Properties

Label 21.14.a.a
Level $21$
Weight $14$
Character orbit 21.a
Self dual yes
Analytic conductor $22.518$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(22.5184950799\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 7 \)
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 \(\beta = 28\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 72) q^{2} + 729 q^{3} + (144 \beta - 656) q^{4} + ( - 740 \beta - 26280) q^{5} + (729 \beta + 52488) q^{6} + 117649 q^{7} + (1520 \beta - 298368) q^{8} + 531441 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 72) q^{2} + 729 q^{3} + (144 \beta - 656) q^{4} + ( - 740 \beta - 26280) q^{5} + (729 \beta + 52488) q^{6} + 117649 q^{7} + (1520 \beta - 298368) q^{8} + 531441 q^{9} + ( - 79560 \beta - 3632640) q^{10} + ( - 93940 \beta - 4296798) q^{11} + (104976 \beta - 478224) q^{12} + (218448 \beta - 18991006) q^{13} + (117649 \beta + 8470728) q^{14} + ( - 539460 \beta - 19158120) q^{15} + ( - 1368576 \beta - 12533504) q^{16} + (2589212 \beta + 8321544) q^{17} + (531441 \beta + 38263752) q^{18} + ( - 4963752 \beta - 105597652) q^{19} + ( - 3298880 \beta - 233389440) q^{20} + 85766121 q^{21} + ( - 11060478 \beta - 530316336) q^{22} + (12654916 \beta - 97498926) q^{23} + (1108080 \beta - 217510272) q^{24} + (38894400 \beta + 757890475) q^{25} + ( - 3262750 \beta - 853562736) q^{26} + 387420489 q^{27} + (16941456 \beta - 77177744) q^{28} + (85588632 \beta - 1740821274) q^{29} + ( - 57999240 \beta - 2648194560) q^{30} + ( - 90054936 \beta - 4196898760) q^{31} + ( - 123522816 \beta - 1677072384) q^{32} + ( - 68482260 \beta - 3132365742) q^{33} + (194744808 \beta + 6688977792) q^{34} + ( - 87060260 \beta - 3091815720) q^{35} + (76527504 \beta - 348625296) q^{36} + (143632656 \beta - 11168973358) q^{37} + ( - 462987796 \beta - 19277775648) q^{38} + (159248592 \beta - 13844443374) q^{39} + (180846720 \beta + 5195581440) q^{40} + (505089780 \beta - 6253701660) q^{41} + (85766121 \beta + 6175160712) q^{42} + ( - 379718208 \beta + 13031373164) q^{43} + ( - 557114272 \beta - 28997651232) q^{44} + ( - 393266340 \beta - 13966269480) q^{45} + (813655026 \beta + 22744439760) q^{46} + ( - 1620335768 \beta + 14611329468) q^{47} + ( - 997691904 \beta - 9136924416) q^{48} + 13841287201 q^{49} + (3558287275 \beta + 146047743000) q^{50} + (1887535548 \beta + 6066405576) q^{51} + ( - 2878006752 \beta + 86443816160) q^{52} + (97048544 \beta + 261955934154) q^{53} + (387420489 \beta + 27894275208) q^{54} + (5648373720 \beta + 276420542640) q^{55} + (178826480 \beta - 35102696832) q^{56} + ( - 3618575208 \beta - 76980688308) q^{57} + (4421560230 \beta + 75965330736) q^{58} + (432706712 \beta - 47222290512) q^{59} + ( - 2404883520 \beta - 170140901760) q^{60} + ( - 4959846216 \beta + 230860277126) q^{61} + ( - 10680854152 \beta - 513985920192) q^{62} + 62523502209 q^{63} + (640659456 \beta - 308600410112) q^{64} + (8312531000 \beta + 118879262640) q^{65} + ( - 8063088462 \beta - 386600608944) q^{66} + ( - 2390900328 \beta - 618939971032) q^{67} + ( - 500220736 \beta + 871476100992) q^{68} + (9225433764 \beta - 71076717054) q^{69} + ( - 9360154440 \beta - 427376463360) q^{70} + (1476015420 \beta + 980563309902) q^{71} + (807790320 \beta - 158564988288) q^{72} + (4871692872 \beta - 1203895891642) q^{73} + ( - 827422126 \beta - 466342074864) q^{74} + (28354017600 \beta + 552502156275) q^{75} + ( - 11949840576 \beta - 1611891177664) q^{76} + ( - 11051947060 \beta - 505513987902) q^{77} + ( - 2378544750 \beta - 622247234544) q^{78} + ( - 3091143384 \beta - 1311247111924) q^{79} + (45240970240 \beta + 2711359641600) q^{80} + 282429536481 q^{81} + (30112762500 \beta + 737704643040) q^{82} + ( - 33572095344 \beta + 2244780907956) q^{83} + (12350321424 \beta - 56262575376) q^{84} + ( - 74202433920 \beta - 4725161878080) q^{85} + ( - 14308337812 \beta + 45161642592) q^{86} + (62394112728 \beta - 1269058708746) q^{87} + (21497556960 \beta + 946187768064) q^{88} + (10542875524 \beta + 260463920292) q^{89} + ( - 42281445960 \beta - 1930533834240) q^{90} + (25700188752 \beta - 2234272864894) q^{91} + ( - 22341470240 \beta + 4350027485664) q^{92} + ( - 65650048344 \beta - 3059539196040) q^{93} + ( - 102052845828 \beta - 2759014004640) q^{94} + (208589665040 \beta + 11414417375520) q^{95} + ( - 90048132864 \beta - 1222585767936) q^{96} + (3400065144 \beta - 6513978315802) q^{97} + (13841287201 \beta + 996572678472) q^{98} + ( - 49923567540 \beta - 2283494625918) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 144 q^{2} + 1458 q^{3} - 1312 q^{4} - 52560 q^{5} + 104976 