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Properties

Label 185.2.e.a
Level $185$
Weight $2$
Character orbit 185.e
Analytic conductor $1.477$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(26,185)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("185.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 11 x^{12} - 2 x^{11} + 86 x^{10} - 18 x^{9} + 332 x^{8} - 110 x^{7} + 935 x^{6} - 290 x^{5} + \cdots + 81 \) 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{2} - \beta_{6} q^{3} + ( - \beta_{11} + \beta_{9} + \beta_{2}) q^{4} + \beta_{9} q^{5} + (\beta_{10} + \beta_{8} + \beta_{4}) q^{6} + \beta_{12} q^{7} + (\beta_{7} - \beta_{4}) q^{8}+ \cdots + (3 \beta_{12} + 10 \beta_{9} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{3} - 8 q^{4} - 7 q^{5} - 4 q^{6} + 6 q^{8} - 13 q^{9} - 2 q^{11} + 6 q^{12} + 4 q^{13} + 20 q^{14} - 2 q^{15} + 2 q^{16} - 3 q^{17} - 6 q^{18} - 14 q^{19} - 8 q^{20} + q^{21} + 7 q^{22} + 15 q^{24}+ \cdots + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 11 x^{12} - 2 x^{11} + 86 x^{10} - 18 x^{9} + 332 x^{8} - 110 x^{7} + 935 x^{6} - 290 x^{5} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2670419840 \nu^{13} + 16828949606 \nu^{12} - 28279168338 \nu^{11} + \cdots + 31759807309383 ) / 10453756973887 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 80473223464 \nu^{13} + 844809598044 \nu^{12} - 1122751342940 \nu^{11} + \cdots + 81063179735616 ) / 282251438294949 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 44281820858 \nu^{13} + 24033778560 \nu^{12} + 335639482984 \nu^{11} + 165948873326 \nu^{10} + \cdots + 32248507246209 ) / 94083812764983 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 69948557836 \nu^{13} - 110309843682 \nu^{12} - 183603049745 \nu^{11} + \cdots - 159295800167685 ) / 94083812764983 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23755117228 \nu^{13} - 177567239937 \nu^{12} + 256491222628 \nu^{11} - 1922434962752 \nu^{10} + \cdots - 4252568622777 ) / 31361270921661 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 221409104290 \nu^{13} + 120168892800 \nu^{12} + 1678197414920 \nu^{11} + \cdots + 161242536231045 ) / 94083812764983 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1014885940486 \nu^{13} - 2768279154039 \nu^{12} + 9319950683612 \nu^{11} + \cdots - 11\!\cdots\!95 ) / 282251438294949 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 398129719089 \nu^{13} + 44281820858 \nu^{12} - 4355393131419 \nu^{11} + \cdots - 3804839669664 ) / 94083812764983 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1901026964449 \nu^{13} + 2415296331603 \nu^{12} - 16090713026354 \nu^{11} + \cdots + 10\!\cdots\!40 ) / 282251438294949 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 398129719089 \nu^{13} + 44281820858 \nu^{12} - 4355393131419 \nu^{11} + \cdots + 90278973095319 ) / 31361270921661 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1623720025947 \nu^{13} + 1210196845127 \nu^{12} + 17985822533784 \nu^{11} + \cdots + 51356133090279 ) / 94083812764983 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1905323225295 \nu^{13} - 1131014883515 \nu^{12} - 21689315420742 \nu^{11} + \cdots - 53528910123807 ) / 94083812764983 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 3\beta_{9} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - 6\beta_{11} + 13\beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + 6\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + 7\beta_{5} + 25\beta_{4} - 25\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{8} + 9\beta_{7} - 9\beta_{4} - 11\beta_{3} - 34\beta_{2} + 62 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{13} + 10 \beta_{12} - 10 \beta_{11} + \beta_{9} - 43 \beta_{7} - \beta_{6} - 43 \beta_{5} + \cdots + 128 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{13} + 52 \beta_{12} + 192 \beta_{11} - \beta_{10} - 308 \beta_{9} + 52 \beta_{8} + 86 \beta_{6} + \cdots - 308 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 86\beta_{10} + 74\beta_{8} + 257\beta_{7} - 670\beta_{4} - 15\beta_{3} - 76\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 15 \beta_{13} - 316 \beta_{12} - 1087 \beta_{11} + 1570 \beta_{9} - 419 \beta_{7} - 589 \beta_{6} + \cdots + 403 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 589 \beta_{13} - 486 \beta_{12} + 521 \beta_{11} - 589 \beta_{10} - 55 \beta_{9} - 486 \beta_{8} + \cdots - 55 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 148\beta_{10} - 1859\beta_{8} + 2631\beta_{7} - 2470\beta_{4} - 3774\beta_{3} - 6170\beta_{2} + 8166 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3774 \beta_{13} + 3002 \beta_{12} - 3390 \beta_{11} + 289 \beta_{9} - 8949 \beta_{7} + \cdots + 19363 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(\beta_{9}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.20043 + 2.07920i
−1.05891 + 1.83409i
−0.560420 + 0.970676i
