↑

Properties

Label 1175.4.a.f
Level $1175$
Weight $4$
Character orbit 1175.a
Self dual yes
Analytic conductor $69.327$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1175,4,Mod(1,1175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1175 = 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1175.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: \(69.3272442567\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 7 x^{14} - 70 x^{13} + 546 x^{12} + 1677 x^{11} - 15964 x^{10} - 14393 x^{9} + 219060 x^{8} + \cdots + 208512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 235)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{4} - 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{13} + \beta_{4} + \beta_1 + 2) q^{6} + ( - \beta_{5} - \beta_{4} + \beta_1 - 2) q^{7} + ( - \beta_{3} - 5 \beta_1 - 2) q^{8} + ( - \beta_{6} - \beta_{5} + \beta_1 + 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{4} - 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{13} + \beta_{4} + \beta_1 + 2) q^{6} + ( - \beta_{5} - \beta_{4} + \beta_1 - 2) q^{7} + ( - \beta_{3} - 5 \beta_1 - 2) q^{8} + ( - \beta_{6} - \beta_{5} + \beta_1 + 15) q^{9} + ( - \beta_{13} - \beta_{7} - \beta_{3} + \cdots + 7) q^{11}+ \cdots + (10 \beta_{14} - 19 \beta_{13} + \cdots + 168) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 7 q^{2} - 8 q^{3} + 69 q^{4} + 30 q^{6} - 13 q^{7} - 63 q^{8} + 239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 7 q^{2} - 8 q^{3} + 69 q^{4} + 30 q^{6} - 13 q^{7} - 63 q^{8} + 239 q^{9} + 87 q^{11} - 73 q^{12} - 132 q^{13} - 166 q^{14} + 349 q^{16} - 67 q^{17} - 236 q^{18} + 119 q^{19} + 386 q^{21} + 266 q^{22} - 155 q^{23} + 161 q^{24} + 681 q^{26} - 152 q^{27} + 5 q^{28} + 955 q^{29} + 388 q^{31} - 356 q^{32} + 10 q^{33} - 160 q^{34} + 1671 q^{36} - 8 q^{37} - 568 q^{38} + 294 q^{39} + 1206 q^{41} + 370 q^{42} - 146 q^{43} + 1074 q^{44} + 1513 q^{46} - 705 q^{47} - 898 q^{48} + 2220 q^{49} - 277 q^{51} - 1663 q^{52} - 300 q^{53} + 1722 q^{54} - 1121 q^{56} - 621 q^{57} + 1059 q^{58} + 1273 q^{59} + 821 q^{61} + 331 q^{62} + 3348 q^{63} + 533 q^{64} - 6255 q^{66} + 836 q^{67} + 1123 q^{68} + 1272 q^{69} - 978 q^{71} + 2215 q^{72} - 337 q^{73} - 1159 q^{74} - 3254 q^{76} - 1160 q^{77} + 7257 q^{78} + 1076 q^{79} + 379 q^{81} + 100 q^{82} + 2195 q^{83} - 4616 q^{84} - 3084 q^{86} - 923 q^{87} + 9210 q^{88} + 2330 q^{89} - 3566 q^{91} + 1891 q^{92} + 2902 q^{93} + 329 q^{94} - 3442 q^{96} + 1897 q^{97} - 2422 q^{98} + 2550 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 7 x^{14} - 70 x^{13} + 546 x^{12} + 1677 x^{11} - 15964 x^{10} - 14393 x^{9} + 219060 x^{8} + \cdots + 208512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 21\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 820523201438519 \nu^{14} + \cdots - 38\!\cdots\!24 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!53 \nu^{14} + \cdots - 67\!\cdots\!36 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!89 \nu^{14} + \cdots + 13\!\cdots\!20 ) / 35\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!73 \nu^{14} + \cdots + 80\!\cdots\!00 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55\!\cdots\!55 \nu^{14} + \cdots + 15\!\cdots\!04 ) / 71\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!85 \nu^{14} + \cdots - 11\!\cdots\!36 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\!\cdots\!49 \nu^{14} + \cdots + 21\!\cdots\!08 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!85 \nu^{14} + \cdots - 41\!\cdots\!04 ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!13 \nu^{14} + \cdots - 66\!\cdots\!84 ) / 95\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!11 \nu^{14} + \cdots + 15\!\cdots\!12 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26\!\cdots\!31 \nu^{14} + \cdots - 39\!\cdots\!64 ) / 14\!