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Properties

Label 900.4.j.b
Level $900$
Weight $4$
Character orbit 900.j
Analytic conductor $53.102$
Analytic rank $0$
Dimension $12$
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] = [900,4,Mod(557,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.557");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.j (of order \(4\), degree \(2\), 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: \(53.1017190052\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 40x^{10} + 607x^{8} - 3980x^{6} + 10171x^{4} + 2180x^{2} + 7225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2 \beta_1 + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2 \beta_1 + 2) q^{7} + (\beta_{11} + \beta_{10} + \cdots + 2 \beta_{6}) q^{11}+ \cdots + ( - 11 \beta_{5} - 47 \beta_{3} + \cdots + 243) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{7} + 84 q^{13} + 144 q^{31} + 564 q^{37} + 960 q^{43} + 1920 q^{61} + 2256 q^{67} + 5076 q^{73} + 4944 q^{91} + 2916 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 40x^{10} + 607x^{8} - 3980x^{6} + 10171x^{4} + 2180x^{2} + 7225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -112\nu^{10} + 4251\nu^{8} - 59662\nu^{6} + 351136\nu^{4} - 859020\nu^{2} + 30265 ) / 745470 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5609\nu^{10} + 121113\nu^{8} + 672748\nu^{6} - 35468782\nu^{4} + 227845017\nu^{2} - 156810535 ) / 19754955 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -15163\nu^{10} + 737997\nu^{8} - 13093372\nu^{6} + 96635062\nu^{4} - 208300629\nu^{2} - 171900455 ) / 19754955 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -54877\nu^{10} + 2356437\nu^{8} - 39188800\nu^{6} + 290640862\nu^{4} - 965215023\nu^{2} + 783531745 ) / 19754955 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 112225\nu^{10} - 4317447\nu^{8} + 63619696\nu^{6} - 401170222\nu^{4} + 930976971\nu^{2} + 771343985 ) / 19754955 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35204 \nu^{11} + 935934 \nu^{9} - 4715135 \nu^{7} - 49157461 \nu^{5} + 309654879 \nu^{3} - 1851042625 \nu ) / 335834235 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -139\nu^{11} + 5730\nu^{9} - 89728\nu^{7} + 617650\nu^{5} - 1799499\nu^{3} + 1161190\nu ) / 405450 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 57207 \nu^{11} + 2198010 \nu^{9} - 31637874 \nu^{7} + 189227820 \nu^{5} + \cdots - 443967180 \nu ) / 124383050 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -13705\nu^{11} + 588932\nu^{9} - 10091236\nu^{7} + 82664002\nu^{5} - 314902881\nu^{3} + 182990300\nu ) / 24876610 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 583807 \nu^{11} - 24208128 \nu^{9} + 379462288 \nu^{7} - 2536560058 \nu^{5} + \cdots + 10545511100 \nu ) / 671668470 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 317905 \nu^{11} + 12012876 \nu^{9} - 168159283 \nu^{7} + 973129021 \nu^{5} + \cdots - 121441475 \nu ) / 335834235 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{11} - 4\beta_{8} - 2\beta_{7} - 3\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 3\beta_{4} + 3\beta_{3} - 3\beta_{2} + 8\beta _1 + 160 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 51\beta_{11} - 18\beta_{10} + 6\beta_{9} - 82\beta_{8} - 76\beta_{7} - 33\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 49\beta_{4} + 33\beta_{3} - 87\beta_{2} + 864\beta _1 + 1544 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 783\beta_{11} - 300\beta_{10} + 144\beta_{9} - 1566\beta_{8} - 988\beta_{7} - 213\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 117\beta_{5} - 353\beta_{4} + 270\beta_{3} - 648\beta_{2} + 10796\beta _1 + 6200 