LMFDB - Embedded newform 7168.2.a.bf.1.6

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:	$0$
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			1.6
Root			$2.21236$ 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)

$f(q)$	$=$	$q+1.78579 q^{3} +4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} +O(q^{10})$	$q+1.78579 q^{3} +4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} +4.50362 q^{11} +4.84427 q^{13} +7.46903 q^{15} +5.13834 q^{17} -2.12835 q^{19} +1.78579 q^{21} -7.11888 q^{23} +12.4932 q^{25} -5.01978 q^{27} +5.43932 q^{29} -0.831138 q^{31} +8.04251 q^{33} +4.18248 q^{35} -7.98492 q^{37} +8.65084 q^{39} -2.22639 q^{41} -2.28804 q^{43} +0.790669 q^{45} +7.83759 q^{47} +1.00000 q^{49} +9.17599 q^{51} -7.90235 q^{53} +18.8363 q^{55} -3.80079 q^{57} +2.62811 q^{59} +2.33370 q^{61} +0.189043 q^{63} +20.2611 q^{65} -8.16822 q^{67} -12.7128 q^{69} -6.04851 q^{71} -7.67177 q^{73} +22.3101 q^{75} +4.50362 q^{77} -1.90198 q^{79} -9.53139 q^{81} -11.2843 q^{83} +21.4910 q^{85} +9.71349 q^{87} -2.49938 q^{89} +4.84427 q^{91} -1.48424 q^{93} -8.90180 q^{95} -1.98784 q^{97} +0.851377 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.78579	1.03103	0.515513	−	0.856882i	$-0.327601\pi$
$3$	1.78579	1.03103	0.515513	+	0.856882i	$0.327601\pi$
$4$	0	0
$5$	4.18248	1.87046	0.935231	−	0.354037i	$-0.115191\pi$
$5$	4.18248	1.87046	0.935231	+	0.354037i	$0.115191\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.189043	0.0630143
$10$	0	0
$11$	4.50362	1.35789	0.678946	−	0.734188i	$-0.262437\pi$
$11$	4.50362	1.35789	0.678946	+	0.734188i	$0.262437\pi$
$12$	0	0
$13$	4.84427	1.34356	0.671779	−	0.740752i	$-0.265531\pi$
$13$	4.84427	1.34356	0.671779	+	0.740752i	$0.265531\pi$
$14$	0	0
$15$	7.46903	1.92850
$16$	0	0
$17$	5.13834	1.24623	0.623115	−	0.782130i	$-0.285867\pi$
$17$	5.13834	1.24623	0.623115	+	0.782130i	$0.285867\pi$
$18$	0	0
$19$	−2.12835	−0.488278	−0.244139	−	0.969740i	$-0.578505\pi$
$19$	−2.12835	−0.488278	−0.244139	+	0.969740i	$0.578505\pi$
$20$	0	0
$21$	1.78579	0.389691
$22$	0	0
$23$	−7.11888	−1.48439	−0.742195	−	0.670184i	$-0.766215\pi$
$23$	−7.11888	−1.48439	−0.742195	+	0.670184i	$0.766215\pi$
$24$	0	0
$25$	12.4932	2.49863
$26$	0	0
$27$	−5.01978	−0.966056
$28$	0	0
$29$	5.43932	1.01006	0.505029	−	0.863103i	$-0.331482\pi$
$29$	5.43932	1.01006	0.505029	+	0.863103i	$0.331482\pi$
$30$	0	0
$31$	−0.831138	−0.149277	−0.0746384	−	0.997211i	$-0.523780\pi$
$31$	−0.831138	−0.149277	−0.0746384	+	0.997211i	$0.523780\pi$
$32$	0	0
$33$	8.04251	1.40002
$34$	0	0
$35$	4.18248	0.706969
$36$	0	0
$37$	−7.98492	−1.31271	−0.656357	−	0.754451i	$-0.727903\pi$
$37$	−7.98492	−1.31271	−0.656357	+	0.754451i	$0.727903\pi$
$38$	0	0
$39$	8.65084	1.38524
$40$	0	0
$41$	−2.22639	−0.347704	−0.173852	−	0.984772i	$-0.555621\pi$
$41$	−2.22639	−0.347704	−0.173852	+	0.984772i	$0.555621\pi$
$42$	0	0
$43$	−2.28804	−0.348923	−0.174462	−	0.984664i	$-0.555818\pi$
$43$	−2.28804	−0.348923	−0.174462	+	0.984664i	$0.555818\pi$
$44$	0	0
$45$	0.790669	0.117866
$46$	0	0
$47$	7.83759	1.14323	0.571615	−	0.820522i	$-0.306317\pi$
$47$	7.83759	1.14323	0.571615	+	0.820522i	$0.306317\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.17599	1.28490
$52$	0	0
$53$	−7.90235	−1.08547	−0.542736	−	0.839903i	$-0.682612\pi$
$53$	−7.90235	−1.08547	−0.542736	+	0.839903i	$0.682612\pi$
$54$	0	0
$55$	18.8363	2.53989
$56$	0	0
$57$	−3.80079	−0.503427
$58$	0	0
$59$	2.62811	0.342150	0.171075	−	0.985258i	$-0.445276\pi$
$59$	2.62811	0.342150	0.171075	+	0.985258i	$0.445276\pi$
$60$	0	0
$61$	2.33370	0.298799	0.149400	−	0.988777i	$-0.452266\pi$
$61$	2.33370	0.298799	0.149400	+	0.988777i	$0.452266\pi$
$62$	0	0
$63$	0.189043	0.0238172
$64$	0	0
$65$	20.2611	2.51307
$66$	0	0
$67$	−8.16822	−0.997907	−0.498954	−	0.866629i	$-0.666282\pi$
$67$	−8.16822	−0.997907	−0.498954	+	0.866629i	$0.666282\pi$
$68$	0	0
$69$	−12.7128	−1.53044
$70$	0	0
$71$	−6.04851	−0.717826	−0.358913	−	0.933371i	$-0.616853\pi$
$71$	−6.04851	−0.717826	−0.358913	+	0.933371i	$0.616853\pi$
$72$	0	0
$73$	−7.67177	−0.897913	−0.448957	−	0.893554i	$-0.648204\pi$
$73$	−7.67177	−0.897913	−0.448957	+	0.893554i	$0.648204\pi$
$74$	0	0
$75$	22.3101	2.57615
$76$	0	0
$77$	4.50362	0.513235
$78$	0	0
$79$	−1.90198	−0.213989	−0.106995	−	0.994260i	$-0.534123\pi$
$79$	−1.90198	−0.213989	−0.106995	+	0.994260i	$0.534123\pi$
$80$	0	0
$81$	−9.53139	−1.05904
$82$	0	0
$83$	−11.2843	−1.23861	−0.619306	−	0.785149i	$-0.712586\pi$
$83$	−11.2843	−1.23861	−0.619306	+	0.785149i	$0.712586\pi$
$84$	0	0
$85$	21.4910	2.33103
