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Properties

Label 2075.4.a.e
Level $2075$
Weight $4$
Character orbit 2075.a
Self dual yes
Analytic conductor $122.429$
Analytic rank $1$
Dimension $15$
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] = [2075,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2075.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2075 = 5^{2} \cdot 83 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(122.428963262\)
Analytic rank: \(1\)
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: \( x^{15} - 3 x^{14} - 69 x^{13} + 180 x^{12} + 1884 x^{11} - 4048 x^{10} - 25928 x^{9} + 42526 x^{8} + \cdots + 130944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 415)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} + 1) q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{12} + 2 \beta_1 - 1) q^{6} + (\beta_{10} + \beta_{5} - 2 \beta_1 + 3) q^{7} + (\beta_{7} - \beta_{6} + \beta_{2} + 5) q^{8} + (\beta_{13} + \beta_{10} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} + 1) q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{12} + 2 \beta_1 - 1) q^{6} + (\beta_{10} + \beta_{5} - 2 \beta_1 + 3) q^{7} + (\beta_{7} - \beta_{6} + \beta_{2} + 5) q^{8} + (\beta_{13} + \beta_{10} + \cdots - \beta_1) q^{9}+ \cdots + (7 \beta_{14} + 7 \beta_{13} + \cdots + 284) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 3 q^{2} + 12 q^{3} + 27 q^{4} - 11 q^{6} + 37 q^{7} + 60 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 3 q^{2} + 12 q^{3} + 27 q^{4} - 11 q^{6} + 37 q^{7} + 60 q^{8} - 9 q^{9} - 108 q^{11} + 146 q^{12} + 63 q^{13} - 247 q^{14} - 177 q^{16} + 357 q^{17} - 27 q^{18} - 237 q^{19} - 327 q^{21} + 38 q^{22} + 28 q^{23} + 164 q^{24} - 41 q^{26} + 171 q^{27} - 31 q^{28} - 632 q^{29} - 279 q^{31} - 127 q^{32} + 423 q^{33} - 623 q^{34} - 642 q^{36} + 162 q^{37} - 90 q^{38} - 700 q^{39} - 1150 q^{41} - 434 q^{42} + 165 q^{43} - 610 q^{44} - 53 q^{46} + 424 q^{47} + 359 q^{48} - 978 q^{49} - 707 q^{51} + 604 q^{52} + 1815 q^{53} - 453 q^{54} - 498 q^{56} - 132 q^{57} + 767 q^{58} - 2482 q^{59} - 1698 q^{61} + 1279 q^{62} + 1122 q^{63} - 2310 q^{64} - 2964 q^{66} - 71 q^{67} + 860 q^{68} - 2550 q^{69} - 2270 q^{71} + 836 q^{72} + 1496 q^{73} - 1504 q^{74} - 2842 q^{76} + 1189 q^{77} + 97 q^{78} - 1528 q^{79} - 3889 q^{81} - 389 q^{82} + 1245 q^{83} - 1049 q^{84} - 2312 q^{86} - 1321 q^{87} + 1078 q^{88} - 4212 q^{89} - 2856 q^{91} - 1405 q^{92} - 518 q^{93} - 273 q^{94} + 330 q^{96} + 4066 q^{97} + 130 q^{98} + 3367 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 69 x^{13} + 180 x^{12} + 1884 x^{11} - 4048 x^{10} - 25928 x^{9} + 42526 x^{8} + \cdots + 130944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 111273717 \nu^{14} - 488303868 \nu^{13} - 6796366317 \nu^{12} + 28937413401 \nu^{11} + \cdots + 1536110297472 ) / 153149179904 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 82844383 \nu^{14} + 446029936 \nu^{13} + 4739266011 \nu^{12} - 26468149627 \nu^{11} + \cdots - 4106448953984 ) / 76574589952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 185361667 \nu^{14} + 953570980 \nu^{13} + 10727065611 \nu^{12} - 56410231679 \nu^{11} + \cdots - 13010325371520 ) / 153149179904 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24519765 \nu^{14} - 112303694 \nu^{13} - 1470783347 \nu^{12} + 6600657793 \nu^{11} + \cdots + 1178522225024 ) / 19143647488 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24519765 \nu^{14} - 112303694 \nu^{13} - 1470783347 \nu^{12} + 6600657793 \nu^{11} + \cdots + 1274240462464 ) / 19143647488 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 281075601 \nu^{14} - 1436660340 \nu^{13} - 16352110753 \nu^{12} + 85198182917 \nu^{11} + \cdots + 20093875818368 ) / 153149179904 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41122475 \nu^{14} + 231957692 \nu^{13} + 2321454107 \nu^{12} - 13839585767 \nu^{11} + \cdots - 3719123716736 ) / 21878454272 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 339855205 \nu^{14} - 1659916180 \nu^{13} - 19994684197 \nu^{12} + 98142011305 \nu^{11} + \cdots + 19834480418176 ) / 153149179904 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 391867673 \nu^{14} + 1977936924 \nu^{13} + 22772849025 \nu^{12} - 116988888013 \nu^{11} + \cdots - 23170833256320 ) / 153149179904 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 397485979 \nu^{14} + 2062889412 \nu^{13} + 23045131619 \nu^{12} - 122295894071 \nu^{11} + \cdots - 24118848943744 ) / 153149179904 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 215382243 \nu^{14} - 1055867668 \nu^{13} - 12734503835 \nu^{12} + 62409180095 \nu^{11} + \cdots + 10262244986496 ) / 