Properties

Label 21.10.a.c
Level $21$
Weight $10$
Character orbit 21.a
Self dual yes
Analytic conductor $10.816$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,10,Mod(1,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

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: yes
Analytic conductor: \(10.8157525594\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 15) q^{2} - 81 q^{3} + ( - 30 \beta + 58) q^{4} + ( - 70 \beta + 564) q^{5} + (81 \beta - 1215) q^{6} + 2401 q^{7} + (4 \beta + 3540) q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 15) q^{2} - 81 q^{3} + ( - 30 \beta + 58) q^{4} + ( - 70 \beta + 564) q^{5} + (81 \beta - 1215) q^{6} + 2401 q^{7} + (4 \beta + 3540) q^{8} + 6561 q^{9} + ( - 1614 \beta + 32610) q^{10} + ( - 1718 \beta + 36642) q^{11} + (2430 \beta - 4698) q^{12} + ( - 528 \beta + 70550) q^{13} + ( - 2401 \beta + 36015) q^{14} + (5670 \beta - 45684) q^{15} + (11880 \beta + 22024) q^{16} + (14830 \beta - 50892) q^{17} + ( - 6561 \beta + 98415) q^{18} + ( - 2892 \beta + 240872) q^{19} + ( - 20980 \beta + 757212) q^{20} - 194481 q^{21} + ( - 62412 \beta + 1142340) q^{22} + (95690 \beta - 491106) q^{23} + ( - 324 \beta - 286740) q^{24} + ( - 78960 \beta + 55471) q^{25} + ( - 78470 \beta + 1240410) q^{26} - 531441 q^{27} + ( - 72030 \beta + 139258) q^{28} + (142260 \beta - 1275462) q^{29} + (130734 \beta - 2641410) q^{30} + ( - 196092 \beta - 2467924) q^{31} + (154128 \beta - 5580720) q^{32} + (139158 \beta - 2968002) q^{33} + (273342 \beta - 5879730) q^{34} + ( - 168070 \beta + 1354164) q^{35} + ( - 196830 \beta + 380538) q^{36} + (442296 \beta - 8128258) q^{37} + ( - 284252 \beta + 4610820) q^{38} + (42768 \beta - 5714550) q^{39} + ( - 245544 \beta + 1899960) q^{40} + ( - 196590 \beta + 24353928) q^{41} + (194481 \beta - 2917215) q^{42} + (168744 \beta + 3994820) q^{43} + ( - 1198904 \beta + 19906536) q^{44} + ( - 459270 \beta + 3700404) q^{45} + (1926456 \beta - 40379640) q^{46} + (768164 \beta + 42786204) q^{47} + ( - 962280 \beta - 1783944) q^{48} + 5764801 q^{49} + ( - 1239871 \beta + 28073265) q^{50} + ( - 1201230 \beta + 4122252) q^{51} + ( - 2147124 \beta + 9556700) q^{52} + (3803152 \beta + 13282662) q^{53} + (531441 \beta - 7971615) q^{54} + ( - 3533892 \beta + 62155788) q^{55} + (9604 \beta + 8499540) q^{56} + (234252 \beta - 19510632) q^{57} + (3409362 \beta - 68211630) q^{58} + (1648036 \beta + 57600480) q^{59} + (1699380 \beta - 61334172) q^{60} + (6373980 \beta - 71410102) q^{61} + ( - 473456 \beta + 30632880) q^{62} + 15752961 q^{63} + (1810080 \beta - 148161248) q^{64} + ( - 5236292 \beta + 52541400) q^{65} + (5055372 \beta - 92529540) q^{66} + ( - 5935140 \beta + 13760696) q^{67} + (2386900 \beta - 156442236) q^{68} + ( - 7750890 \beta + 39779586) q^{69} + ( - 3875214 \beta + 78296610) q^{70} + ( - 19134282 \beta + 19035090) q^{71} + (26244 \beta + 23225940) q^{72} + ( - 7076172 \beta - 1047658) q^{73} + (14762698 \beta - 274515990) q^{74} + (6395760 \beta - 4493151) q^{75} + ( - 7393896 \beta + 43902776) q^{76} + ( - 4124918 \beta + 87977442) q^{77} + (6356070 \beta - 100473210) q^{78} + (20624004 \beta + 217548524) q^{79} + (5158640 \beta - 274480464) q^{80} + 43046721 q^{81} + ( - 27302778 \beta + 433132470) q^{82} + (34502856 \beta + 132144372) q^{83} + (5834430 \beta - 11279898) q^{84} + (11926560 \beta - 386847588) q^{85} + ( - 1463660 \beta + 1705620) q^{86} + ( - 11523060 \beta + 103312422) q^{87} + ( - 5935152 \beta + 127341840) q^{88} + ( - 11924350 \beta + 321336888) q^{89} + ( - 10589454 \beta + 213954210) q^{90} + ( - 1267728 \beta + 169390550) q^{91} + (20283200 \beta - 1018875648) q^{92} + (15883452 \beta + 199901844) q^{93} + ( - 31263744 \beta + 376776480) q^{94} + ( - 18492128 \beta + 205693608) q^{95} + ( - 12484368 \beta + 452038320) q^{96} + (38859852 \beta + 172680614) q^{97} + ( - 5764801 \beta + 86472015) q^{98} + ( - 11271798 \beta + 240408162) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 30 q^{2} - 162 q^{3} + 116 q^{4} + 1128 q^{5} - 2430 q^{6} + 4802 q^{7} + 7080 q^{8} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 30 