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Properties

Label 315.8.a.c.1.1
Level $315$
Weight $8$
Character 315.1
Self dual yes
Analytic conductor $98.401$
Analytic rank $1$
Dimension $2$
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 = [2,-16] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31662\) 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)
 
\(f(q)\) \(=\) \(q-14.6332 q^{2} +86.1320 q^{4} -125.000 q^{5} -343.000 q^{7} +612.665 q^{8} +1829.16 q^{10} +6473.63 q^{11} -11681.7 q^{13} +5019.20 q^{14} -19990.2 q^{16} -13460.5 q^{17} +34955.5 q^{19} -10766.5 q^{20} -94730.3 q^{22} -77831.4 q^{23} +15625.0 q^{25} +170941. q^{26} -29543.3 q^{28} +221135. q^{29} -23222.3 q^{31} +214100. q^{32} +196971. q^{34} +42875.0 q^{35} -422392. q^{37} -511512. q^{38} -76583.1 q^{40} -191818. q^{41} +310754. q^{43} +557587. q^{44} +1.13893e6 q^{46} +240747. q^{47} +117649. q^{49} -228645. q^{50} -1.00617e6 q^{52} +1.06654e6 q^{53} -809204. q^{55} -210144. q^{56} -3.23592e6 q^{58} -451838. q^{59} -831659. q^{61} +339818. q^{62} -574238. q^{64} +1.46021e6 q^{65} +2.26405e6 q^{67} -1.15938e6 q^{68} -627401. q^{70} +2.22036e6 q^{71} +4.99377e6 q^{73} +6.18096e6 q^{74} +3.01078e6 q^{76} -2.22046e6 q^{77} -2.72773e6 q^{79} +2.49877e6 q^{80} +2.80693e6 q^{82} +6.38392e6 q^{83} +1.68256e6 q^{85} -4.54734e6 q^{86} +3.96617e6 q^{88} +7.32978e6 q^{89} +4.00682e6 q^{91} -6.70378e6 q^{92} -3.52291e6 q^{94} -4.36943e6 q^{95} -2.38676e6 q^{97} -1.72159e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8} + 2000 q^{10} + 7906 q^{11} - 17818 q^{13} + 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 3612 q^{19} + 5000 q^{20} - 96688 q^{22} - 13844 q^{23}+ \cdots - 1882384 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.6332 −1.29341 −0.646704 − 0.762741i \(-0.723853\pi\)
−0.646704 + 0.762741i \(0.723853\pi\)
\(3\) 0 0
\(4\) 86.1320 0.672906
\(5\) −125.000 −0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 612.665 0.423066
\(9\) 0 0
\(10\) 1829.16 0.578430
\(11\) 6473.63 1.46647 0.733236 − 0.679974i \(-0.238009\pi\)
0.733236 + 0.679974i \(0.238009\pi\)
\(12\) 0 0
\(13\) −11681.7 −1.47470 −0.737351 − 0.675510i \(-0.763924\pi\)
−0.737351 + 0.675510i \(0.763924\pi\)
\(14\) 5019.20 0.488863
\(15\) 0 0
\(16\) −19990.2 −1.22010
\(17\) −13460.5 −0.664491 −0.332246 − 0.943193i \(-0.607806\pi\)
−0.332246 + 0.943193i \(0.607806\pi\)
\(18\) 0 0
\(19\) 34955.5 1.16917 0.584585 − 0.811333i \(-0.301257\pi\)
0.584585 + 0.811333i \(0.301257\pi\)
\(20\) −10766.5 −0.300933
\(21\) 0 0
\(22\) −94730.3 −1.89675
\(23\) −77831.4 −1.33385 −0.666926 − 0.745124i \(-0.732390\pi\)
−0.666926 + 0.745124i \(0.732390\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 170941. 1.90739
\(27\) 0 0
\(28\) −29543.3 −0.254335
\(29\) 221135. 1.68370 0.841848 − 0.539715i \(-0.181468\pi\)
0.841848 + 0.539715i \(0.181468\pi\)
\(30\) 0 0
\(31\) −23222.3 −0.140004 −0.0700018 − 0.997547i \(-0.522301\pi\)
−0.0700018 + 0.997547i \(0.522301\pi\)
