Properties

Label 74.3.d.b
Level $74$
Weight $3$
Character orbit 74.d
Analytic conductor $2.016$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [74,3,Mod(31,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 74.d (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(2.01635395627\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + 4 i q^{3} + 2 i q^{4} + ( - 3 i + 3) q^{5} + (4 i - 4) q^{6} - 4 q^{7} + (2 i - 2) q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + 4 i q^{3} + 2 i q^{4} + ( - 3 i + 3) q^{5} + (4 i - 4) q^{6} - 4 q^{7} + (2 i - 2) q^{8} - 7 q^{9} + 6 q^{10} + 4 i q^{11} - 8 q^{12} + ( - 3 i + 3) q^{13} + ( - 4 i - 4) q^{14} + (12 i + 12) q^{15} - 4 q^{16} + ( - 23 i + 23) q^{17} + ( - 7 i - 7) q^{18} + ( - 10 i + 10) q^{19} + (6 i + 6) q^{20} - 16 i q^{21} + (4 i - 4) q^{22} + (10 i - 10) q^{23} + ( - 8 i - 8) q^{24} + 7 i q^{25} + 6 q^{26} + 8 i q^{27} - 8 i q^{28} + ( - 19 i - 19) q^{29} + 24 i q^{30} + ( - 18 i - 18) q^{31} + ( - 4 i - 4) q^{32} - 16 q^{33} + 46 q^{34} + (12 i - 12) q^{35} - 14 i q^{36} - 37 i q^{37} + 20 q^{38} + (12 i + 12) q^{39} + 12 i q^{40} + 74 i q^{41} + ( - 16 i + 16) q^{42} + ( - 42 i + 42) q^{43} - 8 q^{44} + (21 i - 21) q^{45} - 20 q^{46} - 44 q^{47} - 16 i q^{48} - 33 q^{49} + (7 i - 7) q^{50} + (92 i + 92) q^{51} + (6 i + 6) q^{52} - 80 q^{53} + (8 i - 8) q^{54} + (12 i + 12) q^{55} + ( - 8 i + 8) q^{56} + (40 i + 40) q^{57} - 38 i q^{58} + (54 i - 54) q^{59} + (24 i - 24) q^{60} + ( - 3 i - 3) q^{61} - 36 i q^{62} + 28 q^{63} - 8 i q^{64} - 18 i q^{65} + ( - 16 i - 16) q^{66} - 12 i q^{67} + (46 i + 46) q^{68} + ( - 40 i - 40) q^{69} - 24 q^{70} + 124 q^{71} + ( - 14 i + 14) q^{72} - 10 i q^{73} + ( - 37 i + 37) q^{74} - 28 q^{75} + (20 i + 20) q^{76} - 16 i q^{77} + 24 i q^{78} + ( - 14 i + 14) q^{79} + (12 i - 12) q^{80} - 95 q^{81} + (74 i - 74) q^{82} - 64 q^{83} + 32 q^{84} - 138 i q^{85} + 84 q^{86} + ( - 76 i + 76) q^{87} + ( - 8 i - 8) q^{88} + (17 i + 17) q^{89} - 42 q^{90} + (12 i - 12) q^{91} + ( - 20 i - 20) q^{92} + ( - 72 i + 72) q^{93} + ( - 44 i - 44) q^{94} - 60 i q^{95} + ( - 16 i + 16) q^{96} + ( - 129 i + 129) q^{97} + ( - 33 i - 33) q^{98} - 28 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 6 q^{5} - 8 q^{6} - 8 q^{7} - 4 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 6 q^{5} - 8 q^{6} - 8 q^{7} - 4 q^{8} - 14 q^{9} + 12 q^{10} - 16 q^{12} + 6 q^{13} - 8 q^{14} + 24 q^{15} - 8 q^{16} + 46 q^{17} - 14 q^{18} + 20 q^{19} + 12 q^{20} - 8 q^{22} - 20 q^{23} - 16 q^{24} + 12 q^{26} - 38 q^{29} - 36 q^{31} - 8 q^{32} - 32 q^{33} + 92 q^{34} - 24 q^{35} + 40 q^{38} + 24 q^{39} + 32 q^{42} + 84 q^{43} - 16 q^{44} - 42 q^{45} - 40 q^{46} - 88 q^{47} - 66 q^{49} - 14 q^{50} + 184 q^{51} + 12 q^{52} - 160 q^{53} - 16 q^{54} + 24 q^{55} + 16 q^{56} + 80 q^{57} - 108 q^{59} - 48 q^{60} - 6 q^{61} + 56 q^{63} - 32 q^{66} + 92 q^{68} - 80 q^{69} - 48 q^{70} + 248 q^{71} + 28 q^{72} + 74 q^{74} - 56 q^{75} + 40 q^{76} + 28 q^{79} - 24 q^{80} - 190 q^{81} - 148 q^{82} - 128 q^{83} + 64 q^{84} + 168 q^{86} + 152 q^{87} - 16 q^{88} + 34 q^{89} - 84 q^{90} - 24 q^{91} - 40 q^{92} + 144 q^{93} - 88 q^{94} + 32 q^{96} + 258 q^{97} - 66 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.00000i
1.00000i
1.00000 + 1.00000i 4.00000i 2.00000i 3.00000 3.00000i −4.00000 + 4.00000i −4.00000 −2.00000 + 2.00000i −7.00000 6.00000
43.1 1.00000 1.00000i 4.00000i 2.00000i 3.00000 + 3.00000i −4.00000 4.00000i −4.00000 −2.00000 2.00000i −7.00000 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.3.d.b 2
3.b odd 2 1 666.3.i.a 2
4.b odd 2 1 592.3.k.c 2
37.d odd 4 1 inner 74.3.d.b 2
111.g even 4 1 666.3.i.a 2
148.g even 4 1 592.3.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.3.d.b 2 1.a even 1 1 trivial
74.3.d.b 2 37.d odd 4 1 inner
592.3.k.c 2 4.b odd 2 1
592.3.k.c 2 148.g even 4 1
666.3.i.a 2 3.b odd 2 1
666.3.i.a 2 111.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(74, [\chi])\):

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{2} - 6T_{5} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} - 46T + 1058 \) Copy content Toggle raw display
$19$ \( T^{2} - 20T + 200 \) Copy content Toggle raw display
$23$ \( T^{2} + 20T + 200 \) Copy content Toggle raw display
$29$ \( T^{2} + 38T + 722 \) Copy content Toggle raw display
$31$ \( T^{2} + 36T + 648 \) Copy content Toggle raw display
$37$ \( T^{2} + 1369 \) Copy content Toggle raw display
$41$ \( T^{2} + 5476 \) Copy content Toggle raw display
$43$ \( T^{2} - 84T + 3528 \) Copy content Toggle raw display
$47$ \( (T + 44)^{2} \) Copy content Toggle raw display
$53$ \( (T + 80)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 108T + 5832 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T - 124)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( T^{2} - 28T + 392 \) Copy content Toggle raw display
$83$ \( (T + 64)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 34T + 578 \) Copy content Toggle raw display
$97$ \( T^{2} - 258T + 33282 \) Copy content Toggle raw display
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