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Properties

Label 9300.2.g.q.3349.3
Level $9300$
Weight $2$
Character 9300.3349
Analytic conductor $74.261$
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] = [9300,2,Mod(3349,9300)] 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("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9300.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-6,0,4,0,0,0,0,0,0,0,0] 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: \(74.2608738798\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2611456.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{3} + 16x^{2} - 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1860)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.3
Root \(0.258652 + 0.258652i\) of defining polynomial
Character \(\chi\) \(=\) 9300.3349
Dual form 9300.2.g.q.3349.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.34889i q^{7} -1.00000 q^{9} +4.69779 q^{11} +2.38350i q^{13} -1.21509i q^{17} -1.03461 q^{19} +1.34889 q^{21} -6.24970i q^{23} +1.00000i q^{27} +5.59859 q^{29} +1.00000 q^{31} -4.69779i q^{33} +5.41811i q^{37} +2.38350 q^{39} +1.30221 q^{41} -5.73240i q^{43} +3.48270i q^{47} +5.18048 q^{49} -1.21509 q^{51} -2.51730i q^{53} +1.03461i q^{57} -7.86620 q^{59} -5.46479 q^{61} -1.34889i q^{63} -1.41811i q^{67} -6.24970 q^{69} +3.79698 q^{71} +1.41811i q^{73} +6.33682i q^{77} +1.81952 q^{79} +1.00000 q^{81} -13.2151i q^{83} -5.59859i q^{87} +2.20302 q^{89} -3.21509 q^{91} -1.00000i q^{93} +6.69779i q^{97} -4.69779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + 4 q^{11} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 4 q^{39} + 32 q^{41} + 10 q^{49} + 20 q^{51} - 32 q^{59} + 28 q^{61} - 4 q^{69} + 20 q^{71} + 32 q^{79} + 6 q^{81} + 16 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\) \(4651\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.34889i 0.509834i 0.966963 + 0.254917i \(0.0820482\pi\)
−0.966963 + 0.254917i \(0.917952\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.69779 1.41644 0.708218 − 0.705994i \(-0.249499\pi\)
0.708218 + 0.705994i \(0.249499\pi\)
\(12\) 0 0
\(13\) 2.38350i 0.661065i 0.943795 + 0.330532i \(0.107228\pi\)
−0.943795 + 0.330532i \(0.892772\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.21509i − 0.294703i −0.989084 − 0.147352i \(-0.952925\pi\)
0.989084 − 0.147352i \(-0.0470749\pi\)
\(18\) 0 0
\(19\) −1.03461 −0.237355 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(20\) 0 0
\(21\) 1.34889 0.294353
\(22\) 0 0
\(23\) − 6.24970i − 1.30315i −0.758583 − 0.651576i \(-0.774108\pi\)
0.758583 − 0.651576i \(-0.225892\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 5.59859 1.03963 0.519816 − 0.854278i \(-0.326000\pi\)
0.519816 + 0.854278i \(0.326000\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) − 4.69779i − 0.817780i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.41811i 0.890732i 0.895349 + 0.445366i \(0.146926\pi\)
−0.895349 + 0.445366i \(0.853074\pi\)
\(38\) 0 0
\(39\) 2.38350 0.381666
\(40\) 0 0
\(41\) 1.30221 0.203371 0.101686 − 0.994817i \(-0.467576\pi\)
0.101686 + 0.994817i \(0.467576\pi\)
\(42\) 0 0
\(43\) − 5.73240i − 0.874182i −0.899417 − 0.437091i \(-0.856009\pi\)
0.899417 − 0.437091i \(-0.143991\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.48270i 0.508003i 0.967204 + 0.254002i \(0.0817469\pi\)
−0.967204 + 0.254002i \(0.918253\pi\)
\(48\) 0 0
\(49\) 5.18048 0.740069
\(50\) 0 0
\(51\) −1.21509 −0.170147
\(52\) 0 0
\(53\) − 2.51730i − 0.345778i −0.984941 − 0.172889i \(-0.944690\pi\)
0.984941 − 0.172889i \(-0.0553102\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.03461i 0.137037i
\(58\) 0 0
\(59\) −7.86620 −1.02409 −0.512046 − 0.858958i \(-0.671112\pi\)
−0.512046 + 0.858958i \(0.671112\pi\)
\(60\) 0 0
\(61\) −5.46479 −0.699695 −0.349848 − 0.936807i \(-0.613767\pi\)
−0.349848 + 0.936807i \(0.613767\pi\)
\(62\) 0 0
\(63\) − 1.34889i − 0.169945i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.41811i − 0.173250i −0.996241 − 0.0866249i \(-0.972392\pi\)
0.996241 − 0.0866249i \(-0.0276082\pi\)
\(68\) 0 0
\(69\) −6.24970 −0.752376
\(70\) 0 0
\(71\) 3.79698 0.450619 0.225309 − 0.974287i \(-0.427661\pi\)
0.225309 + 0.974287i \(0.427661\pi\)
\(72\) 0 0
\(73\) 1.41811i 0.165977i 0.996550 + 0.0829886i \(0.0264465\pi\)
−0.996550 + 0.0829886i \(0.973553\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.33682i 0.722148i
\(78\) 0 0
\(79\) 1.81952 0.204711 0.102356 − 0.994748i \(-0.467362\pi\)
0.102356 + 0.994748i \(0.467362\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 13.2151i − 1.45054i −0.688462 − 0.725272i \(-0.741714\pi\)
0.688462 − 0.725272i \(-0.258286\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 5.59859i − 0.600232i
\(88\) 0 0
\(89\) 2.20302 0.233519 0.116760 − 0.993160i \(-0.462749\pi\)
0.116760 + 0.993160i \(0.462749\pi\)
\(90\) 0 0
\(91\) −3.21509 −0.337033
\(92\) 0 0
\(93\) − 1.00000i − 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.69779i 0.680057i 0.940415 + 0.340029i \(0.110437\pi\)
−0.940415 + 0.340029i \(0.889563\pi\)
\(98\) 0 0
\(99\) −4.69779 −0.472146
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 9300.2.g.q.3349.3 6
5.2 odd 4 9300.2.a.u.1.1 3
5.3 odd 4 1860.2.a.g.1.3 ✓ 3
5.4 even 2 inner 9300.2.g.q.3349.4 6
15.8 even 4 5580.2.a.j.1.3 3
20.3 even 4 7440.2.a.bn.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.g.1.3 ✓ 3 5.3 odd 4
5580.2.a.j.1.3 3 15.8 even 4
7440.2.a.bn.1.1 3 20.3 even 4
9300.2.a.u.1.1 3 5.2 odd 4
9300.2.g.q.3349.3 6 1.1 even 1 trivial
9300.2.g.q.3349.4 6 5.4 even 2 inner