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Properties

Label 462.4.i.g
Level $462$
Weight $4$
Character orbit 462.i
Analytic conductor $27.259$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [462,4,Mod(67,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.67");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.i (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(27.2588824227\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 409 x^{10} - 1168 x^{9} + 132481 x^{8} - 260920 x^{7} + 13887112 x^{6} + 2274848 x^{5} + \cdots + 118041719184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{2} - 2) q^{2} - 3 \beta_{2} q^{3} - 4 \beta_{2} q^{4} - \beta_1 q^{5} + 6 q^{6} + (\beta_{8} - \beta_{6} - \beta_{3} + \cdots - 1) q^{7}+ \cdots + (9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} - 2) q^{2} - 3 \beta_{2} q^{3} - 4 \beta_{2} q^{4} - \beta_1 q^{5} + 6 q^{6} + (\beta_{8} - \beta_{6} - \beta_{3} + \cdots - 1) q^{7}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 18 q^{3} - 24 q^{4} + 72 q^{6} - 8 q^{7} + 96 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 18 q^{3} - 24 q^{4} + 72 q^{6} - 8 q^{7} + 96 q^{8} - 54 q^{9} + 66 q^{11} - 72 q^{12} + 48 q^{13} - 16 q^{14} - 96 q^{16} + 2 q^{17} - 108 q^{18} - 120 q^{19} + 48 q^{21} - 264 q^{22} + 36 q^{23} - 144 q^{24} - 68 q^{25} - 48 q^{26} + 324 q^{27} + 64 q^{28} + 632 q^{29} - 392 q^{31} - 192 q^{32} + 198 q^{33} - 8 q^{34} - 318 q^{35} + 432 q^{36} - 348 q^{37} - 240 q^{38} - 72 q^{39} + 388 q^{41} - 48 q^{42} + 1360 q^{43} + 264 q^{44} + 72 q^{46} - 68 q^{47} + 576 q^{48} - 546 q^{49} + 272 q^{50} + 6 q^{51} - 96 q^{52} - 400 q^{53} - 324 q^{54} - 64 q^{56} + 720 q^{57} - 632 q^{58} - 340 q^{59} - 232 q^{61} + 1568 q^{62} - 72 q^{63} + 768 q^{64} - 1428 q^{65} + 396 q^{66} + 362 q^{67} + 8 q^{68} - 216 q^{69} - 1152 q^{70} - 40 q^{71} - 432 q^{72} - 1156 q^{73} - 696 q^{74} - 204 q^{75} + 960 q^{76} - 176 q^{77} + 288 q^{78} - 3336 q^{79} - 486 q^{81} - 388 q^{82} + 668 q^{83} - 96 q^{84} + 8000 q^{85} - 1360 q^{86} - 948 q^{87} + 528 q^{88} - 16 q^{89} + 1796 q^{91} - 288 q^{92} - 1176 q^{93} - 136 q^{94} - 404 q^{95} - 576 q^{96} - 1092 q^{97} + 768 q^{98} - 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 409 x^{10} - 1168 x^{9} + 132481 x^{8} - 260920 x^{7} + 13887112 x^{6} + 2274848 x^{5} + \cdots + 118041719184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!78 \nu^{11} + \cdots - 16\!\cdots\!80 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!20 \nu^{11} + \cdots + 52\!\cdots\!92 ) / 64\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 84\!\cdots\!95 \nu^{11} + \cdots - 20\!\cdots\!96 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!35 \nu^{11} + \cdots + 45\!\cdots\!96 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!21 \nu^{11} + \cdots + 44\!\cdots\!56 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29\!\cdots\!79 \nu^{11} + \cdots - 73\!\cdots\!20 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32\!\cdots\!03 \nu^{11} + \cdots + 27\!\cdots\!16 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\!\cdots\!01 \nu^{11} + \cdots + 39\!\cdots\!88 ) / 79\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!88 \nu^{11} + \cdots - 31\!\cdots\!16 ) / 39\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 36\!\cdots\!28 \nu^{11} + \cdots - 35\!\cdots\!28 ) / 39\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta_{3} - 137\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 5 \beta_{11} - 8 \beta_{10} - 13 \beta_{9} + 2 \beta_{8} + 27 \beta_{7} + 27 \beta_{6} - 5 \beta_{4} + \cdots + 290 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 802 \beta_{11} + 4 \beta_{10} - 39 \beta_{8} - 806 \beta_{7} - 300 \beta_{6} + 490 \beta_{5} + \cdots - 32625 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9138 \beta_{11} + 3895 \beta_{10} + 5975 \beta_{9} - 13143 \beta_{8} - 2495 \beta_{7} + 3590 \beta_{6} + \cdots - 2495 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1284 \beta_{11} + 102451 \beta_{10} - 138402 \beta_{9} + 105019 \beta_{8} + 274186 \beta_{7} + \cdots + 9087321 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4242537 \beta_{11} - 844119 \beta_{10} + 2966865 \beta_{8} - 3398418 \beta_{7} - 4541487 \beta_{6} + \cdots - 78684783 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 89106446 \beta_{11} - 27518227 \beta_{10} + 43444130 \beta_{9} - 53706780 \beta_{8} - 2011116 \beta_{7} + \cdots - 2011116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 260974643 \beta_{11} + 51677656 \beta_{10} - 780015307 \beta_{9} + 573626942 \beta_{8} + \cdots + 30952499318 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 30221925898 \beta_{11} - 1214404772 \beta_{10} + 8547279705 \beta_{8} - 29007521126 \beta_{7} + \cdots - 851676902973 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 430321504242 \beta_{11} + 39773006743 \beta_{10} + 265674209855 \beta_{9} - 482113940367 \beta_{8} + \cdots - 79366661495 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
7.67189 + 13.2881i
5.84699 + 10.1273i
1.83143 + 3.17213i
−1.53569 − 2.65990i
−4.63571 − 8.02929i
−9.17891 − 15.8983i
7.67189 − 13.2881i
5.84699 − 10.1273i
1.83143 − 3.17213i
−1.53569 + 2.65990i
−4.63571 + 8.02929i
−9.17891 + 15.8983i
−1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −7.67189 − 13.2881i 6.00000 11.3408 + 14.6419i 8.00000 −4.50000 − 7.79423i −15.3438 + 26.5762i
67.2 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −5.84699 − 10.1273i 6.00000 7.66818 − 16.8582i 8.00000 −4.50000 − 7.79423i −11.6940 + 20.2546i
67.3 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −1.83143 − 3.17213i 6.00000 −18.4635 + 1.44891i 8.00000 −4.50000 − 7.79423i −3.66286 + 6.34427i
67.4 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 1.53569 + 2.65990i 6.00000 9.92779 − 15.6345i 8.00000 −4.50000 − 7.79423i 3.07139 − 5.31980i
67.5 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 4.63571 + 8.02929i 6.00000 1.73233 + 18.4391i 8.00000 −4.50000 − 7.79423i 9.27143 − 16.0586i
67.6 −1.00000 − 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 9.17891 + 15.8983i 6.00000 −16.2056 − 8.96536i 8.00000 −4.50000 − 7.79423i 18.3578 − 31.7967i
331.1 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −7.67189 + 13.2881i 6.00000 11.3408 − 14.6419i 8.00000 −4.50000 + 7.79423i −15.3438 − 26.5762i
331.2 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −5.84699 + 10.1273i 6.00000 7.66818 + 16.8582i 8.00000 −4.50000 + 7.79423i −11.6940 − 20.2546i
331.3 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i −1.83143 + 3.17213i 6.00000 −18.4635 − 1.44891i 8.00000 −4.50000 + 7.79423i −3.66286 − 6.34427i
331.4 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 1.53569 − 2.65990i 6.00000 9.92779 + 15.6345i 8.00000 −4.50000 + 7.79423i 3.07139 + 5.31980i
331.5 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 4.63571 − 8.02929i 6.00000 1.73233 − 18.4391i 8.00000 −4.50000 + 7.79423i 9.27143 + 16.0586i
331.6 −1.00000 + 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 9.17891 − 15.8983i 6.00000 −16.2056 + 8.96536i 8.00000 −4.50000 + 7.79423i 18.3578 + 31.7967i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.4.i.g ✓ 12
7.c even 3 1 inner 462.4.i.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.i.g ✓ 12 1.a even 1 1 trivial
462.4.i.g ✓ 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 409 T_{5}^{10} + 1168 T_{5}^{9} + 132481 T_{5}^{8} + 260920 T_{5}^{7} + 13887112 T_{5}^{6} + \cdots + 118041719184 \) acting on \(S_{4}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 118041719184 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 11 T + 121)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} - 24 T^{5} + \cdots - 4608125424)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 85\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( (T^{6} - 316 T^{5} + \cdots + 8527920732)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 335155496652252)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 21242454194876)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 443585869516671)^{2} \) Copy content Toggle raw display
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