LMFDB - Embedded newform 725.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(1,725)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.792287$ 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)

$f(q)$	$=$	\(q-1.73205 q^{2} -0.792287 q^{3} +1.00000 q^{4} +1.37228 q^{6} -5.04868 q^{7} +1.73205 q^{8} -2.37228 q^{9} +0.627719 q^{11} -0.792287 q^{12} -4.25639 q^{13} +8.74456 q^{14} -5.00000 q^{16} +1.58457 q^{17} +4.10891 q^{18} +4.00000 q^{19} +4.00000 q^{21} -1.08724 q^{22} -3.46410 q^{23} -1.37228 q^{24} +7.37228 q^{26} +4.25639 q^{27} -5.04868 q^{28} -1.00000 q^{29} -3.37228 q^{31} +5.19615 q^{32} -0.497333 q^{33} -2.74456 q^{34} -2.37228 q^{36} -3.16915 q^{37} -6.92820 q^{38} +3.37228 q^{39} -4.74456 q^{41} -6.92820 q^{42} +10.8896 q^{43} +0.627719 q^{44} +6.00000 q^{46} +10.8896 q^{47} +3.96143 q^{48} +18.4891 q^{49} -1.25544 q^{51} -4.25639 q^{52} -4.25639 q^{53} -7.37228 q^{54} -8.74456 q^{56} -3.16915 q^{57} +1.73205 q^{58} -10.7446 q^{59} +6.00000 q^{61} +5.84096 q^{62} +11.9769 q^{63} +1.00000 q^{64} +0.861407 q^{66} +1.87953 q^{67} +1.58457 q^{68} +2.74456 q^{69} +6.74456 q^{71} -4.10891 q^{72} +6.92820 q^{73} +5.48913 q^{74} +4.00000 q^{76} -3.16915 q^{77} -5.84096 q^{78} +11.3723 q^{79} +3.74456 q^{81} +8.21782 q^{82} +9.80240 q^{83} +4.00000 q^{84} -18.8614 q^{86} +0.792287 q^{87} +1.08724 q^{88} -0.744563 q^{89} +21.4891 q^{91} -3.46410 q^{92} +2.67181 q^{93} -18.8614 q^{94} -4.11684 q^{96} +6.92820 q^{97} -32.0241 q^{98} -1.48913 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 14 q^{11} + 12 q^{14} - 20 q^{16} + 16 q^{19} + 16 q^{21} + 6 q^{24} + 18 q^{26} - 4 q^{29} - 2 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} + 4 q^{41} + 14 q^{44} + 24 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) $ 4 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 + 14 * q^11 + 12 * q^14 - 20 * q^16 + 16 * q^19 + 16 * q^21 + 6 * q^24 + 18 * q^26 - 4 * q^29 - 2 * q^31 + 12 * q^34 + 2 * q^36 + 2 * q^39 + 4 * q^41 + 14 * q^44 + 24 * q^46 + 28 * q^49 - 28 * q^51 - 18 * q^54 - 12 * q^56 - 20 * q^59 + 24 * q^61 + 4 * q^64 - 54 * q^66 - 12 * q^69 + 4 * q^71 - 24 * q^74 + 16 * q^76 + 34 * q^79 - 8 * q^81 + 16 * q^84 - 18 * q^86 + 20 * q^89 + 40 * q^91 - 18 * q^94 + 18 * q^96 + 40 * 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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	−0.792287	−0.457427	−0.228714	−	0.973494i	$-0.573452\pi$
$3$	−0.792287	−0.457427	−0.228714	+	0.973494i	$0.573452\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.37228	0.560232
$7$	−5.04868	−1.90822	−0.954110	−	0.299456i	$-0.903195\pi$
$7$	−5.04868	−1.90822	−0.954110	+	0.299456i	$0.903195\pi$
$8$	1.73205	0.612372
$9$	−2.37228	−0.790760
$10$	0	0
$11$	0.627719	0.189264	0.0946322	−	0.995512i	$-0.469833\pi$
$11$	0.627719	0.189264	0.0946322	+	0.995512i	$0.469833\pi$
$12$	−0.792287	−0.228714
$13$	−4.25639	−1.18051	−0.590255	−	0.807217i	$-0.700973\pi$
$13$	−4.25639	−1.18051	−0.590255	+	0.807217i	$0.700973\pi$
$14$	8.74456	2.33708
$15$	0	0
$16$	−5.00000	−1.25000
$17$	1.58457	0.384316	0.192158	−	0.981364i	$-0.438451\pi$
$17$	1.58457	0.384316	0.192158	+	0.981364i	$0.438451\pi$
$18$	4.10891	0.968480
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	−1.08724	−0.231800
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	−1.37228	−0.280116
$25$	0	0
$26$	7.37228	1.44582
$27$	4.25639	0.819142
$28$	−5.04868	−0.954110
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.37228	−0.605680	−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228	−0.605680	−0.302840	+	0.953041i	$0.597935\pi$
$32$	5.19615	0.918559
$33$	−0.497333	−0.0865746
$34$	−2.74456	−0.470689
$35$	0	0
$36$	−2.37228	−0.395380
$37$	−3.16915	−0.521005	−0.260502	−	0.965473i	$-0.583888\pi$
$37$	−3.16915	−0.521005	−0.260502	+	0.965473i	$0.583888\pi$
$38$	−6.92820	−1.12390
$39$	3.37228	0.539997
$40$	0	0
$41$	−4.74456	−0.740976	−0.370488	−	0.928837i	$-0.620810\pi$
$41$	−4.74456	−0.740976	−0.370488	+	0.928837i	$0.620810\pi$
$42$	−6.92820	−1.06904
$43$	10.8896	1.66065	0.830327	−	0.557276i	$-0.188154\pi$
$43$	10.8896	1.66065	0.830327	+	0.557276i	$0.188154\pi$
$44$	0.627719	0.0946322
$45$	0	0
$46$	6.00000	0.884652
$47$	10.8896	1.58842	0.794208	−	0.607645i	$-0.207886\pi$
$47$	10.8896	1.58842	0.794208	+	0.607645i	$0.207886\pi$
$48$	3.96143	0.571784
$49$	18.4891	2.64130
$50$	0	0
$51$	−1.25544	−0.175796
$52$	−4.25639	−0.590255
$53$	−4.25639	−0.584660	−0.292330	−	0.956318i	$-0.594431\pi$
$53$	−4.25639	−0.584660	−0.292330	+	0.956318i	$0.594431\pi$
$54$	−7.37228	−1.00324
$55$	0	0
$56$	−8.74456	−1.16854
$57$	−3.16915	−0.419764
$58$	1.73205	0.227429
$59$	−10.7446	−1.39882	−0.699411	−	0.714719i	$-0.746554\pi$
$59$	−10.7446	−1.39882	−0.699411	+	0.714719i	$0.746554\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	5.84096	0.741803
$63$	11.9769	1.50894
$64$	1.00000	0.125000
$65$	0	0
$66$	0.861407	0.106032
$67$	1.87953	0.229621	0.114810	−	0.993387i	$-0.463374\pi$
$67$	1.87953	0.229621	0.114810	+	0.993387i	$0.463374\pi$
$68$	1.58457	0.192158
$69$	2.74456	0.330407
$70$	0	0
$71$	6.74456	0.800432	0.400216	−	0.916421i	$-0.368935\pi$
$71$	6.74456	0.800432	0.400216	+	0.916421i	$0.368935\pi$
