LMFDB - Embedded newform 5456.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5456,2,Mod(1,5456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5456, 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("5456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5456 = 2^{4} \cdot 11 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5456.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:	$43.5663793428$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 341)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.381966 q^{5} -3.85410 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.381966 q^{5} -3.85410 q^{7} -2.00000 q^{9} -1.00000 q^{11} -5.47214 q^{13} -0.381966 q^{15} -4.23607 q^{17} +7.23607 q^{19} -3.85410 q^{21} +3.23607 q^{23} -4.85410 q^{25} -5.00000 q^{27} -1.00000 q^{31} -1.00000 q^{33} +1.47214 q^{35} +0.236068 q^{37} -5.47214 q^{39} +10.0902 q^{41} +7.38197 q^{43} +0.763932 q^{45} -1.09017 q^{47} +7.85410 q^{49} -4.23607 q^{51} -4.09017 q^{53} +0.381966 q^{55} +7.23607 q^{57} -2.23607 q^{59} -11.0902 q^{61} +7.70820 q^{63} +2.09017 q^{65} +7.00000 q^{67} +3.23607 q^{69} +11.0902 q^{71} +10.7082 q^{73} -4.85410 q^{75} +3.85410 q^{77} +5.00000 q^{79} +1.00000 q^{81} -10.3820 q^{83} +1.61803 q^{85} -10.3262 q^{89} +21.0902 q^{91} -1.00000 q^{93} -2.76393 q^{95} -0.944272 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9	$ 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{15} - 4 q^{17} + 10 q^{19} - q^{21} + 2 q^{23} - 3 q^{25} - 10 q^{27} - 2 q^{31} - 2 q^{33} - 6 q^{35} - 4 q^{37} - 2 q^{39} + 9 q^{41} + 17 q^{43} + 6 q^{45} + 9 q^{47} + 9 q^{49} - 4 q^{51} + 3 q^{53} + 3 q^{55} + 10 q^{57} - 11 q^{61} + 2 q^{63} - 7 q^{65} + 14 q^{67} + 2 q^{69} + 11 q^{71} + 8 q^{73} - 3 q^{75} + q^{77} + 10 q^{79} + 2 q^{81} - 23 q^{83} + q^{85} - 5 q^{89} + 31 q^{91} - 2 q^{93} - 10 q^{95} + 16 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^15 - 4 * q^17 + 10 * q^19 - q^21 + 2 * q^23 - 3 * q^25 - 10 * q^27 - 2 * q^31 - 2 * q^33 - 6 * q^35 - 4 * q^37 - 2 * q^39 + 9 * q^41 + 17 * q^43 + 6 * q^45 + 9 * q^47 + 9 * q^49 - 4 * q^51 + 3 * q^53 + 3 * q^55 + 10 * q^57 - 11 * q^61 + 2 * q^63 - 7 * q^65 + 14 * q^67 + 2 * q^69 + 11 * q^71 + 8 * q^73 - 3 * q^75 + q^77 + 10 * q^79 + 2 * q^81 - 23 * q^83 + q^85 - 5 * q^89 + 31 * q^91 - 2 * q^93 - 10 * q^95 + 16 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−0.381966	−0.170820	−0.0854102	−	0.996346i	$-0.527220\pi$
$5$	−0.381966	−0.170820	−0.0854102	+	0.996346i	$0.527220\pi$
$6$	0	0
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	−0.381966	−0.0986232
$16$	0	0
$17$	−4.23607	−1.02740	−0.513699	−	0.857971i	$-0.671725\pi$
$17$	−4.23607	−1.02740	−0.513699	+	0.857971i	$0.671725\pi$
$18$	0	0
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0	0
$21$	−3.85410	−0.841034
$22$	0	0
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	0	0
$25$	−4.85410	−0.970820
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	1.47214	0.248836
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	−5.47214	−0.876243
$40$	0	0
$41$	10.0902	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$41$	10.0902	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$42$	0	0
$43$	7.38197	1.12574	0.562870	−	0.826546i	$-0.309697\pi$
$43$	7.38197	1.12574	0.562870	+	0.826546i	$0.309697\pi$
$44$	0	0
$45$	0.763932	0.113880
$46$	0	0
$47$	−1.09017	−0.159018	−0.0795088	−	0.996834i	$-0.525335\pi$
$47$	−1.09017	−0.159018	−0.0795088	+	0.996834i	$0.525335\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	0	0
$51$	−4.23607	−0.593168
$52$	0	0
$53$	−4.09017	−0.561828	−0.280914	−	0.959733i	$-0.590638\pi$
$53$	−4.09017	−0.561828	−0.280914	+	0.959733i	$0.590638\pi$
$54$	0	0
$55$	0.381966	0.0515043
$56$	0	0
$57$	7.23607	0.958441
$58$	0	0
$59$	−2.23607	−0.291111	−0.145556	−	0.989350i	$-0.546497\pi$
$59$	−2.23607	−0.291111	−0.145556	+	0.989350i	$0.546497\pi$
$60$	0	0
$61$	−11.0902	−1.41995	−0.709975	−	0.704226i	$-0.751294\pi$
$61$	−11.0902	−1.41995	−0.709975	+	0.704226i	$0.751294\pi$
$62$	0	0
$63$	7.70820	0.971142
$64$	0	0
$65$	2.09017	0.259254
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	3.23607	0.389577
$70$	0	0
$71$	11.0902	1.31616	0.658081	−	0.752948i	$-0.271369\pi$
$71$	11.0902	1.31616	0.658081	+	0.752948i	$0.271369\pi$
$72$	0	0
$73$	10.7082	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$73$	10.7082	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$74$	0	0
$75$	−4.85410	−0.560503
