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Properties

Label 35.10.a.b.1.1
Level $35$
Weight $10$
Character 35.1
Self dual yes
Analytic conductor $18.026$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [35,10,Mod(1,35)] 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("35.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.0262542657\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 35.1

$q$-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)
 
\(f(q)\) \(=\) \(q-14.8284 q^{2} -239.735 q^{3} -292.118 q^{4} +625.000 q^{5} +3554.89 q^{6} +2401.00 q^{7} +11923.8 q^{8} +37789.9 q^{9} -9267.77 q^{10} -16523.6 q^{11} +70030.9 q^{12} +26311.4 q^{13} -35603.1 q^{14} -149834. q^{15} -27246.9 q^{16} -144003. q^{17} -560365. q^{18} -159710. q^{19} -182574. q^{20} -575604. q^{21} +245019. q^{22} +2.07393e6 q^{23} -2.85855e6 q^{24} +390625. q^{25} -390156. q^{26} -4.34086e6 q^{27} -701375. q^{28} -4.94938e6 q^{29} +2.22181e6 q^{30} +4.22040e6 q^{31} -5.70096e6 q^{32} +3.96128e6 q^{33} +2.13533e6 q^{34} +1.50062e6 q^{35} -1.10391e7 q^{36} -1.29081e7 q^{37} +2.36824e6 q^{38} -6.30776e6 q^{39} +7.45238e6 q^{40} -2.87518e7 q^{41} +8.53530e6 q^{42} +3.54825e7 q^{43} +4.82683e6 q^{44} +2.36187e7 q^{45} -3.07532e7 q^{46} +5.95633e7 q^{47} +6.53205e6 q^{48} +5.76480e6 q^{49} -5.79235e6 q^{50} +3.45225e7 q^{51} -7.68602e6 q^{52} +2.31161e6 q^{53} +6.43681e7 q^{54} -1.03272e7 q^{55} +2.86290e7 q^{56} +3.82880e7 q^{57} +7.33915e7 q^{58} -1.68651e8 q^{59} +4.37693e7 q^{60} -6.70167e7 q^{61} -6.25819e7 q^{62} +9.07336e7 q^{63} +9.84867e7 q^{64} +1.64446e7 q^{65} -5.87395e7 q^{66} -1.56259e8 q^{67} +4.20657e7 q^{68} -4.97195e8 q^{69} -2.22519e7 q^{70} +6.95067e7 q^{71} +4.50599e8 q^{72} -7.83438e7 q^{73} +1.91407e8 q^{74} -9.36465e7 q^{75} +4.66540e7 q^{76} -3.96731e7 q^{77} +9.35342e7 q^{78} -4.26957e8 q^{79} -1.70293e7 q^{80} +2.96838e8 q^{81} +4.26344e8 q^{82} -5.31242e8 q^{83} +1.68144e8 q^{84} -9.00016e7 q^{85} -5.26149e8 q^{86} +1.18654e9 q^{87} -1.97024e8 q^{88} -1.14168e8 q^{89} -3.50228e8 q^{90} +6.31736e7 q^{91} -6.05833e8 q^{92} -1.01178e9 q^{93} -8.83231e8 q^{94} -9.98185e7 q^{95} +1.36672e9 q^{96} -1.46573e9 q^{97} -8.54829e7 q^{98} -6.24424e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{2} - 174 q^{3} - 720 q^{4} + 1250 q^{5} + 2952 q^{6} + 4802 q^{7} + 20544 q^{8} + 22428 q^{9} - 15000 q^{10} + 18566 q^{11} + 41904 q^{12} - 51090 q^{13} - 57624 q^{14} - 108750 q^{15} + 112768 q^{16}+ \cdots - 1163466540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.8284 −0.655330 −0.327665 − 0.944794i \(-0.606262\pi\)
