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Properties

Label 77.4.b.b
Level $77$
Weight $4$
Character orbit 77.b
Analytic conductor $4.543$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [77,4,Mod(76,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.76");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.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: \(4.54314707044\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 524 x^{18} + 106456 x^{16} + 11083128 x^{14} + 645045870 x^{12} + 21267440656 x^{10} + \cdots + 20988034812900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_{11} q^{3} + (\beta_{3} - 5) q^{4} - \beta_{8} q^{5} - \beta_{2} q^{6} + \beta_{16} q^{7} + (\beta_{12} - 5 \beta_{10}) q^{8} + (\beta_1 - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + \beta_{11} q^{3} + (\beta_{3} - 5) q^{4} - \beta_{8} q^{5} - \beta_{2} q^{6} + \beta_{16} q^{7} + (\beta_{12} - 5 \beta_{10}) q^{8} + (\beta_1 - 13) q^{9} + (\beta_{7} + \beta_{2}) q^{10} + ( - \beta_{14} + 2 \beta_{10} + \cdots + 2) q^{11}+ \cdots + ( - 35 \beta_{17} - 35 \beta_{16} + \cdots - 158) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 96 q^{4} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 96 q^{4} - 252 q^{9} + 52 q^{11} + 92 q^{14} + 312 q^{15} + 568 q^{16} - 396 q^{22} + 360 q^{23} - 1260 q^{25} + 1920 q^{36} - 1280 q^{37} - 2140 q^{42} + 1156 q^{44} + 940 q^{49} + 4632 q^{53} + 872 q^{56} - 2960 q^{58} - 4488 q^{60} + 4760 q^{64} - 4032 q^{67} - 212 q^{70} + 3816 q^{71} - 624 q^{77} - 5000 q^{78} + 3236 q^{81} - 8872 q^{86} + 5284 q^{88} + 2472 q^{91} - 9560 q^{92} - 2920 q^{93} - 2932 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 524 x^{18} + 106456 x^{16} + 11083128 x^{14} + 645045870 x^{12} + 21267440656 x^{10} + \cdots + 20988034812900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 52\!\cdots\!11 \nu^{18} + \cdots + 32\!\cdots\!00 ) / 11\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 72\!\cdots\!93 \nu^{18} + \cdots - 45\!\cdots\!60 ) / 34\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!53 \nu^{18} + \cdots + 80\!\cdots\!04 ) / 34\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!91 \nu^{18} + \cdots - 87\!\cdots\!04 ) / 30\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!49 \nu^{18} + \cdots + 88\!\cdots\!40 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37\!\cdots\!71 \nu^{18} + \cdots + 23\!\cdots\!60 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!31 \nu^{18} + \cdots + 85\!\cdots\!20 ) / 77\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!32 \nu^{19} + \cdots + 68\!\cdots\!80 \nu ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28\!\cdots\!53 \nu^{19} + \cdots + 19\!\cdots\!00 \nu ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!39 \nu^{19} + \cdots + 81\!\cdots\!00 \nu ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!39 \nu^{19} + \cdots + 81\!\cdots\!60 \nu ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!91 \nu^{19} + \cdots - 75\!\cdots\!00 \nu ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 70\!\cdots\!74 \nu^{19} + \cdots + 45\!\cdots\!60 \nu ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32\!\cdots\!51 \nu^{19} + \cdots - 38\!\cdots\!60 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32\!\cdots\!51 \nu^{19} + \cdots - 38\!\cdots\!60 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 61\!\cdots\!57 \nu^{19} + \cdots - 17\!\cdots\!60 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 61\!\cdots\!57 \nu^{19} + \cdots + 17\!\cdots\!60 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 48\!\cdots\!61 \nu^{19} + \cdots + 50\!\cdots\!96 ) / 10\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 16\!\cdots\!69 \nu^{19} + \cdots - 25\!\cdots\!80 ) / 52\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{11} - \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} + \beta _1 - 53 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} + 12 \beta_{17} + 12 \beta_{16} + 3 \beta_{15} - 3 \beta_{14} + \cdots - 9 \beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 24 \beta_{17} - 24 \beta_{16} - 38 \beta_{15} - 38 \beta_{14} - 56 \beta_{7} + 32 \beta_{6} + \cdots + 6282 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 233 \beta_{19} - 227 \beta_{18} - 2864 \beta_{17} - 2864 \beta_{16} - 857 \beta_{15} + 863 \beta_{14} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5328 \beta_{17} + 5328 \beta_{16} + 10736 \beta_{15} + 10736 \beta_{14} + 14816 \beta_{7} + \cdots - 1001087 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48887 \beta_{19} + 48739 \beta_{18} + 605604 \beta_{17} + 605604 \beta_{16} + 196481 \beta_{15} + \cdots - 74 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1017008 \beta_{17} - 1017008 \beta_{16} - 2406866 \beta_{15} - 2406866 \beta_{14} - 3220656 \beta_{7} + \cdots + 183380074 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9977337 \beta_{19} - 10056555 \beta_{18} - 124428192 \beta_{17} - 124428192 \beta_{16} - 41728841 \beta_{15} + \cdots - 39609 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 194320544 \beta_{17} + 194320544 \beta_{16} + 505490668 \beta_{15} + 505490668 \beta_{14} + \cdots - 35657245863 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2020189851 \beta_{19} + 2043662959 \beta_{18} + 25316012476 \beta_{17} + 25316012476 \beta_{16} + \cdots + 11736554 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 37937003720 \beta_{17} - 37937003720 \beta_{16} - 103765299926 \beta_{15} - 103765299926 \beta_{14} + \cdots + 7100591544598 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 408309806037 \beta_{19} - 413413008887 \beta_{18} - 5131674397904 \beta_{17} - 5131674397904 \beta_{16} + \cdots - 2551601425 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7529701384176 \beta_{17} + 7529701384176 \beta_{16} + 21113570009336 \beta_{15} + \cdots - 14\!