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Properties

Label 7203.2.a.b
Level $7203$
Weight $2$
Character orbit 7203.a
Self dual yes
Analytic conductor $57.516$
Analytic rank $1$
Dimension $12$
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] = [7203,2,Mod(1,7203)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7203, 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("7203.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7203 = 3 \cdot 7^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7203.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: \(57.5162445759\)
Analytic rank: \(1\)
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: \( x^{12} - 3 x^{11} - 12 x^{10} + 38 x^{9} + 48 x^{8} - 168 x^{7} - 69 x^{6} + 308 x^{5} + 12 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{9} - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{9} - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{10} - \beta_{9} - \beta_{8} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 12 q^{3} + 9 q^{4} - 3 q^{6} - 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} + 12 q^{3} + 9 q^{4} - 3 q^{6} - 9 q^{8} + 12 q^{9} - 6 q^{10} - 6 q^{11} + 9 q^{12} - 7 q^{13} + 3 q^{16} + q^{17} - 3 q^{18} - 13 q^{19} + 3 q^{20} - 12 q^{22} - 12 q^{23} - 9 q^{24} + 25 q^{26} + 12 q^{27} - 18 q^{29} - 6 q^{30} - 18 q^{31} - 21 q^{32} - 6 q^{33} - 23 q^{34} + 9 q^{36} - 25 q^{37} - 23 q^{38} - 7 q^{39} - 14 q^{40} - 19 q^{41} - 18 q^{43} - 15 q^{44} + 6 q^{46} + 3 q^{47} + 3 q^{48} - 24 q^{50} + q^{51} - 10 q^{52} - 21 q^{53} - 3 q^{54} - 22 q^{55} - 13 q^{57} - 33 q^{58} + 53 q^{59} + 3 q^{60} - 19 q^{61} + 11 q^{62} - 3 q^{64} - 49 q^{65} - 12 q^{66} - 17 q^{67} + 11 q^{68} - 12 q^{69} + 5 q^{71} - 9 q^{72} - 17 q^{73} + 10 q^{74} + 21 q^{76} + 25 q^{78} - 17 q^{79} - 4 q^{80} + 12 q^{81} - 49 q^{82} - 21 q^{83} - 4 q^{85} + 20 q^{86} - 18 q^{87} - 5 q^{88} - 5 q^{89} - 6 q^{90} - 7 q^{92} - 18 q^{93} + 10 q^{94} - 19 q^{95} - 21 q^{96} + 32 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 12 x^{10} + 38 x^{9} + 48 x^{8} - 168 x^{7} - 69 x^{6} + 308 x^{5} + 12 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 397 \nu^{11} + 2079 \nu^{10} - 10864 \nu^{9} - 31151 \nu^{8} + 89181 \nu^{7} + 173254 \nu^{6} + \cdots - 37762 ) / 24919 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 457 \nu^{11} + 824 \nu^{10} - 11941 \nu^{9} - 3345 \nu^{8} + 88097 \nu^{7} - 33369 \nu^{6} + \cdots + 19613 ) / 24919 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 829 \nu^{11} + 6957 \nu^{10} + 3667 \nu^{9} - 99279 \nu^{8} + 38235 \nu^{7} + 507056 \nu^{6} + \cdots - 101229 ) / 24919 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 829 \nu^{11} - 6957 \nu^{10} - 3667 \nu^{9} + 99279 \nu^{8} - 38235 \nu^{7} - 507056 \nu^{6} + \cdots + 26472 ) / 24919 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2519 \nu^{11} + 13234 \nu^{10} + 21543 \nu^{9} - 176443 \nu^{8} - 22602 \nu^{7} + 840604 \nu^{6} + \cdots - 110016 ) / 24919 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3270 \nu^{11} - 6100 \nu^{10} - 46237 \nu^{9} + 70125 \nu^{8} + 239950 \nu^{7} - 259215 \nu^{6} + \cdots - 397 ) / 24919 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3508 \nu^{11} - 6925 \nu^{10} - 53001 \nu^{9} + 92375 \nu^{8} + 297117 \nu^{7} - 436740 \nu^{6} + \cdots + 15379 ) / 24919 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3659 \nu^{11} + 12160 \nu^{10} + 42006 \nu^{9} - 156539 \nu^{8} - 151520 \nu^{7} + 704644 \nu^{6} + \cdots - 53867 ) / 24919 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4013 \nu^{11} - 7105 \nu^{10} - 55836 \nu^{9} + 81372 \nu^{8} + 279687 \nu^{7} - 296509 \nu^{6} + \cdots + 31054 ) / 24919 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12014 \nu^{11} - 31175 \nu^{10} - 148370 \nu^{9} + 377891 \nu^{8} + 623548 \nu^{7} - 1561387 \nu^{6} + \cdots + 87945 ) / 24919 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} + 3\beta_{9} - \beta_{6} + 7\beta_{5} + 6\beta_{4} - \beta_{2} + 2\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 9 \beta_{10} + 13 \beta_{9} + \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 11 \beta_{5} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + 22 \beta_{10} + 34 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 10 \beta_{6} + 50 \beta_{5} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{11} + 71 \beta_{10} + 119 \beta_{9} + 15 \beta_{8} - 70 \beta_{7} - 55 \beta_{6} + 95 \beta_{5} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{11} + 189 \beta_{10} + 301 \beta_{9} + 31 \beta_{8} - 75 \beta_{7} - 80 \beta_{6} + 358 \beta_{5} + \cdots + 519 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 109 \beta_{11} + 541 \beta_{10} + 966 \beta_{9} + 149 \beta_{8} - 525 \beta_{7} - 363 \beta_{6} + \cdots + 775 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 178 \beta_{11} + 1488 \beta_{10} + 2439 \beta_{9} + 327 \beta_{8} - 781 \beta_{7} - 592 \beta_{6} + \cdots + 3320 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 896 \beta_{11} + 4057 \beta_{10} + 7432 \beta_{9} + 1277 \beta_{8} - 3883 \beta_{7} - 2366 \beta_{6} + \cdots + 5890 \) 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
2.66044
2.42787
1.91962
1.42661
1.22142
0.496072
−0.0489541
−0.256459
−1.09639
−1.55290
−1.90005
−2.29727
−2.66044 1.00000 5.07794 2.88028 −2.66044 0 −8.18867 1.00000 −7.66282
1.2 −2.42787 1.00000 3.89457 −2.27752 −2.42787 0 −4.59978 1.00000 5.52952
1.3 −1.91962 1.00000 1.68494 −1.70541 −1.91962 0 0.604794 1.00000 3.27374
1.4 −1.42661 1.00000 0.0352044 3.65641 −1.42661 0 2.80299 1.00000 −5.21626
1.5 −1.22142 1.00000 −0.508141 0.263371 −1.22142 0 3.06349 1.00000 −0.321686
1.6 −0.496072 1.00000 −1.75391 0.815014 −0.496072 0 1.86221 1.00000 −0.404306
1.7 0.0489541 1.00000 −1.99760 −3.42902 0.0489541 0 −0.195699 1.00000 −0.167864
1.8 0.256459 1.00000 −1.93423 −1.67550 0.256459 0 −1.00897 1.00000 −0.429697
1.9 1.09639 1.00000 −0.797933 3.51349 1.09639 0 −3.06762 1.00000 3.85215
1.10 1.55290 1.00000 0.411505 −0.142448 1.55290 0 −2.46678 1.00000 −0.221207
1.11 1.90005 1.00000 1.61020 −0.327772 1.90005 0 −0.740637 1.00000 −0.622785
1.12 2.29727 1.00000 3.27746 −1.57090 2.29727 0 2.93467 1.00000 −3.60879
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

