LMFDB - Embedded newform 725.2.b.g.349.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(349,725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("725.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-8,0,-2,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.88858223543296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 14x^{8} + 63x^{6} + 99x^{4} + 55x^{2} + 9 $ x^10 + 14x^8 + 63x^6 + 99x^4 + 55x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.7
Root			$0.794018i$ of defining polynomial
Character	$\chi$	$=$	725.349
Dual form			725.2.b.g.349.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.794018i q^{2} -2.48525i q^{3} +1.36954 q^{4} +1.97334 q^{6} +3.07654i q^{7} +2.67547i q^{8} -3.17649 q^{9} +4.64881 q^{11} -3.40365i q^{12} +5.59590i q^{13} -2.44283 q^{14} +0.614699 q^{16} +5.43031i q^{17} -2.52219i q^{18} -1.36954 q^{19} +7.64598 q^{21} +3.69124i q^{22} -3.46949i q^{23} +6.64923 q^{24} -4.44325 q^{26} +0.438627i q^{27} +4.21343i q^{28} +1.00000 q^{29} -8.27137 q^{31} +5.83903i q^{32} -11.5535i q^{33} -4.31176 q^{34} -4.35032 q^{36} -7.88607i q^{37} -1.08744i q^{38} +13.9072 q^{39} +3.73021 q^{41} +6.07105i q^{42} -9.31338i q^{43} +6.36671 q^{44} +2.75484 q^{46} -3.63833i q^{47} -1.52768i q^{48} -2.46509 q^{49} +13.4957 q^{51} +7.66379i q^{52} +1.17324i q^{53} -0.348278 q^{54} -8.23119 q^{56} +3.40365i q^{57} +0.794018i q^{58} +6.45859 q^{59} +1.83342 q^{61} -6.56762i q^{62} -9.77260i q^{63} -3.40689 q^{64} +9.17366 q^{66} -9.44462i q^{67} +7.43700i q^{68} -8.62257 q^{69} -14.1565 q^{71} -8.49861i q^{72} -9.80146i q^{73} +6.26168 q^{74} -1.87563 q^{76} +14.3022i q^{77} +11.0426i q^{78} +6.60077 q^{79} -8.43937 q^{81} +2.96186i q^{82} +8.95528i q^{83} +10.4714 q^{84} +7.39499 q^{86} -2.48525i q^{87} +12.4378i q^{88} +8.90284 q^{89} -17.2160 q^{91} -4.75159i q^{92} +20.5565i q^{93} +2.88890 q^{94} +14.5115 q^{96} +8.39270i q^{97} -1.95732i q^{98} -14.7669 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{4} - 2 q^{6} - 14 q^{9} + 4 q^{11} + 26 q^{14} + 12 q^{16} + 8 q^{19} + 30 q^{21} + 38 q^{24} - 8 q^{26} + 10 q^{29} + 10 q^{31} + 18 q^{34} + 70 q^{36} - 8 q^{39} - 6 q^{41} + 38 q^{44} - 26 q^{49}+ \cdots + 24 q^{99}+O(q^{100}) $ 10 * q - 8 * q^4 - 2 * q^6 - 14 * q^9 + 4 * q^11 + 26 * q^14 + 12 * q^16 + 8 * q^19 + 30 * q^21 + 38 * q^24 - 8 * q^26 + 10 * q^29 + 10 * q^31 + 18 * q^34 + 70 * q^36 - 8 * q^39 - 6 * q^41 + 38 * q^44 - 26 * q^49 - 14 * q^51 + 116 * q^54 - 16 * q^56 + 30 * q^59 - 14 * q^61 + 18 * q^64 + 70 * q^66 - 36 * q^69 - 34 * q^71 - 24 * q^74 - 68 * q^76 + 4 * q^79 + 42 * q^81 - 76 * q^84 + 62 * q^86 - 28 * q^89 - 30 * q^91 - 54 * q^94 + 12 * q^96 + 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$.

$n$	$176$	$552$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.794018i	0.561455i	0.959787	+	0.280728i	$0.0905758\pi$
$2$	0.794018i	0.561455i	−0.959787	+	0.280728i	$0.909424\pi$
$3$	− 2.48525i	− 1.43486i	−0.696629	−	0.717431i	$-0.745318\pi$
$3$	− 2.48525i	− 1.43486i	0.696629	−	0.717431i	$-0.254682\pi$
$4$	1.36954	0.684768
$5$	0	0
$5$	0	0
$6$	1.97334	0.805611
$7$	3.07654i	1.16282i	0.813610	+	0.581411i	$0.197499\pi$
$7$	3.07654i	1.16282i	−0.813610	+	0.581411i	$0.802501\pi$
$8$	2.67547i	0.945922i
$9$	−3.17649	−1.05883
$10$	0	0
$11$	4.64881	1.40167	0.700834	−	0.713324i	$-0.252811\pi$
$11$	4.64881	1.40167	0.700834	+	0.713324i	$0.252811\pi$
$12$	− 3.40365i	− 0.982548i
$13$	5.59590i	1.55202i	0.630718	+	0.776012i	$0.282760\pi$
$13$	5.59590i	1.55202i	−0.630718	+	0.776012i	$0.717240\pi$
$14$	−2.44283	−0.652873
$15$	0	0
$16$	0.614699	0.153675
$17$	5.43031i	1.31704i	0.752562	+	0.658522i	$0.228818\pi$
$17$	5.43031i	1.31704i	−0.752562	+	0.658522i	$0.771182\pi$
$18$	− 2.52219i	− 0.594486i
$19$	−1.36954	−0.314193	−0.157097	−	0.987583i	$-0.550213\pi$
$19$	−1.36954	−0.314193	−0.157097	+	0.987583i	$0.550213\pi$
$20$	0	0
$21$	7.64598	1.66849
$22$	3.69124i	0.786974i
$23$	− 3.46949i	− 0.723439i	−0.932287	−	0.361719i	$-0.882190\pi$
$23$	− 3.46949i	− 0.723439i	0.932287	−	0.361719i	$-0.117810\pi$
$24$	6.64923	1.35727
$25$	0	0
$26$	−4.44325	−0.871392
$27$	0.438627i	0.0844139i
$28$	4.21343i	0.796263i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.27137	−1.48558	−0.742791	−	0.669523i	$-0.766499\pi$
$31$	−8.27137	−1.48558	−0.742791	+	0.669523i	$0.766499\pi$
$32$	5.83903i	1.03220i
$33$	− 11.5535i	− 2.01120i
$34$	−4.31176	−0.739461
$35$	0	0
$36$	−4.35032	−0.725053
$37$	− 7.88607i	− 1.29646i	−0.761444	−	0.648231i	$-0.775509\pi$
$37$	− 7.88607i	− 1.29646i	0.761444	−	0.648231i	$-0.224491\pi$
$38$	− 1.08744i	− 0.176405i
$39$	13.9072	2.22694
$40$	0	0
$41$	3.73021	0.582562	0.291281	−	0.956638i	$-0.405919\pi$
$41$	3.73021	0.582562	0.291281	+	0.956638i	$0.405919\pi$
$42$	6.07105i	0.936783i
$43$	− 9.31338i	− 1.42028i	−0.704062	−	0.710139i	$-0.748632\pi$
$43$	− 9.31338i	− 1.42028i	0.704062	−	0.710139i	$-0.251368\pi$
$44$	6.36671	0.959817
$45$	0	0
$46$	2.75484	0.406179
$47$	− 3.63833i	− 0.530705i	−0.964151	−	0.265353i	$-0.914512\pi$
$47$	− 3.63833i	− 0.530705i	0.964151	−	0.265353i	$-0.0854884\pi$
$48$	− 1.52768i	− 0.220502i
$49$	−2.46509	−0.352155
$50$	0	0
$51$	13.4957	1.88978
$52$	7.66379i	1.06278i
$53$	1.17324i	0.161158i	0.996748	+	0.0805788i	$0.0256769\pi$
$53$	1.17324i	0.161158i	−0.996748	+	0.0805788i	$0.974323\pi$
$54$	−0.348278	−0.0473946
$55$	0	0
$56$	−8.23119	−1.09994
$57$	3.40365i	0.450824i
$58$	0.794018i	0.104260i
$59$	6.45859	0.840837	0.420419	−	0.907330i	$-0.361883\pi$
$59$	6.45859	0.840837	0.420419	+	0.907330i	$0.361883\pi$
