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Properties

Label 882.2.u.d.631.2
Level $882$
Weight $2$
Character 882.631
Analytic conductor $7.043$
Analytic rank $0$
Dimension $12$
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] = [882,2,Mod(127,882)] 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("882.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.u (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-2,0,-2,-1] 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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: 12.0.7877952219361.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 294)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 631.2
Root \(-1.41140 + 0.0891373i\) of defining polynomial
Character \(\chi\) \(=\) 882.631
Dual form 882.2.u.d.253.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.623490 - 0.781831i) q^{2} +(-0.222521 - 0.974928i) q^{4} +(1.71127 + 0.824106i) q^{5} +(0.253488 + 2.63358i) q^{7} +(-0.900969 - 0.433884i) q^{8} +(1.71127 - 0.824106i) q^{10} +(-1.71706 + 2.15313i) q^{11} +(2.60600 - 3.26782i) q^{13} +(2.21706 + 1.44383i) q^{14} +(-0.900969 + 0.433884i) q^{16} +(-1.64481 + 7.20637i) q^{17} +7.81704 q^{19} +(0.422650 - 1.85175i) q^{20} +(0.612813 + 2.68491i) q^{22} +(1.08696 + 4.76227i) q^{23} +(-0.868144 - 1.08862i) q^{25} +(-0.930071 - 4.07491i) q^{26} +(2.51114 - 0.833159i) q^{28} +(2.07633 - 9.09698i) q^{29} +0.458576 q^{31} +(-0.222521 + 0.974928i) q^{32} +(4.60865 + 5.77906i) q^{34} +(-1.73656 + 4.71568i) q^{35} +(-1.84011 + 8.06206i) q^{37} +(4.87384 - 6.11161i) q^{38} +(-1.18424 - 1.48499i) q^{40} +(-0.630444 - 0.303606i) q^{41} +(5.56270 - 2.67885i) q^{43} +(2.48123 + 1.19490i) q^{44} +(4.40100 + 2.11941i) q^{46} +(-4.44107 + 5.56892i) q^{47} +(-6.87149 + 1.33516i) q^{49} -1.39239 q^{50} +(-3.76578 - 1.81350i) q^{52} +(0.614243 + 2.69117i) q^{53} +(-4.71277 + 2.26955i) q^{55} +(0.914283 - 2.48276i) q^{56} +(-5.81774 - 7.29522i) q^{58} +(7.31701 - 3.52368i) q^{59} +(2.11063 - 9.24728i) q^{61} +(0.285918 - 0.358530i) q^{62} +(0.623490 + 0.781831i) q^{64} +(7.15261 - 3.44451i) q^{65} +13.7352 q^{67} +7.39170 q^{68} +(2.60413 + 4.29787i) q^{70} +(-1.50083 - 6.57555i) q^{71} +(-9.74755 - 12.2230i) q^{73} +(5.15588 + 6.46527i) q^{74} +(-1.73946 - 7.62105i) q^{76} +(-6.10569 - 3.97623i) q^{77} -4.74202 q^{79} -1.89937 q^{80} +(-0.630444 + 0.303606i) q^{82} +(-2.44596 - 3.06714i) q^{83} +(-8.75353 + 10.9766i) q^{85} +(1.37387 - 6.01933i) q^{86} +(2.48123 - 1.19490i) q^{88} +(-5.34803 - 6.70622i) q^{89} +(9.26666 + 6.03476i) q^{91} +(4.40100 - 2.11941i) q^{92} +(1.58500 + 6.94433i) q^{94} +(13.3771 + 6.44207i) q^{95} -8.03695 q^{97} +(-3.24043 + 6.20480i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 2 q^{4} - q^{5} - 2 q^{8} - q^{10} + 6 q^{11} + q^{13} - 2 q^{16} + 7 q^{17} + 22 q^{19} + 6 q^{20} - q^{22} - 21 q^{23} - 15 q^{25} + 22 q^{26} + 7 q^{28} - 3 q^{29} + 2 q^{31} - 2 q^{32}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{1}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.623490 − 0.781831i 0.440874 − 0.552838i
