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Properties

Label 375.2.i.b
Level $375$
Weight $2$
Character orbit 375.i
Analytic conductor $2.994$
Analytic rank $0$
Dimension $16$
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] = [375,2,Mod(49,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.i (of order \(10\), degree \(4\), not 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: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + ( - \beta_{13} - \beta_{6} + \cdots - \beta_1) q^{3}+ \cdots - \beta_{9} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{2} + ( - \beta_{13} - \beta_{6} + \cdots - \beta_1) q^{3}+ \cdots + ( - 2 \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 32 q^{11} + 16 q^{14} - 34 q^{16} + 10 q^{19} - 22 q^{21} - 60 q^{24} + 12 q^{26} - 10 q^{29} - 38 q^{31} - 24 q^{34} - 18 q^{36} + 16 q^{39} - 28 q^{41} + 6 q^{44} + 32 q^{46} - 32 q^{49} + 8 q^{51} + 2 q^{54} - 30 q^{56} - 60 q^{59} - 28 q^{61} + 88 q^{64} - 14 q^{66} + 16 q^{69} + 42 q^{71} + 76 q^{74} + 160 q^{76} + 60 q^{79} - 4 q^{81} - 16 q^{84} - 68 q^{86} + 42 q^{91} + 66 q^{94} + 68 q^{96} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{15} - \nu^{13} + 14\nu^{11} + 44\nu^{9} - \nu^{7} - 108\nu^{5} - 288\nu^{3} - 192\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 9\nu^{12} - 10\nu^{10} + 12\nu^{8} + 63\nu^{6} + 76\nu^{4} + 48\nu^{2} + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + \nu^{13} - 46\nu^{11} - 76\nu^{9} + 65\nu^{7} + 364\nu^{5} + 256\nu^{3} - 192\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 9\nu^{12} - 26\nu^{10} - 4\nu^{8} + 95\nu^{6} + 204\nu^{4} + 32\nu^{2} - 128 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} - 5\nu^{13} + 10\nu^{11} + 100\nu^{9} + 175\nu^{7} - 112\nu^{5} - 720\nu^{3} - 960\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{13} - 3\nu^{11} - 4\nu^{9} + 8\nu^{7} + 23\nu^{5} + 18\nu^{3} + 8\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{14} - 11\nu^{12} - 30\nu^{10} - 12\nu^{8} + 93\nu^{6} + 180\nu^{4} + 320\nu^{2} + 192 ) / 192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{14} + 7\nu^{12} + 26\nu^{10} + 68\nu^{8} + 83\nu^{6} - 184\nu^{4} - 752\nu^{2} - 960 ) / 192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{14} + 9\nu^{12} - 18\nu^{10} - 100\nu^{8} - 139\nu^{6} + 96\nu^{4} + 512\nu^{2} + 512 ) / 192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{15} + 5\nu^{13} + 3\nu^{11} - 22\nu^{9} - 58\nu^{7} - 21\nu^{5} + 96\nu^{3} + 224\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{14} - 3\nu^{12} + 2\nu^{10} + 30\nu^{8} + 51\nu^{6} - 14\nu^{4} - 162\nu^{2} - 200 ) / 24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -5\nu^{15} - 33\nu^{13} - 38\nu^{11} + 84\nu^{9} + 363\nu^{7} + 392\nu^{5} - 192\nu^{3} - 832\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -5\nu^{15} - 15\nu^{13} - 4\nu^{11} + 72\nu^{9} + 147\nu^{7} + 26\nu^{5} - 216\nu^{3} - 224\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -3\nu^{15} - 6\nu^{13} + 15\nu^{11} + 86\nu^{9} + 113\nu^{7} - 103\nu^{5} - 508\nu^{3} - 448\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\nu^{14} + 49\nu^{12} - 2\nu^{10} - 356\nu^{8} - 707\nu^{6} + 96\nu^{4} + 2016\nu^{2} + 2304 ) / 192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + 2\beta_{13} - \beta_{12} + 3\beta_{10} + 2\beta_{6} - \beta_{5} + \beta_{3} - \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + 3\beta_{11} + \beta_{9} + \beta_{8} + 3\beta_{7} - 3\beta_{4} + 4\beta_{2} - 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{14} + \beta_{13} + 2\beta_{12} - \beta_{10} - 9\beta_{6} + 2\beta_{5} - 2\beta_{3} - 8\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{11} - \beta_{9} - \beta_{7} + 