LMFDB - Embedded newform 7360.2.a.ct.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.406264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 5x^{2} + 7x - 4 $ x^5 - x^4 - 6x^3 + 5x^2 + 7*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.50467$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.264024 q^{3} -1.00000 q^{5} +2.69898 q^{7} -2.93029 q^{9} +O(q^{10})$	$q+0.264024 q^{3} -1.00000 q^{5} +2.69898 q^{7} -2.93029 q^{9} -0.153728 q^{11} +3.71947 q^{13} -0.264024 q^{15} -1.55100 q^{17} +2.13324 q^{19} +0.712596 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.56574 q^{27} -4.73170 q^{29} -4.53302 q^{31} -0.0405877 q^{33} -2.69898 q^{35} -5.51941 q^{37} +0.982028 q^{39} -11.0223 q^{41} +6.27046 q^{43} +2.93029 q^{45} -12.5956 q^{47} +0.284507 q^{49} -0.409500 q^{51} -8.58441 q^{53} +0.153728 q^{55} +0.563227 q^{57} -0.242538 q^{59} +7.24424 q^{61} -7.90881 q^{63} -3.71947 q^{65} +14.7026 q^{67} +0.264024 q^{69} -14.7460 q^{71} +6.89904 q^{73} +0.264024 q^{75} -0.414908 q^{77} +8.28913 q^{79} +8.37748 q^{81} +3.40124 q^{83} +1.55100 q^{85} -1.24928 q^{87} +0.953672 q^{89} +10.0388 q^{91} -1.19683 q^{93} -2.13324 q^{95} +6.49560 q^{97} +0.450466 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 2 * q^9	$ 5 q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 2 q^{9} - q^{11} - q^{13} - 3 q^{15} - 8 q^{17} + 9 q^{19} - 8 q^{21} + 5 q^{23} + 5 q^{25} + 12 q^{27} + q^{29} - 6 q^{31} + 15 q^{33} + 8 q^{35} - q^{37} - 12 q^{39} + 6 q^{41} + 12 q^{43} - 2 q^{45} - 32 q^{47} + 5 q^{49} + 3 q^{51} + 3 q^{53} + q^{55} - 2 q^{57} - 5 q^{59} - 7 q^{61} - 25 q^{63} + q^{65} + 17 q^{67} + 3 q^{69} - 16 q^{71} + 20 q^{73} + 3 q^{75} + 10 q^{77} - 4 q^{79} - 3 q^{81} + q^{83} + 8 q^{85} - 28 q^{87} - 2 q^{89} + 7 q^{91} + 24 q^{93} - 9 q^{95} - 17 q^{97} + q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 2 * q^9 - q^11 - q^13 - 3 * q^15 - 8 * q^17 + 9 * q^19 - 8 * q^21 + 5 * q^23 + 5 * q^25 + 12 * q^27 + q^29 - 6 * q^31 + 15 * q^33 + 8 * q^35 - q^37 - 12 * q^39 + 6 * q^41 + 12 * q^43 - 2 * q^45 - 32 * q^47 + 5 * q^49 + 3 * q^51 + 3 * q^53 + q^55 - 2 * q^57 - 5 * q^59 - 7 * q^61 - 25 * q^63 + q^65 + 17 * q^67 + 3 * q^69 - 16 * q^71 + 20 * q^73 + 3 * q^75 + 10 * q^77 - 4 * q^79 - 3 * q^81 + q^83 + 8 * q^85 - 28 * q^87 - 2 * q^89 + 7 * q^91 + 24 * q^93 - 9 * q^95 - 17 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.264024	0.152434	0.0762171	−	0.997091i	$-0.475716\pi$
$3$	0.264024	0.152434	0.0762171	+	0.997091i	$0.475716\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.69898	1.02012	0.510060	−	0.860139i	$-0.329623\pi$
$7$	2.69898	1.02012	0.510060	+	0.860139i	$0.329623\pi$
$8$	0	0
$9$	−2.93029	−0.976764
$10$	0	0
$11$	−0.153728	−0.0463506	−0.0231753	−	0.999731i	$-0.507378\pi$
$11$	−0.153728	−0.0463506	−0.0231753	+	0.999731i	$0.507378\pi$
$12$	0	0
$13$	3.71947	1.03159	0.515797	−	0.856711i	$-0.327496\pi$
$13$	3.71947	1.03159	0.515797	+	0.856711i	$0.327496\pi$
$14$	0	0
$15$	−0.264024	−0.0681707
$16$	0	0
$17$	−1.55100	−0.376172	−0.188086	−	0.982153i	$-0.560228\pi$
$17$	−1.55100	−0.376172	−0.188086	+	0.982153i	$0.560228\pi$
$18$	0	0
$19$	2.13324	0.489400	0.244700	−	0.969599i	$-0.421311\pi$
$19$	2.13324	0.489400	0.244700	+	0.969599i	$0.421311\pi$
$20$	0	0
$21$	0.712596	0.155501
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.56574	−0.301326
$28$	0	0
$29$	−4.73170	−0.878654	−0.439327	−	0.898327i	$-0.644783\pi$
$29$	−4.73170	−0.878654	−0.439327	+	0.898327i	$0.644783\pi$
$30$	0	0
$31$	−4.53302	−0.814155	−0.407078	−	0.913394i	$-0.633452\pi$
$31$	−4.53302	−0.814155	−0.407078	+	0.913394i	$0.633452\pi$
$32$	0	0
$33$	−0.0405877	−0.00706542
$34$	0	0
$35$	−2.69898	−0.456211
$36$	0	0
$37$	−5.51941	−0.907385	−0.453693	−	0.891158i	$-0.649894\pi$
$37$	−5.51941	−0.907385	−0.453693	+	0.891158i	$0.649894\pi$
$38$	0	0
$39$	0.982028	0.157250
$40$	0	0
$41$	−11.0223	−1.72139	−0.860694	−	0.509122i	$-0.829970\pi$
$41$	−11.0223	−1.72139	−0.860694	+	0.509122i	$0.829970\pi$
$42$	0	0
$43$	6.27046	0.956236	0.478118	−	0.878295i	$-0.341319\pi$
$43$	6.27046	0.956236	0.478118	+	0.878295i	$0.341319\pi$
$44$	0	0
$45$	2.93029	0.436822
$46$	0	0
$47$	−12.5956	−1.83725	−0.918625	−	0.395130i	$-0.870700\pi$
$47$	−12.5956	−1.83725	−0.918625	+	0.395130i	$0.870700\pi$
$48$	0	0
$49$	0.284507	0.0406439
$50$	0	0
$51$	−0.409500	−0.0573414
$52$	0	0
$53$	−8.58441	−1.17916	−0.589580	−	0.807710i	$-0.700707\pi$
$53$	−8.58441	−1.17916	−0.589580	+	0.807710i	$0.700707\pi$
$54$	0	0
$55$	0.153728	0.0207286
$56$	0	0
$57$	0.563227	0.0746013
$58$	0	0
$59$	−0.242538	−0.0315758	−0.0157879	−	0.999875i	$-0.505026\pi$
$59$	−0.242538	−0.0315758	−0.0157879	+	0.999875i	$0.505026\pi$
$60$	0	0
$61$	7.24424	0.927530	0.463765	−	0.885958i	$-0.346498\pi$
