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Properties

Label 3150.2.bf.a.1151.2
Level $3150$
Weight $2$
Character 3150.1151
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [3150,2,Mod(1151,3150)] 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("3150.1151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3150, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,4,0,0,-4,0,0,0,0] 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: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1151.2
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 3150.1151
Dual form 3150.2.bf.a.1601.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.62132 + 2.09077i) q^{7} +1.00000i q^{8} +(-2.59808 - 1.50000i) q^{11} +2.44949i q^{13} +(-2.44949 - 1.00000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(0.507306 - 0.878680i) q^{17} +(-0.878680 + 0.507306i) q^{19} +3.00000 q^{22} +(3.67423 - 2.12132i) q^{23} +(-1.22474 - 2.12132i) q^{26} +(2.62132 - 0.358719i) q^{28} -1.24264i q^{29} +(4.86396 + 2.80821i) q^{31} +(0.866025 + 0.500000i) q^{32} +1.01461i q^{34} +(4.12132 + 7.13834i) q^{37} +(0.507306 - 0.878680i) q^{38} +2.02922 q^{41} -8.24264 q^{43} +(-2.59808 + 1.50000i) q^{44} +(-2.12132 + 3.67423i) q^{46} +(0.507306 + 0.878680i) q^{47} +(-1.74264 + 6.77962i) q^{49} +(2.12132 + 1.22474i) q^{52} +(-1.07616 - 0.621320i) q^{53} +(-2.09077 + 1.62132i) q^{56} +(0.621320 + 1.07616i) q^{58} +(-5.76500 + 9.98528i) q^{59} +(5.12132 - 2.95680i) q^{61} -5.61642 q^{62} -1.00000 q^{64} +(-5.00000 + 8.66025i) q^{67} +(-0.507306 - 0.878680i) q^{68} -10.2426i q^{71} +(-7.24264 - 4.18154i) q^{73} +(-7.13834 - 4.12132i) q^{74} +1.01461i q^{76} +(-1.07616 - 7.86396i) q^{77} +(5.62132 + 9.73641i) q^{79} +(-1.75736 + 1.01461i) q^{82} +3.16693 q^{83} +(7.13834 - 4.12132i) q^{86} +(1.50000 - 2.59808i) q^{88} +(5.19615 + 9.00000i) q^{89} +(-5.12132 + 3.97141i) q^{91} -4.24264i q^{92} +(-0.878680 - 0.507306i) q^{94} +3.76127i q^{97} +(-1.88064 - 6.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 4 q^{7} - 4 q^{16} - 24 q^{19} + 24 q^{22} + 4 q^{28} - 12 q^{31} + 16 q^{37} - 32 q^{43} + 20 q^{49} - 12 q^{58} + 24 q^{61} - 8 q^{64} - 40 q^{67} - 24 q^{73} + 28 q^{79} - 48 q^{82} + 12 q^{88}+ \cdots - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.62132 + 2.09077i 0.612801 + 0.790237i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −2.59808 − 1.50000i −0.783349 − 0.452267i 0.0542666 − 0.998526i \(-0.482718\pi\)
−0.837616 + 0.546259i \(0.816051\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) −2.44949 − 1.00000i −0.654654 − 0.267261i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0.507306 − 0.878680i 0.123040 − 0.213111i −0.797925 − 0.602756i \(-0.794069\pi\)
0.920965 + 0.389645i \(0.127402\pi\)
\(18\) 0 0
\(19\) −0.878680 + 0.507306i −0.201583 + 0.116384i −0.597394 − 0.801948i \(-0.703797\pi\)
0.395811 + 0.918332i \(0.370464\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 3.67423 − 2.12132i 0.766131 − 0.442326i −0.0653618 − 0.997862i \(-0.520820\pi\)
