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 = [4,33] 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: \(0\)
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: \( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-662.567\) 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)
 
\(f(q)\) \(=\) \(q-654.567 q^{2} +6561.00 q^{3} +297386. q^{4} -4.29461e6 q^{6} -359681. q^{7} -1.08863e8 q^{8} +4.30467e7 q^{9} +1.12504e9 q^{11} +1.95115e9 q^{12} +3.56920e9 q^{13} +2.35435e8 q^{14} +3.22794e10 q^{16} -4.86046e10 q^{17} -2.81770e10 q^{18} +5.83337e10 q^{19} -2.35987e9 q^{21} -7.36413e11 q^{22} +1.76838e11 q^{23} -7.14252e11 q^{24} -2.33628e12 q^{26} +2.82430e11 q^{27} -1.06964e11 q^{28} +2.78200e12 q^{29} -6.55778e9 q^{31} -6.86008e12 q^{32} +7.38138e12 q^{33} +3.18150e13 q^{34} +1.28015e13 q^{36} +2.71386e13 q^{37} -3.81833e13 q^{38} +2.34175e13 q^{39} +1.87190e12 q^{41} +1.54469e12 q^{42} +1.01899e14 q^{43} +3.34570e14 q^{44} -1.15752e14 q^{46} -2.15275e14 q^{47} +2.11785e14 q^{48} -2.32501e14 q^{49} -3.18895e14 q^{51} +1.06143e15 q^{52} +8.28889e14 q^{53} -1.84869e14 q^{54} +3.91561e13 q^{56} +3.82728e14 q^{57} -1.82100e15 q^{58} +1.11484e15 q^{59} +2.77493e14 q^{61} +4.29251e12 q^{62} -1.54831e13 q^{63} +2.59456e14 q^{64} -4.83160e15 q^{66} +5.09386e15 q^{67} -1.44543e16 q^{68} +1.16023e15 q^{69} -7.44809e15 q^{71} -4.68621e15 q^{72} -3.88769e14 q^{73} -1.77640e16 q^{74} +1.73476e16 q^{76} -4.04655e14 q^{77} -1.53283e16 q^{78} -1.71928e16 q^{79} +1.85302e15 q^{81} -1.22528e15 q^{82} -7.42835e15 q^{83} -7.01791e14 q^{84} -6.66995e16 q^{86} +1.82527e16 q^{87} -1.22475e17 q^{88} +1.06547e16 q^{89} -1.28377e15 q^{91} +5.25891e16 q^{92} -4.30256e13 q^{93} +1.40912e17 q^{94} -4.50090e16 q^{96} +4.95369e15 q^{97} +1.52188e17 q^{98} +4.84292e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 33 q^{2} + 26244 q^{3} + 439357 q^{4} + 216513 q^{6} - 17583104 q^{7} - 63651621 q^{8} + 172186884 q^{9} - 575495184 q^{11} + 2882621277 q^{12} + 5049645832 q^{13} - 6699316032 q^{14} + 7512683905 q^{16}+ \cdots - 24\!\cdots\!64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −654.567 −1.80800 −0.904001 − 0.427530i \(-0.859384\pi\)
−0.904001 + 0.427530i \(0.859384\pi\)
\(3\) 6561.00 0.577350
\(4\) 297386. 2.26887
\(5\) 0 0
\(6\) −4.29461e6 −1.04385
\(7\) −359681. −0.0235822 −0.0117911 − 0.999930i \(-0.503753\pi\)
−0.0117911 + 0.999930i \(0.503753\pi\)
\(8\) −1.08863e8 −2.29412
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 1.12504e9 1.58245 0.791224 − 0.611526i \(-0.209444\pi\)
0.791224 + 0.611526i \(0.209444\pi\)
\(12\) 1.95115e9 1.30993
\(13\) 3.56920e9 1.21353 0.606767 − 0.794880i \(-0.292466\pi\)
0.606767 + 0.794880i \(0.292466\pi\)
\(14\) 2.35435e8 0.0426367
\(15\) 0 0
\(16\) 3.22794e10 1.87891
\(17\) −4.86046e10 −1.68990 −0.844951 − 0.534844i \(-0.820371\pi\)
−0.844951 + 0.534844i \(0.820371\pi\)
\(18\) −2.81770e10 −0.602667
\(19\) 5.83337e10 0.787978 0.393989 − 0.919115i \(-0.371095\pi\)
0.393989 + 0.919115i \(0.371095\pi\)
\(20\) 0 0
\(21\) −2.35987e9 −0.0136152
\(22\) −7.36413e11 −2.86107
\(23\) 1.76838e11 0.470857 0.235429 − 0.971892i \(-0.424351\pi\)
0.235429 + 0.971892i \(0.424351\pi\)
\(24\) −7.14252e11 −1.32451
\(25\) 0 0
\(26\) −2.33628e12 −2.19407
\(27\) 2.82430e11 0.192450
\(28\) −1.06964e11 −0.0535050
