Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [40,3,Mod(13,40)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40.13");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 40 = 2^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 40.i (of order \(4\), 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: | \(1.08992105744\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{20} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2 \nu^{19} + 19 \nu^{17} - 7 \nu^{15} - 93 \nu^{13} - 35 \nu^{11} - 35 \nu^{9} + 132 \nu^{7} + \cdots + 1408 \nu ) / 9600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11 \nu^{19} + 5 \nu^{17} - 185 \nu^{15} + 177 \nu^{13} + 755 \nu^{11} - 160 \nu^{9} + \cdots - 13312 \nu ) / 9600 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 75 \nu^{19} - 16 \nu^{18} - 75 \nu^{17} + 152 \nu^{16} - 225 \nu^{15} - 56 \nu^{14} + 825 \nu^{13} + \cdots + 11264 ) / 38400 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 75 \nu^{19} + 16 \nu^{18} - 75 \nu^{17} - 152 \nu^{16} - 225 \nu^{15} + 56 \nu^{14} + 825 \nu^{13} + \cdots - 11264 ) / 38400 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17 \nu^{18} + 13 \nu^{16} + 71 \nu^{14} + 33 \nu^{12} + 115 \nu^{10} - 140 \nu^{8} + 972 \nu^{6} + \cdots - 2048 ) / 4800 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 17 \nu^{18} + 13 \nu^{16} + 71 \nu^{14} + 33 \nu^{12} + 115 \nu^{10} - 140 \nu^{8} + 972 \nu^{6} + \cdots - 2048 ) / 4800 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 105 \nu^{19} - 122 \nu^{18} - 489 \nu^{17} + 250 \nu^{16} + 357 \nu^{15} + 590 \nu^{14} + \cdots + 54784 ) / 38400 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 5 \nu^{19} + 234 \nu^{18} - 215 \nu^{17} + 246 \nu^{16} - 805 \nu^{15} - 1278 \nu^{14} + \cdots - 118272 ) / 38400 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 5 \nu^{19} - 234 \nu^{18} - 215 \nu^{17} - 246 \nu^{16} - 805 \nu^{15} + 1278 \nu^{14} + \cdots + 118272 ) / 38400 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 105 \nu^{19} - 122 \nu^{18} + 489 \nu^{17} + 250 \nu^{16} - 357 \nu^{15} + 590 \nu^{14} + \cdots + 54784 ) / 38400 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 25 \nu^{19} + 82 \nu^{18} + 45 \nu^{17} - 50 \nu^{16} - 185 \nu^{15} + 10 \nu^{14} - 95 \nu^{13} + \cdots + 16896 ) / 12800 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 165 \nu^{19} + 166 \nu^{18} - 183 \nu^{17} - 950 \nu^{16} + 1179 \nu^{15} + 830 \nu^{14} + \cdots + 130048 ) / 38400 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( \nu^{18} - \nu^{16} - 3\nu^{14} + 11\nu^{12} + \nu^{10} - 40\nu^{8} + 4\nu^{6} + 176\nu^{4} - 192\nu^{2} - 256 ) / 128 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 5 \nu^{18} - 9 \nu^{16} + 37 \nu^{14} + 19 \nu^{12} - 55 \nu^{10} - 60 \nu^{8} + 100 \nu^{6} + \cdots + 3456 ) / 640 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 25 \nu^{19} + 82 \nu^{18} - 45 \nu^{17} - 50 \nu^{16} + 185 \nu^{15} + 10 \nu^{14} + 95 \nu^{13} + \cdots + 16896 ) / 12800 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 165 \nu^{19} + 166 \nu^{18} + 183 \nu^{17} - 950 \nu^{16} - 1179 \nu^{15} + 830 \nu^{14} + \cdots + 130048 ) / 38400 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 95 \nu^{19} + 553 \nu^{17} + 11 \nu^{15} - 843 \nu^{13} + 775 \nu^{11} + 5290 \nu^{9} + \cdots - 8192 \nu ) / 19200 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 17 \nu^{19} + 13 \nu^{17} + 71 \nu^{15} + 33 \nu^{13} + 115 \nu^{11} - 140 \nu^{9} + \cdots - 2048 \nu ) / 2400 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{19} + 2 \beta_{18} - \beta_{17} + \beta_{16} + \beta_{13} - \beta_{12} - \beta_{11} + \cdots + 2 \beta_{3} ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{17} + \beta_{16} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 