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Properties

Label 882.2.e.s.373.4
Level $882$
Weight $2$
Character 882.373
Analytic conductor $7.043$
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] = [882,2,Mod(373,882)] 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("882.373"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8,0,8,0,0,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 882.373
Dual form 882.2.e.s.655.4

$q$-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+1.00000 q^{2} +(1.67303 - 0.448288i) q^{3} +1.00000 q^{4} +(-1.93185 - 3.34607i) q^{5} +(1.67303 - 0.448288i) q^{6} +1.00000 q^{8} +(2.59808 - 1.50000i) q^{9} +(-1.93185 - 3.34607i) q^{10} +(1.86603 - 3.23205i) q^{11} +(1.67303 - 0.448288i) q^{12} +(-3.34607 + 5.79555i) q^{13} +(-4.73205 - 4.73205i) q^{15} +1.00000 q^{16} +(-2.70831 - 4.69093i) q^{17} +(2.59808 - 1.50000i) q^{18} +(1.48356 - 2.56961i) q^{19} +(-1.93185 - 3.34607i) q^{20} +(1.86603 - 3.23205i) q^{22} +(0.732051 + 1.26795i) q^{23} +(1.67303 - 0.448288i) q^{24} +(-4.96410 + 8.59808i) q^{25} +(-3.34607 + 5.79555i) q^{26} +(3.67423 - 3.67423i) q^{27} +(2.00000 + 3.46410i) q^{29} +(-4.73205 - 4.73205i) q^{30} +1.79315 q^{31} +1.00000 q^{32} +(1.67303 - 6.24384i) q^{33} +(-2.70831 - 4.69093i) q^{34} +(2.59808 - 1.50000i) q^{36} +(-0.267949 + 0.464102i) q^{37} +(1.48356 - 2.56961i) q^{38} +(-3.00000 + 11.1962i) q^{39} +(-1.93185 - 3.34607i) q^{40} +(0.637756 - 1.10463i) q^{41} +(-1.86603 - 3.23205i) q^{43} +(1.86603 - 3.23205i) q^{44} +(-10.0382 - 5.79555i) q^{45} +(0.732051 + 1.26795i) q^{46} +10.5558 q^{47} +(1.67303 - 0.448288i) q^{48} +(-4.96410 + 8.59808i) q^{50} +(-6.63397 - 6.63397i) q^{51} +(-3.34607 + 5.79555i) q^{52} +(1.46410 + 2.53590i) q^{53} +(3.67423 - 3.67423i) q^{54} -14.4195 q^{55} +(1.33013 - 4.96410i) q^{57} +(2.00000 + 3.46410i) q^{58} -8.62398 q^{59} +(-4.73205 - 4.73205i) q^{60} +6.96953 q^{61} +1.79315 q^{62} +1.00000 q^{64} +25.8564 q^{65} +(1.67303 - 6.24384i) q^{66} +5.53590 q^{67} +(-2.70831 - 4.69093i) q^{68} +(1.79315 + 1.79315i) q^{69} +2.53590 q^{71} +(2.59808 - 1.50000i) q^{72} +(3.41542 + 5.91567i) q^{73} +(-0.267949 + 0.464102i) q^{74} +(-4.45069 + 16.6102i) q^{75} +(1.48356 - 2.56961i) q^{76} +(-3.00000 + 11.1962i) q^{78} -4.92820 q^{79} +(-1.93185 - 3.34607i) q^{80} +(4.50000 - 7.79423i) q^{81} +(0.637756 - 1.10463i) q^{82} +(8.95215 + 15.5056i) q^{83} +(-10.4641 + 18.1244i) q^{85} +(-1.86603 - 3.23205i) q^{86} +(4.89898 + 4.89898i) q^{87} +(1.86603 - 3.23205i) q^{88} +(-3.53553 + 6.12372i) q^{89} +(-10.0382 - 5.79555i) q^{90} +(0.732051 + 1.26795i) q^{92} +(3.00000 - 0.803848i) q^{93} +10.5558 q^{94} -11.4641 q^{95} +(1.67303 - 0.448288i) q^{96} +(2.94855 + 5.10703i) q^{97} -11.1962i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{11} - 24 q^{15} + 8 q^{16} + 8 q^{22} - 8 q^{23} - 12 q^{25} + 16 q^{29} - 24 q^{30} + 8 q^{32} - 16 q^{37} - 24 q^{39} - 8 q^{43} + 8 q^{44} - 8 q^{46} - 12 q^{50}+ \cdots - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.67303 − 0.448288i 0.965926 − 0.258819i
\(4\) 1.00000 0.500000
\(5\) −1.93185 − 3.34607i −0.863950 − 1.49641i −0.868086 − 0.496414i \(-0.834650\pi\)
0.00413535 − 0.999991i \(-0.498684\pi\)
\(6\) 1.67303 − 0.448288i 0.683013 − 0.183013i
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 2.59808 − 1.50000i 0.866025 − 0.500000i
\(10\) −1.93185 − 3.34607i −0.610905 − 1.05812i
\(11\) 1.86603 − 3.23205i 0.562628 − 0.974500i −0.434638 − 0.900605i \(-0.643124\pi\)
0.997266 − 0.0738948i \(-0.0235429\pi\)
\(12\) 1.67303 − 0.448288i 0.482963 − 0.129410i
\(13\) −3.34607 + 5.79555i −0.928032 + 1.60740i −0.141420 + 0.989950i \(0.545167\pi\)
−0.786612 + 0.617448i \(0.788167\pi\)
\(14\) 0 0
\(15\) −4.73205 − 4.73205i −1.22181 − 1.22181i
\(16\) 1.00000 0.250000
\(17\) −2.70831 − 4.69093i −0.656861 − 1.13772i −0.981424 − 0.191853i \(-0.938550\pi\)
0.324562 − 0.945864i \(-0.394783\pi\)
\(18\) 2.59808 − 1.50000i 0.612372 − 0.353553i
\(19\) 1.48356 − 2.56961i 0.340353 − 0.589509i −0.644145 − 0.764903i \(-0.722787\pi\)
0.984498 + 0.175395i \(0.0561201\pi\)
\(20\) −1.93185 − 3.34607i −0.431975 − 0.748203i
\(21\) 0 0
\(22\) 1.86603 − 3.23205i 0.397838 − 0.689076i
\(23\) 0.732051 + 1.26795i 0.152643 + 0.264386i 0.932198 − 0.361948i \(-0.117888\pi\)
−0.779555 + 0.626334i \(0.784555\pi\)
\(24\) 1.67303 − 0.448288i 0.341506 − 0.0915064i
\(25\) −4.96410 + 8.59808i −0.992820 + 1.71962i
\(26\) −3.34607 + 5.79555i −0.656217 + 1.13660i
\(27\) 3.67423 − 3.67423i 0.707107 − 0.707107i
\(28\) 0 0
\(29\) 2.00000 + 3.46410i 0.371391 + 0.643268i 0.989780 − 0.142605i \(-0.0455477\pi\)
−0.618389 + 0.785872i \(0.712214\pi\)
\(30\) −4.73205 − 4.73205i −0.863950 − 0.863950i
\(31\) 1.79315 0.322059 0.161030 − 0.986950i \(-0.448519\pi\)
0.161030 + 0.986950i \(0.448519\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.67303 − 6.24384i 0.291238 − 1.08691i
\(34\) −2.70831 − 4.69093i −0.464471 − 0.804488i
\(35\) 0 0
\(36\) 2.59808 − 1.50000i 0.433013 − 0.250000i
\(37\) −0.267949 + 0.464102i −0.0440506 + 0.0762978i −0.887210 − 0.461366i \(-0.847360\pi\)
0.843159 + 0.537664i \(0.180693\pi\)
\(38\) 1.48356 − 2.56961i 0.240666 − 0.416845i
