Properties

Label 363.4.d.c
Level $363$
Weight $4$
Character orbit 363.d
Analytic conductor $21.418$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(362,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.362");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.d (of order \(2\), degree \(1\), 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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 82x^{10} + 2949x^{8} - 55732x^{6} + 613132x^{4} - 3655440x^{2} + 13220496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_1 q^{3} + (\beta_{4} + \beta_{2} - \beta_1 + 8) q^{4} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{5} + ( - \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} - 3 \beta_{6}) q^{6} + (2 \beta_{11} + 2 \beta_{8} + 5 \beta_{6}) q^{7} + (\beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - 5 \beta_{7}) q^{8} + (3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_1 q^{3} + (\beta_{4} + \beta_{2} - \beta_1 + 8) q^{4} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{5} + ( - \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} - 3 \beta_{6}) q^{6} + (2 \beta_{11} + 2 \beta_{8} + 5 \beta_{6}) q^{7} + (\beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - 5 \beta_{7}) q^{8} + (3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3) q^{9} + ( - \beta_{11} - 7 \beta_{9} - \beta_{8} + 24 \beta_{6}) q^{10} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{2} - 7 \beta_1 + 22) q^{12} + ( - 3 \beta_{11} + 8 \beta_{9} - 3 \beta_{8} + 19 \beta_{6}) q^{13} + ( - 4 \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 2) q^{14} + (2 \beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{2} - 14) q^{15} + (4 \beta_{4} + 13 \beta_{2} - 13 \beta_1 + 11) q^{16} + ( - 3 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 10 \beta_{7}) q^{17} + (6 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} - 3 \beta_{8} - 15 \beta_{7} - 36 \beta_{6}) q^{18} + (\beta_{11} + 16 \beta_{9} + \beta_{8} + 11 \beta_{6}) q^{19} + (2 \beta_{5} + \beta_{4} - 24 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 1) q^{20} + ( - 5 \beta_{11} + 12 \beta_{9} + 6 \beta_{8} + 12 \beta_{7} - 54 \beta_{6}) q^{21} + ( - 11 \beta_{2} - 11 \beta_1) q^{23} + (6 \beta_{11} - 6 \beta_{10} + 15 \beta_{9} + 6 \beta_{8} - 33 \beta_{7} + 9 \beta_{6}) q^{24} + ( - 14 \beta_{4} + 3 \beta_{2} - 3 \beta_1 - 15) q^{25} + (6 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} + 22 \beta_{2} + 22 \beta_1 + 3) q^{26} + ( - 6 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} + 42) q^{27} + (23 \beta_{11} + 2 \beta_{9} + 23 \beta_{8} - 49 \beta_{6}) q^{28} + ( - 4 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 18 \beta_{7}) q^{29} + ( - 24 \beta_{11} - 7 \beta_{10} + \beta_{9} - 17 \beta_{8} + 8 \beta_{7} + 6 \beta_{6}) q^{30} + (16 \beta_{4} + 11 \beta_{2} - 11 \beta_1 + 28) q^{31} + ( - 13 \beta_{11} - 10 \beta_{10} - 5 \beta_{9} + 13 \beta_{8} - 9 \beta_{7}) q^{32} + ( - 27 \beta_{4} - 40 \beta_{2} + 40 \beta_1 - 141) q^{34} + ( - 19 \beta_{11} - 16 \beta_{10} - 8 \beta_{9} + 19 \beta_{8} + 28 \beta_{7}) q^{35} + ( - 3 \beta_{5} + 9 \beta_{4} + 36 \beta_{3} + 27 \beta_{2} - 15 \beta_1 + 180) q^{36} + (2 \beta_{4} - 7 \beta_{2} + 7 \beta_1 - 220) q^{37} + ( - 2 \beta_{5} - \beta_{4} + 23 \beta_{3} + 30 \beta_{2} + 30 \beta_1 - 1) q^{38} + ( - 19 \beta_{11} + 8 \beta_{10} - 26 \beta_{9} + 7 \beta_{8} - 34 \beta_{7} + 105 \beta_{6}) q^{39} + ( - 30 \beta_{11} - 12 \beta_{9} - 30 \beta_{8} + 144 \beta_{6}) q^{40} + (30 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} - 30 \beta_{8} - 22 \beta_{7}) q^{41} + ( - \beta_{5} - 29 \beta_{4} + 79 \beta_{3} + 22 \beta_{2} + 24 \beta_1 - 143) q^{42} + (6 \beta_{11} - 4 \beta_{9} + 6 \beta_{8} + 73 \beta_{6}) q^{43} + (21 \beta_{4} - 33 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 210) q^{45} + (11 \beta_{11} - 33 \beta_{9} + 11 \beta_{8} - 66 \beta_{6}) q^{46} + ( - 44 \beta_{3} - 12 \beta_{2} - 12 \beta_1) q^{47} + ( - 4 \beta_{5} + 31 \beta_{4} + 31 \beta_{3} + 55 \beta_{2} - 7 \beta_1 + 358) q^{48} + ( - 48 \beta_{4} + 16 \beta_{2} - 16 \beta_1 - 187) q^{49} + ( - 31 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 31 \beta_{8} + 51 \beta_{7}) q^{50} + ( - 12 \beta_{11} + 12 \beta_{10} - 36 \beta_{9} - 15 \beta_{8} + 60 \beta_{7} - 45 \beta_{6}) q^{51} + ( - 61 \beta_{11} - 40 \beta_{9} - 61 \beta_{8} + 115 \beta_{6}) q^{52} + ( - 4 \beta_{5} - 2 \beta_{4} - 54 \beta_{3} + 42 \beta_{2} + 42 \beta_1 - 2) q^{53} + (45 \beta_{11} - 6 \beta_{10} + 24 \beta_{9} + 51 \beta_{8} - 51 \beta_{7} + 63 \beta_{6}) q^{54} + ( - 14 \beta_{5} - 7 \beta_{4} + 107 \beta_{3} - 10 \beta_{2} - 10 \beta_1 - 7) q^{56} + ( - 11 \beta_{11} + 16 \beta_{10} - 10 \beta_{9} + 35 \beta_{8} - 26 \beta_{7} + 21 \beta_{6}) q^{57} + ( - 38 \beta_{4} - 46 \beta_{2} + 46 \beta_1 - 260) q^{58} + ( - 12 \beta_{5} - 6 \beta_{4} - 70 \beta_{3} + 27 \beta_{2} + 27 \beta_1 - 6) q^{59} + (25 \beta_{5} + 8 \beta_{4} - 46 \beta_{3} + 56 \beta_{2} - 12 \beta_1 + 50) q^{60} + (43 \beta_{11} + 76 \beta_{9} + 43 \beta_{8} - 23 \beta_{6}) q^{61} + (21 \beta_{11} - 22 \beta_{10} - 11 \beta_{9} - 21 \beta_{8} - 98 \beta_{7}) q^{62} + (54 \beta_{11} + 24 \beta_{10} - 18 \beta_{9} - 3 \beta_{8} - 42 \beta_{7} - 90 \beta_{6}) q^{63} + ( - 42 \beta_{4} + 21 \beta_{2} - 21 \beta_1 + 193) q^{64} + (18 \beta_{11} + 40 \beta_{10} + 20 \beta_{9} - 18 \beta_{8} - 32 \beta_{7}) q^{65} + (44 \beta_{4} + 33 \beta_{2} - 33 \beta_1 - 460) q^{67} + (10 \beta_{11} + 48 \beta_{10} + 24 \beta_{9} - 10 \beta_{8} + 222 \beta_{7}) q^{68} + (33 \beta_{4} + 33 \beta_{3} + 33 \beta_{2} - 264) q^{69} + ( - 53 \beta_{4} + 154 \beta_{2} - 154 \beta_1 - 245) q^{70} + ( - 12 \beta_{5} - 6 \beta_{4} + 82 \beta_{3} + 83 \beta_{2} + 83 \beta_1 - 6) q^{71} + (9 \beta_{11} - 18 \beta_{10} + 9 \beta_{9} + 81 \beta_{8} - 135 \beta_{7} - 216 \beta_{6}) q^{72} + (45 \beta_{11} - 8 \beta_{9} + 45 \beta_{8} + \beta_{6}) q^{73} + (11 \beta_{11} + 14 \beta_{10} + 7 \beta_{9} - 11 \beta_{8} + 228 \beta_{7}) q^{74} + (14 \beta_{5} + 37 \beta_{4} + 37 \beta_{3} - 47 \beta_{2} + \beta_1 + 202) q^{75} + (3 \beta_{11} + 16 \beta_{9} + 3 \beta_{8} - 253 \beta_{6}) q^{76} + (12 \beta_{5} - 15 \beta_{4} - 177 \beta_{3} - 60 \beta_{2} - 36 \beta_1 + 651) q^{78} + (4 \beta_{11} - 68 \beta_{9} + 4 \beta_{8} - 151 \beta_{6}) q^{79} + (44 \beta_{5} + 22 \beta_{4} - 36 \beta_{3} + 68 \beta_{2} + 68 \beta_1 + 22) q^{80} + ( - 63 \beta_{4} + 99 \beta_{3} + 18 \beta_{2} - 9 \beta_1 + 99) q^{81} + (120 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 90) q^{82} + ( - 31 \beta_{11} - 32 \beta_{10} - 16 \beta_{9} + 31 \beta_{8} + 176 \beta_{7}) q^{83} + (49 \beta_{11} + 2 \beta_{10} + 136 \beta_{9} + 73 \beta_{8} + 134 \beta_{7} - 615 \beta_{6}) q^{84} + (53 \beta_{11} + 24 \beta_{9} + 53 \beta_{8} - 318 \beta_{6}) q^{85} + ( - 12 \beta_{5} - 6 \beta_{4} - 69 \beta_{3} - 20 \beta_{2} - 20 \beta_1 - 6) q^{86} + ( - 12 \beta_{11} + 20 \beta_{10} - 26 \beta_{9} - 26 \beta_{8} + 62 \beta_{7} - 48 \beta_{6}) q^{87} + (64 \beta_{5} + 32 \beta_{4} + 14 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 32) q^{89} + (15 \beta_{11} + 9 \beta_{10} - 66 \beta_{9} - 78 \beta_{8} - 264 \beta_{7} + 486 \beta_{6}) q^{90} + (160 \beta_{4} + 18 \beta_{2} - 18 \beta_1 + 810) q^{91} + ( - 22 \beta_{5} - 11 \beta_{4} + 22 \beta_{3} - 11) q^{92} + ( - 16 \beta_{5} + \beta_{4} + \beta_{3} + 97 \beta_{2} - 12 \beta_1 + 202) q^{93} + ( - 32 \beta_{11} - 124 \beta_{9} - 32 \beta_{8} + 588 \beta_{6}) q^{94} + (14 \beta_{11} + 24 \beta_{10} + 12 \beta_{9} - 14 \beta_{8} - 144 \beta_{7}) q^{95} + (30 \beta_{11} - 14 \beta_{10} - \beta_{9} - 40 \beta_{8} - 257 \beta_{7} - 393 \beta_{6}) q^{96} + (30 \beta_{4} + 97 \beta_{2} - 97 \beta_1 - 830) q^{97} + ( - 112 \beta_{11} - 32 \beta_{10} - 16 \beta_{9} + 112 \beta_{8} + 299 \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 92 q^{4} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 92 q^{4} + 24 q^{9} + 264 q^{12} - 180 q^{15} + 116 q^{16} - 124 q^{25} + 540 q^{27} + 272 q^{31} - 1584 q^{34} + 2136 q^{36} - 2648 q^{37} - 1596 q^{42} + 2436 q^{45} + 4188 q^{48} - 2052 q^{49} - 2968 q^{58} + 468 q^{60} + 2484 q^{64} - 5696 q^{67} - 3300 q^{69} - 2728 q^{70} + 2220 q^{75} + 7824 q^{78} + 1440 q^{81} + 600 q^{82} + 9080 q^{91} + 2484 q^{93} - 10080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 82x^{10} + 2949x^{8} - 55732x^{6} + 613132x^{4} - 3655440x^{2} + 13220496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -479\nu^{10} + 68478\nu^{8} - 3088095\nu^{6} + 64823564\nu^{4} - 632945160\nu^{2} + 3221616888 ) / 225491040 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 235\nu^{10} - 14998\nu^{8} + 266667\nu^{6} + 1676516\nu^{4} - 74924504\nu^{2} + 382824648 ) / 75163680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{10} - 1159\nu^{8} + 35135\nu^{6} - 506787\nu^{4} + 4966170\nu^{2} - 20493684 ) / 3131820 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -227\nu^{10} + 17523\nu^{8} - 559221\nu^{6} + 8298209\nu^{4} - 56113068\nu^{2} + 116709930 ) / 14093190 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5053\nu^{10} - 351660\nu^{8} + 10236621\nu^{6} - 146337430\nu^{4} + 1233982188\nu^{2} - 4703607792 ) / 112745520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2453\nu^{11} - 180542\nu^{9} + 5829189\nu^{7} - 94126976\nu^{5} + 889786952\nu^{3} - 2947796280\nu ) / 3795765840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2453 \nu^{11} + 180542 \nu^{9} - 5829189 \nu^{7} + 