$86$	0	0
$87$	9.71349	1.04139
$88$	0	0
$89$	−2.49938	−0.264934	−0.132467	−	0.991187i	$-0.542290\pi$
$89$	−2.49938	−0.264934	−0.132467	+	0.991187i	$0.542290\pi$
$90$	0	0
$91$	4.84427	0.507817
$92$	0	0
$93$	−1.48424	−0.153908
$94$	0	0
$95$	−8.90180	−0.913306
$96$	0	0
$97$	−1.98784	−0.201834	−0.100917	−	0.994895i	$-0.532178\pi$
$97$	−1.98784	−0.201834	−0.100917	+	0.994895i	$0.532178\pi$
$98$	0	0
$99$	0.851377	0.0855666
$100$	0	0
$101$	−15.7459	−1.56678	−0.783390	−	0.621531i	$-0.786511\pi$
$101$	−15.7459	−1.56678	−0.783390	+	0.621531i	$0.786511\pi$
$102$	0	0
$103$	15.2106	1.49874	0.749371	−	0.662150i	$-0.230356\pi$
$103$	15.2106	1.49874	0.749371	+	0.662150i	$0.230356\pi$
$104$	0	0
$105$	7.46903	0.728903
$106$	0	0
$107$	1.26941	0.122718	0.0613592	−	0.998116i	$-0.480456\pi$
$107$	1.26941	0.122718	0.0613592	+	0.998116i	$0.480456\pi$
$108$	0	0
$109$	4.02383	0.385413	0.192707	−	0.981256i	$-0.438273\pi$
$109$	4.02383	0.385413	0.192707	+	0.981256i	$0.438273\pi$
$110$	0	0
$111$	−14.2594	−1.35344
$112$	0	0
$113$	1.66487	0.156618	0.0783089	−	0.996929i	$-0.475048\pi$
$113$	1.66487	0.156618	0.0783089	+	0.996929i	$0.475048\pi$
$114$	0	0
$115$	−29.7746	−2.77650
$116$	0	0
$117$	0.915774	0.0846634
$118$	0	0
$119$	5.13834	0.471031
$120$	0	0
$121$	9.28258	0.843871
$122$	0	0
$123$	−3.97586	−0.358492
$124$	0	0
$125$	31.3400	2.80313
$126$	0	0
$127$	7.86069	0.697523	0.348762	−	0.937211i	$-0.386602\pi$
$127$	7.86069	0.697523	0.348762	+	0.937211i	$0.386602\pi$
$128$	0	0
$129$	−4.08596	−0.359749
$130$	0	0
$131$	−7.70009	−0.672760	−0.336380	−	0.941726i	$-0.609203\pi$
$131$	−7.70009	−0.672760	−0.336380	+	0.941726i	$0.609203\pi$
$132$	0	0
$133$	−2.12835	−0.184552
$134$	0	0
$135$	−20.9951	−1.80697
$136$	0	0
$137$	−17.6977	−1.51201	−0.756007	−	0.654564i	$-0.772852\pi$
$137$	−17.6977	−1.51201	−0.756007	+	0.654564i	$0.772852\pi$
$138$	0	0
$139$	−16.1930	−1.37348	−0.686738	−	0.726905i	$-0.740958\pi$
$139$	−16.1930	−1.37348	−0.686738	+	0.726905i	$0.740958\pi$
$140$	0	0
$141$	13.9963	1.17870
$142$	0	0
$143$	21.8167	1.82441
$144$	0	0
$145$	22.7499	1.88927
$146$	0	0
$147$	1.78579	0.147289
$148$	0	0
$149$	12.1806	0.997874	0.498937	−	0.866638i	$-0.333724\pi$
$149$	12.1806	0.997874	0.498937	+	0.866638i	$0.333724\pi$
$150$	0	0
$151$	−17.7449	−1.44406	−0.722029	−	0.691863i	$-0.756790\pi$
$151$	−17.7449	−1.44406	−0.722029	+	0.691863i	$0.756790\pi$
$152$	0	0
$153$	0.971366	0.0785303
$154$	0	0
$155$	−3.47622	−0.279217
$156$	0	0
$157$	−13.9789	−1.11564	−0.557818	−	0.829963i	$-0.688361\pi$
$157$	−13.9789	−1.11564	−0.557818	+	0.829963i	$0.688361\pi$
$158$	0	0
$159$	−14.1119	−1.11915
$160$	0	0
$161$	−7.11888	−0.561047
$162$	0	0
$163$	−13.5962	−1.06494	−0.532468	−	0.846450i	$-0.678735\pi$
$163$	−13.5962	−1.06494	−0.532468	+	0.846450i	$0.678735\pi$
$164$	0	0
$165$	33.6377	2.61869
$166$	0	0
$167$	−6.70735	−0.519030	−0.259515	−	0.965739i	$-0.583563\pi$
$167$	−6.70735	−0.519030	−0.259515	+	0.965739i	$0.583563\pi$
$168$	0	0
$169$	10.4669	0.805147
$170$	0	0
$171$	−0.402350	−0.0307685
$172$	0	0
$173$	20.8358	1.58411	0.792057	−	0.610448i	$-0.209010\pi$
$173$	20.8358	1.58411	0.792057	+	0.610448i	$0.209010\pi$
$174$	0	0
$175$	12.4932	0.944394
$176$	0	0
$177$	4.69325	0.352766
$178$	0	0
$179$	−22.4086	−1.67490	−0.837448	−	0.546517i	$-0.815953\pi$
$179$	−22.4086	−1.67490	−0.837448	+	0.546517i	$0.815953\pi$
$180$	0	0
$181$	4.42942	0.329236	0.164618	−	0.986357i	$-0.447361\pi$
$181$	4.42942	0.329236	0.164618	+	0.986357i	$0.447361\pi$
$182$	0	0
$183$	4.16749	0.308070
$184$	0	0
$185$	−33.3968	−2.45538
$186$	0	0
$187$	23.1411	1.69225
$188$	0	0
$189$	−5.01978	−0.365135
$190$	0	0
$191$	−7.38976	−0.534704	−0.267352	−	0.963599i	$-0.586149\pi$
$191$	−7.38976	−0.534704	−0.267352	+	0.963599i	$0.586149\pi$
$192$	0	0
$193$	0.139138	0.0100154	0.00500769	−	0.999987i	$-0.498406\pi$
$193$	0.139138	0.0100154	0.00500769	+	0.999987i	$0.498406\pi$
$194$	0	0
$195$	36.1820	2.59104
$196$	0	0
$197$	10.4057	0.741377	0.370688	−	0.928757i	$-0.379122\pi$
$197$	10.4057	0.741377	0.370688	+	0.928757i	$0.379122\pi$
$198$	0	0
$199$	5.09550	0.361211	0.180605	−	0.983556i	$-0.442194\pi$
$199$	5.09550	0.361211	0.180605	+	0.983556i	$0.442194\pi$
$200$	0	0
$201$	−14.5867	−1.02887
$202$	0	0
$203$	5.43932	0.381766
$204$	0	0
$205$	−9.31184	−0.650367
$206$	0	0
$207$	−1.34577	−0.0935378
$208$	0	0
$209$	−9.58529	−0.663029
$210$	0	0
$211$	12.5589	0.864592	0.432296	−	0.901732i	$-0.357704\pi$
$211$	12.5589	0.864592	0.432296	+	0.901732i	$0.357704\pi$
$212$	0	0
$213$	−10.8014	−0.740097
$214$	0	0
