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Properties

Label 1375.4.a.c
Level $1375$
Weight $4$
Character orbit 1375.a
Self dual yes
Analytic conductor $81.128$
Analytic rank $1$
Dimension $30$
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] = [1375,4,Mod(1,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, 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("1375.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1375.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: \(81.1276262579\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 15 q^{2} - 24 q^{3} + 97 q^{4} + 13 q^{6} - 26 q^{7} - 156 q^{8} + 256 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 15 q^{2} - 24 q^{3} + 97 q^{4} + 13 q^{6} - 26 q^{7} - 156 q^{8} + 256 q^{9} + 330 q^{11} - 273 q^{12} - 96 q^{13} + 12 q^{14} + 409 q^{16} - 554 q^{17} - 574 q^{18} + 30 q^{19} + 271 q^{21} - 165 q^{22} - 844 q^{23} + 18 q^{24} - 84 q^{26} - 945 q^{27} - 727 q^{28} - 54 q^{29} - 44 q^{31} - 1183 q^{32} - 264 q^{33} + 47 q^{34} - 42 q^{36} - 896 q^{37} + 525 q^{38} + 208 q^{39} + 108 q^{41} - 703 q^{42} - 522 q^{43} + 1067 q^{44} + 1758 q^{46} - 1502 q^{47} + 556 q^{48} + 740 q^{49} + 185 q^{51} - 494 q^{52} - 2526 q^{53} + 2106 q^{54} - 822 q^{56} - 2355 q^{57} - 1933 q^{58} + 798 q^{59} - 1390 q^{61} - 829 q^{62} - 1749 q^{63} + 2778 q^{64} + 143 q^{66} - 4849 q^{67} - 2500 q^{68} - 2784 q^{69} + 206 q^{71} - 1830 q^{72} - 2571 q^{73} - 1482 q^{74} + 2455 q^{76} - 286 q^{77} - 3799 q^{78} + 3102 q^{79} + 3370 q^{81} - 1112 q^{82} - 1922 q^{83} + 5311 q^{84} - 1751 q^{86} - 3953 q^{87} - 1716 q^{88} + 74 q^{89} + 652 q^{91} - 6763 q^{92} - 4363 q^{93} - 820 q^{94} + 73 q^{96} - 4676 q^{97} - 3560 q^{98} + 2816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.54431 5.24682 22.7394 0 −29.0900 16.0991 −81.7197 0.529155 0
1.2 −5.43411 −6.31756 21.5295 0 34.3303 −2.80959 −73.5207 12.9115 0
1.3 −5.20301 −3.10505 19.0713 0 16.1556 2.41861 −57.6042 −17.3587 0
1.4 −4.59963 6.92716 13.1566 0 −31.8624 −14.7308 −23.7186 20.9855 0
1.5 −4.54633 −8.94046 12.6691 0 40.6463 −24.0321 −21.2274 52.9318 0
1.6 −4.12420 0.850012 9.00903 0 −3.50562 −28.7441 −4.16145 −26.2775 0
1.7 −4.00019 −9.43951 8.00151 0 37.7598 −18.2370 −0.00605241 62.1043 0
1.8 −3.44471 6.59933 3.86601 0 −22.7327 14.2142 14.2404 16.5511 0
1.9 −3.06009 −2.55332 1.36417 0 7.81339 29.3542 20.3063 −20.4806 0
1.10 −2.64204 −1.03210 −1.01961 0 2.72685 6.51087 23.8302 −25.9348 0
1.11 −2.60495 3.52678 −1.21426 0 −9.18707 19.1563 24.0026 −14.5618 0
1.12 −2.10856 −10.0676 −3.55399 0 21.2280 33.3106 24.3622 74.3559 0
1.13 −1.60678 7.89057 −5.41825 0 −12.6784 −25.0315 21.5602 35.2611 0
1.14 −1.30148 −7.84504 −6.30616 0 10.2101 3.73025 18.6191 34.5447 0
1.15 −1.11372 −3.91992 −6.75963 0 4.36568 −36.3645 16.4381 −11.6343 0
1.16 −0.0603385 5.22812 −7.99636 0 −0.315457 −10.1593 0.965197 0.333277 0
1.17 0.185722 9.75891 −7.96551 0 1.81245 13.6636 −2.96515 68.2362 0
1.18 0.537789 −0.152810 −7.71078 0 −0.0821796 22.1147 −8.44909 −26.9766 0
1.19 0.793575 −3.93610 −7.37024 0 −3.12359 −24.9174 −12.1974 −11.5071 0
1.20 1.15341 2.20268 −6.66965 0 2.54058 −3.27601 −16.9201 −22.1482 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)

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 1375.4.a.c ✓ 30
5.b even 2 1 1375.4.a.f yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.4.a.c ✓ 30 1.a even 1 1 trivial
1375.4.a.f yes 30 5.b even 2 1