↑

Properties

Label 75.18.a.d.1.3
Level $75$
Weight $18$
Character 75.1
Self dual yes
Analytic conductor $137.417$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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(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, 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, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,253] 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: \(137.416565508\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 182396x + 3921120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(416.408\) 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)
 
\(f(q)\) \(=\) \(q+500.408 q^{2} -6561.00 q^{3} +119336. q^{4} -3.28318e6 q^{6} -8.90512e6 q^{7} -5.87255e6 q^{8} +4.30467e7 q^{9} +4.48684e8 q^{11} -7.82967e8 q^{12} +3.89738e9 q^{13} -4.45620e9 q^{14} -1.85803e10 q^{16} +5.96526e8 q^{17} +2.15409e10 q^{18} -4.36939e10 q^{19} +5.84265e10 q^{21} +2.24525e11 q^{22} -7.23343e10 q^{23} +3.85298e10 q^{24} +1.95028e12 q^{26} -2.82430e11 q^{27} -1.06271e12 q^{28} +1.82054e12 q^{29} +5.27128e12 q^{31} -8.52803e12 q^{32} -2.94382e12 q^{33} +2.98507e11 q^{34} +5.13704e12 q^{36} +1.62659e13 q^{37} -2.18648e13 q^{38} -2.55707e13 q^{39} +9.78891e12 q^{41} +2.92371e13 q^{42} -1.46155e14 q^{43} +5.35444e13 q^{44} -3.61967e13 q^{46} -2.43226e14 q^{47} +1.21906e14 q^{48} -1.53329e14 q^{49} -3.91381e12 q^{51} +4.65100e14 q^{52} -6.84721e14 q^{53} -1.41330e14 q^{54} +5.22958e13 q^{56} +2.86676e14 q^{57} +9.11015e14 q^{58} +9.83635e14 q^{59} -1.84021e15 q^{61} +2.63779e15 q^{62} -3.83336e14 q^{63} -1.83214e15 q^{64} -1.47311e15 q^{66} +2.46536e15 q^{67} +7.11874e13 q^{68} +4.74585e14 q^{69} +9.43154e14 q^{71} -2.52794e14 q^{72} -1.15522e16 q^{73} +8.13959e15 q^{74} -5.21428e15 q^{76} -3.99559e15 q^{77} -1.27958e16 q^{78} +1.58855e16 q^{79} +1.85302e15 q^{81} +4.89845e15 q^{82} -1.62194e16 q^{83} +6.97241e15 q^{84} -7.31369e16 q^{86} -1.19446e16 q^{87} -2.63492e15 q^{88} -4.03683e16 q^{89} -3.47067e16 q^{91} -8.63212e15 q^{92} -3.45849e16 q^{93} -1.21712e17 q^{94} +5.59524e16 q^{96} +7.85724e16 q^{97} -7.67273e16 q^{98} +1.93144e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 253 q^{2} - 19683 q^{3} - 7087 q^{4} - 1659933 q^{6} + 4332484 q^{7} + 16188513 q^{8} + 129140163 q^{9} + 943563680 q^{11} + 46497807 q^{12} - 4257013150 q^{13} + 1847483988 q^{14} - 21943871359 q^{16}+ \cdots + 40\!\cdots\!80 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 500.408 1.38220 0.691098 − 0.722761i \(-0.257127\pi\)
0.691098 + 0.722761i \(0.257127\pi\)
\(3\) −6561.00 −0.577350
\(4\) 119336. 0.910465
\(5\) 0 0
\(6\) −3.28318e6 −0.798011
\(7\) −8.90512e6 −0.583857 −0.291929 − 0.956440i \(-0.594297\pi\)
−0.291929 + 0.956440i \(0.594297\pi\)
\(8\) −5.87255e6 −0.123755
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 4.48684e8 0.631107 0.315554 − 0.948908i \(-0.397810\pi\)
0.315554 + 0.948908i \(0.397810\pi\)
\(12\) −7.82967e8 −0.525657
\(13\) 3.89738e9 1.32512 0.662559 − 0.749010i \(-0.269470\pi\)
0.662559 + 0.749010i \(0.269470\pi\)
\(14\) −4.45620e9 −0.807005
\(15\) 0 0
\(16\) −1.85803e10 −1.08152
\(17\) 5.96526e8 0.0207402 0.0103701 − 0.999946i \(-0.496699\pi\)
0.0103701 + 0.999946i \(0.496699\pi\)
\(18\) 2.15409e10 0.460732
\(19\) −4.36939e10 −0.590222 −0.295111 − 0.955463i \(-0.595357\pi\)
−0.295111 + 0.955463i \(0.595357\pi\)
\(20\) 0 0
\(21\) 5.84265e10 0.337090
\(22\) 2.24525e11 0.872314
\(23\) −7.23343e10 −0.192601 −0.0963003 − 0.995352i \(-0.530701\pi\)
