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Properties

Label 612.2.k.e.217.1
Level $612$
Weight $2$
Character 612.217
Analytic conductor $4.887$
Analytic rank $0$
Dimension $4$
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] = [612,2,Mod(217,612)] 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("612.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(612, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 612.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.88684460370\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 217.1
Root \(-2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 612.217
Dual form 612.2.k.e.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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.00000i) q^{5} +(-1.30278 + 1.30278i) q^{7} +(-3.30278 + 3.30278i) q^{11} -4.60555 q^{13} +(3.60555 + 2.00000i) q^{17} +6.60555i q^{19} +(0.697224 - 0.697224i) q^{23} -3.00000i q^{25} +(-5.60555 - 5.60555i) q^{29} +(0.697224 + 0.697224i) q^{31} -2.60555 q^{35} +(3.00000 + 3.00000i) q^{37} +(-1.00000 + 1.00000i) q^{41} +10.6056i q^{43} +4.00000 q^{47} +3.60555i q^{49} +9.21110i q^{53} -6.60555 q^{55} -1.39445i q^{59} +(8.21110 - 8.21110i) q^{61} +(-4.60555 - 4.60555i) q^{65} -5.21110 q^{67} +(7.90833 + 7.90833i) q^{71} +(-7.00000 - 7.00000i) q^{73} -8.60555i q^{77} +(3.30278 - 3.30278i) q^{79} -3.81665i q^{83} +(1.60555 + 5.60555i) q^{85} +13.8167 q^{89} +(6.00000 - 6.00000i) q^{91} +(-6.60555 + 6.60555i) q^{95} +(-0.394449 - 0.394449i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 2 q^{7} - 6 q^{11} - 4 q^{13} + 10 q^{23} - 8 q^{29} + 10 q^{31} + 4 q^{35} + 12 q^{37} - 4 q^{41} + 16 q^{47} - 12 q^{55} + 4 q^{61} - 4 q^{65} + 8 q^{67} + 10 q^{71} - 28 q^{73} + 6 q^{79}+ \cdots - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/612\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\) \(307\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(5\) 1.00000 + 1.00000i 0.447214 + 0.447214i 0.894427 − 0.447214i \(-0.147584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.30278 + 1.30278i −0.492403 + 0.492403i −0.909063 − 0.416660i \(-0.863201\pi\)
0.416660 + 0.909063i \(0.363201\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.30278 + 3.30278i −0.995824 + 0.995824i −0.999991 − 0.00416699i \(-0.998674\pi\)
0.00416699 + 0.999991i \(0.498674\pi\)
\(12\) 0 0
\(13\) −4.60555 −1.27735 −0.638675 − 0.769477i \(-0.720517\pi\)
−0.638675 + 0.769477i \(0.720517\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.60555 + 2.00000i 0.874475 + 0.485071i
\(18\) 0 0
\(19\) 6.60555i 1.51542i 0.652593 + 0.757709i \(0.273681\pi\)
−0.652593 + 0.757709i \(0.726319\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.697224 − 0.697224i 0.145381 − 0.145381i −0.630670 − 0.776051i \(-0.717220\pi\)
0.776051 + 0.630670i \(0.217220\pi\)
\(24\) 0 0
\(25\) − 3.00000i − 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.60555 − 5.60555i −1.04092 − 1.04092i −0.999126 − 0.0417987i \(-0.986691\pi\)
−0.0417987 − 0.999126i \(-0.513309\pi\)
\(30\) 0 0
\(31\) 0.697224 + 0.697224i 0.125225 + 0.125225i 0.766942 − 0.641717i \(-0.221777\pi\)
−0.641717 + 0.766942i \(0.721777\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.60555 −0.440419
\(36\) 0 0
\(37\) 3.00000 + 3.00000i 0.493197 + 0.493197i 0.909312 − 0.416115i \(-0.136609\pi\)
−0.416115 + 0.909312i \(0.636609\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 + 1.00000i −0.156174 + 0.156174i −0.780869 − 0.624695i \(-0.785223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 10.6056i 1.61733i 0.588268 + 0.808666i \(0.299810\pi\)
−0.588268 + 0.808666i \(0.700190\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 3.60555i 0.515079i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.21110i 1.26524i 0.774461 + 0.632621i \(0.218021\pi\)
−0.774461 + 0.632621i \(0.781979\pi\)
\(54\) 0 0
\(55\) −6.60555 −0.890692