94126976 \nu^{5} - 889786952 \nu^{3} + 6743562120 \nu ) / 3795765840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13909 \nu^{11} + 1245073 \nu^{9} - 41632125 \nu^{7} + 546525619 \nu^{5} - 1000385830 \nu^{3} - 19889581932 \nu ) / 11387297520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14119 \nu^{11} - 1487624 \nu^{9} + 61851879 \nu^{7} - 1263134902 \nu^{5} + 11991954332 \nu^{3} - 46434918672 \nu ) / 11387297520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32573 \nu^{11} - 2835818 \nu^{9} + 86207853 \nu^{7} - 1228173524 \nu^{5} + 11958663944 \nu^{3} - 138499336584 \nu ) / 22774595040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 48962 \nu^{11} - 3263343 \nu^{9} + 87466146 \nu^{7} - 978054059 \nu^{5} + 4338617598 \nu^{3} + 14162248980 \nu ) / 11387297520 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{7} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{11} + 2\beta_{10} - 5\beta_{9} - 2\beta_{8} + 15\beta_{7} + 43\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{5} + 12\beta_{4} + 76\beta_{3} + 33\beta_{2} - 17\beta _1 + 139 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -164\beta_{11} + 34\beta_{10} - 223\beta_{9} - 166\beta_{8} + 89\beta_{7} + 1219\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -364\beta_{5} - 268\beta_{4} + 2118\beta_{3} + 215\beta_{2} + 81\beta _1 - 1359 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4926\beta_{11} - 594\beta_{10} - 6261\beta_{9} - 5700\beta_{8} - 5537\beta_{7} + 29791\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -10032\beta_{5} - 18866\beta_{4} + 46808\beta_{3} - 20207\beta_{2} + 21263\beta _1 - 156859 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -108340\beta_{11} - 61534\beta_{10} - 138395\beta_{9} - 123326\beta_{8} - 328479\beta_{7} + 560959\beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -170132\beta_{5} - 666720\beta_{4} + 679402\beta_{3} - 1130361\beta_{2} + 1067569\beta _1 - 6625907 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1110074 \beta_{11} - 2538194 \beta_{10} - 2050933 \beta_{9} - 1175968 \beta_{8} - 11842537 \beta_{7} + 4420219 \beta_{6} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1
5.18970 1.41421i
5.18970 + 1.41421i
3.79577 + 1.41421i
3.79577 1.41421i
2.37889 1.41421i
2.37889 + 1.41421i
−2.37889 + 1.41421i
−2.37889 1.41421i
−3.79577 1.41421i
−3.79577 + 1.41421i
−5.18970 + 1.41421i
−5.18970 1.41421i
−5.18970 4.90350 1.71922i 18.9330 11.2403i −25.4477 + 8.92224i 20.6673i −56.7391 21.0886 16.8604i 58.3336i
362.2 −5.18970 4.90350 + 1.71922i 18.9330 11.2403i −25.4477 8.92224i 20.6673i −56.7391 21.0886 + 16.8604i 58.3336i
362.3 −3.79577 −4.37965 2.79619i 6.40789 16.3285i 16.6241 + 10.6137i 31.8461i 6.04330 11.3626 + 24.4927i 61.9791i
362.4 −3.79577 −4.37965 + 2.79619i 6.40789 16.3285i 16.6241 10.6137i 31.8461i 6.04330 11.3626 24.4927i 61.9791i
362.5 −2.37889 −0.523851 5.16968i −2.34090 3.61085i 1.24618 + 12.2981i 10.0344i 24.5998 −26.4512 + 5.41628i 8.58981i
362.6 −2.37889 −0.523851 + 5.16968i −2.34090 3.61085i 1.24618 12.2981i 10.0344i 24.5998 −26.4512 5.41628i 8.58981i
362.7 2.37889 −0.523851 5.16968i −2.34090 3.61085i −1.24618 12.2981i 10.0344i −24.5998 −26.4512 + 5.41628i 8.58981i