$61$	7.24424	0.927530	0.463765	+	0.885958i	$0.346498\pi$
$62$	0	0
$63$	−7.90881	−0.996416
$64$	0	0
$65$	−3.71947	−0.461343
$66$	0	0
$67$	14.7026	1.79621	0.898103	−	0.439786i	$-0.144946\pi$
$67$	14.7026	1.79621	0.898103	+	0.439786i	$0.144946\pi$
$68$	0	0
$69$	0.264024	0.0317847
$70$	0	0
$71$	−14.7460	−1.75003	−0.875015	−	0.484097i	$-0.839148\pi$
$71$	−14.7460	−1.75003	−0.875015	+	0.484097i	$0.839148\pi$
$72$	0	0
$73$	6.89904	0.807471	0.403736	−	0.914876i	$-0.367712\pi$
$73$	6.89904	0.807471	0.403736	+	0.914876i	$0.367712\pi$
$74$	0	0
$75$	0.264024	0.0304868
$76$	0	0
$77$	−0.414908	−0.0472831
$78$	0	0
$79$	8.28913	0.932600	0.466300	−	0.884627i	$-0.345587\pi$
$79$	8.28913	0.932600	0.466300	+	0.884627i	$0.345587\pi$
$80$	0	0
$81$	8.37748	0.930831
$82$	0	0
$83$	3.40124	0.373335	0.186667	−	0.982423i	$-0.440231\pi$
$83$	3.40124	0.373335	0.186667	+	0.982423i	$0.440231\pi$
$84$	0	0
$85$	1.55100	0.168229
$86$	0	0
$87$	−1.24928	−0.133937
$88$	0	0
$89$	0.953672	0.101089	0.0505445	−	0.998722i	$-0.483904\pi$
$89$	0.953672	0.101089	0.0505445	+	0.998722i	$0.483904\pi$
$90$	0	0
$91$	10.0388	1.05235
$92$	0	0
$93$	−1.19683	−0.124105
$94$	0	0
$95$	−2.13324	−0.218866
$96$	0	0
$97$	6.49560	0.659528	0.329764	−	0.944063i	$-0.393031\pi$
$97$	6.49560	0.659528	0.329764	+	0.944063i	$0.393031\pi$
$98$	0	0
$99$	0.450466	0.0452736
$100$	0	0
$101$	0.0190213	0.00189269	0.000946343	−	1.00000i	$-0.499699\pi$
$101$	0.0190213	0.00189269	0.000946343		1.00000i	$0.499699\pi$
$102$	0	0
$103$	15.1742	1.49516	0.747580	−	0.664171i	$-0.231215\pi$
$103$	15.1742	1.49516	0.747580	+	0.664171i	$0.231215\pi$
$104$	0	0
$105$	−0.712596	−0.0695422
$106$	0	0
$107$	9.37999	0.906798	0.453399	−	0.891308i	$-0.350211\pi$
$107$	9.37999	0.906798	0.453399	+	0.891308i	$0.350211\pi$
$108$	0	0
$109$	−9.24898	−0.885891	−0.442946	−	0.896548i	$-0.646067\pi$
$109$	−9.24898	−0.885891	−0.442946	+	0.896548i	$0.646067\pi$
$110$	0	0
$111$	−1.45726	−0.138317
$112$	0	0
$113$	−4.76082	−0.447860	−0.223930	−	0.974605i	$-0.571889\pi$
$113$	−4.76082	−0.447860	−0.223930	+	0.974605i	$0.571889\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−10.8991	−1.00762
$118$	0	0
$119$	−4.18611	−0.383740
$120$	0	0
$121$	−10.9764	−0.997852
$122$	0	0
$123$	−2.91014	−0.262399
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.2930	−1.62324	−0.811622	−	0.584183i	$-0.801415\pi$
$127$	−18.2930	−1.62324	−0.811622	+	0.584183i	$0.801415\pi$
$128$	0	0
$129$	1.65555	0.145763
$130$	0	0
$131$	−21.4285	−1.87222	−0.936108	−	0.351714i	$-0.885599\pi$
$131$	−21.4285	−1.87222	−0.936108	+	0.351714i	$0.885599\pi$
$132$	0	0
$133$	5.75759	0.499246
$134$	0	0
$135$	1.56574	0.134757
$136$	0	0
$137$	11.4016	0.974101	0.487050	−	0.873374i	$-0.338073\pi$
$137$	11.4016	0.974101	0.487050	+	0.873374i	$0.338073\pi$
$138$	0	0
$139$	−10.9095	−0.925332	−0.462666	−	0.886533i	$-0.653107\pi$
$139$	−10.9095	−0.925332	−0.462666	+	0.886533i	$0.653107\pi$
$140$	0	0
$141$	−3.32553	−0.280060
$142$	0	0
$143$	−0.571784	−0.0478150
$144$	0	0
$145$	4.73170	0.392946
$146$	0	0
$147$	0.0751167	0.00619552
$148$	0	0
$149$	−1.10066	−0.0901693	−0.0450846	−	0.998983i	$-0.514356\pi$
$149$	−1.10066	−0.0901693	−0.0450846	+	0.998983i	$0.514356\pi$
$150$	0	0
$151$	−15.3634	−1.25026	−0.625129	−	0.780521i	$-0.714954\pi$
$151$	−15.3634	−1.25026	−0.625129	+	0.780521i	$0.714954\pi$
$152$	0	0
$153$	4.54487	0.367431
$154$	0	0
$155$	4.53302	0.364101
$156$	0	0
$157$	14.9770	1.19530	0.597648	−	0.801759i	$-0.296102\pi$
$157$	14.9770	1.19530	0.597648	+	0.801759i	$0.296102\pi$
$158$	0	0
$159$	−2.26649	−0.179744
$160$	0	0
$161$	2.69898	0.212710
$162$	0	0
$163$	−14.1336	−1.10703	−0.553515	−	0.832839i	$-0.686714\pi$
$163$	−14.1336	−1.10703	−0.553515	+	0.832839i	$0.686714\pi$
$164$	0	0
$165$	0.0405877	0.00315975
$166$	0	0
$167$	−2.81683	−0.217973	−0.108987	−	0.994043i	$-0.534761\pi$
$167$	−2.81683	−0.217973	−0.108987	+	0.994043i	$0.534761\pi$
$168$	0	0
$169$	0.834427	0.0641867
$170$	0	0
$171$	−6.25103	−0.478028
$172$	0	0
$173$	2.58314	0.196393	0.0981963	−	0.995167i	$-0.468693\pi$
$173$	2.58314	0.196393	0.0981963	+	0.995167i	$0.468693\pi$
$174$	0	0
$175$	2.69898	0.204024
$176$	0	0
$177$	−0.0640358	−0.00481323
$178$	0	0
$179$	19.1239	1.42939	0.714696	−	0.699436i	$-0.246565\pi$
$179$	19.1239	1.42939	0.714696	+	0.699436i	$0.246565\pi$
$180$	0	0
$181$	−1.92926	−0.143401	−0.0717005	−	0.997426i	$-0.522843\pi$
$181$	−1.92926	−0.143401	−0.0717005	+	0.997426i	$0.522843\pi$
$182$	0	0
$183$	1.91265	0.141387
$184$	0	0
$185$	5.51941	0.405795
$186$	0	0
$187$	0.238431	0.0174358
$188$	0	0
$189$	−4.22590	−0.307389
$190$	0	0
$191$	−23.7456	−1.71817	−0.859087	−	0.511830i	$-0.828968\pi$
$191$	−23.7456	−1.71817	−0.859087	+	0.511830i	$0.828968\pi$
$192$	0	0
$193$	−22.0492	−1.58714	−0.793568	−	0.608482i	$-0.791779\pi$
