
$99$	−4.44549	−0.446788
$100$	5.55765	0.555765
$101$	−4.11882	−0.409838	−0.204919	−	0.978779i	$-0.565693\pi$
$101$	−4.11882	−0.409838	−0.204919	+	0.978779i	$0.565693\pi$
$102$	13.6006	1.34666
$103$	8.16804	0.804821	0.402411	−	0.915459i	$-0.368172\pi$
$103$	8.16804	0.804821	0.402411	+	0.915459i	$0.368172\pi$
$104$	−35.3763	−3.46893
$105$	0	0
$106$	19.3298	1.87748
$107$	−8.52951	−0.824579	−0.412289	−	0.911053i	$-0.635271\pi$
$107$	−8.52951	−0.824579	−0.412289	+	0.911053i	$0.635271\pi$
$108$	5.55765	0.534785
$109$	−3.65685	−0.350263	−0.175132	−	0.984545i	$-0.556035\pi$
$109$	−3.65685	−0.350263	−0.175132	+	0.984545i	$0.556035\pi$
$110$	12.2212	1.16524
$111$	−2.16804	−0.205782
$112$	0	0
$113$	12.6085	1.18611	0.593056	−	0.805161i	$-0.297921\pi$
$113$	12.6085	1.18611	0.593056	+	0.805161i	$0.297921\pi$
$114$	−7.54469	−0.706625
$115$	4.36147	0.406709
$116$	3.67018	0.340768
$117$	−3.61706	−0.334397
$118$	38.0063	3.49876
$119$	0	0
$120$	−9.78039	−0.892823
$121$	8.76235	0.796577
$122$	−25.8808	−2.34314
$123$	5.77568	0.520775
$124$	−6.97815	−0.626656
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.0562783	−0.00499389	−0.00249695	−	0.999997i	$-0.500795\pi$
$127$	−0.0562783	−0.00499389	−0.00249695	+	0.999997i	$0.500795\pi$
$128$	45.5459	4.02572
$129$	−9.11529	−0.802557
$130$	9.94372	0.872122
$131$	−19.6631	−1.71797	−0.858987	−	0.511997i	$-0.828906\pi$
$131$	−19.6631	−1.71797	−0.858987	+	0.511997i	$0.828906\pi$
$132$	−24.7064	−2.15042
$133$	0	0
$134$	8.42900	0.728155
$135$	−1.00000	−0.0860663
$136$	48.3861	4.14907
$137$	−18.7507	−1.60198	−0.800989	−	0.598679i	$-0.795693\pi$
$137$	−18.7507	−1.60198	−0.800989	+	0.598679i	$0.795693\pi$
$138$	−11.9902	−1.02067
$139$	6.14657	0.521345	0.260673	−	0.965427i	$-0.416056\pi$
$139$	6.14657	0.521345	0.260673	+	0.965427i	$0.416056\pi$
$140$	0	0
$141$	−6.48528	−0.546159
$142$	0.762725	0.0640065
$143$	16.0796	1.34464
$144$	15.7721	1.31435
$145$	−0.660384	−0.0548419
$146$	29.1778	2.41478
$147$	0	0
$148$	−12.0492	−0.990440
$149$	20.2306	1.65735	0.828677	−	0.559727i	$-0.189094\pi$
$149$	20.2306	1.65735	0.828677	+	0.559727i	$0.189094\pi$
$150$	2.74912	0.224465
$151$	21.1082	1.71776	0.858882	−	0.512174i	$-0.171160\pi$
$151$	21.1082	1.71776	0.858882	+	0.512174i	$0.171160\pi$
$152$	−26.8414	−2.17712
$153$	4.94725	0.399962
$154$	0	0
$155$	1.25559	0.100852
$156$	−20.1023	−1.60948
$157$	−1.96021	−0.156441	−0.0782207	−	0.996936i	$-0.524924\pi$
$157$	−1.96021	−0.156441	−0.0782207	+	0.996936i	$0.524924\pi$
$158$	−15.2247	−1.21121
$159$	7.03127	0.557616
$160$	−23.7987	−1.88145
$161$	0	0
$162$	2.74912	0.215991
$163$	−8.18137	−0.640814	−0.320407	−	0.947280i	$-0.603820\pi$
$163$	−8.18137	−0.640814	−0.320407	+	0.947280i	$0.603820\pi$
$164$	32.0992	2.50653
$165$	4.44549	0.346081
$166$	19.7155	1.53022
$167$	3.76235	0.291139	0.145570	−	0.989348i	$-0.453498\pi$
$167$	3.76235	0.291139	0.145570	+	0.989348i	$0.453498\pi$
$168$	0	0
$169$	0.0831193	0.00639379
$170$	−13.6006	−1.04312
$171$	−2.74441	−0.209870
$172$	−50.6596	−3.86276
$173$	15.7061	1.19411	0.597055	−	0.802200i	$-0.296337\pi$
$173$	15.7061	1.19411	0.597055	+	0.802200i	$0.296337\pi$
$174$	1.81547	0.137631
$175$	0	0
$176$	−70.1149	−5.28511
$177$	13.8249	1.03914
$178$	2.27744	0.170702
$179$	−1.28097	−0.0957444	−0.0478722	−	0.998853i	$-0.515244\pi$
$179$	−1.28097	−0.0957444	−0.0478722	+	0.998853i	$0.515244\pi$
$180$	−5.55765	−0.414243
$181$	−16.5599	−1.23089	−0.615443	−	0.788181i	$-0.711023\pi$
$181$	−16.5599	−1.23089	−0.615443	+	0.788181i	$0.711023\pi$
$182$	0	0
$183$	−9.41421	−0.695919
$184$	−42.6568	−3.14470
$185$	2.16804	0.159398
$186$	−3.45178	−0.253097
$187$	−21.9929	−1.60828
$188$	−36.0429	−2.62870
$189$	0	0
$190$	7.54469	0.547350
$191$	8.76272	0.634049	0.317024	−	0.948417i	$-0.397316\pi$
$191$	8.76272	0.634049	0.317024	+	0.948417i	$0.397316\pi$
$192$	33.8812	2.44516
$193$	26.4986	1.90741	0.953706	−	0.300741i	$-0.0972340\pi$
$193$	26.4986	1.90741	0.953706	+	0.300741i	$0.0972340\pi$
$194$	−17.0661	−1.22527
$195$	3.61706	0.259023
$196$	0	0
$197$	4.11439	0.293138	0.146569	−	0.989200i	$-0.453177\pi$
$197$	4.11439	0.293138	0.146569	+	0.989200i	$0.453177\pi$
$198$	−12.2212	−0.868520
$199$	7.63538	0.541258	0.270629	−	0.962684i	$-0.412768\pi$
$199$	7.63538	0.541258	0.270629	+	0.962684i	$0.412768\pi$
$200$	9.78039	0.691578
$201$	3.06608	0.216264
$202$	−11.3231	−0.796693
$203$	0	0
$204$	27.4951	1.92504
$205$	−5.77568	−0.403391
$206$	22.4549	1.56451
$207$	−4.36147	−0.303143
$208$	−57.0488	−3.95562
$209$	12.2002	0.843907
$210$	0	0
$211$	−0.317238	−0.0218396	−0.0109198	−	0.999940i	$-0.503476\pi$
$211$	−0.317238	−0.0218396	−0.0109198	+	0.999940i	$0.503476\pi$