\(39\) −3.00000 + 11.1962i −0.480384 + 1.79282i
\(40\) −1.93185 − 3.34607i −0.305453 − 0.529059i
\(41\) 0.637756 − 1.10463i 0.0996008 − 0.172514i −0.811919 − 0.583771i \(-0.801577\pi\)
0.911519 + 0.411257i \(0.134910\pi\)
\(42\) 0 0
\(43\) −1.86603 − 3.23205i −0.284566 − 0.492883i 0.687938 − 0.725770i \(-0.258516\pi\)
−0.972504 + 0.232887i \(0.925183\pi\)
\(44\) 1.86603 − 3.23205i 0.281314 − 0.487250i
\(45\) −10.0382 − 5.79555i −1.49641 − 0.863950i
\(46\) 0.732051 + 1.26795i 0.107935 + 0.186949i
\(47\) 10.5558 1.53973 0.769863 − 0.638209i \(-0.220324\pi\)
0.769863 + 0.638209i \(0.220324\pi\)
\(48\) 1.67303 − 0.448288i 0.241481 − 0.0647048i
\(49\) 0 0
\(50\) −4.96410 + 8.59808i −0.702030 + 1.21595i
\(51\) −6.63397 − 6.63397i −0.928942 − 0.928942i
\(52\) −3.34607 + 5.79555i −0.464016 + 0.803699i
\(53\) 1.46410 + 2.53590i 0.201110 + 0.348332i 0.948886 − 0.315618i \(-0.102212\pi\)
−0.747776 + 0.663951i \(0.768879\pi\)
\(54\) 3.67423 − 3.67423i 0.500000 − 0.500000i
\(55\) −14.4195 −1.94433
\(56\) 0 0
\(57\) 1.33013 − 4.96410i 0.176180 − 0.657511i
\(58\) 2.00000 + 3.46410i 0.262613 + 0.454859i
\(59\) −8.62398 −1.12275 −0.561373 − 0.827563i \(-0.689727\pi\)
−0.561373 + 0.827563i \(0.689727\pi\)
\(60\) −4.73205 − 4.73205i −0.610905 − 0.610905i
\(61\) 6.96953 0.892357 0.446179 − 0.894944i \(-0.352785\pi\)
0.446179 + 0.894944i \(0.352785\pi\)
\(62\) 1.79315 0.227730
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 25.8564 3.20709
\(66\) 1.67303 − 6.24384i 0.205936 − 0.768564i
\(67\) 5.53590 0.676318 0.338159 − 0.941089i \(-0.390196\pi\)
0.338159 + 0.941089i \(0.390196\pi\)
\(68\) −2.70831 − 4.69093i −0.328431 − 0.568859i
\(69\) 1.79315 + 1.79315i 0.215870 + 0.215870i
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 − 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 2.59808 − 1.50000i 0.306186 − 0.176777i
\(73\) 3.41542 + 5.91567i 0.399744 + 0.692377i 0.993694 − 0.112125i \(-0.0357656\pi\)
−0.593950 + 0.804502i \(0.702432\pi\)
\(74\) −0.267949 + 0.464102i −0.0311485 + 0.0539507i
\(75\) −4.45069 + 16.6102i −0.513922 + 1.91798i
\(76\) 1.48356 − 2.56961i 0.170176 − 0.294754i
\(77\) 0 0
\(78\) −3.00000 + 11.1962i −0.339683 + 1.26771i
\(79\) −4.92820 −0.554466 −0.277233 − 0.960803i \(-0.589417\pi\)
−0.277233 + 0.960803i \(0.589417\pi\)
\(80\) −1.93185 − 3.34607i −0.215988 − 0.374101i
\(81\) 4.50000 − 7.79423i 0.500000 − 0.866025i
\(82\) 0.637756 − 1.10463i 0.0704284 − 0.121986i
\(83\) 8.95215 + 15.5056i 0.982626 + 1.70196i 0.652043 + 0.758182i \(0.273912\pi\)
0.330583 + 0.943777i \(0.392755\pi\)
\(84\) 0 0
