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Properties

Label 261.2.k.d.181.1
Level $261$
Weight $2$
Character 261.181
Analytic conductor $2.084$
Analytic rank $0$
Dimension $24$
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] = [261,2,Mod(82,261)] 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("261.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(261, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 181.1
Character \(\chi\) \(=\) 261.181
Dual form 261.2.k.d.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.398448 - 1.74571i) q^{2} +(-1.08681 + 0.523382i) q^{4} +(0.325426 + 1.42578i) q^{5} +(3.26031 + 1.57008i) q^{7} +(-0.886136 - 1.11118i) q^{8} +(2.35934 - 1.13620i) q^{10} +(2.92355 - 3.66602i) q^{11} +(2.18995 - 2.74611i) q^{13} +(1.44185 - 6.31716i) q^{14} +(-3.09092 + 3.87589i) q^{16} -1.80531 q^{17} +(-3.31361 + 1.59575i) q^{19} +(-1.09991 - 1.37924i) q^{20} +(-7.56469 - 3.64296i) q^{22} +(-0.727145 + 3.18583i) q^{23} +(2.57789 - 1.24145i) q^{25} +(-5.66650 - 2.72884i) q^{26} -4.36511 q^{28} +(4.43812 + 3.05009i) q^{29} +(-0.810216 - 3.54979i) q^{31} +(5.43675 + 2.61820i) q^{32} +(0.719321 + 3.15155i) q^{34} +(-1.17761 + 5.15944i) q^{35} +(-5.00851 - 6.28048i) q^{37} +(4.10602 + 5.14879i) q^{38} +(1.29593 - 1.62504i) q^{40} -7.92055 q^{41} +(-1.46918 + 6.43691i) q^{43} +(-1.25863 + 5.51442i) q^{44} +5.85127 q^{46} +(-0.251056 + 0.314815i) q^{47} +(3.80003 + 4.76509i) q^{49} +(-3.19436 - 4.00561i) q^{50} +(-0.942805 + 4.13070i) q^{52} +(1.67750 + 7.34959i) q^{53} +(6.17834 + 2.97533i) q^{55} +(-1.14444 - 5.01410i) q^{56} +(3.55622 - 8.96299i) q^{58} +8.64336 q^{59} +(6.28281 + 3.02564i) q^{61} +(-5.87408 + 2.82881i) q^{62} +(0.198095 - 0.867913i) q^{64} +(4.62802 + 2.22874i) q^{65} +(-4.05513 - 5.08498i) q^{67} +(1.96204 - 0.944867i) q^{68} +9.47611 q^{70} +(-2.49139 + 3.12411i) q^{71} +(-2.76168 + 12.0997i) q^{73} +(-8.96828 + 11.2459i) q^{74} +(2.76609 - 3.46857i) q^{76} +(15.2876 - 7.36214i) q^{77} +(-9.18139 - 11.5131i) q^{79} +(-6.53204 - 3.14567i) q^{80} +(3.15592 + 13.8270i) q^{82} +(-13.8834 + 6.68591i) q^{83} +(-0.587494 - 2.57398i) q^{85} +11.8224 q^{86} -6.66427 q^{88} +(4.04949 + 17.7420i) q^{89} +(11.4515 - 5.51477i) q^{91} +(-0.877135 - 3.84298i) q^{92} +(0.649608 + 0.312835i) q^{94} +(-3.35352 - 4.20519i) q^{95} +(-4.08674 + 1.96807i) q^{97} +(6.80437 - 8.53241i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{4} - 4 q^{10} + 18 q^{13} - 22 q^{16} - 40 q^{22} + 18 q^{25} - 24 q^{28} - 28 q^{31} - 50 q^{34} - 40 q^{37} + 30 q^{43} + 40 q^{46} + 36 q^{49} - 52 q^{52} + 10 q^{55} + 54 q^{58} + 64 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(e\left(\frac{3}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.398448 − 1.74571i −0.281745 − 1.23441i −0.895555 − 0.444951i \(-0.853221\pi\)