q^{2} - 162 q^{3} + 116 q^{4} + 1128 q^{5} - 2430 q^{6} + 4802 q^{7} + 7080 q^{8} + 13122 q^{9} + 65220 q^{10} + 73284 q^{11} - 9396 q^{12} + 141100 q^{13} + 72030 q^{14} - 91368 q^{15} + 44048 q^{16} - 101784 q^{17} + 196830 q^{18} + 481744 q^{19} + 1514424 q^{20} - 388962 q^{21} + 2284680 q^{22} - 982212 q^{23} - 573480 q^{24} + 110942 q^{25} + 2480820 q^{26} - 1062882 q^{27} + 278516 q^{28} - 2550924 q^{29} - 5282820 q^{30} - 4935848 q^{31} - 11161440 q^{32} - 5936004 q^{33} - 11759460 q^{34} + 2708328 q^{35} + 761076 q^{36} - 16256516 q^{37} + 9221640 q^{38} - 11429100 q^{39} + 3799920 q^{40} + 48707856 q^{41} - 5834430 q^{42} + 7989640 q^{43} + 39813072 q^{44} + 7400808 q^{45} - 80759280 q^{46} + 85572408 q^{47} - 3567888 q^{48} + 11529602 q^{49} + 56146530 q^{50} + 8244504 q^{51} + 19113400 q^{52} + 26565324 q^{53} - 15943230 q^{54} + 124311576 q^{55} + 16999080 q^{56} - 39021264 q^{57} - 136423260 q^{58} + 115200960 q^{59} - 122668344 q^{60} - 142820204 q^{61} + 61265760 q^{62} + 31505922 q^{63} - 296322496 q^{64} + 105082800 q^{65} - 185059080 q^{66} + 27521392 q^{67} - 312884472 q^{68} + 79559172 q^{69} + 156593220 q^{70} + 38070180 q^{71} + 46451880 q^{72} - 2095316 q^{73} - 549031980 q^{74} - 8986302 q^{75} + 87805552 q^{76} + 175954884 q^{77} - 200946420 q^{78} + 435097048 q^{79} - 548960928 q^{80} + 86093442 q^{81} + 866264940 q^{82} + 264288744 q^{83} - 22559796 q^{84} - 773695176 q^{85} + 3411240 q^{86} + 206624844 q^{87} + 254683680 q^{88} + 642673776 q^{89} + 427908420 q^{90} + 338781100 q^{91} - 2037751296 q^{92} + 399803688 q^{93} + 753552960 q^{94} + 411387216 q^{95} + 904076640 q^{96} + 345361228 q^{97} + 172944030 q^{98} + 480816324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.78709
−8.78709
−3.57418 −81.0000 −499.225 −736.192 289.508 2401.00 3614.30 6561.00 2631.28
1.2 33.5742 −81.0000 615.225 1864.19 −2719.51 2401.00 3465.70 6561.00 62588.7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.10.a.c 2
3.b odd 2 1 63.10.a.b 2
4.b odd 2 1 336.10.a.l 2
7.b odd 2 1 147.10.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.10.a.c 2 1.a even 1 1 trivial
63.10.a.b 2 3.b odd 2 1
147.10.a.e 2 7.b odd 2 1
336.10.a.l 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 30T_{2} - 120 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 30T - 120 \) Copy content Toggle raw display
$3$ \( (T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 1128 T - 1372404 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 73284 T + 324360384 \) Copy content Toggle raw display
$13$ \( T^{2} - 141100 T + 4881122020 \) Copy content Toggle raw display
$17$ \( T^{2} + 101784 T - 73285474836 \) Copy content Toggle raw display
$19$ \( T^{2} - 481744 T + 55133856304 \) Copy content Toggle raw display
$23$ \( T^{2} + 982212 T - 2917833651264 \) Copy content Toggle raw display
$29$ \( T^{2} + 2550924 T - 5355274808556 \) Copy content Toggle raw display
$31$ \( T^{2} + 4935848 T - 7175316130304 \) Copy content Toggle raw display
$37$ \( T^{2} + 16256516 T - 1422306192956 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 579780377334684 \) Copy content Toggle raw display
$43$ \( T^{2} - 7989640 T + 6134871382480 \) Copy content Toggle raw display
$47$ \( T^{2} - 85572408 T + 16\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} - 26565324 T - 48\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{2} - 115200960 T + 23\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{2} + 142820204 T - 89\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{2} - 27521392 T - 11\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} - 38070180 T - 12\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{2} + 2095316 T - 17\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{2} - 435097048 T - 99\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{2} - 264288744 T - 39\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} - 642673776 T + 54\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{2} - 345361228 T - 49\!\cdots\!84 \) Copy content Toggle raw display
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