Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,2,Mod(17,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("348.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 348.l (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: | \(2.77879399034\) |
| 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: |
\( x^{20} - 2 x^{17} + 10 x^{16} - 24 x^{15} + 2 x^{14} - 44 x^{13} + 153 x^{12} - 102 x^{11} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| 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
\( x^{20} - 2 x^{17} + 10 x^{16} - 24 x^{15} + 2 x^{14} - 44 x^{13} + 153 x^{12} - 102 x^{11} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{18} + 2 \nu^{15} - 10 \nu^{14} + 24 \nu^{13} - 2 \nu^{12} + 44 \nu^{11} - 153 \nu^{10} + \cdots + 4374 \nu ) / 6561 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{19} - 2 \nu^{16} + 10 \nu^{15} - 24 \nu^{14} + 2 \nu^{13} - 44 \nu^{12} + 153 \nu^{11} + \cdots - 4374 \nu^{2} ) / 19683 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 6181 \nu^{19} + 69473 \nu^{18} - 158223 \nu^{17} + 269087 \nu^{16} - 483365 \nu^{15} + \cdots + 846021267 ) / 15274008 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 6467 \nu^{19} + 29517 \nu^{18} - 103401 \nu^{17} + 194345 \nu^{16} - 267901 \nu^{15} + \cdots + 954054693 ) / 15274008 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 10265 \nu^{19} + 5262 \nu^{18} + 24786 \nu^{17} - 54476 \nu^{16} + 31897 \nu^{15} + \cdots - 550336680 ) / 15274008 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 196 \nu^{19} + 486 \nu^{18} - 630 \nu^{17} + 1013 \nu^{16} - 2608 \nu^{15} + 7908 \nu^{14} + \cdots - 2499741 ) / 236196 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 18989 \nu^{19} - 6075 \nu^{18} - 44883 \nu^{17} + 121889 \nu^{16} - 79273 \nu^{15} + \cdots + 1062744219 ) / 22911012 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 39011 \nu^{19} + 6087 \nu^{18} + 94815 \nu^{17} - 192680 \nu^{16} + 128347 \nu^{15} + \cdots - 1889686098 ) / 45822024 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 16157 \nu^{19} - 6467 \nu^{18} - 29517 \nu^{17} + 71087 \nu^{16} - 32775 \nu^{15} + \cdots + 785345139 ) / 15274008 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 9839 \nu^{19} + 34467 \nu^{18} - 69093 \nu^{17} + 110857 \nu^{16} - 217463 \nu^{15} + \cdots + 127289961 ) / 5091336 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 94417 \nu^{19} + 62496 \nu^{18} - 480915 \nu^{17} + 978160 \nu^{16} - 1011179 \nu^{15} + \cdots + 6716193894 ) / 45822024 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 119699 \nu^{19} + 145413 \nu^{18} - 58203 \nu^{17} - 26255 \nu^{16} - 557207 \nu^{15} + \cdots - 3679323507 ) / 45822024 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 120749 \nu^{19} + 155913 \nu^{18} - 84051 \nu^{17} + 52741 \nu^{16} - 608633 \nu^{15} + \cdots - 3282671691 ) / 45822024 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 43032 \nu^{19} + 82769 \nu^{18} - 109308 \nu^{17} + 148713 \nu^{16} - 454108 \nu^{15} + \cdots - 721611585 ) / 15274008 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 130889 \nu^{19} - 342285 \nu^{18} + 571050 \nu^{17} - 862609 \nu^{16} + 2017355 \nu^{15} + \cdots + 593225937 ) / 45822024 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 156134 \nu^{19} - 343311 \nu^{18} + 510597 \nu^{17} - 711895 \nu^{16} + 1886216 \nu^{15} + \cdots + 2045358945 ) / 45822024 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 1771 \nu^{19} - 4116 \nu^{18} + 6507 \nu^{17} - 9725 \nu^{16} + 23431 \nu^{15} - 72378 \nu^{14} + \cdots + 17458821 ) / 472392 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{18} - \beta_{17} - \beta_{16} - \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{18} - 2 \beta_{17} - \beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{12} + \cdots - 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{19} - 2 \beta_{18} + 2 \beta_{17} + 3 \beta_{14} + 4 \beta_{13} - \beta_{12} - 8 \beta_{11} + \cdots + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{19} - 6 \beta_{18} + 6 \beta_{17} - \beta_{15} + 4 \beta_{14} + \beta_{9} + 6 \beta_{6} + \cdots + 5 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 6 \beta_{19} + \beta_{18} + 2 \beta_{17} + 2 \beta_{16} + 2 \beta_{15} - 6 \beta_{12} + 4 \beta_{11} + \cdots + 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 12 \beta_{17} - 15 \beta_{16} - \beta_{15} + 28 \beta_{14} + 15 \beta_{13} - 15 \beta_{12} + \cdots - 36 \)
|
| \(\nu^{9}\) | \(=\) |
\( 12 \beta_{19} - 21 \beta_{18} - 5 \beta_{17} + 23 \beta_{14} + 11 \beta_{13} - 13 \beta_{12} + 66 \beta_{11} + \cdots - 77 \)
|
| \(\nu^{10}\) | \(=\) |
\( 16 \beta_{19} - 36 \beta_{18} + 36 \beta_{17} + 28 \beta_{16} + 12 \beta_{15} - 18 \beta_{14} + \cdots - 38 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( 20 \beta_{19} + 28 \beta_{18} - 22 \beta_{17} - 22 \beta_{16} - 2 \beta_{15} + 40 \beta_{12} + \cdots - 116 \)
|
| \(\nu^{12}\) | \(=\) |
\( 128 \beta_{18} + 132 \beta_{17} + 96 \beta_{16} - 12 \beta_{15} + 106 \beta_{14} - 96 \beta_{13} + \cdots + 145 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 132 \beta_{19} + 318 \beta_{17} + 210 \beta_{14} - 42 \beta_{13} + 60 \beta_{12} + 354 \beta_{11} + \cdots - 312 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 360 \beta_{19} + 396 \beta_{18} - 396 \beta_{17} - 156 \beta_{16} - 156 \beta_{15} + 78 \beta_{14} + \cdots - 534 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 84 \beta_{19} + 415 \beta_{18} + 217 \beta_{17} + 217 \beta_{16} + 138 \beta_{15} + 799 \beta_{12} + \cdots + 648 \)
|
| \(\nu^{16}\) | \(=\) |
\( 355 \beta_{18} + 578 \beta_{17} + 415 \beta_{16} + 113 \beta_{15} - 1547 \beta_{14} - 415 \beta_{13} + \cdots - 2888 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 578 \beta_{19} + 2786 \beta_{18} - 1292 \beta_{17} - 2763 \beta_{14} - 3250 \beta_{13} + 2251 \beta_{12} + \cdots + 5444 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 1958 \beta_{19} + 1734 \beta_{18} - 1734 \beta_{17} - 3168 \beta_{16} + 2119 \beta_{15} + \cdots + 1723 \beta_1 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 4602 \beta_{19} + 4493 \beta_{18} - 524 \beta_{17} - 524 \beta_{16} - 2312 \beta_{15} - 1338 \beta_{12} + \cdots - 592 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/348\mathbb{Z}\right)^\times\).
| \(n\) | \(175\) | \(205\) | \(233\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{11}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −1.50924 | − | 0.849825i | 0 | 0.674282 | 0 | −3.61408 | 0 | 1.55560 | + | 2.56517i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −1.42360 | − | 0.986598i | 0 | −4.10381 | 0 | 1.51330 | 0 | 1.05325 | + | 2.80903i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | −1.41904 | + | 0.993135i | 0 | 2.55342 | 0 | 0.195301 | 0 | 1.02737 | − | 2.81860i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | −0.993135 | + | 1.41904i | 0 | −2.55342 | 0 | 0.195301 | 0 | −1.02737 | − | 2.81860i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | −0.382493 | − | 1.68929i | 0 | 1.89955 | 0 | 3.85056 | 0 | −2.70740 | + | 1.29228i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 0.564214 | − | 1.63758i | 0 | −1.60493 | 0 | −1.94508 | 0 | −2.36332 | − | 1.84789i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.7 | 0 | 0.849825 | + | 1.50924i | 0 | −0.674282 | 0 | −3.61408 | 0 | −1.55560 | + | 2.56517i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 0 | 0.986598 | + | 1.42360i | 0 | 4.10381 | 0 | 1.51330 | 0 | −1.05325 | + | 2.80903i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.9 | 0 | 1.63758 | − | 