Properties

Label 175.4.b.d
Level $175$
Weight $4$
Character orbit 175.b
Analytic conductor $10.325$
Analytic rank $0$
Dimension $4$
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] = [175,4,Mod(99,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, 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("175.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 2 \beta_{2} + \beta_1) q^{3} + (\beta_{3} - 3) q^{4} + (3 \beta_{3} - 13) q^{6} - 7 \beta_{2} q^{7} + (10 \beta_{2} + 5 \beta_1) q^{8} + (5 \beta_{3} + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 2 \beta_{2} + \beta_1) q^{3} + (\beta_{3} - 3) q^{4} + (3 \beta_{3} - 13) q^{6} - 7 \beta_{2} q^{7} + (10 \beta_{2} + 5 \beta_1) q^{8} + (5 \beta_{3} + 8) q^{9} + (10 \beta_{3} - 33) q^{11} + (14 \beta_{2} - 5 \beta_1) q^{12} + (30 \beta_{2} + 8 \beta_1) q^{13} + (7 \beta_{3} - 7) q^{14} + (3 \beta_{3} - 69) q^{16} + (54 \beta_{2} + 5 \beta_1) q^{17} + (50 \beta_{2} + 8 \beta_1) q^{18} + (7 \beta_{3} + 25) q^{19} + (7 \beta_{3} - 21) q^{21} + (100 \beta_{2} - 33 \beta_1) q^{22} + ( - 13 \beta_{2} + 5 \beta_1) q^{23} + (5 \beta_{3} - 35) q^{24} + ( - 22 \beta_{3} - 58) q^{26} + ( - 30 \beta_{2} + 25 \beta_1) q^{27} + (14 \beta_{2} - 7 \beta_1) q^{28} + ( - 27 \beta_{3} + 220) q^{29} + ( - 66 \beta_{3} - 48) q^{31} + (110 \beta_{2} - 29 \beta_1) q^{32} + (146 \beta_{2} - 53 \beta_1) q^{33} + ( - 49 \beta_{3} - 1) q^{34} + ( - 2 \beta_{3} + 26) q^{36} + ( - 9 \beta_{2} + 57 \beta_1) q^{37} + (70 \beta_{2} + 25 \beta_1) q^{38} + ( - 6 \beta_{3} - 14) q^{39} + (61 \beta_{3} - 283) q^{41} + (70 \beta_{2} - 21 \beta_1) q^{42} + ( - 75 \beta_{2} - 77 \beta_1) q^{43} + ( - 53 \beta_{3} + 199) q^{44} + (18 \beta_{3} - 68) q^{46} + ( - 58 \beta_{2} + 108 \beta_1) q^{47} + (162 \beta_{2} - 75 \beta_1) q^{48} - 49 q^{49} + ( - 39 \beta_{3} + 97) q^{51} + (20 \beta_{2} + 6 \beta_1) q^{52} + ( - 174 \beta_{2} - 86 \beta_1) q^{53} + (55 \beta_{3} - 305) q^{54} + (35 \beta_{3} + 35) q^{56} + (6 \beta_{2} + 11 \beta_1) q^{57} + ( - 270 \beta_{2} + 220 \beta_1) q^{58} + (210 \beta_{3} - 200) q^{59} + ( - 54 \beta_{3} + 522) q^{61} + ( - 660 \beta_{2} - 48 \beta_1) q^{62} + ( - 91 \beta_{2} - 35 \beta_1) q^{63} + ( - 115 \beta_{3} - 123) q^{64} + ( - 199 \beta_{3} + 729) q^{66} + ( - 537 \beta_{2} - 166 \beta_1) q^{67} + ( - 58 \beta_{2} + 39 \beta_1) q^{68} + (28 \beta_{3} - 104) q^{69} + (303 \beta_{3} - 88) q^{71} + (380 \beta_{2} + 90 \beta_1) q^{72} + ( - 34 \beta_{2} + 269 \beta_1) q^{73} + (66 \beta_{3} - 636) q^{74} + (11 \beta_{3} - 5) q^{76} + (161 \beta_{2} - 70 \beta_1) q^{77} + ( - 60 \beta_{2} - 14 \beta_1) q^{78} + ( - 41 \beta_{3} + 580) q^{79} + (240 \beta_{3} - 199) q^{81} + (610 \beta_{2} - 283 \beta_1) q^{82} + ( - 724 \beta_{2} + 69 \beta_1) q^{83} + ( - 35 \beta_{3} + 133) q^{84} + ( - 2 \beta_{3} + 772) q^{86} + ( - 656 \beta_{2} + 274 \beta_1) q^{87} + (270 \beta_{2} - 65 \beta_1) q^{88} + ( - 17 \beta_{3} + 865) q^{89} + (56 \beta_{3} + 154) q^{91} + (76 \beta_{2} - 28 \beta_1) q^{92} + ( - 432 \beta_{2} + 84 \beta_1) q^{93} + (166 \beta_{3} - 1246) q^{94} + ( - 197 \beta_{3} + 707) q^{96} + ( - 1018 \beta_{2} - 272 \beta_1) q^{97} - 49 \beta_1 q^{98} + ( - 35 \beta_{3} + 236) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} - 46 q^{6} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} - 46 q^{6} + 42 q^{9} - 112 q^{11} - 14 