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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 161.2
Root \(1.52179 + 1.90827i\) of defining polynomial
Character \(\chi\) \(=\) 464.161
Dual form 464.2.u.h.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.52179 - 1.90827i) q^{3} +(2.60002 + 1.25211i) q^{5} +(1.89765 - 2.37957i) q^{7} +(-0.658071 - 2.88320i) q^{9} +(-1.22038 + 5.34685i) q^{11} +(0.0239308 - 0.104847i) q^{13} +(6.34605 - 3.05610i) q^{15} +0.816005 q^{17} +(-1.27358 - 1.59701i) q^{19} +(-1.65304 - 7.24243i) q^{21} +(-8.25746 + 3.97659i) q^{23} +(2.07491 + 2.60185i) q^{25} +(0.0938049 + 0.0451741i) q^{27} +(-5.37657 + 0.304047i) q^{29} +(-3.10086 - 1.49330i) q^{31} +(8.34605 + 10.4656i) q^{33} +(7.91340 - 3.81089i) q^{35} +(-1.31456 - 5.75946i) q^{37} +(-0.163659 - 0.205223i) q^{39} -2.43376 q^{41} +(4.94123 - 2.37957i) q^{43} +(1.89907 - 8.32035i) q^{45} +(1.31510 - 5.76182i) q^{47} +(-0.503658 - 2.20667i) q^{49} +(1.24179 - 1.55716i) q^{51} +(-3.55945 - 1.71414i) q^{53} +(-9.86785 + 12.3739i) q^{55} -4.98565 q^{57} -1.13359 q^{59} +(5.09132 - 6.38431i) q^{61} +(-8.10956 - 3.90536i) q^{63} +(0.193501 - 0.242642i) q^{65} +(0.212822 + 0.932434i) q^{67} +(-4.97776 + 21.8090i) q^{69} +(0.531778 - 2.32987i) q^{71} +(-8.01331 + 3.85900i) q^{73} +8.12261 q^{75} +(10.4074 + 13.0504i) q^{77} +(-0.934726 - 4.09530i) q^{79} +(8.22238 - 3.95969i) q^{81} +(9.64738 + 12.0974i) q^{83} +(2.12163 + 1.02172i) q^{85} +(-7.60183 + 10.7226i) q^{87} +(-4.99275 - 2.40438i) q^{89} +(-0.204080 - 0.255908i) q^{91} +(-7.56848 + 3.64479i) q^{93} +(-1.31170 - 5.74692i) q^{95} +(7.43305 + 9.32075i) q^{97} +16.2191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - q^{7} - 11 q^{9} + 2 q^{11} + q^{13} + 9 q^{15} - 12 q^{17} + 6 q^{19} - 13 q^{21} - 35 q^{23} - 6 q^{25} - 39 q^{27} - 14 q^{29} + 8 q^{31} + 33 q^{33} + 18 q^{35} + 31 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.52179 − 1.90827i 0.878608 − 1.10174i −0.115496 − 0.993308i \(-0.536846\pi\)
0.994104 − 0.108431i \(-0.0345828\pi\)
\(4\) 0 0
\(5\) 2.60002 + 1.25211i 1.16277 + 0.559959i 0.912844 − 0.408307i \(-0.133881\pi\)
0.249922 + 0.968266i \(0.419595\pi\)
\(6\) 0 0
\(7\) 1.89765 − 2.37957i 0.717242 − 0.899394i −0.280936 − 0.959727i \(-0.590645\pi\)
0.998178 + 0.0603330i \(0.0192162\pi\)
\(8\) 0 0
\(9\) −0.658071 − 2.88320i −0.219357 − 0.961065i
\(10\) 0 0
