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Properties

Label 175.8.a.b.1.1
Level $175$
Weight $8$
Character 175.1
Self dual yes
Analytic conductor $54.667$
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] = [175,8,Mod(1,175)] 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("175.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(54.6673794597\)
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 \(\chi\) \(=\) 175.1

$q$-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)
 
\(f(q)\) \(=\) \(q-14.6332 q^{2} +54.7995 q^{3} +86.1320 q^{4} -801.895 q^{6} +343.000 q^{7} +612.665 q^{8} +815.985 q^{9} -6473.63 q^{11} +4719.99 q^{12} +11681.7 q^{13} -5019.20 q^{14} -19990.2 q^{16} -13460.5 q^{17} -11940.5 q^{18} +34955.5 q^{19} +18796.2 q^{21} +94730.3 q^{22} -77831.4 q^{23} +33573.7 q^{24} -170941. q^{26} -75130.9 q^{27} +29543.3 q^{28} -221135. q^{29} -23222.3 q^{31} +214100. q^{32} -354752. q^{33} +196971. q^{34} +70282.4 q^{36} +422392. q^{37} -511512. q^{38} +640151. q^{39} +191818. q^{41} -275050. q^{42} -310754. q^{43} -557587. q^{44} +1.13893e6 q^{46} +240747. q^{47} -1.09545e6 q^{48} +117649. q^{49} -737628. q^{51} +1.00617e6 q^{52} +1.06654e6 q^{53} +1.09941e6 q^{54} +210144. q^{56} +1.91554e6 q^{57} +3.23592e6 q^{58} +451838. q^{59} -831659. q^{61} +339818. q^{62} +279883. q^{63} -574238. q^{64} +5.19117e6 q^{66} -2.26405e6 q^{67} -1.15938e6 q^{68} -4.26512e6 q^{69} -2.22036e6 q^{71} +499925. q^{72} -4.99377e6 q^{73} -6.18096e6 q^{74} +3.01078e6 q^{76} -2.22046e6 q^{77} -9.36749e6 q^{78} -2.72773e6 q^{79} -5.90170e6 q^{81} -2.80693e6 q^{82} +6.38392e6 q^{83} +1.61896e6 q^{84} +4.54734e6 q^{86} -1.21181e7 q^{87} -3.96617e6 q^{88} -7.32978e6 q^{89} +4.00682e6 q^{91} -6.70378e6 q^{92} -1.27257e6 q^{93} -3.52291e6 q^{94} +1.17326e7 q^{96} +2.38676e6 q^{97} -1.72159e6 q^{98} -5.28239e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 30 q^{3} - 40 q^{4} - 768 q^{6} + 686 q^{7} + 960 q^{8} - 756 q^{9} - 7906 q^{11} + 7848 q^{12} + 17818 q^{13} - 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 9792 q^{18} - 3612 q^{19} + 10290 q^{21}+ \cdots - 3030732 q^{99}+O(q^{100}) \) 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 \(0.723853\pi\)
\(3\) 54.7995 1.17180 0.585898 − 0.810385i \(-0.300742\pi\)
0.585898 + 0.810385i \(0.300742\pi\)
\(4\) 86.1320 0.672906
\(5\) 0 0
\(6\) −801.895 −1.51561
\(7\) 343.000 0.377964
\(8\) 612.665 0.423066
\(9\) 815.985 0.373107
\(10\) 0 0
\(11\) −6473.63 −1.46647 −0.733236 − 0.679974i \(-0.761991\pi\)
−0.733236 + 0.679974i \(0.761991\pi\)
\(12\) 4719.99 0.788509
\(13\) 11681.7 1.47470 0.737351 − 0.675510i \(-0.236076\pi\)
0.737351 + 0.675510i \(0.236076\pi\)
\(14\) −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 \(0.607806\pi\)
\(18\) −11940.5 −0.482580
\(19\) 34955.5 1.16917 0.584585 − 0.811333i \(-0.301257\pi\)
0.584585 + 0.811333i \(0.301257\pi\)
\(20\) 0 0
\(21\) 18796.2 0.442897
\(22\) 94730.3 1.89675
\(23\) −77831.4 −1.33385 −0.666926 − 0.745124i \(-0.732390\pi\)
−0.666926 + 0.745124i \(0.732390\pi\)
\(24\) 33573.7 0.495747
\(25\) 0 0
\(26\) −170941. −1.90739
\(27\) −75130.9 −0.734591
\(28\) 29543.3 0.254335
