Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,3,Mod(33,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, 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("320.33");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.m (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: | \(8.71936845953\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 25x^{14} + 469x^{12} - 3562x^{10} + 20062x^{8} - 23914x^{6} + 20917x^{4} - 8281x^{2} + 2401 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{22} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 25x^{14} + 469x^{12} - 3562x^{10} + 20062x^{8} - 23914x^{6} + 20917x^{4} - 8281x^{2} + 2401 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 10367509 \nu^{14} - 255525612 \nu^{12} + 4774951307 \nu^{10} - 35309568009 \nu^{8} + \cdots - 51984257437 ) / 14532759570 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 570206 \nu^{14} - 13632515 \nu^{12} + 252162638 \nu^{10} - 1745839186 \nu^{8} + \cdots + 8126372781 ) / 593173860 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 10367509 \nu^{15} + 24277981 \nu^{14} + 255525612 \nu^{13} - 580582821 \nu^{12} + \cdots + 74298425698 ) / 58131038280 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10367509 \nu^{15} + 24277981 \nu^{14} - 255525612 \nu^{13} - 580582821 \nu^{12} + \cdots + 74298425698 ) / 58131038280 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3803 \nu^{15} + 63274 \nu^{13} - 1006418 \nu^{11} - 922826 \nu^{9} + 28627326 \nu^{7} + \cdots - 201917044 \nu ) / 13932660 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 128107772 \nu^{15} + 273797279 \nu^{14} + 2712669506 \nu^{13} - 6928984125 \nu^{12} + \cdots - 1099167409158 ) / 406917267960 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 128107772 \nu^{15} + 273797279 \nu^{14} - 2712669506 \nu^{13} - 6928984125 \nu^{12} + \cdots - 1099167409158 ) / 406917267960 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 87083321 \nu^{15} - 3809454607 \nu^{14} - 199644268 \nu^{13} + 95591472389 \nu^{12} + \cdots + 16600936855202 ) / 1220751803880 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 87083321 \nu^{15} - 3809454607 \nu^{14} + 199644268 \nu^{13} + 95591472389 \nu^{12} + \cdots + 16600936855202 ) / 1220751803880 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 684543659 \nu^{15} + 1537975649 \nu^{14} + 16744088992 \nu^{13} - 38287605328 \nu^{12} + \cdots - 7118545323379 ) / 1220751803880 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 684543659 \nu^{15} + 1537975649 \nu^{14} - 16744088992 \nu^{13} - 38287605328 \nu^{12} + \cdots - 7118545323379 ) / 1220751803880 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 2529293 \nu^{14} + 60496841 \nu^{12} - 1118531189 \nu^{10} + 7744111483 \nu^{8} + \cdots - 8751053100 ) / 296586930 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 240651 \nu^{15} - 5954976 \nu^{13} + 111279336 \nu^{11} - 827173328 \nu^{9} + \cdots - 534116856 \nu ) / 200353160 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 158150451 \nu^{15} - 3905523960 \nu^{13} + 72981673310 \nu^{11} - 541068294392 \nu^{9} + \cdots - 350296320010 \nu ) / 20345863398 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2710701513 \nu^{15} + 67070805600 \nu^{13} - 1253337496600 \nu^{11} + \cdots + 6015750158600 \nu ) / 203458633980 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{11} - 3\beta_{10} + 2\beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 7\beta _1 + 12 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} - \beta_{14} + 14\beta_{13} + 12\beta_{4} - 