LMFDB - Embedded newform 7225.2.a.by.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	7225.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30578 q^{2} -1.87767 q^{3} +3.31660 q^{4} -4.32949 q^{6} +1.54574 q^{7} +3.03579 q^{8} +0.525655 q^{9} +4.61783 q^{11} -6.22750 q^{12} -3.51782 q^{13} +3.56413 q^{14} +0.366652 q^{16} +1.21204 q^{18} -6.62312 q^{19} -2.90239 q^{21} +10.6477 q^{22} -2.59608 q^{23} -5.70023 q^{24} -8.11130 q^{26} +4.64601 q^{27} +5.12660 q^{28} -0.979000 q^{29} +1.22437 q^{31} -5.22617 q^{32} -8.67078 q^{33} +1.74339 q^{36} +0.407676 q^{37} -15.2714 q^{38} +6.60531 q^{39} -10.2338 q^{41} -6.69226 q^{42} +4.24674 q^{43} +15.3155 q^{44} -5.98598 q^{46} -3.99531 q^{47} -0.688453 q^{48} -4.61069 q^{49} -11.6672 q^{52} +1.75770 q^{53} +10.7127 q^{54} +4.69254 q^{56} +12.4361 q^{57} -2.25736 q^{58} +1.56173 q^{59} -13.9932 q^{61} +2.82312 q^{62} +0.812524 q^{63} -12.7837 q^{64} -19.9929 q^{66} -8.84212 q^{67} +4.87459 q^{69} -1.29542 q^{71} +1.59578 q^{72} +9.77702 q^{73} +0.940010 q^{74} -21.9663 q^{76} +7.13796 q^{77} +15.2304 q^{78} +10.9166 q^{79} -10.3007 q^{81} -23.5969 q^{82} +12.6903 q^{83} -9.62608 q^{84} +9.79203 q^{86} +1.83824 q^{87} +14.0188 q^{88} -8.62631 q^{89} -5.43762 q^{91} -8.61017 q^{92} -2.29897 q^{93} -9.21229 q^{94} +9.81303 q^{96} +5.92877 q^{97} -10.6312 q^{98} +2.42739 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q + 8 q^{4} - 8 q^{9} - 8 q^{16} - 16 q^{19} - 32 q^{21} - 48 q^{26} + 8 q^{36} - 56 q^{49} - 64 q^{59} - 104 q^{64} - 16 q^{66} - 32 q^{69} - 48 q^{76} - 88 q^{81} - 96 q^{84} - 112 q^{89} - 128 q^{94}+O(q^{100})$ 24 * q + 8 * q^4 - 8 * q^9 - 8 * q^16 - 16 * q^19 - 32 * q^21 - 48 * q^26 + 8 * q^36 - 56 * q^49 - 64 * q^59 - 104 * q^64 - 16 * q^66 - 32 * q^69 - 48 * q^76 - 88 * q^81 - 96 * q^84 - 112 * q^89 - 128 * q^94

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

)

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

$2$	2.30578	1.63043	0.815215	−	0.579158i	$-0.196619\pi$
$2$	2.30578	1.63043	0.815215	+	0.579158i	$0.196619\pi$
$3$	−1.87767	−1.08407	−0.542037	−	0.840354i	$-0.682347\pi$
$3$	−1.87767	−1.08407	−0.542037	+	0.840354i	$0.682347\pi$
$4$	3.31660	1.65830
$5$	0	0
$5$	0	0
$6$	−4.32949	−1.76751
$7$	1.54574	0.584234	0.292117	−	0.956383i	$-0.405640\pi$
$7$	1.54574	0.584234	0.292117	+	0.956383i	$0.405640\pi$
$8$	3.03579	1.07331
$9$	0.525655	0.175218
$10$	0	0
$11$	4.61783	1.39233	0.696165	−	0.717882i	$-0.254888\pi$
$11$	4.61783	1.39233	0.696165	+	0.717882i	$0.254888\pi$
$12$	−6.22750	−1.79772
$13$	−3.51782	−0.975667	−0.487834	−	0.872937i	$-0.662213\pi$
$13$	−3.51782	−0.975667	−0.487834	+	0.872937i	$0.662213\pi$
$14$	3.56413	0.952553
$15$	0	0
$16$	0.366652	0.0916631
$17$	0	0
$17$	0	0
$18$	1.21204	0.285681
$19$	−6.62312	−1.51945	−0.759724	−	0.650245i	$-0.774666\pi$
$19$	−6.62312	−1.51945	−0.759724	+	0.650245i	$0.774666\pi$
$20$	0	0
$21$	−2.90239	−0.633353
$22$	10.6477	2.27010
$23$	−2.59608	−0.541320	−0.270660	−	0.962675i	$-0.587242\pi$
$23$	−2.59608	−0.541320	−0.270660	+	0.962675i	$0.587242\pi$
$24$	−5.70023	−1.16355
$25$	0	0
$26$	−8.11130	−1.59076
$27$	4.64601	0.894125
$28$	5.12660	0.968836
$29$	−0.979000	−0.181796	−0.0908979	−	0.995860i	$-0.528974\pi$
$29$	−0.979000	−0.181796	−0.0908979	+	0.995860i	$0.528974\pi$
$30$	0	0
$31$	1.22437	0.219903	0.109952	−	0.993937i	$-0.464930\pi$
$31$	1.22437	0.219903	0.109952	+	0.993937i	$0.464930\pi$
$32$	−5.22617	−0.923865
$33$	−8.67078	−1.50939
$34$	0	0
$35$	0	0
$36$	1.74339	0.290565
$37$	0.407676	0.0670216	0.0335108	−	0.999438i	$-0.489331\pi$
$37$	0.407676	0.0670216	0.0335108	+	0.999438i	$0.489331\pi$
$38$	−15.2714	−2.47735
$39$	6.60531	1.05770
$40$	0	0
$41$	−10.2338	−1.59826	−0.799129	−	0.601160i	$-0.794705\pi$
$41$	−10.2338	−1.59826	−0.799129	+	0.601160i	$0.794705\pi$
$42$	−6.69226	−1.03264
$43$	4.24674	0.647622	0.323811	−	0.946122i	$-0.395036\pi$
$43$	4.24674	0.647622	0.323811	+	0.946122i	$0.395036\pi$
$44$	15.3155	2.30890
$45$	0	0
$46$	−5.98598	−0.882584
$47$	−3.99531	−0.582776	−0.291388	−	0.956605i	$-0.594117\pi$
$47$	−3.99531	−0.582776	−0.291388	+	0.956605i	$0.594117\pi$
$48$	−0.688453	−0.0993696
$49$	−4.61069	−0.658671
$50$	0	0
$51$	0	0
$52$	−11.6672	−1.61795
$53$	1.75770	0.241439	0.120719	−	0.992687i	$-0.461480\pi$
$53$	1.75770	0.241439	0.120719	+	0.992687i	$0.461480\pi$
$54$	10.7127	1.45781
$55$	0	0
$56$	4.69254	0.627067
$57$	12.4361	1.64720
$58$	−2.25736	−0.296405
$59$	1.56173	0.203320	0.101660	−	0.994819i	$-0.467585\pi$
$59$	1.56173	0.203320	0.101660	+	0.994819i	$0.467585\pi$
$60$	0	0
$61$	−13.9932	−1.79165	−0.895824	−	0.444409i	$-0.853414\pi$
$61$	−13.9932	−1.79165	−0.895824	+	0.444409i	$0.853414\pi$
$62$	2.82312	0.358537
$63$	0.812524	0.102368
$64$	−12.7837	−1.59796
$65$	0	0
$66$	−19.9929	−2.46095
$67$	−8.84212	−1.08024	−0.540119	−	0.841589i	$-0.681621\pi$
$67$	−8.84212	−1.08024	−0.540119	+	0.841589i	$0.681621\pi$
$68$	0	0
$69$	4.87459	0.586831
$70$	0	0
$71$	−1.29542	−0.153738	−0.0768689	−	0.997041i	$-0.524492\pi$
$71$	−1.29542	−0.153738	−0.0768689	+	0.997041i	$0.524492\pi$
