Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2075,2,Mod(1,2075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2075, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2075.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2075 = 5^{2} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2075.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: | \(16.5689584194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - 35 x^{17} - 2 x^{16} + 512 x^{15} + 54 x^{14} - 4061 x^{13} - 568 x^{12} + 18955 x^{11} + \cdots - 192 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - 35 x^{17} - 2 x^{16} + 512 x^{15} + 54 x^{14} - 4061 x^{13} - 568 x^{12} + 18955 x^{11} + \cdots - 192 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13708 \nu^{18} + 277439 \nu^{17} - 1689896 \nu^{16} - 7926929 \nu^{15} + 44995150 \nu^{14} + \cdots - 354503248 ) / 53829472 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2594261 \nu^{18} + 723990 \nu^{17} + 86880251 \nu^{16} - 16822880 \nu^{15} + \cdots - 1184128224 ) / 215317888 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 751545 \nu^{18} - 585267 \nu^{17} - 25504047 \nu^{16} + 19169735 \nu^{15} + 360740550 \nu^{14} + \cdots - 880196624 ) / 53829472 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1581611 \nu^{18} - 2677966 \nu^{17} + 57927917 \nu^{16} + 93620528 \nu^{15} + \cdots - 723843744 ) / 107658944 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5335325 \nu^{18} + 9098242 \nu^{17} - 190324163 \nu^{16} - 309591464 \nu^{15} + \cdots + 1800329952 ) / 215317888 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 7540039 \nu^{18} - 3163222 \nu^{17} + 258545433 \nu^{16} + 130935912 \nu^{15} + \cdots - 2815219232 ) / 215317888 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3975979 \nu^{18} + 773072 \nu^{17} - 136840685 \nu^{16} - 35539926 \nu^{15} + 1957522868 \nu^{14} + \cdots - 837188800 ) / 107658944 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 8007327 \nu^{18} + 3866794 \nu^{17} - 275842529 \nu^{16} - 156861428 \nu^{15} + \cdots + 3795735904 ) / 215317888 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 505985 \nu^{18} + 263918 \nu^{17} - 17249815 \nu^{16} - 9733291 \nu^{15} + 243547775 \nu^{14} + \cdots + 103379600 ) / 13457368 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 5090423 \nu^{18} - 3314770 \nu^{17} + 175633433 \nu^{16} + 124572444 \nu^{15} + \cdots - 1962930656 ) / 107658944 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 5875741 \nu^{18} + 4603250 \nu^{17} - 202707747 \nu^{16} - 169530656 \nu^{15} + 2904635576 \nu^{14} + \cdots + 467541856 ) / 107658944 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 13725647 \nu^{18} - 7198438 \nu^{17} + 474471745 \nu^{16} + 278381832 \nu^{15} + \cdots + 2215588576 ) / 215317888 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 14780011 \nu^{18} - 13192226 \nu^{17} + 513377893 \nu^{16} + 475253412 \nu^{15} + \cdots - 1986952672 ) / 215317888 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 4079623 \nu^{18} + 1863503 \nu^{17} - 140098997 \nu^{16} - 72906475 \nu^{15} + 1996225174 \nu^{14} + \cdots + 120741008 ) / 53829472 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 5785733 \nu^{18} - 5037544 \nu^{17} + 200064311 \nu^{16} + 183531582 \nu^{15} + \cdots - 2969156096 ) / 