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Properties

Label 9680.2.a.dd.1.5
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $6$
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] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,3,0,-6,0,7,0,5,0,0,0,1,0,-3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.25903625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.29082\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.29082 q^{3} -1.00000 q^{5} +4.66366 q^{7} +2.24785 q^{9} -5.94269 q^{13} -2.29082 q^{15} +3.30331 q^{17} -4.19195 q^{19} +10.6836 q^{21} +1.71184 q^{23} +1.00000 q^{25} -1.72304 q^{27} +7.76702 q^{29} +2.97204 q^{31} -4.66366 q^{35} -5.68500 q^{37} -13.6136 q^{39} +6.14758 q^{41} -4.42353 q^{43} -2.24785 q^{45} -4.13243 q^{47} +14.7497 q^{49} +7.56730 q^{51} +8.10104 q^{53} -9.60300 q^{57} +6.21515 q^{59} +1.52758 q^{61} +10.4832 q^{63} +5.94269 q^{65} +7.46447 q^{67} +3.92151 q^{69} +2.23747 q^{71} +7.49400 q^{73} +2.29082 q^{75} +17.5316 q^{79} -10.6907 q^{81} +11.0785 q^{83} -3.30331 q^{85} +17.7928 q^{87} +12.8171 q^{89} -27.7147 q^{91} +6.80840 q^{93} +4.19195 q^{95} -9.53760 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} + q^{13} - 3 q^{15} + 6 q^{17} + 7 q^{19} + 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} + 10 q^{29} - q^{31} - 7 q^{35} + 3 q^{37} + q^{39} - 6 q^{41} - 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.29082 1.32260 0.661302 − 0.750119i \(-0.270004\pi\)
0.661302 + 0.750119i \(0.270004\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.66366 1.76270 0.881348 − 0.472467i \(-0.156637\pi\)
0.881348 + 0.472467i \(0.156637\pi\)
\(8\) 0 0
\(9\) 2.24785 0.749284
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.94269 −1.64821 −0.824103 − 0.566440i \(-0.808320\pi\)
−0.824103 + 0.566440i \(0.808320\pi\)
\(14\) 0 0
\(15\) −2.29082 −0.591487
\(16\) 0 0
\(17\) 3.30331 0.801171 0.400586 − 0.916259i \(-0.368807\pi\)
0.400586 + 0.916259i \(0.368807\pi\)
\(18\) 0 0
\(19\) −4.19195 −0.961699 −0.480850 − 0.876803i \(-0.659672\pi\)
−0.480850 + 0.876803i \(0.659672\pi\)
\(20\) 0 0
\(21\) 10.6836 2.33135
\(22\) 0 0
\(23\) 1.71184 0.356942 0.178471 − 0.983945i \(-0.442885\pi\)
0.178471 + 0.983945i \(0.442885\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.72304 −0.331598
\(28\) 0 0
\(29\) 7.76702 1.44230 0.721149 − 0.692780i \(-0.243614\pi\)
0.721149 + 0.692780i \(0.243614\pi\)
\(30\) 0 0
\(31\) 2.97204 0.533794 0.266897 − 0.963725i \(-0.414002\pi\)
0.266897 + 0.963725i \(0.414002\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.66366 −0.788302
\(36\) 0 0
\(37\) −5.68500 −0.934608 −0.467304 − 0.884097i \(-0.654775\pi\)
