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Properties

Label 21.12.e.b
Level $21$
Weight $12$
Character orbit 21.e
Analytic conductor $16.135$
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] = [21,12,Mod(4,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.4");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 21.e (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: \(16.1352067918\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 13382 x^{14} + 112147 x^{13} + 125048871 x^{12} + 1181867298 x^{11} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{9}\cdot 7^{10} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 7 \beta_{2} - 7) q^{2} - 243 \beta_{2} q^{3} + ( - \beta_{6} + 1346 \beta_{2} + 7 \beta_1) q^{4} + ( - 26 \beta_{4} - \beta_{3} + \cdots - 273) q^{5}+ \cdots + ( - 59049 \beta_{2} - 59049) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 7 \beta_{2} - 7) q^{2} - 243 \beta_{2} q^{3} + ( - \beta_{6} + 1346 \beta_{2} + 7 \beta_1) q^{4} + ( - 26 \beta_{4} - \beta_{3} + \cdots - 273) q^{5}+ \cdots + (59049 \beta_{15} + 177147 \beta_{14} + \cdots - 7903354356) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 55 q^{2} + 1944 q^{3} - 10757 q^{4} - 2156 q^{5} - 26730 q^{6} - 6560 q^{7} + 302094 q^{8} - 472392 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 55 q^{2} + 1944 q^{3} - 10757 q^{4} - 2156 q^{5} - 26730 q^{6} - 6560 q^{7} + 302094 q^{8} - 472392 q^{9} - 718815 q^{10} + 1072828 q^{11} + 2613951 q^{12} - 2844856 q^{13} + 1059682 q^{14} - 1047816 q^{15} - 11955833 q^{16} - 3790296 q^{17} - 3247695 q^{18} + 2759108 q^{19} + 38210918 q^{20} - 5319756 q^{21} + 95137446 q^{22} - 24960960 q^{23} + 36704421 q^{24} - 175384580 q^{25} - 146707378 q^{26} - 229582512 q^{27} + 208799011 q^{28} + 254232472 q^{29} + 174672045 q^{30} + 607263128 q^{31} - 733869731 q^{32} - 260697204 q^{33} - 1637121120 q^{34} + 627607136 q^{35} + 1270380186 q^{36} + 781724012 q^{37} + 1659771566 q^{38} - 345650004 q^{39} - 1224123621 q^{40} - 3506571616 q^{41} + 55626831 q^{42} + 4671710936 q^{43} + 518697653 q^{44} - 127309644 q^{45} - 7349869416 q^{46} - 2356198440 q^{47} - 5810534838 q^{48} + 2659533472 q^{49} + 26148044044 q^{50} + 921041928 q^{51} + 9635762924 q^{52} - 9647676276 q^{53} + 789189885 q^{54} - 16959187272 q^{55} - 37701118269 q^{56} + 1340926488 q^{57} + 8372495583 q^{58} + 19783259052 q^{59} + 4642626537 q^{60} + 11562666392 q^{61} - 11653849134 q^{62} - 905339268 q^{63} - 2861922158 q^{64} + 12440755432 q^{65} + 11559199689 q^{66} + 6525128516 q^{67} + 34775041176 q^{68} - 12131026560 q^{69} - 97836316839 q^{70} + 1322680368 q^{71} - 8919174303 q^{72} - 11555112460 q^{73} + 69076106718 q^{74} + 42618452940 q^{75} + 133073390744 q^{76} - 56594995972 q^{77} - 71299785708 q^{78} - 51076215280 q^{79} - 140049596215 q^{80} - 27894275208 q^{81} + 79124576244 q^{82} + 191396510088 q^{83} - 10412243820 q^{84} - 30338237520 q^{85} - 142834897676 q^{86} + 30889245348 q^{87} - 101794917285 q^{88} - 103693786472 q^{89} + 84890613870 q^{90} + 112259205116 q^{91} + 431278491168 q^{92} - 147564940104 q^{93} + 9192309402 q^{94} - 53428267184 q^{95} + 178330344633 q^{96} + 320469297032 q^{97} - 374440923193 q^{98} - 126698841144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 13382 x^{14} + 112147 x^{13} + 125048871 x^{12} + 1181867298 x^{11} + \cdots + 30\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!17 \nu^{15} + \cdots - 43\!\cdots\!76 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 71\!\cdots\!54 \nu^{15} + \cdots - 66\!\cdots\!96 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 63\!