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(99,175)] 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("175.99"); 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([1, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,1240] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{193})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 97x^{2} + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-6.44622i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.10.b.b.99.3

$q$-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)
 
\(f(q)\) \(=\) \(q-10.8924i q^{2} -195.817i q^{3} +393.355 q^{4} -2132.92 q^{6} +2401.00i q^{7} -9861.52i q^{8} -18661.3 q^{9} +63864.3 q^{11} -77025.5i q^{12} -164679. i q^{13} +26152.8 q^{14} +93981.5 q^{16} +362910. i q^{17} +203267. i q^{18} +436498. q^{19} +470156. q^{21} -695638. i q^{22} +918199. i q^{23} -1.93105e6 q^{24} -1.79375e6 q^{26} -200076. i q^{27} +944445. i q^{28} +3.68643e6 q^{29} +3.47629e6 q^{31} -6.07279e6i q^{32} -1.25057e7i q^{33} +3.95298e6 q^{34} -7.34049e6 q^{36} -1.88149e7i q^{37} -4.75453e6i q^{38} -3.22469e7 q^{39} +2.40714e6 q^{41} -5.12115e6i q^{42} -1.25306e7i q^{43} +2.51213e7 q^{44} +1.00014e7 q^{46} +5.54509e7i q^{47} -1.84032e7i q^{48} -5.76480e6 q^{49} +7.10639e7 q^{51} -6.47772e7i q^{52} -9.26889e7i q^{53} -2.17931e6 q^{54} +2.36775e7 q^{56} -8.54737e7i q^{57} -4.01542e7i q^{58} +2.52600e7 q^{59} +6.93275e7 q^{61} -3.78653e7i q^{62} -4.48057e7i q^{63} -1.80290e7 q^{64} -1.36218e8 q^{66} +2.33494e7i q^{67} +1.42752e8i q^{68} +1.79799e8 q^{69} -1.06194e8 q^{71} +1.84028e8i q^{72} -2.10115e8i q^{73} -2.04940e8 q^{74} +1.71699e8 q^{76} +1.53338e8i q^{77} +3.51247e8i q^{78} +149606. q^{79} -4.06488e8 q^{81} -2.62197e7i q^{82} +5.21565e8i q^{83} +1.84938e8 q^{84} -1.36489e8 q^{86} -7.21865e8i q^{87} -6.29799e8i q^{88} -2.98587e8 q^{89} +3.95394e8 q^{91} +3.61178e8i q^{92} -6.80716e8i q^{93} +6.03996e8 q^{94} -1.18915e9 q^{96} +8.95983e8i q^{97} +6.27928e7i q^{98} -1.19179e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1240 q^{4} - 7976 q^{6} - 22076 q^{9} + 70632 q^{11} - 28812 q^{14} - 1504 q^{16} + 1850852 q^{19} + 412972 q^{21} - 6602592 q^{24} - 8254848 q^{26} + 20007168 q^{29} + 4934520 q^{31} + 4493352 q^{34}+ \cdots - 2818835720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 10.8924i − 0.481383i −0.970602 − 0.240691i \(-0.922626\pi\)
0.970602 − 0.240691i \(-0.0773741\pi\)
\(3\) − 195.817i − 1.39574i −0.716225 − 0.697870i \(-0.754131\pi\)
0.716225 − 0.697870i \(-0.245869\pi\)
\(4\) 393.355 0.768271
\(5\) 0 0
\(6\) −2132.92 −0.671885
\(7\) 2401.00i 0.377964i
\(8\) − 9861.52i − 0.851215i
\(9\) −18661.3 −0.948090
\(10\) 0 0
\(11\) 63864.3 1.31520 0.657599 − 0.753369i \(-0.271572\pi\)
0.657599 + 0.753369i \(0.271572\pi\)
\(12\) − 77025.5i − 1.07231i
\(13\) − 164679.i − 1.59916i −0.600558 − 0.799581i \(-0.705055\pi\)
0.600558 − 0.799581i \(-0.294945\pi\)
\(14\) 26152.8 0.181946
\(15\) 0 0
\(16\) 93981.5 0.358511
\(17\) 362910.i 1.05385i 0.849912 + 0.526925i \(0.176655\pi\)
−0.849912 + 0.526925i \(0.823345\pi\)
\(18\) 203267.i 0.456394i
\(19\) 436498. 0.768406 0.384203 − 0.923249i \(-0.374476\pi\)
0.384203 + 0.923249i \(0.374476\pi\)
\(20\) 0 0
\(21\) 470156. 0.527540
\(22\) − 695638.i − 0.633113i
\(23\) 918199.i 0.684166i 0.939670 + 0.342083i \(0.111132\pi\)
−0.939670 + 0.342083i \(0.888868\pi\)
\(24\) −1.93105e6 −1.18807
\(25\) 0 0
\(26\) −1.79375e6 −0.769809
\(27\) − 200076.i − 0.0724531i
\(28\) 944445.i 0.290379i
\(29\) 3.68643e6 0.967865 0.483932 − 0.875105i \(-0.339208\pi\)
0.483932 + 0.875105i \(0.339208\pi\)
\(30\) 0 0
