Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3] 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: \(38.6276877123\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 469x^{2} + 4449x - 5580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(13.6993\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q+20.9703 q^{2} +81.0000 q^{3} -72.2477 q^{4} +1698.59 q^{6} +3575.78 q^{7} -12251.8 q^{8} +6561.00 q^{9} -74517.5 q^{11} -5852.07 q^{12} -34478.5 q^{13} +74985.1 q^{14} -219933. q^{16} +372733. q^{17} +137586. q^{18} -835114. q^{19} +289638. q^{21} -1.56265e6 q^{22} +31807.7 q^{23} -992398. q^{24} -723024. q^{26} +531441. q^{27} -258342. q^{28} -6.73894e6 q^{29} -4.04277e6 q^{31} +1.66087e6 q^{32} -6.03591e6 q^{33} +7.81631e6 q^{34} -474017. q^{36} -6.29831e6 q^{37} -1.75126e7 q^{38} -2.79276e6 q^{39} +1.10746e7 q^{41} +6.07379e6 q^{42} +1.42409e7 q^{43} +5.38372e6 q^{44} +667017. q^{46} -2.76408e7 q^{47} -1.78146e7 q^{48} -2.75674e7 q^{49} +3.01914e7 q^{51} +2.49100e6 q^{52} -8.30946e7 q^{53} +1.11445e7 q^{54} -4.38099e7 q^{56} -6.76443e7 q^{57} -1.41317e8 q^{58} +1.04437e8 q^{59} +3.99467e7 q^{61} -8.47779e7 q^{62} +2.34607e7 q^{63} +1.47435e8 q^{64} -1.26575e8 q^{66} +1.98351e8 q^{67} -2.69291e7 q^{68} +2.57643e6 q^{69} +4.52976e7 q^{71} -8.03843e7 q^{72} -3.64162e8 q^{73} -1.32077e8 q^{74} +6.03351e7 q^{76} -2.66458e8 q^{77} -5.85650e7 q^{78} -4.55156e8 q^{79} +4.30467e7 q^{81} +2.32238e8 q^{82} -3.16569e7 q^{83} -2.09257e7 q^{84} +2.98635e8 q^{86} -5.45854e8 q^{87} +9.12975e8 q^{88} +2.77919e8 q^{89} -1.23288e8 q^{91} -2.29804e6 q^{92} -3.27464e8 q^{93} -5.79636e8 q^{94} +1.34531e8 q^{96} +1.06978e9 q^{97} -5.78096e8 q^{98} -4.88909e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 324 q^{3} + 597 q^{4} + 243 q^{6} - 9834 q^{7} - 7671 q^{8} + 26244 q^{9} - 35994 q^{11} + 48357 q^{12} - 79998 q^{13} - 208182 q^{14} - 752815 q^{16} - 667878 q^{17} + 19683 q^{18} - 425792 q^{19}+ \cdots - 236156634 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.9703 0.926764 0.463382 − 0.886159i \(-0.346636\pi\)
0.463382 + 0.886159i \(0.346636\pi\)
\(3\) 81.0000 0.577350
\(4\) −72.2477 −0.141109
\(5\) 0 0
\(6\) 1698.59 0.535067
\(7\) 3575.78 0.562898 0.281449 − 0.959576i \(-0.409185\pi\)
0.281449 + 0.959576i \(0.409185\pi\)
\(8\) −12251.8 −1.05754
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) −74517.5 −1.53458 −0.767292 − 0.641297i \(-0.778397\pi\)
−0.767292 + 0.641297i \(0.778397\pi\)
\(12\) −5852.07 −0.0814692
\(13\) −34478.5 −0.334814 −0.167407 − 0.985888i \(-0.553539\pi\)
−0.167407 + 0.985888i \(0.553539\pi\)
\(14\) 74985.1 0.521674
\(15\) 0 0
\(16\) −219933. −0.838979
\(17\) 372733. 1.08237 0.541187 − 0.840902i \(-0.317975\pi\)
0.541187 + 0.840902i \(0.317975\pi\)
\(18\) 137586. 0.308921
\(19\) −835114. −1.47013 −0.735063 − 0.677998i \(-0.762848\pi\)
−0.735063 + 0.677998i \(0.762848\pi\)
\(20\) 0 0
\(21\) 289638. 0.324989
\(22\) −1.56265e6 −1.42220
\(23\) 31807.7 0.0237005 0.0118503 − 0.999930i \(-0.496228\pi\)
0.0118503 + 0.999930i \(0.496228\pi\)
\(24\) −992398. −0.610570
\(25\) 0 0
\(26\) −723024. −0.310294
\(27\) 531441. 0.192450
\(28\) −258342. −0.0794299
\(29\) −6.73894e6 −1.76930 −0.884649 − 0.466258i \(-0.845602\pi\)
−0.884649 + 0.466258i \(0.845602\pi\)
\(30\) 0 0
\(31\) −4.04277e6 −0.786232 −0.393116 − 0.919489i \(-0.628603\pi\)
−0.393116 + 0.919489i \(0.628603\pi\)