$193$	−22.0492	−1.58714	−0.793568	+	0.608482i	$0.791779\pi$
$194$	0	0
$195$	−0.982028	−0.0703245
$196$	0	0
$197$	18.2752	1.30205	0.651027	−	0.759055i	$-0.274339\pi$
$197$	18.2752	1.30205	0.651027	+	0.759055i	$0.274339\pi$
$198$	0	0
$199$	−3.71519	−0.263363	−0.131681	−	0.991292i	$-0.542038\pi$
$199$	−3.71519	−0.263363	−0.131681	+	0.991292i	$0.542038\pi$
$200$	0	0
$201$	3.88183	0.273803
$202$	0	0
$203$	−12.7708	−0.896332
$204$	0	0
$205$	11.0223	0.769828
$206$	0	0
$207$	−2.93029	−0.203669
$208$	0	0
$209$	−0.327938	−0.0226840
$210$	0	0
$211$	2.54999	0.175549	0.0877744	−	0.996140i	$-0.472025\pi$
$211$	2.54999	0.175549	0.0877744	+	0.996140i	$0.472025\pi$
$212$	0	0
$213$	−3.89330	−0.266764
$214$	0	0
$215$	−6.27046	−0.427642
$216$	0	0
$217$	−12.2346	−0.830535
$218$	0	0
$219$	1.82151	0.123086
$220$	0	0
$221$	−5.76888	−0.388057
$222$	0	0
$223$	−25.4275	−1.70275	−0.851374	−	0.524559i	$-0.824230\pi$
$223$	−25.4275	−1.70275	−0.851374	+	0.524559i	$0.824230\pi$
$224$	0	0
$225$	−2.93029	−0.195353
$226$	0	0
$227$	−17.8560	−1.18514	−0.592572	−	0.805517i	$-0.701887\pi$
$227$	−17.8560	−1.18514	−0.592572	+	0.805517i	$0.701887\pi$
$228$	0	0
$229$	−16.8467	−1.11326	−0.556629	−	0.830761i	$-0.687906\pi$
$229$	−16.8467	−1.11326	−0.556629	+	0.830761i	$0.687906\pi$
$230$	0	0
$231$	−0.109546	−0.00720757
$232$	0	0
$233$	−1.11671	−0.0731579	−0.0365790	−	0.999331i	$-0.511646\pi$
$233$	−1.11671	−0.0731579	−0.0365790	+	0.999331i	$0.511646\pi$
$234$	0	0
$235$	12.5956	0.821644
$236$	0	0
$237$	2.18853	0.142160
$238$	0	0
$239$	−0.421027	−0.0272340	−0.0136170	−	0.999907i	$-0.504335\pi$
$239$	−0.421027	−0.0272340	−0.0136170	+	0.999907i	$0.504335\pi$
$240$	0	0
$241$	−15.7250	−1.01293	−0.506467	−	0.862260i	$-0.669049\pi$
$241$	−15.7250	−1.01293	−0.506467	+	0.862260i	$0.669049\pi$
$242$	0	0
$243$	6.90907	0.443217
$244$	0	0
$245$	−0.284507	−0.0181765
$246$	0	0
$247$	7.93453	0.504862
$248$	0	0
$249$	0.898009	0.0569090
$250$	0	0
$251$	−17.1810	−1.08445	−0.542227	−	0.840232i	$-0.682419\pi$
$251$	−17.1810	−1.08445	−0.542227	+	0.840232i	$0.682419\pi$
$252$	0	0
$253$	−0.153728	−0.00966477
$254$	0	0
$255$	0.409500	0.0256439
$256$	0	0
$257$	26.7315	1.66747	0.833734	−	0.552167i	$-0.186199\pi$
$257$	26.7315	1.66747	0.833734	+	0.552167i	$0.186199\pi$
$258$	0	0
$259$	−14.8968	−0.925642
$260$	0	0
$261$	13.8653	0.858238
$262$	0	0
$263$	−13.0981	−0.807666	−0.403833	−	0.914833i	$-0.632322\pi$
$263$	−13.0981	−0.807666	−0.403833	+	0.914833i	$0.632322\pi$
$264$	0	0
$265$	8.58441	0.527336
$266$	0	0
$267$	0.251792	0.0154094
$268$	0	0
$269$	15.9041	0.969693	0.484846	−	0.874599i	$-0.338876\pi$
$269$	15.9041	0.969693	0.484846	+	0.874599i	$0.338876\pi$
$270$	0	0
$271$	−14.2217	−0.863905	−0.431952	−	0.901896i	$-0.642175\pi$
$271$	−14.2217	−0.863905	−0.431952	+	0.901896i	$0.642175\pi$
$272$	0	0
$273$	2.65048	0.160414
$274$	0	0
$275$	−0.153728	−0.00927012
$276$	0	0
$277$	10.4249	0.626374	0.313187	−	0.949691i	$-0.398603\pi$
$277$	10.4249	0.626374	0.313187	+	0.949691i	$0.398603\pi$
$278$	0	0
$279$	13.2831	0.795237
$280$	0	0
$281$	15.0340	0.896852	0.448426	−	0.893820i	$-0.351985\pi$
$281$	15.0340	0.896852	0.448426	+	0.893820i	$0.351985\pi$
$282$	0	0
$283$	−20.0888	−1.19415	−0.597077	−	0.802184i	$-0.703671\pi$
$283$	−20.0888	−1.19415	−0.597077	+	0.802184i	$0.703671\pi$
$284$	0	0
$285$	−0.563227	−0.0333627
$286$	0	0
$287$	−29.7489	−1.75602
$288$	0	0
$289$	−14.5944	−0.858495
$290$	0	0
$291$	1.71499	0.100535
$292$	0	0
$293$	−14.4205	−0.842456	−0.421228	−	0.906955i	$-0.638401\pi$
$293$	−14.4205	−0.842456	−0.421228	+	0.906955i	$0.638401\pi$
$294$	0	0
$295$	0.242538	0.0141211
$296$	0	0
$297$	0.240697	0.0139667
$298$	0	0
$299$	3.71947	0.215102
$300$	0	0
$301$	16.9239	0.975475
$302$	0	0
$303$	0.00502207	0.000288510 0
$304$	0	0
$305$	−7.24424	−0.414804
$306$	0	0
$307$	14.2224	0.811714	0.405857	−	0.913937i	$-0.366973\pi$
$307$	14.2224	0.811714	0.405857	+	0.913937i	$0.366973\pi$
$308$	0	0
$309$	4.00636	0.227914
$310$	0	0
$311$	16.5205	0.936793	0.468396	−	0.883518i	$-0.344832\pi$
$311$	16.5205	0.936793	0.468396	+	0.883518i	$0.344832\pi$
$312$	0	0
$313$	−14.7489	−0.833654	−0.416827	−	0.908986i	$-0.636858\pi$
$313$	−14.7489	−0.833654	−0.416827	+	0.908986i	$0.636858\pi$
$314$	0	0
$315$	7.90881	0.445611
$316$	0	0
$317$	−11.2952	−0.634400	−0.317200	−	0.948359i	$-0.602743\pi$
$317$	−11.2952	−0.634400	−0.317200	+	0.948359i	$0.602743\pi$
$318$	0	0
$319$	0.727392	0.0407261
$320$	0	0
$321$	2.47654	0.138227
$322$	0	0
$323$	−3.30865	−0.184098
$324$	0	0
$325$	3.71947	0.206319
$326$	0	0
$327$	−2.44195	−0.135040
$328$	0	0
$329$	−33.9952	−1.87422
$330$	0	0
$331$	2.00517	0.110214	0.0551071	−	0.998480i	$-0.482450\pi$
$331$	2.00517	0.110214	0.0551071	+	0.998480i	$0.482450\pi$
$332$	0	0
$333$	16.1735	0.886301
$334$	0	0
$335$	−14.7026	−0.803288
$336$	0	0
