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Properties

Label 75.18.b.f.49.8
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 49.8
Root \(662.567i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.f.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+654.567i q^{2} +6561.00i q^{3} -297386. q^{4} -4.29461e6 q^{6} +359681. i q^{7} -1.08863e8i q^{8} -4.30467e7 q^{9} +1.12504e9 q^{11} -1.95115e9i q^{12} +3.56920e9i q^{13} -2.35435e8 q^{14} +3.22794e10 q^{16} +4.86046e10i q^{17} -2.81770e10i q^{18} -5.83337e10 q^{19} -2.35987e9 q^{21} +7.36413e11i q^{22} +1.76838e11i q^{23} +7.14252e11 q^{24} -2.33628e12 q^{26} -2.82430e11i q^{27} -1.06964e11i q^{28} -2.78200e12 q^{29} -6.55778e9 q^{31} +6.86008e12i q^{32} +7.38138e12i q^{33} -3.18150e13 q^{34} +1.28015e13 q^{36} -2.71386e13i q^{37} -3.81833e13i q^{38} -2.34175e13 q^{39} +1.87190e12 q^{41} -1.54469e12i q^{42} +1.01899e14i q^{43} -3.34570e14 q^{44} -1.15752e14 q^{46} +2.15275e14i q^{47} +2.11785e14i q^{48} +2.32501e14 q^{49} -3.18895e14 q^{51} -1.06143e15i q^{52} +8.28889e14i q^{53} +1.84869e14 q^{54} +3.91561e13 q^{56} -3.82728e14i q^{57} -1.82100e15i q^{58} -1.11484e15 q^{59} +2.77493e14 q^{61} -4.29251e12i q^{62} -1.54831e13i q^{63} -2.59456e14 q^{64} -4.83160e15 q^{66} -5.09386e15i q^{67} -1.44543e16i q^{68} -1.16023e15 q^{69} -7.44809e15 q^{71} +4.68621e15i q^{72} -3.88769e14i q^{73} +1.77640e16 q^{74} +1.73476e16 q^{76} +4.04655e14i q^{77} -1.53283e16i q^{78} +1.71928e16 q^{79} +1.85302e15 q^{81} +1.22528e15i q^{82} -7.42835e15i q^{83} +7.01791e14 q^{84} -6.66995e16 q^{86} -1.82527e16i q^{87} -1.22475e17i q^{88} -1.06547e16 q^{89} -1.28377e15 q^{91} -5.25891e16i q^{92} -4.30256e13i q^{93} -1.40912e17 q^{94} -4.50090e16 q^{96} -4.95369e15i q^{97} +1.52188e17i q^{98} -4.84292e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 878714 q^{4} + 433026 q^{6} - 344373768 q^{9} - 1150990368 q^{11} + 13398632064 q^{14} + 15025367810 q^{16} - 397827528736 q^{19} - 230725490688 q^{21} + 835236570762 q^{24} - 1055057698044 q^{26}+ \cdots + 49\!\cdots\!28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 654.567i 1.80800i 0.427530 + 0.904001i \(0.359384\pi\)
−0.427530 + 0.904001i \(0.640616\pi\)
\(3\) 6561.00i 0.577350i
\(4\) −297386. −2.26887
\(5\) 0 0
\(6\) −4.29461e6 −1.04385
\(7\) 359681.i 0.0235822i 0.999930 + 0.0117911i \(0.00375331\pi\)
−0.999930 + 0.0117911i \(0.996247\pi\)
\(8\) − 1.08863e8i − 2.29412i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 1.12504e9 1.58245 0.791224 − 0.611526i \(-0.209444\pi\)
0.791224 + 0.611526i \(0.209444\pi\)
\(12\) − 1.95115e9i − 1.30993i
\(13\) 3.56920e9i 1.21353i 0.794880 + 0.606767i \(0.207534\pi\)
−0.794880 + 0.606767i \(0.792466\pi\)
\(14\) −2.35435e8 −0.0426367
\(15\) 0 0
\(16\) 3.22794e10 1.87891
\(17\) 4.86046e10i 1.68990i 0.534844 + 0.844951i \(0.320371\pi\)
−0.534844 + 0.844951i \(0.679629\pi\)
\(18\) − 2.81770e10i − 0.602667i
\(19\) −5.83337e10 −0.787978 −0.393989 − 0.919115i \(-0.628905\pi\)
−0.393989 + 0.919115i \(0.628905\pi\)
\(20\) 0 0
\(21\) −2.35987e9 −0.0136152
\(22\) 7.36413e11i 2.86107i
\(23\) 1.76838e11i 0.470857i 0.971892 + 0.235429i \(0.0756494\pi\)
−0.971892 + 0.235429i \(0.924351\pi\)
\(24\) 7.14252e11 1.32451
\(25\) 0 0
\(26\) −2.33628e12 −2.19407
\(27\) − 2.82430e11i − 0.192450i
\(28\) − 1.06964e11i − 0.0535050i
\(29\) −2.78200e12 −1.03270 −0.516349 − 0.856378i \(-0.672709\pi\)
