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Properties

Label 2100.2.bo.f.1349.1
Level $2100$
Weight $2$
Character 2100.1349
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [2100,2,Mod(1349,2100)] 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("2100.1349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bo (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,12,0,18,0,0,0,0,0,0,0,-6] 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1349.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1349
Dual form 2100.2.bo.f.1949.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +(-1.73205 - 2.00000i) q^{7} +3.00000 q^{9} +(4.50000 - 2.59808i) q^{11} +(-2.59808 + 1.50000i) q^{17} +(-1.50000 - 0.866025i) q^{19} +(3.00000 + 3.46410i) q^{21} +(2.59808 - 4.50000i) q^{23} -5.19615 q^{27} +(-1.50000 + 0.866025i) q^{31} +(-7.79423 + 4.50000i) q^{33} +(6.06218 + 3.50000i) q^{37} +6.00000 q^{41} +4.00000i q^{43} +(2.59808 + 1.50000i) q^{47} +(-1.00000 + 6.92820i) q^{49} +(4.50000 - 2.59808i) q^{51} +(-2.59808 - 4.50000i) q^{53} +(2.59808 + 1.50000i) q^{57} +(-1.50000 - 2.59808i) q^{59} +(-10.5000 - 6.06218i) q^{61} +(-5.19615 - 6.00000i) q^{63} +(-4.33013 + 2.50000i) q^{67} +(-4.50000 + 7.79423i) q^{69} -10.3923i q^{71} +(-6.06218 - 10.5000i) q^{73} +(-12.9904 - 4.50000i) q^{77} +(-0.500000 + 0.866025i) q^{79} +9.00000 q^{81} -12.0000i q^{83} +(-4.50000 + 7.79423i) q^{89} +(2.59808 - 1.50000i) q^{93} -6.92820 q^{97} +(13.5000 - 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} + 18 q^{11} - 6 q^{19} + 12 q^{21} - 6 q^{31} + 24 q^{41} - 4 q^{49} + 18 q^{51} - 6 q^{59} - 42 q^{61} - 18 q^{69} - 2 q^{79} + 36 q^{81} - 18 q^{89} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −1.73205 −1.00000
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.73205 − 2.00000i −0.654654 − 0.755929i
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.50000 − 2.59808i 1.35680 − 0.783349i 0.367610 − 0.929980i \(-0.380176\pi\)
0.989191 + 0.146631i \(0.0468429\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.59808 + 1.50000i −0.630126 + 0.363803i −0.780801 − 0.624780i \(-0.785189\pi\)
0.150675 + 0.988583i \(0.451855\pi\)
\(18\) 0 0
\(19\) −1.50000 − 0.866025i −0.344124 − 0.198680i 0.317970 − 0.948101i \(-0.396999\pi\)
−0.662094 + 0.749421i \(0.730332\pi\)
\(20\) 0 0
\(21\) 3.00000 + 3.46410i 0.654654 + 0.755929i
\(22\) 0 0
\(23\) 2.59808 − 4.50000i 0.541736 − 0.938315i −0.457068 − 0.889432i \(-0.651100\pi\)
0.998805 − 0.0488832i \(-0.0155662\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.50000 + 0.866025i −0.269408 + 0.155543i −0.628619 − 0.777714i \(-0.716379\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −7.79423 + 4.50000i −1.35680 + 0.783349i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.06218 + 3.50000i 0.996616 + 0.575396i 0.907245 − 0.420602i \(-0.138181\pi\)
0.0893706 + 0.995998i \(0.471514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.59808 + 1.50000i 0.378968 + 0.218797i 0.677369 − 0.735643i \(-0.263120\pi\)
−0.298401 + 0.954441i \(0.596453\pi\)
\(48\) 0 0
\(49\) −1.00000 + 6.92820i −0.142857 + 0.989743i
\(50\) 0 0
\(51\) 4.50000 − 2.59808i 0.630126 − 0.363803i
\(52\) 0 0
\(53\) −2.59808 − 4.50000i −0.356873 − 0.618123i 0.630563 − 0.776138i \(-0.282824\pi\)
−0.987437 + 0.158015i \(0.949491\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.59808 + 1.50000i 0.344124 + 0.198680i
\(58\) 0 0
\(59\) −1.50000 − 2.59808i −0.195283 − 0.338241i 0.751710 − 0.659494i \(-0.229229\pi\)
−0.946993 + 0.321253i \(0.895896\pi\)
\(60\) 0 0
\(61\) −10.5000 − 6.06218i −1.34439 − 0.776182i −0.356939 − 0.934128i \(-0.616180\pi\)
−0.987448 + 0.157945i \(0.949513\pi\)
\(62\) 0 0
\(63\) −5.19615 − 6.00000i −0.654654 − 0.755929i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.33013 + 2.50000i −0.529009 + 0.305424i −0.740613 − 0.671932i \(-0.765465\pi\)
