Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1775,4,Mod(1,1775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1775, 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("1775.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1775 = 5^{2} \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1775.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: | \(104.728390260\) |
| 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: |
\( x^{12} - 2 x^{11} - 73 x^{10} + 130 x^{9} + 1904 x^{8} - 2704 x^{7} - 22040 x^{6} + 19157 x^{5} + \cdots + 205032 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 71) |
| 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
\( x^{12} - 2 x^{11} - 73 x^{10} + 130 x^{9} + 1904 x^{8} - 2704 x^{7} - 22040 x^{6} + 19157 x^{5} + \cdots + 205032 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 17407672325 \nu^{11} + 77624347231 \nu^{10} + 1101209576126 \nu^{9} + \cdots - 20\!\cdots\!24 ) / 579110577808 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 806330173 \nu^{11} + 3363077090 \nu^{10} + 51716387199 \nu^{9} - 217365365024 \nu^{8} + \cdots - 79771802112452 ) / 20682520636 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 15139718809 \nu^{11} + 61862305511 \nu^{10} + 975523233702 \nu^{9} - 4001364656756 \nu^{8} + \cdots - 14\!\cdots\!96 ) / 289555288904 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 53003171975 \nu^{11} - 222244180633 \nu^{10} - 3394147947102 \nu^{9} + 14367020066320 \nu^{8} + \cdots + 53\!\cdots\!44 ) / 579110577808 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 66861232997 \nu^{11} + 272096970799 \nu^{10} + 4308602403222 \nu^{9} + \cdots - 62\!\cdots\!04 ) / 579110577808 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 24529681565 \nu^{11} + 100271633688 \nu^{10} + 1577761006823 \nu^{9} + \cdots - 22\!\cdots\!28 ) / 144777644452 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 24529681565 \nu^{11} + 100271633688 \nu^{10} + 1577761006823 \nu^{9} + \cdots - 22\!\cdots\!04 ) / 144777644452 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 43538377662 \nu^{11} - 178484976353 \nu^{10} - 2799016737243 \nu^{9} + 11523627577794 \nu^{8} + \cdots + 40\!\cdots\!28 ) / 144777644452 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 188803192411 \nu^{11} + 778917276337 \nu^{10} + 12128414598946 \nu^{9} + \cdots - 18\!\cdots\!92 ) / 579110577808 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 56035405383 \nu^{11} + 230901854039 \nu^{10} + 3599953749194 \nu^{9} + \cdots - 53\!\cdots\!80 ) / 144777644452 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{8} + \beta_{7} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{6} - \beta_{5} + 2\beta_{3} + 2\beta_{2} + 18\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{11} - 5 \beta_{10} - 4 \beta_{9} - 31 \beta_{8} + 30 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 253 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 64 \beta_{11} - 48 \beta_{10} - 34 \beta_{9} + 67 \beta_{8} + 8 \beta_{7} + 71 \beta_{6} - 43 \beta_{5} + \cdots + 28 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 46 \beta_{11} - 222 \beta_{10} - 164 \beta_{9} - 871 \beta_{8} + 837 \beta_{7} + 194 \beta_{6} + \cdots + 6237 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1688 \beta_{11} - 1705 \beta_{10} - 880 \beta_{9} + 1898 \beta_{8} + 483 \beta_{7} + 2147 \beta_{6} + \cdots - 536 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1356 \beta_{11} - 7581 \beta_{10} - 5179 \beta_{9} - 24421 \beta_{8} + 23232 \beta_{7} + \cdots + 161584 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 42040 \beta_{11} - 54503 \beta_{10} - 20257 \beta_{9} + 51808 \beta_{8} + 20197 \beta_{7} + \cdots - 37192 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 33666 \beta_{11} - 237138 \beta_{10} - 151416 \beta_{9} - 688923 \beta_{8} + 645434 \beta_{7} + \cdots + 4283793 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1022551 \beta_{11} - 1662978 \beta_{10} - 429021 \beta_{9} + 1400830 \beta_{8} + 730408 \beta_{7} + \cdots - 1396773 \)
