LMFDB - Embedded newform 735.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(1,735)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.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:	$5.86900454856$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +1.41421 q^{10} +0.585786 q^{11} -4.41421 q^{13} +1.00000 q^{15} -4.00000 q^{16} +2.24264 q^{17} -1.41421 q^{18} +4.65685 q^{19} -0.828427 q^{22} +2.24264 q^{23} -2.82843 q^{24} +1.00000 q^{25} +6.24264 q^{26} -1.00000 q^{27} +8.24264 q^{29} -1.41421 q^{30} -5.82843 q^{31} -0.585786 q^{33} -3.17157 q^{34} -8.41421 q^{37} -6.58579 q^{38} +4.41421 q^{39} -2.82843 q^{40} -6.24264 q^{41} -7.58579 q^{43} -1.00000 q^{45} -3.17157 q^{46} -13.3137 q^{47} +4.00000 q^{48} -1.41421 q^{50} -2.24264 q^{51} +6.82843 q^{53} +1.41421 q^{54} -0.585786 q^{55} -4.65685 q^{57} -11.6569 q^{58} -1.41421 q^{59} -4.48528 q^{61} +8.24264 q^{62} +8.00000 q^{64} +4.41421 q^{65} +0.828427 q^{66} -13.7279 q^{67} -2.24264 q^{69} -0.585786 q^{71} +2.82843 q^{72} -12.0711 q^{73} +11.8995 q^{74} -1.00000 q^{75} -6.24264 q^{78} +6.65685 q^{79} +4.00000 q^{80} +1.00000 q^{81} +8.82843 q^{82} +2.58579 q^{83} -2.24264 q^{85} +10.7279 q^{86} -8.24264 q^{87} +1.65685 q^{88} -12.2426 q^{89} +1.41421 q^{90} +5.82843 q^{93} +18.8284 q^{94} -4.65685 q^{95} -5.17157 q^{97} +0.585786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{16} - 4 q^{17} - 2 q^{19} + 4 q^{22} - 4 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{27} + 8 q^{29} - 6 q^{31} - 4 q^{33} - 12 q^{34} - 14 q^{37} - 16 q^{38} + 6 q^{39} - 4 q^{41} - 18 q^{43} - 2 q^{45} - 12 q^{46} - 4 q^{47} + 8 q^{48} + 4 q^{51} + 8 q^{53} - 4 q^{55} + 2 q^{57} - 12 q^{58} + 8 q^{61} + 8 q^{62} + 16 q^{64} + 6 q^{65} - 4 q^{66} - 2 q^{67} + 4 q^{69} - 4 q^{71} - 10 q^{73} + 4 q^{74} - 2 q^{75} - 4 q^{78} + 2 q^{79} + 8 q^{80} + 2 q^{81} + 12 q^{82} + 8 q^{83} + 4 q^{85} - 4 q^{86} - 8 q^{87} - 8 q^{88} - 16 q^{89} + 6 q^{93} + 32 q^{94} + 2 q^{95} - 16 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 + 4 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^16 - 4 * q^17 - 2 * q^19 + 4 * q^22 - 4 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^27 + 8 * q^29 - 6 * q^31 - 4 * q^33 - 12 * q^34 - 14 * q^37 - 16 * q^38 + 6 * q^39 - 4 * q^41 - 18 * q^43 - 2 * q^45 - 12 * q^46 - 4 * q^47 + 8 * q^48 + 4 * q^51 + 8 * q^53 - 4 * q^55 + 2 * q^57 - 12 * q^58 + 8 * q^61 + 8 * q^62 + 16 * q^64 + 6 * q^65 - 4 * q^66 - 2 * q^67 + 4 * q^69 - 4 * q^71 - 10 * q^73 + 4 * q^74 - 2 * q^75 - 4 * q^78 + 2 * q^79 + 8 * q^80 + 2 * q^81 + 12 * q^82 + 8 * q^83 + 4 * q^85 - 4 * q^86 - 8 * q^87 - 8 * q^88 - 16 * q^89 + 6 * q^93 + 32 * q^94 + 2 * q^95 - 16 * q^97 + 4 * q^99

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.41421	0.577350
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	1.00000	0.333333
$10$	1.41421	0.447214
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	0	0
$13$	−4.41421	−1.22428	−0.612141	−	0.790748i	$-0.709692\pi$
$13$	−4.41421	−1.22428	−0.612141	+	0.790748i	$0.709692\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	2.24264	0.543920	0.271960	−	0.962309i	$-0.412328\pi$
$17$	2.24264	0.543920	0.271960	+	0.962309i	$0.412328\pi$
$18$	−1.41421	−0.333333
$19$	4.65685	1.06836	0.534178	−	0.845372i	$-0.320621\pi$
$19$	4.65685	1.06836	0.534178	+	0.845372i	$0.320621\pi$
$20$	0	0
$21$	0	0
$22$	−0.828427	−0.176621
$23$	2.24264	0.467623	0.233811	−	0.972282i	$-0.424880\pi$
$23$	2.24264	0.467623	0.233811	+	0.972282i	$0.424880\pi$
$24$	−2.82843	−0.577350
$25$	1.00000	0.200000
$26$	6.24264	1.22428
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.24264	1.53062	0.765310	−	0.643662i	$-0.222586\pi$
$29$	8.24264	1.53062	0.765310	+	0.643662i	$0.222586\pi$
$30$	−1.41421	−0.258199
$31$	−5.82843	−1.04682	−0.523408	−	0.852082i	$-0.675340\pi$
$31$	−5.82843	−1.04682	−0.523408	+	0.852082i	$0.675340\pi$
$32$	0	0
$33$	−0.585786	−0.101972
$34$	−3.17157	−0.543920
$35$	0	0
$36$	0	0
$37$	−8.41421	−1.38329	−0.691644	−	0.722238i	$-0.743113\pi$
$37$	−8.41421	−1.38329	−0.691644	+	0.722238i	$0.743113\pi$
$38$	−6.58579	−1.06836
$39$	4.41421	0.706840
$40$	−2.82843	−0.447214
$41$	−6.24264	−0.974937	−0.487468	−	0.873141i	$-0.662080\pi$
$41$	−6.24264	−0.974937	−0.487468	+	0.873141i	$0.662080\pi$
$42$	0	0
$43$	−7.58579	−1.15682	−0.578411	−	0.815746i	$-0.696327\pi$
$43$	−7.58579	−1.15682	−0.578411	+	0.815746i	$0.696327\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	−3.17157	−0.467623
$47$	−13.3137	−1.94200	−0.971002	−	0.239071i	$-0.923157\pi$
$47$	−13.3137	−1.94200	−0.971002	+	0.239071i	$0.923157\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	−1.41421	−0.200000
$51$	−2.24264	−0.314033
$52$	0	0
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	1.41421	0.192450
$55$	−0.585786	−0.0789874
$56$	0	0
$57$	−4.65685	−0.616815
$58$	−11.6569	−1.53062
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	−4.48528	−0.574281	−0.287141	−	0.957888i	$-0.592705\pi$
$61$	−4.48528	−0.574281	−0.287141	+	0.957888i	$0.592705\pi$
$62$	8.24264	1.04682
$63$	0	0
$64$	8.00000	1.00000
$65$	4.41421	0.547516
$66$	0.828427	0.101972
$67$	−13.7279	−1.67713	−0.838566	−	0.544800i	$-0.816606\pi$
$67$	−13.7279	−1.67713	−0.838566	+	0.544800i	$0.816606\pi$
$68$	0	0
$69$	−2.24264	−0.269982
$70$	0	0