\(3\) 0 0
\(4\) −0.222521 − 0.974928i −0.111260 − 0.487464i
\(5\) 1.71127 + 0.824106i 0.765305 + 0.368551i 0.775460 − 0.631397i \(-0.217518\pi\)
−0.0101550 + 0.999948i \(0.503233\pi\)
\(6\) 0 0
\(7\) 0.253488 + 2.63358i 0.0958093 + 0.995400i
\(8\) −0.900969 − 0.433884i −0.318541 − 0.153401i
\(9\) 0 0
\(10\) 1.71127 − 0.824106i 0.541152 − 0.260605i
\(11\) −1.71706 + 2.15313i −0.517714 + 0.649193i −0.970122 − 0.242618i \(-0.921994\pi\)
0.452408 + 0.891811i \(0.350565\pi\)
\(12\) 0 0
\(13\) 2.60600 − 3.26782i 0.722774 − 0.906330i −0.275717 − 0.961239i \(-0.588915\pi\)
0.998491 + 0.0549085i \(0.0174867\pi\)
\(14\) 2.21706 + 1.44383i 0.592535 + 0.385879i
\(15\) 0 0
\(16\) −0.900969 + 0.433884i −0.225242 + 0.108471i
\(17\) −1.64481 + 7.20637i −0.398924 + 1.74780i 0.232720 + 0.972544i \(0.425238\pi\)
−0.631644 + 0.775258i \(0.717620\pi\)
\(18\) 0 0
\(19\) 7.81704 1.79335 0.896676 − 0.442687i \(-0.145975\pi\)
0.896676 + 0.442687i \(0.145975\pi\)
\(20\) 0.422650 − 1.85175i 0.0945073 − 0.414064i
\(21\) 0 0
\(22\) 0.612813 + 2.68491i 0.130652 + 0.572424i
\(23\) 1.08696 + 4.76227i 0.226646 + 0.993001i 0.952353 + 0.304997i \(0.0986556\pi\)
−0.725707 + 0.688004i \(0.758487\pi\)
\(24\) 0 0
\(25\) −0.868144 − 1.08862i −0.173629 − 0.217724i
\(26\) −0.930071 − 4.07491i −0.182402 − 0.799155i
\(27\) 0 0
\(28\) 2.51114 − 0.833159i 0.474562 − 0.157452i
\(29\) 2.07633 − 9.09698i 0.385564 − 1.68927i −0.294125 − 0.955767i \(-0.595028\pi\)
0.679689 − 0.733501i \(-0.262115\pi\)
\(30\) 0 0
\(31\) 0.458576 0.0823628 0.0411814 − 0.999152i \(-0.486888\pi\)
0.0411814 + 0.999152i \(0.486888\pi\)
\(32\) −0.222521 + 0.974928i −0.0393365 + 0.172345i
\(33\) 0 0
\(34\) 4.60865 + 5.77906i 0.790377 + 0.991101i
\(35\) −1.73656 + 4.71568i −0.293533 + 0.797095i
\(36\) 0 0
\(37\) −1.84011 + 8.06206i −0.302513 + 1.32539i 0.563808 + 0.825906i \(0.309336\pi\)
−0.866321 + 0.499488i \(0.833521\pi\)
\(38\) 4.87384 − 6.11161i 0.790642 − 0.991434i
\(39\) 0 0
\(40\) −1.18424 − 1.48499i −0.187244 − 0.234797i
\(41\) −0.630444 − 0.303606i −0.0984587 − 0.0474152i 0.384006 − 0.923331i \(-0.374544\pi\)
−0.482465 + 0.875915i \(0.660258\pi\)
\(42\) 0 0
\(43\) 5.56270 − 2.67885i 0.848304 − 0.408521i 0.0413560 − 0.999144i \(-0.486832\pi\)
0.806948 + 0.590623i \(0.201118\pi\)
\(44\) 2.48123 + 1.19490i 0.374059 + 0.180137i
\(45\) 0 0
\(46\) 4.40100 + 2.11941i 0.648891 + 0.312490i
\(47\) −4.44107 + 5.56892i −0.647796 + 0.812311i −0.991954 − 0.126601i \(-0.959593\pi\)
0.344157 + 0.938912i \(0.388165\pi\)
\(48\) 0 0