\(11\) −1.22038 + 5.34685i −0.367959 + 1.61214i 0.364417 + 0.931236i \(0.381268\pi\)
−0.732377 + 0.680900i \(0.761589\pi\)
\(12\) 0 0
\(13\) 0.0239308 − 0.104847i 0.00663720 − 0.0290795i −0.971501 − 0.237036i \(-0.923824\pi\)
0.978138 + 0.207956i \(0.0666812\pi\)
\(14\) 0 0
\(15\) 6.34605 − 3.05610i 1.63854 − 0.789081i
\(16\) 0 0
\(17\) 0.816005 0.197910 0.0989551 − 0.995092i \(-0.468450\pi\)
0.0989551 + 0.995092i \(0.468450\pi\)
\(18\) 0 0
\(19\) −1.27358 − 1.59701i −0.292178 − 0.366380i 0.613978 − 0.789323i \(-0.289568\pi\)
−0.906156 + 0.422943i \(0.860997\pi\)
\(20\) 0 0
\(21\) −1.65304 − 7.24243i −0.360722 − 1.58043i
\(22\) 0 0
\(23\) −8.25746 + 3.97659i −1.72180 + 0.829175i −0.732946 + 0.680286i \(0.761855\pi\)
−0.988854 + 0.148889i \(0.952430\pi\)
\(24\) 0 0
\(25\) 2.07491 + 2.60185i 0.414981 + 0.520370i
\(26\) 0 0
\(27\) 0.0938049 + 0.0451741i 0.0180528 + 0.00869375i
\(28\) 0 0
\(29\) −5.37657 + 0.304047i −0.998405 + 0.0564602i
\(30\) 0 0
\(31\) −3.10086 − 1.49330i −0.556931 − 0.268204i 0.134174 − 0.990958i \(-0.457162\pi\)
−0.691106 + 0.722754i \(0.742876\pi\)
\(32\) 0 0
\(33\) 8.34605 + 10.4656i 1.45286 + 1.82183i
\(34\) 0 0
\(35\) 7.91340 − 3.81089i 1.33761 − 0.644158i
\(36\) 0 0
\(37\) −1.31456 − 5.75946i −0.216112 − 0.946850i −0.960320 − 0.278901i \(-0.910030\pi\)
0.744207 − 0.667949i \(-0.232827\pi\)
\(38\) 0 0
\(39\) −0.163659 − 0.205223i −0.0262065 − 0.0328619i
\(40\) 0 0
\(41\) −2.43376 −0.380090 −0.190045 − 0.981775i \(-0.560863\pi\)
−0.190045 + 0.981775i \(0.560863\pi\)
\(42\) 0 0
\(43\) 4.94123 − 2.37957i 0.753531 − 0.362881i −0.0173595 − 0.999849i \(-0.505526\pi\)
0.770890 + 0.636968i \(0.219812\pi\)
\(44\) 0 0
\(45\) 1.89907 − 8.32035i 0.283096 − 1.24033i
\(46\) 0 0
\(47\) 1.31510 − 5.76182i 0.191827 − 0.840448i −0.783800 − 0.621013i \(-0.786721\pi\)
0.975627 − 0.219435i \(-0.0704214\pi\)
\(48\) 0 0
\(49\) −0.503658 − 2.20667i −0.0719512 − 0.315239i
\(50\) 0 0
\(51\) 1.24179 − 1.55716i 0.173885 − 0.218045i
\(52\) 0 0
\(53\) −3.55945 − 1.71414i −0.488929 − 0.235456i 0.173142 − 0.984897i \(-0.444608\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(54\) 0 0
\(55\) −9.86785 + 12.3739i −1.33058 + 1.66849i
\(56\) 0 0
\(57\) −4.98565 −0.660365
\(58\) 0 0
\(59\) −1.13359 −0.147581 −0.0737904 − 0.997274i \(-0.523510\pi\)
−0.0737904 + 0.997274i \(0.523510\pi\)
\(60\) 0 0
\(61\) 5.09132 − 6.38431i 0.651876 − 0.817427i −0.340555 − 0.940224i \(-0.610615\pi\)
