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Properties

Label 735.2.y.i.263.7
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 263.7
Character \(\chi\) \(=\) 735.263
Dual form 735.2.y.i.422.7

$q$-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+(0.0799329 - 0.298314i) q^{2} +(-0.540759 + 1.64547i) q^{3} +(1.64945 + 0.952310i) q^{4} +(-0.596180 - 2.15513i) q^{5} +(0.447643 + 0.292843i) q^{6} +(0.852694 - 0.852694i) q^{8} +(-2.41516 - 1.77961i) q^{9} +(-0.690558 + 0.00558322i) q^{10} +(0.660315 + 0.381233i) q^{11} +(-2.45895 + 2.19915i) q^{12} +(2.27077 + 2.27077i) q^{13} +(3.86859 + 0.184406i) q^{15} +(1.71841 + 2.97637i) q^{16} +(4.69471 - 1.25794i) q^{17} +(-0.723932 + 0.578226i) q^{18} +(1.41761 - 0.818455i) q^{19} +(1.06898 - 4.12252i) q^{20} +(0.166508 - 0.166508i) q^{22} +(7.39003 + 1.98015i) q^{23} +(0.941983 + 1.86419i) q^{24} +(-4.28914 + 2.56969i) q^{25} +(0.858909 - 0.495891i) q^{26} +(4.23432 - 3.01174i) q^{27} -4.94251 q^{29} +(0.364238 - 1.13931i) q^{30} +(-2.96413 + 5.13403i) q^{31} +(3.35485 - 0.898930i) q^{32} +(-0.984380 + 0.880375i) q^{33} -1.50105i q^{34} +(-2.28894 - 5.23535i) q^{36} +(3.41587 + 0.915280i) q^{37} +(-0.130843 - 0.488313i) q^{38} +(-4.96442 + 2.50855i) q^{39} +(-2.34602 - 1.32930i) q^{40} -4.35963i q^{41} +(2.69037 + 2.69037i) q^{43} +(0.726104 + 1.25765i) q^{44} +(-2.39541 + 6.26594i) q^{45} +(1.18141 - 2.04627i) q^{46} +(-1.10971 + 4.14148i) q^{47} +(-5.82678 + 1.21809i) q^{48} +(0.423730 + 1.48491i) q^{50} +(-0.468795 + 8.40527i) q^{51} +(1.58304 + 5.90798i) q^{52} +(-1.79889 - 6.71354i) q^{53} +(-0.559982 - 1.50389i) q^{54} +(0.427939 - 1.65035i) q^{55} +(0.580162 + 2.77522i) q^{57} +(-0.395069 + 1.47442i) q^{58} +(-3.84501 + 6.65975i) q^{59} +(6.20543 + 3.98826i) q^{60} +(2.19699 + 3.80529i) q^{61} +(1.29462 + 1.29462i) q^{62} +5.80098i q^{64} +(3.54000 - 6.24757i) q^{65} +(0.183944 + 0.364025i) q^{66} +(0.0126297 + 0.0471345i) q^{67} +(8.94164 + 2.39591i) q^{68} +(-7.25451 + 11.0893i) q^{69} -12.4172i q^{71} +(-3.57685 + 0.541931i) q^{72} +(1.34043 - 0.359168i) q^{73} +(0.546081 - 0.945840i) q^{74} +(-1.90896 - 8.44724i) q^{75} +3.11769 q^{76} +(0.351513 + 1.68147i) q^{78} +(3.66808 - 2.11777i) q^{79} +(5.38997 - 5.47784i) q^{80} +(2.66599 + 8.59607i) q^{81} +(-1.30054 - 0.348478i) q^{82} +(5.05351 - 5.05351i) q^{83} +(-5.50993 - 9.36774i) q^{85} +(1.01762 - 0.587525i) q^{86} +(2.67271 - 8.13276i) q^{87} +(0.888122 - 0.237971i) q^{88} +(0.453600 + 0.785658i) q^{89} +(1.67774 + 1.21544i) q^{90} +(10.3038 + 10.3038i) q^{92} +(-6.84502 - 7.65367i) q^{93} +(1.14676 + 0.662081i) q^{94} +(-2.60902 - 2.56717i) q^{95} +(-0.335002 + 6.00642i) q^{96} +(-3.73061 + 3.73061i) q^{97} +(-0.916321 - 2.09584i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0799329 − 0.298314i 0.0565211 − 0.210940i −0.931890 − 0.362741i \(-0.881841\pi\)
