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Properties

Label 21.7.f.a
Level $21$
Weight $7$
Character orbit 21.f
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
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] = [21,7,Mod(10,21)]
 
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, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.10");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\), 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: \(4.83113575602\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1 - 1) q^{2} + (9 \beta_{2} - 18) q^{3} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + ( - 243 \beta_{2} + 243) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1 - 1) q^{2} + (9 \beta_{2} - 18) q^{3} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + (486 \beta_{7} + 729 \beta_{6} + \cdots - 14094) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} - 108 q^{3} - 173 q^{4} - 294 q^{5} - 656 q^{7} + 3326 q^{8} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} - 108 q^{3} - 173 q^{4} - 294 q^{5} - 656 q^{7} + 3326 q^{8} + 972 q^{9} - 3411 q^{10} - 314 q^{11} + 4671 q^{12} - 5360 q^{14} + 5292 q^{15} - 12721 q^{16} - 5532 q^{17} + 1215 q^{18} - 18234 q^{19} + 9342 q^{21} + 86106 q^{22} + 3928 q^{23} - 44901 q^{24} - 17038 q^{25} + 12366 q^{26} + 85037 q^{28} - 8300 q^{29} + 30699 q^{30} - 89508 q^{31} - 186207 q^{32} + 8478 q^{33} - 25860 q^{35} - 84078 q^{36} + 64706 q^{37} - 77136 q^{38} + 29106 q^{39} + 221823 q^{40} + 173583 q^{42} + 45740 q^{43} + 92529 q^{44} - 71442 q^{45} - 111504 q^{46} + 483276 q^{47} - 310684 q^{49} + 967216 q^{50} + 49788 q^{51} - 1673988 q^{52} - 540974 q^{53} - 32805 q^{54} - 241885 q^{56} + 328212 q^{57} + 539799 q^{58} - 181770 q^{59} - 146367 q^{60} + 418224 q^{61} - 92826 q^{63} + 2378626 q^{64} - 414204 q^{65} - 1162431 q^{66} - 1158902 q^{67} - 821250 q^{68} + 1087917 q^{70} + 1442344 q^{71} + 404109 q^{72} - 378666 q^{73} - 432940 q^{74} + 460026 q^{75} + 1065994 q^{77} - 222588 q^{78} + 611452 q^{79} - 2094945 q^{80} - 236196 q^{81} - 1561266 q^{82} - 1579716 q^{84} - 275112 q^{85} + 816224 q^{86} + 112050 q^{87} - 366441 q^{88} - 989196 q^{89} + 304446 q^{91} + 678720 q^{92} + 805572 q^{93} - 716148 q^{94} - 591792 q^{95} + 5027589 q^{96} + 3509629 q^{98} - 152604 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 405518177 \nu^{7} - 12595814413 \nu^{6} - 65484675874 \nu^{5} - 2673339327091 \nu^{4} + \cdots - 10\!\cdots\!70 ) / 11\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + \cdots - 5583985297290 ) / 7542500798721 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30359760961 \nu^{7} - 17272473736 \nu^{6} + 12542253415247 \nu^{5} - 45647117796292 \nu^{4} + \cdots - 88\!\cdots\!20 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36613259519 \nu^{7} + 484838322556 \nu^{6} + 240706081393 \nu^{5} + 69967211645572 \nu^{4} + \cdots - 10\!\cdots\!80 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20960695316 \nu^{7} - 201659710511 \nu^{6} + 3951590320162 \nu^{5} - 21226630078907 \nu^{4} + \cdots - 12\!\cdots\!60 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14593340681 \nu^{7} + 40700920519 \nu^{6} + 3013487903677 \nu^{5} + 21592891249843 \nu^{4} + \cdots + 66\!