$337$	32.8623	1.79012	0.895061	−	0.445944i	$-0.147132\pi$
$337$	32.8623	1.79012	0.895061	+	0.445944i	$0.147132\pi$
$338$	0	0
$339$	−1.25697	−0.0682692
$340$	0	0
$341$	0.696850	0.0377366
$342$	0	0
$343$	−18.1250	−0.978658
$344$	0	0
$345$	−0.264024	−0.0142146
$346$	0	0
$347$	29.9407	1.60730	0.803650	−	0.595103i	$-0.202889\pi$
$347$	29.9407	1.60730	0.803650	+	0.595103i	$0.202889\pi$
$348$	0	0
$349$	−2.20073	−0.117802	−0.0589011	−	0.998264i	$-0.518760\pi$
$349$	−2.20073	−0.117802	−0.0589011	+	0.998264i	$0.518760\pi$
$350$	0	0
$351$	−5.82371	−0.310847
$352$	0	0
$353$	8.02040	0.426883	0.213441	−	0.976956i	$-0.431533\pi$
$353$	8.02040	0.426883	0.213441	+	0.976956i	$0.431533\pi$
$354$	0	0
$355$	14.7460	0.782637
$356$	0	0
$357$	−1.10523	−0.0584951
$358$	0	0
$359$	−7.12463	−0.376024	−0.188012	−	0.982167i	$-0.560204\pi$
$359$	−7.12463	−0.376024	−0.188012	+	0.982167i	$0.560204\pi$
$360$	0	0
$361$	−14.4493	−0.760488
$362$	0	0
$363$	−2.89802	−0.152107
$364$	0	0
$365$	−6.89904	−0.361112
$366$	0	0
$367$	−21.9695	−1.14680	−0.573399	−	0.819276i	$-0.694376\pi$
$367$	−21.9695	−1.14680	−0.573399	+	0.819276i	$0.694376\pi$
$368$	0	0
$369$	32.2984	1.68139
$370$	0	0
$371$	−23.1692	−1.20288
$372$	0	0
$373$	14.4053	0.745878	0.372939	−	0.927856i	$-0.378350\pi$
$373$	14.4053	0.745878	0.372939	+	0.927856i	$0.378350\pi$
$374$	0	0
$375$	−0.264024	−0.0136341
$376$	0	0
$377$	−17.5994	−0.906415
$378$	0	0
$379$	2.29333	0.117800	0.0589002	−	0.998264i	$-0.481241\pi$
$379$	2.29333	0.117800	0.0589002	+	0.998264i	$0.481241\pi$
$380$	0	0
$381$	−4.82980	−0.247438
$382$	0	0
$383$	−23.5860	−1.20519	−0.602594	−	0.798048i	$-0.705866\pi$
$383$	−23.5860	−1.20519	−0.602594	+	0.798048i	$0.705866\pi$
$384$	0	0
$385$	0.414908	0.0211457
$386$	0	0
$387$	−18.3743	−0.934017
$388$	0	0
$389$	−4.35107	−0.220608	−0.110304	−	0.993898i	$-0.535182\pi$
$389$	−4.35107	−0.220608	−0.110304	+	0.993898i	$0.535182\pi$
$390$	0	0
$391$	−1.55100	−0.0784372
$392$	0	0
$393$	−5.65763	−0.285390
$394$	0	0
$395$	−8.28913	−0.417071
$396$	0	0
$397$	−3.55037	−0.178188	−0.0890940	−	0.996023i	$-0.528397\pi$
$397$	−3.55037	−0.178188	−0.0890940	+	0.996023i	$0.528397\pi$
$398$	0	0
$399$	1.52014	0.0761022
$400$	0	0
$401$	13.6149	0.679897	0.339949	−	0.940444i	$-0.389590\pi$
$401$	13.6149	0.679897	0.339949	+	0.940444i	$0.389590\pi$
$402$	0	0
$403$	−16.8604	−0.839878
$404$	0	0
$405$	−8.37748	−0.416280
$406$	0	0
$407$	0.848485	0.0420579
$408$	0	0
$409$	33.6893	1.66583	0.832915	−	0.553402i	$-0.186671\pi$
$409$	33.6893	1.66583	0.832915	+	0.553402i	$0.186671\pi$
$410$	0	0
$411$	3.01028	0.148486
$412$	0	0
$413$	−0.654606	−0.0322110
$414$	0	0
$415$	−3.40124	−0.166960
$416$	0	0
$417$	−2.88037	−0.141052
$418$	0	0
$419$	−36.3063	−1.77368	−0.886839	−	0.462079i	$-0.847104\pi$
$419$	−36.3063	−1.77368	−0.886839	+	0.462079i	$0.847104\pi$
$420$	0	0
$421$	−27.0642	−1.31903	−0.659514	−	0.751692i	$-0.729238\pi$
$421$	−27.0642	−1.31903	−0.659514	+	0.751692i	$0.729238\pi$
$422$	0	0
$423$	36.9087	1.79456
$424$	0	0
$425$	−1.55100	−0.0752343
$426$	0	0
$427$	19.5521	0.946191
$428$	0	0
$429$	−0.150965	−0.00728864
$430$	0	0
$431$	36.0771	1.73777	0.868887	−	0.495011i	$-0.164836\pi$
$431$	36.0771	1.73777	0.868887	+	0.495011i	$0.164836\pi$
$432$	0	0
$433$	35.6727	1.71432	0.857161	−	0.515049i	$-0.172226\pi$
$433$	35.6727	1.71432	0.857161	+	0.515049i	$0.172226\pi$
$434$	0	0
$435$	1.24928	0.0598984
$436$	0	0
$437$	2.13324	0.102047
$438$	0	0
$439$	3.38908	0.161752	0.0808759	−	0.996724i	$-0.474228\pi$
$439$	3.38908	0.161752	0.0808759	+	0.996724i	$0.474228\pi$
$440$	0	0
$441$	−0.833689	−0.0396995
$442$	0	0
$443$	7.19176	0.341691	0.170845	−	0.985298i	$-0.445350\pi$
$443$	7.19176	0.341691	0.170845	+	0.985298i	$0.445350\pi$
$444$	0	0
$445$	−0.953672	−0.0452084
$446$	0	0
$447$	−0.290600	−0.0137449
$448$	0	0
$449$	−30.6295	−1.44550	−0.722748	−	0.691112i	$-0.757121\pi$
$449$	−30.6295	−1.44550	−0.722748	+	0.691112i	$0.757121\pi$
$450$	0	0
$451$	1.69443	0.0797874
$452$	0	0
$453$	−4.05631	−0.190582
$454$	0	0
$455$	−10.0388	−0.470625
$456$	0	0
$457$	−17.0591	−0.797993	−0.398997	−	0.916952i	$-0.630641\pi$
$457$	−17.0591	−0.797993	−0.398997	+	0.916952i	$0.630641\pi$
$458$	0	0
$459$	2.42845	0.113350
$460$	0	0
$461$	−16.1480	−0.752086	−0.376043	−	0.926602i	$-0.622716\pi$
$461$	−16.1480	−0.752086	−0.376043	+	0.926602i	$0.622716\pi$
$462$	0	0
$463$	−15.3168	−0.711832	−0.355916	−	0.934518i	$-0.615831\pi$
$463$	−15.3168	−0.711832	−0.355916	+	0.934518i	$0.615831\pi$
$464$	0	0
$465$	1.19683	0.0555015
$466$	0	0
$467$	−8.57861	−0.396971	−0.198485	−	0.980104i	$-0.563602\pi$
$467$	−8.57861	−0.396971	−0.198485	+	0.980104i	$0.563602\pi$
$468$	0	0
$469$	39.6820	1.83234
$470$	0	0
$471$	3.95429	0.182204
$472$	0	0
$473$	−0.963943	−0.0443221
$474$	0	0
$475$	2.13324	0.0978800