q^{14} - 270 q^{16} + 114 q^{19} - 70 q^{21} - 130 q^{24} - 276 q^{26} + 826 q^{29} - 324 q^{31} - 102 q^{34} + 100 q^{36} - 68 q^{39} - 1010 q^{41} + 690 q^{44} - 236 q^{46} - 196 q^{49} + 310 q^{51} - 1110 q^{54} + 210 q^{56} - 380 q^{59} + 1980 q^{61} - 722 q^{64} + 2518 q^{66} - 360 q^{69} + 254 q^{71} - 2412 q^{74} + 2 q^{76} + 2238 q^{79} - 316 q^{81} + 462 q^{84} + 3084 q^{86} + 3426 q^{89} + 728 q^{91} - 4652 q^{94} + 2434 q^{96} + 874 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{2} - 11\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
3.70156i
2.70156i
2.70156i
3.70156i
3.70156i 5.70156i −5.70156 0 −21.1047 7.00000i 8.50781i −5.50781 0
99.2 2.70156i 0.701562i 0.701562 0 −1.89531 7.00000i 23.5078i 26.5078 0
99.3 2.70156i 0.701562i 0.701562 0 −1.89531 7.00000i 23.5078i 26.5078 0
99.4 3.70156i 5.70156i −5.70156 0 −21.1047 7.00000i 8.50781i −5.50781 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 175.4.b.d 4
5.b even 2 1 inner 175.4.b.d 4
5.c odd 4 1 175.4.a.d 2
5.c odd 4 1 175.4.a.e yes 2
15.e even 4 1 1575.4.a.s 2
15.e even 4 1 1575.4.a.v 2
35.f even 4 1 1225.4.a.r 2
35.f even 4 1 1225.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.d 2 5.c odd 4 1
175.4.a.e yes 2 5.c odd 4 1
175.4.b.d 4 1.a even 1 1 trivial
175.4.b.d 4 5.b even 2 1 inner
1225.4.a.r 2 35.f even 4 1
1225.4.a.t 2 35.f even 4 1
1575.4.a.s 2 15.e even 4 1
1575.4.a.v 2 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 \(S_{4}^{\mathrm{new}}(175, [\chi])\):

\( T_{2}^{4} + 21T_{2}^{2} + 100 \) Copy content Toggle raw display
\( T_{3}^{4} + 33T_{3}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 21T^{2} + 100 \) Copy content Toggle raw display
$3$ \( T^{4} + 33T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 56 T - 241)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 2664T^{2} + 400 \) Copy content Toggle raw display
$17$ \( T^{4} + 5817 T^{2} + \cdots + 5740816 \) Copy content Toggle raw display
$19$ \( (T^{2} - 57 T + 310)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 993T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 413 T + 35170)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 162 T - 38088)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 69417 T^{2} + \cdots + 1017354816 \) Copy content Toggle raw display
$41$ \( (T^{2} + 505 T + 25616)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 124209 T^{2} + \cdots + 3533113600 \) Copy content Toggle raw display
$47$ \( T^{4} + 264200 T^{2} + \cdots + 11451568144 \) Copy content Toggle raw display
$53$ \( T^{4} + 185940 T^{2} + \cdots + 3439587904 \) Copy content Toggle raw display
$59$ \( (T^{2} + 190 T - 443000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 990 T + 215136)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 977130 T^{2} + \cdots + 5826726889 \) Copy content Toggle raw display
$71$ \( (T^{2} - 127 T - 937010)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 1540185 T^{2} + \cdots + 508808302864 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1119 T + 295810)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1248245 T^{2} + \cdots + 277225416484 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1713 T + 730630)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3072520 T^{2} + \cdots + 383689744 \) Copy content Toggle raw display
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