LMFDB - Embedded newform 7168.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.9433055232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 $ x^8 - 4x^7 - 6x^6 + 32x^5 + 9x^4 - 76x^3 - 4x^2 + 48*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1792)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.822080$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.88693 q^{3} -0.992279 q^{5} +1.00000 q^{7} +5.33435 q^{9} +O(q^{10})$	$q-2.88693 q^{3} -0.992279 q^{5} +1.00000 q^{7} +5.33435 q^{9} +3.42224 q^{11} -2.77325 q^{13} +2.86464 q^{15} +6.93050 q^{17} +1.96503 q^{19} -2.88693 q^{21} -2.05612 q^{23} -4.01538 q^{25} -6.73909 q^{27} -7.55775 q^{29} -5.23708 q^{31} -9.87975 q^{33} -0.992279 q^{35} -9.31119 q^{37} +8.00617 q^{39} +0.949797 q^{41} +8.41942 q^{43} -5.29316 q^{45} +4.64785 q^{47} +1.00000 q^{49} -20.0079 q^{51} -10.2501 q^{53} -3.39582 q^{55} -5.67289 q^{57} +12.1346 q^{59} -3.98105 q^{61} +5.33435 q^{63} +2.75184 q^{65} -12.8297 q^{67} +5.93586 q^{69} +3.60510 q^{71} +6.53179 q^{73} +11.5921 q^{75} +3.42224 q^{77} +13.7819 q^{79} +3.45223 q^{81} -6.35508 q^{83} -6.87699 q^{85} +21.8187 q^{87} -0.428825 q^{89} -2.77325 q^{91} +15.1191 q^{93} -1.94986 q^{95} +14.6339 q^{97} +18.2554 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9	$ 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9} - 12 q^{11} - 20 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 12 q^{25} + 12 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} - 52 q^{45} + 20 q^{47} + 8 q^{49} - 32 q^{51} - 40 q^{53} + 24 q^{55} - 4 q^{57} - 4 q^{59} + 8 q^{61} + 12 q^{63} + 36 q^{65} - 28 q^{67} - 4 q^{69} - 16 q^{71} + 16 q^{73} + 28 q^{75} - 12 q^{77} + 20 q^{81} + 8 q^{83} - 16 q^{85} + 20 q^{87} + 16 q^{89} - 20 q^{91} - 16 q^{93} - 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9 - 12 * q^11 - 20 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 12 * q^25 + 12 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 16 * q^39 - 12 * q^41 + 4 * q^43 - 52 * q^45 + 20 * q^47 + 8 * q^49 - 32 * q^51 - 40 * q^53 + 24 * q^55 - 4 * q^57 - 4 * q^59 + 8 * q^61 + 12 * q^63 + 36 * q^65 - 28 * q^67 - 4 * q^69 - 16 * q^71 + 16 * q^73 + 28 * q^75 - 12 * q^77 + 20 * q^81 + 8 * q^83 - 16 * q^85 + 20 * q^87 + 16 * q^89 - 20 * q^91 - 16 * q^93 - 40 * q^95 - 36 * q^97 + 4 * 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
$3$	−2.88693	−1.66677	−0.833384	−	0.552694i	$-0.813600\pi$
$3$	−2.88693	−1.66677	−0.833384	+	0.552694i	$0.813600\pi$
$4$	0	0
$5$	−0.992279	−0.443761	−0.221880	−	0.975074i	$-0.571219\pi$
$5$	−0.992279	−0.443761	−0.221880	+	0.975074i	$0.571219\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.33435	1.77812
$10$	0	0
$11$	3.42224	1.03184	0.515922	−	0.856636i	$-0.327449\pi$
$11$	3.42224	1.03184	0.515922	+	0.856636i	$0.327449\pi$
$12$	0	0
$13$	−2.77325	−0.769161	−0.384581	−	0.923091i	$-0.625654\pi$
$13$	−2.77325	−0.769161	−0.384581	+	0.923091i	$0.625654\pi$
$14$	0	0
$15$	2.86464	0.739646
$16$	0	0
$17$	6.93050	1.68089	0.840447	−	0.541894i	$-0.182293\pi$
$17$	6.93050	1.68089	0.840447	+	0.541894i	$0.182293\pi$
$18$	0	0
$19$	1.96503	0.450808	0.225404	−	0.974265i	$-0.427630\pi$
$19$	1.96503	0.450808	0.225404	+	0.974265i	$0.427630\pi$
$20$	0	0
$21$	−2.88693	−0.629979
$22$	0	0
$23$	−2.05612	−0.428730	−0.214365	−	0.976754i	$-0.568768\pi$
$23$	−2.05612	−0.428730	−0.214365	+	0.976754i	$0.568768\pi$
$24$	0	0
$25$	−4.01538	−0.803076
$26$	0	0
$27$	−6.73909	−1.29694
$28$	0	0
$29$	−7.55775	−1.40344	−0.701720	−	0.712453i	$-0.747584\pi$
$29$	−7.55775	−1.40344	−0.701720	+	0.712453i	$0.747584\pi$
$30$	0	0
$31$	−5.23708	−0.940607	−0.470304	−	0.882505i	$-0.655856\pi$
$31$	−5.23708	−0.940607	−0.470304	+	0.882505i	$0.655856\pi$
$32$	0	0
$33$	−9.87975	−1.71984
$34$	0	0
$35$	−0.992279	−0.167726
$36$	0	0
$37$	−9.31119	−1.53075	−0.765375	−	0.643584i	$-0.777447\pi$
$37$	−9.31119	−1.53075	−0.765375	+	0.643584i	$0.777447\pi$
$38$	0	0
$39$	8.00617	1.28201
$40$	0	0
$41$	0.949797	0.148333	0.0741667	−	0.997246i	$-0.476370\pi$
$41$	0.949797	0.148333	0.0741667	+	0.997246i	$0.476370\pi$
$42$	0	0
$43$	8.41942	1.28395	0.641975	−	0.766726i	$-0.278115\pi$
$43$	8.41942	1.28395	0.641975	+	0.766726i	$0.278115\pi$
$44$	0	0
$45$	−5.29316	−0.789058
$46$	0	0
$47$	4.64785	0.677959	0.338980	−	0.940794i	$-0.389918\pi$
$47$	4.64785	0.677959	0.338980	+	0.940794i	$0.389918\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−20.0079	−2.80166
$52$	0	0
$53$	−10.2501	−1.40796	−0.703979	−	0.710221i	$-0.748595\pi$
$53$	−10.2501	−1.40796	−0.703979	+	0.710221i	$0.748595\pi$
$54$	0	0
$55$	−3.39582	−0.457892
$56$	0	0
$57$	−5.67289	−0.751393
$58$	0	0
$59$	12.1346	1.57979	0.789897	−	0.613239i	$-0.210134\pi$
$59$	12.1346	1.57979	0.789897	+	0.613239i	$0.210134\pi$
$60$	0	0
$61$	−3.98105	−0.509721	−0.254860	−	0.966978i	$-0.582030\pi$