$476$	0	0
$477$	25.1548	1.15176
$478$	0	0
$479$	17.4527	0.797433	0.398716	−	0.917074i	$-0.369456\pi$
$479$	17.4527	0.797433	0.398716	+	0.917074i	$0.369456\pi$
$480$	0	0
$481$	−20.5293	−0.936054
$482$	0	0
$483$	0.712596	0.0324242
$484$	0	0
$485$	−6.49560	−0.294950
$486$	0	0
$487$	−26.5568	−1.20341	−0.601703	−	0.798720i	$-0.705511\pi$
$487$	−26.5568	−1.20341	−0.601703	+	0.798720i	$0.705511\pi$
$488$	0	0
$489$	−3.73161	−0.168749
$490$	0	0
$491$	−12.6077	−0.568977	−0.284488	−	0.958680i	$-0.591824\pi$
$491$	−12.6077	−0.568977	−0.284488	+	0.958680i	$0.591824\pi$
$492$	0	0
$493$	7.33884	0.330525
$494$	0	0
$495$	−0.450466	−0.0202470
$496$	0	0
$497$	−39.7992	−1.78524
$498$	0	0
$499$	2.82919	0.126652	0.0633260	−	0.997993i	$-0.479829\pi$
$499$	2.82919	0.126652	0.0633260	+	0.997993i	$0.479829\pi$
$500$	0	0
$501$	−0.743711	−0.0332266
$502$	0	0
$503$	−1.40760	−0.0627617	−0.0313809	−	0.999507i	$-0.509990\pi$
$503$	−1.40760	−0.0627617	−0.0313809	+	0.999507i	$0.509990\pi$
$504$	0	0
$505$	−0.0190213	−0.000846435 0
$506$	0	0
$507$	0.220309	0.00978425
$508$	0	0
$509$	−12.1105	−0.536788	−0.268394	−	0.963309i	$-0.586493\pi$
$509$	−12.1105	−0.536788	−0.268394	+	0.963309i	$0.586493\pi$
$510$	0	0
$511$	18.6204	0.823717
$512$	0	0
$513$	−3.34010	−0.147469
$514$	0	0
$515$	−15.1742	−0.668656
$516$	0	0
$517$	1.93628	0.0851577
$518$	0	0
$519$	0.682011	0.0299369
$520$	0	0
$521$	2.81909	0.123507	0.0617533	−	0.998091i	$-0.480331\pi$
$521$	2.81909	0.123507	0.0617533	+	0.998091i	$0.480331\pi$
$522$	0	0
$523$	7.39692	0.323445	0.161722	−	0.986836i	$-0.448295\pi$
$523$	7.39692	0.323445	0.161722	+	0.986836i	$0.448295\pi$
$524$	0	0
$525$	0.712596	0.0311002
$526$	0	0
$527$	7.03070	0.306262
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.710707	0.0308421
$532$	0	0
$533$	−40.9969	−1.77577
$534$	0	0
$535$	−9.37999	−0.405532
$536$	0	0
$537$	5.04918	0.217888
$538$	0	0
$539$	−0.0437366	−0.00188387
$540$	0	0
$541$	−40.9323	−1.75982	−0.879908	−	0.475144i	$-0.842396\pi$
$541$	−40.9323	−1.75982	−0.879908	+	0.475144i	$0.842396\pi$
$542$	0	0
$543$	−0.509371	−0.0218592
$544$	0	0
$545$	9.24898	0.396183
$546$	0	0
$547$	23.4426	1.00233	0.501167	−	0.865351i	$-0.332904\pi$
$547$	23.4426	1.00233	0.501167	+	0.865351i	$0.332904\pi$
$548$	0	0
$549$	−21.2277	−0.905977
$550$	0	0
$551$	−10.0939	−0.430013
$552$	0	0
$553$	22.3722	0.951364
$554$	0	0
$555$	1.45726	0.0618571
$556$	0	0
$557$	−41.7113	−1.76736	−0.883682	−	0.468087i	$-0.844943\pi$
$557$	−41.7113	−1.76736	−0.883682	+	0.468087i	$0.844943\pi$
$558$	0	0
$559$	23.3228	0.986448
$560$	0	0
$561$	0.0629514	0.00265781
$562$	0	0
$563$	−29.3789	−1.23817	−0.619087	−	0.785322i	$-0.712497\pi$
$563$	−29.3789	−1.23817	−0.619087	+	0.785322i	$0.712497\pi$
$564$	0	0
$565$	4.76082	0.200289
$566$	0	0
$567$	22.6107	0.949559
$568$	0	0
$569$	24.5675	1.02992	0.514962	−	0.857213i	$-0.327806\pi$
$569$	24.5675	1.02992	0.514962	+	0.857213i	$0.327806\pi$
$570$	0	0
$571$	15.3602	0.642806	0.321403	−	0.946942i	$-0.395846\pi$
$571$	15.3602	0.642806	0.321403	+	0.946942i	$0.395846\pi$
$572$	0	0
$573$	−6.26941	−0.261908
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−25.9471	−1.08019	−0.540096	−	0.841603i	$-0.681612\pi$
$577$	−25.9471	−1.08019	−0.540096	+	0.841603i	$0.681612\pi$
$578$	0	0
$579$	−5.82151	−0.241934
$580$	0	0
$581$	9.17989	0.380846
$582$	0	0
$583$	1.31966	0.0546547
$584$	0	0
$585$	10.8991	0.450623
$586$	0	0
$587$	3.21587	0.132733	0.0663667	−	0.997795i	$-0.478859\pi$
$587$	3.21587	0.132733	0.0663667	+	0.997795i	$0.478859\pi$
$588$	0	0
$589$	−9.67005	−0.398447
$590$	0	0
$591$	4.82508	0.198477
$592$	0	0
$593$	−11.4891	−0.471800	−0.235900	−	0.971777i	$-0.575804\pi$
$593$	−11.4891	−0.471800	−0.235900	+	0.971777i	$0.575804\pi$
$594$	0	0
$595$	4.18611	0.171614
$596$	0	0
$597$	−0.980898	−0.0401455
$598$	0	0
$599$	24.5297	1.00226	0.501128	−	0.865373i	$-0.332919\pi$
$599$	24.5297	1.00226	0.501128	+	0.865373i	$0.332919\pi$
$600$	0	0
$601$	20.9423	0.854254	0.427127	−	0.904192i	$-0.359526\pi$
$601$	20.9423	0.854254	0.427127	+	0.904192i	$0.359526\pi$
$602$	0	0
$603$	−43.0828	−1.75447
$604$	0	0
$605$	10.9764	0.446253
$606$	0	0
$607$	−3.34842	−0.135908	−0.0679540	−	0.997688i	$-0.521647\pi$
$607$	−3.34842	−0.135908	−0.0679540	+	0.997688i	$0.521647\pi$
$608$	0	0
$609$	−3.37179	−0.136632
$610$	0	0
$611$	−46.8487	−1.89530
$612$	0	0
$613$	22.4906	0.908388	0.454194	−	0.890903i	$-0.349927\pi$
$613$	22.4906	0.908388	0.454194	+	0.890903i	$0.349927\pi$
$614$	0	0
$615$	2.91014	0.117348
$616$	0	0
$617$	14.6942	0.591568	0.295784	−	0.955255i	$-0.404419\pi$
$617$	14.6942	0.591568	0.295784	+	0.955255i	$0.404419\pi$
$618$	0	0
$619$	−6.96400	−0.279907	−0.139953	−	0.990158i	$-0.544695\pi$
$619$	−6.96400	−0.279907	−0.139953	+	0.990158i	$0.544695\pi$
$620$	0	0
$621$	−1.56574	−0.0628309