$215$	−9.56970	−0.652648
$216$	0	0
$217$	−0.831138	−0.0564213
$218$	0	0
$219$	−13.7002	−0.925772
$220$	0	0
$221$	24.8915	1.67438
$222$	0	0
$223$	9.66949	0.647517	0.323758	−	0.946140i	$-0.395054\pi$
$223$	9.66949	0.647517	0.323758	+	0.946140i	$0.395054\pi$
$224$	0	0
$225$	2.36174	0.157450
$226$	0	0
$227$	−2.99594	−0.198847	−0.0994237	−	0.995045i	$-0.531700\pi$
$227$	−2.99594	−0.198847	−0.0994237	+	0.995045i	$0.531700\pi$
$228$	0	0
$229$	8.99569	0.594452	0.297226	−	0.954807i	$-0.403939\pi$
$229$	8.99569	0.594452	0.297226	+	0.954807i	$0.403939\pi$
$230$	0	0
$231$	8.04251	0.529158
$232$	0	0
$233$	7.53066	0.493350	0.246675	−	0.969098i	$-0.420662\pi$
$233$	7.53066	0.493350	0.246675	+	0.969098i	$0.420662\pi$
$234$	0	0
$235$	32.7806	2.13837
$236$	0	0
$237$	−3.39653	−0.220629
$238$	0	0
$239$	−1.87072	−0.121007	−0.0605034	−	0.998168i	$-0.519271\pi$
$239$	−1.87072	−0.121007	−0.0605034	+	0.998168i	$0.519271\pi$
$240$	0	0
$241$	14.4911	0.933454	0.466727	−	0.884401i	$-0.345433\pi$
$241$	14.4911	0.933454	0.466727	+	0.884401i	$0.345433\pi$
$242$	0	0
$243$	−1.96173	−0.125845
$244$	0	0
$245$	4.18248	0.267209
$246$	0	0
$247$	−10.3103	−0.656029
$248$	0	0
$249$	−20.1514	−1.27704
$250$	0	0
$251$	6.33974	0.400161	0.200080	−	0.979779i	$-0.435880\pi$
$251$	6.33974	0.400161	0.200080	+	0.979779i	$0.435880\pi$
$252$	0	0
$253$	−32.0607	−2.01564
$254$	0	0
$255$	38.3784	2.40335
$256$	0	0
$257$	12.1594	0.758483	0.379241	−	0.925298i	$-0.376185\pi$
$257$	12.1594	0.758483	0.379241	+	0.925298i	$0.376185\pi$
$258$	0	0
$259$	−7.98492	−0.496159
$260$	0	0
$261$	1.02827	0.0636480
$262$	0	0
$263$	0.0299529	0.00184698	0.000923488	−	1.00000i	$-0.499706\pi$
$263$	0.0299529	0.00184698	0.000923488		1.00000i	$0.499706\pi$
$264$	0	0
$265$	−33.0514	−2.03033
$266$	0	0
$267$	−4.46336	−0.273153
$268$	0	0
$269$	−18.0622	−1.10127	−0.550637	−	0.834745i	$-0.685615\pi$
$269$	−18.0622	−1.10127	−0.550637	+	0.834745i	$0.685615\pi$
$270$	0	0
$271$	10.0906	0.612958	0.306479	−	0.951877i	$-0.400849\pi$
$271$	10.0906	0.612958	0.306479	+	0.951877i	$0.400849\pi$
$272$	0	0
$273$	8.65084	0.523573
$274$	0	0
$275$	56.2644	3.39287
$276$	0	0
$277$	8.86651	0.532737	0.266368	−	0.963871i	$-0.414176\pi$
$277$	8.86651	0.532737	0.266368	+	0.963871i	$0.414176\pi$
$278$	0	0
$279$	−0.157121	−0.00940657
$280$	0	0
$281$	−11.6731	−0.696356	−0.348178	−	0.937428i	$-0.613200\pi$
$281$	−11.6731	−0.696356	−0.348178	+	0.937428i	$0.613200\pi$
$282$	0	0
$283$	−12.5547	−0.746297	−0.373149	−	0.927772i	$-0.621722\pi$
$283$	−12.5547	−0.746297	−0.373149	+	0.927772i	$0.621722\pi$
$284$	0	0
$285$	−15.8967	−0.941642
$286$	0	0
$287$	−2.22639	−0.131420
$288$	0	0
$289$	9.40252	0.553090
$290$	0	0
$291$	−3.54986	−0.208096
$292$	0	0
$293$	24.3152	1.42051	0.710256	−	0.703944i	$-0.248579\pi$
$293$	24.3152	1.42051	0.710256	+	0.703944i	$0.248579\pi$
$294$	0	0
$295$	10.9920	0.639980
$296$	0	0
$297$	−22.6072	−1.31180
$298$	0	0
$299$	−34.4858	−1.99436
$300$	0	0
$301$	−2.28804	−0.131881
$302$	0	0
$303$	−28.1189	−1.61539
$304$	0	0
$305$	9.76065	0.558893
$306$	0	0
$307$	27.2536	1.55544	0.777722	−	0.628608i	$-0.216375\pi$
$307$	27.2536	1.55544	0.777722	+	0.628608i	$0.216375\pi$
$308$	0	0
$309$	27.1629	1.54524
$310$	0	0
$311$	−5.78650	−0.328122	−0.164061	−	0.986450i	$-0.552459\pi$
$311$	−5.78650	−0.328122	−0.164061	+	0.986450i	$0.552459\pi$
$312$	0	0
$313$	3.03673	0.171646	0.0858231	−	0.996310i	$-0.472648\pi$
$313$	3.03673	0.171646	0.0858231	+	0.996310i	$0.472648\pi$
$314$	0	0
$315$	0.790669	0.0445491
$316$	0	0
$317$	4.69828	0.263882	0.131941	−	0.991258i	$-0.457879\pi$
$317$	4.69828	0.263882	0.131941	+	0.991258i	$0.457879\pi$
$318$	0	0
$319$	24.4966	1.37155
$320$	0	0
$321$	2.26690	0.126526
$322$	0	0
$323$	−10.9362	−0.608507
$324$	0	0
$325$	60.5202	3.35706
$326$	0	0
$327$	7.18572	0.397371
$328$	0	0
$329$	7.83759	0.432101
$330$	0	0
$331$	−36.0648	−1.98230	−0.991151	−	0.132738i	$-0.957623\pi$
$331$	−36.0648	−1.98230	−0.991151	+	0.132738i	$0.957623\pi$
$332$	0	0
$333$	−1.50949	−0.0827197
$334$	0	0
$335$	−34.1634	−1.86655
$336$	0	0
$337$	−31.3282	−1.70656	−0.853279	−	0.521454i	$-0.825390\pi$
$337$	−31.3282	−1.70656	−0.853279	+	0.521454i	$0.825390\pi$
$338$	0	0
$339$	2.97311	0.161477
$340$	0	0
$341$	−3.74313	−0.202702
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−53.1712	−2.86264
$346$	0	0
$347$	34.1006	1.83062	0.915308	−	0.402755i	$-0.131947\pi$
$347$	34.1006	1.83062	0.915308	+	0.402755i	$0.131947\pi$
$348$	0	0
$349$	29.6604	1.58769	0.793843	−	0.608123i	$-0.208077\pi$
$349$	29.6604	1.58769	0.793843	+	0.608123i	$0.208077\pi$
