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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [464,2,Mod(49,464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("464.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(464, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.u (of order \(7\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 65.1
Root \(-1.56920 + 0.755686i\) of defining polynomial
Character \(\chi\) \(=\) 464.65
Dual form 464.2.u.h.257.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.56920 - 0.755686i) q^{3} +(0.110081 + 0.482295i) q^{5} +(2.82760 + 1.36170i) q^{7} +(0.0208506 + 0.0261458i) q^{9} +(-0.870839 + 1.09200i) q^{11} +(-3.56353 + 4.46852i) q^{13} +(0.191725 - 0.840003i) q^{15} +5.31800 q^{17} +(4.47471 - 2.15491i) q^{19} +(-3.40804 - 4.27355i) q^{21} +(-0.181045 + 0.793210i) q^{23} +(4.28435 - 2.06324i) q^{25} +(1.14972 + 5.03725i) q^{27} +(5.38501 + 0.0414712i) q^{29} +(1.41596 + 6.20373i) q^{31} +(2.19173 - 1.05548i) q^{33} +(-0.345477 + 1.51363i) q^{35} +(5.56630 + 6.97992i) q^{37} +(8.96867 - 4.31909i) q^{39} -4.01226 q^{41} +(-0.310799 + 1.36170i) q^{43} +(-0.0103147 + 0.0129343i) q^{45} +(6.42079 - 8.05142i) q^{47} +(1.77665 + 2.22785i) q^{49} +(-8.34499 - 4.01873i) q^{51} +(-0.944288 - 4.13720i) q^{53} +(-0.622528 - 0.299794i) q^{55} -8.65014 q^{57} -11.1598 q^{59} +(-4.38454 - 2.11149i) q^{61} +(0.0233543 + 0.102322i) q^{63} +(-2.54742 - 1.22677i) q^{65} +(-4.45072 - 5.58102i) q^{67} +(0.883513 - 1.10789i) q^{69} +(-3.76423 + 4.72019i) q^{71} +(-2.73238 + 11.9713i) q^{73} -8.28215 q^{75} +(-3.94935 + 1.90191i) q^{77} +(5.86220 + 7.35096i) q^{79} +(2.02476 - 8.87107i) q^{81} +(11.4508 - 5.51441i) q^{83} +(0.585409 + 2.56484i) q^{85} +(-8.41880 - 4.13445i) q^{87} +(-0.398796 - 1.74724i) q^{89} +(-16.1610 + 7.78272i) q^{91} +(2.46615 - 10.8049i) q^{93} +(1.53188 + 1.92092i) q^{95} +(-3.42035 + 1.64715i) q^{97} -0.0467086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - q^{7} - 11 q^{9} + 2 q^{11} + q^{13} + 9 q^{15} - 12 q^{17} + 6 q^{19} - 13 q^{21} - 35 q^{23} - 6 q^{25} - 39 q^{27} - 14 q^{29} + 8 q^{31} + 33 q^{33} + 18 q^{35} + 31 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{7}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.56920 − 0.755686i −0.905977 − 0.436295i −0.0779327 − 0.996959i \(-0.524832\pi\)
−0.828044 + 0.560663i \(0.810546\pi\)
\(4\) 0 0
\(5\) 0.110081 + 0.482295i 0.0492296 + 0.215689i 0.993560 − 0.113311i \(-0.0361456\pi\)
−0.944330 + 0.329000i \(0.893288\pi\)
\(6\) 0 0
