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Properties

Label 735.2.y.i.263.4
Level $735$
Weight $2$
Character 735.263
Analytic conductor $5.869$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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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] = [735,2,Mod(128,735)] 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("735.128"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 263.4
Character \(\chi\) \(=\) 735.263
Dual form 735.2.y.i.422.4

$q$-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+(-0.340162 + 1.26950i) q^{2} +(0.664627 + 1.59946i) q^{3} +(0.236127 + 0.136328i) q^{4} +(-1.25032 + 1.85383i) q^{5} +(-2.25660 + 0.299670i) q^{6} +(-2.11207 + 2.11207i) q^{8} +(-2.11654 + 2.12609i) q^{9} +(-1.92813 - 2.21789i) q^{10} +(3.38224 + 1.95274i) q^{11} +(-0.0611145 + 0.468282i) q^{12} +(1.56642 + 1.56642i) q^{13} +(-3.79613 - 0.767733i) q^{15} +(-1.69017 - 2.92747i) q^{16} +(2.58656 - 0.693065i) q^{17} +(-1.97911 - 3.41017i) q^{18} +(1.61097 - 0.930096i) q^{19} +(-0.547963 + 0.267285i) q^{20} +(-3.62951 + 3.62951i) q^{22} +(-2.38315 - 0.638564i) q^{23} +(-4.78191 - 1.97443i) q^{24} +(-1.87339 - 4.63578i) q^{25} +(-2.52141 + 1.45574i) q^{26} +(-4.80730 - 1.97227i) q^{27} +0.513153 q^{29} +(2.26594 - 4.55804i) q^{30} +(4.29138 - 7.43289i) q^{31} +(-1.47892 + 0.396276i) q^{32} +(-0.875396 + 6.70760i) q^{33} +3.51939i q^{34} +(-0.789616 + 0.213483i) q^{36} +(-6.60698 - 1.77034i) q^{37} +(0.632766 + 2.36152i) q^{38} +(-1.46434 + 3.54652i) q^{39} +(-1.27465 - 6.55619i) q^{40} +0.308469i q^{41} +(7.60892 + 7.60892i) q^{43} +(0.532425 + 0.922186i) q^{44} +(-1.29505 - 6.58201i) q^{45} +(1.62131 - 2.80820i) q^{46} +(1.36920 - 5.10994i) q^{47} +(3.55903 - 4.64904i) q^{48} +(6.52238 - 0.801352i) q^{50} +(2.82762 + 3.67646i) q^{51} +(0.156327 + 0.583421i) q^{52} +(0.498259 + 1.85953i) q^{53} +(4.13906 - 5.43199i) q^{54} +(-7.84894 + 3.82855i) q^{55} +(2.55835 + 1.95852i) q^{57} +(-0.174555 + 0.651448i) q^{58} +(-0.259114 + 0.448799i) q^{59} +(-0.791703 - 0.698800i) q^{60} +(2.55451 + 4.42454i) q^{61} +(7.97631 + 7.97631i) q^{62} -8.77299i q^{64} +(-4.86242 + 0.945351i) q^{65} +(-8.21753 - 3.39299i) q^{66} +(2.34332 + 8.74539i) q^{67} +(0.705238 + 0.188968i) q^{68} +(-0.562551 - 4.23616i) q^{69} -15.3749i q^{71} +(-0.0201641 - 8.96073i) q^{72} +(-2.79871 + 0.749913i) q^{73} +(4.49489 - 7.78538i) q^{74} +(6.16963 - 6.07747i) q^{75} +0.507191 q^{76} +(-4.00420 - 3.06538i) q^{78} +(4.37551 - 2.52620i) q^{79} +(7.54030 + 0.526980i) q^{80} +(-0.0405048 - 8.99991i) q^{81} +(-0.391602 - 0.104930i) q^{82} +(-9.16088 + 9.16088i) q^{83} +(-1.94920 + 5.66159i) q^{85} +(-12.2478 + 7.07127i) q^{86} +(0.341055 + 0.820767i) q^{87} +(-11.2678 + 3.01921i) q^{88} +(5.67519 + 9.82972i) q^{89} +(8.79640 + 0.594879i) q^{90} +(-0.475671 - 0.475671i) q^{92} +(14.7408 + 1.92379i) q^{93} +(6.02133 + 3.47641i) q^{94} +(-0.289995 + 4.14939i) q^{95} +(-1.61676 - 2.10210i) q^{96} +(6.81964 - 6.81964i) q^{97} +(-11.3103 + 3.05789i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 2 q^{3} + 24 q^{6} + 8 q^{10} + 10 q^{12} + 16 q^{13} + 4 q^{15} - 8 q^{16} + 14 q^{18} - 8 q^{22} + 4 q^{25} - 40 q^{27} + 40 q^{30} + 24 q^{31} + 4 q^{33} + 8 q^{36} + 4 q^{37} + 16 q^{40} + 16 q^{43}+ \cdots + 120 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.340162 + 1.26950i −0.240531 + 0.897673i 0.735046 + 0.678017i \(0.237160\pi\)
−0.975577 + 0.219656i \(0.929506\pi\)
\(3\) 0.664627 + 1.59946i 0.383723 + 0.923448i
\(4\) 0.236127 + 0.136328i 0.118063 + 0.0681639i
\(5\) −1.25032 + 1.85383i −0.559161 + 0.829059i
\(6\) −2.25660 + 0.299670i −0.921252 + 0.122340i
\(7\) 0 0
\(8\) −2.11207 + 2.11207i −0.746729 + 0.746729i
\(9\) −2.11654 + 2.12609i −0.705514 + 0.708696i
\(10\) −1.92813 − 2.21789i −0.609728 − 0.701358i
\(11\) 3.38224 + 1.95274i 1.01978 + 0.588773i 0.914041 − 0.405621i \(-0.132945\pi\)
0.105743 + 0.994394i \(0.466278\pi\)
\(12\) −0.0611145 + 0.468282i −0.0176422 + 0.135181i
\(13\) 1.56642 + 1.56642i 0.434448 + 0.434448i 0.890138 − 0.455691i \(-0.150608\pi\)
−0.455691 + 0.890138i \(0.650608\pi\)
\(14\) 0 0
\(15\) −3.79613 − 0.767733i −0.980156 − 0.198228i
\(16\) −1.69017 − 2.92747i −0.422544 − 0.731867i
\(17\) 2.58656 − 0.693065i 0.627332 − 0.168093i 0.0688731 − 0.997625i \(-0.478060\pi\)
0.558459 + 0.829532i \(0.311393\pi\)
\(18\) −1.97911 − 3.41017i −0.466480 − 0.803784i
\(19\) 1.61097 − 0.930096i 0.369582 − 0.213379i −0.303694 − 0.952770i \(-0.598220\pi\)
0.673276 + 0.739391i \(0.264887\pi\)
\(20\) −0.547963 + 0.267285i −0.122528 + 0.0597668i
\(21\) 0 0
\(22\) −3.62951 + 3.62951i −0.773815 + 0.773815i
\(23\) −2.38315 − 0.638564i −0.496921 − 0.133150i 0.00164943 − 0.999999i \(-0.499475\pi\)
−0.498571 + 0.866849i \(0.666142\pi\)
\(24\) −4.78191 − 1.97443i −0.976103 − 0.403029i
\(25\) −1.87339 − 4.63578i −0.374677 − 0.927155i
\(26\) −2.52141 + 1.45574i −0.494490 + 0.285494i
\(27\) −4.80730 − 1.97227i −0.925166 − 0.379563i
\(28\) 0 0
\(29\) 0.513153 0.0952901 0.0476450 − 0.998864i \(-0.484828\pi\)
0.0476450 + 0.998864i \(0.484828\pi\)
\(30\) 2.26594 − 4.55804i 0.413702 − 0.832180i
\(31\) 4.29138 − 7.43289i 0.770755 − 1.33499i −0.166394 − 0.986059i \(-0.553212\pi\)
0.937150 − 0.348928i \(-0.113454\pi\)
\(32\) −1.47892 + 0.396276i −0.261439 + 0.0700524i
\(33\) −0.875396 + 6.70760i −0.152387 + 1.16764i
\(34\) 3.51939i 0.603570i
\(35\) 0 0
\(36\) −0.789616 + 0.213483i −0.131603 + 0.0355804i
\(37\) −6.60698 − 1.77034i −1.08618 − 0.291041i −0.329056 − 0.944311i \(-0.606730\pi\)
−0.757126 + 0.653269i \(0.773397\pi\)
\(38\) 0.632766 + 2.36152i 0.102648 + 0.383088i
\(39\) −1.46434 + 3.54652i −0.234483 + 0.567897i
\(40\) −1.27465 − 6.55619i −0.201540 − 1.03662i
