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Properties

Label 225.12.b.p
Level $225$
Weight $12$
Character orbit 225.b
Analytic conductor $172.877$
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] = [225,12,Mod(199,225)]
 
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([0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.b (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: \(172.877215626\)
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: \( x^{12} + 8236125x^{8} + 4753901602500x^{4} + 650506771600000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{16}\cdot 5^{22}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} - 522) q^{4} + (\beta_{9} + \beta_{6} + 17 \beta_{3}) q^{7} + ( - \beta_{7} + 177 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} - 522) q^{4} + (\beta_{9} + \beta_{6} + 17 \beta_{3}) q^{7} + ( - \beta_{7} + 177 \beta_1) q^{8} + \beta_{10} q^{11} + (46 \beta_{9} - 66 \beta_{6} - 541 \beta_{3}) q^{13} + ( - 2 \beta_{10} + \beta_{8} - \beta_{5}) q^{14} + ( - 4 \beta_{4} + 446 \beta_{2} - 614276) q^{16} + ( - 3 \beta_{11} - 22 \beta_{7} + 1502 \beta_1) q^{17} + (53 \beta_{4} - 1841 \beta_{2} + 3179079) q^{19} + (540 \beta_{9} - 144 \beta_{6} - 524 \beta_{3}) q^{22} + (7 \beta_{11} - 204 \beta_{7} + 39120 \beta_1) q^{23} + ( - 92 \beta_{10} + \cdots + 87 \beta_{5}) q^{26}+ \cdots + ( - 15320 \beta_{11} + \cdots - 1485725098 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6264 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6264 q^{4} - 7371312 q^{16} + 38148948 q^{19} - 439482204 q^{31} + 46294080 q^{34} + 1206188160 q^{46} + 8855893416 q^{49} + 27445422324 q^{61} - 8440465056 q^{64} + 77322646344 q^{76} + 54885978192 q^{79} - 371882923500 q^{91} - 105907791360 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 8236125x^{8} + 4753901602500x^{4} + 650506771600000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -771\nu^{11} - 6511360375\nu^{7} - 5528135671627500\nu^{3} - 1929720304813000000\nu ) / 1929720304813000000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} - 11548575\nu^{4} - 12263185153750 ) / 5981477375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -771\nu^{10} - 6511360375\nu^{6} - 5528135671627500\nu^{2} ) / 3087552487700800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -659\nu^{8} - 5217919975\nu^{4} - 1512879636965000 ) / 2392590950 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -771\nu^{11} - 6511360375\nu^{7} - 5528135671627500\nu^{3} + 1929720304813000000\nu ) / 3087552487700800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -400149\nu^{10} - 3379396034625\nu^{6} - 2097214291649472500\nu^{2} ) / 77188812192520000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3294483 \nu^{11} - 322616000 \nu^{9} - 27823042882375 \nu^{7} + \cdots - 72\!\cdots\!00 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 162681 \nu^{11} + 16130800 \nu^{9} - 1373897039125 \nu^{7} + 186287753610000 \nu^{5} + \cdots + 35\!\cdots\!00 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5748161\nu^{10} - 45684764112125\nu^{6} - 14135432612098102500\nu^{2} ) / 55134865851800000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20359501 \nu^{11} - 10694720400 \nu^{9} + 161931474509625 \nu^{7} + \cdots - 25\!\cdots\!00 \nu ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 509582737 \nu^{11} + 267368010000 \nu^{9} + \cdots + 64\!\cdots\!00 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 625\beta_1 ) / 1250 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 25\beta_{6} - 519\beta_{3} ) / 250 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 25\beta_{8} + 125\beta_{7} - 844\beta_{5} - 534125\beta_1 ) / 500 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} - 3295\beta_{2} - 5490750 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -100\beta_{11} + 500\beta_{10} + 16575\beta_{8} - 82875\beta_{7} - 437208\beta_{5} + 277570175\beta_1 ) / 100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 192750\beta_{9} - 9466875\beta_{6} + 116058071\beta_{3} ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 77100 \beta_{11} - 385500 \beta_{10} - 9466875 \beta_{8} - 47334375 \beta_{7} + \cdots + 151301510675 \beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -23097150\beta_{4} + 26089599875\beta_{2} + 38883967873750 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 46194300 \beta_{11} - 230971500 \beta_{10} - 5264114275 \beta_{8} + 26320571375 \beta_{7} + \cdots - 83322317103275 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -325568018750\beta_{9} + 14556186531375\beta_{6} - 174269092877395\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 130227207500 \beta_{11} + 651136037500 \beta_{10} + 14556186531375 \beta_{8} + \cdots - 22\!\cdots\!75 \beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
37.1559 − 37.1559i
−37.1559 − 37.1559i
17.7602 − 17.7602i
−17.7602 − 17.7602i
−15.2158 − 15.2158i
15.2158 − 15.2158i
15.2158 + 15.2158i
−15.2158 + 15.2158i
−17.7602 + 17.7602i
17.7602 + 17.7602i
−37.1559 + 37.1559i
37.1559 + 37.1559i
− 74.3117i 0 −3474.23 0 0 − 7094.45i 105985.i 0 0
199.2 − 74.3117i 0 −3474.23 0 0 7094.45i 105985.i 0 0
199.3 − 35.5203i 0 786.308 0 0 − 18581.0i − 100675.i 0 0
199.4 − 35.5203i 0 786.308 0 0 18581.0i − 100675.i 0 0
199.5 − 30.4316i 0 1121.92 0 0 − 57640.5i − 96465.6i 0 0
199.6 − 30.4316i 0 1121.92 0 0 57640.5i − 96465.6i 0 0
199.7 30.4316i 0 1121.92 0 0 − 57640.5i 96465.6i 0 0
199.8 30.4316i 0 1121.92 0 0 57640.5i 96465.6i 0 0
199.9 35.5203i 0 786.308 0 0 − 18581.0i 100675.i 0 0
199.10 35.5203i 0 786.308 0 0 18581.0i 100675.i 0 0
199.11 74.3117i 0 −3474.23 0 0 − 7094.45i − 105985.i 0 0
199.12 74.3117i 0 −3474.23 0 0 7094.45i − 105985.i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.12
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 225.12.b.p 12
3.b odd 2 1 inner 225.12.b.p 12
5.b even 2 1 inner 225.12.b.p 12
5.c odd 4 1 225.12.a.w ✓ 6
5.c odd 4 1 225.12.a.x yes 6
15.d odd 2 1 inner 225.12.b.p 12
15.e even 4 1 225.12.a.w ✓ 6
15.e even 4 1 225.12.a.x yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.12.a.w ✓ 6 5.c odd 4 1
225.12.a.w ✓ 6 15.e even 4 1
225.12.a.x yes 6 5.c odd 4 1
225.12.a.x yes 6 15.e even 4 1
225.12.b.p 12 1.a even 1 1 trivial
225.12.b.p 12 3.b odd 2 1 inner
225.12.b.p 12 5.b even 2 1 inner
225.12.b.p 12 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{6} + 7710T_{2}^{4} + 13249800T_{2}^{2} + 6452320000 \) Copy content Toggle raw display
\( T_{11}^{6} - 742702914000T_{11}^{4} + 136807837996572240000000T_{11}^{2} - 328309984689267868000000000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 7710 T^{4} + \cdots + 6452320000)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 57\!\cdots\!25)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 11\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 15\!\cdots\!39)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 25\!\cdots\!13)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 10\!\cdots\!25)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 36\!\cdots\!17)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 21\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 45\!\cdots\!96)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 28\!\cdots\!25)^{2} \) Copy content Toggle raw display
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