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Properties

Label 75.18.b.f.49.3
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [75,18,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 18); 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, 1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-878714] 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: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 963373x^{6} + 304121840676x^{4} + 35407691713753600x^{2} + 1297706673500416000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{8}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(-359.218i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.f.49.6

$q$-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)
 
\(f(q)\) \(=\) \(q-367.218i q^{2} +6561.00i q^{3} -3777.06 q^{4} +2.40932e6 q^{6} -1.91248e7i q^{7} -4.67450e7i q^{8} -4.30467e7 q^{9} -1.08277e9 q^{11} -2.47813e7i q^{12} +4.07421e9i q^{13} -7.02299e9 q^{14} -1.76607e10 q^{16} -1.03327e9i q^{17} +1.58075e10i q^{18} +8.27218e10 q^{19} +1.25478e11 q^{21} +3.97612e11i q^{22} +5.01376e11i q^{23} +3.06694e11 q^{24} +1.49612e12 q^{26} -2.82430e11i q^{27} +7.22357e10i q^{28} -3.86913e12 q^{29} +8.19265e12 q^{31} +3.58356e11i q^{32} -7.10405e12i q^{33} -3.79437e11 q^{34} +1.62590e11 q^{36} +1.23893e13i q^{37} -3.03770e13i q^{38} -2.67309e13 q^{39} -7.08479e13 q^{41} -4.60778e13i q^{42} +3.21849e13i q^{43} +4.08968e12 q^{44} +1.84114e14 q^{46} +1.79604e13i q^{47} -1.15872e14i q^{48} -1.33129e14 q^{49} +6.77931e12 q^{51} -1.53885e13i q^{52} +1.35368e14i q^{53} -1.03713e14 q^{54} -8.93991e14 q^{56} +5.42738e14i q^{57} +1.42081e15i q^{58} +1.26236e15 q^{59} +2.30641e15 q^{61} -3.00849e15i q^{62} +8.23262e14i q^{63} -2.18322e15 q^{64} -2.60873e15 q^{66} -4.62557e15i q^{67} +3.90274e12i q^{68} -3.28953e15 q^{69} +1.04536e16 q^{71} +2.01222e15i q^{72} -6.57154e15i q^{73} +4.54956e15 q^{74} -3.12445e14 q^{76} +2.07078e16i q^{77} +9.81605e15i q^{78} -7.36663e15 q^{79} +1.85302e15 q^{81} +2.60166e16i q^{82} +6.77164e15i q^{83} -4.73938e14 q^{84} +1.18189e16 q^{86} -2.53854e16i q^{87} +5.06140e16i q^{88} -6.56709e15 q^{89} +7.79186e16 q^{91} -1.89373e15i q^{92} +5.37520e16i q^{93} +6.59539e15 q^{94} -2.35117e15 q^{96} +2.22965e16i q^{97} +4.88875e16i q^{98} +4.66097e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 878714 q^{4} + 433026 q^{6} - 344373768 q^{9} - 1150990368 q^{11} + 13398632064 q^{14} + 15025367810 q^{16} - 397827528736 q^{19} - 230725490688 q^{21} + 835236570762 q^{24} - 1055057698044 q^{26}+ \cdots + 49\!\cdots\!28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 367.218i − 1.01431i −0.861856 − 0.507153i \(-0.830698\pi\)
0.861856 − 0.507153i \(-0.169302\pi\)
\(3\) 6561.00i 0.577350i
\(4\) −3777.06 −0.0288167
\(5\) 0 0
\(6\) 2.40932e6 0.585610
\(7\) − 1.91248e7i − 1.25391i −0.779057 − 0.626953i \(-0.784302\pi\)
0.779057 − 0.626953i \(-0.215698\pi\)
\(8\) − 4.67450e7i − 0.985077i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −1.08277e9 −1.52299 −0.761497 − 0.648169i \(-0.775535\pi\)
−0.761497 + 0.648169i \(0.775535\pi\)
\(12\) − 2.47813e7i − 0.0166373i
\(13\) 4.07421e9i 1.38524i 0.721304 + 0.692619i \(0.243543\pi\)
−0.721304 + 0.692619i \(0.756457\pi\)
\(14\) −7.02299e9 −1.27184
\(15\) 0 0
\(16\) −1.76607e10 −1.02799
\(17\) − 1.03327e9i − 0.0359252i −0.999839 − 0.0179626i \(-0.994282\pi\)
0.999839 − 0.0179626i \(-0.00571799\pi\)
\(18\) 1.58075e10i 0.338102i
\(19\) 8.27218e10 1.11742 0.558708 − 0.829365i \(-0.311297\pi\)
0.558708 + 0.829365i \(0.311297\pi\)
\(20\) 0 0
\(21\) 1.25478e11 0.723942
\(22\) 3.97612e11i 1.54478i
