Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,22,Mod(1,25)] 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("25.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2910] 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: \(69.8693360718\)
Analytic rank: \(1\)
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: \( x^{4} - x^{3} - 1929606x^{2} - 743130000x + 239341586400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1521.87\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2315.74 q^{2} -45067.6 q^{3} +3.26550e6 q^{4} -1.04365e8 q^{6} +6.93632e8 q^{7} +2.70560e9 q^{8} -8.42927e9 q^{9} -1.34852e10 q^{11} -1.47168e11 q^{12} -7.82310e11 q^{13} +1.60627e12 q^{14} -5.82789e11 q^{16} +3.22221e12 q^{17} -1.95200e13 q^{18} +9.87395e12 q^{19} -3.12603e13 q^{21} -3.12281e13 q^{22} -2.96069e14 q^{23} -1.21935e14 q^{24} -1.81163e15 q^{26} +8.51310e14 q^{27} +2.26506e15 q^{28} +2.05398e15 q^{29} +5.62290e15 q^{31} -7.02364e15 q^{32} +6.07744e14 q^{33} +7.46180e15 q^{34} -2.75258e16 q^{36} -5.49371e16 q^{37} +2.28655e16 q^{38} +3.52568e16 q^{39} -1.44556e17 q^{41} -7.23908e16 q^{42} -2.52052e17 q^{43} -4.40359e16 q^{44} -6.85618e17 q^{46} -7.91015e15 q^{47} +2.62649e16 q^{48} -7.74210e16 q^{49} -1.45217e17 q^{51} -2.55464e18 q^{52} +2.16231e17 q^{53} +1.97141e18 q^{54} +1.87669e18 q^{56} -4.44995e17 q^{57} +4.75648e18 q^{58} -2.98356e18 q^{59} +2.24873e18 q^{61} +1.30212e19 q^{62} -5.84681e18 q^{63} -1.50427e19 q^{64} +1.40738e18 q^{66} -5.10149e18 q^{67} +1.05221e19 q^{68} +1.33431e19 q^{69} +2.66169e19 q^{71} -2.28062e19 q^{72} -2.85617e19 q^{73} -1.27220e20 q^{74} +3.22434e19 q^{76} -9.35374e18 q^{77} +8.16457e19 q^{78} +8.75536e19 q^{79} +4.98066e19 q^{81} -3.34754e20 q^{82} +1.76821e20 q^{83} -1.02081e20 q^{84} -5.83687e20 q^{86} -9.25679e19 q^{87} -3.64855e19 q^{88} -3.17772e20 q^{89} -5.42635e20 q^{91} -9.66813e20 q^{92} -2.53410e20 q^{93} -1.83179e19 q^{94} +3.16539e20 q^{96} -4.33412e20 q^{97} -1.79287e20 q^{98} +1.13670e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2910 q^{2} - 83240 q^{3} + 9165268 q^{4} - 158524712 q^{6} - 512613800 q^{7} - 5167363080 q^{8} + 21732888532 q^{9} + 33727076448 q^{11} + 142435377680 q^{12} - 863532165080 q^{13} + 2725405637616 q^{14}+ \cdots - 15\!\cdots\!16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2315.74 1.59910 0.799549 − 0.600601i \(-0.205072\pi\)
0.799549 + 0.600601i \(0.205072\pi\)
\(3\) −45067.6 −0.440647 −0.220324 − 0.975427i \(-0.570711\pi\)
−0.220324 + 0.975427i \(0.570711\pi\)
\(4\) 3.26550e6 1.55711
\(5\) 0 0
\(6\) −1.04365e8 −0.704638
\(7\) 6.93632e8 0.928110 0.464055 − 0.885806i \(-0.346394\pi\)
0.464055 + 0.885806i \(0.346394\pi\)
\(8\) 2.70560e9 0.890879
\(9\) −8.42927e9 −0.805830
\(10\) 0 0
\(11\) −1.34852e10 −0.156759 −0.0783796 − 0.996924i \(-0.524975\pi\)
−0.0783796 + 0.996924i \(0.524975\pi\)
\(12\) −1.47168e11 −0.686138
\(13\) −7.82310e11 −1.57389 −0.786944 − 0.617025i \(-0.788338\pi\)
−0.786944 + 0.617025i \(0.788338\pi\)
\(14\) 1.60627e12 1.48414
\(15\) 0 0
\(16\) −5.82789e11 −0.132511
\(17\) 3.22221e12 0.387650 0.193825 − 0.981036i \(-0.437911\pi\)
0.193825 + 0.981036i \(0.437911\pi\)
\(18\) −1.95200e13 −1.28860
\(19\) 9.87395e12 0.369469 0.184734 − 0.982788i \(-0.440857\pi\)
0.184734 + 0.982788i \(0.440857\pi\)
\(20\) 0 0
\(21\) −3.12603e13 −0.408969
\(22\) −3.12281e13 −0.250673
\(23\) −2.96069e14 −1.49022 −0.745109 − 0.666942i \(-0.767603\pi\)
−0.745109 + 0.666942i \(0.767603\pi\)
\(24\) −1.21935e14 −0.392564