$350$	0	0
$351$	−24.3171	−1.29795
$352$	0	0
$353$	−0.605671	−0.0322366	−0.0161183	−	0.999870i	$-0.505131\pi$
$353$	−0.605671	−0.0322366	−0.0161183	+	0.999870i	$0.505131\pi$
$354$	0	0
$355$	−25.2978	−1.34267
$356$	0	0
$357$	9.17599	0.485645
$358$	0	0
$359$	11.6214	0.613354	0.306677	−	0.951814i	$-0.400783\pi$
$359$	11.6214	0.613354	0.306677	+	0.951814i	$0.400783\pi$
$360$	0	0
$361$	−14.4701	−0.761585
$362$	0	0
$363$	16.5767	0.870052
$364$	0	0
$365$	−32.0871	−1.67951
$366$	0	0
$367$	13.1299	0.685376	0.342688	−	0.939449i	$-0.388663\pi$
$367$	13.1299	0.685376	0.342688	+	0.939449i	$0.388663\pi$
$368$	0	0
$369$	−0.420883	−0.0219103
$370$	0	0
$371$	−7.90235	−0.410270
$372$	0	0
$373$	18.4801	0.956863	0.478431	−	0.878125i	$-0.341206\pi$
$373$	18.4801	0.956863	0.478431	+	0.878125i	$0.341206\pi$
$374$	0	0
$375$	55.9666	2.89010
$376$	0	0
$377$	26.3495	1.35707
$378$	0	0
$379$	4.13943	0.212628	0.106314	−	0.994333i	$-0.466095\pi$
$379$	4.13943	0.212628	0.106314	+	0.994333i	$0.466095\pi$
$380$	0	0
$381$	14.0375	0.719164
$382$	0	0
$383$	34.3667	1.75606	0.878029	−	0.478608i	$-0.158858\pi$
$383$	34.3667	1.75606	0.878029	+	0.478608i	$0.158858\pi$
$384$	0	0
$385$	18.8363	0.959987
$386$	0	0
$387$	−0.432538	−0.0219872
$388$	0	0
$389$	−10.5339	−0.534089	−0.267044	−	0.963684i	$-0.586047\pi$
$389$	−10.5339	−0.534089	−0.267044	+	0.963684i	$0.586047\pi$
$390$	0	0
$391$	−36.5792	−1.84989
$392$	0	0
$393$	−13.7507	−0.693633
$394$	0	0
$395$	−7.95499	−0.400259
$396$	0	0
$397$	−11.2088	−0.562554	−0.281277	−	0.959627i	$-0.590758\pi$
$397$	−11.2088	−0.562554	−0.281277	+	0.959627i	$0.590758\pi$
$398$	0	0
$399$	−3.80079	−0.190278
$400$	0	0
$401$	−15.4031	−0.769192	−0.384596	−	0.923085i	$-0.625659\pi$
$401$	−15.4031	−0.769192	−0.384596	+	0.923085i	$0.625659\pi$
$402$	0	0
$403$	−4.02625	−0.200562
$404$	0	0
$405$	−39.8649	−1.98090
$406$	0	0
$407$	−35.9610	−1.78252
$408$	0	0
$409$	22.6029	1.11764	0.558820	−	0.829289i	$-0.311254\pi$
$409$	22.6029	1.11764	0.558820	+	0.829289i	$0.311254\pi$
$410$	0	0
$411$	−31.6043	−1.55892
$412$	0	0
$413$	2.62811	0.129321
$414$	0	0
$415$	−47.1964	−2.31678
$416$	0	0
$417$	−28.9174	−1.41609
$418$	0	0
$419$	12.7468	0.622720	0.311360	−	0.950292i	$-0.399215\pi$
$419$	12.7468	0.622720	0.311360	+	0.950292i	$0.399215\pi$
$420$	0	0
$421$	30.9413	1.50799	0.753994	−	0.656882i	$-0.228125\pi$
$421$	30.9413	1.50799	0.753994	+	0.656882i	$0.228125\pi$
$422$	0	0
$423$	1.48164	0.0720399
$424$	0	0
$425$	64.1941	3.11387
$426$	0	0
$427$	2.33370	0.112936
$428$	0	0
$429$	38.9601	1.88101
$430$	0	0
$431$	−13.1089	−0.631434	−0.315717	−	0.948853i	$-0.602245\pi$
$431$	−13.1089	−0.631434	−0.315717	+	0.948853i	$0.602245\pi$
$432$	0	0
$433$	35.1859	1.69093	0.845463	−	0.534033i	$-0.179324\pi$
$433$	35.1859	1.69093	0.845463	+	0.534033i	$0.179324\pi$
$434$	0	0
$435$	40.6265	1.94789
$436$	0	0
$437$	15.1515	0.724795
$438$	0	0
$439$	4.82251	0.230166	0.115083	−	0.993356i	$-0.463287\pi$
$439$	4.82251	0.230166	0.115083	+	0.993356i	$0.463287\pi$
$440$	0	0
$441$	0.189043	0.00900204
$442$	0	0
$443$	−17.7618	−0.843888	−0.421944	−	0.906622i	$-0.638652\pi$
$443$	−17.7618	−0.843888	−0.421944	+	0.906622i	$0.638652\pi$
$444$	0	0
$445$	−10.4536	−0.495548
$446$	0	0
$447$	21.7520	1.02883
$448$	0	0
$449$	−29.9204	−1.41203	−0.706015	−	0.708197i	$-0.749509\pi$
$449$	−29.9204	−1.41203	−0.706015	+	0.708197i	$0.749509\pi$
$450$	0	0
$451$	−10.0268	−0.472144
$452$	0	0
$453$	−31.6886	−1.48886
$454$	0	0
$455$	20.2611	0.949853
$456$	0	0
$457$	29.1293	1.36261	0.681305	−	0.732000i	$-0.261413\pi$
$457$	29.1293	1.36261	0.681305	+	0.732000i	$0.261413\pi$
$458$	0	0
$459$	−25.7933	−1.20393
$460$	0	0
$461$	10.1181	0.471247	0.235624	−	0.971844i	$-0.424287\pi$
$461$	10.1181	0.471247	0.235624	+	0.971844i	$0.424287\pi$
$462$	0	0
$463$	−40.1547	−1.86615	−0.933074	−	0.359686i	$-0.882884\pi$
$463$	−40.1547	−1.86615	−0.933074	+	0.359686i	$0.882884\pi$
$464$	0	0
$465$	−6.20779	−0.287880
$466$	0	0
$467$	−40.3127	−1.86545	−0.932726	−	0.360587i	$-0.882577\pi$
$467$	−40.3127	−1.86545	−0.932726	+	0.360587i	$0.882577\pi$
$468$	0	0
$469$	−8.16822	−0.377174
$470$	0	0
$471$	−24.9633	−1.15025
$472$	0	0
$473$	−10.3045	−0.473800
$474$	0	0
$475$	−26.5899	−1.22003
$476$	0	0
$477$	−1.49388	−0.0684002
$478$	0	0
$479$	12.6994	0.580249	0.290124	−	0.956989i	$-0.406303\pi$
$479$	12.6994	0.580249	0.290124	+	0.956989i	$0.406303\pi$
$480$	0	0
$481$	−38.6811	−1.76371
$482$	0	0
$483$	−12.7128	−0.578454
$484$	0	0
$485$	−8.31409	−0.377523
$486$	0	0
$487$	−34.2373	−1.55144	−0.775721	−	0.631076i	$-0.782614\pi$