\(7\) 2.82760 + 1.36170i 1.06873 + 0.514674i 0.883697 − 0.468059i \(-0.155046\pi\)
0.185033 + 0.982732i \(0.440761\pi\)
\(8\) 0 0
\(9\) 0.0208506 + 0.0261458i 0.00695019 + 0.00871526i
\(10\) 0 0
\(11\) −0.870839 + 1.09200i −0.262568 + 0.329250i −0.895587 − 0.444887i \(-0.853244\pi\)
0.633019 + 0.774136i \(0.281815\pi\)
\(12\) 0 0
\(13\) −3.56353 + 4.46852i −0.988344 + 1.23934i −0.0174472 + 0.999848i \(0.505554\pi\)
−0.970897 + 0.239497i \(0.923018\pi\)
\(14\) 0 0
\(15\) 0.191725 − 0.840003i 0.0495032 − 0.216888i
\(16\) 0 0
\(17\) 5.31800 1.28980 0.644902 − 0.764265i \(-0.276898\pi\)
0.644902 + 0.764265i \(0.276898\pi\)
\(18\) 0 0
\(19\) 4.47471 − 2.15491i 1.02657 − 0.494370i 0.156697 − 0.987647i \(-0.449915\pi\)
0.869872 + 0.493277i \(0.164201\pi\)
\(20\) 0 0
\(21\) −3.40804 − 4.27355i −0.743695 − 0.932565i
\(22\) 0 0
\(23\) −0.181045 + 0.793210i −0.0377505 + 0.165396i −0.990289 − 0.139021i \(-0.955604\pi\)
0.952539 + 0.304417i \(0.0984616\pi\)
\(24\) 0 0
\(25\) 4.28435 − 2.06324i 0.856871 − 0.412647i
\(26\) 0 0
\(27\) 1.14972 + 5.03725i 0.221263 + 0.969419i
\(28\) 0 0
\(29\) 5.38501 + 0.0414712i 0.999970 + 0.00770101i
\(30\) 0 0
\(31\) 1.41596 + 6.20373i 0.254314 + 1.11422i 0.927226 + 0.374501i \(0.122186\pi\)
−0.672912 + 0.739722i \(0.734957\pi\)
\(32\) 0 0
\(33\) 2.19173 − 1.05548i 0.381530 − 0.183735i
\(34\) 0 0
\(35\) −0.345477 + 1.51363i −0.0583962 + 0.255851i
\(36\) 0 0
\(37\) 5.56630 + 6.97992i 0.915095 + 1.14749i 0.988655 + 0.150202i \(0.0479923\pi\)
−0.0735606 + 0.997291i \(0.523436\pi\)
\(38\) 0 0
\(39\) 8.96867 − 4.31909i 1.43614 − 0.691607i
\(40\) 0 0
\(41\) −4.01226 −0.626610 −0.313305 − 0.949652i \(-0.601436\pi\)
−0.313305 + 0.949652i \(0.601436\pi\)
\(42\) 0 0
\(43\) −0.310799 + 1.36170i −0.0473964 + 0.207657i −0.993082 − 0.117427i \(-0.962535\pi\)
0.945685 + 0.325084i \(0.105393\pi\)
\(44\) 0 0
\(45\) −0.0103147 + 0.0129343i −0.00153763 + 0.00192813i
\(46\) 0 0
\(47\) 6.42079 − 8.05142i 0.936569 − 1.17442i −0.0478986 − 0.998852i \(-0.515252\pi\)
0.984467 − 0.175568i \(-0.0561761\pi\)
\(48\) 0 0
\(49\) 1.77665 + 2.22785i 0.253807 + 0.318264i
\(50\) 0 0
\(51\) −8.34499 − 4.01873i −1.16853 − 0.562735i
\(52\) 0 0
\(53\) −0.944288 − 4.13720i −0.129708 − 0.568288i −0.997456 − 0.0712835i \(-0.977291\pi\)
0.867748 − 0.497004i \(-0.165567\pi\)
\(54\) 0 0
\(55\) −0.622528 − 0.299794i −0.0839416 − 0.0404241i
\(56\) 0 0
\(57\) −8.65014 −1.14574
\(58\) 0 0
\(59\) −11.1598 −1.45288 −0.726438 − 0.687232i \(-0.758826\pi\)
−0.726438 + 0.687232i \(0.758826\pi\)
