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Properties

Label 75.18.b.c.49.3
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,-353032] 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: \(4\)
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: \( x^{4} + 7285x^{2} + 13264164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(59.8511i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.c.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q+65.1063i q^{2} +6561.00i q^{3} +126833. q^{4} -427163. q^{6} -2.28730e7i q^{7} +1.67913e7i q^{8} -4.30467e7 q^{9} -5.40173e8 q^{11} +8.32152e8i q^{12} -3.45398e7i q^{13} +1.48918e9 q^{14} +1.55311e10 q^{16} +9.43866e9i q^{17} -2.80261e9i q^{18} +2.25061e9 q^{19} +1.50070e11 q^{21} -3.51687e10i q^{22} -3.45854e11i q^{23} -1.10167e11 q^{24} +2.24876e9 q^{26} -2.82430e11i q^{27} -2.90106e12i q^{28} -5.11838e11 q^{29} +1.14436e11 q^{31} +3.21203e12i q^{32} -3.54407e12i q^{33} -6.14516e11 q^{34} -5.45975e12 q^{36} +1.56972e13i q^{37} +1.46529e11i q^{38} +2.26616e11 q^{39} -8.00761e13 q^{41} +9.77050e12i q^{42} -3.66737e13i q^{43} -6.85118e13 q^{44} +2.25173e13 q^{46} +1.17577e14i q^{47} +1.01899e14i q^{48} -2.90545e14 q^{49} -6.19270e13 q^{51} -4.38080e12i q^{52} -3.90043e14i q^{53} +1.83880e13 q^{54} +3.84067e14 q^{56} +1.47663e13i q^{57} -3.33239e13i q^{58} -1.82562e15 q^{59} +1.41053e15 q^{61} +7.45051e12i q^{62} +9.84609e14i q^{63} +1.82656e15 q^{64} +2.30742e14 q^{66} +1.47114e15i q^{67} +1.19713e15i q^{68} +2.26915e15 q^{69} -7.31441e15 q^{71} -7.22809e14i q^{72} +1.34580e16i q^{73} -1.02199e15 q^{74} +2.85452e14 q^{76} +1.23554e16i q^{77} +1.47541e13i q^{78} +8.37779e15 q^{79} +1.85302e15 q^{81} -5.21346e15i q^{82} -2.55978e16i q^{83} +1.90338e16 q^{84} +2.38769e15 q^{86} -3.35817e15i q^{87} -9.07018e15i q^{88} -4.47540e16 q^{89} -7.90031e14 q^{91} -4.38657e16i q^{92} +7.50815e14i q^{93} -7.65502e15 q^{94} -2.10742e16 q^{96} -7.41658e16i q^{97} -1.89163e16i q^{98} +2.32527e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 353032 q^{4} + 7794468 q^{6} - 172186884 q^{9} - 1975107024 q^{11} + 871130592 q^{14} + 101222647840 q^{16} - 160107084368 q^{19} + 321115915296 q^{21} - 1710324524304 q^{24} - 3271074310152 q^{26}+ \cdots + 85\!\cdots\!04 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 65.1063i 0.179833i 0.995949 + 0.0899163i \(0.0286599\pi\)
−0.995949 + 0.0899163i \(0.971340\pi\)
\(3\) 6561.00i 0.577350i
\(4\) 126833. 0.967660
\(5\) 0 0
\(6\) −427163. −0.103826
\(7\) − 2.28730e7i − 1.49965i −0.661636 − 0.749825i \(-0.730137\pi\)
0.661636 − 0.749825i \(-0.269863\pi\)
\(8\) 1.67913e7i 0.353849i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −5.40173e8 −0.759792 −0.379896 − 0.925029i \(-0.624040\pi\)
−0.379896 + 0.925029i \(0.624040\pi\)
\(12\) 8.32152e8i 0.558679i
\(13\) − 3.45398e7i − 0.0117436i −0.999983 − 0.00587181i \(-0.998131\pi\)
0.999983 − 0.00587181i \(-0.00186907\pi\)
\(14\) 1.48918e9 0.269686
\(15\) 0 0
\(16\) 1.55311e10 0.904027
\(17\) 9.43866e9i 0.328167i 0.986446 + 0.164083i \(0.0524666\pi\)
−0.986446 + 0.164083i \(0.947533\pi\)
\(18\) − 2.80261e9i − 0.0599442i
\(19\) 2.25061e9 0.0304015 0.0152007 − 0.999884i \(-0.495161\pi\)
0.0152007 + 0.999884i \(0.495161\pi\)
\(20\) 0 0
\(21\) 1.50070e11 0.865824
\(22\) − 3.51687e10i − 0.136635i
\(23\) − 3.45854e11i − 0.920886i −0.887689 − 0.460443i \(-0.847690\pi\)
0.887689 − 0.460443i \(-0.152310\pi\)
\(24\) −1.10167e11 −0.204295
\(25\) 0 0
\(26\) 2.24876e9 0.00211188
\(27\) − 2.82430e11i − 0.192450i