$76$	0	0
$77$	3.85410	0.439216
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.3820	−1.13957	−0.569784	−	0.821794i	$-0.692973\pi$
$83$	−10.3820	−1.13957	−0.569784	+	0.821794i	$0.692973\pi$
$84$	0	0
$85$	1.61803	0.175500
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.3262	−1.09458	−0.547290	−	0.836943i	$-0.684340\pi$
$89$	−10.3262	−1.09458	−0.547290	+	0.836943i	$0.684340\pi$
$90$	0	0
$91$	21.0902	2.21085
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−2.76393	−0.283573
$96$	0	0
$97$	−0.944272	−0.0958763	−0.0479381	−	0.998850i	$-0.515265\pi$
$97$	−0.944272	−0.0958763	−0.0479381	+	0.998850i	$0.515265\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	18.1803	1.80901	0.904506	−	0.426461i	$-0.140240\pi$
$101$	18.1803	1.80901	0.904506	+	0.426461i	$0.140240\pi$
$102$	0	0
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	0	0
$105$	1.47214	0.143666
$106$	0	0
$107$	−8.32624	−0.804928	−0.402464	−	0.915436i	$-0.631846\pi$
$107$	−8.32624	−0.804928	−0.402464	+	0.915436i	$0.631846\pi$
$108$	0	0
$109$	−3.29180	−0.315297	−0.157648	−	0.987495i	$-0.550391\pi$
$109$	−3.29180	−0.315297	−0.157648	+	0.987495i	$0.550391\pi$
$110$	0	0
$111$	0.236068	0.0224066
$112$	0	0
$113$	12.9443	1.21769	0.608847	−	0.793287i	$-0.291632\pi$
$113$	12.9443	1.21769	0.608847	+	0.793287i	$0.291632\pi$
$114$	0	0
$115$	−1.23607	−0.115264
$116$	0	0
$117$	10.9443	1.01180
$118$	0	0
$119$	16.3262	1.49662
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	10.0902	0.909800
$124$	0	0
$125$	3.76393	0.336656
$126$	0	0
$127$	5.29180	0.469571	0.234785	−	0.972047i	$-0.424561\pi$
$127$	5.29180	0.469571	0.234785	+	0.972047i	$0.424561\pi$
$128$	0	0
$129$	7.38197	0.649946
$130$	0	0
$131$	16.0902	1.40580	0.702902	−	0.711286i	$-0.251887\pi$
$131$	16.0902	1.40580	0.702902	+	0.711286i	$0.251887\pi$
$132$	0	0
$133$	−27.8885	−2.41824
$134$	0	0
$135$	1.90983	0.164372
$136$	0	0
$137$	9.70820	0.829428	0.414714	−	0.909952i	$-0.363882\pi$
$137$	9.70820	0.829428	0.414714	+	0.909952i	$0.363882\pi$
$138$	0	0
$139$	−8.41641	−0.713870	−0.356935	−	0.934129i	$-0.616178\pi$
$139$	−8.41641	−0.713870	−0.356935	+	0.934129i	$0.616178\pi$
$140$	0	0
$141$	−1.09017	−0.0918089
$142$	0	0
$143$	5.47214	0.457603
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.85410	0.647795
$148$	0	0
$149$	2.23607	0.183186	0.0915929	−	0.995797i	$-0.470804\pi$
$149$	2.23607	0.183186	0.0915929	+	0.995797i	$0.470804\pi$
$150$	0	0
$151$	17.2705	1.40545	0.702727	−	0.711460i	$-0.251966\pi$
$151$	17.2705	1.40545	0.702727	+	0.711460i	$0.251966\pi$
$152$	0	0
$153$	8.47214	0.684932
$154$	0	0
$155$	0.381966	0.0306802
$156$	0	0
$157$	−9.56231	−0.763155	−0.381578	−	0.924337i	$-0.624619\pi$
$157$	−9.56231	−0.763155	−0.381578	+	0.924337i	$0.624619\pi$
$158$	0	0
$159$	−4.09017	−0.324372
$160$	0	0
$161$	−12.4721	−0.982942
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	0.381966	0.0297360
$166$	0	0
$167$	−9.38197	−0.725998	−0.362999	−	0.931789i	$-0.618247\pi$
$167$	−9.38197	−0.725998	−0.362999	+	0.931789i	$0.618247\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	−14.4721	−1.10671
$172$	0	0
$173$	10.9098	0.829459	0.414730	−	0.909945i	$-0.363876\pi$
$173$	10.9098	0.829459	0.414730	+	0.909945i	$0.363876\pi$
$174$	0	0
$175$	18.7082	1.41421
$176$	0	0
$177$	−2.23607	−0.168073
$178$	0	0
$179$	17.7639	1.32774	0.663869	−	0.747849i	$-0.268913\pi$
$179$	17.7639	1.32774	0.663869	+	0.747849i	$0.268913\pi$
$180$	0	0
$181$	−6.09017	−0.452679	−0.226339	−	0.974049i	$-0.572676\pi$
$181$	−6.09017	−0.452679	−0.226339	+	0.974049i	$0.572676\pi$
$182$	0	0
$183$	−11.0902	−0.819809
$184$	0	0
$185$	−0.0901699	−0.00662943
$186$	0	0
$187$	4.23607	0.309772
$188$	0	0
$189$	19.2705	1.40172
$190$	0	0
$191$	−8.18034	−0.591909	−0.295954	−	0.955202i	$-0.595638\pi$
$191$	−8.18034	−0.591909	−0.295954	+	0.955202i	$0.595638\pi$
$192$	0	0
$193$	2.81966	0.202964	0.101482	−	0.994837i	$-0.467642\pi$
$193$	2.81966	0.202964	0.101482	+	0.994837i	$0.467642\pi$
$194$	0	0
$195$	2.09017	0.149680
$196$	0	0
$197$	9.70820	0.691681	0.345840	−	0.938293i	$-0.387594\pi$
$197$	9.70820	0.691681	0.345840	+	0.938293i	$0.387594\pi$
$198$	0	0
$199$	−22.8885	−1.62253	−0.811263	−	0.584682i	$-0.801219\pi$
$199$	−22.8885	−1.62253	−0.811263	+	0.584682i	$0.801219\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.85410	−0.269182
$206$	0	0
$207$	−6.47214	−0.449845
$208$	0	0
$209$	−7.23607	−0.500529
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	0	0
$213$	11.0902	0.759886