\(49\) −6.87149 + 1.33516i −0.981641 + 0.190737i
\(50\) −1.39239 −0.196914
\(51\) 0 0
\(52\) −3.76578 − 1.81350i −0.522220 − 0.251488i
\(53\) 0.614243 + 2.69117i 0.0843727 + 0.369661i 0.999434 − 0.0336549i \(-0.0107147\pi\)
−0.915061 + 0.403316i \(0.867858\pi\)
\(54\) 0 0
\(55\) −4.71277 + 2.26955i −0.635470 + 0.306026i
\(56\) 0.914283 − 2.48276i 0.122176 − 0.331772i
\(57\) 0 0
\(58\) −5.81774 − 7.29522i −0.763907 − 0.957909i
\(59\) 7.31701 − 3.52368i 0.952593 − 0.458745i 0.107999 − 0.994151i \(-0.465556\pi\)
0.844595 + 0.535406i \(0.179842\pi\)
\(60\) 0 0
\(61\) 2.11063 − 9.24728i 0.270239 − 1.18399i −0.639493 − 0.768797i \(-0.720856\pi\)
0.909732 − 0.415196i \(-0.136287\pi\)
\(62\) 0.285918 − 0.358530i 0.0363116 − 0.0455333i
\(63\) 0 0
\(64\) 0.623490 + 0.781831i 0.0779362 + 0.0977289i
\(65\) 7.15261 − 3.44451i 0.887172 − 0.427239i
\(66\) 0 0
\(67\) 13.7352 1.67802 0.839009 − 0.544117i \(-0.183135\pi\)
0.839009 + 0.544117i \(0.183135\pi\)
\(68\) 7.39170 0.896375
\(69\) 0 0
\(70\) 2.60413 + 4.29787i 0.311254 + 0.513694i
\(71\) −1.50083 − 6.57555i −0.178115 − 0.780375i −0.982500 − 0.186265i \(-0.940362\pi\)
0.804384 − 0.594110i \(-0.202495\pi\)
\(72\) 0 0
\(73\) −9.74755 − 12.2230i −1.14086 − 1.43060i −0.886039 − 0.463611i \(-0.846554\pi\)
−0.254825 − 0.966987i \(-0.582018\pi\)
\(74\) 5.15588 + 6.46527i 0.599359 + 0.751572i
\(75\) 0 0
\(76\) −1.73946 − 7.62105i −0.199529 − 0.874194i
\(77\) −6.10569 − 3.97623i −0.695808 − 0.453134i
\(78\) 0 0
\(79\) −4.74202 −0.533519 −0.266759 − 0.963763i \(-0.585953\pi\)
−0.266759 + 0.963763i \(0.585953\pi\)
\(80\) −1.89937 −0.212356
\(81\) 0 0
\(82\) −0.630444 + 0.303606i −0.0696208 + 0.0335276i
\(83\) −2.44596 − 3.06714i −0.268479 − 0.336662i 0.629256 − 0.777198i \(-0.283360\pi\)
−0.897735 + 0.440536i \(0.854788\pi\)
\(84\) 0 0
\(85\) −8.75353 + 10.9766i −0.949453 + 1.19058i
\(86\) 1.37387 − 6.01933i 0.148149 − 0.649081i
\(87\) 0 0
\(88\) 2.48123 − 1.19490i 0.264500 − 0.127376i
\(89\) −5.34803 − 6.70622i −0.566890 − 0.710857i 0.412925 − 0.910765i \(-0.364507\pi\)
−0.979815 + 0.199908i \(0.935936\pi\)
\(90\) 0 0
\(91\) 9.26666 + 6.03476i 0.971409 + 0.632614i
\(92\) 4.40100 − 2.11941i 0.458836 − 0.220964i
\(93\) 0 0
\(94\) 1.58500 + 6.94433i 0.163480 + 0.716253i
\(95\) 13.3771 + 6.44207i 1.37246 + 0.660942i
\(96\) 0 0
\(97\) −8.03695 −0.816029 −0.408014 − 0.912975i \(-0.633779\pi\)
−0.408014 + 0.912975i \(0.633779\pi\)
\(98\) −3.24043 + 6.20480i −0.327333 + 0.626780i
\(99\) 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 882.2.u.d.631.2 12
3.2 odd 2 294.2.i.b.43.1 ✓ 12
49.8 even 7 inner 882.2.u.d.253.2 12
147.8 odd 14 294.2.i.b.253.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.i.b.43.1 ✓ 12 3.2 odd 2
294.2.i.b.253.1 yes 12 147.8 odd 14
882.2.u.d.253.2 12 49.8 even 7 inner
882.2.u.d.631.2 12 1.1 even 1 trivial