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Properties

Label 975.4.a.j.1.2
Level $975$
Weight $4$
Character 975.1
Self dual yes
Analytic conductor $57.527$
Analytic rank $0$
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] = [975,4,Mod(1,975)] 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("975.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,6,14,0,-6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.5268622556\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) of defining polynomial
Character \(\chi\) \(=\) 975.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.74166 q^{2} +3.00000 q^{3} -0.483315 q^{4} +8.22497 q^{6} -7.48331 q^{7} -23.2583 q^{8} +9.00000 q^{9} +22.8999 q^{11} -1.44994 q^{12} +13.0000 q^{13} -20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} +24.6749 q^{18} +16.5167 q^{19} -22.4499 q^{21} +62.7836 q^{22} +175.600 q^{23} -69.7750 q^{24} +35.6415 q^{26} +27.0000 q^{27} +3.61680 q^{28} +291.800 q^{29} +117.283 q^{31} +21.8418 q^{32} +68.6997 q^{33} -183.783 q^{34} -4.34983 q^{36} +154.766 q^{37} +45.2831 q^{38} +39.0000 q^{39} -251.716 q^{41} -61.5501 q^{42} +502.566 q^{43} -11.0679 q^{44} +481.434 q^{46} +281.733 q^{47} -179.700 q^{48} -287.000 q^{49} -201.100 q^{51} -6.28309 q^{52} -366.999 q^{53} +74.0247 q^{54} +174.049 q^{56} +49.5501 q^{57} +800.015 q^{58} -79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} -67.3498 q^{63} +539.082 q^{64} +188.351 q^{66} -400.082 q^{67} +32.3982 q^{68} +526.799 q^{69} +528.299 q^{71} -209.325 q^{72} +734.366 q^{73} +424.316 q^{74} -7.98276 q^{76} -171.367 q^{77} +106.925 q^{78} +113.266 q^{79} +81.0000 q^{81} -690.118 q^{82} +933.466 q^{83} +10.8504 q^{84} +1377.86 q^{86} +875.399 q^{87} -532.613 q^{88} +1190.91 q^{89} -97.2831 q^{91} -84.8699 q^{92} +351.849 q^{93} +772.415 q^{94} +65.5253 q^{96} -557.165 q^{97} -786.856 q^{98} +206.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{3} + 14 q^{4} - 6 q^{6} - 54 q^{8} + 18 q^{9} - 44 q^{11} + 42 q^{12} + 26 q^{13} - 56 q^{14} - 30 q^{16} - 164 q^{17} - 18 q^{18} + 48 q^{19} + 380 q^{22} - 8 q^{23} - 162 q^{24} - 26 q^{26}+ \cdots - 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.74166 0.969322 0.484661 − 0.874702i \(-0.338943\pi\)
0.484661 + 0.874702i \(0.338943\pi\)
\(3\) 3.00000 0.577350
\(4\) −0.483315 −0.0604143
\(5\) 0 0
\(6\) 8.22497 0.559638
\(7\) −7.48331 −0.404061 −0.202031 − 0.979379i \(-0.564754\pi\)
−0.202031 + 0.979379i \(0.564754\pi\)
\(8\) −23.2583 −1.02788
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 22.8999 0.627689 0.313844 − 0.949474i \(-0.398383\pi\)
0.313844 + 0.949474i \(0.398383\pi\)
\(12\) −1.44994 −0.0348802
\(13\) 13.0000 0.277350
\(14\) −20.5167 −0.391665
\(15\) 0 0
\(16\) −59.8999 −0.935936
\(17\) −67.0334 −0.956352 −0.478176 − 0.878264i \(-0.658702\pi\)
−0.478176 + 0.878264i \(0.658702\pi\)
\(18\) 24.6749 0.323107
\(19\) 16.5167 0.199431 0.0997155 − 0.995016i \(-0.468207\pi\)
0.0997155 + 0.995016i \(0.468207\pi\)
\(20\) 0 0
\(21\) −22.4499 −0.233285
\(22\) 62.7836 0.608433
\(23\) 175.600 1.59196 0.795979 − 0.605324i \(-0.206956\pi\)
0.795979 + 0.605324i \(0.206956\pi\)
\(24\) −69.7750 −0.593449
\(25\) 0 0
\(26\) 35.6415 0.268842
\(27\) 27.0000 0.192450
\(28\) 3.61680 0.0244111
\(29\) 291.800 1.86848 0.934239 − 0.356648i \(-0.116080\pi\)
0.934239 + 0.356648i \(0.116080\pi\)
\(30\) 0 0
\(31\) 117.283 0.679505 0.339753 − 0.940515i \(-0.389657\pi\)
