Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [175,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,6] 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: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.44622\) 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)
 
\(f(q)\) \(=\) \(q-10.8924 q^{2} +195.817 q^{3} -393.355 q^{4} -2132.92 q^{6} +2401.00 q^{7} +9861.52 q^{8} +18661.3 q^{9} +63864.3 q^{11} -77025.5 q^{12} +164679. q^{13} -26152.8 q^{14} +93981.5 q^{16} +362910. q^{17} -203267. q^{18} -436498. q^{19} +470156. q^{21} -695638. q^{22} -918199. q^{23} +1.93105e6 q^{24} -1.79375e6 q^{26} -200076. q^{27} -944445. q^{28} -3.68643e6 q^{29} +3.47629e6 q^{31} -6.07279e6 q^{32} +1.25057e7 q^{33} -3.95298e6 q^{34} -7.34049e6 q^{36} -1.88149e7 q^{37} +4.75453e6 q^{38} +3.22469e7 q^{39} +2.40714e6 q^{41} -5.12115e6 q^{42} +1.25306e7 q^{43} -2.51213e7 q^{44} +1.00014e7 q^{46} +5.54509e7 q^{47} +1.84032e7 q^{48} +5.76480e6 q^{49} +7.10639e7 q^{51} -6.47772e7 q^{52} +9.26889e7 q^{53} +2.17931e6 q^{54} +2.36775e7 q^{56} -8.54737e7 q^{57} +4.01542e7 q^{58} -2.52600e7 q^{59} +6.93275e7 q^{61} -3.78653e7 q^{62} +4.48057e7 q^{63} +1.80290e7 q^{64} -1.36218e8 q^{66} +2.33494e7 q^{67} -1.42752e8 q^{68} -1.79799e8 q^{69} -1.06194e8 q^{71} +1.84028e8 q^{72} +2.10115e8 q^{73} +2.04940e8 q^{74} +1.71699e8 q^{76} +1.53338e8 q^{77} -3.51247e8 q^{78} -149606. q^{79} -4.06488e8 q^{81} -2.62197e7 q^{82} -5.21565e8 q^{83} -1.84938e8 q^{84} -1.36489e8 q^{86} -7.21865e8 q^{87} +6.29799e8 q^{88} +2.98587e8 q^{89} +3.95394e8 q^{91} +3.61178e8 q^{92} +6.80716e8 q^{93} -6.03996e8 q^{94} -1.18915e9 q^{96} +8.95983e8 q^{97} -6.27928e7 q^{98} +1.19179e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 86 q^{3} - 620 q^{4} - 3988 q^{6} + 4802 q^{7} - 2616 q^{8} + 11038 q^{9} + 35316 q^{11} - 52136 q^{12} + 26530 q^{13} + 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 332042 q^{18} - 925426 q^{19}+ \cdots + 1409417860 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.8924 −0.481383 −0.240691 − 0.970602i \(-0.577374\pi\)
−0.240691 + 0.970602i \(0.577374\pi\)
\(3\) 195.817 1.39574 0.697870 − 0.716225i \(-0.254131\pi\)
0.697870 + 0.716225i \(0.254131\pi\)
\(4\) −393.355 −0.768271
\(5\) 0 0
\(6\) −2132.92 −0.671885
\(7\) 2401.00 0.377964
\(8\) 9861.52 0.851215
\(9\) 18661.3 0.948090
\(10\) 0 0
\(11\) 63864.3 1.31520 0.657599 − 0.753369i \(-0.271572\pi\)
0.657599 + 0.753369i \(0.271572\pi\)
\(12\) −77025.5 −1.07231
\(13\) 164679. 1.59916 0.799581 − 0.600558i \(-0.205055\pi\)
0.799581 + 0.600558i \(0.205055\pi\)
\(14\) −26152.8 −0.181946
\(15\) 0 0
\(16\) 93981.5 0.358511
\(17\) 362910. 1.05385 0.526925 − 0.849912i \(-0.323345\pi\)
0.526925 + 0.849912i \(0.323345\pi\)
\(18\) −203267. −0.456394
\(19\) −436498. −0.768406 −0.384203 − 0.923249i \(-0.625524\pi\)
−0.384203 + 0.923249i \(0.625524\pi\)
\(20\) 0 0
\(21\) 470156. 0.527540
\(22\) −695638. −0.633113
\(23\) −918199. −0.684166 −0.342083 − 0.939670i \(-0.611132\pi\)
−0.342083 + 0.939670i \(0.611132\pi\)
\(24\) 1.93105e6 1.18807
\(25\) 0 0
\(26\) −1.79375e6 −0.769809
\(27\) −200076. −0.0724531
\(28\) −944445. −0.290379
\(29\) −3.68643e6 −0.967865 −0.483932 − 0.875105i \(-0.660792\pi\)
−0.483932 + 0.875105i \(0.660792\pi\)
\(30\) 0 0
\(31\) 3.47629e6 0.676064 0.338032 − 0.941135i \(-0.390239\pi\)
0.338032 + 0.941135i \(0.390239\pi\)
\(32\) −6.07279e6 −1.02380