0.613810 − 0.789454i \(-0.289636\pi\)
\(3\) 0 0
\(4\) −1.08681 + 0.523382i −0.543407 + 0.261691i
\(5\) 0.325426 + 1.42578i 0.145535 + 0.637629i 0.994093 + 0.108528i \(0.0346137\pi\)
−0.848559 + 0.529101i \(0.822529\pi\)
\(6\) 0 0
\(7\) 3.26031 + 1.57008i 1.23228 + 0.593435i 0.932706 − 0.360637i \(-0.117441\pi\)
0.299575 + 0.954073i \(0.403155\pi\)
\(8\) −0.886136 − 1.11118i −0.313297 − 0.392861i
\(9\) 0 0
\(10\) 2.35934 − 1.13620i 0.746089 − 0.359298i
\(11\) 2.92355 − 3.66602i 0.881484 − 1.10535i −0.112262 − 0.993679i \(-0.535810\pi\)
0.993746 − 0.111667i \(-0.0356190\pi\)
\(12\) 0 0
\(13\) 2.18995 − 2.74611i 0.607383 − 0.761635i −0.379125 − 0.925345i \(-0.623775\pi\)
0.986508 + 0.163711i \(0.0523464\pi\)
\(14\) 1.44185 − 6.31716i 0.385351 − 1.68833i
\(15\) 0 0
\(16\) −3.09092 + 3.87589i −0.772730 + 0.968973i
\(17\) −1.80531 −0.437852 −0.218926 − 0.975741i \(-0.570255\pi\)
−0.218926 + 0.975741i \(0.570255\pi\)
\(18\) 0 0
\(19\) −3.31361 + 1.59575i −0.760194 + 0.366090i −0.773479 − 0.633822i \(-0.781485\pi\)
0.0132853 + 0.999912i \(0.495771\pi\)
\(20\) −1.09991 − 1.37924i −0.245947 − 0.308407i
\(21\) 0 0
\(22\) −7.56469 − 3.64296i −1.61280 − 0.776683i
\(23\) −0.727145 + 3.18583i −0.151620 + 0.664291i 0.840794 + 0.541355i \(0.182088\pi\)
−0.992415 + 0.122937i \(0.960769\pi\)
\(24\) 0 0
\(25\) 2.57789 − 1.24145i 0.515578 − 0.248289i
\(26\) −5.66650 − 2.72884i −1.11129 − 0.535170i
\(27\) 0 0
\(28\) −4.36511 −0.824928
\(29\) 4.43812 + 3.05009i 0.824139 + 0.566388i
\(30\) 0 0
\(31\) −0.810216 − 3.54979i −0.145519 − 0.637561i −0.994097 − 0.108491i \(-0.965398\pi\)
0.848578 − 0.529070i \(-0.177459\pi\)
\(32\) 5.43675 + 2.61820i 0.961091 + 0.462837i
\(33\) 0 0
\(34\) 0.719321 + 3.15155i 0.123363 + 0.540487i
\(35\) −1.17761 + 5.15944i −0.199052 + 0.872104i
\(36\) 0 0
\(37\) −5.00851 − 6.28048i −0.823394 − 1.03250i −0.998847 − 0.0480159i \(-0.984710\pi\)
0.175452 − 0.984488i \(-0.443861\pi\)
\(38\) 4.10602 + 5.14879i 0.666084 + 0.835243i
\(39\) 0 0
\(40\) 1.29593 − 1.62504i 0.204904 − 0.256942i
\(41\) −7.92055 −1.23698 −0.618491 − 0.785792i \(-0.712256\pi\)
−0.618491 + 0.785792i \(0.712256\pi\)
\(42\) 0 0
\(43\) −1.46918 + 6.43691i −0.224048 + 0.981620i 0.730347 + 0.683076i \(0.239358\pi\)
−0.954396 + 0.298544i \(0.903499\pi\)
\(44\) −1.25863 + 5.51442i −0.189746 + 0.831329i
\(45\) 0 0
\(46\) 5.85127 0.862723
\(47\) −0.251056 + 0.314815i −0.0366203 + 0.0459204i −0.799804 − 0.600261i \(-0.795064\pi\)
0.763184 + 0.646181i \(0.223635\pi\)
\(48\) 0 0
\(49\) 3.80003 + 4.76509i 0.542862 + 0.680728i
\(50\) −3.19436 − 4.00561i −0.451751 − 0.566478i
\(51\) 0 0
\(52\) −0.942805 + 4.13070i −0.130743 + 0.572825i
\(53\) 1.67750 + 7.34959i 0.230422 + 1.00954i 0.949291 + 0.314398i \(0.101803\pi\)
−0.718869 + 0.695145i \(0.755340\pi\)
\(54\) 0 0
\(55\) 6.17834 + 2.97533i 0.833087 + 0.401194i