$214$	0	0
$215$	−2.81966	−0.192299
$216$	0	0
$217$	3.85410	0.261633
$218$	0	0
$219$	10.7082	0.723593
$220$	0	0
$221$	23.1803	1.55928
$222$	0	0
$223$	9.29180	0.622225	0.311112	−	0.950373i	$-0.399298\pi$
$223$	9.29180	0.622225	0.311112	+	0.950373i	$0.399298\pi$
$224$	0	0
$225$	9.70820	0.647214
$226$	0	0
$227$	29.3607	1.94874	0.974368	−	0.224958i	$-0.0722246\pi$
$227$	29.3607	1.94874	0.974368	+	0.224958i	$0.0722246\pi$
$228$	0	0
$229$	−25.8541	−1.70849	−0.854244	−	0.519873i	$-0.825979\pi$
$229$	−25.8541	−1.70849	−0.854244	+	0.519873i	$0.825979\pi$
$230$	0	0
$231$	3.85410	0.253581
$232$	0	0
$233$	−20.4721	−1.34117	−0.670587	−	0.741831i	$-0.733958\pi$
$233$	−20.4721	−1.34117	−0.670587	+	0.741831i	$0.733958\pi$
$234$	0	0
$235$	0.416408	0.0271635
$236$	0	0
$237$	5.00000	0.324785
$238$	0	0
$239$	−18.6180	−1.20430	−0.602150	−	0.798383i	$-0.705689\pi$
$239$	−18.6180	−1.20430	−0.602150	+	0.798383i	$0.705689\pi$
$240$	0	0
$241$	3.90983	0.251854	0.125927	−	0.992039i	$-0.459809\pi$
$241$	3.90983	0.251854	0.125927	+	0.992039i	$0.459809\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−39.5967	−2.51948
$248$	0	0
$249$	−10.3820	−0.657930
$250$	0	0
$251$	−15.0902	−0.952483	−0.476242	−	0.879315i	$-0.658001\pi$
$251$	−15.0902	−0.952483	−0.476242	+	0.879315i	$0.658001\pi$
$252$	0	0
$253$	−3.23607	−0.203450
$254$	0	0
$255$	1.61803	0.101325
$256$	0	0
$257$	27.0689	1.68851	0.844255	−	0.535941i	$-0.180043\pi$
$257$	27.0689	1.68851	0.844255	+	0.535941i	$0.180043\pi$
$258$	0	0
$259$	−0.909830	−0.0565341
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.3820	−1.25681	−0.628403	−	0.777888i	$-0.716291\pi$
$263$	−20.3820	−1.25681	−0.628403	+	0.777888i	$0.716291\pi$
$264$	0	0
$265$	1.56231	0.0959717
$266$	0	0
$267$	−10.3262	−0.631955
$268$	0	0
$269$	−13.7426	−0.837904	−0.418952	−	0.908008i	$-0.637602\pi$
$269$	−13.7426	−0.837904	−0.418952	+	0.908008i	$0.637602\pi$
$270$	0	0
$271$	22.2705	1.35284	0.676419	−	0.736517i	$-0.263531\pi$
$271$	22.2705	1.35284	0.676419	+	0.736517i	$0.263531\pi$
$272$	0	0
$273$	21.0902	1.27644
$274$	0	0
$275$	4.85410	0.292713
$276$	0	0
$277$	−13.7082	−0.823646	−0.411823	−	0.911264i	$-0.635108\pi$
$277$	−13.7082	−0.823646	−0.411823	+	0.911264i	$0.635108\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−22.2148	−1.32053	−0.660266	−	0.751032i	$-0.729556\pi$
$283$	−22.2148	−1.32053	−0.660266	+	0.751032i	$0.729556\pi$
$284$	0	0
$285$	−2.76393	−0.163721
$286$	0	0
$287$	−38.8885	−2.29552
$288$	0	0
$289$	0.944272	0.0555454
$290$	0	0
$291$	−0.944272	−0.0553542
$292$	0	0
$293$	0.0557281	0.00325567	0.00162783	−	0.999999i	$-0.499482\pi$
$293$	0.0557281	0.00325567	0.00162783	+	0.999999i	$0.499482\pi$
$294$	0	0
$295$	0.854102	0.0497277
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	−17.7082	−1.02409
$300$	0	0
$301$	−28.4508	−1.63988
$302$	0	0
$303$	18.1803	1.04443
$304$	0	0
$305$	4.23607	0.242557
$306$	0	0
$307$	−25.6869	−1.46603	−0.733015	−	0.680213i	$-0.761887\pi$
$307$	−25.6869	−1.46603	−0.733015	+	0.680213i	$0.761887\pi$
$308$	0	0
$309$	−9.00000	−0.511992
$310$	0	0
$311$	28.4508	1.61330	0.806650	−	0.591030i	$-0.201278\pi$
$311$	28.4508	1.61330	0.806650	+	0.591030i	$0.201278\pi$
$312$	0	0
$313$	−11.1246	−0.628800	−0.314400	−	0.949291i	$-0.601803\pi$
$313$	−11.1246	−0.628800	−0.314400	+	0.949291i	$0.601803\pi$
$314$	0	0
$315$	−2.94427	−0.165891
$316$	0	0
$317$	−17.1246	−0.961814	−0.480907	−	0.876772i	$-0.659693\pi$
$317$	−17.1246	−0.961814	−0.480907	+	0.876772i	$0.659693\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−8.32624	−0.464725
$322$	0	0
$323$	−30.6525	−1.70555
$324$	0	0
$325$	26.5623	1.47341
$326$	0	0
$327$	−3.29180	−0.182037
$328$	0	0
$329$	4.20163	0.231643
$330$	0	0
$331$	11.8197	0.649667	0.324834	−	0.945771i	$-0.394692\pi$
$331$	11.8197	0.649667	0.324834	+	0.945771i	$0.394692\pi$
$332$	0	0
$333$	−0.472136	−0.0258729
$334$	0	0
$335$	−2.67376	−0.146083
$336$	0	0
$337$	28.8541	1.57178	0.785892	−	0.618364i	$-0.212204\pi$
$337$	28.8541	1.57178	0.785892	+	0.618364i	$0.212204\pi$
$338$	0	0
$339$	12.9443	0.703036
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	−1.23607	−0.0665477
$346$	0	0
$347$	4.88854	0.262431	0.131215	−	0.991354i	$-0.458112\pi$
$347$	4.88854	0.262431	0.131215	+	0.991354i	$0.458112\pi$
$348$	0	0
$349$	25.9787	1.39061	0.695304	−	0.718715i	$-0.255270\pi$
$349$	25.9787	1.39061	0.695304	+	0.718715i	$0.255270\pi$
$350$	0	0
$351$	27.3607	1.46041
$352$	0	0
$353$	−29.7426	−1.58304	−0.791521	−	0.611142i	$-0.790710\pi$