2\beta_{4} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{14} - 8\beta_{13} + 9\beta_{12} + 3\beta_{10} - 8\beta_{6} - 16\beta_{5} + 6\beta_{3} + 9\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{15} - 7\beta_{11} - 14\beta_{9} + 16\beta_{8} + 18\beta_{7} - 3\beta_{4} + 9\beta_{2} + 12 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{14} - 4\beta_{13} - 3\beta_{12} - \beta_{10} + 6\beta_{6} + 2\beta_{5} - 7\beta_{3} - 58\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{15} + 10\beta_{11} + 9\beta_{9} - 4\beta_{8} - 2\beta_{7} - 2\beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -19\beta_{14} + 42\beta_{13} + 14\beta_{12} + 38\beta_{10} - 78\beta_{6} + 19\beta_{5} + 26\beta_{3} + 49\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 27\beta_{15} - 27\beta_{11} - 114\beta_{9} + 51\beta_{8} + 3\beta_{7} + 27\beta_{4} + 114\beta_{2} - 13 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 73 \beta_{14} - 139 \beta_{13} + 62 \beta_{12} - 51 \beta_{10} + 11 \beta_{6} - 133 \beta_{5} + \cdots - 133 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10\beta_{15} + 18\beta_{11} + 5\beta_{9} + 5\beta_{8} + 18\beta_{7} + 3\beta_{4} - 31\beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 31\beta_{14} + 67\beta_{13} - 31\beta_{12} + 68\beta_{10} - 243\beta_{6} - \beta_{5} + 136\beta_{3} - 206\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 62\beta_{15} + 143\beta_{11} + 241\beta_{9} + 81\beta_{8} - 62\beta_{7} - 98\beta_{4} + 434\beta_{2} + 62 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 117 \beta_{14} + 31 \beta_{13} + 162 \beta_{12} + 279 \beta_{10} + 31 \beta_{6} + 317 \beta_{5} + \cdots + 162 \beta_1 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(1 - \beta_{2} - \beta_{4} + \beta_{9}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.41395 + 0.0272949i
−0.462894 + 1.33631i
0.462894 − 1.33631i
1.41395 − 0.0272949i
−1.41395 − 0.0272949i
−0.462894 − 1.33631i
0.462894 + 1.33631i
1.41395 + 0.0272949i
−0.132563 − 1.40799i
−0.720348 − 1.21700i
0.720348 + 1.21700i
0.132563 + 1.40799i
−0.132563 + 1.40799i
−0.720348 + 1.21700i
0.720348 − 1.21700i
0.132563 − 1.40799i
−1.59076 + 2.18949i −0.951057 + 0.309017i −1.64533 − 5.06380i 0 0.836312 − 2.57390i − 0.470294i 8.55667 + 2.78023i 0.809017 − 0.587785i 0
49.2 −1.00297 + 1.38048i 0.951057 − 0.309017i −0.281722 − 0.867051i 0 −0.527295 + 1.62285i − 3.94243i −1.76619 − 0.573870i 0.809017 − 0.587785i 0
49.3 1.00297 − 1.38048i −0.951057 + 0.309017i −0.281722 − 0.867051i 0 −0.527295 + 1.62285i 3.94243i 1.76619 + 0.573870i 0.809017 − 0.587785i 0
49.4 1.59076 − 2.18949i 0.951057 − 0.309017i −1.64533 − 5.06380i 0 0.836312 − 2.57390i 0.470294i −8.55667 − 2.78023i 0.809017 − 0.587785i 0
199.1 −1.59076 − 2.18949i −0.951057 − 0.309017i −1.64533 + 5.06380i 0 0.836312 + 2.57390i 0.470294i 8.55667 − 2.78023i 0.809017 + 0.587785i 0
199.2 −1.00297 − 1.38048i 0.951057 + 0.309017i −0.281722 + 0.867051i 0 −0.527295 − 1.62285i 3.94243i −1.76619 + 0.573870i 0.809017 + 0.587785i 0
199.3 1.00297 + 1.38048i −0.951057 − 0.309017i −0.281722 + 0.867051i 0 −0.527295 − 1.62285i − 3.94243i 1.76619 − 0.573870i 0.809017 + 0.587785i 0
199.4 1.59076 + 2.18949i 0.951057 + 0.309017i −1.64533 + 5.06380i 0 0.836312 + 2.57390i − 0.470294i −8.55667 + 2.78023i 0.809017 + 0.587785i 0
274.1 −2.01846 − 0.655837i 0.587785 − 0.809017i 2.02602 + 1.47199i 0 −1.71700 + 1.24748i − 4.35840i −0.629102 − 0.865884i −0.309017 − 0.951057i 0
274.2 −1.06740 − 0.346820i −0.587785 + 0.809017i −0.598970 − 0.435177i 0 0.907987 − 0.659691i 1.11373i 1.80780 + 2.48822i −0.309017 − 0.951057i 0
