↑

Properties

Label 372.3.d.a
Level $372$
Weight $3$
Character orbit 372.d
Analytic conductor $10.136$
Analytic rank $0$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [372,3,Mod(125,372)] 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("372.125"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(372, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 372 = 2^{2} \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 372.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1362658342\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 12 x^{18} - 36 x^{17} + 123 x^{16} - 324 x^{15} + 416 x^{14} - 92 x^{13} + \cdots + 3486784401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{6} q^{5} + \beta_{5} q^{7} - \beta_{8} q^{9} - \beta_{18} q^{11} + (\beta_{4} + \beta_{3} + 2) q^{13} + (\beta_{7} - \beta_{2}) q^{15} + (\beta_{10} + \beta_{4}) q^{17} + ( - \beta_{19} - \beta_{18} - \beta_{17} + \cdots + 1) q^{19}+ \cdots + ( - 4 \beta_{18} + \beta_{17} + \cdots + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} - 8 q^{9} + 28 q^{13} + 2 q^{15} - 4 q^{19} + 38 q^{21} - 192 q^{25} + 28 q^{27} - 36 q^{33} + 16 q^{37} - 136 q^{39} + 148 q^{43} - 120 q^{45} + 244 q^{49} + 200 q^{51} - 100 q^{55} + 174 q^{57}+ \cdots + 208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 12 x^{18} - 36 x^{17} + 123 x^{16} - 324 x^{15} + 416 x^{14} - 92 x^{13} + \cdots + 3486784401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13297 \nu^{19} - 3506924 \nu^{18} + 16323693 \nu^{17} - 120996684 \nu^{16} + \cdots - 15\!\cdots\!32 ) / 23\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 906475 \nu^{19} - 1467436 \nu^{18} + 22615137 \nu^{17} - 69392304 \nu^{16} + \cdots - 16\!\cdots\!44 ) / 435997852714080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{19} - 4 \nu^{18} + 12 \nu^{17} - 36 \nu^{16} + 123 \nu^{15} - 324 \nu^{14} + \cdots - 1549681956 ) / 387420489 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31151575 \nu^{19} - 79098052 \nu^{18} + 117991749 \nu^{17} + 389908422 \nu^{16} + \cdots + 58\!\cdots\!02 ) / 58\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8814288 \nu^{19} - 19545679 \nu^{18} + 130391308 \nu^{17} - 379766937 \nu^{16} + \cdots - 45\!\cdots\!87 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26626108 \nu^{19} - 76435040 \nu^{18} + 96890775 \nu^{17} - 1833728319 \nu^{16} + \cdots - 17\!\cdots\!19 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4 \nu^{19} - 7 \nu^{18} + 12 \nu^{17} - 36 \nu^{16} + 168 \nu^{15} - 189 \nu^{14} + \cdots - 1549681956 ) / 387420489 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61662421 \nu^{19} - 138207694 \nu^{18} + 889079853 \nu^{17} - 2326161438 \nu^{16} + \cdots - 35\!\cdots\!38 ) / 58\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 79938641 \nu^{19} + 185126228 \nu^{18} - 1637888181 \nu^{17} + 760136904 \nu^{16} + \cdots + 21\!\cdots\!04 ) / 58\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 257135149 \nu^{19} - 268713446 \nu^{18} - 1033729383 \nu^{17} - 4108809366 \nu^{16} + \cdots + 60\!\cdots\!94 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 70039489 \nu^{19} - 208241056 \nu^{18} + 897838647 \nu^{17} - 401872932 \nu^{16} + \cdots + 51\!\cdots\!08 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 251074427 \nu^{19} + 531861284 \nu^{18} - 3210483813 \nu^{17} - 3546308880 \nu^{16} + \cdots + 16\!\cdots\!00 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 111545668 \nu^{19} + 39339455 \nu^{18} + 353204940 \nu^{17} + 1853397009 \nu^{16} + \cdots - 16\!\cdots\!81 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 199014772 \nu^{19} - 101748719 \nu^{18} - 362076312 \nu^{17} - 1217065437 \nu^{16} + \cdots + 22\!