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10491\beta_{11} - 4494\beta_{10} + 1350\beta_{9} - 26148\beta_{8} - 7814\beta_{7} + 1185\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 580\beta_{5} - 724\beta_{4} + 943\beta_{3} - 1285\beta_{2} + 32928\beta _1 + 2738 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 117543\beta_{11} - 64116\beta_{10} - 1356\beta_{9} - 394264\beta_{8} + 21298\beta_{7} + 74229\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 144053\beta_{5} - 84285\beta_{4} + 222297\beta_{3} - 144645\beta_{2} + 5983208\beta _1 - 1738400 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 917367 \beta_{11} - 842094 \beta_{10} - 365166 \beta_{9} - 5303962 \beta_{8} + 2429084 \beta_{7} + 1750683 \beta_{6} ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−0.539657 − 0.707107i
0.539657 + 0.707107i
3.81960 − 0.707107i
−3.81960 + 0.707107i
2.57283 + 0.707107i
−2.57283 − 0.707107i
0.539657 − 0.707107i
−0.539657 + 0.707107i
−3.81960 − 0.707107i
3.81960 + 0.707107i
−2.57283 + 0.707107i
2.57283 − 0.707107i
0 0 0 0 0 −7.36478 − 7.36478i 0 0 0
557.2 0 0 0 0 0 −7.36478 − 7.36478i 0 0 0
557.3 0 0 0 0 0 −3.42828 − 3.42828i 0 0 0
557.4 0 0 0 0 0 −3.42828 − 3.42828i 0 0 0
557.5 0 0 0 0 0 16.7931 + 16.7931i 0 0 0
557.6 0 0 0 0 0 16.7931 + 16.7931i 0 0 0
593.1 0 0 0 0 0 −7.36478 + 7.36478i 0 0 0
593.2 0 0 0 0 0 −7.36478 + 7.36478i 0 0 0
593.3 0 0 0 0 0 −3.42828 + 3.42828i 0 0 0
593.4 0 0 0 0 0 −3.42828 + 3.42828i 0 0 0
593.5 0 0 0 0 0 16.7931 − 16.7931i 0 0 0
593.6 0 0 0 0 0 16.7931 − 16.7931i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.4.j.b 12
3.b odd 2 1 inner 900.4.j.b 12
5.b even 2 1 180.4.j.a ✓ 12
5.c odd 4 1 180.4.j.a ✓ 12
5.c odd 4 1 inner 900.4.j.b 12
15.d odd 2 1 180.4.j.a ✓ 12
15.e even 4 1 180.4.j.a ✓ 12
15.e even 4 1 inner 900.4.j.b 12
20.d odd 2 1 720.4.w.e 12
20.e even 4 1 720.4.w.e 12
60.h even 2 1 720.4.w.e 12
60.l odd 4 1 720.4.w.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.j.a ✓ 12 5.b even 2 1
180.4.j.a ✓ 12 5.c odd 4 1
180.4.j.a ✓ 12 15.d odd 2 1
180.4.j.a ✓ 12 15.e even 4 1
720.4.w.e 12 20.d odd 2 1
720.4.w.e 12 20.e even 4 1
720.4.w.e 12 60.h even 2 1
720.4.w.e 12 60.l odd 4 1
900.4.j.b 12 1.a even 1 1 trivial
900.4.j.b 12 3.b odd 2 1 inner
900.4.j.b 12 5.c odd 4 1 inner
900.4.j.b 12 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 12T_{7}^{5} + 72T_{7}^{4} + 5440T_{7}^{3} + 97344T_{7}^{2} + 529152T_{7} + 1438208 \) acting on \(S_{4}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 12 T^{5} + \cdots + 1438208)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 3384 T^{4} + \cdots + 430592)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 42 T^{5} + \cdots + 25956444168)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 97\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 1591876796416)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 4562580070472)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 36 T^{2} + \cdots - 1213888)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 26307385494408)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 52806 T^{4} + \cdots + 168618248)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 480 T^{5} + \cdots + 537477120000)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 201231329403392)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 480 T^{2} + \cdots + 116380800)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 50813522444288)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 46\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 97\!\cdots\!28)^{2} \) Copy content Toggle raw display
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