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Properties

Label 2100.3.be.c.649.1
Level $2100$
Weight $3$
Character 2100.649
Analytic conductor $57.221$
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] = [2100,3,Mod(649,2100)] 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("2100.649"); S:= CuspForms(chi, 3); 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, 0, 3, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2100.be (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,-6,0,6,0,0,0,0,0,0,0,-36] 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: \(57.2208555157\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 649.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.649
Dual form 2100.3.be.c.1249.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 1.50000i) q^{3} +(2.59808 + 6.50000i) q^{7} +(-1.50000 + 2.59808i) q^{9} +(1.50000 + 2.59808i) q^{11} -6.92820 q^{13} +(-8.66025 - 15.0000i) q^{17} +(-9.00000 - 5.19615i) q^{19} +(7.50000 - 9.52628i) q^{21} +(31.1769 + 18.0000i) q^{23} +5.19615 q^{27} +51.0000 q^{29} +(-10.5000 + 6.06218i) q^{31} +(2.59808 - 4.50000i) q^{33} +(-19.0526 - 11.0000i) q^{37} +(6.00000 + 10.3923i) q^{39} -24.2487i q^{41} -10.0000i q^{43} +(-45.0333 + 78.0000i) q^{47} +(-35.5000 + 33.7750i) q^{49} +(-15.0000 + 25.9808i) q^{51} +(-44.1673 + 25.5000i) q^{53} +18.0000i q^{57} +(-64.5000 + 37.2391i) q^{59} +(60.0000 + 34.6410i) q^{61} +(-20.7846 - 3.00000i) q^{63} +(58.8897 - 34.0000i) q^{67} -62.3538i q^{69} +(10.3923 + 18.0000i) q^{73} +(-12.9904 + 16.5000i) q^{77} +(62.5000 - 108.253i) q^{79} +(-4.50000 - 7.79423i) q^{81} -154.153 q^{83} +(-44.1673 - 76.5000i) q^{87} +(-63.0000 - 36.3731i) q^{89} +(-18.0000 - 45.0333i) q^{91} +(18.1865 + 10.5000i) q^{93} -147.224 q^{97} -9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + 6 q^{11} - 36 q^{19} + 30 q^{21} + 204 q^{29} - 42 q^{31} + 24 q^{39} - 142 q^{49} - 60 q^{51} - 258 q^{59} + 240 q^{61} + 250 q^{79} - 18 q^{81} - 252 q^{89} - 72 q^{91} - 36 q^{99}+O(q^{100}) \) 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
\(3\) −0.866025 − 1.50000i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.59808 + 6.50000i 0.371154 + 0.928571i
\(8\) 0 0
\(9\) −1.50000 + 2.59808i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.50000 + 2.59808i 0.136364 + 0.236189i 0.926118 − 0.377235i \(-0.123125\pi\)
−0.789754 + 0.613424i \(0.789792\pi\)
\(12\) 0 0
\(13\) −6.92820 −0.532939 −0.266469 − 0.963843i \(-0.585857\pi\)
−0.266469 + 0.963843i \(0.585857\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −8.66025 − 15.0000i −0.509427 − 0.882353i −0.999940 − 0.0109194i \(-0.996524\pi\)
0.490514 − 0.871433i \(-0.336809\pi\)
\(18\) 0 0
\(19\) −9.00000 − 5.19615i −0.473684 − 0.273482i 0.244096 − 0.969751i \(-0.421509\pi\)
−0.717781 + 0.696269i \(0.754842\pi\)
\(20\) 0 0
\(21\) 7.50000 − 9.52628i 0.357143 − 0.453632i
\(22\) 0 0
\(23\) 31.1769 + 18.0000i 1.35552 + 0.782609i 0.989016 − 0.147809i \(-0.0472219\pi\)
0.366502 + 0.930417i \(0.380555\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 0.192450
\(28\) 0 0
\(29\) 51.0000 1.75862 0.879310 − 0.476249i \(-0.158004\pi\)
0.879310 + 0.476249i \(0.158004\pi\)
\(30\) 0 0
\(31\) −10.5000 + 6.06218i −0.338710 + 0.195554i −0.659701 − 0.751528i \(-0.729317\pi\)
0.320992 + 0.947082i \(0.395984\pi\)
\(32\) 0 0
\(33\) 2.59808 − 4.50000i 0.0787296 − 0.136364i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −19.0526 − 11.0000i −0.514934 − 0.297297i 0.219925 − 0.975517i \(-0.429419\pi\)
−0.734859 + 0.678219i \(0.762752\pi\)
\(38\) 0 0
\(39\) 6.00000 + 10.3923i 0.153846 + 0.266469i
\(40\) 0 0
\(41\) − 24.2487i − 0.591432i −0.955276 − 0.295716i \(-0.904442\pi\)
0.955276 − 0.295716i \(-0.0955582\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 0.232558i −0.993217 − 0.116279i \(-0.962903\pi\)