$72$	−4.10891	−0.484240
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	5.48913	0.638098
$75$	0	0
$76$	4.00000	0.458831
$77$	−3.16915	−0.361158
$78$	−5.84096	−0.661359
$79$	11.3723	1.27948	0.639741	−	0.768591i	$-0.279042\pi$
$79$	11.3723	1.27948	0.639741	+	0.768591i	$0.279042\pi$
$80$	0	0
$81$	3.74456	0.416063
$82$	8.21782	0.907507
$83$	9.80240	1.07595	0.537976	−	0.842960i	$-0.319189\pi$
$83$	9.80240	1.07595	0.537976	+	0.842960i	$0.319189\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−18.8614	−2.03388
$87$	0.792287	0.0849421
$88$	1.08724	0.115900
$89$	−0.744563	−0.0789235	−0.0394617	−	0.999221i	$-0.512564\pi$
$89$	−0.744563	−0.0789235	−0.0394617	+	0.999221i	$0.512564\pi$
$90$	0	0
$91$	21.4891	2.25267
$92$	−3.46410	−0.361158
$93$	2.67181	0.277054
$94$	−18.8614	−1.94541
$95$	0	0
$96$	−4.11684	−0.420174
$97$	6.92820	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$97$	6.92820	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$98$	−32.0241	−3.23492
$99$	−1.48913	−0.149663
$100$	0	0
$101$	−3.25544	−0.323928	−0.161964	−	0.986797i	$-0.551783\pi$
$101$	−3.25544	−0.323928	−0.161964	+	0.986797i	$0.551783\pi$
$102$	2.17448	0.215306
$103$	−3.46410	−0.341328	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−3.46410	−0.341328	−0.170664	+	0.985329i	$0.554591\pi$
$104$	−7.37228	−0.722912
$105$	0	0
$106$	7.37228	0.716059
$107$	5.04868	0.488074	0.244037	−	0.969766i	$-0.421528\pi$
$107$	5.04868	0.488074	0.244037	+	0.969766i	$0.421528\pi$
$108$	4.25639	0.409571
$109$	−16.1168	−1.54371	−0.771857	−	0.635796i	$-0.780672\pi$
$109$	−16.1168	−1.54371	−0.771857	+	0.635796i	$0.780672\pi$
$110$	0	0
$111$	2.51087	0.238322
$112$	25.2434	2.38528
$113$	15.4410	1.45257	0.726283	−	0.687396i	$-0.241246\pi$
$113$	15.4410	1.45257	0.726283	+	0.687396i	$0.241246\pi$
$114$	5.48913	0.514104
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	10.0974	0.933500
$118$	18.6101	1.71320
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−10.6060	−0.964179
$122$	−10.3923	−0.940875
$123$	3.75906	0.338943
$124$	−3.37228	−0.302840
$125$	0	0
$126$	−20.7446	−1.84807
$127$	−3.46410	−0.307389	−0.153695	−	0.988118i	$-0.549117\pi$
$127$	−3.46410	−0.307389	−0.153695	+	0.988118i	$0.549117\pi$
$128$	−12.1244	−1.07165
$129$	−8.62772	−0.759628
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−0.497333	−0.0432873
$133$	−20.1947	−1.75110
$134$	−3.25544	−0.281227
$135$	0	0
$136$	2.74456	0.235344
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	−4.75372	−0.404664
$139$	−2.74456	−0.232791	−0.116395	−	0.993203i	$-0.537134\pi$
$139$	−2.74456	−0.232791	−0.116395	+	0.993203i	$0.537134\pi$
$140$	0	0
$141$	−8.62772	−0.726585
$142$	−11.6819	−0.980325
$143$	−2.67181	−0.223428
$144$	11.8614	0.988451
$145$	0	0
$146$	−12.0000	−0.993127
$147$	−14.6487	−1.20820
$148$	−3.16915	−0.260502
$149$	−3.88316	−0.318121	−0.159060	−	0.987269i	$-0.550846\pi$
$149$	−3.88316	−0.318121	−0.159060	+	0.987269i	$0.550846\pi$
$150$	0	0
$151$	13.4891	1.09773	0.548865	−	0.835911i	$-0.315060\pi$
$151$	13.4891	1.09773	0.548865	+	0.835911i	$0.315060\pi$
$152$	6.92820	0.561951
$153$	−3.75906	−0.303902
$154$	5.48913	0.442326
$155$	0	0
$156$	3.37228	0.269999
$157$	−13.8564	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$158$	−19.6974	−1.56704
$159$	3.37228	0.267439
$160$	0	0
$161$	17.4891	1.37834
$162$	−6.48577	−0.509570
$163$	16.2333	1.27149	0.635744	−	0.771900i	$-0.280693\pi$
$163$	16.2333	1.27149	0.635744	+	0.771900i	$0.280693\pi$
$164$	−4.74456	−0.370488
$165$	0	0
$166$	−16.9783	−1.31777
$167$	−16.7306	−1.29465	−0.647326	−	0.762213i	$-0.724113\pi$
$167$	−16.7306	−1.29465	−0.647326	+	0.762213i	$0.724113\pi$
$168$	6.92820	0.534522
$169$	5.11684	0.393603
$170$	0	0
$171$	−9.48913	−0.725652
$172$	10.8896	0.830327
$173$	−23.9538	−1.82117	−0.910585	−	0.413321i	$-0.864369\pi$
$173$	−23.9538	−1.82117	−0.910585	+	0.413321i	$0.864369\pi$
$174$	−1.37228	−0.104032
$175$	0	0
$176$	−3.13859	−0.236580
$177$	8.51278	0.639860
$178$	1.28962	0.0966611
$179$	5.25544	0.392810	0.196405	−	0.980523i	$-0.437073\pi$
$179$	5.25544	0.392810	0.196405	+	0.980523i	$0.437073\pi$
$180$	0	0
$181$	14.8614	1.10464	0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614	1.10464	0.552320	+	0.833632i	$0.313743\pi$
$182$	−37.2203	−2.75895
$183$	−4.75372	−0.351405
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−4.62772	−0.339321
$187$	0.994667	0.0727372
$188$	10.8896	0.794208
$189$	−21.4891	−1.56310
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	−0.792287	−0.0571784
$193$	−6.92820	−0.498703	−0.249351	−	0.968413i	$-0.580217\pi$
$193$	−6.92820	−0.498703	−0.249351	+	0.968413i	$0.580217\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	18.4891	1.32065
$197$	−0.589907	−0.0420292	−0.0210146	−	0.999779i	$-0.506690\pi$
$197$	−0.589907	−0.0420292	−0.0210146	+	0.999779i	$0.506690\pi$
$198$	2.57924	0.183299
$199$	13.4891	0.956219	0.478109	−	0.878300i	$-0.341322\pi$
$199$	13.4891	0.956219	0.478109	+	0.878300i	$0.341322\pi$
$200$	0	0
$201$	−1.48913	−0.105035
$202$	5.63858	0.396729
$203$	5.04868	0.354348
$204$	−1.25544	−0.0878982
$205$	0	0
$206$	6.00000	0.418040
$207$	8.21782	0.571178
$208$	21.2819	1.47564
$209$	2.51087	0.173681