$353$	−29.7426	−1.58304	−0.791521	+	0.611142i	$0.790710\pi$
$354$	0	0
$355$	−4.23607	−0.224827
$356$	0	0
$357$	16.3262	0.864076
$358$	0	0
$359$	20.1246	1.06214	0.531068	−	0.847329i	$-0.321791\pi$
$359$	20.1246	1.06214	0.531068	+	0.847329i	$0.321791\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−4.09017	−0.214089
$366$	0	0
$367$	−7.14590	−0.373013	−0.186506	−	0.982454i	$-0.559717\pi$
$367$	−7.14590	−0.373013	−0.186506	+	0.982454i	$0.559717\pi$
$368$	0	0
$369$	−20.1803	−1.05055
$370$	0	0
$371$	15.7639	0.818423
$372$	0	0
$373$	17.0902	0.884895	0.442448	−	0.896794i	$-0.354110\pi$
$373$	17.0902	0.884895	0.442448	+	0.896794i	$0.354110\pi$
$374$	0	0
$375$	3.76393	0.194369
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−14.2705	−0.733027	−0.366513	−	0.930413i	$-0.619449\pi$
$379$	−14.2705	−0.733027	−0.366513	+	0.930413i	$0.619449\pi$
$380$	0	0
$381$	5.29180	0.271107
$382$	0	0
$383$	6.52786	0.333558	0.166779	−	0.985994i	$-0.446663\pi$
$383$	6.52786	0.333558	0.166779	+	0.985994i	$0.446663\pi$
$384$	0	0
$385$	−1.47214	−0.0750270
$386$	0	0
$387$	−14.7639	−0.750493
$388$	0	0
$389$	−32.2361	−1.63443	−0.817217	−	0.576330i	$-0.804484\pi$
$389$	−32.2361	−1.63443	−0.817217	+	0.576330i	$0.804484\pi$
$390$	0	0
$391$	−13.7082	−0.693254
$392$	0	0
$393$	16.0902	0.811642
$394$	0	0
$395$	−1.90983	−0.0960940
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	−27.8885	−1.39617
$400$	0	0
$401$	−29.1803	−1.45720	−0.728598	−	0.684941i	$-0.759828\pi$
$401$	−29.1803	−1.45720	−0.728598	+	0.684941i	$0.759828\pi$
$402$	0	0
$403$	5.47214	0.272587
$404$	0	0
$405$	−0.381966	−0.0189800
$406$	0	0
$407$	−0.236068	−0.0117015
$408$	0	0
$409$	30.1246	1.48957	0.744783	−	0.667307i	$-0.232553\pi$
$409$	30.1246	1.48957	0.744783	+	0.667307i	$0.232553\pi$
$410$	0	0
$411$	9.70820	0.478870
$412$	0	0
$413$	8.61803	0.424066
$414$	0	0
$415$	3.96556	0.194662
$416$	0	0
$417$	−8.41641	−0.412153
$418$	0	0
$419$	14.4721	0.707010	0.353505	−	0.935433i	$-0.384990\pi$
$419$	14.4721	0.707010	0.353505	+	0.935433i	$0.384990\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	0	0
$423$	2.18034	0.106012
$424$	0	0
$425$	20.5623	0.997418
$426$	0	0
$427$	42.7426	2.06846
$428$	0	0
$429$	5.47214	0.264197
$430$	0	0
$431$	26.0902	1.25672	0.628360	−	0.777923i	$-0.283727\pi$
$431$	26.0902	1.25672	0.628360	+	0.777923i	$0.283727\pi$
$432$	0	0
$433$	−8.43769	−0.405490	−0.202745	−	0.979232i	$-0.564986\pi$
$433$	−8.43769	−0.405490	−0.202745	+	0.979232i	$0.564986\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.4164	1.12016
$438$	0	0
$439$	−6.18034	−0.294972	−0.147486	−	0.989064i	$-0.547118\pi$
$439$	−6.18034	−0.294972	−0.147486	+	0.989064i	$0.547118\pi$
$440$	0	0
$441$	−15.7082	−0.748010
$442$	0	0
$443$	−20.8328	−0.989797	−0.494898	−	0.868951i	$-0.664795\pi$
$443$	−20.8328	−0.989797	−0.494898	+	0.868951i	$0.664795\pi$
$444$	0	0
$445$	3.94427	0.186976
$446$	0	0
$447$	2.23607	0.105762
$448$	0	0
$449$	−19.7984	−0.934343	−0.467172	−	0.884167i	$-0.654727\pi$
$449$	−19.7984	−0.934343	−0.467172	+	0.884167i	$0.654727\pi$
$450$	0	0
$451$	−10.0902	−0.475128
$452$	0	0
$453$	17.2705	0.811439
$454$	0	0
$455$	−8.05573	−0.377658
$456$	0	0
$457$	−28.5066	−1.33348	−0.666741	−	0.745290i	$-0.732311\pi$
$457$	−28.5066	−1.33348	−0.666741	+	0.745290i	$0.732311\pi$
$458$	0	0
$459$	21.1803	0.988614
$460$	0	0
$461$	−14.1803	−0.660444	−0.330222	−	0.943903i	$-0.607124\pi$
$461$	−14.1803	−0.660444	−0.330222	+	0.943903i	$0.607124\pi$
$462$	0	0
$463$	0.472136	0.0219420	0.0109710	−	0.999940i	$-0.496508\pi$
$463$	0.472136	0.0219420	0.0109710	+	0.999940i	$0.496508\pi$
$464$	0	0
$465$	0.381966	0.0177132
$466$	0	0
$467$	7.85410	0.363444	0.181722	−	0.983350i	$-0.441833\pi$
$467$	7.85410	0.363444	0.181722	+	0.983350i	$0.441833\pi$
$468$	0	0
$469$	−26.9787	−1.24576
$470$	0	0
$471$	−9.56231	−0.440608
$472$	0	0
$473$	−7.38197	−0.339423
$474$	0	0
$475$	−35.1246	−1.61163
$476$	0	0
$477$	8.18034	0.374552
$478$	0	0
$479$	−6.38197	−0.291599	−0.145800	−	0.989314i	$-0.546576\pi$
$479$	−6.38197	−0.291599	−0.145800	+	0.989314i	$0.546576\pi$
$480$	0	0
$481$	−1.29180	−0.0589008
$482$	0	0
$483$	−12.4721	−0.567502
$484$	0	0
$485$	0.360680	0.0163776
$486$	0	0
$487$	14.0344	0.635961	0.317981	−	0.948097i	$-0.396995\pi$
$487$	14.0344	0.635961	0.317981	+	0.948097i	$0.396995\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.763932	−0.0343362
$496$	0	0
$497$	−42.7426	−1.91727
$498$	0	0
$499$	39.3951	1.76357	0.881784	−	0.471654i	$-0.156343\pi$