\(41\) 0.308469i 0.0481748i 0.999710 + 0.0240874i \(0.00766800\pi\)
−0.999710 + 0.0240874i \(0.992332\pi\)
\(42\) 0 0
\(43\) 7.60892 + 7.60892i 1.16035 + 1.16035i 0.984399 + 0.175950i \(0.0562999\pi\)
0.175950 + 0.984399i \(0.443700\pi\)
\(44\) 0.532425 + 0.922186i 0.0802660 + 0.139025i
\(45\) −1.29505 − 6.58201i −0.193055 − 0.981188i
\(46\) 1.62131 − 2.80820i 0.239050 − 0.414046i
\(47\) 1.36920 − 5.10994i 0.199719 − 0.745361i −0.791276 − 0.611460i \(-0.790583\pi\)
0.990995 − 0.133902i \(-0.0427506\pi\)
\(48\) 3.55903 − 4.64904i 0.513702 − 0.671031i
\(49\) 0 0
\(50\) 6.52238 − 0.801352i 0.922404 − 0.113328i
\(51\) 2.82762 + 3.67646i 0.395947 + 0.514807i
\(52\) 0.156327 + 0.583421i 0.0216787 + 0.0809059i
\(53\) 0.498259 + 1.85953i 0.0684411 + 0.255426i 0.991666 − 0.128834i \(-0.0411234\pi\)
−0.923225 + 0.384260i \(0.874457\pi\)
\(54\) 4.13906 − 5.43199i 0.563254 − 0.739200i
\(55\) −7.84894 + 3.82855i −1.05835 + 0.516242i
\(56\) 0 0
\(57\) 2.55835 + 1.95852i 0.338861 + 0.259412i
\(58\) −0.174555 + 0.651448i −0.0229202 + 0.0855393i
\(59\) −0.259114 + 0.448799i −0.0337338 + 0.0584287i −0.882399 − 0.470501i \(-0.844073\pi\)
0.848666 + 0.528930i \(0.177407\pi\)
\(60\) −0.791703 − 0.698800i −0.102208 − 0.0902146i
\(61\) 2.55451 + 4.42454i 0.327071 + 0.566504i 0.981929 − 0.189248i \(-0.0606049\pi\)
−0.654858 + 0.755752i \(0.727272\pi\)
\(62\) 7.97631 + 7.97631i 1.01299 + 1.01299i
\(63\) 0 0
\(64\) − 8.77299i − 1.09662i
\(65\) −4.86242 + 0.945351i −0.603109 + 0.117256i
\(66\) −8.21753 − 3.39299i −1.01151 − 0.417648i
\(67\) 2.34332 + 8.74539i 0.286282 + 1.06842i 0.947897 + 0.318576i \(0.103205\pi\)
−0.661615 + 0.749844i \(0.730129\pi\)
\(68\) 0.705238 + 0.188968i 0.0855227 + 0.0229157i
\(69\) −0.562551 − 4.23616i −0.0677232 − 0.509974i
\(70\) 0 0
\(71\) − 15.3749i − 1.82467i −0.409448 − 0.912333i \(-0.634279\pi\)
0.409448 − 0.912333i \(-0.365721\pi\)
\(72\) −0.0201641 − 8.96073i −0.00237637 − 1.05603i
\(73\) −2.79871 + 0.749913i −0.327565 + 0.0877707i −0.418854 − 0.908054i \(-0.637568\pi\)
0.0912890 + 0.995824i \(0.470901\pi\)
\(74\) 4.49489 − 7.78538i 0.522520 − 0.905032i
\(75\) 6.16963 − 6.07747i 0.712408 − 0.701766i
\(76\) 0.507191 0.0581788
\(77\) 0 0
\(78\) −4.00420 − 3.06538i −0.453386 − 0.347086i
\(79\) 4.37551 − 2.52620i 0.492284 − 0.284220i −0.233238 − 0.972420i \(-0.574932\pi\)
0.725521 + 0.688200i \(0.241599\pi\)
\(80\) 7.54030 + 0.526980i 0.843031 + 0.0589182i
\(81\) −0.0405048 − 8.99991i −0.00450054 − 0.999990i
\(82\) −0.391602 − 0.104930i −0.0432452 − 0.0115875i
\(83\) −9.16088 + 9.16088i −1.00554 + 1.00554i −0.00555287 + 0.999985i \(0.501768\pi\)
−0.999985 + 0.00555287i \(0.998232\pi\)
\(84\) 0 0
\(85\) −1.94920 + 5.66159i −0.211421 + 0.614086i
\(86\) −12.2478 + 7.07127i −1.32071 + 0.762515i
\(87\) 0.341055 + 0.820767i 0.0365650 + 0.0879955i
\(88\) −11.2678 + 3.01921i −1.20116 + 0.321849i
