Properties

Label 1617.4.a.l
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{37})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} - 3 q^{3} + (3 \beta + 2) q^{4} + ( - 4 \beta + 1) q^{5} + ( - 3 \beta - 3) q^{6} + 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} - 3 q^{3} + (3 \beta + 2) q^{4} + ( - 4 \beta + 1) q^{5} + ( - 3 \beta - 3) q^{6} + 21 q^{8} + 9 q^{9} + ( - 7 \beta - 35) q^{10} - 11 q^{11} + ( - 9 \beta - 6) q^{12} + ( - 4 \beta + 69) q^{13} + (12 \beta - 3) q^{15} + ( - 3 \beta + 5) q^{16} + (22 \beta + 2) q^{17} + (9 \beta + 9) q^{18} + (18 \beta - 85) q^{19} + ( - 17 \beta - 106) q^{20} + ( - 11 \beta - 11) q^{22} + (10 \beta + 84) q^{23} - 63 q^{24} + (8 \beta + 20) q^{25} + (61 \beta + 33) q^{26} - 27 q^{27} + ( - 60 \beta + 125) q^{29} + (21 \beta + 105) q^{30} + ( - 6 \beta - 170) q^{31} + ( - \beta - 190) q^{32} + 33 q^{33} + (46 \beta + 200) q^{34} + (27 \beta + 18) q^{36} + (64 \beta - 135) q^{37} + ( - 49 \beta + 77) q^{38} + (12 \beta - 207) q^{39} + ( - 84 \beta + 21) q^{40} + (60 \beta - 152) q^{41} + ( - 110 \beta + 176) q^{43} + ( - 33 \beta - 22) q^{44} + ( - 36 \beta + 9) q^{45} + (104 \beta + 174) q^{46} + (14 \beta - 467) q^{47} + (9 \beta - 15) q^{48} + (36 \beta + 92) q^{50} + ( - 66 \beta - 6) q^{51} + (187 \beta + 30) q^{52} + ( - 46 \beta - 252) q^{53} + ( - 27 \beta - 27) q^{54} + (44 \beta - 11) q^{55} + ( - 54 \beta + 255) q^{57} + (5 \beta - 415) q^{58} + (50 \beta + 181) q^{59} + (51 \beta + 318) q^{60} + ( - 128 \beta - 62) q^{61} + ( - 182 \beta - 224) q^{62} + ( - 168 \beta - 239) q^{64} + ( - 264 \beta + 213) q^{65} + (33 \beta + 33) q^{66} + ( - 38 \beta + 27) q^{67} + (116 \beta + 598) q^{68} + ( - 30 \beta - 252) q^{69} + ( - 140 \beta - 216) q^{71} + 189 q^{72} + ( - 212 \beta + 619) q^{73} + ( - 7 \beta + 441) q^{74} + ( - 24 \beta - 60) q^{75} + ( - 165 \beta + 316) q^{76} + ( - 183 \beta - 99) q^{78} + (12 \beta - 32) q^{79} + ( - 11 \beta + 113) q^{80} + 81 q^{81} + ( - 32 \beta + 388) q^{82} + (226 \beta - 182) q^{83} + ( - 74 \beta - 790) q^{85} + ( - 44 \beta - 814) q^{86} + (180 \beta - 375) q^{87} - 231 q^{88} + ( - 204 \beta - 654) q^{89} + ( - 63 \beta - 315) q^{90} + (302 \beta + 438) q^{92} + (18 \beta + 510) q^{93} + ( - 439 \beta - 341) q^{94} + (286 \beta - 733) q^{95} + (3 \beta + 570) q^{96} + ( - 6 \beta + 556) q^{97} - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 6 q^{3} + 7 q^{4} - 2 q^{5} - 9 q^{6} + 42 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - 6 q^{3} + 7 q^{4} - 2 q^{5} - 9 q^{6} + 42 q^{8} + 18 q^{9} - 77 q^{10} - 22 q^{11} - 21 q^{12} + 134 q^{13} + 6 q^{15} + 7 q^{16} + 26 q^{17} + 27 q^{18} - 152 q^{19} - 229 q^{20} - 33 q^{22} + 178 q^{23} - 126 q^{24} + 48 q^{25} + 127 q^{26} - 54 q^{27} + 190 q^{29} + 231 q^{30} - 346 q^{31} - 381 q^{32} + 