$487$	−34.2373	−1.55144	−0.775721	+	0.631076i	$0.782614\pi$
$488$	0	0
$489$	−24.2799	−1.09798
$490$	0	0
$491$	−30.2843	−1.36671	−0.683356	−	0.730085i	$-0.739480\pi$
$491$	−30.2843	−1.36671	−0.683356	+	0.730085i	$0.739480\pi$
$492$	0	0
$493$	27.9491	1.25876
$494$	0	0
$495$	3.56087	0.160049
$496$	0	0
$497$	−6.04851	−0.271313
$498$	0	0
$499$	−22.2171	−0.994574	−0.497287	−	0.867586i	$-0.665670\pi$
$499$	−22.2171	−0.994574	−0.497287	+	0.867586i	$0.665670\pi$
$500$	0	0
$501$	−11.9779	−0.535134
$502$	0	0
$503$	−28.6238	−1.27627	−0.638137	−	0.769923i	$-0.720294\pi$
$503$	−28.6238	−1.27627	−0.638137	+	0.769923i	$0.720294\pi$
$504$	0	0
$505$	−65.8571	−2.93060
$506$	0	0
$507$	18.6917	0.830128
$508$	0	0
$509$	21.2622	0.942431	0.471215	−	0.882018i	$-0.343815\pi$
$509$	21.2622	0.942431	0.471215	+	0.882018i	$0.343815\pi$
$510$	0	0
$511$	−7.67177	−0.339379
$512$	0	0
$513$	10.6839	0.471704
$514$	0	0
$515$	63.6180	2.80334
$516$	0	0
$517$	35.2975	1.55238
$518$	0	0
$519$	37.2083	1.63326
$520$	0	0
$521$	−23.7033	−1.03846	−0.519230	−	0.854635i	$-0.673781\pi$
$521$	−23.7033	−1.03846	−0.519230	+	0.854635i	$0.673781\pi$
$522$	0	0
$523$	19.6640	0.859847	0.429923	−	0.902865i	$-0.358541\pi$
$523$	19.6640	0.859847	0.429923	+	0.902865i	$0.358541\pi$
$524$	0	0
$525$	22.3101	0.973694
$526$	0	0
$527$	−4.27067	−0.186033
$528$	0	0
$529$	27.6785	1.20341
$530$	0	0
$531$	0.496825	0.0215604
$532$	0	0
$533$	−10.7852	−0.467160
$534$	0	0
$535$	5.30928	0.229540
$536$	0	0
$537$	−40.0170	−1.72686
$538$	0	0
$539$	4.50362	0.193985
$540$	0	0
$541$	−12.2602	−0.527106	−0.263553	−	0.964645i	$-0.584894\pi$
$541$	−12.2602	−0.527106	−0.263553	+	0.964645i	$0.584894\pi$
$542$	0	0
$543$	7.91002	0.339451
$544$	0	0
$545$	16.8296	0.720901
$546$	0	0
$547$	−5.75986	−0.246274	−0.123137	−	0.992390i	$-0.539295\pi$
$547$	−5.75986	−0.246274	−0.123137	+	0.992390i	$0.539295\pi$
$548$	0	0
$549$	0.441169	0.0188286
$550$	0	0
$551$	−11.5768	−0.493188
$552$	0	0
$553$	−1.90198	−0.0808804
$554$	0	0
$555$	−59.6396	−2.53156
$556$	0	0
$557$	2.57936	0.109291	0.0546455	−	0.998506i	$-0.482597\pi$
$557$	2.57936	0.109291	0.0546455	+	0.998506i	$0.482597\pi$
$558$	0	0
$559$	−11.0839	−0.468798
$560$	0	0
$561$	41.3252	1.74475
$562$	0	0
$563$	−15.1195	−0.637212	−0.318606	−	0.947887i	$-0.603215\pi$
$563$	−15.1195	−0.637212	−0.318606	+	0.947887i	$0.603215\pi$
$564$	0	0
$565$	6.96329	0.292948
$566$	0	0
$567$	−9.53139	−0.400281
$568$	0	0
$569$	11.0034	0.461288	0.230644	−	0.973038i	$-0.425917\pi$
$569$	11.0034	0.461288	0.230644	+	0.973038i	$0.425917\pi$
$570$	0	0
$571$	40.9982	1.71572	0.857861	−	0.513881i	$-0.171793\pi$
$571$	40.9982	1.71572	0.857861	+	0.513881i	$0.171793\pi$
$572$	0	0
$573$	−13.1966	−0.551294
$574$	0	0
$575$	−88.9373	−3.70894
$576$	0	0
$577$	2.85596	0.118895	0.0594476	−	0.998231i	$-0.481066\pi$
$577$	2.85596	0.118895	0.0594476	+	0.998231i	$0.481066\pi$
$578$	0	0
$579$	0.248472	0.0103261
$580$	0	0
$581$	−11.2843	−0.468152
$582$	0	0
$583$	−35.5892	−1.47395
$584$	0	0
$585$	3.83021	0.158360
$586$	0	0
$587$	24.8280	1.02476	0.512381	−	0.858758i	$-0.328763\pi$
$587$	24.8280	1.02476	0.512381	+	0.858758i	$0.328763\pi$
$588$	0	0
$589$	1.76895	0.0728885
$590$	0	0
$591$	18.5824	0.764379
$592$	0	0
$593$	−7.72713	−0.317315	−0.158658	−	0.987334i	$-0.550717\pi$
$593$	−7.72713	−0.317315	−0.158658	+	0.987334i	$0.550717\pi$
$594$	0	0
$595$	21.4910	0.881045
$596$	0	0
$597$	9.09950	0.372418
$598$	0	0
$599$	7.28771	0.297768	0.148884	−	0.988855i	$-0.452432\pi$
$599$	7.28771	0.297768	0.148884	+	0.988855i	$0.452432\pi$
$600$	0	0
$601$	20.0313	0.817095	0.408547	−	0.912737i	$-0.366035\pi$
$601$	20.0313	0.817095	0.408547	+	0.912737i	$0.366035\pi$
$602$	0	0
$603$	−1.54414	−0.0628824
$604$	0	0
$605$	38.8242	1.57843
$606$	0	0
$607$	−17.2219	−0.699014	−0.349507	−	0.936934i	$-0.613651\pi$
$607$	−17.2219	−0.699014	−0.349507	+	0.936934i	$0.613651\pi$
$608$	0	0
$609$	9.71349	0.393610
$610$	0	0
$611$	37.9674	1.53600
$612$	0	0
$613$	−37.3993	−1.51054	−0.755271	−	0.655412i	$-0.772495\pi$
$613$	−37.3993	−1.51054	−0.755271	+	0.655412i	$0.772495\pi$
$614$	0	0
$615$	−16.6290	−0.670545
$616$	0	0
$617$	−22.1036	−0.889856	−0.444928	−	0.895566i	$-0.646771\pi$
$617$	−22.1036	−0.889856	−0.444928	+	0.895566i	$0.646771\pi$
$618$	0	0
$619$	30.4908	1.22553	0.612764	−	0.790266i	$-0.290058\pi$
$619$	30.4908	1.22553	0.612764	+	0.790266i	$0.290058\pi$
$620$	0	0
$621$	35.7352	1.43400
$622$	0	0
$623$	−2.49938	−0.100135
$624$	0	0
$625$	68.6132	2.74453
$626$	0	0
$627$	−17.1173	−0.683600
$628$	0	0
$629$	−41.0292	−1.63594
