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Properties

Label 315.3.w.c.271.2
Level $315$
Weight $3$
Character 315.271
Analytic conductor $8.583$
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] = [315,3,Mod(136,315)] 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("315.136"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.w (of order \(6\), degree \(2\), 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: \(8.58312832735\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.2
Root \(-0.925400 - 1.60284i\) of defining polynomial
Character \(\chi\) \(=\) 315.271
Dual form 315.3.w.c.136.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.925400 + 1.60284i) q^{2} +(0.287270 + 0.497567i) q^{4} +(1.93649 + 1.11803i) q^{5} +(-6.88972 - 1.23763i) q^{7} -8.46656 q^{8} +(-3.58406 + 2.06926i) q^{10} +(-3.93873 - 6.82208i) q^{11} -2.73368i q^{13} +(8.35947 - 9.89782i) q^{14} +(6.68587 - 11.5803i) q^{16} +(-4.29344 + 2.47882i) q^{17} +(-25.3152 - 14.6158i) q^{19} +1.28471i q^{20} +14.5796 q^{22} +(-18.0840 + 31.3224i) q^{23} +(2.50000 + 4.33013i) q^{25} +(4.38165 + 2.52975i) q^{26} +(-1.36341 - 3.78363i) q^{28} -0.399005 q^{29} +(17.4396 - 10.0688i) q^{31} +(-4.55891 - 7.89627i) q^{32} -9.17558i q^{34} +(-11.9582 - 10.0996i) q^{35} +(3.61299 - 6.25789i) q^{37} +(46.8534 - 27.0508i) q^{38} +(-16.3954 - 9.46590i) q^{40} -71.1139i q^{41} -51.7627 q^{43} +(2.26296 - 3.91956i) q^{44} +(-33.4699 - 57.9715i) q^{46} +(19.9766 + 11.5335i) q^{47} +(45.9365 + 17.0538i) q^{49} -9.25400 q^{50} +(1.36019 - 0.785306i) q^{52} +(-26.4177 - 45.7568i) q^{53} -17.6145i q^{55} +(58.3322 + 10.4785i) q^{56} +(0.369239 - 0.639542i) q^{58} +(-73.6658 + 42.5310i) q^{59} +(75.4986 + 43.5891i) q^{61} +37.2705i q^{62} +70.3622 q^{64} +(3.05635 - 5.29375i) q^{65} +(-34.7881 - 60.2548i) q^{67} +(-2.46675 - 1.42418i) q^{68} +(27.2541 - 9.82087i) q^{70} +26.4541 q^{71} +(-114.650 + 66.1933i) q^{73} +(6.68693 + 11.5821i) q^{74} -16.7947i q^{76} +(18.6935 + 51.8769i) q^{77} +(-53.6505 + 92.9253i) q^{79} +(25.8943 - 14.9501i) q^{80} +(113.984 + 65.8087i) q^{82} +53.9702i q^{83} -11.0856 q^{85} +(47.9012 - 82.9674i) q^{86} +(33.3475 + 57.7595i) q^{88} +(56.2194 + 32.4583i) q^{89} +(-3.38328 + 18.8343i) q^{91} -20.7800 q^{92} +(-36.9727 + 21.3462i) q^{94} +(-32.6818 - 56.6066i) q^{95} +109.504i q^{97} +(-69.8442 + 57.8473i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 10 q^{4} - 2 q^{7} + 4 q^{8} + 14 q^{11} + 2 q^{14} - 22 q^{16} - 48 q^{17} - 30 q^{19} - 88 q^{22} + 14 q^{23} + 30 q^{25} - 66 q^{26} + 202 q^{28} - 64 q^{29} + 132 q^{31} + 54 q^{32}+ \cdots - 898 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 −0.999094 − 0.0425476i \(-0.986453\pi\)
0.536395 + 0.843967i \(0.319786\pi\)
\(3\) 0 0
\(4\) 0.287270 + 0.497567i 0.0718176 + 0.124392i
\(5\) 1.93649 + 1.11803i 0.387298 + 0.223607i
\(6\) 0 0
\(7\) −6.88972 − 1.23763i −0.984246 − 0.176804i
\(8\) −8.46656 −1.05832
\(9\) 0 0
\(10\) −3.58406 + 2.06926i −0.358406 + 0.206926i
\(11\) −3.93873 − 6.82208i −0.358066 − 0.620189i 0.629572 − 0.776942i \(-0.283230\pi\)
−0.987638 + 0.156754i \(0.949897\pi\)
\(12\) 0 0
\(13\) − 2.73368i − 0.210283i −0.994457 − 0.105142i \(-0.966470\pi\)
0.994457 − 0.105142i \(-0.0335296\pi\)
\(14\) 8.35947 − 9.89782i 0.597105 − 0.706987i
\(15\) 0 0
\(16\) 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 \(0.379913\pi\)
\(18\) 0 0
\(19\) −25.3152 − 14.6158i −1.33238 − 0.769250i −0.346716 − 0.937970i \(-0.612703\pi\)
−0.985664 + 0.168720i \(0.946037\pi\)
\(20\) 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 \(0.454653\pi\)
−0.928243 + 0.371975i \(0.878681\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.100000 + 0.173205i