$61$	−3.98105	−0.509721	−0.254860	+	0.966978i	$0.582030\pi$
$62$	0	0
$63$	5.33435	0.672065
$64$	0	0
$65$	2.75184	0.341324
$66$	0	0
$67$	−12.8297	−1.56740	−0.783700	−	0.621140i	$-0.786670\pi$
$67$	−12.8297	−1.56740	−0.783700	+	0.621140i	$0.786670\pi$
$68$	0	0
$69$	5.93586	0.714593
$70$	0	0
$71$	3.60510	0.427847	0.213923	−	0.976850i	$-0.431376\pi$
$71$	3.60510	0.427847	0.213923	+	0.976850i	$0.431376\pi$
$72$	0	0
$73$	6.53179	0.764488	0.382244	−	0.924061i	$-0.375151\pi$
$73$	6.53179	0.764488	0.382244	+	0.924061i	$0.375151\pi$
$74$	0	0
$75$	11.5921	1.33854
$76$	0	0
$77$	3.42224	0.390000
$78$	0	0
$79$	13.7819	1.55059	0.775293	−	0.631602i	$-0.217602\pi$
$79$	13.7819	1.55059	0.775293	+	0.631602i	$0.217602\pi$
$80$	0	0
$81$	3.45223	0.383581
$82$	0	0
$83$	−6.35508	−0.697561	−0.348780	−	0.937205i	$-0.613404\pi$
$83$	−6.35508	−0.697561	−0.348780	+	0.937205i	$0.613404\pi$
$84$	0	0
$85$	−6.87699	−0.745915
$86$	0	0
$87$	21.8187	2.33921
$88$	0	0
$89$	−0.428825	−0.0454554	−0.0227277	−	0.999742i	$-0.507235\pi$
$89$	−0.428825	−0.0454554	−0.0227277	+	0.999742i	$0.507235\pi$
$90$	0	0
$91$	−2.77325	−0.290716
$92$	0	0
$93$	15.1191	1.56777
$94$	0	0
$95$	−1.94986	−0.200051
$96$	0	0
$97$	14.6339	1.48584	0.742922	−	0.669378i	$-0.233439\pi$
$97$	14.6339	1.48584	0.742922	+	0.669378i	$0.233439\pi$
$98$	0	0
$99$	18.2554	1.83474
$100$	0	0
$101$	15.8402	1.57616	0.788078	−	0.615575i	$-0.211076\pi$
$101$	15.8402	1.57616	0.788078	+	0.615575i	$0.211076\pi$
$102$	0	0
$103$	−18.4673	−1.81964	−0.909819	−	0.415005i	$-0.863780\pi$
$103$	−18.4673	−1.81964	−0.909819	+	0.415005i	$0.863780\pi$
$104$	0	0
$105$	2.86464	0.279560
$106$	0	0
$107$	0.206717	0.0199841	0.00999205	−	0.999950i	$-0.496819\pi$
$107$	0.206717	0.0199841	0.00999205	+	0.999950i	$0.496819\pi$
$108$	0	0
$109$	−0.437577	−0.0419123	−0.0209561	−	0.999780i	$-0.506671\pi$
$109$	−0.437577	−0.0419123	−0.0209561	+	0.999780i	$0.506671\pi$
$110$	0	0
$111$	26.8807	2.55141
$112$	0	0
$113$	−16.7193	−1.57282	−0.786409	−	0.617706i	$-0.788062\pi$
$113$	−16.7193	−1.57282	−0.786409	+	0.617706i	$0.788062\pi$
$114$	0	0
$115$	2.04024	0.190254
$116$	0	0
$117$	−14.7935	−1.36766
$118$	0	0
$119$	6.93050	0.635318
$120$	0	0
$121$	0.711717	0.0647016
$122$	0	0
$123$	−2.74200	−0.247237
$124$	0	0
$125$	8.94578	0.800135
$126$	0	0
$127$	4.91639	0.436259	0.218130	−	0.975920i	$-0.430004\pi$
$127$	4.91639	0.436259	0.218130	+	0.975920i	$0.430004\pi$
$128$	0	0
$129$	−24.3063	−2.14005
$130$	0	0
$131$	−1.56133	−0.136414	−0.0682069	−	0.997671i	$-0.521728\pi$
$131$	−1.56133	−0.136414	−0.0682069	+	0.997671i	$0.521728\pi$
$132$	0	0
$133$	1.96503	0.170390
$134$	0	0
$135$	6.68706	0.575531
$136$	0	0
$137$	−0.184924	−0.0157991	−0.00789957	−	0.999969i	$-0.502515\pi$
$137$	−0.184924	−0.0157991	−0.00789957	+	0.999969i	$0.502515\pi$
$138$	0	0
$139$	−5.78000	−0.490253	−0.245126	−	0.969491i	$-0.578829\pi$
$139$	−5.78000	−0.490253	−0.245126	+	0.969491i	$0.578829\pi$
$140$	0	0
$141$	−13.4180	−1.13000
$142$	0	0
$143$	−9.49073	−0.793654
$144$	0	0
$145$	7.49940	0.622791
$146$	0	0
$147$	−2.88693	−0.238110
$148$	0	0
$149$	7.59491	0.622199	0.311100	−	0.950377i	$-0.399303\pi$
$149$	7.59491	0.622199	0.311100	+	0.950377i	$0.399303\pi$
$150$	0	0
$151$	11.4266	0.929880	0.464940	−	0.885342i	$-0.346076\pi$
$151$	11.4266	0.929880	0.464940	+	0.885342i	$0.346076\pi$
$152$	0	0
$153$	36.9697	2.98882
$154$	0	0
$155$	5.19665	0.417405
$156$	0	0
$157$	−13.3194	−1.06300	−0.531502	−	0.847057i	$-0.678372\pi$
$157$	−13.3194	−1.06300	−0.531502	+	0.847057i	$0.678372\pi$
$158$	0	0
$159$	29.5913	2.34674
$160$	0	0
$161$	−2.05612	−0.162045
$162$	0	0
$163$	17.6451	1.38207	0.691035	−	0.722822i	$-0.257155\pi$
$163$	17.6451	1.38207	0.691035	+	0.722822i	$0.257155\pi$
$164$	0	0
$165$	9.80348	0.763200
$166$	0	0
$167$	16.0783	1.24418	0.622088	−	0.782947i	$-0.286284\pi$
$167$	16.0783	1.24418	0.622088	+	0.782947i	$0.286284\pi$
$168$	0	0
$169$	−5.30908	−0.408391
$170$	0	0
$171$	10.4821	0.801590
$172$	0	0
$173$	20.4009	1.55105	0.775526	−	0.631315i	$-0.217485\pi$
$173$	20.4009	1.55105	0.775526	+	0.631315i	$0.217485\pi$
$174$	0	0
$175$	−4.01538	−0.303534
$176$	0	0
$177$	−35.0318	−2.63315
$178$	0	0
$179$	−4.70472	−0.351648	−0.175824	−	0.984422i	$-0.556259\pi$
$179$	−4.70472	−0.351648	−0.175824	+	0.984422i	$0.556259\pi$
$180$	0	0
$181$	−14.5522	−1.08166	−0.540828	−	0.841134i	$-0.681889\pi$
$181$	−14.5522	−1.08166	−0.540828	+	0.841134i	$0.681889\pi$
$182$	0	0
$183$	11.4930	0.849586
$184$	0	0
$185$	9.23930	0.679287
$186$	0	0
$187$	23.7178	1.73442
$188$	0	0
$189$	−6.73909	−0.490197
$190$	0	0
$191$	−26.6094	−1.92539	−0.962693	−	0.270595i	$-0.912780\pi$
$191$	−26.6094	−1.92539	−0.962693	+	0.270595i	$0.912780\pi$
$192$	0	0
$193$	−25.2624	−1.81843	−0.909215	−	0.416327i	$-0.863317\pi$
$193$	−25.2624	−1.81843	−0.909215	+	0.416327i	$0.863317\pi$