0.831493 + 0.555536i \(0.187487\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.22474 − 2.12132i −0.240192 − 0.416025i
\(27\) 0 0
\(28\) 2.62132 − 0.358719i 0.495383 − 0.0677916i
\(29\) − 1.24264i − 0.230753i −0.993322 − 0.115376i \(-0.963193\pi\)
0.993322 − 0.115376i \(-0.0368074\pi\)
\(30\) 0 0
\(31\) 4.86396 + 2.80821i 0.873593 + 0.504369i 0.868541 − 0.495618i \(-0.165058\pi\)
0.00505256 + 0.999987i \(0.498392\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 1.01461i 0.174005i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.12132 + 7.13834i 0.677541 + 1.17354i 0.975719 + 0.219025i \(0.0702877\pi\)
−0.298178 + 0.954510i \(0.596379\pi\)
\(38\) 0.507306 − 0.878680i 0.0822959 − 0.142541i
\(39\) 0 0
\(40\) 0 0
\(41\) 2.02922 0.316912 0.158456 − 0.987366i \(-0.449348\pi\)
0.158456 + 0.987366i \(0.449348\pi\)
\(42\) 0 0
\(43\) −8.24264 −1.25699 −0.628495 − 0.777813i \(-0.716329\pi\)
−0.628495 + 0.777813i \(0.716329\pi\)
\(44\) −2.59808 + 1.50000i −0.391675 + 0.226134i
\(45\) 0 0
\(46\) −2.12132 + 3.67423i −0.312772 + 0.541736i
\(47\) 0.507306 + 0.878680i 0.0739982 + 0.128169i 0.900650 − 0.434545i \(-0.143091\pi\)
−0.826652 + 0.562713i \(0.809757\pi\)
\(48\) 0 0
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) 0 0
\(51\) 0 0
\(52\) 2.12132 + 1.22474i 0.294174 + 0.169842i
\(53\) −1.07616 − 0.621320i −0.147822 − 0.0853449i 0.424265 − 0.905538i \(-0.360533\pi\)
−0.572087 + 0.820193i \(0.693866\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.09077 + 1.62132i −0.279391 + 0.216658i
\(57\) 0 0
\(58\) 0.621320 + 1.07616i 0.0815834 + 0.141307i
\(59\) −5.76500 + 9.98528i −0.750540 + 1.29997i 0.197022 + 0.980399i \(0.436873\pi\)
−0.947561 + 0.319574i \(0.896460\pi\)
\(60\) 0 0
\(61\) 5.12132 − 2.95680i 0.655718 − 0.378579i −0.134926 − 0.990856i \(-0.543080\pi\)
0.790643 + 0.612277i \(0.209746\pi\)
\(62\) −5.61642 −0.713286
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) −0.507306 − 0.878680i −0.0615199 − 0.106556i
\(69\) 0 0
\(70\) 0 0
\(71\) − 10.2426i − 1.21558i −0.794099 − 0.607789i \(-0.792057\pi\)
0.794099 − 0.607789i \(-0.207943\pi\)
\(72\) 0 0
\(73\) −7.24264 − 4.18154i −0.847687 − 0.489412i 0.0121828 − 0.999926i \(-0.496122\pi\)
−0.859870 + 0.510513i \(0.829455\pi\)
\(74\) −7.13834 − 4.12132i −0.829815 − 0.479094i
\(75\) 0 0
\(76\) 1.01461i 0.116384i
\(77\) −1.07616 − 7.86396i −0.122640 − 0.896182i
\(78\) 0 0
\(79\) 5.62132 + 9.73641i 0.632448 + 1.09543i 0.987050 + 0.160415i \(0.0512831\pi\)
−0.354602 + 0.935017i \(0.615384\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.75736 + 1.01461i −0.194068 + 0.112045i
\(83\) 3.16693 0.347616 0.173808 − 0.984780i \(-0.444393\pi\)
0.173808 + 0.984780i \(0.444393\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.13834 − 4.12132i 0.769747 − 0.444413i
\(87\) 0 0
\(88\) 1.50000 − 2.59808i 0.159901 − 0.276956i
\(89\) 5.19615 + 9.00000i 0.550791 + 0.953998i 0.998218 + 0.0596775i \(0.0190072\pi\)
−0.447427 + 0.894321i \(0.647659\pi\)
\(90\) 0 0
