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Properties

Label 3525.2.a.bi
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3525,2,Mod(1,3525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.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: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 17 x^{11} + 51 x^{10} + 106 x^{9} - 316 x^{8} - 288 x^{7} + 852 x^{6} + 309 x^{5} + \cdots - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 705)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + \beta_{10} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + \beta_{10} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9} + ( - \beta_{12} + 1) q^{11} + (\beta_{2} + 1) q^{12} + (\beta_{5} - 1) q^{13} + (\beta_{12} - \beta_{7} + \beta_{4} - \beta_1) q^{14} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots + 2) q^{16}+ \cdots + ( - \beta_{12} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 3 q^{2} + 13 q^{3} + 17 q^{4} + 3 q^{6} - 4 q^{7} + 15 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 3 q^{2} + 13 q^{3} + 17 q^{4} + 3 q^{6} - 4 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{11} + 17 q^{12} - 8 q^{13} - 4 q^{14} + 29 q^{16} + 12 q^{17} + 3 q^{18} + 28 q^{19} - 4 q^{21} + 6 q^{23} + 15 q^{24} + 4 q^{26} + 13 q^{27} - 20 q^{28} + 12 q^{29} + 26 q^{31} + 53 q^{32} + 16 q^{33} + 8 q^{34} + 17 q^{36} - 4 q^{37} + 2 q^{38} - 8 q^{39} + 24 q^{41} - 4 q^{42} - 6 q^{43} + 4 q^{44} + 16 q^{46} + 13 q^{47} + 29 q^{48} + 21 q^{49} + 12 q^{51} - 32 q^{52} + 6 q^{53} + 3 q^{54} + 28 q^{57} - 4 q^{58} + 34 q^{59} + 24 q^{61} + 30 q^{62} - 4 q^{63} + 13 q^{64} - 24 q^{67} + 44 q^{68} + 6 q^{69} + 20 q^{71} + 15 q^{72} - 6 q^{73} + 20 q^{74} + 66 q^{76} - 2 q^{77} + 4 q^{78} + 6 q^{79} + 13 q^{81} + 20 q^{82} + 14 q^{83} - 20 q^{84} + 48 q^{86} + 12 q^{87} - 22 q^{88} + 36 q^{89} + 4 q^{91} + 4 q^{92} + 26 q^{93} + 3 q^{94} + 53 q^{96} - 32 q^{97} - 39 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 3 x^{12} - 17 x^{11} + 51 x^{10} + 106 x^{9} - 316 x^{8} - 288 x^{7} + 852 x^{6} + 309 x^{5} + \cdots - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - 2\nu^{8} - 13\nu^{7} + 22\nu^{6} + 55\nu^{5} - 72\nu^{4} - 79\nu^{3} + 68\nu^{2} + 32\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} - 15 \nu^{10} + 28 \nu^{9} + 78 \nu^{8} - 142 \nu^{7} - 158 \nu^{6} + 326 \nu^{5} + \cdots + 10 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{11} + \nu^{10} + 17 \nu^{9} - 12 \nu^{8} - 105 \nu^{7} + 46 \nu^{6} + 281 \nu^{5} - 62 \nu^{4} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{12} + 2 \nu^{11} + 19 \nu^{10} - 36 \nu^{9} - 134 \nu^{8} + 238 \nu^{7} + 430 \nu^{6} + \cdots - 50 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{12} + 2 \nu^{11} + 17 \nu^{10} - 32 \nu^{9} - 104 \nu^{8} + 186 \nu^{7} + 268 \nu^{6} - 466 \nu^{5} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} - 17 \nu^{10} + 32 \nu^{9} + 104 \nu^{8} - 186 \nu^{7} - 268 \nu^{6} + 470 \nu^{5} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{12} - 4 \nu^{11} - 17 \nu^{10} + 68 \nu^{9} + 110 \nu^{8} - 418 \nu^{7} - 326 \nu^{6} + 1106 \nu^{5} + \cdots + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} - 17 \nu^{10} + 32 \nu^{9} + 106 \nu^{8} - 190 \nu^{7} - 290 \nu^{6} + 506 \nu^{5} + \cdots - 2 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{4} + 8\beta_{3} + 10\beta_{2} + 28\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} + 10\beta_{4} + 10\beta_{3} + 46\beta_{2} + 14\beta _1 + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{12} - 2 \beta_{11} + 13 \beta_{10} + 11 \beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{6} + \cdots + 111 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{12} - 4 \beta_{11} + 17 \beta_{10} + 15 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} - 17 \beta_{6} + \cdots + 622 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 21 \beta_{12} - 34 \beta_{11} + 126 \beta_{10} + 96 \beta_{9} + 17 \beta_{8} + 17 \beta_{7} - 51 \beta_{6} + \cdots + 926 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 160 \beta_{12} - 76 \beta_{11} + 202 \beta_{10} + 162 \beta_{9} + 126 \beta_{8} + 38 \beta_{7} + \cdots + 4125 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 278 \beta_{12} - 396 \beta_{11} + 1102 \beta_{10} + 786 \beta_{9} + 200 \beta_{8} + 196 \beta_{7} + \cdots + 7325 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1498 \beta_{12} - 952 \beta_{11} + 2058 \beta_{10} + 1538 \beta_{9} + 1100 \beta_{8} + 472 \beta_{7} + \cdots + 28020 \) 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
