LMFDB - Embedded newform 725.2.a.i.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(1,725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("725.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-6,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.78915414654$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.294577.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - x^{2} + 7x + 3 $ x^5 - 7x^3 - x^2 + 7x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.45914$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.45914 q^{2} -3.33648 q^{3} +4.04739 q^{4} -8.20489 q^{6} -3.50903 q^{7} +5.03483 q^{8} +8.13211 q^{9} -3.17006 q^{11} -13.5040 q^{12} -4.61115 q^{13} -8.62920 q^{14} +4.28660 q^{16} +1.29916 q^{17} +19.9980 q^{18} -4.04739 q^{19} +11.7078 q^{21} -7.79563 q^{22} -2.57569 q^{23} -16.7986 q^{24} -11.3395 q^{26} -17.1232 q^{27} -14.2024 q^{28} -1.00000 q^{29} -0.423684 q^{31} +0.471703 q^{32} +10.5768 q^{33} +3.19482 q^{34} +32.9138 q^{36} +3.71028 q^{37} -9.95312 q^{38} +15.3850 q^{39} -10.3443 q^{41} +28.7912 q^{42} +4.57932 q^{43} -12.8305 q^{44} -6.33399 q^{46} +5.35453 q^{47} -14.3022 q^{48} +5.31326 q^{49} -4.33462 q^{51} -18.6631 q^{52} +0.0412739 q^{53} -42.1084 q^{54} -17.6674 q^{56} +13.5040 q^{57} -2.45914 q^{58} +2.86841 q^{59} -1.63042 q^{61} -1.04190 q^{62} -28.5358 q^{63} -7.41322 q^{64} +26.0100 q^{66} +0.464958 q^{67} +5.25821 q^{68} +8.59374 q^{69} -4.33807 q^{71} +40.9438 q^{72} +15.1207 q^{73} +9.12413 q^{74} -16.3814 q^{76} +11.1238 q^{77} +37.8340 q^{78} +10.2945 q^{79} +32.7348 q^{81} -25.4382 q^{82} -11.2606 q^{83} +47.3860 q^{84} +11.2612 q^{86} +3.33648 q^{87} -15.9607 q^{88} +16.6474 q^{89} +16.1806 q^{91} -10.4248 q^{92} +1.41361 q^{93} +13.1676 q^{94} -1.57383 q^{96} -3.96628 q^{97} +13.0661 q^{98} -25.7792 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 6 q^{3} + 4 q^{4} - q^{6} - 10 q^{7} + 3 q^{8} + 7 q^{9} + 2 q^{11} - 18 q^{12} - 8 q^{13} - 13 q^{14} + 6 q^{16} - 3 q^{17} + 16 q^{18} - 4 q^{19} + 15 q^{21} - 16 q^{22} - 3 q^{23} - 19 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) $ 5 * q - 6 * q^3 + 4 * q^4 - q^6 - 10 * q^7 + 3 * q^8 + 7 * q^9 + 2 * q^11 - 18 * q^12 - 8 * q^13 - 13 * q^14 + 6 * q^16 - 3 * q^17 + 16 * q^18 - 4 * q^19 + 15 * q^21 - 16 * q^22 - 3 * q^23 - 19 * q^24 - 4 * q^26 - 21 * q^27 - 10 * q^28 - 5 * q^29 + 5 * q^31 - 4 * q^32 - 3 * q^33 - 9 * q^34 + 35 * q^36 - 4 * q^37 - 3 * q^38 + 4 * q^39 - 3 * q^41 + 16 * q^42 - 12 * q^43 - 19 * q^44 + q^47 - 20 * q^48 + 13 * q^49 - 7 * q^51 - 26 * q^52 - 12 * q^53 - 58 * q^54 - 8 * q^56 + 18 * q^57 - 15 * q^59 - 7 * q^61 - 10 * q^62 - 33 * q^63 - 9 * q^64 + 35 * q^66 - 17 * q^67 + 43 * q^68 + 18 * q^69 - 17 * q^71 + 49 * q^72 - 10 * q^73 + 12 * q^74 - 34 * q^76 + 8 * q^77 + 39 * q^78 - 2 * q^79 + 21 * q^81 - 31 * q^82 - 19 * q^83 + 38 * q^84 + 31 * q^86 + 6 * q^87 + 3 * q^88 + 14 * q^89 - 15 * q^91 + q^92 + 15 * q^93 + 27 * q^94 + 6 * q^96 + 9 * q^97 + 51 * q^98 - 12 * q^99

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 $

$2$	2.45914	1.73888	0.869439	−	0.494040i	$-0.164481\pi$
$2$	2.45914	1.73888	0.869439	+	0.494040i	$0.164481\pi$
$3$	−3.33648	−1.92632	−0.963159	−	0.268932i	$-0.913329\pi$
$3$	−3.33648	−1.92632	−0.963159	+	0.268932i	$0.913329\pi$
$4$	4.04739	2.02370
$5$	0	0
$5$	0	0
$6$	−8.20489	−3.34963
$7$	−3.50903	−1.32629	−0.663143	−	0.748492i	$-0.730778\pi$
$7$	−3.50903	−1.32629	−0.663143	+	0.748492i	$0.730778\pi$
$8$	5.03483	1.78008
$9$	8.13211	2.71070
$10$	0	0
$11$	−3.17006	−0.955808	−0.477904	−	0.878412i	$-0.658603\pi$
$11$	−3.17006	−0.955808	−0.477904	+	0.878412i	$0.658603\pi$
$12$	−13.5040	−3.89828
$13$	−4.61115	−1.27890	−0.639451	−	0.768831i	$-0.720838\pi$
$13$	−4.61115	−1.27890	−0.639451	+	0.768831i	$0.720838\pi$
$14$	−8.62920	−2.30625
$15$	0	0
$16$	4.28660	1.07165
$17$	1.29916	0.315092	0.157546	−	0.987512i	$-0.449642\pi$
$17$	1.29916	0.315092	0.157546	+	0.987512i	$0.449642\pi$
$18$	19.9980	4.71358
$19$	−4.04739	−0.928536	−0.464268	−	0.885695i	$-0.653682\pi$
$19$	−4.04739	−0.928536	−0.464268	+	0.885695i	$0.653682\pi$
$20$	0	0
$21$	11.7078	2.55485
$22$	−7.79563	−1.66203
$23$	−2.57569	−0.537068	−0.268534	−	0.963270i	$-0.586539\pi$
$23$	−2.57569	−0.537068	−0.268534	+	0.963270i	$0.586539\pi$
$24$	−16.7986	−3.42901
$25$	0	0
$26$	−11.3395	−2.22386
$27$	−17.1232	−3.29536
$28$	−14.2024	−2.68400
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.423684	−0.0760959	−0.0380480	−	0.999276i	$-0.512114\pi$
$31$	−0.423684	−0.0760959	−0.0380480	+	0.999276i	$0.512114\pi$
$32$	0.471703	0.0833861
$33$	10.5768	1.84119
$34$	3.19482	0.547907
$35$	0	0
$36$	32.9138	5.48564
$37$	3.71028	0.609967	0.304984	−	0.952358i	$-0.401349\pi$
$37$	3.71028	0.609967	0.304984	+	0.952358i	$0.401349\pi$
$38$	−9.95312	−1.61461
$39$	15.3850	2.46357
$40$	0	0
$41$	−10.3443	−1.61551	−0.807757	−	0.589516i	$-0.799319\pi$
$41$	−10.3443	−1.61551	−0.807757	+	0.589516i	$0.799319\pi$
$42$	28.7912	4.44257
$43$	4.57932	0.698340	0.349170	−	0.937059i	$-0.386464\pi$
$43$	4.57932	0.698340	0.349170	+	0.937059i	$0.386464\pi$
$44$	−12.8305	−1.93426
$45$	0	0
$46$	−6.33399	−0.933896
$47$	5.35453	0.781039	0.390519	−	0.920595i	$-0.372295\pi$
$47$	5.35453	0.781039	0.390519	+	0.920595i	$0.372295\pi$
$48$	−14.3022	−2.06434
$49$	5.31326	0.759037
$50$	0	0
$51$	−4.33462	−0.606968
$52$	−18.6631	−2.58811
$53$	0.0412739	0.00566941	0.00283471	−	0.999996i	$-0.499098\pi$