\cdots\!68 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 21\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} + \beta_{13} - 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \cdots + 269 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{14} + 2 \beta_{13} + 4 \beta_{12} - \beta_{11} - 4 \beta_{9} - 3 \beta_{8} - 4 \beta_{6} + \cdots + 69 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 80 \beta_{14} + 28 \beta_{13} + 16 \beta_{12} - 85 \beta_{11} + 40 \beta_{10} - 51 \beta_{9} + \cdots + 6319 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 131 \beta_{14} + 89 \beta_{13} + 168 \beta_{12} - 26 \beta_{11} + 51 \beta_{10} - 177 \beta_{9} + \cdots + 1703 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2459 \beta_{14} + 509 \beta_{13} + 837 \beta_{12} - 2789 \beta_{11} + 1364 \beta_{10} - 1821 \beta_{9} + \cdots + 157352 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4205 \beta_{14} + 2882 \beta_{13} + 5440 \beta_{12} - 156 \beta_{11} + 3284 \beta_{10} - 5778 \beta_{9} + \cdots + 40429 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 69618 \beta_{14} + 4515 \beta_{13} + 31621 \beta_{12} - 83998 \beta_{11} + 44432 \beta_{10} + \cdots + 4039347 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 121595 \beta_{14} + 81435 \beta_{13} + 162464 \beta_{12} + 16461 \beta_{11} + 144809 \beta_{10} + \cdots + 1010067 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1913556 \beta_{14} - 144606 \beta_{13} + 1055301 \beta_{12} - 2438357 \beta_{11} + 1413531 \beta_{10} + \cdots + 105551199 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3381302 \beta_{14} + 2123162 \beta_{13} + 4697101 \beta_{12} + 1033055 \beta_{11} + 5460147 \beta_{10} + \cdots + 27027996 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 52109758 \beta_{14} - 11010655 \beta_{13} + 33107073 \beta_{12} - 69510145 \beta_{11} + \cdots + 2790654953 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
5.35940
5.06114
4.78431
3.80493
3.67516
1.81393
1.53261
0.781713
−0.107843
−0.738362
−2.82702
−3.04816
−3.08416
−4.77053
−5.23712
−5.35940 4.70137 20.7231 0 −25.1965 −13.4584 −68.1884 −4.89715 0
1.2 −5.06114 −8.93256 17.6152 0 45.2090 32.9626 −48.6638 52.7906 0
1.3 −4.78431 −4.10756 14.8896 0 19.6519 −11.7314 −32.9620 −10.1279 0
1.4 −3.80493 9.60502 6.47747 0 −36.5464 29.2757 5.79313 65.2563 0
1.5 −3.67516 −5.91708 5.50681 0 21.7462 −8.40558 9.16289 8.01183 0
1.6 −1.81393 2.90596 −4.70967 0 −5.27119 −12.0868 23.0544 −18.5554 0
1.7 −1.53261 −9.68379 −5.65110 0 14.8415 −25.8463 20.9218 66.7758 0
1.8 −0.781713 3.08901 −7.38893 0 −2.41472 −17.6792 12.0297 −17.4580 0
1.9 0.107843 7.26532 −7.98837 0 0.783517 35.0393 −1.72424 25.7849 0
1.10 0.738362 −7.39642 −7.45482 0 −5.46123 19.9224 −11.4113 27.7070 0
1.11 2.82702 −0.142923 −0.00794605 0 −0.404045 22.4205 −22.6386 −26.9796 0
1.12 3.04816 −6.06684 1.29126 0 −18.4927 −29.9736 −20.4493 9.80656 0
1.13 3.08416 4.29788 1.51206 0 13.2554 −29.1690 −20.0099 −8.52823 0
1.14 4.77053 8.95581 14.7579 0 42.7239 0.996696 32.2388 53.2065 0
1.15 5.23712 −6.57319 19.4274 0 −34.4246 −5.26705 59.8466 16.2068 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(47\) \(1\)

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 1175.4.a.f 15
5.b even 2 1 235.4.a.d ✓ 15
15.d odd 2 1 2115.4.a.o 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
235.4.a.d ✓ 15 5.b even 2 1
1175.4.a.f 15 1.a even 1 1 trivial
2115.4.a.o 15 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 7 T_{2}^{14} - 70 T_{2}^{13} - 546 T_{2}^{12} + 1677 T_{2}^{11} + 15964 T_{2}^{10} + \cdots - 208512 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 7 T^{14} + \cdots - 208512 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 10046592848 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 66\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 48\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 46\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 48\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 80\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T + 47)^{15} \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 26\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 48\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 56\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 82\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
show more
show less