\cdots\!55 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 82515536771591 \beta_{19} + 83549431428307 \beta_{18} + \cdots + 516947328358 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 15\!\cdots\!60 \beta_{17} + \cdots + 28\!\cdots\!74 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 16\!\cdots\!49 \beta_{19} + \cdots - 103328073331065 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 30\!\cdots\!64 \beta_{17} + \cdots - 58\!\cdots\!79 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 33\!\cdots\!75 \beta_{19} + \cdots + 20\!\cdots\!30 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
1.82933i
8.33786i
− 4.72650i
14.2202i
− 1.20516i
8.52898i
− 6.97456i
9.33135i
− 1.83079i
3.64885i
− 3.64885i
1.83079i
− 9.33135i
6.97456i
− 8.52898i
1.20516i
− 14.2202i
4.72650i
− 8.33786i
− 1.82933i
− 5.08360i − 3.25427i −17.8429 − 17.0127i −16.5434 13.6486 + 12.5186i 50.0376i 16.4097 −86.4856
76.2 − 5.08360i 3.25427i −17.8429 17.0127i 16.5434 −13.6486 + 12.5186i 50.0376i 16.4097 86.4856
76.3 − 4.74684i − 9.47334i −14.5325 16.6325i −44.9684 9.96355 − 15.6118i 31.0088i −62.7441 78.9516
76.4 − 4.74684i 9.47334i −14.5325 − 16.6325i 44.9684 −9.96355 − 15.6118i 31.0088i −62.7441 −78.9516
76.5 − 3.66191i − 4.86707i −5.40959 − 5.34505i −17.8228 −18.2546 + 3.12554i − 9.48586i 3.31167 −19.5731
76.6 − 3.66191i 4.86707i −5.40959 5.34505i 17.8228 18.2546 + 3.12554i − 9.48586i 3.31167 19.5731
76.7 − 1.17840i − 8.15296i 6.61138 − 5.71956i −9.60741 17.7500 + 5.28544i − 17.2180i −39.4707 −6.73991
76.8 − 1.17840i 8.15296i 6.61138 5.71956i 9.60741 −17.7500 + 5.28544i − 17.2180i −39.4707 6.73991
76.9 − 0.909032i − 2.73982i 7.17366 17.6818i −2.49058 6.41465 + 17.3739i − 13.7933i 19.4934 16.0734
76.10 − 0.909032i 2.73982i 7.17366 − 17.6818i 2.49058 −6.41465 + 17.3739i − 13.7933i 19.4934 −16.0734
76.11 0.909032i − 2.73982i 7.17366 17.6818i 2.49058 −6.41465 − 17.3739i 13.7933i 19.4934 −16.0734
76.12 0.909032i 2.73982i 7.17366 − 17.6818i −2.49058 6.41465 − 17.3739i 13.7933i 19.4934 16.0734
76.13 1.17840i − 8.15296i 6.61138 − 5.71956i 9.60741 −17.7500 − 5.28544i 17.2180i −39.4707 6.73991
76.14 1.17840i 8.15296i 6.61138 5.71956i −9.60741 17.7500 − 5.28544i 17.2180i −39.4707 −6.73991
76.15 3.66191i − 4.86707i −5.40959 − 5.34505i 17.8228 18.2546 − 3.12554i 9.48586i 3.31167 19.5731
76.16 3.66191i 4.86707i −5.40959 5.34505i −17.8228 −18.2546 − 3.12554i 9.48586i 3.31167 −19.5731
76.17 4.74684i − 9.47334i −14.5325 16.6325i 44.9684 −9.96355 + 15.6118i − 31.0088i −62.7441 −78.9516
76.18 4.74684i 9.47334i −14.5325 − 16.6325i −44.9684 9.96355 + 15.6118i − 31.0088i −62.7441 78.9516
76.19 5.08360i − 3.25427i −17.8429 − 17.0127i 16.5434 −13.6486 − 12.5186i − 50.0376i 16.4097 86.4856
76.20 5.08360i 3.25427i −17.8429 17.0127i −16.5434 13.6486 − 12.5186i − 50.0376i 16.4097 −86.4856
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.4.b.b ✓ 20
7.b odd 2 1 inner 77.4.b.b ✓ 20
11.b odd 2 1 inner 77.4.b.b ✓ 20
77.b even 2 1 inner 77.4.b.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.b.b ✓ 20 1.a even 1 1 trivial
77.4.b.b ✓ 20 7.b odd 2 1 inner
77.4.b.b ✓ 20 11.b odd 2 1 inner
77.4.b.b ✓ 20 77.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 64T_{2}^{8} + 1369T_{2}^{6} + 10606T_{2}^{4} + 18708T_{2}^{2} + 8960 \) acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 64 T^{8} + \cdots + 8960)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} + 198 T^{8} + \cdots + 11233640)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + 940 T^{8} + \cdots + 23396051360)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 41\!\cdots\!51)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 60\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 46\!\cdots\!04)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{5} - 90 T^{4} + \cdots - 948911040)^{4} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 55\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 42\!\cdots\!40)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 320 T^{4} + \cdots + 34705363120)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 4705615918080)^{4} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 40\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 1008 T^{4} + \cdots - 1698798640)^{4} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 21552062376096)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 46\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 83\!\cdots\!60)^{2} \) Copy content Toggle raw display
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