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Properties

Label 700.2.bj.a.323.39
Level $700$
Weight $2$
Character 700.323
Analytic conductor $5.590$
Analytic rank $0$
Dimension $720$
Inner twists $4$

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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] = [700,2,Mod(127,700)] 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("700.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.bj (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(90\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 323.39
Character \(\chi\) \(=\) 700.323
Dual form 700.2.bj.a.687.39

$q$-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+(-0.315016 + 1.37868i) q^{2} +(1.45856 + 2.86259i) q^{3} +(-1.80153 - 0.868615i) q^{4} +(1.35907 + 1.77565i) q^{5} +(-4.40607 + 1.10913i) q^{6} +(0.707107 + 0.707107i) q^{7} +(1.76506 - 2.21011i) q^{8} +(-4.30364 + 5.92345i) q^{9} +(-2.87619 + 1.31436i) q^{10} +(-2.04399 - 2.81331i) q^{11} +(-0.141155 - 6.42396i) q^{12} +(0.0115670 + 0.0730309i) q^{13} +(-1.19763 + 0.752125i) q^{14} +(-3.10067 + 6.48034i) q^{15} +(2.49102 + 3.12967i) q^{16} +(-2.40861 - 1.22725i) q^{17} +(-6.81084 - 7.79934i) q^{18} +(0.280546 + 0.863431i) q^{19} +(-0.906046 - 4.37939i) q^{20} +(-0.992796 + 3.05551i) q^{21} +(4.52255 - 1.93177i) q^{22} +(0.183083 - 1.15594i) q^{23} +(8.90106 + 1.82904i) q^{24} +(-1.30587 + 4.82646i) q^{25} +(-0.104330 - 0.00705877i) q^{26} +(-13.7139 - 2.17207i) q^{27} +(-0.659670 - 1.88808i) q^{28} +(7.67871 + 2.49497i) q^{29} +(-7.95757 - 6.31625i) q^{30} +(-5.36384 + 1.74282i) q^{31} +(-5.09953 + 2.44842i) q^{32} +(5.07205 - 9.95446i) q^{33} +(2.45073 - 2.93410i) q^{34} +(-0.294568 + 2.21658i) q^{35} +(12.8983 - 6.93307i) q^{36} +(3.47004 - 0.549601i) q^{37} +(-1.27877 + 0.114789i) q^{38} +(-0.192186 + 0.139631i) q^{39} +(6.32321 + 0.130431i) q^{40} +(8.36936 + 6.08070i) q^{41} +(-3.89983 - 2.33129i) q^{42} +(4.55361 - 4.55361i) q^{43} +(1.23862 + 6.84369i) q^{44} +(-16.3669 + 0.408618i) q^{45} +(1.53600 + 0.616552i) q^{46} +(-11.8098 + 6.01737i) q^{47} +(-5.32565 + 11.6956i) q^{48} +1.00000i q^{49} +(-6.24279 - 3.32079i) q^{50} -8.68486i q^{51} +(0.0425975 - 0.141615i) q^{52} +(9.84755 - 5.01758i) q^{53} +(7.31470 - 18.2229i) q^{54} +(2.21753 - 7.45288i) q^{55} +(2.81087 - 0.314701i) q^{56} +(-2.06245 + 2.06245i) q^{57} +(-5.85868 + 9.80055i) q^{58} +(-1.40535 - 1.02104i) q^{59} +(11.2149 - 8.98124i) q^{60} +(-3.22028 + 2.33967i) q^{61} +(-0.713094 - 7.94405i) q^{62} +(-7.23165 + 1.14538i) q^{63} +(-1.76916 - 7.80193i) q^{64} +(-0.113957 + 0.119793i) q^{65} +(12.1263 + 10.1286i) q^{66} +(3.00737 - 5.90229i) q^{67} +(3.27317 + 