0.339753 + 0.940515i \(0.389657\pi\)
\(32\) 21.8418 0.120660
\(33\) 68.6997 0.362396
\(34\) −183.783 −0.927013
\(35\) 0 0
\(36\) −4.34983 −0.0201381
\(37\) 154.766 0.687661 0.343830 − 0.939032i \(-0.388276\pi\)
0.343830 + 0.939032i \(0.388276\pi\)
\(38\) 45.2831 0.193313
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −251.716 −0.958815 −0.479407 − 0.877592i \(-0.659148\pi\)
−0.479407 + 0.877592i \(0.659148\pi\)
\(42\) −61.5501 −0.226128
\(43\) 502.566 1.78234 0.891170 − 0.453669i \(-0.149885\pi\)
0.891170 + 0.453669i \(0.149885\pi\)
\(44\) −11.0679 −0.0379214
\(45\) 0 0
\(46\) 481.434 1.54312
\(47\) 281.733 0.874361 0.437181 − 0.899374i \(-0.355977\pi\)
0.437181 + 0.899374i \(0.355977\pi\)
\(48\) −179.700 −0.540363
\(49\) −287.000 −0.836735
\(50\) 0 0
\(51\) −201.100 −0.552150
\(52\) −6.28309 −0.0167559
\(53\) −366.999 −0.951154 −0.475577 − 0.879674i \(-0.657761\pi\)
−0.475577 + 0.879674i \(0.657761\pi\)
\(54\) 74.0247 0.186546
\(55\) 0 0
\(56\) 174.049 0.415328
\(57\) 49.5501 0.115141
\(58\) 800.015 1.81116
\(59\) −79.6663 −0.175791 −0.0878955 − 0.996130i \(-0.528014\pi\)
−0.0878955 + 0.996130i \(0.528014\pi\)
\(60\) 0 0
\(61\) −194.865 −0.409016 −0.204508 − 0.978865i \(-0.565559\pi\)
−0.204508 + 0.978865i \(0.565559\pi\)
\(62\) 321.550 0.658660
\(63\) −67.3498 −0.134687
\(64\) 539.082 1.05289
\(65\) 0 0
\(66\) 188.351 0.351279
\(67\) −400.082 −0.729519 −0.364759 − 0.931102i \(-0.618849\pi\)
−0.364759 + 0.931102i \(0.618849\pi\)
\(68\) 32.3982 0.0577774
\(69\) 526.799 0.919117
\(70\) 0 0
\(71\) 528.299 0.883065 0.441532 − 0.897245i \(-0.354435\pi\)
0.441532 + 0.897245i \(0.354435\pi\)
\(72\) −209.325 −0.342628
\(73\) 734.366 1.17741 0.588706 − 0.808347i \(-0.299638\pi\)
0.588706 + 0.808347i \(0.299638\pi\)
\(74\) 424.316 0.666565
\(75\) 0 0
\(76\) −7.98276 −0.0120485
\(77\) −171.367 −0.253625
\(78\) 106.925 0.155216
\(79\) 113.266 0.161309 0.0806545 − 0.996742i \(-0.474299\pi\)
0.0806545 + 0.996742i \(0.474299\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −690.118 −0.929400
\(83\) 933.466 1.23447 0.617236 − 0.786778i \(-0.288252\pi\)
0.617236 + 0.786778i \(0.288252\pi\)
\(84\) 10.8504 0.0140937
\(85\) 0 0
\(86\) 1377.86 1.72766
\(87\) 875.399 1.07877
\(88\) −532.613 −0.645191
\(89\) 1190.91 1.41839 0.709195 − 0.705012i \(-0.249059\pi\)
0.709195 + 0.705012i \(0.249059\pi\)
\(90\) 0 0
\(91\) −97.2831 −0.112066
\(92\) −84.8699 −0.0961771
\(93\) 351.849 0.392313
\(94\) 772.415 0.847538
\(95\) 0 0
\(96\) 65.5253 0.0696630
\(97\) −557.165 −0.583211 −0.291606 − 0.956539i \(-0.594189\pi\)
−0.291606 + 0.956539i \(0.594189\pi\)
\(98\) −786.856 −0.811066
\(99\) 206.099 0.209230
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 975.4.a.j.1.2 2
5.4 even 2 39.4.a.b.1.1 ✓ 2
15.14 odd 2 117.4.a.c.1.2 2
20.19 odd 2 624.4.a.r.1.2 2
35.34 odd 2 1911.4.a.h.1.1 2
40.19 odd 2 2496.4.a.s.1.1 2
40.29 even 2 2496.4.a.bc.1.1 2
60.59 even 2 1872.4.a.t.1.1 2
65.34 odd 4 507.4.b.f.337.2 4
65.44 odd 4 507.4.b.f.337.3 4
65.64 even 2 507.4.a.f.1.2 2
195.194 odd 2 1521.4.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 ✓ 2 5.4 even 2
117.4.a.c.1.2 2 15.14 odd 2
507.4.a.f.1.2 2 65.64 even 2
507.4.b.f.337.2 4 65.34 odd 4
507.4.b.f.337.3 4 65.44 odd 4
624.4.a.r.1.2 2 20.19 odd 2
975.4.a.j.1.2 2 1.1 even 1 trivial
1521.4.a.s.1.1 2 195.194 odd 2
1872.4.a.t.1.1 2 60.59 even 2
1911.4.a.h.1.1 2 35.34 odd 2
2496.4.a.s.1.1 2 40.19 odd 2
2496.4.a.bc.1.1 2 40.29 even 2