Properties

Label 4225.2.a.bv
Level $4225$
Weight $2$
Character orbit 4225.a
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4225,2,Mod(1,4225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

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: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 16x^{8} + 84x^{6} - 163x^{4} + 118x^{2} - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 325)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{8} - \beta_{3} + \beta_1) q^{6} - \beta_{8} q^{7} + (\beta_{8} + \beta_{6} - \beta_{3} + \beta_1) q^{8} + (\beta_{7} + \beta_{4} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{8} - \beta_{3} + \beta_1) q^{6} - \beta_{8} q^{7} + (\beta_{8} + \beta_{6} - \beta_{3} + \beta_1) q^{8} + (\beta_{7} + \beta_{4} - \beta_{2} + 2) q^{9} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{12} + ( - \beta_{7} - \beta_{4} - 1) q^{14} + (\beta_{4} + \beta_{2} + 2) q^{16} + ( - \beta_{7} + \beta_{4} + 1) q^{17} + (\beta_{9} + 2 \beta_{8} - \beta_{6} - 2 \beta_{3} + \beta_1) q^{18} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \beta_1) q^{19} + ( - \beta_{8} + \beta_{6} - 2 \beta_1) q^{21} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + 2) q^{22} + (\beta_{7} - \beta_{5} + 3) q^{23} + (2 \beta_{9} + \beta_{8} - 4 \beta_{3} + 4 \beta_1) q^{24} + (2 \beta_{7} - \beta_{2} + 4) q^{27} + ( - \beta_{9} - \beta_{8} + 3 \beta_{3} - 2 \beta_1) q^{28} + (3 \beta_{7} - \beta_{2}) q^{29} + (\beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{3} + 2 \beta_1) q^{31} + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{3} + 3 \beta_1) q^{32} + ( - \beta_{9} + \beta_{6} - 2 \beta_1) q^{33} + (\beta_{9} + \beta_{8} - \beta_{3} + 2 \beta_1) q^{34} + (3 \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{36} + ( - 2 \beta_{9} - \beta_{8} + \beta_{3} + \beta_1) q^{37} + (\beta_{7} - \beta_{5} + 3 \beta_{2} + 6) q^{38} + ( - \beta_{9} - 2 \beta_{8} + 2 \beta_{3}) q^{41} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} - 7) q^{42} + (2 \beta_{7} + \beta_{5} - 2 \beta_{2} + 2) q^{43} + ( - 2 \beta_{9} - \beta_{8} + 2 \beta_{6} + 4 \beta_{3}) q^{44} + ( - \beta_{9} + \beta_{6} + 3 \beta_{3} + 2 \beta_1) q^{46} + ( - \beta_{9} + \beta_{8} + 2 \beta_{3} - \beta_1) q^{47} + (\beta_{7} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 5) q^{48} + (2 \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{49} + (2 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2} - 1) q^{51} + ( - 2 \beta_{7} + \beta_{4} + 2 \beta_{2} + 1) q^{53} + (\beta_{8} - \beta_{6} - \beta_{3} + 2 \beta_1) q^{54} + (2 \beta_{7} - 3 \beta_{5} - 2 \beta_{2} - 3) q^{56} + (2 \beta_{9} + \beta_{8} - 2 \beta_{6} + 5 \beta_1) q^{57} + (2 \beta_{8} - \beta_{6} - 2 \beta_{3} - 2 \beta_1) q^{58} + ( - 2 \beta_{9} - \beta_{8} - 2 \beta_{3} - \beta_1) q^{59} + (2 \beta_{7} - 3 \beta_{5} + \beta_{2} - 2) q^{61} + ( - \beta_{7} + 3 \beta_{5} + 3) q^{62} + (\beta_{9} - \beta_{8} + 2 \beta_{6} - 4 \beta_1) q^{63} + (\beta_{7} + 2 \beta_{5} - \beta_{2} + 4) q^{64} + ( - \beta_{5} - \beta_{4} - 4) q^{66} + (\beta_{8} + \beta_{6} + \beta_1) q^{67} + (2 \beta_{7} + \beta_{5} + 2 \beta_{2} + 3) q^{68} + (4 \beta_{7} - \beta_{5} - 2 \beta_{2} + 6) q^{69} + ( - 3 \beta_{8} - \beta_{6} + 3 \beta_{3} + 2 \beta_1) q^{71} + (2 \beta_{9} + 2 \beta_{8} - 11 \beta_{3} + 3 \beta_1) q^{72} + (\beta_{9} - 3 \beta_{8} - 2 \beta_{6} + 4 \beta_{3} + 2 \beta_1) q^{73} + (\beta_{7} - \beta_{5} - 3 \beta_{4} + \beta_{2} + 6) q^{74} + (\beta_{9} + \beta_{8} + 2 \beta_{6} + 9 \beta_1) q^{76} + (\beta_{5} - \beta_{4} + 5) q^{77} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{79} + (3 \beta_{7} - \beta_{5} - 2 \beta_{4} + 5) q^{81} + ( - \beta_{7} - 2 \beta_{5} - 3 \beta_{4}) q^{82} + ( - \beta_{8} - \beta_{6} + \beta_{3} - 4 \beta_1) q^{83} + ( - 2 \beta_{9} - 3 \beta_{8} - \beta_{6} + 8 \beta_{3} - 5 \beta_1) q^{84} + (\beta_{9} + \beta_{8} - 3 \beta_{6} - 4 \beta_{3} - \beta_1) q^{86} + (2 \beta_{7} - \beta_{5} + 2 \beta_{4} - 4 \beta_{2} + 11) q^{87} + (\beta_{7} - 4 \beta_{5} - \beta_{4} + 2 \beta_{2} - 1) q^{88} + (2 \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{3} + \beta_1) q^{89} + ( - 2 \beta_{7} - 2 \beta_{5} - \beta_{4} + 4 \beta_{2} + 2) q^{92} + (2 \beta_{8} + 2 \beta_{6} - 5 \beta_{3}) q^{93} + (2 \beta_{7} - 2 \beta_{5} - \beta_{2}) q^{94} + ( - \beta_{9} + 5 \beta_{8} - 5 \beta_{3} + 4 \beta_1) q^{96} + (\beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{3} + \beta_1) q^{97} + ( - 2 \beta_{6} - 3 \beta_{3} - \beta_1) q^{98} + (2 \beta_{9} - \beta_{6} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9} + 28 q^{12} - 8 q^{14} + 24 q^{16} + 16 q^{17} + 24 q^{22} + 26 q^{23} + 30 q^{27} - 14 q^{29} - 6 q^{36} + 62 q^{38} - 64 q^{42} + 8 q^{43} + 52 q^{48} - 2 q^{49} - 16 q^{51} + 24 q^{53} - 42 q^{56} - 26 q^{61} + 34 q^{62} + 34 q^{64} - 42 q^{66} + 26 q^{68} + 40 q^{69} + 52 q^{74} + 48 q^{77} + 4 q^{79} + 34 q^{81} - 2 q^{82} + 98 q^{87} - 12 q^{88} + 34 q^{92} - 10 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 16x^{8} + 84x^{6} - 163x^{4} + 118x^{2} - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + 13\nu^{7} - 45\nu^{5} + 13\nu^{3} + 41\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 13\nu^{6} + 50\nu^{4} - 53\nu^{2} + 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{9} - 31\nu^{7} + 150\nu^{5} - 226\nu^{3} + 68\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{8} + 31\nu^{6} - 150\nu^{4} + 231\nu^{2} - 93 ) / 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{9} + 106\nu^{7} - 495\nu^{5} + 706\nu^{3} - 238\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4\nu^{9} - 62\nu^{7} + 305\nu^{5} - 497\nu^{3} + 206\nu ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 9\beta_{8} + 7\beta_{6} - 9\beta_{3} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + 2\beta_{5} + 10\beta_{4} + 45\beta_{2} + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{9} + 68\beta_{8} + 43\beta_{6} - 74\beta_{3} + 202\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{7} + 31\beta_{5} + 80\beta_{4} + 288\beta_{2} + 650 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 111\beta_{9} + 492\beta_{8} + 257\beta_{6} - 585\beta_{3} + 1337\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.63252
−2.44399
−1.36551
−0.887996
−0.666048
0.666048
0.887996
1.36551
2.44399
2.63252
−2.63252 2.69905 4.93016 0 −7.10530 2.74073 −7.71370 4.28488 0
1.2 −2.44399 0.121859 3.97307 0 −0.297823 −0.414112 −4.82215 −2.98515 0
1.3 −1.36551 −0.399918 −0.135381 0 0.546092 −3.64365 2.91589 −2.84007 0
1.4 −0.887996 3.06193 −1.21146 0 −2.71898 3.56304 2.85177 6.37543 0
1.5 −0.666048 −2.48292 −1.55638 0 1.65375 −0.587744 2.36872 3.16491 0
1.6 0.666048 −2.48292 −1.55638 0 −1.65375 0.587744 −2.36872 3.16491 0
1.7 0.887996 3.06193 −1.21146 0 2.71898 −3.56304 −2.85177 6.37543 0