$72$	1.59578	0.188064
$73$	9.77702	1.14431	0.572157	−	0.820144i	$-0.306107\pi$
$73$	9.77702	1.14431	0.572157	+	0.820144i	$0.306107\pi$
$74$	0.940010	0.109274
$75$	0	0
$76$	−21.9663	−2.51970
$77$	7.13796	0.813446
$78$	15.2304	1.72450
$79$	10.9166	1.22821	0.614107	−	0.789223i	$-0.289516\pi$
$79$	10.9166	1.22821	0.614107	+	0.789223i	$0.289516\pi$
$80$	0	0
$81$	−10.3007	−1.14452
$82$	−23.5969	−2.60585
$83$	12.6903	1.39294	0.696471	−	0.717585i	$-0.254752\pi$
$83$	12.6903	1.39294	0.696471	+	0.717585i	$0.254752\pi$
$84$	−9.62608	−1.05029
$85$	0	0
$86$	9.79203	1.05590
$87$	1.83824	0.197080
$88$	14.0188	1.49441
$89$	−8.62631	−0.914387	−0.457193	−	0.889367i	$-0.651145\pi$
$89$	−8.62631	−0.914387	−0.457193	+	0.889367i	$0.651145\pi$
$90$	0	0
$91$	−5.43762	−0.570018
$92$	−8.61017	−0.897672
$93$	−2.29897	−0.238392
$94$	−9.21229	−0.950175
$95$	0	0
$96$	9.81303	1.00154
$97$	5.92877	0.601975	0.300988	−	0.953628i	$-0.402684\pi$
$97$	5.92877	0.601975	0.300988	+	0.953628i	$0.402684\pi$
$98$	−10.6312	−1.07392
$99$	2.42739	0.243961
$100$	0	0
$101$	−5.79390	−0.576515	−0.288257	−	0.957553i	$-0.593076\pi$
$101$	−5.79390	−0.576515	−0.288257	+	0.957553i	$0.593076\pi$
$102$	0	0
$103$	−6.99141	−0.688885	−0.344442	−	0.938807i	$-0.611932\pi$
$103$	−6.99141	−0.688885	−0.344442	+	0.938807i	$0.611932\pi$
$104$	−10.6794	−1.04720
$105$	0	0
$106$	4.05286	0.393649
$107$	1.39751	0.135102	0.0675510	−	0.997716i	$-0.478481\pi$
$107$	1.39751	0.135102	0.0675510	+	0.997716i	$0.478481\pi$
$108$	15.4090	1.48273
$109$	−10.8493	−1.03917	−0.519585	−	0.854419i	$-0.673914\pi$
$109$	−10.8493	−1.03917	−0.519585	+	0.854419i	$0.673914\pi$
$110$	0	0
$111$	−0.765482	−0.0726564
$112$	0.566748	0.0535527
$113$	−3.98090	−0.374491	−0.187246	−	0.982313i	$-0.559956\pi$
$113$	−3.98090	−0.374491	−0.187246	+	0.982313i	$0.559956\pi$
$114$	28.6748	2.68564
$115$	0	0
$116$	−3.24696	−0.301472
$117$	−1.84916	−0.170955
$118$	3.60100	0.331499
$119$	0	0
$120$	0	0
$121$	10.3244	0.938581
$122$	−32.2652	−2.92116
$123$	19.2158	1.73263
$124$	4.06075	0.364666
$125$	0	0
$126$	1.87350	0.166905
$127$	−12.4761	−1.10708	−0.553539	−	0.832823i	$-0.686723\pi$
$127$	−12.4761	−1.10708	−0.553539	+	0.832823i	$0.686723\pi$
$128$	−19.0240	−1.68150
$129$	−7.97399	−0.702070
$130$	0	0
$131$	7.78675	0.680331	0.340166	−	0.940365i	$-0.389517\pi$
$131$	7.78675	0.680331	0.340166	+	0.940365i	$0.389517\pi$
$132$	−28.7575	−2.50302
$133$	−10.2376	−0.887714
$134$	−20.3880	−1.76125
$135$	0	0
$136$	0	0
$137$	15.2375	1.30182	0.650912	−	0.759153i	$-0.274387\pi$
$137$	15.2375	1.30182	0.650912	+	0.759153i	$0.274387\pi$
$138$	11.2397	0.956787
$139$	0.0460990	0.00391006	0.00195503	−	0.999998i	$-0.499378\pi$
$139$	0.0460990	0.00391006	0.00195503	+	0.999998i	$0.499378\pi$
$140$	0	0
$141$	7.50189	0.631773
$142$	−2.98694	−0.250659
$143$	−16.2447	−1.35845
$144$	0.192732	0.0160610
$145$	0	0
$146$	22.5436	1.86572
$147$	8.65737	0.714048
$148$	1.35210	0.111142
$149$	1.49474	0.122454	0.0612268	−	0.998124i	$-0.480499\pi$
$149$	1.49474	0.122454	0.0612268	+	0.998124i	$0.480499\pi$
$150$	0	0
$151$	5.28147	0.429800	0.214900	−	0.976636i	$-0.431057\pi$
$151$	5.28147	0.429800	0.214900	+	0.976636i	$0.431057\pi$
$152$	−20.1064	−1.63085
$153$	0	0
$154$	16.4585	1.32627
$155$	0	0
$156$	21.9072	1.75398
$157$	−12.6990	−1.01349	−0.506745	−	0.862096i	$-0.669152\pi$
$157$	−12.6990	−1.01349	−0.506745	+	0.862096i	$0.669152\pi$
$158$	25.1712	2.00252
$159$	−3.30038	−0.261737
$160$	0	0
$161$	−4.01286	−0.316258
$162$	−23.7510	−1.86605
$163$	−4.06067	−0.318056	−0.159028	−	0.987274i	$-0.550836\pi$
$163$	−4.06067	−0.318056	−0.159028	+	0.987274i	$0.550836\pi$
$164$	−33.9416	−2.65039
$165$	0	0
$166$	29.2610	2.27110
$167$	−16.1680	−1.25112	−0.625561	−	0.780176i	$-0.715130\pi$
$167$	−16.1680	−1.25112	−0.625561	+	0.780176i	$0.715130\pi$
$168$	−8.81106	−0.679788
$169$	−0.624957	−0.0480736
$170$	0	0
$171$	−3.48148	−0.266235
$172$	14.0848	1.07395
$173$	−1.57629	−0.119843	−0.0599216	−	0.998203i	$-0.519085\pi$
$173$	−1.57629	−0.119843	−0.0599216	+	0.998203i	$0.519085\pi$
$174$	4.23857	0.321326
$175$	0	0
$176$	1.69314	0.127625
$177$	−2.93242	−0.220414
$178$	−19.8903	−1.49084
$179$	3.44789	0.257708	0.128854	−	0.991664i	$-0.458870\pi$
$179$	3.44789	0.257708	0.128854	+	0.991664i	$0.458870\pi$
$180$	0	0
$181$	−14.7265	−1.09461	−0.547306	−	0.836933i	$-0.684347\pi$
$181$	−14.7265	−1.09461	−0.547306	+	0.836933i	$0.684347\pi$
$182$	−12.5379	−0.929374
$183$	26.2747	1.94228
$184$	−7.88116	−0.581007
$185$	0	0
$186$	−5.30090	−0.388681
$187$	0	0
$188$	−13.2509	−0.966418
$189$	7.18152	0.522378
$190$	0	0
$191$	−15.5715	−1.12671	−0.563356	−	0.826214i	$-0.690490\pi$
$191$	−15.5715	−1.12671	−0.563356	+	0.826214i	$0.690490\pi$
$192$	24.0036	1.73231
$193$	−0.106013	−0.00763099	−0.00381549	−	0.999993i	$-0.501215\pi$
$193$	−0.106013	−0.00763099	−0.00381549	+	0.999993i	$0.501215\pi$
$194$	13.6704	0.981478
$195$	0	0
$196$	−15.2918	−1.09227
$197$	−10.5446	−0.751272	−0.375636	−	0.926767i	$-0.622576\pi$
$197$	−10.5446	−0.751272	−0.375636	+	0.926767i	$0.622576\pi$