\(56\) −1.14444 − 5.01410i −0.152932 − 0.670037i
\(57\) 0 0
\(58\) 3.55622 − 8.96299i 0.466955 − 1.17690i
\(59\) 8.64336 1.12527 0.562635 − 0.826705i \(-0.309788\pi\)
0.562635 + 0.826705i \(0.309788\pi\)
\(60\) 0 0
\(61\) 6.28281 + 3.02564i 0.804431 + 0.387394i 0.790464 − 0.612509i \(-0.209840\pi\)
0.0139674 + 0.999902i \(0.495554\pi\)
\(62\) −5.87408 + 2.82881i −0.746009 + 0.359259i
\(63\) 0 0
\(64\) 0.198095 − 0.867913i 0.0247619 − 0.108489i
\(65\) 4.62802 + 2.22874i 0.574036 + 0.276441i
\(66\) 0 0
\(67\) −4.05513 − 5.08498i −0.495413 − 0.621229i 0.469774 − 0.882787i \(-0.344335\pi\)
−0.965188 + 0.261558i \(0.915764\pi\)
\(68\) 1.96204 − 0.944867i 0.237932 − 0.114582i
\(69\) 0 0
\(70\) 9.47611 1.13261
\(71\) −2.49139 + 3.12411i −0.295674 + 0.370764i −0.907372 − 0.420328i \(-0.861915\pi\)
0.611698 + 0.791091i \(0.290487\pi\)
\(72\) 0 0
\(73\) −2.76168 + 12.0997i −0.323231 + 1.41617i 0.508536 + 0.861040i \(0.330187\pi\)
−0.831767 + 0.555125i \(0.812670\pi\)
\(74\) −8.96828 + 11.2459i −1.04254 + 1.30731i
\(75\) 0 0
\(76\) 2.76609 − 3.46857i 0.317292 − 0.397872i
\(77\) 15.2876 − 7.36214i 1.74219 − 0.838993i
\(78\) 0 0
\(79\) −9.18139 − 11.5131i −1.03299 − 1.29532i −0.954436 − 0.298414i \(-0.903542\pi\)
−0.0785504 − 0.996910i \(-0.525029\pi\)
\(80\) −6.53204 − 3.14567i −0.730304 − 0.351696i
\(81\) 0 0
\(82\) 3.15592 + 13.8270i 0.348513 + 1.52694i
\(83\) −13.8834 + 6.68591i −1.52391 + 0.733874i −0.993496 − 0.113868i \(-0.963676\pi\)
−0.530409 + 0.847742i \(0.677962\pi\)
\(84\) 0 0
\(85\) −0.587494 − 2.57398i −0.0637227 − 0.279187i
\(86\) 11.8224 1.27484
\(87\) 0 0
\(88\) −6.66427 −0.710413
\(89\) 4.04949 + 17.7420i 0.429245 + 1.88065i 0.472065 + 0.881564i \(0.343509\pi\)
−0.0428200 + 0.999083i \(0.513634\pi\)
\(90\) 0 0
\(91\) 11.4515 − 5.51477i 1.20045 − 0.578105i
\(92\) −0.877135 − 3.84298i −0.0914477 − 0.400658i
\(93\) 0 0
\(94\) 0.649608 + 0.312835i 0.0670020 + 0.0322665i
\(95\) −3.35352 − 4.20519i −0.344064 − 0.431443i
\(96\) 0 0
\(97\) −4.08674 + 1.96807i −0.414946 + 0.199827i −0.629696 − 0.776842i \(-0.716821\pi\)
0.214750 + 0.976669i \(0.431106\pi\)
\(98\) 6.80437 − 8.53241i 0.687345 − 0.861903i
\(99\) 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 261.2.k.d.181.1 ✓ 24
3.2 odd 2 inner 261.2.k.d.181.4 yes 24
29.5 even 14 7569.2.a.br.1.3 12
29.24 even 7 7569.2.a.bq.1.10 12
29.25 even 7 inner 261.2.k.d.199.1 yes 24
87.5 odd 14 7569.2.a.br.1.10 12
87.53 odd 14 7569.2.a.bq.1.3 12
87.83 odd 14 inner 261.2.k.d.199.4 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
261.2.k.d.181.1 ✓ 24 1.1 even 1 trivial
261.2.k.d.181.4 yes 24 3.2 odd 2 inner
261.2.k.d.199.1 yes 24 29.25 even 7 inner
261.2.k.d.199.4 yes 24 87.83 odd 14 inner
7569.2.a.bq.1.3 12 87.53 odd 14
7569.2.a.bq.1.10 12 29.24 even 7
7569.2.a.br.1.3 12 29.5 even 14
7569.2.a.br.1.10 12 87.5 odd 14