$194$	0	0
$195$	−7.94436	−0.568908
$196$	0	0
$197$	−9.85072	−0.701835	−0.350917	−	0.936406i	$-0.614130\pi$
$197$	−9.85072	−0.701835	−0.350917	+	0.936406i	$0.614130\pi$
$198$	0	0
$199$	15.2541	1.08133	0.540666	−	0.841238i	$-0.318172\pi$
$199$	15.2541	1.08133	0.540666	+	0.841238i	$0.318172\pi$
$200$	0	0
$201$	37.0385	2.61249
$202$	0	0
$203$	−7.55775	−0.530450
$204$	0	0
$205$	−0.942464	−0.0658246
$206$	0	0
$207$	−10.9680	−0.762332
$208$	0	0
$209$	6.72480	0.465164
$210$	0	0
$211$	−3.49273	−0.240449	−0.120225	−	0.992747i	$-0.538361\pi$
$211$	−3.49273	−0.240449	−0.120225	+	0.992747i	$0.538361\pi$
$212$	0	0
$213$	−10.4077	−0.713121
$214$	0	0
$215$	−8.35442	−0.569767
$216$	0	0
$217$	−5.23708	−0.355516
$218$	0	0
$219$	−18.8568	−1.27422
$220$	0	0
$221$	−19.2200	−1.29288
$222$	0	0
$223$	−12.2245	−0.818610	−0.409305	−	0.912398i	$-0.634229\pi$
$223$	−12.2245	−0.818610	−0.409305	+	0.912398i	$0.634229\pi$
$224$	0	0
$225$	−21.4194	−1.42796
$226$	0	0
$227$	−15.0878	−1.00141	−0.500707	−	0.865617i	$-0.666927\pi$
$227$	−15.0878	−1.00141	−0.500707	+	0.865617i	$0.666927\pi$
$228$	0	0
$229$	−9.55003	−0.631084	−0.315542	−	0.948912i	$-0.602186\pi$
$229$	−9.55003	−0.631084	−0.315542	+	0.948912i	$0.602186\pi$
$230$	0	0
$231$	−9.87975	−0.650040
$232$	0	0
$233$	7.42241	0.486258	0.243129	−	0.969994i	$-0.421826\pi$
$233$	7.42241	0.486258	0.243129	+	0.969994i	$0.421826\pi$
$234$	0	0
$235$	−4.61197	−0.300852
$236$	0	0
$237$	−39.7874	−2.58447
$238$	0	0
$239$	3.44262	0.222685	0.111342	−	0.993782i	$-0.464485\pi$
$239$	3.44262	0.222685	0.111342	+	0.993782i	$0.464485\pi$
$240$	0	0
$241$	−23.0386	−1.48404	−0.742022	−	0.670376i	$-0.766133\pi$
$241$	−23.0386	−1.48404	−0.742022	+	0.670376i	$0.766133\pi$
$242$	0	0
$243$	10.2510	0.657599
$244$	0	0
$245$	−0.992279	−0.0633944
$246$	0	0
$247$	−5.44952	−0.346744
$248$	0	0
$249$	18.3466	1.16267
$250$	0	0
$251$	17.0883	1.07861	0.539303	−	0.842112i	$-0.318688\pi$
$251$	17.0883	1.07861	0.539303	+	0.842112i	$0.318688\pi$
$252$	0	0
$253$	−7.03652	−0.442382
$254$	0	0
$255$	19.8534	1.24327
$256$	0	0
$257$	2.15842	0.134638	0.0673191	−	0.997731i	$-0.478555\pi$
$257$	2.15842	0.134638	0.0673191	+	0.997731i	$0.478555\pi$
$258$	0	0
$259$	−9.31119	−0.578569
$260$	0	0
$261$	−40.3157	−2.49548
$262$	0	0
$263$	−0.124319	−0.00766581	−0.00383290	−	0.999993i	$-0.501220\pi$
$263$	−0.124319	−0.00766581	−0.00383290	+	0.999993i	$0.501220\pi$
$264$	0	0
$265$	10.1710	0.624797
$266$	0	0
$267$	1.23799	0.0757636
$268$	0	0
$269$	11.2239	0.684334	0.342167	−	0.939639i	$-0.388839\pi$
$269$	11.2239	0.684334	0.342167	+	0.939639i	$0.388839\pi$
$270$	0	0
$271$	0.326600	0.0198395	0.00991976	−	0.999951i	$-0.496842\pi$
$271$	0.326600	0.0198395	0.00991976	+	0.999951i	$0.496842\pi$
$272$	0	0
$273$	8.00617	0.484556
$274$	0	0
$275$	−13.7416	−0.828649
$276$	0	0
$277$	21.2121	1.27451	0.637255	−	0.770653i	$-0.280070\pi$
$277$	21.2121	1.27451	0.637255	+	0.770653i	$0.280070\pi$
$278$	0	0
$279$	−27.9364	−1.67251
$280$	0	0
$281$	−27.8495	−1.66136	−0.830680	−	0.556750i	$-0.812048\pi$
$281$	−27.8495	−1.66136	−0.830680	+	0.556750i	$0.812048\pi$
$282$	0	0
$283$	−23.2706	−1.38329	−0.691645	−	0.722237i	$-0.743114\pi$
$283$	−23.2706	−1.38329	−0.691645	+	0.722237i	$0.743114\pi$
$284$	0	0
$285$	5.62910	0.333439
$286$	0	0
$287$	0.949797	0.0560648
$288$	0	0
$289$	31.0318	1.82540
$290$	0	0
$291$	−42.2469	−2.47656
$292$	0	0
$293$	−28.3586	−1.65672	−0.828362	−	0.560193i	$-0.810727\pi$
$293$	−28.3586	−1.65672	−0.828362	+	0.560193i	$0.810727\pi$
$294$	0	0
$295$	−12.0409	−0.701051
$296$	0	0
$297$	−23.0628	−1.33824
$298$	0	0
$299$	5.70213	0.329763
$300$	0	0
$301$	8.41942	0.485287
$302$	0	0
$303$	−45.7294	−2.62709
$304$	0	0
$305$	3.95031	0.226194
$306$	0	0
$307$	26.5949	1.51785	0.758926	−	0.651177i	$-0.225724\pi$
$307$	26.5949	1.51785	0.758926	+	0.651177i	$0.225724\pi$
$308$	0	0
$309$	53.3138	3.03292
$310$	0	0
$311$	−5.25843	−0.298178	−0.149089	−	0.988824i	$-0.547634\pi$
$311$	−5.25843	−0.298178	−0.149089	+	0.988824i	$0.547634\pi$
$312$	0	0
$313$	7.82442	0.442262	0.221131	−	0.975244i	$-0.429025\pi$
$313$	7.82442	0.442262	0.221131	+	0.975244i	$0.429025\pi$
$314$	0	0
$315$	−5.29316	−0.298236
$316$	0	0
$317$	−25.5817	−1.43681	−0.718407	−	0.695623i	$-0.755128\pi$
$317$	−25.5817	−1.43681	−0.718407	+	0.695623i	$0.755128\pi$
$318$	0	0
$319$	−25.8644	−1.44813
$320$	0	0
$321$	−0.596777	−0.0333089
$322$	0	0
$323$	13.6186	0.757761
$324$	0	0
$325$	11.1357	0.617695
$326$	0	0
$327$	1.26325	0.0698581
$328$	0	0
$329$	4.64785	0.256244
$330$	0	0
$331$	0.994122	0.0546419	0.0273210	−	0.999627i	$-0.491302\pi$
$331$	0.994122	0.0546419	0.0273210	+	0.999627i	$0.491302\pi$
$332$	0	0
$333$	−49.6691	−2.72185
$334$	0	0
$335$	12.7307	0.695551
$336$	0	0
$337$	−13.4691	−0.733710	−0.366855	−	0.930278i	$-0.619566\pi$
$337$	−13.4691	−0.733710	−0.366855	+	0.930278i	$0.619566\pi$
