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Properties

Label 1059.2.a.f
Level $1059$
Weight $2$
Character orbit 1059.a
Self dual yes
Analytic conductor $8.456$
Analytic rank $0$
Dimension $17$
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] = [1059,2,Mod(1,1059)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1059, 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("1059.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1059 = 3 \cdot 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1059.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: \(8.45615757405\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 24 x^{15} + 72 x^{14} + 232 x^{13} - 690 x^{12} - 1167 x^{11} + 3374 x^{10} + \cdots + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 24 x^{15} + 72 x^{14} + 232 x^{13} - 690 x^{12} - 1167 x^{11} + 3374 x^{10} + \cdots + 72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 144967 \nu^{16} + 2202733 \nu^{15} + 3349686 \nu^{14} - 62932512 \nu^{13} - 32926540 \nu^{12} + \cdots - 1270483100 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 596631 \nu^{16} - 565185 \nu^{15} - 19511454 \nu^{14} + 20733688 \nu^{13} + 254065172 \nu^{12} + \cdots + 1082483356 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1604330 \nu^{16} + 3859009 \nu^{15} + 43760148 \nu^{14} - 102809842 \nu^{13} + \cdots - 882195876 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2216237 \nu^{16} + 7788751 \nu^{15} + 50819042 \nu^{14} - 190058736 \nu^{13} + \cdots - 1030179700 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1585599 \nu^{16} - 3817602 \nu^{15} - 42532846 \nu^{14} + 99403030 \nu^{13} + 464012614 \nu^{12} + \cdots + 900021560 ) / 28222328 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3228249 \nu^{16} + 10834649 \nu^{15} + 72850690 \nu^{14} - 254732884 \nu^{13} + \cdots - 645080252 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4378151 \nu^{16} - 15461935 \nu^{15} - 95149646 \nu^{14} + 361327932 \nu^{13} + 786691616 \nu^{12} + \cdots + 821069524 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4474229 \nu^{16} - 17034704 \nu^{15} - 94812346 \nu^{14} + 396258390 \nu^{13} + 756143710 \nu^{12} + \cdots + 1181065144 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1262397 \nu^{16} + 4148036 \nu^{15} + 28646472 \nu^{14} - 97778338 \nu^{13} - 254698046 \nu^{12} + \cdots - 407040380 ) / 14111164 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5124578 \nu^{16} + 15637485 \nu^{15} + 125870576 \nu^{14} - 387089730 \nu^{13} + \cdots - 2178772780 ) / 56444656 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3443933 \nu^{16} + 12613812 \nu^{15} + 75606374 \nu^{14} - 298751926 \nu^{13} + \cdots - 991192816 ) / 28222328 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3604377 \nu^{16} - 11401882 \nu^{15} - 85278318 \nu^{14} + 275987282 \nu^{13} + 804010338 \nu^{12} + \cdots + 1266790976 ) / 28222328 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 12332047 \nu^{16} + 38605089 \nu^{15} + 287112110 \nu^{14} - 917220856 \nu^{13} + \cdots - 3692900732 ) / 56444656 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{16} - \beta_{15} + \beta_{14} - \beta_{7} - \beta_{5} - \beta_{4} + 2\beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{6} + 9\beta_{3} + \beta_{2} + 29\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{16} - 8 \beta_{15} + 11 \beta_{14} + \beta_{12} - \beta_{9} - 3 \beta_{8} - 10 \beta_{7} + \cdots + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{16} + 14 \beta_{15} + 16 \beta_{14} + 2 \beta_{13} + \beta_{12} + 13 \beta_{11} + 12 \beta_{9} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 81 \beta_{16} - 49 \beta_{15} + 98 \beta_{14} + 2 \beta_{13} + 13 \beta_{12} - \beta_{11} + \cdots + 639 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 31 \beta_{16} + 145 \beta_{15} + 176 \beta_{14} + 32 \beta_{13} + 16 \beta_{12} + 123 \beta_{11} + \cdots + 597 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 616 \beta_{16} - 257 \beta_{15} + 819 \beta_{14} + 34 \beta_{13} + 121 \beta_{12} - 13 \beta_{11} + \cdots + 4312 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 337 \beta_{16} + 1330 \beta_{15} + 1674 \beta_{14} + 344 \beta_{13} + 177 \beta_{12} + 1028 \beta_{11} + \cdots + 4768 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4571 \beta_{16} - 1069 \beta_{15} + 6654 \beta_{14} + 386 \beta_{13} + 1002 \beta_{12} - 103 \beta_{11} + \cdots + 29734 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3189 \beta_{16} + 11465 \beta_{15} + 14798 \beta_{14} + 3130 \beta_{13} + 1691 \beta_{12} + \cdots + 37724 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 33556 \beta_{16} - 1740 \beta_{15} + 53227 \beta_{14} + 3696 \beta_{13} + 7900 \beta_{12} + \cdots + 208174 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 28145 \beta_{16} + 95454 \beta_{15} + 125398 \beta_{14} + 26070 \beta_{13} + 14983 \beta_{12} + \cdots + 295876 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 245153 \beta_{16} + 32145 \beta_{15} + 421667 \beta_{14} + 32294 \beta_{13} + 60938 \beta_{12} + \cdots + 1474299 \) 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