$210$	0	0
$211$	6.11684	0.421101	0.210550	−	0.977583i	$-0.432474\pi$
$211$	6.11684	0.421101	0.210550	+	0.977583i	$0.432474\pi$
$212$	−4.25639	−0.292330
$213$	−5.34363	−0.366139
$214$	−8.74456	−0.597766
$215$	0	0
$216$	7.37228	0.501620
$217$	17.0256	1.15577
$218$	27.9152	1.89066
$219$	−5.48913	−0.370921
$220$	0	0
$221$	−6.74456	−0.453688
$222$	−4.34896	−0.291883
$223$	10.3923	0.695920	0.347960	−	0.937509i	$-0.386874\pi$
$223$	10.3923	0.695920	0.347960	+	0.937509i	$0.386874\pi$
$224$	−26.2337	−1.75281
$225$	0	0
$226$	−26.7446	−1.77902
$227$	−0.294954	−0.0195768	−0.00978838	−	0.999952i	$-0.503116\pi$
$227$	−0.294954	−0.0195768	−0.00978838	+	0.999952i	$0.503116\pi$
$228$	−3.16915	−0.209882
$229$	−12.7446	−0.842184	−0.421092	−	0.907018i	$-0.638353\pi$
$229$	−12.7446	−0.842184	−0.421092	+	0.907018i	$0.638353\pi$
$230$	0	0
$231$	2.51087	0.165203
$232$	−1.73205	−0.113715
$233$	−0.497333	−0.0325814	−0.0162907	−	0.999867i	$-0.505186\pi$
$233$	−0.497333	−0.0325814	−0.0162907	+	0.999867i	$0.505186\pi$
$234$	−17.4891	−1.14330
$235$	0	0
$236$	−10.7446	−0.699411
$237$	−9.01011	−0.585270
$238$	13.8564	0.898177
$239$	28.2337	1.82629	0.913143	−	0.407640i	$-0.133648\pi$
$239$	28.2337	1.82629	0.913143	+	0.407640i	$0.133648\pi$
$240$	0	0
$241$	6.62772	0.426929	0.213464	−	0.976951i	$-0.431525\pi$
$241$	6.62772	0.426929	0.213464	+	0.976951i	$0.431525\pi$
$242$	18.3701	1.18087
$243$	−15.7359	−1.00946
$244$	6.00000	0.384111
$245$	0	0
$246$	−6.51087	−0.415118
$247$	−17.0256	−1.08331
$248$	−5.84096	−0.370901
$249$	−7.76631	−0.492170
$250$	0	0
$251$	7.37228	0.465334	0.232667	−	0.972556i	$-0.425255\pi$
$251$	7.37228	0.465334	0.232667	+	0.972556i	$0.425255\pi$
$252$	11.9769	0.754472
$253$	−2.17448	−0.136708
$254$	6.00000	0.376473
$255$	0	0
$256$	19.0000	1.18750
$257$	−17.5229	−1.09305	−0.546524	−	0.837444i	$-0.684049\pi$
$257$	−17.5229	−1.09305	−0.546524	+	0.837444i	$0.684049\pi$
$258$	14.9436	0.930351
$259$	16.0000	0.994192
$260$	0	0
$261$	2.37228	0.146841
$262$	6.92820	0.428026
$263$	−9.89497	−0.610150	−0.305075	−	0.952328i	$-0.598682\pi$
$263$	−9.89497	−0.610150	−0.305075	+	0.952328i	$0.598682\pi$
$264$	−0.861407	−0.0530159
$265$	0	0
$266$	34.9783	2.14465
$267$	0.589907	0.0361017
$268$	1.87953	0.114810
$269$	23.4891	1.43216	0.716079	−	0.698020i	$-0.245935\pi$
$269$	23.4891	1.43216	0.716079	+	0.698020i	$0.245935\pi$
$270$	0	0
$271$	−8.86141	−0.538292	−0.269146	−	0.963099i	$-0.586741\pi$
$271$	−8.86141	−0.538292	−0.269146	+	0.963099i	$0.586741\pi$
$272$	−7.92287	−0.480395
$273$	−17.0256	−1.03043
$274$	12.0000	0.724947
$275$	0	0
$276$	2.74456	0.165203
$277$	6.92820	0.416275	0.208138	−	0.978100i	$-0.433260\pi$
$277$	6.92820	0.416275	0.208138	+	0.978100i	$0.433260\pi$
$278$	4.75372	0.285109
$279$	8.00000	0.478947
$280$	0	0
$281$	−1.37228	−0.0818634	−0.0409317	−	0.999162i	$-0.513033\pi$
$281$	−1.37228	−0.0818634	−0.0409317	+	0.999162i	$0.513033\pi$
$282$	14.9436	0.889881
$283$	−4.05401	−0.240986	−0.120493	−	0.992714i	$-0.538448\pi$
$283$	−4.05401	−0.240986	−0.120493	+	0.992714i	$0.538448\pi$
$284$	6.74456	0.400216
$285$	0	0
$286$	4.62772	0.273643
$287$	23.9538	1.41395
$288$	−12.3267	−0.726360
$289$	−14.4891	−0.852301
$290$	0	0
$291$	−5.48913	−0.321778
$292$	6.92820	0.405442
$293$	2.17448	0.127035	0.0635173	−	0.997981i	$-0.479768\pi$
$293$	2.17448	0.127035	0.0635173	+	0.997981i	$0.479768\pi$
$294$	25.3723	1.47974
$295$	0	0
$296$	−5.48913	−0.319049
$297$	2.67181	0.155034
$298$	6.72582	0.389616
$299$	14.7446	0.852700
$300$	0	0
$301$	−54.9783	−3.16889
$302$	−23.3639	−1.34444
$303$	2.57924	0.148174
$304$	−20.0000	−1.14708
$305$	0	0
$306$	6.51087	0.372202
$307$	−14.0588	−0.802377	−0.401189	−	0.915995i	$-0.631403\pi$
$307$	−14.0588	−0.802377	−0.401189	+	0.915995i	$0.631403\pi$
$308$	−3.16915	−0.180579
$309$	2.74456	0.156133
$310$	0	0
$311$	26.9783	1.52980	0.764898	−	0.644151i	$-0.222789\pi$
$311$	26.9783	1.52980	0.764898	+	0.644151i	$0.222789\pi$
$312$	5.84096	0.330679
$313$	14.3537	0.811321	0.405661	−	0.914024i	$-0.367041\pi$
$313$	14.3537	0.811321	0.405661	+	0.914024i	$0.367041\pi$
$314$	24.0000	1.35440
$315$	0	0
$316$	11.3723	0.639741
$317$	22.3692	1.25638	0.628189	−	0.778061i	$-0.283796\pi$
$317$	22.3692	1.25638	0.628189	+	0.778061i	$0.283796\pi$
$318$	−5.84096	−0.327545
$319$	−0.627719	−0.0351455
$320$	0	0
$321$	−4.00000	−0.223258
$322$	−30.2921	−1.68811
$323$	6.33830	0.352672
$324$	3.74456	0.208031
$325$	0	0
$326$	−28.1168	−1.55725
$327$	12.7692	0.706136
$328$	−8.21782	−0.453753
$329$	−54.9783	−3.03105
$330$	0	0
$331$	28.8614	1.58637	0.793183	−	0.608983i	$-0.208422\pi$
$331$	28.8614	1.58637	0.793183	+	0.608983i	$0.208422\pi$
$332$	9.80240	0.537976
$333$	7.51811	0.411990
$334$	28.9783	1.58562
$335$	0	0
$336$	−20.0000	−1.09109
$337$	7.92287	0.431586	0.215793	−	0.976439i	$-0.430766\pi$
$337$	7.92287	0.431586	0.215793	+	0.976439i	$0.430766\pi$
$338$	−8.86263	−0.482064
$339$	−12.2337	−0.664443
$340$	0	0
$341$	−2.11684	−0.114634
$342$	16.4356	0.888738
$343$	−58.0049	−3.13197
$344$	18.8614	1.01694
$345$	0	0
$346$	41.4891	2.23047
$347$	19.8997	1.06827	0.534137	−	0.845398i	$-0.320637\pi$
$347$	19.8997	1.06827	0.534137	+	0.845398i	$0.320637\pi$
$348$	0.792287	0.0424710
