↑

Properties

Label 19.8.c.a
Level $19$
Weight $8$
Character orbit 19.c
Analytic conductor $5.935$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,8,Mod(7,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.7");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 19.c (of order \(3\), 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: \(5.93531548420\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 874 x^{18} + 315 x^{17} + 507687 x^{16} + 417048 x^{15} + 164584585 x^{14} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{7} + 7 \beta_{2}) q^{3} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 48) q^{4}+ \cdots + ( - \beta_{19} - 6 \beta_{12} + \cdots - 601) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{7} + 7 \beta_{2}) q^{3} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 48) q^{4}+ \cdots + ( - 2947 \beta_{19} + 679 \beta_{17} + \cdots - 2648674) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 9 q^{2} - 68 q^{3} - 475 q^{4} + 545 q^{6} - 1456 q^{7} + 2322 q^{8} - 6052 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 9 q^{2} - 68 q^{3} - 475 q^{4} + 545 q^{6} - 1456 q^{7} + 2322 q^{8} - 6052 q^{9} - 1732 q^{10} + 7248 q^{11} + 30354 q^{12} - 12888 q^{13} - 7578 q^{14} - 13244 q^{15} - 1027 q^{16} - 63024 q^{17} - 93132 q^{18} + 72752 q^{19} + 156588 q^{20} - 18148 q^{21} + 23317 q^{22} + 56532 q^{23} - 45783 q^{24} - 86450 q^{25} + 198888 q^{26} - 58424 q^{27} + 287154 q^{28} - 178704 q^{29} - 384728 q^{30} + 292880 q^{31} - 640377 q^{32} - 408506 q^{33} + 262520 q^{34} + 621336 q^{35} + 486770 q^{36} - 1408096 q^{37} + 205086 q^{38} + 815736 q^{39} + 925470 q^{40} - 926994 q^{41} - 936278 q^{42} - 1837988 q^{43} - 511371 q^{44} + 1933888 q^{45} + 5286604 q^{46} + 1416996 q^{47} + 144447 q^{48} + 681652 q^{49} - 4138710 q^{50} - 2505268 q^{51} - 3586666 q^{52} - 356328 q^{53} + 228251 q^{54} + 865080 q^{55} - 3446916 q^{56} + 1706388 q^{57} + 11381072 q^{58} - 2297052 q^{59} - 2670886 q^{60} - 1496632 q^{61} + 1024512 q^{62} - 88344 q^{63} + 1773542 q^{64} + 585816 q^{65} - 2197457 q^{66} - 6475124 q^{67} + 4032876 q^{68} + 6329016 q^{69} - 12864588 q^{70} - 8042868 q^{71} + 9440430 q^{72} - 485798 q^{73} + 6232812 q^{74} + 20769104 q^{75} - 6953413 q^{76} + 8525208 q^{77} - 7305352 q^{78} - 5604964 q^{79} - 448650 q^{80} + 7714382 q^{81} + 2321819 q^{82} - 14535552 q^{83} - 42654252 q^{84} - 12944460 q^{85} + 2747550 q^{86} + 6535176 q^{87} + 18704514 q^{88} + 5024424 q^{89} - 15096092 q^{90} + 28222480 q^{91} - 30094896 q^{92} + 26255544 q^{93} + 51863760 q^{94} - 43373748 q^{95} + 116808910 q^{96} - 12341194 q^{97} + 1851183 q^{98} - 26743912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 874 x^{18} + 315 x^{17} + 507687 x^{16} + 417048 x^{15} + 164584585 x^{14} + \cdots + 60\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!29 \nu^{19} + \cdots - 15\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 54\!\cdots\!51 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 72\!\cdots\!03 \nu^{19} + \cdots + 55\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!31 \nu^{19} + \cdots - 25\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!88 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36\!\cdots\!76 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 43\!\cdots\!33 \nu^{19} + \cdots - 22\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!77 \nu^{19} + \cdots - 33\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 74\!\cdots\!93 \nu^{19} + \cdots - 74\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 92\!\cdots\!28 \nu^{19} + \cdots + 22\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!05 \nu^{19} + \cdots - 81\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!13 \nu^{19} + \cdots + 49\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 23\!