\cdots\!80 ) / 13\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 106\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{7} - 12\beta_{6} - 10\beta_{5} - 6\beta_{4} + 165\beta_{3} - 4\beta_{2} - 161\beta _1 - 204 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 213\beta_{7} - 217\beta_{6} - 8\beta_{5} - 205\beta_{4} + 4\beta_{3} + 17990\beta_{2} - 718\beta _1 - 17986 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2462\beta_{7} + 930\beta_{6} + 766\beta_{5} + 2544\beta_{4} - 30357\beta_{3} + 77878\beta_{2} - 848\beta _1 - 1614 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4432 \beta_{7} + 6648 \beta_{6} + 45813 \beta_{5} + 43597 \beta_{4} - 195138 \beta_{3} + 2216 \beta_{2} + \cdots + 3368601 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 221730 \beta_{7} + 385038 \beta_{6} + 326616 \beta_{5} - 104886 \beta_{4} - 163308 \beta_{3} + \cdots + 20878866 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
7.29767 + 12.6399i
2.26350 + 3.92050i
−2.75320 − 4.76869i
−6.30797 − 10.9257i
7.29767 − 12.6399i
2.26350 − 3.92050i
−2.75320 + 4.76869i
−6.30797 + 10.9257i
−7.79767 − 13.5060i −13.5000 − 7.79423i −89.6073 + 155.204i 22.3721 − 12.9165i 243.107i −203.121 − 276.389i 1796.81 121.500 + 210.444i −348.901 − 201.438i
10.2 −2.76350 − 4.78652i −13.5000 − 7.79423i 16.7261 − 28.9705i −57.9943 + 33.4830i 86.1574i −240.457 + 244.600i −538.619 121.500 + 210.444i 320.535 + 185.061i
10.3 2.25320 + 3.90266i −13.5000 − 7.79423i 21.8461 − 37.8386i 53.9244 − 31.1333i − 70.2479i 218.833 − 264.124i 485.305 121.500 + 210.444i 243.005 + 140.299i
10.4 5.80797 + 10.0597i −13.5000 − 7.79423i −35.4650 + 61.4271i −165.302 + 95.4373i − 181.074i −103.254 + 327.089i −80.4975 121.500 + 210.444i −1920.14 − 1108.59i
19.1 −7.79767 + 13.5060i −13.5000 + 7.79423i −89.6073 − 155.204i 22.3721 + 12.9165i − 243.107i −203.121 + 276.389i 1796.81 121.500 − 210.444i −348.901 + 201.438i
19.2 −2.76350 + 4.78652i −13.5000 + 7.79423i 16.7261 + 28.9705i −57.9943 − 33.4830i − 86.1574i −240.457 − 244.600i −538.619 121.500 − 210.444i 320.535 − 185.061i
19.3 2.25320 − 3.90266i −13.5000 + 7.79423i 21.8461 + 37.8386i 53.9244 + 31.1333i 70.2479i 218.833 + 264.124i 485.305 121.500 − 210.444i 243.005 − 140.299i
19.4 5.80797 − 10.0597i −13.5000 + 7.79423i −35.4650 − 61.4271i −165.302 − 95.4373i 181.074i −103.254 − 327.089i −80.4975 121.500 − 210.444i −1920.14 + 1108.59i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.7.f.a ✓ 8
3.b odd 2 1 63.7.m.d 8
4.b odd 2 1 336.7.bh.d 8
7.b odd 2 1 147.7.f.d 8
7.c even 3 1 147.7.d.b 8
7.c even 3 1 147.7.f.d 8
7.d odd 6 1 inner 21.7.f.a ✓ 8
7.d odd 6 1 147.7.d.b 8
21.g even 6 1 63.7.m.d 8
21.g even 6 1 441.7.d.c 8
21.h odd 6 1 441.7.d.c 8
28.f even 6 1 336.7.bh.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a ✓ 8 1.a even 1 1 trivial
21.7.f.a ✓ 8 7.d odd 6 1 inner
63.7.m.d 8 3.b odd 2 1
63.7.m.d 8 21.g even 6 1
147.7.d.b 8 7.c even 3 1
147.7.d.b 8 7.d odd 6 1
147.7.f.d 8 7.b odd 2 1
147.7.f.d 8 7.c even 3 1
336.7.bh.d 8 4.b odd 2 1
336.7.bh.d 8 28.f even 6 1
441.7.d.c 8 21.g even 6 1
441.7.d.c 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5T_{2}^{7} + 227T_{2}^{6} - 442T_{2}^{5} + 37712T_{2}^{4} + 12248T_{2}^{3} + 992080T_{2}^{2} - 1281408T_{2} + 20358144 \) acting on \(S_{7}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{7} + \cdots + 20358144 \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 422734160250000 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 73\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 19\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 77\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 99\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
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