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Properties

Label 175.4.k.b
Level $175$
Weight $4$
Character orbit 175.k
Analytic conductor $10.325$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,4,Mod(74,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.74");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.k (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} - 6 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} - 21 \zeta_{12}^{3} q^{8} + (23 \zeta_{12}^{2} - 23) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \zeta_{12} q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} - 6 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} - 21 \zeta_{12}^{3} q^{8} + (23 \zeta_{12}^{2} - 23) q^{9} + 45 \zeta_{12}^{2} q^{11} - 2 \zeta_{12} q^{12} + 59 \zeta_{12}^{3} q^{13} + (63 \zeta_{12}^{2} - 21) q^{14} + ( - 71 \zeta_{12}^{2} + 71) q^{16} + ( - 54 \zeta_{12}^{3} + 54 \zeta_{12}) q^{17} + (69 \zeta_{12}^{3} - 69 \zeta_{12}) q^{18} + (121 \zeta_{12}^{2} - 121) q^{19} + ( - 14 \zeta_{12}^{2} - 28) q^{21} + 135 \zeta_{12}^{3} q^{22} - 69 \zeta_{12} q^{23} + 42 \zeta_{12}^{2} q^{24} + (177 \zeta_{12}^{2} - 177) q^{26} - 100 \zeta_{12}^{3} q^{27} + (21 \zeta_{12}^{3} - 7 \zeta_{12}) q^{28} + 162 q^{29} + 88 \zeta_{12}^{2} q^{31} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{32} - 90 \zeta_{12} q^{33} + 162 q^{34} - 23 q^{36} - 259 \zeta_{12} q^{37} + (363 \zeta_{12}^{3} - 363 \zeta_{12}) q^{38} - 118 \zeta_{12}^{2} q^{39} + 195 q^{41} + ( - 42 \zeta_{12}^{3} - 84 \zeta_{12}) q^{42} - 286 \zeta_{12}^{3} q^{43} + (45 \zeta_{12}^{2} - 45) q^{44} - 207 \zeta_{12}^{2} q^{46} + 45 \zeta_{12} q^{47} + 142 \zeta_{12}^{3} q^{48} + (392 \zeta_{12}^{2} - 245) q^{49} + (108 \zeta_{12}^{2} - 108) q^{51} + (59 \zeta_{12}^{3} - 59 \zeta_{12}) q^{52} + ( - 597 \zeta_{12}^{3} + 597 \zeta_{12}) q^{53} + ( - 300 \zeta_{12}^{2} + 300) q^{54} + ( - 294 \zeta_{12}^{2} + 441) q^{56} - 242 \zeta_{12}^{3} q^{57} + 486 \zeta_{12} q^{58} - 360 \zeta_{12}^{2} q^{59} + (392 \zeta_{12}^{2} - 392) q^{61} + 264 \zeta_{12}^{3} q^{62} + (322 \zeta_{12}^{3} - 483 \zeta_{12}) q^{63} - 433 q^{64} - 270 \zeta_{12}^{2} q^{66} + ( - 280 \zeta_{12}^{3} + 280 \zeta_{12}) q^{67} + 54 \zeta_{12} q^{68} + 138 q^{69} + 48 q^{71} + 483 \zeta_{12} q^{72} + ( - 668 \zeta_{12}^{3} + 668 \zeta_{12}) q^{73} - 777 \zeta_{12}^{2} q^{74} - 121 q^{76} + (945 \zeta_{12}^{3} - 315 \zeta_{12}) q^{77} - 354 \zeta_{12}^{3} q^{78} + ( - 782 \zeta_{12}^{2} + 782) q^{79} - 421 \zeta_{12}^{2} q^{81} + 585 \zeta_{12} q^{82} + 768 \zeta_{12}^{3} q^{83} + ( - 42 \zeta_{12}^{2} + 14) q^{84} + ( - 858 \zeta_{12}^{2} + 858) q^{86} + (324 \zeta_{12}^{3} - 324 \zeta_{12}) q^{87} + ( - 945 \zeta_{12}^{3} + 945 \zeta_{12}) q^{88} + (1194 \zeta_{12}^{2} - 1194) q^{89} + (826 \zeta_{12}^{2} - 1239) q^{91} - 69 \zeta_{12}^{3} q^{92} - 176 \zeta_{12} q^{93} + 135 \zeta_{12}^{2} q^{94} + (90 \zeta_{12}^{2} - 90) q^{96} - 902 \zeta_{12}^{3} q^{97} + (1176 \zeta_{12}^{3} - 735 \zeta_{12}) q^{98} - 1035 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 24 q^{6} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 24 q^{6} - 46 q^{9} + 90 q^{11} + 42 q^{14} + 142 q^{16} - 242 q^{19} - 140 q^{21} + 84 q^{24} - 354 q^{26} + 648 q^{29} + 176 q^{31} + 648 q^{34} - 92 q^{36} - 236 q^{39} + 780 q^{41} - 90 q^{44} - 414 q^{46} - 196 q^{49} - 216 q^{51} + 600 q^{54} + 1176 q^{56} - 720 q^{59} - 784 q^{61} - 1732 q^{64} - 540 q^{66} + 552 q^{69} + 192 q^{71} - 1554 q^{74} - 484 q^{76} + 1564 q^{79} - 842 q^{81} - 28 q^{84} + 1716 q^{86} - 2388 q^{89} - 3304 q^{91} + 270 q^{94} - 180 q^{96} - 4140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−2.59808 + 1.50000i 1.73205 + 1.00000i 0.500000 − 0.866025i 0 −6.00000 −12.1244 + 14.0000i − 21.0000i −11.5000 − 19.9186i 0