4.30307i) q^{68} +(3.57601 - 1.16192i) q^{69} +(-2.96317 - 1.10437i) q^{70} +(12.5751 + 4.08589i) q^{71} +(5.49531 + 19.9667i) q^{72} +(10.3532 + 1.63978i) q^{73} +(-0.335395 + 4.95722i) q^{74} +(-15.7208 + 3.30153i) q^{75} +(0.244577 - 1.79918i) q^{76} +(0.543991 - 3.43463i) q^{77} +(-0.131966 - 0.308950i) q^{78} +(3.55666 - 10.9463i) q^{79} +(-2.17174 + 8.67661i) q^{80} +(-6.99715 - 21.5350i) q^{81} +(-11.0198 + 9.62317i) q^{82} +(3.82180 + 1.94731i) q^{83} +(4.44261 - 4.64224i) q^{84} +(-1.09430 - 5.94476i) q^{85} +(4.84352 + 7.71244i) q^{86} +(4.05781 + 25.6200i) q^{87} +(-9.82546 - 0.448208i) q^{88} +(0.877888 + 1.20831i) q^{89} +(4.59249 - 22.6935i) q^{90} +(-0.0434616 + 0.0598197i) q^{91} +(-1.33389 + 1.92343i) q^{92} +(-12.8125 - 12.8125i) q^{93} +(-4.57578 - 18.1775i) q^{94} +(-1.15187 + 1.67161i) q^{95} +(-14.4468 - 11.0267i) q^{96} +(-6.30893 - 12.3820i) q^{97} +(-1.37868 - 0.315016i) q^{98} +25.4611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 720 q + 16 q^{10} + 16 q^{12} + 4 q^{13} + 20 q^{17} - 28 q^{18} - 20 q^{20} - 4 q^{22} + 20 q^{25} + 4 q^{30} - 20 q^{37} - 64 q^{40} - 80 q^{42} - 140 q^{44} - 20 q^{45} - 236 q^{48} - 40 q^{50} - 16 q^{52}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.315016 + 1.37868i −0.222750 + 0.974876i
\(3\) 1.45856 + 2.86259i 0.842100 + 1.65271i 0.754256 + 0.656580i \(0.227998\pi\)
0.0878436 + 0.996134i \(0.472002\pi\)
\(4\) −1.80153 − 0.868615i −0.900765 − 0.434307i
\(5\) 1.35907 + 1.77565i 0.607794 + 0.794095i
\(6\) −4.40607 + 1.10913i −1.79877 + 0.452800i
\(7\) 0.707107 + 0.707107i 0.267261 + 0.267261i
\(8\) 1.76506 − 2.21011i 0.624041 − 0.781391i
\(9\) −4.30364 + 5.92345i −1.43455 + 1.97448i
\(10\) −2.87619 + 1.31436i −0.909530 + 0.415639i
\(11\) −2.04399 − 2.81331i −0.616285 − 0.848244i 0.380791 − 0.924661i \(-0.375652\pi\)
−0.997076 + 0.0764173i \(0.975652\pi\)
\(12\) −0.141155 − 6.42396i −0.0407479 − 1.85444i
\(13\) 0.0115670 + 0.0730309i 0.00320810 + 0.0202551i 0.989240 − 0.146299i \(-0.0467361\pi\)
−0.986032 + 0.166554i \(0.946736\pi\)
\(14\) −1.19763 + 0.752125i −0.320079 + 0.201014i
\(15\) −3.10067 + 6.48034i −0.800589 + 1.67322i
\(16\) 2.49102 + 3.12967i 0.622754 + 0.782418i
\(17\) −2.40861 − 1.22725i −0.584173 − 0.297651i 0.136817 − 0.990596i \(-0.456313\pi\)
−0.720990 + 0.692945i \(0.756313\pi\)
\(18\) −6.81084 − 7.79934i −1.60533 − 1.83832i
\(19\) 0.280546 + 0.863431i 0.0643616 + 0.198085i 0.978066 − 0.208295i \(-0.0667912\pi\)
−0.913705 + 0.406379i \(0.866791\pi\)
\(20\) −0.906046 − 4.37939i −0.202598 − 0.979262i
\(21\) −0.992796 + 3.05551i −0.216646 + 0.666767i
\(22\) 4.52255 − 1.93177i 0.964210 − 0.411855i
\(23\) 0.183083 − 1.15594i 0.0381754 − 0.241030i −0.961221 − 0.275781i \(-0.911064\pi\)