$349$	26.8614	1.43786	0.718929	−	0.695083i	$-0.244633\pi$
$349$	26.8614	1.43786	0.718929	+	0.695083i	$0.244633\pi$
$350$	0	0
$351$	−18.1168	−0.967006
$352$	3.26172	0.173850
$353$	10.6873	0.568825	0.284413	−	0.958702i	$-0.408201\pi$
$353$	10.6873	0.568825	0.284413	+	0.958702i	$0.408201\pi$
$354$	−14.7446	−0.783665
$355$	0	0
$356$	−0.744563	−0.0394617
$357$	6.33830	0.335458
$358$	−9.10268	−0.481092
$359$	−13.8832	−0.732725	−0.366362	−	0.930472i	$-0.619397\pi$
$359$	−13.8832	−0.732725	−0.366362	+	0.930472i	$0.619397\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−25.7407	−1.35290
$363$	8.40297	0.441042
$364$	21.4891	1.12634
$365$	0	0
$366$	8.23369	0.430382
$367$	10.3923	0.542474	0.271237	−	0.962513i	$-0.412567\pi$
$367$	10.3923	0.542474	0.271237	+	0.962513i	$0.412567\pi$
$368$	17.3205	0.902894
$369$	11.2554	0.585935
$370$	0	0
$371$	21.4891	1.11566
$372$	2.67181	0.138527
$373$	−24.4511	−1.26603	−0.633015	−	0.774140i	$-0.718183\pi$
$373$	−24.4511	−1.26603	−0.633015	+	0.774140i	$0.718183\pi$
$374$	−1.72281	−0.0890846
$375$	0	0
$376$	18.8614	0.972703
$377$	4.25639	0.219215
$378$	37.2203	1.91440
$379$	1.48913	0.0764912	0.0382456	−	0.999268i	$-0.487823\pi$
$379$	1.48913	0.0764912	0.0382456	+	0.999268i	$0.487823\pi$
$380$	0	0
$381$	2.74456	0.140608
$382$	27.7128	1.41791
$383$	−1.28962	−0.0658965	−0.0329483	−	0.999457i	$-0.510490\pi$
$383$	−1.28962	−0.0658965	−0.0329483	+	0.999457i	$0.510490\pi$
$384$	9.60597	0.490203
$385$	0	0
$386$	12.0000	0.610784
$387$	−25.8333	−1.31318
$388$	6.92820	0.351726
$389$	8.74456	0.443367	0.221683	−	0.975119i	$-0.428845\pi$
$389$	8.74456	0.443367	0.221683	+	0.975119i	$0.428845\pi$
$390$	0	0
$391$	−5.48913	−0.277597
$392$	32.0241	1.61746
$393$	3.16915	0.159862
$394$	1.02175	0.0514750
$395$	0	0
$396$	−1.48913	−0.0748314
$397$	15.9383	0.799921	0.399961	−	0.916532i	$-0.369024\pi$
$397$	15.9383	0.799921	0.399961	+	0.916532i	$0.369024\pi$
$398$	−23.3639	−1.17112
$399$	16.0000	0.801002
$400$	0	0
$401$	4.11684	0.205585	0.102793	−	0.994703i	$-0.467222\pi$
$401$	4.11684	0.205585	0.102793	+	0.994703i	$0.467222\pi$
$402$	2.57924	0.128641
$403$	14.3537	0.715011
$404$	−3.25544	−0.161964
$405$	0	0
$406$	−8.74456	−0.433985
$407$	−1.98933	−0.0986076
$408$	−2.17448	−0.107653
$409$	38.4674	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$409$	38.4674	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$410$	0	0
$411$	5.48913	0.270759
$412$	−3.46410	−0.170664
$413$	54.2458	2.66926
$414$	−14.2337	−0.699548
$415$	0	0
$416$	−22.1168	−1.08437
$417$	2.17448	0.106485
$418$	−4.34896	−0.212715
$419$	13.2554	0.647570	0.323785	−	0.946131i	$-0.395044\pi$
$419$	13.2554	0.647570	0.323785	+	0.946131i	$0.395044\pi$
$420$	0	0
$421$	24.9783	1.21737	0.608683	−	0.793414i	$-0.291698\pi$
$421$	24.9783	1.21737	0.608683	+	0.793414i	$0.291698\pi$
$422$	−10.5947	−0.515741
$423$	−25.8333	−1.25606
$424$	−7.37228	−0.358030
$425$	0	0
$426$	9.25544	0.448427
$427$	−30.2921	−1.46594
$428$	5.04868	0.244037
$429$	2.11684	0.102202
$430$	0	0
$431$	−22.7446	−1.09557	−0.547784	−	0.836620i	$-0.684528\pi$
$431$	−22.7446	−1.09557	−0.547784	+	0.836620i	$0.684528\pi$
$432$	−21.2819	−1.02393
$433$	40.9793	1.96934	0.984670	−	0.174427i	$-0.0558074\pi$
$433$	40.9793	1.96934	0.984670	+	0.174427i	$0.0558074\pi$
$434$	−29.4891	−1.41552
$435$	0	0
$436$	−16.1168	−0.771857
$437$	−13.8564	−0.662842
$438$	9.50744	0.454283
$439$	−4.23369	−0.202063	−0.101031	−	0.994883i	$-0.532214\pi$
$439$	−4.23369	−0.202063	−0.101031	+	0.994883i	$0.532214\pi$
$440$	0	0
$441$	−43.8614	−2.08864
$442$	11.6819	0.555653
$443$	−20.4897	−0.973493	−0.486746	−	0.873543i	$-0.661816\pi$
$443$	−20.4897	−0.973493	−0.486746	+	0.873543i	$0.661816\pi$
$444$	2.51087	0.119161
$445$	0	0
$446$	−18.0000	−0.852325
$447$	3.07657	0.145517
$448$	−5.04868	−0.238528
$449$	−34.4674	−1.62662	−0.813308	−	0.581833i	$-0.802336\pi$
$449$	−34.4674	−1.62662	−0.813308	+	0.581833i	$0.802336\pi$
$450$	0	0
$451$	−2.97825	−0.140240
$452$	15.4410	0.726283
$453$	−10.6873	−0.502131
$454$	0.510875	0.0239765
$455$	0	0
$456$	−5.48913	−0.257052
$457$	−37.2203	−1.74109	−0.870545	−	0.492089i	$-0.836234\pi$
$457$	−37.2203	−1.74109	−0.870545	+	0.492089i	$0.836234\pi$
$458$	22.0742	1.03146
$459$	6.74456	0.314809
$460$	0	0
$461$	27.4891	1.28030	0.640148	−	0.768252i	$-0.278873\pi$
$461$	27.4891	1.28030	0.640148	+	0.768252i	$0.278873\pi$
$462$	−4.34896	−0.202332
$463$	7.22316	0.335689	0.167844	−	0.985814i	$-0.446319\pi$
$463$	7.22316	0.335689	0.167844	+	0.985814i	$0.446319\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	0.861407	0.0399039
$467$	−3.37153	−0.156016	−0.0780078	−	0.996953i	$-0.524856\pi$
$467$	−3.37153	−0.156016	−0.0780078	+	0.996953i	$0.524856\pi$
$468$	10.0974	0.466750
$469$	−9.48913	−0.438167
$470$	0	0
$471$	10.9783	0.505851
$472$	−18.6101	−0.856601
$473$	6.83563	0.314303
$474$	15.6060	0.716806
$475$	0	0
$476$	−8.00000	−0.366679
$477$	10.0974	0.462326
$478$	−48.9022	−2.23673
$479$	−4.62772	−0.211446	−0.105723	−	0.994396i	$-0.533716\pi$
$479$	−4.62772	−0.211446	−0.105723	+	0.994396i	$0.533716\pi$
$480$	0	0
$481$	13.4891	0.615051
$482$	−11.4795	−0.522879
$483$	−13.8564	−0.630488
$484$	−10.6060	−0.482090
$485$	0	0
