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Properties

Label 882.6.a.bh.1.2
Level $882$
Weight $6$
Character 882.1
Self dual yes
Analytic conductor $141.459$
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] = [882,6,Mod(1,882)] 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("882.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8,0,32,53,0,0,-128,0,-212,-191] 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: \(141.458529075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{9601}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-48.4923\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} +75.4923 q^{5} -64.0000 q^{8} -301.969 q^{10} +149.462 q^{11} +349.416 q^{13} +256.000 q^{16} +1149.85 q^{17} -2795.20 q^{19} +1207.88 q^{20} -597.847 q^{22} -1813.97 q^{23} +2574.09 q^{25} -1397.66 q^{26} +759.033 q^{29} +9031.74 q^{31} -1024.00 q^{32} -4599.39 q^{34} +7794.89 q^{37} +11180.8 q^{38} -4831.51 q^{40} -7640.49 q^{41} +12188.8 q^{43} +2391.39 q^{44} +7255.88 q^{46} -24598.8 q^{47} -10296.4 q^{50} +5590.65 q^{52} -13596.2 q^{53} +11283.2 q^{55} -3036.13 q^{58} +26358.8 q^{59} +35321.8 q^{61} -36127.0 q^{62} +4096.00 q^{64} +26378.2 q^{65} +54371.9 q^{67} +18397.6 q^{68} +70145.7 q^{71} -44468.8 q^{73} -31179.6 q^{74} -44723.2 q^{76} +61612.5 q^{79} +19326.0 q^{80} +30562.0 q^{82} +87142.0 q^{83} +86804.6 q^{85} -48755.3 q^{86} -9565.55 q^{88} -98569.4 q^{89} -29023.5 q^{92} +98395.2 q^{94} -211016. q^{95} +32342.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 32 q^{4} + 53 q^{5} - 128 q^{8} - 212 q^{10} - 191 q^{11} - 379 q^{13} + 512 q^{16} + 340 q^{17} - 1769 q^{19} + 848 q^{20} + 764 q^{22} - 3236 q^{23} - 45 q^{25} + 1516 q^{26} - 4459 q^{29}+ \cdots - 129423 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 75.4923 1.35045 0.675224 − 0.737613i \(-0.264047\pi\)
0.675224 + 0.737613i \(0.264047\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) −301.969 −0.954911
\(11\) 149.462 0.372433 0.186217 − 0.982509i \(-0.440377\pi\)
0.186217 + 0.982509i \(0.440377\pi\)
\(12\) 0 0
\(13\) 349.416 0.573435 0.286717 − 0.958015i \(-0.407436\pi\)
0.286717 + 0.958015i \(0.407436\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1149.85 0.964979 0.482489 − 0.875902i \(-0.339733\pi\)
0.482489 + 0.875902i \(0.339733\pi\)
\(18\) 0 0
\(19\) −2795.20 −1.77635 −0.888176 − 0.459503i \(-0.848028\pi\)
−0.888176 + 0.459503i \(0.848028\pi\)
\(20\) 1207.88 0.675224
\(21\) 0 0
\(22\) −597.847 −0.263350
\(23\) −1813.97 −0.715007 −0.357504 − 0.933912i \(-0.616372\pi\)
−0.357504 + 0.933912i \(0.616372\pi\)
\(24\) 0 0
\(25\) 2574.09 0.823710
\(26\) −1397.66 −0.405480
\(27\) 0 0
\(28\) 0 0
\(29\) 759.033 0.167597 0.0837984 − 0.996483i \(-0.473295\pi\)
0.0837984 + 0.996483i \(0.473295\pi\)
\(30\) 0 0
\(31\) 9031.74 1.68798 0.843990 − 0.536359i \(-0.180201\pi\)
0.843990 + 0.536359i \(0.180201\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) −4599.39 −0.682343
\(35\) 0 0
\(36\) 0 0
\(37\) 7794.89 0.936064 0.468032 − 0.883711i \(-0.344963\pi\)