$198$	5.59701	0.397762
$199$	1.15122	0.0816077	0.0408038	−	0.999167i	$-0.487008\pi$
$199$	1.15122	0.0816077	0.0408038	+	0.999167i	$0.487008\pi$
$200$	0	0
$201$	16.6026	1.17106
$202$	−13.3594	−0.939967
$203$	−1.51328	−0.106211
$204$	0	0
$205$	0	0
$206$	−16.1206	−1.12318
$207$	−1.36464	−0.0948491
$208$	−1.28982	−0.0894327
$209$	−30.5845	−2.11557
$210$	0	0
$211$	−4.18759	−0.288286	−0.144143	−	0.989557i	$-0.546042\pi$
$211$	−4.18759	−0.288286	−0.144143	+	0.989557i	$0.546042\pi$
$212$	5.82959	0.400378
$213$	2.43237	0.166663
$214$	3.22234	0.220274
$215$	0	0
$216$	14.1043	0.959678
$217$	1.89256	0.128475
$218$	−25.0160	−1.69429
$219$	−18.3580	−1.24052
$220$	0	0
$221$	0	0
$222$	−1.76503	−0.118461
$223$	−7.64436	−0.511904	−0.255952	−	0.966689i	$-0.582389\pi$
$223$	−7.64436	−0.511904	−0.255952	+	0.966689i	$0.582389\pi$
$224$	−8.07829	−0.539753
$225$	0	0
$226$	−9.17906	−0.610582
$227$	23.8653	1.58400	0.791998	−	0.610524i	$-0.209041\pi$
$227$	23.8653	1.58400	0.791998	+	0.610524i	$0.209041\pi$
$228$	41.2455	2.73155
$229$	13.7741	0.910216	0.455108	−	0.890436i	$-0.349601\pi$
$229$	13.7741	0.910216	0.455108	+	0.890436i	$0.349601\pi$
$230$	0	0
$231$	−13.4028	−0.881837
$232$	−2.97204	−0.195124
$233$	5.81269	0.380802	0.190401	−	0.981706i	$-0.439021\pi$
$233$	5.81269	0.380802	0.190401	+	0.981706i	$0.439021\pi$
$234$	−4.26374	−0.278730
$235$	0	0
$236$	5.17965	0.337166
$237$	−20.4978	−1.33147
$238$	0	0
$239$	−5.09183	−0.329363	−0.164682	−	0.986347i	$-0.552660\pi$
$239$	−5.09183	−0.329363	−0.164682	+	0.986347i	$0.552660\pi$
$240$	0	0
$241$	−5.31230	−0.342195	−0.171098	−	0.985254i	$-0.554731\pi$
$241$	−5.31230	−0.342195	−0.171098	+	0.985254i	$0.554731\pi$
$242$	23.8057	1.53029
$243$	5.40322	0.346617
$244$	−46.4100	−2.97109
$245$	0	0
$246$	44.3073	2.82493
$247$	23.2989	1.48248
$248$	3.71693	0.236026
$249$	−23.8283	−1.51005
$250$	0	0
$251$	−28.2032	−1.78017	−0.890084	−	0.455797i	$-0.849354\pi$
$251$	−28.2032	−1.78017	−0.890084	+	0.455797i	$0.849354\pi$
$252$	2.69482	0.169758
$253$	−11.9883	−0.753696
$254$	−28.7672	−1.80501
$255$	0	0
$256$	−18.2976	−1.14360
$257$	−3.29342	−0.205438	−0.102719	−	0.994710i	$-0.532754\pi$
$257$	−3.29342	−0.205438	−0.102719	+	0.994710i	$0.532754\pi$
$258$	−18.3862	−1.14468
$259$	0.630161	0.0391563
$260$	0	0
$261$	−0.514616	−0.0318539
$262$	17.9545	1.10923
$263$	27.8375	1.71653	0.858267	−	0.513203i	$-0.171541\pi$
$263$	27.8375	1.71653	0.858267	+	0.513203i	$0.171541\pi$
$264$	−26.3227	−1.62005
$265$	0	0
$266$	−23.6056	−1.44736
$267$	16.1974	0.991264
$268$	−29.3258	−1.79136
$269$	−14.6617	−0.893939	−0.446970	−	0.894549i	$-0.647497\pi$
$269$	−14.6617	−0.893939	−0.446970	+	0.894549i	$0.647497\pi$
$270$	0	0
$271$	2.50962	0.152449	0.0762244	−	0.997091i	$-0.475713\pi$
$271$	2.50962	0.152449	0.0762244	+	0.997091i	$0.475713\pi$
$272$	0	0
$273$	10.2101	0.617942
$274$	35.1342	2.12253
$275$	0	0
$276$	16.1671	0.973144
$277$	23.9658	1.43996	0.719982	−	0.693993i	$-0.244150\pi$
$277$	23.9658	1.43996	0.719982	+	0.693993i	$0.244150\pi$
$278$	0.106294	0.00637508
$279$	0.643596	0.0385311
$280$	0	0
$281$	14.2207	0.848338	0.424169	−	0.905583i	$-0.360566\pi$
$281$	14.2207	0.848338	0.424169	+	0.905583i	$0.360566\pi$
$282$	17.2977	1.03006
$283$	10.0111	0.595095	0.297548	−	0.954707i	$-0.403831\pi$
$283$	10.0111	0.595095	0.297548	+	0.954707i	$0.403831\pi$
$284$	−4.29638	−0.254944
$285$	0	0
$286$	−37.4566	−2.21486
$287$	−15.8188	−0.933756
$288$	−2.74716	−0.161878
$289$	0	0
$290$	0	0
$291$	−11.1323	−0.652586
$292$	32.4265	1.89762
$293$	−11.2943	−0.659821	−0.329910	−	0.944012i	$-0.607019\pi$
$293$	−11.2943	−0.659821	−0.329910	+	0.944012i	$0.607019\pi$
$294$	19.9620	1.16421
$295$	0	0
$296$	1.23762	0.0719352
$297$	21.4545	1.24492
$298$	3.44653	0.199652
$299$	9.13253	0.528148
$300$	0	0
$301$	6.56435	0.378363
$302$	12.1779	0.700759
$303$	10.8791	0.624985
$304$	−2.42838	−0.139277
$305$	0	0
$306$	0	0
$307$	−25.1396	−1.43479	−0.717397	−	0.696664i	$-0.754667\pi$
$307$	−25.1396	−1.43479	−0.717397	+	0.696664i	$0.754667\pi$
$308$	23.6738	1.34894
$309$	13.1276	0.746802
$310$	0	0
$311$	−22.3273	−1.26606	−0.633032	−	0.774126i	$-0.718190\pi$
$311$	−22.3273	−1.26606	−0.633032	+	0.774126i	$0.718190\pi$
$312$	20.0524	1.13524
$313$	−29.7287	−1.68037	−0.840184	−	0.542301i	$-0.817553\pi$
$313$	−29.7287	−1.68037	−0.840184	+	0.542301i	$0.817553\pi$
$314$	−29.2811	−1.65243
$315$	0	0
$316$	36.2060	2.03675
$317$	−11.8957	−0.668130	−0.334065	−	0.942550i	$-0.608420\pi$
$317$	−11.8957	−0.668130	−0.334065	+	0.942550i	$0.608420\pi$
$318$	−7.60995	−0.426745
$319$	−4.52086	−0.253120
$320$	0	0
$321$	−2.62406	−0.146461
$322$	−9.25275	−0.515636
$323$	0	0
$324$	−34.1632	−1.89795
$325$	0	0
$326$	−9.36300	−0.518569
$327$	20.3714	1.12654
$328$	−31.0678	−1.71543
$329$	−6.17570	−0.340478
$330$	0	0
$331$	9.14800	0.502820	0.251410	−	0.967881i	$-0.419106\pi$
$331$	9.14800	0.502820	0.251410	+	0.967881i	$0.419106\pi$
$332$	42.0887	2.30992
$333$	0.214297	0.0117434
$334$	−37.2799	−2.03987
$335$	0	0
$336$	−1.06417	−0.0580551