$71$	−0.585786	−0.0695201	−0.0347600	−	0.999396i	$-0.511067\pi$
$71$	−0.585786	−0.0695201	−0.0347600	+	0.999396i	$0.511067\pi$
$72$	2.82843	0.333333
$73$	−12.0711	−1.41281	−0.706406	−	0.707807i	$-0.749685\pi$
$73$	−12.0711	−1.41281	−0.706406	+	0.707807i	$0.749685\pi$
$74$	11.8995	1.38329
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	−6.24264	−0.706840
$79$	6.65685	0.748955	0.374477	−	0.927236i	$-0.377822\pi$
$79$	6.65685	0.748955	0.374477	+	0.927236i	$0.377822\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	8.82843	0.974937
$83$	2.58579	0.283827	0.141913	−	0.989879i	$-0.454675\pi$
$83$	2.58579	0.283827	0.141913	+	0.989879i	$0.454675\pi$
$84$	0	0
$85$	−2.24264	−0.243249
$86$	10.7279	1.15682
$87$	−8.24264	−0.883704
$88$	1.65685	0.176621
$89$	−12.2426	−1.29772	−0.648859	−	0.760909i	$-0.724753\pi$
$89$	−12.2426	−1.29772	−0.648859	+	0.760909i	$0.724753\pi$
$90$	1.41421	0.149071
$91$	0	0
$92$	0	0
$93$	5.82843	0.604380
$94$	18.8284	1.94200
$95$	−4.65685	−0.477783
$96$	0	0
$97$	−5.17157	−0.525094	−0.262547	−	0.964919i	$-0.584562\pi$
$97$	−5.17157	−0.525094	−0.262547	+	0.964919i	$0.584562\pi$
$98$	0	0
$99$	0.585786	0.0588738
$100$	0	0
$101$	2.24264	0.223151	0.111576	−	0.993756i	$-0.464410\pi$
$101$	2.24264	0.223151	0.111576	+	0.993756i	$0.464410\pi$
$102$	3.17157	0.314033
$103$	7.24264	0.713639	0.356819	−	0.934173i	$-0.383861\pi$
$103$	7.24264	0.713639	0.356819	+	0.934173i	$0.383861\pi$
$104$	−12.4853	−1.22428
$105$	0	0
$106$	−9.65685	−0.937957
$107$	12.5858	1.21671	0.608357	−	0.793664i	$-0.291829\pi$
$107$	12.5858	1.21671	0.608357	+	0.793664i	$0.291829\pi$
$108$	0	0
$109$	3.48528	0.333829	0.166915	−	0.985971i	$-0.446620\pi$
$109$	3.48528	0.333829	0.166915	+	0.985971i	$0.446620\pi$
$110$	0.828427	0.0789874
$111$	8.41421	0.798642
$112$	0	0
$113$	−15.6569	−1.47287	−0.736436	−	0.676507i	$-0.763493\pi$
$113$	−15.6569	−1.47287	−0.736436	+	0.676507i	$0.763493\pi$
$114$	6.58579	0.616815
$115$	−2.24264	−0.209127
$116$	0	0
$117$	−4.41421	−0.408094
$118$	2.00000	0.184115
$119$	0	0
$120$	2.82843	0.258199
$121$	−10.6569	−0.968805
$122$	6.34315	0.574281
$123$	6.24264	0.562880
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.92893	−0.348636	−0.174318	−	0.984689i	$-0.555772\pi$
$127$	−3.92893	−0.348636	−0.174318	+	0.984689i	$0.555772\pi$
$128$	−11.3137	−1.00000
$129$	7.58579	0.667891
$130$	−6.24264	−0.547516
$131$	6.48528	0.566622	0.283311	−	0.959028i	$-0.408567\pi$
$131$	6.48528	0.566622	0.283311	+	0.959028i	$0.408567\pi$
$132$	0	0
$133$	0	0
$134$	19.4142	1.67713
$135$	1.00000	0.0860663
$136$	6.34315	0.543920
$137$	−17.0711	−1.45848	−0.729240	−	0.684258i	$-0.760126\pi$
$137$	−17.0711	−1.45848	−0.729240	+	0.684258i	$0.760126\pi$
$138$	3.17157	0.269982
$139$	−5.48528	−0.465255	−0.232628	−	0.972566i	$-0.574732\pi$
$139$	−5.48528	−0.465255	−0.232628	+	0.972566i	$0.574732\pi$
$140$	0	0
$141$	13.3137	1.12122
$142$	0.828427	0.0695201
$143$	−2.58579	−0.216234
$144$	−4.00000	−0.333333
$145$	−8.24264	−0.684514
$146$	17.0711	1.41281
$147$	0	0
$148$	0	0
$149$	6.34315	0.519651	0.259825	−	0.965656i	$-0.416335\pi$
$149$	6.34315	0.519651	0.259825	+	0.965656i	$0.416335\pi$
$150$	1.41421	0.115470
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	13.1716	1.06836
$153$	2.24264	0.181307
$154$	0	0
$155$	5.82843	0.468151
$156$	0	0
$157$	12.1421	0.969048	0.484524	−	0.874778i	$-0.338993\pi$
$157$	12.1421	0.969048	0.484524	+	0.874778i	$0.338993\pi$
$158$	−9.41421	−0.748955
$159$	−6.82843	−0.541529
$160$	0	0
$161$	0	0
$162$	−1.41421	−0.111111
$163$	3.31371	0.259550	0.129775	−	0.991543i	$-0.458575\pi$
$163$	3.31371	0.259550	0.129775	+	0.991543i	$0.458575\pi$
$164$	0	0
$165$	0.585786	0.0456034
$166$	−3.65685	−0.283827
$167$	20.2426	1.56642	0.783211	−	0.621756i	$-0.213580\pi$
$167$	20.2426	1.56642	0.783211	+	0.621756i	$0.213580\pi$
$168$	0	0
$169$	6.48528	0.498868
$170$	3.17157	0.243249
$171$	4.65685	0.356119
$172$	0	0
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	11.6569	0.883704
$175$	0	0
$176$	−2.34315	−0.176621
$177$	1.41421	0.106299
$178$	17.3137	1.29772
$179$	8.34315	0.623596	0.311798	−	0.950148i	$-0.399069\pi$
$179$	8.34315	0.623596	0.311798	+	0.950148i	$0.399069\pi$
$180$	0	0
$181$	−10.6569	−0.792118	−0.396059	−	0.918225i	$-0.629622\pi$
$181$	−10.6569	−0.792118	−0.396059	+	0.918225i	$0.629622\pi$
$182$	0	0
$183$	4.48528	0.331562
$184$	6.34315	0.467623
$185$	8.41421	0.618625
$186$	−8.24264	−0.604380
$187$	1.31371	0.0960679
$188$	0	0
$189$	0	0
$190$	6.58579	0.477783
$191$	−14.9706	−1.08323	−0.541616	−	0.840626i	$-0.682187\pi$
$191$	−14.9706	−1.08323	−0.541616	+	0.840626i	$0.682187\pi$
$192$	−8.00000	−0.577350
$193$	−15.2426	−1.09719	−0.548595	−	0.836088i	$-0.684837\pi$
$193$	−15.2426	−1.09719	−0.548595	+	0.836088i	$0.684837\pi$
$194$	7.31371	0.525094
$195$	−4.41421	−0.316108
$196$	0	0
$197$	−25.5563	−1.82081	−0.910407	−	0.413713i	$-0.864232\pi$
$197$	−25.5563	−1.82081	−0.910407	+	0.413713i	$0.864232\pi$
$198$	−0.828427	−0.0588738
$199$	22.4853	1.59394	0.796970	−	0.604019i	$-0.206435\pi$
$199$	22.4853	1.59394	0.796970	+	0.604019i	$0.206435\pi$
$200$	2.82843	0.200000
$201$	13.7279	0.968293
$202$	−3.17157	−0.223151
$203$	0	0
$204$	0	0
$205$	6.24264	0.436005
$206$	−10.2426	−0.713639
$207$	2.24264	0.155874
$208$	17.6569	1.22428
$209$	2.72792	0.188694
$210$	0	0