\cdots\!73 ) / 58\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 23294150 \nu^{19} - 18721930 \nu^{18} - 226707081 \nu^{17} + 107562861 \nu^{16} + \cdots + 36\!\cdots\!87 ) / 588597101164008 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 598339333 \nu^{19} - 253180274 \nu^{18} - 2749188927 \nu^{17} - 5555184030 \nu^{16} + \cdots + 67\!\cdots\!90 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 6269919 \nu^{19} + 11737472 \nu^{18} - 79596347 \nu^{17} + 161605140 \nu^{16} + \cdots + 65\!\cdots\!52 ) / 130799355814224 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 145820837 \nu^{19} - 309370838 \nu^{18} + 1392421611 \nu^{17} - 1242209016 \nu^{16} + \cdots - 12\!\cdots\!56 ) / 19\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} + \beta_{14} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{19} - 2 \beta_{18} - 2 \beta_{17} + \beta_{16} + \beta_{15} + 2 \beta_{14} - \beta_{13} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - 7 \beta_{15} - 3 \beta_{14} - 11 \beta_{13} + \cdots - 29 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{19} - 9 \beta_{18} - 8 \beta_{17} + 3 \beta_{16} + 15 \beta_{15} + 4 \beta_{14} - 24 \beta_{13} + \cdots - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{19} - 26 \beta_{18} - 32 \beta_{17} + 10 \beta_{16} + 34 \beta_{15} - 10 \beta_{14} + \cdots + 432 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 56 \beta_{19} - 52 \beta_{18} - 110 \beta_{17} + 122 \beta_{16} + 200 \beta_{15} + 88 \beta_{13} + \cdots - 380 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12 \beta_{19} - 228 \beta_{18} + 562 \beta_{17} + 168 \beta_{16} - 318 \beta_{15} - 260 \beta_{14} + \cdots + 2519 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1118 \beta_{19} - 824 \beta_{18} - 962 \beta_{17} + 376 \beta_{16} + 1312 \beta_{15} + 596 \beta_{14} + \cdots - 5589 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 974 \beta_{19} - 5320 \beta_{18} - 2555 \beta_{17} + 4724 \beta_{16} + 1424 \beta_{15} - 399 \beta_{14} + \cdots + 39823 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 843 \beta_{19} + 3474 \beta_{18} - 3302 \beta_{17} + 3 \beta_{16} - 3927 \beta_{15} + 5890 \beta_{14} + \cdots + 337211 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6482 \beta_{19} - 38609 \beta_{18} + 26851 \beta_{17} - 5525 \beta_{16} + 27331 \beta_{15} + \cdots - 38979 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 55208 \beta_{19} - 164815 \beta_{18} - 42104 \beta_{17} + 128471 \beta_{16} + 252911 \beta_{15} + \cdots + 1033234 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 48300 \beta_{19} - 100956 \beta_{18} + 34600 \beta_{17} + 207240 \beta_{16} - 473160 \beta_{15} + \cdots - 1226368 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 860744 \beta_{19} + 1309972 \beta_{18} + 1525744 \beta_{17} - 904124 \beta_{16} - 1693100 \beta_{15} + \cdots - 5214636 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 234988 \beta_{19} - 732844 \beta_{18} - 1702592 \beta_{17} + 483128 \beta_{16} + 2172296 \beta_{15} + \cdots + 40475779 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 4459632 \beta_{19} + 2182572 \beta_{18} - 6998984 \beta_{17} - 3654060 \beta_{16} + 4321860 \beta_{15} + \cdots + 88605413 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 31996780 \beta_{19} - 17625356 \beta_{18} - 831839 \beta_{17} - 7000712 \beta_{16} + 11030128 \beta_{15} + \cdots - 157446081 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 7924975 \beta_{19} - 63129094 \beta_{18} + 8609038 \beta_{17} + 10740677 \beta_{16} - 1038211 \beta_{15} + \cdots + 1453055893 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/372\mathbb{Z}\right)^\times\).