$337$	21.5292	1.17277	0.586385	−	0.810033i	$-0.300551\pi$
$337$	21.5292	1.17277	0.586385	+	0.810033i	$0.300551\pi$
$338$	−1.44101	−0.0783807
$339$	7.47482	0.405977
$340$	0	0
$341$	5.65394	0.306178
$342$	−8.02750	−0.434078
$343$	−17.9471	−0.969052
$344$	12.8922	0.695102
$345$	0	0
$346$	−3.63457	−0.195396
$347$	−5.46122	−0.293174	−0.146587	−	0.989198i	$-0.546829\pi$
$347$	−5.46122	−0.293174	−0.146587	+	0.989198i	$0.546829\pi$
$348$	6.09672	0.326819
$349$	−33.0246	−1.76777	−0.883883	−	0.467708i	$-0.845080\pi$
$349$	−33.0246	−1.76777	−0.883883	+	0.467708i	$0.845080\pi$
$350$	0	0
$351$	−16.3438	−0.872369
$352$	−24.1336	−1.28632
$353$	12.4962	0.665103	0.332552	−	0.943085i	$-0.392090\pi$
$353$	12.4962	0.665103	0.332552	+	0.943085i	$0.392090\pi$
$354$	−6.76151	−0.359370
$355$	0	0
$356$	−28.6100	−1.51633
$357$	0	0
$358$	7.95007	0.420174
$359$	−18.1881	−0.959932	−0.479966	−	0.877287i	$-0.659351\pi$
$359$	−18.1881	−0.959932	−0.479966	+	0.877287i	$0.659351\pi$
$360$	0	0
$361$	24.8658	1.30872
$362$	−33.9560	−1.78469
$363$	−19.3858	−1.01749
$364$	−18.0344	−0.945262
$365$	0	0
$366$	60.5835	3.16675
$367$	22.7681	1.18849	0.594243	−	0.804285i	$-0.297452\pi$
$367$	22.7681	1.18849	0.594243	+	0.804285i	$0.297452\pi$
$368$	−0.951859	−0.0496191
$369$	−5.37946	−0.280044
$370$	0	0
$371$	2.71694	0.141057
$372$	−7.62476	−0.395325
$373$	−24.5550	−1.27141	−0.635705	−	0.771932i	$-0.719291\pi$
$373$	−24.5550	−1.27141	−0.635705	+	0.771932i	$0.719291\pi$
$374$	0	0
$375$	0	0
$376$	−12.1289	−0.625502
$377$	3.44394	0.177372
$378$	16.5590	0.851701
$379$	18.6607	0.958538	0.479269	−	0.877668i	$-0.340902\pi$
$379$	18.6607	0.958538	0.479269	+	0.877668i	$0.340902\pi$
$380$	0	0
$381$	23.4261	1.20016
$382$	−35.9043	−1.83703
$383$	25.7152	1.31398	0.656991	−	0.753898i	$-0.271829\pi$
$383$	25.7152	1.31398	0.656991	+	0.753898i	$0.271829\pi$
$384$	35.7208	1.82287
$385$	0	0
$386$	−0.244442	−0.0124418
$387$	2.23232	0.113475
$388$	19.6634	0.998256
$389$	23.9816	1.21592	0.607958	−	0.793969i	$-0.291989\pi$
$389$	23.9816	1.21592	0.607958	+	0.793969i	$0.291989\pi$
$390$	0	0
$391$	0	0
$392$	−13.9971	−0.706961
$393$	−14.6210	−0.737530
$394$	−24.3135	−1.22490
$395$	0	0
$396$	8.05067	0.404562
$397$	−14.3157	−0.718486	−0.359243	−	0.933244i	$-0.616965\pi$
$397$	−14.3157	−0.718486	−0.359243	+	0.933244i	$0.616965\pi$
$398$	2.65445	0.133056
$399$	19.2229	0.962348
$400$	0	0
$401$	2.95294	0.147463	0.0737315	−	0.997278i	$-0.476509\pi$
$401$	2.95294	0.147463	0.0737315	+	0.997278i	$0.476509\pi$
$402$	38.2819	1.90933
$403$	−4.30711	−0.214553
$404$	−19.2161	−0.956036
$405$	0	0
$406$	−3.48928	−0.173170
$407$	1.88258	0.0933161
$408$	0	0
$409$	−2.54353	−0.125769	−0.0628846	−	0.998021i	$-0.520030\pi$
$409$	−2.54353	−0.125769	−0.0628846	+	0.998021i	$0.520030\pi$
$410$	0	0
$411$	−28.6110	−1.41128
$412$	−23.1878	−1.14238
$413$	2.41403	0.118787
$414$	−3.14656	−0.154645
$415$	0	0
$416$	18.3847	0.901384
$417$	−0.0865588	−0.00423880
$418$	−70.5210	−3.44929
$419$	5.75101	0.280955	0.140478	−	0.990084i	$-0.455136\pi$
$419$	5.75101	0.280955	0.140478	+	0.990084i	$0.455136\pi$
$420$	0	0
$421$	−19.5960	−0.955050	−0.477525	−	0.878618i	$-0.658466\pi$
$421$	−19.5960	−0.955050	−0.477525	+	0.878618i	$0.658466\pi$
$422$	−9.65565	−0.470030
$423$	−2.10015	−0.102113
$424$	5.33601	0.259140
$425$	0	0
$426$	5.60850	0.271733
$427$	−21.6299	−1.04674
$428$	4.63497	0.224040
$429$	30.5022	1.47266
$430$	0	0
$431$	25.8174	1.24358	0.621789	−	0.783185i	$-0.286406\pi$
$431$	25.8174	1.24358	0.621789	+	0.783185i	$0.286406\pi$
$432$	1.70347	0.0819583
$433$	−38.8428	−1.86666	−0.933332	−	0.359013i	$-0.883113\pi$
$433$	−38.8428	−1.86666	−0.933332	+	0.359013i	$0.883113\pi$
$434$	4.36381	0.209470
$435$	0	0
$436$	−35.9827	−1.72326
$437$	17.1942	0.822508
$438$	−42.3295	−2.02258
$439$	−3.41822	−0.163143	−0.0815714	−	0.996668i	$-0.525994\pi$
$439$	−3.41822	−0.163143	−0.0815714	+	0.996668i	$0.525994\pi$
$440$	0	0
$441$	−2.42363	−0.115411
$442$	0	0
$443$	19.8457	0.942899	0.471449	−	0.881893i	$-0.343731\pi$
$443$	19.8457	0.942899	0.471449	+	0.881893i	$0.343731\pi$
$444$	−2.53880	−0.120486
$445$	0	0
$446$	−17.6262	−0.834624
$447$	−2.80663	−0.132749
$448$	−19.7602	−0.933583
$449$	6.01997	0.284100	0.142050	−	0.989859i	$-0.454631\pi$
$449$	6.01997	0.284100	0.142050	+	0.989859i	$0.454631\pi$
$450$	0	0
$451$	−47.2582	−2.22530
$452$	−13.2031	−0.621020
$453$	−9.91688	−0.465936
$454$	55.0280	2.58259
$455$	0	0
$456$	37.7533	1.76796
$457$	28.5879	1.33728	0.668642	−	0.743584i	$-0.266876\pi$
$457$	28.5879	1.33728	0.668642	+	0.743584i	$0.266876\pi$
$458$	31.7599	1.48404
$459$	0	0
$460$	0	0
$461$	7.69935	0.358595	0.179297	−	0.983795i	$-0.442618\pi$
$461$	7.69935	0.358595	0.179297	+	0.983795i	$0.442618\pi$
$462$	−30.9038	−1.43777
$463$	−26.7823	−1.24468	−0.622339	−	0.782748i	$-0.713817\pi$
$463$	−26.7823	−1.24468	−0.622339	+	0.782748i	$0.713817\pi$
$464$	−0.358953	−0.0166640
$465$	0	0
$466$	13.4028	0.620871
$467$	−10.3379	−0.478380	−0.239190	−	0.970973i	$-0.576882\pi$
$467$	−10.3379	−0.478380	−0.239190	+	0.970973i	$0.576882\pi$