\(91\) −5.12132 + 3.97141i −0.536860 + 0.416317i
\(92\) − 4.24264i − 0.442326i
\(93\) 0 0
\(94\) −0.878680 − 0.507306i −0.0906289 − 0.0523246i
\(95\) 0 0
\(96\) 0 0
\(97\) 3.76127i 0.381900i 0.981600 + 0.190950i \(0.0611568\pi\)
−0.981600 + 0.190950i \(0.938843\pi\)
\(98\) −1.88064 − 6.74264i −0.189973 − 0.681110i
\(99\) 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 3150.2.bf.a.1151.2 8
3.2 odd 2 inner 3150.2.bf.a.1151.4 8
5.2 odd 4 3150.2.bp.b.899.1 8
5.3 odd 4 3150.2.bp.e.899.4 8
5.4 even 2 126.2.k.a.17.4 yes 8
7.5 odd 6 inner 3150.2.bf.a.1601.4 8
15.2 even 4 3150.2.bp.e.899.1 8
15.8 even 4 3150.2.bp.b.899.4 8
15.14 odd 2 126.2.k.a.17.1 ✓ 8
20.19 odd 2 1008.2.bt.c.17.4 8
21.5 even 6 inner 3150.2.bf.a.1601.2 8
35.4 even 6 882.2.d.a.881.5 8
35.9 even 6 882.2.k.a.215.2 8
35.12 even 12 3150.2.bp.b.1349.4 8
35.19 odd 6 126.2.k.a.89.1 yes 8
35.24 odd 6 882.2.d.a.881.8 8
35.33 even 12 3150.2.bp.e.1349.1 8
35.34 odd 2 882.2.k.a.521.3 8
45.4 even 6 1134.2.l.f.269.4 8
45.14 odd 6 1134.2.l.f.269.1 8
45.29 odd 6 1134.2.t.e.1025.4 8
45.34 even 6 1134.2.t.e.1025.1 8
60.59 even 2 1008.2.bt.c.17.1 8
105.44 odd 6 882.2.k.a.215.3 8
105.47 odd 12 3150.2.bp.e.1349.4 8
105.59 even 6 882.2.d.a.881.1 8
105.68 odd 12 3150.2.bp.b.1349.1 8
105.74 odd 6 882.2.d.a.881.4 8
105.89 even 6 126.2.k.a.89.4 yes 8
105.104 even 2 882.2.k.a.521.2 8
140.19 even 6 1008.2.bt.c.593.1 8
140.39 odd 6 7056.2.k.f.881.1 8
140.59 even 6 7056.2.k.f.881.7 8
315.124 odd 6 1134.2.l.f.215.3 8
315.194 even 6 1134.2.t.e.593.1 8
315.229 odd 6 1134.2.t.e.593.4 8
315.299 even 6 1134.2.l.f.215.2 8
420.59 odd 6 7056.2.k.f.881.2 8
420.179 even 6 7056.2.k.f.881.8 8
420.299 odd 6 1008.2.bt.c.593.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 15.14 odd 2
126.2.k.a.17.4 yes 8 5.4 even 2
126.2.k.a.89.1 yes 8 35.19 odd 6
126.2.k.a.89.4 yes 8 105.89 even 6
882.2.d.a.881.1 8 105.59 even 6
882.2.d.a.881.4 8 105.74 odd 6
882.2.d.a.881.5 8 35.4 even 6
882.2.d.a.881.8 8 35.24 odd 6
882.2.k.a.215.2 8 35.9 even 6
882.2.k.a.215.3 8 105.44 odd 6
882.2.k.a.521.2 8 105.104 even 2
882.2.k.a.521.3 8 35.34 odd 2
1008.2.bt.c.17.1 8 60.59 even 2
1008.2.bt.c.17.4 8 20.19 odd 2
1008.2.bt.c.593.1 8 140.19 even 6
1008.2.bt.c.593.4 8 420.299 odd 6
1134.2.l.f.215.2 8 315.299 even 6
1134.2.l.f.215.3 8 315.124 odd 6
1134.2.l.f.269.1 8 45.14 odd 6
1134.2.l.f.269.4 8 45.4 even 6
1134.2.t.e.593.1 8 315.194 even 6
1134.2.t.e.593.4 8 315.229 odd 6
1134.2.t.e.1025.1 8 45.34 even 6
1134.2.t.e.1025.4 8 45.29 odd 6
3150.2.bf.a.1151.2 8 1.1 even 1 trivial
3150.2.bf.a.1151.4 8 3.2 odd 2 inner
3150.2.bf.a.1601.2 8 21.5 even 6 inner
3150.2.bf.a.1601.4 8 7.5 odd 6 inner
3150.2.bp.b.899.1 8 5.2 odd 4
3150.2.bp.b.899.4 8 15.8 even 4
3150.2.bp.b.1349.1 8 105.68 odd 12
3150.2.bp.b.1349.4 8 35.12 even 12
3150.2.bp.e.899.1 8 15.2 even 4
3150.2.bp.e.899.4 8 5.3 odd 4
3150.2.bp.e.1349.1 8 35.33 even 12
3150.2.bp.e.1349.4 8 105.47 odd 12
7056.2.k.f.881.1 8 140.39 odd 6
7056.2.k.f.881.2 8 420.59 odd 6
7056.2.k.f.881.7 8 140.59 even 6
7056.2.k.f.881.8 8 420.179 even 6