\(85\) −10.4641 + 18.1244i −1.13499 + 1.96586i
\(86\) −1.86603 − 3.23205i −0.201219 − 0.348521i
\(87\) 4.89898 + 4.89898i 0.525226 + 0.525226i
\(88\) 1.86603 − 3.23205i 0.198919 − 0.344538i
\(89\) −3.53553 + 6.12372i −0.374766 + 0.649113i −0.990292 − 0.139003i \(-0.955610\pi\)
0.615526 + 0.788116i \(0.288944\pi\)
\(90\) −10.0382 − 5.79555i −1.05812 − 0.610905i
\(91\) 0 0
\(92\) 0.732051 + 1.26795i 0.0763216 + 0.132193i
\(93\) 3.00000 − 0.803848i 0.311086 − 0.0833551i
\(94\) 10.5558 1.08875
\(95\) −11.4641 −1.17619
\(96\) 1.67303 − 0.448288i 0.170753 − 0.0457532i
\(97\) 2.94855 + 5.10703i 0.299379 + 0.518540i 0.975994 − 0.217797i \(-0.0698870\pi\)
−0.676615 + 0.736337i \(0.736554\pi\)
\(98\) 0 0
\(99\) − 11.1962i − 1.12526i
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 882.2.e.s.373.4 8
3.2 odd 2 2646.2.e.q.1549.4 8
7.2 even 3 882.2.f.q.589.2 yes 8
7.3 odd 6 882.2.h.q.67.4 8
7.4 even 3 882.2.h.q.67.1 8
7.5 odd 6 882.2.f.q.589.3 yes 8
7.6 odd 2 inner 882.2.e.s.373.1 8
9.2 odd 6 2646.2.h.t.667.1 8
9.7 even 3 882.2.h.q.79.2 8
21.2 odd 6 2646.2.f.r.1765.4 8
21.5 even 6 2646.2.f.r.1765.1 8
21.11 odd 6 2646.2.h.t.361.1 8
21.17 even 6 2646.2.h.t.361.4 8
21.20 even 2 2646.2.e.q.1549.1 8
63.2 odd 6 2646.2.f.r.883.4 8
63.5 even 6 7938.2.a.ci.1.4 4
63.11 odd 6 2646.2.e.q.2125.4 8
63.16 even 3 882.2.f.q.295.2 ✓ 8
63.20 even 6 2646.2.h.t.667.4 8
63.23 odd 6 7938.2.a.ci.1.1 4
63.25 even 3 inner 882.2.e.s.655.4 8
63.34 odd 6 882.2.h.q.79.3 8
63.38 even 6 2646.2.e.q.2125.1 8
63.40 odd 6 7938.2.a.cp.1.1 4
63.47 even 6 2646.2.f.r.883.1 8
63.52 odd 6 inner 882.2.e.s.655.1 8
63.58 even 3 7938.2.a.cp.1.4 4
63.61 odd 6 882.2.f.q.295.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.1 8 7.6 odd 2 inner
882.2.e.s.373.4 8 1.1 even 1 trivial
882.2.e.s.655.1 8 63.52 odd 6 inner
882.2.e.s.655.4 8 63.25 even 3 inner
882.2.f.q.295.2 ✓ 8 63.16 even 3
882.2.f.q.295.3 yes 8 63.61 odd 6
882.2.f.q.589.2 yes 8 7.2 even 3
882.2.f.q.589.3 yes 8 7.5 odd 6
882.2.h.q.67.1 8 7.4 even 3
882.2.h.q.67.4 8 7.3 odd 6
882.2.h.q.79.2 8 9.7 even 3
882.2.h.q.79.3 8 63.34 odd 6
2646.2.e.q.1549.1 8 21.20 even 2
2646.2.e.q.1549.4 8 3.2 odd 2
2646.2.e.q.2125.1 8 63.38 even 6
2646.2.e.q.2125.4 8 63.11 odd 6
2646.2.f.r.883.1 8 63.47 even 6
2646.2.f.r.883.4 8 63.2 odd 6
2646.2.f.r.1765.1 8 21.5 even 6
2646.2.f.r.1765.4 8 21.2 odd 6
2646.2.h.t.361.1 8 21.11 odd 6
2646.2.h.t.361.4 8 21.17 even 6
2646.2.h.t.667.1 8 9.2 odd 6
2646.2.h.t.667.4 8 63.20 even 6
7938.2.a.ci.1.1 4 63.23 odd 6
7938.2.a.ci.1.4 4 63.5 even 6
7938.2.a.cp.1.1 4 63.40 odd 6
7938.2.a.cp.1.4 4 63.58 even 3