\cdots\!82 \nu^{15} + \cdots - 65\!\cdots\!88 ) / 60\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!04 \nu^{15} + \cdots - 18\!\cdots\!23 ) / 60\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58\!\cdots\!55 \nu^{15} + \cdots - 18\!\cdots\!76 ) / 56\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!61 \nu^{15} + \cdots + 18\!\cdots\!76 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 74\!\cdots\!37 \nu^{15} + \cdots - 21\!\cdots\!40 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\!\cdots\!72 \nu^{15} + \cdots - 97\!\cdots\!52 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!05 \nu^{15} + \cdots - 38\!\cdots\!44 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!65 \nu^{15} + \cdots + 67\!\cdots\!84 ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{15} + \cdots - 35\!\cdots\!20 ) / 47\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 44\!\cdots\!66 \nu^{15} + \cdots + 11\!\cdots\!12 ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 45\!\cdots\!24 \nu^{15} + \cdots + 56\!\cdots\!80 ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 92\!\cdots\!19 \nu^{15} + \cdots + 41\!\cdots\!24 ) / 47\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + 7\beta_{4} - 3345\beta_{2} - 3345 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{9} + \beta_{8} - 4\beta_{5} + 5325\beta_{4} - 5325\beta _1 - 22873 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{15} + 6 \beta_{14} + 4 \beta_{13} + 15 \beta_{12} - 5 \beta_{11} + 6 \beta_{10} + \cdots - 69647 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 180 \beta_{15} - 204 \beta_{14} + 180 \beta_{13} + 1328 \beta_{12} + 7084 \beta_{11} - 68 \beta_{10} + \cdots + 229068124 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 58256 \beta_{15} + 43624 \beta_{14} - 43624 \beta_{13} - 182840 \beta_{12} + 73264 \beta_{11} + \cdots + 105516532305 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 346892 \beta_{15} + 346892 \beta_{14} + 464100 \beta_{13} - 61914721 \beta_{12} - 50402913 \beta_{11} + \cdots - 3552936 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 187641012 \beta_{15} - 1009372038 \beta_{14} + 187641012 \beta_{13} + 1022043966 \beta_{12} + \cdots - 643420555195200 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 13947048120 \beta_{15} + 21782866848 \beta_{14} - 21782866848 \beta_{13} + 315616819180 \beta_{12} + \cdots - 12\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5142555832008 \beta_{15} + 5142555832008 \beta_{14} + 935573656648 \beta_{13} + 4982329776192 \beta_{12} + \cdots - 3369490487184 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 86261764521192 \beta_{15} - 140198773334856 \beta_{14} + 86261764521192 \beta_{13} + \cdots + 88\!\cdots\!81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 25\!\cdots\!02 \beta_{15} + \cdots + 24\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 45\!\cdots\!64 \beta_{15} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 97\!\cdots\!64 \beta_{15} + \cdots - 15\!\cdots\!89 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 76\!\cdots\!36 \beta_{15} + \cdots - 39\!\cdots\!33 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−39.6429 + 68.6635i
−33.3463 + 57.7574i
−17.6939 + 30.6467i
−13.4491 + 23.2946i
8.08390 − 14.0017i
17.1336 − 29.6762i
38.9943 − 67.5401i
40.4204 − 70.0101i
−39.6429 − 68.6635i
−33.3463 − 57.7574i
−17.6939 − 30.6467i
−13.4491 − 23.2946i
8.08390 + 14.0017i
17.1336 + 29.6762i
38.9943 + 67.5401i
40.4204 + 70.0101i
−43.1429 − 74.7256i 121.500 − 210.444i −2698.61 + 4674.14i −5852.82 − 10137.4i −20967.4 −41570.9 + 15785.6i 288991. −29524.5 − 51137.9i −505015. + 874711.i