274.3 1.06740 + 0.346820i 0.587785 − 0.809017i −0.598970 − 0.435177i 0 0.907987 − 0.659691i − 1.11373i −1.80780 − 2.48822i −0.309017 − 0.951057i 0
274.4 2.01846 + 0.655837i −0.587785 + 0.809017i 2.02602 + 1.47199i 0 −1.71700 + 1.24748i 4.35840i 0.629102 + 0.865884i −0.309017 − 0.951057i 0
349.1 −2.01846 + 0.655837i 0.587785 + 0.809017i 2.02602 − 1.47199i 0 −1.71700 − 1.24748i 4.35840i −0.629102 + 0.865884i −0.309017 + 0.951057i 0
349.2 −1.06740 + 0.346820i −0.587785 − 0.809017i −0.598970 + 0.435177i 0 0.907987 + 0.659691i − 1.11373i 1.80780 − 2.48822i −0.309017 + 0.951057i 0
349.3 1.06740 − 0.346820i 0.587785 + 0.809017i −0.598970 + 0.435177i 0 0.907987 + 0.659691i 1.11373i −1.80780 + 2.48822i −0.309017 + 0.951057i 0
349.4 2.01846 − 0.655837i −0.587785 − 0.809017i 2.02602 − 1.47199i 0 −1.71700 − 1.24748i − 4.35840i 0.629102 − 0.865884i −0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.i.b 16
5.b even 2 1 inner 375.2.i.b 16
5.c odd 4 1 75.2.g.b ✓ 8
5.c odd 4 1 375.2.g.b 8
15.e even 4 1 225.2.h.c 8
25.d even 5 1 inner 375.2.i.b 16
25.d even 5 1 1875.2.b.c 8
25.e even 10 1 inner 375.2.i.b 16
25.e even 10 1 1875.2.b.c 8
25.f odd 20 1 75.2.g.b ✓ 8
25.f odd 20 1 375.2.g.b 8
25.f odd 20 1 1875.2.a.e 4
25.f odd 20 1 1875.2.a.h 4
75.l even 20 1 225.2.h.c 8
75.l even 20 1 5625.2.a.i 4
75.l even 20 1 5625.2.a.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b ✓ 8 5.c odd 4 1
75.2.g.b ✓ 8 25.f odd 20 1
225.2.h.c 8 15.e even 4 1
225.2.h.c 8 75.l even 20 1
375.2.g.b 8 5.c odd 4 1
375.2.g.b 8 25.f odd 20 1
375.2.i.b 16 1.a even 1 1 trivial
375.2.i.b 16 5.b even 2 1 inner
375.2.i.b 16 25.d even 5 1 inner
375.2.i.b 16 25.e even 10 1 inner
1875.2.a.e 4 25.f odd 20 1
1875.2.a.h 4 25.f odd 20 1
1875.2.b.c 8 25.d even 5 1
1875.2.b.c 8 25.e even 10 1
5625.2.a.i 4 75.l even 20 1
5625.2.a.n 4 75.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 3T_{2}^{14} + 48T_{2}^{12} - 341T_{2}^{10} + 1475T_{2}^{8} - 2801T_{2}^{6} + 11828T_{2}^{4} - 19723T_{2}^{2} + 14641 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 3 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 36 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 16 T^{7} + \cdots + 11881)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 22 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( T^{16} - 13 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$19$ \( (T^{8} - 5 T^{7} + \cdots + 75625)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 67 T^{14} + \cdots + 130321 \) Copy content Toggle raw display
$29$ \( (T^{8} + 5 T^{7} + \cdots + 164025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 19 T^{7} + \cdots + 505521)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 153 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} + 7 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 134 T^{6} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 9354951841 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 4640470641 \) Copy content Toggle raw display
$59$ \( (T^{8} + 30 T^{7} + \cdots + 13286025)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 14 T^{7} + \cdots + 81)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 855036081 \) Copy content Toggle raw display
$71$ \( (T^{8} - 21 T^{7} + \cdots + 829921)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 562029482679921 \) Copy content Toggle raw display
$79$ \( (T^{8} - 30 T^{7} + \cdots + 96924025)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 918609150481 \) Copy content Toggle raw display
$89$ \( (T^{8} + 255 T^{6} + \cdots + 10923025)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 19485170468401 \) Copy content Toggle raw display
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