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Properties

Label 3025.2.a.bo
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
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] = [3025,2,Mod(1,3025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, 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("3025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.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: \(24.1547466114\)
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: \( x^{12} - 2 x^{11} - 34 x^{10} + 28 x^{9} + 340 x^{8} + 44 x^{7} - 884 x^{6} - 132 x^{5} + 761 x^{4} + \cdots - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 605)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + ( - \beta_{10} + 2) q^{4} + \beta_{2} q^{6} + ( - \beta_{5} + \beta_1) q^{7} + (\beta_{9} + 2 \beta_1) q^{8} + ( - \beta_{10} - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + ( - \beta_{10} + 2) q^{4} + \beta_{2} q^{6} + ( - \beta_{5} + \beta_1) q^{7} + (\beta_{9} + 2 \beta_1) q^{8} + ( - \beta_{10} - \beta_{6} + 2) q^{9} + (\beta_{8} - \beta_{7} - 3 \beta_{4}) q^{12} + (\beta_{9} - \beta_{5} + \beta_1) q^{13} + ( - \beta_{10} - \beta_{6} + 3) q^{14} + (2 \beta_{6} + 3) q^{16} - \beta_{5} q^{17} + ( - \beta_{9} - \beta_{5} + 3 \beta_1) q^{18} + ( - \beta_{11} - \beta_{3}) q^{19} + ( - 2 \beta_{3} + \beta_{2}) q^{21} + ( - \beta_{8} - \beta_{7}) q^{23} + (\beta_{11} + 3 \beta_{3} + \beta_{2}) q^{24} + ( - \beta_{10} + \beta_{6} + 2) q^{26} + (\beta_{8} + \beta_{7} - \beta_{4}) q^{27} + ( - \beta_{9} + \beta_{5} + 2 \beta_1) q^{28} + ( - \beta_{3} - \beta_{2}) q^{29} + ( - \beta_{10} - 2 \beta_{6} + 1) q^{31} + (2 \beta_{9} + 2 \beta_{5} + \beta_1) q^{32} + ( - \beta_{6} - 1) q^{34} + ( - \beta_{10} - \beta_{6} + 8) q^{36} + ( - \beta_{8} - \beta_{4}) q^{37} + ( - \beta_{8} + 3 \beta_{7} + 2 \beta_{4}) q^{38} + (\beta_{11} + \beta_{3}) q^{39} + ( - \beta_{11} + 2 \beta_{3}) q^{41} + (\beta_{8} + \beta_{7} - 5 \beta_{4}) q^{42} + (2 \beta_{9} + \beta_{5} - \beta_1) q^{43} + (\beta_{11} + 5 \beta_{3}) q^{46} + \beta_{4} q^{47} + ( - 4 \beta_{7} - \beta_{4}) q^{48} + (\beta_{10} - \beta_{6}) q^{49} - 2 \beta_{3} q^{51} + (\beta_{9} + 3 \beta_{5} + 3 \beta_1) q^{52} + (\beta_{7} - \beta_{4}) q^{53} + ( - \beta_{11} - 5 \beta_{3} + \beta_{2}) q^{54} + (\beta_{6} + 4) q^{56} + ( - 3 \beta_{9} - 3 \beta_1) q^{57} + ( - \beta_{8} + 2 \beta_{7} + 5 \beta_{4}) q^{58} + ( - \beta_{10} + 2 \beta_{6} + 3) q^{59} + (\beta_{11} + \beta_{3} - \beta_{2}) q^{61} + ( - 3 \beta_{9} - 2 \beta_{5} + \beta_1) q^{62} + ( - 3 \beta_{9} + 3 \beta_1) q^{63} + ( - \beta_{10} + 2 \beta_{6} - 2) q^{64} + ( - 2 \beta_{8} + 2 \beta_{7} + \beta_{4}) q^{67} + ( - 2 \beta_{9} + \beta_{5} - 2 \beta_1) q^{68} + (\beta_{10} + 4 \beta_{6} + 1) q^{69} + (\beta_{10} - 2 \beta_{6} + 9) q^{71} + (\beta_{9} + \beta_{5} + 3 \beta_1) q^{72} + (2 \beta_{5} + 4 \beta_1) q^{73} + (\beta_{3} + \beta_{2}) q^{74} + ( - \beta_{11} - 9 \beta_{3} - 2 \beta_{2}) q^{76} + (\beta_{8} - 3 \beta_{7} - 2 \beta_{4}) q^{78} + (2 \beta_{3} - 2 \beta_{2}) q^{79} + (\beta_{10} - 2 \beta_{6} - 2) q^{81} + ( - \beta_{8} + 2 \beta_{4}) q^{82} + ( - \beta_{9} - 4 \beta_{5} + \beta_1) q^{83} + ( - \beta_{11} - \beta_{3} + 3 \beta_{2}) q^{84} + (\beta_{10} + 5 \beta_{6} - 5) q^{86} - 6 \beta_1 q^{87} + 9 q^{89} + (3 \beta_{10} + 2 \beta_{6} + 5) q^{91} + (3 \beta_{8} - 5 \beta_{7} - 2 \beta_{4}) q^{92} + (\beta_{8} + 3 \beta_{7} - 4 \beta_{4}) q^{93} - \beta_{2} q^{94} + (2 \beta_{11} + 10 \beta_{3} - \beta_{2}) q^{96} + ( - 2 \beta_{8} - 3 \beta_{7} + \beta_{4}) q^{97} + ( - 3 \beta_{9} - \beta_{5} - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 20 q^{4} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 20 q^{4} + 20 q^{9} + 32 q^{14} + 36 q^{16} + 20 q^{26} + 