−0.327665 + 0.944794i \(0.606262\pi\)
\(3\) −239.735 −1.70878 −0.854390 − 0.519633i \(-0.826069\pi\)
−0.854390 + 0.519633i \(0.826069\pi\)
\(4\) −292.118 −0.570542
\(5\) 625.000 0.447214
\(6\) 3554.89 1.11981
\(7\) 2401.00 0.377964
\(8\) 11923.8 1.02922
\(9\) 37789.9 1.91993
\(10\) −9267.77 −0.293073
\(11\) −16523.6 −0.340280 −0.170140 − 0.985420i \(-0.554422\pi\)
−0.170140 + 0.985420i \(0.554422\pi\)
\(12\) 70030.9 0.974931
\(13\) 26311.4 0.255505 0.127752 − 0.991806i \(-0.459224\pi\)
0.127752 + 0.991806i \(0.459224\pi\)
\(14\) −35603.1 −0.247691
\(15\) −149834. −0.764189
\(16\) −27246.9 −0.103939
\(17\) −144003. −0.418167 −0.209084 − 0.977898i \(-0.567048\pi\)
−0.209084 + 0.977898i \(0.567048\pi\)
\(18\) −560365. −1.25819
\(19\) −159710. −0.281151 −0.140576 − 0.990070i \(-0.544895\pi\)
−0.140576 + 0.990070i \(0.544895\pi\)
\(20\) −182574. −0.255154
\(21\) −575604. −0.645858
\(22\) 245019. 0.222996
\(23\) 2.07393e6 1.54533 0.772663 − 0.634817i \(-0.218925\pi\)
0.772663 + 0.634817i \(0.218925\pi\)
\(24\) −2.85855e6 −1.75872
\(25\) 390625. 0.200000
\(26\) −390156. −0.167440
\(27\) −4.34086e6 −1.57195
\(28\) −701375. −0.215645
\(29\) −4.94938e6 −1.29945 −0.649725 − 0.760169i \(-0.725116\pi\)
−0.649725 + 0.760169i \(0.725116\pi\)
\(30\) 2.22181e6 0.500796
\(31\) 4.22040e6 0.820779 0.410389 − 0.911910i \(-0.365393\pi\)
0.410389 + 0.911910i \(0.365393\pi\)
\(32\) −5.70096e6 −0.961110
\(33\) 3.96128e6 0.581464
\(34\) 2.13533e6 0.274038
\(35\) 1.50062e6 0.169031
\(36\) −1.10391e7 −1.09540
\(37\) −1.29081e7 −1.13228 −0.566142 − 0.824308i \(-0.691565\pi\)
−0.566142 + 0.824308i \(0.691565\pi\)
\(38\) 2.36824e6 0.184247
\(39\) −6.30776e6 −0.436601
\(40\) 7.45238e6 0.460283
\(41\) −2.87518e7 −1.58905 −0.794525 − 0.607231i \(-0.792280\pi\)
−0.794525 + 0.607231i \(0.792280\pi\)
\(42\) 8.53530e6 0.423250
\(43\) 3.54825e7 1.58273 0.791363 − 0.611347i \(-0.209372\pi\)
0.791363 + 0.611347i \(0.209372\pi\)
\(44\) 4.82683e6 0.194144
\(45\) 2.36187e7 0.858617
\(46\) −3.07532e7 −1.01270
\(47\) 5.95633e7 1.78049 0.890243 − 0.455485i \(-0.150534\pi\)
0.890243 + 0.455485i \(0.150534\pi\)
\(48\) 6.53205e6 0.177608
\(49\) 5.76480e6 0.142857
\(50\) −5.79235e6 −0.131066
\(51\) 3.45225e7 0.714555
\(52\) −7.68602e6 −0.145776