$468$	−6.13292	−0.283494
$469$	−13.6676	−0.631112
$470$	0	0
$471$	23.8446	1.09870
$472$	4.74109	0.218227
$473$	19.6107	0.901703
$474$	−47.2633	−2.17088
$475$	0	0
$476$	0	0
$477$	0.923943	0.0423044
$478$	−11.7406	−0.537004
$479$	−25.8769	−1.18234	−0.591172	−	0.806545i	$-0.701335\pi$
$479$	−25.8769	−1.18234	−0.591172	+	0.806545i	$0.701335\pi$
$480$	0	0
$481$	−1.43413	−0.0653907
$482$	−12.2490	−0.557925
$483$	7.53484	0.342847
$484$	34.2419	1.55645
$485$	0	0
$486$	12.4586	0.565134
$487$	−23.7435	−1.07592	−0.537960	−	0.842970i	$-0.680805\pi$
$487$	−23.7435	−1.07592	−0.537960	+	0.842970i	$0.680805\pi$
$488$	−42.4805	−1.92300
$489$	7.62461	0.344797
$490$	0	0
$491$	40.0485	1.80736	0.903682	−	0.428205i	$-0.140854\pi$
$491$	40.0485	1.80736	0.903682	+	0.428205i	$0.140854\pi$
$492$	63.7312	2.87322
$493$	0	0
$494$	53.7221	2.41707
$495$	0	0
$496$	0.448918	0.0201570
$497$	−2.00238	−0.0898188
$498$	−54.9426	−2.46204
$499$	−3.63650	−0.162792	−0.0813961	−	0.996682i	$-0.525938\pi$
$499$	−3.63650	−0.162792	−0.0813961	+	0.996682i	$0.525938\pi$
$500$	0	0
$501$	30.3583	1.35631
$502$	−65.0302	−2.90244
$503$	35.3068	1.57425	0.787127	−	0.616791i	$-0.211568\pi$
$503$	35.3068	1.57425	0.787127	+	0.616791i	$0.211568\pi$
$504$	2.46666	0.109874
$505$	0	0
$506$	−27.6423	−1.22885
$507$	1.17347	0.0521154
$508$	−41.3784	−1.83587
$509$	−8.67454	−0.384492	−0.192246	−	0.981347i	$-0.561577\pi$
$509$	−8.67454	−0.384492	−0.192246	+	0.981347i	$0.561577\pi$
$510$	0	0
$511$	15.1127	0.668547
$512$	−4.14235	−0.183068
$513$	−30.7711	−1.35858
$514$	−7.59390	−0.334953
$515$	0	0
$516$	−26.4466	−1.16424
$517$	−18.4497	−0.811416
$518$	1.45301	0.0638416
$519$	2.95976	0.129919
$520$	0	0
$521$	14.1978	0.622019	0.311009	−	0.950407i	$-0.399333\pi$
$521$	14.1978	0.622019	0.311009	+	0.950407i	$0.399333\pi$
$522$	−1.18659	−0.0519356
$523$	28.1516	1.23098	0.615492	−	0.788143i	$-0.288957\pi$
$523$	28.1516	1.23098	0.615492	+	0.788143i	$0.288957\pi$
$524$	25.8256	1.12819
$525$	0	0
$526$	64.1871	2.79869
$527$	0	0
$528$	−3.17916	−0.138355
$529$	−16.2604	−0.706973
$530$	0	0
$531$	0.820931	0.0356254
$532$	−33.9541	−1.47210
$533$	36.0008	1.55937
$534$	37.3475	1.61619
$535$	0	0
$536$	−26.8429	−1.15944
$537$	−6.47401	−0.279374
$538$	−33.8066	−1.45751
$539$	−21.2914	−0.917086
$540$	0	0
$541$	20.0836	0.863463	0.431732	−	0.902002i	$-0.357903\pi$
$541$	20.0836	0.863463	0.431732	+	0.902002i	$0.357903\pi$
$542$	5.78663	0.248557
$543$	27.6515	1.18664
$544$	0	0
$545$	0	0
$546$	23.5422	1.00751
$547$	2.00586	0.0857644	0.0428822	−	0.999080i	$-0.486346\pi$
$547$	2.00586	0.0857644	0.0428822	+	0.999080i	$0.486346\pi$
$548$	50.5366	2.15882
$549$	−7.35560	−0.313929
$550$	0	0
$551$	6.48404	0.276229
$552$	14.7982	0.629855
$553$	16.8742	0.717564
$554$	55.2597	2.34776
$555$	0	0
$556$	0.152892	0.00648407
$557$	16.9217	0.716994	0.358497	−	0.933531i	$-0.383289\pi$
$557$	16.9217	0.716994	0.358497	+	0.933531i	$0.383289\pi$
$558$	1.48399	0.0628222
$559$	−14.9393	−0.631863
$560$	0	0
$561$	0	0
$562$	32.7898	1.38316
$563$	−42.3789	−1.78606	−0.893030	−	0.449997i	$-0.851425\pi$
$563$	−42.3789	−1.78606	−0.893030	+	0.449997i	$0.851425\pi$
$564$	24.8808	1.04767
$565$	0	0
$566$	23.0832	0.970261
$567$	−15.9221	−0.668666
$568$	−3.93262	−0.165009
$569$	−41.5417	−1.74152	−0.870760	−	0.491708i	$-0.836373\pi$
$569$	−41.5417	−1.74152	−0.870760	+	0.491708i	$0.836373\pi$
$570$	0	0
$571$	19.0990	0.799266	0.399633	−	0.916675i	$-0.369138\pi$
$571$	19.0990	0.799266	0.399633	+	0.916675i	$0.369138\pi$
$572$	−53.8772	−2.25272
$573$	29.2381	1.22144
$574$	−36.4747	−1.52242
$575$	0	0
$576$	−6.71980	−0.279992
$577$	3.81985	0.159023	0.0795113	−	0.996834i	$-0.474664\pi$
$577$	3.81985	0.159023	0.0795113	+	0.996834i	$0.474664\pi$
$578$	0	0
$579$	0.199058	0.00827256
$580$	0	0
$581$	19.6159	0.813805
$582$	−25.6686	−1.06400
$583$	8.11676	0.336162
$584$	29.6810	1.22821
$585$	0	0
$586$	−26.0422	−1.07579
$587$	−6.93889	−0.286399	−0.143199	−	0.989694i	$-0.545739\pi$
$587$	−6.93889	−0.286399	−0.143199	+	0.989694i	$0.545739\pi$
$588$	28.7131	1.18411
$589$	−8.10916	−0.334132
$590$	0	0
$591$	19.7993	0.814435
$592$	0.149475	0.00614340
$593$	37.2097	1.52802	0.764010	−	0.645205i	$-0.223228\pi$
$593$	37.2097	1.52802	0.764010	+	0.645205i	$0.223228\pi$
$594$	49.4693	2.02975
$595$	0	0
$596$	4.95745	0.203065
$597$	−2.16161	−0.0884688
$598$	21.0576	0.861109
$599$	−20.3199	−0.830249	−0.415124	−	0.909765i	$-0.636262\pi$
$599$	−20.3199	−0.830249	−0.415124	+	0.909765i	$0.636262\pi$
$600$	0	0
$601$	24.1279	0.984196	0.492098	−	0.870540i	$-0.336230\pi$
$601$	24.1279	0.984196	0.492098	+	0.870540i	$0.336230\pi$
$602$	15.1359	0.616894
$603$	−4.64790	−0.189277
$604$	17.5166	0.712738
$605$	0	0
$606$	25.0847	1.01899
$607$	1.59852	0.0648821	0.0324410	−	0.999474i	$-0.489672\pi$
$607$	1.59852	0.0648821	0.0324410	+	0.999474i	$0.489672\pi$
$608$	34.6136	1.40377
$609$	2.84144	0.115141
$610$	0	0
$611$	14.0548	0.568595
$612$	0	0
$613$	29.2631	1.18192	0.590962	−	0.806699i	$-0.298748\pi$
$613$	29.2631	1.18192	0.590962	+	0.806699i	$0.298748\pi$