$60$	0	0
$61$	1.83342	0.234745	0.117372	−	0.993088i	$-0.462553\pi$
$61$	1.83342	0.234745	0.117372	+	0.993088i	$0.462553\pi$
$62$	− 6.56762i	− 0.834088i
$63$	− 9.77260i	− 1.23123i
$64$	−3.40689	−0.425862
$65$	0	0
$66$	9.17366	1.12920
$67$	− 9.44462i	− 1.15384i	−0.816799	−	0.576922i	$-0.804254\pi$
$67$	− 9.44462i	− 1.15384i	0.816799	−	0.576922i	$-0.195746\pi$
$68$	7.43700i	0.901869i
$69$	−8.62257	−1.03803
$70$	0	0
$71$	−14.1565	−1.68007	−0.840033	−	0.542535i	$-0.817465\pi$
$71$	−14.1565	−1.68007	−0.840033	+	0.542535i	$0.817465\pi$
$72$	− 8.49861i	− 1.00157i
$73$	− 9.80146i	− 1.14717i	−0.819144	−	0.573587i	$-0.805551\pi$
$73$	− 9.80146i	− 1.14717i	0.819144	−	0.573587i	$-0.194449\pi$
$74$	6.26168	0.727906
$75$	0	0
$76$	−1.87563	−0.215149
$77$	14.3022i	1.62989i
$78$	11.0426i	1.25033i
$79$	6.60077	0.742645	0.371322	−	0.928504i	$-0.378904\pi$
$79$	6.60077	0.742645	0.371322	+	0.928504i	$0.378904\pi$
$80$	0	0
$81$	−8.43937	−0.937708
$82$	2.96186i	0.327082i
$83$	8.95528i	0.982970i	0.870886	+	0.491485i	$0.163546\pi$
$83$	8.95528i	0.982970i	−0.870886	+	0.491485i	$0.836454\pi$
$84$	10.4714	1.14253
$85$	0	0
$86$	7.39499	0.797423
$87$	− 2.48525i	− 0.266447i
$88$	12.4378i	1.32587i
$89$	8.90284	0.943699	0.471850	−	0.881679i	$-0.343587\pi$
$89$	8.90284	0.943699	0.471850	+	0.881679i	$0.343587\pi$
$90$	0	0
$91$	−17.2160	−1.80473
$92$	− 4.75159i	− 0.495387i
$93$	20.5565i	2.13161i
$94$	2.88890	0.297967
$95$	0	0
$96$	14.5115	1.48107
$97$	8.39270i	0.852150i	0.904688	+	0.426075i	$0.140104\pi$
$97$	8.39270i	0.852150i	−0.904688	+	0.426075i	$0.859896\pi$
$98$	− 1.95732i	− 0.197719i
$99$	−14.7669	−1.48413
$100$	0	0
$101$	−0.841314	−0.0837139	−0.0418570	−	0.999124i	$-0.513327\pi$
$101$	−0.841314	−0.0837139	−0.0418570	+	0.999124i	$0.513327\pi$
$102$	10.7158i	1.06103i
$103$	− 9.87056i	− 0.972575i	−0.873799	−	0.486287i	$-0.838351\pi$
$103$	− 9.87056i	− 0.972575i	0.873799	−	0.486287i	$-0.161649\pi$
$104$	−14.9717	−1.46809
$105$	0	0
$106$	−0.931577	−0.0904828
$107$	11.2762i	1.09012i	0.838399	+	0.545058i	$0.183492\pi$
$107$	11.2762i	1.09012i	−0.838399	+	0.545058i	$0.816508\pi$
$108$	0.600716i	0.0578039i
$109$	−3.72560	−0.356847	−0.178424	−	0.983954i	$-0.557100\pi$
$109$	−3.72560	−0.356847	−0.178424	+	0.983954i	$0.557100\pi$
$110$	0	0
$111$	−19.5989	−1.86025
$112$	1.89114i	0.178696i
$113$	7.27299i	0.684186i	0.939666	+	0.342093i	$0.111136\pi$
$113$	7.27299i	0.684186i	−0.939666	+	0.342093i	$0.888864\pi$
$114$	−2.70256	−0.253117
$115$	0	0
$116$	1.36954	0.127158
$117$	− 17.7753i	− 1.64333i
$118$	5.12824i	0.472093i
$119$	−16.7065	−1.53149
$120$	0	0
$121$	10.6114	0.964674
$122$	1.45576i	0.131799i
$123$	− 9.27053i	− 0.835896i
$124$	−11.3279	−1.01728
$125$	0	0
$126$	7.75962	0.691282
$127$	− 11.0106i	− 0.977037i	−0.872554	−	0.488519i	$-0.837537\pi$
$127$	− 11.0106i	− 0.977037i	0.872554	−	0.488519i	$-0.162463\pi$
$128$	8.97292i	0.793101i
$129$	−23.1461	−2.03790
$130$	0	0
$131$	6.69910	0.585303	0.292652	−	0.956219i	$-0.405462\pi$
$131$	6.69910	0.585303	0.292652	+	0.956219i	$0.405462\pi$
$132$	− 15.8229i	− 1.37721i
$133$	− 4.21343i	− 0.365351i
$134$	7.49920	0.647832
$135$	0	0
$136$	−14.5286	−1.24582
$137$	− 22.2380i	− 1.89992i	−0.312368	−	0.949961i	$-0.601122\pi$
$137$	− 22.2380i	− 1.89992i	0.312368	−	0.949961i	$-0.398878\pi$
$138$	− 6.84647i	− 0.582810i
$139$	9.13003	0.774399	0.387200	−	0.921996i	$-0.373442\pi$
$139$	9.13003	0.774399	0.387200	+	0.921996i	$0.373442\pi$
$140$	0	0
$141$	−9.04218	−0.761489
$142$	− 11.2405i	− 0.943282i
$143$	26.0143i	2.17542i
$144$	−1.95259	−0.162716
$145$	0	0
$146$	7.78254	0.644087
$147$	6.12637i	0.505294i
$148$	− 10.8003i	− 0.887776i
$149$	−7.38488	−0.604993	−0.302497	−	0.953150i	$-0.597820\pi$
$149$	−7.38488	−0.604993	−0.302497	+	0.953150i	$0.597820\pi$
$150$	0	0
$151$	−23.1130	−1.88091	−0.940454	−	0.339922i	$-0.889599\pi$
$151$	−23.1130	−1.88091	−0.940454	+	0.339922i	$0.889599\pi$
$152$	− 3.66415i	− 0.297202i
$153$	− 17.2493i	− 1.39453i
$154$	−11.3562	−0.915111
$155$	0	0
$156$	19.0465	1.52494
$157$	18.7526i	1.49662i	0.663349	+	0.748310i	$0.269134\pi$
$157$	18.7526i	1.49662i	−0.663349	+	0.748310i	$0.730866\pi$
$158$	5.24113i	0.416962i
$159$	2.91581	0.231239
$160$	0	0
$161$	10.6740	0.841230
$162$	− 6.70101i	− 0.526481i
$163$	− 21.6256i	− 1.69384i	−0.531716	−	0.846922i	$-0.678453\pi$
$163$	− 21.6256i	− 1.69384i	0.531716	−	0.846922i	$-0.321547\pi$
$164$	5.10866	0.398919
$165$	0	0
$166$	−7.11065	−0.551894
$167$	8.63122i	0.667904i	0.942590	+	0.333952i	$0.108382\pi$
$167$	8.63122i	0.667904i	−0.942590	+	0.333952i	$0.891618\pi$
$168$	20.4566i	1.57826i
$169$	−18.3141	−1.40878
$170$	0	0
$171$	4.35032	0.332677
$172$	− 12.7550i	− 0.972560i
$173$	− 7.66150i	− 0.582493i	−0.956648	−	0.291246i	$-0.905930\pi$
$173$	− 7.66150i	− 0.582493i	0.956648	−	0.291246i	$-0.0940700\pi$
$174$	1.97334	0.149598
$175$	0	0
$176$	2.85762	0.215401
$177$	− 16.0512i	− 1.20649i
$178$	7.06902i	0.529845i
$179$	17.5070	1.30854	0.654268	−	0.756263i	$-0.272977\pi$
$179$	17.5070	1.30854	0.654268	+	0.756263i	$0.272977\pi$
$180$	0	0
$181$	1.69103	0.125693	0.0628467	−	0.998023i	$-0.479982\pi$
$181$	1.69103	0.125693	0.0628467	+	0.998023i	$0.479982\pi$
$182$	− 13.6698i	− 1.01327i
$183$	− 4.55650i	− 0.336827i
$184$	9.28252	0.684316
$185$	0	0
$186$	−16.3222	−1.19680
$187$	25.2445i	1.84606i
$188$	− 4.98282i	− 0.363410i
$189$	−1.34945	−0.0981583
$190$	0	0
$191$	20.4892	1.48255	0.741273	−	0.671203i	$-0.234222\pi$
$191$	20.4892	1.48255	0.741273	+	0.671203i	$0.234222\pi$
$192$	8.46700i	0.611053i
$193$	− 6.82676i	− 0.491401i	−0.969346	−	0.245700i	$-0.920982\pi$