74.2 2.59808 − 1.50000i −1.73205 − 1.00000i 0.500000 − 0.866025i 0 −6.00000 12.1244 − 14.0000i 21.0000i −11.5000 − 19.9186i 0
149.1 −2.59808 − 1.50000i 1.73205 − 1.00000i 0.500000 + 0.866025i 0 −6.00000 −12.1244 − 14.0000i 21.0000i −11.5000 + 19.9186i 0
149.2 2.59808 + 1.50000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0 −6.00000 12.1244 + 14.0000i − 21.0000i −11.5000 + 19.9186i 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.k.b 4
5.b even 2 1 inner 175.4.k.b 4
5.c odd 4 1 35.4.e.a ✓ 2
5.c odd 4 1 175.4.e.b 2
7.c even 3 1 inner 175.4.k.b 4
15.e even 4 1 315.4.j.b 2
20.e even 4 1 560.4.q.b 2
35.f even 4 1 245.4.e.a 2
35.j even 6 1 inner 175.4.k.b 4
35.k even 12 1 245.4.a.f 1
35.k even 12 1 245.4.e.a 2
35.k even 12 1 1225.4.a.a 1
35.l odd 12 1 35.4.e.a ✓ 2
35.l odd 12 1 175.4.e.b 2
35.l odd 12 1 245.4.a.e 1
35.l odd 12 1 1225.4.a.b 1
105.w odd 12 1 2205.4.a.g 1
105.x even 12 1 315.4.j.b 2
105.x even 12 1 2205.4.a.e 1
140.w even 12 1 560.4.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a ✓ 2 5.c odd 4 1
35.4.e.a ✓ 2 35.l odd 12 1
175.4.e.b 2 5.c odd 4 1
175.4.e.b 2 35.l odd 12 1
175.4.k.b 4 1.a even 1 1 trivial
175.4.k.b 4 5.b even 2 1 inner
175.4.k.b 4 7.c even 3 1 inner
175.4.k.b 4 35.j even 6 1 inner
245.4.a.e 1 35.l odd 12 1
245.4.a.f 1 35.k even 12 1
245.4.e.a 2 35.f even 4 1
245.4.e.a 2 35.k even 12 1
315.4.j.b 2 15.e even 4 1
315.4.j.b 2 105.x even 12 1
560.4.q.b 2 20.e even 4 1
560.4.q.b 2 140.w even 12 1
1225.4.a.a 1 35.k even 12 1
1225.4.a.b 1 35.l odd 12 1
2205.4.a.e 1 105.x even 12 1
2205.4.a.g 1 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 9T_{2}^{2} + 81 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 98 T^{2} + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} - 45 T + 2025)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3481)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2916 T^{2} + 8503056 \) Copy content Toggle raw display
$19$ \( (T^{2} + 121 T + 14641)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 4761 T^{2} + 22667121 \) Copy content Toggle raw display
$29$ \( (T - 162)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 88 T + 7744)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 4499860561 \) Copy content Toggle raw display
$41$ \( (T - 195)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 81796)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 2025 T^{2} + 4100625 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 127027375281 \) Copy content Toggle raw display
$59$ \( (T^{2} + 360 T + 129600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 392 T + 153664)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 6146560000 \) Copy content Toggle raw display
$71$ \( (T - 48)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 199115858176 \) Copy content Toggle raw display
$79$ \( (T^{2} - 782 T + 611524)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 589824)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1194 T + 1425636)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 813604)^{2} \) Copy content Toggle raw display
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