Properties

Label 1617.4.a.i
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

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: yes
Analytic conductor: \(95.4060884793\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta - 3) q^{4} + (3 \beta + 8) q^{5} + (3 \beta + 3) q^{6} + (5 \beta - 1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta - 3) q^{4} + (3 \beta + 8) q^{5} + (3 \beta + 3) q^{6} + (5 \beta - 1) q^{8} + 9 q^{9} + ( - 14 \beta - 20) q^{10} + 11 q^{11} + ( - 9 \beta + 9) q^{12} + ( - 23 \beta + 6) q^{13} + ( - 9 \beta - 24) q^{15} + ( - 33 \beta + 5) q^{16} + (2 \beta + 72) q^{17} + ( - 9 \beta - 9) q^{18} + ( - 31 \beta + 66) q^{19} + (24 \beta + 12) q^{20} + ( - 11 \beta - 11) q^{22} + (16 \beta - 100) q^{23} + ( - 15 \beta + 3) q^{24} + (57 \beta - 25) q^{25} + (40 \beta + 86) q^{26} - 27 q^{27} + (19 \beta - 118) q^{29} + (42 \beta + 60) q^{30} + ( - 32 \beta + 6) q^{31} + (21 \beta + 135) q^{32} - 33 q^{33} + ( - 76 \beta - 80) q^{34} + (27 \beta - 27) q^{36} + ( - 121 \beta - 26) q^{37} + ( - 4 \beta + 58) q^{38} + (69 \beta - 18) q^{39} + (52 \beta + 52) q^{40} + (140 \beta - 40) q^{41} + (12 \beta - 88) q^{43} + (33 \beta - 33) q^{44} + (27 \beta + 72) q^{45} + (68 \beta + 36) q^{46} + (167 \beta + 150) q^{47} + (99 \beta - 15) q^{48} + ( - 89 \beta - 203) q^{50} + ( - 6 \beta - 216) q^{51} + (18 \beta - 294) q^{52} + (74 \beta + 126) q^{53} + (27 \beta + 27) q^{54} + (33 \beta + 88) q^{55} + (93 \beta - 198) q^{57} + (80 \beta + 42) q^{58} + (101 \beta + 340) q^{59} + ( - 72 \beta - 36) q^{60} + (128 \beta + 10) q^{61} + (58 \beta + 122) q^{62} + (87 \beta - 259) q^{64} + ( - 235 \beta - 228) q^{65} + (33 \beta + 33) q^{66} + ( - 167 \beta + 320) q^{67} + (216 \beta - 192) q^{68} + ( - 48 \beta + 300) q^{69} + (380 \beta - 568) q^{71} + (45 \beta - 9) q^{72} + (13 \beta + 100) q^{73} + (268 \beta + 510) q^{74} + ( - 171 \beta + 75) q^{75} + (198 \beta - 570) q^{76} + ( - 120 \beta - 258) q^{78} + ( - 488 \beta + 328) q^{79} + ( - 348 \beta - 356) q^{80} + 81 q^{81} + ( - 240 \beta - 520) q^{82} + (84 \beta + 1006) q^{83} + (238 \beta + 600) q^{85} + (64 \beta + 40) q^{86} + ( - 57 \beta + 354) q^{87} + (55 \beta - 11) q^{88} + ( - 588 \beta - 74) q^{89} + ( - 126 \beta - 180) q^{90} + ( - 300 \beta + 492) q^{92} + (96 \beta - 18) q^{93} + ( - 484 \beta - 818) q^{94} + ( - 143 \beta + 156) q^{95} + ( - 63 \beta - 405) q^{96} + (294 \beta - 466) q^{97} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 6 q^{3} - 3 q^{4} + 19 q^{5} + 9 q^{6} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 6 q^{3} - 3 q^{4} + 19 q^{5} + 9 q^{6} + 3 q^{8} + 18 q^{9} - 54 q^{10} + 22 q^{11} + 9 q^{12} - 11 q^{13} - 57 q^{15} - 23 q^{16} + 146 q^{17} - 27 q^{18} + 101 q^{19} + 48 q^{20} - 33 q^{22} - 184 q^{23} - 9 