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Properties

Label 4074.2.a.bf
Level $4074$
Weight $2$
Character orbit 4074.a
Self dual yes
Analytic conductor $32.531$
Analytic rank $0$
Dimension $10$
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] = [4074,2,Mod(1,4074)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4074, 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("4074.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4074 = 2 \cdot 3 \cdot 7 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4074.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: \(32.5310537835\)
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: \( x^{10} - 3x^{9} - 33x^{8} + 92x^{7} + 330x^{6} - 828x^{5} - 972x^{4} + 2176x^{3} + 152x^{2} - 864x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + \beta_1 q^{10} + (\beta_{3} + 1) q^{11} - q^{12} + ( - \beta_{4} + 1) q^{13} + q^{14} - \beta_1 q^{15} + q^{16} + \beta_{2} q^{17} + q^{18} - \beta_{8} q^{19} + \beta_1 q^{20} - q^{21} + (\beta_{3} + 1) q^{22} + (\beta_{6} + 1) q^{23} - q^{24} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + 3) q^{25}+ \cdots + (\beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 10 q^{3} + 10 q^{4} + 3 q^{5} - 10 q^{6} + 10 q^{7} + 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 10 q^{3} + 10 q^{4} + 3 q^{5} - 10 q^{6} + 10 q^{7} + 10 q^{8} + 10 q^{9} + 3 q^{10} + 12 q^{11} - 10 q^{12} + 6 q^{13} + 10 q^{14} - 3 q^{15} + 10 q^{16} + 4 q^{17} + 10 q^{18} + q^{19} + 3 q^{20} - 10 q^{21} + 12 q^{22} + 10 q^{23} - 10 q^{24} + 25 q^{25} + 6 q^{26} - 10 q^{27} + 10 q^{28} + 10 q^{29} - 3 q^{30} + 3 q^{31} + 10 q^{32} - 12 q^{33} + 4 q^{34} + 3 q^{35} + 10 q^{36} + 15 q^{37} + q^{38} - 6 q^{39} + 3 q^{40} + 10 q^{41} - 10 q^{42} + 17 q^{43} + 12 q^{44} + 3 q^{45} + 10 q^{46} - 10 q^{48} + 10 q^{49} + 25 q^{50} - 4 q^{51} + 6 q^{52} + 13 q^{53} - 10 q^{54} + 10 q^{56} - q^{57} + 10 q^{58} - 20 q^{59} - 3 q^{60} + 9 q^{61} + 3 q^{62} + 10 q^{63} + 10 q^{64} + 22 q^{65} - 12 q^{66} + 26 q^{67} + 4 q^{68} - 10 q^{69} + 3 q^{70} + 18 q^{71} + 10 q^{72} + 6 q^{73} + 15 q^{74} - 25 q^{75} + q^{76} + 12 q^{77} - 6 q^{78} + 12 q^{79} + 3 q^{80} + 10 q^{81} + 10 q^{82} - 10 q^{83} - 10 q^{84} + 16 q^{85} + 17 q^{86} - 10 q^{87} + 12 q^{88} + q^{89} + 3 q^{90} + 6 q^{91} + 10 q^{92} - 3 q^{93} + 18 q^{95} - 10 q^{96} + 10 q^{97} + 10 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 33x^{8} + 92x^{7} + 330x^{6} - 828x^{5} - 972x^{4} + 2176x^{3} + 152x^{2} - 864x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 197 \nu^{9} + 2978 \nu^{8} - 738 \nu^{7} - 106339 \nu^{6} + 141318 \nu^{5} + 1167808 \nu^{4} + \cdots + 1261872 ) / 232528 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1383 \nu^{9} - 1357 \nu^{8} - 45205 \nu^{7} + 21874 \nu^{6} + 447188 \nu^{5} + 148148 \nu^{4} + \cdots + 2445440 ) / 465056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2361 \nu^{9} + 10387 \nu^{8} + 73263 \nu^{7} - 317410 \nu^{6} - 663928 \nu^{5} + 2830732 \nu^{4} + \cdots + 2499456 ) / 465056 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3266 \nu^{9} - 8133 \nu^{8} - 108529 \nu^{7} + 242897 \nu^{6} + 1108906 \nu^{5} - 2066888 \nu^{4} + \cdots - 689056 ) / 232528 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1780 \nu^{9} + 3596 \nu^{8} + 63341 \nu^{7} - 103235 \nu^{6} - 725941 \nu^{5} + 813920 \nu^{4} + \cdots + 513592 ) / 116264 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3898 \nu^{9} - 9867 \nu^{8} - 132129 \nu^{7} + 288369 \nu^{6} + 1376584 \nu^{5} - 2358488 \nu^{4} + \cdots - 1006224 ) / 232528 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4389 \nu^{9} + 11904 \nu^{8} + 149544 \nu^{7} - 358281 \nu^{6} - 1578242 \nu^{5} + 3087248 \nu^{4} + \cdots + 746000 ) / 232528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11963 \nu^{9} + 30569 \nu^{8} + 401281 \nu^{7} - 900834 \nu^{6} - 4120476 \nu^{5} + \cdots + 3566112 ) / 465056 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{2} + \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{3} - \beta_{2} + 13\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} + 14\beta_{8} + 19\beta_{7} - 17\beta_{6} - 17\beta_{5} + 4\beta_{3} - 18\beta_{2} + 20\beta _1 + 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 21 \beta_{9} + 2 \beta_{8} - 28 \beta_{7} - 26 \beta_{6} - 39 \beta_{5} - 6 \beta_{4} + 23 \beta_{3} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 97 \beta_{9} + 206 \beta_{8} + 325 \beta_{7} - 285 \beta_{6} - 276 \beta_{5} + 14 \beta_{4} + \cdots + 1630 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 396 \beta_{9} + 84 \beta_{8} - 579 \beta_{7} - 519 \beta_{6} - 667 \beta_{5} - 164 \beta_{4} + \cdots + 1232 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1841 \beta_{9} + 3222 \beta_{8} + 5426 \beta_{7} - 4796 \beta_{6} - 4459 \beta_{5} + 538 \beta_{4} + \cdots + 25734 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7231 \beta_{9} + 2258 \beta_{8} - 10889 \beta_{7} - 9611 \beta_{6} - 10978 \beta_{5} - 3114 \beta_{4} + \cdots + 23534 \) 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
−4.08009
−3.25276
−2.05509
−0.651184
0.0374129
0.827764
1.27996
2.72682
4.00271
4.16447
1.00000 −1.00000 1.00000 −4.08009 −1.00000 1.00000 1.00000 1.00000 −4.08009
1.2 1.00000 −1.00000 1.00000 −3.25276 −1.00000 1.00000 1.00000 1.00000 −3.25276
1.3 1.00000 −1.00000 1.00000 −2.05509 −1.00000 1.00000 1.00000 1.00000 −2.05509
1.4 1.00000 −1.00000 1.00000 −0.651184 −1.00000 1.00000 1.00000 1.00000 −0.651184
1.5 1.00000 −1.00000 1.00000 0.0374129 −1.00000 1.00000 1.00000 1.00000 0.0374129
1.6 1.00000 −1.00000 1.00000 0.827764 −1.00000 1.00000 1.00000 1.00000 0.827764
1.7 1.00000 −1.00000 1.00000 1.27996 −1.00000 1.00000 1.00000 1.00000 1.27996
1.8 1.00000 −1.00000 1.00000 2.72682 −1.00000 1.00000 1.00000 1.00000 2.72682
1.9 1.00000 −1.00000 1.00000 4.00271 −1.00000 1.00000 1.00000 1.00000 4.00271
1.10 1.00000 −1.00000 1.00000 4.16447 −1.00000 1.00000 1.00000 1.00000 4.16447
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(97\) \(-1\)

