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Properties

Label 315.8.a.h.1.2
Level $315$
Weight $8$
Character 315.1
Self dual yes
Analytic conductor $98.401$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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] = [315,8,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 8); 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, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4] 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: yes
Analytic conductor: \(98.4012830275\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 267x^{2} + 1358x + 2744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.55329\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-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)
 
\(f(q)\) \(=\) \(q-11.5387 q^{2} +5.14154 q^{4} +125.000 q^{5} -343.000 q^{7} +1417.63 q^{8} -1442.34 q^{10} -8585.82 q^{11} -7495.09 q^{13} +3957.77 q^{14} -17015.7 q^{16} +19158.5 q^{17} -58457.6 q^{19} +642.693 q^{20} +99069.2 q^{22} +39617.1 q^{23} +15625.0 q^{25} +86483.6 q^{26} -1763.55 q^{28} +167401. q^{29} -211782. q^{31} +14882.6 q^{32} -221065. q^{34} -42875.0 q^{35} -259819. q^{37} +674525. q^{38} +177203. q^{40} -606191. q^{41} +710355. q^{43} -44144.3 q^{44} -457130. q^{46} -637850. q^{47} +117649. q^{49} -180292. q^{50} -38536.3 q^{52} +370888. q^{53} -1.07323e6 q^{55} -486246. q^{56} -1.93159e6 q^{58} -1.71158e6 q^{59} -2.95930e6 q^{61} +2.44369e6 q^{62} +2.00628e6 q^{64} -936886. q^{65} +1.96892e6 q^{67} +98504.4 q^{68} +494722. q^{70} +675275. q^{71} +839284. q^{73} +2.99797e6 q^{74} -300562. q^{76} +2.94494e6 q^{77} -4.35501e6 q^{79} -2.12696e6 q^{80} +6.99465e6 q^{82} -5.32857e6 q^{83} +2.39482e6 q^{85} -8.19657e6 q^{86} -1.21715e7 q^{88} +6.66419e6 q^{89} +2.57082e6 q^{91} +203693. q^{92} +7.35996e6 q^{94} -7.30721e6 q^{95} -9.40124e6 q^{97} -1.35752e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 106 q^{4} + 500 q^{5} - 1372 q^{7} - 12 q^{8} - 500 q^{10} - 3032 q^{11} + 952 q^{13} + 1372 q^{14} - 58494 q^{16} + 50464 q^{17} - 15672 q^{19} + 13250 q^{20} - 40964 q^{22} + 89656 q^{23}+ \cdots - 470596 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.5387 −1.01989 −0.509943 − 0.860208i \(-0.670334\pi\)
−0.509943 + 0.860208i \(0.670334\pi\)
\(3\) 0 0
\(4\) 5.14154 0.0401683
\(5\) 125.000 0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 1417.63 0.978919
\(9\) 0 0
\(10\) −1442.34 −0.456107
\(11\) −8585.82 −1.94494 −0.972472 − 0.233018i \(-0.925140\pi\)
−0.972472 + 0.233018i \(0.925140\pi\)
\(12\) 0 0
\(13\) −7495.09 −0.946183 −0.473091 − 0.881013i \(-0.656862\pi\)
−0.473091 + 0.881013i \(0.656862\pi\)
\(14\) 3957.77 0.385481
\(15\) 0 0
\(16\) −17015.7 −1.03855
\(17\) 19158.5 0.945782 0.472891 − 0.881121i \(-0.343210\pi\)
0.472891 + 0.881121i \(0.343210\pi\)
\(18\) 0 0
\(19\) −58457.6 −1.95526 −0.977628 − 0.210341i \(-0.932543\pi\)
−0.977628 + 0.210341i \(0.932543\pi\)
\(20\) 642.693 0.0179638
\(21\) 0 0
\(22\) 99069.2 1.98362
\(23\) 39617.1 0.678947 0.339473 − 0.940616i \(-0.389751\pi\)
