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Properties

Label 315.3.e.c.244.2
Level $315$
Weight $3$
Character 315.244
Analytic conductor $8.583$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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(244,315)] 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("315.244"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-20,0,0,0,0,0,0,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 244.2
Root \(1.58114 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 315.244
Dual form 315.3.e.c.244.4

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} -5.00000 q^{4} +(1.58114 + 4.74342i) q^{5} +(-6.32456 - 3.00000i) q^{7} +3.00000i q^{8} +(14.2302 - 4.74342i) q^{10} -14.0000 q^{11} -3.16228 q^{13} +(-9.00000 + 18.9737i) q^{14} -11.0000 q^{16} -6.32456 q^{17} +28.4605i q^{19} +(-7.90569 - 23.7171i) q^{20} +42.0000i q^{22} +12.0000i q^{23} +(-20.0000 + 15.0000i) q^{25} +9.48683i q^{26} +(31.6228 + 15.0000i) q^{28} -14.0000 q^{29} -37.9473i q^{31} +45.0000i q^{32} +18.9737i q^{34} +(4.23025 - 34.7434i) q^{35} +18.0000i q^{37} +85.3815 q^{38} +(-14.2302 + 4.74342i) q^{40} -18.9737i q^{41} -42.0000i q^{43} +70.0000 q^{44} +36.0000 q^{46} -44.2719 q^{47} +(31.0000 + 37.9473i) q^{49} +(45.0000 + 60.0000i) q^{50} +15.8114 q^{52} -54.0000i q^{53} +(-22.1359 - 66.4078i) q^{55} +(9.00000 - 18.9737i) q^{56} +42.0000i q^{58} +9.48683i q^{59} +66.4078i q^{61} -113.842 q^{62} +91.0000 q^{64} +(-5.00000 - 15.0000i) q^{65} -102.000i q^{67} +31.6228 q^{68} +(-104.230 - 12.6907i) q^{70} +16.0000 q^{71} +63.2456 q^{73} +54.0000 q^{74} -142.302i q^{76} +(88.5438 + 42.0000i) q^{77} -76.0000 q^{79} +(-17.3925 - 52.1776i) q^{80} -56.9210 q^{82} -72.7324 q^{83} +(-10.0000 - 30.0000i) q^{85} -126.000 q^{86} -42.0000i q^{88} +56.9210i q^{89} +(20.0000 + 9.48683i) q^{91} -60.0000i q^{92} +132.816i q^{94} +(-135.000 + 45.0000i) q^{95} -69.5701 q^{97} +(113.842 - 93.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{4} - 56 q^{11} - 36 q^{14} - 44 q^{16} - 80 q^{25} - 56 q^{29} - 40 q^{35} + 280 q^{44} + 144 q^{46} + 124 q^{49} + 180 q^{50} + 36 q^{56} + 364 q^{64} - 20 q^{65} - 360 q^{70} + 64 q^{71} + 216 q^{74}+ \cdots - 540 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.50000i −0.661438 − 0.750000i \(-0.730053\pi\)
0.661438 − 0.750000i \(-0.269947\pi\)
\(3\) 0 0
\(4\) −5.00000 −1.25000
\(5\) 1.58114 + 4.74342i 0.316228 + 0.948683i
\(6\) 0 0
\(7\) −6.32456 − 3.00000i −0.903508 − 0.428571i
\(8\) 3.00000i 0.375000i
\(9\) 0 0
\(10\) 14.2302 − 4.74342i 1.42302 − 0.474342i
\(11\) −14.0000 −1.27273 −0.636364 − 0.771389i \(-0.719562\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(12\) 0 0