$53$	0.0412739	0.00566941	0.00283471	+	0.999996i	$0.499098\pi$
$54$	−42.1084	−5.73022
$55$	0	0
$56$	−17.6674	−2.36090
$57$	13.5040	1.78865
$58$	−2.45914	−0.322902
$59$	2.86841	0.373435	0.186718	−	0.982414i	$-0.440215\pi$
$59$	2.86841	0.373435	0.186718	+	0.982414i	$0.440215\pi$
$60$	0	0
$61$	−1.63042	−0.208754	−0.104377	−	0.994538i	$-0.533285\pi$
$61$	−1.63042	−0.208754	−0.104377	+	0.994538i	$0.533285\pi$
$62$	−1.04190	−0.132321
$63$	−28.5358	−3.59517
$64$	−7.41322	−0.926652
$65$	0	0
$66$	26.0100	3.20160
$67$	0.464958	0.0568037	0.0284018	−	0.999597i	$-0.490958\pi$
$67$	0.464958	0.0568037	0.0284018	+	0.999597i	$0.490958\pi$
$68$	5.25821	0.637651
$69$	8.59374	1.03456
$70$	0	0
$71$	−4.33807	−0.514834	−0.257417	−	0.966300i	$-0.582871\pi$
$71$	−4.33807	−0.514834	−0.257417	+	0.966300i	$0.582871\pi$
$72$	40.9438	4.82527
$73$	15.1207	1.76974	0.884872	−	0.465835i	$-0.154246\pi$
$73$	15.1207	1.76974	0.884872	+	0.465835i	$0.154246\pi$
$74$	9.12413	1.06066
$75$	0	0
$76$	−16.3814	−1.87907
$77$	11.1238	1.26768
$78$	37.8340	4.28385
$79$	10.2945	1.15822	0.579108	−	0.815251i	$-0.303401\pi$
$79$	10.2945	1.15822	0.579108	+	0.815251i	$0.303401\pi$
$80$	0	0
$81$	32.7348	3.63721
$82$	−25.4382	−2.80918
$83$	−11.2606	−1.23601	−0.618007	−	0.786173i	$-0.712059\pi$
$83$	−11.2606	−1.23601	−0.618007	+	0.786173i	$0.712059\pi$
$84$	47.3860	5.17024
$85$	0	0
$86$	11.2612	1.21433
$87$	3.33648	0.357708
$88$	−15.9607	−1.70142
$89$	16.6474	1.76463	0.882313	−	0.470663i	$-0.155985\pi$
$89$	16.6474	1.76463	0.882313	+	0.470663i	$0.155985\pi$
$90$	0	0
$91$	16.1806	1.69619
$92$	−10.4248	−1.08686
$93$	1.41361	0.146585
$94$	13.1676	1.35813
$95$	0	0
$96$	−1.57383	−0.160628
$97$	−3.96628	−0.402714	−0.201357	−	0.979518i	$-0.564535\pi$
$97$	−3.96628	−0.402714	−0.201357	+	0.979518i	$0.564535\pi$
$98$	13.0661	1.31987
$99$	−25.7792	−2.59091
$100$	0	0
$101$	−2.82323	−0.280922	−0.140461	−	0.990086i	$-0.544858\pi$
$101$	−2.82323	−0.280922	−0.140461	+	0.990086i	$0.544858\pi$
$102$	−10.6595	−1.05544
$103$	−7.04988	−0.694645	−0.347323	−	0.937746i	$-0.612909\pi$
$103$	−7.04988	−0.694645	−0.347323	+	0.937746i	$0.612909\pi$
$104$	−23.2164	−2.27655
$105$	0	0
$106$	0.101499	0.00985841
$107$	3.25962	0.315120	0.157560	−	0.987509i	$-0.449637\pi$
$107$	3.25962	0.315120	0.157560	+	0.987509i	$0.449637\pi$
$108$	−69.3042	−6.66880
$109$	−0.560643	−0.0536999	−0.0268499	−	0.999639i	$-0.508548\pi$
$109$	−0.560643	−0.0536999	−0.0268499	+	0.999639i	$0.508548\pi$
$110$	0	0
$111$	−12.3793	−1.17499
$112$	−15.0418	−1.42132
$113$	−13.3505	−1.25591	−0.627953	−	0.778252i	$-0.716107\pi$
$113$	−13.3505	−1.25591	−0.627953	+	0.778252i	$0.716107\pi$
$114$	33.2084	3.11025
$115$	0	0
$116$	−4.04739	−0.375791
$117$	−37.4984	−3.46673
$118$	7.05383	0.649358
$119$	−4.55878	−0.417903
$120$	0	0
$121$	−0.950746	−0.0864314
$122$	−4.00944	−0.362998
$123$	34.5137	3.11199
$124$	−1.71482	−0.153995
$125$	0	0
$126$	−70.1736	−6.25156
$127$	16.3437	1.45027	0.725135	−	0.688607i	$-0.241777\pi$
$127$	16.3437	1.45027	0.725135	+	0.688607i	$0.241777\pi$
$128$	−19.1736	−1.69472
$129$	−15.2788	−1.34522
$130$	0	0
$131$	7.10277	0.620571	0.310286	−	0.950643i	$-0.399575\pi$
$131$	7.10277	0.620571	0.310286	+	0.950643i	$0.399575\pi$
$132$	42.8086	3.72601
$133$	14.2024	1.23150
$134$	1.14340	0.0987746
$135$	0	0
$136$	6.54105	0.560890
$137$	−0.338571	−0.0289261	−0.0144630	−	0.999895i	$-0.504604\pi$
$137$	−0.338571	−0.0289261	−0.0144630	+	0.999895i	$0.504604\pi$
$138$	21.1332	1.79898
$139$	−13.1480	−1.11520	−0.557599	−	0.830110i	$-0.688277\pi$
$139$	−13.1480	−1.11520	−0.557599	+	0.830110i	$0.688277\pi$
$140$	0	0
$141$	−17.8653	−1.50453
$142$	−10.6679	−0.895234
$143$	14.6176	1.22239
$144$	34.8591	2.90492
$145$	0	0
$146$	37.1840	3.07737
$147$	−17.7276	−1.46215
$148$	15.0170	1.23439
$149$	−14.2553	−1.16784	−0.583919	−	0.811812i	$-0.698482\pi$
$149$	−14.2553	−1.16784	−0.583919	+	0.811812i	$0.698482\pi$
$150$	0	0
$151$	6.02626	0.490410	0.245205	−	0.969471i	$-0.421145\pi$
$151$	6.02626	0.490410	0.245205	+	0.969471i	$0.421145\pi$
$152$	−20.3779	−1.65287
$153$	10.5649	0.854122
$154$	27.3550	2.20433
$155$	0	0
$156$	62.2692	4.98553
$157$	−22.7417	−1.81499	−0.907494	−	0.420065i	$-0.862007\pi$
$157$	−22.7417	−1.81499	−0.907494	+	0.420065i	$0.862007\pi$
$158$	25.3156	2.01400
$159$	−0.137710	−0.0109211
$160$	0	0
$161$	9.03816	0.712307
$162$	80.4997	6.32466
$163$	−24.5842	−1.92558	−0.962791	−	0.270247i	$-0.912895\pi$
$163$	−24.5842	−1.92558	−0.962791	+	0.270247i	$0.912895\pi$
$164$	−41.8676	−3.26931
$165$	0	0
$166$	−27.6915	−2.14928
$167$	12.1715	0.941860	0.470930	−	0.882170i	$-0.343918\pi$
$167$	12.1715	0.941860	0.470930	+	0.882170i	$0.343918\pi$
$168$	58.9468	4.54785
$169$	8.26271	0.635593
$170$	0	0
$171$	−32.9138	−2.51698
$172$	18.5343	1.41323
$173$	−10.3682	−0.788280	−0.394140	−	0.919050i	$-0.628957\pi$
$173$	−10.3682	−0.788280	−0.394140	+	0.919050i	$0.628957\pi$
$174$	8.20489	0.622011
$175$	0	0
$176$	−13.5888	−1.02429
$177$	−9.57039	−0.719355
$178$	40.9385	3.06847
$179$	5.56095	0.415645	0.207822	−	0.978167i	$-0.433362\pi$
$179$	5.56095	0.415645	0.207822	+	0.978167i	$0.433362\pi$
$180$	0	0
$181$	−18.2192	−1.35422	−0.677111	−	0.735881i	$-0.736768\pi$
$181$	−18.2192	−1.35422	−0.677111	+	0.735881i	$0.736768\pi$
$182$	39.7905	2.94947
$183$	5.43987	0.402127
$184$	−12.9682	−0.956026
$185$	0	0
$186$	3.47628	0.254893
$187$	−4.11841	−0.301168
$188$	21.6719	1.58059