$486$	27.2554	1.23633
$487$	29.0024	1.31423	0.657113	−	0.753792i	$-0.271777\pi$
$487$	29.0024	1.31423	0.657113	+	0.753792i	$0.271777\pi$
$488$	10.3923	0.470438
$489$	−12.8614	−0.581613
$490$	0	0
$491$	−23.3723	−1.05478	−0.527388	−	0.849624i	$-0.676829\pi$
$491$	−23.3723	−1.05478	−0.527388	+	0.849624i	$0.676829\pi$
$492$	3.75906	0.169471
$493$	−1.58457	−0.0713656
$494$	29.4891	1.32678
$495$	0	0
$496$	16.8614	0.757100
$497$	−34.0511	−1.52740
$498$	13.4516	0.602783
$499$	5.25544	0.235266	0.117633	−	0.993057i	$-0.462469\pi$
$499$	5.25544	0.235266	0.117633	+	0.993057i	$0.462469\pi$
$500$	0	0
$501$	13.2554	0.592209
$502$	−12.7692	−0.569916
$503$	−3.37153	−0.150329	−0.0751645	−	0.997171i	$-0.523948\pi$
$503$	−3.37153	−0.150329	−0.0751645	+	0.997171i	$0.523948\pi$
$504$	20.7446	0.924036
$505$	0	0
$506$	3.76631	0.167433
$507$	−4.05401	−0.180045
$508$	−3.46410	−0.153695
$509$	−22.8614	−1.01331	−0.506657	−	0.862148i	$-0.669119\pi$
$509$	−22.8614	−1.01331	−0.506657	+	0.862148i	$0.669119\pi$
$510$	0	0
$511$	−34.9783	−1.54735
$512$	−8.66025	−0.382733
$513$	17.0256	0.751697
$514$	30.3505	1.33870
$515$	0	0
$516$	−8.62772	−0.379814
$517$	6.83563	0.300631
$518$	−27.7128	−1.21763
$519$	18.9783	0.833053
$520$	0	0
$521$	2.86141	0.125360	0.0626802	−	0.998034i	$-0.480035\pi$
$521$	2.86141	0.125360	0.0626802	+	0.998034i	$0.480035\pi$
$522$	−4.10891	−0.179842
$523$	15.1460	0.662290	0.331145	−	0.943580i	$-0.392565\pi$
$523$	15.1460	0.662290	0.331145	+	0.943580i	$0.392565\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	17.1386	0.747278
$527$	−5.34363	−0.232772
$528$	2.48667	0.108218
$529$	−11.0000	−0.478261
$530$	0	0
$531$	25.4891	1.10613
$532$	−20.1947	−0.875551
$533$	20.1947	0.874730
$534$	−1.02175	−0.0442154
$535$	0	0
$536$	3.25544	0.140613
$537$	−4.16381	−0.179682
$538$	−40.6844	−1.75403
$539$	11.6060	0.499904
$540$	0	0
$541$	23.2554	0.999829	0.499915	−	0.866075i	$-0.333365\pi$
$541$	23.2554	0.999829	0.499915	+	0.866075i	$0.333365\pi$
$542$	15.3484	0.659271
$543$	−11.7745	−0.505292
$544$	8.23369	0.353016
$545$	0	0
$546$	29.4891	1.26202
$547$	−15.1460	−0.647597	−0.323799	−	0.946126i	$-0.604960\pi$
$547$	−15.1460	−0.647597	−0.323799	+	0.946126i	$0.604960\pi$
$548$	−6.92820	−0.295958
$549$	−14.2337	−0.607479
$550$	0	0
$551$	−4.00000	−0.170406
$552$	4.75372	0.202332
$553$	−57.4150	−2.44153
$554$	−12.0000	−0.509831
$555$	0	0
$556$	−2.74456	−0.116395
$557$	−17.6155	−0.746391	−0.373196	−	0.927753i	$-0.621738\pi$
$557$	−17.6155	−0.746391	−0.373196	+	0.927753i	$0.621738\pi$
$558$	−13.8564	−0.586588
$559$	−46.3505	−1.96042
$560$	0	0
$561$	−0.788061	−0.0332720
$562$	2.37686	0.100262
$563$	−17.8178	−0.750932	−0.375466	−	0.926836i	$-0.622517\pi$
$563$	−17.8178	−0.750932	−0.375466	+	0.926836i	$0.622517\pi$
$564$	−8.62772	−0.363292
$565$	0	0
$566$	7.02175	0.295146
$567$	−18.9051	−0.793939
$568$	11.6819	0.490163
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−2.67181	−0.111714
$573$	12.6766	0.529572
$574$	−41.4891	−1.73172
$575$	0	0
$576$	−2.37228	−0.0988451
$577$	15.4410	0.642816	0.321408	−	0.946941i	$-0.395844\pi$
$577$	15.4410	0.642816	0.321408	+	0.946941i	$0.395844\pi$
$578$	25.0959	1.04385
$579$	5.48913	0.228120
$580$	0	0
$581$	−49.4891	−2.05315
$582$	9.50744	0.394096
$583$	−2.67181	−0.110655
$584$	12.0000	0.496564
$585$	0	0
$586$	−3.76631	−0.155585
$587$	−16.1407	−0.666198	−0.333099	−	0.942892i	$-0.608094\pi$
$587$	−16.1407	−0.666198	−0.333099	+	0.942892i	$0.608094\pi$
$588$	−14.6487	−0.604102
$589$	−13.4891	−0.555810
$590$	0	0
$591$	0.467376	0.0192253
$592$	15.8457	0.651256
$593$	9.01011	0.370001	0.185000	−	0.982738i	$-0.440771\pi$
$593$	9.01011	0.370001	0.185000	+	0.982738i	$0.440771\pi$
$594$	−4.62772	−0.189878
$595$	0	0
$596$	−3.88316	−0.159060
$597$	−10.6873	−0.437400
$598$	−25.5383	−1.04434
$599$	30.3505	1.24009	0.620045	−	0.784567i	$-0.287115\pi$
$599$	30.3505	1.24009	0.620045	+	0.784567i	$0.287115\pi$
$600$	0	0
$601$	−7.25544	−0.295955	−0.147978	−	0.988991i	$-0.547276\pi$
$601$	−7.25544	−0.295955	−0.147978	+	0.988991i	$0.547276\pi$
$602$	95.2251	3.88109
$603$	−4.45877	−0.181575
$604$	13.4891	0.548865
$605$	0	0
$606$	−4.46738	−0.181475
$607$	17.2279	0.699260	0.349630	−	0.936888i	$-0.386307\pi$
$607$	17.2279	0.699260	0.349630	+	0.936888i	$0.386307\pi$
$608$	20.7846	0.842927
$609$	−4.00000	−0.162088
$610$	0	0
$611$	−46.3505	−1.87514
$612$	−3.75906	−0.151951
$613$	−16.9330	−0.683917	−0.341958	−	0.939715i	$-0.611090\pi$
$613$	−16.9330	−0.683917	−0.341958	+	0.939715i	$0.611090\pi$
$614$	24.3505	0.982707
$615$	0	0
$616$	−5.48913	−0.221163
$617$	37.8102	1.52218	0.761090	−	0.648646i	$-0.224665\pi$
$617$	37.8102	1.52218	0.761090	+	0.648646i	$0.224665\pi$
$618$	−4.75372	−0.191223
$619$	31.3723	1.26096	0.630479	−	0.776206i	$-0.282858\pi$
$619$	31.3723	1.26096	0.630479	+	0.776206i	$0.282858\pi$
$620$	0	0
$621$	−14.7446	−0.591679
$622$	−46.7277	−1.87361
$623$	3.75906	0.150603
$624$	−16.8614	−0.674996
$625$	0	0
$626$	−24.8614	−0.993662
$627$	−1.98933	−0.0794463
$628$	−13.8564	−0.552931
$629$	−5.02175	−0.200230
$630$	0	0
$631$	−28.2337	−1.12397	−0.561983	−	0.827149i	$-0.689961\pi$
$631$	−28.2337	−1.12397	−0.561983	+	0.827149i	$0.689961\pi$
$632$	19.6974	0.783519
