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Properties

Label 9300.2.g.r.3349.2
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,-6,0,12,0,0,0,0,0,0,0,-24] 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: 6.0.932935936.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 32x^{4} + 256x^{2} + 324 \) 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 3349.2
Root \(1.24586i\) of defining polynomial
Character \(\chi\) \(=\) 9300.3349
Dual form 9300.2.g.r.3349.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.24586i q^{7} -1.00000 q^{9} +2.00000 q^{11} +1.24586i q^{13} -5.95610i q^{17} -4.00000 q^{19} +1.24586 q^{21} -5.95610i q^{23} +1.00000i q^{27} -9.20197 q^{29} -1.00000 q^{31} -2.00000i q^{33} -3.24586i q^{37} +1.24586 q^{39} +4.00000 q^{41} -2.00000i q^{43} +10.4478i q^{47} +5.44783 q^{49} -5.95610 q^{51} +8.44783i q^{53} +4.00000i q^{57} +3.20197 q^{59} +13.9122 q^{61} -1.24586i q^{63} -12.6663i q^{67} -5.95610 q^{69} -11.6937 q^{71} +1.73759i q^{73} +2.49172i q^{77} -5.46438 q^{79} +1.00000 q^{81} +9.95610i q^{83} +9.20197i q^{87} -16.7102 q^{89} -1.55217 q^{91} +1.00000i q^{93} +15.9122i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + 12 q^{11} - 24 q^{19} - 8 q^{29} - 6 q^{31} + 24 q^{41} - 22 q^{49} + 4 q^{51} - 28 q^{59} + 4 q^{61} + 4 q^{69} - 8 q^{71} - 8 q^{79} + 6 q^{81} - 68 q^{89} - 64 q^{91} - 12 q^{99}+O(q^{100}) \) 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
\(7\) 1.24586i 0.470892i 0.971887 + 0.235446i \(0.0756550\pi\)
−0.971887 + 0.235446i \(0.924345\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 1.24586i 0.345540i 0.984962 + 0.172770i \(0.0552717\pi\)
−0.984962 + 0.172770i \(0.944728\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.95610i − 1.44457i −0.691597 − 0.722284i \(-0.743093\pi\)
0.691597 − 0.722284i \(-0.256907\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 1.24586 0.271869
\(22\) 0 0
\(23\) − 5.95610i − 1.24193i −0.783837 − 0.620967i \(-0.786740\pi\)
0.783837 − 0.620967i \(-0.213260\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −9.20197 −1.70876 −0.854381 − 0.519647i \(-0.826063\pi\)
−0.854381 + 0.519647i \(0.826063\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) − 2.00000i − 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.24586i − 0.533616i −0.963750 − 0.266808i \(-0.914031\pi\)
0.963750 − 0.266808i \(-0.0859690\pi\)
\(38\) 0 0
\(39\) 1.24586 0.199498
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 − 0.152499i \(-0.951268\pi\)
0.988304 − 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.4478i 1.52397i 0.647593 + 0.761986i \(0.275776\pi\)
−0.647593 + 0.761986i \(0.724224\pi\)
\(48\) 0 0
\(49\) 5.44783 0.778261
\(50\) 0 0
\(51\) −5.95610 −0.834021
\(52\) 0 0
\(53\) 8.44783i 1.16040i 0.814475 + 0.580199i \(0.197025\pi\)
−0.814475 + 0.580199i \(0.802975\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 3.20197 0.416860 0.208430 − 0.978037i \(-0.433165\pi\)
0.208430 + 0.978037i \(0.433165\pi\)
\(60\) 0 0
\(61\) 13.9122 1.78128 0.890638 − 0.454713i \(-0.150258\pi\)
0.890638 + 0.454713i \(0.150258\pi\)
\(62\) 0 0
\(63\) − 1.24586i − 0.156964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.6663i − 1.54744i −0.633527 − 0.773720i \(-0.718394\pi\)
0.633527 − 0.773720i \(-0.281606\pi\)
\(68\) 0 0
\(69\) −5.95610 −0.717031
\(70\) 0 0
\(71\) −11.6937 −1.38779 −0.693893 − 0.720078i \(-0.744106\pi\)
−0.693893 + 0.720078i \(0.744106\pi\)
\(72\) 0 0
\(73\) 1.73759i 0.203369i 0.994817 + 0.101685i \(0.0324232\pi\)
−0.994817 + 0.101685i \(0.967577\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.49172i 0.283958i
\(78\) 0 0
\(79\) −5.46438 −0.614791 −0.307395 − 0.951582i \(-0.599457\pi\)
−0.307395 + 0.951582i \(0.599457\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.95610i 1.09282i 0.837516 + 0.546412i \(0.184007\pi\)
−0.837516 + 0.546412i \(0.815993\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.20197i 0.986554i
\(88\) 0 0
\(89\) −16.7102 −1.77128 −0.885641 − 0.464370i \(-0.846281\pi\)
−0.885641 + 0.464370i \(0.846281\pi\)
\(90\) 0 0
\(91\) −1.55217 −0.162712
\(92\) 0 0
\(93\) 1.00000i 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.9122i 1.61564i 0.589429 + 0.807820i \(0.299353\pi\)
−0.589429 + 0.807820i \(0.700647\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
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.r.3349.2 6
5.2 odd 4 9300.2.a.t.1.2 3
5.3 odd 4 1860.2.a.h.1.2 ✓ 3
5.4 even 2 inner 9300.2.g.r.3349.5 6
15.8 even 4 5580.2.a.i.1.2 3
20.3 even 4 7440.2.a.bq.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.h.1.2 ✓ 3 5.3 odd 4
5580.2.a.i.1.2 3 15.8 even 4
7440.2.a.bq.1.2 3 20.3 even 4
9300.2.a.t.1.2 3 5.2 odd 4
9300.2.g.r.3349.2 6 1.1 even 1 trivial
9300.2.g.r.3349.5 6 5.4 even 2 inner