Properties

Label 275.4.b.c
Level $275$
Weight $4$
Character orbit 275.b
Analytic conductor $16.226$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,4,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, 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("275.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(16.2255252516\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{2} + \beta_1) q^{3} + (2 \beta_{3} + 4) q^{4} + (5 \beta_{3} - 13) q^{6} + (4 \beta_{2} + 10 \beta_1) q^{7} + ( - 10 \beta_{2} + 6 \beta_1) q^{8} + (8 \beta_{3} - 22) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{2} + \beta_1) q^{3} + (2 \beta_{3} + 4) q^{4} + (5 \beta_{3} - 13) q^{6} + (4 \beta_{2} + 10 \beta_1) q^{7} + ( - 10 \beta_{2} + 6 \beta_1) q^{8} + (8 \beta_{3} - 22) q^{9} - 11 q^{11} + ( - 14 \beta_{2} - 20 \beta_1) q^{12} + ( - 20 \beta_{2} - 40 \beta_1) q^{13} + (6 \beta_{3} + 2) q^{14} + (32 \beta_{3} - 4) q^{16} + ( - 12 \beta_{2} - 62 \beta_1) q^{17} + (30 \beta_{2} - 46 \beta_1) q^{18} + (60 \beta_{3} - 36) q^{19} + (36 \beta_{3} + 38) q^{21} + (11 \beta_{2} - 11 \beta_1) q^{22} + ( - 36 \beta_{2} + 49 \beta_1) q^{23} + (34 \beta_{3} - 126) q^{24} + ( - 20 \beta_{3} - 20) q^{26} + ( - 12 \beta_{2} - 91 \beta_1) q^{27} + (36 \beta_{2} + 64 \beta_1) q^{28} + ( - 56 \beta_{3} - 72) q^{29} + ( - 28 \beta_{3} - 17) q^{31} + ( - 44 \beta_{2} - 52 \beta_1) q^{32} + (44 \beta_{2} - 11 \beta_1) q^{33} + ( - 50 \beta_{3} + 26) q^{34} + ( - 12 \beta_{3} - 40) q^{36} + (8 \beta_{2} + 27 \beta_1) q^{37} + (96 \beta_{2} - 216 \beta_1) q^{38} + ( - 140 \beta_{3} - 200) q^{39} + (4 \beta_{3} + 268) q^{41} + ( - 2 \beta_{2} - 70 \beta_1) q^{42} + ( - 16 \beta_{2} + 30 \beta_1) q^{43} + ( - 22 \beta_{3} - 44) q^{44} + (85 \beta_{3} - 157) q^{46} + (120 \beta_{2} - 136 \beta_1) q^{47} + (48 \beta_{2} - 388 \beta_1) q^{48} + ( - 80 \beta_{3} + 195) q^{49} + ( - 236 \beta_{3} - 82) q^{51} + ( - 160 \beta_{2} - 280 \beta_1) q^{52} + ( - 56 \beta_{2} + 246 \beta_1) q^{53} + ( - 79 \beta_{3} + 55) q^{54} + (76 \beta_{3} + 60) q^{56} + (204 \beta_{2} - 756 \beta_1) q^{57} + (16 \beta_{2} + 96 \beta_1) q^{58} + ( - 132 \beta_{3} - 317) q^{59} + ( - 184 \beta_{3} + 420) q^{61} + ( - 11 \beta_{2} + 67 \beta_1) q^{62} + ( - 8 \beta_{2} - 124 \beta_1) q^{63} + (248 \beta_{3} - 112) q^{64} + ( - 55 \beta_{3} + 143) q^{66} + (20 \beta_{2} + 377 \beta_1) q^{67} + ( - 172 \beta_{2} - 320 \beta_1) q^{68} + (232 \beta_{3} - 481) q^{69} + ( - 76 \beta_{3} - 339) q^{71} + (268 \beta_{2} - 372 \beta_1) q^{72} + ( - 468 \beta_{2} + 200 \beta_1) q^{73} + (19 \beta_{3} - 3) q^{74} + (168 \beta_{3} + 216) q^{76} + ( - 44 \beta_{2} - 110 \beta_1) q^{77} + (60 \beta_{2} + 220 \beta_1) q^{78} + (656 \beta_{3} - 158) q^{79} + ( - 136 \beta_{3} - 647) q^{81} + ( - 264 \beta_{2} + 256 \beta_1) q^{82} + (120 \beta_{2} - 234 \beta_1) q^{83} + (220 \beta_{3} + 368) q^{84} + (46 \beta_{3} - 78) q^{86} + (232 \beta_{2} + 600 \beta_1) q^{87} + (110 \beta_{2} - 66 \beta_1) q^{88} + ( - 328 \beta_{3} + 921) q^{89} + (360 \beta_{3} + 640) q^{91} + ( - 46 \beta_{2} - 20 \beta_1) q^{92} + (40 \beta_{2} + 319 \beta_1) q^{93} + ( - 256 \beta_{3} + 496) q^{94} + ( - 164 \beta_{3} - 476) q^{96} + ( - 144 \beta_{2} + 1097 \beta_1) q^{97} + ( - 275 \beta_{2} + 435 \beta_1) q^{98} + ( - 88 \beta_{3} + 242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
