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Properties

Label 6034.2.a.k
Level $6034$
Weight $2$
Character orbit 6034.a
Self dual yes
Analytic conductor $48.182$
Analytic rank $1$
Dimension $20$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, 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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 32 x^{18} + 106 x^{17} + 382 x^{16} - 1495 x^{15} - 1963 x^{14} + 10784 x^{13} + \cdots - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{15} q^{5} - \beta_1 q^{6} + q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{15} q^{5} - \beta_1 q^{6} + q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + \beta_{15} q^{10} + ( - \beta_{16} - \beta_{10} - \beta_{4} - 1) q^{11} + \beta_1 q^{12} + (\beta_{15} - \beta_{3}) q^{13} - q^{14} + ( - \beta_{19} - \beta_{18} + \beta_{17} + \cdots - 2) q^{15}+ \cdots + ( - \beta_{19} - \beta_{17} + \beta_{15} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 4 q^{13} - 20 q^{14} - 25 q^{15} + 20 q^{16} + 9 q^{17} - 13 q^{18} - 14 q^{19} - 3 q^{20} + 3 q^{21} + 8 q^{22} - 23 q^{23} - 3 q^{24} + 31 q^{25} + 4 q^{26} - 21 q^{27} + 20 q^{28} - 48 q^{29} + 25 q^{30} - q^{31} - 20 q^{32} - 29 q^{33} - 9 q^{34} - 3 q^{35} + 13 q^{36} - q^{37} + 14 q^{38} - q^{39} + 3 q^{40} - 27 q^{41} - 3 q^{42} - 3 q^{43} - 8 q^{44} - 12 q^{45} + 23 q^{46} - 26 q^{47} + 3 q^{48} + 20 q^{49} - 31 q^{50} - 17 q^{51} - 4 q^{52} - 43 q^{53} + 21 q^{54} - 16 q^{55} - 20 q^{56} - 25 q^{57} + 48 q^{58} - 19 q^{59} - 25 q^{60} + 9 q^{61} + q^{62} + 13 q^{63} + 20 q^{64} - 87 q^{65} + 29 q^{66} + 32 q^{67} + 9 q^{68} - 23 q^{69} + 3 q^{70} - 63 q^{71} - 13 q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - 14 q^{76} - 8 q^{77} + q^{78} - 51 q^{79} - 3 q^{80} + 4 q^{81} + 27 q^{82} - 24 q^{83} + 3 q^{84} + 31 q^{85} + 3 q^{86} - 33 q^{87} + 8 q^{88} - 35 q^{89} + 12 q^{90} - 4 q^{91} - 23 q^{92} + 17 q^{93} + 26 q^{94} - 30 q^{95} - 3 q^{96} + 5 q^{97} - 20 q^{98} - 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 32 x^{18} + 106 x^{17} + 382 x^{16} - 1495 x^{15} - 1963 x^{14} + 10784 x^{13} + \cdots - 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!15 \nu^{19} + \cdots + 67\!\cdots\!98 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16\!\cdots\!53 \nu^{19} + \cdots - 70\!\cdots\!04 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\!\cdots\!55 \nu^{19} + \cdots + 83\!\cdots\!76 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!71 \nu^{19} + \cdots + 53\!\cdots\!66 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!21 \nu^{19} + \cdots + 58\!\cdots\!96 ) / 11\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!73 \nu^{19} + \cdots - 10\!\cdots\!52 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!00 \nu^{19} + \cdots + 11\!\cdots\!28 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\!\cdots\!71 \nu^{19} + \cdots + 35\!\cdots\!26 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31\!\cdots\!27 \nu^{19} + \cdots - 22\!\cdots\!86 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!25 \nu^{19} + \cdots - 24\!\cdots\!34 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!35 \nu^{19} + \cdots + 25\!\cdots\!84 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!50 \nu^{19} + \cdots + 12\!\cdots\!36 ) / 56\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 37\!\cdots\!05 \nu^{19} + \cdots + 18\!\cdots\!62 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!12 \nu^{19} + \cdots - 29\!\cdots\!48 ) / 28\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 45\!