\(89\) 5.67519 + 9.82972i 0.601569 + 1.04195i 0.992584 + 0.121564i \(0.0387910\pi\)
−0.391014 + 0.920385i \(0.627876\pi\)
\(90\) 8.79640 + 0.594879i 0.927222 + 0.0627058i
\(91\) 0 0
\(92\) −0.475671 − 0.475671i −0.0495922 − 0.0495922i
\(93\) 14.7408 + 1.92379i 1.52855 + 0.199488i
\(94\) 6.02133 + 3.47641i 0.621052 + 0.358565i
\(95\) −0.289995 + 4.14939i −0.0297528 + 0.425719i
\(96\) −1.61676 − 2.10210i −0.165010 − 0.214545i
\(97\) 6.81964 − 6.81964i 0.692430 − 0.692430i −0.270336 − 0.962766i \(-0.587135\pi\)
0.962766 + 0.270336i \(0.0871349\pi\)
\(98\) 0 0
\(99\) −11.3103 + 3.05789i −1.13673 + 0.307330i
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 735.2.y.i.263.4 48
3.2 odd 2 inner 735.2.y.i.263.9 48
5.2 odd 4 inner 735.2.y.i.557.4 48
7.2 even 3 inner 735.2.y.i.128.9 48
7.3 odd 6 735.2.j.g.638.4 24
7.4 even 3 735.2.j.e.638.4 24
7.5 odd 6 105.2.x.a.23.9 yes 48
7.6 odd 2 105.2.x.a.53.4 yes 48
15.2 even 4 inner 735.2.y.i.557.9 48
21.2 odd 6 inner 735.2.y.i.128.4 48
21.5 even 6 105.2.x.a.23.4 yes 48
21.11 odd 6 735.2.j.e.638.9 24
21.17 even 6 735.2.j.g.638.9 24
21.20 even 2 105.2.x.a.53.9 yes 48
35.2 odd 12 inner 735.2.y.i.422.9 48
35.12 even 12 105.2.x.a.2.9 yes 48
35.13 even 4 525.2.bf.f.32.9 48
35.17 even 12 735.2.j.g.197.9 24
35.19 odd 6 525.2.bf.f.443.4 48
35.27 even 4 105.2.x.a.32.4 yes 48
35.32 odd 12 735.2.j.e.197.9 24
35.33 even 12 525.2.bf.f.107.4 48
35.34 odd 2 525.2.bf.f.368.9 48
105.2 even 12 inner 735.2.y.i.422.4 48
105.17 odd 12 735.2.j.g.197.4 24
105.32 even 12 735.2.j.e.197.4 24
105.47 odd 12 105.2.x.a.2.4 ✓ 48
105.62 odd 4 105.2.x.a.32.9 yes 48
105.68 odd 12 525.2.bf.f.107.9 48
105.83 odd 4 525.2.bf.f.32.4 48
105.89 even 6 525.2.bf.f.443.9 48
105.104 even 2 525.2.bf.f.368.4 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.x.a.2.4 ✓ 48 105.47 odd 12
105.2.x.a.2.9 yes 48 35.12 even 12
105.2.x.a.23.4 yes 48 21.5 even 6
105.2.x.a.23.9 yes 48 7.5 odd 6
105.2.x.a.32.4 yes 48 35.27 even 4
105.2.x.a.32.9 yes 48 105.62 odd 4
105.2.x.a.53.4 yes 48 7.6 odd 2
105.2.x.a.53.9 yes 48 21.20 even 2
525.2.bf.f.32.4 48 105.83 odd 4
525.2.bf.f.32.9 48 35.13 even 4
525.2.bf.f.107.4 48 35.33 even 12
525.2.bf.f.107.9 48 105.68 odd 12
525.2.bf.f.368.4 48 105.104 even 2
525.2.bf.f.368.9 48 35.34 odd 2
525.2.bf.f.443.4 48 35.19 odd 6
525.2.bf.f.443.9 48 105.89 even 6
735.2.j.e.197.4 24 105.32 even 12
735.2.j.e.197.9 24 35.32 odd 12
735.2.j.e.638.4 24 7.4 even 3
735.2.j.e.638.9 24 21.11 odd 6
735.2.j.g.197.4 24 105.17 odd 12
735.2.j.g.197.9 24 35.17 even 12
735.2.j.g.638.4 24 7.3 odd 6
735.2.j.g.638.9 24 21.17 even 6
735.2.y.i.128.4 48 21.2 odd 6 inner
735.2.y.i.128.9 48 7.2 even 3 inner
735.2.y.i.263.4 48 1.1 even 1 trivial
735.2.y.i.263.9 48 3.2 odd 2 inner
735.2.y.i.422.4 48 105.2 even 12 inner
735.2.y.i.422.9 48 35.2 odd 12 inner
735.2.y.i.557.4 48 5.2 odd 4 inner
735.2.y.i.557.9 48 15.2 even 4 inner