Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [58,4,Mod(7,58)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([6]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.d (of order \(7\), degree \(6\), 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: | \(3.42211078033\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{7})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 2 x^{17} + 93 x^{16} - 100 x^{15} + 3420 x^{14} + 9302 x^{13} + 57475 x^{12} + \cdots + 19384714441 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 2 x^{17} + 93 x^{16} - 100 x^{15} + 3420 x^{14} + 9302 x^{13} + 57475 x^{12} + \cdots + 19384714441 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 28\!\cdots\!88 \nu^{17} + \cdots + 11\!\cdots\!76 ) / 11\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 47\!\cdots\!01 \nu^{17} + \cdots + 58\!\cdots\!81 ) / 11\!\cdots\!16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\!\cdots\!91 \nu^{17} + \cdots + 93\!\cdots\!22 ) / 23\!\cdots\!16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\!\cdots\!04 \nu^{17} + \cdots + 39\!\cdots\!67 ) / 11\!\cdots\!16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 78\!\cdots\!94 \nu^{17} + \cdots + 12\!\cdots\!41 ) / 56\!\cdots\!08 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 47\!\cdots\!98 \nu^{17} + \cdots + 19\!\cdots\!41 ) / 23\!\cdots\!16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 23\!\cdots\!42 \nu^{17} + \cdots - 47\!\cdots\!63 ) / 11\!\cdots\!16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 23\!\cdots\!12 \nu^{17} + \cdots + 92\!\cdots\!81 ) / 11\!\cdots\!16 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 80\!\cdots\!50 \nu^{17} + \cdots + 12\!\cdots\!17 ) / 23\!\cdots\!16 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 80\!\cdots\!71 \nu^{17} + \cdots + 93\!\cdots\!92 ) / 23\!\cdots\!16 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 12\!\cdots\!79 \nu^{17} + \cdots + 67\!\cdots\!08 ) / 23\!\cdots\!16 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 48\!\cdots\!70 \nu^{17} + \cdots - 62\!\cdots\!86 ) / 56\!\cdots\!08 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 15\!\cdots\!87 \nu^{17} + \cdots - 45\!\cdots\!73 ) / 11\!\cdots\!08 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 19\!\cdots\!61 \nu^{17} + \cdots + 33\!\cdots\!58 ) / 11\!\cdots\!16 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 23\!\cdots\!73 \nu^{17} + \cdots - 11\!\cdots\!73 ) / 11\!\cdots\!16 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 37\!\cdots\!68 \nu^{17} + \cdots + 50\!\cdots\!61 ) / 11\!\cdots\!16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( - \beta_{17} + \beta_{15} + \beta_{13} + \beta_{12} + \beta_{11} + 7 \beta_{9} - 9 \beta_{8} + 2 \beta_{7} + \cdots + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{17} + \beta_{16} - \beta_{15} - 2 \beta_{14} - 3 \beta_{13} - 11 \beta_{10} - 18 \beta_{9} + \cdots - 49 \)
|
| \(\nu^{4}\) | \(=\) |