\(25\) 0 0
\(26\) −1.81163e15 −2.51680
\(27\) 8.51310e14 0.795734
\(28\) 2.26506e15 1.44517
\(29\) 2.05398e15 0.906603 0.453301 − 0.891357i \(-0.350246\pi\)
0.453301 + 0.891357i \(0.350246\pi\)
\(30\) 0 0
\(31\) 5.62290e15 1.23215 0.616073 − 0.787689i \(-0.288723\pi\)
0.616073 + 0.787689i \(0.288723\pi\)
\(32\) −7.02364e15 −1.10278
\(33\) 6.07744e14 0.0690755
\(34\) 7.46180e15 0.619890
\(35\) 0 0
\(36\) −2.75258e16 −1.25477
\(37\) −5.49371e16 −1.87823 −0.939113 − 0.343607i \(-0.888351\pi\)
−0.939113 + 0.343607i \(0.888351\pi\)
\(38\) 2.28655e16 0.590817
\(39\) 3.52568e16 0.693529
\(40\) 0 0
\(41\) −1.44556e17 −1.68192 −0.840960 − 0.541098i \(-0.818009\pi\)
−0.840960 + 0.541098i \(0.818009\pi\)
\(42\) −7.23908e16 −0.653982
\(43\) −2.52052e17 −1.77857 −0.889286 − 0.457352i \(-0.848798\pi\)
−0.889286 + 0.457352i \(0.848798\pi\)
\(44\) −4.40359e16 −0.244092
\(45\) 0 0
\(46\) −6.85618e17 −2.38301
\(47\) −7.91015e15 −0.0219360 −0.0109680 − 0.999940i \(-0.503491\pi\)
−0.0109680 + 0.999940i \(0.503491\pi\)
\(48\) 2.62649e16 0.0583906
\(49\) −7.74210e16 −0.138612
\(50\) 0 0
\(51\) −1.45217e17 −0.170817
\(52\) −2.55464e18 −2.45072
\(53\) 2.16231e17 0.169833 0.0849163 − 0.996388i \(-0.472938\pi\)
0.0849163 + 0.996388i \(0.472938\pi\)
\(54\) 1.97141e18 1.27246
\(55\) 0 0
\(56\) 1.87669e18 0.826834
\(57\) −4.44995e17 −0.162806
\(58\) 4.75648e18 1.44975
\(59\) −2.98356e18 −0.759957 −0.379979 − 0.924995i \(-0.624069\pi\)
−0.379979 + 0.924995i \(0.624069\pi\)
\(60\) 0 0
\(61\) 2.24873e18 0.403621 0.201811 − 0.979425i \(-0.435317\pi\)
0.201811 + 0.979425i \(0.435317\pi\)
\(62\) 1.30212e19 1.97032
\(63\) −5.84681e18 −0.747899
\(64\) −1.50427e19 −1.63094
\(65\) 0 0
\(66\) 1.40738e18 0.110459
\(67\) −5.10149e18 −0.341910 −0.170955 − 0.985279i \(-0.554685\pi\)
−0.170955 + 0.985279i \(0.554685\pi\)
\(68\) 1.05221e19 0.603615
\(69\) 1.33431e19 0.656661
\(70\) 0 0
\(71\) 2.66169e19 0.970388 0.485194 − 0.874407i \(-0.338749\pi\)
0.485194 + 0.874407i \(0.338749\pi\)
\(72\) −2.28062e19 −0.717897
\(73\) −2.85617e19 −0.777846 −0.388923 − 0.921270i \(-0.627153\pi\)
−0.388923 + 0.921270i \(0.627153\pi\)
\(74\) −1.27220e20 −3.00347
\(75\) 0 0
\(76\) 3.22434e19 0.575305
\(77\) −9.35374e18 −0.145490
\(78\) 8.16457e19 1.10902
\(79\) 8.75536e19 1.04037 0.520187 − 0.854052i \(-0.325862\pi\)
0.520187 + 0.854052i \(0.325862\pi\)
\(80\) 0 0
\(81\) 4.98066e19 0.455192
\(82\) −3.34754e20 −2.68955
\(83\) 1.76821e20 1.25088 0.625438 − 0.780274i \(-0.284920\pi\)
0.625438 + 0.780274i \(0.284920\pi\)
\(84\) −1.02081e20 −0.636812
\(85\) 0 0
\(86\) −5.83687e20 −2.84411
\(87\) −9.25679e19 −0.399492
\(88\) −3.64855e19 −0.139653
\(89\) −3.17772e20 −1.08024 −0.540121 − 0.841588i \(-0.681621\pi\)
−0.540121 + 0.841588i \(0.681621\pi\)
\(90\) 0 0
\(91\) −5.42635e20 −1.46074
\(92\) −9.66813e20 −2.32044
\(93\) −2.53410e20 −0.542942
\(94\) −1.83179e19 −0.0350778
\(95\) 0 0
\(96\) 3.16539e20 0.485936
\(97\) −4.33412e20 −0.596757 −0.298379 − 0.954448i \(-0.596446\pi\)
−0.298379 + 0.954448i \(0.596446\pi\)
\(98\) −1.79287e20 −0.221654
\(99\) 1.13670e20 0.126321
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 25.22.a.c.1.4 4
5.2 odd 4 25.22.b.c.24.6 8
5.3 odd 4 25.22.b.c.24.3 8
5.4 even 2 5.22.a.b.1.1 ✓ 4
15.14 odd 2 45.22.a.f.1.4 4
20.19 odd 2 80.22.a.g.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.a.b.1.1 ✓ 4 5.4 even 2
25.22.a.c.1.4 4 1.1 even 1 trivial
25.22.b.c.24.3 8 5.3 odd 4
25.22.b.c.24.6 8 5.2 odd 4
45.22.a.f.1.4 4 15.14 odd 2
80.22.a.g.1.3 4 20.19 odd 2