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Properties

Label 237.2.e.b
Level $237$
Weight $2$
Character orbit 237.e
Analytic conductor $1.892$
Analytic rank $0$
Dimension $14$
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] = [237,2,Mod(55,237)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(237, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("237.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 237 = 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 237.e (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: \(1.89245452790\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 12 x^{12} - 10 x^{11} + 72 x^{10} - 55 x^{9} + 251 x^{8} - 110 x^{7} + 527 x^{6} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{11} - \beta_{9} + \cdots - \beta_{4}) q^{4}+ \cdots - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{11} - \beta_{9} + \cdots - \beta_{4}) q^{4}+ \cdots + ( - \beta_{3} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 7 q^{3} - 6 q^{4} - q^{5} - 2 q^{6} - 4 q^{7} + 18 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 7 q^{3} - 6 q^{4} - q^{5} - 2 q^{6} - 4 q^{7} + 18 q^{8} - 7 q^{9} + 10 q^{10} + q^{11} + 12 q^{12} - 8 q^{14} + 2 q^{15} - 4 q^{16} + 2 q^{17} + 4 q^{18} - 10 q^{19} - 8 q^{20} + 8 q^{21} + 14 q^{22} - 2 q^{23} - 9 q^{24} - 10 q^{25} - 8 q^{26} + 14 q^{27} + q^{28} - 5 q^{29} - 5 q^{30} + 8 q^{31} - 35 q^{32} - 2 q^{33} - 20 q^{34} - 29 q^{35} - 6 q^{36} - 7 q^{37} + 62 q^{38} - 14 q^{40} + 8 q^{41} + 4 q^{42} + 14 q^{43} + 12 q^{44} - q^{45} + 20 q^{46} + 10 q^{47} - 4 q^{48} - 13 q^{49} + 7 q^{50} - q^{51} - 48 q^{52} - 21 q^{53} - 2 q^{54} + 3 q^{55} - 16 q^{56} + 20 q^{57} - 22 q^{58} - 16 q^{59} - 8 q^{60} + 50 q^{61} - 24 q^{62} - 4 q^{63} + 78 q^{64} + 24 q^{65} - 7 q^{66} + 46 q^{67} + 9 q^{68} + 4 q^{69} + 12 q^{70} - 28 q^{71} - 9 q^{72} - 32 q^{73} + 8 q^{74} - 10 q^{75} - 36 q^{76} + 16 q^{77} + 16 q^{78} - 11 q^{79} + 18 q^{80} - 7 q^{81} + 29 q^{82} - 10 q^{83} + q^{84} - 6 q^{85} + 56 q^{86} + 10 q^{87} + 43 q^{88} + 26 q^{89} - 5 q^{90} - 54 q^{91} - 35 q^{92} - 16 q^{93} - 46 q^{94} + 6 q^{95} + 70 q^{96} + 4 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 12 x^{12} - 10 x^{11} + 72 x^{10} - 55 x^{9} + 251 x^{8} - 110 x^{7} + 527 x^{6} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21271857541 \nu^{13} + 20704581534 \nu^{12} + 104734050900 \nu^{11} + 377731195486 \nu^{10} + \cdots - 3131573433696 ) / 9804095041232 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24835709545 \nu^{13} - 127632461026 \nu^{12} + 6142341148 \nu^{11} - 1564467930614 \nu^{10} + \cdots - 7636235056832 ) / 9804095041232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20847675275 \nu^{13} + 56126912814 \nu^{12} - 276355024300 \nu^{11} + \cdots + 2159473262784 ) / 4902047520616 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25489016819 \nu^{13} - 86501600121 \nu^{12} - 111389695882 \nu^{11} + \cdots - 15098097609352 ) / 4902047520616 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51289625765 \nu^{13} + 92808736802 \nu^{12} - 690922593452 \nu^{11} + \cdots + 7568400302912 ) / 9804095041232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33741769731 \nu^{13} + 46635864187 \nu^{12} - 348774323958 \nu^{11} + 61062673010 \nu^{10} + \cdots + 55977693184 ) / 4902047520616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8740278407 \nu^{13} + 14117913784 \nu^{12} + 71886646115 \nu^{11} + 241670665522 \nu^{10} + \cdots + 4170090831584 ) / 1225511880154 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35279237539 \nu^{13} - 82309833814 \nu^{12} + 428518447500 \nu^{11} - 445616816606 \nu^{10} + \cdots - 15531364607032 ) / 4902047520616 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13487563681 \nu^{13} - 