$499$	39.3951	1.76357	0.881784	+	0.471654i	$0.156343\pi$
$500$	0	0
$501$	−9.38197	−0.419155
$502$	0	0
$503$	−6.76393	−0.301589	−0.150794	−	0.988565i	$-0.548183\pi$
$503$	−6.76393	−0.301589	−0.150794	+	0.988565i	$0.548183\pi$
$504$	0	0
$505$	−6.94427	−0.309016
$506$	0	0
$507$	16.9443	0.752522
$508$	0	0
$509$	−3.81966	−0.169303	−0.0846517	−	0.996411i	$-0.526978\pi$
$509$	−3.81966	−0.169303	−0.0846517	+	0.996411i	$0.526978\pi$
$510$	0	0
$511$	−41.2705	−1.82570
$512$	0	0
$513$	−36.1803	−1.59740
$514$	0	0
$515$	3.43769	0.151483
$516$	0	0
$517$	1.09017	0.0479456
$518$	0	0
$519$	10.9098	0.478888
$520$	0	0
$521$	17.4508	0.764536	0.382268	−	0.924052i	$-0.375143\pi$
$521$	17.4508	0.764536	0.382268	+	0.924052i	$0.375143\pi$
$522$	0	0
$523$	−18.2705	−0.798914	−0.399457	−	0.916752i	$-0.630801\pi$
$523$	−18.2705	−0.798914	−0.399457	+	0.916752i	$0.630801\pi$
$524$	0	0
$525$	18.7082	0.816493
$526$	0	0
$527$	4.23607	0.184526
$528$	0	0
$529$	−12.5279	−0.544690
$530$	0	0
$531$	4.47214	0.194074
$532$	0	0
$533$	−55.2148	−2.39162
$534$	0	0
$535$	3.18034	0.137498
$536$	0	0
$537$	17.7639	0.766570
$538$	0	0
$539$	−7.85410	−0.338300
$540$	0	0
$541$	39.3607	1.69225	0.846124	−	0.532986i	$-0.178930\pi$
$541$	39.3607	1.69225	0.846124	+	0.532986i	$0.178930\pi$
$542$	0	0
$543$	−6.09017	−0.261354
$544$	0	0
$545$	1.25735	0.0538591
$546$	0	0
$547$	−4.38197	−0.187359	−0.0936797	−	0.995602i	$-0.529863\pi$
$547$	−4.38197	−0.187359	−0.0936797	+	0.995602i	$0.529863\pi$
$548$	0	0
$549$	22.1803	0.946634
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−19.2705	−0.819465
$554$	0	0
$555$	−0.0901699	−0.00382750
$556$	0	0
$557$	1.41641	0.0600151	0.0300076	−	0.999550i	$-0.490447\pi$
$557$	1.41641	0.0600151	0.0300076	+	0.999550i	$0.490447\pi$
$558$	0	0
$559$	−40.3951	−1.70853
$560$	0	0
$561$	4.23607	0.178847
$562$	0	0
$563$	41.1246	1.73320	0.866598	−	0.499007i	$-0.166302\pi$
$563$	41.1246	1.73320	0.866598	+	0.499007i	$0.166302\pi$
$564$	0	0
$565$	−4.94427	−0.208007
$566$	0	0
$567$	−3.85410	−0.161857
$568$	0	0
$569$	28.4164	1.19128	0.595639	−	0.803252i	$-0.296899\pi$
$569$	28.4164	1.19128	0.595639	+	0.803252i	$0.296899\pi$
$570$	0	0
$571$	12.2705	0.513505	0.256752	−	0.966477i	$-0.417348\pi$
$571$	12.2705	0.513505	0.256752	+	0.966477i	$0.417348\pi$
$572$	0	0
$573$	−8.18034	−0.341739
$574$	0	0
$575$	−15.7082	−0.655077
$576$	0	0
$577$	−24.0344	−1.00057	−0.500283	−	0.865862i	$-0.666771\pi$
$577$	−24.0344	−1.00057	−0.500283	+	0.865862i	$0.666771\pi$
$578$	0	0
$579$	2.81966	0.117181
$580$	0	0
$581$	40.0132	1.66003
$582$	0	0
$583$	4.09017	0.169398
$584$	0	0
$585$	−4.18034	−0.172836
$586$	0	0
$587$	18.9098	0.780492	0.390246	−	0.920711i	$-0.372390\pi$
$587$	18.9098	0.780492	0.390246	+	0.920711i	$0.372390\pi$
$588$	0	0
$589$	−7.23607	−0.298157
$590$	0	0
$591$	9.70820	0.399342
$592$	0	0
$593$	25.5066	1.04743	0.523715	−	0.851894i	$-0.324546\pi$
$593$	25.5066	1.04743	0.523715	+	0.851894i	$0.324546\pi$
$594$	0	0
$595$	−6.23607	−0.255654
$596$	0	0
$597$	−22.8885	−0.936766
$598$	0	0
$599$	15.6525	0.639543	0.319771	−	0.947495i	$-0.396394\pi$
$599$	15.6525	0.639543	0.319771	+	0.947495i	$0.396394\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	0	0
$605$	−0.381966	−0.0155291
$606$	0	0
$607$	−33.1246	−1.34449	−0.672243	−	0.740330i	$-0.734669\pi$
$607$	−33.1246	−1.34449	−0.672243	+	0.740330i	$0.734669\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.96556	0.241341
$612$	0	0
$613$	−28.2361	−1.14044	−0.570222	−	0.821491i	$-0.693143\pi$
$613$	−28.2361	−1.14044	−0.570222	+	0.821491i	$0.693143\pi$
$614$	0	0
$615$	−3.85410	−0.155412
$616$	0	0
$617$	−33.3050	−1.34081	−0.670403	−	0.741997i	$-0.733879\pi$
$617$	−33.3050	−1.34081	−0.670403	+	0.741997i	$0.733879\pi$
$618$	0	0
$619$	13.6180	0.547355	0.273677	−	0.961822i	$-0.411760\pi$
$619$	13.6180	0.547355	0.273677	+	0.961822i	$0.411760\pi$
$620$	0	0
$621$	−16.1803	−0.649295
$622$	0	0
$623$	39.7984	1.59449
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	−7.23607	−0.288981
$628$	0	0
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	27.2705	1.08562	0.542811	−	0.839855i	$-0.317360\pi$
$631$	27.2705	1.08562	0.542811	+	0.839855i	$0.317360\pi$
$632$	0	0
$633$	18.0000	0.715436
$634$	0	0
$635$	−2.02129	−0.0802123
$636$	0	0
$637$	−42.9787	−1.70288
$638$	0	0
$639$	−22.1803	−0.877441
$640$	0	0
$641$	13.9098	0.549405	0.274703	−	0.961529i	$-0.411421\pi$
$641$	13.9098	0.549405	0.274703	+	0.961529i	$0.411421\pi$
$642$	0	0
$643$	28.5623	1.12639	0.563194	−	0.826325i	$-0.309572\pi$