53829472 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{17} + \beta_{15} - \beta_{13} - \beta_{11} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} + 8\beta_{2} + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + 10\beta_{3} + 40\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{17} + \beta_{16} + 11 \beta_{15} + \beta_{14} - 11 \beta_{13} - 11 \beta_{11} + \beta_{10} + \cdots + 181 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3 \beta_{18} - 2 \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{12} + \cdots + 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{18} + 80 \beta_{17} + 15 \beta_{16} + 96 \beta_{15} + 16 \beta_{14} - 97 \beta_{13} - 98 \beta_{11} + \cdots + 1295 \)
|
| \(\nu^{9}\) | \(=\) |
\( 53 \beta_{18} - 42 \beta_{17} - 14 \beta_{16} - 18 \beta_{15} - 13 \beta_{14} + 18 \beta_{13} + 39 \beta_{12} + \cdots + 1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 23 \beta_{18} + 594 \beta_{17} + 154 \beta_{16} + 778 \beta_{15} + 179 \beta_{14} - 797 \beta_{13} + \cdots + 9407 \)
|
| \(\nu^{11}\) | \(=\) |
\( 628 \beta_{18} - 564 \beta_{17} - 135 \beta_{16} - 216 \beta_{15} - 112 \beta_{14} + 216 \beta_{13} + \cdots - 404 \)
|
| \(\nu^{12}\) | \(=\) |
\( 334 \beta_{18} + 4260 \beta_{17} + 1351 \beta_{16} + 6113 \beta_{15} + 1731 \beta_{14} - 6343 \beta_{13} + \cdots + 68983 \)
|
| \(\nu^{13}\) | \(=\) |
\( 6303 \beta_{18} - 6247 \beta_{17} - 1125 \beta_{16} - 2186 \beta_{15} - 783 \beta_{14} + 2182 \beta_{13} + \cdots - 6206 \)
|
| \(\nu^{14}\) | \(=\) |
\( 3927 \beta_{18} + 29970 \beta_{17} + 10917 \beta_{16} + 47338 \beta_{15} + 15532 \beta_{14} + \cdots + 509080 \)
|
| \(\nu^{15}\) | \(=\) |
\( 58007 \beta_{18} - 62329 \beta_{17} - 8743 \beta_{16} - 20222 \beta_{15} - 4594 \beta_{14} + \cdots - 69534 \)
|
| \(\nu^{16}\) | \(=\) |
\( 40926 \beta_{18} + 208356 \beta_{17} + 84059 \beta_{16} + 363927 \beta_{15} + 133484 \beta_{14} + \cdots + 3773818 \)
|
| \(\nu^{17}\) | \(=\) |
\( 507365 \beta_{18} - 583280 \beta_{17} - 65573 \beta_{16} - 177383 \beta_{15} - 20919 \beta_{14} + \cdots - 683900 \)
|
| \(\nu^{18}\) | \(=\) |
\( 395058 \beta_{18} + 1437060 \beta_{17} + 627650 \beta_{16} + 2787532 \beta_{15} + 1116829 \beta_{14} + \cdots + 28068710 \)
|
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 |
|
−2.75520 | 0.977224 | 5.59112 | 0 | −2.69245 | 0.767286 | −9.89426 | −2.04503 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.72178 | −2.14654 | 5.40810 | 0 | 5.84243 | 3.26443 | −9.27611 | 1.60765 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.30246 | −2.46160 | 3.30134 | 0 | 5.66773 | −4.25460 | −2.99629 | 3.05945 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.21717 | 0.721511 | 2.91584 | 0 | −1.59971 | −4.60713 | −2.03058 | −2.47942 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.81407 | −2.79557 | 1.29086 | 0 | 5.07137 | 0.363345 | 1.28643 | 4.81521 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.71747 | 1.96457 | 0.949697 | 0 | −3.37409 | 4.67293 | 1.80386 | 0.859549 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.34217 | 3.09791 | −0.198588 | 0 | −4.15791 | −1.93773 | 2.95087 | 6.59704 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.10403 | 1.62594 | −0.781109 | 0 | −1.79510 | −3.08135 | 3.07044 | −0.356310 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.179883 | −1.80190 | −1.96764 | 0 | 0.324132 | 4.64791 | 0.713713 | 0.246837 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.100681 | −3.12184 | −1.98986 | 0 | 0.314311 | −1.11423 | 0.401704 | 6.74590 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.669689 | −1.42350 | −1.55152 | 0 | −0.953300 | −4.14577 | −2.37841 | −0.973656 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.701065 | 3.34923 | −1.50851 | 0 | 2.34803 | 2.22256 | −2.45969 | 8.21731 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0.876535 | −0.234822 | −1.23169 | 0 | −0.205830 | 2.78414 | −2.83269 | −2.94486 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.72975 | −1.80865 | 0.992025 | 0 | −3.12850 | −2.54553 | −1.74354 | 0.271206 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.83329 | 2.34463 | 1.36094 | 0 | 4.29838 | 4.59389 | −1.17158 | 2.49728 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.29167 | 1.21088 | 3.25174 | 0 | 2.77494 | −0.429740 | 2.86856 | −1.53376 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.63441 | 3.06634 | 4.94011 | 0 | 8.07801 | −1.23091 | 7.74547 | 6.40246 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.75235 | −3.30879 | 5.57545 | 0 | −9.10696 | −4.53031 | 9.84089 | 7.94809 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2.76617 | −0.255036 | 5.65169 | 0 | −0.705473 | 3.56080 | 10.1012 | −2.93496 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(83\) | \( +1 \) |
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 | 2075.2.a.l | ✓ | 19 |
| 5.b | even | 2 | 1 | 2075.2.a.m | yes | 19 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2075.2.a.l | ✓ | 19 | 1.a | even | 1 | 1 | trivial |
| 2075.2.a.m | yes | 19 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2075))\):
|
\( T_{2}^{19} - 35 T_{2}^{17} - 2 T_{2}^{16} + 512 T_{2}^{15} + 54 T_{2}^{14} - 4061 T_{2}^{13} + \cdots - 192 \)
|
|
\( T_{3}^{19} + T_{3}^{18} - 46 T_{3}^{17} - 45 T_{3}^{16} + 879 T_{3}^{15} + 822 T_{3}^{14} - 9079 T_{3}^{13} + \cdots - 8624 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} - 35 T^{17} + \cdots - 192 \)
|
| $3$ |
\( T^{19} + T^{18} + \cdots - 8624 \)
|
| $5$ |
\( T^{19} \)
|
| $7$ |
\( T^{19} + T^{18} + \cdots - 6598496 \)
|
| $11$ |
\( T^{19} + \cdots + 168842928 \)
|
| $13$ |
\( T^{19} - 7 T^{18} + \cdots + 16384 \)
|
| $17$ |
\( T^{19} + 9 T^{18} + \cdots - 99990012 \)
|
| $19$ |
\( T^{19} + \cdots - 302120960 \)
|
| $23$ |
\( T^{19} + \cdots + 66877054656 \)
|
| $29$ |
\( T^{19} + \cdots - 19790330700 \)
|
| $31$ |
\( T^{19} + \cdots + 462997242304 \)
|
| $37$ |
\( T^{19} + \cdots - 10403448125372 \)
|
| $41$ |
\( T^{19} + \cdots - 819583070353848 \)
|
| $43$ |
\( T^{19} + \cdots + 17181366747136 \)
|
| $47$ |
\( T^{19} + \cdots - 1299904954368 \)
|
| $53$ |
\( T^{19} + \cdots - 325402156253184 \)
|
| $59$ |
\( T^{19} + \cdots + 74039071505220 \)
|
| $61$ |
\( T^{19} + \cdots - 14518771325114 \)
|
| $67$ |
\( T^{19} + \cdots + 4311664001024 \)
|
| $71$ |
\( T^{19} + \cdots - 76\!\cdots\!64 \)
|
| $73$ |
\( T^{19} + \cdots - 221151912361984 \)
|
| $79$ |
\( T^{19} + \cdots - 20208805406720 \)
|
| $83$ |
\( (T + 1)^{19} \)
|
| $89$ |
\( T^{19} + \cdots - 952412706570240 \)
|
| $97$ |
\( T^{19} + \cdots - 78815447515136 \)
|
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