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Properties

Label 16.5.f.a
Level $16$
Weight $5$
Character orbit 16.f
Analytic conductor $1.654$
Analytic rank $0$
Dimension $14$
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] = [16,5,Mod(3,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 16.f (of order \(4\), degree \(2\), 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: \(1.65391940934\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{21} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{6} q^{3} + ( - \beta_{11} + 2 \beta_{3} - 1) q^{4} + ( - \beta_{7} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{12} - \beta_{10} - \beta_{3} + \cdots + 5) q^{6}+ \cdots + (2 \beta_{13} + 3 \beta_{11} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{6} q^{3} + ( - \beta_{11} + 2 \beta_{3} - 1) q^{4} + ( - \beta_{7} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{12} - \beta_{10} - \beta_{3} + \cdots + 5) q^{6}+ \cdots + (244 \beta_{13} + 32 \beta_{12} + \cdots + 3370) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 64 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 64 q^{6} - 4 q^{7} - 92 q^{8} - 100 q^{10} + 94 q^{11} - 332 q^{12} - 2 q^{13} + 44 q^{14} - 168 q^{16} - 4 q^{17} + 1390 q^{18} - 706 q^{19} + 1900 q^{20} - 164 q^{21} + 900 q^{22} + 1148 q^{23} - 1872 q^{24} - 3416 q^{26} - 1664 q^{27} - 3784 q^{28} + 862 q^{29} - 3740 q^{30} + 3208 q^{32} - 4 q^{33} + 7508 q^{34} + 1340 q^{35} + 11468 q^{36} - 1826 q^{37} + 3568 q^{38} + 2684 q^{39} - 5144 q^{40} - 17064 q^{42} + 1694 q^{43} - 14636 q^{44} + 1410 q^{45} - 5316 q^{46} + 6888 q^{48} + 682 q^{49} + 20070 q^{50} - 3012 q^{51} + 20452 q^{52} - 482 q^{53} + 10784 q^{54} - 11780 q^{55} - 6952 q^{56} - 20456 q^{58} - 2786 q^{59} - 29920 q^{60} - 3778 q^{61} - 11472 q^{62} + 15808 q^{64} - 2020 q^{65} + 30148 q^{66} + 7998 q^{67} + 18032 q^{68} + 9628 q^{69} + 15296 q^{70} + 19964 q^{71} - 17708 q^{72} - 23780 q^{74} + 17570 q^{75} - 23996 q^{76} - 9508 q^{77} - 8052 q^{78} + 1384 q^{80} + 1454 q^{81} + 16016 q^{82} - 17282 q^{83} + 19624 q^{84} + 9948 q^{85} - 4796 q^{86} - 49284 q^{87} + 7288 q^{88} - 5416 q^{90} - 28036 q^{91} - 14632 q^{92} + 8896 q^{93} + 432 q^{94} + 6064 q^{96} - 4 q^{97} - 12246 q^{98} + 49214 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2291 \nu^{13} - 13836 \nu^{12} + 73187 \nu^{11} - 134754 \nu^{10} - 13546 \nu^{9} + \cdots - 8127774720 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2559 \nu^{13} + 209196 \nu^{12} - 627631 \nu^{11} + 1042042 \nu^{10} - 73358 \nu^{9} + \cdots + 48311304192 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2875 \nu^{13} - 13444 \nu^{12} + 26581 \nu^{11} - 16062 \nu^{10} + 24954 \nu^{9} + \cdots - 1279524864 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7471 \nu^{13} - 6884 \nu^{12} + 4513 \nu^{11} - 41366 \nu^{10} + 334706 \nu^{9} + \cdots + 2696151040 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 739 \nu^{13} + 1452 \nu^{12} - 2221 \nu^{11} + 4102 \nu^{10} - 25690 \nu^{9} + \cdots - 198180864 ) / 48496640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11609 \nu^{13} + 81068 \nu^{12} - 216535 \nu^{11} + 92170 \nu^{10} - 445086 \nu^{9} + \cdots + 7904428032 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17845 \nu^{13} + 27996 \nu^{12} - 118459 \nu^{11} + 66818 \nu^{10} + 149274 \nu^{9} + \cdots + 1436286976 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19011 \nu^{13} - 91628 \nu^{12} + 527155 \nu^{11} - 1059490 \nu^{10} + 1147766 \nu^{9} + \cdots - 36868718592 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28401 \nu^{13} - 32588 \nu^{12} - 184641 \nu^{11} + 518662 \nu^{10} - 277010 \nu^{9} + \cdots + 19838795776 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30299 \nu^{13} - 175428 \nu^{12} + 396085 \nu^{11} - 537790 \nu^{10} + 1159226 \nu^{9} + \cdots - 20694433792 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8885 \nu^{13} + 54092 \nu^{12} - 127643 \nu^{11} + 205586 \nu^{10} - 440742 \nu^{9} + \cdots + 5876088832 ) / 169738240 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 73363 \nu^{13} + 121444 \nu^{12} + 210563 \nu^{11} - 703506 \nu^{10} - 613450 \nu^{9} + \cdots - 37401919488 ) / 678952960 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 74425 \nu^{13} + 92076 \nu^{12} + 124041 \nu^{11} - 284342 \nu^{10} + 496674 \nu^{9} + \cdots - 48499523584 ) / 678952960 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{12} - 2\beta_{10} - 2\beta_{9} + \beta_{8} - 2\beta_{4} + 4\beta_{3} + 3\beta _1 - 7 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{13} + 4 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} - \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + \cdots - 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{13} + 12 \beta_{11} + 32 \beta_{10} - 4 \beta_{9} - \beta_{8} - 12 \beta_{7} + 10 \beta_{6} + \cdots - 5 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{13} - 8 \beta_{12} + 52 \beta_{11} + 24 \beta_{10} - 4 \beta_{9} - 7 \beta_{8} + 12 \beta_{7} + \cdots + 765 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 26 \beta_{13} + 12 \beta_{12} + 84 \beta_{11} - 52 \beta_{10} - 16 \beta_{9} + 35 \beta_{8} + \cdots - 1585 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34 \beta_{13} - 136 \beta_{12} - 28 \beta_{11} - 344 \beta_{10} - 260 \beta_{9} + 185 \beta_{8} + \cdots - 1579 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 74 \beta_{13} + 268 \beta_{12} + 516 \beta_{11} + 1100 \beta_{10} + 224 \beta_{9} + 343 \beta_{8} + \cdots + 8371 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1946 \beta_{13} - 528 \beta_{12} - 380 \beta_{11} + 2848 \beta_{10} + 1364 \beta_{9} + 1437 \beta_{8} + \cdots + 4841 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4046 \beta_{13} - 3372 \beta_{12} + 3460 \beta_{11} - 5708 \beta_{10} - 248 \beta_{9} - 157 \beta_{8} + \cdots + 84351 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7230 \beta_{13} + 3624 \beta_{12} + 4260 \beta_{11} - 13896 \beta_{10} - 10932 \beta_{9} + \cdots - 143499 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 15734 \beta_{13} - 5300 \beta_{12} - 8476 \beta_{11} + 1484 \beta_{10} - 56128 \beta_{9} - 1297 \beta_{8} + \cdots + 245835 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 23574 \beta_{13} - 6464 \beta_{12} + 178052 \beta_{11} + 253168 \beta_{10} + 23716 \beta_{9} + \cdots + 313681 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.24452 − 1.72109i
0.153862 − 2.82424i
2.79265 + 0.448449i
−2.15805 − 1.82834i
1.03712 + 2.63142i
0.336831 + 2.80830i
−2.40693 + 1.48549i
2.24452 + 1.72109i
0.153862 + 2.82424i
2.79265 − 0.448449i
−2.15805 + 1.82834i
1.03712 − 2.63142i
0.336831 − 2.80830i
−2.40693 − 1.48549i
−3.96560 + 0.523430i 5.54016 − 5.54016i 15.4520 − 4.15143i 21.7374 − 21.7374i −19.0702 + 24.8700i −6.62054 −59.1037 + 24.5510i 19.6133i −74.8239 + 97.5799i
3.2 −2.97810 − 2.67038i −9.42589 + 9.42589i 1.73818 + 15.9053i −2.84710 + 2.84710i 53.2419 − 2.90058i −76.7794 37.2967 − 52.0092i − 96.6949i 16.0818 − 0.876123i
3.3 −2.34420 + 3.24110i −4.63552 + 4.63552i −5.00945 − 15.1956i −29.2002 + 29.2002i −4.15759 − 25.8908i 59.6196 60.9935 + 19.3854i 38.0239i −26.1896 − 163.092i