362.8 2.37889 −0.523851 + 5.16968i −2.34090 3.61085i −1.24618 + 12.2981i 10.0344i −24.5998 −26.4512 5.41628i 8.58981i
362.9 3.79577 −4.37965 2.79619i 6.40789 16.3285i −16.6241 10.6137i 31.8461i −6.04330 11.3626 + 24.4927i 61.9791i
362.10 3.79577 −4.37965 + 2.79619i 6.40789 16.3285i −16.6241 + 10.6137i 31.8461i −6.04330 11.3626 24.4927i 61.9791i
362.11 5.18970 4.90350 1.71922i 18.9330 11.2403i 25.4477 8.92224i 20.6673i 56.7391 21.0886 16.8604i 58.3336i
362.12 5.18970 4.90350 + 1.71922i 18.9330 11.2403i 25.4477 + 8.92224i 20.6673i 56.7391 21.0886 + 16.8604i 58.3336i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.d.c 12
3.b odd 2 1 inner 363.4.d.c 12
11.b odd 2 1 inner 363.4.d.c 12
33.d even 2 1 inner 363.4.d.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.d.c 12 1.a even 1 1 trivial
363.4.d.c 12 3.b odd 2 1 inner
363.4.d.c 12 11.b odd 2 1 inner
363.4.d.c 12 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 47T_{2}^{4} + 622T_{2}^{2} - 2196 \) acting on \(S_{4}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 47 T^{4} + 622 T^{2} - 2196)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 6 T^{4} - 90 T^{3} - 162 T^{2} + \cdots + 19683)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 406 T^{4} + 38809 T^{2} + \cdots + 439200)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 1542 T^{4} + 578316 T^{2} + \cdots + 43617800)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + 11186 T^{4} + \cdots + 48896899200)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 15888 T^{4} + \cdots - 838705104)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 16946 T^{4} + \cdots + 25545232512)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 18150 T^{4} + \cdots + 70026263208)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 25796 T^{4} + \cdots - 86606726400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 68 T^{2} - 19343 T + 942534)^{4} \) Copy content Toggle raw display
$37$ \( (T^{3} + 662 T^{2} + 140953 T + 9560340)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 251508 T^{4} + \cdots - 18846444441600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 48806 T^{4} + \cdots + 214604209800)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 171904 T^{4} + \cdots + 4188098764800)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 674848 T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 837022 T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 783426 T^{4} + \cdots + 470853594906752)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 1424 T^{2} + 507937 T + 34020990)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 1363294 T^{4} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 739458 T^{4} + \cdots + 61220210739200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 447830 T^{4} + \cdots + 535099888316232)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 2210348 T^{4} + \cdots - 76\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 5828838 T^{4} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 2520 T^{2} + 1358037 T - 274893310)^{4} \) Copy content Toggle raw display
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