0.468032 + 0.883711i \(0.344963\pi\)
\(38\) 11180.8 1.25607
\(39\) 0 0
\(40\) −4831.51 −0.477456
\(41\) −7640.49 −0.709842 −0.354921 − 0.934896i \(-0.615492\pi\)
−0.354921 + 0.934896i \(0.615492\pi\)
\(42\) 0 0
\(43\) 12188.8 1.00529 0.502645 − 0.864493i \(-0.332360\pi\)
0.502645 + 0.864493i \(0.332360\pi\)
\(44\) 2391.39 0.186217
\(45\) 0 0
\(46\) 7255.88 0.505586
\(47\) −24598.8 −1.62431 −0.812156 − 0.583441i \(-0.801706\pi\)
−0.812156 + 0.583441i \(0.801706\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −10296.4 −0.582451
\(51\) 0 0
\(52\) 5590.65 0.286717
\(53\) −13596.2 −0.664858 −0.332429 − 0.943128i \(-0.607868\pi\)
−0.332429 + 0.943128i \(0.607868\pi\)
\(54\) 0 0
\(55\) 11283.2 0.502952
\(56\) 0 0
\(57\) 0 0
\(58\) −3036.13 −0.118509
\(59\) 26358.8 0.985816 0.492908 − 0.870081i \(-0.335934\pi\)
0.492908 + 0.870081i \(0.335934\pi\)
\(60\) 0 0
\(61\) 35321.8 1.21540 0.607698 − 0.794168i \(-0.292093\pi\)
0.607698 + 0.794168i \(0.292093\pi\)
\(62\) −36127.0 −1.19358
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 26378.2 0.774394
\(66\) 0 0
\(67\) 54371.9 1.47975 0.739874 − 0.672746i \(-0.234885\pi\)
0.739874 + 0.672746i \(0.234885\pi\)
\(68\) 18397.6 0.482489
\(69\) 0 0
\(70\) 0 0
\(71\) 70145.7 1.65141 0.825706 − 0.564101i \(-0.190777\pi\)
0.825706 + 0.564101i \(0.190777\pi\)
\(72\) 0 0
\(73\) −44468.8 −0.976671 −0.488335 − 0.872656i \(-0.662396\pi\)
−0.488335 + 0.872656i \(0.662396\pi\)
\(74\) −31179.6 −0.661897
\(75\) 0 0
\(76\) −44723.2 −0.888176
\(77\) 0 0
\(78\) 0 0
\(79\) 61612.5 1.11071 0.555355 − 0.831613i \(-0.312582\pi\)
0.555355 + 0.831613i \(0.312582\pi\)
\(80\) 19326.0 0.337612
\(81\) 0 0
\(82\) 30562.0 0.501934
\(83\) 87142.0 1.38846 0.694228 − 0.719755i \(-0.255746\pi\)
0.694228 + 0.719755i \(0.255746\pi\)
\(84\) 0 0
\(85\) 86804.6 1.30315
\(86\) −48755.3 −0.710847
\(87\) 0 0
\(88\) −9565.55 −0.131675
\(89\) −98569.4 −1.31907 −0.659534 − 0.751675i \(-0.729246\pi\)
−0.659534 + 0.751675i \(0.729246\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −29023.5 −0.357504
\(93\) 0 0
\(94\) 98395.2 1.14856
\(95\) −211016. −2.39887
\(96\) 0 0
\(97\) 32342.3 0.349013 0.174507 − 0.984656i \(-0.444167\pi\)
0.174507 + 0.984656i \(0.444167\pi\)
\(98\) 0 0
\(99\) 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 882.6.a.bh.1.2 2
3.2 odd 2 294.6.a.r.1.1 2
7.2 even 3 126.6.g.h.109.1 4
7.4 even 3 126.6.g.h.37.1 4
7.6 odd 2 882.6.a.bb.1.1 2
21.2 odd 6 42.6.e.c.25.2 ✓ 4
21.5 even 6 294.6.e.s.67.1 4
21.11 odd 6 42.6.e.c.37.2 yes 4
21.17 even 6 294.6.e.s.79.1 4
21.20 even 2 294.6.a.w.1.2 2
84.11 even 6 336.6.q.f.289.2 4
84.23 even 6 336.6.q.f.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.e.c.25.2 ✓ 4 21.2 odd 6
42.6.e.c.37.2 yes 4 21.11 odd 6
126.6.g.h.37.1 4 7.4 even 3
126.6.g.h.109.1 4 7.2 even 3
294.6.a.r.1.1 2 3.2 odd 2
294.6.a.w.1.2 2 21.20 even 2
294.6.e.s.67.1 4 21.5 even 6
294.6.e.s.79.1 4 21.17 even 6
336.6.q.f.193.2 4 84.23 even 6
336.6.q.f.289.2 4 84.11 even 6
882.6.a.bb.1.1 2 7.6 odd 2
882.6.a.bh.1.2 2 1.1 even 1 trivial