$338$	0	0
$339$	48.2674	2.62152
$340$	0	0
$341$	−17.9225	−0.970560
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.89003	−0.317109
$346$	0	0
$347$	3.59625	0.193057	0.0965283	−	0.995330i	$-0.469226\pi$
$347$	3.59625	0.193057	0.0965283	+	0.995330i	$0.469226\pi$
$348$	0	0
$349$	2.56209	0.137146	0.0685728	−	0.997646i	$-0.478155\pi$
$349$	2.56209	0.137146	0.0685728	+	0.997646i	$0.478155\pi$
$350$	0	0
$351$	18.6892	0.997556
$352$	0	0
$353$	−4.72060	−0.251252	−0.125626	−	0.992078i	$-0.540094\pi$
$353$	−4.72060	−0.251252	−0.125626	+	0.992078i	$0.540094\pi$
$354$	0	0
$355$	−3.57727	−0.189862
$356$	0	0
$357$	−20.0079	−1.05893
$358$	0	0
$359$	16.9233	0.893177	0.446589	−	0.894739i	$-0.352639\pi$
$359$	16.9233	0.893177	0.446589	+	0.894739i	$0.352639\pi$
$360$	0	0
$361$	−15.1387	−0.796772
$362$	0	0
$363$	−2.05468	−0.107842
$364$	0	0
$365$	−6.48136	−0.339250
$366$	0	0
$367$	−19.8872	−1.03810	−0.519051	−	0.854743i	$-0.673714\pi$
$367$	−19.8872	−1.03810	−0.519051	+	0.854743i	$0.673714\pi$
$368$	0	0
$369$	5.06655	0.263754
$370$	0	0
$371$	−10.2501	−0.532158
$372$	0	0
$373$	9.68661	0.501554	0.250777	−	0.968045i	$-0.419314\pi$
$373$	9.68661	0.501554	0.250777	+	0.968045i	$0.419314\pi$
$374$	0	0
$375$	−25.8258	−1.33364
$376$	0	0
$377$	20.9595	1.07947
$378$	0	0
$379$	16.8921	0.867688	0.433844	−	0.900988i	$-0.357157\pi$
$379$	16.8921	0.867688	0.433844	+	0.900988i	$0.357157\pi$
$380$	0	0
$381$	−14.1933	−0.727143
$382$	0	0
$383$	24.7947	1.26695	0.633476	−	0.773763i	$-0.281628\pi$
$383$	24.7947	1.26695	0.633476	+	0.773763i	$0.281628\pi$
$384$	0	0
$385$	−3.39582	−0.173067
$386$	0	0
$387$	44.9121	2.28301
$388$	0	0
$389$	−3.09208	−0.156775	−0.0783874	−	0.996923i	$-0.524977\pi$
$389$	−3.09208	−0.156775	−0.0783874	+	0.996923i	$0.524977\pi$
$390$	0	0
$391$	−14.2499	−0.720649
$392$	0	0
$393$	4.50744	0.227370
$394$	0	0
$395$	−13.6755	−0.688089
$396$	0	0
$397$	0.384971	0.0193211	0.00966057	−	0.999953i	$-0.496925\pi$
$397$	0.384971	0.0193211	0.00966057	+	0.999953i	$0.496925\pi$
$398$	0	0
$399$	−5.67289	−0.284000
$400$	0	0
$401$	14.6370	0.730937	0.365468	−	0.930824i	$-0.380909\pi$
$401$	14.6370	0.730937	0.365468	+	0.930824i	$0.380909\pi$
$402$	0	0
$403$	14.5237	0.723479
$404$	0	0
$405$	−3.42557	−0.170218
$406$	0	0
$407$	−31.8651	−1.57950
$408$	0	0
$409$	−5.20342	−0.257292	−0.128646	−	0.991691i	$-0.541063\pi$
$409$	−5.20342	−0.257292	−0.128646	+	0.991691i	$0.541063\pi$
$410$	0	0
$411$	0.533863	0.0263335
$412$	0	0
$413$	12.1346	0.597106
$414$	0	0
$415$	6.30601	0.309550
$416$	0	0
$417$	16.6864	0.817138
$418$	0	0
$419$	−14.0184	−0.684843	−0.342421	−	0.939546i	$-0.611247\pi$
$419$	−14.0184	−0.684843	−0.342421	+	0.939546i	$0.611247\pi$
$420$	0	0
$421$	8.30911	0.404961	0.202481	−	0.979286i	$-0.435100\pi$
$421$	8.30911	0.404961	0.202481	+	0.979286i	$0.435100\pi$
$422$	0	0
$423$	24.7933	1.20549
$424$	0	0
$425$	−27.8286	−1.34989
$426$	0	0
$427$	−3.98105	−0.192656
$428$	0	0
$429$	27.3990	1.32284
$430$	0	0
$431$	18.4777	0.890039	0.445019	−	0.895521i	$-0.353197\pi$
$431$	18.4777	0.890039	0.445019	+	0.895521i	$0.353197\pi$
$432$	0	0
$433$	−8.69984	−0.418088	−0.209044	−	0.977906i	$-0.567035\pi$
$433$	−8.69984	−0.418088	−0.209044	+	0.977906i	$0.567035\pi$
$434$	0	0
$435$	−21.6502	−1.03805
$436$	0	0
$437$	−4.04033	−0.193275
$438$	0	0
$439$	2.08886	0.0996959	0.0498479	−	0.998757i	$-0.484126\pi$
$439$	2.08886	0.0996959	0.0498479	+	0.998757i	$0.484126\pi$
$440$	0	0
$441$	5.33435	0.254017
$442$	0	0
$443$	2.22591	0.105756	0.0528780	−	0.998601i	$-0.483161\pi$
$443$	2.22591	0.105756	0.0528780	+	0.998601i	$0.483161\pi$
$444$	0	0
$445$	0.425515	0.0201713
$446$	0	0
$447$	−21.9260	−1.03706
$448$	0	0
$449$	−22.6235	−1.06767	−0.533835	−	0.845589i	$-0.679250\pi$
$449$	−22.6235	−1.06767	−0.533835	+	0.845589i	$0.679250\pi$
$450$	0	0
$451$	3.25043	0.153057
$452$	0	0
$453$	−32.9876	−1.54989
$454$	0	0
$455$	2.75184	0.129008
$456$	0	0
$457$	22.9357	1.07289	0.536443	−	0.843937i	$-0.319768\pi$
$457$	22.9357	1.07289	0.536443	+	0.843937i	$0.319768\pi$
$458$	0	0
$459$	−46.7053	−2.18002
$460$	0	0
$461$	13.8235	0.643827	0.321913	−	0.946769i	$-0.395674\pi$
$461$	13.8235	0.643827	0.321913	+	0.946769i	$0.395674\pi$
$462$	0	0
$463$	−22.4440	−1.04306	−0.521531	−	0.853233i	$-0.674639\pi$
$463$	−22.4440	−1.04306	−0.521531	+	0.853233i	$0.674639\pi$
$464$	0	0
$465$	−15.0023	−0.695717
$466$	0	0
$467$	2.20341	0.101961	0.0509807	−	0.998700i	$-0.483765\pi$
$467$	2.20341	0.101961	0.0509807	+	0.998700i	$0.483765\pi$
$468$	0	0
$469$	−12.8297	−0.592421
$470$	0	0
$471$	38.4522	1.77178
$472$	0	0
$473$	28.8133	1.32484
$474$	0	0
$475$	−7.89034	−0.362034
$476$	0	0
$477$	−54.6775	−2.50351
$478$	0	0
$479$	−36.0952	−1.64923	−0.824615	−	0.565694i	$-0.808608\pi$
$479$	−36.0952	−1.64923	−0.824615	+	0.565694i	$0.808608\pi$
$480$	0	0
$481$	25.8223	1.17739