3.4 0.329715 − 3.98639i 3.91498 − 3.91498i −15.7826 − 2.62875i 4.72348 − 4.72348i −14.3158 − 16.8975i 45.3712 −15.6830 + 62.0487i 50.3458i −17.2722 − 20.3870i
3.5 1.59430 + 3.66854i 11.5209 − 11.5209i −10.9164 + 11.6975i −14.6016 + 14.6016i 60.6325 + 23.8971i −24.0210 −60.3169 − 21.3980i − 184.461i −76.8459 − 30.2872i
3.6 2.47147 + 3.14513i −7.86839 + 7.86839i −3.78368 + 15.5462i 27.2309 − 27.2309i −44.1936 − 5.30063i 50.3097 −58.2460 + 26.5217i − 42.8233i 152.945 + 18.3444i
3.7 3.89242 − 0.921438i −0.0461995 + 0.0461995i 14.3019 − 7.17325i −8.04297 + 8.04297i −0.137258 + 0.222398i −49.8797 49.0594 − 41.0996i 80.9957i −23.8955 + 38.7177i
11.1 −3.96560 − 0.523430i 5.54016 + 5.54016i 15.4520 + 4.15143i 21.7374 + 21.7374i −19.0702 − 24.8700i −6.62054 −59.1037 − 24.5510i − 19.6133i −74.8239 − 97.5799i
11.2 −2.97810 + 2.67038i −9.42589 − 9.42589i 1.73818 − 15.9053i −2.84710 − 2.84710i 53.2419 + 2.90058i −76.7794 37.2967 + 52.0092i 96.6949i 16.0818 + 0.876123i
11.3 −2.34420 − 3.24110i −4.63552 − 4.63552i −5.00945 + 15.1956i −29.2002 − 29.2002i −4.15759 + 25.8908i 59.6196 60.9935 − 19.3854i − 38.0239i −26.1896 + 163.092i
11.4 0.329715 + 3.98639i 3.91498 + 3.91498i −15.7826 + 2.62875i 4.72348 + 4.72348i −14.3158 + 16.8975i 45.3712 −15.6830 − 62.0487i − 50.3458i −17.2722 + 20.3870i
11.5 1.59430 − 3.66854i 11.5209 + 11.5209i −10.9164 − 11.6975i −14.6016 − 14.6016i 60.6325 − 23.8971i −24.0210 −60.3169 + 21.3980i 184.461i −76.8459 + 30.2872i
11.6 2.47147 − 3.14513i −7.86839 − 7.86839i −3.78368 − 15.5462i 27.2309 + 27.2309i −44.1936 + 5.30063i 50.3097 −58.2460 − 26.5217i 42.8233i 152.945 − 18.3444i
11.7 3.89242 + 0.921438i −0.0461995 − 0.0461995i 14.3019 + 7.17325i −8.04297 − 8.04297i −0.137258 − 0.222398i −49.8797 49.0594 + 41.0996i − 80.9957i −23.8955 − 38.7177i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.5.f.a ✓ 14
3.b odd 2 1 144.5.m.a 14
4.b odd 2 1 64.5.f.a 14
8.b even 2 1 128.5.f.b 14
8.d odd 2 1 128.5.f.a 14
12.b even 2 1 576.5.m.a 14
16.e even 4 1 64.5.f.a 14
16.e even 4 1 128.5.f.a 14
16.f odd 4 1 inner 16.5.f.a ✓ 14
16.f odd 4 1 128.5.f.b 14
48.i odd 4 1 576.5.m.a 14
48.k even 4 1 144.5.m.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.5.f.a ✓ 14 1.a even 1 1 trivial
16.5.f.a ✓ 14 16.f odd 4 1 inner
64.5.f.a 14 4.b odd 2 1
64.5.f.a 14 16.e even 4 1
128.5.f.a 14 8.d odd 2 1
128.5.f.a 14 16.e even 4 1
128.5.f.b 14 8.b even 2 1
128.5.f.b 14 16.f odd 4 1
144.5.m.a 14 3.b odd 2 1
144.5.m.a 14 48.k even 4 1
576.5.m.a 14 12.b even 2 1
576.5.m.a 14 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 268435456 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 2016379008 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 95\!\cdots\!08 \) Copy content Toggle raw display
$7$ \( (T^{7} + 2 T^{6} + \cdots - 82884464768)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 47\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 24\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 37\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 54\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 27\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 51\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 35\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 40\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 38\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 67\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 30\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 50\!\cdots\!96)^{2} \) Copy content Toggle raw display
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