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Properties

Label 475.2.j.c
Level $475$
Weight $2$
Character orbit 475.j
Analytic conductor $3.793$
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] = [475,2,Mod(49,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.j (of order \(6\), degree \(2\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 11x^{14} + 82x^{12} - 337x^{10} + 1006x^{8} - 1596x^{6} + 1765x^{4} - 414x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + (\beta_{13} + \beta_{7}) q^{3} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{4} + (\beta_{9} + \beta_{6} - \beta_1) q^{6} + ( - \beta_{15} + \beta_{8}) q^{7} + ( - 2 \beta_{15} + \beta_{14} + \cdots + 2 \beta_{8}) q^{8}+ \cdots + (\beta_{9} - \beta_{4} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{2} + (\beta_{13} + \beta_{7}) q^{3} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{4} + (\beta_{9} + \beta_{6} - \beta_1) q^{6} + ( - \beta_{15} + \beta_{8}) q^{7} + ( - 2 \beta_{15} + \beta_{14} + \cdots + 2 \beta_{8}) q^{8}+ \cdots + (5 \beta_{9} - 12 \beta_{4} - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{4} - 4 q^{6} + 2 q^{9} - 8 q^{11} - 2 q^{14} - 14 q^{16} - 10 q^{19} + 8 q^{21} + 46 q^{24} + 12 q^{26} - 2 q^{29} + 30 q^{34} + 14 q^{36} - 60 q^{39} + 16 q^{41} - 24 q^{44} + 48 q^{46} + 40 q^{49} - 44 q^{51} - 68 q^{54} - 164 q^{56} - 10 q^{59} - 224 q^{64} + 62 q^{66} + 36 q^{69} - 40 q^{71} + 50 q^{74} + 126 q^{76} + 34 q^{79} - 24 q^{81} + 80 q^{84} - 16 q^{86} + 22 q^{89} - 12 q^{91} + 124 q^{94} + 84 q^{96} + 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 82x^{12} - 337x^{10} + 1006x^{8} - 1596x^{6} + 1765x^{4} - 414x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17 \nu^{14} - 281 \nu^{12} + 3853 \nu^{10} - 26683 \nu^{8} + 96538 \nu^{6} - 201735 \nu^{4} + \cdots - 46638 ) / 73569 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 105339 \nu^{14} + 1015482 \nu^{12} - 7295401 \nu^{10} + 25578470 \nu^{8} - 71187724 \nu^{6} + \cdots - 284972250 ) / 104051089 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 261339 \nu^{14} - 2789005 \nu^{12} + 18099401 \nu^{10} - 63458470 \nu^{8} + 140708376 \nu^{6} + \cdots + 45708276 ) / 104051089 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3020113 \nu^{14} - 32273192 \nu^{12} + 238509928 \nu^{10} - 952119472 \nu^{8} + \cdots - 1141592625 ) / 936459801 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 383448 \nu^{14} + 3735013 \nu^{12} - 26556232 \nu^{10} + 93109040 \nu^{8} - 239139277 \nu^{6} + \cdots - 184780641 ) / 104051089 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1440028 \nu^{14} - 17040962 \nu^{12} + 129078913 \nu^{10} - 568442422 \nu^{8} + 1740211588 \nu^{6} + \cdots - 733877370 ) / 312153267 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 750126 \nu^{15} + 7539500 \nu^{13} - 51951034 \nu^{11} + 182145980 \nu^{9} + \cdots - 931665523 \nu ) / 312153267 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1066143 \nu^{15} + 10585946 \nu^{13} - 73837237 \nu^{11} + 258881390 \nu^{9} + \cdots - 1786582273 \nu ) / 312153267 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2704096 \nu^{14} - 29226746 \nu^{12} + 216623725 \nu^{10} - 875384062 \nu^{8} + \cdots - 1060049574 ) / 312153267 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 488787 \nu^{15} + 4750495 \nu^{13} - 33851633 \nu^{11} + 118687510 \nu^{9} + \cdots - 573803980 \nu ) / 104051089 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1500252 \nu^{15} + 15079000 \nu^{13} - 103902068 \nu^{11} + 364291960 \nu^{9} + \cdots - 1239024512 \nu ) / 312153267 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30472184 \nu^{15} + 349306330 \nu^{13} - 2649325358 \nu^{11} + 11246493662 \nu^{9} + \cdots + 14234057118 \nu ) / 2809379403 