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Properties

Label 245.3.h.c.166.5
Level $245$
Weight $3$
Character 245.166
Analytic conductor $6.676$
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] = [245,3,Mod(31,245)] 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("245.31"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-2] 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: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 166.5
Root \(-0.925400 - 1.60284i\) of defining polynomial
Character \(\chi\) \(=\) 245.166
Dual form 245.3.h.c.31.5

$q$-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+(0.925400 - 1.60284i) q^{2} +(-0.731043 + 0.422068i) q^{3} +(0.287270 + 0.497567i) q^{4} +(1.93649 + 1.11803i) q^{5} +1.56233i q^{6} +8.46656 q^{8} +(-4.14372 + 7.17713i) q^{9} +(3.58406 - 2.06926i) q^{10} +(3.93873 + 6.82208i) q^{11} +(-0.420014 - 0.242495i) q^{12} +2.73368i q^{13} -1.88755 q^{15} +(6.68587 - 11.5803i) q^{16} +(-4.29344 + 2.47882i) q^{17} +(7.66919 + 13.2834i) q^{18} +(25.3152 + 14.6158i) q^{19} +1.28471i q^{20} +14.5796 q^{22} +(18.0840 - 31.3224i) q^{23} +(-6.18942 + 3.57346i) q^{24} +(2.50000 + 4.33013i) q^{25} +(4.38165 + 2.52975i) q^{26} -14.5929i q^{27} +0.399005 q^{29} +(-1.74673 + 3.02543i) q^{30} +(-17.4396 + 10.0688i) q^{31} +(4.55891 + 7.89627i) q^{32} +(-5.75876 - 3.32482i) q^{33} +9.17558i q^{34} -4.76147 q^{36} +(3.61299 - 6.25789i) q^{37} +(46.8534 - 27.0508i) q^{38} +(-1.15380 - 1.99844i) q^{39} +(16.3954 + 9.46590i) q^{40} -71.1139i q^{41} -51.7627 q^{43} +(-2.26296 + 3.91956i) q^{44} +(-16.0485 + 9.26563i) q^{45} +(-33.4699 - 57.9715i) q^{46} +(19.9766 + 11.5335i) q^{47} +11.2876i q^{48} +9.25400 q^{50} +(2.09246 - 3.62424i) q^{51} +(-1.36019 + 0.785306i) q^{52} +(26.4177 + 45.7568i) q^{53} +(-23.3902 - 13.5043i) q^{54} +17.6145i q^{55} -24.6754 q^{57} +(0.369239 - 0.639542i) q^{58} +(-73.6658 + 42.5310i) q^{59} +(-0.542236 - 0.939181i) q^{60} +(-75.4986 - 43.5891i) q^{61} +37.2705i q^{62} +70.3622 q^{64} +(-3.05635 + 5.29375i) q^{65} +(-10.6583 + 6.15358i) q^{66} +(-34.7881 - 60.2548i) q^{67} +(-2.46675 - 1.42418i) q^{68} +30.5307i q^{69} -26.4541 q^{71} +(-35.0830 + 60.7656i) q^{72} +(114.650 - 66.1933i) q^{73} +(-6.68693 - 11.5821i) q^{74} +(-3.65522 - 2.11034i) q^{75} +16.7947i q^{76} -4.27090 q^{78} +(-53.6505 + 92.9253i) q^{79} +(25.8943 - 14.9501i) q^{80} +(-31.1342 - 53.9261i) q^{81} +(-113.984 - 65.8087i) q^{82} +53.9702i q^{83} -11.0856 q^{85} +(-47.9012 + 82.9674i) q^{86} +(-0.291690 + 0.168407i) q^{87} +(33.3475 + 57.7595i) q^{88} +(56.2194 + 32.4583i) q^{89} +34.2977i q^{90} +20.7800 q^{92} +(8.49941 - 14.7214i) q^{93} +(36.9727 - 21.3462i) q^{94} +(32.6818 + 56.6066i) q^{95} +(-6.66552 - 3.84834i) q^{96} -109.504i q^{97} -65.2839 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 6 q^{3} - 10 q^{4} - 4 q^{8} + 14 q^{9} - 14 q^{11} - 18 q^{12} - 20 q^{15} - 22 q^{16} - 48 q^{17} + 64 q^{18} + 30 q^{19} - 88 q^{22} - 14 q^{23} + 36 q^{24} + 30 q^{25} - 66 q^{26} + 64 q^{29}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.925400 − 1.60284i 0.462700 − 0.801420i −0.536395 − 0.843967i \(-0.680214\pi\)
0.999094 + 0.0425476i \(0.0135474\pi\)
