Properties

Label 75.4.a.c
Level $75$
Weight $4$
Character orbit 75.a
Self dual yes
Analytic conductor $4.425$
Analytic rank $1$
Dimension $2$
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] = [75,4,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, 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("75.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(4.42514325043\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} + (6 \beta - 6) q^{7} + ( - \beta - 25) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} + (6 \beta - 6) q^{7} + ( - \beta - 25) q^{8} + 9 q^{9} + ( - 6 \beta - 18) q^{11} + ( - 9 \beta - 9) q^{12} + (6 \beta - 42) q^{13} + ( - 6 \beta - 54) q^{14} + (3 \beta + 11) q^{16} + ( - 10 \beta - 46) q^{17} + ( - 9 \beta - 9) q^{18} + ( - 24 \beta + 40) q^{19} + ( - 18 \beta + 18) q^{21} + (30 \beta + 78) q^{22} + ( - 8 \beta + 28) q^{23} + (3 \beta + 75) q^{24} + (30 \beta - 18) q^{26} - 27 q^{27} + (18 \beta + 162) q^{28} + (42 \beta - 180) q^{29} + ( - 12 \beta + 32) q^{31} + ( - 9 \beta + 159) q^{32} + (18 \beta + 54) q^{33} + (66 \beta + 146) q^{34} + (27 \beta + 27) q^{36} + ( - 54 \beta - 126) q^{37} + (8 \beta + 200) q^{38} + ( - 18 \beta + 126) q^{39} + ( - 12 \beta - 198) q^{41} + (18 \beta + 162) q^{42} + ( - 96 \beta - 12) q^{43} + ( - 90 \beta - 234) q^{44} + ( - 12 \beta + 52) q^{46} + (92 \beta - 136) q^{47} + ( - 9 \beta - 33) q^{48} + ( - 36 \beta + 53) q^{49} + (30 \beta + 138) q^{51} + ( - 90 \beta + 54) q^{52} + (82 \beta - 242) q^{53} + (27 \beta + 27) q^{54} + ( - 150 \beta + 90) q^{56} + (72 \beta - 120) q^{57} + (96 \beta - 240) q^{58} + ( - 6 \beta - 90) q^{59} + (96 \beta + 122) q^{61} + ( - 8 \beta + 88) q^{62} + (54 \beta - 54) q^{63} + ( - 165 \beta - 157) q^{64} + ( - 90 \beta - 234) q^{66} + ( - 60 \beta - 336) q^{67} + ( - 198 \beta - 438) q^{68} + (24 \beta - 84) q^{69} + (180 \beta - 108) q^{71} + ( - 9 \beta - 225) q^{72} + ( - 108 \beta - 612) q^{73} + (234 \beta + 666) q^{74} + ( - 24 \beta - 600) q^{76} + ( - 108 \beta - 252) q^{77} + ( - 90 \beta + 54) q^{78} + (300 \beta + 40) q^{79} + 81 q^{81} + (222 \beta + 318) q^{82} + (208 \beta + 388) q^{83} + ( - 54 \beta - 486) q^{84} + (204 \beta + 972) q^{86} + ( - 126 \beta + 540) q^{87} + (174 \beta + 510) q^{88} + ( - 144 \beta + 630) q^{89} + ( - 252 \beta + 612) q^{91} + (36 \beta - 156) q^{92} + (36 \beta - 96) q^{93} + ( - 48 \beta - 784) q^{94} + (27 \beta - 477) q^{96} + (240 \beta + 264) q^{97} + (19 \beta + 307) q^{98} + ( - 54 \beta - 162) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 6 q^{7} - 51 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 6 q^{7} - 51 q^{8} + 18 q^{9} - 42 q^{11} - 27 q^{12} - 78 q^{13} - 114 q^{14} + 25 q^{16} - 102 q^{17} - 27 q^{18} + 56 q^{19} + 18 q^{21} + 186 q^{22} + 48 q^{23} + 153 q^{24} - 6 q^{26} - 54 q^{27} + 342 q^{28} - 318 q^{29} + 52 q^{31} + 309 q^{32} + 126 q^{33} + 358 q^{34} + 81 q^{36} - 306 q^{37} + 408 q^{38} + 234 q^{39} - 408 q^{41} + 342 q^{42} - 