$193$	− 6.82676i	− 0.491401i	0.969346	−	0.245700i	$-0.0790179\pi$
$194$	−6.66396	−0.478444
$195$	0	0
$196$	−3.37602	−0.241145
$197$	− 6.46787i	− 0.460817i	−0.973094	−	0.230408i	$-0.925994\pi$
$197$	− 6.46787i	− 0.460817i	0.973094	−	0.230408i	$-0.0740062\pi$
$198$	− 11.7252i	− 0.833273i
$199$	19.0874	1.35307	0.676535	−	0.736410i	$-0.263481\pi$
$199$	19.0874	1.35307	0.676535	+	0.736410i	$0.263481\pi$
$200$	0	0
$201$	−23.4723	−1.65561
$202$	− 0.668019i	− 0.0470016i
$203$	3.07654i	0.215931i
$204$	18.4828	1.29406
$205$	0	0
$206$	7.83740	0.546057
$207$	11.0208i	0.765999i
$208$	3.43980i	0.238507i
$209$	−6.36671	−0.440394
$210$	0	0
$211$	10.9062	0.750817	0.375408	−	0.926860i	$-0.377502\pi$
$211$	10.9062	0.750817	0.375408	+	0.926860i	$0.377502\pi$
$212$	1.60680i	0.110355i
$213$	35.1825i	2.41066i
$214$	−8.95354	−0.612051
$215$	0	0
$216$	−1.17354	−0.0798489
$217$	− 25.4472i	− 1.72747i
$218$	− 2.95819i	− 0.200354i
$219$	−24.3591	−1.64604
$220$	0	0
$221$	−30.3875	−2.04408
$222$	− 15.5619i	− 1.04444i
$223$	− 16.3598i	− 1.09553i	−0.836631	−	0.547766i	$-0.815478\pi$
$223$	− 16.3598i	− 1.09553i	0.836631	−	0.547766i	$-0.184522\pi$
$224$	−17.9640	−1.20027
$225$	0	0
$226$	−5.77489	−0.384140
$227$	9.76086i	0.647851i	0.946083	+	0.323926i	$0.105003\pi$
$227$	9.76086i	0.647851i	−0.946083	+	0.323926i	$0.894997\pi$
$228$	4.66141i	0.308710i
$229$	−1.09068	−0.0720744	−0.0360372	−	0.999350i	$-0.511473\pi$
$229$	−1.09068	−0.0720744	−0.0360372	+	0.999350i	$0.511473\pi$
$230$	0	0
$231$	35.5447	2.33867
$232$	2.67547i	0.175653i
$233$	− 7.64981i	− 0.501156i	−0.968096	−	0.250578i	$-0.919379\pi$
$233$	− 7.64981i	− 0.501156i	0.968096	−	0.250578i	$-0.0806207\pi$
$234$	14.1139	0.922657
$235$	0	0
$236$	8.84527	0.575778
$237$	− 16.4046i	− 1.06559i
$238$	− 13.2653i	− 0.859862i
$239$	−7.63023	−0.493558	−0.246779	−	0.969072i	$-0.579372\pi$
$239$	−7.63023	−0.493558	−0.246779	+	0.969072i	$0.579372\pi$
$240$	0	0
$241$	−17.1686	−1.10593	−0.552963	−	0.833206i	$-0.686503\pi$
$241$	−17.1686	−1.10593	−0.552963	+	0.833206i	$0.686503\pi$
$242$	8.42566i	0.541622i
$243$	22.2899i	1.42990i
$244$	2.51093	0.160746
$245$	0	0
$246$	7.36097	0.469318
$247$	− 7.66379i	− 0.487635i
$248$	− 22.1298i	− 1.40525i
$249$	22.2562	1.41043
$250$	0	0
$251$	−7.57023	−0.477829	−0.238914	−	0.971041i	$-0.576792\pi$
$251$	−7.57023	−0.477829	−0.238914	+	0.971041i	$0.576792\pi$
$252$	− 13.3839i	− 0.843108i
$253$	− 16.1290i	− 1.01402i
$254$	8.74265	0.548563
$255$	0	0
$256$	−13.9384	−0.871153
$257$	2.78632i	0.173806i	0.996217	+	0.0869028i	$0.0276970\pi$
$257$	2.78632i	0.173806i	−0.996217	+	0.0869028i	$0.972303\pi$
$258$	− 18.3784i	− 1.14419i
$259$	24.2618	1.50756
$260$	0	0
$261$	−3.17649	−0.196620
$262$	5.31921i	0.328622i
$263$	11.7231i	0.722877i	0.932396	+	0.361438i	$0.117714\pi$
$263$	11.7231i	0.722877i	−0.932396	+	0.361438i	$0.882286\pi$
$264$	30.9110	1.90244
$265$	0	0
$266$	3.34554	0.205128
$267$	− 22.1258i	− 1.35408i
$268$	− 12.9347i	− 0.790115i
$269$	−23.3577	−1.42415	−0.712073	−	0.702106i	$-0.752243\pi$
$269$	−23.3577	−1.42415	−0.712073	+	0.702106i	$0.752243\pi$
$270$	0	0
$271$	−19.2624	−1.17011	−0.585055	−	0.810994i	$-0.698927\pi$
$271$	−19.2624	−1.17011	−0.585055	+	0.810994i	$0.698927\pi$
$272$	3.33800i	0.202396i
$273$	42.7862i	2.58954i
$274$	17.6574	1.06672
$275$	0	0
$276$	−11.8089	−0.710813
$277$	13.7271i	0.824781i	0.911007	+	0.412390i	$0.135306\pi$
$277$	13.7271i	0.824781i	−0.911007	+	0.412390i	$0.864694\pi$
$278$	7.24941i	0.434791i
$279$	26.2740	1.57298
$280$	0	0
$281$	−29.1540	−1.73918	−0.869589	−	0.493775i	$-0.835616\pi$
$281$	−29.1540	−1.73918	−0.869589	+	0.493775i	$0.835616\pi$
$282$	− 7.17965i	− 0.427542i
$283$	− 12.4693i	− 0.741224i	−0.928788	−	0.370612i	$-0.879148\pi$
$283$	− 12.4693i	− 0.741224i	0.928788	−	0.370612i	$-0.120852\pi$
$284$	−19.3878	−1.15046
$285$	0	0
$286$	−20.6558	−1.22140
$287$	11.4761i	0.677415i
$288$	− 18.5476i	− 1.09293i
$289$	−12.4882	−0.734603
$290$	0	0
$291$	20.8580	1.22272
$292$	− 13.4235i	− 0.785548i
$293$	6.96852i	0.407105i	0.979064	+	0.203553i	$0.0652488\pi$
$293$	6.96852i	0.407105i	−0.979064	+	0.203553i	$0.934751\pi$
$294$	−4.86445	−0.283700
$295$	0	0
$296$	21.0990	1.22635
$297$	2.03909i	0.118320i
$298$	− 5.86373i	− 0.339677i
$299$	19.4149	1.12279
$300$	0	0
$301$	28.6530	1.65153
$302$	− 18.3521i	− 1.05605i
$303$	2.09088i	0.120118i
$304$	−0.841852	−0.0482835
$305$	0	0
$306$	13.6963	0.782964
$307$	9.36932i	0.534735i	0.963595	+	0.267368i	$0.0861538\pi$
$307$	9.36932i	0.534735i	−0.963595	+	0.267368i	$0.913846\pi$
$308$	19.5874i	1.11610i
$309$	−24.5308	−1.39551
$310$	0	0
$311$	25.9487	1.47141	0.735707	−	0.677300i	$-0.236850\pi$
$311$	25.9487	1.47141	0.735707	+	0.677300i	$0.236850\pi$
$312$	37.2084i	2.10651i
$313$	23.9833i	1.35562i	0.735238	+	0.677809i	$0.237070\pi$
$313$	23.9833i	1.35562i	−0.735238	+	0.677809i	$0.762930\pi$
$314$	−14.8899	−0.840285
$315$	0	0
$316$	9.03999	0.508539
$317$	20.6136i	1.15778i	0.815407	+	0.578888i	$0.196513\pi$
$317$	20.6136i	1.15778i	−0.815407	+	0.578888i	$0.803487\pi$
$318$	2.31521i	0.129830i
$319$	4.64881	0.260283
$320$	0	0
$321$	28.0243	1.56417
$322$	8.47536i	0.472313i
$323$	− 7.43700i	− 0.413806i
$324$	−11.5580	−0.642112
$325$	0	0
$326$	17.1711	0.951018
$327$	9.25905i	0.512027i
$328$	9.98008i	0.551058i
$329$	11.1935	0.617116
$330$	0	0
$331$	22.6558	1.24527	0.622637	−	0.782510i	$-0.286061\pi$
$331$	22.6558	1.24527	0.622637	+	0.782510i	$0.286061\pi$
$332$	12.2646i	0.673106i
$333$	25.0500i	1.37273i
$334$	−6.85334	−0.374998
$335$	0	0
$336$	4.69998	0.256405