\cdots\!89 \nu^{19} + \cdots - 12\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 95\!\cdots\!83 \nu^{19} + \cdots - 54\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 75\!\cdots\!63 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 27\!\cdots\!11 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 10\!\cdots\!32 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 27\!\cdots\!01 \nu^{19} + \cdots - 23\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - 175\beta_{2} + \beta _1 - 175 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{12} - 2\beta_{9} + 2\beta_{8} - 2\beta_{4} + 291\beta_{3} - 148 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 12 \beta_{13} - 4 \beta_{11} + 15 \beta_{10} - 8 \beta_{9} - 18 \beta_{7} + \cdots - 324 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{19} - 7 \beta_{17} - 499 \beta_{16} + 48 \beta_{14} + 48 \beta_{13} - 983 \beta_{12} + \cdots + 45543 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{19} - 16 \beta_{18} - 580 \beta_{17} + 2750 \beta_{16} + 580 \beta_{15} - 6368 \beta_{14} + \cdots + 17286379 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4096 \beta_{18} + 2494 \beta_{15} - 33400 \beta_{13} - 211537 \beta_{11} + 42270 \beta_{10} + \cdots + 36298925 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10440 \beta_{19} + 250587 \beta_{17} - 1465128 \beta_{16} + 2727084 \beta_{14} + 2727084 \beta_{13} + \cdots - 6298963738 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2318148 \beta_{19} - 2318148 \beta_{18} + 445209 \beta_{17} + 86009527 \beta_{16} - 445209 \beta_{15} + \cdots + 10197897767 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4467912 \beta_{18} - 98723374 \beta_{15} - 1104979824 \beta_{13} - 716000362 \beta_{11} + \cdots + 42086007393 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1097301328 \beta_{19} + 75371960 \beta_{17} - 34597977109 \beta_{16} + 9216752304 \beta_{14} + \cdots - 7879958202660 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1765793488 \beta_{19} - 1765793488 \beta_{18} - 37712395837 \beta_{17} + 335040550820 \beta_{16} + \cdots + 916247510709124 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 482583058276 \beta_{18} - 126289771541 \beta_{15} - 4388664161440 \beta_{13} + \cdots + 20\!\cdots\!92 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 751597666080 \beta_{19} + 14322029310312 \beta_{17} - 152612703463638 \beta_{16} + \cdots - 35\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 204770333592768 \beta_{19} - 204770333592768 \beta_{18} - 85032480318366 \beta_{17} + \cdots + 23\!\cdots\!06 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 357209623435512 \beta_{18} + \cdots + 65\!\cdots\!10 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 85\!\cdots\!96 \beta_{19} + \cdots - 11\!\cdots\!39 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 18\!\cdots\!20 \beta_{19} + \cdots + 55\!\cdots\!97 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 35\!\cdots\!48 \beta_{18} + \cdots + 12\!\cdots\!27 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−10.2356 − 17.7286i
−6.49656 − 11.2524i
−6.16734 − 10.6821i
−4.83664 − 8.37730i
−1.39885 − 2.42288i
1.77243 + 3.06995i
2.77185 + 4.80098i
6.41342 + 11.1084i
9.25701 + 16.0336i
9.42029 + 16.3164i
−10.2356 + 17.7286i
−6.49656 + 11.2524i
−6.16734 + 10.6821i
−4.83664 + 8.37730i
−1.39885 + 2.42288i
1.77243 − 3.06995i
2.77185 − 4.80098i
6.41342 − 11.1084i
9.25701 − 16.0336i
9.42029 − 16.3164i
−10.7356 − 18.5946i −13.9072 − 24.0880i −166.507 + 288.398i −4.29374 − 7.43697i −298.605 + 517.199i −779.113 4401.90 706.679 − 1224.00i −92.1919 + 159.681i
7.2 −6.99656 − 12.1184i 27.3662 + 47.3997i −33.9038 + 58.7232i −193.286 − 334.782i 382.939 − 663.270i −228.443 −842.279 −404.321 + 700.305i −2704.68 + 4684.65i
