Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1997,4,Mod(1,1997)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1997, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1997.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1997 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1997.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: | \(117.826814281\) |
Analytic rank: | \(1\) |
Dimension: | \(239\) |
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.
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 | −5.59440 | −8.36152 | 23.2974 | −4.51522 | 46.7777 | −9.81959 | −85.5797 | 42.9151 | 25.2599 | ||||||||||||||||||
1.2 | −5.52780 | 3.43977 | 22.5565 | −10.5187 | −19.0144 | −34.6164 | −80.4656 | −15.1679 | 58.1450 | ||||||||||||||||||
1.3 | −5.51428 | 7.20768 | 22.4073 | 9.62817 | −39.7451 | −3.48165 | −79.4456 | 24.9506 | −53.0924 | ||||||||||||||||||
1.4 | −5.50614 | −8.56073 | 22.3176 | 13.6270 | 47.1366 | 0.910634 | −78.8349 | 46.2860 | −75.0320 | ||||||||||||||||||
1.5 | −5.50179 | 8.24154 | 22.2697 | −8.43995 | −45.3433 | −6.56247 | −78.5092 | 40.9229 | 46.4349 | ||||||||||||||||||
1.6 | −5.43887 | 0.991773 | 21.5813 | −3.96164 | −5.39413 | −25.9935 | −73.8672 | −26.0164 | 21.5469 | ||||||||||||||||||
1.7 | −5.32569 | −8.11542 | 20.3630 | −6.55697 | 43.2203 | 14.6874 | −65.8417 | 38.8601 | 34.9204 | ||||||||||||||||||
1.8 | −5.28431 | 4.93207 | 19.9239 | 14.3183 | −26.0626 | −11.8358 | −63.0097 | −2.67464 | −75.6624 | ||||||||||||||||||
1.9 | −5.28041 | −3.44790 | 19.8827 | 16.8405 | 18.2063 | 27.1027 | −62.7455 | −15.1120 | −88.9245 | ||||||||||||||||||
1.10 | −5.27417 | −3.32423 | 19.8169 | −16.6866 | 17.5325 | −23.2918 | −62.3242 | −15.9495 | 88.0082 | ||||||||||||||||||
1.11 | −5.23446 | 3.65257 | 19.3996 | −6.07203 | −19.1192 | 18.8571 | −59.6706 | −13.6587 | 31.7838 | ||||||||||||||||||
1.12 | −5.19350 | 0.362833 | 18.9725 | 7.86550 | −1.88438 | 18.4887 | −56.9857 | −26.8684 | −40.8495 | ||||||||||||||||||
1.13 | −5.12919 | −8.38676 | 18.3086 | −1.93936 | 43.0173 | −30.3660 | −52.8747 | 43.3378 | 9.94735 | ||||||||||||||||||
1.14 | −5.08140 | 3.08170 | 17.8206 | −3.39990 | −15.6594 | 5.43909 | −49.9023 | −17.5031 | 17.2762 | ||||||||||||||||||
1.15 | −5.05732 | −7.84414 | 17.5765 | 13.2678 | 39.6703 | 9.68917 | −48.4315 | 34.5305 | −67.0996 | ||||||||||||||||||
1.16 | −5.02812 | −2.19694 | 17.2820 | −12.8877 | 11.0465 | −2.11320 | −46.6709 | −22.1734 | 64.8011 | ||||||||||||||||||
1.17 | −4.91538 | −4.80472 | 16.1610 | 1.32248 | 23.6170 | 24.0734 | −40.1143 | −3.91469 | −6.50051 | ||||||||||||||||||
1.18 | −4.88824 | 6.13395 | 15.8948 | 2.37453 | −29.9842 | −2.66042 | −38.5919 | 10.6254 | −11.6073 | ||||||||||||||||||
1.19 | −4.85005 | −9.72090 | 15.5230 | −21.5228 | 47.1469 | −0.473007 | −36.4868 | 67.4960 | 104.387 | ||||||||||||||||||
1.20 | −4.81468 | −1.36926 | 15.1811 | −10.0416 | 6.59255 | −5.82889 | −34.5748 | −25.1251 | 48.3471 | ||||||||||||||||||
See next 80 embeddings (of 239 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(1997\) | \(-1\) |
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 | 1997.4.a.a | ✓ | 239 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1997.4.a.a | ✓ | 239 | 1.a | even | 1 | 1 | trivial |