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Properties

Label 13.11.d.a
Level $13$
Weight $11$
Character orbit 13.d
Analytic conductor $8.260$
Analytic rank $0$
Dimension $20$
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] = [13,11,Mod(5,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.5");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 13.d (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: \(8.25964428476\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} + 18 x^{18} + 32284 x^{17} + 16288321 x^{16} - 1399914 x^{15} + 236338034 x^{14} + \cdots + 84\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{6}\cdot 13^{6} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2 \beta_{2} - 2) q^{2} + (\beta_{6} - \beta_{3} - \beta_1) q^{3} + ( - \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + (\beta_{16} + 2 \beta_{10} + \cdots + 11933) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2 \beta_{2} - 2) q^{2} + (\beta_{6} - \beta_{3} - \beta_1) q^{3} + ( - \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + ( - 7443 \beta_{19} + \cdots + 3231666779) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 34 q^{2} - 4 q^{3} - 7792 q^{5} - 19264 q^{6} - 38312 q^{7} - 1200 q^{8} + 236192 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 34 q^{2} - 4 q^{3} - 7792 q^{5} - 19264 q^{6} - 38312 q^{7} - 1200 q^{8} + 236192 q^{9} + 331856 q^{11} - 445380 q^{13} + 2007244 q^{14} + 379412 q^{15} - 1736492 q^{16} - 5990914 q^{18} - 2277428 q^{19} + 3611836 q^{20} - 9480856 q^{21} + 5001216 q^{22} + 41307672 q^{24} + 6769646 q^{26} - 4427116 q^{27} - 113669380 q^{28} + 37095424 q^{29} + 17376308 q^{31} - 1963204 q^{32} - 118752028 q^{33} - 32678040 q^{34} + 283757932 q^{35} + 138434576 q^{37} - 180959428 q^{39} - 8799876 q^{40} - 431088064 q^{41} + 1089534556 q^{42} + 102766076 q^{44} - 551620384 q^{45} - 682329276 q^{46} - 316356304 q^{47} - 332833468 q^{48} + 653613022 q^{50} - 1360524100 q^{52} + 882722032 q^{53} + 206912984 q^{54} - 1205505576 q^{55} + 2161977296 q^{57} + 450412980 q^{58} + 1130230868 q^{59} + 4118247964 q^{60} - 1735259352 q^{61} - 3186516460 q^{63} + 2833033736 q^{65} + 1165082704 q^{66} - 626922128 q^{67} - 11230951788 q^{68} - 3144181512 q^{70} - 196931776 q^{71} + 614552052 q^{72} + 5765831656 q^{73} + 10475267944 q^{74} - 5350301912 q^{76} + 8069244872 q^{78} + 1110987384 q^{79} + 870087872 q^{80} - 21624920380 q^{81} + 13391403584 q^{83} - 25726991224 q^{84} + 1992219096 q^{85} + 32878542048 q^{86} + 194656048 q^{87} + 5940615128 q^{89} + 670762144 q^{91} - 12619717320 q^{92} - 27707434216 q^{93} - 31866209148 q^{94} + 34182859808 q^{96} - 14392811236 q^{97} - 58567835410 q^{98} + 64175005436 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 18 x^{18} + 32284 x^{17} + 16288321 x^{16} - 1399914 x^{15} + 236338034 x^{14} + \cdots + 84\!\cdots\!04 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!75 \nu^{19} + \cdots + 19\!\cdots\!40 ) / 57\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 66\!\cdots\!19 \nu^{19} + \cdots + 79\!\cdots\!00 ) / 32\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 44\!\cdots\!73 \nu^{19} + \cdots - 44\!\cdots\!24 ) / 32\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!17 \nu^{19} + \cdots - 46\!\cdots\!12 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 56\!\cdots\!81 \nu^{19} + \cdots - 45\!\cdots\!48 ) / 84\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23\!\cdots\!27 \nu^{19} + \cdots + 27\!\cdots\!00 ) / 57\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68\!\cdots\!73 \nu^{19} + \cdots - 41\!\cdots\!24 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 46\!