$337$	25.4057i	1.38393i	0.721929	+	0.691967i	$0.243256\pi$
$337$	25.4057i	1.38393i	−0.721929	+	0.691967i	$0.756744\pi$
$338$	− 14.5417i	− 0.790966i
$339$	18.0752	0.981713
$340$	0	0
$341$	−38.4520	−2.08229
$342$	3.45423i	0.186783i
$343$	13.9518i	0.753328i
$344$	24.9177	1.34347
$345$	0	0
$346$	6.08337	0.327044
$347$	− 21.8396i	− 1.17241i	−0.810162	−	0.586206i	$-0.800621\pi$
$347$	− 21.8396i	− 1.17241i	0.810162	−	0.586206i	$-0.199379\pi$
$348$	− 3.40365i	− 0.182455i
$349$	−20.3895	−1.09143	−0.545713	−	0.837972i	$-0.683741\pi$
$349$	−20.3895	−1.09143	−0.545713	+	0.837972i	$0.683741\pi$
$350$	0	0
$351$	−2.45452	−0.131012
$352$	27.1445i	1.44681i
$353$	− 27.5752i	− 1.46768i	−0.679322	−	0.733841i	$-0.737726\pi$
$353$	− 27.5752i	− 1.46768i	0.679322	−	0.733841i	$-0.262274\pi$
$354$	12.7450	0.677388
$355$	0	0
$356$	12.1928	0.646215
$357$	41.5200i	2.19747i
$358$	13.9009i	0.734684i
$359$	8.60568	0.454190	0.227095	−	0.973873i	$-0.427077\pi$
$359$	8.60568	0.454190	0.227095	+	0.973873i	$0.427077\pi$
$360$	0	0
$361$	−17.1244	−0.901283
$362$	1.34271i	0.0705713i
$363$	− 26.3721i	− 1.38418i
$364$	−23.5779	−1.23582
$365$	0	0
$366$	3.61795	0.189113
$367$	15.6980i	0.819431i	0.912213	+	0.409716i	$0.134372\pi$
$367$	15.6980i	0.819431i	−0.912213	+	0.409716i	$0.865628\pi$
$368$	− 2.13269i	− 0.111174i
$369$	−11.8490	−0.616834
$370$	0	0
$371$	−3.60953	−0.187398
$372$	28.1528i	1.45966i
$373$	− 11.4031i	− 0.590433i	−0.955430	−	0.295216i	$-0.904608\pi$
$373$	− 11.4031i	− 0.590433i	0.955430	−	0.295216i	$-0.0953917\pi$
$374$	−20.0446	−1.03648
$375$	0	0
$376$	9.73425	0.502006
$377$	5.59590i	0.288204i
$378$	− 1.07149i	− 0.0551115i
$379$	−24.6645	−1.26693	−0.633464	−	0.773772i	$-0.718368\pi$
$379$	−24.6645	−1.26693	−0.633464	+	0.773772i	$0.718368\pi$
$380$	0	0
$381$	−27.3643	−1.40191
$382$	16.2688i	0.832384i
$383$	− 4.26414i	− 0.217888i	−0.994048	−	0.108944i	$-0.965253\pi$
$383$	− 4.26414i	− 0.217888i	0.994048	−	0.108944i	$-0.0347469\pi$
$384$	22.3000	1.13799
$385$	0	0
$386$	5.42057	0.275900
$387$	29.5839i	1.50383i
$388$	11.4941i	0.583525i
$389$	−7.53209	−0.381892	−0.190946	−	0.981601i	$-0.561156\pi$
$389$	−7.53209	−0.381892	−0.190946	+	0.981601i	$0.561156\pi$
$390$	0	0
$391$	18.8404	0.952800
$392$	− 6.59527i	− 0.333111i
$393$	− 16.6490i	− 0.839830i
$394$	5.13560	0.258728
$395$	0	0
$396$	−20.2238	−1.01628
$397$	16.6272i	0.834497i	0.908792	+	0.417248i	$0.137005\pi$
$397$	16.6272i	0.834497i	−0.908792	+	0.417248i	$0.862995\pi$
$398$	15.1557i	0.759689i
$399$	−10.4714	−0.524228
$400$	0	0
$401$	12.1008	0.604283	0.302141	−	0.953263i	$-0.402299\pi$
$401$	12.1008	0.604283	0.302141	+	0.953263i	$0.402299\pi$
$402$	− 18.6374i	− 0.929550i
$403$	− 46.2858i	− 2.30566i
$404$	−1.15221	−0.0573246
$405$	0	0
$406$	−2.44283	−0.121235
$407$	− 36.6608i	− 1.81721i
$408$	36.1074i	1.78758i
$409$	−14.5486	−0.719382	−0.359691	−	0.933071i	$-0.617118\pi$
$409$	−14.5486	−0.719382	−0.359691	+	0.933071i	$0.617118\pi$
$410$	0	0
$411$	−55.2671	−2.72613
$412$	− 13.5181i	− 0.665988i
$413$	19.8701i	0.977744i
$414$	−8.75072	−0.430074
$415$	0	0
$416$	−32.6746	−1.60200
$417$	− 22.6905i	− 1.11116i
$418$	− 5.05528i	− 0.247262i
$419$	26.8362	1.31103	0.655517	−	0.755181i	$-0.272451\pi$
$419$	26.8362	1.31103	0.655517	+	0.755181i	$0.272451\pi$
$420$	0	0
$421$	33.1582	1.61603	0.808016	−	0.589161i	$-0.200541\pi$
$421$	33.1582	1.61603	0.808016	+	0.589161i	$0.200541\pi$
$422$	8.65975i	0.421550i
$423$	11.5571i	0.561927i
$424$	−3.13898	−0.152442
$425$	0	0
$426$	−27.9355	−1.35348
$427$	5.64057i	0.272966i
$428$	15.4432i	0.746476i
$429$	64.6521	3.12143
$430$	0	0
$431$	18.6127	0.896540	0.448270	−	0.893898i	$-0.352040\pi$
$431$	18.6127	0.896540	0.448270	+	0.893898i	$0.352040\pi$
$432$	0.269624i	0.0129723i
$433$	− 26.6321i	− 1.27986i	−0.768434	−	0.639929i	$-0.778964\pi$
$433$	− 26.6321i	− 1.27986i	0.768434	−	0.639929i	$-0.221036\pi$
$434$	20.2055	0.969896
$435$	0	0
$436$	−5.10233	−0.244358
$437$	4.75159i	0.227299i
$438$	− 19.3416i	− 0.924177i
$439$	−34.9051	−1.66593	−0.832965	−	0.553325i	$-0.813359\pi$
$439$	−34.9051	−1.66593	−0.832965	+	0.553325i	$0.813359\pi$
$440$	0	0
$441$	7.83033	0.372873
$442$	− 24.1282i	− 1.14766i
$443$	− 1.19808i	− 0.0569227i	−0.999595	−	0.0284613i	$-0.990939\pi$
$443$	− 1.19808i	− 0.0569227i	0.999595	−	0.0284613i	$-0.00906075\pi$
$444$	−26.8414	−1.27384
$445$	0	0
$446$	12.9900	0.615093
$447$	18.3533i	0.868082i
$448$	− 10.4814i	− 0.495201i
$449$	38.4108	1.81272	0.906359	−	0.422509i	$-0.138850\pi$
$449$	38.4108	1.81272	0.906359	+	0.422509i	$0.138850\pi$
$450$	0	0
$451$	17.3411	0.816558
$452$	9.96062i	0.468508i
$453$	57.4416i	2.69884i
$454$	−7.75030	−0.363739
$455$	0	0
$456$	−9.10636	−0.426444
$457$	− 32.4656i	− 1.51868i	−0.650695	−	0.759339i	$-0.725522\pi$
$457$	− 32.4656i	− 1.51868i	0.650695	−	0.759339i	$-0.274478\pi$
$458$	− 0.866022i	− 0.0404665i
$459$	−2.38188	−0.111177
$460$	0	0
$461$	−11.1460	−0.519121	−0.259560	−	0.965727i	$-0.583578\pi$
$461$	−11.1460	−0.519121	−0.259560	+	0.965727i	$0.583578\pi$
$462$	28.2231i	1.31306i
$463$	− 21.4773i	− 0.998136i	−0.866563	−	0.499068i	$-0.833676\pi$
$463$	− 21.4773i	− 0.998136i	0.866563	−	0.499068i	$-0.166324\pi$
$464$	0.614699	0.0285367
$465$	0	0
$466$	6.07409	0.281377
$467$	− 2.30347i	− 0.106592i	−0.998579	−	0.0532961i	$-0.983027\pi$
$467$	− 2.30347i	− 0.106592i	0.998579	−	0.0532961i	$-0.0169727\pi$
$468$	− 24.3440i	− 1.12530i
$469$	29.0567	1.34172
$470$	0	0
$471$	46.6050	2.14744
$472$	17.2798i	0.795366i
$473$	− 43.2961i	− 1.99076i
$474$	13.0255	0.598283
$475$	0	0
$476$	−22.8802	−1.04871
