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Properties

Label 475.2.l.b
Level $475$
Weight $2$
Character orbit 475.l
Analytic conductor $3.793$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(101,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.l (of order \(9\), degree \(6\), 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: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} + \cdots + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + (\beta_{15} + \beta_{7} + \cdots + \beta_{4}) q^{3} + (\beta_{10} - \beta_{9} + \cdots + \beta_{2}) q^{4} + (\beta_{16} + \beta_{15} + \cdots + \beta_{3}) q^{6} + ( - \beta_{17} - \beta_{16} - \beta_{14} + \cdots + 1) q^{7}+ \cdots + (\beta_{15} - 3 \beta_{12} + \beta_{10} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 6 q^{6} + 12 q^{8} - 21 q^{9} - 6 q^{12} + 3 q^{13} + 24 q^{14} + 21 q^{16} + 24 q^{17} + 12 q^{18} - 12 q^{19} + 3 q^{21} - 15 q^{22} - 21 q^{23} + 21 q^{24} - 21 q^{26}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} + \cdots + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 136164871502623 \nu^{17} + \cdots - 12\!\cdots\!58 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 305669388244104 \nu^{17} + 493873766181026 \nu^{16} + \cdots + 47\!\cdots\!16 ) / 78\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!19 \nu^{17} + \cdots + 40\!\cdots\!81 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!95 \nu^{17} + \cdots + 49\!\cdots\!59 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!82 \nu^{17} + \cdots + 73\!\cdots\!88 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!56 \nu^{17} + 305669388244104 \nu^{16} + \cdots - 15\!\cdots\!37 ) / 78\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 42\!\cdots\!70 \nu^{17} + \cdots - 79\!\cdots\!20 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 116934812256397 \nu^{17} - 400329673137974 \nu^{16} + \cdots - 19\!\cdots\!38 ) / 63\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!03 \nu^{17} + \cdots + 52\!\cdots\!54 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 61\!\cdots\!77 \nu^{17} + 828266836359098 \nu^{16} + \cdots + 60\!\cdots\!85 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 84\!\cdots\!25 \nu^{17} + \cdots - 58\!\cdots\!05 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 84\!\cdots\!44 \nu^{17} + \cdots - 15\!\cdots\!70 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 87\!\cdots\!64 \nu^{17} + \cdots + 47\!\cdots\!08 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 92\!\cdots\!56 \nu^{17} + \cdots + 30\!\cdots\!25 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 316478299376213 \nu^{17} - 588887179277519 \nu^{16} + \cdots - 38\!\cdots\!43 ) / 63\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 17\!\cdots\!19 \nu^{17} + \cdots - 11\!\cdots\!95 ) / 23\!\cdots\!13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{16} + 3\beta_{7} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{12} - 2\beta_{10} + \beta_{9} - \beta_{4} - 3\beta_{3} - 2\beta_{2} - 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + 6 \beta_{16} + \beta_{14} - \beta_{13} + 2 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - 7 \beta_{11} + 9 \beta_{10} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{17} - 8 \beta_{16} + 10 \beta_{15} - 9 \beta_{14} + 10 \beta_{13} - 12 \beta_{12} - 33 \beta_{11} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 46 \beta_{17} + 69 \beta_{16} - 10 \beta_{15} + 22 \beta_{14} - 12 \beta_{13} + 77 \beta_{12} + \cdots - 112 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 52 \beta_{17} - 176 \beta_{16} - 63 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} - 76 \beta_{12} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 252 \beta_{17} - 53 \beta_{16} + 181 \beta_{15} - 105 \beta_{14} + 181 \beta_{13} - 417 \beta_{12} + \cdots + 518 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 682 \beta_{17} + 1497 \beta_{16} - 133 \beta_{15} + 550 \beta_{14} - 417 \beta_{13} + 1349 \beta_{12} + \cdots - 2209 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 487 \beta_{17} - 3061 \beta_{16} - 816 \beta_{15} - 533 \beta_{14} - 533 \beta_{13} - 1081 \beta_{12} + \cdots + 1560 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2804 \beta_{17} - 2093 \beta_{16} + 3829 \beta_{15} - 2748 \beta_{14} + 3829 \beta_{13} - 5996 \beta_{12} + \cdots + 11782 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 13860 \beta_{17} + 24807 \beta_{16} - 3646 \beta_{15} + 9642 \beta_{14} - 5996 \beta_{13} + \cdots - 36433 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13621 \beta_{17} - 54352 \beta_{16} - 18275 \beta_{15} - 8140 \beta_{14} - 8140 \beta_{13} + \cdots + 20828 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 65420 \beta_{17} - 28968 \beta_{16} + 67593 \beta_{15} - 42795 \beta_{14} + 67593 \beta_{13} + \cdots + 185488 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 239831 \beta_{17} + 463488 \beta_{16} - 58869 \beta_{15} + 181667 \beta_{14} - 122798 \beta_{13} + \cdots - 669036 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 208482 \beta_{17} - 982637 \beta_{16} - 300623 \beta_{15} - 168782 \beta_{14} - 168782 \beta_{13} + \cdots + 436673 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(\beta_{4} + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−0.908512 − 1.57359i
0.296923 + 0.514286i
0.785237 + 1.36007i
1.01081 − 1.75077i
0.597039 − 1.03410i
−0.841804 + 1.45805i
−1.31112 + 2.27092i
−0.359728 + 0.623068i
0.731154 − 1.26640i
1.01081 + 1.75077i
0.597039 + 1.03410i
−0.841804 − 1.45805i
−0.908512 + 1.57359i
0.296923 − 0.514286i
0.785237 − 1.36007i
−1.31112 − 2.27092i
−0.359728 − 0.623068i
0.731154 + 1.26640i
−1.39192 + 1.16796i 0.166424 + 0.0605732i 0.226016 − 1.28180i 0 −0.302396 + 0.110063i 0.536732 − 0.929646i −0.634528 − 1.09903i −2.27411 − 1.90820i 0
101.2 0.454913 − 0.381717i −1.81364 − 0.660111i −0.286059 + 1.62232i 0 −1.07702 + 0.392004i 0.530259 − 0.918436i 1.08298 + 1.87578i 0.555408 + 0.466042i 0
101.3 1.20305 − 1.00948i 2.14722 + 0.781523i 0.0809872 − 0.459301i 0 3.37214 − 1.22736i −2.00668 + 3.47568i 1.20425 + 2.08582i 1.70162 + 1.42783i 0
176.1 −1.89970 + 0.691434i 0.330026 − 1.87167i 1.59869 − 1.34146i 0 0.667187 + 3.78381i 1.54609 + 2.67790i −0.0878797 + 0.152212i −0.575162 − 0.209342i 0
176.2 −1.12207 + 0.408399i −0.394733 + 2.23864i −0.439845 + 0.369074i 0 −0.471342 − 2.67312i −1.09825 − 1.90222i 1.53688 − 2.66196i −2.03663 − 0.741274i 0
176.3 1.58207 − 0.575828i 0.564707 − 3.20261i 0.639290 − 0.536428i 0 −0.950745 − 5.39194i −0.274194 − 0.474919i −0.981094 + 1.69930i −7.11876 − 2.59101i 0
226.1 −0.455347 + 2.58240i 0.711484 + 0.597006i −4.58206 − 1.66773i 0 −1.86568 + 1.56549i −1.91579 − 3.31824i 3.77094 − 6.53146i −0.371151 − 2.10490i 0
226.2 −0.124932 + 0.708527i 0.945515 + 0.793382i 1.39298 + 0.507004i 0 −0.680257 + 0.570804i 0.645970 + 1.11885i −1.25271 + 2.16976i −0.256400 − 1.45411i 0
226.3 0.253927 − 1.44009i −1.15700 − 0.970838i −0.130002 − 0.0473169i 0 −1.69189 + 1.41966i 2.03586 + 3.52622i 1.36116 − 2.35759i −0.124823 − 0.707907i 0
251.1 −1.89970 − 0.691434i 0.330026 + 1.87167i 1.59869 + 1.34146i 0 0.667187 − 3.78381i 1.54609 − 2.67790i −0.0878797 − 0.152212i −0.575162 + 0.209342i 0
251.2 −1.12207 − 0.408399i −0.394733 − 2.23864i −0.439845 − 0.369074i 0 −0.471342 + 2.67312i −1.09825 + 1.90222i 1.53688 + 2.66196i −2.03663 + 0.741274i 0