\cdots\!19 \nu^{19} + \cdots + 13\!\cdots\!58 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 14\!\cdots\!37 \nu^{19} + \cdots + 72\!\cdots\!14 ) / 22\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 87\!\cdots\!19 \nu^{19} + \cdots + 23\!\cdots\!06 ) / 11\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \cdots + 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} - 2 \beta_{17} - \beta_{16} + 2 \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + 13 \beta_{15} + 14 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 25 \beta_{11} + 14 \beta_{10} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} - 19 \beta_{18} - 32 \beta_{17} - 16 \beta_{16} + 34 \beta_{15} + 20 \beta_{14} + 32 \beta_{13} + \cdots + 271 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27 \beta_{19} - 2 \beta_{18} - \beta_{17} + 3 \beta_{16} + 133 \beta_{15} + 169 \beta_{14} + 176 \beta_{13} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 21 \beta_{19} - 283 \beta_{18} - 374 \beta_{17} - 194 \beta_{16} + 437 \beta_{15} + 293 \beta_{14} + \cdots + 2653 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 442 \beta_{19} - 55 \beta_{18} - 4 \beta_{17} + 52 \beta_{16} + 1278 \beta_{15} + 1948 \beta_{14} + \cdots + 1355 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 311 \beta_{19} - 3711 \beta_{18} - 3982 \beta_{17} - 2151 \beta_{16} + 5059 \beta_{15} + 3749 \beta_{14} + \cdots + 26891 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5973 \beta_{19} - 950 \beta_{18} + 144 \beta_{17} + 629 \beta_{16} + 12135 \beta_{15} + 21873 \beta_{14} + \cdots + 16814 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4036 \beta_{19} - 45126 \beta_{18} - 41127 \beta_{17} - 22989 \beta_{16} + 55796 \beta_{15} + \cdots + 276951 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 73359 \beta_{19} - 13355 \beta_{18} + 3932 \beta_{17} + 6655 \beta_{16} + 115919 \beta_{15} + \cdots + 194430 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 48819 \beta_{19} - 523562 \beta_{18} - 421467 \beta_{17} - 241793 \beta_{16} + 600443 \beta_{15} + \cdots + 2875172 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 853415 \beta_{19} - 167406 \beta_{18} + 67283 \beta_{17} + 66789 \beta_{16} + 1120251 \beta_{15} + \cdots + 2153054 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 564122 \beta_{19} - 5892624 \beta_{18} - 4322419 \beta_{17} - 2526440 \beta_{16} + 6377698 \beta_{15} + \cdots + 29977633 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 9603713 \beta_{19} - 1952160 \beta_{18} + 967181 \beta_{17} + 662764 \beta_{16} + 10957986 \beta_{15} + \cdots + 23175369 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 6307049 \beta_{19} - 64988626 \beta_{18} - 44493263 \beta_{17} - 26344094 \beta_{16} + 67260110 \beta_{15} + \cdots + 313328762 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 105786370 \beta_{19} - 21660576 \beta_{18} + 12709219 \beta_{17} + 6661747 \beta_{16} + 108325526 \beta_{15} + \cdots + 244640595 \) 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
−3.26119
−2.73475
−2.57684
−2.11643
−1.69612
−1.63154
−0.937572
−0.141161
−0.0460830
0.509862
0.751545
0.814827
1.20597
1.26295
1.67912
1.69438
2.00937
2.16480
2.82470
3.22416
−1.00000 −3.26119 1.00000 −1.57113 3.26119 1.00000 −1.00000 7.63538 1.57113
1.2 −1.00000 −2.73475 1.00000 3.55748 2.73475 1.00000 −1.00000 4.47885 −3.55748
1.3 −1.00000 −2.57684 1.00000 4.04233 2.57684 1.00000 −1.00000 3.64010 −4.04233
1.4 −1.00000 −2.11643 1.00000 −3.30251 2.11643 1.00000 −1.00000 1.47927 3.30251