$211$	−28.1421	−1.93738	−0.968692	−	0.248265i	$-0.920140\pi$
$211$	−28.1421	−1.93738	−0.968692	+	0.248265i	$0.920140\pi$
$212$	0	0
$213$	0.585786	0.0401374
$214$	−17.7990	−1.21671
$215$	7.58579	0.517346
$216$	−2.82843	−0.192450
$217$	0	0
$218$	−4.92893	−0.333829
$219$	12.0711	0.815687
$220$	0	0
$221$	−9.89949	−0.665912
$222$	−11.8995	−0.798642
$223$	22.8284	1.52870	0.764352	−	0.644799i	$-0.223059\pi$
$223$	22.8284	1.52870	0.764352	+	0.644799i	$0.223059\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	22.1421	1.47287
$227$	8.92893	0.592634	0.296317	−	0.955090i	$-0.404242\pi$
$227$	8.92893	0.592634	0.296317	+	0.955090i	$0.404242\pi$
$228$	0	0
$229$	8.31371	0.549385	0.274693	−	0.961532i	$-0.411424\pi$
$229$	8.31371	0.549385	0.274693	+	0.961532i	$0.411424\pi$
$230$	3.17157	0.209127
$231$	0	0
$232$	23.3137	1.53062
$233$	−5.51472	−0.361281	−0.180641	−	0.983549i	$-0.557817\pi$
$233$	−5.51472	−0.361281	−0.180641	+	0.983549i	$0.557817\pi$
$234$	6.24264	0.408094
$235$	13.3137	0.868491
$236$	0	0
$237$	−6.65685	−0.432409
$238$	0	0
$239$	−5.31371	−0.343715	−0.171858	−	0.985122i	$-0.554977\pi$
$239$	−5.31371	−0.343715	−0.171858	+	0.985122i	$0.554977\pi$
$240$	−4.00000	−0.258199
$241$	−21.6569	−1.39504	−0.697520	−	0.716565i	$-0.745713\pi$
$241$	−21.6569	−1.39504	−0.697520	+	0.716565i	$0.745713\pi$
$242$	15.0711	0.968805
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	−8.82843	−0.562880
$247$	−20.5563	−1.30797
$248$	−16.4853	−1.04682
$249$	−2.58579	−0.163868
$250$	1.41421	0.0894427
$251$	−2.58579	−0.163213	−0.0816067	−	0.996665i	$-0.526005\pi$
$251$	−2.58579	−0.163213	−0.0816067	+	0.996665i	$0.526005\pi$
$252$	0	0
$253$	1.31371	0.0825921
$254$	5.55635	0.348636
$255$	2.24264	0.140440
$256$	0	0
$257$	−10.1005	−0.630052	−0.315026	−	0.949083i	$-0.602013\pi$
$257$	−10.1005	−0.630052	−0.315026	+	0.949083i	$0.602013\pi$
$258$	−10.7279	−0.667891
$259$	0	0
$260$	0	0
$261$	8.24264	0.510207
$262$	−9.17157	−0.566622
$263$	6.82843	0.421059	0.210529	−	0.977588i	$-0.432481\pi$
$263$	6.82843	0.421059	0.210529	+	0.977588i	$0.432481\pi$
$264$	−1.65685	−0.101972
$265$	−6.82843	−0.419467
$266$	0	0
$267$	12.2426	0.749237
$268$	0	0
$269$	−20.1421	−1.22809	−0.614044	−	0.789272i	$-0.710458\pi$
$269$	−20.1421	−1.22809	−0.614044	+	0.789272i	$0.710458\pi$
$270$	−1.41421	−0.0860663
$271$	−8.14214	−0.494600	−0.247300	−	0.968939i	$-0.579543\pi$
$271$	−8.14214	−0.494600	−0.247300	+	0.968939i	$0.579543\pi$
$272$	−8.97056	−0.543920
$273$	0	0
$274$	24.1421	1.45848
$275$	0.585786	0.0353243
$276$	0	0
$277$	13.3848	0.804213	0.402107	−	0.915593i	$-0.368278\pi$
$277$	13.3848	0.804213	0.402107	+	0.915593i	$0.368278\pi$
$278$	7.75736	0.465255
$279$	−5.82843	−0.348939
$280$	0	0
$281$	−1.65685	−0.0988396	−0.0494198	−	0.998778i	$-0.515737\pi$
$281$	−1.65685	−0.0988396	−0.0494198	+	0.998778i	$0.515737\pi$
$282$	−18.8284	−1.12122
$283$	−4.75736	−0.282796	−0.141398	−	0.989953i	$-0.545160\pi$
$283$	−4.75736	−0.282796	−0.141398	+	0.989953i	$0.545160\pi$
$284$	0	0
$285$	4.65685	0.275848
$286$	3.65685	0.216234
$287$	0	0
$288$	0	0
$289$	−11.9706	−0.704151
$290$	11.6569	0.684514
$291$	5.17157	0.303163
$292$	0	0
$293$	7.31371	0.427271	0.213636	−	0.976913i	$-0.431469\pi$
$293$	7.31371	0.427271	0.213636	+	0.976913i	$0.431469\pi$
$294$	0	0
$295$	1.41421	0.0823387
$296$	−23.7990	−1.38329
$297$	−0.585786	−0.0339908
$298$	−8.97056	−0.519651
$299$	−9.89949	−0.572503
$300$	0	0
$301$	0	0
$302$	−14.8284	−0.853280
$303$	−2.24264	−0.128836
$304$	−18.6274	−1.06836
$305$	4.48528	0.256826
$306$	−3.17157	−0.181307
$307$	4.41421	0.251932	0.125966	−	0.992035i	$-0.459797\pi$
$307$	4.41421	0.251932	0.125966	+	0.992035i	$0.459797\pi$
$308$	0	0
$309$	−7.24264	−0.412019
$310$	−8.24264	−0.468151
$311$	−2.92893	−0.166085	−0.0830423	−	0.996546i	$-0.526464\pi$
$311$	−2.92893	−0.166085	−0.0830423	+	0.996546i	$0.526464\pi$
$312$	12.4853	0.706840
$313$	−16.2132	−0.916424	−0.458212	−	0.888843i	$-0.651510\pi$
$313$	−16.2132	−0.916424	−0.458212	+	0.888843i	$0.651510\pi$
$314$	−17.1716	−0.969048
$315$	0	0
$316$	0	0
$317$	19.8995	1.11767	0.558833	−	0.829280i	$-0.311249\pi$
$317$	19.8995	1.11767	0.558833	+	0.829280i	$0.311249\pi$
$318$	9.65685	0.541529
$319$	4.82843	0.270340
$320$	−8.00000	−0.447214
$321$	−12.5858	−0.702470
$322$	0	0
$323$	10.4437	0.581100
$324$	0	0
$325$	−4.41421	−0.244857
$326$	−4.68629	−0.259550
$327$	−3.48528	−0.192737
$328$	−17.6569	−0.974937
$329$	0	0
$330$	−0.828427	−0.0456034
$331$	17.6274	0.968890	0.484445	−	0.874822i	$-0.339021\pi$
$331$	17.6274	0.968890	0.484445	+	0.874822i	$0.339021\pi$
$332$	0	0
$333$	−8.41421	−0.461096
$334$	−28.6274	−1.56642
$335$	13.7279	0.750037
$336$	0	0
$337$	18.8995	1.02952	0.514761	−	0.857334i	$-0.327881\pi$
$337$	18.8995	1.02952	0.514761	+	0.857334i	$0.327881\pi$
$338$	−9.17157	−0.498868
$339$	15.6569	0.850364
$340$	0	0
$341$	−3.41421	−0.184890
$342$	−6.58579	−0.356119
$343$	0	0
$344$	−21.4558	−1.15682
$345$	2.24264	0.120740
$346$	−4.00000	−0.215041
$347$	8.97056	0.481565	0.240783	−	0.970579i	$-0.422596\pi$
$347$	8.97056	0.481565	0.240783	+	0.970579i	$0.422596\pi$
$348$	0	0
$349$	28.6274	1.53239	0.766195	−	0.642608i	$-0.222148\pi$
$349$	28.6274	1.53239	0.766195	+	0.642608i	$0.222148\pi$
$350$	0	0
$351$	4.41421	0.235613
$352$	0	0
$353$	−13.8995	−0.739795	−0.369898	−	0.929072i	$-0.620607\pi$