\(26\) 4.38165 + 2.52975i 0.168525 + 0.0972980i
\(27\) 0 0
\(28\) −1.36341 − 3.78363i −0.0486932 − 0.135130i
\(29\) −0.399005 −0.0137588 −0.00687940 − 0.999976i \(-0.502190\pi\)
−0.00687940 + 0.999976i \(0.502190\pi\)
\(30\) 0 0
\(31\) 17.4396 − 10.0688i 0.562568 − 0.324799i −0.191607 − 0.981472i \(-0.561370\pi\)
0.754176 + 0.656673i \(0.228037\pi\)
\(32\) −4.55891 − 7.89627i −0.142466 − 0.246758i
\(33\) 0 0
\(34\) − 9.17558i − 0.269870i
\(35\) −11.9582 − 10.0996i −0.341662 − 0.288560i
\(36\) 0 0
\(37\) 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
\(39\) 0 0
\(40\) −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
\(45\) 0 0
\(46\) −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 \(0.587750\pi\)
\(48\) 0 0
\(49\) 45.9365 + 17.0538i 0.937481 + 0.348038i
\(50\) −9.25400 −0.185080
\(51\) 0 0
\(52\) 1.36019 − 0.785306i 0.0261575 − 0.0151020i
\(53\) −26.4177 − 45.7568i −0.498447 − 0.863336i 0.501551 − 0.865128i \(-0.332763\pi\)
−0.999998 + 0.00179226i \(0.999430\pi\)
\(54\) 0 0
\(55\) − 17.6145i − 0.320264i
\(56\) 58.3322 + 10.4785i 1.04165 + 0.187115i
\(57\) 0 0
\(58\) 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 0
\(61\) 75.4986 + 43.5891i 1.23768 + 0.714576i 0.968620 − 0.248547i \(-0.0799532\pi\)
0.269061 + 0.963123i \(0.413287\pi\)
\(62\) 37.2705i 0.601138i
\(63\) 0 0
\(64\) 70.3622 1.09941
\(65\) 3.05635 − 5.29375i 0.0470208 − 0.0814423i
\(66\) 0 0
\(67\) −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
\(69\) 0 0
\(70\) 27.2541 − 9.82087i 0.389345 − 0.140298i
\(71\) 26.4541 0.372593 0.186297 − 0.982494i \(-0.440351\pi\)
0.186297 + 0.982494i \(0.440351\pi\)
\(72\) 0 0
\(73\) −114.650 + 66.1933i −1.57055 + 0.906758i −0.574450 + 0.818540i \(0.694784\pi\)
−0.996101 + 0.0882179i \(0.971883\pi\)
\(74\) 6.68693 + 11.5821i 0.0903639 + 0.156515i
\(75\) 0 0
\(76\) − 16.7947i − 0.220983i
\(77\) 18.6935 + 51.8769i 0.242773 + 0.673726i
\(78\) 0 0
\(79\) −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
\(81\) 0 0
\(82\) 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 0
\(88\) 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 \(0.547838\pi\)
\(90\) 0 0
\(91\) −3.38328 + 18.8343i −0.0371790 + 0.206970i
\(92\) −20.7800 −0.225870
\(93\) 0 0
\(94\) −36.9727 + 21.3462i −0.393326 + 0.227087i
\(95\) −32.6818 − 56.6066i −0.344019 − 0.595859i
\(96\) 0 0
\(97\) 109.504i 1.12891i 0.825463 + 0.564456i \(0.190914\pi\)
−0.825463 + 0.564456i \(0.809086\pi\)
\(98\) −69.8442 + 57.8473i −0.712696 + 0.590278i
\(99\) 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 315.3.w.c.271.2 12
3.2 odd 2 35.3.h.a.26.5 ✓ 12
7.3 odd 6 inner 315.3.w.c.136.2 12
12.11 even 2 560.3.bx.c.481.2 12
15.2 even 4 175.3.j.b.124.9 24
15.8 even 4 175.3.j.b.124.4 24
15.14 odd 2 175.3.i.d.26.2 12
21.2 odd 6 245.3.d.a.146.4 12
21.5 even 6 245.3.d.a.146.3 12
21.11 odd 6 245.3.h.c.31.5 12
21.17 even 6 35.3.h.a.31.5 yes 12
21.20 even 2 245.3.h.c.166.5 12
84.59 odd 6 560.3.bx.c.241.2 12
105.17 odd 12 175.3.j.b.24.4 24
105.38 odd 12 175.3.j.b.24.9 24
105.59 even 6 175.3.i.d.101.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.h.a.26.5 ✓ 12 3.2 odd 2
35.3.h.a.31.5 yes 12 21.17 even 6
175.3.i.d.26.2 12 15.14 odd 2
175.3.i.d.101.2 12 105.59 even 6
175.3.j.b.24.4 24 105.17 odd 12
175.3.j.b.24.9 24 105.38 odd 12
175.3.j.b.124.4 24 15.8 even 4
175.3.j.b.124.9 24 15.2 even 4
245.3.d.a.146.3 12 21.5 even 6
245.3.d.a.146.4 12 21.2 odd 6
245.3.h.c.31.5 12 21.11 odd 6
245.3.h.c.166.5 12 21.20 even 2
315.3.w.c.136.2 12 7.3 odd 6 inner
315.3.w.c.271.2 12 1.1 even 1 trivial
560.3.bx.c.241.2 12 84.59 odd 6
560.3.bx.c.481.2 12 12.11 even 2