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1.39445i − 0.181542i −0.995872 − 0.0907709i \(-0.971067\pi\)
0.995872 − 0.0907709i \(-0.0289331\pi\)
\(60\) 0 0
\(61\) 8.21110 − 8.21110i 1.05132 − 1.05132i 0.0527143 − 0.998610i \(-0.483213\pi\)
0.998610 − 0.0527143i \(-0.0167873\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.60555 − 4.60555i −0.571248 − 0.571248i
\(66\) 0 0
\(67\) −5.21110 −0.636638 −0.318319 − 0.947984i \(-0.603118\pi\)
−0.318319 + 0.947984i \(0.603118\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.90833 + 7.90833i 0.938546 + 0.938546i 0.998218 − 0.0596723i \(-0.0190056\pi\)
−0.0596723 + 0.998218i \(0.519006\pi\)
\(72\) 0 0
\(73\) −7.00000 − 7.00000i −0.819288 − 0.819288i 0.166717 − 0.986005i \(-0.446683\pi\)
−0.986005 + 0.166717i \(0.946683\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.60555i − 0.980694i
\(78\) 0 0
\(79\) 3.30278 − 3.30278i 0.371591 − 0.371591i −0.496465 − 0.868057i \(-0.665369\pi\)
0.868057 + 0.496465i \(0.165369\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 3.81665i − 0.418932i −0.977816 − 0.209466i \(-0.932827\pi\)
0.977816 − 0.209466i \(-0.0671726\pi\)
\(84\) 0 0
\(85\) 1.60555 + 5.60555i 0.174146 + 0.608007i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.8167 1.46456 0.732281 − 0.681002i \(-0.238456\pi\)
0.732281 + 0.681002i \(0.238456\pi\)
\(90\) 0 0
\(91\) 6.00000 − 6.00000i 0.628971 − 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.60555 + 6.60555i −0.677715 + 0.677715i
\(96\) 0 0
\(97\) −0.394449 − 0.394449i −0.0400502 − 0.0400502i 0.686798 − 0.726848i \(-0.259016\pi\)
−0.726848 + 0.686798i \(0.759016\pi\)
\(98\) 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 612.2.k.e.217.1 4
3.2 odd 2 68.2.e.a.13.2 ✓ 4
4.3 odd 2 2448.2.be.u.1441.2 4
12.11 even 2 272.2.o.g.81.1 4
15.2 even 4 1700.2.m.a.149.2 4
15.8 even 4 1700.2.m.b.149.1 4
15.14 odd 2 1700.2.o.c.1101.1 4
17.4 even 4 inner 612.2.k.e.361.1 4
24.5 odd 2 1088.2.o.t.897.1 4
24.11 even 2 1088.2.o.s.897.2 4
51.2 odd 8 1156.2.a.h.1.3 4
51.5 even 16 1156.2.h.e.733.2 16
51.8 odd 8 1156.2.b.a.577.2 4
51.11 even 16 1156.2.h.e.757.2 16
51.14 even 16 1156.2.h.e.1001.2 16
51.20 even 16 1156.2.h.e.1001.3 16
51.23 even 16 1156.2.h.e.757.3 16
51.26 odd 8 1156.2.b.a.577.3 4
51.29 even 16 1156.2.h.e.733.3 16
51.32 odd 8 1156.2.a.h.1.2 4
51.38 odd 4 68.2.e.a.21.2 yes 4
51.41 even 16 1156.2.h.e.977.3 16
51.44 even 16 1156.2.h.e.977.2 16
51.47 odd 4 1156.2.e.c.905.1 4
51.50 odd 2 1156.2.e.c.829.1 4
68.55 odd 4 2448.2.be.u.1585.2 4
204.83 even 8 4624.2.a.bq.1.3 4
204.155 even 8 4624.2.a.bq.1.2 4
204.191 even 4 272.2.o.g.225.1 4
255.38 even 4 1700.2.m.a.1449.2 4
255.89 odd 4 1700.2.o.c.701.1 4
255.242 even 4 1700.2.m.b.1449.1 4
408.293 odd 4 1088.2.o.t.769.1 4
408.395 even 4 1088.2.o.s.769.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.e.a.13.2 ✓ 4 3.2 odd 2
68.2.e.a.21.2 yes 4 51.38 odd 4
272.2.o.g.81.1 4 12.11 even 2
272.2.o.g.225.1 4 204.191 even 4
612.2.k.e.217.1 4 1.1 even 1 trivial
612.2.k.e.361.1 4 17.4 even 4 inner
1088.2.o.s.769.2 4 408.395 even 4
1088.2.o.s.897.2 4 24.11 even 2
1088.2.o.t.769.1 4 408.293 odd 4
1088.2.o.t.897.1 4 24.5 odd 2
1156.2.a.h.1.2 4 51.32 odd 8
1156.2.a.h.1.3 4 51.2 odd 8
1156.2.b.a.577.2 4 51.8 odd 8
1156.2.b.a.577.3 4 51.26 odd 8
1156.2.e.c.829.1 4 51.50 odd 2
1156.2.e.c.905.1 4 51.47 odd 4
1156.2.h.e.733.2 16 51.5 even 16
1156.2.h.e.733.3 16 51.29 even 16
1156.2.h.e.757.2 16 51.11 even 16
1156.2.h.e.757.3 16 51.23 even 16
1156.2.h.e.977.2 16 51.44 even 16
1156.2.h.e.977.3 16 51.41 even 16
1156.2.h.e.1001.2 16 51.14 even 16
1156.2.h.e.1001.3 16 51.20 even 16
1700.2.m.a.149.2 4 15.2 even 4
1700.2.m.a.1449.2 4 255.38 even 4
1700.2.m.b.149.1 4 15.8 even 4
1700.2.m.b.1449.1 4 255.242 even 4
1700.2.o.c.701.1 4 255.89 odd 4
1700.2.o.c.1101.1 4 15.14 odd 2
2448.2.be.u.1441.2 4 4.3 odd 2
2448.2.be.u.1585.2 4 68.55 odd 4
4624.2.a.bq.1.2 4 204.155 even 8
4624.2.a.bq.1.3 4 204.83 even 8