$353$	−13.8995	−0.739795	−0.369898	+	0.929072i	$0.620607\pi$
$354$	−2.00000	−0.106299
$355$	0.585786	0.0310903
$356$	0	0
$357$	0	0
$358$	−11.7990	−0.623596
$359$	−29.4142	−1.55242	−0.776211	−	0.630473i	$-0.782861\pi$
$359$	−29.4142	−1.55242	−0.776211	+	0.630473i	$0.782861\pi$
$360$	−2.82843	−0.149071
$361$	2.68629	0.141384
$362$	15.0711	0.792118
$363$	10.6569	0.559340
$364$	0	0
$365$	12.0711	0.631829
$366$	−6.34315	−0.331562
$367$	−22.4142	−1.17001	−0.585006	−	0.811029i	$-0.698908\pi$
$367$	−22.4142	−1.17001	−0.585006	+	0.811029i	$0.698908\pi$
$368$	−8.97056	−0.467623
$369$	−6.24264	−0.324979
$370$	−11.8995	−0.618625
$371$	0	0
$372$	0	0
$373$	10.7574	0.556995	0.278497	−	0.960437i	$-0.410164\pi$
$373$	10.7574	0.556995	0.278497	+	0.960437i	$0.410164\pi$
$374$	−1.85786	−0.0960679
$375$	1.00000	0.0516398
$376$	−37.6569	−1.94200
$377$	−36.3848	−1.87391
$378$	0	0
$379$	−24.7990	−1.27384	−0.636919	−	0.770931i	$-0.719792\pi$
$379$	−24.7990	−1.27384	−0.636919	+	0.770931i	$0.719792\pi$
$380$	0	0
$381$	3.92893	0.201285
$382$	21.1716	1.08323
$383$	−4.48528	−0.229187	−0.114594	−	0.993412i	$-0.536557\pi$
$383$	−4.48528	−0.229187	−0.114594	+	0.993412i	$0.536557\pi$
$384$	11.3137	0.577350
$385$	0	0
$386$	21.5563	1.09719
$387$	−7.58579	−0.385607
$388$	0	0
$389$	−36.8701	−1.86939	−0.934693	−	0.355456i	$-0.884326\pi$
$389$	−36.8701	−1.86939	−0.934693	+	0.355456i	$0.884326\pi$
$390$	6.24264	0.316108
$391$	5.02944	0.254350
$392$	0	0
$393$	−6.48528	−0.327139
$394$	36.1421	1.82081
$395$	−6.65685	−0.334943
$396$	0	0
$397$	18.2132	0.914094	0.457047	−	0.889442i	$-0.348907\pi$
$397$	18.2132	0.914094	0.457047	+	0.889442i	$0.348907\pi$
$398$	−31.7990	−1.59394
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−16.4853	−0.823236	−0.411618	−	0.911357i	$-0.635036\pi$
$401$	−16.4853	−0.823236	−0.411618	+	0.911357i	$0.635036\pi$
$402$	−19.4142	−0.968293
$403$	25.7279	1.28160
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−4.92893	−0.244318
$408$	−6.34315	−0.314033
$409$	−3.68629	−0.182275	−0.0911377	−	0.995838i	$-0.529050\pi$
$409$	−3.68629	−0.182275	−0.0911377	+	0.995838i	$0.529050\pi$
$410$	−8.82843	−0.436005
$411$	17.0711	0.842054
$412$	0	0
$413$	0	0
$414$	−3.17157	−0.155874
$415$	−2.58579	−0.126931
$416$	0	0
$417$	5.48528	0.268615
$418$	−3.85786	−0.188694
$419$	−8.82843	−0.431297	−0.215648	−	0.976471i	$-0.569187\pi$
$419$	−8.82843	−0.431297	−0.215648	+	0.976471i	$0.569187\pi$
$420$	0	0
$421$	10.5147	0.512456	0.256228	−	0.966616i	$-0.417520\pi$
$421$	10.5147	0.512456	0.256228	+	0.966616i	$0.417520\pi$
$422$	39.7990	1.93738
$423$	−13.3137	−0.647335
$424$	19.3137	0.937957
$425$	2.24264	0.108784
$426$	−0.828427	−0.0401374
$427$	0	0
$428$	0	0
$429$	2.58579	0.124843
$430$	−10.7279	−0.517346
$431$	31.1716	1.50148	0.750741	−	0.660597i	$-0.229697\pi$
$431$	31.1716	1.50148	0.750741	+	0.660597i	$0.229697\pi$
$432$	4.00000	0.192450
$433$	6.55635	0.315078	0.157539	−	0.987513i	$-0.449644\pi$
$433$	6.55635	0.315078	0.157539	+	0.987513i	$0.449644\pi$
$434$	0	0
$435$	8.24264	0.395204
$436$	0	0
$437$	10.4437	0.499588
$438$	−17.0711	−0.815687
$439$	11.6569	0.556351	0.278176	−	0.960530i	$-0.410270\pi$
$439$	11.6569	0.556351	0.278176	+	0.960530i	$0.410270\pi$
$440$	−1.65685	−0.0789874
$441$	0	0
$442$	14.0000	0.665912
$443$	−16.4853	−0.783239	−0.391620	−	0.920127i	$-0.628085\pi$
$443$	−16.4853	−0.783239	−0.391620	+	0.920127i	$0.628085\pi$
$444$	0	0
$445$	12.2426	0.580357
$446$	−32.2843	−1.52870
$447$	−6.34315	−0.300020
$448$	0	0
$449$	26.9706	1.27282	0.636410	−	0.771351i	$-0.280419\pi$
$449$	26.9706	1.27282	0.636410	+	0.771351i	$0.280419\pi$
$450$	−1.41421	−0.0666667
$451$	−3.65685	−0.172195
$452$	0	0
$453$	−10.4853	−0.492641
$454$	−12.6274	−0.592634
$455$	0	0
$456$	−13.1716	−0.616815
$457$	−1.10051	−0.0514795	−0.0257397	−	0.999669i	$-0.508194\pi$
$457$	−1.10051	−0.0514795	−0.0257397	+	0.999669i	$0.508194\pi$
$458$	−11.7574	−0.549385
$459$	−2.24264	−0.104678
$460$	0	0
$461$	−12.9289	−0.602160	−0.301080	−	0.953599i	$-0.597347\pi$
$461$	−12.9289	−0.602160	−0.301080	+	0.953599i	$0.597347\pi$
$462$	0	0
$463$	−7.58579	−0.352541	−0.176271	−	0.984342i	$-0.556403\pi$
$463$	−7.58579	−0.352541	−0.176271	+	0.984342i	$0.556403\pi$
$464$	−32.9706	−1.53062
$465$	−5.82843	−0.270287
$466$	7.79899	0.361281
$467$	29.3137	1.35648	0.678238	−	0.734842i	$-0.262744\pi$
$467$	29.3137	1.35648	0.678238	+	0.734842i	$0.262744\pi$
$468$	0	0
$469$	0	0
$470$	−18.8284	−0.868491
$471$	−12.1421	−0.559480
$472$	−4.00000	−0.184115
$473$	−4.44365	−0.204319
$474$	9.41421	0.432409
$475$	4.65685	0.213671
$476$	0	0
$477$	6.82843	0.312652
$478$	7.51472	0.343715
$479$	10.6274	0.485579	0.242790	−	0.970079i	$-0.421938\pi$
$479$	10.6274	0.485579	0.242790	+	0.970079i	$0.421938\pi$
$480$	0	0
$481$	37.1421	1.69354
$482$	30.6274	1.39504
$483$	0	0
$484$	0	0
$485$	5.17157	0.234829
$486$	1.41421	0.0641500
$487$	−18.8995	−0.856418	−0.428209	−	0.903680i	$-0.640855\pi$
$487$	−18.8995	−0.856418	−0.428209	+	0.903680i	$0.640855\pi$
$488$	−12.6863	−0.574281
$489$	−3.31371	−0.149851
$490$	0	0
$491$	18.1421	0.818743	0.409372	−	0.912368i	$-0.365748\pi$
$491$	18.1421	0.818743	0.409372	+	0.912368i	$0.365748\pi$
$492$	0	0
$493$	18.4853	0.832535
$494$	29.0711	1.30797
$495$	−0.585786	−0.0263291
$496$	23.3137	1.04682
$497$	0	0
