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Properties

Label 2520.2.bi.q.1801.1
Level $2520$
Weight $2$
Character 2520.1801
Analytic conductor $20.122$
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] = [2520,2,Mod(361,2520)] 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("2520.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2520, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.bi (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,3,0,6,0,0,0,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11337408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1801.1
Root \(-0.391571i\) of defining polynomial
Character \(\chi\) \(=\) 2520.1801
Dual form 2520.2.bi.q.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-0.292113 - 2.62958i) q^{7} +(0.839111 + 1.45338i) q^{11} -4.84667 q^{13} +(1.00000 + 1.73205i) q^{17} +(-3.42334 + 5.92939i) q^{19} +(-1.13122 + 1.95934i) q^{23} +(-0.500000 - 0.866025i) q^{25} -3.32178 q^{29} +(-4.58423 - 7.94011i) q^{31} +(-2.42334 - 1.06181i) q^{35} +(1.42334 - 2.46529i) q^{37} -9.52489 q^{41} +6.58423 q^{43} +(-6.10156 + 10.5682i) q^{47} +(-6.82934 + 1.53627i) q^{49} +(3.74511 + 6.48673i) q^{53} +1.67822 q^{55} +(4.00000 + 6.92820i) q^{59} +(3.24511 - 5.62070i) q^{61} +(-2.42334 + 4.19734i) q^{65} +(2.87634 + 4.98196i) q^{67} +(5.84667 + 10.1267i) q^{73} +(3.57666 - 2.63106i) q^{77} +(-2.84667 + 4.93058i) q^{79} +12.5842 q^{83} +2.00000 q^{85} +(-2.92334 + 5.06337i) q^{89} +(1.41577 + 12.7447i) q^{91} +(3.42334 + 5.92939i) q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 6 q^{7} + 3 q^{11} + 6 q^{13} + 6 q^{17} - 3 q^{19} + 3 q^{23} - 3 q^{25} - 24 q^{29} - 12 q^{31} + 3 q^{35} - 9 q^{37} - 18 q^{41} + 24 q^{43} - 15 q^{47} - 12 q^{49} + 9 q^{53} + 6 q^{55}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 0 0
\(7\) −0.292113 − 2.62958i −0.110408 − 0.993886i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.839111 + 1.45338i 0.253001 + 0.438211i 0.964351 − 0.264627i \(-0.0852490\pi\)
−0.711349 + 0.702839i \(0.751916\pi\)
\(12\) 0 0
\(13\) −4.84667 −1.34422 −0.672112 − 0.740449i \(-0.734613\pi\)
−0.672112 + 0.740449i \(0.734613\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 + 1.73205i 0.242536 + 0.420084i 0.961436 − 0.275029i \(-0.0886875\pi\)
−0.718900 + 0.695113i \(0.755354\pi\)
\(18\) 0 0
\(19\) −3.42334 + 5.92939i −0.785367 + 1.36030i 0.143412 + 0.989663i \(0.454192\pi\)
−0.928779 + 0.370633i \(0.879141\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.13122 + 1.95934i −0.235876 + 0.408550i −0.959527 − 0.281617i \(-0.909129\pi\)
0.723651 + 0.690166i \(0.242463\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.32178 −0.616839 −0.308419 − 0.951250i \(-0.599800\pi\)
−0.308419 + 0.951250i \(0.599800\pi\)
\(30\) 0 0
\(31\) −4.58423 − 7.94011i −0.823351 − 1.42609i −0.903173 − 0.429277i \(-0.858768\pi\)
0.0798217 − 0.996809i \(-0.474565\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.42334 − 1.06181i −0.409619 − 0.179479i
\(36\) 0 0
\(37\) 1.42334 − 2.46529i 0.233995 − 0.405291i −0.724985 − 0.688765i \(-0.758153\pi\)
0.958980 + 0.283473i \(0.0914867\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.52489 −1.48754 −0.743769 − 0.668437i \(-0.766964\pi\)
−0.743769 + 0.668437i \(0.766964\pi\)
\(42\) 0 0
