↑

Properties

Label 59.7.b.c
Level $59$
Weight $7$
Character orbit 59.b
Analytic conductor $13.573$
Analytic rank $0$
Dimension $26$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,7,Mod(58,59)]
 
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([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 59.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(13.5731909336\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 10 q^{3} - 1090 q^{4} + 142 q^{5} + 406 q^{7} + 5432 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 10 q^{3} - 1090 q^{4} + 142 q^{5} + 406 q^{7} + 5432 q^{9} - 1124 q^{12} + 14982 q^{15} + 12734 q^{16} - 9108 q^{17} + 3850 q^{19} - 46896 q^{20} - 49034 q^{21} + 11238 q^{22} + 18792 q^{25} - 64590 q^{26} + 3550 q^{27} - 45542 q^{28} - 31730 q^{29} + 163558 q^{35} - 325266 q^{36} + 91914 q^{41} + 736396 q^{45} + 287148 q^{46} + 479572 q^{48} - 462900 q^{49} + 329932 q^{51} + 8238 q^{53} - 187506 q^{57} + 326182 q^{59} - 970064 q^{60} + 630140 q^{62} + 630508 q^{63} - 1800262 q^{64} - 869200 q^{66} - 319586 q^{68} + 1763840 q^{71} - 2294090 q^{74} + 354736 q^{75} + 247144 q^{76} - 375064 q^{78} + 4702 q^{79} + 1920984 q^{80} - 2435946 q^{81} - 1007672 q^{84} + 864044 q^{85} + 5031110 q^{86} - 1519202 q^{87} + 725994 q^{88} + 2835768 q^{94} - 2396490 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 − 15.6634i −23.5580 −181.343 −4.53524 368.999i 95.4017 1837.99i −174.022 71.0374i
58.2 − 14.7162i 41.3628 −152.567 214.730 − 608.704i 125.837 1303.37i 981.885 − 3160.01i
58.3 − 13.0886i 33.4993 −107.312 −173.560 − 438.461i −128.659 566.899i 393.206 2271.67i
58.4 − 12.6464i 0.494651 −95.9324 19.8525 − 6.25557i −195.165 403.831i −728.755 − 251.064i
58.5 − 11.1082i −46.4934 −59.3927 157.957 516.459i 407.985 − 51.1790i 1432.64 − 1754.62i
58.6 − 11.0322i −38.3464 −57.7084 −194.793 423.043i −255.913 − 69.4101i 741.443 2148.98i
58.7 − 10.3981i 12.0734 −44.1211 −27.1374 − 125.541i 631.586 − 206.703i −583.232 282.178i
58.8 − 9.83054i −11.7050 −32.6396 162.986 115.067i −407.577 − 308.290i −591.992 − 1602.24i
58.9 − 7.19418i 44.5169 12.2438 52.8479 − 320.263i −372.902 − 548.511i 1252.76 − 380.197i
58.10 − 4.84601i 33.0733 40.5162 102.756 − 160.273i 212.228 − 506.486i 364.843 − 497.957i
58.11 − 4.70683i −15.6097 41.8457 −148.856 73.4721i 233.894 − 498.198i −485.338 700.642i
58.12 − 4.66346i −37.4041 42.2521 9.17391 174.432i 221.670 − 495.503i 670.066 − 42.7822i
58.13 − 3.85246i 13.0960 49.1585 −100.422 − 50.4519i −365.384 − 435.939i −557.495 386.871i
58.14 3.85246i 13.0960 49.1585 −100.422 50.4519i −365.384 435.939i −557.495 − 386.871i
58.15 4.66346i −37.4041 42.2521 9.17391 − 174.432i 221.670 495.503i 670.066 42.7822i
58.16 4.70683i −15.6097 41.8457 −148.856 − 73.4721i 233.894 498.198i −485.338 − 700.642i
58.17 4.84601i 33.0733 40.5162 102.756 160.273i 212.228 506.486i 364.843 497.957i
58.18 7.19418i 44.5169 12.2438 52.8479 320.263i −372.902 548.511i 1252.76 380.197i
58.19 9.83054i −11.7050 −32.6396 162.986 − 115.067i −407.577 308.290i −591.992 1602.24i
58.20 10.3981i 12.0734 −44.1211 −27.1374 125.541i 631.586 206.703i −583.232 − 282.178i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.7.b.c ✓ 26
59.b odd 2 1 inner 59.7.b.c ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.7.b.c ✓ 26 1.a even 1 1 trivial
59.7.b.c ✓ 26 59.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(59, [\chi])\):

\( T_{2}^{26} + 1377 T_{2}^{24} + 839313 T_{2}^{22} + 298766727 T_{2}^{20} + 69000111702 T_{2}^{18} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
\( T_{3}^{13} - 5 T_{3}^{12} - 6084 T_{3}^{11} + 27440 T_{3}^{10} + 13887738 T_{3}^{9} + \cdots - 45\!\cdots\!40 \) Copy content Toggle raw display