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Properties

Label 350.4.c.a.99.1
Level $350$
Weight $4$
Character 350.99
Analytic conductor $20.651$
Analytic rank $0$
Dimension $2$
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] = [350,4,Mod(99,350)] 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("350.99"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,-40,0,0,-146,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.4.c.a.99.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -10.0000i q^{3} -4.00000 q^{4} -20.0000 q^{6} +7.00000i q^{7} +8.00000i q^{8} -73.0000 q^{9} +9.00000 q^{11} +40.0000i q^{12} +52.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +96.0000i q^{17} +146.000i q^{18} +10.0000 q^{19} +70.0000 q^{21} -18.0000i q^{22} -75.0000i q^{23} +80.0000 q^{24} +104.000 q^{26} +460.000i q^{27} -28.0000i q^{28} -189.000 q^{29} -232.000 q^{31} -32.0000i q^{32} -90.0000i q^{33} +192.000 q^{34} +292.000 q^{36} +305.000i q^{37} -20.0000i q^{38} +520.000 q^{39} -438.000 q^{41} -140.000i q^{42} -353.000i q^{43} -36.0000 q^{44} -150.000 q^{46} -486.000i q^{47} -160.000i q^{48} -49.0000 q^{49} +960.000 q^{51} -208.000i q^{52} +354.000i q^{53} +920.000 q^{54} -56.0000 q^{56} -100.000i q^{57} +378.000i q^{58} +672.000 q^{59} +206.000 q^{61} +464.000i q^{62} -511.000i q^{63} -64.0000 q^{64} -180.000 q^{66} +599.000i q^{67} -384.000i q^{68} -750.000 q^{69} -471.000 q^{71} -584.000i q^{72} -614.000i q^{73} +610.000 q^{74} -40.0000 q^{76} +63.0000i q^{77} -1040.00i q^{78} -743.000 q^{79} +2629.00 q^{81} +876.000i q^{82} -996.000i q^{83} -280.000 q^{84} -706.000 q^{86} +1890.00i q^{87} +72.0000i q^{88} -180.000 q^{89} -364.000 q^{91} +300.000i q^{92} +2320.00i q^{93} -972.000 q^{94} -320.000 q^{96} -184.000i q^{97} +98.0000i q^{98} -657.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 40 q^{6} - 146 q^{9} + 18 q^{11} + 28 q^{14} + 32 q^{16} + 20 q^{19} + 140 q^{21} + 160 q^{24} + 208 q^{26} - 378 q^{29} - 464 q^{31} + 384 q^{34} + 584 q^{36} + 1040 q^{39} - 876 q^{41}+ \cdots - 1314 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 10.0000i − 1.92450i −0.272166 − 0.962250i \(-0.587740\pi\)
0.272166 − 0.962250i \(-0.412260\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −20.0000 −1.36083
\(7\) 7.00000i 0.377964i
\(8\) 8.00000i 0.353553i
\(9\) −73.0000 −2.70370
\(10\) 0 0
\(11\) 9.00000 0.246691 0.123346 − 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(12\) 40.0000i 0.962250i
\(13\) 52.0000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 96.0000i 1.36961i 0.728725 + 0.684806i \(0.240113\pi\)
−0.728725 + 0.684806i \(0.759887\pi\)
\(18\) 146.000i 1.91181i
\(19\) 10.0000 0.120745 0.0603726 − 0.998176i \(-0.480771\pi\)
0.0603726 + 0.998176i \(0.480771\pi\)
\(20\) 0 0
\(21\) 70.0000 0.727393
\(22\) − 18.0000i − 0.174437i
\(23\) − 75.0000i − 0.679938i −0.940437 − 0.339969i \(-0.889583\pi\)
0.940437 − 0.339969i \(-0.110417\pi\)
\(24\) 80.0000 0.680414
\(25\) 0 0
\(26\) 104.000 0.784465
\(27\) 460.000i 3.27878i
\(28\) − 28.0000i − 0.188982i
\(29\) −189.000 −1.21022 −0.605111 − 0.796141i \(-0.706871\pi\)
−0.605111 + 0.796141i \(0.706871\pi\)
\(30\) 0 0
\(31\) −232.000 −1.34414 −0.672071 − 0.740486i \(-0.734595\pi\)