2.73205i 7.92820i 0.535898 0 −21.6603 3.07180i 23.3205i −35.8564 0
199.2 0.732051i 5.92820i 7.46410 0 −4.33975 16.9282i 11.3205i −8.14359 0
199.3 0.732051i 5.92820i 7.46410 0 −4.33975 16.9282i 11.3205i −8.14359 0
199.4 2.73205i 7.92820i 0.535898 0 −21.6603 3.07180i 23.3205i −35.8564 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 275.4.b.c 4
5.b even 2 1 inner 275.4.b.c 4
5.c odd 4 1 11.4.a.a 2
5.c odd 4 1 275.4.a.b 2
15.e even 4 1 99.4.a.c 2
15.e even 4 1 2475.4.a.q 2
20.e even 4 1 176.4.a.i 2
35.f even 4 1 539.4.a.e 2
40.i odd 4 1 704.4.a.p 2
40.k even 4 1 704.4.a.n 2
55.e even 4 1 121.4.a.c 2
55.k odd 20 4 121.4.c.c 8
55.l even 20 4 121.4.c.f 8
60.l odd 4 1 1584.4.a.bc 2
65.h odd 4 1 1859.4.a.a 2
165.l odd 4 1 1089.4.a.v 2
220.i odd 4 1 1936.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 5.c odd 4 1
99.4.a.c 2 15.e even 4 1
121.4.a.c 2 55.e even 4 1
121.4.c.c 8 55.k odd 20 4
121.4.c.f 8 55.l even 20 4
176.4.a.i 2 20.e even 4 1
275.4.a.b 2 5.c odd 4 1
275.4.b.c 4 1.a even 1 1 trivial
275.4.b.c 4 5.b even 2 1 inner
539.4.a.e 2 35.f even 4 1
704.4.a.n 2 40.k even 4 1
704.4.a.p 2 40.i odd 4 1
1089.4.a.v 2 165.l odd 4 1
1584.4.a.bc 2 60.l odd 4 1
1859.4.a.a 2 65.h odd 4 1
1936.4.a.w 2 220.i odd 4 1
2475.4.a.q 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 8T_{2}^{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 98T^{2} + 2209 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 296T^{2} + 2704 \) Copy content Toggle raw display
$11$ \( (T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 5600 T^{2} + 160000 \) Copy content Toggle raw display
$17$ \( T^{4} + 8552 T^{2} + 11641744 \) Copy content Toggle raw display
$19$ \( (T^{2} + 72 T - 9504)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12578 T^{2} + 2211169 \) Copy content Toggle raw display
$29$ \( (T^{2} + 144 T - 4224)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 34 T - 2063)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 1842 T^{2} + 288369 \) Copy content Toggle raw display
$41$ \( (T^{2} - 536 T + 71776)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 3336 T^{2} + 17424 \) Copy content Toggle raw display
$47$ \( T^{4} + 123392 T^{2} + 610287616 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 2612027664 \) Copy content Toggle raw display
$59$ \( (T^{2} + 634 T + 48217)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 840 T + 74832)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19860983041 \) Copy content Toggle raw display
$71$ \( (T^{2} + 678 T + 97593)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 380777853184 \) Copy content Toggle raw display
$79$ \( (T^{2} + 316 T - 1266044)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 195912 T^{2} + 133541136 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1842 T + 525489)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 1302339722401 \) Copy content Toggle raw display
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