Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [825,4,Mod(1,825)] 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("825.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,-6,1] 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: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.37228 q^{2} -3.00000 q^{3} -2.37228 q^{4} -7.11684 q^{6} -6.74456 q^{7} -24.6060 q^{8} +9.00000 q^{9} +11.0000 q^{11} +7.11684 q^{12} +60.9783 q^{13} -16.0000 q^{14} -39.3940 q^{16} +99.1684 q^{17} +21.3505 q^{18} +24.7011 q^{19} +20.2337 q^{21} +26.0951 q^{22} -112.000 q^{23} +73.8179 q^{24} +144.658 q^{26} -27.0000 q^{27} +16.0000 q^{28} -21.1249 q^{29} -318.717 q^{31} +103.394 q^{32} -33.0000 q^{33} +235.255 q^{34} -21.3505 q^{36} +150.380 q^{37} +58.5979 q^{38} -182.935 q^{39} -252.745 q^{41} +48.0000 q^{42} -214.016 q^{43} -26.0951 q^{44} -265.696 q^{46} -105.870 q^{47} +118.182 q^{48} -297.511 q^{49} -297.505 q^{51} -144.658 q^{52} -325.652 q^{53} -64.0516 q^{54} +165.957 q^{56} -74.1032 q^{57} -50.1143 q^{58} +196.000 q^{59} -402.641 q^{61} -756.087 q^{62} -60.7011 q^{63} +560.432 q^{64} -78.2853 q^{66} -27.4132 q^{67} -235.255 q^{68} +336.000 q^{69} -300.500 q^{71} -221.454 q^{72} -427.815 q^{73} +356.745 q^{74} -58.5979 q^{76} -74.1902 q^{77} -433.973 q^{78} +97.5488 q^{79} +81.0000 q^{81} -599.581 q^{82} -1104.62 q^{83} -48.0000 q^{84} -507.707 q^{86} +63.3748 q^{87} -270.666 q^{88} +463.022 q^{89} -411.272 q^{91} +265.696 q^{92} +956.152 q^{93} -251.152 q^{94} -310.182 q^{96} +1798.36 q^{97} -705.779 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 6 q^{3} + q^{4} + 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9} + 22 q^{11} - 3 q^{12} + 76 q^{13} - 32 q^{14} - 119 q^{16} + 26 q^{17} - 9 q^{18} - 54 q^{19} + 6 q^{21} - 11 q^{22} - 224 q^{23}+ \cdots + 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37228 0.838728 0.419364 − 0.907818i \(-0.362253\pi\)
0.419364 + 0.907818i \(0.362253\pi\)
\(3\) −3.00000 −0.577350
\(4\) −2.37228 −0.296535
\(5\) 0 0
\(6\) −7.11684 −0.484240
\(7\) −6.74456 −0.364172 −0.182086 − 0.983283i \(-0.558285\pi\)
−0.182086 + 0.983283i \(0.558285\pi\)
\(8\) −24.6060 −1.08744
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 7.11684 0.171205
\(13\) 60.9783 1.30095 0.650474 − 0.759529i \(-0.274570\pi\)
0.650474 + 0.759529i \(0.274570\pi\)
\(14\) −16.0000 −0.305441
\(15\) 0 0
\(16\) −39.3940 −0.615532
\(17\) 99.1684 1.41482 0.707408 − 0.706805i \(-0.249864\pi\)
0.707408 + 0.706805i \(0.249864\pi\)
\(18\) 21.3505 0.279576
\(19\) 24.7011 0.298253 0.149127 − 0.988818i \(-0.452354\pi\)
0.149127 + 0.988818i \(0.452354\pi\)
\(20\) 0 0
\(21\) 20.2337 0.210255
\(22\) 26.0951 0.252886
\(23\) −112.000 −1.01537 −0.507687 − 0.861541i \(-0.669499\pi\)
−0.507687 + 0.861541i \(0.669499\pi\)
\(24\) 73.8179 0.627834
\(25\) 0 0
\(26\) 144.658 1.09114
\(27\) −27.0000 −0.192450
\(28\) 16.0000 0.107990
\(29\) −21.1249 −0.135269 −0.0676345 − 0.997710i \(-0.521545\pi\)
−0.0676345 + 0.997710i \(0.521545\pi\)
\(30\) 0 0
\(31\) −318.717 −1.84656 −0.923279 − 0.384130i \(-0.874502\pi\)
−0.923279 + 0.384130i \(0.874502\pi\)
\(32\) 103.394 0.571177
\(33\) −33.0000 −0.174078
\(34\) 235.255 1.18665
\(35\) 0 0
\(36\) −21.3505 −0.0988451
\(37\) 150.380 0.668172 0.334086 − 0.942543i \(-0.391572\pi\)
