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Properties

Label 675.4.b.n.649.1
Level $675$
Weight $4$
Character 675.649
Analytic conductor $39.826$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [675,4,Mod(649,675)] 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("675.649"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-34,0,0,0,0,0,0,10,0,0,-120,0,322,0,0,100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12559936.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{3} + 49x^{2} - 42x + 18 \) 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 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-2.05655 - 2.05655i\) of defining polynomial
Character \(\chi\) \(=\) 675.649
Dual form 675.4.b.n.649.6

$q$-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)
 
\(f(q)\) \(=\) \(q-5.45876i q^{2} -21.7980 q^{4} +11.8065i q^{7} +75.3201i q^{8} -56.2376 q^{11} +34.5961i q^{13} +64.4489 q^{14} +236.770 q^{16} -39.2675i q^{17} +146.561 q^{19} +306.987i q^{22} -23.5777i q^{23} +188.851 q^{26} -257.359i q^{28} -161.003 q^{29} -29.5465 q^{31} -689.908i q^{32} -214.352 q^{34} +217.688i q^{37} -800.039i q^{38} +142.290 q^{41} -468.030i q^{43} +1225.87 q^{44} -128.705 q^{46} -394.318i q^{47} +203.606 q^{49} -754.126i q^{52} -134.780i q^{53} -889.268 q^{56} +878.875i q^{58} +131.195 q^{59} +259.801 q^{61} +161.287i q^{62} -1871.88 q^{64} -445.244i q^{67} +855.954i q^{68} +560.841 q^{71} -88.6681i q^{73} +1188.30 q^{74} -3194.74 q^{76} -663.970i q^{77} -450.342 q^{79} -776.726i q^{82} +284.295i q^{83} -2554.86 q^{86} -4235.82i q^{88} -625.305 q^{89} -408.459 q^{91} +513.948i q^{92} -2152.49 q^{94} +193.261i q^{97} -1111.44i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 34 q^{4} + 10 q^{11} - 120 q^{14} + 322 q^{16} + 100 q^{19} + 370 q^{26} - 230 q^{29} - 230 q^{31} - 826 q^{34} + 1160 q^{41} + 2830 q^{44} - 570 q^{46} - 1154 q^{49} - 4380 q^{56} - 760 q^{59} - 304 q^{61}+ \cdots - 7666 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.45876i − 1.92996i −0.262320 − 0.964981i \(-0.584487\pi\)
0.262320 − 0.964981i \(-0.415513\pi\)
\(3\) 0 0
\(4\) −21.7980 −2.72475
\(5\) 0 0
\(6\) 0 0
\(7\) 11.8065i 0.637492i 0.947840 + 0.318746i \(0.103262\pi\)
−0.947840 + 0.318746i \(0.896738\pi\)
\(8\) 75.3201i 3.32871i
\(9\) 0 0
\(10\) 0 0
\(11\) −56.2376 −1.54148 −0.770740 − 0.637150i \(-0.780113\pi\)
−0.770740 + 0.637150i \(0.780113\pi\)
\(12\) 0 0
\(13\) 34.5961i 0.738094i 0.929411 + 0.369047i \(0.120316\pi\)
−0.929411 + 0.369047i \(0.879684\pi\)
\(14\) 64.4489 1.23034
\(15\) 0 0
\(16\) 236.770 3.69953
\(17\) − 39.2675i − 0.560221i −0.959968 − 0.280111i \(-0.909629\pi\)
0.959968 − 0.280111i \(-0.0903712\pi\)
\(18\) 0 0
\(19\) 146.561 1.76965 0.884825 − 0.465924i \(-0.154278\pi\)
0.884825 + 0.465924i \(0.154278\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 306.987i 2.97500i
\(23\) − 23.5777i − 0.213752i −0.994272 − 0.106876i \(-0.965915\pi\)
0.994272 − 0.106876i \(-0.0340848\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 188.851 1.42449
\(27\) 0 0
\(28\) − 257.359i − 1.73701i
\(29\) −161.003 −1.03095 −0.515473 − 0.856906i \(-0.672384\pi\)
−0.515473 + 0.856906i \(0.672384\pi\)
\(30\) 0 0
\(31\) −29.5465 −0.171184 −0.0855921 − 0.996330i \(-0.527278\pi\)
−0.0855921 + 0.996330i \(0.527278\pi\)
\(32\) − 689.908i − 3.81124i
\(33\) 0 0