$212$	39.0773	2.68384
$213$	0.277444	0.0190101
$214$	−23.4486	−1.60291
$215$	9.11529	0.621658
$216$	9.78039	0.665471
$217$	0	0
$218$	−10.0531	−0.680883
$219$	10.6135	0.717196
$220$	24.7064	1.66571
$221$	−17.8945	−1.20371
$222$	−5.96021	−0.400023
$223$	−29.0590	−1.94594	−0.972968	−	0.230941i	$-0.925820\pi$
$223$	−29.0590	−1.94594	−0.972968	+	0.230941i	$0.925820\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	34.6624	2.30571
$227$	20.8910	1.38658	0.693291	−	0.720657i	$-0.256160\pi$
$227$	20.8910	1.38658	0.693291	+	0.720657i	$0.256160\pi$
$228$	−15.2524	−1.01012
$229$	−4.92893	−0.325713	−0.162857	−	0.986650i	$-0.552071\pi$
$229$	−4.92893	−0.325713	−0.162857	+	0.986650i	$0.552071\pi$
$230$	11.9902	0.790609
$231$	0	0
$232$	6.45881	0.424042
$233$	−1.09382	−0.0716585	−0.0358292	−	0.999358i	$-0.511407\pi$
$233$	−1.09382	−0.0716585	−0.0358292	+	0.999358i	$0.511407\pi$
$234$	−9.94372	−0.650041
$235$	6.48528	0.423053
$236$	76.8339	5.00146
$237$	−5.53803	−0.359734
$238$	0	0
$239$	25.6725	1.66062	0.830309	−	0.557303i	$-0.188164\pi$
$239$	25.6725	1.66062	0.830309	+	0.557303i	$0.188164\pi$
$240$	−15.7721	−1.01809
$241$	−24.7468	−1.59408	−0.797040	−	0.603927i	$-0.793602\pi$
$241$	−24.7468	−1.59408	−0.797040	+	0.603927i	$0.793602\pi$
$242$	24.0887	1.54848
$243$	1.00000	0.0641500
$244$	−52.3209	−3.34950
$245$	0	0
$246$	15.8780	1.01235
$247$	9.92668	0.631619
$248$	−12.2802	−0.779794
$249$	7.17157	0.454480
$250$	−2.74912	−0.173869
$251$	−0.405692	−0.0256070	−0.0128035	−	0.999918i	$-0.504076\pi$
$251$	−0.405692	−0.0256070	−0.0128035	+	0.999918i	$0.504076\pi$
$252$	0	0
$253$	19.3888	1.21897
$254$	−0.154716	−0.00970772
$255$	−4.94725	−0.309809
$256$	57.4486	3.59054
$257$	10.8910	0.679360	0.339680	−	0.940541i	$-0.389681\pi$
$257$	10.8910	0.679360	0.339680	+	0.940541i	$0.389681\pi$
$258$	−25.0590	−1.56011
$259$	0	0
$260$	20.1023	1.24669
$261$	0.660384	0.0408767
$262$	−54.0562	−3.33961
$263$	8.76716	0.540606	0.270303	−	0.962775i	$-0.412876\pi$
$263$	8.76716	0.540606	0.270303	+	0.962775i	$0.412876\pi$
$264$	−43.4786	−2.67592
$265$	−7.03127	−0.431928
$266$	0	0
$267$	0.828427	0.0506989
$268$	17.0402	1.04089
$269$	−22.8347	−1.39226	−0.696128	−	0.717918i	$-0.745095\pi$
$269$	−22.8347	−1.39226	−0.696128	+	0.717918i	$0.745095\pi$
$270$	−2.74912	−0.167306
$271$	−10.6748	−0.648448	−0.324224	−	0.945980i	$-0.605103\pi$
$271$	−10.6748	−0.648448	−0.324224	+	0.945980i	$0.605103\pi$
$272$	78.0288	4.73119
$273$	0	0
$274$	−51.5478	−3.11412
$275$	−4.44549	−0.268073
$276$	−24.2395	−1.45905
$277$	−12.4361	−0.747211	−0.373605	−	0.927588i	$-0.621879\pi$
$277$	−12.4361	−0.747211	−0.373605	+	0.927588i	$0.621879\pi$
$278$	16.8976	1.01345
$279$	−1.25559	−0.0751705
$280$	0	0
$281$	12.9169	0.770556	0.385278	−	0.922800i	$-0.374105\pi$
$281$	12.9169	0.770556	0.385278	+	0.922800i	$0.374105\pi$
$282$	−17.8288	−1.06169
$283$	−25.6139	−1.52259	−0.761294	−	0.648407i	$-0.775436\pi$
$283$	−25.6139	−1.52259	−0.761294	+	0.648407i	$0.775436\pi$
$284$	1.54193	0.0914969
$285$	2.74441	0.162565
$286$	44.2047	2.61388
$287$	0	0
$288$	23.7987	1.40235
$289$	7.47530	0.439723
$290$	−1.81547	−0.106608
$291$	−6.20784	−0.363910
$292$	58.9863	3.45191
$293$	1.68276	0.0983080	0.0491540	−	0.998791i	$-0.484347\pi$
$293$	1.68276	0.0983080	0.0491540	+	0.998791i	$0.484347\pi$
$294$	0	0
$295$	−13.8249	−0.804917
$296$	−21.2043	−1.23248
$297$	−4.44549	−0.257953
$298$	55.6163	3.22176
$299$	15.7757	0.912331
$300$	5.55765	0.320871
$301$	0	0
$302$	58.0290	3.33919
$303$	−4.11882	−0.236620
$304$	−43.2852	−2.48257
$305$	9.41421	0.539056
$306$	13.6006	0.777493
$307$	7.31371	0.417415	0.208708	−	0.977978i	$-0.433074\pi$
$307$	7.31371	0.417415	0.208708	+	0.977978i	$0.433074\pi$
$308$	0	0
$309$	8.16804	0.464664
$310$	3.45178	0.196048
$311$	−11.2019	−0.635204	−0.317602	−	0.948224i	$-0.602878\pi$
$311$	−11.2019	−0.635204	−0.317602	+	0.948224i	$0.602878\pi$
$312$	−35.3763	−2.00279
$313$	31.4857	1.77967	0.889837	−	0.456278i	$-0.150818\pi$
$313$	31.4857	1.77967	0.889837	+	0.456278i	$0.150818\pi$
$314$	−5.38883	−0.304110
$315$	0	0
$316$	−30.7784	−1.73142
$317$	8.45048	0.474626	0.237313	−	0.971433i	$-0.423733\pi$
$317$	8.45048	0.474626	0.237313	+	0.971433i	$0.423733\pi$
$318$	19.3298	1.08396
$319$	−2.93573	−0.164369
$320$	−33.8812	−1.89401
$321$	−8.52951	−0.476071
$322$	0	0
$323$	−13.5773	−0.755459
$324$	5.55765	0.308758
$325$	−3.61706	−0.200638
$326$	−22.4915	−1.24569
$327$	−3.65685	−0.202225
$328$	56.4884	3.11905
$329$	0	0
$330$	12.2212	0.672753
$331$	16.6533	0.915349	0.457675	−	0.889120i	$-0.348682\pi$
$331$	16.6533	0.915349	0.457675	+	0.889120i	$0.348682\pi$
$332$	39.8571	2.18744