\(60\) 0 0
\(61\) −4.38454 − 2.11149i −0.561383 − 0.270348i 0.131598 − 0.991303i \(-0.457989\pi\)
−0.692982 + 0.720955i \(0.743703\pi\)
\(62\) 0 0
\(63\) 0.0233543 + 0.102322i 0.00294237 + 0.0128913i
\(64\) 0 0
\(65\) −2.54742 − 1.22677i −0.315969 − 0.152163i
\(66\) 0 0
\(67\) −4.45072 − 5.58102i −0.543742 − 0.681830i 0.431718 − 0.902009i \(-0.357908\pi\)
−0.975460 + 0.220178i \(0.929336\pi\)
\(68\) 0 0
\(69\) 0.883513 − 1.10789i 0.106362 − 0.133374i
\(70\) 0 0
\(71\) −3.76423 + 4.72019i −0.446732 + 0.560184i −0.953304 − 0.302014i \(-0.902341\pi\)
0.506572 + 0.862198i \(0.330913\pi\)
\(72\) 0 0
\(73\) −2.73238 + 11.9713i −0.319801 + 1.40114i 0.518103 + 0.855319i \(0.326639\pi\)
−0.837903 + 0.545819i \(0.816219\pi\)
\(74\) 0 0
\(75\) −8.28215 −0.956341
\(76\) 0 0
\(77\) −3.94935 + 1.90191i −0.450070 + 0.216743i
\(78\) 0 0
\(79\) 5.86220 + 7.35096i 0.659549 + 0.827048i 0.993294 − 0.115615i \(-0.0368840\pi\)
−0.333745 + 0.942663i \(0.608313\pi\)
\(80\) 0 0
\(81\) 2.02476 − 8.87107i 0.224974 − 0.985675i
\(82\) 0 0
\(83\) 11.4508 − 5.51441i 1.25689 − 0.605285i 0.317538 − 0.948246i \(-0.397144\pi\)
0.939350 + 0.342960i \(0.111430\pi\)
\(84\) 0 0
\(85\) 0.585409 + 2.56484i 0.0634965 + 0.278196i
\(86\) 0 0
\(87\) −8.41880 − 4.13445i −0.902590 − 0.443259i
\(88\) 0 0
\(89\) −0.398796 − 1.74724i −0.0422722 − 0.185207i 0.949384 − 0.314118i \(-0.101709\pi\)
−0.991656 + 0.128911i \(0.958852\pi\)
\(90\) 0 0
\(91\) −16.1610 + 7.78272i −1.69413 + 0.815851i
\(92\) 0 0
\(93\) 2.46615 − 10.8049i 0.255728 − 1.12042i
\(94\) 0 0
\(95\) 1.53188 + 1.92092i 0.157168 + 0.197082i
\(96\) 0 0
\(97\) −3.42035 + 1.64715i −0.347283 + 0.167243i −0.599391 − 0.800456i \(-0.704591\pi\)
0.252108 + 0.967699i \(0.418876\pi\)
\(98\) 0 0
\(99\) −0.0467086 −0.00469439
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 464.2.u.h.65.1 12
4.3 odd 2 58.2.d.b.7.2 ✓ 12
12.11 even 2 522.2.k.h.181.1 12
29.25 even 7 inner 464.2.u.h.257.1 12
116.27 even 28 1682.2.b.i.1681.8 12
116.31 even 28 1682.2.b.i.1681.5 12
116.63 odd 14 1682.2.a.q.1.5 6
116.83 odd 14 58.2.d.b.25.2 yes 12
116.111 odd 14 1682.2.a.t.1.2 6
348.83 even 14 522.2.k.h.199.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.2 ✓ 12 4.3 odd 2
58.2.d.b.25.2 yes 12 116.83 odd 14
464.2.u.h.65.1 12 1.1 even 1 trivial
464.2.u.h.257.1 12 29.25 even 7 inner
522.2.k.h.181.1 12 12.11 even 2
522.2.k.h.199.1 12 348.83 even 14
1682.2.a.q.1.5 6 116.63 odd 14
1682.2.a.t.1.2 6 116.111 odd 14
1682.2.b.i.1681.5 12 116.31 even 28
1682.2.b.i.1681.8 12 116.27 even 28