2 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{18} - \beta_{16} + \beta_{12} - \beta_{11} + \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} + 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 2 \beta_{8} + \cdots - 8 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( \beta_{19} - 2 \beta_{17} - 2 \beta_{16} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \cdots + 20 \beta_{2} ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - \beta_{17} + 3 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - \beta_{13} + 3 \beta_{12} + 5 \beta_{11} + \cdots - 12 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 2 \beta_{19} - 4 \beta_{18} + \beta_{17} - 5 \beta_{16} - \beta_{13} + 5 \beta_{12} + \cdots - 50 \beta_{2} ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 4 \beta_{17} + 8 \beta_{16} - 7 \beta_{15} - 9 \beta_{14} + 4 \beta_{13} + 8 \beta_{12} + \beta_{11} + \cdots + 11 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 14 \beta_{19} - 4 \beta_{18} + 12 \beta_{17} + 6 \beta_{16} - 12 \beta_{13} - 6 \beta_{12} + \cdots - 28 \beta_{2} ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 8 \beta_{17} - 28 \beta_{16} + 16 \beta_{15} + 48 \beta_{14} - 8 \beta_{13} - 28 \beta_{12} + \cdots + 96 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 5 \beta_{19} - 2 \beta_{18} - 11 \beta_{17} + 15 \beta_{16} + 11 \beta_{13} - 15 \beta_{12} + \cdots - 184 \beta_{2} ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 31 \beta_{17} + 7 \beta_{16} + 64 \beta_{15} - 16 \beta_{14} - 31 \beta_{13} + 7 \beta_{12} + \cdots - 162 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 4 \beta_{19} + 51 \beta_{18} - 6 \beta_{17} + 51 \beta_{16} + 6 \beta_{13} - 51 \beta_{12} + \cdots - 85 \beta_{2} ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 122 \beta_{17} - 36 \beta_{16} + 108 \beta_{15} + 200 \beta_{14} - 122 \beta_{13} - 36 \beta_{12} + \cdots + 208 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 87 \beta_{19} + 40 \beta_{18} + 46 \beta_{17} - 34 \beta_{16} - 46 \beta_{13} + 34 \beta_{12} + \cdots + 460 \beta_{2} ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 7 \beta_{17} + 205 \beta_{16} + 74 \beta_{15} + 74 \beta_{14} - 7 \beta_{13} + 205 \beta_{12} + \cdots - 1132 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 234 \beta_{19} + 244 \beta_{18} - 25 \beta_{17} + 453 \beta_{16} + 25 \beta_{13} + \cdots + 978 \beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−1.96423 | − | 0.376547i | 0.977390 | − | 0.977390i | 3.71643 | + | 1.47925i | −0.801246 | − | 4.93538i | −2.28785 | + | 1.55179i | 8.39950 | − | 8.39950i | −6.74292 | − | 4.30500i | 7.08942i | −0.284568 | + | 9.99595i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.2 | −1.60823 | − | 1.18894i | −2.52630 | + | 2.52630i | 1.17282 | + | 3.82420i | 3.09141 | + | 3.92978i | 7.06650 | − | 1.05925i | −5.20520 | + | 5.20520i | 2.66059 | − | 7.54462i | − | 3.76437i | −0.299408 | − | 9.99552i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | −1.46837 | + | 1.35790i | −2.57493 | + | 2.57493i | 0.312234 | − | 3.98780i | −4.90427 | − | 0.973739i | 0.284467 | − | 7.27744i | −4.07624 | + | 4.07624i | 4.95654 | + | 6.27955i | − | 4.26050i | 8.52353 | − | 5.22967i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | −1.35790 | + | 1.46837i | 2.57493 | − | 2.57493i | −0.312234 | − | 3.98780i | 4.90427 | + | 0.973739i | 0.284467 | + | 7.27744i | −4.07624 | + | 4.07624i | 6.27955 | + | 4.95654i | − | 4.26050i | −8.08930 | + | 5.87905i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.5 | −0.665418 | − | 1.88606i | 3.60765 | − | 3.60765i | −3.11444 | + | 2.51004i | −2.34539 | + | 4.41578i | −9.20485 | − | 4.40365i | 1.47907 | − | 1.47907i | 6.80648 | + | 4.20379i | − | 17.0303i | 9.88909 | + | 