$333$	−2.16804	−0.118808
$334$	10.3431	0.565952
$335$	−3.06608	−0.167518
$336$	0	0
$337$	31.9178	1.73867	0.869337	−	0.494220i	$-0.164546\pi$
$337$	31.9178	1.73867	0.869337	+	0.494220i	$0.164546\pi$
$338$	0.228505	0.0124290
$339$	12.6085	0.684802
$340$	−27.4951	−1.49113
$341$	5.58173	0.302268
$342$	−7.54469	−0.407970
$343$	0	0
$344$	−89.1511	−4.80671
$345$	4.36147	0.234813
$346$	43.1778	2.32126
$347$	−7.91282	−0.424783	−0.212391	−	0.977185i	$-0.568125\pi$
$347$	−7.91282	−0.424783	−0.212391	+	0.977185i	$0.568125\pi$
$348$	3.67018	0.196742
$349$	−25.1525	−1.34638	−0.673190	−	0.739469i	$-0.735076\pi$
$349$	−25.1525	−1.34638	−0.673190	+	0.739469i	$0.735076\pi$
$350$	0	0
$351$	−3.61706	−0.193064
$352$	−105.797	−5.63899
$353$	20.5667	1.09465	0.547327	−	0.836919i	$-0.315645\pi$
$353$	20.5667	1.09465	0.547327	+	0.836919i	$0.315645\pi$
$354$	38.0063	2.02001
$355$	−0.277444	−0.0147252
$356$	4.60411	0.244017
$357$	0	0
$358$	−3.52155	−0.186120
$359$	−14.1023	−0.744293	−0.372147	−	0.928174i	$-0.621378\pi$
$359$	−14.1023	−0.744293	−0.372147	+	0.928174i	$0.621378\pi$
$360$	−9.78039	−0.515472
$361$	−11.4682	−0.603591
$362$	−45.5251	−2.39274
$363$	8.76235	0.459904
$364$	0	0
$365$	−10.6135	−0.555538
$366$	−25.8808	−1.35281
$367$	−0.386844	−0.0201931	−0.0100965	−	0.999949i	$-0.503214\pi$
$367$	−0.386844	−0.0201931	−0.0100965	+	0.999949i	$0.503214\pi$
$368$	−68.7897	−3.58591
$369$	5.77568	0.300670
$370$	5.96021	0.309856
$371$	0	0
$372$	−6.97815	−0.361800
$373$	−10.6345	−0.550632	−0.275316	−	0.961354i	$-0.588783\pi$
$373$	−10.6345	−0.550632	−0.275316	+	0.961354i	$0.588783\pi$
$374$	−60.4612	−3.12637
$375$	−1.00000	−0.0516398
$376$	−63.4286	−3.27108
$377$	−2.38865	−0.123022
$378$	0	0
$379$	−26.4290	−1.35757	−0.678783	−	0.734339i	$-0.737492\pi$
$379$	−26.4290	−1.35757	−0.678783	+	0.734339i	$0.737492\pi$
$380$	15.2524	0.782434
$381$	−0.0562783	−0.00288322
$382$	24.0898	1.23254
$383$	−5.14567	−0.262931	−0.131466	−	0.991321i	$-0.541968\pi$
$383$	−5.14567	−0.262931	−0.131466	+	0.991321i	$0.541968\pi$
$384$	45.5459	2.32425
$385$	0	0
$386$	72.8478	3.70785
$387$	−9.11529	−0.463356
$388$	−34.5010	−1.75152
$389$	−12.9080	−0.654462	−0.327231	−	0.944944i	$-0.606116\pi$
$389$	−12.9080	−0.654462	−0.327231	+	0.944944i	$0.606116\pi$
$390$	9.94372	0.503520
$391$	−21.5773	−1.09121
$392$	0	0
$393$	−19.6631	−0.991873
$394$	11.3109	0.569837
$395$	5.53803	0.278649
$396$	−24.7064	−1.24155
$397$	22.7627	1.14243	0.571214	−	0.820801i	$-0.306473\pi$
$397$	22.7627	1.14243	0.571214	+	0.820801i	$0.306473\pi$
$398$	20.9906	1.05216
$399$	0	0
$400$	15.7721	0.788607
$401$	−9.85080	−0.491926	−0.245963	−	0.969279i	$-0.579104\pi$
$401$	−9.85080	−0.491926	−0.245963	+	0.969279i	$0.579104\pi$
$402$	8.42900	0.420400
$403$	4.54156	0.226231
$404$	−22.8910	−1.13887
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	9.63801	0.477738
$408$	48.3861	2.39547
$409$	14.6742	0.725594	0.362797	−	0.931868i	$-0.381822\pi$
$409$	14.6742	0.725594	0.362797	+	0.931868i	$0.381822\pi$
$410$	−15.8780	−0.784159
$411$	−18.7507	−0.924903
$412$	45.3951	2.23646
$413$	0	0
$414$	−11.9902	−0.589285
$415$	−7.17157	−0.352039
$416$	−86.0813	−4.22049
$417$	6.14657	0.300999
$418$	33.5398	1.64049
$419$	9.89450	0.483378	0.241689	−	0.970354i	$-0.422299\pi$
$419$	9.89450	0.483378	0.241689	+	0.970354i	$0.422299\pi$
$420$	0	0
$421$	35.4388	1.72718	0.863591	−	0.504193i	$-0.168210\pi$
$421$	35.4388	1.72718	0.863591	+	0.504193i	$0.168210\pi$
$422$	−0.872125	−0.0424544
$423$	−6.48528	−0.315325
$424$	68.7686	3.33970
$425$	4.94725	0.239977
$426$	0.762725	0.0369541
$427$	0	0
$428$	−47.4040	−2.29136
$429$	16.0796	0.776330
$430$	25.0590	1.20845
$431$	−16.6931	−0.804079	−0.402040	−	0.915622i	$-0.631699\pi$
$431$	−16.6931	−0.804079	−0.402040	+	0.915622i	$0.631699\pi$
$432$	15.7721	0.758838
$433$	13.0363	0.626483	0.313241	−	0.949674i	$-0.398585\pi$
$433$	13.0363	0.626483	0.313241	+	0.949674i	$0.398585\pi$
$434$	0	0
$435$	−0.660384	−0.0316630
$436$	−20.3235	−0.973319
$437$	11.9696	0.572585
$438$	29.1778	1.39417
$439$	−22.7114	−1.08396	−0.541979	−	0.840392i	$-0.682325\pi$
$439$	−22.7114	−1.08396	−0.541979	+	0.840392i	$0.682325\pi$
$440$	43.4786	2.07276
$441$	0	0
$442$	−49.1941	−2.33992
$443$	8.24891	0.391917	0.195959	−	0.980612i	$-0.437218\pi$
$443$	8.24891	0.391917	0.195959	+	0.980612i	$0.437218\pi$
$444$	−12.0492	−0.571831
$445$	−0.828427	−0.0392712
$446$	−79.8867	−3.78274
$447$	20.2306	0.956874
$448$	0	0
$449$	24.5212	1.15723	0.578613	−	0.815602i	$-0.303594\pi$
$449$	24.5212	1.15723	0.578613	+	0.815602i	$0.303594\pi$
$450$	2.74912	0.129595
$451$	−25.6757	−1.20902
$452$	70.0738	3.29599
$453$	21.1082	0.991751
