Properties

Label 448.4.p.f
Level $448$
Weight $4$
Character orbit 448.p
Analytic conductor $26.433$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.12258833328.1
Defining polynomial: \(x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \beta_{3} + \beta_{5} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + ( -4 + 9 \beta_{3} - \beta_{4} ) q^{7} + ( -25 - \beta_{1} + \beta_{2} - 25 \beta_{3} - 2 \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( 2 \beta_{3} + \beta_{5} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + ( -4 + 9 \beta_{3} - \beta_{4} ) q^{7} + ( -25 - \beta_{1} + \beta_{2} - 25 \beta_{3} - 2 \beta_{4} ) q^{9} + ( 22 + \beta_{1} + 11 \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 23 + \beta_{2} + 46 \beta_{3} - \beta_{4} ) q^{13} + ( 19 + \beta_{1} - 7 \beta_{2} + 38 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{15} + ( 11 \beta_{1} - 5 \beta_{4} + 8 \beta_{5} ) q^{17} + ( 49 - 2 \beta_{1} + \beta_{2} + 49 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + ( 1 - 2 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + \beta_{4} - 16 \beta_{5} ) q^{21} + ( 5 \beta_{1} + 6 \beta_{2} + \beta_{5} ) q^{23} + ( -94 \beta_{3} - 24 \beta_{5} ) q^{25} + ( 53 + 15 \beta_{1} + 7 \beta_{2} + 7 \beta_{4} ) q^{27} + ( 51 + 22 \beta_{1} - \beta_{2} - \beta_{4} ) q^{29} + ( \beta_{1} + 2 \beta_{2} + 77 \beta_{3} - \beta_{4} - 23 \beta_{5} ) q^{31} + ( -51 - 4 \beta_{1} + 4 \beta_{2} + 51 \beta_{3} + 8 \beta_{5} ) q^{33} + ( -200 - 22 \beta_{1} - 8 \beta_{2} - 19 \beta_{3} + 9 \beta_{4} - \beta_{5} ) q^{35} + ( 73 + 26 \beta_{1} - 2 \beta_{2} + 73 \beta_{3} + 4 \beta_{4} + 24 \beta_{5} ) q^{37} + ( -54 - 37 \beta_{1} - 27 \beta_{3} + 7 \beta_{4} - 22 \beta_{5} ) q^{39} + ( 141 - 8 \beta_{1} - 9 \beta_{2} + 282 \beta_{3} + 9 \beta_{4} - 16 \beta_{5} ) q^{41} + ( 20 - 22 \beta_{1} - 16 \beta_{2} + 40 \beta_{3} + 16 \beta_{4} - 44 \beta_{5} ) q^{43} + ( -438 - 73 \beta_{1} - 219 \beta_{3} - 25 \beta_{4} - 24 \beta_{5} ) q^{45} + ( -33 + 22 \beta_{1} - \beta_{2} - 33 \beta_{3} + 2 \beta_{4} + 21 \beta_{5} ) q^{47} + ( 126 + 26 \beta_{1} + 19 \beta_{2} + 28 \beta_{3} - \beta_{4} + 40 \beta_{5} ) q^{49} + ( -289 + 32 \beta_{1} + 11 \beta_{2} + 289 \beta_{3} - 21 \beta_{5} ) q^{51} + ( -6 \beta_{1} - 12 \beta_{2} - 237 \beta_{3} + 6 \beta_{4} + 48 \beta_{5} ) q^{53} + ( -162 + 11 \beta_{1} + 16 \beta_{2} + 16 \beta_{4} ) q^{55} + ( 7 - 32 \beta_{1} + 8 \beta_{2} + 8 \beta_{4} ) q^{57} + ( -2 \beta_{1} - 