0.339473 + 0.940616i \(0.389751\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 86483.6 0.964999
\(27\) 0 0
\(28\) −1763.55 −0.0151822
\(29\) 167401. 1.27458 0.637288 − 0.770626i \(-0.280056\pi\)
0.637288 + 0.770626i \(0.280056\pi\)
\(30\) 0 0
\(31\) −211782. −1.27680 −0.638400 − 0.769705i \(-0.720403\pi\)
−0.638400 + 0.769705i \(0.720403\pi\)
\(32\) 14882.6 0.0802886
\(33\) 0 0
\(34\) −221065. −0.964591
\(35\) −42875.0 −0.169031
\(36\) 0 0
\(37\) −259819. −0.843266 −0.421633 − 0.906767i \(-0.638543\pi\)
−0.421633 + 0.906767i \(0.638543\pi\)
\(38\) 674525. 1.99414
\(39\) 0 0
\(40\) 177203. 0.437786
\(41\) −606191. −1.37362 −0.686809 − 0.726838i \(-0.740989\pi\)
−0.686809 + 0.726838i \(0.740989\pi\)
\(42\) 0 0
\(43\) 710355. 1.36250 0.681249 − 0.732052i \(-0.261437\pi\)
0.681249 + 0.732052i \(0.261437\pi\)
\(44\) −44144.3 −0.0781251
\(45\) 0 0
\(46\) −457130. −0.692449
\(47\) −637850. −0.896140 −0.448070 − 0.893998i \(-0.647888\pi\)
−0.448070 + 0.893998i \(0.647888\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) −180292. −0.203977
\(51\) 0 0
\(52\) −38536.3 −0.0380065
\(53\) 370888. 0.342197 0.171099 − 0.985254i \(-0.445268\pi\)
0.171099 + 0.985254i \(0.445268\pi\)
\(54\) 0 0
\(55\) −1.07323e6 −0.869806
\(56\) −486246. −0.369997
\(57\) 0 0
\(58\) −1.93159e6 −1.29992
\(59\) −1.71158e6 −1.08496 −0.542481 − 0.840068i \(-0.682515\pi\)
−0.542481 + 0.840068i \(0.682515\pi\)
\(60\) 0 0
\(61\) −2.95930e6 −1.66930 −0.834651 − 0.550779i \(-0.814331\pi\)
−0.834651 + 0.550779i \(0.814331\pi\)
\(62\) 2.44369e6 1.30219
\(63\) 0 0
\(64\) 2.00628e6 0.956670
\(65\) −936886. −0.423146
\(66\) 0 0
\(67\) 1.96892e6 0.799773 0.399887 − 0.916565i \(-0.369049\pi\)
0.399887 + 0.916565i \(0.369049\pi\)
\(68\) 98504.4 0.0379905
\(69\) 0 0
\(70\) 494722. 0.172392
\(71\) 675275. 0.223912 0.111956 − 0.993713i \(-0.464288\pi\)
0.111956 + 0.993713i \(0.464288\pi\)
\(72\) 0 0
\(73\) 839284. 0.252510 0.126255 − 0.991998i \(-0.459704\pi\)
0.126255 + 0.991998i \(0.459704\pi\)
\(74\) 2.99797e6 0.860035
\(75\) 0 0
\(76\) −300562. −0.0785393
\(77\) 2.94494e6 0.735120
\(78\) 0 0
\(79\) −4.35501e6 −0.993789 −0.496894 − 0.867811i \(-0.665526\pi\)
−0.496894 + 0.867811i \(0.665526\pi\)
\(80\) −2.12696e6 −0.464456
\(81\) 0 0
\(82\) 6.99465e6 1.40093
\(83\) −5.32857e6 −1.02291 −0.511455 − 0.859310i \(-0.670893\pi\)
−0.511455 + 0.859310i \(0.670893\pi\)
\(84\) 0 0
\(85\) 2.39482e6 0.422967
\(86\) −8.19657e6 −1.38959
\(87\) 0 0
\(88\) −1.21715e7 −1.90394
\(89\) 6.66419e6 1.00203 0.501017 − 0.865438i \(-0.332960\pi\)
0.501017 + 0.865438i \(0.332960\pi\)
\(90\) 0 0
\(91\) 2.57082e6 0.357623
\(92\) 203693. 0.0272721
\(93\) 0 0
\(94\) 7.35996e6 0.913961
\(95\) −7.30721e6 −0.874417
\(96\) 0 0
\(97\) −9.40124e6 −1.04589 −0.522943 − 0.852368i \(-0.675166\pi\)
−0.522943 + 0.852368i \(0.675166\pi\)
\(98\) −1.35752e6 −0.145698
\(99\) 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 315.8.a.h.1.2 4
3.2 odd 2 105.8.a.g.1.3 ✓ 4
15.14 odd 2 525.8.a.j.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.8.a.g.1.3 ✓ 4 3.2 odd 2
315.8.a.h.1.2 4 1.1 even 1 trivial
525.8.a.j.1.2 4 15.14 odd 2