Properties

Label 237.2.e.a
Level $237$
Weight $2$
Character orbit 237.e
Analytic conductor $1.892$
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] = [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: \(12\)
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: \( x^{12} - 2x^{11} + 11x^{10} + 49x^{8} + 5x^{7} + 137x^{6} + 76x^{5} + 189x^{4} + 14x^{3} + 46x^{2} + 6x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1 + 1) q^{2} + (\beta_{4} + 1) q^{3} + ( - \beta_{8} + \beta_{6} + 2 \beta_{4} - \beta_1) q^{4} - \beta_{10} q^{5} + (\beta_{6} + \beta_{4} - \beta_1) q^{6} + (\beta_{10} + \beta_{9}) q^{7} + ( - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} - 2) q^{8} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1 + 1) q^{2} + (\beta_{4} + 1) q^{3} + ( - \beta_{8} + \beta_{6} + 2 \beta_{4} - \beta_1) q^{4} - \beta_{10} q^{5} + (\beta_{6} + \beta_{4} - \beta_1) q^{6} + (\beta_{10} + \beta_{9}) q^{7} + ( - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} - 2) q^{8} + \beta_{4} q^{9} + ( - 2 \beta_{10} - \beta_{9} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 1) q^{10} + ( - \beta_{11} - \beta_{6} + \beta_1) q^{11} + (\beta_{6} + \beta_{3} - 2) q^{12} + ( - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{13} + ( - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{14}+ \cdots + ( - \beta_{7} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 6 q^{3} - 8 q^{4} + q^{5} - 4 q^{6} - 18 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 6 q^{3} - 8 q^{4} + q^{5} - 4 q^{6} - 18 q^{8} - 6 q^{9} + 10 q^{10} - q^{11} - 16 q^{12} + 5 q^{13} + 4 q^{14} + 2 q^{15} - 8 q^{16} - 6 q^{17} - 8 q^{18} + 14 q^{19} - 2 q^{20} + 22 q^{22} + 12 q^{23} - 9 q^{24} - 3 q^{25} + 8 q^{26} - 12 q^{27} + 5 q^{28} - 7 q^{29} + 5 q^{30} + q^{31} + 17 q^{32} - 2 q^{33} - 8 q^{34} + 17 q^{35} - 8 q^{36} - 16 q^{37} + 66 q^{38} - 5 q^{39} - 14 q^{40} - 32 q^{41} + 2 q^{42} - 13 q^{43} + q^{45} + 12 q^{46} - 10 q^{47} + 8 q^{48} - 4 q^{49} + 23 q^{50} - 3 q^{51} - 44 q^{52} - 5 q^{53} - 4 q^{54} + 3 q^{55} - 34 q^{56} + 28 q^{57} + 2 q^{58} + 14 q^{59} + 2 q^{60} - 18 q^{61} - 28 q^{62} + 6 q^{64} - 28 q^{65} + 11 q^{66} - 48 q^{67} + 31 q^{68} + 24 q^{69} - 60 q^{70} + 44 q^{71} + 9 q^{72} - 14 q^{73} + 20 q^{74} + 3 q^{75} + 68 q^{76} - 16 q^{77} + 16 q^{78} + 8 q^{79} - 22 q^{80} - 6 q^{81} - 15 q^{82} - 12 q^{83} - 5 q^{84} + 10 q^{85} + 26 q^{86} - 14 q^{87} - 13 q^{88} + 10 q^{89} - 5 q^{90} + 34 q^{91} + 11 q^{92} + 2 q^{93} + 18 q^{94} - 22 q^{95} + 34 q^{96} - 38 q^{97} + 54 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 11x^{10} + 49x^{8} + 5x^{7} + 137x^{6} + 76x^{5} + 189x^{4} + 14x^{3} + 46x^{2} + 6x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5295523 \nu^{11} + 13452272 \nu^{10} - 22915263 \nu^{9} + 345399173 \nu^{8} - 114273838 \nu^{7} + 1280218292 \nu^{6} - 434295829 \nu^{5} + \cdots + 575507286 ) / 468655901 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8262330 \nu^{11} + 12040263 \nu^{10} - 83511323 \nu^{9} - 40560107 \nu^{8} - 429951151 \nu^{7} - 212803159 \nu^{6} - 1200935436 \nu^{5} + \cdots + 1223523408 ) / 468655901 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27445483 \nu^{11} + 88260248 \nu^{10} - 360056585 \nu^{9} + 345026619 \nu^{8} - 1250336017 \nu^{7} + 1470679732 \nu^{6} + \cdots + 96593619 ) / 1405967703 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32855483 \nu^{11} - 132362092 \nu^{10} + 503940199 \nu^{9} - 738156510 \nu^{8} + 1668143813 \nu^{7} - 2994838769 \nu^{6} + \cdots - 2481716196 ) / 1405967703 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11123094 \nu^{11} + 19385424 \nu^{10} - 115008873 \nu^{9} - 31497550 \nu^{8} - 535969049 \nu^{7} - 161633368 \nu^{6} - 1472694087 \nu^{5} + \cdots - 82336449 ) / 468655901 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43756787 \nu^{11} - 110676361 \nu^{10} + 518198233 \nu^{9} - 301054383 \nu^{8} + 2135773451 \nu^{7} - 1399337192 \nu^{6} + \cdots - 1872430146 ) / 1405967703 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16322389 \nu^{11} + 68874824 \nu^{10} - 245047712 \nu^{9} + 376524169 \nu^{8} - 714366968 \nu^{7} + 1632313100 \nu^{6} - 1802436980 \nu^{5} + \cdots + 178930068 ) / 468655901 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 150127913 \nu^{11} + 350901874 \nu^{10} - 1728005356 \nu^{9} + 509053902 \nu^{8} - 7127972642 \nu^{7} + 1847157203 \nu^{6} + \cdots - 470600175 ) / 1405967703 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 66912061 \nu^{11} - 158678566 \nu^{10} + 796372108 \nu^{9} - 281923803 \nu^{8} + 3332143262 \nu^{7} - 633415025 \nu^{6} + 9115512771 \nu^{5} + \cdots + 255939213 ) / 468655901 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 207104953 \nu^{11} - 314859584 \nu^{10} + 2061015383 \nu^{9} + 1096305378 \nu^{8} + 10109466199 \nu^{7} + 5229304208 \nu^{6} + \cdots + 1893702039 ) / 1405967703 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - \beta_{6} + 3\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 5\beta_{6} + \beta_{5} + 2\beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{8} + \beta_{7} + 3\beta_{5} - 19\beta_{4} + 9\beta_{3} - 10\beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} - \beta_{10} - 13\beta_{9} + 25\beta_{8} + 35\beta_{6} - 46\beta_{4} - 35\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{11} - 2\beta_{10} - 40\beta_{9} - 13\beta_{7} + 93\beta_{6} - 40\beta_{5} - 85\beta_{3} - 2\beta_{2} + 158 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -256\beta_{8} - 40\beta_{7} - 136\beta_{5} + 455\beta_{4} - 256\beta_{3} - 11\beta_{2} + 296\beta _1 + 455 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 136\beta_{11} + 29\beta_{10} + 421\beta_{9} - 813\beta_{8} - 871\beta_{6} + 1445\beta_{4} + 871\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 421 \beta_{11} + 107 \beta_{10} + 1341 \beta_{9} + 421 \beta_{7} - 2708 \beta_{6} + 1341 \beta_{5} + 2497 \beta_{3} + 107 \beta_{2} - 4375 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7780 \beta_{8} + 1341 \beta_{7} + 4152 \beta_{5} - 13617 \beta_{4} + 7780 \beta_{3} + 314 \beta_{2} - 8239 \beta _1 - 13617 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4152 \beta_{11} - 1027 \beta_{10} - 12959 \beta_{9} + 24009 \beta_{8} + 25484 \beta_{6} - 41828 \beta_{4} - 25484 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(80\) \(82\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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
1.54518 2.67634i
0.928387 1.60801i
0.254077 0.440074i
−0.233732 + 0.404836i
−0.659142 + 1.14167i
−0.834775 + 1.44587i
1.54518 + 2.67634i
0.928387 + 1.60801i
0.254077 + 0.440074i
−0.233732 0.404836i
−0.659142 1.14167i
−0.834775 1.44587i
−1.04518 + 1.81031i 0.500000 0.866025i −1.18482 2.05217i −0.929299 1.60959i 1.04518 + 1.81031i −0.641869 1.11175i 0.772692 −0.500000 0.866025i 3.88515
55.2 −0.428387 + 0.741988i 0.500000 0.866025i 0.632970 + 1.09634i 0.782924 + 1.35606i 0.428387 + 0.741988i 1.24913 + 2.16356i −2.79817 −0.500000 0.866025i −1.34158
55.3 0.245923 0.425951i 0.500000 0.866025i 0.879044 + 1.52255i 0.715982 + 1.24012i −0.245923 0.425951i −0.761138 1.31833i 1.84840 −0.500000 0.866025i 0.704305
55.4 0.733732 1.27086i 0.500000 0.866025i −0.0767255 0.132892i −1.45453 2.51933i −0.733732 1.27086i 0.0229355 + 0.0397254i 2.70974 −0.500000 0.866025i −4.26895
55.5 1.15914 2.00769i 0.500000 0.866025i −1.68722 2.92235i 1.95463 + 3.38552i −1.15914 2.00769i −2.04868 3.54841i −3.18634 −0.500000 0.866025i 9.06277
55.6 1.33477 2.31190i 0.500000 0.866025i −2.56325 4.43967i −0.569703 0.986755i −1.33477 2.31190i 2.17962 + 3.77521i −8.34632 −0.500000 0.866025i −3.04170