$498$	3.65685	0.163868
$499$	−30.7990	−1.37875	−0.689376	−	0.724404i	$-0.742115\pi$
$499$	−30.7990	−1.37875	−0.689376	+	0.724404i	$0.742115\pi$
$500$	0	0
$501$	−20.2426	−0.904374
$502$	3.65685	0.163213
$503$	−1.51472	−0.0675380	−0.0337690	−	0.999430i	$-0.510751\pi$
$503$	−1.51472	−0.0675380	−0.0337690	+	0.999430i	$0.510751\pi$
$504$	0	0
$505$	−2.24264	−0.0997962
$506$	−1.85786	−0.0825921
$507$	−6.48528	−0.288021
$508$	0	0
$509$	−25.7990	−1.14352	−0.571760	−	0.820421i	$-0.693739\pi$
$509$	−25.7990	−1.14352	−0.571760	+	0.820421i	$0.693739\pi$
$510$	−3.17157	−0.140440
$511$	0	0
$512$	22.6274	1.00000
$513$	−4.65685	−0.205605
$514$	14.2843	0.630052
$515$	−7.24264	−0.319149
$516$	0	0
$517$	−7.79899	−0.342999
$518$	0	0
$519$	−2.82843	−0.124154
$520$	12.4853	0.547516
$521$	23.1127	1.01259	0.506293	−	0.862362i	$-0.331015\pi$
$521$	23.1127	1.01259	0.506293	+	0.862362i	$0.331015\pi$
$522$	−11.6569	−0.510207
$523$	−10.4142	−0.455382	−0.227691	−	0.973733i	$-0.573118\pi$
$523$	−10.4142	−0.455382	−0.227691	+	0.973733i	$0.573118\pi$
$524$	0	0
$525$	0	0
$526$	−9.65685	−0.421059
$527$	−13.0711	−0.569385
$528$	2.34315	0.101972
$529$	−17.9706	−0.781329
$530$	9.65685	0.419467
$531$	−1.41421	−0.0613716
$532$	0	0
$533$	27.5563	1.19360
$534$	−17.3137	−0.749237
$535$	−12.5858	−0.544131
$536$	−38.8284	−1.67713
$537$	−8.34315	−0.360033
$538$	28.4853	1.22809
$539$	0	0
$540$	0	0
$541$	9.68629	0.416446	0.208223	−	0.978081i	$-0.433232\pi$
$541$	9.68629	0.416446	0.208223	+	0.978081i	$0.433232\pi$
$542$	11.5147	0.494600
$543$	10.6569	0.457329
$544$	0	0
$545$	−3.48528	−0.149293
$546$	0	0
$547$	7.79899	0.333461	0.166730	−	0.986003i	$-0.446679\pi$
$547$	7.79899	0.333461	0.166730	+	0.986003i	$0.446679\pi$
$548$	0	0
$549$	−4.48528	−0.191427
$550$	−0.828427	−0.0353243
$551$	38.3848	1.63525
$552$	−6.34315	−0.269982
$553$	0	0
$554$	−18.9289	−0.804213
$555$	−8.41421	−0.357163
$556$	0	0
$557$	−19.6569	−0.832888	−0.416444	−	0.909161i	$-0.636724\pi$
$557$	−19.6569	−0.832888	−0.416444	+	0.909161i	$0.636724\pi$
$558$	8.24264	0.348939
$559$	33.4853	1.41628
$560$	0	0
$561$	−1.31371	−0.0554648
$562$	2.34315	0.0988396
$563$	42.2843	1.78207	0.891035	−	0.453935i	$-0.149980\pi$
$563$	42.2843	1.78207	0.891035	+	0.453935i	$0.149980\pi$
$564$	0	0
$565$	15.6569	0.658689
$566$	6.72792	0.282796
$567$	0	0
$568$	−1.65685	−0.0695201
$569$	16.8701	0.707230	0.353615	−	0.935391i	$-0.384952\pi$
$569$	16.8701	0.707230	0.353615	+	0.935391i	$0.384952\pi$
$570$	−6.58579	−0.275848
$571$	−5.97056	−0.249860	−0.124930	−	0.992166i	$-0.539871\pi$
$571$	−5.97056	−0.249860	−0.124930	+	0.992166i	$0.539871\pi$
$572$	0	0
$573$	14.9706	0.625404
$574$	0	0
$575$	2.24264	0.0935246
$576$	8.00000	0.333333
$577$	−24.5563	−1.02229	−0.511147	−	0.859493i	$-0.670779\pi$
$577$	−24.5563	−1.02229	−0.511147	+	0.859493i	$0.670779\pi$
$578$	16.9289	0.704151
$579$	15.2426	0.633463
$580$	0	0
$581$	0	0
$582$	−7.31371	−0.303163
$583$	4.00000	0.165663
$584$	−34.1421	−1.41281
$585$	4.41421	0.182505
$586$	−10.3431	−0.427271
$587$	21.7574	0.898022	0.449011	−	0.893526i	$-0.351776\pi$
$587$	21.7574	0.898022	0.449011	+	0.893526i	$0.351776\pi$
$588$	0	0
$589$	−27.1421	−1.11837
$590$	−2.00000	−0.0823387
$591$	25.5563	1.05125
$592$	33.6569	1.38329
$593$	10.5858	0.434706	0.217353	−	0.976093i	$-0.430258\pi$
$593$	10.5858	0.434706	0.217353	+	0.976093i	$0.430258\pi$
$594$	0.828427	0.0339908
$595$	0	0
$596$	0	0
$597$	−22.4853	−0.920261
$598$	14.0000	0.572503
$599$	−13.8579	−0.566217	−0.283108	−	0.959088i	$-0.591366\pi$
$599$	−13.8579	−0.566217	−0.283108	+	0.959088i	$0.591366\pi$
$600$	−2.82843	−0.115470
$601$	−29.8284	−1.21673	−0.608363	−	0.793659i	$-0.708174\pi$
$601$	−29.8284	−1.21673	−0.608363	+	0.793659i	$0.708174\pi$
$602$	0	0
$603$	−13.7279	−0.559044
$604$	0	0
$605$	10.6569	0.433263
$606$	3.17157	0.128836
$607$	7.24264	0.293970	0.146985	−	0.989139i	$-0.453043\pi$
$607$	7.24264	0.293970	0.146985	+	0.989139i	$0.453043\pi$
$608$	0	0
$609$	0	0
$610$	−6.34315	−0.256826
$611$	58.7696	2.37756
$612$	0	0
$613$	−18.1421	−0.732754	−0.366377	−	0.930467i	$-0.619402\pi$
$613$	−18.1421	−0.732754	−0.366377	+	0.930467i	$0.619402\pi$
$614$	−6.24264	−0.251932
$615$	−6.24264	−0.251728
$616$	0	0
$617$	−3.17157	−0.127683	−0.0638414	−	0.997960i	$-0.520335\pi$
$617$	−3.17157	−0.127683	−0.0638414	+	0.997960i	$0.520335\pi$
$618$	10.2426	0.412019
$619$	6.02944	0.242344	0.121172	−	0.992632i	$-0.461335\pi$
$619$	6.02944	0.242344	0.121172	+	0.992632i	$0.461335\pi$
$620$	0	0
$621$	−2.24264	−0.0899941
$622$	4.14214	0.166085
$623$	0	0
$624$	−17.6569	−0.706840
$625$	1.00000	0.0400000
$626$	22.9289	0.916424
$627$	−2.72792	−0.108943
$628$	0	0
$629$	−18.8701	−0.752398
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	18.8284	0.748955
$633$	28.1421	1.11855
$634$	−28.1421	−1.11767
$635$	3.92893	0.155915
$636$	0	0
$637$	0	0
$638$	−6.82843	−0.270340
$639$	−0.585786	−0.0231734
$640$	11.3137	0.447214
$641$	29.2132	1.15385	0.576926	−	0.816796i	$-0.304252\pi$
$641$	29.2132	1.15385	0.576926	+	0.816796i	$0.304252\pi$
$642$	17.7990	0.702470
$643$	−6.55635	−0.258557	−0.129279	−	0.991608i	$-0.541266\pi$
$643$	−6.55635	−0.258557	−0.129279	+	0.991608i	$0.541266\pi$
$644$	0	0
$645$	−7.58579	−0.298690
$646$	−14.7696	−0.581100
$647$	46.3848	1.82357	0.911787	−	0.410664i	$-0.134703\pi$