$633$	−4.84630	−0.192623
$634$	−38.7446	−1.53874
$635$	0	0
$636$	3.37228	0.133720
$637$	−78.6969	−3.11808
$638$	1.08724	0.0430443
$639$	−16.0000	−0.632950
$640$	0	0
$641$	−19.4891	−0.769774	−0.384887	−	0.922964i	$-0.625760\pi$
$641$	−19.4891	−0.769774	−0.384887	+	0.922964i	$0.625760\pi$
$642$	6.92820	0.273434
$643$	15.1460	0.597301	0.298650	−	0.954363i	$-0.403464\pi$
$643$	15.1460	0.597301	0.298650	+	0.954363i	$0.403464\pi$
$644$	17.4891	0.689168
$645$	0	0
$646$	−10.9783	−0.431933
$647$	14.1514	0.556347	0.278174	−	0.960531i	$-0.410271\pi$
$647$	14.1514	0.556347	0.278174	+	0.960531i	$0.410271\pi$
$648$	6.48577	0.254785
$649$	−6.74456	−0.264747
$650$	0	0
$651$	−13.4891	−0.528681
$652$	16.2333	0.635744
$653$	30.8820	1.20850	0.604252	−	0.796793i	$-0.293472\pi$
$653$	30.8820	1.20850	0.604252	+	0.796793i	$0.293472\pi$
$654$	−22.1168	−0.864837
$655$	0	0
$656$	23.7228	0.926220
$657$	−16.4356	−0.641216
$658$	95.2251	3.71226
$659$	10.3505	0.403199	0.201600	−	0.979468i	$-0.435386\pi$
$659$	10.3505	0.403199	0.201600	+	0.979468i	$0.435386\pi$
$660$	0	0
$661$	−39.4891	−1.53595	−0.767974	−	0.640480i	$-0.778735\pi$
$661$	−39.4891	−1.53595	−0.767974	+	0.640480i	$0.778735\pi$
$662$	−49.9894	−1.94289
$663$	5.34363	0.207529
$664$	16.9783	0.658884
$665$	0	0
$666$	−13.0217	−0.504583
$667$	3.46410	0.134131
$668$	−16.7306	−0.647326
$669$	−8.23369	−0.318333
$670$	0	0
$671$	3.76631	0.145397
$672$	20.7846	0.801784
$673$	39.8921	1.53773	0.768863	−	0.639413i	$-0.220823\pi$
$673$	39.8921	1.53773	0.768863	+	0.639413i	$0.220823\pi$
$674$	−13.7228	−0.528583
$675$	0	0
$676$	5.11684	0.196802
$677$	45.7330	1.75766	0.878832	−	0.477132i	$-0.158324\pi$
$677$	45.7330	1.75766	0.878832	+	0.477132i	$0.158324\pi$
$678$	21.1894	0.813773
$679$	−34.9783	−1.34234
$680$	0	0
$681$	0.233688	0.00895494
$682$	3.66648	0.140397
$683$	−14.5561	−0.556974	−0.278487	−	0.960440i	$-0.589833\pi$
$683$	−14.5561	−0.556974	−0.278487	+	0.960440i	$0.589833\pi$
$684$	−9.48913	−0.362826
$685$	0	0
$686$	100.467	3.83586
$687$	10.0974	0.385238
$688$	−54.4482	−2.07582
$689$	18.1168	0.690197
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−23.9538	−0.910585
$693$	7.51811	0.285589
$694$	−34.4674	−1.30836
$695$	0	0
$696$	1.37228	0.0520162
$697$	−7.51811	−0.284769
$698$	−46.5253	−1.76101
$699$	0.394031	0.0149036
$700$	0	0
$701$	0.116844	0.00441314	0.00220657	−	0.999998i	$-0.499298\pi$
$701$	0.116844	0.00441314	0.00220657	+	0.999998i	$0.499298\pi$
$702$	31.3793	1.18434
$703$	−12.6766	−0.478107
$704$	0.627719	0.0236580
$705$	0	0
$706$	−18.5109	−0.696666
$707$	16.4356	0.618126
$708$	8.51278	0.319930
$709$	−35.0951	−1.31802	−0.659012	−	0.752132i	$-0.729025\pi$
$709$	−35.0951	−1.31802	−0.659012	+	0.752132i	$0.729025\pi$
$710$	0	0
$711$	−26.9783	−1.01176
$712$	−1.28962	−0.0483306
$713$	11.6819	0.437492
$714$	−10.9783	−0.410851
$715$	0	0
$716$	5.25544	0.196405
$717$	−22.3692	−0.835392
$718$	24.0463	0.897401
$719$	14.7446	0.549879	0.274940	−	0.961461i	$-0.411342\pi$
$719$	14.7446	0.549879	0.274940	+	0.961461i	$0.411342\pi$
$720$	0	0
$721$	17.4891	0.651329
$722$	5.19615	0.193381
$723$	−5.25106	−0.195289
$724$	14.8614	0.552320
$725$	0	0
$726$	−14.5544	−0.540163
$727$	13.5615	0.502966	0.251483	−	0.967862i	$-0.419082\pi$
$727$	13.5615	0.502966	0.251483	+	0.967862i	$0.419082\pi$
$728$	37.2203	1.37947
$729$	1.23369	0.0456921
$730$	0	0
$731$	17.2554	0.638215
$732$	−4.75372	−0.175703
$733$	−27.7128	−1.02360	−0.511798	−	0.859106i	$-0.671020\pi$
$733$	−27.7128	−1.02360	−0.511798	+	0.859106i	$0.671020\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	−18.0000	−0.663489
$737$	1.17981	0.0434590
$738$	−19.4950	−0.717620
$739$	15.3723	0.565479	0.282739	−	0.959197i	$-0.408757\pi$
$739$	15.3723	0.565479	0.282739	+	0.959197i	$0.408757\pi$
$740$	0	0
$741$	13.4891	0.495535
$742$	−37.2203	−1.36640
$743$	17.3205	0.635428	0.317714	−	0.948187i	$-0.397085\pi$
$743$	17.3205	0.635428	0.317714	+	0.948187i	$0.397085\pi$
$744$	4.62772	0.169660
$745$	0	0
$746$	42.3505	1.55056
$747$	−23.2540	−0.850821
$748$	0.994667	0.0363686
$749$	−25.4891	−0.931352
$750$	0	0
$751$	48.4674	1.76860	0.884300	−	0.466919i	$-0.154636\pi$
$751$	48.4674	1.76860	0.884300	+	0.466919i	$0.154636\pi$
$752$	−54.4482	−1.98552
$753$	−5.84096	−0.212857
$754$	−7.37228	−0.268483
$755$	0	0
$756$	−21.4891	−0.781552
$757$	36.2256	1.31664	0.658321	−	0.752738i	$-0.271267\pi$
$757$	36.2256	1.31664	0.658321	+	0.752738i	$0.271267\pi$
$758$	−2.57924	−0.0936822
$759$	1.72281	0.0625342
$760$	0	0
$761$	−30.4674	−1.10444	−0.552221	−	0.833698i	$-0.686219\pi$
$761$	−30.4674	−1.10444	−0.552221	+	0.833698i	$0.686219\pi$
$762$	−4.75372	−0.172209
$763$	81.3687	2.94575
$764$	−16.0000	−0.578860
$765$	0	0
$766$	2.23369	0.0807064
$767$	45.7330	1.65132
$768$	−15.0535	−0.543195
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	13.8832	0.499990
$772$	−6.92820	−0.249351
$773$	−38.4001	−1.38115	−0.690577	−	0.723259i	$-0.742643\pi$
$773$	−38.4001	−1.38115	−0.690577	+	0.723259i	$0.742643\pi$
$774$	44.7446	1.60831
$775$	0	0
$776$	12.0000	0.430775
$777$	−12.6766	−0.454770
$778$	−15.1460	−0.543011
$779$	−18.9783	−0.679966
$780$	0	0
$781$	4.23369	0.151493
$782$	9.50744	0.339985
