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Properties

Label 49.6.g.a
Level $49$
Weight $6$
Character orbit 49.g
Analytic conductor $7.859$
Analytic rank $0$
Dimension $264$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,6,Mod(2,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.2");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.g (of order \(21\), degree \(12\), 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: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(22\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9} - 792 q^{10} - 635 q^{11} + 3605 q^{12} + 1834 q^{13} - 875 q^{14} - 2572 q^{15} + 3735 q^{16} - 1023 q^{17} - 1336 q^{18} + 9741 q^{19} + 8330 q^{20} + 1512 q^{21} + 842 q^{22} - 1553 q^{23} + 1756 q^{24} + 13144 q^{25} - 1946 q^{26} - 19849 q^{27} - 31584 q^{28} - 30356 q^{29} + 6129 q^{30} + 43903 q^{31} + 27725 q^{32} - 3412 q^{33} - 73182 q^{34} - 17360 q^{35} - 102868 q^{36} - 22105 q^{37} + 19116 q^{38} + 109165 q^{39} + 77354 q^{40} + 47446 q^{41} + 157738 q^{42} + 6674 q^{43} + 15154 q^{44} - 130923 q^{45} - 136952 q^{46} - 195163 q^{47} - 151484 q^{48} - 131194 q^{49} + 263000 q^{50} - 84276 q^{51} - 58758 q^{52} - 110577 q^{53} + 170313 q^{54} + 290245 q^{55} + 79688 q^{56} + 47310 q^{57} + 252156 q^{58} + 127308 q^{59} + 86254 q^{60} + 166863 q^{61} - 63578 q^{62} - 243033 q^{63} - 293812 q^{64} - 16702 q^{65} - 661788 q^{66} + 46195 q^{67} + 393869 q^{68} + 321394 q^{69} - 64330 q^{70} - 61793 q^{71} + 333944 q^{72} + 21699 q^{73} + 133514 q^{74} - 431136 q^{75} + 3087 q^{76} - 48993 q^{77} - 450576 q^{78} - 181 q^{79} + 264228 q^{80} + 142090 q^{81} - 512442 q^{82} + 102424 q^{83} + 344932 q^{84} - 205502 q^{85} + 611991 q^{86} + 1427200 q^{87} + 685993 q^{88} + 145800 q^{89} + 677844 q^{90} + 732956 q^{91} + 10018 q^{92} - 1233536 q^{93} - 601938 q^{94} - 945265 q^{95} - 2372958 q^{96} - 1932952 q^{97} - 672973 q^{98} + 1090736 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −10.4274 − 3.21642i −3.43400 + 8.74968i 71.9454 + 49.0516i 12.8738 − 1.94041i 63.9503 − 80.1911i −82.8526 − 99.7118i −374.716 − 469.879i 113.367 + 105.189i −140.481 − 21.1741i
2.2 −10.0399 − 3.09691i 9.56163 − 24.3626i 64.7699 + 44.1593i −25.7914 + 3.88743i −171.447 + 214.988i 24.4890 + 127.308i −303.901 − 381.080i −323.981 − 300.611i 270.983 + 40.8441i
2.3 −7.64989 − 2.35968i 2.34348 − 5.97110i 26.5131 + 18.0763i −0.467538 + 0.0704700i −32.0173 + 40.1484i 116.464 − 56.9478i −0.443581 − 0.556232i 147.969 + 137.296i 3.74290 + 0.564151i
2.4 −7.58622 − 2.34004i −8.77900 + 22.3685i 25.6353 + 17.4778i 44.1683 − 6.65731i 118.943 − 149.149i 109.919 + 68.7371i 4.81872 + 6.04248i −245.149 − 227.465i −350.649 − 52.8518i
2.5 −7.56205 − 2.33258i −5.53040 + 14.0912i 25.3041 + 17.2520i −100.121 + 15.0908i 74.6902 − 93.6586i −91.8316 + 91.5093i 6.78132 + 8.50350i 10.1540 + 9.42155i 792.323 + 119.423i