\(34\) −214.352 −1.08121
\(35\) 0 0
\(36\) 0 0
\(37\) 217.688i 0.967233i 0.875280 + 0.483617i \(0.160677\pi\)
−0.875280 + 0.483617i \(0.839323\pi\)
\(38\) − 800.039i − 3.41536i
\(39\) 0 0
\(40\) 0 0
\(41\) 142.290 0.541999 0.270999 − 0.962580i \(-0.412646\pi\)
0.270999 + 0.962580i \(0.412646\pi\)
\(42\) 0 0
\(43\) − 468.030i − 1.65986i −0.557869 − 0.829929i \(-0.688381\pi\)
0.557869 − 0.829929i \(-0.311619\pi\)
\(44\) 1225.87 4.20015
\(45\) 0 0
\(46\) −128.705 −0.412533
\(47\) − 394.318i − 1.22377i −0.790946 − 0.611886i \(-0.790411\pi\)
0.790946 − 0.611886i \(-0.209589\pi\)
\(48\) 0 0
\(49\) 203.606 0.593604
\(50\) 0 0
\(51\) 0 0
\(52\) − 754.126i − 2.01112i
\(53\) − 134.780i − 0.349311i −0.984630 − 0.174655i \(-0.944119\pi\)
0.984630 − 0.174655i \(-0.0558811\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −889.268 −2.12203
\(57\) 0 0
\(58\) 878.875i 1.98969i
\(59\) 131.195 0.289495 0.144747 − 0.989469i \(-0.453763\pi\)
0.144747 + 0.989469i \(0.453763\pi\)
\(60\) 0 0
\(61\) 259.801 0.545313 0.272657 − 0.962111i \(-0.412098\pi\)
0.272657 + 0.962111i \(0.412098\pi\)
\(62\) 161.287i 0.330379i
\(63\) 0 0
\(64\) −1871.88 −3.65602
\(65\) 0 0
\(66\) 0 0
\(67\) − 445.244i − 0.811869i −0.913902 − 0.405935i \(-0.866946\pi\)
0.913902 − 0.405935i \(-0.133054\pi\)
\(68\) 855.954i 1.52647i
\(69\) 0 0
\(70\) 0 0
\(71\) 560.841 0.937459 0.468729 − 0.883342i \(-0.344712\pi\)
0.468729 + 0.883342i \(0.344712\pi\)
\(72\) 0 0
\(73\) − 88.6681i − 0.142162i −0.997471 − 0.0710809i \(-0.977355\pi\)
0.997471 − 0.0710809i \(-0.0226448\pi\)
\(74\) 1188.30 1.86672
\(75\) 0 0
\(76\) −3194.74 −4.82186
\(77\) − 663.970i − 0.982681i
\(78\) 0 0
\(79\) −450.342 −0.641360 −0.320680 − 0.947188i \(-0.603911\pi\)
−0.320680 + 0.947188i \(0.603911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 776.726i − 1.04604i
\(83\) 284.295i 0.375969i 0.982172 + 0.187985i \(0.0601955\pi\)
−0.982172 + 0.187985i \(0.939804\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2554.86 −3.20346
\(87\) 0 0
\(88\) − 4235.82i − 5.13114i
\(89\) −625.305 −0.744744 −0.372372 − 0.928083i \(-0.621455\pi\)
−0.372372 + 0.928083i \(0.621455\pi\)
\(90\) 0 0
\(91\) −408.459 −0.470529
\(92\) 513.948i 0.582421i
\(93\) 0 0
\(94\) −2152.49 −2.36183
\(95\) 0 0
\(96\) 0 0
\(97\) 193.261i 0.202296i 0.994871 + 0.101148i \(0.0322516\pi\)
−0.994871 + 0.101148i \(0.967748\pi\)
\(98\) − 1111.44i − 1.14563i
\(99\) 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 675.4.b.n.649.1 6
3.2 odd 2 675.4.b.m.649.6 6
5.2 odd 4 135.4.a.h.1.3 yes 3
5.3 odd 4 675.4.a.p.1.1 3
5.4 even 2 inner 675.4.b.n.649.6 6
15.2 even 4 135.4.a.e.1.1 ✓ 3
15.8 even 4 675.4.a.s.1.3 3
15.14 odd 2 675.4.b.m.649.1 6
20.7 even 4 2160.4.a.bq.1.2 3
45.2 even 12 405.4.e.v.271.3 6
45.7 odd 12 405.4.e.q.271.1 6
45.22 odd 12 405.4.e.q.136.1 6
45.32 even 12 405.4.e.v.136.3 6
60.47 odd 4 2160.4.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.e.1.1 ✓ 3 15.2 even 4
135.4.a.h.1.3 yes 3 5.2 odd 4
405.4.e.q.136.1 6 45.22 odd 12
405.4.e.q.271.1 6 45.7 odd 12
405.4.e.v.136.3 6 45.32 even 12
405.4.e.v.271.3 6 45.2 even 12
675.4.a.p.1.1 3 5.3 odd 4
675.4.a.s.1.3 3 15.8 even 4
675.4.b.m.649.1 6 15.14 odd 2
675.4.b.m.649.6 6 3.2 odd 2
675.4.b.n.649.1 6 1.1 even 1 trivial
675.4.b.n.649.6 6 5.4 even 2 inner
2160.4.a.bi.1.2 3 60.47 odd 4
2160.4.a.bq.1.2 3 20.7 even 4