\(3\) −0.731043 + 0.422068i −0.243681 + 0.140689i −0.616867 − 0.787067i \(-0.711599\pi\)
0.373186 + 0.927756i \(0.378265\pi\)
\(4\) 0.287270 + 0.497567i 0.0718176 + 0.124392i
\(5\) 1.93649 + 1.11803i 0.387298 + 0.223607i
\(6\) 1.56233i 0.260388i
\(7\) 0 0
\(8\) 8.46656 1.05832
\(9\) −4.14372 + 7.17713i −0.460413 + 0.797459i
\(10\) 3.58406 − 2.06926i 0.358406 − 0.206926i
\(11\) 3.93873 + 6.82208i 0.358066 + 0.620189i 0.987638 − 0.156754i \(-0.0501029\pi\)
−0.629572 + 0.776942i \(0.716770\pi\)
\(12\) −0.420014 − 0.242495i −0.0350012 − 0.0202079i
\(13\) 2.73368i 0.210283i 0.994457 + 0.105142i \(0.0335296\pi\)
−0.994457 + 0.105142i \(0.966470\pi\)
\(14\) 0 0
\(15\) −1.88755 −0.125836
\(16\) 6.68587 − 11.5803i 0.417867 − 0.723767i
\(17\) −4.29344 + 2.47882i −0.252555 + 0.145813i −0.620934 − 0.783863i \(-0.713246\pi\)
0.368379 + 0.929676i \(0.379913\pi\)
\(18\) 7.66919 + 13.2834i 0.426066 + 0.737968i
\(19\) 25.3152 + 14.6158i 1.33238 + 0.769250i 0.985664 − 0.168720i \(-0.0539633\pi\)
0.346716 + 0.937970i \(0.387297\pi\)
\(20\) 1.28471i 0.0642356i
\(21\) 0 0
\(22\) 14.5796 0.662709
\(23\) 18.0840 − 31.3224i 0.786261 − 1.36184i −0.141982 − 0.989869i \(-0.545347\pi\)
0.928243 − 0.371975i \(-0.121319\pi\)
\(24\) −6.18942 + 3.57346i −0.257893 + 0.148894i
\(25\) 2.50000 + 4.33013i 0.100000 + 0.173205i
\(26\) 4.38165 + 2.52975i 0.168525 + 0.0972980i
\(27\) − 14.5929i − 0.540480i
\(28\) 0 0
\(29\) 0.399005 0.0137588 0.00687940 − 0.999976i \(-0.497810\pi\)
0.00687940 + 0.999976i \(0.497810\pi\)
\(30\) −1.74673 + 3.02543i −0.0582245 + 0.100848i
\(31\) −17.4396 + 10.0688i −0.562568 + 0.324799i −0.754176 − 0.656673i \(-0.771963\pi\)
0.191607 + 0.981472i \(0.438630\pi\)
\(32\) 4.55891 + 7.89627i 0.142466 + 0.246758i
\(33\) −5.75876 − 3.32482i −0.174508 − 0.100752i
\(34\) 9.17558i 0.269870i
\(35\) 0 0
\(36\) −4.76147 −0.132263
\(37\) 3.61299 − 6.25789i 0.0976485 − 0.169132i −0.813062 − 0.582177i \(-0.802201\pi\)
0.910711 + 0.413044i \(0.135535\pi\)
\(38\) 46.8534 − 27.0508i 1.23298 − 0.711864i
\(39\) −1.15380 − 1.99844i −0.0295846 − 0.0512420i
\(40\) 16.3954 + 9.46590i 0.409885 + 0.236648i
\(41\) − 71.1139i − 1.73448i −0.497887 − 0.867242i \(-0.665890\pi\)
0.497887 − 0.867242i \(-0.334110\pi\)
\(42\) 0 0
\(43\) −51.7627 −1.20378 −0.601892 − 0.798577i \(-0.705586\pi\)
−0.601892 + 0.798577i \(0.705586\pi\)
\(44\) −2.26296 + 3.91956i −0.0514309 + 0.0890810i
\(45\) −16.0485 + 9.26563i −0.356634 + 0.205903i
\(46\) −33.4699 − 57.9715i −0.727606 − 1.26025i
\(47\) 19.9766 + 11.5335i 0.425034 + 0.245393i 0.697229 − 0.716849i \(-0.254416\pi\)
−0.272195 + 0.962242i \(0.587750\pi\)
\(48\) 11.2876i 0.235158i
\(49\) 0 0
\(50\) 9.25400 0.185080
\(51\) 2.09246 − 3.62424i 0.0410286 − 0.0710636i
\(52\) −1.36019 + 0.785306i −0.0261575 + 0.0151020i
\(53\) 26.4177 + 45.7568i 0.498447 + 0.863336i 0.999998 − 0.00179226i \(-0.000570495\pi\)
−0.501551 + 0.865128i \(0.667237\pi\)
\(54\) −23.3902 − 13.5043i −0.433151 − 0.250080i
\(55\) 17.6145i 0.320264i
\(56\) 0 0
\(57\) −24.6754 −0.432901
\(58\) 0.369239 − 0.639542i 0.00636620 − 0.0110266i
\(59\) −73.6658 + 42.5310i −1.24857 + 0.720864i −0.970825 − 0.239790i \(-0.922921\pi\)
−0.277748 + 0.960654i \(0.589588\pi\)