\(29\) −221135. −1.68370 −0.841848 − 0.539715i \(-0.818532\pi\)
−0.841848 + 0.539715i \(0.818532\pi\)
\(30\) 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
\(33\) −354752. −1.71841
\(34\) 196971. 0.859459
\(35\) 0 0
\(36\) 70282.4 0.251066
\(37\) 422392. 1.37091 0.685456 − 0.728114i \(-0.259603\pi\)
0.685456 + 0.728114i \(0.259603\pi\)
\(38\) −511512. −1.51221
\(39\) 640151. 1.72805
\(40\) 0 0
\(41\) 191818. 0.434657 0.217329 − 0.976099i \(-0.430266\pi\)
0.217329 + 0.976099i \(0.430266\pi\)
\(42\) −275050. −0.572847
\(43\) −310754. −0.596042 −0.298021 − 0.954559i \(-0.596327\pi\)
−0.298021 + 0.954559i \(0.596327\pi\)
\(44\) −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 \(0.445908\pi\)
\(48\) −1.09545e6 −1.42971
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) −737628. −0.778649
\(52\) 1.00617e6 0.992336
\(53\) 1.06654e6 0.984040 0.492020 − 0.870584i \(-0.336259\pi\)
0.492020 + 0.870584i \(0.336259\pi\)
\(54\) 1.09941e6 0.950126
\(55\) 0 0
\(56\) 210144. 0.159904
\(57\) 1.91554e6 1.37003
\(58\) 3.23592e6 2.17771
\(59\) 451838. 0.286418 0.143209 − 0.989692i \(-0.454258\pi\)
0.143209 + 0.989692i \(0.454258\pi\)
\(60\) 0 0
\(61\) −831659. −0.469127 −0.234564 − 0.972101i \(-0.575366\pi\)
−0.234564 + 0.972101i \(0.575366\pi\)
\(62\) 339818. 0.181082
\(63\) 279883. 0.141021
\(64\) −574238. −0.273818
\(65\) 0 0
\(66\) 5.19117e6 2.22260
\(67\) −2.26405e6 −0.919654 −0.459827 − 0.888009i \(-0.652088\pi\)
−0.459827 + 0.888009i \(0.652088\pi\)
\(68\) −1.15938e6 −0.447140
\(69\) −4.26512e6 −1.56300
\(70\) 0 0
\(71\) −2.22036e6 −0.736241 −0.368120 − 0.929778i \(-0.619999\pi\)
−0.368120 + 0.929778i \(0.619999\pi\)
\(72\) 499925. 0.157849
\(73\) −4.99377e6 −1.50244 −0.751222 − 0.660049i \(-0.770535\pi\)
−0.751222 + 0.660049i \(0.770535\pi\)
\(74\) −6.18096e6 −1.77315
\(75\) 0 0
\(76\) 3.01078e6 0.786742
\(77\) −2.22046e6 −0.554274
\(78\) −9.36749e6 −2.23508
\(79\) −2.72773e6 −0.622452 −0.311226 − 0.950336i \(-0.600740\pi\)
−0.311226 + 0.950336i \(0.600740\pi\)
\(80\) 0 0
\(81\) −5.90170e6 −1.23390
\(82\) −2.80693e6 −0.562189
\(83\) 6.38392e6 1.22550 0.612751 − 0.790276i \(-0.290063\pi\)
0.612751 + 0.790276i \(0.290063\pi\)
\(84\) 1.61896e6 0.298028
\(85\) 0 0
\(86\) 4.54734e6 0.770926
\(87\) −1.21181e7 −1.97295
\(88\) −3.96617e6 −0.620414
\(89\) −7.32978e6 −1.10211 −0.551056 − 0.834468i \(-0.685775\pi\)
−0.551056 + 0.834468i \(0.685775\pi\)
\(90\) 0 0
\(91\) 4.00682e6 0.557385
\(92\) −6.70378e6 −0.897557
\(93\) −1.27257e6 −0.164056
\(94\) −3.52291e6 −0.437476
\(95\) 0 0
\(96\) 1.17326e7 1.35346
\(97\) 2.38676e6 0.265526 0.132763 − 0.991148i \(-0.457615\pi\)
0.132763 + 0.991148i \(0.457615\pi\)
\(98\) −1.72159e6 −0.184773
\(99\) −5.28239e6 −0.547151
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 175.8.a.b.1.1 2
5.2 odd 4 175.8.b.c.99.1 4
5.3 odd 4 175.8.b.c.99.4 4
5.4 even 2 35.8.a.a.1.2 ✓ 2
15.14 odd 2 315.8.a.c.1.1 2
20.19 odd 2 560.8.a.i.1.2 2
35.34 odd 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 5.4 even 2
175.8.a.b.1.1 2 1.1 even 1 trivial
175.8.b.c.99.1 4 5.2 odd 4
175.8.b.c.99.4 4 5.3 odd 4
245.8.a.b.1.2 2 35.34 odd 2
315.8.a.c.1.1 2 15.14 odd 2
560.8.a.i.1.2 2 20.19 odd 2