\cdots\!25 \nu^{19} + \cdots - 25\!\cdots\!84 ) / 83\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27\!\cdots\!28 \nu^{19} + \cdots - 63\!\cdots\!56 ) / 41\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46\!\cdots\!03 \nu^{19} + \cdots - 10\!\cdots\!08 ) / 62\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22\!\cdots\!02 \nu^{19} + \cdots - 50\!\cdots\!16 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 51\!\cdots\!15 \nu^{19} + \cdots - 11\!\cdots\!60 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25\!\cdots\!55 \nu^{19} + \cdots - 12\!\cdots\!88 ) / 37\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!23 \nu^{19} + \cdots - 30\!\cdots\!56 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11\!\cdots\!27 \nu^{19} + \cdots - 30\!\cdots\!16 ) / 84\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14\!\cdots\!71 \nu^{19} + \cdots + 16\!\cdots\!96 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 60\!\cdots\!09 \nu^{19} + \cdots - 16\!\cdots\!44 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 26\!\cdots\!97 \nu^{19} + \cdots + 29\!\cdots\!44 ) / 62\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 3\beta_{3} - 1375\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + 6\beta_{7} + 25\beta_{6} + 25\beta_{5} - 6\beta_{4} - 2405\beta_{3} - 4137\beta_{2} - 4137 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{18} + 2 \beta_{16} + 32 \beta_{15} - 12 \beta_{14} + 32 \beta_{13} + 12 \beta_{12} + \cdots - 3286822 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 143 \beta_{19} - 143 \beta_{18} + 258 \beta_{17} + 258 \beta_{16} + 736 \beta_{13} + 3991 \beta_{12} + \cdots - 22250539 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 17787 \beta_{19} + 13026 \beta_{17} - 142400 \beta_{15} + 68438 \beta_{14} + 142400 \beta_{13} + \cdots + 68438 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 973789 \beta_{19} + 973789 \beta_{18} + 1400214 \beta_{17} - 1400214 \beta_{16} - 4739680 \beta_{15} + \cdots + 101738863204 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79431465 \beta_{18} - 65135670 \beta_{16} - 522853824 \beta_{15} + 303204428 \beta_{14} + \cdots + 27531726540224 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4549544333 \beta_{19} + 4549544333 \beta_{18} - 5804227750 \beta_{17} - 5804227750 \beta_{16} + \cdots + 427673845492033 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 323495184543 \beta_{19} - 285895425690 \beta_{17} + 1838867313664 \beta_{15} - 1226131240650 \beta_{14} + \cdots - 1226131240650 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18472055908717 \beta_{19} - 18472055908717 \beta_{18} - 21986533385174 \beta_{17} + \cdots - 17\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12\!\cdots\!89 \beta_{18} + \cdots - 27\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 70\!\cdots\!49 \beta_{19} + \cdots - 66\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 47\!\cdots\!79 \beta_{19} + \cdots + 17\!\cdots\!18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 25\!\cdots\!49 \beta_{19} + \cdots + 24\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 17\!\cdots\!41 \beta_{18} + \cdots + 30\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 93\!\cdots\!61 \beta_{19} + \cdots + 93\!\cdots\!61 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 66\!\cdots\!43 \beta_{19} + \cdots - 24\!\cdots\!38 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 33\!\cdots\!85 \beta_{19} + \cdots - 34\!\cdots\!72 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−37.3633 − 37.3633i
−33.8970 − 33.8970i
−20.7752 − 20.7752i
−14.5424 − 14.5424i
−6.55552 − 6.55552i
2.27503 + 2.27503i