0.564214i | 0 | 1.60493 | 0 | −1.94508 | 0 | 2.36332 | − | 1.84789i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.10 | 0 | 1.68929 | + | 0.382493i | 0 | −1.89955 | 0 | 3.85056 | 0 | 2.70740 | + | 1.29228i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | 0 | −1.50924 | + | 0.849825i | 0 | 0.674282 | 0 | −3.61408 | 0 | 1.55560 | − | 2.56517i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.2 | 0 | −1.42360 | + | 0.986598i | 0 | −4.10381 | 0 | 1.51330 | 0 | 1.05325 | − | 2.80903i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.3 | 0 | −1.41904 | − | 0.993135i | 0 | 2.55342 | 0 | 0.195301 | 0 | 1.02737 | + | 2.81860i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.4 | 0 | −0.993135 | − | 1.41904i | 0 | −2.55342 | 0 | 0.195301 | 0 | −1.02737 | + | 2.81860i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.5 | 0 | −0.382493 | + | 1.68929i | 0 | 1.89955 | 0 | 3.85056 | 0 | −2.70740 | − | 1.29228i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.6 | 0 | 0.564214 | + | 1.63758i | 0 | −1.60493 | 0 | −1.94508 | 0 | −2.36332 | + | 1.84789i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.7 | 0 | 0.849825 | − | 1.50924i | 0 | −0.674282 | 0 | −3.61408 | 0 | −1.55560 | − | 2.56517i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.8 | 0 | 0.986598 | − | 1.42360i | 0 | 4.10381 | 0 | 1.51330 | 0 | −1.05325 | − | 2.80903i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.9 | 0 | 1.63758 | + | 0.564214i | 0 | 1.60493 | 0 | −1.94508 | 0 | 2.36332 | + | 1.84789i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.10 | 0 | 1.68929 | − | 0.382493i | 0 | −1.89955 | 0 | 3.85056 | 0 | 2.70740 | − | 1.29228i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 29.c | odd | 4 | 1 | inner |
| 87.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 348.2.l.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 348.2.l.a | ✓ | 20 |
| 29.c | odd | 4 | 1 | inner | 348.2.l.a | ✓ | 20 |
| 87.f | even | 4 | 1 | inner | 348.2.l.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 348.2.l.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 348.2.l.a | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 348.2.l.a | ✓ | 20 | 29.c | odd | 4 | 1 | inner |
| 348.2.l.a | ✓ | 20 | 87.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(348, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} - 2 T^{17} + \cdots + 59049 \)
|
| $5$ |
\( (T^{10} - 30 T^{8} + \cdots - 464)^{2} \)
|
| $7$ |
\( (T^{5} - 17 T^{3} - 2 T^{2} + \cdots - 8)^{4} \)
|
| $11$ |
\( T^{20} + \cdots + 1787767524 \)
|
| $13$ |
\( (T^{10} + 48 T^{8} + \cdots + 576)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 21884133090304 \)
|
| $19$ |
\( (T^{10} + 2 T^{9} + \cdots + 1152)^{2} \)
|
| $23$ |
\( (T^{10} + 128 T^{8} + \cdots + 118784)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 420707233300201 \)
|
| $31$ |
\( (T^{10} + 8 T^{9} + \cdots + 366368)^{2} \)
|
| $37$ |
\( (T^{10} + 2 T^{9} + \cdots + 2048)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 220463104 \)
|
| $43$ |
\( (T^{10} + 4 T^{9} + \cdots + 323208)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 2379293284 \)
|
| $53$ |
\( (T^{10} + 354 T^{8} + \cdots + 235222016)^{2} \)
|
| $59$ |
\( (T^{10} + 480 T^{8} + \cdots + 3923799296)^{2} \)
|
| $61$ |
\( (T^{10} - 10 T^{9} + \cdots + 1525286912)^{2} \)
|
| $67$ |
\( (T^{10} + 414 T^{8} + \cdots + 399424)^{2} \)
|
| $71$ |
\( (T^{10} - 336 T^{8} + \cdots - 9621504)^{2} \)
|
| $73$ |
\( (T^{10} + 14 T^{9} + \cdots + 165888)^{2} \)
|
| $79$ |
\( (T^{10} + 8 T^{9} + \cdots + 648)^{2} \)
|
| $83$ |
\( (T^{10} + 292 T^{8} + \cdots + 475136)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 39\!\cdots\!04 \)
|
| $97$ |
\( (T^{10} - 22 T^{9} + \cdots + 310303872)^{2} \)
|
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