Properties

Label 294.4.a.j.1.1
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [294,4,Mod(1,294)] 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("294.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,-6,8,12,12,0,-16,18,-24,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.3465615417\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -3.89949 q^{5} +6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +7.79899 q^{10} -61.3970 q^{11} -12.0000 q^{12} +53.6985 q^{13} +11.6985 q^{15} +16.0000 q^{16} +32.1005 q^{17} -18.0000 q^{18} -55.7990 q^{19} -15.5980 q^{20} +122.794 q^{22} -94.6030 q^{23} +24.0000 q^{24} -109.794 q^{25} -107.397 q^{26} -27.0000 q^{27} +138.191 q^{29} -23.3970 q^{30} +132.603 q^{31} -32.0000 q^{32} +184.191 q^{33} -64.2010 q^{34} +36.0000 q^{36} +149.206 q^{37} +111.598 q^{38} -161.095 q^{39} +31.1960 q^{40} +427.497 q^{41} +437.588 q^{43} -245.588 q^{44} -35.0955 q^{45} +189.206 q^{46} +57.0051 q^{47} -48.0000 q^{48} +219.588 q^{50} -96.3015 q^{51} +214.794 q^{52} -263.588 q^{53} +54.0000 q^{54} +239.417 q^{55} +167.397 q^{57} -276.382 q^{58} +451.799 q^{59} +46.7939 q^{60} +579.307 q^{61} -265.206 q^{62} +64.0000 q^{64} -209.397 q^{65} -368.382 q^{66} +309.588 q^{67} +128.402 q^{68} +283.809 q^{69} -1058.98 q^{71} -72.0000 q^{72} +1193.66 q^{73} -298.412 q^{74} +329.382 q^{75} -223.196 q^{76} +322.191 q^{78} +1319.56 q^{79} -62.3919 q^{80} +81.0000 q^{81} -854.995 q^{82} +1190.33 q^{83} -125.176 q^{85} -875.176 q^{86} -414.573 q^{87} +491.176 q^{88} -233.085 q^{89} +70.1909 q^{90} -378.412 q^{92} -397.809 q^{93} -114.010 q^{94} +217.588 q^{95} +96.0000 q^{96} -1609.44 q^{97} -552.573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 12 q^{5} + 12 q^{6} - 16 q^{8} + 18 q^{9} - 24 q^{10} - 4 q^{11} - 24 q^{12} + 48 q^{13} - 36 q^{15} + 32 q^{16} + 84 q^{17} - 36 q^{18} - 72 q^{19} + 48 q^{20} + 8 q^{22}+ \cdots - 36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −3.89949 −0.348781 −0.174391 − 0.984677i \(-0.555796\pi\)
−0.174391 + 0.984677i \(0.555796\pi\)
\(6\) 6.00000 0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 7.79899 0.246626
\(11\) −61.3970 −1.68290 −0.841449 − 0.540336i \(-0.818297\pi\)
−0.841449 + 0.540336i \(0.818297\pi\)
\(12\) −12.0000 −0.288675
\(13\) 53.6985 1.14564 0.572818 − 0.819682i \(-0.305850\pi\)
0.572818 + 0.819682i \(0.305850\pi\)
\(14\) 0 0
\(15\) 11.6985 0.201369
\(16\) 16.0000 0.250000
\(17\) 32.1005 0.457972 0.228986 − 0.973430i \(-0.426459\pi\)
0.228986 + 0.973430i \(0.426459\pi\)
\(18\) −18.0000 −0.235702
\(19\) −55.7990 −0.673746 −0.336873 − 0.941550i \(-0.609369\pi\)
−0.336873 + 0.941550i \(0.609369\pi\)
\(20\) −15.5980 −0.174391
\(21\) 0 0
\(22\) 122.794 1.18999
\(23\) −94.6030 −0.857656 −0.428828 − 0.903386i \(-0.641073\pi\)
−0.428828 + 0.903386i \(0.641073\pi\)
\(24\) 24.0000 0.204124
\(25\) −109.794 −0.878352
\(26\) −107.397 −0.810088
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 138.191 0.884876 0.442438 − 0.896799i \(-0.354114\pi\)
0.442438 + 0.896799i \(0.354114\pi\)
\(30\) −23.3970 −0.142389
\(31\) 132.603 0.768265 0.384132 − 0.923278i \(-0.374501\pi\)
0.384132 + 0.923278i \(0.374501\pi\)
\(32\) −32.0000 −0.176777
\(33\) 184.191 0.971622
\(34\) −64.2010 −0.323835
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 149.206 0.662955 0.331477 − 0.943463i \(-0.392453\pi\)