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31047565 \nu^{15} - 339328241 \nu^{13} + 2507751769 \nu^{11} - 10118919067 \nu^{9} + \cdots - 2156816727 \nu ) / 2809379403 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15452485 \nu^{15} + 171076316 \nu^{13} - 1275617842 \nu^{11} + 5283598924 \nu^{9} + \cdots + 6523373457 \nu ) / 936459801 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 48633295 \nu^{15} - 534513455 \nu^{13} + 3989805988 \nu^{11} - 16358061805 \nu^{9} + \cdots - 20082252600 \nu ) / 2809379403 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 2\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + 3\beta_{4} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} + 8\beta_{13} + 3\beta_{12} + 3\beta_{11} + 2\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{9} + \beta_{6} + 12\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -12\beta_{15} - 2\beta_{14} + 34\beta_{13} + 11\beta_{12} + 34\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} + \beta_{3} - 23\beta_{2} - 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -47\beta_{11} + 16\beta_{10} - 60\beta_{8} + 148\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 104\beta_{9} - 38\beta_{6} + 38\beta_{5} - 222\beta_{4} + 11\beta_{3} - 104\beta_{2} + 11\beta _1 - 222 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 284\beta_{15} + 98\beta_{14} - 652\beta_{13} - 217\beta_{12} - 217\beta_{11} + 98\beta_{10} - 284\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 468\beta_{9} - 191\beta_{6} - 978\beta_{4} + 82\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1318\beta_{15} + 546\beta_{14} - 2892\beta_{13} - 1033\beta_{12} - 2892\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -932\beta_{5} - 519\beta_{3} + 2105\beta_{2} + 4338 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4963\beta_{11} - 2902\beta_{10} + 6074\beta_{8} - 12886\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 9480 \beta_{9} + 4488 \beta_{6} - 4488 \beta_{5} + 19329 \beta_{4} - 3008 \beta_{3} + 9480 \beta_{2} + \cdots + 19329 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 27936 \beta_{15} - 14992 \beta_{14} + 57618 \beta_{13} + 23865 \beta_{12} + 23865 \beta_{11} + \cdots + 27936 \beta_{8} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.426014 + 0.245959i
−1.87040 + 1.07988i
1.77290 − 1.02359i
1.19454 − 0.689667i
−1.19454 + 0.689667i
−1.77290 + 1.02359i
1.87040 − 1.07988i
0.426014 − 0.245959i
−0.426014 − 0.245959i
−1.87040 − 1.07988i
1.77290 + 1.02359i
1.19454 + 0.689667i
−1.19454 − 0.689667i
−1.77290 − 1.02359i
1.87040 + 1.07988i
0.426014 + 0.245959i
−2.38851 − 1.37901i 1.29204 + 0.745959i 2.80333 + 4.85550i 0 −2.05737 − 3.56347i − 2.84864i − 9.94721i −0.387090 − 0.670459i 0
49.2 −1.44154 − 0.832272i 1.00438 + 0.579878i 0.385355 + 0.667454i 0 −0.965233 − 1.67183i 2.43525i 2.04621i −0.827483 − 1.43324i 0
49.3 −1.03136 − 0.595455i −2.63893 − 1.52359i −0.290867 − 0.503797i 0 1.81445 + 3.14272i 0.609175i 3.07461i 3.14263 + 5.44319i 0
49.4 −0.950409 − 0.548719i −0.328513 − 0.189667i −0.397815 − 0.689035i 0 0.208148 + 0.360522i 1.89307i 3.06803i −1.42805 − 2.47346i 0
49.5 0.950409 + 0.548719i 0.328513 + 0.189667i −0.397815 − 0.689035i 0 0.208148 + 0.360522i − 1.89307i − 3.06803i −1.42805 − 2.47346i 0
49.6 1.03136 + 0.595455i 2.63893 + 1.52359i −0.290867 − 0.503797i 0 1.81445 + 3.14272i − 0.609175i − 3.07461i 3.14263 + 5.44319i 0
49.7 1.44154 + 0.832272i −1.00438 − 0.579878i 0.385355 + 0.667454i 0 −0.965233 − 1.67183i − 2.43525i − 2.04621i −0.827483 − 1.43324i 0
49.8 2.38851 + 1.37901i −1.29204 − 0.745959i 2.80333 + 4.85550i 0 −2.05737 − 3.56347i 2.84864i 9.94721i −0.387090 − 0.670459i 0
349.1 −2.38851 + 1.37901i 1.29204 − 0.745959i 2.80333 − 4.85550i 0 −2.05737 + 3.56347i 2.84864i 9.94721i −0.387090 + 0.670459i 0
