Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [363,4,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,6,1,16,-3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.37228 q^{2} +3.00000 q^{3} -2.37228 q^{4} +19.4891 q^{5} +7.11684 q^{6} -6.74456 q^{7} -24.6060 q^{8} +9.00000 q^{9} +46.2337 q^{10} -7.11684 q^{12} +60.9783 q^{13} -16.0000 q^{14} +58.4674 q^{15} -39.3940 q^{16} +99.1684 q^{17} +21.3505 q^{18} -24.7011 q^{19} -46.2337 q^{20} -20.2337 q^{21} +112.000 q^{23} -73.8179 q^{24} +254.826 q^{25} +144.658 q^{26} +27.0000 q^{27} +16.0000 q^{28} +21.1249 q^{29} +138.701 q^{30} -318.717 q^{31} +103.394 q^{32} +235.255 q^{34} -131.446 q^{35} -21.3505 q^{36} -150.380 q^{37} -58.5979 q^{38} +182.935 q^{39} -479.549 q^{40} +252.745 q^{41} -48.0000 q^{42} -214.016 q^{43} +175.402 q^{45} +265.696 q^{46} +105.870 q^{47} -118.182 q^{48} -297.511 q^{49} +604.519 q^{50} +297.505 q^{51} -144.658 q^{52} +325.652 q^{53} +64.0516 q^{54} +165.957 q^{56} -74.1032 q^{57} +50.1143 q^{58} +196.000 q^{59} -138.701 q^{60} +402.641 q^{61} -756.087 q^{62} -60.7011 q^{63} +560.432 q^{64} +1188.41 q^{65} +27.4132 q^{67} -235.255 q^{68} +336.000 q^{69} -311.826 q^{70} -300.500 q^{71} -221.454 q^{72} -427.815 q^{73} -356.745 q^{74} +764.478 q^{75} +58.5979 q^{76} +433.973 q^{78} -97.5488 q^{79} -767.755 q^{80} +81.0000 q^{81} +599.581 q^{82} -1104.62 q^{83} +48.0000 q^{84} +1932.71 q^{85} -507.707 q^{86} +63.3748 q^{87} +463.022 q^{89} +416.103 q^{90} -411.272 q^{91} -265.696 q^{92} -956.152 q^{93} +251.152 q^{94} -481.402 q^{95} +310.182 q^{96} -1798.36 q^{97} -705.779 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9} + 58 q^{10} + 3 q^{12} + 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} + 26 q^{17} - 9 q^{18} + 54 q^{19} - 58 q^{20}+ \cdots + 375 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37228 0.838728 0.419364 − 0.907818i \(-0.362253\pi\)
0.419364 + 0.907818i \(0.362253\pi\)
\(3\) 3.00000 0.577350
\(4\) −2.37228 −0.296535
\(5\) 19.4891 1.74316 0.871580 − 0.490253i \(-0.163096\pi\)
0.871580 + 0.490253i \(0.163096\pi\)
\(6\) 7.11684 0.484240
\(7\) −6.74456 −0.364172 −0.182086 − 0.983283i \(-0.558285\pi\)
−0.182086 + 0.983283i \(0.558285\pi\)
\(8\) −24.6060 −1.08744
\(9\) 9.00000 0.333333
\(10\) 46.2337 1.46204
\(11\) 0 0
\(12\) −7.11684 −0.171205
\(13\) 60.9783 1.30095 0.650474 − 0.759529i \(-0.274570\pi\)
0.650474 + 0.759529i \(0.274570\pi\)
\(14\) −16.0000 −0.305441
\(15\) 58.4674 1.00641
\(16\) −39.3940 −0.615532
\(17\) 99.1684 1.41482 0.707408 − 0.706805i \(-0.249864\pi\)
0.707408 + 0.706805i \(0.249864\pi\)
\(18\) 21.3505 0.279576
\(19\) −24.7011 −0.298253 −0.149127 − 0.988818i \(-0.547646\pi\)
−0.149127 + 0.988818i \(0.547646\pi\)
\(20\) −46.2337 −0.516908
\(21\) −20.2337 −0.210255
\(22\) 0 0
\(23\) 112.000 1.01537 0.507687 − 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) −73.8179 −0.627834
\(25\) 254.826 2.03861
\(26\) 144.658 1.09114
\(27\) 27.0000 0.192450
\(28\) 16.0000 0.107990
\(29\) 21.1249 0.135269 0.0676345 − 0.997710i \(-0.478455\pi\)
0.0676345 + 0.997710i \(0.478455\pi\)
\(30\) 138.701 0.844108
\(31\) −318.717 −1.84656 −0.923279 − 0.384130i \(-0.874502\pi\)
−0.923279 + 0.384130i \(0.874502\pi\)
\(32\) 103.394 0.571177
\(33\) 0 0
\(34\) 235.255 1.18665
\(35\) −131.446 −0.634810
\(36\) −21.3505 −0.0988451
\(37\) −150.380 −0.668172 −0.334086 − 0.942543i \(-0.608428\pi\)
