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Properties

Label 245.3.h.c.166.4
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 166.4
Root \(0.242987 + 0.420865i\) of defining polynomial
Character \(\chi\) \(=\) 245.166
Dual form 245.3.h.c.31.4

$q$-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+(-0.242987 + 0.420865i) q^{2} +(4.74894 - 2.74180i) q^{3} +(1.88192 + 3.25957i) q^{4} +(1.93649 + 1.11803i) q^{5} +2.66488i q^{6} -3.77301 q^{8} +(10.5350 - 18.2471i) q^{9} +(-0.941083 + 0.543334i) q^{10} +(3.29893 + 5.71391i) q^{11} +(17.8742 + 10.3197i) q^{12} -3.29523i q^{13} +12.2617 q^{15} +(-6.61087 + 11.4504i) q^{16} +(-9.44949 + 5.45566i) q^{17} +(5.11970 + 8.86759i) q^{18} +(-8.27840 - 4.77954i) q^{19} +8.41618i q^{20} -3.20638 q^{22} +(5.80641 - 10.0570i) q^{23} +(-17.9178 + 10.3449i) q^{24} +(2.50000 + 4.33013i) q^{25} +(1.38685 + 0.800696i) q^{26} -66.1866i q^{27} -23.6149 q^{29} +(-2.97943 + 5.16052i) q^{30} +(-9.22362 + 5.32526i) q^{31} +(-10.7587 - 18.6347i) q^{32} +(31.3328 + 18.0900i) q^{33} -5.30261i q^{34} +79.3035 q^{36} +(31.3288 - 54.2630i) q^{37} +(4.02308 - 2.32273i) q^{38} +(-9.03486 - 15.6488i) q^{39} +(-7.30641 - 4.21836i) q^{40} +49.7889i q^{41} -7.82740 q^{43} +(-12.4166 + 21.5062i) q^{44} +(40.8017 - 23.5569i) q^{45} +(2.82176 + 4.88743i) q^{46} +(-26.4161 - 15.2513i) q^{47} +72.5028i q^{48} -2.42987 q^{50} +(-29.9167 + 51.8172i) q^{51} +(10.7410 - 6.20134i) q^{52} +(31.0976 + 53.8627i) q^{53} +(27.8556 + 16.0824i) q^{54} +14.7532i q^{55} -52.4182 q^{57} +(5.73811 - 9.93870i) q^{58} +(39.0701 - 22.5571i) q^{59} +(23.0755 + 39.9679i) q^{60} +(-35.0955 - 20.2624i) q^{61} -5.17587i q^{62} -42.4300 q^{64} +(3.68418 - 6.38118i) q^{65} +(-15.2269 + 8.79125i) q^{66} +(-26.3967 - 45.7204i) q^{67} +(-35.5663 - 20.5342i) q^{68} -63.6802i q^{69} -48.2295 q^{71} +(-39.7485 + 68.8464i) q^{72} +(25.2610 - 14.5844i) q^{73} +(15.2249 + 26.3704i) q^{74} +(23.7447 + 13.7090i) q^{75} -35.9787i q^{76} +8.78140 q^{78} +(5.47308 - 9.47965i) q^{79} +(-25.6038 + 14.7824i) q^{80} +(-86.6559 - 150.092i) q^{81} +(-20.9544 - 12.0980i) q^{82} +14.0223i q^{83} -24.3985 q^{85} +(1.90195 - 3.29428i) q^{86} +(-112.146 + 64.7474i) q^{87} +(-12.4469 - 21.5586i) q^{88} +(-91.7289 - 52.9597i) q^{89} +22.8960i q^{90} +43.7087 q^{92} +(-29.2016 + 50.5787i) q^{93} +(12.8375 - 7.41174i) q^{94} +(-10.6874 - 18.5111i) q^{95} +(-102.185 - 58.9966i) q^{96} +118.638i q^{97} +139.016 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.242987 + 0.420865i −0.121493 + 0.210433i −0.920357 − 0.391080i \(-0.872102\pi\)
0.798863 + 0.601512i \(0.205435\pi\)
\(3\) 4.74894 − 2.74180i 1.58298 − 0.913934i 0.588558 − 0.808455i \(-0.299696\pi\)
0.994422 − 0.105479i \(-0.0336376\pi\)
\(4\) 1.88192 + 3.25957i 0.470479 + 0.814893i
\(5\) 1.93649 + 1.11803i 0.387298 + 0.223607i
\(6\) 2.66488i 0.444147i
\(7\) 0 0
\(8\) −3.77301 −0.471627
\(9\) 10.5350 − 18.2471i 1.17055 − 2.02745i
\(10\) −0.941083 + 0.543334i −0.0941083 + 0.0543334i
