Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,4,Mod(1,161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 161.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: | \(9.49930751092\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1057087 \nu^{11} - 736072 \nu^{10} - 81046848 \nu^{9} + 59112285 \nu^{8} + 2160058053 \nu^{7} + \cdots + 27581677952 ) / 466611584 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1117105 \nu^{11} - 226867 \nu^{10} - 85952171 \nu^{9} + 21399060 \nu^{8} + 2305747495 \nu^{7} + \cdots + 19319250176 ) / 233305792 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1519271 \nu^{11} - 498841 \nu^{10} - 117032797 \nu^{9} + 43441900 \nu^{8} + 3145054901 \nu^{7} + \cdots + 27799261696 ) / 233305792 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1605177 \nu^{11} - 326483 \nu^{10} - 123035627 \nu^{9} + 30155740 \nu^{8} + 3282153271 \nu^{7} + \cdots + 31578353216 ) / 233305792 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 99065 \nu^{11} - 21862 \nu^{10} - 7609238 \nu^{9} + 2030749 \nu^{8} + 203556131 \nu^{7} + \cdots + 1798578304 ) / 6964352 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7097167 \nu^{11} - 2425124 \nu^{10} - 546101432 \nu^{9} + 210879061 \nu^{8} + \cdots + 160329330944 ) / 466611584 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 5914582 \nu^{11} + 1189665 \nu^{10} + 454454473 \nu^{9} - 112071589 \nu^{8} + \cdots - 101406339072 ) / 233305792 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 7229129 \nu^{11} - 1769452 \nu^{10} - 555592220 \nu^{9} + 160203215 \nu^{8} + \cdots + 135239697088 ) / 233305792 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 22127067 \nu^{11} - 4919124 \nu^{10} - 1700971564 \nu^{9} + 457650357 \nu^{8} + \cdots + 427632581376 ) / 466611584 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 23\beta _1 - 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + 2\beta_{9} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} + 6\beta_{4} + \beta_{3} + 31\beta_{2} - 2\beta _1 + 315 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{11} - 6 \beta_{10} + 36 \beta_{9} + 40 \beta_{7} + 44 \beta_{6} + 44 \beta_{5} + 14 \beta_{4} + \cdots - 185 \)
|
| \(\nu^{6}\) | \(=\) |
\( 27 \beta_{11} - 10 \beta_{10} + 75 \beta_{9} + 16 \beta_{8} - 65 \beta_{7} + 83 \beta_{6} - 109 \beta_{5} + \cdots + 8130 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 58 \beta_{11} - 354 \beta_{10} + 1026 \beta_{9} + 48 \beta_{8} + 1242 \beta_{7} + 1570 \beta_{6} + \cdots - 6210 \)
|
| \(\nu^{8}\) | \(=\) |
\( 476 \beta_{11} - 748 \beta_{10} + 2172 \beta_{9} + 1184 \beta_{8} - 1532 \beta_{7} + 2692 \beta_{6} + \cdots + 224802 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2552 \beta_{11} - 15264 \beta_{10} + 26977 \beta_{9} + 3408 \beta_{8} + 36409 \beta_{7} + \cdots - 202781 \)
|
| \(\nu^{10}\) | \(=\) |