0.334086 + 0.942543i \(0.391572\pi\)
\(38\) 58.5979 0.250153
\(39\) −182.935 −0.751103
\(40\) 0 0
\(41\) −252.745 −0.962733 −0.481367 − 0.876519i \(-0.659859\pi\)
−0.481367 + 0.876519i \(0.659859\pi\)
\(42\) 48.0000 0.176347
\(43\) −214.016 −0.759004 −0.379502 − 0.925191i \(-0.623905\pi\)
−0.379502 + 0.925191i \(0.623905\pi\)
\(44\) −26.0951 −0.0894087
\(45\) 0 0
\(46\) −265.696 −0.851623
\(47\) −105.870 −0.328567 −0.164284 − 0.986413i \(-0.552531\pi\)
−0.164284 + 0.986413i \(0.552531\pi\)
\(48\) 118.182 0.355377
\(49\) −297.511 −0.867379
\(50\) 0 0
\(51\) −297.505 −0.816845
\(52\) −144.658 −0.385777
\(53\) −325.652 −0.843995 −0.421998 − 0.906597i \(-0.638671\pi\)
−0.421998 + 0.906597i \(0.638671\pi\)
\(54\) −64.0516 −0.161413
\(55\) 0 0
\(56\) 165.957 0.396016
\(57\) −74.1032 −0.172197
\(58\) −50.1143 −0.113454
\(59\) 196.000 0.432492 0.216246 − 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) 0 0
\(61\) −402.641 −0.845130 −0.422565 − 0.906333i \(-0.638870\pi\)
−0.422565 + 0.906333i \(0.638870\pi\)
\(62\) −756.087 −1.54876
\(63\) −60.7011 −0.121391
\(64\) 560.432 1.09459
\(65\) 0 0
\(66\) −78.2853 −0.146004
\(67\) −27.4132 −0.0499860 −0.0249930 − 0.999688i \(-0.507956\pi\)
−0.0249930 + 0.999688i \(0.507956\pi\)
\(68\) −235.255 −0.419543
\(69\) 336.000 0.586227
\(70\) 0 0
\(71\) −300.500 −0.502292 −0.251146 − 0.967949i \(-0.580807\pi\)
−0.251146 + 0.967949i \(0.580807\pi\)
\(72\) −221.454 −0.362480
\(73\) −427.815 −0.685917 −0.342959 − 0.939351i \(-0.611429\pi\)
−0.342959 + 0.939351i \(0.611429\pi\)
\(74\) 356.745 0.560415
\(75\) 0 0
\(76\) −58.5979 −0.0884426
\(77\) −74.1902 −0.109802
\(78\) −433.973 −0.629971
\(79\) 97.5488 0.138925 0.0694627 − 0.997585i \(-0.477872\pi\)
0.0694627 + 0.997585i \(0.477872\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −599.581 −0.807472
\(83\) −1104.62 −1.46082 −0.730408 − 0.683011i \(-0.760670\pi\)
−0.730408 + 0.683011i \(0.760670\pi\)
\(84\) −48.0000 −0.0623480
\(85\) 0 0
\(86\) −507.707 −0.636598
\(87\) 63.3748 0.0780976
\(88\) −270.666 −0.327876
\(89\) 463.022 0.551463 0.275732 − 0.961235i \(-0.411080\pi\)
0.275732 + 0.961235i \(0.411080\pi\)
\(90\) 0 0
\(91\) −411.272 −0.473769
\(92\) 265.696 0.301094
\(93\) 956.152 1.06611
\(94\) −251.152 −0.275578
\(95\) 0 0
\(96\) −310.182 −0.329769
\(97\) 1798.36 1.88243 0.941214 − 0.337810i \(-0.109686\pi\)
0.941214 + 0.337810i \(0.109686\pi\)
\(98\) −705.779 −0.727495
\(99\) 99.0000 0.100504
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 825.4.a.k.1.2 2
3.2 odd 2 2475.4.a.o.1.1 2
5.2 odd 4 825.4.c.i.199.3 4
5.3 odd 4 825.4.c.i.199.2 4
5.4 even 2 33.4.a.d.1.1 ✓ 2
15.14 odd 2 99.4.a.e.1.2 2
20.19 odd 2 528.4.a.o.1.2 2
35.34 odd 2 1617.4.a.j.1.1 2
40.19 odd 2 2112.4.a.bh.1.1 2
40.29 even 2 2112.4.a.ba.1.1 2
55.54 odd 2 363.4.a.j.1.2 2
60.59 even 2 1584.4.a.x.1.1 2
165.164 even 2 1089.4.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.1 ✓ 2 5.4 even 2
99.4.a.e.1.2 2 15.14 odd 2
363.4.a.j.1.2 2 55.54 odd 2
528.4.a.o.1.2 2 20.19 odd 2
825.4.a.k.1.2 2 1.1 even 1 trivial
825.4.c.i.199.2 4 5.3 odd 4
825.4.c.i.199.3 4 5.2 odd 4
1089.4.a.t.1.1 2 165.164 even 2
1584.4.a.x.1.1 2 60.59 even 2
1617.4.a.j.1.1 2 35.34 odd 2
2112.4.a.ba.1.1 2 40.29 even 2
2112.4.a.bh.1.1 2 40.19 odd 2
2475.4.a.o.1.1 2 3.2 odd 2