$783$	−4.25639	−0.152111
$784$	−92.4456	−3.30163
$785$	0	0
$786$	−5.48913	−0.195791
$787$	−0.294954	−0.0105140	−0.00525698	−	0.999986i	$-0.501673\pi$
$787$	−0.294954	−0.0105140	−0.00525698	+	0.999986i	$0.501673\pi$
$788$	−0.589907	−0.0210146
$789$	7.83966	0.279099
$790$	0	0
$791$	−77.9565	−2.77181
$792$	−2.57924	−0.0916493
$793$	−25.5383	−0.906893
$794$	−27.6060	−0.979699
$795$	0	0
$796$	13.4891	0.478109
$797$	28.7075	1.01687	0.508436	−	0.861100i	$-0.330224\pi$
$797$	28.7075	1.01687	0.508436	+	0.861100i	$0.330224\pi$
$798$	−27.7128	−0.981023
$799$	17.2554	0.610453
$800$	0	0
$801$	1.76631	0.0624096
$802$	−7.13058	−0.251790
$803$	4.34896	0.153472
$804$	−1.48913	−0.0525174
$805$	0	0
$806$	−24.8614	−0.875706
$807$	−18.6101	−0.655108
$808$	−5.63858	−0.198365
$809$	12.7446	0.448075	0.224037	−	0.974581i	$-0.428076\pi$
$809$	12.7446	0.448075	0.224037	+	0.974581i	$0.428076\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	5.04868	0.177174
$813$	7.02078	0.246229
$814$	3.44563	0.120769
$815$	0	0
$816$	6.27719	0.219745
$817$	43.5586	1.52392
$818$	−66.6274	−2.32957
$819$	−50.9783	−1.78132
$820$	0	0
$821$	17.3723	0.606297	0.303148	−	0.952943i	$-0.401962\pi$
$821$	17.3723	0.606297	0.303148	+	0.952943i	$0.401962\pi$
$822$	−9.50744	−0.331610
$823$	33.1662	1.15610	0.578051	−	0.816000i	$-0.303813\pi$
$823$	33.1662	1.15610	0.578051	+	0.816000i	$0.303813\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−93.9565	−3.26916
$827$	16.8232	0.584999	0.292500	−	0.956266i	$-0.405513\pi$
$827$	16.8232	0.584999	0.292500	+	0.956266i	$0.405513\pi$
$828$	8.21782	0.285589
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	−5.48913	−0.190416
$832$	−4.25639	−0.147564
$833$	29.2974	1.01509
$834$	−3.76631	−0.130417
$835$	0	0
$836$	2.51087	0.0868404
$837$	−14.3537	−0.496138
$838$	−22.9591	−0.793109
$839$	−19.3723	−0.668805	−0.334403	−	0.942430i	$-0.608535\pi$
$839$	−19.3723	−0.668805	−0.334403	+	0.942430i	$0.608535\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−43.2636	−1.49096
$843$	1.08724	0.0374466
$844$	6.11684	0.210550
$845$	0	0
$846$	44.7446	1.53835
$847$	53.5461	1.83987
$848$	21.2819	0.730825
$849$	3.21194	0.110233
$850$	0	0
$851$	10.9783	0.376330
$852$	−5.34363	−0.183070
$853$	−25.5383	−0.874416	−0.437208	−	0.899360i	$-0.644033\pi$
$853$	−25.5383	−0.874416	−0.437208	+	0.899360i	$0.644033\pi$
$854$	52.4674	1.79540
$855$	0	0
$856$	8.74456	0.298883
$857$	−13.3591	−0.456337	−0.228169	−	0.973622i	$-0.573274\pi$
$857$	−13.3591	−0.456337	−0.228169	+	0.973622i	$0.573274\pi$
$858$	−3.66648	−0.125172
$859$	51.6060	1.76077	0.880386	−	0.474257i	$-0.157283\pi$
$859$	51.6060	1.76077	0.880386	+	0.474257i	$0.157283\pi$
$860$	0	0
$861$	−18.9783	−0.646777
$862$	39.3947	1.34179
$863$	−6.63325	−0.225798	−0.112899	−	0.993606i	$-0.536014\pi$
$863$	−6.63325	−0.225798	−0.112899	+	0.993606i	$0.536014\pi$
$864$	22.1168	0.752430
$865$	0	0
$866$	−70.9783	−2.41194
$867$	11.4795	0.389866
$868$	17.0256	0.577885
$869$	7.13859	0.242160
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−27.9152	−0.945328
$873$	−16.4356	−0.556262
$874$	24.0000	0.811812
$875$	0	0
$876$	−5.48913	−0.185460
$877$	8.42020	0.284330	0.142165	−	0.989843i	$-0.454594\pi$
$877$	8.42020	0.284330	0.142165	+	0.989843i	$0.454594\pi$
$878$	7.33296	0.247476
$879$	−1.72281	−0.0581090
$880$	0	0
$881$	36.9783	1.24583	0.622914	−	0.782290i	$-0.285948\pi$
$881$	36.9783	1.24583	0.622914	+	0.782290i	$0.285948\pi$
$882$	75.9702	2.55805
$883$	1.28962	0.0433992	0.0216996	−	0.999765i	$-0.493092\pi$
$883$	1.28962	0.0433992	0.0216996	+	0.999765i	$0.493092\pi$
$884$	−6.74456	−0.226844
$885$	0	0
$886$	35.4891	1.19228
$887$	37.4226	1.25653	0.628265	−	0.778000i	$-0.283766\pi$
$887$	37.4226	1.25653	0.628265	+	0.778000i	$0.283766\pi$
$888$	4.34896	0.145942
$889$	17.4891	0.586566
$890$	0	0
$891$	2.35053	0.0787458
$892$	10.3923	0.347960
$893$	43.5586	1.45763
$894$	−5.32878	−0.178221
$895$	0	0
$896$	61.2119	2.04495
$897$	−11.6819	−0.390048
$898$	59.6992	1.99219
$899$	3.37228	0.112472
$900$	0	0
$901$	−6.74456	−0.224694
$902$	5.15848	0.171759
$903$	43.5586	1.44954
$904$	26.7446	0.889511
$905$	0	0
$906$	18.5109	0.614983
$907$	33.1662	1.10127	0.550634	−	0.834747i	$-0.314386\pi$
$907$	33.1662	1.10127	0.550634	+	0.834747i	$0.314386\pi$
$908$	−0.294954	−0.00978838
$909$	7.72281	0.256150
$910$	0	0
$911$	−26.1168	−0.865290	−0.432645	−	0.901564i	$-0.642420\pi$
$911$	−26.1168	−0.865290	−0.432645	+	0.901564i	$0.642420\pi$
$912$	15.8457	0.524705
$913$	6.15315	0.203639
$914$	64.4674	2.13239
$915$	0	0
$916$	−12.7446	−0.421092
$917$	20.1947	0.666888
$918$	−11.6819	−0.385561
$919$	52.2337	1.72303	0.861515	−	0.507732i	$-0.169516\pi$
$919$	52.2337	1.72303	0.861515	+	0.507732i	$0.169516\pi$
$920$	0	0
$921$	11.1386	0.367029
$922$	−47.6126	−1.56804
$923$	−28.7075	−0.944918
$924$	2.51087	0.0826017
$925$	0	0
$926$	−12.5109	−0.411133
$927$	8.21782	0.269909
$928$	−5.19615	−0.170572
$929$	−47.4891	−1.55807	−0.779034	−	0.626982i	$-0.784290\pi$
$929$	−47.4891	−1.55807	−0.779034	+	0.626982i	$0.784290\pi$
$930$	0	0
$931$	73.9565	2.42383
$932$	−0.497333	−0.0162907
$933$	−21.3745	−0.699770
$934$	5.83966	0.191079
$935$	0	0
$936$	17.4891	0.571650