\(n\) \(125\) \(187\) \(313\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
−2.95249 + 0.531799i
−2.95249 − 0.531799i
−2.11207 + 2.13053i
−2.11207 − 2.13053i
−2.03692 + 2.20249i
−2.03692 − 2.20249i
−1.02561 + 2.81924i
−1.02561 − 2.81924i
−0.344858 + 2.98011i
−0.344858 − 2.98011i
1.11762 + 2.78405i
1.11762 − 2.78405i
1.41103 + 2.64745i
1.41103 − 2.64745i
2.32313 + 1.89817i
2.32313 − 1.89817i
2.62034 + 1.46077i
2.62034 − 1.46077i
2.99982 + 0.0326151i
2.99982 − 0.0326151i
0 −2.95249 − 0.531799i 0 − 7.94974i 0 −2.64422 0 8.43438 + 3.14026i 0
125.2 0 −2.95249 + 0.531799i 0 7.94974i 0 −2.64422 0 8.43438 − 3.14026i 0
125.3 0 −2.11207 − 2.13053i 0 6.36056i 0 −11.1545 0 −0.0783385 + 8.99966i 0
125.4 0 −2.11207 + 2.13053i 0 − 6.36056i 0 −11.1545 0 −0.0783385 − 8.99966i 0
125.5 0 −2.03692 − 2.20249i 0 − 1.56310i 0 8.85401 0 −0.701916 + 8.97259i 0
125.6 0 −2.03692 + 2.20249i 0 1.56310i 0 8.85401 0 −0.701916 − 8.97259i 0
125.7 0 −1.02561 − 2.81924i 0 − 3.88631i 0 −6.33831 0 −6.89625 + 5.78289i 0
125.8 0 −1.02561 + 2.81924i 0 3.88631i 0 −6.33831 0 −6.89625 − 5.78289i 0
125.9 0 −0.344858 − 2.98011i 0 7.78728i 0 6.16406 0 −8.76215 + 2.05543i 0
125.10 0 −0.344858 + 2.98011i 0 − 7.78728i 0 6.16406 0 −8.76215 − 2.05543i 0
125.11 0 1.11762 − 2.78405i 0 1.30490i 0 0.963710 0 −6.50183 − 6.22304i 0
125.12 0 1.11762 + 2.78405i 0 − 1.30490i 0 0.963710 0 −6.50183 + 6.22304i 0
125.13 0 1.41103 − 2.64745i 0 − 9.32271i 0 12.0099 0 −5.01801 − 7.47125i 0
125.14 0 1.41103 + 2.64745i 0 9.32271i 0 12.0099 0 −5.01801 + 7.47125i 0
125.15 0 2.32313 − 1.89817i 0 6.34142i 0 −4.11205 0 1.79390 − 8.81941i 0
125.16 0 2.32313 + 1.89817i 0 − 6.34142i 0 −4.11205 0 1.79390 + 8.81941i 0
125.17 0 2.62034 − 1.46077i 0 − 5.50897i 0 −10.6731 0 4.73233 − 7.65539i 0
125.18 0 2.62034 + 1.46077i 0 5.50897i 0 −10.6731 0 4.73233 + 7.65539i 0
125.19 0 2.99982 − 0.0326151i 0 − 2.23125i 0 6.93051 0 8.99787 − 0.195679i 0
125.20 0 2.99982 + 0.0326151i 0 2.23125i 0 6.93051 0 8.99787 + 0.195679i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 372.3.d.a ✓ 20
3.b odd 2 1 inner 372.3.d.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
372.3.d.a ✓ 20 1.a even 1 1 trivial
372.3.d.a ✓ 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(372, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 5144753822976 \) Copy content Toggle raw display
$7$ \( (T^{10} - 306 T^{8} + \cdots - 35919216)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{10} - 14 T^{9} + \cdots + 2195445864)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{10} + 2 T^{9} + \cdots + 5490350800)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{2} - 31)^{10} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 25434859394304)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 85\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 45\!\cdots\!40)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 386053039872)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 50\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 10\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less