Properties

Label 59.4.a.b
Level $59$
Weight $4$
Character orbit 59.a
Self dual yes
Analytic conductor $3.481$
Analytic rank $1$
Dimension $2$
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] = [59,4,Mod(1,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 59.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: \(3.48111269034\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 3 \beta + 1) q^{3} + (\beta - 4) q^{4} + (3 \beta - 17) q^{5} + ( - 2 \beta - 12) q^{6} + (2 \beta - 5) q^{7} + ( - 11 \beta + 4) q^{8} + (3 \beta + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - 3 \beta + 1) q^{3} + (\beta - 4) q^{4} + (3 \beta - 17) q^{5} + ( - 2 \beta - 12) q^{6} + (2 \beta - 5) q^{7} + ( - 11 \beta + 4) q^{8} + (3 \beta + 10) q^{9} + ( - 14 \beta + 12) q^{10} + (\beta + 6) q^{11} + (10 \beta - 16) q^{12} + (9 \beta - 52) q^{13} + ( - 3 \beta + 8) q^{14} + (45 \beta - 53) q^{15} + ( - 15 \beta - 12) q^{16} + ( - 3 \beta + 86) q^{17} + (13 \beta + 12) q^{18} + (27 \beta - 67) q^{19} + ( - 26 \beta + 80) q^{20} + (11 \beta - 29) q^{21} + (7 \beta + 4) q^{22} + ( - 24 \beta - 36) q^{23} + (10 \beta + 136) q^{24} + ( - 93 \beta + 200) q^{25} + ( - 43 \beta + 36) q^{26} + (45 \beta - 53) q^{27} + ( - 11 \beta + 28) q^{28} + ( - 61 \beta - 123) q^{29} + ( - 8 \beta + 180) q^{30} + ( - 48 \beta + 52) q^{31} + (61 \beta - 92) q^{32} + ( - 20 \beta - 6) q^{33} + (83 \beta - 12) q^{34} + ( - 43 \beta + 109) q^{35} + (\beta - 28) q^{36} + (53 \beta - 172) q^{37} + ( - 40 \beta + 108) q^{38} + (138 \beta - 160) q^{39} + (166 \beta - 200) q^{40} + ( - 58 \beta + 255) q^{41} + ( - 18 \beta + 44) q^{42} + (53 \beta - 44) q^{43} + (3 \beta - 20) q^{44} + ( - 12 \beta - 134) q^{45} + ( - 60 \beta - 96) q^{46} + (106 \beta - 40) q^{47} + (66 \beta + 168) q^{48} + ( - 16 \beta - 302) q^{49} + (107 \beta - 372) q^{50} + ( - 252 \beta + 122) q^{51} + ( - 79 \beta + 244) q^{52} + ( - 151 \beta - 245) q^{53} + ( - 8 \beta + 180) q^{54} + (4 \beta - 90) q^{55} + (41 \beta - 108) q^{56} + (147 \beta - 391) q^{57} + ( - 184 \beta - 244) q^{58} + 59 q^{59} + ( - 188 \beta + 392) q^{60} + ( - 120 \beta - 242) q^{61} + (4 \beta - 192) q^{62} + (11 \beta - 26) q^{63} + (89 \beta + 340) q^{64} + ( - 282 \beta + 992) q^{65} + ( - 26 \beta - 80) q^{66} + (166 \beta - 746) q^{67} + (95 \beta - 356) q^{68} + (156 \beta + 252) q^{69} + (66 \beta - 172) q^{70} + (441 \beta - 76) q^{71} + ( - 131 \beta - 92) q^{72} + (244 \beta - 98) q^{73} + ( - 119 \beta + 212) q^{74} + ( - 414 \beta + 1316) q^{75} + ( - 148 \beta + 376) q^{76} + (9 \beta - 22) q^{77} + ( - 22 \beta + 552) q^{78} + ( - 26 \beta - 99) q^{79} + (174 \beta + 24) q^{80} + ( - 12 \beta - 863) q^{81} + (197 \beta - 232) q^{82} + (75 \beta + 944) q^{83} + ( - 62 \beta + 160) q^{84} + (300 \beta - 1498) q^{85} + (9 \beta + 212) q^{86} + (491 \beta + 609) q^{87} + ( - 73 \beta - 20) q^{88} + (356 \beta + 166) q^{89} + ( - 146 \beta - 48) q^{90} + ( - 131 \beta + 332) q^{91} + (36 \beta + 48) q^{92} + ( - 60 \beta + 628) q^{93} + (66 \beta + 424) q^{94} + ( - 579 \beta + 1463) q^{95} + (154 \beta - 824) q^{96} + ( - 672 \beta + 514) q^{97} + ( - 318 \beta - 64) q^{98} + (31 \beta + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - 7 q^{4} - 31 q^{5} - 26 q^{6} - 8 q^{7} - 3 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - 7 