\(11\) 3.29893 + 5.71391i 0.299902 + 0.519446i 0.976113 − 0.217262i \(-0.0697126\pi\)
−0.676211 + 0.736708i \(0.736379\pi\)
\(12\) 17.8742 + 10.3197i 1.48952 + 0.859973i
\(13\) − 3.29523i − 0.253479i −0.991936 − 0.126740i \(-0.959549\pi\)
0.991936 − 0.126740i \(-0.0404513\pi\)
\(14\) 0 0
\(15\) 12.2617 0.817447
\(16\) −6.61087 + 11.4504i −0.413179 + 0.715648i
\(17\) −9.44949 + 5.45566i −0.555852 + 0.320921i −0.751479 − 0.659757i \(-0.770659\pi\)
0.195627 + 0.980678i \(0.437326\pi\)
\(18\) 5.11970 + 8.86759i 0.284428 + 0.492644i
\(19\) −8.27840 − 4.77954i −0.435705 − 0.251555i 0.266069 − 0.963954i \(-0.414275\pi\)
−0.701774 + 0.712399i \(0.747608\pi\)
\(20\) 8.41618i 0.420809i
\(21\) 0 0
\(22\) −3.20638 −0.145744
\(23\) 5.80641 − 10.0570i 0.252453 − 0.437261i −0.711748 − 0.702435i \(-0.752096\pi\)
0.964201 + 0.265174i \(0.0854294\pi\)
\(24\) −17.9178 + 10.3449i −0.746575 + 0.431035i
\(25\) 2.50000 + 4.33013i 0.100000 + 0.173205i
\(26\) 1.38685 + 0.800696i 0.0533403 + 0.0307960i
\(27\) − 66.1866i − 2.45135i
\(28\) 0 0
\(29\) −23.6149 −0.814308 −0.407154 − 0.913360i \(-0.633479\pi\)
−0.407154 + 0.913360i \(0.633479\pi\)
\(30\) −2.97943 + 5.16052i −0.0993143 + 0.172017i
\(31\) −9.22362 + 5.32526i −0.297536 + 0.171783i −0.641336 − 0.767261i \(-0.721619\pi\)
0.343799 + 0.939043i \(0.388286\pi\)
\(32\) −10.7587 − 18.6347i −0.336210 − 0.582333i
\(33\) 31.3328 + 18.0900i 0.949479 + 0.548182i
\(34\) − 5.30261i − 0.155959i
\(35\) 0 0
\(36\) 79.3035 2.20288
\(37\) 31.3288 − 54.2630i 0.846724 − 1.46657i −0.0373921 − 0.999301i \(-0.511905\pi\)
0.884116 − 0.467268i \(-0.154762\pi\)
\(38\) 4.02308 − 2.32273i 0.105871 − 0.0611244i
\(39\) −9.03486 − 15.6488i −0.231663 − 0.401252i
\(40\) −7.30641 − 4.21836i −0.182660 − 0.105459i
\(41\) 49.7889i 1.21436i 0.794563 + 0.607182i \(0.207700\pi\)
−0.794563 + 0.607182i \(0.792300\pi\)
\(42\) 0 0
\(43\) −7.82740 −0.182033 −0.0910163 − 0.995849i \(-0.529012\pi\)
−0.0910163 + 0.995849i \(0.529012\pi\)
\(44\) −12.4166 + 21.5062i −0.282195 + 0.488777i
\(45\) 40.8017 − 23.5569i 0.906704 − 0.523486i
\(46\) 2.82176 + 4.88743i 0.0613426 + 0.106249i
\(47\) −26.4161 − 15.2513i −0.562044 − 0.324496i 0.191921 − 0.981410i \(-0.438528\pi\)
−0.753966 + 0.656914i \(0.771862\pi\)
\(48\) 72.5028i 1.51047i
\(49\) 0 0
\(50\) −2.42987 −0.0485973
\(51\) −29.9167 + 51.8172i −0.586602 + 1.01602i
\(52\) 10.7410 − 6.20134i 0.206558 − 0.119257i
\(53\) 31.0976 + 53.8627i 0.586748 + 1.01628i 0.994655 + 0.103254i \(0.0329254\pi\)
−0.407907 + 0.913023i \(0.633741\pi\)
\(54\) 27.8556 + 16.0824i 0.515845 + 0.297823i
\(55\) 14.7532i 0.268241i
\(56\) 0 0
\(57\) −52.4182 −0.919617
\(58\) 5.73811 − 9.93870i 0.0989329 − 0.171357i
\(59\) 39.0701 − 22.5571i 0.662205 − 0.382324i −0.130911 − 0.991394i \(-0.541790\pi\)
0.793117 + 0.609070i \(0.208457\pi\)
\(60\) 23.0755 + 39.9679i 0.384592 + 0.666132i
\(61\) −35.0955 − 20.2624i −0.575336 − 0.332170i 0.183942 − 0.982937i \(-0.441114\pi\)
−0.759278 + 0.650767i \(0.774448\pi\)