\( 3329 \beta_{11} - 36960 \beta_{10} + 56722 \beta_{9} + 57632 \beta_{8} - 29538 \beta_{7} + \cdots + 6459459 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 100129 \beta_{11} - 579662 \beta_{10} + 683620 \beta_{9} + 162544 \beta_{8} + 1056368 \beta_{7} + \cdots - 6540689 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.60949 | −7.98435 | 23.4664 | −5.27021 | 44.7882 | 7.00000 | −86.7589 | 36.7498 | 29.5632 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.89764 | 7.57994 | 15.9869 | −14.0191 | −37.1238 | 7.00000 | −39.1171 | 30.4555 | 68.6607 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.54240 | −2.15594 | 4.54860 | 6.95559 | 7.63719 | 7.00000 | 12.2262 | −22.3519 | −24.6395 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.06154 | 5.71887 | 1.37302 | 21.1978 | −17.5086 | 7.00000 | 20.2888 | 5.70552 | −64.8978 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.718333 | −2.25536 | −7.48400 | −17.9779 | 1.62010 | 7.00000 | 11.1227 | −21.9133 | 12.9141 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.285899 | 8.01264 | −7.91826 | 1.18183 | 2.29081 | 7.00000 | −4.55102 | 37.2024 | 0.337884 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.40034 | −4.22464 | −6.03905 | 13.7756 | −5.91594 | 7.00000 | −19.6594 | −9.15238 | 19.2904 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.11879 | −9.53944 | −3.51074 | −13.8523 | −20.2120 | 7.00000 | −24.3888 | 64.0009 | −29.3501 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 3.76511 | 9.31158 | 6.17602 | 1.25334 | 35.0591 | 7.00000 | −6.86747 | 59.7056 | 4.71897 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 3.76781 | 2.58468 | 6.19640 | 21.4417 | 9.73858 | 7.00000 | −6.79562 | −20.3194 | 80.7883 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4.95487 | 3.24611 | 16.5508 | −2.30377 | 16.0841 | 7.00000 | 42.3679 | −16.4628 | −11.4149 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 5.53660 | −9.29409 | 22.6539 | 3.61751 | −51.4577 | 7.00000 | 81.1328 | 59.3802 | 20.0287 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 161.4.a.d | ✓ | 12 |
| 3.b | odd | 2 | 1 | 1449.4.a.o | 12 | ||
| 7.b | odd | 2 | 1 | 1127.4.a.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.4.a.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 1127.4.a.h | 12 | 7.b | odd | 2 | 1 | ||
| 1449.4.a.o | 12 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 4 T_{2}^{11} - 76 T_{2}^{10} + 311 T_{2}^{9} + 1979 T_{2}^{8} - 8466 T_{2}^{7} + \cdots - 70656 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(161))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 4 T^{11} + \cdots - 70656 \)
|
| $3$ |
\( T^{12} + \cdots + 394600512 \)
|
| $5$ |
\( T^{12} + \cdots - 9891773184 \)
|
| $7$ |
\( (T - 7)^{12} \)
|
| $11$ |
\( T^{12} + \cdots - 23\!\cdots\!04 \)
|
| $13$ |
\( T^{12} + \cdots + 82\!\cdots\!04 \)
|
| $17$ |
\( T^{12} + \cdots - 14\!\cdots\!72 \)
|
| $19$ |
\( T^{12} + \cdots + 63\!\cdots\!52 \)
|
| $23$ |
\( (T - 23)^{12} \)
|
| $29$ |
\( T^{12} + \cdots + 52\!\cdots\!36 \)
|
| $31$ |
\( T^{12} + \cdots - 33\!\cdots\!64 \)
|
| $37$ |
\( T^{12} + \cdots - 28\!\cdots\!52 \)
|
| $41$ |
\( T^{12} + \cdots + 32\!\cdots\!52 \)
|
| $43$ |
\( T^{12} + \cdots + 75\!\cdots\!08 \)
|
| $47$ |
\( T^{12} + \cdots + 36\!\cdots\!48 \)
|
| $53$ |
\( T^{12} + \cdots + 39\!\cdots\!48 \)
|
| $59$ |
\( T^{12} + \cdots - 41\!\cdots\!32 \)
|
| $61$ |
\( T^{12} + \cdots - 60\!\cdots\!16 \)
|
| $67$ |
\( T^{12} + \cdots - 10\!\cdots\!64 \)
|
| $71$ |
\( T^{12} + \cdots + 32\!\cdots\!88 \)
|
| $73$ |
\( T^{12} + \cdots - 13\!\cdots\!24 \)
|
| $79$ |
\( T^{12} + \cdots - 26\!\cdots\!72 \)
|
| $83$ |
\( T^{12} + \cdots + 76\!\cdots\!04 \)
|
| $89$ |
\( T^{12} + \cdots - 81\!\cdots\!12 \)
|
| $97$ |
\( T^{12} + \cdots + 22\!\cdots\!52 \)
|
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