$454$	57.4317	2.69541
$455$	0	0
$456$	−26.8414	−1.25696
$457$	−28.3468	−1.32601	−0.663004	−	0.748616i	$-0.730719\pi$
$457$	−28.3468	−1.32601	−0.663004	+	0.748616i	$0.730719\pi$
$458$	−13.5502	−0.633160
$459$	4.94725	0.230918
$460$	24.2395	1.13017
$461$	−38.3727	−1.78720	−0.893598	−	0.448868i	$-0.851827\pi$
$461$	−38.3727	−1.78720	−0.893598	+	0.448868i	$0.851827\pi$
$462$	0	0
$463$	2.38530	0.110855	0.0554273	−	0.998463i	$-0.482348\pi$
$463$	2.38530	0.110855	0.0554273	+	0.998463i	$0.482348\pi$
$464$	10.4157	0.483536
$465$	1.25559	0.0582268
$466$	−3.00704	−0.139298
$467$	19.8079	0.916598	0.458299	−	0.888798i	$-0.348459\pi$
$467$	19.8079	0.916598	0.458299	+	0.888798i	$0.348459\pi$
$468$	−20.1023	−0.929231
$469$	0	0
$470$	17.8288	0.822381
$471$	−1.96021	−0.0903215
$472$	135.213	6.22368
$473$	40.5219	1.86320
$474$	−15.2247	−0.699293
$475$	−2.74441	−0.125922
$476$	0	0
$477$	7.03127	0.321940
$478$	70.5768	3.22811
$479$	3.43959	0.157159	0.0785795	−	0.996908i	$-0.474962\pi$
$479$	3.43959	0.157159	0.0785795	+	0.996908i	$0.474962\pi$
$480$	−23.7987	−1.08626
$481$	7.84194	0.357562
$482$	−68.0318	−3.09876
$483$	0	0
$484$	48.6981	2.21355
$485$	6.20784	0.281883
$486$	2.74912	0.124703
$487$	−7.12235	−0.322745	−0.161372	−	0.986894i	$-0.551592\pi$
$487$	−7.12235	−0.322745	−0.161372	+	0.986894i	$0.551592\pi$
$488$	−92.0747	−4.16803
$489$	−8.18137	−0.369974
$490$	0	0
$491$	−31.6896	−1.43013	−0.715066	−	0.699057i	$-0.753603\pi$
$491$	−31.6896	−1.43013	−0.715066	+	0.699057i	$0.753603\pi$
$492$	32.0992	1.44714
$493$	3.26709	0.147142
$494$	27.2896	1.22782
$495$	4.44549	0.199810
$496$	−19.8034	−0.889200
$497$	0	0
$498$	19.7155	0.883473
$499$	−39.0027	−1.74600	−0.873001	−	0.487718i	$-0.837829\pi$
$499$	−39.0027	−1.74600	−0.873001	+	0.487718i	$0.837829\pi$
$500$	−5.55765	−0.248546
$501$	3.76235	0.168089
$502$	−1.11529	−0.0497780
$503$	18.5308	0.826247	0.413123	−	0.910675i	$-0.364438\pi$
$503$	18.5308	0.826247	0.413123	+	0.910675i	$0.364438\pi$
$504$	0	0
$505$	4.11882	0.183285
$506$	53.3022	2.36957
$507$	0.0831193	0.00369146
$508$	−0.312775	−0.0138771
$509$	39.8053	1.76434	0.882169	−	0.470934i	$-0.156083\pi$
$509$	39.8053	1.76434	0.882169	+	0.470934i	$0.156083\pi$
$510$	−13.6006	−0.602244
$511$	0	0
$512$	66.8412	2.95399
$513$	−2.74441	−0.121168
$514$	29.9406	1.32062
$515$	−8.16804	−0.359927
$516$	−50.6596	−2.23016
$517$	28.8302	1.26795
$518$	0	0
$519$	15.7061	0.689420
$520$	35.3763	1.55135
$521$	−26.3594	−1.15483	−0.577413	−	0.816452i	$-0.695938\pi$
$521$	−26.3594	−1.15483	−0.577413	+	0.816452i	$0.695938\pi$
$522$	1.81547	0.0794611
$523$	24.3727	1.06574	0.532872	−	0.846196i	$-0.321113\pi$
$523$	24.3727	1.06574	0.532872	+	0.846196i	$0.321113\pi$
$524$	−109.281	−4.77395
$525$	0	0
$526$	24.1019	1.05089
$527$	−6.21174	−0.270588
$528$	−70.1149	−3.05136
$529$	−3.97762	−0.172940
$530$	−19.3298	−0.839633
$531$	13.8249	0.599949
$532$	0	0
$533$	−20.8910	−0.904888
$534$	2.27744	0.0985546
$535$	8.52951	0.368763
$536$	29.9874	1.29526
$537$	−1.28097	−0.0552781
$538$	−62.7753	−2.70643
$539$	0	0
$540$	−5.55765	−0.239163
$541$	24.3627	1.04744	0.523718	−	0.851892i	$-0.324545\pi$
$541$	24.3627	1.04744	0.523718	+	0.851892i	$0.324545\pi$
$542$	−29.3463	−1.26053
$543$	−16.5599	−0.710652
$544$	117.738	5.04798
$545$	3.65685	0.156642
$546$	0	0
$547$	10.1421	0.433646	0.216823	−	0.976211i	$-0.430430\pi$
$547$	10.1421	0.433646	0.216823	+	0.976211i	$0.430430\pi$
$548$	−104.210	−4.45161
$549$	−9.41421	−0.401789
$550$	−12.2212	−0.521112
$551$	−1.81236	−0.0772092
$552$	−42.6568	−1.81560
$553$	0	0
$554$	−34.1882	−1.45252
$555$	2.16804	0.0920283
$556$	34.1605	1.44873
$557$	−14.5560	−0.616756	−0.308378	−	0.951264i	$-0.599786\pi$
$557$	−14.5560	−0.616756	−0.308378	+	0.951264i	$0.599786\pi$
$558$	−3.45178	−0.146125
$559$	32.9706	1.39451
$560$	0	0
$561$	−21.9929	−0.928543
$562$	35.5100	1.49790
$563$	36.5845	1.54185	0.770926	−	0.636925i	$-0.219794\pi$
$563$	36.5845	1.54185	0.770926	+	0.636925i	$0.219794\pi$
$564$	−36.0429	−1.51768
$565$	−12.6085	−0.530445
$566$	−70.4156	−2.95979
$567$	0	0
$568$	2.71351	0.113856
$569$	−3.64687	−0.152885	−0.0764424	−	0.997074i	$-0.524356\pi$
$569$	−3.64687	−0.152885	−0.0764424	+	0.997074i	$0.524356\pi$
$570$	7.54469	0.316012
$571$	−38.5933	−1.61508	−0.807540	−	0.589812i	$-0.799202\pi$
$571$	−38.5933	−1.61508	−0.807540	+	0.589812i	$0.799202\pi$
$572$	89.3647	3.73653
$573$	8.76272	0.366068
$574$	0	0
$575$	−4.36147	−0.181886
$576$	33.8812	1.41171
$577$	−14.0523	−0.585006	−0.292503	−	0.956265i	$-0.594488\pi$
$577$	−14.0523	−0.585006	−0.292503	+	0.956265i	$0.594488\pi$
$578$	20.5505	0.854787
$579$	26.4986	1.10124
$580$	−3.67018	−0.152396
