LMFDB - Embedded newform 725.2.a.g.1.4

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			1.4
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	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0.792287	0.457427	0.228714	−	0.973494i	$-0.426548\pi$
$3$	0.792287	0.457427	0.228714	+	0.973494i	$0.426548\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.37228	0.560232
$7$	5.04868	1.90822	0.954110	−	0.299456i	$-0.0968053\pi$
$7$	5.04868	1.90822	0.954110	+	0.299456i	$0.0968053\pi$
$8$	−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	$-0.299027\pi$
$13$	4.25639	1.18051	0.590255	+	0.807217i	$0.299027\pi$
$14$	8.74456	2.33708
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−1.58457	−0.384316	−0.192158	−	0.981364i	$-0.561549\pi$
$17$	−1.58457	−0.384316	−0.192158	+	0.981364i	$0.561549\pi$
$18$	−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	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	−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	$-0.416112\pi$
$37$	3.16915	0.521005	0.260502	+	0.965473i	$0.416112\pi$
$38$	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	$-0.811846\pi$
$43$	−10.8896	−1.66065	−0.830327	+	0.557276i	$0.811846\pi$
$44$	0.627719	0.0946322
$45$	0	0
$46$	6.00000	0.884652
$47$	−10.8896	−1.58842	−0.794208	−	0.607645i	$-0.792114\pi$
$47$	−10.8896	−1.58842	−0.794208	+	0.607645i	$0.792114\pi$
$48$	−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	$-0.405569\pi$
$53$	4.25639	0.584660	0.292330	+	0.956318i	$0.405569\pi$
$54$	−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	$-0.536626\pi$
$67$	−1.87953	−0.229621	−0.114810	+	0.993387i	$0.536626\pi$
$68$	−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	$-0.632883\pi$
$73$	−6.92820	−0.810885	−0.405442	+	0.914121i	$0.632883\pi$
$74$	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	$-0.680811\pi$
$83$	−9.80240	−1.07595	−0.537976	+	0.842960i	$0.680811\pi$
$84$	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	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	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	$-0.445409\pi$
$103$	3.46410	0.341328	0.170664	+	0.985329i	$0.445409\pi$
$104$	−7.37228	−0.722912
$105$	0	0
$106$	7.37228	0.716059
$107$	−5.04868	−0.488074	−0.244037	−	0.969766i	$-0.578472\pi$
$107$	−5.04868	−0.488074	−0.244037	+	0.969766i	$0.578472\pi$
$108$	−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	$-0.758754\pi$
$113$	−15.4410	−1.45257	−0.726283	+	0.687396i	$0.758754\pi$
$114$	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	$-0.450883\pi$
$127$	3.46410	0.307389	0.153695	+	0.988118i	$0.450883\pi$
$128$	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	$-0.404361\pi$
$137$	6.92820	0.591916	0.295958	+	0.955201i	$0.404361\pi$
$138$	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	$-0.313509\pi$
$157$	13.8564	1.10586	0.552931	+	0.833227i	$0.313509\pi$
$158$	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	$-0.719307\pi$
$163$	−16.2333	−1.27149	−0.635744	+	0.771900i	$0.719307\pi$
$164$	−4.74456	−0.370488
$165$	0	0
$166$	−16.9783	−1.31777
$167$	16.7306	1.29465	0.647326	−	0.762213i	$-0.275887\pi$
$167$	16.7306	1.29465	0.647326	+	0.762213i	$0.275887\pi$
$168$	−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	$-0.135631\pi$
$173$	23.9538	1.82117	0.910585	+	0.413321i	$0.135631\pi$
$174$	−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	$-0.419783\pi$
$193$	6.92820	0.498703	0.249351	+	0.968413i	$0.419783\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	18.4891	1.32065
$197$	0.589907	0.0420292	0.0210146	−	0.999779i	$-0.493310\pi$
$197$	0.589907	0.0420292	0.0210146	+	0.999779i	$0.493310\pi$
$198$	−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	$-0.613126\pi$
$223$	−10.3923	−0.695920	−0.347960	+	0.937509i	$0.613126\pi$
$224$	−26.2337	−1.75281
$225$	0	0
$226$	−26.7446	−1.77902
$227$	0.294954	0.0195768	0.00978838	−	0.999952i	$-0.496884\pi$
$227$	0.294954	0.0195768	0.00978838	+	0.999952i	$0.496884\pi$
$228$	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	$-0.494814\pi$
$233$	0.497333	0.0325814	0.0162907	+	0.999867i	$0.494814\pi$
$234$	−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	$-0.315951\pi$
$257$	17.5229	1.09305	0.546524	+	0.837444i	$0.315951\pi$
$258$	−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	$-0.401318\pi$
$263$	9.89497	0.610150	0.305075	+	0.952328i	$0.401318\pi$
$264$	−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	$-0.566740\pi$