$647$	46.3848	1.82357	0.911787	+	0.410664i	$0.134703\pi$
$648$	2.82843	0.111111
$649$	−0.828427	−0.0325186
$650$	6.24264	0.244857
$651$	0	0
$652$	0	0
$653$	16.2426	0.635624	0.317812	−	0.948154i	$-0.397052\pi$
$653$	16.2426	0.635624	0.317812	+	0.948154i	$0.397052\pi$
$654$	4.92893	0.192737
$655$	−6.48528	−0.253401
$656$	24.9706	0.974937
$657$	−12.0711	−0.470937
$658$	0	0
$659$	−4.68629	−0.182552	−0.0912760	−	0.995826i	$-0.529095\pi$
$659$	−4.68629	−0.182552	−0.0912760	+	0.995826i	$0.529095\pi$
$660$	0	0
$661$	9.68629	0.376753	0.188377	−	0.982097i	$-0.439677\pi$
$661$	9.68629	0.376753	0.188377	+	0.982097i	$0.439677\pi$
$662$	−24.9289	−0.968890
$663$	9.89949	0.384465
$664$	7.31371	0.283827
$665$	0	0
$666$	11.8995	0.461096
$667$	18.4853	0.715753
$668$	0	0
$669$	−22.8284	−0.882598
$670$	−19.4142	−0.750037
$671$	−2.62742	−0.101430
$672$	0	0
$673$	27.7279	1.06883	0.534416	−	0.845221i	$-0.320531\pi$
$673$	27.7279	1.06883	0.534416	+	0.845221i	$0.320531\pi$
$674$	−26.7279	−1.02952
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−15.8995	−0.611067	−0.305534	−	0.952181i	$-0.598835\pi$
$677$	−15.8995	−0.611067	−0.305534	+	0.952181i	$0.598835\pi$
$678$	−22.1421	−0.850364
$679$	0	0
$680$	−6.34315	−0.243249
$681$	−8.92893	−0.342157
$682$	4.82843	0.184890
$683$	23.4142	0.895920	0.447960	−	0.894054i	$-0.352151\pi$
$683$	23.4142	0.895920	0.447960	+	0.894054i	$0.352151\pi$
$684$	0	0
$685$	17.0711	0.652252
$686$	0	0
$687$	−8.31371	−0.317188
$688$	30.3431	1.15682
$689$	−30.1421	−1.14832
$690$	−3.17157	−0.120740
$691$	7.68629	0.292400	0.146200	−	0.989255i	$-0.453296\pi$
$691$	7.68629	0.292400	0.146200	+	0.989255i	$0.453296\pi$
$692$	0	0
$693$	0	0
$694$	−12.6863	−0.481565
$695$	5.48528	0.208069
$696$	−23.3137	−0.883704
$697$	−14.0000	−0.530288
$698$	−40.4853	−1.53239
$699$	5.51472	0.208586
$700$	0	0
$701$	−1.21320	−0.0458221	−0.0229110	−	0.999738i	$-0.507293\pi$
$701$	−1.21320	−0.0458221	−0.0229110	+	0.999738i	$0.507293\pi$
$702$	−6.24264	−0.235613
$703$	−39.1838	−1.47784
$704$	4.68629	0.176621
$705$	−13.3137	−0.501423
$706$	19.6569	0.739795
$707$	0	0
$708$	0	0
$709$	51.1127	1.91958	0.959789	−	0.280723i	$-0.0905742\pi$
$709$	51.1127	1.91958	0.959789	+	0.280723i	$0.0905742\pi$
$710$	−0.828427	−0.0310903
$711$	6.65685	0.249652
$712$	−34.6274	−1.29772
$713$	−13.0711	−0.489515
$714$	0	0
$715$	2.58579	0.0967029
$716$	0	0
$717$	5.31371	0.198444
$718$	41.5980	1.55242
$719$	34.4853	1.28608	0.643042	−	0.765831i	$-0.277672\pi$
$719$	34.4853	1.28608	0.643042	+	0.765831i	$0.277672\pi$
$720$	4.00000	0.149071
$721$	0	0
$722$	−3.79899	−0.141384
$723$	21.6569	0.805427
$724$	0	0
$725$	8.24264	0.306124
$726$	−15.0711	−0.559340
$727$	13.2426	0.491142	0.245571	−	0.969379i	$-0.421024\pi$
$727$	13.2426	0.491142	0.245571	+	0.969379i	$0.421024\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−17.0711	−0.631829
$731$	−17.0122	−0.629219
$732$	0	0
$733$	−19.2426	−0.710743	−0.355372	−	0.934725i	$-0.615646\pi$
$733$	−19.2426	−0.710743	−0.355372	+	0.934725i	$0.615646\pi$
$734$	31.6985	1.17001
$735$	0	0
$736$	0	0
$737$	−8.04163	−0.296217
$738$	8.82843	0.324979
$739$	−6.85786	−0.252271	−0.126135	−	0.992013i	$-0.540257\pi$
$739$	−6.85786	−0.252271	−0.126135	+	0.992013i	$0.540257\pi$
$740$	0	0
$741$	20.5563	0.755156
$742$	0	0
$743$	20.7279	0.760434	0.380217	−	0.924897i	$-0.375849\pi$
$743$	20.7279	0.760434	0.380217	+	0.924897i	$0.375849\pi$
$744$	16.4853	0.604380
$745$	−6.34315	−0.232395
$746$	−15.2132	−0.556995
$747$	2.58579	0.0946090
$748$	0	0
$749$	0	0
$750$	−1.41421	−0.0516398
$751$	−25.6274	−0.935158	−0.467579	−	0.883951i	$-0.654874\pi$
$751$	−25.6274	−0.935158	−0.467579	+	0.883951i	$0.654874\pi$
$752$	53.2548	1.94200
$753$	2.58579	0.0942313
$754$	51.4558	1.87391
$755$	−10.4853	−0.381598
$756$	0	0
$757$	−37.5980	−1.36652	−0.683261	−	0.730174i	$-0.739439\pi$
$757$	−37.5980	−1.36652	−0.683261	+	0.730174i	$0.739439\pi$
$758$	35.0711	1.27384
$759$	−1.31371	−0.0476846
$760$	−13.1716	−0.477783
$761$	−35.0711	−1.27133	−0.635663	−	0.771967i	$-0.719273\pi$
$761$	−35.0711	−1.27133	−0.635663	+	0.771967i	$0.719273\pi$
$762$	−5.55635	−0.201285
$763$	0	0
$764$	0	0
$765$	−2.24264	−0.0810828
$766$	6.34315	0.229187
$767$	6.24264	0.225409
$768$	0	0
$769$	17.9706	0.648035	0.324018	−	0.946051i	$-0.394966\pi$
$769$	17.9706	0.648035	0.324018	+	0.946051i	$0.394966\pi$
$770$	0	0
$771$	10.1005	0.363761
$772$	0	0
$773$	39.0711	1.40529	0.702644	−	0.711541i	$-0.252003\pi$
$773$	39.0711	1.40529	0.702644	+	0.711541i	$0.252003\pi$
$774$	10.7279	0.385607
$775$	−5.82843	−0.209363
$776$	−14.6274	−0.525094
$777$	0	0
$778$	52.1421	1.86939
$779$	−29.0711	−1.04158
$780$	0	0
$781$	−0.343146	−0.0122787
$782$	−7.11270	−0.254350
$783$	−8.24264	−0.294568
$784$	0	0
$785$	−12.1421	−0.433371
$786$	9.17157	0.327139
$787$	18.8284	0.671161	0.335580	−	0.942012i	$-0.391068\pi$
$787$	18.8284	0.671161	0.335580	+	0.942012i	$0.391068\pi$
$788$	0	0
$789$	−6.82843	−0.243098
$790$	9.41421	0.334943
$791$	0	0
$792$	1.65685	0.0588738
$793$	19.7990	0.703083
$794$	−25.7574	−0.914094
$795$	6.82843	0.242179
$796$	0	0
$797$	41.5563	1.47200	0.736001	−	0.676981i	$-0.236712\pi$
$797$	41.5563	1.47200	0.736001	+	0.676981i	$0.236712\pi$
$798$	0	0
$799$	−29.8579	−1.05630
$800$	0	0
$801$	−12.2426	−0.432572
$802$	23.3137	0.823236
$803$	−7.07107	−0.249533
$804$	0	0
$805$	0	0