4 \beta_{2} - 108 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{59} + ( 189 + 2 \beta_{1} - 6 \beta_{2} - 189 \beta_{3} - 8 \beta_{5} ) q^{61} + ( 487 + 13 \beta_{1} - 15 \beta_{2} + 481 \beta_{3} + 32 \beta_{4} + 20 \beta_{5} ) q^{63} + ( -219 - \beta_{1} - 23 \beta_{2} - 219 \beta_{3} + 46 \beta_{4} - 24 \beta_{5} ) q^{65} + ( 88 - 8 \beta_{1} + 44 \beta_{3} + 42 \beta_{4} - 25 \beta_{5} ) q^{67} + ( 66 + 8 \beta_{1} - 39 \beta_{2} + 132 \beta_{3} + 39 \beta_{4} + 16 \beta_{5} ) q^{69} + ( 322 - 26 \beta_{1} + 14 \beta_{2} + 644 \beta_{3} - 14 \beta_{4} - 52 \beta_{5} ) q^{71} + ( 190 - 58 \beta_{1} + 95 \beta_{3} - 42 \beta_{4} - 8 \beta_{5} ) q^{73} + ( 1340 + 70 \beta_{1} - 24 \beta_{2} + 1340 \beta_{3} + 48 \beta_{4} + 46 \beta_{5} ) q^{75} + ( 33 + 11 \beta_{1} + 25 \beta_{2} + 246 \beta_{3} - 30 \beta_{4} + 32 \beta_{5} ) q^{77} + ( 372 + 33 \beta_{1} - 14 \beta_{2} - 372 \beta_{3} - 47 \beta_{5} ) q^{79} + ( -23 \beta_{1} - 46 \beta_{2} - 122 \beta_{3} + 23 \beta_{4} + 72 \beta_{5} ) q^{81} + ( 42 - 92 \beta_{1} - 22 \beta_{2} - 22 \beta_{4} ) q^{83} + ( -639 - 16 \beta_{1} + 40 \beta_{2} + 40 \beta_{4} ) q^{85} + ( -17 \beta_{1} - 34 \beta_{2} + 1197 \beta_{3} + 17 \beta_{4} ) q^{87} + ( -291 + 4 \beta_{1} - 28 \beta_{2} + 291 \beta_{3} - 32 \beta_{5} ) q^{89} + ( -525 + 54 \beta_{1} + 47 \beta_{2} - 210 \beta_{3} - 15 \beta_{4} + 68 \beta_{5} ) q^{91} + ( 1007 - 90 \beta_{1} - 30 \beta_{2} + 1007 \beta_{3} + 60 \beta_{4} - 120 \beta_{5} ) q^{93} + ( -476 - 6 \beta_{1} - 238 \beta_{3} + 44 \beta_{4} - 25 \beta_{5} ) q^{95} + ( 45 + 48 \beta_{1} - \beta_{2} + 90 \beta_{3} + \beta_{4} + 96 \beta_{5} ) q^{97} + ( -113 - 4 \beta_{1} + 23 \beta_{2} - 226 \beta_{3} - 23 \beta_{4} - 8 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 7q^{3} + 3q^{5} - 52q^{7} - 78q^{9} + O(q^{10}) \) \( 6q - 7q^{3} + 3q^{5} - 52q^{7} - 78q^{9} + 99q^{11} + 9q^{17} + 143q^{19} + 15q^{21} + 15q^{23} + 306q^{25} + 362q^{27} + 348q^{29} - 205q^{31} - 471q^{33} - 1185q^{35} + 249q^{37} - 288q^{39} - 2118q^{45} - 75q^{47} + 702q^{49} - 2505q^{51} + 645q^{53} - 918q^{55} - 6q^{57} + 321q^{59} + 1707q^{61} + 1502q^{63} - 612q^{65} + 447q^{67} + 705q^{73} + 4138q^{75} - 555q^{77} + 3447q^{79} + 225q^{81} + 24q^{83} - 3786q^{85} - 3642q^{87} - 2607q^{89} - 2448q^{91} + 2991q^{93} - 2085q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 29 \nu^{5} - 841 \nu^{4} + 1629 \nu^{3} - 23432 \nu^{2} + 19488 \nu - 428592 \)\()/45520\)