$581$	0	0
$582$	−17.0661	−0.707412
$583$	−31.2574	−1.29455
$584$	103.804	4.29546
$585$	3.61706	0.149547
$586$	4.62611	0.191103
$587$	27.0402	1.11607	0.558034	−	0.829818i	$-0.311556\pi$
$587$	27.0402	1.11607	0.558034	+	0.829818i	$0.311556\pi$
$588$	0	0
$589$	3.44586	0.141984
$590$	−38.0063	−1.56469
$591$	4.11439	0.169243
$592$	−34.1947	−1.40539
$593$	−21.6764	−0.890145	−0.445073	−	0.895495i	$-0.646822\pi$
$593$	−21.6764	−0.890145	−0.445073	+	0.895495i	$0.646822\pi$
$594$	−12.2212	−0.501440
$595$	0	0
$596$	112.434	4.60550
$597$	7.63538	0.312495
$598$	43.3692	1.77350
$599$	25.9898	1.06191	0.530957	−	0.847399i	$-0.321833\pi$
$599$	25.9898	1.06191	0.530957	+	0.847399i	$0.321833\pi$
$600$	9.78039	0.399283
$601$	−12.0746	−0.492533	−0.246267	−	0.969202i	$-0.579204\pi$
$601$	−12.0746	−0.492533	−0.246267	+	0.969202i	$0.579204\pi$
$602$	0	0
$603$	3.06608	0.124860
$604$	117.312	4.77336
$605$	−8.76235	−0.356240
$606$	−11.3231	−0.459971
$607$	−41.7390	−1.69413	−0.847067	−	0.531486i	$-0.821634\pi$
$607$	−41.7390	−1.69413	−0.847067	+	0.531486i	$0.821634\pi$
$608$	−65.3133	−2.64880
$609$	0	0
$610$	25.8808	1.04788
$611$	23.4576	0.948995
$612$	27.4951	1.11142
$613$	−11.8016	−0.476662	−0.238331	−	0.971184i	$-0.576600\pi$
$613$	−11.8016	−0.476662	−0.238331	+	0.971184i	$0.576600\pi$
$614$	20.1062	0.811422
$615$	−5.77568	−0.232898
$616$	0	0
$617$	12.2913	0.494829	0.247415	−	0.968910i	$-0.420419\pi$
$617$	12.2913	0.494829	0.247415	+	0.968910i	$0.420419\pi$
$618$	22.4549	0.903269
$619$	−24.3772	−0.979801	−0.489900	−	0.871778i	$-0.662967\pi$
$619$	−24.3772	−0.979801	−0.489900	+	0.871778i	$0.662967\pi$
$620$	6.97815	0.280249
$621$	−4.36147	−0.175020
$622$	−30.7955	−1.23479
$623$	0	0
$624$	−57.0488	−2.28378
$625$	1.00000	0.0400000
$626$	86.5578	3.45954
$627$	12.2002	0.487230
$628$	−10.8941	−0.434723
$629$	−10.7259	−0.427668
$630$	0	0
$631$	−43.3200	−1.72454	−0.862271	−	0.506448i	$-0.830958\pi$
$631$	−43.3200	−1.72454	−0.862271	+	0.506448i	$0.830958\pi$
$632$	−54.1641	−2.15453
$633$	−0.317238	−0.0126091
$634$	23.2314	0.922635
$635$	0.0562783	0.00223334
$636$	39.0773	1.54952
$637$	0	0
$638$	−8.07066	−0.319521
$639$	0.277444	0.0109755
$640$	−45.5459	−1.80036
$641$	18.5408	0.732316	0.366158	−	0.930553i	$-0.380673\pi$
$641$	18.5408	0.732316	0.366158	+	0.930553i	$0.380673\pi$
$642$	−23.4486	−0.925443
$643$	−8.55489	−0.337372	−0.168686	−	0.985670i	$-0.553952\pi$
$643$	−8.55489	−0.337372	−0.168686	+	0.985670i	$0.553952\pi$
$644$	0	0
$645$	9.11529	0.358914
$646$	−37.3255	−1.46855
$647$	−43.0886	−1.69399	−0.846994	−	0.531603i	$-0.821590\pi$
$647$	−43.0886	−1.69399	−0.846994	+	0.531603i	$0.821590\pi$
$648$	9.78039	0.384210
$649$	−61.4584	−2.41245
$650$	−9.94372	−0.390025
$651$	0	0
$652$	−45.4692	−1.78071
$653$	−14.0277	−0.548948	−0.274474	−	0.961595i	$-0.588504\pi$
$653$	−14.0277	−0.548948	−0.274474	+	0.961595i	$0.588504\pi$
$654$	−10.0531	−0.393108
$655$	19.6631	0.768302
$656$	91.0949	3.55666
$657$	10.6135	0.414073
$658$	0	0
$659$	−19.1488	−0.745932	−0.372966	−	0.927845i	$-0.621659\pi$
$659$	−19.1488	−0.745932	−0.372966	+	0.927845i	$0.621659\pi$
$660$	24.7064	0.961697
$661$	26.5717	1.03352	0.516759	−	0.856131i	$-0.327138\pi$
$661$	26.5717	1.03352	0.516759	+	0.856131i	$0.327138\pi$
$662$	45.7819	1.77937
$663$	−17.8945	−0.694965
$664$	70.1408	2.72199
$665$	0	0
$666$	−5.96021	−0.230953
$667$	−2.88024	−0.111523
$668$	20.9098	0.809025
$669$	−29.0590	−1.12349
$670$	−8.42900	−0.325641
$671$	41.8508	1.61563
$672$	0	0
$673$	46.6900	1.79977	0.899883	−	0.436132i	$-0.143652\pi$
$673$	46.6900	1.79977	0.899883	+	0.436132i	$0.143652\pi$
$674$	87.7458	3.37984
$675$	1.00000	0.0384900
$676$	0.461948	0.0177672
$677$	−41.6953	−1.60248	−0.801240	−	0.598343i	$-0.795826\pi$
$677$	−41.6953	−1.60248	−0.801240	+	0.598343i	$0.795826\pi$
$678$	34.6624	1.33120
$679$	0	0
$680$	−48.3861	−1.85552
$681$	20.8910	0.800544
$682$	15.3448	0.587584
$683$	−21.5430	−0.824321	−0.412160	−	0.911111i	$-0.635226\pi$
$683$	−21.5430	−0.824321	−0.412160	+	0.911111i	$0.635226\pi$
$684$	−15.2524	−0.583192
$685$	18.7507	0.716426
$686$	0	0
$687$	−4.92893	−0.188050
$688$	−143.768	−5.48110
$689$	−25.4325	−0.968902
$690$	11.9902	0.456458
$691$	−22.9224	−0.872010	−0.436005	−	0.899944i	$-0.643607\pi$
$691$	−22.9224	−0.872010	−0.436005	+	0.899944i	$0.643607\pi$
$692$	87.2888	3.31822
$693$	0	0
$694$	−21.7533	−0.825743
$695$	−6.14657	−0.233153
$696$	6.45881	0.244821
$697$	28.5737	1.08231
$698$	−69.1471	−2.61726
$699$	−1.09382	−0.0413720
$700$	0	0
$701$	−51.4317	−1.94255	−0.971275	−	0.237960i	$-0.923521\pi$
$701$	−51.4317	−1.94255	−0.971275	+	0.237960i	$0.923521\pi$
$702$	−9.94372	−0.375302