0.992432 + 0.122797i \(0.0391865\pi\)
\(62\) 0 0
\(63\) −8.10956 − 3.90536i −1.02171 − 0.492029i
\(64\) 0 0
\(65\) 0.193501 − 0.242642i 0.0240008 − 0.0300961i
\(66\) 0 0
\(67\) 0.212822 + 0.932434i 0.0260003 + 0.113915i 0.986263 − 0.165183i \(-0.0528216\pi\)
−0.960263 + 0.279098i \(0.909964\pi\)
\(68\) 0 0
\(69\) −4.97776 + 21.8090i −0.599252 + 2.62549i
\(70\) 0 0
\(71\) 0.531778 − 2.32987i 0.0631104 − 0.276505i −0.933520 − 0.358525i \(-0.883280\pi\)
0.996631 + 0.0820198i \(0.0261371\pi\)
\(72\) 0 0
\(73\) −8.01331 + 3.85900i −0.937886 + 0.451662i −0.839423 − 0.543478i \(-0.817107\pi\)
−0.0984633 + 0.995141i \(0.531393\pi\)
\(74\) 0 0
\(75\) 8.12261 0.937918
\(76\) 0 0
\(77\) 10.4074 + 13.0504i 1.18603 + 1.48723i
\(78\) 0 0
\(79\) −0.934726 − 4.09530i −0.105165 − 0.460758i −0.999900 − 0.0141598i \(-0.995493\pi\)
0.894735 − 0.446598i \(-0.147364\pi\)
\(80\) 0 0
\(81\) 8.22238 − 3.95969i 0.913598 − 0.439965i
\(82\) 0 0
\(83\) 9.64738 + 12.0974i 1.05894 + 1.32787i 0.942329 + 0.334688i \(0.108631\pi\)
0.116609 + 0.993178i \(0.462798\pi\)
\(84\) 0 0
\(85\) 2.12163 + 1.02172i 0.230123 + 0.110822i
\(86\) 0 0
\(87\) −7.60183 + 10.7226i −0.815002 + 1.14959i
\(88\) 0 0
\(89\) −4.99275 − 2.40438i −0.529231 − 0.254864i 0.150133 − 0.988666i \(-0.452030\pi\)
−0.679364 + 0.733802i \(0.737744\pi\)
\(90\) 0 0
\(91\) −0.204080 − 0.255908i −0.0213934 − 0.0268265i
\(92\) 0 0
\(93\) −7.56848 + 3.64479i −0.784815 + 0.377947i
\(94\) 0 0
\(95\) −1.31170 − 5.74692i −0.134577 − 0.589622i
\(96\) 0 0
\(97\) 7.43305 + 9.32075i 0.754712 + 0.946379i 0.999732 − 0.0231388i \(-0.00736597\pi\)
−0.245020 + 0.969518i \(0.578795\pi\)
\(98\) 0 0
\(99\) 16.2191 1.63008
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.161.2 12
4.3 odd 2 58.2.d.b.45.1 ✓ 12
12.11 even 2 522.2.k.h.451.1 12
29.20 even 7 inner 464.2.u.h.49.2 12
116.3 even 28 1682.2.b.i.1681.6 12
116.7 odd 14 1682.2.a.t.1.1 6
116.51 odd 14 1682.2.a.q.1.6 6
116.55 even 28 1682.2.b.i.1681.7 12
116.107 odd 14 58.2.d.b.49.1 yes 12
348.107 even 14 522.2.k.h.397.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.1 ✓ 12 4.3 odd 2
58.2.d.b.49.1 yes 12 116.107 odd 14
464.2.u.h.49.2 12 29.20 even 7 inner
464.2.u.h.161.2 12 1.1 even 1 trivial
522.2.k.h.397.1 12 348.107 even 14
522.2.k.h.451.1 12 12.11 even 2
1682.2.a.q.1.6 6 116.51 odd 14
1682.2.a.t.1.1 6 116.7 odd 14
1682.2.b.i.1681.6 12 116.3 even 28
1682.2.b.i.1681.7 12 116.55 even 28