\(32\) 214100. 1.15503
\(33\) 0 0
\(34\) 196971. 0.859459
\(35\) 42875.0 0.169031
\(36\) 0 0
\(37\) −422392. −1.37091 −0.685456 − 0.728114i \(-0.740397\pi\)
−0.685456 + 0.728114i \(0.740397\pi\)
\(38\) −511512. −1.51221
\(39\) 0 0
\(40\) −76583.1 −0.189201
\(41\) −191818. −0.434657 −0.217329 − 0.976099i \(-0.569734\pi\)
−0.217329 + 0.976099i \(0.569734\pi\)
\(42\) 0 0
\(43\) 310754. 0.596042 0.298021 − 0.954559i \(-0.403673\pi\)
0.298021 + 0.954559i \(0.403673\pi\)
\(44\) 557587. 0.986798
\(45\) 0 0
\(46\) 1.13893e6 1.72522
\(47\) 240747. 0.338235 0.169117 − 0.985596i \(-0.445908\pi\)
0.169117 + 0.985596i \(0.445908\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) −228645. −0.258682
\(51\) 0 0
\(52\) −1.00617e6 −0.992336
\(53\) 1.06654e6 0.984040 0.492020 − 0.870584i \(-0.336259\pi\)
0.492020 + 0.870584i \(0.336259\pi\)
\(54\) 0 0
\(55\) −809204. −0.655826
\(56\) −210144. −0.159904
\(57\) 0 0
\(58\) −3.23592e6 −2.17771
\(59\) −451838. −0.286418 −0.143209 − 0.989692i \(-0.545742\pi\)
−0.143209 + 0.989692i \(0.545742\pi\)
\(60\) 0 0
\(61\) −831659. −0.469127 −0.234564 − 0.972101i \(-0.575366\pi\)
−0.234564 + 0.972101i \(0.575366\pi\)
\(62\) 339818. 0.181082
\(63\) 0 0
\(64\) −574238. −0.273818
\(65\) 1.46021e6 0.659507
\(66\) 0 0
\(67\) 2.26405e6 0.919654 0.459827 − 0.888009i \(-0.347912\pi\)
0.459827 + 0.888009i \(0.347912\pi\)
\(68\) −1.15938e6 −0.447140
\(69\) 0 0
\(70\) −627401. −0.218626
\(71\) 2.22036e6 0.736241 0.368120 − 0.929778i \(-0.380001\pi\)
0.368120 + 0.929778i \(0.380001\pi\)
\(72\) 0 0
\(73\) 4.99377e6 1.50244 0.751222 − 0.660049i \(-0.229465\pi\)
0.751222 + 0.660049i \(0.229465\pi\)
\(74\) 6.18096e6 1.77315
\(75\) 0 0
\(76\) 3.01078e6 0.786742
\(77\) −2.22046e6 −0.554274
\(78\) 0 0
\(79\) −2.72773e6 −0.622452 −0.311226 − 0.950336i \(-0.600740\pi\)
−0.311226 + 0.950336i \(0.600740\pi\)
\(80\) 2.49877e6 0.545647
\(81\) 0 0
\(82\) 2.80693e6 0.562189
\(83\) 6.38392e6 1.22550 0.612751 − 0.790276i \(-0.290063\pi\)
0.612751 + 0.790276i \(0.290063\pi\)
\(84\) 0 0
\(85\) 1.68256e6 0.297170
\(86\) −4.54734e6 −0.770926
\(87\) 0 0
\(88\) 3.96617e6 0.620414
\(89\) 7.32978e6 1.10211 0.551056 − 0.834468i \(-0.314225\pi\)
0.551056 + 0.834468i \(0.314225\pi\)
\(90\) 0 0
\(91\) 4.00682e6 0.557385
\(92\) −6.70378e6 −0.897557
\(93\) 0 0
\(94\) −3.52291e6 −0.437476
\(95\) −4.36943e6 −0.522869
\(96\) 0 0
\(97\) −2.38676e6 −0.265526 −0.132763 − 0.991148i \(-0.542385\pi\)
−0.132763 + 0.991148i \(0.542385\pi\)
\(98\) −1.72159e6 −0.184773
\(99\) 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.c.1.1 2
3.2 odd 2 35.8.a.a.1.2 ✓ 2
12.11 even 2 560.8.a.i.1.2 2
15.2 even 4 175.8.b.c.99.4 4
15.8 even 4 175.8.b.c.99.1 4
15.14 odd 2 175.8.a.b.1.1 2
21.20 even 2 245.8.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.2 ✓ 2 3.2 odd 2
175.8.a.b.1.1 2 15.14 odd 2
175.8.b.c.99.1 4 15.8 even 4
175.8.b.c.99.4 4 15.2 even 4
245.8.a.b.1.2 2 21.20 even 2
315.8.a.c.1.1 2 1.1 even 1 trivial
560.8.a.i.1.2 2 12.11 even 2