0.988411 + 0.151802i \(0.0485075\pi\)
\(3\) −0.540759 + 1.64547i −0.312207 + 0.950014i
\(4\) 1.64945 + 0.952310i 0.824725 + 0.476155i
\(5\) −0.596180 − 2.15513i −0.266620 − 0.963802i
\(6\) 0.447643 + 0.292843i 0.182749 + 0.119553i
\(7\) 0 0
\(8\) 0.852694 − 0.852694i 0.301473 − 0.301473i
\(9\) −2.41516 − 1.77961i −0.805053 − 0.593203i
\(10\) −0.690558 + 0.00558322i −0.218374 + 0.00176557i
\(11\) 0.660315 + 0.381233i 0.199092 + 0.114946i 0.596232 − 0.802812i \(-0.296664\pi\)
−0.397140 + 0.917758i \(0.629997\pi\)
\(12\) −2.45895 + 2.19915i −0.709839 + 0.634841i
\(13\) 2.27077 + 2.27077i 0.629797 + 0.629797i 0.948017 − 0.318220i \(-0.103085\pi\)
−0.318220 + 0.948017i \(0.603085\pi\)
\(14\) 0 0
\(15\) 3.86859 + 0.184406i 0.998866 + 0.0476133i
\(16\) 1.71841 + 2.97637i 0.429602 + 0.744092i
\(17\) 4.69471 − 1.25794i 1.13864 − 0.305096i 0.360233 − 0.932862i \(-0.382697\pi\)
0.778402 + 0.627766i \(0.216030\pi\)
\(18\) −0.723932 + 0.578226i −0.170632 + 0.136289i
\(19\) 1.41761 − 0.818455i 0.325221 − 0.187767i −0.328496 − 0.944505i \(-0.606542\pi\)
0.653717 + 0.756739i \(0.273209\pi\)
\(20\) 1.06898 − 4.12252i 0.239031 − 0.921823i
\(21\) 0 0
\(22\) 0.166508 − 0.166508i 0.0354996 − 0.0354996i
\(23\) 7.39003 + 1.98015i 1.54093 + 0.412890i 0.926566 − 0.376133i \(-0.122747\pi\)
0.614363 + 0.789024i \(0.289413\pi\)
\(24\) 0.941983 + 1.86419i 0.192281 + 0.380525i
\(25\) −4.28914 + 2.56969i −0.857828 + 0.513937i
\(26\) 0.858909 − 0.495891i 0.168446 − 0.0972523i
\(27\) 4.23432 − 3.01174i 0.814894 − 0.579610i
\(28\) 0 0
\(29\) −4.94251 −0.917801 −0.458900 − 0.888488i \(-0.651757\pi\)
−0.458900 + 0.888488i \(0.651757\pi\)
\(30\) 0.364238 − 1.13931i 0.0665005 − 0.208009i
\(31\) −2.96413 + 5.13403i −0.532374 + 0.922099i 0.466911 + 0.884304i \(0.345367\pi\)
−0.999286 + 0.0377949i \(0.987967\pi\)
\(32\) 3.35485 − 0.898930i 0.593060 − 0.158910i
\(33\) −0.984380 + 0.880375i −0.171358 + 0.153254i
\(34\) − 1.50105i − 0.257428i
\(35\) 0 0
\(36\) −2.28894 − 5.23535i −0.381491 − 0.872559i
\(37\) 3.41587 + 0.915280i 0.561566 + 0.150471i 0.528426 − 0.848980i \(-0.322783\pi\)
0.0331401 + 0.999451i \(0.489449\pi\)
\(38\) −0.130843 − 0.488313i −0.0212255 − 0.0792148i
\(39\) −4.96442 + 2.50855i −0.794943 + 0.401689i
\(40\) −2.34602 − 1.32930i −0.370939 − 0.210181i
\(41\) − 4.35963i − 0.680860i −0.940270 − 0.340430i \(-0.889427\pi\)
0.940270 − 0.340430i \(-0.110573\pi\)
\(42\) 0 0
\(43\) 2.69037 + 2.69037i 0.410277 + 0.410277i 0.881835 − 0.471558i \(-0.156308\pi\)
−0.471558 + 0.881835i \(0.656308\pi\)
\(44\) 0.726104 + 1.25765i 0.109464 + 0.189598i