0.211604 + 0.977356i \(0.432131\pi\)
\(68\) 0 0
\(69\) −4.50000 + 7.79423i −0.541736 + 0.938315i
\(70\) 0 0
\(71\) − 10.3923i − 1.23334i −0.787222 − 0.616670i \(-0.788481\pi\)
0.787222 − 0.616670i \(-0.211519\pi\)
\(72\) 0 0
\(73\) −6.06218 − 10.5000i −0.709524 − 1.22893i −0.965034 − 0.262126i \(-0.915577\pi\)
0.255510 − 0.966807i \(-0.417757\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.9904 − 4.50000i −1.48039 − 0.512823i
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.0562544 + 0.0974355i −0.892781 − 0.450490i \(-0.851249\pi\)
0.836527 + 0.547926i \(0.184582\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.50000 + 7.79423i −0.476999 + 0.826187i −0.999653 − 0.0263586i \(-0.991609\pi\)
0.522654 + 0.852545i \(0.324942\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.59808 − 1.50000i 0.269408 − 0.155543i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.92820 −0.703452 −0.351726 − 0.936103i \(-0.614405\pi\)
−0.351726 + 0.936103i \(0.614405\pi\)
\(98\) 0 0
\(99\) 13.5000 − 7.79423i 1.35680 − 0.783349i
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 2100.2.bo.f.1349.1 4
3.2 odd 2 2100.2.bo.a.1349.2 4
5.2 odd 4 84.2.k.b.5.1 yes 2
5.3 odd 4 2100.2.bi.e.1601.1 2
5.4 even 2 inner 2100.2.bo.f.1349.2 4
7.3 odd 6 2100.2.bo.a.1949.1 4
15.2 even 4 84.2.k.a.5.1 ✓ 2
15.8 even 4 2100.2.bi.f.1601.1 2
15.14 odd 2 2100.2.bo.a.1349.1 4
20.7 even 4 336.2.bc.b.257.1 2
21.17 even 6 inner 2100.2.bo.f.1949.2 4
35.2 odd 12 588.2.f.a.293.1 2
35.3 even 12 2100.2.bi.f.101.1 2
35.12 even 12 588.2.f.c.293.2 2
35.17 even 12 84.2.k.a.17.1 yes 2
35.24 odd 6 2100.2.bo.a.1949.2 4
35.27 even 4 588.2.k.c.509.1 2
35.32 odd 12 588.2.k.d.521.1 2
45.2 even 12 2268.2.w.f.1349.1 2
45.7 odd 12 2268.2.w.a.1349.1 2
45.22 odd 12 2268.2.bm.f.593.1 2
45.32 even 12 2268.2.bm.a.593.1 2
60.47 odd 4 336.2.bc.d.257.1 2
105.2 even 12 588.2.f.c.293.1 2
105.17 odd 12 84.2.k.b.17.1 yes 2
105.32 even 12 588.2.k.c.521.1 2
105.38 odd 12 2100.2.bi.e.101.1 2
105.47 odd 12 588.2.f.a.293.2 2
105.59 even 6 inner 2100.2.bo.f.1949.1 4
105.62 odd 4 588.2.k.d.509.1 2
140.47 odd 12 2352.2.k.a.881.1 2
140.87 odd 12 336.2.bc.d.17.1 2
140.107 even 12 2352.2.k.d.881.2 2
315.52 even 12 2268.2.bm.a.1025.1 2
315.122 odd 12 2268.2.w.a.269.1 2
315.157 even 12 2268.2.w.f.269.1 2
315.227 odd 12 2268.2.bm.f.1025.1 2
420.47 even 12 2352.2.k.d.881.1 2
420.107 odd 12 2352.2.k.a.881.2 2
420.227 even 12 336.2.bc.b.17.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.k.a.5.1 ✓ 2 15.2 even 4
84.2.k.a.17.1 yes 2 35.17 even 12
84.2.k.b.5.1 yes 2 5.2 odd 4
84.2.k.b.17.1 yes 2 105.17 odd 12
336.2.bc.b.17.1 2 420.227 even 12
336.2.bc.b.257.1 2 20.7 even 4
336.2.bc.d.17.1 2 140.87 odd 12
336.2.bc.d.257.1 2 60.47 odd 4
588.2.f.a.293.1 2 35.2 odd 12
588.2.f.a.293.2 2 105.47 odd 12
588.2.f.c.293.1 2 105.2 even 12
588.2.f.c.293.2 2 35.12 even 12
588.2.k.c.509.1 2 35.27 even 4
588.2.k.c.521.1 2 105.32 even 12
588.2.k.d.509.1 2 105.62 odd 4
588.2.k.d.521.1 2 35.32 odd 12
2100.2.bi.e.101.1 2 105.38 odd 12
2100.2.bi.e.1601.1 2 5.3 odd 4
2100.2.bi.f.101.1 2 35.3 even 12
2100.2.bi.f.1601.1 2 15.8 even 4
2100.2.bo.a.1349.1 4 15.14 odd 2
2100.2.bo.a.1349.2 4 3.2 odd 2
2100.2.bo.a.1949.1 4 7.3 odd 6
2100.2.bo.a.1949.2 4 35.24 odd 6
2100.2.bo.f.1349.1 4 1.1 even 1 trivial
2100.2.bo.f.1349.2 4 5.4 even 2 inner
2100.2.bo.f.1949.1 4 105.59 even 6 inner
2100.2.bo.f.1949.2 4 21.17 even 6 inner
2268.2.w.a.269.1 2 315.122 odd 12
2268.2.w.a.1349.1 2 45.7 odd 12
2268.2.w.f.269.1 2 315.157 even 12
2268.2.w.f.1349.1 2 45.2 even 12
2268.2.bm.a.593.1 2 45.32 even 12
2268.2.bm.a.1025.1 2 315.52 even 12
2268.2.bm.f.593.1 2 45.22 odd 12
2268.2.bm.f.1025.1 2 315.227 odd 12
2352.2.k.a.881.1 2 140.47 odd 12
2352.2.k.a.881.2 2 420.107 odd 12
2352.2.k.d.881.1 2 420.47 even 12
2352.2.k.d.881.2 2 140.107 even 12