$189$	60.0857	4.37059
$190$	0	0
$191$	−9.51629	−0.688574	−0.344287	−	0.938864i	$-0.611879\pi$
$191$	−9.51629	−0.688574	−0.344287	+	0.938864i	$0.611879\pi$
$192$	24.7341	1.78503
$193$	−7.95873	−0.572882	−0.286441	−	0.958098i	$-0.592472\pi$
$193$	−7.95873	−0.572882	−0.286441	+	0.958098i	$0.592472\pi$
$194$	−9.75365	−0.700271
$195$	0	0
$196$	21.5048	1.53606
$197$	18.3498	1.30737	0.653686	−	0.756766i	$-0.273222\pi$
$197$	18.3498	1.30737	0.653686	+	0.756766i	$0.273222\pi$
$198$	−63.3949	−4.50528
$199$	−18.8066	−1.33317	−0.666583	−	0.745431i	$-0.732244\pi$
$199$	−18.8066	−1.33317	−0.666583	+	0.745431i	$0.732244\pi$
$200$	0	0
$201$	−1.55132	−0.109422
$202$	−6.94273	−0.488489
$203$	3.50903	0.246285
$204$	−17.5439	−1.22832
$205$	0	0
$206$	−17.3367	−1.20790
$207$	−20.9458	−1.45583
$208$	−19.7662	−1.37054
$209$	12.8305	0.887501
$210$	0	0
$211$	4.73351	0.325868	0.162934	−	0.986637i	$-0.447904\pi$
$211$	4.73351	0.325868	0.162934	+	0.986637i	$0.447904\pi$
$212$	0.167052	0.0114732
$213$	14.4739	0.991734
$214$	8.01588	0.547955
$215$	0	0
$216$	−86.2124	−5.86601
$217$	1.48672	0.100925
$218$	−1.37870	−0.0933775
$219$	−50.4499	−3.40909
$220$	0	0
$221$	−5.99062	−0.402973
$222$	−30.4425	−2.04317
$223$	7.31989	0.490176	0.245088	−	0.969501i	$-0.421183\pi$
$223$	7.31989	0.490176	0.245088	+	0.969501i	$0.421183\pi$
$224$	−1.65522	−0.110594
$225$	0	0
$226$	−32.8307	−2.18387
$227$	15.8457	1.05172	0.525858	−	0.850572i	$-0.323744\pi$
$227$	15.8457	1.05172	0.525858	+	0.850572i	$0.323744\pi$
$228$	54.6562	3.61969
$229$	16.0440	1.06021	0.530107	−	0.847931i	$-0.322152\pi$
$229$	16.0440	1.06021	0.530107	+	0.847931i	$0.322152\pi$
$230$	0	0
$231$	−37.1144	−2.44195
$232$	−5.03483	−0.330553
$233$	−18.9965	−1.24450	−0.622251	−	0.782818i	$-0.713782\pi$
$233$	−18.9965	−1.24450	−0.622251	+	0.782818i	$0.713782\pi$
$234$	−92.2139	−6.02821
$235$	0	0
$236$	11.6096	0.755719
$237$	−34.3473	−2.23109
$238$	−11.2107	−0.726682
$239$	−20.1139	−1.30106	−0.650531	−	0.759480i	$-0.725454\pi$
$239$	−20.1139	−1.30106	−0.650531	+	0.759480i	$0.725454\pi$
$240$	0	0
$241$	−16.6785	−1.07435	−0.537177	−	0.843470i	$-0.680509\pi$
$241$	−16.6785	−1.07435	−0.537177	+	0.843470i	$0.680509\pi$
$242$	−2.33802	−0.150294
$243$	−57.8497	−3.71106
$244$	−6.59896	−0.422455
$245$	0	0
$246$	84.8742	5.41138
$247$	18.6631	1.18751
$248$	−2.13318	−0.135457
$249$	37.5708	2.38095
$250$	0	0
$251$	5.87363	0.370740	0.185370	−	0.982669i	$-0.440652\pi$
$251$	5.87363	0.370740	0.185370	+	0.982669i	$0.440652\pi$
$252$	−115.495	−7.27553
$253$	8.16508	0.513334
$254$	40.1915	2.52184
$255$	0	0
$256$	−32.3242	−2.02026
$257$	13.5371	0.844421	0.422210	−	0.906498i	$-0.361254\pi$
$257$	13.5371	0.844421	0.422210	+	0.906498i	$0.361254\pi$
$258$	−37.5728	−2.33918
$259$	−13.0195	−0.808991
$260$	0	0
$261$	−8.13211	−0.503365
$262$	17.4667	1.07910
$263$	5.19055	0.320063	0.160031	−	0.987112i	$-0.448840\pi$
$263$	5.19055	0.320063	0.160031	+	0.987112i	$0.448840\pi$
$264$	53.2526	3.27747
$265$	0	0
$266$	34.9258	2.14144
$267$	−55.5439	−3.39923
$268$	1.88187	0.114953
$269$	−14.6610	−0.893894	−0.446947	−	0.894560i	$-0.647489\pi$
$269$	−14.6610	−0.893894	−0.446947	+	0.894560i	$0.647489\pi$
$270$	0	0
$271$	11.6971	0.710546	0.355273	−	0.934763i	$-0.384388\pi$
$271$	11.6971	0.710546	0.355273	+	0.934763i	$0.384388\pi$
$272$	5.56898	0.337669
$273$	−53.9864	−3.26741
$274$	−0.832596	−0.0502989
$275$	0	0
$276$	34.7822	2.09364
$277$	12.9375	0.777341	0.388670	−	0.921377i	$-0.372934\pi$
$277$	12.9375	0.777341	0.388670	+	0.921377i	$0.372934\pi$
$278$	−32.3328	−1.93919
$279$	−3.44544	−0.206273
$280$	0	0
$281$	−7.94275	−0.473825	−0.236912	−	0.971531i	$-0.576135\pi$
$281$	−7.94275	−0.473825	−0.236912	+	0.971531i	$0.576135\pi$
$282$	−43.9333	−2.61619
$283$	−28.3465	−1.68502	−0.842512	−	0.538677i	$-0.818924\pi$
$283$	−28.3465	−1.68502	−0.842512	+	0.538677i	$0.818924\pi$
$284$	−17.5579	−1.04187
$285$	0	0
$286$	35.9468	2.12558
$287$	36.2985	2.14264
$288$	3.83594	0.226035
$289$	−15.3122	−0.900717
$290$	0	0
$291$	13.2334	0.775756
$292$	61.1994	3.58142
$293$	24.8491	1.45170	0.725849	−	0.687854i	$-0.241447\pi$
$293$	24.8491	1.45170	0.725849	+	0.687854i	$0.241447\pi$
$294$	−43.5947	−2.54249
$295$	0	0
$296$	18.6807	1.08579
$297$	54.2814	3.14973
$298$	−35.0558	−2.03073
$299$	11.8769	0.686858
$300$	0	0
$301$	−16.0689	−0.926199
$302$	14.8194	0.852763
$303$	9.41966	0.541145
$304$	−17.3496	−0.995065
$305$	0	0
$306$	25.9806	1.48521
$307$	−9.99873	−0.570658	−0.285329	−	0.958430i	$-0.592103\pi$
$307$	−9.99873	−0.570658	−0.285329	+	0.958430i	$0.592103\pi$
$308$	45.0224	2.56539
$309$	23.5218	1.33811
$310$	0	0
$311$	1.63825	0.0928968	0.0464484	−	0.998921i	$-0.485210\pi$
$311$	1.63825	0.0928968	0.0464484	+	0.998921i	$0.485210\pi$
$312$	77.4610	4.38537
$313$	−1.93025	−0.109104	−0.0545522	−	0.998511i	$-0.517373\pi$
$313$	−1.93025	−0.109104	−0.0545522	+	0.998511i	$0.517373\pi$
$314$	−55.9252	−3.15604
$315$	0	0
$316$	41.6657	2.34388
$317$	−29.1121	−1.63510	−0.817549	−	0.575859i	$-0.804668\pi$
$317$	−29.1121	−1.63510	−0.817549	+	0.575859i	$0.804668\pi$
$318$	−0.338648	−0.0189904
$319$	3.17006	0.177489
$320$	0	0
$321$	−10.8757	−0.607021
$322$	22.2261	1.23861
$323$	−5.25821	−0.292574
$324$	132.491	7.36060
$325$	0	0
$326$	−60.4561	−3.34835
$327$	1.87058	0.103443
$328$	−52.0820	−2.87575
$329$	−18.7892	−1.03588
$330$	0	0
$331$	−3.88586	−0.213586	−0.106793	−	0.994281i	$-0.534058\pi$