\( 53 \beta_{17} - 53 \beta_{16} - 13 \beta_{15} + 13 \beta_{14} - 11 \beta_{13} - 62 \beta_{12} + \cdots - 921 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 83 \beta_{17} - 69 \beta_{16} + 181 \beta_{15} + 83 \beta_{14} + 928 \beta_{13} + 69 \beta_{12} + \cdots - 1871 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1138 \beta_{17} + 3747 \beta_{16} - 3048 \beta_{14} - 4847 \beta_{13} + 3048 \beta_{12} + \cdots + 23970 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13443 \beta_{17} - 13443 \beta_{15} - 4790 \beta_{14} - 146873 \beta_{13} - 7447 \beta_{12} + \cdots + 2589 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 228812 \beta_{16} + 79561 \beta_{15} + 228812 \beta_{14} - 79561 \beta_{12} + 186841 \beta_{11} + \cdots - 1129369 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 947642 \beta_{17} + 344156 \beta_{16} + 602755 \beta_{15} + 8497357 \beta_{13} + 947642 \beta_{12} + \cdots + 8497357 \)
|
| \(\nu^{10}\) | \(=\) |
\( 5255332 \beta_{17} + 11095482 \beta_{16} - 11095482 \beta_{15} - 14095383 \beta_{14} + 5679289 \beta_{13} + \cdots + 81042897 \)
|
| \(\nu^{11}\) | \(=\) |
\( 45255747 \beta_{17} - 45255747 \beta_{16} - 24683480 \beta_{15} + 24683480 \beta_{14} - 279865059 \beta_{13} + \cdots - 807719451 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 685386954 \beta_{17} - 340824186 \beta_{16} + 874492833 \beta_{15} + 685386954 \beta_{14} + \cdots - 5659507627 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1745100266 \beta_{17} + 4465217747 \beta_{16} - 3246671489 \beta_{14} + 6959657150 \beta_{13} + \cdots + 40700245245 \)
|
| \(\nu^{14}\) | \(=\) |
\( 54599522482 \beta_{17} - 54599522482 \beta_{15} - 21955425466 \beta_{14} - 138170179438 \beta_{13} + \cdots + 219201306759 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 301854352134 \beta_{16} + 121472271554 \beta_{15} + 301854352134 \beta_{14} - 121472271554 \beta_{12} + \cdots - 1078106322810 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 3428781484341 \beta_{17} + 1411053136154 \beta_{16} + 2690399575859 \beta_{15} + 9158931971631 \beta_{13} + \cdots + 9158931971631 \)
|
| \(\nu^{17}\) | \(=\) |
\( 8341270626933 \beta_{17} + 15501794777815 \beta_{16} - 15501794777815 \beta_{15} - 20256598190062 \beta_{14} + \cdots + 59873154651895 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) |
| \(\chi(n)\) | \(\beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
0.445042 | + | 1.94986i | −4.37285 | − | 2.10585i | −3.60388 | + | 1.73553i | 2.71647 | + | 11.9016i | 2.16001 | − | 9.46361i | −29.8445 | − | 14.3724i | −4.98792 | − | 6.25465i | −2.14706 | − | 2.69233i | −21.9975 | + | 10.5935i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | 0.445042 | + | 1.94986i | −3.43816 | − | 1.65573i | −3.60388 | + | 1.73553i | −3.85753 | − | 16.9009i | 1.69831 | − | 7.44078i | −0.479890 | − | 0.231103i | −4.98792 | − | 6.25465i | −7.75473 | − | 9.72413i | 31.2376 | − | 15.0433i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | 0.445042 | + | 1.94986i | 5.50907 | + | 2.65303i | −3.60388 | + | 1.73553i | 1.12413 | + | 4.92514i | −2.72125 | + | 11.9226i | 5.64568 | + | 2.71881i | −4.98792 | − | 6.25465i | 6.47703 | + | 8.12194i | −9.10303 | + | 4.38379i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.1 | −1.24698 | + | 1.56366i | −1.59387 | + | 