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Properties

Label 225.5.d.c
Level $225$
Weight $5$
Character orbit 225.d
Analytic conductor $23.258$
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] = [225,5,Mod(224,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.224");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 225.d (of order \(2\), degree \(1\), 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: \(23.2582416939\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 44 x^{10} - 24 x^{9} + 968 x^{8} - 132 x^{7} - 10486 x^{6} + 2904 x^{5} + 56980 x^{4} + \cdots + 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{10}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_1 + 15) q^{4} + (\beta_{10} - 2 \beta_{7} + 2 \beta_{6}) q^{7} + ( - \beta_{4} - 17 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_1 + 15) q^{4} + (\beta_{10} - 2 \beta_{7} + 2 \beta_{6}) q^{7} + ( - \beta_{4} - 17 \beta_{2}) q^{8} + (\beta_{11} - 2 \beta_{9} + 2 \beta_{8}) q^{11} + ( - 8 \beta_{10} + 9 \beta_{7} - 2 \beta_{6}) q^{13} + (2 \beta_{11} - 4 \beta_{9} - 7 \beta_{8}) q^{14} + ( - 4 \beta_{3} + 16 \beta_1 + 296) q^{16} + ( - \beta_{5} + \beta_{4} - 20 \beta_{2}) q^{17} + (3 \beta_{3} - 7 \beta_1 + 45) q^{19} + ( - 2 \beta_{10} + 47 \beta_{7} + 27 \beta_{6}) q^{22} + (\beta_{5} + 2 \beta_{4} - 53 \beta_{2}) q^{23} + ( - 2 \beta_{11} - 10 \beta_{9} + 35 \beta_{8}) q^{26} + (35 \beta_{10} - 182 \beta_{7} + 77 \beta_{6}) q^{28} + ( - 10 \beta_{11} - 7 \beta_{9} + 6 \beta_{8}) q^{29} + ( - 11 \beta_{3} - 21 \beta_1 + 311) q^{31} + ( - 8 \beta_{5} - 20 \beta_{4} - 336 \beta_{2}) q^{32} + ( - \beta_{3} + 615) q^{34} + (266 \beta_{7} + 2 \beta_{6}) q^{37} + (6 \beta_{5} + 22 \beta_{4} + 99 \beta_{2}) q^{38} + ( - 14 \beta_{11} + \cdots + 42 \beta_{8}) q^{41}+ \cdots + (20 \beta_{5} + 134 \beta_{4} + 1002 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 184 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 184 q^{4} + 3632 q^{16} + 500 q^{19} + 3692 q^{31} + 7384 q^{34} + 19512 q^{46} + 6464 q^{49} + 21676 q^{61} + 72288 q^{64} - 49560 q^{76} - 7808 q^{79} + 34612 q^{91} - 132728 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 44 x^{10} - 24 x^{9} + 968 x^{8} - 132 x^{7} - 10486 x^{6} + 2904 x^{5} + 56980 x^{4} + \cdots + 162 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 92\!\cdots\!64 \nu^{11} + \cdots - 76\!\cdots\!79 ) / 27\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 42\!\cdots\!30 \nu^{11} + \cdots - 20\!\cdots\!68 ) / 82\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34\!\cdots\!96 \nu^{11} + \cdots - 25\!\cdots\!27 ) / 39\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!42 \nu^{11} + \cdots + 95\!\cdots\!36 ) / 82\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33\!\cdots\!52 \nu^{11} + \cdots - 14\!\cdots\!92 ) / 39\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!20 \nu^{11} + \cdots - 28\!\cdots\!53 ) / 35\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 728211395480 \nu^{11} - 98179240470 \nu^{10} - 32041148924650 \nu^{9} + \cdots + 651112431395355 ) / 42822470560023 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 54\!\cdots\!10 \nu^{11} + \cdots + 63\!\cdots\!50 ) / 24\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 74912206398590 \nu^{11} - 10368114558450 \nu^{10} + \cdots + 88\!\cdots\!18 ) / 810126637938499 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 77\!\cdots\!72 \nu^{11} + \cdots + 68\!\cdots\!83 ) / 82\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 71\!\cdots\!86 \nu^{11} + \cdots + 84\!\cdots\!50 ) / 24\!\cdots\!39 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 10\beta_{9} - 54\beta_{8} - 9\beta_{7} - 45\beta_{6} - 9\beta_{3} + 18\beta _1 - 9 ) / 540 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 45 \beta_{11} - 135 \beta_{10} - 145 \beta_{9} + 18 \beta_{8} + 720 \beta_{7} - 45 \beta_{6} + \cdots - 3933 ) / 540 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 70 \beta_{9} + 297 \beta_{8} + 261 \beta_{7} + 495 \beta_{6} + 42 \beta_{5} + 42 \beta_{4} + \cdots + 1818 ) / 270 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -990\beta_{11} + 4455\beta_{10} + 3250\beta_{9} - 720\beta_{8} - 23841\beta_{7} + 1080\beta_{6} ) / 270 