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 7203.2.a.b 12
7.b odd 2 1 7203.2.a.a 12
49.f odd 14 2 147.2.i.a ✓ 24
147.k even 14 2 441.2.u.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.i.a ✓ 24 49.f odd 14 2
441.2.u.c 24 147.k even 14 2
7203.2.a.a 12 7.b odd 2 1
7203.2.a.b 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7203))\):

\( T_{2}^{12} + 3 T_{2}^{11} - 12 T_{2}^{10} - 38 T_{2}^{9} + 48 T_{2}^{8} + 168 T_{2}^{7} - 69 T_{2}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{12} - 30 T_{5}^{10} - 17 T_{5}^{9} + 303 T_{5}^{8} + 347 T_{5}^{7} - 1057 T_{5}^{6} - 1832 T_{5}^{5} + \cdots - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 30 T^{10} + \cdots - 13 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots - 1079 \) Copy content Toggle raw display
$13$ \( T^{12} + 7 T^{11} + \cdots + 30667 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + \cdots - 742147 \) Copy content Toggle raw display
$19$ \( T^{12} + 13 T^{11} + \cdots - 190331 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots - 878423 \) Copy content Toggle raw display
$29$ \( T^{12} + 18 T^{11} + \cdots - 2339 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 197686097 \) Copy content Toggle raw display
$37$ \( T^{12} + 25 T^{11} + \cdots - 23741549 \) Copy content Toggle raw display
$41$ \( T^{12} + 19 T^{11} + \cdots + 18529931 \) Copy content Toggle raw display
$43$ \( T^{12} + 18 T^{11} + \cdots - 41511919 \) Copy content Toggle raw display
$47$ \( T^{12} - 3 T^{11} + \cdots - 4131491 \) Copy content Toggle raw display
$53$ \( T^{12} + 21 T^{11} + \cdots - 82412777 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 247797017 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 110933353 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 4671276623 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 3176608169 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 305659901 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 12150054181 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 35141255023 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 121518313021 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 14480424637 \) Copy content Toggle raw display
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