−0.672071 + 0.740486i \(0.734595\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 90.0000i − 0.474757i
\(34\) 192.000 0.968463
\(35\) 0 0
\(36\) 292.000 1.35185
\(37\) 305.000i 1.35518i 0.735439 + 0.677590i \(0.236976\pi\)
−0.735439 + 0.677590i \(0.763024\pi\)
\(38\) − 20.0000i − 0.0853797i
\(39\) 520.000 2.13504
\(40\) 0 0
\(41\) −438.000 −1.66839 −0.834196 − 0.551467i \(-0.814068\pi\)
−0.834196 + 0.551467i \(0.814068\pi\)
\(42\) − 140.000i − 0.514344i
\(43\) − 353.000i − 1.25191i −0.779860 − 0.625953i \(-0.784710\pi\)
0.779860 − 0.625953i \(-0.215290\pi\)
\(44\) −36.0000 −0.123346
\(45\) 0 0
\(46\) −150.000 −0.480789
\(47\) − 486.000i − 1.50831i −0.656699 − 0.754153i \(-0.728048\pi\)
0.656699 − 0.754153i \(-0.271952\pi\)
\(48\) − 160.000i − 0.481125i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 960.000 2.63582
\(52\) − 208.000i − 0.554700i
\(53\) 354.000i 0.917465i 0.888574 + 0.458732i \(0.151696\pi\)
−0.888574 + 0.458732i \(0.848304\pi\)
\(54\) 920.000 2.31845
\(55\) 0 0
\(56\) −56.0000 −0.133631
\(57\) − 100.000i − 0.232374i
\(58\) 378.000i 0.855756i
\(59\) 672.000 1.48283 0.741415 − 0.671047i \(-0.234155\pi\)
0.741415 + 0.671047i \(0.234155\pi\)
\(60\) 0 0
\(61\) 206.000 0.432387 0.216193 − 0.976351i \(-0.430636\pi\)
0.216193 + 0.976351i \(0.430636\pi\)
\(62\) 464.000i 0.950453i
\(63\) − 511.000i − 1.02190i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −180.000 −0.335704
\(67\) 599.000i 1.09223i 0.837710 + 0.546116i \(0.183894\pi\)
−0.837710 + 0.546116i \(0.816106\pi\)
\(68\) − 384.000i − 0.684806i
\(69\) −750.000 −1.30854
\(70\) 0 0
\(71\) −471.000 −0.787288 −0.393644 − 0.919263i \(-0.628786\pi\)
−0.393644 + 0.919263i \(0.628786\pi\)
\(72\) − 584.000i − 0.955904i
\(73\) − 614.000i − 0.984428i −0.870474 − 0.492214i \(-0.836188\pi\)
0.870474 − 0.492214i \(-0.163812\pi\)
\(74\) 610.000 0.958258
\(75\) 0 0
\(76\) −40.0000 −0.0603726
\(77\) 63.0000i 0.0932405i
\(78\) − 1040.00i − 1.50970i
\(79\) −743.000 −1.05815 −0.529076 − 0.848574i \(-0.677461\pi\)
−0.529076 + 0.848574i \(0.677461\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) 876.000i 1.17973i
\(83\) − 996.000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) −280.000 −0.363696
\(85\) 0 0
\(86\) −706.000 −0.885232
\(87\) 1890.00i 2.32907i
\(88\) 72.0000i 0.0872185i
\(89\) −180.000 −0.214382 −0.107191 − 0.994238i \(-0.534186\pi\)
−0.107191 + 0.994238i \(0.534186\pi\)
\(90\) 0 0
\(91\) −364.000 −0.419314
\(92\) 300.000i 0.339969i
\(93\) 2320.00i 2.58680i
\(94\) −972.000 −1.06653
\(95\) 0 0
\(96\) −320.000 −0.340207
\(97\) − 184.000i − 0.192602i −0.995352 − 0.0963009i \(-0.969299\pi\)
0.995352 − 0.0963009i \(-0.0307011\pi\)
\(98\) 98.0000i 0.101015i
\(99\) −657.000 −0.666980
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 350.4.c.a.99.1 2
5.2 odd 4 350.4.a.k.1.1 yes 1
5.3 odd 4 350.4.a.j.1.1 ✓ 1
5.4 even 2 inner 350.4.c.a.99.2 2
35.13 even 4 2450.4.a.a.1.1 1
35.27 even 4 2450.4.a.bp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.4.a.j.1.1 ✓ 1 5.3 odd 4
350.4.a.k.1.1 yes 1 5.2 odd 4
350.4.c.a.99.1 2 1.1 even 1 trivial
350.4.c.a.99.2 2 5.4 even 2 inner
2450.4.a.a.1.1 1 35.13 even 4
2450.4.a.bp.1.1 1 35.27 even 4