\(13\) −3.16228 −0.243252 −0.121626 − 0.992576i \(-0.538811\pi\)
−0.121626 + 0.992576i \(0.538811\pi\)
\(14\) −9.00000 + 18.9737i −0.642857 + 1.35526i
\(15\) 0 0
\(16\) −11.0000 −0.687500
\(17\) −6.32456 −0.372033 −0.186016 − 0.982547i \(-0.559558\pi\)
−0.186016 + 0.982547i \(0.559558\pi\)
\(18\) 0 0
\(19\) 28.4605i 1.49792i 0.662615 + 0.748960i \(0.269447\pi\)
−0.662615 + 0.748960i \(0.730553\pi\)
\(20\) −7.90569 − 23.7171i −0.395285 − 1.18585i
\(21\) 0 0
\(22\) 42.0000i 1.90909i
\(23\) 12.0000i 0.521739i 0.965374 + 0.260870i \(0.0840093\pi\)
−0.965374 + 0.260870i \(0.915991\pi\)
\(24\) 0 0
\(25\) −20.0000 + 15.0000i −0.800000 + 0.600000i
\(26\) 9.48683i 0.364878i
\(27\) 0 0
\(28\) 31.6228 + 15.0000i 1.12938 + 0.535714i
\(29\) −14.0000 −0.482759 −0.241379 − 0.970431i \(-0.577600\pi\)
−0.241379 + 0.970431i \(0.577600\pi\)
\(30\) 0 0
\(31\) − 37.9473i − 1.22411i −0.790816 − 0.612054i \(-0.790344\pi\)
0.790816 − 0.612054i \(-0.209656\pi\)
\(32\) 45.0000i 1.40625i
\(33\) 0 0
\(34\) 18.9737i 0.558049i
\(35\) 4.23025 − 34.7434i 0.120864 − 0.992669i
\(36\) 0 0
\(37\) 18.0000i 0.486486i 0.969965 + 0.243243i \(0.0782113\pi\)
−0.969965 + 0.243243i \(0.921789\pi\)
\(38\) 85.3815 2.24688
\(39\) 0 0
\(40\) −14.2302 + 4.74342i −0.355756 + 0.118585i
\(41\) − 18.9737i − 0.462772i −0.972862 − 0.231386i \(-0.925674\pi\)
0.972862 − 0.231386i \(-0.0743261\pi\)
\(42\) 0 0
\(43\) − 42.0000i − 0.976744i −0.872635 − 0.488372i \(-0.837591\pi\)
0.872635 − 0.488372i \(-0.162409\pi\)
\(44\) 70.0000 1.59091
\(45\) 0 0
\(46\) 36.0000 0.782609
\(47\) −44.2719 −0.941955 −0.470978 − 0.882145i \(-0.656099\pi\)
−0.470978 + 0.882145i \(0.656099\pi\)
\(48\) 0 0
\(49\) 31.0000 + 37.9473i 0.632653 + 0.774435i
\(50\) 45.0000 + 60.0000i 0.900000 + 1.20000i
\(51\) 0 0
\(52\) 15.8114 0.304065
\(53\) − 54.0000i − 1.01887i −0.860510 − 0.509434i \(-0.829855\pi\)
0.860510 − 0.509434i \(-0.170145\pi\)
\(54\) 0 0
\(55\) −22.1359 − 66.4078i −0.402472 − 1.20742i
\(56\) 9.00000 − 18.9737i 0.160714 − 0.338815i
\(57\) 0 0
\(58\) 42.0000i 0.724138i
\(59\) 9.48683i 0.160794i 0.996763 + 0.0803969i \(0.0256188\pi\)
−0.996763 + 0.0803969i \(0.974381\pi\)
\(60\) 0 0
\(61\) 66.4078i 1.08865i 0.838873 + 0.544326i \(0.183215\pi\)
−0.838873 + 0.544326i \(0.816785\pi\)
\(62\) −113.842 −1.83616
\(63\) 0 0
\(64\) 91.0000 1.42188
\(65\) −5.00000 − 15.0000i −0.0769231 − 0.230769i
\(66\) 0 0
\(67\) − 102.000i − 1.52239i −0.648524 − 0.761194i \(-0.724613\pi\)
0.648524 − 0.761194i \(-0.275387\pi\)
\(68\) 31.6228 0.465041
\(69\) 0 0
\(70\) −104.230 − 12.6907i −1.48900 − 0.181296i
\(71\) 16.0000 0.225352 0.112676 − 0.993632i \(-0.464058\pi\)
0.112676 + 0.993632i \(0.464058\pi\)