$477$	− 3.72680i	− 0.170639i
$478$	− 6.05854i	− 0.277111i
$479$	26.9013	1.22915	0.614576	−	0.788858i	$-0.289327\pi$
$479$	26.9013	1.22915	0.614576	+	0.788858i	$0.289327\pi$
$480$	0	0
$481$	44.1297	2.01214
$482$	− 13.6322i	− 0.620928i
$483$	− 26.5277i	− 1.20705i
$484$	14.5327	0.660578
$485$	0	0
$486$	−17.6986	−0.802823
$487$	− 3.03012i	− 0.137308i	−0.997641	−	0.0686539i	$-0.978130\pi$
$487$	− 3.03012i	− 0.137308i	0.997641	−	0.0686539i	$-0.0218704\pi$
$488$	4.90525i	0.222050i
$489$	−53.7450	−2.43043
$490$	0	0
$491$	9.14435	0.412679	0.206339	−	0.978481i	$-0.433845\pi$
$491$	9.14435	0.412679	0.206339	+	0.978481i	$0.433845\pi$
$492$	− 12.6963i	− 0.572395i
$493$	5.43031i	0.244569i
$494$	6.08518	0.273785
$495$	0	0
$496$	−5.08440	−0.228297
$497$	− 43.5530i	− 1.95362i
$498$	17.6718i	0.791892i
$499$	−1.87658	−0.0840072	−0.0420036	−	0.999117i	$-0.513374\pi$
$499$	−1.87658	−0.0840072	−0.0420036	+	0.999117i	$0.513374\pi$
$500$	0	0
$501$	21.4508	0.958350
$502$	− 6.01090i	− 0.268279i
$503$	38.2346i	1.70480i	0.522894	+	0.852398i	$0.324852\pi$
$503$	38.2346i	1.70480i	−0.522894	+	0.852398i	$0.675148\pi$
$504$	26.1463	1.16465
$505$	0	0
$506$	12.8067	0.569328
$507$	45.5153i	2.02140i
$508$	− 15.0795i	− 0.669044i
$509$	−3.56666	−0.158089	−0.0790447	−	0.996871i	$-0.525187\pi$
$509$	−3.56666	−0.158089	−0.0790447	+	0.996871i	$0.525187\pi$
$510$	0	0
$511$	30.1546	1.33396
$512$	6.87846i	0.303988i
$513$	− 0.600716i	− 0.0265223i
$514$	−2.21239	−0.0975841
$515$	0	0
$516$	−31.6994	−1.39549
$517$	− 16.9139i	− 0.743873i
$518$	19.2643i	0.846425i
$519$	−19.0408	−0.835797
$520$	0	0
$521$	5.73623	0.251309	0.125654	−	0.992074i	$-0.459897\pi$
$521$	5.73623	0.251309	0.125654	+	0.992074i	$0.459897\pi$
$522$	− 2.52219i	− 0.110393i
$523$	21.1604i	0.925282i	0.886546	+	0.462641i	$0.153098\pi$
$523$	21.1604i	0.925282i	−0.886546	+	0.462641i	$0.846902\pi$
$524$	9.17466	0.400797
$525$	0	0
$526$	−9.30834	−0.405863
$527$	− 44.9161i	− 1.95658i
$528$	− 7.10191i	− 0.309071i
$529$	10.9626	0.476637
$530$	0	0
$531$	−20.5157	−0.890304
$532$	− 5.77044i	− 0.250180i
$533$	20.8739i	0.904150i
$534$	17.5683	0.760255
$535$	0	0
$536$	25.2688	1.09145
$537$	− 43.5094i	− 1.87757i
$538$	− 18.5464i	− 0.799594i
$539$	−11.4597	−0.493605
$540$	0	0
$541$	−30.7581	−1.32239	−0.661197	−	0.750212i	$-0.729951\pi$
$541$	−30.7581	−1.32239	−0.661197	+	0.750212i	$0.729951\pi$
$542$	− 15.2947i	− 0.656964i
$543$	− 4.20265i	− 0.180353i
$544$	−31.7077	−1.35946
$545$	0	0
$546$	−33.9730	−1.45391
$547$	14.4043i	0.615882i	0.951405	+	0.307941i	$0.0996400\pi$
$547$	14.4043i	0.615882i	−0.951405	+	0.307941i	$0.900360\pi$
$548$	− 30.4558i	− 1.30101i
$549$	−5.82383	−0.248555
$550$	0	0
$551$	−1.36954	−0.0583442
$552$	− 23.0694i	− 0.981900i
$553$	20.3075i	0.863564i
$554$	−10.8996	−0.463078
$555$	0	0
$556$	12.5039	0.530283
$557$	13.7463i	0.582450i	0.956655	+	0.291225i	$0.0940629\pi$
$557$	13.7463i	0.582450i	−0.956655	+	0.291225i	$0.905937\pi$
$558$	20.8620i	0.883158i
$559$	52.1168	2.20430
$560$	0	0
$561$	62.7389	2.64884
$562$	− 23.1488i	− 0.976471i
$563$	− 16.3491i	− 0.689034i	−0.938780	−	0.344517i	$-0.888043\pi$
$563$	− 16.3491i	− 0.689034i	0.938780	−	0.344517i	$-0.111957\pi$
$564$	−12.3836	−0.521443
$565$	0	0
$566$	9.90087	0.416164
$567$	− 25.9641i	− 1.09039i
$568$	− 37.8753i	− 1.58921i
$569$	−12.4255	−0.520904	−0.260452	−	0.965487i	$-0.583872\pi$
$569$	−12.4255	−0.520904	−0.260452	+	0.965487i	$0.583872\pi$
$570$	0	0
$571$	1.84163	0.0770698	0.0385349	−	0.999257i	$-0.487731\pi$
$571$	1.84163	0.0770698	0.0385349	+	0.999257i	$0.487731\pi$
$572$	35.6275i	1.48966i
$573$	− 50.9209i	− 2.12725i
$574$	−9.11227	−0.380339
$575$	0	0
$576$	10.8220	0.450915
$577$	19.7519i	0.822280i	0.911572	+	0.411140i	$0.134869\pi$
$577$	19.7519i	0.822280i	−0.911572	+	0.411140i	$0.865131\pi$
$578$	− 9.91589i	− 0.412447i
$579$	−16.9662	−0.705092
$580$	0	0
$581$	−27.5513	−1.14302
$582$	16.5616i	0.686502i
$583$	5.45419i	0.225889i
$584$	26.2235	1.08514
$585$	0	0
$586$	−5.53313	−0.228571
$587$	− 15.5449i	− 0.641606i	−0.947146	−	0.320803i	$-0.896047\pi$
$587$	− 15.5449i	− 0.641606i	0.947146	−	0.320803i	$-0.103953\pi$
$588$	8.39028i	0.346009i
$589$	11.3279	0.466760
$590$	0	0
$591$	−16.0743	−0.661208
$592$	− 4.84756i	− 0.199234i
$593$	0.959978i	0.0394216i	0.999806	+	0.0197108i	$0.00627454\pi$
$593$	0.959978i	0.0394216i	−0.999806	+	0.0197108i	$0.993725\pi$
$594$	−1.61908	−0.0664316
$595$	0	0
$596$	−10.1139	−0.414280
$597$	− 47.4371i	− 1.94147i
$598$	15.4158i	0.630399i
$599$	−30.3109	−1.23847	−0.619235	−	0.785206i	$-0.712557\pi$
$599$	−30.3109	−1.23847	−0.619235	+	0.785206i	$0.712557\pi$
$600$	0	0
$601$	−23.6651	−0.965321	−0.482660	−	0.875808i	$-0.660329\pi$
$601$	−23.6651	−0.965321	−0.482660	+	0.875808i	$0.660329\pi$
$602$	22.7510i	0.927261i
$603$	30.0008i	1.22173i
$604$	−31.6540	−1.28798
$605$	0	0
$606$	−1.66020	−0.0674409
$607$	− 6.31841i	− 0.256456i	−0.991745	−	0.128228i	$-0.959071\pi$
$607$	− 6.31841i	− 0.256456i	0.991745	−	0.128228i	$-0.0409290\pi$
$608$	− 7.99675i	− 0.324311i
$609$	7.64598	0.309831
$610$	0	0
$611$	20.3597	0.823667
$612$	− 23.6236i	− 0.954926i
$613$	19.0182i	0.768137i	0.923305	+	0.384069i	$0.125477\pi$
$613$	19.0182i	0.768137i	−0.923305	+	0.384069i	$0.874523\pi$
$614$	−7.43941	−0.300230
$615$	0	0
$616$	−38.2652	−1.54175
$617$	− 19.4944i	− 0.784815i	−0.919791	−	0.392408i	$-0.871642\pi$
$617$	− 19.4944i	− 0.784815i	0.919791	−	0.392408i	$-0.128358\pi$
$618$	− 19.4779i	− 0.783517i
$619$	−23.3633	−0.939049	−0.469524	−	0.882920i	$-0.655575\pi$
$619$	−23.3633	−0.939049	−0.469524	+	0.882920i	$0.655575\pi$
