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Properties

Label 309.6.a.d
Level $309$
Weight $6$
Character orbit 309.a
Self dual yes
Analytic conductor $49.559$
Analytic rank $0$
Dimension $25$
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] = [309,6,Mod(1,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 309.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: \(49.5586003222\)
Analytic rank: \(0\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 14 q^{2} + 225 q^{3} + 486 q^{4} + 47 q^{5} + 126 q^{6} + 402 q^{7} + 342 q^{8} + 2025 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 14 q^{2} + 225 q^{3} + 486 q^{4} + 47 q^{5} + 126 q^{6} + 402 q^{7} + 342 q^{8} + 2025 q^{9} + 693 q^{10} + 1470 q^{11} + 4374 q^{12} + 2515 q^{13} + 254 q^{14} + 423 q^{15} + 11542 q^{16} + 880 q^{17} + 1134 q^{18} + 7412 q^{19} + 1927 q^{20} + 3618 q^{21} + 5461 q^{22} + 5567 q^{23} + 3078 q^{24} + 31584 q^{25} + 18502 q^{26} + 18225 q^{27} + 25011 q^{28} + 17230 q^{29} + 6237 q^{30} + 22821 q^{31} + 50233 q^{32} + 13230 q^{33} + 38342 q^{34} + 30664 q^{35} + 39366 q^{36} + 13342 q^{37} + 25860 q^{38} + 22635 q^{39} + 40701 q^{40} + 36374 q^{41} + 2286 q^{42} + 48371 q^{43} - 4133 q^{44} + 3807 q^{45} + 30489 q^{46} + 17740 q^{47} + 103878 q^{48} + 119201 q^{49} - 9505 q^{50} + 7920 q^{51} + 50699 q^{52} - 52204 q^{53} + 10206 q^{54} + 90638 q^{55} - 80285 q^{56} + 66708 q^{57} + 15313 q^{58} + 34099 q^{59} + 17343 q^{60} + 71175 q^{61} - 92130 q^{62} + 32562 q^{63} + 289374 q^{64} - 32899 q^{65} + 49149 q^{66} + 85201 q^{67} - 41169 q^{68} + 50103 q^{69} - 92312 q^{70} + 102652 q^{71} + 27702 q^{72} + 186396 q^{73} - 258113 q^{74} + 284256 q^{75} + 148369 q^{76} - 109016 q^{77} + 166518 q^{78} + 210994 q^{79} + 17955 q^{80} + 164025 q^{81} + 635103 q^{82} + 68429 q^{83} + 225099 q^{84} + 375692 q^{85} + 360833 q^{86} + 155070 q^{87} + 556985 q^{88} + 163508 q^{89} + 56133 q^{90} + 591882 q^{91} + 388500 q^{92} + 205389 q^{93} + 205288 q^{94} + 87988 q^{95} + 452097 q^{96} + 385683 q^{97} - 61147 q^{98} + 119070 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.9182 9.00000 87.2065 −42.1419 −98.2635 136.700 −602.754 81.0000 460.113
1.2 −10.1365 9.00000 70.7479 87.9714 −91.2282 145.240 −392.767 81.0000 −891.719
1.3 −9.32103 9.00000 54.8817 −99.7771 −83.8893 −63.0953 −213.281 81.0000 930.026
1.4 −8.41150 9.00000 38.7533 −69.1144 −75.7035 151.167 −56.8051 81.0000 581.355
1.5 −8.29891 9.00000 36.8718 32.2581 −74.6901 −139.028 −40.4308 81.0000 −267.707
1.6 −7.82591 9.00000 29.2448 102.916 −70.4332 −33.8830 21.5617 81.0000 −805.408
1.7 −5.42827 9.00000 −2.53390 −49.9590 −48.8544 −257.124 187.459 81.0000 271.191
1.8 −5.24523 9.00000 −4.48757 −21.2020 −47.2071 −27.3773 191.386 81.0000 111.209
1.9 −2.97521 9.00000 −23.1481 36.7294 −26.7769 67.5957 164.077 81.0000 −109.278
1.10 −2.95711 9.00000 −23.2555 59.9608 −26.6140 248.119 163.397 81.0000 −177.311
1.11 −0.666354 9.00000 −31.5560 −57.3315 −5.99719 172.727 42.3508 81.0000 38.2031
1.12 0.860947 9.00000 −31.2588 −61.7793 7.74852 −150.744 −54.4625 81.0000 −53.1887
1.13 0.865827 9.00000 −31.2503 26.2215 7.79244 −186.935 −54.7638 81.0000 22.7033
1.14 2.23990 9.00000 −26.9828 82.0013 20.1591 −9.63459 −132.116 81.0000 183.675
1.15 3.58267 9.00000 −19.1644 88.5240 32.2441 119.852 −183.306 81.0000 317.153
1.16 3.82060 9.00000 −17.4030 −16.4672 34.3854 229.289 −188.749 81.0000 −62.9147
1.17 4.64560 9.00000 −10.4184 −79.9870 41.8104 −231.697 −197.059 81.0000 −371.587
1.18 5.85389 9.00000 2.26804 −40.0668 52.6850 −60.9454 −174.048 81.0000 −234.547
1.19 6.30173 9.00000 7.71183 −103.382 56.7156 116.004 −153.058 81.0000 −651.483
1.20 7.20633 9.00000 19.9312 88.7546 64.8570 −52.9231 −86.9717 81.0000 639.595
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(103\) \(1\)

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 309.6.a.d ✓ 25
3.b odd 2 1 927.6.a.f 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.6.a.d ✓ 25 1.a even 1 1 trivial
927.6.a.f 25 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 14 T_{2}^{24} - 545 T_{2}^{23} + 8132 T_{2}^{22} + 123739 T_{2}^{21} + \cdots + 28\!\cdots\!68 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(309))\). Copy content Toggle raw display