Properties

Label 294.4.a.j.1.2
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 1.2
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +15.8995 q^{5} +6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} -31.7990 q^{10} +57.3970 q^{11} -12.0000 q^{12} -5.69848 q^{13} -47.6985 q^{15} +16.0000 q^{16} +51.8995 q^{17} -18.0000 q^{18} -16.2010 q^{19} +63.5980 q^{20} -114.794 q^{22} -213.397 q^{23} +24.0000 q^{24} +127.794 q^{25} +11.3970 q^{26} -27.0000 q^{27} -218.191 q^{29} +95.3970 q^{30} +251.397 q^{31} -32.0000 q^{32} -172.191 q^{33} -103.799 q^{34} +36.0000 q^{36} +386.794 q^{37} +32.4020 q^{38} +17.0955 q^{39} -127.196 q^{40} +328.503 q^{41} -37.5879 q^{43} +229.588 q^{44} +143.095 q^{45} +426.794 q^{46} +254.995 q^{47} -48.0000 q^{48} -255.588 q^{50} -155.698 q^{51} -22.7939 q^{52} +211.588 q^{53} +54.0000 q^{54} +912.583 q^{55} +48.6030 q^{57} +436.382 q^{58} +412.201 q^{59} -190.794 q^{60} +836.693 q^{61} -502.794 q^{62} +64.0000 q^{64} -90.6030 q^{65} +344.382 q^{66} -165.588 q^{67} +207.598 q^{68} +640.191 q^{69} -465.015 q^{71} -72.0000 q^{72} -449.658 q^{73} -773.588 q^{74} -383.382 q^{75} -64.8040 q^{76} -34.1909 q^{78} -343.558 q^{79} +254.392 q^{80} +81.0000 q^{81} -657.005 q^{82} -1502.33 q^{83} +825.176 q^{85} +75.1758 q^{86} +654.573 q^{87} -459.176 q^{88} +341.085 q^{89} -286.191 q^{90} -853.588 q^{92} -754.191 q^{93} -509.990 q^{94} -257.588 q^{95} +96.0000 q^{96} +865.437 q^{97} +516.573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 12 q^{5} + 12 q^{6} - 16 q^{8} + 18 q^{9} - 24 q^{10} - 4 q^{11} - 24 q^{12} + 48 q^{13} - 36 q^{15} + 32 q^{16} + 84 q^{17} - 36 q^{18} - 72 q^{19} + 48 q^{20} + 8 q^{22}+ \cdots - 36 q^{99}+O(q^{100}) \) 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
\(5\) 15.8995 1.42209 0.711047 − 0.703144i \(-0.248221\pi\)
0.711047 + 0.703144i \(0.248221\pi\)
\(6\) 6.00000 0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −31.7990 −1.00557
\(11\) 57.3970 1.57326 0.786629 − 0.617426i \(-0.211824\pi\)
0.786629 + 0.617426i \(0.211824\pi\)
\(12\) −12.0000 −0.288675
\(13\) −5.69848 −0.121575 −0.0607875 − 0.998151i \(-0.519361\pi\)
−0.0607875 + 0.998151i \(0.519361\pi\)
\(14\) 0 0
\(15\) −47.6985 −0.821046
\(16\) 16.0000 0.250000
\(17\) 51.8995 0.740440 0.370220 − 0.928944i \(-0.379282\pi\)
0.370220 + 0.928944i \(0.379282\pi\)
\(18\) −18.0000 −0.235702
\(19\) −16.2010 −0.195619 −0.0978096 − 0.995205i \(-0.531184\pi\)
−0.0978096 + 0.995205i \(0.531184\pi\)
\(20\) 63.5980 0.711047
\(21\) 0 0
\(22\) −114.794 −1.11246
\(23\) −213.397 −1.93462 −0.967312 − 0.253590i \(-0.918389\pi\)
−0.967312 + 0.253590i \(0.918389\pi\)
\(24\) 24.0000 0.204124
\(25\) 127.794 1.02235
\(26\) 11.3970 0.0859665
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −218.191 −1.39714 −0.698570 − 0.715542i \(-0.746180\pi\)
−0.698570 + 0.715542i \(0.746180\pi\)
\(30\) 95.3970 0.580567
\(31\) 251.397 1.45652 0.728262 − 0.685299i \(-0.240329\pi\)
0.728262 + 0.685299i \(0.240329\pi\)
\(32\) −32.0000 −0.176777
\(33\) −172.191 −0.908321
\(34\) −103.799 −0.523570
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 386.794 1.71861 0.859304 − 0.511464i \(-0.170897\pi\)
0.859304 + 0.511464i \(0.170897\pi\)
\(38\) 32.4020 0.138324
\(39\) 17.0955 0.0701914
\(40\) −127.196 −0.502786