$622$	0	0
$623$	2.57394	0.103123
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.0865835	−0.00345781
$628$	0	0
$629$	8.56058	0.341333
$630$	0	0
$631$	25.2402	1.00480	0.502399	−	0.864636i	$-0.332451\pi$
$631$	25.2402	1.00480	0.502399	+	0.864636i	$0.332451\pi$
$632$	0	0
$633$	0.673259	0.0267596
$634$	0	0
$635$	18.2930	0.725937
$636$	0	0
$637$	1.05821	0.0419280
$638$	0	0
$639$	43.2101	1.70937
$640$	0	0
$641$	0.953672	0.0376678	0.0188339	−	0.999823i	$-0.494005\pi$
$641$	0.953672	0.0376678	0.0188339	+	0.999823i	$0.494005\pi$
$642$	0	0
$643$	2.40695	0.0949208	0.0474604	−	0.998873i	$-0.484887\pi$
$643$	2.40695	0.0949208	0.0474604	+	0.998873i	$0.484887\pi$
$644$	0	0
$645$	−1.65555	−0.0651873
$646$	0	0
$647$	−37.4388	−1.47187	−0.735935	−	0.677052i	$-0.763257\pi$
$647$	−37.4388	−1.47187	−0.735935	+	0.677052i	$0.763257\pi$
$648$	0	0
$649$	0.0372848	0.00146356
$650$	0	0
$651$	−3.23021	−0.126602
$652$	0	0
$653$	13.9148	0.544530	0.272265	−	0.962222i	$-0.412227\pi$
$653$	13.9148	0.544530	0.272265	+	0.962222i	$0.412227\pi$
$654$	0	0
$655$	21.4285	0.837280
$656$	0	0
$657$	−20.2162	−0.788709
$658$	0	0
$659$	−17.9813	−0.700453	−0.350227	−	0.936665i	$-0.613895\pi$
$659$	−17.9813	−0.700453	−0.350227	+	0.936665i	$0.613895\pi$
$660$	0	0
$661$	−30.1698	−1.17347	−0.586735	−	0.809779i	$-0.699587\pi$
$661$	−30.1698	−1.17347	−0.586735	+	0.809779i	$0.699587\pi$
$662$	0	0
$663$	−1.52312	−0.0591531
$664$	0	0
$665$	−5.75759	−0.223270
$666$	0	0
$667$	−4.73170	−0.183212
$668$	0	0
$669$	−6.71345	−0.259557
$670$	0	0
$671$	−1.11364	−0.0429915
$672$	0	0
$673$	−24.9054	−0.960031	−0.480016	−	0.877260i	$-0.659369\pi$
$673$	−24.9054	−0.960031	−0.480016	+	0.877260i	$0.659369\pi$
$674$	0	0
$675$	−1.56574	−0.0602653
$676$	0	0
$677$	45.9616	1.76645	0.883223	−	0.468953i	$-0.155369\pi$
$677$	45.9616	1.76645	0.883223	+	0.468953i	$0.155369\pi$
$678$	0	0
$679$	17.5315	0.672797
$680$	0	0
$681$	−4.71441	−0.180657
$682$	0	0
$683$	−19.6055	−0.750182	−0.375091	−	0.926988i	$-0.622389\pi$
$683$	−19.6055	−0.750182	−0.375091	+	0.926988i	$0.622389\pi$
$684$	0	0
$685$	−11.4016	−0.435631
$686$	0	0
$687$	−4.44792	−0.169699
$688$	0	0
$689$	−31.9294	−1.21641
$690$	0	0
$691$	26.5179	1.00879	0.504393	−	0.863474i	$-0.331716\pi$
$691$	26.5179	1.00879	0.504393	+	0.863474i	$0.331716\pi$
$692$	0	0
$693$	1.21580	0.0461845
$694$	0	0
$695$	10.9095	0.413821
$696$	0	0
$697$	17.0955	0.647538
$698$	0	0
$699$	−0.294837	−0.0111518
$700$	0	0
$701$	44.3748	1.67601	0.838006	−	0.545661i	$-0.183721\pi$
$701$	44.3748	1.67601	0.838006	+	0.545661i	$0.183721\pi$
$702$	0	0
$703$	−11.7742	−0.444074
$704$	0	0
$705$	3.32553	0.125247
$706$	0	0
$707$	0.0513380	0.00193077
$708$	0	0
$709$	−24.8433	−0.933010	−0.466505	−	0.884518i	$-0.654487\pi$
$709$	−24.8433	−0.933010	−0.466505	+	0.884518i	$0.654487\pi$
$710$	0	0
$711$	−24.2896	−0.910930
$712$	0	0
$713$	−4.53302	−0.169763
$714$	0	0
$715$	0.571784	0.0213835
$716$	0	0
$717$	−0.111161	−0.00415139
$718$	0	0
$719$	27.4010	1.02189	0.510943	−	0.859614i	$-0.329296\pi$
$719$	27.4010	1.02189	0.510943	+	0.859614i	$0.329296\pi$
$720$	0	0
$721$	40.9550	1.52524
$722$	0	0
$723$	−4.15176	−0.154406
$724$	0	0
$725$	−4.73170	−0.175731
$726$	0	0
$727$	9.86650	0.365928	0.182964	−	0.983120i	$-0.441431\pi$
$727$	9.86650	0.365928	0.182964	+	0.983120i	$0.441431\pi$
$728$	0	0
$729$	−23.3083	−0.863270
$730$	0	0
$731$	−9.72546	−0.359709
$732$	0	0
$733$	43.9579	1.62362	0.811812	−	0.583919i	$-0.198482\pi$
$733$	43.9579	1.62362	0.811812	+	0.583919i	$0.198482\pi$
$734$	0	0
$735$	−0.0751167	−0.00277072
$736$	0	0
$737$	−2.26019	−0.0832552
$738$	0	0
$739$	−16.0165	−0.589177	−0.294589	−	0.955624i	$-0.595183\pi$
$739$	−16.0165	−0.589177	−0.294589	+	0.955624i	$0.595183\pi$
$740$	0	0
$741$	2.09490	0.0769583
$742$	0	0
$743$	−6.24941	−0.229269	−0.114634	−	0.993408i	$-0.536570\pi$
$743$	−6.24941	−0.229269	−0.114634	+	0.993408i	$0.536570\pi$
$744$	0	0
$745$	1.10066	0.0403249
$746$	0	0
$747$	−9.96663	−0.364660
$748$	0	0
$749$	25.3164	0.925043
$750$	0	0
$751$	−24.1986	−0.883020	−0.441510	−	0.897256i	$-0.645557\pi$
$751$	−24.1986	−0.883020	−0.441510	+	0.897256i	$0.645557\pi$
$752$	0	0
$753$	−4.53619	−0.165308
$754$	0	0
$755$	15.3634	0.559133
$756$	0	0
$757$	5.92947	0.215510	0.107755	−	0.994177i	$-0.465634\pi$
$757$	5.92947	0.215510	0.107755	+	0.994177i	$0.465634\pi$
$758$	0	0
$759$	−0.0405877	−0.00147324
$760$	0	0
$761$	2.14382	0.0777134	0.0388567	−	0.999245i	$-0.487628\pi$
$761$	2.14382	0.0777134	0.0388567	+	0.999245i	$0.487628\pi$
$762$	0	0
$763$	−24.9628	−0.903715
$764$	0	0
$765$	−4.54487	−0.164320
$766$	0	0
$767$	−0.902112	−0.0325734
$768$	0	0
$769$	21.5137	0.775804	0.387902	−	0.921701i	$-0.373200\pi$
$769$	21.5137	0.775804	0.387902	+	0.921701i	$0.373200\pi$
$770$	0	0
$771$	7.05776	0.254179
$772$	0	0