$643$	28.5623	1.12639	0.563194	+	0.826325i	$0.309572\pi$
$644$	0	0
$645$	−2.81966	−0.111024
$646$	0	0
$647$	−34.5066	−1.35659	−0.678297	−	0.734788i	$-0.737282\pi$
$647$	−34.5066	−1.35659	−0.678297	+	0.734788i	$0.737282\pi$
$648$	0	0
$649$	2.23607	0.0877733
$650$	0	0
$651$	3.85410	0.151054
$652$	0	0
$653$	19.4508	0.761171	0.380585	−	0.924746i	$-0.375723\pi$
$653$	19.4508	0.761171	0.380585	+	0.924746i	$0.375723\pi$
$654$	0	0
$655$	−6.14590	−0.240140
$656$	0	0
$657$	−21.4164	−0.835534
$658$	0	0
$659$	5.12461	0.199627	0.0998133	−	0.995006i	$-0.468175\pi$
$659$	5.12461	0.199627	0.0998133	+	0.995006i	$0.468175\pi$
$660$	0	0
$661$	20.5410	0.798953	0.399477	−	0.916743i	$-0.369192\pi$
$661$	20.5410	0.798953	0.399477	+	0.916743i	$0.369192\pi$
$662$	0	0
$663$	23.1803	0.900250
$664$	0	0
$665$	10.6525	0.413085
$666$	0	0
$667$	0	0
$668$	0	0
$669$	9.29180	0.359242
$670$	0	0
$671$	11.0902	0.428131
$672$	0	0
$673$	−3.36068	−0.129545	−0.0647723	−	0.997900i	$-0.520632\pi$
$673$	−3.36068	−0.129545	−0.0647723	+	0.997900i	$0.520632\pi$
$674$	0	0
$675$	24.2705	0.934172
$676$	0	0
$677$	1.74265	0.0669753	0.0334877	−	0.999439i	$-0.489339\pi$
$677$	1.74265	0.0669753	0.0334877	+	0.999439i	$0.489339\pi$
$678$	0	0
$679$	3.63932	0.139664
$680$	0	0
$681$	29.3607	1.12510
$682$	0	0
$683$	−31.3607	−1.19998	−0.599992	−	0.800006i	$-0.704829\pi$
$683$	−31.3607	−1.19998	−0.599992	+	0.800006i	$0.704829\pi$
$684$	0	0
$685$	−3.70820	−0.141683
$686$	0	0
$687$	−25.8541	−0.986396
$688$	0	0
$689$	22.3820	0.852685
$690$	0	0
$691$	34.1803	1.30028	0.650141	−	0.759814i	$-0.274710\pi$
$691$	34.1803	1.30028	0.650141	+	0.759814i	$0.274710\pi$
$692$	0	0
$693$	−7.70820	−0.292810
$694$	0	0
$695$	3.21478	0.121944
$696$	0	0
$697$	−42.7426	−1.61899
$698$	0	0
$699$	−20.4721	−0.774327
$700$	0	0
$701$	−31.0902	−1.17426	−0.587130	−	0.809493i	$-0.699742\pi$
$701$	−31.0902	−1.17426	−0.587130	+	0.809493i	$0.699742\pi$
$702$	0	0
$703$	1.70820	0.0644261
$704$	0	0
$705$	0.416408	0.0156828
$706$	0	0
$707$	−70.0689	−2.63521
$708$	0	0
$709$	33.5410	1.25966	0.629830	−	0.776733i	$-0.283125\pi$
$709$	33.5410	1.25966	0.629830	+	0.776733i	$0.283125\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	−3.23607	−0.121192
$714$	0	0
$715$	−2.09017	−0.0781679
$716$	0	0
$717$	−18.6180	−0.695303
$718$	0	0
$719$	−22.8885	−0.853599	−0.426799	−	0.904346i	$-0.640359\pi$
$719$	−22.8885	−0.853599	−0.426799	+	0.904346i	$0.640359\pi$
$720$	0	0
$721$	34.6869	1.29181
$722$	0	0
$723$	3.90983	0.145408
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.2918	−0.418790	−0.209395	−	0.977831i	$-0.567149\pi$
$727$	−11.2918	−0.418790	−0.209395	+	0.977831i	$0.567149\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−31.2705	−1.15658
$732$	0	0
$733$	−27.8328	−1.02803	−0.514014	−	0.857782i	$-0.671842\pi$
$733$	−27.8328	−1.02803	−0.514014	+	0.857782i	$0.671842\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−7.00000	−0.257848
$738$	0	0
$739$	28.4164	1.04531	0.522657	−	0.852543i	$-0.324941\pi$
$739$	28.4164	1.04531	0.522657	+	0.852543i	$0.324941\pi$
$740$	0	0
$741$	−39.5967	−1.45462
$742$	0	0
$743$	−4.85410	−0.178080	−0.0890399	−	0.996028i	$-0.528380\pi$
$743$	−4.85410	−0.178080	−0.0890399	+	0.996028i	$0.528380\pi$
$744$	0	0
$745$	−0.854102	−0.0312919
$746$	0	0
$747$	20.7639	0.759713
$748$	0	0
$749$	32.0902	1.17255
$750$	0	0
$751$	−40.5410	−1.47936	−0.739681	−	0.672957i	$-0.765024\pi$
$751$	−40.5410	−1.47936	−0.739681	+	0.672957i	$0.765024\pi$
$752$	0	0
$753$	−15.0902	−0.549916
$754$	0	0
$755$	−6.59675	−0.240080
$756$	0	0
$757$	19.5836	0.711778	0.355889	−	0.934528i	$-0.384178\pi$
$757$	19.5836	0.711778	0.355889	+	0.934528i	$0.384178\pi$
$758$	0	0
$759$	−3.23607	−0.117462
$760$	0	0
$761$	−19.9098	−0.721731	−0.360865	−	0.932618i	$-0.617519\pi$
$761$	−19.9098	−0.721731	−0.360865	+	0.932618i	$0.617519\pi$
$762$	0	0
$763$	12.6869	0.459297
$764$	0	0
$765$	−3.23607	−0.117000
$766$	0	0
$767$	12.2361	0.441819
$768$	0	0
$769$	45.0000	1.62274	0.811371	−	0.584532i	$-0.198722\pi$
$769$	45.0000	1.62274	0.811371	+	0.584532i	$0.198722\pi$
$770$	0	0
$771$	27.0689	0.974862
$772$	0	0
$773$	30.0557	1.08103	0.540515	−	0.841335i	$-0.318230\pi$
$773$	30.0557	1.08103	0.540515	+	0.841335i	$0.318230\pi$
$774$	0	0
$775$	4.85410	0.174364
$776$	0	0
$777$	−0.909830	−0.0326400
$778$	0	0
$779$	73.0132	2.61597
$780$	0	0
$781$	−11.0902	−0.396837
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.65248	0.130362
$786$	0	0