$331$	−3.88586	−0.213586	−0.106793	+	0.994281i	$0.534058\pi$
$332$	−45.5761	−2.50132
$333$	30.1724	1.65344
$334$	29.9315	1.63778
$335$	0	0
$336$	50.1867	2.73791
$337$	−4.54398	−0.247527	−0.123763	−	0.992312i	$-0.539496\pi$
$337$	−4.54398	−0.247527	−0.123763	+	0.992312i	$0.539496\pi$
$338$	20.3192	1.10522
$339$	44.5435	2.41927
$340$	0	0
$341$	1.34310	0.0727331
$342$	−80.9399	−4.37673
$343$	5.91882	0.319586
$344$	23.0561	1.24310
$345$	0	0
$346$	−25.4969	−1.37072
$347$	−13.3348	−0.715848	−0.357924	−	0.933751i	$-0.616515\pi$
$347$	−13.3348	−0.715848	−0.357924	+	0.933751i	$0.616515\pi$
$348$	13.5040	0.723893
$349$	32.8243	1.75704	0.878522	−	0.477703i	$-0.158530\pi$
$349$	32.8243	1.75704	0.878522	+	0.477703i	$0.158530\pi$
$350$	0	0
$351$	78.9576	4.21444
$352$	−1.49532	−0.0797011
$353$	18.4743	0.983286	0.491643	−	0.870797i	$-0.336397\pi$
$353$	18.4743	0.983286	0.491643	+	0.870797i	$0.336397\pi$
$354$	−23.5350	−1.25087
$355$	0	0
$356$	67.3788	3.57107
$357$	15.2103	0.805014
$358$	13.6752	0.722756
$359$	−25.8079	−1.36209	−0.681043	−	0.732243i	$-0.738474\pi$
$359$	−25.8079	−1.36209	−0.681043	+	0.732243i	$0.738474\pi$
$360$	0	0
$361$	−2.61861	−0.137822
$362$	−44.8036	−2.35483
$363$	3.17215	0.166494
$364$	65.4894	3.43258
$365$	0	0
$366$	13.3774	0.699250
$367$	13.7885	0.719753	0.359876	−	0.933000i	$-0.382819\pi$
$367$	13.7885	0.719753	0.359876	+	0.933000i	$0.382819\pi$
$368$	−11.0410	−0.575549
$369$	−84.1213	−4.37918
$370$	0	0
$371$	−0.144831	−0.00751927
$372$	5.72145	0.296643
$373$	11.7048	0.606053	0.303026	−	0.952982i	$-0.402003\pi$
$373$	11.7048	0.606053	0.303026	+	0.952982i	$0.402003\pi$
$374$	−10.1278	−0.523694
$375$	0	0
$376$	26.9592	1.39031
$377$	4.61115	0.237486
$378$	147.759	7.59992
$379$	−10.4270	−0.535598	−0.267799	−	0.963475i	$-0.586296\pi$
$379$	−10.4270	−0.535598	−0.267799	+	0.963475i	$0.586296\pi$
$380$	0	0
$381$	−54.5305	−2.79368
$382$	−23.4019	−1.19735
$383$	−12.0391	−0.615170	−0.307585	−	0.951521i	$-0.599521\pi$
$383$	−12.0391	−0.615170	−0.307585	+	0.951521i	$0.599521\pi$
$384$	63.9723	3.26457
$385$	0	0
$386$	−19.5717	−0.996171
$387$	37.2395	1.89299
$388$	−16.0531	−0.814972
$389$	16.5646	0.839858	0.419929	−	0.907557i	$-0.362055\pi$
$389$	16.5646	0.839858	0.419929	+	0.907557i	$0.362055\pi$
$390$	0	0
$391$	−3.34623	−0.169226
$392$	26.7514	1.35115
$393$	−23.6982	−1.19542
$394$	45.1249	2.27336
$395$	0	0
$396$	−104.339	−5.24322
$397$	14.1044	0.707878	0.353939	−	0.935268i	$-0.384842\pi$
$397$	14.1044	0.707878	0.353939	+	0.935268i	$0.384842\pi$
$398$	−46.2482	−2.31821
$399$	−47.3860	−2.37227
$400$	0	0
$401$	11.7748	0.588007	0.294003	−	0.955804i	$-0.405012\pi$
$401$	11.7748	0.588007	0.294003	+	0.955804i	$0.405012\pi$
$402$	−3.81493	−0.190271
$403$	1.95367	0.0973193
$404$	−11.4267	−0.568501
$405$	0	0
$406$	8.62920	0.428260
$407$	−11.7618	−0.583011
$408$	−21.8241	−1.08045
$409$	−32.5326	−1.60864	−0.804318	−	0.594199i	$-0.797469\pi$
$409$	−32.5326	−1.60864	−0.804318	+	0.594199i	$0.797469\pi$
$410$	0	0
$411$	1.12964	0.0557209
$412$	−28.5336	−1.40575
$413$	−10.0653	−0.495282
$414$	−51.5087	−2.53152
$415$	0	0
$416$	−2.17509	−0.106643
$417$	43.8680	2.14823
$418$	31.5520	1.54326
$419$	6.80130	0.332265	0.166133	−	0.986103i	$-0.446872\pi$
$419$	6.80130	0.332265	0.166133	+	0.986103i	$0.446872\pi$
$420$	0	0
$421$	−5.08494	−0.247825	−0.123912	−	0.992293i	$-0.539544\pi$
$421$	−5.08494	−0.247825	−0.123912	+	0.992293i	$0.539544\pi$
$422$	11.6404	0.566645
$423$	43.5436	2.11716
$424$	0.207807	0.0100920
$425$	0	0
$426$	35.5934	1.72450
$427$	5.72119	0.276868
$428$	13.1930	0.637706
$429$	−48.7714	−2.35470
$430$	0	0
$431$	17.0921	0.823295	0.411648	−	0.911343i	$-0.364953\pi$
$431$	17.0921	0.823295	0.411648	+	0.911343i	$0.364953\pi$
$432$	−73.4002	−3.53147
$433$	7.10031	0.341219	0.170610	−	0.985339i	$-0.445426\pi$
$433$	7.10031	0.341219	0.170610	+	0.985339i	$0.445426\pi$
$434$	3.65605	0.175496
$435$	0	0
$436$	−2.26914	−0.108672
$437$	10.4248	0.498687
$438$	−124.064	−5.92799
$439$	12.1883	0.581718	0.290859	−	0.956766i	$-0.406059\pi$
$439$	12.1883	0.581718	0.290859	+	0.956766i	$0.406059\pi$
$440$	0	0
$441$	43.2080	2.05752
$442$	−14.7318	−0.700720
$443$	0.994508	0.0472505	0.0236252	−	0.999721i	$-0.492479\pi$
$443$	0.994508	0.0472505	0.0236252	+	0.999721i	$0.492479\pi$
$444$	−50.1039	−2.37782
$445$	0	0
$446$	18.0007	0.852357
$447$	47.5625	2.24963
$448$	26.0132	1.22901
$449$	−17.6484	−0.832881	−0.416441	−	0.909163i	$-0.636723\pi$
$449$	−17.6484	−0.832881	−0.416441	+	0.909163i	$0.636723\pi$
$450$	0	0
$451$	32.7921	1.54412
$452$	−54.0345	−2.54157
$453$	−20.1065	−0.944686
$454$	38.9669	1.82881
$455$	0	0
$456$	67.9906	3.18395
$457$	−4.27450	−0.199953	−0.0999763	−	0.994990i	$-0.531877\pi$
$457$	−4.27450	−0.199953	−0.0999763	+	0.994990i	$0.531877\pi$
$458$	39.4544	1.84358
$459$	−22.2457	−1.03834
$460$	0	0
$461$	1.36149	0.0634108	0.0317054	−	0.999497i	$-0.489906\pi$
$461$	1.36149	0.0634108	0.0317054	+	0.999497i	$0.489906\pi$
$462$	−91.2696	−4.24625
$463$	23.9028	1.11086	0.555429	−	0.831564i	$-0.312554\pi$
$463$	23.9028	1.11086	0.555429	+	0.831564i	$0.312554\pi$
$464$	−4.28660	−0.199000
$465$	0	0
$466$	−46.7151	−2.16404
$467$	−13.0810	−0.605317	−0.302658	−	0.953099i	$-0.597874\pi$
$467$	−13.0810	−0.605317	−0.302658	+	0.953099i	$0.597874\pi$
$468$	−151.771	−7.01560
$469$	−1.63155	−0.0753379
$470$	0	0
$471$	75.8774	3.49624
$472$	14.4420	0.664745
$473$	−14.5167	−0.667479
$474$	−84.4649	−3.87960