−2.32477
−2.24873
−2.12231
−1.22883
−0.697808
−0.105426
0.0655745
0.747203
1.32490
1.67335
2.57687
2.59761
2.74237
−2.32477 1.00000 3.40453 0 −2.32477 −4.29166 −3.26521 1.00000 0
1.2 −2.24873 1.00000 3.05678 0 −2.24873 −1.38491 −2.37640 1.00000 0
1.3 −2.12231 1.00000 2.50418 0 −2.12231 4.63433 −1.07002 1.00000 0
1.4 −1.22883 1.00000 −0.489976 0 −1.22883 −1.54483 3.05976 1.00000 0
1.5 −0.697808 1.00000 −1.51306 0 −0.697808 3.46613 2.45144 1.00000 0
1.6 −0.105426 1.00000 −1.98889 0 −0.105426 −3.91364 0.420531 1.00000 0
1.7 0.0655745 1.00000 −1.99570 0 0.0655745 1.54896 −0.262016 1.00000 0
1.8 0.747203 1.00000 −1.44169 0 0.747203 0.654660 −2.57164 1.00000 0
1.9 1.32490 1.00000 −0.244653 0 1.32490 1.72082 −2.97393 1.00000 0
1.10 1.67335 1.00000 0.800100 0 1.67335 −2.75180 −2.00785 1.00000 0
1.11 2.57687 1.00000 4.64024 0 2.57687 1.51741 6.80354 1.00000 0
1.12 2.59761 1.00000 4.74755 0 2.59761 −4.83808 7.13706 1.00000 0
1.13 2.74237 1.00000 5.52059 0 2.74237 1.18262 9.65475 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(47\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3525.2.a.bi 13
5.b even 2 1 3525.2.a.bh 13
5.c odd 4 2 705.2.c.c ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.c ✓ 26 5.c odd 4 2
3525.2.a.bh 13 5.b even 2 1
3525.2.a.bi 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2}^{13} - 3 T_{2}^{12} - 17 T_{2}^{11} + 51 T_{2}^{10} + 106 T_{2}^{9} - 316 T_{2}^{8} - 288 T_{2}^{7} + \cdots - 2 \) Copy content Toggle raw display
\( T_{7}^{13} + 4 T_{7}^{12} - 48 T_{7}^{11} - 172 T_{7}^{10} + 842 T_{7}^{9} + 2388 T_{7}^{8} + \cdots - 24064 \) Copy content Toggle raw display
\( T_{11}^{13} - 16 T_{11}^{12} + 17 T_{11}^{11} + 892 T_{11}^{10} - 3548 T_{11}^{9} - 15976 T_{11}^{8} + \cdots - 2451456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 3 T^{12} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} \) Copy content Toggle raw display
$7$ \( T^{13} + 4 T^{12} + \cdots - 24064 \) Copy content Toggle raw display
$11$ \( T^{13} - 16 T^{12} + \cdots - 2451456 \) Copy content Toggle raw display
$13$ \( T^{13} + 8 T^{12} + \cdots - 892224 \) Copy content Toggle raw display
$17$ \( T^{13} - 12 T^{12} + \cdots - 60736 \) Copy content Toggle raw display
$19$ \( T^{13} - 28 T^{12} + \cdots - 899200 \) Copy content Toggle raw display
$23$ \( T^{13} - 6 T^{12} + \cdots - 60416 \) Copy content Toggle raw display
$29$ \( T^{13} - 12 T^{12} + \cdots + 17120 \) Copy content Toggle raw display
$31$ \( T^{13} - 26 T^{12} + \cdots + 17920 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 389395456 \) Copy content Toggle raw display
$41$ \( T^{13} - 24 T^{12} + \cdots + 58496 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 308908544 \) Copy content Toggle raw display
$47$ \( (T - 1)^{13} \) Copy content Toggle raw display
$53$ \( T^{13} - 6 T^{12} + \cdots + 25854912 \) Copy content Toggle raw display
$59$ \( T^{13} - 34 T^{12} + \cdots - 178880 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 27648112896 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 2592738688 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 9039415936 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 594804992 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 953661440 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 35253854208 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 461035624960 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 27544367104 \) Copy content Toggle raw display
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