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 270 \beta_{11} - 1215 \beta_{10} - 7030 \beta_{9} + 26244 \beta_{8} + 23706 \beta_{7} + 32265 \beta_{6} + \cdots - 222579 ) / 540 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 27225 \beta_{11} - 114345 \beta_{10} - 91475 \beta_{9} + 28710 \beta_{8} + 617400 \beta_{7} + \cdots + 3352419 ) / 270 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25245 \beta_{11} + 106920 \beta_{10} + 370090 \beta_{9} - 1236087 \beta_{8} - 1329156 \beta_{7} + \cdots + 2398185 ) / 270 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -234272\beta_{5} - 715052\beta_{4} - 383520\beta_{3} + 958844\beta_{2} - 1276275\beta _1 - 60245535 ) / 90 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1421145 \beta_{11} - 6027615 \beta_{10} - 15480925 \beta_{9} + 46889172 \beta_{8} + 60802713 \beta_{7} + \cdots + 114116886 ) / 270 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 38600595 \beta_{11} + 163719765 \beta_{10} + 135546145 \beta_{9} - 65506824 \beta_{8} + \cdots + 4853831166 ) / 270 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 58986495 \beta_{11} + 250332930 \beta_{10} + 532313455 \beta_{9} - 1478794806 \beta_{8} + \cdots - 25140551940 ) / 540 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
224.1
2.27380 + 5.48943i
2.27380 − 5.48943i
−2.41063 − 0.998514i
−2.41063 + 0.998514i
0.136829 − 0.330335i
0.136829 + 0.330335i
0.136829 + 0.0566766i
0.136829 − 0.0566766i
−2.41063 + 5.81977i
−2.41063 − 5.81977i
2.27380 − 0.941838i
2.27380 + 0.941838i
−7.84548 0 45.5516 0 0 − 71.8372i −231.847 0 0
224.2 −7.84548 0 45.5516 0 0 71.8372i −231.847 0 0
224.3 −5.40407 0 13.2040 0 0 − 16.7236i 15.1100 0 0
224.4 −5.40407 0 13.2040 0 0 16.7236i 15.1100 0 0
224.5 −1.80123 0 −12.7556 0 0 − 12.1136i 51.7953 0 0
224.6 −1.80123 0 −12.7556 0 0 12.1136i 51.7953 0 0
224.7 1.80123 0 −12.7556 0 0 − 12.1136i −51.7953 0 0
224.8 1.80123 0 −12.7556 0 0 12.1136i −51.7953 0 0
224.9 5.40407 0 13.2040 0 0 − 16.7236i −15.1100 0 0
224.10 5.40407 0 13.2040 0 0 16.7236i −15.1100 0 0
224.11 7.84548 0 45.5516 0 0 − 71.8372i 231.847 0 0
224.12 7.84548 0 45.5516 0 0 71.8372i 231.847 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 224.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.5.d.c 12
3.b odd 2 1 inner 225.5.d.c 12
5.b even 2 1 inner 225.5.d.c 12
5.c odd 4 1 225.5.c.c ✓ 6
5.c odd 4 1 225.5.c.d yes 6
15.d odd 2 1 inner 225.5.d.c 12
15.e even 4 1 225.5.c.c ✓ 6
15.e even 4 1 225.5.c.d yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.5.c.c ✓ 6 5.c odd 4 1
225.5.c.c ✓ 6 15.e even 4 1
225.5.c.d yes 6 5.c odd 4 1
225.5.c.d yes 6 15.e even 4 1
225.5.d.c 12 1.a even 1 1 trivial
225.5.d.c 12 3.b odd 2 1 inner
225.5.d.c 12 5.b even 2 1 inner
225.5.d.c 12 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 94T_{2}^{4} + 2092T_{2}^{2} - 5832 \) acting on \(S_{5}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 94 T^{4} + \cdots - 5832)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 5587 T^{4} + \cdots + 211789809)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 45574 T^{4} + \cdots + 598579922952)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 29082216340849)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 8541059417352)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 125 T^{2} + \cdots + 8576225)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 429606 T^{4} + \cdots - 26533786248)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 34\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 923 T^{2} + \cdots + 974080359)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 54\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 22\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 80\!\cdots\!21)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 16\!\cdots\!12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 5419 T^{2} + \cdots - 1000775657)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 26\!\cdots\!01)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 1952 T^{2} + \cdots - 2952585216)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 28\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 78\!\cdots\!21)^{2} \) Copy content Toggle raw display
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