$475$	0	0
$476$	−18.4512	−0.845708
$477$	0.335644	0.0153681
$478$	−49.4631	−2.26239
$479$	34.7395	1.58729	0.793643	−	0.608383i	$-0.208182\pi$
$479$	34.7395	1.58729	0.793643	+	0.608383i	$0.208182\pi$
$480$	0	0
$481$	−17.1087	−0.780089
$482$	−41.0147	−1.86817
$483$	−30.1556	−1.37213
$484$	−3.84804	−0.174911
$485$	0	0
$486$	−142.261	−6.45308
$487$	−18.3895	−0.833308	−0.416654	−	0.909065i	$-0.636797\pi$
$487$	−18.3895	−0.833308	−0.416654	+	0.909065i	$0.636797\pi$
$488$	−8.20890	−0.371600
$489$	82.0247	3.70928
$490$	0	0
$491$	42.5512	1.92031	0.960155	−	0.279469i	$-0.0901583\pi$
$491$	42.5512	1.92031	0.960155	+	0.279469i	$0.0901583\pi$
$492$	139.690	6.29773
$493$	−1.29916	−0.0585112
$494$	45.8953	2.06493
$495$	0	0
$496$	−1.81616	−0.0815482
$497$	15.2224	0.682818
$498$	92.3921	4.14019
$499$	−17.4618	−0.781699	−0.390849	−	0.920455i	$-0.627819\pi$
$499$	−17.4618	−0.781699	−0.390849	+	0.920455i	$0.627819\pi$
$500$	0	0
$501$	−40.6100	−1.81432
$502$	14.4441	0.644672
$503$	−14.4442	−0.644036	−0.322018	−	0.946734i	$-0.604361\pi$
$503$	−14.4442	−0.644036	−0.322018	+	0.946734i	$0.604361\pi$
$504$	−143.673	−6.39970
$505$	0	0
$506$	20.0791	0.892625
$507$	−27.5684	−1.22435
$508$	66.1494	2.93491
$509$	−1.83780	−0.0814591	−0.0407295	−	0.999170i	$-0.512968\pi$
$509$	−1.83780	−0.0814591	−0.0407295	+	0.999170i	$0.512968\pi$
$510$	0	0
$511$	−53.0589	−2.34719
$512$	−41.1426	−1.81827
$513$	69.3042	3.05986
$514$	33.2897	1.46834
$515$	0	0
$516$	−61.8394	−2.72233
$517$	−16.9742	−0.746523
$518$	−32.0168	−1.40674
$519$	34.5933	1.51848
$520$	0	0
$521$	36.8440	1.61417	0.807083	−	0.590439i	$-0.201045\pi$
$521$	36.8440	1.61417	0.807083	+	0.590439i	$0.201045\pi$
$522$	−19.9980	−0.875290
$523$	−30.1607	−1.31883	−0.659417	−	0.751777i	$-0.729197\pi$
$523$	−30.1607	−1.31883	−0.659417	+	0.751777i	$0.729197\pi$
$524$	28.7477	1.25585
$525$	0	0
$526$	12.7643	0.556550
$527$	−0.550433	−0.0239772
$528$	45.3387	1.97311
$529$	−16.3658	−0.711558
$530$	0	0
$531$	23.3262	1.01227
$532$	57.4827	2.49219
$533$	47.6993	2.06609
$534$	−136.590	−5.91085
$535$	0	0
$536$	2.34099	0.101115
$537$	−18.5540	−0.800664
$538$	−36.0534	−1.55437
$539$	−16.8433	−0.725493
$540$	0	0
$541$	30.2558	1.30080	0.650400	−	0.759592i	$-0.274601\pi$
$541$	30.2558	1.30080	0.650400	+	0.759592i	$0.274601\pi$
$542$	28.7648	1.23555
$543$	60.7880	2.60866
$544$	0.612817	0.0262743
$545$	0	0
$546$	−132.760	−5.68162
$547$	22.2206	0.950084	0.475042	−	0.879963i	$-0.342433\pi$
$547$	22.2206	0.950084	0.475042	+	0.879963i	$0.342433\pi$
$548$	−1.37033	−0.0585376
$549$	−13.2588	−0.565870
$550$	0	0
$551$	4.04739	0.172425
$552$	43.2681	1.84161
$553$	−36.1235	−1.53613
$554$	31.8153	1.35170
$555$	0	0
$556$	−53.2151	−2.25682
$557$	−31.1531	−1.32000	−0.659999	−	0.751267i	$-0.729443\pi$
$557$	−31.1531	−1.32000	−0.659999	+	0.751267i	$0.729443\pi$
$558$	−8.47285	−0.358684
$559$	−21.1159	−0.893109
$560$	0	0
$561$	13.7410	0.580145
$562$	−19.5324	−0.823923
$563$	20.2319	0.852675	0.426337	−	0.904564i	$-0.359804\pi$
$563$	20.2319	0.852675	0.426337	+	0.904564i	$0.359804\pi$
$564$	−72.3079	−3.04471
$565$	0	0
$566$	−69.7082	−2.93005
$567$	−114.867	−4.82398
$568$	−21.8415	−0.916447
$569$	23.1088	0.968773	0.484386	−	0.874854i	$-0.339043\pi$
$569$	23.1088	0.968773	0.484386	+	0.874854i	$0.339043\pi$
$570$	0	0
$571$	−41.9425	−1.75524	−0.877620	−	0.479356i	$-0.840870\pi$
$571$	−41.9425	−1.75524	−0.877620	+	0.479356i	$0.840870\pi$
$572$	59.1632	2.47374
$573$	31.7509	1.32641
$574$	89.2634	3.72578
$575$	0	0
$576$	−60.2851	−2.51188
$577$	−19.8997	−0.828436	−0.414218	−	0.910178i	$-0.635945\pi$
$577$	−19.8997	−0.828436	−0.414218	+	0.910178i	$0.635945\pi$
$578$	−37.6549	−1.56624
$579$	26.5541	1.10355
$580$	0	0
$581$	39.5138	1.63931
$582$	32.5429	1.34895
$583$	−0.130841	−0.00541887
$584$	76.1302	3.15029
$585$	0	0
$586$	61.1075	2.52433
$587$	−3.03862	−0.125417	−0.0627086	−	0.998032i	$-0.519974\pi$
$587$	−3.03862	−0.125417	−0.0627086	+	0.998032i	$0.519974\pi$
$588$	−71.7505	−2.95894
$589$	1.71482	0.0706577
$590$	0	0
$591$	−61.2239	−2.51841
$592$	15.9045	0.653671
$593$	−40.8122	−1.67596	−0.837979	−	0.545703i	$-0.816263\pi$
$593$	−40.8122	−1.67596	−0.837979	+	0.545703i	$0.816263\pi$
$594$	133.486	5.47699
$595$	0	0
$596$	−57.6967	−2.36335
$597$	62.7479	2.56810
$598$	29.2070	1.19436
$599$	15.7500	0.643530	0.321765	−	0.946820i	$-0.395724\pi$
$599$	15.7500	0.643530	0.321765	+	0.946820i	$0.395724\pi$
$600$	0	0
$601$	1.00387	0.0409489	0.0204745	−	0.999790i	$-0.493482\pi$
$601$	1.00387	0.0409489	0.0204745	+	0.999790i	$0.493482\pi$
$602$	−39.5159	−1.61055
$603$	3.78109	0.153978
$604$	24.3906	0.992441
$605$	0	0
$606$	23.1643	0.940985
$607$	30.6666	1.24472	0.622360	−	0.782731i	$-0.286174\pi$
$607$	30.6666	1.24472	0.622360	+	0.782731i	$0.286174\pi$
$608$	−1.90917	−0.0774269
$609$	−11.7078	−0.474424
$610$	0	0
$611$	−24.6905	−0.998873
$612$	42.7603	1.72848
$613$	3.13840	0.126759	0.0633793	−	0.997990i	$-0.479812\pi$
$613$	3.13840	0.126759	0.0633793	+	0.997990i	$0.479812\pi$
$614$	−24.5883	−0.992304
$615$	0	0
$616$	56.0065	2.25657
$617$	48.0571	1.93471	0.967353	−	0.253434i	$-0.0815602\pi$
$617$	48.0571	1.93471	0.967353	+	0.253434i	$0.0815602\pi$
$618$	57.8435	2.32681
$619$	42.0099	1.68852	0.844260	−	0.535934i	$-0.180040\pi$
$619$	42.0099	1.68852	0.844260	+	0.535934i	$0.180040\pi$
$620$	0	0
$621$	44.1040	1.76983
$622$	4.02870	0.161536
$623$	−58.4163	−2.34040
$624$	65.9494	2.64009
$625$	0	0