\(\beta_{2}\)\(=\)\((\)\( 64 \nu^{5} + 989 \nu^{4} + 5459 \nu^{3} + 30793 \nu^{2} - 82172 \nu + 596328 \)\()/68280\)
\(\beta_{3}\)\(=\)\((\)\( -203 \nu^{5} + 197 \nu^{4} - 5713 \nu^{3} - 986 \nu^{2} - 159176 \nu - 4176 \)\()/136560\)
\(\beta_{4}\)\(=\)\((\)\( -133 \nu^{5} + 1012 \nu^{4} + 4792 \nu^{3} + 24959 \nu^{2} + 35804 \nu + 543504 \)\()/68280\)
\(\beta_{5}\)\(=\)\((\)\( 581 \nu^{5} + 221 \nu^{4} + 16351 \nu^{3} + 2822 \nu^{2} + 481472 \nu + 11952 \)\()/45520\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-12 \beta_{5} + 2 \beta_{4} - 111 \beta_{3} - \beta_{2} - 11 \beta_{1} - 111\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(29 \beta_{4} + 29 \beta_{2} + 46 \beta_{1} - 51\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(116 \beta_{5} - 11 \beta_{4} + 1029 \beta_{3} + 22 \beta_{2} + 11 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(396 \beta_{5} - 1642 \beta_{4} + 1851 \beta_{3} + 821 \beta_{2} - 425 \beta_{1} + 1851\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1 + \beta_{3}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
−2.61524 + 4.52973i
2.68858 4.65676i
0.426664 0.739004i
−2.61524 4.52973i
2.68858 + 4.65676i
0.426664 + 0.739004i
0 −4.64712 + 8.04905i 0 17.1915 9.92549i 0 −18.3972 2.13127i 0 −29.6915 51.4271i 0
255.2 0 −2.38417 + 4.12950i 0 −14.6315 + 8.44749i 0 8.89981 + 16.2417i 0 2.13148 + 3.69182i 0
255.3 0 3.53129 6.11637i 0 −1.05999 + 0.611983i 0 −16.5026 + 8.40621i 0 −11.4400 19.8147i 0
383.1 0 −4.64712 8.04905i 0 17.1915 + 9.92549i 0 −18.3972 + 2.13127i 0 −29.6915 + 51.4271i 0
383.2 0 −2.38417 4.12950i 0 −14.6315 8.44749i 0 8.89981 16.2417i 0 2.13148 3.69182i 0
383.3 0 3.53129 + 6.11637i 0 −1.05999 0.611983i 0 −16.5026 8.40621i 0 −11.4400 + 19.8147i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.p.f 6
4.b odd 2 1 448.4.p.g 6
7.d odd 6 1 448.4.p.g 6
8.b even 2 1 112.4.p.g yes 6
8.d odd 2 1 112.4.p.f 6
28.f even 6 1 inner 448.4.p.f 6
56.j odd 6 1 112.4.p.f 6
56.j odd 6 1 784.4.f.h 6
56.k odd 6 1 784.4.f.h 6
56.m even 6 1 112.4.p.g yes 6
56.m even 6 1 784.4.f.g 6
56.p even 6 1 784.4.f.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.f 6 8.d odd 2 1
112.4.p.f 6 56.j odd 6 1
112.4.p.g yes 6 8.b even 2 1
112.4.p.g yes 6 56.m even 6 1
448.4.p.f 6 1.a even 1 1 trivial
448.4.p.f 6 28.f even 6 1 inner
448.4.p.g 6 4.b odd 2 1
448.4.p.g 6 7.d odd 6 1
784.4.f.g 6 56.m even 6 1
784.4.f.g 6 56.p even 6 1
784.4.f.h 6 56.j odd 6 1