\(28\) − 2.90106e12i − 1.45115i
\(29\) −5.11838e11 −0.189998 −0.0949992 − 0.995477i \(-0.530285\pi\)
−0.0949992 + 0.995477i \(0.530285\pi\)
\(30\) 0 0
\(31\) 1.14436e11 0.0240984 0.0120492 − 0.999927i \(-0.496165\pi\)
0.0120492 + 0.999927i \(0.496165\pi\)
\(32\) 3.21203e12i 0.516423i
\(33\) − 3.54407e12i − 0.438666i
\(34\) −6.14516e11 −0.0590150
\(35\) 0 0
\(36\) −5.45975e12 −0.322553
\(37\) 1.56972e13i 0.734697i 0.930083 + 0.367349i \(0.119734\pi\)
−0.930083 + 0.367349i \(0.880266\pi\)
\(38\) 1.46529e11i 0.00546717i
\(39\) 2.26616e11 0.00678018
\(40\) 0 0
\(41\) −8.00761e13 −1.56617 −0.783087 − 0.621912i \(-0.786356\pi\)
−0.783087 + 0.621912i \(0.786356\pi\)
\(42\) 9.77050e12i 0.155703i
\(43\) − 3.66737e13i − 0.478489i −0.970959 − 0.239245i \(-0.923100\pi\)
0.970959 − 0.239245i \(-0.0768998\pi\)
\(44\) −6.85118e13 −0.735221
\(45\) 0 0
\(46\) 2.25173e13 0.165605
\(47\) 1.17577e14i 0.720263i 0.932901 + 0.360132i \(0.117268\pi\)
−0.932901 + 0.360132i \(0.882732\pi\)
\(48\) 1.01899e14i 0.521940i
\(49\) −2.90545e14 −1.24895
\(50\) 0 0
\(51\) −6.19270e13 −0.189467
\(52\) − 4.38080e12i − 0.0113638i
\(53\) − 3.90043e14i − 0.860534i −0.902702 − 0.430267i \(-0.858419\pi\)
0.902702 − 0.430267i \(-0.141581\pi\)
\(54\) 1.83880e13 0.0346088
\(55\) 0 0
\(56\) 3.84067e14 0.530651
\(57\) 1.47663e13i 0.0175523i
\(58\) − 3.33239e13i − 0.0341679i
\(59\) −1.82562e15 −1.61870 −0.809352 − 0.587323i \(-0.800182\pi\)
−0.809352 + 0.587323i \(0.800182\pi\)
\(60\) 0 0
\(61\) 1.41053e15 0.942059 0.471029 − 0.882118i \(-0.343883\pi\)
0.471029 + 0.882118i \(0.343883\pi\)
\(62\) 7.45051e12i 0.00433368i
\(63\) 9.84609e14i 0.499884i
\(64\) 1.82656e15 0.811157
\(65\) 0 0
\(66\) 2.30742e14 0.0788865
\(67\) 1.47114e15i 0.442607i 0.975205 + 0.221304i \(0.0710311\pi\)
−0.975205 + 0.221304i \(0.928969\pi\)
\(68\) 1.19713e15i 0.317554i
\(69\) 2.26915e15 0.531674
\(70\) 0 0
\(71\) −7.31441e15 −1.34426 −0.672130 − 0.740433i \(-0.734620\pi\)
−0.672130 + 0.740433i \(0.734620\pi\)
\(72\) − 7.22809e14i − 0.117950i
\(73\) 1.34580e16i 1.95315i 0.215169 + 0.976577i \(0.430970\pi\)
−0.215169 + 0.976577i \(0.569030\pi\)
\(74\) −1.02199e15 −0.132122
\(75\) 0 0
\(76\) 2.85452e14 0.0294183
\(77\) 1.23554e16i 1.13942i
\(78\) 1.47541e13i 0.00121930i
\(79\) 8.37779e15 0.621297 0.310648 − 0.950525i \(-0.399454\pi\)
0.310648 + 0.950525i \(0.399454\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 5.21346e15i − 0.281649i
\(83\) − 2.55978e16i − 1.24750i −0.781625 − 0.623748i \(-0.785609\pi\)
0.781625 − 0.623748i \(-0.214391\pi\)
\(84\) 1.90338e16 0.837823
\(85\) 0 0
\(86\) 2.38769e15 0.0860480
\(87\) − 3.35817e15i − 0.109696i
\(88\) − 9.07018e15i − 0.268852i
\(89\) −4.47540e16 −1.20508 −0.602541 − 0.798088i \(-0.705845\pi\)
−0.602541 + 0.798088i \(0.705845\pi\)
\(90\) 0 0
\(91\) −7.90031e14 −0.0176113
\(92\) − 4.38657e16i − 0.891105i
\(93\) 7.50815e14i 0.0139132i
\(94\) −7.65502e15 −0.129527
\(95\) 0 0
\(96\) −2.10742e16 −0.298157
\(97\) − 7.41658e16i − 0.960824i −0.877043 − 0.480412i \(-0.840487\pi\)
0.877043 − 0.480412i \(-0.159513\pi\)
\(98\) − 1.89163e16i − 0.224602i
\(99\) 2.32527e16 0.253264
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.c.49.3 4
5.2 odd 4 3.18.a.b.1.1 ✓ 2
5.3 odd 4 75.18.a.b.1.2 2
5.4 even 2 inner 75.18.b.c.49.2 4
15.2 even 4 9.18.a.c.1.2 2
20.7 even 4 48.18.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.b.1.1 ✓ 2 5.2 odd 4
9.18.a.c.1.2 2 15.2 even 4
48.18.a.h.1.2 2 20.7 even 4
75.18.a.b.1.2 2 5.3 odd 4
75.18.b.c.49.2 4 5.4 even 2 inner
75.18.b.c.49.3 4 1.1 even 1 trivial