−0.467304 + 0.884097i \(0.654775\pi\)
\(38\) 0 0
\(39\) −13.6136 −2.17993
\(40\) 0 0
\(41\) 6.14758 0.960090 0.480045 − 0.877244i \(-0.340620\pi\)
0.480045 + 0.877244i \(0.340620\pi\)
\(42\) 0 0
\(43\) −4.42353 −0.674582 −0.337291 − 0.941401i \(-0.609511\pi\)
−0.337291 + 0.941401i \(0.609511\pi\)
\(44\) 0 0
\(45\) −2.24785 −0.335090
\(46\) 0 0
\(47\) −4.13243 −0.602776 −0.301388 − 0.953502i \(-0.597450\pi\)
−0.301388 + 0.953502i \(0.597450\pi\)
\(48\) 0 0
\(49\) 14.7497 2.10710
\(50\) 0 0
\(51\) 7.56730 1.05963
\(52\) 0 0
\(53\) 8.10104 1.11276 0.556381 − 0.830927i \(-0.312189\pi\)
0.556381 + 0.830927i \(0.312189\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −9.60300 −1.27195
\(58\) 0 0
\(59\) 6.21515 0.809144 0.404572 − 0.914506i \(-0.367420\pi\)
0.404572 + 0.914506i \(0.367420\pi\)
\(60\) 0 0
\(61\) 1.52758 0.195587 0.0977934 − 0.995207i \(-0.468822\pi\)
0.0977934 + 0.995207i \(0.468822\pi\)
\(62\) 0 0
\(63\) 10.4832 1.32076
\(64\) 0 0
\(65\) 5.94269 0.737100
\(66\) 0 0
\(67\) 7.46447 0.911931 0.455965 − 0.889998i \(-0.349294\pi\)
0.455965 + 0.889998i \(0.349294\pi\)
\(68\) 0 0
\(69\) 3.92151 0.472094
\(70\) 0 0
\(71\) 2.23747 0.265539 0.132770 − 0.991147i \(-0.457613\pi\)
0.132770 + 0.991147i \(0.457613\pi\)
\(72\) 0 0
\(73\) 7.49400 0.877106 0.438553 − 0.898705i \(-0.355491\pi\)
0.438553 + 0.898705i \(0.355491\pi\)
\(74\) 0 0
\(75\) 2.29082 0.264521
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 17.5316 1.97246 0.986230 − 0.165378i \(-0.0528845\pi\)
0.986230 + 0.165378i \(0.0528845\pi\)
\(80\) 0 0
\(81\) −10.6907 −1.18786
\(82\) 0 0
\(83\) 11.0785 1.21603 0.608014 − 0.793926i \(-0.291966\pi\)
0.608014 + 0.793926i \(0.291966\pi\)
\(84\) 0 0
\(85\) −3.30331 −0.358295
\(86\) 0 0
\(87\) 17.7928 1.90759
\(88\) 0 0
\(89\) 12.8171 1.35861 0.679305 − 0.733856i \(-0.262281\pi\)
0.679305 + 0.733856i \(0.262281\pi\)
\(90\) 0 0
\(91\) −27.7147 −2.90529
\(92\) 0 0
\(93\) 6.80840 0.705998
\(94\) 0 0
\(95\) 4.19195 0.430085
\(96\) 0 0
\(97\) −9.53760 −0.968396 −0.484198 − 0.874958i \(-0.660889\pi\)
−0.484198 + 0.874958i \(0.660889\pi\)
\(98\) 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 9680.2.a.dd.1.5 6
4.3 odd 2 4840.2.a.ba.1.2 6
11.7 odd 10 880.2.bo.i.401.1 12
11.8 odd 10 880.2.bo.i.801.1 12
11.10 odd 2 9680.2.a.dc.1.5 6
44.7 even 10 440.2.y.c.401.3 yes 12
44.19 even 10 440.2.y.c.361.3 ✓ 12
44.43 even 2 4840.2.a.bb.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.361.3 ✓ 12 44.19 even 10
440.2.y.c.401.3 yes 12 44.7 even 10
880.2.bo.i.401.1 12 11.7 odd 10
880.2.bo.i.801.1 12 11.8 odd 10
4840.2.a.ba.1.2 6 4.3 odd 2
4840.2.a.bb.1.2 6 44.43 even 2
9680.2.a.dc.1.5 6 11.10 odd 2
9680.2.a.dd.1.5 6 1.1 even 1 trivial