\(43\) 6.58423 1.00408 0.502042 − 0.864843i \(-0.332582\pi\)
0.502042 + 0.864843i \(0.332582\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.10156 + 10.5682i −0.890004 + 1.54153i −0.0501344 + 0.998742i \(0.515965\pi\)
−0.839869 + 0.542789i \(0.817368\pi\)
\(48\) 0 0
\(49\) −6.82934 + 1.53627i −0.975620 + 0.219466i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.74511 + 6.48673i 0.514431 + 0.891021i 0.999860 + 0.0167445i \(0.00533020\pi\)
−0.485429 + 0.874276i \(0.661336\pi\)
\(54\) 0 0
\(55\) 1.67822 0.226291
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 + 6.92820i 0.520756 + 0.901975i 0.999709 + 0.0241347i \(0.00768307\pi\)
−0.478953 + 0.877841i \(0.658984\pi\)
\(60\) 0 0
\(61\) 3.24511 − 5.62070i 0.415494 − 0.719657i −0.579986 − 0.814627i \(-0.696942\pi\)
0.995480 + 0.0949692i \(0.0302753\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.42334 + 4.19734i −0.300578 + 0.520616i
\(66\) 0 0
\(67\) 2.87634 + 4.98196i 0.351401 + 0.608644i 0.986495 − 0.163791i \(-0.0523722\pi\)
−0.635094 + 0.772434i \(0.719039\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 5.84667 + 10.1267i 0.684301 + 1.18524i 0.973656 + 0.228022i \(0.0732260\pi\)
−0.289355 + 0.957222i \(0.593441\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.57666 − 2.63106i 0.407599 − 0.299837i
\(78\) 0 0
\(79\) −2.84667 + 4.93058i −0.320276 + 0.554734i −0.980545 − 0.196295i \(-0.937109\pi\)
0.660269 + 0.751029i \(0.270442\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.5842 1.38130 0.690649 − 0.723190i \(-0.257325\pi\)
0.690649 + 0.723190i \(0.257325\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.92334 + 5.06337i −0.309873 + 0.536716i −0.978334 − 0.207031i \(-0.933620\pi\)
0.668461 + 0.743747i \(0.266953\pi\)
\(90\) 0 0
\(91\) 1.41577 + 12.7447i 0.148414 + 1.33601i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.42334 + 5.92939i 0.351227 + 0.608343i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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 2520.2.bi.q.1801.1 6
3.2 odd 2 280.2.q.e.121.1 yes 6
7.4 even 3 inner 2520.2.bi.q.361.1 6
12.11 even 2 560.2.q.l.401.3 6
15.2 even 4 1400.2.bh.i.849.2 12
15.8 even 4 1400.2.bh.i.849.5 12
15.14 odd 2 1400.2.q.j.401.3 6
21.2 odd 6 1960.2.a.w.1.3 3
21.5 even 6 1960.2.a.v.1.1 3
21.11 odd 6 280.2.q.e.81.1 ✓ 6
21.17 even 6 1960.2.q.w.361.3 6
21.20 even 2 1960.2.q.w.961.3 6
84.11 even 6 560.2.q.l.81.3 6
84.23 even 6 3920.2.a.cc.1.1 3
84.47 odd 6 3920.2.a.cb.1.3 3
105.32 even 12 1400.2.bh.i.249.5 12
105.44 odd 6 9800.2.a.ce.1.1 3
105.53 even 12 1400.2.bh.i.249.2 12
105.74 odd 6 1400.2.q.j.1201.3 6
105.89 even 6 9800.2.a.cf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.e.81.1 ✓ 6 21.11 odd 6
280.2.q.e.121.1 yes 6 3.2 odd 2
560.2.q.l.81.3 6 84.11 even 6
560.2.q.l.401.3 6 12.11 even 2
1400.2.q.j.401.3 6 15.14 odd 2
1400.2.q.j.1201.3 6 105.74 odd 6
1400.2.bh.i.249.2 12 105.53 even 12
1400.2.bh.i.249.5 12 105.32 even 12
1400.2.bh.i.849.2 12 15.2 even 4
1400.2.bh.i.849.5 12 15.8 even 4
1960.2.a.v.1.1 3 21.5 even 6
1960.2.a.w.1.3 3 21.2 odd 6
1960.2.q.w.361.3 6 21.17 even 6
1960.2.q.w.961.3 6 21.20 even 2
2520.2.bi.q.361.1 6 7.4 even 3 inner
2520.2.bi.q.1801.1 6 1.1 even 1 trivial
3920.2.a.cb.1.3 3 84.47 odd 6
3920.2.a.cc.1.1 3 84.23 even 6
9800.2.a.ce.1.1 3 105.44 odd 6
9800.2.a.cf.1.3 3 105.89 even 6