\(31\) 3.47629e6 0.676064 0.338032 − 0.941135i \(-0.390239\pi\)
0.338032 + 0.941135i \(0.390239\pi\)
\(32\) − 6.07279e6i − 1.02380i
\(33\) − 1.25057e7i − 1.83567i
\(34\) 3.95298e6 0.507305
\(35\) 0 0
\(36\) −7.34049e6 −0.728390
\(37\) − 1.88149e7i − 1.65042i −0.564826 − 0.825210i \(-0.691057\pi\)
0.564826 − 0.825210i \(-0.308943\pi\)
\(38\) − 4.75453e6i − 0.369897i
\(39\) −3.22469e7 −2.23201
\(40\) 0 0
\(41\) 2.40714e6 0.133038 0.0665188 − 0.997785i \(-0.478811\pi\)
0.0665188 + 0.997785i \(0.478811\pi\)
\(42\) − 5.12115e6i − 0.253949i
\(43\) − 1.25306e7i − 0.558938i −0.960155 − 0.279469i \(-0.909842\pi\)
0.960155 − 0.279469i \(-0.0901584\pi\)
\(44\) 2.51213e7 1.01043
\(45\) 0 0
\(46\) 1.00014e7 0.329346
\(47\) 5.54509e7i 1.65756i 0.559577 + 0.828779i \(0.310964\pi\)
−0.559577 + 0.828779i \(0.689036\pi\)
\(48\) − 1.84032e7i − 0.500388i
\(49\) −5.76480e6 −0.142857
\(50\) 0 0
\(51\) 7.10639e7 1.47090
\(52\) − 6.47772e7i − 1.22859i
\(53\) − 9.26889e7i − 1.61356i −0.590849 − 0.806782i \(-0.701207\pi\)
0.590849 − 0.806782i \(-0.298793\pi\)
\(54\) −2.17931e6 −0.0348777
\(55\) 0 0
\(56\) 2.36775e7 0.321729
\(57\) − 8.54737e7i − 1.07250i
\(58\) − 4.01542e7i − 0.465913i
\(59\) 2.52600e7 0.271393 0.135696 − 0.990750i \(-0.456673\pi\)
0.135696 + 0.990750i \(0.456673\pi\)
\(60\) 0 0
\(61\) 6.93275e7 0.641093 0.320547 − 0.947233i \(-0.396133\pi\)
0.320547 + 0.947233i \(0.396133\pi\)
\(62\) − 3.78653e7i − 0.325446i
\(63\) − 4.48057e7i − 0.358344i
\(64\) −1.80290e7 −0.134326
\(65\) 0 0
\(66\) −1.36218e8 −0.883661
\(67\) 2.33494e7i 0.141559i 0.997492 + 0.0707796i \(0.0225487\pi\)
−0.997492 + 0.0707796i \(0.977451\pi\)
\(68\) 1.42752e8i 0.809643i
\(69\) 1.79799e8 0.954918
\(70\) 0 0
\(71\) −1.06194e8 −0.495950 −0.247975 − 0.968766i \(-0.579765\pi\)
−0.247975 + 0.968766i \(0.579765\pi\)
\(72\) 1.84028e8i 0.807028i
\(73\) − 2.10115e8i − 0.865974i −0.901400 − 0.432987i \(-0.857460\pi\)
0.901400 − 0.432987i \(-0.142540\pi\)
\(74\) −2.04940e8 −0.794483
\(75\) 0 0
\(76\) 1.71699e8 0.590344
\(77\) 1.53338e8i 0.497098i
\(78\) 3.51247e8i 1.07445i
\(79\) 149606. 0.000432144 0 0.000216072 − 1.00000i \(-0.499931\pi\)
0.000216072 1.00000i \(0.499931\pi\)
\(80\) 0 0
\(81\) −4.06488e8 −1.04922
\(82\) − 2.62197e7i − 0.0640420i
\(83\) 5.21565e8i 1.20630i 0.797626 + 0.603152i \(0.206089\pi\)
−0.797626 + 0.603152i \(0.793911\pi\)
\(84\) 1.84938e8 0.405294
\(85\) 0 0
\(86\) −1.36489e8 −0.269063
\(87\) − 7.21865e8i − 1.35089i
\(88\) − 6.29799e8i − 1.11952i
\(89\) −2.98587e8 −0.504448 −0.252224 − 0.967669i \(-0.581162\pi\)
−0.252224 + 0.967669i \(0.581162\pi\)
\(90\) 0 0
\(91\) 3.95394e8 0.604426
\(92\) 3.61178e8i 0.525625i
\(93\) − 6.80716e8i − 0.943610i
\(94\) 6.03996e8 0.797919
\(95\) 0 0
\(96\) −1.18915e9 −1.42895
\(97\) 8.95983e8i 1.02761i 0.857908 + 0.513803i \(0.171764\pi\)
−0.857908 + 0.513803i \(0.828236\pi\)
\(98\) 6.27928e7i 0.0687689i
\(99\) −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.b.b.99.2 4
5.2 odd 4 7.10.a.a.1.2 ✓ 2
5.3 odd 4 175.10.a.b.1.1 2
5.4 even 2 inner 175.10.b.b.99.3 4
15.2 even 4 63.10.a.d.1.1 2
20.7 even 4 112.10.a.e.1.2 2
35.2 odd 12 49.10.c.c.18.1 4
35.12 even 12 49.10.c.b.18.1 4
35.17 even 12 49.10.c.b.30.1 4
35.27 even 4 49.10.a.b.1.2 2
35.32 odd 12 49.10.c.c.30.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.2 ✓ 2 5.2 odd 4
49.10.a.b.1.2 2 35.27 even 4
49.10.c.b.18.1 4 35.12 even 12
49.10.c.b.30.1 4 35.17 even 12
49.10.c.c.18.1 4 35.2 odd 12
49.10.c.c.30.1 4 35.32 odd 12
63.10.a.d.1.1 2 15.2 even 4
112.10.a.e.1.2 2 20.7 even 4
175.10.a.b.1.1 2 5.3 odd 4
175.10.b.b.99.2 4 1.1 even 1 trivial
175.10.b.b.99.3 4 5.4 even 2 inner