$482$	0	0
$483$	5.93586	0.270091
$484$	0	0
$485$	−14.5209	−0.659359
$486$	0	0
$487$	−40.4748	−1.83409	−0.917044	−	0.398785i	$-0.869432\pi$
$487$	−40.4748	−1.83409	−0.917044	+	0.398785i	$0.869432\pi$
$488$	0	0
$489$	−50.9400	−2.30359
$490$	0	0
$491$	−38.3653	−1.73140	−0.865701	−	0.500562i	$-0.833127\pi$
$491$	−38.3653	−1.73140	−0.865701	+	0.500562i	$0.833127\pi$
$492$	0	0
$493$	−52.3790	−2.35903
$494$	0	0
$495$	−18.1145	−0.814185
$496$	0	0
$497$	3.60510	0.161711
$498$	0	0
$499$	1.96133	0.0878010	0.0439005	−	0.999036i	$-0.486022\pi$
$499$	1.96133	0.0878010	0.0439005	+	0.999036i	$0.486022\pi$
$500$	0	0
$501$	−46.4169	−2.07375
$502$	0	0
$503$	−16.6445	−0.742140	−0.371070	−	0.928605i	$-0.621009\pi$
$503$	−16.6445	−0.742140	−0.371070	+	0.928605i	$0.621009\pi$
$504$	0	0
$505$	−15.7179	−0.699436
$506$	0	0
$507$	15.3269	0.680692
$508$	0	0
$509$	−24.6463	−1.09243	−0.546214	−	0.837646i	$-0.683931\pi$
$509$	−24.6463	−1.09243	−0.546214	+	0.837646i	$0.683931\pi$
$510$	0	0
$511$	6.53179	0.288949
$512$	0	0
$513$	−13.2425	−0.584671
$514$	0	0
$515$	18.3247	0.807484
$516$	0	0
$517$	15.9061	0.699548
$518$	0	0
$519$	−58.8960	−2.58525
$520$	0	0
$521$	2.69192	0.117935	0.0589676	−	0.998260i	$-0.481219\pi$
$521$	2.69192	0.117935	0.0589676	+	0.998260i	$0.481219\pi$
$522$	0	0
$523$	15.5510	0.679997	0.339998	−	0.940426i	$-0.389573\pi$
$523$	15.5510	0.679997	0.339998	+	0.940426i	$0.389573\pi$
$524$	0	0
$525$	11.5921	0.505921
$526$	0	0
$527$	−36.2956	−1.58106
$528$	0	0
$529$	−18.7724	−0.816191
$530$	0	0
$531$	64.7303	2.80906
$532$	0	0
$533$	−2.63403	−0.114092
$534$	0	0
$535$	−0.205121	−0.00886816
$536$	0	0
$537$	13.5822	0.586115
$538$	0	0
$539$	3.42224	0.147406
$540$	0	0
$541$	9.14601	0.393218	0.196609	−	0.980482i	$-0.437007\pi$
$541$	9.14601	0.393218	0.196609	+	0.980482i	$0.437007\pi$
$542$	0	0
$543$	42.0111	1.80287
$544$	0	0
$545$	0.434199	0.0185990
$546$	0	0
$547$	−29.0039	−1.24012	−0.620059	−	0.784555i	$-0.712891\pi$
$547$	−29.0039	−1.24012	−0.620059	+	0.784555i	$0.712891\pi$
$548$	0	0
$549$	−21.2363	−0.906343
$550$	0	0
$551$	−14.8512	−0.632682
$552$	0	0
$553$	13.7819	0.586066
$554$	0	0
$555$	−26.6732	−1.13221
$556$	0	0
$557$	15.6343	0.662448	0.331224	−	0.943552i	$-0.392538\pi$
$557$	15.6343	0.662448	0.331224	+	0.943552i	$0.392538\pi$
$558$	0	0
$559$	−23.3492	−0.987565
$560$	0	0
$561$	−68.4716	−2.89087
$562$	0	0
$563$	−21.7492	−0.916619	−0.458309	−	0.888793i	$-0.651545\pi$
$563$	−21.7492	−0.916619	−0.458309	+	0.888793i	$0.651545\pi$
$564$	0	0
$565$	16.5902	0.697955
$566$	0	0
$567$	3.45223	0.144980
$568$	0	0
$569$	9.21322	0.386238	0.193119	−	0.981175i	$-0.438140\pi$
$569$	9.21322	0.386238	0.193119	+	0.981175i	$0.438140\pi$
$570$	0	0
$571$	−14.6331	−0.612375	−0.306187	−	0.951971i	$-0.599053\pi$
$571$	−14.6331	−0.612375	−0.306187	+	0.951971i	$0.599053\pi$
$572$	0	0
$573$	76.8193	3.20917
$574$	0	0
$575$	8.25609	0.344303
$576$	0	0
$577$	2.75852	0.114839	0.0574193	−	0.998350i	$-0.481713\pi$
$577$	2.75852	0.114839	0.0574193	+	0.998350i	$0.481713\pi$
$578$	0	0
$579$	72.9308	3.03090
$580$	0	0
$581$	−6.35508	−0.263653
$582$	0	0
$583$	−35.0783	−1.45279
$584$	0	0
$585$	14.6793	0.606913
$586$	0	0
$587$	−5.92116	−0.244392	−0.122196	−	0.992506i	$-0.538994\pi$
$587$	−5.92116	−0.244392	−0.122196	+	0.992506i	$0.538994\pi$
$588$	0	0
$589$	−10.2910	−0.424034
$590$	0	0
$591$	28.4383	1.16980
$592$	0	0
$593$	3.11656	0.127982	0.0639908	−	0.997950i	$-0.479617\pi$
$593$	3.11656	0.127982	0.0639908	+	0.997950i	$0.479617\pi$
$594$	0	0
$595$	−6.87699	−0.281929
$596$	0	0
$597$	−44.0373	−1.80233
$598$	0	0
$599$	−5.12382	−0.209354	−0.104677	−	0.994506i	$-0.533381\pi$
$599$	−5.12382	−0.209354	−0.104677	+	0.994506i	$0.533381\pi$
$600$	0	0
$601$	28.1920	1.14997	0.574987	−	0.818162i	$-0.305007\pi$
$601$	28.1920	1.14997	0.574987	+	0.818162i	$0.305007\pi$
$602$	0	0
$603$	−68.4382	−2.78702
$604$	0	0
$605$	−0.706222	−0.0287120
$606$	0	0
$607$	−40.0403	−1.62518	−0.812592	−	0.582832i	$-0.801944\pi$
$607$	−40.0403	−1.62518	−0.812592	+	0.582832i	$0.801944\pi$
$608$	0	0
$609$	21.8187	0.884137
$610$	0	0
$611$	−12.8897	−0.521460
$612$	0	0
$613$	17.9351	0.724391	0.362196	−	0.932102i	$-0.382027\pi$
$613$	17.9351	0.724391	0.362196	+	0.932102i	$0.382027\pi$
$614$	0	0
$615$	2.72083	0.109714
$616$	0	0
$617$	−35.2965	−1.42098	−0.710491	−	0.703706i	$-0.751527\pi$
$617$	−35.2965	−1.42098	−0.710491	+	0.703706i	$0.751527\pi$
$618$	0	0
$619$	−9.06462	−0.364338	−0.182169	−	0.983267i	$-0.558312\pi$
$619$	−9.06462	−0.364338	−0.182169	+	0.983267i	$0.558312\pi$
$620$	0	0
$621$	13.8564	0.556037
$622$	0	0
$623$	−0.428825	−0.0171805
$624$	0	0
$625$	11.2002	0.448008
$626$	0	0
$627$	−19.4140	−0.775320
$628$	0	0
$629$	−64.5312	−2.57303
$630$	0	0
$631$	−8.85344	−0.352450	−0.176225	−	0.984350i	$-0.556389\pi$
$631$	−8.85344	−0.352450	−0.176225	+	0.984350i	$0.556389\pi$
