Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [370,2,Mod(23,370)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(370, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([9, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("370.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 370.r (of order \(12\), degree \(4\), 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: | \(2.95446487479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} + \cdots + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} + \cdots + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} + \cdots + 46\!\cdots\!94 ) / 51\!\cdots\!90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 123434729989149 \nu^{15} + 492304781090243 \nu^{14} + \cdots + 195044141171618 ) / 51\!\cdots\!90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 137809854283897 \nu^{15} + 666808683532056 \nu^{14} + \cdots + 69\!\cdots\!44 ) / 10\!\cdots\!78 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} + \cdots - 16575048540328 ) / 28585593772190 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + \cdots + 3631882693804 ) / 28585593772190 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 14\!\cdots\!27 \nu^{15} + \cdots + 35\!\cdots\!44 ) / 51\!\cdots\!90 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + \cdots - 77929949288 ) / 18265555126 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + \cdots + 65633529574178 ) / 16530327389030 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 20\!\cdots\!87 \nu^{15} + \cdots - 28\!\cdots\!14 ) / 51\!\cdots\!90 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + \cdots - 330526984 ) / 63923990 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 567889621787826 \nu^{15} + \cdots - 77\!\cdots\!96 ) / 10\!\cdots\!78 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 52\!\cdots\!58 \nu^{15} + \cdots - 60\!\cdots\!06 ) / 51\!\cdots\!90 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} + \cdots + 258618081490 ) / 18265555126 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} + \cdots + 791533522 ) / 63923990 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 69\!\cdots\!34 \nu^{15} + \cdots - 83\!\cdots\!68 ) / 51\!\cdots\!90 \)
|
| \(\nu\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{3} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{15} + 4\beta_{14} + 2\beta_{13} + 3\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{6} - 5\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} + \cdots + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} + \cdots - 37 \)
|
| \(\nu^{6}\) | \(=\) |
\( 28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} + \cdots + 93 \)
|
| \(\nu^{7}\) | \(=\) |
\( 163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} + \cdots + 985 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + \cdots + 9 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} + \cdots - 5566 \)
|
| \(\nu^{10}\) | \(=\) |
\( 6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} + \cdots + 17408 \)
|
| \(\nu^{11}\) | \(=\) |
\( 8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} + \cdots + 77085 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + \cdots - 166538 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + \cdots - 469625 \)
|
| \(\nu^{14}\) | \(=\) |
\( 880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + \cdots + 2650257 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} + \cdots + 3840272 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(297\) |
| \(\chi(n)\) | \(-\beta_{5}\) | \(\beta_{13}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
0.866025 | + | 0.500000i | −2.83195 | − | 0.758819i | 0.500000 | + | 0.866025i | −1.91159 | − | 1.16009i | −2.07313 | − | 2.07313i | 2.40553 | + | 0.644560i | 1.00000i | 4.84607 | + | 2.79788i | −1.07544 | − | 1.96047i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.2 | 0.866025 | + | 0.500000i | −2.83195 | − | 0.758819i | 0.500000 | + | 0.866025i | 2.17041 | − | 0.537883i | −2.07313 | − | 2.07313i | −1.69842 | − | 0.455091i | 1.00000i | 4.84607 | + | 2.79788i | 2.14857 | + | 0.619385i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.3 | 0.866025 | + | 0.500000i | −0.900100 | − | 0.241181i | 0.500000 | + | 0.866025i | −1.62401 | − | 1.53707i | −0.658919 | − | 0.658919i | −2.22532 | − | 0.596272i | 1.00000i | −1.84607 | − | 1.06583i | −0.637899 | − | 2.14315i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.4 | 0.866025 | + | 0.500000i | −0.900100 | − | 0.241181i | 0.500000 | + | 0.866025i | 1.36519 | + | 1.77095i | −0.658919 | − | 0.658919i | 1.51821 | + | 0.406803i | 1.00000i | −1.84607 | − | 1.06583i | 0.296818 | + | 2.21628i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 177.1 | 0.866025 | − | 0.500000i | −2.83195 | + | 0.758819i | 0.500000 | − | 0.866025i | −1.91159 | + | 1.16009i | −2.07313 | + | 2.07313i | 2.40553 | − | 0.644560i | − | 1.00000i | 4.84607 | − | 2.79788i | −1.07544 | + | 1.96047i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 177.2 | 0.866025 | − | 