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Properties

Label 462.4.i.c
Level $462$
Weight $4$
Character orbit 462.i
Analytic conductor $27.259$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 59x^{6} - 70x^{5} + 2930x^{4} - 2716x^{3} + 32980x^{2} + 31872x + 248004 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} + ( - 3 \beta_{2} + 3) q^{3} + (4 \beta_{2} - 4) q^{4} + \beta_{5} q^{5} - 6 q^{6} + (3 \beta_{7} + \beta_{4} + 2 \beta_{2} - 6) q^{7} + 8 q^{8} - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} + ( - 3 \beta_{2} + 3) q^{3} + (4 \beta_{2} - 4) q^{4} + \beta_{5} q^{5} - 6 q^{6} + (3 \beta_{7} + \beta_{4} + 2 \beta_{2} - 6) q^{7} + 8 q^{8} - 9 \beta_{2} q^{9} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{10} + ( - 11 \beta_{2} + 11) q^{11} + 12 \beta_{2} q^{12} + ( - \beta_{7} - \beta_{6} - 5 \beta_{4} + \cdots + 22) q^{13}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 12 q^{3} - 16 q^{4} - 48 q^{6} - 27 q^{7} + 64 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 12 q^{3} - 16 q^{4} - 48 q^{6} - 27 q^{7} + 64 q^{8} - 36 q^{9} + 44 q^{11} + 48 q^{12} + 130 q^{13} + 36 q^{14} - 64 q^{16} - 37 q^{17} - 72 q^{18} - 50 q^{19} - 27 q^{21} - 176 q^{22} + 57 q^{23} + 96 q^{24} + 156 q^{25} - 130 q^{26} - 216 q^{27} + 36 q^{28} - 778 q^{29} + 59 q^{31} - 128 q^{32} - 132 q^{33} + 148 q^{34} + 42 q^{35} + 288 q^{36} + 514 q^{37} - 100 q^{38} + 195 q^{39} + 424 q^{41} + 162 q^{42} - 1192 q^{43} + 176 q^{44} + 114 q^{46} - 689 q^{47} - 384 q^{48} + 365 q^{49} - 624 q^{50} + 111 q^{51} - 260 q^{52} + 110 q^{53} + 216 q^{54} - 216 q^{56} - 300 q^{57} + 778 q^{58} - 43 q^{59} - 198 q^{61} - 236 q^{62} + 162 q^{63} + 512 q^{64} + 168 q^{65} - 264 q^{66} + 699 q^{67} - 148 q^{68} + 342 q^{69} + 420 q^{70} - 1966 q^{71} - 288 q^{72} - 957 q^{73} + 1028 q^{74} - 468 q^{75} + 400 q^{76} - 99 q^{77} - 780 q^{78} - 341 q^{79} - 324 q^{81} - 424 q^{82} + 368 q^{83} - 216 q^{84} - 868 q^{85} + 1192 q^{86} - 1167 q^{87} + 352 q^{88} + 86 q^{89} - 495 q^{91} - 456 q^{92} - 177 q^{93} - 1378 q^{94} + 1246 q^{95} + 384 q^{96} - 290 q^{97} - 482 q^{98} - 792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 59x^{6} - 70x^{5} + 2930x^{4} - 2716x^{3} + 32980x^{2} + 31872x + 248004 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29043299 \nu^{7} + 242270753 \nu^{6} + 8021632661 \nu^{5} + 21965525492 \nu^{4} + \cdots + 11113172329824 ) / 5543622753804 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1084859 \nu^{7} + 7968879 \nu^{6} - 60603183 \nu^{5} + 415393530 \nu^{4} + \cdots + 203961217644 ) / 205319361252 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 480385 \nu^{7} + 6724552 \nu^{6} + 23687950 \nu^{5} + 1772455 \nu^{4} + 517126684 \nu^{3} + \cdots - 317526517008 ) / 33395317794 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 154858379 \nu^{7} + 183382889 \nu^{6} + 11026832411 \nu^{5} + 5838707030 \nu^{4} + \cdots + 6567811998984 ) / 2771811376902 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 182126713 \nu^{7} + 1147251626 \nu^{6} - 8724828202 \nu^{5} + 86889429575 \nu^{4} + \cdots + 29363582830536 ) / 2771811376902 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2634041 \nu^{7} - 2599694 \nu^{6} - 129885470 \nu^{5} - 9718703 \nu^{4} + \cdots - 55723583250 ) / 33395317794 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3753731 \nu^{7} - 3153275 \nu^{6} - 185097770 \nu^{5} - 13849973 \nu^{4} + \cdots - 99484625592 ) / 33395317794 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{2} - \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{6} - 6\beta_{5} + 2\beta_{4} - 6\beta_{3} + 89\beta_{2} + \beta _1 - 89 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -19\beta_{7} - 19\beta_{6} - 83\beta_{4} + 9\beta_{3} - 83\beta_{2} + 