$703$	5.94999	0.224408
$704$	−150.618	−5.67664
$705$	6.48528	0.244250
$706$	56.5402	2.12792
$707$	0	0
$708$	76.8339	2.88759
$709$	−34.4682	−1.29448	−0.647241	−	0.762286i	$-0.724077\pi$
$709$	−34.4682	−1.29448	−0.647241	+	0.762286i	$0.724077\pi$
$710$	−0.762725	−0.0286246
$711$	−5.53803	−0.207692
$712$	8.10234	0.303648
$713$	5.47623	0.205086
$714$	0	0
$715$	−16.0796	−0.601343
$716$	−7.11920	−0.266057
$717$	25.6725	0.958759
$718$	−38.7690	−1.44685
$719$	−42.0751	−1.56914	−0.784568	−	0.620043i	$-0.787115\pi$
$719$	−42.0751	−1.56914	−0.784568	+	0.620043i	$0.787115\pi$
$720$	−15.7721	−0.587793
$721$	0	0
$722$	−31.5275	−1.17333
$723$	−24.7468	−0.920342
$724$	−92.0340	−3.42042
$725$	0.660384	0.0245260
$726$	24.0887	0.894017
$727$	−36.6722	−1.36010	−0.680048	−	0.733168i	$-0.738041\pi$
$727$	−36.6722	−1.36010	−0.680048	+	0.733168i	$0.738041\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−29.1778	−1.07992
$731$	−45.0957	−1.66792
$732$	−52.3209	−1.93384
$733$	46.3696	1.71270	0.856350	−	0.516397i	$-0.172727\pi$
$733$	46.3696	1.71270	0.856350	+	0.516397i	$0.172727\pi$
$734$	−1.06348	−0.0392538
$735$	0	0
$736$	−103.797	−3.82602
$737$	−13.6302	−0.502075
$738$	15.8780	0.584478
$739$	1.35846	0.0499719	0.0249859	−	0.999688i	$-0.492046\pi$
$739$	1.35846	0.0499719	0.0249859	+	0.999688i	$0.492046\pi$
$740$	12.0492	0.442938
$741$	9.92668	0.364666
$742$	0	0
$743$	−26.2748	−0.963930	−0.481965	−	0.876191i	$-0.660077\pi$
$743$	−26.2748	−0.963930	−0.481965	+	0.876191i	$0.660077\pi$
$744$	−12.2802	−0.450214
$745$	−20.2306	−0.741191
$746$	−29.2354	−1.07039
$747$	7.17157	0.262394
$748$	−122.229	−4.46914
$749$	0	0
$750$	−2.74912	−0.100384
$751$	−25.2271	−0.920548	−0.460274	−	0.887777i	$-0.652249\pi$
$751$	−25.2271	−0.920548	−0.460274	+	0.887777i	$0.652249\pi$
$752$	−102.287	−3.73002
$753$	−0.405692	−0.0147842
$754$	−6.56668	−0.239144
$755$	−21.1082	−0.768207
$756$	0	0
$757$	6.02331	0.218921	0.109460	−	0.993991i	$-0.465088\pi$
$757$	6.02331	0.218921	0.109460	+	0.993991i	$0.465088\pi$
$758$	−72.6564	−2.63900
$759$	19.3888	0.703770
$760$	26.8414	0.973638
$761$	30.8214	1.11727	0.558637	−	0.829412i	$-0.311325\pi$
$761$	30.8214	1.11727	0.558637	+	0.829412i	$0.311325\pi$
$762$	−0.154716	−0.00560476
$763$	0	0
$764$	48.7001	1.76191
$765$	−4.94725	−0.178868
$766$	−14.1460	−0.511117
$767$	−50.0055	−1.80559
$768$	57.4486	2.07300
$769$	31.8736	1.14939	0.574695	−	0.818367i	$-0.305121\pi$
$769$	31.8736	1.14939	0.574695	+	0.818367i	$0.305121\pi$
$770$	0	0
$771$	10.8910	0.392229
$772$	147.270	5.30036
$773$	−17.0912	−0.614727	−0.307364	−	0.951592i	$-0.599447\pi$
$773$	−17.0912	−0.614727	−0.307364	+	0.951592i	$0.599447\pi$
$774$	−25.0590	−0.900728
$775$	−1.25559	−0.0451023
$776$	−60.7151	−2.17954
$777$	0	0
$778$	−35.4857	−1.27222
$779$	−15.8508	−0.567914
$780$	20.1023	0.719779
$781$	−1.23337	−0.0441335
$782$	−59.3184	−2.12122
$783$	0.660384	0.0236002
$784$	0	0
$785$	1.96021	0.0699627
$786$	−54.0562	−1.92812
$787$	28.1617	1.00386	0.501929	−	0.864909i	$-0.332624\pi$
$787$	28.1617	1.00386	0.501929	+	0.864909i	$0.332624\pi$
$788$	22.8663	0.814580
$789$	8.76716	0.312119
$790$	15.2247	0.541670
$791$	0	0
$792$	−43.4786	−1.54494
$793$	34.0518	1.20921
$794$	62.5774	2.22079
$795$	−7.03127	−0.249374
$796$	42.4347	1.50406
$797$	−9.60058	−0.340070	−0.170035	−	0.985438i	$-0.554388\pi$
$797$	−9.60058	−0.340070	−0.170035	+	0.985438i	$0.554388\pi$
$798$	0	0
$799$	−32.0843	−1.13506
$800$	23.7987	0.841411
$801$	0.828427	0.0292710
$802$	−27.0810	−0.956264
$803$	−47.1823	−1.66503
$804$	17.0402	0.600960
$805$	0	0
$806$	12.4853	0.439775
$807$	−22.8347	−0.803819
$808$	−40.2837	−1.41718
$809$	−21.0331	−0.739485	−0.369742	−	0.929134i	$-0.620554\pi$
$809$	−21.0331	−0.739485	−0.369742	+	0.929134i	$0.620554\pi$
$810$	−2.74912	−0.0965941
$811$	−0.770313	−0.0270493	−0.0135247	−	0.999909i	$-0.504305\pi$
$811$	−0.770313	−0.0270493	−0.0135247	+	0.999909i	$0.504305\pi$
$812$	0	0
$813$	−10.6748	−0.374382
$814$	26.4960	0.928685
$815$	8.18137	0.286581
$816$	78.0288	2.73155
$817$	25.0161	0.875201
$818$	40.3412	1.41050
$819$	0	0
$820$	−32.0992	−1.12095
$821$	−16.4782	−0.575094	−0.287547	−	0.957767i	$-0.592840\pi$
$821$	−16.4782	−0.575094	−0.287547	+	0.957767i	$0.592840\pi$
$822$	−51.5478	−1.79794
$823$	−16.0563	−0.559687	−0.279843	−	0.960046i	$-0.590283\pi$
$823$	−16.0563	−0.559687	−0.279843	+	0.960046i	$0.590283\pi$
$824$	79.8867	2.78298
$825$	−4.44549	−0.154772
$826$	0	0
$827$	−1.11736	−0.0388545	−0.0194272	−	0.999811i	$-0.506184\pi$
$827$	−1.11736	−0.0388545	−0.0194272	+	0.999811i	$0.506184\pi$
$828$	−24.2395	−0.842380
$829$	52.5724	1.82592	0.912958	−	0.408054i	$-0.133793\pi$