12\beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{12} - 39 \beta_{11} - 39 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 37 \beta_{7} + 37 \beta_{6} + \cdots - 144 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 26 \beta_{15} - 13 \beta_{14} + 324 \beta_{13} - 39 \beta_{11} + 39 \beta_{10} - 78 \beta_{9} + \cdots - 163 \beta_{3} ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -13\beta_{12} - 234\beta_{4} - 234\beta_{3} + 88\beta_{2} - 991 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 494 \beta_{15} + 176 \beta_{14} - 5916 \beta_{13} - 528 \beta_{11} + 528 \beta_{10} + \cdots + 2365 \beta_{3} ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 494 \beta_{12} + 7623 \beta_{11} + 7623 \beta_{10} - 1482 \beta_{9} - 1482 \beta_{8} - 10016 \beta_{7} + \cdots - 29016 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -8463\beta_{15} + 2541\beta_{14} - 99722\beta_{13} - 35676\beta_{4} + 35676\beta_{3} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 8463 \beta_{12} + 114651 \beta_{11} + 114651 \beta_{10} - 25389 \beta_{9} - 25389 \beta_{8} + \cdots + 439956 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 138856 \beta_{15} + 38217 \beta_{14} - 1624224 \beta_{13} + 114651 \beta_{11} + \cdots + 549919 \beta_{3} ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( 69428\beta_{12} + 1047264\beta_{4} + 1047264\beta_{3} - 294068\beta_{2} + 3400153 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 2233384 \beta_{15} - 588136 \beta_{14} + 26032896 \beta_{13} + 1764408 \beta_{11} + \cdots - 8574217 \beta_{3} ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 2233384 \beta_{12} - 27487059 \beta_{11} - 27487059 \beta_{10} + 6700152 \beta_{9} + 6700152 \beta_{8} + \cdots + 106181100 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 35579105\beta_{15} - 9162353\beta_{14} + 414006590\beta_{13} + 134481420\beta_{4} - 134481420\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(257\) | \(261\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{13}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | −2.81877 | + | 2.81877i | 0 | −0.666369 | + | 4.95540i | 0 | −2.47156 | + | 2.47156i | 0 | − | 6.89092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | −2.81877 | + | 2.81877i | 0 | 0.666369 | − | 4.95540i | 0 | 2.47156 | − | 2.47156i | 0 | − | 6.89092i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | −0.871242 | + | 0.871242i | 0 | −4.78740 | + | 1.44248i | 0 | −6.00687 | + | 6.00687i | 0 | 7.48188i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | −0.871242 | + | 0.871242i | 0 | 4.78740 | − | 1.44248i | 0 | 6.00687 | − | 6.00687i | 0 | 7.48188i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 0.717526 | − | 0.717526i | 0 | −3.64925 | − | 3.41803i | 0 | −3.20606 | + | 3.20606i | 0 | 7.97031i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.6 | 0 | 0.717526 | − | 0.717526i | 0 | 3.64925 | + | 3.41803i | 0 | 3.20606 | − | 3.20606i | 0 | 7.97031i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.7 | 0 | 3.97248 | − | 3.97248i | 0 | −4.72437 | − | 1.63717i | 0 | 5.52540 | − | 5.52540i | 0 | − | 22.5613i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.8 | 0 | 3.97248 | − | 3.97248i | 0 | 4.72437 | + | 1.63717i | 0 | −5.52540 | + | 5.52540i | 0 | − | 22.5613i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −2.81877 | − | 2.81877i | 0 | −0.666369 | − | 4.95540i | 0 | −2.47156 | − | 2.47156i | 0 | 6.89092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −2.81877 | − | 2.81877i | 0 | 0.666369 | + | 4.95540i | 0 | 2.47156 | + | 2.47156i | 0 | 6.89092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | −0.871242 | − | 0.871242i | 0 | −4.78740 | − | 1.44248i | 0 | −6.00687 | − | 6.00687i | 0 | − | 7.48188i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | −0.871242 | − | 0.871242i | 0 | 4.78740 | + | 1.44248i | 0 | 6.00687 | + | 6.00687i | 0 | − | 7.48188i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | 0.717526 | + | 0.717526i | 0 | −3.64925 | + | 3.41803i | 0 | −3.20606 | − | 3.20606i | 0 | − | 7.97031i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0 | 0.717526 | + | 0.717526i | 0 | 3.64925 | − | 3.41803i | 0 | 3.20606 | + | 3.20606i | 0 | − | 7.97031i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 0 | 3.97248 | + | 3.97248i | 0 | −4.72437 | + | 1.63717i | 0 | 5.52540 | + | 5.52540i | 0 | 22.5613i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.8 | 0 | 3.97248 | + | 3.97248i | 0 | 4.72437 | − | 1.63717i | 0 | −5.52540 | − | 5.52540i | 0 | 22.5613i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
| 20.e | even | 4 | 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 | 320.3.m.d | yes | 16 |
| 4.b | odd | 2 | 1 | 320.3.m.c | ✓ | 16 | |
| 5.c | odd | 4 | 1 | 320.3.m.c | ✓ | 16 | |
| 8.b | even | 2 | 1 | 320.3.m.c | ✓ | 16 | |
| 8.d | odd | 2 | 1 | inner | 320.3.m.d | yes | 16 |
| 20.e | even | 4 | 1 | inner | 320.3.m.d | yes | 16 |
| 40.i | odd | 4 | 1 | inner | 320.3.m.d | yes | 16 |
| 40.k | even | 4 | 1 | 320.3.m.c | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 320.3.m.c | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 320.3.m.c | ✓ | 16 | 5.c | odd | 4 | 1 | |
| 320.3.m.c | ✓ | 16 | 8.b | even | 2 | 1 | |
| 320.3.m.c | ✓ | 16 | 40.k | even | 4 | 1 | |
| 320.3.m.d | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 320.3.m.d | yes | 16 | 8.d | odd | 2 | 1 | inner |
| 320.3.m.d | yes | 16 | 20.e | even | 4 | 1 | inner |
| 320.3.m.d | yes | 16 | 40.i | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} + 52T_{3}^{5} + 520T_{3}^{4} + 152T_{3}^{3} + 8T_{3}^{2} - 112T_{3} + 784 \)
acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( (T^{8} - 2 T^{7} + \cdots + 784)^{2} \)
|
| $5$ |
\( T^{16} + \cdots + 152587890625 \)
|
| $7$ |
\( T^{16} + \cdots + 1224793743616 \)
|
| $11$ |
\( (T^{8} + 732 T^{6} + \cdots + 7225344)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 66\!\cdots\!36 \)
|
| $17$ |
\( (T^{8} - 40 T^{7} + \cdots + 1699747984)^{2} \)
|
| $19$ |
\( (T^{4} + 4 T^{3} + \cdots - 16496)^{4} \)
|
| $23$ |
\( T^{16} + \cdots + 62\!\cdots\!00 \)
|
| $29$ |
\( (T^{8} - 5184 T^{6} + \cdots + 723860640000)^{2} \)
|
| $31$ |
\( (T^{8} - 3452 T^{6} + \cdots + 8620751104)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 36\!\cdots\!36 \)
|
| $41$ |
\( (T^{4} + 26 T^{3} + \cdots + 309904)^{4} \)
|
| $43$ |
\( (T^{8} + 70 T^{7} + \cdots + 323838388624)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 80\!\cdots\!76 \)
|
| $53$ |
\( T^{16} + \cdots + 13\!\cdots\!00 \)
|
| $59$ |
\( (T^{4} - 56 T^{3} + \cdots + 34864)^{4} \)
|
| $61$ |
\( (T^{8} + 7412 T^{6} + \cdots + 826615545856)^{2} \)
|
| $67$ |
\( (T^{8} + \cdots + 150031827567504)^{2} \)
|
| $71$ |
\( (T^{8} - 17804 T^{6} + \cdots + 410030192896)^{2} \)
|
| $73$ |
\( (T^{8} + \cdots + 20542540982544)^{2} \)
|
| $79$ |
\( (T^{8} + \cdots + 173089597947904)^{2} \)
|
| $83$ |
\( (T^{8} + \cdots + 1729488010000)^{2} \)
|
| $89$ |
\( (T^{8} + \cdots + 19002345652224)^{2} \)
|
| $97$ |
\( (T^{8} - 204 T^{7} + \cdots + 312020553744)^{2} \)
|
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