\(29\) 2.78200e12 1.03270 0.516349 − 0.856378i \(-0.327291\pi\)
0.516349 + 0.856378i \(0.327291\pi\)
\(30\) 0 0
\(31\) −6.55778e9 −0.00138097 −0.000690483 − 1.00000i \(-0.500220\pi\)
−0.000690483 1.00000i \(0.500220\pi\)
\(32\) −6.86008e12 −1.10295
\(33\) 7.38138e12 0.913627
\(34\) 3.18150e13 3.05535
\(35\) 0 0
\(36\) 1.28015e13 0.756291
\(37\) 2.71386e13 1.27020 0.635101 − 0.772429i \(-0.280959\pi\)
0.635101 + 0.772429i \(0.280959\pi\)
\(38\) −3.81833e13 −1.42467
\(39\) 2.34175e13 0.700634
\(40\) 0 0
\(41\) 1.87190e12 0.0366117 0.0183059 − 0.999832i \(-0.494173\pi\)
0.0183059 + 0.999832i \(0.494173\pi\)
\(42\) 1.54469e12 0.0246163
\(43\) 1.01899e14 1.32950 0.664748 − 0.747068i \(-0.268539\pi\)
0.664748 + 0.747068i \(0.268539\pi\)
\(44\) 3.34570e14 3.59037
\(45\) 0 0
\(46\) −1.15752e14 −0.851311
\(47\) −2.15275e14 −1.31875 −0.659374 − 0.751815i \(-0.729179\pi\)
−0.659374 + 0.751815i \(0.729179\pi\)
\(48\) 2.11785e14 1.08479
\(49\) −2.32501e14 −0.999444
\(50\) 0 0
\(51\) −3.18895e14 −0.975665
\(52\) 1.06143e15 2.75335
\(53\) 8.28889e14 1.82874 0.914369 − 0.404881i \(-0.132687\pi\)
0.914369 + 0.404881i \(0.132687\pi\)
\(54\) −1.84869e14 −0.347950
\(55\) 0 0
\(56\) 3.91561e13 0.0541005
\(57\) 3.82728e14 0.454939
\(58\) −1.82100e15 −1.86712
\(59\) 1.11484e15 0.988487 0.494244 − 0.869323i \(-0.335445\pi\)
0.494244 + 0.869323i \(0.335445\pi\)
\(60\) 0 0
\(61\) 2.77493e14 0.185331 0.0926655 − 0.995697i \(-0.470461\pi\)
0.0926655 + 0.995697i \(0.470461\pi\)
\(62\) 4.29251e12 0.00249679
\(63\) −1.54831e13 −0.00786073
\(64\) 2.59456e14 0.115221
\(65\) 0 0
\(66\) −4.83160e15 −1.65184
\(67\) 5.09386e15 1.53254 0.766269 − 0.642520i \(-0.222111\pi\)
0.766269 + 0.642520i \(0.222111\pi\)
\(68\) −1.44543e16 −3.83417
\(69\) 1.16023e15 0.271850
\(70\) 0 0
\(71\) −7.44809e15 −1.36883 −0.684414 − 0.729093i \(-0.739942\pi\)
−0.684414 + 0.729093i \(0.739942\pi\)
\(72\) −4.68621e15 −0.764708
\(73\) −3.88769e14 −0.0564218 −0.0282109 − 0.999602i \(-0.508981\pi\)
−0.0282109 + 0.999602i \(0.508981\pi\)
\(74\) −1.77640e16 −2.29653
\(75\) 0 0
\(76\) 1.73476e16 1.78782
\(77\) −4.04655e14 −0.0373176
\(78\) −1.53283e16 −1.26675
\(79\) −1.71928e16 −1.27502 −0.637509 − 0.770443i \(-0.720035\pi\)
−0.637509 + 0.770443i \(0.720035\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −1.22528e15 −0.0661941
\(83\) −7.42835e15 −0.362017 −0.181008 − 0.983482i \(-0.557936\pi\)
−0.181008 + 0.983482i \(0.557936\pi\)
\(84\) −7.01791e14 −0.0308911
\(85\) 0 0
\(86\) −6.66995e16 −2.40373
\(87\) 1.82527e16 0.596229
\(88\) −1.22475e17 −3.63033
\(89\) 1.06547e16 0.286897 0.143449 − 0.989658i \(-0.454181\pi\)
0.143449 + 0.989658i \(0.454181\pi\)
\(90\) 0 0
\(91\) −1.28377e15 −0.0286178
\(92\) 5.25891e16 1.06831
\(93\) −4.30256e13 −0.000797301 0
\(94\) 1.40912e17 2.38430
\(95\) 0 0
\(96\) −4.50090e16 −0.636786
\(97\) 4.95369e15 0.0641754 0.0320877 − 0.999485i \(-0.489784\pi\)
0.0320877 + 0.999485i \(0.489784\pi\)
\(98\) 1.52188e17 1.80700
\(99\) 4.84292e16 0.527483
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.f.1.1 4
5.2 odd 4 75.18.b.f.49.1 8
5.3 odd 4 75.18.b.f.49.8 8
5.4 even 2 15.18.a.d.1.4 ✓ 4
15.14 odd 2 45.18.a.f.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.4 ✓ 4 5.4 even 2
45.18.a.f.1.1 4 15.14 odd 2
75.18.a.f.1.1 4 1.1 even 1 trivial
75.18.b.f.49.1 8 5.2 odd 4
75.18.b.f.49.8 8 5.3 odd 4