Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [349,2,Mod(1,349)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(349, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("349.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 349 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 349.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: | \(2.78677903054\) |
| Analytic rank: | \(0\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - 5 x^{16} - 14 x^{15} + 102 x^{14} + 26 x^{13} - 792 x^{12} + 474 x^{11} + 2887 x^{10} + \cdots - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 5 x^{16} - 14 x^{15} + 102 x^{14} + 26 x^{13} - 792 x^{12} + 474 x^{11} + 2887 x^{10} + \cdots - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 6881 \nu^{16} + 3046496 \nu^{15} - 12143208 \nu^{14} - 53358172 \nu^{13} + 248098852 \nu^{12} + \cdots - 349096 ) / 13854332 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 129907 \nu^{16} - 673231 \nu^{15} - 842951 \nu^{14} + 11316825 \nu^{13} - 15879081 \nu^{12} + \cdots + 33470372 ) / 13854332 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 342344 \nu^{16} - 1539407 \nu^{15} - 4675055 \nu^{14} + 29695925 \nu^{13} + 6728395 \nu^{12} + \cdots - 16651472 ) / 13854332 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 954477 \nu^{16} - 6048190 \nu^{15} - 9436362 \nu^{14} + 122225562 \nu^{13} - 56644828 \nu^{12} + \cdots - 60860676 ) / 13854332 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1423805 \nu^{16} - 5123424 \nu^{15} - 25701018 \nu^{14} + 104173792 \nu^{13} + 157565816 \nu^{12} + \cdots - 38280072 ) / 13854332 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1624725 \nu^{16} + 8295022 \nu^{15} + 22421616 \nu^{14} - 168218710 \nu^{13} - 38246072 \nu^{12} + \cdots - 16113024 ) / 13854332 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 2320607 \nu^{16} + 13871476 \nu^{15} + 23761238 \nu^{14} - 276884452 \nu^{13} + \cdots + 100990532 ) / 13854332 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1177751 \nu^{16} - 4876786 \nu^{15} - 19898736 \nu^{14} + 100486682 \nu^{13} + 101420316 \nu^{12} + \cdots - 38449256 ) / 6927166 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1430016 \nu^{16} - 7943686 \nu^{15} - 16588569 \nu^{14} + 158865051 \nu^{13} - 33046398 \nu^{12} + \cdots - 53459238 ) / 6927166 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 2212232 \nu^{16} - 10666320 \nu^{15} - 31922034 \nu^{14} + 216378870 \nu^{13} + 78707123 \nu^{12} + \cdots - 54985350 ) / 6927166 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 4464588 \nu^{16} + 22316577 \nu^{15} + 62453009 \nu^{14} - 453309389 \nu^{13} + \cdots + 161461920 ) / 13854332 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2258564 \nu^{16} + 11405411 \nu^{15} + 30768077 \nu^{14} - 230738128 \nu^{13} - 42311710 \nu^{12} + \cdots + 41232606 ) / 6927166 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 5588767 \nu^{16} + 28694781 \nu^{15} + 74918371 \nu^{14} - 580575835 \nu^{13} + \cdots + 155763180 ) / 13854332 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{13} - \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{16} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + \cdots + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{16} - 2 \beta_{14} - 11 \beta_{13} - 2 \beta_{11} - 8 \beta_{10} - 8 \beta_{9} + \beta_{8} + \cdots + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 15 \beta_{16} + \beta_{15} - 14 \beta_{14} - 16 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} + \cdots + 32 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 17 \beta_{16} - 3 \beta_{15} - 32 \beta_{14} - 96 \beta_{13} - 7 \beta_{12} - 36 \beta_{11} + \cdots + 538 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 160 \beta_{16} + 13 \beta_{15} - 139 \beta_{14} - 179 \beta_{13} - 126 \beta_{12} - 148 \beta_{11} + \cdots + 352 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 202 \beta_{16} - 53 \beta_{15} - 354 \beta_{14} - 785 \beta_{13} - 129 \beta_{12} - 433 \beta_{11} + \cdots + 3572 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1481 \beta_{16} + 105 \beta_{15} - 1229 \beta_{14} - 1730 \beta_{13} - 1102 \beta_{12} - 1406 \beta_{11} + \cdots + 3364 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 2070 \beta_{16} - 622 \beta_{15} - 3377 \beta_{14} - 6286 \beta_{13} - 1573 \beta_{12} - 4385 \beta_{11} + \cdots + 24725 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 12736 \beta_{16} + 616 \beta_{15} - 10358 \beta_{14} - 15482 \beta_{13} - 9192 \beta_{12} + \cdots + 30022 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 19563 \beta_{16} - 6149 \beta_{15} - 29869 \beta_{14} - 50019 \beta_{13} - 16060 \beta_{12} + \cdots + 176300 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 105024 \beta_{16} + 2128 \beta_{15} - 85333 \beta_{14} - 132384 \beta_{13} - 74733 \beta_{12} + \cdots + 257719 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 175725 \beta_{16} - 55584 \beta_{15} - 252927 \beta_{14} - 397350 \beta_{13} - 149019 \beta_{12} + \cdots + 1284423 \)