−0.516349 + 0.856378i \(0.672709\pi\)
\(30\) 0 0
\(31\) −6.55778e9 −0.00138097 −0.000690483 − 1.00000i \(-0.500220\pi\)
−0.000690483 1.00000i \(0.500220\pi\)
\(32\) 6.86008e12i 1.10295i
\(33\) 7.38138e12i 0.913627i
\(34\) −3.18150e13 −3.05535
\(35\) 0 0
\(36\) 1.28015e13 0.756291
\(37\) − 2.71386e13i − 1.27020i −0.772429 − 0.635101i \(-0.780959\pi\)
0.772429 − 0.635101i \(-0.219041\pi\)
\(38\) − 3.81833e13i − 1.42467i
\(39\) −2.34175e13 −0.700634
\(40\) 0 0
\(41\) 1.87190e12 0.0366117 0.0183059 − 0.999832i \(-0.494173\pi\)
0.0183059 + 0.999832i \(0.494173\pi\)
\(42\) − 1.54469e12i − 0.0246163i
\(43\) 1.01899e14i 1.32950i 0.747068 + 0.664748i \(0.231461\pi\)
−0.747068 + 0.664748i \(0.768539\pi\)
\(44\) −3.34570e14 −3.59037
\(45\) 0 0
\(46\) −1.15752e14 −0.851311
\(47\) 2.15275e14i 1.31875i 0.751815 + 0.659374i \(0.229179\pi\)
−0.751815 + 0.659374i \(0.770821\pi\)
\(48\) 2.11785e14i 1.08479i
\(49\) 2.32501e14 0.999444
\(50\) 0 0
\(51\) −3.18895e14 −0.975665
\(52\) − 1.06143e15i − 2.75335i
\(53\) 8.28889e14i 1.82874i 0.404881 + 0.914369i \(0.367313\pi\)
−0.404881 + 0.914369i \(0.632687\pi\)
\(54\) 1.84869e14 0.347950
\(55\) 0 0
\(56\) 3.91561e13 0.0541005
\(57\) − 3.82728e14i − 0.454939i
\(58\) − 1.82100e15i − 1.86712i
\(59\) −1.11484e15 −0.988487 −0.494244 − 0.869323i \(-0.664555\pi\)
−0.494244 + 0.869323i \(0.664555\pi\)
\(60\) 0 0
\(61\) 2.77493e14 0.185331 0.0926655 − 0.995697i \(-0.470461\pi\)
0.0926655 + 0.995697i \(0.470461\pi\)
\(62\) − 4.29251e12i − 0.00249679i
\(63\) − 1.54831e13i − 0.00786073i
\(64\) −2.59456e14 −0.115221
\(65\) 0 0
\(66\) −4.83160e15 −1.65184
\(67\) − 5.09386e15i − 1.53254i −0.642520 − 0.766269i \(-0.722111\pi\)
0.642520 − 0.766269i \(-0.277889\pi\)
\(68\) − 1.44543e16i − 3.83417i
\(69\) −1.16023e15 −0.271850
\(70\) 0 0
\(71\) −7.44809e15 −1.36883 −0.684414 − 0.729093i \(-0.739942\pi\)
−0.684414 + 0.729093i \(0.739942\pi\)
\(72\) 4.68621e15i 0.764708i
\(73\) − 3.88769e14i − 0.0564218i −0.999602 − 0.0282109i \(-0.991019\pi\)
0.999602 − 0.0282109i \(-0.00898101\pi\)
\(74\) 1.77640e16 2.29653
\(75\) 0 0
\(76\) 1.73476e16 1.78782
\(77\) 4.04655e14i 0.0373176i
\(78\) − 1.53283e16i − 1.26675i
\(79\) 1.71928e16 1.27502 0.637509 − 0.770443i \(-0.279965\pi\)
0.637509 + 0.770443i \(0.279965\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 1.22528e15i 0.0661941i
\(83\) − 7.42835e15i − 0.362017i −0.983482 − 0.181008i \(-0.942064\pi\)
0.983482 − 0.181008i \(-0.0579361\pi\)
\(84\) 7.01791e14 0.0308911
\(85\) 0 0
\(86\) −6.66995e16 −2.40373
\(87\) − 1.82527e16i − 0.596229i
\(88\) − 1.22475e17i − 3.63033i
\(89\) −1.06547e16 −0.286897 −0.143449 − 0.989658i \(-0.545819\pi\)
−0.143449 + 0.989658i \(0.545819\pi\)
\(90\) 0 0
\(91\) −1.28377e15 −0.0286178
\(92\) − 5.25891e16i − 1.06831i
\(93\) − 4.30256e13i 0 0.000797301i
\(94\) −1.40912e17 −2.38430
\(95\) 0 0
\(96\) −4.50090e16 −0.636786
\(97\) − 4.95369e15i − 0.0641754i −0.999485 − 0.0320877i \(-0.989784\pi\)
0.999485 − 0.0320877i \(-0.0102156\pi\)
\(98\) 1.52188e17i 1.80700i
\(99\) −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.b.f.49.8 8
5.2 odd 4 75.18.a.f.1.1 4
5.3 odd 4 15.18.a.d.1.4 ✓ 4
5.4 even 2 inner 75.18.b.f.49.1 8
15.8 even 4 45.18.a.f.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.4 ✓ 4 5.3 odd 4
45.18.a.f.1.1 4 15.8 even 4
75.18.a.f.1.1 4 5.2 odd 4
75.18.b.f.49.1 8 5.4 even 2 inner
75.18.b.f.49.8 8 1.1 even 1 trivial