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Properties

Label 29.6.b.a
Level $29$
Weight $6$
Character orbit 29.b
Analytic conductor $4.651$
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] = [29,6,Mod(28,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.28");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.b (of order \(2\), degree \(1\), 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: \(4.65113077458\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 278x^{10} + 28285x^{8} + 1260472x^{6} + 22944832x^{4} + 140087936x^{2} + 966400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{14}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 14) q^{4} + ( - \beta_{6} + 4) q^{5} + ( - \beta_{7} + 2) q^{6} + (\beta_{3} - 2 \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{4} - 10 \beta_1) q^{8} + ( - \beta_{10} + \beta_{6} + \cdots - 131) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 14) q^{4} + ( - \beta_{6} + 4) q^{5} + ( - \beta_{7} + 2) q^{6} + (\beta_{3} - 2 \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{4} - 10 \beta_1) q^{8} + ( - \beta_{10} + \beta_{6} + \cdots - 131) q^{9}+ \cdots + (49 \beta_{11} + 111 \beta_{9} + \cdots + 11378 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 172 q^{4} + 46 q^{5} + 24 q^{6} + 20 q^{7} - 1574 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 172 q^{4} + 46 q^{5} + 24 q^{6} + 20 q^{7} - 1574 q^{9} + 1362 q^{13} + 340 q^{16} - 4508 q^{20} + 11376 q^{22} + 5852 q^{23} - 6292 q^{24} + 12678 q^{25} - 25056 q^{28} + 11328 q^{29} + 14952 q^{30} - 22694 q^{33} - 22504 q^{34} + 4532 q^{35} + 22840 q^{36} - 43408 q^{38} + 8280 q^{42} - 52816 q^{45} + 102836 q^{49} + 58540 q^{51} + 15172 q^{52} + 25650 q^{53} - 89080 q^{54} - 32824 q^{57} + 4960 q^{58} - 3900 q^{59} + 37720 q^{62} - 146616 q^{63} + 252276 q^{64} + 169574 q^{65} - 28264 q^{67} - 286832 q^{71} - 263072 q^{74} + 519072 q^{78} - 230964 q^{80} - 24084 q^{81} - 178008 q^{82} + 85692 q^{83} - 126624 q^{86} - 137716 q^{87} - 83604 q^{88} - 182372 q^{91} - 5664 q^{92} + 377966 q^{93} + 192144 q^{94} - 415284 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 278x^{10} + 28285x^{8} + 1260472x^{6} + 22944832x^{4} + 140087936x^{2} + 966400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{10} - 18002\nu^{8} - 3564589\nu^{6} - 193044948\nu^{4} - 2100465264\nu^{2} + 3169589568 ) / 37604096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -125\nu^{11} - 35271\nu^{9} - 3622703\nu^{7} - 161356045\nu^{5} - 2877987192\nu^{3} - 17080930576\nu ) / 56406144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -125\nu^{11} - 35271\nu^{9} - 3622703\nu^{7} - 161356045\nu^{5} - 2821581048\nu^{3} - 12906875920\nu ) / 56406144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -543\nu^{10} - 114438\nu^{8} - 7870715\nu^{6} - 185889684\nu^{4} - 714248592\nu^{2} + 7095849024 ) / 75208192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -521\nu^{10} - 87078\nu^{8} - 3797045\nu^{6} - 9883192\nu^{4} + 430061424\nu^{2} + 233612288 ) / 56406144 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4129 \nu^{11} - 1048734 \nu^{9} - 96066205 \nu^{7} - 3784789952 \nu^{5} + \cdots - 302055473792 \nu ) / 225624576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3967 \nu^{11} - 1087632 \nu^{9} - 109815991 \nu^{7} - 4927586774 \nu^{5} + \cdots - 609397253600 \nu ) / 112812288 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4237 \nu^{10} + 1022802 \nu^{8} + 86899681 \nu^{6} + 2985321308 \nu^{4} + 32933911152 \nu^{2} + 30413683520 ) / 112812288 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7485 \nu^{11} - 2070898 \nu^{9} - 209723921 \nu^{7} - 9293509212 \nu^{5} + \cdots - 1002999387456 \nu ) / 75208192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 74\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{7} + 4\beta_{6} + 3\beta_{3} - 94\beta_{2} + 3420 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{11} - 4\beta_{9} - \beta_{8} - 108\beta_{5} + 45\beta_{4} + 5977\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -114\beta_{10} - 222\beta_{7} - 320\beta_{6} - 426\beta_{3} + 8139\beta_{2} - 276642 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -534\beta_{11} + 624\beta_{9} + 226\beta_{8} + 10067\beta_{5} + 2147\beta_{4} - 494572\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11423\beta_{10} + 31751\beta_{7} + 20044\beta_{6} + 48149\beta_{3} - 697028\beta_{2} + 22902656 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68477\beta_{11} - 73452\beta_{9} - 30623\beta_{8} - 913598\beta_{5} - 743283\beta_{4} + 41324323\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1097336\beta_{10} - 3816044\beta_{7} - 1093800\beta_{6} - 4999668\beta_{3} + 59790281\beta_{2} - 1914098054 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7718376 \beta_{11} + 7804664 \beta_{9} + 3381832 \beta_{8} + 82417837 \beta_{5} + \cdots - 3475181502 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
− 9.47123i
− 9.02264i
− 8.60273i
− 4.44887i
− 3.61683i
− 0.0831044i
0.0831044i
3.61683i
4.44887i
8.60273i
9.02264i
9.47123i
− 9.47123i − 19.9099i −57.7042 −4.84615 −188.571 219.131 243.450i −153.403 45.8990i
28.2 − 9.02264i 26.5755i −49.4080 71.1177 239.781 131.830 157.067i −463.255 − 641.670i
28.3 − 8.60273i 4.65015i −42.0069 −58.8511 40.0039 −192.738 86.0865i 221.376 506.279i
28.4 − 4.44887i − 21.5070i 12.2075 90.0922 −95.6821 −211.678 − 196.674i −219.553 − 400.809i
28.5 − 3.61683i 5.13082i 18.9185 15.9022 18.5573 69.7674 − 184.164i 216.675 − 57.5154i
28.6 − 0.0831044i − 25.1364i 31.9931 −90.4149 −2.08895 −6.31225 − 5.31810i −388.840 7.51387i
28.7 0.0831044i 25.1364i 31.9931 −90.4149 −2.08895 −6.31225 5.31810i −388.840 − 7.51387i
28.8 3.61683i − 5.13082i 18.9185 15.9022 18.5573 69.7674 184.164i 216.675 57.5154i
28.9 4.44887i 21.5070i 12.2075 90.0922 −95.6821 −211.678 196.674i −219.553 400.809i
28.10 8.60273i − 4.65015i −42.0069 −58.8511 40.0039 −192.738 − 86.0865i 221.376 − 506.279i
28.11 9.02264i − 26.5755i −49.4080 71.1177 239.781 131.830 − 157.067i −463.255 641.670i
28.12 9.47123i 19.9099i −57.7042 −4.84615 −188.571 219.131 − 243.450i −153.403 − 45.8990i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.6.b.a ✓ 12
3.b odd 2 1 261.6.c.b 12
4.b odd 2 1 464.6.e.c 12
29.b even 2 1 inner 29.6.b.a ✓ 12
29.c odd 4 2 841.6.a.d 12
87.d odd 2 1 261.6.c.b 12
116.d odd 2 1 464.6.e.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.b.a ✓ 12 1.a even 1 1 trivial
29.6.b.a ✓ 12 29.b even 2 1 inner
261.6.c.b 12 3.b odd 2 1
261.6.c.b 12 87.d odd 2 1
464.6.e.c 12 4.b odd 2 1
464.6.e.c 12 116.d odd 2 1
841.6.a.d 12 29.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(29, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 278 T^{10} + \cdots + 966400 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 46577165867100 \) Copy content Toggle raw display
$5$ \( (T^{6} - 23 T^{5} + \cdots - 2627317458)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 10 T^{5} + \cdots - 519034134784)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 15\!\cdots\!82)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 31\!\cdots\!08)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 74\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 28\!\cdots\!50)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
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