0.500000i | −2.83195 | + | 0.758819i | 0.500000 | − | 0.866025i | 2.17041 | + | 0.537883i | −2.07313 | + | 2.07313i | −1.69842 | + | 0.455091i | − | 1.00000i | 4.84607 | − | 2.79788i | 2.14857 | − | 0.619385i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 177.3 | 0.866025 | − | 0.500000i | −0.900100 | + | 0.241181i | 0.500000 | − | 0.866025i | −1.62401 | + | 1.53707i | −0.658919 | + | 0.658919i | −2.22532 | + | 0.596272i | − | 1.00000i | −1.84607 | + | 1.06583i | −0.637899 | + | 2.14315i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 177.4 | 0.866025 | − | 0.500000i | −0.900100 | + | 0.241181i | 0.500000 | − | 0.866025i | 1.36519 | − | 1.77095i | −0.658919 | + | 0.658919i | 1.51821 | − | 0.406803i | − | 1.00000i | −1.84607 | + | 1.06583i | 0.296818 | − | 2.21628i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.1 | −0.866025 | + | 0.500000i | −0.392794 | − | 1.46593i | 0.500000 | − | 0.866025i | −1.19306 | + | 1.89119i | 1.07313 | + | 1.07313i | −0.552037 | − | 2.06023i | 1.00000i | 0.603425 | − | 0.348387i | 0.0876265 | − | 2.23435i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.2 | −0.866025 | + | 0.500000i | −0.392794 | − | 1.46593i | 0.500000 | − | 0.866025i | 2.15899 | + | 0.582041i | 1.07313 | + | 1.07313i | −0.155070 | − | 0.578728i | 1.00000i | 0.603425 | − | 0.348387i | −2.16076 | + | 0.575432i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.3 | −0.866025 | + | 0.500000i | 0.124844 | + | 0.465926i | 0.500000 | − | 0.866025i | −1.99671 | + | 1.00656i | −0.341081 | − | 0.341081i | −0.510450 | − | 1.90503i | 1.00000i | 2.39658 | − | 1.38366i | 1.22592 | − | 1.87006i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.4 | −0.866025 | + | 0.500000i | 0.124844 | + | 0.465926i | 0.500000 | − | 0.866025i | 1.03078 | + | 1.98431i | −0.341081 | − | 0.341081i | 1.21756 | + | 4.54399i | 1.00000i | 2.39658 | − | 1.38366i | −1.88484 | − | 1.20307i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 347.1 | −0.866025 | − | 0.500000i | −0.392794 | + | 1.46593i | 0.500000 | + | 0.866025i | −1.19306 | − | 1.89119i | 1.07313 | − | 1.07313i | −0.552037 | + | 2.06023i | − | 1.00000i | 0.603425 | + | 0.348387i | 0.0876265 | + | 2.23435i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 347.2 | −0.866025 | − | 0.500000i | −0.392794 | + | 1.46593i | 0.500000 | + | 0.866025i | 2.15899 | − | 0.582041i | 1.07313 | − | 1.07313i | −0.155070 | + | 0.578728i | − | 1.00000i | 0.603425 | + | 0.348387i | −2.16076 | − | 0.575432i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 347.3 | −0.866025 | − | 0.500000i | 0.124844 | − | 0.465926i | 0.500000 | + | 0.866025i | −1.99671 | − | 1.00656i | −0.341081 | + | 0.341081i | −0.510450 | + | 1.90503i | − | 1.00000i | 2.39658 | + | 1.38366i | 1.22592 | + | 1.87006i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 347.4 | −0.866025 | − | 0.500000i | 0.124844 | − | 0.465926i | 0.500000 | + | 0.866025i | 1.03078 | − | 1.98431i | −0.341081 | + | 0.341081i | 1.21756 | − | 4.54399i | − | 1.00000i | 2.39658 | + | 1.38366i | −1.88484 | + | 1.20307i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 185.u | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 370.2.r.e | yes | 16 |
| 5.c | odd | 4 | 1 | 370.2.q.e | ✓ | 16 | |
| 37.g | odd | 12 | 1 | 370.2.q.e | ✓ | 16 | |
| 185.u | even | 12 | 1 | inner | 370.2.r.e | yes | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 370.2.q.e | ✓ | 16 | 5.c | odd | 4 | 1 | |
| 370.2.q.e | ✓ | 16 | 37.g | odd | 12 | 1 | |
| 370.2.r.e | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 370.2.r.e | yes | 16 | 185.u | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 8T_{3}^{7} + 26T_{3}^{6} + 48T_{3}^{5} + 62T_{3}^{4} + 48T_{3}^{3} + 20T_{3}^{2} + 8T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{8} + 8 T^{7} + 26 T^{6} + \cdots + 4)^{2} \)
|
| $5$ |
\( T^{16} - 8 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 12 T^{14} + \cdots + 35344 \)
|
| $11$ |
\( T^{16} + 104 T^{14} + \cdots + 28558336 \)
|
| $13$ |
\( T^{16} + \cdots + 154157056 \)
|
| $17$ |
\( T^{16} + 16 T^{15} + \cdots + 591361 \)
|
| $19$ |
\( T^{16} + \cdots + 1413760000 \)
|
| $23$ |
\( T^{16} + \cdots + 3175998736 \)
|
| $29$ |
\( T^{16} + \cdots + 217903173601 \)
|
| $31$ |
\( T^{16} + \cdots + 2606287360000 \)
|
| $37$ |
\( (T^{8} + 24 T^{6} + \cdots + 1874161)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 6483309361 \)
|
| $43$ |
\( T^{16} + \cdots + 429815824 \)
|
| $47$ |
\( T^{16} + \cdots + 34744328602624 \)
|
| $53$ |
\( T^{16} + \cdots + 29243736064 \)
|
| $59$ |
\( T^{16} + \cdots + 178794124017664 \)
|
| $61$ |
\( T^{16} + \cdots + 693034605169 \)
|
| $67$ |
\( T^{16} + \cdots + 133593174016 \)
|
| $71$ |
\( T^{16} + \cdots + 31\!\cdots\!64 \)
|
| $73$ |
\( T^{16} + \cdots + 32262400000000 \)
|
| $79$ |
\( T^{16} + \cdots + 2852045440000 \)
|
| $83$ |
\( T^{16} + 48 T^{15} + \cdots + 234256 \)
|
| $89$ |
\( T^{16} + \cdots + 10645493329 \)
|
| $97$ |
\( (T^{8} - 8 T^{7} + \cdots + 50233)^{2} \)
|
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