4.2 −36.8463 − 63.8196i 121.500 − 210.444i −1691.29 + 2929.41i 5402.58 + 9357.55i −17907.3 20587.0 − 39414.5i 98349.0 −29524.5 − 51137.9i 398130. − 689581.i
4.3 −21.1939 − 36.7089i 121.500 − 210.444i 125.640 − 217.615i 1738.76 + 3011.61i −10300.2 634.714 + 44462.6i −97461.3 −29524.5 − 51137.9i 73701.9 − 127655.i
4.4 −16.9491 − 29.3567i 121.500 − 210.444i 449.454 − 778.478i −5866.37 − 10160.9i −8237.27 43905.4 − 7045.50i −99895.0 −29524.5 − 51137.9i −198860. + 344435.i
4.5 4.58390 + 7.93955i 121.500 − 210.444i 981.976 − 1700.83i −166.478 − 288.348i 2227.78 −897.453 − 44458.1i 36780.8 −29524.5 − 51137.9i 1526.23 − 2643.51i
4.6 13.6336 + 23.6140i 121.500 − 210.444i 652.251 − 1129.73i 3269.71 + 5663.30i 6625.92 −29346.7 + 33408.1i 91413.2 −29524.5 − 51137.9i −89155.6 + 154422.i
4.7 35.4943 + 61.4779i 121.500 − 210.444i −1495.69 + 2590.60i −3663.30 − 6345.02i 17250.2 −40877.1 − 17504.0i −66968.7 −29524.5 − 51137.9i 260052. − 450424.i
4.8 36.9204 + 63.9480i 121.500 − 210.444i −1702.23 + 2948.34i 4059.92 + 7031.99i 17943.3 44285.0 + 4020.21i −100162. −29524.5 − 51137.9i −299788. + 519248.i
16.1 −43.1429 + 74.7256i 121.500 + 210.444i −2698.61 − 4674.14i −5852.82 + 10137.4i −20967.4 −41570.9 − 15785.6i 288991. −29524.5 + 51137.9i −505015. − 874711.i
16.2 −36.8463 + 63.8196i 121.500 + 210.444i −1691.29 − 2929.41i 5402.58 − 9357.55i −17907.3 20587.0 + 39414.5i 98349.0 −29524.5 + 51137.9i 398130. + 689581.i
16.3 −21.1939 + 36.7089i 121.500 + 210.444i 125.640 + 217.615i 1738.76 − 3011.61i −10300.2 634.714 − 44462.6i −97461.3 −29524.5 + 51137.9i 73701.9 + 127655.i
16.4 −16.9491 + 29.3567i 121.500 + 210.444i 449.454 + 778.478i −5866.37 + 10160.9i −8237.27 43905.4 + 7045.50i −99895.0 −29524.5 + 51137.9i −198860. − 344435.i
16.5 4.58390 − 7.93955i 121.500 + 210.444i 981.976 + 1700.83i −166.478 + 288.348i 2227.78 −897.453 + 44458.1i 36780.8 −29524.5 + 51137.9i 1526.23 + 2643.51i
16.6 13.6336 − 23.6140i 121.500 + 210.444i 652.251 + 1129.73i 3269.71 − 5663.30i 6625.92 −29346.7 − 33408.1i 91413.2 −29524.5 + 51137.9i −89155.6 − 154422.i
16.7 35.4943 − 61.4779i 121.500 + 210.444i −1495.69 − 2590.60i −3663.30 + 6345.02i 17250.2 −40877.1 + 17504.0i −66968.7 −29524.5 + 51137.9i 260052. + 450424.i
16.8 36.9204 − 63.9480i 121.500 + 210.444i −1702.23 − 2948.34i 4059.92 − 7031.99i 17943.3 44285.0 − 4020.21i −100162. −29524.5 + 51137.9i −299788. − 519248.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.12.e.b ✓ 16
3.b odd 2 1 63.12.e.d 16
7.c even 3 1 inner 21.12.e.b ✓ 16
7.c even 3 1 147.12.a.k 8
7.d odd 6 1 147.12.a.l 8
21.h odd 6 1 63.12.e.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.12.e.b ✓ 16 1.a even 1 1 trivial
21.12.e.b ✓ 16 7.c even 3 1 inner
63.12.e.d 16 3.b odd 2 1
63.12.e.d 16 21.h odd 6 1
147.12.a.k 8 7.c even 3 1
147.12.a.l 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 55 T_{2}^{15} + 15083 T_{2}^{14} + 523222 T_{2}^{13} + 135476162 T_{2}^{12} + \cdots + 14\!\cdots\!84 \) acting on \(S_{12}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$3$ \( (T^{2} - 243 T + 59049)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 47\!\cdots\!12)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 80\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 30\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 78\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 88\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 73\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 24\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
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