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Properties

Label 3484.2.a.e
Level $3484$
Weight $2$
Character orbit 3484.a
Self dual yes
Analytic conductor $27.820$
Analytic rank $1$
Dimension $17$
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] = [3484,2,Mod(1,3484)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3484, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3484.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3484 = 2^{2} \cdot 13 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3484.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: \(27.8198800642\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 35 x^{15} - x^{14} + 490 x^{13} + 19 x^{12} - 3568 x^{11} - 118 x^{10} + 14753 x^{9} + \cdots - 1856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{9} q^{5} - \beta_{6} q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{9} q^{5} - \beta_{6} q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{5} - 1) q^{11} - q^{13} + (\beta_{15} - \beta_{13} + \beta_{9} + \cdots - 2) q^{15}+ \cdots + ( - \beta_{15} + \beta_{14} + 2 \beta_{12} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 7 q^{5} - 3 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 7 q^{5} - 3 q^{7} + 19 q^{9} - 14 q^{11} - 17 q^{13} - 16 q^{15} + 3 q^{17} - 5 q^{19} - 23 q^{21} - 13 q^{23} + 28 q^{25} + 3 q^{27} - 24 q^{29} - 18 q^{31} - 4 q^{33} - 10 q^{35} - 8 q^{37} - 7 q^{41} - 29 q^{43} - 36 q^{45} - 27 q^{47} + 16 q^{49} - 22 q^{51} - 23 q^{53} - 31 q^{55} - 10 q^{57} - 45 q^{59} - 11 q^{61} - 43 q^{63} + 7 q^{65} + 17 q^{67} - 37 q^{69} - 72 q^{71} - 13 q^{73} - 26 q^{75} - 16 q^{79} + 13 q^{81} - 24 q^{83} - 30 q^{85} - q^{87} + 24 q^{89} + 3 q^{91} - 20 q^{93} - 19 q^{95} - 5 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 35 x^{15} - x^{14} + 490 x^{13} + 19 x^{12} - 3568 x^{11} - 118 x^{10} + 14753 x^{9} + \cdots - 1856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1250999 \nu^{16} - 2709522 \nu^{15} - 45036471 \nu^{14} + 82232315 \nu^{13} + 655628238 \nu^{12} + \cdots + 6299218288 ) / 86311504 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40408485 \nu^{16} - 26647496 \nu^{15} + 1357844737 \nu^{14} + 902852041 \nu^{13} + \cdots - 144289549328 ) / 949426544 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4947801 \nu^{16} + 20208825 \nu^{15} + 162112743 \nu^{14} - 656859890 \nu^{13} + \cdots + 16378322032 ) / 86311504 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 187375633 \nu^{16} + 583555572 \nu^{15} + 6104043951 \nu^{14} - 18831648331 \nu^{13} + \cdots + 498007390368 ) / 1898853088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 259509403 \nu^{16} + 51040760 \nu^{15} + 8400970005 \nu^{14} - 1522063853 \nu^{13} + \cdots + 178436623296 ) / 1898853088 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 72684089 \nu^{16} - 230014900 \nu^{15} + 2364407303 \nu^{14} + 7618052507 \nu^{13} + \cdots - 171657402440 ) / 474713272 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 158243551 \nu^{16} - 316863191 \nu^{15} - 5148176103 \nu^{14} + 10116191902 \nu^{13} + \cdots - 224024211104 ) / 949426544 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 203500661 \nu^{16} - 375052622 \nu^{15} - 6657832215 \nu^{14} + 11984289689 \nu^{13} + \cdots - 224437930544 ) / 949426544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 224837467 \nu^{16} + 429232301 \nu^{15} + 7273092395 \nu^{14} - 13799433588 \nu^{13} + \cdots + 412961859488 ) / 949426544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 290554518 \nu^{16} + 464979503 \nu^{15} + 9492686468 \nu^{14} - 14901511667 \nu^{13} + \cdots + 459507742400 ) / 949426544 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 67847167 \nu^{16} - 133748232 \nu^{15} - 2212873345 \nu^{14} + 4311494729 \nu^{13} + \cdots - 113407797120 ) / 172623008 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 816698877 \nu^{16} - 785471924 \nu^{15} - 26827888411 \nu^{14} + 24954530327 \nu^{13} + \cdots - 643131470304 ) / 1898853088 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1010263125 \nu^{16} - 1731105010 \nu^{15} - 32869238383 \nu^{14} + 55531023201 \nu^{13} + \cdots - 1540631409920 ) / 1898853088 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1048036991 \nu^{16} - 2074023236 \nu^{15} - 34225026177 \nu^{14} + 66610807261 \nu^{13} + \cdots - 1662677145088 ) / 1898853088 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{13} - \beta_{12} + 2\beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} - \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + \beta_{9} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{16} - 3 \beta_{15} - 15 \beta_{14} + 18 \beta_{13} - 13 \beta_{12} + 28 \beta_{11} - 8 \beta_{10} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{16} + 22 \beta_{15} - 37 \beta_{14} + 55 \beta_{13} - 18 \beta_{12} + 62 \beta_{11} + \cdots + 250 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41 \beta_{16} - 56 \beta_{15} - 188 \beta_{14} + 251 \beta_{13} - 140 \beta_{12} + 334 \beta_{11} + \cdots + 110 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 88 \beta_{16} + 189 \beta_{15} - 520 \beta_{14} + 775 \beta_{13} - 248 \beta_{12} + 785 \beta_{11} + \cdots + 2469 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 612 \beta_{16} - 779 \beta_{15} - 2242 \beta_{14} + 3204 \beta_{13} - 1461 \beta_{12} + 3842 \beta_{11} + \cdots + 1782 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1355 \beta_{16} + 1463 \beta_{15} - 6630 \beta_{14} + 9946 \beta_{13} - 3122 \beta_{12} + 9428 \beta_{11} + \cdots + 25490 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8088 \beta_{16} - 9745 \beta_{15} - 26216 \beta_{14} + 39210 \beta_{13} - 15295 \beta_{12} + \cdots + 25602 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18254 \beta_{16} + 10136 \beta_{15} - 80860 \beta_{14} + 122158 \beta_{13} - 37802 \beta_{12} + \cdots + 269580 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 100633 \beta_{16} - 115838 \beta_{15} - 303753 \beta_{14} + 468685 \beta_{13} - 162186 \beta_{12} + \cdots + 344840 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 231126 \beta_{16} + 55594 \beta_{15} - 963917 \beta_{14} + 1465469 \beta_{13} - 448976 \beta_{12} + \cdots + 2897144 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1210639 \beta_{16} - 1338699 \beta_{15} - 3503953 \beta_{14} + 5523304 \beta_{13} - 1746128 \beta_{12} + \cdots + 4460074 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2834249 \beta_{16} + 82774 \beta_{15} - 11346110 \beta_{14} + 17343003 \beta_{13} - 5275679 \beta_{12} + \cdots + 31512763 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−3.16048
−3.02778
−2.23941
−1.74589
−1.61705
−1.54176
−1.51231
−0.679048
0.381468
0.415908
1.35993
1.47806
1.56920
1.63683
2.49724
2.79025
3.39484
0 −3.16048 0 1.51029 0 −1.10867 0 6.98862 0
1.2 0 −3.02778 0 −1.25997 0 4.51199 0 6.16742 0
1.3 0 −2.23941 0 −3.79365 0 −3.63727 0 2.01495 0
1.4 0 −1.74589 0 4.07288 0 −4.09513 0 0.0481467 0
1.5 0 −1.61705 0 −4.04956 0 0.847931 0 −0.385160 0
1.6 0 −1.54176 0 1.39961 0 0.579497 0 −0.622965 0
1.7 0 −1.51231 0 −0.227301 0 4.20645 0 −0.712917 0
1.8 0 −0.679048 0 4.12546 0 1.67716 0 −2.53889 0
1.9 0 0.381468 0 −0.958071 0 −2.97930 0 −2.85448 0
1.10 0 0.415908 0 −0.459712 0 −0.448330 0 −2.82702 0
1.11 0 1.35993 0 −3.67598 0 3.86644 0 −1.15059 0
1.12 0 1.47806 0 −0.0917601 0 2.17438 0 −0.815353 0
1.13 0 1.56920 0 2.34985 0 −1.64615 0 −0.537600 0
1.14 0 1.63683 0 −0.325631 0 1.80521 0 −0.320773 0
1.15 0 2.49724 0 −4.10896 0 −1.89558 0 3.23620 0
1.16 0 2.79025 0 0.733583 0 −3.51365 0 4.78547 0
1.17 0 3.39484 0 −2.24107 0 −3.34498 0 8.52494 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)
\(67\) \(-1\)