7.3 −6.66734 − 11.5482i 12.5396 + 21.7192i −24.9069 + 43.1400i 183.518 + 317.862i 167.212 − 289.619i 823.002 −1042.59 779.017 − 1349.30i 2447.15 − 4238.59i
7.4 −5.33664 − 9.24333i −38.4779 − 66.6457i 7.04059 − 12.1947i −118.331 − 204.955i −410.685 + 711.328i 1248.71 −1516.47 −1867.60 + 3234.77i −1262.98 + 2187.54i
7.5 −1.89885 − 3.28891i −19.2878 − 33.4074i 56.7887 − 98.3610i 90.1006 + 156.059i −73.2491 + 126.871i −1568.65 −917.439 349.465 − 605.291i 342.175 − 592.665i
7.6 1.27243 + 2.20392i 41.0456 + 71.0931i 60.7618 − 105.243i 90.3664 + 156.519i −104.456 + 180.923i −488.213 635.005 −2275.99 + 3942.12i −229.971 + 398.321i
7.7 2.27185 + 3.93496i 1.65941 + 2.87418i 53.6774 − 92.9720i −124.311 − 215.312i −7.53984 + 13.0594i 506.955 1069.38 1087.99 − 1884.46i 564.830 − 978.314i
7.8 5.91342 + 10.2423i −28.8741 − 50.0114i −5.93714 + 10.2834i 248.329 + 430.118i 341.490 − 591.477i 1049.81 1373.40 −573.927 + 994.071i −2936.95 + 5086.94i
7.9 8.75701 + 15.1676i −34.2821 − 59.3783i −89.3705 + 154.794i −194.797 − 337.399i 600.417 − 1039.95i −1273.58 −888.681 −1257.02 + 2177.22i 3411.68 − 5909.21i
7.10 8.92029 + 15.4504i 18.2181 + 31.5548i −95.1432 + 164.793i 22.7050 + 39.3262i −325.022 + 562.955i −18.4771 −1111.23 429.698 − 744.259i −405.071 + 701.603i
11.1 −10.7356 + 18.5946i −13.9072 + 24.0880i −166.507 − 288.398i −4.29374 + 7.43697i −298.605 − 517.199i −779.113 4401.90 706.679 + 1224.00i −92.1919 − 159.681i
11.2 −6.99656 + 12.1184i 27.3662 − 47.3997i −33.9038 − 58.7232i −193.286 + 334.782i 382.939 + 663.270i −228.443 −842.279 −404.321 − 700.305i −2704.68 − 4684.65i
11.3 −6.66734 + 11.5482i 12.5396 − 21.7192i −24.9069 − 43.1400i 183.518 − 317.862i 167.212 + 289.619i 823.002 −1042.59 779.017 + 1349.30i 2447.15 + 4238.59i
11.4 −5.33664 + 9.24333i −38.4779 + 66.6457i 7.04059 + 12.1947i −118.331 + 204.955i −410.685 − 711.328i 1248.71 −1516.47 −1867.60 − 3234.77i −1262.98 − 2187.54i
11.5 −1.89885 + 3.28891i −19.2878 + 33.4074i 56.7887 + 98.3610i 90.1006 − 156.059i −73.2491 − 126.871i −1568.65 −917.439 349.465 + 605.291i 342.175 + 592.665i
11.6 1.27243 − 2.20392i 41.0456 − 71.0931i 60.7618 + 105.243i 90.3664 − 156.519i −104.456 − 180.923i −488.213 635.005 −2275.99 − 3942.12i −229.971 − 398.321i
11.7 2.27185 − 3.93496i 1.65941 − 2.87418i 53.6774 + 92.9720i −124.311 + 215.312i −7.53984 − 13.0594i 506.955 1069.38 1087.99 + 1884.46i 564.830 + 978.314i
11.8 5.91342 − 10.2423i −28.8741 + 50.0114i −5.93714 − 10.2834i 248.329 − 430.118i 341.490 + 591.477i 1049.81 1373.40 −573.927 − 994.071i −2936.95 − 5086.94i
11.9 8.75701 − 15.1676i −34.2821 + 59.3783i −89.3705 − 154.794i −194.797 + 337.399i 600.417 + 1039.95i −1273.58 −888.681 −1257.02 − 2177.22i 3411.68 + 5909.21i
11.10 8.92029 − 15.4504i 18.2181 − 31.5548i −95.1432 − 164.793i 22.7050 − 39.3262i −325.022 − 562.955i −18.4771 −1111.23 429.698 + 744.259i −405.071 − 701.603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.8.c.a ✓ 20
19.c even 3 1 inner 19.8.c.a ✓ 20
19.c even 3 1 361.8.a.f 10
19.d odd 6 1 361.8.a.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.8.c.a ✓ 20 1.a even 1 1 trivial
19.8.c.a ✓ 20 19.c even 3 1 inner
361.8.a.d 10 19.d odd 6 1
361.8.a.f 10 19.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(19, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 86\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 32\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 32\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 44\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 88\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 19\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 78\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 60\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 50\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 97\!\cdots\!89 \) Copy content Toggle raw display
show more
show less