\(53\) 2.31161e6 0.0402414 0.0201207 − 0.999798i \(-0.493595\pi\)
0.0201207 + 0.999798i \(0.493595\pi\)
\(54\) 6.43681e7 1.03015
\(55\) −1.03272e7 −0.152178
\(56\) 2.86290e7 0.389010
\(57\) 3.82880e7 0.480425
\(58\) 7.33915e7 0.851569
\(59\) −1.68651e8 −1.81198 −0.905992 − 0.423296i \(-0.860873\pi\)
−0.905992 + 0.423296i \(0.860873\pi\)
\(60\) 4.37693e7 0.436002
\(61\) −6.70167e7 −0.619725 −0.309863 − 0.950781i \(-0.600283\pi\)
−0.309863 + 0.950781i \(0.600283\pi\)
\(62\) −6.25819e7 −0.537881
\(63\) 9.07336e7 0.725664
\(64\) 9.84867e7 0.733783
\(65\) 1.64446e7 0.114265
\(66\) −5.87395e7 −0.381051
\(67\) −1.56259e8 −0.947343 −0.473671 − 0.880702i \(-0.657072\pi\)
−0.473671 + 0.880702i \(0.657072\pi\)
\(68\) 4.20657e7 0.238582
\(69\) −4.97195e8 −2.64062
\(70\) −2.22519e7 −0.110771
\(71\) 6.95067e7 0.324612 0.162306 − 0.986740i \(-0.448107\pi\)
0.162306 + 0.986740i \(0.448107\pi\)
\(72\) 4.50599e8 1.97603
\(73\) −7.83438e7 −0.322888 −0.161444 − 0.986882i \(-0.551615\pi\)
−0.161444 + 0.986882i \(0.551615\pi\)
\(74\) 1.91407e8 0.742020
\(75\) −9.36465e7 −0.341756
\(76\) 4.66540e7 0.160409
\(77\) −3.96731e7 −0.128614
\(78\) 9.35342e7 0.286118
\(79\) −4.26957e8 −1.23328 −0.616641 − 0.787245i \(-0.711507\pi\)
−0.616641 + 0.787245i \(0.711507\pi\)
\(80\) −1.70293e7 −0.0464828
\(81\) 2.96838e8 0.766190
\(82\) 4.26344e8 1.04135
\(83\) −5.31242e8 −1.22869 −0.614343 − 0.789039i \(-0.710579\pi\)
−0.614343 + 0.789039i \(0.710579\pi\)
\(84\) 1.68144e8 0.368489
\(85\) −9.00016e7 −0.187010
\(86\) −5.26149e8 −1.03721
\(87\) 1.18654e9 2.22047
\(88\) −1.97024e8 −0.350225
\(89\) −1.14168e8 −0.192881 −0.0964404 − 0.995339i \(-0.530746\pi\)
−0.0964404 + 0.995339i \(0.530746\pi\)
\(90\) −3.50228e8 −0.562678
\(91\) 6.31736e7 0.0965716
\(92\) −6.05833e8 −0.881674
\(93\) −1.01178e9 −1.40253
\(94\) −8.83231e8 −1.16681
\(95\) −9.98185e7 −0.125735
\(96\) 1.36672e9 1.64232
\(97\) −1.46573e9 −1.68105 −0.840524 − 0.541774i \(-0.817753\pi\)
−0.840524 + 0.541774i \(0.817753\pi\)
\(98\) −8.54829e7 −0.0936186
\(99\) −6.24424e8 −0.653313
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 35.10.a.b.1.1 ✓ 2
3.2 odd 2 315.10.a.b.1.2 2
5.2 odd 4 175.10.b.c.99.1 4
5.3 odd 4 175.10.b.c.99.4 4
5.4 even 2 175.10.a.c.1.2 2
7.6 odd 2 245.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.1 ✓ 2 1.1 even 1 trivial
175.10.a.c.1.2 2 5.4 even 2
175.10.b.c.99.1 4 5.2 odd 4
175.10.b.c.99.4 4 5.3 odd 4
245.10.a.c.1.1 2 7.6 odd 2
315.10.a.b.1.2 2 3.2 odd 2