120 q^{43} - 558 q^{44} + 92 q^{46} - 180 q^{47} - 75 q^{48} + 70 q^{49} + 306 q^{51} + 18 q^{52} - 402 q^{53} + 81 q^{54} + 30 q^{56} - 168 q^{57} - 384 q^{58} - 186 q^{59} + 340 q^{61} + 168 q^{62} - 54 q^{63} - 479 q^{64} - 558 q^{66} - 732 q^{67} - 1074 q^{68} - 144 q^{69} - 36 q^{71} - 459 q^{72} - 1332 q^{73} + 1566 q^{74} - 1224 q^{76} - 612 q^{77} + 18 q^{78} + 380 q^{79} + 162 q^{81} + 858 q^{82} + 984 q^{83} - 1026 q^{84} + 2148 q^{86} + 954 q^{87} + 1194 q^{88} + 1116 q^{89} + 972 q^{91} - 276 q^{92} - 156 q^{93} - 1616 q^{94} - 927 q^{96} + 768 q^{97} + 633 q^{98} - 378 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
3.70156
−2.70156
−4.70156 −3.00000 14.1047 0 14.1047 16.2094 −28.7016 9.00000 0
1.2 1.70156 −3.00000 −5.10469 0 −5.10469 −22.2094 −22.2984 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)

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 75.4.a.c 2
3.b odd 2 1 225.4.a.o 2
4.b odd 2 1 1200.4.a.bt 2
5.b even 2 1 75.4.a.f 2
5.c odd 4 2 15.4.b.a 4
15.d odd 2 1 225.4.a.i 2
15.e even 4 2 45.4.b.b 4
20.d odd 2 1 1200.4.a.bn 2
20.e even 4 2 240.4.f.f 4
40.i odd 4 2 960.4.f.q 4
40.k even 4 2 960.4.f.p 4
60.l odd 4 2 720.4.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 5.c odd 4 2
45.4.b.b 4 15.e even 4 2
75.4.a.c 2 1.a even 1 1 trivial
75.4.a.f 2 5.b even 2 1
225.4.a.i 2 15.d odd 2 1
225.4.a.o 2 3.b odd 2 1
240.4.f.f 4 20.e even 4 2
720.4.f.j 4 60.l odd 4 2
960.4.f.p 4 40.k even 4 2
960.4.f.q 4 40.i odd 4 2
1200.4.a.bn 2 20.d odd 2 1
1200.4.a.bt 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T - 360 \) Copy content Toggle raw display
$11$ \( T^{2} + 42T + 72 \) Copy content Toggle raw display
$13$ \( T^{2} + 78T + 1152 \) Copy content Toggle raw display
$17$ \( T^{2} + 102T + 1576 \) Copy content Toggle raw display
$19$ \( T^{2} - 56T - 5120 \) Copy content Toggle raw display
$23$ \( T^{2} - 48T - 80 \) Copy content Toggle raw display
$29$ \( T^{2} + 318T + 7200 \) Copy content Toggle raw display
$31$ \( T^{2} - 52T - 800 \) Copy content Toggle raw display
$37$ \( T^{2} + 306T - 6480 \) Copy content Toggle raw display
$41$ \( T^{2} + 408T + 40140 \) Copy content Toggle raw display
$43$ \( T^{2} + 120T - 90864 \) Copy content Toggle raw display
$47$ \( T^{2} + 180T - 78656 \) Copy content Toggle raw display
$53$ \( T^{2} + 402T - 28520 \) Copy content Toggle raw display
$59$ \( T^{2} + 186T + 8280 \) Copy content Toggle raw display
$61$ \( T^{2} - 340T - 65564 \) Copy content Toggle raw display
$67$ \( T^{2} + 732T + 97056 \) Copy content Toggle raw display
$71$ \( T^{2} + 36T - 331776 \) Copy content Toggle raw display
$73$ \( T^{2} + 1332 T + 324000 \) Copy content Toggle raw display
$79$ \( T^{2} - 380T - 886400 \) Copy content Toggle raw display
$83$ \( T^{2} - 984T - 201392 \) Copy content Toggle raw display
$89$ \( T^{2} - 1116T + 98820 \) Copy content Toggle raw display
$97$ \( T^{2} - 768T - 442944 \) Copy content Toggle raw display
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