Properties

Label 175.4.b.b
Level $175$
Weight $4$
Character orbit 175.b
Analytic conductor $10.325$
Analytic rank $0$
Dimension $2$
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: \(2\)
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: no (minimal twist has level 7)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - 2 i q^{3} + 7 q^{4} + 2 q^{6} + 7 i q^{7} + 15 i q^{8} + 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - 2 i q^{3} + 7 q^{4} + 2 q^{6} + 7 i q^{7} + 15 i q^{8} + 23 q^{9} - 8 q^{11} - 14 i q^{12} + 28 i q^{13} - 7 q^{14} + 41 q^{16} - 54 i q^{17} + 23 i q^{18} + 110 q^{19} + 14 q^{21} - 8 i q^{22} + 48 i q^{23} + 30 q^{24} - 28 q^{26} - 100 i q^{27} + 49 i q^{28} + 110 q^{29} + 12 q^{31} + 161 i q^{32} + 16 i q^{33} + 54 q^{34} + 161 q^{36} + 246 i q^{37} + 110 i q^{38} + 56 q^{39} + 182 q^{41} + 14 i q^{42} + 128 i q^{43} - 56 q^{44} - 48 q^{46} - 324 i q^{47} - 82 i q^{48} - 49 q^{49} - 108 q^{51} + 196 i q^{52} - 162 i q^{53} + 100 q^{54} - 105 q^{56} - 220 i q^{57} + 110 i q^{58} - 810 q^{59} - 488 q^{61} + 12 i q^{62} + 161 i q^{63} + 167 q^{64} - 16 q^{66} - 244 i q^{67} - 378 i q^{68} + 96 q^{69} - 768 q^{71} + 345 i q^{72} - 702 i q^{73} - 246 q^{74} + 770 q^{76} - 56 i q^{77} + 56 i q^{78} - 440 q^{79} + 421 q^{81} + 182 i q^{82} - 1302 i q^{83} + 98 q^{84} - 128 q^{86} - 220 i q^{87} - 120 i q^{88} - 730 q^{89} - 196 q^{91} + 336 i q^{92} - 24 i q^{93} + 324 q^{94} + 322 q^{96} - 294 i q^{97} - 49 i q^{98} - 184 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 4 q^{6} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} + 4 q^{6} + 46 q^{9} - 16 q^{11} - 14 q^{14} + 82 q^{16} + 220 q^{19} + 28 q^{21} + 60 q^{24} - 56 q^{26} + 220 q^{29} + 24 q^{31} + 108 q^{34} + 322 q^{36} + 112 q^{39} + 364 q^{41} - 112 q^{44} - 96 q^{46} - 98 q^{49} - 216 q^{51} + 200 q^{54} - 210 q^{56} - 1620 q^{59} - 976 q^{61} + 334 q^{64} - 32 q^{66} + 192 q^{69} - 1536 q^{71} - 492 q^{74} + 1540 q^{76} - 880 q^{79} + 842 q^{81} + 196 q^{84} - 256 q^{86} - 1460 q^{89} - 392 q^{91} + 648 q^{94} + 644 q^{96} - 368 q^{99}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 2.00000i 7.00000 0 2.00000 7.00000i 15.0000i 23.0000 0
99.2 1.00000i 2.00000i 7.00000 0 2.00000 7.00000i 15.0000i 23.0000 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 175.4.b.b 2
5.b even 2 1 inner 175.4.b.b 2
5.c odd 4 1 7.4.a.a 1
5.c odd 4 1 175.4.a.b 1
15.e even 4 1 63.4.a.b 1
15.e even 4 1 1575.4.a.e 1
20.e even 4 1 112.4.a.f 1
35.f even 4 1 49.4.a.b 1
35.f even 4 1 1225.4.a.j 1
35.k even 12 2 49.4.c.b 2
35.l odd 12 2 49.4.c.c 2
40.i odd 4 1 448.4.a.i 1
40.k even 4 1 448.4.a.e 1
55.e even 4 1 847.4.a.b 1
60.l odd 4 1 1008.4.a.c 1
65.h odd 4 1 1183.4.a.b 1
85.g odd 4 1 2023.4.a.a 1
105.k odd 4 1 441.4.a.i 1
105.w odd 12 2 441.4.e.e 2
105.x even 12 2 441.4.e.h 2
140.j odd 4 1 784.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 5.c odd 4 1
49.4.a.b 1 35.f even 4 1
49.4.c.b 2 35.k even 12 2
49.4.c.c 2 35.l odd 12 2
63.4.a.b 1 15.e even 4 1
112.4.a.f 1 20.e even 4 1
175.4.a.b 1 5.c odd 4 1
175.4.b.b 2 1.a even 1 1 trivial
175.4.b.b 2 5.b even 2 1 inner
441.4.a.i 1 105.k odd 4 1
441.4.e.e 2 105.w odd 12 2
441.4.e.h 2 105.x even 12 2
448.4.a.e 1 40.k even 4 1
448.4.a.i 1 40.i odd 4 1
784.4.a.g 1 140.j odd 4 1
847.4.a.b 1 55.e even 4 1
1008.4.a.c 1 60.l odd 4 1
1183.4.a.b 1 65.h odd 4 1
1225.4.a.j 1 35.f even 4 1
1575.4.a.e 1 15.e even 4 1
2023.4.a.a 1 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 784 \) Copy content Toggle raw display
$17$ \( T^{2} + 2916 \) Copy content Toggle raw display
$19$ \( (T - 110)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2304 \) Copy content Toggle raw display
$29$ \( (T - 110)^{2} \) Copy content Toggle raw display
$31$ \( (T - 12)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 60516 \) Copy content Toggle raw display
$41$ \( (T - 182)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16384 \) Copy content Toggle raw display
$47$ \( T^{2} + 104976 \) Copy content Toggle raw display
$53$ \( T^{2} + 26244 \) Copy content Toggle raw display
$59$ \( (T + 810)^{2} \) Copy content Toggle raw display
$61$ \( (T + 488)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 59536 \) Copy content Toggle raw display
$71$ \( (T + 768)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 492804 \) Copy content Toggle raw display
$79$ \( (T + 440)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1695204 \) Copy content Toggle raw display
$89$ \( (T + 730)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 86436 \) Copy content Toggle raw display
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