1.5 −1.00000 −1.69612 1.00000 −1.00716 1.69612 1.00000 −1.00000 −0.123166 1.00716
1.6 −1.00000 −1.63154 1.00000 1.13171 1.63154 1.00000 −1.00000 −0.338090 −1.13171
1.7 −1.00000 −0.937572 1.00000 2.71354 0.937572 1.00000 −1.00000 −2.12096 −2.71354
1.8 −1.00000 −0.141161 1.00000 −1.57700 0.141161 1.00000 −1.00000 −2.98007 1.57700
1.9 −1.00000 −0.0460830 1.00000 −3.28191 0.0460830 1.00000 −1.00000 −2.99788 3.28191
1.10 −1.00000 0.509862 1.00000 0.844791 −0.509862 1.00000 −1.00000 −2.74004 −0.844791
1.11 −1.00000 0.751545 1.00000 2.13649 −0.751545 1.00000 −1.00000 −2.43518 −2.13649
1.12 −1.00000 0.814827 1.00000 0.275205 −0.814827 1.00000 −1.00000 −2.33606 −0.275205
1.13 −1.00000 1.20597 1.00000 2.17113 −1.20597 1.00000 −1.00000 −1.54564 −2.17113
1.14 −1.00000 1.26295 1.00000 −3.04196 −1.26295 1.00000 −1.00000 −1.40496 3.04196
1.15 −1.00000 1.67912 1.00000 −1.69524 −1.67912 1.00000 −1.00000 −0.180554 1.69524
1.16 −1.00000 1.69438 1.00000 −4.04996 −1.69438 1.00000 −1.00000 −0.129071 4.04996
1.17 −1.00000 2.00937 1.00000 2.56029 −2.00937 1.00000 −1.00000 1.03758 −2.56029
1.18 −1.00000 2.16480 1.00000 2.31808 −2.16480 1.00000 −1.00000 1.68636 −2.31808
1.19 −1.00000 2.82470 1.00000 −1.73446 −2.82470 1.00000 −1.00000 4.97893 1.73446
1.20 −1.00000 3.22416 1.00000 −3.48974 −3.22416 1.00000 −1.00000 7.39519 3.48974
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(431\) \(-1\)

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 6034.2.a.k ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.k ✓ 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3}^{20} - 3 T_{3}^{19} - 32 T_{3}^{18} + 106 T_{3}^{17} + 382 T_{3}^{16} - 1495 T_{3}^{15} + \cdots - 44 \) Copy content Toggle raw display
\( T_{5}^{20} + 3 T_{5}^{19} - 61 T_{5}^{18} - 177 T_{5}^{17} + 1557 T_{5}^{16} + 4327 T_{5}^{15} + \cdots + 966416 \) Copy content Toggle raw display
\( T_{11}^{20} + 8 T_{11}^{19} - 80 T_{11}^{18} - 733 T_{11}^{17} + 2216 T_{11}^{16} + 26931 T_{11}^{15} + \cdots - 13416128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 3 T^{19} + \cdots - 44 \) Copy content Toggle raw display
$5$ \( T^{20} + 3 T^{19} + \cdots + 966416 \) Copy content Toggle raw display
$7$ \( (T - 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + \cdots - 13416128 \) Copy content Toggle raw display
$13$ \( T^{20} + 4 T^{19} + \cdots + 13471952 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 237058568 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 621266000 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 1978772800 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 41980145036 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 346906240 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 9056480934400 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 1339736384768 \) Copy content Toggle raw display
$43$ \( T^{20} + 3 T^{19} + \cdots + 10600000 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 193000802551040 \) Copy content Toggle raw display
$53$ \( T^{20} + 43 T^{19} + \cdots + 82232576 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 161413934606608 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 20\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 13\!\cdots\!50 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 74021183683168 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 11\!\cdots\!78 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 65\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 47\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
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