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Properties

Label 2075.4.a.m
Level $2075$
Weight $4$
Character orbit 2075.a
Self dual yes
Analytic conductor $122.429$
Analytic rank $1$
Dimension $62$
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: \(62\)
Twist minimal: no (minimal twist has level 415)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 62 q - 10 q^{2} - 18 q^{3} + 252 q^{4} - 98 q^{7} - 120 q^{8} + 544 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 62 q - 10 q^{2} - 18 q^{3} + 252 q^{4} - 98 q^{7} - 120 q^{8} + 544 q^{9} + 30 q^{11} - 216 q^{12} - 442 q^{13} - 62 q^{14} + 1040 q^{16} - 136 q^{17} - 450 q^{18} - 62 q^{19} + 12 q^{21} - 921 q^{22} - 322 q^{23} - 184 q^{24} - 88 q^{26} - 1656 q^{27} - 1176 q^{28} - 104 q^{29} + 186 q^{31} - 2395 q^{32} - 1056 q^{33} - 248 q^{34} + 2296 q^{36} - 3088 q^{37} - 608 q^{38} + 296 q^{39} - 192 q^{41} - 2894 q^{42} - 1462 q^{43} + 318 q^{44} - 608 q^{46} - 1228 q^{47} - 2016 q^{48} + 3252 q^{49} - 348 q^{51} - 5241 q^{52} - 2650 q^{53} - 4 q^{54} - 664 q^{56} - 2696 q^{57} - 3132 q^{58} + 534 q^{59} + 476 q^{61} - 2040 q^{62} - 2646 q^{63} + 4590 q^{64} - 142 q^{66} - 4582 q^{67} - 1360 q^{68} - 1748 q^{69} - 1488 q^{71} - 6148 q^{72} - 3796 q^{73} + 866 q^{74} + 402 q^{76} - 4020 q^{77} - 3744 q^{78} - 540 q^{79} + 4350 q^{81} - 3653 q^{82} - 5146 q^{83} - 3376 q^{84} - 1346 q^{86} - 2332 q^{87} - 9832 q^{88} - 416 q^{89} + 732 q^{91} - 3468 q^{92} - 11318 q^{93} - 1310 q^{94} + 2946 q^{96} - 6372 q^{97} - 7174 q^{98} - 1334 q^{99}+O(q^{100}) \) 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 −5.59070 −9.32458 23.2559 0 52.1309 14.9769 −85.2909 59.9478 0
1.2 −5.58218 4.95572 23.1608 0 −27.6637 −27.5915 −84.6302 −2.44088 0
1.3 −5.43165 5.71216 21.5028 0 −31.0264 16.9337 −73.3426 5.62872 0
1.4 −5.32811 −1.15648 20.3887 0 6.16183 −32.9123 −66.0084 −25.6626 0
1.5 −5.06299 −4.85010 17.6339 0 24.5560 −6.05924 −48.7763 −3.47650 0
1.6 −4.98044 −7.53464 16.8048 0 37.5259 −14.6496 −43.8520 29.7709 0
1.7 −4.82727 2.71547 15.3025 0 −13.1083 13.1614 −35.2511 −19.6262 0
1.8 −4.78707 8.94432 14.9160 0 −42.8171 26.2645 −33.1076 53.0009 0
1.9 −4.53810 −9.82601 12.5943 0 44.5914 −18.0658 −20.8496 69.5504 0
1.10 −4.41429 −4.21802 11.4859 0 18.6196 21.5411 −15.3879 −9.20830 0
1.11 −4.19599 7.96065 9.60634 0 −33.4028 −20.2067 −6.74017 36.3720 0
1.12 −3.94391 5.36094 7.55441 0 −21.1431 6.20225 1.75738 1.73968 0
1.13 −3.83927 −1.26925 6.74001 0 4.87298 19.0163 4.83746 −25.3890 0
1.14 −3.51762 2.06014 4.37362 0 −7.24677 −26.6515 12.7562 −22.7558 0
1.15 −3.37957 1.60215 3.42147 0 −5.41457 31.2181 15.4734 −24.4331 0
1.16 −3.33904 −7.14813 3.14921 0 23.8679 −33.3386 16.1970 24.0957 0
1.17 −3.28998 5.08135 2.82398 0 −16.7175 7.39064 17.0290 −1.17992 0
1.18 −2.92221 −0.0289585 0.539306 0 0.0846229 −26.9683 21.8017 −26.9992 0
1.19 −2.53995 −8.75820 −1.54863 0 22.2454 −18.3262 24.2531 49.7060 0
1.20 −2.42307 −7.12404 −2.12875 0 17.2620 12.7165 24.5426 23.7520 0
See all 62 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.62
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.m 62
5.b even 2 1 2075.4.a.n 62
5.c odd 4 2 415.4.b.a ✓ 124
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.4.b.a ✓ 124 5.c odd 4 2
2075.4.a.m 62 1.a even 1 1 trivial
2075.4.a.n 62 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{62} + 10 T_{2}^{61} - 324 T_{2}^{60} - 3480 T_{2}^{59} + 48832 T_{2}^{58} + 571967 T_{2}^{57} + \cdots - 16\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2075))\). Copy content Toggle raw display