784.4.f.h 6 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 7 T_{3}^{5} + 104 T_{3}^{4} + 241 T_{3}^{3} + 5216 T_{3}^{2} + 17215 T_{3} + 97969 \) acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 97969 + 17215 T + 5216 T^{2} + 241 T^{3} + 104 T^{4} + 7 T^{5} + T^{6} \)
$5$ \( 168507 + 241029 T + 115632 T^{2} + 1017 T^{3} - 336 T^{4} - 3 T^{5} + T^{6} \)
$7$ \( 40353607 + 6117748 T + 343343 T^{2} + 14056 T^{3} + 1001 T^{4} + 52 T^{5} + T^{6} \)
$11$ \( 4876875 - 2765475 T + 648954 T^{2} - 71577 T^{3} + 3990 T^{4} - 99 T^{5} + T^{6} \)
$13$ \( 2167603200 + 7385616 T^{2} + 5304 T^{4} + T^{6} \)
$17$ \( 710833347 - 677307393 T + 214982352 T^{2} + 132003 T^{3} - 14640 T^{4} - 9 T^{5} + T^{6} \)
$19$ \( 1106959441 - 184221527 T + 25900616 T^{2} - 725249 T^{3} + 14912 T^{4} - 143 T^{5} + T^{6} \)
$23$ \( 8915001507 - 1848154239 T + 126894906 T^{2} + 169515 T^{3} - 11226 T^{4} - 15 T^{5} + T^{6} \)
$29$ \( ( 4622400 - 27504 T - 174 T^{2} + T^{3} )^{2} \)
$31$ \( 2349411462841 + 44559418309 T + 1159342736 T^{2} - 2893997 T^{3} + 71096 T^{4} + 205 T^{5} + T^{6} \)
$37$ \( 14539922265625 - 127835015625 T + 2073393750 T^{2} + 721475 T^{3} + 95526 T^{4} - 249 T^{5} + T^{6} \)
$41$ \( 38591270836992 + 14002779024 T^{2} + 235176 T^{4} + T^{6} \)
$43$ \( 83474849412288 + 16477161840 T^{2} + 251796 T^{4} + T^{6} \)
$47$ \( 20686781241 + 4969435779 T + 1204558776 T^{2} - 2303667 T^{3} + 40176 T^{4} + 75 T^{5} + T^{6} \)
$53$ \( 3742269164979921 - 7129289506149 T + 53039092086 T^{2} - 47179233 T^{3} + 532566 T^{4} - 645 T^{5} + T^{6} \)
$59$ \( 825904716849 - 27948111129 T + 654024456 T^{2} - 8054127 T^{3} + 72288 T^{4} - 321 T^{5} + T^{6} \)
$61$ \( 912934776095523 - 16058508837363 T + 123934119804 T^{2} - 523791243 T^{3} + 1278132 T^{4} - 1707 T^{5} + T^{6} \)
$67$ \( 34861164967167075 - 166194940681935 T + 215916796746 T^{2} + 229717323 T^{3} - 447306 T^{4} - 447 T^{5} + T^{6} \)
$71$ \( 1026472074810048 + 436555595376 T^{2} + 1416948 T^{4} + T^{6} \)
$73$ \( 48340453761263403 - 236639118427497 T + 296644791996 T^{2} + 438086295 T^{3} - 455724 T^{4} - 705 T^{5} + T^{6} \)
$79$ \( 6567218609777523 + 126963434082393 T + 656914190058 T^{2} - 3117945933 T^{3} + 4865142 T^{4} - 3447 T^{5} + T^{6} \)
$83$ \( ( 308212992 - 856512 T - 12 T^{2} + T^{3} )^{2} \)
$89$ \( 6090737949537003 - 59194781610489 T + 74300935212 T^{2} + 1141639191 T^{3} + 2703396 T^{4} + 2607 T^{5} + T^{6} \)
$97$ \( 19259496900926208 + 308008959120 T^{2} + 1038696 T^{4} + T^{6} \)
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