35392017772 \nu^{12} + 165040600681 \nu^{11} - 203376072684 \nu^{10} + \cdots + 512591051776 ) / 1225511880154 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 86793746929 \nu^{13} + 113724671561 \nu^{12} - 894159548674 \nu^{11} + \cdots - 13203958264696 ) / 4902047520616 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 435279367113 \nu^{13} + 619538266394 \nu^{12} - 4703492440284 \nu^{11} + \cdots + 8244308376640 ) / 9804095041232 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 477123133845 \nu^{13} - 799545128324 \nu^{12} + 5405747649808 \nu^{11} + \cdots - 19580842893248 ) / 9804095041232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} - 3\beta_{7} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - \beta_{9} + \beta_{6} - 5\beta_{4} + \beta_{2} - 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{11} - 2\beta_{10} + 14\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} - 8\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{13} - \beta_{12} - 12\beta_{11} + 12\beta_{9} + 15\beta_{7} - 9\beta_{6} + 29\beta_{4} + 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{13} - 2 \beta_{12} + 23 \beta_{10} + 49 \beta_{9} - 2 \beta_{8} - 23 \beta_{6} - 12 \beta_{5} + \cdots + 81 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 108 \beta_{11} + 74 \beta_{10} - 12 \beta_{8} - 143 \beta_{7} - 35 \beta_{5} - 47 \beta_{3} - 47 \beta_{2} + \cdots + 143 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 109 \beta_{13} + 27 \beta_{12} + 357 \beta_{11} - 357 \beta_{9} - 534 \beta_{7} + 202 \beta_{6} + \cdots - 96 \beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 311 \beta_{13} + 106 \beta_{12} - 589 \beta_{10} - 886 \beta_{9} + 106 \beta_{8} + 589 \beta_{6} + \cdots - 1195 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2666 \beta_{11} - 1633 \beta_{10} + 259 \beta_{8} + 3794 \beta_{7} + 900 \beta_{5} + 780 \beta_{3} + \cdots - 3794 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2533 \beta_{13} - 853 \beta_{12} - 7002 \beta_{11} + 7002 \beta_{9} + 9503 \beta_{7} + \cdots + 2407 \beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7138 \beta_{13} - 2198 \beta_{12} + 12795 \beta_{10} + 20168 \beta_{9} - 2198 \beta_{8} + \cdots + 28028 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 54429 \beta_{11} + 35653 \beta_{10} - 6646 \beta_{8} - 74026 \beta_{7} - 19933 \beta_{5} - 17970 \beta_{3} + \cdots + 74026 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/237\mathbb{Z}\right)^\times\).

\(n\) \(80\) \(82\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
1.38396 − 2.39708i
0.947806 − 1.64165i
0.555519 − 0.962187i
0.291179 − 0.504336i
−0.355822 + 0.616302i
−0.859718 + 1.48908i
−0.962920 + 1.66783i
1.38396 + 2.39708i
0.947806 + 1.64165i
0.555519 + 0.962187i
0.291179 + 0.504336i
−0.355822 − 0.616302i
−0.859718 − 1.48908i
−0.962920 − 1.66783i
−1.38396 + 2.39708i −0.500000 + 0.866025i −2.83068 − 4.90288i −0.681396 − 1.18021i −1.38396 − 2.39708i −0.872527 − 1.51126i 10.1343 −0.500000 − 0.866025i 3.77209
55.2 −0.947806 + 1.64165i −0.500000 + 0.866025i −0.796671 − 1.37987i 0.312321 + 0.540957i −0.947806 − 1.64165i 2.33497 + 4.04428i −0.770866 −0.500000 − 0.866025i −1.18408
55.3 −0.555519 + 0.962187i −0.500000 + 0.866025i 0.382797 + 0.663025i −1.90100 − 3.29263i −0.555519 − 0.962187i −1.43654 − 2.48815i −3.07268 −0.500000 − 0.866025i 4.22416
55.4 −0.291179 + 0.504336i −0.500000 + 0.866025i 0.830430 + 1.43835i 1.84000 + 3.18697i −0.291179 − 0.504336i −0.776427 − 1.34481i −2.13193 −0.500000 − 0.866025i −2.14307
55.5 0.355822 − 0.616302i −0.500000 + 0.866025i 0.746781 + 1.29346i −0.0113917 − 0.0197309i 0.355822 + 0.616302i 0.0974861 + 0.168851i 2.48617 −0.500000 − 0.866025i −0.0162136