1.48520i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.6 | 0.376547 | + | 1.96423i | −0.977390 | + | 0.977390i | −3.71643 | + | 1.47925i | 0.801246 | + | 4.93538i | −2.28785 | − | 1.55179i | 8.39950 | − | 8.39950i | −4.30500 | − | 6.74292i | 7.08942i | −9.39254 | + | 3.43224i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.7 | 0.764474 | − | 1.84813i | −0.130791 | + | 0.130791i | −2.83116 | − | 2.82569i | 4.38731 | − | 2.39823i | 0.141733 | + | 0.341706i | −1.59713 | + | 1.59713i | −7.38659 | + | 3.07218i | 8.96579i | −1.07825 | − | 9.94170i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.8 | 1.18894 | + | 1.60823i | 2.52630 | − | 2.52630i | −1.17282 | + | 3.82420i | −3.09141 | − | 3.92978i | 7.06650 | + | 1.05925i | −5.20520 | + | 5.20520i | −7.54462 | + | 2.66059i | − | 3.76437i | 2.64449 | − | 9.64400i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.9 | 1.84813 | − | 0.764474i | 0.130791 | − | 0.130791i | 2.83116 | − | 2.82569i | −4.38731 | + | 2.39823i | 0.141733 | − | 0.341706i | −1.59713 | + | 1.59713i | 3.07218 | − | 7.38659i | 8.96579i | −6.27494 | + | 7.78621i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.10 | 1.88606 | + | 0.665418i | −3.60765 | + | 3.60765i | 3.11444 | + | 2.51004i | 2.34539 | − | 4.41578i | −9.20485 | + | 4.40365i | 1.47907 | − | 1.47907i | 4.20379 | + | 6.80648i | − | 17.0303i | 7.36189 | − | 6.76776i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.1 | −1.96423 | + | 0.376547i | 0.977390 | + | 0.977390i | 3.71643 | − | 1.47925i | −0.801246 | + | 4.93538i | −2.28785 | − | 1.55179i | 8.39950 | + | 8.39950i | −6.74292 | + | 4.30500i | − | 7.08942i | −0.284568 | − | 9.99595i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.2 | −1.60823 | + | 1.18894i | −2.52630 | − | 2.52630i | 1.17282 | − | 3.82420i | 3.09141 | − | 3.92978i | 7.06650 | + | 1.05925i | −5.20520 | − | 5.20520i | 2.66059 | + | 7.54462i | 3.76437i | −0.299408 | + | 9.99552i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.3 | −1.46837 | − | 1.35790i | −2.57493 | − | 2.57493i | 0.312234 | + | 3.98780i | −4.90427 | + | 0.973739i | 0.284467 | + | 7.27744i | −4.07624 | − | 4.07624i | 4.95654 | − | 6.27955i | 4.26050i | 8.52353 | + | 5.22967i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.4 | −1.35790 | − | 1.46837i | 2.57493 | + | 2.57493i | −0.312234 | + | 3.98780i | 4.90427 | − | 0.973739i | 0.284467 | − | 7.27744i | −4.07624 | − | 4.07624i | 6.27955 | − | 4.95654i | 4.26050i | −8.08930 | − | 5.87905i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.5 | −0.665418 | + | 1.88606i | 3.60765 | + | 3.60765i | −3.11444 | − | 2.51004i | −2.34539 | − | 4.41578i | −9.20485 | + | 4.40365i | 1.47907 | + | 1.47907i | 6.80648 | − | 4.20379i | 17.0303i | 9.88909 | − | 1.48520i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.6 | 0.376547 | − | 1.96423i | −0.977390 | − | 0.977390i | −3.71643 | − | 1.47925i | 0.801246 | − | 4.93538i | −2.28785 | + | 1.55179i | 8.39950 | + | 8.39950i | −4.30500 | + | 6.74292i | − | 7.08942i | −9.39254 | − | 3.43224i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.7 | 0.764474 | + | 1.84813i | −0.130791 | − | 0.130791i | −2.83116 | + | 2.82569i | 4.38731 | + | 2.39823i | 0.141733 | − | 0.341706i | −1.59713 | − | 1.59713i | −7.38659 | − | 3.07218i | − | 8.96579i | −1.07825 | + | 9.94170i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.8 | 1.18894 | − | 1.60823i | 2.52630 | + | 2.52630i | −1.17282 | − | 3.82420i | −3.09141 | + | 3.92978i | 7.06650 | − | 1.05925i | −5.20520 | − | 5.20520i | −7.54462 | − | 2.66059i | 3.76437i | 2.64449 | + | 9.64400i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.9 | 1.84813 | + | 0.764474i | 0.130791 | + | 0.130791i | 2.83116 | + | 2.82569i | −4.38731 | − | 2.39823i | 0.141733 | + | 0.341706i | −1.59713 | − | 1.59713i | 3.07218 | + | 7.38659i | − | 8.96579i | −6.27494 | − | 7.78621i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.10 | 1.88606 | − | 0.665418i | −3.60765 | − | 3.60765i | 3.11444 | − | 2.51004i | 2.34539 | + | 4.41578i | −9.20485 | − | 4.40365i | 1.47907 | + | 1.47907i | 4.20379 | − | 6.80648i | 17.0303i | 7.36189 | + | 6.76776i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 40.i | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 40.3.i.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | 360.3.u.b | 20 | ||