$632$	0	0
$633$	10.0832	0.400773
$634$	0	0
$635$	−4.87844	−0.193595
$636$	0	0
$637$	−2.77325	−0.109880
$638$	0	0
$639$	19.2309	0.760761
$640$	0	0
$641$	−0.523959	−0.0206951	−0.0103476	−	0.999946i	$-0.503294\pi$
$641$	−0.523959	−0.0206951	−0.0103476	+	0.999946i	$0.503294\pi$
$642$	0	0
$643$	−10.6478	−0.419909	−0.209954	−	0.977711i	$-0.567332\pi$
$643$	−10.6478	−0.419909	−0.209954	+	0.977711i	$0.567332\pi$
$644$	0	0
$645$	24.1186	0.949669
$646$	0	0
$647$	−17.9059	−0.703955	−0.351977	−	0.936009i	$-0.614491\pi$
$647$	−17.9059	−0.703955	−0.351977	+	0.936009i	$0.614491\pi$
$648$	0	0
$649$	41.5276	1.63010
$650$	0	0
$651$	15.1191	0.592563
$652$	0	0
$653$	−10.7527	−0.420784	−0.210392	−	0.977617i	$-0.567474\pi$
$653$	−10.7527	−0.420784	−0.210392	+	0.977617i	$0.567474\pi$
$654$	0	0
$655$	1.54927	0.0605351
$656$	0	0
$657$	34.8428	1.35935
$658$	0	0
$659$	−17.9988	−0.701133	−0.350567	−	0.936538i	$-0.614011\pi$
$659$	−17.9988	−0.701133	−0.350567	+	0.936538i	$0.614011\pi$
$660$	0	0
$661$	31.5748	1.22812	0.614058	−	0.789261i	$-0.289536\pi$
$661$	31.5748	1.22812	0.614058	+	0.789261i	$0.289536\pi$
$662$	0	0
$663$	55.4868	2.15493
$664$	0	0
$665$	−1.94986	−0.0756122
$666$	0	0
$667$	15.5396	0.601696
$668$	0	0
$669$	35.2911	1.36443
$670$	0	0
$671$	−13.6241	−0.525952
$672$	0	0
$673$	−29.7030	−1.14497	−0.572483	−	0.819917i	$-0.694020\pi$
$673$	−29.7030	−1.14497	−0.572483	+	0.819917i	$0.694020\pi$
$674$	0	0
$675$	27.0600	1.04154
$676$	0	0
$677$	−1.46765	−0.0564063	−0.0282031	−	0.999602i	$-0.508979\pi$
$677$	−1.46765	−0.0564063	−0.0282031	+	0.999602i	$0.508979\pi$
$678$	0	0
$679$	14.6339	0.561596
$680$	0	0
$681$	43.5575	1.66913
$682$	0	0
$683$	8.35850	0.319829	0.159915	−	0.987131i	$-0.448878\pi$
$683$	8.35850	0.319829	0.159915	+	0.987131i	$0.448878\pi$
$684$	0	0
$685$	0.183497	0.00701104
$686$	0	0
$687$	27.5702	1.05187
$688$	0	0
$689$	28.4261	1.08295
$690$	0	0
$691$	−8.93094	−0.339749	−0.169874	−	0.985466i	$-0.554336\pi$
$691$	−8.93094	−0.339749	−0.169874	+	0.985466i	$0.554336\pi$
$692$	0	0
$693$	18.2554	0.693466
$694$	0	0
$695$	5.73537	0.217555
$696$	0	0
$697$	6.58257	0.249333
$698$	0	0
$699$	−21.4280	−0.810480
$700$	0	0
$701$	−49.9961	−1.88833	−0.944163	−	0.329479i	$-0.893127\pi$
$701$	−49.9961	−1.88833	−0.944163	+	0.329479i	$0.893127\pi$
$702$	0	0
$703$	−18.2968	−0.690075
$704$	0	0
$705$	13.3144	0.501450
$706$	0	0
$707$	15.8402	0.595731
$708$	0	0
$709$	36.2179	1.36019	0.680096	−	0.733123i	$-0.261938\pi$
$709$	36.2179	1.36019	0.680096	+	0.733123i	$0.261938\pi$
$710$	0	0
$711$	73.5175	2.75712
$712$	0	0
$713$	10.7680	0.403267
$714$	0	0
$715$	9.41745	0.352193
$716$	0	0
$717$	−9.93859	−0.371163
$718$	0	0
$719$	−12.8390	−0.478816	−0.239408	−	0.970919i	$-0.576953\pi$
$719$	−12.8390	−0.478816	−0.239408	+	0.970919i	$0.576953\pi$
$720$	0	0
$721$	−18.4673	−0.687759
$722$	0	0
$723$	66.5106	2.47356
$724$	0	0
$725$	30.3473	1.12707
$726$	0	0
$727$	−50.1537	−1.86010	−0.930049	−	0.367436i	$-0.880236\pi$
$727$	−50.1537	−1.86010	−0.930049	+	0.367436i	$0.880236\pi$
$728$	0	0
$729$	−39.9504	−1.47965
$730$	0	0
$731$	58.3508	2.15818
$732$	0	0
$733$	30.3047	1.11933	0.559665	−	0.828719i	$-0.310930\pi$
$733$	30.3047	1.11933	0.559665	+	0.828719i	$0.310930\pi$
$734$	0	0
$735$	2.86464	0.105664
$736$	0	0
$737$	−43.9064	−1.61731
$738$	0	0
$739$	3.34736	0.123135	0.0615673	−	0.998103i	$-0.480390\pi$
$739$	3.34736	0.123135	0.0615673	+	0.998103i	$0.480390\pi$
$740$	0	0
$741$	15.7324	0.577943
$742$	0	0
$743$	4.56306	0.167403	0.0837013	−	0.996491i	$-0.473326\pi$
$743$	4.56306	0.167403	0.0837013	+	0.996491i	$0.473326\pi$
$744$	0	0
$745$	−7.53627	−0.276108
$746$	0	0
$747$	−33.9002	−1.24034
$748$	0	0
$749$	0.206717	0.00755328
$750$	0	0
$751$	6.04590	0.220618	0.110309	−	0.993897i	$-0.464816\pi$
$751$	6.04590	0.220618	0.110309	+	0.993897i	$0.464816\pi$
$752$	0	0
$753$	−49.3328	−1.79779
$754$	0	0
$755$	−11.3383	−0.412644
$756$	0	0
$757$	35.4956	1.29011	0.645054	−	0.764137i	$-0.276835\pi$
$757$	35.4956	1.29011	0.645054	+	0.764137i	$0.276835\pi$
$758$	0	0
$759$	20.3139	0.737349
$760$	0	0
$761$	16.2508	0.589089	0.294545	−	0.955638i	$-0.404832\pi$
$761$	16.2508	0.589089	0.294545	+	0.955638i	$0.404832\pi$
$762$	0	0
$763$	−0.437577	−0.0158414
$764$	0	0
$765$	−36.6843	−1.32632
$766$	0	0
$767$	−33.6524	−1.21512
$768$	0	0
$769$	−18.6812	−0.673660	−0.336830	−	0.941565i	$-0.609355\pi$
$769$	−18.6812	−0.673660	−0.336830	+	0.941565i	$0.609355\pi$
$770$	0	0
$771$	−6.23119	−0.224411
$772$	0	0
$773$	−15.0314	−0.540643	−0.270322	−	0.962770i	$-0.587130\pi$
$773$	−15.0314	−0.540643	−0.270322	+	0.962770i	$0.587130\pi$
$774$	0	0
$775$	21.0289	0.755380
$776$	0	0
$777$	26.8807	0.964341
$778$	0	0
$779$	1.86638	0.0668700
$780$	0	0
$781$	12.3375	0.441471
$782$	0	0
$783$	50.9324	1.82018
$784$	0	0
$785$	13.2166	0.471720
$786$	0	0