\(45\) −2.39541 + 6.26594i −0.357087 + 0.934071i
\(46\) 1.18141 − 2.04627i 0.174190 − 0.301706i
\(47\) −1.10971 + 4.14148i −0.161867 + 0.604097i 0.836552 + 0.547888i \(0.184568\pi\)
−0.998419 + 0.0562089i \(0.982099\pi\)
\(48\) −5.82678 + 1.21809i −0.841023 + 0.175817i
\(49\) 0 0
\(50\) 0.423730 + 1.48491i 0.0599244 + 0.209998i
\(51\) −0.468795 + 8.40527i −0.0656444 + 1.17697i
\(52\) 1.58304 + 5.90798i 0.219528 + 0.819290i
\(53\) −1.79889 − 6.71354i −0.247096 − 0.922176i −0.972318 − 0.233661i \(-0.924929\pi\)
0.725222 − 0.688515i \(-0.241737\pi\)
\(54\) −0.559982 − 1.50389i −0.0762039 − 0.204654i
\(55\) 0.427939 − 1.65035i 0.0577032 − 0.222533i
\(56\) 0 0
\(57\) 0.580162 + 2.77522i 0.0768444 + 0.367587i
\(58\) −0.395069 + 1.47442i −0.0518751 + 0.193601i
\(59\) −3.84501 + 6.65975i −0.500577 + 0.867026i 0.499422 + 0.866359i \(0.333546\pi\)
−1.00000 0.000666931i \(0.999788\pi\)
\(60\) 6.20543 + 3.98826i 0.801118 + 0.514883i
\(61\) 2.19699 + 3.80529i 0.281295 + 0.487218i 0.971704 − 0.236202i \(-0.0759026\pi\)
−0.690409 + 0.723420i \(0.742569\pi\)
\(62\) 1.29462 + 1.29462i 0.164417 + 0.164417i
\(63\) 0 0
\(64\) 5.80098i 0.725122i
\(65\) 3.54000 − 6.24757i 0.439083 − 0.774916i
\(66\) 0.183944 + 0.364025i 0.0226419 + 0.0448084i
\(67\) 0.0126297 + 0.0471345i 0.00154296 + 0.00575840i 0.966693 − 0.255939i \(-0.0823845\pi\)
−0.965150 + 0.261697i \(0.915718\pi\)
\(68\) 8.94164 + 2.39591i 1.08433 + 0.290546i
\(69\) −7.25451 + 11.0893i −0.873341 + 1.33500i
\(70\) 0 0
\(71\) − 12.4172i − 1.47365i −0.676082 − 0.736826i \(-0.736324\pi\)
0.676082 − 0.736826i \(-0.263676\pi\)
\(72\) −3.57685 + 0.541931i −0.421536 + 0.0638672i
\(73\) 1.34043 − 0.359168i 0.156886 − 0.0420374i −0.179521 − 0.983754i \(-0.557455\pi\)
0.336407 + 0.941717i \(0.390788\pi\)
\(74\) 0.546081 − 0.945840i 0.0634806 − 0.109952i
\(75\) −1.90896 − 8.44724i −0.220428 − 0.975403i
\(76\) 3.11769 0.357624
\(77\) 0 0
\(78\) 0.351513 + 1.68147i 0.0398010 + 0.190389i
\(79\) 3.66808 − 2.11777i 0.412692 − 0.238268i −0.279254 − 0.960217i \(-0.590087\pi\)
0.691946 + 0.721950i \(0.256754\pi\)
\(80\) 5.38997 − 5.47784i 0.602617 − 0.612441i
\(81\) 2.66599 + 8.59607i 0.296221 + 0.955119i
\(82\) −1.30054 − 0.348478i −0.143620 − 0.0384830i
\(83\) 5.05351 − 5.05351i 0.554695 − 0.554695i −0.373097 − 0.927792i \(-0.621704\pi\)
0.927792 + 0.373097i \(0.121704\pi\)
\(84\) 0 0
\(85\) −5.50993 − 9.36774i −0.597635 − 1.01607i
\(86\) 1.01762 − 0.587525i 0.109733 − 0.0633544i
\(87\) 2.67271 − 8.13276i 0.286544 − 0.871924i
\(88\) 0.888122 − 0.237971i 0.0946741 − 0.0253678i
\(89\) 0.453600 + 0.785658i 0.0480815 + 0.0832796i 0.889065 − 0.457782i \(-0.151356\pi\)
−0.840983 + 0.541061i \(0.818023\pi\)
\(90\) 1.67774 + 1.21544i 0.176850 + 0.128118i