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 3484.2.a.e ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3484.2.a.e ✓ 17 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3484))\):

\( T_{3}^{17} - 35 T_{3}^{15} - T_{3}^{14} + 490 T_{3}^{13} + 19 T_{3}^{12} - 3568 T_{3}^{11} - 118 T_{3}^{10} + \cdots - 1856 \) Copy content Toggle raw display
\( T_{5}^{17} + 7 T_{5}^{16} - 32 T_{5}^{15} - 306 T_{5}^{14} + 144 T_{5}^{13} + 4564 T_{5}^{12} + \cdots + 120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} \) Copy content Toggle raw display
$3$ \( T^{17} - 35 T^{15} + \cdots - 1856 \) Copy content Toggle raw display
$5$ \( T^{17} + 7 T^{16} + \cdots + 120 \) Copy content Toggle raw display
$7$ \( T^{17} + 3 T^{16} + \cdots + 192030 \) Copy content Toggle raw display
$11$ \( T^{17} + 14 T^{16} + \cdots + 4220670 \) Copy content Toggle raw display
$13$ \( (T + 1)^{17} \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 5243251593 \) Copy content Toggle raw display
$19$ \( T^{17} + 5 T^{16} + \cdots + 54852320 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 2345045699 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 27825269499 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 156687443088 \) Copy content Toggle raw display
$37$ \( T^{17} + 8 T^{16} + \cdots - 83419552 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 20043844490 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 652761061696 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 10052125379680 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 18595273160256 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 154368862880 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 33473135936 \) Copy content Toggle raw display
$67$ \( (T - 1)^{17} \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 77190465360000 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 83263730336 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 18093720863744 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 358983614655840 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
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