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Properties

Label 3381.2.a.bk.1.8
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $10$
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] = [3381,2,Mod(1,3381)] 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("3381.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3381, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,4,-10,8,-4,-4,0,12,10,-8,2] 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: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.94871\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.94871 q^{2} -1.00000 q^{3} +1.79746 q^{4} -1.40111 q^{5} -1.94871 q^{6} -0.394699 q^{8} +1.00000 q^{9} -2.73035 q^{10} +4.53162 q^{11} -1.79746 q^{12} -2.35527 q^{13} +1.40111 q^{15} -4.36406 q^{16} -1.10811 q^{17} +1.94871 q^{18} +0.638333 q^{19} -2.51843 q^{20} +8.83079 q^{22} -1.00000 q^{23} +0.394699 q^{24} -3.03689 q^{25} -4.58973 q^{26} -1.00000 q^{27} +2.63096 q^{29} +2.73035 q^{30} -5.19267 q^{31} -7.71488 q^{32} -4.53162 q^{33} -2.15939 q^{34} +1.79746 q^{36} +4.53435 q^{37} +1.24392 q^{38} +2.35527 q^{39} +0.553017 q^{40} -1.45216 q^{41} +1.49386 q^{43} +8.14538 q^{44} -1.40111 q^{45} -1.94871 q^{46} -6.43916 q^{47} +4.36406 q^{48} -5.91801 q^{50} +1.10811 q^{51} -4.23350 q^{52} -11.1364 q^{53} -1.94871 q^{54} -6.34929 q^{55} -0.638333 q^{57} +5.12697 q^{58} -6.20679 q^{59} +2.51843 q^{60} -0.821232 q^{61} -10.1190 q^{62} -6.30591 q^{64} +3.29999 q^{65} -8.83079 q^{66} -3.33164 q^{67} -1.99179 q^{68} +1.00000 q^{69} -14.4996 q^{71} -0.394699 q^{72} -1.99401 q^{73} +8.83611 q^{74} +3.03689 q^{75} +1.14738 q^{76} +4.58973 q^{78} +4.18578 q^{79} +6.11453 q^{80} +1.00000 q^{81} -2.82984 q^{82} +8.69016 q^{83} +1.55259 q^{85} +2.91110 q^{86} -2.63096 q^{87} -1.78862 q^{88} -7.56109 q^{89} -2.73035 q^{90} -1.79746 q^{92} +5.19267 q^{93} -12.5480 q^{94} -0.894375 q^{95} +7.71488 q^{96} -9.73998 q^{97} +4.53162 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} - 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9} - 8 q^{10} + 2 q^{11} - 8 q^{12} + 4 q^{15} + 4 q^{16} - 12 q^{17} + 4 q^{18} - 26 q^{19} - 24 q^{20} - 8 q^{22} - 10 q^{23}+ \cdots + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.94871 1.37794 0.688972 − 0.724788i \(-0.258062\pi\)
0.688972 + 0.724788i \(0.258062\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.79746 0.898728
\(5\) −1.40111 −0.626595 −0.313298 − 0.949655i \(-0.601434\pi\)
−0.313298 + 0.949655i \(0.601434\pi\)
\(6\) −1.94871 −0.795556
\(7\) 0 0
\(8\) −0.394699 −0.139547
\(9\) 1.00000 0.333333
\(10\) −2.73035 −0.863413
\(11\) 4.53162 1.36633 0.683167 − 0.730262i \(-0.260602\pi\)
0.683167 + 0.730262i \(0.260602\pi\)
\(12\) −1.79746 −0.518881
\(13\) −2.35527 −0.653235 −0.326617 − 0.945157i \(-0.605909\pi\)
−0.326617 + 0.945157i \(0.605909\pi\)
\(14\) 0 0
\(15\) 1.40111 0.361765
\(16\) −4.36406 −1.09102
\(17\) −1.10811 −0.268757 −0.134379 − 0.990930i \(-0.542904\pi\)
−0.134379 + 0.990930i \(0.542904\pi\)
\(18\) 1.94871 0.459314
\(19\) 0.638333 0.146444 0.0732218 − 0.997316i \(-0.476672\pi\)
0.0732218 + 0.997316i \(0.476672\pi\)
\(20\) −2.51843 −0.563139
\(21\) 0 0
\(22\) 8.83079 1.88273
\(23\) −1.00000 −0.208514
\(24\) 0.394699 0.0805676