$829$	52.5724	1.82592	0.912958	+	0.408054i	$0.133793\pi$
$830$	−19.7155	−0.684335
$831$	−12.4361	−0.431402
$832$	−122.550	−4.24866
$833$	0	0
$834$	16.8976	0.585117
$835$	−3.76235	−0.130202
$836$	67.8045	2.34507
$837$	−1.25559	−0.0433997
$838$	27.2012	0.939648
$839$	−8.65667	−0.298861	−0.149431	−	0.988772i	$-0.547744\pi$
$839$	−8.65667	−0.298861	−0.149431	+	0.988772i	$0.547744\pi$
$840$	0	0
$841$	−28.5639	−0.984962
$842$	97.4254	3.35750
$843$	12.9169	0.444881
$844$	−1.76310	−0.0606884
$845$	−0.0831193	−0.00285939
$846$	−17.8288	−0.612967
$847$	0	0
$848$	110.898	3.80826
$849$	−25.6139	−0.879067
$850$	13.6006	0.466496
$851$	9.45584	0.324142
$852$	1.54193	0.0528258
$853$	21.9361	0.751078	0.375539	−	0.926807i	$-0.377458\pi$
$853$	21.9361	0.751078	0.375539	+	0.926807i	$0.377458\pi$
$854$	0	0
$855$	2.74441	0.0938567
$856$	−83.4219	−2.85130
$857$	−25.1253	−0.858263	−0.429132	−	0.903242i	$-0.641180\pi$
$857$	−25.1253	−0.858263	−0.429132	+	0.903242i	$0.641180\pi$
$858$	44.2047	1.50912
$859$	−8.09183	−0.276090	−0.138045	−	0.990426i	$-0.544082\pi$
$859$	−8.09183	−0.276090	−0.138045	+	0.990426i	$0.544082\pi$
$860$	50.6596	1.72748
$861$	0	0
$862$	−45.8913	−1.56307
$863$	23.0674	0.785222	0.392611	−	0.919705i	$-0.371572\pi$
$863$	23.0674	0.785222	0.392611	+	0.919705i	$0.371572\pi$
$864$	23.7987	0.809649
$865$	−15.7061	−0.534023
$866$	35.8382	1.21783
$867$	7.47530	0.253874
$868$	0	0
$869$	24.6192	0.835150
$870$	−1.81547	−0.0615503
$871$	−11.0902	−0.375776
$872$	−35.7655	−1.21117
$873$	−6.20784	−0.210103
$874$	32.9059	1.11306
$875$	0	0
$876$	58.9863	1.99296
$877$	−4.32351	−0.145994	−0.0729972	−	0.997332i	$-0.523256\pi$
$877$	−4.32351	−0.145994	−0.0729972	+	0.997332i	$0.523256\pi$
$878$	−62.4364	−2.10713
$879$	1.68276	0.0567581
$880$	70.1149	2.36357
$881$	19.9929	0.673579	0.336790	−	0.941580i	$-0.390659\pi$
$881$	19.9929	0.673579	0.336790	+	0.941580i	$0.390659\pi$
$882$	0	0
$883$	1.89450	0.0637551	0.0318776	−	0.999492i	$-0.489851\pi$
$883$	1.89450	0.0637551	0.0318776	+	0.999492i	$0.489851\pi$
$884$	−99.4513	−3.34491
$885$	−13.8249	−0.464719
$886$	22.6772	0.761856
$887$	5.22525	0.175447	0.0877234	−	0.996145i	$-0.472041\pi$
$887$	5.22525	0.175447	0.0877234	+	0.996145i	$0.472041\pi$
$888$	−21.2043	−0.711570
$889$	0	0
$890$	−2.27744	−0.0763401
$891$	−4.44549	−0.148929
$892$	−161.500	−5.40741
$893$	17.7982	0.595595
$894$	55.6163	1.86009
$895$	1.28097	0.0428182
$896$	0	0
$897$	15.7757	0.526735
$898$	67.4116	2.24955
$899$	−0.829175	−0.0276545
$900$	5.55765	0.185255
$901$	34.7855	1.15887
$902$	−70.5855	−2.35024
$903$	0	0
$904$	123.316	4.10144
$905$	16.5599	0.550469
$906$	58.0290	1.92789
$907$	−5.41921	−0.179942	−0.0899709	−	0.995944i	$-0.528677\pi$
$907$	−5.41921	−0.179942	−0.0899709	+	0.995944i	$0.528677\pi$
$908$	116.105	3.85307
$909$	−4.11882	−0.136613
$910$	0	0
$911$	0.226686	0.00751043	0.00375522	−	0.999993i	$-0.498805\pi$
$911$	0.226686	0.00751043	0.00375522	+	0.999993i	$0.498805\pi$
$912$	−43.2852	−1.43332
$913$	−31.8811	−1.05511
$914$	−77.9287	−2.57765
$915$	9.41421	0.311224
$916$	−27.3933	−0.905099
$917$	0	0
$918$	13.6006	0.448886
$919$	−8.10624	−0.267400	−0.133700	−	0.991022i	$-0.542686\pi$
$919$	−8.10624	−0.267400	−0.133700	+	0.991022i	$0.542686\pi$
$920$	42.6568	1.40635
$921$	7.31371	0.240995
$922$	−105.491	−3.47417
$923$	−1.00353	−0.0330316
$924$	0	0
$925$	−2.16804	−0.0712848
$926$	6.55748	0.215492
$927$	8.16804	0.268274
$928$	15.7163	0.515912
$929$	−2.29945	−0.0754424	−0.0377212	−	0.999288i	$-0.512010\pi$
$929$	−2.29945	−0.0754424	−0.0377212	+	0.999288i	$0.512010\pi$
$930$	3.45178	0.113188
$931$	0	0
$932$	−6.07906	−0.199126
$933$	−11.2019	−0.366735
$934$	54.4541	1.78179
$935$	21.9929	0.719246
$936$	−35.3763	−1.15631
$937$	16.1023	0.526041	0.263020	−	0.964790i	$-0.415281\pi$
$937$	16.1023	0.526041	0.263020	+	0.964790i	$0.415281\pi$
$938$	0	0
$939$	31.4857	1.02750
$940$	36.0429	1.17559
$941$	−13.5247	−0.440893	−0.220446	−	0.975399i	$-0.570751\pi$
$941$	−13.5247	−0.440893	−0.220446	+	0.975399i	$0.570751\pi$
$942$	−5.38883	−0.175578
$943$	−25.1904	−0.820313
$944$	218.048	7.09687
$945$	0	0
$946$	111.400	3.62191
$947$	6.85381	0.222719	0.111359	−	0.993780i	$-0.464480\pi$
$947$	6.85381	0.222719	0.111359	+	0.993780i	$0.464480\pi$
$948$	−30.7784	−0.999637
$949$	−38.3898	−1.24618
$950$	−7.54469	−0.244782
$951$	8.45048	0.274026
$952$	0	0
$953$	−44.9483	−1.45602	−0.728009	−	0.685568i	$-0.759554\pi$
$953$	−44.9483	−1.45602	−0.728009	+	0.685568i	$0.759554\pi$
$954$	19.3298	0.625825
$955$	−8.76272	−0.283555
$956$	142.679	4.61457
$957$	−2.93573	−0.0948986
$958$	9.45584	0.305504
$959$	0	0
$960$	−33.8812	−1.09351
$961$	−29.4235	−0.949145