$614$	−57.9664	−2.33933
$615$	0	0
$616$	21.6694	0.873084
$617$	−17.8941	−0.720390	−0.360195	−	0.932877i	$-0.617290\pi$
$617$	−17.8941	−0.720390	−0.360195	+	0.932877i	$0.617290\pi$
$618$	30.2693	1.21761
$619$	−3.11518	−0.125210	−0.0626048	−	0.998038i	$-0.519941\pi$
$619$	−3.11518	−0.125210	−0.0626048	+	0.998038i	$0.519941\pi$
$620$	0	0
$621$	−12.0614	−0.484008
$622$	−51.4817	−2.06423
$623$	−13.3340	−0.534216
$624$	2.42185	0.0969517
$625$	0	0
$626$	−68.5478	−2.73972
$627$	57.4277	2.29344
$628$	−42.1175	−1.68067
$629$	0	0
$630$	0	0
$631$	−36.4404	−1.45067	−0.725335	−	0.688396i	$-0.758315\pi$
$631$	−36.4404	−1.45067	−0.725335	+	0.688396i	$0.758315\pi$
$632$	33.1405	1.31826
$633$	7.86292	0.312523
$634$	−27.4289	−1.08934
$635$	0	0
$636$	−10.9461	−0.434040
$637$	16.2196	0.642643
$638$	−10.4241	−0.412694
$639$	−0.680942	−0.0269376
$640$	0	0
$641$	−3.93744	−0.155520	−0.0777598	−	0.996972i	$-0.524777\pi$
$641$	−3.93744	−0.155520	−0.0777598	+	0.996972i	$0.524777\pi$
$642$	−6.05049	−0.238794
$643$	14.9025	0.587698	0.293849	−	0.955852i	$-0.405064\pi$
$643$	14.9025	0.587698	0.293849	+	0.955852i	$0.405064\pi$
$644$	−13.3091	−0.524451
$645$	0	0
$646$	0	0
$647$	28.8503	1.13422	0.567110	−	0.823642i	$-0.308061\pi$
$647$	28.8503	1.13422	0.567110	+	0.823642i	$0.308061\pi$
$648$	−31.2706	−1.22843
$649$	7.21182	0.283089
$650$	0	0
$651$	−3.55360	−0.139277
$652$	−13.4676	−0.527434
$653$	14.1367	0.553212	0.276606	−	0.960983i	$-0.410790\pi$
$653$	14.1367	0.553212	0.276606	+	0.960983i	$0.410790\pi$
$654$	46.9718	1.83674
$655$	0	0
$656$	−3.75226	−0.146501
$657$	5.13933	0.200505
$658$	−14.2398	−0.555125
$659$	−38.8064	−1.51168	−0.755840	−	0.654756i	$-0.772771\pi$
$659$	−38.8064	−1.51168	−0.755840	+	0.654756i	$0.772771\pi$
$660$	0	0
$661$	−17.6631	−0.687016	−0.343508	−	0.939150i	$-0.611615\pi$
$661$	−17.6631	−0.687016	−0.343508	+	0.939150i	$0.611615\pi$
$662$	21.0932	0.819812
$663$	0	0
$664$	38.5252	1.49507
$665$	0	0
$666$	0.494121	0.0191468
$667$	2.54156	0.0984097
$668$	−53.6230	−2.07474
$669$	14.3536	0.554943
$670$	0	0
$671$	−64.6184	−2.49456
$672$	15.1684	0.585133
$673$	−38.9804	−1.50258	−0.751291	−	0.659971i	$-0.770569\pi$
$673$	−38.9804	−1.50258	−0.751291	+	0.659971i	$0.770569\pi$
$674$	49.6415	1.91212
$675$	0	0
$676$	−2.07274	−0.0797206
$677$	46.3086	1.77978	0.889892	−	0.456172i	$-0.150780\pi$
$677$	46.3086	1.77978	0.889892	+	0.456172i	$0.150780\pi$
$678$	17.2353	0.661916
$679$	9.16432	0.351694
$680$	0	0
$681$	−44.8112	−1.71717
$682$	13.0367	0.499202
$683$	29.0957	1.11332	0.556658	−	0.830742i	$-0.312083\pi$
$683$	29.0957	1.11332	0.556658	+	0.830742i	$0.312083\pi$
$684$	−11.5467	−0.441498
$685$	0	0
$686$	−41.3820	−1.57997
$687$	−25.8632	−0.986742
$688$	1.55708	0.0593630
$689$	−6.18327	−0.235564
$690$	0	0
$691$	7.75676	0.295081	0.147541	−	0.989056i	$-0.452864\pi$
$691$	7.75676	0.295081	0.147541	+	0.989056i	$0.452864\pi$
$692$	−5.22793	−0.198736
$693$	3.75210	0.142531
$694$	−12.5924	−0.477999
$695$	0	0
$696$	5.58052	0.211529
$697$	0	0
$698$	−76.1473	−2.88222
$699$	−10.9143	−0.412818
$700$	0	0
$701$	−4.74948	−0.179386	−0.0896928	−	0.995969i	$-0.528589\pi$
$701$	−4.74948	−0.179386	−0.0896928	+	0.995969i	$0.528589\pi$
$702$	−37.6852	−1.42234
$703$	−2.70009	−0.101836
$704$	−59.0329	−2.22489
$705$	0	0
$706$	28.8133	1.08440
$707$	−8.95586	−0.336820
$708$	−9.72568	−0.365513
$709$	43.4694	1.63253	0.816264	−	0.577679i	$-0.196042\pi$
$709$	43.4694	1.63253	0.816264	+	0.577679i	$0.196042\pi$
$710$	0	0
$711$	5.73836	0.215205
$712$	−26.1877	−0.981425
$713$	−3.17856	−0.119038
$714$	0	0
$715$	0	0
$716$	11.4353	0.427357
$717$	9.56080	0.357055
$718$	−41.9377	−1.56510
$719$	38.0357	1.41849	0.709246	−	0.704961i	$-0.249036\pi$
$719$	38.0357	1.41849	0.709246	+	0.704961i	$0.249036\pi$
$720$	0	0
$721$	−10.8069	−0.402470
$722$	57.3349	2.13378
$723$	9.97476	0.370965
$724$	−48.8420	−1.81520
$725$	0	0
$726$	−44.6994	−1.65895
$727$	32.4604	1.20389	0.601944	−	0.798538i	$-0.294393\pi$
$727$	32.4604	1.20389	0.601944	+	0.798538i	$0.294393\pi$
$728$	−16.5075	−0.611809
$729$	20.7565	0.768758
$730$	0	0
$731$	0	0
$732$	87.1427	3.22089
$733$	39.4649	1.45767	0.728834	−	0.684690i	$-0.240063\pi$
$733$	39.4649	1.45767	0.728834	+	0.684690i	$0.240063\pi$
$734$	52.4982	1.93774
$735$	0	0
$736$	13.5675	0.500106
$737$	−40.8315	−1.50405
$738$	−12.4038	−0.456592
$739$	9.30119	0.342150	0.171075	−	0.985258i	$-0.445276\pi$
$739$	9.30119	0.342150	0.171075	+	0.985258i	$0.445276\pi$
$740$	0	0
$741$	−43.7478	−1.60712
$742$	6.26466	0.229983
$743$	41.6442	1.52778	0.763889	−	0.645348i	$-0.223288\pi$
$743$	41.6442	1.52778	0.763889	+	0.645348i	$0.223288\pi$
$744$	−6.97919	−0.255869
$745$	0	0
$746$	−56.6184	−2.07295
$747$	6.67072	0.244069
$748$	0	0
$749$	2.16018	0.0789312
$750$	0	0
$751$	15.5526	0.567524	0.283762	−	0.958895i	$-0.408417\pi$
$751$	15.5526	0.567524	0.283762	+	0.958895i	$0.408417\pi$
$752$	−1.46489	−0.0534190
$753$	52.9563	1.92983
$754$	7.94097	0.289193
$755$	0	0
$756$	23.8182	0.866261
$757$	41.2769	1.50023	0.750117	−	0.661306i	$-0.229997\pi$