181.1 −1.04518 1.81031i 0.500000 + 0.866025i −1.18482 + 2.05217i −0.929299 + 1.60959i 1.04518 1.81031i −0.641869 + 1.11175i 0.772692 −0.500000 + 0.866025i 3.88515
181.2 −0.428387 0.741988i 0.500000 + 0.866025i 0.632970 1.09634i 0.782924 1.35606i 0.428387 0.741988i 1.24913 2.16356i −2.79817 −0.500000 + 0.866025i −1.34158
181.3 0.245923 + 0.425951i 0.500000 + 0.866025i 0.879044 1.52255i 0.715982 1.24012i −0.245923 + 0.425951i −0.761138 + 1.31833i 1.84840 −0.500000 + 0.866025i 0.704305
181.4 0.733732 + 1.27086i 0.500000 + 0.866025i −0.0767255 + 0.132892i −1.45453 + 2.51933i −0.733732 + 1.27086i 0.0229355 0.0397254i 2.70974 −0.500000 + 0.866025i −4.26895
181.5 1.15914 + 2.00769i 0.500000 + 0.866025i −1.68722 + 2.92235i 1.95463 3.38552i −1.15914 + 2.00769i −2.04868 + 3.54841i −3.18634 −0.500000 + 0.866025i 9.06277
181.6 1.33477 + 2.31190i 0.500000 + 0.866025i −2.56325 + 4.43967i −0.569703 + 0.986755i −1.33477 + 2.31190i 2.17962 3.77521i −8.34632 −0.500000 + 0.866025i −3.04170
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.6
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.a 12
3.b odd 2 1 711.2.f.b 12
79.c even 3 1 inner 237.2.e.a 12
237.g odd 6 1 711.2.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.2.e.a 12 1.a even 1 1 trivial
237.2.e.a 12 79.c even 3 1 inner
711.2.f.b 12 3.b odd 2 1
711.2.f.b 12 237.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 4 T_{2}^{11} + 18 T_{2}^{10} - 34 T_{2}^{9} + 100 T_{2}^{8} - 153 T_{2}^{7} + 373 T_{2}^{6} - 344 T_{2}^{5} + 475 T_{2}^{4} - 156 T_{2}^{3} + 321 T_{2}^{2} - 120 T_{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(237, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{11} + 18 T^{10} - 34 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + 17 T^{10} + 6 T^{9} + \cdots + 2916 \) Copy content Toggle raw display
$7$ \( T^{12} + 23 T^{10} + 16 T^{9} + 437 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + 28 T^{10} + \cdots + 169744 \) Copy content Toggle raw display
$13$ \( T^{12} - 5 T^{11} + 50 T^{10} + \cdots + 522729 \) Copy content Toggle raw display
$17$ \( (T^{6} + 3 T^{5} - 29 T^{4} - 31 T^{3} + \cdots - 378)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{11} + 185 T^{10} + \cdots + 71824 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + 107 T^{10} + \cdots + 26244 \) Copy content Toggle raw display
$29$ \( T^{12} + 7 T^{11} + 53 T^{10} + 110 T^{9} + \cdots + 324 \) Copy content Toggle raw display
$31$ \( T^{12} - T^{11} + 78 T^{10} + \cdots + 2181529 \) Copy content Toggle raw display
$37$ \( T^{12} + 16 T^{11} + \cdots + 119180889 \) Copy content Toggle raw display
$41$ \( (T^{6} + 16 T^{5} + 45 T^{4} - 97 T^{3} + \cdots + 224)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 13 T^{11} + 152 T^{10} + \cdots + 37249 \) Copy content Toggle raw display
$47$ \( T^{12} + 10 T^{11} + 151 T^{10} + \cdots + 7054336 \) Copy content Toggle raw display
$53$ \( T^{12} + 5 T^{11} + 212 T^{10} + \cdots + 173343556 \) Copy content Toggle raw display
$59$ \( T^{12} - 14 T^{11} + 218 T^{10} + \cdots + 5531904 \) Copy content Toggle raw display
$61$ \( (T^{6} + 9 T^{5} - 99 T^{4} - 715 T^{3} + \cdots - 58554)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 24 T^{5} + 100 T^{4} + \cdots - 66393)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 22 T^{5} - 23 T^{4} + \cdots + 321198)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 14 T^{11} + \cdots + 579172356 \) Copy content Toggle raw display
$79$ \( T^{12} - 8 T^{11} + \cdots + 243087455521 \) Copy content Toggle raw display
$83$ \( T^{12} + 12 T^{11} + \cdots + 1622317284 \) Copy content Toggle raw display
$89$ \( (T^{6} - 5 T^{5} - 189 T^{4} + 1801 T^{3} + \cdots + 7078)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 19 T^{5} - 4 T^{4} - 1408 T^{3} + \cdots + 7168)^{2} \) Copy content Toggle raw display
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