$787$	17.1246	0.610426	0.305213	−	0.952284i	$-0.401272\pi$
$787$	17.1246	0.610426	0.305213	+	0.952284i	$0.401272\pi$
$788$	0	0
$789$	−20.3820	−0.725617
$790$	0	0
$791$	−49.8885	−1.77383
$792$	0	0
$793$	60.6869	2.15506
$794$	0	0
$795$	1.56231	0.0554093
$796$	0	0
$797$	50.4853	1.78828	0.894140	−	0.447787i	$-0.147788\pi$
$797$	50.4853	1.78828	0.894140	+	0.447787i	$0.147788\pi$
$798$	0	0
$799$	4.61803	0.163374
$800$	0	0
$801$	20.6525	0.729719
$802$	0	0
$803$	−10.7082	−0.377884
$804$	0	0
$805$	4.76393	0.167907
$806$	0	0
$807$	−13.7426	−0.483764
$808$	0	0
$809$	−24.5967	−0.864776	−0.432388	−	0.901688i	$-0.642329\pi$
$809$	−24.5967	−0.864776	−0.432388	+	0.901688i	$0.642329\pi$
$810$	0	0
$811$	5.36068	0.188239	0.0941195	−	0.995561i	$-0.469996\pi$
$811$	5.36068	0.188239	0.0941195	+	0.995561i	$0.469996\pi$
$812$	0	0
$813$	22.2705	0.781061
$814$	0	0
$815$	−4.20163	−0.147177
$816$	0	0
$817$	53.4164	1.86880
$818$	0	0
$819$	−42.1803	−1.47390
$820$	0	0
$821$	45.5410	1.58939	0.794696	−	0.607007i	$-0.207630\pi$
$821$	45.5410	1.58939	0.794696	+	0.607007i	$0.207630\pi$
$822$	0	0
$823$	36.4508	1.27060	0.635298	−	0.772267i	$-0.280877\pi$
$823$	36.4508	1.27060	0.635298	+	0.772267i	$0.280877\pi$
$824$	0	0
$825$	4.85410	0.168998
$826$	0	0
$827$	11.3475	0.394592	0.197296	−	0.980344i	$-0.436784\pi$
$827$	11.3475	0.394592	0.197296	+	0.980344i	$0.436784\pi$
$828$	0	0
$829$	30.1246	1.04627	0.523136	−	0.852250i	$-0.324762\pi$
$829$	30.1246	1.04627	0.523136	+	0.852250i	$0.324762\pi$
$830$	0	0
$831$	−13.7082	−0.475532
$832$	0	0
$833$	−33.2705	−1.15275
$834$	0	0
$835$	3.58359	0.124015
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	28.4164	0.981043	0.490522	−	0.871429i	$-0.336806\pi$
$839$	28.4164	0.981043	0.490522	+	0.871429i	$0.336806\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	12.0000	0.413302
$844$	0	0
$845$	−6.47214	−0.222648
$846$	0	0
$847$	−3.85410	−0.132429
$848$	0	0
$849$	−22.2148	−0.762409
$850$	0	0
$851$	0.763932	0.0261873
$852$	0	0
$853$	−4.94427	−0.169289	−0.0846443	−	0.996411i	$-0.526975\pi$
$853$	−4.94427	−0.169289	−0.0846443	+	0.996411i	$0.526975\pi$
$854$	0	0
$855$	5.52786	0.189049
$856$	0	0
$857$	43.2492	1.47737	0.738683	−	0.674053i	$-0.235448\pi$
$857$	43.2492	1.47737	0.738683	+	0.674053i	$0.235448\pi$
$858$	0	0
$859$	33.9443	1.15816	0.579082	−	0.815269i	$-0.303411\pi$
$859$	33.9443	1.15816	0.579082	+	0.815269i	$0.303411\pi$
$860$	0	0
$861$	−38.8885	−1.32532
$862$	0	0
$863$	−16.0344	−0.545819	−0.272909	−	0.962040i	$-0.587986\pi$
$863$	−16.0344	−0.545819	−0.272909	+	0.962040i	$0.587986\pi$
$864$	0	0
$865$	−4.16718	−0.141689
$866$	0	0
$867$	0.944272	0.0320692
$868$	0	0
$869$	−5.00000	−0.169613
$870$	0	0
$871$	−38.3050	−1.29791
$872$	0	0
$873$	1.88854	0.0639175
$874$	0	0
$875$	−14.5066	−0.490412
$876$	0	0
$877$	14.0557	0.474628	0.237314	−	0.971433i	$-0.423733\pi$
$877$	14.0557	0.474628	0.237314	+	0.971433i	$0.423733\pi$
$878$	0	0
$879$	0.0557281	0.00187966
$880$	0	0
$881$	−26.0902	−0.879000	−0.439500	−	0.898243i	$-0.644844\pi$
$881$	−26.0902	−0.879000	−0.439500	+	0.898243i	$0.644844\pi$
$882$	0	0
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	0	0
$885$	0.854102	0.0287103
$886$	0	0
$887$	10.0902	0.338795	0.169397	−	0.985548i	$-0.445818\pi$
$887$	10.0902	0.338795	0.169397	+	0.985548i	$0.445818\pi$
$888$	0	0
$889$	−20.3951	−0.684030
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−7.88854	−0.263980
$894$	0	0
$895$	−6.78522	−0.226805
$896$	0	0
$897$	−17.7082	−0.591260
$898$	0	0
$899$	0	0
$900$	0	0
$901$	17.3262	0.577221
$902$	0	0
$903$	−28.4508	−0.946785
$904$	0	0
$905$	2.32624	0.0773268
$906$	0	0
$907$	0.416408	0.0138266	0.00691330	−	0.999976i	$-0.497799\pi$
$907$	0.416408	0.0138266	0.00691330	+	0.999976i	$0.497799\pi$
$908$	0	0
$909$	−36.3607	−1.20601
$910$	0	0
$911$	−10.8197	−0.358471	−0.179236	−	0.983806i	$-0.557362\pi$
$911$	−10.8197	−0.358471	−0.179236	+	0.983806i	$0.557362\pi$
$912$	0	0
$913$	10.3820	0.343593
$914$	0	0
$915$	4.23607	0.140040
$916$	0	0
$917$	−62.0132	−2.04785
$918$	0	0
$919$	57.8115	1.90703	0.953513	−	0.301351i	$-0.0974377\pi$
$919$	57.8115	1.90703	0.953513	+	0.301351i	$0.0974377\pi$
$920$	0	0
$921$	−25.6869	−0.846413
$922$	0	0
$923$	−60.6869	−1.99753
$924$	0	0
$925$	−1.14590	−0.0376769
$926$	0	0
$927$	18.0000	0.591198
$928$	0	0
$929$	−52.0344	−1.70719	−0.853597	−	0.520933i	$-0.825584\pi$
$929$	−52.0344	−1.70719	−0.853597	+	0.520933i	$0.825584\pi$
$930$	0	0
$931$	56.8328	1.86262
$932$	0	0
$933$	28.4508	0.931439