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 4074.2.a.bf ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4074.2.a.bf ✓ 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4074))\):

\( T_{5}^{10} - 3 T_{5}^{9} - 33 T_{5}^{8} + 92 T_{5}^{7} + 330 T_{5}^{6} - 828 T_{5}^{5} - 972 T_{5}^{4} + \cdots + 32 \) Copy content Toggle raw display
\( T_{11}^{10} - 12 T_{11}^{9} - 15 T_{11}^{8} + 594 T_{11}^{7} - 940 T_{11}^{6} - 8312 T_{11}^{5} + \cdots - 1280 \) Copy content Toggle raw display
\( T_{13}^{10} - 6 T_{13}^{9} - 63 T_{13}^{8} + 372 T_{13}^{7} + 1238 T_{13}^{6} - 6700 T_{13}^{5} + \cdots - 46208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 3 T^{9} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 12 T^{9} + \cdots - 1280 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots - 46208 \) Copy content Toggle raw display
$17$ \( T^{10} - 4 T^{9} + \cdots + 545344 \) Copy content Toggle raw display
$19$ \( T^{10} - T^{9} + \cdots + 418304 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + \cdots + 317440 \) Copy content Toggle raw display
$29$ \( T^{10} - 10 T^{9} + \cdots - 43072 \) Copy content Toggle raw display
$31$ \( T^{10} - 3 T^{9} + \cdots + 55552 \) Copy content Toggle raw display
$37$ \( T^{10} - 15 T^{9} + \cdots + 99714128 \) Copy content Toggle raw display
$41$ \( T^{10} - 10 T^{9} + \cdots - 391360 \) Copy content Toggle raw display
$43$ \( T^{10} - 17 T^{9} + \cdots - 78592 \) Copy content Toggle raw display
$47$ \( T^{10} - 286 T^{8} + \cdots + 52428800 \) Copy content Toggle raw display
$53$ \( T^{10} - 13 T^{9} + \cdots - 15568640 \) Copy content Toggle raw display
$59$ \( T^{10} + 20 T^{9} + \cdots - 3112960 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 552624640 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 102617344 \) Copy content Toggle raw display
$71$ \( T^{10} - 18 T^{9} + \cdots - 49008640 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 1613965696 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 140288000 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 2272137728 \) Copy content Toggle raw display
$89$ \( T^{10} - T^{9} + \cdots + 1316416 \) Copy content Toggle raw display
$97$ \( (T - 1)^{10} \) Copy content Toggle raw display
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