$620$	0	0
$621$	1.52181	0.0610683
$622$	20.6037i	0.826134i
$623$	27.3899i	1.09735i
$624$	8.54877	0.342225
$625$	0	0
$626$	−19.0432	−0.761119
$627$	15.8229i	0.631906i
$628$	25.6823i	1.02484i
$629$	42.8238	1.70750
$630$	0	0
$631$	14.1691	0.564061	0.282031	−	0.959405i	$-0.408992\pi$
$631$	14.1691	0.564061	0.282031	+	0.959405i	$0.408992\pi$
$632$	17.6602i	0.702484i
$633$	− 27.1048i	− 1.07732i
$634$	−16.3676	−0.650040
$635$	0	0
$636$	3.99331	0.158345
$637$	− 13.7944i	− 0.546553i
$638$	3.69124i	0.146137i
$639$	44.9680	1.77891
$640$	0	0
$641$	23.1269	0.913457	0.456729	−	0.889606i	$-0.349021\pi$
$641$	23.1269	0.913457	0.456729	+	0.889606i	$0.349021\pi$
$642$	22.2518i	0.878209i
$643$	− 28.8094i	− 1.13613i	−0.822984	−	0.568065i	$-0.807692\pi$
$643$	− 28.8094i	− 1.13613i	0.822984	−	0.568065i	$-0.192308\pi$
$644$	14.6184	0.576047
$645$	0	0
$646$	5.90511	0.232333
$647$	5.41457i	0.212869i	0.994320	+	0.106434i	$0.0339434\pi$
$647$	5.41457i	0.212869i	−0.994320	+	0.106434i	$0.966057\pi$
$648$	− 22.5793i	− 0.886999i
$649$	30.0248	1.17857
$650$	0	0
$651$	−63.2428	−2.47868
$652$	− 29.6170i	− 1.15989i
$653$	23.4806i	0.918866i	0.888213	+	0.459433i	$0.151947\pi$
$653$	23.4806i	0.918866i	−0.888213	+	0.459433i	$0.848053\pi$
$654$	−7.35185	−0.287480
$655$	0	0
$656$	2.29296	0.0895250
$657$	31.1343i	1.21466i
$658$	8.88781i	0.346483i
$659$	−42.6581	−1.66172	−0.830862	−	0.556478i	$-0.812152\pi$
$659$	−42.6581	−1.66172	−0.830862	+	0.556478i	$0.812152\pi$
$660$	0	0
$661$	−32.8525	−1.27781	−0.638907	−	0.769284i	$-0.720613\pi$
$661$	−32.8525	−1.27781	−0.638907	+	0.769284i	$0.720613\pi$
$662$	17.9891i	0.699166i
$663$	75.5206i	2.93298i
$664$	−23.9596	−0.929813
$665$	0	0
$666$	−19.8902	−0.770729
$667$	− 3.46949i	− 0.134339i
$668$	11.8208i	0.457359i
$669$	−40.6583	−1.57194
$670$	0	0
$671$	8.52320	0.329034
$672$	44.6451i	1.72222i
$673$	27.7081i	1.06807i	0.845463	+	0.534033i	$0.179324\pi$
$673$	27.7081i	1.06807i	−0.845463	+	0.534033i	$0.820676\pi$
$674$	−20.1725	−0.777017
$675$	0	0
$676$	−25.0818	−0.964686
$677$	− 42.4769i	− 1.63252i	−0.577686	−	0.816259i	$-0.696044\pi$
$677$	− 42.4769i	− 1.63252i	0.577686	−	0.816259i	$-0.303956\pi$
$678$	14.3521i	0.551188i
$679$	−25.8205	−0.990899
$680$	0	0
$681$	24.2582	0.929577
$682$	− 30.5316i	− 1.16912i
$683$	12.4049i	0.474661i	0.971429	+	0.237330i	$0.0762724\pi$
$683$	12.4049i	0.474661i	−0.971429	+	0.237330i	$0.923728\pi$
$684$	5.95792	0.227807
$685$	0	0
$686$	−11.0780	−0.422960
$687$	2.71063i	0.103417i
$688$	− 5.72493i	− 0.218261i
$689$	−6.56536	−0.250120
$690$	0	0
$691$	−20.8110	−0.791686	−0.395843	−	0.918318i	$-0.629548\pi$
$691$	−20.8110	−0.791686	−0.395843	+	0.918318i	$0.629548\pi$
$692$	− 10.4927i	− 0.398872i
$693$	− 45.4309i	− 1.72578i
$694$	17.3411	0.658257
$695$	0	0
$696$	6.64923	0.252038
$697$	20.2562i	0.767259i
$698$	− 16.1896i	− 0.612787i
$699$	−19.0117	−0.719090
$700$	0	0
$701$	−17.4913	−0.660636	−0.330318	−	0.943870i	$-0.607156\pi$
$701$	−17.4913	−0.660636	−0.330318	+	0.943870i	$0.607156\pi$
$702$	− 1.94893i	− 0.0735576i
$703$	10.8003i	0.407339i
$704$	−15.8380	−0.596917
$705$	0	0
$706$	21.8952	0.824038
$707$	− 2.58834i	− 0.0973444i
$708$	− 21.9828i	− 0.826162i
$709$	−24.7888	−0.930963	−0.465482	−	0.885057i	$-0.654119\pi$
$709$	−24.7888	−0.930963	−0.465482	+	0.885057i	$0.654119\pi$
$710$	0	0
$711$	−20.9673	−0.786335
$712$	23.8193i	0.892666i
$713$	28.6974i	1.07473i
$714$	−32.9677	−1.23378
$715$	0	0
$716$	23.9765	0.896043
$717$	18.9631i	0.708189i
$718$	6.83306i	0.255008i
$719$	34.6651	1.29279	0.646394	−	0.763004i	$-0.276276\pi$
$719$	34.6651	1.29279	0.646394	+	0.763004i	$0.276276\pi$
$720$	0	0
$721$	30.3671	1.13093
$722$	− 13.5971i	− 0.506030i
$723$	42.6683i	1.58685i
$724$	2.31593	0.0860709
$725$	0	0
$726$	20.9399	0.777153
$727$	2.12509i	0.0788152i	0.999223	+	0.0394076i	$0.0125471\pi$
$727$	2.12509i	0.0788152i	−0.999223	+	0.0394076i	$0.987453\pi$
$728$	− 46.0609i	− 1.70713i
$729$	30.0779	1.11400
$730$	0	0
$731$	50.5745	1.87057
$732$	− 6.24030i	− 0.230648i
$733$	28.8575i	1.06588i	0.846154	+	0.532939i	$0.178912\pi$
$733$	28.8575i	1.06588i	−0.846154	+	0.532939i	$0.821088\pi$
$734$	−12.4645	−0.460074
$735$	0	0
$736$	20.2584	0.746736
$737$	− 43.9062i	− 1.61731i
$738$	− 9.40831i	− 0.346325i
$739$	−0.767740	−0.0282418	−0.0141209	−	0.999900i	$-0.504495\pi$
$739$	−0.767740	−0.0282418	−0.0141209	+	0.999900i	$0.504495\pi$
$740$	0	0
$741$	−19.0465	−0.699689
$742$	− 2.86603i	− 0.105215i
$743$	− 41.0261i	− 1.50510i	−0.658535	−	0.752550i	$-0.728824\pi$
$743$	− 41.0261i	− 1.50510i	0.658535	−	0.752550i	$-0.271176\pi$
$744$	−54.9983	−2.01633
$745$	0	0
$746$	9.05430	0.331502
$747$	− 28.4464i	− 1.04080i
$748$	34.5732i	1.26412i
$749$	−34.6918	−1.26761
$750$	0	0
$751$	−30.2586	−1.10415	−0.552076	−	0.833794i	$-0.686164\pi$
$751$	−30.2586	−1.10415	−0.552076	+	0.833794i	$0.686164\pi$
$752$	− 2.23648i	− 0.0815560i
$753$	18.8140i	0.685618i
$754$	−4.44325	−0.161813
$755$	0	0
$756$	−1.84813	−0.0672157
$757$	− 20.0163i	− 0.727504i	−0.931496	−	0.363752i	$-0.881496\pi$
$757$	− 20.0163i	− 0.727504i	0.931496	−	0.363752i	$-0.118504\pi$
$758$	− 19.5840i	− 0.711324i
$759$	−40.0847	−1.45498
$760$	0	0
$761$	−19.2752	−0.698724	−0.349362	−	0.936988i	$-0.613602\pi$
$761$	−19.2752	−0.698724	−0.349362	+	0.936988i	$0.613602\pi$
$762$	− 21.7277i	− 0.787112i
$763$	− 11.4619i	− 0.414950i
$764$	28.0607	1.01520
$765$	0	0
$766$	3.38581	0.122334
$767$	36.1416i	1.30500i
$768$	34.6406i	1.24998i
$769$	−34.0946	−1.22948	−0.614741	−	0.788729i	$-0.710739\pi$