| 4.b | odd | 2 | 1 | 160.3.m.a | 20 | ||
| 5.b | even | 2 | 1 | 200.3.i.b | 20 | ||
| 5.c | odd | 4 | 1 | inner | 40.3.i.a | ✓ | 20 |
| 5.c | odd | 4 | 1 | 200.3.i.b | 20 | ||
| 8.b | even | 2 | 1 | inner | 40.3.i.a | ✓ | 20 |
| 8.d | odd | 2 | 1 | 160.3.m.a | 20 | ||
| 15.e | even | 4 | 1 | 360.3.u.b | 20 | ||
| 20.d | odd | 2 | 1 | 800.3.m.b | 20 | ||
| 20.e | even | 4 | 1 | 160.3.m.a | 20 | ||
| 20.e | even | 4 | 1 | 800.3.m.b | 20 | ||
| 24.h | odd | 2 | 1 | 360.3.u.b | 20 | ||
| 40.e | odd | 2 | 1 | 800.3.m.b | 20 | ||
| 40.f | even | 2 | 1 | 200.3.i.b | 20 | ||
| 40.i | odd | 4 | 1 | inner | 40.3.i.a | ✓ | 20 |
| 40.i | odd | 4 | 1 | 200.3.i.b | 20 | ||
| 40.k | even | 4 | 1 | 160.3.m.a | 20 | ||
| 40.k | even | 4 | 1 | 800.3.m.b | 20 | ||
| 120.w | even | 4 | 1 | 360.3.u.b | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.3.i.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 40.3.i.a | ✓ | 20 | 5.c | odd | 4 | 1 | inner |
| 40.3.i.a | ✓ | 20 | 8.b | even | 2 | 1 | inner |
| 40.3.i.a | ✓ | 20 | 40.i | odd | 4 | 1 | inner |
| 160.3.m.a | 20 | 4.b | odd | 2 | 1 | ||
| 160.3.m.a | 20 | 8.d | odd | 2 | 1 | ||
| 160.3.m.a | 20 | 20.e | even | 4 | 1 | ||
| 160.3.m.a | 20 | 40.k | even | 4 | 1 | ||
| 200.3.i.b | 20 | 5.b | even | 2 | 1 | ||
| 200.3.i.b | 20 | 5.c | odd | 4 | 1 | ||
| 200.3.i.b | 20 | 40.f | even | 2 | 1 | ||
| 200.3.i.b | 20 | 40.i | odd | 4 | 1 | ||
| 360.3.u.b | 20 | 3.b | odd | 2 | 1 | ||
| 360.3.u.b | 20 | 15.e | even | 4 | 1 | ||
| 360.3.u.b | 20 | 24.h | odd | 2 | 1 | ||
| 360.3.u.b | 20 | 120.w | even | 4 | 1 | ||
| 800.3.m.b | 20 | 20.d | odd | 2 | 1 | ||
| 800.3.m.b | 20 | 20.e | even | 4 | 1 | ||
| 800.3.m.b | 20 | 40.e | odd | 2 | 1 | ||
| 800.3.m.b | 20 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(40, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 2 T^{19} + \cdots + 1048576 \)
|
| $3$ |
\( T^{20} + 1020 T^{16} + \cdots + 82944 \)
|
| $5$ |
\( T^{20} + \cdots + 95367431640625 \)
|
| $7$ |
\( (T^{10} + 2 T^{9} + \cdots + 5671712)^{2} \)
|
| $11$ |
\( (T^{10} + 552 T^{8} + \cdots + 473497600)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 23\!\cdots\!00 \)
|
| $17$ |
\( (T^{10} + 6 T^{9} + \cdots + 2344207392)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots - 384717505536)^{2} \)
|
| $23$ |
\( (T^{10} + 2 T^{9} + \cdots + 415411488)^{2} \)
|
| $29$ |
\( (T^{10} - 2548 T^{8} + \cdots - 31360000)^{2} \)
|
| $31$ |
\( (T^{5} + 34 T^{4} + \cdots - 1252704)^{4} \)
|
| $37$ |
\( T^{20} + \cdots + 59\!\cdots\!44 \)
|
| $41$ |
\( (T^{5} + 2 T^{4} + \cdots + 74680800)^{4} \)
|
| $43$ |
\( T^{20} + \cdots + 81\!\cdots\!64 \)
|
| $47$ |
\( (T^{10} + \cdots + 418650446283552)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 30\!\cdots\!84 \)
|
| $59$ |
\( (T^{10} + \cdots - 26\!\cdots\!16)^{2} \)
|
| $61$ |
\( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 23\!\cdots\!44 \)
|
| $71$ |
\( (T^{5} - 62 T^{4} + \cdots + 423110304)^{4} \)
|
| $73$ |
\( (T^{10} + \cdots + 37\!\cdots\!88)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 66\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 23\!\cdots\!64 \)
|
| $89$ |
\( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 90\!\cdots\!32)^{2} \)
|
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