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Properties

Label 2075.4.a.n
Level $2075$
Weight $4$
Character orbit 2075.a
Self dual yes
Analytic conductor $122.429$
Analytic rank $0$
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: \(0\)
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 −5.36371 −4.49952 20.7693 0 24.1341 33.2479 −68.4909 −6.75431 0
1.2 −5.22457 9.01343 19.2961 0 −47.0912 8.67367 −59.0173 54.2418 0
1.3 −5.17649 1.39860 18.7961 0 −7.23982 10.8286 −55.8857 −25.0439 0
1.4 −5.09381 −2.66448 17.9469 0 13.5724 −10.1095 −50.6679 −19.9006 0
1.5 −5.04149 −0.170131 17.4166 0 0.857713 10.4782 −47.4735 −26.9711 0
1.6 −4.54206 3.36464 12.6303 0 −15.2824 −11.6054 −21.0312 −15.6792 0
1.7 −4.48211 9.62254 12.0894 0 −43.1293 −31.5948 −18.3290 65.5932 0
1.8 −4.35680 6.61019 10.9817 0 −28.7993 −14.5063 −12.9909 16.6946 0
1.9 −4.12476 3.06656 9.01365 0 −12.6488 7.17192 −4.18107 −17.5962 0
1.10 −4.04522 −6.62870 8.36380 0 26.8146 28.3059 −1.47163 16.9397 0
1.11 −4.03404 −2.64287 8.27345 0 10.6614 −24.5523 −1.10313 −20.0152 0
1.12 −4.02495 −6.18996 8.20019 0 24.9142 −8.20343 −0.805735 11.3156 0
1.13 −3.79200 −3.45358 6.37926 0 13.0960 5.61221 6.14583 −15.0728 0
1.14 −3.44929 −8.73212 3.89758 0 30.1196 24.5491 14.1504 49.2499 0
1.15 −3.40125 −8.90834 3.56852 0 30.2995 5.50871 15.0726 52.3586 0
1.16 −3.28181 8.63699 2.77030 0 −28.3450 23.7479 17.1629 47.5976 0
1.17 −2.95046 6.24280 0.705239 0 −18.4191 35.5812 21.5229 11.9725 0
1.18 −2.46783 8.88714 −1.90981 0 −21.9320 0.0768410 24.4557 51.9813 0
1.19 −2.40766 −1.65271 −2.20318 0 3.97916 −1.85096 24.5658 −24.2685 0
1.20 −2.17267 −4.78409 −3.27952 0 10.3942 −25.0570 24.5066 −4.11244 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.n 62
5.b even 2 1 2075.4.a.m 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 5.b even 2 1
2075.4.a.n 62 1.a even 1 1 trivial

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