$769$	−34.0946	−1.22948	−0.614741	+	0.788729i	$0.710739\pi$
$770$	0	0
$771$	6.92471	0.249387
$772$	− 9.34948i	− 0.336495i
$773$	− 9.88199i	− 0.355430i	−0.984082	−	0.177715i	$-0.943129\pi$
$773$	− 9.88199i	− 0.355430i	0.984082	−	0.177715i	$-0.0568706\pi$
$774$	−23.4901	−0.844335
$775$	0	0
$776$	−22.4544	−0.806067
$777$	− 60.2968i	− 2.16313i
$778$	− 5.98061i	− 0.214415i
$779$	−5.10866	−0.183037
$780$	0	0
$781$	−65.8108	−2.35490
$782$	14.9596i	0.534955i
$783$	0.438627i	0.0156753i
$784$	−1.51529	−0.0541174
$785$	0	0
$786$	13.2196	0.471527
$787$	20.0540i	0.714849i	0.933942	+	0.357425i	$0.116345\pi$
$787$	20.0540i	0.714849i	−0.933942	+	0.357425i	$0.883655\pi$
$788$	− 8.85798i	− 0.315552i
$789$	29.1349	1.03723
$790$	0	0
$791$	−22.3756	−0.795586
$792$	− 39.5084i	− 1.40387i
$793$	10.2596i	0.364330i
$794$	−13.2023	−0.468533
$795$	0	0
$796$	26.1409	0.926539
$797$	− 17.5895i	− 0.623053i	−0.950237	−	0.311527i	$-0.899160\pi$
$797$	− 17.5895i	− 0.623053i	0.950237	−	0.311527i	$-0.100840\pi$
$798$	− 8.31451i	− 0.294331i
$799$	19.7573	0.698962
$800$	0	0
$801$	−28.2798	−0.999218
$802$	9.60822i	0.339278i
$803$	− 45.5651i	− 1.60796i
$804$	−32.1461	−1.13371
$805$	0	0
$806$	36.7518	1.29453
$807$	58.0499i	2.04345i
$808$	− 2.25091i	− 0.0791868i
$809$	43.2338	1.52002	0.760010	−	0.649911i	$-0.225194\pi$
$809$	43.2338	1.52002	0.760010	+	0.649911i	$0.225194\pi$
$810$	0	0
$811$	−31.0124	−1.08899	−0.544496	−	0.838763i	$-0.683279\pi$
$811$	−31.0124	−1.08899	−0.544496	+	0.838763i	$0.683279\pi$
$812$	4.21343i	0.147862i
$813$	47.8721i	1.67895i
$814$	29.1094	1.02028
$815$	0	0
$816$	8.29579	0.290411
$817$	12.7550i	0.446241i
$818$	− 11.5518i	− 0.403901i
$819$	54.6865	1.91090
$820$	0	0
$821$	−1.37077	−0.0478401	−0.0239201	−	0.999714i	$-0.507615\pi$
$821$	−1.37077	−0.0478401	−0.0239201	+	0.999714i	$0.507615\pi$
$822$	− 43.8831i	− 1.53060i
$823$	10.9713i	0.382436i	0.981548	+	0.191218i	$0.0612438\pi$
$823$	10.9713i	0.382436i	−0.981548	+	0.191218i	$0.938756\pi$
$824$	26.4084	0.919980
$825$	0	0
$826$	−15.7772	−0.548960
$827$	26.2960i	0.914403i	0.889363	+	0.457202i	$0.151148\pi$
$827$	26.2960i	0.914403i	−0.889363	+	0.457202i	$0.848852\pi$
$828$	15.0934i	0.524531i
$829$	−1.95965	−0.0680616	−0.0340308	−	0.999421i	$-0.510834\pi$
$829$	−1.95965	−0.0680616	−0.0340308	+	0.999421i	$0.510834\pi$
$830$	0	0
$831$	34.1153	1.18345
$832$	− 19.0646i	− 0.660947i
$833$	− 13.3862i	− 0.463804i
$834$	18.0166	0.623865
$835$	0	0
$836$	−8.71943	−0.301568
$837$	− 3.62805i	− 0.125404i
$838$	21.3084i	0.736087i
$839$	−53.2443	−1.83820	−0.919098	−	0.394028i	$-0.871081\pi$
$839$	−53.2443	−1.83820	−0.919098	+	0.394028i	$0.871081\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	26.3282i	0.907330i
$843$	72.4550i	2.49548i
$844$	14.9365	0.514135
$845$	0	0
$846$	−9.17657	−0.315497
$847$	32.6464i	1.12174i
$848$	0.721192i	0.0247658i
$849$	−30.9894	−1.06356
$850$	0	0
$851$	−27.3606	−0.937911
$852$	48.1837i	1.65075i
$853$	− 29.3456i	− 1.00478i	−0.864643	−	0.502388i	$-0.832455\pi$
$853$	− 29.3456i	− 1.00478i	0.864643	−	0.502388i	$-0.167545\pi$
$854$	−4.47872	−0.153258
$855$	0	0
$856$	−30.1693	−1.03116
$857$	− 6.64960i	− 0.227146i	−0.993530	−	0.113573i	$-0.963770\pi$
$857$	− 6.64960i	− 0.227146i	0.993530	−	0.113573i	$-0.0362296\pi$
$858$	51.3349i	1.75255i
$859$	6.67954	0.227903	0.113952	−	0.993486i	$-0.463649\pi$
$859$	6.67954	0.227903	0.113952	+	0.993486i	$0.463649\pi$
$860$	0	0
$861$	28.5211	0.971998
$862$	14.7788i	0.503367i
$863$	45.5888i	1.55186i	0.630818	+	0.775931i	$0.282720\pi$
$863$	45.5888i	1.55186i	−0.630818	+	0.775931i	$0.717280\pi$
$864$	−2.56116	−0.0871323
$865$	0	0
$866$	21.1464	0.718583
$867$	31.0365i	1.05405i
$868$	− 34.8508i	− 1.18291i
$869$	30.6857	1.04094
$870$	0	0
$871$	52.8512	1.79079
$872$	− 9.96772i	− 0.337550i
$873$	− 26.6594i	− 0.902282i
$874$	−3.77285	−0.127618
$875$	0	0
$876$	−33.3607	−1.12715
$877$	− 14.1694i	− 0.478466i	−0.970962	−	0.239233i	$-0.923104\pi$
$877$	− 14.1694i	− 0.478466i	0.970962	−	0.239233i	$-0.0768960\pi$
$878$	− 27.7153i	− 0.935346i
$879$	17.3185	0.584140
$880$	0	0
$881$	13.3321	0.449170	0.224585	−	0.974455i	$-0.427897\pi$
$881$	13.3321	0.449170	0.224585	+	0.974455i	$0.427897\pi$
$882$	6.21742i	0.209351i
$883$	16.3432i	0.549993i	0.961445	+	0.274997i	$0.0886767\pi$
$883$	16.3432i	0.549993i	−0.961445	+	0.274997i	$0.911323\pi$
$884$	−41.6167	−1.39972
$885$	0	0
$886$	0.951300	0.0319595
$887$	30.4667i	1.02297i	0.859292	+	0.511486i	$0.170905\pi$
$887$	30.4667i	1.02297i	−0.859292	+	0.511486i	$0.829095\pi$
$888$	− 52.4363i	− 1.75965i
$889$	33.8747	1.13612
$890$	0	0
$891$	−39.2330	−1.31436
$892$	− 22.4053i	− 0.750186i
$893$	4.98282i	0.166744i
$894$	−14.5729	−0.487389
$895$	0	0
$896$	−27.6055	−0.922236
$897$	− 48.2510i	− 1.61106i
$898$	30.4989i	1.01776i
$899$	−8.27137	−0.275866
$900$	0	0
$901$	−6.37108	−0.212251
$902$	13.7691i	0.458461i
$903$	− 71.2099i	− 2.36972i
$904$	−19.4587	−0.647186
$905$	0	0
$906$	−45.6097	−1.51528
$907$	− 5.02171i	− 0.166743i	−0.996519	−	0.0833716i	$-0.973431\pi$
$907$	− 5.02171i	− 0.166743i	0.996519	−	0.0833716i	$-0.0265688\pi$
$908$	13.3678i	0.443628i
$909$	2.67243	0.0886388
$910$	0	0
$911$	15.7871	0.523051	0.261526	−	0.965197i	$-0.415774\pi$
$911$	15.7871	0.523051	0.261526	+	0.965197i	$0.415774\pi$
$912$	2.09222i	0.0692802i
$913$	41.6314i	1.37780i
$914$	25.7783	0.852670
$915$	0	0
$916$	−1.49373	−0.0493542
$917$	20.6100i	0.680604i
$918$	− 1.89126i	− 0.0624208i
$919$	34.5950	1.14118	0.570592	−	0.821233i	$-0.306714\pi$
$919$	34.5950	1.14118	0.570592	+	0.821233i	$0.306714\pi$
$920$	0	0
$921$	23.2851	0.767272