\(60\) −0.542236 − 0.939181i −0.00903727 − 0.0156530i
\(61\) −75.4986 − 43.5891i −1.23768 − 0.714576i −0.269061 − 0.963123i \(-0.586713\pi\)
−0.968620 + 0.248547i \(0.920047\pi\)
\(62\) 37.2705i 0.601138i
\(63\) 0 0
\(64\) 70.3622 1.09941
\(65\) −3.05635 + 5.29375i −0.0470208 + 0.0814423i
\(66\) −10.6583 + 6.15358i −0.161490 + 0.0932361i
\(67\) −34.7881 − 60.2548i −0.519226 − 0.899325i −0.999750 − 0.0223441i \(-0.992887\pi\)
0.480525 − 0.876981i \(-0.340446\pi\)
\(68\) −2.46675 − 1.42418i −0.0362758 − 0.0209438i
\(69\) 30.5307i 0.442474i
\(70\) 0 0
\(71\) −26.4541 −0.372593 −0.186297 − 0.982494i \(-0.559649\pi\)
−0.186297 + 0.982494i \(0.559649\pi\)
\(72\) −35.0830 + 60.7656i −0.487264 + 0.843966i
\(73\) 114.650 − 66.1933i 1.57055 − 0.906758i 0.574450 − 0.818540i \(-0.305216\pi\)
0.996101 − 0.0882179i \(-0.0281172\pi\)
\(74\) −6.68693 − 11.5821i −0.0903639 − 0.156515i
\(75\) −3.65522 − 2.11034i −0.0487362 − 0.0281379i
\(76\) 16.7947i 0.220983i
\(77\) 0 0
\(78\) −4.27090 −0.0547552
\(79\) −53.6505 + 92.9253i −0.679120 + 1.17627i 0.296127 + 0.955149i \(0.404305\pi\)
−0.975246 + 0.221121i \(0.929028\pi\)
\(80\) 25.8943 − 14.9501i 0.323678 − 0.186876i
\(81\) −31.1342 − 53.9261i −0.384373 − 0.665754i
\(82\) −113.984 − 65.8087i −1.39005 − 0.802546i
\(83\) 53.9702i 0.650244i 0.945672 + 0.325122i \(0.105405\pi\)
−0.945672 + 0.325122i \(0.894595\pi\)
\(84\) 0 0
\(85\) −11.0856 −0.130419
\(86\) −47.9012 + 82.9674i −0.556991 + 0.964737i
\(87\) −0.291690 + 0.168407i −0.00335276 + 0.00193572i
\(88\) 33.3475 + 57.7595i 0.378948 + 0.656358i
\(89\) 56.2194 + 32.4583i 0.631679 + 0.364700i 0.781402 − 0.624028i \(-0.214505\pi\)
−0.149723 + 0.988728i \(0.547838\pi\)
\(90\) 34.2977i 0.381085i
\(91\) 0 0
\(92\) 20.7800 0.225870
\(93\) 8.49941 − 14.7214i 0.0913915 − 0.158295i
\(94\) 36.9727 − 21.3462i 0.393326 − 0.227087i
\(95\) 32.6818 + 56.6066i 0.344019 + 0.595859i
\(96\) −6.66552 − 3.84834i −0.0694325 − 0.0400869i
\(97\) − 109.504i − 1.12891i −0.825463 − 0.564456i \(-0.809086\pi\)
0.825463 − 0.564456i \(-0.190914\pi\)
\(98\) 0 0
\(99\) −65.2839 −0.659433
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 245.3.h.c.166.5 12
7.2 even 3 245.3.d.a.146.3 12
7.3 odd 6 inner 245.3.h.c.31.5 12
7.4 even 3 35.3.h.a.31.5 yes 12
7.5 odd 6 245.3.d.a.146.4 12
7.6 odd 2 35.3.h.a.26.5 ✓ 12
21.11 odd 6 315.3.w.c.136.2 12
21.20 even 2 315.3.w.c.271.2 12
28.11 odd 6 560.3.bx.c.241.2 12
28.27 even 2 560.3.bx.c.481.2 12
35.4 even 6 175.3.i.d.101.2 12
35.13 even 4 175.3.j.b.124.4 24
35.18 odd 12 175.3.j.b.24.9 24
35.27 even 4 175.3.j.b.124.9 24
35.32 odd 12 175.3.j.b.24.4 24
35.34 odd 2 175.3.i.d.26.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.h.a.26.5 ✓ 12 7.6 odd 2
35.3.h.a.31.5 yes 12 7.4 even 3
175.3.i.d.26.2 12 35.34 odd 2
175.3.i.d.101.2 12 35.4 even 6
175.3.j.b.24.4 24 35.32 odd 12
175.3.j.b.24.9 24 35.18 odd 12
175.3.j.b.124.4 24 35.13 even 4
175.3.j.b.124.9 24 35.27 even 4
245.3.d.a.146.3 12 7.2 even 3
245.3.d.a.146.4 12 7.5 odd 6
245.3.h.c.31.5 12 7.3 odd 6 inner
245.3.h.c.166.5 12 1.1 even 1 trivial
315.3.w.c.136.2 12 21.11 odd 6
315.3.w.c.271.2 12 21.20 even 2
560.3.bx.c.241.2 12 28.11 odd 6
560.3.bx.c.481.2 12 28.27 even 2