\(62\) − 5.17587i − 0.0834817i
\(63\) 0 0
\(64\) −42.4300 −0.662969
\(65\) 3.68418 − 6.38118i 0.0566797 − 0.0981721i
\(66\) −15.2269 + 8.79125i −0.230711 + 0.133201i
\(67\) −26.3967 − 45.7204i −0.393981 − 0.682395i 0.598990 − 0.800757i \(-0.295569\pi\)
−0.992970 + 0.118362i \(0.962236\pi\)
\(68\) −35.5663 − 20.5342i −0.523033 − 0.301973i
\(69\) − 63.6802i − 0.922901i
\(70\) 0 0
\(71\) −48.2295 −0.679289 −0.339644 − 0.940554i \(-0.610307\pi\)
−0.339644 + 0.940554i \(0.610307\pi\)
\(72\) −39.7485 + 68.8464i −0.552063 + 0.956200i
\(73\) 25.2610 − 14.5844i 0.346040 − 0.199787i −0.316900 − 0.948459i \(-0.602642\pi\)
0.662940 + 0.748673i \(0.269308\pi\)
\(74\) 15.2249 + 26.3704i 0.205742 + 0.356356i
\(75\) 23.7447 + 13.7090i 0.316596 + 0.182787i
\(76\) − 35.9787i − 0.473404i
\(77\) 0 0
\(78\) 8.78140 0.112582
\(79\) 5.47308 − 9.47965i 0.0692795 − 0.119996i −0.829305 − 0.558796i \(-0.811263\pi\)
0.898584 + 0.438801i \(0.144597\pi\)
\(80\) −25.6038 + 14.7824i −0.320047 + 0.184779i
\(81\) −86.6559 − 150.092i −1.06983 − 1.85299i
\(82\) −20.9544 − 12.0980i −0.255542 − 0.147537i
\(83\) 14.0223i 0.168943i 0.996426 + 0.0844717i \(0.0269203\pi\)
−0.996426 + 0.0844717i \(0.973080\pi\)
\(84\) 0 0
\(85\) −24.3985 −0.287041
\(86\) 1.90195 − 3.29428i 0.0221157 − 0.0383056i
\(87\) −112.146 + 64.7474i −1.28903 + 0.744223i
\(88\) −12.4469 − 21.5586i −0.141442 − 0.244985i
\(89\) −91.7289 − 52.9597i −1.03066 − 0.595053i −0.113487 − 0.993539i \(-0.536202\pi\)
−0.917174 + 0.398487i \(0.869535\pi\)
\(90\) 22.8960i 0.254400i
\(91\) 0 0
\(92\) 43.7087 0.475095
\(93\) −29.2016 + 50.5787i −0.313996 + 0.543857i
\(94\) 12.8375 − 7.41174i 0.136569 − 0.0788483i
\(95\) −10.6874 − 18.5111i −0.112499 − 0.194853i
\(96\) −102.185 − 58.9966i −1.06443 − 0.614548i
\(97\) 118.638i 1.22307i 0.791218 + 0.611534i \(0.209447\pi\)
−0.791218 + 0.611534i \(0.790553\pi\)
\(98\) 0 0
\(99\) 139.016 1.40420
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.166.4 12
7.2 even 3 245.3.d.a.146.6 12
7.3 odd 6 inner 245.3.h.c.31.4 12
7.4 even 3 35.3.h.a.31.4 yes 12
7.5 odd 6 245.3.d.a.146.5 12
7.6 odd 2 35.3.h.a.26.4 ✓ 12
21.11 odd 6 315.3.w.c.136.3 12
21.20 even 2 315.3.w.c.271.3 12
28.11 odd 6 560.3.bx.c.241.6 12
28.27 even 2 560.3.bx.c.481.6 12
35.4 even 6 175.3.i.d.101.3 12
35.13 even 4 175.3.j.b.124.7 24
35.18 odd 12 175.3.j.b.24.6 24
35.27 even 4 175.3.j.b.124.6 24
35.32 odd 12 175.3.j.b.24.7 24
35.34 odd 2 175.3.i.d.26.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.h.a.26.4 ✓ 12 7.6 odd 2
35.3.h.a.31.4 yes 12 7.4 even 3
175.3.i.d.26.3 12 35.34 odd 2
175.3.i.d.101.3 12 35.4 even 6
175.3.j.b.24.6 24 35.18 odd 12
175.3.j.b.24.7 24 35.32 odd 12
175.3.j.b.124.6 24 35.27 even 4
175.3.j.b.124.7 24 35.13 even 4
245.3.d.a.146.5 12 7.5 odd 6
245.3.d.a.146.6 12 7.2 even 3
245.3.h.c.31.4 12 7.3 odd 6 inner
245.3.h.c.166.4 12 1.1 even 1 trivial
315.3.w.c.136.3 12 21.11 odd 6
315.3.w.c.271.3 12 21.20 even 2
560.3.bx.c.241.6 12 28.11 odd 6
560.3.bx.c.481.6 12 28.27 even 2