|
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 |
|
−2.51266 | 2.89863 | 4.31346 | 2.77794 | −7.28327 | −0.223033 | −5.81294 | 5.40206 | −6.98003 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.36348 | −2.15844 | 3.58604 | −0.000712583 | 0 | 5.10144 | −2.22015 | −3.74859 | 1.65887 | 0.00168418 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.99653 | 1.39055 | 1.98612 | −2.55061 | −2.77626 | −4.72921 | 0.0277164 | −1.06638 | 5.09235 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.82251 | −0.490735 | 1.32154 | 3.61587 | 0.894370 | 2.81440 | 1.23649 | −2.75918 | −6.58995 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.09071 | 2.75693 | −0.810342 | 1.35282 | −3.00702 | 0.0387215 | 3.06528 | 4.60064 | −1.47554 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.358260 | −1.20564 | −1.87165 | −1.57855 | 0.431934 | 1.55949 | 1.38706 | −1.54642 | 0.565531 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.207529 | 2.64177 | −1.95693 | −1.74345 | −0.548246 | 3.85565 | 0.821180 | 3.97897 | 0.361817 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.107621 | −2.82440 | −1.98842 | −3.72669 | 0.303965 | −3.10404 | 0.429238 | 4.97724 | 0.401070 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.226775 | −1.51708 | −1.94857 | 2.45530 | −0.344035 | 0.487144 | −0.895438 | −0.698475 | 0.556801 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.695551 | 1.81098 | −1.51621 | 3.87525 | 1.25963 | 2.52014 | −2.44570 | 0.279648 | 2.69543 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.36177 | 1.04922 | −0.145589 | 0.274763 | 1.42880 | 2.68858 | −2.92179 | −1.89913 | 0.374163 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.43824 | 3.12218 | 0.0685351 | 1.24047 | 4.49045 | −3.90458 | −2.77791 | 6.74802 | 1.78410 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.93329 | −3.25199 | 1.73762 | 0.302138 | −6.28704 | 2.99039 | −0.507260 | 7.57541 | 0.584120 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.18773 | 1.16554 | 2.78617 | 0.568409 | 2.54988 | 0.610781 | 1.71993 | −1.64153 | 1.24353 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.36706 | 2.46697 | 3.60297 | −3.97349 | 5.83947 | −0.131055 | 3.79433 | 3.08596 | −9.40548 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.45146 | −0.386523 | 4.00965 | 3.21890 | −0.947545 | −4.43315 | 4.92658 | −2.85060 | 7.89101 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.79743 | −1.46796 | 5.82560 | −1.10838 | −4.10651 | 2.17992 | 10.7018 | −0.845091 | −3.10061 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(349\) | \(-1\) |
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 | 349.2.a.b | ✓ | 17 |
| 3.b | odd | 2 | 1 | 3141.2.a.e | 17 | ||
| 4.b | odd | 2 | 1 | 5584.2.a.m | 17 | ||
| 5.b | even | 2 | 1 | 8725.2.a.m | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 349.2.a.b | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 3141.2.a.e | 17 | 3.b | odd | 2 | 1 | ||
| 5584.2.a.m | 17 | 4.b | odd | 2 | 1 | ||
| 8725.2.a.m | 17 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 5 T_{2}^{16} - 14 T_{2}^{15} + 102 T_{2}^{14} + 26 T_{2}^{13} - 792 T_{2}^{12} + 474 T_{2}^{11} + \cdots - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(349))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 5 T^{16} + \cdots - 4 \)
|
| $3$ |
\( T^{17} - 6 T^{16} + \cdots - 5056 \)
|
| $5$ |
\( T^{17} - 5 T^{16} + \cdots + 2 \)
|
| $7$ |
\( T^{17} - T^{16} + \cdots + 142 \)
|
| $11$ |
\( T^{17} - 39 T^{16} + \cdots - 778568 \)
|
| $13$ |
\( T^{17} + 6 T^{16} + \cdots - 896 \)
|
| $17$ |
\( T^{17} - T^{16} + \cdots - 3937 \)
|
| $19$ |
\( T^{17} - 7 T^{16} + \cdots - 1169728 \)
|
| $23$ |
\( T^{17} + \cdots - 1001171008 \)
|
| $29$ |
\( T^{17} + \cdots - 29830145783 \)
|
| $31$ |
\( T^{17} + \cdots + 6892082176 \)
|
| $37$ |
\( T^{17} + \cdots - 8613499783 \)
|
| $41$ |
\( T^{17} + \cdots + 43720841546 \)
|
| $43$ |
\( T^{17} + \cdots - 1270947093868 \)
|
| $47$ |
\( T^{17} + \cdots - 92686819988 \)
|
| $53$ |
\( T^{17} + \cdots - 207131776 \)
|
| $59$ |
\( T^{17} + \cdots + 29595049814498 \)
|
| $61$ |
\( T^{17} + \cdots - 25827233152 \)
|
| $67$ |
\( T^{17} + \cdots - 677114962112 \)
|
| $71$ |
\( T^{17} + \cdots + 40841913506 \)
|
| $73$ |
\( T^{17} + \cdots - 339242114153 \)
|
| $79$ |
\( T^{17} + \cdots + 110475548 \)
|
| $83$ |
\( T^{17} + \cdots - 3124825344832 \)
|
| $89$ |
\( T^{17} + \cdots + 1626180608 \)
|
| $97$ |
\( T^{17} + \cdots + 82696354304 \)
|
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