$277$	−6.92820	−0.416275	−0.208138	+	0.978100i	$0.566740\pi$
$278$	−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	$-0.461552\pi$
$283$	4.05401	0.240986	0.120493	+	0.992714i	$0.461552\pi$
$284$	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	$-0.520232\pi$
$293$	−2.17448	−0.127035	−0.0635173	+	0.997981i	$0.520232\pi$
$294$	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	$-0.368597\pi$
$307$	14.0588	0.802377	0.401189	+	0.915995i	$0.368597\pi$
$308$	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	$-0.632959\pi$
$313$	−14.3537	−0.811321	−0.405661	+	0.914024i	$0.632959\pi$
$314$	24.0000	1.35440
$315$	0	0
$316$	11.3723	0.639741
$317$	−22.3692	−1.25638	−0.628189	−	0.778061i	$-0.716204\pi$
$317$	−22.3692	−1.25638	−0.628189	+	0.778061i	$0.716204\pi$
$318$	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	$-0.569234\pi$
$337$	−7.92287	−0.431586	−0.215793	+	0.976439i	$0.569234\pi$
$338$	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	$-0.679363\pi$
$347$	−19.8997	−1.06827	−0.534137	+	0.845398i	$0.679363\pi$
$348$	−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	$-0.591799\pi$
$353$	−10.6873	−0.568825	−0.284413	+	0.958702i	$0.591799\pi$
$354$	−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	$-0.587433\pi$
$367$	−10.3923	−0.542474	−0.271237	+	0.962513i	$0.587433\pi$
$368$	−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	$-0.281817\pi$
$373$	24.4511	1.26603	0.633015	+	0.774140i	$0.281817\pi$
$374$	−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	$-0.489510\pi$
$383$	1.28962	0.0658965	0.0329483	+	0.999457i	$0.489510\pi$
$384$	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	$-0.630976\pi$
$397$	−15.9383	−0.799921	−0.399961	+	0.916532i	$0.630976\pi$
$398$	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	$-0.944193\pi$
$433$	−40.9793	−1.96934	−0.984670	+	0.174427i	$0.944193\pi$
$434$	−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	$-0.338184\pi$
$443$	20.4897	0.973493	0.486746	+	0.873543i	$0.338184\pi$
$444$	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	$-0.163766\pi$
$457$	37.2203	1.74109	0.870545	+	0.492089i	$0.163766\pi$
$458$	−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	$-0.553681\pi$
$463$	−7.22316	−0.335689	−0.167844	+	0.985814i	$0.553681\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	0.861407	0.0399039
$467$	3.37153	0.156016	0.0780078	−	0.996953i	$-0.475144\pi$
$467$	3.37153	0.156016	0.0780078	+	0.996953i	$0.475144\pi$
$468$	−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	$-0.728223\pi$
$487$	−29.0024	−1.31423	−0.657113	+	0.753792i	$0.728223\pi$
$488$	−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	$-0.476052\pi$
$503$	3.37153	0.150329	0.0751645	+	0.997171i	$0.476052\pi$
$504$	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	$-0.607435\pi$
$523$	−15.1460	−0.662290	−0.331145	+	0.943580i	$0.607435\pi$
$524$	−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	$-0.395040\pi$
$547$	15.1460	0.647597	0.323799	+	0.946126i	$0.395040\pi$
$548$	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	$-0.378262\pi$
$557$	17.6155	0.746391	0.373196	+	0.927753i	$0.378262\pi$
$558$	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	$-0.377483\pi$
$563$	17.8178	0.750932	0.375466	+	0.926836i	$0.377483\pi$
$564$	−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	$-0.604156\pi$
$577$	−15.4410	−0.642816	−0.321408	+	0.946941i	$0.604156\pi$
$578$	−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	$-0.391906\pi$
$587$	16.1407	0.666198	0.333099	+	0.942892i	$0.391906\pi$
$588$	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	$-0.559229\pi$
$593$	−9.01011	−0.370001	−0.185000	+	0.982738i	$0.559229\pi$
$594$	−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	$-0.613693\pi$
$607$	−17.2279	−0.699260	−0.349630	+	0.936888i	$0.613693\pi$
$608$	−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	$-0.388910\pi$
$613$	16.9330	0.683917	0.341958	+	0.939715i	$0.388910\pi$
$614$	24.3505	0.982707
$615$	0	0
$616$	−5.48913	−0.221163
$617$	−37.8102	−1.52218	−0.761090	−	0.648646i	$-0.775335\pi$
$617$	−37.8102	−1.52218	−0.761090	+	0.648646i	$0.775335\pi$
$618$	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	$-0.596536\pi$
$643$	−15.1460	−0.597301	−0.298650	+	0.954363i	$0.596536\pi$
$644$	17.4891	0.689168
$645$	0	0
$646$	−10.9783	−0.431933
$647$	−14.1514	−0.556347	−0.278174	−	0.960531i	$-0.589729\pi$