$626$	−4.74677	−0.189719
$627$	−42.8086	−1.70961
$628$	−92.0447	−3.67298
$629$	4.82025	0.192196
$630$	0	0
$631$	−20.9064	−0.832271	−0.416135	−	0.909303i	$-0.636616\pi$
$631$	−20.9064	−0.832271	−0.416135	+	0.909303i	$0.636616\pi$
$632$	51.8309	2.06172
$633$	−15.7933	−0.627726
$634$	−71.5909	−2.84324
$635$	0	0
$636$	−0.557365	−0.0221010
$637$	−24.5002	−0.970734
$638$	7.79563	0.308632
$639$	−35.2776	−1.39556
$640$	0	0
$641$	17.6324	0.696439	0.348220	−	0.937413i	$-0.386786\pi$
$641$	17.6324	0.696439	0.348220	+	0.937413i	$0.386786\pi$
$642$	−26.7448	−1.05553
$643$	−9.33296	−0.368056	−0.184028	−	0.982921i	$-0.558914\pi$
$643$	−9.33296	−0.368056	−0.184028	+	0.982921i	$0.558914\pi$
$644$	36.5810	1.44149
$645$	0	0
$646$	−12.9307	−0.508751
$647$	−6.08656	−0.239287	−0.119644	−	0.992817i	$-0.538175\pi$
$647$	−6.08656	−0.239287	−0.119644	+	0.992817i	$0.538175\pi$
$648$	164.815	6.47453
$649$	−9.09302	−0.356932
$650$	0	0
$651$	−4.96041	−0.194414
$652$	−99.5018	−3.89679
$653$	−15.8120	−0.618770	−0.309385	−	0.950937i	$-0.600123\pi$
$653$	−15.8120	−0.618770	−0.309385	+	0.950937i	$0.600123\pi$
$654$	4.60002	0.179875
$655$	0	0
$656$	−44.3420	−1.73127
$657$	122.963	4.79725
$658$	−46.2053	−1.80127
$659$	38.2114	1.48850	0.744252	−	0.667899i	$-0.232806\pi$
$659$	38.2114	1.48850	0.744252	+	0.667899i	$0.232806\pi$
$660$	0	0
$661$	−32.2183	−1.25315	−0.626573	−	0.779362i	$-0.715543\pi$
$661$	−32.2183	−1.25315	−0.626573	+	0.779362i	$0.715543\pi$
$662$	−9.55590	−0.371401
$663$	19.9876	0.776253
$664$	−56.6953	−2.20021
$665$	0	0
$666$	74.1984	2.87513
$667$	2.57569	0.0997311
$668$	49.2629	1.90604
$669$	−24.4227	−0.944235
$670$	0	0
$671$	5.16853	0.199529
$672$	5.52260	0.213039
$673$	31.8086	1.22613	0.613065	−	0.790033i	$-0.289936\pi$
$673$	31.8086	1.22613	0.613065	+	0.790033i	$0.289936\pi$
$674$	−11.1743	−0.430418
$675$	0	0
$676$	33.4424	1.28625
$677$	−10.6030	−0.407507	−0.203754	−	0.979022i	$-0.565314\pi$
$677$	−10.6030	−0.407507	−0.203754	+	0.979022i	$0.565314\pi$
$678$	109.539	4.20682
$679$	13.9178	0.534115
$680$	0	0
$681$	−52.8689	−2.02594
$682$	3.30288	0.126474
$683$	−12.2980	−0.470572	−0.235286	−	0.971926i	$-0.575603\pi$
$683$	−12.2980	−0.470572	−0.235286	+	0.971926i	$0.575603\pi$
$684$	−133.215	−5.09361
$685$	0	0
$686$	14.5552	0.555722
$687$	−53.5304	−2.04231
$688$	19.6297	0.748376
$689$	−0.190320	−0.00725063
$690$	0	0
$691$	20.5877	0.783193	0.391596	−	0.920137i	$-0.371923\pi$
$691$	20.5877	0.783193	0.391596	+	0.920137i	$0.371923\pi$
$692$	−41.9642	−1.59524
$693$	90.4600	3.43629
$694$	−32.7921	−1.24477
$695$	0	0
$696$	16.7986	0.636750
$697$	−13.4389	−0.509036
$698$	80.7196	3.05528
$699$	63.3815	2.39731
$700$	0	0
$701$	−31.8857	−1.20431	−0.602153	−	0.798380i	$-0.705690\pi$
$701$	−31.8857	−1.20431	−0.602153	+	0.798380i	$0.705690\pi$
$702$	194.168	7.32840
$703$	−15.0170	−0.566376
$704$	23.5003	0.885701
$705$	0	0
$706$	45.4309	1.70981
$707$	9.90679	0.372583
$708$	−38.7351	−1.45576
$709$	−37.0325	−1.39079	−0.695393	−	0.718630i	$-0.744769\pi$
$709$	−37.0325	−1.39079	−0.695393	+	0.718630i	$0.744769\pi$
$710$	0	0
$711$	83.7156	3.13958
$712$	83.8171	3.14118
$713$	1.09128	0.0408687
$714$	37.4043	1.39982
$715$	0	0
$716$	22.5073	0.841139
$717$	67.1098	2.50626
$718$	−63.4653	−2.36850
$719$	2.30674	0.0860268	0.0430134	−	0.999074i	$-0.486304\pi$
$719$	2.30674	0.0860268	0.0430134	+	0.999074i	$0.486304\pi$
$720$	0	0
$721$	24.7382	0.921299
$722$	−6.43955	−0.239655
$723$	55.6474	2.06955
$724$	−73.7402	−2.74053
$725$	0	0
$726$	7.80077	0.289514
$727$	−30.0583	−1.11480	−0.557401	−	0.830243i	$-0.688201\pi$
$727$	−30.0583	−1.11480	−0.557401	+	0.830243i	$0.688201\pi$
$728$	81.4668	3.01936
$729$	94.8098	3.51147
$730$	0	0
$731$	5.94926	0.220042
$732$	22.0173	0.813783
$733$	−49.7450	−1.83738	−0.918688	−	0.394985i	$-0.870750\pi$
$733$	−49.7450	−1.83738	−0.918688	+	0.394985i	$0.870750\pi$
$734$	33.9079	1.25156
$735$	0	0
$736$	−1.21496	−0.0447840
$737$	−1.47394	−0.0542934
$738$	−206.866	−7.61486
$739$	−46.1269	−1.69681	−0.848404	−	0.529350i	$-0.822436\pi$
$739$	−46.1269	−1.69681	−0.848404	+	0.529350i	$0.822436\pi$
$740$	0	0
$741$	−62.2692	−2.28752
$742$	−0.356161	−0.0130751
$743$	14.0745	0.516345	0.258172	−	0.966099i	$-0.416880\pi$
$743$	14.0745	0.516345	0.258172	+	0.966099i	$0.416880\pi$
$744$	7.11731	0.260933
$745$	0	0
$746$	28.7839	1.05385
$747$	−91.5725	−3.35046
$748$	−16.6688	−0.609472
$749$	−11.4381	−0.417939
$750$	0	0
$751$	6.40760	0.233817	0.116908	−	0.993143i	$-0.462702\pi$
$751$	6.40760	0.233817	0.116908	+	0.993143i	$0.462702\pi$
$752$	22.9527	0.837000
$753$	−19.5972	−0.714163
$754$	11.3395	0.412960
$755$	0	0
$756$	243.190	8.84474
$757$	−37.6194	−1.36730	−0.683651	−	0.729809i	$-0.739609\pi$
$757$	−37.6194	−1.36730	−0.683651	+	0.729809i	$0.739609\pi$
$758$	−25.6415	−0.931339
$759$	−27.2426	−0.988845
$760$	0	0
$761$	−7.98770	−0.289554	−0.144777	−	0.989464i	$-0.546246\pi$
$761$	−7.98770	−0.289554	−0.144777	+	0.989464i	$0.546246\pi$
$762$	−134.098	−4.85787
$763$	1.96731	0.0712214
$764$	−38.5161	−1.39347
$765$	0	0
$766$	−29.6059	−1.06970
$767$	−13.2267	−0.477587
$768$	107.849	3.89166
$769$	−12.3260	−0.444486	−0.222243	−	0.974991i	$-0.571338\pi$
$769$	−12.3260	−0.444486	−0.222243	+	0.974991i	$0.571338\pi$
$770$	0	0
$771$	−45.1663	−1.62662
$772$	−32.2121	−1.15934
$773$	46.9639	1.68917	0.844587	−	0.535419i	$-0.179846\pi$
$773$	46.9639	1.68917	0.844587	+	0.535419i	$0.179846\pi$
$774$	91.5774	3.29168