q^{6} + 235298 q^{7} - 596736 q^{8} + 1062882 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 144 q^{2} + 1458 q^{3} - 1312 q^{4} - 52560 q^{5} + 104976 q^{6} + 235298 q^{7} - 596736 q^{8} + 1062882 q^{9} - 7265280 q^{10} - 8593596 q^{11} - 956448 q^{12} - 37982012 q^{13} + 16941456 q^{14} - 38316240 q^{15} - 25067008 q^{16} + 16643088 q^{17} + 76527504 q^{18} - 211195304 q^{19} - 466778880 q^{20} + 171532242 q^{21} - 1060632672 q^{22} - 194997852 q^{23} - 435020544 q^{24} + 1515780950 q^{25} - 1707125472 q^{26} + 774840978 q^{27} - 154355488 q^{28} - 3481642548 q^{29} - 5296389120 q^{30} - 8393797520 q^{31} - 3354144768 q^{32} - 6264731484 q^{33} + 13377955584 q^{34} - 6183631440 q^{35} - 697250592 q^{36} - 22337946716 q^{37} - 38555551296 q^{38} - 27688886748 q^{39} + 10391162880 q^{40} - 12507403320 q^{41} + 12350321424 q^{42} + 26062746328 q^{43} - 57995302464 q^{44} - 27932538960 q^{45} + 45488879520 q^{46} + 29222658936 q^{47} - 18273848832 q^{48} + 27682574402 q^{49} + 292095486000 q^{50} + 12132811152 q^{51} + 172887632320 q^{52} + 523911868308 q^{53} + 55788550416 q^{54} + 552841085280 q^{55} - 70205393664 q^{56} - 153961376616 q^{57} + 151930661472 q^{58} - 94444581024 q^{59} - 340281803520 q^{60} + 461720554252 q^{61} - 1027971840384 q^{62} + 125047004418 q^{63} - 617200820224 q^{64} + 237758525280 q^{65} - 773201217888 q^{66} - 1237879942064 q^{67} + 1742952201984 q^{68} - 142153434108 q^{69} - 854752926720 q^{70} + 1961126619804 q^{71} - 317129976576 q^{72} - 2407791783284 q^{73} - 932684149728 q^{74} + 1105004312550 q^{75} - 3223782355328 q^{76} - 1011027975804 q^{77} - 1244494469088 q^{78} - 2622494223848 q^{79} + 5422719283200 q^{80} + 564859072962 q^{81} + 1475409286080 q^{82} + 4489561815912 q^{83} - 112525150752 q^{84} - 9450323756160 q^{85} + 90323285184 q^{86} - 2538117417492 q^{87} + 1892375536128 q^{88} + 520927840584 q^{89} - 3861067668480 q^{90} - 4468545729788 q^{91} + 8700054971328 q^{92} - 6119078392080 q^{93} - 5518028009280 q^{94} + 22828834751040 q^{95} - 2445171535872 q^{96} - 13027956631604 q^{97} + 1993145356944 q^{98} - 4566989251836 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
−1.73205
1.73205
23.5026 729.000 −7639.63 9608.09 17133.4 117649. −372084. 531441. 225815.
1.2 120.497 729.000 6327.63 −62168.1 87842.6 117649. −224652. 531441. −7.49109e6
\(n\): 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 21.14.a.a 2
3.b odd 2 1 63.14.a.a 2
7.b odd 2 1 147.14.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.14.a.a 2 1.a even 1 1 trivial
63.14.a.a 2 3.b odd 2 1
147.14.a.c 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 144T_{2} + 2832 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 144T + 2832 \) Copy content Toggle raw display
$3$ \( (T - 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 52560 T - 597316800 \) Copy content Toggle raw display
$7$ \( (T - 117649)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8593596 T - 2293276854396 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 248421977380228 \) Copy content Toggle raw display
$17$ \( T^{2} - 16643088 T - 15\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{2} + 211195304 T - 46\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{2} + 194997852 T - 36\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{2} + 3481642548 T - 14\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{2} + 8393797520 T - 14\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{2} + 22337946716 T + 76\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{2} + 12507403320 T - 56\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{2} - 26062746328 T - 16\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{2} - 29222658936 T - 59\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} - 523911868308 T + 68\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{2} + 94444581024 T + 17\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{2} - 461720554252 T - 45\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} + 1237879942064 T + 36\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} - 1961126619804 T + 95\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{2} + 2407791783284 T + 13\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{2} + 2622494223848 T + 16\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{2} - 4489561815912 T + 23\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} - 520927840584 T - 19\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{2} + 13027956631604 T + 42\!\cdots\!32 \) Copy content Toggle raw display
show more
show less