Properties

Label 3484.2.a.d
Level $3484$
Weight $2$
Character orbit 3484.a
Self dual yes
Analytic conductor $27.820$
Analytic rank $1$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} - 13 x^{14} + 127 x^{13} + 6 x^{12} - 1031 x^{11} + 590 x^{10} + 4092 x^{9} - 3353 x^{8} - 8208 x^{7} + 7674 x^{6} + 7534 x^{5} - 7680 x^{4} - 2216 x^{3} + \cdots - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{13} q^{5} - \beta_{6} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{13} q^{5} - \beta_{6} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{5} - 1) q^{11} + q^{13} + ( - \beta_{13} - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} + \beta_1 - 2) q^{15} + ( - \beta_{14} + \beta_{6} + \beta_1 - 1) q^{17} + (\beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{8} + \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{19} + ( - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{2} + \cdots + 1) q^{21}+ \cdots + (2 \beta_{15} - 3 \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + 3 \beta_{11} + 2 \beta_{10} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{3} - 5 q^{5} - q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{3} - 5 q^{5} - q^{7} + 14 q^{9} - 16 q^{11} + 16 q^{13} - 10 q^{15} - 12 q^{17} - 10 q^{19} - q^{21} - 28 q^{23} + 13 q^{25} - 33 q^{27} - 15 q^{29} - 8 q^{31} + 2 q^{33} - 38 q^{35} + 7 q^{37} - 6 q^{39} + 3 q^{41} - 11 q^{43} - 10 q^{45} - 16 q^{47} + 19 q^{49} - 34 q^{51} - 15 q^{53} + 21 q^{55} + 2 q^{57} - 44 q^{59} - 15 q^{61} + 15 q^{63} - 5 q^{65} - 16 q^{67} + 7 q^{69} - 36 q^{71} + 2 q^{73} - 6 q^{75} - 44 q^{77} - 20 q^{79} + 20 q^{81} - 20 q^{83} + 12 q^{85} - 23 q^{87} - 11 q^{89} - q^{91} - 22 q^{93} - 21 q^{95} + 17 q^{97} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} - 13 x^{14} + 127 x^{13} + 6 x^{12} - 1031 x^{11} + 590 x^{10} + 4092 x^{9} - 3353 x^{8} - 8208 x^{7} + 7674 x^{6} + 7534 x^{5} - 7680 x^{4} - 2216 x^{3} + \cdots - 192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1090921 \nu^{15} + 6675442 \nu^{14} + 18374929 \nu^{13} - 168206317 \nu^{12} - 46849202 \nu^{11} + 1641239141 \nu^{10} - 779755000 \nu^{9} + \cdots - 924916456 ) / 189112568 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13511479 \nu^{15} + 75201791 \nu^{14} + 246770989 \nu^{13} - 1922136470 \nu^{12} - 922993759 \nu^{11} + 19001042731 \nu^{10} + \cdots - 6028692224 ) / 1891125680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16505038 \nu^{15} - 39800417 \nu^{14} - 548236188 \nu^{13} + 1260761595 \nu^{12} + 6851050173 \nu^{11} - 14404998772 \nu^{10} + \cdots - 2069299472 ) / 1891125680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38843367 \nu^{15} + 200395618 \nu^{14} + 624093087 \nu^{13} - 4116650425 \nu^{12} - 3215664662 \nu^{11} + 32033800573 \nu^{10} + \cdots - 15557245792 ) / 3782251360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20070363 \nu^{15} - 142602997 \nu^{14} - 47967303 \nu^{13} + 2328516040 \nu^{12} - 3330793697 \nu^{11} - 11679639557 \nu^{10} + \cdots - 5179903792 ) / 1891125680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28683647 \nu^{15} - 150366643 \nu^{14} - 484891557 \nu^{13} + 3313803150 \nu^{12} + 2340810667 \nu^{11} - 27897523743 \nu^{10} + \cdots + 876810832 ) / 1891125680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14588331 \nu^{15} + 69084862 \nu^{14} + 295749919 \nu^{13} - 1591477361 \nu^{12} - 2300265574 \nu^{11} + 14288405053 \nu^{10} + \cdots - 6642684928 ) / 756450272 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 85227277 \nu^{15} - 535978408 \nu^{14} - 929428877 \nu^{13} + 10882327005 \nu^{12} - 2522124168 \nu^{11} - 83433450863 \nu^{10} + \cdots - 8453244768 ) / 3782251360 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46068828 \nu^{15} + 263714917 \nu^{14} + 605670138 \nu^{13} - 5226081965 \nu^{12} - 1328519393 \nu^{11} + 38837549742 \nu^{10} + \cdots - 3563858368 ) / 