Properties

Label 35.6.k.a
Level $35$
Weight $6$
Character orbit 35.k
Analytic conductor $5.613$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,6,Mod(3,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.3");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.k (of order \(12\), degree \(4\), 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: \(5.61343369345\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{2} - 6 q^{3} + 60 q^{5} + 190 q^{7} + 356 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{2} - 6 q^{3} + 60 q^{5} + 190 q^{7} + 356 q^{8} - 1494 q^{10} - 364 q^{11} + 186 q^{12} - 2972 q^{15} + 5380 q^{16} - 1608 q^{17} - 1916 q^{18} + 6132 q^{21} + 1048 q^{22} - 9670 q^{23} + 328 q^{25} + 2628 q^{26} + 35018 q^{28} + 34228 q^{30} - 2472 q^{31} - 10518 q^{32} - 72180 q^{33} - 30648 q^{35} - 114320 q^{36} - 15820 q^{37} + 44208 q^{38} + 140118 q^{40} + 67334 q^{42} + 74708 q^{43} - 156 q^{45} + 29956 q^{46} - 139536 q^{47} - 259532 q^{50} + 81176 q^{51} + 68424 q^{52} - 19600 q^{53} - 50204 q^{56} + 285176 q^{57} - 40026 q^{58} + 11774 q^{60} + 172728 q^{61} + 144636 q^{63} + 23516 q^{65} - 97860 q^{66} - 125934 q^{67} - 394512 q^{68} - 287480 q^{70} - 348616 q^{71} - 75608 q^{72} - 295764 q^{73} + 315834 q^{75} - 130280 q^{77} + 1029472 q^{78} + 594756 q^{80} + 249560 q^{81} + 141174 q^{82} - 423272 q^{85} - 203704 q^{86} - 425382 q^{87} + 32332 q^{88} - 23768 q^{91} + 596044 q^{92} + 115864 q^{93} + 62528 q^{95} - 391800 q^{96} + 345498 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −9.95978 2.66872i −4.43456 16.5500i 64.3624 + 37.1597i 55.6892 + 4.86925i 176.669i 127.528 23.3159i −308.553 308.553i −43.7930 + 25.2839i −541.658 197.115i
3.2 −8.82949 2.36586i −1.01039 3.77082i 44.6498 + 25.7786i −51.5659 + 21.5861i 35.6848i −68.0178 + 110.366i −126.411 126.411i 197.246 113.880i 506.370 68.5968i
3.3 −8.64179 2.31556i 7.51818 + 28.0582i 41.6059 + 24.0212i −15.7494 + 53.6373i 259.882i 119.105 51.1954i −101.488 101.488i −520.297 + 300.393i 260.303 427.054i
3.4 −8.12393 2.17680i 3.17140 + 11.8358i 33.5470 + 19.3683i 32.4351 45.5298i 103.057i −127.425 + 23.8695i −40.0639 40.0639i 80.4154 46.4279i −362.610 + 299.276i
3.5 −5.77451 1.54727i −5.02978 18.7714i 3.23807 + 1.86950i −46.7278 30.6841i 116.178i 36.5834 124.373i 119.466 + 119.466i −116.622 + 67.3320i 222.353 + 249.486i
3.6 −4.18974 1.12264i 0.900875 + 3.36211i −11.4192 6.59287i 36.1647 + 42.6276i 15.0977i −63.7723 112.872i 138.589 + 138.589i 199.952 115.442i −103.665 219.199i
3.7 −3.55905 0.953643i 3.79917 + 14.1787i −15.9554 9.21188i −11.6079 54.6832i 54.0857i 120.817 + 47.0133i 131.374 + 131.374i 23.8423 13.7654i −10.8354 + 205.690i
3.8 −2.46827 0.661371i −7.06878 26.3810i −22.0579 12.7351i 47.7429 29.0794i 69.7907i −83.3911 + 99.2619i 103.843 + 103.843i −435.548 + 251.464i −137.075 + 40.2001i
3.9 −1.58619 0.425019i −2.61466 9.75805i −25.3775 14.6517i −7.86751 + 55.3453i 16.5894i 65.3428 + 111.970i 71.1838 + 71.1838i 122.061 70.4720i 36.0022 84.4444i
3.10 0.911102 + 0.244129i 4.40748 + 16.4489i −26.9423 15.5551i −55.1076 + 9.38874i 16.0626i −68.6887 109.949i −42.0929 42.0929i −40.6973 + 23.4966i −52.5007 4.89927i
3.11 1.93900 + 0.519555i 6.66647 + 24.8796i −24.2230 13.9852i 54.2832 + 13.3542i 51.7053i −57.1121 + 116.384i −85.1248 85.1248i −364.109 + 210.219i 98.3171 + 54.0970i
3.12 3.92292 + 1.05114i −1.12570 4.20116i −13.4284 7.75291i 47.6558 29.2220i 17.6641i 109.646 69.1724i −136.426 136.426i 194.062 112.042i 217.666 64.5422i
3.13 4.04554 + 1.08400i −2.28884 8.54208i −12.5215 7.22927i −38.9420 40.1063i 37.0384i −111.439 + 66.2439i −137.589 137.589i 142.716 82.3971i −114.066 204.465i
3.14 5.18499 + 1.38931i −6.95065 25.9402i −2.75889 1.59285i −15.4213 + 53.7325i 144.156i −8.69466 129.350i −133.553 133.553i −414.137 + 239.102i −154.611 + 257.177i
3.15 7.58665 + 2.03284i 3.15524 + 11.7755i 25.7121 + 14.8449i −36.2352 + 42.5677i 95.7510i 105.610 + 75.1901i −12.8308 12.8308i 81.7366 47.1907i −361.437 + 249.286i
3.16 8.99064 + 2.40904i 6.51729 + 24.3229i 47.3154 + 27.3176i −2.18478 55.8590i 234.379i −13.7803 128.907i 148.975 + 148.975i −338.682 + 195.538i 114.924 507.471i
3.17 9.05026 + 2.42501i −0.494625 1.84597i 48.3138 + 27.8940i 47.5992 + 29.3141i 17.9059i −129.513 + 5.78206i 157.601 + 157.601i 207.281 119.674i 359.698 + 380.729i
3.18 10.1356 + 2.71583i −5.75210 21.4671i 67.6421 + 39.0532i −25.1607 49.9193i 233.204i 112.890 + 63.7413i 342.099 + 342.099i −217.307 + 125.463i −119.446 574.295i
12.1 −9.95978 + 2.66872i −4.43456 + 16.5500i 64.3624 37.1597i 55.6892 4.86925i 176.669i 127.528 + 23.3159i −308.553 + 308.553i −43.7930 25.2839i −541.658 + 197.115i
12.2 −8.82949 + 2.36586i −1.01039 + 3.77082i 44.6498 25.7786i −51.5659 21.5861i 35.6848i −68.0178 110.366i −126.411 + 126.411i 197.246 + 113.880i 506.370 + 68.5968i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.6.k.a 72
5.c odd 4 1 inner 35.6.k.a 72
7.d odd 6 1 inner 35.6.k.a 72
35.k even 12 1 inner 35.6.k.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.k.a 72 1.a even 1 1 trivial
35.6.k.a 72 5.c odd 4 1 inner
35.6.k.a 72 7.d odd 6 1 inner
35.6.k.a 72 35.k even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(35, [\chi])\).