83\beta _1 + 110 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -116\beta_{7} + 103\beta_{6} + 339\beta_{5} - 13\beta_{4} - 3538\beta_{2} - 116\beta _1 + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3779\beta_{7} - 3226\beta_{6} - 567\beta_{5} + 3226\beta_{4} - 567\beta_{3} + 6835\beta_{2} - 553\beta _1 - 6835 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -407\beta_{7} - 407\beta_{6} - 7009\beta_{4} + 16683\beta_{3} - 7009\beta_{2} + 7009\beta _1 + 166252 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 156058 \beta_{7} + 177431 \beta_{6} + 33417 \beta_{5} + 21373 \beta_{4} - 240860 \beta_{2} + \cdots - 21373 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
3.29382 + 5.70507i
−3.53627 − 6.12499i
2.04752 + 3.54641i
−1.30508 − 2.26046i
3.29382 − 5.70507i
−3.53627 + 6.12499i
2.04752 − 3.54641i
−1.30508 + 2.26046i
−1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −5.31223 − 9.20105i −6.00000 −8.45434 − 16.4780i 8.00000 −4.50000 − 7.79423i −10.6245 + 18.4021i
67.2 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i −3.90747 − 6.76794i −6.00000 −5.16807 + 17.7846i 8.00000 −4.50000 − 7.79423i −7.81494 + 13.5359i
67.3 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 4.53740 + 7.85901i −6.00000 18.4875 − 1.10081i 8.00000 −4.50000 − 7.79423i 9.07481 − 15.7180i
67.4 −1.00000 − 1.73205i 1.50000 − 2.59808i −2.00000 + 3.46410i 4.68229 + 8.10997i −6.00000 −18.3651 + 2.39230i 8.00000 −4.50000 − 7.79423i 9.36459 − 16.2199i
331.1 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −5.31223 + 9.20105i −6.00000 −8.45434 + 16.4780i 8.00000 −4.50000 + 7.79423i −10.6245 − 18.4021i
331.2 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i −3.90747 + 6.76794i −6.00000 −5.16807 − 17.7846i 8.00000 −4.50000 + 7.79423i −7.81494 − 13.5359i
331.3 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 4.53740 − 7.85901i −6.00000 18.4875 + 1.10081i 8.00000 −4.50000 + 7.79423i 9.07481 + 15.7180i
331.4 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 − 3.46410i 4.68229 − 8.10997i −6.00000 −18.3651 − 2.39230i 8.00000 −4.50000 + 7.79423i 9.36459 + 16.2199i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
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.c ✓ 8
7.c even 3 1 inner 462.4.i.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.i.c ✓ 8 1.a even 1 1 trivial
462.4.i.c ✓ 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 172T_{5}^{6} - 72T_{5}^{5} + 22528T_{5}^{4} - 6192T_{5}^{3} + 1214928T_{5}^{2} + 254016T_{5} + 49787136 \) acting on \(S_{4}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 172 T^{6} + \cdots + 49787136 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{2} - 11 T + 121)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 65 T^{3} + \cdots - 739508)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 322776241956 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 38490769952649 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 153857826755136 \) Copy content Toggle raw display
$29$ \( (T^{4} + 389 T^{3} + \cdots - 137533086)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 37\!\cdots\!89 \) Copy content Toggle raw display
$41$ \( (T^{4} - 212 T^{3} + \cdots + 744681168)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 596 T^{3} + \cdots - 1908835321)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 64\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 87\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} + 983 T^{3} + \cdots - 3250107972)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{4} - 184 T^{3} + \cdots + 25677357696)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{4} + 145 T^{3} + \cdots + 148697495142)^{2} \) Copy content Toggle raw display
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