0.999396 + 0.0347507i \(0.0110637\pi\)
\(24\) 8.90106 + 1.82904i 1.81692 + 0.373352i
\(25\) −1.30587 + 4.82646i −0.261173 + 0.965292i
\(26\) −0.104330 − 0.00705877i −0.0204608 − 0.00138434i
\(27\) −13.7139 − 2.17207i −2.63924 − 0.418015i
\(28\) −0.659670 − 1.88808i −0.124666 − 0.356813i
\(29\) 7.67871 + 2.49497i 1.42590 + 0.463303i 0.917472 − 0.397800i \(-0.130226\pi\)
0.508429 + 0.861104i \(0.330226\pi\)
\(30\) −7.95757 − 6.31625i −1.45285 − 1.15318i
\(31\) −5.36384 + 1.74282i −0.963375 + 0.313019i −0.748138 − 0.663543i \(-0.769052\pi\)
−0.215236 + 0.976562i \(0.569052\pi\)
\(32\) −5.09953 + 2.44842i −0.901478 + 0.432824i
\(33\) 5.07205 − 9.95446i 0.882931 − 1.73285i
\(34\) 2.45073 − 2.93410i 0.420297 − 0.503194i
\(35\) −0.294568 + 2.21658i −0.0497910 + 0.374671i
\(36\) 12.8983 − 6.93307i 2.14972 − 1.15551i
\(37\) 3.47004 − 0.549601i 0.570472 − 0.0903538i 0.135467 − 0.990782i \(-0.456747\pi\)
0.435005 + 0.900428i \(0.356747\pi\)
\(38\) −1.27877 + 0.114789i −0.207444 + 0.0186212i
\(39\) −0.192186 + 0.139631i −0.0307744 + 0.0223589i
\(40\) 6.32321 + 0.130431i 0.999787 + 0.0206230i
\(41\) 8.36936 + 6.08070i 1.30708 + 0.949646i 0.999998 − 0.00203903i \(-0.000649043\pi\)
0.307077 + 0.951685i \(0.400649\pi\)
\(42\) −3.89983 − 2.33129i −0.601757 − 0.359725i
\(43\) 4.55361 − 4.55361i 0.694419 − 0.694419i −0.268782 − 0.963201i \(-0.586621\pi\)
0.963201 + 0.268782i \(0.0866212\pi\)
\(44\) 1.23862 + 6.84369i 0.186729 + 1.03173i
\(45\) −16.3669 + 0.408618i −2.43984 + 0.0609132i
\(46\) 1.53600 + 0.616552i 0.226471 + 0.0909057i
\(47\) −11.8098 + 6.01737i −1.72263 + 0.877724i −0.745093 + 0.666961i \(0.767595\pi\)
−0.977537 + 0.210763i \(0.932405\pi\)
\(48\) −5.32565 + 11.6956i −0.768691 + 1.68811i
\(49\) 1.00000i 0.142857i
\(50\) −6.24279 − 3.32079i −0.882863 − 0.469630i
\(51\) − 8.68486i − 1.21612i
\(52\) 0.0425975 − 0.141615i 0.00590721 − 0.0196384i
\(53\) 9.84755 − 5.01758i 1.35267 − 0.689217i 0.380779 − 0.924666i \(-0.375656\pi\)
0.971887 + 0.235449i \(0.0756560\pi\)
\(54\) 7.31470 − 18.2229i 0.995404 − 2.47982i
\(55\) 2.21753 − 7.45288i 0.299012 − 1.00495i
\(56\) 2.81087 − 0.314701i 0.375618 − 0.0420536i
\(57\) −2.06245 + 2.06245i −0.273178 + 0.273178i
\(58\) −5.85868 + 9.80055i −0.769283 + 1.28688i
\(59\) −1.40535 − 1.02104i −0.182960 − 0.132929i 0.492535 − 0.870293i \(-0.336070\pi\)
−0.675496 + 0.737364i \(0.736070\pi\)
\(60\) 11.2149 − 8.98124i 1.44783 − 1.15947i
\(61\) −3.22028 + 2.33967i −0.412315 + 0.299564i −0.774538 − 0.632527i \(-0.782018\pi\)
0.362223 + 0.932091i \(0.382018\pi\)
\(62\) −0.713094 − 7.94405i −0.0905631 − 1.00890i
\(63\) −7.23165 + 1.14538i −0.911102 + 0.144304i
