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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 = [2,-594] 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: \(2\)
Coefficient field: \(\Q(\sqrt{14569}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3642 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-59.8511\) 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)
 
\(f(q)\) \(=\) \(q+65.1063 q^{2} -6561.00 q^{3} -126833. q^{4} -427163. q^{6} -2.28730e7 q^{7} -1.67913e7 q^{8} +4.30467e7 q^{9} -5.40173e8 q^{11} +8.32152e8 q^{12} +3.45398e7 q^{13} -1.48918e9 q^{14} +1.55311e10 q^{16} +9.43866e9 q^{17} +2.80261e9 q^{18} -2.25061e9 q^{19} +1.50070e11 q^{21} -3.51687e10 q^{22} +3.45854e11 q^{23} +1.10167e11 q^{24} +2.24876e9 q^{26} -2.82430e11 q^{27} +2.90106e12 q^{28} +5.11838e11 q^{29} +1.14436e11 q^{31} +3.21203e12 q^{32} +3.54407e12 q^{33} +6.14516e11 q^{34} -5.45975e12 q^{36} +1.56972e13 q^{37} -1.46529e11 q^{38} -2.26616e11 q^{39} -8.00761e13 q^{41} +9.77050e12 q^{42} +3.66737e13 q^{43} +6.85118e13 q^{44} +2.25173e13 q^{46} +1.17577e14 q^{47} -1.01899e14 q^{48} +2.90545e14 q^{49} -6.19270e13 q^{51} -4.38080e12 q^{52} +3.90043e14 q^{53} -1.83880e13 q^{54} +3.84067e14 q^{56} +1.47663e13 q^{57} +3.33239e13 q^{58} +1.82562e15 q^{59} +1.41053e15 q^{61} +7.45051e12 q^{62} -9.84609e14 q^{63} -1.82656e15 q^{64} +2.30742e14 q^{66} +1.47114e15 q^{67} -1.19713e15 q^{68} -2.26915e15 q^{69} -7.31441e15 q^{71} -7.22809e14 q^{72} -1.34580e16 q^{73} +1.02199e15 q^{74} +2.85452e14 q^{76} +1.23554e16 q^{77} -1.47541e13 q^{78} -8.37779e15 q^{79} +1.85302e15 q^{81} -5.21346e15 q^{82} +2.55978e16 q^{83} -1.90338e16 q^{84} +2.38769e15 q^{86} -3.35817e15 q^{87} +9.07018e15 q^{88} +4.47540e16 q^{89} -7.90031e14 q^{91} -4.38657e16 q^{92} -7.50815e14 q^{93} +7.65502e15 q^{94} -2.10742e16 q^{96} -7.41658e16 q^{97} +1.89163e16 q^{98} -2.32527e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 594 q^{2} - 13122 q^{3} + 176516 q^{4} + 3897234 q^{6} - 24471568 q^{7} - 130340232 q^{8} + 86093442 q^{9} - 987553512 q^{11} - 1158121476 q^{12} + 2519398244 q^{13} - 435565296 q^{14} + 50611323920 q^{16}+ \cdots - 42\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 65.1063 0.179833 0.0899163 − 0.995949i \(-0.471340\pi\)
0.0899163 + 0.995949i \(0.471340\pi\)
\(3\) −6561.00 −0.577350
\(4\) −126833. −0.967660
\(5\) 0 0
\(6\) −427163. −0.103826
\(7\) −2.28730e7 −1.49965 −0.749825 − 0.661636i \(-0.769863\pi\)
−0.749825 + 0.661636i \(0.769863\pi\)
\(8\) −1.67913e7 −0.353849
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −5.40173e8 −0.759792 −0.379896 − 0.925029i \(-0.624040\pi\)
−0.379896 + 0.925029i \(0.624040\pi\)
\(12\) 8.32152e8 0.558679
\(13\) 3.45398e7 0.0117436 0.00587181 − 0.999983i \(-0.498131\pi\)
0.00587181 + 0.999983i \(0.498131\pi\)
\(14\) −1.48918e9 −0.269686
\(15\) 0 0
\(16\) 1.55311e10 0.904027
\(17\) 9.43866e9 0.328167 0.164083 − 0.986446i \(-0.447533\pi\)
0.164083 + 0.986446i \(0.447533\pi\)
\(18\) 2.80261e9 0.0599442
\(19\) −2.25061e9 −0.0304015 −0.0152007 − 0.999884i \(-0.504839\pi\)
−0.0152007 + 0.999884i \(0.504839\pi\)
\(20\) 0 0
\(21\) 1.50070e11 0.865824
\(22\) −3.51687e10 −0.136635
\(23\) 3.45854e11 0.920886 0.460443 − 0.887689i \(-0.347690\pi\)
0.460443 + 0.887689i \(0.347690\pi\)
\(24\) 1.10167e11 0.204295
\(25\) 0 0
\(26\) 2.24876e9 0.00211188
\(27\) −2.82430e11 −0.192450
\(28\) 2.90106e12 1.45115