|
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 |
|
−5.34148 | 4.05055 | 20.5314 | 0 | −21.6359 | −4.11311 | −66.9361 | −10.5931 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.81307 | −6.58757 | 15.1656 | 0 | 31.7064 | 11.7471 | −34.4888 | 16.3960 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.78171 | −1.81588 | 6.30134 | 0 | 6.86715 | −19.7961 | 6.42383 | −23.7026 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.41459 | −5.16965 | 3.65941 | 0 | 17.6522 | 20.5921 | 14.8213 | −0.274690 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.36356 | 7.92140 | −6.14070 | 0 | −10.8013 | −35.0778 | 19.2817 | 35.7486 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.33211 | −9.69577 | −6.22549 | 0 | 12.9158 | −29.1008 | 18.9499 | 67.0079 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.28932 | −6.74804 | −6.33766 | 0 | −8.70035 | 8.40758 | −18.4858 | 18.5360 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.12082 | 7.68554 | −3.50214 | 0 | 16.2996 | 33.4857 | −24.3939 | 32.0675 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.12896 | −1.14583 | −3.46754 | 0 | −2.43943 | −28.4733 | −24.4139 | −25.6871 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.20387 | 8.50728 | −3.14296 | 0 | 18.7489 | −17.8630 | −24.5576 | 45.3739 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4.95545 | −7.10589 | 16.5565 | 0 | −35.2129 | −0.895356 | 42.4014 | 23.4936 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 5.34810 | 3.10386 | 20.6022 | 0 | 16.5998 | 13.0868 | 67.3980 | −17.3661 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(71\) | \(1\) |
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 | 1775.4.a.c | 12 | |
| 5.b | even | 2 | 1 | 71.4.a.c | ✓ | 12 | |
| 15.d | odd | 2 | 1 | 639.4.a.h | 12 | ||
| 20.d | odd | 2 | 1 | 1136.4.a.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.4.a.c | ✓ | 12 | 5.b | even | 2 | 1 | |
| 639.4.a.h | 12 | 15.d | odd | 2 | 1 | ||
| 1136.4.a.k | 12 | 20.d | odd | 2 | 1 | ||
| 1775.4.a.c | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 2 T_{2}^{11} - 73 T_{2}^{10} - 130 T_{2}^{9} + 1904 T_{2}^{8} + 2704 T_{2}^{7} + \cdots + 205032 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1775))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 2 T^{11} + \cdots + 205032 \)
|
| $3$ |
\( T^{12} + \cdots - 214513944 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + \cdots - 33734215581696 \)
|
| $11$ |
\( T^{12} + \cdots - 22\!\cdots\!52 \)
|
| $13$ |
\( T^{12} + \cdots - 10\!\cdots\!04 \)
|
| $17$ |
\( T^{12} + \cdots - 30\!\cdots\!16 \)
|
| $19$ |
\( T^{12} + \cdots - 33\!\cdots\!24 \)
|
| $23$ |
\( T^{12} + \cdots - 59\!\cdots\!72 \)
|
| $29$ |
\( T^{12} + \cdots + 43\!\cdots\!76 \)
|
| $31$ |
\( T^{12} + \cdots + 26\!\cdots\!12 \)
|
| $37$ |
\( T^{12} + \cdots - 20\!\cdots\!12 \)
|
| $41$ |
\( T^{12} + \cdots + 14\!\cdots\!52 \)
|
| $43$ |
\( T^{12} + \cdots + 34\!\cdots\!24 \)
|
| $47$ |
\( T^{12} + \cdots - 32\!\cdots\!92 \)
|
| $53$ |
\( T^{12} + \cdots - 97\!\cdots\!84 \)
|
| $59$ |
\( T^{12} + \cdots - 95\!\cdots\!08 \)
|
| $61$ |
\( T^{12} + \cdots - 96\!\cdots\!36 \)
|
| $67$ |
\( T^{12} + \cdots + 43\!\cdots\!84 \)
|
| $71$ |
\( (T + 71)^{12} \)
|
| $73$ |
\( T^{12} + \cdots + 46\!\cdots\!02 \)
|
| $79$ |
\( T^{12} + \cdots + 81\!\cdots\!64 \)
|
| $83$ |
\( T^{12} + \cdots + 26\!\cdots\!56 \)
|
| $89$ |
\( T^{12} + \cdots - 73\!\cdots\!26 \)
|
| $97$ |
\( T^{12} + \cdots + 22\!\cdots\!48 \)
|
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