0.178876 − 0.309823i
0.473849 − 0.820731i
0.985880 − 1.70759i
1.18115 − 2.04582i
−1.20043 − 2.07920i
−1.05891 − 1.83409i
−0.560420 − 0.970676i
0.178876 + 0.309823i
0.473849 + 0.820731i
0.985880 + 1.70759i
1.18115 + 2.04582i
−1.20043 − 2.07920i −0.362403 + 0.627700i −1.88204 + 3.25980i −0.500000 + 0.866025i 1.74015 −0.659966 + 1.14310i 4.23532 1.23733 + 2.14312i 2.40085
26.2 −1.05891 − 1.83409i 1.49035 − 2.58135i −1.24259 + 2.15223i −0.500000 + 0.866025i −6.31258 0.0794638 − 0.137635i 1.02753 −2.94226 − 5.09615i 2.11783
26.3 −0.560420 − 0.970676i −1.67000 + 2.89252i 0.371859 − 0.644078i −0.500000 + 0.866025i 3.74360 −0.612087 + 1.06017i −3.07527 −4.07778 − 7.06292i 1.12084
26.4 0.178876 + 0.309823i −0.0462746 + 0.0801499i 0.936006 − 1.62121i −0.500000 + 0.866025i −0.0331097 1.38534 − 2.39948i 1.38522 1.49572 + 2.59066i −0.357753
26.5 0.473849 + 0.820731i −0.650926 + 1.12744i 0.550934 − 0.954246i −0.500000 + 0.866025i −1.23376 −1.91182 + 3.31137i 2.93963 0.652591 + 1.13032i −0.947698
26.6 0.985880 + 1.70759i 1.32465 − 2.29437i −0.943918 + 1.63491i −0.500000 + 0.866025i 5.22380 −0.391800 + 0.678617i 0.221160 −2.00941 − 3.48041i −1.97176
26.7 1.18115 + 2.04582i −1.08540 + 1.87997i −1.79024 + 3.10080i −0.500000 + 0.866025i −5.12809 2.11087 − 3.65613i −3.73360 −0.856183 − 1.48295i −2.36231
121.1 −1.20043 + 2.07920i −0.362403 − 0.627700i −1.88204 − 3.25980i −0.500000 − 0.866025i 1.74015 −0.659966 − 1.14310i 4.23532 1.23733 − 2.14312i 2.40085
121.2 −1.05891 + 1.83409i 1.49035 + 2.58135i −1.24259 − 2.15223i −0.500000 − 0.866025i −6.31258 0.0794638 + 0.137635i 1.02753 −2.94226 + 5.09615i 2.11783
121.3 −0.560420 + 0.970676i −1.67000 − 2.89252i 0.371859 + 0.644078i −0.500000 − 0.866025i 3.74360 −0.612087 − 1.06017i −3.07527 −4.07778 + 7.06292i 1.12084
121.4 0.178876 − 0.309823i −0.0462746 − 0.0801499i 0.936006 + 1.62121i −0.500000 − 0.866025i −0.0331097 1.38534 + 2.39948i 1.38522 1.49572 − 2.59066i −0.357753
121.5 0.473849 − 0.820731i −0.650926 − 1.12744i 0.550934 + 0.954246i −0.500000 − 0.866025i −1.23376 −1.91182 − 3.31137i 2.93963 0.652591 − 1.13032i −0.947698
121.6 0.985880 − 1.70759i 1.32465 + 2.29437i −0.943918 − 1.63491i −0.500000 − 0.866025i 5.22380 −0.391800 − 0.678617i 0.221160 −2.00941 + 3.48041i −1.97176
121.7 1.18115 − 2.04582i −1.08540 − 1.87997i −1.79024 − 3.10080i −0.500000 − 0.866025i −5.12809 2.11087 + 3.65613i −3.73360 −0.856183 + 1.48295i −2.36231
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.e.a ✓ 14
5.b even 2 1 925.2.e.c 14
5.c odd 4 2 925.2.o.b 28
37.c even 3 1 inner 185.2.e.a ✓ 14
37.c even 3 1 6845.2.a.l 7
37.e even 6 1 6845.2.a.k 7
185.n even 6 1 925.2.e.c 14
185.s odd 12 2 925.2.o.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.e.a ✓ 14 1.a even 1 1 trivial
185.2.e.a ✓ 14 37.c even 3 1 inner
925.2.e.c 14 5.b even 2 1
925.2.e.c 14 185.n even 6 1
925.2.o.b 28 5.c odd 4 2
925.2.o.b 28 185.s odd 12 2
6845.2.a.k 7 37.e even 6 1
6845.2.a.l 7 37.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 11 T_{2}^{12} - 2 T_{2}^{11} + 86 T_{2}^{10} - 18 T_{2}^{9} + 332 T_{2}^{8} - 110 T_{2}^{7} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 11 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{14} + 2 T^{13} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + 22 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{7} + T^{6} - 29 T^{5} + \cdots - 321)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} - 4 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{14} + 3 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 1594963969 \) Copy content Toggle raw display
$23$ \( (T^{7} - 69 T^{5} + \cdots - 1695)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 6 T^{6} + \cdots + 1707)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 131 T^{5} + \cdots - 49465)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 94931877133 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 8431096041 \) Copy content Toggle raw display
$43$ \( (T^{7} - 19 T^{6} + \cdots + 72627)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} - 4 T^{6} + \cdots + 1558197)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 1010284306641 \) Copy content Toggle raw display
$59$ \( T^{14} - 2 T^{13} + \cdots + 205209 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 340660849 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 9134580625 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 995087025 \) Copy content Toggle raw display
$73$ \( (T^{7} - 228 T^{5} + \cdots - 144915)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 898031046025 \) Copy content Toggle raw display
$83$ \( T^{14} + 11 T^{13} + \cdots + 5890329 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 13477120281 \) Copy content Toggle raw display
$97$ \( (T^{7} - 15 T^{6} + \cdots + 270769)^{2} \) Copy content Toggle raw display
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