q^{24} + 7 q^{25} + 212 q^{26} - 54 q^{27} - 217 q^{29} + 162 q^{30} - 20 q^{31} + 291 q^{32} - 66 q^{33} - 236 q^{34} - 27 q^{36} - 173 q^{37} + 112 q^{38} + 33 q^{39} + 156 q^{40} + 60 q^{41} - 164 q^{43} - 33 q^{44} + 171 q^{45} + 140 q^{46} + 467 q^{47} + 69 q^{48} - 495 q^{50} - 438 q^{51} - 570 q^{52} + 326 q^{53} + 81 q^{54} + 209 q^{55} - 303 q^{57} + 164 q^{58} + 781 q^{59} - 144 q^{60} + 148 q^{61} + 302 q^{62} - 431 q^{64} - 691 q^{65} + 99 q^{66} + 473 q^{67} - 168 q^{68} + 552 q^{69} - 756 q^{71} + 27 q^{72} + 213 q^{73} + 1288 q^{74} - 21 q^{75} - 942 q^{76} - 636 q^{78} + 168 q^{79} - 1060 q^{80} + 162 q^{81} - 1280 q^{82} + 2096 q^{83} + 1438 q^{85} + 144 q^{86} + 651 q^{87} + 33 q^{88} - 736 q^{89} - 486 q^{90} + 684 q^{92} + 60 q^{93} - 2120 q^{94} + 169 q^{95} - 873 q^{96} - 638 q^{97} + 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
−3.56155 −3.00000 4.68466 15.6847 10.6847 0 11.8078 9.00000 −55.8617
1.2 0.561553 −3.00000 −7.68466 3.31534 −1.68466 0 −8.80776 9.00000 1.86174
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1617.4.a.i 2
7.b odd 2 1 231.4.a.f 2
21.c even 2 1 693.4.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.f 2 7.b odd 2 1
693.4.a.k 2 21.c even 2 1
1617.4.a.i 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{2} + 3T_{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 19T_{5} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 19T + 52 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 11T - 2218 \) Copy content Toggle raw display
$17$ \( T^{2} - 146T + 5312 \) Copy content Toggle raw display
$19$ \( T^{2} - 101T - 1534 \) Copy content Toggle raw display
$23$ \( T^{2} + 184T + 7376 \) Copy content Toggle raw display
$29$ \( T^{2} + 217T + 10238 \) Copy content Toggle raw display
$31$ \( T^{2} + 20T - 4252 \) Copy content Toggle raw display
$37$ \( T^{2} + 173T - 54742 \) Copy content Toggle raw display
$41$ \( T^{2} - 60T - 82400 \) Copy content Toggle raw display
$43$ \( T^{2} + 164T + 6112 \) Copy content Toggle raw display
$47$ \( T^{2} - 467T - 64006 \) Copy content Toggle raw display
$53$ \( T^{2} - 326T + 3296 \) Copy content Toggle raw display
$59$ \( T^{2} - 781T + 109136 \) Copy content Toggle raw display
$61$ \( T^{2} - 148T - 64156 \) Copy content Toggle raw display
$67$ \( T^{2} - 473T - 62596 \) Copy content Toggle raw display
$71$ \( T^{2} + 756T - 470816 \) Copy content Toggle raw display
$73$ \( T^{2} - 213T + 10624 \) Copy content Toggle raw display
$79$ \( T^{2} - 168 T - 1005056 \) Copy content Toggle raw display
$83$ \( T^{2} - 2096 T + 1068316 \) Copy content Toggle raw display
$89$ \( T^{2} + 736 T - 1333988 \) Copy content Toggle raw display
$97$ \( T^{2} + 638T - 265592 \) Copy content Toggle raw display
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