$806$	−36.3848	−1.28160
$807$	20.1421	0.709037
$808$	6.34315	0.223151
$809$	40.6274	1.42838	0.714192	−	0.699950i	$-0.246794\pi$
$809$	40.6274	1.42838	0.714192	+	0.699950i	$0.246794\pi$
$810$	1.41421	0.0496904
$811$	10.9706	0.385229	0.192614	−	0.981275i	$-0.438303\pi$
$811$	10.9706	0.385229	0.192614	+	0.981275i	$0.438303\pi$
$812$	0	0
$813$	8.14214	0.285557
$814$	6.97056	0.244318
$815$	−3.31371	−0.116074
$816$	8.97056	0.314033
$817$	−35.3259	−1.23590
$818$	5.21320	0.182275
$819$	0	0
$820$	0	0
$821$	−17.5563	−0.612721	−0.306360	−	0.951916i	$-0.599111\pi$
$821$	−17.5563	−0.612721	−0.306360	+	0.951916i	$0.599111\pi$
$822$	−24.1421	−0.842054
$823$	−34.2843	−1.19507	−0.597537	−	0.801841i	$-0.703854\pi$
$823$	−34.2843	−1.19507	−0.597537	+	0.801841i	$0.703854\pi$
$824$	20.4853	0.713639
$825$	−0.585786	−0.0203945
$826$	0	0
$827$	7.17157	0.249380	0.124690	−	0.992196i	$-0.460206\pi$
$827$	7.17157	0.249380	0.124690	+	0.992196i	$0.460206\pi$
$828$	0	0
$829$	−7.34315	−0.255038	−0.127519	−	0.991836i	$-0.540701\pi$
$829$	−7.34315	−0.255038	−0.127519	+	0.991836i	$0.540701\pi$
$830$	3.65685	0.126931
$831$	−13.3848	−0.464313
$832$	−35.3137	−1.22428
$833$	0	0
$834$	−7.75736	−0.268615
$835$	−20.2426	−0.700525
$836$	0	0
$837$	5.82843	0.201460
$838$	12.4853	0.431297
$839$	−32.7279	−1.12989	−0.564947	−	0.825127i	$-0.691103\pi$
$839$	−32.7279	−1.12989	−0.564947	+	0.825127i	$0.691103\pi$
$840$	0	0
$841$	38.9411	1.34280
$842$	−14.8701	−0.512456
$843$	1.65685	0.0570651
$844$	0	0
$845$	−6.48528	−0.223100
$846$	18.8284	0.647335
$847$	0	0
$848$	−27.3137	−0.937957
$849$	4.75736	0.163272
$850$	−3.17157	−0.108784
$851$	−18.8701	−0.646857
$852$	0	0
$853$	−44.0122	−1.50695	−0.753474	−	0.657477i	$-0.771624\pi$
$853$	−44.0122	−1.50695	−0.753474	+	0.657477i	$0.771624\pi$
$854$	0	0
$855$	−4.65685	−0.159261
$856$	35.5980	1.21671
$857$	2.24264	0.0766071	0.0383036	−	0.999266i	$-0.487805\pi$
$857$	2.24264	0.0766071	0.0383036	+	0.999266i	$0.487805\pi$
$858$	−3.65685	−0.124843
$859$	4.34315	0.148186	0.0740931	−	0.997251i	$-0.476394\pi$
$859$	4.34315	0.148186	0.0740931	+	0.997251i	$0.476394\pi$
$860$	0	0
$861$	0	0
$862$	−44.0833	−1.50148
$863$	29.7990	1.01437	0.507185	−	0.861837i	$-0.330686\pi$
$863$	29.7990	1.01437	0.507185	+	0.861837i	$0.330686\pi$
$864$	0	0
$865$	−2.82843	−0.0961694
$866$	−9.27208	−0.315078
$867$	11.9706	0.406542
$868$	0	0
$869$	3.89949	0.132281
$870$	−11.6569	−0.395204
$871$	60.5980	2.05328
$872$	9.85786	0.333829
$873$	−5.17157	−0.175031
$874$	−14.7696	−0.499588
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−16.4853	−0.556351
$879$	−7.31371	−0.246685
$880$	2.34315	0.0789874
$881$	−10.3431	−0.348469	−0.174235	−	0.984704i	$-0.555745\pi$
$881$	−10.3431	−0.348469	−0.174235	+	0.984704i	$0.555745\pi$
$882$	0	0
$883$	−6.07107	−0.204308	−0.102154	−	0.994769i	$-0.532573\pi$
$883$	−6.07107	−0.204308	−0.102154	+	0.994769i	$0.532573\pi$
$884$	0	0
$885$	−1.41421	−0.0475383
$886$	23.3137	0.783239
$887$	−37.2132	−1.24950	−0.624749	−	0.780826i	$-0.714799\pi$
$887$	−37.2132	−1.24950	−0.624749	+	0.780826i	$0.714799\pi$
$888$	23.7990	0.798642
$889$	0	0
$890$	−17.3137	−0.580357
$891$	0.585786	0.0196246
$892$	0	0
$893$	−62.0000	−2.07475
$894$	8.97056	0.300020
$895$	−8.34315	−0.278881
$896$	0	0
$897$	9.89949	0.330535
$898$	−38.1421	−1.27282
$899$	−48.0416	−1.60228
$900$	0	0
$901$	15.3137	0.510174
$902$	5.17157	0.172195
$903$	0	0
$904$	−44.2843	−1.47287
$905$	10.6569	0.354246
$906$	14.8284	0.492641
$907$	−45.3848	−1.50698	−0.753488	−	0.657461i	$-0.771630\pi$
$907$	−45.3848	−1.50698	−0.753488	+	0.657461i	$0.771630\pi$
$908$	0	0
$909$	2.24264	0.0743837
$910$	0	0
$911$	−4.24264	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$912$	18.6274	0.616815
$913$	1.51472	0.0501299
$914$	1.55635	0.0514795
$915$	−4.48528	−0.148279
$916$	0	0
$917$	0	0
$918$	3.17157	0.104678
$919$	−35.3431	−1.16586	−0.582931	−	0.812521i	$-0.698094\pi$
$919$	−35.3431	−1.16586	−0.582931	+	0.812521i	$0.698094\pi$
$920$	−6.34315	−0.209127
$921$	−4.41421	−0.145453
$922$	18.2843	0.602160
$923$	2.58579	0.0851122
$924$	0	0
$925$	−8.41421	−0.276658
$926$	10.7279	0.352541
$927$	7.24264	0.237880
$928$	0	0
$929$	31.0122	1.01748	0.508739	−	0.860921i	$-0.330112\pi$
$929$	31.0122	1.01748	0.508739	+	0.860921i	$0.330112\pi$
$930$	8.24264	0.270287
$931$	0	0
$932$	0	0
$933$	2.92893	0.0958889
$934$	−41.4558	−1.35648
$935$	−1.31371	−0.0429629
$936$	−12.4853	−0.408094
$937$	−22.0711	−0.721030	−0.360515	−	0.932753i	$-0.617399\pi$
$937$	−22.0711	−0.721030	−0.360515	+	0.932753i	$0.617399\pi$
$938$	0	0
$939$	16.2132	0.529098
$940$	0	0
$941$	−36.7279	−1.19730	−0.598648	−	0.801012i	$-0.704295\pi$
$941$	−36.7279	−1.19730	−0.598648	+	0.801012i	$0.704295\pi$
$942$	17.1716	0.559480
$943$	−14.0000	−0.455903
$944$	5.65685	0.184115
$945$	0	0
$946$	6.28427	0.204319
$947$	−19.0711	−0.619726	−0.309863	−	0.950781i	$-0.600283\pi$
$947$	−19.0711	−0.619726	−0.309863	+	0.950781i	$0.600283\pi$
$948$	0	0
$949$	53.2843	1.72968
$950$	−6.58579	−0.213671
$951$	−19.8995	−0.645285
$952$	0	0
$953$	31.5147	1.02086	0.510431	−	0.859919i	$-0.329486\pi$
$953$	31.5147	1.02086	0.510431	+	0.859919i	$0.329486\pi$
$954$	−9.65685	−0.312652
$955$	14.9706	0.484436
$956$	0	0
$957$	−4.82843	−0.156081
$958$	−15.0294	−0.485579
$959$	0	0
$960$	8.00000	0.258199
$961$	2.97056	0.0958246