−0.0963003 + 0.995352i \(0.530701\pi\)
\(24\) 3.85298e10 0.0714499
\(25\) 0 0
\(26\) 1.95028e12 1.83157
\(27\) −2.82430e11 −0.192450
\(28\) −1.06271e12 −0.531582
\(29\) 1.82054e12 0.675800 0.337900 − 0.941182i \(-0.390284\pi\)
0.337900 + 0.941182i \(0.390284\pi\)
\(30\) 0 0
\(31\) 5.27128e12 1.11005 0.555024 − 0.831835i \(-0.312709\pi\)
0.555024 + 0.831835i \(0.312709\pi\)
\(32\) −8.52803e12 −1.37112
\(33\) −2.94382e12 −0.364370
\(34\) 2.98507e11 0.0286671
\(35\) 0 0
\(36\) 5.13704e12 0.303488
\(37\) 1.62659e13 0.761313 0.380656 − 0.924717i \(-0.375698\pi\)
0.380656 + 0.924717i \(0.375698\pi\)
\(38\) −2.18648e13 −0.815803
\(39\) −2.55707e13 −0.765057
\(40\) 0 0
\(41\) 9.78891e12 0.191457 0.0957285 − 0.995407i \(-0.469482\pi\)
0.0957285 + 0.995407i \(0.469482\pi\)
\(42\) 2.92371e13 0.465924
\(43\) −1.46155e14 −1.90691 −0.953455 − 0.301534i \(-0.902501\pi\)
−0.953455 + 0.301534i \(0.902501\pi\)
\(44\) 5.35444e13 0.574601
\(45\) 0 0
\(46\) −3.61967e13 −0.266212
\(47\) −2.43226e14 −1.48997 −0.744987 − 0.667079i \(-0.767544\pi\)
−0.744987 + 0.667079i \(0.767544\pi\)
\(48\) 1.21906e14 0.624415
\(49\) −1.53329e14 −0.659111
\(50\) 0 0
\(51\) −3.91381e12 −0.0119744
\(52\) 4.65100e14 1.20647
\(53\) −6.84721e14 −1.51067 −0.755334 − 0.655340i \(-0.772525\pi\)
−0.755334 + 0.655340i \(0.772525\pi\)
\(54\) −1.41330e14 −0.266004
\(55\) 0 0
\(56\) 5.22958e13 0.0722551
\(57\) 2.86676e14 0.340765
\(58\) 9.11015e14 0.934087
\(59\) 9.83635e14 0.872151 0.436076 − 0.899910i \(-0.356368\pi\)
0.436076 + 0.899910i \(0.356368\pi\)
\(60\) 0 0
\(61\) −1.84021e15 −1.22903 −0.614516 − 0.788904i \(-0.710649\pi\)
−0.614516 + 0.788904i \(0.710649\pi\)
\(62\) 2.63779e15 1.53430
\(63\) −3.83336e14 −0.194619
\(64\) −1.83214e15 −0.813631
\(65\) 0 0
\(66\) −1.47311e15 −0.503631
\(67\) 2.46536e15 0.741729 0.370864 − 0.928687i \(-0.379062\pi\)
0.370864 + 0.928687i \(0.379062\pi\)
\(68\) 7.11874e13 0.0188833
\(69\) 4.74585e14 0.111198
\(70\) 0 0
\(71\) 9.43154e14 0.173335 0.0866676 − 0.996237i \(-0.472378\pi\)
0.0866676 + 0.996237i \(0.472378\pi\)
\(72\) −2.52794e14 −0.0412516
\(73\) −1.15522e16 −1.67657 −0.838286 − 0.545231i \(-0.816442\pi\)
−0.838286 + 0.545231i \(0.816442\pi\)
\(74\) 8.13959e15 1.05228
\(75\) 0 0
\(76\) −5.21428e15 −0.537377
\(77\) −3.99559e15 −0.368477
\(78\) −1.27958e16 −1.05746
\(79\) 1.58855e16 1.17807 0.589035 − 0.808108i \(-0.299508\pi\)
0.589035 + 0.808108i \(0.299508\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 4.89845e15 0.264631
\(83\) −1.62194e16 −0.790442 −0.395221 − 0.918586i \(-0.629332\pi\)
−0.395221 + 0.918586i \(0.629332\pi\)
\(84\) 6.97241e15 0.306909
\(85\) 0 0
\(86\) −7.31369e16 −2.63572
\(87\) −1.19446e16 −0.390173
\(88\) −2.63492e15 −0.0781026
\(89\) −4.03683e16 −1.08699 −0.543495 − 0.839413i \(-0.682899\pi\)
−0.543495 + 0.839413i \(0.682899\pi\)
\(90\) 0 0
\(91\) −3.47067e16 −0.773679
\(92\) −8.63212e15 −0.175356
\(93\) −3.45849e16 −0.640886
\(94\) −1.21712e17 −2.05944
\(95\) 0 0
\(96\) 5.59524e16 0.791614
\(97\) 7.85724e16 1.01791 0.508956 − 0.860792i \(-0.330031\pi\)
0.508956 + 0.860792i \(0.330031\pi\)
\(98\) −7.67273e16 −0.911020
\(99\) 1.93144e16 0.210369
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.a.d.1.3 3
5.2 odd 4 75.18.b.e.49.6 6
5.3 odd 4 75.18.b.e.49.1 6
5.4 even 2 15.18.a.c.1.1 ✓ 3
15.14 odd 2 45.18.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.1 ✓ 3 5.4 even 2
45.18.a.d.1.3 3 15.14 odd 2
75.18.a.d.1.3 3 1.1 even 1 trivial
75.18.b.e.49.1 6 5.3 odd 4
75.18.b.e.49.6 6 5.2 odd 4