\(33\) 1.25057e7 1.83567
\(34\) −3.95298e6 −0.507305
\(35\) 0 0
\(36\) −7.34049e6 −0.728390
\(37\) −1.88149e7 −1.65042 −0.825210 − 0.564826i \(-0.808943\pi\)
−0.825210 + 0.564826i \(0.808943\pi\)
\(38\) 4.75453e6 0.369897
\(39\) 3.22469e7 2.23201
\(40\) 0 0
\(41\) 2.40714e6 0.133038 0.0665188 − 0.997785i \(-0.478811\pi\)
0.0665188 + 0.997785i \(0.478811\pi\)
\(42\) −5.12115e6 −0.253949
\(43\) 1.25306e7 0.558938 0.279469 − 0.960155i \(-0.409842\pi\)
0.279469 + 0.960155i \(0.409842\pi\)
\(44\) −2.51213e7 −1.01043
\(45\) 0 0
\(46\) 1.00014e7 0.329346
\(47\) 5.54509e7 1.65756 0.828779 − 0.559577i \(-0.189036\pi\)
0.828779 + 0.559577i \(0.189036\pi\)
\(48\) 1.84032e7 0.500388
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 7.10639e7 1.47090
\(52\) −6.47772e7 −1.22859
\(53\) 9.26889e7 1.61356 0.806782 − 0.590849i \(-0.201207\pi\)
0.806782 + 0.590849i \(0.201207\pi\)
\(54\) 2.17931e6 0.0348777
\(55\) 0 0
\(56\) 2.36775e7 0.321729
\(57\) −8.54737e7 −1.07250
\(58\) 4.01542e7 0.465913
\(59\) −2.52600e7 −0.271393 −0.135696 − 0.990750i \(-0.543327\pi\)
−0.135696 + 0.990750i \(0.543327\pi\)
\(60\) 0 0
\(61\) 6.93275e7 0.641093 0.320547 − 0.947233i \(-0.396133\pi\)
0.320547 + 0.947233i \(0.396133\pi\)
\(62\) −3.78653e7 −0.325446
\(63\) 4.48057e7 0.358344
\(64\) 1.80290e7 0.134326
\(65\) 0 0
\(66\) −1.36218e8 −0.883661
\(67\) 2.33494e7 0.141559 0.0707796 − 0.997492i \(-0.477451\pi\)
0.0707796 + 0.997492i \(0.477451\pi\)
\(68\) −1.42752e8 −0.809643
\(69\) −1.79799e8 −0.954918
\(70\) 0 0
\(71\) −1.06194e8 −0.495950 −0.247975 − 0.968766i \(-0.579765\pi\)
−0.247975 + 0.968766i \(0.579765\pi\)
\(72\) 1.84028e8 0.807028
\(73\) 2.10115e8 0.865974 0.432987 − 0.901400i \(-0.357460\pi\)
0.432987 + 0.901400i \(0.357460\pi\)
\(74\) 2.04940e8 0.794483
\(75\) 0 0
\(76\) 1.71699e8 0.590344
\(77\) 1.53338e8 0.497098
\(78\) −3.51247e8 −1.07445
\(79\) −149606. −0.000432144 0 −0.000216072 − 1.00000i \(-0.500069\pi\)
−0.000216072 1.00000i \(0.500069\pi\)
\(80\) 0 0
\(81\) −4.06488e8 −1.04922
\(82\) −2.62197e7 −0.0640420
\(83\) −5.21565e8 −1.20630 −0.603152 − 0.797626i \(-0.706089\pi\)
−0.603152 + 0.797626i \(0.706089\pi\)
\(84\) −1.84938e8 −0.405294
\(85\) 0 0
\(86\) −1.36489e8 −0.269063
\(87\) −7.21865e8 −1.35089
\(88\) 6.29799e8 1.11952
\(89\) 2.98587e8 0.504448 0.252224 − 0.967669i \(-0.418838\pi\)
0.252224 + 0.967669i \(0.418838\pi\)
\(90\) 0 0
\(91\) 3.95394e8 0.604426
\(92\) 3.61178e8 0.525625
\(93\) 6.80716e8 0.943610
\(94\) −6.03996e8 −0.797919
\(95\) 0 0
\(96\) −1.18915e9 −1.42895
\(97\) 8.95983e8 1.02761 0.513803 − 0.857908i \(-0.328236\pi\)
0.513803 + 0.857908i \(0.328236\pi\)
\(98\) −6.27928e7 −0.0687689
\(99\) 1.19179e9 1.24693
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 175.10.a.b.1.1 2
5.2 odd 4 175.10.b.b.99.2 4
5.3 odd 4 175.10.b.b.99.3 4
5.4 even 2 7.10.a.a.1.2 ✓ 2
15.14 odd 2 63.10.a.d.1.1 2
20.19 odd 2 112.10.a.e.1.2 2
35.4 even 6 49.10.c.c.30.1 4
35.9 even 6 49.10.c.c.18.1 4
35.19 odd 6 49.10.c.b.18.1 4
35.24 odd 6 49.10.c.b.30.1 4
35.34 odd 2 49.10.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.2 ✓ 2 5.4 even 2
49.10.a.b.1.2 2 35.34 odd 2
49.10.c.b.18.1 4 35.19 odd 6
49.10.c.b.30.1 4 35.24 odd 6
49.10.c.c.18.1 4 35.9 even 6
49.10.c.c.30.1 4 35.4 even 6
63.10.a.d.1.1 2 15.14 odd 2
112.10.a.e.1.2 2 20.19 odd 2
175.10.a.b.1.1 2 1.1 even 1 trivial
175.10.b.b.99.2 4 5.2 odd 4
175.10.b.b.99.3 4 5.3 odd 4