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Properties

Label 28.6.f.a
Level $28$
Weight $6$
Character orbit 28.f
Analytic conductor $4.491$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,6,Mod(3,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.3");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 28.f (of order \(6\), degree \(2\), 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: \(4.49074695476\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{4} - 6 q^{5} - 72 q^{8} - 1136 q^{9} - 804 q^{10} + 1632 q^{12} + 1024 q^{14} + 664 q^{16} - 6 q^{17} - 1780 q^{18} - 4586 q^{21} + 13768 q^{22} + 564 q^{24} + 7612 q^{25} - 8892 q^{26} - 5928 q^{28}+ \cdots + 418676 q^{98}+O(q^{100}) \) 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 −5.54682 − 1.11033i −11.0700 + 19.1738i 29.5343 + 12.3176i −2.22247 + 1.28315i 82.6927 − 94.0624i −89.5633 − 93.7305i −150.145 − 101.116i −123.591 − 214.066i 13.7524 − 4.64969i
3.2 −5.50906 + 1.28461i 13.0818 − 22.6584i 28.6996 − 14.1540i −60.0746 + 34.6841i −42.9616 + 141.632i 33.9161 − 125.127i −139.926 + 114.843i −220.769 − 382.383i 286.399 − 268.249i
3.3 −5.20690 + 2.21092i −0.490819 + 0.850123i 22.2237 − 23.0241i 22.7704 − 13.1465i 0.676095 − 5.51167i −16.6033 + 128.574i −64.8124 + 169.019i 121.018 + 209.610i −89.4976 + 118.796i
3.4 −4.49521 − 3.43411i 0.901244 − 1.56100i 8.41378 + 30.8741i −41.1834 + 23.7773i −9.41192 + 3.92205i 122.887 + 41.3025i 68.2033 − 167.679i 119.876 + 207.630i 266.782 + 34.5447i
3.5 −4.33025 − 3.63991i 10.6204 − 18.3950i 5.50212 + 31.5234i 88.2183 − 50.9329i −112.945 + 40.9978i −128.565 + 16.6726i 90.9169 − 156.532i −104.084 − 180.280i −567.398 − 100.555i
3.6 −2.83899 + 4.89286i −11.9407 + 20.6819i −15.8802 − 27.7816i −26.3592 + 15.2185i −67.2942 − 117.140i 129.440 − 7.22687i 181.016 + 1.17187i −163.661 − 283.469i 0.371546 − 172.177i
3.7 −2.40397 + 5.12064i 4.28235 − 7.41724i −20.4419 − 24.6197i 47.5979 − 27.4807i 27.6864 + 39.7592i −6.17988 − 129.494i 175.210 − 45.4904i 84.8230 + 146.918i 26.2947 + 309.794i
3.8 −0.987129 − 5.57006i −10.6204 + 18.3950i −30.0512 + 10.9967i 88.2183 − 50.9329i 112.945 + 40.9978i 128.565 − 16.6726i 90.9169 + 156.532i −104.084 − 180.280i −370.782 − 441.104i
3.9 −0.726423 − 5.61002i −0.901244 + 1.56100i −30.9446 + 8.15049i −41.1834 + 23.7773i 9.41192 + 3.92205i −122.887 − 41.3025i 68.2033 + 167.679i 119.876 + 207.630i 163.307 + 213.767i
3.10 0.0458201 + 5.65667i 5.07408 − 8.78857i −31.9958 + 0.518378i −73.9432 + 42.6911i 49.9465 + 28.2997i −80.3926 + 101.706i −4.39834 − 180.966i 70.0074 + 121.256i −244.878 − 416.316i
3.11 1.81183 − 5.35885i 11.0700 − 19.1738i −25.4345 − 19.4187i −2.22247 + 1.28315i −82.6927 − 94.0624i 89.5633 + 93.7305i −150.145 + 101.116i −123.591 − 214.066i 2.84943 + 14.2347i
3.12 2.36282 + 5.13976i −9.72416 + 16.8427i −20.8342 + 24.2886i 43.6962 − 25.2280i −109.544 − 10.1835i −129.558 − 4.64978i −174.065 − 49.6932i −67.6187 − 117.119i 232.912 + 164.979i
3.13 3.26975 + 4.61614i 9.72416 − 16.8427i −10.6174 + 30.1872i 43.6962 − 25.2280i 109.544 − 10.1835i 129.558 + 4.64978i −174.065 + 49.6932i −67.6187 − 117.119i 259.332 + 119.218i
3.14 3.86703 − 4.12869i −13.0818 + 22.6584i −2.09210 − 31.9315i −60.0746 + 34.6841i 42.9616 + 141.632i −33.9161 + 125.127i −139.926 − 114.843i −220.769 − 382.383i −89.1108 + 382.154i
3.15 4.51816 − 3.40385i 0.490819 − 0.850123i 8.82757 − 30.7583i 22.7704 − 13.1465i −0.676095 − 5.51167i 16.6033 − 128.574i −64.8124 − 169.019i 121.018 + 209.610i 58.1317 − 136.905i
3.16 4.87591 + 2.86802i −5.07408 + 8.78857i 15.5490 + 27.9684i −73.9432 + 42.6911i −49.9465 + 28.2997i 80.3926 − 101.706i −4.39834 + 180.966i 70.0074 + 121.256i −482.979 − 3.91222i
3.17 5.63659 + 0.478421i −4.28235 + 7.41724i 31.5422 + 5.39333i 47.5979 − 27.4807i −27.6864 + 39.7592i 6.17988 + 129.494i 175.210 + 45.4904i 84.8230 + 146.918i 281.437 − 132.125i
3.18 5.65684 − 0.0122071i 11.9407 − 20.6819i 31.9997 − 0.138107i −26.3592 + 15.2185i 67.2942 − 117.140i −129.440 + 7.22687i 181.016 − 1.17187i −163.661 − 283.469i −148.924 + 86.4103i
19.1 −5.54682 + 1.11033i −11.0700 − 19.1738i 29.5343 − 12.3176i −2.22247 − 1.28315i 82.6927 + 94.0624i −89.5633 + 93.7305i −150.145 + 101.116i −123.591 + 214.066i 13.7524 + 4.64969i
19.2 −5.50906 − 1.28461i 13.0818 + 22.6584i 28.6996 + 14.1540i −60.0746 − 34.6841i −42.9616 − 141.632i 33.9161 + 125.127i −139.926 − 114.843i −220.769 + 382.383i 286.399 + 268.249i
See all 36 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
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.6.f.a ✓ 36
4.b odd 2 1 inner 28.6.f.a ✓ 36
7.c even 3 1 196.6.d.b 36
7.d odd 6 1 inner 28.6.f.a ✓ 36
7.d odd 6 1 196.6.d.b 36
28.f even 6 1 inner 28.6.f.a ✓ 36
28.f even 6 1 196.6.d.b 36
28.g odd 6 1 196.6.d.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.f.a ✓ 36 1.a even 1 1 trivial
28.6.f.a ✓ 36 4.b odd 2 1 inner
28.6.f.a ✓ 36 7.d odd 6 1 inner
28.6.f.a ✓ 36 28.f even 6 1 inner
196.6.d.b 36 7.c even 3 1
196.6.d.b 36 7.d odd 6 1
196.6.d.b 36 28.f even 6 1
196.6.d.b 36 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(28, [\chi])\).