16.4457 + 16.4457i
21.5985 + 21.5985i
33.6121 + 33.6121i
42.2020 + 42.2020i
−37.3633 + 37.3633i
−33.8970 + 33.8970i
−20.7752 + 20.7752i
−14.5424 + 14.5424i
−6.55552 + 6.55552i
2.27503 − 2.27503i
16.4457 − 16.4457i
21.5985 − 21.5985i
33.6121 − 33.6121i
42.2020 − 42.2020i
−39.3633 + 39.3633i 361.049 − 2074.94i −2244.35 + 2244.35i −14212.1 + 14212.1i −23343.5 − 23343.5i 41368.4 + 41368.4i 71307.4 − 176690.i
5.2 −35.8970 + 35.8970i −170.084 − 1553.18i 1962.18 − 1962.18i 6105.50 − 6105.50i 224.952 + 224.952i 18996.1 + 18996.1i −30120.4 140873.i
5.3 −22.7752 + 22.7752i 85.0537 − 13.4171i −1400.68 + 1400.68i −1937.11 + 1937.11i 17603.1 + 17603.1i −23016.2 − 23016.2i −51814.9 − 63801.5i
5.4 −16.5424 + 16.5424i −418.411 476.695i −3347.55 + 3347.55i 6921.53 − 6921.53i −5816.97 − 5816.97i −24825.2 − 24825.2i 116018. − 110753.i
5.5 −8.55552 + 8.55552i 347.925 877.606i 1886.34 − 1886.34i −2976.68 + 2976.68i 1469.56 + 1469.56i −16269.2 − 16269.2i 62003.1 32277.3i
5.6 0.275033 − 0.275033i −117.626 1023.85i 1857.78 − 1857.78i −32.3511 + 32.3511i −14253.6 − 14253.6i 563.227 + 563.227i −45213.1 − 1021.90i
5.7 14.4457 − 14.4457i 144.596 606.641i −4157.09 + 4157.09i 2088.80 − 2088.80i −3753.74 − 3753.74i 23555.8 + 23555.8i −38140.9 120105.i
5.8 19.5985 − 19.5985i −301.530 255.796i 1032.20 − 1032.20i −5909.55 + 5909.55i 17288.9 + 17288.9i 25082.1 + 25082.1i 31871.6 − 40459.2i
5.9 31.6121 − 31.6121i 276.450 − 974.655i 1306.29 − 1306.29i 8739.18 − 8739.18i 3225.46 + 3225.46i 1559.91 + 1559.91i 17375.6 − 82589.2i
5.10 40.2020 − 40.2020i −209.424 − 2208.39i −791.113 + 791.113i −8419.23 + 8419.23i −11800.1 − 11800.1i −47614.9 − 47614.9i −15190.8 63608.6i
8.1 −39.3633 − 39.3633i 361.049 2074.94i −2244.35 − 2244.35i −14212.1 − 14212.1i −23343.5 + 23343.5i 41368.4 − 41368.4i 71307.4 176690.i
8.2 −35.8970 − 35.8970i −170.084 1553.18i 1962.18 + 1962.18i 6105.50 + 6105.50i 224.952 − 224.952i 18996.1 − 18996.1i −30120.4 − 140873.i
8.3 −22.7752 − 22.7752i 85.0537 13.4171i −1400.68 − 1400.68i −1937.11 − 1937.11i 17603.1 − 17603.1i −23016.2 + 23016.2i −51814.9 63801.5i
8.4 −16.5424 − 16.5424i −418.411 − 476.695i −3347.55 − 3347.55i 6921.53 + 6921.53i −5816.97 + 5816.97i −24825.2 + 24825.2i 116018. 110753.i
8.5 −8.55552 − 8.55552i 347.925 − 877.606i 1886.34 + 1886.34i −2976.68 − 2976.68i 1469.56 − 1469.56i −16269.2 + 16269.2i 62003.1 − 32277.3i
8.6 0.275033 + 0.275033i −117.626 − 1023.85i 1857.78 + 1857.78i −32.3511 − 32.3511i −14253.6 + 14253.6i 563.227 − 563.227i −45213.1 1021.90i
8.7 14.4457 + 14.4457i 144.596 − 606.641i −4157.09 − 4157.09i 2088.80 + 2088.80i −3753.74 + 3753.74i 23555.8 − 23555.8i −38140.9 − 120105.i
8.8 19.5985 + 19.5985i −301.530 − 255.796i 1032.20 + 1032.20i −5909.55 − 5909.55i 17288.9 − 17288.9i 25082.1 − 25082.1i 31871.6 40459.2i
8.9 31.6121 + 31.6121i 276.450 974.655i 1306.29 + 1306.29i 8739.18 + 8739.18i 3225.46 − 3225.46i 1559.91 − 1559.91i 17375.6 82589.2i
8.10 40.2020 + 40.2020i −209.424 2208.39i −791.113 − 791.113i −8419.23 − 8419.23i −11800.1 + 11800.1i −47614.9 + 47614.9i −15190.8 − 63608.6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.11.d.a ✓ 20
13.d odd 4 1 inner 13.11.d.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.11.d.a ✓ 20 1.a even 1 1 trivial
13.11.d.a ✓ 20 13.d odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots - 22\!\cdots\!16)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 79\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 65\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 98\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 47\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
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