76574589952 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 501187213 \nu^{14} - 2448361100 \nu^{13} - 29573236629 \nu^{12} + 144825372945 \nu^{11} + \cdots + 24006685787520 ) / 153149179904 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{2} + 16\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + \beta_{12} - 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots + 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} + 27 \beta_{7} - 24 \beta_{6} + \cdots + 150 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 29 \beta_{14} + 6 \beta_{13} + 41 \beta_{12} - 2 \beta_{11} - 62 \beta_{9} - 64 \beta_{8} + \cdots + 3005 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 96 \beta_{14} + 86 \beta_{13} + 144 \beta_{12} - 74 \beta_{11} - 28 \beta_{10} - 76 \beta_{9} + \cdots + 3726 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 660 \beta_{14} + 316 \beta_{13} + 1284 \beta_{12} - 108 \beta_{11} - 44 \beta_{10} - 1524 \beta_{9} + \cdots + 58906 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2368 \beta_{14} + 2800 \beta_{13} + 4896 \beta_{12} - 2084 \beta_{11} - 1296 \beta_{10} + \cdots + 88513 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14041 \beta_{14} + 11292 \beta_{13} + 35925 \beta_{12} - 4156 \beta_{11} - 2660 \beta_{10} + \cdots + 1185434 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 53843 \beta_{14} + 80882 \beta_{13} + 144315 \beta_{12} - 53870 \beta_{11} - 42144 \beta_{10} + \cdots + 2063298 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 293309 \beta_{14} + 342882 \beta_{13} + 945157 \beta_{12} - 136350 \beta_{11} - 103332 \beta_{10} + \cdots + 24207917 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1188448 \beta_{14} + 2183638 \beta_{13} + 3937720 \beta_{12} - 1348614 \beta_{11} - 1191644 \beta_{10} + \cdots + 47536642 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6122812 \beta_{14} + 9541368 \beta_{13} + 23940680 \beta_{12} - 4069192 \beta_{11} - 3311856 \beta_{10} + \cdots + 499147226 \) 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
−4.42293
−4.31309
−3.30810
−2.32602
−2.12604
−2.09699
−0.642633
0.291029
0.951168
1.32876
2.94141
3.03464
4.37185
4.56103
4.75592
−4.42293 1.84865 11.5623 0 −8.17645 17.1863 −15.7559 −23.5825 0
1.2 −4.31309 5.93961 10.6027 0 −25.6180 3.27770 −11.2257 8.27891 0
1.3 −3.30810 −0.698412 2.94350 0 2.31041 11.6104 16.7274 −26.5122 0
1.4 −2.32602 −6.49379 −2.58964 0 15.1047 35.5767 24.6317 15.1692 0
1.5 −2.12604 8.82895 −3.47996 0 −18.7707 1.95854 24.4068 50.9504 0
1.6 −2.09699 −3.42970 −3.60261 0 7.19206 −12.8621 24.3306 −15.2372 0
1.7 −0.642633 2.36505 −7.58702 0 −1.51986 −20.5806 10.0167 −21.4065 0
1.8 0.291029 7.17772 −7.91530 0 2.08893 4.37041 −4.63182 24.5196 0
1.9 0.951168 −7.05585 −7.09528 0 −6.71130 16.7814 −14.3581 22.7851 0
1.10 1.32876 −2.28455 −6.23440 0 −3.03562 −1.10114 −18.9141 −21.7808 0
1.11 2.94141 5.30074 0.651885 0 15.5916 14.3744 −21.6138 1.09780 0
1.12 3.03464 −4.09213 1.20905 0 −12.4182 −20.6088 −20.6081 −10.2545 0
1.13 4.37185 −4.72022 11.1130 0 −20.6361 14.9399 13.6097 −4.71955 0
1.14 4.56103 3.58050 12.8029 0 16.3308 −1.28818 21.9064 −14.1800 0
1.15 4.75592 5.73343 14.6188 0 27.2677 −26.6349 31.4783 5.87220 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
\(5\) \( +1 \)
\(83\) \( -1 \)

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 2075.4.a.e 15
5.b even 2 1 415.4.a.a ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.4.a.a ✓ 15 5.b even 2 1
2075.4.a.e 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 3 T_{2}^{14} - 69 T_{2}^{13} + 180 T_{2}^{12} + 1884 T_{2}^{11} - 4048 T_{2}^{10} + \cdots + 130944 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2075))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 3 T^{14} + \cdots + 130944 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 867300688 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 147937839975360 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 87\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 10\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 23\!\cdots\!40 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 15\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 88\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 61\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 48\!\cdots\!15 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 87\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 82\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 39\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( (T - 83)^{15} \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
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