251.3 1.58207 + 0.575828i 0.564707 + 3.20261i 0.639290 + 0.536428i 0 −0.950745 + 5.39194i −0.274194 + 0.474919i −0.981094 − 1.69930i −7.11876 + 2.59101i 0
301.1 −1.39192 − 1.16796i 0.166424 − 0.0605732i 0.226016 + 1.28180i 0 −0.302396 − 0.110063i 0.536732 + 0.929646i −0.634528 + 1.09903i −2.27411 + 1.90820i 0
301.2 0.454913 + 0.381717i −1.81364 + 0.660111i −0.286059 − 1.62232i 0 −1.07702 − 0.392004i 0.530259 + 0.918436i 1.08298 − 1.87578i 0.555408 − 0.466042i 0
301.3 1.20305 + 1.00948i 2.14722 − 0.781523i 0.0809872 + 0.459301i 0 3.37214 + 1.22736i −2.00668 − 3.47568i 1.20425 − 2.08582i 1.70162 − 1.42783i 0
351.1 −0.455347 − 2.58240i 0.711484 − 0.597006i −4.58206 + 1.66773i 0 −1.86568 − 1.56549i −1.91579 + 3.31824i 3.77094 + 6.53146i −0.371151 + 2.10490i 0
351.2 −0.124932 − 0.708527i 0.945515 − 0.793382i 1.39298 − 0.507004i 0 −0.680257 − 0.570804i 0.645970 − 1.11885i −1.25271 − 2.16976i −0.256400 + 1.45411i 0
351.3 0.253927 + 1.44009i −1.15700 + 0.970838i −0.130002 + 0.0473169i 0 −1.69189 − 1.41966i 2.03586 − 3.52622i 1.36116 + 2.35759i −0.124823 + 0.707907i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.l.b 18
5.b even 2 1 95.2.k.b ✓ 18
5.c odd 4 2 475.2.u.c 36
15.d odd 2 1 855.2.bs.b 18
19.e even 9 1 inner 475.2.l.b 18
19.e even 9 1 9025.2.a.ce 9
19.f odd 18 1 9025.2.a.cd 9
95.o odd 18 1 1805.2.a.u 9
95.p even 18 1 95.2.k.b ✓ 18
95.p even 18 1 1805.2.a.t 9
95.q odd 36 2 475.2.u.c 36
285.bd odd 18 1 855.2.bs.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.b ✓ 18 5.b even 2 1
95.2.k.b ✓ 18 95.p even 18 1
475.2.l.b 18 1.a even 1 1 trivial
475.2.l.b 18 19.e even 9 1 inner
475.2.u.c 36 5.c odd 4 2
475.2.u.c 36 95.q odd 36 2
855.2.bs.b 18 15.d odd 2 1
855.2.bs.b 18 285.bd odd 18 1
1805.2.a.t 9 95.p even 18 1
1805.2.a.u 9 95.o odd 18 1
9025.2.a.cd 9 19.f odd 18 1
9025.2.a.ce 9 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 3 T_{2}^{17} + 6 T_{2}^{16} + 5 T_{2}^{15} - 15 T_{2}^{14} - 24 T_{2}^{13} + 70 T_{2}^{12} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 3 T^{17} + \cdots + 361 \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{17} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + 33 T^{16} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{18} + 36 T^{16} + \cdots + 361 \) Copy content Toggle raw display
$13$ \( T^{18} - 3 T^{17} + \cdots + 98743969 \) Copy content Toggle raw display
$17$ \( T^{18} - 24 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 3993860809 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 492805404001 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 993257404129 \) Copy content Toggle raw display
$37$ \( (T^{9} - 30 T^{8} + \cdots - 27721)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 132479256529 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 101989801 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 32737816559809 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 19250147475001 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 333749999521 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 3468418292161 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 36\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 8590138489 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 1047660743809 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 316957384070209 \) Copy content Toggle raw display
$83$ \( T^{18} + 171 T^{16} + \cdots + 96609241 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 143304841 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 36\!\cdots\!09 \) Copy content Toggle raw display
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