\(64\) −1.76916 − 7.80193i −0.221145 − 0.975241i
\(65\) −0.113957 + 0.119793i −0.0141346 + 0.0148585i
\(66\) 12.1263 + 10.1286i 1.49264 + 1.24674i
\(67\) 3.00737 − 5.90229i 0.367408 − 0.721079i −0.631099 − 0.775703i \(-0.717396\pi\)
0.998507 + 0.0546231i \(0.0173957\pi\)
\(68\) 3.27317 + 4.30307i 0.396930 + 0.521824i
\(69\) 3.57601 − 1.16192i 0.430501 − 0.139878i
\(70\) −2.96317 − 1.10437i −0.354166 − 0.131998i
\(71\) 12.5751 + 4.08589i 1.49239 + 0.484906i 0.937787 − 0.347212i \(-0.112872\pi\)
0.554599 + 0.832118i \(0.312872\pi\)
\(72\) 5.49531 + 19.9667i 0.647629 + 2.35310i
\(73\) 10.3532 + 1.63978i 1.21175 + 0.191922i 0.729418 − 0.684069i \(-0.239791\pi\)
0.482329 + 0.875990i \(0.339791\pi\)
\(74\) −0.335395 + 4.95722i −0.0389889 + 0.576265i
\(75\) −15.7208 + 3.30153i −1.81529 + 0.381228i
\(76\) 0.244577 − 1.79918i 0.0280550 − 0.206380i
\(77\) 0.543991 − 3.43463i 0.0619936 − 0.391412i
\(78\) −0.131966 − 0.308950i −0.0149422 − 0.0349817i
\(79\) 3.55666 − 10.9463i 0.400155 − 1.23155i −0.524718 − 0.851276i \(-0.675829\pi\)
0.924873 − 0.380275i \(-0.124171\pi\)
\(80\) −2.17174 + 8.67661i −0.242808 + 0.970074i
\(81\) −6.99715 − 21.5350i −0.777461 − 2.39278i
\(82\) −11.0198 + 9.62317i −1.21694 + 1.06270i
\(83\) 3.82180 + 1.94731i 0.419497 + 0.213745i 0.650981 − 0.759094i \(-0.274358\pi\)
−0.231483 + 0.972839i \(0.574358\pi\)
\(84\) 4.44261 − 4.64224i 0.484729 − 0.506509i
\(85\) −1.09430 − 5.94476i −0.118694 − 0.644799i
\(86\) 4.84352 + 7.71244i 0.522290 + 0.831654i
\(87\) 4.05781 + 25.6200i 0.435043 + 2.74676i
\(88\) −9.82546 − 0.448208i −1.04740 − 0.0477791i
\(89\) 0.877888 + 1.20831i 0.0930559 + 0.128080i 0.853006 − 0.521901i \(-0.174777\pi\)
−0.759950 + 0.649981i \(0.774777\pi\)
\(90\) 4.59249 − 22.6935i 0.484091 − 2.39211i
\(91\) −0.0434616 + 0.0598197i −0.00455601 + 0.00627081i
\(92\) −1.33389 + 1.92343i −0.139068 + 0.200531i
\(93\) −12.8125 − 12.8125i −1.32859 − 1.32859i
\(94\) −4.57578 − 18.1775i −0.471955 − 1.87486i
\(95\) −1.15187 + 1.67161i −0.118179 + 0.171504i
\(96\) −14.4468 − 11.0267i −1.47447 − 1.12540i
\(97\) −6.30893 − 12.3820i −0.640575 − 1.25720i −0.951758 − 0.306850i \(-0.900725\pi\)
0.311183 − 0.950350i \(-0.399275\pi\)
\(98\) −1.37868 − 0.315016i −0.139268 − 0.0318215i
\(99\) 25.4611 2.55893
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 700.2.bj.a.323.39 ✓ 720
4.3 odd 2 inner 700.2.bj.a.323.43 yes 720
25.12 odd 20 inner 700.2.bj.a.687.43 yes 720
100.87 even 20 inner 700.2.bj.a.687.39 yes 720
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.bj.a.323.39 ✓ 720 1.1 even 1 trivial
700.2.bj.a.323.43 yes 720 4.3 odd 2 inner
700.2.bj.a.687.39 yes 720 100.87 even 20 inner
700.2.bj.a.687.43 yes 720 25.12 odd 20 inner