2.77375
2.72311
2.46640
2.09027
1.85388
1.60982
1.06363
0.748174
−0.177201
−0.250110
−0.460173
−1.27032
−1.34963
−1.60867
−2.25754
−2.38844
−2.56696
−2.77375 −1.00000 5.69367 −1.34227 2.77375 2.31617 −10.2453 1.00000 3.72311
1.2 −2.72311 −1.00000 5.41534 4.37350 2.72311 0.0842452 −9.30035 1.00000 −11.9095
1.3 −2.46640 −1.00000 4.08313 −2.26967 2.46640 −2.68063 −5.13784 1.00000 5.59791
1.4 −2.09027 −1.00000 2.36924 −3.15800 2.09027 2.70839 −0.771820 1.00000 6.60108
1.5 −1.85388 −1.00000 1.43688 1.31946 1.85388 3.38999 1.04396 1.00000 −2.44613
1.6 −1.60982 −1.00000 0.591526 2.95388 1.60982 −4.03330 2.26739 1.00000 −4.75522
1.7 −1.06363 −1.00000 −0.868694 1.33748 1.06363 −3.94005 3.05123 1.00000 −1.42258
1.8 −0.748174 −1.00000 −1.44024 4.18682 0.748174 2.97040 2.57390 1.00000 −3.13247
1.9 0.177201 −1.00000 −1.96860 1.25244 −0.177201 3.67905 −0.703240 1.00000 0.221933
1.10 0.250110 −1.00000 −1.93744 −2.68170 −0.250110 −3.01445 −0.984795 1.00000 −0.670721
1.11 0.460173 −1.00000 −1.78824 0.905361 −0.460173 −2.72082 −1.74325 1.00000 0.416623
1.12 1.27032 −1.00000 −0.386290 3.52865 −1.27032 −2.76257 −3.03135 1.00000 4.48251
1.13 1.34963 −1.00000 −0.178496 −3.34972 −1.34963 1.89975 −2.94017 1.00000 −4.52089
1.14 1.60867 −1.00000 0.587808 3.54269 −1.60867 3.56954 −2.27175 1.00000 5.69901
1.15 2.25754 −1.00000 3.09650 1.16691 −2.25754 3.88073 2.47539 1.00000 2.63435
1.16 2.38844 −1.00000 3.70463 −1.75314 −2.38844 −1.94124 4.07141 1.00000 −4.18728
1.17 2.56696 −1.00000 4.58928 2.98731 −2.56696 −0.405198 6.64657 1.00000 7.66830
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(353\) \(-1\)

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 1059.2.a.f ✓ 17
3.b odd 2 1 3177.2.a.j 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1059.2.a.f ✓ 17 1.a even 1 1 trivial
3177.2.a.j 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} + 3 T_{2}^{16} - 24 T_{2}^{15} - 72 T_{2}^{14} + 232 T_{2}^{13} + 690 T_{2}^{12} - 1167 T_{2}^{11} + \cdots - 72 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1059))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + 3 T^{16} + \cdots - 72 \) Copy content Toggle raw display
$3$ \( (T + 1)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} - 13 T^{16} + \cdots - 714626 \) Copy content Toggle raw display
$7$ \( T^{17} - 3 T^{16} + \cdots - 391168 \) Copy content Toggle raw display
$11$ \( T^{17} - 3 T^{16} + \cdots - 13134208 \) Copy content Toggle raw display
$13$ \( T^{17} - 22 T^{16} + \cdots + 17344 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 1069631424 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 8501193924 \) Copy content Toggle raw display
$23$ \( T^{17} + 12 T^{16} + \cdots - 2971008 \) Copy content Toggle raw display
$29$ \( T^{17} - 28 T^{16} + \cdots - 5185728 \) Copy content Toggle raw display
$31$ \( T^{17} - 13 T^{16} + \cdots - 581632 \) Copy content Toggle raw display
$37$ \( T^{17} - 13 T^{16} + \cdots + 21890304 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 1212996672 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 299344572 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 1928853504 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 463400731974 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 229546332252 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 1751554366 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 12456097286656 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 11204214624 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 251901604994 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 14767743640064 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 40454156713344 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 1490555916 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 45\!\cdots\!86 \) Copy content Toggle raw display
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