−0.334086 + 0.942543i \(0.608428\pi\)
\(38\) −58.5979 −0.250153
\(39\) 182.935 0.751103
\(40\) −479.549 −1.89558
\(41\) 252.745 0.962733 0.481367 − 0.876519i \(-0.340141\pi\)
0.481367 + 0.876519i \(0.340141\pi\)
\(42\) −48.0000 −0.176347
\(43\) −214.016 −0.759004 −0.379502 − 0.925191i \(-0.623905\pi\)
−0.379502 + 0.925191i \(0.623905\pi\)
\(44\) 0 0
\(45\) 175.402 0.581053
\(46\) 265.696 0.851623
\(47\) 105.870 0.328567 0.164284 − 0.986413i \(-0.447469\pi\)
0.164284 + 0.986413i \(0.447469\pi\)
\(48\) −118.182 −0.355377
\(49\) −297.511 −0.867379
\(50\) 604.519 1.70984
\(51\) 297.505 0.816845
\(52\) −144.658 −0.385777
\(53\) 325.652 0.843995 0.421998 − 0.906597i \(-0.361329\pi\)
0.421998 + 0.906597i \(0.361329\pi\)
\(54\) 64.0516 0.161413
\(55\) 0 0
\(56\) 165.957 0.396016
\(57\) −74.1032 −0.172197
\(58\) 50.1143 0.113454
\(59\) 196.000 0.432492 0.216246 − 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) −138.701 −0.298437
\(61\) 402.641 0.845130 0.422565 − 0.906333i \(-0.361130\pi\)
0.422565 + 0.906333i \(0.361130\pi\)
\(62\) −756.087 −1.54876
\(63\) −60.7011 −0.121391
\(64\) 560.432 1.09459
\(65\) 1188.41 2.26776
\(66\) 0 0
\(67\) 27.4132 0.0499860 0.0249930 − 0.999688i \(-0.492044\pi\)
0.0249930 + 0.999688i \(0.492044\pi\)
\(68\) −235.255 −0.419543
\(69\) 336.000 0.586227
\(70\) −311.826 −0.532433
\(71\) −300.500 −0.502292 −0.251146 − 0.967949i \(-0.580807\pi\)
−0.251146 + 0.967949i \(0.580807\pi\)
\(72\) −221.454 −0.362480
\(73\) −427.815 −0.685917 −0.342959 − 0.939351i \(-0.611429\pi\)
−0.342959 + 0.939351i \(0.611429\pi\)
\(74\) −356.745 −0.560415
\(75\) 764.478 1.17699
\(76\) 58.5979 0.0884426
\(77\) 0 0
\(78\) 433.973 0.629971
\(79\) −97.5488 −0.138925 −0.0694627 − 0.997585i \(-0.522128\pi\)
−0.0694627 + 0.997585i \(0.522128\pi\)
\(80\) −767.755 −1.07297
\(81\) 81.0000 0.111111
\(82\) 599.581 0.807472
\(83\) −1104.62 −1.46082 −0.730408 − 0.683011i \(-0.760670\pi\)
−0.730408 + 0.683011i \(0.760670\pi\)
\(84\) 48.0000 0.0623480
\(85\) 1932.71 2.46625
\(86\) −507.707 −0.636598
\(87\) 63.3748 0.0780976
\(88\) 0 0
\(89\) 463.022 0.551463 0.275732 − 0.961235i \(-0.411080\pi\)
0.275732 + 0.961235i \(0.411080\pi\)
\(90\) 416.103 0.487346
\(91\) −411.272 −0.473769
\(92\) −265.696 −0.301094
\(93\) −956.152 −1.06611
\(94\) 251.152 0.275578
\(95\) −481.402 −0.519903
\(96\) 310.182 0.329769
\(97\) −1798.36 −1.88243 −0.941214 − 0.337810i \(-0.890314\pi\)
−0.941214 + 0.337810i \(0.890314\pi\)
\(98\) −705.779 −0.727495
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 363.4.a.j.1.2 2
3.2 odd 2 1089.4.a.t.1.1 2
11.10 odd 2 33.4.a.d.1.1 ✓ 2
33.32 even 2 99.4.a.e.1.2 2
44.43 even 2 528.4.a.o.1.2 2
55.32 even 4 825.4.c.i.199.2 4
55.43 even 4 825.4.c.i.199.3 4
55.54 odd 2 825.4.a.k.1.2 2
77.76 even 2 1617.4.a.j.1.1 2
88.21 odd 2 2112.4.a.ba.1.1 2
88.43 even 2 2112.4.a.bh.1.1 2
132.131 odd 2 1584.4.a.x.1.1 2
165.164 even 2 2475.4.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.1 ✓ 2 11.10 odd 2
99.4.a.e.1.2 2 33.32 even 2
363.4.a.j.1.2 2 1.1 even 1 trivial
528.4.a.o.1.2 2 44.43 even 2
825.4.a.k.1.2 2 55.54 odd 2
825.4.c.i.199.2 4 55.32 even 4
825.4.c.i.199.3 4 55.43 even 4
1089.4.a.t.1.1 2 3.2 odd 2
1584.4.a.x.1.1 2 132.131 odd 2
1617.4.a.j.1.1 2 77.76 even 2
2112.4.a.ba.1.1 2 88.21 odd 2
2112.4.a.bh.1.1 2 88.43 even 2
2475.4.a.o.1.1 2 165.164 even 2