55.6 0.859718 − 1.48908i −0.500000 + 0.866025i −0.478230 − 0.828319i −1.38701 − 2.40237i 0.859718 + 1.48908i −2.36219 − 4.09144i 1.79430 −0.500000 − 0.866025i −4.76975
55.7 0.962920 − 1.66783i −0.500000 + 0.866025i −0.854431 − 1.47992i 1.32847 + 2.30098i 0.962920 + 1.66783i 1.01523 + 1.75843i 0.560686 −0.500000 − 0.866025i 5.11686
181.1 −1.38396 − 2.39708i −0.500000 − 0.866025i −2.83068 + 4.90288i −0.681396 + 1.18021i −1.38396 + 2.39708i −0.872527 + 1.51126i 10.1343 −0.500000 + 0.866025i 3.77209
181.2 −0.947806 − 1.64165i −0.500000 − 0.866025i −0.796671 + 1.37987i 0.312321 − 0.540957i −0.947806 + 1.64165i 2.33497 − 4.04428i −0.770866 −0.500000 + 0.866025i −1.18408
181.3 −0.555519 − 0.962187i −0.500000 − 0.866025i 0.382797 − 0.663025i −1.90100 + 3.29263i −0.555519 + 0.962187i −1.43654 + 2.48815i −3.07268 −0.500000 + 0.866025i 4.22416
181.4 −0.291179 − 0.504336i −0.500000 − 0.866025i 0.830430 − 1.43835i 1.84000 − 3.18697i −0.291179 + 0.504336i −0.776427 + 1.34481i −2.13193 −0.500000 + 0.866025i −2.14307
181.5 0.355822 + 0.616302i −0.500000 − 0.866025i 0.746781 − 1.29346i −0.0113917 + 0.0197309i 0.355822 − 0.616302i 0.0974861 − 0.168851i 2.48617 −0.500000 + 0.866025i −0.0162136
181.6 0.859718 + 1.48908i −0.500000 − 0.866025i −0.478230 + 0.828319i −1.38701 + 2.40237i 0.859718 − 1.48908i −2.36219 + 4.09144i 1.79430 −0.500000 + 0.866025i −4.76975
181.7 0.962920 + 1.66783i −0.500000 − 0.866025i −0.854431 + 1.47992i 1.32847 − 2.30098i 0.962920 − 1.66783i 1.01523 − 1.75843i 0.560686 −0.500000 + 0.866025i 5.11686
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 237.2.e.b ✓ 14
3.b odd 2 1 711.2.f.d 14
79.c even 3 1 inner 237.2.e.b ✓ 14
237.g odd 6 1 711.2.f.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.2.e.b ✓ 14 1.a even 1 1 trivial
237.2.e.b ✓ 14 79.c even 3 1 inner
711.2.f.d 14 3.b odd 2 1
711.2.f.d 14 237.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 2 T_{2}^{13} + 12 T_{2}^{12} + 10 T_{2}^{11} + 72 T_{2}^{10} + 55 T_{2}^{9} + 251 T_{2}^{8} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(237, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + T^{13} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{14} + 4 T^{13} + \cdots + 4624 \) Copy content Toggle raw display
$11$ \( T^{14} - T^{13} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( T^{14} + 46 T^{12} + \cdots + 2879809 \) Copy content Toggle raw display
$17$ \( (T^{7} - T^{6} - 67 T^{5} + \cdots + 6178)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 10 T^{13} + \cdots + 183184 \) Copy content Toggle raw display
$23$ \( T^{14} + 2 T^{13} + \cdots + 2704 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 76422390916 \) Copy content Toggle raw display
$31$ \( T^{14} - 8 T^{13} + \cdots + 64009 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1055535121 \) Copy content Toggle raw display
$41$ \( (T^{7} - 4 T^{6} + \cdots + 7154)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 3423771169 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 25490038336 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 320675303524 \) Copy content Toggle raw display
$59$ \( T^{14} + 16 T^{13} + \cdots + 174724 \) Copy content Toggle raw display
$61$ \( (T^{7} - 25 T^{6} + \cdots - 47272)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} - 23 T^{6} + \cdots + 78799)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + 14 T^{6} + \cdots - 145378)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 330220323904 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 19203908986159 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 12598715536 \) Copy content Toggle raw display
$89$ \( (T^{7} - 13 T^{6} + \cdots + 996028)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} - 2 T^{6} + \cdots + 177152)^{2} \) Copy content Toggle raw display
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