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Properties

Label 672.4.c.a
Level $672$
Weight $4$
Character orbit 672.c
Analytic conductor $39.649$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,4,Mod(337,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("672.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(39.6492835239\)
Analytic rank: \(0\)
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: \( x^{16} - 2 x^{15} + x^{14} + 2 x^{13} + 3 x^{12} - 14 x^{11} - 18 x^{10} + 176 x^{9} - 296 x^{8} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{2} q^{5} - 7 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{2} q^{5} - 7 q^{7} - 9 q^{9} + ( - \beta_{13} - 3 \beta_{5}) q^{11} + (\beta_{8} - \beta_{6} + \cdots + \beta_{2}) q^{13}+ \cdots + (9 \beta_{13} + 27 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 112 q^{7} - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 112 q^{7} - 144 q^{9} - 60 q^{15} - 52 q^{17} + 84 q^{23} - 144 q^{25} - 264 q^{31} - 396 q^{33} + 312 q^{39} - 236 q^{41} + 784 q^{49} + 360 q^{55} + 912 q^{57} + 1008 q^{63} + 1744 q^{65} - 2116 q^{71} + 1304 q^{73} + 2664 q^{79} + 1296 q^{81} + 220 q^{89} + 1240 q^{95} - 2584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + x^{14} + 2 x^{13} + 3 x^{12} - 14 x^{11} - 18 x^{10} + 176 x^{9} - 296 x^{8} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 69 \nu^{15} - 138 \nu^{14} + 1333 \nu^{13} + 2922 \nu^{12} + 5183 \nu^{11} - 8998 \nu^{10} + \cdots - 2211840 ) / 1261568 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 641 \nu^{15} - 2966 \nu^{14} + 671 \nu^{13} + 5446 \nu^{12} + 445 \nu^{11} + 23014 \nu^{10} + \cdots + 45432832 ) / 6307840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 219 \nu^{15} - 2902 \nu^{14} - 2197 \nu^{13} - 6474 \nu^{12} + 8737 \nu^{11} - 8954 \nu^{10} + \cdots - 25395200 ) / 1261568 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41 \nu^{15} - 84 \nu^{14} + 209 \nu^{13} - 956 \nu^{12} + 115 \nu^{11} - 64 \nu^{10} + \cdots + 6448128 ) / 197120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1431 \nu^{15} + 654 \nu^{14} + 1881 \nu^{13} - 3054 \nu^{12} + 15915 \nu^{11} - 13566 \nu^{10} + \cdots + 786432 ) / 6307840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 185 \nu^{15} + 254 \nu^{14} + 847 \nu^{13} + 410 \nu^{12} - 755 \nu^{11} + 13250 \nu^{10} + \cdots + 19423232 ) / 630784 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 439 \nu^{15} + 3342 \nu^{14} - 5767 \nu^{13} + 7442 \nu^{12} + 11019 \nu^{11} - 40062 \nu^{10} + \cdots + 21970944 ) / 1261568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 681 \nu^{15} + 2338 \nu^{14} - 121 \nu^{13} - 12258 \nu^{12} + 30197 \nu^{11} - 32370 \nu^{10} + \cdots - 72876032 ) / 1261568 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 199 \nu^{15} + 106 \nu^{14} + 289 \nu^{13} + 654 \nu^{12} + 3715 \nu^{11} - 7114 \nu^{10} + \cdots - 6152192 ) / 286720 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62 \nu^{15} - 107 \nu^{14} - 276 \nu^{13} - 339 \nu^{12} - 892 \nu^{11} - 2321 \nu^{10} + \cdots + 520192 ) / 78848 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 151 \nu^{15} - 226 \nu^{14} + 2873 \nu^{13} - 3678 \nu^{12} - 4277 \nu^{11} + 12210 \nu^{10} + \cdots + 3915776 ) / 180224 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3349 \nu^{15} - 11146 \nu^{14} + 11781 \nu^{13} + 26666 \nu^{12} - 20945 \nu^{11} + \cdots - 159285248 ) / 3153920 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 210 \nu^{15} + 93 \nu^{14} - 88 \nu^{13} + 653 \nu^{12} - 2136 \nu^{11} + 4111 \nu^{10} + \cdots + 4143104 ) / 157696 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2469 \nu^{15} + 4938 \nu^{14} + 939 \nu^{13} - 426 \nu^{12} - 37407 \nu^{11} - 40986 \nu^{10} + \cdots + 77938688 ) / 1261568 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2539 \nu^{15} + 1082 \nu^{14} - 1765 \nu^{13} + 12550 \nu^{12} + 36273 \nu^{11} + 2486 \nu^{10} + \cdots - 43761664 ) / 1261568 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{15} - \beta_{14} - 2 \beta_{13} + \beta_{12} - 3 \beta_{10} - 2 \beta_{9} + 3 \beta_{7} + \cdots + 16 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{15} - 4 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \cdots + 15 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{15} - \beta_{14} - 3 \beta_{13} - 3 \beta_{11} - 3 \beta_{10} + 3 \beta_{8} - 3 \beta_{7} + \cdots - 11 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{15} + \beta_{12} - 2 \beta_{10} - \beta_{9} - \beta_{8} + 3 \beta_{7} + 9 \beta_{6} + \cdots - 53 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{15} - 5 \beta_{14} - 20 \beta_{13} - 5 \beta_{12} - 6 \beta_{11} + 9 \beta_{10} + 10 \beta_{9} + \cdots + 74 ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 45 \beta_{15} + 3 \beta_{13} + 15 \beta_{12} + 15 \beta_{11} + 66 \beta_{10} - 9 \beta_{9} + 18 \beta_{8} + \cdots + 543 ) / 48 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 75 \beta_{15} + 113 \beta_{14} - 110 \beta_{13} - 29 \beta_{12} + 24 \beta_{11} - 45 \beta_{10} + \cdots - 4436 ) / 96 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 120 \beta_{15} + 104 \beta_{14} - 62 \beta_{13} - 79 \beta_{12} - 6 \beta_{11} + 86 \beta_{10} + \cdots + 575 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 66 \beta_{15} + 97 \beta_{14} - 9 \beta_{13} + 60 \beta_{12} - 105 \beta_{11} + 27 \beta_{10} + \cdots - 16477 ) / 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1114 \beta_{15} - 476 \beta_{14} + 1464 \beta_{13} - 843 \beta_{12} + 448 \beta_{11} + 1018 \beta_{10} + \cdots - 48045 ) / 96 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2817 \beta_{15} + 889 \beta_{14} + 2428 \beta_{13} + 61 \beta_{12} - 1290 \beta_{11} - 21 \beta_{10} + \cdots + 36122 ) / 96 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1017 \beta_{15} + 544 \beta_{14} + 1215 \beta_{13} + 647 \beta_{12} + 99 \beta_{11} - 326 \beta_{10} + \cdots + 9367 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4857 \beta_{15} + 2419 \beta_{14} - 3754 \beta_{13} + 29 \beta_{12} + 5640 \beta_{11} - 3951 \beta_{10} + \cdots - 6688 ) / 96 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2836 \beta_{15} + 4708 \beta_{14} + 2706 \beta_{13} - 5859 \beta_{12} + 2434 \beta_{11} + 1834 \beta_{10} + \cdots + 188727 ) / 96 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 16026 \beta_{15} + 967 \beta_{14} - 5007 \beta_{13} + 1176 \beta_{12} - 3831 \beta_{11} + \cdots - 400279 ) / 48 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.938719 + 1.76601i
1.83143 − 0.803651i
−1.99330 + 0.163621i
−1.35928 + 1.46709i
1.90022 + 0.623843i
−1.28983 − 1.52851i
−0.104269 + 1.99728i
1.07631 − 1.68569i
1.07631 + 1.68569i
−0.104269 − 1.99728i
−1.28983 + 1.52851i
1.90022 − 0.623843i
−1.35928 − 1.46709i
−1.99330 − 0.163621i
1.83143 + 0.803651i
0.938719 − 1.76601i
0 − 3.00000i 0 − 19.8753i 0 −7.00000 0 −9.00000 0
337.2 0 − 3.00000i 0 − 14.1686i 0 −7.00000 0 −9.00000 0
337.3 0 − 3.00000i 0 − 5.55182i 0 −7.00000 0 −9.00000 0
337.4 0 − 3.00000i 0 − 2.56228i 0 −7.00000 0 −9.00000 0
337.5 0 − 3.00000i 0 − 2.03259i 0 −7.00000 0 −9.00000 0
337.6 0 − 3.00000i 0 7.88063i 0 −7.00000 0 −9.00000 0
337.7 0 − 3.00000i 0 9.51544i 0 −7.00000 0 −9.00000 0
337.8 0 − 3.00000i 0 16.7945i 0 −7.00000 0 −9.00000 0
337.9 0 3.00000i 0 − 16.7945i 0 −7.00000 0 −9.00000 0
337.10 0 3.00000i 0 − 9.51544i 0 −7.00000 0 −9.00000 0
337.11 0 3.00000i 0 − 7.88063i 0 −7.00000 0 −9.00000 0
337.12 0 3.00000i 0 2.03259i 0 −7.00000 0 −9.00000 0
337.13 0 3.00000i 0 2.56228i 0 −7.00000 0 −9.00000 0
337.14 0 3.00000i 0 5.55182i 0 −7.00000 0 −9.00000 0
337.15 0 3.00000i 0 14.1686i 0 −7.00000 0 −9.00000 0
337.16 0 3.00000i 0 19.8753i 0 −7.00000 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.c.a 16
3.b odd 2 1 2016.4.c.c 16
4.b odd 2 1 168.4.c.a ✓ 16
8.b even 2 1 inner 672.4.c.a 16
8.d odd 2 1 168.4.c.a ✓ 16
12.b even 2 1 504.4.c.d 16
24.f even 2 1 504.4.c.d 16
24.h odd 2 1 2016.4.c.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.c.a ✓ 16 4.b odd 2 1
168.4.c.a ✓ 16 8.d odd 2 1
504.4.c.d 16 12.b even 2 1
504.4.c.d 16 24.f even 2 1
672.4.c.a 16 1.a even 1 1 trivial
672.4.c.a 16 8.b even 2 1 inner
2016.4.c.c 16 3.b odd 2 1
2016.4.c.c 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 1072 T_{5}^{14} + 430108 T_{5}^{12} + 81495264 T_{5}^{10} + 7645389296 T_{5}^{8} + \cdots + 105152063079424 \) acting on \(S_{4}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 105152063079424 \) Copy content Toggle raw display
$7$ \( (T + 7)^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 84\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 7084413835488)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 70730882424576)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 79\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 18\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 79\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 63\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 35\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 81\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 44\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 12\!\cdots\!44)^{2} \) Copy content Toggle raw display
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