349.2 −1.44154 + 0.832272i 1.00438 − 0.579878i 0.385355 − 0.667454i 0 −0.965233 + 1.67183i − 2.43525i − 2.04621i −0.827483 + 1.43324i 0
349.3 −1.03136 + 0.595455i −2.63893 + 1.52359i −0.290867 + 0.503797i 0 1.81445 − 3.14272i − 0.609175i − 3.07461i 3.14263 − 5.44319i 0
349.4 −0.950409 + 0.548719i −0.328513 + 0.189667i −0.397815 + 0.689035i 0 0.208148 − 0.360522i − 1.89307i − 3.06803i −1.42805 + 2.47346i 0
349.5 0.950409 − 0.548719i 0.328513 − 0.189667i −0.397815 + 0.689035i 0 0.208148 − 0.360522i 1.89307i 3.06803i −1.42805 + 2.47346i 0
349.6 1.03136 − 0.595455i 2.63893 − 1.52359i −0.290867 + 0.503797i 0 1.81445 − 3.14272i 0.609175i 3.07461i 3.14263 − 5.44319i 0
349.7 1.44154 − 0.832272i −1.00438 + 0.579878i 0.385355 − 0.667454i 0 −0.965233 + 1.67183i 2.43525i 2.04621i −0.827483 + 1.43324i 0
349.8 2.38851 − 1.37901i −1.29204 + 0.745959i 2.80333 − 4.85550i 0 −2.05737 + 3.56347i − 2.84864i − 9.94721i −0.387090 + 0.670459i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.c 16
5.b even 2 1 inner 475.2.j.c 16
5.c odd 4 1 95.2.e.c ✓ 8
5.c odd 4 1 475.2.e.e 8
15.e even 4 1 855.2.k.h 8
19.c even 3 1 inner 475.2.j.c 16
20.e even 4 1 1520.2.q.o 8
95.i even 6 1 inner 475.2.j.c 16
95.l even 12 1 1805.2.a.i 4
95.l even 12 1 9025.2.a.bp 4
95.m odd 12 1 95.2.e.c ✓ 8
95.m odd 12 1 475.2.e.e 8
95.m odd 12 1 1805.2.a.o 4
95.m odd 12 1 9025.2.a.bg 4
285.v even 12 1 855.2.k.h 8
380.v even 12 1 1520.2.q.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c ✓ 8 5.c odd 4 1
95.2.e.c ✓ 8 95.m odd 12 1
475.2.e.e 8 5.c odd 4 1
475.2.e.e 8 95.m odd 12 1
475.2.j.c 16 1.a even 1 1 trivial
475.2.j.c 16 5.b even 2 1 inner
475.2.j.c 16 19.c even 3 1 inner
475.2.j.c 16 95.i even 6 1 inner
855.2.k.h 8 15.e even 4 1
855.2.k.h 8 285.v even 12 1
1520.2.q.o 8 20.e even 4 1
1520.2.q.o 8 380.v even 12 1
1805.2.a.i 4 95.l even 12 1
1805.2.a.o 4 95.m odd 12 1
9025.2.a.bg 4 95.m odd 12 1
9025.2.a.bp 4 95.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 13T_{2}^{14} + 119T_{2}^{12} - 504T_{2}^{10} + 1515T_{2}^{8} - 2714T_{2}^{6} + 3529T_{2}^{4} - 2628T_{2}^{2} + 1296 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 13 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$3$ \( T^{16} - 13 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 18 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 25 T^{2} + \cdots + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} - 35 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 136048896 \) Copy content Toggle raw display
$19$ \( (T^{8} + 5 T^{7} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 38 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{8} + T^{7} + \cdots + 19881)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 67 T^{2} + \cdots + 1063)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 66 T^{6} + \cdots + 13924)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 8 T^{7} + \cdots + 5008644)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 397449550096 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 28770951188736 \) Copy content Toggle raw display
$53$ \( T^{16} - 197 T^{14} + \cdots + 8503056 \) Copy content Toggle raw display
$59$ \( (T^{8} + 5 T^{7} + \cdots + 3515625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 130 T^{6} + \cdots + 9296401)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 88 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$71$ \( (T^{8} + 20 T^{7} + \cdots + 59049)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 8874893813776 \) Copy content Toggle raw display
$79$ \( (T^{8} - 17 T^{7} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 125 T^{6} + \cdots + 133956)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 11 T^{7} + \cdots + 14561856)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
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