\(32\) 1.66087e6 0.280003
\(33\) −6.03591e6 −0.885993
\(34\) 7.81631e6 1.00311
\(35\) 0 0
\(36\) −474017. −0.0470363
\(37\) −6.29831e6 −0.552480 −0.276240 − 0.961089i \(-0.589088\pi\)
−0.276240 + 0.961089i \(0.589088\pi\)
\(38\) −1.75126e7 −1.36246
\(39\) −2.79276e6 −0.193305
\(40\) 0 0
\(41\) 1.10746e7 0.612072 0.306036 − 0.952020i \(-0.400997\pi\)
0.306036 + 0.952020i \(0.400997\pi\)
\(42\) 6.07379e6 0.301188
\(43\) 1.42409e7 0.635226 0.317613 − 0.948220i \(-0.397119\pi\)
0.317613 + 0.948220i \(0.397119\pi\)
\(44\) 5.38372e6 0.216543
\(45\) 0 0
\(46\) 667017. 0.0219648
\(47\) −2.76408e7 −0.826249 −0.413124 − 0.910675i \(-0.635563\pi\)
−0.413124 + 0.910675i \(0.635563\pi\)
\(48\) −1.78146e7 −0.484385
\(49\) −2.75674e7 −0.683146
\(50\) 0 0
\(51\) 3.01914e7 0.624909
\(52\) 2.49100e6 0.0472452
\(53\) −8.30946e7 −1.44654 −0.723272 − 0.690564i \(-0.757363\pi\)
−0.723272 + 0.690564i \(0.757363\pi\)
\(54\) 1.11445e7 0.178356
\(55\) 0 0
\(56\) −4.38099e7 −0.595286
\(57\) −6.76443e7 −0.848778
\(58\) −1.41317e8 −1.63972
\(59\) 1.04437e8 1.12207 0.561034 − 0.827793i \(-0.310404\pi\)
0.561034 + 0.827793i \(0.310404\pi\)
\(60\) 0 0
\(61\) 3.99467e7 0.369400 0.184700 − 0.982795i \(-0.440869\pi\)
0.184700 + 0.982795i \(0.440869\pi\)
\(62\) −8.47779e7 −0.728652
\(63\) 2.34607e7 0.187633
\(64\) 1.47435e8 1.09848
\(65\) 0 0
\(66\) −1.26575e8 −0.821106
\(67\) 1.98351e8 1.20254 0.601268 − 0.799047i \(-0.294662\pi\)
0.601268 + 0.799047i \(0.294662\pi\)
\(68\) −2.69291e7 −0.152733
\(69\) 2.57643e6 0.0136835
\(70\) 0 0
\(71\) 4.52976e7 0.211550 0.105775 − 0.994390i \(-0.466268\pi\)
0.105775 + 0.994390i \(0.466268\pi\)
\(72\) −8.03843e7 −0.352513
\(73\) −3.64162e8 −1.50087 −0.750433 − 0.660947i \(-0.770155\pi\)
−0.750433 + 0.660947i \(0.770155\pi\)
\(74\) −1.32077e8 −0.512018
\(75\) 0 0
\(76\) 6.03351e7 0.207448
\(77\) −2.66458e8 −0.863815
\(78\) −5.85650e7 −0.179148
\(79\) −4.55156e8 −1.31473 −0.657367 − 0.753570i \(-0.728330\pi\)
−0.657367 + 0.753570i \(0.728330\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 2.32238e8 0.567246
\(83\) −3.16569e7 −0.0732178 −0.0366089 − 0.999330i \(-0.511656\pi\)
−0.0366089 + 0.999330i \(0.511656\pi\)
\(84\) −2.09257e7 −0.0458589
\(85\) 0 0
\(86\) 2.98635e8 0.588704
\(87\) −5.45854e8 −1.02150
\(88\) 9.12975e8 1.62288
\(89\) 2.77919e8 0.469530 0.234765 − 0.972052i \(-0.424568\pi\)
0.234765 + 0.972052i \(0.424568\pi\)
\(90\) 0 0
\(91\) −1.23288e8 −0.188466
\(92\) −2.29804e6 −0.00334435
\(93\) −3.27464e8 −0.453932
\(94\) −5.79636e8 −0.765737
\(95\) 0 0
\(96\) 1.34531e8 0.161660
\(97\) 1.06978e9 1.22694 0.613468 − 0.789720i \(-0.289774\pi\)
0.613468 + 0.789720i \(0.289774\pi\)
\(98\) −5.78096e8 −0.633115
\(99\) −4.88909e8 −0.511528
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 75.10.a.l.1.3 4
3.2 odd 2 225.10.a.q.1.2 4
5.2 odd 4 15.10.b.a.4.6 yes 8
5.3 odd 4 15.10.b.a.4.3 ✓ 8
5.4 even 2 75.10.a.i.1.2 4
15.2 even 4 45.10.b.c.19.3 8
15.8 even 4 45.10.b.c.19.6 8
15.14 odd 2 225.10.a.u.1.3 4
20.3 even 4 240.10.f.c.49.2 8
20.7 even 4 240.10.f.c.49.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.b.a.4.3 ✓ 8 5.3 odd 4
15.10.b.a.4.6 yes 8 5.2 odd 4
45.10.b.c.19.3 8 15.2 even 4
45.10.b.c.19.6 8 15.8 even 4
75.10.a.i.1.2 4 5.4 even 2
75.10.a.l.1.3 4 1.1 even 1 trivial
225.10.a.q.1.2 4 3.2 odd 2
225.10.a.u.1.3 4 15.14 odd 2
240.10.f.c.49.2 8 20.3 even 4
240.10.f.c.49.6 8 20.7 even 4