\(29\) 5.11838e11 0.189998 0.0949992 − 0.995477i \(-0.469715\pi\)
0.0949992 + 0.995477i \(0.469715\pi\)
\(30\) 0 0
\(31\) 1.14436e11 0.0240984 0.0120492 − 0.999927i \(-0.496165\pi\)
0.0120492 + 0.999927i \(0.496165\pi\)
\(32\) 3.21203e12 0.516423
\(33\) 3.54407e12 0.438666
\(34\) 6.14516e11 0.0590150
\(35\) 0 0
\(36\) −5.45975e12 −0.322553
\(37\) 1.56972e13 0.734697 0.367349 − 0.930083i \(-0.380266\pi\)
0.367349 + 0.930083i \(0.380266\pi\)
\(38\) −1.46529e11 −0.00546717
\(39\) −2.26616e11 −0.00678018
\(40\) 0 0
\(41\) −8.00761e13 −1.56617 −0.783087 − 0.621912i \(-0.786356\pi\)
−0.783087 + 0.621912i \(0.786356\pi\)
\(42\) 9.77050e12 0.155703
\(43\) 3.66737e13 0.478489 0.239245 − 0.970959i \(-0.423100\pi\)
0.239245 + 0.970959i \(0.423100\pi\)
\(44\) 6.85118e13 0.735221
\(45\) 0 0
\(46\) 2.25173e13 0.165605
\(47\) 1.17577e14 0.720263 0.360132 − 0.932901i \(-0.382732\pi\)
0.360132 + 0.932901i \(0.382732\pi\)
\(48\) −1.01899e14 −0.521940
\(49\) 2.90545e14 1.24895
\(50\) 0 0
\(51\) −6.19270e13 −0.189467
\(52\) −4.38080e12 −0.0113638
\(53\) 3.90043e14 0.860534 0.430267 − 0.902702i \(-0.358419\pi\)
0.430267 + 0.902702i \(0.358419\pi\)
\(54\) −1.83880e13 −0.0346088
\(55\) 0 0
\(56\) 3.84067e14 0.530651
\(57\) 1.47663e13 0.0175523
\(58\) 3.33239e13 0.0341679
\(59\) 1.82562e15 1.61870 0.809352 − 0.587323i \(-0.199818\pi\)
0.809352 + 0.587323i \(0.199818\pi\)
\(60\) 0 0
\(61\) 1.41053e15 0.942059 0.471029 − 0.882118i \(-0.343883\pi\)
0.471029 + 0.882118i \(0.343883\pi\)
\(62\) 7.45051e12 0.00433368
\(63\) −9.84609e14 −0.499884
\(64\) −1.82656e15 −0.811157
\(65\) 0 0
\(66\) 2.30742e14 0.0788865
\(67\) 1.47114e15 0.442607 0.221304 − 0.975205i \(-0.428969\pi\)
0.221304 + 0.975205i \(0.428969\pi\)
\(68\) −1.19713e15 −0.317554
\(69\) −2.26915e15 −0.531674
\(70\) 0 0
\(71\) −7.31441e15 −1.34426 −0.672130 − 0.740433i \(-0.734620\pi\)
−0.672130 + 0.740433i \(0.734620\pi\)
\(72\) −7.22809e14 −0.117950
\(73\) −1.34580e16 −1.95315 −0.976577 − 0.215169i \(-0.930970\pi\)
−0.976577 + 0.215169i \(0.930970\pi\)
\(74\) 1.02199e15 0.132122
\(75\) 0 0
\(76\) 2.85452e14 0.0294183
\(77\) 1.23554e16 1.13942
\(78\) −1.47541e13 −0.00121930
\(79\) −8.37779e15 −0.621297 −0.310648 − 0.950525i \(-0.600546\pi\)
−0.310648 + 0.950525i \(0.600546\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −5.21346e15 −0.281649
\(83\) 2.55978e16 1.24750 0.623748 − 0.781625i \(-0.285609\pi\)
0.623748 + 0.781625i \(0.285609\pi\)
\(84\) −1.90338e16 −0.837823
\(85\) 0 0
\(86\) 2.38769e15 0.0860480
\(87\) −3.35817e15 −0.109696
\(88\) 9.07018e15 0.268852
\(89\) 4.47540e16 1.20508 0.602541 − 0.798088i \(-0.294155\pi\)
0.602541 + 0.798088i \(0.294155\pi\)
\(90\) 0 0
\(91\) −7.90031e14 −0.0176113
\(92\) −4.38657e16 −0.891105
\(93\) −7.50815e14 −0.0139132
\(94\) 7.65502e15 0.129527
\(95\) 0 0
\(96\) −2.10742e16 −0.298157
\(97\) −7.41658e16 −0.960824 −0.480412 − 0.877043i \(-0.659513\pi\)
−0.480412 + 0.877043i \(0.659513\pi\)
\(98\) 1.89163e16 0.224602
\(99\) −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.a.b.1.2 2
5.2 odd 4 75.18.b.c.49.3 4
5.3 odd 4 75.18.b.c.49.2 4
5.4 even 2 3.18.a.b.1.1 ✓ 2
15.14 odd 2 9.18.a.c.1.2 2
20.19 odd 2 48.18.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.b.1.1 ✓ 2 5.4 even 2
9.18.a.c.1.2 2 15.14 odd 2
48.18.a.h.1.2 2 20.19 odd 2
75.18.a.b.1.2 2 1.1 even 1 trivial
75.18.b.c.49.2 4 5.3 odd 4
75.18.b.c.49.3 4 5.2 odd 4