$773$	−17.2080	−0.618927	−0.309464	−	0.950911i	$-0.600150\pi$
$773$	−17.2080	−0.618927	−0.309464	+	0.950911i	$0.600150\pi$
$774$	0	0
$775$	−4.53302	−0.162831
$776$	0	0
$777$	−3.93311	−0.141099
$778$	0	0
$779$	−23.5132	−0.842447
$780$	0	0
$781$	2.26687	0.0811149
$782$	0	0
$783$	7.40860	0.264762
$784$	0	0
$785$	−14.9770	−0.534552
$786$	0	0
$787$	−26.9062	−0.959104	−0.479552	−	0.877514i	$-0.659201\pi$
$787$	−26.9062	−0.959104	−0.479552	+	0.877514i	$0.659201\pi$
$788$	0	0
$789$	−3.45822	−0.123116
$790$	0	0
$791$	−12.8494	−0.456871
$792$	0	0
$793$	26.9447	0.956834
$794$	0	0
$795$	2.26649	0.0803841
$796$	0	0
$797$	12.8937	0.456718	0.228359	−	0.973577i	$-0.426664\pi$
$797$	12.8937	0.456718	0.228359	+	0.973577i	$0.426664\pi$
$798$	0	0
$799$	19.5357	0.691122
$800$	0	0
$801$	−2.79454	−0.0987401
$802$	0	0
$803$	−1.06057	−0.0374268
$804$	0	0
$805$	−2.69898	−0.0951266
$806$	0	0
$807$	4.19907	0.147814
$808$	0	0
$809$	3.14370	0.110526	0.0552632	−	0.998472i	$-0.482400\pi$
$809$	3.14370	0.110526	0.0552632	+	0.998472i	$0.482400\pi$
$810$	0	0
$811$	−35.5575	−1.24859	−0.624296	−	0.781188i	$-0.714614\pi$
$811$	−35.5575	−1.24859	−0.624296	+	0.781188i	$0.714614\pi$
$812$	0	0
$813$	−3.75486	−0.131689
$814$	0	0
$815$	14.1336	0.495079
$816$	0	0
$817$	13.3764	0.467982
$818$	0	0
$819$	−29.4165	−1.02790
$820$	0	0
$821$	37.7606	1.31785	0.658926	−	0.752207i	$-0.271011\pi$
$821$	37.7606	1.31785	0.658926	+	0.752207i	$0.271011\pi$
$822$	0	0
$823$	−19.7349	−0.687915	−0.343957	−	0.938985i	$-0.611768\pi$
$823$	−19.7349	−0.687915	−0.343957	+	0.938985i	$0.611768\pi$
$824$	0	0
$825$	−0.0405877	−0.00141308
$826$	0	0
$827$	9.41412	0.327361	0.163680	−	0.986513i	$-0.447663\pi$
$827$	9.41412	0.327361	0.163680	+	0.986513i	$0.447663\pi$
$828$	0	0
$829$	18.7758	0.652109	0.326055	−	0.945351i	$-0.394281\pi$
$829$	18.7758	0.652109	0.326055	+	0.945351i	$0.394281\pi$
$830$	0	0
$831$	2.75243	0.0954808
$832$	0	0
$833$	−0.441269	−0.0152891
$834$	0	0
$835$	2.81683	0.0974805
$836$	0	0
$837$	7.09753	0.245326
$838$	0	0
$839$	50.6175	1.74751	0.873755	−	0.486366i	$-0.161678\pi$
$839$	50.6175	1.74751	0.873755	+	0.486366i	$0.161678\pi$
$840$	0	0
$841$	−6.61104	−0.227967
$842$	0	0
$843$	3.96933	0.136711
$844$	0	0
$845$	−0.834427	−0.0287052
$846$	0	0
$847$	−29.6250	−1.01793
$848$	0	0
$849$	−5.30391	−0.182030
$850$	0	0
$851$	−5.51941	−0.189203
$852$	0	0
$853$	8.42664	0.288523	0.144261	−	0.989540i	$-0.453919\pi$
$853$	8.42664	0.288523	0.144261	+	0.989540i	$0.453919\pi$
$854$	0	0
$855$	6.25103	0.213781
$856$	0	0
$857$	35.5459	1.21422	0.607112	−	0.794617i	$-0.292328\pi$
$857$	35.5459	1.21422	0.607112	+	0.794617i	$0.292328\pi$
$858$	0	0
$859$	44.7019	1.52521	0.762604	−	0.646865i	$-0.223920\pi$
$859$	44.7019	1.52521	0.762604	+	0.646865i	$0.223920\pi$
$860$	0	0
$861$	−7.85442	−0.267678
$862$	0	0
$863$	1.14514	0.0389811	0.0194906	−	0.999810i	$-0.493796\pi$
$863$	1.14514	0.0389811	0.0194906	+	0.999810i	$0.493796\pi$
$864$	0	0
$865$	−2.58314	−0.0878294
$866$	0	0
$867$	−3.85327	−0.130864
$868$	0	0
$869$	−1.27427	−0.0432266
$870$	0	0
$871$	54.6857	1.85296
$872$	0	0
$873$	−19.0340	−0.644203
$874$	0	0
$875$	−2.69898	−0.0912423
$876$	0	0
$877$	−15.2315	−0.514330	−0.257165	−	0.966368i	$-0.582788\pi$
$877$	−15.2315	−0.514330	−0.257165	+	0.966368i	$0.582788\pi$
$878$	0	0
$879$	−3.80736	−0.128419
$880$	0	0
$881$	0.600412	0.0202284	0.0101142	−	0.999949i	$-0.496780\pi$
$881$	0.600412	0.0202284	0.0101142	+	0.999949i	$0.496780\pi$
$882$	0	0
$883$	−36.8711	−1.24081	−0.620406	−	0.784281i	$-0.713032\pi$
$883$	−36.8711	−1.24081	−0.620406	+	0.784281i	$0.713032\pi$
$884$	0	0
$885$	0.0640358	0.00215254
$886$	0	0
$887$	4.86720	0.163425	0.0817123	−	0.996656i	$-0.473961\pi$
$887$	4.86720	0.163425	0.0817123	+	0.996656i	$0.473961\pi$
$888$	0	0
$889$	−49.3726	−1.65590
$890$	0	0
$891$	−1.28785	−0.0431446
$892$	0	0
$893$	−26.8694	−0.899150
$894$	0	0
$895$	−19.1239	−0.639243
$896$	0	0
$897$	0.982028	0.0327890
$898$	0	0
$899$	21.4489	0.715361
$900$	0	0
$901$	13.3144	0.443566
$902$	0	0
$903$	4.46830	0.148696
$904$	0	0
$905$	1.92926	0.0641308
$906$	0	0
$907$	35.8713	1.19109	0.595544	−	0.803323i	$-0.296936\pi$
$907$	35.8713	1.19109	0.595544	+	0.803323i	$0.296936\pi$
$908$	0	0
$909$	−0.0557378	−0.00184871
$910$	0	0
$911$	−0.697042	−0.0230940	−0.0115470	−	0.999933i	$-0.503676\pi$
$911$	−0.697042	−0.0230940	−0.0115470	+	0.999933i	$0.503676\pi$
$912$	0	0
$913$	−0.522864	−0.0173043
$914$	0	0
$915$	−1.91265	−0.0632303
$916$	0	0
$917$	−57.8351	−1.90988
$918$	0	0
$919$	46.5763	1.53641	0.768205	−	0.640204i	$-0.221150\pi$
$919$	46.5763	1.53641	0.768205	+	0.640204i	$0.221150\pi$
$920$	0	0
$921$	3.75505	0.123733
$922$	0	0
$923$	−54.8473	−1.80532
$924$	0	0
$925$	−5.51941	−0.181477
$926$	0	0
$927$	−44.4649	−1.46042
$928$	0	0
$929$	36.7121	1.20448	0.602242	−	0.798313i	$-0.294274\pi$