0.331477 + 0.943463i \(0.392453\pi\)
\(38\) 111.598 0.476410
\(39\) −161.095 −0.661434
\(40\) 31.1960 0.123313
\(41\) 427.497 1.62839 0.814194 − 0.580593i \(-0.197179\pi\)
0.814194 + 0.580593i \(0.197179\pi\)
\(42\) 0 0
\(43\) 437.588 1.55190 0.775948 − 0.630797i \(-0.217272\pi\)
0.775948 + 0.630797i \(0.217272\pi\)
\(44\) −245.588 −0.841449
\(45\) −35.0955 −0.116260
\(46\) 189.206 0.606455
\(47\) 57.0051 0.176916 0.0884579 − 0.996080i \(-0.471806\pi\)
0.0884579 + 0.996080i \(0.471806\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) 219.588 0.621088
\(51\) −96.3015 −0.264410
\(52\) 214.794 0.572818
\(53\) −263.588 −0.683143 −0.341572 − 0.939856i \(-0.610959\pi\)
−0.341572 + 0.939856i \(0.610959\pi\)
\(54\) 54.0000 0.136083
\(55\) 239.417 0.586964
\(56\) 0 0
\(57\) 167.397 0.388987
\(58\) −276.382 −0.625702
\(59\) 451.799 0.996936 0.498468 − 0.866908i \(-0.333896\pi\)
0.498468 + 0.866908i \(0.333896\pi\)
\(60\) 46.7939 0.100685
\(61\) 579.307 1.21594 0.607972 − 0.793958i \(-0.291983\pi\)
0.607972 + 0.793958i \(0.291983\pi\)
\(62\) −265.206 −0.543245
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −209.397 −0.399577
\(66\) −368.382 −0.687040
\(67\) 309.588 0.564510 0.282255 − 0.959339i \(-0.408918\pi\)
0.282255 + 0.959339i \(0.408918\pi\)
\(68\) 128.402 0.228986
\(69\) 283.809 0.495168
\(70\) 0 0
\(71\) −1058.98 −1.77012 −0.885059 − 0.465479i \(-0.845882\pi\)
−0.885059 + 0.465479i \(0.845882\pi\)
\(72\) −72.0000 −0.117851
\(73\) 1193.66 1.91380 0.956898 − 0.290424i \(-0.0937964\pi\)
0.956898 + 0.290424i \(0.0937964\pi\)
\(74\) −298.412 −0.468780
\(75\) 329.382 0.507116
\(76\) −223.196 −0.336873
\(77\) 0 0
\(78\) 322.191 0.467704
\(79\) 1319.56 1.87926 0.939632 − 0.342187i \(-0.111168\pi\)
0.939632 + 0.342187i \(0.111168\pi\)
\(80\) −62.3919 −0.0871954
\(81\) 81.0000 0.111111
\(82\) −854.995 −1.15144
\(83\) 1190.33 1.57417 0.787083 − 0.616847i \(-0.211590\pi\)
0.787083 + 0.616847i \(0.211590\pi\)
\(84\) 0 0
\(85\) −125.176 −0.159732
\(86\) −875.176 −1.09736
\(87\) −414.573 −0.510883
\(88\) 491.176 0.594994
\(89\) −233.085 −0.277607 −0.138803 − 0.990320i \(-0.544326\pi\)
−0.138803 + 0.990320i \(0.544326\pi\)
\(90\) 70.1909 0.0822086
\(91\) 0 0
\(92\) −378.412 −0.428828
\(93\) −397.809 −0.443558
\(94\) −114.010 −0.125098
\(95\) 217.588 0.234990
\(96\) 96.0000 0.102062
\(97\) −1609.44 −1.68468 −0.842338 − 0.538950i \(-0.818821\pi\)
−0.842338 + 0.538950i \(0.818821\pi\)
\(98\) 0 0
\(99\) −552.573 −0.560966
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 294.4.a.j.1.1 ✓ 2
3.2 odd 2 882.4.a.bc.1.2 2
4.3 odd 2 2352.4.a.cd.1.1 2
7.2 even 3 294.4.e.o.67.2 4
7.3 odd 6 294.4.e.n.79.1 4
7.4 even 3 294.4.e.o.79.2 4
7.5 odd 6 294.4.e.n.67.1 4
7.6 odd 2 294.4.a.k.1.2 yes 2
21.2 odd 6 882.4.g.bd.361.1 4
21.5 even 6 882.4.g.y.361.2 4
21.11 odd 6 882.4.g.bd.667.1 4
21.17 even 6 882.4.g.y.667.2 4
21.20 even 2 882.4.a.bi.1.1 2
28.27 even 2 2352.4.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.1 ✓ 2 1.1 even 1 trivial
294.4.a.k.1.2 yes 2 7.6 odd 2
294.4.e.n.67.1 4 7.5 odd 6
294.4.e.n.79.1 4 7.3 odd 6
294.4.e.o.67.2 4 7.2 even 3
294.4.e.o.79.2 4 7.4 even 3
882.4.a.bc.1.2 2 3.2 odd 2
882.4.a.bi.1.1 2 21.20 even 2
882.4.g.y.361.2 4 21.5 even 6
882.4.g.y.667.2 4 21.17 even 6
882.4.g.bd.361.1 4 21.2 odd 6
882.4.g.bd.667.1 4 21.11 odd 6
2352.4.a.bn.1.2 2 28.27 even 2
2352.4.a.cd.1.1 2 4.3 odd 2