1891125680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21972371 \nu^{15} + 105943346 \nu^{14} + 413482003 \nu^{13} - 2313607973 \nu^{12} - 2954957374 \nu^{11} + 19570308217 \nu^{10} + \cdots - 7200192864 ) / 756450272 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13493606 \nu^{15} - 66977027 \nu^{14} - 248247966 \nu^{13} + 1480576025 \nu^{12} + 1631791717 \nu^{11} - 12661064650 \nu^{10} + \cdots + 3042432096 ) / 378225136 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 86384101 \nu^{15} + 402416734 \nu^{14} + 1666364321 \nu^{13} - 8709154995 \nu^{12} - 12553374726 \nu^{11} + 72224493019 \nu^{10} + \cdots - 7446021776 ) / 1891125680 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 208429481 \nu^{15} + 1118999184 \nu^{14} + 3378466541 \nu^{13} - 24063015185 \nu^{12} - 16267075396 \nu^{11} + 199696406039 \nu^{10} + \cdots - 76375772256 ) / 3782251360 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{8} + \beta_{3} + 2\beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} - 3 \beta_{6} + \beta_{4} + 10 \beta_{2} + 11 \beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 13 \beta_{14} + 2 \beta_{13} - 3 \beta_{12} - 9 \beta_{11} - 8 \beta_{10} - 11 \beta_{9} + 23 \beta_{8} - \beta_{7} - 4 \beta_{6} + 3 \beta_{4} + 11 \beta_{3} + 23 \beta_{2} + 58 \beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{15} + 31 \beta_{14} + 31 \beta_{13} - 3 \beta_{12} - 11 \beta_{11} - 9 \beta_{10} - 15 \beta_{9} + 30 \beta_{8} - \beta_{7} - 43 \beta_{6} + \beta_{5} + 17 \beta_{4} + 4 \beta_{3} + 93 \beta_{2} + 111 \beta _1 + 248 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 39 \beta_{15} + 144 \beta_{14} + 48 \beta_{13} - 49 \beta_{12} - 70 \beta_{11} - 56 \beta_{10} - 108 \beta_{9} + 228 \beta_{8} - 14 \beta_{7} - 74 \beta_{6} - 2 \beta_{5} + 55 \beta_{4} + 103 \beta_{3} + 234 \beta_{2} + 513 \beta _1 + 461 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 196 \beta_{15} + 380 \beta_{14} + 379 \beta_{13} - 69 \beta_{12} - 87 \beta_{11} - 56 \beta_{10} - 175 \beta_{9} + 365 \beta_{8} - 22 \beta_{7} - 496 \beta_{6} + 14 \beta_{5} + 231 \beta_{4} + 81 \beta_{3} + 869 \beta_{2} + 1103 \beta _1 + 2277 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 531 \beta_{15} + 1532 \beta_{14} + 761 \beta_{13} - 605 \beta_{12} - 500 \beta_{11} - 348 \beta_{10} - 1009 \beta_{9} + 2195 \beta_{8} - 170 \beta_{7} - 1021 \beta_{6} - 33 \beta_{5} + 766 \beta_{4} + 952 \beta_{3} + \cdots + 4945 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2205 \beta_{15} + 4292 \beta_{14} + 4321 \beta_{13} - 1075 \beta_{12} - 537 \beta_{11} - 173 \beta_{10} - 1834 \beta_{9} + 4112 \beta_{8} - 364 \beta_{7} - 5377 \beta_{6} + 124 \beta_{5} + 2878 \beta_{4} + 1147 \beta_{3} + \cdots + 21703 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6337 \beta_{15} + 16026 \beta_{14} + 10189 \beta_{13} - 6795 \beta_{12} - 3181 \beta_{11} - 1642 \beta_{10} - 9176 \beta_{9} + 21052 \beta_{8} - 2043 \beta_{7} - 12556 \beta_{6} - 426 \beta_{5} + 9578 \beta_{4} + \cdots + 52225 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 23908 \beta_{15} + 46616 \beta_{14} + 47850 \beta_{13} - 14237 \beta_{12} - 1834 \beta_{11} + 2076 \beta_{10} - 18128 \beta_{9} + 44471 \beta_{8} - 5252 \beta_{7} - 56965 \beta_{6} + 711 \beta_{5} + \cdots + 211532 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 71188 \beta_{15} + 166081 \beta_{14} + 124971 \beta_{13} - 73366 \beta_{12} - 15678 \beta_{11} - 465 \beta_{10} - 82194 \beta_{9} + 202618 \beta_{8} - 24362 \beta_{7} - 145433 \beta_{6} + \cdots + 547478 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 254526 \beta_{15} + 495313 \beta_{14} + 521539 \beta_{13} - 173066 \beta_{12} + 15958 \beta_{11} + 56349 \beta_{10} - 173100 \beta_{9} + 469312 \beta_{8} - 69318 \beta_{7} - 597573 \beta_{6} + \cdots + 2091816 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 775476 \beta_{15} + 1710885 \beta_{14} + 1455842 \beta_{13} - 778243 \beta_{12} - 14719 \beta_{11} + 134060 \beta_{10} - 730134 \beta_{9} + 1961101 \beta_{8} - 285820 \beta_{7} + \cdots + 5710960 \) 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