$630$	0	0
$631$	40.3151	1.60492	0.802458	−	0.596708i	$-0.203525\pi$
$631$	40.3151	1.60492	0.802458	+	0.596708i	$0.203525\pi$
$632$	0	0
$633$	22.4276	0.891417
$634$	0	0
$635$	32.8772	1.30469
$636$	0	0
$637$	4.84427	0.191937
$638$	0	0
$639$	−1.14343	−0.0452333
$640$	0	0
$641$	−0.875535	−0.0345815	−0.0172908	−	0.999851i	$-0.505504\pi$
$641$	−0.875535	−0.0345815	−0.0172908	+	0.999851i	$0.505504\pi$
$642$	0	0
$643$	40.4431	1.59492	0.797459	−	0.603373i	$-0.206177\pi$
$643$	40.4431	1.59492	0.797459	+	0.603373i	$0.206177\pi$
$644$	0	0
$645$	−17.0895	−0.672897
$646$	0	0
$647$	3.32973	0.130905	0.0654526	−	0.997856i	$-0.479151\pi$
$647$	3.32973	0.130905	0.0654526	+	0.997856i	$0.479151\pi$
$648$	0	0
$649$	11.8360	0.464603
$650$	0	0
$651$	−1.48424	−0.0581718
$652$	0	0
$653$	29.3589	1.14890	0.574452	−	0.818539i	$-0.305215\pi$
$653$	29.3589	1.14890	0.574452	+	0.818539i	$0.305215\pi$
$654$	0	0
$655$	−32.2055	−1.25837
$656$	0	0
$657$	−1.45029	−0.0565814
$658$	0	0
$659$	29.2836	1.14073	0.570364	−	0.821392i	$-0.306802\pi$
$659$	29.2836	1.14073	0.570364	+	0.821392i	$0.306802\pi$
$660$	0	0
$661$	13.7308	0.534066	0.267033	−	0.963687i	$-0.413957\pi$
$661$	13.7308	0.534066	0.267033	+	0.963687i	$0.413957\pi$
$662$	0	0
$663$	44.4509	1.72633
$664$	0	0
$665$	−8.90180	−0.345197
$666$	0	0
$667$	−38.7219	−1.49932
$668$	0	0
$669$	17.2677	0.667606
$670$	0	0
$671$	10.5101	0.405737
$672$	0	0
$673$	49.3202	1.90115	0.950577	−	0.310490i	$-0.100493\pi$
$673$	49.3202	1.90115	0.950577	+	0.310490i	$0.100493\pi$
$674$	0	0
$675$	−62.7129	−2.41382
$676$	0	0
$677$	−17.3000	−0.664892	−0.332446	−	0.943122i	$-0.607874\pi$
$677$	−17.3000	−0.664892	−0.332446	+	0.943122i	$0.607874\pi$
$678$	0	0
$679$	−1.98784	−0.0762861
$680$	0	0
$681$	−5.35011	−0.205017
$682$	0	0
$683$	−5.37135	−0.205529	−0.102764	−	0.994706i	$-0.532769\pi$
$683$	−5.37135	−0.205529	−0.102764	+	0.994706i	$0.532769\pi$
$684$	0	0
$685$	−74.0202	−2.82816
$686$	0	0
$687$	16.0644	0.612895
$688$	0	0
$689$	−38.2811	−1.45839
$690$	0	0
$691$	44.8853	1.70752	0.853759	−	0.520668i	$-0.174317\pi$
$691$	44.8853	1.70752	0.853759	+	0.520668i	$0.174317\pi$
$692$	0	0
$693$	0.851377	0.0323411
$694$	0	0
$695$	−67.7271	−2.56904
$696$	0	0
$697$	−11.4399	−0.433319
$698$	0	0
$699$	13.4482	0.508657
$700$	0	0
$701$	−8.61013	−0.325200	−0.162600	−	0.986692i	$-0.551988\pi$
$701$	−8.61013	−0.325200	−0.162600	+	0.986692i	$0.551988\pi$
$702$	0	0
$703$	16.9947	0.640969
$704$	0	0
$705$	58.5392	2.20472
$706$	0	0
$707$	−15.7459	−0.592187
$708$	0	0
$709$	3.07942	0.115650	0.0578250	−	0.998327i	$-0.481583\pi$
$709$	3.07942	0.115650	0.0578250	+	0.998327i	$0.481583\pi$
$710$	0	0
$711$	−0.359556	−0.0134844
$712$	0	0
$713$	5.91677	0.221585
$714$	0	0
$715$	91.2481	3.41248
$716$	0	0
$717$	−3.34071	−0.124761
$718$	0	0
$719$	42.5677	1.58751	0.793754	−	0.608239i	$-0.208124\pi$
$719$	42.5677	1.58751	0.793754	+	0.608239i	$0.208124\pi$
$720$	0	0
$721$	15.2106	0.566471
$722$	0	0
$723$	25.8781	0.962416
$724$	0	0
$725$	67.9543	2.52376
$726$	0	0
$727$	−3.44163	−0.127643	−0.0638214	−	0.997961i	$-0.520329\pi$
$727$	−3.44163	−0.127643	−0.0638214	+	0.997961i	$0.520329\pi$
$728$	0	0
$729$	25.0909	0.929294
$730$	0	0
$731$	−11.7567	−0.434839
$732$	0	0
$733$	12.1501	0.448775	0.224388	−	0.974500i	$-0.427962\pi$
$733$	12.1501	0.448775	0.224388	+	0.974500i	$0.427962\pi$
$734$	0	0
$735$	7.46903	0.275499
$736$	0	0
$737$	−36.7866	−1.35505
$738$	0	0
$739$	27.9556	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$739$	27.9556	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$740$	0	0
$741$	−18.4120	−0.676383
$742$	0	0
$743$	−14.0786	−0.516495	−0.258248	−	0.966079i	$-0.583145\pi$
$743$	−14.0786	−0.516495	−0.258248	+	0.966079i	$0.583145\pi$
$744$	0	0
$745$	50.9452	1.86649
$746$	0	0
$747$	−2.13322	−0.0780503
$748$	0	0
$749$	1.26941	0.0463832
$750$	0	0
$751$	27.6318	1.00830	0.504148	−	0.863617i	$-0.331806\pi$
$751$	27.6318	1.00830	0.504148	+	0.863617i	$0.331806\pi$
$752$	0	0
$753$	11.3214	0.412576
$754$	0	0
$755$	−74.2176	−2.70106
$756$	0	0
$757$	−2.76084	−0.100344	−0.0501722	−	0.998741i	$-0.515977\pi$
$757$	−2.76084	−0.100344	−0.0501722	+	0.998741i	$0.515977\pi$
$758$	0	0
$759$	−57.2537	−2.07818
$760$	0	0
$761$	34.5598	1.25279	0.626396	−	0.779505i	$-0.284529\pi$
$761$	34.5598	1.25279	0.626396	+	0.779505i	$0.284529\pi$
$762$	0	0
$763$	4.02383	0.145673
$764$	0	0
$765$	4.06272	0.146888
$766$	0	0
$767$	12.7313	0.459699
$768$	0	0
$769$	49.7370	1.79356	0.896781	−	0.442474i	$-0.145899\pi$
$769$	49.7370	1.79356	0.896781	+	0.442474i	$0.145899\pi$
