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Properties

Label 1175.4.a.g
Level $1175$
Weight $4$
Character orbit 1175.a
Self dual yes
Analytic conductor $69.327$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 100 x^{16} + 101 x^{15} + 4071 x^{14} - 4087 x^{13} - 87059 x^{12} + 85913 x^{11} + \cdots + 2210048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 3) q^{4} + (\beta_{11} - 1) q^{6} + ( - \beta_{15} + \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{8} + (\beta_{9} - \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 3) q^{4} + (\beta_{11} - 1) q^{6} + ( - \beta_{15} + \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{8} + (\beta_{9} - \beta_{2} + 3) q^{9} + ( - \beta_{10} - \beta_{4} - \beta_{2} + \cdots - 1) q^{11}+ \cdots + ( - 3 \beta_{17} - 12 \beta_{16} + \cdots - 71) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} - 7 q^{3} + 57 q^{4} - 14 q^{6} - 6 q^{7} + 18 q^{8} + 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - q^{2} - 7 q^{3} + 57 q^{4} - 14 q^{6} - 6 q^{7} + 18 q^{8} + 57 q^{9} - 31 q^{11} + 31 q^{12} + 16 q^{13} - 229 q^{14} + 41 q^{16} - 5 q^{17} - 109 q^{18} - 283 q^{19} - 338 q^{21} + 214 q^{22} - 160 q^{23} - 324 q^{24} - 197 q^{26} + 11 q^{27} - 264 q^{28} - 464 q^{29} - 390 q^{31} + 168 q^{32} + 237 q^{33} - 824 q^{34} - 621 q^{36} + 362 q^{37} - 19 q^{38} - 378 q^{39} - 591 q^{41} - 79 q^{42} + 54 q^{43} - 859 q^{44} - 1164 q^{46} + 846 q^{47} + 484 q^{48} - 662 q^{49} - 1796 q^{51} - 502 q^{52} + 260 q^{53} - 62 q^{54} - 1261 q^{56} - 759 q^{57} + 805 q^{58} - 44 q^{59} - 1754 q^{61} - 1168 q^{62} + 2133 q^{63} - 1304 q^{64} - 2051 q^{66} - 1593 q^{67} + 1418 q^{68} - 1536 q^{69} - 2214 q^{71} - 1890 q^{72} + 1831 q^{73} + 477 q^{74} - 4034 q^{76} - 1406 q^{77} + 2903 q^{78} - 2268 q^{79} - 2610 q^{81} - 2299 q^{82} + 1794 q^{83} - 611 q^{84} - 3005 q^{86} - 2296 q^{87} + 2793 q^{88} - 1505 q^{89} - 3626 q^{91} - 2379 q^{92} + 1770 q^{93} - 47 q^{94} - 5976 q^{96} - 2436 q^{97} + 3428 q^{98} - 1456 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 100 x^{16} + 101 x^{15} + 4071 x^{14} - 4087 x^{13} - 87059 x^{12} + 85913 x^{11} + \cdots + 2210048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36\!\cdots\!07 \nu^{17} + \cdots - 51\!\cdots\!28 ) / 98\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36\!\cdots\!07 \nu^{17} + \cdots - 61\!\cdots\!76 ) / 98\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 67\!\cdots\!11 \nu^{17} + \cdots - 39\!\cdots\!96 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!57 \nu^{17} + \cdots + 76\!\cdots\!32 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!23 \nu^{17} + \cdots + 63\!\cdots\!72 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24\!\cdots\!07 \nu^{17} + \cdots - 47\!\cdots\!28 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 94\!\cdots\!49 \nu^{17} + \cdots - 10\!\cdots\!84 ) / 16\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!39 \nu^{17} + \cdots + 97\!\cdots\!16 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 88\!\cdots\!13 \nu^{17} + \cdots + 17\!\cdots\!84 ) / 98\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 36\!\cdots\!09 \nu^{17} + \cdots - 11\!\cdots\!56 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!27 \nu^{17} + \cdots - 60\!\cdots\!32 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 84\!\cdots\!49 \nu^{17} + \cdots - 25\!\cdots\!56 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 96\!\cdots\!59 \nu^{17} + \cdots - 23\!\cdots\!16 ) / 49\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 56\!\cdots\!71 \nu^{17} + \cdots - 23\!\cdots\!04 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!37 \nu^{17} + \cdots + 15\!\cdots\!48 ) / 49\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + 18\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 2\beta_{11} + \beta_{10} - \beta_{7} - \beta_{5} - \beta_{3} + 25\beta_{2} - \beta _1 + 203 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 3 \beta_{12} + \beta_{11} - \beta_{9} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{17} + 46 \beta_{15} - 10 \beta_{14} - 4 \beta_{13} - 10 \beta_{12} - 78 \beta_{11} + \cdots + 4342 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 46 \beta_{17} - 38 \beta_{16} - 109 \beta_{15} + 100 \beta_{14} + 8 \beta_{13} + 150 \beta_{12} + \cdots - 1537 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 325 \beta_{17} + 57 \beta_{16} + 1467 \beta_{15} - 518 \beta_{14} - 200 \beta_{13} - 551 \beta_{12} + \cdots + 99090 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1671 \beta_{17} - 1169 \beta_{16} - 4021 \beta_{15} + 3660 \beta_{14} + 476 \beta_{13} + 5381 \beta_{12} + \cdots - 56269 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12089 \beta_{17} + 3859 \beta_{16} + 41625 \beta_{15} - 18992 \beta_{14} - 6964 \beta_{13} + \cdots + 2338431 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 54989 \beta_{17} - 34603 \beta_{16} - 128472 \beta_{15} + 118356 \beta_{14} + 18780 \beta_{13} + \cdots - 1900531 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 388172 \beta_{17} + 171778 \beta_{16} + 1130049 \beta_{15} - 610810 \beta_{14} - 211284 \beta_{13} + \cdots + 56314924 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1704409 \beta_{17} - 1024613 \beta_{16} - 3842901 \beta_{15} + 3594422 \beta_{14} + 624088 \beta_{13} + \cdots - 60716190 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11600761 \beta_{17} + 6373443 \beta_{16} + 30165800 \beta_{15} - 18444384 \beta_{14} + \cdots + 1375433973 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 50781100 \beta_{17} - 30500562 \beta_{16} - 111076196 \beta_{15} + 105318698 \beta_{14} + \cdots - 1864718574 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 333427900 \beta_{17} + 214059640 \beta_{16} + 800808064 \beta_{15} - 538094924 \beta_{14} + \cdots + 33960520367 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1473169424 \beta_{17} - 909159928 \beta_{16} - 3147930032 \beta_{15} + 3018514488 \beta_{14} + \cdots - 55640931349 \) 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
4.87025
4.76474
4.63565
3.49356
3.07949
2.59243
1.76747
0.946591
0.755985
0.129263
−0.795026
−2.23112
−2.53289
−2.64006
−3.61072
−4.14408
−4.88747
−5.19406
−4.87025 8.22936 15.7193 0 −40.0790 1.99204 −37.5949 40.7223 0
1.2 −4.76474 −0.390987 14.7027 0 1.86295 −8.47898 −31.9366 −26.8471 0
1.3 −4.63565 −3.15199 13.4892 0 14.6115 13.2882 −25.4461 −17.0650 0
1.4 −3.49356 −2.44254 4.20494 0 8.53316 1.79210 13.2582 −21.0340 0
1.5 −3.07949 −9.26749 1.48328 0 28.5392 25.7737 20.0682 58.8864 0
1.6 −2.59243 4.60669 −1.27932 0 −11.9425 6.49156 24.0560 −5.77839 0
1.7 −1.76747 −5.12284 −4.87605 0 9.05446 −23.6786 22.7580 −0.756497 0
1.8 −0.946591 7.44306 −7.10397 0 −7.04554 19.3637 14.2973 28.3992 0
1.9 −0.755985 2.75360 −7.42849 0 −2.08168 −27.3808 11.6637 −19.4177 0
1.10 −0.129263 1.21829 −7.98329 0 −0.157480 6.32492 2.06605 −25.5158 0
1.11 0.795026 −7.92555 −7.36793 0 −6.30101 17.9312 −12.2179 35.8143 0
1.12 2.23112 −6.29796 −3.02211 0 −14.0515 −21.3404 −24.5916 12.6643 0
1.13 2.53289 7.27626 −1.58448 0 18.4300 −23.1663 −24.2764 25.9440 0
1.14 2.64006 −5.67970 −1.03011 0 −14.9947 21.3444 −23.8400 5.25897 0
1.15 3.61072 1.69197 5.03733 0 6.10926 22.0006 −10.6974 −24.1372 0
1.16 4.14408 6.21915 9.17336 0 25.7726 −13.4083 4.86250 11.6778 0
1.17 4.88747 −5.65024 15.8873 0 −27.6154 −5.98904 38.5490 4.92526 0
1.18 5.19406 −0.509093 18.9783 0 −2.64426 −18.8601 57.0219 −26.7408 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.g ✓ 18
5.b even 2 1 1175.4.a.h yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1175.4.a.g ✓ 18 1.a even 1 1 trivial
1175.4.a.h yes 18 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + T_{2}^{17} - 100 T_{2}^{16} - 101 T_{2}^{15} + 4071 T_{2}^{14} + 4087 T_{2}^{13} + \cdots + 2210048 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} + \cdots + 2210048 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 8446424341 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 10\!\cdots\!85 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 82\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 28\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 50\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 20\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T - 47)^{18} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 86\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 25\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 55\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 68\!\cdots\!47 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 77\!\cdots\!50 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 40\!\cdots\!80 \) Copy content Toggle raw display
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