Properties

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

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Show commands: Magma / PariGP / SageMath

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: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
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.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - 7 \beta_{3} - 5) q^{4} + (5 \beta_{3} + 7) q^{6} + (4 \beta_{2} - 9 \beta_1) q^{7} + ( - 22 \beta_{2} - 25 \beta_1) q^{8} + ( - 3 \beta_{3} + 22) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - 7 \beta_{3} - 5) q^{4} + (5 \beta_{3} + 7) q^{6} + (4 \beta_{2} - 9 \beta_1) q^{7} + ( - 22 \beta_{2} - 25 \beta_1) q^{8} + ( - 3 \beta_{3} + 22) q^{9} - 11 q^{11} + (26 \beta_{2} + 19 \beta_1) q^{12} + ( - 10 \beta_{2} + 10 \beta_1) q^{13} + (19 \beta_{3} - 15) q^{14} + (63 \beta_{3} + 117) q^{16} + (42 \beta_{2} + 17 \beta_1) q^{17} + (32 \beta_{2} + 10 \beta_1) q^{18} + (45 \beta_{3} - 21) q^{19} + ( - \beta_{3} - 19) q^{21} + ( - 22 \beta_{2} - 11 \beta_1) q^{22} + (11 \beta_{2} - 4 \beta_1) q^{23} + ( - 69 \beta_{3} - 119) q^{24} + ( - 10 \beta_{3} + 50) q^{26} + ( - 40 \beta_{2} - 43 \beta_1) q^{27} + (78 \beta_{2} - 11 \beta_1) q^{28} + (71 \beta_{3} + 75) q^{29} + ( - 117 \beta_{3} + 129) q^{31} + (310 \beta_{2} + 169 \beta_1) q^{32} + (11 \beta_{2} + 11 \beta_1) q^{33} + ( - 135 \beta_{3} - 269) q^{34} + ( - 118 \beta_{3} - 26) q^{36} + (129 \beta_{2} - 43 \beta_1) q^{37} + (138 \beta_{2} + 159 \beta_1) q^{38} + ( - 10 \beta_{3} + 10) q^{39} + (156 \beta_{3} - 150) q^{41} + ( - 42 \beta_{2} - 23 \beta_1) q^{42} + (24 \beta_{2} + 156 \beta_1) q^{43} + (77 \beta_{3} + 55) q^{44} + ( - 10 \beta_{3} - 62) q^{46} + ( - 13 \beta_{2} - 100 \beta_1) q^{47} + ( - 306 \beta_{2} - 243 \beta_1) q^{48} + (225 \beta_{3} - 270) q^{49} + (101 \beta_{3} + 135) q^{51} + ( - 20 \beta_{2} + 90 \beta_1) q^{52} + ( - 13 \beta_{2} - 169 \beta_1) q^{53} + (209 \beta_{3} + 283) q^{54} + (29 \beta_{3} - 577) q^{56} + ( - 114 \beta_{2} - 69 \beta_1) q^{57} + (434 \beta_{2} + 359 \beta_1) q^{58} + ( - 158 \beta_{3} + 122) q^{59} + (119 \beta_{3} - 137) q^{61} + ( - 210 \beta_{2} - 339 \beta_1) q^{62} + (130 \beta_{2} - 222 \beta_1) q^{63} + ( - 623 \beta_{3} - 1093) q^{64} + ( - 55 \beta_{3} - 77) q^{66} + ( - 29 \beta_{2} + 150 \beta_1) q^{67} + ( - 742 \beta_{2} - 673 \beta_1) q^{68} + (18 \beta_{3} + 10) q^{69} + (61 \beta_{3} + 763) q^{71} + ( - 268 \beta_{2} - 418 \beta_1) q^{72} + (32 \beta_{2} + 58 \beta_1) q^{73} + ( - 129 \beta_{3} - 731) q^{74} + ( - 393 \beta_{3} - 1155) q^{76} + ( - 44 \beta_{2} + 99 \beta_1) q^{77} + ( - 20 \beta_{2} - 30 \beta_1) q^{78} + (114 \beta_{3} + 590) q^{79} + ( - 204 \beta_{3} + 385) q^{81} + (324 \beta_{2} + 474 \beta_1) q^{82} + ( - 72 \beta_{2} + 270 \beta_1) q^{83} + (145 \beta_{3} + 123) q^{84} + ( - 516 \beta_{3} - 300) q^{86} + ( - 288 \beta_{2} - 217 \beta_1) q^{87} + (242 \beta_{2} + 275 \beta_1) q^{88} + (43 \beta_{3} + 867) q^{89} + ( - 350 \beta_{3} + 870) q^{91} + ( - 76 \beta_{2} - 134 \beta_1) q^{92} + (222 \beta_{2} + 105 \beta_1) q^{93} + (326 \beta_{3} + 178) q^{94} + (789 \beta_{3} + 1127) q^{96} + (197 \beta_{2} + 454 \beta_1) q^{97} + (360 \beta_{2} + 630 \beta_1) q^{98} + (33 \beta_{3} - 242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 34 q^{4} + 38 q^{6} + 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 34 q^{4} + 38 q^{6} + 82 q^{9} - 44 q^{11} - 22 q^{14} + 594 q^{16} + 6 q^{19} - 78 q^{21} - 614 q^{24} + 180 q^{26} + 442 q^{29} + 282 q^{31} - 1346 q^{34} - 340 q^{36} + 20 q^{39} - 288 q^{41} + 374 q^{44} - 268 q^{46} - 630 q^{49} + 742 q^{51} + 1550 q^{54} - 2250 q^{56} + 172 q^{59} - 310 q^{61} - 5618 q^{64} - 418 q^{66} + 76 q^{69} + 3174 q^{71} - 3182 q^{74} - 5406 q^{76} + 2588 q^{79} + 1132 q^{81} + 782 q^{84} - 2232 q^{86} + 3554 q^{89} + 2780 q^{91} + 1364 q^{94} + 6086 q^{96} - 902 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 5\beta_1 \) 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
1.56155i
2.56155i
2.56155i
1.56155i
5.56155i 3.56155i −22.9309 0 19.8078 6.05398i 83.0388i 14.3153 0
199.2 1.43845i 0.561553i 5.93087 0 −0.807764 31.0540i 20.0388i 26.6847 0
199.3 1.43845i 0.561553i 5.93087 0 −0.807764 31.0540i 20.0388i 26.6847 0
199.4 5.56155i 3.56155i −22.9309 0 19.8078 6.05398i 83.0388i 14.3153 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.b 4
5.b even 2 1 inner 275.4.b.b 4
5.c odd 4 1 55.4.a.b 2
5.c odd 4 1 275.4.a.c 2
15.e even 4 1 495.4.a.e 2
15.e even 4 1 2475.4.a.l 2
20.e even 4 1 880.4.a.r 2
55.e even 4 1 605.4.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.b 2 5.c odd 4 1
275.4.a.c 2 5.c odd 4 1
275.4.b.b 4 1.a even 1 1 trivial
275.4.b.b 4 5.b even 2 1 inner
495.4.a.e 2 15.e even 4 1
605.4.a.g 2 55.e even 4 1
880.4.a.r 2 20.e even 4 1
2475.4.a.l 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 33T_{2}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 33T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1001 T^{2} + 35344 \) Copy content Toggle raw display
$11$ \( (T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2100 T^{2} + 40000 \) Copy content Toggle raw display
$17$ \( T^{4} + 13857 T^{2} + 19998784 \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T - 8604)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1288 T^{2} + 258064 \) Copy content Toggle raw display
$29$ \( (T^{2} - 221 T - 9214)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 141 T - 53208)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 4936748644 \) Copy content Toggle raw display
$41$ \( (T^{2} + 144 T - 98244)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 10511990784 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 1757621776 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 13915033444 \) Copy content Toggle raw display
$59$ \( (T^{2} - 86 T - 104248)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 155 T - 54178)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 6074020096 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1587 T + 613828)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 31044 T^{2} + 170877184 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1294 T + 363376)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 53816576256 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1777 T + 781574)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 719280394816 \) Copy content Toggle raw display
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