6.98319i | −0.890084 | − | 3.89971i | −7.92729 | + | 9.94051i | −8.93184 | − | 11.2002i | 5.00628 | − | 21.9339i | 7.20775 | + | 3.47107i | −21.8984 | − | 10.5457i | −5.65844 | − | 24.7912i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.2 | −1.24698 | + | 1.56366i | 0.0818380 | − | 0.358556i | −0.890084 | − | 3.89971i | 6.93903 | − | 8.70127i | 0.458610 | + | 0.575079i | 2.18340 | − | 9.56612i | 7.20775 | + | 3.47107i | 24.2043 | + | 11.6562i | 4.95303 | + | 21.7006i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.3 | −1.24698 | + | 1.56366i | 0.566988 | − | 2.48414i | −0.890084 | − | 3.89971i | −7.83210 | + | 9.82114i | 3.17733 | + | 3.98424i | −5.44174 | + | 23.8418i | 7.20775 | + | 3.47107i | 18.4767 | + | 8.89791i | −5.59049 | − | 24.4935i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.1 | 0.445042 | − | 1.94986i | −4.37285 | + | 2.10585i | −3.60388 | − | 1.73553i | 2.71647 | − | 11.9016i | 2.16001 | + | 9.46361i | −29.8445 | + | 14.3724i | −4.98792 | + | 6.25465i | −2.14706 | + | 2.69233i | −21.9975 | − | 10.5935i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | 0.445042 | − | 1.94986i | −3.43816 | + | 1.65573i | −3.60388 | − | 1.73553i | −3.85753 | + | 16.9009i | 1.69831 | + | 7.44078i | −0.479890 | + | 0.231103i | −4.98792 | + | 6.25465i | −7.75473 | + | 9.72413i | 31.2376 | + | 15.0433i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.3 | 0.445042 | − | 1.94986i | 5.50907 | − | 2.65303i | −3.60388 | − | 1.73553i | 1.12413 | − | 4.92514i | −2.72125 | − | 11.9226i | 5.64568 | − | 2.71881i | −4.98792 | + | 6.25465i | 6.47703 | − | 8.12194i | −9.10303 | − | 4.38379i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.1 | 1.80194 | + | 0.867767i | −4.40357 | + | 5.52190i | 2.49396 | + | 3.12733i | −5.16532 | − | 2.48749i | −12.7267 | + | 6.12885i | −10.1998 | + | 12.7902i | 1.78017 | + | 7.79942i | −5.09192 | − | 22.3091i | −7.14902 | − | 8.96459i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.2 | 1.80194 | + | 0.867767i | −0.0163817 | + | 0.0205420i | 2.49396 | + | 3.12733i | 8.89395 | + | 4.28310i | −0.0473444 | + | 0.0227998i | 3.52995 | − | 4.42641i | 1.78017 | + | 7.79942i | 6.00791 | + | 26.3224i | 12.3096 | + | 15.4358i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 45.3 | 1.80194 | + | 0.867767i | 5.16693 | − | 6.47913i | 2.49396 | + | 3.12733i | −2.89134 | − | 1.39240i | 14.9329 | − | 7.19129i | 0.100700 | − | 0.126274i | 1.78017 | + | 7.79942i | −9.27385 | − | 40.6314i | −4.00174 | − | 5.01803i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.1 | 1.80194 | − | 0.867767i | −4.40357 | − | 5.52190i | 2.49396 | − | 3.12733i | −5.16532 | + | 2.48749i | −12.7267 | − | 6.12885i | −10.1998 | − | 12.7902i | 1.78017 | − | 7.79942i | −5.09192 | + | 22.3091i | −7.14902 | + | 8.96459i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 1.80194 | − | 0.867767i | −0.0163817 | − | 0.0205420i | 2.49396 | − | 3.12733i | 8.89395 | − | 4.28310i | −0.0473444 | − | 0.0227998i | 3.52995 | + | 4.42641i | 1.78017 | − | 7.79942i | 6.00791 | − | 26.3224i | 12.3096 | − | 15.4358i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 1.80194 | − | 