$770$	0	0
$771$	21.7141	0.782015
$772$	0	0
$773$	14.1791	0.509986	0.254993	−	0.966943i	$-0.417927\pi$
$773$	14.1791	0.509986	0.254993	+	0.966943i	$0.417927\pi$
$774$	0	0
$775$	−10.3835	−0.372988
$776$	0	0
$777$	−14.2594	−0.511553
$778$	0	0
$779$	4.73855	0.169776
$780$	0	0
$781$	−27.2402	−0.974730
$782$	0	0
$783$	−27.3042	−0.975772
$784$	0	0
$785$	−58.4664	−2.08675
$786$	0	0
$787$	−13.1039	−0.467104	−0.233552	−	0.972344i	$-0.575035\pi$
$787$	−13.1039	−0.467104	−0.233552	+	0.972344i	$0.575035\pi$
$788$	0	0
$789$	0.0534896	0.00190428
$790$	0	0
$791$	1.66487	0.0591960
$792$	0	0
$793$	11.3050	0.401454
$794$	0	0
$795$	−59.0229	−2.09333
$796$	0	0
$797$	30.2441	1.07130	0.535651	−	0.844440i	$-0.320066\pi$
$797$	30.2441	1.07130	0.535651	+	0.844440i	$0.320066\pi$
$798$	0	0
$799$	40.2722	1.42473
$800$	0	0
$801$	−0.472490	−0.0166946
$802$	0	0
$803$	−34.5507	−1.21927
$804$	0	0
$805$	−29.7746	−1.04942
$806$	0	0
$807$	−32.2553	−1.13544
$808$	0	0
$809$	−1.36939	−0.0481450	−0.0240725	−	0.999710i	$-0.507663\pi$
$809$	−1.36939	−0.0481450	−0.0240725	+	0.999710i	$0.507663\pi$
$810$	0	0
$811$	−29.7374	−1.04422	−0.522110	−	0.852878i	$-0.674855\pi$
$811$	−29.7374	−1.04422	−0.522110	+	0.852878i	$0.674855\pi$
$812$	0	0
$813$	18.0196	0.631976
$814$	0	0
$815$	−56.8659	−1.99192
$816$	0	0
$817$	4.86976	0.170371
$818$	0	0
$819$	0.915774	0.0319997
$820$	0	0
$821$	−18.1869	−0.634727	−0.317363	−	0.948304i	$-0.602797\pi$
$821$	−18.1869	−0.634727	−0.317363	+	0.948304i	$0.602797\pi$
$822$	0	0
$823$	41.5715	1.44909	0.724546	−	0.689226i	$-0.242049\pi$
$823$	41.5715	1.44909	0.724546	+	0.689226i	$0.242049\pi$
$824$	0	0
$825$	100.476	3.49814
$826$	0	0
$827$	46.0708	1.60204	0.801019	−	0.598639i	$-0.204292\pi$
$827$	46.0708	1.60204	0.801019	+	0.598639i	$0.204292\pi$
$828$	0	0
$829$	−0.574936	−0.0199683	−0.00998417	−	0.999950i	$-0.503178\pi$
$829$	−0.574936	−0.0199683	−0.00998417	+	0.999950i	$0.503178\pi$
$830$	0	0
$831$	15.8337	0.549265
$832$	0	0
$833$	5.13834	0.178033
$834$	0	0
$835$	−28.0534	−0.970827
$836$	0	0
$837$	4.17213	0.144210
$838$	0	0
$839$	28.6700	0.989799	0.494899	−	0.868950i	$-0.335205\pi$
$839$	28.6700	0.989799	0.494899	+	0.868950i	$0.335205\pi$
$840$	0	0
$841$	0.586242	0.0202152
$842$	0	0
$843$	−20.8456	−0.717961
$844$	0	0
$845$	43.7777	1.50600
$846$	0	0
$847$	9.28258	0.318953
$848$	0	0
$849$	−22.4200	−0.769452
$850$	0	0
$851$	56.8437	1.94858
$852$	0	0
$853$	−11.8044	−0.404175	−0.202087	−	0.979367i	$-0.564773\pi$
$853$	−11.8044	−0.404175	−0.202087	+	0.979367i	$0.564773\pi$
$854$	0	0
$855$	−1.68282	−0.0575513
$856$	0	0
$857$	48.6998	1.66355	0.831777	−	0.555110i	$-0.187324\pi$
$857$	48.6998	1.66355	0.831777	+	0.555110i	$0.187324\pi$
$858$	0	0
$859$	0.0381681	0.00130228	0.000651140	−	1.00000i	$-0.499793\pi$
$859$	0.0381681	0.00130228	0.000651140		1.00000i	$0.499793\pi$
$860$	0	0
$861$	−3.97586	−0.135497
$862$	0	0
$863$	−1.64855	−0.0561173	−0.0280587	−	0.999606i	$-0.508933\pi$
$863$	−1.64855	−0.0561173	−0.0280587	+	0.999606i	$0.508933\pi$
$864$	0	0
$865$	87.1452	2.96303
$866$	0	0
$867$	16.7909	0.570250
$868$	0	0
$869$	−8.56579	−0.290574
$870$	0	0
$871$	−39.5690	−1.34075
$872$	0	0
$873$	−0.375786	−0.0127184
$874$	0	0
$875$	31.3400	1.05949
$876$	0	0
$877$	44.2259	1.49340	0.746701	−	0.665160i	$-0.231637\pi$
$877$	44.2259	1.49340	0.746701	+	0.665160i	$0.231637\pi$
$878$	0	0
$879$	43.4219	1.46458
$880$	0	0
$881$	1.02249	0.0344486	0.0172243	−	0.999852i	$-0.494517\pi$
$881$	1.02249	0.0344486	0.0172243	+	0.999852i	$0.494517\pi$
$882$	0	0
$883$	−10.6948	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$883$	−10.6948	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$884$	0	0
$885$	19.6294	0.659836
$886$	0	0
$887$	−4.15374	−0.139469	−0.0697344	−	0.997566i	$-0.522215\pi$
$887$	−4.15374	−0.139469	−0.0697344	+	0.997566i	$0.522215\pi$
$888$	0	0
$889$	7.86069	0.263639
$890$	0	0
$891$	−42.9257	−1.43807
$892$	0	0
$893$	−16.6812	−0.558214
$894$	0	0
$895$	−93.7235	−3.13283
$896$	0	0
$897$	−61.5843	−2.05624
$898$	0	0
$899$	−4.52083	−0.150778
$900$	0	0
$901$	−40.6050	−1.35275
$902$	0	0
$903$	−4.08596	−0.135972
$904$	0	0
$905$	18.5260	0.615825
$906$	0	0
$907$	4.85902	0.161341	0.0806706	−	0.996741i	$-0.474294\pi$
$907$	4.85902	0.161341	0.0806706	+	0.996741i	$0.474294\pi$
$908$	0	0
$909$	−2.97666	−0.0987295
$910$	0	0
$911$	21.2511	0.704082	0.352041	−	0.935985i	$-0.385488\pi$
$911$	21.2511	0.704082	0.352041	+	0.935985i	$0.385488\pi$
$912$	0	0
$913$	−50.8202	−1.68190
$914$	0	0
$915$	17.4305	0.576233
$916$	0	0
$917$	−7.70009	−0.254279