\(91\) 0 0
\(92\) 10.3038 + 10.3038i 1.07424 + 1.07424i
\(93\) −6.84502 − 7.65367i −0.709796 − 0.793649i
\(94\) 1.14676 + 0.662081i 0.118279 + 0.0682884i
\(95\) −2.60902 − 2.56717i −0.267680 − 0.263386i
\(96\) −0.335002 + 6.00642i −0.0341910 + 0.613028i
\(97\) −3.73061 + 3.73061i −0.378786 + 0.378786i −0.870664 − 0.491878i \(-0.836311\pi\)
0.491878 + 0.870664i \(0.336311\pi\)
\(98\) 0 0
\(99\) −0.916321 − 2.09584i −0.0920937 − 0.210640i
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.263.7 48
3.2 odd 2 inner 735.2.y.i.263.6 48
5.2 odd 4 inner 735.2.y.i.557.7 48
7.2 even 3 inner 735.2.y.i.128.6 48
7.3 odd 6 735.2.j.g.638.7 24
7.4 even 3 735.2.j.e.638.7 24
7.5 odd 6 105.2.x.a.23.6 yes 48
7.6 odd 2 105.2.x.a.53.7 yes 48
15.2 even 4 inner 735.2.y.i.557.6 48
21.2 odd 6 inner 735.2.y.i.128.7 48
21.5 even 6 105.2.x.a.23.7 yes 48
21.11 odd 6 735.2.j.e.638.6 24
21.17 even 6 735.2.j.g.638.6 24
21.20 even 2 105.2.x.a.53.6 yes 48
35.2 odd 12 inner 735.2.y.i.422.6 48
35.12 even 12 105.2.x.a.2.6 ✓ 48
35.13 even 4 525.2.bf.f.32.6 48
35.17 even 12 735.2.j.g.197.6 24
35.19 odd 6 525.2.bf.f.443.7 48
35.27 even 4 105.2.x.a.32.7 yes 48
35.32 odd 12 735.2.j.e.197.6 24
35.33 even 12 525.2.bf.f.107.7 48
35.34 odd 2 525.2.bf.f.368.6 48
105.2 even 12 inner 735.2.y.i.422.7 48
105.17 odd 12 735.2.j.g.197.7 24
105.32 even 12 735.2.j.e.197.7 24
105.47 odd 12 105.2.x.a.2.7 yes 48
105.62 odd 4 105.2.x.a.32.6 yes 48
105.68 odd 12 525.2.bf.f.107.6 48
105.83 odd 4 525.2.bf.f.32.7 48
105.89 even 6 525.2.bf.f.443.6 48
105.104 even 2 525.2.bf.f.368.7 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.x.a.2.6 ✓ 48 35.12 even 12
105.2.x.a.2.7 yes 48 105.47 odd 12
105.2.x.a.23.6 yes 48 7.5 odd 6
105.2.x.a.23.7 yes 48 21.5 even 6
105.2.x.a.32.6 yes 48 105.62 odd 4
105.2.x.a.32.7 yes 48 35.27 even 4
105.2.x.a.53.6 yes 48 21.20 even 2
105.2.x.a.53.7 yes 48 7.6 odd 2
525.2.bf.f.32.6 48 35.13 even 4
525.2.bf.f.32.7 48 105.83 odd 4
525.2.bf.f.107.6 48 105.68 odd 12
525.2.bf.f.107.7 48 35.33 even 12
525.2.bf.f.368.6 48 35.34 odd 2
525.2.bf.f.368.7 48 105.104 even 2
525.2.bf.f.443.6 48 105.89 even 6
525.2.bf.f.443.7 48 35.19 odd 6
735.2.j.e.197.6 24 35.32 odd 12
735.2.j.e.197.7 24 105.32 even 12
735.2.j.e.638.6 24 21.11 odd 6
735.2.j.e.638.7 24 7.4 even 3
735.2.j.g.197.6 24 35.17 even 12
735.2.j.g.197.7 24 105.17 odd 12
735.2.j.g.638.6 24 21.17 even 6
735.2.j.g.638.7 24 7.3 odd 6
735.2.y.i.128.6 48 7.2 even 3 inner
735.2.y.i.128.7 48 21.2 odd 6 inner
735.2.y.i.263.6 48 3.2 odd 2 inner
735.2.y.i.263.7 48 1.1 even 1 trivial
735.2.y.i.422.6 48 35.2 odd 12 inner
735.2.y.i.422.7 48 105.2 even 12 inner
735.2.y.i.557.6 48 15.2 even 4 inner
735.2.y.i.557.7 48 5.2 odd 4 inner