0.993217 − 0.116279i \(-0.0370967\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −45.0333 + 78.0000i −0.958156 + 1.65957i −0.231180 + 0.972911i \(0.574259\pi\)
−0.726975 + 0.686664i \(0.759075\pi\)
\(48\) 0 0
\(49\) −35.5000 + 33.7750i −0.724490 + 0.689286i
\(50\) 0 0
\(51\) −15.0000 + 25.9808i −0.294118 + 0.509427i
\(52\) 0 0
\(53\) −44.1673 + 25.5000i −0.833345 + 0.481132i −0.854997 − 0.518634i \(-0.826441\pi\)
0.0216515 + 0.999766i \(0.493108\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 18.0000i 0.315789i
\(58\) 0 0
\(59\) −64.5000 + 37.2391i −1.09322 + 0.631171i −0.934432 − 0.356142i \(-0.884092\pi\)
−0.158788 + 0.987313i \(0.550759\pi\)
\(60\) 0 0
\(61\) 60.0000 + 34.6410i 0.983607 + 0.567886i 0.903357 − 0.428889i \(-0.141095\pi\)
0.0802495 + 0.996775i \(0.474428\pi\)
\(62\) 0 0
\(63\) −20.7846 − 3.00000i −0.329914 − 0.0476190i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 58.8897 − 34.0000i 0.878951 − 0.507463i 0.00863871 − 0.999963i \(-0.497250\pi\)
0.870312 + 0.492500i \(0.163917\pi\)
\(68\) 0 0
\(69\) − 62.3538i − 0.903679i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.3923 + 18.0000i 0.142360 + 0.246575i 0.928385 − 0.371620i \(-0.121197\pi\)
−0.786025 + 0.618195i \(0.787864\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.9904 + 16.5000i −0.168706 + 0.214286i
\(78\) 0 0
\(79\) 62.5000 − 108.253i 0.791139 − 1.37029i −0.134123 − 0.990965i \(-0.542822\pi\)
0.925262 − 0.379329i \(-0.123845\pi\)
\(80\) 0 0
\(81\) −4.50000 − 7.79423i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −154.153 −1.85726 −0.928630 − 0.371008i \(-0.879012\pi\)
−0.928630 + 0.371008i \(0.879012\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −44.1673 − 76.5000i −0.507670 − 0.879310i
\(88\) 0 0
\(89\) −63.0000 − 36.3731i −0.707865 − 0.408686i 0.102405 − 0.994743i \(-0.467346\pi\)
−0.810270 + 0.586057i \(0.800680\pi\)
\(90\) 0 0
\(91\) −18.0000 − 45.0333i −0.197802 − 0.494872i
\(92\) 0 0
\(93\) 18.1865 + 10.5000i 0.195554 + 0.112903i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −147.224 −1.51778 −0.758888 − 0.651221i \(-0.774257\pi\)
−0.758888 + 0.651221i \(0.774257\pi\)
\(98\) 0 0
\(99\) −9.00000 −0.0909091
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 2100.3.be.c.649.1 4
5.2 odd 4 2100.3.bd.b.901.1 2
5.3 odd 4 84.3.m.a.61.1 ✓ 2
5.4 even 2 inner 2100.3.be.c.649.2 4
7.3 odd 6 inner 2100.3.be.c.1249.2 4
15.8 even 4 252.3.z.b.145.1 2
20.3 even 4 336.3.bh.b.145.1 2
35.3 even 12 84.3.m.a.73.1 yes 2
35.13 even 4 588.3.m.a.313.1 2
35.17 even 12 2100.3.bd.b.1501.1 2
35.18 odd 12 588.3.m.a.325.1 2
35.23 odd 12 588.3.d.a.97.2 2
35.24 odd 6 inner 2100.3.be.c.1249.1 4
35.33 even 12 588.3.d.a.97.1 2
60.23 odd 4 1008.3.cg.b.145.1 2
105.23 even 12 1764.3.d.c.685.2 2
105.38 odd 12 252.3.z.b.73.1 2
105.53 even 12 1764.3.z.e.325.1 2
105.68 odd 12 1764.3.d.c.685.1 2
105.83 odd 4 1764.3.z.e.901.1 2
140.3 odd 12 336.3.bh.b.241.1 2
140.23 even 12 2352.3.f.c.97.1 2
140.103 odd 12 2352.3.f.c.97.2 2
420.143 even 12 1008.3.cg.b.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.3.m.a.61.1 ✓ 2 5.3 odd 4
84.3.m.a.73.1 yes 2 35.3 even 12
252.3.z.b.73.1 2 105.38 odd 12
252.3.z.b.145.1 2 15.8 even 4
336.3.bh.b.145.1 2 20.3 even 4
336.3.bh.b.241.1 2 140.3 odd 12
588.3.d.a.97.1 2 35.33 even 12
588.3.d.a.97.2 2 35.23 odd 12
588.3.m.a.313.1 2 35.13 even 4
588.3.m.a.325.1 2 35.18 odd 12
1008.3.cg.b.145.1 2 60.23 odd 4
1008.3.cg.b.577.1 2 420.143 even 12
1764.3.d.c.685.1 2 105.68 odd 12
1764.3.d.c.685.2 2 105.23 even 12
1764.3.z.e.325.1 2 105.53 even 12
1764.3.z.e.901.1 2 105.83 odd 4
2100.3.bd.b.901.1 2 5.2 odd 4
2100.3.bd.b.1501.1 2 35.17 even 12
2100.3.be.c.649.1 4 1.1 even 1 trivial
2100.3.be.c.649.2 4 5.4 even 2 inner
2100.3.be.c.1249.1 4 35.24 odd 6 inner
2100.3.be.c.1249.2 4 7.3 odd 6 inner
2352.3.f.c.97.1 2 140.23 even 12
2352.3.f.c.97.2 2 140.103 odd 12