$918$	0	0
$919$	−14.2571	−0.470297	−0.235149	−	0.971959i	$-0.575558\pi$
$919$	−14.2571	−0.470297	−0.235149	+	0.971959i	$0.575558\pi$
$920$	0	0
$921$	48.6692	1.60370
$922$	0	0
$923$	−29.3006	−0.964441
$924$	0	0
$925$	−99.7569	−3.27999
$926$	0	0
$927$	2.87545	0.0944422
$928$	0	0
$929$	31.9394	1.04790	0.523948	−	0.851750i	$-0.324458\pi$
$929$	31.9394	1.04790	0.523948	+	0.851750i	$0.324458\pi$
$930$	0	0
$931$	−2.12835	−0.0697540
$932$	0	0
$933$	−10.3335	−0.338303
$934$	0	0
$935$	96.7873	3.16528
$936$	0	0
$937$	4.91109	0.160438	0.0802191	−	0.996777i	$-0.474438\pi$
$937$	4.91109	0.160438	0.0802191	+	0.996777i	$0.474438\pi$
$938$	0	0
$939$	5.42296	0.176972
$940$	0	0
$941$	−55.0908	−1.79591	−0.897954	−	0.440090i	$-0.854947\pi$
$941$	−55.0908	−1.79591	−0.897954	+	0.440090i	$0.854947\pi$
$942$	0	0
$943$	15.8494	0.516128
$944$	0	0
$945$	−20.9951	−0.682972
$946$	0	0
$947$	−49.0597	−1.59423	−0.797113	−	0.603830i	$-0.793641\pi$
$947$	−49.0597	−1.59423	−0.797113	+	0.603830i	$0.793641\pi$
$948$	0	0
$949$	−37.1641	−1.20640
$950$	0	0
$951$	8.39014	0.272069
$952$	0	0
$953$	−56.6312	−1.83446	−0.917232	−	0.398354i	$-0.869582\pi$
$953$	−56.6312	−1.83446	−0.917232	+	0.398354i	$0.869582\pi$
$954$	0	0
$955$	−30.9075	−1.00014
$956$	0	0
$957$	43.7458	1.41410
$958$	0	0
$959$	−17.6977	−0.571487
$960$	0	0
$961$	−30.3092	−0.977716
$962$	0	0
$963$	0.239973	0.00773302
$964$	0	0
$965$	0.581943	0.0187334
$966$	0	0
$967$	47.2953	1.52092	0.760458	−	0.649388i	$-0.224975\pi$
$967$	47.2953	1.52092	0.760458	+	0.649388i	$0.224975\pi$
$968$	0	0
$969$	−19.5298	−0.627386
$970$	0	0
$971$	−2.80126	−0.0898966	−0.0449483	−	0.998989i	$-0.514312\pi$
$971$	−2.80126	−0.0898966	−0.0449483	+	0.998989i	$0.514312\pi$
$972$	0	0
$973$	−16.1930	−0.519125
$974$	0	0
$975$	108.076	3.46121
$976$	0	0
$977$	−22.0570	−0.705665	−0.352832	−	0.935687i	$-0.614781\pi$
$977$	−22.0570	−0.705665	−0.352832	+	0.935687i	$0.614781\pi$
$978$	0	0
$979$	−11.2562	−0.359751
$980$	0	0
$981$	0.760677	0.0242865
$982$	0	0
$983$	55.5608	1.77212	0.886058	−	0.463575i	$-0.153434\pi$
$983$	55.5608	1.77212	0.886058	+	0.463575i	$0.153434\pi$
$984$	0	0
$985$	43.5217	1.38672
$986$	0	0
$987$	13.9963	0.445507
$988$	0	0
$989$	16.2883	0.517938
$990$	0	0
$991$	−13.9880	−0.444345	−0.222173	−	0.975007i	$-0.571315\pi$
$991$	−13.9880	−0.444345	−0.222173	+	0.975007i	$0.571315\pi$
$992$	0	0
$993$	−64.4042	−2.04380
$994$	0	0
$995$	21.3119	0.675631
$996$	0	0
$997$	−13.9439	−0.441608	−0.220804	−	0.975318i	$-0.570868\pi$
$997$	−13.9439	−0.441608	−0.220804	+	0.975318i	$0.570868\pi$
$998$	0	0
$999$	40.0825	1.26816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bf.1.6		8
4.3	odd	2		7168.2.a.be.1.3		8
8.3	odd	2		7168.2.a.ba.1.6		8
8.5	even	2		7168.2.a.bb.1.3		8
32.3	odd	8		1792.2.m.g.1345.2	yes	16
32.5	even	8		1792.2.m.h.449.2	yes	16
32.11	odd	8		1792.2.m.g.449.2	yes	16
32.13	even	8		1792.2.m.h.1345.2	yes	16
32.19	odd	8		1792.2.m.f.1345.7	yes	16
32.21	even	8		1792.2.m.e.449.7	✓	16
32.27	odd	8		1792.2.m.f.449.7	yes	16
32.29	even	8		1792.2.m.e.1345.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.7	✓	16	32.21	even	8
1792.2.m.e.1345.7	yes	16	32.29	even	8
1792.2.m.f.449.7	yes	16	32.27	odd	8
1792.2.m.f.1345.7	yes	16	32.19	odd	8
1792.2.m.g.449.2	yes	16	32.11	odd	8
1792.2.m.g.1345.2	yes	16	32.3	odd	8
1792.2.m.h.449.2	yes	16	32.5	even	8
1792.2.m.h.1345.2	yes	16	32.13	even	8
7168.2.a.ba.1.6		8	8.3	odd	2
7168.2.a.bb.1.3		8	8.5	even	2
7168.2.a.be.1.3		8	4.3	odd	2
7168.2.a.bf.1.6		8	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q+1.78579 q^{3} +4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} +O(q^{10})\)	\(q+1.78579 q^{3} +4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} +4.50362 q^{11} +4.84427 q^{13} +7.46903 q^{15} +5.13834 q^{17} -2.12835 q^{19} +1.78579 q^{21} -7.11888 q^{23} +12.4932 q^{25} -5.01978 q^{27} +5.43932 q^{29} -0.831138 q^{31} +8.04251 q^{33} +4.18248 q^{35} -7.98492 q^{37} +8.65084 q^{39} -2.22639 q^{41} -2.28804 q^{43} +0.790669 q^{45} +7.83759 q^{47} +1.00000 q^{49} +9.17599 q^{51} -7.90235 q^{53} +18.8363 q^{55} -3.80079 q^{57} +2.62811 q^{59} +2.33370 q^{61} +0.189043 q^{63} +20.2611 q^{65} -8.16822 q^{67} -12.7128 q^{69} -6.04851 q^{71} -7.67177 q^{73} +22.3101 q^{75} +4.50362 q^{77} -1.90198 q^{79} -9.53139 q^{81} -11.2843 q^{83} +21.4910 q^{85} +9.71349 q^{87} -2.49938 q^{89} +4.84427 q^{91} -1.48424 q^{93} -8.90180 q^{95} -1.98784 q^{97} +0.851377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

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