\(25\) −3.03689 −0.607378
\(26\) −4.58973 −0.900121
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.63096 0.488557 0.244278 − 0.969705i \(-0.421449\pi\)
0.244278 + 0.969705i \(0.421449\pi\)
\(30\) 2.73035 0.498492
\(31\) −5.19267 −0.932632 −0.466316 − 0.884618i \(-0.654419\pi\)
−0.466316 + 0.884618i \(0.654419\pi\)
\(32\) −7.71488 −1.36381
\(33\) −4.53162 −0.788853
\(34\) −2.15939 −0.370332
\(35\) 0 0
\(36\) 1.79746 0.299576
\(37\) 4.53435 0.745442 0.372721 − 0.927943i \(-0.378425\pi\)
0.372721 + 0.927943i \(0.378425\pi\)
\(38\) 1.24392 0.201791
\(39\) 2.35527 0.377145
\(40\) 0.553017 0.0874396
\(41\) −1.45216 −0.226790 −0.113395 − 0.993550i \(-0.536173\pi\)
−0.113395 + 0.993550i \(0.536173\pi\)
\(42\) 0 0
\(43\) 1.49386 0.227812 0.113906 − 0.993492i \(-0.463664\pi\)
0.113906 + 0.993492i \(0.463664\pi\)
\(44\) 8.14538 1.22796
\(45\) −1.40111 −0.208865
\(46\) −1.94871 −0.287321
\(47\) −6.43916 −0.939248 −0.469624 − 0.882866i \(-0.655611\pi\)
−0.469624 + 0.882866i \(0.655611\pi\)
\(48\) 4.36406 0.629898
\(49\) 0 0
\(50\) −5.91801 −0.836933
\(51\) 1.10811 0.155167
\(52\) −4.23350 −0.587080
\(53\) −11.1364 −1.52970 −0.764849 − 0.644209i \(-0.777187\pi\)
−0.764849 + 0.644209i \(0.777187\pi\)
\(54\) −1.94871 −0.265185
\(55\) −6.34929 −0.856138
\(56\) 0 0
\(57\) −0.638333 −0.0845493
\(58\) 5.12697 0.673204
\(59\) −6.20679 −0.808056 −0.404028 − 0.914747i \(-0.632390\pi\)
−0.404028 + 0.914747i \(0.632390\pi\)
\(60\) 2.51843 0.325128
\(61\) −0.821232 −0.105148 −0.0525740 − 0.998617i \(-0.516743\pi\)
−0.0525740 + 0.998617i \(0.516743\pi\)
\(62\) −10.1190 −1.28511
\(63\) 0 0
\(64\) −6.30591 −0.788239
\(65\) 3.29999 0.409314
\(66\) −8.83079 −1.08699
\(67\) −3.33164 −0.407025 −0.203513 − 0.979072i \(-0.565236\pi\)
−0.203513 + 0.979072i \(0.565236\pi\)
\(68\) −1.99179 −0.241540
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −14.4996 −1.72079 −0.860396 − 0.509627i \(-0.829784\pi\)
−0.860396 + 0.509627i \(0.829784\pi\)
\(72\) −0.394699 −0.0465157
\(73\) −1.99401 −0.233381 −0.116691 − 0.993168i \(-0.537229\pi\)
−0.116691 + 0.993168i \(0.537229\pi\)
\(74\) 8.83611 1.02718
\(75\) 3.03689 0.350670
\(76\) 1.14738 0.131613
\(77\) 0 0
\(78\) 4.58973 0.519685
\(79\) 4.18578 0.470938 0.235469 − 0.971882i \(-0.424337\pi\)
0.235469 + 0.971882i \(0.424337\pi\)
\(80\) 6.11453 0.683625
\(81\) 1.00000 0.111111
\(82\) −2.82984 −0.312504
\(83\) 8.69016 0.953869 0.476934 − 0.878939i \(-0.341748\pi\)
0.476934 + 0.878939i \(0.341748\pi\)
\(84\) 0 0
\(85\) 1.55259 0.168402
\(86\) 2.91110 0.313911
\(87\) −2.63096 −0.282068
\(88\) −1.78862 −0.190668
\(89\) −7.56109 −0.801474 −0.400737 − 0.916193i \(-0.631246\pi\)
−0.400737 + 0.916193i \(0.631246\pi\)
\(90\) −2.73035 −0.287804
\(91\) 0 0
\(92\) −1.79746 −0.187398
\(93\) 5.19267 0.538455
\(94\) −12.5480 −1.29423
\(95\) −0.894375 −0.0917609
\(96\) 7.71488 0.787397
\(97\) −9.73998 −0.988945 −0.494473 − 0.869193i \(-0.664639\pi\)
−0.494473 + 0.869193i \(0.664639\pi\)
\(98\) 0 0
\(99\) 4.53162 0.455444
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 3381.2.a.bk.1.8 ✓ 10
7.6 odd 2 3381.2.a.bl.1.8 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.2.a.bk.1.8 ✓ 10 1.1 even 1 trivial
3381.2.a.bl.1.8 yes 10 7.6 odd 2