$934$	0	0
$935$	−1.61803	−0.0529154
$936$	0	0
$937$	−37.8541	−1.23664	−0.618320	−	0.785927i	$-0.712186\pi$
$937$	−37.8541	−1.23664	−0.618320	+	0.785927i	$0.712186\pi$
$938$	0	0
$939$	−11.1246	−0.363038
$940$	0	0
$941$	23.9098	0.779438	0.389719	−	0.920934i	$-0.372572\pi$
$941$	23.9098	0.779438	0.389719	+	0.920934i	$0.372572\pi$
$942$	0	0
$943$	32.6525	1.06331
$944$	0	0
$945$	−7.36068	−0.239443
$946$	0	0
$947$	53.4296	1.73623	0.868114	−	0.496365i	$-0.165332\pi$
$947$	53.4296	1.73623	0.868114	+	0.496365i	$0.165332\pi$
$948$	0	0
$949$	−58.5967	−1.90213
$950$	0	0
$951$	−17.1246	−0.555304
$952$	0	0
$953$	32.8673	1.06467	0.532337	−	0.846532i	$-0.321314\pi$
$953$	32.8673	1.06467	0.532337	+	0.846532i	$0.321314\pi$
$954$	0	0
$955$	3.12461	0.101110
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−37.4164	−1.20824
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	16.6525	0.536619
$964$	0	0
$965$	−1.07701	−0.0346703
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	−30.6525	−0.984699
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	32.4377	1.03990
$974$	0	0
$975$	26.5623	0.850675
$976$	0	0
$977$	−32.7771	−1.04863	−0.524316	−	0.851524i	$-0.675679\pi$
$977$	−32.7771	−1.04863	−0.524316	+	0.851524i	$0.675679\pi$
$978$	0	0
$979$	10.3262	0.330028
$980$	0	0
$981$	6.58359	0.210198
$982$	0	0
$983$	5.67376	0.180965	0.0904825	−	0.995898i	$-0.471159\pi$
$983$	5.67376	0.180965	0.0904825	+	0.995898i	$0.471159\pi$
$984$	0	0
$985$	−3.70820	−0.118153
$986$	0	0
$987$	4.20163	0.133739
$988$	0	0
$989$	23.8885	0.759612
$990$	0	0
$991$	26.8197	0.851955	0.425977	−	0.904734i	$-0.359930\pi$
$991$	26.8197	0.851955	0.425977	+	0.904734i	$0.359930\pi$
$992$	0	0
$993$	11.8197	0.375086
$994$	0	0
$995$	8.74265	0.277161
$996$	0	0
$997$	−14.7639	−0.467578	−0.233789	−	0.972287i	$-0.575113\pi$
$997$	−14.7639	−0.467578	−0.233789	+	0.972287i	$0.575113\pi$
$998$	0	0
$999$	−1.18034	−0.0373443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5456.2.a.r.1.2		2
4.3	odd	2		341.2.a.a.1.1	✓	2
12.11	even	2		3069.2.a.b.1.2		2
20.19	odd	2		8525.2.a.d.1.2		2
44.43	even	2		3751.2.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
341.2.a.a.1.1	✓	2	4.3	odd	2
3069.2.a.b.1.2		2	12.11	even	2
3751.2.a.a.1.2		2	44.43	even	2
5456.2.a.r.1.2		2	1.1	even	1	trivial
8525.2.a.d.1.2		2	20.19	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 5456 = 2^{4} \cdot 11 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5456.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.381966 q^{5} -3.85410 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.381966 q^{5} -3.85410 q^{7} -2.00000 q^{9} -1.00000 q^{11} -5.47214 q^{13} -0.381966 q^{15} -4.23607 q^{17} +7.23607 q^{19} -3.85410 q^{21} +3.23607 q^{23} -4.85410 q^{25} -5.00000 q^{27} -1.00000 q^{31} -1.00000 q^{33} +1.47214 q^{35} +0.236068 q^{37} -5.47214 q^{39} +10.0902 q^{41} +7.38197 q^{43} +0.763932 q^{45} -1.09017 q^{47} +7.85410 q^{49} -4.23607 q^{51} -4.09017 q^{53} +0.381966 q^{55} +7.23607 q^{57} -2.23607 q^{59} -11.0902 q^{61} +7.70820 q^{63} +2.09017 q^{65} +7.00000 q^{67} +3.23607 q^{69} +11.0902 q^{71} +10.7082 q^{73} -4.85410 q^{75} +3.85410 q^{77} +5.00000 q^{79} +1.00000 q^{81} -10.3820 q^{83} +1.61803 q^{85} -10.3262 q^{89} +21.0902 q^{91} -1.00000 q^{93} -2.76393 q^{95} -0.944272 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9	\( 2 q + 2 q^{3} - 3 q^{5} - q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{15} - 4 q^{17} + 10 q^{19} - q^{21} + 2 q^{23} - 3 q^{25} - 10 q^{27} - 2 q^{31} - 2 q^{33} - 6 q^{35} - 4 q^{37} - 2 q^{39} + 9 q^{41} + 17 q^{43} + 6 q^{45} + 9 q^{47} + 9 q^{49} - 4 q^{51} + 3 q^{53} + 3 q^{55} + 10 q^{57} - 11 q^{61} + 2 q^{63} - 7 q^{65} + 14 q^{67} + 2 q^{69} + 11 q^{71} + 8 q^{73} - 3 q^{75} + q^{77} + 10 q^{79} + 2 q^{81} - 23 q^{83} + q^{85} - 5 q^{89} + 31 q^{91} - 2 q^{93} - 10 q^{95} + 16 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 3 * q^5 - q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^15 - 4 * q^17 + 10 * q^19 - q^21 + 2 * q^23 - 3 * q^25 - 10 * q^27 - 2 * q^31 - 2 * q^33 - 6 * q^35 - 4 * q^37 - 2 * q^39 + 9 * q^41 + 17 * q^43 + 6 * q^45 + 9 * q^47 + 9 * q^49 - 4 * q^51 + 3 * q^53 + 3 * q^55 + 10 * q^57 - 11 * q^61 + 2 * q^63 - 7 * q^65 + 14 * q^67 + 2 * q^69 + 11 * q^71 + 8 * q^73 - 3 * q^75 + q^77 + 10 * q^79 + 2 * q^81 - 23 * q^83 + q^85 - 5 * q^89 + 31 * q^91 - 2 * q^93 - 10 * q^95 + 16 * q^97 + 4 * q^99

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