$647$	−14.1514	−0.556347	−0.278174	+	0.960531i	$0.589729\pi$
$648$	−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	$-0.706528\pi$
$653$	−30.8820	−1.20850	−0.604252	+	0.796793i	$0.706528\pi$
$654$	−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	$-0.779177\pi$
$673$	−39.8921	−1.53773	−0.768863	+	0.639413i	$0.779177\pi$
$674$	−13.7228	−0.528583
$675$	0	0
$676$	5.11684	0.196802
$677$	−45.7330	−1.75766	−0.878832	−	0.477132i	$-0.841676\pi$
$677$	−45.7330	−1.75766	−0.878832	+	0.477132i	$0.841676\pi$
$678$	−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	$-0.410167\pi$
$683$	14.5561	0.556974	0.278487	+	0.960440i	$0.410167\pi$
$684$	−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	$-0.580918\pi$
$727$	−13.5615	−0.502966	−0.251483	+	0.967862i	$0.580918\pi$
$728$	−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	$-0.328980\pi$
$733$	27.7128	1.02360	0.511798	+	0.859106i	$0.328980\pi$
$734$	−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	$-0.602915\pi$
$743$	−17.3205	−0.635428	−0.317714	+	0.948187i	$0.602915\pi$
$744$	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	$-0.728733\pi$
$757$	−36.2256	−1.31664	−0.658321	+	0.752738i	$0.728733\pi$
$758$	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	$-0.257357\pi$
$773$	38.4001	1.38115	0.690577	+	0.723259i	$0.257357\pi$
$774$	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	$-0.498327\pi$
$787$	0.294954	0.0105140	0.00525698	+	0.999986i	$0.498327\pi$
$788$	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	$-0.669776\pi$
$797$	−28.7075	−1.01687	−0.508436	+	0.861100i	$0.669776\pi$
$798$	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	$-0.696187\pi$
$823$	−33.1662	−1.15610	−0.578051	+	0.816000i	$0.696187\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−93.9565	−3.26916
$827$	−16.8232	−0.584999	−0.292500	−	0.956266i	$-0.594487\pi$
$827$	−16.8232	−0.584999	−0.292500	+	0.956266i	$0.594487\pi$
$828$	−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	$-0.355967\pi$
$853$	25.5383	0.874416	0.437208	+	0.899360i	$0.355967\pi$
$854$	52.4674	1.79540
$855$	0	0
$856$	8.74456	0.298883
$857$	13.3591	0.456337	0.228169	−	0.973622i	$-0.426726\pi$
$857$	13.3591	0.456337	0.228169	+	0.973622i	$0.426726\pi$
$858$	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	$-0.463986\pi$
$863$	6.63325	0.225798	0.112899	+	0.993606i	$0.463986\pi$
$864$	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	$-0.545406\pi$
$877$	−8.42020	−0.284330	−0.142165	+	0.989843i	$0.545406\pi$
$878$	−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	$-0.506908\pi$
$883$	−1.28962	−0.0433992	−0.0216996	+	0.999765i	$0.506908\pi$
$884$	−6.74456	−0.226844
$885$	0	0
$886$	35.4891	1.19228
$887$	−37.4226	−1.25653	−0.628265	−	0.778000i	$-0.716234\pi$
$887$	−37.4226	−1.25653	−0.628265	+	0.778000i	$0.716234\pi$
$888$	−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	$-0.685614\pi$
$907$	−33.1662	−1.10127	−0.550634	+	0.834747i	$0.685614\pi$
$908$	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	$-0.786074\pi$
$937$	−47.9075	−1.56507	−0.782535	+	0.622606i	$0.786074\pi$
$938$	−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	$-0.616253\pi$
$947$	−21.9817	−0.714308	−0.357154	+	0.934046i	$0.616253\pi$
$948$	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	$-0.552776\pi$
$953$	−10.1899	−0.330084	−0.165042	+	0.986287i	$0.552776\pi$
$954$	−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	$-0.310707\pi$
$967$	34.8434	1.12049	0.560244	+	0.828328i	$0.310707\pi$
$968$	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	$-0.313616\pi$
$977$	34.5484	1.10530	0.552651	+	0.833413i	$0.313616\pi$
$978$	−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	$-0.359795\pi$
$983$	26.7354	0.852726	0.426363	+	0.904552i	$0.359795\pi$
$984$	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	$-0.299369\pi$
$997$	37.2203	1.17878	0.589389	+	0.807850i	$0.299369\pi$
$998$	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.1.4		4
3.2	odd	2		6525.2.a.bk.1.2		4
5.2	odd	4		145.2.b.a.59.3	yes	4
5.3	odd	4		145.2.b.a.59.2	✓	4
5.4	even	2	inner	725.2.a.g.1.1		4
15.2	even	4		1305.2.c.e.784.1		4
15.8	even	4		1305.2.c.e.784.3		4
15.14	odd	2		6525.2.a.bk.1.3		4
20.3	even	4		2320.2.d.c.929.2		4
20.7	even	4		2320.2.d.c.929.3		4

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

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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