2.6 −6.81543 − 2.10228i 3.52552 − 8.98289i 15.5909 + 10.6297i 80.2537 − 12.0963i −42.9125 + 53.8106i −85.3330 + 97.5975i 58.3893 + 73.2178i 109.869 + 101.943i −572.393 − 86.2744i
2.7 −5.24707 − 1.61851i 7.63255 − 19.4474i −1.52748 − 1.04142i −57.8178 + 8.71463i −71.5243 + 89.6886i −83.4438 − 99.2176i 115.884 + 145.314i −141.815 − 131.585i 317.479 + 47.8522i
2.8 −3.14587 − 0.970373i −3.81108 + 9.71048i −17.4847 − 11.9209i 67.5754 − 10.1854i 21.4120 − 26.8498i −81.7407 − 100.625i 109.121 + 136.833i 98.3625 + 91.2671i −222.467 − 33.5315i
2.9 −2.66606 − 0.822370i −6.33899 + 16.1515i −20.0081 − 13.6413i −58.2851 + 8.78507i 30.1826 − 37.8478i 54.7544 − 117.511i 97.7899 + 122.625i −42.5557 − 39.4859i 162.616 + 24.5104i
2.10 −1.52973 − 0.471861i 10.3277 − 26.3145i −24.3222 − 16.5826i 90.2851 − 13.6083i −28.2154 + 35.3810i 124.488 − 36.1916i 61.3216 + 76.8948i −407.661 − 378.255i −144.533 − 21.7849i
2.11 −1.11459 − 0.343805i 1.70006 − 4.33168i −25.3155 − 17.2598i −34.5506 + 5.20767i −3.38412 + 4.24356i 61.0553 + 114.365i 45.5542 + 57.1232i 162.258 + 150.554i 40.3001 + 6.07427i
2.12 −0.768968 − 0.237195i −8.72763 + 22.2376i −25.9046 − 17.6615i 28.5592 − 4.30461i 11.9859 − 15.0299i −107.947 + 71.7951i 31.7861 + 39.8585i −240.209 − 222.882i −22.9821 − 3.46400i
2.13 2.16057 + 0.666447i 6.26230 − 15.9561i −22.2157 − 15.1464i 0.666814 − 0.100506i 24.1640 − 30.3007i −129.491 + 6.24655i −83.0154 − 104.098i −37.2488 − 34.5618i 1.50768 + 0.227246i
2.14 3.73931 + 1.15342i 0.0993195 − 0.253062i −13.7876 − 9.40021i 41.9031 − 6.31588i 0.663274 − 0.831720i 64.1502 − 112.658i −118.788 − 148.955i 178.077 + 165.232i 163.974 + 24.7151i
2.15 4.24738 + 1.31014i −9.79667 + 24.9615i −10.1159 − 6.89687i −27.7351 + 4.18039i −74.3134 + 93.1860i 129.357 − 8.58801i −122.612 − 153.751i −348.971 − 323.797i −123.279 − 18.5812i
2.16 5.22133 + 1.61057i 8.72914 − 22.2415i −1.77125 − 1.20762i −104.732 + 15.7858i 81.3992 − 102.071i 122.487 − 42.4725i −116.321 − 145.862i −240.355 − 223.016i −572.263 − 86.2548i
2.17 5.34627 + 1.64911i −3.67157 + 9.35501i −0.576558 − 0.393090i 96.5461 − 14.5520i −35.0566 + 43.9596i 41.8489 + 122.702i −114.061 − 143.027i 104.096 + 96.5868i 540.160 + 81.4160i
2.18 5.87515 + 1.81224i −3.88671 + 9.90318i 4.79349 + 3.26815i −68.8177 + 10.3726i −40.7820 + 51.1390i −128.506 − 17.1261i −100.429 − 125.934i 95.1652 + 88.3004i −423.112 − 63.7738i
2.19 7.81307 + 2.41001i 8.62663 − 21.9803i 28.7962 + 19.6329i 40.7138 − 6.13662i 120.373 − 150.943i −56.8046 + 116.534i 14.5399 + 18.2325i −230.583 − 213.950i 332.889 + 50.1749i
2.20 9.25626 + 2.85518i 4.05332 − 10.3277i 51.0867 + 34.8304i 20.5483 − 3.09715i 67.0061 − 84.0230i 13.7032 − 128.916i 180.161 + 225.915i 87.8996 + 81.5589i 199.043 + 30.0009i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.g.a ✓ 264
49.g even 21 1 inner 49.6.g.a ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.g.a ✓ 264 1.a even 1 1 trivial
49.6.g.a ✓ 264 49.g even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(49, [\chi])\).