$787$	−24.3102	−0.866566	−0.433283	−	0.901258i	$-0.642645\pi$
$787$	−24.3102	−0.866566	−0.433283	+	0.901258i	$0.642645\pi$
$788$	0	0
$789$	0.358898	0.0127771
$790$	0	0
$791$	−16.7193	−0.594470
$792$	0	0
$793$	11.0404	0.392058
$794$	0	0
$795$	−29.3628	−1.04139
$796$	0	0
$797$	−27.6904	−0.980844	−0.490422	−	0.871485i	$-0.663157\pi$
$797$	−27.6904	−0.980844	−0.490422	+	0.871485i	$0.663157\pi$
$798$	0	0
$799$	32.2120	1.13958
$800$	0	0
$801$	−2.28750	−0.0808250
$802$	0	0
$803$	22.3533	0.788832
$804$	0	0
$805$	2.04024	0.0719091
$806$	0	0
$807$	−32.4026	−1.14063
$808$	0	0
$809$	−52.7688	−1.85525	−0.927626	−	0.373511i	$-0.878154\pi$
$809$	−52.7688	−1.85525	−0.927626	+	0.373511i	$0.878154\pi$
$810$	0	0
$811$	−14.2177	−0.499251	−0.249625	−	0.968342i	$-0.580307\pi$
$811$	−14.2177	−0.499251	−0.249625	+	0.968342i	$0.580307\pi$
$812$	0	0
$813$	−0.942869	−0.0330679
$814$	0	0
$815$	−17.5088	−0.613308
$816$	0	0
$817$	16.5444	0.578815
$818$	0	0
$819$	−14.7935	−0.516926
$820$	0	0
$821$	5.80931	0.202746	0.101373	−	0.994848i	$-0.467676\pi$
$821$	5.80931	0.202746	0.101373	+	0.994848i	$0.467676\pi$
$822$	0	0
$823$	−23.4496	−0.817403	−0.408702	−	0.912668i	$-0.634018\pi$
$823$	−23.4496	−0.817403	−0.408702	+	0.912668i	$0.634018\pi$
$824$	0	0
$825$	39.6710	1.38117
$826$	0	0
$827$	−47.1666	−1.64014	−0.820071	−	0.572262i	$-0.806066\pi$
$827$	−47.1666	−1.64014	−0.820071	+	0.572262i	$0.806066\pi$
$828$	0	0
$829$	30.4350	1.05705	0.528526	−	0.848917i	$-0.322745\pi$
$829$	30.4350	1.05705	0.528526	+	0.848917i	$0.322745\pi$
$830$	0	0
$831$	−61.2377	−2.12431
$832$	0	0
$833$	6.93050	0.240128
$834$	0	0
$835$	−15.9542	−0.552117
$836$	0	0
$837$	35.2932	1.21991
$838$	0	0
$839$	36.6874	1.26659	0.633295	−	0.773911i	$-0.281702\pi$
$839$	36.6874	1.26659	0.633295	+	0.773911i	$0.281702\pi$
$840$	0	0
$841$	28.1196	0.969642
$842$	0	0
$843$	80.3994	2.76910
$844$	0	0
$845$	5.26809	0.181228
$846$	0	0
$847$	0.711717	0.0244549
$848$	0	0
$849$	67.1804	2.30562
$850$	0	0
$851$	19.1449	0.656279
$852$	0	0
$853$	−8.51186	−0.291441	−0.145720	−	0.989326i	$-0.546550\pi$
$853$	−8.51186	−0.291441	−0.145720	+	0.989326i	$0.546550\pi$
$854$	0	0
$855$	−10.4012	−0.355714
$856$	0	0
$857$	−54.0539	−1.84645	−0.923223	−	0.384266i	$-0.874455\pi$
$857$	−54.0539	−1.84645	−0.923223	+	0.384266i	$0.874455\pi$
$858$	0	0
$859$	−28.4333	−0.970131	−0.485066	−	0.874478i	$-0.661204\pi$
$859$	−28.4333	−0.970131	−0.485066	+	0.874478i	$0.661204\pi$
$860$	0	0
$861$	−2.74200	−0.0934470
$862$	0	0
$863$	11.1553	0.379732	0.189866	−	0.981810i	$-0.439195\pi$
$863$	11.1553	0.379732	0.189866	+	0.981810i	$0.439195\pi$
$864$	0	0
$865$	−20.2434	−0.688296
$866$	0	0
$867$	−89.5867	−3.04252
$868$	0	0
$869$	47.1650	1.59996
$870$	0	0
$871$	35.5800	1.20558
$872$	0	0
$873$	78.0621	2.64200
$874$	0	0
$875$	8.94578	0.302422
$876$	0	0
$877$	32.7330	1.10531	0.552657	−	0.833409i	$-0.313614\pi$
$877$	32.7330	1.10531	0.552657	+	0.833409i	$0.313614\pi$
$878$	0	0
$879$	81.8691	2.76138
$880$	0	0
$881$	22.2981	0.751244	0.375622	−	0.926773i	$-0.377429\pi$
$881$	22.2981	0.751244	0.375622	+	0.926773i	$0.377429\pi$
$882$	0	0
$883$	−28.7948	−0.969023	−0.484512	−	0.874785i	$-0.661003\pi$
$883$	−28.7948	−0.969023	−0.484512	+	0.874785i	$0.661003\pi$
$884$	0	0
$885$	34.7613	1.16849
$886$	0	0
$887$	−40.1793	−1.34909	−0.674544	−	0.738234i	$-0.735660\pi$
$887$	−40.1793	−1.34909	−0.674544	+	0.738234i	$0.735660\pi$
$888$	0	0
$889$	4.91639	0.164891
$890$	0	0
$891$	11.8143	0.395795
$892$	0	0
$893$	9.13316	0.305630
$894$	0	0
$895$	4.66840	0.156047
$896$	0	0
$897$	−16.4616	−0.549638
$898$	0	0
$899$	39.5806	1.32009
$900$	0	0
$901$	−71.0382	−2.36663
$902$	0	0
$903$	−24.3063	−0.808861
$904$	0	0
$905$	14.4398	0.479996
$906$	0	0
$907$	54.3734	1.80544	0.902720	−	0.430228i	$-0.141567\pi$
$907$	54.3734	1.80544	0.902720	+	0.430228i	$0.141567\pi$
$908$	0	0
$909$	84.4970	2.80259
$910$	0	0
$911$	−28.5940	−0.947362	−0.473681	−	0.880697i	$-0.657075\pi$
$911$	−28.5940	−0.947362	−0.473681	+	0.880697i	$0.657075\pi$
$912$	0	0
$913$	−21.7486	−0.719773
$914$	0	0
$915$	−11.4043	−0.377013
$916$	0	0
$917$	−1.56133	−0.0515596
$918$	0	0
$919$	14.6386	0.482881	0.241441	−	0.970416i	$-0.422380\pi$
$919$	14.6386	0.482881	0.241441	+	0.970416i	$0.422380\pi$
$920$	0	0
$921$	−76.7776	−2.52991
$922$	0	0
$923$	−9.99785	−0.329083
$924$	0	0
$925$	37.3880	1.22931
$926$	0	0
$927$	−98.5111	−3.23553
$928$	0	0
$929$	48.8608	1.60307	0.801535	−	0.597948i	$-0.204017\pi$
$929$	48.8608	1.60307	0.801535	+	0.597948i	$0.204017\pi$
$930$	0	0
$931$	1.96503	0.0644012
$932$	0	0
$933$	15.1807	0.496994
$934$	0	0
$935$	−23.5347	−0.769667
$936$	0	0
$937$	51.4306	1.68016	0.840081	−	0.542460i	$-0.182507\pi$
$937$	51.4306	1.68016	0.840081	+	0.542460i	$0.182507\pi$
$938$	0	0
$939$	−22.5885	−0.737149
$940$	0	0
$941$	−37.0854	−1.20895	−0.604474	−	0.796624i	$-0.706617\pi$
