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Properties

Label 6045.2.a.be
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
Analytic rank $1$
Dimension $15$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 21 x^{13} + 20 x^{12} + 170 x^{11} - 154 x^{10} - 669 x^{9} + 573 x^{8} + 1326 x^{7} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + ( - \beta_{5} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + ( - \beta_{5} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} - \beta_1 q^{10} + (\beta_{13} + \beta_{11} + \beta_1) q^{11} + ( - \beta_{2} - 1) q^{12} + q^{13} + (\beta_{14} + \beta_{12} + \cdots + 2 \beta_1) q^{14}+ \cdots + (\beta_{13} + \beta_{11} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{2} - 15 q^{3} + 13 q^{4} + 15 q^{5} + q^{6} - 16 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{2} - 15 q^{3} + 13 q^{4} + 15 q^{5} + q^{6} - 16 q^{7} + 15 q^{9} - q^{10} - 8 q^{11} - 13 q^{12} + 15 q^{13} - 3 q^{14} - 15 q^{15} + 9 q^{16} - q^{18} - 12 q^{19} + 13 q^{20} + 16 q^{21} - 11 q^{22} - 26 q^{23} + 15 q^{25} - q^{26} - 15 q^{27} - 38 q^{28} - 3 q^{29} + q^{30} - 15 q^{31} - 11 q^{32} + 8 q^{33} - q^{34} - 16 q^{35} + 13 q^{36} - 33 q^{37} - 2 q^{38} - 15 q^{39} + 16 q^{41} + 3 q^{42} - 24 q^{43} - 15 q^{44} + 15 q^{45} - 26 q^{46} - 13 q^{47} - 9 q^{48} + 25 q^{49} - q^{50} + 13 q^{52} - 3 q^{53} + q^{54} - 8 q^{55} - 11 q^{56} + 12 q^{57} - 46 q^{58} - 14 q^{59} - 13 q^{60} + 5 q^{61} + q^{62} - 16 q^{63} - 2 q^{64} + 15 q^{65} + 11 q^{66} - 44 q^{67} - 35 q^{68} + 26 q^{69} - 3 q^{70} - 12 q^{71} - 20 q^{73} + 24 q^{74} - 15 q^{75} - 22 q^{76} + 11 q^{77} + q^{78} - 8 q^{79} + 9 q^{80} + 15 q^{81} - 9 q^{82} - 29 q^{83} + 38 q^{84} - 6 q^{86} + 3 q^{87} - 31 q^{88} + 17 q^{89} - q^{90} - 16 q^{91} - 11 q^{92} + 15 q^{93} - 4 q^{94} - 12 q^{95} + 11 q^{96} - 32 q^{97} - 43 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 21 x^{13} + 20 x^{12} + 170 x^{11} - 154 x^{10} - 669 x^{9} + 573 x^{8} + 1326 x^{7} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 945 \nu^{14} + 5937 \nu^{13} - 16859 \nu^{12} - 121722 \nu^{11} + 96994 \nu^{10} + 940802 \nu^{9} + \cdots + 22196 ) / 20516 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9633 \nu^{14} - 1121 \nu^{13} + 196995 \nu^{12} + 29566 \nu^{11} - 1524158 \nu^{10} - 257902 \nu^{9} + \cdots - 3448 ) / 20516 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12513 \nu^{14} - 3095 \nu^{13} + 254725 \nu^{12} + 67876 \nu^{11} - 1958486 \nu^{10} + \cdots + 94012 ) / 20516 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7414 \nu^{14} - 2795 \nu^{13} + 150075 \nu^{12} + 60261 \nu^{11} - 1141088 \nu^{10} - 467216 \nu^{9} + \cdots + 5056 ) / 10258 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9401 \nu^{14} - 3441 \nu^{13} + 192487 \nu^{12} + 74778 \nu^{11} - 1491564 \nu^{10} + \cdots + 132598 ) / 10258 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9401 \nu^{14} + 3441 \nu^{13} - 192487 \nu^{12} - 74778 \nu^{11} + 1491564 \nu^{10} + 586976 \nu^{9} + \cdots - 173630 ) / 10258 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19975 \nu^{14} + 10539 \nu^{13} - 406617 \nu^{12} - 223662 \nu^{11} + 3121174 \nu^{10} + \cdots - 215240 ) / 20516 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10427 \nu^{14} + 3439 \nu^{13} - 214015 \nu^{12} - 75924 \nu^{11} + 1666040 \nu^{10} + 602548 \nu^{9} + \cdots - 182064 ) / 10258 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25289 \nu^{14} + 13818 \nu^{13} - 516948 \nu^{12} - 293815 \nu^{11} + 3994290 \nu^{10} + \cdots - 353964 ) / 10258 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 30683 \nu^{14} - 11168 \nu^{13} + 626888 \nu^{12} + 242791 \nu^{11} - 4841858 \nu^{10} + \cdots + 327278 ) / 10258 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 67753 \nu^{14} - 25143 \nu^{13} + 1387521 \nu^{12} + 544096 \nu^{11} - 10752458 \nu^{10} + \cdots + 834684 ) / 20516 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{3} + 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{7} - \beta_{4} + 8 \beta_{3} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} + \beta_{12} - \beta_{11} - 9 \beta_{9} - 11 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} + \cdots + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{13} + 12 \beta_{12} + 13 \beta_{11} - 15 \beta_{10} - 14 \beta_{9} - \beta_{8} - 25 \beta_{7} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{14} + 16 \beta_{12} - 15 \beta_{11} - 3 \beta_{10} - 68 \beta_{9} - 93 \beta_{8} - 15 \beta_{7} + \cdots + 432 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{14} + 96 \beta_{13} + 110 \beta_{12} + 120 \beta_{11} - 152 \beta_{10} - 137 \beta_{9} + \cdots + 106 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 140 \beta_{14} + 2 \beta_{13} + 175 \beta_{12} - 151 \beta_{11} - 54 \beta_{10} - 494 \beta_{9} + \cdots + 2611 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18 \beta_{14} + 771 \beta_{13} + 910 \beta_{12} + 968 \beta_{11} - 1318 \beta_{10} - 1167 \beta_{9} + \cdots + 845 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1224 \beta_{14} + 44 \beta_{13} + 1629 \beta_{12} - 1289 \beta_{11} - 642 \beta_{10} - 3550 \beta_{9} + \cdots + 16349 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 219 \beta_{14} + 5924 \beta_{13} + 7144 \beta_{12} + 7297 \beta_{11} - 10565 \beta_{10} - 9277 \beta_{9} + \cdots + 6479 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9957 \beta_{14} + 614 \beta_{13} + 13890 \beta_{12} - 10091 \beta_{11} - 6376 \beta_{10} - 25434 \beta_{9} + \cdots + 105163 \) 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
2.66987
2.32548
2.10358
1.40187
1.38928
1.11534
0.505694
0.198717
−0.428109
−0.575074
−1.13094
−1.70328
−2.02231
−2.25392
−2.59621
−2.66987 −1.00000 5.12819 1.00000 2.66987 −3.27871 −8.35186 1.00000 −2.66987
1.2 −2.32548 −1.00000 3.40784 1.00000 2.32548 3.35624 −3.27390 1.00000 −2.32548
1.3 −2.10358 −1.00000 2.42507 1.00000 2.10358 −4.09959 −0.894166 1.00000 −2.10358
1.4 −1.40187 −1.00000 −0.0347532 1.00000 1.40187 −1.28467 2.85246 1.00000 −1.40187
1.5 −1.38928 −1.00000 −0.0698923 1.00000 1.38928 0.189602 2.87567 1.00000 −1.38928
1.6 −1.11534 −1.00000 −0.756016 1.00000 1.11534 −5.23737 3.07390 1.00000 −1.11534
1.7 −0.505694 −1.00000 −1.74427 1.00000 0.505694 2.19644 1.89346 1.00000 −0.505694
1.8 −0.198717 −1.00000 −1.96051 1.00000 0.198717 3.35057 0.787022 1.00000 −0.198717
1.9 0.428109 −1.00000 −1.81672 1.00000 −0.428109 −0.204805 −1.63397 1.00000 0.428109
1.10 0.575074 −1.00000 −1.66929 1.00000 −0.575074 −4.01605 −2.11011 1.00000 0.575074
1.11 1.13094 −1.00000 −0.720977 1.00000 −1.13094 −0.596266 −3.07726 1.00000 1.13094
1.12 1.70328 −1.00000 0.901160 1.00000 −1.70328 −1.73192 −1.87163 1.00000 1.70328
1.13 2.02231 −1.00000 2.08974 1.00000 −2.02231 2.03009 0.181485 1.00000 2.02231
1.14 2.25392 −1.00000 3.08014 1.00000 −2.25392 −3.11678 2.43456 1.00000 2.25392
1.15 2.59621 −1.00000 4.74029 1.00000 −2.59621 −3.55680 7.11436 1.00000 2.59621
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(-1\)
\(31\) \(1\)

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 6045.2.a.be ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.be ✓ 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{15} + T_{2}^{14} - 21 T_{2}^{13} - 20 T_{2}^{12} + 170 T_{2}^{11} + 154 T_{2}^{10} - 669 T_{2}^{9} + \cdots - 16 \) Copy content Toggle raw display
\( T_{7}^{15} + 16 T_{7}^{14} + 63 T_{7}^{13} - 257 T_{7}^{12} - 2364 T_{7}^{11} - 2100 T_{7}^{10} + \cdots - 8096 \) Copy content Toggle raw display
\( T_{11}^{15} + 8 T_{11}^{14} - 64 T_{11}^{13} - 604 T_{11}^{12} + 1021 T_{11}^{11} + 15498 T_{11}^{10} + \cdots + 737344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + T^{14} + \cdots - 16 \) Copy content Toggle raw display
$3$ \( (T + 1)^{15} \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 16 T^{14} + \cdots - 8096 \) Copy content Toggle raw display
$11$ \( T^{15} + 8 T^{14} + \cdots + 737344 \) Copy content Toggle raw display
$13$ \( (T - 1)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 138 T^{13} + \cdots - 256 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 184999136 \) Copy content Toggle raw display
$23$ \( T^{15} + 26 T^{14} + \cdots + 6119424 \) Copy content Toggle raw display
$29$ \( T^{15} + 3 T^{14} + \cdots + 18792448 \) Copy content Toggle raw display
$31$ \( (T + 1)^{15} \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 164753872 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 276242868 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 1138414736 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 686732288 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 459305368336 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 65399258016 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 23151188584 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 7873024582336 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 9621651456 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 8284639624 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 2932558848 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 1678712959008 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 405541808 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 7692174004 \) Copy content Toggle raw display
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