\(72\) 0 0
\(73\) 63.2456 0.866377 0.433189 − 0.901303i \(-0.357388\pi\)
0.433189 + 0.901303i \(0.357388\pi\)
\(74\) 54.0000 0.729730
\(75\) 0 0
\(76\) − 142.302i − 1.87240i
\(77\) 88.5438 + 42.0000i 1.14992 + 0.545455i
\(78\) 0 0
\(79\) −76.0000 −0.962025 −0.481013 − 0.876714i \(-0.659731\pi\)
−0.481013 + 0.876714i \(0.659731\pi\)
\(80\) −17.3925 − 52.1776i −0.217407 − 0.652220i
\(81\) 0 0
\(82\) −56.9210 −0.694159
\(83\) −72.7324 −0.876294 −0.438147 − 0.898903i \(-0.644365\pi\)
−0.438147 + 0.898903i \(0.644365\pi\)
\(84\) 0 0
\(85\) −10.0000 − 30.0000i −0.117647 − 0.352941i
\(86\) −126.000 −1.46512
\(87\) 0 0
\(88\) − 42.0000i − 0.477273i
\(89\) 56.9210i 0.639562i 0.947492 + 0.319781i \(0.103609\pi\)
−0.947492 + 0.319781i \(0.896391\pi\)
\(90\) 0 0
\(91\) 20.0000 + 9.48683i 0.219780 + 0.104251i
\(92\) − 60.0000i − 0.652174i
\(93\) 0 0
\(94\) 132.816i 1.41293i
\(95\) −135.000 + 45.0000i −1.42105 + 0.473684i
\(96\) 0 0
\(97\) −69.5701 −0.717218 −0.358609 − 0.933488i \(-0.616749\pi\)
−0.358609 + 0.933488i \(0.616749\pi\)
\(98\) 113.842 − 93.0000i 1.16165 − 0.948980i
\(99\) 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.e.c.244.2 4
3.2 odd 2 35.3.c.c.34.4 yes 4
5.4 even 2 inner 315.3.e.c.244.3 4
7.6 odd 2 inner 315.3.e.c.244.1 4
12.11 even 2 560.3.p.f.209.1 4
15.2 even 4 175.3.d.b.76.1 2
15.8 even 4 175.3.d.h.76.2 2
15.14 odd 2 35.3.c.c.34.1 ✓ 4
21.2 odd 6 245.3.i.c.129.3 8
21.5 even 6 245.3.i.c.129.4 8
21.11 odd 6 245.3.i.c.19.1 8
21.17 even 6 245.3.i.c.19.2 8
21.20 even 2 35.3.c.c.34.3 yes 4
35.34 odd 2 inner 315.3.e.c.244.4 4
60.59 even 2 560.3.p.f.209.3 4
84.83 odd 2 560.3.p.f.209.4 4
105.44 odd 6 245.3.i.c.129.2 8
105.59 even 6 245.3.i.c.19.3 8
105.62 odd 4 175.3.d.b.76.2 2
105.74 odd 6 245.3.i.c.19.4 8
105.83 odd 4 175.3.d.h.76.1 2
105.89 even 6 245.3.i.c.129.1 8
105.104 even 2 35.3.c.c.34.2 yes 4
420.419 odd 2 560.3.p.f.209.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.c.c.34.1 ✓ 4 15.14 odd 2
35.3.c.c.34.2 yes 4 105.104 even 2
35.3.c.c.34.3 yes 4 21.20 even 2
35.3.c.c.34.4 yes 4 3.2 odd 2
175.3.d.b.76.1 2 15.2 even 4
175.3.d.b.76.2 2 105.62 odd 4
175.3.d.h.76.1 2 105.83 odd 4
175.3.d.h.76.2 2 15.8 even 4
245.3.i.c.19.1 8 21.11 odd 6
245.3.i.c.19.2 8 21.17 even 6
245.3.i.c.19.3 8 105.59 even 6
245.3.i.c.19.4 8 105.74 odd 6
245.3.i.c.129.1 8 105.89 even 6
245.3.i.c.129.2 8 105.44 odd 6
245.3.i.c.129.3 8 21.2 odd 6
245.3.i.c.129.4 8 21.5 even 6
315.3.e.c.244.1 4 7.6 odd 2 inner
315.3.e.c.244.2 4 1.1 even 1 trivial
315.3.e.c.244.3 4 5.4 even 2 inner
315.3.e.c.244.4 4 35.34 odd 2 inner
560.3.p.f.209.1 4 12.11 even 2
560.3.p.f.209.2 4 420.419 odd 2
560.3.p.f.209.3 4 60.59 even 2
560.3.p.f.209.4 4 84.83 odd 2