3.21439
3.18799
2.89719
2.48965
1.75278
1.16738
1.01839
0.681028
0.361559
−0.302409
−0.652107
−1.48999
−1.59846
−1.83356
−2.11732
−2.77650
0 −3.21439 0 2.95579 0 1.94061 0 7.33229 0
1.2 0 −3.18799 0 2.04913 0 −1.16411 0 7.16325 0
1.3 0 −2.89719 0 −3.64399 0 3.06957 0 5.39372 0
1.4 0 −2.48965 0 −1.85473 0 −3.91149 0 3.19834 0
1.5 0 −1.75278 0 0.384605 0 −3.35461 0 0.0722277 0
1.6 0 −1.16738 0 −2.53209 0 −1.47509 0 −1.63723 0
1.7 0 −1.01839 0 2.04531 0 1.69338 0 −1.96289 0
1.8 0 −0.681028 0 −2.98343 0 3.54536 0 −2.53620 0
1.9 0 −0.361559 0 0.0841576 0 4.07977 0 −2.86928 0
1.10 0 0.302409 0 0.530429 0 0.114386 0 −2.90855 0
1.11 0 0.652107 0 3.53364 0 −4.46175 0 −2.57476 0
1.12 0 1.48999 0 2.63908 0 −2.27288 0 −0.779936 0
1.13 0 1.59846 0 −1.00607 0 0.534024 0 −0.444933 0
1.14 0 1.83356 0 −3.74118 0 0.127204 0 0.361928 0
1.15 0 2.11732 0 −0.512609 0 −3.61258 0 1.48304 0
1.16 0 2.77650 0 −2.94805 0 4.14820 0 4.70897 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3484.2.a.d 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3484))\):

\( T_{3}^{16} + 6 T_{3}^{15} - 13 T_{3}^{14} - 127 T_{3}^{13} + 6 T_{3}^{12} + 1031 T_{3}^{11} + 590 T_{3}^{10} - 4092 T_{3}^{9} - 3353 T_{3}^{8} + 8208 T_{3}^{7} + 7674 T_{3}^{6} - 7534 T_{3}^{5} - 7680 T_{3}^{4} + 2216 T_{3}^{3} + \cdots - 192 \) Copy content Toggle raw display
\( T_{5}^{16} + 5 T_{5}^{15} - 34 T_{5}^{14} - 188 T_{5}^{13} + 420 T_{5}^{12} + 2734 T_{5}^{11} - 2194 T_{5}^{10} - 19426 T_{5}^{9} + 3154 T_{5}^{8} + 69330 T_{5}^{7} + 10376 T_{5}^{6} - 111436 T_{5}^{5} - 29938 T_{5}^{4} + \cdots + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 6 T^{15} - 13 T^{14} - 127 T^{13} + \cdots - 192 \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} - 34 T^{14} - 188 T^{13} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{16} + T^{15} - 65 T^{14} - 57 T^{13} + \cdots - 3882 \) Copy content Toggle raw display
$11$ \( T^{16} + 16 T^{15} + 33 T^{14} + \cdots - 1436382 \) Copy content Toggle raw display
$13$ \( (T - 1)^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 12 T^{15} - 53 T^{14} + \cdots - 28027 \) Copy content Toggle raw display
$19$ \( T^{16} + 10 T^{15} + \cdots - 115263248928 \) Copy content Toggle raw display
$23$ \( T^{16} + 28 T^{15} + \cdots - 646917073 \) Copy content Toggle raw display
$29$ \( T^{16} + 15 T^{15} + \cdots + 2113031289 \) Copy content Toggle raw display
$31$ \( T^{16} + 8 T^{15} - 174 T^{14} + \cdots - 3361224 \) Copy content Toggle raw display
$37$ \( T^{16} - 7 T^{15} + \cdots - 3708949600 \) Copy content Toggle raw display
$41$ \( T^{16} - 3 T^{15} - 340 T^{14} + \cdots + 67182 \) Copy content Toggle raw display
$43$ \( T^{16} + 11 T^{15} + \cdots + 255043782720 \) Copy content Toggle raw display
$47$ \( T^{16} + 16 T^{15} - 227 T^{14} + \cdots + 18799776 \) Copy content Toggle raw display
$53$ \( T^{16} + 15 T^{15} + \cdots - 665457675968 \) Copy content Toggle raw display
$59$ \( T^{16} + 44 T^{15} + \cdots - 53317261152 \) Copy content Toggle raw display
$61$ \( T^{16} + 15 T^{15} + \cdots - 21206060480 \) Copy content Toggle raw display
$67$ \( (T + 1)^{16} \) Copy content Toggle raw display
$71$ \( T^{16} + 36 T^{15} + \cdots - 259811926656 \) Copy content Toggle raw display
$73$ \( T^{16} - 2 T^{15} + \cdots + 26094391840 \) Copy content Toggle raw display
$79$ \( T^{16} + 20 T^{15} + \cdots - 6849942669888 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 161375752438272 \) Copy content Toggle raw display
$89$ \( T^{16} + 11 T^{15} + \cdots + 82525108819680 \) Copy content Toggle raw display
$97$ \( T^{16} - 17 T^{15} + \cdots - 124406627976 \) Copy content Toggle raw display
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