8 q^{31} - 12 q^{34} + 92 q^{36} + 4 q^{49} + 48 q^{56} + 32 q^{59} - 28 q^{64} + 16 q^{69} + 112 q^{71} - 20 q^{81} - 56 q^{86} + 108 q^{89} + 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 34 x^{10} + 28 x^{9} + 340 x^{8} + 44 x^{7} - 884 x^{6} - 132 x^{5} + 761 x^{4} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 153391689 \nu^{11} + 208588499 \nu^{10} + 5371359931 \nu^{9} - 900487575 \nu^{8} + \cdots - 613807515 ) / 304631027 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11025817 \nu^{11} + 14668720 \nu^{10} + 385385890 \nu^{9} - 49729176 \nu^{8} + \cdots + 137162752 ) / 16466542 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24956495 \nu^{11} + 37487300 \nu^{10} + 866026610 \nu^{9} - 264468764 \nu^{8} + \cdots - 422203120 ) / 32933084 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 500197417 \nu^{11} + 863147766 \nu^{10} + 17255679184 \nu^{9} - 9304386214 \nu^{8} + \cdots - 14983204480 ) / 609262054 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 254231037 \nu^{11} - 459310711 \nu^{10} - 8746107718 \nu^{9} + 5448963459 \nu^{8} + \cdots + 7385941755 ) / 304631027 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15009352 \nu^{11} - 21424894 \nu^{10} - 523036116 \nu^{9} + 121169203 \nu^{8} + 5188967132 \nu^{7} + \cdots + 144735042 ) / 8233271 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2927126331 \nu^{11} + 4229906008 \nu^{10} + 101969772882 \nu^{9} - 25562559320 \nu^{8} + \cdots - 31534669536 ) / 1218524108 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3833949609 \nu^{11} - 6506115372 \nu^{10} - 132303997562 \nu^{9} + 67150231880 \nu^{8} + \cdots + 97627303520 ) / 1218524108 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1340487538 \nu^{11} + 2163331836 \nu^{10} + 46408123246 \nu^{9} - 19623296403 \nu^{8} + \cdots - 30440942833 ) / 304631027 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 36289910 \nu^{11} + 58969591 \nu^{10} + 1255624472 \nu^{9} - 544172476 \nu^{8} + \cdots - 781784115 ) / 8233271 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 254146587 \nu^{11} + 412494360 \nu^{10} + 8795458338 \nu^{9} - 3799513884 \nu^{8} + \cdots - 5439656816 ) / 32933084 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{8} - \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 3\beta_{10} + 4\beta_{8} - 2\beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{3} + 2\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} + 22 \beta_{10} - 7 \beta_{9} + 19 \beta_{8} + \beta_{7} + 5 \beta_{6} + 9 \beta_{5} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18 \beta_{11} + 40 \beta_{10} - 8 \beta_{9} + 50 \beta_{8} - 18 \beta_{7} + 24 \beta_{6} + 34 \beta_{5} + \cdots + 94 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 145 \beta_{11} + 506 \beta_{10} - 215 \beta_{9} + 459 \beta_{8} - 17 \beta_{7} + 143 \beta_{6} + \cdots + 500 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1077 \beta_{11} + 2133 \beta_{10} - 732 \beta_{9} + 2482 \beta_{8} - 710 \beta_{7} + 1126 \beta_{6} + \cdots + 4117 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4998 \beta_{11} + 12487 \beta_{10} - 6010 \beta_{9} + 11946 \beta_{8} - 1188 \beta_{7} + 4093 \beta_{6} + \cdots + 14885 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15544 \beta_{11} + 28950 \beta_{10} - 12588 \beta_{9} + 32112 \beta_{8} - 7776 \beta_{7} + 13829 \beta_{6} + \cdots + 49968 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 153846 \beta_{11} + 324181 \beta_{10} - 166540 \beta_{9} + 321248 \beta_{8} - 43678 \beta_{7} + \cdots + 426447 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 883639 \beta_{11} + 1591321 \beta_{10} - 777520 \beta_{9} + 1715550 \beta_{8} - 370874 \beta_{7} + \cdots + 2572105 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4511661 \beta_{11} + 8685878 \beta_{10} - 4623709 \beta_{9} + 8789643 \beta_{8} - 1377009 \beta_{7} + \cdots + 12060744 ) / 2 \) Copy content Toggle raw display