1.8 1.36551 −0.399918 −0.135381 0 −0.546092 3.64365 −2.91589 −2.84007 0
1.9 2.44399 0.121859 3.97307 0 0.297823 0.414112 4.82215 −2.98515 0
1.10 2.63252 2.69905 4.93016 0 7.10530 −2.74073 7.71370 4.28488 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4225.2.a.bv 10
5.b even 2 1 4225.2.a.bu 10
13.b even 2 1 inner 4225.2.a.bv 10
13.f odd 12 2 325.2.n.e 10
65.d even 2 1 4225.2.a.bu 10
65.o even 12 2 325.2.m.d 20
65.s odd 12 2 325.2.n.f yes 10
65.t even 12 2 325.2.m.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.m.d 20 65.o even 12 2
325.2.m.d 20 65.t even 12 2
325.2.n.e 10 13.f odd 12 2
325.2.n.f yes 10 65.s odd 12 2
4225.2.a.bu 10 5.b even 2 1
4225.2.a.bu 10 65.d even 2 1
4225.2.a.bv 10 1.a even 1 1 trivial
4225.2.a.bv 10 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):

\( T_{2}^{10} - 16T_{2}^{8} + 84T_{2}^{6} - 163T_{2}^{4} + 118T_{2}^{2} - 27 \) Copy content Toggle raw display
\( T_{3}^{5} - 3T_{3}^{4} - 7T_{3}^{3} + 19T_{3}^{2} + 6T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{10} - 34T_{7}^{8} + 381T_{7}^{6} - 1456T_{7}^{4} + 676T_{7}^{2} - 75 \) Copy content Toggle raw display
\( T_{11}^{10} - 45T_{11}^{8} + 687T_{11}^{6} - 4320T_{11}^{4} + 10161T_{11}^{2} - 6075 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 16 T^{8} + 84 T^{6} - 163 T^{4} + \cdots - 27 \) Copy content Toggle raw display
$3$ \( (T^{5} - 3 T^{4} - 7 T^{3} + 19 T^{2} + 6 T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 34 T^{8} + 381 T^{6} + \cdots - 75 \) Copy content Toggle raw display
$11$ \( T^{10} - 45 T^{8} + 687 T^{6} + \cdots - 6075 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( (T^{5} - 8 T^{4} + 61 T^{2} + 10 T - 39)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 136 T^{8} + 6456 T^{6} + \cdots - 531723 \) Copy content Toggle raw display
$23$ \( (T^{5} - 13 T^{4} + 42 T^{3} + 5 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 7 T^{4} - 63 T^{3} - 521 T^{2} + \cdots + 1329)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 154 T^{8} + \cdots - 10546875 \) Copy content Toggle raw display
$37$ \( T^{10} - 289 T^{8} + \cdots - 112914675 \) Copy content Toggle raw display
$41$ \( T^{10} - 172 T^{8} + 9072 T^{6} + \cdots - 45387 \) Copy content Toggle raw display
$43$ \( (T^{5} - 4 T^{4} - 92 T^{3} + 451 T^{2} + \cdots - 725)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 160 T^{8} + 6108 T^{6} + \cdots - 771147 \) Copy content Toggle raw display
$53$ \( (T^{5} - 12 T^{4} - 51 T^{3} + 819 T^{2} + \cdots - 8469)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 402 T^{8} + 44175 T^{6} + \cdots - 3102867 \) Copy content Toggle raw display
$61$ \( (T^{5} + 13 T^{4} - 149 T^{3} + \cdots + 47315)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 102 T^{8} + 3405 T^{6} + \cdots - 16875 \) Copy content Toggle raw display
$71$ \( T^{10} - 325 T^{8} + \cdots - 116251875 \) Copy content Toggle raw display
$73$ \( T^{10} - 490 T^{8} + \cdots - 408916875 \) Copy content Toggle raw display
$79$ \( (T^{5} - 2 T^{4} - 227 T^{3} - 94 T^{2} + \cdots - 9475)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 393 T^{8} + 36615 T^{6} + \cdots - 243 \) Copy content Toggle raw display
$89$ \( T^{10} - 246 T^{8} + \cdots - 45139923 \) Copy content Toggle raw display
$97$ \( T^{10} - 511 T^{8} + 78504 T^{6} + \cdots - 5738067 \) Copy content Toggle raw display
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