$937$	47.9075	1.56507	0.782535	−	0.622606i	$-0.213926\pi$
$937$	47.9075	1.56507	0.782535	+	0.622606i	$0.213926\pi$
$938$	16.4356	0.536643
$939$	−11.3723	−0.371120
$940$	0	0
$941$	25.3723	0.827113	0.413556	−	0.910479i	$-0.364286\pi$
$941$	25.3723	0.827113	0.413556	+	0.910479i	$0.364286\pi$
$942$	−19.0149	−0.619539
$943$	16.4356	0.535218
$944$	53.7228	1.74853
$945$	0	0
$946$	−11.8397	−0.384940
$947$	21.9817	0.714308	0.357154	−	0.934046i	$-0.383747\pi$
$947$	21.9817	0.714308	0.357154	+	0.934046i	$0.383747\pi$
$948$	−9.01011	−0.292635
$949$	−29.4891	−0.957258
$950$	0	0
$951$	−17.7228	−0.574702
$952$	−13.8564	−0.449089
$953$	10.1899	0.330084	0.165042	−	0.986287i	$-0.447224\pi$
$953$	10.1899	0.330084	0.165042	+	0.986287i	$0.447224\pi$
$954$	−17.4891	−0.566231
$955$	0	0
$956$	28.2337	0.913143
$957$	0.497333	0.0160765
$958$	8.01544	0.258967
$959$	34.9783	1.12951
$960$	0	0
$961$	−19.6277	−0.633152
$962$	−23.3639	−0.753281
$963$	−11.9769	−0.385950
$964$	6.62772	0.213464
$965$	0	0
$966$	24.0000	0.772187
$967$	−34.8434	−1.12049	−0.560244	−	0.828328i	$-0.689293\pi$
$967$	−34.8434	−1.12049	−0.560244	+	0.828328i	$0.689293\pi$
$968$	−18.3701	−0.590437
$969$	−5.02175	−0.161322
$970$	0	0
$971$	−6.97825	−0.223943	−0.111971	−	0.993711i	$-0.535716\pi$
$971$	−6.97825	−0.223943	−0.111971	+	0.993711i	$0.535716\pi$
$972$	−15.7359	−0.504730
$973$	13.8564	0.444216
$974$	−50.2337	−1.60959
$975$	0	0
$976$	−30.0000	−0.960277
$977$	−34.5484	−1.10530	−0.552651	−	0.833413i	$-0.686384\pi$
$977$	−34.5484	−1.10530	−0.552651	+	0.833413i	$0.686384\pi$
$978$	22.2766	0.712327
$979$	−0.467376	−0.0149374
$980$	0	0
$981$	38.2337	1.22071
$982$	40.4820	1.29183
$983$	−26.7354	−0.852726	−0.426363	−	0.904552i	$-0.640205\pi$
$983$	−26.7354	−0.852726	−0.426363	+	0.904552i	$0.640205\pi$
$984$	6.51087	0.207559
$985$	0	0
$986$	2.74456	0.0874047
$987$	43.5586	1.38648
$988$	−17.0256	−0.541655
$989$	−37.7228	−1.19952
$990$	0	0
$991$	17.2554	0.548137	0.274069	−	0.961710i	$-0.411630\pi$
$991$	17.2554	0.548137	0.274069	+	0.961710i	$0.411630\pi$
$992$	−17.5229	−0.556352
$993$	−22.8665	−0.725647
$994$	58.9783	1.87068
$995$	0	0
$996$	−7.76631	−0.246085
$997$	−37.2203	−1.17878	−0.589389	−	0.807850i	$-0.700631\pi$
$997$	−37.2203	−1.17878	−0.589389	+	0.807850i	$0.700631\pi$
$998$	−9.10268	−0.288140
$999$	−13.4891	−0.426777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.g.1.1		4
3.2	odd	2		6525.2.a.bk.1.3		4
5.2	odd	4		145.2.b.a.59.2	✓	4
5.3	odd	4		145.2.b.a.59.3	yes	4
5.4	even	2	inner	725.2.a.g.1.4		4
15.2	even	4		1305.2.c.e.784.3		4
15.8	even	4		1305.2.c.e.784.1		4
15.14	odd	2		6525.2.a.bk.1.2		4
20.3	even	4		2320.2.d.c.929.3		4
20.7	even	4		2320.2.d.c.929.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.a.59.2	✓	4	5.2	odd	4
145.2.b.a.59.3	yes	4	5.3	odd	4
725.2.a.g.1.1		4	1.1	even	1	trivial
725.2.a.g.1.4		4	5.4	even	2	inner
1305.2.c.e.784.1		4	15.8	even	4
1305.2.c.e.784.3		4	15.2	even	4
2320.2.d.c.929.2		4	20.7	even	4
2320.2.d.c.929.3		4	20.3	even	4
6525.2.a.bk.1.2		4	15.14	odd	2
6525.2.a.bk.1.3		4	3.2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} -0.792287 q^{3} +1.00000 q^{4} +1.37228 q^{6} -5.04868 q^{7} +1.73205 q^{8} -2.37228 q^{9} +0.627719 q^{11} -0.792287 q^{12} -4.25639 q^{13} +8.74456 q^{14} -5.00000 q^{16} +1.58457 q^{17} +4.10891 q^{18} +4.00000 q^{19} +4.00000 q^{21} -1.08724 q^{22} -3.46410 q^{23} -1.37228 q^{24} +7.37228 q^{26} +4.25639 q^{27} -5.04868 q^{28} -1.00000 q^{29} -3.37228 q^{31} +5.19615 q^{32} -0.497333 q^{33} -2.74456 q^{34} -2.37228 q^{36} -3.16915 q^{37} -6.92820 q^{38} +3.37228 q^{39} -4.74456 q^{41} -6.92820 q^{42} +10.8896 q^{43} +0.627719 q^{44} +6.00000 q^{46} +10.8896 q^{47} +3.96143 q^{48} +18.4891 q^{49} -1.25544 q^{51} -4.25639 q^{52} -4.25639 q^{53} -7.37228 q^{54} -8.74456 q^{56} -3.16915 q^{57} +1.73205 q^{58} -10.7446 q^{59} +6.00000 q^{61} +5.84096 q^{62} +11.9769 q^{63} +1.00000 q^{64} +0.861407 q^{66} +1.87953 q^{67} +1.58457 q^{68} +2.74456 q^{69} +6.74456 q^{71} -4.10891 q^{72} +6.92820 q^{73} +5.48913 q^{74} +4.00000 q^{76} -3.16915 q^{77} -5.84096 q^{78} +11.3723 q^{79} +3.74456 q^{81} +8.21782 q^{82} +9.80240 q^{83} +4.00000 q^{84} -18.8614 q^{86} +0.792287 q^{87} +1.08724 q^{88} -0.744563 q^{89} +21.4891 q^{91} -3.46410 q^{92} +2.67181 q^{93} -18.8614 q^{94} -4.11684 q^{96} +6.92820 q^{97} -32.0241 q^{98} -1.48913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 14 q^{11} + 12 q^{14} - 20 q^{16} + 16 q^{19} + 16 q^{21} + 6 q^{24} + 18 q^{26} - 4 q^{29} - 2 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} + 4 q^{41} + 14 q^{44} + 24 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) \) 4 * q + 4 * q^4 - 6 * q^6 + 2 * q^9 + 14 * q^11 + 12 * q^14 - 20 * q^16 + 16 * q^19 + 16 * q^21 + 6 * q^24 + 18 * q^26 - 4 * q^29 - 2 * q^31 + 12 * q^34 + 2 * q^36 + 2 * q^39 + 4 * q^41 + 14 * q^44 + 24 * q^46 + 28 * q^49 - 28 * q^51 - 18 * q^54 - 12 * q^56 - 20 * q^59 + 24 * q^61 + 4 * q^64 - 54 * q^66 - 12 * q^69 + 4 * q^71 - 24 * q^74 + 16 * q^76 + 34 * q^79 - 8 * q^81 + 16 * q^84 - 18 * q^86 + 20 * q^89 + 40 * q^91 - 18 * q^94 + 18 * q^96 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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