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

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
−0.630269
0.437593
1.40343
0.527636
3.88174
0.338792
5.28545
−1.72161
−2.35143
0.236881
−1.50598
−3.90222
−2.60777 −2.13980 4.80044 0 5.58011 −0.988879 −7.30289 1.57876 0
1.2 −2.60777 2.13980 4.80044 0 −5.58011 −0.988879 −7.30289 1.57876 0
1.3 −2.02281 −2.91475 2.09174 0 5.89598 −3.21128 −0.185581 5.49579 0
1.4 −2.02281 2.91475 2.09174 0 −5.89598 −3.21128 −0.185581 5.49579 0
1.5 −0.328351 −0.962000 −1.89219 0 0.315873 3.27259 1.27800 −2.07456 0
1.6 −0.328351 0.962000 −1.89219 0 −0.315873 3.27259 1.27800 −2.07456 0
1.7 0.328351 −0.962000 −1.89219 0 −0.315873 −3.27259 −1.27800 −2.07456 0
1.8 0.328351 0.962000 −1.89219 0 0.315873 −3.27259 −1.27800 −2.07456 0
1.9 2.02281 −2.91475 2.09174 0 −5.89598 3.21128 0.185581 5.49579 0
1.10 2.02281 2.91475 2.09174 0 5.89598 3.21128 0.185581 5.49579 0
1.11 2.60777 −2.13980 4.80044 0 −5.58011 0.988879 7.30289 1.57876 0
1.12 2.60777 2.13980 4.80044 0 5.58011 0.988879 7.30289 1.57876 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.2.a.bo 12
5.b even 2 1 inner 3025.2.a.bo 12
5.c odd 4 2 605.2.b.h ✓ 12
11.b odd 2 1 inner 3025.2.a.bo 12
55.d odd 2 1 inner 3025.2.a.bo 12
55.e even 4 2 605.2.b.h ✓ 12
55.k odd 20 8 605.2.j.k 48
55.l even 20 8 605.2.j.k 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.h ✓ 12 5.c odd 4 2
605.2.b.h ✓ 12 55.e even 4 2
605.2.j.k 48 55.k odd 20 8
605.2.j.k 48 55.l even 20 8
3025.2.a.bo 12 1.a even 1 1 trivial
3025.2.a.bo 12 5.b even 2 1 inner
3025.2.a.bo 12 11.b odd 2 1 inner
3025.2.a.bo 12 55.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{6} - 11T_{2}^{4} + 29T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{6} - 14T_{3}^{4} + 51T_{3}^{2} - 36 \) Copy content Toggle raw display
\( T_{19}^{6} - 72T_{19}^{4} + 1332T_{19}^{2} - 3888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 11 T^{4} + 29 T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 14 T^{4} + \cdots - 36)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 22 T^{4} + \cdots - 108)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 37 T^{4} + \cdots - 108)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 17 T^{4} + \cdots - 48)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 72 T^{4} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 80 T^{4} + \cdots - 17424)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 75 T^{4} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 2 T^{2} - 38 T - 68)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 51 T^{4} + \cdots - 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 117 T^{4} + \cdots - 2187)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 130 T^{4} + \cdots - 432)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 14 T^{4} + \cdots - 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 47 T^{4} + \cdots - 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 50 T - 24)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} - 90 T^{4} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 174 T^{4} + \cdots - 39204)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 28 T^{2} + \cdots + 48)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 292 T^{4} + \cdots - 442368)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 300 T^{4} + \cdots - 110592)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 296 T^{4} + \cdots - 591408)^{2} \) Copy content Toggle raw display
$89$ \( (T - 9)^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} - 519 T^{4} + \cdots - 4743684)^{2} \) Copy content Toggle raw display
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