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Properties

Label 309.6.a.c
Level $309$
Weight $6$
Character orbit 309.a
Self dual yes
Analytic conductor $49.559$
Analytic rank $1$
Dimension $22$
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: \(1\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 8 q^{2} - 198 q^{3} + 342 q^{4} - 53 q^{5} + 72 q^{6} + 10 q^{7} - 264 q^{8} + 1782 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 8 q^{2} - 198 q^{3} + 342 q^{4} - 53 q^{5} + 72 q^{6} + 10 q^{7} - 264 q^{8} + 1782 q^{9} - 355 q^{10} - 708 q^{11} - 3078 q^{12} - 133 q^{13} - 2748 q^{14} + 477 q^{15} + 3678 q^{16} - 2006 q^{17} - 648 q^{18} - 4788 q^{19} - 2785 q^{20} - 90 q^{21} + 3609 q^{22} - 5695 q^{23} + 2376 q^{24} + 18477 q^{25} + 2432 q^{26} - 16038 q^{27} + 7635 q^{28} - 978 q^{29} + 3195 q^{30} - 6009 q^{31} + 22809 q^{32} + 6372 q^{33} - 4078 q^{34} - 22822 q^{35} + 27702 q^{36} + 13640 q^{37} - 5454 q^{38} + 1197 q^{39} - 13351 q^{40} - 24618 q^{41} + 24732 q^{42} + 1257 q^{43} - 65465 q^{44} - 4293 q^{45} - 6175 q^{46} - 63834 q^{47} - 33102 q^{48} + 18022 q^{49} - 41643 q^{50} + 18054 q^{51} - 40853 q^{52} - 13316 q^{53} + 5832 q^{54} - 35934 q^{55} - 251195 q^{56} + 43092 q^{57} - 103895 q^{58} - 138587 q^{59} + 25065 q^{60} - 53985 q^{61} - 218186 q^{62} + 810 q^{63} + 23758 q^{64} - 114073 q^{65} - 32481 q^{66} - 102785 q^{67} - 338669 q^{68} + 51255 q^{69} - 104184 q^{70} - 108740 q^{71} - 21384 q^{72} + 69762 q^{73} - 221377 q^{74} - 166293 q^{75} - 223267 q^{76} - 140360 q^{77} - 21888 q^{78} - 238938 q^{79} - 864251 q^{80} + 144342 q^{81} - 660293 q^{82} - 305455 q^{83} - 68715 q^{84} - 201204 q^{85} - 794679 q^{86} + 8802 q^{87} - 420823 q^{88} - 438448 q^{89} - 28755 q^{90} - 294186 q^{91} - 1251930 q^{92} + 54081 q^{93} - 826416 q^{94} - 652572 q^{95} - 205281 q^{96} - 284729 q^{97} - 887529 q^{98} - 57348 q^{99}+O(q^{100}) \) 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 −10.4914 −9.00000 78.0691 −3.41958 94.4224 97.3026 −483.328 81.0000 35.8762
1.2 −10.2362 −9.00000 72.7789 −87.0620 92.1254 187.261 −417.419 81.0000 891.180
1.3 −9.32105 −9.00000 54.8820 35.3971 83.8895 19.4648 −213.284 81.0000 −329.939
1.4 −7.76336 −9.00000 28.2698 −63.7534 69.8702 −222.370 28.9593 81.0000 494.940
1.5 −7.71849 −9.00000 27.5752 59.0766 69.4665 −234.823 34.1531 81.0000 −455.982
1.6 −7.25337 −9.00000 20.6114 11.4457 65.2804 191.311 82.6057 81.0000 −83.0197
1.7 −7.12203 −9.00000 18.7233 95.7289 64.0983 177.516 94.5570 81.0000 −681.784
1.8 −5.31841 −9.00000 −3.71447 65.7440 47.8657 −27.6535 189.944 81.0000 −349.654
1.9 −3.10804 −9.00000 −22.3401 −107.957 27.9723 −13.2146 168.891 81.0000 335.534
1.10 −1.37411 −9.00000 −30.1118 53.5472 12.3670 −13.2336 85.3483 81.0000 −73.5796
1.11 −0.949926 −9.00000 −31.0976 −46.7244 8.54933 −212.784 59.9381 81.0000 44.3847
1.12 0.322720 −9.00000 −31.8959 −76.7443 −2.90448 42.9648 −20.6205 81.0000 −24.7669
1.13 1.35356 −9.00000 −30.1679 −30.1704 −12.1820 2.34390 −84.1477 81.0000 −40.8373
1.14 2.91827 −9.00000 −23.4837 76.8195 −26.2644 −66.4325 −161.916 81.0000 224.180
1.15 3.07017 −9.00000 −22.5741 17.3342 −27.6315 72.5706 −167.551 81.0000 53.2189
1.16 4.24339 −9.00000 −13.9936 −90.4056 −38.1905 213.175 −195.169 81.0000 −383.626
1.17 6.96773 −9.00000 16.5493 46.3309 −62.7096 −167.758 −107.657 81.0000 322.821
1.18 7.40263 −9.00000 22.7989 110.720 −66.6237 −55.8072 −68.1123 81.0000 819.619
1.19 7.44776 −9.00000 23.4691 −28.6431 −67.0298 41.5463 −63.5364 81.0000 −213.327
1.20 8.14988 −9.00000 34.4205 −15.8441 −73.3489 122.881 19.7270 81.0000 −129.128
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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.c ✓ 22
3.b odd 2 1 927.6.a.d 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.6.a.c ✓ 22 1.a even 1 1 trivial
927.6.a.d 22 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 8 T_{2}^{21} - 491 T_{2}^{20} - 3840 T_{2}^{19} + 101877 T_{2}^{18} + \cdots - 372963978717184 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(309))\). Copy content Toggle raw display