$962$	21.5584	0.695071
$963$	−8.52951	−0.274860
$964$	−137.534	−4.42967
$965$	−26.4986	−0.853020
$966$	0	0
$967$	12.4245	0.399546	0.199773	−	0.979842i	$-0.435979\pi$
$967$	12.4245	0.399546	0.199773	+	0.979842i	$0.435979\pi$
$968$	85.6992	2.75448
$969$	−13.5773	−0.436165
$970$	17.0661	0.547959
$971$	−20.8158	−0.668012	−0.334006	−	0.942571i	$-0.608401\pi$
$971$	−20.8158	−0.668012	−0.334006	+	0.942571i	$0.608401\pi$
$972$	5.55765	0.178262
$973$	0	0
$974$	−19.5802	−0.627390
$975$	−3.61706	−0.115839
$976$	−148.482	−4.75281
$977$	42.7418	1.36743	0.683716	−	0.729748i	$-0.260363\pi$
$977$	42.7418	1.36743	0.683716	+	0.729748i	$0.260363\pi$
$978$	−22.4915	−0.719200
$979$	−3.68276	−0.117702
$980$	0	0
$981$	−3.65685	−0.116754
$982$	−87.1184	−2.78006
$983$	−5.68982	−0.181477	−0.0907386	−	0.995875i	$-0.528923\pi$
$983$	−5.68982	−0.181477	−0.0907386	+	0.995875i	$0.528923\pi$
$984$	56.4884	1.80078
$985$	−4.11439	−0.131095
$986$	8.98160	0.286033
$987$	0	0
$988$	55.1690	1.75516
$989$	39.7560	1.26417
$990$	12.2212	0.388414
$991$	59.1216	1.87806	0.939029	−	0.343838i	$-0.111727\pi$
$991$	59.1216	1.87806	0.939029	+	0.343838i	$0.111727\pi$
$992$	−29.8815	−0.948739
$993$	16.6533	0.528477
$994$	0	0
$995$	−7.63538	−0.242058
$996$	39.8571	1.26292
$997$	45.5294	1.44193	0.720965	−	0.692972i	$-0.243699\pi$
$997$	45.5294	1.44193	0.720965	+	0.692972i	$0.243699\pi$
$998$	−107.223	−3.39409
$999$	−2.16804	−0.0685938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.a.o.1.4	yes	4
3.2	odd	2		2205.2.a.bg.1.1		4
5.4	even	2		3675.2.a.bk.1.1		4
7.2	even	3		735.2.i.m.361.1		8
7.3	odd	6		735.2.i.n.226.1		8
7.4	even	3		735.2.i.m.226.1		8
7.5	odd	6		735.2.i.n.361.1		8
7.6	odd	2		735.2.a.n.1.4	✓	4
21.20	even	2		2205.2.a.bf.1.1		4
35.34	odd	2		3675.2.a.bl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.a.n.1.4	✓	4	7.6	odd	2
735.2.a.o.1.4	yes	4	1.1	even	1	trivial
735.2.i.m.226.1		8	7.4	even	3
735.2.i.m.361.1		8	7.2	even	3
735.2.i.n.226.1		8	7.3	odd	6
735.2.i.n.361.1		8	7.5	odd	6
2205.2.a.bf.1.1		4	21.20	even	2
2205.2.a.bg.1.1		4	3.2	odd	2
3675.2.a.bk.1.1		4	5.4	even	2
3675.2.a.bl.1.1		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74912 q^{2} +1.00000 q^{3} +5.55765 q^{4} -1.00000 q^{5} +2.74912 q^{6} +9.78039 q^{8} +1.00000 q^{9} -2.74912 q^{10} -4.44549 q^{11} +5.55765 q^{12} -3.61706 q^{13} -1.00000 q^{15} +15.7721 q^{16} +4.94725 q^{17} +2.74912 q^{18} -2.74441 q^{19} -5.55765 q^{20} -12.2212 q^{22} -4.36147 q^{23} +9.78039 q^{24} +1.00000 q^{25} -9.94372 q^{26} +1.00000 q^{27} +0.660384 q^{29} -2.74912 q^{30} -1.25559 q^{31} +23.7987 q^{32} -4.44549 q^{33} +13.6006 q^{34} +5.55765 q^{36} -2.16804 q^{37} -7.54469 q^{38} -3.61706 q^{39} -9.78039 q^{40} +5.77568 q^{41} -9.11529 q^{43} -24.7064 q^{44} -1.00000 q^{45} -11.9902 q^{46} -6.48528 q^{47} +15.7721 q^{48} +2.74912 q^{50} +4.94725 q^{51} -20.1023 q^{52} +7.03127 q^{53} +2.74912 q^{54} +4.44549 q^{55} -2.74441 q^{57} +1.81547 q^{58} +13.8249 q^{59} -5.55765 q^{60} -9.41421 q^{61} -3.45178 q^{62} +33.8812 q^{64} +3.61706 q^{65} -12.2212 q^{66} +3.06608 q^{67} +27.4951 q^{68} -4.36147 q^{69} +0.277444 q^{71} +9.78039 q^{72} +10.6135 q^{73} -5.96021 q^{74} +1.00000 q^{75} -15.2524 q^{76} -9.94372 q^{78} -5.53803 q^{79} -15.7721 q^{80} +1.00000 q^{81} +15.8780 q^{82} +7.17157 q^{83} -4.94725 q^{85} -25.0590 q^{86} +0.660384 q^{87} -43.4786 q^{88} +0.828427 q^{89} -2.74912 q^{90} -24.2395 q^{92} -1.25559 q^{93} -17.8288 q^{94} +2.74441 q^{95} +23.7987 q^{96} -6.20784 q^{97} -4.44549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{5} + 4 q^{6} + 12 q^{8} + 4 q^{9} - 4 q^{10} + 8 q^{11} + 8 q^{12} - 4 q^{15} + 12 q^{16} + 8 q^{17} + 4 q^{18} - 8 q^{19} - 8 q^{20} + 12 q^{24} + 4 q^{25} + 4 q^{27}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^5 + 4 * q^6 + 12 * q^8 + 4 * q^9 - 4 * q^10 + 8 * q^11 + 8 * q^12 - 4 * q^15 + 12 * q^16 + 8 * q^17 + 4 * q^18 - 8 * q^19 - 8 * q^20 + 12 * q^24 + 4 * q^25 + 4 * q^27 + 8 * q^29 - 4 * q^30 - 8 * q^31 + 28 * q^32 + 8 * q^33 - 8 * q^34 + 8 * q^36 + 8 * q^37 + 4 * q^38 - 12 * q^40 - 8 * q^43 - 16 * q^44 - 4 * q^45 + 12 * q^46 + 8 * q^47 + 12 * q^48 + 4 * q^50 + 8 * q^51 - 32 * q^52 + 8 * q^53 + 4 * q^54 - 8 * q^55 - 8 * q^57 - 24 * q^58 + 16 * q^59 - 8 * q^60 - 32 * q^61 - 20 * q^62 + 24 * q^64 + 24 * q^68 - 8 * q^71 + 12 * q^72 - 32 * q^74 + 4 * q^75 + 8 * q^76 - 12 * q^80 + 4 * q^81 - 8 * q^82 + 40 * q^83 - 8 * q^85 - 32 * q^86 + 8 * q^87 - 40 * q^88 - 8 * q^89 - 4 * q^90 - 8 * q^92 - 8 * q^93 - 16 * q^94 + 8 * q^95 + 28 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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