66 q^{33} + 446 q^{34} + 63 q^{36} - 206 q^{37} + 105 q^{38} - 402 q^{39} - 42 q^{40} - 244 q^{41} + 242 q^{43} - 77 q^{44} - 18 q^{45} + 452 q^{46} - 920 q^{47} - 21 q^{48} + 220 q^{50} - 78 q^{51} + 247 q^{52} - 550 q^{53} - 81 q^{54} + 22 q^{55} + 456 q^{57} - 825 q^{58} + 412 q^{59} + 687 q^{60} - 252 q^{61} - 630 q^{62} - 646 q^{64} + 162 q^{65} + 99 q^{66} + 16 q^{67} + 1312 q^{68} - 534 q^{69} - 572 q^{71} + 378 q^{72} + 1026 q^{73} + 875 q^{74} - 144 q^{75} + 467 q^{76} - 381 q^{78} - 52 q^{79} + 215 q^{80} + 162 q^{81} + 744 q^{82} - 138 q^{83} - 1654 q^{85} - 1672 q^{86} - 570 q^{87} - 462 q^{88} - 1512 q^{89} - 693 q^{90} + 1178 q^{92} + 1038 q^{93} - 1121 q^{94} - 1180 q^{95} + 1143 q^{96} + 1106 q^{97} - 198 q^{99}+O(q^{100}) \) 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
−2.54138
3.54138
−1.54138 −3.00000 −5.62414 11.1655 4.62414 0 21.0000 9.00000 −17.2103
1.2 4.54138 −3.00000 12.6241 −13.1655 −13.6241 0 21.0000 9.00000 −59.7897
\(n\): 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.l 2
7.b odd 2 1 231.4.a.i 2
21.c even 2 1 693.4.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.i 2 7.b odd 2 1
693.4.a.h 2 21.c even 2 1
1617.4.a.l 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{2} - 3T_{2} - 7 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} - 147 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T - 7 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 147 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 134T + 4341 \) Copy content Toggle raw display
$17$ \( T^{2} - 26T - 4308 \) Copy content Toggle raw display
$19$ \( T^{2} + 152T + 2779 \) Copy content Toggle raw display
$23$ \( T^{2} - 178T + 6996 \) Copy content Toggle raw display
$29$ \( T^{2} - 190T - 24275 \) Copy content Toggle raw display
$31$ \( T^{2} + 346T + 29596 \) Copy content Toggle raw display
$37$ \( T^{2} + 206T - 27279 \) Copy content Toggle raw display
$41$ \( T^{2} + 244T - 18416 \) Copy content Toggle raw display
$43$ \( T^{2} - 242T - 97284 \) Copy content Toggle raw display
$47$ \( T^{2} + 920T + 209787 \) Copy content Toggle raw display
$53$ \( T^{2} + 550T + 56052 \) Copy content Toggle raw display
$59$ \( T^{2} - 412T + 19311 \) Copy content Toggle raw display
$61$ \( T^{2} + 252T - 135676 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T - 13293 \) Copy content Toggle raw display
$71$ \( T^{2} + 572T - 99504 \) Copy content Toggle raw display
$73$ \( T^{2} - 1026 T - 152563 \) Copy content Toggle raw display
$79$ \( T^{2} + 52T - 656 \) Copy content Toggle raw display
$83$ \( T^{2} + 138T - 467692 \) Copy content Toggle raw display
$89$ \( T^{2} + 1512 T + 186588 \) Copy content Toggle raw display
$97$ \( T^{2} - 1106 T + 305476 \) Copy content Toggle raw display
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