Properties

Label 950.4.b.a
Level $950$
Weight $4$
Character orbit 950.b
Analytic conductor $56.052$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,4,Mod(799,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.799");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), not 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: \(56.0518145055\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 2 \beta q^{3} - 4 q^{4} - 8 q^{6} - 10 \beta q^{7} - 4 \beta q^{8} + 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 2 \beta q^{3} - 4 q^{4} - 8 q^{6} - 10 \beta q^{7} - 4 \beta q^{8} + 11 q^{9} - 44 q^{11} - 8 \beta q^{12} - 21 \beta q^{13} + 40 q^{14} + 16 q^{16} - 43 \beta q^{17} + 11 \beta q^{18} - 19 q^{19} + 80 q^{21} - 44 \beta q^{22} + 82 \beta q^{23} + 32 q^{24} + 84 q^{26} + 76 \beta q^{27} + 40 \beta q^{28} + 162 q^{29} - 312 q^{31} + 16 \beta q^{32} - 88 \beta q^{33} + 172 q^{34} - 44 q^{36} + 113 \beta q^{37} - 19 \beta q^{38} + 168 q^{39} + 34 q^{41} + 80 \beta q^{42} + 216 \beta q^{43} + 176 q^{44} - 328 q^{46} + 290 \beta q^{47} + 32 \beta q^{48} - 57 q^{49} + 344 q^{51} + 84 \beta q^{52} - 253 \beta q^{53} - 304 q^{54} - 160 q^{56} - 38 \beta q^{57} + 162 \beta q^{58} - 364 q^{59} + 518 q^{61} - 312 \beta q^{62} - 110 \beta q^{63} - 64 q^{64} + 352 q^{66} + 462 \beta q^{67} + 172 \beta q^{68} - 656 q^{69} + 320 q^{71} - 44 \beta q^{72} + 271 \beta q^{73} - 452 q^{74} + 76 q^{76} + 440 \beta q^{77} + 168 \beta q^{78} + 1208 q^{79} - 311 q^{81} + 34 \beta q^{82} + 560 \beta q^{83} - 320 q^{84} - 864 q^{86} + 324 \beta q^{87} + 176 \beta q^{88} + 1022 q^{89} - 840 q^{91} - 328 \beta q^{92} - 624 \beta q^{93} - 1160 q^{94} - 128 q^{96} + 583 \beta q^{97} - 57 \beta q^{98} - 484 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 16 q^{6} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 16 q^{6} + 22 q^{9} - 88 q^{11} + 80 q^{14} + 32 q^{16} - 38 q^{19} + 160 q^{21} + 64 q^{24} + 168 q^{26} + 324 q^{29} - 624 q^{31} + 344 q^{34} - 88 q^{36} + 336 q^{39} + 68 q^{41} + 352 q^{44} - 656 q^{46} - 114 q^{49} + 688 q^{51} - 608 q^{54} - 320 q^{56} - 728 q^{59} + 1036 q^{61} - 128 q^{64} + 704 q^{66} - 1312 q^{69} + 640 q^{71} - 904 q^{74} + 152 q^{76} + 2416 q^{79} - 622 q^{81} - 640 q^{84} - 1728 q^{86} + 2044 q^{89} - 1680 q^{91} - 2320 q^{94} - 256 q^{96} - 968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.00000i
1.00000i
2.00000i 4.00000i −4.00000 0 −8.00000 20.0000i 8.00000i 11.0000 0
799.2 2.00000i 4.00000i −4.00000 0 −8.00000 20.0000i 8.00000i 11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.4.b.a 2
5.b even 2 1 inner 950.4.b.a 2
5.c odd 4 1 190.4.a.c 1
5.c odd 4 1 950.4.a.a 1
15.e even 4 1 1710.4.a.b 1
20.e even 4 1 1520.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.4.a.c 1 5.c odd 4 1
950.4.a.a 1 5.c odd 4 1
950.4.b.a 2 1.a even 1 1 trivial
950.4.b.a 2 5.b even 2 1 inner
1520.4.a.g 1 20.e even 4 1
1710.4.a.b 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 400 \) Copy content Toggle raw display
$11$ \( (T + 44)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1764 \) Copy content Toggle raw display
$17$ \( T^{2} + 7396 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 26896 \) Copy content Toggle raw display
$29$ \( (T - 162)^{2} \) Copy content Toggle raw display
$31$ \( (T + 312)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 51076 \) Copy content Toggle raw display
$41$ \( (T - 34)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 186624 \) Copy content Toggle raw display
$47$ \( T^{2} + 336400 \) Copy content Toggle raw display
$53$ \( T^{2} + 256036 \) Copy content Toggle raw display
$59$ \( (T + 364)^{2} \) Copy content Toggle raw display
$61$ \( (T - 518)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 853776 \) Copy content Toggle raw display
$71$ \( (T - 320)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 293764 \) Copy content Toggle raw display
$79$ \( (T - 1208)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1254400 \) Copy content Toggle raw display
$89$ \( (T - 1022)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1359556 \) Copy content Toggle raw display
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