$922$	− 8.85012i	− 0.291463i
$923$	− 79.2183i	− 2.60750i
$924$	48.6797	1.60145
$925$	0	0
$926$	17.0534	0.560409
$927$	31.3537i	1.02979i
$928$	5.83903i	0.191675i
$929$	45.6719	1.49845	0.749223	−	0.662318i	$-0.230427\pi$
$929$	45.6719	1.49845	0.749223	+	0.662318i	$0.230427\pi$
$930$	0	0
$931$	3.37602	0.110645
$932$	− 10.4767i	− 0.343175i
$933$	− 64.4891i	− 2.11128i
$934$	1.82900	0.0598467
$935$	0	0
$936$	47.5574	1.55446
$937$	12.5837i	0.411090i	0.978648	+	0.205545i	$0.0658968\pi$
$937$	12.5837i	0.411090i	−0.978648	+	0.205545i	$0.934103\pi$
$938$	23.0716i	0.753313i
$939$	59.6047	1.94512
$940$	0	0
$941$	−23.0083	−0.750049	−0.375024	−	0.927015i	$-0.622366\pi$
$941$	−23.0083	−0.750049	−0.375024	+	0.927015i	$0.622366\pi$
$942$	37.0052i	1.20569i
$943$	− 12.9419i	− 0.421448i
$944$	3.97009	0.129215
$945$	0	0
$946$	34.3779	1.11772
$947$	− 25.0167i	− 0.812932i	−0.913666	−	0.406466i	$-0.866761\pi$
$947$	− 25.0167i	− 0.812932i	0.913666	−	0.406466i	$-0.133239\pi$
$948$	− 22.4667i	− 0.729684i
$949$	54.8480	1.78044
$950$	0	0
$951$	51.2301	1.66125
$952$	− 44.6979i	− 1.44867i
$953$	12.0176i	0.389289i	0.980874	+	0.194645i	$0.0623553\pi$
$953$	12.0176i	0.389289i	−0.980874	+	0.194645i	$0.937645\pi$
$954$	2.95915	0.0958059
$955$	0	0
$956$	−10.4499	−0.337973
$957$	− 11.5535i	− 0.373471i
$958$	21.3601i	0.690114i
$959$	68.4161	2.20927
$960$	0	0
$961$	37.4156	1.20696
$962$	35.0398i	1.12973i
$963$	− 35.8189i	− 1.15425i
$964$	−23.5130	−0.757303
$965$	0	0
$966$	21.0634	0.677705
$967$	− 0.930539i	− 0.0299241i	−0.999888	−	0.0149621i	$-0.995237\pi$
$967$	− 0.930539i	− 0.0299241i	0.999888	−	0.0149621i	$-0.00476275\pi$
$968$	28.3906i	0.912507i
$969$	−18.4828	−0.593754
$970$	0	0
$971$	−1.92126	−0.0616563	−0.0308282	−	0.999525i	$-0.509814\pi$
$971$	−1.92126	−0.0616563	−0.0308282	+	0.999525i	$0.509814\pi$
$972$	30.5268i	0.979147i
$973$	28.0889i	0.900488i
$974$	2.40597	0.0770922
$975$	0	0
$976$	1.12700	0.0360743
$977$	− 22.8372i	− 0.730626i	−0.930885	−	0.365313i	$-0.880962\pi$
$977$	− 22.8372i	− 0.730626i	0.930885	−	0.365313i	$-0.119038\pi$
$978$	− 42.6745i	− 1.36458i
$979$	41.3876	1.32275
$980$	0	0
$981$	11.8343	0.377841
$982$	7.26077i	0.231701i
$983$	26.6592i	0.850297i	0.905124	+	0.425149i	$0.139778\pi$
$983$	26.6592i	0.850297i	−0.905124	+	0.425149i	$0.860222\pi$
$984$	24.8030	0.790692
$985$	0	0
$986$	−4.31176	−0.137314
$987$	− 27.8186i	− 0.885476i
$988$	− 10.4958i	− 0.333917i
$989$	−32.3127	−1.02748
$990$	0	0
$991$	−8.46554	−0.268917	−0.134458	−	0.990919i	$-0.542929\pi$
$991$	−8.46554	−0.268917	−0.134458	+	0.990919i	$0.542929\pi$
$992$	− 48.2968i	− 1.53342i
$993$	− 56.3054i	− 1.78680i
$994$	34.5818	1.09687
$995$	0	0
$996$	30.4806	0.965815
$997$	− 5.80125i	− 0.183728i	−0.995772	−	0.0918638i	$-0.970718\pi$
$997$	− 5.80125i	− 0.183728i	0.995772	−	0.0918638i	$-0.0292824\pi$
$998$	− 1.49004i	− 0.0471663i
$999$	3.45905	0.109439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.b.g.349.7		10
5.2	odd	4		725.2.a.i.1.2	✓	5
5.3	odd	4		725.2.a.j.1.4	yes	5
5.4	even	2	inner	725.2.b.g.349.4		10
15.2	even	4		6525.2.a.bo.1.4		5
15.8	even	4		6525.2.a.bp.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.i.1.2	✓	5	5.2	odd	4
725.2.a.j.1.4	yes	5	5.3	odd	4
725.2.b.g.349.4		10	5.4	even	2	inner
725.2.b.g.349.7		10	1.1	even	1	trivial
6525.2.a.bo.1.4		5	15.2	even	4
6525.2.a.bp.1.2		5	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.794018i q^{2} -2.48525i q^{3} +1.36954 q^{4} +1.97334 q^{6} +3.07654i q^{7} +2.67547i q^{8} -3.17649 q^{9} +4.64881 q^{11} -3.40365i q^{12} +5.59590i q^{13} -2.44283 q^{14} +0.614699 q^{16} +5.43031i q^{17} -2.52219i q^{18} -1.36954 q^{19} +7.64598 q^{21} +3.69124i q^{22} -3.46949i q^{23} +6.64923 q^{24} -4.44325 q^{26} +0.438627i q^{27} +4.21343i q^{28} +1.00000 q^{29} -8.27137 q^{31} +5.83903i q^{32} -11.5535i q^{33} -4.31176 q^{34} -4.35032 q^{36} -7.88607i q^{37} -1.08744i q^{38} +13.9072 q^{39} +3.73021 q^{41} +6.07105i q^{42} -9.31338i q^{43} +6.36671 q^{44} +2.75484 q^{46} -3.63833i q^{47} -1.52768i q^{48} -2.46509 q^{49} +13.4957 q^{51} +7.66379i q^{52} +1.17324i q^{53} -0.348278 q^{54} -8.23119 q^{56} +3.40365i q^{57} +0.794018i q^{58} +6.45859 q^{59} +1.83342 q^{61} -6.56762i q^{62} -9.77260i q^{63} -3.40689 q^{64} +9.17366 q^{66} -9.44462i q^{67} +7.43700i q^{68} -8.62257 q^{69} -14.1565 q^{71} -8.49861i q^{72} -9.80146i q^{73} +6.26168 q^{74} -1.87563 q^{76} +14.3022i q^{77} +11.0426i q^{78} +6.60077 q^{79} -8.43937 q^{81} +2.96186i q^{82} +8.95528i q^{83} +10.4714 q^{84} +7.39499 q^{86} -2.48525i q^{87} +12.4378i q^{88} +8.90284 q^{89} -17.2160 q^{91} -4.75159i q^{92} +20.5565i q^{93} +2.88890 q^{94} +14.5115 q^{96} +8.39270i q^{97} -1.95732i q^{98} -14.7669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{4} - 2 q^{6} - 14 q^{9} + 4 q^{11} + 26 q^{14} + 12 q^{16} + 8 q^{19} + 30 q^{21} + 38 q^{24} - 8 q^{26} + 10 q^{29} + 10 q^{31} + 18 q^{34} + 70 q^{36} - 8 q^{39} - 6 q^{41} + 38 q^{44} - 26 q^{49}+ \cdots + 24 q^{99}+O(q^{100}) \) 10 * q - 8 * q^4 - 2 * q^6 - 14 * q^9 + 4 * q^11 + 26 * q^14 + 12 * q^16 + 8 * q^19 + 30 * q^21 + 38 * q^24 - 8 * q^26 + 10 * q^29 + 10 * q^31 + 18 * q^34 + 70 * q^36 - 8 * q^39 - 6 * q^41 + 38 * q^44 - 26 * q^49 - 14 * q^51 + 116 * q^54 - 16 * q^56 + 30 * q^59 - 14 * q^61 + 18 * q^64 + 70 * q^66 - 36 * q^69 - 34 * q^71 - 24 * q^74 - 68 * q^76 + 4 * q^79 + 42 * q^81 - 76 * q^84 + 62 * q^86 - 28 * q^89 - 30 * q^91 - 54 * q^94 + 12 * q^96 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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