q^{4} - 31 q^{5} - 26 q^{6} - 8 q^{7} - 3 q^{8} + 23 q^{9} + 10 q^{10} + 13 q^{11} - 22 q^{12} - 95 q^{13} + 13 q^{14} - 61 q^{15} - 39 q^{16} + 169 q^{17} + 37 q^{18} - 107 q^{19} + 134 q^{20} - 47 q^{21} + 15 q^{22} - 96 q^{23} + 282 q^{24} + 307 q^{25} + 29 q^{26} - 61 q^{27} + 45 q^{28} - 307 q^{29} + 352 q^{30} + 56 q^{31} - 123 q^{32} - 32 q^{33} + 59 q^{34} + 175 q^{35} - 55 q^{36} - 291 q^{37} + 176 q^{38} - 182 q^{39} - 234 q^{40} + 452 q^{41} + 70 q^{42} - 35 q^{43} - 37 q^{44} - 280 q^{45} - 252 q^{46} + 26 q^{47} + 402 q^{48} - 620 q^{49} - 637 q^{50} - 8 q^{51} + 409 q^{52} - 641 q^{53} + 352 q^{54} - 176 q^{55} - 175 q^{56} - 635 q^{57} - 672 q^{58} + 118 q^{59} + 596 q^{60} - 604 q^{61} - 380 q^{62} - 41 q^{63} + 769 q^{64} + 1702 q^{65} - 186 q^{66} - 1326 q^{67} - 617 q^{68} + 660 q^{69} - 278 q^{70} + 289 q^{71} - 315 q^{72} + 48 q^{73} + 305 q^{74} + 2218 q^{75} + 604 q^{76} - 35 q^{77} + 1082 q^{78} - 224 q^{79} + 222 q^{80} - 1738 q^{81} - 267 q^{82} + 1963 q^{83} + 258 q^{84} - 2696 q^{85} + 433 q^{86} + 1709 q^{87} - 113 q^{88} + 688 q^{89} - 242 q^{90} + 533 q^{91} + 132 q^{92} + 1196 q^{93} + 914 q^{94} + 2347 q^{95} - 1494 q^{96} + 356 q^{97} - 446 q^{98} + 175 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−1.56155
2.56155
−1.56155 5.68466 −5.56155 −21.6847 −8.87689 −8.12311 21.1771 5.31534 33.8617
1.2 2.56155 −6.68466 −1.43845 −9.31534 −17.1231 0.123106 −24.1771 17.6847 −23.8617
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(59\) \(-1\)

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 59.4.a.b 2
3.b odd 2 1 531.4.a.a 2
4.b odd 2 1 944.4.a.e 2
5.b even 2 1 1475.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.4.a.b 2 1.a even 1 1 trivial
531.4.a.a 2 3.b odd 2 1
944.4.a.e 2 4.b odd 2 1
1475.4.a.a 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(59))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$5$ \( T^{2} + 31T + 202 \) Copy content Toggle raw display
$7$ \( T^{2} + 8T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$13$ \( T^{2} + 95T + 1912 \) Copy content Toggle raw display
$17$ \( T^{2} - 169T + 7102 \) Copy content Toggle raw display
$19$ \( T^{2} + 107T - 236 \) Copy content Toggle raw display
$23$ \( T^{2} + 96T - 144 \) Copy content Toggle raw display
$29$ \( T^{2} + 307T + 7748 \) Copy content Toggle raw display
$31$ \( T^{2} - 56T - 9008 \) Copy content Toggle raw display
$37$ \( T^{2} + 291T + 9232 \) Copy content Toggle raw display
$41$ \( T^{2} - 452T + 36779 \) Copy content Toggle raw display
$43$ \( T^{2} + 35T - 11632 \) Copy content Toggle raw display
$47$ \( T^{2} - 26T - 47584 \) Copy content Toggle raw display
$53$ \( T^{2} + 641T + 5816 \) Copy content Toggle raw display
$59$ \( (T - 59)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 604T + 30004 \) Copy content Toggle raw display
$67$ \( T^{2} + 1326 T + 322456 \) Copy content Toggle raw display
$71$ \( T^{2} - 289T - 805664 \) Copy content Toggle raw display
$73$ \( T^{2} - 48T - 252452 \) Copy content Toggle raw display
$79$ \( T^{2} + 224T + 9671 \) Copy content Toggle raw display
$83$ \( T^{2} - 1963 T + 939436 \) Copy content Toggle raw display
$89$ \( T^{2} - 688T - 420292 \) Copy content Toggle raw display
$97$ \( T^{2} - 356 T - 1887548 \) Copy content Toggle raw display
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