$929$	36.7121	1.20448	0.602242	+	0.798313i	$0.294274\pi$
$930$	0	0
$931$	0.606923	0.0198911
$932$	0	0
$933$	4.36181	0.142799
$934$	0	0
$935$	−0.238431	−0.00779752
$936$	0	0
$937$	−2.26988	−0.0741539	−0.0370769	−	0.999312i	$-0.511805\pi$
$937$	−2.26988	−0.0741539	−0.0370769	+	0.999312i	$0.511805\pi$
$938$	0	0
$939$	−3.89405	−0.127077
$940$	0	0
$941$	−55.0821	−1.79563	−0.897813	−	0.440378i	$-0.854844\pi$
$941$	−55.0821	−1.79563	−0.897813	+	0.440378i	$0.854844\pi$
$942$	0	0
$943$	−11.0223	−0.358934
$944$	0	0
$945$	4.22590	0.137469
$946$	0	0
$947$	8.03176	0.260997	0.130499	−	0.991449i	$-0.458342\pi$
$947$	8.03176	0.260997	0.130499	+	0.991449i	$0.458342\pi$
$948$	0	0
$949$	25.6607	0.832983
$950$	0	0
$951$	−2.98219	−0.0967042
$952$	0	0
$953$	−7.98766	−0.258746	−0.129373	−	0.991596i	$-0.541296\pi$
$953$	−7.98766	−0.258746	−0.129373	+	0.991596i	$0.541296\pi$
$954$	0	0
$955$	23.7456	0.768391
$956$	0	0
$957$	0.192049	0.00620806
$958$	0	0
$959$	30.7726	0.993699
$960$	0	0
$961$	−10.4517	−0.337152
$962$	0	0
$963$	−27.4861	−0.885728
$964$	0	0
$965$	22.0492	0.709788
$966$	0	0
$967$	0.134998	0.00434124	0.00217062	−	0.999998i	$-0.499309\pi$
$967$	0.134998	0.00434124	0.00217062	+	0.999998i	$0.499309\pi$
$968$	0	0
$969$	−0.873563	−0.0280629
$970$	0	0
$971$	44.3940	1.42467	0.712335	−	0.701840i	$-0.247638\pi$
$971$	44.3940	1.42467	0.712335	+	0.701840i	$0.247638\pi$
$972$	0	0
$973$	−29.4446	−0.943949
$974$	0	0
$975$	0.982028	0.0314501
$976$	0	0
$977$	−22.0754	−0.706255	−0.353127	−	0.935575i	$-0.614882\pi$
$977$	−22.0754	−0.706255	−0.353127	+	0.935575i	$0.614882\pi$
$978$	0	0
$979$	−0.146606	−0.00468554
$980$	0	0
$981$	27.1022	0.865306
$982$	0	0
$983$	−17.3210	−0.552453	−0.276227	−	0.961092i	$-0.589084\pi$
$983$	−17.3210	−0.552453	−0.276227	+	0.961092i	$0.589084\pi$
$984$	0	0
$985$	−18.2752	−0.582296
$986$	0	0
$987$	−8.97554	−0.285695
$988$	0	0
$989$	6.27046	0.199389
$990$	0	0
$991$	−35.0345	−1.11291	−0.556453	−	0.830879i	$-0.687838\pi$
$991$	−35.0345	−1.11291	−0.556453	+	0.830879i	$0.687838\pi$
$992$	0	0
$993$	0.529413	0.0168004
$994$	0	0
$995$	3.71519	0.117779
$996$	0	0
$997$	−9.80170	−0.310423	−0.155212	−	0.987881i	$-0.549606\pi$
$997$	−9.80170	−0.310423	−0.155212	+	0.987881i	$0.549606\pi$
$998$	0	0
$999$	8.64195	0.273419

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.ct.1.3		5
4.3	odd	2		7360.2.a.ck.1.3		5
8.3	odd	2		3680.2.a.bb.1.3	yes	5
8.5	even	2		3680.2.a.u.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.u.1.3	✓	5	8.5	even	2
3680.2.a.bb.1.3	yes	5	8.3	odd	2
7360.2.a.ck.1.3		5	4.3	odd	2
7360.2.a.ct.1.3		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q+0.264024 q^{3} -1.00000 q^{5} +2.69898 q^{7} -2.93029 q^{9} +O(q^{10})\)	\(q+0.264024 q^{3} -1.00000 q^{5} +2.69898 q^{7} -2.93029 q^{9} -0.153728 q^{11} +3.71947 q^{13} -0.264024 q^{15} -1.55100 q^{17} +2.13324 q^{19} +0.712596 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.56574 q^{27} -4.73170 q^{29} -4.53302 q^{31} -0.0405877 q^{33} -2.69898 q^{35} -5.51941 q^{37} +0.982028 q^{39} -11.0223 q^{41} +6.27046 q^{43} +2.93029 q^{45} -12.5956 q^{47} +0.284507 q^{49} -0.409500 q^{51} -8.58441 q^{53} +0.153728 q^{55} +0.563227 q^{57} -0.242538 q^{59} +7.24424 q^{61} -7.90881 q^{63} -3.71947 q^{65} +14.7026 q^{67} +0.264024 q^{69} -14.7460 q^{71} +6.89904 q^{73} +0.264024 q^{75} -0.414908 q^{77} +8.28913 q^{79} +8.37748 q^{81} +3.40124 q^{83} +1.55100 q^{85} -1.24928 q^{87} +0.953672 q^{89} +10.0388 q^{91} -1.19683 q^{93} -2.13324 q^{95} +6.49560 q^{97} +0.450466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 2 * q^9	\( 5 q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 2 q^{9} - q^{11} - q^{13} - 3 q^{15} - 8 q^{17} + 9 q^{19} - 8 q^{21} + 5 q^{23} + 5 q^{25} + 12 q^{27} + q^{29} - 6 q^{31} + 15 q^{33} + 8 q^{35} - q^{37} - 12 q^{39} + 6 q^{41} + 12 q^{43} - 2 q^{45} - 32 q^{47} + 5 q^{49} + 3 q^{51} + 3 q^{53} + q^{55} - 2 q^{57} - 5 q^{59} - 7 q^{61} - 25 q^{63} + q^{65} + 17 q^{67} + 3 q^{69} - 16 q^{71} + 20 q^{73} + 3 q^{75} + 10 q^{77} - 4 q^{79} - 3 q^{81} + q^{83} + 8 q^{85} - 28 q^{87} - 2 q^{89} + 7 q^{91} + 24 q^{93} - 9 q^{95} - 17 q^{97} + q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 2 * q^9 - q^11 - q^13 - 3 * q^15 - 8 * q^17 + 9 * q^19 - 8 * q^21 + 5 * q^23 + 5 * q^25 + 12 * q^27 + q^29 - 6 * q^31 + 15 * q^33 + 8 * q^35 - q^37 - 12 * q^39 + 6 * q^41 + 12 * q^43 - 2 * q^45 - 32 * q^47 + 5 * q^49 + 3 * q^51 + 3 * q^53 + q^55 - 2 * q^57 - 5 * q^59 - 7 * q^61 - 25 * q^63 + q^65 + 17 * q^67 + 3 * q^69 - 16 * q^71 + 20 * q^73 + 3 * q^75 + 10 * q^77 - 4 * q^79 - 3 * q^81 + q^83 + 8 * q^85 - 28 * q^87 - 2 * q^89 + 7 * q^91 + 24 * q^93 - 9 * q^95 - 17 * q^97 + q^99

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