$775$	0	0
$776$	−19.9695	−0.716865
$777$	43.4393	1.55837
$778$	40.7347	1.46041
$779$	41.8676	1.50006
$780$	0	0
$781$	13.7519	0.492082
$782$	−8.22886	−0.294264
$783$	17.1232	0.611933
$784$	22.7758	0.813422
$785$	0	0
$786$	−58.2774	−2.07869
$787$	−35.1053	−1.25137	−0.625684	−	0.780076i	$-0.715180\pi$
$787$	−35.1053	−1.25137	−0.625684	+	0.780076i	$0.715180\pi$
$788$	74.2690	2.64572
$789$	−17.3182	−0.616543
$790$	0	0
$791$	46.8471	1.66569
$792$	−129.794	−4.61203
$793$	7.51812	0.266976
$794$	34.6847	1.23091
$795$	0	0
$796$	−76.1177	−2.69792
$797$	36.4490	1.29109	0.645546	−	0.763722i	$-0.276630\pi$
$797$	36.4490	1.29109	0.645546	+	0.763722i	$0.276630\pi$
$798$	−116.529	−4.12509
$799$	6.95639	0.246099
$800$	0	0
$801$	135.379	4.78338
$802$	28.9560	1.02247
$803$	−47.9334	−1.69153
$804$	−6.27882	−0.221437
$805$	0	0
$806$	4.80436	0.169226
$807$	48.9160	1.72192
$808$	−14.2145	−0.500064
$809$	50.3845	1.77142	0.885712	−	0.464235i	$-0.153671\pi$
$809$	50.3845	1.77142	0.885712	+	0.464235i	$0.153671\pi$
$810$	0	0
$811$	44.6328	1.56727	0.783635	−	0.621221i	$-0.213363\pi$
$811$	44.6328	1.56727	0.783635	+	0.621221i	$0.213363\pi$
$812$	14.2024	0.498407
$813$	−39.0271	−1.36874
$814$	−28.9240	−1.01379
$815$	0	0
$816$	−18.5808	−0.650458
$817$	−18.5343	−0.648433
$818$	−80.0025	−2.79722
$819$	131.583	4.59787
$820$	0	0
$821$	25.6437	0.894971	0.447486	−	0.894291i	$-0.352320\pi$
$821$	25.6437	0.894971	0.447486	+	0.894291i	$0.352320\pi$
$822$	2.77794	0.0968918
$823$	4.74708	0.165473	0.0827364	−	0.996571i	$-0.473634\pi$
$823$	4.74708	0.165473	0.0827364	+	0.996571i	$0.473634\pi$
$824$	−35.4950	−1.23653
$825$	0	0
$826$	−24.7521	−0.861235
$827$	16.8954	0.587509	0.293755	−	0.955881i	$-0.405095\pi$
$827$	16.8954	0.587509	0.293755	+	0.955881i	$0.405095\pi$
$828$	−84.7758	−2.94616
$829$	−31.1447	−1.08170	−0.540850	−	0.841119i	$-0.681897\pi$
$829$	−31.1447	−1.08170	−0.540850	+	0.841119i	$0.681897\pi$
$830$	0	0
$831$	−43.1658	−1.49741
$832$	34.1835	1.18510
$833$	6.90277	0.239167
$834$	107.878	3.73550
$835$	0	0
$836$	51.9299	1.79603
$837$	7.25482	0.250763
$838$	16.7254	0.577769
$839$	20.3277	0.701791	0.350896	−	0.936415i	$-0.385877\pi$
$839$	20.3277	0.701791	0.350896	+	0.936415i	$0.385877\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−12.5046	−0.430937
$843$	26.5008	0.912737
$844$	19.1584	0.659458
$845$	0	0
$846$	107.080	3.68149
$847$	3.33619	0.114633
$848$	0.176925	0.00607563
$849$	94.5776	3.24589
$850$	0	0
$851$	−9.55654	−0.327594
$852$	58.5815	2.00697
$853$	48.0623	1.64562	0.822810	−	0.568316i	$-0.192405\pi$
$853$	48.0623	1.64562	0.822810	+	0.568316i	$0.192405\pi$
$854$	14.0692	0.481439
$855$	0	0
$856$	16.4117	0.560939
$857$	24.0426	0.821280	0.410640	−	0.911797i	$-0.365305\pi$
$857$	24.0426	0.821280	0.410640	+	0.911797i	$0.365305\pi$
$858$	−119.936	−4.09454
$859$	−38.8389	−1.32517	−0.662583	−	0.748988i	$-0.730540\pi$
$859$	−38.8389	−1.32517	−0.662583	+	0.748988i	$0.730540\pi$
$860$	0	0
$861$	−121.109	−4.12740
$862$	42.0318	1.43161
$863$	−43.5717	−1.48320	−0.741599	−	0.670843i	$-0.765932\pi$
$863$	−43.5717	−1.48320	−0.741599	+	0.670843i	$0.765932\pi$
$864$	−8.07705	−0.274787
$865$	0	0
$866$	17.4607	0.593338
$867$	51.0888	1.73507
$868$	6.01733	0.204242
$869$	−32.6340	−1.10703
$870$	0	0
$871$	−2.14399	−0.0726464
$872$	−2.82275	−0.0955902
$873$	−32.2542	−1.09164
$874$	25.6362	0.867156
$875$	0	0
$876$	−204.191	−6.89896
$877$	−33.4549	−1.12969	−0.564846	−	0.825197i	$-0.691064\pi$
$877$	−33.4549	−1.12969	−0.564846	+	0.825197i	$0.691064\pi$
$878$	29.9729	1.01154
$879$	−82.9085	−2.79643
$880$	0	0
$881$	−17.1987	−0.579440	−0.289720	−	0.957111i	$-0.593562\pi$
$881$	−17.1987	−0.579440	−0.289720	+	0.957111i	$0.593562\pi$
$882$	106.255	3.57778
$883$	−31.2157	−1.05049	−0.525247	−	0.850950i	$-0.676027\pi$
$883$	−31.2157	−1.05049	−0.525247	+	0.850950i	$0.676027\pi$
$884$	−24.2464	−0.815494
$885$	0	0
$886$	2.44564	0.0821628
$887$	50.3074	1.68916	0.844579	−	0.535430i	$-0.179851\pi$
$887$	50.3074	1.68916	0.844579	+	0.535430i	$0.179851\pi$
$888$	−62.3277	−2.09158
$889$	−57.3505	−1.92347
$890$	0	0
$891$	−103.771	−3.47647
$892$	29.6265	0.991968
$893$	−21.6719	−0.725222
$894$	116.963	3.91183
$895$	0	0
$896$	67.2806	2.24769
$897$	−39.6270	−1.32311
$898$	−43.4001	−1.44828
$899$	0.423684	0.0141307
$900$	0	0
$901$	0.0536214	0.00178639
$902$	80.6406	2.68504
$903$	53.6137	1.78415
$904$	−67.2173	−2.23562
$905$	0	0
$906$	−49.4448	−1.64269
$907$	−40.2681	−1.33708	−0.668540	−	0.743676i	$-0.733080\pi$
$907$	−40.2681	−1.33708	−0.668540	+	0.743676i	$0.733080\pi$
$908$	64.1338	2.12835
$909$	−22.9588	−0.761496
$910$	0	0
$911$	−46.0923	−1.52710	−0.763552	−	0.645746i	$-0.776547\pi$
$911$	−46.0923	−1.52710	−0.763552	+	0.645746i	$0.776547\pi$
$912$	57.8865	1.91681
$913$	35.6968	1.18139
$914$	−10.5116	−0.347693
$915$	0	0
$916$	64.9362	2.14555
$917$	−24.9238	−0.823056
$918$	−54.7055	−1.80555
$919$	−10.9441	−0.361012	−0.180506	−	0.983574i	$-0.557773\pi$
$919$	−10.9441	−0.361012	−0.180506	+	0.983574i	$0.557773\pi$
$920$	0	0
$921$	33.3606	1.09927
$922$	3.34809	0.110264
$923$	20.0035	0.658423
$924$	−150.216	−4.94176
$925$	0	0
$926$	58.7804	1.93165
$927$	−57.3304	−1.88298
$928$	−0.471703	−0.0154844
$929$	−50.6115	−1.66051	−0.830255	−	0.557384i	$-0.811805\pi$
$929$	−50.6115	−1.66051	−0.830255	+	0.557384i	$0.811805\pi$
$930$	0	0
$931$	−21.5048	−0.704793
$932$	−76.8863	−2.51849
$933$	−5.46600	−0.178949