$962$	−52.5269	−1.69354
$963$	12.5858	0.405571
$964$	0	0
$965$	15.2426	0.490678
$966$	0	0
$967$	−13.3848	−0.430425	−0.215213	−	0.976567i	$-0.569044\pi$
$967$	−13.3848	−0.430425	−0.215213	+	0.976567i	$0.569044\pi$
$968$	−30.1421	−0.968805
$969$	−10.4437	−0.335498
$970$	−7.31371	−0.234829
$971$	25.6569	0.823368	0.411684	−	0.911327i	$-0.364941\pi$
$971$	25.6569	0.823368	0.411684	+	0.911327i	$0.364941\pi$
$972$	0	0
$973$	0	0
$974$	26.7279	0.856418
$975$	4.41421	0.141368
$976$	17.9411	0.574281
$977$	−45.6985	−1.46202	−0.731012	−	0.682365i	$-0.760952\pi$
$977$	−45.6985	−1.46202	−0.731012	+	0.682365i	$0.760952\pi$
$978$	4.68629	0.149851
$979$	−7.17157	−0.229204
$980$	0	0
$981$	3.48528	0.111276
$982$	−25.6569	−0.818743
$983$	−57.1543	−1.82294	−0.911470	−	0.411367i	$-0.865052\pi$
$983$	−57.1543	−1.82294	−0.911470	+	0.411367i	$0.865052\pi$
$984$	17.6569	0.562880
$985$	25.5563	0.814293
$986$	−26.1421	−0.832535
$987$	0	0
$988$	0	0
$989$	−17.0122	−0.540956
$990$	0.828427	0.0263291
$991$	−19.3431	−0.614455	−0.307228	−	0.951636i	$-0.599401\pi$
$991$	−19.3431	−0.614455	−0.307228	+	0.951636i	$0.599401\pi$
$992$	0	0
$993$	−17.6274	−0.559389
$994$	0	0
$995$	−22.4853	−0.712831
$996$	0	0
$997$	9.24264	0.292717	0.146359	−	0.989232i	$-0.453245\pi$
$997$	9.24264	0.292717	0.146359	+	0.989232i	$0.453245\pi$
$998$	43.5563	1.37875
$999$	8.41421	0.266214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.a.i.1.1		2
3.2	odd	2		2205.2.a.u.1.2		2
5.4	even	2		3675.2.a.z.1.2		2
7.2	even	3		105.2.i.c.46.2	yes	4
7.3	odd	6		735.2.i.j.226.2		4
7.4	even	3		105.2.i.c.16.2	✓	4
7.5	odd	6		735.2.i.j.361.2		4
7.6	odd	2		735.2.a.j.1.1		2
21.2	odd	6		315.2.j.d.46.1		4
21.11	odd	6		315.2.j.d.226.1		4
21.20	even	2		2205.2.a.s.1.2		2
28.11	odd	6		1680.2.bg.p.961.2		4
28.23	odd	6		1680.2.bg.p.1201.2		4
35.2	odd	12		525.2.r.g.424.1		8
35.4	even	6		525.2.i.g.226.1		4
35.9	even	6		525.2.i.g.151.1		4
35.18	odd	12		525.2.r.g.499.1		8
35.23	odd	12		525.2.r.g.424.4		8
35.32	odd	12		525.2.r.g.499.4		8
35.34	odd	2		3675.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.i.c.16.2	✓	4	7.4	even	3
105.2.i.c.46.2	yes	4	7.2	even	3
315.2.j.d.46.1		4	21.2	odd	6
315.2.j.d.226.1		4	21.11	odd	6
525.2.i.g.151.1		4	35.9	even	6
525.2.i.g.226.1		4	35.4	even	6
525.2.r.g.424.1		8	35.2	odd	12
525.2.r.g.424.4		8	35.23	odd	12
525.2.r.g.499.1		8	35.18	odd	12
525.2.r.g.499.4		8	35.32	odd	12
735.2.a.i.1.1		2	1.1	even	1	trivial
735.2.a.j.1.1		2	7.6	odd	2
735.2.i.j.226.2		4	7.3	odd	6
735.2.i.j.361.2		4	7.5	odd	6
1680.2.bg.p.961.2		4	28.11	odd	6
1680.2.bg.p.1201.2		4	28.23	odd	6
2205.2.a.s.1.2		2	21.20	even	2
2205.2.a.u.1.2		2	3.2	odd	2
3675.2.a.x.1.2		2	35.34	odd	2
3675.2.a.z.1.2		2	5.4	even	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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +1.41421 q^{10} +0.585786 q^{11} -4.41421 q^{13} +1.00000 q^{15} -4.00000 q^{16} +2.24264 q^{17} -1.41421 q^{18} +4.65685 q^{19} -0.828427 q^{22} +2.24264 q^{23} -2.82843 q^{24} +1.00000 q^{25} +6.24264 q^{26} -1.00000 q^{27} +8.24264 q^{29} -1.41421 q^{30} -5.82843 q^{31} -0.585786 q^{33} -3.17157 q^{34} -8.41421 q^{37} -6.58579 q^{38} +4.41421 q^{39} -2.82843 q^{40} -6.24264 q^{41} -7.58579 q^{43} -1.00000 q^{45} -3.17157 q^{46} -13.3137 q^{47} +4.00000 q^{48} -1.41421 q^{50} -2.24264 q^{51} +6.82843 q^{53} +1.41421 q^{54} -0.585786 q^{55} -4.65685 q^{57} -11.6569 q^{58} -1.41421 q^{59} -4.48528 q^{61} +8.24264 q^{62} +8.00000 q^{64} +4.41421 q^{65} +0.828427 q^{66} -13.7279 q^{67} -2.24264 q^{69} -0.585786 q^{71} +2.82843 q^{72} -12.0711 q^{73} +11.8995 q^{74} -1.00000 q^{75} -6.24264 q^{78} +6.65685 q^{79} +4.00000 q^{80} +1.00000 q^{81} +8.82843 q^{82} +2.58579 q^{83} -2.24264 q^{85} +10.7279 q^{86} -8.24264 q^{87} +1.65685 q^{88} -12.2426 q^{89} +1.41421 q^{90} +5.82843 q^{93} +18.8284 q^{94} -4.65685 q^{95} -5.17157 q^{97} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{16} - 4 q^{17} - 2 q^{19} + 4 q^{22} - 4 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{27} + 8 q^{29} - 6 q^{31} - 4 q^{33} - 12 q^{34} - 14 q^{37} - 16 q^{38} + 6 q^{39} - 4 q^{41} - 18 q^{43} - 2 q^{45} - 12 q^{46} - 4 q^{47} + 8 q^{48} + 4 q^{51} + 8 q^{53} - 4 q^{55} + 2 q^{57} - 12 q^{58} + 8 q^{61} + 8 q^{62} + 16 q^{64} + 6 q^{65} - 4 q^{66} - 2 q^{67} + 4 q^{69} - 4 q^{71} - 10 q^{73} + 4 q^{74} - 2 q^{75} - 4 q^{78} + 2 q^{79} + 8 q^{80} + 2 q^{81} + 12 q^{82} + 8 q^{83} + 4 q^{85} - 4 q^{86} - 8 q^{87} - 8 q^{88} - 16 q^{89} + 6 q^{93} + 32 q^{94} + 2 q^{95} - 16 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 + 4 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^16 - 4 * q^17 - 2 * q^19 + 4 * q^22 - 4 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^27 + 8 * q^29 - 6 * q^31 - 4 * q^33 - 12 * q^34 - 14 * q^37 - 16 * q^38 + 6 * q^39 - 4 * q^41 - 18 * q^43 - 2 * q^45 - 12 * q^46 - 4 * q^47 + 8 * q^48 + 4 * q^51 + 8 * q^53 - 4 * q^55 + 2 * q^57 - 12 * q^58 + 8 * q^61 + 8 * q^62 + 16 * q^64 + 6 * q^65 - 4 * q^66 - 2 * q^67 + 4 * q^69 - 4 * q^71 - 10 * q^73 + 4 * q^74 - 2 * q^75 - 4 * q^78 + 2 * q^79 + 8 * q^80 + 2 * q^81 + 12 * q^82 + 8 * q^83 + 4 * q^85 - 4 * q^86 - 8 * q^87 - 8 * q^88 - 16 * q^89 + 6 * q^93 + 32 * q^94 + 2 * q^95 - 16 * q^97 + 4 * q^99

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