$757$	41.2769	1.50023	0.750117	+	0.661306i	$0.229997\pi$
$758$	43.0275	1.56283
$759$	22.5100	0.817062
$760$	0	0
$761$	42.6707	1.54681	0.773407	−	0.633910i	$-0.218551\pi$
$761$	42.6707	1.54681	0.773407	+	0.633910i	$0.218551\pi$
$762$	54.0153	1.95677
$763$	−16.7701	−0.607119
$764$	−51.6444	−1.86843
$765$	0	0
$766$	59.2934	2.14236
$767$	−5.49389	−0.198373
$768$	34.3570	1.23975
$769$	8.29133	0.298993	0.149497	−	0.988762i	$-0.452235\pi$
$769$	8.29133	0.298993	0.149497	+	0.988762i	$0.452235\pi$
$770$	0	0
$771$	6.18397	0.222710
$772$	−0.351603	−0.0126545
$773$	−0.237927	−0.00855765	−0.00427883	−	0.999991i	$-0.501362\pi$
$773$	−0.237927	−0.00855765	−0.00427883	+	0.999991i	$0.501362\pi$
$774$	5.14723	0.185013
$775$	0	0
$776$	17.9985	0.646109
$777$	−1.18324	−0.0424483
$778$	55.2962	1.98246
$779$	67.7800	2.42847
$780$	0	0
$781$	−5.98202	−0.214054
$782$	0	0
$783$	−4.54845	−0.162548
$784$	−1.69052	−0.0603758
$785$	0	0
$786$	−33.7127	−1.20249
$787$	−1.48737	−0.0530191	−0.0265096	−	0.999649i	$-0.508439\pi$
$787$	−1.48737	−0.0530191	−0.0265096	+	0.999649i	$0.508439\pi$
$788$	−34.9723	−1.24584
$789$	−52.2697	−1.86085
$790$	0	0
$791$	−6.15342	−0.218791
$792$	7.36904	0.261847
$793$	49.2256	1.74805
$794$	−33.0089	−1.17144
$795$	0	0
$796$	3.81813	0.135330
$797$	0.564647	0.0200008	0.0100004	−	0.999950i	$-0.496817\pi$
$797$	0.564647	0.0200008	0.0100004	+	0.999950i	$0.496817\pi$
$798$	44.3237	1.56904
$799$	0	0
$800$	0	0
$801$	−4.53446	−0.160217
$802$	6.80883	0.240428
$803$	45.1486	1.59326
$804$	55.0643	1.94197
$805$	0	0
$806$	−9.93123	−0.349813
$807$	27.5299	0.969097
$808$	−17.5891	−0.618782
$809$	46.8966	1.64880	0.824398	−	0.566010i	$-0.191514\pi$
$809$	46.8966	1.64880	0.824398	+	0.566010i	$0.191514\pi$
$810$	0	0
$811$	−6.46841	−0.227137	−0.113568	−	0.993530i	$-0.536228\pi$
$811$	−6.46841	−0.227137	−0.113568	+	0.993530i	$0.536228\pi$
$812$	−5.01894	−0.176130
$813$	−4.71225	−0.165266
$814$	4.34081	0.152145
$815$	0	0
$816$	0	0
$817$	−28.1267	−0.984028
$818$	−5.86480	−0.205058
$819$	−2.85831	−0.0998775
$820$	0	0
$821$	11.7430	0.409835	0.204917	−	0.978779i	$-0.434307\pi$
$821$	11.7430	0.409835	0.204917	+	0.978779i	$0.434307\pi$
$822$	−65.9705	−2.30099
$823$	−24.2294	−0.844584	−0.422292	−	0.906460i	$-0.638774\pi$
$823$	−24.2294	−0.844584	−0.422292	+	0.906460i	$0.638774\pi$
$824$	−21.2245	−0.739390
$825$	0	0
$826$	5.56621	0.193673
$827$	8.28800	0.288202	0.144101	−	0.989563i	$-0.453971\pi$
$827$	8.28800	0.288202	0.144101	+	0.989563i	$0.453971\pi$
$828$	−4.52597	−0.157288
$829$	42.2364	1.46693	0.733466	−	0.679726i	$-0.237902\pi$
$829$	42.2364	1.46693	0.733466	+	0.679726i	$0.237902\pi$
$830$	0	0
$831$	−44.9999	−1.56103
$832$	44.9707	1.55908
$833$	0	0
$834$	−0.199585	−0.00691107
$835$	0	0
$836$	−101.437	−3.50826
$837$	5.68844	0.196621
$838$	13.2605	0.458078
$839$	−7.41774	−0.256089	−0.128044	−	0.991768i	$-0.540870\pi$
$839$	−7.41774	−0.256089	−0.128044	+	0.991768i	$0.540870\pi$
$840$	0	0
$841$	−28.0416	−0.966950
$842$	−45.1840	−1.55714
$843$	−26.7019	−0.919661
$844$	−13.8886	−0.478065
$845$	0	0
$846$	−4.84248	−0.166488
$847$	15.9588	0.548351
$848$	0.644465	0.0221310
$849$	−18.7975	−0.645128
$850$	0	0
$851$	−1.05836	−0.0362801
$852$	8.06720	0.276378
$853$	14.3725	0.492103	0.246052	−	0.969257i	$-0.420867\pi$
$853$	14.3725	0.492103	0.246052	+	0.969257i	$0.420867\pi$
$854$	−49.8736	−1.70664
$855$	0	0
$856$	4.24254	0.145007
$857$	43.3088	1.47940	0.739701	−	0.672936i	$-0.234967\pi$
$857$	43.3088	1.47940	0.739701	+	0.672936i	$0.234967\pi$
$858$	70.3313	2.40107
$859$	13.6870	0.466993	0.233497	−	0.972358i	$-0.424983\pi$
$859$	13.6870	0.466993	0.233497	+	0.972358i	$0.424983\pi$
$860$	0	0
$861$	29.7026	1.01226
$862$	59.5291	2.02757
$863$	6.87984	0.234192	0.117096	−	0.993121i	$-0.462641\pi$
$863$	6.87984	0.234192	0.117096	+	0.993121i	$0.462641\pi$
$864$	−24.2808	−0.826051
$865$	0	0
$866$	−89.5628	−3.04347
$867$	0	0
$868$	6.27686	0.213050
$869$	50.4110	1.71008
$870$	0	0
$871$	31.1050	1.05395
$872$	−32.9361	−1.11536
$873$	3.11648	0.105477
$874$	39.6459	1.34104
$875$	0	0
$876$	−60.8863	−2.05716
$877$	−28.4444	−0.960499	−0.480249	−	0.877132i	$-0.659454\pi$
$877$	−28.4444	−0.960499	−0.480249	+	0.877132i	$0.659454\pi$
$878$	−7.88165	−0.265993
$879$	21.2070	0.715295
$880$	0	0
$881$	20.9435	0.705605	0.352803	−	0.935698i	$-0.385229\pi$
$881$	20.9435	0.705605	0.352803	+	0.935698i	$0.385229\pi$
$882$	−5.58835	−0.188170
$883$	7.88825	0.265461	0.132730	−	0.991152i	$-0.457626\pi$
$883$	7.88825	0.265461	0.132730	+	0.991152i	$0.457626\pi$
$884$	0	0
$885$	0	0
$886$	45.7598	1.53733
$887$	5.46346	0.183445	0.0917224	−	0.995785i	$-0.470763\pi$
$887$	5.46346	0.183445	0.0917224	+	0.995785i	$0.470763\pi$
$888$	−2.32385	−0.0779832
$889$	−19.2848	−0.646793
$890$	0	0
$891$	−47.5667	−1.59354
$892$	−25.3533	−0.848892
$893$	26.4614	0.885498
$894$	−6.47145	−0.216438
$895$	0	0
$896$	−29.4061	−0.982388
$897$	−17.1479	−0.572552
$898$	13.8807	0.463205
$899$	−1.19866	−0.0399775
$900$	0	0
$901$	0	0
$902$	−108.967	−3.62820
$903$	−12.3257	−0.410173
$904$	−12.0852	−0.401947