\(23\) 5.01376e11i 1.33499i 0.744615 + 0.667494i \(0.232633\pi\)
−0.744615 + 0.667494i \(0.767367\pi\)
\(24\) 3.06694e11 0.568735
\(25\) 0 0
\(26\) 1.49612e12 1.40506
\(27\) − 2.82430e11i − 0.192450i
\(28\) 7.22357e10i 0.0361334i
\(29\) −3.86913e12 −1.43625 −0.718126 − 0.695914i \(-0.755000\pi\)
−0.718126 + 0.695914i \(0.755000\pi\)
\(30\) 0 0
\(31\) 8.19265e12 1.72524 0.862621 − 0.505851i \(-0.168821\pi\)
0.862621 + 0.505851i \(0.168821\pi\)
\(32\) 3.58356e11i 0.0576156i
\(33\) − 7.10405e12i − 0.879301i
\(34\) −3.79437e11 −0.0364392
\(35\) 0 0
\(36\) 1.62590e11 0.00960556
\(37\) 1.23893e13i 0.579870i 0.957046 + 0.289935i \(0.0936336\pi\)
−0.957046 + 0.289935i \(0.906366\pi\)
\(38\) − 3.03770e13i − 1.13340i
\(39\) −2.67309e13 −0.799768
\(40\) 0 0
\(41\) −7.08479e13 −1.38568 −0.692842 − 0.721089i \(-0.743642\pi\)
−0.692842 + 0.721089i \(0.743642\pi\)
\(42\) − 4.60778e13i − 0.734299i
\(43\) 3.21849e13i 0.419923i 0.977710 + 0.209962i \(0.0673339\pi\)
−0.977710 + 0.209962i \(0.932666\pi\)
\(44\) 4.08968e12 0.0438876
\(45\) 0 0
\(46\) 1.84114e14 1.35409
\(47\) 1.79604e13i 0.110023i 0.998486 + 0.0550117i \(0.0175196\pi\)
−0.998486 + 0.0550117i \(0.982480\pi\)
\(48\) − 1.15872e14i − 0.593508i
\(49\) −1.33129e14 −0.572278
\(50\) 0 0
\(51\) 6.77931e12 0.0207414
\(52\) − 1.53885e13i − 0.0399180i
\(53\) 1.35368e14i 0.298657i 0.988788 + 0.149328i \(0.0477112\pi\)
−0.988788 + 0.149328i \(0.952289\pi\)
\(54\) −1.03713e14 −0.195203
\(55\) 0 0
\(56\) −8.93991e14 −1.23519
\(57\) 5.42738e14i 0.645140i
\(58\) 1.42081e15i 1.45680i
\(59\) 1.26236e15 1.11928 0.559642 − 0.828734i \(-0.310939\pi\)
0.559642 + 0.828734i \(0.310939\pi\)
\(60\) 0 0
\(61\) 2.30641e15 1.54040 0.770200 − 0.637803i \(-0.220157\pi\)
0.770200 + 0.637803i \(0.220157\pi\)
\(62\) − 3.00849e15i − 1.74992i
\(63\) 8.23262e14i 0.417968i
\(64\) −2.18322e15 −0.969546
\(65\) 0 0
\(66\) −2.60873e15 −0.891880
\(67\) − 4.62557e15i − 1.39165i −0.718212 − 0.695825i \(-0.755039\pi\)
0.718212 − 0.695825i \(-0.244961\pi\)
\(68\) 3.90274e12i 0.00103525i
\(69\) −3.28953e15 −0.770755
\(70\) 0 0
\(71\) 1.04536e16 1.92119 0.960597 − 0.277946i \(-0.0896536\pi\)
0.960597 + 0.277946i \(0.0896536\pi\)
\(72\) 2.01222e15i 0.328359i
\(73\) − 6.57154e15i − 0.953724i −0.878978 − 0.476862i \(-0.841774\pi\)
0.878978 − 0.476862i \(-0.158226\pi\)
\(74\) 4.54956e15 0.588165
\(75\) 0 0
\(76\) −3.12445e14 −0.0322002
\(77\) 2.07078e16i 1.90969i
\(78\) 9.81605e15i 0.811209i
\(79\) −7.36663e15 −0.546310 −0.273155 − 0.961970i \(-0.588067\pi\)
−0.273155 + 0.961970i \(0.588067\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 2.60166e16i 1.40551i
\(83\) 6.77164e15i 0.330012i 0.986293 + 0.165006i \(0.0527644\pi\)
−0.986293 + 0.165006i \(0.947236\pi\)
\(84\) −4.73938e14 −0.0208616
\(85\) 0 0
\(86\) 1.18189e16 0.425931
\(87\) − 2.53854e16i − 0.829220i
\(88\) 5.06140e16i 1.50027i
\(89\) −6.56709e15 −0.176831 −0.0884153 − 0.996084i \(-0.528180\pi\)
−0.0884153 + 0.996084i \(0.528180\pi\)
\(90\) 0 0
\(91\) 7.79186e16 1.73696
\(92\) − 1.89373e15i − 0.0384699i
\(93\) 5.37520e16i 0.996069i
\(94\) 6.59539e15 0.111597
\(95\) 0 0
\(96\) −2.35117e15 −0.0332644
\(97\) 2.22965e16i 0.288853i 0.989516 + 0.144426i \(0.0461337\pi\)
−0.989516 + 0.144426i \(0.953866\pi\)
\(98\) 4.88875e16i 0.580465i
\(99\) 4.66097e16 0.507665
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 75.18.b.f.49.3 8
5.2 odd 4 75.18.a.f.1.3 4
5.3 odd 4 15.18.a.d.1.2 ✓ 4
5.4 even 2 inner 75.18.b.f.49.6 8
15.8 even 4 45.18.a.f.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.2 ✓ 4 5.3 odd 4
45.18.a.f.1.3 4 15.8 even 4
75.18.a.f.1.3 4 5.2 odd 4
75.18.b.f.49.3 8 1.1 even 1 trivial
75.18.b.f.49.6 8 5.4 even 2 inner