0.867767i | 5.16693 | + | 6.47913i | 2.49396 | − | 3.12733i | −2.89134 | + | 1.39240i | 14.9329 | + | 7.19129i | 0.100700 | + | 0.126274i | 1.78017 | − | 7.79942i | −9.27385 | + | 40.6314i | −4.00174 | + | 5.01803i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.1 | −1.24698 | − | 1.56366i | −1.59387 | − | 6.98319i | −0.890084 | + | 3.89971i | −7.92729 | − | 9.94051i | −8.93184 | + | 11.2002i | 5.00628 | + | 21.9339i | 7.20775 | − | 3.47107i | −21.8984 | + | 10.5457i | −5.65844 | + | 24.7912i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.2 | −1.24698 | − | 1.56366i | 0.0818380 | + | 0.358556i | −0.890084 | + | 3.89971i | 6.93903 | + | 8.70127i | 0.458610 | − | 0.575079i | 2.18340 | + | 9.56612i | 7.20775 | − | 3.47107i | 24.2043 | − | 11.6562i | 4.95303 | − | 21.7006i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.3 | −1.24698 | − | 1.56366i | 0.566988 | + | 2.48414i | −0.890084 | + | 3.89971i | −7.83210 | − | 9.82114i | 3.17733 | − | 3.98424i | −5.44174 | − | 23.8418i | 7.20775 | − | 3.47107i | 18.4767 | − | 8.89791i | −5.59049 | + | 24.4935i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.d | even | 7 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 58.4.d.a | ✓ | 18 |
| 29.d | even | 7 | 1 | inner | 58.4.d.a | ✓ | 18 |
| 29.d | even | 7 | 1 | 1682.4.a.o | 9 | ||
| 29.e | even | 14 | 1 | 1682.4.a.p | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.4.d.a | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 58.4.d.a | ✓ | 18 | 29.d | even | 7 | 1 | inner |
| 1682.4.a.o | 9 | 29.d | even | 7 | 1 | ||
| 1682.4.a.p | 9 | 29.e | even | 14 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{18} + 5 T_{3}^{17} + 44 T_{3}^{16} + 154 T_{3}^{15} + 3447 T_{3}^{14} + 25285 T_{3}^{13} + \cdots + 1366561 \)
acting on \(S_{4}^{\mathrm{new}}(58, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 64)^{3} \)
|
| $3$ |
\( T^{18} + 5 T^{17} + \cdots + 1366561 \)
|
| $5$ |
\( T^{18} + \cdots + 11\!\cdots\!81 \)
|
| $7$ |
\( T^{18} + \cdots + 79715094445561 \)
|
| $11$ |
\( T^{18} + \cdots + 32\!\cdots\!61 \)
|
| $13$ |
\( T^{18} + \cdots + 17\!\cdots\!01 \)
|
| $17$ |
\( (T^{9} + \cdots - 142668951628288)^{2} \)
|
| $19$ |
\( T^{18} + \cdots + 18\!\cdots\!29 \)
|
| $23$ |
\( T^{18} + \cdots + 12\!\cdots\!69 \)
|
| $29$ |
\( T^{18} + \cdots + 30\!\cdots\!09 \)
|
| $31$ |
\( T^{18} + \cdots + 51\!\cdots\!29 \)
|
| $37$ |
\( T^{18} + \cdots + 18\!\cdots\!29 \)
|
| $41$ |
\( (T^{9} + \cdots + 14\!\cdots\!88)^{2} \)
|
| $43$ |
\( T^{18} + \cdots + 23\!\cdots\!09 \)
|
| $47$ |
\( T^{18} + \cdots + 95\!\cdots\!69 \)
|
| $53$ |
\( T^{18} + \cdots + 48\!\cdots\!89 \)
|
| $59$ |
\( (T^{9} + \cdots - 63\!\cdots\!32)^{2} \)
|
| $61$ |
\( T^{18} + \cdots + 12\!\cdots\!49 \)
|
| $67$ |
\( T^{18} + \cdots + 46\!\cdots\!81 \)
|
| $71$ |
\( T^{18} + \cdots + 31\!\cdots\!61 \)
|
| $73$ |
\( T^{18} + \cdots + 16\!\cdots\!89 \)
|
| $79$ |
\( T^{18} + \cdots + 15\!\cdots\!76 \)
|
| $83$ |
\( T^{18} + \cdots + 27\!\cdots\!01 \)
|
| $89$ |
\( T^{18} + \cdots + 10\!\cdots\!41 \)
|
| $97$ |
\( T^{18} + \cdots + 13\!\cdots\!04 \)
|
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