\(41\) 328.503 1.25130 0.625652 − 0.780102i \(-0.284833\pi\)
0.625652 + 0.780102i \(0.284833\pi\)
\(42\) 0 0
\(43\) −37.5879 −0.133305 −0.0666523 − 0.997776i \(-0.521232\pi\)
−0.0666523 + 0.997776i \(0.521232\pi\)
\(44\) 229.588 0.786629
\(45\) 143.095 0.474031
\(46\) 426.794 1.36799
\(47\) 254.995 0.791379 0.395690 − 0.918384i \(-0.370506\pi\)
0.395690 + 0.918384i \(0.370506\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) −255.588 −0.722912
\(51\) −155.698 −0.427493
\(52\) −22.7939 −0.0607875
\(53\) 211.588 0.548374 0.274187 − 0.961676i \(-0.411591\pi\)
0.274187 + 0.961676i \(0.411591\pi\)
\(54\) 54.0000 0.136083
\(55\) 912.583 2.23732
\(56\) 0 0
\(57\) 48.6030 0.112941
\(58\) 436.382 0.987927
\(59\) 412.201 0.909559 0.454780 − 0.890604i \(-0.349718\pi\)
0.454780 + 0.890604i \(0.349718\pi\)
\(60\) −190.794 −0.410523
\(61\) 836.693 1.75619 0.878095 − 0.478486i \(-0.158814\pi\)
0.878095 + 0.478486i \(0.158814\pi\)
\(62\) −502.794 −1.02992
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −90.6030 −0.172891
\(66\) 344.382 0.642280
\(67\) −165.588 −0.301937 −0.150969 − 0.988539i \(-0.548239\pi\)
−0.150969 + 0.988539i \(0.548239\pi\)
\(68\) 207.598 0.370220
\(69\) 640.191 1.11696
\(70\) 0 0
\(71\) −465.015 −0.777284 −0.388642 − 0.921389i \(-0.627056\pi\)
−0.388642 + 0.921389i \(0.627056\pi\)
\(72\) −72.0000 −0.117851
\(73\) −449.658 −0.720938 −0.360469 − 0.932771i \(-0.617383\pi\)
−0.360469 + 0.932771i \(0.617383\pi\)
\(74\) −773.588 −1.21524
\(75\) −383.382 −0.590255
\(76\) −64.8040 −0.0978096
\(77\) 0 0
\(78\) −34.1909 −0.0496328
\(79\) −343.558 −0.489282 −0.244641 − 0.969614i \(-0.578670\pi\)
−0.244641 + 0.969614i \(0.578670\pi\)
\(80\) 254.392 0.355524
\(81\) 81.0000 0.111111
\(82\) −657.005 −0.884806
\(83\) −1502.33 −1.98677 −0.993387 − 0.114812i \(-0.963373\pi\)
−0.993387 + 0.114812i \(0.963373\pi\)
\(84\) 0 0
\(85\) 825.176 1.05298
\(86\) 75.1758 0.0942606
\(87\) 654.573 0.806639
\(88\) −459.176 −0.556231
\(89\) 341.085 0.406236 0.203118 − 0.979154i \(-0.434893\pi\)
0.203118 + 0.979154i \(0.434893\pi\)
\(90\) −286.191 −0.335191
\(91\) 0 0
\(92\) −853.588 −0.967312
\(93\) −754.191 −0.840924
\(94\) −509.990 −0.559590
\(95\) −257.588 −0.278189
\(96\) 96.0000 0.102062
\(97\) 865.437 0.905895 0.452947 − 0.891537i \(-0.350373\pi\)
0.452947 + 0.891537i \(0.350373\pi\)
\(98\) 0 0
\(99\) 516.573 0.524419
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.1.2 ✓ 2
3.2 odd 2 882.4.a.bc.1.1 2
4.3 odd 2 2352.4.a.cd.1.2 2
7.2 even 3 294.4.e.o.67.1 4
7.3 odd 6 294.4.e.n.79.2 4
7.4 even 3 294.4.e.o.79.1 4
7.5 odd 6 294.4.e.n.67.2 4
7.6 odd 2 294.4.a.k.1.1 yes 2
21.2 odd 6 882.4.g.bd.361.2 4
21.5 even 6 882.4.g.y.361.1 4
21.11 odd 6 882.4.g.bd.667.2 4
21.17 even 6 882.4.g.y.667.1 4
21.20 even 2 882.4.a.bi.1.2 2
28.27 even 2 2352.4.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.2 ✓ 2 1.1 even 1 trivial
294.4.a.k.1.1 yes 2 7.6 odd 2
294.4.e.n.67.2 4 7.5 odd 6
294.4.e.n.79.2 4 7.3 odd 6
294.4.e.o.67.1 4 7.2 even 3
294.4.e.o.79.1 4 7.4 even 3
882.4.a.bc.1.1 2 3.2 odd 2
882.4.a.bi.1.2 2 21.20 even 2
882.4.g.y.361.1 4 21.5 even 6
882.4.g.y.667.1 4 21.17 even 6
882.4.g.bd.361.2 4 21.2 odd 6
882.4.g.bd.667.2 4 21.11 odd 6
2352.4.a.bn.1.1 2 28.27 even 2
2352.4.a.cd.1.2 2 4.3 odd 2