$934$	−32.1681	−1.05257
$935$	0	0
$936$	−188.798	−6.17106
$937$	−18.9658	−0.619585	−0.309793	−	0.950804i	$-0.600260\pi$
$937$	−18.9658	−0.619585	−0.309793	+	0.950804i	$0.600260\pi$
$938$	−4.01222	−0.131003
$939$	6.44025	0.210170
$940$	0	0
$941$	−37.2758	−1.21516	−0.607579	−	0.794260i	$-0.707859\pi$
$941$	−37.2758	−1.21516	−0.607579	+	0.794260i	$0.707859\pi$
$942$	186.593	6.07954
$943$	26.6438	0.867642
$944$	12.2957	0.400192
$945$	0	0
$946$	−35.6987	−1.16066
$947$	−25.3754	−0.824591	−0.412296	−	0.911050i	$-0.635273\pi$
$947$	−25.3754	−0.824591	−0.412296	+	0.911050i	$0.635273\pi$
$948$	−139.017	−4.51506
$949$	−69.7238	−2.26333
$950$	0	0
$951$	97.1320	3.14972
$952$	−22.9527	−0.743902
$953$	−43.6572	−1.41419	−0.707097	−	0.707116i	$-0.749996\pi$
$953$	−43.6572	−1.41419	−0.707097	+	0.707116i	$0.749996\pi$
$954$	0.825397	0.0267232
$955$	0	0
$956$	−81.4090	−2.63295
$957$	−10.5768	−0.341900
$958$	85.4294	2.76010
$959$	1.18806	0.0383643
$960$	0	0
$961$	−30.8205	−0.994209
$962$	−42.0727	−1.35648
$963$	26.5076	0.854195
$964$	−67.5043	−2.17417
$965$	0	0
$966$	−74.1571	−2.38597
$967$	−28.4146	−0.913751	−0.456875	−	0.889531i	$-0.651032\pi$
$967$	−28.4146	−0.913751	−0.456875	+	0.889531i	$0.651032\pi$
$968$	−4.78685	−0.153855
$969$	17.5439	0.563591
$970$	0	0
$971$	6.34773	0.203708	0.101854	−	0.994799i	$-0.467523\pi$
$971$	6.34773	0.203708	0.101854	+	0.994799i	$0.467523\pi$
$972$	−234.140	−7.51005
$973$	46.1366	1.47907
$974$	−45.2225	−1.44902
$975$	0	0
$976$	−6.98897	−0.223711
$977$	6.78708	0.217138	0.108569	−	0.994089i	$-0.465373\pi$
$977$	6.78708	0.217138	0.108569	+	0.994089i	$0.465373\pi$
$978$	201.711	6.44999
$979$	−52.7733	−1.68664
$980$	0	0
$981$	−4.55921	−0.145564
$982$	104.640	3.33918
$983$	33.2974	1.06202	0.531011	−	0.847365i	$-0.321812\pi$
$983$	33.2974	1.06202	0.531011	+	0.847365i	$0.321812\pi$
$984$	173.771	5.53961
$985$	0	0
$986$	−3.19482	−0.101744
$987$	62.6898	1.99544
$988$	75.5370	2.40315
$989$	−11.7949	−0.375056
$990$	0	0
$991$	23.2716	0.739247	0.369623	−	0.929182i	$-0.379487\pi$
$991$	23.2716	0.739247	0.369623	+	0.929182i	$0.379487\pi$
$992$	−0.199853	−0.00634534
$993$	12.9651	0.411435
$994$	37.4341	1.18734
$995$	0	0
$996$	152.064	4.81833
$997$	18.5645	0.587945	0.293973	−	0.955814i	$-0.405023\pi$
$997$	18.5645	0.587945	0.293973	+	0.955814i	$0.405023\pi$
$998$	−42.9412	−1.35928
$999$	−63.5319	−2.01006

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.i.1.5	✓	5
3.2	odd	2		6525.2.a.bo.1.1		5
5.2	odd	4		725.2.b.g.349.10		10
5.3	odd	4		725.2.b.g.349.1		10
5.4	even	2		725.2.a.j.1.1	yes	5
15.14	odd	2		6525.2.a.bp.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.i.1.5	✓	5	1.1	even	1	trivial
725.2.a.j.1.1	yes	5	5.4	even	2
725.2.b.g.349.1		10	5.3	odd	4
725.2.b.g.349.10		10	5.2	odd	4
6525.2.a.bo.1.1		5	3.2	odd	2
6525.2.a.bp.1.5		5	15.14	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45914 q^{2} -3.33648 q^{3} +4.04739 q^{4} -8.20489 q^{6} -3.50903 q^{7} +5.03483 q^{8} +8.13211 q^{9} -3.17006 q^{11} -13.5040 q^{12} -4.61115 q^{13} -8.62920 q^{14} +4.28660 q^{16} +1.29916 q^{17} +19.9980 q^{18} -4.04739 q^{19} +11.7078 q^{21} -7.79563 q^{22} -2.57569 q^{23} -16.7986 q^{24} -11.3395 q^{26} -17.1232 q^{27} -14.2024 q^{28} -1.00000 q^{29} -0.423684 q^{31} +0.471703 q^{32} +10.5768 q^{33} +3.19482 q^{34} +32.9138 q^{36} +3.71028 q^{37} -9.95312 q^{38} +15.3850 q^{39} -10.3443 q^{41} +28.7912 q^{42} +4.57932 q^{43} -12.8305 q^{44} -6.33399 q^{46} +5.35453 q^{47} -14.3022 q^{48} +5.31326 q^{49} -4.33462 q^{51} -18.6631 q^{52} +0.0412739 q^{53} -42.1084 q^{54} -17.6674 q^{56} +13.5040 q^{57} -2.45914 q^{58} +2.86841 q^{59} -1.63042 q^{61} -1.04190 q^{62} -28.5358 q^{63} -7.41322 q^{64} +26.0100 q^{66} +0.464958 q^{67} +5.25821 q^{68} +8.59374 q^{69} -4.33807 q^{71} +40.9438 q^{72} +15.1207 q^{73} +9.12413 q^{74} -16.3814 q^{76} +11.1238 q^{77} +37.8340 q^{78} +10.2945 q^{79} +32.7348 q^{81} -25.4382 q^{82} -11.2606 q^{83} +47.3860 q^{84} +11.2612 q^{86} +3.33648 q^{87} -15.9607 q^{88} +16.6474 q^{89} +16.1806 q^{91} -10.4248 q^{92} +1.41361 q^{93} +13.1676 q^{94} -1.57383 q^{96} -3.96628 q^{97} +13.0661 q^{98} -25.7792 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 6 q^{3} + 4 q^{4} - q^{6} - 10 q^{7} + 3 q^{8} + 7 q^{9} + 2 q^{11} - 18 q^{12} - 8 q^{13} - 13 q^{14} + 6 q^{16} - 3 q^{17} + 16 q^{18} - 4 q^{19} + 15 q^{21} - 16 q^{22} - 3 q^{23} - 19 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) 5 * q - 6 * q^3 + 4 * q^4 - q^6 - 10 * q^7 + 3 * q^8 + 7 * q^9 + 2 * q^11 - 18 * q^12 - 8 * q^13 - 13 * q^14 + 6 * q^16 - 3 * q^17 + 16 * q^18 - 4 * q^19 + 15 * q^21 - 16 * q^22 - 3 * q^23 - 19 * q^24 - 4 * q^26 - 21 * q^27 - 10 * q^28 - 5 * q^29 + 5 * q^31 - 4 * q^32 - 3 * q^33 - 9 * q^34 + 35 * q^36 - 4 * q^37 - 3 * q^38 + 4 * q^39 - 3 * q^41 + 16 * q^42 - 12 * q^43 - 19 * q^44 + q^47 - 20 * q^48 + 13 * q^49 - 7 * q^51 - 26 * q^52 - 12 * q^53 - 58 * q^54 - 8 * q^56 + 18 * q^57 - 15 * q^59 - 7 * q^61 - 10 * q^62 - 33 * q^63 - 9 * q^64 + 35 * q^66 - 17 * q^67 + 43 * q^68 + 18 * q^69 - 17 * q^71 + 49 * q^72 - 10 * q^73 + 12 * q^74 - 34 * q^76 + 8 * q^77 + 39 * q^78 - 2 * q^79 + 21 * q^81 - 31 * q^82 - 19 * q^83 + 38 * q^84 + 31 * q^86 + 6 * q^87 + 3 * q^88 + 14 * q^89 - 15 * q^91 + q^92 + 15 * q^93 + 27 * q^94 + 6 * q^96 + 9 * q^97 + 51 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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