$941$	−37.0854	−1.20895	−0.604474	+	0.796624i	$0.706617\pi$
$942$	0	0
$943$	−1.95289	−0.0635950
$944$	0	0
$945$	6.68706	0.217530
$946$	0	0
$947$	−12.1662	−0.395347	−0.197674	−	0.980268i	$-0.563339\pi$
$947$	−12.1662	−0.395347	−0.197674	+	0.980268i	$0.563339\pi$
$948$	0	0
$949$	−18.1143	−0.588015
$950$	0	0
$951$	73.8526	2.39484
$952$	0	0
$953$	−26.1027	−0.845549	−0.422775	−	0.906235i	$-0.638944\pi$
$953$	−26.1027	−0.845549	−0.422775	+	0.906235i	$0.638944\pi$
$954$	0	0
$955$	26.4039	0.854411
$956$	0	0
$957$	74.6687	2.41370
$958$	0	0
$959$	−0.184924	−0.00597152
$960$	0	0
$961$	−3.57299	−0.115258
$962$	0	0
$963$	1.10270	0.0355340
$964$	0	0
$965$	25.0674	0.806948
$966$	0	0
$967$	−8.74115	−0.281097	−0.140548	−	0.990074i	$-0.544887\pi$
$967$	−8.74115	−0.281097	−0.140548	+	0.990074i	$0.544887\pi$
$968$	0	0
$969$	−39.3160	−1.26301
$970$	0	0
$971$	42.2285	1.35518	0.677589	−	0.735441i	$-0.263025\pi$
$971$	42.2285	1.35518	0.677589	+	0.735441i	$0.263025\pi$
$972$	0	0
$973$	−5.78000	−0.185298
$974$	0	0
$975$	−32.1478	−1.02955
$976$	0	0
$977$	24.3299	0.778383	0.389191	−	0.921157i	$-0.372755\pi$
$977$	24.3299	0.778383	0.389191	+	0.921157i	$0.372755\pi$
$978$	0	0
$979$	−1.46754	−0.0469029
$980$	0	0
$981$	−2.33419	−0.0745249
$982$	0	0
$983$	−36.0892	−1.15107	−0.575534	−	0.817778i	$-0.695206\pi$
$983$	−36.0892	−1.15107	−0.575534	+	0.817778i	$0.695206\pi$
$984$	0	0
$985$	9.77466	0.311447
$986$	0	0
$987$	−13.4180	−0.427100
$988$	0	0
$989$	−17.3113	−0.550468
$990$	0	0
$991$	44.3921	1.41016	0.705080	−	0.709127i	$-0.250911\pi$
$991$	44.3921	1.41016	0.705080	+	0.709127i	$0.250911\pi$
$992$	0	0
$993$	−2.86996	−0.0910754
$994$	0	0
$995$	−15.1363	−0.479852
$996$	0	0
$997$	2.03994	0.0646055	0.0323027	−	0.999478i	$-0.489716\pi$
$997$	2.03994	0.0646055	0.0323027	+	0.999478i	$0.489716\pi$
$998$	0	0
$999$	62.7490	1.98529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bb.1.1		8
4.3	odd	2		7168.2.a.ba.1.8		8
8.3	odd	2		7168.2.a.be.1.1		8
8.5	even	2		7168.2.a.bf.1.8		8
32.3	odd	8		1792.2.m.g.1345.7	yes	16
32.5	even	8		1792.2.m.h.449.7	yes	16
32.11	odd	8		1792.2.m.g.449.7	yes	16
32.13	even	8		1792.2.m.h.1345.7	yes	16
32.19	odd	8		1792.2.m.f.1345.2	yes	16
32.21	even	8		1792.2.m.e.449.2	✓	16
32.27	odd	8		1792.2.m.f.449.2	yes	16
32.29	even	8		1792.2.m.e.1345.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.2	✓	16	32.21	even	8
1792.2.m.e.1345.2	yes	16	32.29	even	8
1792.2.m.f.449.2	yes	16	32.27	odd	8
1792.2.m.f.1345.2	yes	16	32.19	odd	8
1792.2.m.g.449.7	yes	16	32.11	odd	8
1792.2.m.g.1345.7	yes	16	32.3	odd	8
1792.2.m.h.449.7	yes	16	32.5	even	8
1792.2.m.h.1345.7	yes	16	32.13	even	8
7168.2.a.ba.1.8		8	4.3	odd	2
7168.2.a.bb.1.1		8	1.1	even	1	trivial
7168.2.a.be.1.1		8	8.3	odd	2
7168.2.a.bf.1.8		8	8.5	even	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-2.88693 q^{3} -0.992279 q^{5} +1.00000 q^{7} +5.33435 q^{9} +O(q^{10})\)	\(q-2.88693 q^{3} -0.992279 q^{5} +1.00000 q^{7} +5.33435 q^{9} +3.42224 q^{11} -2.77325 q^{13} +2.86464 q^{15} +6.93050 q^{17} +1.96503 q^{19} -2.88693 q^{21} -2.05612 q^{23} -4.01538 q^{25} -6.73909 q^{27} -7.55775 q^{29} -5.23708 q^{31} -9.87975 q^{33} -0.992279 q^{35} -9.31119 q^{37} +8.00617 q^{39} +0.949797 q^{41} +8.41942 q^{43} -5.29316 q^{45} +4.64785 q^{47} +1.00000 q^{49} -20.0079 q^{51} -10.2501 q^{53} -3.39582 q^{55} -5.67289 q^{57} +12.1346 q^{59} -3.98105 q^{61} +5.33435 q^{63} +2.75184 q^{65} -12.8297 q^{67} +5.93586 q^{69} +3.60510 q^{71} +6.53179 q^{73} +11.5921 q^{75} +3.42224 q^{77} +13.7819 q^{79} +3.45223 q^{81} -6.35508 q^{83} -6.87699 q^{85} +21.8187 q^{87} -0.428825 q^{89} -2.77325 q^{91} +15.1191 q^{93} -1.94986 q^{95} +14.6339 q^{97} +18.2554 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9	\( 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9} - 12 q^{11} - 20 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 12 q^{25} + 12 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} - 52 q^{45} + 20 q^{47} + 8 q^{49} - 32 q^{51} - 40 q^{53} + 24 q^{55} - 4 q^{57} - 4 q^{59} + 8 q^{61} + 12 q^{63} + 36 q^{65} - 28 q^{67} - 4 q^{69} - 16 q^{71} + 16 q^{73} + 28 q^{75} - 12 q^{77} + 20 q^{81} + 8 q^{83} - 16 q^{85} + 20 q^{87} + 16 q^{89} - 20 q^{91} - 16 q^{93} - 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9 - 12 * q^11 - 20 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 12 * q^25 + 12 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 16 * q^39 - 12 * q^41 + 4 * q^43 - 52 * q^45 + 20 * q^47 + 8 * q^49 - 32 * q^51 - 40 * q^53 + 24 * q^55 - 4 * q^57 - 4 * q^59 + 8 * q^61 + 12 * q^63 + 36 * q^65 - 28 * q^67 - 4 * q^69 - 16 * q^71 + 16 * q^73 + 28 * q^75 - 12 * q^77 + 20 * q^81 + 8 * q^83 - 16 * q^85 + 20 * q^87 + 16 * q^89 - 20 * q^91 - 16 * q^93 - 40 * q^95 - 36 * q^97 + 4 * q^99

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