$905$	0	0
$906$	−22.8661	−0.759675
$907$	−14.3557	−0.476673	−0.238337	−	0.971183i	$-0.576602\pi$
$907$	−14.3557	−0.476673	−0.238337	+	0.971183i	$0.576602\pi$
$908$	79.1517	2.62674
$909$	−3.04559	−0.101016
$910$	0	0
$911$	−23.3285	−0.772908	−0.386454	−	0.922309i	$-0.626300\pi$
$911$	−23.3285	−0.772908	−0.386454	+	0.922309i	$0.626300\pi$
$912$	4.55971	0.150987
$913$	58.6018	1.93944
$914$	65.9172	2.18035
$915$	0	0
$916$	45.6831	1.50941
$917$	12.0363	0.397473
$918$	0	0
$919$	−27.6524	−0.912170	−0.456085	−	0.889936i	$-0.650749\pi$
$919$	−27.6524	−0.912170	−0.456085	+	0.889936i	$0.650749\pi$
$920$	0	0
$921$	47.2040	1.55542
$922$	17.7530	0.584663
$923$	4.55704	0.149997
$924$	−44.4516	−1.46235
$925$	0	0
$926$	−61.7539	−2.02936
$927$	−3.67507	−0.120705
$928$	5.11642	0.167955
$929$	0.973310	0.0319333	0.0159666	−	0.999873i	$-0.494917\pi$
$929$	0.973310	0.0319333	0.0159666	+	0.999873i	$0.494917\pi$
$930$	0	0
$931$	30.5372	1.00082
$932$	19.2784	0.631484
$933$	41.9233	1.37251
$934$	−23.8368	−0.779965
$935$	0	0
$936$	−5.61366	−0.183488
$937$	4.08311	0.133390	0.0666948	−	0.997773i	$-0.478755\pi$
$937$	4.08311	0.133390	0.0666948	+	0.997773i	$0.478755\pi$
$938$	−31.5144	−1.02898
$939$	55.8209	1.82164
$940$	0	0
$941$	16.3402	0.532674	0.266337	−	0.963880i	$-0.414187\pi$
$941$	16.3402	0.532674	0.266337	+	0.963880i	$0.414187\pi$
$942$	54.9802	1.79135
$943$	26.5679	0.865169
$944$	0.572613	0.0186370
$945$	0	0
$946$	45.2180	1.47016
$947$	−16.5623	−0.538203	−0.269102	−	0.963112i	$-0.586727\pi$
$947$	−16.5623	−0.538203	−0.269102	+	0.963112i	$0.586727\pi$
$948$	−67.9831	−2.20799
$949$	−34.3938	−1.11647
$950$	0	0
$951$	22.3363	0.724303
$952$	0	0
$953$	57.0108	1.84676	0.923380	−	0.383887i	$-0.125415\pi$
$953$	57.0108	1.84676	0.923380	+	0.383887i	$0.125415\pi$
$954$	2.13041	0.0689744
$955$	0	0
$956$	−16.8876	−0.546184
$957$	8.48870	0.274401
$958$	−59.6663	−1.92773
$959$	23.5531	0.760570
$960$	0	0
$961$	−29.5009	−0.951643
$962$	−3.30678	−0.106615
$963$	0.734605	0.0236723
$964$	−17.6188	−0.567463
$965$	0	0
$966$	17.3736	0.558988
$967$	−10.1747	−0.327196	−0.163598	−	0.986527i	$-0.552310\pi$
$967$	−10.1747	−0.327196	−0.163598	+	0.986527i	$0.552310\pi$
$968$	31.3427	1.00739
$969$	0	0
$970$	0	0
$971$	−15.0671	−0.483527	−0.241763	−	0.970335i	$-0.577726\pi$
$971$	−15.0671	−0.483527	−0.241763	+	0.970335i	$0.577726\pi$
$972$	17.9203	0.574795
$973$	0.0712569	0.00228439
$974$	−54.7472	−1.75421
$975$	0	0
$976$	−5.13065	−0.164228
$977$	−18.0224	−0.576586	−0.288293	−	0.957542i	$-0.593088\pi$
$977$	−18.0224	−0.576586	−0.288293	+	0.957542i	$0.593088\pi$
$978$	17.5807	0.562167
$979$	−39.8349	−1.27313
$980$	0	0
$981$	−5.70296	−0.182082
$982$	92.3429	2.94678
$983$	−23.9812	−0.764881	−0.382440	−	0.923980i	$-0.624916\pi$
$983$	−23.9812	−0.764881	−0.382440	+	0.923980i	$0.624916\pi$
$984$	58.3352	1.85966
$985$	0	0
$986$	0	0
$987$	11.5960	0.369103
$988$	77.2734	2.45839
$989$	−11.0249	−0.350571
$990$	0	0
$991$	35.2870	1.12093	0.560465	−	0.828178i	$-0.310622\pi$
$991$	35.2870	1.12093	0.560465	+	0.828178i	$0.310622\pi$
$992$	−6.39876	−0.203161
$993$	−17.1769	−0.545094
$994$	−4.61703	−0.146443
$995$	0	0
$996$	−79.0289	−2.50413
$997$	51.2819	1.62411	0.812057	−	0.583577i	$-0.198348\pi$
$997$	51.2819	1.62411	0.812057	+	0.583577i	$0.198348\pi$
$998$	−8.38497	−0.265421
$999$	1.89407	0.0599257

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.by.1.23		24
5.2	odd	4		1445.2.b.i.579.24		24
5.3	odd	4		1445.2.b.i.579.1		24
5.4	even	2	inner	7225.2.a.by.1.2		24
17.10	odd	16		425.2.m.e.151.1		24
17.12	odd	16		425.2.m.e.76.1		24
17.16	even	2	inner	7225.2.a.by.1.24		24
85.12	even	16		85.2.m.a.59.1	yes	24
85.27	even	16		85.2.m.a.49.6	yes	24
85.29	odd	16		425.2.m.e.76.6		24
85.33	odd	4		1445.2.b.i.579.2		24
85.44	odd	16		425.2.m.e.151.6		24
85.63	even	16		85.2.m.a.59.6	yes	24
85.67	odd	4		1445.2.b.i.579.23		24
85.78	even	16		85.2.m.a.49.1	✓	24
85.84	even	2	inner	7225.2.a.by.1.1		24
255.182	odd	16		765.2.bh.b.739.6		24
255.197	odd	16		765.2.bh.b.559.1		24
255.233	odd	16		765.2.bh.b.739.1		24
255.248	odd	16		765.2.bh.b.559.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.m.a.49.1	✓	24	85.78	even	16
85.2.m.a.49.6	yes	24	85.27	even	16
85.2.m.a.59.1	yes	24	85.12	even	16
85.2.m.a.59.6	yes	24	85.63	even	16
425.2.m.e.76.1		24	17.12	odd	16
425.2.m.e.76.6		24	85.29	odd	16
425.2.m.e.151.1		24	17.10	odd	16
425.2.m.e.151.6		24	85.44	odd	16
765.2.bh.b.559.1		24	255.197	odd	16
765.2.bh.b.559.6		24	255.248	odd	16
765.2.bh.b.739.1		24	255.233	odd	16
765.2.bh.b.739.6		24	255.182	odd	16
1445.2.b.i.579.1		24	5.3	odd	4
1445.2.b.i.579.2		24	85.33	odd	4
1445.2.b.i.579.23		24	85.67	odd	4
1445.2.b.i.579.24		24	5.2	odd	4
7225.2.a.by.1.1		24	85.84	even	2	inner
7225.2.a.by.1.2		24	5.4	even	2	inner
7225.2.a.by.1.23		24	1.1	even	1	trivial
7225.2.a.by.1.24		24	17.16	even	2	inner

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7225.2.a.by.1.23

Level	$7225$
Weight	$2$
Character	7225.1
Self dual	yes
Analytic conductor	$57.692$
Analytic rank	$1$
Dimension	$24$
Inner twists	$4$